JPRS ID: 10556 USSR REPORT EARTH SCIENCES

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• Content Type: REPORTS

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JPRS L/10556
1 June 1982
USSR Re ort
p
EARTH SCIENCES
(FOUO 3/82)
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- JPRS L/10556
1 June 1982
USSR REPORT
EARTH SCIENCES
C'k'OUO 3/821
CONTENTS
METEOROLOGY
Articles on Short-Range Forecasting of Meteorological
Elements and Dangeroas Weather Phenomena 1
OCEANOGRAPHY
Collection of Papers on Marine I~ydrological Computations
and Forecasts 6
Model of Nonstationary Turbulent Heat and Mass Exchange
in Fluid With Highly Stable Stratification 12
Experiment in Machine P~ocessing of Satellite Oceanological
Information 21
Collection of Articles on Optics of Ocean and Atmosphere..... 31
Some Optical Methods for Investigating Wave-Covered Wa+er
Surface 34
Propagation of Packet of Slightly Nonlinear Internal Waves
in Medium With Constant Vaisala ~equency Lt6
Some Properties of Optical Transfer F~nction of Wave-Covered
Sea Surface 53
Qne Mechanism for Forming of Oceanic Electric Fields 58
Generation of Internal Waves by Bottom Irregularity at
IJiscontinuity of Two Fluids Flowing at Angle to One
Another 63
Associated Internal Waves in Fluid With Exponeiltial
Den sity 1}istribution 70
_ - a- [III - USSR - 21K S&T FOUOJ
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Paxametric Resonance in Stratified F'luid 80
Spectra of Current Fields in Ocean Determined Along
Trajectories of ~eely Drifting SOFAR System Buoys 90
TERRESTRIAL GEOPHYSICS
Ra.diation of Elastic Waves in Unvented Explosion 95
New Developments in Gravimetric Methods and Instrumentation.. 105
Multisided Investigations of the Earth's Crust and Upper
Mantle: Results and Prospects 110
PHYSICS OF ATMOSPHERE
Articles on Structure of Auroral Substorm 116
-b-
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METEOROLOGY
UDC 551.509.32
.4RTICLES ON SHORT-RAPdGE FORECASTING OF METEOROLOGICAL ELEMENTS APiD DANGEROUS
- ti~IEATIiER PHENOMENA
I~eningrad TRUDY C?RDENA LENINA GIDROMETEOROLOGICHESKOGO NAUCHNO-ISSLEDOVATEL'SKOGO
~SFNTRA SSSR: KRATKOSROCHidYY PROGi10Z METEOROLOGICHESKIKH ELEMENTOV I OPASNYKH
YAVLENIY POGODY in Russian No 233, 1981 (signed to press 5 Jun 81) pp 129-135
[i~bstracts from collection "Short-Range Forecasting of Meteorological Etements and
- Dangerous Weather Phenomena," edited by E. N. Novikova, candidate of geographical
sciences, and B. Ye. Peskov, Gidrometeoizdat, 880 copies, 135 pages]
UDC 551.577.1+551.578.7
GONDITIONS FOR FALLING OF IiEAVY SHO~~JERS AND HAIL
,
[Article by Glushkova, N. I.]
[Text] The author sets forth the conditions governing the falling of heavy show-
ers and hail. The dependences between the quantity of falling precipitation, macro-
::cale verticai movements and mesoscale convective Currents in a cloud are determin-
` ed. Expressions are a~so derived for compu~ing a precipitation sum greater than 50
mm and hail, which can inflict losses on agricultural crops over a great area. Ex-
pressions are given which represent the relationship between the product of tae
radar. pa.rameters of the cloud H~~gzm and the maximum velocity of the ascending flow
in the cloud, which make it possible to compute the quantity of precipitation over
~ great area on the basis of observational data from the network of ineteorological
r.adars. The results of the investigation can be used in the diagnosis and prediction
of precipitation and heavy hail inflicting great losses on the national economy and
also for evaluating the effect exerted on hail processes. Tables 5, referenc~s 8.
UDC 551.515.4
SOME t2rSULTS OF IIdVESTIGATIOi~ OF SYNOPTIC-DYNAMIC CONDITIONS FOR DEVELOPMENT OF
CONV~CTIVE CLUUDS AND PHEPtOMENA ASSOCIATED WITH THEt4 OBTAINED DURING MONEX
[Article b~~ Peskov, B. Ye., Zhelnin, A. A., Shupyatskiy, A. B., Khamarina, T. V.
- r~nd Casova, It. I. ]
[Text] The authors have found the dependence of convective activity on the con-
vergence of flows in the lower layers of the troposphere under favorable conditions
1
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of stratification and humidity on the basis of data for the equatorial zone of the
Soutl~ China Sea. It was possible to ascertain the physical reasons for the influ-
ence of convergence. The article gives the diurnal variation of cenvergence, the
pressure field, precipitation, altitude of the cloud tops (on the basis of radar
data) at sea at a distance of 100-300 km from major land masses. An analysis is
given of the role of the mesoscale pressure field in determining the divergence
of surface currents in the equatorial zone and its macroscale characteristics in
a forecast of convective activity. The vertical profiles of temperature, humidity,
instability energy, divergence and vertical movements, averaged for different
classes of cases, are biven, as we11 at the mean values of the altitudes of the
"tops" of radioechoes and the quantity of precipitation. Figures 1, tables 4, ref-
srence~ i7.
UDC 551.588.7
RESULTS OF ROUTINE TESTING OF METHOD FOR SHORT-RANGE FORECASTING OF METEOROLOGICAL
CONDITIONS FOR CONTAPiINATION OF SURFACE AIR LAYER
[Article by Neronova, L. M. and Ponomarenko, S. I.]
[Text] The article gives the results of the probable success of experimental
forecasts of the meteorological conditions for the accumulation and scattering of
effluents on the basis of data for 1978 in the Moscow region. It was possible to
deterniine the critical values of the parameters of the tr.Armodynamic state of the
atmospheric boundary layer and the characteristic meteorological conditions recom-
mended for preparation of forecasts of the meteorological conditions for the con- ,
tamination of urban air. Tables 5, references 3.
UDC 551.509.323
RECOMMENDATIONS ON REFINING FORECAST OF VERTICAL DISTRIBUTION OF TEtiPERATURE IN
ATMOSPHERIC BOUNDARY LAYER
[Article by IVovikova, E. N.]
(']'ext) Pressure pattern maps for the standard isobaric surfaces, situated at a
distance of 150-200 mbar f rom one another, cannct be used in a reliab~~ analysis
and prediction of the temperature fields in thinner layers of the atmosphere. The
aixthor gives practical recommendations on refining the forecast of the vertical
distribution of temperature in the lower 500-m layer of the atmosphere on the basis
of empirical data obtained from high structures, in the example of use of observa-
tional data from the television tower at Ostankino (Moscow). Tables 3, references
6.
UDC 551.50q.5
- MF_"HOD FOR EVALUATING FORECASTS OF CONVECTIVE LdEATHER PI3ENOMEI3A AND PROPOSALS F~R
IT~ IMPROVEMENT
[Article by Lapcheva, V. F.]
[Text] The article is an analysis of the existing method for evaluating operational
methods for predicting convective weather phenomena, especially forecasts of
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precipitation,in specific examples and proposals are given on improvement of the
evaluation method. It is shewn that the evaluation principle used in the Instruc-
~iotis (involving use of a smal.l number of ineteorological s~ations) applicable to
Eorecasts of convective weather phenomena does not agree :aith the nature of forma-
tlon of these phenomena and the probability of their detection using the meteoro-
l.ogical network of observations. As a rule, the forecasts are not compared with
ttle real, but with some arrificial weather pattern in the forecast region: The
need for using data obtained using artificial earth satellites aiid meteorological
radars in the evaluation of forecasts is pointed out. Tables 2, references 12.
UDC 551.501(776+777)
COM~'T.EX MAPS OF CLOUD COVER AND ATMOSPHERIC PHENOMENA AND THEIR USE IN SYNOPTIC .
PItAC'1 I CE
[Article by Minakova, N. Ye.]
[Text] 7'he information content of a complex map of cloud cover and atmospheric
phenomena, compiled using data from radar and visual observations, is discussed.
An evaluation of the effectiveness of radar data, ma.king use of the complex maps,
again ~~n~irmed the high reliability of information on thunderstorms (82%) and
rc~ins (65%). Ti1e complex maps contain a great volume of information (especially
on convective phenom~na), more than "microring" charts (by 35%) or "ring" charts
(by 57%j. The problem of the use of data from complex maps for determining the na-
ture of an air mass and a].so the degree of atmospheric instability is examined in
detail. Figures 1, tables 4, references 8.
UDC 551.509.52
~ QUl~NTITATIVE ESTIMATE OF WIND VELOCITY jdITH ALLOWANCE FOR BREEZE AIR CIRCULATION
[Article by Masterskikh, M. A.]
[Text] A method f.or computing the pressure gradient governing the wind on the
shores o~ seas, lakes and large reservoirs in summer during weather with few clouds
is examined. The influence of breeze cir~~ilation an intensification (weakening) of
the wind in the shore zone is demonstr~cted. Figures 1, references 4.
UDC 551.524.31
T)IURNAL VARIATION OF AIR TEMPERATURE AT MOSCO[J AP1D ITS SUBURBS UNDER DIFFERENT
WEATHP:R CONDTTIUNS
, [Article by Gerburt-Geybovich, A. A., Bakhareva, G. M. and Remizov, G. A.]
~Text) Tl~e authors computed tlie diur.nal variation of air temperature with dif-
ferent wind directions and during a calm with different cloud cover or precipita-
tion on the basis of data from the suburban meteorological station Nebol'sina and
meteorological station Balchug, situated at the center of the city (observatians
- made eight times a day during the period I966-1975). The results can be used in
. 3
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short-range weather forecasting for Moscow, and in particular, for separate allow-
ance for the diurnal variation of air temperature in the city and in its suburbs..
- Figures 3, tables 3, references 7.
UDC 551.510.522
CHANGE IN METEOROLOGICAL ELEMENTS IN LOWER LAYER OF TROPOSPHERE DURING
THUNDERSTORi3~S AND SHOWERS
[Article by Klinov, F. Ya.]
[Text] The principal characteristics of inesoscale changes in temperature, ~~ind
velocity and direction in the lower layer of the atmosphere during thunderstorms
and showers are considered. Using samples of ineasurement data from the meteorolog-
ical mast at Obninsk and the television tower at Moscow, for each of the considered
elements the author gives quantitative estimates af their mesoscale changes. These
~ characteristics are indicative for possible deviations of the value~ of inereorolog-
ical elements from the diurnal variation. Tables 3, references l.
UDC 551.524.31
COMPUTATION OF THERt$AI. TRANSFORMATION FOR MOSCOW AURING CONSIDERAhLE ANOMALIES OF
MEAtl DAILY TEMPERATURE '
[Article by Sokolova, N. G.]
[Text] Computations of the thermal transformation were made for cases of con-
siderable anomalies and sharp day-to-day changes in mean daily temperature at Mos-
cow. The movement of pressure formations was taken into account in constructing an
air particle trajectory. The computed temperatures in the cold half-year, in compar-
ison with the actual temperatures, were exaggerated on the average by 2�C. This val-
ue can bs used as an additional correction in the computations. Figures 1, tables 1,
references 2.
UDC 551.584.2
MESOSCt1LE REGIONALIZATION OF MOSCOW AND ITS SUBURBS WITH R~SPECT TO TEMPERATURE AND
_ WIND
[Article by Gerburt-Geybovich, A. A.]
