JPRS ID: 8888 USSR REPORT METEORLOGY AND HYDROLOGY NO.11, OCTOBER 1979

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I~OR OFI~IC'IAI. US1~: ONI.Y
JPRS L/8~888
28 January 1980
~ IJSSR Re ort
p
METEOROLOGY AND HYDROLOGY
No. 11, November 1978 _
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JPRS L/8888
28 January 1980
USSR REPORT
~ METEOROLOGY AND HYDROLOGY
No. 11, November 1979
Selected articles from the Russian-language journal METEOROLOGIYA
I GIDROLOGIYA, Moscow.
CONTENTS PAGE -
Moistening of the Continents and Intensity of Summer Monsoonal
Circulation
(G. P. Kurbatkin, et al.) 1
Construction of a Model of the AtmosphPric Boundary Layer tor the
Equatorial Zone
(Ye. M. Dobryshman) 10
Small Oscillations of the Polytropic Atmosphere and the Filtering Rale of
the Hydrostatic Approximation
(V. M. Kadyshnikov) 2!?
Correlation Between Minimum Pressure ~nd Maximum Wind Velocity in
Trbpical Cyclones
(V. M. Radikevich, G. G. Tarakanov) 37
Possible Mechanism of Transfer of Disturbances from the Lower Thermosphere
into the Meso-Stratosphere
(V. I. Bekaryukov, et al.) 48 -
Characteristic Diurnal Varialtions of Winds ~n the Upper Mesopause Region -
Over Central Europe and Eastern Siberia
(R. Schminder, et al.) 58
Evaluation of Errors in Computing Effective :Zadiation
~ (A. I. Budagovskiy, L. Ya. Dzhogan) G4
~ -a- CIII -USSR- 335&TFOUO]
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CONTENTS (Continued) page
� Optical Characteristics of the Atmosphere in the Tropical Zone of
_ the Atlantic Ocean
(V. N. Adnashkin, et al.)
72
Present Status of Research on Sea Surface Temperature
(F. S. Terziyev, et al.) 82
Maximum Possible Heights of Wind Waves in the Oceans and Seas
(G. V. Matushevskiy) 93
Evaluation of Accuracy in Determining Water Discharge in Streams
(G. V. Zheleznyakov, B. B. Danilevich) 99
Status and Prospects for the Development of Agroclimatic Investiga-
- tions -
(I. G. Gringof, Yu. Z~ Chirkov) 104
Degree of Activity of Winter Wheat During Winter Thaws
(I. v. svisyuk) 114
- Scientific Center of Soviet Hydrology (Sixtieth Anniversary of the
State Hydrological Institute)
(V. I. Korzun) 120
Patterns in the Redistribution of Ice in the Waters of the Foreign
Arctic
(V. I. Smirnov) 135
The Magnus Effect for a Spherical Particle During Deta~ament from a ~
Solid Surface
(N. N. Grishin) 141
natermination of Mean Annual Runoff from Slopes When Taking
Antierosion Measures
(A. I. Gonchar, I. K. Sribnyy) 145
Ensuring the Uniformity of Measurements in the Spstem Operated by
the State Committee on Hydrometeorology and Environmental
Monitoring
(G. N. Kondrashov, L. V. Selivanov) 149
Review of Monograph by V. R. Alekseyev: NALEDI I NALEDNYYE PROTSE�SY
(VOPROSY KLASSIFIKATSII I TERMINOLOGII) (Ice Encrustations and I;.e -
Encrustation Processes (Problems in Classification and Terminology)),
Novosibirsk, Nauka, 1978, 192 pages
(B. M. Krivonosov) 154
b
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. CONTENYS (Continued) Page
Sixtieth $irthday of Sergey Konstantinovich Cherkavskiy 158
Seventieth Birthday of Andrey Anisimovich Glomozda .............o.... 161
Sixtieth ~irthday of Konstantin Petrovich Vasil~yev 165 =
Sixtieth Birthday of Yuriy Ivanovich Chirkov 168
Conferences, Meetings and Seminars
fM. A. Butuzova, et al.) 171
Notes from Abroad
(V. I. Silkin) 177
- c
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PUBLICATION DATA
_ English title . METEOROLOGY AND HYDROLOGY
Russian title . MET~EOROLOGIYA I GIDROLOGIYA
Author (s) ;
Editor (s) . Ye. I. Tolstikov
Publishing House ; Gidrometeoizdat
Place of Publication , Moscow
Date of Publication ; November 1979
Signed to press ' ; 23 Oct 79
Copies ~ 3870
COPYRIGHT : "Meteorologiya i gidrologiya",
1979
ci ~
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J ~
UDC 551.(513:571)
MOIST'ENING OF TI~ CONTINENTS AND INTENSITY OF SUMMER MONSOONAL CIRCULATION
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No I1, Nov 79 pp 5-11
[Article by Correspending Member USSR Academy of Sciences G. P. Kurbatkin,
Professor S. Manabe and Doctor D. G. Hahn, Computation Center Siberian De-
partment USSR Academy of Sciences and Geophysical Fluid Dynamics Laboratory,
Princeton, New Jersey, suhmi.tted for publication 2~ June 1979]
Abstract: Using the ~pecr_ral model of the
atm~sphere developed at the Geophysical
Fluid Dynamics Laboratory (Princeton, New
Jersey), a study was made of the influ- -
ence of changes in moistening of the con-
tinents on the intensity of swmmer mon-
soonal clrculation in the middle latitudes.
The mc~del includes the annual cycle of cli-
mate, the hydrology of the atmosphere and
continents. An analysis of the numerical
experiments indicated that the drying out
of the continents can lead to a decrease
of precipitation not only over the contin-
ents, but also over the ocean; drying-out �
of the continents simultaneously can inten-
sify planetary summer monsoonal circulation
in th~ middle latitudes, which can be an
; i~nportant condition in the annual cycle of _
~ climate for summer radiation heating o~
- the ocean.
jText] Without allo~vance for the annual variation of solar radiation it
is evidently impossible to detect and understand the rela~ive importance
_ of different interacting physical processes determining stable and un-
stable "weather systems" and in the long run the forming climates. The
annual cycle of climate can be quantitatively explained by solution of
the problem of intEraction between the atmosphere, continents and ocean
in accordance with the characteristic times of the processes participatin~
in this interaction. But how and why the physical processes forming the
annual cycle of climate interact with one another to produce climatic,
1
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seasonal and year-to-year fluctuations is almost unluiown. For example, one
of the unstudied aspects of this prohlem is the influence of changes in
t:lie mo:istening of the continents on the intensity of summer atmoapheric
~�Ir~~iilr~t lun.
As is well known, planetary monsoonal circulation is secondary circulacion
of the convective type. It is not described by the stream function, but is
manifested only in the wind velocity potential field a small-scale com-
ponent of large-scale horizontal motion of the atmosphere, not subject to
direct instrumental measurements. At the present time it is unknown to
what extent there is a change in summer planetary monsoonal circ.ulation
in the middle latitudes from y~ar to year and what the reasons for these
changes are. It is postulated that the reason for its changes may be a
change in the moistening of the continents. However, at the present time
not even mean seasonal maps of moistening of the continents for each year
are being compiled. We have only mean long-term (climatic) seasonal maps
of moistening of the continents [1]. Moreover, at the present time mean
seasonal maps of the wind velocity potential field are also not being com-
piled. This does not make it possible to judge the nature of the changes
in summer monsoonal circulation of each year.
- Thus, at the present time it is virtually impossible to investigate this
~~rol~Lem by means of a diagnostic analysis of observational data.
For studying summer planetary monsoonal circulation in the middle lati-
tudes we used a complex spectral model of general circulation of the at-
mosphere developed at the Fluid Dynamics Laboratory located at Prince-
- ton University in the United States [2]. ,
Spectral models differ from grid (finite-difference) models in that the
dynamic variables in them are represented by a synthesis of a finite sum -
of spherical harmonics, and not by the values in a grid of discrete points.
The equations of the model predict the spectral components, and not the
variables at the grid points.
The predicte~ variables in this model are the following: d~ of the stream
function, ~ of wind velocity potential (~7 2 is the horizontal Laplace
operator), temperature, mixture ratio and logarithm of pressure at the
level of the earth's surface. These variables are scalars (the spectral
representations of vector values introduce singularities at the poles). In
tlie model use was made of a hydrostatic approximation and in the vertical
direction use is made of the d coordinate, equal to the ratio of pressure
to pressure at the earth's surface in order to introduce topographic ef-
fects. SimPle liorizontal viscosity was introduced by attenuation of the
model variables by a constant multiplied by 'd 4 of the hydrothermodynamic
elements. A simple diffusion scheme wi~ch the mixing length is used vertic-
ally.
2
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The model has nine vertical levels (d = 0.025, 0.095, 0.205, 0.350, 0.515,
0.680, 0.830, 0.940, 0.99~). These were selected in such a way as to de-
scribe the lower stratosphere and the Ekman boundary layer. The liorizontal
rliomboidal resolution of the model is M= 21. Tiie prognostic equations Qf
the model are integrated in time using a quasi-implicit scheme: the linear
and nonlinear components of the trends are split and integrated in time
implicitly and explicitly respectively. Use is made of temporal smoothing
� with oC = 0.01 in each time interval.
Ttie model includes the annual cycle, the hydrology of the atnosphere and
continents. In order to compute tlie flux of solar radiation tliere is stip-
ulation of the seasonal ctiange in insolation at the upper boundary of the
model atmosphere. The attenuation of sular radiation and the transfer of
long-wave radiation emitted by the earth and atmosphere are computed tak-
ing ir~to account the effects of clouds, water vapor, carbon dioxide and -
ozone. The carbon dioxide mixing ratio is everywhere assumed to be con-
_ stant. The zonally homogeneous distribution of ozone is stipulated as a
function of latitud e, altitude and season. The tirie-dependent spatial dis-
tribution of water vapor is fosnd as a result of integration in.time for
the prognostic equa tion for water vapor, includ ing: three-dimensional advec-
tion of water vapor, vertical mixing of water vapor in tlie planetary boun-
dary layer, evaporation, nonc~nvective condensation and moist convective
ada~tation. In computing radiation fluxes an allowance is made for the
time-variable distribution of cloud cover at three levels in dependence on
the change in water vapor and air temperature.
Ttie temperature of the earth's surface over the continents is determined by
tlie boundary condition expressing the accumulation of heat in the soil
(tliat is, the fluxes of solar and long-wave radiation and the turbulent
fluxes of apparent and latent heat locally together are equal to zero).
Over the oceanic part the seasonal change in temperature of the ocean sur-
face is stipulated. It is determined by interpolation in time between the
four observed distr ibutions of the mean monthly temperature fields of the -
ocean surface. In o rder to compute the descendin g flux of sol.ar radiation
the albedo of the earth's surface i.s stipulated as a function of latitude
over the ocean and as a function of latitude and longitude over the con-
tinents; in places where as a result of the computations there is a repro-
duction of snow cover or sea ice, the albedo is replaced by higher values.
The rates of change in moistening of the continents and thickness of the
snow cover are determined by the budget of water, snow and heat at the
land surface.
Numerical integration in time was carried out for two years and eight model
montlis. In this control variant the distribution of moistening of the con-
tinents was determined at each moment in time from the budget of evapora-
tion, precipitation, melting of snow and river runoff.
The model quite precisely reproduces climate and its seasonal changes. Fig-
ure 1 shocas the field of atmospheric pressure at sea level (a) and the
field of moistening of the continents (b) reproduce3 by this model in the
3
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control variant, averaged for July and August (here the moistening values
beyond the continents must not be taken into account). Ttiey can be compar-
ed with the corresponding fields constructed on the basis of long-term ob-
servations [1, 3], which, unfortunately, are not cited here in order to
save space.
Q)
:1: ::i,~:..;.': '
~
- ~ I~.
I
, I
~ I
I
~ ~ i
~ \ .
I
i: I .
-
. \
r: \ \
:i: i : ` \ \
~ ~
1 .
':I~�'.~~
I� ~ :i
.
�`I'.
�
~.i r'� ' .
' � ~~I : - ;
: \
: , ; ~ . .
+ \ \
~T; :rl;:::.~.:::::,::~::�`:r:;�>::.
