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ADVANCES IN

G E O P H Y S I C S

VOLUME 30

This Page Intentionally Left Blank

Advances in

GEOPHYSICS Edited by

BARRY SALTZMAN Department of Geology and Geophysics Yale University New Haven. Connecticut

VOLUME 30

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

San Diego New York Berkeley Boston London Sydney Tokyo Toronto

COPYRIGHT 0

1988 BY ACADEMICPRESS, INC.

ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS. ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING. OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM. WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. 24-28 Oval Road, London NW1 7DX

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ISBN 0-12-018830-9 (alk. paper)

PRINTED IN THE UNITED STATES OF AMERICA

88 89 90 91

9

8

7 6 5 4 3 2

1

CONTENTS

Selected P Wave Problems

MARKUS BATH 1 . Multiple P Waves .............................................. 2. P Wave Reflections ............................................. 3 . Travel Times of Diffracted P Waves .............................. Appendix 1 . Multiple P Observations ............................. Appendix 2. Diffracted P Observations ........................... References.....................................................

4 35 52 74 80

90

Ocean Currents Over The Continental Slope G. T. CSANADY

1. 2. 3. 4. 5. 6. 7. 8.

Introduction ................................................... The Observational Evidence ..................................... The Fundamental Slope Effect ................................... Vortex 'hbe Stretching Versus Vorticity Advection . . . . . . . . . . . . . . . . . . Topographic Waves ............................................. Pressure Torque Versus Bottom Stress Curl ........................ Pressure Torque and Planetary Vorticity Advection . . . . . . . . . . . . . . . . . Conclusion .................................................... References .....................................................

95 96 132 138 150 174 193 198 199

Obtaining Attractor Dimensions from Meteorological Time Series

HARRY W. HENDERSON AND ROBERT WELLS 1. 2. 3. 4.

Introduction ................................................... Basis of Model Reconstruction .................................. Calculation of Attractor Dimensions ............................. Conclusion .................................................... Appendix ..................................................... References .....................................................

INDEX.......................................................... V

205 206 221 234 236 237

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ADVANCES IN GEOPHYSICS.VOL. 30

SELECTED P WAVE PROBLEMS MARKUS BATH Department of Seismology Uppsala University Uppsala, Sweden

This article consists of three independent parts with the common purpose of fillingsome gaps in our knowledge about P wave generation and propagation. Section 1 gives a statistical and physical study of multiple, especially double, P waves generated by successive events. The frequency of occurrence of double P waves is independent of focal depth, of magnitude, and mostly of region. The time lag between the two arrivals is independent of focal depth, but it decreases with increasing magnitude. The amplitude ratio between the second, larger event and the first, smaller event is also independent of focal depth, but it increases with increasing magnitude. Section 2 is a theoretical investigation of P wave reflection, especially pP and PP, from a free surface as compared with an ocean bottom, as well as compared with the MohoroviEiE(Moho)discontinuity.The amplitude ratio of free surface/ocean bottom reflection lies generally in the range of 1-2, i.e., the ocean bottom-reflected P wave shows somewhat lower amplitudes. The amplitude ratio of free surface/Moho discontinuity lies in the range of 1-5, increasing rapidly with the angle of emergence, i.e., the earlier arrival of the Moho reflected wave is mostly relatively weak. Section 3 is an empirical determination of travel times of P waves diffracted around the outer core boundary. The derived travel times have a standard deviation of about f2 sec. They may be taken as representative of modern, sensitive recordings on 'fypical continental bedrock without sediments, especially on shield structures. As a by-product, our observations yield a P wave velocity of 13.62 km/sec at the base of the mantle. The sudden slips between adjacent rock masses, which we call earthquakes, generate a series of elastic or seismic waves that propagate through and over all parts of the earth. The waves are of two main types: body waves that propagate through the earth's interior and surface waves that propagate along the surface layers of the earth. Each of these types can be divided into two kinds of wave motion: body waves consist partly of longitudinal or P waves and partly of transverse or S waves, while surface waves consist of Love waves (Q)and Rayleigh waves (R).The different waves are characterized by different 1 Copyright @ 1988 by Academic Press, Inc. All rights of reproduction in any form

2

MARKUS BATH

particle motions, amplitudes, and periods as well as different speeds of propagation. As a consequence, they arrive in the order P S Q R, thus constituting the typical picture of a seismogram. By analyzing their seismograms, seismologists can make deductions about the earth’s internal constitution as well as about the earthquake properties. All these matters are well known to geophysicists or can be found in any introductory book on seismology. Among the different seismic waves, there is no doubt that P waves have attracted a great amount of attention throughout seismological history. The obvious reasons are their relative simplicity in the records and the reliability of the information they may furnish. As the first wave to arrive on a record, the P wave is undisturbed by any previous earthquake waves and is only subject to possible disturbance from the background noise or microseisms. Therefore, the arrival time, the amplitude, and the phase can generally be read visually with greater accuracy for the P wave than for any of the following waves in a seismic record of an earthquake. We may point to several facts in present-day seismology which all illustrate the importance attached to the P waves. Most modern seismograph stations operate not only long-period seismographs but also short-period ones, generally in three components. The main purpose of the latter is to provide good records of longitudinal waves, primarily P and PKP. Supplementary stations with less equipment often operate only one seismograph which is a short-period vertical component one, in order to obtain the necessary P wave information. Daily or weekly reports from stations around the world concentrate almost exclusively on P wave readings. These also constitute the only essential data used by worldwide or regional seismological centers in their calculations of source data, i.e., epicenter location, focal depth, and origin time. In addition, the amplitudes of P waves are practically the only ones used for the calculation of body wave magnitude (m), although methods exist for its calculation also from PP, S, PKP, and SKP waves. The direction of initial motion of the P waves, i.e., compression or dilatation, constituted for a long time the only reliable way of determining the mechanism of earthquakes. No doubt, the accuracy of our knowledge of earthquake properties as well as of the earth’s internal constitution depends very much on the accuracy with which we are able to record and read P waves. It is therefore no wonder that much effort has gone into the study of P waves all through the history of instrumental seismology in order to improve recordings and their interpretations. Still, there may be some gaps left which could be filled with more knowledge. It is the purpose of this article to report some original research concerned with such problems, conducted during the last 2 years by the author. Although different problems are dealt with, they are all concerned with P waves, concentrating on properties which may need more information.

SELECTED P WAVE PROBLEMS

3

Accurate readings of arrival time, amplitude, and phase of P waves presuppose simple wave shapes, which originate from a single slip in the source region. However, double or multiple slips occur frequently which give rise to double or multiple P waves within a time span of a few seconds. Sensitive stations provide more complete and accurate recordings of such multiple arrivals than less sensitive ones. It is natural that a combination of many stations of different sensitivity will affect the accuracy of the results, whether these concern earthquake locations and timing or the velocity distribution in the earth's interior. Related aspects are discussed in Section 1, Multiple P Waves. This is a statistical and physical study of source properties that lead to multiple, especially double, P arrivals. It is an empirical investigation, based mostly on records from the Swedish seismograph network. But even in the case of a single slip and a single P wave, complications may arise during the propagation due to multiple wave paths, also leading to multiple P waves with similar complications as just mentioned. One aspect of this problem concerns reflected P waves, such as P P and pP, for which the nature of the reflecting surface is of importance. For a single discontinuity, an incident simple wave gives rise to an equally simple reflected wave, but for more complicated reflectors, the reflected wave may show multiplicity and thus complicate the record analysis. One example is offered by the reflection against the earth's surface, where the MohoroviEii: discontinuity offers complications, leading to smaller, early arrivals of PP and pP waves. The time interval between p P and Pis often used for the estimation of focal depth, which could be disturbed by an unreliable early p P arrival. These problems can only be investigated in a theoretical fashion which is done in Section 2, P Wave Reflections. In spite of such complications which may affect the arrival of longitudinal waves, it is true that their travel times, especially of the P wave itself, have been determined with greater accuracy than for any other wave, both globally and regionally. The corresponding velocity distribution in the earth is also known to a good accuracy nowadays. However, while there are such reliable travel time determinations for P waves up to around 100" distance, there are remarkably few determinations for greater distance, i.e., for the P waves diffracted around the outer core boundary. In Section 3 of this article, Travel Times of Diffracted P Waves, we have aimed at a remedy on this point. This is an empirical investigation, based on records of the Swedish seismograph network. Besides travel times, it also gives information on the longitudinal wave velocity at the base of the mantle. The present article with its three different parts may seem rather heterogeneous to any reader. However, as our aim has been to fill some gaps in our knowledge about P waves, it is natural that some heterogeneity is involved. We have to consider that over nearly the past hundred years of instrumental

4

MARKUS BATH

seismology, a very large number of investigationshave been done on P waves, thus leaving only a few scattered gaps to be filled. Our main theme is to improve the knowledge about P waves, and thus to assist in improving the reading and interpretation of seismograph records. In turn, this is hoped to lead to increased accuracy in studies both of earthquakes and of the earth's internal structure. Each section is self-contained and can be read independently of the other two. The waves that we record are affected by source-path-receiver properties, in this order. This is also the order followed here: Section 1 concerns source effects, Section 2 and Section 3 path effects. For an efficient solution of the wave problems, we need a combination of theory and observations. Some problems are best approached by observations, such as those reported in Sections 1 and 3, while others require theoretical calculations for their solution, as in Section 2. 1. MULTIPLE P WAVES'

Multiple arrivals,especially of P waves, are familiar to all seismologistswho regularly read and analyze seismograms. Although well known since the early days of instrumental seismology, it is practically only in the last two decades that this topic has received more attention and been more examined. The

'

Notation used in this section: 1 and 2, subscripts referring to first and second event or alternatively to first and second P wave (Pl, P2); a, ground amplitude (pm); A, fault plane area (cm'); b, coefficient of m, or M in recurrence relations; D, fault displacement (cm); E, seismic wave energy (erg); h, focal depth (km); log, decadic logarithm; m, body wave magnitude = m(UPP,KIR), calculated from the maximum amplitude in the P wave group, usually P2; m(US), body wave magnitude according to USGS; M,surface wave magnitude, corresponding to m; M,,, seismic moment (dynes cm);N, number of events, number of data points, or frequency ( N , refers to double P, N2 to earthquakes, whenever needed); 0,origin time (hr, min, sec); pP - P = T(pP) - T(P) (in sec); q, calibration term in formula for m; SD, standard deviation of a single observation; t , travel time (min sec); T, arrival time (hr, min, sec); a and 8. constants in formula for M,,; 6, relatively small but finite difference; 6h = h, - h,, focal depth difference between events 2 and 1 (km); 6m = m2 - m,, magnitude difference between events 2 and 1; 6 0 = 0,- 0,, origin time difference between events 2 and I (sec); 6T= - T,, arrival time difference between events 2 and 1 = time lag (in sec) (occasionally, 6 T = T3 - TI); A, epicentral distance (deg); SA = A, - A,, distance difference between events 2 and 1, also vectorial epicenter shift (deg); p, modulus of rigidity (dynes/cm2); T, wave period (sec); IASPEI, International Association of Seismology and Physics of the Earth's Interior; ISC, International Seismological Center; PDE, Preliminary Determination of Epicenters, issued by USGS;USGS, United States Geological Survey (although USGS does not appear on the PDE bulletins before June, 1973, we use this notation for simplicity for the preceding bulletins also). Although SI units are recommended for use in many modern texts, I find it most convenient for the reader to keep the traditional units still used in seismological books and papers. This rule is followed throughout this article.

SELECTED P WAVE PROBLEMS

5

purpose of the present study is to investigatestatistical and physical properties of multiple, especially double, P waves, generated by double events, whereas multiple paths or other path effects are not included. The literature on multiple events can be conveniently divided into two categories. In the first category we find a number of studies of individual earthquakes in which multiple rupture was ascertained. Examples of such studies are offered by Stoneley (1937), Usami (1956), Ryall(1962), Florensov and Solonenko (1963), Gupta (1964), Wyss and Brune (1967), Niazi (1969), Trifunac and Brune (1970), Chandra (1970), Gupta et al. (1971), Wu and Kanamori (1973),Fukao and Furumoto (1975),Kanamori and Stewart (1978), Rial (1978), Deschamps et al. (1982), and Abrahamson and Darragh (1985). Several of these papers also contain brief comments or explanations of the findings. The other category of relevant seismological publications is concentrated on detailed discussions of the underlying phenomena, includingfault structure and fault mechanism, in several cases illustrated from observations. In this category there are papers by Miyamura et al. (1 965), Savage (1965),Aki (1979), Lomnitz-Adler (1985), and Schwartz and Ruff (1985). For empirical studies of multiple P waves there are essentially three ways to collect observational data: 1. For one or a few earthquakes, to collect data from a global network of seismograph stations. This method allows estimation of the shifts in time and location between primary and secondary events. 2. For numerous earthquakes, distributed worldwide, to collect data from a limited group of stations. This procedure permits statistical studies of the properties of multiple P waves. 3. As a combination of methods 1 and 2, i.e., for numerous earthquakes, in all parts of the world, to collect data from a global network of stations. For this study, I have chosen method 2 for several reasons. Partly, the relevant observations are immediately available for the seismograph network operated by our institute at Uppsala. Partly, our purpose is to investigate the general behavior of multiple P waves, for which a statisticallyreliable material is needed. Method 3 is naturally the most complete attack on the problem, but it is so comprehensive that it would need the cooperation of some international organization such as the IASPEI. 1.1. Data Collection 1.1.1. Stations, Instruments, Bulletins

The stations and instruments used in this study are summarized in Table I. All readings were published in the monthly seismological bulletins issued

6

MARKUS BATH TABLE 1.

STATIONS AND RECORDS

Records used'

Coordinates (deg)

Elevation (m)

Uppsala, UPP

59.86N 17.63E

14

Beniof SP

Kiruna, KIR

67.84N 20.42E

390

Grenet-Coulomb SP

Skalstugan, SKA Goteborg, GOT U m d , UME

63.58N 12.28E 57.70N 11.98E 63.82N 20.24E

580 66 16

Grenet-Coulomb SP

Karlskrona, KLS Uddeholm. UDD

56.17N 15.59E 60.09N 13.61E

11 240

Benioff SP

Delary, DEL

56.47N 13.87E

150

Grenet-Coulomb SP

Station, code

Multiple P

-

Benioff SP -

Diffracted P Benioff SP LP, Press-Ewing LP Grenet-Coulomb S1 Galitzin LP Grenet-Coulomb S1 Grenet-Coulomb S1 Benioff SP, Press-Ewing LP Grenet-Coulomb S1 Grenet-Coulomb S1 Benioff SP Grenet-Coulomb SI

a SP, short-period vertical component; LP, long-period vertical component. Multiple P:All stations use were in continuous operation throughout the investigated period 1969-1974. Diffracted P: Some of tt stations were in operation for only part of the investigated period 1965-1974 (for details, see BBth, 197' pp 57-58).

from the Uppsala institute and they were all done in connection with the regular bulletin work. The recording paper speed of 60 mm/min defines the time resolution obtainable at teleseismic distances. Quite clearly, sensitive recordings with high time resolution in the epicentral area of an earthquake could reveal many more details in the development of the focal process. On the other hand, our data could be representative of the achievement in this respect of fairly sensitive stations at teleseismic distances. This means that our readings could probably be verified by many other stations around the world. Only shortperiod or broad-band seismographs will do for the present type of study, while medium- or long-period seismographs have generally too small a resolution. The observational material covers the 6 years 1969-1974. This period is selected for several reasons. It is sufficiently long to provide enough material for reliable statistical investigations of the multiple P wave properties. Moreover, the station network and the instrumentation were unchanged in this period and the readings for the bulletin have been made in a consistent manner. All record readings have been made or examined by the author. Taken together, these aspects guarantee some degree of homogeneity of the observational material. Only visual record readings are used, but more sophisticated analysis methods could be recommended for a further development of this project (Flinn et al., 1973). In our bulletins, double or multiple P waves are denoted in various ways, as

SELECTED P WAVE PROBLEMS

7

follows (to be read vertically within each group): iP1 iP2

iP iX

iP i

i(P) iP

iP il i2

As a general rule, the existence of multiple P waves is specially remarked on in the bulletins. Later P waves should show up as discontinuities (generally increases) of amplitude, whereas cases exhibiting a gradual amplitude increase are not included. 1.1.2. Criteria

Only reliable readings from our records are included in the bulletins as well as in this study. But for our present purpose, this is not enough. As we are concerned with multiple P waves generated only by multiple events (multiple rupture) and not by multiple path propagation and, moreover, as a clear separation has to be made from several other waves (PP, Pn, pP, PcP), our material is subject to a number of criteria, as follows. 1. Only multiple P waves are included, not multiple PKP. Even though multiple PKP waves due to successive events may be as common as multiple P waves, several complications in the wave propagation also lead to multiple P K P arrivals (see, for example, Payo Subiza and Bath, 1964).Therefore, PKP waves are not included in this study. 2. At least two of our stations should have recorded each incident of clear multiple P waves with reasonably agreeing time lags. In this way, we avoid any possibly spurious readings appearing at one station only. 3. The time lag is restricted to 10 sec:

TI 10sec

(1.1)

Naturally, in a succession of events there may be time lags from fractions of a second to much longer intervals. From this viewpoint, the restriction in Eq. (1.1) imposed on our material may be considered as artificial and arbitrary. It is a matter of definition of what should be regarded as a one-event succesion of ruptures or as essentially new events in the same sequence. 4. Events at epicentral distances (A) below 30" are excluded, i.e., our data should fulfill the following condition: A 2 30"

(1.2)

Multiple P waves are frequently recorded at our stations from events within 30" distance, e.g., from the North Atlantic, Iceland, Svalbard, Rumania, Greece, Caucasus. But in this range, especially around 20" distance, there are multiple P arrivals due to structure, especially the asthenosphere low velocity

MARKUS BATH

8

ao"N

80"N

6O0

60"

0"

0"

60"s

60"S

FIG.I . World map (Mercator's projection) showing the distance limits of 30" and 100" from the Uppsala station.

layer (Bath, 1957). Because there could be problems in separating multipleevent P waves from multiple-path P waves in this distance range, except possibly by amplitude ratios, they all have to be excluded. Moreover, by this rule we avoid complication with PP waves at shorter distances as well as with teleseismic Pn waves. In combination with criterion 1, this rule implies a restriction to the general distance range of 30-100" from our stations (Fig. 1). 5. The pP wave offersmore problems than meeting the criteria 1 to 4. The following rule for the time lag has been adopted as a condition for inclusion among our material: 67'5 0.14h N 0.6(pP - P)

(1.3)

Comparison with standard travel time tables (Gutenberg-Richter or Jeffreys-Bullen) demonstrates that by travel time of Eq. (1.3), the time lag 6T is short enough to guarantee that the multiple P wave arrives well before the true pP wave. In a majority of our cases, the factor of (pP - P) in Eq. (1.3) lies in the range 0.3 to 0.5. Quite clearly, for h 5 70 km this is a further restriction on 6T beyond the statement in Eq. (1.1). In applying the condition in Eq. (1.3),we require it to be fulfilled by focal depths given by both USGS and ISC, especially if based upon pP - P, whereas events that fulfill only one of the given data are discarded. Among the items listed here, the pP wave offers the greatest risk of mixing with multiple P waves. We have adopted the time lag 6Tcompared with pP - P as the most reliable criterion. Amplitude ratios and phase changes are not conclusive. Byerly et a!. (1949) developed a method to calculate the amplitude ratio PP/P, which I applied to corresponding calculations of pP/P (unpublished). The results demonstrate large variations of this amplitude

SELECTED P WAVE PROBLEMS

9

ratio, primarily due to the source mechanism, as well as phase changes. This method does therefore not allow any separation between p P and multiple P. Another method could be envisaged, based on the different angles of incidence of P and pP at a station. But this method would need combination of all three short-period components, which in our network are available only at U P P and UME. Moreover, the difference of the angles of incidence is not large enough to provide a reliable discrimination. 6. Confusion with PcP waves must be avoided at the greatest distances, about 80" and over. In general, it is easy to exclude events where a suspected multiple P wave may instead be a true PcP wave just by using available travel time tables. 1.1.3. Observutional Material In a first data sampling from our bulletins for the period 1969-1974, we arrived at a list of 444 earthquakes with suspected multiple P waves. Each one of these events was then subjected to the criteria in the previous section. The final list includes 153 earthquakes which passed all the criteria, This list could be considered to be as reliable as possible, including only multiples due to multiple rupture, and can be used for further analysis. There are probably several more multiple rupture P waves among the 444 earthquakes, but these have to be excluded because of unsatisfactory identification. The accepted 153 earthquakes are reported in full in Appendix 1. A typical case is reproduced in Fig. 2. Some explanations of Appendix 1 will be given here. Origin times and epicentral coordinates are quoted from USGS, whose data generally show good agreement with those of ISC. The focal depths are determined from pP - P at our stations, if available. If not, the depths from USGS are quoted. They are given to the nearest 5 km for h I 60 km and to the nearest 10 km for greater depths. For magnitude, the body wave estimate m is preferred since we are concerned with P waves. It is determined as follows: 1. If rn is available from our bulletins (an average of UPP and KIR), this value is reported. 2. If m(UPP,KIR) is not available, but M is given in our bulletin (an average of U P P and KIR), then m is calculated by the following formula, provided h I 50 km (Bdth, 1977):

m

= 0.55M

+ 2.94

(1.4)

3. In all other cases, rn is obtained from m(US) by the following expression, valid for all h (Bath, 1977): rn = 1.24m(US)- 0.70 (1.5)

10

MARKUS BATH

FIG.2. Enlarged record of double P (Pl, P2) recorded at UDD from an earthquake in China on July 18, 1969 (No. 10 in Appendix 1).

Equations (1.4) and (1.5) are both orthogonal regressions, which means that they can be inverted. Of the data tabulated in Appendix 1, epicentral coordinates, focal depth, and origin time are assumed to refer to the first arriving P wave, while magnitude refers to the maximum P in the group of P arrivals. In some cases there are more than two P arrivals. All these were subject to the same criteria cited in the previous section. It is to be expected that most arrivals later than the second P wave do not pass the criteria. In fact, only three such events remained (Nos. 76, 87, and 100 in Appendix 1). Therefore, our following analysis will only be concerned with the first multiple. Moreover, we may use the term “double P as synonymous with “multiple P.” 1.2. Data Analysis and Results Our observational data consist of some dependent parameters, i.e.,the time lag GTand its standard deviation SD, to which we shall add the magnitude difference 6m.The occurrence frequency and the numerical value of these parameters will be discussed as functions of several independent earthquake parameters, i.e., focal depth h, magnitude m, and epicentral region. First of all, we give some statistics of the stations involved.

SELECTED P WAVE PROBLEMS

11

1.2.1. Station Statistics

It is evident from Table I1 and Fig. 3 that a double P wave is usually recorded by four to six stations in our network, in fact in 77% of the investigated cases. On the other hand, the minimum number of stations recording, i.e., only two, occurs in less than 5% of the cases. There are several reasons behind this result. One is the event magnitude. It is quite clear from Table I1 that as the magnitude increases, more and more stations will show a given double P wave. Generally speaking, it is the signalto-noise ratio that is of crucial importance. Not only does the noise background vary from time to time but at any given time, it may vary considerably from station to station. TABLE 11. STATION STATISTICS (FIG.3) Number of stations recording

Percentage frequency, N = 153

Average magnitude

2 3 4

4.6 18.3

5.1 f 0.5 6.0 k 0.5 6.0 k 0.7 6.2 _+ 0.5 6.3 f 0.4

5

26.1 22.9

6

28.1

Station recording

rn

1

Uppsala Kiruna Skalstugan UmeA Uddeholm Delary

i

I

30

20 10

0

ill--!/ NUMBER OF STATIONS

100 - N%

--

-

50 -

0

- --

7

-

UPP KIR SKA UME UDD DEL

Percentage frequency, N = 153

83.0 19.I 58.8 83.7

92.2 54.9

12

MARKUS BATH

The frequency of occurrence of double P with regard to the stations (righthand part of Table I1 and lower part of Fig. 3) clearly demonstrates that the stations can be divided into three groups of different sensitivity: Highest sensitivity: Uddeholm Medium sensitivity: Uppsala, Kiruna, Umel Lowest sensitivity: Skalstugan, Delary It is of interest to note that this scale agrees well with a special investigation of the signal sensitivity of Swedish seismograph stations (Bath, 1965; see also Bath, 1979, pp. 249-251, where explanations for the different sensitivities are also provided). In the following, all stations are treated together for each event, i.e., the time lags 6T are averages for each event (mean i-SD in Appendix l), being more accurate than single station data. 1.2.2. Time Lag 6T 1.2.2.1. Frequency of occurrence. Double P is generally recognized as a common phenomenon, even though USGS reports “complex multiple rupture” only occasionally. Thus, for the period 1969-1974, we find such USGS remarks in the PDE bulletins for the following events:

Mar 31,1969, Sea of Japan (No. 5 in Appendix 1) Aug 11, 1969, Kurile Islands Jun 11, 1970, Macquarie Islands Nov 18,1970, Bismarck Sea Sep 30, 1971, Gulf of California Jul 30, 1972, Alaska (“aftershocks indicate at least 75 km of rupture SE from hypocenter along Fairweather fault,” No. 74 in Appendix 1) Jun 15,1973, off east coast of Kamchatka Except for Nos. 5 and 74, these events are excluded from our list as they do not pass all our criteria. It is immediately obvious from our data that double P waves are a frequent occurrence. Appendix 1 reports 153 cases for 6 years, i.e., on average about two clear events per month. There are probably many more which could not be included here because they would need a more detailed examination at numerous stations. However, triple P waves or greater are much less frequent than double P. The frequency distribution of the time lag ST for our sample is presented in Table I11 and Fig. 4. Clearly, most of the 6 T values lie in the range of 1.6-4.0 sec, in fact for about two-thirds of our data sample. The maximum frequency is found in the range of 2.6-3.0 sec. The total average with its standard

13

SELECTED P WAVE PROBLEMS TABLE 111. FREQUENCY DISTRIBUTION OF THE TIME LAG (FIG.4) Time lag ST(sec)

Percentage frequency, N = 153

Time lag 6T(sec)

Percentage frequency, N = 153

0.1-0.5 0.6-1.0 1.1-1.5 1.6-2.0 2.1-2.5 2.6-3.0 3.1-3.5 3.6-4.0 4.1-4.5 4.6-5.0

0.0 5.2 7.8 13.1 12.4 16.3 11.8 14.4 7.2 4.6

5.1-5.5 5.6-6.0 6.1-6.5 6.6-7.0 7.1-7.5 7.6-8.0 8.1-8.5 8.6-9.0 9.1-9.5 9.6-10.0

3.3 2.0 0.7 0.0 0.0 0.6 0.0 0.0 0.0 0.6

I

I

5

io

N%

0

FIG.4. Histograms showing the percentage frequency distributions of the time lag 6T (Table 111) and of its standard deviation SD (Table VIII). Mean values are indicated by dots.

14

MARKUS BATH

deviation is 6T = 3.0 & 1.4 sec,

N = 153

(1.4)

There are no observations for 0.1-0.5 sec, which is due to the limited time resolution of our records. It is important to observe that this frequency distribution applies only to our material, i.e., a data set that has passed all the criteria in Section 1.1.2. Any other selection of material would naturally lead to other frequency distributions. The possible bias of our data selection is an important point to which we shall return in Section 1.3. 1.2.2.2. Relation to focal depth. The frequency distribution of the double P waves with regard to focal depth at first appears to show a strong weighting in the depth range h = 0-100 km (Table IV). The most impressive dominance is found for h = 26-50 km (62.1%), followed by h = 51-75 km (15.7%) and h = 0-25 km (7.2%). However, for a correct interpretation of this frequency distribution it should be compared with the general depth distribution of earthquakes. For this purpose, two time intervals are used (Jan-Feb 1969 and Jan-Feb 1974), for which we collected from our bulletins all events with assigned focal depths in the general distance range of 30-100" (Table IV). This selection of comparison material appears most appropriate, using the same station network, the same epicentral areas, and the same magnitude range. It is then obvious that the frequency-depth distribution of the double P waves of our sample shows no significant deviation from the corresponding distribution of all earthquakes. In fact, the similarity between the double P distribution and the earthquake distribution for Jan-Feb 1969 is even

Frequency distribution (%) Earthquakes (Uppsala bulletin) Depth h(km)

Double P,

0-100 101-200 201-300 301-400 401-so0 501-600 601-700

88.2 5.9 0.0 2.0 1.3 2.0 0.6

N,

= 153

Jan-Feb 1969

Jan-Feb 1974

N , = 224

N2 = 221

88.4 6.3 0.9 0.9 2.2 0.9 0.4

81.0

11.9 3.1 2.2 0.9 0.9

0.0

Depth h (km)

No.of double P in N ,

ST (sec

0-25 26-50 51-75 76-100 101-200 201-300 301-500

11 95 24 5

2.3 f 0. 3.0 2 1. 3.3 & 2. 3.7 f 1. 2.7 f 1. 3.5 & 2. 3.4 5 1.

Sol-700

9

0 5 4

Time la

SELECTED P WAVE PROBLEMS I

6

I

I

I

I

15

I

I

I

I

T

& T sec

I

I

1

I

600 FIG.5. Values of the time lag 6T in relation to focal depth h (Table IV). The vertical bars indicate respective standard deviations. 200

400

striking. The conclusion is therefore that the proportion of double P, i.e., the number of cases with double P relative to the total population, is independent of focal depth. In addition, the average values of the time lag 6T are calculated for different depth ranges (right-hand part of Table IV and Fig. 5). There appears to be an increase of 6T with h for h I 100km, but at least down to k = 70 km this is just a spurious effect of our data selection, especially the restriction expressed in Eq. ( I .3). Disregarding this feature, the time lag is independent of focal depth, uninfluenced by any p P waves. The average of the values in Table IV and Fig. 5 is 6T = 3.1 t- 0.5 sec (N = 7), to be compared with the total average in Eq. (1.6). 1.2.2.3. Relation to Magnitude. The frequency distribution of double P waves with regard to magnitude is presented in Table Va and Fig. 6. There is an obvious discontinuity of the material at m = 6.3-6.4. For lower m, the frequency drops off for several reasons: not all earthquakes are recorded, and even if recorded, the multiple P will appear less clear due to weak records. Moreover, smaller events are more seldom multiple. For larger m,our data set of double P waves is expected to be complete. A least squares solution for m 2 6.5 for the frequency N1 of double P waves yields the following result (Fig. 6):

log N1

= (9.54 f

0.16) - (1.23 f 0 . 0 2 ) ~

(1.7)

[56 events, 4 data points] Eliminating m from this equation by means of Eq. (1.4), we find log N ,

=

5.92 - 0.68M

(1.8)

MARKUS BATH

16

TABLE V. RELATION TO MAGN~NDE a. Double P frequency and time lag (Figs. 6 and 7 a ) ; earthquake frequency

Magnitude m

4.9-5.1 5.2-5.4 5.5-5.7 5.8-6.0 6.1-6.3 6.4-6.6 6.7-6.9 7.0-7.2 7.3-7.5

Earthquake frequency 1969-1974 in N z

No. of double P in N,

Time lag 6T (sec)

Magnitude

M

Worldwide

8 9 25 30 25 33 11 10 2

3.4 f 0.7 4.6 f 2.4 3.3 f 1.4 2.8 f 1.0 2.9 f 1.4 2.6 1.2 3.5 f 1.4 2.9 f 0.8 1.6 f 0.6

7.0-7.2 7.3-7.5 7.6-7.8 7.9-8.1 8.2-8.4 8.5-8.7

55 29 26 12 1 1

A

= 30-100"

35 16 21 9 1 1

b. Frequency of events with assigned magnitudes in the distance range 30-100" in the Uppsala seismological bulletin for 1969 and 1974

Magnitude

Earthquake frequency in

Magnitude

Earthquake frequency in

m

N2

m

N2

5.2 5.3 5.4 5.5 5.6 5.7 5.8

1 2 3 9 28 42 36

5.9 6.0 6.1 6.2 6.3 6.4

105 92 91 68

40 36

Compared to the b coefficient of 1.00 for all earthquakes of M 2 7.0 as found by BAth and Duda (1979), it appears as if the decrease of log N, with increasing M in Eq. (1.8) would be significantly slower. However, instead of comparing with a result valid for all earthquakes of M 2 7.0 in the world and for a long period of time, we should make a more specific comparison (righthand part of Table Va). Thus, the catalog of BAth and Duda (1979), for the interval 1969-1974 and M = 7.0-8.0 only, yields the following least squares solution: lOgN2 = (6.51 f 0.08) - (0.67 f 0.01)M

(1.9)

[122 events, 4 data points]

After an additional restriction to events within the distance range of 30-100"

SELECTED P WAVE PROBLEMS

17

FIG.6. (a) Logarithmic frequency distribution of the time lag 6T in relation to magnitude M (Table V); (b) Schematic graph demonstrating the relative slopes of log N for double P (full lines) and earthquakes (dotted lines).

from our network, the Bath and Duda (1979)data give the following solution: lOgN2 = (5.41 f 0.12) - (0.55 & 0.02)M

(1.10)

[81 events, 4 data points]

Then it appears that the b coefficient for the double P, Eq. (1.8), agrees with the b coefficient for the earthquake material, as defined. The agreement with Eq. (1.9) is almost perfect. The conclusion is that the proportion of double P, i.e., the number of events with clear double P relative to the total population, is independent of magnitude, at least for larger m. The relatively low value of the b coefficient in Eqs. (1.9) and (1.10)could be explained as due to a superposition of several log N vs M curves, each of a more complicated shape, as developed for example for Turkey in BAth (1981a). The development so far concerns only m 2 6.5. For lower m, some other earthquake material must be used for comparison with the double P frequency. We have chosen the frequency of events in the distance range 30100" with assigned magnitudes in the Uppsala bulletin for 1969 and 1974

18

MARKUS BATH

(Table Vb). With N, equal to the double P frequency and N, representing earthquake frequency, the following results are obtained by plotting log N, vs rn (Fig. 6b): for rn = 6.0-6.4 (bulletin data complete), N1/N2 increases with increasing m; rn = 5.1-5.9 (bulletin data incomplete), N1/N2 decreases with increasing m. The increase of N1/N2with m in the range m = 6.0-6.4 could be a result of two cooperating factors: partly a true increase of double P frequency for larger earthquakes due to their fault mechanism, partly an easier discovery of double P for larger events on the records. For the range rn = 5.1-5.9, the decrease of N1/N2with rn reflects a more rapid increase of the total number of recorded events, which can be tentatively explained by the difficulty of discovering double P for low magnitude earthquakes. This is a suggestion based on an extensive experience from record readings and measurements. It is a general impression, also stated in the literature, that double and multiple P waves are more frequent for large magnitude earthquakes. However, besides a true source effect there is also an increasing frequency of double P due to the clarity of the records, which is also magnitude dependent.

-

-\* 3Eq. (1.12)

6

7

FIG.7. Values of the time lag ST in relation to magnitude m: (a) from Table V; (b) from Table VI.

19

SELECTED P WAVE PROBLEMS

This latter effect influences the results up to m = 6.4. Beyond this limit, we have found that N J N , remains constant, independent of magnitude. In addition to frequencies, Table Va also presents the average values of the time lag for a sequence of magnitude ranges (Fig. 7a). In spite of some scatter due to various other influences, there is a clear decrease of 6T with increasing m. A least squares solution for the data in Table Va and Fig. 7a yields the following equation: 6T

= (7.23

f 0.59) - (0.67 f 0.10)m,

N

=9

(1.11)

A probable explanation could be that increasing magnitude implies increasing stress with the consequence that sequences of shocks occur at shorter time intervals. At the same time, the dependence of 6T on m is evidence against explanations in terms of path effects. The rupture velocity, beyond our present control, is usually assumed to be about 3 km/sec. However, the rupture propagation would deserve a more detailed investigation, especially with regard to its possible dependence on magnitude, source displacements, and other parameters. 1.2.2.4. Relation to region. When studying relations of double P and its 6T to focal depth and magnitude, we divided our data in Appendix 1 into seven to nine groups (Tables IV and V). This was found to be an appropriate and natural subdivision of our data in order to yield reliable and representative values for each group. Consistently with this procedure, our material is now divided into nine larger regions (Table VI). Table VI gives reference numbers to Appendix 1, and thus it also serves as a regional index of Appendix 1. In order to judge the frequency of occurrence of double P with regard to region, we collected from the Uppsala bulletin the corresponding information for all earthquakes, partly for Jan-Jun 1969, partly for Jan-Jun 1974 (Table VT). As can be seen by a comparison of the two data sets, there are quite large variations from time to time, and much longer intervals would be needed for more stable results. In a comparison of the double P frequency with the earthquake frequencies of 1969 and 1974, there are clear indications of a higher proportion of double P for regions 2 and 4, whereas the other regions exhibit double P frequencies in fairly good agreement with the overall seismic activity. With regard to the average regional values of the time lag 6T, Table VI exhibits some differences from region to region. However, these differences are just an expression of the regional averages of the magnitude m. Relating the average 6T to the average m for each region by the least squares method, the following solution is obtained (Table VT and Fig. 7b):

6T= (6.47 & 0.41) - (0.57 & 0.06)m,

N =9

(1.12)

TABLE VI. RELATION TO

&GlON

(FIG.

7b) Frequency (%)

Region

No. of double P in N ,

1. Atlantic 2. Middle East

5 20

3. Central Asia

21

4. China

14

5. Indonesia-

26

Philippines 6. Japan

27

7. Kurile-

16

Kamchatkil 8. AleutiansAlaska 9. America a

13

Appendix 1 no. 7,31,90, €16,128 4, 6,26,42,48,49, 58, 59, 62, 70, 71, 72, 88, 93,94,95, 101, 114, 121, 130 17, 18,21, 30,41,44,45, 55, 65,68, 98, 125, 133,139, 140, 141, 142, 143, 144, 151, 153 10,23,46, 51,87, 89, 102, 104, 110, 112, 113, 115,131,132 1, 2, 15, 20, 27,29, 33, 37, 38, 47, 61,63,64, 66,67,79, 80, 81, 84, 92, 97, 118, 127, 129, 145, 146 5, 22.28, 32, 36, 50, 52, 53, 56, 73, 77, 78, H3,86, 100, 106, 107, 108, 117,122, 123, 124, 126, 138, 148, 149, 150 11, 12, 13, 14, 16, 19, 24, 34,76,82, 105, 111, 119, 135, 136, 137 8, 25, 35, 39,40, 54,60, 74, 75, 96, 103, 120,

Double P, inN, = 153

Earthquakes" 1969

1974

Magnitude m

3.3 13.1

4.6 4.5

7.4 4.1

5.5 f 0.5

3.5 0.8 3.2 f 1.1

13.7

10.7

13.9

5.7 kO.5

3.2 f 1.3

9.1

0.8

3.7

6.1

0.4

2.8 f 1.0

17.0

30.8

19.9

6.5 f 0.4

3.3 f 1.3

11.6

17.8

21.0

6.3 f 0.5

3.2

+ 2.0

10.5

8.2

11.2

6.3 f 0.5

2.5

1.3

8.5

11.0

10.7

6.4 f 0.4

2.1

7.2

11.6

8.1

k 0.5

3.3

5.7

152 11

3,9,43, 57, 69, 85.91.99, 109, 134, 147

Earthquakes in the Uppsala bulletin for Jan-Jun 1969 (N2= 764) and Jan-Jun 1974 (NZ= 811).

6.2

*

0.2

Time lag 6T (sec)

*

* *

1.2 1.2

TABLE VII. DOUBLE P: ALMOST IDENTICAL EPICENTERS Depth Appendix 1 no. 4 6 48 49 72 58 59 62 88 94 95 139 140 141 142 143 144

44 45 23 113 89 115 132 102 104 20 27 80 81 61 63 121 56 83 100

I48 149 52 53 135 136 85 91

Magn. rn

Timelag 6T(sec)

7.0 5.3 5.5 5.6 5.5

2.4 4.8 4.3 2.9 3.6 2.9 2. I 2.0 4.0 3.1 4.1 3.4 2.4 3.2 2.9 2.4 3.4 3.7

Epicenter

h(km)

27.7'" 34.O"E 27.6 34.0

35 35

30.2 30.1 30.1 28.4 28.4 28.3 32.0 32.1 32.0 39.3 39.5 39.5 39.4 39.3 39.4 41.5 41.3 23.1 23.2 31.5 31.5 31.8 41.0 41.0 6.8 6.8 6.5 6.5 13.4 13.5 13.8 33.3 33.2 33.4 41.0 40.9 22.5 23.0 46.3 46.2

55 45 30 35 35 35 50 35 35 35 35 35 35 35 35 35 45 35 35 35

5.7 6.5 5.3 6.2 5.6 6.5

35

5.8

35 35 35 70 55 35 55 55 55 I 20 55 55 70 55 45 35 35 60 40 45 35

6.0 5.8

50.7 50.8 50.8 52.9 53.0 53.0 49.4 49.4 49.4 73.8 73.9 13.1 13.1 14.0 73.9 79.3 79.3 100.8 100.9 100.3 100.0 100.1 82.2 82.3 126.7 126.7 126.6 126.6 120.6 120.5 120.6 140.8 140.8 140.7 143.1 143.1 122.3 122.2 153.3 153.1

18.5"N 103.0"W 18.9 103.5

5.1

5.6 5.4 5.7 5.4 4.9 5.8 5.3 6.1 5.0 5.5

5.5

7.1 7.1 7.2 6.5 5.7 6.1 5.6 1.4 5.6 6.0 6.4 6.6 7.2 6.8 1.5 6.1

6.9 6.1

21

Region Red Sea Iran

Iran

Iran

Tadzhik- Sinkiang

Kirghiz- Sinkiang

5.8

2.7 3.9 3.8 3.0 3.8 2.0 2.0 3.4 3.2 2.9 3.2 5.1 1.2 1.8 2.0 3.4 1.o 2.0 5.0 2.2 2.2 1.2 3.3 3.8 3.2

China China

China Mindanao

Mindoro

Japan

Japan Taiwan Kurile Islands Mexico

22

MARKUS BATH

The coefficient of rn shows good agreement with Eq. (1.1 l), which is natural as they are both based on the same data, only differently arranged. On the other hand, there is no consistent variation of ST with epicentral distance. As variations exist even within a given region, it becomes especially interesting to compare results for identical or almost identical epicenters (Table VII). The obvious and significant variations of 6T within each group are due to multiple events beyond doubt, as multiplicity due solely to multiple paths or any other path effect would result in constant 6T for any given source location. Nor is it possible to explain the observations in terms of pP. Local or regional structural properties may influence 6T. For example, in Iran, the second group shows a lower 6T than the first and third groups. China is another example of clear differences between the first two groups and the third one. 1.2.2.5. Standard deoiation. Besides 6T itself, its standard deviation (SD) may also provide some interesting information, summarized in Table VIII. In each individual case, SD is a measure of the scatter of the observations. This scatter is due partly to unavoidable measuring errors, partly to real differences between stations. With a recording paper speed of 60 mm/min, sharp onsets can be measured to the nearest 0.1-0.2 sec. Hence, the difference 6T between two measurements on a record may even in the best case be in error by 0.2-0.3 sec. Beyond this limit, somewhat less evident onsets may contribute to the SD, in addition to real differences due to shifts of hypocenter location between the first and second events. The frequency distribution of SD (Table VIII and Fig. 4) shows a rather strong concentration in the range of 0.1-0.4 sec with a clear maximum at 0.2 sec, suggesting that the majority of our SD is due to inevitable measuring errors. The total average is SD = 0.36 & 0.26 sec (N = 153). Only a smaller percentage of SD is due to real differences. A contributing factor is naturally the relatively limited extent of the station network used, globally speaking, and only more significant hypocenter shifts will result in noticeable differences of 6T from station to station. Table VIII further shows that the value of SD is practically independent of focal depth, except for some indication of an increase with h for h 5 75 km. In contrast, there is a clear increase of SD with increasing time lag 6T, which is best represented by two straight lines of different slope (Fig. 8). The following least squares solutions are obtained: 6 T I 2.55 sec SD = (0.120 f 0.000)6T+ (0.074 f O.OO0)

[84 events, 3 data points]

(1.13)

TABLE VIII. STANDARD DEVIATION SD OF THE TIMELAG(FIGS.4

(sec)

Percentage frequency in N = 153

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o 1.1 1.2 1.3

0.7 15.0 26.1 19.6 14.4 7.2 3.3 4.6 2.6 1.3 1.3 2.0 1.3 0.6

SD

Depth

SD

Magnitude

SD

8)

SD

h (km)

N

(sec)

m

N

(sec)

Region

0-25 26-50 51-75 76-100 101-200 201-300 301-500 501-700

11 95 24 5 9 0 5 4

0.31 0.36 0.40 0.36 0.26 0.52 0.28

4.9-5.1 5.2-5.4 5.5-5.7 5.8-6.0 6.1-6.3 6.4-6.6 6.7-6.9 7.0-7.2 7.3-7.5

8 9 25 30 25 33

1 2 3 4

10 2

0.33 0.36 0.35 0.34 0.39 0.38 0.26 0.54 0.15

76-700

23

0.34 6.7-7.5

23

0.37

11

AND

5

6 7 8

N

(sec)

5

0.66 0.32 0.27 0.39

20 21 14 26 27 16 13

0.44 0.36 0.38 0.21

Time lag 6T(sec)

0.1-1.0 1.1-2.0 2.1-3.0 3.1-4.0 4.1-5.0 5.1-6.0

SD N

(sec)

(%)

8 32

0.14 0.26 0.38 0.40 0.42 0.46

25.5 16.8 14.9 11.3 9.2 8.3

44 40 18

8

24

MARKUS BATH

I

I

0

I

I

2

I

I

6

4

FIG.8. Values of the standard deviation SD of 6T in relation to the time lag 6T (Table VIII).

6T 2 2.55

sec

SD = (0.026 f 0.001)bT+ (0.310 f 0.006)

(1.14)

[llO events, 4 data points] Equations (I. 13) and (1.14)provide an explanation of the slight increase of SD with h for h < 75 km, as just mentioned. When SD is expressed as percentages of dT, there is a clear decrease with increasing 6T (Table VIII). Table VIII further suggests that SD is practically independent of magnitude m. However, this is most likely an apparent independence, resulting from cancellation of opposing effects. SD is a function f of time and space shifts, i.e., 6T and dA:

SD = f(dT, 6A)

(1.15)

Differentiating Eq. (1.15) with regard to m, we find dSD L-,--.-2-

>o

->o

(1.16)

>o

With larger m,a larger volume is subject to increased stress (Bith and Duda, 1964),and then the hypocentral shift is expected to be generally larger than for smaller rn, i.e., d6A/dm > 0. The net result of Eq. (1.16)is that different effects tend to cancel each other at least partially, leading to an apparent SD independence of m. With regard to epicentral regions, Table VIII demonstrates a fairly uniform distribution of the standard deviation, apart from its maximum of SD = 0.66 in region 1 and its minimum of SD = 0.21 in region 8. But there are variations

SELECTED P WAVE PROBLEMS

25

also within the assigned regions. For example, Mindanao in region 5 exhibits an SD = 0.64 (N = 8). The regional variations no doubt reflect a complicated dependence on source and receiver properties, whose detailed exploration would only be possible by method 3 as mentioned in the beginning of Section I. 1.2.3. Magnitude Difference 6m Besides the time lag 6T, the corresponding magnitude difference 6m is of importance. However, with a few exceptions the Uppsala bulletin reports amplitude and period only for the largest P wave in the group. Therefore, for a solution of this problem we use the annual seismological bulletin of the Moxa station (50.65"N, 11.62"E) for the interval 1970-1974, produced by Stelzner et al. (1976-1979), one of the few bulletins nowadays with detailed information. Searching the Moxa bulletin for our 134 events in the period 1970-1974, we find common reports of double P in 33 cases (25%)and of these, the time lags 6T agree to the nearest sec in 76%. On the other hand, the Moxa bulletin reports several other cases of multiple P, some of which are excluded by our criteria. Nevertheless, it is obvious also from the Moxa bulletin that multiple P is of frequent occurrence. The earthquakes common to our Appendix 1 and the Moxa bulletin, and with magnitude data (24 in all), are reported in Table IX. The values of 6m range from 0.0 to 1.0 and the frequency distribution is given in Table IX. Typically, there are no negative am,i.e., practically all cases included show a smaller amplitude of the first onset followed by a larger amplitude of the second one. The limit, i.e., 6rn = 0 or same amplitude of P1

No."

Sm

No."

Srn

Srn

N

Region

N

20 27 30 36 38

0.5 0.5 0.3 0.1 0.5 0.6 0.6 0.0

72 74 76 80 82 85 89 103 118 I34 141 149

0.6 0.0 0.3 0.6 0.7 0.3 0.2 1.0 0.8 0.4 0.4 0.9

0.0 0.1 0.2 0.3

2 2 1 3 2 4

1 2 3

0 2 4 2

41

43 46 55 57 66 71 a

0.9 0.9 0.5 0.1

Refers to number in Appendix 1

0.4

0.5 0.6 0.7 0.8 0.9 1.0

4

1 1

3 1

4 5

6 7 8 9

6

2 2 1 5

Sm

m

-

5.7 5.5 5.7 6.1 6.5 6.3 6.3 6.4 6.2

0.35 0.55 0.10 0.57 0.50 0.50

1.00 0.44

MARKUS BATH

26

and P2, is a rare occurrence. The average 6m is as follows:

6m = 0.5 k 0.3,

N = 24

(1.17)

An explanation of the double P in terms of Ps is excluded by this result (see Section 2), as well as by the observed time lags. An explanation in terms of pP is equally excluded, because 6rn is then expected to center around zero. With the numerical values of 6m,the corresponding values of the amplitude, energy, and moment ratios between events 2 and 1 can be deduced. Starting from the following equation for m,

m = log(a/r) + q(A,h)

(1.18)

we get

6m = m, - rn, = log(a,/r,)

-

log(a,/z,)

N

log(a,/a,)

as z1 N z,

(1.19)

The material in Table IX yields the following average amplitude ratio: a,fa,

=

3.8,

N = 24

range = 1-10

(1.20)

This average ratio agrees well with the Uppsala bulletin in the few cases where both amplitudes are given. For the corresponding energy ratio, we combine the following well-known equation, log E = 12.24 + 1.44M (1.21) with Eq. (1,4),leading to the following relation: log E = 4.54 + 2.62m

( 1.22)

Equation (1.22)gives the desired expression for the energy ratio: log(E,/E,) = 2.62 6m

(1.23)

The following average holds for our material in Table IX:

N = 24 E , / E , = 66, range = 1-417

( 1.24)

This result suggests quite a significant average energy ratio between the two successive events. Finally, for the seismic moment M, we combine the following equation, log M, = log(pDA) = a + PM

(1.25)

with Eq. (1.4),which leads to the following expression, log[(DA),/(DA),] = 1.83p 6m = 3.0 6m

(1.26)

SELECTED P WAVE PROBLEMS

27

'1

I

I/,]

0

Eq. (1.28)

0.5

m

0

0

5 6 7 FIG.9. Values of the magnitude difference 6m in relation to magnitude m (Table IX).

if B = 1.7 (see Bith, 1981c, pp. 370-371). Table IX yields the following average ratio of the seismic moments: (DA),/(DA), = 139,

range

=

N

= 24

1-1000

(1.27)

This result gives an idea of the ratio of the fault plane areas A,/A, for a constant displacement D , = D,,or alternatively the ratio of the displacements D,/D, for a constant fault plane area A, = A , . In general, both A and D are expected to vary from event 1 to event 2. Plotting individual values of 6m against focal depth h, we find no indication of any relation. In other words, am is independent of h. On the other hand, 6m exhibits a dependence on magnitude m,but the scatter is large due to various other influences. For this reason, individual observations could hardly be used, but instead the average values of 6rn and m are related for the nine regions of Table VI. From the data of Table IX the following least squares solution is obtained (Fig. 9):

6m = (0.28 & 0.04)m - (1.23 f 0.23) [24 events, 8 data points]

(1.28)

Some dependence on region, exhibited by 6m,is explained by this relation of 6m to m, whereas path effects or influence of other waves are excluded. Numerical values are summarized in Table X. The following comments need to be made regarding the development in this section: ( 1 ) The observational material ( N = 24) may seem too small to get representative results. Therefore, a test was made with the Moxa data in 19701974 with reported m, and m,, guided by our criteria of Section 1.1.2. In all, 151 cases were thus collected, yielding an average of 6m = 0.5 0.3, i.e., in exact agreement with Eq. (1.17). Moreover, 95.4% of the larger material ( N = 151) belong to the range of 6m = 0.0-1.0. Therefore, our results for the

28

MARKUS BATH

DIFFERENCE, AMPLITUDE, ENERGY, AND SEISMIC MOMENT TABLEX . TIMELAG, MAGNITUDE RATIO.CALCULATED AS FUNCTIONS OF MAGNITUDE (FIG.1 1) Magnitude difference 6m Eq. (1.28)

%la,

E2I4

(W*I(W,

m

Time lag 6T (sec) Eq. (1.11)

Eq. (1.19)

Eq. (1.23)

Eq. (1.26)

5.0 5.5 6.0 6.5 7.0 7.5

3.9 3.6 3.2 2.9 2.5 2.2

0.2 0.3 0.5 0.6 0.7 0.9

1.5 2.0 2.8 3.9 5.4 7.4

2.8 6.5 15 35 82 190

3.2 8.5 22 59 155 407

Magnitude

smaller material ( N = 24) can be considered as truly representative. (2) The magnitude difference 6m is determined from Moxa records, while m is identical with m(UPP, KIR). However, to judge by the response curves used at the stations mentioned, there is no reason to expect any significant influence on the magnitude differences, i.e., 6m determined from Moxa records should be essentially the same as determined from UPP, KIR. Gupta and Rastogi (1972) studied the relation between m and M , corresponding to our Eq. (1.4), in dependence upon source multiplicity. It is then important to know if their m refer to the first P or the maximum in the P group. If m refers to the first P, larger ratios of M / m are natural for multiple events. We have consistently used m as referring to the maximum in the P wave group.

