JPRS ID: 9917 USSR REPORT EARTH SCIENCES

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JPRS L/9917
18 August 1981
USSR Report
EARTH SCIENCES
(FOUO 6/81)
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JPRS L/9917
18 August 1981
USSR REPORT
, EARTH SC I ENCES
- (FOUO 6/81)
CONTENTS
OCF.ANOGRAPHY
Yastrebov
Outlines Progress in Uiiderwater
Research
1
Influence
of Internai Gravitational Waves
on the Spectrum of Wind Waves.
9
Approximate Solutions for Internai Waves in a Medium With an Arbitrary
Stability Distribution 20
Investigation of Internal Waves and Mesoscale Variability of Currents in
the Equatorial Atlantic 29
Table of Contents From 'Marine Hydrophysical Research' No 1(88), 1980 36
Large-Scale and Synoptic Variability of Fields in the Ocean 38
Collection of Articles on Marine Geology, Geophysics, Physics and Biology 40
Several Applications of Parametric Antennas in Oceanographic Research... 44
TERRESTRIAL GEOPHYSICS
Gravimetric Studies of the Oceanic Earth's Crust 48
PHYSICS OF ATMOSPHERE
Spectral and Integral Clearness of Atmosphere Over Lake Baykal.......... 50
- a- [III - USSR - 21K S&T FOUO]
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OCEANOGRAPHY
_ YASTREBOV OUTLINES PROGRESS IN UNDERWATER RESEARCH
Moscow SOVIET UNION in English No 5(374), May 81 pp 16-19
[Interview with Prof Vyacheslav Yastrebov, doctor of technical sciences, deputy
director of the Institute of Oceanology, USSR Academy of Sciences, by SOVIET UNION
correspondent A. Chuba; date and place not specified: "The Ocean: 56 Steps Into
the Deep"J
[Text] The testing of the "Oceanolog", a new manned underwater
vessel, was completed in the autumn of 1980 near Novorossiisk.
It moved about easily in the water, manoeuvred and handled well,
and, when necessary, could hover over a g3ven point. Its arm--
an electro-hydraulic trailing manipulator--confidently held a
geologist's hammer, scoop, ground pipe for taking samples, and
other instruments. Duriag one of their descents the hydronauts
--Comma.nder illadimir Gromov, mechanic Nikolai Tokovenko and
observer Yuri Belyayev--reached a depth of 609 metres, a record
for this type of apparatus. The "Oceanolog", developed by de-
signers from the USSR Min?stry of the Fishing Industry in ful-
fillment of an order of the Far Eastern Scientific Centre of
the USSR Academy of Sciences, became the 56th device built in
this country for underwater experiments and projects.
Soviet Union correspondent A. Chuba interviewed Professor
Vyacheslav Yastrebov, D.Sc. (Techn.), deputy director of the
Institute of Oceanology of the USSR Acauemy of Sciences, who
discusses the tasks facing researchers of the ocean depths,
- and how modern technology is making it possible to tackle them.
The exploration of the ocean dept}is is often compared to the conquest of outer
space. There is a measure of logic to this comparison, as the world ocean, like
the expanse of outer space, has an enormous influence on many earthly processes,
and it is just as difficult for man to penetrate it; as is the case with outer
space, he has to create special apparatuses with reliable life-support systems.
Regrettably, we know the ocean's depths, which man has been trying to penptrate
since time immemorial, only s.lightly better than outer space, exploration o2' which
was launched a quarter of a century ago. Meanwhile, ma.n has no less a vital in-
terest in the ocean than in the extraterrestrial expanses. The exploration of the
1
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' world ocean is directly connected with the solutian of such crucial issues facing
the planet today as the food, energy and raw materials problems.
From the standpoint of science, priority in researching the ocean depths should'
be relegated to the prominent oceanologist Jacques-Yves Cousteau, the inventor of
the aqualung and zhe famous Divins Saucer research minisubmarine. From 1962 to
1965 he headed the Precontinent Programcne, which launched underwater research
projects according to plan. Groups of aquanauts who lived on the ocean floor in
special houses (the idea of the diving bell, known since the times of Alexander
of Macedon, was used in their construction), conducted systematic observations of
the inhabitants of the sea, collected geological specimens and carried out trial
assembly and exploitation of equipment for sea oil rigs.