[Text] On the basis of processing and generalization of inesometeorological ob-
servations for 1975-1978 at 50 points in Moscow and in its suburbs it was possible
to develop a mesoclimatic regionalization of the Moscow metropolitan area and also
validate the desirability of clifferentiation of the background temperature forecast
for Moscow for 6-8 hours for sectors of the city and suburbs. Figures 1, tables 1,
references 4.
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UDC 551.515.5
INT~RTROPICAL CONVERGENCE ZONE ACCORDING TO DATA FROM THE 30th VOYAGE OF THE
SCIENTIFIC RESEARCH SHiP 'YU. M. SHOKAL~SKIY'
- [Article by Belinskiy, 0. N. and V~selov, Ye. P.]
� [Text] The authors dePine two intertropical convergence zones(ICZ) in the western
part of the Pacific Ocean in the northern and southern hemispheres and a zone
of divergence of wind flows. A study is made of the correlation of their migration
with pulsations of the subtropical anticyclones. It was established that ICZ move-
ment occurs in the direction of the resultant pressure gradient (sub tropical anti-
cyclone - equator). A scheme of the development of processes in the ICZ is propos-
ed. In the ICZ regions and beyond their limit the authors computed the velocities
of ordered vertical movements, vorticity components and ,~Z, Richardson number,
therr.~dynamic temperature gradients and other parameters. It is shown that for de-
l:Lneaeing the I~Z boundaries it is sufficient to use the vertical spatial-temporal
sections of the meridional wind component, ordered vertical movements and dew point
spreads. Figures 2, tables 7, references 10.
UDC 5510509.32
CORREI,ATION BETWEEN LOWER-LEVEL CLOUD COVER AND DEW POINT SPREAD AND AIR
TENiPERATURE
[Article by Alekseyeva-Obukhov, I. A.]
[Text] On tiie basis of statistical processing of data from aircraft ascents dur-
ir.g tlle sprin~ and autumn jeasons of 1948-1951 it was possibl_e to ob tain expres-
sions for computing the quantity of low clouds on the basis of the dew point spread
at the 850-mbar l.evel. A checking of the results of computations on the basis of in-
dependent material indicated that the proposed method has a good probable success
(the mean error in computing the quantity of clouds is from 0.8 to 2.0 scale units
(tenths)) for a11 months of the transitional seasons, except for March and November
- monthG of the cold half-year, for which the mean error in computations is 3.8-4.9
scale units (tenths) respectively. Tables 2, references 9.
COPYP.IGIi't': Gidrometeorologicheskiy nauchno-issledovatpl'skiy tsentr SSSR (Gidro-
mettsentr SSSK), 1981
5303
C~O: 1865/76
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OCEANOGR.APHY
' ' UDC 551.46
COLLECTION OF PAPERS ON MARINE HYDROL~~~CAL COMPUTATIONS AND FORECASTS
Leningrad TRUDY ORDENA LENINA GIDROMETEOROL~GICHESKOGO NAUCHNO-ISSLEDOVATEL'SKOGO
TSENTRA SSSR: MORSKIYE GIDROLOGICHESKIYE RASCHETY I PROGNOZY in Russian No 241,
1981 (signed to press 25 May 81) pp 1Q7-112
[Abstracts from collection of articles "Marine Hydrological Computations and Fore-
casts", edited by V. S. Krasyuk, candidate of geographical sciences, and K. M.
Sirotov, candidate of geographical sciences, Gidrometeoizdat, 510 copies, 112
pages]
UDC�551.463.6(261)
MACROSCALE WATER SURFACE TEMPERATURE ANOMALIES IN THE PACIFIC OCEAN
~Article by Karasev, Ye. V. and Ugryumov, A. I.]
[Text] The article gives a statistical analysis of the f ield of water surface
temperature anomalies in the northern part of the Pacific Ocean (20-50�N) on the
basis of data for 5� grid squares during the period from 1949 through 1976. Using
the information content index In, the spatial correlation and autocorrelation func-
tions it was possible to discriminate four informative regions of the ocean, arbi-
trarily called: northern, central, southwestern and southeastern. It is proposed
that the use of informa.tion from the ocean areas of these regions will be most use-
ful for investigating the macroscale interaction between the ocean and atmosphere
for the purposes of long-range weather forecasting. Figures 3, references 17.
UDC 551.526.6:551.465.635
CY~:,LICITY OF VARIATIONS IN WATER SURFACE TEMPERATURE ANOMALIES IN PACYFIC OCEAN
[Article by Karasev, Ye. V. and Ugryumov, A. I.] �
[Text] The authors analyze the results of computations of the sper.tra]. density
runctions for water surface temperature anomalies in four informative regions of
the Pacific Ocean: northern, central and two southern (southwestern and southeast-
ern) regions. The basis for the computations was time series Qtw for the period
from 1949 through 1962. As a result of the analysis it was demonstrated that along
~
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the North Pacific Ocean subtropical circulation there is a predominance of long-
per.iod variations ~aith a period of 4.2 years, coinciding with the full period of
circul
r~~ (1-s) cT t ~rt t 4+ '
where keff is the effective thermal conductivity coefficient in the medium of the
laminar layers; its determination will be discussed below.
~de note that the parameter (1 - s) the relative volume of the laminar layers
is small.. The atcT~, parameter has the same order of magnitude as the parameter
~ tcTs, which, in accordance with (2.1) is close to q since s is close to unity.
Accordingly, the rate of change of the heat content in the layers the left-hand
side of equation (2.2) is small in comparison with the intensity of heat flow
q entering into the right-hand side and equation (2.2) can be simplified:
k~~ca==T++q=0. (2 . 3)
Finally, it is natural to assume tha" IZV. Add SSSR: FAO,
Vol 16, No 4, pp 433-435, 1980.
COPYRIGHT: Izdatel'stvo "Nauka", "Izvestiya AN SSSR, Fizika atmosfery i okeana",
1982
5303
- CSO: 1865/134
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FdR OFFIClAL l: _
EXPERIMENT IN MACHINE PROCESSING OF SATELLITE OCEANOLOGICAL INFORMATION
Leningrad UCHENYYE ZAPISKI LGU, SERIYA GEOGRAFICHESKIKH NAUK: SPUTNIKOVAYA
OKEANOLOGIYA, CFIAST' 2 in Russian No 27(403), 1980 (signed to vress 16 Apr 80)
pp 118-119 ~
[Article by V. V. Vinogradov, I. V. Likhachev and D. K. Staritsyn]
[Text] The article examines the results of processing of satellite infrared in-
formation obtained from artificial earth satellites in a direct transmission
regime. In the processing use was made of data from "NOAA-2" artificial earth
satellites during their flight over the polygon of the international tropical
experiment GATE-74.
The results of the processing are presented in the form of profiles and temper-
atur.e maps of. the ocean. We compared the results of processing of artificial
earth satellite data with synchronous data from radiation measurements from an
IL-18 aircraft of the Main Geophysical Observatory imeni A. I. Voyeykov and
subsatellite measurements from scientific research ships in the GATE-74 poly-
gon. Aboard the aircra.ft and on the scientific research ship "Akademik Koro-
l.ev" synchronous measurements were made using identical apparatus: IR radio-
~ meters in the spectral range 9-12 N.,m.
Maps of radiation temperature were obtained which have a greater information
content than the mean long-term maps.
The temperature of the ocean surface is one of the principal physical charac-
teristics determining the intensity of dynamic processes in the ocean and ex-
- erting an influence on formation of atmospheric circulation.
It is entirely obvious that there is a need for obtaining mass operational data
concerning the surface of the world ocean in order to meet a whole series of
needs and requirements on the part of the national economy and science. In con-
- nection wj.th the development of new technical means for the noncontact measure-
ment of thermal radiation in the IR spectral region it is now possible to ob-
tain data en the temperature of the water surface from artificial earth satel-
lites. From each quasipolar orbit, depending on the flight altitude and tech-
nical specificacions of the vehicle, the artificial satellite scans a part of
the ocean with a total breadth up to 3000 km [I. P. Velov, 1967; K. Ya. Kon-
drat`yev and Yu. M. Timofeyev, 1970~ U. Bandin, 1975].
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Since in a direc~ transmission regime an artificial earth satellite gives high-
er-quality information, it was received in the ocean under conditions of an in-
significant cloud cover aboard the scientific research ship "Akademik Korolev."
outfitted for this purpose with special receiving-recording apparatus.
The principal tasks in the reception of temperature data were the checking of
the reliability of information and study of the possibility of its use in
oceanological practice.
The work, for which about 40 scientific research ships, located in the GATE-74
polygon,were used in a comparison of the measured parameters, was carried out
for the first time for oceanological purposes. Such a combination of subsatel- .
lite support made possible a better study of the problems related to the tie-
in of the collected information, an evaluation of its reliability and planning
a measurement method and operational use.
The maps of absolute temperature of the ocean surface were plotted without tak-
ing absorption into account on the basis of synchronous reference measurements
in the IR range from the scientific research vessel "Akademik Korolev." .
In an operational analysis the great volume of satellite information requires
the use of high-capacity electronic computers. Experience in the use of digital
computers indicated that using electronic computers outfitted with appropriate
input-output devices it is possible to solve a wide range of problems related
to the collection, registry, transmission and analysis of TV and IR informa-
t ion .
Now we will examine the problems involved in the preparation and input of data
obtained from artificial earth satellites using special apparatus two-chan-
nel scannin~ radiometers with the information being renresented in the form of
an analog signal in the memory of an M-222__electron~.c computer for_subsequent
analysis and constructing maps of temperature distribution in digital form.
TI~e use of two-channel scanning radiometers on "NOAA" artificial earth satel-
- lites, operating in the visible and IR spectral regions, on the one hand r,?ade
it possible to obtain a line image of the earth during both daytime and night-
time, and on the other hand, created the fundamental possibility of represent-
ing the information in digital form, its coding and input into an electronic
computer.
An individual scanning line for the "NOAA-2" aztificial earth satellite begins
with a group of synchronizing pulses which can serve as time reference points,
this being followed by useful information from the IR radiometer; then comes
the calibr.ation step (the step interval corresponds to a change in the abserved
br.ightness temperature by 20�K with a temperature interval from 310�K black
to 180�K white); the first half of the line is closed with a signal on
the total amplitude value (indicator of system operating stability). The sec-
ond half of the line also begins with a group of synchronizing pulses, follow-
ed by useful information in the visible part of the spectrum, "calibration
wedge" and total signal (100%) (Fig. 1).
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N ~ ~
O V
1 h
. ~
~
, ~ ' , P
Z~
Fig. 1. Observation diagram.
The observation process essentially involves measurement (in the plane z3) of the
brightness Bi(r) = Bi(r, z~) in the direction o� a unit fixed vector z~*. The object
is situated in a medium with the refractive index n~l and the "observer" is situ-
ated in the air (n~ = 1). The image formation process is considered taking into ac-
count~the variation in slopes of the interface, stipulated in the plane z2 of the
vector-function q(r, t) the projection of the unit vector of the normal to the
surface onto the plane z= 0.
As indicated in [4], the expression determining the relationship between the bright-
ness distribution Bi in the image and the brightness distribution Bp in the object
has the following form:
13~ ~r; t) =13o Iru=r-I-aq(r; t) ~ + (1)
where a= h(n - 1)/n, h is the distance from the object plane to the interface. This
expression is also correct in a case when the "observer" and the test object are in
the air. In this case the image is formed by light rays reflected from the wave-
covered surface; the a coefficient has the form a=-2h. We note that in the deriv-
ation of formula (1) no allowance was made for the scattering properties of water
and air and it was also assumed that the surface slopes are small, that~is, q2~1.