1.2.4. Miscellaneous Aspects In addition to the results already reported in Section 1, there are a number of additional aspects on multiple or double P, which will be summarized here. 1.2.4.1. Wave periods and phases. For events within about 30" from our stations (North Atlantic, Iceland, Svalbard, Rumania, Greece, Caucasus), double P is a general feature, with the first P of longer period and lower amplitude than the second one. These cases are probably due to path effects and are excluded by our criteria. For double P due to double events, on the other hand, there is no general difference of the wave period between the first and second P, as witnessed by both our data and those of the Moxa bulletins. Of the data in Appendix 1, only Nos. 28 and 108 show definitely a longer period of the first P than of the second one. Comparison of phase (compression vs dilatation) of the first and second P

SELECTED P WAVE PROBLEMS

29

can be made reliably only in a few cases because of the general weakness of the first onset. A priori, phases would be expected to be the same in successive events repeated at close intervals. However, event No. 135 appears to be an exception where the initial relatively weak compression is followed after 1.2 sec on average by a strong dilatation. The few relevant cases reported in the Moxa bulletin suggest that phase reversals are equally frequent as unchanged phases (see also Savage, 1965). 1.2.4.2. Later multiples. Double or multiple events are expected to produce double or multiple onsets not only for P but for all following waves as well. However, there are remarkably few cases where double or multiple onsets are clearly seen for the later waves. A contributing reason is naturally the already existing motion in a seismogram which tends to hide small amplitude precursors. Another contributing factor is the generally longer periods of the following waves which tend to smooth out minor details. Replacing the presently used visual examination and reading of records by more sophisticated techniques could possibly lead to an improvement in this respect (Flinn et al., 1973). In a comparison of different double waves at a given station or the same double wave at different stations, it is helpful to start from the following obvious equation: (1.29) T = 0 t(A,h)

+

A small shift in origin time and hypocenter location is expressed by the following difference equation, which applies Taylor’s series expansion:

6~ = 60 + (at/aA), 6~ + (at/ah), 6 h (1.30) Assuming unchanged focal depth, 6h = 0, Eq. (1.30)simplifies to the following: 6T = 6 0

+ (at/aA),sA

(1.31)

An interesting example is offered by event No. 1 in Appendix 1. Besides double P, there are double P P with the following time lags: UPP 5.0 sec, KIR 5.7 sec, UME 6.0 sec, on average 5.6 sec. Application of Eq. (1.31)to this case with 6 0 and 6 A as the unknowns (for simplicity, 6A is assumed to be constant for our network) yields the solution 6 0 = 0.1 sec and 6A = 0.7”. Another example is offered by No. 5, where in addition to double P at all our stations, U P P and KIR show a double S with a time lag practically the same as for P, i.e., nearly 3.7 sec. Equation (1.31) leads to the solution 6 0 = 3.7 sec and dA = O.W, i.e., a second shock 3.7 sec later in the same location (actually, at the same distance). This conclusion is supported by the USGS in the PDE bulletin: “Two shocks apparently occurred within extremely small intervals of time and space. The smaller event occurred about 3.5 sec before the larger quake.” (See also Abrahamson and Darragh, 1985.)

30

MARKUS BATH

FIG.10. Sketch demonstrating the epicentral shift from Epl to Ep2 and the ensuing change of epicentral distance. N, North.

1.2.4.3. Epicenter shifts. In several cases, e.g., Nos. 19 and 130, the time lag exhibits a fairly regular trend over our network (generally increasing or decreasing with epicentral distance), which indicates a slight shift in epicenter location. In contrast, constant time lags suggest another event in the same location or at least at the same distance. In favorable cases, approximate solutions may be obtained from our network by the least squares method, using individual station data instead of averages as above. For this purpose, we develop Eq. (1.31) with sufficient accuracy as follows, guided by the sketch in Fig. 10:

6T = 6 0 + - 6XCOS(€ - y)

(1.32)

(::)h

= d o + - cosE6Xcosy+ (::)h

Clearly, in this equation there are three unknowns: 60, dxcosy, and 6 X sin y, which can be calculated from our station data, provided at least three stations have reliable readings. As an example we select No. 74 (Alaska). Our results by this method can be summarized as follows: Event 1:

Event 2:

56.8"N, 135.7"W

57.3"N,135.8"W

0 = 21 45 14.1

0 = 21 45 20.5

This means that the second event occurred 6.4sec after the first one in a location situated at a distance of 0.5" toward N 7.9"W from the epicenter of the first event. All 6T values calculated from this solution agree well with the observed 6T. Instead of working with Eq. (1.32)new solutions for each P wave could be made by the method of Shapira and BAth (1976).

SELECTED P WAVE PROBLEMS

31

None of these results permits any calculation of rupture velocity, due to their uncertainty and the probability that a simple rupture propagation does not always explain the sequence of events (Section 1.2.2.3). Quite clearly, calculations of this type from a limited network have only a limited validity beyond some indications. For reliable quantitative conclusions, global networks must be involved; alternatively dense local networks should be available. With a sufficient number of well-distributed readings, a reliable solution could be obtained by the least squares method. The second and possibly following P wave readings could also be solved as new earthquakes by the methods regularly used for this purpose. There are numerous papers in the seismological literature which develop sophisticated methods for the calculation of source location. In the present context, it may be appropriate to refer to Jordan and Sverdrup (1981). 1.2.4.4. Eflects on source calculation. The existence of double or multiple arrivals may have significant effects on the accuracy of earthquake hypocenter and origin time calculations at data centers. For example, event No. 7 shows a double P a t UME and UDD, while DEL reports only the first and UPP, KIR, and SKA only the second onset. Another example, among many others, is No. 33 with a double P at UPP, KIR, and UDD, while SKA, UME, and DEL record only the second onset. Combination of such divergent data will naturally influence the accuracy of source determination. A problem is that data centers collecting P wave readings for source calculations cannot decide if a reading belongs to the first or later onsets in a P wave group, only that more sensitive stations will in general record more multiples than less sensitive ones. A reliable decision would need direct visual inspection of all records used. Not only data centers but also record interpreters at individual stations may run into trouble in spite of visual record examination. The calculation of focal depth from pP - P is frequently complicated by a double P but apparently only a single pP. It is then open to question if pP should be combined with the first or later P waves in the h calculation. As a rule, it appears most correct to connect pP with the later, larger P onset. There are several examples of this problem, such as Nos. 8, 12, 73, 83, 87, 103, for which special comments are given in the Uppsala bulletin. The general rule to combine later single onsets with the largest wave in the P group can be extended to all other waves, not only pP. An example with an S wave is No. 10, which is specially commented on in the Uppsala bulletin. The existence of a double P wave has significant influence also on the determination of magnitude. By definition, m(US) mostly refers to P1, while m = m(UPP,KIR) refers to P2 (or later P waves if of larger amplitude) (see BBth, 1981c, pp. 329, 335). It is therefore of interest to compare our

32

MARKUS BATH

6m with m - m(US). Solved for rn - m(US), Eq. (1.5) yields the following expression:

m

- rn(US) = 0.19m - 0.56

(1.33)

Equation (1.33) should be compared with Eq. (1.213, both demonstrating that the magnitude difference increases with m. Moreover, the difference m - m(US) is 0.5 for m = 5.5 and 0.7 for m = 6.7, in fairly good agreement with Eq. (1.17). It should be added that the difference m - m(US) is not restricted to double P but is a general phenomenon due to the different ways of measuring records and that it is instrumentally affected. 1.2.4.5. Fennoscandian earthquakes. Earthquakes in our own region very seldom show double arrivals. This result agrees with the opinion that small magnitude events are rarely double or multiple. An earthquake in Dalecarlia (Dalarna),Sweden, on Jun 12,1972,is practically the only fairly reliable case. It shows double Pgl with time lags of 2.7 sec at UPP and 2.4 sec at UDD. Moreover, it shows double Sgl with time lags of 2.6 sec at UME and 2.5 sec at UDD. Altogether, these data suggest two events separated by about 2.6 sec in the same location. On the other hand, double Pn and double Sn, sometimes observed, are to be referred to as multiple paths, rather than multiple events. The complicated structure of the MohoroviEii: discontinuity is probably responsible for such observations (Bath, 1981b). Examples of double Pn and Sn waves are found in the events of Apr 21,1974, epicenter off west coast of Norway, and of Dec 18,1974, epicenter in the Norwegian Sea.

1.3. Discussion There are a number of points concerning applied methods that need to be commented on as follows. 1.3.1. Data Selection

In order to get reliable and representative results, it is important that observational data are unbiased. In this connection, there are two aspects to be considered. The first is that in our observations the initial P has the lowest amplitude, which increases significantly at the second P and so on at the following P, where more than two P are observed. The reverse case, i.e., decreasing amplitudes along the sequence, is not observed here. Either this situation is very rare or it escapes visual detection on records because of the already existing motion in the seismogram. Anyway, we have to be clear about the fact

SELECTED P WAVE PROBLEMS

33

that our results refer only to increasing amplitudes along the P wave sequence, or constant amplitudes in the limit (Trifunac and Brune, 1970). The second aspect concerns our criteria by which reliable data due to multiple events are selected (Section 1.1.2). A strict selection of events to be studied is absolutely necessary to guarantee observational material due solely to multiple events, excluding effects of multiple paths or other waves. To decide the true nature of the rejected cases, records at numerous stations worldwide would be needed. Our restriction to P arrivals within a few seconds only may at first seem to be serious. Fortunately, this restriction has no influence on our results. The only effect is a spurious increase of 6T for h 5 100 km, which is ignored. Therefore, there is every reason to consider our results as generally valid for multiple P due to multiple events, at least in a qualitative sense. 1.3.2. Data Grouping

The magnitude m is a parameter of importance for both 6T and 6m.This result expresses no doubt an influence of local or regional stress and strain conditions on the properties of multiple P. Because such conditions vary considerably from case to case, as well as the fault structure and mechanism, our observations have to be collected in groups to arrive at reliable results. Working with individual data leads just to indications, often with no obvious statistical significance, due to the scatter of the data. By forming averages for groups of data, various individual effects are eliminated as much as possible. For our material of totally 153 observations, it appears that seven to nine groups are an optimum: seven or eight for h, nine for m. A larger number of groups would entail too much of regional influences with less representative averages for each group. On the other hand, fewer groups would lead to too few data points for analysis and would smooth out significant variations too much. In our study of regional differences, nine groups are also used. This is the only case where a more detailed subdivision of a larger observational material could be warranted to better reveal the regional details. 1.3.3. Least Squares Solutions

The number of events entering each group, as just described, will vary from group to group as a natural consequence of the earthquake distribution with regard to h, m, and region. In performing least squares analysis of such data, there are essentially two procedures to consider: (1) attach to each data group a weight equal to the number of individual observations entering the group, or (2) treat all data groups as of equal weight, independent of the number of entering individual observations. In our study the simpler second approach is adopted, as only the general trends and their significance are of interest, at least at this stage of investigation.

MARKUS BATH

34

I .3.4. Possible Relations between Earthquake Parameters Let us consider any three quantities, A, B, and C. Assume that A is s i g nificantly related to B, and B significantly related to C. Then, A is also found to be significantly related to C, although this latter relation is due to the circumstance that A depends on B and that B varies with C. In our case, focal depth h and magnitude m are used as earthquake parameters. It is then important to investigate whether our observational material shows any relation between h and rn. Plotting these two quantities against each other for our total data set, it appears very clearly that there is no such relation. As h varies along its range from 15 to 610 km, there is no consistent variation of m. Likewise, when m varies from 4.9 to 7.5, there is practically no variation of the average h for each m. Therefore, there are no spurious relations of 6T, SD, and Sm to h and rn, which could have resulted from some relation between h and rn of our observational material. However, entering region 1 to 9 as an independent parameter, we find some relation of 6T and Srn to region. But these relations are due to the relations of 6T and 6rn to m.In fact, the average rn varies with region, and the only natural explanation of the regional dependence of 6T and 6m is to be found in the magnitude dependence of these quantities. 1.4. Conclusions The main results concerning double P waves, generated by double events, obtained from our study of 153 selected events recorded by the Swedish seismograph station network in the period 1969-1974 can be summarized in the following points. 1. The frequency of occurrence of double P, expressed as the proportion of double P to the corresponding total earthquake population, is independent of focal depth, of larger magnitude, and mostly of region. 2. The value of the time lag 6T is independent of focal depth, but it decreases with increasing magnitude. Its variation with region is explained by its magnitude dependence. 3. The magnitude difference 6m is independent of focal depth, but it increases with increasing magnitude. On average, 6m = 0.5. 4. The magnitude m is clearly of decisive importance, and the effects of changes in rn on other variables can be summarized as follows (Fig. 11 and Table X):

Trend

m

ST

dm

SA

SD(ST)

Up/down

Down/up Eq. ( 1 . 1 1)

Up/down

Up/down

Eq.(1.28)

Constant Eq. (1.16)

SELECTED P WAVE PROBLEMS m.5.0

.

p1

6Tz3.9 sec

4

4

PP

a1

6T13.2 sec

t

35

-

a2=1.5al

I

FIG.11. Schematic diagram summarizing the main features of double P, i.e., the variation of the time lag ST and the amplitude ratio +/a, with magnitude m. The sketched amplitudes are comparable within each one of the three graphs, but not between graphs. See also Table X.

Larger m implies a larger volume of rock in stressed condition, but it could also affect the rupture propagation. The interplay of stress and strength is probably of significance. 5. Considering the relatively frequent occurrence of multiple, especially double, P, its importance for accurate source data calculations and for a more detailed understanding of the focal processes, as well as its engineering significance (Trifunac and Brune, 1970), it is recommended that the necessary investigations, involving numerous earthquakes worldwide and a global station network, coupled with more sophisticated analysis methods be undertaken by an international organization, preferably the IASPEI. 2. P WAVEREFLECTIONS~ Multiple arrivals on seismic records may arise due to source effects, investigated in Section 1, or due to path effects. Among the tatter, we can distinguish between effects of multiple path propagation and effects of

* Notation used in this section: A, A , , A', amplitude factors (=wave function amplitudes) of incident, reflected, and refracted P (see further Fig. 13); B , , B', amplitude factors of reflected and refracted S; c, wave velocity (P and S)measured along the x axis; e, angle of emergence (incident continues

36

MARKUS BATH

reflection and transmission of seismic waves. A remarkable example is offered by the Sp wave, arriving a few seconds before the proper S wave, found by Bath and Stefansson (1966), confirmed by Leong (1976), and also reported by several other authors. The early PP in the distance range around 100” is another instance of multiplicity (Meyer, 1979), as well as the multiple core waves (see, for example, Payo Subiza and BAth, 1964). In this section, we will investigate how the properties of reflecting boundaries affect the P waves, especially their amplitudes, but also their phases and multiplicity. Different structures will be compared, as well as different reflectors in the same structure. Reflection at the free surface is thus compared with ocean bottom reflection. Moreover, the free surface reflection is investigated in relation to the reflection at the MohoroviEik (Moho) discontinuity.The results are primarily applied to pP and PP waves, but could also be used for interpretation of pPKP and PKPPKP waves. Moreover, P-S conversions upon reflection/refraction are included both theoretically and numerically, permitting studies of waves such as pS, PS, PSKS, etc. Our treatment refers to short-period seismic records with wave periods around 1 sec. Core reflections of P waves (PcP)were studied earlier with special reference to phase changes and the density ratio at the core boundary (Bath, 1954). In this case too, multiple arrivals could be expected, especially because of the layering at the core boundary that Ibrahim (1971) has investigated in detail. 2.1. Theory of Reflection and Refraction Our base consists of the theory of reflection and refraction of plane wave fronts at horizontal, plane, parallel, and sharp boundaries. The media are assumed to be elastic, homogeneous, and isotropic. Originally developed by Knott (1899) and Zoppritz (1919), the theory is treated in many papers, notably by Jeffreys (1926) and Gutenberg (1944), as well as in numerous

Footnote continued P);h, focal depth;H, crustal thickness; i, ( - 1)”’; pzz,pzr,normal and tangential stresses (actingon the plane z = 0);r, radius; t , travel time; up, us. wave velocities for P and S ; w, u, normal and tangential displacements (referred to the plane z = 0); z, x , vertical and horizontal coordinates; a, tane for incident P; /3, tan (emergence angle) for reflected S; 4 epicentral distance; K, wave number = 27r/L, where L = wave length measured along the x axis; A, p, Lam6 elastic parameters; p , density; u, Poisson ratio; a, Y , wave functions (displacement potentials) for P and S. Structural properties: quantities referring to the lower medium are denoted as above, upper medium with a prime. Wave properties: quantities referring to reflected waves have suffix 1, refracted waves a prime.

SELECTED P WAVE PROBLEMS

37

textbooks, more recently in Aki and Richards (1980), Ben-Menahem and Singh (1981), and Bath and Berkhout (1984). Although well known to many readers, we consider it important for the sake of completeness to reproduce formulas to the extent that they are used in our discussion. 2.1.1. Fundamental Equations There are some equations of fundamental importance that we need in every application. They are summarized in the following points. 2.1.1.1. Wave equations. The differential equations for the wave propagation constitute the start of the treatment. They are as follows:

P wave a2a/at2 = u;

v2a

2.1.1.2. Snell's law. For our present purposes, the law of reflection and refraction is preferably written in the following form: (1 + a2)uF = (1 + a")uL2 = (1 + p')ui

= (1

+ p")vL2

= c2

(2.2)

2.1.1.3. Displacement and stress. The boundary conditions are expressed in terms of displacement and stress, obtained from the wave functions by the following relations:

2.1.1.4. True amplitude. The true amplitudes of the respective waves are not identical with the amplitude factors appearing in the wave functions. Instead, the true amplitude of any wave is obtained from the expression ( w 2 + u2)'I2 by inserting the formulas for w and u of the last section. However, the true amplitude ratio of reflected to incident P is given by A , / A , as we are then concerned with the same wave and the same medium for reflection as for incidence.

38

MARKUS BATH

2.1.1.5. Energy. The energy of an incident P is proportional to paA'. Likewise, for reflected P, refracted P, reflected S, and refracted S, the corresponding expressions are respectively paA:, p'u' A", pflB:, and p'f"B'', the factor of proportionality being the same in all cases. In the reflection/ refraction of seismic waves we have an energy condition which states that the energy of the incident wave is equal to the sum of the energies of reflected and refracted waves: paA2 = paA:

+ p'a'Ar2 + p f l ~ :+ p ' / ? ' ~ ' *

(2-5)

This condition is very useful as a check on numerical results. Compare Laski (1977), whose equations are identical with Eq. (2.5), only given in a somewhat modified form. The following theoretical developments are classical, relatively simple, and straightforward. The solutions of the wave equations, i.e., the wave functions, are inserted into the boundary conditions, thus yielding a system of linear equations whose solutions consist of ratios of the respective amplitude factors. These ratios may easily be transformed into the true amplitude ratios or the energy ratios.

2.1.2. Rejection at a Free Surface (Fig. I2a) A P wave incident at the free surface of a solid gives rise to a reflected P wave and a reflected SV wave with the following wave functions:

Incident and reflected P (D = A exp[iK(x

+

CIZ -

ct)] + A , exp[iK(x - tlz

-

ct)]

(2.6)

Reflected SV 4' ' = B, exp[iK(x - jz - ct)]

The surface is assumed to be stress free; i.e., both normal and tangential stresses vanish at z = 0, which gives the following boundary conditions: Pzz =

0

atz=O

Pz* = 0

Substituting the wave functions in Eqs. (2.6) into the boundary conditions in Eqs. (2.7), we arrive at the following system of linear equations: A,/A - [2j/(fl' A,/A

-

l)]B,/A = - 1

+ [(S2 - 1)/2CI]BI/A

=1

(2.8)

For a given direction of the incident P wave, i.e., a given angle of emergence e, the amplitude factor ratios A , / A and B , / A depend only on j,i.e., on the ratio

39

SELECTED P WAVE PROBLEMS

i

a

xxb v VACUUM

SOLID

xxLIQUID

SOLID

A

01

A1

SOLID

cA*x-

01

A1 SOLID

FIG. 12. Structural models used in the theoretical development of P wave reflectionsas a basis for the numerical applications.

of up to us or, in other words, on the Poisson ratio CI of this medium. The ratios A , / A and B J A are easily obtained by solving Eqs. (2.8). The condition for energy conservation states that the energy of the incident P is equal to the sum of the energies of the emergent P and S:

(A1/4*+ ( B / w l / 4 2

(2.9)

=1

2.1.3. Refection and Refraction at a Solid-Liquid Boundary (Fig.12b) In this case, a P wave incident from the solid medium gives rise to a reflected P, a refracted P, and a reflected SV. The respective wave functions are immediately available: Incident and reflected P 0 = A exp[irc(x

+ az - ct)] + A, exp[irc(x

Refracted P

+

0’ = A’exp[i~(x a’z - ct)]

Reflected SV Y = B,exp[irc(x - flz - ct)]

-

az - ct)]

(2.10)

40

MARKUS BATH

The boundary conditions imply continuity of normal displacement and normal and tangential stress (in fact, the latter vanishes at the boundary): w, pz,, p z x continuous at z = 0

(2.11)

Inserting the wave functions, Eqs. (2.10), into the boundary conditions, Eq. (2.1l), we find the following equations for the three unknowns, i.e., A , / A , A’/A, B,/A: A,/A A,/A - [ p ’ ( p z

+ (a’/a)A’/A - (l/a)B,/A = 1

+ l)/p(p2 - l)]A’/A - [2/?/p2 - l)]B,/A = - 1 A,/A + [(pz - 1)/2a]B,/A = 1

(2.12)

Obviously, for a given direction of the incident P wave only two velocity ratios and one density ratio are of influence o n the results, not the individual values of these quantities. In this case, the following energy equation has to be fulfilled by our solutions: (Al/42

+ (p‘a’/pa)(A’/AY + ( P / w 4 / A ) 2 = 1

(2.13)

2.1.4. Rejection and Refraction at a Solid-Solid Boundary (Fig. 12c)

A P wave impinging on a solid-solid boundary leads to a reflected P, a refracted P, a reflected SV, and a refracted SV with the following wave functions: Incident and reflected P 0 = Aexp[il~(x+ az - ct)]

+ A, exp[iK(x - az - ct)]

Refracted P

0‘= A‘exp[i~(x+ a’z

- ct)]

Reflected SV Y

= B,

(2.14)

exp[ilc(x - pz - ct)]

Refracted SV

+

Y’ = B’exp[i~(x p‘z - ct)]

Assuming a welded contact between the two solids, there is continuity of both normal and tangential displacements and stresses at the boundary. Thus, we get the following boundary conditions: w, u,p z z , pzx continuous at z = 0

(2.1 5 )

41

SELECTED P WAVE PROBLEMS

With the expressions for the wave functions, Eqs. (2.14), the boundary conditions lead to the following system of equations: A,/A

+ (a’/a)A’/A- ( l / a ) B , / A + ( l / a ) B ’ / A= 1 A , / A - A’/A + BB,/A + B’B’/A = - 1

A l / A - (Y’/Y)A’/A- (&P/Y)Bl/A - (&’B’/Y)B’/A A,/A

(2.16)

= -1

+ (p’a’/pa)A’/A+ (y/2pa)B,/A - (yf/2p4B’/A = 1

where

+ a 2 )+ 2pa2 = p(P2 - 1) = A‘(1 + a’Z) + 2p‘a’2 = p’(P’2

y = l(1

y’

-

1)

The four linear Eqs. (2.16) permit solutions of the four unknowns A,/A, A’/A, & / A , and B’fA for any incident P wave and for any given properties of the two media. There is a minimum requirement to make Eqs. (2.16) numerically solvable, namely that three velocity ratios and one density ratio are given. In the case of a reversed z axis, i.e., waves propagating toward negative z, Eqs. (2.16) remain the same with the only difference that the signs of B, and B‘ are reversed. In the present case, we have the following energy equation (for any direction of the axes): (A1/AI2 + ( P ‘ ~ ’ / P ~ ) ( A ’ / A+) (2P / w 4 / 4 2 + (P’B’/PB)(wA)2= 1 (2.17) 2.2. Numerical Applications

The development of the fundamental reflection/refraction theory in Section 2.1 will now be applied to some cases of seismological interest. We shall compare P wave reflections partly at a free surface and an ocean bottom, partly at a free surface and the Moho discontinuity. 2.2.1. Free Surface versus an Ocean Bottom (Fig. 13a) The numerical values for our assumed structures are summarized in Table XI. Four different models are used in our calculations (Models a-d). Just by way of example, the models are given in more detail than the theory requires. As is evident from Sections 2.1.2 and 2.1.3, only velocity and density ratios need to be given numerically. With application of the equations developed in Sections 2.1.2 and 2.1.3 we calculate the ratios of the amplitude factors, summarized in Table XII. In the calculations, we consistently use four decimals, but the final results are

MARKUS BATH

42 a

VACUUM

1

VACUUM

CRUST

b

CRUST

VACUUM F

F

/ /

/

\ \

A

A MANTLE

/

A,

FIG. 13. Numerical applications of P wave reflections: (a) free surface (FF) versus ocean bottom (00);(b) free surface (FF) versus Moho (MM). For clarity, only P waves are shown. Subscript w in (a) stands for “water” to mark the distinction from the free surface. Reflections are indicated by subscript 1, refractions by prime; e.g., A’,, is a P wave reflected twice and refracted once, A’,’ is a P wave reflected once and refracted twice. Analogous notation is used for S, substituting E for A.

TABLE XI. FREESURFACE VERSUSOCEANBOTTOM: STRUCWRAL MODELS(FIG. 13a)

~~

a b C

d

Crust Ocean Crust Ocean Crust Ocean Crust Ocean

6.50 1.50 7.00 I .so 6.50 1.50 7.00 1 .so

3.75 -

4.04 -

3.47 -

3.74 -

3.00 1.OO 3.00 1.oo 3.00

0.25 0.50 0.25 0.50 0.30

1

0.50

3.00 1.00

0.30 0.50

.oo

rounded off to two decimals. Checks of the numerical results by means of the energy equations have been performed in all cases. Besides furnishing results for the amplitude and energy ratios, the signs indicate the phases. For the P waves, a positive sign of a ratio means unchanged phase, while a negative sign implies a change of phase, i.e., that an incident compressive P becomes

43

SELECTED P WAVE PROBLEMS TABLE XII. FREESURFACE VERSUS OCEAN BOTTOM:AMPLITUDE FACIYIRS (FIG. 13a)" Models c and d

Models a and b

e (deg)

A,IA

BilA

AiIA

B,IA

Free surface 0 10 20 30 40 50 60 70 80 90

-1 - 0.08 0.07 -0.00 -0.17 -0.40 - 0.63 -0.82 -0.95 -I

0 0.35 0.48 0.58 0.63 0.63 0.56 0.42 0.23 0

-1

-0.39 -0.21 - 0.23 -0.36 -0.53 -0.70 - 0.86 -0.96 -1

0 0.30 0.45 0.53 0.57

Model a

Ocean boi'tom -1 0 - 0.02 10 0.14 20 0.07 30 -0.09 40 50 -0.30 -0.51 60 -0.69 70 80 -0.81 90 -0.86

0 0.12 0.17 0.22 0.27 0.32 0.36 0.40 0.42 0.43

Model b

0 0.33 0.45 0.54 0.59 0.59 0.52 0.39 0.2 1 0

-1 -0.02 0.13 0.07 -0.09 -0.30 -0.52 -0.70 -0.82 -0.87

Model c

Ocean bottom 0 -1 10 -0.33 20 -0.14 30 -0.15 40 -0.26 50 -0.42 60 -0.58 70 -0.73 80 -0.82 90 -0.86

0 0.12 0.19 0.24 0.29 0.33 0.37 0.40 0.42 0.43

0.56

0.49 0.37 0.20 0

0 0.1 1 0.16 0.20 0.25 0.30 0.34 0.37 0.39 0.40

0 0.33 0.45 0.54 0.59 0.59 0.53 0.40 0.2 1 0

Model d

0 0.29 0.42 0.50 0.53 0.52 0.46 0.34 0.18 0

-1 -0.33 -0.14 -0.15 -0.26 -0.42 -0.59 -0.73 -0.83 -0.87

0 0.11 0.17 0.22 0.27 0.31 0.35 0.38 0.39 0.40

0 0.29 0.42 0.50 0.54 0.52 0.46 0.34 0.18 0

Free surface: Models a and b, A,/A = 0 for e = 12.8" and 30.0". Ocean bottom: Model a, A,JA = 0 fore = 10.5" and 35.1"; Model b, A , , / A = 0 fore = 10.7" and 34.7".

44

MARKUS

BATH

TABLE XIII. FREESURFACE/~CEAN BOTTOM:AMPLITUDE RATIO (FIGS.13a AND 14) e

Amplitude ratio A J A , , for following models

(deg)

a

b

C

d

0 10 20 30 40

1 4.25 0.53 -0.00 1.98 1.35 1.23 1.19 1.17 1.17

1 3.45 0.55 -0.00 1.86 1.32 1.21 1.18 1.16 1.15

1 1.20 1.55

1 1.18 1S O 1.51 1.34 1.24 1.19 1.17 1.16 1.15

50

60 70 80 90

1.57

1.38 1.27 1.21 1.18 1.17 1.17

FIG.14. An example of calculated amplitude ratios for free surface/ocean bottom A I / A , , (Model c) and free surface/Moho A ; / A , (see Tables XI11 and XVI).

dilatational or vice versa upon the reflection/refraction (see also Ergin, 1952; Ingram and Hodgson, 1956). The ratio Al/Al, constitutes our final result from this section (Table XI11 and Fig. 14), giving direct information on the effect of the free surface versus the ocean bottom on the amplitude of the reflected P wave.

2.2.2. Free Surface versus the MohoroviciC Boundary (Fig.13b) The numerical values assumed for this structure, approximately typical of continents, are given in Table XIV. The equations of Sections 2.1.2 and 2.1.4

45

SELECTED P WAVE PROBLEMS TABLE XIV. FREESURFACE VERSUS MOHO:STRUCTURAL MODEL (FIG. 13b) Medium

(km/sec)

9 (kmisec)

(g/cm3)

Ub

Mantle Crust

8.15 6.50

4.57 3.85

3.32 2.87

0.27 0.23

UP

Pa

See Bullen (1963, pp. 214-215). ’See Gutenberg (1959, p. 181).

are used in the calculation of the ratios of the amplitude factors, collected in Table XV. As already stated, these may be used for evaluating any amplitude or energy ratio involved. Also in this case, the original computations are made to four decimals, but the results are rounded off to two decimals. As a consequence, there may be some seeming discrepancies between tabulated values. Checks by means of the energy equations have been made in every individual case. Phase changes of P waves upon reflection/refraction are indicated by negative ratios (see also Nafe, 1957; McCamy et al., 1962). The ratio A J A for the Moho reflected P is given immediately by the solutions (Table XV). However, the corresponding ratio for the surfacereflected P, i.e., A’JA, has to be calculated by combination of the results of upward Moho refraction, free surface reflection, and downward Moho refraction: A;‘/A = (A’/A)(A;/A’)(A;’/A;)

(2.18)

The ratio A;’/A, provides our final result, i.e., a comparison of free surface and Moho reflection for a given incident P wave (Table XVI and Fig. 14). The Moho reflected P arrives earlier than the surface-reflectedone and the travel time difference 6t is estimated by the following formula (in our model, 6t is the time difference in passing from one dashed line to the other by the two alternative routes depicted in Fig. 13):

6t

= (2H/vb) sin e’

(2.19)

Table XVI gives numerical values of the time difference for an assumed crustal thickness of H = 33 km.

2.2.3. The Function e(A, h)

In the calculations and results we have so far used the angle of emergence (e) of the incident P wave as our entry. The advantage is that our results are valid for any type of incident P wave, whether pP, PP, pPKP, PKPPKP, etc. However, the use of e as entry is inconvenient in seismological applications. For this purpose, we evaluate e(A, h) for pP and PP.

46

MARKUS BATH

TABLE XV. FREESURFACE VERSUS MOHO: AMPLITUDE FACTORS(FIG.13b)" Moho from below e

(deg)

A,IA

A'IA

BIIA

0 10 20 30 40 50 60 70 80 90

-1 -0.50 -0.26 -0.15 -0.12 -0.12 -0.14 -0.16 -0.18 -0.18

0 0.43 0.65 0.77 0.84 0.89 0.91 0.93 0.94 0.94

0 0.02 0.04 0.06 0.08 0.08 0.08 0.06 0.03 0

B'IA -

0

-0.05 -0.08 -0.08 -0.08 - 0.01 -0.05

-0.04 -0.02 0

Free surface e' (deg) 37.1 38.2 41.5 46.3 52.3 59.2 66.5 74.2 82.0 90

A',/A'

B',/A'

-0.04 -0.07 -0.14 -0.26

0.64

0.64 0.66 0.66 0.65 0.60

-0.41 -0.58

-0.74 -0.88 -0.97 -1

0.50

0.36 0.19 0

Moho {rom above By//,

A,'fA',

37.1 38.2 41.5 46.3 52.3 59.2 66.5 74.2 82.0 90

0.99 0.51

0.29 0.18 0.15 0.14 0.15 0.17 0.18 0.19

2.23 1.67 1.37 1.21 1.13 1.08 1.05 1.03 1.03 1.02

-0.07 -0.00 0.04 0.06 0.07 0.08 0.07 0.05

0.03 0

-0.08 -0.10 -0.10 -0.10 -0.08 -0.07 -0.05 -0.04 - 0.02 0

e' is the angle of emergence at the free surface corresponding to the angle e at the Moho (Fig. 13b).

SELECTED P WAVE PROBLEMS

47

TABLEXVI. FREESURFACE/MOHO: AMPLITUDE RATIOA N D TRAVEL TIMEDIFFERENCE (FIGS.13b AND 14)

0 10 20 30 40 50 60 70 80 90

0 0.10 0.48 1.55 3.27 4.59 5.14 5.39 5.29 5.24

6.1 6.3 6.7 7.3 8.o 8.7 9.3 9.8 10.1

10.2

TABLEXVII. ANGLEOF EMERGENCE I N RELATION TO EPICENTRAL DISTANCE AND FOCAL DEPTH"

4& h) (deg) PP h = 0.01 (96 km)

A (deg)

FF

MM

10 20 30 40 50 60 70 80 90 100

45 58 61 63 66 69 72 74 75

28 48 52 55 60 63 67 70 71

PP h = 0.12 (794)

FF

62 64 68 71 73 75

h = 0.00 (33)

h = 0.12 (794)

MM

FF

MM

FF

MM

53 51 62 66 69 71

34 37 40 51 56 59 60 61 62 63

38 45 49 51 52 54 55

59 60 61 62 63 64

50 51 52 54 55 57

FF, free surface; MM, Moho.

The function e(A,h) is determined by the following well-known equation (Bullen, 1963, p. 110): cos e = (up/r) dt/dA (2.20) For our present purpose, it is sufficient to calculate e for two different levels: the free surface (FF), up = 6.50 km/sec, r = 6371 km;and the Moho (MM), up = 8.15 km/sec, r = 6338 km (H = 33 km). The dependence on A and h is given by the travel time derivative dt/dA (sec/radian). It is evaluated for p P and PP by means of the Jeffreys-Bullen (1967) travel time tables with assistance of Ritsema's (1958) graphs (Table XVII).

48

MARKUS BATH

It is found that the angle of emergence varies only slightly with focal depth, and therefore it is sufficient to perform the calculations for small and large depths only. Values of e for intermediate depths are obtained with sufficient accuracy by simple interpolation between the tabulated values. The variation of e with epicentral distance A is more marked, as is evident from Table XVII. There might be a small inconsistency in Table XVII because of the fact that our models do not agree exactly with the model of the travel time tables used. However, any possible discrepancy is considered insignificant for our problem. In practical applications of our results, we first read off the value of e from Table XVII for the given wave (pP or PP), given distance, and focal depth. After that, we proceed to Tables XI11 or XVI to see the effects of ocean bottom or Moho on the studied waves,

2.3. Conclusions Correct interpretation of arrival times, amplitudes, and phases in seismic records is a prerequisite for reliable conclusions about the source location, magnitude, and mechanism as well as about the earth's structure. Tables XI11 and XVI, combined with Table XVII, permit us to formulate the following conclusions. 2.3.1. Free Surface/Ocean Bottom As expected, the P wave reflected at the free surface is generally somewhat larger than the P wave reflected at the ocean bottom. From Table XIII, we see that the amplitude ratio A1/AI, depends strongly on the Poisson ratio, especially for e 40".In this range, Q = 0.25 (Models a and b) leads to rapidly varying amplitude ratios, which would need to be calculated at least for each full degree for clarification. For e > 40", there is a better agreement between Models a and b on the one hand and Models c and d on the other, i.e., the value of the Poisson ratio is then of less significance. In this range, the amplitude ratio decreases gradually from nearly 2 to about 1.2, equivalent to a magnitude difference of 0.3-0.1. The effect of the chosen wave velocities is relatively small, as can be seen by a comparison of Model a with b or of Model c with d. Nor is there any change of the initial phase, With regard to the effects on particular P waves, pP will not be affected by the irregular behavior of Models a and b for e < 40", because in general e 2 45" (Table XVII). For PP, however, there might be an effect for A < 30" (shallow depth). The ratio AL/A for the refracted P wave, passing into the water, is generally

-=

SELECTED P WAVE PROBLEMS

49

somewhat smaller than A,,/A. Even though returned by total reflection at the ocean surface and by partial refraction to the solid medium, it could be hard to detect visually on seismic records, as it arrives later than the P reflected at the bottom. In addition, its amplitude will generally be smaller than that of the bottom-reflected P wave. Although this appears as a general result of the theoretical calculations, there are some important empirical modifications to this rule. For example, in the epicentral area the P wave refracted into the water may give rise to T phases recorded at great distance. It may also be strong enough to impinge against ships in the area to such an extent that the crew fears the ship has gone aground. Moreover, referring to an organ-pipe effect in the water layer, Shurbet (1965) observed amplitude ratios pP/P > 1 for reflection in some water-covered areas. Likewise, Forsyth (1982)reported a number of observations in which the amplitude of pP reflected at the ocean surface exceeds that of p P reflected at the ocean bottom. The last-mentioned two cases may be typical for areas of shallower water with resonance effects, whereas our theory assumes a water layer of a thickness many times the wave length. 2.3.2. Free SurfaceJMohoroviZiC Boundary The P wave amplitude ratio free surface/Moho, i.e., A;'/A,, increases rapidly with increasing values of the angle of emergence e (Table XVI and Fig. 14). The ratio is < 1 for e < 27" approximately, while for e > 27" it increases to around 5 at e = 90". In this range, there is thus a rather strong dominance of the surface-reflected P over the Moho reflected one. Moreover, with A;'/A, 2 0 in all cases, the initial phase is the same for the two kinds of reflection. Applying these results to p P and PP (Table XVII), we find that in all listed cases, e > 27" at Moho (MM); thus in all cases we have A'JA, > 1. This result means that p P and PP may be preceded by a smaller wave with a time lead 6t of a few seconds (Table XVI). In many, perhaps most, cases the Moho reflected wave entailing this precursor is hard to detect visually on seismic records, partly due to its relatively small amplitude, partly due to the already existing motion in the record. However, if detected and, for example, misinterpreted as a true pP, it would yield too low a value of the focal depth. With regard to observations of Moho reflected waves, an interesting comment by Jeffreys (1973, p. 35) may herewith have its quantitative explanation. In analogy to the Sp wave, we could expect a Ps wave, i.e., a conversion of P into S at Moho. However, such a wave would arrive later than P and thus be hidden by the motion on a record. Moreover, it would be far too weak to be easily detected, except possibly by special techniques, as can be seen from the ratios B'/A in Table XV. Our tabulated data permit some further studies as well, e.g., a comparison of PS reflected at the free surface and at the Moho.

50

MARKUS BATH

(For a useful review of PS studies, see Cook et al., 1962; cf. also Jacob and Booth, 1977.) 2.3.3. ContinentlOcean It is of particular interest to compare the reflection of the free surface of a continent with that of the ocean bottom. The continental reflection is expected to dominate for two reasons: first, the absence of water, and second, the steeper incidence of P at the surface due to lower crustal velocities. Combination of the data of Tables XI1 and X V gives at least an approximate idea of the difference in reflectivity. We may compare the ocean bottom reflection from Table XI1 with the free surface reflection A;/A' of Table XV,or for the latter we may instead use the ratio A'JA corresponding to the wave emerging below the Moho discontinuity. In either case, the continental reflection exceeds the oceanic reflection by a factor usually between 2 and 1.2 (except at short distances), decreasing slightly with increasing angle of emergence e. The amplitude ratios PP/P and pP/P have been used for a variety of investigations. It is to be observed that these ratios do not depend only on the reflection properties (continental or oceanic),but also on the focal mechanism (the waves leaving the focus in different directions) and on absorption (being larger for the shallower propagation of PP and pP than for the deeper one of P). Gutenberg and Richter (1935)found the ratio PP/P to be less for reflections under the Pacific than under other ocean areas and continents, a result that is not confirmed by Mei (1943).Byerly et al. (1949)concludedthat the ratio PP/P depends on too many factors to be very useful for determining any one of them, while Papazachos (1964) considered it difficult, but not impossible, to use the amplitude ratio PP/P for structural studies. Mooney (1951)found that the energy ratio pP/P decreases with increasing focal depth more than available theory could explain, and Pearce (1977) presented an efficient method for fault plane solutions based on the amplitude ratio pP/P (see also Shurbet, 1965, and Forsyth, 1982, referred to in Section 2.3.1). 2.4. Discussion

Our calculations and deductions are strictly valid only for the models considered. Compared to the true conditions in the earth, these models include a number of simplifications.In this section we will discuss the applicability of our models and results to the real earth. 2.4.1. Structural Properties 2.4.1.1. Plane boundaries. The boundaries or discontinuity surfaces are assumed to be plane; i.e., the earth's curvature is neglected.This assumption is

SELECTED P WAVE PROBLEMS

51

permitted since in all individual cases we are only concerned with a very small portion of each surface. 2.4.1.2. Stress-Jree surface. In reflection/refraction theory, it is generally assumed that the free surface is stress free. However, direct measurements indicate that large horizontal stresses are common in the crust. A theory to include such stresses in the wave propagation was developed by Biot (1965). His theory has been applied in many different cases, notably in a series of doctoral theses from the Indian School of Mines at Dhanbad in India under the guidance of Professor S. Dey. In their recent paper, Dey et al. (1985) demonstrate that initial stresses may have a large influence on the seismic wave propagation. However, as existing stresses are not known to sufficient accuracy, we refrain from including them in our present calculations. Their effect is probably smaller on amplitude ratios as given in Tables XI11 and XVI than on the amplitudes themselves. 2.4.1.3. Absorption. We are only concerned with amplitude changes due to reflection/refraction, whereas other effects on amplitudes, such as absorption and scattering, are left out. Applications of our model in Fig. 13b to the earth, including effects of absorption as well as of downward increasing velocity in the mantle, are expected to yield smaller amplitude ratios A'JA, and larger time differences 6r. However, these effectsare estimated to be minor compared to those calculated here. 2.4.1.4. Typical models. We have aimed at studying models that are typical of true earth conditions, but still they can only be considered as examples. In order to explore the effects of various structures in more detail, it would be necessary to supplement the present effort by a large variety of numerical models-a major undertaking beyond our present ambition. 2.4.2. Wave Properties 2.4.2.1. Plane wave fronts. It may appear more serious to assume plane wave fronts than to assume a flat earth, especially in the case of p P at close distance. Spherical wave fronts could appear more adequate, although even those are an approximation due to the spatial variation of wave velocity. However, in this case also we are concerned only with a very small portion of the wave front which could be considered as quasi-plane. Moreover, we are not dealing with individual amplitudes, but with amplitude ratios (Tables XI11 and XVI), for which the effect of the wavefront curvature would be small (E. R. Lapwood, personal communication). In a detailed theoretical investigation, Richards (1976) demonstrated that corrections of the plane wave results are needed only at or near grazing incidence, i.e., for e = 0" (see especially

52

MARKUS BATH

Table XII). And, by virtue of Table XVII, it is clear that such cases do not arise in the practical application of our results (see also Officer, 1958, p. 192). The incident as well as the emergent waves at the Moho (Fig. 13b) are assumed to be parallel, i.e., to have the same angle of emergence eat the Moho. This is a permissible assumption because over the short distance intervals considered in each case the angle e exhibits only a small variation.

2.4.2.2. Reverberations. Our calculations of amplitudes, phases, and arrival times refer only to the initial waves (pulses)of each kind. In other words, we are not concerned with the following wave train, which may be expected to show up frequency-dependent reverberation effects, characteristic of any layered medium (cf. BAth, 1974, pp. 297-301). Moreover, the wavelengths and the wave trains involved are very short compared to the thickness of the top layer, whether oceanic or crustal, thus excluding resonance effects.

3. TRAVEL TIMESOF DIFFRACTED P WAVES3 Well-defined onsets on seismic records and accurate arrival times of the various waves are apt to facilitate the determination of reliable travel time tables. In Sections 1 and 2 we studied several effects, both of source and path, that may lead to complications in these respects. Nevertheless, reliable record readings and careful analysis have made it possible to determine longitudinal travel times, especially of the P wave itself, with rather great accuracy. But this is true only up to about 100". Beyond this distance, i.e., for diffracted P waves, there are only a few determinations so far available. Diffraction is a well-known phenomenon, common to all kinds of wave propagation. Seismic wave propagation in the earth is no exception. Among various types of diffraction encountered in the earth, the one exhibited by P waves traveling around the outer core surface is probably the most spectacular one. The interest in this wave type derives from the fact that it may provide important information on the physical properties of the mantle base and the core-mantle boundary. Both the dynamics and the kinematics of the

Notation used in this section: a, slowness = l/u, (sec/deg);b,, b , , b 2 ,constants in travel time formulas (sec, sec/km, sec/km2, respectively);J(h), function of h, appearing in the formula for t ; h, focal depth (km);ha,, average focal depth (km);i, angle of incidence (deg); N,number of observations; Pdif, diffracted P wave; r, radial coordinate in the earth (km); r z , outer core radius = 3471 km, alternatively = 3473 km; R, earth's radius = 6371 km; t, travel time (sec in formulas, min sec in Table XXIV, min in Fig. 19); u, u(r), P wave velocity (km/sec)at radius r (km) u2, P wave velocity (deg/sec or km/sec), at the base of the mantle of radius r2 (km); A, geocentric distance, (deg); ISC, International SeismologicalCenter; USGS, United States Geological Survey.

SELECTED P WAVE PROBLEMS

53

diffracted P wave have been investigated in the literature, theoretically as well as observationally. Dynamical properties, especially amplitude attenuation, and velocity structure at the mantle base have been studied extensively. Fundamental theoretical developments are given in several books (Aki and Richards, 1980; Ben-Menahem and Singh, 1981; Hanyga, 1985). Among the numerous special papers dealing with these problems, we may mention those of Gutenberg (1960), Knopoff and Gilbert (1961), Alexander and Phinney (1966), Phinney and Alexander (1966), Phinney and Cathles (1969), Ansell (1974), Doornbos and Mondt (1979a,b), Mula (1981), Doornbos (1983),and Menke (1986).This list is by no means exhaustive, and many useful additional references can be found in these publications. On the other hand, kinematic properties, especially travel times of diffracted P waves, appear only sparsely in the literature. For example, the tables of Jeffreys and Bullen (1967) or of Herrin (1968) do not report travel times of Pdif, However, Gutenberg and Richter (1936) list Pdif travel times up to a distance of 150" and for focal depths of 100-800 km. This table was found to be a most useful reference in our own work. As a contribution to travel time determination in general and to the study of mantle base properties in particular, the purpose of this section is to develop a travel time table of Pdif on the basis of records of the Swedish seismograph network. In spite of the fact that the theory, outlined below, suggests a simple formula for the travel time, its practical evaluation meets with several problems. These will be discussed in detail in the following. 3.1. Theoretical Background 3.1 . l . Trauel Time Equation

The propagation path of a P wave diffracted around the outer core boundary is schematically depicted in Fig. 15. As indicated in this figure, it is natural to divide the path into three segments, 1,2, and 3, of which 1 and 3 refer to propagation through the mantle and 2 to propagation along the core boundary. The corresponding divisions of the total travel time t and the total angular distance A become

+ + 1, A = A, -I-A2 + t = t,

t2

A3

By means of the expression for t , , t2

= A2102

(3.3)

54

MARKUS BATH

MANTLE CORE

‘2

* SOURCE

R

STATION

FIG. 15. Propagation path of diffracted Pin a spherical earth model. The small square in (a) is reproduced on a larger scale in (b).

combined with Eq. (3.2), we rewrite Eq. (3.1): t = ( t l - A1/v2)

+ A h 2 + @3 - A 3 1 0 2 1

(3.4)

where t , - A,/u2 is a function of h, A/v2 is a function of A, and t 3 - A 3 / ~ 2is a constant. Introducing the slowness a = 1/u2, we simplify Eq. (3.4): t = aA

+f(h)

(3.5)

3.1.2. Evaluation of f ( h )

The function of h as well as the constant terms in Eq. (3.4)can be evaluated by ray theory. The ray parameter, (r sin i ) / u = r2/u2

(i2 = 90”)

(3.6)

yields

cos i = [I

- (r2u/u2r)Z]”2

tan i = (rzu/u2r)/[1

- (rzu/vzr)2 3 112

(3.7)

The geometry depicted in Fig. 15b gives immediately the following expressions: tl

= lr;-h(vcosi)-’ dr

(3.8)

(3.9)

SELECTED P WAVE PROBLEMS

55

t3 and A3 are expressed by the same integrals as in Eqs. (3.8) and (3.9), respectively, with the only differencethat the upper limit is then R. Introducing cosi and tani from Eq. (3.7) and a function v(r) for the P wave velocity distribution in the mantle, the integrals can be solved. Formulas are needed for u(r) like those developed for the lower mantle by Bith (1966, 1967), but now for the whole mantle and the crust. It is most convenient to divide the mantle into a sequence of sections in carrying out the integration. An equivalent summation procedure could also be envisaged. 3.1.3. Travel Time Methods

With this background, at least three methods to determine the travel times of diffracted P can be suggested: (1) with a given function v(r) by means of Eq. (3.4), including the integrations explained above; (2) with given travel times of P or PcP together with a given value of v,; (3) by a purely empirical procedure from records of diffracted P. This method is the only one that will be applied in this investigation, being independent of any prior knowledge of velocities or travel times. Methods 1 and 3 are to some extent mutually inverse. Method I starts from given velocities u(r) and calculates travel times t , while method 3 starts from travel times t and could calculate velocities. This latter procedure is applicable at least to v 2 , while mantle velocities are reliably determined only from P waves at shorter distances than Pdif. Equation (3.5) demonstrates a simple shape of the Pdif travel time formula: A and h appear separated from each other and the formula is linear in A. Such features are typical of waves that have propagated along a surface with a constant velocity. The crustal waves, such as Pn or Sn, offer well-known examples of this type of travel time equation. The equations for other waves differ markedly from these characteristics. Even though Pdif has a simple travel time equation and therefore may seem to offer an easy and attractive investigation, there are still a number of problems involved. In the following section the problems are explained in detail using available observations.

3.2. Observational Material 3.2.1. Stations

All reliable readings of diffracted P waves were collected from the monthly bulletins of the Swedish seismograph network, operated by the Uppsala institute, for the 10-year interval 1965-1974. The network is presented in Table I (Section 1). All stations are located on a typical continental crust without sediments, most of them on the Baltic shield.

56

MARKUS BATH

There are several reasons behind the choice of the interval 1965-1974 for this investigation. A period of 10 years is deemed necessary to yield sufficient material for our analysis. Moreover, in this period our station network was fully developed and equipped with modern instruments. Finally, all readings were made or checked by the author, which ought to guarantee some degree of homogeneity among the observations. 3.2.2. Records

Both short-period and long-period seismograph readings were used (Table I), in both cases from the vertical components. While the short-period readings dominate at shorter distances (up to around 125- 13o"), the longperiod ones prevail toward longer distances. The short- and long-period readings were treated together. No difference is obvious from plots of t versus A; i.e., no dispersion or any other period effect appears, at least not with the timing accuracy achieved here. In fact, there is good agreement between data for shorter distance (short period) and longer distance (long period). Moreover, there are several advantages of treating long-period and shortperiod readings jointly. For one thing, a sufficiently large body of observational data becomes available; for another, the largest possible distance range is covered. Figure 16 shows a typical case demonstrating the long-period Pdif. 3.2.3. Earthquakes

In the selection of earthquakes from our bulletins, A 2 100' is the adopted distance limit. At this limit the travel times of P and PcP merge into each other and the t - A curve has assumed its straight line characteristics. At A = loo", Jeffreys and Bullen's (1967) tables show a difference of only 0.1 sec between the travel times for P and PcP for a surface focus. In Richter's (1958, p. 257) graph

Pdif

PKP

hl

H 1 min

FIG. 16. Typical long-period vertical component record (Press-Ewing) of Pdif and PKP at Uppsala from an earthquake in the Tonga Islands (A = 144, h = 50 km) on Dec 18, 1963. The wave period of Pdif is about 30 sec and it is recorded only by long-period seismographs. Another instructive case is reproduced in BHth (1979, p. 73).

57

SELECTED P WAVE PROBLEMS

of wave propagation, the ray for A = 100" just touches the core boundary. Moreover, from our plots of t versus A, the travel time curves are found to be strictly linear, confirmed by Gutenberg and Richter (1936).For earthquakes at greater depth, the limit occurs at shorter distances, but the limit A = 100" is kept for all focal depths. Figure 1 (Section 1) demonstrates the distance limit of 100" from Uppsala. Obviously, there are two earthquake regions that dominate our observations, i.e., southwest Pacific and South America. In fact, of the total number of 187 investigated earthquakes, 149 events or 80% derive from the southwest Pacific and 38 events or 20% from South America, the latter group also including two events in the South Sandwich Islands. For distances beyond loo", Pdif may be observed from earthquakes of any epicentral location and any focal depth (possibly with some relative bias toward greater depths), provided only that the magnitude is sufficientlylarge and the focal mechanism is favorably orientated. 3.2.4. Explanation of Appendix 2

All observations are collected in Appendix 2. In all cases, t is given to 0.1 sec and A to 0.01", which certainly exceed the true accuracy in many cases but which are chosen simply for computational purposes. In collecting the material from our bulletins, all readings are excluded that are deemed unreliable. Some of these were already considered unreliable in the bulletin work and marked by (P) in the bulletins. In addition, on the basis of our present plots of t versus A, readings are excluded that diverge strongly and are obviously incorrect. The data that remain and that were used in our analysis are as reliable as ever possible from our station network. Reliability is particularly important in view of the rather uneven distribution of the observations with regard to both epicentral distance and focal depth (Table XVIII). TABLE XVIII. NUMBER OF OBSERVATIONS FOR GIVEN DISTANCES AND FOCAL DEPTHS Distance Adeg 100-110 110-120 120-130 130-140 140- 150 100-150

Number of observations at following focal depths h (km) 0-100

100-200

225 102 27

8

9

69 21 7 2

2 -

-

-

-

-

13

11

15

99

365

200-300

300-400 11

400-500 9 6

3 2

500-600

600-700

0-700

35 4 3

12

369 133

42

40 3 -1 16

16 3 561"

-

~~

The 561 observations are based on 187 earthquakes, i.e., on the average each earthquake has produced 'dif records at three of our stations.