In the Soviet Union, a large part of whose borders lies on seas and oceans and
which therefore has a vital stake in their exploration, such resedreh began ap-
proxima.tely at that time, in 1966, with the Ikhtiandr programme. Fur three years
in a row various types of underwater abodes were tested along the Crimean shores,
and medical experiments made. Later other Soviet programmes, including the
- Chernomor programme, prepared by scientists from our institute, were implemented
as well. It had a purely oceanological thrust. The research project, in which
oceanologists, marine biologists, geologists, hydrophysicists and hydro-opticians
took part, showed the high level of the efficiency of the underwater structures
_ for conducting a wide range of operations and experiments.
To be used for these purposes now is a,diving complex which is installed on the
"Vityaz" research vessel, the successor of the famous "Vityaz", aboard which
Soviet scientists discovered life at a depth of some ten kilometres and pinpoint-
ed the deepest spot in the Mariana Trench near the Island of Guam (11,022 metres),
and carried out other important research pro3ects. The new "Vityaz" will make
its first trip this year.
A special arm of underwater r_echnology is equipment which is capable of moiring
about underwater, descending to the bottom and doing various types of work there.
The underwater apparatus enables the scientist to be an eyewitness to processes
occurring in the deep, to the life of its denizens, and to collect specimens and
take photographs. Many interesting structures have been built in this country and
abroad over the past 20 years. These are the American underwater research ship,
the "Ben Franklin", which completed a drift in the depths of the Gulfstream, the
- French bathyscaph "Archimede", the Japanese "Shinkai" and many others. Several
of them, such as the "Trieste", which reached the bottom of the Mariana Trench,
were "preprogrammed" to conquer record dept?-s and can function solely as a kind
of lift. Meanwhile the range of tasks the underwater devices have to hand'Le is
growing from one year to the next. They are now servicing ocean oil rigs, pros-
pecting minerals and fish resources, doing rescue and repair work, and carrying
out a series of oceanologa.cal research projects. It is only naturai that like
"land" equipment, they should have their own trucks and passenger vehicles,
mobile laboratories and all-purpose vehicles, and other specialised means.
A whole series of manned underwater apparatuses has been developed and is being
used in the Soviet Union. These are the "Sever-2", "Shelf-1", "Tetis", "Argus",
"Tinro-2", "Bentos-300", "Atlant-1" and "Atlant-2", "Oceanolog" and others.
2
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The "Tinro-2" undeYwater
research boat, designed
for scientif.ic work and
observation of schools
of fish and of commercial
fishing nets.
4
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Once every two years countrywide seminars for designers of
underwater equipment for research are held in Gelendzhik, on
the Black Sea. Above: the "Argus" manned undezwater apparatus,
and the "Skat" second-generation aqua-robot. Below: the
"~ianta-1.5" remote-controlled manipulator.
S
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All of them iire different In terms of size, depth of descent and self-sufficiency.
Some, for instance, are designed for plar.ting underwater "orchards" of seaweed;
others help fishermen; still others arE adapted for taking precise measurements
and doing scientific work.
The "Bentos-300", for example, is one of the world's largest underwater research
vessels, with a 14-man capacity. It has a displacement of up to 500 tons, and cazi
operate underwater independently for 14 aays. It is equipped with a device for
ejecting a diver deep below the surface. The "Bentos" makes it possible to do
extensive research for the fishing industry, and it is used in various fields of
scientific research as well.
The deep-underwater apparatus "Sever-2" possesses great possibilities. Five re-
searchers can descend to a depth of 2,000 m and, taking advantage of the great
manoeuvrability of the vessel, observe the movement of schools of fish and the
operation of fish nets, and conduct various tests. The apparatus is equipped with
five independent emergency ascent systems, and an effective fire-fighting system.
The actual presence of a scientist in the ur.derwater world has done a great deal
to change and enrich our conception of the life of the ocean. It has turned out,
for example, that trawlers catch far from all the types of fish inhabiting the
deep, and the devices lowered from a vessel to take samples of the bottom give an
incomplete picture of the composition of the ocean floor. Underwater apparatuses
have enabled scientists to conduct research impossible with other methods, and to
obtain new information about the structure of the earth's crust, the movement of
mainlands and the earth's geological and historical past. Small apparatuses, such
as the "Argus", designed at our institute, are usually used for doing such re-
search. The "Argus" is six metres long and has a displacement of 8.5 tons. The
three-man crew can descend to a depth of 600 m and work underwater for eight hours.
