HORIZONS IN WORLD PHYSICS
HORIZONS IN WORLD PHYSICS. VOLUME 271
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HORIZONS IN WORLD PHYSICS
HORIZONS IN WORLD PHYSICS. VOLUME 271
ALBERT REIMER EDITOR
Nova Science Publishers, Inc. New York
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Published by Nova Science Publishers, Inc. † New York
CONTENTS vii
Preface Chapter 1
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications, and Applications V. G. Polnikov
1 71
Chapter 2
An Overview of Plasma Confinement in Toroidal Systems Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
Chapter 3
Cosmic Rays and Safety Neïla Zarrouk and Raouf Bennaceur
187
Chapter 4
Laser Physics Ahmed Safwat
229
Chapter 5
Expression of Full Vector Vertex Function in QED A. D. Bao
247
Chapter 6
Condensate Fraction in Metallic Superconductor and Ultracold Atomic Vapors Luca Salasnich
259
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons Trapped in Double-Well Potentials S. K. Adhikari, B.A. Malomed, L. Salasnich and F. Toigo
271
Chapter 7
Chapter 8
Alegbra and Thermodynamics of q-Deformed Fermion Oscillators A. Lavagno and P. Narayana Swamy
291
Chapter 9
Spacetime Fermion Manifolds Bernd Schmeikal
319
Index
329
PREFACE Chapter 1 - Due to stochastic feature of a wind-wave field, the time-space evolution of the field is described by the transport equation for the 2-dimensional wave energy spectrum density, S (σ , θ ; x, t ) , spread in the space, x, and time, t. This equation has the forcing
named the source function, F, depending on both the wave spectrum, S , and the external wave-making factors: local wind, W(x, t), and local current, U(x, t). The source function, F, is the “heart” of any numerical wind wave model, as far as it contains certain physical mechanisms responsible for a wave spectrum evolution. It is used to distinguish three terms in function F: the wind-wave energy exchange mechanism, In; the energy conservative mechanism of nonlinear wave-wave interactions, Nl; and the wave energy loss mechanism, Dis, related, mainly, to the wave breaking and interaction of waves with the turbulence of water upper layer and with the bottom. Differences in mathematical representation of the source function terms determine general differences between wave models. The problem is to derive analytical representations for the source function terms said above from the fundamental wave equations. Basing on publications of numerous authors and on the last two decades studies of the author, the optimized versions of the all principal terms for the source function, F, have been constructed. Detailed description of these results is presented in this chapter. The final version of the source function is tested in academic test tasks and verified by implementing it into numerical shells of the well known wind wave models: WAM and WAVEWATCH. Procedures of testing and verification are presented and described in details. The superiority of the proposed new source function in accuracy and speed of calculations is shown. Finally, the directions of future developments in this topic are proposed, and some possible applications of numerical wind wave models are shown, aimed to study both the wind wave physics and global wind-wave variability at the climate scale, including mechanical energy exchange between wind, waves, and upper water layer. Chapter 2- This overview presents a tutorial introduction to the theory of magnetic plasma confinement in toroidal confinement systems with particular emphasis on axisymmetric equilibrium geometries, and tokamaks. The discussion covers three important aspects of plasma physics: Equilibrium, Stability, and Transport. The section on equilibrium will go through an introduction to ideal magnetohydrodynamics, curvilinear system of
viii
Albert Reimer
coordinates, flux coordinates, extensions to axisymmetric equilibrium, Grad-Shafranov Equation (GSE), Green’s function formalism, as well as analytical and numerical solutions to GSE. The section on stability will address topics including Lyapunov Stability in nonlinear systems, energy principle, modal analysis, and simplifications for axisymmetric machines. The final section will consider transport in toroidal systems. The authors present the fluxsurface-averaged system of equations describing classical and non-classical transport phenomena. Applications to the small-sized high-aspect-ratio Damavand tokamak will be described. Chapter 3 - Aircraft crew and frequent flyers are exposed to high levels of ionizing radiation principally from cosmic radiations of galactic and solar origin and from secondary radiation produced in the atmosphere. The need to assess the dose received by aircrew and frequent flyers has arisen following Recommendations of the International Commission on Radiological Protection in publication 60 ICRP 60. In 1996 the European Union introduced a revised Basic Safety Standards Directive that included exposure to natural sources of ionising radiations, including cosmic radiation as occupational exposure. Several equipments were used for both neutron and non neutron components of the onboard radiation field produced by cosmic rays. Such a field is very complex, therefore dose measurement is complex and the use of appropriate computer programs for dose calculation is essential. The experimental results were often confronted with calculations using transport codes. A reasonable agreement of measured and calculated data was observed. Particular attention was devoted to the results obtained during some extreme situations: intense solar flare and “forbush decreases” The authors’ results concerning effective doses received by Tunisian flights, computed with CARI-6, EPCARD 3.2, PCAIRE, and SIEVERT codes, show a mean effective dose rate ranging between 3 and 4 mSv/h. However, the majority of codes stay unpredictable, thus the authors have used the Neural Network system NNT associated with CARI code to predict values of effective doses and heliocentric potentials (Hp) which the authors have obtained at least for some months ahead. According original, Morlet reconstructed and extrapolated Hp variations functioning as a measure of solar activity they have shown 8 to 13 years cycles. The first next maximum of Hp≈1400MV is located around 2022-2024. The minima of Hp corresponding to highest fluxes of cosmic rays are located around 2015 and 2035 years. Two classes of big periods are also found around 20-33 years and 75 years. Especially Morlet monthly analysis showed two main periods of 6 and 12 months, long periods of 5-6.25 and 11 years. Short structures are also detected Since the Earth is permanently bombarded with energetic cosmic rays particles, cosmic ray flux has been monitored by ground based neutron monitors for decades. Thus the authors give their investigations about decomposition provided by Morlet wavelets technique, using data series of cosmic rays variabilities. The wavelet analysis constitute an input data base for NNT system with which the authors can then predict decomposition coefficients and all related parameters for other points on the earth, they have studied the Mediterranean case in which the authors don't have any information about cosmic rays intensities. NNT associated with wavelets seem to be very suitable, the authors have now a kind of virtual NM for these locations on the earth.
Preface
ix
Chapter 4 - The word “laser” is an acronym which stands for “Light Amplification by Stimulated Emission of Radiation”. Laser is light energy that is part of the electromagnetic wave spectrum. Most of the commercially available lasers are either in the visible or infrared spectrum of light. Albert Einstein originally described the theoretical basis of stimulated emission in 1917. The ruby laser was the first to be built. This device was designed and constructed by Maiman in 1969 at Hughes Laboratories. The decade of the 1960s saw the development of most lasers that are commercially available today. The rapidity of development attests to the fact that the state of the art in physics and engineering had progressed to the point where new laser sources could be developed with existing technology. Chapter 5 - The complete expressions of the full fermion-boson vertex functions with transverse component in four dimensional QED are presented by solving a complete set of the Ward Takahashi type’s identities in the momentum space without considering the constraint imposing any Ansatz. In the colculation of reducing vertex function , the topological singularity of the various fermion currents coupling gauge field is taken fully into account.The computation shows that there is no anomaly for the transverse Ward-Takahashi relation for the vector vertex and axial-vector vertex. Chapter 6 - The authors investigate the condensate density and the condensate fraction of conduction electrons in weak-coupling superconductors by using the BCS theory and the concept of off-diagonal-long-range-order. The authors discuss the analytical formula of the zerotemperature condensate density of Cooper pairs as a function of Debye frequency and energy gap, and calculate the condensate fraction for some metals. The authors study the den- sity of Cooper pairs also at finite temperature showing its connection with the gap order parameter and the effects of the electron-phonon coupling. Finally, the authors analyze similarities and differences between superconductors and ultracold Fermi atoms in the determination of their condensate density by using the BCS theory. PACS numbers: 74.20.Fg; 74.70.Aq; 03.75.Ss. Chapter 7 - The authors study the spontaneous symmetry breaking (SSB) of a superfluid Bose-Fermi (BF)mixture loaded into a double-well potential (DWP), in the effectively onedimensional setting. The mixture is described by the Gross-Pitaevskii equation (GPE) for the bosons, which is coupled to an equation for the order parameter of the Fermi superfluid, which is derived from the respective density-functional model in the unitarity limit (a similar model may apply to the Bardeen-Cooper-Schriefer (BCS) regime too). Straightforward SSB in the quantum Fermi gas loaded into a DWP is impossible, as it requires an attractive selfinteraction acting in the medium, while the intrinsic nonlinearity in the Fermi gas may only be repulsive. However, the authors demonstrate that the symmetry breaking can be made possible in the mixture, provided that interaction between the fermions and bosons is attractive (a real example is themixture of potassium and rubidium atoms, which represent fermions and bosons, respectively). Numerical results for the SSB are represented by dependencies of asymmetry parameters for both components on numbers of particles in the mixture, and by phase diagrams in the plane of these two numbers. The diagrams display regions of symmetric and asymmetric ground states of the mixture. Dynamical pictures of the SSB, induced by a gradual transition from the single-well potential into the DWP, are reported too. In addition to the systematic numerically generated results, an analytical approximation is elaborated for the case when the GPE for the boson wave function is amenable to the application of the Thomas-Fermi (TF) approximation. Under a special linear relation between the numbers of fermions and bosons, the TF approximation makes it
x
Albert Reimer
possible to reduce the model to a single equation for the fermionic function, which includes competing repulsive and attractive nonlinear terms, of powers 7/3 and 3, respectively. The latter terms directly illustrates the generation of the effective attraction in the Fermi superfluid, mediated by the bosonic component of the mixture, whose density is “enslaved” to the fermion density, in that case. Chapter 8 - The formulation of the theory of q-deformed fermions has been of considerable interest in the literature. The authors have formulated the theory of q-deformed fermions in considerable detail and investigated the thermodynamics of such systems. The algebra, Fock space and the thermodynamics of q-deformed fermions has been fully investigated. The distribution function of such systems has been studied as a function of the deformation parameter and the behavior of the ideal q-fermion gas has been compared with that of the ordinary fermions. More recently, the interpolating statistics of q-fermions have been studied in terms of B-type and F-type interpolating statistics. The distribution function of such systems has been determined in terms of their analytic forms and has also been expressed as infinite continued fractions. The advantage of such infinite continued fractions is in clarifying the nature of the approximations. Moreover, the statistical mechanics of particles obeying interpolating statistics has been formulated in terms of q-deformed oscillator algebras of q-bosons and q-fermions on the basis of Feynman’s method of Detailed Balance. This formulation describes the connection between anyons (statistics which interpolates between standard bosons and fermions) and the principle of Detailed Balance and investigates the distribution function and other thermodynamic functions as infinite continued fractions. This formulation of interpolating statistics has also been studied in the context of Haldane and Gentile statistics. The formulation of interpolating statistics or intermediate statistics has also been shown to be linked to deformed oscillator algebras. Deformed permutation in terms of a parameter _ has been shown to imply the existence of the basic number which is shown in turn to imply the deformed algebra. In this formulation, the occupation number is generalized to the basic number N which is expressed in terms of the parameter _ which in turn leads to the deformed algebra corresponding to intermediate statistics. The authors thus find that the subject of q-deformed fermions has been investigated rather thoroughly – not only a formulation in terms of the Fock space of states but also the consequences for the various thermodynamic property of the particles obeying such statistics. Chapter 9 - The authors derive the natural embedding of fermion manifolds {|u }, {|d }, {|s }, {|c }, {|b }, {|t } into the Minkowski algebra. Using six isomorphic Cartan subalgebras and a generalization of Cartan’s concept of isotropic vector fields, the authors obtain the natural spinor manifolds of the spacetime-SU(3) calculated from the Clifford algebra Cℓ3,1 by the aid of minimal left ideals. Together with the previously constructed spacetime group this concept removes the necessity for auxiliary bundles that Yang- Mills theories presently require.
In: Horizons in World Physics. Volume 271 Editor: Albert Reimer
ISBN: 978-1-61761-884-0 © 2011 Nova Science Publishers, Inc.
Chapter 1
NUMERICAL MODELLING OF WIND WAVES: PROBLEMS, SOLUTIONS, VERIFICATIONS, AND APPLICATIONS V. G. Polnikov Obukhov Institute for Physics of Atmosphere of Russian Academy of Sciences, Moscow, Russia
ABSTRACT Due to stochastic feature of a wind-wave field, the time-space evolution of the field is described by the transport equation for the 2-dimensional wave energy spectrum density, S (σ , θ ; x, t ) , spread in the space, x, and time, t. This equation has the forcing named the source function, F, depending on both the wave spectrum, S , and the external wave-making factors: local wind, W(x, t), and local current, U(x, t). The source function, F, is the “heart” of any numerical wind wave model, as far as it contains certain physical mechanisms responsible for a wave spectrum evolution. It is used to distinguish three terms in function F: the wind-wave energy exchange mechanism, In; the energy conservative mechanism of nonlinear wave-wave interactions, Nl; and the wave energy loss mechanism, Dis, related, mainly, to the wave breaking and interaction of waves with the turbulence of water upper layer and with the bottom. Differences in mathematical representation of the source function terms determine general differences between wave models. The problem is to derive analytical representations for the source function terms said above from the fundamental wave equations. Basing on publications of numerous authors and on the last two decades studies of the author, the optimized versions of the all principal terms for the source function, F, have been constructed. Detailed description of these results is presented in this chapter. The final version of the source function is tested in academic test tasks and verified by implementing it into numerical shells of the well known wind wave models: WAM and WAVEWATCH. Procedures of testing and verification are presented and described in details. The superiority of the proposed new source function in accuracy and speed of calculations is shown.
2
V. G. Polnikov Finally, the directions of future developments in this topic are proposed, and some possible applications of numerical wind wave models are shown, aimed to study both the wind wave physics and global wind-wave variability at the climate scale, including mechanical energy exchange between wind, waves, and upper water layer.
Keywords: wind waves, numerical model, source function, evolution mechanisms, buoy data, fitting the numerical model, validation, accuracy estimation, inter-comparison of models.
1. INTRODUCTION This chapter deals with theoretical description of wind wave phenomenon taking place at the air-sea interface. Herewith, the main aim of this description is directed to numerical simulation of the wind wave field evolution in space and time. As an introduction to the problem, consider a typical scheme of the air-sea interface. In simplified approach it consists of three items (Figure 1): • • •
Turbulent air boundary layer with the shear mean wind flow having a velocity value W10(x) at the fixed horizon z =10m; Wavy water surface; Thing water upper layer where the turbulent motions and mean shear currents are present.
The main source of all mechanical motions of different space-time scales at the air-sea interface is a mean wind flow above the surface, which has variability scales of the order of thousand meters and thousand seconds. The turbulent part of a near-water layer (boundary layer) has scales smaller than a meter and a second. Variability of the wavy surface has scales of tens meters and ten seconds, whilst the upper water motions have a wide range of scales covering all mentioned values. Thus, the wind impacts on the water upper layer indirectly via the middle scale motions of wind waves, and this impact is spread through a wide range of scales, providing the great importance of wind wave motion on the global scale. Besides of the said, this phenomenon has its own scientific and practical interest. The former is provided by a physical complexity of the system, whilst the latter is due to dangerous feature of the phenomenon. All these features justify the long period interest to the problem of wind wave modeling, staring from the well know paper by Stokes (1847). From scientific point of view it is important to describe in a clear mathematical form a whole system of mechanical interactions between items mentioned above, responsible for the exchange processes at the air-sea interface. This is the main aim of the interface hydrodynamics. From practical point of view, a mathematical description of these processes permits to solve a lot of certain problems. As an example of such problems one may point out an improvement of wave and wind forecasting, calculation of heat and gas exchange between atmosphere and ocean, surface pollution mixing and diffusion, and so on.
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
3
Figure 1. The air-sea interface system
Direct mathematical description of mechanical exchange processes in the system considered is very complicated due to multi-scale and stochastic nature of them (for example, see Kitaigorodskii & Lamly, 1983). It can not be done in an exact form. Nevertheless, real advantage in this point can be reached by consideration of the problem in a spectral representation. Up to the date, a principal physical understanding exchange processes at the air-sea interface was achieved to some extent (Proceedings of the symposium on the wind driven air-sea interface, 1994; 1999), and mathematical tool for their description in spectral representation was constructed (for example, see Hasselmann, 1962; Zakharov, 1974; Phillips, 1977). Thus, one may try to make description of main processes at the air-sea interface from the united point of view. Below, we consider the main theoretical procedures needed to manage this problem.
2. FUNDAMENTAL EQUATIONS AND CONCEPTIONS From mathematical point of view, a wind wave field is a stochastic dynamical process, and the properties of this field should be governed by a proper statistical ensemble. Therefore, the best way of the phenomenon description lies in the domain of statistical characteristics, the main of which for a non-stationary field is the two-dimensional spatial wave energy spectrum, S(k, x,t) ≡ S , spread in the space, x, and time, t . Traditionally, the space-time evolution of this characteristic is described by the so called transport equation written in the following spectral representation (Komen et al, 1994)
∂S ∂S ∂S + C gx + C gy = F ≡ N l + In − D is . ∂t ∂x ∂ y
(2.1)
Here, the left-hand side is the full time-derivative of the spectrum, and the right-hand side is the so called source function (“forcing”), F. Vector ( C gx ,C gy ) is the group velocity one, corresponding to a wave component with wave vector k, which is defined by the ratio
4
V. G. Polnikov
Cg =
∂σ (k ) k = ( C gx , C gy ) ∂k k
(2.2)
Dependence of frequency σ (k ) on the wave vector k is given by the expression
σ = gk ,
(2.3)
known as the dispersion relation for the case of deep water, considered below. The left-hand side of equation (2.1) is responsible for the “mathematical” part of model. The physical essence of model is held by the source function, F, depending on both the wave spectrum, S , and the external wave-making factors: local wind, W(x,t), and local current, U(x,t). At present, it is widely recognized that F can be written as a sum of three terms – three parts of the united evolution mechanism for wind waves: • • •
The rate of conservative nonlinear energy transfer through a wave spectrum, Nl , (“nonlinear-term”); The rate of energy transfer from wind to waves, In , (“input-term”); The rate of wave energy loss due to numerous dissipative processes, Dis , (“dissipation-term”).
The source function is the “heart” of the model. It describes certain physical processes included in the model representation, which determine mechanisms responsible for the wave spectrum evolution (Efimov& Polnikov, 1991; Komen et al, 1994). Differences in representation of the source function terms mentioned above determine general differences between different wave models. In particular, the models are classified with the category of generations, by means of ranging the parameterization for Nl-term (The SWAMP group, 1985). This classification could be extended, taking into account all source function terms (for example, see Polnikov, 2005, 2009; Polnikov&Tkalich, 2006). The worldwide spread models WAM (The WAMDI group, 1988) and WAWEWATCH (WW) (Tolman&Chalikov, 1996) are the representatives of such a kind models, which are classified as the third generation ones. Differences in representation of the left hand side of evolution equation (2.1) and in realization of its numerical solution are mainly related to the mathematics of the wave model. Such a kind representation determines specificity of the model as well. But it is mainly related to the category of variation the applicability range of the models (i.e. accounting for a sphericity of the Earth, wave refraction on the bottom or current inhomogeneity, and so on). We will not dwell on this issue more in this chapter. Note that equation (2.1) has a meaning of the energy conservation law applied to each spectral component of wave field. Nevertheless, to have any physical meaning, this equation should be derived from the principal physical equations. By this way, the most general expressions for the source terms could be found. And this is the main problem of the task considered here.
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
5
Since pioneering paper by Stokes (1847), the basic hydrodynamic equations, describing the wave dynamics at the interface of an ideal liquid, are as follows
ρ
r du = −∇ 3 P − ρ g + f (x, t ); dt
z =η ( x ,t )
,
∂ρ r + ∇ 3 ( ρu ) = 0 , ∂t
uz
z =η ( x ,t )
uz
=
r ∂η + (u∇ 2η ) , ∂t
z =−∞
=0
(2.4)
(2.5)
(2.6)
(2.7)
Here, the following designations are used:
ρ ( z , t ) is the fluid density; u( x, z, t ) = ( u x , u y , u z ) is the velocity field; P ( x, z , t ) is the atmospheric pressure; g is the acceleration due to gravity; f ( x, z , t ) is the external forcing (viscosity, surface tension, wind stress and so on); η ( x, t ) is the surface elevation field; x = ( x, y ) is the horizontal coordinates vector; z is the vertical coordinate up-directed; r ∂ ∂ ∇ 2 = ( , ) is the horizontal gradient vector; ∂x ∂y r r ∂ ∇ 3 = (∇ 2 , ) is the full gradient, ∂z and the full time-derivative operator is defined as
r ⎞ d ⎛∂ (...) = ⎜ + u∇ 3 ⎟(...) . dt ⎝ ∂t ⎠
We remind that Eq. (2.4) is the main dynamic equation used at the water surface z =
η ( x , t ) , Eq. (2.5) is the mass conservation law, Eq. (2.6) is the kinematical boundary condition at surface η ( x , t ) , and Eq. (2.7) is the boundary condition at the bottom. Note that Eqs. (2.4) and (2.6) are principally nonlinear. General problem is to derive all source terms from the set of equations (2.4)-(2.7), taking into account a stochastic feature for motions near interface. It is easy to understand that the posed problem is quite complicated. Nevertheless, it can be solved under some approximations, if one takes into account each evolution mechanism separately. The history of such investigations is described in quite numerous papers, the main results of which are
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V. G. Polnikov
accumulated in numerous books (Komen et al, 1994; Young, 1999; and others). Below we reconstruct some principal results of these papers, permitting us to show the state-of-the-art in this field of hydrophysics. To this end, first of all, one should introduce the rules of transition form physical fields variables, u( x, z , t ) , η ( x, t ) , and f ( x, z , t ) , to its spectral representation. To do this, the so called Fourier-Stiltjes decomposition is introduced for each of the fields mentioned. As far as the main equations are used at the interface surface, we demonstrate this decomposition procedure on the example of surface elevation field. In such a case, one writes
η (x, t ) = const ⋅ ∫ exp[i (kx )]ηk (t )dk
(2.8)
k
Here,
ηk (t ) is the so called Fourier-amplitude of the field η ( x, t ) , taking in mind that
this field is non-stationary, but homogeneous. In such a case, only, the exponential decomposition is effective in a further simplification of the equations (for details, see, Monin&Yaglom, 1971). By substitution of the decompositions of the kind (2.8) for each field into the system of Eqs. (2.4)-(2.7), one could get the final equation for the main variable, ηk (t ) , in the form
∂ηk / ∂t = func1[ηk , uk , f k ]
(2.9)
where the right hand side of (2.9) represents a complicated functional having as an arguments the Fourier-amplitudes for each field variables. Then, one introduces the wave energy spectrum S(k) by the rule '
<< ηks (t )ηks ' (t ) >>= S (k , t )δ ( k − k ' )δ ( s + s ' ) .
(2.10)
Here, the brackets <<…>> means the wave statistical ensemble averaging, indexes s and s means the sign of imaginary part of the complex conjugated form for ηk (t ) , and δ(…) is ’
the Dirak’s delta-function. After this, one should execute some procedure, including multiplication of Eq. (2.9) by the complex conjugated amplitude, ηk * (t ) , and summation the resulting equation with the complex conjugated one. Finally, one gets the final equation of the form
∂S ( k ) / ∂t = func 2[ηk ,ηk * , uk , f k ]
(2.11)
resembling Eq. (2.1), but written without spatial derivatives (i.e. for a homogeneous field). The right hand side of Eq. (2.11), if specified, is the sought source function. Thus, with some simplifications and introducing certain rules of statistical averaging, the theoretical way of derivation for the spectrum evolution equation is found. Generalization of the procedure described above for the case of inhomogeneous wave field, η ( x, t ) , is less
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
7
important for our further consideration. The readers could find this generalization in paper (Zaslavskii& Lavrenov, 2005). So, we could start to consider description and parameterizations for the definite source function terms responsible for certain physical evolution mechanisms of wind wave spectrum, which are needed further for a wind wave numerical simulation.
3. WAVE EVOLUTION MECHANISM DUE TO NONLINEARITY 3.1. General Grounds Nonlinear feature of fundamental equations (4) and (6) leads to a certain nonlinear mechanism of evolution for wave spectrum. This mechanism, described by the source term Nl, is theoretically the most investigated among others evolution mechanisms. For the first time the Nl term was derived by Hasselmann (1962), under some reasonable theoretical suggestions. Later it was rederived in the terms of very elegant Hamiltonian formalism by Zakharov (1968, 1974). Following to the latter papers, we show several points of this derivation. First of all, we point out that the nonlinear theory is constructed under very certain suggestions: potential motions, no external forcing, and weak nonlinearity. The latter means existence of small values for wave steepness parameter defined by
ε = k p << η 2 >>1/2
(3.1)
where k p is the modulus of the dominant wave vector corresponding to the peak of spectrum, and << η >> 2
1/2
is the mean wave amplitude.
For the ideal fluid without forcing and the potential motion approximation, it was shown (Zakharov, 1968) that the system of Eqs. (2.4)-(2.7) is the Hamiltonian one. It means that the system can be replaces by classical Hamiltonian equations of the form
∂η δ H , = ∂t δΦ
(3.2)
∂Φ δH =− ∂t δη
(3.3)
where H (η , Φ ) is the Hamiltonian function of two canonical variables. Zakharov has found that canonical variables are the surface elevation, η ( x, t ) , and the velocity potential at this
r
surface, Φ (x, t ) ≡ ϕ (x, z, t ) z =η ( x ,t ) , where ∇3ϕ ( x, z, t ) = u( x, z, t ) is the wave velocity potential. Eventually, in terms of Hamiltonian formalism, the system of nonlinear equations (2.4)-(2.7) can be reduced to one of the form
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V. G. Polnikov
i
∂a (k ) = σ ( k )a ( k ) + ∂t
[
]
+ ∫ U 0(1,1), 2 a(k1 )a(k 2 )δ 0−1−2 + 2U1(,10), 2 a(k1 )a* (k 2 )δ1−0−2 + U 0(3,1), 2 a* (k1 )a* (k 2 )δ 0+1+2 dk 1dk 2 +
∫
+ V0,1, 2 ,3 a (k 1 )a (k 2 )a (k 3 )δ 0 +1−2 −3 dk 1dk 2 dk 3 ( n)
( n)
( 2)
*
where U 0,1,2 , V0,1,2 ,3 are known functions of ki (i = 0,1,2,3), and
(3.4)
δ 0−1−2 = δ ( k − k1 − k 2 )
are the Dirak’s delta-functions. For us it is important that equation (3.4) is an analog of eq. (2.9) with the only difference that Fourier-amplitude a(k) is the specially defined so called “normal” variable related linearly to the Fourier-amplitudes for elevation, ηk , and for wave velocity potential at the surface, Φ k . The spectrum defined for amplitude a(k) by the rule (2.10) is called as the wave action spectrum, N ( k ) , which is linearly related to the wave energy spectrum, S ( k ) , by definition of variable a(k). Details of this theory are not principal for us here. The most important fact is that the standard procedure described above results in the final evolution equation for spectrum N ( k ) with the right hand side provided by the nonlinearity of the system under consideration. Just this part is the nonlinear evolution term, Nl(k). At present, it is well recognized that the exact theoretical result for the nonlinear source term has the kind of the so called four-wave kinetic integral
∂N ( k 4 ) / ∂t ≡ Nl[ N ( k 4 )] = 4π ∫ dk 1 ∫ dk 2 ∫ dk 3 M 2 ( k1 , k 2 , k 3 , k 4 ) ×
⎡⎣ N (k1 ) N (k 2 ) ( N (k 3 ) + N (k 4 ) ) − N (k 3 ) N ( k 4 ) ( N ( k1 ) + N ( k 2 ) ) ⎤⎦ × (3.5) ×δ [σ (k1 ) + σ (k2 ) − σ (k3 ) − σ (k4 )]δ ( k1 + k 2 − k 3 − k 4 ) Here, ki is the wave vector having to a proper frequency-angular wave component
(σ i ,θ i ) (i = 1,2,3,4), М(…) are the matrix elements describing intensity of interactions between four waves, and δ (...) is the Dirak’s delta-function providing the resonance feature of interactions. The properties of kinetic integral (3.5) were studied numerically in numerous papers for very different spectral shapes (for references, see Polnikov&Farina, 2002; or, in more details, Polnikov, 2007). It was found that the main feature of Nl (σ , θ ) is a permanent shift of the spectrum to lower frequencies with a very certain changing the shape of spectrum (Figure 2). Additionally it should be noted that Nl-mechanism is proportional to the 6th power of small parameter of nonlinearity, ε . For this reason, the rate of nonlinear energy transfer through the spectrum is proportional to ε 4 , i.e. it is very slow mechanism. Nevertheless, just Nl term is responsible for wave length enlarging in the course of wind wave evolution, permitting waves to grow to large heights, but maintaining small value of wave steepness, ε .
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
9
S(σ)
0.001
t t t t t t t
=0 = 250 = 1000 = 10e4 = 10e5 = 10e6 = 10e7 Y = 2.788E-010* pow(X,16.08) Y = 0.04252* pow(X,-5.869) Y = 0.7133e-3*pow(X,-2.596)
Y = 2.788E-010* pow(X,16.08)
0.0001
Spectrum
1e-005
1e-006
Y = 0.7133e-3*pow(X,-2.596) 1e-007
1e-008 Y = 0.04252* pow(X,-5.869) 1e-009
1e-010 0
4
8
12
16
W
20
σ
Figure 2. Example of numerical solution of wave evolution due to nonlinear mechanism (following to Lavrenov& Polnikov, 2001)
Regarding to practical use of these theoretical results, due to great complicity of term (3.5), it needs to find an optimal approximation of the kinetic integral, which conserves all its properties. This point was a subject of numerous investigations, among which there is a special research executed in Polnikov&Farina (2002) and Polnikov (2003). By comparing several theoretically justified approximations for Nl (σ , θ ) , it was found that the most efficient one (in terms of a specially defined criterion for accuracy and speed of calculation) is the discrete interaction approximation (DIA) proposed by Hasselmann et al. (1985). Just this approximation is used in widely spread models WAM and WAVEWATCH. Due to this, it is worth wile to dwell in more details on these results.
3.2. The Most Effective Approximation for Nl Term First of all, let us say several words about DIA approximation. This approximation means replacing the whole continuum of summands under kinetic integral (3.5) by the single one. This single summand is especially chosen, to get the best accuracy against exact kinetic integral, and the choice is defined by the geometry of four interacting wave vectors (ki where i = 1,2,3,4). The optimal configuration of four wave
10
V. G. Polnikov
vectors in DIA, proposed in original paper (Hasselmann et al, 1985) is shown in Figure 3. In the polar coordinates, (σ ,θ ) , the original configuration is governed by the following rations: 1.
k1 = k 2 = k , where the arbitrary wave vector k is represented by σ and θ ;
2.
k 3 = k + , where k + is represented by σ + = σ (1 + λ ) and θ + = θ + Δθ + ;
3.
k 4 = k − , where k − is represented by σ − = σ (1 − λ ) and θ − = θ − Δθ − ;
4. Parameters of the configuration are
λ = 0.25, Δθ + = 11.5 o, and Δθ − = 33.6o.
(3.6)
In such a case, the nonlinear term Nl(k) at all mentioned k-points takes the form
Nl (k − ) = I (k , k + , k − ) , Nl ( k + ) = I ( k, k + , k − ) , Nl ( k ) = −2 I ( k, k + , k − ) , (3.7) where
I ( k , k + , k − ) = C NL g −8σ 19 ⎡⎣ N 2 ( k ) ( N ( k + ) + N ( k − ) ) − 2 N ( k ) N ( k + ) N ( k − ) ⎤⎦ .(3.8)
Figure 3. The original optimal configuration of four wave vectors in DIA (Hasselmann et al, 1985)
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
11
Figure 4. The configuration of Fast DIA is shown by dashed lines
In (1.19) CNL is the fitting constant. The net nonlinear term at any fixed (σ ,θ ) -point is found by the procedure of running of Eqs. (3.7)-(3.8) through all points of the frequencyangle integration grid σ i ,θ j .
{
}
According to study Polnikov&Farina (2002), features of the original DIA are as follows. F1. A mean relative error (over the set of 16 representative spectral shapes) is about 60%. F2. Relative consuming time for calculation of Nl term (in WAM with DIA approximation) is about 48% of the CPU. Despite of such a kind features, in the study mentioned it was found that DIA is the best among others approximations. Herewith, a simple consideration has shown that the reason of small accuracy is due to replacing the exact kinetic integral by the only summand. The reason of high consuming time is provided by necessity to make the spectrum interpolation for wave vectors k 3 and k 4 , which are not allocated at the frequency-angle integration grid σ i ,θ j .
{
}
In the studies mentioned it was found that the shortages said could be reduced to some extent. The first step of approximation improvement was a proposal to refuse of the resonant feature of four-wave interaction, and to allocate all four vectors at the integration grid σ i ,θ j . In such a case, the new configuration was constructed, which is shown in Figure 4
{
}
by dashed lines. (Moreover, the vectors k1 and k2 are meant to be allocated on the integration grid, but they are not necessarily equal each to other.) It allows us to exclude the interpolation procedure and make calculation of Nl faster. This version of DIA was named as FDIA. The relative consuming time in FDIA is reduced twice (improvement of feature F2). The second step of DIA improvement was a choice of more efficient configuration. After some attempts, the best configuration was found (Polnikov&Farina, 2002; Polnikov, 2003). This has permitted us to reduce the relative error of approximation to 40% (improvement of feature F1). Taking into account these two advantages, we propose to use the following new version of FDIA for a numerical representation of Nl term. (1) The calculating frequency-angular grid, exponential frequency grid
{σ i , θ j } , is defined typically:
12
V. G. Polnikov
σ ( i ) = σ 0 ei −1
(1 ≤ i ≤ N),
(3.9a)
and equidistant angular grid
θ ( j ) = −π + ( j − 1 ) ⋅ Δθ (1 ≤ j ≤ M),
(3.9b)
σ 0 , e , and Δθ are the grid parameters specified in a certain numerical
where model.
(2) The reference wave component,
(σ 4 , θ4 ) , is allocated at any current node of the
grid (1.20). (3) The other 3 waves have the components allocated at nodes of the same grid and defined by the ratios
σ 1 = σ 4 e m1 , σ 2 = σ 4 e m 2 , σ 3 = σ 4 e m 3 ,
(3.10a)
Δ θ 1 = n1 Δθ , Δ θ 2 = n2 Δθ , Δθ 3 = n3 Δθ ,(where Δθi ≡ θi − θ 4 ) .
(3.10b)
Thus, the optimal configuration of four interacting waves is given by the certain set of integers: m1, m2, m3; n1, n2, n3, which, in turn, are to be especially calculated, in dependence on the grid parameters, e and Δθ (Polnikov&Farina, 2002). For the frequency-angle grid with parameters e = 1.1 and Δθ = π / 12 , which are typical for the models WAM and WW, the most effective configuration is given by the following parameters (Polnikov, 2003) m1 = 3, m2 = 3, m3 = 5; n1 = n2 = 2, n3 = 3.
(3.11)
Finally, Nl-term is calculated by the standard formulas (3.7)-(3.8), making the loops for the reference components,
( σ , θ ) , arranged through the grid (3.9) . For completeness,
these formulas for the energy spectrum representation are as follows
Nl (σ 4 , θ 4 ) = Nl (σ 3 , θ 3 ) = I (σ 1 , θ1 , σ 2 , θ 2 , σ 3 , θ 3 , σ 4 ,θ 4 ) ,
(3.12a)
Nl (σ 1 , θ1 ) = Nl (σ 2 , θ 2 ) = − I (σ 1 , θ1 , σ 2 , θ 2 , σ 3 , θ 3 , σ 4 ,θ 4 ) ,
(3.12b)
where
I (...) = Cnl g −4σ 411 ⎡⎣ S1S2 ( S3 + (σ 3 / σ 4 )4 S4 ) − S3S4 ((σ 2 / σ 4 )4 S1 + (σ 1 / σ 4 )4 S2 )⎤⎦ . (3.13)
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
13
Here, С nl is the only fitting dimensionless coefficient, and notation Si ≡ S (σ i , θi ) is used. The coefficient С nl should be fitted together with other source terms simultaneously. Hereby, the modern and comprehensive description of Nl term parameterization is finished, and we can go to the next terms.
4. WIND WAVE ENERGY PUMPING MECHANISM 4.1. General Grounds This mechanism is studied rather well both theoretically and empirically (for references, see recent review paper, The WISE group, 2007). Nevertheless, it is known much more less than Nl-mechanism described above in previous section 3. The reason of such a situation is a high complexity of processes taking place in a boundary layer of atmosphere located in the vicinity of waving surface. To get the main understanding of the problem, let us consider the simplest approach proposed in the earlier papers by Phillips (1957) and Miles (1960) and summarized in book (Phillips, 1977). To the purpose said, let us rewrite the system of Eqs. (2.4)-(2.7) in a linear approximation and potential motion approach. We have
∂Φ = − gη + P ( a p ) ∂t
(4.1)
∂η ∂Φ = ∂t ∂z
(4.2)
Δϕ = 0
(4.3)
∂ϕ ∂z
z =−∞
= 0.
(4.4)
In this representation, the first two equations are written at the surface η ( x, t ) , whilst two equations, (4.3) and (4.4), done in the whole water layer. The last term in the r. h. s. of (4.1), P ( a p ) , means the result of transition to the potential representation for the atmospheric pressure forcing, real and unequivocal representation of which is not specified yet. Here, P ( a p ) is normalized by the water density, ρ w , and ap is the formal argument of the forcing function. For more certainty, note that the first two equations, (4.1) and (4.2), are the main ones, as far as they determine unknown fields, η ( x, t ) and Φ ( x , t ) . Equations (4.3) and (4.4) are auxiliary ones; they are used for determination of the vertical structure for velocity potential ϕ ( x, z , t ) , only.
14
V. G. Polnikov
To make a transition into the spectral representation, one introduces the following Fourier-decompositions
η (x, t ) = const ⋅ ∫ exp[i (kx )]ηk (t )dk ,
(4.5)
ϕ ( x, z, t ) = const ⋅ ∫ exp[i ( kx )] f ( z )ϕ k (t )dk .
(4.6)
k
k
Hereafter, the wave vector, k, as well as the spatial vector, x, has the horizontal components, only, and f(z) is the so called vertical structure function, determined from Eqs. (4.3) and (4.4), solved analytically. In our case, f(z)=exp(-kz), what dose not play any principal role later. After substitution of representations (4.5), (4.6) into the system of Eqs. (4.1)-(4.4), equations (4.3) and (4.4) give the solution for function f(z) mentioned above, and the other two equations get the kind
& + gη = Π (k, a ) Φ k k p
(4.7)
η&k = k Φ k
(4.8)
The solution of system (4.7)-(4.8) radically depends on representation of the atmospheric pressure forcing, Π ( k , a p ) . This representation is the main theoretical problem which is not solved wholly till present. There are some simple solutions of the system (4.7)-(4.8), resulting in two principle mechanisms of wave pumping: so called Philips’ and Miles’ mechanisms. Their simplified treatments are as follows (for details, see Phillips, 1977). The atmospheric pressure forcing can be represented in the form
Π(k, a p ) = (α + i β )c 2kηk (t ) + γ k (t )
(4.9)
where the first term in the right hand side means the part of pressure oscillations induced by waves and the second term means the pure turbulent oscillations uncorrelated with wave motions. Besides, the induced oscillations have two constituents: one of them, having intensity coefficient α , is in phase with the surface elevations; and the other one, having intensity coefficient β , is in phase with the wave slope. As usually, c = c ( k ) = σ ( k ) / k ) is the phase velocity of the wave component with vector k. Substitution of (4.9) into system (4.7)-(4.8) with some rather complicated mathematics (via getting an equation of the form 2.9) leads to the following general solution for the energy wave spectrum S(k,t) defined by rule (2.10):
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
⎛ sh( βσ t ) ⎞ S (k , t ) = Cin A(k , σ ) ⎜ ⎟ ⎝ βσ ⎠
15
(4.10)
Here, CIN is the fitting coefficient of the theory, and A( k , σ ) is the 3-dimensional spectrum of the turbulent part of atmospheric oscillations, the kind of which it the main problem of the theory. But the temporal behavior of wave spectrum evolution is clear from result (4.10). Really, on the small time scale, when βσ t << 1, wave spectrum is proportional to time
S ( k, t ) = Cin A(k , σ )t
(4.11)
what is equivalent to the evolution equation
∂S ( k , t ) / ∂t = Cin A( k , σ ) .
(4.12)
In such a case the sought input term, In, is independent of wave spectrum. Such mechanism was called as the Phillips’ one (after paper Phillips, 1957). This mechanism is working on very small time scales, and for this reason it is not used in numerical simulation on oceanic scales. The role of result (4.11) is an explanation of the wind wave origin. The opposite case, when βσ t >>1, solution (4.10) results in the exponential growth of wave spectrum, what is equivalent to evolution equation of the form
∂S ( k, t ) / ∂t = Cin βσ S ( k, t ) = Cinσβ ( k, W) S ( k, t )
(4.13)
where the final expression of In term is written in the commonly used form. Dimensionless function β ( k , W ) , depending of wave vector and wind at the reference horizon, is often called as the Miles’ increment coefficient (or function). In accordance to the said earlier, specification of this function is the main difficulty of the interface dynamic theory and experiment. Do not dwelling on this problem in details, below we shall barely mention some experimental results (Snyder et al, 1981; Plant, 1982) and numerical simulation ones (Chalikov, 1980, 1998, 2002), which gave the grounds for practically used parameterizations of β ( k , W ) .
4.2. Effective Approximations for in Term Understanding the difficulty of measurements for wind-wave energy exchange is following from the fact that the first more or less accurate experimental results have been obtained in famous work by Snyder et al.(1981), only, i.e. much later than the first theoretical results mentioned above. On the basis of their results, Snyder and coauthors have constructed the following simplest parameterization
16
V. G. Polnikov
⎡
β( σ, θ, U) = max ⎢ 0, a
⎣
⎞⎤ ρ a ⎛ W5σ cos(θ − θ u ) − b ⎟ ⎥ , ⎜ ρb ⎝ g ⎠⎦
(4.14)
where ρa and ρw is the air and water density, respectively, g is the gravity acceleration, a and b are the fitting parameters, and W5 and
θ u is the local wind at horizon z=5m and its
direction, respectively. Parameters a and b are varying in the following intervals: a ≅ 0.2-0.3 и b ≅ 0.9-1. The main disadvantage of result (4.14) was a very short range of validity: 1 < W5σ / g < 3. Only one year later, Plant (1982) have accumulated a lot of experimental data and proposed an alternative parameterization
⎛u σ β = ( 0.04 ± 0.02 )⎜⎜ * ⎝ g
2
⎞ ⎟⎟ cos( θ − θ u ) ⎠
(4.15)
where
u* = Cd1/2 ( z )W ( z )
(4.16)
is the so called friction velocity, and Cd ( z ) was the drag coefficient for the horizon z. Advantage of this result is a much wider range of validity: 2.5 < W10σ / g < 75.
(4.17)
But the behavior of β in the vicinity of W10σ / g ≅ 1 was remaining unclear. This uncertainty was vanished in numerical simulation of air-sea interface dynamics, started by Chalikov (1980). By direct numerical solution of proper equations for wave and air motions in the interface boundary zone, he have found the rate of wind-wave energy exchange and proposed very effective parameterization of the kind
β = 10−4 ( aσ~ 2 + bσ~ + c ) .
(4.18a)
Here, a, b, c are the fitting parameters depending (in a rather complicated manner) on the
value of the drag coefficient, Cd ( z ) , and the non-dimensional frequency, σ~ , defined as
σ% = σ W (λ p ) / g
(4.18b)
where λp is the wave length corresponding to the peak frequency of the wave spectrum, The main advantage of the Chalikov’s result has two items:
σp.
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17
(1) Direct proof of the quadratic power dependence of β on W; (2) Evidence that in the case W10σ / g < 1, i.e. when waves overcome the local wind, β becomes negative with the order of magnitude of -10-5. Disadvantage of numerical simulations is a rather short range of validity, provided by a growing complexity of calculations with an increase of frequency under consideration. Thus, due to extreme complexity of the problem, the best way is to combine the most reliable experimental and numerical results. Just this way was used by the author (Polnikov, 2005). First of all, we have accepted the unified empirical result prepared by Yan (1987) from (4.14) and (4.15) in the form
βYAN
2 ⎧⎪ ⎡ ⎫⎪ ⎤ ⎛ u*σ ⎞ u*σ = ⎨ ⎢0.04 ⎜ + + − − 0.00544 0.000055 cos( ) 0.00031 θ θ ⎥ ⎬ .(4.19a) u ⎟ g ⎝ g ⎠ ⎥⎦ ⎪⎩ ⎢⎣ ⎪⎭
Secondly, we add a theoretically found and physically justified negative value of β for waves running faster than wind. Finally, we have got representation for β of the kind
β = max {β L , βYAN } where
(4.19b)
β L is the fitting parameter having the default value β L = 5 ⋅ 10−6
and
(4.19c)
βYAN is given by (4.19a).
Eventually, the most effective and physically justified parameterization of In term has the kind
In = Cinσ β ( k, W) S ( k, t )
(4.20)
with β ( k , W ) given by formulas (4.19). This parameterization has only two uncertainties: principal fitting coefficient, Cin , and auxiliary fitting parameter
β L having the default value
(4.19c). With respect to previous parameterizations, the proposed parameterization (4.20) has the triple advantage: (1) Relatively large frequency range of validity, given by rations (4.17); (2) Negative value of In for waves running faster than local wind; (3) Rather simple and reasonable mathematical representation.
18
V. G. Polnikov
As a conclusion of the section, and for completeness of our consideration, we could say that in the model WAM they use In term in form (4.20) with β ( k , W ) corresponding to formula (4.14), and in the model WW they use (4.20) with the Chalikov’s result (4.18) for β ( k , W ) . There are a lot of others parameterizations of β ( k , W ) (see review, The WISE group, 2007), but no of them has any superiority with respect to (4.19). Our experience of wind wave model constructions ( Polnikov, 1985, 1991) have showed that the approach for β ( k , W ) used in WAM is too simplified, but that one used in WW is too complicated, with no necessity. Thus, the parameterization of In proposed above in (4.19) is the best compromise of our present understanding the problem of wind-wave energy exchange.
4.3. Choice of the Wind Representation Finally, we touch a point dealing with the wind representation used in the input term: W10 , or u*. The principal difference between these too representations is realized only in the case of existing dependence Cd ( z ) (or u*) on the wave-origin parameters: wind W10 , wave age A = c p / W10 = g / σ pW10 , and the other, may be implicit parameters of the wind-waves system. In the case of constant value of Cd ( z ) , there is no principal difference in the wind representation. Thus, the question is the following: can one justify using the complicated form of Cd ( z,W10 , A,...) in a practical numerical modelling for wind waves? In this aspect we note that in such widely used models as WAM and WW, they apply one or another simplified form of dependence Cd ( z ) on wind W10 and wave age A, basing on empirical ratios for the former. But there are at least three reasons which give rise doubts of effectiveness of using dependencies of such a kind. They are as follows: (1) As it was shown in Banner and Young (1994), the impact of using a sea-state dependent function for u* in the input term is negligible for the forecast of wind waves. Very similar result was found in our recent calculations (Polnikov et al., 2002). (2) More over, it is well known (Donelan et al., 1993; Drennan et al., 1999) that empirical data for the drag coefficient, Cd ( z,W10 , A,...) , and for the sea roughness height, z0 (W10 , A,...) , have a very large scattering which provides a great uncertainty in empirical parameterizations for them. The problem of this uncertainty could be solved by constructing a special dynamic boundary layer (DBL) theory permitting to relate the wind-wave state, i.e. local 2D wave spectrum, S(k), with the parameters of the boundary layer of atmosphere, Cd , u*, and z0 . Our recent study (Polnikov et al, 2003), based on the DBL-theory, proposed in Makin& Kudryavtzev(1999), have shown that all present empirical parameterizations for the
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
19
dependencies mentioned are not correct. They should be replaced by more accurate (and more complicated) parameterizations, which are not established yet. (3) Finally, the necessity to use theoretically well justified dependencies
Cd ( z,W10 , A,...) and z0 (W10 , A,...) results in numerous additional calculations, what provides a great time consuming by numerical model, decreasing the practical effectiveness of the forecast. Nevertheless, the latter circumstance does not mean that one should refuse of calculations for the dependencies said above. But, to our mind, they are rather needed for estimation of the state of atmospheric boundary layer than for the state of wind waves. This point of view is based on the experience of our recent investigations (Polnikov, et al., 2002, 2003) where a preliminary version of a wind wave model with the DBL-block was considered. Just such a kind version of the model, with the DBL-block “switched on”, could be classified as the model of new (fourth) generation. There are attempts to make models of such a kind (Janssen, 1991, Polnikov et al, 2002), but the final solution of this problem is waiting its time. For the completeness of our consideration, we reproduce shortly the main ideas of constructing the DBL-block, which could be “switched on” optionally, if needed in a certain version of a wind wave model.
4.4. The Dynamic Boundary Block Construction There are three principal postulates making the basis of the DBL-block construction. They are as follows. (1) One supposes that at the local space point and time moment, the boundary layer system characterized with the constant value of the wind stress
τ = − ρ a < u x' u z' >≡ ρ a u*2 = const
(4.21)
what means that there is no dependence of total stress τ on vertical coordinate z. In turn, the total stress, τ , can be shared into two parts:
τ = τ t ( z ) + τ w ( z ) = ρ a u*2 Here, the first part,
(4.22)
τ t , does not supplies energy to waves, whilst the second part,
τ w , corresponds to the stress which is responsible for the energy pumping of waves. In terms of surface drag (see 4.16), they say that drag, and
τ t is responsible for the “skin”
τ w does for the “form” drag ( Donelan, 2003). In Russian literature τ w is
called as the “wave induced” stress, and
τ t is called as the “turbulent” or
20
V. G. Polnikov “tangential” stress. Each of these parts can depend on z under the condition of constant value for the total stress, τ . (2) By definition, the wave induced stress can be expresses via wave spectrum by the formula
τ w ( z = 0) ≡ τ w (0) = ρ w g ∫
k cos(θ )
ω
In ( S ,W , ω ,θ ) d ωdθ .
(4.23)
where In ( S ,W , ω, θ ) is the well know input term, discussed in subsection 4.2. (3) Dependence of all boundary layer parameters, namely, Cd , u*, and W(z), on the sea state can be found from dependence of friction velocity u* on wave spectrum S(k). To specify these dependencies, one should define dependence
τ t on Cd , u*, W(z),
or S(k), and to solve equation (4.22). Herewith, accepting the logarithmic law for vertical profile of the mean wind
W ( z) =
u*
κ
ln
z z0
(4.24)
is the additional condition of this approach. There are several approaches to construct the DBL-block (Polnikov, 2009b). Non of them are unequivocally acceptable. As an example of a DBL-model which could be a basis of the DBL-block in a wind wave model, we could refer to paper Makin&Kudryavtzev(1999) and give some ideas, following to this paper . As a particular approach, they accept the following dependence τ t on W ( z ) τ t ( z) = K
∂W ( z ) , ∂z
(4.25)
where K is the vertical mixing coefficient, specification of which was given by a special consideration. Finally, they found 3/4 z⎡ u z ⎡ τ ( z) ⎤ τ w ( z ) ⎤ −1 W ( z ) = u ∫ ⎢1 − 2 ⎥ K dz = * ∫ ⎢1 − w 2 ⎥ d (ln z ) . * ν⎢ u ⎥ u ⎥ κ zν ⎢⎣ z0 ⎣ * ⎦ * ⎦ 0 2
(4.26)
This principal result permits to specify all other characteristics of the boundary layer. Namely, after specification of dependence τ w ( z ) , they found
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
z
⎡ J (z) ⎤ 1− W ( z) = ⎢ ∫ κ zν ⎣ 1 + J (0) ⎥⎦ 0 u*
3/ 4
21
u d (ln z ) = * F ( z ) ,
κ
where
J ( z) =
ωmax
[exp( −10 zk ) cos(5πzk )] ⋅ k ∫ ∫ ω min
θ
2
⋅ β (...) S (ω, θ ) cos(θ )dσdθ , (4.27)
and some fitting theoretical parameters are introduced. Equation (4.27) gives the sought boundary layer characteristics:
u* = κW10 / F (10) , Cd (10) = (u* / W10 ) 2 z0 = 10 / exp[κU10 / u* ] .
(4.28)
Here we will not stay more on details of this DBL-model, which has its own advances and shortages. By this consideration we would like to state that a principal way for constriction of the DBL-block is known. Here, it is important to point out state that final solution of this problem permits to solve a lot of additional tasks besides simple wave forecast, basing on a wind wave models of fourth generation (see section 7 below).
5. WIND WAVE DISSIPATION MECHANISM 5.1. Overview of the Problem Let us shortly consider the present state of our knowledge about the dissipation term, Dis. First of all, we should state that there is no any widely recognized spectral representation for function DIS(S) (see the most recent review of the state-of-the-art, The WISE group, 2007). In particular, there is not understanding what is the power of spectrum in function DIS: the first power (as used in WAM and WW, according to Hasselmann’s theory, 1974), the second power (as proposed in Polnikov 1994, 1995, 2005), or the higher power (according to Phillips 1985; Donelan 2001; Hwang&Wang 2004). Second, there is not any theory where the issue of wind wave dissipation is considered from the most general point of view. All of innumerous theoretical papers, including direct derivations (Hasselmann 1974), dimensional considerations (Phillips, 1985), numerical simulations (Chalikov&Shanin, 1998; Zakharov et al., 2007), all of them do not take into account a presence of small-scale turbulence in the water upper layer, which interacts strongly with waves in reality. They consider, mainly, different aspects of wave breaking processes or accompanying effects, like white-capping in (Hasselmann 1974). Besides earlier papers by the author (Polnikov, 1994, 1995, 2005), the only exclusion is the paper (Tolman&Chalikov, 1996), where these authors tried to estimate the role of turbulent viscosity in the wave energy dissipation. But they have restricted themselves with a
22
V. G. Polnikov
very certain kind of parameterization for the viscosity term, what have led them to numerous, very particular parameterizations vulnerable from a theoretical point of view. The author’s earlier papers mentioned above, dealing with the dissipation of wind wave due to turbulence, are also rather particular and unsystematic. Thus, the problem of construction of more general and logically self consistent theory is strongly in demand at present. Third, some words about experimental results. Here we will not dwell on review of experimental researches in this field, as far as it is well done in the last paper (Babanin 2007). It needs only to mention, that exhaustive experimental studies of the dissipation mechanism for wind waves are hardly possible, especially in a spectral presentation, due to presence a lot of invisible and immeasurable processes in the water upper layer, governing the dissipation of waves. Therefore, as far as the empirical measurements are restricted by study of different aspects of the wave breaking only, they do not fully correspond to reality. Nevertheless, for the last 5-10 year, the efforts of experimenters are rather fruitful, and some established empirical effects related to the wind wave breaking process are quite interesting. In particular, following the papers (Banner&Tiang, 1998; Babanin et al., 2001) and some others (see references in Young & Babanin, 2006), we have to mention the following empirical effects related to the wave dissipation process: E1) Threshold feature of the wave breaking; E2) Influence of the breaking for long waves on the intensity of breaking for shorter waves; E3) Different feature of the dissipation rate for dominant waves and for waves in the tail part of wave spectrum; E4) More intensive breaking of waves running at some angle to the mean wind direction (i.e. two-lobe feature of the angular function for breaking intensity and for wave dissipation rate, consequently). All of these effects can serve as guidelines, to get a true theoretical representation of DIS. Below we reproduce a short version of the paper (Polnikov, submitted) which is directly devoted to construction the most general and logically self consistent theory of dissipation mechanism for wind wave.
5.2. Basic Statements The main fundamental of the theory states that on the scales of Eq. (2.1) validity, the most general reason for wind wave dissipation is a turbulence of the water upper layer. Herewith, specification of processes producing the turbulence is quite unprincipled. Really, it is evident that a reasonable part of the turbulence intensity is provided by the wave breaking processes. Thus, the breaking processes are taken into account in our statement, though by an implicit way. Herewith, it is equally clear that accompanying processes, as like as sprinkling, white capping, results of shear currents and wave orbital motions instabilities, taking place both in an atmosphere and in a water boundary layer, bubble clouds, and so on, all of them, in terms of hydrodynamics, are chaotic motions without
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
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any determined scale, i.e. the turbulence motions with respect to waves. Contribution of these motions to the wind wave dissipation is to be taken into account, as well. In other words, the proposed theoretical approach is based on including into considerations all dissipative processes leading to production of turbulence in the water upper layer. According to the said, without any restriction of the consideration generality, the current field in a waving water layer can be written in the form of two constituents
u( x, z , t ) = u w ( x, z , t ) + u ' ( x, z , t ) .
(5.1)
The first summand, u w , in the right-hand side (further, the r.h.s.) of (5.1), we treat as the potential motion attributed to wind waves. This motion is governed by the system of equations corresponding to one given by Eqs. (2.4)-(2.7), but written in the potential approximation. The second summand, u ' , is treated as the turbulent constituent of full velocity, totally uncorrelated with u w in statistical sense. As regards to the surface elevation, η ( x, t ) , we do not introduce analogous decomposition, attracting the well known Hasselmann’s hypothesis of “a small distortion in mean” for the surface profile, induced by the processes said above (Hasselmann 1974). This allows us to use conception of the surface elevation field, η ( x, t ) , in a traditional, commonly used sense.
5.3. Reynolds Stress Following to the said, let us rewrite the basic equations, (2.4) and (2.6), without external force but in the standard tensor kind
∂ui ∂u + ∑ u j i = − gδ i ,3 ∂t ∂x j j
∂η ∂η = u3 − ∑ ui ∂t ∂xi i =1,2
(5.2)
z =η ( ξ ,t )
(5.3)
z =η ( ξ ,t )
Here, the indexes i, j take values 1, 2, 3, corresponding to two horizontal and one vertical coordinates, respectively, and the sub-index,
z =η ( ξ ,t )
, means that the both equations are
written at the interface. Further the latter sub-index will be omitted for simplicity. Substitution of ratio (5.1) into Eqs. (5.2) and (5.3), with the next averaging of them over the turbulent scales and accounting the condition of incompressibility, , and ∂u / ∂ x = 0
∑
j
j
j
absence of a fluctuation part for η ( x, t ) , permits us to write the following system of equations valid for the wave motions only
24
V. G. Polnikov
∂ < ui' u 'j > ∂ui ∂u , + ∑ u j i = − gδ i ,3 − ∑ ∂t ∂x j ∂x j j j
∂η ∂η = u3 − ∑ ui ∂t ∂xi i =1,2
(5.4)
(5.5)
η and ui , are denoted with the bar which will be
Here, the mean wave variables,
omitted later with the aim of simplicity, and the additional term in the r. h. s. of Eq. (5.4), appearing due to nonlinearity of the system, is
∑ j
∂ < ui'u 'j > ∂x j
≡ Pi .
(5.6)
Physically, this term represents the disturbing force, Р, providing for the wave motion dissipation. To convince in this, it is enough to accept a simplest parameterization of the force in terms of wave velocity, u, of the kind
Pi ≡ ∑ j
with a constant value of factor
∂ < ui' u 'j > ∂x j
= −ν T
∂ 2ui ∑j ∂x 2 j
(5.7)
ν T . In such a case, term (5.7) is absolutely equal to expression
for the typical molecular viscosity force, what, by solution of the system (5.4)-(5.5) in a linear approximation and in a spectral representation for variables η and u , leads to the well known equation for the temporal evolution of wave spectrum of the kind (Hasselmann 1960)
% k S ( k , t ) ,(ν% = ν T k 2 / σ k ) ∂S ( k , t ) / ∂t = −2νσ
(5.8)
having an exponentially decay solution (details of Eq. (5.8) derivation can be found in Efimov&Polnikov 1991). Taking into account that dimension of the introduced constant, ν T , has one of viscosity, and the solution of (5.8) has a decay feature for the wave spectrum, with no doubt one can state that the additional term (5.6) in Eq. (5.4) has meaning of the dissipative term provided by turbulence in the water upper layer. This simple consideration helps us to draw further very important consequences. Really, the numerator under the sum in expression (5.6) is a very well known magnitude in the turbulence theory, which is called as the Reynolds stress (Monin&Yaglom, 1971)
< ui' u 'j >≡ τ ij
(5.9)
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which, in our representation, is normalized by the water density. Methods of parameterizations for τ ij are well developed. Therefore, to complete the theory, it needs to specify a representation for τ ij in terms of wave variables, η and u, to find evolution equation of the form (5.8), and to ascribe to the r.h.s. of final equation the physical meaning of dissipation term, DIS. To get final result, it is reasonable to use some principles of the Hasselmann’s approach to this problem (Hasselmann 1974), while finding the final evolution equation for wave spectrum. Just this program will be realized below.
5.4. Phenomenological Closure of Reynolds Stress First of all, let us formulate the main grounds of our concept for a procedure of Reynolds stress closure, the aim of which is to express the turbulent characteristic, τ ij , via the wave variables, η and u. One of them is the hypotheses of “a small distortion in mean”, mentioned above. This hypotheses permits to safe commonly used conception of the wave profile, η ( x, t ) , and introduce all derivatives of η ( x, t ) , if needed. The second ground is an assumption that the relative value of Reynolds stress term, Pi , is much greater of the dynamical nonlinear term described by the second summand,
∑u j
j
∂ui , in ∂x j
the l. h. s. of Eq. (5.4). Thereby, we postulate the statement of “strong” nonlinearity of the turbulent type, permitting to neglect the dynamical nonlinearity in this problem. Thus, we should solve, in fact, the following system of equations
∂ui + gδ i ,3 = − Pi (u,η ) ∂t
(5.10)
∂η = u3 ∂t
(5.11)
Third. The kind of closure (5.7) is too much simple, reflecting the meaning of “forcing” term (5.6) qualitatively, only. The principle shortage of this closure is reducing the nonlinear dynamics to the linear one. It is evident that more complex, nonlinear closure of the forcing term is more adequate to physics of the processes considered. The problem is to find such a closure which would be more general and have a reasonable physical treatment. Below we try to do it. One of such a kind version of the stress closure is related to using concept of the Prandtle’s mixing length, L, allowing to express the turbulent fluctuation velocity, u ' , via derivative of the wave velocity field, u , in the form (Monin&Yaglom, 1971)
ui' = Li ∂ui / ∂xi
(5.12)
26
V. G. Polnikov In such a case, Reynolds stress becomes a nonlinear function of wave variables
τ ij =< Li Li (∂ui / ∂xi )(∂u j / ∂x j ) > ,
(5.13)
what changes radically both the structure of final equations and solution of them. Note that closure (5.12), related to spatial derivatives of the wave velocity, u , is quite adequate for a horizontally homogeneous, near-wall turbulence. In such a case, values Li are ascribed to spatial scales of turbulent eddies, magnitudes of which may be postulated. In our case, the turbulence is realized under conditions of instability of waving interface, therefore, some modifications of the approach are needed. In particular, in the case of waving interface, there are not only possible but needed the closure versions related to derivatives of the wave profile, η ( x, t ) , for example, of the kind
ui' = Ci ∂η / ∂xi
(5.14)
where values Ci (by analogy to Li ) have meaning of scales of mixing velocity. Another important point, which should be modified in the closure approach, is the averaging procedure over the turbulent scales, applied for the turbulent velocities production,
ui' u 'j , to get a proper component of stress τ ij . With the aim to point out this circumstance, we did save the averaging brackets, <.>, in formula (5.13). Meaningfulness of these signs consists in the fact that the wave-like phase structure of those wave variables, η ( x, t ) and u(x,z,t ), which are used for closuring the turbulent value τ ij , should be destroyed (or smoothed) to a certain extent, during the process of averaging over turbulent scales. Here we mean the phase factors alike exp[i ( kx )] , standing under integral in the Fourierrepresentation of the fields η ( x, t ) and u(x, z, t). Such a kind representation is inevitable in the course of constructing a theory for wave spectrum evolution from dynamic equations (5.10)-(5.11) (see below). The assumption said above is rather extraordinary; nevertheless it has some physical grounds. Really, the wave-like phase information is inappropriate in a turbulent motion. This information should be suppressed to some extent by the averaging procedure over the turbulent scales. Herewith, this reasonable assumption is very fruitful, as far as it allows to manipulate more or less free with the phase factors in the Fourier-representation for the turbulent characteristic, τ ij , and, consequently, for the forcing function, Pi ( u,η ) . On the basis theoretical modifications postulated above, it is quite substantiated to represent the forcing term in the form of rather general quadratic function − Pi ( u,η ) =< ∑ j
∂ {[ Li ( ∂ui / ∂xi ) + Ci ( ∂η / ∂xi ) ] ⎡⎣ L j ( ∂u j / ∂x j ) + C j ( ∂η / ∂x j ) ⎤⎦} > . (5.15) ∂x j
Taking into account the presumptions done above, we should here emphasize that closure (5.15) maintains the following principal features of the problem:
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
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(a) Nonlinear nature of the dissipation process; (b) Dependence of the turbulent forcing on gradients of both surface elevation field, η ( x, t ) , and velocity one, u(x,z,t). Moreover, we have a freedom for manipulation with the phase factors in summand Pi ( u,η ) , while making transition to the Fourier-representation for dynamic equations (5.10)-(5.11). All these theoretical grounds have an evident physical meaning. Besides the physical content, closure (5.15) has an important technical advantage. The latter consists in the fact that the technique of derivation a spectrum evolution equation from dynamic equations (5.10)-(5.11) needs an introduction of generalized Fourier-variable ak represented by a linear combination of wave variables ηk and Φ k corresponding to the Fourier–transforms of the elevation and velocity fields (see below). The proposed closure of the kind of (5.15) allows existence a set of stochastic coefficients Li,j and Ci,j, providing for the Fourier-representation of forcing term, Pi ( ak ) , in a simple quadratic form of generalized variables. Just this form will be realized below. The said above allows to state that further specification of coefficients Li,j and Ci,j in form (5.15) in not principal at the moment. Moreover, as far as we do not know real processes generating turbulence of the water upper layer, there is no sense to construct any more complicated and detailed approximation for the forcing term, Pi ( u,η ) , in the physical space (as they have been done in earlier papers by the author, Polnikov 1993, 1995). At present stage of the theory derivation, it is the most important to take account the nonlinear feature of forcing term only. As it will be shown below, this fact itself gives sufficient grounds for a further finding the general kind of the sought function DIS(S). Thus, the approach proposed permits to transfer the whole difficulty of choosing specification of the forcing term in a physical space, Pi ( u,η ) , to the choice of it in a spectral representation, Pi ( ak ) .
5.5. General Kind of the Wave Dissipation Term in a Spectral Form Now, return to initial system of equations, (2.4)-(2.7), and rewrite it in the linear and potential approximations without any external force, excluding the turbulent one, P(η , u ) , introduced in the previous subsection. Accepting the following definitions
r u w (x, z, t ) = ∇3ϕ (x, z, t ) Φ ( x, t ) ≡ ϕ ( x, t )
z =η ( x )
(5.16) (5.17)
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V. G. Polnikov
one finds that two unknown functions: the surface elevation field, η ( x, t ) , and the velocity potential at the surface, Φ ( x, t ) , are described by the following equations
∂Φ + gη = − Pˆ (η , Φ ) , ∂t
(5.18)
∂η ∂Φ , = ∂t ∂z
(5.19)
Δϕ = 0 and
∂ϕ ∂z
z =−∞
=0.
(5.20)
Note that the system (5.19)-(5.20) has the same kind as the system (4.1)-(4.4), except that the last term in the r. h. s. of (5.18) means the result of transition to the potential
v
representation for the turbulence forcing, i.e. Pˆ (η , Φ ) = (∇3 ) [P(η , u)] . To make a −1
transition into the spectral representation, we introduce, as we done in section 4.1, the following Fourier-decompositions
η (x, t ) = const ⋅ ∫ exp[i (kx )]ηk (t )dk ,
(5.21)
ϕ ( x, z, t ) = const ⋅ ∫ exp[i ( kx )] f ( z )ϕ k (t )dk
.(5.22)
k
k
After substitution of representations (5.22) into the system of Eqs. (5.18)-(5.20), equations (5.20) give the solution for the potential structure function: f ( z ) = exp( − kz ) , and the other two equations get the kind
& + gη = −Π ( k ,η , Φ ) Φ k k k k
(5.22)
η&k = k Φ k
(5.23)
Here, the point above wave variables means the partial derivative in time, and
Π(k,ηk , Φ k ) ≡ F −1[ Pˆ (η , Φ )] is the new denotation of forcing function where the operator F-1 means the inverse Fourier-transition applied to the forcing function, Pˆ (η , Φ ) (see technical details in Hasselmann, 1974; Polnikov, 2007). System (5.22)-(5.23) is easily reduced to one equation having a sense of the well know equation for harmonic oscillator with a forcing
η&&k + gkηk = −k Π ( k ,ηk ,η&k )
(5.24)
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Solution of (5.24), written in the kind of evolution equation for the wave spectrum, can be carried out with the technique used in (Hasselmann 1974). Following to this technique, introduce the generalized variables
ak s = 0.5(ηk + s
i η& ) ,(where s = ± and σ ( k ) = ( gk )1/2 ) σ (k ) k
(5.25)
and rewrite Eq. (5.24) in the kind
a&ks + isσ (k )aks = −isσ ( k ) Π ( k ,ηk ,η&k ) / 2 g .
(5.26)
Now, accept the definition of the wave spectrum, used in (Hasselmann 1974) s'
2 << ak ak >>= S ( k )δ ( s + s ) s
'
(5.27)
where the doubled brackets <<.>> mean averaging over the statistical ensemble for wind waves. To finish the evolution equation derivation, one needs to do the following steps: −s
(1) to multiply Eq. (5.26) by the complex conjugated component, ak ; (2) to sum the newly obtained equation with the original one, (35); (3) to make ensemble averaging the resulting summarized equation. Finally, one gets the most general evolution equation for wave spectrum of the kind
2σ S& (k , t ) = k Im << Π (k ,ηk ,η&k )ak− >>≡ − Dis( S ) g
(5.28)
General kind of the sought dissipation term, Dis (S), can be found after specification of the forcing function Π ( k ,ηk ,η&k ) based, for example, on the closure formula given by (5.15). Due to qualitative feature of closure (5.15), there is no need to reproduce here all mathematical procedures explicitly. It is important, only, to take into account the main theoretical grounds providing for the sought final result: the dissipation term as a function of wave spectrum, Dis (S). For more clarity, list below the proper grounds: (a) The structure of generalized variables (5.25) includes a sum of Fourier-components for elevation variable, ηk , and for velocity potential one, η&k ∝ Φ k ; (b) The initial representation of forcing term (5.15) includes analogous sums for derivatives, what means that the forcing term can be expressed via the generalized variables in the form
30
V. G. Polnikov
Π ( k ,ηk ,η&k ) = function( aks , ak− s ) ;
(5.29)
(c). Due to averaging over turbulent scales, the exponential phase factors in the Fourierrepresentation for Π ( k ,ηk ,η&k ) can be arbitrary combined (or simply omitted). It needs to mention especially that just the item (c) allows executing the inverse Fouriertransitions in the nonlinear summands of forcing term P (η , Φ ) without appearance of residual integral-like convolutions containing the resonance-like factors for a set of wave vectors, which are typical in the conservative nonlinear theories (see technical details, for example, in Krasitskii, 1994; Polnikov, 2007). Thus, on basis of the grounds mentioned, it is quite reasonable (and sufficient for the aim posed) to represent the final expression for Π ( k ,ηk ,η&k ) in the most simple kind s Π (k,ηk ,η&k ) = ∑ Tij ( k )aksi ak j
.
(5.30)
si , s j
This form of function Π ( k ,ηk ,η&k ) has the main feature of the forcing: nonlinearity in wave amplitudes aks . Herewith, both the explicit kind of multipliers Tij ( k ) and the certain representation of the quadratic form in the r. h. s. of (5.30) are not principle, as far as the main physical feature is here conserved. Now, one can get a general kind of the r. h. s. in evolution equation (5.28), using the procedure of multiplication and averaging Eq. (5.26), described above in items 1)-3). First result of this procedure can be found by the following way. Substitution of (5.30) into (5.28) results in a sum of the third statistical moments of the kind << aks 1aks 2 aks 3 >> in the r. h. s. of (5.28). Due to an even power in wave amplitudes for the wave spectrum (by definition (5.27)), any third moment can not be directly expressed via the spectrum function, S ( k ) . In such a case, according to a common technique of the nonlinear theory (see, for example, Krasitskii, 1994; Polnikov, 2007), one should use the main equation (5.26), to write and solve equations for each kind of the third moments,
<< aks1aks 2 aks 3 >> , and to put these solutions into the spectrum evolution equation (5.28). From the kind of the r. h. s. of Eq. (5.26), it is clear that any third moment will be expressed via a set of the fourth moments of the kind << aks1aks 2 aks 3aks 4 >> , having a lot of combinations for the superscripts, si. A part of these moments, for which the condition s1+s2+s3+s4 ≠ 0 is fulfilled, must be put zero, according to definition (5.27). Residual fourth moments can be split into a sum of products of the second moments, << aksi ak− si >> , each of which corresponds to the spectrum definition (5.27). By this way, the first nonvanishing summand appears in the r. h. s. of spectrum evolution equation (5.28), and this summand is proportional to the second power in spectrum S ( k ) . The procedure described can be continued for a part of the fourth moments, what, through the chain of actions described above, results in a sum of terms of the third power in
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
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spectrum, in the r. h. s. of evolution equation (5.28). Eventually, the procedure mentioned provides for the power series in spectrum S ( k ) in the r.h.s. of (5.28), starting from the quadratic term. As far as the whole r. h. s. of Eq. (5.28) has, by origin, a meaning of the dissipative evolution mechanism for a wave spectrum, the proposed theory results in function DIS(S,k,W) of the following general kind1: N
Dis ( S , k , W ) = ∑ cn ( k , W ) S n ( k ) .
(5.31)
n =2
Specification of the decomposition coefficients, cn , including their dependence on the wave-origin factors, and determination of the final value of N in series (5.31), is based on principles not related to hydrodynamic equations. Therefore, these points will be specified below, by a separate way. As a conclusion of this section, it is worth while to emphasize that the main fundamental of the theory, providing for result (40), is nothing else as nonlinear feature of the Reynolds stress closure, substantiated physically in subsection 2.3. Consequently, the nonlinear feature of result (40) is substantiated at an equal extent.
5.6. Parameterization of the Dissipation Term and Its Properties In this section, using ideology of the earlier papers (Polnikov 1995, 2005), we will consider the following points: (a) Certain specification of the dissipation term, Dis(S,k,W), of the kind (5.31); (b) Physical meaning of the parameters introduced; (c) Correspondence of the parameterization for Dis(S,k,W) to experimental effects E1E4 mentioned in subsection 5.1; (d) Evidence of effectiveness of the proposed version for Dis(S,k,W).
5.6.1. Specification of function Dis(S,k,W) First of all, one should estimate the value of power N, which can limit the general representation of Dis(S,k,W) in the kind of series (5.31). To do it, let us use the following well known fact of existence of a stable and equilibrium spectral shape, Seq(σ), usually attributed to a fully developed sea (Komen et al. 1994). Not addressing to discussion about a falling law for the tail part of the wave spectrum, accept here that in the tail part, i.e. under the condition
σ > 2.5σ p
1
(5.32)
In a more detailed pose of the problem, instead of simple powers of the spectrum, function Dis(S) could include a set of integral-like convolutions of the same powers in S(k). This point is related to a future elaboration of the theory.
32
V. G. Polnikov ( σ p is the peak frequency of the spectrum S (σ , θ ) ), the equilibrium spectrum has the
shape Seq(σ) = αp g2σ -5 ( αp ≈ 0.01)
(5.33)
corresponding to the standard Phillips’ spectrum (Komen et al, 1994). This assumption gives us a possibility to introduce a small parameter, α , defined by the spectral function, S ( k ) ∝ S (σ , θ ) ∝ S (σ ) , in the whole frequency band:
α = max[ S (σ ,θ )σ 5 / g 2 ] << 1 (0 < σ < ∞).
(5.34)
It is worth while to mention that parameter α , defined by (5.34), has the order of the second power of mean wave slope, i.e. it is really quite small ( α ≈ 0.01) . Existence of a small parameter in a spectral representation of the wave field means that any series in spectrum, related to real wave physics, is the series in a small parameter. Hereof, one should immediately conclude that series (5.31) can be restricted by the first term, i.e. N = 2, with no lose of theoretical accuracy. Hence, after some algebra, the dissipation function can be written in the form
Dis (...) ≅ c2 (...) S 2 (k ) = γ (σ ,θ ,W )
σ6 g
2
S 2 (σ , θ )
(5.35)
where the unknown dimensional factor, c2 , is changed by the unknown dimensionless function, γ (σ ,θ ,W ) , all arguments of which a written explicitly. Besides, in the r.h.s. of (5.35), all powers of frequency are brought together to a single one. It is left to define explicit expression for function γ (σ ,θ ,W ) . General kind of γ (σ ,θ ,W ) can be easily defined on the basis of assumption for existence of equilibrium spectrum, accepted before. By definition, an equilibrium at the tail part of wave spectrum means that for a fixed spectrum shape, Seq(σ,θ), the balance of terms in the source function F is close to zero, i.e.
F
S = Seq
= [ Nl + In − Dis ]
S = Seq
≈ 0.
(5.36)
Now, take into account that the relative contribution of the nonlinear term NL to the source function, in the high frequency domain defined by ratio (5.32), is less than 10%. Then, ratio (5.36) gets the form
[ In − Dis ]
S = Seq
≈ 0.
(5.37)
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Accepting ratio (4.20) as the basic parameterization for the input term, IN, by using ratios (5.35) and (5.37) one can easily find a formal expression for function γ (σ ,θ ,W ) , eventually resulting in the following expression for the dissipation term
Dis (σ , θ , S , W )
S = Seq ≈ β (σ , θ , W )
σ6 g
2
S 2 (σ ,θ )
(5.38)
which is valid in the spectrum tail domain corresponding to ineqality (5.32). To get final specification of Dis (σ , θ , S , W ) , valid in the whole frequency domain, one should take into account the following points: (a) Specific character of dissipation processes in the energy containing domain, i.e. in the vicinity of the spectral peak where σ ≈ σ p ; (b) “Background” dissipation taking place when β (σ , θ , W ) ≤ 0 ; (c) Two-lobe feature of the angular spreading function, T (σ , θ , θ w ) , describing an increase of the dissipation rate whilst growing the angular difference, θ − θ w , i.e. waves do not propagate in the mean wind direction,
θw .
(d) With account of the said above, finally we have the following specification
Dis (σ , θ , S , W ) = c(σ , θ , σ p ) max [ β L , β (σ , θ , W ) ]
σ6 g2
S 2 (σ , θ )
(5.39)
Here, the well known increment, β (σ , θ , W ) , is given by a certain empirical formula the kind of which is not principal at the moment (for more details, see Polnikov 2005); the “background” dissipation parameter, the default value of which is
β L is
β L = 0.00005 ; and
c( σ ,θ ,σ p ) is the dimensionless fitting function describing peculiarities of the dissipation rate in the vicinity of spectrum peak frequency,
σ p . According to (Polnikov 2005), the latter
is given by the following phenomenological formula
c(σ , θ , σ p ) = Cdis max ⎡⎣ 0, (1 − cσ (σ p / σ ) ⎤⎦ T (σ , θ , σ p )
(5.40)
where the angular spreading factor is accepted in the form
⎧⎪ σ θ − θ w ⎫⎪ T (σ , θ , θ w , σ p ) = ⎨1 + 4 sin 2 ( ) ⎬ max [1, 1 − cos(θ − θ w ) ] , σp 2 ⎪⎭ ⎪⎩
(5.41)
34
V. G. Polnikov
Cdis and cσ are the fitting parameters, and the standard designation, max[a, b], means a choice of maximum value among two ones under the brackets. Hereby, the sought parameterization of Dis (σ , θ , S , W ) is wholly defined, and a general semi phenomenological theoretical substantiation for the dissipation term is finished. It is left to add that in the course of specification of function γ (σ ,θ ,W ) , one has a certain arbitrariness related to a choice of the equilibrium spectrum shape, Seq(σ,θ,W), and the kind for the angular form, T (σ , θ , θ w ) , as functions of their arguments. This arbitrariness is justified to some extent by uncertainty of the proper functions obtained experimentally (Rodrigues & Soares, 1999; Young & Babanin, 2006). Nevertheless, the general kind of parameterization (5.35) is robust to the uncertainties of such a kind. Just this circumstance allows hoping on the universality of its application in different numerical models for wind waves.