~ Z, S ~3?6 -
Fig. 1. Atmospheric nressure at sea level (a) and moistening of the contin-
ents (b), reproduced by Geophysical Hy3rodynamics Laborator}r model (Prince-
ton, United States). Isolines are drawn each 10 mb (a) and each 4 cm (b).
1) less than 1000 mb, 2) 1000-1020 mb, 3) more than 1020 mb, 4) less than
- 2 cm, 5) 2-10 cm, 6) more than ld cm
From an analysis of these maps it is possible to n.ote a coincidence of the
principal regions of high and low pressure. However, they are more strong-
ly expressed in the model climate than in the real climate. In an analysis
of the maps for moistening of the cpntinents it is possible to note a
4
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coincidence in the control variant with the real climate of extensive re-
gions of quite high moistening ~more than 10 cm, shaded regions) in the
tropical zone of Africa, in India, on the islands of Indonesia and in the
middl.e latituaes of the F.urasian continent.
Q~ f / i ~ i /`'i ~ . / , ; ~ / /
~tY
. ~ ~ '~j i~ '
~ ~ j' i~` ' / / /
~ `~i' / I
' ~ y/
~ ' i/
0 /
~
_ i ~ / ' /
/ ' ! ,
, ,
. . . ;
i ,
_ ,
~ ~ 1' ; ~ i.
~ / ' i /
' ~ ~ : ~i
i i 'i!.',~i/, /Y ~
_ . ^
- ~ / ~1,
'/'j/,~y', i
4~,~~'~
~ ; l.~i/~' f/ 'il I i
`.~~r y
~ ~ ~ ~ % .
, t ~
~ , Y ~ ~ y, ; ~ 'i /
. / 1 ~'>/~y ~ ; / / /j: �"V~ .i . ,'?j' % ~ / ~ ~
~S, _ _ ~
I ' ~ i . ~A77`-~-:�' : ~ j ~ ' ~ ~ i
l ~
/ i , /
'Y ~ ~ i. -
" . , , ~ ' ~ -
~ / : ~
~
/ ~
. ~ j ~ / . i
_ ~ ;
Fig. 2. Difference in rate of evaporation (a) and in rate of precipitation
(b) between "Arid" and control variants.
Of the two simglest nwnerical experiments with a change in moistening of
the continents total moistening of the continents or their total des-
sication it was the second which could be favorable for a quantitative
, study of the lLydro*_hermodynamic interactions between the atmosphere and the
continents in the middle latitudes. Therefore, after carrying out and anal-
~zing the control experiment the last three model months (June, July and
August) were recalcuiated witfi the regtriction that the continents remain-
ed completely waterless during the entire three-month summer period ("Arid"
vartant). -
5 _
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' tri/tym cm/day
, q~0
6)
Q)
- a4 ' 1~ ~ '/n ~ ~
~ ~ i r
~ ~ 4r~, i ; ,
~ ~ ~`'ii~.i v ~
~ ' 4S'GUI. ~ 0 45 Kt m. S' ~ 4S G u!. N D 45~0. ur $
_ Fig. 3. Total quantity of precipitation per day, averaged for July-August
over the continents (a) and over the ocean (b).
~
Q) i ~ � ; ~ i, ; . -
' / o $ ` _ ~ ~ ~ 'r' ~ !
` ~ ~ I ,S~ f ; ~
~ ,
~ ~
/ , , i
- ' ~ ~ ~
' . ' . O ~ ~ Q-~
: ~7 ~ I ~ ii, -
� O I I ~y ' ~ ~
/ ~ % Q /'i ~ ~ o f ~1" t!/ Q
, /r ~ ~i / / (wr_ ~ :
i
/ ~ _ ~ ~
J~ ~ i ~ / ~ ~ 11~ / -
~ 'i. ~ r -
i
, ~ ~
~
- , ' - --r--
0 . ~ ~
, j" ~i. ~ t~
i �
' /',I /
/ . ~ ~ I,'
~ / / ~ � ~ 1/ .
' ~ ~ . I. . i
/ / / i,1; ,%i
/ � -
/ / / ~ ' i
. ,./i I ~ /
� ~ '
i
i
~
Fig. 4. Difference between "Arid" and control variants of horizontally
smoothed vertical velocities in isobaric coordinate system at S00-mb level
(a) and atmospheric pressure at sea level (b), aver~~ed t'or July-August. �
6 ;
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'['fie dessication of the continents r,aturally led to a decrease in evapora- `
r~Ion Erom the surface of the cont~nents and 3ccordingly to a decrease in
~~rrc~f~~itriC[~~n, rigure 2.i show~ the differences in the rnte of evap~ration
b~t;wc~~n ~I~~~ "Ar(c1" .u~d c~unlrc~l vurlr~nLH, wi~~rrut~ rig. 2b ~~I~uw4 HimLl~tr cllf-
ferences in the rate of precipitatioti. (In Fig. 2 shaded regions corres- '
pond to ne~ative differences, whereas regions without shading correspond
_ to positive differences; the isulines are drawn at a logarithmic sca2e: t0.2
_ and fl cm/day) .
_ However, precipitation decreased not only over the continents, but also ~ver
the oceans. (Fi.gure 3 shows the total quantity of precipitation per day,
averaged during July-August over the continents (a) and over the ocean (b);
_ 1) control variant, 2) "Arid"). This is associated with an intensification
of descending vextical movements over the ocean in the "Arid" variant. Fig-
ure 4a shows the differences in vertical velocities ir. an isobaric coordin-
ate system at the 500-mb level between the "Arid" and control variants; the
unshaded areas show regions of inter.sification of descending (or weakening
of ascending) vertical movements. Figure 4a shows an intensification of as-
cending (or weakening of descending) movements over the continents in the -
"Arid" variant (shading; the isolines are drawn each 20 mb/day). _
Tl~e intensification of ascending vertical currents over the continents in '
the "Arid" variant caused a pressure decrease over the continents as a re-
sult of an increase in the temperature of the continental surfaces (and the
lower troposphere over them) due to a decrease in the heat loss on evapora-
ti.oti and a simultaneous decrease in the cloud cover, increasing the radia-
tion heat influx. Figure 4b shows the difference in atmospheric pressure at
_ sea level between the "Arid" and control va-riants (shading region~ of
pressure decrease; isolines drawn each 4 mb).
The intensification of ascending vertical currents over the continents in
t11is "Arid" variant must not only intensify the descending vertical cur-
rents over the ocean and the inflow of moister air onro the continents
from the ocean (intensification of planetary monsoonal circulation in the
middle latitudes), but also somewhat increase precipitation over the con-
' tinents. However, a decrease in evaporation ovzr the continents led to both
an increase in air temperature and to a decrease in specific humidity and
both factors decreased relative humidity to such an extent that this ef�-
fect exceeded the preceding effect of a possib"le increase in precipitation
over the continents as a result of an intensirication of ascending ver-
tical currents over the continents and led to a general decrease in pre-
_ cipitation. Figure 5 shows a map of the influence of drying-out of the
continents on the intensification and weakening of different interacting
processes.
This numerical experiment demonstrated that the dry~ing-out of the continents -
can lead to a decrease in precipitation not only over the continents, but
~ 11so over the ocean; the drying-out of the continents can simultaneou~ly
inr.ensify the planetary summer monsoonal circulation in the middle latitudes,
7
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which can be an important condition in the annual cycle of climate for sum- _
mer radiation heating of the ocean.
� ~ . : - / ~ i-T'/~ ~ - CXOpOClil b UCAOp~NUA C -
/AIIIMOC~EpHOE (JIlO/IfHfle~ TeMnepamypa Bo3ayxa nobepxHOCmu KoHmuHeH-. ~
~ Ha ypOdHB MOJ~R ' Ha ypoBHe Mopa Z %~;moB u oKear+vi -
//.iii~..'i'i ii i ri}~ /iiii;~~"~ ,i .
Bme?raHUe Bo~dyxa Ho noy- A6conwmHOR OnanrHOCme
murrzHmbi OcneBcmGue do3ayxa g
MyccoNxou uupxynxuuu ~ ~
BOCXOaAUSUP dBUM(eNlIR ObAQ9Hb1U 0lIiHOCU/OPllbl.'OA B/IQMfNOCT
- bo3ayxQ 5 / ~
j o0 : 3a xa j
ir
CKOpocmn oca8rro0 6 crropocme ocaaKOB -
; ~
~
r
06waA cKOpncmb oca8xo8
i ii~i
Fig. 5. Diagram of influence of drying-out of the continents on intensif-
ication (unshaded rectangles) and weakening (shaded rectangles) of differ-
ent interacting processes.
KEY:
1. Atmospheric pressure at sea level
7_. Air temperature at sea level
3. Rate of evaporation from surface of continents and ocean
4. Inflow of air onto continents as a result of monsoonal circulation
5. Ascending air movements
6. Rate of precipitation
7. Cloud cover
8. Absolute humidity
9. Relative humidity
10. Rate of precipitation
' 11. Total rate of precipitation
BIBLIOGRAPHY
1. Budyko, M. I., ATLAS TEPLOVOGO BALANSA (Heat Balance Atlas), Moscow.
Gidrometeoizdat, 1963.
2. Manabe, S., Ha~in, D. G., Holloway, S. L., "Climate Simulations With
~GFDL Spectral Models of the Atmosphere: Effect of Spectral Truncation,"
PROC. OF GARP CONFERENCE ON CLIMATE MODELS, Washington, April 1978.
8
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- 3. Taljaard, J. J., Loon van H., Crutcher, H. L., Jenne, R. L., "Climat~e
- of the Upper Air: Part I. Southern Hemi~phere Temperatures, Dew Point~s
and Heights at Selected Pressures," NAVAIR 50-IC-55, Superintendent of
_ Documents, Washington, D. C., Vol 1, 1969.
9
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~
UDC 551.510.522(-062.4)
CONSTRUCTION OF A MODEL OF THE ATMOSPHERIC BOUNDARY LAYER FOR THE
EQUATORIAL ZONE
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 79 pp 12-22
[Article by Professor Ye. M. Dobryshman, Institute of Atmospheric Physics,
submitted for publication 22 May 1979]
Abstract: The author analyzes the difficulties
involved in constructing a model of the plan-
etary boundary l~yer for a narrow equatorial
zone. The well-known Kuo model of a plane
boundary layer is refined and a model of the
three-dimensional boundary layer is construct-
ed within the framework of the linear theory.
In the latter case the vertical velocity max-
imwn is in.the layer 1.5-3.0 km, which agrees
~~ith observational data: the angle between the
outer zonal flow and the wind velocity vector
near the surface is less (31�) than accord-
ing to the classical Ekman formula (45�). ~
[Text] The classical formula for the thickness of the planetary boundary
layer (PBL), determined from the ratio of the viscosity coefficient v to
the main Coriolis parameter .~1, loses sense at the equator. This occurs
because the values ~,1 = 2 ulsin ~ with 0 become equal to zero, which
reflects, in particular, the fact of an impossibility of using the quasi-
geostrophic approximation in a narrow equatorial zone with a width of ap- -
proximately 500 km on each side of the equator [3, 4] (~1 = 7.29�10-5 sec'1
is the angular velocity~of the earth's rotatio~, ~ is geographic latitude).
In the equatorial zone the relationships between the wind field and the .
pressure gradient are extremely complex, not fitting within the framework -
of such a simple linear operator as the operator of geostrophic corres-
pondence. The fundamental difficulties in formulating a model of the boun-
dary layer (BL) for the equatorial zone are as follows: first, it is neces-
sary to be able to "splice" the model with the Ekman model of the PBL; sec-
ond, the model must take into account all three wind velocity components
in the equatorial zone. The latter circumstance indicates the necessity for
drawing upon models of the three-dimensional BL, which, as is well known,
10
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is more complex, since these have a number of specif.tc peculiarities [11].
In this article we will refine one of the models of a plane BL and will
examine a variant of a model of a spatial BL.
In attempts to formulate a PBL model for the low latitudes using both anal-
- ytical methods [13] and numerical methods [10], emphasis is on allowance
for the vertical wind velocity component w-- one of the principal factors
in virtually any circulation mechanism in the equatorial zone. "Qualitative"
considerations on the role of w can lead to contradictory results. For ex-
ample, if we use as a point of departure the general idea of a PBL in which
the wind changes with altitude in such a way that there is "pumping" of air,
it must be assumed that on the BL boundary near the equator there is a pos-
itive vertical velocity component (it gives rise to or favors the convection
process). On the other hand, a very simple model of a one-dimensional BL
[13], not taking into account the principal peculiarities in the dynamics
of the equatorial atmosphere, leads to a solution with general descending .