There are situations, however, when using a manned vessel is either impossible or
irrational. In such instances its functions are assumed by an automatic device,
a kind of oceanic robot. The underwater robots have invariably been a part of
Soviet oceanological expeditions in recent years. Our institute's remote con-
trolled apparatuses the "Crab", "Manta-1.5", "Zvuk-4m", "Zvuk-6" and others can
operate at great depths. The underwater devices successfully showed what they
- can do during last year's expedition, which analysed zones of fractures in the
earth's crust (rift zones) in the Red Sea. They enabled scientists to put togeth-
er a unique collection of samplPS of Red Sea basalts, take more than 4,000 photo-
graphs and shoot a long-length video film. The scientists discovered stones on
the bottom on which there is practically no sediment, "pipes" of fresh volcanic
lava, and mountains split from foot to peak. This means that today, too, move-
ments of the ear`th's crust continue, and processes are taking place which are
shaping the face of the planet.
Our institute has at its disposal all types of underwater vessels: manned, towed,
remote controlled with manipulators, robots. We are soon to add the "Skat"--a
second-generation aqua-robot. For the time being there is no information about
the existence of such robots in other countries. Unlike the remote-controlled
apparatusAs, there is no cable between the mothership and the "Skat". S1.1 the
robot's actions, depending on the floor's relief and the outside situation, are
7
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- guided by a built-in computer. The "Skat" has a"memory"; it can acquire elemen-
tary skills, independently adapt to the atmosphere on the bottom and take photo-
graphs.
The "Skat" is the latest rung in the la3der of man's exploration of the ocean
~ depths. He has made a start, but there is still a long way to go.
COPYRIGHT: "Soviet Union" 19�31 CSO: 1852/5
8
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UDC 551.466.2
INFLUENCE OF INTERNAL GRAVITATIONAL iJAVES ON THE SPECTRUM OF TJItdD WAVES
Sevastopol' MORSKIYE GIDP.OFIZICHESKIYE ISSLEDOVANIYA in Russian No 1(88),
Jan-Mar 80 (manuscript received 17 riar 80) pp 44-55
[Article by Yu. M. Kuftarkov and V. N. I:udryavtsev]
[Text] Abstract: The transformation of the spectrum
cf wind waves under the influence of a vari-
able current induced by an internal wave is
examined. The spectral density of the wave
effect is described by a kinetic equation
whose right-hand side includes a Miles gen-
eration source with nonlinear limitation of
amplitude. Under definite conditions it is
possible to obtain an analytical solution in
the phase plane region corresponding to trapp-
ed waves. The transformation of the Phillips
spectrum under the influence of an internal
gravitational wave is illustrated.
Interest in study of evolution of the spectrum under the influence of a current
induced by an internal wave is associated both with the problem of sensing of in-
ternal waves by remote methods and the need for a detailed investigation of the
mechanisms of generation of wind waves. At the present time there is no finalized
theory which could predict the evolution of the spectrum of wind waves in the field
of internal gravitational waves. Theoretical. investigations in this direction have
been limited to one degree or another. For example, in [1] in an adiabatic approx-
imation a study was made of the effect of modulation of the parameters of surface
waves on a current induced by a long wave. In this study it was shown that the
most significant changes in the parameters of the wave packet occur in a case when
its group velocity cg is close to the phase velocity c of the internal wave. The
concept of blocking points, in which c$ = c- U(U is the velocity induced by the
internal wave) is introduced. These points are kinematically limited, beyond which
propagation of the wave packet is inipossible. It was postulated that at these
points the packet is present for an unlimited length of time, so that the energy
ir.creases expoLtentially without limitation due to the work of radiation stresses.
In later studies [4, 5] it w3s shown that the blocking points are the points of
reflection of wave packets and therefore the singularity associated with an unlim-
ited increase in energy is excluded. A detailed analysis of evolution of the para-
meters of an adiabatic wave packet with allowance for reflection and capture by
an internal wave was given in [3]. A more realistic situation is examined in [7],
9
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where a wave generation source (Miles generation mechanism) was added to the spec-
tral equation. However,' no consideration is given to wind waves with a group velo-
city close to the phase velocity of the internal wave.
This study is devoted to an investigation of the reaction wtiicii can be nctecl in a
wind wave spectrum in the gravitational interval to internal waves. In contrast
to the mentioned studies, there are no limitations on the region of wave numbers
of surface waves, on the one hand, and on the other hand energy gains and losses
itave been included in the spectral equation. Equations of P4odel
~ The evolution of the spatial spectrum of the wind effect ~v'on a variable current
_ caused by internal waves is described by the equation
- aw + 032 vO/ _ a4 aw _ q . (i)
'
dt dk~ d,r~ dxA d,~~ W
uzhere yJ= T/liJ ;9 is the spectral density of energy; Wis frequency in the reckon-
ing system in which the medium is locally at rest; R (k, ; t) is the known disper-
sion relationship; Q is a function modeling concentrated effects for a particular
spectral compo nent.