5.6.2. Physical meaning of the dissipation term parameters and correspondence to the empirics For completeness of the theory, it is important to reveal physical meaning of all innumerous parameters which are used in the proposed version of Dis (σ , θ , S , W ) , given by formulas (5.39)-(5.41). Without taking into account the fully phenomenological angular spreading function, T (σ , θ , θ w ) , note that the theory has only three fitting parameters: Cdis ,
cσ and β L . Meaning of coefficient Cdis is evident and simple. It regulates the dissipation intensity. This parameter is inevitable in any approach to the problem of the source function construction. Moreover, just Cdis is strongly varied while fitting any numerical model of the kind of (2.1), having representation of the source function as a sum of several separate evolution mechanisms. Meaning of parameter cσ consists, mainly, in separation of dissipation features in too frequency domains: the vicinity of spectral peak, and the spectrum tail. In our representation, in the vicinity of spectral peak, it regulates an extent of suppression of the “pure turbulent” dissipation intensity described by ratio (5.38). Here one can see manifestation of the empirical effects E2 and E3, mentioned in subsection 5.1. Really, parameter cσ variation does impact at some extent on the dissipation rate in the spectrum tail domain (a weak analogue of effect E2), what result in the variation of such integral characteristic as the mean wave period, Tm . For example, decreasing cσ results in lowering the rate of the spectrum peak growth in the course of wave evolution and, consequently, to increasing a relative contribution of the spectrum tail part to the value of mean period, Tm , estimated by the formula
Tm =
2π ∫ σ −1S (σ )dσ
∫ S (σ )dσ
(5.42)
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
35
With a glance that a value of the dominant frequency, σ p , is mainly defined by the nonlinear mechanism of evolution (Komen et al. 1994, Polnikov 2005, 2009) and weakly depend on the value of cσ , the decreasing cσ provides for a mean period reducing, without a remarkable change of the dominant period, Tp ∝ σ p . Naturally, an increase of cσ results in −1
the inverse effect. This feature of parameter cσ was effectively used in the tasks of fitting and verification of new source function (Polnikov at al., 2008; Polnikov&Innocentini, 2008). Finally, we should say some words about a meaning of parameter β L . Its main role is to regulate the dissipation rate at the moments of sharp changing the local wind (falling or turning). It is clear that at these moments, the value of increment β (σ , θ , W ) , corresponding to the former wind direction, is radically reduced, resulting in reducing the rate of breaking for the wave components running along the former wind direction. But in reality, certain (background) turbulence retains, as far as it “lives” in the water upper layer for long time. Thus, the background turbulence should provide remarkable attenuation of the wave components running the former wind direction, which are becoming now a swell. The said explains both a meaning of introducing parameter β L and its role in general. Initial choice of the value for
β L is based on numerous empirical and theoretical estimations (see references
in Polnikov 2005). But the final value should be found during fitting a whole numerical model, in a concert with a choice of all others parameters. The joint dynamics of the input term and the dissipation mechanism including a background constituent gives rise a quicker wind sea accommodation to a new wind direction. Just this effect is not described by the numerical models with a traditional dissipation term without a background constituent (WAM or WW), what was explicitly shown in papers (Polnikov at al. 2008, Polnikov & Innocentini 2008) (see below). Thus, the said above allows stating that all fitting parameters introduced into the proposed version of dissipation term function, Dis (σ , θ , S , W ) , have both purpose-oriented and physical meaning features. Additionally, it is worth while to emphasize the important role of quadratic dependence of function Dis(S) in the spectrum, S . Just this feature allows easy regulating the modeled dependence of the equilibrium spectrum, Seq(σ, θ), on frequency, σ, by means of varying the frequency power in ratio (5.38). It is caused by the fact that an expression for an equilibrium spectrum shape follows directly from the commonly accepted balance condition (5.37). Really, a substituting the linear in spectrum input term, IN(S), and quadratic in spectrum dissipation term, DIS(S), into Eq. (5.37) gives simply an equation for a shape of spectrum, Seq(σ, θ). Thus, the theory has no restrictions for the shape of equilibrium spectrum. Particularly, if one wants to postulate the equilibrium spectrum of the Toba’s shape (Komen et al.1994) Seq,T(σ) = αTu* gσ -4,
(5.43)
36
V. G. Polnikov
he should separately extract the dimensionless multiplier, g/u*σ, from function γ (σ ,θ ,W ) in (3.35), which should be saved in the r.h.s. of ratios (5.38) and (5.39). In such a case, the balance condition (5.37) results in the sought equilibrium spectrum of the kind (5.43).2 A positive comparison of the theoretical version for dissipation term with the well established empirical effects mentioned in subsection 5.1 is finished by the evident fact that the accepted angular spreading function for Dis (σ , θ , S , W ) of the kind (5.41) is directly corresponding to the recently revealed empirical fact of two-lobe feature for the angular function discussed (effect E4 found in Young & Babanin 2006). Herewith, regarding to the threshold feature of breaking phenomenon (effect E1), it is easily understood that, in a spectral representation for the dissipation term, this effect is smoothed due to statistical distribution of breaking events in a stochastic wave field alike a wind sea.
6. VERIFICATION OF NEW SOURCE FUNCTION In this section will dwell on the problem of estimation the performances of the source function which terms have been discussed previously. This point is directly related to the one for wind wave numerical model as a whole. There are two approaches to study the numerical model performance: testing and validation processes. The former is based on execution of academic testing tasks, and the latter does on validation of models against natural observation data. In our study, we dealt with the both approaches. As far as the basic principles of these processes have their own specifications, it is worth while to remind them briefly, following to Efimov&Polnikov(1991) and Komen et al. (1994).
6.1. Main Regulations for Testing and Verification of Models There are three principal features providing for an importance of the testing process. They are as follows: I.
Possibility to reveal numerical features of the model by means of simplified consideration based on using the fully controlled wind and boundary conditions. II. Message comprehensibility and predictability of the testing tasks. III. Simple and narrow aimed posing the testing tasks. There is a long list of testing tasks which could be used for a model properties evaluation (for example, see: The SWAMP group, 1985; Efimov&Polnikov, 1991; Komen et al., 1994; or Polnikov, 2005). But execution of all of them is out of our main aim. At present stage of studying, we have used the following list of tests.
2
Note that by this way one changes the dependence of Dis on wind W. Thus, the way shown could be used get the best balance between input and dissipation term as functions of the wind (if anybody knows this dependence for equilibrium spectrum).
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
37
#1. Straight fetch test (the wave development or tuning test). #2. Swell decay test (the dissipation test). In general, it is possible to distinguish three levels of adequacy of numerical wind wave models, which are defined by the proper choice of reference parameters used for comparison against observations (Efimov&Polnikov, 1991). But, here we restrict ourselves by the first level only, as far as the checking of the second and third level of adequacy needs much more time and efforts. Example of such a kind testing can be found in Polnikov(2005). The first level reference parameters are of the most importance, as far they are used in the test #1 which is, in turn, of the to principal importance. They are as follows: dimensionless wave energy
Eg 2 ~ Eg 2 E = 4 (or E * = 4 ) W10 u*
(6.1)
and dimensionless peak frequency of the wave spectrum
σ~p = where
the
dimensional
energy,
σ pW10 g E,
(or σ *p = is
σ p u* g
calculated
),
(6.2) by
the
ordinary
formula,
E = ∫∫ S (σ , θ )dσdθ , and σ p is the peak frequency of the spectrum S (σ , θ ) . Both ~ values, E and σ~ p , estimated from simulations for a stationary stage of the wind wave field, are considered as functions of the non-dimensional fetch,
~ X = Xg / W102 .
(6.3)
~ ~ ~ Numerical dependences E ( X ) and σ~ p ( X ) , found in simulations, are to be compared with the reference empirical ratios of the kind (Komen et al, 1994): (a) For the stable atmospheric stratification:
~ ~ −0.24 ~ ~ ~ E( X ) = 9.3 ⋅10−7 X 0.77 ; σ~ p ( X ) = 12 X
(6.4)
(b) For the unstable atmospheric stratification:
~ ~ ~ ~ ~ E( X ) = 5.4 ⋅ 10−7 X 0.94 ; σ~ p ( X ) = 14 X −0.28 For the test #2, the proper reference parameters are specified below.
(6.5)
38
V. G. Polnikov On the basis of this comparison, the first tuning of the unknown coefficients in the source
function, Сin , С dis , С nl , is done. Thus, the sense of these tests is to tune the model. But, here we should to say that the results of this tuning is not an unequivocal (see below), and, in principle, it needs to use more complicated tasks to make a sophisticated tuning. The validation process is one of these tasks. The only convincing way to prove the superiority of new model (or source function) in solving numerical simulation tasks for wind waves is to carry out the so called procedure of “comparative verification”. According to papers (Polnikov at al. 2008, Polnikov & Innocentini 2008), the regulations of comparative verification procedure demand a fulfillment of the following series of conditions: (a) Reasonable data base including accurate and frequent wave observations; (b) Reliable wind field given on a rather fine space-time grid for the whole period of wave observations; (c) Properly elaborated mathematical part for a numerical model of the kind (2.1); (d) Certain numerical wind wave model, chosen for comparison as a reference one. In papers (Polnikov at al., 2008; Polnikov&Innocentini, 2008), the last two requirements were satisfied by the choice of the model WAM and WW, respectively. Due to very similar results, below we dwell on reproduction the most interesting results of the second paper, only. Just in it, the other conditions, a) and b), were met by the following way. A. Two oceanic areas were chosen, for which the wave observation data were available: Western and Eastern parts of the North Atlantic (hereafter is referred as NA). At the first stage of validation process, we have used the one-month data (January, 2006) for 19 buoys located both in the Western and in the Eastern parts of NA (Figure 5) These data have a time discretization of 1 hour what corresponds to more than 700 points of observations on each buoy. B. As the wind field, we have used a reanalysis made in NCEP/NCAR with a spatial resolution of 1.00 both in longitude and in latitude. The time resolution for the wind was 3 hours. To exclude uncertainties with the boundary conditions, the simulation region was restricted by the following coordinates: 780S – 780N in latitudes and 1000W – 200E in longitudes, and the ice covering fields were included into consideration. The first stage of validation has been executed, basing on the data said above. These calculations resulted in a sophisticated choice of the fitting coefficients, Сin , С dis , С nl , found for the default values of the other fitting parameters mentioned above: bL,
β dis , cσ
(see below). At the second stage of validation, we have used the long-period historical data of the National Buoy Data Centrum (NBDC) (covering October-May period of 2005-2006 years) for 12 buoys located in the Western part of NA. The wind fields and the time-space resolution
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
39
were of the same features as at the first stage. Basing on these data, the standard validation of the both models has been done without changing any coefficients. 70 65 60 55 50 45 40 35 30 25 20 15 -80
-75
-70
-65
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
Figure 5. A sketch of simulating region in the North Atlantic, pointing some buoys locations
As far as the most reliable buoy data are related mainly to the observations of a significant wave height, H s , an estimation of simulation accuracy for this wave characteristic was done in each validation cases. Similar error estimations for a peak wave period, T p , and mean period, Tm , were executed for the 2 cases of the long-period simulations with the aim of the work completeness, only. Detailed analysis of the latter estimations is postponed for future investigations.
6.2. Specification of Numerical Simulations and Error Estimations In our calculation, we have used the frequency-angle grid of the kind (3.9), having parameters
σ 0 = 2π ⋅ 0.04 rad, e = 1.1 and Δθ = π / 12 (or Δθ = 15o)
(6.6)
with the number of frequency bins N =24 and number of angle bins M =24. In the case of model testing, the spatial grid was taken in Cartesian coordinates, including 100 points in the x-direction and 21 points in the y-direction. In the case of model validation in oceanic regions, the grid was taken in spherical coordinates. The space and time steps of calculations, ΔX , ΔY , Δt , were varying in accordance with the tasks and the numerical stability conditions. In Cartesian coordinates they varied in limits
ΔX = ΔY = 10 3 − 90 ⋅ 10 3 m and ΔT = 300 − 900 s, in spherical coordinates they were ΔX = ΔY = 10 and ΔT = 1200 s. Every time, an initial spectrum was taken in the frame of WW codes.
40
V. G. Polnikov
To assess an accuracy of simulating a time-series of a certain wave parameter P(t), we have used the following error estimates: (a) The root-mean-square error, δP , given by the formula
⎛ 1 δP = ⎜ ⎜ N obs ⎝
∑ (P n =1
and (b) The relative root-mean-square error,
⎛ 1 ρP = ⎜ ⎜ N obs ⎝
2 1/ 2
N obs
num
⎞ ( n) − Pobs ( n ) ) ⎟ ⎟ ⎠
(6.7)
ρP , defined as
N obs
⎛ Pnum ( n ) − Pobs ( n ) ⎞ ⎜⎜ ⎟⎟ ∑ Pobs ( n ) n =1 ⎝ ⎠
2 1/ 2
⎞ ⎟ ⎟ ⎠
.
(6.8)
Here N obs is the total number of observation points taken into consideration, and the evident sub-indexes are used. In addition to this, the following arithmetic error was used for analysis:
⎛ 1 αP = ⎜⎜ ⎝ N obs
Nobs
∑ (P n =1
num
⎞ (n ) − Pobs (n ) )⎟ . ⎟ ⎠
(6.9)
Here we remind that the first two errors describe statistical scattering of the simulating results (or the errors of the input fields, like a wind), whilst the latter one does the mean shift of numerical results with respect to observations. There are several other statistical characteristics which could be useful for assessment of a numerical model quality (correlation coefficient, probability function, and so on, for example, see Tolman et al, 2002). But at this stage of validation they are omitted, for the sake of more clearness of the primary analysis of the results presented below. The comparison of error was carried out between numerical results obtained with the original models, WAM and WW, and analogous results done with modifications of the both models, realized by means of replacing the source functions, only. The role of new source function was attributed to one proposed in paper (Polnikov 2005) and described above in sections 3, 4, and 5. This version of modified numerical model (in both cases, WAM and WW) was denoted as the model NEW. For a further understanding, it is important to note that in both cases, the modification of source functions consists in replacing the terms Nl and In, represented in the forms (3.103.13) and (4.19,4.20), respectively, maintaining the physics enclosed in them. The modification of Dis was done in accordance to the version described by formulas (5.39)-(5.41), changing the physics involved radically. Therefore, the whole deference in accuracy of these calculations was ascribed to changing the term Dis in the modified source
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
41
function. So, in the case of accuracy enhancement, the said permits to say about effectiveness of new version for Dis, and vice-verse.
6.3. Results of Testing New Source Function 6.3.1. Straight fetch test Pose of the task. Spatially homogeneous and invariable in time wind, W(x, t) = W10= const, is blowing normally to a very long straight shore line. Initial conditions are given by a homogeneous wave field with a wave spectrum of small intensity. Boundary conditions are invariable in time and correspond to the initial wave state. The purpose of the test is to check correspondence of the wind wave growing curves,
~ ~ ~ E ( X ) , σ~p ( X ) , provided by the model, to the reference empirical growing curves for the
stationary state of developed wind waves, given by ratios (6.4), (6.5). As far as the results of this test are typical and well predicted, here we show only some examples of testing results of the model NEW for different wind values, W10 =10-30 m/s. They are presented in Figures 6-8 for values of С nl = 9 ⋅ 10 7 , Сin = 0.4, С dis = 60, and the default values for the other fitting parameters (see Sects. 3, 4). The proper results for original WW are presented, for example, in Tolman and Chalikov (1996).
~ ~ ~ From figures 6-8 one can see that curves E ( X ) and σ~ p ( X ) corresponding to the
modified model are in a good accordance with empirical ratios (6.4), (6.5). It permits to state a good degree of tuning of the model, securing the first level of its adequacy, at least. For completeness of treating the results shown, it is worth while to note the following. First, the jumps between simulation curves, presented in Figures 3-5, are provided by a change of the spatial step, ΔX = ΔY , in 10 times. This change of the spatial step was done
~
in our calculations with the aim to cover a large range of dimensionless fetches, X , for a
~ ~ ~ fixed wind velocity, W10.3 Such a jump for E ( X ) and σ~ p ( X ) is a typical feature of any numerical scheme used in the model, consisting in an inevitable dependence of numerical errors on the value of spatial step. Usually, these errors are exaggerated at the points located near the shoreline.4 Additionally, in our presentation of the reference parameters, the location of curve
~ ~ ~ E ( X ) is shifted for the same fetch, X , while changing values of W10. This result is also typical (see Komen et al, 1994), taking into account dependence of friction velocity, u* , on
W10, realized in WW. The shifting effect is much more less expressed, if one represents *
*
dimensionless parameters in terms of u* (i.e. the dependencies E ( X ) and
3
4
3 For ΔX = 10 m and W10=10m/s, the range of the non-dimensional fetch
ΔX = 10 4 is was 103
5 ~ ≤ X ≤ 10 .
~ X was 102
σ *p(X *) .
~ ≤ X ≤ 104, and for
This point is mainly related to the mathematical part of the model, which is not discussed here. Evidently that it should be elaborated further in more details.
42
V. G. Polnikov
But, this artificial effect is not so principal, to dwell on it (for details, one could be referred to Komen et al, 1994; Tolman and Chalikov, 1996).
Figure 6. Dependence of dimensionless energy on dimensionless fetch, model NEW; 2 – Stable stratification ; 3 – Unstable stratification
~ ~ E ( X ) , for W10= 10m/s 1 –
1-NEW 2-OMp-stab 3-OMp-unst
OMp(X) 10
1
X 0,1 1,E+02
1,E+03
1,E+04
1,E+05
~
Figure 7. Dependence of dimensionless peak frequency on dimensionless fetch, see Figure 6
σ~ p ( X )
. For legend
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
43
1-NEW 2-En-stab 3-En-unst
En(X) 1,E-02
1,E-03
1,E-04
X 1,E-05 1,E+02
1,E+03
1,E+04
Figure 8. Dependence of dimensionless energy on dimensionless fetch, legend, see Figure 6.
1,E+05
~ ~ E( X )
for W10 = 30m/s, for
Second, it should be taken into account that the empirical dependences (6.4), (6.5) are
~
valid for dimensionless fetches of the range 102 ≤ X ≤ 104 with the errors of the order of 10-15% (Komen et al, 1994). This natural scattering feature of empirical data provides for a possibility to fit a lot of different models to the dependences (6.4), (6.5) with the same accuracy. Third (and it is of the most importance), a good correspondence of numerical and
~ ~ ~ empirical dependences E ( X ) , σ~ p ( X ) does not secure an unequivocal choice of the fitting
parameters. Coincidence with the root-mean-square errors of the order of 10-15% can be achieved for a continuum of values for the fitting parameters alike of Сin , С dis , С nl and the others mentioned in Sections 2.2-2.4. This result is provided for the simplified meteorological conditions used in the testing task. The sophisticated fitting of the model could be achieved only by means of the model validation against observations executed for a rather long period of wave evolution under well controlled but varying meteorological conditions. This point will be discussed in details below.
6.3.2. Swell decay test Pose of the task. Forcing wind of the fixed values is present in the first part of the testing area: W(X) = W10 at points 0 ≤ X ≤ Xm. In the second part of the area, the wind is absent: W(X) = 0 at Xm < X ≤ 3Xm. Initial wave state and boundary conditions are typical (see above the pose of test #1). Numerical evolution is continued for the period T securing a full development of waves at the fetch X=Xm and getting a stable state of the decaying swell field, taking place in the
~
second part of the testing area. Corresponding value of dimensionless time, T = Tg / W10 , should be about several units of 105.
44
V. G. Polnikov The aim of the test is to reveal quantitative features of the swell decay process, starting
from the fully developed sea with different peak frequencies, f sw = fp (Xm). The latter is considered as a principal initial characteristic of the swell. (Here we take into account that the initial intensity of the swell is mainly provided by f sw ). To reach the aim posed, different values of W10, Xm, and T should be taken into consideration. In our calculations, for a force of wind W10 = 10 m/s, we took: ΔX = 10 km, Xm = 240 km, T = 48 h; and for W10 = 20 m/s, we did: ΔX = 40 km, Xm = 760 km, T = 72 h. In the second part of the area, the following reference parameters are checked: •
the relative energy lost parameter given by the ration Ren(X) = E(X-Xm)/E(Xm);
•
(6.10)
the relative frequency shift parameter defined as Rfp(X) = fp (X-Xm)/ fp (Xm).
(6.11)
As far as there are no widely recognized empirical dependences Ren(X) and Rfp(X), the found ones are evaluated at the expert level, only. The latter means a qualitative physical analysis of the numerical results. Results of our simulation are shown in Figures 9, 10, representing the swell decay process for values W10 = 10 and 20 m/s. The correspondent values of the initial swell frequency, f sw , are 0.18 Hz and 0.085Hz, respectively. From these figures one can draw the following conclusions. (1) The rate of swell energy dissipation depends strongly on the initial peak frequency of swell, f sw . This rate is quickly going down with the distance of swell propagation (Figure 9). (2) The swell dissipation rate for the model NEW is faster than for WW (Figure 9). (3) The rate of peak frequency shifting to lower values, provided by the nonlinear interaction between waves, depends strongly on the initial value of peak frequency,
f sw (Figure 10). The greater f sw the greater rate of frequency shifting. This is well understood, taking into account formula (3.13) for the nonlinear evolution term. (4) The model NEW has practically the same rate of the peak frequency shifting, in contrast to the rate of the relative energy loss (Figure 10). This test is very instructive in the physical aspect. Really, from the results obtained, one can draw the following consequences.
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
45 1 2 3 4
swell decay En(X)/Enmax 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0,E+00
X,m 1,E+05
2,E+05
3,E+05
4,E+05
5,E+05
6,E+05
Figure 9. Dependence Ren(X) for two values of initial peak frequency of swell: 1, 2 – original model WW; 3, 4 – model NEW; 1, 3 -
Fp(x)/Fpmax 1 0,99 0,98 0,97 0,96 0,95 0,94 0,93 0,92 0,91 0,9 0,E+00
f sw =0.18Hz ; 2, 4 - f sw =0.085Hz 1 2 3 4
Swell decay (Fp relative)
X, m 1,E+05
2,E+05
3,E+05
4,E+05
5,E+05
6,E+05
Figure 10. Dependence Rfp(X) for two values of initial peak frequency of swell: For the legend, see Figure 9.
First, from the conclusion 2), one can state that the new dissipation term is more intensive than one used in the original model WW. Second, from conclusion 4), one can state the fact of very close similarity of the nonlinear terms in the both models. Third, from previous two consequences, one could state that the main qualitative difference of the numerical results obtained for these two models is mainly provided by the new parameterization of the Dis-term. Herewith, we note that though the new parameterization of the In-term has a feature of additional background dissipation, in this test it is two small to play any remarkable role, especially at the initial stage of swell decay.
46
V. G. Polnikov
As one could see later, the last consequence is of the most importance for understanding and treatment of difference between these models, which will be found during validation.
6.3. Results of Comparative Validation of the Models WW and NEW 6.3.1. One-month simulations in the North Atlantic After several runs of the model NEW, intended to a sophisticated choice of the fitting coefficients Сin , С dis , and С nl , we have found that the best results (i.e. minimum errors
δH s for the major part of buoys) are obtained for the following values: Сnl = 9 ⋅ 10 7 , Сin = 0.4; С dis = 70, and cσ =0.7
(6.12)
with the default values of the other fitting parameters. A typical time history of the significant wave height, H s (t ) , obtained in these simulations is shown in Figure 11 for buoy 41001 chosen as an example. From this figure, in particular, one can see that the model NEW follows the extreme values of real waves better than it is done by the model WW. Visual analysis of all proper curves has showed that this feature of the model NEW is typical for the majority of buoys taken into consideration. More detailed and quantitative analysis needs using the statistical procedures based on the error measures described above in Sec 3.4. At this stage of validation, the properly estimated errors have been found for the significant wave height, H s , only. They are presented in Tables 1, 2 for two parts of NA separately. For a quickness of general (visual) evaluating the results, we have shaded sells corresponding to the cases when the model NEW has a loss of accuracy. Table 1. Root-mean-square errors of simulations in the Eastern part of NA Eastern NA, No of buoy 62029 62081 62090 62092 62105 62108 64045 64046
Model WW
δH s ,m 0.57 0.67 0.66 0.58 0.79 0.99 0.71 0.72
Model NEW
(δH s )WW
ρH s ,% δH s ,m ρH s ,% (δH s ) NEW 14 15 14 14 18 15 12 15
0.54 0.56 0.57 0.53 0.68 0.84 0.61 0.76
13 13 14 14 15 13 12 15
1.05 1.20 1.16 1.09 1.16 1.18 1.16 0.95
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
47
Figure 11. Time history of the observed and simulated wave heights, H s (t ) , on buoy 41001 for January 2006. 1- wave heights measured on the buoy, 2- wave heights simulated by the model WW, 3wave heights simulated by the model NEW
Analysis of these results leads to the following conclusions: First, the accuracy of the model NEW is regularly better with respect to one of the original WW. Such a kind result is revealed for more than 70% of buoys considered. Second, discrepancy of the r.m.s. errors for the both models is remarkable. Typical winning of accuracy for the model NEW is of the order of 15-20%, but sometime it can reach 70% (buoy 44142). Third, the relative error,
ρH s , calculated by taking into account each point of
observations, is not so small (15-27%). It has a tendency of reducing for the model NEW, but this is not so well expressed. Table 2. Root-mean-square errors of simulations in the Western part of NA Western NA No of buoy 41001 41002 44004 44008 44011 44137 44138 44139 44140 44141 44142
Model WW
δH s ,m 0.81 0.52 0.82 0.83 0.82 0.58 0.70 0.63 0.78 0.64 0.81
Model NEW
(δH s )WW
ρH s ,% δH s ,m ρH s ,% (δH s ) NEW 22 18 25 27 23 19 19 19 19 20 27
0.66 0.47 0.68 0.61 0.55 0.51 0.74 0.69 0.80 0.68 0.48
20 18 26 24 18 17 19 20 19 20 18
1.23 1.11 1.21 1.36 1.49 1.14 0.95 0.91 0.97 0.94 1.69
48
V. G. Polnikov
Figure 12. Time history of the observation and simulation wind,
W10 (t ) , on buoy 41001 for January
2006. 1- wind measured on the buoy, 2 – wind used in the modeling simulation
Basing on the above, we should note that in the present statistical form of consideration,
ρH s is not so sensitive to the specificity of the model, as it could be expected. It seems that the effect of more increasing sensitivity of ρH s could arise, if we the relative error
introduce the lower limit of the wave heights, taken into the procedure of error estimation. For example, the proper error estimations could be done, if one excludes the time-series points H s (t ) with the wave values below than 2m. But, a role of introduction of limiting values for H s (and for T p ) is not so evident, therefore this issue should be especially studied later. In this connection, it is worth while to mention about an accuracy of the input wind. The proper time history for W10 (t ) is shown in Figure 12. From the first sight, the correspondence between the simulation wind and the observed wind seems to be rather well. But direct calculations of the errors δW10 and
ρW10 , made, for
example, for buoy 41001, give the values
δW10 =1.56 m/sand ρW10 = 32% .
(6.13)
The first value is more or less reasonable, taking into account that the input wind is calculated by the reanalysis for a very large domain covering the whole Earth. But the last value in (6.13), error
ρW10 , seems to be fairly great with respect to the corresponding relative
ρH s (Table 1). Due to an arbitrary choice of the buoy considered, one can expect that
such a kind mismatch between values of
ρH s and ρW10 is typical for the present
consideration, what, in turn, needs its understanding and explanation.
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications… This mismatch of values for
49
ρW10 and ρH s leads to a pose of the following new task:
how to treat the present inconsistency between these errors. To solve this task, first of all, it needs to have a large statistics of the errors. A part of such a kind statistics will be presented below in Table 3. Besides, physically it is reasonable to introduce the lower limiting values for wind, W10 , and wave heights, H s , which restrict the proper time-series points involved into the procedure of error estimation. In such a way, one could find a physically expected, unequivocal interrelation between errors
ρW10 and ρH s . If it is found, this relation permits
to make a proper physical treatment of the errors and to clarify prospective for numerical modeling improvements. Such a kind work is postponed for a future investigation.
6.3.2. Long-period simulations in the Western part of the North Atlantic Simulating results for the second stage of validation are very similar to the ones presented above. The proper errors are shown the Table 3, where the shaded sells correspond to the cases of less accuracy of H s for the model NEW. From this table, in general, one can state a reasonable advantage of the new model with respect to WW, in the aspect of simulation accuracy for the wave heights, which is defined by the values of r.m.s. error
δH s . The winning in accuracy is varying in the limits of 1.1-1.5
times. More detailed analysis results in the following. Arithmetic errors for WW are regularly greater then ones for the model NEW. Herewith, from table 3 it is seen that the model WW gives permanent underestimation of the wave heights, H s , whilst the model NEW has more symmetrical and smaller arithmetic errors. These facts allows us to conclude that the model NEW (and the new source function, consequently) has apparently better physical grounds. In the aspect of accuracy for the wave periods, Tm and T p , we should confess that the model NEW has less accuracy of calculation for the mean wave period, Tm , but, herewith, it has practically the same (or even better) accuracy for the peak wave period, T p (Table 3). Regarding to the wave periods, we should note a very specific feature, consisting in the fact that the both models show a certain overestimation of the mean wave period, Tm , whilst the peak period, T p , is permanently underestimated. The most probable reason of such a behavior of models could be related to an insufficient accuracy for calculation the 2D-shape of wave spectrum, S (σ , θ ) , taking place for the both models (for details, see Polniko&Innocentitni, 2008). Thus, the definite conclusion about superiority of one model against the other can not be drawn at present. Nevertheless, in principle, it could be done later, when the proper criteria will be formulated. This point is only posed here, and we plan to solve it in our future work.
Table 3. Consolidated input and output errors for the 8-months simulations in the Western part of NA No of buy/model
41001/WW /NEW 41002/WW /NEW 41004/WW /NEW 41010/WW /NEW 41025/WW /NEW 41040/WW /NEW 41041/WW /NEW 44004/WW /NEW 44005/WW /NEW 44008/WW /NEW 44014/WW /NEW 44018/WW /NEW
δW10 , m/s 2.01
ρW10
δH s
ρH s
δTm
ρTm
δT p
ρT p
, %
, m 0.68 0.48 0.48 0.44 0.97 0.64 0.40 0.37 0.44 0.54 0.22 0.25 0.20 0.23 0.72 0.57 0.58 0.45 0.70 0.50 0.49 0.35 0.44 0.55
%
, s 0.93 1.23 1.20 1.58 1.33 1.36 1.61 2.07 1.47 1.82 1.78 2.11 1.92 2.26 1.13 1.32 1.44 1.78 1.11 1.30 1.17 1.27 1.14 1.37
%
, s 2.02 2.13 2.01 2.22 2.40 2.38 2.06 2.34 2.23 2.23 1.87 1.96 2.22 2.20 1.96 1.88 2.27 2.24 2.01 1.88 2.46 2.37 2.03 1.88
, %
40
1.77
48
2.54
36
1.24
32
2.18
50
0.91
20
0.96
22
1.91
40
2.28
59
2.35
51
Bad information 3.01
43
22 18 19 20 51 36 19 20 24 31 10 11 09 10 24 24 25 27 25 21 26 23 23 33
17 22 22 30 31 32 33 43 30 38 30 35 32 38 21 25 30 37 21 25 21 24 22 28
24 30 27 35 36 38 29 41 30 35 18 22 21 24 24 26 38 43 26 28 31 33 25 28
αW10 , m/s 0.58 0.25 -1.48 0.09 0.47 0.08 0.17 0.16 1.19 0.69 Bad info 0.61
αH s
αTm
αT p
, m -0.45 -0.22 -0.23 -0.3 -0.97 -0.47 -0.19 -0.08 -0.05 0.17 -0.10 -0.07 -0.06 0.04 -0.38 -0.04 -0.30 0.03 -0.43 -0.09 -0.27 -0.11 -0.03 0.29
, s 0.46 0.79 0.78 1.11 0.63 0.73 1.25 1.69 1.09 1.48 1.64 1.90 1.78 2.06 0.52 0.82 0.84 1.37 0.60 0.91 0.46 0.83 0.72 1.03
, s -1.32 -0.88 -1.05 -0.57 -1.26 -1.10 -0.88 -0.21 -1.16 -0.57 -0.90 -0.53 -0.90 -0.54 -1.34 -1.00 -0.84 -0.19 -1.31 -0.90 -1.66 -1.23 -1.25 -0.85
(δH s ) ww (δH s ) new
(αH s ) ww (αH s ) new
1.42
2.04
1.09
7.67
1.52
2.06
1.08
2.37
0.81
0.29
0.88
1.43
0.87
1.50
1.26
9.5
1.29
10.0
1.4
4.78
1.4
2.45
0.80
0.10
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
51
6.3.4. Point of the speed of calculation By using the numerical procedure PROFILE, we have checked the speed of calculation, realized while execution of all main numerical subroutines used in the models. In terms of central processor consuming-time, the proper time distributions among the main subroutines are shown in Table 4 for both models. These distributions are corresponding to the case of execution the task of 24-hours simulation of the wave evolution in the whole Atlantic. Table 4. Distribution of the central processor consuming-time, realized by the two versions of WW Model Original WW
Modified WW
Name of procedure (explanation) w3snl1md_w3snl1 (Nl-term calculation) w3pro3md_w3xyp3 (space propagation scheme) w3uqckmd_w3qck3 (time evolution scheme-3) w3iogomd_w3outg (output of results) w3src2md_w3sin2 (In-term calculation) w3uqckmd_w3qck1 (time evolution scheme-1) w3srcemd_w3srce (integration subroutine) w3src2md_w3sds2 (Dis-term calculation) others All procedures w3pro3md_w3xyp3 w3uqckmd_w3qck3 w3snl1md_w3snl1 (Nl-term) w3iogomd_w3outg w3uqckmd_w3qck1 w3srcemd_w3srce w3src2md_w3sds2(Dis-term) w3src2md_w3sin2 (In-term) others All procedures
Time, s
Time, %
123.41 87.01 68.58 37.73 21.99 17.66 13.29 2.75 … 455.9 89.72 71.29 70.97 38.60 17.97 12.15 7.68 6.04 … 398.8
27.06 19.08 15.04 8.27 4.82 3.87 2.91 0.60 … 100 22.52 17.88 17.80 9.68 4.51 3.05 1.93 1.52 … 100
From this table one can see that in the model NEW, the nonlinear term is calculated in 1.73 times faster than in the original WW. It leads to the consuming-time winning of the order of 60 seconds, which results in 15%-winning of the total consuming time. The acceleration effect is provided by using the fast DIA approximation mentioned above in Sec. 3. Additional 3%-winning of time is gained due to new parameterization of the input term. But, in turn, the new approximation of Dis-term results in a lost of calculation speed on 2%. Nevertheless, as we said above, just this parameterization provides the better accuracy of the model NEW, because of the physics of NL-term and In-term in both models is very similar.
6.3.5. Conclusion for verification Thus, the new source function was tested and validated by means of its incorporating into the mathematical shell of the reference model WW. Results of the test #1 are typical for any
52
V. G. Polnikov
modern numerical model and have a technical feature of the primary tuning. But, the test #2 is more physical, as far as it testifies specific properties of the proposed dissipation term. The real performance of new model was checked during the comparative validation process, which was executed in three steps differing both by duration of simulations and by regions of the World Ocean, taken into consideration. In general, we may state that the both models have rather high performance, which are apparently the best among present models, taking into account the results of WW’s validation represented in Tolman et al. (2002). Herewith, the comparative validation has showed a real advantage of the model NEW with respect to the original WW, especially in the accuracy of the wave heights calculation. The advantage consists in reduction of the simulation errors for significant wave height, H s , in 1.2-1.5 times and increasing the speed of calculation on 15%. Analysis of the curves, like presented in Figure 11, shows that the greatest percentage into the r.m.s. error is contributed at the time-series points with extreme values of the wave heights and at the points corresponding to the phases of wave dissipation, at which the wave intensity is going down. Both of these features are controlled by the dissipation mechanism of wave evolution. On these grounds, we conclude that the dissipation term is parameterized more efficiently in the new model than in the original WW. This property of the model NEW is very important in a sense of application it for the tasks of risk assessment. In our study, the relative r.m.s. error,
ρH s , is introduced, as one of the most instructive
measure for estimation an accuracy of the wave heights simulations. We have found that this parameter has mean values of the order of 12-35% for both models. It is naturally to suppose that magnitudes of
ρH s should be related to the value of inaccuracy for the wind field used
as an input. Regarding to this, the new task is posed, consisting in a search for a quantitative relation between the errors for waves,
ρH s , and the errors of input wind, ρW10 . This
relation is quite expected, taking into account the experimental ratios likes (6.4), (6.5). The proper study is planned to be done in a future work. There are several another tasks related to the further validation and elaboration of the numerical wind wave models. One of them consists in seeking for a certain upper limits of inaccuracy for wind field and for wave observations, which are requested for a further progress in the wind wave modeling. Estimation of these limits is the primary future task. At present it seems that the main requirement, which define the limits of the further elaboration of the numerical wind wave models, consists in using the wind field having inaccuracy below the limits mentioned above.
7. FUTURE APPLICATIONS In this section we will point out several possible application of any modern wind wave model, to solve the tasks which are not simply the wave forecast.
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
53
7.1. A Role of Wind Waves in Dynamics of Air-Sea Interface 7.1.1. Introductory words As we have already said in the INTRODUCTION, wind waves are considered as an intermediate-scale stochastic dynamic process at the air-see interface, which modulates radically the large-scale dynamic processes and small-scale turbulence in the boundary layers of water and air. Interaction of a large-scale atmospheric air flux (simply say, a wind) with the water upper layer, resulting in the drift currents, is realized at several temporal and spatial scales. Intermediate result of such a kind interaction is the wind sea (wind waves). Here we mean that the large-scale motion of atmosphere and water has variability scales of the order of 103104 m and 103-104 s, whilst the wind wave processes have the scales of the order of 10-102 m and 10s. In turn, the wave motion of surface interface is realized on the background of smallscale turbulent pulsations of velocity in air and water with characteristics of variability less then 1m and 1s. All these motions characterize dynamics of the air-sea interface. Therefore, the momentum and energy transfer from the wind to currents are realized by means of different direct and inverse cascade transfers in the considered band of temporal and spatial scales. It is physically evident that the wind waves play a radical modulating role in dynamics of the air-sea interface. But an exact mathematical description of this dynamics meets insuperable difficulties related to multi-scale feature of the processes under consideration. Below we state that mathematical formalization of description for processes of momentum and energy transfer through the air-sea boundary could be done on the basis of well recognized representations of physical mechanisms for wind wave evolution. Addressing to Figure 1, we distinguish in the air-sea interface 3 constituents: (1) The turbulent atmospheric boundary layer (ABL) with a mean flux of local wind, W(x), which is characterized by a certain vertical distribution, W(x, z), and a certain value at a fixed standard horizon, for example, by the value W10(x); (2) Waving interface surface, given by the function of wave elevations η (x); (3) Turbulent water upper layer (WUL) with a mean local flux of drift current, U(x, z), distributed by a certain manner through the vertical coordinate z. The vertical dimensions of ABL and WUL are of the order of characteristic (dominant) wave length on the waving interface. The horizontal and temporal scales of variability for all kind of motions under consideration are mentioned above. One of the main problems of interface dynamics description is the problem of calculation large and scale characteristics of ABL (including wind profile W(z), and wind stress τ) and characteristics of WUL (including drift current vector U and the vertical exchange coefficient K). It is evident that such a kind detailed description of interface dynamics needs a complicated system of equations, whilst each of them is valid for a certain time-space scale. Attempts of writing such a kind system were undertaken by various authors (for example, see reviews: Monin&Krasitskii, 1985; The WISE group, 2007). In some partial cases, the selfconsistent solutions were found (Zaslavskii, 1995; Makin&Kudryavtzev, 1999). But in the
54
V. G. Polnikov
general case, a detailed consideration of the problem was not succeeded. An example of such a try is the famous paper by Kitaigorodskii&Lamley 1983. Usually, in the pose of global circulation tasks, characteristics of wind waves are frequently omitted from the consideration due to small-scale feature of this process (Pedloskii, 1984). However, as it was shown by the practice of theoretical considerations for the wind wave evolution mechanisms, some important particular solutions of the general problem for interface dynamics description, formulated above, can be found just on the basis of well known representations for these mechanisms (Qiao et al, 2004). From scientific point of view, solution of the circulation problem as a whole is interesting for understanding the general role of dynamics for all items of interface: ABL + wind waves + WUL. On the other hand, from practical point of view, a clear mathematical description of the interface system gives possibility to solve numerous tasks of air-sea dynamics, including diffusion and exchange processes at the interface. Examples of such a kind tasks are as follows: • • • •
self-consistent calculation of wind, waves, and currents; calculation of heat, gas, and passive impurity exchange between atmosphere and ocean; calculation of mixing in the WUL, including tasks of impurity diffusion and transportation, air bubbles layer generation, and so on.
It is easy to see that all of these large and middle-scale processes are in some manner modulated by the wind wave dynamics. By this way the role of wind waves in dynamics of the interface is displayed. The further matching the middle-scale motion with the large-scale one permit to spread a study the wind wave impact on the ocean and atmosphere circulation in a whole. As the world wide experience shows, attempts of construction a multi-scale equations system in physical variables do not succeed in a full extent, whilst description for dynamics of the system in a spectral representation is more prospective ((Proceedings of AIR-sea interface symposium, 1995, 1999). Herewith, the whole experience gotten in the course of investigation of the wind wave evolution by numerical methods has an essential superiority. Up-to-date, one may state that a principal physical understanding the exchange processes at the air-sea interface is achieved. Numerous results of scientific investigations published, for example, in proceedings of special conferences can be seen as the evidence of this statement (last references). AS we have shown in sections 3, 4, 5 above, the mathematical tool for spectral description of wind waves evolution processes is also well developed. Thus, one may try to construct a series of particular solutions of the general problem formulated above, in the frame of mathematical terms derived already for evolution mechanisms of wind waves.
7.1.2. The Role of wind wave evolution mechanisms First of all, we remind that the waving surface of air-sea interface has a stochastic feature. For this reason, the most adequate description of its motion could be given in terms of twodimensional wave energy spectrum, S ≡ S (σ , θ , x, t ) , spread in space and time and
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
55
governed by the proper evolution equation (2.1). The heart of this equation is the source function F describing a set of evolution mechanism for wind wave in spectral representation: • • •
Mechanism of nonlinear energy transfer through the wave spectrum, Nl (“nonlinearity-term”); Mechanism of energy transfer from the wind to waves, In (“input-term”); Mechanism of wave energy loss due to interaction of waves with the turbulence of upper layer, Dis (“dissipation-term”).
Let us summarize the roles of these mechanisms in the air-sea interface dynamics, i.e. the role of them in distribution of energy between ABL and WUL. As we have seen, the input mechanism, described by formulas (4.19-4.20) is responsible for energy exchange between ABL and wind waves. The reference wind variable of this process is the friction velocity, u*, the value of which is directly related to the wind stress, τ (z), corresponding to the vertical turbulent flux of the horizontal momentum, provided by the shear of the wind profile, W(z). The essential factor of the wind-wave energy exchange process is dependence for both friction velocity and local wind profile, W (z ) , on the wave state, i.e. on the shape of spectrum S ≡ S (σ , θ , x, t ) . Just a determination of this dependence is the task of the feed back influence of wind waves on the state of ABL. In section 4, we have shown that this point can be self-consistently solved by means of construction the model of dynamic boundary layer (DBL). An example of such kind DBL-model was there represented. By this manner, the system of joint description of waves and a boundary layer wind profile becomes closed, and the first part of the problem for interface dynamics, formulated above, does solved. Secondly, discuss now the role of nonlinear interactions among waves, responsible for the nonlinear evolution mechanism of waves, Nl. In section 4, it was shown that nonlinear energy transfer through the wave spectrum has a conservative feature and ensures, mainly, the transfer of wave energy from the high frequency domain of the wave spectrum into the domain located below peak frequency of the spectrum, σ p . Such a kind transfer results in diminishing frequency
σp
with the time of evolution. It means that during wave evolution
due to Nl- mechanism, the dominant wave length increasing takes place. By this manner, the energy, supplied from wind to waves in the high frequency domain mainly, is accumulated by Nl-mechanism in the domain below the dominant frequency σ p , which is slowly shifting down, whilst the high frequency spectrum tail is practically left unchangeable in intensity (due to balance F=0, in this domain) . Consequently, just the nonlinear mechanism of evolution procures a significant growth of the total wave energy in the course of their development. Herewith, due to progressive feature of waves, the main part of wave energy is running from the region of its origin into the region of wave propagation. Thus, due to nonlinear feature of wave evolution, the wave energy is not determined by the local wind, only, but it is provided by the whole dynamics of energy exchange during wave propagation through the wave evolution space. Just this fact displays the principal role of term Nl in the wave dynamics and, consequently, in the dynamics of energy distributions between items of the interface.
56
V. G. Polnikov
Thirdly, consider the role of wave dissipation mechanism in the air-sea interface dynamics. The direct role of this mechanism is description of the wave energy loss. But the question under consideration is the following: where to and how manner by, the energy lost by waves is distributed? The answer to the first part of the question is rather clear. The energy lost by waves is shared in two parts. One part of the energy is consumed for generating the WUL turbulence. This part has no any mean momentum, though it has a feature of vertical flux of small-scale turbulent motions, which is spent to the work against buoyancy. The second part of the energy lost by waves is consumed by drift currents. They take a horizontal momentum lost by waves, but it is not the whole momentum taken by waves from wind, due to fact that a part of the whole momentum is gone by waves running to the space of their propagation. The question of consumption for the energy lost by waves should be considered by the following manner. Estimation of energy partition to the turbulence of WUL and to the drift currents occurrence should be done by the special block of dynamic upper layer (DUL). In such a block, the tasks of joint description for the state of wave, turbulence, and currents in WUL should be solved in a self-consistent manner, in terms of the wave spectrum shape, S . In a final form, such a block in not present in modern models till now. Expedience and possibility of its construction is demonstrated by the fact that this point is actively developing last years (see, Ardhuin et al, 2004; Fomin&Cherkesov, 2006). Thus, the role and importance of adequate representation for all the terms of source function F becomes clear and evident.
7.1.3. Energy and momentum balance at the air-sea interface The wind is the energy source for all kinds of mechanical motions realized in the items of air-sea interface. Denoting the local large-scale wind at the standard horizon as W, let us write expressions for the key characteristics of the wind flux. In particular, a surface density of the energy flux of local wind is
FWE = ρ aW 2 W / 2 ,
(7.1)
and local density of the flux of horizontal momentum in given by the ratio
FWM = ρ aWW Here
(7.2)
ρ a is the air density and W is the local wind speed. Both these fluxes correspond to
the unit of air volume located at the standard horizon. Due to turbulent feature of the air motion in ABL, the vertical flux of horizontal momentum to the interface, τ , does take place in the system considered. As we said earlier in section 4, τ can be expressed directly via characteristics of the boundary layer by the formula
τ = ρ au*2 = ρ aCd ( z )W 2 ( z ) = τ w + τ t = const .
(7.3)
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
57
Physically, the energy transfer from wind to waves is just done by the wave part of momentum flux, τ w , depending on spectrum S . On the other hand, the tangent (or turbulent) component,
τ t , give rise to drift current, directly. The task of ABL, wind waves, and UWL
dynamics becomes closed in terms of wave spectrum, if one takes into account the circumstance mentioned above, and realizes them as the proper blocks of the generalized wind wave model. Mathematical aspect of this approach is the following. First, the total value of local density for the wave energy per unit of surface and at the fixed time moment, E (x, t ) , has the kind 2π
∞
E (x, t ) = ρ w g ∫ dσ ∫ dθ S (σ ,θ , x, t ) 0
where
(7.4)
0
ρ w is the water density, and g is the gravity acceleration.
Second, the rate of energy transfer from wind to waves is unequivocally following from the kind of the input term, In, used in the source function of model (2.1). Thus, the total energy flux from wind to waves (per unit of surface) is given by expression ∞
2π
0
0
I E = ρ w g ∫ dσ
∫ dθ In[ W, S (σ ,θ )] .
(7.5)
This energy flux is corresponded by a proper momentum flux from wind to waves, which just determines a certain wave component,
τ w , of the total flux, τ . Formula for τ w has the
kind ∞
2π
τ w = ρ w g ∫ d σ ∫ dθ 0
k cos(θ − θ w )
σ
0
In[ W, S (σ , θ )] .
(7.6)
Further construction of the scheme for energy and momentum balance at the air-sea interface, related to the energy transfer from waves into WUL, can be made by analogy with the consideration given above for ABL. Namely, the rate of energy transfer from waves into the water upper layer is determined by the expression for the dissipation term in the source function of model (2.1), i.е. it has the kind ∞
2π
0
0
D E ( x , t ) = ρ w g ∫ dσ ∫ dθ Dis[ W , S (σ , θ , x, t )] .
(7.7)
58
V. G. Polnikov
Figure 13. Scheme of energy redistribution in the air-sea interface
This energy flux is corresponded by the local momentum flux from waves into the WUL
r
∞
2π
0
0
τ d = ρ w g ∫ dσ ∫ dθ
k
σ
Dis[ W , S (σ , θ )] .
(7.8)
These two fluxes regulate both the WUL turbulence and drift currents origin. The letter are characterized by the surface density flux of kinetic energy due to currents, EC=
ρ w UU 2 / 2 , and the WUL turbulence do by the rate of turbulent energy production per unit surface square, ET. Ratio between these values is not known in advance, as this needs construction of the dynamic model for WUL (which is hereafter called the block of DUL).
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
59
Nevertheless, “a priori” one can say that the total momentum flux due to dissipation is shared into the pure turbulent and wave components. Further, with the account of the conservation laws, the energy and momentum fluxes are redistributed among different kinds of motion. Eventually, this mechanics determines the feature of dynamics for the upper mixed layer, which, in the feed back regime, in turn, can modulate the processes of wind wave dissipation. In such a case, the dissipation term Dis becomes to be dependent not only on the local wind vector, W , but on the current vector, U, as well. In full details, the DUL-block construction needs a separate multi-steps consideration. Nevertheless, a whole picture of redistribution of energy and momentum incoming into WUL is rather clear. Herewith, it is evident that just the wind wave dissipation mechanism does play the crucial role in the WUL dynamics, but not the local wind does. The letter is only one of parameters of this mechanism. Eventually, a general scheme of energy redistribution from the wind to waves, and further into WUL, is presented in Figure 13.
7.2. Examples for Estimation of Wave Impact on Parameters of the ABL and WUL It is principally clear that in addition to the wind wave impact on the state of ABL via the energy supply processes, the dissipation processes in waves lead to the origin of turbulent and current motions in the water upper layer. Each of these mechanisms of wave evolution determines radically an intensity of different processes at the interface. In particular, the interface turbulence is responsible to the rate of heat and gas exchange between air and water, to the rate of passive impurity mixing and diffusion in the WUL, and so on. Besides, the wave crests breaking results in an origin of the air bubble layer in the WUL. Study each the processes mentioned above needs a separate consideration what is out range of this chapter. Therefore, here we present only two examples for quantitative estimations of wave state impact on the parameters of ABL and WUL. One of them will touch the calculation of the friction coefficient, C d , as a function of the wind, inverse wave age, A, and the tail shape of 2D-spectrum for wind waves, S (σ , θ ) . Herewith, the magnitude
А is defined by the formula
A = u*σ p / g = u* / c p , and the spectrum tail shape in a wide range for frequencies with values up to the value
(7.9)
σ > σ p , spreading
σ max having the order of 80 rad/s, is given by the ratio S (ω, θ ) ∝ σ − n cos2 (θ − θ w ) .
(7.10)
The second example deals with the calculation for dependence of the acoustic noise intensity, Ia, provided by air bubbles in the WUL, on the local wind speed, W .
60
V. G. Polnikov
Examples of wave state impact on the drift current in a shallow water basin and for ocean circulation one may find in paper (Fomin&Cherkesov, 2006) and in (Qiao et al., 2004), respectively.
7.2.1. Wave state impact on the value of friction coefficient in the ABL This issue was studied in details in paper (Polnikov et al, 2003). First of all, it was noted there that an experimental variability of values for friction coefficient, C d , measured at the horizon z =10m has a dynamical range of variability in the limits of (0.5-2.5)×10-3 units, for the fixed values of local wind,
W . Herewith, in the case of swell, the meaning of Cd can get
the negative values. The last property of the magnitude Cd , as it is clear at present, is totally secured by the inverse energy transfer from waves into the ABL (see section 4). For this reason, below we will not dwell on this point, paying attention on the first point. For the better understanding the physics of such a kind feature of atmosphere and ocean interaction, the following question should be answered:
C d , observable for the same
•
What is the reason of strong variability for values of
•
wind speed W ? Is this effect a result of measurements errors or it is provided by physical reasons?
The answer to this question was found in (Polnikov et al, 2003) where the calculation of values of C d were executed with the use of the DBL-model described earlier in subsection 4.4. Results of these calculations for different values of wave age, А, and different laws for the spectrum tail fall of the kind of (7.10) are presented in Figure 14. Analysis of the results shown in Figure 14 permits to draw the following conclusions. (1) The scattering of friction coefficient values is secured by the physics of wind-wave interaction process. The value of C d is determined not only by the local wind speed, W, and the current value of wave age, А, but by the shape of tail of 2D-spectrum for wind waves, S (σ , θ ) , too. That explains the wide range for variability of C d , realizing in experimental observations. (2) When the falling laws for the spectrum tail are faster than fifth order in frequency, one may expect the decreasing Cd with decreasing the value of А. This effect is frequently observed in experiments (for references, see Polnikov et al, 2003). But for a more weak dependence of the spectrum tail on frequency, it is probable a slow increasing the value of C d in the course of wave development, which can finish itself by fixing the final value of coefficient, growing with the growth of wind speed, W. Consequently, to distinguish the real dependencies C d (A, u*), it needs to know (or to calculate) the high frequency tail for 2-D wind wave spectrum, the shape of which is determined by numerous factors including an impact of the surface currents, as well.
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
61
(3) A simple regression dependence of Cd on parameters of the system considered, alike W and A, frequently used in the wave modeling practice (including WAM and WW), is the too rough approximation. Such a kind dependence should be determined by means of the DBL-model, i.e. by means of more elaborated wind wave models (of fourth generation).
Figure 14. Dependence of
C d on inverse wave age for series of values for wind, W, and spectrum
shape parameter, n: 1 - W = 5 m/s, n = 4; 2 - W = 10 m/s, n = 4; 3 - W = 10 m/s, n = 5; 4 - W = 20 m/s, n = 4; 5 - W = 20 m/s, n = 5
Thus, taking into account that the main parameters for a wind profile in the ABL (see formulas 4.28) are calculated with the model of DBL (i.e. without attracting the hypotheses of logarithmic wind profile), one may state that just the presence of the DBL-block in a model leads to appearance a new quality of the latter. This new quality permits to solve the applied tasks for wave forecast and atmosphere circulation with more accuracy and completeness. To the completeness of consideration, we say several words about redistribution of turbulent and wave components of the vertical momentum flux in the ABL. Numerical estimations of the profile τ w (z), made with the DBL-model said above, show that at the mean air-sea surface, the ratio of components
τ t and τ w of the total momentum flux, τ , has a
meaning of the following order
τ t (0) ≅ 0.5 – 0.6, τ w (0)
(7.11)
where the value z = 0 means a mean level of the waving surface. Such a kind redistribution for the components of τ leads to a disturbance of the standard logarithmic profile for wind speed, which depends on the wave state. By other words, the use of the standard logarithmic profile for wind speed and estimation of the roughness height z 0 with formula (4.28) is the
62
V. G. Polnikov
rough approximation to the real situation. Consideration of this issue in more details needs a separate research.
7.2.2. Estimation of acoustic noise intensity dependence on the wind speed One of important practical task is an estimation of acoustic noise level produced by the air bubbles origin in the WUL due to wave crests breaking. In particular, it is very desirable to know the dependence of bubble noise intensity on the wind speed. Estimation of relative integral rate of wave energy dissipation, DRE, defined by
DRE = DE / Ef p
(7.12)
permits to give a theoretical solution of the question posed. Here, DE is the integral rate of dissipation given by (7.7), E is the total wave energy (7.4), and fp is the peak frequency. In Polnikov(2009) such estimations were done in a series of simplest cases. It was found that typical value of DRE is of the order of DRE ≈ 0,001
(7.13)
what leads to the following solution of the problem posed. Let us rewrite the formulas derived in Tkalich&Chan(2002), where the physical model for acoustic noise of the bubble layer in WUL was constructed. In this paper, it is shown that under certain assumptions, the intensity of bubble noise, Ia, is described by the ratio
I a = CT R(W , H S ) ⋅ ( f r / f 0 ) −2 ,
(7.14)
where CT is the theoretical coefficient, R(W , H S ) is the radius of bubble cloud as a function of the local wind speed, W, and significant wave height, H S ; ( f 0 / f r )
2
is the non-
dimensional frequency for the bubble acoustic oscillations, depending on a structure of the cloud. Further it is significant only that Ia is linearly dependent on the cloud radius, R(W), the value of which contains a whole information about the wind speed determining the dependence sought. In Tkalich&Chan(2002), it was shown that the radius value, R(W , H S ) , is linearly dependent on the rate of wave energy dissipation in accordance with the ratio
R (W , H S ) = cb c t
DE Bh
(7.15)
Here DE is the rate of energy income into the WUL due to wave energy dissipation,
cb ~ (0.3 − 0.5) is the empirical coefficient defining the fraction of value for DE spent to the bubble cloud origin, ct ≈ 0.5 is the fraction of DE spent to the turbulence production in the WUL. B is the void fraction, and h is the characteristic depth of the bubble cloud (mass
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
63
center). Values of cb and B are determined from experimental observations, and the value of
ct can be estimated theoretically with the physical models of DBL, discussed above in Sec. 3. Further we will suppose that the values mentioned have a weak dependence on wind. In such a case, the sought dependence takes the kind, I a (W ) ∝ R(W ) , so it can be determined on the basis of calculation for D E (W ) and on physical models describing the dependence
h(W ) . According to definition (7.12), the dependence D E (W ) can be found from calculations for the magnitude E(W) and from tabulated rate of the non-dimensional dissipation for wave energy, DRE(W). In general case, for the wind field prescribed, this task is solved by means of numerical simulation a wind wave spectrum evolution at the fixed point in the basin under consideration. But in the simplest case of constant and homogeneous wind field, the sought dependences can be obtained with the use of result (7.13) and well known empirical
~
dependences of E (W ) for the fully developed sea, (6.4), (6.5). Let us consider the fully developed sea. In such a case, with the account of ratios (7.12) and (7.13), one may write
DE (W ) = 0.001 Ew (W ) f p (W ) .
(7.16)
As far as for the fully developed sea, dependences E w (W ) and f p (W ) are given by the well known ratios (6.4), (6.5), we have
Ew ≈ 3 ⋅10−3W 4 g 2
(7.17)
f p ≈ g / W 2π .
(7.18)
and
Then, under the assumption of the lack of dependence B(W ) , for the acoustic noise intensity due to bubbles we have
I a (W ) ∝ W 3 / h(W ) .
(7.19)
Thus, in the case considered, the final result is determined by the model for a depth of the bubbles cloud center, h(W) . There are possible the following cases here. I.
In the case of weak wave sea, the assumption that h = const is quite reasonable, due to small dependence of any mechanical parameters for WUL on the wind (including the bubble cloud deepening). In such a case, the dependence I a (W ) is determined by the ratio
64
V. G. Polnikov 3
Ia ~ W .
(7.20)
II. In the case of rather visible waves which are far from their extreme development, it is widely used the following empirical formula (see Tkalich&Chan, 2002) h ≈ 0.35HS
(7.21)
where HS is the significant wave height. With the account of definition HS =1,4(E)1/2, the sought dependence (7.19) takes the kind Ia ~ W .
(7.22)
III. And finally, in the case of high winds and fully developed sea, it is more reasonable to put that the bubble cloud depth is linearly related to the radius of the cloud, i.e. h ∝ R . Under such an assumption and with the account of ratios (7.16)-(7.18), the solution of equation (7.15) reads
h(W ) ∝ W 3 / 2 .
(7.23)
Consequently, in this case, the sought dependence takes the kind Ia ~ W
3/ 2
.
(7.24)
Table 5. Empirical estimations for dependence I a (W ) Wind speed (m/s)
Dependence
I II
5<W<10 10<W<15
~⎨
III
W>15
~W1.5
I a (W )
⎧0.004W3 - 0.049W2 ⎫ ⎬ ⎩+ 0.463W - 1.5 ⎭
Wave state Gravity-capillary and developing waves Fully developed sea
It is interesting to note that all three types of dependences I a (W ) , i.e. formulas (7.20), (7.22) and (7.24), do well correspond to generalized observation data presented in Table 5, which is taken from paper Tkalich&Chan(2002). It means that the physical assumptions, used above for constructions the models of wind sea and WUL, are fairly adequate to the real processes, and the models themselves can be widely used for solution of practical tasks. Certainly, the solution of acoustic noise problem is still far from its completion. Nevertheless, the main step in this direction has been already done. And it lies in the topic of construction the DBL-model based on the energy and momentum balance established above for the system containing wind, waves, and upper water layer.
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
65
7.2.3. Intermediate conclusion Considerations, presented above, permit to look at the whole problem of small-scale and large-scale circulation in atmosphere and upper ocean from the new point of view. Really, up to the present, in the frame of commonly used approximations of geophysical hydrodynamics, solution of the circulation tasks was being executed without account of the waving surface state (Pedloskii, 1984). In such an approach the water surface was considered as an undisturbed one (hard cover approximation), and the momentum transfer from wind into WUL was unequivocally determined by the local wind. On the basis of the said above about the role of wind waves in dynamics of the air-sea interface, the commonly used approach should be radically sophisticated by means of account of the wave state, basing on numerical models including the proper blocks for dynamic ABL and WUL. A general scheme of the energy and momentum fluxes redistribution in the dynamical system of air-sea interface is presented in Figure 13. In the frame of this scheme, there is a fairly certain clarity for the wind wave model itself and for DBL-block: there are certain versions of them. Naturally, some details of these models can be sophisticated during their verification, but the principal approach will not get radical changes. About the model of DBL there is not such a clarity. In particular, there is not any estimation for the fractions of energy and momentum fluxes, going from waving surface to the drift currents and turbulence in the WUL. It needs strong efforts to specify a model of DUL, permitting to close the task of fluxes redistribution and to approach to solution of the circulation task. A series of studies in this direction have been already done (Ardhuin et al., 2004; Fomin& Cherkesov, 2006). And this fact gives a basis to expect an appearance nearest time of new wind wave models installed with the DBL-block. In relation to the said, it becomes possible to formulate the following sophistication of the present classification for wind wave models, started in (The SWAMP group, 1985). As it is well recognized in the world practice, the models of the third generation are ones which calculate a full 2-D wind wave spectrum, S (σ , θ ) , and have source functions operating with no limits for the wave spectrum shape (mainly, this request touches a parameterization for the term NL, see, The SWAMP group, 1985). WAM and WW are the most widely used representatives of the models of third generation. The model of the next generation should have new quality. Such a kind model can be one which is installed with a special DBL-block, permitting to describe a dynamic fitting of the atmospheric boundary layer to the state of wind sea, including calculation of the wind profile without attracting the hypotheses of the logarithmic friction law. The model of such a level can be named as the model of forth generation. Up to date versions of models WAM and WW, in which the friction velocity is calculated and dynamics of ABL takes place, nevertheless, use the hypotheses of logarithmic ABL and some few-parametric representations for dependences of ABL’s parameters on the wave state’s ones. Consequently, they do not meet the request formulated for the forth generation model. But the model described in paper (Polnikov, 2005) does meet the request formulated above. Following to this logics, one may state that the model installed with the block of DUL, permitting to describe dynamics of the water upper layer (the coefficients of turbulent mixing and drift currents, at least), without attraction of few-parametric dependences of them on
66
V. G. Polnikov
wind and wave age, will have additional, new quality. Therefore, such a model, installed with the block of DUL, can be classified as the model of fifth generation. With the account of the said, the classification of wind wave models does get its logical completeness from the theoretical point of view. It is only left to realize the whole chain of models in the practice. A completeness of this chain can be considered as one of the most important theoretical and practical task in the problem of joint description for atmosphere and ocean circulation at the synoptic scales. We suppose that the model based on results of Polnikov (2005) can serve as a basis of the task solution.
7.3. Using wind wave Models for Studying Long-Term Mechanical Energy Exchange in the System: Wind-Wave-Upper Ocean As an example of alternative application of modern wind wave model, let us consider the following possible project with the conditional title “Wind and wave climate study in Atlantic ocean, based on numerical simulations with a modern model of the third generation”.
7.3.1. The Main tasks On the basis of the hind-casting numerical simulations for the 20-years period (19902010yy), with the aim of wind and wave climate variability, the following tasks could be done. (1) Calculation and tabulation of the seasonal statistics for wind and wave mechanical 5 energy accumulated in the atmosphere and ocean for 5 regions of Atlantic ocean . (2) Annual and seasonal statistics (histograms) of the maximum values of remarkable wave heights (with the threshold H s > 3m) in 5 regions of the Atlantic. (3) Determination of the spatial and temporal distribution of local domains with the extreme wave heights ( H s > 10m) in 5 regions of the Atlantic. (4) Annual and seasonal statistics of durations of the extreme waves (with the threshold
H s > 15 m) for 5 different regions of the Atlantic. (5) Making electronic maps of wave heights distributions for the extraordinary events in the whole Atlantic (i.e. wind speed is more 30 m/s, or wave heights are of H s > 15m).
7.3.2. Method of study Having a modern wind wave model (for example, WAM with the optimized source function, as it was done in Polnikov et al, 2008), one could make numerical simulations of wave evolution in the whole Atlantic Ocean for the period of 20 years, to get a good statistics 0 0 of waves. The proper wind field data are available for us on the space grid 1 x1 with the time discrete of 3h. 5
Partition means fixing the numbers of events in the following 5 regions: Western part of North Atlantic, Eastern part of North Atlantic, Tropical (near-equatorial) part of Atlantic, Western part of Southern Atlantic, and Eastern part of Southern Atlantic.
Numerical Modelling of Wind Waves. Problems, Solutions, Verifications…
67
Method of the wave climate study includes the following actions. A. One makes a spatial partition of the whole Atlantic into 5 parts having, for example, the following boundaries: (X- longitudes, Y – latitudes) 1. Western part of the North Atlantic (WNA): 100W < X < 40W, 20N < Y < 78N; 2. Eastern part of the North Atlantic (ENA): 40W < X < 20E, 20N < Y < 78N; 3. Tropical part of the Atlantic (TA): 100W < X < 20 E,20S < Y < 20N; 4. Western part of the South Atlantic (WSA): 100W < X < 40W, 78S < Y < 20S; 5. Eastern part of the South Atlantic (ESA): 40W < X < 20E, 78S < Y < 20S. B. One introduces 3 reference values of significant wave height, H s , which distinguish description of different meteorological events: • Ordinary waves heights (with Hs > 3m); • Extreme wave heights (Hs > 10m); • Extraordinary wave heights (Hs > 15m). C. In each region of the Atlantics, description of the following events is of interest: (a) Distribution in space and time of the mechanical energy accumulated in atmosphere, E A (t ) , (wind) and in ocean, E w (t ) , (wind waves) (task 1). Atmosphere energy time history, E A (t ) , is calculated by the formula
E A (t ) = ΔS ∑
ρa
i , j ,n
where
2
Wi ,3j (tn )Δtn
(7.25)
ρ a is the air density, and Wi , j (t ) is the wind at the standard horizon (z =
10m) and at each space-time grid points, (i,j) and time moment tn. Each term under the sum in (7.25) is the density of the kinetic energy flux over a unit of the surface, Δ S . Wind waves energy analog is calculated by the formula
E w (t ) = ΔS ∑ i, j
where
ρwg 16
H i2, j (t )
(7.26)
ρ w is the water density. Each term under the sum in (7.26) is the density
of the mechanical energy of waves over unit of the surface. Study of these values is important for understanding of the mechanical energy exchange between atmosphere and ocean and their climate variability. 20-years historical series of such values could be needed for estimation of the wave climate variability in time and space. (b) Statistics of the maximum waves. It includes seasonal, annual and total (20years) histogram of the maximum wave heights, obtained by simulations of wave evolution for 20 years (task 2). This information is important for understanding a regional distribution of wind waves by their strength.
68
V. G. Polnikov (c) Registration of domains with the extreme waves in each region of the Atlantic, and making comparison the numbers of events among the regions (task 3). This information is important for determination of the most dangerous region in the Atlantic. (d) Registration of the extreme wave’s duration in the regions (task 3). This information gives more details of the previous study (task 4). It is important for evaluation of the time variability of the extreme events. (e) Making the atlas of maps and seasonal-annual statistics of the extraordinary waves (number of the events in each region) (task 5). This is important for understating of the extraordinary events distribution among regions for the long period. There is no map of such a kind, and for this reason they are of great scientific and practical interest.
The above does clarify the purpose and the method of executing the project proposed. We sure that the work drafty described in this subsection will be very fruitful in many aspects.
REFERENCES Ardhuin, F; Chapron, B; Elfouhaily, T. J Phys Oceanogr., 2004, 34, 1741-1755. Babanin, AV. Acta Physica Slovaca, 2009, 59, 305-535. Babanin,VA; Young, IR; Banner, ML. J Geophys Res., 2001, 106C, 11659–11676. Banner, ML; Young, IR. J. Phys. Oceanogr., 1994, 24, 1550-1571. Banner, ML; Tian, X. J Fluid Mech., 1998, 367, 107-137. Chalikov, DV. Boundary Layer Meteorology, 1980, 34, 63-98. Chalikov, D; Sheinin, D. Advances in Fluid Mechanics, 1998, 17, 207-222. Drennan, WM; Kahma, KK; Donelan, MA. Boundary-Layer Meteorology, 1999, 92, 489515. Donelan, MA; Dobson, FW; Smith, SD; et al. J Phys Oceanogr., 1993, 23, 2143-2149. Donelan, MA. Coastal and Estuarine Studies, 1998, 54, 19-36. Donelan, MA. Proc. ECMWF Workshop on Ocean Wave Forecasting, Reading, UK, ECMWF., 2001, 87–94. Efimov, VV; Polnikov, VG. Oceanology, 1985, 25, 725-732 (in Russian). Efimov, VV; Polnikov, VG. Numerical modelling of wind waves. Naukova dumka Publishing house. Kiev. UA. 1991, 240 (in Russian). Fomin, VN; Cherkesov, LV. Izvestiya, Atmospheric and Oceanic Phys., 2006, 42, 393-402 (in Russian). Hasselmann, K. Shifttechnik, 1960, 7, 191-195 (in German). Hasselmann, K. J Fluid Mech., 1962, 12, 481-500. Hasselmann, K. Boundary Layer Meteorology, 1974, 6, 107-127. Hwang, PA; Wang DW. Geophys Res Letters, 2004, 31, L15301, doi:10.1029/2004GL020080. Janssen, PEAM. J Phys Oceanogr., 1991, 21, 1389-1405. Kitaigorodskii, A; Lumley, JL. J. Physical Oceanography, 1983, 13, 1977-1987.
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Komen, GL; Cavaleri, L; Donelan, M; et al. Dynamics and Modelling of Ocean Waves, Cambridge University Press. UK. 1994, 532. Krasitskii, VP. J Fluid Mech., 1994, 272, 1-20. Lavrenov, IV; Polnikov, VG. Izvestiya, Atmospheric and Oceanic Phys., 2001, 37, 661-670 (English transl). Makin, VK; Kudryavtsev, VN. J Geophys Res, 1999, 104, 7613-7623. Miles, JW. J. Fluid Mech., 1960, 7, 469-478. Monin, SA; Krasitskii, VP. Phenomena on the ocean surface. Hydrometeoizdat. Leningrad. RU. 1985, 375. Monin, AS; Yaglom, AM. Statistical Fluid Mechanics: Mechanics of Turbulence.. The MIT Press, Cambridge, Massachusets, and London, UK. 1971, v.1, 769. Pedloskii. J. Geophysical Fluid dynamics. Springer Verlag. N.Y. 1984, V.1. 350. Phillips, OM. J. Fluid Mech., 1957, 2, 417-445. Phillips, OM. Dynamics of the Upper Ocean. Second ed; Cambridge University Press, UK. 1977, 261. Phillips, OM. J. Fluid Mech., 1985, 156, 505-631. Plant, WJ. J Geophys Res., 1982, 87, 1961-1967. Polnikov, VG. Izvestiya, Atmospheric and Oceanic Physics, 1991, 27, 615-623 (English transl.). Polnikov, VG. Proceedings of Air-Sea Interface Symposium, Marseilles, France. Marseilles University. 1994, 227-282. Polnikov, VG. The study of nonlinear interactions in wind wave spectrum. Doctor of Science dissertation. Marine Hydrophysical Institute of NASU. Sebastopol. UA. 1995, 271, (in Russian). Polnikov, VG. Nonlinear Processes in Geophysics, 2003, 10, 425-434. Polnikov, VG. Izvestiya, Atmospheric and Oceanic Physics, 2005, 41, 594–610 (English transl.). Polnikov, VG. Nonlinear theory for stochastic wave field in water. LENAND publishing house. Moscow. RU. 2007, 404, (in Russian). Polnikov, VG. Izvestiya, Atmospheric and Oceanic Physics, 2009a, 45, 346–356 (English transl). Polnikov, VG. Izvestiya, Atmospheric and Oceanic Physics, 2009b, 45, 583–597 (English transl.). Polnikov, VG; Dymov, VI; Pasechnik, TA; et al. Oceanology, 2008, 48, 7–14 (English transl.). Polnikov, VG; Farina, L. Nonlinear Processes in Geophysics, 2002, 9, 497-512. Polnikov, VG; Innocentini, V. Engineering Applications of Computational Fluid Mechanics, 2008, 2, 466-481. Polnikov, VG; Tkalich, P. Ocean Modelling, 2006, 11, 193-213. Polnikov, VG; Volkov, YuA; Pogarskii, FA. Nonlinear Processes in Geophysics, 2002, 9, 367-371. Polnikov, VG; Volkov YuA; Pogarskii, FA. Izvestiya, Atmospheric and Oceanic Phys., 2003, 39, 369-379 (English transl.). Proceedings of the symposium on the wind driven air-sea interface. Ed. by M. Donelan. The University of Marseilles, France. 1994, 550.
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Proceedings of the symposium on the wind driven air-sea interface. Ed. by M. Banner. The University of New South Wales, Sydney, Australia. 1999, 452. Qiao, F; Yuan, Y; Yang, Y; et al. Geophysical Research Letter 2004, 31, L11303, doi: 10.1029/2004GL019824. Rodriguez, G; Soares, CG. Uncertainty in the estimation of the slope of the high frequency tail of wave spectra. Applied Ocean research., 1999 , 21, 207-213. Snyder, RL; Dobson, FW; Elliott, JA; Long, RB. J Fluid Mech., 1981, 102, 1-59. Stokes, GG. Transaction of Cambridge Phys Soc., 1947, 8, 441 -455. The SWAMP group. Ocean wave modelling. Plenum press. N.Y. & L. 1985, 256. The WAMDI Group. J Phys Oceanogr., 1988, 18, 1775-1810. The WISE group. Progress in oceanography, 2007, 75, 603-674. Tkalich, P; Chan, ES. J Acoustical Society of America, 2002, 112, 456-483. Tolman, HL; Chalikov. DV. J Phys Oceanogr., 1996, 26, 2497-2518. Yan, L. Report No. 87-8. 1987. Royal Dutch Meteorological Inst; NL. 20. Young, IR; Babanin, AV. J Phys Oceanogr., 2006, 36, 376–394. Zakharov, VE. Applied mechanics and technical physics, 1968, 2, 86-94 (in Russian) Zakharov, VE. Izvestiya VUZov, Radiofizika, 1974, 17, 431-453 (in Russian). Zakharov, VE; Korotkevich, AO; Pushkarev, A; Resio, D. Phys Rev Letters, 2007, 99, 16-21 Zaslavskii, MM; Lavrenov, IV. Izvestiya, Atmospheric and Oceanic Phys., 2005, 45, 645-654 (in Russian).
In: Horizons in World Physics. Volume 271 Editor: Albert Reimer
ISBN: 978-1-61761-884-0 © 2011 Nova Science Publishers, Inc.
Chapter 2
AN OVERVIEW OF PLASMA CONFINEMENT IN TOROIDAL SYSTEMS Fatemeh Dini1, Reza Baghdadi1, Reza Amrollahi1* and Sina Khorasani2 1
Department of Physics and Nuclear Engineering, Amirkabir University of Technology, Tehran, Iran 2 School of Electrical Engineering, Sharif University of Technology, Tehran, Iran
ABSTRACT This overview presents a tutorial introduction to the theory of magnetic plasma confinement in toroidal confinement systems with particular emphasis on axisymmetric equilibrium geometries, and tokamaks. The discussion covers three important aspects of plasma physics: Equilibrium, Stability, and Transport. The section on equilibrium will go through an introduction to ideal magnetohydrodynamics, curvilinear system of coordinates, flux coordinates, extensions to axisymmetric equilibrium, Grad-Shafranov Equation (GSE), Green’s function formalism, as well as analytical and numerical solutions to GSE. The section on stability will address topics including Lyapunov Stability in nonlinear systems, energy principle, modal analysis, and simplifications for axisymmetric machines. The final section will consider transport in toroidal systems. We present the flux-surface-averaged system of equations describing classical and nonclassical transport phenomena. Applications to the small-sized high-aspect-ratio Damavand tokamak will be described.
Keywords: Plasma Confinement, Axisymmetric Equilibrium, Stability, Transport, Nuclear Fusion.
*
Corresponding author: Dr. Reza Amrollahi, Professor of Physics and Chair, Department of Physics and Nuclear Engineering, Amirkabir University of Technology, Tehran, Iran, Cell: +98-912-159-2837, Office: +98-2164542572 +98-21-66419506, Fax: +98-21-6649-5519, Email:
[email protected] [email protected]
72
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
I. INTRODUCTION The increasing worldwide energy demand asks for new solutions and changes in the energy policy of the developed world, but the challenges are even greater for the emerging economies. Saving energy and using renewable energy sources will not be sufficient. Nuclear energy using fission is an important part of the worldwide energy mixture and has great potential, but there are concerns in many countries. A future possibility is the nuclear reaction of fusion, the source of solar energy. Though many scientific and technical issues have still to be resolved, controlled fusion is becoming more and more realistic. Two methods of nuclear reactions can be used to produce energy: fission – gaining energy through the break-up of heavy elements like uranium; and fusion – gaining energy by merging light elements such as deuterium and tritium. The fusion option is still far on the horizon, but international exploration has started in earnest these years. Nuclear fusion promises some welcome characteristics: an inexhaustible source of energy in light nucleus atoms; the inherent safety of a nuclear reaction that cannot be sustained in a non-controlled reaction; and few negative environmental implications. Research in controlled nuclear fusion has a self-sustainable burning plasma as its goal, and good progress has been made in recent years towards this objective by using both laser power and radiation to merge the light nuclei (inertial confinement) or using magnetic fields (magnetic confinement) to confine and merge deuterium and tritium. Large new facilities are currently under construction, the most prominent using magnetic confinement is ITER, which is seen as the international way towards the peaceful use of controlled nuclear fusion.
I.1. Energy Crisis The daily increasing demand for energy in the world points out the growing need of the mankind to the various sources of energy. Renewable energies, despite their compatibility with the environment, are economical only in small scales of power delivery. On the other hand, reserves of fossil fuels are limited too, and also the obtained energy from burning fossil fuels causes the emission of carbon dioxide and particles, which in turn leads to the rise of the average temperature and air pollution. In Figure I.1.1, it can be seen that the gap between the demand and delivery of crude oil is rapidly widening, as predicted over the next two decades. In the year 2030, the daily available access to the crude oil will be amounted to about 65 million barrels, and this is while there would be an extra 60 million barrels which should be replaced by other energy resources. Currently, more than 440 nuclear fission reactors around the globe produce 16% of the total spent energy by the mankind. The United States of America and France, respectively, with capacities of 98 and 63 giga Watts out of 104 and 59 nuclear reactors are the largest suppliers of nuclear electricity. With the completion of the Busheher nuclear reactors, Iran would join to the 33 countries in the world which are capable of producing nuclear electricity.
An Overview of Plasma Confinement in Toroidal Systems
73
Figure I.1.1. The widening gap between oil delivery and demand (red) [1]
Figure I.1.2. Predicted energy demand till 2100 based on three different scenarios (billion tone crude oil equivalent) [1]
74
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani Table I.1.1. Nuclear Reactors around the globe [2]
I.2. Nuclear Fission In all of the nuclear reactors in the world, fission of heavy and unstable isotopes of Uranium makes the nuclear energy available, which is usually extract through a thermal cycle after first transforming into mechanical and subsequently electrical forms. The corresponding reactions are: 235
U + 1n → fission products+ neturons+ energy ( ~ 200MeV )
(I.1.1)
U + 1n → 239U + gamma rays
(I.1.2)
238
239
U → 239 Np → 239 Pu
( a series of beta − decays )
(I.1.3)
In (I.1.1), the number of emitted neutrons and daughter nuclei might be different and range from 2 to 4. But the average number of neutrons is equal to 2.43. Neutrons may cause a chain reaction of (I.1.1) and transformation into 238U, or through continued reactions (I.1.2)
An Overview of Plasma Confinement in Toroidal Systems
75
and (I.1.3) produce the heavier element 239Pu. As we know, only 0.7% of the uranium in the nature is fissionable via thermal neutrons. The rest of existing uranium is in the form of 235U, which cannot be broken up by thermalized (slow) neutrons. Hence, the fuel used in the nuclear reactors is usually in the form of U2O, with 235U isotope enriched up to 4%. 239Pu can be fissioned by neutrons, and 232Th by absorption of one neutron transforms into 233U, which in turn is highly fissionable by thermal neutrons. The fission of uranium causes a large energy density. The fission of only one gram of 235 U per day can generate an average power of 1MW. This is equivalent to burning of 3 tones of coal and more than 600 gallons of oil product, emitting 250 Kg of carbon dioxide. The released energy is carried by the kinetic energy of daughter nuclei, which is absorbed in the water pool of the reactor. In some designs such as pressurized water reactors (PWR), the thermal energy is exploited for evaporation of water in a separate cycle. But in boiling water reactors (BWR), the water in contact with the nuclear fuel is directly evaporated and used for driving turbines. Also, there exists the possibility of using fast neutrons in place of thermal neutrons with 239Pu fuel. Therefore, 235U reactors produce the necessary fuel for the former kind of reactors. Annually, about 100 tones of 239Pu is obtained worldwide. Considering the daily need to the production of nuclear electricity and applications of radioactive materials in various areas of energy, medicine, industry, agriculture, and research in countries, the use of nuclear energy is inevitable. Despite the advantages of using fission energy, many drawbacks are also associated with this nuclear technology as well, the problem of wastes being the most important. The transmutation of nuclear wastes containing or contaminated by radioactive materials is among the most important unsolved problems of this technology. It seems that simple methods are only explored for this purpose, and no acceptable plan for long time isolation or transmutation of nuclear wastes exists to date. Until the early 1950s, dilution, air release, submerging in ocean floors, and concealing over deserts have been used. Since then other methods such as concealing in multilayer undergrounds and vacant mines are also proposed. But through the time, the production of nuclear wastes has raised so much that none of the mentioned methods would work in the long run. Five decades of exploiting nuclear reactors in the United States, only, has produced 50,000 million tones of spent nuclear fuel. It is anticipated that this trend would increase all the way to 20,000 million tones annually. For achievement of a permanent solution, the Yukka mountain project with the capacity of 70,000 million tones has been under way, which clearly is insufficient (Figure I.2.1). But even this project has been stopped due to its extraordinary cost of 6 billion dollars. The alternative proposed solution is irradiation of radioactive wastes by neutrons obtained from accelerated protons (Figure I.2.2). In this method, nuclear wastes with long life times are converted into short-lived radioisotopes. Also, the generated heat from many of the burning isotopes such as 129I, 99Tc, 237Np, 90Sr, and 137Cs can be extracted by Pb and exploited for production of electricity needed to run the accelerators. Other replacements include Fission-Fusion hybrids and also using thermonuclear plasmas of tokamaks as neutron sources, both of which are based on the Fusion technology. Therefore, the nuclear fusion once completed can be used for energy production as well as transmutation of the nuclear waste from fission reactors. It should be mentioned, however, that still the cheapest energy in the world is not from nuclear, but rather coal resources (Figure I.2.3).