' movement. The idea behin~ the formulation of such models is as follows. We
will examine a constant zonal flow in the free atmosphc:re (u = Up = const).
The system of equations for the BL is taken in the form
a?, a~ � av v a~ v av am
7�' az az, ~ w ds + 2~� ro u=~ az-; dy -F- ~s = 1
- Here z is the vertical coordinate; y is the horizontal cornponPnt, reckoned
from the equator to the north; r~ = 6.37�106 m is the earth's mean radius.
System (1) must describe the process in the three-dimensional boundary layer
because all three wind velocity components are present. In order to reduce -
_ the problem to a plane BL system (1) is integrated for y from the equator -
(y = 0) to some value y= L(N 500-1,000 km), where the BL structure is
close to the structure of the Ekman BL. With y= L the zonal component of
wind velocity outside the BL can be determined approximately from the geo-
strophic relationship. By postulating the nondependence of each of the aver-
aged components on y, we find that the nonlinear terms of the equations of
system (1), integrated for y, remain unchanged in form. Thus, for the values
c t ~ t
- u- ~ udy; v- L f vdy; w= wdy
0 0 0
we obtain the system of equations (the lines over u, v and w have been dropp-
ed)
da d"u dv 2 w L d~ v dur
~ ~ 2 )
. w dz dz1 ' ce dz r~ a tt='~ dz~ ' L ds '
The ~X-parameter has been introduced for the sake of universality. Its value
is dependent on the hypothesis determining the "averaged" value of the zon-
al velocity component. In [13] it is assumed t?~at OC = 1. A more accurate
value is oC= 1/2; in this case the procedure of removing the mean values
from t}ie integral sign is identical for all the terma in yystem (1).
11
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If we exclude u and v, we obtain one nonlinear equation for w in the form
d~ ri~tr~ rf~ w d d~w d~ w
, ~ dz~ ~.w d:~ ~is~ ~ w ~is I.~ ds~ - v ds~ 1~ ( 3)
It must be solved with the following boundary conditions: ~
~ dw d~w this is e uivalent to u= v=
z-~' dz - dz~~ Q
Lt = 0~�
z oo v 0 = = ~s = ~ ~4~ -
u= U~ u' - W~ 2 w jo UW = ~ -
where we use Woo to denote the w value when z--~ oo ; this value is deter-
mined frora the expression
~ a Va, v d~ w l ~ 5)
~ ro ds3 s~m
d~~ -
dz= I Z
(The operator
d~ w d= w
v--~~--
dz3 dz=
is typical for houndary layer prohlems [11]). Equation (3) cannot be pre-
:~isely integrated; it is reduced to a nonlinear integrodifferential equa-
tion
~ f se~d:
� dw-~~i~:-C,~e`. dz-}-C.
(6)
When C2 =~0 and C1 = 80~? equation (6) has the partial solution w=
-Sv/z.]
It can be seen from this form of the formula that when w< 0 equation (6) ~
is conveniently integrated approximately, for example, by iterations.
In the case of westerly flow (u~ 0=~' du/dz> 0 in the lower layer of the
atmosphere) the turbulence is compensated by the vertical flow u, as fol-
lows from the first equaCion of system (2) or (1). This means that with
u) 0 we should have w< 0.
As a first approximation it is proposed in [13] that it be asstnned that
- w=-wp = const, which after substitution into system (Z) reduces it to
an easily integrable linear equation. The general integral for the deriv-
ed equation has the form
_w~= _ _ -
w_(C, + C., z) e � -I- Cq (;a z-F~ C:, ZZ� .
If there is rigorous satisfaction of all the conditions (4), then C4 = C5
= 0; C3 =-W and for the remaining two ~rbitrary constants a contradic-
tory system is obtained. Accordingly, Kuo [13] satisfies only one of the
conditions and a"correction" is introduced into the second. As a result,
for w a solution is obtained which in the adopted notations can be written
as follows:
1z
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_ ~n s
~c+ _ W�� ~ ~ Zl e ~ - 1 . ~
/
~~1
In thia case w~ must be expressed through the parameters of the problem,
nut lepending on the zero approximation wp), and specifically U~,
2 w/r0. Using the methods of dimensionality theory [1], it can be
shown that W oo is not uniquely determined through these parameters. In ac- _
tuality, the combination
i U1_3pV^p(2l' lp $
u' o ~ ~
for any p has the dimensionality of velocity, which in this case can be
interpreted as w~. In [13] a value was adopted for w~ corresponding to _
- p= 1/4 in formula (8). This gives
' (9)
W~, _ [ ro' U~ ~z]' �
Assuming 10 m2 �sec-1; U ov = 4 m�sec'1 we obtain [,1~ N 10-z m� sec 1.
It is easy to see that
s
p= ~ W~ =~~U~~)z ~o ~ 4~ 10-s m�sec'2; .
when .
~ 6
when p= b W~ = YU~ 1' v~ 2ro ~ 5� 1(1-~ m� sec 1.
With other "reasonable" assumptions the W~ value is unnaturally small
(when p= 1/2 W~ ^~2�10-5 m�sec l, when p= 1 Wa,^~10-1U m�sec 1, etc.). -
The continuous curves in Fig. 1 reproduce the results from [13] for u, v,
w; the last two functions are easily found from system (2) after substit-
ution there of the zero approximation for w:
u,,, ~ i Q�o
1 - ~ (10)
n- U~ 1- e v= L r`~ U~, ze .
~ i ~
The structure of these formulas is similar to the Ekman formulas for the
PBL. However, it must be remembered that they were derived at the expense
of "coarsening" of both the physical model and the mathematical methods
for analysis of the initial system of equations. In addition, the model
does not take into account the dynamics of the processes transpiring in
the equatorial zone, it is unsuitable with Ua, ~ 0-- an easterly flow,
which is typical for the equatorial zone [6, 12]. The Kuo model is not
13
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ciCed for critical purposes, but as an illustration of the difficulties
arising in formulating a BL model in the low latitudes in general and in
the equatorial zone in particular. -
8
_ y Iv y I ~
1
.
,
,
o ~ ` ~
0 1 -
Fig. 1. Vertical profiles of wind velocity components in dimensionless form
using the Kuo model [13~. The dashed curve shows the proper value w for
small z. .
The model can be made more precise, for example, by introducing a parametric
dependence on y and stipulating the zero approximation~for w not in the farm
of a constant, but selecting a function of altitude which is more real for
the lower layer, for example, a power-law function. We will assume
2Q1 = - ?Qlo (
h i~~ ~ 11 ~
\ /
where h is the characteristic height whicui must be determined proceeding '
on the basis of dimensionality considerations. As in the case nf velocity,
the representation of h is unambiguous through the parameters of the prob-
lem 2~�J /rp; v; U ~,p . Specifically for any p ~
- p ,,,+2 p - -'-3 p
h-r f~ 1 U� . ~ ,
~ ~
The h values "reasonable" from the point of view of interpret~~tion of BI,
theory are obtained, for example, for .
_ - - - - .
p-- 4 h-Y 2~W " U~ 10~~t; for p=- ~ 1i=Y 2~ "a ~
W U:,o
^'0.3�103 m. Nonallowance for the Rossby parameter (p = 0) leads to the un-
expected result: h= y/U~ 2 m. It is probably possible to interpret this
result as follows: this is the characteristic thickness of the Prandtl
boundary layer, which is unrelated to the macrocirculation mechanism. Here
tne scales are completely different: W^~0(u). ~
Then, the parameter r) 1. In actuality, from the continuity equation
_ 14
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_ :
~ _ - ` dy dz
0
and by virtue of the condition v~ z~0 = 0 for the function w the point z~
c) must be zero of a higher order than for v; > r~1.
Substituting (11) inro the first equation of system (1), we obtain
z wo sl~r -
- h; z~ aZ = Y as u= C, + CZ ~ e Y h~ ~~+r) dx.
_ o
Satisfying the boundary conditions2 we find u_in the form
ar~� z i t, 1
Y C vh~ (I -r r) � r�4- l, (j2) -
tt = Voo (y) ~ ,
~~r+l)
where y(p,`q) and r(p) are the incomplete and complete Euler gamma func- _
tions respectively [2].
Substituting (11) and (12) into the second equation of system (1), we ob-
- tain for v a linear inhomogeneous equation whose solution can be repre-
sented in the form
- m~
v=[ V�~)') - A~Y)] u(Y~ z) t~ e ~ I~~ (rt 1) dzi ~13)
[10, (Y )
z~ wo Z.+t
z e�~~Hj hr Z dz.,,
X Q ~ z, Y )
0
where, for the sake of brevity, we have introduced the notations
v~ (Y) = v (y, z) ~
W t�o si~'1 i~ wn s~+l (14 )
A(Y) = bf e Y(r+ 1) ?i= dZ~ f Q(zz, Y) e v h~ r+ 1 dzz~
Q(z, Y) - 2~' a u~Y~ z).
(It is easy to confirm that all the integrals entering into (13) and (14)
converge: wp, h, r- 1> 0) .
Differentiating (13) for y and then integrating for z, we obtain a more pre-
cise formula for w(z) than (11):
v� (y)-A' (Y) r wozi+i 1 1 da _
w=- r 1 ,~T`vh'(r+l)' r+l~
. (r+l) (15)
. . mo ~~1 Z , mo t2+~
's r' e ~ hr ~r t 1~~' ~Q (~y ~ Y) e v hr (r+1) dia dzZ dz,.
~
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rux ~rrt~ttu, u~r, uNLx
An anal.ysis of formulas (12), (13) and (14) shows that with small z uNz;
vN z; w N z2. Tficrefore, evidently, as a zero approximation it i~ naCural
to take w=-wp(z/h)2, that is, assume r= 2. In this case the gamma func-
tLons ;uil.t be oE the parameter 1/3.
Strictly speaking, the dependence of the functions on y is not "purely"
parametric the derivatives of y enter into the expressior., for (15).
As Ucao and Voo it is possible to take a suitable solution corresponding to
a stationary model of circulation; in the simplest case it is possible to
obtain a solution, for example, from [3]:
(1a, = uo - cu ~ , _ -V~ _ j
01 uo ~ 1 - r _
0 0 o~y
(up < 0 is easterly flow).
As can be seen from the solution found (13)-(15), allowance for the nonlin-
ear terms, even in very approximate form, by the stipulation of w in the
_ form (11), which actually brought about linearization, to some degree
made ir possible to reflect the interrelationship between the boundary lay-
er and the external circulation mechanism.
Now we will proceed to an examination of another model in which the spatial
- structure of the BL is manifested more clearly than in the case considered
ahove. This linear model is based on the same Ekman idea a correspond- _
ence between dissipative forces and Coriolis accelerations with a stipul-
ated pressure gradient 2�/d y. We will write the initial system of equa-
tions in the form _
y di 2 w w- 0
Y
~~Z 2 w ro tt a~ = 0 . ~ 16 ~ ~
tly + �
~r' = 0
Here in the first equation we have omitted the term - 2~ y/rp v for two
reasons. The basic reason is a peculiarity characteristic for any BL: the
~ basic wall (underlying surface) effect is the appearance of a mass flow
in the direction perpendicular to the wall. Accordingly, in the BL the
terms with w are more important than the terms with v-- velocity "across"
is less than the velocity "along" the main external flow ~v ~u~. Sec-
ond, allowance for the discarded term complicates the algebraic part of
formulation of the model, which is scarcely justified in such a rough
(linear) model, presented for the most part to illustrate the overall
qualitative picture and the difficulties in formulating a good model. This
~ simplification should not have a strong effect on the characteristic of
particular interest w. We can formally introduce a small parameter in
front of the term _2 v
rp
16 ~
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and seek a solution in the form of a series in powers of this small para-
meter: in the zero ap~roximation thia term disappears.