The explicit form of Q is determined in each specific case in dependence on the de-
gree of interaction of the wind wave components with one another and with different
- movements both in the ocean and in the near-water layer of the atmosphere.
In accordance with [7], we will represent the source Q in the form
a = d~ - d wz , (2)
(d ~1o
where J( is the wind-wave interaction coefficient. The second term in (2) is a non-
linear limitation of the exponential energy increase and determines the existence
of some stationary spectrum 9/c(k).
For a full
waves it is
to equation
wind waves in a nonuniform current induced by internal
a kinematic relationshig for the wave vectors be added
dkQ as dA,6
t
at a~r~ axA ax~ aX,, ax~
(3)
Then U= U(x - ct) and as a simplification we examine a one-dimensional case: the
x-axis is directed along the propagation of surface waves and the internal waves
can be both codirectional (phase velocity c> 0) and oppositely directed to the
surface waves (c < 0). The case of codirectionality is most complex for analysis
in connection with the phenomenon of capture of surface waves by internal waves.
The opposite.variant causes no analytical difficulties and will be examined below
for a possibly greater completeness of the model.
Instead of equations in partial derivatives (1), (3) we will write an equivalent
characteris tic sys tem of ordinary differential equations [expression (6) is given
in a coordinate system movicig with the phase velocity of an internal wave]
10
description of
necessary that
(1)
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~
A~o(yr-
dt y/o (4)
dk 03? dx 10 8 (5)
,
dt dx dt dA
Sl=(gk)'2 + A (U r) � (6)
In completing formulation of the problem we will write the initial. conditions
S=So~k~~ ~ k'A,� x-x* ' (7)
Leaving to one side for the moment the problems associated with the kinematics of
the wave packets, we will write a solution for
the spectral density of the wave
effect ,
t
d(
~ lh
't
d
f
4(A
[
fd~h,x, t~
f
p dk
dt
t . ~8~
-
eV
)a
2 ]
,
t~
With (5) and (6) taken into account,
the latter formula assumes the form
~
+
(
t
dU
f
~
d
/
!
t
] dt . (9)
) exp f d(k )dt
A)
f~(k,x~ t
Wo
o dX
d
k
(k)

z i
o
t,
This solution was represented
in [7],
but, as was noted above, it was analyzed
only for the region of wave numbers s
atisfying
the condition
G
~k~-C
I
y
I
N 1 .
0
Kinematics of Wave Packets
The problem of the motion of a wave packet in phase space includes both a deter-
mination of the trajectory of motion and the position of the packet on the trajec-
tory at each moment in time. Along the characteristics (S) dj`l/dt = a~/a t`= 0
and accordingly the equation
1/1 + (10)
determines the type of trajectory in phase space.
lde will eYamine a case when the velocity U induced by the internal wave has the
form
U= U,J c o s
(11)
Using formula (11), using expression (10).it is easy to canstruct the trajectory of
- wave packets for the case c.> 0 in the phase plane (k, x) (Fig. 1). The closed curves
correspond to trapped waves and the unclosed curves correspond to untrapped waves.
The phase plane region situated above the dashed line correspcnds to "retrograde"
waves (0y'c/d k< 0), that is, wave packets which in the considezt~d coordinate system
- move in the negative direction of the x-axis. The region situateki below the dashed
line corresponds to direct waves (cJR/ a k> 0) which are propagated in a positive
11
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direction. The intersecttons of the dashed line, determined by the equation
~ \ j~\1/2 _ c+ Um co3 kix = 0,
i
with the trajectories of the wave packets are the "turning points," that is, the
_ points of inerging of the straight lines and the "return" branches of equation (10).
L'/G.
Fig. l.
A detailed analysis of the linearized dispersion relationship for trapped waves
was Kiven earlier in [3]. However, the linearization carried out for the Um/c
parameter in tlie last analysis led to great restrictions of the Um/c value (Um/
- c< 10-2). Internal ocean waves are characterized by Um/c, 10'"1. Accordingly, in a
;,eneral case in the analysis of the kinematics of wave packets in the velocity
12
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f:iel.d induceci by aii inLernal wave it is necessary to use a nonlinear dispersion
relationship.
