Veibel turbulence

Plasma Physics (Journal of Nuclear Energy Part C) 1963, Vol. 5. pp. 43 t o 41. Pergamon Press Ltd. Printed in Northern Ireland

COLLISIONLESS SHOCK WAVES IN A PLASMA IN A WEAK MAGNETIC FIELD* S. S. MOISEEV and R. Z. SAGDEEV Institute of Nuclear Physics, Siberian Branch, U.S.S.R. Academy of Sciences, Novosibirsk, U.S.S.R. (Received 20 September 1962)

Abstract-The relationship is investigated between laminar and turbulent theories of the structure of a collisionless shock wave. Forp, >p,, wherep,,p, are the ion and electron pressures, a laminar structure of the shock wave may be constructed in terms of the ‘solitary’ wave with thickness of the order of the Debye length. Whenp, 2 pethe principal part within the shock front is played by scattering of ions by small-scale field fluctuations which result from anisotropic instability. The thickness of the front is in order of magnitude

where c is the velocity of light, R, number.

= 47~ine*/m,,m,,

m, are the ion and electron masses, and M is the Mach

1. AT THE present time a detailed study has been made of what are called ‘collisionless’ shock waves in a plasma, propagated transversely to a strong magnetic field (SAGDEEV,1962). A magnetic field parallel to the plane of the shock front contains the ‘hotter’ particles and prevents a broadening of the transition region between the undisturbed ‘cold’ plasma in front of the shock wave and the ‘heated’ plasma behind it. At high Mach numbers the thickness of the front of such a shock is in order of magnitude close to the Larmor radius of the ions. Various authors (KAHN, 1958; PARKER,1958) have discussed the possibility of the existence of collisionless shock waves in a plasma without a magnetic field also. Here the mechanism which prevents the broadening of the transition region is the beam instability of two inter-penetrating plasmas. With this approach, however, no account is taken of the thermal spread within each ‘beam’. A more accurate discussion which takes account of the thermal motion does not give instability for Mach numbers from unity up to -t/(m,//wp) if the electron ‘temperatures’ are comparable with or less than the ion ‘temperatures’ (NOERDLINGER, 1961). Here / ? I , is the ion mass and n7, the electron mass.

np = no exp (e+/O). On the above assumptions the one-dimensional motion can be described by the equations

Assuming that all quantities depend on x and t only through 6 = ,Y - ut, we can reduce equations (1) to a single second-order differential equation for the potential:

where V ( 4 )is the ‘effective potential energy’. We shall suppose that the ordinary dissipation due to particle collisions is absent, but we shall take into account the reflection of ions from the forward front of the shock, which forms a ‘collisionless’ dissipation. The structure of the resulting collisionless shock wave can be described on the basis of the following simple picture. In the absence of any dissipation, a solitary wave forming a symmetrical potentia1 hump can be propagated in the plasma. I n reality the plasma

2 . In the particular casep, pL(where p,, p 1 are the electron and ion pressures), we can set up a laminar theory of the collisionless shock wave, i.e. no instability mechanism is needed (VEDENOV et d., 1961). Since in this case the shock-wave velocity U

the ion thermal spread may be neglected. For the electrons, however, the motion is quasi-steady, i.e. we can use the Boltzmann distribution

> .t’[(p,. -: P,>/n~7~,l

considerably exceeds the thermal velocity of the ions,

’Translated from the Russian by J. B. SYKES. 43

44

S. S. MOISEEV and R. Z. SAGDEEV

z

FIG.1

contains at least a few ions which are reflected from the moving potential hump, and this destroys the symmetry; periodic oscillations appear beyond the hump, so that a kind of shock wave is formed which links two different states of the plasma: undisturbed plasma in front of the shock and a state modulated by intense ordered oscillations behind the shock. The corresponding ‘shock adiabatic’ must take account of further terms in the expressions for the energy and momentum flux densities behind the front, due to such ordered oscillations. It should be noted, however, that the distribution of energy between thermal motion and oscillations depends on the actual mechanism of collisionless dissipation. If the number of reflected particles is small, we can find the shape of the profile of such a shock wave. Figure 1 shows the form of the potential in the wave, In the absence of dissipation 4, = 42and I. = co-a symmetrical solitary wave. When allowance is made for reflected ions the potential in region I satisfies an equation which differs from (2) by extra terms on the right:

By solving the equation for the potential in regions I and I1 with the boundary conditions of continuity of 4 and d‘, we find the potential profile (Fig. 1). If we again consider the analogy with the motion of a ‘particle’ in a potential well V(+),we can say that the effect of reflected ions is simply that the ‘total energy’ E becomes negative. This leads to a periodic motion (a periodic structure behind the shock front). The decrease in the energy E is proportional to the number of reflected ions: (4)

Since the ‘potential energy’ “(4) varies quadratically for small 4, the ‘turning point’ 4%is proportional to the square root of the energy - E : $2

-

2/-%

and the period of the oscillations increases logarithmically with decreasing energy :

1

N

In (--I/&).

Thus the minimum value of the potential 42behind the shock wave front is

M 2 = miu2/i3. (5) The value of qhax differs only slightly from the corresponding value in the solitary wave with the same Mach number. The length of the oscillations at the front is

The first term corresponds to subtracting the reflected In an actual plasma the oscillations are finally damped ions from the total number of ions no; the second term by collisions. is the contribution of the reflected ions. noF(4) is the total density of reflected ions at a point where the 3. In the case p c G pi the number of ions reflected potential is 4. The actual form of F is easily found from the moving potential hump is not small and the if we know the unperturbed ion velocity distribusolution of equations (1) and (1’) can not now be tion. is due to ions sought in the form of a dependence on x and t only The discontinuity of the potential going away to infinity which have been reflected from through the combination x - ut. In other words, the potential barrier, and in the present case, where the there is no frame of reference in which the motion of number of reflected ions is small ( F < I), the discon- the shock appears steady. We shall show below, however, that in this case the tinuity is proportional to F. The value of 4-,, IS, ’ as we shall see, proportional to the square root of the number ‘anisotropic’ instability must lead, even at low Mach of reflected particles, so that dl C d2. The state of the numbers, to the formation of a collisionless shock plasma behind the front (region 11) is described by the wave. First of all we may mention some features of the inquantities c $ and ~ &, ~ which ~ determine the amplitude stability of a plasma with an anisotropic particle velocof the oscillations, and by the length 3, of the oscillaity distribution when there is not a magnetic field Ho tions; in this region equation (2) holds.

Collisionless shock waves in a plasma in a weak magnetic field

45

where v(k) < 0, i.e. the fluctuations are damped. Thus a steady value of fi is set up, determined from the condition of balance between the increase due to instability and the non-linearity, where

H

f i e being 'the correction to the electron distribution FIG.2.

in the undisturbed plasma. The instability mechanism is easily understood from the following model. Let us consider two groups of ions moving in opposite directions along the x-axis (Fig. 2 gives a plan view). We shall assume that the electron velocity distribution is isotropic, so that the initial current in the plasma is zero. Let a transverse electromagnetic perturbation of the form N exp i(kz - at) occur in the plasma. Then the redistribution of the particle motion in the inhomogeneous magnetic field of the perturbation causes an electric current in the direction opposite to the electric field, and this brings about an increase in the perturbation if the current due to the electric field itself is smaller. The instability is aperiodic and increasing solutions occur for perturbations of wave number k < k, (STEFANOVICH, 1962), where

Here (uO: = 4n-e2n/m,,n is the particle density,fi(v) the given anisotropic ion velocity distribution. The y-axis is taken along H. The minimum rise time of the perturbations is shown by the linear theory (STEFANOVICH, 1962; ZASLAVSKI~ et al., 1962) to be

function which describes the effect of the perturbations. On the right of (9) is a non-linear term which, as is easily seen, leads to the generation of new perturbation waves. The increase of this term is due mainly to the waves with the smallest build-up time ( T is ~reached ~ ~ a t k = 0*6k,). From (9) we have for the pulsation amplitude

fl

-

where rHeis the Larmor radius of the electrons. Hence we can see the physical reason why fields considerably exceeding (IO) are impossible : the drift approximation would be applicable to electrons in such fields, and would lead to a very great heating of the electrons-the electron energy might exceed the energy stored in the ions. As a result of the feedback of the perturbation the anisotropic distribution fi(v) varies with time. Equation (1 l ) shows that a moving ion 'collides' with smallscale pulsations and so is slightly deflected; the Fokker-Planck equation therefore holds for fi. The terms responsible for the isotropy of the initial distribution give in order of magnitude I