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
Figure I.2.1. The six billion dollar Yukka mountain project [6,7]
Figure I.2.2. Transmutation of nuclear wastes from fission reactors via proton accelerators [6,7]
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77
Figure I.2.3. Cost of electricity produced from different resources (cent per KWh)
I.3. Nuclear Fusion As discussed above, the nuclear energy can be either obtained from the fission of heavy elements, or fusion of light elements. Generally speaking, whenever the heavier element has a lower potential energy compared to the sum of potential energies of two separate nuclei, the fusion reaction is plausible. The experiment reveals that iron with the atomic number 26 has the lowest level of potential energy, and therefore it would be the most stable nucleas. This shows that fusion of lighter elements than iron always generates energy, as the fission of heavier elements than iron does. But the released energy depends on the reaction cross section as well as the energy obtained from every individual reaction.
Figure I.3.1. First generation nuclear fusion reactions
Figure I.3.2. Second (top) and third (bottom) generations of nuclear fusion reactions
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
The nuclear fusion reactions take place in the universe in the center of stars among the nuclei of hydrogen and helium, and in white dwarfs among nuclei lighter than iron. The simplest nuclear fusion reactions which can be achieved on the earth are among the four lightest elements of the periodic table, and their isotopes. These include hydrogen (and its isotopes: deuterium and tritium), helium, lithium, and beryllium, each generating an enormous amount of energy. But the H-H reaction has a very small cross section and hence very small probability for taking place. In contrast, heavier elements than hydrogen or its isotopes can be used to obtain the four nuclear fusion reactions corresponding to three distinct generations. The reactions belonging to the first generation occur between the isotopes of hydrogen, namely deuterium D=2H, and tritium T=3H. A significant amount of deuterium can be found on the surface of earth, and via industrial methods can be obtained from water in the form of heavy water D2O. But tritium is the unstable and radioactive isotope of hydrogen with a life time of about 12 years, and therefore does not exits naturally. To produce tritium, reactions of fast neutrons with isotopes of lithium can be exploited as follows
Li + 1n → T + 4 He + 4.8MeV
(I.3.1)
Li + 1n + 2.5MeV → T + 4 He + 1n
(I.3.2)
6
7
The reactions belonging to the second generation does not produce neutrons, and therefore have the advantage that the collection of resulting energy, which in the first generation usually escapes in the form of kinetic energy of fast neutrons, would be much simpler. Among the reactions of first to third generations, the D-T reaction has the highest cross section, but this is maximized at the temperature of 100keV. However, experiments and theoretical calculations show that sustainable chain reactions could be achieved at much lower temperatures, being around 10keV. In other words, self-sustaining nuclear fusion reactions require a temperature of around 120×106K. At such elevated temperatures, matter could exist only in the form of plasma, and all atoms become fully ionized. Clearly, under such severe conditions, the problem of confinement and heating of thermonuclear plasmas forms the bottleneck of nuclear fusion technology; thermonuclear plasmas cannot be simply confined in a manner comparable to gases and liquids. In stars, the thermonuclear plasma exists in the center and is inertially confined through the force of gravity (Figure I.3.3). The strength of gravitational force and temperature is so high in the core, that nuclear fusion reactions take place on their own. When the fusion reactions among all of the light elements stop due to the termination of nuclear fuel, the star undergoes either a collapse or expansion depending on its mass. Heavy stars form white dwarfs with extremely high mass densities where nuclear fusion reactions continue until all of the fuel is transformed into iron, while lighter stars expand and continue to faintly radiate as a red giant. On the earth, the time needed for confinement of thermonuclear plasma in order to achieve self-sustained reactions depends on the plasma density. For this reason, the plasma can be confined using ultra strong magnetic fields obtained by superconducting coils. The technique is referred to as the Magnetic Confinement Fusion (MCF). In the other approach, plasma is confined by pettawatt laser pulses having energies exceeding 10MJ, or accelerated
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79
particles, which uniformly irradiate a solidified spherical micro target. This technique is referred to as the Inertial Confinement Fusion (ICF). In MCF, the mean plasma density should be of the order of 1019cm-3, and its temperature peaks at 10keV. In ICF, the mean plasma density should exceed 1028cm-3, which is at least four orders of magnitude, or 10,000 times, higher than the density of solids under standard conditions. In this way, nuclear fusion reactions require the triple product of density, temperature, and confinement time to obey the following inequality, widely known as Lawson’s criterion
n × T × t > 1020 cm−3 ⋅ keV ⋅ s
(I.3.3)
where ⋅ sign represents the average. Therefore, the thermonuclear plasma in MCF should be kept at the temperature of several keVs for several seconds. Similarly, the confinement time in inertial fusion should be at least of the order of few nanoseconds. In practice, the confinement of plasma for such time intervals is so difficult due to many instabilities, that the experimental thermonuclear plasmas have been heated only up to the ignition point. Under such circumstances, the ratio of output to input plasma power Q is around 3, at which the heat generated by nuclear fusion reactions balances the plasma natural losses through plasma-wall interactions, radiations, and escape of energetic particles. But in order to obtain useful electrical power, this ratio should exceed 10. Currently, the largest existing project to achieve thermonuclear fusion is ITER (Figure I.3.4), which is to be built in the city of Caradache, France. The multi-billion dollar ITER project is scheduled for operation by 2025, and is funded by many countries including the United States, Russia, European Union, China, South Korea, India, and Japan, and each country is responsible for fabrication of one or several parts of the project. ITER is based on a machine named Tokamak which benefits from the technology evolved from decades of research in MCF science and technology. In tokamak, plasma is confined in the form of a torus by very strong magnetic fields in a vacuum vessel. To attain the plasma stability conditions a large unidirectional toroidal electrical current should be maintained in the plasma. For the case of ITER tokamak, this toroidal current should be about 15MA. The plasma confinement time is designed to be at least 400sec. Calculations predict that the plasma passes the ignition point and Q factor reaches 10. The total plasma volume in this giant machine amount to 840m3. The cross section of toroidal plasma in ITER tokamak has been shown in Figure I.3.5, indicating the dimensions as well.
Figure I.3.3. Gravitational force of the sun makes the thermonuclear Fusion process gets rolling
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
Figure I.3.4. The International thermonuclear Experimental Reactor (ITER) located in Caradache, France [3]
Figure I.3.5. Cross section of toroidal plasma in ITER tokamak [3]
I.4. Other Fusion Concepts Tokamak is not the only proposed path to the controlled nuclear fusion. There are other designs, among them spherical tokamaks, stellarators, and laser fusion could be named out. Spherical tokamaks are much similar to tokamaks in the concepts of design and operation, with the main difference being their tight aspect ratio. Aspect ratio is defined as the ratio of plasma's major radius to its minor radius; for tokamaks this ratio is within the range 3-5, while for spherical tokamaks is typically less than 1.5. This allows operation at higher magnetic pressures, which results in better confinement properties. Currently, there exist two major spherical tokamak experiments in the world: the Mega-Ampere Spherical Tokamak (MAST) in Culham, United Kingdom, where the Joint European Torus (JET) tokamak resides, and the National Spherical Tokamak eXperiment (NSTX) at Princeton Plasma Physics Laboratory (PPPL) in New Jersey, the United States. Photographs of NSTX plasma and facility are seen in Figure I.4.1.
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Figure I.4.1. The National Spherical Tokamak eXperiment (NSTX); top: facility; bottom: plasma in the vacuum vessel [4]
Figure I.4.2. Typical stellarator configuration
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
Figure I.4.3. National Compact Stellarator eXperiment at PPPL [4]
The stellarator concept is also similar to tokamaks, in the aspect that in both designs the plasma is produced and maintained in a toroidal vacuum vessel. However, while the stability of tokamak plasma is provided through the establishment of a DC toroidal plasma current, the stellarator plasma is stable without need to such a toroidal plasma current. The reason is that the stability is obtained by a complex topology of magnetic field windings which produce both the toroidal and poloidal magnetic fields. Stellarators are usually considered as too complex for realistic reactor designs, but they offer unlimited possibilities in plasma confinement. Wendelstein stellarator at Max Planck Institute of Plasma Physics (Figure I.4.2), Germany, and National Compact Stellarator eXperiment (NCSX) (Figure I.4.3) are examples of advanced stellarator configurations around the globe. It is also worth mentioning about the modern fusion-fission hybrid concept [5], which connects the possibilities of both technologies, combining the benefits and eliminating the drawbacks. In this design, a fission reactor produces the output electrical power, which is also used to run a tokamak or z-pinch (another MCF concept) and a proton accelerator. Both of auxiliary systems produce fast neutrons to keep the fission energy yield as high as possible. Because of the energetic neutrons used, heavy elements such as uranium may break up into much smaller elements, releasing even more energy and much less radioactive waste. These designs [6,7] are nowadays bringing attractions up as the progress in controlled nuclear fusion has slightly slowed down.
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Figure I.4.5. Hybrid fission-fusion design to produce clean and efficient nuclear energy
II. PLASMA EQUILIBRIUM II.1. Ideal Magnetohydrodynamics (MHD) Plasma is often misinterpreted as a "hot gas," but its conductivity and dynamic response to electricity and magnetism recognize it from a gas. The shape of the plasma and location of the plasma boundary deeply affect its stability. Since the electromagnetic fields control the movement of the plasma which itself induces electromagnetic fields, determining this shape may quickly lead to nonlinear equations. One simple way of studying magnetically confined plasmas with an emphasis on the shaping magnetic field topology is magnetohydrodynamics (MHD) model. MHD model first proposed by Hannes Olof Gösta Alfvén (Figure II.1.1). The word magnetohydrodynamic (MHD) is derived from magneto- meaning magnetic field, and hydro- meaning liquid, and -dynamics meaning movement.
Figure II.1.1. Hannes Alfvén, the father of modern plasma science, receives Nobel Prize from the King of Sweden in 1970 [9]
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
MHD equations consist of the equation of fluid dynamics and Maxwell’s equations that should be solved simultaneously. The MHD model is composed of the following relations
∂ρ + ∇ ⋅ (ρ V) = 0 ∂t ρ
∂V + ∇p = J × B ∂t
1 J = E + V×B σ ∇×E = −
∂B ∂t
1 ∂D ∇×B = J + μ0 ∂t ∇⋅D = ρ ∇⋅B = 0
⎪⎫⎪ ⎪⎪⎪ ⎪⎪ ⎪⎪ ⎬ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎭
Continuity
Momentum
Ohm's Law
(II.1.1)
(II.1.2)
(II.1.3)
Maxwell's Equations
(II.1.4)
In the above equations, plasmas are described as magnetohydrodynamic fluids with mass density ρ , current density J , mass flow velocity V , and local electric E and magnetic B fields. As in a plasma we have both the ion and electron species, we should write MHD equations for both ions and electrons, separately, but charge neutrality of plasma enables us to approximate the plasma as a neutral fluid with zero local electric charge density. Furthermore, since the mass of ions is much larger than the mass of electrons (the ion-to-electron mass ratio is mi me = 1836A , where A is the atomic weight of the ion) the contribution of ions govern the mass density of the plasma. MHD establishes a relationship between the magnetic field B and plasma motion V . Let us examine the relationship of these two parameters by applying curl operator to equation (II.1.3), which results in: ⎛J⎞ ∇ × ⎜⎜ ⎟⎟⎟ = ∇ × E + ∇ × ( V × B ) ⎝σ ⎠
(II.1.5)
Now by using equation (II.1.4) we get:
⎛ ∂B 1 J⎞ = ∇ × ⎜⎜ V × E − ⎟⎟⎟ = ∇ × (V × B ) − ∇×J ⎟ ⎜ σ⎠ σμ 0 ∂t ⎝ 1 = ∇ × (V × B ) − ∇ × (∇ × B ) σμ 0 1 = ∇ × (V × B ) − ∇ 2B σμ 0
(II.1.6)
An Overview of Plasma Confinement in Toroidal Systems
(
85
)
Equation (II.1.6) consists of two terms: the first term ∇ × V × B , is the convection 2
term and the second term proportional to ∇ B , represents the diffusion. Rate of change of the magnetic field is controlled by these two terms. Assume that the velocity of plasma V is zero everywhere so that the plasma does not move, therefore the first term in equation (II.1.6) vanishes, and we get:
∂B 1 = ∇2B = Dm ∇2B ∂t (σμ0 )
(II.1.7)
where Dm is called the diffusion coefficient of plasma. If the resistivity is finite, the magnetic field diffuses into the plasma to remove local magnetic inhomogeneities, e.g., curves in the field, etc. Ideal magnetohydrodynamics (MHD) describes the plasma as a single fluid with infinite conductivity. Hence by putting σ = ∞ in the Ohm’s law (II.1.3), we obtain
E + V×B = 0
(II.1.8)
In case of ideal MHD, σ → ∞ , the diffusion is very slow and the evolution of magnetic field B is solely determined by the plasma flow. For this reason the equation (II.1.7) recasts into the form
∂B = ∇× ( V × B) ∂t
(II.1.9)
The measure of the relative strengths of convection and diffusion is the magnetic Reynolds number Rm . Hence magnetic Reynolds number is a representation of combination of quantities that indicate the dynamic behavior of plasma. Reynolds number is the ratio of the first term to the second term on the right-hand-side of (II.1.6)
∇ × (V × B) 1 σμ0∇2B
VB L ≈ = μ0VLσ ≡ Rm ⎛ B ⎞⎟ ⎛⎜ 1 ⎞⎟ ⎜⎜ ⎟⎜ ⎟ ⎜⎝ L2 ⎠⎟⎟ ⎜⎜⎝ σμ ⎠⎟⎟ 0
(II.1.10)
where L is the typical plasma dimension. In (II.1.10) the magnetic Reynolds number is equal to the ratio of the magnetic diffusion time τ R = μ0L2 σ to the Alfven transit time
τH = L V , that is Rm = τR τHc . The magnetic field in a plasma changes according to a
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
diffusion equation, when Rm plasma, when Rm
1 while the lines of magnetic force are frozen in the
1.
We can demonstrate frozen-In theorem in integral form as below:
dΦ d = dt dt
∫∫ B ⋅ dS = 0
(II.1.11)
S
Hence, magnetic flux passing through any surface S with the plasma motion is constant. When Rm → ∞ , σ → ∞ the rate of change of the flux becomes zero. This means the magnetic flux is frozen in the plasma.
II.2. Curvilinear System of Coordinates Using curvilinear system of coordinates in analytical and numerical computations of equilibrium, stability and transport of toroidal plasmas is vital. The purpose of this section is to review a few fundamental ideas about curvilinear system of coordinate in general. Flux, Boozer and Hamada coordinates are typical coordinate systems in study of magnetic fusion plasmas. By definition, the position vector r in Cartesian coordinate system has three components
(x, y, z ) along its basis vectors (xˆ, yˆ, zˆ) , so that r = xxˆ + yyˆ + zzˆ . We may represent 1
the components of the position vector x with x , y with x
2
3
and z with x , and similarly
the basis vectors xˆ = ∇x with xˆ1 = ∇x1 , yˆ = ∇y with xˆ2 = ∇x 2 and zˆ = ∇z with
xˆ3 = ∇x 3 , to get r = x 1xˆ1 + x 2xˆ2 + x 3xˆ3 , or simply r = x i xˆi where the Einstein summation convention on repeated indices is adopted. The vectors xˆj , j = 1,2, 3 are called contravariant basis vectors, while x j , j = 1,2, 3 are referred to as the covariant components. Similarly, the components x j , j = 1, 2, 3 are called contravariant components of position vector, and while xˆj , j = 1, 2, 3 are referred to as the covariant bases. For the case of Cartesian coordinates, there is no distinction between contravariant r = x i xˆi and covariant
r = x i xˆi representations, in the sense that x i = x i and xˆi = xˆi .
(ζ , ζ , ζ ) uniquely establishes a one-to-one correspondence to the Cartesian coordinates (x , x , x ) through the set of analytic relations A curvilinear system of coordinates
1
1
2
2
3
3
ζ j = ζ j (x 1, x 2, x 3 ), j = 1,2, 3
(II.2.1)
An Overview of Plasma Confinement in Toroidal Systems For instance, suppose that
(ξ , ξ , ξ ) 1
2
3
87
are components of the standard spherical
coordinate system. Then
ξ1 =
2
(x 1 )
2
2
+ (x 2 ) + (x 3 )
(
)
⎛ ξ 3 = cos−1 ⎜⎜x 3 ⎝
(x 1 )
ξ 2 = − tan−1 x 2 x 1
2
2 2⎞ + (x 2 ) + (x 3 ) ⎟⎟ ⎠
(II.2.2)
In Cartesian coordinates, we represent any arbitrary vector quantity A with respect to its basis as
A = A1xˆ1 + A2xˆ2 + A3xˆ3 = Aj xˆj
(II.2.3)
where the components Aj , j = 1,2, 3 of the vector A , are called covariant components of A j . In a curvilinear system of coordinates, we must use different basis vectors ζˆ , j = 1,2, 3 ,
given by
⎛ ∂ ⎞ ∂ ∂ ∂ζ j ζˆj = ∇ζ j = ⎜⎜ 1 xˆ1 + 2 xˆ2 + 3 xˆ3 ⎟⎟⎟ ζ j = i xˆi ⎝ ∂x ⎠ ∂x ∂x ∂x
(II.2.4)
Note that unlike the Cartesian coordinates, where the covariant bases xˆ j , j = 1, 2, 3 are j physically dimensionless, ζˆ , j = 1,2, 3 may take on non-trivial physical dimensions. In
general, these basis vectors need not to be unit vectors. The condition for one-to-one correspondence of the curvilinear system of coordinates is that the basis vectors ζˆ , j = 1,2, 3 construct a parallelepiped with non-vanishing volume. j
Mathematically, the Jacobian determinant defined as
∂ζ 1 ∂x 1 ∂ζ 2 J = ζˆ1 ⋅ ζˆ2 × ζˆ3 = ∇ζ 1 ⋅ ∇ζ 2 × ∇ζ 3 = ∂x 1 ∂ζ 3 ∂x 1
∂ζ 1 ∂x 2 ∂ζ 2 ∂x 2 ∂ζ 3 ∂x 2
∂ζ 1 ∂x 3 ∂ζ 2 ∂x 3 ∂ζ 3 ∂x 3
(II.2.5)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
should not vanish. Furthermore, we suppose that the order of coordinates is chosen in such a way that the Jacobian J is always positive, which corresponds to a right handed system. As examples, the values of Jacobian in spherical and cylindrical coordinate systems are
1 R sin θ and 1 r , respectively. Since J > 0 , any vector such as A can be expanded in terms of the linearly independent bases as
A = Aj ζˆj
(II.2.6)
where components are in covariant forms, and hence (II.2.6) is a covariant representation of
A . In order to find A1 one may perform a dot product on both sides by ζˆ2 × ζˆ3 to find: A ⋅ (ζˆ2 × ζˆ3 ) = AJ 1
(II.2.7)
By cyclic permutation of indices we get the relation
Ai =
εijk ˆj ˆk ζ ×ζ ⋅ A 2J
(II.2.8)
Here, εijk is Levi-Civita pseudo-tensor symbol and is given by:
εijk
⎧⎪+1 ⎪⎪ ⎪ = xˆi ⋅ xˆj × xˆk = ⎪⎨−1 ⎪⎪ ⎪⎪0 ⎪⎩
If i,j,k is an even permutation of 1,2,3 If i,j,k is an odd permutation of 1,2,3 Otherwise
(II.2.9)
so that only nonzero components are
ε123 = ε231 = ε312 = 1
ε132 = ε321 = ε213 = −1
(II.2.10)
The factor 2 in the denominator of (II.2.8) comes from the fact that a summation convention is adopted on the right-hand-side because of the repeating indices. An intelligent fellow could have chosen an alternative set of basis vectors derived from the covariant bases ζˆ = ∇ζ , j = 1, 2, 3 , simply given by j
j
ε ζˆi = ijk ζˆj × ζˆk , i = 1,2, 3 2J
(II.2.11)
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It is easy to verify that the new contravariant bases ζˆj , j = 1,2, 3 construct a parallelepiped with non-vanishing volume equal to 1 J = ζˆ1 ⋅ ζˆ2 × ζˆ3 , and are hence linearly independent. It should be noted that covariant and contravariant bases usually have physically different dimensions, while in the Cartesian coordinates they coincide. Now (II.2.8) together with (II.2.11) gives the relation for covariant components of A as
Aj = A ⋅ ζˆj
(II.2.12)
Similarly, any vector such as A can be expanded in terms of the contravariant bases like (II.2.6) as
ε Aj A = A j ζˆj = jkl ζˆk × ζˆl 2J
(II.2.13)
Hence, (II.2.12) is a contravariant representation of A . In contrast to (II.2.12), the contravariant components A j , j = 1, 2, 3 can now be easily found by performing an inner product with ζˆj on both sides to give
Ai = Ai δij = Ai ζˆi ⋅ ζˆj = A ⋅ ζˆj
(II.2.14)
in which we have used the relation ζˆi ⋅ ζˆj = δij . Hence, we get the fairly easy relation for the contravariant components
Aj = A ⋅ ζˆj
(II.2.15)
Figure II.2.1. Geometrical representation of contravariant and covariant bases in two-dimensional plane
90
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani Figure II.2.1 shows the typical construction of contravariant and convariant bases in the 1
two-dimensional plane and the curvilinear coordinates for which ζ = ζ
1
(x 1, x 2 ) ,
ζ 2 = ζ 2 (x 1, x 2 ) , and ζ 3 = x 3 . In addition, suppose that we have positive Jacobian J > 0 everywhere. As it can be seen, the covariant bases ζˆ are by their definition always normal to their j
respective
constant
contours
given
ζ j = cte ,
by
while
contravariant
bases
(ζˆ , ζˆ ) (ζˆ × ζˆ , ζˆ × ζˆ ) are respectively tangent to the contours (ζ = cte, ζ = cte) . Note that in this example, all vectors (ζˆ , ζˆ ) and (ζˆ , ζˆ ) lie on the two-dimensional plane 1
2
2
3
3
2
1
1
1
(x 1, x 2 )
1
2
2
1
because of the special choice of ζ = ζ
1
(x 1, x 2 ) ,
ζ 2 = ζ 2 (x 1, x 2 ) , and
ζ 3 = x 3 . Hence, the orthogonality relationship holds as ζˆ1 ⋅ ζˆ2 = ζˆ2 ⋅ ζˆ1 = 0 . However, the normality of covariant bases and tangential property of contravariant bases to contours are quite universal rules, and applicable everywhere the Jacobian does not vanish. We stress again that in general these two sets of bases need to have neither similar physical dimensions nor identical directions. The only system of coordinates for which both covariant and contravariant vectors share the same physical dimensions and directions is the rectangular Cartesian system of coordinates, and only for the square Cartesian coordinates two bases are equal. On the other hand, orthogonal systems of coordinates are marked with covariant and contravariant vectors pointing to the same directions, while having different physical dimensions. For this to happen, covariant vectors need to be mutually orthogonal. Examples of orthogonal coordinate systems include spherical and cylindrical coordinates. Hence, for an orthogonal coordinate system, we have two further orthogonality relationships given by ζˆi ⋅ ζˆj = 0 and
ζˆi ⋅ ζˆj = 0 , only if i ≠ j . For these two coordinate systems, contravariant (and hence covariant) bases are always mutually normal, yet position dependent and changing direction from point to point. In contrast, for Cartesian coordinates the direction (as well as the length of) contravariant bases are fixed throughout the space.
II.2.1. Transformation of Coordinates We can transform components of an arbitrary vector from a given coordinate system
(ζ 1, ζ 2, ζ 3 ) into another coordinate system (ζ 1, ζ 2, ζ 3 ) , related by
( ) (ζ , ζ , ζ ) (ζ , ζ , ζ )
ζ 1 ζ 1, ζ 2 , ζ 3 ζ
2
ζ3
1
2
3
1
2
3
By using the covariant representation of the vector A , we get
(II.2.16)
An Overview of Plasma Confinement in Toroidal Systems
A = Aj ∇ζ j = Ai ∇ζ i = Aj
∂ζ j ∇ζ i i ∂ζ
91
(II.2.17)
from which it can be concluded that
∂ζ j Ai = Aj ∂ζ i
(II.2.18)
It is easy to verify that transformation law for contravariant components is given by
Ai = A j
∂ζ i ∂ζ j
(II.2.19)
II.2.2. Metric Tensor In order to get contravariant components of vector A from its covariant components one should multiply the covariant components by the elements g
A j = Ai g ij
ij
(II.2.20)
where g = ζˆ ⋅ ζˆ is the symmetric metric tensor and includes all necessary information ij
i
j
about the coordinate system. In particular, if the coordinate system is orthogonal, the metric tensor will be diagonal. Determinant of metric tensor has a relation to Jacobian of coordinate system as
g ij = J 2
(II.2.21)
From equation (II.2.20) one can obtain the relation of covariant Ai and contravariant components Ai as −1
Ai = ⎡⎢g ij ⎤⎥ Aj = gij Aj ⎣ ⎦
(II.2.22)
where gij is the covariant form of metric tensor. We furthermore define the determinant of the covariant metric tensor g as
g ≡ gij =
1 J2
(II.2.23)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
The relations (II.2.20) and (II.2.23) are frequently referred to as laws of raising and lowering indices, respectively. Together, these relations constitute the concept of ‘index gymnastics’.
II.2.3. Volume and Surface Elements Volume element is defined as
dV = dx 1dx 2dx 3 =
1 1 2 3 d ζ d ζ d ζ = g d ζ 1d ζ 2d ζ 3 J
(
(II.2.24)
)
Besides, the square line element for coordinate system ζ1, ζ 2 , ζ 3 is
ds 2 = dr ⋅ dr = ζˆi ⋅ ζˆjd ζid ζ j = g ijd ζid ζ j = g ijd ζ id ζ j
(II.2.25)
More often, metrics are represented by their respective line elements in the form (II.2.25), instead of expressing all independent components in matrix form.
II.2.4. Dot and Cross Product The inner (dot) product of two arbitrary vectors A and B may be easily found if one is represented in covariant and the other in contravariant forms
A ⋅ B = (Ai ζˆi ) ⋅ (B j ζˆj ) = Ai B j ζˆi ⋅ ζˆj = Ai B j δij = Ai Bi = Ai B i
(II.2.26)
In order to obtain the cross product, both vectors may be expressed in covariant form. Therefore W = A × B = (Ai ζˆi ) × (B j ζˆj ) = (A2B3 − A3B2 ) ζˆ2 × ζˆ3 + (A3B1 − A1B3 ) ζˆ3 × ζˆ1 + (A1B2 − A2B1 ) ζˆ1 × ζˆ2 = J (A2B3 − A3B2 ) ζˆ1 + J (A3B1 − A1B3 ) ζˆ2 + J (A1B2 − A2B1 ) ζˆ3
(II.2.27)
Through comparison to W = W ζˆi , the contravariant components of W can be thus i
found as
W i = W ⋅ ζˆi = εijk Aj Bk J where we here define the contravariant Levi-Civita symbol as ε
(II.2.28) ijk
components of W = Wi ζˆ may be directly obtained from (II.2.28) as i
= εijk . The covariant
An Overview of Plasma Confinement in Toroidal Systems
93
1 jk εi Aj Bk J
(II.2.29)
Wi = gijW j = gij ε jkl Ak BlJ = g εikl Ak Bl J = jk
Again, we adopt the definition εi = εijk , that is the Levi-Civita pseudo-tensor does not transform according to the transformation laws of index gymnastics.
II.2.5. Gradient, Divergence and Curl Operator The gradient operator is by definition given by
∇ = ∇ζ i
∂ ∂ = ζˆi i i ∂ζ ∂ζ
(II.2.30)
(
1
2
By applying the gradient operator to a scalar function f ζ , ζ , ζ
3
) we get a vector field
as
∂f S = ∇f (ζ 1, ζ 2, ζ 3 ) = ζˆi i ∂ζ
(II.2.31)
Comparing to (II.2.27), we directly obtain the covariant components of S as
Si =
∂f ∂ζ i
(II.2.32)
Now, using (II.2.22) one can find the contravariant components of S as
S i = g ij
∂f ∂ζ j
(II.2.33)
One can take the directional derivative of any vector A along curvilinear coordinates
∂ ∂ ∂Ai ˆi ∂ζˆi i ˆ A ζ ζ = A = + A ( i ) ∂ζ j i ∂ζ j ∂ζ j ∂ζ j
(II.2.34)
Here, the second term expresses the dependence of basis vectors on coordinates, and identically vanishes for Cartesian coordinates. Hence, we can write the covariant components of directional derivative of vector A , or Ai , j as
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
⎛ ∂ ⎞ ∂ A ⋅ ζˆi Ai, j = ⎜⎜ j A⎟⎟⎟ = ∂ζ j ⎝⎜ ∂ζ ⎠i =
∂Ai + Γkji Ak j ∂ζ
(II.2.35)
On the other hand, the contravariant components of directional derivative of vector A , i ,j
A can be obtained as i
ˆ i ⎛ ∂ ⎞ ⎡ ∂ ⎤ ∂Ai k ∂ ζk A = ⎜⎜ j A⎟⎟⎟ = ⎢ j (Ak ζˆk )⎥ ⋅ ζˆi = A + ⋅ ζˆ ⎜⎝ ∂ζ ⎠ ⎢⎣ ∂ζ ⎥⎦ ∂ζ j ∂ζ j ∂Aj = − Γijk Ak ∂ζ j i ,j
(II.2.36)
In the latter relations, Γ is referred to as the Christoffel Symbol and is defined as
Γkji
∂ζˆk ˆi ∂ζˆi ˆk ζ ⋅ = − ⋅ζ ∂ζ j ∂ζ j
(II.2.37)
The Christoffel Symbol can be presented in terms of the metric tensor and more convenient form
⎛ ∂g ∂g ∂g ⎞ 1 Γijk = − g im ⎜⎜ mij + mji − mjk ⎟⎟⎟ 2 ∂ζ ∂ζ ⎠ ⎝⎜ ∂ζ
(II.2.38)
The divergence operator ∇ ⋅ acts on a vector field and is defined as ∂ζˆj ˆi ∂ ∂A j ˆ ˆi ∂Ai ∂ ⎛ εjkl ˆk ˆl ⎞⎟ ˆi ⎜ ⋅ A j ζˆj ) = ⋅ζ = + Aj ζ j ⋅ ζ + Aj ζ ×ζ ⎟⋅ ζ i ( i i ⎠ ∂ζ ∂ζ ∂ζ ∂ζ i ∂ζ i ⎜⎝ 2J i ε ∂A ∂ ˆk ˆl ˆi = + Aj jkl (ζ × ζ ) ⋅ ζ + Aj εjkl (ζˆk × ζˆl ) ⋅ ζˆi ∂∂ζ i ⎜⎜⎝⎛ 21J ⎟⎠⎟⎟⎞ 2J ∂ζ i ∂ζ i
∇ ⋅ A = ζˆi
=
∂Ai ∂ ⎛ 1 ⎞⎟ ∂Ai ∂ ⎛ 1 ⎞⎟ ⎜ ⎟ ⎜ ⎟= + A j εjkl εiklJ + A j 2J i ∂ζ j ⎜⎝ 2J ⎠⎟ ∂ζ ∂ζ i ⎜⎝ 2J ⎠⎟ ∂ζ i
(II.2.39)
in which we have made use of the identity ∂ζ j ∂ζ i = δ ij and thus ∂ζˆ ∂ζ = 0 , as well j
i
as εjkl εikl = 2δij . Finally the relation of divergence of a vector field in curvilinear coordinates simplifies to the compact form
An Overview of Plasma Confinement in Toroidal Systems ∇⋅ A = J
∂ ∂ζ i
⎛ 1 j ⎞⎟ ⎜ A ⎟⎟ ⎜⎝J ⎠
95
(II.2.40)
Similarly, it is possible to express the rotation or curl of a vector field A , which is the vector product of the operator ∇ and the vector A given by ∇ × A = ζˆi
∂ ×A ∂ζ i
(II.2.41)
The contravariant components of (II.2.41) are
(∇ × A) = ζˆi ⋅ ζˆj × i
= J εijk
ˆk ∂ ˆk ) = ζˆi ⋅ ζˆj × ζˆk ∂Ak + A ζˆi ⋅ ζˆj × ∂ζ A ζ ( k k ∂ζ j ∂ζ j ∂ζ j
∂ Ak ∂ζ j
(II.2.42)
Here, the identity ∂ζˆ ∂ζ = 0 is exploited again. j
i
II.3. Flux Coordinates Many operators take on simple forms in flat coordinate systems such as Cartesian coordinates and are easy to remember and evaluate. However, when problems deal with toroidally symmetric systems it is often helpful to use coordinate systems, which exploit the toroidal symmetry and in particular nested toroidal shape of closed magnetic surfaces. In order to study toroidal devices such as tokamaks, we have to choose the handiest coordinate system so that the equations of equilibrium as well as major plasma parameters become straightforward. At first, we consider the Primitive Toroidal Coordinates, in which an arbitrary point in 3-space can be uniquely identified by a set of one radial and two angle coordinates. Then we proceed to study the Flux Coordinates, in which poloidal cross section of magnetic surfaces look as concentric circles. We also mention Boozer and Hamada systems as particular cases of flux coordinates.
II.3.1. Primitive Toroidal Coordinates Figure II.3.1 shows primitive toroidal coordinates (r0 , θ0 , ζ 0 ) in which r0 is the distance measured from the plasma major axis, θ0 is the poloidal angle and ζ0 is the toroidal angle; axisymmetric equilibrium rules out any dependence on the toroidal angle ζ 0 . The values assumed by these coordinates are physically limited according to 0 ≤ r0 < ∞ , 0 ≤ θ0 < 2π and 0 ≤ ζ 0 < 2π . Relations between primitive toroidal and cylindrical coordinates
(R, ϕ, Z ) are
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
Figure II.3.1. Primitive toroidal coordinates
(R − R )
2
+ Z2 ⎛ Z ⎞⎟ ⎟⎟ θ0 = tan−1 ⎜⎜⎜ ⎝ R − R ⎠⎟ ζ0 = −ϕ
r0 =
0
(II.3.1)
where R0 is major radius of plasma, and the minus sign in the third equation is for maintaining right-handedness of the system. This primitive toroidal coordinates is evidently orthogonal and its metric tensor is therefore diagonal. Consequently the squared differential length is
ds 2 = dR 2 + R 2 d ϕ 2 + dz 2 2
=dr02 + r02d θ02 + (R0 + r0 cos θ0 ) d ζ 02
(II.3.2)
Hence the metric coefficients of primitive toroidal coordinates are given by g rr = 1 ,
g θθ = r0 and gζζ = R = R0 + r0 cos θ0 ; all other metric coefficients are zero. Using equation (II.2.5), the Jacobian determinant of this system is found to be 1 rR0 . The gradient of a scalar field f in primitive coordinates is simply
∇f =
∂f 1 ∂f ˆ 1 ∂f ˆ rˆ0 + θ0 + ζ0 ∂r0 r0 ∂θ0 R0 + r0 cos θ0 ∂ζ0
(II.3.3)
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97
The divergence and curl of a vector field A are
∇⋅A = +
∇×A =
⎫ ⎪⎧ 1 ∂ ⎡ 1 ⎤⎬⎪⎪ + r R r θ A cos ( ) ⎨⎪ 0 0 0 0 r ⎢ ⎥ 0 ⎦ ⎣ ⎪⎭⎪ R0 + r0 cos θ0 ⎪ ⎩⎪r0 ∂r0 ∂Aζ0 ∂ζ 0
+
1 ∂ ⎡ (R0 + r0 cos θ0 ) Aθ0 ⎤⎥⎦ r0 ∂θ0 ⎢⎣
⎧⎪⎪ ∂Aθ0 ⎫⎪ 1 1 ∂ ⎡ − (⎣⎢ R0 + r0 cos θ0 ) Aζ0 ⎤⎦⎥⎪⎬ rˆ0 ⎨ ⎪⎭⎪ (R0 + r0 cos θ0 ) ⎪⎩⎪ ∂ζ 0 r0 ∂θ0
⎡ ∂A ⎤ ⎪⎫ ⎢(R0 + r0 cos θ0 ) Aζ − r0 ⎥ ⎪⎬ θˆ0 0 ⎢ ∂ζ 0 ⎥⎦ ⎪⎭⎪ ⎣ ⎫⎪ ∂ 1 ⎪⎧ ∂Ar r0Aθ0 )⎪⎬ ζˆ0 + ⎪⎨ 0 − ( ⎪⎭⎪ r0 ⎪⎪⎩ ∂θ0 ∂r0
+
(II.3.4)
⎪⎧⎪ ∂ 1 ⎨ (R0 + r0 cos θ0 ) ⎪⎩⎪ ∂r0
(II.3.5)
It can be shown that the non-vanishing Christoffel symbols in primitive toroidal coordinates are 2 Γ221 = −Γ12 =
Γ223 = −Γ232 =
cos θ0 R0 + r0 cos θ0 − sin θ0
R0 + r0 cos θ0 1 3 Γ331 = −Γ13 = r0
(II.3.6)
II.3.2. Flux Coordinates In confined toroidal plasmas, magnetic field lines define closed magnetic surfaces due to a famous ‘hairy ball’ theorem proven by Poincaré, which implies that field lines of a non-zero magnetic field must cover a toroidal surface, as shown in Figure II.3.2. These define surfaces, to which the particles are approximately constrained, known as flux surfaces. The surfaces are mathematically expressed as constant poloidal flux surfaces, denoted by ψ = cte . Magnetic surfaces for equilibrium plasmas with no external current drive coincide with isobar, i.e. constant pressure surfaces. Figure II.3.3 shows magnetic field lines as well as current density lines, which lie on these nested isobaric flux surfaces. The flux coordinates (ψ, χ, ζ ) shown in Figure II.3.4, represent the true complicated physical shape of magnetic surfaces, and are here expressed as functions of the primitive
(
)
toroidal coordinates r0 , θ0 , ζ 0 . ψ denotes the poloidal magnetic flux (or any monotonic function of), and χ and ζ are respectively referred to as poloidal and toroidal angles. The
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
latter two coordinates are not true angles although they have the dimension of radians. Therefore we have
ψ = ψ (r0 , θ0 , ζ 0 ) χ = χ (r0 , θ0 , ζ 0 ) ζ = ζ (r0 , θ0 , ζ 0 )
Figure II.3.2. A hairy doughnot
Figure II.3.3. Both magnetic field and current density lines lie on nested magnetic surfaces
Figure II.3.4. Flux coordinates
(II.3.7)
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99
Since the coordinates χ and ζ are similar to angles, then any physical quantity such as
A = A(ψ, χ, ζ ) in flux coordinates should be periodic as A = A(ψ, χ + 2mπ, ζ + 2nπ) . This necessitates the transformation (II.3.7) to be of the form
ψ = ψ0 (r ) +
∑
ψmn (r )e
i (m θ +nϕ)
m ,n
(m ,n )≠(0,0) i (m θ +nϕ )
χ = θ + ∑ θmn (r )e m ,n
ζ = −ϕ + ∑ ϕmn (r )e
i(m θ +nϕ)
m ,n
(II.3.8)
From (II.2.24) the volume element in Flux coordinate is defined as
dV =
g d ζ 1d ζ 2d ζ 3 = J −1d ψd χd ζ
(II.3.9)
where the Jacobian is
ψx ψy ψz
ˆ × ζˆ > 0 J = χx χy χz = ψˆ ⋅ χ ζ x ζy
ζz
(II.3.10)
ˆ ˆ Here, ψ = ∇ ψ , χˆ = ∇χ , and ζ = ∇ζ are the covariant basis vectors. Hence, the
ˆ ˆ
ˆ
ˆ
ˆ × ζ , ζ × ψ , and ψ × χ ˆ, corresponding contravariant bases in flux coordinate system are χ respectively. Hence, the contravariant representations of magnetic field as well as current density in flux coordinates are
B = B ψ (ψ, χ, ζ )
J = J ψ (ψ, χ, ζ ) where
ˆ × ζˆ ˆ χ ζˆ× ψˆ ψˆ × χ + B χ (ψ, χ, ζ ) + B ζ (ψ, χ, ζ ) J J J (II.3.11) ˆ × ζˆ χ ζˆ× ψˆ ψˆ × χˆ + J χ (ψ, χ, ζ ) + J ζ (ψ, χ, ζ ) J J J
(II.3.12)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
B ψ = B ⋅ ψˆ
J ψ = J ⋅ ψˆ
ˆ Bχ = B ⋅ χ
J χ = J ⋅ χˆ
B ζ = B ⋅ ζˆ
J ζ = J ⋅ ζˆ
(II.3.13)
As both magnetic field lines and current density lines lie on magnetic surfaces, we must have
B ψ = B ⋅ ψˆ = 0 J ψ = J ⋅ ψˆ = 0
(II.3.14)
since contravariant field components are tangent to the surfaces and have no component along the corresponding covariant bases. On the other from Maxwell’s equation we know that ∇ ⋅ B = 0 , and therefore by applying divergence operator we obtain
⎡ ∂ ⎛B ζ ⎞ ∂ ⎜⎛ B χ ⎞⎟⎥⎤ ∇ ⋅ B = J ⎢ ⎜⎜ ⎟⎟ + =0 ⎢ ∂ζ ⎜⎝ J ⎟⎠ ∂χ ⎜⎜⎝ J ⎠⎟⎟⎥ ⎣ ⎦
(II.3.15)
∂ ⎛⎜ B ζ ⎞⎟ ∂ ⎜⎛ B χ ⎞⎟ ⎜⎜ ⎟⎟ = − ⎜ ⎟ ∂ζ ⎝ J ⎠ ∂χ ⎜⎝ J ⎠⎟
(II.3.16)
which results in
The continuity equation for electric charge also reads
∇⋅ J = −
∂ρ ∂t
(II.3.17)
But the time derivative ∂ ∂t vanishes under steady-state assumption, therefore
∇ ⋅ J = 0 , and similar to (II.3.16) we get ∂ ⎛⎜J ζ ⎞⎟ ∂ ⎛⎜J χ ⎞⎟ ⎜ ⎟=− ⎜ ⎟ ∂ζ ⎜⎝ J ⎠⎟ ∂χ ⎜⎝ J ⎠⎟ Now we adopt the notations (II.3.16) allows us to write down
χ χ bζ = B ζ J b = B J
,
, and
(II.3.18) jζ = J ζ J j χ = J χ J , , which by
An Overview of Plasma Confinement in Toroidal Systems
b χ (ψ, χ, ζ ) = b χ (ψ ) − b (ψ ) χ −
∂b (ψ, χ, ζ ) ∂ζ
b ζ (ψ, χ, ζ ) = b ζ (ψ ) + b (ψ ) ζ +
∂b (ψ, χ, ζ ) ∂χ
j χ (ψ, χ, ζ ) = j χ (ψ ) − j (ψ ) χ −
∂j (ψ, χ, ζ ) ∂ζ
j ζ (ψ, χ, ζ ) = j ζ (ψ ) + j (ψ ) ζ +
∂j (ψ, χ, ζ ) ∂χ
101
(II.3.19)
(II.3.20)
Because of periodicity with respect to the angular coordinates χ and ζ , we need
b (ψ) = j (ψ) = 0 . On the other hand the auxiliary functions need to obey b (ψ, χ, ζ ) = b (ψ, χ + 2m π, ζ + 2n π) j (ψ, χ, ζ ) = j (ψ, χ + 2m π, ζ + 2n π )
(II.3.21)
Hence, (II.3.11) and (II.3.12) can be rewritten as
B = b χ (ψ, χ, ζ ) ζˆ× ψˆ + b ζ (ψ, χ, ζ ) ψˆ × χˆ ⎡ ∂b (ψ, χ, ζ ) ⎤⎥ ˆ ˆ ⎡⎢ ζ ∂b (ψ, χ, ζ ) ⎤⎥ ˆ ˆ = ⎢b χ (ψ) − ζ × ψ + b (ψ ) + ψ×χ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ζ χ ⎣ ⎦ ⎣ ⎦ (II.3.21)
ˆ J = j χ (ψ, χ, ζ ) ζˆ× ψˆ + j ζ (ψ, χ, ζ ) ψˆ × χ ⎡ ∂j (ψ, χ, ζ ) ⎤ ˆ ˆ ⎡ ζ ∂ ψ, χ, ζ ) ⎤ ˆ ⎥ ζ × ψ + ⎢ j (ψ ) + j ( ⎥ ψ×χ ˆ = ⎢ j χ (ψ ) − ⎢ ⎥ ⎢ ⎥ ∂ζ ∂χ ⎣ ⎦ ⎣ ⎦ (II.3.22) But, the current density J and magnetic field B are further interrelated by the Ampere’s law ∇× B = μ0 J , and thus noting (II.3.22) we get the alternative form
J= for which
1 ∇× B = ∇× ⎡⎢−c (ψ, χ, ζ ) ψˆ + cχ (ψ ) χˆ + cζ (ψ ) ζˆ⎤⎥ ⎣ ⎦ μ0
(II.3.23)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
j χ (ψ ) = + j ζ (ψ ) = −
dcζ (ψ)
dψ dcχ (ψ) dψ
c (ψ, χ, ζ ) = c (ψ, χ + 2m π, ζ + 2n π)
(II.3.24) (II.3.25)
must hold according to (II.2.42) and (II.3.8). The magnetic field can be thus derived from
B = μ0 ⎡⎢−c (ψ, χ, ζ ) ψˆ + bχ (ψ ) χˆ + bζ (ψ ) ζˆ + ∇g (ψ, χ, ζ )⎤⎥ ⎣ ⎦
(II.3.26)
where g (ψ, χ, ζ ) is an arbitrary function given by
g (ψ, χ, ζ ) = g (ψ, χ, ζ ) + aχχ + aζ ζ + a g (ψ, χ, ζ ) = g (ψ, χ + 2mπ, ζ + 2n π)
(II.3.27)
in which a, a χ and aζ are constants. It is obvious that the latter two constants are trivial and may be respectively absorbed in the functions bχ (ψ ) and bζ (ψ ) , and thus may be safely ignored.
II.3.3. Boozer Coordinates Allen Boozer showed that by a proper transformation of coordinates, one could even get
(
)
rid of g ψ, χ, ζ . This transformation leads us to the so-called Boozer coordinate system. The required transformation is
ψ′ = ψ χ′ = χ + Aχ (ψ )G (ψ, χ, ζ ) + C χ (ψ ) F (ψ, χ, ζ ) ζ ′ = ζ + Aζ (ψ )G (ψ, χ, ζ ) + C ζ (ψ ) F (ψ, χ, ζ )
with
Aχ
,
Aζ
,
Cχ
,
Cζ
, and G
and F
(II.3.28)
being arbitrary functions satisfying
G = F = 0 ; here, ⋅ stands for angular average. Appropriate candidate for these functions are
An Overview of Plasma Confinement in Toroidal Systems
103
ψ′ = ψ χ′ = χ + ζ′ = ζ +
1 (b χg + bcζ ) b cζ + b χcχ ζ
1 b ζ g − bcχ ) ( χ b cζ + b cχ ζ
(II.3.29)
Finally, one can find covariant and contravariant forms of the magnetic field as follows
B = b χζˆ× ψˆ + b ζ ψˆ × χˆ ˆ + Bζ ζˆ = c ψˆ + Bχ χ
(II.3.30)
It may be shown that
c (ψ, χ, ζ ) = (b χg + Bζb )
dBχ dψ
+ (b ζ g − Bχb )
dBζ dψ
(II.3.31)
b ζ (ψ ) = q (ψ ) b χ (ψ ) = 1 Bχ (ψ ) = μ0 I t (ψ) 2π
it
Bζ (ψ ) = μ0 ⎣⎡I p coil (ψ) − I p plasma (ψ)⎦⎤ 2π
ip ( ψ )
(II.3.32)
( )
Here, q ψ is the safety factor of plasma, which is number of turns the helical magnetic field lines in a tokamak makes round the major circumference per each turn of the minor circumference, I t (ψ) is the toroidal current within the magnetic surface ψ , and ip (ψ ) is the poloidal current difference between poloidal coils and plasma within the magnetic surface ψ. Upon substituting (II.3.31) and (II.3.32) into (II.3.30) we obtain the fairly simple forms of contravariant and covariant representations of the magnetic field as
B = q (ψ ) ψˆ × χˆ + ζˆ× ψˆ = it (ψ ) χˆ + ip (ψ ) ζˆ + c ψˆ
(II.3.33)
This shows that the covariant and contravariant components of the magnetic field are given as
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
Bψ = 0
Bψ = c
Bχ = J
Bχ = it (ψ )
B ζ = Jq (ψ )
Bζ = ip (ψ )
(II.3.34)
Equation (II.3.34) displays both covariant and contravariant components of magnetic 2
field in flux coordinates. The squared magnitude of magnetic field B = B ⋅ B , may be 2
i
readily found from B = Bi B as
B ⋅ B = ⎢⎡k (ψ, χ, ζ ) ψˆ + it (ψ ) χˆ + ip (ψ ) ζˆ⎥⎤ ⋅ ⎢⎡q (ψ ) ψˆ × χˆ + ζˆ× ψˆ⎥⎤ ⎣ ⎦ ⎣ ⎦ = Jqip (ψ ) + Jit (ψ )
(II.3.35)
Jacobian can thus be determined as
Bp2 + Bt2 B2 = J = q (ψ )ip (ψ ) + it (ψ ) q (ψ ) ip (ψ ) + it (ψ)
(II.3.36)
II.3.4. Hamada Coordinates In general, the Jacobian vary as a function of coordinates like J = J (ψ, χ, ζ ) . Hamada coordinates (ψH , χH , ζ H ) are chosen in such a way that the Jacobian J is made a flux label; a scalar flux label function has the characteristic that its gradient is always parallel to
ψˆ and hence a function of only ψ . For the particular choice of ψH = V (ψ ) (2π )2 , χH = χ , and ζH = ζ it can be shown that J = 1 , where V (ψ) is the volume of magnetic flux tube bounded by the poloidal flux ψ . By virtue of Hamada coordinates, (II.3.34), (II.3.12) and (II.3.14) we also readily obtain
Bψ = 0
Jψ = 0
B χ = B χ (ψ ) = 1
J χ = J χ (ψ )
B ζ = B ζ (ψ ) = q (ψ )
J ζ = J ζ (ψ )
(II.3.35)
In other words, all contravariant components of fields become flux functions. Use of Hamada coordinates also implies many other attractive features, some of which will be discussed in the following. It can be furthermore shown that for a toroidal plasma equilibrium Hamada coordinates exists either in absence of pressure anisotropy or under axisymmetry. The former condition is automatically met in most practical situations where no external heating mechanism is used.
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105
II. 4. Extensions to Axisymmetric Equilibrium In order to have steady state fusion energy, the hot plasma of Tokamak or other promising toroidal devices such as stellarator should be kept away at equilibrium from the first wall. Without use of strong magnetic field, the confinement of this hot plasma is out of reach. Tokamaks are axisymmetric machines which make their analysis much easier. Although recent progress in this field has resulted in some novel equilibrium configurations [37-40], however, we limit the discussion to the well-established cases.
II.4.1. MHD Equilibrium From MHD momentum balance equation (II.1.2) we have: ρ
dV = −∇p + J × B dt
(II.4.1)
By using Ampere’s law, we can eliminate the current density J from the J × B force term to get J×B =
1 (∇ × B)× B μ0
(II.4.2)
Now by means of vector identity, ∇ (A ⋅ B) = (A ⋅ ∇) B + A × (∇× B) + (B ⋅ ∇) A + B × (∇× A) , one can rewrite (II.4.2) as J×B =
⎛ B 2 ⎞⎟ 1 ⎟ (B ⋅ ∇) B − ∇ ⎜⎜ ⎜⎝ 2μ0 ⎠⎟⎟ μ0
(II.4.3)
The left-hand-side of (II.4.1) under equilibrium vanishes and therefore by substituting (II.4.3) in (II.4.1) we have ∇p = J × B =
⎛ B 2 ⎞⎟ 1 ⎟⎟ (B ⋅ ∇) B − ∇ ⎜⎜⎜ μ0 ⎝ 2μ0 ⎠⎟
(II.4.4)
This is equilibrium equation which states that under equilibrium, the pressure gradient is balanced by forces due to magnetic field curvature and pressure gradient. The next thing that may be inferred from (II.4.4) is that the current and magnetic field lie on isobaric surfaces. We accepted this fact without proof, but this consequence arises from the fact J ⋅ ∇p = B ⋅ ∇p = 0 while ∇ p is normal to isobar surfaces. Now rewriting (II.4.4) gives
⎛ B 2 ⎞⎟ 1 ⎟ = (B ⋅ ∇) B ∇ ⎜⎜p + ⎜⎝ 2μ0 ⎠⎟⎟ μ0
(II.4.5)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
When the field lines are straight and parallel (with no curvature), the right-hand-side of (II.4.5) vanishes and it reduces to a simple form that the total pressure is constant everywhere within a plasma
p+
B2 B2 = (1 + β ) = cte 2μ0 2μ0
(II.4.6)
where β is here defined as
β
p B 2μ0 2
(II.4.7)
According to (II.4.7), β is the ratio of plasma pressure to magnetic field pressure and is one of the figures of merit for magnetic confinement devices. It should be mentioned that for practical confinement geometries (II.4.7) is not applicable and an averaged definition for β is needed. In view of the fact that fusion reactivity increases with plasma pressure, a high value of beta is a sign of good performance. The highest value of beta achieved in a large tokamak is about 13%, though higher values are theoretically possible at lower aspect ratio. There is a theoretical limit on the maximum β that can be achieved in a magnetic plasma and is due to deterioration in the confinement. The Troyon β limit which states that for a stable plasma operation β cannot be greater than gI aB where g is Troyon coefficient and has a value of about 3.5 for conventional tokamaks, I is the plasma current in Mega Amperes, a is the minor radius in meters and B is the toroidal field in Tesla.
II.4.2. Z-pinch Equilibrium As an example, we are going to evaluate equilibrium of an ideal Z-pinch, a conceptual one-dimensional magnetic confinement device, which confines the plasma in cylindrical geometry by using an axial current and poloidal magnetic windings. In cylindrical coordinates (r, ϕ, z ) the equilibrium equation (II.4.4), for Z-pinch takes the form
∂p = −J z Bϕ ∂r
(II.4.8)
Using Ampere’s law,
Jz = Substituting (II.4.9) into (II.4.8) gives
1 ∂ (rBϕ ) μ0 ∂r
(II.4.9)
An Overview of Plasma Confinement in Toroidal Systems 2 2 ∂p ∂ Bϕ ⎞⎟ 1 ⎛B ⎟ = − ⎜⎜⎜ ϕ + μ0 ⎝⎜ r ∂r ∂r 2 ⎠⎟⎟
107
(II.4.10)
Assuming that a uniform current distribution J z = const flows along the z-axis for
r ≤ a , and by integrating (II.4.9) one can obtain
μ0 ⎧ ⎪ J z r, ⎪ ⎪ 2 ⎪ Bϕ (r ) = ⎨ μ0 a 2 ⎪ ⎪ Jz , ⎪ ⎪ r ⎩2
r ≤a r >a (II.4.11)
Now (II.4.11) can be integrated to give equilibrium pressure distribution for r ≤ a as follows
p (r ) =
1 μ0J z2 a 2 − r 2 4
(
)
(II.4.12)
Magnetic field and pressure Profiles of Z-pinch for uniform current density are depicted in Figure II.4.1.
II.4.3. θ-pinch Equilibrium Another conceptual one-dimensional magnetic confinement device, which confines the plasma in cylindrical geometry is θ-pinch. Due to the fact that in a θ-pinch, the current is azimuthal and the magnetic field is axial, the equilibrium Equation (II.4.4) for a θ-pinch becomes
∂p = J ϕ Bz ∂r
Figure II.4.1. Z-pinch profiles
(II.4.13)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani We can eliminate J ϕ in (I.4.13) by using Ampere’s Law
Jϕ = −
1 ∂Bz μ0 ∂r
(II.4.14)
Substituting (II.4.14) in (II.4.13) yields
B ∂ ∂p =− z B ∂r μ0 ∂r z
(II.4.15)
Rewriting (II.4.15) gives 2 ⎞⎟ ∂ ⎛⎜ Bz ⎜⎜ + p ⎟⎟ = 0 ∂r ⎜⎝ 2μ0 ⎟⎠⎟
(II.4.16)
Hence the solution is
Bz2 B2 + p = ext = const 2μ0 2μ0
(II.4.17)
where the first term is magnetic pressure, the second term is plasma pressure, and Bext represents the external magnetic field on the plasma edge. So the total pressure for a θ-pinch is constant. θ-pinch profiles are illustrated in Figure II.4.2.
Figure II.4.2. θ-pinch profiles
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109
II.4.4. Screw Pinch Equilibrium Here, we present the case which the cylindrical plasma column contains both axial and azimuthal current density distributions which leads us to axial as well as azimuthal magnetic fields. This configuration is known as Screw Pinch, or also sometimes known as straight tokamak. For a screw pinch which the magnetic field lines are helical, from the fourth of Maxwell’s Equations (II.1.4), we get:
1 ∂Bϕ ∂Bz + =0 ∂z r ∂ϕ
(II.4.18)
From Ampere’s law one has
μ0 (J ϕϕˆ + J z zˆ) = −
1 ∂ ∂Bz ϕˆ + (rBϕ ) zˆ r ∂r ∂r
(II.4.19)
Using equilibrium equation gives 2 2 ⎛ 2⎞ ∂p ∂ ⎛⎜ Bz ⎞⎟⎟ ∂ ⎜⎜ Bϕ ⎟⎟ Bϕ = J ϕBz − J z Bϕ = − ⎜⎜ ⎟− ⎟− ⎜ ∂r ∂r ⎝⎜ 2μ0 ⎠⎟⎟ ∂r ⎝⎜⎜ 2μ0 ⎠⎟⎟ μ0r
(II.4.20)
Rewriting the above equation gives rise to the governing equation for a screw pinch as ⎛ Bϕ2 ⎞⎟ Bϕ2 Bz2 ∂ ⎜⎜ ⎟⎟ = + ⎜p + ∂r ⎜⎜⎝ 2μ0 2μ0 ⎠⎟⎟ μ0r
(II.4.21)
This result reveals that knowledge of equilibrium configuration of a screw pinch requires the solution of an ordinary differential equation with three unknowns. Hence, we normally require information about the profiles of two parameters at least, which might be extracted from transport equations or experimental measurements. The same situation applies to tokamaks and will be discussed later.
II.4.5. Force Free Equilibrium An equilibrium is called to be force free, if J and B are parallel; as a result, plasma pressure should have zero gradient. In cases where β 1 and one with good approximation could ignore ∇ p , the force free equilibrium can be achieved. In this situation, current flows along field lines, so we have: J = k (r ) B
()
in which k r is constant along field lines. Taking divergence from the above gives
(II.4.22)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
∇ ⋅ J = ∇ ⋅ [k (r ) B] = k (r ) ∇ ⋅ B + (B ⋅ ∇) k (r ) = 0
(II.4.23)
Since from Maxwell’s equation ∇ ⋅ B = 0 , one can conclude that:
(B ⋅ ∇) k (r ) = 0
(II.4.24)
Now, we substitute (II.4.22) into Ampere’s law ∇× B = μ0 J , which gives
∇ × B = μ0k (r ) B
(II.4.25)
Applying the curl operator on (II.4.25) results in
∇× (∇× B) = ∇× [μ0k (r ) B]
(II.4.26)
By using vector identities ∇× (kB) = k ∇× B + ∇k × B and
∇× (∇× B) = ∇ (∇ ⋅ B) − ∇2B , (II.4.25) recasts into 2
∇2B + [μ0k (r )] B = −μ0∇k (r ) × B
(II.4.27)
The above differential equation may be easily solved if one neglects the radial dependence of k (r ) . Expanding the above in terms of axial Bz and azimuthal Bϕ fields gives the set of linear differential equations
r
2
d 2Bϕ dr
r2
2
+r
d 2Bz dr
2
dBϕ dr
+r
(
)
+ K 2r 2 − 1 Bϕ = 0
dBz dr
+ K 2r 2Bz = 0
(II.4.28)
(II.4.29)
Here, K = μ0k . Solutions of (II.4.28) and (II.4.29) are simply given by Bessel’s functions of the first kind and integer order as
Bϕ (r ) = B0J 1 (μ0kr )
(II.4.30)
Bz (r ) = B0J 0 (μ0kr )
(II.4.31)
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in which B0 is the maximum axial field on the plasma axis.
II.5. Grad-Shafranov Equation (GSE) The ideal MHD of axisymmetric toroidal plasma in tokamaks is described by GradShafranov Equation (GSE) that was first proposed by H. Grad and H. Rubin (1958) and Shafranov (1966) for poloidal flux function. Here, we derive the GSE in flux coordinate system. In tokamaks it is convenient to express magnetic field in mixed covariant-contravariant representation. As current lines lie on constant magnetic flux surfaces we have
J ⋅ ∇ψ = J ψ =
1 ψ (∇ × B) = 0 μ0
(II.5.1)
Or equivalently
⎛ ∂B ∂B ⎞ J ⎜⎜ ζ − χ ⎟⎟⎟ = 0 ∂ζ ⎠ ⎝⎜ ∂χ
(II.5.2)
Axisymmetry requires that ∂ ∂ζ = 0 , and furthermore J > 0 . Thus
∂B χ ∂ζ
=
∂ Bζ ∂χ
=0 (II.5.3)
It is clear that Bζ is only function of ψ . More often in the context of tokamaks, the notation Bζ (ψ ) = I (ψ ) is adopted. We also notice that Bζ is not the same as the magnitude of toroidal magnetic field Bt ; these two parameters even do not share the same physical dimensions, while they point to the same direction, i.e. Bt
Bζ ζˆ . Anyway, the
magnetic field of tokamak may be decomposed into its toroidal and poloidal field components as
B = Bt + B p
(II.5.4)
ˆ = I (ψ ) ψˆ and B = ζˆ × ψˆ . Hence rewriting (II.5.4) gives the mixed where Bt = Bζ ψ p covariant-contravariant representation of magnetic field in tokamaks as
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
B = Bt + B p = I (ψ ) ζˆ + ζˆ × ψˆ
(II.5.5)
The magnitude of toroidal magnetic field is hence
I (ψ ) Bt = Bt = I (ψ ) ζˆ = R
(II.5.6)
Here, we have noticed that ζ is simply the angular coordinate of cylindrical system due to axisymmetry. Therefore, ζˆ =
g ζζ = 1 R , and I (ψ ) = Bζ (ψ ) = RBt
(II.5.7)
This shows that the magnitude of toroidal field is not a flux function, while the covariant component Bζ (ψ ) is. Similarly, the magnitude of poloidal magnetic field is given as
Bp = Bp = ζˆ × ψˆ = ζˆ ψˆ sin (ζˆ, ψˆ) Here,
(ζˆ, ψˆ)
(II.5.8)
ˆ . Axisymmetry of is the angle made by the basis vectors ζˆ and ψ
tokamaks excludes dependence on ζ coordinate, and hence this would be a right angle. This would mean that four elements of the metric tensor should vanish, that is ψζ χζ ζψ ζχ g ψζ = g χζ = g ζψ = g ζχ = 0 , and g = g = g = g = 0 . Another conclusion is
that gζζ g
ζζ
= 1 . Therefore 1 Bp = ζˆ ψˆ = ψˆ R
(II.5.9)
In order to derive Grad-Shafranov Equation we begin with equilibrium equation (II.4.4). Upon substitution of J by the rotation of magnetic field we get
μ0 J = ∇ × ⎡⎢I (ψ ) ζˆ + ζˆ× ψˆ⎤⎥ ⎣ ⎦ = ∇I (ψ ) × ζˆ + I (ψ ) ∇× ζˆ + ∇ × (ζˆ× ψˆ) =
∂I (ψ ) ˆ ˆ ψ × ζ + I (ψ ) ∇ × ζˆ + ∇× (ζˆ× ψˆ) ∂ψ
(II.5.10)
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113
ˆ in the cylindrical coordinates (R, ϕ, Z ) is given through the definition of covariant where ψ bases as
⎛ ∂ ∂ 1 ∂ ψˆ = ∇ψ = ⎜⎜ ϕˆ + rˆ − ∂Z R ∂ϕ ⎝⎜ ∂R ∂ψ ∂ψ = rˆ + zˆ ∂R ∂Z
⎞ zˆ⎟⎟⎟ ψ ⎠ (II.5.11)
Here, we have taken the fact into account that from axisymmetry we have ∂ ∂ϕ = 0 . Since ζ in axisymmetric flux coordinates is the same as −ϕ in cylindrical coordinates, we can write
1 ζˆ = ∇ζ = − ϕˆ R
(II.5.12)
By substitution of (II.5.12) and (II.5.11) we get the expression for contravariant component of the magnetic field as
1 ∂ψ ∂ψ ζˆ× ψˆ = ∇ψ × ϕˆ = zˆ − rˆ R ∂R ∂Z
(II.5.13)
Now we are able to simplify (II.5.10) as follows
∂I (ψ) ˆ ˆ ⎛ ∂ 1 ∂ψ ⎞⎟ ∂ ⎞⎟ ⎛⎜ 1 ∂ψ rˆ + zˆ⎟⎟ × ⎜− zˆ + rˆ⎟ ψ × ζ − ⎜⎜ ⎝ ∂R R ∂Z ⎠⎟ ∂ψ ∂Z ⎠ ⎝ R ∂R ∂I (ψ ) ˆ ˆ ∂ ⎛ 1 ∂ψ ⎞⎟ ∂ ⎛⎜ 1 ∂ψ ⎞⎟ ⎜⎜− ψ ×ζ − = ⎟⎟ rˆ× zˆ + ⎟ zˆ× rˆ ⎜ ∂ψ ∂R ⎝ R ∂ R ⎠ ∂Z ⎝ R ∂Z ⎠⎟ ⎡ ∂ ⎛1 ∂ ⎞ ∂I ( ψ ) ˆ ˆ ∂ ⎛⎜ 1 ∂ ⎞⎟⎤ ⎜ ψ × ζ − ϕˆ ⎢ = ⎟⎟⎟ + ⎟⎥ ψ ⎜ ⎢⎣ ∂R ⎝ R ∂R ⎠ ∂Z ⎝⎜ R ∂Z ⎠⎟⎥⎦ ∂ψ
μ0 J =
∂I (ψ ) ˆ ˆ ϕˆ ⎧⎪⎪ ∂ ⎛ 1 ∂ ⎞⎟ ∂2 ⎫⎪⎪ ⎜ ψ × ζ − ⎨R = ⎬ψ ⎟+ ⎜ R ⎪⎩⎪ ∂R ⎝ R ∂R ⎠⎟ ∂Z 2 ⎪⎭⎪ ∂ψ
(II.5.14)
Since the first term on the right-hand-side of (II.5.14) is not in toroidal direction, we can write the toroidal current density as
Jt = J t ϕˆ = −
1 ϕˆ * Δψ μ0 R
(II.5.15)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
where Δ* = R
∂ ⎛⎜ 1 ∂ ⎞⎟ ∂2 ∂2 1 ∂ ∂2 + = − + ⎟ ∂R ⎜⎝ R ∂R ⎠⎟ ∂Z 2 ∂R 2 R ∂R ∂ Z 2
(II.5.16)
is so-called Grad-Shafranov operator. Hence, the GSE reads
1 * Δ ψ = −μ0J t R
(II.5.17)
It is now very instructive to go back to (II.4.4) to find out
J × B = ∇p =
dp ˆ ψ dψ
=
⎫⎪ ⎡ ⎪ 1 ⎧ Δ*ψ ⎤⎥ ⎪ ϕˆ × (I (ψ ) ζˆ + ζˆ× ψˆ)⎪⎬ ⎨− ⎢⎢I ′ (ψ )ζˆ× ψˆ + ⎥ ⎪⎭⎪ r μ0 ⎪ ⎦ ⎪ ⎣ ⎩
=
I (ψ ) I ′ (ψ ) ˆ I ′ (ψ ) ˆ 1 ⎧ ⎪ ⎪ ψ × ϕˆ) × ϕˆ + ⎨ ( (ψ × ϕˆ)× (ϕˆ × ψˆ) 2 2 r r μ0 ⎪ ⎪ ⎩ +
⎫ Δ* ψ Δ* ψ ˆ)⎪⎪⎬ ˆ ˆ ˆ ˆ I ϕ ψ ϕ ϕ ϕ ψ × + × × ( ) ( ⎪⎪⎭ r2 r2
* ⎡ I (ψ ) I ′ (ψ ) ⎤ ˆ × ϕˆ) × ϕˆ + Δ ψ ϕˆ × (ϕˆ × ψˆ)⎥ ⎢ ψ ( ⎢ ⎥ R2 R2 ⎣ ⎦ * 1 ⎡ I (ψ ) I ′ ( ψ ) ˆ Δ ψ ˆ ⎤ ⎢− ψ + 2 ψ⎥ = ⎥ R2 μ0 ⎢⎣ R ⎦
=
1 μ0
(II.5.18)
Or equivalently
⎪ I (ψ ) I ′ (ψ ) Δ*ψ ⎫⎪⎪ dp 1 ⎧ ⎪ = + 2 ⎬ ⎨− d ψ μ0 ⎪⎪⎩ R2 R ⎪⎭⎪
(II.5.19)
By rearranging (II.5.19) we arrive at the alternative form of GSE given by
dp (ψ ) I (ψ ) dI (ψ ) 1 * Δ ψ = μ0R + R dψ R dψ
(II.5.20)
This form of GSE has the advantage that the right-hand-side is also given in terms of the poloidal flux ψ. This however makes the above equation nonlinear in terms of the poloidal flux. More often, profiles of pressure p (ψ ) and toroidal field function I (ψ) = RBt are either
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known from experiment, or self-consistent solution of transport equations. Alternatively, prescribed polynomial forms are assumed for these two functions. The nature of the axisymmetric equilibrium in a tokamak is thus to a large extent determined by the choice of
( ) ( )
the free functions p ψ , I ψ and the boundary conditions. We may notice the right-hand-side of (II.5.20) is because of (II.5.17) actually the toroidal current density J t and hence a flux function. If the toroidal current density is assumed to have a linear dependence on flux as J t (ψ ) ≈ J 0 + J 1ψ , then (II.5.20) allows exact solutions in terms of Bessel’s functions. But numerical solution is inevitable for more complicated profiles. A straightforward solution of (II.5.20) with the special choice of p (ψ) = ψ and I (ψ) = I 0 , known as Solov’ev solution has been shown to exist by Vitali Shafranov, which is given by
ψ (R, Z ) =
2 2 1 4 4b 2 2 2 R − R − 4 RZ a2 a4 a
(II.5.21)
where a and b are constants which determine the final equilibrium configuration. This solution is noticeably useful in description of a wide range of plasma equilibria in tokamaks.
II.6. Green’s Function Formalism In previous section we derived GSE from MHD equilibrium equation of a toroidal plasma. By solving Grad-Shafranov equation one can find flux distribution of magnetic flux. Once the flux distribution is known, it is easy to reconstruct the plasma boundary and the shape of nested magnetic surfaces. There are lots of different methods which have been proposed to solve GSE, which are categorized in Figure II.6.1. In this section we are going to study Green’s Function Method as an analytical solution to GSE.
II.6.1. Green’s Function for GSE The axisymmetric magnetostatics in cylindrical coordinates is described by the GSE equation:
Δ*ψ = −μ0rJ t
(II.6.1)
in which ψ = rAϕ is the magnetic poloidal flux, and where Aϕ is the toroidal component of the magnetic vector potential. Also, J t is the toroidal current density and Δ* is the elliptic Grad-Shafranov operator which is already defined in (II.5.16). This concept is shown graphically in Figure II.6.2. From now on, the cylindrical coordinates are represented by the set of coordinates (r, ϕ, z ) .
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Figure II.6.1. Different solutions to GSE
Figure II.6.2. Green’s Function Method
In (II.6.1), we disregard the inherent dependence of J t on the poloidal flux ψ , making the GSE a linear differential equation. From a system engineering point of view, GSE represents a Linear Time Invariant (LTI) system, whose impulse response is given by its associated Green’ function. Here in order to find flux function, we seek solutions of the form
ψ (r , z ) =
∞ ∞
∫ ∫ G (r, r ′, z, z ′)J (r ′, z ′)dr ′dz ′ t
−∞ 0
(II.6.2)
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117
or equivalently
ψ (r, z ) =
∞ ∞
∫ ∫ G (r,r′)J (r′)dr ′dz ′ t
−∞ 0
where
(II.6.3)
r = (r, z ) is the two-dimensional position vector on the constant ϕ-plane, and
G (r,r ′) is referred to as the Green’s function, obtained through the solution of the following equation:
Δ*ψ = μ0δ ( r − r ′) = μ0δ (r − r ′) δ (z − z ′)
(II.6.4)
with δ (⋅) being the Dirac’s delta function. At first, we first examine the solution to the homogeneous Grad–Shafranov equation,
Δ*ψ = 0
(II.6.5)
and then proceed to construct the Green’s function. Now let ψ(r , z ) = R(r )Z (z ) and using separation of variables we get
⎛ ∂2 ∂2 ⎞ 1 ∂ Δ* [R(r )Z (z )] = ⎜⎜ 2 − + 2 ⎟⎟⎟ R(r )Z (z ) = 0 ⎜⎝ ∂r r ∂r ∂z ⎠
(II.6.6)
i
For the sake of simplicity, let ∂ ∂r = (⋅)′ and ∂ ∂z = (.) , and rewrite (II.6.6) as
ZR '' − Z
ii 1 ' R + RZ = 0 r
(II.6.7)
() ()
Dividing both sides by Z z R r yields
''
'
ii
R 1R Z − + =0 R r R Z
(II.6.8)
As R and Z are only function of coordinates r and z , respectively, one can write
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani ii
R '' 1 R ' Z − = − = −k 2 R r R Z
(II.6.9)
2
where k is a real-valued constant. Therefore we should have ii
Z = k2 Z
(II.6.10)
R'' 1 R ' − = −k 2 R r R
(II.6.11)
and
Bounded solutions of (II.6.10) require that k 2 < 0 , and are in the form
Z (z ) = ak cos(kz ) + bk sin(kz )
()
(II.6.12)
()
Letting R r = rA r in (II.6.11) gives the Modified Bessel Function with the general solution
R (r ) = r [ck K1 (kr ) + dk I 1(kr )]
()
(II.6.13)
()
in which ak , bk , ck , and dk are constants, and K 1 . and I 1 . are the first order modified
( )
() ()
Bessel functions. Hence, the proposed eigen-solution ψ r , z = Z z R r becomes
ψk (r, z ) = r [ak cos(kz ) + bk sin(kz )][ck K1 (kr ) + dk I 1(kr )]
(II.6.14)
Now since for the modified Bessel functions we have
lim I 1 (kr ) = ∞ r →0
(II.6.15)
dk should be zero, and hence ψk (r , z ) = r ⎡⎢ak cos(kz ) + bk sin(kz )⎤⎥ K 1 (kr ) ⎣ ⎦
(II.6.16)
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119
Green’s function may be constituted from a proper superposition of eigen-functions (II.6.16) using integration on all k, given by
ψ (r, z ) =
∞
∫ 0
∞
ψk (r, z )dk = ∫ r ⎡⎢⎣ak cos(kz ) + bk sin(kz )⎤⎥⎦ K1 (kr )dk 0
(II.6.17)
The reciprocity property of the Green’s function as understood from linearity of the GSE system requires that
G (r − r ′) ≡ G ( r ′ − r )
(II.6.18)
Consequently, symmetry with regard to the change of arguments reduces Green’s function to
G ( r − r ′) =
∞
∫ ra
k
0
cos ⎡⎣⎢k (z − z ′)⎤⎦⎥ K 1 ⎡⎣⎢k (r − r ′)⎤⎦⎥ dk
(II.6.19)
Substituting (II.6.19) in (II.6.4) gives
Δ G (r, r ′) = Δ *
*
∞
∫ a (r, r ′) cos ⎡⎣⎢k (z − z ′)⎤⎦⎥ dk = −rμ δ (r − r ′) k
0
0
(II.6.20)
where ak (r , r ′) = rak K 1 ⎡⎢k (r − r ′)⎤⎥ . Applying the expanded form of Grad-Shafranov
⎣
⎦
operator on (II.6.20) yields Δ*G ( r, r ′) =
∞
⎛ ∂2
∫ ⎜⎜⎜⎝ ∂r
2
−
0
∂2 1 ∂ ⎞⎟ ⎟⎟ ak (r, r ′) cos ⎡⎣⎢k (z − z ′)⎤⎦⎥ + ak (r, r ′) 2 cos ⎡⎢⎣k (z − z ′)⎤⎦⎥ dk r ∂r ⎠ ∂z
(II.6.21)
which may be rewritten in the form
Δ G (r, r ′) = *
∞
∫ 0
⎡⎛ ∂2 ⎤ ⎞ ⎢⎜⎜ 2 − 1 ∂ − k 2 ⎟⎟ak (r, r ′)⎥ cos ⎡⎢k (z − z ′)⎤⎥ dk ⎟⎠ ⎣ ⎦ ⎢⎜⎝ ∂r ⎥ r ∂r ⎣ ⎦
(II.6.22)
Now we adopt the definition
1 ∂ ∂2 Δ = − − k2 2 ∂r r ∂r * k
(II.6.23)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
and rewrite (I.6.22) as
Δ G (r, r ′) = *
∞
∫ ⎡⎣⎢Δ a (r, r ′)⎤⎦⎥ cos ⎡⎣⎢k (z − z ′)⎤⎦⎥ dk * k k
0
= −r μ0δ (r − r ′) = r μ0δ (r − r ′) δ (z − z ′)
(II.6.24)
But Dirac’s delta function may be defined as ∞
2πδ (z )
∞
∫e
jkz
−∞
dk = ∫ cos (kz ) + j sin (kz )dk −∞
(II.6.25)
Since the right-hand-side of (II.6.24) is real, (II.6.25) simplifies as ∞
∫ cos (kz ) = πδ (z ) (II.6.26)
0
Comparing (II.6.24) and (II.6.26) results in the linear ordinary differential equation
∂2 1 ∂ a r , r ′) − ak (r , r ′) − k 2ak (r , r ′) 2 k ( ∂r r ∂r r = − μ0δ (r − r ′) π
Δk*ak (r , r ′) =
(II.6.27)
which has the solution
⎧ rr ′ ⎪ ⎪⎪ K 1 (kr ) I 1 (kr ′) ⎪ π A ⎪ ak = ⎨ ⎪ ⎪ rr ′ K (kr ′) I (kr ) ⎪ 1 1 ⎪ ⎪ ⎩ πA
r > r′ r < r′ (II.6.28)
This can be written in the more compact and convenient form
ak (r , r ′) =
r>r< I 1 (kr< ) K 1 (kr> ) πA
(II.6.29)
An Overview of Plasma Confinement in Toroidal Systems
( )
in which A is the Wronskian of functions r ′I 1 kr ′
121
( )
and r ′K 1 kr ′ ; furthermore,
r< = min(r , r ′) and r> = max (r, r ′) . Since the unknown coefficients ak (r , r ′) are determined, one can obtain the integral form of Green’s Function as
G (r, r ′) = μ0
∞
rr ′ I 1 (kr< ) K1 (kr> ) cos ⎡⎢⎣k (z − z ′)⎤⎥⎦ dk π ∫0
(II.6.30)
Surprisingly, (II.6.30) allows a very simple closed form integral given by
⎡ r 2 + r ′2 + (z − z ′)2 ⎤ ⎢ ⎥ G ( r, r ′) = μ0 ⎢ ⎥ 2rr ′ ⎢⎣ ⎥⎦ 2 ⎛ ⎞⎟ rr ′ ⎜⎜ r − r ′ Q1 ⎜ = μ0 + 1⎟⎟⎟ 2π 2 ⎜⎜⎝ 2rr ′ ⎠⎟ rr ′ Q1 2π 2
(II.6.31)
where Q1 2 (⋅) is the Legendre function of the second kind, satisfying
(1 − x )y ′′ (x ) − 2xy ′ (x ) + ν (ν + 1)y (x ) = 0 2
(II.6.32)
with ν = 1 2 . It is noticeable that the latter result justifies the requirement for the reciprocity property of the Green’s function as stated above. This completes our assertion. The asymptotic behavior of Legendre functions Q ν (x ) at large x may be studied by the integral ∞
Qν (x ) =
dθ
∫ (1 + cosh θ)
ν +1
x −(1+ν )
ν > −1
0
≈ x −(1+ν )
Hence for
ν =12
π Γ (ν + 1) 2 Γ ⎛⎜ν + 3 ⎞⎟ ⎜⎝ 2 ⎠⎟ 1+ν
(II.6.33) −1 2
the asymptotic expansion is
G (r, r ′) ≈ μ0
Q1 2 ≈ 32
πx
−3 2
. Therefore
r 2r ′2 3
4 ⎡⎣r 2 + r ′2 (z − z ′)2 ⎦⎤ 2
(II.6.34)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
which satisfies all of the requirements on the boundaries as A contour plot of the Green’s function is illustrated in Figure II.6.3.