Excluding v and w from system (16), we obtain one equation for u
� a;" 4 � ? ri = - 2 ~ dy . (17)
dz~ w- ay ru
In the BL the pressure gradient is naturally considered to be independent
of altitude. Assuming that the solution (17) can b~ represented in the
form of the product
u= ui (J) u~ (z) . ~18~
we find
1_ y (l-I~ro)*
u C ~lg')
[It will be assumed that ul is a dimensionless function and u2 and there-
~ after v2 and w2 are functions having the dimensionality of velocity.]
where ~ is the separation constant. We will determine it from the condi- ~
tion that outside the BL the velocity of the zonal flow should be depend-
ent on y in the same way that this is determined in the problem of the
circulation mechanism without taking dissipative forces into account. In
the simplest case, with a symmetric distribution of pressure u~ y2. Hence
~ _ -3/rp.
This means,
_ u~ - L? . (19)
where L corresponds to the latitude at which u can be determined from the
geostrophic relationship. As is easily checked, in this case the pressure
gradient will be a function of the type
d ~ y3 .
~y
For u2(z) w.e obtain the equation
dQz~ ~~ro u== ~ . ~20) .
The roots of the characteristic equation are S5 + 12 w 2 _ p~
y r0
Sk_~ Y,~o [cos 5 (2k-1)-}-isin j (2k-1),,k-l,?,3,4,5. ~21~
17
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It can be seen from this ~ormula that S1 and S5 have a positive real pa~rt.
Accordingly, the arbitrary constants with these so]:utions must be assumed
equal to zero.
Thus, the solution (20) can be written in the fonn
u: ~z) = U~. -F- Cs es' "-I- Ca es3 Z-~- C~ .es` Z,
where U~ corresponds to a partial solution of the. inhomogeneous equation.
Hence we find : 12 _ � ~2 ~o;
- v~ r Voo - ~-J r ~Ia..
0 0
In order to determine the integration constants it is necessary to stip-
ulate three conditions;
1)
z=0 u=0~uz=0~
which gives C2'I'C3"I"C4=-U~,.
2~ z=0 w=0-> aZ'-0,
~~e obtain ~ -
CZ Cy S3 C4 S4 = O. '
3) From ttie continuity equation dv
dy I:_o = 0 ~
a~ I ~
as - ~ ~ az~ I:=o= 0.
:-u
Thus, the third equation will be
C= SZ C~ S3 -F C~ S4 = O.
Solving a system of three equations relative to C2, C3, C4, taking into ac-
count the values of the roots Sk, we find their powers S7~ and make use of
the Euler formulas
et's = cos x�i sin x
sin ~
' Cx=C~=-U~ _
5 3 ~ ~ -.0,'l76 U�
2 sln 5 T sin 5 � '
sin 3
"
C3 U�� 5 s,~ - 0~447 U...
2 sin 5-~ sin 5 ~
18
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~ Combining the reaults, we write u2 in tlie form
1 '
s
uz = Um 1- 2 sin cos lU e- t coS 10 ( 2>
2
+ 2 e C atn 10 sin 5~os (t cos .
where, fo r the sake of brevity, we introduce the notation
_ _ . C - s I'l z
v~ rp
Assuming the value 'V = 10 m2�sec-1 to be usual for the turbulence coef-
ficient, we obtain
Z.: 10-3z ( z in m) .
The working formula for computing u(y, z) can now be represented as
u- U~, 1- 0,447 e- s- 0,553 e- = cos (0,951 z)} ,
where z, y are in km, L^'S00 km.
From the first equa~ion in system (16) we find .
w � d~ r~
2 W ds~ '
This means that the dependence of w on altitude will be determined using
the formula
v - 12 w-
~ 5 ~ ^ _ _ _ 1 ( cos " � e- c -
w - U~, -
. 2~ ro 2 sin 5-~- cos i~ l 10
( 23)
~
- 2 sIn 5 e 10 cos (~-I- C cos i~ .
From the continuity equation we find the dependence of v on altitude:
s
9s = U~ L 1 ( - cos" e- ~
~ "ro 2 sin ~ cosiu ~ !0 (24)
~ 3 r, a
2 sin 5 e~ sin sin ('1U C cos la
19
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Expanding (22), (23) and (24) into a series of powers of z it is easy to .
show that with small z we will have
uZ I~V zJ v~ N Zl iQ12 Z~I. _
2
. S.
4
7.
3
_ 4 .
- , I 2
3 w
I ~
I ~ ~
11
u 2 ; t t
~ i~ ~ i ,~y y
~ ~ii~
~
U 0, 5 0,4 0,3 0, 2 y ' j~ I i �
~ ~ ~
'Q2-Q1 �0,1 0,2 !{J w I
i'~3
~ _
Z. .
x '
Fig. 2. Vertical profiles of wind Fig. 3. Hodograph of wind velocity
velocity components in dimension- (1), its projection onto the hori-
less form using model (16). In order zontal plane (2) and the Ekman spir-
to facilitate reading of the dia- al (3). The small cross indicates
gram the positive direction for u the limiting value (with z--s c~o) of
is indicated to the left. the velocity h~dograph projection.
It can be seen that in accordance
~~ith model (16) a tendency to the
limiting value occurs far more slow-
ly than according to the Ackerblom
~ model.
With respect to the dependence on y, for w it evidently will be the same
as for u, but for 3
Y
~ N 3L3 .
20
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In general, from the point of view of BL theo ry the function U oo(y) is
considered stipulated and it can be selected, to a certain degree, arbi-
trarily. If in (18') it is assumed that 1- ~ r0 = 0, that is, if it is _
assumed that Un, = const, then only the one function v will be dependen.t
on y. (In this case d~/~ y ti y). The dependence of the solutions on z does
not change. Such a model is ~ustified by the fact that when y= 0, that
_ is, on the equator, the boundary layer does not disappear: the components
u and w are not different from zero.
In such models w-?0 when z-y ~~o, which cax?no t be the case for a plane boun-
dary layer; in a three-dimensional case there is a possibility for compen-
sating the vertical flow due to a change in the second component of hori- -
zontal velocity. In these models u can be of any sign. Figure 2 shows the
u2( v2( and w2(~ ) profiles in dimensionless values. Normalization
was carried out for the corresponding factors
'r 12 ,or -,r ~ - 3-3
~ Um, uco ~-ro ~I v~!'u ) ' ~ vr3 .
~ 0
Figure 3 is a velocity hodograph for y= 1 and its pro~ection onto the plane
xoy. It can be seen clearly from the figure that w-r0 when z-+~o due to the
fact that aloft v fluctuates near zero with an attenuating amplitude.
It should be noted that the main maximum w falls on the value ~ N 2.0. In
dimensional values this mr:ans that in the layer 1.5 to 3.0 km vertical
movements are particularly appreciable. This agrees fairly well with the
results obtained during GATE (Atlantic Tropical Experiment - an internation-
al program carried out in the summer of 1974 [12]).
One of the characteristics which is usually determined in models of the at-
mospheric boundary layer is the angle between the isobar and the limiting
- position of the velocity vector when z= 0, that is, at the earth's sur-
= face. In this case i~ is correct to speak of the angle between the wind di- ~
rection outside the BL and the limiting value when z= 0, although actual- ~
ly these concepts coincide here: the isobars run parallel to the equator
and the principal external flow is zonal. In the classical Ackerblom model
[7J, as is well known (when y= const) this. angle is a= 45�. However, in
general it is dependent on the turbulence regime (on stratification) in the -
boundary layer [9], but it is less than 45�. It is easy to compute this
angle for the considered model as well:
5 l - sin-"
tga = lim v= lim v.~ (z) = L~ 54 ~ lu
- x..o u z~o u: (Z) ~ r~~ 2 tg ~ sin Z'~ .
lU 10
, Digressing from the dimensionless factor, which with the selected values of
the parameters is the value 0(1), we find a~ 31�. To be sure, this is
only an approximate value whose main sense is that it is less than 45�
the model value for the~PBL characteristic outside the equatorial zone.
21 ~
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llespite a number of shortcomings the considered model of a three-dimension-
al boundary layer reflects some qualitative aspects of the processes trans- '
piring in.the lower layers of the troposphere in the equatorial zone: a
maximum of vertical movements in the layer 1.5-3.0 km; a tendency to its
limiting values of the ~aind velocity components with altitude slower in
camparison with the PBL; a lesser an gle between the main flow outside the
BL and flow at the earth's surface.
I consider it my duty to express appreciation to I. B. Kazitskaya for ae-
sistance in finalizing the article.
BIBLIOGRAPHY
1. Barenblatt, G. I.~ PODOBIYE, AVTOMODEL'NOST', PROMEZHUTOCHNAYA ASIMPTO-
TIKA (Similarity, Self-Similarity, Intermediate Asymptotic Behavior),
Ltningrad, Gidrometeoizdat, 1978. ~
2. Gradshteyn, I. S., Ryzhik, I. M., TABLITSY INTEGRALOV, SjJr4f, F,YADOV I
PROIZVEDENIY (Tables of Integrals, Sums, Series and Products), Moscow, .
Fizmatgiz, 1962. ~ _
3. Dobryshman, Ye. M., "Some Peculiarities of the Pr~ssure and Wind Fields
in the Equatorial Region," METEOROLOGICHESKIYE ISSLEDOVANIYA (Meteoro-
logical Investigatians), No 16, Moscow, 1968.
4. Dobryshman, Ye. M., "Determination of the Width of the Equatorial Zone,"
ME'iEOROLOGIYA I GIDROLOGIYA (Meteorology and Hydrology), No 12, 1973.
5. Dobryshman, Ye. M., "Dynamics of Atmospheric Processes in the Tropical
Zone 5-15� in Latitude," METEORO~OGIYA I GIDROLOGIYA, No 4, 1975.
' 6. Zaychikov, B. P., Romanov, Yu. A., "Peculiarities of the Wind Regime,
Temperature and Air Humidity in the Equatorial Region of Central Atlan-
tic," TROPEKS-74 (TROPEX-74), Vol I, Leningrad, Gidrometeoizdat, 1974.
7. Koshmider, G., DINAMICHESKAYA METEOROLOGIYA (Dynamic Meteorology), Mos-
cow, Gosizdat, 1938.
8. Krivelevich, L. M., Laykhtman, D. L., "Meridional Structure of the Plan- ~
etary Boundary Layer of the AtmosphQre in the Low Latitudes," IZV. AN
SSSR, FIZIKA ATMOSFERY I OKEANA (News of the USSR Academy of Sciences, -
_ Physics of the Atmosphere and Ocean), Vol 13, No 7[year not given]. =
9. rionin, A. S., Yaglom, A. M., STATISTICHESKAYA GIDROMEKHANIKA (Statis-
tical Hydromechanics), Moscow, Nauka., Part I, 1966, Part II, 1967.
10. Pushistov, P. Yu., "Results of Numerical Modeling of Stationary Circula-
tion of the Atmosphere in the Equatorial Region," IZV. AN SSSR, FIZIKA
ATMOSFERY I OKEANA, Vol 9, No 3, 1973.
22 -
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~
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11. Roze, Kibel', Kochin, GIDRODINAMIKA (Hydrodynamics), Vol II, Moscow,
Gostekhizdat, 1954.
12. TROPEKS-72, T':OPEKS-74 (Tropex-72~ Tropex-74), edited by M. A. P~etro-
syants, Lenin�rad, Gidrometeoizdat, 1974, 1976.
13. Kuo, H. L., "On the Planetary Boundary Layer of the Equator," J. ATMOS.
SCI., Vol 30, No 1, 1973.
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UDC 551.511.12
SMALL OSCILLATIONS OF THE POLYTROPIC ATMOSPHERE AND THE FILTERING ROLE
OF THE IiYDROSTATIC APPROXIMATION
rioscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 79 pp 23-33
[Article by Candidate of Physical and Mathematical Sciences V. M. Kadysh- ' -
nikov, USSR Hydrometeorological Scientific Research Center, submitted for
publication 23 May 1979]
Abstract: The article gives a comparison of ~
the frequency spectra of wave moven~ents of
the polytropic atmosphere in nonhydrostatic
and hydrostatic cases. In the first case the
_ dependence of the solution on altitude.is
descrilied by confluent hypergeometric func-
tions, and in the second case by Bessel
functions. It is shown.that the hydrostatic
approxitnation, as in an isothermic atmosphere,
filters out acoustic oscillations and consid-
erably distorts the first modes (that is, those
changing least with altitude) of gravitational
oscillations only with a wavelength shorter
than 100 km.