D

4. Going on now to consider the non-linear case, let us estimate the amplitude of the pulsations of the magnetic field of the perturbation. Firstly, the instability leads to an increase in the pulsation amplitude dHJdt v(k)H,; secondly, the non-linearity leads to a transfer of energy along the spectrum to shorter waves,

(1.0)

where U, is the mean thermal velocity of the electrons. It may be noted that (10) corresponds to magnetic fields for which rHe llkoi, ( 1 1)

where are the mean energies along the where E , , and corresponding axes, AE = E~ - E,,> 0. From (8) we easily see that it is legitimate to neglect collisions in cases of practical interest.

(koim,c/e)tle,

-

e2a2/m:c2k02Tn,

(1 3)

-

with T , the characteristic time of flight of a particle in the perturbation field l/k,p). D can be obtained both from dimensional arguments and from more rigorous arguments. The coefficient D corresponds in the equation forfi to the term

which is obtained from the Boltzmann equation by

46

S. S. MOISEEV and R. Z. SAGDEEV

averaging in a quasi-steady state over small-scale spatial pulsations of the terms (e/mc)v x H, . afi/av, which are related only t o the magnetic field of the perturbation (and are responsible for the isotropy of the distribution: VEDENOV et al., 1961). In (14) we have used the fact that 101 < k.v (STEFANOVICH, 1962; ZASLAVSKI~ et al., 1962). From equation (14), assuming that the non-linear state is determined by perturbations with the maximum increment, we can easily obtain the expression for D given above. Using (13), (7) and (lo), let us estimate the time needed to make the ions isotropic ( T ?/io) by ‘collision’ with field inhomogeneities in the nonlinear state: ~ - - -c l mi Jzgdv+ 1

-

Qoi

me

where cTi is the mean thermal velocity of the ions. Using (15) we can easily show that the time to make the electrons isotropic is (milme)’ times less than for ions, and this justifies the use of the isotropic distribution function for electrons.

It is known that flows which are described by a ‘displaced’ Maxwellian distribution function are isentropic. In the present case the zero-order approximationf(O’(lcl), with c = v - V and V the mean velocity of the flow, for the equation

similarly gives on taking moments the equations of hydrodynamics: p dV/dt = -Vp, (20) where p is the mass density and p the pressure. The third moments vanish, sincef(O)(lcl) is isotropic. We may determinef‘l) and find the viscosity term in the equation of motion, taking for simplicity the onedimensional case, aye) fa) E - = - (21) ax T Then the equation of motion, with the correction term f“’, becomes p;iT = - G -

5. The microfluctuations of the magnetic field give the ‘collision’ (Stoss) term (14) in the averaged Boltzmann equation. It is of interest to derive a hydrodynamics from the Boltzmann equation with this St&), by analogy with the derivation of ordinary hydrodynamics. As usual (CHAPMAN et al., 1952) we represent the distribution function f as an expansion

f=f‘O’

+f(l’

4-

,

.. ,

(16)

wheref‘”’ is the zero-order approximation, satisfying the equation (17) St,(f‘O’) = 0. In the ordinary case the solution of the equation in the zero-order approximation St(f‘O)) = 0 is the Maxwellian distribution function. In our case it is clear from (14) that the equation (17) corresponds to any isotropic function f‘O)(lvl). However, as we shall see below, this does not prevent us from setting up a hydrodynamicsusing the ordinarymethod of moments, since to ‘cut off’ the sequence of equations for the moments we need only use the isotropy off(O)(r). The form (14) for St,,(f) is somewhat inconvenient. We may write it, as is often done in the kinetic theory, i n the ‘7 approximation’:

-

(18) St,,(f) (fo - f ) / T . This approximation leads to reasonable results, and here it is certainly at least as good as the estimate of the pulsation amplitude due to instability.

ax.