1
0.5
0
-0.5
-1 0
0.5
1
1.5
2
Figure II.6.3. Contour plot of the Green’s function as given by (15) limr →0 G (r − r ′) = 0 limr →∞ G (r − r ′) = 0 lim r→r ′ G (r − r ′) = ∞
(II.6.35)
II.6.2. Application of Green’s Function to Different Current Density Distributions As it was mentioned the poloidal flux function ψ can be accurately obtained by through the Green’s function formalism once the toroidal current density profile is known, following (II.6.2). As examples, we discuss the resultant magnetic fields due to a current loop and a solenoid. a) Current loop In order to find poloidal flux of a current loop placed at the equatorial plane z = 0 , we may immediately use the Green’s function with the solution ψ (r, z ) = G (r, a; z, 0) , in which a is the radius of loop. But for illustration purposes, we use the asymptotic expression
( )
of Green’s function. This results in the following for ψ r , z ∞ ∞
ψ (r, z ) ≈ μ0 ∫
∫
−∞ 0
r 2r ′2 3 2 2
4 ⎣⎡r + r ′ (z − z ′) ⎤⎦ 2
2
J t (r ′)dr ′dz ′ (II.6.36)
An Overview of Plasma Confinement in Toroidal Systems
123
For the current loop, the corresponding toroidal current density is
J t (r′) = I 0δ (r ′ − a ) δ (z ′)
(II.6.37)
where I 0 is the current passing throught the loop. As a result, the magnetic poloidal flux will be ∞ ∞
ψ (r , z ) ≈ μ0 ∫
∫
−∞ 0
= μ0I 0
r 2r ′2 3
4 ⎡⎣r 2 + r ′2 (z − z ′)2 ⎦⎤ 2 r 2a 2
I 0δ (r ′ − a ) δ (z ′)dr ′dz ′
3
4(r 2 + a 2 + z 2 )2
Since
ψ (r , z ) = rAϕ
(II.6.38)
we have:
B=−
1 ∂ψ (r , z ) 1 ∂ψ (r , z ) rˆ + zˆ r ∂z r ∂r
(II.6.39)
or
⎫ ⎪⎧ a 2 ⎡⎣2(r 2 + a 2 + z 2 ) − 3⎤⎦ ⎪⎪⎪ μ0 ⎪⎪ 3ra 2z ˆ+ B= I0 ⎨ zˆ⎬ 5 r 5 ⎪⎪ 4 ⎪⎪ 2 2 2 2 2 2 2 2 (r + a + z ) ⎪ ⎩(r + a + z ) ⎭⎪
(II.6.40)
b) Solenoid with toroidal current density In this case, we consider the toroidal current distribution of a cylinder with radius a, which carries uniform current density J 0 in the poloidal direction. We furthermore postulate that the cylinder’s axis coincides with the z-axis. Hence, the corresponding current density is
J (r , z ) = J t (r , z ) ϕˆ = J 0δ (r − a ) ϕˆ Using (II.6.36) yields:
(II.6.41)
124
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani ∞ ∞
ψ (r , z ) ≈ μ0 ∫
∫
r 2r ′2 3
4 ⎡⎣r 2 + r ′2 + (z − r ′)2 ⎤⎦ 2 ∞ J 0 μ0r 2a 2 ′ =∫ 3 dz 2⎤2 2 2 ⎡ −∞ 4 ⎢r + a + (z − z ′) ⎥ ⎣ ⎦ 2 2 J μr a = 02 0 2 2(r + a )
J 0δ (r ′ − a )dr ′dz ′
−∞ 0
(II.6.42)
Using (II.6.39) for determining the magnetic field results in
B=−
J μ a4 1 ∂ψ (r , z ) 1 ∂ψ (r , z ) rˆ + zˆ = 2 0 0 2 2 zˆ r r ∂z ∂r (r + a )
(II.6.43)
Finally, the on-axis magnetic field is given by the well-known expression
B = zˆ
μ0J 0b 2 2b 2 limr →0+ 2 = zˆμ0J 0 2 2 2 + r b ( )
(II.6.44)
II.7. Analytical and Numerical Solutions to GSE II.7.1. Analytical Solution Grad-Shafranov equation (I.6.1) is normally expressed in cylindrical coordinates. This equation can be written in primitive toroidal coordinates by applying the transformations
r = R0 + r0 cos θ0 z = r0 sin θ0 ϕ = −ζ0
(II.7.1)
Therefore we get 2 ⎤ ⎡ ⎛ ⎞ ⎛ ⎞ 1 ⎜cos θ ∂ − sin θ0 ∂ ⎟⎟ ψ ⎢ 1 ∂ ⎜⎜r ∂ ⎟⎟ + 1 ∂ ⎥ ψ − ⎜ ⎟ 0 ⎢ r ∂r ⎜⎜ 0 ∂r ⎟⎟ r 2 ∂θ 2 ⎥ ∂r0 R0 + r0 cos θ0 ⎜⎜⎝ r0 ∂θ0 ⎟⎠ ⎢⎣ 0 0 ⎝ 0⎠ 0 0 ⎥⎦ ∂ I (ψ ) 2 ∂ p (ψ ) = −μ0 (R0 + r0 cos θ0 ) − μ02I (ψ ) ∂ψ ∂ψ (II.7.2)
An Overview of Plasma Confinement in Toroidal Systems
125
Now, we assume that the poloidal flux is composed of circular and non-circular contributions as
ψ (r0 , θ0 ) = ψ0 (r0 ) + ψ1 (r0 , θ0 )
(
Here, ψ1 r0 , θ0
(II.7.3)
) represents the deviation from the concentric nested circular magnetic
surfaces in tokamaks, and hence is responsible for characteristics such as Shafranov Shift,
( )
Triangularity and Elongation. It is evident that the dominant term in (II.7.3) is due to ψ r0 , which plays a significant role in construction of magnetic flux surfaces.
(
For large aspect ratio tokamaks, ψ1 r0 , θ0
( )
satisfying ψ0 r0
) may be treated as a perturbation function
ψ1 (r0 , θ0 ) , so that (II.7.2) for ψ0 (r0 ) could be written as
∂p (ψo ) ∂I (ψo ) 1 ∂ ⎜⎛ ∂ ⎞⎟ 2 − μ02I (ψo ) ⎜⎜r ⎟⎟ ψo (r ) = −μ0R0 r ∂r ⎝ ∂r ⎟⎠ ∂ψo ∂ψo
(II.7.4)
Similarly, (II.7.2) for the perturbation term ψ1 (r0, θ0 ) may be approximated as
⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ⎤ ∂p (ψ0 ) ⎜⎜r ⎢ ⎥ ψ (r , θ ) ≈ cos θ ∂ ψ − 2μ R r cos θ ⎟⎟⎟ + 2 − 0 0 2⎥ 1 ⎢ ⎜ ⎟ R0 ∂r 0 ∂ψ0 ⎣⎢ r ∂r ⎝ ∂r ⎠ r ∂θ ⎦⎥ ⎡ ∂p ( ψ ) ∂I (ψ0 )⎤⎥ 1 ∂ ⎢ 2 2 R I μ − μ ψ ( ) ⎢ 0 0 ⎥ ψ (r , θ ) 0 0 ⎛ ⎞ ∂ψ0 ∂ψ0 ⎥ 1 ⎜⎜ ∂ψ0 ⎟⎟ ∂r ⎢⎣ ⎦ ⎜⎜ ∂r ⎟⎟ ⎝ ⎠ (II.7.5) Hereinafter, we drop the subscript “0” denoting the primitive toroidal coordinates for the sake of convenience. Input
pressure
and
toroidal
field
profiles
given
by
∂p (ψ0 ) ∂ψ0
and
I (ψ0 ) ∂I (ψ0 ) ∂ψ0 on the right-hand-side of (II.5.20) can be expanded as Taylor series ∞ ∂p (ψ0 ) = ∑ Pn ψ0n = P0 +P1ψ0 + P2ψ02 + ∂ψ0 n =0
+ Pn ψ0n +
∞ ∂I (ψ0 ) I (ψ0 ) = ∑ I n ψ0n = I 0 +I 1ψ0 + I 2ψ02 + ∂ψ0 n =0
Substituting (II.7.6) into (II.7.4) and (II.7.5) results in
I n ψ0n + (II.7.6)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
1 ∂ ⎜⎛ ∂ ⎞⎟ r ⎟ ψo (r ) = A0 + A1ψ0 + A2 ψ02 + r ∂r ⎜⎝ ∂r ⎠⎟
(II.7.7)
and
⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ⎤ cos θ ∂ ⎜r ⎟⎟ + 2 ⎢ ⎥ ψ1 (r , θ ) ≈ ψ0 2 ⎟ ⎜ R0 ∂r ⎣⎢ r ∂r ⎝ ∂r ⎠ r ∂θ ⎦⎥ − (A0 + 2A1ψ0 + ...) ψ1 − r cos θ (B0 + B1ψ0 + B2 ψ02 + ...)
(II.7.8)
In order to solve GSE analytically, one should discard all those terms that makes GSE nonlinear. Hence after retaining linear terms (II.7.7) reduces to
1 ∂ ⎛ ∂ ⎞⎟ ⎜⎜r ⎟ ψ0 (r ) = −A0 − A1ψ0 r ∂r ⎝ ∂r ⎠⎟
(II.7.9)
∂ 2 ψ0 1 ∂ψ0 + + A1ψ0 = −A0 2 ∂r r ∂r
(II.7.10)
or equivalently
Since the poloidal flux ψ0 is a potential function, whose gradient is physically important giving rise to magnetic field, we may freely set A0 = 0 for the moment. Now, letting
A1 = k 2 and A0 = 0 gives
∂ 2 ψ0 1 ∂ψ0 + + k 2ψ0 = 0 2 ∂r r ∂r
(II.7.11)
The solution of homogenous equation is
ψ0h = ψcJ 0 (kr )
(II.7.12)
( ) is dimensionless, and hence ψ
No need to mention that J 0 kr
c
appearing in (II.7.12)
is a constant with the dimension of Weber. Now we are ready to add up the particular solution of (II.7.10) in the form of ψ0 p = A , which by putting in (II.7.12) and solving equation for
A , yields
An Overview of Plasma Confinement in Toroidal Systems
ψ0 p = A =
127
−A0 k2
(II.7.13)
Now the general solution is
ψ0 = ψ0h + ψ0 p = ψcJ 0 (kr ) −
A0 k2
(II.7.14)
There are three unknowns ψc , k and A0 which can be determined by imposing plasma constraints. The (arbitrary) choice of ψ0 (r = 0) = 0 , gives
ψc =
A0 k2
(II.7.15)
Now by rewriting (II.7.14) we obtain the solution given by
ψ0 = ψc [J 0 (kr ) − 1]
(
(II.7.16)
)
Now we turn to the perturbation function ψ1 r0 , θ0 ; by inserting (II.7.16) into (II.7.8) and neglecting terms An , n ≥ 1 we get
⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ⎤ 2⎥ ⎜r ⎟⎟ + ⎢ k + ⎜ ⎢ r ∂r ⎜⎝ ∂r ⎠⎟⎟ r 2 ∂θ 2 ⎥ ψ1 (r, θ ) = f (r , θ ) ⎣⎢ ⎦⎥
(II.7.17)
in which
f (r , θ ) =
cos θ ∂ψo − r cos θ (B0 + B1ψ0 ) R0 ∂r
(II.7.18)
The equation (II.7.17) can be analytically solved by means of Green’s function technique. For this purpose we first define the Helmholtz operator in two-dimensional polar coordinates as
L=
1 ∂ ⎛ ∂ ⎞⎟ 1 ∂2 ⎜r ⎟ + + k2 r ∂r ⎜⎝ ∂r ⎠⎟ r 2 ∂θ 2
The appropriate Green’s function for solution of (II.7.17) hence obeys
(II.7.19)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
1 LkG (r , r ′; θ, θ ′) = −δ (r, r ′) = − δ (r, r ′) δ (θ, θ ′) r
(II.7.20)
in which r = r cos θxˆ + r sin θyˆ is the two-dimensional position vector. The Green’s function is well-known to be
⎡j ⎤ G ( r − r ′) = Re ⎢ H 0(1) (k r − r ′ )⎥ ⎢⎣ 4 ⎥⎦ (1)
where H 0
(⋅)
(II.7.21)
( )
is Hankel’s function of the first kind and zeroth order. Now ψ1 r , θ can be
readily determined by the convolution integral 2π ∞
ψ1 (r, θ ) =
∫ ∫ G (r, r ′)f (r′) r ′dr ′d θ ′ 0
0
(II.7.22)
where
⎧⎪k ψ ⎫ ⎪ cos θ f (r) = ⎪⎨ c J 1 (kr ) + r ⎡⎢(B0 − B1ψc ) + B1ψcJ 0 (kr )⎤⎥⎪ ⎣ ⎦⎬ ⎪⎪ R0 ⎪ ⎪ ⎩ ⎭
(II.7.23)
As it will be discussed later, the sawtooth instability causes the safety factor on the plasma axis to be fixed to unity, that is q (0) = 1 . This may be used to obtain the other constraint to find the remaining unknown coefficient. Hence, we first develop an expression for safety factor. Starting with the toroidal flux φ(r ) for approximately circular cross section we have r
φ(r ) =
∫ ∫ B (ρ, θ)ρd θd ρ t
0
( )
Here, Bt r , θ
2π
0
(II.7.24)
is toroidal magnetic field across the poloidal cross section of plasma.
Solov’ev equilibrium allows us to make the very good approximation
Bt (r , θ ) ≈
Substituting (II.7.25) in (II.7.24) yields
Bt 0 r 1+ cos θ R0
(II.7.25)
An Overview of Plasma Confinement in Toroidal Systems r
2π
Bto
∫∫
φ(r ) =
0
r 1+ cos θ R0
0
129
ρd θd ρ (II.7.26)
Since the term r cos θ R0 in denominator is always less than unity, one can use the binomial expansion theorem to obtain r
2π
φ(r ) = ∫
∫
0
0
n
⎛−ρ ⎞ Bt 0 ∑ ⎜⎜ cos θ⎟⎟⎟ ρd θd ρ ⎜ ⎠⎟ n =0 ⎝ R0 ∞
r
∞
−1 Bt 0 ∫ =∑ n =0 R0 0
2π
∫ 0
n
⎛−ρ ⎞ ⎜⎜ cos θ ⎟⎟⎟ ρd θd ρ ⎜⎝ R ⎠⎟ 0
(II.7.27)
Using the identity 2π
∫ 0
⎧⎪ 2 π ⎛ 1⎞ ⎪⎪ Γ ⎜⎜⎜n + ⎟⎟⎟ ⎪ (cos θ ) d θ = ⎨⎪ n ! ⎝ 2 ⎟⎠ ⎪⎪0 ⎪⎩
n = 0,2, 4,...
n
n = 1, 3, 5,...
(II.7.28)
we reach at the expression for toroidal flux
⎛ 1⎞ Γ ⎜⎜n + ⎟⎟ r ⎝ 2⎠ φ(r ) = 2 πBt 0 ∑ ∫ n! n =0 0 ∞
2n
⎛ρ⎞ ⎜⎜⎜ ⎟⎟⎟ ρd ρ ⎝ R0 ⎠⎟
⎛ 1⎞ Γ ⎜⎜n + ⎟⎟ ⎛ ⎞2n ⎝ 2 ⎠ ⎜ r ⎟⎟ = πBt 0r 2 ∑ ⎜⎜ ⎟ ⎟ n =0 (n + 1) ! ⎝ R0 ⎠ ∞
⎡ = πr 2Bt 0 ⎢⎢1 + ⎢⎣
2
4
1 ⎛⎜ r ⎞⎟ 3⎛r ⎞ ⎜ ⎟⎟ + ⎜⎜ ⎟⎟⎟ + 4 ⎜⎝ R0 ⎠⎟ 24 ⎜⎝ R0 ⎠⎟
⎤ ⎥ ⎥ ⎥⎦
(II.7.29) 2
As it can be seen here, within zeroth-order approximation we have φ(r ) ≈ πr Bt 0 , which shows that the toroidal magnetic flux is approximately equal to the product of crosssectional area of the outermost magnetic surface and toroidal magnetic field on the plasma axis. Now, the safety factor is defined as
q=
φ′ 1 =− ι ψ′
(II.7.30)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
in which ι is called the rotational transform. Hence, we get
⎛ 1⎞ Γ ⎜n + ⎟⎟ 2n +1 2 πBt 0 ∞ ⎜⎝ 2⎠ r q (r ) = ∑ k ψcJ 1 (kr ) n =0 n! Ro2n
(II.7.31)
Here, Bt 0 and R0 are design parameters of the tokamak machine. It is instructive if we take a look at safety factor on the plasma axis in the limit of r → 0 . Under these assumptions, we have J 1 (kr ) ∼ 21 kr by the corresponding asymptotic expansion near origin. Together with the condition imposed by sawtooth instability q (0) = 1 , we obtain one missing equation to determine the unknown coefficients
q (0) =
4πBt 0 =1 k 2 ψc
(II.7.32)
and finally
k 2 = A1 =
4πBt 0 ψc
(II.7.33)
Other unknown parameters can be found by having the toroidal current density function, integration of which gives the total plasma current. By substitution of magnetic flux (II.7.16) into GSE we obtain the following
J t (r ) = =
−1 ⎛⎜ 1 d d2 ⎞ + 2 ⎟⎟⎟ ψ0 (r ) ⎜⎜ μ0R0 ⎝ r dr dr ⎠
−ψc μ0R0
⎡1 ′ ⎤ ⎢ J 0 (kr ) + J 0′′ (kr )⎥ ⎥⎦ ⎣⎢ r
(II.7.34)
After some manipulation we get the fairly convenient form
J t (r ) =
ψc {2J 1 (kr ) + kr [J 0 (kr ) − J 2 (kr )]} 2r μ0R0
(II.7.35)
It can be readily seen that the maximum plasma current occurs on the plasma axis and is 2
given by J t (0) = k ψc μ0R0 . Now the plasma current can be computed as
An Overview of Plasma Confinement in Toroidal Systems a
Ip ≈
∫ J (r ) 2πrdr =
2πk ψcaJ 1 (ka )
t
0
μ0R0
131
(I.7.36)
As I p is one of the design parameters of tokamaks, (II.7.33) can be simultaneously solved with (II.7.36) to determine the equilibrium. Table I.7.1. Main parameters of Damavand tokamak Parameter Major Radius Minor Radius Aspect Ratio Toroidal Magnetic Field Elongation Peak Plasma Current Peak Plasma Density Peak Electron Temperature Peak Ion Temperature Number of Toroidal Field Coils Discharge Duration
Value 37cm 7cm 5.1 1.2T 1.2 40kA 1019cm3 300eV 150eV 20 25ms
For example, the unknowns k and ψ0 for Damavand Tokamak with the main parameters listed in Table II.7.1 are found as k = 41.614 m −1 ψc = 7.98223 × 10−3 Wb
(II.7.37)
Figure II.7.1 shows the plasma configuration together with poloidal and toroidal coils in the large-aspect-ratio Damavand tokamak. In Figure II.7.2, variations of poloidal flux versus minor radius is demonstrated. As it is normally expected, the poloidal flux on the plasma axis is zero. This is due to the convention used here for definition of poloidal flux; one could equivalently use any other reference for poloidal flux ψ, as only derivatives of this function are important for determination of magnetic fields which are real physical quantities. Figure II.7.3 shows the variations of safety factor versus plasma minor axis. As the boundary conditions for safety factor on the plasma axis and edge require, safety factor is a monotonic increasing function of plasma minor radius and reaches from the minimum of 1 on the axis to a maximum of 4 on the boundary. Figure II.7.4 illustrates the toroidal current density function versus minor radius. As it is expected, the toroidal current density reaches its maximum on the plasma axis.
132
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
Figure II.7.1. Cross section of Damavand tokamak facility
Figure II.7.2. Poloidal magnetic flux versus minor radius in Damavand
An Overview of Plasma Confinement in Toroidal Systems
133
Figure II.7.3. Safety factor versus minor radius in Damavand
Figure II.7.4. Toroidal current density versus minor radius in Damavand
II.7.2. Numerical Solution As GSE intrinsically is a non-linear partial differential equation (PDE), the use of numerical solution is inevitable for description of axisymmetric plasma equilibria. Various numerical methods have been proposed to solve GSE, which could be found in the literature. The Finite Element method (FEM) is the most popular general purpose technique for computing accurate solutions to PDEs, which we hereby exploit to solve GSE. The family of FEMs may be divided into Galerkin and variational approaches, in both of which the solution is expanded on a set of eigenfunctions. In this section, the variational formulation of FEM, based on first-order triangular elements is presented. The GSE (II.6.1) is here redisplayed for the sake of convenience
⎛ ∂2 1 ∂ ∂2 ⎞ Δ* ψ = ⎜⎜ 2 − + 2 ⎟⎟⎟ ψ = −μ0rJ t ⎜⎝ ∂r r ∂r ∂z ⎠
(II.7.38)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
It can be shown that (II.7.38) may be regarded as an Euler-Ostogradskii equation of the functional
Π (ψ ) =
∫∫
⎛1 ⎞ ⎜⎜ ∇ψ 2 − μ J ψ⎟⎟dr dz ⎟ 0 t ⎟ ⎜⎝ 2r ⎠
(II.7.39)
where the integration is taken over a domain Ω in the two-dimensional (r, z ) plane, illustrated in Figure II.7.5, and ∇ = ∂ ∂r rˆ + ∂ ∂z zˆ is the two-dimensional gradient.
z Poloida Coil Plasma
r
l
Solution Region
Ω Figure II.7.5. Typical solution region for numerical methods
The basic idea of the FEM is to make a piecewise approximation, that is the solution of a problem is achieved by dividing the region of interest into small regions called elements, and approximating the solution over each element by simple function with prescribed forms. The functions used to represent the behavior of the solution within an element are called interpolation functions; the simplest choice is linear dependence to coordinates referring to first-order elements. For example, the simplex element in two dimensions is a triangle with three nodes (corners). Nodes are usually shared by more than one element and it is desirable to find the nodal values of unknown functions through a set of algebraic operations which simultaneously extremize (II.7.39). The choice of simplex triangle elements, allows us to express the variations of discretized function over the element with index e as
An Overview of Plasma Confinement in Toroidal Systems
ψe (r , z ) = a e + be r + ce z
135
(II.7.40)
where superscript e refer to element e , and unknown constants a, b and c are easily determined from:
⎡a e ⎤ ⎡1 r e i ⎢ ⎥ ⎢ ⎢be ⎥ = ⎢1 r e ⎢ ⎥ ⎢ j ⎢ e⎥ ⎢ e ⎢c ⎥ ⎢1 rk ⎣ ⎦ ⎣
z ie ⎤⎥ z ej ⎥⎥ ⎥ z ke ⎥ ⎦
−1
⎡ψ ⎤ ⎢ i⎥ e e ⎢ ⎥ ⎢ψj ⎥ = D Ψ ⎢ ⎥ ψ ⎣⎢ k ⎦⎥
(II.7.41)
Here, i, j, and k refer to indices of nodes of element e . Furthermore, rle and zle correspond to radial and longitudinal coordinates of node l , belonging to element e with l standing either of i, j, or k . It is also customary to define the shape functions N le , l = i, j , k for the element e as
⎡N e (r , z )⎤ ⎢ i ⎥ ⎢ e ⎥ T e N (r , z ) ≡ ⎢N j (r , z )⎥ ≡ De ⎢ ⎥ ⎢N e (r , z )⎥ ⎢⎣ k ⎥⎦
⎡1 ⎤ ⎢ ⎥ ⎢r ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣z ⎥⎦
(II.7.42)
Therefore we have:
ψe (r , z ) = Ne (r , z ) Ψe T
(II.7.43)
Gradient of Ψ is needed in (II.7.39), so one can approximate the gradient of unknown function over the element e as
⎡D e ∇ψ = ∇N Ψ = ⎢⎢ eji ⎢⎣Dki eT
e
e
Dejj Dkje
D ejk ⎤⎥ e Ψ ≡ Be Ψe e ⎥ Dkk ⎥ ⎦
(II.7.44)
e where Drse refers to the elements of matrix D .
Now we can substitute (II.7.43) and (II.7.44) into the functional (II.7.39), which leads us to
⎛1 T T ⎞ T T Π (ψ ) ≈ ∑ Πe ψe = ∑ ∫∫ ⎜⎜ Ψe Be Be Ψe − μ0 Jet Ne Ne Ψe ⎟⎟⎟dr dz ⎜⎝ 2 ⎠⎟ e e Se
( )
(II.7.45)
136
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani Here, the summation is applied over all elements and Jet is the array of nodal values of
toroidal current density function J t over the nodes i, j, and k of element e , and S e is the area of element e , which is obtained from −1 1 det (De ) 2
Se =
(II.7.46)
The variational property of (II.7.39) requires that the functional (II.7.45) with respect to the array Ψ of the nodal values of the unknown function be stationary. Therefore, we have
∂ ∂Ψe
∑ Π (ψ ) = 0 e
e
(II.7.47)
e
which turns into the set of linear algebraic equations
1
∑ ∫∫ r drdz B e
eT
Be Ψe = μ0 ∑ ∫∫ Ne Nedrdz Jet T
e
Se
Se
(II.7.48)
Here, the partial stiffness matrix Ke and partial force vector Fe are defined as:
Ke =
1
∫∫ r drdz B
eT
Be
Se
(II.7.49)
T
Fe = μ0 ∫∫ Ne Nedrdz Jet ≡ μ0 Ee Jet Se
(II.7.50)
e e It should be noted K and E are both symmetric, and fortunately there are simple
closed form expressions for evaluation of Ee . As well, the double integral in Ke can be directly evaluated through algebraic expansion of integral region. For instance, the basic triangular elements A- and B-type as illustrated in Figure II.7.6, yields the following expression for A-type
∫∫ Se
and similarly for B-type
⎡ r ⎤ r 1 drdz = (z k − z i ) ⎢⎢ j ln j − 1⎥⎥ r ⎢⎣ rj − ri ri ⎥⎦
(II.7.51)
An Overview of Plasma Confinement in Toroidal Systems
∫∫ Se
⎡ r ⎤ r 1 drdz = (z k − z i ) ⎢⎢ i ln j + 1⎥⎥ r ⎢⎣ ri − rj ri ⎥⎦
137
(II.7.52)
elements. For other triangular elements which are not in the form of A- or B-type elements, one can always present them in combinations of A- and B-type, as any arbitrary triangle can be set in rectangle, surrounded by A- and B-type triangles, as illustrated in Figure II.7.7.
z
i
B − Type
k
k B − Type
j
k
i k
j A − Type
j
A − Type
i
j
i
r
Figure II.7.6. Elementary triangular of A- and B-type
B-type A-type
B-type
Figure II.7.7. Arbitrary triangle can be enclosed by three A- and B-type elements, forming a rectangle
Therefore, by subtraction of integrals belonging to the basic type elements from the surface integral on the rectangle, the unknown surface integral of the triangle is found. This technique helps us to get relieved from the excessive two-dimensional numerical integration needed over each elemental area, thus speeding up the calculations significantly. The final system of equations by (II.7.48) can be hence written as
Kψ = F
(II.7.53)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
where the overall stiffness matrix K and force vector F , have the dimensions N × N and
N × 1 (where N is the number of nodes), respectively, and are generated by (II.7.49) and −1 (II.7.50). The N × 1 vector ψ = K F also denotes the array of unknown nodal values of the poloidal flux function.
II.7.2.1 Problems with the formulation
a) Singularity of (II.7.53) At first glance, the set of linear algebraic (II.7.53) due to the fact that the stiffness matrix K is singular cannot be solved. Because according to the GSE, the poloidal flux function ψ is a potential and thus insensitive to the choice of an absolute reference. Therefore, at least one node must be subject to a boundary condition of Dirichlet type, so that K is not singular. It is now shown that ψ must take on zero value on the z-axis. As stated earlier, the GSE (II.6.1) allows Green’s function solutions having the form
ψ (r , z ) =
∞ ∞
∫ ∫ G (r, r′)J (r ′, z ′)dr ′dz ′ t
−∞ 0
(
(II.7.54)
)
in which the Green’s function G r, r ′ has the asymptotic expansion near the z-axis given by
r 2r ′2
G (r, r ′ ) ≈ μ0
3
4(r + r ′2 (z − z ′)2 )2 2
(II.7.55)
from which we readily obtain the required boundary condition
limr →0 G (r, r ′) = 0
( )
Accordingly, the poloidal flux function ψ r , z
(II.7.56)
has to take on zero value at r = 0 .
This shows that the zero-boundary condition of Dirichlet type over the symmetry axis must be imposed to the system of equations (II.7.53), that is
ψ (0, z ) = 0 This elevates the singularity of K .
(II.7.57)
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139
b) Non-physical Neumann boundary condition Another problem with the system (II.7.53) is the occurrence of a non-physical boundary condition of homogeneous Neumann type over the boundary of the solution region. This difficulty happens in the form of normal magnetic surfaces or poloidal flux contours at the boundaries in the numerical solution. Mathematically it can be represented as
∂ ψ = nˆ ⋅ ∇ψ = nˆ ⋅ ψˆ = 0 ∂n
(II.7.58)
where nˆ stands for the normal vector to the boundaries. To show how this boundary condition implicitly appears in the variational formulation of the GSE (II.7.39), we directly take the variation of ψ in (II.7.39), which yields
δ Π (ψ ) =
⎛1
⎞
∫∫ ⎜⎜⎝ r ∇ψ ⋅ ∇δψ − μ J ψ⎟⎠⎟drdz 0 t
(II.7.59)
Using the identity
⎛ δψ ⎞ δψ * 1 ∇ψ ⋅ ∇δψ = ∇ ⋅ ⎜⎜ ∇ψ ⎟⎟⎟ − Δψ ⎝r ⎠ r r
(II.7.60)
equation (II.7.59) turns into
⎛ δψ ⎞ ⎛1 ⎞ δ Π (ψ ) = −∫∫ ⎜⎜ Δ* ψ + μ0J t ⎟⎟ δψ drdz + ∫∫ ∇ ⋅ ⎜⎜ ∇ψ ⎟⎟⎟drdz ⎝r ⎠ ⎝r ⎠ (II.7.61) The second integral in (II.7.61) can be written as
⎛ δψ
∫∫ ∇ ⋅ ⎜⎜⎝ r
⎞ ∇ψ ⎟⎟⎟drdz = ⎠
∫
δψ ∂ψ ds r ∂n
(II.7.62)
( )
where the contour integration is done in a counter clockwise sense in the r , z plane. Setting (II.7.61) to zero requires that the GSE hold. Therefore, in order to prevent the effect of (II.7.62) entering the solution, either ψ should be fixed over the boundary, that is the case only for the left boundary at r = 0 with (II.7.57), or its normal derivate should vanish, as stated in (II.7.58). Physically, if the system is symmetric with respect to its equatorial plane at z = 0 , the solution region can be halved at the equatorial plane z = 0 . In this case, (II.7.58) must hold at the bottom of the solution region in order to maintain the mirror symmetry. However, the numerical solution over the right and upper borders would be meaningless, because of the fact that (II.7.58) is here non-physical. To stay away from this problem, the infinite elements
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
provide excellent solution when used over the upper and right boundaries. The infinite elements virtually extend the solution region to infinity, where both ψ and ∇ψ approach to zero and therefore (II.7.58) is automatically satisfied.
Figure II.7.8. Infinite element used in computation of magnetic poloidal flux
A typical infinite element is illustrated in Figure II.7.8. The definition of an infinite element relies on taking three fixed reference points, which are not in a straight line. The first point can be chosen to be the origin of the system of coordinates at (0, 0) . However, the second and third points vary with the position of the infinite element. In order to preserve the continuity of the solution, it is necessary to choose two consecutive boundary nodes to serve
(
)
(
) The triangular system of coordinates (ρ, ξ ) for the infinite element e
as these two points, e.g. at r1, z 1 and r2 , z 2 .
( (
r = ρ ⎡⎢r1e + ξ r2e − r1e ⎣ z = ρ ⎡⎢z1e + ξ z 2e − z1e ⎣
)⎤⎦⎥ )⎤⎥⎦
are defined as
(II.7.63)
This coordinate transformation will be utilized for mapping the infinite element into a rectangular region, so that the infinite element e occupies the area extended from ρ = 1 to
ρ = ∞ , and from ξ = 0 to ξ = 1 . This technique simplifies the evaluation of integrals. Moreover, the flux function is assumed to behave as
An Overview of Plasma Confinement in Toroidal Systems
ψe (ρ, ξ ) =
1⎡ e ξψi + (1 − ξ ) ψej ⎤⎥ ⎢ ⎣ ⎦ ρ
141
(II.7.64)
within the finite element. This special definition of variation of the unknown function on the infinite element guarantees continuity of the solution on all three borders of the element, as well as decaying the solution and its derivative at infinity. Now the contribution of the element integrals corresponding to infinite elements should be added to (II.7.48). Since J t = 0 outside the solution region where the infinite elements are, therefore the infinite elements only affect the stiffness matrix K . Hence, it would be necessary to compute only the corresponding partial stiffness matrices Ke . One can show that 1
Ke =
∞
∫∫ 0
0
1
T
(
)
ρ ⎡⎢r1e + ξ r2e − r1e ⎤⎥ ⎣ ⎦
Be Be
∂ (r , z ) ∂ (ρ, ξ )
d ρd ξ (II.7.65)
where the Jacobian of the triangular system of coordinates is given by:
∂ (r , z ) ∂ (ρ, ξ )
= 2Ae ρ (II.7.66)
in which Ae is the area of the triangle formed by the three reference points. Note that the corresponding triangle should be formed in a counter-clock-wise sense ao that Ae be positive. Also, the matrix Be as a function of coordinates is given by
⎡ ξ − 1 ∂ρ 1 ∂ ξ ⎢ − ⎢ ρ ∂r ρ ∂r e ⎢ B = ⎢ ξ − 1 ∂ρ 1 ∂ξ ⎢ − ⎢⎣ ρ ∂z ρ ∂z
ξ ρ2 ξ − 2 ρ
−
∂ρ 1 ∂ξ ⎤⎥ + ∂r ρ ∂r ⎥⎥ ∂ρ 1 ∂ξ ⎥ ⎥ + ∂z ρ ∂z ⎥⎦
(II.7.67)
Thus, the evaluation of partial stiffness matrix needs numerical integration, but it is carried out only on the nodes over the boundary shared by infinite elements.
II.7.2.2. Example In this section, the flux resulting from a magnetic quadrupole consisting of four poloidal
(
)
coils located at (r, z ) = (1,2) , (2,1) , (1, −2) , and 2, −1 with toroidal currents +1 , -1 ,
+1 and −1 , respectively, is considered. In Figure II.7.9, the computation is done by the Variational Axisymmetric Finite Element Method (VAFEM). It should be mentioned that since the system is symmetric with respect to the equatorial plane z = 0 , only the upper half
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
is shown. The resulting poloidal flux by the Green’s function formalism through (II.6.3) is also illustrated in Figure II.7.10 for comparison.
Figure II.7.9. Constant contours of the poloidal flux of the magnetic quadrupole computed by VAFEM
Figure II.7.10. Constant contours of the poloidal flux of the magnetic quadrupole computed by Green’s function method
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143
III. PLASMA STABILITY The most challenging problem in magnetic confinement of plasmas is instabilities. In order to achieve confinement, the plasma needs to be in equilibrium as well as in stable state. Otherwise, small perturbations would grow immoderately, causing catastrophic instabilities. Apart from the consideration of stability or instability, several classifications exist for plasma oscillation modes as follows: (a) (b) (c) (d)
Ideal and resistive MHD modes Internal and external modes Pressure-driven and current-driven modes Micro and macro instabilities
The first classification deals with the finite resistivity of plasmas. Ideal MHD modes are described with the approximation of infinite conductivity for plasma, and therefore do not trigger tearing of magnetic surfaces. Most ideal MHD modes occur on short time scales, typically under 10μsec, and are normally controlled via passive mechanisms. In contrast, finite resistivity of plasma is usually responsible to cause major instabilities, which are accompanied with change of topology of magnetic surfaces and birth and growth of islands. As the growth rates of these instabilities are slow, they do not lead to a macroscopic loss of plasma, but instead they increase transport losses. Resistive MHD modes are associated with a typical time scale of 100μsec or larger, and need stabilization via active electronic control systems. A second classification is based on the location of the instability where the instability starts to develop. If the corresponding mode grows without perturbing the plasma surface then it is referred to as internal modes; internal modes thus by definition affect the shape and location of closed magnetic surfaces inside the plasma, but do not cause change of topology. On the other hand, those modes that perturb the plasma boundary are called external modes. External kink modes cause large distortions in the shape of plasma column and need feedback control stabilization, otherwise they can easily lead to disruptions. Another way to classify plasma instabilities is to notice the driving source of the plasma instability. In general, instabilities are driven by gradients in the pressure or the current density profiles. Pressure-driven modes have little role in equilibrium and stability of plasmas, while current-driven modes are usually responsible for nearly all ideal MHD instabilities. Finally, one could categorize the instabilities with regard to the plasma volume affected. Instabilities that only affect a small portion of the plasma volume are called micro instabilities, while those associated with a large portion of the plasma volume are called macro instabilities. Due to the fact that plasmas of thermonuclear fusion reactors can be seen as strongly nonlinear, it is possible to make use of the infamous Lyapunov Stability Theorem to deal with such systems. In the next section we will assess this method.
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III.1. Lyapunov Stability in Nonlinear Systems Any nonlinear system is subject to instability, even though it might be under equilibrium. In theory, there are several types of stability such as input-output stability, stability of periodic orbits, and the most important of all, stability of equilibrium points. Not every equilibrium configuration would result stable operation. The purpose of study of stability is to decide whether a given plasma equilibrium is stable or not, which modes are not stable and what methods should be employed to stabilize those. In the context of nonlinear systems, Lyapunov stability occurs when all solutions of dynamical system which start near an equilibrium point req in the corresponding phase space, stay near it forever. Mathematically it can be written as
∀ ε > 0, ∃ δ = δ (ε) > 0;
req (0) < δ ⇒ req (t ) < ε ∀ t ≥ 0
(III.1.1)
The nonlinear system at the equilibrium point req , is said to be asymptotically stable, if all solutions that start out near re converge to req . Equivalently
∀ req (0) < δ;
lim req (t ) = 0
t →∞
(III.1.2)
For an asymptotic stable nonlinear system, the state may initially tend away from the equilibrium state but subsequently return to it. It should be noted that asymptotic stability does not imply anything about how long it takes to converge to a prescribed neighborhood of equilibrium point.
III.1.1. Intuitive interpretation (Ball and wall analogy) Simple notions of stability often use the paradigm of the ball and curved surface as illustrated in Figure III.1.1. This idea employs the concept of potential energy, which states that physical systems are stable when they are at their lowest energy. As illustrated in Figure III.1.1, various configurations of ball and curved surfaces lead to different states in stability and equilibrium of ball, which are listed in Table III.1.1. This mechanical system is analogous to plasma in magnetic thermonuclear fusion, in which ball represents the plasma and form of the curve is a symbol of potential energy due to magnetic field configuration. When the ball is in stable position, any perturbation causes the ball to oscillate with reference to its equilibrium position. In contrast to ball in stable position, any small perturbation causes an unstable ball to incessantly move farther from the equilibrium point. When ball is marginally stable, it is on the border between stability and instability; any perturbation may cause switching between these two states. When the ball is linearly stable but non-linearly unstable, a small perturbation leaves the system at rest, but large perturbations kick the ball out of equilibrium. On the other hand, when the ball is linearly unstable but non-linearly stable, a large perturbation drives the system toward a stable state.
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145
Figure III.1.1. Stability and equilibrium of different mechanical system consist of ball and curved surfaces
Table III.1.1. Mechanical equilibrium and stability of a ball in a curve surface Configuration (a) (b) (c) (d) (e) (f)
Equilibrium ☺ ☺ ☺ No Equilibrium ☺ ☺
Stability Marginally Stable Stable Unstable Unstable Linearly Stable, Non-linearly Unstable Linearly Unstable, Non-linearly Stable
Difference of energy levels of ball between the initial and final states determines the stability of the ball, while its slope determines the equilibrium. Hence, the concept of energy principle has been evolved as a powerful mathematical tool to study the stability of equilibrium configurations.
III.2. Energy Principle As stated earlier in discussion of MHD, the forces are in balance under equilibrium condition. Now, suppose that magnetic plasma is in its equilibrium state, where the potential energy of system is at a minimum. Let fluctuations cause the plasma to be physically displaced by an infinitesimal vector field ξ out of its equilibrium point. Due to this fact, the net applied force F, on magnetic plasma is no longer equal to zero, the system is no more in equilibrium. Assume the displacement ξ and the force F are not in the same direction, so the force F tends to bring the plasma back to equilibrium. In this case, the net change in potential energy δW is positive and the system is stable. Mathematically the extremum of the energy is a local minimum. Now if both the force F and displacement ξ are in the same direction, then the force tends to move the system farther from its equilibrium position. One can conclude that the change in potential energy is negative and consequently the system is unstable. In this situation the extremum of the energy corresponds to a local maximum, or an inflection point.
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Now, we exploit MHD theory to develop an expression for the change in potential energy δW of plasma, when displaced from an equilibrium. We start with MHD equations
ρ
∂V 1 + V ⋅ ∇V = −∇p + (∇ × B) × B ∂t μ0 ∂B = ∇ × ( V × B) ∂t ∂ρ + ∇ ⋅ (ρ V ) = 0 ∂t ⎛ ∂ρ ⎞ p ⎜⎜ + V ⋅ ∇⎟⎟⎟ γ = 0 ⎝ ∂t ⎠ρ
(III.2.1)
As MHD stability analysis is a complex nonlinear problem, linear perturbation method is the best mathematical tool that helps us to simplify the stability problem through linearization. The perturbation method leads us to an expression for the desired solution in terms of a power series in some small parameter, call perturbation. Due to the fact that the amplitude of the perturbation is infinitesimal, one can obtain the linear perturbation solution by truncating the series, usually by retaining the first two terms, referring to as the equilibrium solution and the first order perturbation correction. Hence we have
ρ (r, t ) = ρ0 ( r) + ρ1 ( r, t ) p (r, t ) = p0 (r) + p1 ( r, t ) J ( r, t ) = J0 (r) + J1 ( r, t ) B (r, t ) = B0 ( r) + B1 ( r, t ) in which the terms marked with zero index ρ0 (r) ,
(III.2.2)
p (r) , J0 (r) , and B0 (r) are
respectively the mass density, pressure, current density and magnetic field, respectively; the zero subscript denotes the equilibrium values. Also, the terms marked with unity index,
ρ1 (r,t ) , p1 ( r, t ) , J1 ( r,t ) , and B1 (r,t ) , represent the infinitesimal perturbation values.
Assume that the perturbed displacement from equilibrium position is represented by oscillatory time-dependent vector field d (r, t ) = ξ ( r) exp (−i ωt ) , so that the velocity and all other perturbed quantities such as mass density, current density, pressure and magnetic field can be written as
An Overview of Plasma Confinement in Toroidal Systems
V=
147
∂d (r, t ) = −i ωξ (r) exp (−i ωt ) ∂t δρ = ρ1 (r) exp (−i ωt ) δ p = p1 (r) exp (−i ωt ) δ B = B1 (r) exp (−i ωt )
(III.2.3)
where
ρ1 = −∇ ⋅ (ρ0ξ ) p1 = − (ξ ⋅ ∇) p0 − γ p0∇ ⋅ ξ ⎛1⎞ J1 = ⎜⎜ ⎟⎟⎟ ∇ × [∇ × (ξ × B)] ⎜⎝ μ0 ⎠⎟ B1 = ∇ × (ξ × B0 )
(III.2.4)
The angular frequency ω in (III.2.3) may taken on complex values and appears as an eigenvalue in the formulation. It can be shown that the final eigenvalue problem appears as an eigenfunction problem belonging to the force field F(ξ), which is a self-adjoint operator and thus has real eigenvalues given by ω 2 ∈
. Hence we have either non-negative ω 2
corresponding to stable and oscillatory motion of the perturbation, or negative ω 2 corresponding to a purely imaginary angular frequency ω , thus exponentially growing perturbations and unstable equilibrium. A given equilibrium may be stable with regard to a some perturbation modes, while being unstable with regard to the rest. In practice for stable modes with real-valued ω , some energy is lost along with the oscillations by various energy loss mechanisms of plasma, thereby damping the oscillation amplitudes gradually towards equilibrium. We furthermore note that perturbation method requires smallness of perturbation amplitudes, that is
ρ0 (r)
ρ1 (r, t )
p0 ( r )
p1 ( r, t )
J0 ( r)
J1 (r, t )
B0 ( r )
B1 (r, t )
(III.2.5)
Along with MHD equations (III.2.1), and perturbation expansions (III.2.2), one can easily obtain linear stability equations given by
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
∂B1 = ∇ × ( V × B0 ) ∂t ∇ × B1 = μ0 J1 ∂ρ1 + ∇ ⋅ (ρo V1 ) = 0 ∂t ∂V1 + ∇p = J0 × B1 + J1 × B1 ∂t ⎞ γ p ⎛ ∂ρ ∂p1 + V ⋅ ∇p0 + 0 ⎜⎜ 1 + V ⋅ ∇ρ0 ⎟⎟⎟ = 0 ⎠ ρ 0 ⎝ ∂t ∂t ρ0
(III.2.6)
One can decide on the stability of system with regard to a given perturbation or mode, by knowing the sign of δW as
δW > 0
Stable
δW < 0
Unstable
(III.2.7)
in which, the change in potential of system δW caused by perturbation (here physical displacement) ξ is equal to
δW =
−1 ξ ⋅ F (ξ)d τ = δWP + δWV + δWS 2 ∫
(III.2.8)
where δW P , δWV and δW S are changes in the potential energy of the plasma, the vacuum magnetic field around the plasma and the plasma surface, given respectively by
(III.2.9)
δWV =
∫ Vacuum Region
B12 2μ0
dτ (III.2.10)
⎡ ⎛ 1 B02 ⎞⎟⎤⎥ 1 ⎜ ⎢ ⎟⎟ ⋅ dS= δWS = ξn ⎢∇ ⎜p0 + ξn2 ⎡⎣(B0 ⋅ ∇) B0 ⎤⎦ ⋅ dS ∫ ∫ ⎥ ⎜ ⎟ 2 Plasma 2μ0 ⎠⎦ 2 Plasma ⎣ ⎝ Interface
Interface
(III.2.11)
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149
In (III.2.9), the first term is the change in magnetic field energy caused by perturbation ξ , and the forthcoming terms correspond to changes in energy due to perturbation in pressure and the work done against magnetic forces. As it can be seen, the two first terms in (III.2.9) are always positive, while the remaining two terms can take on negative values. Change in vacuum energy given by (III.2.10) is always positive and hence it contributes to stabilization of plasma. However, the interface energy between plasma and vacuum (III.2.11), which is due to surface current could have a destabilizing role.
III.2.1. Application of Energy Principle The simple configuration between plasma and vacuum is illustrated in Figure III.2.1, where the magnetic field of plasma vanishes and pressure profile is uniform; on the other hand, pressure in vacuum is effectively zero.
Figure III.2.1. Plasma–vacuum interface
Potential energy inside the plasma is determined by (III.2.9), where in this situation the non-vanishing term is
δWP =
1 2 γ p0 (∇ ⋅ ξ) d τ ∫ 2 Plasma Volume
(III.2.12)
It can be easily seen that
δWP ≥ 0
(III.2.13)
For those modes satisfying ∇ ⋅ ξ = 0 then the total energy becomes
δW = δWV + δWS
(III.2.14)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
in which according to (III.2.10) and (III.2.11) we obtain
⎛ B 2 ⎞⎟ 2 1 ⎜⎜ 0v ⎟ds + 1 2 ˆ A δW = ξ n dτ ⋅ ∇ ∇ × ⎟ ∫ n ⎜⎜⎝ 2μ ⎠⎟⎟ ∫ v 2 Interface 2μ0 Vacuum 0
(III.2.15)
As you can see, the stability is determined by the first term on the right-hand-sine of 2
(III.2.15). Equivalently the sign of expression nˆ ⋅ ∇B0v
interface
= ∂B02v ∂n is the stability
criterion. Therefore one can conclude that the system can be unstable when
∂B02v ∂n
<0 (III.2.16)
We notice that ∇ B ov2 plays an important role in the stability of system. Using the vector identity
∇ (A ⋅ B) = (A ⋅ ∇) B + (B ⋅ ∇) A + A × (∇ × B) + B × (∇ × A)
(III.2.17)
where by putting A = B = B 0v we get
∇B02v = 2 (B0v ⋅ ∇) B0v + 2B0v × (∇× B0v )
(III.2.18)
In vacuum region, we have μ0 J = ∇ × B0v = 0 , and hence
∇B02v = 2 (B0v ⋅ ∇) B0v
(III.2.19)
∂B02v B2 = nˆ ⋅ ∇B02v = nˆ ⋅ (B0v ⋅ ∇) B0v = 2 (nˆ ⋅ Rc ) 02v ∂n Rc
(III.2.20)
One can show that
Here, the so-called curvature vector Rc points from the interface to the center of curvature, illustrated in Figure III.2.2. Substitution of (III.2.20) in (III.2.11) yields
δWs =
1 4μ0
∫
Plasma Interface
2
(nˆ ⋅ ξ) (nˆ ⋅ Rc )
B02v ds R2 (III.2.21)
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151
According to (III.2.21), the dot product of normal and curvature vectors nˆ ⋅ R c determines the stability as follows: Case 1: nˆ ⋅ R c > 0 and surface energy is stabilizing. The plasma and vacuum configuration at the interface is shown in Figure III.2.3, in which is known as good curvature. Case 2: nˆ ⋅ R c < 0 and surface energy is destabilizing. The plasma and vacuum configuration at the interface is shown in Figure III.2.4, in which is known as bad curvature. In the next section we exploit energy principle to analyze the stability properties of the θ-pinch, the z-pinch, and the general screw pinch.
Vacuum
nˆ
Rc
Plasma
C
Figure III.2.2. Plasma-vacuum interface curvature and the curvature vector R
c
Rc
Plasma Figure III.2.3. Good curvature
Figure III.2.4. Bad curvature
B
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
III.3. Modal Analysis In this section we present the application of energy principle to analyze the stability characteristics of θ-pinch and z-pinch. With the same method The kink instability is being studied.
III.3.1. θ-pinch Since the equilibrium is symmetric with respect to both θ- and z-coordinates, the perturbation can have the following form
ξ ( r) = ξ (r ) exp [i (mθ + kz )]
(III.3.1)
where m and k are called poloidal and toroidal (or axial) mode numbers, respectively. While
m must be an integer, k is a continuous variable if the system be infinitively long. For a cylinder with finite length k can take on discrete values. Different values of mode numbers m and k lead to various perturbations, as illustrated in Figure III.3.1. According to Figure III.3.1, the mode with m = 0 and k ≠ 0 , called sausage mode, usually arise from thermal disturbances, which can cause the incompressible plasma to develop axially periodic constrictions and bulges. The m = 1 and k = 0 mode, only shifts the plasma column with respect to its axis. Helical kink instabilities occurs in mode with m = 1 and k ≠ 0 . In this instability, the concave surfaces of the plasma experience concentration of the azimuthal field resulting in a magnetic pressure that increases the concavity. Likewise at the convex surfaces, the azimuthal field is weaker so that the convex bulge will tend to increase. The plasma cross section at m = 2 mode becomes elliptical , while for m = 3 mode, the cross section becomes triangular, and so on.
Equilibrium
m=0 k≠0
m =1
m =1
m =2
m=3
k =0
k≠0
k=0
k =0
Figure III.3.1. Different perturbation correspond with various values of m and n
One can obtain the expression for potential energy when k ≠ 0 as
δW π = L μ0
a
∫ 0
⎡ ⎢ ∂ξr ⎢r 2 2 2 m + k r ⎢ ∂r ⎢⎣ k 2Bz2
2
(
2
2 2
+ m +k r
)
⎤ ⎥ ξr ⎥ rdr ⎥ ⎥⎦ 2
(III.3.2)
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153
It can be understood from (III.3.2) that for every choice of mode numbers, we have
δW > 0 ; therefore θ-pinch is stable with regard to all MHD modes having finite k. One reason that θ-pinch is stable for all MHD modes, is that θ-pinch has no curvature field lines. Another important factor that makes θ-pinch so much resistant to MHD modes is that there is no axial current, i. e. Jz = 0 , and hence no current driven instabilities. The magnetic field lines of a typical θ-pinch is depicted in Figure III.3.2. According to Figure III.3.2a, magnetic field lines in θ-pinch are straight, bending them, Figure III.3.2b, will lead to a magnetic field tension, and consequently to a force that makes the field straight again. Meanwhile, squeezing field lines as in Figure III.3.2c, will lead to an increase in the magnetic field pressure and consequently to a force that prevents further squeezing.
(a )
(b )
(c )
Figure III.3.2. magnetic field lines of a typical θ-pinch
III.3.2. z-Pinch III.3.2.1. z-Pinch, m ≠ 0 Modes The equilibrium condition for z-pinch was mentioned in (II.4.10), where we display it here again
∂p0 ∂r
=−
Bθ ∂ (rBθ ) μ0 ∂r
(III.3.3)
154
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani The potential energy of a z-pinch with m ≠ 0 condition may be shown to be
δW π = μ0 L
a
∫ 0
2 2 2 2 ⎡⎛ ⎤ ⎢⎜2μ r ∂p + m 2B 2 ⎞⎟⎟ ξr + m r Bθ ∂ ⎛⎜ 1 ξ ⎞⎟⎟ ⎥ r dr θ r 0 ⎜ ⎢⎝ ⎠⎟ r m 2 + r 2k 2 ∂r ⎜⎝ r ⎠ ⎥⎦⎥ ∂r ⎣⎢
(III.3.4)
The worst situation is achieved by letting k → ∞ . Therefore the stability is determined by
δW π = L μ0
a
∫ 0
⎛ ⎞ ⎜⎜2μ r ∂p + m 2B 2 ⎟⎟ ξ 2dr r θ⎟ ⎜⎝ 0 ∂r ⎠⎟
(III.3.5)
In order for the system to be stable for all point inside the plasma the integrand should be positive, hence
m 2Bθ2 > −2μ0r
∂p 0 ∂r
(III.3.6)
Substituting (III.3.3) in (III.3.6) gives
m 2Bθ2 > 2Bθ
∂ (rBθ ) ∂r
(III.3.7)
The right-hand-side of (III.3.7) can be written as
Bθ
∂ ∂ ⎛⎜ 2 Bθ ⎞⎟⎟ ∂ ⎛⎜ Bθ ⎞⎟⎟ 2 ⎜⎜r ⎜⎜ ⎟ + 2Bθ2 rBθ ) = Bθ ( ⎟ = r Bθ ∂r ∂r ⎜⎝ r ⎠⎟ ∂r ⎝⎜ r ⎠⎟
(III.3.8)
or equivalently
Bθ
2 2 2 ∂ ∂ ⎜⎛ Bθ ⎞⎟⎟ ∂ ⎛⎜rBθ ⎞⎟⎟ Bθ 2 ⎜ ⎜ = + = + rB r B ( θ ) ∂r ⎜⎜ 2 ⎟⎟⎟ θ ∂r ⎜⎜ 2 ⎟⎟⎟ 2 ∂r ⎝ ⎠ ⎝ ⎠
(III.3.9)
Therefore by using (III.3.8) or (III.3.9) and substitution in the stability criterion (III.3.7), we arrive at
r 2 ∂ ⎛⎜ Bθ ⎞⎟⎟ 1 2 ⎜ ⎟ m −4 > Bθ ∂r ⎜⎜⎝ r ⎠⎟ 2
(
)
(III.3.10)
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155
or
1 2 ∂ m − 1 > Bθ−2 rBθ2 2 ∂r
(
)
( )
(III.3.11)
Typical magnetic field of a z-pinch is illustrated in Figure III.3.3. According to Figure III.3.3, for r → 0 the magnetic field in z-pinch is proportional to r . Therefore the stability condition (III.3.10) simply becomes
m2 > 4 This is while for r → ∞ we have Bθ ∼ r
(III.3.12) −1
, and the stability condition (III.3.10)
becomes
∂ ⎛ 1 ⎞⎟ 1 2 r 2 ∂ ⎛ Bθ ⎞⎟ ⎜⎜ ⎟ > r 3 ⎜ ⎟ = −2 (m − 4) > ⎟ 2 Bθ ∂r ⎝ r ⎠ ∂r ⎜⎝ r 2 ⎠
(III.3.13)
or m 2 > 0 . Hence the stability condition m > 2 is dominant. Similarly, the stability condition (III.3.11) for r → 0 and m = 1 gives
r>0
(III.3.14)
In which, it means that for core plasma with small r , z-pinch is unstable. For plasma boundary of a thick z-pinch with r → ∞ , the stability condition is simply
m2 > 1
(II.3.1)
Bθ
Bθ ∼ r
Bθ ∼ r −1
r
Figure III.3.3. z-pinch profile
156
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani As in z-pinch the azimuthal current is zero Jθ = 0 , the instability for m = 1 is caused
by bad curvature of magnetic field lines.
III.3.2.1. z-pinch, m = 0 Mode Potential energy of the z-pinch for m = 0 mode equals to
δW π = L μ0
a
∫ 0
∂p0 ⎤⎥ ξr2 ⎡⎢ r γ p0Bθ2 r dr + 2 r ⎢⎢⎣ γμ0 p0 + Bθ2 ∂r ⎥⎥⎦
(III.3.16)
where
⎡ rB 2 ∂ ⎛ ξ ⎞ γ p ∂ ⎤ i ⎜ r ⎟⎟ + 0 ⎢ θ ⎥ ⎜ r ξz = ξ ( ) ⎟ r ∂r r ⎥⎥ γ p0 + Bθ2 μ0 ⎢⎢⎣ μ0 ∂r ⎝⎜⎜ r ⎠⎟ ⎦
(III.3.17)
In order for the z-pinch to be stable for m = 0 mode, the integrand of (III.3.16) should be positive, that is
−
r ∂p0 4γ < p0 ∂r 2 + γ 2μ0 p0 Bθ2
(
)
(III.3.18)
At the plasma edge the pressure rapidly goes to zero that makes the radial pressure gradient ∂p0 ∂r to increase dramatically. This situation does not satisfy the stability condition (III.3.18). If the plasma is to be confined well by magnetic field, the upper limit on which the pressure can be decreased becomes
r dp < 2γ p dr
(III.3.19)
dp dr < 2γ p r
(III.3.20)
− or equivalently
−
Integration of both sides of (III.3.20) and noting that γ ≈ 5 3 gives
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157
10
p (r ) > r − 3
(III.3.21)
The above result states that, pressure must vary no faster than r
−10 3
.
III.3.3. Kink Instability The kink instability is an ideal MHD instability which at low β is driven by the current gradient and at high β , by pressure gradients. It usually happens when between the plasma and the conducting wall there is a vacuum region. As stated earlier, in order to examine stability of plasma we perturb plasma from its equilibrium position, and determine whether a small perturbation will grow to disrupt the plasma or tends back to equilibrium. The perturbation in primitive toroidal coordinates may be written as
ξ (r, t ) = ξ (r ) exp [i (mθ − nϕ − ωt )]
(III.3.22)
in which ϕ and θ are the toroidal and poloidal angles, respectively. Under equilibrium, the plasma region is located at r < a , and the vacuum region is a < r < b , where b is the radius of perfectly conducting wall. Plasma potential energy δWp for this configuration becomes 2 2⎤ ⎡ 2 ⎛n 1 ⎞⎟ 2 2 ⎛ ∂ξ ⎞ ⎜ ⎟ ⎢ ⎥ ⎜ δW = (m − 1) ξ + r ⎝⎜ ⎠⎟⎟ ⎥ ⎜⎜ − ⎟⎟ rdr + ∂r ⎦ ⎝ m q ⎠ μ0R0 ∫0 ⎣⎢ 2 Bϕ2a 2ξa2 ⎡⎢ 2 ⎛⎜ n 1 ⎞⎟ ⎛⎜ n 1 ⎞⎟ ⎤⎥ + ⎟+⎜ − ⎟ ⎜ − μ0R0 ⎢⎢⎣ q (a ) ⎝⎜ m q (a )⎠⎟ ⎝⎜ m q (a )⎠⎟ ⎥⎦⎥
Bϕ2
a
(III.3.23)
where q is the safety factor. On the other hand, the potential energy of vacuum is obtained as
δWv =
2 π2R ⎡ n 1 ⎤ ⎢ − ⎥ mλa 2ξa2 μ0 ⎢⎣ m q (a ) ⎥⎦
(III.3.24)
where
( ) λ= 1 − (a b ) 1+ a b
2m
2m
(III.3.25)
Using (III.3.24) and (III.3.23), one can obtain the total change in potential energy as
158
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani 2 ⎡⎛ ∂ξ ⎞2 ⎤⎛ 1 ⎟⎞ 2 2⎥⎜ n ⎟ ⎢ ⎜ δW = ⎜r ⎟ + (m − 1) ξ ⎥ ⎜⎜ − ⎟⎟ rdr μ0R ∫0 ⎢⎣⎝ ∂r ⎟⎠ ⎦ ⎝m q ⎠ ⎞ ⎛ ⎞⎤ 2π2Bϕ2a 2ξa2 ⎡⎛⎜ n ⎢⎜ − 1 ⎟⎟ + (1 + mλ )⎜⎜ n − 1 ⎟⎟⎥ + ⎜⎝m q (a )⎟⎠⎥ μ0q (a ) R ⎢⎣⎜⎝m q (a )⎟⎠ ⎦
π2Bϕ2
a
(II.3.2)
From (III.3.26) one can conclude that if the vacuum region could be removed and the conducting wall would touch the plasma boundary, then ξa would vanish, the potential energy difference would become positive, and in this case the plasma column would be
(
stable.; clearly, this condition is not practical. Otherwise the stability condition for m, n
)
mode is satisfied by qa > m n .
III.3.3. Interchange Instability When two types of fluids in contact are situated with an external force such that the potential energy is not a minimum, interchange instability occurs and the two fluids will then interchange locations to bring the potential energy to a minimum. In plasmas with magnetic fields, the plasma may interchange position with the magnetic field. A prime example is the flute instability in mirror machines, in which the perturbation is uniform parallel to the magnetic field. Two neighboring magnetic flux tubes with p1 and p2 as initial pressures, and V1 and V2 as volumes of tubes are shown in Figure III.3.4. As magnetic fluxes are assumed to be equal, we have:
φ = B1A1 = B2A2
(III.3.27)
where B1 and B2 are the magnetic fields, and A1 and A2 are the cross sections of two flutes. Plasma of volume V is adiabatic when
pV γ = cte
(III.3.28)
After interchanging the new pressures will be γ
⎛V ⎞⎟ p1′ = p1 ⎜⎜⎜ 1 ⎟⎟ ⎜⎝V2 ⎠⎟
(III.3.29) γ
⎛V ⎞ p2′ = p2 ⎜⎜⎜ 2 ⎟⎟⎟ ⎜⎝V1 ⎠⎟
(III.3.30)
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159
The difference in final and initial potential energy and of two tubes is therefore γ γ ⎡ ⎤ ⎛V ⎞⎟ 1 ⎢ ⎛⎜V1 ⎞⎟⎟ ⎥ ⎜ 2⎟ δW = − p2V2 ⎥ ⎢ p1 ⎜⎜ ⎟ V2 + p2 ⎜⎜ ⎟ V1 − pV 1 1 ⎟ ⎟ γ − 1 ⎢ ⎝⎜V2 ⎠ ⎥ ⎝⎜V1 ⎠ ⎣⎢ ⎦⎥
(III.3.31)
Now let
δ p = p2 − p1 δV = V2 −V1
(III.3.32)
Using (III.2.32), the change in potential energy becomes
δW = δ p δV + γ p
1 δV 2 V
(III.3.33)
Pressure p1 Cross Section A1 Volume V1
Pressure p2 Cross Section A2 Volume V2 Figure III.3.4. Two adjacent magnetic flux tubes
The second term in right-hand-side of (III.3.33) is always positive, and it can be ignored at plasma edge where the pressure is too small. Therefore the stability condition simply becomes
δ p δV > 0
(III.3.34)
δ p for a confined plasma is negative because of outward decay pressure profile. Therefore in order to make the plasma stable, it is required to have negative δV as well. But δV can be written as
δV = δ
(∫ Adl )
(III.3.35)
160
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani Using φ = AB , one can rewrite (III.3.35) as
δV = δ
⎛ dl ⎞ Adl = φ δ ⎜⎜∫ ⎟⎟⎟ ⎜⎝ B ⎠⎟
(∫ )
(III.3.36)
Hence for stability we need to have
⎛ dl ⎞ δ ⎜⎜⎜ ∫ ⎟⎟⎟ < 0 ⎝ B ⎠⎟
(III.3.37)
III.4. Simplifications for Axisymmetric Toroidal Machines The Change in potential energy, which determines the stability of system (III.2.8), can also be evaluated in axisymmetric toroidal system. To derive δW in axisymmetric system, it
(
)
is convenient to employ flux coordinate system ψ, ζ , ϕ , which is shown in Figure (III.4.1).
ψ is the flux function which is defined by
ψ = −RAϕ
(III.4.1)
Also ζ and ϕ are poloidal and toroidal angels, respectively. Magnetic field and a field line in flux coordinates can be written as
B = ϕˆ × ψˆ + I (ψ ) ϕˆ
(III.4.2)
Z
ψˆ
ϕˆ
ζˆ
R Figure III.4.1. Flux orthogonal coordinate system
An Overview of Plasma Confinement in Toroidal Systems
I (ψ ) B Rdϕ = ϕ = JBζd ζ Bζ RBζ
161
(III.4.3)
( )
where J ψ is the Jacobian determinant, which is obtained by using (II.2.5) and the flux function I (ψ ) is defined as
I (ψ ) =
μ0I (ψ ) 2π
(III.4.4)
Safety factor in flux coordinates can also be defined using the path integral as q (ψ ) = ∫
Bϕ RBζ
ds
(III.4.5)
in which the integral is taken along a closed path encircling the minor axis and lying on a specific magnetic surface. Substituting (III.4.3) in (III.4.5) yields
q (ψ ) =
J (ψ ) I (ψ ) 1 dζ 2π ∫ R2
(III.4.6)
The change in the potential energy of system can be written as
1 W = ∫ 2V
⎡ B12 ⎤ 2 ⎢ + γ p ∇ ⋅ ξ + (ξ.∇p) (∇ ⋅ ξ* ) − ξ* ⋅ ( J × B1 )⎥ d τ ⎢ μ0 ⎥ ⎣ ⎦
(III.4.7)
But the perturbation vector in flux coordinates can be represented in the covariant form of
ξ = ξψ ψˆ + ξζ ζˆ + ξϕϕˆ
(III.4.8)
with the components ξψ =
K RBζ
ξζ = Bζ L ξϕ = RM +
I L R
(III.4.9)
162
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani Here, M defined as M ≡
1 (B ξ − Bϕ ξζ ) RBζ ζ ϕ
(III.4.10)
The first term in (III.4.7) can be written as
B ⋅B B12 1 = 1 1 = 2μ0 2μ0 2μ0
⎡ B 2 + B 2⎤ 1ϕ ⎥ ⎢⎣ 1ζ ⎦
(III.4.11)
One should thus obtain expressions for B12ζ and B12ϕ in flux coordinates. We first note by (III.2.4) that
B1 = ∇× (ξ × B) = ∇× ⎡⎢(ξζ Bϕ − ξϕBζ ) ψˆ − ξψBϕζˆ + ξψBζ ϕˆ⎤⎥ ⎣ ⎦ ˆ ˆ = B1ψ ψ + B1ζ ζ + B1ϕϕˆ
(III.4.12)
Where
B1ψ =
i Bk K Bζ R
⎛ ∂K ⎟⎞ B1ζ = −Bζ ⎜⎜inM + ⎟ ∂ψ ⎠⎟ ⎝⎜ R ⎡⎢ ∂ ⎛⎜JI ⎞⎟ ∂M ⎤⎥ − ⎜ K⎟ + J ⎢⎣ ∂ψ ⎜⎝ R 2 ⎠⎟⎟ ∂ζ ⎥⎦
B1ϕ =
(III.4.13)
Consequently, we have the followings 2
B1ζ
=
2μ0
Bζ2 2μ0
2
inM + K ′ =
+ (inMK * − inM *K )
B1ϕ 2μ0 −
2
Bζ2 2μ0 Jϕ 2R
inM + K ′ −
μ0J ϕ RBζ2
+ (K ′K * + K * ) ′
Jϕ R
K −
μ0J ϕ2 2R 2Bζ2
KK * (III.4.14)
2
⎛JK ⎞′ JK ⎛JK ⎞′ R 2 ∂M R 2 ∂M I ′2 KK * = − I ⎜⎜ 2 ⎟⎟⎟ − 2 I ′ = − I ⎜⎜ 2 ⎟⎟⎟ + 2 2 ⎝R ⎠ ⎝R ⎠ 2μ0J ∂ζ 2μ0J ∂ζ μ0R 2 R
I′ 2μ0J
⎛ ∂M * ∂M * ⎞⎟ II ′ II ′ ⎛⎜ J ′ 2R ′ ⎞⎟ ′ * ⎜⎜ K − K ⎟⎟ + K ′K * + K * K ) + ⎜ − 3 J ⎟⎟ KK 2 ( ⎟ ⎜⎝ ∂ζ μ0J ⎜⎝ R2 R ⎠ ∂ζ ⎠ 2μ0R (III.4.15)
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163
Using (III.4.8), the term ∇ ⋅ ξ in (III.4.7) in flux coordinate system can be expressed as
∇⋅ξ = =
1 J
1 J
⎡ ∂ ∂ ⎛⎜ ξζ ⎞⎟⎟ ∂ ⎛⎜J ζϕ ⎞⎟⎤⎥ ⎢ ⎜ JB R ξ + + ⎢ ∂ψ ( ζ ψ ) ∂ζ ⎜⎜ B ⎟⎟ ∂ϕ ⎜⎝⎜ R ⎠⎟⎟⎥ ⎝ ζ⎠ ⎣⎢ ⎦⎥
⎡ ∂ I ∂ ∂ ⎛⎜ ⎢ JK ) + L) + J ⎜M + 2 ( ( ⎢ ∂ψ ⎜ ∂ζ ∂ϕ ⎝ R ⎣
()
(
)
(
⎞⎤ L ⎟⎟⎟⎥ ⎠⎟⎥⎦
(III.4.16)
)
Using (III.4.9) and letting ξ r = ξ ψ, ζ exp inϕ , (III.4.16) turns into
∇.ξ =
1 (JK )′ + iBk L + inM J
(III.4.17)
where
⎛ I 1 ∂ ⎞⎟ ⎟ k = −⎜⎜ 2 n + i ⎜⎝ BR JB ∂ζ ⎠⎟⎟
(III.4.18)
The term ξ ⋅ ∇p in (III.4.7) in flux coordinates also takes the form
ξ ⋅ ∇p = ξψRBξ p′ = Kp′
(III.4.19)
From GSE (II.5.20), one can obtain
⎛J II ′ ⎞⎟ ⎟ ξ ⋅ ∇p = Kp ′ = −K ⎜⎜ ϕ + ⎜⎝ R μ0R2 ⎠⎟⎟
(III.4.20)
Multiplying (III.4.17) by (III.4.20) yields
⎛J ϕ II ′ ⎞⎟ ⎡ 1 ⎤ ⎟ ⎢ (JK )′ + iBk L + inM ⎥ + 2⎟ ⎜⎝ R ⎥⎦ μ0R ⎠⎟ ⎢⎣J
(III.4.21)
⎛J ϕ ⎤ II ′ ⎟⎞ ⎡ 1 * ′ * * ⎢ ⎥ ⎟ + − − JK iBk L inM ( ) ⎥⎦ ⎜⎝ R μ0R2 ⎟⎟⎠ ⎢⎣J
(III.4.22)
(ξ ⋅ ∇p )(∇ ⋅ ξ) = −K ⎜⎜⎜ Therefore, we similarly obtain
(ξ ⋅ ∇p )(∇ ⋅ ξ* ) = −K ⎜⎜⎜
164
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
The next term in (III.4.7) that should be manipulated in order to be expressible in flux coordinates is ξ* ⋅ ( J × B1 ) . Using the vector identity (A × B) ⋅ C ≡ (B × C) ⋅ A ≡ (C × A) ⋅ B we have
ξ* ⋅ ( J × B1 ) = J ⋅ (B1 × ξ* )
(III.4.23)
Substituting (III.4.12) and (III.4.8) in (II.4.23) yields
J × (B1 × ξ* ) =
I′ μ0J
* ⎛ IJ ⎞⎟′ * ⎜⎜ K ⎟ K − I ′ ∂M K * + K K ′ J ϕ 2 R μ0J ∂ζ ⎝⎜ R ⎠⎟⎟
⎡ * I ′ ⎛⎜ I * ⎞⎟⎤⎥ nMK * * ⎢ K L jϕ + ⎜RM + L ⎟⎟ + i Jϕ +i ⎢ R R ⎠⎟⎥⎦ R μ0 ⎜⎝ ⎣ (III.4.24) Bk
where J is the Jacobian determinant and J ϕ is toroidal current. Now, by substituting (II.4.14), (II.4.15), (II.4.22), and (II.4.24) in (II.4.7) we get:
W =
∫ V
2 ⎡ ⎢ 1 B 2k 2 ⎛JK ⎞⎟′ 1 R 2 ∂M 2 ⎢ K + − I ⎜⎜ 2 ⎟⎟ − UKK * ⎝R ⎠ ⎢ 2μ0 Bζ2R 2 2μ0 J 2 ∂ζ ⎣⎢
+
Bζ2 2μ0
μ0J ϕ
inM + K ′ −
2 ζ
RB
2
K +
2⎤ 1 1 γ p (JK )′ + iBk L + inM ⎥⎥ d τ 2 J ⎥ ⎦ (III.4.25)
where γ is the plasma density, and d τ in flux coordinates is
d τ = Jd ψd ζd ϕ
(III.4.26)
Also, U in (III.4.25) is defined as
U≡
II ′ R′ J ϕ ⎜⎛J ′ μ0J ϕ ⎟⎟⎞ II ′ R′ J ϕ ⎜⎛J ′ Bζ′ ⎟⎟⎞ ⎜ + + + ⎜⎜ + ⎟ ⎟= μ0R2 R 2R ⎜⎝⎜ J RBζ2 ⎠⎟ μ0R2 R R ⎝⎜ J Bζ ⎠⎟
(III.4.27)
IV. PLASMA TRANSPORT In sections II and III, we studied equilibrium and stability of a plasma, respectively. The quality of plasma confinement with regard to the maximum plasma temperature, density, and confinement time is limited by the transport of heat and particles across the magnetic
An Overview of Plasma Confinement in Toroidal Systems
165
surfaces. In most equilibrium configurations transport contributes to significant loss of energy from the plasma core. Gradients in particle density, as well as electron and ion temperatures known as potentials, drive fluxes known as transport, in such a way to counter the gradients, thus lowering maximum achievable performance of plasma confinement. Moreover, the toroidal shape of magnetic surfaces result in excessive transport than what is predicted by the so-called classical transport for cylindrical plasma of the hypothetical straight tokamak with zero curvature, which is normally referred to neo-classical transport. It is known that even the theory of neo-classical transport fails to describe the confinement behavior of thermonuclear plasmas where other mechanisms, such as turbulence, play a dominant role. In inertially confined plasmas, radiation transport adds up to the major transport mechanisms, which needs a very detailed and elaborate consideration. In this section we limit the discussion to classical and neo-classical transport and leave the discussion of turbulence and radiation transport to references. The Boltzmann transport equation in phase space can be derived by considering how a distribution function changes in time. The classical and neo-classical theories of transport are best understood when their respective formulations are based on Boltzmann equation.
IV.1. Boltzmann Equation Plasma consists of numerous charged and uncharged particles. At any given moment, every particle has a precise position r and velocity v in the phase space (r, v) , and hence follows a trajectory expressible via a parametric curve as C (t ) = [ r (t ), v (t )] . Knowing the exact trajectory C i ,s (t ) for all particles indexed by i belonging to the species s enable us to characterize the plasma accurately at all times. This can be only done through extensive particle simulations; even though powerful supercomputers are utilized for this purpose, it is impossible to simulate a real thermonuclear plasma with its full number of particles. The alternative solution is to make a local average over all particles belonging to the species s at a given time and within the neighborhood of a given phase space point (r, v) .
(
)
This averaged quantity known as the distribution function fs r, v, t thus gives information about the phase-space density of species s at the time t; hence, dn = fs ( r, v, t )dr dv 3
3
represents the time-dependent number of particles which at the neighborhood of r have velocities close to v. Since plasma can be considered almost free of neutral particles, the
(
)
governing equation for the evolution of distribution function fs r, v, t , or the so-called Boltzmann’s equation, is only written for ions and electrons. Boltzmann’s equation is
⎛df ⎞ Df ∂f ∂f ∂f = + v⋅ +a⋅ = ⎜⎜ ⎟⎟⎟ = Cˆ [ f ] ⎝ ⎠ Dt dt collision ∂t ∂r ∂v
(IV.1.1)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
where D Dt is total time derivative, and a is the particle acceleration which is given by Lorentz force as
a=
(
q (E + v × B) m
(IV.1.2)
)
and Cˆ ⎡⎢ f ⎤⎥ in (IV.1.1) represent the collision term and collision ⎣ ⎦ operator, respectively. Inserting Coulomb collision in a plasma leads to the Fokker-Plank’s Also df dt
collision
(
equation. On the other hand, in a collisionless plasma the collision term df dt
)
collision
becomes zero and Boltzmann’s equation turns into the Vlasov’s equation
∂f ∂f ∂f + v⋅ +a⋅ =0 ∂t ∂r ∂v
(IV.1.3)
which is valid for high temperatures and low densities.
(
In a fluid description of a plasma motion, the distribution function fs r, v, t
) can be
used to define a number of macroscopic quantities as follows 1. Density of species s
ns (r, t ) ≡
∫ f (r, v, t )d v
(IV.1.4)
Vs ≡ ns−1 ∫ vfs (r, v, t )d 3v
(IV.1.5)
3
s
2. Average velocity of species s
3. Pressure tensor
ps ≡
∫ m f (r, v, t )(v − V )( v − V )d v 3
s s
s
s
(IV.1.6)
4. Trace of pressure tensor, or simply the isotropic pressure
ps ≡
2 1 ms v − Vs fs (r, v, t )d 3v ∫ 3
5. Kinetic temperature of species s
(IV.1.7)
An Overview of Plasma Confinement in Toroidal Systems
Ts ≡
167
ps ns
(IV.1.8)
6. Stress Tensor
Ps ≡
∫ m vvf (r, v, t )d v 3
s
Ps
where the relation between
and
s
ps
(IV.1.9)
is
Ps = ps + ms ns Vs Vs
(IV.1.10)
7. Energy flux of species s
Qs =
1
∫ 2 m v vf (r, v, t )d v 2
s
3
s
(IV.1.11)
8. Heat flux of species s
qs ≡
1
∫ 2m
s
where the relation between
v − Vs
Qs
and
(v − V ) f (r, v, t )d v
2
3
s
qs
Qs = qs + Vs ⋅ ps +
s
(III.1.1)
is
3 1 ps Vs + msnsVs2 Vs 2 2
(IV.1.13)
9. Energy-weighted stress
Rs ≡
1 ms v 2 vvfs (r, v, t )d 3v ∫ 2
(IV.1.14)
1 msv 2 vCˆ ⎡⎣⎢ fs ⎤⎦⎥ d 3v ∫ 2
(IV.1.15)
10. Energy-weighted friction
Gs =
where Cˆ ⎡⎢ f ⎤⎥ is the collision operator.
⎣ ⎦
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani 11. Energy exchange
1 2 ms v − Vc Cˆ [ f ] ∫ 2
(IV.1.16)
Fs =
∫ m vCˆ ⎡⎣⎢ f ⎤⎦⎥ d v
(IV.1.17)
Rsn =
∫ C z ( v )d v
(IV.1.18)
W = 12. Friction force
3
s
13. Collisional friction
where
zn
3
s n
is defined as
z0 = 1
z1 = mv
1 1 z 2 = m ( v ⋅ v) z 3 = m ( v ⋅ v ) v 2 2
(IV.1.19)
IV.1.1. Moments Equations While the microscopic distribution depends on r , v , and t , macroscopic physical parameters such as density or temperature, depend only on r and t , and consequently are obtained by integration over the entire velocity space, which are called as moments. The i-th moment is defined as
Μi (r, t ) =
∫ f (r, v, t ) v d v, i ∈ i
3
+
(IV.1.20)
in which v i = v.v....v denotes the i-fold dyadic product. The zeroth-order moment of (IV.1.1) yields the equation of continuity
∂n + ∇ ⋅ (nv) = 0 ∂t
(IV.1.21)
First- and second-order moments of the Boltzmann equation yield
F=
∂ mnv + ∇ ⋅ P − en (E + v × B) ∂t
(IV.1.22)
An Overview of Plasma Confinement in Toroidal Systems
∂ ⎛3 1 ⎞ ⎜⎜ p + mnv 2 ⎟⎟ + ∇ ⋅ Q = W + v ⋅ (F + neE ) ⎝ ⎠ ∂t 2 2
169
(IV.1.23)
The fourth moment equation is obtain by multiplying Boltzmann equation by v 3 and integrating
∂Q 3 e 1 e e + ∇⋅R − pE − env 2E − E ⋅ P − Q×B = G ∂t 2m 2 m mc (IV.1.24) IV.1.2. Application of Boltzmann Equation Consider a distribution function with x-direction dependence in position and velocity
f (x, vx , t ) . The Boltzmann equation then becomes ⎡ ∂f (x , vx , t )⎤ ∂f (x , vx , t ) ∂x ∂f (x , vx , t ) ⎢ ⎥ = = vx ⎢ ⎥ ∂t ∂t ∂x ∂x ⎣ ⎦ collision
(IV.1.25)
in which vx = ∂x ∂t . But the left-hand-side of (IV.1.25) equals to
⎡ ∂f (x , vx , t )⎤ f (x , vx , t ) − feq (x , vx ) ⎢ ⎥ =− ⎢ ⎥ τ ∂t ⎣ ⎦ collision
(
where feq x , vx
(IV.1.26)
) is the time-independent distribution function in equilibrium and τ
is the
relaxation time. Thus
vx
⎡ f (x, vx , t ) − feq (x , vx )⎤ ∂f (x , vx , t ) ⎥ = −⎢ ⎢ ⎥ ∂x τ ⎣ ⎦
(IV.1.27)
The first order solution to (IV.1.27) is hence
f1 (x, vx ) = feq (x, vx ) − vx τ
∂feq ∂x
(IV.1.28)
Higher order solutions can be obtained by iterating. Hence the second order solution is
f2 (x , vx ) = feq (x , vx ) − vx τ
∂f1 ∂x
= feq − vx τ
∂feq ∂x
2 x
+v τ
2
∂ 2 feq ∂x 2
(IV.1.29)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani The iteration is useful in considering nonlinear effects.
IV.2. Flux-Surface-Average Operator A flux-surface averaged of some quantity such as particle flux and heat flux, is a very useful concept for transport analysis of a toroidal plasma. The flux-surface average of a function is defined by the volume average over an infinitesimally small shell with volume ΔV as,
A = lim
ΔV → 0
1 ΔV
∫ Ad r 3
ΔV
(IV.2.1)
where ΔV lies between two neighboring flux surfaces with volume V and V + ΔV . To be strict, V denotes the volume, while v represents the velocity coordinate in phase space. It is physically more appealing to take average over a flux layer instead of taking average over geometric surface. Labeling flux surface by ψ , leads to:
A =
dψ dV
∫
fdS dψ = dV ∇ψ
∫
fdS ψˆ
(IV.2.2)
Here, V is the volume enclosed by the flux surface. One can rewrite (IV.2.1) in flux coordinate as
f =
1 V′ ∫
gd θd ζ (IV.2.3)
where g is by (II.2.23) equal to the inverse of square of Jacobian. There a number of important properties associated with the flux-surface average operator as 1. The flux-surface average of the divergence of a vector A
1 ΔV →0 ΔV
∇ ⋅ A = lim
where
V ′ = dV d ψ
1 d
∫ A ⋅ dS = V ′ d ψ {V ′ S
.