58

MARKUS BATH

TABLE XIX. FOCAL DEPTHDIFFERENCE^ h(I8C) - h(USGS) FOR OURMATERIAL ~~

~

h(1SC) (km) 0-50 50-100 100-200 200-300 300-500 500-700 0- 700

~

~~~~~

N -

~~

~

h(ISC) - h(USGS) (km)

Ih(lSC) - h(USGS)I,, (am)

-4 13 + 5 f 23 + O f 13 -6 f 13 -3 f 12 -8 f 22 -2 k 16

9 17 9 I1 7

91 35 30 6 5 14 181

18 -

11

TABLExx. ANALYSIS OF 10 EARTHQUAKES FROM THE ~~~~~~~

~

Isc BULLETINS

~

a. Source data

No.

Date

Origin time (GMT)

Epicenter (deg)

Depth h (km)

Region

07 19 51.4 05 34 06.2 20 23 28.4 17 08 05.4 10 22 20

25.858 177.29W 15.78N 121.71E 9.15s 78.83W 1.46s 72.56W 14.13s i 6 6 . 5 6 ~

82 40 48 653 20

Kermadec Islands Luzon Peru Colombia New Hebrides Islands

07 17 04.7 03 03 16.9 06 11 28.9 02 02 44.5 17 58 37.9

3.21s 139.69E 32.513 71.21W 5.52s 153.86E 2.79s 77.35W 15.578 167.24E

41 40 43 88 49

West Irian Chile New Ireland Peru-Ecuador New Hebrides Islands

1970

I 2 3 4 5

Jan 20 Apr 7 May 31 Jul31 Aug 1I 1971

6 7 8 9 10

Jan 10 Jul 9 14 27 Oct 27

b. Pdif results ( F i g . 1 7 ) No.

N

Distance A (deg)

Travel time equation ( t in sec, A in deg)

1

44 31 34 25 30 96 93 88 19 37

106-156 104-154 105-160 106-135 105-141 104-153 105- 151 106-141 105-137 107-143

+ 363.7) f 6.0 + 365.4) f 3.6 t = (4.723A + 347.3) f 5.1 t = (4.667A + 293.1) 5.5 t = (4.466A + 380.2) f 3.2 t = (4.6896 + 353.8) 5 4.1 t = (4.501A + 372.8) f 3.5 t = (4.481A + 375.1) f 4.6 t = (4.632A + 355.6) f 3.4 t = (4.341A + 393.1) f 3.5

2 3 4 5 6 7 8 9 10

= (4.547A t = (4.584A t

SELECTED P WAVE PROBLEMS

59

The source data used in this investigation and listed in Appendix 2 are based on the ISC bulletins. The USGS data show in general some minor differences with regard to epicenter location and origin time, which are not expected to be of any significant influence on the analysis. Focal depth differences may be larger in a few exceptional cases, but on the average the agreement is satisfactory also here, i.e., generally within the inevitable focal depth errors (Table XIX). Source inaccuracies will contribute to the scatter of the results (see Section 3.6.1), but they have no biasing effect.

3.3. Preliminary Analysis Before we discuss our own observational material, an initial orienting analysis will be made of independent observational data in order to clarify the functional dependence of travel times of Pdif on epicentral distance and focal depth. 3.3.1. Relation to A

The ISC bulletins contain much information on arrival times of Pdif. Ten earthquakes have been selected from these bulletins under the condition that the records of each quake cover as extensive a distance range as possible (Table XX and Fig. 17). For each earthquake, h is naturally a constant which simplifies the travel time equation [Eq. (3.5)]: f =aA

+ b,,

h =constant

(3.10)

The least squares solutions of Eq. (3.10) with standard deviations of t are collected in Table XXb. With the guidance of f-A plots, a few divergent and obviously incorrect data are excluded. Some ISC data were clearly in error by a full minute and could be included after correction. The linear relation between t and A is confirmed by the observations for the entire extent of Ay as expected. On the other hand, the a coefficient shows unexpected variations from case to case. Some of these variations certainly fall within the limits of statistical significance, but there are also significant differences. Take for example the largest, a = 4.723 (event No. 3), and compare with the lowest, a = 4.341 (No. 10). This difference is no doubt statistically significant. Converting the a values into u2 may be even more instructive: u,(deg/sec) = l/a u,(km/sec) = (7r/180°)(r2/a)

(3.1 1)

With r2 = 3471 km we find u2 = 12.83 km/sec (No. 3) and u2 = 13.96 km/sec

60

MARKUS BATH

I

I

I

I

I *

DISTANCE A DEG 750 100

I

1

120

I

I

140

FIG.17. Travel time graphs of Pdif for two earthquakes (Nos. 4 and 5 in Table XX), based on data from the ISC bulletins.

(No. 10).A weighted average of all 10 earthquakes with weights equal to the number of observations yields u2 = 13.29 km/sec. A far more extensive investigation would be required to clarify the exact reason behind these variations of u 2 , and at present only some speculations could be made. For instance, there could be lateral inhomogeneities at the mantle base; alternatively there could be variations of the core radius, causing diffracted P from different events to propagate at slightly different depths. In this connection, it is of interest to note that attenuation investigations of longperiod Pdif seem to indicate considerable lateral variation of the core boundary properties (see Bath, 1974, p. 340, for further references; also Doornbos and Mondt, 1979b). Gradually less clear onsets with increasing distance could be another contributing factor. (See also Section 3.5.2.) The data of Table XX suggest a well-defined relation between bo and a in Eq. (3.10), provided the focal depth is constant. Selecting those events in Table XX that have nearly the same depth, h = 40-50 km, i.e., Nos. 2,3,6,7,8, and 10, the following relation holds: bo = (890.5 - 114.8~1) f. 1.4

(3.12)

SELECTED P WAVE PROBLEMS

61

If this equation could be taken as an expression of the true conditions, it would speak in favor of variations of the core radius and against lateral heterogeneities alone at the mantle base. Apart from these speculations, the result is still of importance for the analysis of our own material. Initially, I planned to determine a worldwide average a value by the 10 earthquakes from the ISC bulletins and then to use this a value in the analysis. This method could have facilitated the following analysis with regard to h. However, the obvious variability of the a values prohibits this procedure. Instead, the a values have to be determined from our own observational material.

3.3.2. Relation to h In order to get some empirical guidance on the shape of the depth function f ( h ) in Eq. ( 3 . 9 , resort is taken to the Gutenberg and Richter (1936) table of Pdif travel times, here reproduced in Table XXI. By a search procedure, formulas are found that yield a satisfactory fit to these data: t = 4.4773A + 0.00005h2- 0.135h + 382.0 for h = 100-400 km

t = 4.4550A - 0.090h for h = 500-800 km

(3.13)

+ 374.0

(3.14)

Table XXII demonstrates the difference between these equations and the travel times of Table XXI. The average difference amounts to 0.35 sec(N = 42) in absolute value, i.e., considerably less than the time error of any single TABLE XXI. PDIFTRAVEL TIMES ACCORDING TO GUTENBERG AND RICHTER (1936) Distance A(deg)

Travel time (min sec) at following focal depths h (km) 100"

200

300

400

500

600

37 46 55 04 13 22

13 25 13 34 13 43 13 52 14 01 14 10

13 13 13 13 13 13

14 23 32 41 50 59

13 13 13 13 13 13

04 13 22 31 40 49

12 13 13 13 13 13

55 04 13 22 30 39

12 12 13 13 13 13

44 06 28 51 36 17 21

14 32 14 54 15 16 15 39 16 24 17 09

14 14 15 15 16 16

21 43 05 28 13 58

14 14 14 15 16 16

11

33 55 18 03 48

14 14 14 15 15 16

01 23 45 08 53 38

13 14 14 14 15 16

100 102 104 106 108 110

13 13 13 14 14 14

115 120 125 130 140 150

14 15 15 15 16

' Gutenberg and Richter (1936) gave no data for h < 100 km

46

55 04 13

21 30 52 14 36 59 44

29

700

800

12 12 12 13 13 13

37 46 55 04 13 22

12 12 12 12 13 13

29 37 46 05 14

13 14 14 14 15 16

43 05 27 50 35 20

13 13 14 14 15 16

35 57 19 42 27 12

56

62

MARKUS BATH TABLE XXII. TRAVEL TIMEDIFFERENCES: EQS.(3.13) AND (3.14) MINUS THE GUTENBERG-RICHTER DATA(TABLE XXI) ~

Distance A (deg) 100 110 120 130 140 150

____~

~~

____

____~

~-

Travel time difference (sec) at following focal depths h (km) 100

200

300

-0.3

-0.3

-0.5

-0.5

+0.3 0.0 -0.2 -0.4

+0.3 0.0 -0.2

+0.3 0.0 -0.2 - 0.4

-0.4

400

500

600

700

-0.3

-0.3

-0.5

-0.5

-0.5

-0.5

+0.1 0.6

+0.1 0.6

-0.3 -0.7

+0.2 -0.3 -0.7

-0.5 - 0.9 + 0.6 +0.2 -0.3

+0.3

0.0

-0.2 -0.4

+ +0.2

+

-0.7

measurement of Pdif (see Section 3.6). The agreement must be considered satisfactory, noting also that the Gutenberg-Richter times are given only to the nearest full second. The problem with the numerical value of the a coefficient, encountered in Section 3.3.1, is also met with here. In fact, it proves easier to get an accurate fit to h than to A, contrary to the original expectations. Both linear and quadratic h formulas are worth testing for different h ranges.

3.4. Travel Time Equations Based on Swedish Data There are problems connected not only with the methods used, as explained in the preceding section, but also concerning our observational data. These exhibit a rather uneven distribution with regard to focal depth and distance. Most of the observations are concentrated at shallow depth and close to 100" (Table XVIII). It may seem that the simplest way to achieve results would be t o calculate travel times in each individual case, or to calculate residuals versus any given travel time table, e.g., the one by Gutenberg and Richter (1936).This method can be considered nonanalytical, and would work well if an even distribution of a sufficient number of observations were available for all distances and depths of interest. As this condition is not fulfilled, analytical solutionsmust be sought that permit extrapolations with a better accuracy.

3.4.1. Solutions t(A) Solutions of the type expressed in Eq. (3.10) can be applied provided h = constant. In the limit, this method implies investigation of individual earthquakes as in Section 3.3.1. However, in our case this method is not recommendable due to the few data available for each event as well as the relatively limited distance interval covered by each event.

63

SELECTED P WAVE PROBLEMS

TABLE XXIII. TRAVEL TIMEEQUATIONS BASEDON

DATA

SWEDISH

a. Solutions [(A) = aA t b, (Fig. 18)

Depth" (km) h

ha,

N

Distance A (deg)

0-10 10-20 20-30 30-40 40-60 60-80 80-100

5 16 25 35 48 71 88

28 30 58 75 99 63 42

100-120 102-145 100-131 100-135 101-131 103- 130 101-1 12

Travel time equation ( t in sec, A in deg)

+ 379.4) k 2.6 + 382.4) f 2.9 t = (4.509A + 372.8) f 2.4 t = (4.467A + 376.7) k 3.2 f = (4.491A + 371.6) k 3.4 t = (4.506A + 366.9) k 3.0 t = (4.368A + 379.4) 1.7 t = (4.469A t = (4.434A

b. Solutions t(A, h) = aA + b,h + b, and f(A,h2,h)= aA + h,h2 + b,h + 6 ,

Depth" h(km)

N

Distance A(deg)

0-50

251

100-145

Travel time equation (t in sec, A in deg) t = (4.4628 -

0.1252h t 381.0) f 2.8

+ 380.5) f 2.8 + 382.3) k 2.4 t = (4.430A - 0.001125h2+ 0.0367h + 378.4) 2.4 1 = (4.452A - 0.1050h + 379.0) f 2.0 t = (4.474A - 0.000565h' + 0.0628h + 364.7) f 2.0 t = (4.4608 - 0.001419h2 - 0.0480h

50-100

120

101-130

100-200

100

103-130

300-500

26

101-120

500-700

58

100-141

t = (4.446A - 0.1256h

t = (4.461A - 0.0887h t 370.6) 1.6 t = (4.426A - 0.000131h2 0.0166h t = (4.419A - 0.0912h 375.3) 1.8 t = (4.419A - 0.000092h2 0.0141h

+

+ +

+ 353.6) k 1.5 + 345.5) k 1.8

Earthquakes with h exactly on the limit between adjacent h ranges are included in both ranges, i.e., for h = 10.20, 30, etc. in (a) and for h = 50, 100, and 500 in (b).

Instead, our observations are grouped into a sequence of h intervals, narrow enough to let h be considered as practically constant in each interval (Table XXIIIa). Least squares solutions are given for each h range. However, there are several inconveniences connected with this procedure. For example, h ranges as short as 10 km are practically narrower than the general accuracy of individual h determinations (see Table XIX). There is some inconsistency between the solutions for h = 0-10 km and h = 10-20 km for this reason. In Fig. 18, the 10-km intervals are therefore consistently replaced by 20-km intervals. Even though most of the relations show good agreement, the result for h = 80- 100 km seems to deviate, as shown in an exaggerated way by the reduced time scale in Fig. 18. Another problem with the ?(A) method is its general inapplicability for h > 100 km due to the scarcity of observations. For the method to work

64

MARKUS BATH 385

I

1

I

I

I

I

1

t-4.46h

SEC h KM

0-20

380

375

370

365

100

120

140

FIG.18. Travel time graphs of Pdif based on Swedish observational material for selected depth ranges between h = 0 and h = 100 km with a reduced time scale (Table XXIIIa).

satisfactorily, many observations would be needed in each short interval of h, which is not the case for h > 100 km. Therefore, we have to proceed to larger h intervals to obtain enough observations and then at the same time to include functions of h in the t formula. 3.4.2. Solutions t(A, h)

Table XXIIIb gives the results of the t(A, h) calculation, includingboth types of h formulas developed in Section 3.3.2. The formulas are obtained by the least squares method and include the standard deviations of t. The solutions for h = 0-100 km in Table XXIIIb have standard deviations which on the average are only slightly lower than those of the formulas in Table XXIIIa. One could expect a priori that the formulas including h2 would lead to a higher accuracy than the more simple h formulas. But from Table XXIIIb the standard deviations of t are seen to be virtually as large for the hZ formulas as for the simpler h formulas. This result means that the h2 formulas do not offer any better fit to our observations than the h formulas do. This fact could also be expressed in saying that our observations are too inaccurate to allow any more sophisticated representation than done by the simple h formulas. Even though the accuracy of our observations thus puts a certain limit on the analytical solutions, it is gratifying to see that the standard deviations of our observations (Table XXIIIb) are consistently lower than the standard deviations of the worldwide observations provided by ISC (Table XXb). Our

SELECTED P WAVE PROBLEMS

65

observations do reflect a better homogeneity. This fact is particularly important in view of the rather uneven distribution of our observations with regard to distance and focal depth (Table XVIII). The standard deviation of t of about + 2 sec is no doubt the best possible achievement of standard seismograph equipment, especially of long period, for this type of wave, which often has a rather poorly defined onset. Further discussion of this point follows in Section 3.6. As can be seen in Table XXIIIb there are no solutions provided for h = 200-300 km. This is due to the scarcity of observations. The available 13 observations in this depth range yield the following formula: t = (4.397A - 0.1389h

+ 392.8) & 1.5

(3.15)

Most of the data (N = 11) exist for h = 202-214 km, h,, = 209 km, with A = 102-137" and a t(A) solution: t = (4.419A

+ 361.4) _+ 1.5

The two remaining observations, h = 229 and 242 km, A give a t(A) solution: t = (4.370A

+ 362.9) f 0.0

(3.16) =

108 and 123",also (3.17)

However, none of the Eqs. (3.15)-(3.17) is representative enough to be of use in the final evaluation of t.

3.5. Results Based on Swedish Data 3.5.1. Pdif Travel Times

Our aim has been to develop formulas for t that correspond to the accuracy of our observations and at the same time are as simple as possible. There is no doubt that the linear h formulas in Table XXIIIb meet these criteria. They are therefore used in the calculation of the final travel times. The values for h = 50, 100, 250, and 500 km are interpolated between results of adjacent formulas. Their agreement is generally well within the standard deviations. The travel times, thus calculated to 0.1 sec, are rounded off to the nearest full second. This accuracy is sufficient considering the standard deviations in Table XXIIIb, whereas offering a higher accuracy would be completely illusory. As a final operation, a certain smoothing is done, essentially to remove some apparent inconsistencies due to the rounding off to full seconds. Beyond this, no further smoothing is done. This aspect is important because the results ought to giye a true picture of the observations and real features should not be removed by overdoing the smoothing. The resulting travel times are collected in Table XXIV and graphically demonstrated in Fig. 19. Table XXIV can be

TABLE XXIV. RESULTING TRAVEL TIMESBASEDON SWEDISHDATA(FIG.19) Distance A(deg)

Travel time (min sec) at following focal depths h (km) 0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

106 108 110

13 47 13 56 14 05 14 14 14 23 14 32

13 41 13 50 13 59 1408 14 16 1425

13 35 1343 13 52 1401 14 10 14 19

13 29 13 37 1346 13 55 1404 14 13

13 23 13 32 1341 1349 13 58 1407

13 17 1326 13 35 1343 13 52 1401

13 11 13 20 13 29 13 38 13 46 13 55

13 06 13 15 13 24 13 33 13 41 13 50

13 01 13 10 13 19 13 28 13 37 13 46

1257 1306 13 I5 1324 13 33 1341

12 52 1301 13 10 13 19 13 28 13 36

12 47 12 56 13 05 13 14 13 23 13 31

12 43 12 51 13 00 13 09 13 18 13 27

12 38 1247 1256 1304 13 13 13 22

12 33 1242 1251 1300 1309 13 18

112 114 116 118 120

14 41 14 50 14 59 15 08 15 17

1434 1443 14 52 1501 15 10

1428 14 37 1446 14 55 1504

1422 14 31 1440 1449 14 58

14 16 1425 14 34 1443 14 52

14 10 14 19 1428 14 37 1446

14 04 14 13 14 22 14 31 14 40

13 59 14 08 14 17 14 26 14 35

13 55 14 04 14 13 14 22 14 31

13 50 13 59 14 08 14 17 14 26

1345 13 54 1403 14 12 1421

13 40 13 49 13 58 14 07 14 16

13 36 13 44 13 53 14 02 14 11

I3 31 1340 1349 13 58 1406

1326 13 35 1344 13 53 1402

122 124 126 128 130

15 25 15 34 15 43 15 52 16 01

15 19 15 28 1537 1546 1554

15 12

15 06 15 15 1524 1533 1542

15 01 15 10 15 19 1527 1536

14 55 1504 15 13 1521 1530

14 49 14 58 15 07 15 16 15 24

14 44 14 53 15 02 15 11 15 20

14 39 14 48 14 57 15 06 15 15

14 35 1444 14 52 1501 15 10

14 30 14 39 1447 1456 1505

14 25 14 34 14 42 14 51 I5 00

14 20 14 29 14 37 14 46 14 55

14 15 1424 14 33 1442 1451

14 11 14 19 14 28 14 37 1446

100 102 104

15 21

1530 1539 1548

m

4

132 134 136 138 140

16 10 16 19 16 28 16 37 16 46

16 03 16 12 16 21 16 30 16 39

1557 1606 16 15 16 24 1633

15 51 1600 1609 16 18 1627

15 45

15 54 16 03 16 12 16 21

15 39 15 48 15 57 16 06 16 15

15 33 1542 15 51 1600 1609

1528 15 37 1546 15 55 1604

1524 1533 1542 15 51 1600

15 19 1528 15 37 1546 15 55

15 23 15 32 15 41 15 50

15 09 15 18 15 27 15 35 15 44

15 39

142 144 146 148 150

16 55 I7 04 17 13 17 22 17 30

16 48 16 57 17 06 17 15 17 24

1642 1650 16 59 1708 I 7 17

16 36 1644 16 53 1702 17 11

16 30 16 39 16 48 16 57 17 06

16 24 16 33 16 42 16 51 17 00

16 18 1627 16 36 1645 16 54

16 13 1622 16 31 1640 1649

1609 16 18 1626 16 35 1644

1604 16 13 1621 16 30 1639

15 58 16 07 16 16 16 25 16 34

15 53 16 02 16 I 1 16 20 16 28

15 48 15 57 16 06 16 15 16 23

16 10 16 19

15 39 15 48 15 57 16 06 16 14

152 I54 156 158 160

17 39 17 48 17 57 18 06 18 15

17 32 17 41 17 50 17 59 18 08

1726 17 35 1744 I7 53 1802

1720 17 29 1738 1747 17 56

17 15 17 23 17 32 17 41 17 50

17 08 17 17 17 26 17 35 17 44

1703 17 12 17 21 17 29 17 38

1658 1707 17 16 17 24 1733

16 53 1702 17 11 17 20 17 29

1648 1657 1706 17 15 17 24

16 43 16 52 17 01 17 10 17 18

16 37 16 46 16 55 17 04 17 13

16 32 16 41 16 50 16 59 17 08

16 28 16 37 16 45 16 54 17 03

16 23 16 32 16 41 16 50 16 59

I62 164 166 168 I70

18 24 18 33 18 42 18 51 19 00

18 17 18 26 18 35 18 44 18 53

I8 11 18 19 18 28 18 37 1846

1805 18 13 18 22 18 31 1840

17 59 18 08 18 17 18 26 18 35

17 53 18 02 18 11 18 20 18 29

1747 17 56 1805 18 14 18 23

1742 I7 51 1800 1809 18 18

1738 1747 17 56 18 05 18 14

1733 1742 17 51 18 00 1809

17 27 17 36 17 45 17 54 18 03

17 22 17 30 17 39 17 48 17 57

17 17 17 25 17 34 17 43 17 52

17 12 17 21 17 30 17 38 17 47

17 07 17 16 17 25 17 34 17 43

15 14

15 04 15 13

15 22 15 30

14 59

15 08 15 17 15 26 15 35 15 44

15 52 16 01

14 55 15 04 15 12 15 21 15 30

68

MARKUS BATH

200

400

600

FIG.19. Travel time graphs corresponding to our final results based on Swedish data (Table XXIV). Each isoiine for t (isochron)consists essentially of two straight lines,joined by a smooth curvature in the middle of the figure; i.e., in three dimensions, the t surface resembles a mansard roof.

used for any application, including linear interpolation to get values between those listed. As an alternative to this procedure, the final travel times could have been calculated from the t(A,hz,h) formulas in Table XXIIIb instead of from the t(A, h) formulas. Tests demonstrate that t values from the two approaches agree in general well within the standard deviations of t. Hence, results from the t ( A , h 2 , h ) formulas cannot be considered as more accurate. On the contrary, their accuracy is the same as the one achieved by the simpler t(A, h) formulas, as suggested by their practically identical standard deviations. The residuals versus the Gutenberg-Richter (1936) travel times for Pdif are summarized in Table XXV. It is obvious that our times are consistently a few seconds less than those of Gutenberg and Richter. The average difference is - 3.4 f 1.0 sec (N = 42). There is some tendency for the residuals to increase in absolute value with greater distance and greater depth. This tendency shows some correlation with the frequency distribution of our observations (Table XVIII). It is not unlikely that this correlation has some significance. Anyway, the slightly earlier times now deduced are probably typical for sensitive instruments on a shield structure versus the records available when Gutenberg and Richter (1936) made their determination. Table XXV includes also a comparison with the Jeffreys-Bullen (1967) travel times, as far as this is possible.

69

SELECTED P WAVE PROBLEMS TABLE XXV. TRAVEL TIMEDIFFERENCES a. Our data (Table X X I V ) minus the Gutenberg-Richter (1936) data (Table X X l )

Distance A(deg) 100 110 120 130 140 150

Travel time difference (sec) at following focal depths h (km) 100

200

300

400

500

600

700

-2

-2

-3

-3

-4

-2

-3

-3 -3 -3

-4

-3 -3 -2 -3 -3 -4

-4 -4

-3 -3 -4

-3 -3 -2 -3

-3

-3 -2

-4 -4

-3 -4

-3 -3 -4

-5 -6

-3 -4 -5 -6

h. Our data (Table XXIV) minus the Jeffreys-Bullen (1967) data

Focal depth h (km)

Distance A (deg)

0

100

350

600

100 102 104

- 1.4 - 1.4 - 1.2

-0.5 - 1.4 - 1.2

- 1.1 -0.9

-0.4

+ 0.4

3.5.2. Pdif Velocity The primary result of our investigation consists of the travel time table (Table XXIV). A secondary result concerns the P wave velocity u2 at the base of the mantle. In Section 3.3.1 we determined o2 on the basis of the ISC data for 10 earthquakes. The t(A, h) formulas in Table XXIIIb, used in the derivation of our travel time table, give a = 4.452 as a weighted average (weighted by respective number of observations). With r2 = 3471 km, this a value yields u2 = 13.61 km/sec by Eq. (3.11), or with r2 = 3473 km, yields u2 = 13.62 km/sec. The c(A, h2, h) formulas in Table XXIIIb lead to a = 4.450 as a weighted average and the same values of u2 as those already obtained from t(A, h). These results should be compared with the P wave velocity at the mantle base = 13.64 km/sec (Bullen, 1963, p. 223). The agreement between the u2 values must be regarded as very satisfactory. This result indicates that our observations are more homogeneous than those obtained from the ISC bulletins. The difference between the respective standard deviations of the

70

MARKUS BATH

travel time equations supports this conclusion. Moreover, the a values in Table XXIIIb are not significantly different, contrary to those of Table XXb (Section 3.3.1). 3.6. Discussion of Methods and Results

Most of the methodical problems connected with the development of the Pdif travel time table have already been discussed in preceding sections. Therefore, only a few supplementary points will be included in this section. 3.6.1. Scatter of Data: Source Eflects In our solutions, early and late arrivals of Pdif are treated alike, i.e., early arrivals are not considered more accurate than later ones. One could suspect that early arrivals would be more reliable and representative than late ones. Even though this may be true in some cases, it is no general rule. Apart from the clearness and sharpness of an arriving phase, there are other factors that may cause apparently early or late arrivals, among them the accuracy of the source parameters used. In particular, the focal depth may have a great effect in some individual cases (Table XIX). Say that the depth is assumed to be 100 km but the true depth is 75 km. Then the Pdif arrival will appear to be 3.1 sec too late. The average depth effect, however, is smaller. A combination of Table XIX (righthand column) with the t(A, h) formulas in Table XXIIIb results in a travel time error of & 1.3 sec as a weighted average, due to focal depth errors. Inaccuracies of origin time and epicenter location also cause apparent divergences. Comparing ISC and USGS source data, we find as averages for 1965 and 1974 (first and last year in our investigation) a difference of about - 1 sec of origin time and a difference of +0.09" of epicenter location, the latter equivalent to a travel time difference of only 0.2 sec.

+

3.6.2. Scatter of Data: Path and Receiver Effects

In addition to the reason for early or late arrivals just mentioned, regional variations of structural properties affecting the wave velocities will cause deviations from the average. Regional corrections (path corrections) could be derived to correct for structural variations as well as for ellipticity. In the present investigation, I refrain from the derivation of regional corrections for several reasons. Mainly, there are no further methodical problems involved in their determination. Also, they are not expected to provide any significant improvement of the travel times, due to the influence of other independent factors.

SELECTED P WAVE PROBLEMS

71

With 80%of our investigated earthquakes located in the southwest Pacific, we have to expect that regional corrections (structural properties and ellipticity) for our stations are approximately constant and therefore constitute only a minor contribution to the observed scatter. For our source and station locations, the ellipticity correction is mostly insignificant, amounting on average to 0.2 sec in absolute value (Comrie, 1938; Jeffreys and Bullen, 1967, p. 50). Because of the relative insignificanceof ellipticity, no correction is made for this effect on our travel times. Neither, therefore, should any ellipticity correction be applied in the application of our travel time table. The measuring accuracy (for clear onsets about f0.1-0.2 sec on shortperiod seismograms and about & 1 sec on long-period ones) introduces a scatter in addition to the source effects mentioned above. Because of the different possible reasons for apparently early or late arrivals, I consider it most correct to treat all such arrivals alike, i.e., as scattered observations. And the standard deviation of & 2 sec of the final t values must be regarded as satisfactory under the prevailing observational conditions. Numerically, it agrees well with the error analysis above if partial compensation of different errors is taken into account. 3.6.3. Number of Data On the basis of Table XVIII it may be suspected that our data are too incomplete for several distance and depth ranges. It could be surmised that the standard deviations (as a true measure of observational scatter) would decrease, i.e., the reliability would increase, with an increasing number of observations. However, this belief is contradicted by Table XXIIIb, which demonstrates increasing standard deviations with increasing number of observations. Improved results depend not only on the number of observations. Probably even more important is the quality of the data. All unreliable observations are excluded, keeping only those that can be considered as fully reliable and homogeneous. Comparison of the standard deviations in Tables XXIIIb and XXb testifies to this effect. In view of the combined effect of reliability and number of observations, the results are as reliable as possible from our records. This statement is verified by the fact that the results for those h ranges that have few observations conform very well to adjacent ranges with numerous data. The relatively simple character of the t(A, h) function is an important factor in this connection. It permits both interpolations and extrapolations to be made with confidence, and by this property it compensates for the uneven distribution of our observations. Introducing a greater amount of data is therefore not expected to lead to any significant change of the resulting travel times (Table XXIV).

72

MARKUS BATH

3.6.4. Diffracted S Waves

The present investigation is restricted to the Pdif wave. Travel times of Sdif could be derived by methods analogous to those used here, but there might be data problems. There is a scarcity of sufficiently reliable Sdif observations. For example, our bulletins for 1965-1974, used for Pdif, contain insufficient data for a fully reliable analysis of Sdif. Within the distance range of 100-126" there are about 100 observations of Sdif in our bulletins for 1965-1974, three-quarters of which belong to the range 100-1 lo". They exhibit SH motion and, recorded only by long-period seismographs,they have wave periods of 20-25 sec. The ISC bulletins do not seem to provide any data on Sdif, and Gutenberg and Richter (1936) listed travel times for Sdif up to A = 120" only. But with sufficientlynumerous and reliable readings of Sdif available,the development of the corresponding travel times would not offer any additional methodological problems. An interesting study of core-diffracted SH waves with the purpose of determining the S wave velocity at the mantle base was presented by Bolt and Niazi (1984)(see also Doornbos and Mondt, 1979a,b;Okal and GeHer, 1979; Mula, 1981, all with useful additional references to papers on Sdif research). 3.7. Conclusions

1 . Theoretically, the Pdif travel time equation is of a simple form, analogous to that of crustal waves. It is linear in A, and A and h appear in separate terms. 2. Observationally, using ISC data, the linearity of the t-A relation is certainly confirmed, but the slope, i.e., the P wave velocity at the mantle base, varies from case to case, in part suggesting significant differences. Data from the Swedish seismograph network in the period 1965-1974, analyzed here, are more homogeneous, yielding a P wave velocity of 13.62 km/sec at the base of the mantle. This value is in good agreement with 13.64 km/sec to be found in the literature. 3. For the analysis of the Swedish data, a number of t(A, h) relations were tested. It transpires that a relation, linear in both A and h and evaluated for a sequence of h intervals, combines accuracy and simplicity in the best possible way. This type of relation is therefore used in the calculation of our travel time table. 4. Resulting travel times, based on Swedish data, were tabulated for A = 100-170" and h = 0-700 km. Their standard deviation with respect to the observations is about 1 2 sec. This is the maximum possible accuracy achievable with our observational material and must be considered satis-

SELECTED P WAVE PROBLEMS

73

factory in view of various influences: source effects (errors on source parameters, especially focal depth), path effects (structural variations, affecting wave velocity; ellipticity), and receiver effects (sharpness of recorded Pdif). 5. Compared to the Gutenberg and Richter (1936) Pdif travel times, our times are in general 2-4 sec less. Our results can be regarded as applicable to modern, sensitive recordings at stations on typical continental crust without sediments, especially in shield areas. In this sense our results constitute an extension to the P wave travel times determined for the same area by Enayatollah (1972). 6. The travel time table (Table XXIV) and the corresponding formulas (Table XXIIIb) supplement each other in the sense that the former is most convenient in working with seismic records, while the latter may be useful in further calculations, theoretical or empirical. ACKNOWLEDGMENTS I want to express my most sincere thanks and appreciation for the assistance oRered by personnel of the Seismological Section, Uppsala, during the preparation of this manuscript. Mrs. Britta Wallwork contributed to the typing and Mr. Conny Holmqvist did the final drawing of all figures. Finally, I want to thank Professor James Brown (Calgary, Alberta, Canada) for his critical and constructive reading of the manuscript.

APPENDIX1. MULTIPLEP OBSERVATIONS Time lag 6T(sec) No.

1

2 3 4

1969 Feb 3 M= m

19

21 31 31 Apr 17 19 Jun 18 23 Jul 18 Aug 12 I2 12 13 19 21 28 Nov 24 Dec 18

20 21 22 23

I970 Jan 10 19 20 Feb 6

5

2

Date

6 7 8

9 10 II

12 13 14 15

16 17 18

Origin time (GMT)

Epicenter (deg)

Depth h(km)

Magn. m

UPP

KIR

7.0 6.7 5.9 7.0 6.5 5.3 5.5

3.1

4.5 4.2 2.3 2.4 3.6 4.4

21 41 41.9 16 18 56.4 04 56 20.3 07 15 54.4 19 25 27.2 08 01 04.1 08 16 06.5 01 38 46.4 07 08 27.1 05 24 48.0 03 58 19.9 05 53 28.2 23 05 57.1 08 31 32.2 01 39 08.3 03 32 11.5 03 58 34.8 I7 23 20.2 13 32 05.2

4.9N 8.7N 31.2N 27.7N 38.3N 27.6N 25.2N 59.5N 18.4N 38.3N 43.1N 43.7N 43.3N 44.ON 6.1s 43.2N 39.1N 37.2N 46.3N

127.4E 127.3E 114.2W 34.OE 134.6E 34.OE 46.7W 145.0W 104.5W I19.4E 148.7E 148.5E 147.18 147.7E 105.3E l47.OE 73.68 71.7E 142.5E

35 35 35 35 410 35 35 30 35 35

12 07 08.6

6.8N 41.IN 42.5N 23.IN

126.78 69.38 143.OE 100.8E

00 31 52.6

17 33 05.4 22 10 41.6

35 50

35 35 50 45 20

2.4 2.6 4.0 4.5

5.8

2.2

6.0 6.9 4.9 6.6 6.2 6.3 6. I 6. I

4.0 2.8 2.3 2.5 3. I 7.0 5.6

5.8

4.5 2.9 1.5

3.3

2.2 4.0 5. I 2.9 2.5 3.0

2.2 3.4 4.5

6.9 5.7 2.9

7.2 6.3 2.2

3.0

2.3

0.9

370

6.2 6.2

70 45 45 35

7.1 5.5 7.0 6.2

4.8 4.6

2.6

1.1

1.6 2.9

140

SKA

2.6 2.6

1.1

UME

2.3 4.3 4.9 2.6 3.9 5.4 5.2 2.3 3.4 4.3 3.2 2.1 2.7 I .6 5.5

5.3 3.0 0.8

UDD

2.5 2.7 4.0 4.1 3.6 2.0 4.4 2.9 3.0 2.9 1.6 7.1 6.9

DEL

4.6 I .4 3.6

5.1

2.5 2.5 4.8

1.1 I .5

2.7 5.0 1.5

5.0 1.1

2.5

4.6 I .o

MeanfSD

Region

3.3 f 1.1 4.4 f 0.2 2.7 f 1.2 2.4 f 0.5 3.7 f 0.3 4.8 f 0.5 4.4 f 1.1 2.2 f 0.1 3.6 f 0.3 4.6 f 0.4 3.0 f 0.2 2.5 f 0.3 2.7 f 0.2 2.1 f 0.9 6 . 4 f 1.0 6.0 f 0.6 2.7 f 0.4 1 .o f 0.2 2.3 f 0.8

Talaud Islands Mindanao Gulf of California Red Sea Sea of Japan Red sea North Atlantic Ocean Alaska Mexico China Kurile Islands Kurile Islands Kurile Islands Kurile Islands Sunda Strait Kurile Islands Tadzhik-Sinkiang Pamir Sakhalin

3.4 f 1.2 4.8 f 0.2 1.3 f 0.3 2.7 f 0.3

Mindanao Kirghiz SSR Japan China

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

v, 4

39 40 41 42 43 44 45 46 47 48 49 50 51 52

18

19 58 48.1 16 46 45.6 14 23 25.1 05 18 06.8 20 12 16.9 15 53 11.2 16 33 29.2 16 28 47. I 07 52 27.9 00 57 42.8 00 26 21.6 12 47 00.3 23 50 12.2

47.2N 52.7N 27.8N 6.8N 39.8N 15.1N 50.2N 0.0s 27.3N 19.9N 52.2N 54.2N 34.6N 4.2s 5.IN

154.1E 175.1W 56.3E 126.78 141.8E 122.1E 91.3E 17.9W 141.7E 121.8E I51.4E 164.6W 136.7E 103.4E 123.58

35 I70 35 55 80 35 35 45 35 35 610 25 350 100 520

6.3 7.1 6.0 7.1 6.5 6.3 5.7 6.0 6.4 6.0 6.3 5.9 5.7 6.0 6.6

1971 Feh 7 Apr 30 May 22 Jun 2 II 15 22 28 Jul 17 Aug 27 Sep 2 Oct 9

02 42 04.5 15 48 06.5 20 03 32.4 10 05 09.3 12 56 04.3 22 04 13.4 10 25 32.9 05 01 48.6 05 32 42.9 05 20 15.1 18 24 47.3 13 15 36.4

51.2N 52.8N 32.4N 29.4N 18.ON 41.5N 41.3N 37.9N 7.ON 30.2N 30.1N 24.9N

177.IW 112.5E 92.1E 51.6E 69.8W 79.38 79.3E 106.2E 94.7E 50.7E 50.8E 122.1E

50

45 35 I40 55 45 90

6.5 6.3 6.2 5.3 6.9 6.5 5.3 6.0 6.4 5.5 5.6 6.1

1.8 4.2 2.9 4.9

02 06 01.2 02 06 23.3

23.6N 102.7E 22.5N 122.38

35 35

6.0 7.2

2.0

7 28 28 Mar 30 Apr I I2 May 15 Jun 26 Aug 7 21

kP5 Oct 29 Nov 10 Dec 13

I972 Jan 23 25

10 01 05.4 10 52 31.2

35 35 35 60 35

3.1 4.5 2.0 4.2 2.6

2.3 3.6 2.9 7.8 3.7 4.9 1.0 1.5 3.6 2.5 5.3 3.7 5.4 3.9

4.4 4.6 4.5 4.1 2.8

4.2 3.7 3.0

3.5 4.5 4.9 2.4 4.5 3.0

8.0

3.2 3.2 9.1

4.3

5.6

0.9 1.5 4.4

3.5 2.6 7.6 3.1 4.9 0.9 1.4 4.1 1.6 4.5

5.2 4.6 3.9 5.0 2.7 3.5 3.8 1.8

1.0

1.1

1.6 4.6

5.7

5.5

2.5 2.1

2.0

2.7 2.5

2.8 2.3

4.4

3.1 3.9 2.3

4.1

5.3

3.5

3.6 4.6

4.5 3.5 3.1 7.0 4.2 5.2

1.2 3.8 2.9 5.4 3.3 6. I 3.8 1.9 4.4 2.8 5.6

5.3 4.0 6.1 3.5 1.7

1.6 4.6 2.9 5.3 3.7 5.4 3.4 1.5

3.4 4.5 4.7 3.0 4.0 3.0 3.5 3.5 2.2

3.1 3.2

5.5 3.6 3.0 1.7 4.0

3.9 f 0.7 4.5 f 0.1 4.7 f 0.2 3.2 f 1.0 4.4 f 0.4 2.8 f 0.2 3.4 f 0.2 3.7 0.2 2.2 f 0.2 4.1 f 0.5 3.4 f 0.2 3.0 f 0.2 7.9 f 0.8 3.9 f 0.3 4.8 f 0.6

Kurile Islands Aleutian Islands Iran Mindanao Japan Luzon USSR- Mongolia North of Ascension Isl. Bonin Islands Luzon Okhotsk Sea Unimak Island Japan Sumatra Mindanao

1.0 f 0.1 1.5 f 0.2 4.2 f 0.4 2.5 f 0.6 5.2 f 0.4 3.7 f 0.3 5.8 f 0.4 3.5 f 0.3 I .7 0.2 4.3 f 0.2 2.9 f 0. I 5.4 f 0.3

Aleutian Islands Aleutian Islands Tibet Iran Dominican Republic Kirghiz-Sinkiang Kirghiz-Sinkiang China Nicohar Islands Iran Iran Taiwan

2.5 f 0.4 2.2 f 0.2

China Taiwan

*

*

continued

APPENDIX1. (Continued) Time lag 6T (a) No.

4

m

53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71

72 73 74 75 76

Date Jan 25 Feb 21 26 29 Mar 20 Apr I0 12 21 26 29 30 May 17 17 22 Jun 2 8 10 13 28 Jul 2 27 30 Aug 3 4

Origin time (GMT) 03 41 23.7 19 34 50.9 23 31 09.6 09 22 59.8 07 33 49.6 20 27 07.5 23 07 49.9 01 28 09.5 17 35 08.4 16 04 21.1 15 15 34.3 05 27 51.6 10 06 05.8 06 04 00.1 22 50 47.4 23 10 12.0 03 31 24.1 00 55 37.3 09 49 34.9 I 2 56 06.7 16 41 30.2 21 45 14.1 04 40 54.9 17 51 12.9

Epicenter (ded 23.0N 55.9N 50.6N 33.3N 6.8s 28.4N 28.4N 54.ON 13.4N 28.3N 13.5N 13.4N 33.5N 16.6N 3.6N 29.5N 61.5N 33.1N 27.6N 30.IN 25.4N 56.8N 51.2N 49.2N

Depth h(km)

Magn. m

l22.2E 158.3W 97.3E

35 60 35

140.8E

55

76.8W 52.98 53.OE 166.9W 120.6E 53.OE 120.5E 119.9E 71.5E I22.3E 96.7E 92.38 140.2W 46.38 33.8E 50.8E 130.5E 135.7W 178.IW I56.1E

60 35 35 120

6.8 6.7 6.0 1.4 6.6 5.1

55 35 55 40 35 35 55

5.6 6.1 5.7 5.4 6.1 6.0 5.6 6.8 6.1

60

5.1

35 25 15 30 20 25 50

6.2

55

UPP

KIR

SKA

2.6

2.2 1.3 2.5

1.0

1.4 3.6

3.0 3.7

3.0 3.6 2.2 2.0 4.8 1.9 1.3 2.3 2.1 1.3 2.2 3.8 2.5

1.0

2.8 2.0 4.5 2.1 2.2 1.7 4.6 1.8

1.4 2.2

6.1 5.5 5.9 7.1 6.4 6.0

3.6 2.4 3.2 0.7 1.1

3.4 77 78 79

9 Sep 2 24

15 34 49.6 01 49 36.8 20 09 35.6

26.4N 140.5E 29.4N 130.5E 6.3s 131.2E

470 40 35

5.6 6.5 6.9

3.6 2.7 1.9 1.8 5.8

1.9 1.2

1.5

5.5

1.0

1.7

UDD

1.9

2.1 1.3 2.4 2.1 4.2 2.4 1.9 0.9 5.2 2.3 1.2 1.8

2.0

0.6

1.8

2.6 4.3 2.0

UME

1.9 4.2 2.9 0.8 0.7 3.9 4.0 2.5 0.6 0.7 3.5 1.6 0.9 2.7

1.9 2.3 3.7

1.1

1.3 3.9 3.0

1.2 3.9 2.8 2.7

1.0

3.6 1.7

1.0

2.7

DEL

1.4 2.5 2.0 4.4

1.6

3.3 1.1

3.4 2.6 2.6 0.7 0.9

1.4

2.5 2.4 1.0 2.9 3.0

I .2 0.7 3.0

2.2 f 0.3 1.2 f 0.2 2.8 f 0.5 2.0 f 0.1 3.8 f 0.7 2.9 f 0.7 2.1 f 0.2 1.6 f 0.4 5.1 f 0.5

1.8

2.0 2.4

Meanf SD

2.0 f 0.2 1.2 f 0.3 2.1 f 0.3 1.9 f 0.2 I .7 f 0.3 2.3 f 0.2 4.0 f 0.3 2.6 f 0.5 1.0 f 0.1 1.1 f 0.3 3.6 f 0.4 3.0 f 0.7 2.8 f 0.3 0.7 f 0.1 0.9 f 0.2 3.5 f 0.1 1.3 f 0.3 1.1 f 0.5 2.8 f 0.2

Region Taiwan Alaska USSR-Mongolia Japan Peru Iran Iran Aleutian Islands Mindoro Iran Mindoro Mindoro Pakistan Luzon Sumatra Tibet Canada Iran-Iraq Egypt Iran Ryukyu Islands Alaska Aleutian Islands Kurile Islands Bonin Islands Ryukyu Islands Tanimbar Islands

80 81

Dec 2 2

82 83 84

15 28

85 86 87

1973 Jan 30 31 Feb 6

10

-

1 4 .1

88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 I05 I06 I07 108 109 110

7 7 8 10

Mar 9 12 13 14 19 20 22 Apr 14 25 26 May 17 29 Jun 9 11

17

22 30 Jul 8 Aug 2

00 19 47.2

05 54 10.6

6.5N 6.5N 44.8N 33.2N 5.6N

21 01 12.5 20 55 53.1 10 37 10.1

18.5N 103.OW 28.2N 139.2E 31.4N 100.6E

45 520

40

6.9 6.5 6.6

05 27 20.0 16 06 25.0 19 05 21.9 I I 53 27.5 10 06 37.7 15 39 58.1 06 03 49.0 03 45 41.7 I I 41 07.7 19 09 06.6 01 06 57.2 08 34 00.1 1421 13.2

32.0N 49.48 31.5N 100.3E 10.4s 13.0W 18.9N 103.5W 6.3N 127.3E 31.9N 49.IE 32.1N 49.48 32.ON 49.4E 52.8N 173.88 8.3s 117.4E 28.IN 87.OE 10.7N 84.8W 33.4N 140.7E

50 35 35 35 55 35 35 35 100 180 35 25 70

5.7 6.5 5.9 6.1 6.8 4.9 5.4 4.9 6.4 6.4 5.7 6. I 6.0

14 30 05.4 09 38 09.9 01 46 44.9 04 19 14.3 08 42 04.0 13 33 28.3 06 07 37.9 17 43 27.6 04 03 34.5 08 58 15.1

27.1N 60.8E 41.ON 82.28 51.7N 176.2E 41.ON 82.3E 53.7N 161.6E 43.1N 145.48 42.9N 146.38 22.9N 121.4E 6.8N 73.0W 27.8N 104.5E

45 35

5.8

01 40 47.8 18 26 07.1 01 51 58.3

126.68 126.68 149.4E 140.8E 127.IE

35 55 I5 55 70

7.2 6.5 6.4 5.6 6.5

2.8 3.O

3.4 3.3

1.9

1.9

3.7 2.9 1 .o 3.1 4. I 3.6 4.1

55

35 30 45 50 35 160 35

5.8 6.2

3.3

2.6 3.0 1.7 4.0

2.9 3.1 1.5 3.1 3.6

3.8 3.2

3.9 3.0 1.3 4.4 3.9 3.9 3.9

2.1

6.0 3.6 3.7 4.0 1.1

4.8 4.2 2.8 1.O 5.0 3.0 1.8

4.3

5.5

6.6 6.4 6.5 5.9 5.8 6.4

3.0 2.3 3.3

1.8

2.0 4.0 0.8

I .5

3.7 2.9

5.0 3.9 3.3 5.5

I .o 4.0 3.3 I .o 5.0 3.2 2.2 3.7 1.9 2.1 4.9 3. I 3.4 1.5

3.7 3.7 4.1 3.6 5.1

4.1 4.0 3.4 2.8 5. I

I .o

1.1

4.1 3.0 1.1

3.5 2.1 4.2

5.0 2.7 4.0 1.7

5.3 3.1 4.3

4.2 2.4 3.3

3. I

1.3

5.3 3.9 3.7 4.2 I .4 4.4 4.5 3.0 I .o 5.3 2.8 I .8 3.7 2.0 I .8 4.5 2.0 3.8 0.9 I .3

3.7 1.8

3.7 2.9 1.1 3.7 2.6 3.2 5.1 3.8 3.7 1.5 4.9 3.3

2.9 f 0.3 3.2 f 0.3 1.9 f 0.3 3.4 f 0.4 2.9 f 1.1

Mindanao Mindanao Kurile Islands Japan Mindanao

3.8 f 0.1 3.0 f 0.1 1.1 f 0.2 3.7 f 0.7 4.0 f 0.2 3.8 f 0.8 3.9 f 0.3 3.2 f 0.3 5.4 f 0.4 3.8 f 0.2 3.7 f 0.0 4.1 f 0.1 1.2 0.2 4.5 f 0.4 4.3 f 0.2 3.1 f 0.2

Mexico Bonin Islands China

*

1.0 f 0.1 5.1 f 0.2 3.0 f 0.3 2.0 f 0.2

3.7 2.0 5.4 2.9 3.4 0.8 0.6

3.9 f 0.3 2.0 f 0. I 1.9 f 0.2 4.9 f 0.5 2.6 f 0.5 3.7 f 0.4 0.8 f 0. I 1.6 f 0.8

Iran China Ascension Island Mexim Mindanao Iran Iran Iran Aleutian Islands Sumbawa Island Tibet

Costa Rica Japan Iran China Aleutian Islands China Kamchatka Japan Japan Taiwan Colombia China

continued

APPENDIX 1. (Continued) Time lag 6T(sec)

m 4

No.

Date

Ill 112 1 I3 1 I4 115 116 117 118 1 I9 I20

Aug 9

121 I22 123 I24 I25 126 127 I28 I29 130

11

16 24 SeP 9 9 9 20 Nov 3 6

-

1974 Jan 7 Feb 22 Mar 3 12 24 Apr I I 16 17 May I I 17

Origin time (GMT)

Epicenter (deg)

10 44 26.5 07 15 39.7 06 05 28.2 02 06 01.6 02 42 33.1 08 32 14.8 18 25 49.4 20 43 39.8 00 19 51.5 18 26 35.1

43.4N 33.ON 23.2N 27.8N 31.5N 7.1s 39.5N 9.0N 54.6N 51.6N

146.4E 104.OE 100.9E 52.7E 100.OE 12.8W 143.1E 123.8E 161.4E 175.2W

15 24 38.2

33.3N 33.2N 35.6N 23.6N 27.7N 42.4N 13.8N 35.2N I .7N 31.3N

47.9E 136.9E 140.6E 125.4E 86.IE 144.4E 120.6E 35.3W 126.48 51.1E

00 36 53.8 04 50 48.9 18 07 56.0

14 16 03.1 21 37 52.9 I I 22 52.9 00 32 21.4 00 43 44.9 19 46 20.2

Depth h(km)

Magn. rn

40 35 35 35 35 25 35

40

6.4 6.4 5.6 5.6 5.8 5.7 6.5 6.5 5.9 6.4

30 380 45 70 35 80 I20 35 35 35

5.6 6.5 6.0 5.3 6.3 5.9 5.6 5.6 6.6 4.9

570

60

UPP

KIR

SKA

UME

UDD

DEL

MeaniSD

2.1 2.9 4.0 3.6 3.0

I .9 3.0 4.1 3.2 3.1

I .4 2.0 3.4 3.3 2.6

I .9 3.6 3.9

I .8 2.2 3.9 4.5 3.2 2.6 2.1 2.6 0.9

2.6 1.7

1.1

1.5

2.0 + 0.4 2.6 f 0.7 3.9 f 0.3 3.7 f 0.6 3.0 f 0.2 2.3 i 0.4 1.8 f 0.4 2.5 f 0.2 0.8 f 0.2 1.3 f 0.2

Kurile Islands China China Iran China Ascension Island Japan Negros Kamchatka Aleutian Islands

3.6 2.2 I .7 9.8 2.0 4.0 1.9

3.9 2.1 I .8

3.8 f 0.1 2.4 i 0.4 1.7 f 0.1 9.6 f 0.2 1.7 i 0.2 3.4 f 0.6 1.8 f 0.1 3.3 f 1.3 3.1 f 0.1 4.1 f 0.9

Iran Japan Japan Ryukyu Islands Nepal Japan Mindoro North Atlantic Ocean Molucca Passage Iran

2.6

I .2 3.8 2. I 1.6 9.4 I .6 3.6 I .8 2.3 3. I 3.6

1.4 2.5 0.6 1.2 3.1 2.6 1.8

I .8 2.5 4.7 3.2

2.2

3.0 1.7 9.5 1.5 2.8 I .7 3.3 3. I

4.4

3.0 2.0 I .9 2.6 I .o

2.2 1.8

1.5

3.5 1.8 4.5 3.3 5.4

3.1 3.2

2.4

4.1 I .9

3.0 3.4

4. I

Region

131 I32 133 I34 135 I36 I37 138 139 I40 141 I42 I43 144

4 W

145 146 147 148 149 150 I51 152 153

Jun 5 15

Jul 2 13 28 29 Aug I 3 II II II II 27 27 Sep 3 7 27 oct 10 10 Nov 8

Dec2 7 28

12 30 20.2 10 19 29.2 16 25 53.3 23 08 41.9 I I 34 59.7 03 15 16.7 22 39 21 .o 18 I6 34.0 05 12 33.3 13 59 23.8 20 05 30.1 22 10 27.1 05 43 33.7 17 33 58.1 05 59 41.0 2043 11.5 12 06 57.0 06 48 14.0 06 56 49.0 21 23 21.8 01 08 45.9 07 34 1 1 .O 12 I I 43.7

28.2N 104.OE 31.8N 100.18 35.6N 80.9E 7.IN 77.7W 46.3N 153.3E 46.2N 153.1E 49.8N 156.OE 36.ON 139.88 39.3N 73.88 39.5N 73.9E 39.5N 73.7E 39.4N 73.78 39.3N 74.OE 39.4N 73.9E 18.2N 119.1E 9.8s 108.4E 2.5N 71.8W 41.ON 143.IE 40.9N 143.1E 42.5N 141.8E 24.5N 95.3E 51.9N 170.8W 35.IN 72.9E

35 35 35 25

5.7 6.0 5.1

60

5.9 7.5 6.7 6.0 6.2

35

5.8

35 35 35 35 35

5.3 6.1

60 40

40

35 35 70 55 45 I30 130 45 20

5.0 5.5

5.7 5.9 6.8 5.3 6.4 6.6 7.0 5.3 6.3 6.5

3.1 3.9 2.1

4.2 2.0

3.3 3.3

1.3 3.2 2.3 3.1 4.4 3.1 2.8

1.7 3.7 5.0 2.0 4.5 3.6

1.4 3.2

1.5

2.6 3.7 2.0 3.0 1.1

3.2

2.8 3.2 2.4 2.6

2.6 3.2 2.0 3.0

2.1 3.6

2.7 3.3

3.4 2.3 3.5

3.0 2.4 3.2 2.9 2.4 3.4 1.8

4. I 6.3 1.8 4.9 3.1

3.6 4.6

3.4

5.6 2.7

1.7 1.2

2.0 1.2

5.6 3.7 5.9 2.2

1.8

5.1

2.2

2.3 3.8 2.3 2.6

1.2

3.8 2.4

0.9 3.5

1.4 2.9 3.1 2.5 3.7 2.9 2.3

2.8 3.2 2.5 3.5 3.0 3.4

1.8 3.3 4.9 2.2 5.5

3.3 5.7 1.7 1.3

3.6 4.9 3.9 2.9 2.4

2.7 f 0.4 3.8 f 0.3 2.1 f 0.1 2.8 f 0.4 1.2 f 0.2 3.3 f 0. I 2.0 f 0.5 3.0 f 0.3 3.4 f 0.5 2.4 f 0.2 3.2 f 0.4 2.9 f 0.1 2.4 f 0.3 3.4 f 0.1 1.8 f 0.1 3.6 f 0.3 5.1 f 0.7 2.0 f 0.2 5.0 f 0.7 3.2 i 0.4 5.6 f 0.4 2.0 f 0.3 1.3 f 0.1

China China Kashmir-Tibet Panama-Colombia Kurile Islands Kurile Islands Kurile Islands Japan Tadzhik4inkiang Tadzhik-Sinkiang Tadzhik- Sinkiang Tadzhik-Sinkiang Tadzhik-Sinkiang Tadzhik-Sinkiang Luzon South of Java Colombia Japan Japan Japan Burma Aleutian Islands Pakistan

APPENDIX 2. DIFFRACTED P OBSERVATIONS Date

Origin t h e (GMT)

Epicenter (deg)

Depth h (W

1965 Jan 10 24

13 36 30.8 00 11 12.0

13.498 166.54E 2.40s 125.98E

32 6

Mar3 28 Apr 9 9 May 12 31 Aug5 11 11 11 12 12 20

15 1409.3 16 33 15.2 22 52 24.6 23 47 12.9 10 33 44.3 11 3825.9 0007 52.0 03 40 55.5 19 52 29.2 2231 49.1 0801 44.0 12 57 10.0 05 54 50.6

5.36s 151.9E 32.428 71.10W 4.20s 134.05E 4.01s 134.228 6.20s 130.33E 7.53s 128.60E 5.21s 151.65E 15.473 166.91E 15.648 167.00E 15.753 167.12E 15.86s 167.368 5.25s 152.138 5.748 128.63E

33 68 33 33 136 19 59 14 23 31 26 38 328

Oct 4 Nov 21

00 13 26.6 10 31 51.0

6.383 147.40E 6.21s 130.39E

82 101

1966 Feb 22 May 16

05 02 40.7 02 46 39.2

5.41s 151.57E 6.958 129.37E

59 182

Pdif: Station and t/A (secldeg) UME 940.2/124.52 UPP 830.7J101.22 KLS 842.1/103.25 KIR 867.7/109.64 UPP 893.811 16.82 UME 838.9/104.11 KIR 835.0J102.53 UPP 839.1/106.63 KIR 842.1/103.76 UPP 888.9/115.29 UPP 959.5/130.33 UPP 958.8/130.52 UPP 956.9/130.66 UME 940.0/127.01 UPP 890.0/115.51 UPP 812.8/105.41 UME 801.3/103.20 KIR 857.6/109.20 UPP 842.5/106.67 GOT 859.0/110.16

KIR 865.5/109.58 UPP 833.7/106.82 UME 824.4/104.60

SKA 838.7J102.52 GOT 841.01104.67 UME 874.7/112.13 UME 900.8/119.14 UME 839.7/104.01 KIR 823.4/103.16 UME 828.7/104.33

KIR 856.0/109.42

UME 870.0/111.90 KIR 926.5/123.38 UME 941.5/126.51 UME 944.8/126.70 KIR 923.9/123.71

UME 874.0/112.11 KIR 797.2/102.12 SKA 818.4/106.64 KLS 819.4/107.47 KIR 828.4/103.19 SKA 849.0/107.80 UME 832.8/104.37 UME 873.2/112.06 KIR 819.2/103.50 SKA 840.5/108.04

Region New Hebrides Islands" Gram Sea New Britain Chile New Guinea New Guinea Banda Sea Banda Sea New Britain New Hebrides Islands New Hebrides Islands New Hebrides Islands New Hebrides Islands New Britain Banda Sea New Guinea Banda Sea

New Britain Banda Sea

03

Jun 7 13 14 15 22

00 59 41 18 08 36.6 16 39 47.9 00 59 46.1 20 29 05.3

14.823 75.87W 12.23s 167.02E 5.39s 124.41E 10.43s 160.89E 7.21s 124.69E

2 242 616 34 523

Aug 22 Sep 4 14 Oct 12 17

17 02 05 0941 22 23 18 41.9 00 06 38.9 21 41 56.6

1.73s 134.19E 2.51s 138.72E 60.338 27.25W 11.94s 121.77E 10.74s 78.63W

27 23 27 37 38

Dec 14 20 23 28

21 07 52.5 12 26 53.6 15 50 21.3 08 18 05

4.898 144.06E 26.06s 63.10W 7.11s 148.31E 25.518 70.74W

70 571 46 23

1967 Feb 19 Mar 19

22 14 36.4 01 10 48.2

9.12s 113.04E 6.733 129.84E

88 80

Apr 4 9 21 Jun 23

00 37 28.2 00 05 08.2 08 14 24.5 05 05 05.3

2.478 138.55E 3.978 135.73E 5.458 126.77E 5.853 130.42E

16 14 25 89

26 Aug 9 22 29

17 36 09.8 08 20 03.9 13 02 06.8 07 27 37

22.588 69.68W 6.538 130.53E 60.84s 24.33W 6.8s 123.5E

79 91 33 33

L

New Hebrides Islands now called Vanuatu.

UPP UME KIR UPP UPP GOT KIR KIR UME KIR UPP KLS KIR UPP KIR UPP GOT

847.0/104.43 902.4/123.46 761.1/100.23 925.9/123.58 795.7/104.75 811.7/108.11 825.3/100.40 836.0/102.71 957.1/128.92 846.8/105.26 832.8/102.35 828.3/101.18 849.1/106.72 802.6/107.70 866.0/110.18 874.2/110.90 863.0/107.45

UPP UPP UDD UME KIR UME UPP UDD UDD KIR UME KIR UDD

818.1/100.54 845.8/106.86 857.8/108.71 842.6/104.39 837.8/103.03 833.1/102.13 843.0/106.37 851.7/108.21 839.7/106.04 830.5/103.54 957.2/128.80 830.4/101.19 850.0/105.73

KIR 855.0/105.98 UME 854.0/105.87

KIR 900.9/117.02 UME 907.9/119.91 KIR 777.3/102.01 SKA 796.5/106.30 UME 780.6/102.80 KLS 798.1/106.56

KIR UDD UME GOT

833.4/103.29 UME 836.4/103.48 827.4/100.34 858.5/108.78 784.0/104.14 UDD 792.4/105.93

KIR 888.0/113.78 SKA 873.1/109.51 UME 882.0/113.00 UDD 870.0/109.03 SKA 828.8/102.90 UDD 828.6/102.55 KIR 831.3/103.47 UME 836.1/104.60

UME 847.9/104.60 UDD 867.8/108.99 KIR 828.1/102.87 UME 833.0/104.06

UME 835.2/104.72 UDD 854.4/108.86 SKA 850.0/105.43 UME 833.0/101.91

Peru Santa Cruz Islands Banda Sea Solomon Islands Banda Sea New Guinea New Guinea South Sandwich Islands Southwest of Timor Peru New Guinea Argentina New Guinea Chile

Java Banda Sea New Guinea New Guinea Banda Sea Banda Sea Chile Banda Sea South Sandwich Islands Banda Sea

continued

APPENDIX 2. (Continued) Date

Origin time (GMT)

Sep 3

21 07 30

10.59s 79.67W

29

03 36 12 10 06 44.5

6.98s 129.37E 27.628 63.15W

97 577

18 28 Oct 4 9 12

15 33 06.6 04 56 53.3 17 21 20.4 17 21 46.2 18 31 39.0

5.973 146.59E 6.598 153.47E 5.668 153.92E 21.10s 179.18W 7.15s 129.83E

Nov9 Dec 21

02 18 47.3 02 25 21

7.18s 123.72E 21.89s 70.07W

580 20

25 27

01 23 33.3

09 17 50.3

5.258 153.70 E 21.293 68.20W

91

1968 Jan 6 14

23 21 22.4 12 25 06.2

27.903 70.97W 7.53s 127.91E

49 80

30 Feb 12

03 44 24.8 05 44 45.1

6.10s 113.36E 5.543 153.36E

599 46

Mar 26

00 41 57.8

6.59s 116.18E

528

27

22 36 43.4

4.19s 133.35E

33

8 9

m

Epicenter (ded

Depth h (km)

41 20

44 605

60

55

Pdif: Station and t/A (sec/deg) UPP UME UME UPP

830.1/102.75 841.5/103.81 834.3/104.63 802.5/109.05 GOT 790.6/105.47 KIR 858.4/108.56 KIR 871.7/111.27 UME 881.6/113.12 KIR 899.8/131.67 UPP 850.9/107.22 UME 840.3/104.98 KIR 770.2/101.62 UPP 860.0/107.53 UME 866.0/109.54 UPP 892.7/116.13 SKA 837.4/104.70 UME UPP UME SKA UPP GOT UDD SKA UDD UPP

890.6/115.18 846.3/106.61 834.5/104.49 763.7/100.36 896.0/116.26 910.9/119.90 900.2/117.63 777.2/102.05 776.8/101.95 854.9/106.32

Region

KIR 838.1/103.55 SKA 827.5/100.27 KLS 829.0/101.64 UDD 828.1/100.73 KIR 821.5/112.89 SKA 805.5/108.21 UME 814.5/111.59 UDD 798.7/107.29

UME KIR KLS UME KIR

917.8/135.35 835.0/103.86 867.8/109.29 773.2/102.35 871.0/110.24

SKA UDD UDD SKA

856.7/108.42 858.4/109.07 790.1/106.17 856.7/106.03

KIR 868.7/110.07 UME 873.7/112.66 GOT 828.5/102.66 UME 851.7/108.19

KIR UDD GOT KIR UME

831.4/103.50 SKA 851.9/107.96 852.1/108.49 768.3/101.2 1 866.2/110.24 SKA 896.0/115.61 879.1/112.82 KLS 914.7/119.45

Peru

Banda Sea Argentina New Guinea New Britain New Ireland Fiji Islands Banda Sea Banda Sea Chile New Ireland Chile- Bolivia Chile Banda Sea

Java New Ireland

GOT 781.9/103.12 KLS 773.5/101.34

Bali Sea

UME 840.9/103.81 UDD 862.5/108.11

New Guinea

Apr 8 11 May 1 2

w m

04 31 10 23 26 02.9

6.80s 3.90s 2.923 6.348

129.52E 127.69E 128.03E 130.00E

136 38 29 119

24

15 43 54.8

6.843 118.91E

618

28

13 27 19.8

2.983 139.34E

73

Jun 3 12 Jul 17 25 29

09 17 45.2 20 15 44 05 24 15.2 07 23 02 23 52 17

5.463 146.91E 0.63s 132.81E 8.66s 125.03E 30.979 178.13W 0.27s 133.47E

182 7 17 17 25

30 Aug 18

20 38 42.3 18 38 30.3

6.86s 80.42W 10.20s 159.90E

36 534

23

22 36 49.8

21.959 63.64W

513

15 12 24.4

3.748 143.01E

32

16 26 27

13 55 35.7 18 02 47 03 58 58

6.08s 148.77E 30.525: 178.01W 6.893 129.21E

49 12 151

27 Oct 23 28

19 06 44 21 04 42.9 23 32 27.7

3.658 143.37E 3.383 143.29E 12.41s 166.43E

12 21 49

Sep 8

14 50 18

00 19 51

UME 830.2/104.53 KIR 821.4/100.07 UME 824.6/100.41 UPP 840.7/106.60 UDD 848.9/108.44 UPP 767.5/101.55 KLS 774.4/103.06 UPP 852.8/108.03 GOT 873.0/111.65 UDD 860.3/109.69 KIR 840.1/108.18 UPP 838.8/102.95 KIR 838.7/103.47 UME 1020.0/145.14 UPP 836.1/102.94 UME 822.6/100.32 KIR 824.0/100.41 UPP 871.7/123.01 UME 855.6/119.37 UPP 789.8/104.46 UDD 781.5/102.64 UPP 867.6/110.34 GOT 884.6/113.98 UDD 875.7/111.93 UPP 885.3/114.90 UME 1030.0/144.73 UPP 837.0/106.69 GOT 853.1/110.15 UDD 845.0/108.55 KIR 849.0/105.33 UPP 869.1/110.14 UME 923.3/123.47

UME 826.0/101.14 UDD 843.6/105.22 UDD 843.0/104.52 KIR 825.2/103.17 UME 830.3/104.32

SKA UDD KIR UME

?76.2/103.47 GOT 782.5/104.78 776.4/103.51 831.2/103.36 SKA 854.3/108.39 839.9/105.17 KLS 864.5/110.68

UME 826.7/100.37 UDD 846.4/104.71 UME 845.1/104.24 UDD 866.9/108.08

SKA 838.8/103.64 KLS 846.4/105.39 UME 825.7/100.83 KIR 836.9/116.53 UDD 875.0/124.24 SKA 785.2/103.38

GOT 852.3/106.52 UDD 843.8/104.69

Banda Sea Banda Sea Ceram Sea Banda Sea Flores Sea New Guinea

New Guinea New Guinea Timor Kermadec Islands New Guinea

SKA 866.3/121.95

Peru Solomon Islands

GOT 775.6/100.94

Bolivia- Argentina

KIR 846.0/105.30 SKA 872.6/110.44 UME 849.6/107.32 KLS 881.5/113.13

New Guinea

KIR 860.3/109.36 UME 871.3/111.66

New Britain Kermadec Islands Banda Sea

KIR 822.5/103.39 SKA 843.3/107.92 UME 827.3/104.48 KLS 848.2/108.74

KIR 848.1/105.05

UME 856.1/107.10

New Guinea New Guinea Santa Cruz Islands continued

APPENDIX 2. (Continued) Origin time (GMT)

Epicenter (deg)

Depth h (km)

22 36 47.9 13 26 42.8 18 50 52.4 04 10 16 22 58 03.3 22 16 11.5

6.80s 129.79E 8.03s 358.948 14.899 167.22E 8.07s 80.09W 22.759 178.76E 6.768 126.74E

106 71 114 27 635 425

00 08 46

6.238 131.01E

48

Mar 10

06 54 16.3

5.60s 147.29E

194

Apr 13

23 33 17.3

6.11s 129.918

170

May 13

14 30 20.7

7.223 120.90E

627

Jun 9 16 24 Jul 19

0651 15 15 45 53.0 03 29 17.8 04 54 53.6

3.268 142.91E 4.988 125.80E 5.85s 146.79E 17.30s 72.48W

11 38 117

24 Aug 3 4

02 59 20.9 00 22 32.0 17 19 19.6

11.84s 75.10W 4.2s 153.0E 5.7s 125.3E

1 65 521

16 32 25.8

5.2s 153.8E

Date 1969 Jan 4 5

19 Feb 4 10 11 24 P 00

5

54

69

Pdif: Station and t/A (sec/deg) UDD 851.6/108.75 UME 887.2/117.03 UPP 948.6/129.88 KIR 830.0/101.39 UPP 938.7/140.60 UPP 803.9/105.37 UME 793.8/103.29 UPP 851.4/106.99 UME 841.3/104.66 KIR 840.8/108.44 UDD 872.0/115.36 UPP 834.7/106.35 UME 824.1/104.08 UPP 772.6/102.87 UME 764.3/101.15 KIR 843.0/104.82 KIR 830.6/100.37 KIR 849.5/108.51 UPP 837.4/104.84 UDD 835.4/102.89 UPP 834.3/101.51 KIR 857.1/108.86 UPP 784.8/103.75 UME 776.0/101.72 KIR 865.2/110.05

UME 928.6/126.05 UME 831.2/101.76 UME 920.7/136.52 KIR 791.0/102.36 UDD 811.7/107.26 KIR 836.9/103.44 UDD 859.1/108.82 SKA 864.9/113.68 KIR UDD KIR UDD

SKA 81 1.0/106.77 DEL 818.2/108.15 SKA 860.0/108.07 UME 850.4/110.66

818.6/102.93 SKA 842.7/107.51 842.0/108.20 762.0/100.60 SKA 781.8/104.68 781.1/104.82 DEL 784.7/105.37

UME 833.6/101.29 UDD 851.4/105.26 SKA 834.3/103.02 UME 847.4/106.56 DEL 830.4/102.5 KIR 841.1/102.96 UME 842.4/102.89 KIR 772.2/100.85 SKA 791.8/105.21 UDD 793.5/105.65 DEL 798.6/106.49 SKA 893.2/115.42 UME 871.2/112.65

Region Banda Sea Solomon Islands New Hebrides Islands Peru Tonga- Kermadec Islands Banda Sea Tanimbar Islands New Guinea Banda Sea Flores Sea New Guinea Banda Sea New Guinea Peru Peru New Ireland (USGS)* Banda Sea (USGS)b New Ireland (USGS)b

8

20 44 21.0

6.1s 129.7E

196

Oct 1 Dec 31

05 05 50.0 05 58 14 23 38 52.4

11.75s 75.15W 11.67s 74.97W 7.03s 1 l7.75E

43 9 487

1970 May 9 31 Jun 11 19 28

18 00 49.4 20 23 28.4 06 02 52.4 10 56 13.5 01 30 13.8

4.373 151.81E 9.15s 78.83W 24.478 68.46W 22.283 70.55W 8.75s 324.04E

196 48 87

Aug 1 1 28

10 22 20 01 02 47.6

14.13s 166.56E 4.60s 153.228

20 76

Oct 13 31 Nov 8

18 53 30.5 17 53 10.5 22 35 46.4

4.10s 143.10E 4.978 145.45E 3.43s 135.65E

124 45 33

28 Dec 28

20 22 51.2 20 03 25.5

4.11s 142.88E 5.238 153.59E

119 63

1971 Jan 10

07 17 04.7

3.21s 139.69E

41

21

17 19 36.8

7.883 122.66E

210

21 04 19.1 10 35 19.7

63,438 61.36W 23.818 67.20W

12 166

1

vI m

Feb 8 21

44

50

UPP UDD KIR KIR UPP

830.1/106.24 838.6/108.09 835.2/102.90 838.9/102.76 776.9/101.13

KIR 840.5/108.67 UPP 825.2/101.10 KIR 864.7/111.98 SKA 851.5/106.58 UPP 845.2/105.75 UME 836.3/103.89 UPP 955.0/128.96 UPP 888.4/115.35 UME 870.9/111.89 KIR 837.6/105.67 KIR 853.2/107.25 UPP 853.2/106.74 UDD 864.6/108.47 UPP 860.8/110.61 UPP 898.4/116.07 UPP UME UPP UDD UPP UPP UME

855.3/108.39 843.3/105.52 823.2/104.31 832.2/106.25 989.9/136.70 843.6/107.75 860.3/109.98

Source data of USGS (or its predecessors) used for lack of ISC data.

KIR 815.4/102.84 UME 820.3/103.98 DEL 843.6/109.16

UDD 786.5/103.10 DEL 788.9/103.49 UME KIR UME UDD KIR UDD UME KIR UDD UDD

85241 11.18 829.0/101.91 863.6/111.10 849.9/106.20 833.3/103.19 853.8/107.68 940.0/125.14 858.0/109.31 891.1/116.71 865.7/112.29

UDD UME DEL DEL SKA DEL

874.6/115.98 829.6/102.16 844.0/105.87 845.3/105.28 853.5/107.40 858.1/108.35

SKA 883.0/114.68 DEL 902.4/119.08

Banda Sea (USGS)b Peru Peru Bali Sea New Britain Peru Chile- Argentina Chile Timor New Hebrides Islands New Ireland

KIR 833.1/102.50 UME 841.4/104.08

New Guinea New Guinea New Guinea

UDD 866.7/112.21 DEL 878.3/114.06 KIR 870.1/110.02 SKA 891.5/115.39

New Guinea New Ireland

KIR UDD KIR DEL

New Guinea

836.9/103.69 SKA 864.3/108.73 868.8/110.05 DEL 876.2/111.75 810.3/101.87 UME 814.8/102.51 834.2/106.88

KIR 860.3/110.92 SKA 838.3/106.51 UDD 835.8/105.91 DEL 831.1/104.78

Flores Sea South Shetland Islands Chile-Argentina continued

APPENDIX 2. (Continued) ~