We will determine the position of the wave packets on the trajectory in tjme. [n ii
general case it is impossible to make an analytical determination of k(t) and x(t)
from system (5), (6). However, some asymptotic solutions meriting special atten-
tion can be found.
We will examine the region of the phase plane corresponding to the condition [in a
- case when surface and internal waves have opposite directions the entire phase
plane satisfies this condition] I ~ ~1 1 ` ,f - G
N ! .
~
Using an ordinary asympt.otic scheme, we represent k and x in the form
k(t), x~~) = XW x ct~ ~ c12,
where~x (t) = xo +[cg(k0) - c] t. It is easy to demonstrate that the disturbances
k and x have the order of magnitude 0(U/cg - c), and equations (5), (6) can be
written in the form
dk dU ~ U
dt � (dx ) X [ C - C '
C J
9 (13)
N ~
dt 1 o w.R (c
12 J.
9
Substituting the solution of this system into the expansions (12), we obtain
k(t)=Ao{I- U'n [cosk~x-coS~ x,9 -
t c9 (k, ~ C
A Um A
x(t x* Csin hi x- sihA~ x, ~
t c9 Um cos h` xC. .
l 1 C9~~o~ -C J Z Cy(ko) -L
Formulas (14), (15) give the position of the wave packet on the phase plai.ie in
time. Expressions (14), (15) were cited earlier (but with a different order of
accuracy) in [7]. With cg c), which corresponds to fast surface waves on a slow,
long internal wave, the degree of variability of the wave number, as can be seen
from (14), is equal to U/cg [6].
13
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It was not possible to obtain an analytical solution in the region of the phase
p].ane corresponding to trapped waves. However, as will be demonstrated below,
this circumstance in a number of cases is not fundamental in an investigation
of evolution of the spectrum of surface waves interacting with the wind.
- Evolution of Wave Packet Spectral Density
The spectral density of the energy of wave packets in the region of untrapped
waves (lcg - c/clw 1) is determined from expression (9) with the use of formulas
(14), (15). For a specific field U= Um cos ki x the expression for spectral den-
sity (with x t-y oo) is given in [7] and has the form
r~� U d(ho)IAIX. cos(k~x+B). j(~)21
+d K Gk� 1+0 , (16)
where d (ko)/C~1i
B= arct y
is a value determining the phase shift associated with interaction of the wave
- packet and the wind; CJi is the frequency of the internal wave. In expression
(16) the variables k and x are functions of time and determine the position of
the packet on its trajectory. We note that in the case of an opposite direction
of surface and internal waves the solution (16) is correct for any k.
We will ascertain the spectral density (P of wave packets trapped by the internal
wave. For nonblocking wave packets the wave number disturbances are small
[0(U
~8C This was taken into account in the derivation of expressions (13), (16). For trapp-
ed waves (Fig. 1) the wave niunber of the packet can change its order of magnitude.
Accordingly, the asymptotic scheme of the expansion (12) is not applicable for the
evolution of the spectral density of trapped waves. However, this difficulty can
be overcome if use is made of physical considerations associated with the relation-
ship between the rates of change in the energy of the wave packet during its inter-
action with the wind and an internal wave.
An analysis of the integrand in (9) gives basis for assuming that in a number of
cases the adiabatic function
4t X
~ _ ~ aU d ( 1
~ x dk \ wo ( )
a
is slow relative to the fast exponent. In other words, the adiabatic changes in spec-
tral density for the region of trapped waves are slow in comparison with the char-
acteristic time of surface wave generation.
Assume that T and aC-1 are the characteristic times of adiabatic and wind variabil-
ity of the packet respectively. Then the preceding assertion is correct with satis-
faction of the condition 0/_-1/T � 1.
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For an evaluation of the coeff.icient of interaction of the wind with surface waves
we will use the empirical approximation
10-2 (g/k-`~ r.�-- [7].
We will determine the evaluation 0C -1 and the understated evaluation T:
oC/w ~ /0-j G" . T - wj~~ .
Then with 41i:< 10-4�gW/c2 the ratio 0(--1/T � l. In the latter expressions W is
the wind velocity related to d namic velocity U* in the follocaing way: U2 10' 3,J2.
ror the phase velocity value c~ ^'103cm2�sec-~, charactEristic for internal ocean
waves and wind Wr-5�102 cm�sec'1, the upper value of the frequency is4liP.r 5�102
rad� sec-l.