2. The flux-surface average annihilates the operator B ⋅ ∇
A ⋅ ψˆ
} (IV.2.4)
An Overview of Plasma Confinement in Toroidal Systems
B ⋅ ∇A ≡ 0
171 (IV.2.5)
3. The identity of flux-surface average
∇ψ ⋅ ∇× G ≡ 0
(IV.2.6)
which holds for any vector field G. In order to achieve the flux-surface averaged form of the equation of Continuity (IV.1.19), we apply the flux-surface average operator to obtain
∂n + ∇ ⋅ (nv) = 0 ∂t
(IV.2.7)
or equivalently
∂ d 1 d n =− nv ⋅ ∇V = − nv ⋅ ∇V ∂t dV V ′ dψ
(IV.2.8)
One also can rewrite (IV.2.8) as
∂n 1 + (V ′ nv ψ )′ = 0 ∂t V ′ In which the nv
ψ
(IV.2.9)
is the contravariant component of particle flux in direction of ψ and
its flux-surface average is radial particle flux, usually denoted by Γ . Moreover, prime denotes differentiation with respect to the magnetic poloidal flux. In the next section we will study classical and non-classical transport in axisymmetric toroidal system.
IV.3. Classical and Non-Classical Transport Classical transport refers to those transport fluxes that happen in straight and uniform magnetic field lines. Classical transport of particles is due to Coulomb collisions and one should take into the account the gyrations of particles in the magnetic field. But when the geometry change into torus the dominant diffusive transport is most due to drifts across particle guiding center orbits. In particular, the collision and particle displacements are enhanced because the gyrocenter displacement from the magnetic surface gets larger than the gyroradius itself. This type of transport is faster than classical transport and is called Neoclassical (non-classical) Transport. Therefore, geometrical effects cause to complicate
172
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
particle orbits and drifts in neoclassical model, where they are routinely ignored in the classical model. Banana orbits, potato orbits, and bootstrap current arise from the neoclassical transport model. We first study the classical theory of collisions in cylindrical plasma and next we consider the neoclassical transport.
IV.3.1. Classical Collisional Transport Equations of transport are
∂ns + ∇ ⋅ Γs = Source- Sink ∂t ∂ 3 ( 2 nskTs ) + ∇ ⋅ qs = Source- Sink ∂t ∂v p + ∇ ⋅ Π = Source- Sink ∂t J E + v × B = η J = η⊥ J⊥ + η B B
(IV.3.1)
which become complete along with Maxwell’s equations
∂Bϕ ∂Bz ∂E z 1 ∂ =− = rEϕ ) ( ∂t ∂t ∂r r ∂r ∂B 1 ∂ μ0J z = rBϕ ) μ0J ϕ = − z ( ∂r r ∂r
(IV.3.2)
Here, the subscript s refers to ion or electron species, and Γs , q s and Π are particle flux, heat flux and viscous tensor respectively, defined as
Γs = −D ∇ns + ns Vc 5 q s = −ns χs ∇Ts + Γs kTj + q conv 2
(IV.3.3)
Also J⊥ and J in (III.3.1) are given by:
J⊥ =
J ϕ Bz − J z Bϕ B
2
1 ∂p (Bz ϕˆ − Bϕzˆ) ∂r
(Bz ϕˆ − Bϕzˆ) = B 2
J =
(IV.3.4)
J ϕBϕ − J z Bz B
(IV.3.5)
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173
IV.3.1.1. Random walk model Random Walk Model is the simplest model that can be used to determine transport coefficients, and it is dependent on the mean collision time and the mean free path associated with the random motion of particles. The random motion of a particle is shown in Figure IV.3.1. In this model the diffusion coefficient is simply given by
D = l2 τ
(IV.3.6)
where l and τ are the average step size and average time between collisions, respectively.
IV.3.1.2. Particle diffusion in fluid picture We may take the cross product of Ohm’s law with magnetic field B to yield
E × B + ( v × B ) × B = η J × B = η ⊥ ∇p
(IV.3.7)
which upon simplification takes the form
E × B − v ⊥B 2 = η ⊥ ∇ p
(IV.3.8)
with the perpendicular velocity given by
v⊥ =
Figure IV.3.1. Random walk
( E × B) B2
⎛ η ⎞⎟ − ⎜⎜⎜ ⊥2 ⎟⎟ ∇p ⎜⎝ B ⎠⎟
(IV.3.9)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
The first term on the right-hand-side of (IV.3.9) is E × B drift of particles and the second term is diffusion velocity in direction of ∇ p . Now, letting T to be constant, we get
∇p = T ∇n
(IV.3.10)
Hence, the radial particle flux is derived as
(
)
Γ⊥ = nv⊥ = ηnT B 2 ∇n = D⊥∇n
(IV.3.11)
where D⊥ = ηnT B 2 is the particle diffusion coefficient. When the electric field is applied to the plasma, electrons accelerate to the drift velocity vd . In this situation the force of electric field is balanced by collision force, in this manner we have:
eE = mevd τc
(IV.3.12)
Here, τc is momentum loss time. Hence the scalar resistivity is obtained as
η = E J = me nee 2 τc ≈ me nee 2 τe
(IV.3.13)
with τe being the electron collision time. Substituting (IV.3.13) in (IV.3.6) yields the expression for electron diffusion coefficient (the perpendicular subscript denotes transport across magnetic surfaces) as
D⊥ =
p me B 2 nee 2 τe
(IV.3.14)
IV.3.2. Neoclassical Collisional Transport IV.3.2.1. Trapped particles and banana orbit Since the toroidal field cannot individually confine the plasma of tokamak at equilibrium, a combination of toroidal and poloidal magnetic fields, together with a toroidal current, is necessary to form closed magnetic surfaces. Therefore, the magnetic field lines are helically wound on toroidally nested surfaces and charged particles follow helical field lines. Now, let R be the distance from the major axis in toroidal geometry; then the magnitude of toroidal magnetic field falls off with distance from the major axis of torus R , according to the Solov’ev equilibrium (II.7.25). Therefore, the guiding centers of particles as they follow along the magnetic field feel a change in the strength of the magnetic field. This means that particles moving slowly along the magnetic field are reflected and subsequently, when they attempt to travel across the torus in the reverse direction, they are reflected back again. These
An Overvieew of Plasma Confinement in Toroidal Syystems
175
arre the trapped particles in thhe so-called Banana B orbits. The name off Banana comees from the faact that poloid dal projectionss of trapped paarticle onto coonstant ζ-surfaace are similarr to Banana ass shown in Figgure IV.3.2. On the othher hand, we have passingg particles in contrast to traapped particlees. Passing paarticles are no ot trapped annd thus not reeflected, and follow spiral paths aroundd the torus foollowing the helical h path off the field linees. Hence the particles whoose velocity components c allong the field are low contribute to the population p off trapped prticcles, while parrticles with hiigher velocitiees parallel to the field cyccle around thee torus and inncrese the poppulation of paassing particlees. The condittion for particcles to be trappped in a largee aspect ratio tokamak is obtained o by ussing the conseervation of eneergy and magnnetic moment as
v v2
< 1−
Bmin Bmax
(IV.3.15)
Fiigure IV.3.2. Trrapped particless in Banana orbits and passing particles [12]
Now, sincee according to (II.7.25) we roughly r have B ∼ 1 R wee have:
Bmin Bmax
=
R0 − r R0 + r
≈1+
Thus, requiirement for traapping simplyy becomes
2r R0
(IV.3.16)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
v v2
< 2ε (IV.3.17)
where ε = r R0 is the inverse aspect ratio. Integration of the equation of motion leads us to the Banana width orbit Δb
Δb =
mv 1/2 ρε−1/2 ε = qBθ ι
(IV.3.18)
in which ι is the rotational transform and is given by
ι=
RBθ rBϕ
(IV.3.19)
Similarly the displacement of the guiding centre from the flux surface for passing particles is
Δp=
mv ι q Bϕ
(IV.3.20)
One can illustrate the boundary between trapped and untrapped particles in velocity phase space as shown in Figure IV.3.3.
V⊥
θcritical
θcritical
Figure IV.3.3. Boundary between trapped and untrapped particles
V
An Overview of Plasma Confinement in Toroidal Systems
177
Critical angle θc in Figure IV.3.3 is determined by
θc = cos−1
v v
≈ cos−1 2r R0
(IV.3.21)
For a Maxwellian distribution function, one can then easily obtain the fraction of trapped particles, as
2π f = n
π−θc ∞
∫ ∫ F (v )v M
θc
2
sin θ dv d θ = cos θc =
0
2r = 2ε Ro
(IV.3.22)
IV.3.2.2. Different regimes Diffusion coefficients in neo-classical transport significantly vary in Banana, Plateau and Pfirsch-Schlüter regimes, depending on the strength of collisionality as illustrated in Figure IV.3.4. The dimensionless collisionality ν * in Banana regime is defined as
νe* =
τb τe
=
Roq τevth ε 3/2
(IV.3.23)
D DPS
Plateau Regime
DPlat
Db
ν e*
1
ε − 3/2
Figure IV.3.4. Different transport regimes *
In Banana regime where νe < 1 electrons can complete their Banana orbits many times before colliding; hence, only trapped particles contribute to the transport. Therefore one can use Banana-orbit width Δb as the step size in random walk model and obtain
178
Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
Db =
ftrapΔb 2 τe
2 2 1 q λe 1 ∼ 3/2 ∼ 3/2 q 2Dclassic τe ε ε
(IV.3.24)
Pfirsch-Schlüter transport arises from E × B term for v⊥ in (IV.3.9). When
νe* > ε−3 /2 collisions prevent the particles completing Banana orbits and Pfirsch–Schlüter diffusion reads
(
)
DPS = 1 + αq 2 Dclassic
(IV.3.25)
where α is a numerical factor having the order of unity. The intermediate regime bounded by Banana and Pfirsch–Schlüter regimes is Plateau *
−3/2
regime, for which we have 1 < νe < ε
. In this regime, particles make about one
collision after completing one Banana orbit. One determines the plateau diffusion as
DPlat ∼ qTeλe
(IV.3.26)
IV.3.2.3. Transport matrix The current density, particle, electron and ion heat transport fluxes are functions of driving
gradients (∇n, ∇Ti , ∇Te , ∇Vl )
in
which
the
parallel
electric
field
is
E = −∇Vl , and Vl is the plasma’s electric potential around the torus. The neoclassical transport is described by a transport matrix as below:
⎛D M 12 ⎛Γ ⎞⎟ ⎜⎜ ⎜⎜ ⎟ ⎜⎜ M ⎜q ⎟⎟ n χe ⎜⎜⎜ e ⎟⎟⎟ = − ⎜⎜ 21 ⎜⎜q i ⎟⎟ ⎜⎜⎜ M 31 M 32 ⎜⎜ ⎟⎟ ⎜⎜ ⎜⎜⎝J ⎠⎟⎟ ⎜⎝ bn b τe where ω ∼ ε
M 13 M 23 n χi b τi
ω ⎞⎛ ⎟⎟⎜⎜∇n ⎞⎟⎟ ⎟ ⎟ M 24 ⎟⎟⎜⎜⎜∇Te ⎟⎟ ⎟⎟ ⎜ ⎟⎟ ⎜ T ⎟⎟ M 34 ⎟⎟⎟⎜∇ ⎟⎟⎜⎜ i ⎟⎟⎟ ⎜ σ ⎠⎝ ⎟⎟⎜∇Vl ⎟⎠⎟
(IV.3.27)
1/ 2
n Bθ . The above equation reveals that every type of transport can be driven
by any of the potential gradients. This fact complicates the study of neo-classical transport phenomena in plasmas. This minus sign stresses on the fact that transport opposes gradients. The above can also be written as
{Fj } = − ⎡⎣Oij ⎤⎦ ∇ {Pi }
(IV.3.28)
An Overview of Plasma Confinement in Toroidal Systems
179
in which Fj , Oij , and Pi are respectively transport fluxes, Onsager coefficients, and potential functions. Onsager coefficients are functions of magnetic field B and may be shown to satisfy the symmetry Oij (B) = ±O ji (−B)
(IV.3.29)
Hinton and Hazeltine gave mathematical derivation of neo-classical flux parameters as
(IV.3.28) Here, r superscript denotes the radial contravariant component obtained by inner product with rˆ . Also, v f is the radial flux surface velocity, and N = lnn and T s = lnTs , s = e, i are dimensionless density and species temperature. Furthremore, σ is Spitzer conductivity given by
σ = 1.98 τee 2n me
(IV.3.29)
Hence, (IV.3.28) can be written in a similar form to (IV.3.27) as
(IV.3.30)
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
in which γie = Ti Te . The Onsager symmetry in (IV.3.30) is not apparent since the radial r
velocity v f should also first be expressed in terms of other potential gradients. However, the above form is more preferred in computations where fluxes across magnetic surfaces are required. Typical solution of plasma equilibrium and transport for Damavand tokamak is depicted in Figure IV.3.5.
IV.3.3. Bootstrap Current In the Banana region, radial diffusion induces a current in the toroidal direction known as the Bootstrap current J BS , which is in parallel to the magnetic field. Unlike the Ohmic current J Ohmic , this current does not require any external electric field and occurs naturally due to gradients in plasma profiles of temperature and density. From the fourth equation of (IV.3.30) we have J
r
=−
pe Bp
ε ⎡⎣2.44 (1 + γie ) ∇N − 0.69∇T e + 0.42γie ∇T ⎤⎦ + (1 − 1.95 ε ) σ 0.42γ ie E
= J BS + J Ohmic
Figure IV.3.5. Separatrix plasma configuration in Damavand tokamak
(IV.3.29)
An Overview of Plasma Confinement in Toroidal Systems
181
In fact, there is a fraction ε1/2 of trapped particles having a parallel velocity as
ε1/2vth = ε1/2 kBTe 3me where execute a Banana orbit of width wb = q ρL ε−1/2 . Therefore, when a radial density gradient exists, these particles produce a current analogous to the diamagnetic current of untrapped, which reads as
J trapped ~ −ewb
dn 1/ 2 1/ 2 ε1/ 2 dn ε (ε vth ) ~ −q T dr B dr
(IV.3.31)
There is a momentum transfer from the trapped to passing particles of both ions and electrons, due to this fact that both species produce such a current, which modifies the velocity of the passing particles. The difference in modified velocities of passing particles produces the toroidal bootstrap current J BS . Now, the momentum exchange between passing ions and electrons is meJ BS eτei . The passing electrons are affected by a momentum exchange with the trapped electrons. The trapped electrons are localized to a part ~ ε1/2 of velocity space and the effective collision frequency is ascertained by the time needed to scatter out of this region as τeff ~ ετee . Thus, the momentum exchange rate between trapped and passing electrons is meJ trapped eετee . The bootstrap current originates form balancing the momentum exchange of passing electrons with passing ions and with trapped electrons, approximately given by
J BS
τei J trapped τee ε
−
ε1/ 2 dn T Bθ dr
(IV.3.32)
( )
This is while the precise expression to Ο ε1/2 according to (IV.3.29) is
J BS
ε1/2n ⎡ 1 dn dT dT ⎤ =− + 0.69 e − 0.42 i ⎥ ⎢2.44 (Te + Ti ) Bθ ⎢⎣ n dr dr dr ⎦⎥
(IV.3.33)
which indicates that the bootstrap current fraction of the total current scales as
I BS = c ε1/ 2 βp I
(IV.3.34)
with c being a dimensionless constant about 13 . In the low-aspect-ratio limit ε → 1 , when most particles are trapped, the bootstrap current is however determined by
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Fatemeh Dini, Reza Baghdadi, Reza Amrollahi and Sina Khorasani
J BS ≈ −
1 dp Bθ dr
(IV.3.35)
Here, the bootstrap current is driven entirely by the pressure rather than the density gradient.
IV.3.4. Confinement Times The particle confinement time for ions can be defined as τp ≡
Number of Ions in Plasma
Number of Ions in Plasma
Ion Loss Rate
Ion Production Rate at Equilibrium
(IV.3.36)
where
Number of Ions in Plasma =
∫
ndr 3
Plasma
Ion Loss Rate =
∫
Γ ⋅ dS (IV.3.37)
Surface
If the plasma is at the steady state equilibrium then the production rate equals the loss rate. Then the electron particle confinement time is the same due to quasi-neutrality condition. The energy confinement time for electrons τEe is obtained by τEe ≡
Electron Energy in Plasma
Electron Energy in Plasma
Electron Energy Loss Rate
Electron Heating Rate at Equilibrium
(IV.3.38)
where Electron Energy in Plasma =
3 3 ∫ nTedr 2 Plasma
Electron Energy Loss Rate =
∫
(qe + 2.5Te Γe ) ⋅dS +
Surface
∫
Praddr 3
Plasma
(IV.3.39)
For the whole plasma, the energy confinement time is τE =
where
Plasma Energy
Plasma Energy
Energy Loss Rate
Plasma Heating Rate at Equilibrium
(IV.3.40)
An Overview of Plasma Confinement in Toroidal Systems Plasma Energy =
183
3 3 ∫ n (Te + Ti )dr 2 Plasma
Ion Energy Loss Rate =
∫
(q i + 2.5Ti Γi ) ⋅ dS +
Surface
∫
n n n i σx v i
Volume
Plasma Energy Loss = Electron Energy Loss+ Ion Energy Loss
3 3 Tdr i 2
(IV.3.41)
V. CONCLUSION In summary, physical and technological studies and surveys considering the daily growing need of mankind to inexhaustible and clean energy, directs the researches towards nuclear fusion, where a bright future is seen for the life of the man on the earth. Fusion can be however reached only in extremely hot plasmas, which are normally confined either magnetically by strong magnetic fields, or inertially by powerful radiations of photons or energetic ions. Various plasma confinement technologies have been developed, among which tokamaks as magnetic plasma confinement machines have produced the most successful fusion experiments. At the moment, the only known promising candidate for a nuclear fusion power reactor is tokamak. The detailed theory behind the operation of magnetically confined hot plasmas was discussed in this tutorial, addressing important aspects related to the plasma equilibrium, stability, and transport. Comparing to the nuclear fission reactions, nuclear fusion reactions enjoy an inherent safety, which is due to the fact that in case of any serious instability or runaway plasma disrupts and reactions automatically stop. In contrast, fission reactions would lead to disaster if their control is lost. From this point of view, fusion science and technology is almost entirely declassified and all its documents are openly accessible to all nations. On the other hand, it is necessary that developing countries diversify their energy resources, and assign larger budget volumes and human taskforce to investigate active areas in nuclear fusion. Since the funding needed to realize a full-size thermonuclear fusion machine is normally out of reach of developing countries, appropriate actions and decisions should be taken to minimize the technological and scientific gap between advanced and developing states in the future. Calculations show that fission of the available uranium on earth is sufficient only for the next 300 years, while fusion of naturally abundant deuterium on the earth and oceans, should provide the necessary energy for more than a million years, or so. That is why nuclear fusion is called as the ‘Tomorrow’s Energy’.
ACKNOWLEDGMENT One of the authors (F. Dini) would like to acknowledge insightful discussions with Prof. Vladimir Shafranov at the Russian Research Center Kurchatov Institute, Moscow, Prof. Weston Stacey and Dr. John Mandrekas at Georgia Institute of Technology, Atlanta, and Prof. Thomas Dolan at University of Illinois at Urbana-Champaign. The authors are indebted to Mr. Mehdi Baghdadi for proofreading the manuscript and illustration of diagrams. They
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also wish to thank fruitful discussions with students, including Miss Shiva Shahshenas, Mr. Mohsen Mardani and Mr. Ahmad Abrishami. This work grew out of the lecture notes of an advanced graduate course on Magnetic Confinement Fusion offered at Amirkabir University of Technology by S. Khorasani.
REFERENCES Abramowitz, M; Stegun, IA. Handbook of Mathematical Functions, Dover, 1965. Amrollahi, R; Khorasani, S; Dini, F. J. Plasma Fusion Res. SERIES 2000, vol. 3, 161. Arfken, G. Mathematical Methods for Physicists, 3rd ed., Academic Press: Orlando, 1985. Bateman, G. MHD Instabilities, 1978, MIT Press: Cambridge, MA, 1978. Berkowitz, J; Grad, H; Rubin, H. in Proc. 2nd United Nations Conf. Peaceful Use At. Energy, 1958, vol. 31, 177-189. Bethe, HA. Phys. Today, May 1979, 44-51. Boozer, A. Phys. Plasmas, 1998, vol. 5, 1647-1655. Borisenko, AI; Tarapov, IE. Vector and Tensor Analysis with Application; Dover Publication: New York, 1979. Braginski, SI; in Reviews of Plasma Phys; Leontovich, M. A; ed; Consultants Bureau: New York, 1965, vol. 1, 205-311. Chen, FF. Introduction to Plasma Physics; 2nd Ed., Plenum Press: London, 1984. Cheng, DK. Field and Wave Electromagnetics; 2nd ed., Addision-Wesley; Reading,1989. Connor, JW; Hastie, RJ; Taylor, JB. Phys. Rev. Lett, 1978, vol. 40, 396-399. Dini, F; Khorasani, S. J. Fusion Energy, 2009, vol. 28, 282-289. Dini, F; Khorasani, S. J. Fusion Energy, 2009, vol. 28, 282-289. Dini, F; Khorasani, S. J. Nucl. Sci. Technol., 2009, no. 48, 1-12. Dini, F; Khorasani, S. Proc. Int. Multi-conf. Role of Isfahan Develop. Islamic Sci., 2006, 224. Dini, F; Khorasani, S; Amrollahi, R. Iranian J. Sci. Technol. A, 2003, vol. 28, 197-204. Dini, F; Khorasani, S; Amrollahi, R. Proc. 6th Nat. Energy Cong., 2007. Dini, F; Khorasani, S; Amrollahi, R. Scientia Iranica, 2003, 419-425. Dinklage, A; Klinger, T; Marx, G; Schweikhard, L. Lecture Notes in Physic, Plasma Physics Confinement, Transport and Collective Effectss; Springer: Berlin, 2005. Dolan, T. J. Fusion Research; Pergamon Press, 1982. Dolan, TJ. Fusion Research, Pergamon Press, 1982. Erdélyi, A. Tables of Integral Transforms, McGraw-Hill: New York, 1954. Fälthammar, CG; Dessler, AJ. Eos, Trans. Am. Geophys. Union, 1995, vol. 76, 385-387. Freidberg, JP. Ideal Magnetohydrodynamics, Clarendon Press: Oxford, 1987. Freidberg, JP. Plasma Physics and Fusion Energy; Cambridge University Press: New York, 2007. Freidberg, JP. Plasma Physics and Fusion Energy; Cambridge University Press: New York, 2007. Garnier, DT; Kesner, J; Mauel, ME. Phys. Plasmas, 1999, vol. 6, 3431. Grad, H; Rubin, H. in Proc. 2nd United Nations Conf. Peaceful Use At. Energy, 1958, vol. 31, 190. Guo, Y. Commun. Pure Appl. Math, 1997, vol. 50, 891-933.
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Hazeltine, RD; Meiss, JD. Plasma Confinement, Addison-Wesley: Redwood City, 1992. Hazeltine, RD; Meiss, JD. Plasma Confinement; Addison-Wesley: Redwood City, 1992. Hinton, FL; Hazeltine, RD. Rev. Mod. Phys., 1976, vol. 48, 239-308. Hinton, FL; Hazeltine, RD. Rev. Mod. Phys., 1976, vol. 48, 239-308. http://iter.rma.ac.be/en/physics/tokamak/index.php http://www.iaea.org/ http://www.iter.org/ http://www.pppl.gov/ http://www.worldenergy.org/ Hunter, DB. Numerical Algorithms, 1995, vol. 10, 41. Jackson, JD. Classical Electrodynamics; Wiley; New York, 1962. Khorasani, S; Rashidian, B. Scientia Iranica, 2002, vol. 9, 404-408. Kissick, MW; Leboeuf, JN; Cowley, SC; Dawson, JM. Phys. Plasmas, 2001, vol. 8, 174-179. Kittel, C. Introduction to Solid State Physics, 7th ed., Wiley, 1996. Kruskal, MD; Oberman, R; in Proc. 2nd United Nations Conf. Peaceful Use At. Energy, 1958, vol. 31, 137-143. Lorenzini, R; Martines, E; Piovesan, P; Terranova, D; Zanca, P; Zuin, M; Alfier, A; Bonfiglio, D; Bonomo, F; Canton, A; Cappello, S; Carraro, L; Cavazzana, R; Escande, DF; Fassina, A; Franz, P; Gobbin, M; Innocente, P; Marrelli, L; Pasqualotto, R; Puiatti, ME; Spolaore, M; Valisa, M; Vianello, N; Martin P. RFX-mod team and collaborators Nature Phys., 2009, vol. 5, 570-574. Lüst, R; Schlüter, AZ. Naturforsch, 1958, vol. 12a, 850Z. Miyamoto, K. Introduction to Plasma Physics; 2nd Ed., Springer-Verlag: Berlin, 2005. Miyamoto, K. Plasma Physics and Controlled Nuclear Fusion; Springer-Verlag: Berlin, 2005. Reusch, WR; Neilson, GH. J. Comp. Phys., 1986, vol. 64, 416. Schultz, JH; Kesner, J; Minervini, JV; Radovinsky, A; Pourrahimi, S; Smith, B; Thomas, P; Wang, PW; Zhukovsky, A; Myatt, RL; Kochan, S; Mauel, M; Garnier, D. IEEE Trans. Appl. Supercond, 1999, vol. 9, 378-381. Shafranov, VD. Reviews of Plasma Physics; Leontovich, M. A; ed., Consultants Bureau, New York, 1966, vol. 2, 103. Sigmar, DJ; Helander, P. Collisional Transport in Magnetized Plasmas; Cambridge University Press: Cambridge, 2002. Smythe, WR. Static and Dynamic Electricity; 3rd ed., McGraw-Hill: New York, 1989. Solov’ev, LS. Sov. Physics JETP 1968, vol. 26, 400. Solov'ev, LS. in Reviews of Plasma Physics; Leontovich, M. A; ed., Consultants Bureau: New York, 1975, vol. 6, 239. Stacey, WM. Fusion Plasma Physics, Wiley-VCH: Weinheim, 2005. Stacey, WM. Fusion Plasma Physics, Wiley-VCH: Weinheim, 2005. Wesson, J. Tokamaks, Clarendon Press: Oxford, 2003. Wesson, J. Tokamaks; Clarendon: Oxford, 2003. Zakharov, LE; Shafranov, VD. in Reviews of Plasma Physics; MA. Leontovich, ed., Consultants Bureau: New York, 1986, vol. 11, 153. Zueva, NM; Solov’ev, LS. Atomnya Energia, 1968, vol. 24, 453.
In: Horizons in World Physics. Volume 271 Editor: Albert Reimer
ISBN: 978-1-61761-884-0 © 2011 Nova Science Publishers, Inc.
Chapter 3
COSMIC RAYS AND SAFETY Neïla Zarrouk* and Raouf Bennaceur Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis, Tunisia
ABSTRACT Aircraft crew and frequent flyers are exposed to high levels of ionizing radiation principally from cosmic radiations of galactic and solar origin and from secondary radiation produced in the atmosphere. The need to assess the dose received by aircrew and frequent flyers has arisen following Recommendations of the International Commission on Radiological Protection in publication 60 ICRP 60. In 1996 the European Union introduced a revised Basic Safety Standards Directive that included exposure to natural sources of ionising radiations, including cosmic radiation as occupational exposure. Several equipments were used for both neutron and non neutron components of the onboard radiation field produced by cosmic rays. Such a field is very complex, therefore dose measurement is complex and the use of appropriate computer programs for dose calculation is essential. The experimental results were often confronted with calculations using transport codes. A reasonable agreement of measured and calculated data was observed. Particular attention was devoted to the results obtained during some extreme situations: intense solar flare and “forbush decreases” Our results concerning effective doses received by Tunisian flights, computed with CARI-6, EPCARD 3.2, PCAIRE, and SIEVERT codes, show a mean effective dose rate ranging between 3 and 4 mSv/h. However majority of codes stay unpredictable, thus we have used the Neural Network system NNT associated with CARI code to predict values of effective doses and heliocentric potentials (Hp) which we have obtained at least for some months ahead. According original, Morlet reconstructed and extrapolated Hp variations functioning as a measure of solar activity we have showed 8 to 13 years cycles. The first next maximum of Hp≈1400MV is located around 2022-2024. The minima of Hp corresponding to *
Corresponding author: E-mail adress:
[email protected]
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Neïla Zarrouk and Raouf Bennaceur highest fluxes of cosmic rays are located around 2015 and 2035 years. Two classes of big periods are also found around 20-33 years and 75 years. Especially Morlet monthly analysis showed two main periods of 6 and 12 months, long periods of 5-6.25 and 11 years. Short structures are also detected Since the Earth is permanently bombarded with energetic cosmic rays particles, cosmic ray flux has been monitored by ground based neutron monitors for decades. Thus we give our investigations about decomposition provided by Morlet wavelets technique, using data series of cosmic rays variabilities. The wavelet analysis constitute an input data base for NNT system with which we can then predict decomposition coefficients and all related parameters for other points on the earth, we have studied the Mediterranean case in which we don't have any information about cosmic rays intensities. NNT associated with wavelets seem to be very suitable, we have now a kind of virtual NM for these locations on the earth.
INTRODUCTION The growing apprehension concerning the radiation safety of crew members had as consequence that a number of studies on cosmic radiation exposure of local airlines crews and on the development of appropriate dosimetry methods, have been undertaken in many countries in recent years [1-4] In this chapter we present and compare at first step different results of other works concerning dose measurements and calculations due to cosmic rays on board aircrafts. In another part we show our calculations results for effective doses of cosmic rays received by Tunisian flights, computed with CARI-6, EPCARD 3.2, PCAIRE, and SIEVERT codes. The calculations were performed on mostly regular passenger flights of the Nouvelair Tunisian Company. The heliocentric potential is introduced to account for the cosmic ray modulation induced by solar activity. An approach seems to be more effective is to use wavelets associated to Neural Network. Thus we have proceeded in a following step to Morlet decomposition and reconstruction in order to analyze and reveal cycles and structures hidden in the time dependence of heliocentric potentials. These results specially the Morlet extrapolations are discussed and compared with the Neural network predictions of monthly heliocentric potentials where the training samples were the last heliocentric potentials used in the previous sequence. During high solar activity , emissions of matter and electromagnetic fields from the sun namely solar wind increase making it difficult for GCRs to penetrate the inner solar system and then reach the Earth. The GCR intensity is low when the solar activity is high and viceversa constituting an approximately 11 year periodicity [5]. Articulating the significance of neutron monitors to the field of radiation safety. We have used in a previous work the real time data series of cosmic rays variabilities of Moscow neutron monitor, we present in this chapter our improved Morlet wavelets analysis of these data series for cosmic rays variabilities relative to different stations [6-12] from the network of neutrons monitors. Thus knowing the characteristics of the different NM stations we have built the training inputs base of neural network system. We have then obtained the Morlet decomposition coefficients and reconstructions of cosmic rays variabilities curves in the Mediterranean case.
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1. MEASUREMENTS AND CALCULATIONS OF COSMIC RAYS EFFECTIVE DOSES ON BOARD AIRCRAFTS 1.1. Brief Description of Research Performed: Measurements and Calculations from Other Works 1.1.1. Cosmic particles and electromagnetic field In the atmosphere and at ground level, the flux of cosmic ray particles is mostly due to galactic protons incident on the atmosphere. At typical cruising altitudes, aircrews are exposed to higher levels of radiation from galactic cosmic rays than persons on the ground receive from natural background radiation. Primary cosmic rays (85% protons, 12% alpha particles, 1% heavy nuclei ranging from carbon to iron, and 2% electrons and positrons) arrive at the heliosphere isotropically. Their sources are thought to include supernovae, pulsar acceleration, and explosion of galactic nuclei. [30-32].These particles can have energies in excess of 1020 eV. The penetrating ability of an ionised particle is directly affected by the magnetic rigidity, which is the ratio of momentum to charge.[33] The particle rigidity is influenced in an anticoincident manner with the solar cycle due to the changing solar modulation; the galactic radiation contribution reaches a maximum during solar minimum conditions. Cosmic ray doses and doses rates in the atmosphere are also affected by geomagnetic shielding. Lorentz force deflects charged particles moving through the earth’s magnetic field. Particles that enter near the poles experience little deflection while those entering near the equator approach at right angles so that they are deflected more strongly. The specific rigidity required to enter the atmosphere at a particular point in a particular direction is referred to as the geomagnetic cutoff rigidity. Cut-off rigidities are lowest at high latitudes and highest at low latitudes. Particles that penetrate the upper layers of the atmosphere interact with atmosphere nuclei resulting in cascades of secondary particles including neutrons, pions, muons and gamma radiations. These processes lead to a variation of dose with the altitude, longitude, and latitude of the aircraft. At typical commercial flight altitudes (33000-43000 ft) most of the radiation dose originates from the secondary particles. 1.1.2. Choice of materials and methods For on-board dosimetry is still under debate. Indeed while a broad range of measurements and calculations of air-crew radiation exposure exist, it is difficult to apply them in general, as various national airlines perform their regular flights along specific routes, at various destinations, frequency patterns, different altitude profiles, etc. Measurements on board aircraft have been performed with many different types of instruments. Some are electronic instruments measuring the dose continuously during a flight either as dose rate (dose per hour, µSv /h) or as the dose for the complete route (route dose, µSv). The result is basically available immediately after the flight. Among these detectors : the Tissue Equivalent Proportional Counter (TEPCs), considered as a reference instrument for air-crew dosimetry, being sensitive to environmental conditions (such as vibrations, noise, change of pressure, etc.) and requiring specialised service and maintenance, is not applicable as a routine monitoring instrument, ionisation chambers, neutron monitors, Geiger-Muller-(GM)-counters or detectors based on semiconductor
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Neïla Zarrouk and Raouf Bennaceur
techniques. Such detectors are detecting the electric charge that ionising radiation creates, when passing a material. The electric current or electric charge generated in many of those detectors is extremely small and the detectors themselves are often fragile. The equipment has then to be handled with great care. Another feature is that they need power supply (either a battery or a connection to the power line onboard the aircraft). As such installations have to follow certain regulations or routines special permissions are usually requested. Other detectors are passive in the sense that they store the dose a particle deposit when passing the detector. Here the radiation produces a reversible or non-reversible effect in the detector. The result is evaluated after the flight with special equipment. Such detectors are without electronic components and are rugged and usually quite small. For that reason they are very easy to use on board. However the sensitivity is usually low and to improve it with counting statistics, several detectors are often stacked together and /or could be flown several times before being evaluated. Examples of such detectors are thermoluminescence detectors (TLDs), bubble detectors and track etched detectors (a common material is PADC). Detectors based on neutron-induced fissions in Bismuth and Gold have been developed ad hoc for cosmic ray dosimetry, which make it possible to measure the component of high energy neutrons selectively. Some detectors are sensitive to only a part of the radiation qualities present onboard aircraft and several different detectors are then needed. All instruments need to be calibrated carefully and traceability to international dose standard needs to be established. According to the detectable component of radiation, the instruments have been divided into those designed to measure the non- neutron (some authors use “ionising”) and the neutron components of the cosmic radiation. The non neutron component approximately corresponds to the low-LET component (<10 keV µm-1) and the neutron component accounts the neutron and the nuclear interaction of the high- energy proton component of the field.
1.1.3. Transport code calculations comparison There are a number of methods in current use to compute dose to aircraft crew , for example CARI , EPCARD, FREE, PC-AIRE, SIEVERT (using the data from CARI , and since January 2004, EPCARD), and the algorithm of Peliccioni. The programs use the results of radiation transport calculations together with details of an aircraft’s flight path to calculate the route dose, which can be used for the dose assessment for a crew member. None of the programs take the influence of the aircraft itself into account. EPCARD 3.2 [13] was developed during EC projects, and was kindly made available from Dr Hans Schraube from the GSF-National research centre for Environmental and Health, Institute of Radiation Protection (Neuherberg, Germany ) , EPCARD allows to calculate the effective dose E or the ambient doses equivalent H*(10) and to determine the contribution of the different field components . PCAIRE provides the total ambient equivalent or the effective dose and CARI the total effective dose The PCAIRE model is effectively an encapsulation of the experimental H*(10) measurements made with the TEPC. The CARI–6 model was developed in USA and largely used by different groups in several countries cannot output in ambient doses equivalent units. However, all codes can provide an output of the route dose in effective dose (E) units, which is the regulatory quantity of interest.
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These codes [14] calculate the expected effective dose rate as a function of altitude, geomagnetic latitude and longitude or cut-off, and phase of solar cycle, combined with flight profiles. As reported in the paper of Lewis. B.J. et al.[15] , when a comparison between the three codes has been performed for a complete database of 62 flights details in the reference of this work, both PCAIRE and EPCARD give comparable results in terms of the route dose in units of ambient dose equivalent H*(10). The CARI-6 model cannot output in this dose units. However , all codes can provide an output of the route dose in effective dose units (E), when the E/H*(10) ratio of LUIN is chosen for the effective dose calculation of PCAIRE, both PCAIRE and CARI-6 are in good agreement. However when the alternative FLUKA option is chosen for PCAIRE model, the agreement is better between PCAIRE and EPCARD. FLUKA uses an older cut-off rigidity model for the IGRF (compared to the 1995 model used by PCAIRE and CARI-6/LUIN). In the same way in the work of Saez.J.C et al.[16], the comparison of effective doses estimates reported by EPCARD 3.2 and CARI-6 shows an excellent agreement (1.00±0.11). However a clear influence of the flight operation area can be distinguished on these results. While the ratio EPCARD/CARI is clearly lower than unity for southern destination (a mean value of 0.9 for the flights from/to South America), Northern destinations show that the ratio is greater than unity, reaching up to 1.15 for the flights from/to North America. The reasons for this discrepancy are probably due to the basic differences between the two codes, in particular the models to consider the geomagnetic shield and the solar activity influence In general all codes are consistent with one another; they typically deviate by less than 20% at subsonic altitudes, which is sufficient for radiation protection purposes and regulatory applications. However as previously discussed, further validation is required over the solar cycle.
1.1.4. Calculation models and measurement comparison As applied and analysed in the work of van Dijk.J.W.E [17], the terminology and recommendations of the guide to the Expression of uncertainty in measurement (GUM) have been followed also in many works. In this guide the evaluation of uncertainties is classified into type A and type B. the type A evaluation is of the standard uncertainty is based on statistical techniques that give a standard deviation. Such type of uncertainty is then given the same numerical value as the standard deviation. The GUM states “scientific judgment based on all of the available information on the possible variability “and it is in this manner that the type B of standard uncertainty is evaluated. Indeed the term “scientific judgment” is well appropriate considering the evaluation of uncertainty due to differences between planned and actual flight and due to the dose calculation model. Assuming that these sources of uncertainty are independent, the resulting combined uncertainty is calculated using the following equation: u C =
u A2 + u B2 , f + u B2 , m
The differences distribution between flight plan and actual flights suggests a triangular distribution from which a standard uncertainty of 6% can be estimated. However, this property of the distribution is not used because there seems to be a systematic difference between continental and intercontinental flights. The estimate of the uncertainty resulting from the computer model is based on the range of values found with several models and is about 10%. A number of other sources of uncertainty could be named such as the shielding by
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and backscatter of the aircraft hull, equipment, cargo, and passengers. Although this is not substantiated by data it is believed that these contributions are small compared with the uncertainty due to the models As reported in DOSMAX consortium[18], the agreement between dose data measured by a wide range of instruments at aircraft altitudes and dose values predicted by the computer code EPCARD 3.2 for two different stages of the solar cycle and also for other stages as reported in [EUR04] [19], is quite good. The results are for a specific flight (11277m), there are several reasons why this is not always the case. However the agreement between measured and calculated dose rate results is typically within 25% for both stages of solar activity [19] Indeed different experimental use different calibration methods, then they have generally different procedures for the measurements and analysis, which already may lead to small systematic differences in the results. The statistical uncertainty varies typically between 5% and 15% in a one –hour measurement depending on instrument type. The measurements are also affected by the date through both the 11 –year and daily variations of solar activity. Such influences parameters are not taken into account by the programs for route dose or dose rate calculations. These variations may influence the dose rate significantly (at least 10%) and is another analysis for differences, both between measurements and calculation on a hand and between different measurement results on another hand. The codes for doses calculation are all based generally on different models which have limitations. The uncertainty in these programs is difficult to evaluate. Some of the uncertainties can be reduced if several instruments are measuring on the same flight. As an example the route dose observed by the four different proportional counters agreed within ±25%. The mean value for the round trip agreed with the calculated value within ±25% for both stages of solar activity. Also reported by Spurny. F [25], when using an equipment of active detectors (bubble detectors, superheated detectors...), the ratio of experimental values of H*(10) and CARI calculated E-values was in average equal to-(9±6) %. The values of E should be in the fields on board higher than the values of H*(10), about 20% in the case of “northern routes”, a little less for the routes close to the equator. CARI underestimated a little the actual exposure level in the value of effective dose. Based on the current requirements on accuracy for mixed fields [20], as described in the work of Saez.J.C et al.[16], a performance ratio was defined for each flight as the “computed value from the code” divided by “measured value with a reference instrument “. Acceptable values for this ratio lie in the range 0.77-1.30 for a 95% confidence level. As expected , the ratio in terms of H*(10) of ion chamber and neutron monitor is about 10% lower than the ratio to the TEPC results due to the double account of some protons . For this reason, the TEPC was selected as the reference instrument for integrated route doses. Using the discussed ratio E/H*(10) obtained with EPCARD 3.2 code, the performance ratio in terms of effective dose for CARI is found dependent with the latitude. As reported by Bottollier. J.F et al.[22] and considering the average on the estimations of the total ambient dose equivalent given by PCAIRE and EPCARD, calculated and TEPC results are found to be consistent also. looking to GM devices results , one can see that readings from GM-based overestimated the low LET component average values. This could be due to a non negligible GM response to high energy neutrons. for passive detectors ,
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estimation of both neutron and non-neutron component in term of ambient dose equivalent respectively performed with etched track detectors or bubble detectors and TLDs are found to be very close to average value of ambient dose equivalent whether calculated or measured. As described in the work of Bilski. P et al.[21], comparison of dose values measured with TLD dosimeters with those calculated using CARI code show an agreement typically better than 15%. Results of calculations performed with the CARI-6 code and measurements performed with TL and track detectors are consistent. The proposed dosimetry package with TLDs for measuring the ionising component and CR-39 for measuring the neutron component is suitable for environmental dosimetry on board of commercial aircraft. The aim of such a comparison is not to verify the correctness of the calculation methods (because this would require active systems of dosimetry, such as TEPC or recombination chambers), but rather to identify cases where, over some monitoring period, for one or more aircraft, any significant discrepancy is observed between measurement and calculation. An example of reason for such a discrepancy could be the occurrence, during flight of a large solar event which cannot be accounted for by CARI-6 or EPCARD calculations. Concerning the Mobile Dosimetry Model MDU, Spurny. F et al.[23] has reported in his paper that when using the MDU, the data sets agree very well, only the values of effective dose are, when EPCARD code is used, a little in average by 12%, higher. Such difference is from the point of view of radiation protection largely acceptable [24]. Again the values of total apparent H*(10) are in rather good agreement with the E-values calculated by means of CARI code. The average relative uncertainties of the differences between calculated and treated measured data did not exceeded 10%. Total exposure for GLE 60 about 44% higher, for forbush decrease 16% lower. When analysing results in greater detail, the data calculated by EPCARD are for flights to equatorial regions lower than MDU’s and CARI. The relatively more important decrease in exposure level with increasing geomagnetic cut-off is observed for EPCARD 3.2 data when compared to those calculated by CARI-6. While MDU data are closer to CARI-6 results, other experimental methods give the influence of geomagnetic cutoff closer to EPCARD 3.2 data
1.2. Estimates of Cosmic Radiation Exposure on Tunisian Passenger Aircraft 1.2.1. Numerical codes The LUIN code is a deterministic treatment based on an analytical 2 component solution (longitudinal and transverse components) of the Boltzmann transport equation that uses the Garcia-Munoz and Peters equations for the primary nucleon fluence rate at the top of the atmosphere as a boundary condition.[42] Dose contributions from vertical and nonvertical geomagnetic cut-offs rigidities are used. The LUIN code is the basis of the CARI aircrew exposure code, [34, 35] which was developed by the US Federal Aviation Administration. CARI calculates the effective dose of galactic cosmic radiation received during a flight. The CARI program [26] requires: time and altitude profile of the flight, geographic locations of starting and destination airports, and solar activity during the flight. The most commonly used version of the CARI code assumes a geodesic route between airports. The CARI-6M version calculates dose according to a user-entered route plan consisting of geographic coordinates
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and altitudes. The model allows calculations of the effective dose rate at specific locations as well as the determination of the effective dose for an individual during a flight. The heliocentric potential, functioning as a measure of solar activity, is needed for the calculations and is available monthly. In CARI-6, the heliocentric potential model of O’Brien has been employed, which is characterised by a heliocentric potential U (in MV) that is tabulated by the FAA from daily ground level neutron monitoring. [26, 48, 49]. Both CARI-6 and CARI6M use ICRP-60 recommended radiation weighting factors. CARI-6P, which is not widely distributed, allows the use of either the NCRP ( wR =2 for protons) or ICRP ( wR = 5 for protons) weighting factors. The Elsevier description of ICRP 92 states, “Thus, while the report [ICRP 92] suggests some future modifications, the wR values given in the 1990 recommendations are still valid at this time. The report provides a scientific background and suggests how the ICRP might proceed with the derivation of wR values ahead of its forthcoming recommendations.” [47] In this study, we used the monthly average of the solar modulation parameter although some reservations have been expressed by P. Wollenberg about the validity of monthly averages.[50] K. O’Brien states: “With respect to estimates of career doses, the use of monthly heliocentric potentials or daily values should be unimportant over a period of several years.” [41] Also, it has been reported that the heliocentric potentials used by the FAA as input to CARI are in error, by up to 40% (K. O’Brien, E. Felsberger, P. Kindl, Cosmic-ray Propagation and the Calculation of Cosmic-ray Doses to Air Crew, Human Performance Committee of the International Federation of Airline Pilots Associations, Singapore, July 11, 2006). Monte Carlo analysis with the FLUKA code has led to the aircrew exposure computer program EPCARD 3.2. [28,36] Only the vertical geomagnetic cut-off rigidities are used. EPCARD is derived from FLUKA computations that employ the primary spectra of Badhwar, where the solar modulation of these spectra is determined by a diffusion-convection model developed by the National Aeronautics and Space Administration (NASA), Johnson Space Center. [43, 44] It provides the effective dose or the ambient dose equivalent and the contribution of the different field components. Its final version is described in a GSF Report. [36] EPCARD uses ICRP-60 radiation weighting factors.[52] The PCAIRE code is an experimentally based treatment that enables the interpolation of the dose rate for any global position (vertical cut-off rigidity), altitude (atmospheric depth) and date (solar modulation) based on an extensive set of TEPC measurements [27,45]. For PCAIRE, a great circle route is produced between the 2 airports, and the latitude and longitude are calculated for every minute of the flight. The vertical cut-off rigidity is calculated from either a 3-epoch average or interpolated from IGRF-1995 tabulated data for the given geographical coordinates along the flight path. This semi-empirical model, which is expected to agree with measurements, provides the total ambient dose equivalent or the effective dose.[37] PCAIRE uses ICRP-60 radiation weighting factors.[51] The SIEVERT system (Système d’information et d’Evaluation par Vol de l’Exposition au Rayonnement cosmique dans les Transports aériens)[29] has been developed on behalf of the French Aviation Administration (DGAC). The flight plan of each flight is sent by the companies to the server, operated on behalf of DGAC, Institute for Radioprotection and Nuclear Safety (IRSN), French Polar Institute (IPEV) and Paris Observatory. The server returns to the companies the effective radiation dose for each flight computed using a 3-D
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world map of effective dose rates. The SIEVERT program relies on CARI or EPCARD for the basic input data and is therefore not an independent program. [36] For the SIEVERT application the heliocentric potential is obtained from measurements of French neutron monitors located at Port-aux-Français (Kergulen Islands in the Indian Ocean) and at Dumon d’Urville (Terre Adelie in Antarctica). Both are locations of low geomagnetic cut-off rigidity: 1,1 GV for Kergulen and 0,0 GV for Terre Adelie. Heliocentric potentials are calculated from a quadratic fit of the IPEV neutron monitor count rates to past heliocentric potential values (from 1964 to 1997) given by the authors of the CARI program. [46] Data from the 2 stations are received on a daily basis via satellite links. SIEVERT probably uses ICRP-60 radiation weighting factors. [52, 53] No program has yet adopted ICRP-92 recommendations. Using CARI, EPCARD, PCAIRE, and SIEVERT codes, we calculated the effective dose from galactic radiation.
1.2.2. Passengers, aircrafts movements and flight data The collection of information was an important part of the present work, particularly statistics on aircraft movements, numbers of passengers travelling from or to Tunisia; flight profile data and airport location data. It is difficult to obtain straightforward information on flight profiles. Instead of cruising at one altitude only, pilots make use of different flying altitudes. The airplanes flying east make use of different flight levels than those flying west. For short distances, pilots will more often fly at lower altitudes than for intercontinental flights. Furthermore, every type of aircraft has its optimum flying altitude, depending on weight and fuel consumption, which again depends on distance flown. Flight profiles may also be affected by weather conditions. The collective dose received on a flight to or from a location on earth, can be estimated when we know the number of passengers travelling on the flights. For the year 2006 the total number of passengers was 10130000, the total number of flights in the same year was 93107. Monastir airport is the second in Tunisia for traffic, with 4133768 passengers in 2006. Airport location data is needed for a proper calculation of the effective dose using CARI, EPCARD, PCAIRE, or SIEVERT codes. These data collected from various sources [38, 39] include airport code (using IATA or ICAO coding),[40] geographical coordinates and elevation above sea level. 1.2.3. Results and discussion Dependence of effective dose rate on altitude above Monastir airport in Tunisia There are 7 airports in Tunisia. This work was performed for Monastir airport located at 35°45’ North, 10°45’ East. CARI and EPCARD models were used to compute variation of effective dose rates at Monastir airport versus altitude. As shown in Figure 1, effective dose rate initially increases with depth in atmosphere reaching a maximum at ~ 60000 ft, the Pfotzer maximum (20 km or 50 g cm-2), then slowly drops off to sea level. This is clearly seen in the results obtained by EPCARD code. CARI-6 is limited to altitudes up to 60000 ft. The contribution to dose from each particle type depends on altitude, cut-off, and phase of the solar cycle.
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Figure 1. Variation of effective dose rate with altitude calculated with CARI-6 and EPCARD codes, for coordinates of Monastir airport
Dose rates calculated by EPCARD are lower than those computed by CARI for altitudes greater than 30000 ft. The maximum effective dose rate was about 6,2 µSv/h.
Effective doses received on Tunisian flights The 4 programs were used to calculate route doses for 13 different flights [38] during the period from 23 June to 20 July 2007. In table 1 we summarize the effective dose rates for different destinations from or to Tunisia (Monastir or Djerba airport). Because most of the destinations are in Europe (within a few flying hours), climbing and landing are relatively important since they both take roughly 1 hour. The CARI program takes climbing and descending into account in estimating flight dose. For flights of much longer duration, the flying time on cruising altitude is the most important factor in determining the overall dose. Therefore, average dose rates on short distances will be highly affected by climbing and landing. One can see that the values given by PCAIRE are the highest. For PCAIRE and SIEVERT the highest effective dose was received on the 20 July 2007 flight from Dublin to Monastir; the effective dose was 15 µSv for PCAIRE and 13,4 µSv for SIEVERT. For CARI and EPCARD the highest effective dose was received on the 23 June 2007 flight from Bristol to Monastir; the effective dose was 12,3 µSv by CARI and 15 µSv by EPCARD. Generally, data calculated by EPCARD are in a good agreement with those of CARI. However an influence of the flight operation area can be seen on these results, they are often greater than those calculated by CARI where the ratio EPCARD / CARI effective dose range between 0,98 to 1,21. This difference is probably due to the basic differences between the 2 codes, in particular the models used to account for the geomagnetic shielding and the sun’s activity influence.
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Table 1. Effective doses received on different Tunisian flights computed by: CARI, EPCARD, PCAIRE and SIEVERT
Date, time of the flight and aircraft 20.7.2007 :03h14 A320-212 23.6.2007 :02h44 A320-211 19.7.2007 :02h39 A320-212 10.7.2007 :02h12 A320-212 12.7.2007 :02h39 A320-211 14.7.2007 :02h29 A320-211 26.6.2007 :02h17 A320-211 18.7.2007: 02h30 A320-212 27.6.2007:22mn A321-211 06.7.2007:02h06 A320-212 01.7.2007 :02h09 A321-211 29.6.2007 :02h14 A320-212 28.6.2007 :02h13 A320-211
Destination DUBLIN, IRELAND-EIDW to MONASTIR, TUNISIADTMB BRISTOL, JOHNSON/ KINGSPORT- TN-TRI to MONASTIR,TUNISIA-DTMB WARSZAWA, WARSAW/ POLAND -WAW to MONASTIR,TUNISIA-DTMB MONASTIR, TUNISIADTMB to KOSICE, SLOVAKIA-KSC KATOWICE, POLAND-EPKT to MONASTIR, TUNISIADTMB LIEGE, BELGIUM-EBLG to MONASTIR, TUNISIADTMB BUDAPES, HUNGARYLHBP to MONASTIR, TUNISIA-DTMB BRUSSELS, BELGIUMEBBR to MONASTIR,TUNISIA-DTMB DJERBA, TUNISIA-DTTJ to MONASTIR, TUNISIADTMB VIENNA, AUSTRIA-VIE to MONASTIR, TUNISIADTMB BORDEAUX, FRANCELFBD to DJERBA, MELLITA/TUNISIA-DTTJ PARIS, FRANCE-LFPG to MONASTIR, TUNISIADTMB KATOWICE, POLAND-KTW to MONASTIR, TUNISIADTMB
Effective doses in µSv calculated by : CARI EPCARD PCAIRE SIEVERT 11.3
13
15
13.4
12.3
15
13
9.60
8.7
10
12
9.90
6.4
7
9
6.40
8.2
9
11
9.70 (WARSAW)*
8.8
9
12
7.90 (BRUSSELS)*
6.8
7
9
6.70
8.2
9
11
8.10
0.087
0 less than 0.5
0
0.40
6.3
7
8
5.90
5.9
6
8
5.30
7.1
7
10
7.30
7.1
7
12
7.20 (WARSAW)*
*SIEVERT system contains only principal cities of country, calculation with SIEVERT were performed for this cities
The average dose over the month for CARI code would be adequate, but as Wollenberg showed, different crews fly at different times during the month, so that the effective dose gotten by successive crews flying the same route might well be quite different [41]. It is surprising how different programs give very close results, considering the great difference between the approaches. The differences are much smaller than one could expect
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from so different programs. One reason the codes would give different results, even if all radiation transport details were the same (which they most definitely are not) is that they use different GCR primary spectra. Indeed LUIN2000 uses a combination of the Peters 1958 spectrum and the more recent Garcia-Munoz, Mason, and Simpson (1975) spectrum while EPCARD uses the latest spectrum from NASA. The FLUKA code used by EPCARD is based on a Monte Carlo simulation using the environmental model of Badhwar for a given boundary condition [43, 44]. In addition FLUKA uses an older cut-off rigidity model for the IGFR (compared to the 1995 model used by PCAIRE and CARI-6/LUIN). In comparison, the PCAIRE model is effectively an encapsulation of the experimental measurements made with TEPC, either the NASA-JSC deceleration parameter or the climax count rates are used to describe the effect of solar modulation. The vertical cut-off rigidity is calculated from either a three-epoch average or interpolated from IGRF-1995 tabulated data for the given geographical coordinates along the flight path. Therefore there is no reason one would expect CARI, EPCARD and PCAIRE to give exactly the same answers. For the SIEVERT, the server returns to the companies the effective radiation dose for the flight, computed using a 3-D world map of effective dose rates. A quadratic fit of past heliocentric potential values (from 1964 to 1997) given by the authors of the CARI software vs. neutron monitor counts appears to be sufficient [46]. The discrepancy may also result from the use of different energy-dependent conversion coefficients (fluence to effective dose) in each of the codes. Nearly all of the codes (but not CARI and not FREE) use the vertical cut-off for all zenith and azimuth angles, and that will lead to an overestimate of the dose rates. We think that the differences in the doses calculated by the various programs are considerably less important than the uncertainty in the biological effects of the radiation [47].
Individual yearly effective doses Professionally exposed group (aircrews) and passengers are the exposed population, but the focus on aircrews is important because they receive relatively high doses. Thus for pilots, flight engineers and cabin crew, there are two important parameters. The number of flying hours per year and the specific flight schedules. The mean effective dose rate for each flight was calculated as the corresponding route effective dose divided by the flight duration. The annual effective dose is then calculated as this mean flight dose rate multiplied by the effective flying hours. For the Tunisian republic the mean number of flying hours must be less than 900 hours/year, the actual mean number is of 750h/y.[38] Thus, the annual effective dose for the different regions is about: 2.27mSv calculated by CARI, 2.45 by EPCARD, 3.04 by PCAIRE and 2,32 by SIEVERT. Effect of the 11-year solar cycle on cosmic radiation levels at 36000 ft above monastir airport The galactic radiation levels at 36000 ft above Monastir airport are depicted in Figure 2. The effect of the 11-year solar cycle on cosmic radiation levels can be clearly seen. The minimum exposure received at this altitude is of 2,4 µSv when calculated with EPCARD, for the year 1990. The highest value of effective dose is of 3,4 µSv, calculated with CARI, for the
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summer of 2007. At this location, the modulation of solar activity is more apparent with EPCARD. Different models have been used in CARI-6 and EPCARD to account for the changing solar modulation. For instance, in CARI-6, the heliocentric potential model of O’Brien was adopted. On the other hand for EPCARD, FLUKA computations employ the primary spectra of Badhwar where the solar modulation of these spectra is determined via a diffusionconvection model as developed by the National Aeronautics and Space Administration (NASA), Johnson Space Center. FLUKA uses an older cut-off rigidity model for the IGRF (compared to the 1995 model used by PCAIRE and CARI-6/LUIN). Some codes (EPCARD, probably) uses the deceleration potential, which is in error. Some use different radiation weighting factors (CARI, for instance) and do solar modulation wrong (CARI, for instance)[41,48]. Many use the monthly average of the solar modulation parameter and as Wollenberg has shown , that can lead to serious errors in calculating the flight dose to a particular flight crew[41].
Effective dose received an a flight versus time Figure 3 shows the effect of the 11-year solar cycle on cosmic exposure for the flight on 23 June 2007 from Bristol to Monastir. One can see that the lowest effective dose for this flight, 8,9 µSv, was obtained in 1990 with both CARI and EPCARD. The other minima ascribed to the other solar maxima are about 10 µSv. The most recent solar activity maximum occurred in early 2000. The year 2007 corresponds to a period of solar activity minimum. As consequence, the effective dose for this flight reached the maximum value compared to other solar minima: 12,3 µSv for CARI and 15 µSv for EPCARD. The PCAIRE code follows the same modulation with CARI and EPCARD, almost always having the highest values.
Figure 2. Effect of the 11-year solar cycle on effective dose rates (at 36000 ft above Monastir airport) calculated with CARI and EPCARD
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Figure 3. Modulation of solar activity for the flight Bristol/Monastir computed by EPCARD, CARI, and PCAIRE
Effective dose for single altitude flights In table 2 we compare for 4 destinations the effective dose received for the actual flight profile and a virtual single altitude flight profile, the lowest or the highest altitude of the real flight profile. According to CARI code, the actual effective dose is approximately the mean of both values of effective doses for the maximum and minimum altitude, while for PCAIRE it can be one of both values. For EPCARD the value of dose for the real flight profile is often the upper bound. Table 2. Comparison of effective doses received on real flights and those received on virtual single altitude flights Effective dose in µSv for Monastir/Kosice
Effective dose in µSv for Dublin/Monastir Effective dose in µSv for Bristol/Monastir Effective dose in µSv for Warszawa/Monastir
Real flight Maximum altitude flight Minimum altitude flight Real flight Maximum altitude flight Minimum altitude flight Real flight Maximum altitude flight Minimum altitude flight Real flight Maximum altitude flight Minimum altitude flight
CARI 6,4 7,4 6,0 11,3 12,1 10,4 12,3 13,3 9,9 8,7 8,9 8,3
EPCARD 7 8 6 13 13 12 15 16 12 10 10 9
PCAIRE 9 10 8 15 16 13 13 19 13 12 12 11
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Effect of aircraft type Additional radiation exposure may occur as a result of sporadic solar flare activity, particularly at high (supersonic) jet altitudes and at latitudes close to the magnetic poles. Sporadic solar flares due to magnetic energy release from the sun can send a large number of charged particles (mainly protons, some alpha particles and a few heavier nuclei) into the atmosphere with typically maximum energies between ~ 10 and 700 MeV, which is the energy range of NOAA’s GOES satellites. It is worth noting that the particle fluence rates and spectra are highly variable, particles can be accelerated to GeV energies within 10 seconds or less [55]. The energy of these solar particles is much smaller than those of galactic origin; they are expected to induce a significant dose only at the higher supersonic altitudes Indeed and as shown in table 3, the effective dose received onboard supersonic aircrafts is generally lower than the effective dose received onboard subsonic aircrafts. Although for the longest flight Dublin-Monastir the effective dose increases onboard supersonic aircraft.
1.3. Wavelets and Neural Network Based Study of CARI Heliocentric Potentials as a Measure Of Solar Activity 1.3.1. Wavelets and neural network in cosmic rays study Neural network [56] models can be used to infer a function from observations and also to use it. This is the utility of artificial neural network models. This is especially useful in applications where the complexity of the data or task makes the design of such a function by hand impractical, such as the complexity of radiation field produced by cosmic radiations at aircraft flight altitudes, dose measurements or computations of effective doses and heliocentric potential Hp the essential parameter used by the data base of CARI code [57, 6872] Neural networks are universal models which can approximate any functions, under condition that one proceed to their training with a sufficiently consistent data base made up of inputs (flight profile corresponding altitudes and times, departures and destinations: corresponding geomagnetic co-ordinates, date of the flight, year, month, day, and duration,) [60]. The training consists in adjusting the internal parameters of NN so that this one approximates correctly the physical behavior of the system. The results of this simulation tool are compared with those of CARI and those of wavelets analysis. Table 3. Effect of aircraft type on the effective doses during flights Destination Dublin-Monastir Budapest-Monastir Bruxelles-Monastir Bordeaux-Djerba Djerba-Monastir
Effective dose in µSv received on Subsonic aircraft Supersonic aircraft 13,40 21,90 6,30 5,90 8,10 7,60 5,30 5,00 0,40 0,40
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The wavelet theory involves representing general functions in terms of simple, fixed building blocks at different scales and positions. We use translations and dilations of one fixed function for wavelet expansion. Sophisticated wavelets such as Morlet wavelet are more powerful in revealing hidden detailed structures. This type of wavelets has been used to examine the processes, models and structures of the solar activity [61] variability. Using Morlet wavelet expansion we present in this part a mathematical zoom to discover the hidden structures and to extrapolate unknown aspects in CARI heliocentric potentials variability in time The Continuous Wavelet Transform (CWT) is an ideal tool for mapping the changing properties of non-stationary signals and also to determine whether or not a signal is stationary in a global sense. CWT is then used to build a time-frequency representation [67] of a signal that offers very good time and frequency localization [62,63]. An adjustable parameter controlling the properties of the localized oscillation is generally included in the mother wavelet which can be complex or real. In this study we have used Morlet wavelet [64] which is defined as a complex sine wave, localized with a Gaussian. The frequency domain representation is a single symmetric Gaussian peak. The frequency localization is very good. This wavelet has the advantage of incorporating a wave of a certain period, as well as being finite in extent. Associating NN system and Morlet wavelets to CARI code, we aimed to predict the heliocentric potentials, to zoom and repair CARI errors [59,75] , and to discover the Hp modulations provided by extrapolation phase .
1.3.2. Results and discussion Neural network study of monthly heliocentric potentials Training samples were prepared using the data related to June and July of the year 2007 for heliocentric potentials H.P available from CARI data base or from our last paper containing also values of effective doses derived with CARI and other codes for different flight destinations from or to Tunisia[58]. Since we had only the results until 2007 July month given by CARI at that time, which is unable to predict results for Hp or for effective doses already like the other codes, thus we have investigated these results in N.N system to know the extrapolation or prediction period we can reach and to which level of accuracy we can succeed these prediction comparing the results with CARI available data taking account of it’s proper errors and on the other hand with wavelets results. Neural Network was trained with a training base of 13 examples. Each example contains all flight characteristics as inputs (profile, duration, destination, date) and outputs : effective dose E and heliocentric potential U, MLP incorporated a non-linear activation function, allowing them to learn non-linear relationships. This flexibility was useful when trying to learn complex relationships between flight profiles (altitudes latitudes ,geomagnetic coordinates , dates durations) and outputs effective doses and heliocentric potentials on the other hand, we are just interested with results of Hp in this work. In this study the activation function φ was sigmoidal, linear over a small range of values close to zero, but saturated for large values (Eq. 1).
Cosmic Rays and Safety
φ (ν j ( n ) ) =
1 1 + exp ( −αν j ( n ) )
203 (1)
Where ν was the sum on node j for case n, and α was a constant value >0 Once the values on the output nodes had been calculated, they were compared with the desired values and a back propagation algorithm was used to adjust the weights to decrease the difference between the actual and desired predictions. The process was repeated iteratively using all cases in the training set until it met the least mean square error (MSE) between the target and actual output values When NN approximate correctly the outputs contained in the test base, the input data of various flights of the next dates to which CARI cannot derive heliocentric potentials for next months, are then presented to NN inputs in order to evaluate corresponding heliocentric potentials on the desired time interval: [August 2007... March 2008]. (Table 4). We have used a four layers Neural Network, an input layer containing 14 to 17 neurons dependently on the number of cruising altitudes for the corresponding flight, two hidden layers composed respectively of 40 neurons for U and 40 neurons for E and an output layer composed of two neurons. For processing data a supervised learning was applied with a gradient backpropagation algorithm. We have obtained heliocentric potentials for months: August, September, December of year 2007 and January..,March of 2008, with Neural network method trained with last CARI results and compared them with CARI results for each current month when heliocentric potentials becomes available in CARI data base. Daily heliocentric potentials are obtained for each flight and then averaged on each month. It is worth noting that neural network overestimates Hp values with respect to CARI values. Daily values of Neural Network HP are kept roughly constant during the period o1 to 20 of each month then decrease to the end of month nearly by 20 MV (table 4, figure 4)
Figure 4. CARI and averaged NNetwork Hp variations from August 2007to March 2008
Table 4. Heliocentric potentials for period August 2007-March 2008 estimated by N.N and compared with CARI values
August 07 September 07 October 07 November 07 December 07 January 08 February 08 March 08
01 318 318 317 317 317 321 321 321
06 317 316 315 314 312 327 326 324
10 318 317 317 316 315 323 323 322
12 319 318 318 318 318 320 320 320
Heliocentric potentials in MV Predicted by N.N for different days of the month 14 18 19 20 23 26 318 319 317 319 300 298 318 318 315 318 299 296 317 318 314 318 299 295 317 318 312 318 299 294 317 318 311 317 299 306 321 320 329 319 301 305 321 320 328 319 300 304 321 319 326 320 300 303
27 300 299 299 299 299 301 300 300
28 299 299 299 299 298 301 301 300
29 299 299 298 298 297 301 301 301
Av 310.8 310 309.5 309.1 309.5 314.5 314.1 313.6
Computed by CARI 312 300 307 291 287 298 312 321
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The prediction of solar modulation and other cycles was also tried with neural network system for heliocentric potentials and for next years, unlike the Morlet wavelets analysis, neural network method didn’t reproduce the 11-year cycle, in fact it represents a singularity which must be studied in the corresponding interval and also for a longer learning period. Thus these are generally the limits of NNT in our framework which will be held and repaired in following section through wavelets study
Morlet wavelet analysis of yearly heliocentric potentials Wavelet analysis is a useful tool both to find the dominant mode of variation and also to study how it varies with time [73]. As we have already used the Morlet wavelet variety in our previous work[74], we have used this type of wavelets in this paper to study and to test the prediction or extrapolation phase provided to CARI heliocentric potentials Hp. Solar or other modulations and variations in time of CARI Hp occur through Morlet analysis. The wavelet transform of a function y (t) uses spatially localized functions called wavelets and is given by w(a, b ) = a −1 2
+∞
∫ y(t )g
∗
−∞
⎛t −b⎞ ⎟dt , ⎜ ⎝ a ⎠
(2)
Where a is the scale dilation (compressing and stretching) of the wavelet g used to change the scale, a determines the characteristic wavelength; b is the translation parameter (the shifting of g used to slide in time), and g* is the complex conjugate of g. Morlet wavelet is a complex sine wave multiplied by a Gaussian envelop and given by (Eq.3)
⎛ t2 g (t ) = exp⎜⎜ iω 0 t − 2 ⎝
⎞ ⎟⎟ ⎠
(3)
Where we have taken the phase constant ω 0 = 2π . The period was fixed to T = 1year The frequency domain transform of a real wavelet is symmetric around frequency 0 and contains two peaks. For analysis of a discrete signal
y(ti ) we need to sample the continuous
wavelet transform on a grid in the time scale plane (b, a) by setting a = j and b = k .The wavelet coefficients w j ,k are given by:
w j ,k = j
−1 2
+∞
∫ y(t )g
−∞
∗
⎛t −k ⎞ ⎟⎟dt ⎜⎜ ⎝ j ⎠
(4)
In order to separate independent components of the signal characterised by different wavelet coefficient magnitudes, we have followed a principal reconstruction strategy in which we have used a kind of band pass filter by considering signal components due to integer and fractional scale parameter j in another terms considering the integer scales j from which we deduce respectively the called “approximation “ or the deterministic component associated to
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Neïla Zarrouk and Raouf Bennaceur
high periods and low frequency so it is the low pass filter and on another hand the stochastic component for low periods and high frequencies. It is then the high pass filter. In this work we were interested particularly in the deterministic component. Real and imaginary decomposition coefficients components were calculated and plotted versus scale dilation and translation parameters j and k tf
w R j , k = j −1 2 ∫ y (t )g Rj , k (t )dt , ti
⎛t −k ⎞ ⎟⎟ g Rj , k (t ) = g R ⎜⎜ ⎝ j ⎠
(5)
tf
w I j,k
⎛t −k ⎞ ⎟⎟ = − j −1 2 ∫ y (t )g Ij , k (t )dt , g Ij , k (t ) = g I ⎜⎜ ⎝ j ⎠ ti
[t , t ] is the study time interval i
f
We have manipulated limited and discrete time series, for a 50 years period of study. We need to discretize expression of Eq.5 as follows 50
w R j , k = j −1 2 ∑ y (t l )g Rj , k (t l )
(6)
l =1
After decomposing the CARI heliocentric potentials variation in time for the period from 1958 to 2008 in Morlet wavelets, we have proceeded to reconstruction and to the extrapolation. We have reconstructed (Figure 5) the heliocentric potentials variations function using the real component of reconstructed function. y (t ) = ∑ w j ,k g j ,k (t )
(7)
j ,k
In this reconstruction case, scaling and shift parameter j and k vary from 1 to 100 with a step of 1 year We can notice that CARI Hp variations similarly to Morlet reconstructed variations show the well known11-years modulation often related to solar activity [65,66] varying here from 8 to 12 years. The magnitudes of harmonics as we have mentioned for Cosmic rays and solar spots variations, in our previous paper [74] are not the same. From The extrapolation phase we can point out that the length of cycles has changed. Indeed the first maximum of Hp showed by Morlet extrapolation occurs around 2009 after 9 years (2000-2009), the next maximum is around 2022 and the last one occurs after a 25 years period around 2047. The length of known 11-years cycle given by the extrapolation is varying from 9 years around the beginning of extrapolation region to 13 years, a longer cycle of 25 years is also present. The minima show also a first extrapolation minimum around 2015 and a second one around 2035. The Morlet extrapolation showed the already found [74] modulation of 11-years varying here between 8, 9 to 13 years and also the big periods ranging from 20 to 30 years. The most important value of Hp is detected around 2022 year.
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(a)
(b)
(c) Figure 5. (a) yearly heliocentric potentials plotted from CARI data base, (b) Morlet reconstructed Hp, from 1958 to 2008, (c) Morlet reconstructed and extrapolated Hp for the whole period 1958-2057
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(a)
(b)
(c) Figure 6. Most important periods for yearly Hp (a) in reconstruction interval time 0-200 years, (b) in reconstruction interval time 5-50 years, (c) in whole reconstruction and extrapolation interval time 0100 years
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Main periods for yearly heliocentric potentials Studying the Morlet most important decomposition coefficients we have illustrated the existence of main periods of 11-12 years clearly occurring in reconstruction decomposition coefficients (Figure 6a,b). For highest decomposition coefficients occurring in whole phase of reconstruction and extrapolation (Figure 6c), the absolute maximum of decomposition coefficients corresponds to cycle varying from 11 to 14 years The second period denoted for Hp is around 75 years (Figure 6c). We point out the presence of 2 to 3 years small structures. Morlet wavelet analysis of monthly heliocentric potentials For a 600 months period of study, the decomposition coefficients and respectively reconstructed variation function are given by equations 8 and 9. 600
w R j , k = j −1 2 ∑ y (t l )g Rj , k (t l )
(8)
y (t ) = ∑ w j ,k g j ,k (t )
(9)
l =1
j ,k
For this reconstruction case, scaling and shift parameter j and k vary from 1 to 600 with a step of 1 month. In this section we present monthly heliocentric potentials analysis especially that the heliocentric potentials used by the FAA as input to CARI are in error, by up to 40% as we have mentioned above. Thus Morlet analysis is a mathematical zoom not only for decomposition and extrapolation of Hp variations but also for the CARI error. We have taken the phase constant ω0
= 24π , the period was fixed to T = 1 12 year ≈ 1 month .
Monthly heliocentric potentials variations are more accurately reconstructed by Morlet wavelets. Similarly to CARI Hp modulations, a main reconstructed Hp cycle is varying here from 110 months (9 years ) to 140 months (11.6 years ) (Figure 7a,b). Results of extrapolation show that the length of cycles as shown in yearly variations has changed to 170 months corresponding then to 14 years cycle. A secondary minimum is showed just before month 600 corresponding to 2008 year which is not shown in yearly variations. The minima show also a first extrapolation minimum around 2016 and a second one around 2033. The most important value of Hp is detected around 2022-2024 year (Figure 7c)
Main periods The highest monthly decomposition coefficients corresponding to reconstruction phase for Hp are found around 6 months then 12 months (Figure 8a), For longer periods we have denoted three main maxima, the first is around 75 months (6.25 years), then a second maximum for 60 months (5 years) and a third maximum occurring around a cycle of 137 months corresponding to the well known 11 years cycles (Figure 8.b) The main periods appearing for the whole period of reconstruction and extrapolation (Figure 8.c) are around 120, 150 months (10, 12.5 years). The big period of 400 months (33 years) already detected in yearly variations is illustrated here as a second maximum, a third cycle of higher coefficient and also detected in yearly variations is present around 900 months
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(75 years), other periods are also present around 20 months (1.6 years) , 80 months (6,6 years) , we notice small regular structures around 2.5 to 2.85 months.
(a)
(b)
(c) Figure 7. (a) Monthly heliocentric potentials (1958-2008) plotted from CARI data base, (b) Morlet reconstructed Hp,600 months after 1958, (c) Morlet reconstructed and extrapolated HpP for the whole period of 1000 months after 1958.
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(a)
(b)
(c) Figure 8. Most important periods for monthly Hp (a) in reconstruction interval time 0-50 months, (b) in reconstruction interval time 50-200 months, (c) in whole reconstruction and extrapolation interval time 0-1000 months.
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2. NEURAL NETWORK AND WAVELETS IN PREDICTION OF COSMIC RAY VARIABILITY: THE MEDITERRANEAN AS STUDY CASE 2.1. Theoritical Methods and Data: Stations In July 1997 the Moscow NM station was the first one, in the world presenting real time data to the internet, and then the number of other stations increases operating in various latitudes around the world. There are now 25 stations providing real or quasi real time data (in digital and/or graphical forms Mavromichalaki et al, 2001 [76]) elaborated automatically to ensure compatibility with other stations. The use of all stations as a unified multidirectional detector repair the accuracy of the measurements and made them higher (<0.1%for hourly data). This system has been designed with the capability to support a large number of stations. It’s worth noting that the collection system is able to provide reliable data, based on the issue that there are independent programs collecting simultaneously data from different stations in different ways. Nowadays there are 21 stations from which the described system collects data. The early detection of earth directed solar energetic particles (SEP) [5] event by NMs offers a very good chance of preventive prognosis of dangerous particles flux and can alert with a very low probability of false alarm.
2.2. Results and Discussions 2.2.1. Preparation of training samples: Decomposition in morlet wavelets As we have operated in previous sequence for heliocentric potentials and in our previous work [81] concerning cosmic rays variation versus time given by the curve relative to Moscow Neutron Monitor where the Morlet decomposition coefficients relative to cosmic rays variations for 49 years period of study are given by: 49
w R j , k = j −1 2 ∑ y (t l )g Rj ,k (t l )
(10)
l =1
Table 5. List of neutron monitors investigated in calculations Stations Moscow Jungfraujoch Irkutsk Oulu Lomnický Climax Sanae Potchefstroom Tsumeb Hermanus
Study periods 1958-2007 1992-2007 1958-2007 1964-2007 1982-2008 1953-2007 1977-2002 1972-2002 1977-2002 1973-2002
Latitude in ° dl 55.47 N 46.55N 52.28N 65.05N 49.20N 39.37N 70.19S 26.41S 19.12S 34.25S
Longitude in ° dl 37.32 E 7.98 E 104.02E 25.47 E 20.22 E 106.18W 02. 21 W 27. 06 E 17 .35 E 19 .13 E
Altitude in m 200 3570 475 15 2634 3400 52 1351 1240 26
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Thus we have investigated data corresponding to yearly cosmic ray variations measured by ten neutron monitors stations: Moscow, Climax, Oulu, Jungfraujoch, Irkutsk, Lomnický, Sanae, Potchefstroom, Tsumeb and Hermanus [6-12] among the twenty five known stations. These data constitute the training samples we have reproduced in main details all curves of yearly cosmic ray variations relative to different stations then we have decomposed the different cosmic ray variations functions in Morlet wavelets. The list of neutron monitors used for calculations is brought in table 5.
2.2.2. Training Training on main periods In a first step and once we have decomposed the different cosmic ray variations relative to 10 different stations in Morlet wavelets, we have also derived the main periods studying and plotting the decomposition coefficients versus time, then we have notice the main three periods T1,T2,T3 corresponding to three first maxima coefficients for the 10 stations. We present the periods for each station as outputs to Neural Network system, the inputs are geographic coordinates of stations Training on decomposition coefficients Once the tables of imaginary and real components of Morlet decomposition coefficients corresponding to each station are obtained, we present a certain number of coefficients necessary for reconstruction as outputs to Neural Network system. The matrix of real coefficients is reduced to a vector for a single value of scaling parameter, the inputs are the geographic coordinates of corresponding neutron monitor stations. After the derivation of values on the output nodes, the latter were compared with the desired values and a back propagation algorithm was applied to adjust the weights to decrease the difference between the actual and desired predictions. The process was repeated iteratively using all cases in the training set until it met the least mean square error (MSE) between the target and actual output values,this work present the training phase of our Neural Network study. We have used two training stages. The training test of which samples are all examples of stations except one example on which we try the credibility of test loading the results of training and then, comparing the reconstructed signal to observed signal of corresponding station. On the other side the main training in which the complete stations unit was used as samples unit, the results of this training are used for our aim of extrapolating numerically neutrons monitors’ stations network. 2.2.3. Prediction of main periods Once the system trained on main periods and in the same training interval for latitude and longitude, we have interpolated and plotted in 3D dimensions space for each couple of (lat, long) T1, T2, and T3 versus latitude, longitude and for a fixed altitude of 26m which may describe any station or position on the Earth, the results of interpolation are as shown in Figure 9.
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Figure 9. the main periods T1,T2,T3 variations versus latitude longitude and for an altitude of 26m
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The main known period of 11-12 years modulating the curves of relative cosmic rays intensities is illustrated and found here. In fact the values of T1 calculated with neural network system are ranging from 9.52 to 15.94 when varying latitude and longitude. The less important period T2 found with neural network system and corresponding then to a lower decomposition coefficient according to periods training is found fluctuating between 11.58 and 38.56. For T3 the period which must correspond to the less frequent cycle seems to oscillate here from 40.82 to 80.64 years and we notice a certain singularity occurring in the interval of longitude between 12° and 23° in which we can intercalate the Tsumeb NM station longitude then for which we have found indeed the lowest value of T3, 33 years with respect to other stations of training. This singularity occurs as a fluctuation for T1 but it is deeply seen in T3, it is probably related to Earth’s atmosphere phenomena. Indeed in the context of finding incident cosmophysical periodicities that may modulate terrestrial phenomena, little attention has been given to the large scale climatic phenomena: the Atlantic Multidecadal Oscillation (AMO). Common periodicities were analysed [83,84] between phenomena presumably associated to hurricanes: the Atlantic Multidecadal Oscillation (AMO) and the Sea Surface Temperature (SST) versus cosmic rays, and on the other hand cosmic rays versus Atlantic hurricanes. A common period of 30±2 years was found for total Cyclonal Energy, the total number of Tropical Storms landing in the Atlantic coast of Mexico and others. Thus these terrestrial phenomena indexes modulate or at least influence at the first order the less frequent period due to cosmic rays modulation. We show then a competitive mechanism measuring the relevance of cosmophysical phenomena with respect to terrestrial sources of affectation. It is worth noting that we have used for the training the period T3 found by Morlet decomposition, the study and training periods of different NMs stations were different thus we expect that T3 don’t appears for all Mediterranean cases of applications or appears with the presence of certain singularity. The maximum of T3 is reached for values of longitude between 40° and 57°, the longitude of Moscow NM of 37° is the nearest. From the curves of cosmic ray variations of Moscow and Tsumeb NMs we can see the difference of variations values for the cosmic rays intensities (Figure 10). We have on another hand studied the variation in 2D dimensional space of main periods T1,T2,T3 separately with altitude, this is shown in Figure 11. The most dominant period T1 decreases with altitude and also with latitude parameter for a fixed longitude and for all the latitudes varying from 0 to π/2 rd keeping also the same shape of variation. T1 is ranging from 8.08 to 15.17 years. The decrease of the most dominant period T1 with altitude is expected. Indeed this known cycle of 11 years mean value is mostly related to extragalactic component of cosmic rays which increase in intensity with altitude and is the most energetic and important near the top of the atmosphere. This is the atmospheric shielding provided by the earth’s atmosphere as a shielding mass, at sea level of about 1033 grams of matter per cm3. Indeed high energy protons will generate an atmospheric nuclear cascade, and these high-energy cascading particles have enough energy to continue the process to sea level. The cycles are then more entertained more transparent near the top of the atmosphere reflecting more solar activity and not masked nor dilated with other phenomena. The decrease of T1 with latitude is related to the configuration of the Earth’s magnetic field having a kind of shield less important on higher latitudes which are then more sensitive to solar phenomena particles and activity.