(Text] In [6] the authors obtained an anal~~i,;~1 s~Iu~~~i~ of the problem of
small oscillations of a nonhydrostatic atmosphere under the condition that
the linearization of the equations of hydrothermodynamics is carried out
relative to an isothermic state. An analysis of this solution indicated
that the atmosphere is characterized by wave processes of different nature,
to wit: there are high-frequency acoustic and low-frequency gravitational �
oscillationS. It was also demonstrated that the hydrostaticity hypothesis
filters out acoustic wave~ and the spectrum of gravitational oscillations
is distorted the lesser ths longer the corresponding wave. In.[2] a sim-
ilar problem was examined for a model of the atmosphere with a real te~-
perature stratification, but the solution obtained to a considerable de-
gree is qualitative. In our article [4] we gave an analytical solution for
a neutrally stratified atmosphere. In such an atmosphere there are no grav-
itational waves. The objectivE of this article is solution of the problem
of small oscillations in a stably stratified, nonhydrostatic, polytropic
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_ atmosphere and a quantitative evaluation of the role of a hydrostatic ap-
proximation in this case.
A~ d~~monRtr~itc~d in (fi~~ the :;yatem of equatione tn hydrothermodynamice J.in-
e,irL~~~~ rel.utlvc lu :i :~L.tCc ~~1~ t'cal h~tv lltr 1'ur.m
rc~ pX lv, v~ py - lu, ~~e P: - P g,
Pe = - (ux + vy ~s), Pe = - 2ro - e" (us vy w:), (1) .
where u, v, w are F u*, P v*, ~ w*:/~ is the density of the main state (its
_ pressure and temperature are p and T; all these three functions are depend-
ent only on z), and uk, v*, w* are velocity vector components; p and Y are
- the deviations of pressure and density from the corresponding values of
the main state; 1~ is the Coriolis parameter (constant), g is the accelera-
tion of free falling, c2 p/p ,~.is the ratio of heat capacities, (r =
i ?~R( ~'a - r) is the stratification parameter, R is the gas constant, Ya is
- the dry adiabatic vertical temperature gradient, ~'=-dT/dz.
We will solve the problem with initial conditions periodic relative to x,
y. Now we will consider formulation of boundary conditions for z. We will
a~sume that the underlying surface is impermeable, that is, at the earth
w = 0. This gives
~=0 (z=0). (2)
For formulation of the second condition we note that the following formula
follows from system (1) ~6]: ~ - ~
~r + ~u# P)X + Iv'~ P),, -'t- P): _ ~
where the quadratic form (energy) is
~x + v~ ~ L 1 , ( )
E 2 0 2-c: [P'" B~P - P)",. 3
P
If, accordingly, from the initial data it is required that
, ~
_ ~ Edx dy ds < oo (4)
(D is the periodicity region), then with condition (2) and the condition
~~"~*P~ z-r a~ ~ that is
~p ~ (Z -s c~o ( 5 )
the energy of the svstem will be conserved. Moreover, below we will confirm
that if we take (4), rather than (5) for all t~ 0 as the second condition,
condition (S) will be automatically satisfied, that is, from the limitation
of energy follows its conservation.
_ Thus, we will seek a periodic solution of the system of equations (1) under
conditions (2) and (5). As already mentioned, the y value falling between
0 and ya is considered constant, so that r"= const. The altitude of the
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polytropic atmosphere Ii is finite and equal to Tp/r , where T~ = Tlz a 0�
We will represent all the functions in the form
f~x, y, z, t) =f ~z)expi (k jz+k~ y-~. t).
It follows from the equations of horizontal m~tion that the amplitude of
plane divergence is, t7lk': ~
- ,,1 _ P, ,
where k2 = 1cX + ky. We introduce three-dimensional divergence ~G= uX + vy +
wZ. We have
- . _ 11? k= ~ .
Y
z= a- l, P-F- ( 6)
The third equation of motion (with the continuity equation taken into ac-
count) and the heat influx equation give, respectively
-i?.z~-{-pr- ~ ~=U,
~ ~ ~ (8)
-ilp-~I'w-}-c'X-O.
Lde have a system of three ordinary differential equations (6)-(8) for the
functions 'C, p and w. We will write an equation for some one of them.
Since the coefficients of the system are variable (c2 is a function of 2),
for each function there will be a,specific equation. It is convenient to
examine the equation for '~C:; its eigenfunctions are single-term functions
(it goes without saying that the final solution of the problem is not de-
pendent on the choice).
We will express p from (8). Then we differentiate this expression and we
will substitute the p and pZ values into (6) and (7). We will solve the
derived system of two equations as an algebraic system relative to w and
wz. In particular, we have .(c? = s),. ~
xRY(a=--1=) ,g+1' `
~ a~ (a~-n~-k~r�- xR7~ - 1 - ~9~
xRY -l'-') S~ ~-s%~'.,
Differentiating this expression and comparing it with the already deter-
_ mined wZ, we obtain
g + ~ ~ k~ . ~k= I'
S7,~s- ( Sl - 2~ Si ~a=-t=) S + s2 + Sl ~A:_~_~,"/. _ (10)
:
where ~
_ s-c2=so-s~z (so=xR3`o, s~=xRy). .
In the derivation of this equation we had to exclude from consideration the
roots of the equation
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~.(7~~-l') ~~2~~2-1~)-k~I'2] =0. (11)
They must be investigated separately. It is easy to show that the frequency
0 is the solution of our problem. It corresponds to a geostrophic hy-
drostatic case. Howe^er, the ~lvalues, bringing the terms in parentheses
and brackets to 0 in (11), are not solutions: in the first case from (7)
and (8) for w and }
we obtain a homogeneous algebraic system of equations
with a determinant different from zer.o only at the one point where the
linear function c2 is g(-/,~ 2, whereas according to the equations of motion
p t'0; in the second case, from (6)-(8) we obtain a similar system for Y
and xs which can be different from 0 also only at one point where c2 is
r 2/ it therefore follows that there are no continuou~ nontrivial solu-
tions in the two cases.
The solution of the fundamental equation (10) of the problem is
x=CiJi~S)+CsJs~s), ~12)
where C1 and C2 are integration constants and the linearly independent
special solutions are [5] _ as `
Y~ = e ~(1 - 2- 3, 2- 6, as
. aS ~ 13~
_ Y= = S�-~ e 1~( 2 b+ as),
cind the function of the three arguments ~(r, m, x) is a confl�ient hyper-
geometric function (8), determined by a Pochhammer series
~~=1 ~ ~ r(r-i-1). . .(r+n-1) x" .
m (m-r 1). . . (nc+n-1) n!' (14~
� n-1
Here we have introduced the nota~ions
a= 2 x k S~~ a= ~ a= r-1 +~k= r b_ g+ r (15 )
S Si 1 ~i:= - ' 'l A ks, ) ' s~
Using (9), and in particular, the fact that when s~0 wNs ,L S- (b - 1)~1',
and also that p�~~w (this, incidentally, confirms the advantage of reduction
of system (6)-(8) to an equation specifically for x.), we substitute the
resulting solution into condition (5), in a polytropic atmosphere exist-
ing, as already noted, when s= 0. Since P-~ sb, this gives
Qi~i +a~zC~C~+asC2 =0, _
and al is a linear combination of the powers s2-b, sl-b, s-b, a12 the
powers sl, s~, s'1; a2 tlie powers sb, sb'1, sb-2. Since
x 7=-1= ~ Yrt _~]2~
b - x-1 ~ 7
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C1 = 0 and the C2 coefficient can be arbitrary. Thus, with an accuracy
to a constant,factor the solution of equation (10), satisfying condition
(5), has the form
a
% =S�-1 e `s ~ ( 2 - P~ b, a:)� (16)
IE we found the remaining amplitudes from the function x, determined by
Eormulas (12)-(15), and formed energy E from formula (3), condition (4)
Cor all t> 0 would yield this same solution. Accordingly, we see that
its con~ervation will follow from the condition of restriction on the
Lot~l energy.
We wi11 make two coimnents. First: the solution (16) was obtained proceed-
ing on the basis that (13) are special solutions. But the latter is cor-
rect under the condition that the difference in the roots of the determin-
ing equation for (10), but the roots in this ca~e are b- 1 and 0.,
is not a whole number. Otherwise the second argument of the ~ function
for one of yi, in accordance with (13), is a whole negative number, and
- in accordance with (14) the eigenfunction does not exist. But, for e~
ample, with Y close to t:he real value 5.83 K/km, this is a whole number.
Nevertheless, i.n this case the solution of equation (10) under the condi-
tion (S) is the function (16). In actuality, in the neighborhood of the
regular singularity s= 0 in this exceptional case as one independent so-
luti.on, as before it is necessary to take Y2 (this corresponds to the
greater root of the dete~nining equation), and as the other, not yl, but
the expression P(s) + ayl ln s, where a is some number and P(s) is a power
_ series with a free term [7]. But only this, and specifically the presence
of a free term in the fun~ction yl, was used in the proof that C1 = 0.
The second remark is as follows: we will assume that 0< Y< Y. When Y=
ya there is also a solution of the formulated problem [4). At the same
time, oscillations of the isothermic atmosphere with restricted energy are
not possible. In actuality, in this case c~ = const and the equation for
p is the same as for ~C,:
g-F~ 1' k~ g i'
Pu P: [ (1- P= 0.
We denote g+ f/2 c2 by r and introduce R in such a way that the character-
istic equation assumes the form x2 + 2rx + r2 - R= 0. There can be three
forms of the dependence of p on z:
e-~= (C, stn z j~- R-E- C, cos z~) (R < 0),
e-.:(C~+C:z) (R=~)~
e-?z I C, es C= e- : vR ~~R ~ p~, .
2s
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Taking into account that w-~ pZ + g/c2 p, and J~ e'2rZ, we conclude that con-
dition (5) in the first two cases is not satisfied by the two apecial solu-
tions, and in the third case, although it is satistied, the lower boundary
condition (2) requires from it that
r(2-~~x12 _ Ig+rl~ _ k~a: gr -
L 2c' ~ ~ l~' ~ +~'-C'(1-~,~,1.
/
If this condition is satisfied, then w= 0. The corresponding ~l values can
be found. However, it is clear that it is impossible to solve the problem
with any general initial conditions.
Now we substitute solution (16) into the lower boundary condition (2). Us-
ing (9), we obtain the dispersion expression in the form
~1-2ti)~r? -3~-1, b~-l,al-(1- St~ c~(2 b~ al, ~17~
~ i 1 \ /
where ~
St = Si V~S~ ~)2 - g r, a=' as ~ s=so � ~18~ .
In this case we considered that, in accordance with (15),
i' - a S . k - YR~ -1~ . (19) -
-"2H ' -2H ~
Equation (17) determines in the plane of the variables a, ~ two families
of curves corresponding to the values S+ and S_. The first family corres-
ponds to acoustic waves because it owes its existence to atmospheric com-
pressibility. In actuality, we will assume that the atmosphere is incom-
pressible, that is, X--~ ~[6]. This means that the parameters r, sp, sl
also tend to infinity. This is in no way reflected in determination of
the ~~G para~qeter in (15) . At the same time
gk ( 7~ 1~1
~ ~ 2~7i~ e=-l~
Accordingly, in accordance with (18), S+ ->c~o. Thus, in accordance with
(19), ~~oo, that is, the assumption made actually filters out oscillations
of this family. In this case
_ S_~
k
The second family corresponds to gravitational waves because it owes its
existence to the presence of stratification. In actuality, with 0 the
parameter S= 0, so that a= 0.
It follows from the conservation of energy that all ~1 are real. Below it
will be dempnstrated that ~l 2~~,2, that is, oC and Q are also real. It can
be seen from (15) that oe~9 > 0. Now we will first examine the case a~ 0,
0, and then we will show that the case a< 0, ,8 0 is a sufficiently large whole number.
We note that the several first solutions of equations (Zl) and (22) can be
parasitic because sufficiently small a values under the condition ~~i b
can lead to ~ values which are too small. The hyperbolic equations approx-
imately following from (21) and (22) can be ~oined into one:
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r +c / . 20~ I 11~
- L 2`n 4 = 9t�
(23)
Here and in the text which follows the subscripts and will relate ~
to the acoustic and gravitational families respectively.
