JPRS ID: 10119 USSR REPORT METEORLOGY AND HYDROLOGY NO. 7, JULY 1981

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JPRS L/ 101 19
17 November 1981
USS~ Re ort
_ p
METEOROLOGY AND HYDROLOGY
No. 7, July 1981
;
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JPRS I,/10119
. 17 Noveriber 1981
USSP REPO4T
h~ETEOROLOGY AND NYDROLOGY
No. 7, July 1981
Translations or abstracts of all articles of the Russian-language
monthly journal t~iETE~ROLOGIYA I GIDROLOGIYA published in Moscow by
Gidrometeoizdat.
CONTENTS
Short-Range Precipitation For~casting 1
*Nonadiabatic Model of Atmosphere in Primitive Equations for Predicting
Meteorological Elements Over Europe 16
*Investigation of Patterns of Movement of Macroscale Vortices Relative to a
Purely Zonal Flow 17
Informativeness of Global Systems for Observing Total Ozone Content............ 18
Axisymmetric Problem of Free Convection and Numerical Experiments for Dynamic
Modification of a Cumulus Cloud 30
, Cloud Extent in Zone 45�N-45�S Over Planet 39
Results of Checking Methods for Determining Water Surface Temperature From
'Meteor' Artificial Earth Sat~ilites 47
Comparative Analysis of Methods for Galculating Turbulent Heat and Moisture
Flows From the Ocean to the Atmosphere 59
Numeri.cal Experiments Using a Mfldel of the Ocean`s Upper Layer 68
, Hydrological Structure and Energy Reserves of Rings in the Main Black Sea
Current 7~3
* Numerical Modeling of Wind-Driven Currents in Lakes 87
~ Structure of Atmospheric Pres~ure and Wind IVear the Equator in the Central Part
of the Pacific Ocean 88
* Denotes items which have been abstracted.
- a- [III - USSR - 33 S&T FOUO]
Gnv nFFTr~ ~ i 1~c~ n~'? v
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*Role af Hydrogen Peroxide (H2O2) in the Formation of ~lesospheric Clouds........ 89
: Influence of Meteorological Factors on the Yield of Spring Wheat 90
Experimental Investigations of Ultraviolet Radiation in the Lower Atmosphere... 96
*Meteorological Work of I. N. U1'yanov (150th Anniversa-ry of His Birth)........ 106
Review of Collection of Articles on Atmospheric Physics and Climate........... 107
*Seventy-Fifth Birthday of Ida Arturovna Gol'tsberg 110
*Govern~ent Awards to Soviet Hydrometeorologists.,. 111
*New International Hydrometeorological Code 112
* Notes From Abroad ll3
*Obituary of N3.kolay Sergeyevich Shishkin (1912-1981) 115
* Denotes items which have been abstracted.
~
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_ UDC 551.509.324
SHORT-RANGE PR.ECIPITATION FORECASTING
Moscow METEOROLOGIYA I GIDROLOGI~A in Russian No 7, Jul 81 (manuscript received
15 Dec 80) pp 5-17
[Article by A. I. Snitkovskiy, candidate of geographical sciences, USSR Hydro-
meteorologa_cal Scientific Research Center]
[Text] Abstract: A method is proposed for predict-
ing the fact and quantity of ~teady precip-
- itation for 24 and 36 hours by application
of the MOS concept. The possibilities for
predicting summPr precipitation are consid-
er ed .
- The theoretical principles for pr~dicting steady precipitation, developed b}~ A. F.
Dyubyuk, have been set forth in the 1~FANUAL (6]. The computation formula for the
quantity of steady precipit~.tion has the form
ra
- ~ - ~ ~ ~J ~t~ dP dt, (1)
J
where Q is the quantity of moisture concentrated in a column of the atmosphere be-
, tween the levels P~ and P during the time Q t: q is specific humidity at maximum
_ saturation; t is time; g is the acceleration of free falling.