The author of [4], within the framework of the already described image transfer mod-
el, derived expressions for the first (mi) and the second (Mi) statistical distribu-
tion moments for brightness in the image Bi(r; t) of a test ob~ect with a determined
luminosity distribution function B~(r). We will cite these expressions here:
.
m~~r)== (2~)' f f Fo~k)6~~ak)er'`~dk~ ~2~
M~~r~; rz; i)==
(3)
1 ~ ~cr ~ +k . i ~,~t,
(2~)~~ I'o~k~)ro~k=)~:~ak>> ak~; P, i)e ~ ~
-
* Such an idealization is correct for image detectors with a high resolution and a
small angular field of view.
35
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~ .
F� ~k) - J J Bo (r) e_~k~ dr~
~3)
�
whe~e 6~ ~k) =J J w, ~9) e_tt9 dQ
_m
is a single-point characteristic function of surface slopes; wl(�) is a single-
point distribution function of the probabilities of slopes;
m
Oa ~k~; kz~ P; = f... f wx ~9~,; 9z; P~ ti) e`~~`~Q~+~`iQ1~ dq+ dq2 .
is a two-point characteristic function of slopes; w2(�) is a two-point distribution
function of the probabilities of slopes (the surface is assumed to be statistical-
ly homogeneous and stationary); r2 - rl; 'G = t2 - tl. For a normal (Gaussian)
water-air interface (the "traditional" model of well-developed wind waves [5]) the
expressions for e 1(�) and e 2(�) have the following form [4]:
~ 6, (k) =exp [-0,5 ~aEZkxz+o�Zk~z) ) ; (4 )
8: ~k~; k:; P; ti) =exp {-0,5 [QEZ ( k,XZ+Ic~Z) +
- -F-Un2(k1yZ-I-k2~Z~-I-2ME~P~ ~~k~:kx:-I-21~1n~P, ~~k~ukzv~-
+2Man~P; i) (Ic,Yk2�+kzx~~~u)~}, (5)
where M~, Mrt are the autocorrelation functions of slopes in two mutually perpen-
dicular directions; d~2, U,,~2 are the corresponding dispersions; M~,~ is the cross-
correlation function of slopes of the wave-covered water surface.
2. The simplest test object by means of which it is possible to have direct registry
of a ser.ies of slopes of the wave-covered sea is an optical wedge, a test-object
with a linear luminosity distribution function:
Bo (r) =A (1+~,x) . ~6)
The brightness distribution in the image Bi contains a component proportional to
the surface slopes in the direction of the x-axis:
A 1-I-~,x-4-~,aq: (r; t) ] . ~ 7 )
B,(r; t)= [
Information on the spatial structure of the random field of slopes qX(r) can be ob-
tained, for example, by photographing the test ob~ect (6) with a small exposure and
subsequent analysis (photometric measurements) of the photograph. In actual practice
it is far simpler to measure the function qX(t). Information on the temporal vari-
- ability of slopes at a fixed point in space can be obtained using a lucimeter or-
iented at the nadir (downwards). The constant cor~ponent present in the signal
Bi(t) can be easily filtered out.
- In actuality the same principle of use of a test object of the type (6) for the pur-
pose of "diagnosis" of the wave-covered surface lies at the basis of the method for
determining the slopes of the sea surface on the basis of the characteristics of
36
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the scattered light field under natural illumination conditions [2].
3. We will examine a test object with a brightness distribution in the form:
Bo(r)=Bo-I-B, cos(kox-I-~). (g)
Substituting the expression for the spatial spectrum of the test object (8) into
formula (2) and taking (4) into account, we will compute the first moment of the
image Bi:
m; (r) =Bo-f-B,e-a~= cos (kox-~-~) , (9 )
where a2(.~~ 2k~2 is a parameter characterizing the degree of image distortions.
Qn the basis of the contrast of the averaged image (9)
a
4-~mi max - mi min~~~mi max mi min~ -
Q~e- ~/2(Q~ = Bl/B~)
it is nossible to judge the dispersion of slopes of the wave-covered surface.
The expression for the second moment'~ of the image Bi can be obtained using for-
mulas (3) and (8):
dl r, ; r~i) = 2~~Z {0:~ ~ P; ti) cos kop:-I-O2" P; ti) cos [ lco (x,-f-xz) -f-
(10)
+z,~ ] } +ao +BoB,e, ( [ ~og (ka~,+,~) +~og (koxZ+~l~) ] ,
here ni:. ~ nn-R
9,~')=e- ~ ex ~')=e' ,
~2ii/ . ~=E-Hll+~rE(D~*)1~ ~BIPi T~=-12 Mt~Pi ti~�
~ 6t
The expression for the secon3 time moment of the "image" Bi(t) is obtained from (10)
Witt, p = o: 1
It~l,(x; i)= 2 B~ZL~:~~~; i)+OZ�~~~ ~)cos 2(ka~+~p)~+C
(all tlie terms in (10) not dependent on time are denoted by C). This expression as-
sumes a particularly simple form when x= 0, T!/2:
Mt ~i) =B,Ze-n Sh ((~Ra ~ +~o+
wtiere C~ = B~Z = Mi ( ~G .
Ilence it is already simple to express the normalized correlation function of slopes:
RE(T)= p Arslir M~(ti)-M,(�O) sh~~ (11)
L M~(~)-M~(��)
We emPh~isize that all the described operations over the "image" Bi(t) are technic-
ally relatively easy to carry out.
~
In this article in all cases we examine the second moments, not the correlation
- functions, since the measurement of the latter in actual practice is more difficult.
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A somewhat different approach to the problem of studying the statistical properties
of the wave-covered surface involves an examination of the spectral characteristics
of the test object image.
We will examine the time energy spectrum of the function Bi(t):
cI~, ( c~ ) =B,Ze'"~ sh I ~Ra ~i) ~ e'rm~ d~t-i-2nCo8 ~ W ) � . .
We will investigate this expression in two limiting cases.
a) Case of weak distortions ~C 1; in this case e-~ sh R~)~ ~R~
~ ~ ~ W ) a2ko~13,'~e ( c~ ) -l-2n=B,~S ( u~) ,
where
~e = f Me ~z) e-rm~ di
.
is the time energy spectrum of the slopes qX(t). As indicated by the derived expres-
sion, the spectrum of the "image" Bi(t) is simply proportional to the spectrum of
slopes.
b) Case of strong distortions 1; in this case:
~
1 ~ , 2n -
~i 2 B~ ZRg� ~~R~"~ 2nBo'8 (w);
where R~" _-d2R~ ( ti)/d 2~-~ _ p> 0. It follows from this expression that the image
- spectrum is described by a Gaussian function, on the basis of whose characteristic
_ width it is possible to determine the~arameter R" and accordingly the time cor-
relation interval of slopes 'G~= ~
R~".
Now we will proceed to an examination of the spatial characteristics of images of
the "sinusoid." The expression for the second space moment of the Bi(r) image can
be obtained using formula (10) with 'C = 0. Averaging expression (10) for ~(in
practice this operation may not be so simple), we obtain the following dependence
ofMionR~:
Mr(P) =13oz+0,5B,Z0,' P) cos kap,~.
We will examine the space energy spectrtim of the Bi(r) image averaged for ~[4]:
Z� 9[t-rt (P)1 ttp t i
tt?; (k) = 2 B, ~ f e- e cos ka p=e- dp~-4n Bo 8(k) .
_
In the case of weak distortions ( p3' 1) we obtain:
( 1 [ k-ko k-I-Ic ) +
k) = 4 B~Za=koZ ~e ~ ) t~ o
where tt' ~'~~B,z(1-~) [S(k-ko)-1-5(k-F-ko) ]-I-4~tZB aS(k),
~E ~k~ ~ J ~ Me ~P) e_`~P dp.
As indicated by the derived expression, the energy spectrum of the image Bi(r) in
the neighborhood of the frequencies �k~ with an accuracy to the factor coincides
witli the space spectrum of slopes of an uneven 'nterface*.
*
This is correct in a case when the spectra, spaced by �k~, do not overlap one
another. ,
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In the case of strong distortions (~3~1) the image spectrum is written in the fol-
lowing way:
~~(k)=~~ (k-k~,)+~1 (k+k,)+4~=Bo~S(k),
where
~ {a (1`) - ~Bsz eXp r _ ~uiik~2-I-Rvv"kx2-2R:u"k=k~ 1
2~yR,~�Rv~'-Rxr" { 2~~R~~~R~ry"-R:~.~~~ j~ ~12~
R~i� ~-B=Rt ~P) /aPt~Pi ~v-o 1 ^'x, y) �
The coefficients Ri " are related to the dispersions of the second space derivatives
of the rises ~(r) of an uneven interface in the following way:
~zz,= ~ a2~ Z Qu2 . R�_ i aZg 2_ as~= .
Q;Z ~ ~ ax� ~ ~ - QEz ~ vv 6E2 ~ ~ax a~~ ~ a,2 '
RX~" = 12 ~ a ~ .
oE clx- JxBy =0.
Accordingly, expression (12) can be transformed to the form:
o~~ ~B~Z p r_ 1 r kxZ + ky~ 11 ,
k 2aZko2~~6~` ex ~ ~a'`ko2 ~ 6x:~ QxyZ ~ ~
It follows from this expression that by measuring (in the case of strong distor-
tions) the width of the energy spectrum of the image of the '.'sinusoid" in the form
. of two sections kz = 0 and k= 0, it is possible to compute the values of the dis-
persions of the curvature ofythe wave-covered surface O'~2 and 0'Xy2.
4. The principal shortcoming of the methods for measuring the characteristics of
waves from the practical point of view is the need for fabricating, and most im-
portantly, the placement of light sources having large a d� x a a'r~) dimensions
iinder or over the water. This circumstance is not particularly important when carry-
ing out a model experiment; it is a different matter when making in situ measure-
ments. Here it is desirable to employ test objects of the smallest area possible.
This requirement is met by a test object with a brightness distribution of the fol-
lowing form: . . . _
Bo(r)=I3~8(x). (13)
The task of determining the statisti.cal characteristics of images of a test object
(13) for a random interface, not changing in time, was examined in [4]. We will
cite an expression for the second statistical moment of an image, also taking into
account the "dynamics" of the surface [4]:
B,Z q r x,Z+x=Z-2REx~x= J
M, (r,; r2j ti) _ � exp I - , (14 )
~ 2naZv~~ V1-REZ ~ 2a=aEZ(1-RE=)
where R~~ R~(P ; 'C ) .
The second time moment of the "image" Bi(t) is obtained from (14) with rl = r2 = 0:
39
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M~(T) =13o=/2na'otZY1-R~=(T). (15)
ThP expression for the second space moment of the image Bi(r) is obtained by assum-
ing in (14) that = 0, rl = 0 (r2 - D):
/
13oZ 1 Pss (16)
M, ~P) _ ,w , eXp .
na`aE' 1' 1-R~' (P) 2a-aEZ[ 1-Ra" ~P) ~
It is easy to confirm that in the region of values PX
dRF 2rc1'1-R;Z(~: T) ~ 2aZQ;z~1-Re'~P; T)
The time correlation function of the "image" Bi(t) is obtained from the differential
equation (18) with rl = r2 = 0; the expression for it has the following form:
M~(T)--~~~~`/~n) ~resinR;(T). , (19)
A similar solution of the differential equation (18) can be obtained for the space
correlation function of the image Bi(r), if the condition P~X~2ad~ is satisfied
_ (see preceding problem). The solution for this case is written in the following way:
1~0
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Mr(P) =(BoZ/2n) aresin Rt(P) (r,=0; rZ=p; Px~P~; P~~Pav) � (20)
We note that in the practical realization of this method there is no need to use a
test object of the type (17). An image of an object in the form (13) can be subjeCt-
ed co correlation analysis if we first carry out the operation of "filling" of the
intervals between contour lines "alternately" with black and white colors (Fig. 2)
on this image. The result of the analysis of the images "processed" in tl-~is way
� is described by formulas (19) and (20). y�~~
'
I
I ~ �
~ ~
I y~ B ~o s
i ~
i , 4,
/
i
i ~
~ ~_0 i a=0
Q Fig. 2. Fig. 3.