In case 2 we make direct use of the series (14). There are two possibilit-
ies: is limited and ~--;~~o . For the first for acoustic waves there
is no solution, whereas for gravitational waves we obtain f3 = b/2. And in
general, if S= S, the entire straight line j3 = b/2 forroally satisfies
equation (17). But the corresponding frequency curve, reducing the bracket-
ed term in (11) to zero, as already noted, is not a solution. With the sec-
ond possibility for acoustic waves there is again no salution, whereas for
gravitational waves we have
a~b+t~r
a~ - g+( b-I- 1)1" ( 24 )
= In this case it is assumed that ~p ~ 1, that is (~-s 0.
In case 3, in accordance with [8], we have
~ b b~ Q ~ (b-111 + (b-11! e� .
,
Z ~ 2 +a+l~~ (_a~~ ( ~ -~1--~~~ sb +a
~
Therefore, from equation (17)bwe obtain
(-1)~ 1 a2 q 8a
(25)
( 2 +~-t~i s+r ~l -~-i~!
In order for the riglit-hand side of quation (25) with oc.-~ro to remain, like
ttie left-hand side, finite (the a 2~ factor on the left-hand side cannot
ensure the corresponding increase), it is necessary that the denominator
on the right-hand side increase without limit, compensating the increase
in the numerator. Hence -
P=Pn-` I~R - ~ -}-iyl~ ~26~
~ 1
where n is any whole non-negative number and I�I ~ 1. In this case
b _ 1 ~ ~ ~n
\l - n!` '
Substituting (26) into equation (25), we find
e-a Qo+1R s_r
E- (b+n-1)! n! S+~'
With ~n ~ '
St =g 2 s, n�~(g 2 ~ I~ S,n (g i') 7. (s,~y)',
/
Therefore, for any n, taking into account that g>(', we have S+, Since,
moreover, the root is a value not less than g-(~/2 + sln, for n~ 1 we have,
on the contrary, S . This means that when a-s~~3+ --y ~n - 0, but
-s ~n + 0 � -
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r~rA vrrl~,lnL UDr, VriLi
Since the different modes of the long-wave part of the spectrum are charac-
terized by the parameter q, it is convenient to find 'lfrom the equation
obtained by multiplying the first two formulas (15). Since k is small, we
obtain
' - s' q+
+ - H ~ (27)
= 1= + g ~ k3.
S~ 9_ (28)
y - r= .
In the short-wave part of the spectrum the modes are characterized by a
constancy of the parameter ~(~~~n); using this parameter, directly from ~
(15) we have
- l + 1' 4 +S�kz '
If, in particular, n is sufficiently large, then
~ l/ ~
= 1-~ Y q~- sl ~rt 2 g I' j k=, -
~2 _ l + ~ 4 + 4 t 'l k., .
t
We assumed OL ~ 0, p? 0. It follows from (15) that a simultaneous change in �
t;l~e signs on oc. and ,B is equivalent to a replacement of a by - a. Since ~
a11 our formulas for a(k) contain ~ 2(that is, the frequency was deter-
mined with an accur,3cy to the sign), we evidently obtain no new solutions.
We note that the eigenfunction (~.6) in this case.also remains unchanged
because ~(r, m, x) = eX ~(m - r, m, -x) [8].
We also assumed that a2 ~,Q.2. We will confirm this. We will assume that
this is not so. Then a=-i o~ 1, ~_-i ~1~ 0~1 and ~1 are real positive
numbers. It can be shown that if the wavelength does not exceed several
tens of thousands of kilometers, even for very small f~ 0 O~C
that is, it is possible to use the asymptotic form (20).. We again obtain
the equations (21) and (22). Their left-hand side is complex, whereas the ~
right-hand side in the first case is real, but in the second case pure-
- ly fictitious, because S= 2i ?crll~. Thus, in actuality, ~ 2 7.~-2~ It was
demonstrated in [4] that this inequality also is observed with r= 0.
Now we will proceed to a clarification of the problem of how the frequency �
spectrum changes if the hydrostaticity hypothesis is~adopted. Here it is
possible to use the results in [3J, where the corresponding spectrum was
constructed. However, we will proceed differently and construct it by pro-
ceeding directly from equations (1) and the boundary conditions (2j, (5).
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W~~ wlll NuPply th~~ I~~ft-i~rind 9ide oE the thlyd c~quatiun of mc~t[an in (1)
witli tiie Eactur E�1 ;;o ll~at Lhe transitton to liydroetatice will cnrrespond
to the limiting transition ~2 -s0. First, the frequency ~1 = 0 as before is
a solution. Second, we will again obtain equation (10) and the dispersion
expression (17), but formulas (15) assume the form
2~ca a E2 + gk~ f
as = S~ ~ _ , (15' )
sl y~~~ - . 2 ~c~ ks~ -
that is, seemingly not entering into the combination ~2 -,Q,2, is sup-
plied with the factor � 2. This means that in any range of wavelengths with
- E-2 ~ 1 we have the case 1, that is, we have formulas (21)-(23). Similarly
to (27), (28) we obtain
. 2 _ S~ 4+
_ Hc~ , ~27~)
_ l-}- s~ 9 g r k~' ( 28' )
~ - c~
With E2 t0 the frequencies (27') tend to infinity, that is, the correspond-
ing oscillations are filtered out. Thus, in a hydrostatic polytropic atmo-
sphere all the solutions are given by the formula
~ a -
~a j~ S
9_ k, ~ 28� ~
H
_ However, for its derivation it is not necessary to assume a smallness of
k. It is sufficient from (15') to form q=� 8, assume E2 = 0 and solve
the derived formula for a2. The q values can be found from equation (22), -
bearing in mind that q= S 2/4. However, it is better (at the limit a, f 0
and the asymptotic formula (20) even with ~~-~-oo, generally speaking, is noe
true) , to discard i12, except the combination -~Q,2, directl;~ in
equation (10) and obtain a corresponding solution. Instead of (10), we ob-
tain the Bessel equation
~ _ gk, r ~
s~s-(b-2)~+ S~ ~~~-t:~ ~=0.
Its solutioii, satisfying the upper boundary condition, is
b-1 "
r-
X�S ~ ~�-12 s, y ~,g ~ S~,
and the lower boundary condition gives a dispersion equation in the form
/ _Stz ( 2k ~gf'so~
~6 lz~ - 2
~ ?b-: ~Z~ lz = Sl _ . (17' )
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rvcc urrtt,tr~t~ u~~ ULVLT
If, in particular, z~l, using the corresponding asymptotic formula [8],
from (17') we obtain equation (22) for z. And formula (28") follows f rom
tlia der~rmi,,,lti~~~ of z.
We see that it virtually coincides with formula (28) for the frequency of
long gravitational waves because the parameter ~,2 in the denominator of
the latter can b~ neglected: for example, if T~ = 273 K, r= 5.83 K/km,
then slq
/H = 0.5�10-4 q; c~ > 1, whereas ~t2 = 10-8. A comparison of for-
mulas (28) and (28") shows that the influence of the hydrostatic approx-
imation decreases not only with an increase in wavelength, but also with
an increase in the number of the mode denoting tlie depree of "vertical"
variability of the solution.
All the waves with the frequencies (28"), except the zero mode, which is
designated so because ~j~ corresponds to its short waves, whereas the long =
waves of all other modes are described by formulas (23), beginning with n
= 1, are gravitational. In actuality, according to (28") with ~'~0 a2~.~2,
but since these frequencies are not solutions, the corresponding oscilla-
tions are filtered out. The zero mode, which in a general case consists of
two branches (the short waves are acoustic, the long waves are gravitation-
a1) cannot be regarded as purely gravitational: with j~=~ 0 it is gravita-
t1UI1~11~ since in the long-wave range it corresponds to gravitational waves
i~i a rionhydrostatic atmosphere, but with (--i0 it is not filtered out, but
E,~~sses into a solution corresponding to acoustic waves in a nonhydrostatic
- atmosphere. Now we will consider how this occurs. If r--3 0, then it also
tlas one value q-~ 0, and specifically (24). Therefore, it follows from (28")
that
Az _ ls + BTo (r. - l) kZ t29~
y' ~
~ i
t}iat is, this mode actually exists also when ~'-i ~'a, but in this case it
already becomes acoustic. In actuality, if 0, then, in accordance with
(15'), the hydrostatic approximation means a s~ 1�a ~ 1. It follows from
(17), in accordance with (14), that .
b
4~b+1~ a.
Therefore ~ ~ _ _
b 2 b a~ Nx
9= 4~b~-1) a- b+ 1~2-[- k2E~'
~ Substituting this into formula (27') for acoustic frequencies; we obtain
(29) with the replacement of y by Ya. Thus, this mode must be considered
acoustic-gravitational.
Thus, the hydrostatic approximation filters out the acoustic waves and the ~
frequencies of the gravitational aald acoustic-gravitational waves are dis-
torted the less the longer the waue and the higher the number of the mode.
34
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. ~7/C io
t
_ ~ Figure 1 shows the frequency curves
600 i computed using the dispersion equa- -
~ tions: (17) for a nonhydrostatic
0 i ~ atmosphere (acoustic part of the
i
i spectrum is tlot represented) and
' (17') for a hydrostatic atmo-
S00- ~ sphere. In complete accordance with
~ the asymptotic formulas (28) and
i
~ (28") the difFerence in the frequen-
i cy spectrum decreases with an in- -
y00 ~ crease in the mode nwnber and an
increase in wavelength. Only the
zero mode in a considerable part of
~ the spectrum is raore hydrostatic
than, for example, the first mode.
300 ` But with the selected Y(5.83 K/km)
~
` it is acoustic to L= 187 km, so -
~1 - ~ that its behavior is not describ-
ed Uy formula (28). We see that -
2~ ~pi~~ gravitational waves longer than ap-
1~~~~, Proximately 100 km with great ac-
curacy are hydrostatic. _
~
9 ~ -
0
` `
900 ~
~ ~
� ~
_
,
0 SO 1J^ ;50 G r,rf
Fig. 1. Wave frequencies of gravi-
tational (1) and (for the zero mode)
- the acoustic (2) nonhydrostatic os-
cillations and wave frequencies of
fiydrostatic oscillaLiuns (3) in de-
pendence on wavelength in a polytropic
( Y= 5.83 K/km) atmosphere. The fig- .
ures on the curves denote the mode
number.
BIBLIOGRAPHY ~
1. Gradshteyn, I. S., Ryzhik, I. M., TABLITSY INTEGRALOV, SUNihf, RYADOV I
- PROIZVEDENIY (Tables of Integrals, Sums, Series and Derivatives), Mos-
cow, Fizmatgiz, 1352.
2. Dikiy, L. A., TEORIYA KGLEBANIY ZEMNOY ATMOSFERY (Theory of Os~illa-
tions of the ~arth's Atmo~phere), Leningrad, Gidrometeoizdat, 1969.
35 -
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- 3. Kadyshnikov, V. M., "Reason for Closeness of the Wind to Geostrophic,"
MET~OROLOGIYA I GIUROLOG.T..YA (Meteo~ology and Hydrology), No 9, 1977.
4. Kadyshnikov, V. M., "Modeling of Hydrostaticity of Large-Scale Atmo-
spheric Processes," METEOROLOGIYA I GIDROLOGIYA, No 1, 1979.
5. Kamke, E., SPRAVOCHNIK PO OBYKNOVENNYM DIFFERENTSIAL'NYM URAVNENIYAM -
(Handbook on Ordinary Differential Equations), Translated from German,
_ Moscow, Nauka, 1971.
6. Monin, A. S., Obukhov, A. M., "Small Atmospheric Oscil?ations and Adap-
tation of Meteorological Fields," IZV. AN SSSR, SER. GEOFIZ. (News ot
the USSR Academy of Sciences, Geophysical Series), No 11, 1958. _
7. Smirnov, V. I., KUItS VYSSHEY MATEMATIKI (Course in Higher Mathematics),
~~ol 3, Part 2, Moscow, Gostoptekhizdat, 1957.
8. .t~hnke, F., Emde, F., Loesch, F., SPETSIAL'NYYE FUNKTSII (Special Func-
tions), translated from German, Moscow, Nauka, 1968.