It follows from formula (1) that the quantity of precipitation is determined by
the product of the baric thickness of air and the individual change of specific
humidity in it. The individual change in specific humidity is associated first ,
of all with the sign and intensity of vertical air movements. Since in a column
of air with a section 1 cm2 witn a thickness of 100 mb the condensation of 1 g of
moisture in 1 kg of air is equivalent to the falling of precipitation in a quan-
tity of 1 mm, the working formula for predicting the quantity of steady precipi-
tation during some time interval will be as follows:
Q~ = 1~5 ~ Q830~ ~.8 :~1~700`,q50U, ` 2 l
where ~ qg50~ d q700 and l~Q500 are the individual changes of specific humidity
in the case of ascent from the levels 850, 700 and 500 mb.
Accordingly, knowing the actual distribution of specific humidity with altitude
and in this same time interval the vertical movements, it is possible to use (2)
to compute the quantity of steady precipitation and compare it with the actual
1
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quantity, thereby in a diagnostic pian evaluating the reliability of the working
- formula. However, when we proceed to a forecast, '~t i.s necessary to have inform-
ation on the prognostic values of specif ic h:m?idity ai~d de~tical ai.r movements. The
- prediction of tropospheric humidity is accomplished in the usual way: on the basie
of a model of air particle trajectories and the advective 1-~umidity values, with
transformation taken into account. However, due to imperfection of objective anal-
ysis of the humidity field and errors in computations of air particle trajectories,
the prediction of humidity also contains errors. Nevertheless, the fundamental
error in the prediction of steady precipitation arises due to a failure to know the
real distribution of orderen vertical air movements, which are computed from the
equations of thermohydrodynamics, and naturally, the impossibility of comparing
them with the prognosti.c vertical movements. However, it is not for these reasons
~ alone that synoptic forecasts of precipitation are better than the forecasts pre-
pared by numerical and computation methods. It is most important that weathermen
in forecast~ use information on weather phenomena and elements for the region
from which the transport of air masses occurs, whereas in numerical and computation
methods infor~nation on weather nhenomena and elements are not used at all. Such
phenomena as steady rain, showers, thunderstorms, squalls and a number of others
are computed in numerical models on the basis of ph~~i~~-cheoretical concepts or
empirical dependences.
With these circumstances taken into a~count, in this article, in additi~n to known
atmospheric parameters, determining the formation of steady precipitation, as the
predictors, the same as in done by weathermen in the case of a real forecast, ex-
tensive use is made of information on advective precipitation quantities.
In this article, which is a natural continuation of [8], we propose a method for
predicting the fact and quantity of steady precipitation in the cold half-year for
24 and 36 hours on the basis of a synoptic-statistical approach; the possibilities ,
of predicting summer precipitation are also examined. The forecasting method is
developed in the example of Moscow. In contrast to [8], where use was made of two
forecasting concepts PP (perfect prognosis) and MOS (model output statistics)
here we examine only the MOS concept as indicating, in accordance with [8, 9,
10, 13, 14],the best results in operational forecasting.
Init.ial Data and Processing Method
Data on steady precipitation were taken for the period from October through March
1974-1978. The predictant was the quantity of precipitation, averaged for all
mQteorological stations of Moscow and Moscow Oblast during the period of day from
2100 to 0900 hours (for a 24-hour period) and from 0900 to 2100 hours (for a 36-
hour forecast).
The predictors were selected on the basis of general concepts concerning the phys-
ics of formation of steady precipitation and taking into account the lat~st re-
sults of investigations in this field [5, 6, 8, 10-14]. The list of potential
_ predictors, selected for the prediction of precipitation, is given below.