Fig. 2. Images: a) "linear" and b) "stepp- Fig. 3. Diagram of reflection of waves
ed" test objects obtained with observation from fixed rigid screen.
through uneven surface.
, A fur.ther modification of this method involves the following. 4:e will assume that a
light s~urce with a distribution of luminosity in the form
. 130 (r) =Bo exp ~ r ~ Z/ro2)
moves over (or under) a water surface with the velocity v relative to some observa-
~ tion system. We will also assume that the image of this source is integrated over a
long time interval. The brightness distribution in the image, "fixed" in such a way,
is described by the following eKpression:
I3,(r)= f Bo[r~-vt-t-aq(r;t)]dt. . ~21~
We will determine the second statistical moment of this image. Substituting (21)
into formula (3) and assuming v= y~v, we obtain the expression:
M~ ra) _ ~2n~, ~ ~ ~ Fo -ku) Fo ~kz:; kp) X
,
1 ak ak T)ex i k +k +k (~-vT) ]}X ~22~
X-0:lak,x; akx:; v; - P~ P{~~ z~a v P~
v
X dk,r dkZx dk~ dT.
Witl~ a great velocity of motion of the source the time variability of the waves can
be neglected, that is, the interface can be considered "frozen-in." In this case
ex~~ression (22) (with r~ -~0) is identical to expression (16).
1~1
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M r+nalysis of the conditions under which it is possible to adopt the hypothesis
of a"frozen-in" character of the surface directly on the basis of expression (22)
is difficult primarily due to a lack of knowledge concerning the precise form of
the ft~nctions M~ (R ;~C Mrt ~j~ ? M~7 ~f~ ~ t)� However, such an analysis can be
carried outby examining the expression for dispersion following from (22) with
rl = r2 = 0. Carrying out integration for the variables klX, k2X, ky, as a result
we obtain the expression:
~
Q,== ~ ~ f~(T)f=(ti)fo(T)di~
~ 9
where
f, (ti) _ (a--1) [a~-RE=(ti) ] -o,s~
f:~i)=~a-1)�'6~a-Rn~ti)~-0'�, .
io =exp {-pti2l [a-Rn~ti~ ~ } ~
OC = 1+ rp2/2a2 Clq 2, p= v2/4a2 0'q2 (as a simplification we will assume that the
waves are isotropic: ~2 = ~"~2 2)� The functions fn(~ entering intc this ex-
pression, have a monitonicallv dec~easing character; the integration region is de-
termined by the function f~( The characteristic scales of change in the func-
tions fn, determined using the formula
,~�=Y'L/ f,,,, ~ fn� _-dZfnldtiZ I t_o,
where
have the following values: ~
_ ro
i~=i:=iqra/aoQ, io=y2-, T,= y2/R~".
v
When there is satisfaction of the condition C"~1 (the function f~ is "narrow"
in comparison with fl and f2), the integration region -rrp/v ~~Gq and the depend-
ence of the functions R~ and R,~ under the integral on t can be neglected, which
is equivalent to the nondependence of the function Q 2 in expression (22) on Z�
Thus, the hypothesis of a"frozen-in" character of the wave-covered surface is
correct with satisfaction of the condition:
v~ y 2aaql~cq.
6. The single-point characteristlcs of waves are measured most simply by optical .
(as, however, by other) methods. In this section we will examine a method for de-
termining the space correlation function of rises or slopes of the wave-covered
surface on the basis of the results of ineasurements of the corresponding disper-
sions.
Assume that the random watzr-air interface is described by the function of rises
~,C +(r; t). This function can be represented in the form of a linear cotpbination of
elementary monochromatic plane waves with a random complex amplitude dAk, propagat-
~ ing in different directions with the wave vectors k and the frequencies c~k = ~1(k)*:
t(kr-wkt)
l)= Itc d~fke ' .
k
[de wi11 assume that a rigid screen (a wall reflecting waves) has been placed in the
water. The orientation of the screen is stipulated by the vector x~' the normal
to its plane (Fig. 3). The function of rises of the "disturbed" interface is repre-
sented in the form of the sum of the functions and describing the random
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wave fields of the "incident" and "reflected" systems of waves respectively:
~ ~r; t) ~r; t) ~r; t) ; ~r; t) =Re ~ dl~.e{~"'_��"`~.
The condition for the reflection of an elementary plane wave from a rigid wall is
written in the following way:
~ d~�=d~� wi th x=~ (k) =k-2 (kx; ) x,'.
The expression for the correlation function of surface rises ~
Mc~r~~ r:; t~; t:)=~~~r~; tt)~~rz; ta)>
has the following form:
1
Mz ( � ) = 2 Re f f { + -I- + (23)
+ + + (k,) d [xz-x (k,) J dk,dxs;
here is the energy spectrum of rises of the undisturbed surface S+(r; t).
Carrying out integration in {23), we obtain:
Mv~r~; r:; T)=2M~+~P, t)+Mc+~P~)~'Mc+~Pz)~ ~24)
where M~ + is the correlation function of surface rises
p=r2-r,; i=tZ-tt;
~ p,x=x,+x2 cos 20-t-yz sin 26;
P~ ~
p,v=~,-y_ cos 20+xZ sin 20;
_ ~ pu=-x=-x, cos 26-~, sin 28;
~
P=~=-yz~'J~ cos 20-x, sin 28.
- The dispersion of rises of a random surface ~ is:
. ~25~
Qc= (r) ='Mc (r; r; 0) =2Q~+z-F-2Mc+ ~Po) ~
where
p,==2 (x cos A-I-y sin 6) cos 6;
p� po~=2 (x cos O+y sin A) sin 6.
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As indicated by (25), the space correlation function of the "undisturbed" surface
is represented in the dispersion Cl~ 2(r) by its section along the straight line
POy -/~Ox tg e� BY measuring the position of the screen (angle B) and measur-
ing the dispersion O'~ 2 it is possible to obtain complete information on the form
of the correlation function M~ +(P
In most cases when reference is to optical methods for investigating waves it is
- simpler to make measurements of the dispersion of slopes, not water surface rises.
The expression for the dispersion of slopes and the direction of the x-axis can be
obtained from (24) using the formula
Q;z (r) _ ~ Mc ~r~ ~ r:~ 0)
8x~ 8zZ
It has the following form:
aEz(r)=2vE+Z-2ML+(Po) cos2A-2Ma~+(Pa) sin29, ~26~
wher.e M~+, M~~ + are the correlation functions of slopes of the "undisturbed"
interface.
The expression for the dispersion of slopes in the direction of the y-axis is deter-
mined similarly and has the form;
- v,,z(r) =2vn+Z+2Mn+(Po) cos 20-2Mt�+(Po) sin 2A. (27)
It follows from expressions (26) and (27) that a"pure" measurement of the functions
M~+ and M y~+ can be carried out only for the sections 8= 0 and D=~'1 /2. whereas
"pure" measurement of the function M~~ can be accomplished only for the sections
e=+~ /4 . For measuring the dispersions of slopes O'~ or o'n it is possible to use
the optical methods described in sections 2, 3, 4 of this work.
We note in conclusion that for practical purposes this method for determining the
space correlation functions of waves can be convenient in measurements made from
aboard a ship or from stationary structures in the sea.
The author expresses appreciation to L. S. Dolin and T. G. Talipova for an extreme-
ly useful discussion of the results of this study.
BIBLIOGRAPHY
1. Cox, C. and Munk, W. H., "Statistics of the Sea Surface Derived From Sun
Glitter," J. MAR. RES., Vol 13, No 2, pp 198-204, 1954.
2. Stilwell, D. J., "Directional Energy Spectra of the Sea From Pho tographs," J.
GEOPHYS. RES., Vol 74, No 8, pp 1974-1986, 1969.
3. Titov, V. I., "Determination of the Spectrum of Sea Waves by a Spectral Analysis
of Aerial Photographs," manuscript deposited at the All-Union Institute of Sci-
entific and Technical Information, 3 September 1981, No 4324-81 DEP.
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4. Veber, V. L., "Statistical Characteristics of Images Obtained in Observations
Through an Uneven Interface of Media With Different Refractive Indices," IZV.
VUZOV: RADIOFIZIKA (News of Higher Educational Institutions: Radiophysics),
Vol 22, No 8, pp 989-1001, 1979.
5. Longuet-Higgins, M. S., "Statistical Analysis of a Random Moving Surface,"
VETROVYYE VOLNY (Wind Waves), Moscow, IL, pp 125-218, 1962.
6. Sinitsyn, Yu. A., Leykin, I. A. and Rozenberg, A. D., "Spatial-Temporal Charac-
teristics of Ripples in the Presence of Long Waves," IZV. AN SSSR: FAO (News of
the USSR Academy of Sciences: Physics of the Atmosphere and Ocean), Vol 9, No 5,
pp 511-519, 1973.
COPYRIGHT: Izdatel'stvo "Nauka", "Izvestiya AN SSSR, Fizika atmosfery i okeana",
1982
5303
CSO: 1865/134
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UDC 551.466.81
PROPAGATION OF PACKET OF SLIGHTLY NO;~L,INEAR INTERNAL WAVES IN MEDIUM WITH CONSTANT
V9ISALA FREQUENCY
Moscow IZVESTIYA AKADEMII NAUK SSSR: FIZIKA ATMOSFERY I OKEANA in Russian Vol 18,
No 3, Ma.r 82 (manuscript received 8 Dec 80) pp 320-324
[Article by A. G. Voronovich, Institute of Oceanology, USSR Academy of Sciences]
[Text] It was demonstrated in [1] that during the propagation of a packet of
slightly nonlinear internal waves (IW) belonging to a mode with some f ixed number
the evolution of the packet envelope is described by a nonlinear Schrodinger equa-
tion. The packet generates some mean flows and corrections to density not disap-
pearing with averaging for the wave period. It is not impossible that these cor-
rections are related to the fine structure of hydrophysical fields in the ocean
[2, 3]. It is of interest to examine this problem applicable to a three-dimensional
packet of IW not having a modal structure. A similar problem for a compressible
medium was examined in [4], where, in particular, it was demonstrated that a packet
of powerful acoustic waves generates internal waves during its propagation (Cheren-
kov radiation).