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T2 increases with altitude up to a certain value of latitude around 1rd ( 57°) and then T2 remains constant versus altitude, T2 decreases with latitude and, it is generally varying between 11.39 and 39.60 years. The T3 cycle increases with altitude with the same shape of variation, T3 increase with latitude parameter remaining in the interval 53.20, 90.93 years. Unlike T1, T2 and T3 seem to have the opposite behaviour, they generally increase with altitude. Indeed the cycles are more dilated with altitude indicating lower sensitivity to certain Earth’s atmospheric phenomena which may be then most intensive on the Earth’s level. However we can notice that T2 has a mixing behaviour between T1 and T3 combining the reflection of solar and Earth’s phenomena. Besides the cycle T3 is longer with higher latitude excluding the Earth’s magnetic field as effect, this may be so purely related to another Earth’s phenomena decreasing or shielding from the cosmic rays intensities and even surmounting the opposite effect of Earth’s magnetic field.
Figure 10. CR Tsumeb variation corresponding for the interval of minimum for T3 and CR Moscow variations nearest to the maximum of T3
Table 6. Neural Network main periods for Mediterranean cases Main periods in years T1 T2 T3
Rabat
Main periods in years T1 T2 T3
Alger
18.01 24.50 82.55
Tunis
14.04 21.39 58.56
Athens
12.42 20.90 46.33
Ankara
11.68 20.89 49.16
Tripoli
12.19 21.12 44.29
Damascus
12.21 22.52 58.56
12.66 21.61 66.39
Cairo
Paris
12.13 20.05 58.35
Monaco
13.67 18.22 62.97
Beirout
Jerusalem
12.44 21.61 64.29
12.70 19.47 51.85 Madrid
12.63 27.02 71.61
Rome
Ljubljana
11.98 20.03 45.90 Valetta
16.27 21.30 76.28
Zagreb
11.63 19.74 45.48
11.53 19.97 45.19
Nicosia
11.97 20.76 44.21
Sarajevo
Tirana
12.21 27.00 69.27
11.54 20.68 45.89
Gibraltar
12.30 21.55 62.15
Londres
17.26 23.24 79.41
14.25 17.59 69.45
Table 7. minima and maxima years of cosmic rays variations for Mediterranean cases Year of
Rabat
1st Minimum
1985
2d Minimum
19921993 1989
Maximum
Year of 1st minimum 2d minimum Maximum
Alger 19841985 19921993 19881989
Athens 1985 1994 1989
Tunis 19841985 19921993 1988
Ankara 1986 1994 1990
Tripoli
Cairo
Paris
Monaco
Rome
Ljubljana
Zagreb
Sarajevo
Tirana
19841985 1993
1986
1986
1985
1985
1983
1985
1986
1985
1994
1992-1993
1992-1993
1992-1993
1993
1994
1993
1988
1990
19921993 1989
1989
1989
1989
1989
1990
1989
Damascus 1986 1994 1990
Beirout 1986 1994 1990
Jerusalem 1986 1994 1990
Madrid 1985 1992-1993 1989
Valetta 1984-1985 1993 1988
Nicosia 1986 1994 1990
Gibraltar 1985 1992-1993 1989
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Figure 11. T1,T2,T3 variations with altitude for a longitude of 0.0348rd and a group of latitude values varying from 0 to π/2rd
We have proceeded in another step to calculate the first three main periods for the Mediterranean case: Tunisia, Egypt, Algeria, Maroc, Libya,..(Table 6). We have found a first period T1 corresponding to the higher decomposition coefficients according to the corresponding training, T1 is ranging from 12 to 18 years for all the studied countries. The second cycle T2 is varying from 20 to 24 years. The less frequent period T3 is ranging from 44 to 82 years. Thus we illustrate in this first attempt of prediction, the known 11-12 cycle, with the single exception of Maroc country for all periods T1, T2, T3 which seems to be overestimated by neural network system and constitute then a singularity.
2.2.4. Decomposition and reconstruction of cosmic ray variation for virtual stations Morlet reconstruction test As we have mentioned above we carried out the training test on the totality of stations except one station on which we apply the results of training and confront them after reconstruction of the signal to measured cosmic rays variations of corresponding station. We have repeated this procedure for all examples of stations. Thus These Morlet test reconstructions are used as test as well for suitability of Morlet reconstruction base as for credibility of neural network prediction. In following curves of Figure 12 we showed Morlet reconstructed tests for cosmic rays variations compared to corresponding measured CR variations for the period 1982-2000 In the second stage of training we have then used the totality of stations examples in a main training. Once the NN system has converged with a satisfactory precision with respect
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Figure 12. Continued
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Figure 12. Morlet reconstructions tests on the right compared to measured signals for cosmic rays variations on the left
to original decomposition coefficients, we have determined then the inputs corresponding to Mediterranean countries geographic coordinates of virtual neutron monitors stations where we want to derive cosmic ray variation. At the end we have presented these inputs to Neural Network system already trained, we reach then our purpose when we obtain the unknown decomposition coefficients. We have proceeded to reconstruction in order to have the desired variation cosmic ray curves corresponding then to our virtual neutron monitor stations We have used a four layers Network, an input layer containing 3 neurons corresponding to geographic coordinates, two hidden layers composed of 16 neurons for wR and an output layer composed of 16 neurons. For processing data a supervised learning was applied with a gradient back propagation algorithm. As shown in any examples of Morlet reconstructed curves (Figure 13), they seem to have the same shape with the original curve given by Moscow NM, and also similar shapes for the common 20 years period corresponding to the training period between 1982 and 2002, especially having a minimum corresponding to the lowest relative cosmic rays intensities for the period 1992-1994 and a maximum indicating a peak for the cosmic rays intensities for years 1988-1990, a secondary minimum is also appearing for the period 1984-1986. The main 11 years cycle is appearing between 1983-1994, the length of the cycle is shorter in this case and it ranges from 7 to 10 years for the Mediterranean case, the shortest cycles are found for Paris, Rabat and the longest one is of Ljubljana. The cosmic rays variations details of Morlet analysis are given in table 7.
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Figure 13. Examples of Morlet reconstructed CR variations curves for Mediterranean case
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GENERAL CONCLUSION The cosmic radiations field received at aircraft flight altitudes is complex, thus dose measurement is a difficult problem. Therefore the use of an appropriate program for dose calculation is expected to be of a great interest. Measurements were often confronted with calculations using transport codes. In all cases, agreement between measured and calculated values was within 25%. Experimental methods, particularly active detectors give the influence of geomagnetic cut-off closer to EPCARD 3.2 data In general, all codes are consistent with one another. they typically deviate by less than 20% at subsonic altitudes, which is sufficient for radiation protection purposes and regulatory applications. However, as previously discussed, further validation is required over the solar cycle. Values of yearly effective dose received by Tunisian crew members regularly flying A320-321 aircraft range between 2 and 3 mSv when calculated by CARI-6, EPCARD v.3.2, PCAIRE and SIEVERT. These doses depend mainly on their flying frequency, flight profile (mainly ascent and descent) and on solar activity. The mean value of effective dose rate onboard aircraft ranges between 3 and 4 µSv/h (for all codes) The modulation of solar activity is relatively more obvious with EPCARD v.3.2 and PCAIRE than with CARI-6. Doses received by aircrew due to their exposure to cosmic radiation are not negligible and it is important that realistic assessments be considered to ensure appropriate protection for crew who are deemed to be occupationally exposed. The most important aspect of the exposure of aircrew to cosmic radiation is the provision of appropriate protection for female crew during pregnancy. Otherwise Operators should make arrangements to provide education regarding the risks of occupational exposure to radiation to their aircrew, which are defined as flight crew, cabin crew and any person employed by the aircraft operator to perform a function onboard the aircraft while it is in flight. Female aircrew should be made aware of the need to control doses during pregnancy and to notify their employer if they become pregnant so that any necessary dose control measures can be introduced. Although majority of codes stay unpredictable, thus we have used Neural Network (NNT) and wavelets associated with CARI code to predict values of heliocentric potentials. This simulation tool provides us a reconstruction and an extrapolation of CARI heliocentric potentials through the Morlet wavelet analysis. According original, reconstructed and extrapolated Hp variations functioning as a measure of solar activity we have showed 8 to 13 years cycles. The first next maximum of Hp≈1400MV is located around 2022-2024. The minima of Hp corresponding to highest fluxes of cosmic rays are located around 2015 and 2035 years. Two classes of big periods are also found around 20-33 years and 75 years. Especially Morlet monthly analysis showed two main periods of 6 and 12 months, long periods of 5-6.25 and 11 years. Short structures are detected for 2-3 months, 20 months (1.6 years). Neutron radiation is a hazard in nuclear reactors. Neutron detectors used for radiation safety must take into account the way damage caused by neutrons varies with energy, describing in our case variations in cosmic rays flux. We have succeeded to extrapolate numerically the network of neutron monitors stations to other points on the Earth’s surface
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such the Mediterranean case and other points which just must be in the training interval that we have used in our neural network system. Through this work we have found the aspect of very known 11-years cycle: T1, we have revealed the variation type of T2 and especially T3 cycles which seem to be induced by particular Earth’s phenomena that will be studied in next works. Thus in this manner we have derived the Morlet decomposition coefficients corresponding to geographic coordinates of different points. Consequently we have plotted relative cosmic rays intensities curves for a certain period suitable to give a good precision in the training phase 16 or 50 years The main periods modulating galactic cosmic rays variations detected according to Morlet decomposition coefficients and reconstructions for all Mediterranean virtual stations are around 7-10 years. Neural network in combination with Morlet wavelets seems to construct a very suitable tool for our case of numerical extrapolation from NM stations network
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[37] Green, A. R., Lewis, B. J., Bennett, L. G. I., McCall, M, J. & Ellaschuk, B. (2001). Cosmic Radiation Exposure of aircrew-phase II, volume 1 : visual PC-AIRE theory manual and volume 2: visual PC-AIRE user’s manual. Project Report to Transport Canada, March. [38] Nouvelair Civil Aviation Company, Monastir/Tunisia. [39] OACA, Office de l’Aviation Civile et des Aéroports. [40] ICAO, (2001). Outlook for air transport to the year 2010, International Civil Aviation Organisation, ICAO Circular, 281. [41] O’Brien, K. Private communication. [42] O’Brien, K. The theory of Cosmic-ray and high energy Solar-particle Transport in the Atmosphere. (to be published). [43] Roesler, S., Heinrich, W. & Schraube, H. (1998). Calculation of Radiation Fields in the Atmosphere and Comparison to Experimental Data. Radiat. Res., 149, 87-97. [44] Badhwar, G. D. (1997). The radiation Environment in Low –Earth Orbit. Radiat. Res., 148, 3-10. [45] Lewis, B. J., McCall, M. J., Green, A. R., Bennett, L. G. I., Pierre, M., Schrewe. U. J., O’Brien, K. & Felsberg, E. (2001). Aircrew Exposure from Cosmic Radiation on Commercial Airline Flights. Radiat. Prot. Dosim., 93(4), 293-314. [46] Lantos, P., Fuller. N. & Bottollier, J. F. (2003). Methods for Estimating Radiation Doses Received by Commercial Aircrew Aviation. Space and Environmental Medicine., 74, 7. [47] Friedberg, W. Private communication. [48] O’Brien, K. & de P. Burke, G. (1973). Calculated Cosmic Neutron Monitor Response to Solar Modulation of Galactic Cosmic Rays. J. Geophys. Res., 78, 283-289. [49] Wilson, M. Bartol Research Institute, and Vashenyuk, E. Polar Geophysical Institute, Russia, in http://www.cami.jccbi.gov/aam-600/610/600radio.html. [50] Wollenberg, P. (2006). GLOBALOG. The Flight and Radiation Logbook for Flight Crew Members, Human performance Committee of the International Federation of Airline pilots Associations , Singapore, July 11. [51] McCall, M. Private communication. [52] Fasso, A. Private communication. [53] Bottolier, J. F. & Biau. A, (2002). Cosmic radiation dosimetry on board an aircraft: the SIEVERT SYSTEM. Scientific and technical report. [54] Lindborg, L., Bartlett, D., Beck, P., Schraube, H. & Spurny, F. (2004). Cosmic Radiation Exposure of Aircraft Crew: Compilation of Measured and Calculated Data. European Commission: Luxembourg [55] Sakurai, K. (1989). Space Science Reviews, 51, 1-9. [56] Matlab Neural Network toolbox. [57] Attolini, M. R. (1975). et al., Planetary Space Sci., 23, 1603. [58] Zarrouk, N. & Bennaceur, R. (2008). Estimates Of Cosmic Radiation Exposure on Tunisian Passenger Aircraft Radiation Protection Dosimetry, 1–8. [59] Wollenberg, P. (2006). GLOBALOG. The Flight and Radiation Logbook for Flight Crew Members. (Singapore: Human performance Committee of the International Federation of Airline pilots Associations) (July 11). [60] Hormik, K., Stinchcombe, M. & White, H. (1989). Multilayer feedforward networks are universal approximators . neural network , vol 2, 359-366.
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[61] Kozlov, V. I. & Markov, V. V. (2007). Wavelet image of the fine structure of the 11year cycle based on studying cosmic ray fluctuations during cycles 20-23., Geomagnetism and Aeronomy., 47, 43-51. [62] Kudela, K. (1991). et al., J Geophys. Res., 96, 15871. [63] Kudela, K., Storini, M., Antalova, A. & Rybak, J. (2001). On the wavelet approach to cosmic ray variability., proceeding of ICRC 2001: 3773 c Copernicus Gesellschaft. [64] Morlet, J., Arehs, G., Forgeau, I. & Giard, D. (1982). Wave propagation and sampling theory Geophys., 47, 203. [65] O’Brien, K. (2005). The theory of Cosmic-rays and high energy Solar-particles Transport in the Atmosphere. In: McLaughlin, J. P., Simopoulos, E. S. and Steinhausler, F., Eds. The Naturel Radiation Environment VII, Vol.7 (Amsterdam: Elsevier Press), ISBN 0080441378. [66] O’Brien, K., Friedberg, W., Sauer, H. S. & Smart, D. F. (1996). Atmospheric cosmic rays and solar energetic particles at aircraft altitudes. Environ. Int. 22(Suppl. 1), S9– S44. [67] O’Brien, K. & LUIN, A. (1978). Code for the Calculation of Cosmic Ray Propagation In The Atmosphere. Update of HASL-275, EML-338. [68] Federal Aviation Administration, Civil Aeromedical Institute. CARI-6 Computer Program. http://www.cami.jccbi.gov/Aam-600/Radiation/600radio.html. [69] O’Brien, K. & de P Burke, G. (1973). Calculated cosmic neutron monitor response to solar modulation of galactic cosmic rays. J. Geophys. Res., 78, 283–289. [70] Wilson, M. & Vashenyuk, E. Bartol Research Institute Polar Geophysical Institute, Russia. Available on http:// www.cami.jccbi.gov/aam-600/610/600radio.html. [71] Friedberg, W. Private communication. [72] O’Brien, K., Felsberger, E. & Kindl, P. (2006). Cosmic-ray Propagation and the Calculation of Cosmic-ray Doses to Air Crew, Human Performance Committee of the International Federation of Airline Pilots Associations, Singapore, July 11). [73] Chao, B. F. & Naito, I. (1995). Wavelet analysis provides a new tool for studying Earth’s rotation. EOS, 76, 161-165. [74] Zarrouk, N. & Bennaceur, R. (2009). A wavelet analysis of cosmic rays modulation., Acta Astronautica(2009)., doi10.1016/j.actaastro. 01.20. [75] O’Brien, K., Felsberger, E. & Kindl, P. (2006). Cosmic-ray Propagation and the Calculation of Cosmic-ray Doses to Air Crew, Human Performance Committee of the International Federation of Airline Pilots Associations, Singapore, July 11). [76] Mavromichalaki, H., Sarlanis, C., Souvatzoglou, G., Tatsis, S., Belov, A., Eroshenko, E., Yanke, V. G. & Pchelkin, A. (2001). Athens neutron monitor and its aspects in cosmic ray variations studies. Proc.27th ICRC, 4099. [77] Demuth, H., Beale, M. & Hagan, M. Neural Network Toolbox 5 User’s Guide. [78] Kozlov, V. I. & Markov, V. V. (2007). Wavelet image of the fine structure of the 11year cycle based on studying cosmic ray fluctuations during cycles 20-23., Geomagnetism and Aeronomy., 47, 43-51. [79] Attolini, M. R. et al., (1975). Planetary Space Sci., 23, 1603. [80] Chao, B. F. & Naito, I. (1995). Wavelet analysis provides a new tool for studying Earth’s rotation. EOS, 76, 161-165. [81] Zarrouk, N. & Bennaceur, R. (2009). A wavelet based analysis of cosmic rays modulation, Acta Astronautica, 65, 262-272.
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[82] Torres-Moreno, J. M. (1997). Apprentissage et généralisation par des réseaux de neurones : étude de nouveaux algorithmes constructifs. Thèse, Institut National Polytechnique de Grenoble. [83] Pérez-Peraza, J., Velasco, V. & Kavlakov, S. (2008). Wavelet coherence analysis of Atlantic hurricanes and cosmic rays. Geofisica International, 47(3), 231-244. [84] Pérez-Peraza, J. et al. (2008). On the trend of Atlantic Hurricane with cosmic rays. Proceedings of the 30th International Cosmic Ray Conference, Mexico.
Inn: Horizons in World Physiccs. Volume 2771 Editor: Albert Reimer R
IS SBN: 978-1-611761-884-0 © 2011 Novaa Science Publlishers, Inc.
C Chapter 4
LASE ER PHYSIICS Ahm med Safwaat Urology Department, D A Assiut Univerrsity, Assiut, Egypt E
The word "laser" is an acronym which stands forr "Light Ampplification by Stimulated Emission of Radiation". E R Laaser is light energy that is i part of thee electromagnnetic wave sppectrum. Mosst of the com mmercially avaailable lasers are either inn the visible or o infrared sppectrum of light. nstein originaally describedd the theoretical basis of stimulated em mission in Albert Ein 19917.[1] The ru uby laser was the first to be built. This deevice was desiigned and constructed by M Maiman in 196 69 at Hughes Laboratories. The decade of the 1960s saw the devellopment of m lasers thaat are commerccially availablle today. The rapidity of deevelopment atttests to the most faact that the staate of the art inn physics and engineering had h progressedd to the point where new laaser sources co ould be develooped with exissting technologgy.[2] Reflector
partially reflective end face
totally reflective end face RUBY ROD
Lamp
Fiigure 1. The firsst laser.[3]
LASER OUTPUT
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1.1. Principlees of Spontaaneous and Stimulated Emission In a normaal population of atoms, electrons occupyy certain enerrgy states. Moost of them occcupy the low west energy state s "ground state" (E0), less l number of o electrons occupy o the hiigher energy state s (E1), less occupy E2 and so forth.[4] When an atom m becomes exiited, one or m more of its eleectrons move into a state of o higher enerrgy and as excitation continnues, more ellectrons will move m into higgher energy sttates. When thhe number of electrons occcupying the hiigher energy states s is more than those occcupying the lower l energy states, the population of thhe atom will be called an invversion populaation. As the staate of populaation inversionn is an unstaable state eacch excited eleectron will evventually drop p down to the ground state. As it is jumpping from the higher energyy level to a loower one, it sp pontaneously emits energy,, a process knnown as sponttaneous emisssion. When thhis takes placee, the amountt of the emitteed energy (phhoton) will bee equal to the difference beetween the eneergy of the uppper level and that of the low wer level. (Figgure 2) When the spontaneously s y emitted photoon strikes anoother excited atom it forces the t latter to giive off its exxtra photon ennergy and retturn to the reesting state, this t is called stimulated em mission. The resulting emiitted energy is i equal to thhe sum of thee energy of thhe incident phhoton and the energy of thee atom struck by the incidennt photon. As this process is i repeated, m more photons of o energy are discharged d andd thus amplificcation takes place.[5]
Vab
Vab
Vab Electron fall back to the lower orbital
Vab Excited particle at its highest energy level is left by another photon
Two photons are produced with the same phase and wavelength
Ea
Ea
Vab
Vab Vab Eb
Eb
Fiigure 2. Diagram m to show sponntaneous emission (left) and stiimulated emissiion (right).[6]
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Figure 3. Components of a laser.[7]
1.2. Basic Laser Design Any laser producing device consists of: (Figure 3). 1. A closed cavity containing the device medium. 2. An energy pump to create the population inversion, and 3 A heat sink.
1.2.1. The Active medium It is the suitable material, the atoms of which are excited by energy pumping, thus acquiring the state of population inversion and so stimulated emission takes place. The active medium is placed in the laser cavity. A wide variety of active media are available these days. They could be: • • • •
Gasses: e.g. Noble gases, molecular gases and vapours. Fluids: e.g. organic dyes in solution. Solids: e.g. - crystal and glasses doped with metal atoms. - rare earths. Semiconductor elements (laser diodes).
1.2.2. The Laser cavity The laser cavity is a cylinder that is closed at both ends by mirrors. The mirror in the back is a fully reflecting mirror, whereas the one in the front has a small aperture in the middle, through which the laser beam can be released. The walls of the cavity are usually treated with materials that minimize energy loss. 1.2.3. The Energy pump The energy pump supply for the laser machine could be from: • • • •
Chemical reaction Electric gas discharge Electromagnetic radiation, or Electric current.
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1.2.4. Mechanism of the laser machine function By switching on the energy pump, the followings occur: 1. Atoms of the active medium are raised to higher energy levels (i.e. a state of population inversion is created). 2. As spontaneous emission occurs, photons of energy will be discharged striking other excited atoms discharging more photons of energy (stimulated emission). 3. The photons begin to bounce around within the laser cavity. Part of the energy escapes into the interstices surrounding the laser cavity and must be removed by a heat sink. 4. The rest of photons of energy will be trapped between the excited atoms by the process of stimulated emission and thus amplifying the energy value of the developing beam of laser. 5. The trapped photons continue to recruit more and more of the excited atoms by the process of stimulated emission and thus amplifying the energy value of the developing beam of laser. 6. At the desired time, determined by the operator and according to the mode, the machine is set, the aperture is opened, and photons in the direction of the aperture will be released as a laser beam.
1.3. Modes of Laser Transmission 1.3.1. Continuous wave lasers In a Continuous wave (CW) laser, a constant beam is produced for as long as the laser is engaged. The energy source is continually supplying a population of excited state electrons that are stimulated to decay, creating a laser beam.[8] All medical Nd: YAG, KPT, argon, and CO2 lasers are continuous wave. 1.3.2. Pulsed lasers Pulsed lasers emit far less total energy than continuous wave lasers. However because all the energy is emitted in a very short duration, pulsed lasers currently used in urologic applications involve Q-switching. Other methods can be combined with Q-switching in order to drastically shorten the pulse duration.[8]
Figure 4. Schematic illustration of the coherence.[6]
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1.4. Propertiies of the Laaser Beam Three propperties make laaser unique ovver ordinary ligght. These aree:
1..4.1. Coheren nce Normal lig ght (setting ligght from a tunngsten lamp as a an examplee) is noncoherrent, which m means that thee peaks and valleys v of thee sine wave do d not necessarily coincidee perfectly. B Because of the properties of stimulated em mission and sinnce the photonns of energy forming f the laaser beam are of the same poower value, laaser light is peerfectly in phasse; that is, eacch peak and vaalley of the sine s wave currves align exaactly. Coherennce can be eiither temporall or spatial cooherence. (Fig gure 4) Tempooral coherencee implies that the t relative phhases between two points inn time remain constant. Spattial coherencee implies that the t relative phhases between two points inn space remainn constant. 1..4.2. Monoch hromaticity Laser emitss light over a very narrow, well defined wavelength w (tuungsten lamp emits light ovver the entirre visible sppectrum; that is to say, it has variaable wavelenggths). The m monochromatic city of a laser beam b is attribuutable to its hiigh temporal coherence. c 1..4.3. Collimation (non divvergence) Laser radiaation forms a beam with alm most no diverrgence, that iss, the rays are parallel to eaach other. Thiis is attributabble to its high spatial coherence. Althouggh the power output o of a fluorescent lam mp could be more m than thaat of a laser, the irradiancee (power per unit crossseectional area) of the laser beam, owingg to its high collimation, is i much higheer and can reemain so for greater g distancces. This is duue to the fact that t fluorescennt lamp emits light in all diirections as an a approximattely sphericall wave, the irradiance of which decreaases as the diistance from the lamp inccreases. Owinng to the higgh collimationn of laser, itt is almost im mpossible to arrange a a convventional lightt source to prooduce a beam m of light withh irradiance coomparable to that of laser. The three chaaracters of laser: coherence, monochromaaticity, and coollimation maake it possible to focus a lasser beam into a very small cross-sectiona c al area with veery high energgy densities.
Fiigure 5. Total in nternal reflectioon of light withiin a glass cylindder surrounded by air. The anggle a is the accceptance anglee of the cylinderr.[3]
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1.5. Fiber Optic Wave Guides The development of effective laser delivery systems has been as important to medical applications as the development of the laser itself. Fiber optic technology has been developed largely in response to demand from the communications industry. Optical wave-guides used in medical applications are manufactured from fused silica glass. The propagation of light through the core fiber occurs by a series of reflections from one sidewall to the opposite wall. A layer of lower refractive index glass or plastic called "cladding" covers the fiber core.8 When light obliquely enters the core fiber, it is reflected from the cladding interface back into the core. The degree to which an oblique beam can be propagated through the core fiber is determined by the difference in refractive indices of the core and cladding material. The maximum angle to which an incident beam can enter the fiber and be transmitted is termed the "acceptance angle". (Figure 5)
1.6. Coupling Systems The launch window is the coupling between the laser and the fiber optic wave-guide. Medical lasers have a secure attachment to decrease the possibility of beam exposure to operating room personnel and also to prevent loss of laser efficiency.
1.7. Aiming Beam An aiming beam allows accurate targeting of the optical fiber before laser discharge. Helium: neon (He-Ne), diode lasers, or white flash lamp light may be used for this task. He: Ne lasers produce continuous red (633nm) light. Laser sources are favored over flash lamp light for precise targeting due to less beam divergence.[8]
1.8. Laser Focus Like other electromatic radiation, a laser beam may be manipulated by refractive and reflective surfaces. Laser hand pieces utilize these systems to direct energy to a focal point or to expand the beam.[9]
1.9. Laser- Tissue Interaction The laser beam as it contacts tissues, part of it becomes reflected, and part becomes absorbed by tissues. The effects of laser on tissue are the end result of a complex interaction of variables involving both properties of laser and properties of the tissue being irradiated.[10]
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1.9.1. The Laser beam produces one or more of the following effects 1.9.1.1. Thermal effect Heat created by the laser beam, as it collides with tissues, is responsible for most of the therapeutic values of available lasers. When tissues are heated up to 600 c denaturation of cell enzymes and loosening of cell membranes will take place but no permanent damage to the tissues occurs. (i.e. reversible changes).[10] Above 600 c, irreversible cell protein denaturation takes place that is to say coagulative necrosis. With protein denaturation, the tissues become white in color and firm in consistency. Although cell death occurs, still (and up to 900c) no loss of structural or architectural integrity of the irradiated tissue is encountered.[8 8] When tissues are heated from 900 c to 1000 c, tissue drying and shrinkage starts. Here, definitive architectural permanent damage happens. Above 1000 c, the process of drying out continues and carbonization (blackening) of tissues occurs usually above 1500 c, while tissue pyrolosis and vaporization occur at 3000 c.[10] 1.9.1.2. Mechanical effect The laser energy is transferred into mechanical energy that could produce certain damage to tissues. A good example for this mechanism is using pulsed laser beams in stone disintegration. The laser energy as it is applied in rapid successions (pulses), the instantaneous flux of the emitted photons will generate plasma around the stone. The mechanical effect and pressure variation created by the rapid expansion of the plasma will result in stone disruption.[8 8] 1.9.1.3. Photochemical effect This denotes the effect of some lasers which when applied to photosensitized tissue, produce certain chemical reactions that end in permanent damage of this irradiated tissue.[8 8] 1.9.2. Factors determining the effect of laser on tissues 1.9.2.1. Wavelength and its relation to optical absorption of water and haemoglobin Water absorbs light with wavelengths of 1µ (1000 nm) and above. The capacity of water to absorb light increases as the wavelength of the light increases. The maximum absorptive capacity of water is for light energies of more than 10µm (10000 nm) wavelength.[8 8] The maximum absorptive capacity of haemoglobin is for light energies of 0.6µm (600 nm) or less. As the wavelength of light increases, haemoglobin absorptive capacity decreases.[10] The argon laser operates in the area of the optical spectrum, in which the haemoglobin absorption curve is at its maximum, so this laser is highly absorbed by the pigment haemoglobin in preference to absorption by surrounding tissues. The CO2 laser in the far infrared region of the spectrum lies at the point at which the optical absorption of the water curve is at its maximum. The Nd: YAG laser operating in the near infrared region lies at the point at which both the haemoglobin and water curves are at their lowest. Thus, a YAG laser will penetrate both haemoglobin and water without significant decrease in its energy. (Figure 6).
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Figure 6. Different lasers absorption as regard water and hemoglobin.
1.9.2.2. Power density The power measured in watts divided by the area over which the energy is delivered is known as the power density or power intensity of the laser beam. When the time factor is considered the energy density or irradiation dose is determined. Thus the power density is defined as the concentration of energy of the laser beam and is expressed in watt divided by centimeters squared of the spot size. 1.9.2.3. Depth of penetration The depth of penetration is commonly misunderstood to be the depth to which the laser is functional. This is not correct. The depth of penetration or extinction length is the tissue thickness in which 90% of the laser beam has been absorbed. This does not imply that the laser is still active enough at 90% of its absorption to produce a significant temperature rise to create tissue effects.[10] Therefore, the depth of penetration is based on the calculation of the laser energy penetration, not laser effect. It was found that: For Co2 laser, 97% of its energy is absorbed at the point of contact resulting in strong vaporization effect, and this is why the CO2 laser is called the cutting laser or the laser scalpel. As the exposure time increases; the vaporization effect increases yet the penetrating depth is not more than 0.1 mm at any time.[10] For Nd: YAG laser, 50% of its power is reflected and absorbed while the other 50% will penetrate accounting for the thermal effect previously mentioned. The depth of penetration is around 4-5 mm. For argon laser, 60% of its power is reflected while 40% usually accounts for not more that 1.0 mm penetration (due to absorption by many tissue pigments).[8 8] 1.9.3. Determinants of penetrative power 1.9.3.1. Scattering effect When a laser beam, penetrates into tissue, diffusion takes place laterally and the amount of temperature rise decreases proportionally with penetration. Thus, there will be an area at which penetration is still occurring but significant temperature rise is not achieved.
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1.9.3.2. Wavelength The shorter the wavelength of the laser, the greater the depth of penetration. 1.9.3.3. Power density The higher the power density, the more its penetrative power. 1.9.3.4. Duration of exposure If laser is applied in the form of 25 watts over 4 seconds, the depth of penetration will be more than if 100 watts are applied over one second. This is explained by the fact that surface cooling occurs in the first situation but not in the second. 1.9.3.5. Surface cooling Surface cooling will decrease surface tissue vaporization, thus allowing more tissue penetration. This is called the blooming effect.[11] 1.9.3.6. Mode of exposure Multiple intermittent exposures showed increased depth of penetration than one continuous exposure (if both have the same power setting and total exposure time). 1.9.3.7. Tissue vasculature The higher the tissue vascularity, the less the depth of penetration for two reasons: 1. Increased tissue vascularity will provide tissue with higher concentrations of the pigment haemoglobin which increases the optical absorptive capacity of the irradiated tissue. 2. The increased blood flow produces a cooling effect, thus decreasing the temperature to which the tissue should rise.
2. LASER USE IN UROLOGY Many types of medical lasers are commercially available today, all named according to the medium in the laser cavity.
2.1. Nd: YAG (1064 Nm Wavelength) The solid state Nd: YAG laser is the most frequently used urologic laser. It is within the near infrared region. About 50% of its energy is reflected back from the tissue, while the other 50% is absorbed. Its penetration is 4.0 mm into tissue, and its energy can easily be transmitted through thin quartz fibers.[8]
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2.2. KTP (532 Nm Wavelength) A KPT (potassium titanyl phosphate) laser is simply an Nd: YAG laser with a KPT frequency doubling crystal. It produces green light, which experiences stronger tissue absorption than infrared Nd: YAG energy. Consequently, tissue coagulation depth is decreased.
2.3. Argon (488 Or 515 Nm Wavelength) It emits light with a wavelength in the blue/green portion of the electromagnetic spectrum. The argon is absorbed by haemoglobin or melanin selectively and does not penetrate more than 1.0 mm. Fiber optic transmission of the argon laser is possible.
2.4. CO2 (10,500 Nm Wavelength) It has a wavelength in the far infrared region of the spectrum. Nearly 97% of its energy is absorbed at the point of contact, with a penetration of only 0.1 mm. Laser beam must be directed by a series of mirrors, and can not be transmitted by fiber optic technology.
2.5. Pulsed Dye Laser (504 Nm Wavelength): This type of laser uses a flash lamp to pump a resonator medium filled with coumarin green dye. The pulsed dye laser has successfully been used for ureteral and renal stone fragmentation.[8]
2.6. Alexandrite Laser (577 Nm Wavelength) This device uses a synthetic alexandrite rod. This pulsed solid state laser has been developed for ureteral and renal stone fragmentation.[8]
2.7. Holmium: YAG Laser (2100 Nm Wavelength)
3. HOLMIUM LASER The Ho: YAG laser produces light of 2,100 nm in a pulsed fashion; (a wavelength in the infrared part of the electromagnetic spectrum). This wavelength is strongly absorbed by water, with a depth of penetration of < 0.5 mm Therefore, It effectively can ablate tissue through vaporization. The laser can also coagulate blood vessels as large as 1 mm in diameter
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annd fragment all a types of calculi.[12] Thhe energy of each e pulse annd the frequenncy can be vaaried. The eneergy can be vaaried from 0.22 to 2.8 J per pulse p and the frequency cann be varied frrom 5 to 30 Hz. H The light produced p can be b carried alonng low water density fibers ranging in siize from 200 to 1000 um. Lateral L firing fibers are alsso available. It I is different from other laasers because it does not caause forward scatter s such ass the neodymium-YAG andd unlike the C 2 laser can be CO b carried throough a flexiblee fiber.[13]
3.1. Applicattions The mechaanism of actiion of the hoolmium laser is based on superheating water and crreating a micrroscopic vaporrization bubble at the tip off a low water density d quartz fiber. This bu ubble has suffficient powerr to destabilizze or vaporizee any biologiical material it i contacts. Thermal effectss can be locallized to an areea a few milliimeters from the t fiber tip as a long as a gant is appliedd.[14] The hoolmium laser is i very effectivve for the fraggmentation coontinuous irrig off calculi andd ablation of tissue includding neoplasm m or endolum minal narrowiing. These caapabilities maake it an excellent single lasser for urologiic use becausee these three applications a acccount for the majority of nondiagnostic urinary u tract inndications for endoscopy.[115] Tunable Helium Dye Neon Laser Laser
Argon Laser
400
500
600
Ultraviolet X-Ray
700
Nd YAG Laser
1000
Holmium Laser
1100
(Nanometers)
Visible
11000
Microwave TV & FM Radio AM Radio
Cosmic
Ultraviolet Ionizing
CO2 Laser
Infrared Non-Ionizing
Fiigure 7. Electro omagnetic radiattion spectrum.[7]
Fiigure 8. Ureteraal stone fragmennted under direcct vision with fiber fi applied dirrectly to stone. [7] [
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3.2. Holmium Laser Lithotripsy Stone treatment remains a major indication for both rigid and flexible ureteroscopy. Most of ureteral calculi can be accessed ureteroscopically. Although some calculi can be removed intact most require fragmentation before removal. (Figure 8) Lasers have played a major role in expanding the therapeutic role of ureteroscopy. The pulsed dye laser was the first laser widely accepted and applicable for stone fragmentation. It represented a major improvement that allowed smaller endoscopes to provide therapeutic role. The holmium laser unlike the pulsed dye laser is able to successfully fragment any stone. It does not discriminate against composition, color, size or location and truly can be called the “workhorse of the laser world”.[13]
3.3. Physics of Stone Fragmentation Several lasers currently available for intracorporeal stone fragmentation have been found to fragment stones through generation of a shock wave. These short pulsed lasers induce the rapid formation of a spherical plasma cavitations bubble that expands symmetrically to a maximum size and then collapses violently. Bubble collapse leads to the generation of a strong shock wave that comprises the primary mechanism of fragmentation.[16] The holmium laser mechanism of action, with its unique characteristics of longer pulse duration and wavelength, is related to a photothermal mechanism that occurs by direct absorption of the holmium energy by the stone. In other words, the stone is literally melted. Support for this theory arises from the finding that efficiency of holmium laser stone fragmentation increases with increased stone temperature and that thermal byproducts for all stone compositions tested are formed on the surface of craters that are produced during holmium fragmentation. Moreover, thermal byproducts were not recovered on nontreated stone surfaces, whereas soluble thermal byproducts were recovered from the solution. These findings suggest that a thermo chemical reaction generated by the laser occurs on the stone surface.[16] This different mechanism of action has several clinical implications, most of which favor the holmium laser over other intracorporeal lithotripsy devices. First the absence of a strong shock wave avoids the proximal migration of stone fragments and minimizes the risk of scatter damage to adjacent tissues and endoscopic equipment, which is more common with EHL and pulsed dye laser energies. It has been suggested, however, that this weak shock wave is indeed strong enough to dissipate stone dust created by stone breakdown. Second, studies have demonstrated that holmium laser stone fragmentation yields smaller fragments than those produced by pulsed lasers, EHL, or pneumatic lithotripsy, when fired on a wide variety of stones of different compositions. Moreover the holmium laser was the only lithotrite that did not yield fragments greater than 4 mm, and consistently produced a significantly greater number of fragments less than 1 mm in size. Clinically these findings imply less need for auxillary procedures to remove residual fragments, and therefore less operative time (less need for stone retrieval). Furthermore, the ability to literally vaporize a stone to dust particles (an event that is not possible with other devices) combined with the minimal trauma associated with the use of currently available endoscopes has recently
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quuestioned the need for stentting after uretteroscopy. Thiird, because all types of stoones absorb thhe holmium energy, this laaser can be ussed to fragmennt all stone tyypes, includinng the hard cyystine and calccium oxalate monohydrate m s stones.[16]
3.4. Patternss of Stone Frragmentatioon Several diffferent patternns of fragmenntation can bee employed too achieve stonne removal inncluding “drilll and core”, “ablate “ and ch hip” or ”direcct fragmentatiion”.[1 13] ”Drill and d core” includees application of the fiber too the calculus to drill a hole toward the ceenter. The lasser fiber tip is then left within w the cennter of the sttone while acctivation is coontinued. In this t way the central portioon of the calcculus or coree is removed. The outer suurface of the stone is thenn fragmented directly withh removal of the larger fraagments or coomplete ablatiion of the calcculus. (Figure 9) Care shoulld be taken to drill a hole larrge enough too allow the heeated irrigant and a any steam m formed to escape from thhe center of thhe calculus. O Overheating thaat area may melt m the fiber. The fiber shouuld not be advvanced too farr because it m exit the farr side of the caalculus and daamage the uretter.[1 may 13]
Fiigure 9. The driill and core techhnique of fragm mentation.[7]
Fiigure 10. Ablatee and chip technnique.[7]
Fiigure 11. Directt fragmentationn technique.[7]
242
Ahmed Safwat
The pattern described as “ablate and chip” is started with a hole drilled in the surface. Circular movement of the fiber around this hole will enlarge it. The fiber is continually applied to the edge of the crater. If it is applied 1-3 mm away from the lumen of the crater then chips are formed thus increasing the rate of stone removal. In this way a large central crater is formed. The edges of the calculus can be removed circumferentially.[13] “Direct fragmentation” can be employed for stones and fragments of larger calculi. It is helpful to apply the fiber along cleavage planes on the surface of the stone. As the laser fiber is applied and activated to the smaller fragments they may break into two or more smaller fragments. Fragments small enough to pass from the ureter spontaneously may be left in place, whereas larger ones may be removed with graspers or baskets.[13]
4. FUTURE OF LASER LITHOTRIPSY The development of laser technology for use in intracorporeal lithotripsy was stimulated by the need to have a device that could be placed in the urinary tract through small instruments as well as is used with minimal surrounding tissue damage. The development of instrumentation specific for laser use has improved the success of laser lithotripsy. The currently used 7.2 F and 6.9 F semi rigid ureteroscopes have the advantages of a rigid instrument that has a flexible optical fiber so that it may be bent without losing vision. A second advantage is that these ureteroscopes may be passed directly into the ureter without the necessity of tunnel dilation. Small baskets can be passed through the working channels of these miniscopes. As described earlier, the laser is particularly useful for impacted calculi in the ureter. The 6.9- F semi rigid ureteroscope is passed into the ureteral orifice without dilation to the impacted stone and the laser is introduced under direct vision and fragmentation carried out. Fragmentation of stones with the currently available laser lithotripters has comparable clinical results; however each of the lasers has its specific drawbacks. The decision as to which laser would best treat calculi in the future will be based on the ability to be used with less invasive approaches (fluoroscopic control under local anesthesia), cost efficiency, and the development of smaller specific ureteroscopic instrumentation.[17]
5. LASER SAFETY Safety is an issue of paramount importance in all areas of surgery but it takes on greater relevance when discussing new technology. When used with proper training and experience, laser surgery can be performed with a high margin of safety. In fact, the potential for fewer complications and less morbidity than alternative procedures is one of the advantages of lasers in medicine. On the other hand, failure to follow certain well-established guidelines for safety can have drastic implications.[8]
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5.1. Laser Injuries If a person is accidentally exposed to a laser beam, two organs are at great risk for various injuries: the eye and the skin.
(A) Laser injuries to the eyes The potential injury to personnel that requires great attention during laser surgery is damage to the eye. Both direct and reflected laser beam can cause eye injury. Powerful reflecting surfaces as mirrors and reflecting metallic surfaces should be avoided in furnishing a laser theatre. Laser eye injury can be in the form of corneal burns up to corneal ulceration that may predispose to a large corneal scar as with CO2 laser. Also Nd: YAG laser and argon laser can result in retinal damage leading to defects in the field of vision (i.e. scotomas). Avoiding these serious complications is possible through: 1. Wearing special goggles by all personnel in the room during the use of laser 2. The patient's eye should be protected by a double layer of moistened eye pads. 3. Careful handling of the laser fiber and caution during operating the laser.
(B) Laser injuries to the skin If the laser beam inadvertently comes in contact with the skin of the operating surgeon or an assistant, a thermal injury can occur. Because of the slight amount of beam divergence, the target tissue must be in relatively close proximity to the end of the fiber or hand piece for the thermal effect to be injurious. The severity of burn depends also on the power of the beam and the duration of exposure. Most often, laser burns occur on the fingers or hands of the surgeon. The injuries are mostly not serious but are completely avoidable through careful handling of the laser fiber or hand piece. Protection of the patient's skin with moist towels is required as well. It should be noted that laser use in urology has not exceeded two decades thus; time may uncover other hazards of laser. For this reason, it is that extreme caution should be followed during the use of laser. Holmium: YAG lithotripsy of uric acid calculi risks production of cyanide, which raises significant safety issues.[18]
5.2. Practical Guidelines for Laser Use 5.2.1. Environmental Safety Prior to their entering into a controlled laser area, proper precautions are necessary to protect the patient and personnel. In some cases, door interlocks may be appropriate (as, for example, on a laser laboratory door that opens into a public corridor next to a pediatric clinic where individuals may inadvertently wander into the control area).[8 8] It must be noted that if door interlocks are implemented, it is an advantage to have them wired to a standby mode and not a laser “shut off" mode. Alternatives to this would be placing proper signs on the outside of the control area, having a flashing sign activated when the laser is in the operating mode, and providing proper eye wear outside of the entryway. The
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Ahmed Safwat
following recommendations, taken from a draft copy of the document [American National Standards Institute (ANSI Z136.1)], are applicable to most medical laser treatment areas: • • • • •
Appropriate warning signs. Supervised and only be occupied by authorized personnel. Be under the direct supervision of an individual knowledgeable in laser safety. Should be a controlled area. Any potentially hazardous beam terminated in a beam stop of an appropriate material. • Only diffused and reflective material in or near the beam path when feasible. • Personnel who regularly require entry into the laser treatment control area are adequately trained and provided with appropriate protective equipment. To ensure that all of the appropriate administrative and procedure controls are performed, all windows, doorways, and other ports are covered appropriately. (31) Where safety latches or interlocks are not feasible or inappropriate, the following shall be observed: A door, blocking barrier, screen or curtain, etc., shall be used to block or attenuate the laser radiation at the entryway. Outside the entryway shall be a visible or audible signal indicating that the laser is energized. A lighted laser warning sign or flashing light are appropriate methods to accomplish this.[8 8] (Figure 12) Since medical lasers often use electricity for power, electrical water hazards may be present within the immediate environment. Isolated transformers and Hubble lock receptacles should be installed initially in the treatment areas to help prevent electrical problems. To prevent exit of the laser beam from the immediate operating area, windows should be protected from wavelengths that are transmitted through glass and plastic by covering the windows with blinds or some other protective material.[8]
Figure 12. Warning sign indicating laser in use in the operating room
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245
5.2.2. Equipment safety Installation of laser equipment by electrical engineers or bioengineer within the hospital should be coordinated with engineers of the laser manufacturer. (31) Power requirements and water requirements should be well reviewed before the laser is on site for installation. It must be noted that there are specific parameters for water pressure. Pressure or water temperature below manufacturers' specifications can place the machine on a water interlock. During a hot summer month the cold water temperature may exceed the range of the machine‘s specifications. Improper calibration of a laser because of problems with water temperature could alter power output and therefore tissue effects. Periodic safety inspections and equipment maintenance should be performed either by the manufacturer or by the hospital‘s engineering department.[8]
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
Einstein, A. (1971). Zur quanten Theorie der Strahlung. Phys Zeit, 18, 121. Maiman, T. (1960). Stimulated optical radiation in ruby. Nature, 187, 493. Mc Nicholas, T. A. (1990). Lasers in urology: Principles and practice, SpringerVerlag London Limited, 83-118. Stein, B. S. (1986). Laser physics and Tissue Interaction, Urologic Clinic of North America, 13, 3. Stein, B. S. & Kendall A. R. (1984). Lasers in urology: I Laser physics and safety. Urology, 23, 405. (1993). Berlin, University: Lasers in surgery, Germany. Demetrius, H. (2001). Bagley: Endourologic use of the Holmium Laser, Teton Newmedia, Ch.2: 8; Ch.4: 23-28. Smith, J. A., Stein, B. S. & Benson, R. C. (1994). Lasers in Urologic Surgery Third Edition Mosby-Year Book, 1-9. Stein, B. S. (1986). Laser physics and Tissue Interaction, Urologic Clinic of North America, 13, 3. Schaeffer, A. J. (1986). Use of Co2 laser in urology. Urol. Clin N Amer, 13, 297. Mills. T. N. (1990). An introduction to lasers and laser physics, Springer-Verlag London Limited, Ch.1, p 5. Erhard, M. J. & Bagley, D. H. (1995). Urologic Applications of the Holmium Laser: Preliminary Experience: J.Endourol, 9, 5, 383-386. Bagley, D. H. & Erhard, M. (1995). Use of the Holmium Laser in the Upper Urinary Tract. Technique in Urology, 1, 25-30. Webb, D. R., Kockelburgh, R. & Johnson, W. F. (1993). The Versapulse holmium surgical laser in clinical urology: a pilot study. Min Invas Ther, 2, 23-6. Abdel-Razzak, O. M. & Bagley, D. H. (1992). Clinical experience with flexible ureteropyeloscopy. J. Urol, 184, 1788-1792. Delvecchio, F. C., Preminger, G. M. (2000). Endoscopic management of urologic disease with the holmium laser. Current Opinion in Urology, 10, 233-237.
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[17] Bennett, J. P. & Dretler, S. P. (1994). Laser in Urologic Surgery Third Edition Ch 15 Mosby-Year Book, 190-198. [18] Teichman, J. M., et al. (1998). Holmium: YAG lithotripsy: photothermal mechanism converts uric acid calculi to cyanide: J Urol, Aug; 160(2), 320-4.
In: Horizons in World Physics. Volume 271 Editor: Albert Reimer
ISBN: 978-1-61761-884-0 © 2011 Nova Science Publishers, Inc.
Chapter 5
EXPRESSION OF FULL VECTOR VERTEX FUNCTION IN QED A.D. Bao* Department of Physics, Northeast Normal University, Chang chun 130024,P.R. China and Center for Theoretical Physics, College of Physics, Jilin University, Chang Chun 130023, P. R.China
ABSTRACT The complete expressions of the full fermion-boson vertex functions with transverse component in four dimensional QED are presented by solving a complete set of the Ward Takahashi type’s identities in the momentum space without considering the constraint imposing any Ansatz. In the colculation of reducing vertex function , the topological singularity of the various fermion currents coupling gauge field is taken fully into account.The computation shows that there is no anomaly for the transverse WardTakahashi relation for the vector vertex and axial-vector vertex.
Keywords: Green function; Ward-Takahashi identity; Abelian gauge field; Anomaly PACS number: 11.30 Na, 11.15 Tk, 12.20. Ds, 11.15Wx..
I. INTRODUCTION The knowledge of the fermion-boson vertex function (three-point vertex ) is crucial in the nonperturbative studies of the gauge theories, such as the dynamical chiral symmetry breaking, confinement, through the use of the Dyson-Schwinger equations [1].In Abelian gauge theories, QED may have a quite surpring nonperturbative behavior if the couping is much larger than the dimensionless fine-structure constant [2,3].To learn whther such results are real consequences of such a four-dimensional field theory requires a systematic *
Corresponding author: E-mail address: Baoad433 @ nenu.edu.cn
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A.D. Bao
investigation needed to make such nonperturbative studies tractable [4]. In the past years, much effort has been devoted to constructing the transverse part of the vertex based on perturbative constraints in ensuring multiplicative renormalizability and in determining the propagator.The built transverse part depends on the Ansatz not coming from the constraint imposed by the symmetry of the gauge invariance [5,6]. Takahashi first discussed the constraint relation for the transverse part of the vertex from symmetry in the conventional field theory, which is called the transverse WT relations [7]. Recently some progress on the problem has been made by using formal operator approach grounded on the first principles in Abelian gauge theory [8]. The transverse part of GF as well as the full vertices are determined by the WT ralations without any Ansatz, in which the complete sets of Ward-Takahashi identities without considering anomaly are solved by computing the curl of the time-ordered products of three-point Green functions involving the vector, the axial-vector and the tensor current operators respectively. In order to derive the WT relations in a general way, the pathintegral formalism has been attempted in Abelian gauge theory [9]. However, in the computation of the vertex functions the differential equations of motion for the fermions can not be used to reduce fermion-photon vertex in QED in the most straightforward way, since the product of the local operators is often singular,which can destroy symmetries of the conservation current and the classical equations of motion.For the reason, the singgularity of fermion current couping to gauge field has to be examined through topological mathed, when using the equations of fermion motion to reduce vertex [10]. Therefore the presence of anomalous terms associated with the WTIs has been taken into account in a model of the quantized Dirac field with arbitrary internal degrees of freedom having arbitrary nonderivative coupling to external scalar、pseudo scalar、vector and axial-vector fields by operator approach [11]. The chiral-symmetry-breaking anomaly enters only in the integration measure, which is is first realized by Fujikawa in the path –integral formulation of quantum field theory [12]. Apart from the anomaly, it is still desirable to know what types of fermion current can give rise to anomaly coming from the symmetry of quantized gauge theory. As for the issue, Jacobian factor of the integration measure under the various fermion transformations have displayed by generalizing the use of the Fujikawa’s method [9]. In this paper, in order to derive the WT relations, we introduce new local transfornations of field variables preserving the invariance of generating functional itself in QED theory. This fact holds irrespective whether these re-naming field variables take the form of symmetry of action, or not.The local transformation of fields can induce the transverse part of the vector vertex function through the transverse WT identities for the vector, axial-vector in the theory. In secton II, In the gauge theory the group parameter being a scalar function can be expanded as a Dirac matrix series, so that the local transformations naturally provide different types of interaction currents in the variation of the Lagrangian Leff without loss of any rigorousness. The introduced postulate for transformation parameter is a key to the derivation of various WT relations associating with anomaly terms. Furthermore, the collection of the local transformations with the parameter forms a connected Lie group [13]. In section III, we are devoted to deriving the anomaly term arised from the quantum measure due to the transformations of field variables by generalizing Fujikawa’s method. The regulator
2 ⎛ ⎛D / ⎞ ⎞ f ⎜ − ⎜ ⎟ ⎟ to the integration measure is modified under the transverse ⎜ ⎝M ⎠ ⎟ ⎝ ⎠
Expression of Full Vector Vertex Function in QED
249
transformation of fermion variables.The trivial or non-trivial Jacobian due to the transformations are evaluated respectively. In order to reduce the vector vertex in computations by using the equations of fermion motion, we develop a topological approch to examine the singularity of the product of quantum operators of fermion currents coupling to gauge field .As a consequence, these WT relations give the complete expressions of the full vector, the full axial-vector vertex functions. In the last section, some remarks on the comparison between our results and the previous investigations will be made.
II. SYMMETRY TRANSFORMATION ON GENERATING FUNCTION The symmetry transformation preserving the invariance of generating functional itself leads to a variety of exact relations amang the Green’s functions [14]. From the mathematical viewpoint, the changing variables in feild transformation in an integral never affects its value. Therefore we now pay attention to the structure of the parameter θ ( x ) in the transformations. Taken broadly, the changing variables of matter field in theory can be transformed as [15].
φα' = φα + δφα
(2.1)
δφα = θ[ν ] (x )Hα[ν ] (φβ ,φβλ ) + ∂ μθ [ν ] (x )hαμ [ν ] (φβ ,φβλ )
(2.2)
[ν ]
μ [ν ]
where the matrices H α , hα
are the matrix functions composed of both Dirac matrices
(Dirac algebra is the algebra of 4×4 complex matrices) and field variables. The dummy indices [ν ] denote the index set for tensor rank. As viewed from mathematics, a infinitesimal scalar function
θ ( x ) can be expanded as a
series with tensor functions in matrix form
θ (x ) = θ c I + θ (x )I + θ μ (x )iγ μ + θ 5 (x )γ 5 + θ μ 5 (x )γ μ γ 5 + + θ μν 5 (x )iσ μν γ 5 + θ μν ( x )iσ μν + θ μνκ 5 ( x )ε μνκρ γ ρ γ 5 + LL where
(2.3)
θ [ν ] ( x ) are a set of arbitrary real tensor functions of x , The above idea comes into
being a physical interpretation tractable. The various couplings currents may occur in the process of mutual interaction of particles. Subsequently it is necessary to examine the property of the gauge transformations with the continuous parameter θ ( x ) . By taking an operator U (T (θ ( x ))) as a representation for the symmetry transformations, the transformations T (θ ( x )) , which are described by a finite set
of real continuous parameter, induce the linear transformation on the matter field in physical Hilbert space
250
A.D. Bao
ψ → U (T (θ (x )))ψ
( 2.4 )
Thus the operators U (T (θ ( x ))) can be represented in at least a finite neighborhood of the identity by a power series
U (T (θ ( x ))) = 1 + iθ a T a + where
1 2 a b a b i θ θ T T + LL , 2
θ a (x) are a set of continuous parameters, T a
and obey commutation relations
[T
a
]
(2.5)
are generators relating to electric charge
T b = 0 . Connecting with the parameterization Eq.
(2.3), the operators U (θ ( x )) change into
U (θ ( x )) = e iθ
a
( x )T a
=e
(
)
i θ [ν ]H [ν ] T a a
(2.6)
The group multiplication law then takes the form
U (θ1 )U (θ 2 ) = U ( f (θ1 ,θ 2 )) , the law can be performed (for infinitesimal
e
iθ [ μ ] ( x )Γ [ μ ]
e
iθ [ν ] ( x )Γ [ν ]
=e
iθ [λ ] ( x )Γ [ λ ]
θ (x ) ) e
[
]
1 − θ [ μ ] ( x )θ [ν ] ( x ) Γ [ μ ] , Γ [ν ] − ( higher − order ) 2
where we write the transformation parameter as order
(2.7)
θ [ν ]Γ [ν ] ,
(2.8)
the factors including the higher
θ μθν should be neglected. In virtue of the property of Dirac gamma matrices, a
collection of the operators U (θ ( x )) meet the group properties such as closure, associativity, an identity element and an inverse, which is called a matrix Lie group. Based on the above argument, it is expected that the invariance of the generating functional itself under the group transformations relating to the postulate Eq(2.3) will lead to a set of Ward-Takahashi type’s identities.
III. FULL FERMION-BOSON VERTEX FUNCTION According to Fujikawa’s interpretation, it is argued that the appearance of the anomaly in WTI is a symptom of the impossibility of defining a suitably invariant functional integral measure with respect to the relevant transformations on fermionic field variables. The analysis based on the use of path integrals provides access to anomaly objects for other WT identities. The other hand, the singularity of the operator product of fermion currents has to be examined in the reduction of vertex.
Expression of Full Vector Vertex Function in QED
251
Let us consider the transformation
ψ ' (x ) = e
− iθ ( x )ε μν σ μν
ψ ' (x ) = ψ γ 0 e
ψ
iθ ( x )ε μν σ μν
+
γ0 ,
（3.1）
B ' μ (x ) = Bμ (x ) − ∂ν θ (x )ε μν where
ε μν stands for the antisymmetry tensor.
The change of the function integral due to the transformation is given by WT relation
(
)
(
)
∂ λ 0 T ψ (x )g λμ γ νψ (x )ψ (x1 )ψ (x 2 ) 0 + ∂ λ 0 T ψ (x )g λν γ μψ (x )ψ (x1 )ψ (x 2 ) 0
(
)
(
)
− i∂ λ 0 T ψ (x )ε λμνρ γ ρ γ 5ψ (x )ψ ( x1 )ψ ( x 2 ) 0 + i∂ν 0 T ψ (x )γ μψ (x )ψ (x1 )ψ (x 2 ) 0
(
)
+ 2i 0 T ψ ( x )TB ψ (x )ψ (x1 )ψ (x 2 ) 0 + iδ ( x − x 2 )σ [μν ]
μν
0 T (ψ (x1 )ψ (x )) 0
(3.2)
− iδ (x − x1 )σ μν 0 T (ψ (x )ψ (x 2 )) 0 = 0
where
TB[μν ] = B μ γ ν − Bν γ μ .
In momentum space, it changes to
− iq μ ΓVν ( p1 , p 2 ) − iqν ΓVμ ( p1 , p 2 ) − ε λμνρ q λ ΓAρ ( p1 , p 2 ) + qν ΓVμ ( p1 , p 2 ) − 2igΓB[μν ] + S −1 ( p 2 )σ μν − S −1 ( p1 )σ μν = 0
(3.3)
Obviously, the full vector functions and the full axial –vector functions are coupled with μ
each other. The apparent feature of this transverse identity is that the vertex function ΓV (fermion’s three point function) has the transverse component of itself. Completely analogous to the calculations above, consider the other transformation
ψ ' (x ) = e
−θ ( x )ε μν σ μν γ 5
ψ ' (x ) = ψ γ 0 e
ψ
iθ ( x )ε μν γ 5 +σ μν
+
γ0
B ' μ (x ) = Bμ (x ) − ∂ν θ (x )ε μν we have the identity
(3.4)
252
A.D. Bao
( ) ( − ∂ 0 T (ψ ( x )ε γ ψ ( x )ψ ( x )ψ ( x )) 0 + i∂ 0 T (ψ ( x )γ ψ ( x )ψ ( x )ψ ( x )) 0 [ ] − 2 0 T (ψ ( x )T ψ ( x )ψ ( x )ψ ( x )) 0 + δ ( x − x )σ γ 0 T (ψ ( x )ψ ( x )) 0
)
− i∂ λ 0 T ψ ( x )g λμ γ ν γ 5ψ ( x )ψ ( x1 )ψ ( x2 ) 0 − i∂ λ 0 T ψ ( x )g λν γ μ γ 5ψ ( x )ψ ( x1 )ψ ( x2 ) 0 λμνρ
λ
ν
ρ
1
μ
2
1
μν
μν
B
− δ ( x − x2 )σ
μν
1
2
2
(3.5)
5
1
2
γ 0 T (ψ ( x1 )ψ ( x )) 0 = 0 5
The identity for the axial –vector current is rewritten as in momentum space,
− q μ ΓAν ( p1 , p 2 ) − qν ΓAμ ( p1 , p 2 ) + iε λμνρ q λ ΓVρ ( p1 , p 2 ) + qν ΓVμ ( p1 , p 2 ) + 2 gΓB[μν ] + iS −1 ( p1 )σ μν γ 5 − iS −1 ( p 2 )σ μν γ 5 = 0
(3.6)
Being similar to Eq.(3.3), the axial-vertex functions in Eq.(3.6) couple with the vector vertex functions. Clearly, no amomaly term contributes to the transverse WT identity for the axial-vector vertex in the QED(Appendix A). As a straightforward application of these WTIs, we can now give the expressions of the full vector axial-vector and tensor vertex functions by solving the above set of coupled equations .In order to illustrate clearly the physical meaning of the identity Eq.(3.3), the full vector vertex function is decomposed into transverse and longitudinal modes
ΓVμ ( p1 p2 ) = ΓVμ( L ) ( p1 p2 ) + ΓVμ(T ) ( p1 p2 )
(3.7)
It goes without saying that, Considering the antisymmetry property of ε λμνρ and σ μν , the full vector vertex gives naturally the identity
qμ ΓVμ(T ) ( p1 p2 ) = 0
(3.8)
The longirudinal mode is just the well-known form
qμ ΓVμ( L ) ( p1 p2 ) = S −1 ( p1 ) − S −1 ( p2 )
(3.9)
We can derive the full fermion vertex as a consequence of the fundamental relations (3.3) and (3.6).
(
(
)
)
ΓVν = − q −2 qν − ε λμνρ q μ q λ q −4 − q ρ γ 5 + q ρ + iσ αρ q α γ 5 + q −2 q μ σ μν S −1 ( p1 ) +
(
ν
+ −q q −ε −2
λμνρ
q μ qλ q
−4
(− q γ ρ
5
)
)
− q ρ − iσ αρ q γ 5 − q q μ σ μν S −1 ( p 2 ) − α
−2
⎞ ⎛ q2 F ( p1 p 2 ) + 2q α TBA[αρ ] ⎟⎟ + 2iq μ q −2TB[μν ] − ε λμνρ q μ q λ q −4 ⎜⎜ 2mΓ5 ( p1 , p 2 ) + i 2 16π ⎠ ⎝
In the calculation, the axial-vector identity is used
, (3.10)
Expression of Full Vector Vertex Function in QED
q λ Γ Aμ ( p1 p 2 ) = 2 mΓ5 ( p1 , p 2 ) + γ 5 S −1 ( p 2 ) + γ 5 S −1 ( p1 ) + i
253
q2 F ( p1 , p 2 ) 16π 2
and
∫d
4
xd 4 x1 d 4 x 2 e i ( p1x − p2 x2 − qx ) 0 T (ψ ( X )ψ (x1 )ψ ( x 2 )) ε μνλρ Fμν ( x )Fλρ ( x ) 0
= (2π ) δ 4 ( p1 − p 2 − q )iS F ( p1 )F ( p1 p 2 )(q )iS F ( p 2 ) 4
It is shown that the full vector and the full axial-vector vertex functions can be expressed entirely and rigorously in terms of two –point functions, the scalar and the pseudo –scalar vertex functions, respectively. The vertex function is very complicated. It is also shown that the contribution of anomaly is indispensable for the full vertex.
CONCLUSION In this paper, we have obtained successfully the presentation of the transverse part and longitudinal part of full vector vertex functions by solving the complete set of WT type’s identities in a quantized gauge theory (QED). It should be emphasiszed the WT relations holds true for the transformations with generalized parameter θ ( x ) which may or may not leave the action invariant. By introducing a postulate of the parameter of transformation, such a collection of local transformations is confirmed to meet the properties of connected Lie group U (θ ( x )) . As already described, the path-integral method provides a general regularization procedure handling the anomaly factor associated with the WT identities. In particular, the calculation shows that no quantum anomaly for the transverse Ward-Takahashi relation to the axial-vecter vertex arises. As a consequence, the full vector, the full axial-vector vertex functions exactly and completely derived from a set of WT relations without any Ansatz. There are many advantages of the derivation given in this paper. First the transvers WT relations can be obtained under the Lie group transformation in the path-integral formalism. Second Jacbian factor coming from integral measure duo to the group transformations can be evaluated throuhg the use of Fujikawa’s methed, which is relating to the anomaly term.Third in the reduction of the vertex, the singularity of operator productor of various fermion current coupling gauge field is examined.Therefore there is a viable probability to derive Full Green function in the case of effective QCD with Fadeev –popov ghost field.
REFERENCES [1] [2] [3]
Roberts, C. D. & Williams, A. G. (1994). Prog.Part.Nucl.Phys., Vol 33, 477 . Booth, S. P., Kenway, R. D. & Pendleton, B. J. (1989). Phys.Lett., Vol B228, 115. Kondo, K.. I. (1991). Nucl.Phys, VOL B351, 259.
254 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
A.D. Bao Curtis, D. C. & Pennington, M. R. (1991). Phys. Rev., Vol D 44, 536 Ball, J. S. & Chiu, T. W. (1980). Phys.Rev., Vol D22, 2542. Maris, P. & Roberts, C. D. (1997). Phys.Rev., Vol C56, 3369. Takahashi, Y. (1978). Phys. Rev., 1976, Vol D 15, 1589 ; Nuovo.cimemto , Vol A 47, 392. He, H. X. (2001). Phys. Rev., Vol C 63, 025207. Baoand, A. D. & Wu, S. S. (2007). Inte.Jour.Theo. Phys., Vol 46, 12. He, H. X., Khanna, F. C. & Takahashi, Y. (2000). Phys.Lett., Vol B480, 222. Bao, A. D. & Wu, S. S. (2009). Chin.Phys., Vol C33, 177. Bardeen, W. A. (1969). Phys.Rev., Vol 184, 1848. In: R. Jackiw, & K. Johnson, ibid., Vol 182, 1459. Fujikawa, K. (1979). Phys.Rev.Lett., Vol 42, 1195; Phys.Rev.,1980, VOL D21, 2848; ,1984,VOL D29, 285 . Brianc. Hall, Lie group, Lie Algibras, & Representations, (2004). Springer, USA, 30. Itzyksin, C. & zuber, J. B. (1986). Quantum Field Theory M.Graw-Hill，New York, 551. Yong, B. L. (1987). Introduction to quantum field theory, science press, Beijing China, 56.
APPENDIX A: CALCULATON OF ANOMALY FACTOR The usual gauge-invariant QED Lagrangian density Leff is
Leff = ψ iγ μ (∂ μ −iBμ ( x))ψ −ψ mψ + LG + LFG LG = −
(
1 μν 1 μ F Fμν L FG = − ∂ Bμ 4 2ξ
(For convenience, the coupling constant
(A1)
)
2
g = −e is suppressed and repeated indices are
generally summed later on; In the sense of the Minkowski space, metric tensor
g μν = diag(+ 1,−1,−1,−1) and the anticommutation relations γ μ γ ν + γ ν γ μ = 2 g μν ).
In simple case, the fixed gauge term in Eq. (A1) is dropped out by taking the special gauge ( ξ → ∞ ) . We now turn to a calculation of anomaly in the transformation of the measure.The general localized infinitesimal group transformation rules for the fermion and gauge fields now take the form
ψ ' (x ) = e
−iθ [ν ] ( x )Γ [ν ]
ψ ' (x ) = ψ γ 0 e
ψ +
iθ[ν ] ( x )Γ [ν ]
γ0,
(A2)
Expression of Full Vector Vertex Function in QED
255
B ' μ ( x) = Bμ ( x) − ∂ μθ ( x ) [ν ]
where Γ denots a set of an algebra terms being Dirac matrices (herein the repeated index [ν ] is fixed). The change in functional measure varies with the group transformations
(
dμ → dμ = ∏ dc n' ∏ dc n' = (det f nm ) det f ' nm −1
'
n
n
) ∏ dc ∏ dc −1
m
m
m
(A3)
m
The corresponding Jacobian factor becomes i d 4 xA[ν ] ( x )θ[ν ] ( x )
(det f nm )−1 = e ∫
(det f ) '
where anomaly function
−i d =e ∫
−1
nm
4
xγ 0 A [ν ] ( x )γ 0θ [ν ] ( x )
,
(A4)
A[ν ] (x) denotes the trace of Dirac matrix Γ [ν ] in the function
space. As discussed in Ref.［12］, the above anomaly function can be written as the limit of a manifestly convergent integral
A[ν ] ( x ) = lim
M →∞
A [ν ] ( x ) = lim
M →∞
d 4x
∫ (2π ) d 4k
∫ (2π )
4
4
⎛ − Hψ e −ikx Γ [ν ] f d ⎜⎜ 2 ⎝ M
⎞ ikx ⎟⎟e ⎠
⎛ − Hψ + e −ikx γ 0 Γ [ν ] γ 0 f d ⎜⎜ 2 ⎝ M
⎞ ikx ⎟⎟e ⎠
(A5)
(A6)
⎛ ⎛ λn ⎞ 2 ⎞ where power number d of the function f ⎜ − ⎜ ⎟ ⎟ takes an integer 1 (or 2), the former is ⎜ ⎝M ⎠ ⎟ ⎝ ⎠ corresponding to first-rank tensor function θ [ν ] ( x ) , the other is to high rank tensor currents.
In addition the transformation of the field Bμ (x ) is a translation, so that its Jacobian is
trivial. Thus the Jacobian related to general transformations Eq.(3.1) can be put in the form i d 4 xΓ [ν ] ( x )θ [ν ] ( x ) −i d 4 xΓ [ν ] ( x )θ [ν ] ( x )
J =e∫
e ∫
Jacobian of the meaure due to various fermion transformations are evaluated below:
(A7)
256
A.D. Bao i)
Γ [1] = 1 ,,
J [1] = 1 . ii)
Γ[5] = γ [5] , J [5] = e
iii)
(
)
~ ⎞ ⎛ i 2 i d 4 xθ 5 ( x )⎜ tr Fμν Fρσ ⎟ ⎝ 16 π 2 ⎠
∫
.
(A9)
Γ[ν ] = γ μ ,
iv) Γ
v)
(A8)
[μ 5 ]
J [μ ] = 1 .
(A10)
J [μ 5 ] = 1
(A11)
J [λμν ] = 1
(A12)
J [μν ] = 1 ,
(A13)
= γ μγ 5 ,
Γ [λμν 5] = ε λμνρ γ ρ γ 5 ,
vi) Γ [μν ] = σ
viii) Γ
[μν 5 ]
μν
,
= σ μν γ 5 , J [μν 5 ] = 1 . (A14）
In the above calculation, Dirac matrix’s properties are used sufficiently.
APPENDIX B: WARD-TAKAHASHI IDENTITIES We are now in a position to derive general Ward-Takahashi identities associated with the variations of field variables (A2) in a broad sense. In the QED the generating functional with external sources J μ , η and η :
Expression of Full Vector Vertex Function in QED
[
]
[
] {
(
257
)}
Z η η J μ = ∫ D ψ ψ B μ exp i ∫ d 4 x Leff + J μ B μ + η ψ + ψ η ,
(B1)
Hence,the variation of the generating functional with respect to parameter θ [ν ] ( x ) due to the group transformation gives a generation equation ⎛ J + iδI eff + ⎞ μ ⎜ ⎟ ⎧⎪ ⎫ ⎛ ⎞ J B + + η ψ δ (B2) ⎪ μ 4 μ ⎜ ⎟⎬ ⎜ ⎟ [ ] D B iI i d x + ψ ψ exp =0 ⎛ ⎞ J B δ ⎨ μ μ ∫ ∫ 4 ⎜ ⎟ ⎜ ⎟ ⎪⎩ ⎝ +ψ η ⎠⎪⎭ δθ [ν ] (x ) ⎜⎜ + i ∫ d x⎜ ⎟ ⎟⎟ ⎝ + η δψ + δψ η ⎠ ⎠ θ[ν ] ( x )=0 ⎝
where the variation of the generating functional is parameterized by
θ [ν ] ( x ) (all of θ [ν ] ( x )
are regarded as formally independent functions to each other). The sympol
J denotes a
Jacobian factor of the transformation measure, the variation of the action I eff is derived from the variations of field variables.
In: Horizons in World Physics. Volume 271 Editor: Albert Reimer
ISBN: 978-1-61761-884-0 c 2011 Nova Science Publishers, Inc.
Chapter 6
C ONDENSATE F RACTION IN M ETALLIC S UPERCONDUCTORS AND U LTRACOLD ATOMIC VAPORS Luca Salasnich∗ CNR and CNISM, Dipartimento di Fisica “Galileo Galilei”, Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy
Abstract We investigate the condensate density and the condensate fraction of conduction electrons in weak-coupling superconductors by using the BCS theory and the concept of off-diagonal-long-range-order. We discuss the analytical formula of the zerotemperature condensate density of Cooper pairs as a function of Debye frequency and energy gap, and calculate the condensate fraction for some metals. We study the density of Cooper pairs also at finite temperature showing its connection with the gap order parameter and the effects of the electron-phonon coupling. Finally, we analyze similarities and differences between superconductors and ultracold Fermi atoms in the determination of their condensate density by using the BCS theory.
PACS numbers: 74.20.Fg; 74.70.Aq; 03.75.Ss.
1.
Introduction
The condensate fraction of fermionic alkali-metal atoms has been recently investigated [1, 2, 3, 4] by using extended BCS (EBCS) equations [5, 6, 7, 8, 9, 10] from the BCS regime of Cooper-pairs to the BEC regime of molecular dimers [1, 2, 3]. In particular, we have found [1] a remarkable agreement between this simple mean-field theory and the experimental results [11, 12]. These results indicate the presence of a relevant fraction of condensed pairs of 6 Li atoms also on the BCS side of the Feshbach resonance. Monte Carlo calculations [13] have shown that the zero-temperature mean-field predictions [1, 2] slightly overestimate ∗
E-mail:
[email protected]
260
Luca Salasnich
the condensed fraction of Fermi pairs. Very recently it has been reported [14] an accurate measurement of the temperature dependence of the condensate fraction for a fermion pair condensate of 6Li atoms near the unitarity limit of the BCS-BEC crossover. Also these new experimental data [14] are in agreement with mean-field theoretical predictions at finite temperature [3]. In superfluids made of ultracold atoms, the inter-atomic interaction is attractive for all fermions of the system [6]. On the contrary, in metallic superconductors there is an attractive interaction between fermions only near the Fermi surface [15, 16]. As a consequence, the condensate fraction of metallic superconductors has distinctive properties with respect to those of atomic superfluids. Despite the BCS theory is 52 years old [17], the condensate fraction of Cooper pairs in superconductors has been considered only in few papers [18, 20, 19, 21] and in the recent book of Leggett [15]. In fact, in superconductors the condensate fraction has never been measured: only very recently Chakravarty and Kee have proposed to measure it by using magnetic neutron scattering [21]. In this paper we analyze in detail the condensate density of conduction electrons in weak-coupling superconductors at zero and finite temperature by using BCS theory [17] and the concept of off-diagonal-long-range-order [18, 22]. For the first time, we calculate explicitly the density of electronic Cooper pairs and the condensate fraction for various metals and show its dependence on the Debye frequency, the electron-phonon interaction and the energy gap. Another novelty of this paper is the analytical and numerical investigation of the temperature dependence of the condensate fraction, for which we find a power-law behavior. Finally, we compare of the BEC theory of superconductors with the extended BEC theory of ultracold Fermi atoms for obtaining the condensate density and the condensate fraction.
2.
BCS Theory and ODLRO
The BCS Lagrangian density of conduction electrons with spin σ = ↑, ↓ near the Fermi surface is given by Lˆ =
X σ
∂ + ˆ ψσ i~ − (∇) + µ ψˆσ + g ψˆ↑+ ψˆ↓+ ψˆ↓ψˆ↑ , ∂t
(1)
where ψˆσ (r, t) is the electronic field operator which satisfies the familiar equal-time anticommutation rules of fermions. Here (∇) is the differential operator such that (∇)eik·r = k eik·r , where k is the energy spectrum of conduction electrons in a specific metal [23]. The attractive interaction between electrons is described by a contact pseudo-potential of strength g (g > 0). For metals this electron-phonon interaction strength is attractive only for conduction electrons near the Fermi surface [15, 16, 17]. The chemical potential µ fixes the number N of conduction electrons. The Heisenberg equation of motion of the field operator ψˆ↑(r, t) can be immediately derived and reads ∂ (2) i~ ψˆ↑ = [(∇) − µ] ψˆ↑ − g ψˆ↓+ ψˆ↓ψˆ↑ . ∂t
Condensate Fraction in Metallic Superconductors and Ultracold Atomic Vapors 261 In the BCS theory the interaction term of Eq. (2) can be treated within the minimal meanfield approximation ψˆ↓+ψˆ↓ ψˆ↑ ' ψˆ↓+hψˆ↓ψˆ↑ i. In this way Eq. (2) becomes i~
∂ ˆ ψ↑ = [(∇) − µ] ψˆ↑ − ∆ ψˆ↓+ , ∂t
(3)
ˆ↑(r, t)i ∆(r, t) = g hψˆ↓(r, t) ψ
(4)
where is the gap function. The condensate wave function of Cooper pairs [15, 19] is instead given by Ξ(r, r0, t) = hψˆ↓(r, t) ψˆ↑(r0, t)i . (5) As shown by Yang [18], this two-particle wave function is strictly related to the largest eigenvalue N0 of two-body density matrix of the system. N0 gives the number of Fermi pairs in the lowest state, i.e. the condensate number of Fermi pairs [15, 19, 18], and it can be written as Z N0 = |Ξ(r, r0, t)|2 d3r d3r0 . (6) A finite value for the condensate fraction f = N0/(N/2) in the thermodynamic limit N → ∞ implies off-diagonal-long-range-order [18, 22].
3.
Gap Equation and Condensate Density
To investigate the properties of the condensate fraction of electronic pairs we adopt the following Bogoliubov representation of the field operator X uk vk −i(k·r−ωk t)ˆ+ i(k·r−ωk t)ˆ (7) b bk↓ e − e ψˆ↑(r, t) = k↑ 1/2 1/2 V V k in terms of the anti-commuting quasi-particle Bogoliubov operators ˆbkσ , with V the volume of the system and Ek = ~ωk the excitation energies of quasi-particles [15, 16]. The thermal averages of quasi-particle Bogoliubov operators are given by ˆ hˆb+ kσ bk0 σ 0 i =
1 eβEk
+1
¯ k δkk0 δσσ0 , δkk0 δσσ0 = n
(8)
¯k where β = 1/(kB T ) with kB the Boltzmann constant, T the absolute temperature, and n is the thermal Fermi distribution. By using these results, the gap function, Eq. (4), becomes ∆=
g X0 (1 − 2¯ nk )uk vk , V
(9)
k
while the condensate number of conduction electrons, Eq. (6), satisfies this expression [1, 19] X0 N0 = (1 − 2¯ nk )2u2k vk2 . (10) k
262
Luca Salasnich
The ’prime’ restricts the summation to states within a shell of width ~ωD about the Fermi surface. To determine the amplitudes uk and vk of quasi-particles, one inserts Eq. (7) into Eq. (3) and obtains the familiar Bogoliubov-de Gennes equations, which give ξk ξk 1 1 2 2 1+ 1− , vk = , (11) uk = 2 Ek 2 Ek where ξk = k − µ ,
Ek =
q
ξk2 + ∆2 .
(12)
Eqs. (9) and (10) can then be written as ∆=
βEk g X0 ∆ tanh( ) V 2Ek 2
(13)
X0 ∆2 βEk ) tanh2( 2 2 4Ek
(14)
k
N0 =
k
nk . where tanh(βEk /2) = 1 − 2¯ P InRthe thermodynamicR limit, where the volume RV goes to infinity, k can be replaced by V d3 k/(2π)3 = V N (ξ)dξ with N (ξ) = d3k/(2π)3 δ(ξ − ξk ). In metals the condition ~ωD µ R R is always satisfied [16], consequently we can use the approximation N (ξ)dξ ' N (0) dξ, where Z d3 k N (0) = δ(µ − k ) (15) (2π)3 is the density of states at the Fermi surface. In this way the previous equations (13) and (14) become p Z ~ωD tanh( β2 ξ 2 + ∆2 ) 1 p dξ (16) = gN (0) ξ 2 + ∆2 0 p Z ~ωD tanh2 ( β2 ξ 2 + ∆2) 1 2 n0 = N (0)∆ dξ (17) 2 ξ 2 + ∆2 0 where n0 = N0/V is the density of electrons in the condensate.
3.1.
Zero-temperature Condensate
Let us consider first the zero-temperature case (T = 0). From Eqs. (16) and (17) we get the zero temperature energy gap ∆(0): s 2ω2 ~ 1 ~ω D D , (18) = ln + 1+ gN (0) ∆(0) ∆(0)2 and the zero-temperature condensate density: 1 ~ωD n0 (0) = N (0)∆(0) arctan( ). 2 ∆(0)
(19)
Condensate Fraction in Metallic Superconductors and Ultracold Atomic Vapors 263 This expression shows that the condensate density n(0) can be expressed in terms of density of states N (0), energy gap ∆(0) and Debye energy ~ωD . Finally, the zero-temperature condensate fraction f (0) = n0 (0)/(n/2) is given by f (0) =
~ωD 1 N (0) ∆(0) arctan( ), 2 n ∆(0)
(20)
where n is the density of conduction electrons. Under the condition ∆(0) ~ωD , from Eqs. (18) we find the familiar weak-coupling BCS result 1 ) (21) ∆(0) = 2~ωD exp(− gN (0) for the energy-gap order parameter, while the condensate density (19) can be written as [21, 15]: π (22) n0 (0) = N (0)∆(0) 4 We stress that in many real superconductors the simple BCS theory reported above is not accurate, and one has to take into account the retarded electron-electron interaction via phonons [26] and also the Coulomb repulsion [27]. The results obtained above by using the mean-field BCS theory are reliable only in the weak-coupling regime, i.e. for ∆(0) ~ωD , where gN (0) ≤ 0.3. Therefore we will continue our analysis of the condensate fraction only for a class of superconductors which satisfy this condition. Within the free-electrons Sommerfeld approximation, where the energy spectrum k of conduction electrons has the simple quadratic behavior k = ~2k2 /(2m∗), the free particle density of states Nf ree (0) is related to the total density of conduction electrons by n = 4Nf ree (0)µ/3 and the zero-temperature condensate fraction reads f (0) = 3π∆(0)/(8µ). To get a better estimate, we correct the free electron value of N (0) by an effective mass as obtained from specific heat measurements, i.e. we use the expression N (0) = (m∗/m)Nf ree(0). Table 1. BCS predictions for weak-coupling superconducting metals: n0 is the zerotemperature condensate density, obtained with Eq. (22); f (0) = n0 (0)/(n/2) is the zero-temperature condensate fraction; gN (0) is the electron-phonon strength, calculated with Eq. (18). Tc is the critical temperature from Eq. (25).