~~~

Date

Origin time (GMT)

Mar 13

19 12 24.8

5.753 145.39E

114

Jun 17 Jul I

21 00 39.2 01 16 16.4

25.40s 69.06W 6.40s 130.24E

76 129

8

19 07 07.3

7.03s 129.70E

92

9

03 03 16.9

32.51s 71.21W

40

14

06 11 28.9

5.528 153.86E

43

14

18 27 44

5.175 153.32E

32

18

14 31 17.8

4.863 153.32E

48

19 26

14 48 42.4 01 23 21.2

4.90s 144.52E 4.938 153.18E

75 43

27 28

20 47 51.5 01 10 23.1

5.20s 152.85E 5.14s 152.87E

41 24

06 18 15.3

4.638 152,98E 5.563 152.47E 5.883 154.31E 5.683 152.12E 5.743 152.13E

71 24 60 15 43

30 Aug 2 9 9 10

00 19 53 12 12 01.5 2001 37

04 23 57.6

Epicenter (deg)

Depth h (km)

Region

Pdif: Station and t/A (sec/deg) UPP 873.4/113.17 UDD 878.7/114.73 UPP 863.8/109.99 UPP 838.8/106.76 UDD 848.2/108.61 UPP 845.8/107.05 UME 835.1/104.81 UPP 896.1/116.94 UME 907.1/119.27 UPP 891.1/116.44 UME 880.7/112.97 UPP 897.6/115.91 UME 884.4/112.46 UPP 892.8/115.63 UME 875.0/112.17 KIR 846.6/106.88 UPP 889.8/115.63 UME 874.3/112.18 UPP 895.5/115.75 UPP 891.9/115.71 UDD 897.9/117.08 KIR 858.4/109.27 KIR 867.4/110.00 UME 875.5/113.46 KIR 873.0/110.01 KIR 867.4/110.07

KIR 847.6/107.97

UME 857.7/110.08

NewGuinea

SKA 862.8/108.70 UDD 856.8/108.14 KIR 825.1/103.31 UME 829.7/104.48

Chile Banda Sea

KIR UDD KIR UDD KIR UDD KIR UDD KIR UDD

831.2/103.70 854.4/108.90 913.1/120.27 897.3/115.13 866.1/110.37 894.7/117.79 868.4/109.88 901.7/117.27 863.4/109.58 896.2/116.98

Banda Sea

KIR UDD KIR KIR

863.8/109.61 SKA 888.2/114.98 899.2/117.00 871.5/109.77 UDD 902.6/117.13 868.6/109.72 UME 879.6/112.27

SKA DEL SKA DEL SKA DEL SKA

851.7/108.25 859.6/109.95 902.1/115.82 889.8/113.87 897.2/115.74 918.3/120.17 894.0/115.25

SKA 890.2/114.95 DEL 899.0/119.35

Chile New Ireland New Ireland New Ireland New Guinea New Ireland New Britain New Britain New Ireland New Britain Solomon Islands New Britain New Britain

22 08 58.5 04 36 13.7 01 30 33.6 17 58 37.9 05 57 12.0

6.565: 130.67E 6.548 146.64E 5.893 154.14E 15.57s 167.24E 11.87s 166.55E

61 111

00 30 18.0

4.60s 151.86E

166

06 25 46

2.09s 138.96E

12

18 18 19 Feb 14

21 55 15 22 08 14.1 15 01 01.2 23 29 51.6

4.848 145.10E 4.698 145.02E 4.84s 145.14E 11.43s 166.37E

23 33 100 101

Mar8

03 45 25.1

3.743 131.39E

29

Apr 4

22 43 06.7

7.47s 125.56E

375

23 32 10.6

5.13s 154.23E

413

22 Jun 8 Jul30

07 48 17.6 23 16 29.0 20 45 55.1 18 53 42.8 23 17 21.5

15.948 167.53E 4.22s 152.71E 37.763 175.05W 30.343 71.58W 5.753 130.51E

46 37 208 39 76

Aug 14 17 Sep 4

22 29 27.8 23 44 08.6 18 I 1 12.5

6.298 144.46E 6.04s 152.9OE 11.78s 166.31E

43 26 64

Sep 8 25 Oct 4 27 Nov 21

1972 Jan 6 7

3

28

May 4 5

64 49 119

KIR KIR UPP UPP UPP

834.5/103.62 852.0/1O9.11 893.4/116.89 954.1/130.53 931.0/126.84

UPP 875.0/114.81 UDD 881.0/116.21 UPP 856.3/107.07 UDD 863.7/108.73 KIR 856.7/107.02 KIR 854.1/106.85 KIR 844.2/107.03 UPP 930.8/126.36 UME 9 14.41122.52 UPP 845.4/105.00 UME 833.8/102.58 UPP 806.6/105.40 UME 797.9/103.41 UPP 848.4/116.22 UME 835.4/112.74 UPP 952.4/130.97 KIR 869.0/108.80 UPP 966.91136.96 UPP 889.2/115.33 UPP 844.9/106.33 UME 833.7/104.01 KIR 856.9/108.17 UPP 896.4/116.54 UPP 936.5/126.67

KIR 867.41110.80 UME 879.4/113.41 KIR 928.1/123.56 UME 941.1/126.70 UME 915.0/122.99

Banda Sea New Guinea Solomon Islands New Hebrides Islands Santa Cruz Islands

KIR 845.7/108.90 UME 856.9/111.41

New Britain

KIR 835.5/102.40 UME 843.7/104.21 DEL 872.0/110.42

New Guinea

UME 839.7/104.81

KIR 900.4/119.36 DEL 938.4/130.27 KIR 827.9/101.27 UDD 853.1/106.81 KIR 794.2/102.58 UDD 815.4/107.31 KIR 826.2/110.11 UDD 858.3/117.56 KIR 921.4/123.99

SKA 921.4/124.79

New Guinea New Guinea New Guinea Santa Cruz Islands

SKA 849.31105.97

New Guinea

SKA DEL SKA DEL UME

Banda Sea

813.9/106.91 820.0/108.11 852.3/115.49 869.4p19.97 940.4/127.13

UME 947.9/132.80 UME 898.2/117.56 KIR 829.1/102.81 SKA 852.5/107.43 UDD 852.8/108.16 DEL 858.5/109.29

KIR 869.4/110.58 UME 881.4/113.12 KIR 907.5/119.69 UME 920.5/122.83

Solomon Islands New Hebrides Islands New Britain Tonga Islands Chile Banda Sea New Guinea New Britain Santa Cruz Islands continued

APPENDIX 2. (Continued)

m

oo

Date

Origin time (GMT)

Epicenter

Depth h (km)

Sep 18

20 35 35

9.933 119.60E

23

Oct 28 Nov 2 4 5

02 27 10.4 19 55 23.3 21 35 58.5 20 08 03.0

7.333 146.83E 20.03s 168.918 8.19s 112.27E 5.40s 146.70E

2 37 99 229

UPP UDD KIR UPP SKA KIR

1973 Jan 18 Feb 1 Jul31 Aug 1 13 Oct 5 5 25 Nov 30

09 28 13.7 05 14 19.9 10 51 13.8 01 31 31.1 08 28 19.4 05 45 27.3 05 47 50.2 14 08 58.5 08 09 55.5

6.888 150.03E 22.533 66.19W 26.96s 71.11W 14.33s 167.29E 4.50s 144.10E 32.928 71.88W 33.0s 71.9W 21.968 63.65W 15.18s 167.43E

38 214 33 202 109 8 33 517 124

UPP 891.3/116.14 UPP 832.1/106.18 UPP 878.2/112.30 UPP 931.9/129.38 UPP 859.7/111.49 UME 915.7/119.90 UME 915.8/119.98 SKA 785.5/103.39 UPP 947.5/130.22

KIR UDD UME UME KIR

1974 Jan 2

10 42 27.7

22.493 68.26W

83

10

08 51 13.8

14.45s 166.87E

36

UPP 851.3/107.15 UDD 840.5/105.28 UPP 952.2/129.36

SKA 844.9/105.79 UME 859.7/109.28 DEL 838.2/104.25 KIR 921.2/122.39 UME 937.2/125.53

Region

Pdif: Station and t/A (sec/deg) 845.2/104.54 853.9/106.52 871.3/109.92 983.7/135.24 822.8/101.74 835.1/108.06

KIR 835.4/102.60 UME 837.5/102.98 DEL 856.2/106.90 KIR 949.7/128.25 UME 965.7/131.41 UDD 821.9/101.37 DEL 822.5/101.42

867.8/110.50 UME 878.3/112.86 819.1/104.34 886.2/114.42 916.9/125.54 841.1/106.37 UDD 874.5/113.07

UDD 781.8/102.66 DEL 776.6/101.42

Sumba Island New Guinea Loyalty Islands Java New Guinea New Britain Argentina Chile New Hebrides Islands New Guinea Chile Chile Argentina New Hebrides Islands Chile New Hebrides Islands

29 29 30

18 57 10.3 22 37 23.9 09 53 13.9

7.368 128.45E 7.388 128.53E 5.15s 134.lSE

127 150 51

May 12 Jun 24 Aug 18 23

10 05 54.6 21 35 0s 10 44 12.8 04 50 35.1

19.57s 69.05W 2.25s 141.10E 38.5s 73.4W 7.538 l27.48E

107 0 36 139

sep 20

21 20 11.8 14 21 29.3

6.20s 146.10E 12.249 77.58W

105 9

03 14 18.6

6.938 129.52E

156

12 59 51.0

12.448 77.46W

6

04 14 50.1

15.12s 167.16E

62

Oct 3

29

Nov 9 m \o

20

KIR KIR UPP UME UDD KIR UPP KIR UDD KIR UPP UME UPP UME UPP DEL UPP

826.1/103.55 824.4/103.59 854.4/107.54 84 1.8/105.0 1 831.2/103.16 842.3/103.27 924.2/122.84 823.4/103.34 845.7/108.28 851.2/108.62 838.7/103.10 844.4/104.35 838.5/106.87 827.8/104.65 841.4/103.21 832.8/100.99 949.9/130.08

UME UME KIR UDD

830.5/104.57 UDD 848.2/108.60 828.2/104.62 836.4/103.57 SKA 857.7/108.36 862.3/109.32

UME UME SKA DEL

849.7/105.21 936.2/125.29 843.6/107.78 UME 829.7/104.30 851.9/109.19

KIR UDD KIR UDD KIR

847.71104.27 830.9/101.10 822.9/103.54 846.5/108.73 838.0/104.41

UME 932.9/126.25

SKA DEL SKA DEL UDD

832.8/100.80 826.7/100.89 843.9/108.09 851.9/109.77 833.0/ 101.21

Banda Sea Banda Sea Aroe Islands Chile New Guinea Chile (USGS)b Banda Sea New Guinea Peru Banda Sea Peru New Hebrides Islands

90

MARKUS BATH

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Cook, K. L., Algermissen, S. T., and Costain, J. K. (1962).The status of PS converted waves in crustal studies. J. Geophys. Res. 67,4769-4778. Deschamps, A., Gaudemer, Y.,and Cisternas, A. (1982).The El Asnam, Algeria, earthquake of I0 October 1980: Multiple-source mechanism determined from long-period records. Bull. Seismol. SOC.Am. 72, 1 1 1 1-1 128. Dey, S., Roy, N., and Dutta, A. (1985).Propagation of P and S waves and reflection of P wave in a medium under normal initial stresses. Geophys. Res. Bull. (Hyderahad) 23, 1-29. Doornbos, D. J. (1983).Present seismic evidence for a boundary layer at the base of the mantle. J . Geophys. Res. 88,3498-3505. Doornbos, D. J., and Mondt, J. C. (1979a).Attenuation of P and S waves diffracted around the core. Geophys. J. R. Astron. SOC.57,353-379. Doornbos, D. J., and Mondt, J. C. (1979b).P and S waves diffracted around the core and the velocity structure at the base of the mantle. Geophys. J . R. Astron. SOC.57, 381-395. Enayatollah, M. A. (1972).Travel times of P-waves for the Swedish-Finnish seismograph network. Pure Appl. Geophys. 94, 101-135. Ergin, K. (1952).Energy ratio of the seismic waves reflected and refracted at a rock-water boundary. Bull. Seismol. SOC.Am. 42,349- 372. Flinn, E. A., Cohen, T. J., and McCowan, D. W. (1973).Detection and analysis of multiple seismic events. Bull. Seismol. SOC.Am. 63, 1921-1936. Florensov, N. A., and Solonenko, V. P. (1963).“The Gobi-Altai Earthquake.” Izdatel’stvo Akademii Nauk SSR, Moscow. Forsyth, D. W. (1982).Determinations of focal depths of earthquakes associated with the bending of oceanic plates at trenches. Phys. Earth Planet. Inter. 28,141-I60. Fukao, Y., and Furumoto, M. (1975).Foreshocks and multiple shocks of large earthquakes. Phys. Earth Planet. Inter. 10,355-368. Gupta, H.K., and Rastogi, B. K. (1972).Earthquake m, vs M,relations and source multiplicity. Geophys. J . R. Astron. SOC.28, 65-89. Gupta, H. K.,Rastogi, B. K.,and Narain, H. (1971). The Koynaearthquake of December 10,1967: A multiple seismic event. Bull. Seismol. SOC.Am. 61, 167-176. Gupta, 1. N. (1964).A note on the Alaska earthquake of July 10,1958.Bull. Seismol. SOC.Am. 54, 208I -2083. Gutenberg, B. (1944).Energy ratio of reflected and refracted seismic waves. Bull. Seismol. SOC.Am.

~,ns-i02. Gutenberg, B. (1959).“Physics of the Earth’s Interior.” Academic Press, New York. Gutenberg, B. (1960).The shadow at the earth’s core. J . Geophys. Res. 65, 1013-1020. Gutenberg, B., and Richter, C. F. (1935).On seismic waves 11. Gerlands Beitr. Geophys. 45, 280- 360.

Gutenberg, B., and Richter, C. F. (1936).Materials for the study of deep-focus earthquakes. Bull. Seismol. Sor. Am. 26,341-390. Hanyga, A,, ed. (1985).“Seismic Wave Propagation in the Earth.” Elsevier, Amsterdam; Polish Scientific Pub].. Warsaw. Herrin, E., Chairman (1968).1968 seismological tables for P phases. Bull. Seismol. SOC.Am. 58,

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Jeffreys, H., ed. (1973). “Collected Papers of Sir Harold Jeffreys on Geophysics and Other Sciences,” Vol. 2. Gordon & Breach, London. Jeffreys, H., and Bullen, K. E. (1967). “Seismological Tables.” Brit. Assoc. Adv. Sci., London. Jordon, T. H., and Sverdrup, K. A. (1981). Teleseismiclocation techniques and their application to earthquake clusters in the south-central Pacific. Bull. Seismol. SOC.Am. 71,1105-1 130. Kanamori, H., and Stewart, G. S. (1978). Seismological aspects of the Guatemala earthquake of February 4, 1976. J . Geophys. Res. 83,3427-3434. Knopoff, L., and Gilbert, F. (1961). Diffraction of elastic waves by the core of the earth. Bull. Seismol. SOC.Am. 51, 35-49. Knott, C. G. (1899). Reflection and refraction of elastic waves, with seismological applications. Philos. Mag., Ser. 5 48,64-97, 567-569. Laski, J. (1977). Reflection and transmission coefficients-revision of formulae. Pol. Acad. Sci., Publ. Inst. Geophys. A-4 (1 15), 201-212. Leong, L. S. (1976). Sp converted waves, synthetic long-period S waveforms and crustal structure at Umei, Sweden. Seismol. Inst., Uppsaia Rep. 4-76. Lomnitz-Adler, J. (1985). Asperity models and characteristic earthquakes. Geophys. J. R. Astron. SOC.83,435-450.

McCamy, K., Meyer, R. P., and Smith, T. J. (1962). Generally applicable solutions of Zoeppritz’ amplitude equations. Bull. Seismol. Soc. Am. 52,923-955. Mei, A. I., S. J. (1943). The amplitude ratio PP/P as recorded by Galitzin seismographs. Bull. Seismol.SOC.Am. 33, 149-196. Menke, W.(1986). Effect of heterogeneities in D” on the decay rate of PdW. J. Geophys. Res. 91, 1927- 1933.

Meyer, K. (1979). Travel times, amplitudes and periods of precursors to P P at epicentral distances 97”-116”. Seismol. Inst. Uppsala Rep. 3-79. Miyamura, S., Omote, S., Teisseyre, R., and Vesanen, E.(1965). Multiple shocks and earthquake series pattern. Bull. Inr. Inst. Seismol. Earthq. Eng. 2,71-92. Mooney, H. M. (1951). A study of the energy content of the seismic waves P and pP. Bull. Seismol. SOC.Am. 41,13-30. Mula, A. H. G. (1981). Amplitudes of diffracted long-period P and S waves and the veIocities and Q structure at the base of the mantle. J . Geophys. Res. 86,4999-501 1. Nafe, J. E. (1957). Reflection and transmission coefficients at a solid-solid interface of high velocity contrast. Bull. Seismol. Soc. Am. 47,205-219. Niazi, M. (1969). Source dynamics of the Dasht-e BayZiz earthquake of August 31, 1968. Bull. Seismol. SOC.Am. 59, 1843-1861. Officer, C. B. (1958). “Introduction to the Theory of Sound Transmission.” McGraw-Hill, New York. Okal, E. A., and Geller, R. J. (1979). Shear-wave velocity at the base of the mantle from profiles of diffracted SH waves. Bull. Seismol. SOC.Am. 69, 1039-1053. Papazachos, B. (1964). Angle of incidence and amplitude ratio of P and P P waves. Bull. Seismol. SOC.Am. 54, 105-121. Payo Subiza, G,, and BBth, M. (1964). Core phases and the inner core boundary. Geophys. J. R. Astron. SOC.8,496-513. Pearce, R. G. (1977). Fault plane solutions using relative amplitudes of P and pP. Geophys. J. R. Astron. SOC.50, 381-394. Phinney, R. A., and Alexander, S. S. (1966). P-wave diffraction theory and the structure of the core-mantle boundary. J. Geophys. Res. 71, 5959-5975. Phinney, R. A., and Cathles, L. M. (1969). Diffraction of P by the core: A study of long-period amplitudes near the edge of the shadow. J. Geophys. Res. 74, 1556-1574. Rial, J. A. (1978). The Caracas, Venezuela, earthquake of July 1967: A multiple-source event. J . Geophys. Res. 83,5405-5414.

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Richards, P. G . (1976). On the adequacy of plane-wave reflection/transmission coefficients in the analysis of seismic body waves. Bull. Seismol. SOC.Am. 66,701 -717. Richter, C. F. (1958). “Elementary Seismology.” Freeman, San Francisco. Ritsema, A. R. (1958). (i,A)-curvesfor bodily seismic waves of any focal depth. Lembaga Meteor. Geofis.. Verhand. 54. Ryall, A. (1962). The Hebgen Lake, Montana, earthquake of August 18, 1959: P waves. Bull. Seismol. SOC.Am. 52,235-271. Savage, J. C . (1965). The stopping phase on seismograms. Bull. Seismol. SOC.Am. 55,47-58. Schwartz, S. Y., and Ruff, L. J. (1985). The 1968 Tokachi-Oki and the 1969 Kurile Islands earthquakes: Variability in the rupture process. J. Geophys. Res. 90,8613-8626. Shapka, A., and Bith, M. (1976). Location of teleseisms from P-wave arrivals at the Swedish stations. Seismol. lnst. Uppsala Rep. 9-76. Shurbet, D. H. (1965). The pP phase generated in water covered areas. Bull. Seismol.SOC.Am. 55, 721-726. Stelzner, J., Giith, D., and Weyrauch, J. (1976-1979). “Seismological Bulletin Station Moxa (MOX) 1970- 74,” Five Vols. Akademie-Verlag, Berlin. Stoneley, R.(1937). The Mongolian earthquake of 1931 August 10. Br. Assoc. Seismol.Comm. Rep. 42, 5-6. Trifunac, M. D., and Brune, J. N. (1970). Complexity of energy release during the Imperial Valley, California, earthquake of 1940. Bull. Seismol. Soc. Am. 60, 137-160. Usami, T. (1956). Seismometrical study of Boso-Oki earthquake of November 26, 1953. Q. J . Seismol. (Jpn. Meteorol. Agency) 21, 1-1 3. Wu,F. T., and Kanamori, H. (1973). Source mechanism of February 4, 1965, Rat Island earthquake. J. Geophys. Res. 78,6082-6092. Wyss, M.,and Brune, J. N. (1967). The Alaska earthquake of 28 March 1964: A complex multiple rupture. Bull. Seismol. Soc. Am. 57,1017-1023. Zoppritz, K. (1919). Erdbebenwellen VIIb: Uber Reflexion und Durchgang seismischer Wellen durch Unstetigkeitsflachen. Gdttinger Nachr. 1,66-84.

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ADVANCES IN GEOPHYSICS. VOL. 30

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE G. T. CSANADY* Woods Hole Oceanographic Institution Woods Hole, Massachusetts 02543

1. INTRODUCTION

Continental slopes underlie some 12% of the world ocean’s surface area. Their importance to ocean circulation is out of proportion, since most of the major ocean currents flow over them and are in various ways affected by their topography. Yet, until about 25 years ago oceanographers paid no serious attention to the fact that the boundaries of the deep ocean were not vertical walls. The typical continental slope is 30 times as wide as the abyss is deep. It is not that the importance of a sloping seafloor to ocean dynamics was unknown. Sixty-four years ago Ekman wrote Wo der Tiefenstrom iiber wachsender Tiefe stromt, entsteht ein Curl oder Wirbel confra solem; wo er iiber ahnehmender Tiefe stromt, entsteht ein Wirbel cum sole: wo er den Niveaulinien des Bodens folgt, haben die Tiefenunterschiede auf den Strom iiberhaupt keine Wirkung (Ekman, 1923, p. 29). [Where the deep current flows over increasing depth, a cyclonic curl or vortex arises; where it flows over decreasing depth, an anticyclonic vortex arises; where it follows the depth contours, depth variations have no effect at all on the current.]

Clear as Ekman’s conclusions on the dynamical importance of bottom slope were, one finds only cursory references to it in standard texts such as Sverdrup et al. (l942), Proudman (1953), Defant (1961), or Fofonoff’s article in “The Sea” (1962).Modern texts, for example those of Pedlosky (1979), or Gill (1982), treat the dynamical effects of variable depth in general, but not specifically currents over the continental slope. None of the texts mentioned contains even the briefest systematic account of steady or slowly varying currents observed over continental slopes. Given this background, it is not surprising that in an account of current measurements taken south of New England over the continental slope, on the occasion of the first long-term deployment of internally recording current meters, Webster (1969, p. 8 5 ) remarked, “Currents measured at Site D might be representative of general, mid-ocean conditions.” He went on to explain the rationale of the experiment, “It was hoped that the influence of the continental

* Present address: Department of Oceanography, Old Dominion University, Norfolk, Virginia, 23529. 95 Copyripht @ 1988Iby Academic Press. Inc. All rights of reproduction in m y form reserved.

96

G. T. CSANADY

shelf and the Gulf Stream on ocean currents at Site D would be small.” With the clarity of hindsight we now realize that the narrow band of ocean wedged in between the Gulf Stream and the North American continental plate behaves very differently from ocean basins over abyssal plains. Not only is this “slopewater” located over a steeply sloping seafloor, but it is strongly affected by disturbances in the western boundary current flowing just seaward, not over, the continental slope. What this story illustrates is that empirical knowledge of ocean currents over continental slopes was nonexistent two decades ago. The first dynamical study of an observed phenomenon in which the effect of bottom slope was a central element was Warren’s (1963) explanation of Gulf Stream meandering. He showed that the narrow, coherent Stream in contact with sloping bottom will meander about an equilibrium position, if started off at an angle against depth contours. Arthur (1965) gave an explanation of upwelling pockets in an eastern boundary current using much the same ideas, and noted that upwelling accompanies meanders. Robinson (1964) pointed out that the sloping seafloor makes the continental shelf-slope region an effective wave guide, and proposed a theoretical model for the interpretation of Hamon’s (1962) observations on the slow propagation of sea level disturbances along the east coast of Australia. Robinson christened this phenomenon the “continental shelf wave.” Slow oscillations found by Webster (1961a) at Site D over the deep continental slope were shown by Thompson (1971) to be bottom slope-related “topographic waves,” a broader class of oscillation similar to the continental shelf wave. Today the study of steady currents and slow oscillations over the continental slope has become a new frontier of physical oceanography. The empirical base has expanded tremendously following a series of field studies on coastal upwelling along the Oregon, Peru, and North African coasts and sustained observational work over the continental shelf and slope along the east coast of North America from the Florida keys to the Grand Banks. A number of observed phenomena have been explained by simple theoretical models, while others remain puzzling. It seems useful at this time to summarize what progress we have made, pull together apparently disparate ideas into a coherent framework, point out weaknesses in models underlying our present notions on slope currents, and note what we have yet to explain about them.

2. THEOBSERVATIONAL EVIDENCE Chapter 1 of Brunt’s great treatise on physical and dynamical meteorology is headed by the phrase, “The facts which call for explanation.” It is noteworthy that this approach is found in a text described as “an account of

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

97

theoretical meteorology” by an author who states that “no physical theory can be regarded as wholly satisfactory which cannot be expressed in mathematical form.” Brunt’s arrangement of facts before fancy, and his clear intention to explain observations with the aid of theory, rather than merely to build theory, is a model for synthesizing any branch of physical science. Especially important is the discipline of facts in the case of the atmosphere or the ocean, where one proceeds to understanding by examining a number of idealized models, based on a variety of different and possibly conflicting postulates, representing phenomena of widely different time or space scales. Having the facts stored at the back of one’s mind helps guard against the temptation to pursue mathematical argument beyond its range of validity. Guided by this philosophy, the present article starts with a brief summary of what we know about steady and slowly changing motions over the continental slope.

2.1. Investigations in the Mid-Atlantic Bight “Slope Sea” As already mentioned in the introduction, the first long-term observations of currents over steeply sloping bottom were made during the debut of the moored current meter south of New England, more or less by accident (Webster, 1969). Following the initial investigations at the single Site D, a cross-isobath array of moorings was deployed in 1970 near 70”W. Schmitz (1974)analyzed steady and low-frequency motions observed at this array. The location of the moorings, the depth profile, and low-passed currents for the first period of observations are shown in Figs. 1-3. Instrument 3461 was placed 100 m above the bottom, in water 2263 m deep (site 346). It recorded irregular oscillations of a period typically near 15 days. The flow sustained an apparent impulse at the middle of October and the oscillation amplitude decreased afterward. The record at the nearest mooring, 351 1, was similar to that at 3461, but not to the records at the other moorings. However, the two records from 3471 and 3502 were again similar. There was an underlying mean flow westward, at about 5 cm/sec. The amplitude of superimposed oscillations peaked at 20 cm/sec. The phase at site 351 led that at 346 by about 3 days, although the significance of this is questionable, because the time base had to be adjusted. Phase differences between 3471 and 3502 are barely perceptible to the eye, but statistical analysis shows 3471 leading by a little less than a day. These differences imply phase propagation offshore and westward. The along-isobath velocity component was much larger than the cross-isobath one. Schmitz’s studies were extended by Luyten (1977) to the continental rise along almost the same meridian (70’W), as far south as 36’30”. The north wall of the Gulf Stream was at 38”N at the time, about halfway along the

G . T.CSANADY

98 41"

W

n

G

40"

-I

2

39" 72"

70"

?I0 W. LONGITUDE

FIG.I. Location of moorings deployed in early studies of the Slope Sea south of New England. Records mentioned in the text are numbered as the moorings shown here, with a final digit appended designating the current meter (up to three on each mooring, counted from the uppermost one down) (from Schmitz, 1974). loo

-

500

-

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Y

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$ 1500 E Q 8 2000 -

L

2500

3000

I

I

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I

I

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I

I

I

I

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

99

FIG.3. Some low-passed current records from New England array (Schmitz, 1974).

array. Over the upper rise, shoreward of the Gulf Stream, the results were similar to those of Schmitz (1974): weak mean flow westward, and oscillations aligned with the depth contours, at a typical amplitude of 20 cm/sec. The typical period was longer, however: 30 days instead of 15. Offshore phase propagation was clear, at about 8-15 cmjsec. Further south, under the Gulf Stream, the oscillations were dramatically different. They were stronger, with a typical amplitude of 40 cm/sec and velocities were predominantly across, rather than along, isobaths. Figure 4 shows the location of the current meters deployed, and Fig. 5 depicts daily average current vectors (in a “stick plot”). The two mooring lines along 70” and near 69” were only 75-100 km apart along the isobaths, yet the observed currents show little similarity. The records are similar, however, between meters on the same line, except for the abrupt change in character at 38”N,the location of the Gulf Stream north wall. The kinetic energy of the oscillations increases southward along the array (Fig. 6 ) , by a factor of at least 4, suggesting that the oscillations come from the Stream. Luyten also drew attention to the “bottom intensification” of the oscillations, in the sense that

G . T.CSANADY

100

2000 -

H

r '

3000-

-._ .

W 0

4000-

5000

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I

39"

I

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LAT.. N FIG.4. Current meters deployed along the "rise array" of 1974 along 70"W(solid lines, also bottom profile) and approximately 69"W (broken lines; Luyten, 1977).

stronger currents were observed at 200 m than at 1000 m above the bottom. The average ratio of kinetic energy at these two levels calculated from Luyten's data for the 10 current meter pairs was 0.75, with no systematic variation along the array. A vertical decay length scale for energy calculated from this is 800 m/(l - 0.75) = 3.2 km. The recent SEEP, MASAR, and NASACS experiments (Csanady et al., 1987) extended over a broader region of the Slope Sea. Figure 7 shows the location of moorings deployed, and Fig. 8 is a sample daily average current vector stick plot from mooring C, off New Jersey along the 2000-m isobath. The character of the record is very much as found by Schmitz in the same depth range: mean southwestward flow at a velocity near 5 cm/sec, and underlying oscillations of about 15-day period, 20 cm/sec amplitude. The less chaotic parts of the record (from which Fig. 8 was taken) clearly show individual impulses of strong currents, followed by a slow decay of the oscillations. Bottom currents were not stronger than mid-level ones. The SEEP experiment was designed to test the hypothesis of mid-slope quiescence, i.e., weakness of near-bottom currents between about the 1- and 2-km isobaths. This led to a useful general classification of the Slope Sea into upper, mid-, and lower slope regions, according to the character and intensity of near-bottom currents. Figure 9 contrasts current speeds, regardless of direction, observed in two of these regions. Not only are upper slope currents much more intense, but they also vary much more rapidly. Spectra from the

NORTH

534 I (3337 m)

I

527I (1977 rn)

I

5352 (3453 m)

10 cM/sEc

5373 (4262m)

536 I

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APR 74

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MAY

'

JUN ' JULY

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AUG ' SEPT' OCT ' NOV ' M C

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FIG.5. One-day average currents from the rise array (Luyten, 1977).

SEPT OCT

NOV ' DEC

102

G . T. CSANADY

Q

8 60-

c g

X

4

X

40-

20-

+ +

+ +@

0 X

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I

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38’ 3 7’ 36“ FIG.6. Kinetic energy of low-passed currents along the rise array. Circles mark the eastern line observations (Luyten, 1977). 39”

different depth ranges (Fig. 10) give further information. Going from upper slope down, all motions become less intense except low-frequency, alongisobath oscillations. Over the mid-slope, cross-isobath motions at low frequencies are very much weaker than along-isobath ones. Over the lower slope, the discrepancy is much less, and a remarkable spectral gap opens up between the inertial and the low-frequency motions, in the spectral range of the “wind band.” In this band currents on the outer continental shelf are particularly intense. The different character of the three slope regions manifests itself also in the frequency (percentage of time observed) of relatively strong (>20 cm/sec) and especially weak ( < 5 cm/sec) currents (Fig. 11). Some insight into the possible causes of mid-slope quiescence, or its absence, is gained from band-filtering the observed currents into frequency ranges: a 6.6- to 29-day period (“topographic wave band”), a 30-hr to 6.6-day period (“wind band”), a 15- to 30-hr period (“inertial-diurnal band”), a 11- to 15-hr period (“semidiurnal tide band”), and a c 1 1-hr period (“high-frequency band”). The distribution of energy in these bands versus bottom depth is shown in Fig. 12. All bands except the topographic wave band peak over the upper slope. Topographic wave band energy varies irregularly in time and space, and, except over the upper slope, dominates the other bands. Another recent study, by Johns and Watts (1986),has revealed details of the dramatic difference in character between Slope Sea currents and those under the Gulf Stream (Luyten, 1977). Johns and Watts deployed inverted echo sounders, as well as current meters east of Cape Hatteras. Their current meter records are very similar to those of Luyten taken under the Gulf Stream. Johns and Watts were able to separate the low-frequency oscillations into two categories, a slower 12- to 48-day period band, and a faster 4- to 10-day band.

kzzs: MASAR + NASACS WIND

A

FIG 7 Moorings in the Slope Sea, 1984-1985, deployed in the Shelf Edge 'Exchange Processes (SEEP), Mid-Atlantic Slope and Rise (MASAR), and North Atlantic Slope and Canyon Study (NASACS) expenments Repnnted with permission from Csanady et al , Cont Shelj Res SEEP issue, Copyright 1988, Pergamon Journals, Ltd

1800 m -1.

W

1000 m

U U

382 m -U -4. -0.

FIG.8. Stick plot of daily currents from mooring C off New Jersey, for two periods in 1984 (top) and 1985 (bottom).Abscissa gives Julian days. Note the compressed velocity scale at the upper meters. Fluctuations are of about the same amplitude at all levels, except just above the bottom (1900 m), where they are weaker. Along-isobath currents are shown as vertical vectors in this diagram. There is little cross-isobath motion.

105

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

202M ISOBATH

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8

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me YAR FIG.9. Speed of observed currents on upper versus mid-slope near 70"W (Csanady et al., bEPT 8J

NOV

DEC

JAN

a4

1988).

The slower oscillations were aligned with the isobaths, much as the oscillations observed in the Slope Sea. The faster ones were nearly isotropic, but stronger across than along isobaths. The inverted echo sounders also showed that the faster oscillations occurred in tandem with cross-stream meanders of the Gulf Stream, while the slower ones were uncoupled to the meanders, and were presumably the topographic waves also observable north of the Stream.