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UDC 551.(512.2+54+55)
_ �
CORRELATION BETWEEN MINIMUM PRESSURE AND MAXIMUM faIND VELOCITY IN
TROPICAL CYCLONES
Muscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 79 pp 34-41
[Article by Candidate of Physical and Mathematical Sciences V. M. Radike-
cich and Candidate of Geographical Sc3.ences G. G. Tarakanov, Leningrad
Hydrometeorological Institute, submitted for publication 2 April 1979]
The at~,thors propose a model for describing
the dependence of maximum wind velocity V~ �
in a tropical cyclone on the radius of the
region with maximum wind velocity rp, the
Coriolis parameter cJ Z, pressure P~ at the
center of a tropical cyclone and pressure P1
at a great distance from the center rd. In
addition to a precise expression for deter-
mi.ning V~, the authors also derived an ap- ~
proximate and simplified expression. The re-
sults of the computations agree satisfactor- -
ily with observations. The systematic discrep-
ancy between computations and observations
falls within the range of discrepancies be- ~
tween the wind in the free atmosphere and
the surface wind and varies from 0.78 to
0.65. ~
[Text] Tropical cyclones develop in the tropical and equatorial zones be-
tween 22�S and 35�N, except for a narrow equatorial zone (about 2�N-2�S). '
However, the main mass of tr~pical cyclones (87%) is associated with a
narrower zone taking in the region between 3 and 20�N and S[2J. Observa-
tions show that in each tropical cyclone (TC) there is a so-called ring of
naximum wind (if it is sufficiently narrow it is called the maximum wind
circle). The maximum wind velocities can attain 80-100 m/sec. It is evi-
dent that the instrumental measurement of such vElocities involves great
difficulties. Measurements.and the presently known theoretical models o~
a tropical cyclone show that pressure decreases toward the center of the
cyclone in conformity to a parabolic law. Taking this circumstan~e into
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accuunt, and also that pressure measurements are usually inadequate for a
precise determination of the pressure gradient in ~he neighborhood of the
maximum wind zone, there have been numerous attempts to relate the maximum
wind velocity (Vm) to the pressure drop at the center (Pp) and on the peri- ~
phery (P1) of the TC. A physical validation of the correlation between Vm
and (P1 - Pp) can b~ and T as follows:
Z
_ g ~ dz
T~:t-~t~Z~ '
P-poe
. a~,~ R
z= Zoa -I- Tn.~ r(
P 1 g 1. .
a~,2ll P~,21 ~ ~
a, R
zo_O~ x'_ Q~~~PI ~ g _
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_ Computations sho~~ ti~at the relative ciianges in pressure in the mesosphere
~~nd u~~n~~r atr~~t~~:~~h~~re remuin fipproxim~~te].y conHtrint ~ which, r~y inclie�f~te~d
ubuvey i;~ riLtrihutable to the fact that we are solving the Cauchy problem.
The changes in temperature in p-coordinates in the presence of ascending
and descending currents (3-5 cm/sec) are given in Fig. 1; it can be seen
that:
1) considerable changes in the temperature of the mesopause (20 K) (c, d)
cause a weak temperature change 1.5�K at z~75 km and about 0.6�K at
other levels; the geopotential changes insignificantly (0.1-0.4 km);
2) a temperature increase at the stratopause by 10�K (b,c) causes a tem-
perature change in the entire layer by approximately 0.5�K and virtually
does not change tlie geopotential;
3) a temperature increase of the lower stratosphere by 10 K(a, c) does not
change geopotential and weakly disturbs stratospheric temperature (0.3-0.4
K) .
PHb mb ~ ~ '3 ~ ~
9 ~ 9 _
~ ~ ~ J 1 50 a 2~~\ b 2
a) ~ 9 b) ~ ~
Z ~ ~
a 2 b y r,
O,SS / ~ r~. ,
~a same not- =
ations in ~J ~ ' ' ' ' '
~
0,01 ~ Fig. 2 u) ~ 2) ti
BJ ~ Z~ \ J d ~~y 3
d 1 2
~~~2 ~ 2 60- ` ~ ~
0,55 ~
~
- ~o ~ ' yo
210 230 250 270 210 230 ?50 T!f
zn
210 23D 2~ 0 ~~4 230 250 T If
Fig. 1. Temperature stratification of the atmosphere in p-coordinates with
different initial conditions. 1) initial, 2) disturbed by ascending currents,
3) disturbed by descending currents. Fig. 2. Temperature stratification of the
atmosphere in z-coordinates with different initial conditions.
Thus, the maximum variations of temperature and~geopotential, caused by ver- ~
tical currents, are observed in the case of a very warm mesosphere (d), but
they are quite small and it can be asserted that the role of the temperature
gradients in the process of downward propa~ation of disturbances is also
small.
From (III) (2, 3):
t (y-~)
u~ = CaYe~' -au~,
2
1 ~ (y-~)
d'z = 1 g~ e' yz ~-(a w)=1.
1
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r
Here
_ = y
In p-F d-1' Cd - a u? - ut0 ~1 ~
Po ' _
It was demonstrated in I2] that the solution of system (III)-(VI) has the
following peculiarity: all the even terms in the expansion u(u2, u4,...)
and odd t~rms ~ ( ~t~, tY3 , . . . ) and ~ ( cAl, ~ 3, . . . ) become equal to zero .
Therefore from (IV) (2, 3) u2 =~3 = 0.
In the case of a linear stratificat:ion it is impossible to obtain an ana- -
lytical solution for u3 and ~4, but estimates show that the temperature
gradients do not change the behavior of u3, obtained in [~2] (~u3~~~ul ~ and
of the same sign), but decrease ~ u 31 by .'.0-50% with differ~nt initial con- .
ditions, and this means that ~ changes insignificantly. Since the contrib-
ution of the terms A 3u3 and 8~ S~4 is small (less than 15~) and . they can
only intensify the ef~ect, it can be assumed that the vortex for the mnst
- part is determined by the terms ul, ~p and ~2. Thus, the transformation
of the circumpolar vortex described in [2] is correct also for a real tem-
perature stratification.
Now we will cite variations of the temperature profile computed in z-coor-
dinates (Fig. 2).~It can be seen that in this case the role of the t-emper-
ature gradients is quite large. For example, at z~70 km the variations -
4T~2.5 K-- b(1, 2) and [jT=8 K-- d(1, 2); at z=40 km ~ T='15 K
a(1, 2) and Q T= 20 K-- b(1, 2). Moreover, from the figure (b, d)
it can be seen that an increase in the temperature gradient in the meso-
sphere causes a decrease in L~T in it, and as indicated by computations,
with great gradients (a ~ 2 I:/km) the ascending currents cause a.cooling of
the mesosphere and heating of the stratosphere, which agrees well with ex-
perimental data [3, 4).
. BIBLIOGRAPHY
1. Bekaryukov, V. I., Purganskiy, V. S., "Precise Solution of a One-Dimen-
sional System of Equations in Hydrodynamics and Some Possibilities for
its Use," TRUDY TsAO (Transactions of the Central Aerological Observa-
tory), No 115, 1973.
2. Zadvernyuk, V. M., "Some Results of Computation of Changes in the Ther-
modynamic Parameters of tTie Strato-Mesosphere Caused by Disturbance of
the Thermosphere," TRUDY GGO (Transactions of the rfain Geophysical
Observatory), No 429, 1979.
3. Zadvernyuk, V. M., Mikhnevich, V. V., "Some Problems in Solar-Atmo-
spheric Relationships," METEOROLOGIYA I GIDROLOGIYA (Meteorolo~y and
Hydrology), No 9, 1973.
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4. Ivanov-Kholodnyy, G. S., et al., "Discussion of the Results of the Con-
ducted Experiment," ISSLEDOVANIYE ATMOSFERY I IQNOSFERY V PERIOD POVYSH-
ENNOY SOLNECHNOY AK'�IVNOSTI (Investigation of the Atmosphere and Iono-
sphere During a Period of Increased Solar Activity), Leningrad, Gidro-
meteoizdat, 1970.
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UDC 551.557(47-15)(571.5)
CHARACTERISTIC DIURNAL VARIATIONS OF WINDS IN THE UPp~R MESOPAUSE REGION
OVER CENTRAL EUROPE AND EASTERN SIBERIA '
, Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 79 pp 50-54
[Article by Doctor R. Schminder, Doctor of Physical and Mathematical Sci-
ences E. S. Kazimirovskiy, Doctor D. Kurschner and.Candidate of Physical
and Mathematical Sciences V. D. Kokourov and V. F. Petrukhin, :~ollm Geo-
physical Observatory, Leipzig University, and Siberian Institute of Terres-
trial Magnetism, Ionosphere and Radio Wave Propagation, submitted for pub-
lication 26 February 1979]
Abstract: On the basis of an analysis of simul-
taneous wind observations in the upper mesopause
regiQn over Central Europe and Eastern Siberia in
the winter of 1977/1978 and in the spring of 1978
it is demonstrated that together with the coincid-
ing nature of the diurnal variation there are differ-
ences indicating the existence of ma~or regional
structures in circulation and in the systems of
tidal winds. A synoptic a~alysis requires an ade-
quately dense network of observation stations.
[Text] During the winter of 1974/1975 joint work was undertaken by the Kollm
Geophysical Observatory at Leipzig University imeni K. Marks and the Badary
Observatory of the Siberian Institute of Terrestrial Magnetism, Ionosphere
and Radio Wave Propagation Siberian Department USSR,Academy of Sciences in
the upper mesosphere region by the radiophysical :nethod by measuring iono-
spheric drift with the spaced reception of signals of long-wave radio trans-
mitters (D1 method). Continuing this program [1, 2, 4], it was possible to
obtain results of simultaneous measurements for the periods 4-5 - 16-17 De-
cember 1977 and 1-2 - 19-20 March 1978 (the measurement method was such that
the observations were made during the nighttime hours local longitude time
between sunset and sunrise). In addition to the results of ineasurements of
long-wave drifts, the analysis was supplemented by wind measurement data
for this same region obtained by the radiometeor method (D2) by specialists of
the Kuhlungsborn Observatory of the Central Institute of Solar-Terrestrial
Physics Academy of Sciences German Democratic Republic (Heinrich Hertz
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~ Institute). The characteristics of the measurement path~ are given in Table
1.
� Table 1
Wind Measurement Paths in the Upper Mesopause Region at the Observatories in
Kollm and Ku}ilun~sborn (German Democratic Regublic) and Badary (Soviet Union) -
Iis~tcpiirenbxa~ PaccroA- MeTOA ~Iacrora 06o3xa- KoopAH}~ar~t rovKx
Tpacca I Hxe, x.s ~ 3I ~ KexNe ~ orpaxtexaa 6 -
7 llc.i ~ex,~op~-konnM 170 D~ 185 xzy K 185 5.2� c, m, I3� s, A,
$ 3ipKyres-&aAap~t 150 D~ 200 2 fi 200 b2 13 14
9 Bapmasa-KonnM 460 D~ 227 K 227 52 17
10 hY�~yHrc6opH-KonnM 500 D~ 245 K 245 53 12
11 paaHOHeTeopHme us~epe-
xuA eeTpa 150 Dz 32,5 M~y K-PAM 54 ]5
5 56 12
KEY:
1. Measurement path 15. MHz
2. Distance, km
3. Method K = Kuhlungsborn
4. Frequency 6 = Badary
5. Notation PAM =.Radiometeor observations
6. Coordinates of reflection point
7. Zellendorf-Kollm .
3. Irkutsk-Badary -
9. Warsaw-Kollm
10. Kuhlungsbo rn-Kollm
11. Radiometeor wind measurements
- 12. KHz >
13. N
14. E
The selected interval of synchronous measurements did not make it possible
to investigate disruptions of circulation associated with stratospheric
heating, which according to measurements at Kollm reached the upper meso-
pause region over Central Europe only on 1-2 January 1978. And here the
March measurement period coincided with the onset of the.spring restructur- �
ing of circulation, which despite the considerable distance between Kollm
and Badary (about 5,000 km) began virtually simultaneously on 2-3 March 1978
and was manifested as a weakening of the westerly wind, a frequent change
in the directions of zonal movement and an intensification of the easterly �
wind. With respect to the semidiurnal tide there was a decrease in amplitude,
a strong phase instability, and at the beginnin~ of the third 10-day period
in March a rapid change in phase characteristic for the transition from a
winter to a summer type of circulation [5j.