~ 2
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LIST OF POTENTIAL PREDICTORS FOR PRE~ICTION OF FACT AND QUANTITY OF STF.ADY PRECIP-
ITATION IN MOSCOW AND MOSCOW OBLaST FOR 24 AND 36 HOURS
P~ is the pressure at sea level, mb;
H1000 is the geopotential at the 1000-mb level, dam;
H850~ H700' HS00 is the geopotential at the levels 850, 700 and 500 mb, dam;
D700' D500 is wind direction at the levels 700 and 500 mb; degrees;
- ff~pp, ff500 is wind velocity at the levels 700 and 500 mb, m/sec;
j~850~ W700~ W500 are the vertical air movements at the levels 850, 700 and 500 mb9
mb/12 hours;
K is the instability index, K=~H1000+H500~-2H850~ dam;
~ ( T - Tal
hw
is the total dew point spread at the levels 850 and 700 mb, �C; 700
4Q
ax~
is the total specific hun~idity at the levels 850 and 700 mb, g/kg;
,~Af (ground)
is the tutal relative humidity at the ground level and at the surface 850 mb,
T, Td, T- Td are temperature, dew point and dew poir.t spread at the earth's sur-
face, �C;
pW500 is the precipitated water in the layer between the ground and the level 500
gr 500-
~b~ PW r 5 qgr + q500~2~ where qgr and 9500 is specif{.~ humidity in g/kg at
the surfac~ and at the level 500 mb;
Qin and Qadv are the fact and quantity of initial and advective precipitation.
Coding for the fact of precipitation: presence 3, absence 1. Quantity of precip-
itation, mm;
Q~yn~ psyn are the fact ann quantity of precipitation in synoptic predictions of
~M~
precipitation. Coding the same;
Q~,I, Qt,q~ is the actual precipitation (predictants). Coding the same.
The archives of predictors for predicting precipitation for 24 and 36 hours in ac-
cordance with the MOS concept c~ere obtained on the basis of prognostic charts of
pressure, geopotential and ordered vertical air movements, at whos2 initial points
data were read on temperature, humidity and other atmospheric parameters using a
model of air particle trajectories. The fact and quantity of advective precipita-
tion in the corresponding time intervals were determined from the prognostic
trajectory at the 700-mb level. If precipitation was noted at the ground al~ng the
corresponding segment of the trajectory path at the 700-mb level, it was assumed
that the fact of precipitation is established and the mean quantity of precipi-
tation was determined along this very same trajectory on the basis of data for
stations falling in its zone. For ttie fact anc~ quantity of initial precipitation
we took its mean value during the past night for all stations of Moscow or Mos-
cow Oblast in dependence on the territory for which the investigation was made.
The statistical processing of data included the forming of paired correlation
matrices, tlie "sifting" of predictors for the purpose of finding among them those
which are most closely related to the predictant, and at the sa~e time,
3
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independent of one another, the writing of multiple regression equations and
their evaluation. The evaluations of real �orecasts were made using th~ correl -
ation coefficient (r), reliability test (H) and accuracy test (Q) [3]; the
tot~l guaranteed probability, guaranteed prebability of the presence and absence
of precipitation were determined.
The same as in [9], the "sifting" of predictors was carried out using the algo-
rithm in [2J, in accordance with which in the first interval there is selection
of the paired correlation coefficient of maximum value between the predictant
and predictor and then the correlation coefficients between the selected predic-
t~r and the others are found. Then, by means of a definite procedure, the ortho-
gonalization of t:~e values of the predictors is carried out. Among the orthogon-
alized predictors the predictor most closely related to the predictant is again
- selected, etc. Thus, sucn a number of predictors is selected that the difference
" in the accumulated dispersions of the predictant in two adjacent intervalG would
not be less than 0.03.
After ranking c,f the predictors a regression was formed using the algorithm in
[1], in which the number of predictors ensuring the best quality of writing of
the regression equation is dete~mined. An evaluation of the regression is made ~
on the basis of the mean risk (I~K~). Such a number of predictors k is selected
as to guarantee a minimum value of the mean risk in the examination. A compar-
ison of the I~k~ values for different groups of predictors was the basis for
selecting the working regression equations for the purpose of forecasting pre-
cipitation.
The prediction of steady precipitation by statistical methods is an extremely
complex problem because *_he distribution of precipitation does not con~orm to
a normal law. An attempt to normalize p~ecipitation by means of f and lg x did
not lead to positive results. An attempt at si.multaneous prediction of the fact
and quantity of steady precipitation by means of regression equations also was
without success.
Bearing these circumstances in mind, it was decided to predict precipitation in
two successive steps: first prediction of the fact of precipitation and in the
case of its presence prediction of the quantity of steady precipitation.