2.2. Meandering of Boundary Currents over Slope Meanders of the Gulf Stream off New England and Nova Scotia were first studied intensively by Fuglister and Worthington (1951) and Fuglister (1963), whose demonstration of the Stream's meandering path from the temperature

31

G.T.CSANADY

106

- olongslope

- cross-slope

velocity

1o*

10'

10'

10'

0'

10 '

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125m ISOBATH 6m above bottom

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loom above 95% level

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cycles/hour FIG. 10. Energy spectra of currents from upper to lower continental slope along 70"W.Lighter cycle s/houT

lines show along-isobath and heavier lines cross-isobath spectral density (Csanady et al., 1988).

distribution at 200 m, with some help from surface velocity observations and float tracks, was a remarkable achievement at the time. Regarding the character of the meanders, Fuglister pointed out that they "do not suggest a series of waves gradually increasing in amplitude from west to east, but rather a quasi-stationary pattern with an abrupt change, near 62"W, from small

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE (0) Velocity Magnitude > 2 0 cm/s SEEP Winter 0 SEEP Summer A MASAR North Line 7 MASAR South Line + NASACS

107

1

[ ] S (Om FrmElottom

(blVelocity Magnitude < Scm/s A

% 40 6o

t

v

+

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1

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2000

3000

Bottom Depth .(m ) FIG.11. Frequency of occurrenceof (a)strongand(b)weak currents at different bottom depths (1984-1985 Slope Sea experiment; Csanady et al., 1988).

amplitude to very large amplitude waves” (1963, p. 276). He supported this with an illustration of Gulf Stream path in several surveys, all showing sharp northward deflection between 65 and 62”W. Suggestively, he added to this figure the position of the New England seamounts. Also he quoted velocity measurements at a number of depths to show that the Stream reached to the bottom, where its velocity was of the order of 10 cm/sec. Today, satellite images show the meanders in dramatic detail. The strongest temperature contrasts seen in infrared images may be used to define the position of the “north wall” of the Stream. Monthly series of these were used by R. Evans (private communication) to calculate an average path (Fig. 13, solid lines) as well as envelopes of location within which the north wall could be found 50 and 90% of the time (dashed and dotted lines). After leaving Cape Hatteras, the Stream is seen curving northward, staying at first close to the shelf edge, then taking two southward dips before embarking on the larger northward excursion near 62”W noted by Fuglister. As the envelopes show, however, the Stream moves irregularly north and south everywhere, and on

108

G. T. CSANADY

TOPOGRAPHIC WAVE

-

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low

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Bottom Depth

#

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FIG.12. Current velocity variancein spectral bandsdefined in text versus bottom depth(19841985 Slope Sea experiments; Csanady et a/.. 1988).

some images appears almost straight, flowingjust slightly north of east, along approximately 38"N.Along this latitude, the Stream reaches the abyssal plain at about 68"W, where its further movements have presumably nothing to do with bottom slope, at least until it reaches the New England seamounts. Meanders are also observed south of Cape Hatteras, where the north wall stays quite close to the edge of the continental shelf and strong northward flow

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110

G. T. CSANADY

reaches to the steeply sloping seafloor of the continental slope. Satelliteimages of these meanders were analyzed in detail by Legeckis (I 979).The “Charleston Bump” is a major topographic irregularity in the middle of the South Atlantic Bight, marked by a southward dip of the 600-m isobath (Fig. 14). Downstream of this feature the amplitude of the meanders increases considerably. The figure shows a succession of wave patterns, and demonstrates that they progress downstream. The typical wavelength is 150 km, wave amplitude 40 km, and phase velocity 30 km per day. The initial seaward deflection which starts the intense wave pattern occurs near, but somewhat downstream, of the Charleston Bump; as Legeckis observed, “the surfacecurrent overshoots” the Bump by something like 50 km. The maximum seaward deflection angle against the isobaths changes every few days, varying between 30 and 90”.The downstream wave pattern is also variable, and occasionally consists of only a single large deflection and return.

FIG. 14. Wave patterns of the Gulf Stream north wall from April 7, night (7N) to April 17, day (17D).Reprinted with permission from Legeckis, J . Phys. Oceanogr.,9, Copyright 1979, American Meteorological Society. .

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

111

The three-dimensional structure of the meanders was first discussed by Webster (1961a). From current studies and hydrographic surveys he inferred much the same bulk characteristics (wavelength, amplitude, phase velocity) as satellite images later showed. In addition, Webster demonstrated that onshore meanders were associated with upwelling of colder water, which floods the outer shelf and trails behind as the meander passes downstream. This was illustrated in greater detail by Bane et al. (1980;Figs. 15 and 16). The meander at the north end of the study region (box in Fig. 15) curves backward as it leaves behind warmer surface waters, while a vertical section (Fig. 16) shows the cold dome that forms by the upward motion of the lower layers. Meanders also occur upstream of the Charleston Bump, as first noted by von Arx et al. (1955). Their amplitude is much smaller than further north, but their structure is similar. Lee et a!. (1981) illustrated this structure as shown in Fig. 17. It is interesting to note that comparable meanders have also been observed in much weaker eastern boundary currents, such as the California

FIG.15 meanders over cold

G . T.CSANADY

112

0

20

40

-

DISTANCE

60

I60

(km)

FIG. 16. Cross section of folded meander and cold dome along solid line in previous figure. Isotherms of cold dome are displaced upward from their average depth (Bane et al., 1980).

Current, where such currents pass capes and points. Upwelling in the lee of capes is suggested by cool surface temperatures, implying vertical motions according to the scheme of Fig. 17 (Reid et d.,1958).

2.3. Upwelling and Undercurrent Where boundary currents are weak or distant, wind-driven flow events tend to dominate over the upper slope. These were first explored in a series of studies of coastal upwelling along eastern boundaries of ocean basins. They began off the Oregon coast in the early 1960s, shifted to western North Africa, and finished two decades later along the coast of Peru. The Oregon studies dealt almost entirely with the continental shelf, although they led to the discovery of the “poleward undercurrent,” a narrow band of flow which is opposite in direction to the alongshore wind generating the upwelling and is confined to the outer shelf and the upper slope (Smith, 1974; Mooers et al., 1976). A more complete picture of the undercurrent, and its relationship to the upwelling regime, came from investigations of the Northwest African upwelling area (Hughes and Barton, 1974; Mittelstaedt et al., 1975; Barton et al., 1977). Figure 18 shows the mean hydrography of this region, Fig. 19

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

113

KILOMETERS

FIG.17. Schematic illustration of Gulf Stream frontal eddy formed from a backward-folding meander and enclosed cold dome. Reprinted with permission from Lee et al.. Deep-sea Res., 28, Copyright 1981, Pergamon Journals, Ltd.

the distribution of alongshore and cross-shore velocity, and Fig. 20 horizontal property distributions on the 0, = 26.8 isopycnal surface. The 250-fold vertical exaggeration in Fig. 18 is somewhat misleading, but it is still clear that the undercurrent (positive, i.e., northward alongshore velocity; the upwelling occurs in response to southward-directed wind stress) hugs the upper slope, its core being at about 250 m depth. In the density transect (Fig. 18, right), the presence of the undercurrent is marked by the opening up of the isopycnals, a, = 26.8-27.0. The cross-shore velocity is directed shoreward beneath a thin surface layer, and is fairly strong down to 300 m. The steeply sloping seafloor presumably deflects the onshore flow upward, causing the upwelling. The property distributions in Fig. 20 show northward intruding tongues as a result of advection by the undercurrent. The details of the undercurrent-upwelling pattern vary with location. Smith (1981) compared the three upwelling regions studied in detail, Oregon, North Africa, and Peru. While the offshore surface drift under the equatordirected winds is in all areas confined to a layer 25-20 m deep, the depth of the compensating onshore flow varies, and is especially shallow (100 m) off Peru

114

G . T. CSANADY S TATION

L10

0

8

6

4

2 L1

i.2

100 Y

5

‘2 P

200

b 300

a

400 500

I00

0 100 50 0 100 DISTANCE FROM SHORE ( k m l

50

0

50

FIG. 18. Property distributions (average of 16 traverses) in cross-isobath transect off North Africa. Reprinted with permission from Hughes and Barton, Deep-sea Res. 21, Copyright 1974, Pergamon Journals, Ltd.

srmotv

STA rioN 0

FG

FF

L O

UW

R

FG

FF

L O

UW

R

500

100 50 0 0 DISrANCE FROM SHORE (kmJ FIG.19. Mean current velocity component (in cm/sec) alongshore (left) and cross-shore (right) (from Mittelstaedt et al., 1975). 4 00

50

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

15OW

110

1 5 OW

ISOW

FIG. 20. Depth of uo = 26.8 potential density surface (left), salinity distribution (middle), and temperature distribution (right panel) on the same surface, off the North African coast. Reprinted with permission from Barton et al., Deep-sea Res. 24, Copyright 1977, Pergamon Journals, Ltd.

(Brink et al., 1980, 1983). The poleward undercurrent in the same location extends upward to the base of the surface layer, downward to several hundred meters, and has its core at 100 m (Brockmann et al., 1980).

2.4. Coastally Trapped Waves Superimposed on the typical upwelling regime are variations in the strength of the alongshore flow in response to changes of the wind stress (Smith, 1974; Fig. 21). At the shelf edge (the 100-m isobath off Oregon) these are independent of depth and are accompanied by coastal sea level changes of a sign and magnitude required for geostrophic balance of a current extending approximately 60 km from the coast. Essentially the same response is seen over the extent of the wind field, i.e., up to thousands of kilometers (Huyer et al., 1975). Detailed statistical analysis of the Oregon observations by Kundu and Allen (1976) showed further that the disturbances started by wind propagate poleward, conserving their simple structure of unidirectional along-isobath flow, and extending seaward to a distance of order 60 km. Somewhat surprising was the finding that the propagation velocity changed between 1.5 and 6 m/sec from one year to the next. The alongshore propagation of massive flow events is certainly a remarkable phenomenon and has sparked much interest. Hamon (1962) discovered the slow (3 m/sec) propagation of a sea level signal northward along the east coast of Australia. Mooers and Smith (1968) found in sea level records, taken in 1933-1934 along the west coast of North America, evidence for propagation at a speed similar to Hamon's. Presumably this pressure signal is the

116

G. T. CSANADY

-6OL ADJUSTED DEPOE BAY S E A L E V E L

L

10

L IS

A

?O

JULY

L

25

I

A

L I

5

L

~ 111

~ I5

L

20

u

25

-

AUGll 5 1

FIG.21. North component of low-passed wind (upper), currents at three levels along the 100-m isobath (middle) and sea level adjusted for atmospheric pressure (lower panel) off the Oregon coast during the upwelling season. The similarity of records is striking. [From R. L. Smith, “Coastal Upwelling,” (F.A. Richards, ed.) 1974, Copyright by the American Geophysical Union].

W

d

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

117

signature of propagating flow events started by wind change, known today as “coastally trapped waves.” If the flow events in question are indeed wavelike, one would expect them to exist independently of the local winds, perhaps outside their region of generation, or started by some cause other than wind. What may be the most dramatic propagating upper-slope flow events occur along the west coast of South America, notably off Peru, and these are indeed unrelated to the local wind. Figure 22 shows the location of current meter moorings discussed by Smith (1978); Fig. 23 shows winds, sea level, and currents at 80 m, over 120 m depth. Both current and sea level records clearly show poleward propagation of large amplitude events, in that the current events are superimposed on a strong poleward undercurrent, and have no relationship to wind stress fluctuations. From the time lag at maximum correlation between two stations at varying distances a mean propagation velocity of 2.3 m/sec may be inferred. As Brink et al. (1978) pointed out, the relationship of currents to sea level is not quite straightforward, but the sea level signal also gives a similar estimate for the propagation velocity. To the extent that one can associate a typical period with the oscillation, it is about 12 days, implying a wavelength of about 2500 km. The flow events responsible for Hamon’s original discovery of coastally trapped waves were investigated in detail recently in the course of the Australian Coastal Experiment (ACE) and were reported by Freeland et al. (1986) and Church et al. (1986a,b). These studies again demonstrate clearly that propagating motions are confined to the shelf and some of the upper slope, since motions in greater depth and further from shore have an entirely different character. The clear conclusion is that, just as with the poleward undercurrent and upwelling, the coastally trapped wave is an upper slope (and outer shelf) phenomenon.

2.5. Deep Slope Currents in Other Locations Most of the evidence quoted so far came from a few locations, and one wonders just how general the results are. Over the deep slope off the Mid Atlantic Bight currents stronger than 5 cm/sec were associated with Gulf Stream meanders or topographic waves, the latter also presumably radiated by Gulf Stream disturbances. In analogy with this result, one would expect deep slope currents to be feeble where no vigorous boundary current is near, and topographic waves to be in evidence in proximity to such a current. Recent observations on the two sides of the Tasman Sea give weak support to this generalization. Current measurements by Heath (1986) off the South Island of New Zealand showed oscillations of ca. 30-day period over the continental slope, apparently impulsively started (Fig. 24). Current direction

118

G. T. CSANADY

FIG.22. Location of moored meters off Peru (Smith, J . Geophys. Res., 83, 1978, Copyright by the American Geophysical Union).

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

119

CALLAO WIND

SAN JUAN WIND

15

1 -10

19 76

15

-10

FIG.23. Wind, sea level, and currents (stick plot) at moorings Y and M (see previous figure). Maximum winds are about 10m/sec, maximum currents 0.5 m/sec. Note the absence of similarity between wind and current records, in sharp contrast to Fig. 21 (Smith, J . Geophys. Res., 83,1978. Copyright by the American Geophysical Union).

changed little, presumably following the local isobath, which included an angle of 45" with the coast. Current speed was the same at the two levels of observation. In character and amplitude these results strongly resemble Mid Atlantic Bight current records, and it is reasonable to conclude tentatively that the oscillations were topographic waves. Near-bottom currents were not observed however. Near-bottom currents were recorded in the Australian Coastal Experiment already referred to. Figure 25 shows a current vector time series from the Newcastle, New South Wales, transect, 90 m above bottom, in 1990m of water (Freeland et al., 1985).The most significant part of this record is the scale: only

G . T.CSANADY

120

d

-___ -adlam

B3Qn

limo-

FIG.24. Current records in deep water off the New Zealand continental slope (top panel) at the location shown in the bottom sketch. Reprinted with permission from Heath, Continental Shelf Rex, 5, Copyright 1986, Pergamon Journals, Ltd.

u t w u >

-5

,

3 1 RUG

0

2 0 SEP

20

10 O C T

40

30 OCT

60

19 NOV 80

9

DFC

29 D E C

18 JAN

I00

120

140

7

FEE

2 7 FEE

18 MAR

160

180

200

FIG.25. Current vectors over the deep continental slope off Newcastle, NSW, 100 m above the seafloor. Speeds are mostly near 2 cm/sec. Reprinted with permission from Freeland er ul., J . Phys. Oceunogr., 16, Copyright 1985, American Meteorological Society.

122

G.T.CSANADY

once was a speed of 5 cm/sec reached, and the currents were “especially weak” by Mid Atlantic Bight criteria 100% of the time. Two other current meters deployed in this experiment at similar depths yielded similar records. Higher up in the water column currents were much stronger, and were associated with temperature fluctuations. Warm core eddies in this region are common, and are generally thought to be eddies spun off of the East Australian Current. Why the East Australian Current should radiate topographic waves on the eastern side of the Tasman Sea, but not on the western side, is unknown.

2.6. Density-Driven Currents The only steady continental slope currents discussed so far have been the poleward undercurrents of upwelling regions, which are presumably part of the ocean’s response to the wind system which evokes the upwelling. Important upper, middle, and deep slope currents also arise in response to density differences between coastal and abyssal waters, or between any two water masses of different origin. Such differences may be caused, for example, by freshwater runoff, surface heat loss, or ice melting. One good instance is the Labrador Current flowing along the edge of the Grand Banks. Figure 26 shows the mean dynamic topography of the sea surface in this region in April; note that the Labrador Current is marked by strong gradients just inshore of the 1000-m depth contour south of 48”N. Figure 27 shows temperature and salinity in a section at 44”30”, revealing that the density differences responsible for the dynamic height gradients are due to the presence of a fresh and cold surface intrusion, the freshness presumably due to river runoff. The dynamic height gradients imply velocities of order 20 cm/sec in the wedge of fresh surface water, relative t o the deeper waters (which here also move southward at a speed of order 10 cm/sec). Prevailing northwesterly winds presumably help maintain the southward flow, so that the current in the fresh wedge cannot be regarded as entirely density driven. A remarkable example of a density-driven current persisting over long distances in the face of opposing wind is the Leeuwin Current, flowing along the coast of Western Australia (Thompson, 1984; Godfrey and Ridgway, 1985).The fresh and warm water mass leaks in from the Pacific along the north coast of Australia, and flows poleward above an equatorward undercurrent of more saline water. The latter takes the place of the usual equatorward surface currents at other eastern ocean boundaries. Coastal currents driven by density differences due to river runoff are common on continental shelves (Pettigrew and Murray, 1986), but reach to the upper slope only in a few places. Strong surface cooling, in the absence of significant salinity contrasts, produces cold and heavy coastal waters, the

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

123

FIG.26. Mean monthly topography of the sea surface off the Grand Banks in April relative to 1000 m. The Labrador Current flows along the edge of the shelf southward, retroflects, and joins the North Atlantic Current (Scobie and Schultz, 1976).

124

G. T. CSANADY

0

100

200 300

400

g500

r'8600 TOO

000

1000

1100

0

200

400

3

i

H

600

600

1000

FIG.27. Temperature (upper, in "C) and salinity distribution (lower panel, %) in a transect of the Labrador Current at W30'N (Scobie and Schultz, 1976).

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

125

opposite of the coastal freshening case. Shallow water cools off rapidly, as its temperature also becomes uniform due to thermal convection, and tends to underflow less completely cooled deeper water. Convective overturn often extends to outer shelf waters, which then cascade down the upper slope. This occurs in the Gulf of Lions in winter, under the tramontane and mistral winds (Fieux, 1970). Figure 28 shows the temperature in a transect off Banyuls. The isopycnals (not shown) suggest southward flow in the cold water hugging the slope at 200-300 m depth, at a low velocity. It is interesting to note that a heavy bubble of water sits on the bottom of the outer shelf, bottom waters over the upper slope being slightly lighter. Similar bubbles were observed in the Adriatic (Artegiani et al., 1986).The “cold pool” of the Mid Atlantic Bight is analogous: cold water generated mainly over Nantucket Shoals underflows outer shelf waters, establishing a temperature minimum, without, however, cascading down the slope (Houghton et al., 1982). It is not only coastal processes that generate density contrasts; convective overturn occurs in deep water and may reach abyssal depths, as in the Norwegian Sea, or some intermediate deep level, as in the Labrador Sea. The cold water mass produced by convective overturn is drained mainly via deep or mid-level western boundary currents, reviewed recently by Warren (1981). All of these currents are in contact with moderately or steeply sloping bottom. Upon reaching the seafloor or the level of their density, water masses produced by convective overturn presumably drift westward, before being deflected by the slope into an equatorward deep or mid-level boundary current. Westward drift, collection into a boundary current, and flow along the boundary all require pressure gradients for geostrophic balance, which implies that sloping isopycnal surfaces intersect a differently sloping seafloor. Many of the properties of deep or mid-level boundary currents may be inferred from the topography of these isopycnal surfaces. One should remember, however, that uneven isopycnal surfaces are a symptom, not the cause, of the currents. Their generation is due to the injection of fluid into some isopycnal layer by the water mass formation process. Where the injection is into a layer in contact with seafloor of steep enough slope, drainage takes place along depth contours, much as the drainage of freshened water near a coast. A dramatic illustration of water mass drainage, from Talley and McCartney (1982), is that of Fig. 29, which shows the distribution of Labrador Sea Water (LSW) in the North Atlantic. Potential vorticity, at once a measure of layer thickness and a quasi-conservative tracer, is mapped in the figure. The darkshaded minimum in the Labrador Sea marks the generation area of LSW, where convective overturn in winter reaches depths of a kilometer and more. The density of LSW is typically a, = 27.7, and it spreads out in layers at an average depth of 1500 m. As the figure shows, its core hugs the continental slope, and is identifiable as far south as the Antillean arc. South of the Grand Banks the LSW core is only about 200 km wide. A hydrographic section of

126

G . T. CSANADY

i00

OOO

1500

T (Oc)

pot

2000

FIG.28. Temperature and salinity distribution over the continental slope off Banyuls, France, in the Gulf of Lions (Fieux, 1970).

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

127

FIG.29. Potential vorticity ( f d e / d z ) x 10-'2m-' sec-' at the core of the Labrador Sea Water layer. Reprinted with permission from Talley and McCartney,J. Phys. Ocennogr.. 12, Copyright 1982, American Meteorological Society.

Richardson (1977) taken off Cape Hatteras shows the core in contact with the continental slope between 500 and 1500 m depth. MASAR observations off New Jersey gave an average southwestward current velocity of 7 cm/sec in the center of this LSW core. At the sea surface, production and accumulation of warm water masses cause equatorial and subtropical pools to pile up against the continental margin. As a result, the thermocline and the sea surface slope up and down along the coast. Alongshore gradients of thermocline depth and surface dynamic height (steric height) imply onshore or offshore flow, to be absorbed by or supplied from the shelf and slope region. A system of shelf and upper slope currents must be present under such circumstances to maintain mass balance. Even if the distribution of steric height offshore is wind produced, the

128

G . T. CSANADY

proximate cause of the associated boundary currents is the density field. One should note here that coastal sea level differs from the dynamic height offshore by an amount determined by shelf and slope currents. Steric heights off the west coast of North America have been discussed in detail by Reid and Mantyla (1976). Figure 30 shows their steric height distribution versus latitude, for several reference levels. Most of the steric sea level variation is seen to be due to the gradient of a thermocline shallower than 500 m. Between 16 and 38"Nthe surface slope is 1.4 x lo-', implying onshore flow at a velocity of 1.4 cm/sec at 38"N and 3 cm/sec at 16"N. Over a 500-m deep surface layer the total shoreward transport is 22 x lo6 m3/sec, showing the need for substantial upper slope currents to accommodate the inflow. Along other continental margins the pressure gradients are similar or even larger, as Sturges (1974)has demonstrated. Figure 31, taken from Godfrey and Ridgway (1989, is Sturges'comparison of steric heights in the different oceans, with the Indian Ocean data added. Offshore steric heights constitute external forcing of the shelf-slope region on a long, at least seasonal, time scale, affecting primarily the portion of the slope above the thermocline. The response to this forcing is complicated by a

9

OD

5'N

.(oo

.Iso

20"

25O

300

35'

40'

450

500

55-

W N

NORTH LATITUDE FIG.30. Alongshore variation of sea level off the west coast of North America, deduced from steric heights (Reid and Mantyla, J. Geophys. Res.. 81, 1976, Copyright by the American Geophysical Union).

129

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

2.6 2.4

-

*.'

Eastern

p g 2.2 -- a

1.4 1 **

,

t

..*/*\\ ,*'

** ,

- 2.0

Western

Pacific

\

-1.8 0

-1.82: -c -1.45"

*- -3'

a :*

.

I

w est ern

f

a.'

Atlantlc

.....,"' 1

40"s.

1

1

20a

1

Oa

.

20a

.* t.. ..* '* e.4 1

1

-0.8

p"

- 0.6 ~

~

~

40°H.

Latltude

FIG.31. Steric height variations along continental margins. Reprinted with permission from Godfrey and Ridgway, J . Phys. Oceanogr., 15, Copyright 1985, American Meteorological Society.

short-term effects so that upper slope currents often present a confused pattern. A good example was given by Kinder et al. (1975), of the Bering Slope current system. Figure 32 shows the dynamic topography of the sea surface in this area. Fairly strong upper slope currents are present in some locations, forming branches of eddy-like flow structures extending some distance offshore. The detailed discussion of the California Current system by Hickey (1979) makes it clear that a similarlp complex flow structure is the rule along the west coast of North America. Upper slope currents transporting relatively warm water, presumably from a lower latitude offshore pool, have been described by Booth and Ellett (1983) off Scotland, and by Thomson and Emery (1986) off British Columbia. In the Slope Sea off the North American east coast an upper slope current 500 m deep and 100 km wide transports some 3 x lo6 m3/sec of water southwestward along the Mid Atlantic Bight continental slope (Csanady and Hamilton, 1988). The poleward undercurrents of Peru and North Africa have already been mentioned. One may conclude that currents in the upper 300 to 500 m of the water column over the continental slope are ubiquitous. In some places they have complex structure, containing closed streamlines. Everywhere they are bound to be important in accommodating inflow from and outflow to the deep sea arising from the uneven geographical distribution of thermocline depth. A fitting finish to this survey of slope currents is reference to a satellite image of the North American west coast in Fig. 33. The image was color coded to

130

G. T. CSANADY

poo

s Y

FIG.32. Dynamic topography (in units of cm) of sea surface over the continental slope along the northeastern edge of the Aleutian Basin (BeringSea). Reprinted with permission from Kinder er al.. J. Phys. Oceanogr, 5, Copyright 1975, American Meteorological Society.

show chlorophyll pigment concentration arising from biological activity, which is most intense next to the coast of the Baja peninsula. Long curved fingers of high chlorophyll water extend out from the coast several hundred kilometers, demonstrating that surface flow structures are coherent across the entire shelf-slope region, right out to abyssal depths. The image was taken in May; Fig. 34 shows the mean dynamic topography of the sea surface in April from Hickey (1979)for comparison. The eddies of the image relate to the shape of the coastline and cut across the main streamlines of the California Current. Nothing could make clearer the importance and challenge of understanding shelf-slope currents.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

131

FIG.33. Satellite image of surface chlorophyll concentration on the coast of Baja, California, showing large coastal eddies which extend many hundreds of kilometers from the coast. (Courtesy Dennis Clark, NOAA, Miami, Florida.)

132

G.T. CSANADY

FIG.34. Dynamic topography (in units of m) of sea surface in April along the California and Baja coast. Reprinted with permission from Hickey, Prog. Oceanogr., 8, Copyright 1979, Pergamon Journals, Ltd.

3. THE FUNDAMENTAL SLOPEEFFECT

As the survey of the previous section has shown, in all the complexity of ocean currents over continental slopes a few major phenomena stand out. These can be simulated in quantitative detail by simple models and related to the forces that affect a large body of water on a rotating globe. In this sense we may claim to understand them. The simpler the model, the clearer is the

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

133

insight gained from it, as long as the simulation remains realistic. The object of the remaining sections of this article is to review our insight into phenomena observed over the continental slope, an end best served by maximum simplicity, consistent with realism, in the models employed. In the usual approach to ocean dynamics, the pressure field is calculated from the hydrostatic equation, and the density is taken to be the sum of a constant reference density po and a small density perturbation poc(x,y, z, t). The typical horizontal pressure (p) gradient is then (3.1) where [(x, y, t) is surface elevation above a geopotential surface, g is acceleration of gravity, and the pressure is in kinematic units, i.e., divided by the reference density. The equations of motion and continuity may now be written as

where u, v, w are velocity components; djdt = a/at

+ ua/ax + vajay + walaz

is the total derivative following the motion; f is Coriolis parameter, and r,, r y are friction force components related to the Reynolds stress tensor (again in kinematic units) by - - au’2

I%=------

ax

auw

auiw‘

ay

aZ

3.1. Vortex Tube Stretching

The dynamics of steady or low-frequency motions in the ocean usually revolves around the balance of the vertical vorticity component. A sloping bottom contributes to this balance a term which may be interpreted physically either as “vortex tube stretching” or as “pressure torque.” Applied to flow phenomena requiring different idealizations, e.g., a fast boundary current or

134

G. T. CSANADY

slow oscillations of long period, the same term may appear in different forms and may be balanced by different physical effects,such as changing vorticity or bottom friction torque. Sometimes it is realistic to suppose that the distribution of horizontal velocity is “slablike,” i.e., independent of depth between two surfaces zl(x, y) and z2(x,y) defined by some rule such as “the free surface,” “the seafloor,” or “a surface of constant density”:

aulaz = au/az = o

(z2 I

I zI)

(3.3)

Sufficient conditions for ensuring that this assumption is consistent with the equations of motion and continuity are

o =o

aelax = aE/ay =

ar,/az

= ar,/az

awiaz

=

(3.4)

-au/ax - av/ay = (w,

- Wb)/h

where h = z , - z 2 is layer thickness and the subscripts t and b designate the top and bottom surfaces of the layer in question. As the last equation shows, the divergence of the horizontal velocity equals the “stretching rate” of the water column. For the case restricted by Eqs. (3.3) and (3.4) a simple form of the vorticity equation now emerges upon taking curl on Eqs. (3.2), uncomplicated by vortex tilting: (d/dt)(o

+f

) = (0 + f)(w, - W b ) / h

+ ar,/ax

-

(3.5)

where o = av/ax - au/ay

is the vertical component of the vorticity in a frame of reference rotating with the earth, and dldt now includes advection by the horizontal velocity only. In an inertial frame the (“absolute”) vorticity is o + f.Equation (3.5) shows that the absolute vorticity changes, following the motion of a fluid column, at a rate equal to vorticity times the stretching rate of the water column plus the friction force curl. In the absence of friction the result expresses Helmholtz’s theorem: the total vorticity w f changes, as the vortex tubes are stretched, so that the angular momentum of fluid columns remains conserved. The column stretching rate may also be written as

+

(w, - W J / h = (l/h)dh/dt

(3.6)

so that Eq. (3.5) may be put into the alternative form frequently used by C. G. Rossbv:

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

135

The quantity (w + f ) / h is known as the potential vorticity of a slablike layer. According to Eq. (3.7), in the absence of any stresses, this is conserved following the motion of slablike fluid columns. Although of severely restricted validity, this conservation principle is useful in the examination of idealized models. Suppose now that the slablike layer extends from the free surface to the solid bottom, i.e., z1 = [, z2 = -H(x,y), where H is water depth. The velocity component normal to a solid bottom must vanish, so that ubaH/ax

+ ub a H p y +

wb

=

o

(z = -H)

(3.8)

where the subscript b denotes quantities at the bottom. The elevation of the free surface above its static equilibrium position ( is much smaller than water depths of interest, so that h z H. Also, in slow motions over bottom as steep as one finds on continental slopes, the surface velocity w, = dc/dt in steady or slowly varying motion is negligible compared to the typical bottom velocity w b . Neglecting w, and substituting Eq. (3.8) into Eq. ( 3 . 9 , one arrives at d dt

-(w

+ f )=

ar, ~

H

ar,

(3.9)

The term in parentheses on the right is the scalar product of horizontal velocity and the gradient of the seafloor, i.e., the velocity component perpendicular to the depth contours times the depth gradient. One may think of the bottom as deflecting the flow upward or downward. The resulting stretching or squashing of vortex tubes generates vorticity. This term containing the slope of the bottom embodies the fundamental slope effect on ocean circulation, the effect to which Ekman (1923) drew attention in the passage quoted in the Introduction. The stretching or squashing of vortex tubes, especially when most of the absolute vorticity is contributed by the planetary vorticity f,is an abstract notion, and one wonders just how exactly the angular momentum of fluid columns, as seen on a rotatingearth, is changed. In the equations of motion the Coriolis force embodies earth rotation effects. The curl of this force (proportional to the torque about the center of mass of a fluid element) consists of two parts: (1) the advective change df/dt = udf/ax + u @lay, and (2) f times the divergence of the horizontal velocity. The latter, according to the above discussion, can be interpreted as vortex tube stretching. A more direct interpretation is Coriolis force torque associated with the expansion (contraction) of the fluid column as it moves up (down) the slope. Pedlosky (1979,p. 29) gave a clear and detailed discussion of the relationship of Coriolis force torque to Helmholtz’s theorem applied to a rotating fluid. It is illuminating to pursue the point still further for the case of a sloping seafloor.

136

G. T. CSANADY

3.2. Pressure Torque The fundamental slope effect examined above turns up in a different guise if the angular momentum balance of a fluid column fixed in space is considered instead of vorticity, allowing for external torques as well as the torque due to momentum advection. The drastic simplifications of the previous section are not needed, only a depth integration of the equations of motion at fixed x and y. One arrives at the following “transport” equations:

_au_ f V = -gHat

av

-

at

ac - 9

ax

a€ + R, + M, ax

(H+ z ) - ~ z

(3.10)

+ fU =

avlax + a v j a y = -ailat where the transports are defined by

r U = [-nudz

6

V = j-nudz

The last terms on the right of Eq. (3.10) express friction and momentum advection: c c R, = r, dz R, = r,, dz

1.

Taking curl now results in

da t ( av D ; - yav ) + u d x f vafi . y = -af J ( H , p a ) + 2 - - + a- -Rax

aR, ay

aM,

ax

aM, ay (3.1 1)

On the right, apart from friction and momentum advection terms one finds the vector product of depth gradient and pressure gradient at the bottom, or the Jacobian of depth and bottom pressure:

in which the terms in parentheses are the components of the pressure gradient at the bottom [see Eq. (3.1)]. This expression can be interpreted as normalized

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

137

bottom pressure torque. The Jacobian is nonzero if bottom pressure varies along lines of constant depth. Suppose for simplicity that lines of constant depth are parallel to the y axis. The varying bottom pressure then exerts a torque about a vertical axis, through the center of mass of an elementary fluid column of cross section (dx dy):

(See Fig. 35 for illustration.) As foreshadowed, the elementary torque is proportional to the Jacobian of Eq. (3.12). The Jacobian may be rewritten using the geostrophic velocity (Ugh, ugh) at the bottom:

-J(H,

Pb)

= fugb

a H i d x + fVgb aHiay

(3.14)

In case the geostrophic velocity (ugb,ugb)is a good approximation to the total velocity, the results of the previous section are recovered; by Eq. (3.8) the scalar product of depth gradient and bottom horizontal velocity equals the vertical velocity. To the same approximation then, the bottom pressure torque expresses the vortex tube stretching effect. Angular momentum balance of a fluid column is, however, not quite the same thing as vorticity

4 FIG.35. Pressure torque on elementary fluid column, when pressure varies along an isobath. As the plan view (bottom) shows, the varying pressure force tends to impart anticyclonic spin to the fluid (in the illustrated case, the pressure increasing toward positive y).

138

G. T. CSANADY

tendency balance, and vortex stretching may be present in ageostrophic flow in the absence of pressure torque. For an illustration consider the following simple geostrophic adjustment problem, modeling the generation of a coastal jet by onshore or offshore flow:

f V = g H atlax

av/at + f V

=0

(3.15)

au/ax = o The along-isobath flow is geostrophic, and the cross-isobath transport steady and nondivergent, so that the Coriolis force of the cross-isobath flow accelerates the fluid along the isobaths. The vorticity is o = (a/&)( V/H),the rate of change of which is, for constant Coriolis parameter, 1 d H d ( V / H ) fU d H H dx at H Z dx

(3.16)

expressing the vortex tube stretching effect. The angular momentum of the fluid column, however, is proportional to aV/dx, and its rate of change vanishes according to the second of Eqs. (3.15), because all torques vanish. This comes about because the shallower part of a column speeds up more than the deeper part, conserving total angular momentum while increasing vorticity. As may be seen from Eqs. (3.9) or (3.11), the fundamental slope effect, whether expressed as vortex tube stretching or as pressure torque, may be balanced by a combination of local and advective change of (relative) vorticity, planetary vorticity advection, or friction curl. The general case is clearly complex. However, when one of the balancing terms dominates, a simple model can usually be constructed. With any luck, such a model describes an observed phenomenon tolerably well. In the following sections the different balancing terms for the slope effect in the vorticity equation are taken one by one, and their effects exhibited in isolation by means of simple models. It turns out that this approach yields more or less all the insight into the physics of steady or slowly changing slope currents that we possess today; different major phenomena generally correspond to different vorticity balances. More complex interactions have been explored in a few recent studies, but these have not so far contributed important new ideas. 4. VORTEXTUBESTRETCHING VERSUSVORTICITY ADVECTION

One class of phenomena in which vortex tube stretching plays a central role is the steady flow of a current over a deep, sloping seafloor, under circumstances such that the vorticity of a fluid column changes following the

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

139

motion, the vorticity change being the only effect of vortex tube stretching. Naturally irregular topography, curvature, or divergence of isobaths causes misalignment of flow and isobaths, giving rise to vortex tube stretching, which is then reflected in vortex motion. The next few sections discuss different manifestations of this scenario. 4.1. Meandering of a Boundary Current One important and very interesting phenomenon is the meandering of a narrow, coherent boundary current (a “jet,” in the sense of jet stream). When such ajet, in contact with a sloping bottom, flows across isobaths the primary balance to vortex tube stretching may be provided by changing vorticity, in particular changing curvature of the path. This results in a deflection of the jet into a sinuous track, akin to the north-south excursions of the jet stream. The idea that the meandering of a western boundary current may be due to vortex stretching over a sloping bottom was first explored by Warren (1963), in his study of Gulf Stream meanders. This study was the first detailed analysis of bottom slope effects on ocean currents, combining geophysical fluid mechanics with observational oceanography. Warren’s model, reduced to its essentials, is as follows. Consider a deep, slablike fluid column in contact with the seafloor, and suppose friction negligible. Equation (3.7)applies, with the right-hand side set to zero: (4.1) Let streamlines of steady flow originate in a region with parallel isobaths, where the vorticity w is zero. Conservation of potential vorticity along individual streamlines results in w = f ( H - H,)/H,

(4.2)

where H , is the depth in the region of origin and the Coriolis parameter is taken to be constant. The vorticity is conveniently expressed in “natural” coordinates, directed along (axis s) and across (axis n) the stream: o = v/r

-dv/h

(4.3)

where u is velocity magnitude and I is radius of curvature. Substituting into Eq. (4.2)one has

v / r - &/an = f ( H - H J H ,

(4.4)

Let the seafloor be modeled by an inclined plane of slope a. Changes in depth along streamlines are then proportional to the cross-isobath displacement of a streamline from its original position. Laying the x axis across the

140

G. T.CSANADY

isobaths, the y axis along them, with positive x toward increasing depth, the depth change along a streamline is H - H , = a(x - x,)

(4.5)

For a coherent jet it is realistic t o suppose that the locus of maximum velocity is a streamline, For this streamline the previous equations yield the following simple relationship between cross-isobath displacement and curvature: (4.6) u/r = (Ef/Ho)(X - xo) where the normal velocity gradient &/an is zero by hypothesis. In the x , y coordinate system fixed to the isobaths the radius of curvature is expressed by

I/r = -x"/( I

+ x'2)3/2

(4.7)

where primes denote differentiation with respect to y. When substituted into Eq. (4.6) this yields a differential equation for cross-isobath displacement, with the value of u yet to be specified. For small proportionate changes of depth u may be taken to be constant. All dimensional quantities in Eq. (4.6) may then be combined into the length scale:

L = uHo/af

(4.8)

If distances x and y are expressed as multiples of the length scale L, the path equation becomes [Eq. (4.6) with Eq. (4.7) substituted]: x"/(1

+

x'2)3/2

+x =0

(4.9)

where xo has been set zero without loss of generality. Physically, this equation expresses the fact that the displacement of the jet from its equilibrium depth contour causes the path to curve back toward the equilibrium position (as long as the flow has a component toward positive y). The character of the solutions emerges for the case of small departures from the isobaths: if x" << 1, the denominator in Eq. (4.9) may be dropped and the linearized solution found: x = siny

(4.10)

This describes sinusoidal meanders about an equilibrium isobath, of wavelength 2xL. The solution of the full Eq. (4.9) may be expressed as (4.1 1)

where C = l/,/m a parameter , related to the initial inclination xb of the path. Quarter-wavelength integral curves for different values of C show the wavelength diminishing slowly with increasing initial inclination. The shape

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

141

of the path departs slightly from the sine curve, but the character of the solution remains the same as in the linearized case. This simple model leaves open the question of meander initiation: A given initial inclination may be thought due to an unstable wave or to sudden encounter with a topographic irregularity. Once a current toward positive y is impulsively forced to cross isobaths, it oscillatesabout its equilibrium isobath indefinitely. Warren (1963)formulating the problem somewhat more generally, took into account planetary vorticity advection and defined the parameter uH,/for in terms of integrals involving mass and momentum transport. This definition retains the correct quantitative relationship between vortex tube stretching and angular momentum change for the very nonuniform distribution of velocity with depth which characterizes the Gulf Stream. Quantitative transport values Warren used are equivalent in the simple slablike model to Y = 1.1 m/sec, H = 4.5 km, f = 10-4/sec, and c1 = 0.3-1.1 x These give a length scale L of order 70 km and a linearized model wavelength of order 450 km. Warren calculated the path of the Gulf Stream for a few typical cases, for comparison with observed paths, section by section, allowing for changes in bottom slope. One of his comparisons is shown here in Fig. 36. The predicted and observed meander wavelengths are indeed closely similar. 4.2. Upwelling in the Lee of Capes and Points As discussed before, vortex tube stretching is due to vertical motion near the seafloor, with velocity w,, determined by cross-isobath flow and bottom slope. A positive value of w,, means that fluid is brought up from lower lying layers, an occurrence described as upwelling. Positive wb is associated with vortex tube squashing and anticyclonic(negative) vorticity tendency. In the absence of a balancing effect, the vorticity decreases following the motion, while upwelling is active. If changes in au/an are negligible, the vorticity drop is reflected in changing curvature of the streamlines, straightening of a cyclonically curving streamline, or increasing anticyclonic curvature. Vice versa, increasing anticyclonic curvature may be attributed to upwelling. Applying ideas similar to these, Arthur (1965) offered an explanation for the intensification of upwelling south of capes and points of the California coast, discovered by Reid et al. (1958). Southward-directed coastal currents flow around these capes; their vorticity is cyclonic and greatest just off a cape (Fig. 37). The streamlines more or less follow the isobaths, so that their vorticity decreases in the lee of the cape. If vortex tube squashing is responsible for the vorticity change, upwelling should accompany the straightening-out of the streamlines. This implies that the streamline has to overshoot its equilibrium position in order to be squashed before being straightened out.

142

G. T. CSANADY

AVERAGE DEPTH GRADIENT FOR EACH PATH SEGMENT:

--

-

0, OBSERVED PATH CALCULATED PATH. INITIAL DIRECTION! 015.r

a

0, G

A 0

7*10-' 9.10-'

140.7

c

153

n

9.10-1

I

0

9r10-'

170 170

E F

IO~IO'' 3x10"

I65 I35

J K L

a

SxIO-'

120.T

SIIO-'

I20

0 7~10'' 7.10*'

7r10-'

-

16s 165 165

FIG.36. One of Warren's (1963) comparisons of observed Gulf Stream path with theory.

11 1 11

'I'

FIG.37. Arthur's (1965) sketch of flow and vorticity change around California capes. From Arthur, J . Geophys. Res., 70, Copyright 1965 by the American Geophysical Union.

Quantitative estimates are as follows. A current of velocity 0.1 m/sec rounding a cape with a radius of isobath curvature of 30 km has vorticity o = u/r = 3 x 10-6/sec. It loses this in a period of order t = r/u = 3 x lo5 sec, so that dwldt z lo-"

seC2 = fwb/H

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

143

In water 100 m deep this requires an upward-directed bottom velocity of m/sec, which is as high as found in prime upwelling regions of the world. The above argument depends on the assumption that the flow follows isobaths of relatively small radius of curvature. In light of results in the previous section one would expect the flow to meander about the equilibrium isobaths following encounter with the cape. The length scale of such meandering is, from Eq. (4.8) with the quantitative data as used above, L = 3 km, or short compared to the radius of curvature. Any meandering under such circumstances should be of low amplitude and should not affect the conclusions regarding upwelling. Other important conditions for the validity of those conclusions are small av/an and vanishing bottom friction. These are generally satisfied over the upper continental slope in weak coastal currents. Arthur’s ideas were invoked by Blanton et al. (1981) as an explanation of low bottom temperatures on the outer continental shelf in the South Atlantic Bight, in the lee of Cape Canaveral and Cape Lookout. At the edge of the shelf bottom slope changes rapidly and complicatesthe kinematicsof cross-isobath flow. As discussed in the next section, the simple balance of curvaturevorticity advection and vortex tube stretching which underlies Arthur’s ideas does not hold in such more complex regions.

4.3. Irregular Topography: Perturbation Analysis In the meandering jet model the initial inclination of the path is loosely ascribed to a sudden change of isobath orientation or other irregularity on an otherwise smooth slope. In Arthur’s conceptual model of upwelling in the lee of capes the flow is supposed to follow curved isobaths. Ad hoc assumptions of this kind are unsatisfactory, because intuition often proves to be misleading when earth rotation effects are important. A more rigorous analysis of the effects of irregular topography on flow over a steep slope was performed by Janowitz and Pietrafesa (1982), based on a perturbation expansion. A simplified version of their analysis follows. Consider a steady current originating in an upstream region of constant velocity u, directed along isobaths over a seafloor of constant slope a. Downstream, the current encounters irregular topography, where isobaths curve and diverge on a horizontal scale L large enough to keep the Rossby number small:

RO = v,/fL<< 1

(4.12)

In the previously discussed typical cases these parameters were, for Gulf Stream, u = 1 m/sec, f = lOP4/sec, L = 100 km, Ro = 0.1; for California capes, u = 0.1 m/sec, f same, L = 30 km, Ro = 0.3. Over the Charleston Bump

G. T.CSANADY

144

in the South Atlantic Bight u = 0.5 m/sec, L = 100 km, Ro = 0.05. The small Rossby number assumption is thus satisfied in all these cases. A second scaling assumption used by Janowitz and Pietrafesa is that the slope is comparable to the aspect ratio H / L of the topography: a L / H = 0(1)

(4.13)

In the casesjust referred to this is also true: H J L is generally of order 0.01, which is also the typical depth gradient of the continental slope. Neglecting density variations and friction, the equations of motion, Eqs. (3.2),are rescaled by XlL, Y l L

ufu,,,vfu,, + u , u Plf4L

z f H , +z

x, Y

+

+

wL/u,,H,+w

(4.14)

HJH, + H

P

where H , is a suitable reference depth. The scaled equations are (4.15)

aulax

+ au/ay + awlaz = o

The flow variables are now expanded in terms of the Rossby number, on the scheme: u = 00 + RO u1 + . . * *

(4.16)

Substituting such expansions into Eq. (4.15), the zero-order flow is found to be geostrophic and nondivergent:

uo = - a p o / a y

vo = apo/ax

auolax

(4.17)

+ auo/ay = o

Neglecting the very small vertical velocity dlJdt at the surface as before, awJdz = 0 implies w,, = 0, i.e., by Eq. (3.8): u0 dHJdx

+ tr0 dHJdy = 0

(4.18)

so that the zero-order flow indeed follows the isobaths as supposed by Arthur. The zero-order pressure p o , which plays the role of streamfunction,is then also a function of depth only: Po = Po(H)

(4.19)

from which 00

= (dpo/dH)dH/ax

UO =

-(dpo/dH)aH/ay

(4.20)

145

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

The dpo/dH term follows from the upstream conditions:

dpo/dH

= u,/a

(4.21)

The vorticity of the zero-order flow is therefore

mo = avo/ax - auo/ay = ( ~ , / ~ ) V Z H

(4.22)

exhibiting the effects of topography. In terms of curvilinear coordinates along the flow, n positive to deeper water, the Laplacian may be written as

(s) and across (n)isobaths, with s positive in the direction of

V ~ = H (a2H/an2)- ( l / r ) a H / a n

(4.23)

with r positive for cyclonic curvature. The last two equations show that the vorticity wo changes with the convexity of the seafloor as well as with the curvature of the streamlines. Where the convexity does not vary, vorticity changes are as envisaged by Arthur (1965). The velocity magnitude is 40 = ( ~ U / W ~ I (4.24) The rate of change of the vorticity, following the flow along isobaths, is therefore dwo/dto = (v,/a)21VH 1(a/as)(V2H) (4.25) where

didto = qoa/as How are these vorticity changes brought about? The answer is contained in the first-order equations: (4.26)

au,/ax + ao,/ax

+ aw,/az = o

Cross-differentiationyields the rate of change of zero-order vorticity:

dooldto = dw,/dz = - W I L / H

(4.27)

This is the nondimensional version of balance between vorticity tendency and vortex tube stretching due to the first-order pressure torque:

H dooldto = (aPl/ax)/aH/aY- (aPl/aY)/aH/ax

(4.28)

The changes in the vorticity of the zero-order flow over irregular topography thus result from cross-isobath flow superimposed on the basic along-isobath motion. This also implies vertical motion near the seafloor, ie., upwelling or downwelling, as discussed earlier at some length.

146

G . T. CSANADY

Janowitz and Pietrafesa (1982) also draw attention to the importance of isobath divergence, rather than curvature, near the shelf edge, where the seafloor is convex, aZH/an2relatively large. Over diverging isobaths in this region the changes in convexity contribute the major part of the change of V’H. Along the 92-m isobath off Cape Canaveral, for example, VzH = 0(10-6) m-l

(i/r)aH/an < 1 0 - ~ m d l so that vorticity changes are mostly due to changing convexity. Moreover, the divergence of the isobaths is large enough to induce bottom velocities of order 3x m/sec, and correspondingly strong cross-isobath flow. Figure 38 shows the near-bottom current vector estimated by Janowitz and Pietrafesa near the shelf edge, at three locations in the vicinity of Cape Canaveral, compared with the observed direction of flow. Although the theory overestimates the cross-isobath flow, the direction and order of magnitude are correct. The authors emphasized that the large onshore velocities induced by isobath divergence over a seafloor of substantial convexity are the likely cause of the low temperatures observed over the outer shelf (Blanton et al. 1981), rather than the minor changes in isobath curvature. The perturbation analysis gives very good insight into the effects of capes, points, or diverging isobaths in a region of varied topography. It verifies the intuitive arguments of the previous section, and adds the recognition that isobath divergence is also potentially important. The discussion so far, however, has not addressed the meandering of a current initiated by seafloor irregularity. To this topic we turn next. 4.4. Rossby Wake of a Seamount Let it be supposed that seafloor irregularity is concentrated in a limited region, and that elsewhere the isobaths are straight. The prototype problem is a seamount on a sloping plane bottom, immersed in a stream of uniform velocity, except for changes due to the seamount. This is a reasonable idealization of the flow of the Gulf Stream past the Charleston Bump, for example. Of interest is the flow in the wake of the seamount. According to the perturbation analysis of the previous section, the zeroorder flow in the wake is parallel to the straight isobaths, and of uniform velocity 0,. The first-order velocity is then geostrophic and nondivergent: fVl

=w a x

au,/ax

fUl

=-h/aY

+ aV,/ay = o

(4.29)

f 5 0 crn/sec

FIG. 38. Cross-isobath flow of Gulf Stream along the Florida coast. Broken lines: 15°C isotherm (Cape Canaveral) and 16" isotherm, both at the bottom. Arrows: velocity vectors calculated by Janowitz and Pietrafesa (1982). Reprinted with permission from Janowitz and Pietrafesa, Conrinrntal Shelf Res., 1, Copyright 1982, Pergamon Journals, Ltd. 147

148

G . T. CSANADY

neglecting changes of f.The first-order vorticity is o1=

av,/ax - au,/ay

(4.30)

= (i/j)v2pl

Along each streamline of the perturbed flow potential vorticity is conserved:

(f + o l ) / H = f / H ,

= constant

(4.31)

where H , is the depth far upstream on the same streamline. The depth distribution will be taken to be

H = Ho + ax - h ( x , y )

(4.32)

with H, the upstream depth of the center isobath. The seamount occupies the region near the origin of the coordinate system, so that the depth perturbation due to the seamount, h(x,y), vanishes at large x , y . Depth changes are supposed small, h << H, and also ax, << H,, where x , is the typical crossisobath excursion of a streamline. The f / H term in Eq. (4.31) may then be approximated by

(4.33) The upstream value of this term is

f/Zf,= (f/Ho)(l - ax,/Ho)

(4.34)

Substitution into Eq. (4.31) yields w,/H

+ ( f ~ / H ; ) ( x-, X ) = - f h / H i

(4.35)

The streamline displacement xu - x can be found by integration from the second of Eqs. (4.29): x - xu = [ ; m u , d r , =

-7

s

a y d t , = --P1 apl

fv,

(4.36)

because d t , = ds/o,. Putting H z H,, in the o , / H term, permissible in virtue of the small depth variations, and substituting from Eq. (4.30), one finds now V Z P , + (f~/O"HO)Pl= -f zh/Ho

(4.37)

an equation for the pressure perturbation derived by Rooney et al. (1978). The homogeneous equation is almost the same as the linearized Eq. (4.9), and the two are equivalent for av,/ax = 0. The length scale defined in Eq. (4.8) again emerges as characteristic of meanders. Equation (4.37) is more informative, however, mainly because it contains a source term, which serves to quantify meander amplitude.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

149

Rooney et al. (1978) gave the Green’s function solution of Eq. (4.37): sin[(1 - q2)’/’x] cos qy

dq

(4.38)

exp[-(1 - q2)1’21x(]cosqy ( 4 2 - 1)1/2

with X ( y ) the Heaviside unit step function, so that the first integral only affects the region downstream from the seamount. Each element of the seamount, h dx dy, produces its own wake, but far enough downstream only the total volume M matters: m

M

=

JJ hdxdy

(4.39)

-m

as written in Eq. (4.38). Rooney et al. made calculations for Gulf Stream meandering caused by the Charleston Bump, distributing the seamount volume into four point masses, and allowing for the presence of the coast by image sources. Their result is shown in Fig. 39. Meandering continues further downstream, with wavelength 1 = 2L [ L as given by Eq. (4.8)] and amplitude diminishing with y-’/’, i.e., rather slowly. The problem discussed in this section is mathematically identical with that of long, stationary Rossby waves in the atmospheric westerlies in the wake of high mountains, first analyzed by Rossby (1939). White (1971) has called the spatially decaying wave pattern behind the obstacle (an island in his analysis) a “Rossby wake.” Porter and Rattray (1964) and McCartney (1976) treated other oceanic examples of Rossby wakes. McIntyre (1968) gave an exhaustive mathematical discussion of the problem. As discussed in Section 2, the path of the Gulf Stream in the wake of the Charleston Bump is indeed sinuous, with meander amplitude decaying slowly. The flow, however, is not steady, as supposed in the above model; the meanders propagate downstream. If such a flow pattern is viewed as the superposition of a steady current and a wave propagating upstream, it becomes stationary if wave celerity exactly equals the current velocity. In a faster current the wave is swept downstream. Motions akin to Rossby waves are indeed possible over a sloping seafloor, and are called topographic waves. The meandering behavior of the Gulf Stream in the wake of the Charleston Bump is then reasonably thought of as the superposition of the Stream’s own strong current and topographic waves, much as the Rossby wake of a mountain or island is a superposition of an eastward current and Rossby waves. The analogs of the latter, topographic waves, are discussed in the next section.

150

G. T. CSANADY

FIG.39. Rossby wake of Gulf Stream downstream of Charleston Bump, calculation of Rooney et al. (1978). Line A, western front of Gulf Stream; line B, axis of Gulf Stream; line C, eastern wall of Gulf Stream.

5. TOPOGRAPHIC WAVES

When vortex tube stretching causes local rather than advective change of vorticity, the possibility of vorticity propagation arises as the following elementary argument shows. Consider a vortex over bottom slope, which one may imagine to have been generated by a wind stress curl impulse. On its along-isobath flanks the vortex induces cross-isobath velocities (see Fig. 40). These give rise to vortex tube stretching and result by hypothesis in the local growth of positive and negative vorticity, as indicated in the figure. Some time later, the region occupied by fluid with vorticity equal in sign to that of the original vortex has spread along the isobaths toward negative y (in the coordinate system of the figure, supposed to lie in the northern hemisphere), and a region of opposite vorticity has appeared toward positive y. As the newly grown vortices induce their own velocity field, the situation becomes complicated, but the tendency to propagate toward negative y remains.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

151

H, =constant

shollow

=constant do/dt
FIG.40. Vortex over bottom slope. Induced velocities give rise to vortex stretchingand spread the region of vortex motion.