Thus, as~ume that there is an unbounded fluid with a constant Vaisala frequency NO
= const and a packet of internal waves of a small but finite amplitude is propa-
gated in it. An examination of the corresponding slightly nonlinear effects is con-
veniently carried out using Hamiltonian formalism proposed in [5]. We will solve
the problem using the Boussinesq approximation. Applicable to a medium with a con--
stant Vaisala frequency NO the results in [6] give the following. In the equations
of hydrodynamics we will convert from the v values to the variable bk such that
� ~=Vm+~,~(Po+P)~ (la)
~a=-A-' {7~O~Po+P)},
(lb)
p~ (lc )
p = - - (2n,)-y' - bke~~~ dk+x.c. , complex
No ~ 2 conjugate
~,o ~g (Zn)-v~ i b~ert~dk+x.c. (ld)
No ~ y2c,~r
!~6
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i
Here N02 =-g ~P 0~' ~ k- NO I x I/ I k is the frequency of a linear internal wave
with the wave vector k=(~r,,-v where X and~? are the projections of the vector
k onto the harizontal plane and vertical respectively; the prime here and in the
presentation which follows denotes a z. Then in the new variables bk the equa-
tions of motion are Hamiltoniansbk + ic~ H/~ bk'~ = 0_ where
H= r~rbkb~' dk +~IVk~,r,b~�b~,G~,Br-k,-t, dk dk, dkz + x.c.+
J 1 ~
+ 3 r Utti,r,brbr,bt,8k+r,+~, dk dk, dkz + x.c. + ~2~
J
1 ~
+ 2 Wkk~k~k~bk~br~'bk,bk~sn+k~-k3-r, dk dk, dkz dk,
(here only the "significant" term [5] has been left from the fourth-order terms).
The bk values are simply related to the vertical displacement I(r):
~_~2n) `y' No J 1' 2~ b~e{r~ dk + tc.c. 3
~ )
The full expressions for the coefficients V, U and W have the form
y c~r xxi xxz
_ ~ - (2n) - c~~w,~,w~,l vr ( + - +
r U 1~~'~' !NO -v~ ~ 2 L KZ \ c~~~ w~, ~
~ /
4a
ffil f(ilk -filk Wk SC~%2 9C~7S flilt~Wki Wt~ ~ ~
+ ~ ~~v,~ , ~ + ~
a~~ ~ x,Z c~~, c~,~ W~,
u~~, xZx x2x, twr-wt; u~t,
~ v: r xz2 ( wr + W~~ u~~, J}
l.
(ttie upper and lower signs in this formula relate to V and U respectively) and
1
Wkkik~t~ _ ` ~Z~~-9(fk,tk.ti,+fk~r=k,ti,+f-t,-k,t,k + (4b)
k
where fkik~kk~-It~-ki-kk~~"~-k~kt~-k~~+
wk~c~k. 'l~ (kk~) (kk:)
fr~r~t~r~ c~k~wr~ / L~k~kZ> k2 J~�
, k=k, +k,=kZ+k~.
lde will examine the propagation of a spectrally narroca packet of ZW with the char-
ac~er~jtic wave vector k~ such that bk is substantially different from zero in a
linear approxim~tion only when ~k - kd~k~ (here and in the text which follows k~
_;k~~). It is convenient to make a standard canonical replacement of the vari-
ables [7, R], excluding third-order terms in the Hamiltonian H:
vk~ik~ak~a~~dr_r~_k~ vk~tkiak~ak~dk~-ktt~
Gk=a~ f dk1 dkz + 2 r dkt dkZ - (5)
J U)k-Wk~'-(~kt d (Ot~-(Ok-(D~~
. .
_ r Ut~~t,at~at,at+t~+t,
J dk~ dk1 + 0(a').
(il k-I- W Ic, W k~
4~
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In this case additional fourth-order terms appear in H and the value of the W co-
efficient changes. The general formula for the coefficient obtained as a result,
denoted T, is also given in [7]. In the variables ak the equation of motion ~k +
i b Hf b~ ak* = 0 assumes the form
dk+t~tak+i f Ttt,t,t,ak,a~,ar,St+t,-t,-t, dks dk= dh~ ~ 0. (6)
d
The unwieldy expression for the T coefficient is substantially simplif ied if we ex-
amine the following situation. Assume that the wave is modulated only in the direc-
tion of the unit vector e(Fig. 1) ~(r) _~(er). Then ak will contain the
functions for the k components perpendicular to e and in formula (6) in actuality
only a single integration for the k component parallel to e remains. It is easy to
_ confirm that in this case
T!
(2n)'Trr~t,w To - ,
. w.=-[~s(k-kz)~Z
To=~coZ- (koe)'-=koZ (1- (sin a sin 0 cos ~+cos a cos 0)=),
~ T,=Noako~ (cos a ein A-sin a cos 0 eos s,
where ~e = N~ sin ~ is the frequency of the IW traveling in the direction of the
e vec:tor; Cg is the group velocity (the sense of the angles 6 and ~G is clear
from the figure). Using the approximate expression (7), it is convenient to re-
write equation (6) in a different form. We will assume that
~f'(r, t) _ (2~1)-3/2 J ei(k-kp)rak dk.
With an accuracy to terms quadratic in wave amplitude ~ is simply related to the
complex amplitude of the wave of vertical displacements T~:
! ~t~ .
=No y 2 e-~t~~,
In addition, the value P ~ WkOI"~I2 is the energy density in the packet of IW.
Carrying out the Fourier transform in (6), we rewrite it, taking (7) into account,
in the following form: ,
,p+two~V+C�$t- 2 ~�~VEE+iTo~~~2'~-tT~Q$6~~ ~ga~
~ ~Qei+w: Q�I+hl~� (8b)
Here f, is the coordinate in the direction of the e vector: er, k~ is the pro-
jection of k onto this same direction,
z
s �
~ ko
~i~ C,=Cee = ko cos a(cos a sin 0 cos ~-sina cos 9),
(9)
- i e Z Z_~ Zko `~koe~ C. H
-wo
ko e W~~_axw~BkEs k kaa
InIZ a W�
- ~ 1 e~~~t-e'~ ~ Z ~
~ y Q~~)~ 2rc J~~-kZ d~ dk.
cy i Y
i
i
~
!~8
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Transf.orming in equations (8a), (8b) into a coordinate system related to the pack-
- et: x=~, -Cet and substituting ~-?~e-i~Ot, we obtain
i
+ 2 ~�~~-To~$~Z~h+TiQ$�0~ ~10a)
~.=Q~+~: Q-I+PIZ� (lOb)
The Q parameter is related to the low~frequency IW generated by the packet. In ac-
tuality, if formulas (5) and (lc) are used in transforming from the new variable
ak to the initial parameter P, we ob~ain the following quadratic correction to ~o :
c.Ts'~~ aQ
~low frequency = P~Po � - No ax
low frequency obviously has the sense of a vertical displacement of fluid par-
ticles in the generated low-frequency IW). If it is assumed that the packet is
finite in the ~ coordinate and it is assumed that Q= 0 with from equa--
tion (lOb) it is easy to find the Q value and accordingly ~ low frequency With
} -oo:
1 T
~ low f requency - exp ~ i(~ er-~,t 2C N J ~ Z e~p i~ ~ d~'+x.c. (11)
\ s ) . o .
Thus, during its propagation the packet leaves behind itself a wake of low-fre-
quency IW. It is clear that an appreciable amplitude of the generated waves will
- be observed only under the condition e/Ce)L< 2Tt, where L is the dimension of
the packet in the direction e. It therefore follows that generation occurs when
8(2/3)1~2N~.^: 0.$2N~, and the packet must be modulated in a direction adequately
close to the vertical: cos28 > cos2 a'/1 - 3/4 sin2 2oG).
We will examine a packet modulated in an arbitrary way so that its spectrum no long-
er has a one-dimensional character. If the modulation effects in an almost vertical
= direction, considered above, are neglected, in formula (7) we can assume that
~Zn~ ~Tkkik~k~ ~ To-T~/weZ@koZ (wo2~No~) ~iIIZ
In this case, as is easy to confirm by direct computation
a
II ~ f~' (c~o+i(CeO)- 2 tut~,~ azaax )~'dr+
J \ 1
+ 1 hoZ ~~2 ~ a~ Ax-~~~~sdr,
2 Noz 8yz
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where !JH = a~la x2 + a 2/d y2 and ~i~" = d 2~1/8 kid k~. The equations of
motion after transformation to a coordinate system related to the packet acquires
the form of Davey-Stewartson equations [9]
1 8z utoa
i~h~ + 2~~~,. 8z 8x ~-k~Z N~Qs~'
t ~ ox
aZQ + a=Q m a' I+h I z.
~zz 8y~ 8yL
For the considered problem these equations were derived in [10] by the multi.scale
expansions method.
- BIBLIOGRAPHY
1. Borisenko, Yu. D., Voronovich, A. G., Leonov, A. I. and Miropol'skiy, Yu. Z.,
"On the Theory of Nonstationary Slightly Nonlinear Internal Waves in a Strat-
ified Fluid," IZV. AN SSSR: FAO (News of the USSR Academy of Sciences: Physics
_ of the Atmosphere and Ocean), Vol 12, No 3, pp 293-302, 1976.
2. Voronovich, A. G., Leonov, A. I. and Miropol'skiy, Yu. Z., "On the Theory of
Formation of Fine Structure of Hydrophysical Fields in the Ocean." OKEANO-
LOGIYA (Oceanology), Vol 16, No 5, pp 750-759, 1976.
3. Leonov, A. I., Miropol'skiy, Yu. Z. and Tamsalu, R. E., "Computing ~ine Struc-
ture of the Density and Velocity Fields (In the Example of the Baltic Sea),"
- OKEANOLOGIYA, Vol 17, No 3, pp 389-394, 1977.
4. Goncharov, V. P., Krasil'nikov, V. A. and Pavlov, V. I., "Cerenkov Radiation of
Internal Gravitational Waves," IZV. AN SSSR: FAO, Vol 12, No 12, pp 1310-1314,
1976.
S. Zakharov, V. Ye., "Hamiltonian Formalism for Waves in Nonlinear Media With Dis-
persion," I7.V. W ZOV. RADIOFIZIKA (News of Schools of Higher Education. Radio-
physics), Vol 17, No 4, pp 431-453, 1974.
6. Voronovich, A. G., "Hamiltonian Formalism for Internal Waves in the Ocean,"
IZV. AN SSSR: FAO, Vol 15, No 1, pp 82-92, 1979.
7. 7.akharov, B. Ye. and Rubenchik, A. M., "Nonlinear Interaction of High-Frequency
and Low-Frequency Waves," PMTF (Applied Mathematics and Technical Physics), No
5, pp 84-98, 1972.
8. Zakharov, V. Ye., L'vov, V. S. and Starobinets, S. S., "Instability of Mono-
chromatic Spin Waves," FIZIKA TVERDOGO TELA (Solid-State Physics), Vol 11, No
10, pP 2972-2984, 1969.
9. Davey, A. and Stewartson, K., "On Three-Dimensional Packets of Surface Waves,"
= PR~C. ROY. SOC. LONDON, A338, pp 101-110, 1974.
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10. Shrira, V. I., "On the Propagation of a Three-Dimensional Packet of Weakly
Nonlinear Internal Gravity Waves," INT. J. OF NONLINEAR MECHANICS, Vol 16,
No 2, pp 129-138, 1981.
COFYRIGHT: Izdatel'stvo "Nauka", "Izvestiya AN SSSR, Fizika atmosfery i okeana",
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UDC 551.463.5:535.31
SOME PROPERTIES OF OPTICAL TRANSFER FUNCTION OF WAVE-COVERED SEA SURFACE
Moscow I7.VESTIYA AKADEMII NAUK SSSR: FIZIKA ATMOSFERY I OKEANA in Russian Vol 18,
No 3, Mar 82 (manuscript received 18 Nov 80) pp 330--333
[Article by A. G. Luchinin, Institute of Applied Physics, USSR Academ~ of Sciences]
[Text] The influence of waves on the properties of an image formed by an optical
system was investigated in [1-4]. Depending on the purpose of the investigation,
the authors have neglected different factors whose importance to a considerable
degree is determined by the observation conditions. In this article it will be
demonstrated how it is possible to modify the optical transfer function (OTF) of
the wave-covered discontinuity in dependence on the difference in angles between
the radiation incident on the surface and the direction of sighting. In order to
simplify the problem we will neglect the effects of scattering in water, assuming
that the object plane is situated at a small optical depth.