Cd Zn Al Tl In Sn
n0 (0) [10−33 m−3 ] 4.18 7.72 22.4 31.0 62.1 62.5
f (0) [10−5 ] 0.9 1.2 2.5 5.9 10.8 8.4
gN (0) 0.179 0.172 0.168 0.263 0.267 0.254
Tc [K] 0.51 0.79 1.15 2.43 3.46 3.68
In the first two columns of Tab. 1 we show the zero-temperature condensate density n0 and condensate fraction f (0) of simple metals obtained from Eqs. (19) and (20) by
264
Luca Salasnich
using the experimental data of ∆(0) and ωD obtained from Ref. 24 (when the comparison is possible, they agree within a few percent with those reported in Ref. 25). In the third column we report the electron-phonon strength gN (0) calculated with Eq. (18) knowing f (0). The table shows that indeed these simple metals are all in the weak-coupling regime. For completeness, in the forth column we insert the theoretical determination (see Eq. (25)) of the critical temperature Tc , which is very reliable for these simple metals, when compared with the experimental data.
3.2.
Finite-temperature Condensate
Let us now investigate the behavior of the condensate density n0 at finite temperature T . Under the condition ~ωD kB Tc , which is always satisfied, near Tc the energy gap goes to zero according to the power law [16, 15] ∆(T ) = 3.06 kB Tc
T 1− Tc
1/2
.
(23)
Instead for the condensate density n0 (T ), from Eq. (17) and the previous expression, we find near Tc T ∆(T )2 . (24) n0 (T ) = 0.43 N (0) = 4.03 N (0)kBTc 1 − kB Tc Tc For a generic temperature T we solve numerically Eqs. (16) and (17). The theoretical critical temperature Tc , obtained from Eq. (16) setting ∆(Tc) = 0, is given by the wellknown result [16] 1 ). (25) kB Tc = 1.13 ~ωD exp(− gN (0) For simple metals the theoretical critical temperature Tc , reported in the last column of Table 1, is in good agreement with the experimental one Tcexp : the relative difference (Tcexp − Tc )/Tcexp is not large (i.e. within 10%), and for some metals (Tl, In, Sn) it is quite small (i.e. within 2%). Taking into account Eq. (20), the BCS theory predicts that the zero-temperature condensate density in superconductors in the weak-coupling regime, can be written as n0 (0) = 1.39 N (0)kBTc .
(26)
This equation resembles the familiar BCS result ∆(0) = 1.764 kBTc for the zerotemperature energy gap. We stress that the predictions of the BCS theory can be surely improved by using the Eliashberg theory [25, 26]. This more sophisticated approach will not change the order of magnitude of the numbers in the first two columns of Tab. 1 but it could change the last significant figure. Coming back to the study of finite-temperature effects, in the upper panel of Fig. 1 we plot the condensate density n0 (T ) vs electron-phonon strength gN (0) for different values of the temperature T . As expected, by increasing the temperature T it is necessary to increase the strength gN (0) to get the same condensate density.
n0(T)/(N(0)ED)
Condensate Fraction in Metallic Superconductors and Ultracold Atomic Vapors 265
0.08 kBT/ED = 0 kBT/ED = 0.01 kBT/ED = 0.02
0.06 0.04 0.02 0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
g N(0) n0(T)/n0(0)
1 0.8 0.6
numerics power law
0.4 0.2 0
0
0.2
0.4
0.8
0.6
1
T/Tc Figure 1. Upper panel: condensate density n0 (T ) vs electron-phonon strength gN (0) in a superconductor for different values of the temperature T , where N (0) is the density of states and ED = ~ωD is the Debye energy. Lower panel: condensate density n0 (T ) as a function of the temperature T in a superconductor with gN (0) = 0.2. Solid line: numerical solution of Eqs. (16) and (17); dashed line: analytical approximation, Eq. (27).
As it happens for the energy gap ∆(T ), one may show that Eqs. (17) and (17) together also imply that the condensate density may be written as its value at T = 0 times a universal function of T /Tc. In the lower panel of Fig. 1 we plot the condensate density n0 (T )/n0(0) as a function of the temperature T /Tc: in the full range of temperatures the numerical results (solid line) are reasonably well approximated by (dashed line)
n0 (T ) = n0 (0)
with α = 3.16 (best fit).
1−
T α Tc
,
(27)
266
4.
Luca Salasnich
Superconductors vs Ultracold Atoms
In metallic superconductors there is an attractive interaction between fermions only near the Fermi surface [15, 16]. On the contrary, as remarked in the introduction, in superfluid ultracold two-component Fermi atoms, the effective inter-atomic interaction can be made attractive for all atoms of the system by using the technique of Fano-Feshbach resonances [12, 15, 28]. This implies that in the BCS equations for ultracold atoms there is not a natural ultraviolet cutoff. For attractive ultracold atoms the mean-field BCS theory is given by the gap equation (9) and the number equation X vk2 + 2(u2k − vk2 )¯ (28) nk , N= k
while the condensate fraction is given by Eq. (10). But, for ultracold atoms, in these equations the sum over momenta is no more restricted within a thin shell around the Fermi surface. As well known, due to the choice of a contact potential, the gap equation (9) diverges in the ultraviolet. This divergence is logarithmic in two dimensions (2D) and linear in three dimensions (3D). In 3D, a suitable regularization is obtained by introducing the inter-atomic scattering length aF via the equation 1 X 1 1 m + = − , 4π~2aF g V 2k
(29)
k
where k = ~2k2 /(2m) with m the atomic mass, and then subtracting this equation from the gap equation [5, 6, 7]. In this way one obtains the 3D regularized gap equation 1 X tanh (βEk /2) 1 m . (30) = − − 4π~2aF V 2Ek 2k k
In 2D, quite generally the bound-state energy B exists for any value of the interaction strength g between two atoms [10, 29]. For the contact potential the bound-state equation is 1 1 X 1 = , (31) 2 k2 ~ g V 2m + B k and subtracting this equation from the gap equation [10, 29] one obtains a 2D regularized gap equation ! X tanh (βEk /2) 1 − =0. (32) ~2 k 2 2Ek + B k
2m
The number equation (28) and the renormalized gap equation (30) (or Eq. (32) in 2D) are the so-called generalized BCS equations, from which one determines, for a fixed value of the temperature T and the average number of atoms N , the chemical potential µ(T ) and the gap energy ∆(T ) as a function of the scattering length aF (or of the bound-state energy B in 2D). The extended BCS equations can be applied in the full crossover from weak coupling to strong-coupling [15, 28]. In 3D, the crossover is from the BCS state of weaklyinteracting Cooper pairs (with 1/aF −1) to the Bose-Einstein Condensate (BEC) of
Condensate Fraction in Metallic Superconductors and Ultracold Atomic Vapors 267 molecular dimers (with 1/aF 1) across the unitarity limit (1/aF = 0) [6]. In 2D, there is a similar BCS-BEC crossover by increasing the value B of the bound-state energy [10, 4]. R R P At zero-temperature, by using the continuum limit k → V /(2π)3 d3k → V /(2π 2) k2 dk, the 3D condensate density (10) √ has a simple analytical expression [1]. The 3D density of states is N (ξ) = (2m/~2)3/2 ξ + µ/(4π 2) and the 3D condensate density is given by v s u u µ(0) 3/2 m µ(0)2 3/2t ∆(0) 1 + . (33) + n0 (0) = 8π~3 ∆(0) ∆(0)2 In the 3D BCS regime (1/aF −1), where µ(0)/∆(0) 1 and the size of weaklybound Cooper pairs exceeds the typical interparticle spacing kF−1 , µ(0) approaches the non-interacting Fermi energy F = ~2kF2 /(2m) with kF = (3π 2n)1/3 and there is an exponentially small energy gap ∆(0) = 8e−2 F exp (π/(2kF aF )). In this weak-coupling regime the 3D condensate density becomes [1] 1 π 3π . (34) n0 (0) = N (0)∆(0) = 2 n exp π 2e 2kF aF Notice that this is formula is similar to Eq. (22) of weak-coupling superconductors (here aF < 0), but the behavior of ∆(0) is quite different. In 2D, the density of states is constant and reads N (ξ) = N (0) = (2m/~2)/(4π). The zero-temperature 2D condensate density is easily obtained[4] as µ(0) 1 π n0 (0) = N (0)∆(0) + arctan , (35) 4 2 ∆(0) while the zero-temperature 2D energy gap is given by the implicit formula s µ(0) µ(0) 1+ − . ∆(0) = 2F ∆(0) ∆(0)
(36)
From these equations, in the 2D BCS regime (0 ≤ B F ) where µ(0)/∆(0) 1 one finds exactly Eq. (22), but here the energy gap ∆(0) depends on the Fermi energy F and the bound-state energy B according to the formula [29] √ (37) ∆(0) = 2F B , while the chemical potential is µ(0) = F − B /2. It is not surprising that in the BCS regime the condensate density of 2D superfluid atoms is formally equivalent to the Eq. (22) we have found for weak-coupling superconductors. In fact, to obtain Eq. (22) we have R R used the approximation N (ξ)dξ ' N (0) dξ that is exact in the strictly 2D case, and the condition ∆(0) ~ωD which implies that the upper limit of integration is practically +∞. In the previous section we have shown that the BCS equations can be used to determine the (quite small) condensate fraction of superconductors only in the weak-coupling regime. Instead, the extended BCS equations have been used in recent papers [1, 2, 3, 4] to get the
268
Luca Salasnich
condensate fraction of ultracold atoms in the full BCS-BEC crossover. The theory predicts that in the crossover the zero-temperature condensate fraction grows from zero to one. Two experiments [11, 14] have confirmed these predictions for the 3D superfluid two-component Fermi gas.
5.
Conclusions
In this paper we have studied, within the mean-field BCS theory of superconductors, the condensate of electronic Cooper pairs at zero and finite temperature showing the crucial role played by the Debye frequency and by the electron-phonon interaction. We have found that the zero-temperature condensate fraction f (0) of weak-coupling metals is quite small (' 10−5) and the condensate density increases in metals with higher critical temperature Tc , according to the law f (0) = 1.39 N (0)kBTc , where N (0) is the density of states at the Fermi energy. As discussed by Chakravarty and Kee [21], the spin-spin correlation function depends significantly on the condensate density and magnetic neutron scattering can provide a direct measurement of the condensate fraction of a superconductor. In the next future our BCS predictions, which are meaningful for weak-coupling superconductors, could be experimentally tested. In the last part of the paper we have shown similarities and differences between metallic superconductors and atomic Fermi vapors in the determination of the condensate fraction by using the mean-field BCS theory and its extension in the BCSBEC crossover. Acknowledgments The author thanks A.J. Leggett and F. Toigo for useful comments and critical remarks.
References [1] L. Salasnich, N. Manini, and A. Parola, Phys. Rev. A 72 (2005) 023621. [2] G. Ortiz and J. Dukelsky, Phys. Rev. A 72 (2005) 043611. [3] Y. Ohashi and A. Griffin, Phys. Rev. A 72 (2005) 063606; N. Fukushima, Y. Ohashi, E. Taylor, and A. Griffin, Phys. Rev. A 75 (2007) 033609. [4] L. Salasnich, Phys. Rev. A 76 (2007) 015601. [5] D.M. Eagles, Phys. Rev. 186 (1967) 456. [6] A.J. Leggett, J. Phys. (Paris) 41 (1980) C7-19. [7] P. Noziers and S. Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195. [8] C.A.R. Sa de Melo, M. Randeria, and J.R. Engelbrecht, Phys. Rev. Lett. 71 (1993) 3202. [9] J.R. Engelbrecht, M. Randeria, and C.A.R. Sa de Melo, Phys. Rev. B 55 (1997) 15153.
Condensate Fraction in Metallic Superconductors and Ultracold Atomic Vapors 269 [10] M. Marini, F. Pistolesi, and G.C. Strinati, Eur. Phys. J. B 1 (1998) 151. [11] M.W. Zwierlein et al., Phys. Rev. Lett. 92 (2004) 120403; M.W. Zwierlein, C.H. Schunck, C.A. Stan, S.M.F. Raupach, and W. Ketterle, Phys. Rev. Lett. 94 (2005) 180401. [12] W Ketterle and M W Zwierlein, Riv. Nuovo Cimento 13, 31 (2008) 247. [13] G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Phys. Rev. Lett. 95 (2005) 230405. [14] Y. Inada, M. Horikoshi, S. Nakajima, M. Kuwata-Gonokami, M. Ueda, and T. Makaiyama, Phys. Rev. Lett. 101 (2008) 180406. [15] A.J. Leggett, Quantum Liquids. Bose Condensation and Cooper Pairing in Condensed-Matter Systems, Oxford Univ. Press, Oxford, 2006. [16] A.L. Fetter and J.D. Walecka, Quantum Theory of Many Particle Systems , Mc Graw Hill, New York, 1971. [17] J. Bardeen, L.N. Cooper, and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [18] C.N. Yang, Rev. Mod. Phys. 34 (1962) 694. [19] C.E. Campbell, in Condensed Matter Theories, Nova Science, New York, 1997, vol. 12, 131. [20] L.J. Dunne and T.P. Spiller, J. Phys.: Cond. Matter 4 (1992) L563. [21] S. Chakravarty and H-Y. Kee, Phys. Rev. B 61 (2000) 14821. [22] O. Penrose, Phil. Mag. 42, 1373 (1951); O. Penrose and L. Onsager, Phys. Rev. 104 (1956) 576. [23] H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo Field Dynamics and Condensed States, North-Holland, Amsterdam, 1982. [24] N.W. Ashcroft and N.D. Mermin, Solid State Physics, Holt-Sounders, Philadelphia 1976. [25] J.P. Carbotte, Rev. Mod. Phys. 62 (1990) 1027. [26] G.M. Eliashberg, Sov. Phys. JEPT 11 (1960) 696. [27] W.L. McMillan, Phys. Rev. 167 (1967) 331. [28] S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 80 (2008) 1215. [29] M. Randeria, Ji-Min Duan, and Lih-Yir Shieh, Phys. Rev. B 41 (1990) 327.
In: Horizons in World Physics. Volume 271 Editor: Albert Reimer
ISBN: 978-1-61761-884-0 c 2011 Nova Science Publishers, Inc.
Chapter 7
S PONTANEOUS S YMMETRY B REAKING IN A M IXED S UPERFLUID OF F ERMIONS AND B OSONS T RAPPED IN D OUBLE - WELL P OTENTIALS B. A. Malomed1∗, L. Salasnich2†, and F. Toigo2‡ Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 2 CNISM and CNR-INFM, Unit`a di Padova, Dipartimento di Fisica “Galileo Galilei”, Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy 1
Abstract We study the spontaneous symmetry breaking (SSB) of a superfluid Bose-Fermi (BF) mixture loaded into a double-well potential (DWP), in the effectively onedimensional setting. The mixture is described by the Gross-Pitaevskii equation (GPE) for the bosons, which is coupled to an equation for the order parameter of the Fermi superfluid, which is derived from the respective density-functional model in the unitarity limit (a similar model may apply to the Bardeen-Cooper-Schriefer (BCS) regime too). Straightforward SSB in the quantum Fermi gas loaded into a DWP is impossible, as it requires an attractive self-interaction acting in the medium, while the intrinsic nonlinearity in the Fermi gas may only be repulsive. However, we demonstrate that the symmetry breaking can be made possible in the mixture, provided that interaction between the fermions and bosons is attractive (a real example is the mixture of potassium and rubidium atoms, which represent fermions and bosons, respectively). Numerical results for the SSB are represented by dependencies of asymmetry parameters for both components on numbers of particles in the mixture, and by phase diagrams in the plane of these two numbers. The diagrams display regions of symmetric and asymmetric ground states of the mixture. Dynamical pictures of the SSB, induced by a gradual transition from the single-well potential into the DWP, are reported too. In addition to the systematic numerically generated results, an analytical approximation is elaborated for the case when the GPE for the boson wave function is amenable to ∗
email:
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[email protected] ‡ email:
[email protected] †
272
B. A. Malomed, L. Salasnich and F. Toigo the application of the Thomas-Fermi (TF) approximation. Under a special linear relation between the numbers of fermions and bosons, the TF approximation makes it possible to reduce the model to a single equation for the fermionic function, which includes competing repulsive and attractive nonlinear terms, of powers 7/3 and 3, respectively. The latter terms directly illustrates the generation of the effective attraction in the Fermi superfluid, mediated by the bosonic component of the mixture, whose density is “enslaved” to the fermion density, in that case.
PACS 03.75.Ss, 03.75.Hh, 64.75.+g
1.
Introduction
The achievement of quantum degeneracy in bosonic [1] and fermionic [2] gases of alkali atoms has opened the way to the investigation and manipulation of novel states of atomic matter, such as Bose-Einstein condensates (BEC) [3, 4] and superfluid Fermi gases [5]. A simple but reliable theoretical tool for the study of these trapped degenerate gases is the density-functional theory [6]. In particular, the Gross-Pitaevskii equation (GPE), which accurately describes BECs in dilute gases, is the Euler-Lagrange equation produced by the Thomas-Fermi (TF) density functional which takes into regard the inhomogeneity of the condensate [4]. In parallel to that, many properties of superfluid Fermi gases with balanced (equally populated) spin components, and the formation of various patterns in them, can be accurately described, under the conditions of the BCS-BEC crossover, by an extended TF density functional and its time-dependent version, as it has been shown recently [7, 8, 9, 10]. One of the fundamental effects in nonlinear media, including BEC, which has been studied in detail, is the spontaneous symmetry breaking (SSB) in double-well potentials (DWPs). Asymmetric states trapped in symmetric DWPs are generated by symmetrybreaking bifurcations from obvious symmetric or antisymmetric states, in the media with the attractive or repulsive intrinsic nonlinearity, respectively [11] [the SSB under the action of competing attractive (cubic) and repulsive (quintic) terms was studied too [12]]. In terms of BEC and other macroscopic quantum systems, the SSB may be realized as a quantum phase transition, which replaces the original symmetric ground state by a new asymmetric one, when the strength of the self-attractive nonlinearity exceeds a certain critical value. Actually, a transition of this type was predicted earlier in the classical context, viz., in a model of dual-core nonlinear optical fibers with the self-focusing Kerr nonlinearity [13]. Still earlier, the SSB of nonlinear states was studied, in an abstract form, in the context of the nonlinear Schr¨odinger equation (NLSE) with a potential term [14], as well in the discrete self-trapping model [15]. The latter approach to the description of the SSB effects was later developed in many works in the form of the two-mode expansion, each mode representing a mode which is trapped in one of the potential wells (see Refs. [16] and references therein). As concerns the interpretation of the SSB as the phase transition, it may be categorized as belonging to the first or second kind (alias sub- or supercritical SSB mode), depending on the form of the nonlinearity, spatial dimension, and the presence or absence of a periodic external potential (an optical lattice) acting along the additional spatial dimension (if any) [17]. In the experiment, the self-trapping of asymmetric states has been demonstrated in the condensate of 87Rb atoms with repulsive interactions [18].
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons...273 Theoretical studies of the SSB in BECs were extended in various directions. In particular, the symmetry breaking of matter-wave solitons was predicted in various twodimensional (2D) DWP settings [17], including the spontaneous breaking of the skew symmetry of solitons and solitary vortices trapped in double-layer condensates with mutually perpendicular orientations of quasi-one-dimensional optical lattices induced in the two layers [19]. A different variety of the 2D geometry, which gives rise to a specific mode of the SSB, is based on a symmetric set of four potential wells [20] (a three-well system was considered too [21]). Recently, self-trapping of asymmetric states was predicted in the model of the BEC of dipolar atoms, which interact via long-range forces [22]. SSB was also studied in the context of the NLSE with a general nonlinearity [23]. The symmetry breaking is possible not only in linear potentials composed of two wells, but also in a similarly structured pseudopotential, which is produced by a symmetric spatial modulation of the non-linearity coefficient, with two sharp maxima [24]. Another generalization is the study of the SSB in two- [25] and three-component (spinor) [26] BEC mixtures, where the asymmetry of the density profiles in the two wells is coupled to a difference in distributions of the different species. Further, the analysis was extended to a Bose-Fermi (BF) mixture in Ref. [27], where a “frozen” fermion component was treated as a source of an additional potential for bosons. Dynamical manifestations of the symmetry breaking (Josephson oscillations) in a Fermi superfluid trapped in the DWP were recently considered too [28]. In spite of many realizations of the SSB studied in the models of degenerate quantum gases, the self-trapping of stationary asymmetric states has not yet been considered in fermionic systems. An obvious problem is that a Fermi gas, loaded into a DWP, cannot feature a direct self-attractive nonlinearity, which is necessary to induce the SSB in symmetric states. The objective of the present work is to introduce a model in which the SSB in a trapped Fermi superfluid is possible due to an effective attraction mediated by a bosonic component , mixed with the fermionic one. Actually, we consider the SSB in semi-trapped BF mixtures, with the DWP acting on a single species, either the fermionic or bosonic one, as this setting may be sufficient to hold the entire mixture in the trapping potential, and induce the SSB in its fermionic component. The analysis will be performed in the framework of a mean-field model, which couples, via nonlinear attraction terms, the GPE for the bosonic component to an equation for the fermionic order parameter, derived from the respective density functional. Inducing an effective boson-mediated attraction between the fermions requires an attractive BF interaction. For this purpose, we take well-known physical parameters corresponding to the 87Rb −40 K mixture, which features repulsion between rubidium atoms and attraction between the rubidium and potassium, characterized by the respective positive and negative scattering lengths, aB > 0 and aBF < 0 [29]. We consider the case when the spin-balanced fermionic component of the mixture is in the unitary regime, corresponding to a diverging scattering length which accounts for the interaction between the fermionic atoms with opposite orientations of the spin, aF → ±∞ (while the BCS regime corresponds to the vanishingly weak attraction, with aF → −0; in either limit, the effective fermionic Lagrangian does not depend explicitly on aF ). In fact, the same model with a different coefficient of the effective self-repulsion in the Fermi superfluid applies to the description of the BF mixture with the fermionic component falling into the BCS regime. Although
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the self-interaction, induced by the quantum pressure, in the equation for the fermionic order parameter is always repulsive, we demonstrate that the SSB in the fermionic component is indeed possible in the 87Rb −40 K mixture, due to the BF attraction which, as said above, mediates an effective attraction force in the Fermi superfluid. We also conclude that the attraction can induce symmetry-preserving or symmetry-breaking localization of both components in the semi-trapped mixture, depending on the numbers of the bosons and fermions in it. The paper is organized as follows. The model is formulated, in a sufficiently detailed form, in Section II. Results produced by the numerical analysis are reported in section III, for two variants of the model, with the DWP acting either only on the fermions, or on the bosons. In Section IV, we report approximate analytical results, obtained by means of the TF approximation applied to the GPE for the bosonic wave function. In particular, assuming a specific linear relation between the fermion and boson numbers, we can reduce the model to a single equation for the fermionic wave function with competing self-repulsive and self-attractive terms, the latter one explicitly demonstrating the mechanism of the effective attraction between the fermions mediated by “enslaved” bosons. The analytical results offer a qualitative explanation to general findings produced by the numerical analysis. The paper is concluded by Section V.
2.
The Model
Our starting point is a model for the degenerate rarefied quantum gas composed of NB condensed bosons of mass mB and NF fermions of mass mF , in two equally-populated spin components, at zero temperature. The fermionic component is assumed to be in the superfluid state at unitarity or, alternatively, in the BCS regime. The system is made effectively one-dimensional (1D), assuming that the gas is confined in transverse directions by a tight axisymmetric harmonic potential, with trapping frequencies ω⊥B , ω⊥F for the bosons and fermions, respectively. Within the framework of the density-functional theory for superfluids [8, 9], the 3D action of the BF mixture is Z (1) S = (LB + LF + LBF ) d3r dt , where LB is the ordinary bosonic Lagrangian density, ∗ ~2 i ∂ψB ∗ ∂ψB ~ ψB − ψB − LB = |∇ψB |2 2 ∂t ∂t 2mB 2π~2aB − UB |ψB |2 − |ψB|4 , mB
(2)
ψB (r, t) is the macroscopic BEC wave function, and the confining potential for the bosons is 1 2 R2 + VB (z) , (3) UB (r) = mBω⊥B 2 with R the transverse cylindric radial coordinate and VB (z) the potential acting in the axial direction z. The bosonic superfluid velocity is vB (r, t) = (~/mB)∇θB (r, t), where
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons...275 p θB (r, t) is the phase of wave function, ψB(r, t) ≡ nB (r, t)eiθB (r,t), and nB (r, t) is the bosonic density. The Galilean-invariant Lagrangian density LF of the Fermi gas with two equallypopulated spin components is [8, 30] ~2 ~ ∂ψF ∂ψF − ψF − |∇ψF|2 LF = i ψF 4 ∂t ∂t 8mF 3 ~2 − (3π 2)2/3|ψF|10/3 − UF |ψF|2 , (4) ξ 5 2mF where ψF(r, t) is the superfluid order parameter of the Fermi gas at unitary [8], 2mF is the mass of a pair of fermions with spins up and down, and the potential acting on the fermionic atoms is 1 2 UF (r) = mF ω⊥F R2 + VF(z), (5) 2 cf. its bosonic counterpart (3). The fermionic superfluid velocity is vF (r, t) = (~/2m )∇θ (r, t), where θ (r, t) is the phase of the order parameter, ψF (r, t) ≡ F F F p iθ (r,t) F nF (r, t)e , and nF (r, t) is the density of fermionic atoms. Constant ξ in expression (4) is ξ = 1 in the deep BCS regime, and ξ ' 0.4 at the unitarity [5]. In calculations reported below, we fixed ξ = 0.45, assuming the unitarity regime. Lastly, the Lagrangian density LBF in expression (1) accounting for the BF interaction is LBF = −
2π~2aBF |ψB|2 |ψF|2 , mR
(6)
where mR = mB mF /(mB + mF ) is the respective reduced mass, and as said above, aBF < 0 corresponds to an attractive BF interaction, which is necessary to support the SSB of the fermionic component in the presence of the DWP. In the nearly 1D configuration, transverse widths of the atomic distributions are determined by the width of the ground states of the respective harmonic oscillators: p p a⊥B = ~/(mBω⊥B ), a⊥F = ~/(2mFω⊥F ), (7) for the condensed bosons and superfluid fermions at the unitarity. The boson and fermion components exhibit the effective 1D behavior if their chemical potentials are much smaller than the corresponding transverse-trapping energies, ~ω⊥B and ~ω⊥F /2. Under these conditions, we can adopt the known factorized ans¨atze for the 3D wave functions [31], p
2
2
e−R /(2a⊥B) NB ΦB (z, t) , ψB (r, t) = π 1/2a⊥B 2 2 p e−R /(2a⊥F) ψF (r, t) = NF ΦF (z, t) , π 1/2a⊥F
(8) (9)
where ΦB (z) and ΦF (z) are the 1D (axial) wave functions, which are subject to the usual normalization conditions, Z +∞ Z +∞ |ΦB(z, t)|2 dz = |ΦF (z, t)|2 dz = 1. (10) −∞
−∞
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Inserting expressions (8) and (9) into Eq. (1), the action can be written as S = S (1D) − [NB ~ω⊥B − NF ~ω⊥F /2] t, with the effective 1D action S
(1D)
=
Z
dt
Z
+∞ −∞
h i (1D) (1D) (1D) dz LB + LF + LBF ,
(11)
where the usual 1D Gross-Pitaevskii Lagrangian density is (1D) LB
~2 ~ ∂Φ∗B ∗ ∂ΦB − ΦB − = NB i ΦB 2 ∂t ∂t 2mB 1 −VB |ΦB|2 − GB|ΦB |4 , 2
∂ΦB 2 ∂z (12)
where the boson self-interaction strength in 1D is GB ≡ 2NB ~ω⊥B aB .
(13)
Further, the 1D fermionic Lagrangian density in Eq. (11) is (1D) LF
∂Φ∗F ~2 ~ ∗ ∂ΦF = NF i ΦF − ΦF − 4 ∂t ∂t 8mF 3 − A|ΦF |10/3 − VF|ΦF |2 , 5
∂ΦF 2 ∂z (14)
with the effective strength of the fermionic quantum pressure, 2/3
4/3
A = (3π 2)2/3(3ξ/5)~2NF /(2mFa⊥F ),
(15)
emerging as the coefficient in front of the bulk kinetic energy of the Fermi gas in the unitarity limit. Finally, the 1D Lagrangian density of the BF interaction is (1D)
LBF = −NBNF GBF |ΦB|2 |ΦF |2,
(16)
where the corresponding 1D interaction strength is GBF ≡ 2~2aBF /(mRa⊥B a⊥F ).
(17)
For numerical calculations, we set a⊥B = a⊥F ≡ a⊥ and ω⊥B = ω⊥F ≡ ω⊥ , measur−1 ing lengths and time in units of a⊥ and ω⊥ , respectively. This implies that 2mF = mB (hence, mR = mB /3), a condition which is roughly satisfied by the 87 Rb −40 Kb mixture. It is a good candidate for experimental study of the SSB because the BF scattering length in this mixture is negative, as stressed above, aBF ≈ −284a0, where a0 is the Bohr radius [29]. Simultaneously, the scattering length for collisions between rubidium atoms is positive, aB ≈ 108a0 [3, 29, 37]. In all the calculations reported in the present work, this values of aBF and aB were used.
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons...277 The application of the variational procedure to the effective action (11) produces a system of coupled NLSEs, 1 ∂2 − + WB (x) + gB|φB |2 2 ∂x2 ∂ +gBF NF |φF|2 φB = i φB , (18) ∂τ
1 ∂2 + gF |φF |4/3 + WF (x) 8 ∂x2 i ∂ φF , +gBF NB |φB |2 φF = 2 ∂τ −
1/2
(19)
1/2
where x = z/a⊥ , τ = ω⊥ t, and φB = a⊥ ΦB , φF = a⊥ ΦF , WB = VB /(~ω⊥), WF = VF /(~ω⊥), and the renormalized interaction coefficients are derived from expressions (13), (15), and (17), gB = GB /(~ω⊥a⊥ ) ≡ 2 (aB /a⊥ ) NB, gBF = GBF /(~ω⊥a⊥ ) ≡ 6aBF /a⊥ , 2/3 2/3 2/3 (3ξ/5) NF . gF = A/(a⊥ ~ω⊥ ) ≡ 3π 2
(20)
Note that the rescaled wave functions are subject to the same normalization conditions as in Eqs. (10), i.e., Z +∞ Z +∞ 2 |φB (x, τ )| dx = |φF (x, τ )|2 dx = 1. (21) −∞
−∞
If condition 2mF = mB does not hold, the coupled equations can be cast in the same form, with a difference that coefficient 2mF/mB appears in front of the second derivative in Eq. (18). The coupled NLSEs in the form of Eqs. (19) actually generalize the static and dynamical equations for BF systems which were used in various settings in Refs. [32]. In particular, the semi-phenomenological equations used for the study of 1D gap solitons in the BF mixture in Ref. [33] are also tantamount to Eqs. (19), up to a difference in coefficients. To find stationary solutions to Eqs. ((18) and (19), we employed a Crank-Nicolson finite-difference scheme for simulations of the equations in imaginary time, using the Fortran codes provided in Ref. [34]. We employed space and time steps ∆x = 0.025 and ∆t = 0.001, and a sufficiently large number of iterations to ensure the convergence. The stability of the stationary solutions against small perturbations was then tested by simulations in real time. Due to the attractive character of the BF interactions (aBF < 0), the true ground state of the 3D mixture collapses towards energy E = −∞. Nevertheless, because of the strong transverse confinement, the quasi-1D metastable state has an indefinitely long lifetime [35]. Actually, stationary solutions generated by the imaginary-time integration represent the ground state of the effective one-dimensional BF system based on Eqs. (18) and (19) – in the same sense as the famous matter-wave solitons realize the ground state of the quasi-1D condensate of 7Li atoms [36].
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B. A. Malomed, L. Salasnich and F. Toigo
Results of the Numerical Analysis Axially Trapped Fermions and Free Bosons
We start the analysis by considering the configuration with the DWP acting solely on the fermionic component: (22) WF (x) = αF x2 + βF exp −γF x2 , WB (z) = 0, where all constants are positive. An elementary consideration of this potential demonstrates that it features the double-well structure provided that αF < βF γF , with two symmetric potential minima located at points p −1/2 ln (βFγF /αF ). (23) xmin = ±γF We reports results of simulations for αF = 1/2, βF = 16, and γF = 10, which adequately represents the generic situation; in this case, Eq. (23) yields xmin ≈ ±0.76. The purpose of the analysis is to construct x-symmetric and asymmetric ground states of the system, varying the control parameters, and identify the respective SSB bifurcation, i.e., the transition to the asymmetric ground state. The asymmetry of its Fermi and Bose components is characterized by parameters R0 R∞ 2 2 0 |φF,B (x)| dx − −∞ |φF,B (x)| dx θF,B ≡ . (24) R +∞ 2dx |φ (x)| F,B −∞ Recall that the denominator in this expression is actually 1 for both species, as per Eq. (21). In Fig. 1 we display a set of axial (1D) profiles of the densities of both components in the ground state, generated by the integration of Eqs. (18) and (19) in imaginary time, in the case of the BF attraction and weak repulsion between the bosons. The respective values of the interaction coefficients in Eqs. (18) and (19) are taken as per Eqs. (20), with the above-mentioned values of aBF and aB for the 87Rb −40 Kb mixture and a fixed transversetrapping length, a⊥ = 1 µm, substituting various values of atomic numbers NB and NF . The figure clearly shows the transition from the symmetric ground state to the asymmetric one, with the increase of the number of bosons, NB , i.e., as a matter of fact, the strength of both the boson-boson and BF interactions. Accordingly, the increase of NB leads to a stronger overlap between the bosons and fermions, and to the SSB, i.e., the transition from symmetric ground states to an asymmetric one, which happens (simultaneously in both components) between NB = 150 and 170, for a fixed number of fermions, NF = 300. Similarly to what is shown in Fig. 1, the symmetric ground state is replaced by an asymmetric one with the increase of NF at fixed NB, as this implies the strengthening of the BF interaction. The summary of the results for the SSB in the present setting is provided in Fig. 2 by plots of asymmetry parameters (24) versus NB for fixed NF, and vice versa. Naturally, the SSB of both components happens at the same point [for instance, at NB ≈ 105 in panel (a)]. Nevertheless, the resulting bosonic asymmetry is essentially stronger [in Fig. 1(a) – up to a point, NB ≈ 580, at which both θB and θF attain values very close to 1, i.e., practically all the atoms are collected in a single potential well]. In Fig. 1(b), the behavior of the fermionic asymmetry, ΘF , is different: it jumps to a maximum value
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons...279 0.3 φB2(x), φF2(x)
(a)
B F pot
NB = 150 NF = 300
0.2
0.1
0 -12
-8
-4
0 x
4
0.6 φB2(x), φF2(x)
(b)
12
8
12
8
12
B F pot
NB = 170 NF = 300
0.4
8
0.2
0 -12
-8
-4
0 x
4
1.5 φB2(x), φF2(x)
B F pot
NB = 500 NF = 300
(c) 1
0.5
0 -12
-8
-4
0 x
4
Figure 1. (Color online). Density profiles of the 87Rb −40 Kb mixture, φ2B (x) and φ2F (x), marked by the respective labels, in the case of the double-well potential (shown in arbitrary units by curve “pot”) acting on the fermions, while no axial potential is applied to the bosons. The three panels differ by the number of bosons, NB, as indicated in each panel. Recall that the axial coordinate x is measured in units of the transverse-confinement length, a⊥ , while φ2B and φ2F are displayed in units of a−1 ⊥ .
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B. A. Malomed, L. Salasnich and F. Toigo
at the SSB point, and then gradually decreases. This difference between the bosonic and fermionic components is natural, as the intrinsic repulsion in the latter one tends to restore the symmetry between the distributions of atoms in the two potential wells.
1
B F
asymmetry
(a) 0.8 0.6
NF = 300
0.4 0.2 0 100
200
300
NB
400
asymmetry
(b) 1 0.8
500
600
800
1000
B F
0.6
NB = 500
0.4 0.2 0 0
200
400
NF
600
Figure 2. (Color online). Asymmetry parameters (24), for the bosons (B) and fermions (F) in the 87Rb −40 Kb mixture, loaded into potential (22), as functions of: (a) the number of bosons, NB , at a fixed number of fermions, NF = 300; (b) the number of fermions, at fixed NB = 500. The results are further summarized in Fig. 3, which displays the phase diagram of the mixture. There are three regions in the (NF, NB) plane: an area where the attraction to fermions cannot keep bosonic atoms in the trapped state (“free bosons”), the region where the bosons and fermions are trapped in the symmetric state, with respect to the DWP, and the SSB region, where the mixture is trapped in the asymmetric ground state. An example of the dynamical development of the SSB from an initially symmetric configuration, in the case where the ground state is asymmetric, is presented by Fig. 3. Initially, the bosons and fermions form a stable symmetric bound state, via their mutual attraction,
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons...281
1000 free bosons
SSB
NB
100 symmetric 10
1 1
10
NF
100
1000
Figure 3. (Color online). The phase diagram of the 87 Rb −40 Kb mixture in potential (22), which acts only on the fermions. The diagram shows, on the logarithmic scales in the (NB, NF) plane, regions of the symmetric ground state, and of the spontaneous symmetry breaking (SSB). “Free bosons” implies delocalization of the bosonic wave function. in the single-well potential acting on the fermions, which is taken in the form of (22) with βF = 0. Then, βF is ramped linearly in time (0 < t < 80) from βF = 0 to βF = 16, which leads to splitting the single potential well into two, as per Eq. (23). The dynamical picture clearly shows the transition of the initial symmetric state into the symmetry-broken one. Both components get spontaneously collected in one of the wells, where they stay together due to the mutual attraction, approaching an equilibrium configuration.
3.2.
Axially Trapped Bosons and Free Fermions
Now, we consider the action of the DWP on the bosons only, taking the potential as WB (x) = αB x2 + βB exp −γB x2 , WF = 0
(25)
For this setting, numerical results are presented with αB = 1/2, βB = 16, and γB = 10, i.e., the same parameters of the DWP as used above for the trapping the fermions. We again aim to construct the ground state of the system as a function of the control parameters, and investigate its spontaneous transition into an asymmetric shape. Results of the analysis for this setting are summarized in the respective phase diagram of the mixture plotted in Fig. 5 (cf. Fig. 3). In this case too, there are three regions in the (NF , NB) plane: an area where the fermions cannot be held in a localized state by the
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4
0.5
4
3
3 0.4
2 1
1
0.3
0 0.2
-1 -2
0.1
-3
0.4
2
x
x
0.5
0.3
0 0.2
-1 -2
0.1
-3
-4
0 0
10 20 30 40 50 60 70 80 time
-4
0 0
10 20 30 40 50 60 70 80 time
Figure 4. (Color online). Contour plots for the evolution of densities |φB (x, t)|2 and |φF (x, t)|2 of the (a) bosons and (b) fermions in the 87Rb −40 Kb mixture. At t = 0, the fermions are trapped in the single-well potential (22), with βF = 0. Then, βF linearly increases from 0 to 16 by t = 80, which implies the transition to the double-well potential with well-separated symmetric minima. Numbers of particles are NB = 500 and NF = 300. attraction to bosons (“free fermions”), the region where the fermions are trapped, along with the bosons, in a symmetric ground state, and the region where the trapped ground state is asymmetric (“SSB”), for both the fermionic and bosonic components. An example of the transition from the symmetric ground state of the BF mixture to an asymmetric one, caused by the increase of the number of fermions from NF = 10 to NF = 1000, while the number of bosons trapped in potential (25) is kept constant, is displayed in Fig. 6. Although the applicability of the functional-density description for NF = 10 [in panel (a)] may be disputed, this figure adequately shows the transition to the SSB.
3.3.
The Case of the Fermionic Component in the BCS Regime
The analysis presented above pertained to the superfluid BF mixture with the spin-balanced fermion components in the unitarity regime, where the s-wave scattering length, aF , which accounts for the interaction between the spin-up and spin-down fermions is extremely large (ideally, aF → ±∞). Actually, essentially the same Lagrangian (4) also applies to the BF mixture with the fermionic component kept in the BCS regime, with aF is negative and small (ideally, aF → −0). In this regime, one is practically dealing with a gas of ideal fermions, because the respective superfluid energy gap is exponentially small. In practical terms, to study the system whose Fermi component falls into the BCS regime, it is sufficient to set ξ = 1 instead of ξ = 0.45 in Eq. (4) [7]. Obviously, this change of ξ will make the fermions effectively more repulsive [see Eq. (15)], as the Pauli repulsion attains
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons...283
1000
NB
800
SSB
600
symmetric
400 200 free fermions 0 1
10
NF
100
1000
Figure 5. (Color online). The phase diagram of the 87Rb−40 K mixture loaded into potential (25) which acts only on the bosons. its maximum at ξ = 1. We have verified that, as the SSB in the mixture emerges chiefly due to BF attraction, the increase of the intrinsic repulsion in the Fermionic component corresponding to ξ = 1 makes the natural BF attraction in the 87Rb −40 K mixture insufficient for the appearance of the SSB for relatively small values of NF and NB considered above. To achieve the transition to asymmetric ground states (for the same value of a⊥ = 1 µm as taken above), it is necessary to consider values of NB and NF exceeding 1500 [cf. Figs. 3 and 5, which display the phase diagrams of the mixture with the fermionic component in the unitarity regime for NB, NF ≤ 1000].
4. 4.1.
The Analytical Approach: the Thomas-Fermi Approximation for the Bose Component The General Case
For the application of the analytical approach, we use the stationary version of general equations (18) and (19), obtained by the substitution of φB,F (x, τ ) = exp (−iµB,F τ ) uB,F (x), where chemical potentials for the localized states must be non-positive, µB,F ≤ 0, and real functions uB,F obey the following equations (with the prime standing for d/dx): 1 − u00B + −µB + WB (x) + gB u2B + gBFNF u2F uB = 0, 2
(26)
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B. A. Malomed, L. Salasnich and F. Toigo 0.8
B F pot
NB = 1000 NF = 10
0.6
2
φB (x), φF (x)
(a)
2
0.4 0.2 0 -6
-4
-2
0 x
3.2 2.4
4
6
4
6
B F pot
NB = 1000 NF = 100
2
φB (x), φF (x)
(b)
2
2
1.6 0.8 0 -6
-4
-2
0 x
2
Figure 6. (Color online). An example of the spontaneous symmetry breaking in the 87Rb– 40 K mixture trapped in potential (25). The two panels differ by the number of fermions: (a) NF = 10; (b) NF = 1000. h i 1 4/3 − u00F + −µF + gF uF + gBFNB u2B + WF (x) uF = 0. (27) 8 An essential simplification of Eqs. (26) and (27) can be achieved if the TF approximation may be applied to the former equation, i.e., the term with the second derivative may be neglected in it [3, 4, 6]. In the present setting, with the characteristic size of the trapped states ∆x ∼ 1 [see Eqs. (23) and Figs. 1 and 6], and the wave functions subject to normalization conditions (21) [hence the amplitudes of normalized density u2B (x) and u2F (x) are also ∼ 1], a straightforward consideration of Eq. (26) demonstrates that the kinetic energy (the second derivative) is negligible in comparison with either nonlinear term under conditions NB a⊥ /aB or NF a⊥ / (10 |aBF |). For the value of a⊥ = 1 µm adopted above, and the values of aB and |aBF | for the 87Rb −40 K mixture, these conditions reduce to quite realistic inequalities, NB 200, NF 10. If the TF approximation is valid, it allows one to solve Eq. (26) in the following form: −1 gB |gBF | NF u2F (x) − WB (x) + µB , at |x| < x0 , 2 (28) uB (x) = 0, at |x| > x0 , where it is taken into account that we are dealing with gBF < 0, and x0 is a positive root of equation u2F (x0) = (|gBF | NF)−1 [WB (x0) − µB ] (29)
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons...285 Note that the bosonic chemical potential, µB , is not an arbitrary parameter; instead, it must be found from the normalization condition (21), applied to expression (28): Z x0 (30) |gBF | NF u2F (x) − WB (x) + µB dx = gB . 2 0
4.2.
A Tractable Example
The substitution of approximation (28) for u2B into equation (27) for the fermionic function, uF (x), allows one to reduce the underlying system to a single equation for uF (x); however, in the general case this equation is quite complex. In particular, the additional equation (29) for x0 actually makes the resulting equation for uF nonlocal. Thus, in the general case the TF approximation does not yield an explicit analytical solution. Nevertheless, it can be obtained in a special case, when WB = 0 [cf. Eq. (22)] and µB = 0. In this case, Eq. (28) yields a simple local relation, u2B (x) = (|gBF | NF /gB) u2F (x),
(31)
which is valid at all x, and Eq. (30) reduces to a special relation between the boson and fermion numbers, (32) NF = gB / |gBF | ≡ − (aB /3aBF) NB, where we have made use of Eqs. (20); note that this approximation is meaningful only in the case when the signs of aB and aBF are opposite. For the parameters of the 87Rb −40 K mixture Eq. (32) amounts to NF ≈ 0.127NB . Finally, the single equation for the fermionic stationary function takes the following form, upon the substitution of expression (31): 1 − u00F + [WF (x) − µF ] uF 8 18a2BF 2/3 3ξ 2/3 7/3 + 3π 2 NFu3F = 0, NF uF − 5 aB a⊥
(33)
where we have again used Eqs. (20). Note that the last term in Eq. (33) directly illustrates the possibility proposed in this work, namely, that the interaction mediated by the boson field may give rise to an effective attraction in the Fermi component of the BF superfluid mixture: indeed, the coefficient in front of this term is proportional to the square of the scattering length, a2BF , which accounts for the BF attraction. It is also worthy to note that, in the present simplest approximation, which makes it possible to eliminate the boson field and reduce the model to the single equation for the fermionic function, the attractive character of the resulting boson-mediated interaction requires aB > 0, i.e., the repulsive character of the direct interaction between the bosons. Note that in the case when both the BF and boson-boson interactions are attractive, i.e., both aB and aBF are negative, the present approximation is impossible, according to Eq. (32); in the case of the BF repulsion and boson-boson attraction, i.e., aB < 0 and aBF > 0, the approximation is possible, but it leads to an effective boson-mediated repulsion between the fermions. Equation (33) is a variant of the stationary NLSE with two competing nonlinear terms, 2/3 7/3 the self-repulsive one ∼ NF uF , and the self-attractive cubic term. For a⊥ = 1 µm and the scattering lengths corresponding to the 87Rb −40 K mixture, the coefficient in front of
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the cubic term is 18a2BF/ (aB a⊥ ) ≈ 0.71, while, with ξ = 0.4 (recall it corresponds to the unitarity regime for the fermion component), the coefficient in front of the repulsive term is (3π 2)2/3(3ξ/5) ≈ 2. 30. The SSB controlled by competing nonlinearities, viz., self-focusing cubic and selfdefocusing quintic terms, was studied in Refs. [12], where it was concluded that the respective SSB diagrams, showing the asymmetry versus the total norm of the mode (cf. Fig. 2), tend to form a closed loop connecting an initial symmetry-breaking bifurcation and a final symmetry-restoring one. The difference of Eq. (33) is that here the self-focusing (cubic) term has a higher nonlinearity power than its self-defocusing counterpart of power 7/3, therefore no closed-loop bifurcation diagram is expected in the present case. √ The substitution of uF (x) ≡ vF / NF casts Eq. (33) in a parameter-free form [the respective equation for vF seems as Eq. ((33) with NF replaced by 1]; of course, this substitution changes normalization (21) for the fermionic function, as the norm of vF is exactly NF, but not 1. Thus, in the present approximation, the BF mixture is described by the universal equation. This circumstance, along with condition (32) necessary for the applicability of Eq. (33), may explain the fact that the border of the trapped states in Fig. 3 is practically a straight line with the slope equal to 1, which runs into the SSB border at a critical value of NF [the latter one actually corresponds to the critical norm of field vF (x) at which the SSB occurs in the framework of Eq. (33)]. The actual SSB point generated by Eq. (33) can be predicted in an approximate analytical form by means of the two-mode expansion, which, as mentioned above, is commonly used for the description of SSB effects in DWP settings [16], assuming that the stationary solution is approximated by a superposition of two linear modes, u± (x), which are trapped in the left and right potential wells: uF (x) = A+ u+ (x) + A− u− (x).
(34)
Symmetric states correspond to A+ = A− ≡ A0 in Eq. (34). The SSB border can be found by looking for a point where a solution with an infinitely small antisymmetric perturbation (δ), A± = A0 ± δ, branches off from the parent symmetric state. By performing this analysis (we do not display straightforward details here), one finds that the value of A0 for √ SSB scales inversely proportional to NF, hence the increase of the number of particles facilitates the transition to the asymmetric ground state, as one may expect.
5.
Conclusion
The objective of this work is to extend the analysis of the SSB (spontaneously symmetry breaking) in DWP settings (double-well potentials), which was recently studied in BEC and bosonic mixtures, to the BF (Bose-Fermi) mixtures. The system is described by the GPE (Gross-Pitaevskii equation) for the bosons, which is nonlinearly coupled to the equation for the fermionic order parameter derived from the density functional in the unitarity limit (in fact, a similar model also applies to the BF mixture with the fermionic component kept in the BCS regime). Direct symmetry breaking in the Fermi superfluid trapped in the DWP is impossible, as it must be induced by attractive interactions, while density perturbations in the degenerate fermionic gas interact repulsively. Nevertheless, we have demonstrated that
Spontaneous Symmetry Breaking in a Mixed Superfluid of Fermions and Bosons...287 the SSB is possible in the mixture of 87Rb and 40 K atoms, due to the attraction between fermions and bosons. The most interesting case, when the effective SSB in the fermionic component could be studied in the “pure form”, is that when the fermions are subject to the action of the DWP, while there is no potential confining the bosons. We have also investigated the alternative situation, with the DWP acting solely on the bosons. Our phase diagrams in the (NF, NB) plane, produced by means of numerical methods, clearly show that the inter-atomic attractions can produce both symmetric and symmetry-broken localization of the atoms which are not subject to the direct action of the trapping potential. Applying the TF (Thomas-Fermi) approximation to the bosonic equation, we have also developed an analytical approximation, which allows us to reduce the model to the single equation for the fermionic function. In the latter case, the model explicitly demonstrates the generation of the effective attraction between fermions mediated by the bosons. The analysis reported in this work can be extended in other directions. A straightforward generalization may deal with the system including the confining potential in both components, as well as a more general analysis of the TF approximation. A challenging possibility is to predict similar effects in multi-dimensional Bose-Fermi mixtures. The work of S.K.A. was partially supported by CNPq and FAPESP (Brazil). B.A.M. acknowledges hospitality of the Department of Physics “Galileo Galilei” at the University of Padua, Italy.
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In: Horizons in World Physics. Volume 271 Editor: Albert Reimer
ISBN: 978-1-61761-884-0 c 2011 Nova Science Publishers, Inc.
Chapter 8
A LGEBRA AND T HERMODYNAMICS OF q- DEFORMED F ERMION O SCILLATORS 1
A. Lavagno1 ∗, P. Narayana Swamy2 † Dipartimento di Fisica, Politecnico di Torino, I-10129 Torino, Italy and INFN, Sezione di Torino, I-10126 Torino, Italy Department of Physics, Southern Illinois University, Edwardsville, IL, U.S.A.
Abstract The formulation of the theory of q-deformed fermions has been of considerable interest in the literature. We have formulated the theory of q-deformed fermions in considerable detail and investigated the thermodynamics of such systems. The algebra, Fock space and the thermodynamics of q-deformed fermions has been fully investigated. The distribution function of such systems has been studied as a function of the deformation parameter and the behavior of the ideal q-fermion gas has been compared with that of the ordinary fermions. More recently, the interpolating statistics of q-fermions have been studied in terms of B-type and F-type interpolating statistics. The distribution function of such systems has been determined in terms of their analytic forms and have also been expressed as infinite continued fractions. The advantage of such infinite continued fractions is in clarifying the nature of the approximations. Moreover, the statistical mechanics of particles obeying interpolating statistics has been formulated in terms of q-deformed oscillator algebras of q-bosons and q-fermions on the basis of Feynman’s method of Detailed Balance. This formulation describes the connection between anyons (statistics which interpolates between standard bosons and fermions) and the principle of Detailed Balance and investigates the distribution function and other thermodynamic functions as infinite continued fractions. This formulation of interpolating statistics has also been studied in the context of Haldane and Gentile statistics. The formulation of interpolating statistics or intermediate statistics has also been shown to be linked to deformed oscillator algebras. Deformed permutation in terms of a parameter κ has been shown to imply the existence of the basic number which is shown in turn to imply the deformed algebra. In this formulation, the occupation number is generalized to the basic number N which ∗ †
Email address:
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A. Lavagno and P. Narayana Swamy is expressed in terms of the parameter κ which in turn leads to the deformed algebra corresponding to intermediate statistics. We thus find that the subject of q-deformed fermions has been investigated rather thoroughly – not only a formulation in terms of the Fock space of states but also the consequences for the various thermodynamic property of the particles obeying such statistics.
Keywords: fermion oscillators, q-deformation algebra, q-fermion thermostatistics
1.
Introduction
Let us begin with the classical system of ordinary quantum fermion harmonic oscillators with the spectrum 1 En = (n − )~ω, n = 0, 1 , (1) 2 and the Partition function given by Z=
1 X
eβ En = 2 cosh
0
β~ω , 2
(2)
where β = 1/kT and k is the Boltzmann constant. The free energy is 1 1 β~ω F = − ln Z = − ln 2 + ln cosh , β β 2
(3)
from which we obtain the entropy: ∂F ~ω β~ω = −kβF − β tanh . ∂T 2 2 The internal energy of the quantum fermion oscillators is then determined by S=
β~ω 1 . U = F + T S = − ~ω tanh 2 2 We may also express the internal energy in the form 1 1 − ex 1 1 = ~ω − + x U = ~ω 2 1 + ex 2 e +1
(4)
(5)
(6)
where x = β~ω. Here the first term contains the zero point energy and the second is determined by the Fermi distribution. For N non-interacting fermion oscillators, we have U = N ~ωf where f is the probability distribution function and therefore we infer that 1 1 β~ω 1 = − tanh . f =− + x 2 e +1 2 2
(7)
Alternatively we can derive the form of the distribution function as follows. The occupational probability is Pn =
e−βEn e−βEn , = βE Z 2 cosh 2
1 En = (n − )~ω, 2
n = 0, 1.
(8)
Algebra and Thermodynamics of q-deformed Fermion Oscillators Thus we have
1 X β~ω 1 1 f= (n − )Pn = − tanh 2 2 2 0
293
(9)
which can also be expressed as in Eq.(7). We may examine the high temperature limit of the internal energy when β~ω << 1 and thus 1 1 UT →∞ = lim N ~ω − + x 2 e +1 ! 1 1 = lim N ~ω − + = 0, (10) 2 2 + β~ω + 12 (β~ω)2 · · · as expected: the energy of the fermion oscillator indeed vanishes in the classical limit, it is purely a quantum effect, when Pauli exclusion principle prevails. Next, we may consider the low temperature limit, when β~ω >> 1. We then find 1 1 (11) UT →0 = lim N ~ω − + e−β~ω = − N ~ω 2 2 again as expected. Of course in this limit, we expect the internal energy to reduce to the zero point energy, a pure quantum effect. It is useful to consider the development of the ordinary fermion oscillators based on the most intuitive formulation [1]. This theory is described by the algebra of creation and annihilation operators defined by aa† + a†a = 1,
[N, a] = −a,
[N, a†] = a† ,
a2 = (a†)2 = 0,
(12)
in accordance with Pauli exclusion principle, where N is the number operator. It is important to recognize that every deformed version of this theory reduces to the standard fermionic oscillators in the limit when the deformation parameter q tends to the limit q → 1. This basic formulation provides a good contrast with the bosonic oscillators discussed by Das [1]. We have suppressed the quantum number indices of the creation and annihilation operators. The Hamiltonian is given by H=
1 1 ~ω(a†a − aa† ) = ~ω(N − ) 2 2
with the eigenvalues E = ~ω(N − 12 ). The Fock states constructed by √ √ a|n > = n|n − 1 >, a† |n >= n + 1|n + 1 >,
(13)
(14)
obey the Pauli exclusion principle, and Eq.(12) implies N 2 = a† (1−a† a)a = N, N (N − 1) = 0, so that the number operator has n = 0, 1 as the only allowed eigenvalues. The conventional theory of q-deformed fermion oscillators consists of a straightforward extension of the ordinary fermion oscillators, employing the q-analog [2, 3] of the ordinary fermion algebra. This version is defined by aa† + qa† a = q N ,
[N, a] = −a,
[N, a†] = a†,
a2 = (a† )2 = 0.
(15)
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It is well-known that this version of q-fermions such as in the work of Hayashi [4] and others reduces trivially to the ordinary undeformed fermions with Pauli exclusion principle as it can be shown that the deformation can be transformed away. This has been demonstrated by R. Parthasarathy et al [5] (in the following, PVC algebra) and by Jing and Xu [6]. Accordingly, we shall therefore investigate the theory of the generalized q-deformed fermions proposed by Parthasarathy et al [5]. This particular theory allows many fermion states with inclusion principle, i.e., without exclusion principle, but reduces to the ordinary fermions with exclusion principle in the classical limit q → 1. We shall present a brief review of this generalized theory and point out some extraordinary features of this theory before investigating the statistical mechanics of this generalized theory.
2.
The Algebra of Deformed Fermions
The q-deformed theory of generalized fermions is described by the algebra of the creation and annihilation operators. The mathematical framework of q-oscillators is formulated on the basis of q-calculus based on the Jackson derivative and hence we expect q-calculus to play a fundamental role in the thermostatistics of the deformed q-oscillators. Indeed, it has been shown [7, 8] that an internally self-consistent statistical thermodynamics of q-bosons can be successfully formulated using an appropriate prescription of the Jackson derivative and consequently the entire structure of thermodynamics is preserved, in the sense of Legendre transformations, if such a q-calculus is employed. We begin with the algebra describing the q-deformed fermions [c, c]κ = [c†, c†]κ = 0 , cc† − κ q κ c†c = q −N ,
(16)
[N, c†] = c† , [N, c] = −c ,
(17)
where the deformation parameter q is real and [x, y]κ = xy − κyx , where we take the parameter κ = 1 for q-bosons with commutators and κ = −1 for q-fermions with anticommutators. The operators also satisfy the relations c†c = [N ] ,
cc† = [1 + κN ] ,
(18)
where the q-basic number is defined as [x] =
q x − q −x . q − q −1
(19)
We note that the above algebra defining the deformation is symmetric in q → q −1 and thus has many advantages such as the possibility of the choice q > 1. We shall simply refer to this as the symmetric formulation. By using the above algebra it is possible to construct the Fock space, or the number space, valid for both q-bosons and q-fermions. It is, however, important to note that for q-fermions the eigenvalues of the number operator N takes on the values n = 0, 1 only (as in the case of undeformed fermions), in accordance with Pauli principle.
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295
The transformation from Fock space to the configuration space (Bargmann holomorphic representation) may be accomplished by means of the Jackson derivative [9, 10, 11] defined as f (qx) − f (q −1 x) , (20) Dq f (x) = x (q − q −1 ) which reduces to the ordinary derivative in the limit when q goes to unity. Therefore, the q-dependent derivative occurs naturally in q-deformed structures and plays a crucial role in the q-generalization of the thermodynamic relations. The thermal average of any observable can be computed by following the usual prescription of quantum mechanics. The Hamiltonian of the non-interacting q-deformed oscillators for fermions or bosons is expected to have the form X (i − µ) Ni , (21) H= i
in the standard notation of thermodynamics. Despite appearances, we note that the Hamiltonian is deformed and depends implicitly on q, since the number operator is deformed by means of Eq.(3). The thermal average of an operator has the standard form hOi = T r (ρ O) ,
(22)
where ρ is the density operator and Z is the grand canonical partition function, defined by ρ=
e−βH , Z
Z = T r e−βH ,
(23)
and β = 1/kT . We observe that the structure of the density matrix ρ and the thermal average remain undeformed and consequently, the structure of the partition function is also undeformed. We note that this is not a trivial assumption because its validity implicitly amounts to an unmodified structure of the Boltzmann-Gibbs entropy, Sq = log Wq . Obviously the number Wq is modified in the q-deformed case. The above assumptions allow
us to calculate the average occupation number ni defined by [ni ] = T r e−βH c†i ci /Z .
Repeated application of the algebra of c, c† along with the use of the cyclic property of the trace [12], leads us to the expression for the average occupation number −1 βi 1 z e − κ q −κ log , (24) ni = q − q −1 z −1 eβi − κ q κ where z = eβµ is the fugacity. It is easily verified that this reproduces the standard distribution when q → 1. We shall now proceed to deal with the thermodynamic functions describing the behavior of an ideal fermion gas. In the thermodynamic limit, for a large volume and a large number of particles, the sum over states can be replaced by the integral and the thermodynamic relation P V /T = log Z can be expressed as Z 2 gκ ∞ P √ = −κ dx x1/2 log(1 − κ z e−x ) , (25) T π λ3 0
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where gκ is the spin degeneracy factor, x = β, = p2/2m is the kinetic energy and λ = h/(2πmkT )1/2 is the thermal wavelength. Following the prescription of the thermodynamic derivative, we may re-express the above equation as gκ P = 3 hκ5/2 (z, q) , (26) T λ where we have defined the q-deformed hκn (z, q) function as −1 x Z ∞ 1 z e − κ q −κ 1 κ n−1 hn (z, q) = dx x log Γ(n) 0 q − q −1 z −1 ex − κ q κ ! ∞ ∞ X 1 (κ q κ z)i X (κ q −κ z)i ≡ − . (27) q − q −1 in+1 in+1 i=1
i=1
In the limit q → 1, the deformed hκn (z, q) functions reduce to the standard fn (z) functions for fermions [13]. We thus obtain the particle density in the deformed case, gκ N = 3 hκ3/2 (z, q) , V λ
(28)
and the internal energy, 3 gκ V T hκ5/2 (z, q) . (29) 2 λ3 Comparing Eqs.(26) and (29), we see that the following well-known relation is satisfied U=
U=
3 PV . 2
(30)
We can also obtain the entropy per unit volume in the thermodynamic limit, gκ 5 κ S κ = 3 h (z, q) − h3/2 (z, q) log z . V λ 2 5/2
(31)
We may also calculate the specific heat ∂U Cv = . ∂T V,N
(32)
Making use of the Jackson Derivative prescription as before, we determine the specific heat: X ∂αi 1 1 − κ q −κ αi 2 (q) i D log , (33) Cv = −β ∂β q − q −1 αi 1 − κ q κ αi i
where αi = ze−βi . The above equation can be written in the thermodynamic limit as (q)
κ 2 gκ n 15 9 z [Dz h5/2 (z, q)] o (q) κ Cv = 3 z Dz h7/2 (z, q) − . λ 4 4 Dz(q)hκ (z, q) 3/2
(34)
We observe that the above equations have the same structure as the undeformed relation, even though the deformation is contained in the JD and in the h(z, q) functions.
Algebra and Thermodynamics of q-deformed Fermion Oscillators
3.
297
Equation of State in the Semi-classical Limit
We observe that in the classical limit z = eβµ 1, the q-deformed distribution function (24) reduces to the standard Maxwell-Boltzmann distribution and the entropy reduces to P the Boltzmann entropy, S = − i ni log ni , for any value of q. Furthermore, studying the classical limit in Eq.(34), we find that the specific heat goes to its classical value, Cv → 3/2N . Hence the q-deformation in the thermodynamic relations is a pure quantum effect, which is washed up in the classical limit. We shall now investigate the effects of the deformation in the equation of state of an ideal quantum fermion gas in the semiclassical limit. For low values of z, Eq.(27) reduces to n q + q −1 z o T . (35) P ' 3 gκ z 1 + κ λ 2 25/2 which can be expressed in terms of the number of particles considering the semiclassical limit of Eq.(28) n q + q −1 z o V . (36) N ' 3 gκ z 1 + κ λ 2 23/2 Inverting the above equation and inserting the result in Eq.(35), we can determine the equation of state in the semiclassical limit to be h q + q −1 N λ3 o PV ' NT 1 − κ . 2 V gκ 25/2
(37)
When the thermal wavelength λ is much less than the average inter-particle distance V ), the quantum statistics does not have a significant influence on the thermodynamic property of the gas. Otherwise we find that the equation of state is modified by the quantum statistical effect and, at fixed volume, the pressure is increased for q-fermion particles (κ = −1) compared to the classical case. This feature is similar to the standard fermion result, namely, the attractive boson interactions reduce the pressure and the repulsive fermion interactions increase the degeneracy pressure. However, in Eq.(37), this effect is enhanced by the factor (q + q −1 )/2 (always greater than unity). Therefore, the q-deformation of the algebra leads to an enhancement of the quantum statistical behavior of the particles. The ideal q-deformed fermions at low temperatures is examined in ref. [7] and it is shown that the deformation increases the strength of the Pauli repulsion at any finite temperature. This result is similar to the finding by Greenberg [14]. The thermodynamics is thus developed for the q-deformed fermions and it is shown that the structure of thermodynamics is preserved in a self-consistent manner if the ordinary thermodynamic derivatives are replaced by the Jackson Derivatives. The generalized thermodynamics developed in this manner appears to provide a deeper insight into the nature of the deformed fermion algebra. These results could be conceptually significant in many physical situations. For instance, primordial nucleosynthesis can be non-trivially modified by the influence of statistics. Furthermore, finite limits on the violation of Pauli principle by nuclei produced in the core collapse supernovas could be studied.
(λ3
298
4.
A. Lavagno and P. Narayana Swamy
The PVC Algebra, Further Analysis
We shall first consider the deformed algebra of Ref. [5] (PVC algebra) aa† + qa† a = q −N , [N, a] = −a, [N, a†] = a† .
(38)
in terms of the number operator N , the annihilation and creation operators a, a† and the deformation parameter q. It may be noted that, despite appearances, this algebra is quite different from the algebra of Eq.(16). This algebra was introduced by Parthasarathy et al and further investigated by Chaichian et al [15]. We may proceed to consider the action on the number states |ni, thus assuming the form a|ni = Cn |n−1i, ; a†|ni = Cn0 |n+1i where Cn , Cn0 are constants to be determined. We are consistently employing the notation where ˜ N stands for the operator and n refers to the eigenvalue. If we take N|ni = a† a|ni = ˜ a†|ni = αn+1 C 0 |ni. Employing the algebra, we immediately obtain αn ni, the result N n −n αn+1 = q − qαn . Solution of this difference equation leads to the determination αn = q −n − (−1)n q n , thus leading to the basic number form. The operator basic number [N ] is q + q −1 thus recognized as q −N − (−1)N q N . (39) [N ] = q + q −1 We observe that this basic number for generalized fermions differs from the corresponding basic number for q-bosons; it also differs from the one assumed for the q-fermions defined in Ref.[8]. Hence it follows that the algebra, Eq.(38) automatically leads to the basic number. Furthermore, we can determine the results p p Cn = [n], Cn0 = [n + 1], aa† = [N + 1] = q −N − q[N ] . (40) The Fock space of the q-deformed fermions is defined by |ni, n = 0, 1, 2, · · · when q 6= 1 and can be constructed from the vacuum state by the prescription (a† )n |0i , |ni = p [n]!
(41)
where [n]! = [n][n − 1] · · · [2][1], when n > 0. The Hamiltonian of the ordinary fermion harmonic oscillator generalizes to 1 (42) H = (a† a − aa†) = ~ω([N ] − [N + 1]) , 2 which includes the zero point energy. The energy spectrum is given by the eigenvalues 1 ~ω([n] − [n + 1]), n = 0, 1, 2, · · · . (43) 2 It is seen that the above reduces to the case of ordinary fermions when q = 1. We also observe that limq→1 α1 = 1, α2 = 0. Hence, in the limit, all the states vanish from the spectrum and Pauli principle prevails in the undeformed limit. For q 6= 1, the theory describes generalized fermions with no exclusion principle. From the trace formula for the thermal averages, we have Enq =
hq −N i =
T r(q −N e−βH ) T r((−1)N q −N e−βH ) , (−1)N q N = , Z Z
(44)
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299
we obtain the result h[N ]ii = e−β(Ei −µ)h[N +1]i )i. Dropping the subscript for convenience, we may express it in the form [n + 1] , (45) eη = [n] where η = β(E − µ). It is easy to verify that this leads to the standard Fermi distribution, n = 1/(eη + 1), in the undeformed limit. To proceed further, it is best to separate the cases of even and odd n, thus rewriting the above equation as η −1 e − q −1 q − eη 1 1 ln , nO = ln , (46) nE = 2 ln q eη + q 2 ln q eη + q corresponding to the two cases. We may now write the modified form of the q-fermion distribution to be η 1 |e − q −1 | n= ln . (47) 2| ln q| eη + q For plots of the standard Fermi distribution ( q = 1) and the plots for the deformed distribution (q = 1/3, q = 1/2) and for plots of the modified functions h(5/2, z, q), f (5/2, z), one may refer to the figures in PNS-1 [16].
5.
Other Deformed Fermion Algebras Recently Investigated
We shall first consider the algebra defined by aa† + q −1 a† a = q −N ,
(48)
where 0 ≤ q ≤ 1, which reduces to the standard Fermi algebra in the limit q → 1, together with the relations [N, a] = −a, [N, a†] = a† where N is the number operator, and q is ˜ = a† a, with the concomitant action on Fock the deformation parameter. We may define N ˜ states, N |ni = βn |ni, where the constant βn is to be determined. Solving the recurrence relation βn+1 = q −n − q −1 βn , we accordingly obtain the result βn = 0, 1, 0, q −2, 0, q −4, · · · =
1 − (−1)n −n+1 . q 2
(49)
The action of the creation and annihilation operators on the Fock states produces the results a† |0i = 1, ; (a†)2|0i = 0. we may now consider the algebra proposed by Parthasarathy et al [5], namely aa† + qa† a = q −N ,
(50)
where N is the number operator and other variables are as before. This algebra has also been discussed by Chaichian et al [15]. It is evident that this reduces to the standard ermion ˜ = a† a to satisfy the relation algebra in the undeformed limit. We may take the operator N ˜ N|ni = αn |ni and determine the constant αn as follows. If we define a† |ni = Cn0 |n + 1i, a|ni = Cn |n − 1i, we may determine Cn , Cn0 in the standard manner. We also obtain the relation αn+1 = q −n − qαn . Solving this recurrence relation, we immediately obtain α0 = 0, α1 = 1, α2 = q −1 − q, · · · , αn = q −n+1 − q −n+3 + · · · q n−3 − q n−1 ,
(51)
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which is immediately recognized as the basic number defined as αn = [n] =
q −n − (−1)n q n . q + q −1
(52)
This basic number is seen to be slightly different from the standard basic number of bosons as well as fermions obeying a different algebra. This also explains that this introduction of the basic number is a straightforward consequence of the q-fermion algebra. We observe that Pauli exclusion is valid only in the undeformed limit, q → 1. The above basic number can also be generalized to the operator form. For arbitrary values of q, we may construct the Fock states from the vacuum state on up by the prescription (a†)n |0i . |ni = p [n]!
(53)
The exclusion principle follows from the fact that α2 = 0 when q → 1. However, we observe that [n] = 12 (1 − (−1)n ) in the undeformed limit in which case the Fock space breaks up into an infinity of 2-dimensional subspaces, with the exclusion principle valid in each subspace. The basic number here exhibits skew symmetry, i.e., [n] → ±[n] for n = odd, even and this contrasts with the situation in other algebras. The Jackson derivative needs special care in this case. First, we recall that the Jackson derivative in q-boson case [8] reduces to the ordinary derivative in the limit q → 1. Now we seek to determine the Jackson derivative corresponding to the algebra given by Eq.(50).we may invoke the holomorphic representation given by a ⇒ Dx , a† ⇒ x. The algebra thus implies the operator relation Dx x + qxDx = q −N . It may be useful to recall that the holomorphy leads to properties such as [17] q N x = xq N +1, q N xr = q r xr , [N ]x = x[N + 1], [N ]x + qx[N ] = xq −N ,
(54)
etc. We therefore infer the solution of the relation Dx x + qxcalDx = q −N to be Dx =
1 q −N − (−1)N q N , x q + q −1
(55)
which thus defines the appropriate Jack derivative for the fermions deformed in this particular manner. If we now employ the properties q N f (x) = f (qx), q −N f (x) = f (q −1 x), (−q)N f (x) = f (−qx) ,
(56)
this can be expressed as another representation for the Jackson Derivative which may be useful for the above Q-deformed fermions: Dx f (x) =
1 f (q −1 x) − f (−qx) . x q + q −1
(57)
One can investigate many of the properties satisfied by this Jackson Derivative. In particular one may observe that the q-fermion Jackson Derivative does not reduce to the ordinary derivative in the limit q → 1 [18].
Algebra and Thermodynamics of q-deformed Fermion Oscillators
301
We have thus investigated the consequences of the q-deformed fermion algebras aa† + q −1 + a† a = q −N and aa† + qa† a = q −N . One of them obeys the exclusion principle in the undeformed limit. We have presented detailed physical applications of the generalized fermions. For the determination of the various thermodynamic functions such as the partition function, pressure, entropy and etc. we refer to the investigation in ref. [19]. We have also investigated the generalized fermions which do not obey the Pauli exclusion principle. The Fock states of these generalized fermions can be built from the vacuum state in the usual manner. The q-calculus need to investigate these fermions must employ the appropriate Jackson Derivative and this is characteristic of the generalized fermions. We have determined the form of this Jackson Derivative.
6.
Anyon Statistics, Intermediate Statistics from General Principles
The particles which are described by intermediate statistics, i.e., obeying statistics that interpolates between Bose and Fermi statistics, could be the anyons dealt with in the literature. Chern-Simons [20] theory provides a convenient representation of anyons. Such particles are also based on the connection between fractional spin and statistics[21, 22]. Anyons carry electric charge and magnetic flux Φ in two space dimensions, based on the vector potential Φ ij xj . (58) Ak (x) = π |x|2 The Hamiltonian containing non-local effects, via (pi − eAi /c)2 can be reduced to the free Hamiltonian in a special gauge, the anyon gauge. In two dimensional space, the exchange of two particles reduces to a rotation, (plus translation), whereas rotation by an angle is not unique in higher dimensions. Permutation symmetry is then equivalent to the braid group in two space dimensions [22]. The infinite group of N strands, generated by N −1 elementary moves has the property σi σi+1 σi = σi+1 σi σi+1 for i = 1, · · · , N − 2;
σi σj = σj σi for |i − j| ≥ 2 .
(59)
Despite much progress based on the formalism of the braid group, there has been only limited understanding of the thermostatistics of anyons besides the work on the virial coefficients of anyons [23]. The earlier work [24] was an investigation of the statistical mechanics of anyons based on the ansatz N BA = γΣk
1 eβ(E−µ)
−f
;
N F A = γΣk
1 eβ(E−µ)
+g
(60)
which defines the distribution function for boson-like anyons and fermion-like anyons respectively, where γ is the multiplicity and f, g are functions of the statistics determining parameter α = θ/π. The parameters f, g are determined by f (α) = 1 − 4α + 2α2 ; g(α) = 2α2 − 1. The Boson, Fermion and Semion limits thus correspond to α = 0, 1, 12 . The coefficients Bi in the virial expansion, λ2ρ λ2ρ 2 PV (61) |BA = B1 + B2 ( ) + B3 ( ) +··· , N kT γ γ
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are determined and evaluated in Ref.[24]. Here the thermal wavelength is λ2 = 2πβ~2/m. Various thermodynamic quantities such as the specific heat have been expressed [24] in terms of T0, f and the generalized Riemann Zeta function. It is interesting to note that the Aharanov-Bohm [25] scattering cross-section dσ 1 sin2 πα = , dφ 2πk sin2 φ/2
(62)
is non-zero for 0 < α < 1 and could provide a significant test of the interpolating statistics. According to conventional wisdom, there may not be any connection between interpolating statistics and q-oscillator algebra since anyons arise in two dimensions, based on the braid group, while the q-oscillator algebra exists in any number of dimensions, but this argument may not be true. More recently, Frappat et al [26] have analyzed the connection between q-algebra on a two-dimensional lattice and anyons, with q = eiπν . More recently Chaturvedi et al [27] have studied the various forms of interpolation statistics available in the literature. They examine the question as to which of these statistics can be considered as a proper generalization of the permutation group statistics and which ones could be consistent with other theories such as Haldane and Gentile statistics. It must be observed that despite a great deal of theoretical progress, there has not been a satisfactory formulation of the statistical mechanics corresponding to the interpolating statistics. There exists no satisfactory derivation of the distribution function and no clear formulation of the statistical mechanics for the ensemble of such particles. One interesting approach to understand fractional statistics from kinetics, such as the master equation and the Fokker-Planck equation, has been due to G. Kaniadakis, A. Lavagno and P. Quarati [28], which does not seem to restrict the theory to two space dimensions. We may now proceed to formulate a theory of particles obeying intermediate statistics, interpolating between Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. We must build the states of these particles by a generalized procedure of f -symmetrizing in such a way that it will reduce to the standard procedure of symmetrizing for boson and anti-symmetrizing for fermions. Accordingly we shall take the exchange symmetry to be the factor f = eiπα with the property |f |2 = 1 so that the limits f → ±1 correspond to bosons and fermions. Following Feynman [29], we deal with the general two particle scattering amplitude defined by the product a1 b2 = h1|aih2|bi. We shall eventually take the states 1 and 2 to be the same for identical particles. The exchange process corresponding to a → 2, b → 1 is described by the amplitude f a2 b1 due to the exchange symmetry factor f . The total probability amplitude is the sum of the direct and exchange processes. Employing the abbreviation h1|ai = h2|ai = a, we find the probability of this two particle scattering process involving non-identical particles to be (2)
pnon−identical = (1 + |f |2)|a|2|b|2 = 2(|a|2|b|2) ,
(63)
which is the same as for ordinary bosons. However, for identical particles, we need to take account of interference between the two processes and we obtain the probability in this case to be (2) (64) pidentical = (1 + |f |2 + f + f −1 ) |a|2|b|2 ,
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since f ∗ = f −1 . The probability therefore depends on the statistics determining parameter α. In the limit f → 1, this would reduce to the case of bosons and it would be twice as much as in Eq.(63) for the non-identical particles. For arbitrary f the probability for the process involving identical particles relative to that for non-identical particles is given by (2)
P
(2)
=
pidentical (2)
=
pnon−identical
1 2 + f + f −1 |a|2|b|2. 2
(65)
We may omit |a|2, |b|2 etc. since the single particle states would be normalized appropriately. Similarly, we consider the three-particle processes a → 1, b → 2, c → 3 together with the exchange processes with factor f for each exchange operation, thus resulting in the combination abc + f acb + f bac + f 2 bca + f 2 cab + f 3 cba with the same abbreviation as earlier. We now determine the probability of the state with three identical particles as P (3) =
1 (1 + 2f −1 + 2f −2 + f −3 )(1 + 2f + 2f 2 + f 3 ) . 6
(66)
which can be re-expressed as P (3) =
1 6 + 7(f + f −1 ) + 4(f 2 + f −2 + 1) + (f 3 + f −3 + f + f −1 ) . 6
(67)
We observe that the above expressions can be expressed succinctly in terms of the basic numbers [30] defined by f n − f −n . (68) [n] = f − f −1 Here f = eiπα , the BE limit is α → 0, f → 1 and the FD limit is α → 1, f → −1 and f ∗ = f −1 . Our formulation is symmetric under f → f −1 and is the generalization familiar in the literature, of the basic numbers introduced long ago by F. H. Jackson [30]. Studying the limits, we find that the Bose limit gives [n] → n while the Fermi limit is quite different: [n] → (−1)n+1 n which becomes −n for even numbers and +n for odd numbers. However, it can be inferred that the particles obey Pauli exclusion principle in the undeformed limit. Returning to the basic numbers, it is quite evident that the basic numbers arise naturally, automatically, in the theory of particles obeying interpolating statistics. The representation [n] = f n−1 + f n−3 + · · · + f −n+3 + f −n+1 , (69) for the bracket number can be alternatively expressed in the form [n] =
sin nπα . sin πα
(70)
In terms of the basic numbers, the probabilities may now be expressed succinctly as P (2) = P (3) = P (4) =
1 (2 + [2]) , 2 1 (6 + 7[2] + 4[3] + [4]) , 6 1 ( 35 + 54[2] + 52[3] + 36[4] + 18[5] + 6[6] + [7] ) , 4!