The local change of vorticity may be supposed to dominate if the Rossby number is small. When momentum advection and friction terms are deleted from the equations of motion, a set of wavelike solutions are found to describe phenomena of the kind indicated by the preceding qualitative argument; they will be referred to as topographic waves. In the literature they are sometimes called “topographic Rossby waves,” because of the mathematical kinship with Rossby waves. Physically, the two phenomena are quite different. Rossby waves do not involve effects of bottom slope. Furthermore, the equations governing topographic wave motion reduce to those of Rossby waves under special circumstances only.

5.1. Scale Analysis A wide variety of wave like motions are encountered in the shelf-slope region. Topographic waves are singled out by scaling hypotheses: moderate velocity amplitude, u = 0.1 m/sec, horizontal length scale at least comparable to slope width, L = 100 km, Coriolis parameter of order f = sec-’. This gives a Rossby number of

RO = o/fL

= 0.01

which justifies the neglect of momentum advection. The time scale is supposed long compared to l/f, so that the motions to be considered are nearly geostrophic. This implies the order of magnitude of the pressure gradients, hence of the pressure or surface elevation, [:

a P p x z fv

=

1 0 - ~ m/sec2

p = Lap/& = 1 m2/sec2

C = p/g =0.1 m The typical value of the vorticity is w = u/L =

10P sec-’

152

G . T. CSANADY

Neglecting momentum advection and friction, the depth-integrated equations of motion are

+ H aplax = o avlat + f u + H ap/ay = o a u / a x + a v / a y + ( i / g ) a p / a t= o aulat -fv

(5.1)

Upon taking curl and divergence the transports drop out and the following equation for the pressure results: (5.2)

at

On the right one recognizes the pressure torque, and on the left, the time derivative is the local change of depth-integrated angular momentum. The hypothesis of slow motion implies that the last term in parentheses, 9-l d’p/at’, is small compared to the term preceding it, f 2 p / g , which itself is only of order lo-’ m/sec2, given the scales estimated above. Neglecting g-’ a p 2 / 8 t 2 filters out gravity waves. Over the mid- to upper slope, where the depth gradient is of order lo-’, the scalar product of depth gradient and pressure gradient is of order lo-’ m/sec’, or 100 times larger than f ’ p / g . Physically, the latter describes vortex stretching due to surface elevation changes. Its neglect, justifiable in the present context, is known, somewhat inaccurately, as the “rigid lid approximation.” This is a useful simplification, just as readily justifiable as the neglect of advection terms. However, for fully understanding the initial radiation of topographic waves from a disturbance, which involves motions of small spatial scale, it is necessary to retain this term. The remaining (first) term in the parentheses in Eq. (5.2) is also of order lo-’ m/sec2, if the depth is of order 1 km. Therefore the first two terms in the bracket must, in general, be retained. However, over the continental rise, where the depth exceeds 3 km while the depth gradient is only 3 x or less, the second term is 10 times smaller than the first and may be dropped in a first approximation. Taking for simplicity the isobaths to be aligned with the y axis, H = H ( x ) , and writing H ’ = d H / d x , one has the following two equations:

over the slope, and

a H-V’p at

+ H ’ f -a P = 0 aY

(5.4)

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

153

on the continental rise. The last equation is identical with that describing Rossby waves with rigid lid on a beta plane. Equation (5.3)underlies many studies of “continental shelf” waves, or “coastally trapped waves. An alternative formulation of these equations for the transport streamfunction is often useful. Dropping the ap/at term in the third of Eqs. (5.1) (the rigid lid approximation again), that equation is satisfied by

+

Eliminating the pressure from Eq. (5.1) now results in

in lieu of Eq. (5.3). Over the continental rise, dropping the H’a$/axterm results in exactly the same equation for the streamfunction as for the pressure CEq. ( 5 4 1 . 5.2. Long Continental Shelf Waves

Weather systems passing over continental shelves generate sea level and current velocity changes which propagate along the coast in the same direction as a Kelvin wave, i.e., leaving the coast to the right in the northern hemisphere. The first indication of this phenomenon came from Hamon (1962), who thought that the changes were started by atmospheric pressure variations. Robinson (1964) showed that the propagation can be explained by a coastally trapped topographic wave model, and called the waves involved “continental shelf waves.” Adams and Buchwald (1969) demonstrated that these waves are generated by the alongshore component of the wind stress, not by the atmospheric pressure. Many other studies of continental shelf waves have appeared in the literature since. Mysak’s (1980) excellent review of the subject lists 154 references, and there have been quite a few more contributions in the last 6 years (see Brink, 1987). The physical idea underlying Robinson’s (1964) continental shelf wave model is that the strongly sloping seafloor of the shelf-slope region acts as a wave guide, in the sense that motions further offshore are negligible, i.e., that the wave is “trapped” near the coast. This idea is expressed by the postulate, applax = o

( x -,CO)

(5.7) which supplies a boundary condition at infinity for Eq. (5.3). At the coast, the boundary condition is that the cross-shore transport vanishes, which requires the velocity u to remain finite as the depth vanishes. For the streamfunction the

I54

G . T. CSANADY

two boundary conditions are $=O

a$/ax = o

(x=O)

(x -,00)

These conditions select from the set of solutions of Eq. (5.6) a subset modeling coastally trapped topographic waves. One might also note here for later reference that while Eq. (5.7) only requires the vanishing of alongshore velocity far from the coast, the second of Eqs. (5.8) calls for evanescent transport. Their mode of generation implies that continental shelf waves have wavelengths long compared to shelf-slope region width. If cross-shore variations occur on the scale of that width, L , a boundary layer approximation is legitimately made to the effect that the cross-shore velocity component is small compared to the alongshore one:

which may also be expressed as 1L<
where I is reciprocal alongshore length scale (wavenumber). This approximation further restricts the class of solutions of Eq. (5.3) to those with long enough alongshore scales to be called “continental” shelf waves in the sense of extending over a large fraction of a continent’s margin. It was introduced by Adams and Buchwald (1969), effectively exploited by Gill and Schumann (1979),and followed in many later studies of continental shelf waves. With this approximation, Eq. (5.6) reduces to

a -

[ a (_H _ -

at ax

l a $) ] + $ 3 0 ax

(5.10)

As Gill and Schumann (1979) have shown, the solutions of Eq. (5.10) are separable: $(-%Y , t ) = d4Y9 W x )

(5.1 1)

Substitution into Eq. (5.10) results in separate equations for the two functions F ( x ) and +( y, t):

(5.12)

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

155

with c a separation constant of the dimension of velocity, which may be interpreted as a velocity of wave propagation. This is seen at once from the second of Eqs. (5.12),the solution of which, for an initial disturbance, &,(y), is (5.13) 4(Y,t) = 4o(Y + 4 so that the disturbance propagates toward negative y with speed c, without change of shape. The boundary conditions of Eqs. (5.8) translate into conditions on the function F(x): F(x) = 0 at x = 0, dF/dx = 0 as x + 00. The first of Eqs. (5.12) together with the boundary conditions defines an eigenvalue problem of the kind often encountered in wave motion, in that solutions are found only for a specific set of values of c = c l , c2,. . . . The corresponding eigenfunctions, F,, F,, ... define the amplitude distribution of the transport streamfunction over the shelf-slope region.

5.3. Exponential Shelf-Slope Profile To gain further insight it is necessary to examine the character of continental shelf waves over a typical shelf-slope region. Buchwald and Adams (1968) have shown that an exponential depth distribution over shelf and slope, matched to a constant-depth abyss, is a realistic and mathematically convenient model:

H = Hoexp(2ax/L)

(0 _< x IL )

(5.14) H = H, = constant (x 2 L ) Figure 41 shows the comparison between this idealization and the depth distribution off Sydney, New South Wales. The shelf-slope region is here L = 80 km wide, the abyssal depth H is 5 km, and the slope constant ci is 2.7. The depth at x = 0 is then Ho = 23 m, which places this locus on the inner shelf. Application of the boundary condition, I$, = 0 or F = 0 at H = H , instead of H = 0, is justifiable for the reason that the inner shelf can store only little fluid. In very shallow water the neglect of friction makes the calculation unrealistic in any case. At the deep water end of the model, the velocity amplitude of the important wave modes becomes small in depths greater than about 1 km, so that the unrealistic cusp in the depth profile at the junction of slope and abyss should have a negligible effect. Given the exponential depth profile, the first of Eqs. (5.12) has constant coefficients: d2F/dx2 - (2a/L)dF/dx

+ (2aF/cL)F = 0

(5.15)

In the discussion of this or similar models it is often stated or implied that the outer boundary condition [on the streamfunction or its cross-shore

156

G. T. CSANADY

Kilometers FIG.41. Variation of bottom depth with distance from shore off Sydney, NSW ((from Buchwald and Adams, 1968). exponential shelf model of Eq. (5.14) (-)

-

-)

and

amplitude'distribution F(x)] may be applied at x = L, instead of at infinity. Equation (5.15) would seem to imply this because a vanishes over the abyss. The conclusion is invalid; the boundary layer approximation does not hold over the abyss. As Buchwald and Adams (1968) or Caldwell et al. (1972) pointed out, at x = L the shelf-slope solution must be matched to a solution of Eq. (5.6) valid over the abyss, which is evanescent at infinity. For sinusoidal 4(y ct) such a solution is proportional to exp( - Ix), where 1 is the alongshore wave number, and yields for the boundary condition on F ( x ) at x = L

+

d F / d x = -IF

(X

= L)

(5.16)

This condition implies that the two velocity components u,u are of the same order at the edge of the slope. Setting the right-hand side equal to zero unrealistically forces the streamlines at the slope edge to be perpendicular to the isobaths. The calculation of eigenvalues and associated constants is most conveniently carried out using nondimensional variables. To this end let distances x , y be expressed as multiples of the shelf-slope region width L, and let time be in units of f-'. Velocities and celerity c become multiples of fL (6.64 m/sec for the Sydney transect). Equation (5.15) then simplifies to F" - ~ u F + ' ( ~ u / c ) F= 0

(5.15a)

which has the solution satisfying the boundary condition at x = 0: F = Ae" sin fix

with A the streamfunction amplitude and

p = J W

(5.17)

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

The outer boundary condition for sinusoidal # ( y

157

+ c t ) requires now

ccsinp + Bcosa = -1sinp

(5.18)

or tanp = - j / ( a

+ 1)

If a >> 1 (IL in dimensional variables), the result is the same as if the boundary condition d F / d x = 0 had been applied at x = 1 (dimensional L). The long wave approximation thus justifies the use of this simpler boundary condition in the calculation of eigenvalues. Equation (5.18) can be satisfied at an infinite number of values of B = PI, /Iz,. . .,&. . . , bracketed by the inequality (n - j ) n < p, < Ilk

(5.19)

For 1 = 0 and cc = 2.7, the first few values of p, and c, are listed in Table I. Multiplied by the scale value, fL,appropriate to the Sydney transect, the highest celerity is about 2.75 m/sec. To illustrate the structure of the wave modes, Fig. 42 shows the distribution of alongshore velocity amplitude over distance from the coast for the three lowest modes, with data as in Table I. At the shelf edge the velocity amplitudes are zero because of the long wave approximation; more realistically they are small in comparison with upper slope velocities. As usual in problems of this kind, one finds zero, one, and two nodes between the coast and the abyss. Much of the distance spanned is over the continental shelf, however. A better appreciation of what the waves look like over the slope is gained from Fig. 43, TABLE 1. NONDIMENSIONAL EIGENVALUE, CELERITY, AMPLITUDE, AND FORCE COEFFICIENT'

N 1 2 3

4 5 6 7 8 9 10

Eigenvalue

Celerity

Amplitude

P.

c,

A"

Force coeficient b"

2.4124 5.1919 8.1730 11.2315 14.3235 17.4324 20.5510 23.6755 26.8039 29.9351

0.41 19 0.1577 0.9729 0.0405 0.0254 0.01 74 0.0126 0.0095 0.0074 0.0060

0.5542 0.5859 0.5978 0.6025 0.6047 0.6060 0.6067 0.6071 0.6075 0.6077

0.5507 0.4797 0.3561 0.2739 0.2202 0.1833 0.1567 0.1367 0.1211 0.1087 ~

For the first IOeigenmodes of the exponential slope model (a = 2.7).

158

G . T. CSANADY

1.0 V

FIG.42. Distribution of nondimensional alongshore velocity over distance from shore in lowest three eigenmodes.

1.0 V

n=3

FIG.43. Distributions as in Fig. 42, shown against depth.

which shows the same velocity amplitudes as Fig. 42, but against depth. Motions in depths greater than 1 km are much weaker than over the shelf and upper slope.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

159

5.4. Generation of Continental Shelf Waves by Wind

The continental shelf wave model is important because an efficient mechanism exists for the generation of such waves. As Adams and Buchwald (1969) have demonstrated, the dominant driving force is the alongshore component of the wind stress. Adding wind stress components, zx, t y ,to the right-hand side of the first two Eqs. (5.1), dropping the a p / a t term in the third, and taking curl after division by the depth H one arrives at the forced version of Eq. (5.6):

The first forcing term is the alongshore component of the wind stress divided by a width scale H/(dH/dx)of the slope. This scale is typically much smaller than the distance over which the wind stress varies, so that the first term on the right of Eq. (5.20)dwarfs the second. The large spatial scale of the weather systems also justifies the boundary layer or long wave approximation, and allows the wind stress to be supposed constant over the width of the shelfslope region. This then leads to the forced version of Eq. (5.10):

(5.21) where T~ = 7,,(y7t), independent of x. Following Gill and Schumann (1979),the solution of this equation, satisfying the boundary conditions of Eqs. (5.8), may be found by eigenfunction expansions:

+

(x, Y, t ) =

1A(

~7

t)Fn(x)

n

(5.22)

t ) = ~ y ( . ~t)C y bnFn(x)

zy(~,

n

where the forcing coefficients b, are found from

1 b ” W )= 1

(5.23)

n

A convenient normalization scheme for the eigenfunctions Fn(x)is

(5.24) where S, is the Kronecker delta. The factors H, and L have been included to make FJx) nondimensional. Equation (5.23) yields then

3 L

loL$

F,(x) dx = b,

(5.25)

160

G. T. CSANADY

Substituting the expansions in Eq. (5.22) into Eq. (5.21) one finds a forced wave equation for each of the functions 4,,(y, t): (I/c,)a4n/at - a4n/ay = bn r y / f

(5.26)

Characteristic curves of this equation in the y, t plane are s =t

+ y / c , = constant

(5.27)

Transforming to an rl, s coordinate system, with 9 = y, Eq. (5.26) may be written as

a4n/aV

s-v/c~) If zy = 0 for y > 0, integration with respect to q gives = -(bn/f

)~y(q,

(5.28)

(5.29) which can be evaluated for any space-time history of the wind stress. Note that 4nis nonzero only for y < 0. If zy = 0 at y > 0 as supposed, the effect of wind stress propagates to negative y only. To take a simple example, let a constant wind stress be switched on suddenly at t = 0, and switched off at t = T, extending over the portion of the coast from y = 0 to y = - Y. In Fig. 44 the range of action of the wind stress in the t, y plane is the rectangle with sides T, Y. For a given eigenmode n the slope of the characteristics is l/cn, known for a given depth distribution. At a given time t = t l > T, points to the left of yo lie on characteristics that d o not cross

f'

FIG.44. (Top) Characteristics of Eq. (5.26) in t, y plane, for given eigenmode n, showing intersection with the range of action of the wind stress (T,Y rectangle), and a time line, t = t , . (Bottom) Amplitude distribution of eigenfunction &(y, t , ) at time t , .

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

161

the wind stress rectangle, so that, for this mode = 0, because all of the wave has passed to greater negative y. Similarly, points to the right of y3 have not yet been reached by the wave. Between points y , and y, the characteristics intersect the wind stress rectangle over the maximum possible distance. On this stretch Eq. (5.29) gives the constant amplitude

(5.30) As the lower panel of Fig. 44 illustrates, there are linear flanks on 4, between yo and y,, and between y, and y,.

For the higher modes the characteristics become steeper and the wave shape changes to one with shorter flanks but relatively longer top. The maximum amplitude also decreases, in proportion to c,b,, or relatively fast with mode number n. 5.5, Application to the Exponential Shelf

In most of the large literature of continental shelf waves, attention is focused on such properties of individual wave modes as the dispersion relationship or the velocity amplitude distribution. One must remember, however, that the solution of Eq. (5.21) for realistic forcing contains an infinite series of wave modes. There is no more reason to expect such a composite solution to resemble a single wave mode than to suppose the exponential function (say) to be approximated by a single term of its power series. The justification sometimes given for the neglect of higher modes, that friction would preferentially damp these out, fails if one mode tends to cancel parts of another. If damping by friction is significant, it affects the composite solution and its precise influence on any given mode is difficult to foresee. To illustrate the importance of looking at the complete solution, the properties of the combined wave field set up by a wind stress impulse will be examined here. The case discussed in the previous section, impulsively applied wind for a limited section of the coast for a limited period, will be applied to the exponential shelf-slope model of Section(5.3). For the intensity, alongshore extent, and duration of the wind stress, realistic choices are m2/sec2

zy =

y = 12.5

(1000 km)

T = 30

(4.18 days)

A convenient scale for the streamfunction is now

LTr,

= 2.9

x lo6 m3/sec

162

G. T.CSANADY

For the exponential shelf the celerity is, from Eq. (5.17), c, = 2a/(u’

+ b;)

(5.31)

Normalization according to Eq. (5.24) yields the amplitude constant A, : A, = l/JJOc(l +C,/2)

(5.32)

The forcing coefficient b, is now, from Eq. (5.25), b, = A,c,& = - 2~$,/J~t(a’

+ a;)’ + c~*(cI’+ flf)

(5.33)

Quantities derived from the depth distribution, b,, etc., have been listed above in Table I. Together with the parameters Y and T they allow the calculation of the alongshore wave amplitude distribution, 4,,(y,t). Figure 45 shows such a distribution for t = 40 (5.58 days), for the first five modes, n = 1 to 5. The streamfunction at the outer edge of the slope region is $ n ( l , Y , l ) = cCn(y9t)Ane”sinbn

(5.34)

The sum over the first five modes 9 x C: $, is shown in Fig. 46 for the same five modes. Noting that the scale value of the streamfunction is nearly 3 sverdrups, one observes that large onshore-offshore transports occur at the edge of the slope. The whole streamline pattern for the slope region, arising from the first five modes, is shown in Fig. 47. At the edge of the slope the dFldx = 0 boundary condition distorts the streamlines, forcing them to be perpendicular to the slope. As discussed above, this is unrealistic, but is should not affect the flow pattern on the upper slope. Some of the streamlines penetrate to the upper slope, and cause relatively large velocities there, as seen in individual modes in Figs. 42 and 43. Offshore from the shelf-slope region, the bottom is flat and the steamlines close over distances of order Y. Thus the transport pattern-in contrast to velocity amplitudes-is not “coastally trapped” in the sense of being confined to distances from the coast of order L.

FIG.45. Calculated distribution of first five eigenrnode amplitudes at after cessation of wind) over alongshore distance.

t = 5.58 days (1.4 days

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

163

FIG.46. Nondimensional streamfunction amplitude at edge of slope, approximated by the sum of the first five eigenmodes, versus alongshore distance. The dimensional maximum value is about 3.5 sverdrups. Calculated fort = 5.58 days,

2000 Y l km FIG.47. Contours of streamfunction approximated as in previous figure, over shelf-slope region. Cross-shore scale is 10 times alongshore scale. 1000

0

A further important sidelight comes from the solution of the initial value problem, by a Laplace transform method, instead of relying on the Ansatz of the eigenfunction expansions, Eq. (5.22). Without going into the mathematical details one may describe the results of such a calculation as follows. The Laplace transform of the streamfunction,

Y

=

Iffi

$e-P'dt

(5.35)

0

has simple poles at p = 0, as well as where Eq. (5.18) is satisfied. The residue at p = 0 turns out to be zero, verifying that the eigenfunction expansion yields the

164

G . T. CSANADY

complete solution. However, Adams and Buchwald (1969)pointed out that the same calculation for the pressure does yield a nonzero residue at the origin of the transform variable, which adds to the pressure distribution due to the waves a field balancing the wind stress at the slope edge: (5.36) where X is the Heaviside unit step function, and H , is depth at the edge of the slope region. This part of the solution is analogous to the “steady” part of the pressure field in a flat-bottom closed basin under constant wind stress (Csanady, 1982, p. 31), or, even more directly, to the alongshore setup over a coastal region of constant depth, subject to wind along a section of the coast (Crepon and Richez, 1983).That such a pressure gradient must be part of the solution under the assumptions made in the continental shelf wave model follows directly from the second of Eqs. (5. l), with alongshore wind stress added. At the edge of the slope, V = 0 at all t, according to the approximate boundary condition imposed. The cross-shore transport U develops on the time scale Y/c,, as the solution has shown, so that it is initially zero. The left-hand side of the second Eq. (5.1) then vanishes, implying that the pressure gradient must balance the wind stress. This result at once illuminates how the massive propagating vortex seen in Fig. (5.8) is first set up in the model used. The alongshore pressure gradient exerts pressure torque on all of the fluid columns in the slope region and spins them up. An alternative view is that the initial absence of net offshore transport arises from a balance between Ekman transport and geostrophic transport. The latter runs into the slope, produces upwelling or downwelling, and spins up the fluid by vortex tube stretching. The whole scenario is a direct consequence of the boundary condition imposed at infinity; an alternative choice would be to prescribe vanishing alongshore pressure gradient and Ekman transport to balance alongshore wind stress. 5.6. The Effect of Stratification In the scenario just described the entire deep offshore water column runs into the slope and generates vorticity, with flow extending to the deepest layers. This is in clear conflict with the experimental evidence on coastally trapped waves described in Section 2. Coastally trapped waves were seen to be outer shelf and upper slope phenomena, just as upwelling and poleward undercurrent are, in that all three arise in response to alongshore wind and affectingtypically no more than the top 300m of the water column. Intuitively, one is inclined to ascribe the discrepancy to stratification. This is why motions

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

165

generated at the surface do not, as a rule, penetrate to great depth in the ocean. However, in attempting to quantify the effect of stratification on coastally trapped waves over shelf and slope one encounters conceptual and mathematical difficulties which have not so far been resolved. Much of the insight that we do have into the influence of stratification comes from linearized theory based on the assumption that density changes are due only to small vertical displacements of particles from their equilibrium position:

apyat

=

(5.37)

- apjaz

where p(z) is the undisturbed density distribution and p’ is the perturbation due to motion. This approximation not only requires that w ap’laz be small, but also neglects horizontal density advection, u ap’jax + u dp’lay. Motions leading to the development of coastal upwelling, e.g., as observed off North Africa (Fig. 23) are certainly of much greater amplitude than implied by Eq. (5.37).All one can hope from theoretical models based on this approximation is that they predict the structure and at least the initial history of the motion evoked by wind stress with some degree of realism. Linearized equations of motion, the hydrostatic approximation, and the equation of continuity yield four equations in addition to Eq. (5.37) for the variables u, u, w, p’, and p (perturbation pressure): aulat - j-0 = -apyax

+ aT,/az (5.38)

aulax

+ au/ay + awjaz = o

where 7 y are shear stress components in horizontal planes. These are important in a surface mixed layer only, representing forcing by wind stress. Dropping the forcing terms and eliminating four of the five variables one arrives at

(5.39) where the prime has been dropped from the perturbation pressure and N is Vaisala frequency, N Z = -gpO’ apjdz. If isobaths are parallel to the y axis, the boundary condition at the bottom is w = -udH/dx

(Z =

(5.40)

-H)

or, expressed in terms of the pressure,

($+

J 2 ) ( $ 2 )

=-

g(g 2) +f

(5.41)

166

G . T. CSANADY

Other boundary conditions may be imposed as before, at the coast and at “infinity” to represent coastal trapping. Wang and Mooers (1976) gave an illuminating discussion of the physical content of Eq. (5.39).For slow motions the a 2 / d t 2term may be neglected, and horizontal and vertical scales of the motion defined by

(5.42) Equation (5.39) then shows that

(5.43)

HJH = L/R

where R = N H / f is an intrinsic length scale, often called the radius of deformation. The vertical scale H , may be thought of as a “trapping” depth, in the terminology of Rhines (1970); if flow runs into a sloping bottom, the resulting squashing of vortex lines only affects layers in the vicinity of the seafloor, up to a height of order H,, an effect described by Rhines as bottom trapping. When the trapping depth is large, the whole water column is affected, much as without stratification. According to Eq. (5.43),this occurs in motions of a horizontal scale large compared to the radius of deformation. Motions of short vertical scale also have to have small horizontal scales. Solutions of Eq. (5.39)representing coastally trapped waves are found with the aid of the Ansatz: (5.44) p = P ( x , z)exp[i(ot ly)]

+

where w is frequency, 1 is the alongshore wavenumber; c = o/l is phase velocity. Substituting into Eqs. (5.39)and (5.41)and neglecting w 2 compared with f 2 , one arrives at the two-dimensional eigenvalue problem for p(x,z) (Wang and Mooers, 1976; Huthnance, 1978; Brink, 1982):

(z = 0)

The problem may be solved numerically for arbitrary stratification N ( z ) and depth distribution H(x).Such calculations have been reported by Brink (1982) for a model of the Peru shelf-slope region, using smoothed versions of the

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

167

depth and density profiles. The calculations yield the sequence of phase speeds, c,, c 2 ,etc., as well as the pressure amplitude distribution in a transect, P (x, z), for each of the eigenmodes. Figure 48 shows Brink's results for the first two eigenmodes at different values of the stratification parameter S :

s = CZ/(f'L')

(5.46)

0

-

X

h

e

%

. s

(roo

f

,OE

IOU

--

FIG.48. Pressure amplitude distribution (arbitrary units) in cross-shore transect of Brink's (1982) slope model. (Top) First and second modes at high latitude, j s e C ' [low stratification parameter, S , of Eq. (5.46)]. (Bottom) Same at low latitude, f sec-'. Vaisala frequency distribution assumed is shown at bottom right. Reprinted with permission from Brink, J. Phys. Oceanogr. 12, Copyright 1982, American Meteorological Society.

168

G. T. CSANADY

where L = 87 km is the shelf-slope width. Similar distributions for idealized depth profiles were shown before by Wang and Mooers (1976) and Huthnance (1978). At low values of S the amplitude distributions are much the same as without stratification, since the lines of constant pressure are nearly vertical. As S increases, the pressure lines tilt, the second mode in particular becoming much like the internal Kelvin wave of elementary two-layer constant depth models, while the first mode shows strong nearshore trapping. These results are illuminating, and account for several aspects of observed coastally trapped wave behavior (see Brink‘s 1982, for a careful discussion), but must be interpreted with due regard to the nature and limitations of the theoretical model. Much effort has been expended on attempts to account in quantitative detail for observed motions on a continental shelf and slope using the first few wave modes of models similar to Brink’s (see e.g., Freeland er al.. 1986; Church et al., 1986a,b). Such attempts are almost certainly doomed to failure, not so much because of the overidealizations of the theory as because the complete response of the shelf-slope region to given forcing consists of an infinity of modes (not forgetting the zero-frequency mode found in the solution of the initial value problem by Adams and Buchwald, 1969),the sum of which may well turn out to be quite different from any individual mode. Figure 47 has already illustrated this point for a homogeneous fluid continental shelf wave model, showing a horizontal pattern of response rather unlike a single free wave. In a stratified fluid the vertical distribution of pressure or velocity, as described by the sum of modes, can even more strikingly differ from single wave modes. In the simple example of a constant depth basin this is readily demonstrated to be the case (Csanady, 1982, p. 75 et seq.); in response to impulsive wind, outside a coastal boundary layer, the modes sum to zero below the surface mixed layer, while individual mode amplitudes remain large. One surmises that something similar occurs over shelf-slope topography, but the details might be unexpected. For example, an alongshore thermocline setup might develop at some stage of the response, expanding in time in the direction of Kelvin wave propagation, but not budging much from its region of generation, somewhat analogously to the unstratified response shown in Fig. 47. A comprehensive mathematical investigation of this question is certainly very desirable. One’s ardor for such an undertaking would perhaps be somewhat dampened by the limitations Eq. (5.37) places on the results of the theory. As isopycnals of the upper slope rise to the outer shelf, the stratification at fixed x, and therefore fixed H, changes dramatically. The bottom boundary condition in Eqs. (5.44) does not take this into account, so that the model might be seriously in error even at relatively small response amplitudes as the thermocline sweeps shoreward. Neglecting density advection is a much more restrictive assumption in the present context than neglecting momentum ad-

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

169

vection, so that one should regard theoretical results derived using Eq. (5.37) with an especially healthy dose of skepticism. Eigenfunctionexpansions based on this theory, and their manipulation in various ways advocated by Clarke and van Gorder (1986), may in particular be quite misleading.

5.6. Deepwater Topographic Waves As was discussed in Section 2, oscillations with a typical period of 16 days are the dominant motions in the deep “Slope Sea,” the narrow band of ocean between the Gulf Stream and the edge of the continental shelf north of Cape Hatteras. Thompson (1971, 1977) demonstrated that these oscillations are very well described by a topographic wave model akin to those discussed in the preceding sections, but without the coastal trapping hypothesis. The energy of the oscillations was found to propagate shoreward, a fact that led Thompson to conclude that “a significant part of the currents (observed by Webster, 1969) may be topographic waves driven by eddies in the Gulf Stream” (1977, p. 18). Tracing the wave energy backward along ray paths, Hogg (1981) was able to identify Gulf Stream meander formation as the source of topographic wave radiation. Similar work on observations taken south of Nova Scotia by Louis and Smith (1982) and Louis et al. (1982) confirmed this conclusion. The actual source term turns out to be pressure torque on isopycnal surfaces under the meandering surface layer jet of the Gulf Stream (Csanady, 1988). The problem of what boundary conditions to impose to model deepwater topographic waves in the narrow stretch of ocean between the Gulf Stream and the continental shelf has been the subject of debate (Rhines, 1970; Kroll and Niiler, 1976; Louis and Smith, 1982; Ou, 1980; Ou and Beardsley, 1980). One question concerns the degree of reflection by the coast. Calculations of wave propagation upslope invariably show large amplification as the wave approaches the coast, suggesting that the attendant intense friction dissipates the wave energy. This is consistent with the burstlike appearance of topographic wave activity, clearly related to the nearby appearance of Gulf Stream meanders (Hogg, 1981; Louis and Smith, 1982). Presumably the bursts decay as the wave packet climbs the slope, so that they are only noticeable when moving upslope. On the open ocean side the waves are presumably coupled to a circulation pattern similar to that found for continental shelf waves, the function of which is to close the streamlines, without exerting a major influence on what happens over the mid-slope. It is then realistic to neglect reflection and calculate the response to a given forcing impulse as if the slope region extended to infinity on both sides. As mentioned in Section 5.2, the propagation of deepwater topographic waves is well enough described by Eq. (5.4), or the same equation with the streamfunction taking the place of pressure. To understand how they are

170

G. T.CSANADY

generated, it is also necessary to find the important source terms. Gulf Stream meanders give rise to a source term through pressure torque exerted across the main thermocline, analogous to bottom stress torque. This may be shown by a two-layer model of the Gulf Stream. Surface and bottom layer depths in the two-layer model will be written as

+ i- l’ h’ = H - ho + l’ h = ho

(5.47)

where ho is the undisturbed constant depth of the surface layer and i’is thermocline elevation above z = -ho. The equations of motion, depth integrated separately for the two layers, yield after taking curl

where E is the constant density defect of the surface layer (divided by the reference density). Planetary vorticity advection, stress curl (surface, interface, and bottom), and momentum advection have been neglected. These effects may be shown to be dwarfed by pressure torque on the interface in a developing Gulf Stream meander, Substituting for the layer depths in the Jacobians one finds

J(h,l )= -J(l’,0

W’,i)= JP, i)+ W’,0 JW,“1 = J ( H , C’)

(5.49)

The Jacobians containing the depth H are linear in surface or interface displacement and belong to the left-hand side of the transport equations. The remaining pressure torque acts on the interface:

-

gJ(l’,i) = glvCl- IVl‘l s h y = fggs Vl‘ = -fWi

(5.50)

where y is the angle between surface and interface gradient vectors, ups is geostrophic surface layer velocity, and wi is vertical velocity at the interface. The interfacial pressure torque is significant only where and while a Gulf Stream meander is in process of being “pinched off.” Outside such a source region the linear terms in the equations dominate and describe topographic wave propagation. Supposing that the waves have a large enough spatial scale, L>>f

-‘a

(5.51)

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

171

and far enough from the source region, Eqs. (5.48)may be reduced to a single equation for bottom layer streamfunction: (5.52) where c( = d H / d x , and H is supposed to depart only slightly from the reference depth H , . Equation (5.52) is of the form of Eq. (5.4),with a source term on the right. The source term vanishes outside a relatively small source region, and only acts for a limited period. The solution of similar problems is usually found from a concentrated source solution, or Green’s function. The flow field of a concentrated source of vorticity placed at (x’,y ’ ) in an infinite field, and active for an instant at t’, is described by the Green’s function found by Veronis (1958):

where K O @ )is the hyperbolic Bessel function, fi = a i / H o , 5 = x - x’, r] = y - y‘, t* = t - t‘, and r2 = + r]’. For an arbitrary source function the flow field may be obtained by integration:

<’

+’ = {{{q(x’,y’, t ’ ) G [ ( x - x ‘ ) , ( y - y ’ ) , ( t - t ’ ) ] dx’dy’dt‘

(5.54)

where q(x’,y’,t ’ ) stands for the right-hand side of Eq. (5.52).To put this result in perspective, it should be mentioned that an initial establishment phase has been neglected. If the f 2 p / g term in Eq. (5.2) is retained, a more complex expression is found for the Green’s function. As Longuet-Higgins (1965) discusses in detail, the more complete solution contains a set of rapidly propagating waves, which spread the disturbance out over the available space. The wave pattern left behind this “squall line” is nearly nondivergent, described to a good approximation by the Green’s function solution above. The interesting properties of the solution are revealed by the structure of the Green’s function itself, which one may regard as the streamfunction field of a “tweak,” or concentrated source of vorticity. Let distances be measured in multiples of (fit)-’ and the streamfunction in those of the integrated source strength Q : Q

= lll4(.’,y..I.)d..dy’di’

The streamlines of the tweak are described now by

(5.55)

172

G. T. CSANADY

where the new integration variable is p = (z/t)'/'. At large radius r the Green's function simplifies to

w,Y ) = -ccos %/%731/&

(r -b a)

(5.57)

The asymptotic formula is a good approximation for r greater than 20. Calculated contours are shown in Figs. 49 and 50. To interpret the illustrations one must remember that distances are y scaled by (fit)-', and that the solution is a function only of r and y / r . For a fixed location, y / r is constant and r varies as Bt. Consider, for example, the ray drawn from the origin to point A in Fig. 50. This ray encounters the $ = 0 contour (approximately the maximum eastward velocity locus) at point A. For a point at a dimensional distance r = 200 km from the source this means the arrival of a high eastward velocity pulse at t = 35/$ 2 12 days (B = foc/Ho = 1.5 x lo-'' m-l sec-', with CL = 0.5 x lo-') after the time origin at the instant when the $ = 0 contour passes, points B, C, and D being intersected at a longer r. Figure 50 may also be interpreted as the spatial distribution of J/ equal to constant lines at fixed t . If point A still represents a point 200 km from the origin at the instant when the $ = 0 contour passes, point B, C,and D represent points along a ray at a somewhat lower azimuth angle. At the instant shown, the t j = 0 contour is at point D, and it will reach point B later, and

20

15

I0

5

0

-Y FIG.49. Streamfunction contours (in nondimensional units) in the upslope, negative y quadrant from an instantaneous point source of vorticity. close to the source. The distance scale contracts as (bt)-', so that the diagram shown here representsa progressively larger region, i.e.,for fixed scales the pattern shrinks toward the origin.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

-0.10

-

-0.15

-0.20 y

i

80

173

70

60

50

40

30

r

0

20

-Y FIG.50. As in the previous figure, but for a larger region, calculated from the asymptotic formula in Eq. (5.57). As the pattern shrinks toward the origin, the $ = 0 line initially through points D-A reaches first point B and then point C, which is perceived as phase propagation downslope and toward negative y.

point C later still, in the ratio of the distances to the origin, t s / t = OB/OD, t c / t = OC/OD. This is perceived as phase propagation westward (alongisobath, toward negative y) and southward (cross-isobath, to negative x). The velocities at a fixed point may be found by differentiation. The east velocity, for example, is u = (l/H(#$'/dr (5.58) The velocity scale is Q / r H o , or typically 0.5 m/sec (Q = 3 x 10' m3/sec, r = 200 km, H, = 3 km). Note also that, given a disturbance duration of 10 days, the result is only valid in its simplified form for times large compared to 5 days, or, at 200-km distance, with fl as above, R >> 15. Figure 51 shows along-isobath velocities at a fixed point calculated from the asymptotic formula [Eq. (5.57)] for increasing time. This is equivalent to calculating u = r di,b/ax along a ray from the origin outward. The scale of the ordinate is QIrH,,, of the abscissa (/3r)-'. The latter, at a distance of 200 km from source center, is equivalent to 32 days per 100 units, the entire length of the record shown in Fig. 51. The two records shown are at azimuth angles 8 (north of west) such that sin 6' = 0.7 and 0.9, or at 44.4" and 64.2'. The predicted velocity peaks are of order 0.1 units or 5 cm/sec dimensionally. For the first few cycles the apparent period is 50 units, or 16 days at 200 km from the source. The amplitude decreases to half the value of the first cycle (not counting a negative peak a t low r, not shown, where the calculation is invalid) in about 14 cycles. Allowing for some amplification as the wave

174

0.2 0.1

I

G. T.CSANADY

v o - 0.1

1

-0.2

time

FIG.51. Along-isobath velocity at fixed points calculated from asymptotic formula: azimuth angle 0 = sin-' 0.7 (solid line) and B = sin-' 0.9 (dashed line). Maximum velocities are 5-6 cm/ sec, and I 0 0 on the abscissa corresponds to 32 days, at 200 km from the origin, given typical slope conditions.

moves upslope, the theoretical results simulate observed velocity records quite well. The theory yields an impulsively generated pattern of motion which is rather unlike a plane wave of fixed frequency. The perceived frequency is proportional to Br, i.e., low close to the source, high far away. Given random variations in the location of meander pinch-off events, this means that waves of different frequencies are likely to be observed at the same time in different places and at the same place at different times. Although the notion of an impulsive generation is overidealized, the results still suggest that an interpretation of observations in the framework of a plane wave model of constant frequency may be difficult. Much of the earlier work on topographic waves focused on such plane waves. As the exhaustive analysis of LonguetHiggins (1965) has shown, the impulsively generated motion may be interpreted as superposition of many plane waves. It is, however, the combination of these waves which is a realistic model, not the individual components, the centetum censeo of this section. 6. PRESSURE TORQUEVERSUSBOTTOMSTRESSCURL

Another illuminating model arises from the postulate that pressure torque is balanced by bottom stress curl alone. It gives some insight into such steady flow phenomena as the poleward undercurrent, or other upper slope currents

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

175

of short cross-isobath extent, and may well have applications to mid-level or deep boundary currents in contact with sloping bottom. Particularly interesting results arise from this balance in a stratified fluid, when the density is nonuniform along isobaths. Numerical modelers coined the acronym “JEBAR” for the consequences of such a constellation, from Joint Effect of Baroclinity and Relief. The effect arises from pressure torque accompanying the change of density along depth contours, which can be a powerful generator of density-driven flow over the continental slope.

6.1. Equation for Bottom Pressure Taking again isobaths parallel to the y axis, and dropping all terms in the depth-integrated angular momentum balance except pressure torque and bottom stress curl, one has (dH/dX)(dpb/dy) 4- d T b , / d X - &b,/dy

=0

(6.1)

The bottom stress depends on the velocity close to the seafloor, in a relationship that is in general complex. For steady flow perturbed by tides and other transient motions a realistic simple approximation is a linear law: ?bx

= rUgb

zby

= Yogb

(6.2)

where r is a friction coefficient of the dimension of velocity, which one may think of as a drag coefficient times mean speed (of the time-varying perturbed flow), Its typical value is 3-5 x lop4 mjsec. The geostrophic velocities in Eq. (6.2) are those at the bottom:

It is important to realize that the pressure gradients in this equation are evaluated at constant level z = -H, and are not gradients of the bottom pressure pb.Gradients of the latter are also affected by pressure changes in the vertical, as the water gets deeper or shallower; such changes occur in a stratified fluid. There is no distinction when bottom density is constant, in which case a simple linear equation for the bottom pressure is obtained from Eqs. (6.1)and (6.3): (r/f)V:pb

+ a apb/aY =

(6.4)

where a = dH/dx. This equation is of the same form as the heat conduction equation with an advection term along they axis. Multiplying through by f L , where L is a horizontal length scale, the Laplacian comes to be multiplied by rL, the analog of thermometric conductivity, the y derivative by a f L ,

176

G. T. CSANADY

corresponding to velocity toward negative y . The analog of the Peclet number, which reflects the relative importance of advection compared to diffusion, is Pe = afL/r

(6.5)

If the length scale L is much greater than the intrinsic scale defined by rlaf, advection dominates diffusion, meaning that the second y derivative contained in the Laplacian is negligible compared to the “advection” (actually pressure torque) term. Physically, along-isobath velocities are much larger than cross-isobath ones, and the contribution of cross-isobath stress to the curl is small. The resulting simplified equation,

+

( r / f )a z p b / d x 2 a apb/ay = o (6.6) is parabolic, and a close analog of the one-dimensional heat equation, with negative y playing the role of time. This equation has been useful in modeling shelf circulation problems (Csanady, 1982, p. 189 et seq.). Qualitative results are readily transferred to the continental slope, using an order of magnitude The intrinsic larger value of a = d H / d x , typically lo-’, rather than length scale r/af is then a few hundred meters at mid-latitudes, making the Pklet number large for any scale of interest in steady flow problems. 6.2. Mound in Homogeneous Water

One problem taken over from shelf circulation theory should illustrate the character of the solutions of Eq. (6.6),even if it is not particularly realistic for the continental slope, Consider a homogeneous water column overlying an inclined-plane continental slope (a is constant) which extends to the coast, without a shelf of flatter bottom inclination between coast and slope. Let a “mound (surface level perturbation) be present at some point along the coast, at the origin of they axis. The problem is to calculate the flow field induced by the disturbance, supposing that pressure torque is resisted by bottom friction. The equations of motion and continuity appropriately simplified and in a depth-integrated form are

-f = -gH a l l a x + ( d f1w a y f u = -g H w a y - ( r g / f )a u a x aujax

(6.7)

+avpy =o

where gl is pressure, and the linear friction law is introduced. Upon elimination of the transports, Eq. (6.4) is recovered. A transport streamfunction is also useful, defined as before by

u = -a+/ay

v = a+/ax

(6.8)

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

177

The relationship of the streamfunction field to the pressure field is readily written using Eqs. (6.7). For the asymptotic case of a large Pkclet number the relationship is simple: j-a*lax = gH ailax

(6.9) The boundary condition at the coast is U = 0. At infinity the coastal trapping hypothesis is invoked, [ -+ 0 as x -+ a.An integration of the continuity equation then shows that the total alongshore transport is independent of y,

&,

=

lom Vdx

=

-gcrM/f

= constant

(6.10)

and proportional to the volume M of the mound, which remains conserved. If, as supposed, the mound is concentrated at the coast, the pressure field is given by ( = J2/71(~/~,)exp(-x~/2~,2) (6.11) with the local cross-isobath length scale,

L, = J--2ry/af

(6.12)

The solution is valid a t y < 0 only, since the mound only affects that part of the coast lying in the direction of Kelvin or topographic wave propagation. The transport streamfunction field may be calculated from Eq. (6.9),

(6.13) showing that the streamfunction is a function of x / L , alone, i.e., its distribution across isobaths self-similar, with a scale expanding toward negative y. Figure 52 illustrates the pressure and streamfunction fields. Two points are of general significance: (1) the mound affects the half-space y < 0 only; (2) the cross-isobath scale is typically quite small. With r = 3 x m/sec, ct = lop2, f = sec-' one finds at --y = lo00 km, L , = 25 km. The flow tends to follow the depth contours, moving only very slowly deeper. Another important point is that a negative pressure perturbation gives rise to exactly the same fields, with the sign of the pressure or streamfunction, and the arrows on the transport vectors, reversed. The flow drawn into such a suction mound moves up the slope at a small angle to the depth contours, and is collected again from the y < 0 half-space.

178

G. T. CSANADY

y - LONG SHORE

a

y - LONG SHORE

b. -2

n

-- 40 -50

t FIG.52. Pressure (a) and streamfunction (b) field of a coastal mound over a plane inclined seafloor. Distances are given in multiples of K = r/fa (equivalent to conductivity), pressures in multiples of gM/K, and streamfunction values in multiples of agM/f. Typical parameters over the continental slope give K = 300 m, and if the mound is 0.1 m high by 30 km long across-shore, or M = 3000 m2,then ~ M / = K 100 m2/sec and u g M / / = 3 x lo6 m3/sec.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

179

The result in Eq. (6.13) applies to the case of constant bottom slope. Partial integration of Eq. (6.9) gives the more general formula for the calculation of the streamfunction: (6.14) Once the pressure field is known, the streamfunction field is easily found over whatever depth distribution H ( x ) . Note that Eq. (6.14) is true only for large Peclet numbers, however. Under the same restriction one may also derive from Eqs. (6.7)and (6.8) a differential equation for the streamfunction: (6.15) which is slightly more difficult to deal with than Eq. (6.6), although it is of a familiar form, similar to equations applying to topographic waves [see Eqs. (5.10),(5.12),or(5.15)]. 6.3. The Shelf-Edge Boundary Layer Consider now a more realistic shelf-slope region model consisting of a gently inclined shelf, inshore of a steeper slope (Fig. 53). For typical values one may take u 1 = lo-’ and u2 = giving for the intrinsic length scale rluf = 3 km over the shelf, 0.3 km over the slope. At an alongshore length scale L = 1000 km, the corresponding Peclet numbers are then 333 and 3333, large compared to unity in both parts, larger on the slope. With the nondimensional coordinates, x = x / L , y = y/L, and again with p = gc, Eq. (6.6) becomes Pe-’ d2[/dx2

+ a[py =o

(6.16)

This equation is of the form encountered in boundary layer problems; the highest derivative is multiplied by a small number. Neglecting the highest derivative is equivalent to postulating a solution with similar x and y scales, labeled in boundary layer problems the “interior” solution. In the present case that solution is 5 = [(x) or flow following the depth contours. Where the interior solution conflicts with the boundary conditions, a boundary layer solution must be found and patched to the interior solution. The boundary layer solution has a short cross-isobath range. In the system illustrated in Fig. 53 different equations apply on the two sides of the shelf-slope boundary and matching conditions must be satisfied, in that the pressure and its x derivative must be continuous. This, in general, requires boundary layers on both sides of the boundary.

180

G. T.CSANADY

FIG.53. Shelf-slope topography modeled by inclined planes. Given a2 boundary layer on the slope side is much narrower than on the shelf side.

the shelf-edge

Boundary layer width is Pe-’I2 times the length scale L or 17 km on the slope side, and 55 km on the shelf side at -y = 1000 km, with the previously quoted typical numbers. This means that influences from the slope affect a large outer portion of a shelf, but influences from the shelf reach only the upper slope. Suppose that the depth at the edge of the shelf is 100 m.At 17 km seaward over the slope, given a gradient of lo-’, the depth is still only 270 my so that all of the boundary layer flow is accommodated well within what was empirically defined to be the “upper” slope. Shaw (1982) (see also Csanady and Shaw, 1983) and Wang (1982) have given specific illustrations of this “insulating” effect of a steep slope. To the extent that friction torque governs the behavior of currents over the continental slope also at deeper levels, a disturbance at mid-slope does not spread either upward or downward very far, because of the steepness of the bottom. Thus friction over continental slopes does not interfere much with the tendency of the stratified ocean interior to motion along isopycnal surfaces.

6.4. Stratified Water Column The neglect of density variations over the continental slope is unrealistic, and it is necessary to investigate how the pressure torque-bottom friction balance works in a stratified water column. The basic relationship underlying this balance is Eq. (6.I), valid for arbitrary density distribution. Stratification enters the problem through the hydrostatic equation which is used to specify

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

181

bottom pressure: YO

(6.17) where E = ( p - po)/pois nondimensional density excess over reference density po. Bottom geostrophic velocities are therefore

(6.18)

with the pressure gradient evaluated at constant level z = - H(x). Crossdifferentiation gives the divergence (Shaw and Csanady, 1983), au,,,/ax

+ ao,,/ay

(6.19)

= -p/faEb/aY

with c( = d H / d x again and E b the value of the excess density at z = - H ( x ) . Equation (6.19) implies that, if the density varies along isobaths, bottom pressure torque may vanish only under special circumstances. To show this, suppose that the pressure torque does vanish, i.e., Ugb = 0 everywhere, while aEb/dy # 0. Integration of Eq. (6.19),supposing Eb and Vgb to vanish at infinity, yields ygb

(6.20)

= -gaEb/f

for constant a. With bottom friction present, Eqs. (6.1) and (6.2) give

r(augb/ax- au,,/ay) -aj-~,,,

=

o

(6.21)

If Ugh = 0, as supposed, this equation requires Ugb to be independent of x. The result is consistent with Eq. (6.20) only if also Eb is constant across isobaths, E,, = ~,,(y),a very special kind of density distribution. If bottom density varies in both horizontal directions, pressure torque must be present. Suppose now that the PCclet number is large, so that I au,,,/ay in Eq. (6.21)is small compared to a&,,. Dropping the small term and combining Eq. (6.21) with Eq. (6.19) one arrives at ( r / M f) a 2 u g b / a X 2

+

aub/aY

= (ga/f)aEb/ay

(6.22)

This is again Eq. (6.6) or (6.16),with a density-forcing term. For large Peclet numbers one may again try to drop the second derivative to find an interior solution. That solution is Eq. (6.20), valid only if density is constant across isobaths, and characterized by the absence of pressure torque or of crossisobath flow at the bottom. In the general case of arbitrarily varying €,, the

182

G . T. CSANADY

boundary layer part of the solution is not negligible anywhere. However, the interior solution still reveals the tendency of heavy fluid over bottom slope (E,, positive) to cause flow toward negative y, light fluid toward positive y. For typical values, a = lo-’, f = and somewhat large but not atypical density excess, E = the interior solution yields ugb = 1 m/sec. Alongisobath density variations over steep bottom slope are thus seen to be powerful generators of motion. Local density in the ocean is determined by advection and diffusion of heat and salt, together with the equation of state. Neglecting complications arising from the nonlinearity of the equation of state, an advection-turbulent diffusion equation may be supposed to describe the density field:

where K , and K , are horizontal and vertical eddy diffusivities. With few exceptions, horizontal or vertical Peclet numbers u L / K , or w H / K , are large, so that density advection dominates. Because the fluid velocities themselves depend on the density field, density advection considerably complicates the task of determining the coupled density and flow fields. This well-known fundamental difficulty of atmosphere-ocean dynamics forces idealizations one would rather do without, such as the linear internal wave theory already discussed in Section 5.6 in connection with coastally trapped waves. For the analysis of steady, frictionally controlled flow, alternative idealizations are available, such as the diagnostic approach or the hypothesis of superefficient vertical mixing, which allows horizontal density gradients only. Models based on these idealizations have serious shortcomings, just as linear internal wave theory has, but nevertheless they offer some insight into complex density field-slope interactions. 6.5. The Diagnostic Approach

So-called “diagnostic” calculations of pressure or streamline fields make use of a supposedly known density field, either observed or calculated from Eq. (6.23) with simplified advection terms, on the hypothesis that the resulting flow field can coexist with the density field. The approach has long been common in oceanography; calculations of geostrophic flow relative to a supposed “level of no motion” fall in this category. At least in the case of an observed density field what one calculates in this approach is flow consistent with the density distribution, not necessarily driven by it. The interesting interplay of stratification and bottom slope discussed in the last section can be illustrated by calculations using the diagnostic approach.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

183

Substituting for bottom pressure in Eq. (6.1)from Eq. (6.17)and supposing a large Peclet number, one arrives at an expanded form of Eq. (6.16),containing “forcing” terms related to the density distribution:

Wind stress curl may be added to this equation to represent forcing by wind. Because the equation is linear, the effects of wind and density field are uncoupled. With the density field prescribed, wind effects are as in a homogeneous fluid, because there is no mechanism for the generation of a new baroclinic zone. Consider first the case of what was called in the older literature a field of “parallel solenoids,” or constant-density surfaces with generators parallel to those of the slope, E = E(X) only. The first forcing term in Eq. (6.24)drops out, and the second becomes independent of y. The solution is then also independent of y, and may be written as (Csanady, 1979)

(6.25) with the path of integration shown in Fig. 54. H,is water depth at the outer edge of the slope region of interest, This is a calculation of dynamic height relative to a reference level H,, generalized to the case when the level H , runs into the slope. With no external forcing other than a density field with shoreparallel generators, the bottom geostrophic velocity vanishes everywhere, and velocities are simply found from the thermal wind relationship with the bottom as reference. Equation (6.25) gives the pressure field corresponding to this velocity distribution. When the density excess is also a function of the along-isobath coordinate, the solution of Eq. (6.24) is conveniently calculated in two steps: first determining [ = Cl from Eq. (6.25) as if the law of parallel solenoids held (although of course ll is now a function of x and y), then writing where rzis a solution of

i= 1, + 1 2

(6.26) (6.27)

as one readily verifies by substitution of Eq. (6.25) into (6.24). Note that in these expressions an arbitrary depth distribution is allowed, and the integration variable in the source term on the right of Eq. (6.27) is the depth. When dealing with an observed density field, having an integrated distribution in the source term is convenient. Although as calculated from Eq. (6.25),for

184

G . T.CSANADY

FIG.54. Calculation of “dynamic height” component of surface pressure over sloping bottom. Setting[ = Oat theedgeof the slope region of interest, where water depth is H,, thedensity excess E is integrated along the path shown to find surface elevation [(x) at given location x.

the general case of arbitrarily varying e, is a function of both x and y, it is still dynamic height calculated with the bottom as the reference so that it does not contribute to bottom pressure torque, which is all contained in the field. Equation (6.27) is again the same as Eq. (6.16) with a source term, and the general character of the solutions is readily inferred from the heat conduction analogy. Along-isobath gradients of bottom density cause perturbations to the geostrophic flow relative to the bottom. These (barotropic) perturbations affect the half-space toward negative y, and spread only very slowly across isobaths over steep bottom slope. Simple examples below illustrate flow fields arising from these general principles.

c2

6.6. River Inflow Model The first example is again a straight transcription of a case originally meant to model shelf circulation (Csanady, 1984): diffusion of freshwater from an extended coastal source into an alongshore current, with the water column supposed vertically well mixed. As a slope circulation model this is rather unrealistic, but it illustrates in specific detail the influence of along-isobath density variations on the distribution of bottom pressure, which should have some intuition-building value.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

185

Figure 55 illustrates the model. The equation describing the distribution of density is solved supposing uniform alongshore flow with constant velocity. The density distribution so obtained is then taken as prescribed, and the flow field calculated via the diagnostic approach. The density field is a standard alongshore plume. The partial solution according to Eq. (6.25) is, for a vertically well-mixed water column,

cl = - a x e

-a

(6.28)

Edx

The second part of the solution, according to Eq. (6.27) can also be written down in closed form, being the solution of (6.29)

The source term in this equation may be shown to be proportional to all2,so that the effect on bottom pressure is rather stronger over the slope than over the shelf. The distributions of bottom pressure and streamfunction are shown in Figs. 56 and 57 for a typical case, given an underlying current to negative y . The pressure field shows a large cyclonic cell offshore from the inflow region. This is also indicated by an interior solution of Eq. (6.29), which reveals the tendency for a pressure drop toward negative y. From the point of view of mass balance, the cyclonic cell seen in the pressure field has the function of transporting fluid to the source region, to supply the geostrophic flow associated with coastal freshening. The strong cyclonic pressure torque turns the streamlines around; the bottom friction torque in the hairpin turn is strong enough to balance the pressure torque. In the complementary case, when the underlying current is directed to positive y, the cyclonic cell deflects the perturbation flow shoreward (and the combined flow seaward) (see Fig. 58).