An analysis indicated that the "pure" absence of precipitation in Moscow is ob-
served in only 20-30% of the cases. Accordingly, it was decided to regard as a
fact of absence also such~cases when its mean quantity is 0.0 mm (we note that in
. climatology a quantity of 0.1 mm or more is regarded as a day with precipita-
tion). With such an approach the newly formed class of absence of precipitation
has a probability on the average of 63-45%, whereas the class of presence of pre-
cipitation (quantity of precipitation ~ 0.1 mm) is 37-54% respectively, that is,
the classes of presence and absence of precipitation have approximately identical
probabilities.
- As a result, two approximately equiprobable classes were formed:
- presence of precipitation cases when the averaged quantity of precipita-
tion for all the stations of Moscow or Moscow Oblast was 0.1 mm and more;
4
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absence of precipitation cases when precipitation in Moscow and Moscow
Oblast was absent or its mean qvantity was 0.0 mm.
- It is possible to separate these two classes by means of a regression evaluation
of the probabilities of events, which, as was demonstrated in [7], is advantageous
in comparison with linear discriminant analysis. In this case the predictant5 and
predictors should have a binary character. Hence tne following coding was adopted
for the actual, initial and adequate precipitation: precipitation 3, absence
of precipitation 1; for the ascending vertic~l movements 3, for ascending
vertical movements 1. The remaining predictors could not be represented in bi-
nary form.
When the predicted value of the predictant in the regression equation, written
in accordance with a r~gression evaluation of the probabilities of events, ex-
ceeds a definite threshold (in our case two), the fact of precipitation is
predic.t:.d, and then ;.ising another regression equation the quantity of steady
precipitation is computed.
With this approach to the prediction of precipitation it was necessary to prepare
the eight data archives: four archives for predicting the fact of precipitation
for 24 and 36 hours, four for the prediction of the quantity of precipitation
for 24 and 36 hours in Moscow and Moscow Oblast.
Analysis of Results
For each of the indicated eight archives initially in all the potential predictors
we obtained correlation matrices. Analysis of these matrices indicated that an en-
r_ire series of predictor~, including the surface pressure value, geopotential at
the levels 850, 700 and 500 mb, temperature and dew point at the earth, have vir-
tually n~ signific.ant correlation with precipitation. As a result of many numerical
experiments it was possible to select predictors which relatively better than the
ottiers were reiated to the fact and quantity of precipitation. We have
ff500' W~~~' K~ 7~ ~T - Td), PWg~~, Qin and Qadv� The correlations of precipita-
tion with these p8edictors for 24 and 36 hours are given in Table 1. Dependi::g en
the archives and the advance time of the forecast, the degree of closeness of the
correlatiun changes and among the data in this table there are very small correl-
ation coefficients. Naturally, the correlation coefficients for 24 hours are high-
. er tt~an for 36 hours; for an advance time of 36 hours the correlation coefficients
are low, especially for predicting the quantity o� precipitation. ~tao conclusions
can therefore be dr 25 m/sec). In the lat-
ter there is discrimination of very high values of turbulent flows computed by
the method in [1]. However, it is necessary to take into account the approxim:xte
character of the exchange coefficient values for velocities greater than 14 m/sec
_ and a relatively low trequency of recurrence of velocities greater th3n 25 m/sec.
Conclusions
- 1. In ti?e ~egion of low wind velocities the minimum values of the turbulent flows
- ui~heat and moisture are given by the Friehe and Schmitt method, based on the re-
sults of direct measurement of the flows.
2. The assumption of an influence of the surface temperature "film" and the intro-
duction of a corresponding correction leads to some improvement in the consistency
- of the methods.
- 3. In the region of great wind velocities the Friehe and Schmitt parameterization
is doubtful because it is based on the results of only 14 measurements, which give
a great scatter.
4. For wind velocities from 14 to 25 m/sec the parameterizations [1, 6, 9] agree
within the limits of 30% accuracy with an unstable stratification, and in the
case of a stable stratification w�~thin the limits of a 40% accuracy..
BIBLIOGRAPHY
l. Ariel', N. Z., Bortkovskiy, R. S. and Byutner, E. K., "Fundataental Principles
in Constructing Tables for Determining Turbulent Flows in the Lower Air Layer
Over the Sea," METEOROLOGIYA I GIDROLOGIYA (Meteorology and Hydrology), No 11,
1975.