Thus, for the frequencies of internal waves satisfying w.~`JYP the integrand adia-
Latic function Oin (9) is "slow" relative to the "fast" exponent. This makes it
possible to expand 0 (k, x) and
t
10,/(k) dt2
tl
into a Taylor series in the neighborhood of the time t determining the position of
the packet on the phase plane
Ot' ~t - t) -D 01,Y] I
d
clt I t=t ~ \d
~ ~ . (17)
t
S ~(k)dt2 =-d ~,~(t)~ (t~- t) +
d G
_ where tl
~ 11 d xaU d / l dk . (ls)
I
d 0_ vU d (A
- dndACdt
dt dx dk \dx ~'o
Substituting formulas (17) into expression (9), we obtain, after integration and
proceeding to the limit with Oe*t--;~- o0
1 / 0 - / d~ ) + ~ ~C U J.
- ~ ~ c
~ V. Cc 1 / - dm dt
Tlle l:itter fcrmula, with the use of (18), can be written
4/o aU d( C L d ZU d([~/c13 lZl,
--k-(tlz) (-~-I1/7, 1+0~maxd) a (19)
d dx dK ~ d~G /dx 1
ror a specific velocity field stipulated by expression (11) for the spectral den-
sity of the wave effect we obtain the expression
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w=y~,+ ~tld~ \C _~}l zA ~cos(k~ x
dk (20)
This solution is correct for trapped wave packets in a case when (Ji/K� 1. Compar-
ing (20) with the solution obtained for the region of the phase plane of untrapped
waves, it is easy to see that with Wi/x k
0 when k< k
(21)
where B is a constant; k* is the wave number of the spectral peak, which is deter-
mined from the empirical relationship (g/k*)1/2 = W. For this we substitute (21)
into expression (16). As a result, for the relative change in the spectral density
of energy GJ~ we obtain the expression
(k, x ) = BA "3 7 UM cos k~ x+ B) (22)
y [ / - �
1 C d (,t) cy i
d~C
As was demonstrated above, this so].ution is equally suitable in the region of
trapped waves under the condition ct)i/0C� 1.
Figure 2 shows the normalized spectral density of energy for different phases of
an internal wave propagating in the direction of the x-axis (c > 0), computed using
formula (22). The curves were plotted for the following values of the parameters:
Lim/c = 10-1, c= 60 cm�sec-1, k* = 3�10-1 rad�cm 1(W = 6 m�sec-1), wi = 3�10'3
rad�sec-1. In the computations use was made of an empirical expression for the
cioefficient of wind-wave interaction [7]
d= /0-I~'U cos 9~CI+ /,6 (~osB~ ~ U/c~~ /-exp[-B,9CU/c-O,Oi)~1~}
llere c is the phase velocity of the spectral component of surface waves; tg" is the
angle between the vectors c and U*.
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y' (,qI B,r'3
I,J6 1
/,B8
I
091
0,84
K rad�cm 1
y(k)~8,~-9 ~l , ~'paa�uv_
I,16 r .
l,OB
1
0, 91
Q84 ~ _ 1
ad.tM-i
d
Iig. 2. Normalized spectrum of surface waves for different phases ki x of internal
wave.
17
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For tlie shurt-wave spectral region with a high accuracy 99(k) _fo(k). This re-
sult follows from an evaluation of the terms in equation (4). Since in the region
of trapped wa.ves G/i/~X� 1, this inequality is intensified with an increase in k
and accordingly, with movement of the packet with a high accuracy there is satisfac-
tion of the expression
a( (p- sp 2/~p 0)-0,
that is, in each phase of the internal wave Sp(k)= Bk-3. In the case of small wave
numbers close to tha wawe ninnber of the spectral peak the parameter cJi/&'>> 1, that
is, the long waves behave almost as free waves. The relative changes in spectral
density are of the order of Um/cg [6].
In order to comprehend the physical side of the investigated phenomenon and explain
the behavior of the q spectrum in its energy-carxying part we will rewrite equation
(4) in the following way: Z .
(23)
d t 1 dx ' yo
The new tei-m on the right-hand side is related to a different writing of the energy
expression and represents the rate of change in wave packet energy associated with
the work of radiation stresses during its interaction with an internal wave. In the
- internal wave phase from 0 to Sit/ 6 (Fig. 2,a) aU/ a x< 0, and accordingly, the in-
ternal wave simultaneously with the wind is a source of surface wave energy. The
spectral level increases until the energy loss O'.� ,92/Voexceeds the source, after
xahich the spectral level decreases. It is easy to determine the phase of the inter-
nal wave at which for the selected k value the spectrum of surface waves will be
- maximum (the energy sources are equal to the losses). This phase is determined
from the equation
/r c . X + g(k ) _ ,7l (2,1 + .