- V,)’

-

$(v - V)’] dv. (22)

Using (21), we have

Here the brackets () denote averaging over the function i n velocity space. On the other hand, substituting f = f ( O ’ 4-f“’ in equation ( I 5) for T, we find that

f(0)

Hence

and the viscosity term in the equation of motion (22) has the final form

The viscosity coefficient itself depends on the velocity gradient. The hydrodynamics based on St,,(f) becomes valid (see the expansion (16)) if the characteristic dimensions of the problem are c m, L>>--wO1m e

Collisionless shock waves in a plasma in a weak magnetic field

By means of the above hydrodynamic equations it is easy to derive in the usual way the expression for the profile of a weak shock wave front. (Here we neglect thermal conduction, although it may make a comparable contribution.) On account of the non-linearity of the viscosity coefficient it has the form (e.g. for a velocity V(x))

A----

and the maximum increment is obtained, when Ho2
On estimating the amplitude of magnetic field pulsations in long-wave perturbations from the Boltzmann equation for ions, and the time to establish isotropy by analogy with (15), we see that, if

1

c mi

ooim6( M

-

Unlike ordinary hydrodynamics, in our case the dependence on the Mach number is considerable. 6. In the preceding section we have neglected the presence in the plasma of an external magnetic field. When such a field is present, the results may be considerably modified. This is because the linear theory of instability is changed. If HO2< nB, the instability again remains even at low anisotropy (AB < e), but the increments may vary, especially for wavelengths exceeding the Larmor radius of the ions and electrons. In order to assess the effect of the long-wave region, let us first consider the nature of the development of instability in the drift approximation, but with allowance for the finite Larmor radius. For simplicity we shall take only the case where p , ,> p l ( p being the pressure along the magnetic field Ho).The instability develops in perturbations of the Alfvtn-wave type (k H,); the dispersion relation is derived in the usual wav as

I/

where A, = 0 in a plasma with zero Larmor radius, while when the finite ion Larmor radius is taken into account (in an isothermal plasma the finite electron Larmor radius may be neglected) we have (30)

= eHOlmlc"

('Ifo2

Hence we have finally

\'..

f G v ._1,

to=&-

47

1 I1

corr,z

4n Ap

( ~ 3 ; ~

=

4"-1

47"

,

(31)

the long waves have the principal effect on the manner of establishment of isotropy. The subsequent discussion is entirely similar to that in the previous section. Since the dependence of k, on the anisotropy is as before, the viscosity term in the equations of motion along the magnetic field has the form (with H , = Hn)

The thickness of the shock wave front is

A

1 rill,

*

(35)

Zt is known (PIKEL'NER, 1961) that magnetic storms are caused by fluxes of particles from the sun with a sharp forward front of thickness 100-200 thousand km, forming a shock wave in the interplanetary gas. At ion velocities in the flux 10scm]s and interplanetary gas density lo2 ~ m the - ~free path for collisions is 1013kin. This shows that the shock wave can be formed only by self-consistent fields in the ionized interplanetary gas. It is of interest to note that, even assuming no frozen-in fields, we obtain from (28) a reasonable upper limit to the thickness of the shock wave at the density of the interplanetary gas.

REFERENCES S. and COWLINGT. G . (1952) The Mathematical CHAPMAN Theory o j Non-Vtiifurm Gases, 2nd edn., Cambridge. KAHNF. D. (1958) Rev. mod. Phys. 30, 1069 NOERDLINGER P. D. (1961) Astrophys. J . 133, 1034. PARKER E. N. (1958) Phys. Rev. 112, 1429. PIKEL'VER S. B. (1961) Foiindations o j Cosmic Electrodynamics, Moscow. SAGDEEV R. Z (1962) Sou. Phys. Tech. Phjs. 6, 867. STEFANOVICH A. E. (1962) Sou. Phys. Tech. Phys. I , 462. VEDENOV A. A., VELIKHOV E. P. and SAGDEEV R. 2 (1961) Conferetice 011 Plasma Phvsics aiid Cotitrotled Nuclear Firsion R e s k h , Salzburg. ZASLAVSKI~ G.M. and MOISEEV S. S. (1962) Sou. Phys. J.E.T.P. 15, 731.