(71)
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and so on. Employing some trigonometric identities [31], we can derive some useful formulae: (72) [1] + [3] + [5] + · · · + [2n − 1] = ([n])2 and [n − 1] [n] = [2] + [4] + [6] + · · · [2(n − 1)] ,
(73)
so that [2] [3] = [2] + [4], [3] [4] = [2] + [4] + [6], etc. By using the above identities and after some algebra, we can express the probability functions for the states of many identical particles in the following manner: 1 ( 2 + [2] ) , 2! 1 ( 2 + [2] )( 2 + 2[2] + [3] ) , P (3) = 3! 1 P (4) = ( 2 + [2] )( 2 + 2[2] + [3] )( 2 + 2[2] + 2[3] + [4] ) , 4! and so on. Accordingly we obtain the probability for the n-particle state: P (2) =
(74)
1 P (n) = √ (2 + [2]) (2 + 2[2] + [3]) · · · (2 + 2[2] + 2[3] + · · · + 2[n − 1] + [n]) . n! (75) Now we shall employ the principle of Detailed Balance in order to determine the Statistics. From the above form, we can infer the enhancement factor, which measures how much greater the probability of the n + 1-particle state is, compared to the probability of the n-particle state: n! P (n+1) 2 + 2[2] + 2[3] + · · · + 2[n] + [n + 1] = F (n) = (n) (n + 1) ! 2 + 2[2] + 2[3] + · · · + 2[n − 1] + [n] P
(76)
This enhancement factor plays a vital role in the formulation of intermediate statistics: if n1 , n2 represent the average occupation numbers of states 1 and 2 respectively, then the rate for transition 1 → 2 must equal that for 2 → 1. We should stress that the principle of Detailed Balance goes beyond mere time reversal invariance and is valid when thermodynamic equilibrium prevails or microscopic reversibility [32, 33, 34] in the language of statistical physics: it can be seen as a consequence of the second law of thermodynamics. The principle of Detailed Balance accordingly leads to the condition n1 F (n2 )eβE1 = n2 F (n1 )eβE2 ,
(77)
since the population of each level is governed by the Boltzmann factor eβE ; F (n) is the enhancement factor for each level. In the BE limit for instance, the enhancement factor is just n + 1, which immediately leads to the BE distribution, n=
1 z −1 eβE
−1
,
(78)
where z = eβµ is the fugacity of the gas. The enhancement factor for arbitrary f is given by Eq.(20) which can be put in the convenient form F (n) =
1 (2 + 2[2] + 2[3] + · · · 2[n] + [n + 1]) . n+1
(79)
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305
Making use of the identities n X
[k] = 2 cos(πα/2) [n/2] [(n + 1)/2] ,
(80)
k=0
and [n/2] + [n/2 + 1] = 2 cos(πα/2)[(n + 1)/2] ,
(81)
we can reduce the above to the form F (n) =
4 {[(n + 1)/2] cos(πα/2)}2 . n+1
(82)
Invoking the principle of detailed balance, we accordingly obtain the result 4 1 βE e = F (n)/n = {[(n + 1)/2] cos(πα/2)}2 . z n(n + 1)
(83)
We can express it more conveniently as eβ(E−µ) =
1 sin2 (n + 1)πα/2 . n(n + 1) sin2 πα/2
(84)
In the BE limit, F (n) → n + 1 and the distribution reproduces the Bose distribution. The case of Fermi limit, while not straightforward, does lead to the Fermi distribution. Evaluating the limit we find 2 1 1 i(n+1)π/2 lim F (n) = Im e {1, 0, 1, 0, 1, · · ·} , (85) = α→1 n+1 n+1 thus leading to a generalized fermion theory. Determining the distribution function for these particles from a solution of Eq.(28) is a difficult task. We cannot obtain a solution for n of this equation in closed form. However, we can express it as a series in the form eβ(E−µ) =
1 + a0 + a1 n + a2 n2 + · · · , n
(86)
where the coefficients ak are given by trigonometric functions of πα. This series can be reverted to obtain the following solution for the distribution function n=
1 a1 a2 (2a21 + a3) + + + + ··· , g g3 g4 g5
(87)
where g = 1/(eβ(E−µ) − a0 ). we would now like to formulate the theory of interpolating statistics in terms of an infinite continued fraction whose advantage will be discussed later [35]: 1 , (88) n(E) = α3 g g− α4 g g 2 + α3 − α5 g α3 g + α4 − α4 g + α5 − · · ·
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where α3 = a1, α4 = a2 , α5 = 2a21 + a3 , α6 = 5a1a2 + a4 ,
(89)
etc. The first convergent, or the first approximant of such a continued fraction, taken as the first truncation, turns out to be the same as the first term in the series in Eq.(87). This first approximation, n(E) = (eβ(E−µ) − a0 )−1 was successfully used to describe the complete thermostatistics of anyons in our previous work [24]. The form of continued fractions has some advantages. The first convergent reproduces the form of the distribution function which was the ansatz in the earlier work [24] and is also the preferred form based on kinetics [28]. Secondly, the question of what is the best approximation is answered in an explicit manner by the sequence of inequalities involving the various odd and even convergents of the continued fraction [35]. The exact expression for the interpolating distribution function for α = 12 is given by 1 a0 g
n(E)|α= 1 = 2
g+ g 2 − a0 +
,
(90)
(a0 − π 3 /48)g a0 g − (a0 − π 3/48) + · · ·
where a0 ( 12 ) = 12 π − 1. This shows that we can determine the exact form of the distribution function for any value of the parameter in the range 0 < α < 1, although in the form of an infinite continued fraction which is not easily amenable to applications involving analytical calculations. The subject of interpolation between Bose and Fermi statistics is still “work in progress”. The method of obtaining the distribution function and the thermodynamical variables has not been established to everyone’s satisfaction. In this work we shall show that the distribution introduced in our earlier investigation [24], is a first approximant of the complete theory. This earlier investigation contains many results derived from this approximate distribution function. A comparative investigation of various interpolations has recently been made [27] which points out the special features of the interpolation in [24] and its connection with Jack polynomials. The present work deals with the theory of interpolating statistics based on the principle of Detailed Balance where the basic numbers arise automatically and hence the theory is described naturally in terms of the basic numbers. We now present a summary of some of the results which follow from our analysis of the exact theory of interpolating statistics, thus offering a comparison with BE and FD statistics. For large E, as eβ(E−µ) 1, we have n → e−β(E−µ) which is the Classical limit, thus showing that the intermediate statistics behaves just as BE and FD for large E. When E = µ, n = 1/(1 − a0 ) + a1 /(1 − a0 )3 + · · · : in the case of BE limit, when a0 → 1, we have n → ∞; which is the expected result. In the limit T → 0: we have n → 0 if E > µ and n → −1/a0 − a1 /a30 · · · if E < µ and thus in the low temperature limit, the distribution goes to Haldane statistics only in the first order approximation. If n1 , n2 , · · · represent the various convergents, then nF D < n1 < nBE . We can then prove n > nF D . More generally, the theorem for continued fractions, n1 < n3 < · · · < n · · · < n4 < n2 , shows the sequence of approximations and the “best approximation”, thus establishing the advantage of expressing our results in terms of continued fractions.
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One might imagine that the exact distribution function lies between that of BE and FD but this does not follow from the above sequence of inequalities.
7.
Intermediate Statistics based on Deformed Algebra
The intermediate statistics that continuously interpolates between Bose-Einstein statistics (BE) and Fermi-Dirac statistics (FD) is applicable to some particles often referred to as anyons [36]. The ensemble of such particles has been of great interest in the study of many topics such as Chern-Simons [20] gauge theory of fields, Aharonov-Bohm effect [25], the second virial coefficient, the braid group [22], fractional statistics and anyon superconductivity [21]. The study by Arovas in Ref.[23] of the exact two-particle partition function leads to a computation of the second virial coefficient valid for low densities and/or high temperatures. The work in Ref.[24] deals with the consequences of an appropriately approximate distribution function for the intermediate statistics and examines various properties of the thermostatistics of particles obeying interpolating statistics, such as the partition function, entropy, pressure, internal energy, the various virial coefficients and the specific heat of such particles, providing a compareison with the properties of standard bosons or fermions. More recently, the same subject has been examined as a consequence of detailed balance [24]. Employing standard methods of statistical physics, based on the principle of Detailed Balance, the theory of exchange symmetry has been shown in this work to lead to a continuous interpolation between Bose and Fermi statistics. The basic numbers arise naturally in this theory as a consequence of exchange or permutation symmetry. Indeed such a formulation, not restricted to 2+1 dimensions, leads to a mean occupation number expressed as an infinite continued fraction so that the meaning of successive approximations is well clarified. There is another generalization of the standard thermostatistics that has been investigated extensively in the literature which has to do with the theory of q-deformed quantum oscillators or quantum groups [3, 2]. It is evident that a complete formulation of generalized thermostatistics of q-bosons and q-fermions can be established using basic numbers with the base q and employing the q-calculus based on the Jackson Derivative. The thermodynamic functions such as entropy, pressure, internal energy, specific heat etc. of such deformed systems have been investigated and compared with standard bosons and fermions [37, 7, 8, 38, 39, 40, 41, 42, 43, 44]. The method of detailed balance may be employed for the purpose of establishing an intermediate statistics requiring the use of basic numbers or bracket numbers. Such basic numbers arise naturally in the q-deformed algebra of harmonic oscillators so that the limit of q → 1 corresponds to the conventional boson and fermion oscillators. However, the task of establishing the equivalence of a deformed algebra and the description of intermediate statistics has been an outstanding problem. A basic problem, pointed out in the literature [26, 15] is that the oscillator algebra is valid in any number of dimensions while anyons arise only in two space dimensions. However, this argument may be unreliable and it is quite possible that intermediate statistics may indeed be different from that of anyons. The legitimate question accordingly arises naturally: what kind of deformation can successfully describe interpolating statistics? We shall now investigate the basic formulation which is the equivalence of intermediate statistics and the deformed algebra of oscillators. Our formulation may apply to the
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intermediate statistics applicable to particles which might differ from the anyons in the literature. This demonstration has so far been an outstanding problem in the literature. As for the consequences in thermostatistics of such a deformation, we shall refer to earlier work [24] which is based on the first approximation of the occupation number, while the occupation number dealt with in the present work is exact. We shall now demonstrate that deformed κ-permutation implies the existence of the basic number, an important result. We shall begin with a brief summary of the procedure for dealing with many particles and the quantum probabilities, according to the method developed by Feynman [29]. Feynman’s method emphasizes the special rules for the interference that occurs in processes with identical particles and considers in detail the direct and exchange amplitudes. Such attention is necessary for the determination of the enhancement factor of (n + 1) particles for bosons, which measures the probability of an additional boson from a state of n bosons. The permutation or exchange of the coordinates of the n particle wave function results in multiplication by the statistics determining parameter κ, which may be complex in general, with the property |κ|2 = 1. We shall begin with the two particle state wave function, defined in terms of single particle wave functions 1 ψ (2) = √ (ψa(r1)ψb (r2) + κ ψa(r2)ψb(r1)) , N2
(91)
and the corresponding probability density associated with the two particle state can be stated as in the previous section. In the case of identical particles, the process of a going into r1 and b going into r2 cannot be distinguished from the exchange process in which a goes into r2 and b goes into r1. Thus, we set r1 = r2 in the following and the probability density for identical particles reduces to 1 (1 + κ−1 )(1 + κ) Π2 , (92) P (2) =⇒ N2 where the normalization from single particle states is Π2 = |ψa|2|ψb |2. We may express the probability density associated with the two particle state as in the previous section, by employing the basic number or bracket number [x] defined in the symmetric formulation. Concomitantly, we can introduce the operator form [N ] =
κN − κ−N , κ − κ−1
(93)
where [N ] is not the number operator but N is i.e., N |ni = n|ni. It is well-known that the basic number [n] defined above can be expressed as the sum of a geometric series and thus we may also frequently employ the series form [n] = κ−n+1 + κ−n+3 + · · · + κn−3 + κn−1 .
(94)
The parameter κ, generally a complex number, may be expressed as κ = eiα in terms of the real number α. The statistics determining parameter, κ = ±1, takes the values corresponding to α = 0, π which define the Bose-Einstein (BE) and Fermi-Dirac (FD) cases respectively. The parameter has the properties κ∗ = κ−1 , |κ| = 1. The wave function, Eq.(91) contains both the direct and the exchange terms, and it is the exchange
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term which contains the factor κ. In the limit when κ → 1 it reduces to the standard Bose case and reduces to the Fermi case when κ → −1. Next the three particle state can be dealt with as in the previous section. In this case we observe that the normalization is given by N3 = 1 + 2|κ|2 + 2|κ|4 + |κ|6 = 1 + 2 + 2 + 1 = 6 = 3!. Specializing to the case of identical particles, setting r1 = r2 = r3, the probability density reduces to P (3) =
1 (1 + 2κ−1 + 2κ−2 + κ−3 )(1 + 2κ + 2κ2 + κ3 ) Π3 , 6
(95)
where, analogously as before, we have defined Π3 = |a|2|b|2|c|2. Again, we write in terms of the basic numbers by using, [2] = κ+κ−1 , [3] = κ2 +κ−2 +1, [4] = κ3 +κ+κ−1 +κ−3 etc. with [n] given by N → n in Eq.(93). To establish the equivalence of the above two forms, we first need to obtain some formulae. Consider the sum [1] + [3] + [5] + · · · + [2n − 1] .
(96)
Since κ = eiα , we may express the basic number in terms of the trigonometric functions [n] =
κn − κ−n sin nα = . −1 κ−κ sin α
(97)
The above series may accordingly be rewritten as [1] + [3] + [5] + · · · + [2n − 1] =
sin α + sin 3α + · · · + sin(2n − 1)α . sin α
(98)
We can now sum the series of the sine functions by employing a trigonometric identity [31]: n X
sin(2k − 1)α =
k=1
sin2 nα . sin α
(99)
Thus we obtain [1] + [3] + [5] + · · · + [2n − 1] =
sin2 nα = [n][n] , sin2 α
(100)
and hence we derive the important formula: [n] [n] = [1] + [3] + [5] + · · · + [2n − 1] .
(101)
Employing this formula, we see that the form of the probability density P (3) in Eq.(95) is clearly the same as the one discussed in the previous section. In a similar manner, we can also establish the following result: [1][2] = [2], [2][3] = [2] + [4], [3][4] = [2] + [4] + [6], · · · ,
(102)
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and more generally for any n: [n − 1][n] = [2] + [4] + [6] + · · · + [2(n − 1)] .
(103)
Next we can perform the same exercise for the four-particle state. After much algebra, we determine the probability density associated with the four particle states ( a, b, c, d) to be 1 (104) (2 + [2])(2 + 2[2] + [3])(2 + 2[2] + 2[3] + [4]) Π4 , P (4) = 4! where Π4 = |a|2|b|2|c|2|d|2. This enables us to generalize to the case of n-particle states (a, b, c, · · · ). Thus we determine P (N ) =
1 N!
( 2 + [2])(2 + 2[2] + [3])(2 + 2[2] + 2[3] + [4]) · · · ( 2 + 2[2] + 2[3] + · · · + 2[N − 1] + [N ]) ΠN ,
(105)
where ΠN = |a|2|b|2|c|2 · · · and we have thus obtained the probability density for the N -particle state in the desired closed form. Finally, let us consider the BE and FD limits. If we take the limit κ → +1, we find P (2) → 2! Π2; P (3) → 3! Π3; P (4) → 4! Π4; · · · ; P (N ) → N ! ΠN , thus obtaining the true BE limit. Furthermore we also confirm the fact that in the BE limit, the enhancement factor determined by P (N +1)/P (N ) turns out to be proportional to n + 1, as it should exactly be. Now let us consider the Fermi limit when κ → −1. We know that [2] → −2 in this limit. Hence 2 + [2] vanishes. Since this is the common factor for the general probability density P (n) , we conclude that all the probabilities vanish for n = 2, 3, · · ·N in the Fermi limit, which is in accordance with the Pauli exclusion principle for standard fermions. In this manner, we are in a sense confirming the correctness of all the analysis above. It is important to stress that the permutation or exchange of particles implies the existence of the basic number. We have accordingly met the objective, namely, to show explicitly that κ−permutation implies the existence of the basic numbers, or the bracket numbers. The case of such an exchange in two dimensional space corresponds to the symmetry of the braid group [22] but we shall not concern ourselves with this feature at this time. Having established the fact that κ- permutation implies the existence of the basic number, we shall now proceed to establish the connection between the basic number, in the symmetric formulation, and the deformed algebra. Let us begin from the definition of the basic number in the case of particles obeying intermediate statistics, given in Eq.(6), and since the Fock state |ni is an eigenstate of the number operator N , we also have [N ]|ni = [n]|ni, [n] =
κn − κ−n , κ − κ−1
(106)
where n = 0, 1, 2, · · · . It is readily seen from the definition that the basic numbers satisfy the relation (107) [n + 1] = κ[n] + κ−n .
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We may now identify the operator [N ] in terms of the creation and annihilation operators as: (108) [N ] = a†a , where it is expected that the creation operator raises the number of quanta and the annihilation operator lowers the number of the quanta in the Fock state representation, so that we may express the raising and lowering property in the form a†|ni = Cn0 |n + 1i, a|ni = Cn |n + 1i ,
(109)
where Cn , Cn0 are constants to be determined. We observe that Eq.(29) is our second assumption but this may not necessarily be required since [N ] must go to N in the limit when κ → 1. The constants Cn , Cn0 are determined in the customary manner as p p (110) Cn = [n], Cn0 = [n + 1] . and we also obtain the relations [N ]a†|ni = [N ]Cn0 [n + 1]|n + 1i = [n + 1]Cn0 |n + 1i,
(111)
a† [N ]|ni = a†[n]|ni = [n]a†|ni = [n]Cn0 |n + 1i ,
(112)
and in the derivation of which, we observe that eigenvalues commute, while operators in general need not and that the above relation is true for any state |ni. From this we derive the relations, valid for any |ni, ([N ]a† − κa† [N ])|ni = κ−N a† |ni .
(113)
This can be expressed as an operator identity: ([N ]a† − κa† [N ]) = a† κ−N .
(114)
If we employ the definition in Eq.(110), the above can be written in the form a†aa† − κa† a† a = a† κ−N ,
(115)
From this, the deformed algebra follows immediately: aa† − κa† a = κ−N ,
(116)
which is an operator identity. Hence we have established the connection between the basic number and the deformed algebra i.e., the basic number implies that the creation and annihilation operators obey the deformed algebra. At this point, we may make a general remark that what we have presented is the algebra describing deformed oscillators. Thus we infer that the three cases +1, −1, and κ corresponding to BE, FD, and intermediate statistics respectively, are in one-to-one correspondence with the oscillator algebras as follows: aa† − a† a = 1; bb† + b† b = 1; aa† − κ a† a = κ−N ,
(117)
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where the variable parameter κ may generally be complex, with end points (limits) -1 and +1. It is evident that the above third equation is formally equivalent to the symmetric qdeformed oscillator algebra [8, 15]. This might be asserted as our basic premise. We are merely attempting to point out the above correspondence. We just note that this is analogous to the Bargmann-Wigner holomorphic representation [45]. Planar physical systems, in two space and one time dimensions, display many peculiar and interesting quantum properties owing to the unusual structure of rotation, Lorentz and Poincare groups and thus lead to a theory of anyons. That is, in the case of two dimensions, as pointed out extensively by Ref.[22], the exchange or permutation of two objects is described by the direct consequences of the braid group. We do not explicitly deal with the case of two space dimensions. We consider only the deformed algebra without resorting to any specific dimensions and demonstrate that the deformation of the algebra, derived form the properties of the basic numbers and strictly connected to κ-permutations, can lead to intermediate statistics with a deeper physical insight with respect to the standard formulation of the q-deformed oscillators. Now we proceed to study the thermostatistics of the particles obeying the deformed algebra i.e., particles obeying intermediate statistics.
8.
Intermediate statistics: mean occupation number
Let us start from the Hamiltonian H=
X
Ni (Ei − µ) ,
(118)
i
which, in spite of the appearance, does include deformation, as will become evident from the form of the average occupation number. We begin by introducing the mean value, after omitting the subscript for convenience, 1 T r(e−βH q N ) . Z
(119)
P 1 T r(e−β (Ei −µ)(Ni +1) aa† ) . Z
(120)
qn = Next, we consider the expectation value [n] =
Taking account of the cyclic property of the trace, and the relation af (N ) = f (N + 1)a, which can be established in a straightforward manner for any polynomial function f (N ) and proceeding in the standard manner, we obtain [n] = e−η , [n + 1]
(121)
where η = β(E − µ). This is an important step. Proceeding further, we obtain, after some algebra, κ−n [n] = η . (122) e −κ
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From the above equation, we obtain for the occupation number n: (κ2)n =
eη − κ−1 , eη − κ
which leads to the result 1 ln n= 2 ln |κ|
eη − κ−1 eη − κ
(123)
,
(124)
thus expressing the form of the mean occupation number for the particles obeying interpolating statistics. It is now evident that the chosen Hamiltonian does indeed include the deformation. In this context, it is relevant to observe that the statistical origin of such qdeformation lies in the modification, relative to the standard case, of the number of states W of the system corresponding to the set of occupational number ni [7]. In the subject literature, other statistical generalizations are present, such as the so-called nonextensive thermostatistics or superstatistics with a completely different origin [46, 47, 48, 49, 50, 51]. Now we need to further express the occupation number in a useful form. From Eq.(49), we arrive at the result in the form of a power series 1 1 κ 1 κ 2 n= + + 3 + + y 3 y3 2 y2 3 y3 2 y2 1 κ 2 κ 4 + · · · , + + + (125) 5 y5 2 y4 3 y3 2 y2 where y = eη − κ and we have taken κ = 1 − , 1. At this point, if we use the approximation by retaining only the leading term we then arrive at the form n≈
eη
1 . −κ
(126)
This is the form familiar from the work of Ref.[24] which contains many interesting applications. More generally, for not very small, we have the series form in Eq.(50) describing the various powers of the deformation parameter . It might now appear as though we may have obtained the above result for both the boson B-type and the fermion F-type intermediate statistics, accordingly, we need to examine the B-type and F-type cases separately. Let us first consider the B-type case of intermediate statistics. The algebra is as before, with the occupation number given in Eq.(49), but the range may now be taken as 0 ≤ κ ≤ 1 with the understanding that κ = 1 corresponds to the BE case. The series form for n is given by Eq.(50) and the approximate form is given by Eq.(51). Ref. [24] provides many applications to the B-type intermediate statistics. Next, we may consider the F-type particles. Now the range for the F-type particles is −1 ≤ κ ≤ 0. Setting κ = −λ, we may modify Eq.(48) as (λ2)n =
eη + λ−1 . eη + λ
(127)
Expressing the range in the form 0 ≤ λ ≤ 1, the algebra above leads to the relation n=
1 2 ln(1 − η ), 2 ln λ e +λ
(128)
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where we have set κ = −λ = 1 − , κ−1 = −λ−1 = 1 + , which is appropriate for the case of the F-type intermediate statistics. Indeed many relations can be modified by the replacement κ → −λ to go from the B-type to the F-type. For the F-type intermediate statistics, the result as in Eq.(128) is still valid and consequently, the mean occupation number is given by the power series form exactly as in Eq.(50) except that now we have y = eη + λ for the F-type anyons. The approximate form for the occupation number is thus 1 . (129) n≈ η e +λ The detailed thermodynamic properties stemming from this form, such as the equation of state, virial expansion etc. for the F-type intermediate statistics are as described in Ref.[24]. It is also observed that the equality of the specific heats of B-type and F-type intermediate statistics particles [52] also prevails as shown in Ref. [24], if we utilize the approximate forms for the occupation numbers. This is a very interesting result and may be true more generally for the exact forms of the occupation numbers formulated in the present work. We shall now demonstrate that the theory can be formulated in terms of an infinite continued fraction. Let us begin with the series expression which may be expressed conveniently in the form α1 α2 α3 n= + 2 + 3 +··· , (130) y y y where we have set y = eη − κ and α1 , α2 etc. and are determined from the previous sections, specifically Eq.(125), such as 19 5 1 3 5 3 5 3 + − 2 + + O(4 ) , α2 = + − 2 + + O(4(131) ), 12 24 24 24 12 4 4 12 etc. by combining terms containing various powers of in Eq.(125). The method of determining the CF form of a function given by an infinite series is well-known in the literature [53, 54] and there is a standard method by which this infinite series can be put in the form of a CF. We shall briefly summarize the procedure here. The general continued fraction of order r is of the form a1 , (132) Cr = b0 + a2 b1 + a3 b2 + b3 + · · · where the constants b0, b1, · · · , a0, a1, · · · can be determined by a straightforward procedure. The various convergents are C0 , C1 , · · · corresponding to r = 0, 1, 2, · · · , ∞. Accordingly we have α1 =
C0 = b0 =: A0 /B0 ; C1 = b0 + a1 /b1 =
b0b1 + a1 A1 =: ; b1 B1
C2 = b0 + a1 /(b1 + a2 /b2) = b0 + a1 b2/(b1b2 + a2 ) =:
A2 , B2
(133)
etc. The parameters An , Bn satisfy the two-term recurrence relations [54]: An = bn An−1 + an An−2 ; A−1 = 1 ; Bn = bn Bn−1 + an Bn−2 ; B−1 = 0 .
(134)
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By solving the recurrence relations, the general CF can be determined. We may consider two examples of this procedure. The standard sine series may be expressed in the form of a CF as: x . (135) sin x = x2 1+ x2 + 2 · 3x2 2·3− 4 · 5 − x2 + · · · Furthermore, we can also deal with the inverse problem, i.e., given the standard series form of the cosine function, cos x = 1 − x2 /2! + x4 /4! + · · · , (136) we can solve the corresponding recurrence relations and obtain the CF form for the cosine function as: 1 . (137) cos x = x2 1+ x2 + 2x2 2·1− 4 · 3 − x2 + · · · Employing this procedure for our present problem, after some algebra, the final result can be expressed in terms of the various convergents (approximants): n1 =
α1 , y
n2 = − n3
(138) α1
, 2 y y − α α+ α2 1 α1 = − . α2 y y− 2(α1 + α2 )
(139) (140)
In the literature on CF, the convergents, which may be obtained in a straightforward manner after some algebra, play an important role. The general form of the CF is given by the form Cr as in Eq.(132) and the procedure can be extended to many convergents. Now the question which might arise is: what is the advantage of CF? Other than the elegant mathematical form, there is a distinct advantage. The Pade approximant is a well-known application. There is a theorem [53], involving the convergents n1 , n2, n3 · · · which may be stated as: (141) n1 < n3 < · · · < n and n2 > n4 . · · · > n . This immediately provides a clarifying definition of successive approximations i.e., the above inequality tells us how to obtain successive approximations of the quantity n. Indeed, it is evident that the exact form of n lies between n1 and n2 , hence its importance. We can thus establish that the exact n is bigger than the first convergent n1 = α1 /y but smaller than n2 obtained above. We have presented our formulation which leads to the conclusion that the intermediate statistics arises from a deformed algebra, a result we obtain in two steps: first, that the exchange symmetry characterizing the intermediate statistics leads to the existence of basic numbers; secondly the basic numbers naturally lead to the deformed algebra. The
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deformed algebra is signified by the parameter κ, the deformation parameter, the statistics determining parameter such that we recover the standard BE and FD statistics in the limits κ → +1, −1 and the intermediate statistics is obtained for arbitrary values of κ in the range −1 ≤ κ ≤ +1. In order to describe intermediate statistics, continuously interpolating anywhere between BE and FD statistics, we have introduced the parameter κ as a variable parameter, the statistics determining parameter, so that κ = +1, −1 represent the extreme values which correspond to the standard BE and FD statistics (boson and fermion oscillator algebra) in the limits.
References [1] Ashok Das, Field Theory: a path integral approach , World Scientific Pub., Singapore (1953). [2] A. J. Macfarlane, J. Phys. A 22, 4581 (1989). [3] L. Biedenharn, J. Phys. A 22, L873 (1989). [4] T. Hayashi, Commun. Math. Phys. 127, 129 (1990); M. Chaichian and P. Kulish, Phys. Lett. B 234, 72 (1990); M. Chaichian, D. Ellinas and P. Kulish, Phys. Rev. Lett. 65, 980 (1990). [5] R. Parthasarathy and K. Viswanathan, J. Phys. A 24, 613 (1991); K. Viswanathan, R. Parthasarathy and R. Jagannathan, J. Phys. A 25, L335 (1992). [6] S. Jing and J.Xu, J. Phys. A 24, L 891 (1991); M.Ge and G. Su, J. Phys. A 24, L721 (1991). [7] A. Lavagno and P. Narayana Swamy, Phys. Rev. E 65, 036101 (2002). [8] A. Lavagno and P. Narayana Swamy, Phys. Rev. E 61, 1218 (2000). [9] F. Jackson, Mess. Math. 38, 57 (1909). [10] E. Floratos, J. Math. Phys. 24, 4739 (1991). [11] R.Finkelstein, Int. J. Mod. Phys. A 13, 1795 (1998). [12] J. Tuszynski et al, Phys. Lett. A 175, 173 (1993). [13] L. Reichl, A Modern course in Statistical Physics second edition, University of Texas (Austin), 1999. [14] O. Greenberg, Phys. Rev. Lett. 64, 705 (1990); Phys. Rev. D 43, 4111 (1991). [15] M. Chaichianet al, J. Phys. A 26, 4017 (1993). [16] P. Narayana Swamy, Int. J. Mod. Phys. B 20, 2537 (2006). [17] P. Narayana Swamy, Physica A 328, 145 (2003).
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[18] M. Schork, Russian J. Math. Phys. 2, 394 (2005). [19] P. Narayana Swamy, Eur. Phys. Journal. B 50, 291 (2006). [20] S.S. Chern et al(ed.), Physics and Mathematics of Anyons , World Scientific Pub. (1991) Singapore. [21] F. Wilczek (Ed.), Fractional Statistics and Anyon Superconductivity , World Scientific (1992), Singapore. [22] A. Lerda, Anyons, Springer Verlag Pub. (1992), Berlin. [23] D. Arovas et al, Nucl. Phys. B 251 (1985) 117. [24] R. Acharya and P. Narayana Swamy, J. Phys. A A 27 (1994) 7247. [25] Y. Aharanov, D. Bohm, Phys. Rev. 115 (1959) 485. [26] L. Frappat et al, Phys. Lett. B 369 (1996) 313. [27] S. Chaturvedi and V. Srinivasan, Physica A 246 (1997) 576. [28] G. Kaniadakis, A. lavagno and P. Quarati, Nucl. Phys. B 466 (1996) 527; ibid Mod. Phys. Lett. B 10 (1996) 497. [29] R.P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Addison-Wesley Pub. Co.(1965), Reading MA. [30] H. Exton, q-Hypergeometric functions and applications , Ellis Horwood Ltd.(1983) Chichester. [31] I.S.Gradshteyn and I.M.Ryzhik, Table of Integrals, Series and products , Academic press (1980) New York. [32] L.E. Reichl, A Modern Course in Statistical Physics , second edition, John Wiley & sons Inc.(1998), New York. [33] S-K Ma, Statistical Mechanics, World Scientific Publishing (1985) Singapore. [34] P.T. Landsberg, editor, Problems in thermodynamics and statistical physics Pion Limited (1971), London. [35] H.S. Wall, Analytical theory of continued fractions , D. Van Nostrand Co. (1948) New York; G.E. Andrews et al, Special Functions, Cambridge University Press (2001) New York. [36] J. Leinnas, J. Myrheim J, Nuovo Cimento B 37 (1977) 1. [37] P. Narayana Swamy, Int. J. Mod. Phys. B 10 (1996) 683; A. Lavagno, P. Narayana Swamy, Int. J. Mod. Phys. B 23 (2009) 235. [38] A. Lavagno, A.M. Scarfone, P. Narayana Swamy, Eur. Phys. J. C 47 (2006) 253; J. Phys. A: Math. Theor. 40 (2007) 8635.
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[39] A. Lavagno, J. Phys. A: Math. Theor. 41 (2008) 244014. [40] A. Lavagno, Phys. Lett. A 301 (2002) 13; A. Lavagno, Physica A 305 (2002) 238. [41] A. Lavagno, P. Quarati, Phys. Lett. B 489 (2001) 47; A. Lavagno, P. Quarati, Nucl. Phys. B [PS] 87 (2000) 209; G. Kaniadakis, A. Lavagno, P. Quarati, Phys. Lett. A 227 (1997) 227. [42] W.M. Alberico et al., Physica A 387 (2008) 467; M. Di Toro et al., Nucl. Phys. A 775 (2006) 102; A. Drago, A. Lavagno, P. Parenti, Ap. J. 659 (2007) 1519; L. Bonanno, A. Drago, A. Lavagno, Phys. Rev. Lett. 99 (2007) 242301. [43] M.A. Martin-Delgado. J. Phys. A 24 (1991) L1285; M.A. Martin-Delgado, J. Phys. A Math. (1991) L807. [44] V.K. Dobrev, H.-D. Doebner and C. Mrugalla, J. Phys. A 29 (1996) 5909. [45] V. Bargmann, E.P. Wigner, Proc. Nat. Acad. Sci. 34 (1948) 211. [46] C. Tsallis, J. Stat. Phys. 52 (1988) 479. See also http://tsallis.cat.cbpf.br/biblio.htm for a regularly updated bibliography on the subject. [47] M. Gell-Mann, C. Tsallis, eds., Nonextensive Entropy: Interdisciplinary Applications, Oxford University Press, New York, 2004. [48] C. Tsallis, Introduction to Nonextensive Statistical Mechanics, Springer-Verlag, New York, 2009. [49] U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75 (2007) 040106; C. Tsallis et al., Physica A 381 (2007) 143; A. Pluchino, A. Rapisarda, C. Tsallis, Physica A 387 (2008) 3121; F. Caruso, C. Tsallis, Phys. Rev. E 78 (2008) 021102; U. Tirnakli, C. Tsallis, C. Beck, Phys. Rev. E 79 (2009) 056209; U. Tirnakli, D.F. Torres, Physica A 268 (1999) 225. [50] S. Abe, Phys. Lett. A 224 (1997) 326; S. Abe, C. Beck, E.G.D. Cohen, Phys. Rev. E 76 (2007) 031102; S. Abe, Physica A 368 (2006) 430; S. Abe, Physica A 344 (2004) 359; S. Abe, Phys. Rev. E 79 (2009) 041116. [51] C. Beck, E.G.D. Cohen, Physica A 322 (2003) 267; C. Beck, Physica A 342 (2004) 459; C. Beck, Phys. Rev. Lett. 98 (2007) 064502; E. Van der Straeten, C. Beck, Phys. Rev. E 78 (2008) 051101. [52] R. May, Phys. Rev. 135 (1964) A1515. [53] H.S. Wall, Analytic theory of continued fractions, Van Nostrand Company, Princeton, 1948. [54] G. Andrews et al, Special Functions, Cambridge University Press, Cambridge, 1999.
In: Horizons in World Physics. Volume 271 Editor: Albert Reimer
ISBN: 978-1-61761-884-0 c 2011 Nova Science Publishers, Inc.
Chapter 9
S PACETIME F ERMION M ANIFOLDS Bernd Schmeikal∗ University of Vienna, Austria
Abstract We derive the natural embedding of fermion manifolds {|ui}, {|di}, {|si}, {|ci}, {|bi}, {|ti} into the Minkowski algebra. Using six isomorphic Cartan subalgebras and a generalization of Cartan’s concept of isotropic vector fields, we obtain the natural spinor manifolds of the spacetime-SU(3) calculated from the Clifford algebra C ⊗ C`3,1 by the aid of minimal left ideals. Together with the previously constructed spacetime group this concept removes the necessity for auxiliary bundles that YangMills theories presently require.
PACS 3.65-Fd, 02.20.-a, 02.20.Sv, 11.30.-j, 11.30.Pb, 12.38.-t, 12.39.Ki, 12.39.St, 14.20.-c, 14.60.-z, 14.65.-q Keywords: Clifford algebra, geometric flavor, inhomogeneous Lorentz transformation, manifold, minimal ideal, particle theory, pregeometry, primitive idempotent, spacetime fermion, spacetime spinor, natural spinor, spinor, spinor manifold, unfolding AMS Subject Classification: 17C27, 17C36, 15A66, 33D80, 35Q40, 37N20, 46N50, 58A15, 58B32, 81R25, 81V05.
1.
The Natural Approach
The systemic integration of the fundamental forces of nature appears to most of us as bound to the tradition of gauge theories. Originally it has been legitimated by the observation that there are local degrees of freedom to choose freely the scales of intensities in field theories. We denote them as gauge theories or more specifically as local spin gauge theories. The names ”gauge invariance” and ”gauge symmetry” go back to Herman Weyl. Today, gauge ∗
E-mail address:
[email protected]
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theories seem to disclose something rather artificial. This has three reasons. First, the symmetries appear as pressed on to some linear equations of motion such as the Dirac equation. Second, the nonlinear phenomena of quantum interaction suggest ”scale invariance” rather than local gauge invariance. Third, as a matter of fact, spinors are universal concepts. They are not at all restricted to the microscopic, subnuclear level. Therefore, we have asked if there are quite natural, simple reasons for the appearance of the standard model symmetry group SU (3) × SU (2) × U (1) in both space-time and high energy physics. We can show that the high dimensions used in string- and M-theory can be identified with the dimensions of the Clifford algebra of the Minkowski space-time and time-space respectively. At most we need the complex matrix algebra Mat(4, C). This finally leads us to a classification of natural strong force manifolds as represented in this chapter. The result is self-evident and clear. We need only the multivectorial space-time in order to obtain topological types of matter (Schmeikal 2010). Those may remind us of the topological invariants developed by Seiberg (1994) and Seiberg and Witten (1994 a, b).
2.
Reflections and the Spacetime Spinor
The few who followed my rigor during the last ten years or so have comprehended how one can construct the isometric standard model by the aid of spacetime reflections in the Minkowski algebra. Sure it is not easy to fully understand the meaning of a graded Lie algebra within Clifford algebra C ⊗ C`3,1 which otherwise is known as ungraded, namely su(3, C) within sl(4, C). The latter are mere matrix algebras with complex entries. Per se such a matrix algebra does not involve any grading. Phenomenologically this is not trivial, because the graded Lie group is responsible for a dimensional widening of fermion trajectories. This I have decided to call an unfolding. The dimensional broadening involves transitions from dimension 1 (spatial) to 2 (space ×time) and further to 3 (spatial area×time). Mixes of such measures within fixed chromatic spaces and different flavors are possible. So we begin with a quite new concept of space dynam ics without touching the macroscopic appearance of the Minkowski space. At the same time we obtain the structure of the inner spaces of matter from the natural symmetries of the Minkowski algebra. Note, the Minkowski algebra is nothing else than the Clifford algebra C`3,1 of the Minkowski spacetime R3,1. Introducing his paper ”Isometry from Reflections Versus Isom´ J. Cartan proved etry from Bivectors” Zbigniew Oziewicz (2009) reminded us that Elie in 1937 that every isometry can be composed by reflections. This statement is known as the Cartan-Dieudonn´e theorem. Algebraic reflections in the sense of Coxeter are period 2 elements s2 = Id of the Clifford algebra. They can be expressed in terms of primitive idempotens f by the 1-form s = Id − 2f where Id denotes the identity in the Clifford algebra. David Hestenes showed how two reflections give rise to a rotor R, or so called unimodular spinor , which is an element of the even part of the Spin group inside the Clifford algebra and determines a Lorentz transformation of the frame. Bivectors can be used to generate the isometries of a special Lie algebra in a basis-free and coordinate-free way. However, reflections as elements of Clifford algebra lead to a more general approach than bivectors.
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Restricting the Lie group of orthogonal rotations to the even part of the Clifford algebra actually implies a reduction of the mathematical complexity of the problem. Indeed, the whole of mathematical physics gets lost. Namely, we must not restrict our calculation to rotations of vectors. Clearly, as we rotate two orthogonal vectors, their orthogonality is preserved. But when we do quantum physics, we are interested in the orthogonality of primitive idempotents, erzeugende Einheiten as Herrmann Weyl (1931) used to call them. We got to find the most general Lie group that rotates primitive idempotents while preserving their orthogonality. A grade-rotation, or unfolding of a primitive idempotent, - I suggest as a theorem, - preserves the trace of the primitive idempotent equal to unity, provided there is a trace. In the Majorana algebra most evidently there is such a trace. Zbigniew has worked out very clearly the difference between Chevalley’s spinors and Hestene s spinors. We should add to them Cartan’s pure spinors. Lounesto (2003, p. 144) too investigated the unimodular spinors. He pointed out that in this approach the state function is a mapping ψ : R1,3 → C`+ 1,3 . Therefore, in the so called Dirac-Hestenes equation the role of the Dirac columns is taken over by real even multivectors, which are not in any proper left ideal of the Clifford algebra C`1,3. This deficiency is exactly representing that restriction of quantum physics which I mentioned above. We have to construct the primitive idempotents of the standard representation and identify them with densities of fermion states. Next we construct the spinors which give us the above densities. Aditionally we calculate the manifolds for each standard density. By a standard density we should mean the most simple algebraic term for a primitive idempotent in the standard representation of the Clifford algebra. We shall also construct from these primitive idempotents the reflections and from those the flavor- and color rotations and last, but not least their manifolds which are basis free and coordinate free. Let me first give you the 24 standard densities of the real Clifford algebra C`3,1 . We begin with a standard calibration of the quark densities ρu = f13 = |uihu|, ρd = f14 = |dihd|, ρs = f12 = |sihs|: 1 1 (1 − e1 ) (1 + e24 ) , 2 2 1 1 ρs = (1 + e1 ) (1 − e24 ) , 2 2
ρu =
ρd =
1 1 (1 − e1) (1 − e24) , 2 2
represent u, d, and s-quarks.
(1)
These can be collected into one equivalence class of densities to low energy fermions having baryon number 13 connected by a trigonal flavor rotation (Schmeikal 2010, equation 37) from a subalgebra suCl (2). Note, the primitive idempotent f11 represents an electron neutrino. Generally we obtain χµ = 24 standard densities because of all the index- and sign combinations possible, those are fχµ = 14 (1 ± ei )(1 ± ek4 ) with e2i = 1 = e2k4 and i 6= k. The indices run as χ = 1, ..., 6; µ = 1, ..., 4 and thus χ × µ = 24 denote twentyfour equivalence classes of fermion manifolds. The χ = 6 can be collected to color spaces. Total space inversions bring forth the anti state densities. In various papers it has been shown how the standard color- and flavor rotations are constructed from standard reflections and respectively their primitive idempotents. Select the following primitive idempotents from Cartan subalgebras 2 and 6 f23 =
1 (1 − e1 )(1 + e34 ) 4
1 f62 = (1 + e3 )(1 − e24) 4
(2)
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and form ’reflections’ s23 = 1 − 2f23 and s62 = 1 − 2f62 . So you obtain the Clifford number 1 1 (3) T = s23 s62 = (Id + e1 − e34 + e134) (Id − e3 + e24 − e234 ) 2 2 which represents a flavor rotation within the first chromatic space. Tringonal T acts on idempotents in terms of conjugations (4) T −1 f14 T −→ f13 ,
T −1 f13 T −→ f12
T −1 f12 T −→ f14 In the natural spinor approach we essentially need two of Cartan’s important concepts, namely that of the isomorphic Cartan subalgebras of the Clifford algebra, and that of a space of isotropic vectors. The first refers to the existence of isomorphic chromatic spaces, the second to Cartan’s ”The Theory of Spinors” (Cartan 1937). To introduce the spacetimespinor we need a procedure like this: Consider the lepton state f1 ∈ C`3,1 and the corresponding minimal ideal S1 = f1 C`3,1 . Therein we find the space of isotropic spinors which satisfy ξ ◦ ξ = 0. For simplicity we omit the chromatic index thus replacing the f1µ by fµ with µ = 1, ..., 4. We define (denoting by ◦ the Clifford product) Definition 1. isotropic spinors Let S1 = f1 C`3,1
and
H1 = {ξ / ξ ∈ S1 = f1 C`3,1 and ξ ◦ ξ = 0}
and ask for the general solution of the equation φ ◦ φˆ = f1
there follows
φ ∈ H1
where φˆ is the gradeinverse of φ.
(5)
We denote φ as the spacetime spinor of the electron neutrino.
3.
The Natural Strong Force Spinor Manifolds
The natural fermion spinor is a nilpotent element in the 16-dimensional Clifford algebra C × C`3,1 having the general form: ξ =x1 Id + x2e1 + x3 e2 + x4 e3 + x5 e4 + x6 e12 + x7 e13 + x8 e14 + x9e23 + x10 e24+ + x11e34 + x12 e123 + x13 e124 + x14 e134 + x15 e234 + x16e1234
(6)
First we seek the solution to ξ2 ξˆ2 = f2 with ξ2 ∈ f2 C`3,1 and f2 = 14 (1 + e1 )(1 − e24): Choose freely the 12 parameters x2 , x5, x6, x7, x8, x10, x11, x12, x13, x14, x15, x16 and calculate the manifold Ξ2 = {x1, x3, x4, x9} as x1 = −x2 + x10 + x13 x3 = −x5 − x6 − x8 x9 = x11 − x12 + x14 q x4 = Z 2 + (2x7 + 2x15 + 2x16)Z + 1 + x27 + 2x7x15 + 2x7x16 + 2x15x16 + x215 + x216
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The natural strong force spinor manifolds are essentially given by three manifolds Ξs = Ξ2 , Ξu = Ξ3 , Ξd = Ξ4 . We obtain the sign combinations as follows
Ξs : Ξu : Ξd :
Ξs :
x1 = −x2 + x10 + x13
Ξu : Ξd :
x1 = +x2 − x10 + x13 x1 = +x2 + x10 − x13
Ξs :
x3 = −x5 − x6 − x8
Ξu :
x3 = +x5 + x6 − x8
Ξd :
x3 = −x5 + x6 + x8
Ξs :
x9 = +x11 − x12 + x14
Ξu :
x9 = −x11 + x12 + x14
Ξd :
x9 = +x11 + x12 − x14
q
Z 2 + (2x7 + 2x15 + 2x16 )Z + 1 + x27 + 2x7 x15 + 2x7x16 + 2x15 x16 + x215 + x216 q x4 = Z 2 + (−2x7 − 2x15 + 2x16 )Z + 1 + x27 + 2x7 x15 − 2x7 x16 − 2x15 x16 + x215 + x216 q x4 = Z 2 + (−2x7 + 2x15 − 2x16 )Z + 1 + x27 − 2x7 x15 + 2x7x16 − 2x15 x16 + x215 + x216
x4 =
(7)
Analogous expressions are obtained for the |ci, |bi, and |ti quarks by considering the corrsponding Cartan subalgebras of the C`3,1.
4.
Minimal Modules of given Properties
Consider the following statement: In the Clifford algebra C`3,1 there is a lattice manifold of 16 commuting idempotents generated by mutually annihilating (or ”orthogonal”) primitive idempotents f1 , f2 , f3 , f4 . That is, there exists a continuous manifold of commuting 16-tuples generated by a continuous manifold of orthogonal quadruples. The minimal modules of orthogonality are given by the Cartan subalgebras and respectively the color spaces of the Clifford algebra. Therefore the minimal dimension of any primitive idempotent that generates such a 16-element lattice is exactly 4, not more and not less. Now we can take another property such as ”isotropic spinoriality”. Again, as we shall show, the lowest dimension of any subspace that gives rise to isotropic spinors which bring forth a standard density fk , k = 1, ..., 4 is equal to four. Generally, we propose to classify elements of Clifford algebra according to their properties with respect to physics. For example the element J+ = √12 (e1 − e13) represents the property of an angular-momentum shift-operator J+ in a subspace having minimal dimension 2. The reason why this is so is in the fact that - given that we define J3 = 12 e3 - we realize the commutator equation [J3, J+ ] = J+ . This is characteristic for an angular momentum ladder operator in euclidean 3-space. But it is clear that we can easily unfold that ladder-operator into the 16 dimensional algebra. With this little clue in mind we investigate the minimal modules for isotropic fermion spinors.
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5.
Minimal Modules of Natural Isotropic Fermion Spinors
To obtain minimal modules of isotropic space-time spinors we consider the hyperplanes in equation (7) where x1, x3 and x9 vanish. But we let x7, x15, x16 unequal zero in order to get the squareroot x4 6= 0. So we obtain a special minimal solution for the first Cartan subalgebra and respectively chromatic space ch1: i |νe i = y1 = ± √ ϕ ι+ 2 i ˆ ι+ |si = y2 = ± √ ϕ 2 i |ui = y3 = ± √ ϕ ˆ ˆι+ 2 i |di = y4 = ± √ ϕ ˆι+ 2 (8) where ϕ = 12 (e1 +e24 ) is an area extender of direction e1 and ι+ = angular shift operator associated with direction e1 . We have [ϕ, ι+] = ι+
√1 (e3 +e13) 2
a positive
1 [ϕ, ι−] = −ι− with ι− = √ (e3 − e13 ) 2
It can easily be verified that the yi are isotropic direction fields. We have: yi ◦ yi = 0
and yi ◦ yˆi = f1i
i = 1, ...4
(9)
The ’hat’ yˆ denotes the grade involuted y. The first index of f1i equal 1 denotes the first Cartan subalgebra and chromatic space. Evaluating the standard spinor by Clifford multiplying the factors in (8) we obtain rather complicated Clifford numbers in a 4-dimensional subspace i y3 = ∓ √ (e3 − e13 − e234 + j) (10) 2 If we decompose this term into a Clifford product between area extender and angular shift as in equations (8), that decomposition discloses its mystery. The area extender is a linear component of hypercharge. Note that it represents the standard form belonging to the primitive idempotent f13. The spinor y3 can be turned into a manifold by the group action of the constitutive Lie group L(2).
6.
Unfolding Spacetime Spinor Manifolds
In any theory of strong interaction we have to describe movements in the isospin spaces which, at the same time, classify the elementary particles. As we have shown, it is the constitutive Lie algebra l(2) ⊂ C`3,1 which provides the root spaces A2 of the special unitary group SU (3). The rank-2 Lie algebra l(2) gives rise to both color-rotations and
Spacetime Fermion Manifolds
325
flavor rotations as well as Lorentz transformations. Force transformations and motion in spacetime occur in the Minkowski algebra, not in Minkowsi space. This algebra represents so to say the metric surface. We recall the C`3,1-generators of the l(2). These generators of the algebra sl(3) can be written in Clifform, that is, as a multivectorial form of Clifford numbers independent of a definite representation over some determined number field: 1 (e34 − e134) 4 1 λ4 = (e2 + e14 ) 4 1 λ7 = (e13 + j) 4
λ1 =
1 1 (−e23 + e123) λ3 = (e24 − e124) 4 4 1 1 λ5 = − (e4 + e12) λ6 = (e3 − e234) 4 4 1 λ8 = − (e1 + e24) all in C`3,1 4
λ2 =
(11) This we can call an abstract representation of a ’Clifform’ suCl (3) in the standard basis of the Clifford algebra C`3,1. As we know that the real algebra C`3,1 is isomorphic with the Majorana algebra of real 4 × 4 matrices, we can be sure this is a non-compact form of sl(3, R) standing for the compact form su(3, C). In the real representation M at(4, R) the underlined elements are the diagonal matrices corresponding noncompactly with isospin and hypercharge. The λ1 to λ8 are Clifford algebra forms of the familiar algebra sl(3, R). As the algebra is usually described by a list of non-vanishing root commutation relations, we prefer the following linear combinations:
T0,1 = −2λ3
T0,2 = 2λ8
diagonal matrices
T+,1 = −λ1 − λ2
T+,2 = λ4 + λ5
T+,3 = −λ6 − λ7
T−,1 = λ2 − λ1
T−,2 = λ4 − λ5
T−,3 = λ7 − λ6
shift elements (12)
The constitutive group L(2) has a subgroup SUCl (2) which carries out trigonal rotations and thereby executes the Pauli principle in strong force fields. It has a SLCl (2)-component isomorphic with the spin-group Spin(2, 1) responsible for the unfolding of Lorentz boosts. To obtain a compact form of su(3, C) in C ⊗ C`3,1 we just have to construct a linear combination of the diagonal generators and instead use the hypercharge generator. 1 λ8 = √ (−2e1 + e24 + e124) 2 3
(13)
In agreement with the Weyl trick where non-compact real forms are derived from compact complex forms, we invert the procedure and multiply the second, fifth and eight matrix by the unit imaginary. In this way we obtain up to a factor 2 a standard representation within the complex Clifford algebra. The various forms of sl(3, C), sl(3, R) and su(3, C) are equivalent with respect to classification of hadrons since they possess the same root space, namely A2. The Clifform we use is, however, independent from the chosen field of representation. In that sense it is indeterminate where the predicate of compactness is
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concerned. If we do not know the field over which the Clifford algebra is constructed, we cannot decide on the special group algebra that shall be constitutive for the physical phenomena. Anyway, the largest constitutive Lie algebra in C ⊗ C`3,1 is sl(4, C) and respectively su(4, C). The reduction of su(4, C) to su(3, C) is brought on by the fixing of one primitive idempotent as a neutrino state that does not interact with the strong force. In the matrix algebra this means that one row and column, say, row 1, column 1, have zero entries. The exponential map produces a unit at the same entry. In accordance with pregeometric considerations we consider quotient spaces derived from the action of the group L(2) given by the exponential map exp(l(2)). By an Unfolding we denote a rotation that raises the grade of a multivector. We have the isomorphism L(2) ' SLCl (3). The index ’Cl’ tells us that the group L(2) may be either Sl(3, R) or Sl(3, C) depending on the definition of the Clifford algebra, that is, on the field over which the algebra is defined. The equivalence classes are the orbits and leaves derived by conjugation x0 = {g ◦ x ◦ g −1 /g ∈ L(2)} (where ◦ denotes Clifford multiplication). So the L(2) acts like a spin group. However, it properly transforms the elements of the whole Minkowski algebra, that is, its isospin subspaces including the generating Minkowski space. We could calibrate the Lie algebra such that the inner spaces of the low energy quarks u, d, s could be represented in standard notation by the three mutually annihilating (orthogonal) primitive idempotents representing field densities. Those densities then transform like fχk −→ eg fχk e−g
with g ∈ l(2)
(14)
As every such primitive idempotent is decomposed into a product y yˆ the spacetime spinors are transforming like yχk −→ eg yχk e−ˆg
with
gˆ ∈ l(2)
the main involuted of g
(15)
So we can obtain a flavor rotated state such as for example y4 = T y2 Tˆ2
(16)
Consider a Lorentz boost in the plane {e3 , e4} which transforms like eλ1 e3 e−λ1 = (cosh 1) e3 − (sinh 1) e4
(17)
We denote an arbitrary group element as eαλ1 . This carries the spinor y4 to equal to, say, z4 . The Clifford product z4 zˆ4 should be equal to the transformed density eαλ1 f4 e−αλ1 which it actually is. So we have constructed the boosted fermion spinor that preserves the orthogonality relations among the manifolds. ˆ eαλ1 y4 e−αλ1
7.
Conclusion and Prospect
Skipping Langrangians crammed with gauge groups and abandoning auxiliary bundles as introduced by Young-Mills types of theories, we construct the natural space-time spinors in the Minkowski algebra. Thereby we can solve fundamental theoretical problems such as the problem of extra dimensions and supersymmetry. We can answer why nature appears to have more than four spacetime dimensions and where the standard model types of Lie
Spacetime Fermion Manifolds
327
groups have their origin. We can pose such questions as that of cosmic inflation in a new way. We can also reflect anew about unexplained formulae like the mass formula discovered by Yoshio Koide in 1981 that relates the masses of the three charged leptons so well that it predicted the mass of the tauon.
References [1] Abłamowicz, R. CLIFFORD - A Maple 8 Package for Clifford Algebra Computations; http://math.tntech.edu/rafal/, version 8, September 17, 2005 [2] Cartan, A., J. Lecons sur la th´eorie des spineurs vols I, II . Paris 1937, 1938; The Theory of Spinors. New York 1966. [3] Cartan, A., J. La th´eorie des groupes finis et continus et L’Analysis Situs . Memorial des Sciences Mathmatiques, vol. 42, Paris 1930 (Euvres Compl´etes, vol. I, pp. 1165-1225) [4] Chevalley, C. The Algebraic Theory of Spinors. New York 1954. [5] Crumeyrolle, A.Alg´ebres de Clifford et spineurs. Toulouse 1974. [6] Oziewicz, Z. Isometry from Reflections Versus Isometry from Bivector . Adv Appl Cliff Alg, 2009, 19, No. 3-4, 793-817. [7] Schmeikal, B. Lie group Guide to the Universe. In: A. B. Canterra (ed.): Lie Groups - New Research, Mathematics Research Development Series, Nova. New York 2009, 1-59. [8] Schmeikal, B. Primordial Space; Nova Science Publishers: New York 2010. [9] Seiberg, N. Exact Results on the Space of Vacua of Four Dimensional SUSY Gauge Theories ; Phys Rev, D49, 1994, 6857, hep-th/9402044. [10] Seiberg, N. and Witten, E. Electric-Magnetic Duality, Monopole Condensation, and Confinement in N = 2 Supersymmetric Yang-Mills Theory ; Nucl Phys B426, 1994, 485, hepth/9407087. [11] Seiberg, N. and Witten, E. Monopoles, Duality and Chiral Symmetry Breaking in N = 2 Supersymmetric QCD; Nucl Phys B431, 1994, hep-th/9408099. [12] Weyl, H.; Gruppentheorie und Quantenmechanik ; Leipzig, 1931, reprint: Stuttgart 1967.
INDEX A accelerator, 82 access, 72, 250 actual output, 203, 213 airports, 193, 194, 195 Algeria, 218 algorithm, 190, 203, 213, 220 amplitude, 6, 7, 8, 146, 302 anisotropy, 104 annihilation, 293, 294, 298, 299, 311 argon, 232, 235, 236, 238, 243 assessment, 40, 190, 224 asymmetry, ix, 271, 273, 278, 280, 286 atmosphere, viii, 2, 13, 18, 22, 53, 54, 60, 61, 65, 66, 67, 187, 189, 193, 195, 201, 215, 224 atmospheric pressure, 5, 13, 14 atoms, ix, 72, 78, 230, 231, 232, 259, 260, 266, 267, 268, 271, 272, 273, 275, 276, 277, 278, 280, 287
B background radiation, 189 baryon, 321 base, viii, 38, 188, 201, 202, 203, 207, 210, 218, 307 BCS theory, ix, 259, 260, 261, 263, 264, 266, 268 beams, 235 Beijing, 254 bending, 153 benefits, 79, 82 beryllium, 78 blood, 237, 238 blood flow, 237 blood vessels, 238 body density, 261 Bogoliubov-de Gennes equations, 262 Boltzmann constant, 261, 292 Boltzmann distribution, 297 Bose-Einstein condensates, 272
boson, ix, 247, 271, 273, 274, 275, 276, 278, 285, 297, 300, 301, 302, 307, 308, 313, 316 bosons, ix, x, 271, 272, 273, 274, 275, 278, 279, 280, 281, 282, 283, 285, 286, 287, 291, 294, 295, 298, 300, 302, 303, 307, 308 Brazil, 271, 287 breakdown, 240 building blocks, 202 Bulgaria, 224 burn, 243
C Cairo, 217 calcium, 241 calculus, 241, 242, 294, 301, 307 calibration, 192, 245, 321 capillary, 64 carbon, 72, 75, 189 carbon dioxide(CO2), 72, 75, 232, 235, 236, 238, 239, 243 carbonization, 235 casting, 66 cell death, 235 cell membranes, 235 chemical, 235, 240, 260, 266, 267, 275, 283, 285 chemical reactions, 235 China, 79, 247, 254 circulation, 54, 60, 61, 65, 66 cladding, 234 clarity, 29, 65 classes, viii, 188, 222, 321, 326 classification, 4, 65, 66, 143, 320, 325 clean energy, 183 cleavage, 242 Clifford algebra, x, 319, 320, 321, 322, 323, 325, 326 climate, vii, 2, 66, 67, 223 closure, 25, 26, 27, 29, 31, 250
330
Index
coal, 75 coding, 195 coherence, 227, 232, 233 collisions, 171, 172, 173, 276 color, 235, 240, 321, 323, 324 commercial, 189, 193 communication, 225, 226 compatibility, 72, 212 complexity, 2, 13, 17, 201, 321 complications, 242, 243 composition, 240 computation, ix, 140, 141, 247, 248, 307 computer, viii, 187, 191, 192, 194 computing, 133, 248 conception, 23, 25 conduction, ix, 259, 260, 261, 263 conductivity, 83, 85, 143, 179 configuration, 9, 10, 11, 12, 81, 109, 115, 131, 144, 149, 151, 157, 180, 215, 275, 278, 281, 295 confinement, vii, 71, 72, 78, 79, 80, 82, 105, 106, 107, 143, 164, 182, 183, 247, 277, 279 conjugation, 326 conservation, 4, 5, 59, 175, 248 constituents, 14, 23, 53 construction, 19, 22, 34, 54, 55, 56, 57, 58, 59, 64, 72, 90, 125 consumption, 56 contour, 122, 139 control measures, 222 convection model, 194, 199 convention, 86, 88, 131 convergence, 277 cooling, 237 corneal ulcer, 243 correlation, 40, 268 correlation coefficient, 40 correlation function, 268 cosmic ray flux, viii, 188 cosmic rays, viii, 187, 188, 189, 201, 212, 215, 216, 217, 218, 220, 222, 223, 224, 226, 227 cost, 75, 242 covering, 2, 38, 48, 244 CPU, 11 critical value, 272, 286 crude oil, 72, 73 cyanide, 243, 246 cycles, viii, 187, 188, 205, 206, 209, 215, 216, 220, 222, 223, 226 cystine, 241 Czech Republic, 224
D damping, 147 data set, 193 database, 191 decay, 24, 37, 43, 44, 45, 159, 232 decomposition, viii, 6, 23, 31, 188, 206, 209, 212, 213, 215, 218, 220, 223, 324 defects, 243 deficiency, 321 deformation, x, 291, 292, 293, 294, 296, 297, 298, 299, 307, 308, 312, 313, 316 degenerate, 272, 273, 274, 286 denaturation, 235 depth, 62, 63, 64, 194, 195, 236, 237, 238 derivatives, 6, 25, 26, 29, 131, 297 detectable, 190 detection, 212 developing countries, 183 deviation, 125, 191 differential equations, 110, 248 diffusion, 2, 54, 59, 85, 173, 174, 178, 180, 194, 199, 236 diffusion time, 85 dilation, 205, 206, 242 dilute gas, 272 diode laser, 234 diodes, 231 Dirac equation, 320 direct action, 287 direct measure, 268 disaster, 183 discretization, 38 dispersion, 4 displacement, 145, 146, 148, 171, 176 distortions, 143 distribution, x, 36, 53, 55, 66, 67, 68, 107, 115, 123, 165, 166, 168, 169, 177, 191, 261, 291, 292, 297, 299, 301, 302, 304, 305, 306, 307 distribution function, x, 165, 166, 169, 177, 291, 292, 297, 301, 302, 305, 306, 307 divergence, 94, 97, 100, 109, 170, 233, 234, 243, 266 draft, 244 drying, 235 dyes, 231
E Egypt, 218, 229 elaboration, 31, 52 electric charge, 84, 100, 190, 250, 301 electric current, 190
Index electric field, 174, 178, 180 electricity, 72, 75, 77, 83, 244 electromagnetic, ix, 83, 188, 189, 229, 238 electron, ix, 84, 165, 172, 174, 178, 182, 230, 259, 260, 263, 264, 265, 268, 321, 322 electron-phonon coupling, ix electrons, ix, 84, 165, 174, 177, 181, 182, 189, 230, 232, 259, 260, 261, 262, 263 elementary particle, 324 emission, ix, 72, 229, 230, 231, 232, 233 encapsulation, 190, 198 endoscopy, 239 energy density, 75, 236 energy supply, 59 energy transfer, 4, 8, 53, 55, 57, 60 engineering, ix, 116, 229, 245 entropy, 292, 295, 296, 297, 301, 307 environment, 72, 244 environmental conditions, 189 enzymes, 235 equality, 314 equilibrium, vii, 31, 32, 34, 35, 36, 71, 86, 95, 97, 104, 105, 106, 107, 109, 112, 115, 128, 131, 143, 144, 145, 146, 147, 152, 153, 157, 164, 169, 174, 180, 182, 183, 281, 304 equipment, 190, 192, 240, 244, 245 error estimation, 39, 48, 49 Europe, 196 European Commission, 224, 225 European Union, viii, 79, 187 evaporation, 75 evolution, vii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 24, 25, 26, 27, 29, 30, 31, 34, 35, 43, 44, 51, 52, 53, 54, 55, 59, 63, 66, 67, 85, 165, 282 exchange rate, 181 excitation, 230, 261 exclusion, 21, 293, 294, 298, 300, 301, 303, 310 execution, 36, 51 exposure, viii, 187, 188, 189, 192, 193, 194, 198, 199, 201, 222, 223, 224, 234, 236, 237, 243 extinction, 236
F FAA, 194, 209 FEM, 133, 134 Fermi surface, 260, 262, 266 fermions, ix, x, 248, 260, 266, 271, 272, 273, 274, 275, 278, 279, 280, 281, 282, 283, 284, 285, 287, 291, 292, 294, 295, 296, 297, 298, 300, 301, 302, 307, 310, 321 fiber, 234, 238, 239, 241, 242, 243 fibers, 237, 239
331
field theory, 247, 259 first generation, 78 fission, 72, 74, 75, 76, 77, 82, 83, 183 flavor, 319, 321, 322, 325, 326 flexibility, 202 flight, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 222 flights, viii, 187, 188, 189, 191, 193, 195, 196, 197, 200, 201, 203, 223 fluctuations, 145, 226 fluid, 5, 7, 84, 85, 166, 173, 274 Fock space, x, 291, 292, 294, 295, 298, 300 force, 23, 24, 27, 44, 78, 79, 86, 109, 136, 138, 145, 147, 153, 158, 166, 168, 174, 189, 274 forecasting, 2 formation, 240, 272 formula, ix, 18, 20, 26, 29, 33, 34, 37, 40, 44, 56, 59, 61, 64, 67, 259, 267, 298, 309, 327 fragments, 240, 241, 242 France, 69, 72, 79, 80 free energy, 292 friction, 16, 20, 41, 55, 59, 60, 65, 167, 168 fuel consumption, 195 fundamental forces, 319 funding, 183 fusion, 72, 75, 77, 78, 79, 80, 82, 83, 105, 106, 143, 144, 183
G Galileo, 259, 271, 287 gamma radiation, 189 gauge group, 326 gauge theory, 248, 253, 307 geometry, 9, 106, 107, 171, 174, 273 Georgia, 183 Germany, 82, 190, 245 glasses, 231 global scale, 2 grading, 320 gravitational force, 78 gravity, 5, 16, 57, 78 growth, 15, 34, 55, 60, 143 growth rate, 143 guidelines, 22, 242 gymnastics, 92, 93
H hadrons, 325 Hamiltonian, 7, 293, 295, 298, 301, 312, 313 handedness, 96
332
Index
hazards, 243, 244 height, 18, 39, 46, 52, 61, 62, 64, 67 helium, 78 hemoglobin, 236 high winds, 64 Hilbert space, 249 histogram, 66, 67 historical data, 38 history, 5, 46, 47, 48, 67 holmium, 239, 240, 245 Hunter, 185 hurricanes, 215, 227 hybrid, 82 hydrogen, 78 hypothesis, 23
I ideal, vii, x, 5, 7, 71, 85, 106, 111, 143, 157, 202, 282, 291, 295, 297, 319, 321, 322 identity, 94, 95, 105, 129, 139, 150, 164, 171, 247, 250, 251, 252, 309, 311, 320 India, 79 individuals, 243 industry, 75, 234 inequality, 79, 315 inflation, 327 inhomogeneity, 4, 272 injuries, 243 inspections, 245 integration, 11, 51, 119, 130, 134, 137, 139, 141, 168, 248, 267, 277, 278, 319 integrity, 235 interaction process, 60 interface, 2, 3, 5, 6, 15, 16, 23, 26, 53, 54, 55, 56, 57, 58, 59, 65, 69, 70, 149, 150, 151, 234 interface energy, 149 interference, 302, 308 inversion, 230, 231, 232 ionizing radiation, viii, 187 ions, 84, 165, 181, 182, 183 Iran, 71, 72 iron, 77, 78, 189 irradiation, 75, 236 islands, 143 isolation, 75 isospin, 324, 325, 326 isotope, 75, 78 isotropic direction field, 324 Israel, 271 issues, 72, 243 Italy, 259, 271, 287, 291
J Japan, 79, 223 jumping, 230
K kinetics, 302, 306
L Lagrangian density, 254, 260, 274, 275, 276 laser radiation, 244 lasers, ix, 229, 232, 234, 235, 236, 237, 239, 240, 242, 244, 245 laws, 59, 60, 92, 93 lead, 59, 83, 143, 144, 153, 183, 189, 192, 198, 199, 250, 305, 307, 312, 315, 320 learning, 203, 205, 220 legend, 42, 43, 45 lepton, 322 Lie algebra, 320, 324, 326 Lie group, 248, 250, 253, 254, 320, 321, 324, 327 light, ix, 72, 77, 78, 229, 233, 234, 235, 238, 244 linear dependence, 115, 134 liquids, 78 lithium, 78 lithotripsy, 240, 242, 243, 246 local anesthesia, 242 localization, 202, 274 low temperatures, 297 lumen, 242
M magnetic field, 72, 78, 79, 82, 83, 84, 85, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 122, 124, 126, 128, 129, 131, 144, 146, 148, 149, 153, 155, 156, 158, 171, 173, 174, 179, 180, 183, 189, 215, 216 magnetic fusion, 86 magnetic moment, 175 magnetism, 83 magnitude, 17, 24, 59, 60, 63, 79, 104, 111, 112, 174, 264 management, 245 manifolds, x, 319, 320, 321, 323, 326 manipulation, 27, 130, 272 mapping, 140, 202, 321 Marx, 184
Index mass, 5, 62, 78, 84, 146, 215, 263, 266, 274, 275, 327 MAST, 80 master equation, 302 materials, 75, 189, 231 mathematics, 4, 14, 249 matrix, 8, 92, 135, 136, 138, 141, 178, 213, 248, 249, 250, 255, 256, 261, 295, 320, 325, 326 matrix algebra, 320, 326 matter, 78, 188, 215, 249, 272, 273, 277, 278, 320 measurement, viii, 187, 191, 192, 193, 222, 223, 260 measurements, 15, 22, 60, 109, 188, 189, 190, 192, 193, 194, 195, 198, 201, 212, 263 media, 231, 272 medical, 232, 234, 237, 244 medicine, 75, 242 Mediterranean, viii, 188, 212, 215, 217, 218, 220, 221, 223 Mediterranean countries, 220 melanin, 238 melt, 241 metals, ix, 259, 260, 262, 263, 264, 268 meter, 2 Mexico, 215, 227 migration, 240 Minkowski spacetime, 320 mixing, 2, 20, 25, 26, 54, 59, 216 modelling, 18, 68, 70 models, vii, 1, 2, 4, 9, 12, 18, 19, 21, 34, 35, 36, 37, 39, 40, 43, 45, 46, 47, 49, 51, 52, 56, 61, 63, 64, 65, 66, 191, 192, 195, 196, 199, 201, 202, 273 modifications, 26, 40, 194 modules, 323, 324 modulus, 7 momentum, ix, 53, 55, 56, 57, 58, 59, 61, 64, 65, 105, 174, 181, 189, 247, 251, 252, 323 morbidity, 242 Moscow, 1, 69, 183, 188, 212, 213, 215, 216, 220, 223 multiplication, 6, 30, 250, 308, 326 multiplier, 36 muons, 189
N naming, 248 Nanostructures, 287 National Aeronautics and Space Administration, 194, 199 Nd, 232, 235, 236, 237, 238, 243 necrosis, 235 neodymium, 239 neon, 234
333
neoplasm, 239 Netherlands, 224 neural network, 188, 201, 203, 205, 215, 218, 223, 225 neurons, 203, 220 neutral, 84, 165 neutrons, 74, 75, 78, 82, 188, 189, 190, 192, 213, 222, 224 New South Wales, 70 next generation, 65 Nobel Prize, 83 nodes, 12, 134, 135, 136, 138, 140, 141, 203, 213 nonlinear dynamics, 25 nonlinear systems, viii, 71, 144 North America, 191, 245 nuclei, 72, 74, 75, 77, 78, 189, 201, 297 nucleus, 72 numerical analysis, 274 numerical computations, 86
O oceanic areas, 38 oceans, 183 oil, 73, 75 operations, 134 optical fiber, 234, 242, 272 optical lattice, 272, 273 orbit, 174, 176, 177, 178, 181 organs, 243 orthogonality, 90, 321, 323, 326 oscillation, 143, 147, 202 overlap, 278 oxalate, 241
P parallel, 104, 106, 109, 158, 175, 178, 180, 181, 233, 272 partition, 56, 67, 295, 307 path integrals, 250 PDEs, 133 periodicity, 101, 188 permission, iv permit, 54, 65 phase diagram, ix, 271, 280, 281, 283 Philadelphia, 269 phonons, 263 phosphate, 238 photons, 183, 230, 232, 233, 235 physical fields, 6 physical mechanisms, vii, 1, 53
334
Index
physics, vii, ix, 2, 25, 32, 40, 51, 60, 70, 71, 185, 223, 229, 245, 304, 307, 317, 320, 321, 323 pilot study, 245 pions, 189 plasma current, 82, 106, 130 Poincaré, 97 Poincare group, 312 polar, 10, 127 policy, 72 pollution, 2, 72 population, 175, 198, 230, 231, 232, 304 positrons, 189 potassium, ix, 238, 273 Potchefstroom, 212, 213 predicate, 325 predictability, 36 pressure gradient, 105, 156, 157 principles, 25, 31, 36, 248 probability, 40, 78, 212, 253, 292, 302, 303, 304, 308, 309, 310 probability distribution, 292 prognosis, 212 project, 66, 68, 75, 76, 79 propagation, 44, 51, 55, 56, 203, 213, 220, 224, 226, 234 protection, 191, 193, 222, 224 protons, 75, 189, 192, 194, 201, 215 PVC, 294, 298
Q QCD, 253, 327 QED, , ix, 247, 248, 252, 253, 254, 256 quanta, 311 quantitative estimation, 59 quantum field theory, 248, 254 quantum groups, 307 quantum mechanics, 295 quarks, 321, 323, 326 quartz, 237, 239
R radiation, viii, 72, 165, 187, 188, 189, 190, 191, 193, 194, 195, 198, 199, 201, 222, 223, 224, 225, 231, 233, 234, 239, 245 radioactive waste, 75, 82 radius, 62, 64, 80, 96, 106, 122, 123, 131, 132, 133, 157, 276 random walk, 177 reactions, 72, 74, 77, 78, 79, 183 reactivity, 106
real forms, 325 real time, 188, 212, 277 recall, 286, 300, 325 reciprocity, 119, 121 recombination, 193 recommendations, iv, 191, 194, 195, 244 reconstruction, 188, 205, 206, 208, 209, 211, 213, 218, 220, 222 recurrence, 299, 314, 315 redistribution, 58, 59, 61, 65 refractive index, 234 refractive indices, 234 regression, 61 regulations, 38, 190 relaxation, 169 relevance, 215, 242 renewable energy, 72 repair, 202, 212 reproduction, 38 repulsion, 263, 273, 278, 280, 282, 283, 285, 297 requirements, 38, 122, 192, 245 reserves, 72 resolution, 38 resonator, 238 resources, 72, 75, 77, 183 response, 83, 116, 192, 226, 234 restrictions, 35 risk, 52, 240, 243 risk assessment, 52 risks, 222, 243 root, 40, 43, 284, 324, 325 root-mean-square, 40, 43 rotations, 321, 324, 325 roughness, 18, 61 routes, 189, 192 routines, 190 rubidium, ix, 271, 273, 276 rules, 6, 90, 95, 254, 260, 308 Russia, 1, 79, 225, 226
S safety, 72, 103, 128, 129, 130, 131, 157, 183, 188, 222, 242, 243, 244, 245 Sarajevo, 217 scalar field, 96 scaling, 206, 209, 213 scatter, 181, 239, 240 scattering, 18, 40, 43, 60, 260, 266, 268, 273, 276, 282, 285, 302 science, 79, 83, 183, 254 scientific investigations, 54 sea level, 195, 215
Index second generation, 78 second virial coefficient, 307 secondary radiation, viii, 187 semiconductor, 189 sensitivity, 48, 190, 216 shape, 8, 31, 32, 34, 35, 49, 55, 56, 59, 60, 61, 65, 83, 95, 97, 115, 135, 143, 165, 215, 216, 220, 281 shear, 2, 22, 55 shock, 240 shoreline, 41 shortage, 25 showing, ix, 259, 268, 286, 306 signals, 202, 220 signs, 26, 243, 244, 285 silica, 234 simulation, 2, 7, 15, 16, 38, 39, 41, 44, 48, 49, 51, 52, 63, 198, 201, 222 simulations, 17, 21, 37, 39, 46, 47, 49, 50, 52, 66, 67, 165, 277, 278 sine wave, 202, 205, 233 Singapore, 194, 225, 226, 287, 316, 317 skin, 19, 243 solar system, 188 solid state, 237, 238 solitons, 273, 277 solution, 4, 9, 14, 15, 16, 19, 21, 24, 26, 28, 54, 62, 64, 65, 66, 75, 108, 109, 115, 117, 118, 120, 122, 126, 127, 133, 134, 139, 140, 141, 146, 165, 169, 180, 193, 231, 240, 265, 285, 286, 300, 305, 322, 324 South America, 191 South Korea, 79 spacetime, x, 319, 320, 322, 325, 326 space-time, 2, 3, 38, 67 species, 84, 165, 166, 167, 172, 179, 181, 273, 278 specific heat, 263, 296, 297, 302, 307, 314 specifications, 36, 245 spectral component, 4 spin, 260, 268, 272, 273, 274, 275, 282, 296, 301, 319, 325, 326 stability, viii, 39, 71, 79, 82, 83, 86, 143, 144, 145, 146, 147, 148, 150, 151, 152, 154, 155, 156, 157, 158, 159, 160, 164, 183, 277 stabilization, 143, 149 standard deviation, 191 stars, 78 state, ix, 6, 18, 19, 20, 21, 24, 27, 41, 43, 45, 49, 52, 53, 54, 55, 56, 59, 60, 61, 64, 65, 100, 105, 143, 144, 145, 182, 229, 230, 231, 232, 261, 266, 267, 272, 274, 277, 278, 280, 281, 282, 286, 297, 298, 300, 301, 303, 304, 308, 309, 310, 311, 314, 321, 322, 326
335
states, ix, x, 22, 105, 106, 144, 145, 157, 183, 191, 194, 230, 262, 263, 265, 267, 268, 271, 272, 273, 275, 278, 283, 284, 286, 292, 293, 294, 295, 298, 299, 300, 301, 302, 303, 304, 308, 310, 313, 321 statistics, x, 49, 66, 68, 190, 195, 291, 292, 297, 301, 302, 303, 304, 305, 306, 307, 308, 310, 311, 312, 313, 314, 315, 316 stratification, 37, 42 stress, 5, 19, 20, 24, 25, 26, 31, 53, 55, 90, 167, 263, 264, 304, 310 stretching, 205 strong force, 320, 323, 325, 326 strong interaction, 324 structure, 13, 14, 26, 28, 29, 62, 226, 247, 249, 278, 294, 295, 296, 297, 312, 320 substitution, 6, 14, 28, 112, 113, 130, 154, 283, 285, 286 successive approximations, 307, 315 superconductor, 265, 268 superfluid, ix, 266, 267, 268, 271, 272, 273, 274, 275, 282, 285, 286 supernovae, 189 supersymmetry, 326 supervision, 244 suppliers, 72 suppression, 34 surface energy, 151 surface tension, 5 SUSY, 327 Sweden, 83 symmetry, ix, 95, 119, 138, 139, 179, 180, 247, 248, 249, 271, 272, 273, 274, 280, 281, 284, 286, 287, 300, 301, 302, 307, 310, 315, 319, 320
T target, 79, 203, 213, 243 tau, 271 techniques, 190, 191 technologies, 183 technology, ix, 75, 78, 79, 183, 229, 234, 238, 242 temperature, ix, 72, 78, 79, 164, 166, 168, 179, 180, 236, 237, 240, 245, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 274, 293, 306 temperature dependence, 260 tension, 153 testing, vii, 1, 36, 37, 39, 41, 43 theatre, 243 thermal energy, 75 thermodynamic properties, 314 thermodynamics, x, 291, 294, 295, 297, 304, 317 thermoluminescence, 190 time resolution, 38
336
Index
time series, 206 tissue, 234, 235, 236, 237, 238, 239, 242, 243, 245 tokamak, viii, 71, 79, 80, 82, 103, 106, 109, 111, 115, 130, 131, 132, 165, 174, 175, 180, 183, 185 tones, 75 topological invariants, 320 topology, 82, 83, 143 torus, 79, 171, 174, 175, 178 total energy, 57, 149, 232 training, 188, 201, 202, 203, 212, 213, 215, 218, 220, 223, 242 trajectory, 165 transformation, 74, 91, 93, 99, 102, 140, 248, 249, 250, 251, 253, 254, 255, 257, 295, 319, 320 transformations, 124, 248, 249, 250, 253, 255, 294, 325 translation, 205, 206, 255, 301 transmission, 238 transport, vii, viii, 1, 3, 71, 86, 109, 115, 143, 164, 165, 170, 171, 172, 173, 174, 177, 178, 179, 180, 183, 187, 190, 193, 198, 222, 225 transportation, 54 treatment, 25, 46, 49, 193, 194, 240, 244 tungsten, 233 turbulence, vii, 1, 21, 22, 24, 26, 27, 28, 35, 53, 55, 56, 58, 59, 62, 65, 165 turbulent mixing, 65
U uniform, 107, 123, 149, 158, 171 United, 72, 75, 79, 80, 184, 185 United Kingdom (UK), 68, 69, 80 United Nations (UN), 184, 185 United States (USA), 72, 75, 79, 80, 190, 254 universality, 34 universe, 78 uranium, 72, 75, 82, 183 ureter, 241, 242 uric acid, 243, 246 urinary tract, 239, 242
V vacuum, 79, 81, 82, 148, 149, 150, 151, 157, 158, 298, 300, 301 validation, 2, 36, 38, 39, 40, 43, 46, 49, 52, 191, 222 variables, 6, 7, 24, 25, 26, 27, 28, 29, 54, 117, 234, 248, 249, 250, 256, 257, 299, 306 variations, viii, 131, 134, 187, 192, 203, 205, 206, 209, 212, 213, 214, 215, 216, 217, 218, 220, 221, 222, 223, 226, 256, 257 vasculature, 237 vector, ix, x, 5, 14, 53, 59, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97, 105, 110, 115, 117, 128, 136, 138, 139, 145, 146, 150, 151, 161, 164, 170, 171, 213, 247, 248, 249, 251, 252, 253, 301, 319 velocity, 2, 3, 5, 7, 8, 13, 14, 16, 20, 23, 24, 25, 26, 27, 28, 29, 41, 53, 55, 65, 84, 85, 146, 165, 166, 168, 169, 170, 173, 174, 175, 176, 179, 180, 181, 274, 275 vertical dimensions, 53 viscosity, 5, 21, 24 vision, 239, 242, 243
W water, vii, 1, 2, 4, 5, 13, 16, 21, 22, 23, 24, 25, 27, 35, 53, 57, 59, 60, 64, 65, 67, 69, 75, 78, 235, 236, 238, 239, 244, 245 wave propagation, 55 wave vector, 3, 4, 7, 8, 9, 10, 11, 14, 15, 30 wavelengths, 233, 235, 244 wavelet, viii, 188, 202, 205, 209, 222, 226 wavelet analysis, viii, 188, 205, 209, 222, 226 wear, 243 wells, 272, 273, 280, 281, 286
Y yield, 82, 168, 173, 240, 285