_ 2. Bortkovskiy, R. S., Byutner, E. K., Malevskiy-Malevich, S. P. and Preobrazhen-
skiy, L. Yu., PROTSESSY PERENOSA VBLIZI POVERKHNOSTI RAZD'ELA MORYA I ATMO-
SFERY (Transfer Processes Near the Sea-Atmosphere Discontinuity), Leningrad,
Gidrometeoizdat, 1974.
66
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3. Deacon, I. L. and jdebb, I. K., "Microscale Interaction," MORE (The Sea), Len-
ingrad, Gidrometeoizdat, 1965.
4. Kitayborodskiy, S. A., FIZIKA ATMOSFERY I OKEANA (Physics of the Atmosphere
ar_d Ocean), Leningrad, Gidrometeoizdat, 1970.
5. Malkus, Zh., "Macroscale Interaction~" MORE, Leningrad, Gidrometeoizdat, 1965.
6. Arakawa, A., "Design of the UCLA General Circulation Model," Univ. of Califor-
~ nia, Los Angeles, Department of Meteorology, TECHNICAL REPORT No 7, 1972.
7. Friehe, C. A. and Schmitt, K. F., "Parameterisation of Air-Sea Interface Fluxes
of Sensible Heat an3 Moisture by the Bulk Aerodynamic Formulas," J. PHYS.
OCEANOGR., Vol 6, 1975.
8. Garrat, J. R., "Rzview of Drag Coefficients Over Oceans and Continents,"
MON. WEATHER REV., Vo1 105, 1977.
9. Louis, J. F., "Parameterization of the Surface Fluxes," European Center for
Medium-Range Weather Forecasts, Research Dep., Internal Report, No 4, Feb 77.
, 10. Panin, G. N., Volkova, S. V. and Foken, T. H., "On Heat Exchange ef Surface
Layer of Water Reservoir With Atmosphere. XV III Congresso IAHR Italia 1979,
AYDRAULIC Ei�IGINEERING IN WATER RESOURCES DEVELOPMENT AND MANAGEMENT PROCEED-
TNGS, Vol 3, Subject B, Cagliari, Italia, Sep 1979.
67
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UDC 551.465.(152+53)(261.2)
NUMERICAL EXPERIMENTS USING A MODEL OF THE OCEAN'S UPPER LAYER
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 7, Jul 81 (manuscript received
8 Jul 80} pp 77-85
[Article by G. Friedrich, doctor, V. P. Kochergin, professor, V. I. Klimok, A. V.
Protasov and V. A. Sukhorukov, candidates of physical and mathematical sciences
Institute of Oceanography, Hamburg University, and Computation Center, Siberian
Department, USSR Academy of Sciences]
[Text] Abstract: The article presents meteorological
and oceanological data from measurements made
~ in the FLE:{ experiment. On the basis of a mathe-
matical model of the upper layer of the ocean
with the use of FLEX data in the.boundary con-
ditions it was possible to carry out numerical
computations for reconstruction of hydrodynamic
characteristics of the sea. The results of the
computations are compared with the experimental
_ data. The sensitivity of solution of the problem
to the model parameters ls evaluated.
Measurement Data From FLEX .
During 1978 West German oceanographers carried out measurements under the program
of the FLEX experiment in a polygon in the North Sea. The authors had at their dis-
posa~ the following filtered data obtained during the period 6 April-13 June with
a discreteness of 4 hours: wind velocity components ua an~i va, near-water pressure
P, air te.mperature Ta and wet-bulb thermometer TW, tatal flux of short-wave radia-
- tion FI, flux of outgoing long--wave radiation FB and temperature of the sea sur-
face TS. Meteorological measurements were made at a standard horizon. Figure 1
represents rhese data by days corresponding to measurements at midday. A peculiar-
ity of this period is the heating of the upper layer of the sea and the near-water
layer of the atmosphere.
The measurements were made from the scientific research shlps "Meteor" and "Anton
Dorn, which operated alternately. On individual days the measurements were made
outside the polygon: 5-7 May and 5-8 June the "Meteor" operated approximately
100 km and on 12-14 May the "Anton Dorn"' operated 80 km.