If the sea surface is illuminated by an infinitely wide beam of parallel rays and
the image is formed by scanning with a narrow receiving diagram, the following ex-
pression [4j is correct for the optical transfer function in a small-angle approx-
imation:
ro .
S~ J�1 Fu (h, 6zII -r- ~n ) 0((h - k) qL, kqL) e'i p~-~`~ (n+e~~) d p dk '
F~ = Frec~ ~1)
_ � N nrad " 1: (1~) _ ~ ~
n~- = nsight' S S S S FQ (0, k m) 0(- kql., kqL) e'~ ~N+~N~ dp dk
z~ = zrec
where Frec~h~ h~7rec + L/m}) is the frequency--contrast characteristi~c of the re-
ceiving system; ~ is a two-point characteristic function of the sui~face slopes;
~P -~nrad - nsight~L/m, nrad and nsight are the projections of unit vectors de-
scribing the direction of the incident radiation and the direction of sighting
onto the I~orizontal plane; m is the refractive index of water; q=(m - 1)/m, zrec
is the i~eigl~t of the receiver aUove the surface.
As is well known, the function K(h) characterizes the contrast transfer coefficient
when making observations of an infinite sinusoidal mire with a period and direction
described by the vector h and r.epresents the normalized complex spectrum for the
~oint scattering function. Our objective is a study of the behavior of this func-
tion (its modulus and phase) with different values of thell~ parameter. The value
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~f this p:~rrimeter ~ives some idea of how close to one another the rays incident
:ind ticattered on t}ie plane at the depth L, participating in image formation,
are refracted at the surface. Thus, reference is to that influence which is exert-
ed on the OTF by the statistical dependence of the incident radiation and the ra-
diation emanating from beneath the surface.
We will stipulate the function Frec in the form Frec - exp[-(hzrec + kL/m)2/2 ~ 2].
With respect to the surface properties, we will assume that the waves are normal
and uniform and that the correlation function of the surface rises has the form
Be~P=) �at2 e~p (-ap:Z) cos kop:.
Omitting below the subscript indicating the one-dimensionality of the problem,
in this case we write an expression for the OTF following from (1);
~ g-v~eaP r U~p-i(p+~p))Z l dP
l ~ b' f
K(h) ' ~2~
m
r (P+OP)= .
. J B-v~ ~XP { _ ~ } dp
~
where
g~ qzLz (Qe~-Be ~ P) ) +Lz~t/ZmZ,
p~g+L=~~/2?n:+zaL~z/n+,
Be (P) m-d'B~/dpz, oe=�Be �
The denominator in (2) is proportional to P(~ Q)-- the mean power scattered by an
infinite plane with a uniform refl~ction coefficient. As indicated in [S], this
parameter, depending on the observation conditions, may be greater than or less
than the corresponding value in the case of a smooth discontinuity. In essence
this means that the wave-covered surface on the average plays the role of a col-
lecting or scattering lens and our task is an investigation of its resolution.
Here wc will introduce three cases differing qualitatively from one another.
I. 1't~e ~pl~arameter is much greater than the correlation radius of the waves, so
that r~e(op) is close to zero. Then it is easy to see that in the region signif-
icant for integration in (2) the g function can be considered constant, the OTF is
real and is described by the formula
g~h~ ~~Xp (-0,5h=oeZ4~L~),
derived in [1].
IT. ~p = 0. In this case the OTF is also purely real (the imaginary part is rigor-
ousl.y equal to zero), but, as will be demonstrated, its form differs substantial-
ly from case I.
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III. ~pis finite, the imaginary part, and accordingly the phase ~,f the OTF is dif-
ferent from zero.
Lg ~Q~
~~pad rad
' f ! ; ;s
~ ;
a ~ ~ ,
~ z T
' i ; t
Z ; r~ ~ ~ ~ p 1
i ~ % 3 .
~ ~ ~
i ~
~ , i
7 ~ ~i ~S
/ ~
~ i ~ .
~ ~
~ ~
S /0 15 10
a % � ,.s ~o rs n~rQ h/Y
: .
Fig. 1. Fig. 2.
Fig. 1. Modulus of function Q(h) with dP = 0; 2= 3.5�10-4 m2; a= 7.6 m 2; k~
= 1.8 m-1; ~3 = 2.5�10'S; zrec - 100 m: 1) L= 5, 2) 10, 3) 20 m. ~
Fig. 2. Phase of optical transfer function with ap~ = 1. For the remaining para-
meters see Fig. 1.
We will examine cases II and III in greater detail. In order to emphasize the ef-
fect of cross-correlation of fluctuations of incident and emerging radiation we
will compute the function '
Q~h)~R~h)~~~~P)/~~~P-'��)l ezp (0,5haQe�q:L=),
which reveals the difference, first of all, of the mean transfer coefficient (with
h= C}) and, second, the optical transfer function proper from case I. This function
has the form: 1 � (hp-i(p+Ap))=
Q~~t) _ _ r g-~i, exp dp.
2yn ~ ~ 4g
The results of the computations for case II are given in Fig. 1. We note the fol-
lowing significant properties of this function. In some regions (in Fig. 1 they
are bounded by curves with a dashed line) it is negative, which corresponds to a
change in the sign of contrast for the observed mire. This makes it possible to
conclude that the scattering functi.on in this case has a multimode character. The
fart that a change in the contrast sign with an increase in the depth of the object
pl.ane occurs with lesser values of spatial frequency seems rather natural, as the
, scattering function in this case has a general tendency to broadening. The second
peculiarity of the Q(h) function is an increase in the envelope of its extrema
with an increase in h; that is, the presence of the correlation~which was mention-
ed above, leads to some contrast increase in comparison with case I. The true value
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of the determined contrast can be judged by dividing the value Q(h)/Q(0) by
exp(O.Sh2 O'82q2L2) (the dashed curves in Fig. 1 represent the value of this ex-
ponent at a logarithmic scale).
tJ~Q~ ~g ~a~
Z
a z ~
z ~
~
~ .
' 3 .
~
S 10 h/?ra
S 10 15 h/~
Fig. 3. Fig. 4.
Fig. 3. Modulus of function Q(h) with aP.` = 1.
Fig. 4. Modulus of function Q(h) with L= 20 m; ~~2 = 1.~�10-2 m2; a= 0.4 m 2;
k~ = 0.46 u~ 1: 1) ~P 0, 2) l, 3) 2.5.
Now we will turn to case III. As already mentioned above, its most important cri-
terion is a complexity of Q(h). Figure 2 shows the dependence of its phase value,
which is identically equal to the phase of the OTF, on the spatial frequency for
different depths with a fixed value of the 6p parameter. In the case of a small
depth (L = 5 m) the phase increases monotonically in the region of spatial fre-
quencies shown in Fig. 2. In the case of greater depths its increase is accompan-
ied by oscillations whose depth increases with an increase in the depth of the
object plane L. With respect to the modulus of this function, it also increases
with an increase in frequency; its value virtually coincides with the envelope of .
the extrema Q(h) in case II with the exception of the region of small spatial fre-
quencies where it is close to unity (Fig. 3). With an increase in the dp parameter
this region expands and the described effects become appreciable only for large
spatial frequencies for which, however, the absolute value of the contrast is
very small (see Fig. 4, whose curves were computed for larger-scale waves).
We emphasize in conclusion that the results cited here to a certain degree are of
an illustrative character since the number of combinations of parameters exerting
an influence on the magnitude of the effects is extremely great. For a quantita-
tive description of their values under real conditions it is necessary, first of
all, to have information on the form of the correlation functions of slopes for
nonuniform waves for different wind velocities over the surface. It is also neces-
sary to clarify the role of scattering in the water and to svaluate its influence
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with different relationships of the length of the free path of protons and the cor-
relation radii for surface waves.
The author expresses appreciation to I. M. Nefedov for assistance in carrying out
t}le numerical computations and L. S. Dolin for useful discussions.
BIBLIOGRAPHY
1. Mullamaa, Yu.-A. R., "Influence of Wave-Covered Sea Surface on Visibility of
Underwater Objects," IZV. AN SSSR: FAO (News of the USSR Academy of Sciences:
Physics of the Atmosphere and Ocean), Vol 11, No 2, pp 199-206, 1975.
2. Veber, V. L., "Statistical Characteristics of Images Obtained in Observations
Thruiigh an Uneven Discontinuity of Media With Different Refractive Indices,"
IZV. VUZOV. RADIOFIZIKA (News of Higher Educational Institutions. Radiophys-
ics), Vol 22, No 8, p 989, 1979.
3. Veber, V. L. and Dolin, L. S., "Fluctuations of Images During Observations
Through a Randomly Uneven Nonstationary Discontinuity," IZV. AN SSSR: FAO,
Vol 17, No 11, pp 1168-1177, 1981.
4. Luchinin, A. G., "Some Patterns in Forming of the Image of the Shelf During
its Observation Through the Wave-Covered Sea Surface," IZV. AN SSSR: FAO,
Vol 17, No 7, pp 732-736, 1981.
5. Luchinin, A. G., "Influence of Wind Waves on the Characteristics of a Light
Field Backscattered by the Bottom and a Water Layer," IZV. AN SSSR: FAO, Vol
15, No 7, pp 770-775, 1979.
COPYRIGHT: Izdatel'stvo "Nauka", "Izvestiya AN SSSR, Fizika atmosfery i okeana",
1982
5303
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UDC 551.463.7
ONE MECHANISM FOR FORMING OF OCEANIC ELECTRIC FIELDS
Moscow IZVESTIYA AKADEMII NAUK SSSR: FIZIKA ATMOSFERY I OKEANA in Russian Vol 18,
No 3, Mar 82 (manuscript received 30 Dec 80) pp 333-335 '
[Article by A. L. Virovlyanskiy and A. N. Malakhov, Gor'kiy State University]
[Text] In this communication we bring the attention of researchers to one possibil-
ity of the appearance of an electric field in a solution of electrolyte, specific-
ally, sea water. The mechanism of field development, which is under discussion, is
governed by the separation of charges due to different displacements of cations
znd anions during accelerated hydrodynamic motion of the solution. This effect
applicable to the propagation of acoustic waves in an electrolyte solution was for
the f irst time predicted theoretically by Debye [1] and then was confirmed repeat-
edly experimentally [2). For this reason the electric field developing in the solu-
tion will henceforth be called a Debye field. �
Another mechanism of the development of a field during the motion of sea water,
caused by the movement of a conducting fluid in the earth's magnetic f ield (EMF),
is well known [3-5]*. It is understandable that one and the same hydrodynamic mo-
tion is responsible for both of the mentioned effects. Below we will compare the
contributions to the electric field from both effects for the simplest models of
fluid motion. In these examples we will strive to understand in what cases it makes
sense to take the Debye field into account and when it is masked by a field induced
by motion in the EMF. In the computations we will neglect the in~luence of electric
forces on macroscopic motion of the solution.