Y

FIG.55. Model of circulation induced by coastal freshening. Advection of density is attributed to an alongshore current of velocity u,,, ignoring the effect of the perturbation velocities induced by the density field.

G . T.CSANADY

186

0

*

w

K

0 I v)

LL LL

0 10

15 FIG.56. Bottom pressure field offshore from a coastal buoyancy source. Note the cyclonic cell, the function of which is to steer a supply current toward the buoyancy source. The volume transported by this current is many times greater than the freshwater inflow.

-100 -90 1

-80

-70

-60

Y

-50

-40 1

-30

-20

-10 1

0

‘0

x

45

10

-I

4 4 15

-I

20

FIG.57. Transport pattern accompanying pressure field of previous figure. In this case the unperturbed flow was directed toward negative y, as shown in Fig. 55.

6.7. Pycnocline Running into Slope The next example explores the effects of an alongshore pycnocline slope similar to that discussed by Reid and Mantyla (1976) for the west coast of North America (see Fig. 30 above). Figure 59 shows the problem reduced to its essentials. Where the pycnocline intersects the slope, there is sharp along-

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

Y -100 x 0’

15 I-

-75

-50

-25

I

I

0 I

25

50 I

187

-

75

100

-

SOURCE FIG.58. Transport pattern due to coastal buoyancy source in flow toward positive y. Vector addition of the unperturbed flow and the perturbation results in a pattern of flow to positive y, with a seaward deflection upstream of the buoyancy source, and a return deflection downstream.

FIG.59. Model of pycnocline with alongshore tilt, intersecting a plane sloping seafloor. Geostrophic flow is directed shoreward by the pressure field arising from the pycnocline tilt.

isobath change of bottom density. According to the above general results, this may be expected to induce pressure torque and barotropic flow over the slope, but one cannot foresee what the resulting flow pattern would be like. If h ( y ) is the pycnocline depth distribution alongshore, the slope is intersected along the line, x = X(Y)= “)/a

(6.30)

188

G.T.CSANADY

where a is again the constant bottom slope. The bottom density distribution is Eb

= € S ( X - x)

(6.3 1)

with E the constant density excess of the layer above the pycnocline (a negative constant) and %(A) the Heaviside unit step function, % = 1 if 1 > 0, zero otherwise. The dynamic height part of the pressure field [Eq. (6.25)] is now CI =

-MY)

(6.32)

or the same as the dynamic height offshore from the pycnocline-bottom intersection. The density-induced pressure field is calculated from the ea uation

(6.33) after the source term is evaluated in Eq. (6.27). The boundary condition at the coast is (Csanady, 1985) ay,/ax = o

(x = 0)

(6.34)

which is satisfied by the following solution of Eq. (6.33):

The resulting surface pressure field and the transport streamline pattern are shown in Figs. 60 and 61. The onshore baroclinic flow is deflected by the slope toward negative y , well before reaching the coast, and the locus of maximum transport moves slowly offshore from a closest approach at about 300 m depth (see Table I1 for data assumed). Surface elevation at the coast drops much less steeply along negative y than the offshore dynamic height. Thus the slope again “insulates” the coast from offshore effects as it turns onshore flow into an upper slope current. The diagnostic models described in this section and in Section 6.6 have the major flaw that they d o not account for the advection of density by the flow. In the model just discussed, the onshore surface layer flow was conducted away in a barotropic upper slope current deeper than the pycnocline, a nonsensical result unless one assumes that much of the oncoming flow is rapidly cooled as it turns into an upper slope current. The assumption of a frozen density field shaped as shown in Fig. 59 was clearly wrong. If all of the incoming flow indeed turns to negative y , it must all be accommodated above the pycnocline, much of it offshore from the bottom intersection. This is only possible if the pycnocline dips down shoreward. Huthnance (1984)discussed the adjustment problem involved, but his approach is too complex to pursue further here.

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

Ill1

I

'I1

.C

Iill I I

10

1

0

y

189

I 1

-i

FIG.60. Pressure field associated with pycnocline tilting alongshore. Tilted section occupies only the range 0 > y > - 1.

0

0

y

-1

-2

-3

-4

0

-5

FIG.61. Transport streamlines for case illustrated.inprevious figure.

Huthnance's general conclusion appears to be that the necessary density adjustment does take place, so that a baroclinic upper slope current to negative y does form under conditions similar to those envisaged in the last section. To gain more insight into the difficult problem of how the density field behaves in the flow it induces, two rather overidealized models are discussed in

G.T. CSANADY

190

TABLE11. TYPICAL PARAMETERS USED IN CALCULATIONS FOR FIGS.60 AND 61 Scale

Parameter Density perturbation, --E Bottom slope, s Pycnocline depth, h, or h , Coriolis parameter, 1 Acceleration of gravity, g Resistance coefficient, r Pycnocline length scale, Y Pycnocline slope, m Equivalent conductivity,k = r / f s Cross-shore length scale, L , = fi Alongshore length scale, L, Elevation scale, - m Y Transport streamfunction scale, ( g s L , / f ) ( - € ) m y

10-3 10-2

300 or 200 m 10-4 sec-’ 10 m/sec2 2x m/sec 5 x lo5 m 2 x 10-4 200 m lo4 m 5 x IO’m 0.1 m lo6 m’jsec

the next two sections in which density advection is explicitly accounted for. Further theoretical study of problems of the kind discussed here would certainly be useful.

6.8. Self-Advection of Density When the flow is entirely due to density differences, it is possible to imagine that velocities are proportional to the local density perturbation, so that the density field in a manner of speaking advects itself. Consider a mass of fluid of anomalous density suddenly produced for example by surface cooling, sitting over sloping bottom and surrounded by ambient fluid of constant density, E = 0. The density is everywhere uniform in the vertical, maintained that way by vigorous mixing as the anomalous fluid spreads out and mixes horizontally. The problem is to calculate the spreading and mixing, given a certain initial distribution of density anomaly. Depth integration of the density equation Eq. (6.23) yields

In the vertically well-mixed case the baroclinic velocity does not contribute to transport. Therefore, the transport components multiplying the density gradients in this equation are the sum of barotropic and Ekman transports. The simplest flow field consistent with the conditions of this problem is the “interior” regime characterized by vanishing pressure torque and friction torque discussed in connection with Eq. (6.20). As shown there, the regime is

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

possible if

E =~

191

( yonly, ) so that Eq. (6.36) reduces to

aE/at

-

( a g E / f )aE/ay = K , a2E/ay2

(6.37)

having substituted for uBb from Eq. (6.20). Let distance be expressed as multiples of a scale L and time as multiples of T, and choose the scales so that

agT/fL = 1

(6.38)

Equation (6.37) now becomes aejat

+

delay = 1a Z e / a y 2

(6.39)

with 1= K,T/L. This is Burgers’ equation, which was originally introduced as an elementary turbulence model and contains both self-advection, which complicates the turbulence problem so enormously, and diffusion. The character of the solution emerges upon considering a localized initial anomaly. (6.40)

which leads to (6.41) where C = exp(~,/21) - 1. Equation (6.40) specifies the density anomaly integrated over y as cOL,in dimensional variables. If e0 is taken to be the average density anomaly over the initial disturbance, then the length scale L is the alongshore length of that disturbance. In cases of interest 1 is a very small 1 and the following asymptotic solution is sufficiently quantity, ~ ~ / is2 large, accurate: E

z -y/t

E = 0

(0 < y < J 2 E O t )

(6.42)

otherwise

The solution is illustrated in Fig. 62 for E positive, i.e., a batch of heavy fluid. The batch spreads out as a front toward negative y, leaving a trail behind from

FIG.62. Spreading out of heavy fluid along isobaths, as a sharp front, with linear wake trailing.

192

G . T. CSANADY

the origin. The maximum density perturbation, which occurs at the front is, in dimensional variables, Em =

fY/W

(6.43)

f= sec-', a = Taking for typical values L = 100 km, E,, = and t = lo6 sec (10 days) one finds E, = 0.45 x at a distance of y = 450 km from the origin. In 10 days the disturbance has spread out at an average speed of 45 cm/sec, and the maximum value of the anomaly has decreased by 55%. Note that the speed of the spreading depends on the amplitude of the disturbance, and its direction depends on the sign of the anomaly (light fluid spreads to positive y, at a speed proportional to E'"). 6.9. Cross-Isobath Density Advection

The last section showed that density anomaly spreads out rapidly along isobaths. For the same somewhat overidealized case of a vertically well-mixed water column, Vennell(l986) has recently succeeded in assessing the rate of cross-isobath spread. Vennell considered steady flow and neglected horizontal exchange of density due to turbulence or shear. The depth-integrated density equation then reduces to J(E,

*) = 0

(6.44)

where $ is a transport streamfunction:

v = a$/ax

u = -a+/ay

(6.45)

This is satisfied if transport follows density contours, or E = 4(*)

(6.46)

Velocities at the surface or bottom may now be expressed in terms of $. Bottom geostrophic velocities are

where K = d$/d$. The physical meaning of K emerges upon calculating the difference between surface and bottom geostrophic velocities: -(%g

- ubp)/(a$/aY)

= ('sg - vbg)/(a$/ax)

= -gHK/f

(6*48)

Positive K therefore signifies flow stronger at the bottom, and negative K signifies flow stronger at the surface. Substituting Eq. (6.47) into Eq. (6.20) a

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

193

differential equation for the streamfunction is found,

which is again very similar to Eq. (6.6). For constant K and small changes in depth the bracketed quantity drops out, so that the streamline pattern is the same as the pattern of constant pressure contours in a homogenous fluid over shelf or slope for the same boundary conditions. Cross-isobath spreading of density anomaly is thus slow, and the flow advecting the density anomalies behaves much as without such anomalies. These models in which the isopycnal surfaces are forced to remain vertical unfortunately do not tell us anything about the interesting adjustment process that must occur where a more realistically shaped density field intersects a sloping seafloor. A model which contains the essence of that process, while remaining simple enough to be informative, would be a prize achievement.

7. PRESSURE TORQUE AND PLANETARY VORTICITY ADVECTION It remains to consider the possibility of balance between pressure torque and planetary vorticity advection. Because the variation of the Coriolis parameter with latitude is relatively slow, it takes massive flow to make planetary vorticity advection important. For this reason this brief section mainly deals with the vorticity balance of western boundary currents. 7.1. Bottom Pressure Field Calculation Balance between pressure torque and planetary vorticity advection implies the neglect of acceleration and friction in the momentum equations, so that the velocity at any level is in geostrophic balance:

As in Section 6, it is convenient to express pressure as the sum of bottom pressure and dynamic height relative to the bottom:

P(X,Y,Z) = Pb(X,Y) - 9

1:.

EdZ

(7.2)

where E ( X , y , z ) is again the density anomaly. The variation of the Coriolis parameter with latitude will now be taken into account, f = f ( x , y). Also, for the time being the depth distribution will be arbitrary, H = H(x,y). The

194

G. T.CSANADY

bottom geostrophic velocities are then

The “baroclinic” component of the velocity, defined relative to the bottom as in Fofonoffs (1962) article, is

The depth-independentcomponents of (Ugh, Vgb) and (uC,uc),proportional to Eb, are equal and opposite, and drop out when the depth-integrated transport is calculated, a point made by Rattray (1982),

The integrals involving the density distribution may be shown to be derivatives of the anomaly of potential energy (Fofonoff, 1962), ro

so that

f u = - H apb/ay - axfay f v = H apb/ax + aX/ax

(7.7)

The depth-integrated transport in steady flow is nondivergent. Taking curl on Eqs. (7.7) therefore results in

which expresses the balance between pressure torque and planetary vorticity advection, a result that could have been written down at once from Eq. (3.1 l), upon the neglect of acceleration and friction terms. Substituting from Eq. (7.7) into Eq. (7.8)for the transports, the following equation for bottom pressure is

OCEAN CURRENTS OVER THE CONTINENTAL SLOPE

195

obtained: J(Pb,flH)

= -1/H2J(%f)

(7.9)

In the simple case of a homogeneous fluid is constant and the Jacobian on the right vanishes. The pressure is then constant along f / H = constant contours. Whether this is the case or not, Eq. (7.8) shows that fluid columns may cross latitude circles if pressure torque changes their angular momentum to match the local planetary vorticity. Thus a western boundary current, for example, can penetrate poleward or equatorward over bottom slope by moving into deeper or shallower water, without requiring bottom stress curl or some other effect to maintain angular momentum balance. The Jacobian on the right of Eq. (7.9)may be rewritten in terms of baroclinic transport components defined by

v, = -(l/f)ax/aY

K =(l/f)axlax

(7.10)

This is not quite the depth-integrated baroclinic velocity given by Eqs. (7.4), in that the contribution proportional to Eb has been dropped. However, offshore from the main thermocline-bottom intersection the Eb term is small, and the baroclinic transport of a boundary current is well approximated by Eq. (7.10).Independent of the precise physical meaning of V, and V,,from their definition one has J(x9.f)

=

f ( v ,af/ax + v,w a y )

(7.1 1)

or planetary vorticity advection by the baroclinic transport, the latter as defined by Eq. (7.10). Equation (7.9) is a first-order equation for bottom pressure, and has characteristic curves defined by topography: f f H = constant

(7.12)

Choosing the y axis to coincide with a constant f f H contour, one has then

(7.13) Where the Gulf Stream flows over the steep continental slope of the South Atlantic Bight, the f / H contours nearly parallel the isobaths and (7.14) where CI = IVHI. Also V , >> V, and J f / a y z Bcos y where y is the azimuth angle of the coastline measured from North. Equation (7.13) becomes then,

196

G. T. CSANADY

(7.15) so that bottom pressure torque balances planetary vorticity advection by the baroclinic transport. The first-order Equation (7.9)may be solved by integration along characteristics for a prescribed x field, given a pbdistribution along an initial transect across f/H contours. This, of course, is again a diagnostic calculation,because it does not take into account the barotropic flow (associated with the calculated P b field)on the distribution of density, hence on the x field. What the calculation does show is a barotropic current field consistent with an observed density distribution and given seafloor topography. It is a logical extension of the dynamic height method to flow over variable depth, akin to that discussed in Section 6, but now for cases of small bottom friction torque compared to planetary vorticity advection. For some applications of this method, see Rattray (1982). Over the continental slope, the most important application of the above ideas is to questions of boundary current dynamics. Is it reasonable to suppose that the two terms retained in the angular momentum balance, Eq. (7.Q planetary vorticity advection and bottom pressure torque, are indeed dominant in the best explored western boundary current, the Gulf Stream? 7.2. Western Boundary Current over Bottom Slope

In an illuminating discussion of the dynamics of midocean gyres, Stommel (1951)pointed out that advection of planetary vorticity by a vigorous western boundary current such as the Gulf Stream dwarfs wind stress curl. Some other effect must then be present to maintain vorticity balance. In his own earlier ocean circulation model (Stommel, 1948) this effect was taken to be bottom friction, and Munk (1950) assumed it to be horizontal momentum transfer by eddies. Today it is clear that neither assumption is realistic. Where the Gulf Stream touches bottom, so that bottom friction is presumably high, the curl integrates to zero, the velocity becoming negligible on both sides of the Stream’s core. Horizontal momentum transfer, on the other hand, is not viscosity-like, nor, in general, strong enough and in the absence of a vertical wall at the coast its integrated curl must again vanish. It was Holland (1973) who first pointed out that bottom pressure torque under a western boundary current flowing over the continental slope can affect the circulation of a midocean gyre as much as the wind stress exerted over the whole basin. In the numerical circulation study he reported, both cyclonic and anticyclonic pressure torques appear, the anticyclonic torque predominating and adding to the effect of the wind stress curl to produce a particularly intense

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197

gyre. Vorticity balance was, however, preserved by friction, as in the classical models. The Gulf Stream is in contact with the bottom as it flows along the upper continental slope from the Florida Keys to Cape Hatteras, off the South Atlantic Bight. Its dynamics in this region have been intensively studied for more than two decades. Webster (1961b) first noted that horizontal eddy momentum transfer on the cyclonic (inshore) side of the Stream is "upgradient," a fact that makes eddylviscosity models inappropriate. On the other hand, momentum transfer on the anticyclonic side is down-gradient. In terms of the energy balance this is equivalent to eddy-to-mean flow energy conversion on the inshore side, the opposite on the other side. According to Schmitz and Niiler (1969) and Brooks and Niiler (1977), the net total eddy to mean energy conversion, integrated across the Stream, is negligible. Thus eddy transfer, while it redistributes momentum across the Stream, is unimportant either in the vorticity balance or as a dissipation mechanism. Bottom pressure torque, by contrast, is far from negligible. From steric heights Sturges (1974)inferred an alongshore elevation gradient in this region of about 2 x lo-' (2 cm per 100 km). This is presumably present offshore but not necessarily inshore. It is therefore reassuring to find that, from a careful study of the continental shelf in the region, Lee et al. (1984)concluded that the elevation gradient inshore is about 1.7 x lo-', much the same as inferred by Sturges. The sign of this gradient is negative downstream, driving the flow toward Cape Hatteras. The magnitude and sign of the alongshore pressure gradient are also in accord with the results of Dewar and Bane (1985), who have carried out a careful analysis of the energy budget of eddies and mean flow in this portion of the Gulf Stream. The typical bottom gradient over the so that the pressure upper slope (Blake Plateau) is here about 0.8 x torque amounts to adp,/dy

=

1.6 x lo-' m/sec2

where the x axis points offshore, they axis northeastward. The pressure torque is thus positive and two orders of magnitude larger than the typical wind stress curl of lo-'' m/sec'. The data of Dewar and Bane can also be used to estimate the planetary vorticity advection term in the vorticity balance. At 32"N, given a 45" inclination of the isobaths against North, the rate of change of the Coriolis parameter with alongshore distance is Pcos45" = 1.4 x lo-"

m-' sec-'

The baroclinic transport in the Gulf Stream core is of order lo3 m2/sec, so that planetary vorticity advection ~ ( C O y)K S is of order lo-* m/sec2,much the same as the bottom pressure torque, so that the two sides of Eq. (7.15)

198

G.T. CSANADY

approximately balance. One concludes that a diagnostic calculation using Eq. (7.9) should give a realistic picture of Gulf Stream behavior (minus its meanders) over the continental slope off the South Atlantic Bight. This, however, does not tell us how the Stream dissipates some of its energy.

8. CONCLUSION Successive analysis of different balances for pressure torque or vortex stretching over the continental slope has revealed an interesting variety of phenomena and exposed weak links in our chain of understanding them. Theoretical models are complex enough when dealing with one kind of balance only, and yet, two or more of these often act together. Meandering of a boundary current over bottom slope usually involves both local and advective vorticity change, giving rise to a propagation of the meanders. Coastally trapped topographic waves are strongly damped by bottom friction torque. Planetary vorticity advection by a poleward boundary current is enhanced on the cyclonic side of the current by friction in shallow water. More complex models than those treated above are needed to simulate such compound balances. The insight gained from the simpler models should be valuable in interpreting the results. Allowing for compound balances, how complete is our understanding of the facts sketched in Section 2? The meandering of a boundary current over bottom slope was discussed without regard to stratification. However, considerations similar to those in Section 7 show that pressure torque may be balanced by curvature-vorticity change of a baroclinic current. This is, in fact, the basis of Warren’s (1963) comparison of theory and observation. A more important discrepancy between theory and observation arose in regard to wind-driven upwelling, coastal undercurrent, and coastally trapped waves: observation shows all these phenomena to be confined to the outer shelf and the upper slope. Existing theory does not clearly demonstrate this, let alone identify its precise causes. Nor are the coastally trapped surface flow patterns extending to hundreds of kilometers from the coast, seen in satellite images, accounted for by current models. There is every reason to believe that more sophisticated treatment of a stratified fluid over shelf-slope topography is necessary for resolving these questions. Similar comments apply to models involving frictional balance, most emphatically to those attempting to cope with density change along isobaths. All of the models in this category treated so far are overidealized to the point where comparison with observation is premature. At best, they give us hints on important physical relationships. There are many other interesting investigations relevant to the phenomena discussed here which I have not been able to cover, laboratory or numerical

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models, for example. There will have to be other reviews and many further investigations before we understand the physics of currents over the continental slope. The subject is a wonderful present frontier in geophysical fluid dynamics.

ACKNOWLEDGMENT This work was supported by the Department of Energy under contract number DE-ACO279EV1005. Pergamon Journals, Ltd. Kindly gave permission to reproduce Figs. 17,18,20,24,34, and 38; the American Geophysical Union, permission to reproduce Figs. 21-23, 30, and 37; the American Meteorological Society, permission to reproduce Figs. 14, 29, 31, 32, and 42.

REFERENCES Adams, J. K., and Buchwald, V. T. (1969). The generation of continental shelf waves. J . Fluid Mech. 35, 815-826. Artegiani, A,, Azzolini, R., and Salusti, E. (1986). O n the dense water in the Adriatic Sea. Unpublished. Arthur, R. S. (1965). On the calculation of vertical motion in eastern boundary currents from determinations of horizontal motion. J . Geophys. Res. 70, 2799-2803. Bane, J. M., Jr., Brooks, D. A,, and Lorenson, K. R. (1980). Three-dimensional view of a Gulf Stream meander between Savannah, Ga. and Cape Hatteras, N. C. Gulfstream 6.2-7. Barton, E. D., Huyer, A,, and Smith, R. L. (1977). Temporal variation observed in the hydrographic regime near Cab0 Corveiro in the northwest African upwelling region, February to April 1974. Deep-sea Res. 24,7-23. Blanton, J. L., Atkinson, L. P., Pietrafesa, L. J., and Lee, T. N. (1981).The intrusion of Gulf Stream water across the continental shelf due to topographically-induced upwelling. Deep-sea Res. 28A, 393-405. Booth, D. A,, and Ellett, D. J.( 1983).The Scottish continental slope current. Continental Shelf Res. 2,127-146. Brink, K. H. (1982).A comparison of long coastal trapped wave theory with observations off Peru. J . Phys. Oceanogr. 12,897-913. Brink, K. (1987).Coastal ocean physical processes. Rev. Geophys., in press. Brink, K. H., Allen, J. S., and Smith, R. L. (1978).A study of low-frequency fluctuations near the Peru coast. J . Phys. Oceanogr. 8, 1025-1041. Brink, K. H., Halpern, D., and Smith, R. L. (1980).Circulation in the Peruvian upwelling system near 15 ‘S.J . Geophys. Res. 85,4036-4048. Brink, K. H., Halpern, D., Huyer, A,, and Smith, R. L. (1983).The physical environment of the Peruvian upwelling system. Proy. Oceanogr. 12,285-305. Brockman, C., Fahrbach, E., Huyer, A:, and Smith, R. L. (1980).The poleward undercurrent along the Peru coast: 5 to 15”s. Deep-sea Res. 27A, 847-856. Brooks, 1. H., and Niiler, P. P. (1977).Energetics of the Florida Current. J . Mar. Res. 35,163-191. Brunt, D. (1939).“Physical and Dynamical Meteorology.” Cambridge University Press. Buchwald, V. T., and Adams, J. K. (1968).The propagation of continental shelf waves. Proc. R. Soc. Ser. A 305,235-250. Caldwell, D. R., Cutchin, D. L., and Longuet-Higgins, M. S. (1972).Some model experiments on continental shelf waves. J . Mar Res. 30, 39-55.

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Hamon, B. V. (1962). The spectrums of mean sea level at Sydney, Cotl's Harbour and Lord Howe Island. J. Geophys. Res. 67, 5147-5155. Heath, R. A. (1986). One to four weekly currents on the west coast South Island New Zealand continental slope. Continental Shelf Res. 5,645-664. Hickey, B. M. (1979). The California Current System-hypotheses and facts. Prog. Oceanogr. 8, 191-279. Hogg, N. (1981). Topographic waves along 70W on the continental rise. J. Mar. Res. 39,627-649. Holland, W. R. (1973). Baroclinic and topographic influences on the transport in western boundary currents. Geophys. Fluid Dyn. 4, 187-210. Houghton, R. W., Schlitz, R., Beardsley, R. C., Butman, B., and Chamberlin, J. L. (1982). The Middle Atlantic Bight Cold Pool: Evolution of the temperature structure during summer 1979. J . Phys. Oceanogr. 12,1019-1029. Hughes, P., and Barton, E. D. (1974). Stratification and water mass structure in the upwelling area off northwest African in April/May 1969. Deep-sea Res. 21,611-628. Huthnance, J. M. (1978). On coastal trapped waves: Analysis and numerical calculation by inverse iteration. J. Phys. Oceanogr. 8.74-92. Huthnance, J. M. (1984). Slope currents and "JEBAR." J. Phys. Oceanogr. 14,795-810. Huyer, A,, Pillsbury, R. D., and Smith, R. L. (1975). Seasonal variation of the alongshore velocity field over the continental shelf off Oregon. Lirnnol. Oceanogr. 20,90-95. Janowitz, G. S., and Pietrafesa, L. J. (1982). The effects of alongshore variation in bottom topography on a boundary current-(topographically induced upwelling). Continental Sherf Res. I, 123-141. Johns, W. E., and Watts, D. R. (1986). Time scales and structure of topographic Rossby waves and meanders in the deep Gulf Stream. J . Mar Res. 44,267-290. Kinder, T. H., Coachman, L. K., and Galt, J. A. (1975). The Bering Slope current system. J. Phys. Oceanogr. 5,23 1- 244. Kroll, J., and Niiler, P. P. (1976). The transmission and decay of barotropic topographic Rossby waves incident on a continental shelf. J. Phys. Oceanogr. 6,432-450. Kundu, P. K., and Allen, J. S. (1976). Some three-dimensional characteristics of low-frequency current fluctuations near the Oregon coast. J. Phys. Oceanogr. 6, 181-199. Lee, T. N., Atkinson, L. P., and Legeckis, R. (1981). Observations of a Gulf Stream frontal eddy on the Georgia continental shelf, April 1977. Deep-sea Res. 2% 347-378. Lee, T. N., Ho, W.-J., Kourafalou, V., and Wang, J. D. (1984). Circulation on the continental shelf of the southeastern United States. Part I: Subtidal response to wind and Gulf Stream forcing during winter. J. Phys. Oceanogr. 14, 1001-1012. Legeckis, R. V. (1979). Satellite observations of the influence of bottom topography on the seaward deflection of the Gulf Stream off Charleston, South Carolina. J. Phys. Oceanogr. 9, 483-497. Longuet-Higgins, M. S. (1965). The response of a stratified ocean to stationary or moving windsystems. Deep-sea Res. 12,923-973. Louis, J. P., and Smith, P. C. (1982). The development of the barotropic radiation field of an eddy over a slope. J. Phys. Oceanogr. 12,56-73. Louis, J. P., Petrie, B. D., and Smith, P. C. (1982). Observations of topographic Rossby waves on the continental margin off Nova Scotia. J. Phys. Oceanogr. Luyten, J. R. (1977). Scales of motion in the deep Gulf Stream and across the continental rise. J . Mar. Res. 3549-74. McCartney, M. S. (1976). The interaction of zonal currents with topography with applications to the Southern Ocean. Deep-sea Rex 23,413-427. Mclntyre, M. E. (1968). On stationary topography-induced Rossby-wave patterns in a barotropic zonal current. Dtsch. Hydrogr. Z.21,203-214.

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ADVANCES IN GEOPHYSICS. VOL. 30

OBTAINING ATTRACTOR DIMENSIONS FROM METEOROLOGICAL TIME SERIES HARRYW. HENDERSON AND ROBERT WELLS The Pennsylvania State University Departments of Meleorology and Mathematics University Park, Pennsylvania 16802

1. INTRODUCTION

In recent years there has been interest in the application of dynamical systems theory to meteorological processes. Much of the current work in this area may be traced to a paper by Lorenz (1 963). In this work, he gave a picture of a strange attractor for a low-order, three-component system modeling convection. This combination of a relatively simple set of first-order differential equations, a firm physical application, and a good description of the resulting attractor opened up a new field of research into system behavior, although a great deal of the research has been done in fields other than meteorology. The Lorenz attractor was not the first strange attractor to be discovered. In fact, Birkhoff (1932) described an attractor which was later (ca. 1977) shown to have the general appearance of a wrinkled torus. But Lorenz emphasized that dynamic behavior on these attractors is both deterministic and unpredictable, and thereby provides a mathematical model of turbulence. This observation has led to the descriptive, rather synonymous terms “chaotic attractor” and “strange attractor” to describe the resulting behavior of the system. Recent papers have been concerned with the calculation of the dimension of an attractor from a univariate time series of empirical observations (Nicolis and Nicolis, 1984; Fraedrich, 1986). These studies suggested that the attractors associated with some meteorological time series are of reasonably low dimension (of size 3-6), and that a model system of equations might not be very large. It must be noted that a recent paper by Grassberger (1986) emphasized that the data set must be adequately large and free of artificial dependencies introduced by smoothing, interpolation, or other such smoothing procedures. Failure to recognize these problems may result in dimension estimates that are too low, or may result in a dimension estimate where none really exists, Nonetheless, these results are important to the numerical modeling of meteorological processes, since the techniques identify system behavior (attractor type) and suggest something about system size (embedding dimension). Furthermore, these results are also important in 205 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

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HARRY W. HENDERSON A N D ROBERT WELLS

another way; they make it possible to test models derived from physical concepts by comparing their attractors to the empirically reconstructed attractors. However, there seems to be no work in the meteorological literature that explains the role of attractor size in model development. It is the aim of this article to (1) describe the mathematical basis of the problem, and (2) illustrate the numerical techniques that yield (bounded) estimates of the attractor dimension from a time series, Two meteorological processes are studied, and the numerical algorithms suggest that large-scale processes may have a low-order attractor, while thunderstorm dynamics may be of higher order. The method used to find the upper bounds (limit capacity) has apparently not been widely used, even though it may, under favorable circumstances, directly yield the fractal dimension.

2. BASISOF MODELRECONSTRUCTION In model reconstruction we attempt to develop numerical models for physical systems by seeking an algorithm that reads a time series of data from a physical processes and, from that, generates an ad hoc model to describe the future behavior of that process. Clearly such a quest appears to be ambitious and perhaps overoptimistic, and yet a recent theorem of Takens (1981) suggests that it is not entirely hopeless. In fact, the Takens embedding theorem states, in principle, that model reconstruction should succeed in certain cases. An advantage of the resulting procedure to be described below is that, being highly empirical, it is uncontaminated by any particular physical preconceptions that may enter into the derivation of a theoretical model. Thus, comparison of the empirically reconstructed attractor with a theoretically predicted one offers the investigator a stringent test of the theory that makes the prediction. 2.1. Concepts At the most basic level, one applies model reconstruction to a time series ..by finding an integer N = 1,2,3,. ..such that any N consecutive entries xk + xk + z , xk + 3 , . ..,x k + N determine the next entry in a fixed manner,

x l , x2, xj,.

+ 1 = f ( X k + 1, x k + 2 x k + 3 * . x k + N ) (1) Of course, major problems with this procedure are immediately apparent. How do we know such a number N even exists? Even if we know that such a number exists, how do we find it? And granted that it is known, how do we find the function f ? It is with the first two of these problems that we are primarily xk + N

3

9

.?

207

ATTRACTOR DIMENSIONS FROM TIME SERIES

concerned in this article. More particularly, we are concerned with the decisive role that the Takens embedding theorem assigns to the concept of phase space dimension in the resolution of that problem. In general, we may regard a model as comprising three principal items: a phase space of states, a collection of observables, and a law of motion. The phase space is a collection of points representing-by definition-the admissible states of the system. Each observable assigns to each state a real value. And finally, the law of motion assigns to each state that new state into which it evolves after a fixed interval of time. The phase space is most often parameterized locally by d-tuples of real numbers, in such a way that every dtuple of real numbers (within certain bounds) corresponds to precisely one of the states in a neighborhood of a given state. Such a space is called a manifold and the number d is its dimension. Manifolds are generalizations of the familiar surfaces contained in Euclidean three-dimensional space: recall that these surfaces are parameterized locally by pairs of real numbers. The observables are arbitrary real-valued functions on the phase space, but the law of motion is almost always a smoothly invertible transformation of the manifold. Finally, to see how a model time series arises, let the law of motion be denoted by z and consider some initial state s together with its subsequent states: T(S), T2(S)

= T [ T ( S ) ] , T 3 ( S ) = T[T2(S)],.

.., Z k ( S ) = ?[?k-l(S)]

,...

Suppose that the observable assigns the real number $(a) to the state a; then a model time series (xl,x2,. ..,x k , . . .) is given by the numbers $(s), I,b[r(s)],

I,bCz2(sll,.* .

5

I,b[zk(s)l,-.

.

Examples of this type of model abound. One such example is the damped, periodically forced pendulum, whose equation of motion is given by d28/dt2

+ A sin(8) + B d 8 / d t + Csin(ot) = 0

(2)

where A, B, C, and o are nonnegative constants, 0 is the angular displacement of the pendulum, and t is the time. The phase space consists of all pairs (8,u) where the angular displacement 8 is an angle in radians (so that 27c = 0) and the velocity u = r e is a real number ( I is the radius). A natural observable is the kinetic energy given by *(@,u) = v 2 / 2

(3)

The law of motion or transformation z is given by r(OO,uo) = (01,u l ) where

(el,u , ) is the result of integrating the differential Eq. (2)from t = 0 to t = 2413. Finally, the kinetic energies at times t = 0, 2 4 0 , 4a/o, 6n/w,... constitute a time series xo = $(eo,uo), x l = $ [ T ( O o , u o ) ] , x2 = I,b[?(eo, u o ) ] , . . . . Another example is given by a grid-point model for the iterative approximation of solutions of the Navier-Stokes equations with the pressure

208

HARRY W. HENDERSON AND ROBERT WELLS

eliminated. Because there are three velocity components and two thermodynamic components, the phase space is now the Euclidean space of dimension equal to the number of grid-points times 5. The transformation z is the iterative integration step that assigns to an approximate solution at time nh the approximate solution a t time (n + l)h, where h is the time step. An observable is given, for example, by evaluation of a vector component at a grid-point and a model time series by the sequence of such evaluations on an evolving approximate solution. Finally, a model not of the type we are considering here is given by the Navier-Stokes equations themselves, with or without the pressure eliminated. This model is not of our type because the phase space for this model is not finite dimensional; we will disregard such infinite dimensional models. Having established the type of model we intend to discuss, we describe the Takens embedding theorem. Suppose that the d-dimensional manifold M is the phase space for such a model, z the law of motion, and $ an observable. Let N be any integer greater than or equal to 2d + 1. Then we may assign to each state s in M a point in Euclidean N-dimensional space given by

The Takens embedding theorem states that for a generic transformation z and a generic observable t,b this assignment is one-to-one. The term generic may be read informally as "almost any"; that is, only very exceptional (and therefore highly improbable) transformations z and observables $ fail to produce a oneto-one assignment. More precisely, even if a function such as z or $ fails to be generic, suitable but arbitrarily small perturbations of the function will be generic, and if a function is generic, then any sufficiently small perturbation of the function is again generic. Sound modeling principles (Abraham and Marsden, 1964; Dutton, 1987)recommend that an acceptable model be structurally stable; that is, perturbing the law of motion, the observables, and the control parameters slightly should produce an equivalent model. For such a model then, we may assume that the assignment I defined above is one-toone. In that case we may identify the phase space M with its image I ( M ) = M' contained in the Euclidean space RN,but not equal to it. Because the assignment I defines a bijective correspondence between the phase space M and its image M', these two objects are entirely equivalent. In particular, the law of motion z determines an equivalent law of motion z' on M' by the requirement that the equation be satisfied. Upon using Eqs. (4) and ( 5 ) we see that for (xl,x2,. ..,x N )= I($, a point in M',the equation ~ ' ( x l~ ,

2

- .,. ,xN)

=(

. ., xN, x N + 1 )

~ 2 , .

(6)

ATTRACTOR DIMENSIONS FROM TIME SERIES

209

will hold with xN+l

= $bN’l(S)l

(7)

However, as long as it is in M‘, the point (xl,xz,..., xN) uniquely deter~ . the point mines the state s, which then via Eq. (7) determines x ~ +Thus (xl, x2,.. . ,x N )of M’ determines the coordinate x N + and we may write XN+I

= f(Xl,X2>...,XN)

(8)

for some uniquely defined function f . And clearly this function is the one required in Eq. (1). Finally, it follows from the Takens embedding theorem that this function is smooth. Thus the Takens embedding theorem enables us to conclude that if our physical system admits of a model with a d-dimensional manifold M as phase space and a generic transformation z as the law of motion, then for the time series of any generic observable, an equation of the type in Eq. (1) will hold with N greater than or equal to 2d + 1 and f ( x l ,x2,. ..,xN)a suitable smooth function. [In fact, a minor addition to the theory assures us that we may assume that f(x,, x 2 , .. .,x N )is defined for any N-tuple (xl,x2,. . .,xN),not just one in M’.]In this way, the Takens embedding theorem carries us some way toward the resolution of the first of the problems outlined above, provided that we know that our physical system admits of a model of known finite dimension. We turn now to the interesting case in which we have reason to believe that our system does admit of a finite dimensional model, but in which we have almost no information about such a model; in particular, we do not know even the dimension of the model. On the other hand, we assume that we do have available a very long time series of data for a generic observable, as long a series as we may need, and we assume that it is a time series of the form given by xo = $(s), x1 = $[T(s)], x2 = $[z2(s)], x3 = $[z3(s)], ... for some initial state s. We also assume that both z and $ are generic. In a favorable case, the initial state s may satisfy a condition of ergodic type that requires that the sequence of successive states s, z(s), ~’(s), .r3(s),... visits every neighborhood of every state CT in the phase space M . In other words, the sequence s, z(s), z2(s), z3(s), ... “fills up” the phase space M. It then follows that the sequence of N-tuples (xo,xl,. . .,xN-1), (x1,x2,...,xN), (x2,x3,..., x N +1), ... “fills up” the image phase space M‘. To illustrate this phenomenon, we let the phase space M be the unit circle in the complex plane, M

={zlz

complex and IzI = I}

(9)

and we let the transformation 7 be defined by z(z) = .zeZniU

(10)

where i and n have their usual meanings and u is an irrational number. Then

210

HARRY W. HENDERSON A N D ROBERT WELLS

the successive states z, z(z), zz(z), z3(z), . , . do indeed “fill up” the unit complex circle (see Fig. l a and b). For our observable we use the real-part function (1 1)

$(z) = Re(4

so that we generate the model time series xo= $(z), x 1 = $(z(z)), x 2 = $(T~(z)), x3 = $ ( z 3 ( z ) ).,. . Now we may take N = 2 . 1 1 = 3. As the reader may

+

.

check easily, the 3-tuples ( x o ,x I , x 2 ) , ( x , , x 2y x 3 ) ,( x 2 , x 3 , x 4 ).,.. “fill up” a planar ellipse in Euclidian 3-space and that planar ellipse is a distorted copy of the unit circle (Fig. lc). In addition, once we have reconstructed this smooth copy of our original phase space, we may reconstruct the law of motion as the transformation z ’ ( x k + 1I x k + 2 x k

+ 3) = (xk+

(12)

2 9 xk+ 3 x k + 4 )

where xk +4 = f ( x k +

1 9

(13)

x k + 2 9 xk+ 3)

and f ( X 1 9 X 2 x3)

= (x1x3

+

(14)

- xt)/x2

The derivation of Eq. (14) is given in the Appendix. To summarize the process of model reconstruction, we may say that the reconstructed phase space M’ is the subspace of Euclidean N-dimensional space R N “filled up” by the consecutive N-tuples of the given time series and that the reconstructed law of motion is given by t ‘ ( X k + 1Y X k

+2

3

-

*

a

9

xk

+ N ) = [xk + 2

9

xk

+3

9.

*.

t

x k +N,f ( X k +

19%

+Z

9

* * *

,xk

+N)1 (15)

where the function f is the function of Eq. (1) which predicts the next entry in the time series. The Takens theorem assures us that this function exists, and therefore in principle we may approximate on the reconstructed phase space by using the time series as data. In the favorable example above, for the sake of simplicity, we find the function f directly from our knowledge of 7 rather than from an approximate algorithm. A less favorable example than the one above is defined by extending the law of motion to the new phase space consisting of all complex numbers, with the new law of motion given by

This example is more difficult because it is no longer true that the successive states z, z(z), 7*(z), ~ ~ ( z...) ,will “fill up” the phase space M. Instead

1.25

0.75

I

I

I

:a

1 1

I

I

I

I

1 1

I

I

I

I

I

I

I

I

I

I

I

8

1

e4

-

m2

025-

-> 3 0,

mi

-

%-025-

-

-0.75-

-1.25

m3

5.

' ' ' '

I

I

' ' '

I

' '

'

I

I

'

" I

'

' ' '

-0.75 -1.25t' -2

I.o

rn

I

'

I -1.2

I

'

I

'

I ' ' ' ' ' -0.4 0.4 x Value

' '

I

I

'

'

I '

2

1.2

d

*"B

-. 2

I.o

J?

FIG. I . (a) The first five iterates on the unit complex circle with a = and initial point The attracting dimension is 1. (b) The first 150 iterates on the unit circle with a and zo as in (a). The "filling in" of the circle is becoming apparent. (c) A plot of the 3-tuples (Re@,), Re(z,+ ,), Re(z,+ *)) for (b). The 3-tuples form a planar ellipse. zo = (1,O).

212

HARRY W. HENDERSON AND ROBERT WELLS

FIG.2. A more complicated model with a trajectory approaching the unit circle from outside. The attracting dimension is now 2 since the entire plane (except the origin) is now part of the system. The initial point is z = (2,O) and a = &/loo. The first 150 iterates are shown.

they approach the unit circle and visit every state on that circle as closely and as often as one wishes (see Fig. 2). Now we have N = 2 2 1 = 5 (because the new phase space has dimension 2). If we use the same observable )I as before, we obtain a long sequence of 5-tuples (xo, xl, x2,xj, x4), (x1,x2,x3,x4,x5),(x2,x3,x4,x5,x6),.... This sequence “draws” a planar ellipse in Euclidean 5-dimensional space to any level of resolution we choose if we simply continue the sequence long enough and neglect enough initial members of the sequence. Again, the predictor function f may be reconstructed as before, but we have not reconstructed our original process consisting of the complex plane and the law of motion defined by Eq. (16). Instead, we have reconstructed the restriction of that process to the unit circle. The unit circle is the attractor of the process defined by Eq. (16). In general, an attractor in a model with phase space M and law of motion r may be defined to be a subset A of M with three properties:

-

+

i. If the state s is near A, then the sequence of states s, ~(s),z2(s), z3(s), . .. approaches A. ii. If s is in A, then z(s) is in A. iii. For almost any s in A, the sequence of states s, r(s), z2(s), r3(s),. . . fills up A. An attractor in a model represents permanent behavior; that is, it is the behavior that remains after we discount the transient behavior by neglecting the initial terms in a sequence of states s, z(s), z2(s), z3(s), .. .. Equivalently, we

ATTRACTOR DIMENSIONS FROM TIME SERIES

213

neglect the initial terms in a generic time series xo,x1,x2,. . , . Now we may state the principal conclusion that follows from the Takens theorem. Suppose that a physical process has a model ( M , z) with M a d-dimensional manifold and z a generic transformation. Suppose further that $ is a generic observable and that xo, xl, x 2 ,... is the time series of this observable with xo = I&). Then we may conclude that model reconstruction from that time series with N = 2d 1 will generate a model for the permanent behavior approached asymptotically by the succession of states s, ~ ( s ) ,r2(s), z3(s), . , , . We have illustrated the case of a 1-dimensional, periodic attractor with the unit circle, but there are other attractors of different dimension. It will be informative to consider these other cases. The first, and simplest, case to consider is that of steady behavior. Trajectories starting from different initial conditions may converge to a particular point. (The point need not be unique; trajectories that are initially close may proceed to different points.) The dimension of the attracting point(s) is appropriately taken to be zero (Fig. 3a). A little more complicated is the case of the periodic attractor. Trajectories arising nearby will all converge to a closed curve, but instead of becoming steady, the trajectories move along the curve and eventually repeat the circuit more and more closely. The dimension of this attractor is 1. Figure 3b illustrates a trajectory that begins inside the attracting orbit, and subsequently converges to the orbit. A trajectory that begins outside the orbit will also converge if the initial point is close enough to the attractor. (The example of the unit circle discussed in Fig. 2 shows this behavior.) More complicated curves than the one shown can be constructed, but as long as the trajectories complete a circuit the dimension is 1 and the motion is periodic. The next higher dimension, that of 2, arises quite easily by considering periodic motion of two different frequencies. The motion is most easily pictured as trajectories moving about a torus (or surface of a doughnut). The longer period may occur with motion about the center, while the shorter period then occurs with motion around the smaller, outer ring. If the trajectories describe a closed loop about the torus, the attractor is periodic and the dimension is 1. This case is shown in Fig. 3c, in which the trajectory completes 20 smaller loops as it travels around the main circle. But something very different happens when the ratio of the two periods is irrational. For example, suppose the shorter period was 1 time period in length and the longer It is not possible for the trajectory to close the loop. Instead the was trajectory continues around the torus, and eventually it fills the surface. This motion is called quasi-periodic and results in an attractor of dimension 2. Attractors of dimension 3,4,. . are possible if the system contains multiply periodic components with frequencies that lead to higher dimensional toroidal surfaces (and are not easily pictured). The motion over toroidal surfaces

+

a.

Variable I

5-b'

'

'

I

'

I

I

I

'

I

'

'

I

-

2.5 (u 0

-

n

.-0

0

P -2.5

Variable I

FIG.3. Illustrations of some different attractors that are possible within dynamical systems: (a) trajectory spiraling in toward a steady fixed point, (b) trajectory that moves toward a periodic orbit from within the orbit, (c) motion over a torus which is periodic with two frequencies, the higher frequency completing 20 cycles for each slower cycle, and (d) the Lorenz chaotic attractor. The dimension of the fixed point in (a) is 0, and that of the periodic attractors in (b) and (c) is 1, while the dimension of the Lorenz attractor is about 2.06.

ATTRACTOR DIMENSIONS FROM TIME SERIES

215

Variable I

FIG.3. (Continued)

(dimension 2,3,4,.. . .) is the simplest type of bounded nonperiodic motion. The surfaces are evenly covered; however, the system is structurally unstable-small changes to the system may result in periodic behavior, which is fundamentally different from quasi-periodic behavior. Structurally stable attractors of integer dimension do exist; these are certain dynamical systems on smooth manifolds called Anosov attractors. The appearance of these attractors is smooth and unfolded, unlike that of strange attractors. We omit their description because they arise principally in nonfluid mechanical systems, and have not yet been found in atmospheric models. The attractor dimensions that fall between the integer values all belong to strange attractors. The most interesting trait of a system that has such an attractor is related to the structural stability issue. The known attractors with fractional dimensions are associated with systems that are either analytically or numerically structurally stable-small changes to the system do not result in radically different behavior. Once such an attractor is established, it will persist under small changes of the control parameters. A picture of a strange attractor with dimension about 2.06, the Lorenz attractor, is shown in Fig. 3d. The detailed microstructure is not apparent on this scale, but it does have the complex sheeting that is to be found in an attractor that has more structure than a 2-dimensional surface. It is to attractors of this type that the techniques of this article apply. We regard model reconstruction as a procedure for constructing algorithms, rather than as an algorithm itself. One may say it is a metaalgorithm: by neglecting initial terms in the time series, by extending the series far enough, and by choosing a suitable level of resolution, one determines which subspace M' of Euclidean N-space is being traced out by the resulting

216

HARRY W. HENDERSON AND ROBERT WELLS

N-tuples. Then on that space one fits approximations of the predictor function to the data provided by the time series. The resulting system will reproduce an attractor from a possibly unknown finite-dimensional model for the physical process generating the time series. Because model reconstruction produces an attractor of smaller dimension than the whole model, the issue of attractor dimension becomes crucial. In our example above with phase space the complex plane, the attractor was precisely the preceding model, which could be reconstructed in Euclidean 3-dimensional space rather than 5-dimensional space. Thus, in general, we may expect that the condition N 2 2 d + 1 applies to d-attractor dimension rather than dmodel dimension; however, this expectation is a conjecture and not a proved theorem. The validity of this proposition would follow immediately from the Takens theorem if attractors were manifolds, but in general they are much more complicated spaces. Nonetheless, their dimension may be defined in several different ways, and these dimensions provide criteria for deciding when an embedding dimension N is sufficiently large (Grassberger and Procaccia, 1983).Conceptually, this criterion is simple: one applies model reconstruction to the time series x0,x,,x2,x3 ,... with N = 1,2,3,4,..., thus obtaining a sequence of candidates A ; , A;, A ; , A:, ... for the model of the attractor; the space Ah lies in Euclidean N-dimensional space. By using algorithms to be described below, one calculates their dimensions d l,dZ,d3,d4,.... Once N is large enough, the Takens theorem assures us that they will all be copies of the same space, an attractor in the original but unknown model, and therefore they will all have the same dimension. Accordingly, one knows that N must be greater than or equal to the index K for which we have dk = dk+, = d,,, = dk+3= . * * . One hopes that N = K, but again, that hope represents a conjecture, not a proved theorem. Still, the existence of such an index constitutes strong evidence that the correct attractor dimension d and the correct embedding dimension N have been found. Conversely, if the sequence d, 'dz,d3,.. . does not stabilize, then the physical system generating the time series x x 2 ,x3,. . . certainly does not admit of a finite-dimensional model. It is this fact that was emphasized by Grassberger (1986). 2.2. Attractor Definitions

Not only does the concept of attractor dimension serve to select the correct embedding dimension N, but it also serves as a means for illuminating the complexity of attractor structure. In this article we will be concerned with two definitions of dimension that are appropriate for attractors, the fractal and correlation dimensions. We begin with the fractal dimension. To motivate the definition of fractal dimension, we consider a cube in Euclidean 3-space. If we cover this cube with

ATTRACTOR DIMENSIONS FROM TIME SERIES

217

k balls of radius r, we know that the number k4nr3/3 is an overestimate for the volume of the cube; moreover, if k, is the minimum number of balls of radius r necessary to cover the cube, then k,47cr3/3 is a better (smaller) overestimate of the volume of the cube. In fact, we know that the limit of these overestimates as the radius r approaches zero is the volume itself of the cube. Thus limr+okrr3 exists and is finite. It is easy to see then that the number 3 is a critical threshold for the cube: limk,rP = 0

for p > 3

r+O

limk,rP = co

(17)

for p < 3

r-O

The same procedure carried out for a square in the plane produces a threshold of 2. In general, for the hypercube in Euclidean n-space the threshold is n, and for a (compact) d-dimensional manifold, the threshold is d. Thus one is led to the following definition. Let J, be the minimum number of balls of radius r necessary to cover the (compact) space A. If there exists a threshold d, such that limJrrP = 0

for p > dl

r+O

lirn J,rP = 00

for p < dl

r-0

then we say that d, is the fractal dimension of the space A (Mandelbrot 1983). It is a routine matter to show that the number d, determined by Eqs. (19) and (20) is given more directly by the equation

d

- lim ln(Jr)/ln(l / r )

- r-0

The serious difficulty remains that the limits in Eqs. (19)-(20) may not exist and to resolve it we must introduce the limits inferior and superior. The limit inferior of a possibly nonconvergent sequence of real numbers is simply the smallest limit point of the sequence, and the limit superior of a sequence is the largest limit point. Because cc and - co are allowable as such limits, the limit superior and the limit inferior of a sequence always exist; they are equal if and only if the ordinary limit exists. Now we may follow Takens (1981) to define the fractal dimension by setting

dI = lim inf ln(J,)/ln( l/r) r-0

or we may follow Maiie (1981) to define the fractal dimension by setting

d, = lim sup In(J,)/ln( l/r) r-0

Since we have available now two fractal dimensions, we distinguish them by

HARRY W. HENDERSON AND ROBERT WELLS

218

Step I Step2 Step3

Step5

-1111

- -

=

II

II

1111

IIII

Ill1

1111

,111

II

1111

1111

FIG.4. Construction of the Cantor set. Begin with a bar of unit length. In the first step, remove the middle third of the bar. In subsequent steps, remove the middle third of each of the remaining bars. This process is illustrated for the first five steps. The fractal dimension of the final set of points is ln(2)/1n(3) 0.6309.