Figure 2,b illustrates the evolution of the normalized spectrum in the internal
wave phase from r( to 11/6 Tr. In this phase interval dU/a x> 0. Here the internal
wave draws energy from the surface waves and the wind, as before, is a source.
In this case the spectral density in the energy-carrying part of the spectrum al-
most everywhere drops off.
We note in conclusion that the relative changes in the wind wave spectrum under the
influence of an internal wave can attain (Fig. 2) 16%. This circumstance can be de-
cisive in the problem of sensing of internal ocean waves by remote methods.
BIBLIOGRAPHY
1. Ptiillips, 0. M., "Tnteraction Between Internal and Surface Waves," IZV. AN SSSR:
FAO (News of the USSR Academy of Sciences: Physics of the Atmosphere and
Ocean), 9, No 2, pp 954-961, 1973.
2. Pelinovskiy, Ye. N., "Linear Theory of Forming and Variability of Wind Waves
in a Weak Wind," IZV. AN SSSR: FAO, 14, No 11, pp 1167-1176, 1978.
3. Basovich, A. Ya., "Transformation of the Spectrum of Surface Waves Under the
Influence of an Internal Wave," IZV. Atd SSSR: FAO, 15, No 6, pp 655-661, 1979.
18
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4. Voronovich, A. G., "Propagation of Internal and Surface Gravitational Waves
in the Approximation of Geometrical Optics," IZV. AN SSSR: FAO, 12, No 8,
pp 850-857, 1976. 5, Basovich, A. Ya. and Talanov, V. I., "Transformation of Short Surface Waves in
Nonuniform Currents," IZV. AN SSSR: FAO, 13, No 7, pp 766-773, 1977.
6. Kuftarkov, Yu. M. and Kudryavtsev, V. N., "Determination of the Spectral Char-
acteristics of Internal Waves in the Ocean by Remote Sounding Methods," IZV.
AN SSSR: FAO, 1980 (in press).
7. Hughes, B. A., "The Effect of Internal Waves on Surface Wind Waves. 2. Theor-
etical Analysis," J. GEOPHYS. RES., Vol 83, No 1, pp 455-465, 1978.
8. Longuet-Higgins, M. S. and Stewart, R. W., "Changes in the Form of Short Grav-
ity Waves on Tidal Currents," J. FLUID. MECH., Vol 8, No 4, pp 565-583, 1960.
COPYRIGHT: Morskoy gidrofizicheskiy institut AN USSR (MGI AN USSR), 1980
5303
CSO: 1865/163
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UDC 432.593
APPROXIMATE SOLUTIONS FOR INTERNAL WAVES IN A MEDIUM WITH AN ARBITRARY
STABILITY DISTRIBUTION
Sevastopol' MORSKIYE GIDROFIZICHESKIYE ISSLEDOVANIYA in Russian No 1(88),
Jan-Mar 30 (manuscript received 19 Oct 79) pp 56-64
[Article by V. V. Gorbatskiy]
[Text] Abstract: Asymptotic methods were used in inves-
tigating the properties of the dispersion rela-
= tionships for internal waves propagating in a
thermocline with an arbitrary distribution of
the Vaisala-Brent frequency.
Nonstationarv vlane wave movement in a medium characterized by the stability para-
meter E dz
(or the Vaisala-Brent frequency), in a linear approximation with use also of the
Boussinesq approximation, is described by the equation [1]
N(z)= ly ~ + NZ(z) zw atz ax1 dzz aXl
for the vertical component of the velocity of vertical movement.
By the introduction of the dimensionless variables
" , / 9 ) \i/z
t=t( ,
G L ~H /
where L, H are the horizontal and vertical linear scales respectively, equation (1)
for plane internal waves w(z)e i(kx+CJt) is reduced tu the forni
C2i2!(I) d,2;LZ r f(I) - I, W(Z)=O , (2)
d Z z L ~.z6z
where f(t) = E (I)=-~ 9~2 . x=~c'G, ~9 Z y
In combination with the approximate boundary condition at the free surface w= 0
with z= 0 and the boundary condition at the bottom w= 0 with z= -D equatiun
(2) is a boundary-value problem for eigenvalues with tlie parameter A2, whose so-
lution makes it possible to establish the dispersion relatianships 6= 6(k).