We carried out a series of numerical experiments in which the FLEX data were intro~
duced into the boundary conditions of the problem. The parameters of wind frictional
stress at the sea surface Zx~ 2y were computed for the two wind velocity compon~
ents:
68
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=x = C~ �a ua ~ uo = C- Pa `Ua ~ vo (1~
The fluxes of apparent FH and latent heat of evaporation FE at the sea surface can
be computed on the basis of the BULK method:
~
Fy ~Ts~ � ~a CD ~ ~~al ~P ~Ts - Ta~+ ~2~
. FE ( TSl = p~ Co ~ 2'a ~ L~9s (TS) - 9a~� (3)
Here d is air density; c is the specific heat capacity of air; L is the latent
heat of evaporation; qs(TS~ is the specific humidity of saturated air at the tem~
perature of the sea surface TS; qa is specif ic humidity; C,~ , CD are the drag coef-�
ficient and the heat exchange coefficient.
The specific humidity qs (TS) is determined on the basis of the equa.tion of state
- of an ideal gas and Dalton's law 0622es
9s ~ T a) = P- u,378 es
(4)
Here es is the partial pressure of saturated water vapor. The pressure es, accord-
ing to the Clausius-Clapeyron equation with a constant mean latent heat of evapor-
ation L, is approximately determined as follows:
es lT~) = e(~
o) e~P ~ R�., ( To - Ts ll = 6,108 exp 119,8~(1 - Tsl (5)
~ i
Here T~ = 273.16 K, E~ is the gas constant for water vapor, TS is temperature in
Kelvin.
Formula (5) gives the es value in millibars, differing by less than 2% from the pre-
cise values at a temperature below 30�C [2]. Formula (5) does not take into account
, the salinity of sea water, which exaggerates the pressure value es by 2% [2].
The specific humidity qa was determined through the temperatu'ce of the wet-bulb
thermometer TW
T;;, = T� - ~ (SS - S), ~E~
where SS, S are the mixture ratios for air saturated and unsaturated with vapor re-
spectively. According to def inition of the S value, it is the ratio of vapor density
to the density of dry air and the equation of state of an ideal gas
p,E~~~ ec ( 7'w1
P-eslTm) '
By definition, the mixture ratio is related to specific humidity by the expression
_ S
9a = i ~ S�
, 69
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~ Kon/(cni~c)cal/(cm2�~sec)
~2 ;o~' f~
e ~.r~ ' ro � I ~ ~ je n t
~ ~ ' !S I ~!I I ~ ~ ~I
~ ~I ~~~~~1~ ~',T ~ 'lOs ~ I~I~ '~N' I ~I I n ~
w j. ~ ~ I .1
uz /(c,at�cl -~r" i !~I I'~^! ~~1,�..,;~V ''J`'ll
.f0 .~1, f.
. O .
~ fe -z(~ o ~/~y~v/~
_ '
1 ~ ~ ~ 1 /~i i i : L' 1 ~V .
P~ Y~f~ ~~'~t~..% y V
10'~ ~ ' 10 t~.. ,
Iv',I m%sec nic xBr� v/Hz KW.hour/m2
~ ~ ~ ~0 Q~
~ 1
~ ? ~ li ~
mb ro6 ~,i i 1i ii~~'I~~ Ii i ~10
~ i`~~ i~i~ii tl,l I~1~+ N
105(i ~ ~ ~ ~ ~ i ` ~ ~ 4 i ~ ~ ~ ~ f f
~
" ' ~ ;1' 9a
~a~o ' ' ~ ~ :
, ~ ~ ~ ~ ~ ~ 4r
� . . ,
970 ~ O 4.~.2'-v*"""' ~�iLL':.::~JLLb .I, ~
ZO ;~0 60 20 f0 cS'm days
Fig. 1. FLEX data. ~ '
From expressions (6), (7) we obtain
~p o.622 er 1 Tn,l ~9~
_ (Ty-T'~) .