-

referring to the dimension in Eq. (22) as the lower fractal dimension and denoting it by d; , while referring to the dimension in Eq. (23) as the upper fractal dimension and denoting it by d,’. When both are equal, we refer simply to the fractal dimension and denote it by d,. Now, although these dimensions d; and d/’ always exist, they are not always integers. For example, the Cantor set in Euclidean 1-dimensional space has the fractal dimension ln(2)/ln(3). This set is obtained by removing the open middle third from a unit interval, the open middle thirds from the resulting intervals, and so on ad injniturn (see Fig. 4). For any point in the resulting space, there exist other points in the space arbitrarily near to the given point (dense in itself). Between any two points in that space there is an open gap (totally disconnected) and the limit of any convergent sequence of its points lies in the space (closed). In some sense, it is this fractured nature that the fractional part of the fractional dimension measures, and it is a similar fractured nature that is shared by most attractors. The other dimension, the correlation dimension, is defined for a sequence of points a , , u 2 ,u s , ... in the space A. The assumption is that this sequence “fills up” (is dense in) the space A. Then one expects that the proportion of these points within a ball of radius r should be equal to the d-dimensional normalized “volume” of that ball, so that the normalized volume of the whole space A is equal to one. O n the other hand, one also expects that the volume of a ball of radius r should be proportional to rd as r -+ 0. Accordingly, after fixing a point a, in A one is led to define first:

P, = lim {number of integers k I n such that la, - a,l 5 r } / n n+m

(24)

ATTRACTOR DIMENSIONS FROM TIME SERIES

219

so that P, is the proportion of the sequence a,, a,, a 3 , .. . that lies in the ball of radius r centered at the point a,. Then since one expects that this proportion P, is proportional to rd, &*rd we may define a dimension d by setting

(25)

d = lim In(P,)/ln(r) r-0

Next we modify the definition above to match that introduced by Grassberger and Procaccia (1983). The definition we have just given really defines a dimension at the point a,. The dimensions we seek are global; they must apply to the whole space. Therefore, we replace P, with its average over the whole space, A, = lim {number of pairs k , m I n such that (ak- a,,,! I r } / n 2

(27)

n-m

Then the definition of Grassberger and Procaccia for the correlation dimension is given by d, = lim ln(A,)/ln(r) r-0

Any difficulties with the existence of this limit may be eliminated by replacing the limits in Eqs. (27)-(28) with limits superior and limits inferior as appropriate. To decide which of the modified limits is appropriate, we recall that Grassberger and Procaccia gave a heuristic argument to show that d, I df. Here we replace that argument with a short but rigorous proof of the same fact and simultaneously determine which modified limits are appropriate in Eqs. (27)-(28). We begin by expressing A, in terms of the quantity given by N(r,n) = {number of pairs k, m I n such that la, - a,l I r}

(29)

so that A, = limn+mN(r, n)/n2, provided that limit exists. For brevity write p = Jr,2.Then there exist p balls of radius r / 2 covering the space A, and any

two points in a single one of these balls are within distance r of each other. Some ak may lie in the overlap of two or more balls; we choose one of these balls and say ak definitely lies in that ball but not in the others. Let n, be the number of integers k I n such that ak definitely lies in the ith ball. Then clearly we have n: n: . * . ng I N(r,n) (30) and

+ + +

n,

+ n, + . . * + np = n

(3 1)

220

HARRY W. HENDERSON AND ROBERT WELLS

It is easy to see by using the methods of advanced calculus that the absolute minimum of n: + n$ + - .. + n i , subject to the constraint of Eq. (31), is equal to n 2 / p . Upon recalling that p = Jr,2 we see that n Z / J , / zI “3

4

(32)

or 1/Jr/2

I N(r, n)/n2

(33)

It follows that we have the inequality 1/& I lim inf N(r, n)/n2 n+m

(34)

or 1/Jr/2

Ar

(35)

where we define A , = liminf,,, N(r,n)/n2if the actual limit does not exist. By taking the natural logarithm of both sides and carrying out a few elementary operations we see that ln(Ar)/Wr/2) 5 -1n(Jr/d/Mr/2)

(36)

from which it follows immediately that the Grassberger-Procaccia inequality is valid in the form

d: I df’

(37)

if we define df and d; by setting

df

= lim supIn(A,)/ln(r)

(38)

d; = lim inf ln(Ar)/ln(r)

(39)

r-w

and r+O

Having outlined the concepts of model reconstruction, attractors, and dimension, and having established the Grassberger- Procaccia inequality, we are ready to test these concepts by applying them to some empirical time series. However, before doing so, it is convenient to make a few further remarks about the computation of these dimensions. The first remark concerns error estimates: although the theory presented above is rigorously established at a great many points, error estimates are still unavailable. Thus the determination of limits from approximants, as in most empirical work, is ultimately subjective-the user finds the analysis of the data persuasive or not, The second remark concerns the GrassbergerProcaccia inequality: although it is rigorously established that Eq. (37) holds,

ATTRACTOR DIMENSIONS FROM TIME SERIES

22 1

it is an inequality in the limit and it does not follow that it must hold for the approximants - ln(J,)/ln(r) and ln(Ar)/ln(r) of the fractal and correlation dimensions, respectively. However, Eq. (36) is not an equation in the limit and must therefore hold at the approximate level. Thus it provides a further test for the adequacy of the embedding dimension.

3. CALCULATION OF ATTRACTOR DIMENSIONS

Of the various numbers that have been used to measure the size of an attractor, only two seem to be easily calculated from a time series of observed data. The remaining numbers are either inappropriate for meteorological data, or are much too difficult to calculate. The two appropriate numbers, which were defined earlier, will be discussed further because they provide an estimate of the upper and lower bounds of the fractal dimension. 3.1. Correlation Dimension The definition and elementary properties of the correlation dimension of Grassberger and Procaccia (1983) were given in the preceding section. In this section we address its numerical determination from empirical data. We begin with a set of points (xk 1 k = 1,. ..,N ) from a time series. It is assumed that the process has continued “long enough” to have found its way to an attractor, should such an attractor exist. The measurements of xk are evenly spaced, i.e., x k = x ( t kz) where t is the time increment. It is in the nature of most strange attractors that pairs of trajectories that are initially close will eventually become temporally uncorrelated. However, both trajectories will still be on the attractor and remain spatially correlated. This cumulative correlation was defined by Grassberger and Procaccia (1983)to be given by

+

I

N

where H is the Heaviside function H(x)= 0 =1

if

x I 0

if

x>O

(41)

The norm Ixk - xjl may be any of the three usual norms: maximum norm, diamond norm, or the standard Euclidean norm. (The maximum norm is just the maximum absolute difference between the elements of xk and x,, while the

222

HARRY W. HENDERSON A N D ROBERT WELLS

diamond norm is the sum of all the absolute differences.)The Euclidean norm is probably used more often than the first two, as it is the usual way to calculate the distance between two points. While this norm has the advantage of being familiar, it is the most time consuming to calculate. The relationship between the correlation dimension and the cumulative correlation function is based upon the power law

-

C(r) rv

(42)

which is derived from the geometric considerations previously discussed. The result is valid for “small enough” values of r. There are several examples given by Grassberger and Procaccia (1983) that illustrate the region in which the power law holds. A determination of the correlation dimension is found by plotting C(r) vs r on a log-log graph. The region in which the power law is obeyed appears as a straight line, and the slope (which is an estimate of the correlation dimension) is found by fitting a least squares line to this part of the graph. If the sequence of estimates converges, then we have an estimate of the attractor dimension d,. An alternative method that may be used to calculate v is from the direct formula

1%

W,) - 1% C(r2) =v logr, - logr2

(43)

for values of rl and r2 from the appropriate part of the curve. We illustrate the algorithm with the following set of 10 values from a time series. The observations xk are xk=45

51 58 62 45

51 50 49 53 53

Next we form a matrix with the sequence of xk in the first row and then lagged by one period for the second row

45 51 58 62 45 51 50 49 53 51 58 62 45 51 50 49 53 53

[

1

Why just one period, and not two, or five? This lag period depends upon the nature of the data. If adjacent data points are always quite similar, they possess a high degree of autocorrelation. An unmanageably long series would be required to find sufficiently large differences for r. On the other hand, if the lag period is too large, the differences r will be larger and again an unmanageably long series would be required to capture small values of r. Heuristically, this dilemma may be visualized by seeing the attractor as it is covered with consecutive observations. It is necessary to measure the correlations between the various parts of the attractor. Observations sampled too close together may take too long to fill the attractor uniformly; ob-

223

ATTRACTOR DIMENSIONS FROM TIME SERIES

servations sampled too far apart may skip past important portions of the attractor. For the present example, we will use a lag period of one. In practice, it might be best to use the period given by the autocorrelation lag value for which the autocorrelation function first approaches 0. The next step is to form a matrix of the Euclidean distances between the columns. Only the upper triangular part is needed by the algorithm, although we note here that the matrix is symmetric. For the example above we have

8.03 0 6.08 5.39 4.47 8.25 7.03 9.22 8.00 9.06 5.39 5.39 7.46 17.03 13.89 15.26 12.73 10.30 0 18.03 12.08 12.65 15.26 12.04 0 6.08 5.39 4.47 8.25 0 1.41 3.61 3.61 0 4.12 5.00 0 4.00 0

1 9.22 17.03 0 8.06 0

With this matrix available the correlation matrix may be calculated directly from its definition r 0

1

2

3

4

5

6

7

8

9

10

11

12

13

-

14

15

16

17

18

19

We note that the values have been normalized by N (N - 1)/2, the number of elements above the diagonal. This normalizing factor replaces N 2 in the formula of Grassberger and Procaccia because there is no need to consider the zero elements on the diagonal or the symmetric values below the diagonal. The next step is to plot the results on log-log paper, as in Fig. 5. In our illustrative case, there is really insufficient data to identify the region where C(r) follows power law behavior, but we take the liberty of fitting a least squares line to the lower part of the data. The slope of the line is found to be 1.93. If the data were Gaussian, the slope would be 2.0 for a two-row lag matrix, 3.0 for a three-row lag matrix, etc. The series is too short for the technique to distinguish between Gaussian or correlated data, so the value of 2 is to be expected. The distances r are plotted directly on this graph. It may be advantageous to scale these differences to a common value, say 1.0, by

-

20

224

HARRY W. HENDERSON AND ROBERT WELLS

Distance Between Pairs

FIG.5. A log-log plot of the cumulative correlation function as a function of the distance between selected vectors over the (possible)attractor.The embeddingdimension for this case is 2.

dividing by the largest observed difference. This would permit computerdriven plots to be produced more efficiently.The final step is an iterative one; the process is repeated with additional offsets. Thus we would treat next the three-element vector case

1

45 51 58 62 45 51 50 51 58 62 45 51 50 44 58 62 45 51 50 44 53

[

and so on. The results for a five-row lag matrix yielded a least squares estimate of the slope of 4.1, which suggests that there is no convergencetoward an attractor for this simple example. The data appear to be randomly distributed, and it would take a longer series to establish any correlations.

-

3.2. Limit Capacity The second number that is easily calculated is the limit capacity. In contrast to the correlation dimension, which provides a lower bound to the fractal dimension, the limit capacity serves to provide either an upper bound, or a direct estimate of the fractal dimension if coverage of the attractor is uniform. There are at least three ways to define the limit capacity, and the definition that is suitable for numerical analysis is the third definition given by Takens (1981). For completeness the definition is presented, as it appears that the numerical

ATTRACTOR DIMENSIONS FROM TIME SERIES

225

algorithm for limit capacity has not yet been used in meteorology. Then the algorithm is illustrated with the same example used for the correlation dimension. Let (S, p ) be a compact metric space. For e > 0 we make the following definitions: s(S,r) is the maximum cardinality of a subset of S such that no two points have distance less than r; such a set is called a maximal €-separated set; r(S,r) is the minimum cardinality of a subset of S such that S is the union of all the €-neighborhoods of its points; such a set is also called a minimal r-spanning set. Let {bi):= be some countable dense sequence in S. For r > 0 we define the subset J, c N by:

OEJ,;

for i > 0:i E J, if and only if for all j with 0 I j c i and j E Jc, we have p ( b i ,b,) t E. C, denotes the cardinality of J. From these definitions it easily follows that whenever 0 < r < El, r(S. 5 ' )

C, I s(S, r).

Hence we may.. .define D ( S ) by D(S) = l i m i n f { z } . .

(Takens, 1981, p. 374)

<-0

The numerical calculation of the limit capacity begins with the same initial steps as with the correlation exponent. The matrix of distances between the two-row lag matrixs of the previous example is repeated now, but with the symmetric values in place.

r o 9.22 17.03 18.03

D..= 0 V

L

6.08 5.39 4.47 8.25

9.22 17.03 0 8.06 8.06 0 7.03 17.46 9.22 17.03 8.00 13.89 9.06 15.26 5.39 12.73 5.39 10.30

18.03 17.03 17.46 0 18.03 12.08 12.65 15.26 12.04

0 6.08 5.39 4.47 8.25 9.22 8.00 9.06 5.39 5.39 7.03 13.89 15.26 12.73 10.30 8.03 12.08 12.65 15.26 12.04 6.08 5.39 4.47 8.25 0 1.41 3.61 3.61 0 6.08 5.39 1.41 0 4.12 5.00 0 4.47 3.61 4.12 4.00 8.25 3.61 5.00 4.00 0

Whereas the correlation dimension was found by searching for differences that were less than a particular distance, the limit capacity is found by searching for differences that exceed a particular distance. The sequence of the search is listed in Table I, and is performed as follows. Pick an E bigger than any difference in the first row of the difference matrix (we have picked 20). Immediately include the first element (whose difference is zero) as the first entry of the set J,. The cardinality is, of course, one. Next, continue the search

226

HARRY W. HENDERSON AND ROBERT WELLS TABLEI. LISTING OF THE CARDINALITY AND SELECTION ORDERUSEDIN THE CALCULATION OF LIMITCAPACITY FOR THE 2-Row LAGMATRIX CASE €

Cardinality

20 19 18 17 16 15 14 13 I2 11 10 9 8 7 6 5 4 3 2 1 0 -0

1 1 2 3 3 3 3 3 3 3 3 3 4 4 5 5 5

7

I

8 8 9

Column vector

1 1

1-4 1,4,3 L4,3 443 443 1,4,3 L4,3 1,4,3 1,4,3 1,493 1,4939 2 1,4,3, 2 1,4,3, 2 , 6 1,4,3,56 1,4, 3 , 2 , 6 1,4, 3,2,6, 8 , 9 1,4,3,2,6,8,9 44,3,2,6,8,9,7 L4, 3 , 2 , 6 , 8 , 9 , 7 h4.3, 2,6,8,9,7,5

for differencesthat exceed the next smaller value of e. Note that the criterion is “greater than” E: values that equal E are considered to belong to the next (lower) value of E. In the example, the next element to be considered is the fourth one (value of 18.03) for an E of 18. For the next value of E, 17, the third element in the first row must be considered. It is added to the list because it is far enough from the first entry (17.03) and is also far enough from the second entry (1 7.46, which was the fourth element). Each differencemust be more than E from those already selected. It is easiest to check for this condition by scanning down the entire column for differencesto previously added elements. This is the reason for including all the locations in the difference matrix. For example, when the second element is considered for an E of 8, we see that it is far enough from elements 1,4, 3 (differences of 9.22, 17.03, and 8.06) and is included in the J, set. The last step is analogous to that of the correlation dimension. The data are plotted on a log-log graph with the cardinality VSE. Figure 6 shows the results, and once again we have taken the liberty of fitting a least squares straight line

ATTRACTOR DIMENSIONS FROM TIME SERIES

227

Distance Between Pairs

FIG.6. A log-log plot of the cumulative capacity function (cardinality)as a function of the distance between selected vectors over the (possible)attractor.The embeddingdimension for this case is 2.

to the lower part of the curve. The slope of the line is seen to be - 2.7, which is larger (in absolute magnitude) than that expected for Gaussian data. Since the limit capacity provides, at worst, an upper bound to the dimension, this value is reasonable. In contrast to the correlation dimension, the limit capacity estimate follows a relationship of the form

c,

-

(45) so the slopes will be negative for positive values of the limit capacity estimated. The example above may be tested for the limit capacity of 3- and 4-lagged matrices; slopes near - 3 and -4 are found and thus no convergence of d is seen. If there were an attractor evident in the data, the lower part of the curve would reveal this fact by approximating a straight line with absolute value of the slope substantially less than the embedding dimension. (The embedding dimension is just the number of elements in a column of the lagged matrix.) As with the correlation dimension, it may be advantageous to scale the axes for numerical plots. In order to apply these techniques to atmospheric data, we analyze the time series from two different meteorological processes. The first series, shown in Figs. 7 and 8, is that of the 500-mbar height index data for the period Dec. I, 1979 to Nov. 30,1980. Averaged heights at 60"N were subtracted from those at 30"N (Fig. 7) to give index values that provide an estimate of the strength of the westerlies. We next removed the annual cycle from the series, and the result €-d

228

HARRY W. HENDERSON AND ROBERT WELLS

8oo

700

9 L

! 400

Days

FIG.7. The height index (found by taking the difference between averaged 500-mbar heights at W N from 30"N) for each day from Dec. 1, 1979 to Nov. 30, 1980. Larger values of the index correspondto stronger westerlies.

--

-t5OO'

I

'

I00

' '

'

I

200

I

I

'

I

300

I

I

I

400

Days

FIG.8. The same height index data as in Fig. 7, but with the annual cycle removed.

is shown in Fig. 8. The reason for doing this is to obtain a stationary time series. A trend test is applied to the stationary series and found to pass the "no trend" condition at the 95% significance level. A nonparametric test called a "runs test up and down" (which counts the number of turning points in the series) is applied to the series, but this fails at the 95% level. This result indicates that the series was not Gaussian in nature. The next step in the analysis is to find the lag size that would best cover the

229

ATTRACTOR DIMENSIONS FROM TIME SERIES

-0.5

- I t 1I

0

I

I

20

'

'

I

40

' ' '

I

I

60

'

I

'

80

lag

FIG.9. Theestimated autocorrelation function for the height index data of Fig. 8. Note the first zero crossing at a lag of 5 days.

series attractor (if it exists). This step is carried out by computing the autocorrelations, as shown in Fig. 9. It can be seen from the figure that the estimated autocorrelation function first crosses zero at a lag value of 5 days. Thus by choosing a lag value of 5 for the lag vectors, we are (presumably) choosing vectors that best cover the attractor. In Fig. 10 the power spectrum of the series is presented, which shows that the data exhibit the usual behavior

V

0'

I

I I

I

I

l

l

10

I00

I

,

I

I

Days

FIG.10. A power spectrum estimate of the height index data of Fig. 8. A line of slope -3 has been drawn for reference over the higher frequency values. The spectral peak near 25 days is typical for mid-latitude height data.

230

HARRY W. HENDERSON AND ROBERT WELLS

l o p

I

I

I

I

!-

Embedding Dimension FIG. 11. The attractor dimension bounds calculated for the height index data. Limit capacity results are shown by the dashed line, while correlation dimension is shown by the solid line. If the data were random with a Gaussian distribution the calculated dimensionswould follow the solid, straight line.

of global height data with peak power in the 3- to 4-week range, and a high frequency slope of about - 3. The remaining step is to apply the algorithms for correlation dimension and limit capacity to the stationary data. The results are shown in Fig. 11. For embedding dimension up to 9, both algorithms show essentially the same behavior, with similar values but no convergence. Above embedding dimension 9, the limit capacity levels off at a value of d 6.5 while the correlation dimension stabilizes at v 5.0. This indicates a probable attractor for the series that has the fractal dimension d, between df = 5.0 and df = 6.5. The fact that the two techniques bound the dimension estimate suggests that the coverage of the attractor is not fully adequate, and either more data are needed to cover the attractor or the lag size is not quite correct. A time series of 365 points is probably too short to resolve the attractor size in this case, but the techniques do indicate a bound on the size. There may also be times when the correlation dimension will exceed the limit capacity estimate. This result fails the test (correlation dimension Ilimit capacity) and indicates that the embedding dimension is much larger. Such a result can happen when the data is “clumped” or localized on a restricted portion of the attractor and the assumption of uniform coverage is violated. A longer time series may solve the problem. In the second example, the vertical velocity data from a thunderstorm gust front that occurred near Boulder, Colorado (August, 1985) are analyzed (data

-

-

23 1

ATTRACTOR DIMENSIONS FROM TIME SERIES

-5g

'

"

'I

-

shown in Fig. 12). It may be argued that local data are not really suitable for model reconstruction, since the theory is based upon global variables. One way to use local data is to have simultaneous measurements at several sites, calculate the correlation between sites, and then weight each time series for its contribution to the correlation dimension or limit capacity. Another method is to make measurements far enough apart so that the time series are uncorrelated. The global attractor is then the sum of the local attractors from each independent "box." This is more easily said than done. In the case of the wind data (which is local), we should be using data from several sites within the area influenced by the thunderstorm. However, turbulence tends to be ergodic; measurements made at one location over a long enough period of time are representative of the entire process. In general, it may be possible only to check the ergodic assumption by comparing a long time series from a single site with shorter records from several sites. For this thunderstorm case, we have only the single time series. With these reservations in mind, we may look at the results in Figs. 13-16. The estimated autocorrelation function, in Fig. 13, shows that the series is nearly uncorrelated (dynamically) at 50 lags. Since the interval size is 0.4 sec, this lag amounts to 20 sec. Also shown is the power spectrum of the vertical velocity data (Fig. 14). Evident is the - 5/3 slope expected of well-developed, mechanical turbulence. The results of applying the correlation dimension and limit capacity algorithms, with a lag number of 50, are shown in Fig. 15. The correlation dimension flattens rather quickly at a value of v 4. The limit capacity

-

HARRY W. HENDERSON AND ROBERT WELLS

232

- 1 ~ ~ 1 " ' 1 " ' ' ' 1 ' ' ~ 1 1 ' 1 ~ 1 ' 1 ~ ' 1 1 1 ~ ~

0

40

120

80

160

240

200

log

FIG. 13. Estimated autocorrelation function from the vertical velocity data of Fig. 12. Note the first zero crossing at a lag value of about 50. (Lag 50 = 20 sec at a sampling interval of 0.4 sec.)

104

3 c b

:2

0

.-E +

w" -

I

g o 0

g

fn -I

-2r I

I

,

I

I

400

,

1

40

I

I

4

1

.4

Period (Seconds) FIG.14. Power spectrum estimate for the vertical velocity data of Fig. 12. A line of slope - 5/3 has been drawn for reference over the higher frequency values.

-

increases throughout most of the graph, but there is some suggestion that it may settle down at a value of d 5.5. Interestingly, for low-embedding dimensions the correlation dimension exceeds the limit capacity, but perhaps little significance should be given this. The theory behind these algorithms is "in the limit," and the low-embedding dimensions are just the initial stages in the sequences of estimates.

ATTRACTOR DIMENSIONS FROM TIME SERIES

233

u

-

-0

3

6

9

12

15

Embedding Dimension FIG.15. The attractor dimension bounds calculated for the vertical velocity data. Limit capacity results are shown by the dashed line, while correlation dimension results are shown by the solid line. If the data were random with a Gaussian distribution the calculated dimensions would follow the solid, straight line.

An estimate of the global attractor for the gust front may be made as follows: if we assume that the front is about 25 km long, 4 km wide, and 2 km deep, and that measurements become independent at about 2 km, the number of independent “boxes” is 24. If each “box” has a local attractor of size 4, then the global attractor is 4 times 24, or about 100. One may choose other similar estimates here, but the size of the attractor appears to be of order 100 in any case. The two examples just presented suggest that the numerical techniques can find the attractor dimension easily and unambiguously. Frequently this conclusion fails. It can be difficult to decide just where the “tangent line” should be drawn on the log-log plots of the respective cumulative distribution functions. As an example, we may look further at the estimates of correlation dimension from the thunderstorm case. Figure 16 shows a sequence of estimated slopes (based on nine-point regression lines centered at each point of the function) for increasing values of the embedding dimension. The obvious noise makes it difficult to decide if there is convergence in the sequence, but for this case there is a region (which is rapidly decreasing in extent) that seems to be converging to a value of about v 4 around bin number 150. There have been mathematical papers that attempt to quantify this behavior (Takens, 1983),and studies that suggest how long a series should be to make the convergence obvious. For attractor dimension d, a frequently cited estimate is that n lod series values are needed for obvious resolution

-

-

234

HARRY W. HENDERSON AND ROBERT WELLS

Bin Number

FIG.16. A sequence of estimated slopes for the cumulative distribution function of the correlationdimension (embeddingdimensions 2 through 8). There is convergenceof the values to about 4.0 around bin number 150.

(10,000points ford 4), although present experience (Fraedrich, 1986) is that shorter series may be used to obtain less sharp but still unambiguous results. In some cases, such as the index cycle, there is almost certainly not enough data for a definitive result-a series of lo5 points sampled twice a day requires 137 years! N

4. CONCLUSION In recent years there has been steady progress in the study of mathematical dynamic systems, particularly with regard to the long-term behavior of the solutions. It is an important goal of atmospheric research to find corresponding systems that closely approximate the behavior of the atmosphere. At present it appears that the mathematical study of dynamic systems is sufficiently advanced to give a good “catalog” of various types of long-term system behavior, but it remains for meteorologists to identify the different responses of the atmosphere. It has been the intent of this work to provide some background on the state of mathematical and numerical techniques for estimating the dimension of system attractors. This estimate, in turn, allows for inferences to be made about the model size. Verification of the models has been difficult because of the lack of numerical tools to estimate the attractor sizes from observed data. But recent work from the early 1980s has produced several ideas for attacking

ATTRACTOR DIMENSIONS FROM TIME SERIES

235

this problem. The correlation dimension and limit capacity are two such techniques, and have the desirable property of at least bounding the estimates under the proper conditions. It is the longer term hope that progress in understanding attractor sizes can be coupled with the interesting work of Foias et al. (1967,1983) and Constantin et al. (1985) on the finiteness of system sizes to produce tractable models of atmospheric behavior on all scales of motion in time and space. The numerical techniques that were used to estimate the attractor sizes showed that the mathematical concepts are both viable and useful in practice. When applied to the meteorological data, the methods produced bounds on the attractor sizes. The results suggest a low-order attractor for the large-scale flow, of size 5.0-6.5, while the gust front data showed a local attractor of 4.0-5.5. The global attractor for the gust front could be much higher, perhaps of order 100. Both techniques behaved in the limit as they should, with the limit capacity larger than the correlation dimension. As an application of model reconstruction, the following method of producing forecasts is offered. First, the necessary lag interval z is found from a calculation of the autocorrelations of the time series, followed by the calculation of the attractor dimension and embedding dimension. Next, the attractor is simulated by using a phase space whose axes are the lagged elements of the series, i.e., x(t + z), x ( t + 22), x ( t + 3z), . . . .The number of axes is determined by the embedding dimension at which the attractor dimension converges. The prediction function f is gained by tabulating the future value of the lagged series element as a function of position in the phase space. This may be done by dividing the phase space into cells and then building a frequency table of probabilities for “landing” in a particular cell from an initial cell. This cell concept is easily visualized, but may prove to be prohibitive due to computer memory storage. A simplifying procedure such as a calculation of the mean and variance of the future state may prove to be adequate. This method may be tested first on a known system, such as the one studied by Lorenz (l963), and then applied to an atmospheric example such as the index cycle. The advantage of model reconstruction over such statistical forecasting methods such as ARIMA is that the system attractor has been identified and the forecast can be made to remain near it.

ACKNOWLEDGMENTS We would like to thank Dr. John A. Dutton for suggesting the problem and later, improvements to the manuscript, and Victor King for drafting several of the figures. The thunderstorm gust front data was kindly supplied by Mr. John E. Gaynor of the Wave Propagation Laboratory, Boulder,

236

HARRY W. HENDERSON AND ROBERT WELLS

Colorado. This research was supported by the National Aeronautics and Space Administration under Contract No. NAS8-36150 and by the National Science Foundation under Grant No. ATM83-07213.

APPENDIX: MODELRECONSTRUCTION OF THE CIRCLE WITH SMALLIRRATIONALROTATION

A

Our dynamical system has phase space M equal to the set of unit wmplex numbers and law of motion given by (All

z(z) = ze2nk

where a is irrational, and for simplicity 0 < a < 1/4. Consider the trajectory determined by - e2ni~ (A21 %+I - " We obtain a time series of observations by using the real-part function ('43)

s, = Re(z,)

Write z, = s,

+ it, so that s, is the real part and t, is the imaginary part. Then - t,,sin(2na)

(A41

s . + ~ = s, cos(4na) - tnsin(4na)

(AS)

s,,

= s, ws(2na)

and Using sin(4na) = 2 sin(2na)cos(2na), we may eliminate t, from Eqs. (A4) and (AS) to obtain 2sm+,cos(2xa)- s . + ~ = 2s,cos2(2na) - sncos(4sa)

('46)

or (A71

2s,*~cos(2na)-s,+,-s"=O Thus, the reconstructed space M will lie in the plane defined by the equation 2x, cos(2na) - x3 - x I = 0

(A8)

We have sin(2na) = [I - c0s~(2na)]"~and st + t i = 1. From these two relations and Eq.(A4) we see that M' also lies in the elliptic cylinder defined by the equation x:

- ~ x l x z c o s ( ~ n+a )x i = sin2(2za)

('49)

In fact M' is the planar ellipse equal to the intersection of the plane [Eq. (A8)] and the cylinder (A9)]. Equation (A7) will hold also with n replaced by n + 1, so that s n + 3 = 2s,+ z cos(2na) - sn+ I

(AW

Solving Eq. (A7) for 2cos(2na) and substituting the result in Eq. (AlO), we obtain finally

which implies that the function f(x,,xz,x3) is given by Eq. (13) of the text; that is, by f(XI,X,,X3)

= (XIX3

+ x:

- X:)/Xt

(A12)

ATTRACTOR DIMENSIONS FROM TIME SERIES

237

REFERENCES Abraham, R., and Marsden, J. E. (1967). “Foundations of Mechanics.” Benjamin, New York. Birkhoff, G. D. (1932).Sur quelques courbes fermCs remarquables. Bull. Soc. Math. Fr. 60,l-26. Constantin, P., Foias, C., and Temam, R. (1985). Attractors representing turbulent flows. Mem. Am. Math. Soc. 53, (314). Dutton, J. A. (1987). Modeling: A strategy for understanding. Chapter 1 I n “Nonlinear Hydrodynamic Modeling: A Mathematical Introduction” (H. N. Shirer, ed.), Chap. 1. Lect. Notes Phys. 271, 1-21. Foias, C., and Prodi, G.(1967). Sur le comportement global des solutions non stationnaires des equations de Navier-Stokes en dimension 2. C. R. Semin. Mat. Univ. Pudoua 39, 1-34. Foias, C., Manley, 0. P., Temam, R., and Treve, Y. M. (1983). Asymptotic analysis of the Navier-Stokes equations. Physica 9D, 157-188. Fraedrich, K. (1986). Estimating the dimensions of weather and climate attractors. J. Atmos. Sci. 43,419-432. Grassberger, P. (1986). Do climate attractors exist? Nature (London) 323,609-612. Grassberger, P., and Procaccia, 1. (1983).Measuring the strangeness of strange attractors. Physica 9D,189-208. Lorenz, E. N. (1963).Deterministic nonperiodic flow. J . Atmos, Sci. 20, 130-141. Mandelbrot, B. B. (1983).“The Fractal Geometry of Nature.” Freeman, San Francisco. MHne, R. (1981). On the dimension of compact invariant sets of certain non-linear maps. In “Dynamical Systems and Turbulence, Warwick 1980” (D. A. Rand and L.3. Young, eds.). Leer. Notes Math. (898), 230-242. Nicolis, C., and Nicolis, G. (1984). 1s there a climatic attractor? Nature (London) 311, 529-532. Takens, F. (1981). Detecting strange attractors in turbulence. In ^Dynamical Systems and Turbulence, Warwick 1980” (D. A. Rand and L.3. Young, eds.). Lect. Notes Math. (898), 366-381. Takens, F. (1983).On the numerical determination of the dimension of an attractor. Lect. Notes Math. (1125), 99-106.

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A

power law, 222 trajectory pairs temporally uncorrelated, 221 definition and properties, 212 dimensions, 213-215 bounds for height index data, 230 bounds for vertical velocity data, 233 definition, embedding dimension and,

Absorption, effect on P wave amplitudes, 51

ACE, see Australian Coastal Experiment Advection density, cross-isobath, 192-193 self-advection of density, 190-192 Advection-turbulent diffusion equation, for density field, 182 Aleutian Basin, dynamic topography of sea surface over continental slope, 130 Algorithm construction, model reconstruction and, 215-216 Amplitude, true, of reflected P waves, 37 Amplitude factors, P wave reflection at free surface vs. ocean bottom, 43 at MohorovifiC boundary vs. free surface,

216

Lorenz chaotic, 214 model reconstruction and, 216 periodic motion over a torus, 213-214 quasi-periodic motion, 213 trajectory moving toward a periodic orbit, 213-214 trajectory spiraling in toward steady fixed point, 213-214 fractal dimension, 216-218 approximants, Grassberger-Procaccia inequality and, 221 limit capacity, 224-234 limits, superior and inferior, 217-218 permanent behavior, 212-213 structurally stable, of integer dimension,

46

Amplitude ratios multiple P waves, average, 26 reflected P waves free surface/MohoroviEid boundary, 47, 49-50

free surface/ocean bottom, 44 plane wave fronts and, 51-52 reverberations and, 52 Angle of emergence, reflected P waves, 45-48

Angular momentum, fundamental slope effect and, 136-138 Anosov attractors, 215 ARIMA, model reconstruction forecasting and, 234 Attractors Anosov, 215 correlation dimension, 218-221 approximants, 221 cumulative correlation function, 221-222

Grassberger-Procaccia inequality, 220-221

215

Australian Coastal Experiment, 117 near-bottom current off Newcastle (New South Wales), 119-122 Autocorrelation function estimates height index data, 229 vertical velocity data, 232

B Banyuls (France), temperature and salinity distribution over continental slope, 126

Barolinic transport, 196 in Gulf Stream, 197-198 Bottom intensification, Slope Sea oscillations, 99-100

239

240

INDEX

Bottom pressure distribution, effect of along-isobath density variation, 184-186 equation, 175-176 Bottom pressure curl diagnostic models, 182-184 stratified water column, 180-182 Bottom pressure field calculation, 193-196 offshore from a coastal buoyancy source, I86 Bottom pressure torque, normalized, 136-137 Bottom stress curl mound in homogeneous water, 176-179 pressure torque and, 174-193 shelf-edge boundary layer, 179-180 Boundaries Mohorovifid, see MohoroviEiC discontinuity plane, in reflectionlrefraction theory, 50-51 Boundary conditions reflected P waves, 37 solid-liquid boundary, 40 solid-solid boundary. 40-41 Boundary currents meandering over slope, 105-112 vorticity advection and, 141-143 western, over bottom slope, 196-197 Boundary layers, shelf-edge, 179-180 Bulletins, multiple P waves, 6-7 Burgers’ equation, 191

C

California Current meanders, 111-112 offshore steric heights, 129 Capes flow and vorticity change around, 142 upwelling around, 141 Charleston Bump Gulf Stream meandering and, 149-150 Gulf Stream Rossby wake downstream of, I50 upstream meanders. 111-1 12 Chlorophyll concentration, Baja California coast, satellite image, 131

Coastal freshening, circulation induced by, 185 Coastally trapped waves alongshore propagation of massive flow events and, 115-117 Australian Coastal Experiment, 117 propagating upper slope flow events, 117 stratification effect, 164-169 wind stress and, 115 Cold domes, formation, 111 Cold pool, Midatlantic Bight, 125 Continental reflection, P waves, 50 Continental shelf waves, 96 exponential shelf model, 161-164 exponential shelf-slope profile, 155-158 generation by wind, 158-161 long, 153-155 wind stress and, 153 wave modes, structure, 157 Continental slope bottom stress curl, 174-193 coastally trapped waves, 115-117 deep slope currents Newcastle, New South Wales, 119-122 Tasman Sea, 117 density-driven currents, 122-132 fundamental slope effect, 132-133 pressure torque, 136-138 vortex tube stretching, 133-135 meandering of boundary currents over, 105-112 pressure torque, 174-193 pycnoclines running into, 186-190 Slope Sea, 97-105 steep, insulating effect, 180 topographic waves, 150-174 upwelling and undercurrent, 112-115 western boundary current, bottom pressure torque, 196-197 Continuity equation, in ocean dynamics, 133 Convective overturn, density-driven currents and, 125 Coriolis force bottom pressure field calculation and. 193 vortex tube stretching and, 135 Correlation dimension application to atmospheric data 500-mbar height index data, 227-230

241

INDEX

estimates, 233 length of series and, 233-234 vertical velocity from thunderstorm gust, 230-233 attractors, 218-221 approximants, Grassberger-Procaccia inequality and, 221 cumulative correlation function, 221-222

Grassberger-Procaccia inequality, 220 power law. 222 trajectory pairs temporally uncorrelated, 221 lag period, 222-223 Cross-isobath density advection. 192-193 Cumulative correlation function, 221-222 and correlation dimension, power law, 222

D Deepwater topographic waves, 169-174 Density advection, cross-isobath, 192-193 Density, calculation in ocean dynamics, 133 Density-driven currents convective overturn and, 125 due to river runoff, 122, 125 Labrador Current, 122 Leeuwin Current, 122 uneven isopycnal surfaces and, 125 Density field advection-turbulent diffusion equation, 182

cross-isobath density advection, 192-193 self-advection, 190-192 variation along-isobath, effect on distribution of bottom pressure,

E East Australian Current. radiation of topographic waves, 122 Eddys, transfer of momentum, 197 Embedding dimension, attractor definition and, 216 Energy ratio for multiple P waves, 26 reflected P waves, 38 at liquid-solid boundary, 40 at solid-solid boundary, 41 Epicenter distance diffracted P wave travel time and, 59-61

reflected P waves, angle of emergence and, 45-48 location inacuracies, diffracted P wave travel time and, 70 multiple P waves, 30-31 almost identical, 21 determination, 9 shifts, 30-31 time lag (6T) standard deviation and, 24-25

Equation of motion, see Motion equation Error estimates determination of limits from approximants, 220 limit determination from approximants, 220

Error measurement, time lag (bT) standard deviation and, 22 Estimates, limit capacity, 227

F

184-186

Diagnostic models advection of density by flow and, 188 pycnocline running into slope, 186-190 river inflow, 184-186 Dimension, manifold space, 207 Displacement, reflected P waves, 37 Double P waves, see P waves, multiple Dynamic height parallel solenoids, 183 surface pressure component over sloping bottom, 184

Fennoscandian earthquakes, 32 Focal depth diffracted P waves differences in, 58 travel time dependence on, 61-62 travel time errors and, 70 function, evaluation, 54 multiple P waves determination, 9 frequency distribution and, 14-15 magnitude difference (6rn),27

242

INDEX

Focal depth, multiple P waves (conf.) relation to magnitude, 34 time lag (6T) and, 14-15 time lag (ST)standard deviation and. 22-24

Bight, 1954% cross-isobath flow, 147 horizontal eddy momentum transfer, 197 low-frequency oscillations, categories, 102, 105

reflected P waves, angle of emergence and, 45-48 Forecasts. model reconstruction and, 235 Fractal dimension for attractors, 216-218 approximants. Grassberger-Procaccia inequality and, 221 error estimates, 220 limit capacity, 224-234 limits. superior and inferior, 217-218 Free surface/Mohorovi&d boundary, reflected P wave amplitude ratio, 49-40

Free surfacelocean bottom, reflected P wave amplitude ratio, 48-49 Free surfaces P wave reflection, 38-39 vs. MohoroviEiC boundary, 44-45 vs. ocean bottom, 41-44 stress-free, in reflectionhefraction theory, 51

Frequency distribution focal depth and, 14-15 multiple P waves relation to region. 19-22 time lag (6T) and, 12-14 Frequency ranges, Slope Sea currents, bandfiltering, 102 Friction, fundamental slope effect and, 136

G Grand Banks Labrador Current, 122-123 mean monthly topography of sea surface, 123

Grassberger-Procaccia inequality, 220 for approximants of fractal and correlation dimensions, 221 Grid-point model, Navier-Stokes equations, 207-208

Gulf Stream baroclinic transport, 197-198 bottom pressure over South Atlantic

meanders formation, as source of topographic wave radiation, 169 interfacial pressure torque and, 170-171 over slope, 105-109 meandering, 96 caused by Charleston Bump, 149-150 monthly mean path and envelopes, 109 path, observation and theory comparison, 142

Rossby wake downstream of Charleston Bump, 150 Sea Slope currents and, 102, 105 two-layer model, 170 Gutenberg-Richter data (diffracted P waves) travel time differences, 68-69 travel times, 61-62

H Height, dynamic, parallel solenoids, 183 Height index data attractor dimension bounds, 230 autocorrelation function estimate, 229 correlation dimension and limit capacity applications, 227-230 power spectrum estimate, 229-230 Helmholtz’s theorem, vortex tube stretching and, 134, 135 Hypocenter (multiple P waves) calculation, 31-32 shifts, 29

I Inferior limit, fractal dimension, 217-218 Instruments diffracted P wave travel times, 55 multiple P wave detection, 6 Insulating effect, steep slopes, 180 Isobaths divergence, near shelf edge, 146 equilibrium, sinusoidal meanders, 140 Isopycnal surfaces, uneven, 125

243

lNDEX

J Joint effect of Baroclinity and Relief (JEBAR), 175

K Kinetic energy, Slope Sea oscillation, 99-100

L Labrador Current density-driven, 122 temperature and salinity distribution, 124 Labrador Sea, convective overturn, 125 Labrador Sea Water layer distribution in North Atlantic, 125, 127 potential vorticity at core, 127 Lag period, nature of data and, 222 Law of motion extension to phase space with all complex numbers, 210-212 in model reconstruction, 207 reconstructed, 210 Leeuwin Current, density-driven, 122 Limit capacity application to atmospheric data 500-mbar height index data, 227-230 vertical velocity from thunderstorm gust. 230-233 definition by Takens, 224-225 estimate, 227 numerical calculation, 225-227 vertical velocity data from thunderstorm gust, 231-233 Limits, of fractal dimension, 217-218

M Magnitude multiple P waves, 20 determination, 9-10, 31-32 frequency distribution and, 15-19 magnitude difference (Sm)dependence on, 27 relation to focal depth, 34

time lag (6T) and, 15-19 time lag (67') standard deviation and, 24

Magnitude difference (Sm) multiple P waves, 25-28 average, 26 focal depth and, 27 magnitude (m) and, 27 region and, 27 relation to region, 34 Manifold space, 207 Mantle base, P wave velocity at, 69-70 MASAR, see Mid atlantic Slope and Rise experiment Massive flow events, alongshore propagation, 115-117 Meandering boundary currents satellite images. 107-110 over slope, 105-112 vorticity advection and, 141-143 Gulf Stream, 96 caused by Charleston Bump, 149-150 Meanders California Current, 111-112 Charleston Bump, 110 cold dome formation, 111 onshore, 111 sinusoidal, about an equilibrium isobath, 140

three-dimensional structure, 111 upstream of Charleston Bump, 111-112 Mid atlantic Bight cold pool, 125 Slope Sea, 97-105 Mid atlantic Slope and Rise experiment, 100, 103

Mid-slope quiescence, 100-102 Model reconstruction algorithm construction and, 215-216 ARIMA forecasting method and, 234 attractor definition and properties, 212 attractor dimension and, 216 circle with small irrational rotation, 236 forecast production, 235 law of motion, 207, 210 observables, 207 phase space, 207, 210 Takens embedding theorem, 206-209 verification, 234-235

244

INDEX

Models P wave reflectionhefraction at free surface vs. ocean bottom, 42 P wave reflection, true earth conditions and, 51 MohoroviGf discontinuity P wave reflection amplitude factors, 46 amplitude ratio free surface/MohoroviEiC boundary, 49-50 free surfaces and, 44-45 travel time difference (at) between freesurface reflected waves and, 47 Momentum transfer, horizontal eddy, 197 Motion equation damped, periodically forced pendulum, 207

in ocean dynamics, 133 Multiple P waves, see P waves, multiple

N NASACS, see North Atlantic Slope and Canyon Study Navier-Stokes equations, 208 solutions, grid-point model, 207-208 Newcastle (New South Wales), near-bottom currents, 119-122 New England, slope sea, see Slope Sea New Zealand continental slope, current records in deep water, 120 South Island, deep slope currents, 117-119 North Africa, upwelling region, 113-115 North America (west coast) satellite image, 129-131 sea level, alongshore variation, 128 steric heights, 128 North Atlantic, distribution of Labrador Sea Water layer, 125, 127 North Atlantic Slope and Canyon Study, 100

Norwegian Sea, convective overturn, 125

0

Observables generic, 208 in model reconstruction, 207

natural, 207 Ocean bottom P wave reflection, 50 P wave reflection vs. free surfaces. 41-44, 48-49

Oregon, upwelling region, 113-115 Origin time diffracted P waves, inaccuracies in, 70 multiple P waves calculation. 31-32 determination, 9 shifts, 29

P Parallel solenoids, field of, 183 Path effects, travel time of diffracted P waves, 70-71 Ptclet numbers, 176-177, 181, 183 Pendulum, damped and periodically forced, motion equation, 207 Periodic behavior attractor motion, 215 multiple P waves, 28-29 Perturbation analysis, topography effects on flow over slopes, 143-146 Peru propagating upper slope flow events, 117 upwelling region, 113-115 Phases, multiple P waves, 28-29 Phase space all complex numbers, 210-212 filled up, 209-210 model reconstruction and, 207 reconstructed, 210 Plane wave fronts, P wave amplitude ratios and, 51-52 Pn waves, double, multiple P waves and, 32 Pockets, upwelling, 96 Points, upwelling around, 141 Poisson ratio, amplitude ratios of reflected P waves and, 38-39, 48 Poleward undercurrent, 112, 115 Potential vorticity, slablike layer, 135 Power law, correlation dimension and cumulative correlation function, 222 Power spectrum estimates height index data, 229-230 vertical velocities from thunderstorm gust, 232

245

INDEX

Pressure fields calculation, 133 diagnostic models, 182-184 pycnocline tilting alongshore, 189 Pressure torque, 136-138 bottom, 197-198 bottom stress curl and, 174-193 diagnostic models, 182-184 friction, 136 interfacial, Gulf Stream, 170-171 Jacobian of depth and bottom pressure, 136

momentum advection, 136 mound in homogeneous water, 176-179 normalized bottom, 136-137 planetary vorticity advection and, 193-198 balance between, 194 baroclinic transport, 1% shelf-edge boundary layer, 179-180 stratified water column, 180-182 under western boundary current flowing over a continental slope, 196-197 P waves, diffracted earthquake selection, 56-57 focal depth differences. 58 focal depth function, 54 methodological problems number of observations, 71 path effects, 70-71 receiver effects, 70-71 source effects, 70 observations, 57-59, 71, 80-89 propagation path in spherical earth model, 54 seismographic records, 56 stations, 55-56 total angular distance and, 59-61 travel times dependence on focal depth, 61-62 dependence on total angular distance, 59-61

determination methods, 55 differences with Gutenberg-Richter data, 68-69 diffracted S waves and, 72 equation, 53-54 Gutenberg-Richter data, 61-62 summary of findings, 72-73 f(A,h)solutions, 64-65 f ( A ) solutions, 62-64 Swedish data-based, 65-69

velocity at mantle base, 69-70 P waves, multiple amplitude ratio, 26 bulletins, 6-7 criteria, 7-9 data selection and grouping, 32-33 energy ratio, 26 epicenters almost identical, 21 epicenter shifts, 30-31 epicentral coordinates, 9 Fennoscandian earthquakes, 32 focal depth and magnitude relation, 34 focal depths, determination, 9 instruments, 6 later multiples, 29 least squares solutions, 33 magnitude, determination, 9-10 magnitude difference (Sm), 25-28 observational material, 9-10, 74-79 origin times, 9 Pn waves, double, 32 pP waves and, 8-9 rupture velocity, 31 seismic moment ratio, 26-27 shifts in origin time and hypocenter location, 29 signal-to-noise ratio and, 11 Sn waves, double, 32 source calculations and, 31-32 stations, 5-6 statistics from stations, 11-12 summary of results, 34-35 time lag (6") focal depth and, 14-15 frequency of occurrence, 12-14 magnitude and, 15-19 region and, 19-22 standard deviation. 22-25 triple P waves, 12 wave periods and phases, 28-29 P waves, reflected amplitude ratio, Poisson ratio and, 38-39, 48

boundary conditions, 37 condition for energy conservation, 39 continent and ocean reflection, 50 displacement, 37 a t free surface, 38-39 at free surface vs. MohorovifiC discontinuity, 44-45

246

INDEX

P waves, reflected (conf.) at free surface vs. ocean bottom, 41-44 amplitude factors, 43 amplitude ratio, 44 structural models, 42 reflection/refraction theory absorption, 51 models, 51 plane boundaries, 50-51 plane wave fronts, 51-52 reverberations, 52 stress-free surfaces, 51 Snell’s law, 37 at solid-liquid boundary, 39-40 at solid-solid boundary, 40 stress, 37 SV wave propagation, 37 true amplitude, 37 wave propagation equation. 37 P waves, refracted energy, 38 at solid-liquid boundary, 39 at solid-solid boundary, 40 Pycnoclines running into slope, 186-190 tilting alongshore, pressure field. 189

Q Quasi-periodic motion, attracton, 213, 215 Quiescence, mid-slope, 100-102

R Radius of deformation, 166 Receiver effects, on diffracted P wave travel time., 70-71 Reflected P waves, see P waves, reflected Region (multiple P waves) magnitude difference (6m)and, 27, 34 time lag (6T) and, 19-22, 34 Reverberations, P wave calculations and, 52 Rigid lid approximation, 152 River inflow, model, 184-186 River runoff, coastal currents driven by density differences due to, 122, 125

Rossby wake Gulf Stream, downstream of Charleston Bump, 150 seamount, 146-149 Rupture velocity, multiple P waves, 31 time lag (6T) and, 19

S

Salinity distribution over continental slope of Banyuls, 126 Labrador Current, 124 Satellite images Baja California chlorophyll concentration, 131 meandering of boundary currents over slope, 107-110 North American west coast, 129-131 Scale analysis topographic waves, 151-153 transport streamfunction. 153 Sea level, alongshore variation, 128 Seamounts, Rossby wakes, 146-149 Sea surface, Aleutian Basin, dynamic topographic over continental slope, 130 SEEP experiment, see Shelf Edge Exchange Processes experiment Seismic moment ratio, average for multiple P waves, 27 Self-advection, of density, 190-192 Sensitivity, stations recording multiple P waves, 12 Shelf circulation theory, modeling, 176 Shelf Edge Exchange Processes (SEEP) experiment, 100-103 Signal-to-noise ratio, P wave magnitude and, 11 Slope Sea currents along-isobath velocities, 97 amplitude of superimposed oscillations, 97 band-filtering into frequency ranges, 102 bottom intensification of oscillations, 99-100 cross-isobath velocities, 97 frequency of relatively strong and especially weak currents, 102 under Gulf Stream, 99

247

lNDEX Gulf Stream currents and, 102, 105 kinetic energy of oscillations, 99-100 mid-slope quiescence, 100-102 mooring locations, 98 phase differences, 97 Shelf Edge Exchange Processes (SEEP) experiment, 100-103 spectral gap between inertial and lowfrequency motions, 102 upper, mid-, and lower slope regions, 100-101

upper slope current transport, 129 wind band, 102 velocity variance in spectral bands, 102,

models, 208 Superior limit, fractal dimension, 217-218 SV waves, propagation, 37 SV waves, reflected at a free surface, 38-39 at solid-liquid boundary, 39-40 at solid-solid boundary, 40 SV waves, refracted at solid-solid boundaries, 40-41 S waves, diffracted, diffracted P wave travel times and, 72 S waves, reflected, energy, 38 S waves, refracted, energy, 38

108

Snell’s law, 37 Sn waves, double, multiple P waves and, 32 Solid-liquid boundaries, P wave reflection and refraction, 39-40 Solid-solid boundaries, P wave reflection and refraction at, 40-41 Source effects, on travel times of diffracted P waves, 70 South Atlantic Bight Charleston Bump, 110 upstream meanders, 111-112 Gulf Stream bottom pressure, 1954% Squall line, topographic wave pattern behind, 171 Standard deviation, time lag (6T) of multiple P waves, 22-25 Stations diffracted P waves, 55-56 multiple P waves, 5-6, 11-12 Steric height continental slope and, 127-129 forcing of shelf-slope region, 128-129 variations along continental margins, 129 west coast of North America, 128 Stratification and bottom slope, diagnostic approach, 182-184

effect on coastally trapped waves, 164-169 Stratified water columns, pressure torquebottom friction balance, 180-182 Streamline fields, diagnostic models, 182-184

Stress, reflected P waves, 37 Structural stability attractors of integer dimension, 215

T Takens embedding theorem, 206-209, 213, 216

Temperature distribution over continental slope of Banyuls, 126 Labrador Current, 124 Time lag (6T),15-19 multiple P waves focal depth and, 14-15 frequency of occurrence, 12-14 magnitude and, 15-19 region and, 19-22 relation to region, 34 rupture velocity and, 19 standard deviation, 22-25 region and, 19-22 Topographic waves, 96 deepwater, 169-174 generation, 170-1 74 interfacial pressure torque and, 170 velocities at a fixed point, 173-174 wave pattern behind squall line, 171 exponential shelf-slope model, 161-164 exponential shelf-slope profile, 155-1 58 generation of continental shelf waves by wind, 158-161 long continental shelf waves, 153-155 radiation East Australian Current, 122 Gulf Stream meander as source of, 169 scale analysis, 151-153 stratification effects, 164-169 Topography, irregular, vorticity advection and, 143-146

248

INDEX

Toroidal surfaces, higher-dimensional, attractors and, 213, 215 'Itansport streamfunction. mound in homogeneous water, 176-179 Travel time (diffracted P waves) dependence on epicentral distance, 59-61 dependence on focal depth, 61-62 epicenter location inaccuracies and, 70 equation for, 53-54 focal depth errors and, 70 Gutenberg-Richter data. 61-62, 68-69 origin time inaccuracies and, 70 summary of findings, 72-73 Swedish data-based, 65-69 ?(Ah) solutions, 64-65 t(6) solutions, 62-64 Ravel time difference (St) MohoroviEi6 discontinuity-reflected vs. surface-reflected P waves, 45 reflected P waves, free surface/MohoroviW boundary, 47 liveak, streamlines of, 171-172

U Undercurrent poleward, 112, 115 upwelling and, 112-115 Upwelling around capes and points of California coast, 141 Northwest African area, 112-113 Oregon, North Africa, and Peru, comparison, 113-115 Upwelling pockets, 96

V Velocities P waves at mantle base, 69-70 vertical (thunderstorm gust)

correlation dimension and limit capacity algorithms, 231-233 estimated autocorrelation function, 231-232

power spectrum, 231-232 Vortex tube stretching, 133-135 column stretching rate. 134 Coriolis force and, 135 distribution of horizontal velocity, 134 due to surface elevation, 152 fundamental slope effect, 135 Helmholtz's theorem, 134-135 potential vorticity of a slablike layer, 135 vorticity advection and, 138-149 irregular topography and, 143-146 meandering of boundary currents, 139-141

upwelling, 141-143 Vorticity advection planetary pressure t o q u e and, 193-198 in vorticity balance, estimation, 197 vortex tube stretching irregular topography effects, 143-146 meandering of boundary currents, 139-141

Rossby wake of a seamount, 146-149 upwelling, 141-143

W Warren's model, bottom slope effects on ocean currents, 139-141 Water mass drainage, Labrador Sea Water layer, 125, 127 Wave propagation P waves, 37 SV waves, 37 Wind band, Slope Sea, 102 Wind stress coastally trapped waves, 115 generation of continental shelf waves and, 158-161 long continental shelf waves and, 153