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At the present time the dispersion relationships for internal waves in the ocean
are investigated theoretically by two principal methods.
The method of schematization of a specific distribution of stability by uniform
layers separated by infinitely thin density jumps has been developed to the great-
est degree. In this case the finding of the dispersion relationships essentially
involves a determination of the roots of a system of algebraic equations arising
as a result of satisfaction of the boundary conditions at the interfaces of the
layers [2-5].
The method of determining the dispersion relationships based on a schematization of
the density distrib ution by layers with a constant value of the stability para-
meter 1 ~
E_ -p dZ,
separated by density jumps, is more general but requires the carrying out of un-
wieldy computations.
Both methods in actuality consitute an approximation of the dispersion relation-
ships for internal waves in a medium with a real density distribution. They can be
called "a priori" because they provide for a preliminary schematization of density
and subsequent formulation of corresponding precise dispersion relationships.
The generalization of such "a priori" approximations can be accomplished by con-
structing schematic profiles of the vertical distribution of the E parameter of a
more complex form, but also leading to precise dispersion relationships.
As demonstrated in [6], such a class of profiles of the stability parameter E,
described by a nonmonotonic function, includes: a parabolic vertical distribution;
a distribution in the form of hyperbolic functions; a power-law distribution.
A fundamental shortcoming of the indicated methods for determining the dispersion
relationships is the lack of any possibility for evaluating the error in the ap-
proximately determined dispersion relationships due to the arbitrary choice of the
density distribution scheme.
An alternative method for determining the approximate dispersion relationships for
internal waves in the ocean is their formulation by uniform asymptotic solutions
of equation (2) with the arbitrary function f(z).
In such a method for formulating the dispersion relationships an important circum-
stance is the choice of the parameter from which the asymptotic solutions are
sought.
Henceforth we will assume the parameter of the boundary-value problem (2) to be
the parameter b. The parameters making it possible to change the form of the Brent-
Vaisala frequency profile will be determined in the following way.
jJe will construct a universal stability profile characteristic for the layer of the
ocean medium containing the thermocline using the function
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f (x) ~ Ae'~~Z1+ Be-'~1Z1
(3)
By examining the typical profile of the stability parameter in Fig. 1 it can be
established that the function describing it must satisfy at least the conditions
f (zo)=0, (Zf)=4, (zZ) =0, )c
.,(Zz)-Alnux� (4)
With these taken into account the function fl(z) acquires the form
,f (z) =,/S ~ e'xw( z1-zo )-xzo - e-xz21 (5)
~
where N�tcz x Nmaz
e -a-4, tz1 =za )-X io - P-arz1 ~ e-arzL I e xt/-(v)(zi
The following relationship ekists between the parameters df and 5G
.V (Zo-ZZ)=-,e/7,y � (6)
and between the parameters z0, zl, z2 and the 9D parameter there is the dependence
- Plz y 9 (7)
where I - cp
z
-ol Z .
,~6 = Z,z z
p 0,05�!0"2 /�10-I /~`'b1~o ~(Z)
~

{

Z, ~
i
i i
2
ZI i-
i
I
~
I
Fig. 1. Schematic distribution of stability for two extremal values of the para-
meter T: 0.8. well-expressed thermocline; 0.01 blurred thermocline.
22
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I.t fo,llows: from expressl.on (6) that 0< 5P\ 1, since with passage through the point
~P= 1 the Function f( sP, z) changes sign.
Thus, by varying the r parameter, with stipulated zo, zl, z2, NmaX values it is pog-
sible to bring the profile (3) suitably closer to the real stability distribution.
Figure 2 shows the stability profiles corresponding to the values ~P= 0.8 and 0.01
for the special cases zo ~ 0, zl = 0, z2 =-1. Their comparison with the situation
really existing in the ocean (Fig. 1) shows that the adopted approximations have
all the characteristic features of a seasonal thermocline.
/l . , !4 I6 18 10 6_ yur.ed
sec 1
~
~ ' - _
1
~ ~

~
~
1 ~
~ i
.
~ N
Fig. 2. Characteristic stability distributions in Black Sea and their schematiza-
tion by a three-layer model June, October).
If the z values are assumed close to the horizon z2 =-1 for a profile with the
parameter 30--1 or if the profile is approximated by the function f(z, T) with the
value Cp