It was demonstrated in [7] that a Debye field develops with any accelerated motion
of the solution. In [7] a formula was derived which makes it possible to find the
strength of this field for arbitrary hydrodynamic motion:
r (r, t) ~~'a~ e'~`IUS ~r, =t) dtt. ~1)
* In addition, there are a number of reasons for the appearance of an electric
field in the ocean associated with the presence of foreign bodies in the water,
influence of the bottom, etc. [3-6]. These will not be considered here.
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Here F.(r, t) is the strength of the Debye field at the point r at the time t;
N N
a= (1/eeo~ ~~PZVP6Di ~~leeo~ ~epvr?+trba~
pvl pmt
e, mp, bp are the charge, mass and mobility of an ion of the p~th species; ~P
i~ the mean number of ions of the p~-th species in a unit volume; E� ~ is the diel-
ectric constant of the solvent~ N is the number of species of ions in the solu-
tion; A(r, tl) is the eddy--free component of the vector field of accelerations tn
the flu~d, taken at the point r at a moment in time tl< t. In the examples consid-
ered beloca reference will be to the eddy--free motions of a fluid and therefore
there ilg= u= a u/at +(u\7)u is the total acceleration of the solution.
As indicated in [7), the numerical values of DC and /3 can be computed theoretically
- only with a great error which it is difficult even to estimate. This is attributable
to the fact that in determining mp and bP it is necessary to take into account the
phenomenon of hydration of ions, the theory of which at the present time has not
been adequately developed~. Accordingly, a and strictly speaking, must be de-
termined experimentally.
In the evaluation we will use the values of these parameters, computed on the as-
sumption that sea water is a 0.5-molar NaCl solution. In order to determine mP we
will reckon the Na+ and C1- hydration numbers to be equal to 6 and 8 respectively
[8]. In determining the mobilities we will employ the values of the equivalent
conductivities of these ions [9]. Thus, we obtain oG = 0.9-1010 1/sec, 0.7�104
kg/A�sec2.
The principal contribution to the integral (1) is from acceleration values in the
time interval (t - 1/oc, t). Since 1/cr~-10-1~ sec, which is known to be much less
than any time scales of hydrodynamic motion, the factor ug(r, t) can be removed
from beneath the integral. Formula (1) in this case is transformed into
~(r, t)=(~/a)ug(r, t) (~/a=0,8~l0-e kg/A�sec). (la)
As we see, computation of the Debye field for any stipulated motion of sea water
presents no difficulties. The situation is different with computation of the el-
ectric f ield induced by motion in the EMF. However, there are also no fundamental
difficulties here: it is necessary to solve the Maxwell equations in a moving con-
d�cting fluid with corresponding boundary conditions at the water-air and water-
bottom interfaces. However, it is possible to obtain a solution of such a system
ot equations in partial (although linear) derivatives in analytical form only for
tt~e simplc:st motions of a fluid; the corresponding formulas usually have a rather
unwieldy Eorm [5]. Precisely for this reason as examples we selected very simple
models of hydrodynamic motions, especially since we are interested primarily in
an evaluation in order of magnitude.
~ Debye [lJ proposed that the mass of a hydrate shell be determined on the basis
1 of ineasurements of the electric field arising during the propagation of an acous-
tic wave in the electrolyte solution.
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As the first example we will examine a monochromatic surface gravitational wave
propagating in a fluid of infinite depth. Assume that the undisturbed surface co-
incides with the xy-plane and the z-axis is directed vertically dowanward. The
wav~ travels along the x-axis and its amplitude is sufficiently small so that in
the computations it is possible to use the linearized equations of hydrodynamics.
Using the known formulas for the field of velocities in such a wave (see, for ex-
ample [10]) and formula (la), we obtain an expression for the components of
strength of the Debye field
E~=-iwvx~/a, E�=0, E:=-iwvi~/cti, ~2~
where vX(r, t), vZ(r, t) are the projections of velocity onto the x- and z-axes,
~ is the angular frequency of the wave.
The strength of the electric field of the surface wave induced by motion in the
EriF has been computed in many studies (see [5] and the citations given there).
For this case
Es =��ollvv:, E:'=-��ou,7ly, Ev =��o~v:ll:-v=Hx)~ ~3)
where EX~y~z(r; t) are the projections of the electric field onto the coordinate
axes, is the permeability of sea water (usually � is very close to 1), HX~y~Z
= are the projections of EMF strength onto the x-, y- and z-axes.
It is easy to show that EX and EX' (and in exactly the same way, EZ and EZ') are
shifted in phase by n/2 relative to one another.
Since vX and vZ differ from one another only in phase, it follows from (2) and (3)
that:
~ E:/E:' ~ E_/E=~ W~/a) /�o ~ H~ ~4 )
Ttlis ratio is highly dependent on the projection of the EMF onto the y-axis, that
is, on the direction of wave propagation. In evaluating the minimum value of this
ratio we will assume Hy = 0.5 oe. Then
~E:~E:'~�~E:/Ea'~R'm�10-Z, ~5~
where cJ is measured in degrees/sec. The waves really existing in the ocean usual-
ly }~ave ~J~ 1 degree/sec. Thus, it follows from (5) that in this case the Debye
f-ield is greatly masked by a f ield caused by motion in the EMF. Formally the ratio
(4) increases to infinity with Hy-i 0, that is, if the direction of wave propaga-
tion coincides with the projection of the EMF onto the xy-plane, but scarcely
never does the Debye field become "predominant," since, to be sure, there are no
plane surface waves in the ocean.
It can be seen from (2) and (3) that for the given example of fluid motion the
fields caused by the two considered mechanisms differ in directions. This circum-
i stance is characteristic for many fluid motions. Its cause can be seen particular-
ly graptiically from an examination of the following very simple model of a current
[3, 5]. Assume that in a fluid (the positioning of the axes is the same as in the
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preceding example) there is a moving layer of tt-~e thickness h. The motion occurs
along the y-axis; the velocity in the layer varies in conformity to the law vy =
v0(x)ei`'~t. In this case the velocity and acceleration vectors are collinear and
therefore the Debye field is also directed along the y-axis. The f ield induced by
motion in the EMF is perpendicular to this axis since the Lorenz force perpendiC~
ular to velocity does not create separations of charges in the y-direction. Thus,
for this hydrodynamic motion the developing electrical fields are orthogonal to
one another.
In conclusion we will examine the propagation of longitudinal compressional-dila-
tational (acoustic and shock) waves in sea water. As in the preceding example,
acceleration and velocity here are directed in the same direction. Accordingly,
the Debye electric field and the electric f ield induced by motion in the EMF
in this case are also orthogonal to one another.
Assume tl~at a plane acoustic wave, the velocity of the fluid in which changes in
conformity to the law v(r, t) = u~ei(kr- ~ t)~ travels in the water; the vectors u~
and k are collinear. From (la) we obtain the value of strength of the Debye field
r(r, t)=-i(~/a)uoc~exp[~(kr-wt)]� (6)
In the evaluation we take ~.J = 105 rad/sec, u~ = 10-2 m/sec. In this case I E 10-3
V/m.
For computing the field induced by motion in the EMF, we use the general method in
[5J. Omitting the elementary computations, we immediately cite the result:
F~~~~ ~~=(��ozQ[uoxIIl/~k2-tw�oQ))exp[i(kr-t~t)~, ~7)
U is tile conductivity of. sea water. In the evaluations we assume d= 4.5 cm/m. The
rritio of the amplitudes (6) and (7) is equal to = i05 rad/sec): ~E/E'~ _
1. 2� 10~ . In this case ~ E
Since the Debye field is proportional to the acceleration of the fluid, it can at-
tai.n especially high values at t-he front of an explosive wave. It is easy to vis-
ualize tiiat the amplitude of. this field at the shock wave front can be evaluated
using the formula
~~/a)hicoZ/l, (g)
wtiere Pi is the riach number of the wave, c~ is the speed of sound, Q is front
widt}i. In tt~e evaluation taki.ng the real characterisr_ics of the explosive wave:
^f = 1.0-2, c~ = 1.5�10-2, = 1.5 cm, from (8) we find that E~ 1.2 ~I/m.
On the basis of the considered examples some conclusions can be drawn concerning
tl~e role which the Debye effect plays in the general pattern of formation of the
el.ectric field in the ocean. First of all, we note that the Debye field, in ac-
cor.dance with (la), is directly proportional to the eddy-free part of fluid ac-
celeration. For the natural movements of water in the ocean the acceleration will
rare].y be more than 1 m/sec2, which corresponds to an amplitude of the Debye field
of ].~.V/m. 'I'he experimentally measured electric f ields of natural origin in
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the ocean usually have a strength from tenths to tens of � V/m [3-5]. Thus, it can
be said qualitatively that the Debye field introduces a real, although a small con-
tribution to the creation of the general pattern of the natural electromagnetic
field in the ocean. This can also be seen in the considered examples. In addition,
it can be concluded from these examples that the Debye field usually differs from
the field induced by movement in the EMF either in phase or in direction.
Hydrodynamic movements with an acceleration substantially greater than 1 m/sec2
are usually related to man's activity. As indicated by the third example, for such
movements the Debye field in its amplitude considerably exceeds the field induced
by movement in the EMF.
BIBLIOGRAPHY
1. Debye, P. A., "A Method for the Determination of the Mass of Electrolytic
Ions," J. CHEM. PHYS., Vol 1, p 13, 1933.
2. Styuer, Dzh. and Yeger, E., "Propagation of Ultrasonic Waves in Electralyte
Solutions," FIZICHESKAYA AKUSTIKA (Physical Acoustics), edited by U. M. Mason,
Mir, 1968.
3. Shuleykin, V. V., FIZIKA MORYA (Sea Physics), Moscow, Nauka, 1968.
4. Akindinov, V. V., Naryshkin, V. I. and Ryazantsev, A. M., "Electromagnetic
Fields in Sea Water (Review)," RADIOTEKHNIKA I ELEKTRONIKA (Radio Engineering
and Electronics), Vol 21, No 5, pp 913-944, 1976.
5. Sachel'nikov, V. V., OSNOVY TEORII YESTESTVENNOGO ELEKTROMAGNITNOGO POLYA V
MORE (Principles of the Theory nf the Natural Electromagnetic Field in the
Sea), Leningrad, Gidrometeoizdat, 1979.
6. Leybo, A. B., "Electrokinetic Phenomena Associated With Sea Waves," GEOMAG-
NETIZM I AERONOMIYA (Geomagnetism and Aeronomy), Vol 17, No 3, pp 502-506,
1977.
7. Virovlyanskiy, A. L.and P4alakhov, A. N., "Electric Field of an Arbitrarily Piov-
ing Electrolyte," IZV. WZOV: RADIOFIZIKA (News of Higher Educational Institu-
tions: Radiophysics), Vol 24, No 7, pp 851-854, 1981.
8. Izmaylov, N. A., ELEKTROKHIMIYA RASTVOROV (Electrochemistry of Solutions), Mos-
cow, Khimiya, 1966.
9. KRATKIY SPRAVOCHNIK FIZIKO-KHIMICHESKIKH VELICHIN (Concise Handbook of Phys-
icochemical Parameters), Leningrad, Goskhimizdat, 1959.
10. Uizem, Dzh., LINEYNYYE I NELINEYNYYE VOLNY (Linear and Nonlinear Waves), Mos-
cow, Mir, 1977.
COPYRiGHT: Izdatel'stvo "Nauka", "Izvestiya AN SSSR, Fizika atmosfery i okeana",
1982 .