L P-eSlTw)
Substituting th~e S value from expression (9) into formula (8), we find the specific
humidity qa. In computations of the flux of late-~t heat.using formula (3) the LtT)
value was dete:rmined using the expression [2]
L(T)=597,3-0,56(T-To) cal/g. (10)
In numerical experiments the flux of latent heat was computed using the formula
proposed iu [8],
pE(7s)=?aCo~va~j�[9~(Tal -9(Tu)+ arl TQ lT,-T~)~, (li)
which ~Ls expansion of formula (3) into a series in powers of (TS - Ta). Comparative
comput;ations with formulas (3) and (11) indicated that the total heat flux into the
sea over a two-month period r�
Qf~- I (F,-FB-F~-FN)dt
differs from the results of computations using different formulas by less than 1%,
~ although the quantity of latent heat of evaporation at some times differs up to
50% and sometimes also changes sign. During this period of observations the heat
balance at the surface was determined by the flux of short-wave radiation and ~
therefore different methods fnr computing the flux of latent heat were not re-
flected in the total entry of heat into the sea.
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f OR (1F F I('S ~11. t'rF (111.1
Figure 1 shows the change in the heat content of the 80-~n layer in the sea duritig
this period r, y
NQ_ ~ M o~ C aT d~ dt,
r~ o
which was computed by West German s~~cialists using data on the vertical distrib,
ution of temperature. The difference between the entry of heat and heat content
increases with time and was caused by the error in computing the total heat flux
at the surface and horizontal advection. The error in computing the heat flux is
related to the inaccuracy of the computation formulas and the error in measuring
the initial parameters.
In the computations the input constants assumed the f~llowing values:
C, = Cn = 1,3 ~ l~;-~, = 1,2 � 10-3 g/cm3, cP = 0.24 cal/ (g�degree),
C= 1 cal/(g�degree)~ L= 595 cal/g.
- Formulation of Problem
A study was made of the one-dimensional boundary layer of the ocean:
- equations of drift movements
vu a au
vc -rv - a: k as~ (12)
oc~ n dv
-ac ` f`t - az x a:~ (13)
thermal conductivity equation
ar - d~ I,h a: ~ F~ ~ 1' . (14)
1
formula for determining the coefficient of vertical turbulent exchange
~ ~~--(~'h~ d��~ uv= ~ d~ 15
a ) ~dzJ +~dz) -~'rc dx' ~ ~
~
and equation of state
d? _ dT
a: ! ' d~ ~16~
The boundary-value conditions of pro'olem (12)-(14) are as follows:
, du _.Y d~~ "y
_ dz - po ' ~ dz - po '
. ~ 17~
C po l; aJ = FB + FE ; pHI .
if the last term FI e- ~Z is absent in equation (3), then
~
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FOR OFFICIAI. l'~F: UNI.1
C;,a h' dT FB FL FH - Ff Qo~ -
z=N=100 M: u=v=0, T=TH. ~18~
Here u, v are the velocity components of sea currents; T is temperature; f= 2uJsin~
is the Coriolis parameter; ~ is the angular velocity of the earth's rotation; ~ is
latitude = 59�N); h is the depth of the surface turbulent layer, which is deter-
mined from the computation point zk, the first from the surface, at which the fol-
lowing c:ondition is satisfied
, ~ ~
~Ck ~k~~ ~ d~~ + ov) _ P a? ~ min K= Ku~ ~
~ ( d. , . ( d2 , os ~19,
~ is the coefficient of thermal expansion of sea water; [Y = 1.5�10-4 degree'l; ~
is *_~:e ir.u~x ~f absorption of short-wave radiation in the water ~nedium; the z-axis
is directed downward.
The formulated problem was solved numerically on an electronic computer produced by
the "Telefunken" Corporation at Hamburg University. The t~oundary-value conditions
were computed on the basis of FLEX data. The equations of motion (12)-(13) were
approximated in time by an implicit difference scheme and a spatial central-differ-
- ence approximation. The derived system of algebraic equations is solved by matrix
factorization. The thermal conductivity equation (14) was approximated in time us-
ing an implicit scherae and was approximated in space by central differences. The
derived system of algebraic equations is solved by a modif ication of the differEnce
- schemes method [6] which makes possible more careful computation of the difference
_ solution gradients in the neighborhood of the temperature jump. The entire system
of equations (12)-(15) with the boundary conditions (17)-(18) was solved success-
ively in time with a time interval of 2l+ minutes I.0 steps between the measure-
ment data with linear interpolation between them. Vertically use was made of a uni-
form computation grid with an interval of 2.5 m. The details of the solution method
for such a formulation were given in [1].