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UDC 551.466.4
GENERATION OF INTERNAL WAVES BY BOTTOM IRREGULARITY AT DISCONTINUITY OF TWO FLUIDS
FLOWING AT ANGLE TO ONE ANOTHER
Novosibirsk ZHURNAL PRIKLADNOY MEKHANIKI I TEKHNICHESKOY FIZIKI in Russian
No 6, Nov-Dec 81 (manuscript received 12 Sep 80) pp 41-47
[Article by I. V. Sturova, Novosibirsk]
[Text] Abstract: Zhe simplest example of three-dimen-
sional internal waves in a flow whose velocity
changes with depth both in intensity and direc-
tion are waves at t:ie discontinuity of two fluids
flowing at an angle to one another. The investiga-
tion of the kinematic characteristics of wave move-
ment in such a fluid under the condition that the
depth of the lower layer is infinite was made in
[lJ. The asymptotic behavior of the waves at the
discontinuity arising during flow around a body
for the case of infinitely deep layers and an ob-
stacle at the bottom under the condition of an
infinite thickness of the upper layer was examin-
ed in [2]. The stability of waves arising at the
disconCinuity of two unlimited flows flowing at
an angle to one another was investigated in [3].
We will examine flow around a rise described by the function f(x, z), unbounded
in horizontal directions by a flow, in whose upper layer of the thickness H1 the
density of the fluid is equal to f~ l, in che lower layer of the thickness H2
P2 = Pl (1 0) . The velocity of the lower flow is equal to U2 and is di-
rected along the x-axis, the velocity of the upper flow is U~ and forms the angle
~ with the x-axis. The x- and z-axes are situated at the undisturbed discontin-
uity, the y-axis is directed vertically upward, the axis of symmetry of the ob-
stacle passes through the origin of coordinates.
Assuming the motion of the fluid within each layer to be eddy-free and the disturb=
ances at the free surface and the discontinuity are small, the equations for the
velocity potentials of disturbed motion in each layer are written in the form
~cpl = 0 with 0< y C III, ~cps = 0 with ~II~ < ~ (1)
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with the boundary conditions at the free surface (y = H1)
~~r,l~y -f- L,~ = 0, L~~~ _ ~C, (2)
at the discontinuity (y = 0)
8cp1~~3J -I- L?,l ~~fz~~J -I- Ls~l = P:La~Ps - P~L~~~ _ (3)
_ _ ~(P: - P ~)~I ~
_ at the bottom (y = -H2)
a~2~a~ + L~~ 0, ~Pt, ~Pt ~'0 with ~s ZZ -r oo, .
where U2a/ax.
L1= U1(cos a� a~a~ S?n a� a~aZ~, I.z =
Here the functions ~(x, z) and ~j(x, z) describe the vertical displacements of
the free surface and discontinuity respectively, g is the acceleration of gravity.
In the investigation of internal waves in many cases at the free surface instead
of condition (2) use is made of the simpler "solid lid " condition, for which d~l/
ay=0withy=Hl.
We introduce dimensionless variables, as the scale units for length and velocity
using the values h= f(0, 0) (height of rise) and U2 and employing the Fourier
transform ~
~l~, y, y) o-suxdx f o-{v:~ (x~ y~ dz
~ .
for real � and from equation (1) we obtain a system of ordinary differential
equations whose solution gives the following representations for the functions
j'~( f.c, 'V) and Yf*(�, V)~ being the Fourier transform of the functions and
y ~ik`did~/w(1-{-e)e ::(tI1+11�)
6:g ~ I- -'Lkfll/ \1 ~ '=1cHZ~ ' ,
1- c n
2kd~fr ~1 s) c-htt2 ' n
- /1 +e-2lcllz~ D~kdi - A ~li h~li~,
~
D = nDl - ~.a1Dz;
- Dl (k, _[eA th kNs -(1-~- e) kd;] th kHl - k~ th kS~;
D, (k, 9) _(8A - kdi th kHl) th kH, -(1-}- e) kd;;
d1= V sin (A ac); d, = sin A; A= gh/~; V= Ul/U~;
f~ is the Fourier transform f(x, z) and the following substitution is made
k sin A, v= k ~os e.
With use of the "solid lid" condition at the free surface Yj* assumes the form
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2kd=1� (i e) e-kH' th kRl ~
?'1* - (t + e ans~~ D ' '
i
Performing lnverse Fourier transforms, we obtair.
t�x tv: ikr ~In (e+~) `4,
(x, z) = 4R2 f e d� f e r~*dv = Z~z Re ~ d8 f ke �r~,~dk~
o 0
where the substitution x= r cos z= r sin ~ is made. A similar expression ex-
ists for the function ~(x, z) as weZl.
The functions and Yl* have simple poles being roots of the equation D(k, B)
= 0 or D1(k, e) = 0. It can be confirmed that with
sAE H
A[VaHZ sin' (A a) Hl'sin~ Aj - V� sinz 0 sin2 (0 a) < i-}- a~
the equation D(k,8 0 has the tti~o positive roots k1~2(kl< k2), otherwise there
is only root k2. The equation Dl(k, e) = 0 has no moc'p than one positive root
icl and only under the condition
V2HZ sin~ (0 a) (1 -I- e)Hl sinz A~ eAH1Hs.
In selecting the integration contour in the k-plane in (4) we will use the Rayleigh
- method for introducing small dissipative forces [1] proportional to the velocities
of f~uid particles. In the initial problem (1)-(3) changes are experienced only by
the dvnamic conditions at the free surface in (2) and at the discontinuity in (3),,
which now will have tl~.: form
Li4~i ~~Pi = g~, PaLz~Pa PiLiWi -I- ~~Pa~a - P~Q~~) �
~ = B~Pz - Pi)~l+
where 0 is a dissipation coefficient which is small in value. The solution of
problem (1) with these boundary couditions shows that the poles of the integrand
in (4) with 0 have the form k= k+ i~ where, for example, under the "solid
lid" condition
- 2k1 ~Y sin (C a) cth k11I1 (1 e) sin 0 cth k1H~~
~ eA - Ici ~(1 e) Nz sinz A/sh2 kiH2 V ZHl sinz (A a)/ah9 k1A1~ ~
Acc:ordingly, in (4) the integrati.on contour is selected in quadrant I or quadrant
IV r~s a function of the sign on sin ( Q+~); all its real pales are "bypassed"
by semi.circles on which with Y~< 0 Im k:0, with ~~0 Im k~0. In [2] this circum-
- stance was not taken i_nto account and all the poles were "bypassed" from below.
As a result, the integral representations for the sought-for functions can be writ-
ten in the form of. the sum of single integrals governed by the presence of the
poles and the double integrals arising as a result of integration along the imag-
inary axis, which henceforth wilt be discarded from consideration since they de-
scribe local effects in the neig'~tiorhood of the rise and rapidly decrease with an
increase in r.
The final expression for the function Y1(r, (and, similarly for ~(r, cp has
the form ~ b~
r~ (r, n~ f lc; sin (]c; sin~(0 ~)~Res~* ~k� e) ae,
3=i a~
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-40 0 ao 80 ~
-20 -
I/h
r/h 0 0 0 -0,0}
-0,03
-0,/
~ 001 ~
- O~Di~ 0
O~Of
- ,O1 0
0 ~
- ~ 0
30 ~
0,01
O _ p
4 ? ,0! 0
~ o ?
- ~0!' r~
e ~
\
- ,Ol
\ 0
0,01
~
80 _
' ~ 2�28' 4f�57~
0 40 Fig . 2 .
-2a
r~n z~n
0 0
_p~pp1
~ T~/h� ~
-0 006 "0,003 ~ I ~
_ , -o,or ' g~ � I ,
^ ~ ~
0 001 p _000J �~1 ~w~ ~1
-0,00f ~i VI ~-i
-O,f ~
0,00! 0 I 1
20 ~ !1 V 1
~ ~~Z 6 ~ 1 - - 2
. 1 i 3
0,1 ~ ~
-O,ODI ~ ~ ~ 20~ ~ ~1 B0 ^ �
" � BO ~ .
~~~~1 -Z01 ~ � 0 ~ " ~ Z'~1
-0,0 p -0~f i` J N
0~00 0 -0~2
p U
,oo~ ~ Fig. 5.
0
-o,oo
so �
20�
Fig. 3.
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whcre k. is the root of tlie equation D(k, B) = 0 or D1(k, e)= 0 and integration
is car.r~ed out using those intervals of the e values in which sgn (sin(B _
:;gn ( ~y) .
With satisfaction of the specific computations the form of the axisymmetric rise
on the bottom was stipulated in two types:
~~T) = 1- r2/d~= with 0< r C d, = 0 with ~C sin ( ~a sin cp cos;0) cus ( U sin cp -y- sgu (sin q~) lAU sin U ~
+ Q~ ri2~, 1'* -r oo, L~ 0~ 7t.
The second exponential factor leads to a rapid attenuation of (16) with U~-* 0 and
exerts a small influence with large U. In the case of high velocities n+.~ 1/U
with a decrease in velocity the amplitudeli'j0 lof the vertical displacement (16)
has maxima at the points Un, determined with ~l~l~/3Z from the condition
va ~ sin cp cos 9 2-}- ~n, n= 0, 1, 2, ~17~
n
With an increase in ~ the maxima of amplitude are displaced insignificantly in the
direction of greater U.
The dependence of the amplitude of internal waves on the flow-around velocity in
the case of small U, such that Fra = U2/(N2a2)< 1, is attributable to the interfer-
ence of waves from coherent sources (inflow and outflowj, situated at the distance
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2a. In the XZ-plane the maxima of the amplitude are observed with U= Un, deter-
mined from
~ cos 8 a(2n -F- 1), n= 0, 1, 2,
~ ~18)
it therefore follows that
2a~co5 A~ = n~,,, ~,n/2, ~,n = U�T,
which is the interference condition. The appearance of ~n/2 is related to the op-
posite effect of inflow and outflow.
It follows from (18) that in the case of low velocities (Fra /~1) there are several
maxima of amplitude at different 8n angles
cosen = � n (19a
~ ~ Na 2~2~ 1), n= 0, 1, 2,
With an increase in velocity the maxima are displaced toward the wake axis.
With motion of an elongated body with the lengthening a/R the amplitude of the
internal wave is ~j~~ ti 1+~ 2 with Fr~l, which coincides with [6]; with Fra z +~z + ZZ 1 Cos ~ U
~ 1
z 1z s6n (s) 1_ a2 (Uc)3 +
~ [ VJa-F'zz ' 2A ~!y2,+,a1 ( U~-f-zz ~U1)`-I-??2-f-s'.),~~
i
-~-Or~Ut~z+Ua+f~l~ Y~Ut)2-~-~~-{�Z'-~oo, t>0.
1 /
Expression (20) becomes equal to zero for t->c~owhen y= 0 due to a decrease in the
term =a (Ut)z
~1-
y~"'+'s' ~iJt~~'~"r/3+sa?
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with y=~ 0 due to a decrease in the second exponential factor. The frequency
of oscillations with an increase in time tends to
N ~2 Ua ~r s
N
ya+=a LZ + 2A ya...~za yz~,z2 .
The first oscillations have a period which is somewhat greater than T( y+ z2)/z.
The asymptotic solution is applicable at great (in comparison with the wavelength)
distances from the source:
r* � ~o. ~21~
The linearity criterion is the inequality
~~lo~ ~2~~~)-~�
Such an instability can be important for predicting the behavior of a stratified
fluid under weightlessness conditions.
- Allowance for the viscosity of a fluid leads to a replacement of the operator a/a t
heCore tl~c~ velocity components in (1.1) by a i a t- vo, Under the condition of a
constancy of the coefficient of kinematic viscosity V= const in place of (1.4) we
obtain the equation
(o - ~2~4)