Results of Numerical Experiments
There are a number of ~free parameters in the problem~ the Ck coefficients in for-
mula (15) and the absorption coefficient K0, the drag coeff icient C~ and the heat
exchange caefficient CD. The sensitivity of the solution to these parameters was
checked in the numerical e~periments.
In the first series of computations the short-wave radiation was taken into account
by the boundary conditions of the thermal conductivity equation. The following ex-
periments were carried out in this series: with a constant value of the coefficiPnt
of vertical turbulent exchange K, with different values of the coefficients Ck and
K~. In the second series of computations short-wave radiation was taken into ac-
= count as a source on the ri~ht-hand side of the thermal conductivity equation (14).
� The influence of the coefficients ~ and K~ was evaluated in this series.
1) ~igure 2a represents the computed surface temperature with constant values of
the coefficient of turbulent viscosity K. On the basis of tt~ese results it is pos-
sible to estimate the characteristic value of the K coefficient which was equal ap-
proximately to 10 cm2/sec. The use of the model for the K coefficient considerably
improved the result. This can be seen from a comparison of Figures 2a and 2b.
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F'OR OFM'1('1:11, l~~F: l)N1.1
~ . + ~
' ~ i
Qi ~ i i
10 ~ i
~ , j
1 L i l~ J' i
i L J
~ . a.~ ..i
o j . ~ ;
,
,
i��~`~.,~ ~ ~
- i ~ ~ t.:~l'~1,, - _ - i
6 '
10 6~ . j~ I vk
~
~ \ ~ I
f� , �t ~i'�~
9 n:~ N~,~.� D ~
�
' J .~~'~,,-f' . ~ I
~ , z
' 'I
/ s. - - - 4
1 ~ '
6 2;1 t:,~ :y.., days
Fig. 2. Temperature of sea surface. 0) FLEX data, a) computations with constant
K coefficient: 1) K= 50; 2) K= 10; 3) K= I cm2/sec; 4) K variable, C= 0.05,
K~ = 1 cm2/sec, b) computations with value K~ = 8-7.9 (z/H)1~2: 1) Ck = 0.21-0.2 z/
H; 2) Ck = 0.41-0.4 z/H; 3) Ck = 0.11-0.1 z/H, Q z= 10 m; 4) Ck = 0.11-0.1 z/H.
The formula for determining the K coefficient (15) is a special case of the dynam-
ic turbulence equations [1]. The Ck coefficient was found by solution of the dynam-
ic ttirbulence equar.ions and in the case of weak stable stratification can be assum-
ed equal to 0.05 [1]. In a general case the Ck value is dependent on the thickness
of the surface tur~bul2nt layer [3]. The analytical solutions of the stationary
equations (12)-(14), obtained with the a priori coefficient K;~re subetituted into
the left- and right-hand sides of formula (15): K= K1 and K= K1(1-z/H)2 = Klx2
(K1 = const) L4~ and the Ck coefficient is evaluated using this method:
C h au~z+( aU~' ~ d = C h z~` 20
~ k~` ( os f da j P~ dy ] ~ k~ K, ~ ~
f~~3 =~Ck jl~~ : C f;7 ll4
' k-( e ) /t
Substituting the ki~own solutian obtained with a constant value of the K coef-
ficient into the right-hand side of formula (20) [4], we obtain
(2 e-T �-1 11~' (2 e-t )~~2
Ck :t (e-2Tl -e-' )~l4 :a0 r ~ ~21~
' With a variable K coefficient, K= Klx2
~ (2 e-r u'1 )~~2 x312 (2 e-' 1 11~~ ~ )
~k A(KeTI(1-ReTi~') (x`"'-e'")~~4s=o A(her)(1-ReT~~~). 22
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N'c~tt