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. Ol /0211! 1978
Od / 09d~ 9977 ~
- v H~cEelr m~se 1 ~ ~ I ir,~ 1 V"', r.
I ~ l } wr ~
40 - f Ij i.~ ,o, 1 ~ r 1 ,':ti
a , t I B ~1: i
20 ~ _~~'~~~f ^ r' ~ r' ~b 1 l . -
t ~ ~ r ..i�� l ; ~ y... ~ ~ ; ^ ~ ?
W~- ~~'�!r's C 1 1 .
~ '
a ~ . . ~�l ~-'i--
~ ~ A ~ Pj
r 1
-1U ~ ~ ~
~ + ~ r.
-40 ~d ~ 3 I ~ -
ti
W ~ J?
40 � CN 1+.. 1! C~l �
- ~ 6~ t ~'O,'~RI /ry !iM~~''v1 ~ 2~ ~ ' ;
,:u - Pl~~r ;~~e,~ ~
~4~,�r,~;v~ d r~ =i
_ ~ ~I�+~~, i I, I ; ~ f
O ; � w ` - ~ VI , ' ~ , '
-~:l ` '�J ~ r' J ~ ~ ~ ~~r: i ~ -
. ~ ~ , t ~ �'ti�~~~ ~ ~ - 9
- ~%o j y ~ - -
~o S ~ ~ ----~r
s
_ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~6 18 ~ 20 ?2 24 02 09 OB � OB ~7 19 21 23 01 OJ OS 07 Oy
MecmNOe BpeMA Local Time
- Fig. 1. Diurnal variations of wind velocity vector in upner mesopause region
o~~er Central Europe and Eastern Siberia at middle (a, b) and at end (c, d)
of winter. a, b) zonal component, b, d) meridional compoiient. 1) K 185; 2)
B 200; 3) K 227; 4) K 245; 5) K-PAM [K = Kollm; ~j = Badary; PAM = radio-
meteor wind measurementsJ
An analysis of wind measurement data on all paths in December 1977 and in
March 1978 indicated a coincidence of the nature of the diurnal variations
for all stations having an approximately identical geographical latitude, ~
regardless of longitude. However, in details it is possible to see an ex-
cellent correlation and very great difference in both the prevailing and
in the tidal o~inds. With respect to the short-per~od variations, which are
interpreted as the effects of internal gravitational waves, here it is im-
possible to detect any correlation, since the coherence scale for these
variations is less than 200 km [6).
- Figure 1 presents data from simultaneous measurements for one night in Decem-
ber and one night in March, when the agreement of the results was rather
good. The values are given in local longitude time. It must be remembered
60
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that the difference in time between Central Europe and Eastern Siberia is
. 7 hours. Figure la,h, which shows the wind values for all five measure-
- ment paths (including measurements by the D2 method at Kuhlungsborn),
gives some idea concerning the real differences in the results between
the paths w3th reflection points situated very close together. For all the
European paths they are all in a small volume with dimensions in longi-
tude and latitude of the order of several degrees and with respect to al-
titude not more than 10 km. It can be seen that the measured values occupy
some interval and in general it would be very useful in many respects on
the basis of ineasurements in Central Europe to ascertain the center of this
"zone of values" and provide users not only the mean wind value, but also
the width of this interval. Ztaelve years of experience with measurements
at Kollm, where at first the measurements were made only on one path, en-
ables us to note that in the interpretation of such measurements it is
necessary to approach very carefully the evaluation of the accuracy in
the measurements themselves and the representativeness of the results for
one reflecti~n point relative to the mean chax�acteristic of the wind re-
gime on a regional scale. "
Figure l,c,d shows the diurnal variations on the last day of stable win-
ter circulation for the paths K 185, 6 200 and K-PAM.
The data from radiometeor measurements were furnished through the courtesy
of the Kuhlungsborn ionospheric observatory (see Table 1).
Figure lc,d clearly shows the excellent agreement of the results of ineas-
urements by the Dl and D2 methods for Europe. The agreement is not always -
so good, but already available observational data, making it possible to
carry out their comparison, indicated that a high correlation exists with
adherence to the following conditions. First, when the measurement condi-
tions ensure statistical reliability of the result; second, when a stable
circulatiori ensures a spatial uniformity of the wind regime; third, when
there are sufficiently great amplitudes of the tidal winds, which facil-
itates the relative analysis. If at least one of these conditions is vio- -
lated, the correlation rapidly decreases. This means that if we observe
differences in the measurement results by both methods, they can be at-
tributed either to a certain uncertainty in the method.itself (for ex-
ample, due to the lack'of data on the precise reflection altitude) or
to some still not finally clarified pec~iliarities of these methods.
Table 2 gives a sample of results for other days. It also gives some idea
concerning both the coincidences and discrepancies in measurements at two
points. After comparing the results, we must conclude that in addition to
the coinciding nature of the diurnal variation for both the prevailing and
the tidal wind in the upper mesopause region over Central Europe and East-
ern Siberia, t~ere are differences indicating the presence of regional
structures in circulation and in the systems of tidal winds. This means
61
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Table 2
Result:~ of Analysis of Wind Measurements in Upper Mesopause Region at the -
Observatories Kollm (GDR) and Badary (USSR) on Paths With Close Frequenc-
ies (185 and 200 K~Iz) and Approximately Equal Distances from Transmitters
3oxaabx~ii xo~noxexz MepHltFCOeanbxd~
06oaxa- 3 KowrtoHei+r q
HOqb qeaxe
1 2 vo , v: I 1's Yo I vs I T~
5 05/~ ,~exa6pA ] K 185 -05 38 20.45 -03 17 18.00
1977 r. 8 S 200 + 18 11 20.00 + 09 33 15.45 .
OS/09 K 185 + 09 29 21.45 i-00 12 19.i5
S 200 +28 31 2L30 +04 1? �19.15
12/13 K 185 +28 33 21.15 +06 35 17.45
B 200 +32 12 17.30 -17 07 14.45
14/15 K 185 +24 32 20.00 +Ol 23 18.15
� S 200 + 17 20 17.00 -18 21 ! 3.15 . -
Ol/02 uapTa K 185 +21 42 21.15 -11 23 16.30
6 1978 r. & 200 +07 35 21.15 -13 27 18.00
. : ~
O6/O7 K 185 +03 30 20.30 -04 18 15.45
B 200 -17 24 22.15 -15 ~ 34 17.15
C~EY :
1. Night 5. December
2. Notation 6. March
3. Zonal component 7. Kolmm
- 4. Meridional component 8. Badary
Note: V~ prevailing wind, positive to the north or to the east, V2
amplitude of the semidiurnal tidal wind, T2 (local time) phase of sersi-
diurnal tidal wind, moment of maximum wind, directed to the north or east.
the detailed coincidence of the curves of temporal wind variations can be
either random (Fig. la,b) or associated with a marked seasonal change in
the wind fields (Fig. lc,d). A study of changes in the prevailing wind
from day to day could afford a possibility for investigating planetary _
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waves, but this would requira intermediate measurement points [2]. In the
future, evidently, it is necessary to make a detailed study of the dis-
covered regional large-scale structures separately in each regi~n and only
then~ in the second stage, combine the results for constructing a general
model of circulation. Real synoptic investigations require the organiza-
tion of additional measurement points; the density of the network which
exists at the present time for these purposes ts inadequate.
BIBLICJGRAPHY
_ 1. Grayziger, K. M., Kazimirovskiy, E. S., Kokourov, V. D., Petrukhin, "
V. F., Shminder, R., Stirenher, K., "First Results of a Comparison of ~
Measurements of Ionospheric Drifts in the Long-Wave Range Over East-
ern Siberia and Central Europe,".ISSLEDOVANIYA PO GEOMAGNETIZMU, AERO-
NOMII I FIZIKE SOLNTSA (Investigations of Geomagnetism, Aeronomy and
Solar Physics), Vol 38, 1976.
2. Shminder, R., Kyurshner, D., Kazimirovskiy, E. S., Kokourov, V. D.,
"Mean Monthly Diurnal Variations of Zonal and Meridional Components -
of the tidind in the Mesopause Region Over Central Europe and Eastern
Siberia in the Winter of 1976 and the Spring of 1977," GEOMAGNETIZM
- I AERONOMIYA Geoma ~
( gnetism and Aeronomy), No 18, 1978.
3. Mu11er~ I~. G., Nelson, L., "A Travelling Quasi-2-Day Wave in the
Meteor Region," J. ATMOS. TERR. PHYS., Vol 40, 1978.
4. Schminder, R., Kazimirovskiy, E. S., Kurschner, I)., Petruchiny W. F.,
"Ein Vergleich der Ergebnisse von Ionospharendriftmessungen im Lang-
wellenbereich uber Mitteleuropa und Ostsibirien in Winter 1975/76,"
_ Z. MET., b. 28, 1978. _
5. Schminder, R., Kurschner, D., "On the Behavior of Wind Systems in the
Upper Mesopause Region in Winter and During the Transition from Winter
to Summer Conditions," J. ATMOS. TERR. PHYS., Vol 40, 1978.
6. Vidal-Madjar, D., "Gravity Wave Detection in the Lower Thermosphere
~dith the French Incoherent Scatter Facility," J. ATMOS. TERR. PHYS.,
Vol 40, 1978, -
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UDC 551.(521.32:524.7)
EVALUATION OF ERRORS IN COMPUT"ING EFFECTIVE RADIATION
- Moscow METEOROLOGIYA I GIDROLOGIYA in Russian tJo 11, Nov 79 pp 55-61
[Article by Doctor of Geographical Sciences A. I. Budagovskiy and L. Ya.
Dzhogan, Institute of Water Yrohlems, submitted for publication 20 March
1979]
Abstract: The paper describes a method for
the indirect evaluation of errors in comput-
ing effective radiation. It is based on a -
comparison of the differences between the
temperature of the soil surface and the air,
determined by three independent methods. The
merhod is applied to observational data from
Takiatash actinometric station. The authors
give a brief analysis of the results.
[Text] Regular observations of the radiation regime have been made in the
relatively dense network of actinometric stations in the Soviet Union -
since the mid-1950's. The accumulated dat4 are necessary for the use of
modern methods for climatological and hydrological computations, in par-
ticular, for computations of evaporation and irrigation norms. However,
the use of the mentioned materials involves considerable difficulties.
They are caused, in particular, by the fact that the measured values of
the radiation balance to a considerable degree are dependent on the albedo
and temperature of the underlying surface at the measurement site. The
= difficulties mentioned above are usually overcome by the use of the total -
radiation values measured at actinometric stations and accumulated data
on the characteristics of albedo for different underlying surfaces; in -
this case effective radiation is determined by computations. '
It is evident that errors in such computations are in need of evaluation.
It is desirable that this be done on the basis of quite extensive mater-
ial so as to form somp idea concerning the statistical stability of the
results. This problem can be solved indirectly. It involves essentially
the following.
' On the basis of data from observations made at actinometric stations it is
Possible to determine the effective radiation value. It is equal to the
difference between the measured values of absorbed short-wave radiation
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and the radiation balance. The latter is dependent, in particular, on the
temperature of the underlying surface in the measurement sector beneath
Lhe balancemeter. The effective radiation value obtained in this way
wi11 t?~nceforth be ca:Lled the "measured" radiation and in case of necea-
sity it can be assumed conditionally that it does not contain errors.
It is convenient to represent the effective radiation value found in this
way in the form of the sum of two terms. The first of these is equal to
the effective radiation for a case when the temperature of the underlying
surface beneath the balancemeter Tu is equal to the air temperature meas-
ured in the meteorological booth. The second term takes into account the
influence exerted on effective radiation by the difference between the _
iaentioned temperature values.
III=1~+4a S (273--~ T2) 3 (T~-T:) . ~1)
[ i7 = u ]
Here Iu is the effec.tive radiation found by the method indicated above on
the basis of the measurement results, I* is the effective radiation when
Tu = T2.
W* will denote the computed effective radiation by Icomp and assuming that _
I� Icomp~ on the basis of (1) we write
etp=1~-ly =46S(273+Ts)3(T~~-rz)� (2) -
[ p = comp; 1T = u]
Since the computed values of effective radiation in the most general case
can contain both random and systematic errors, they will automatically
enter into Q I~omp . These errors can be evaluated by computing the Q I
values by some other two independent methods.
The first of these methods involves the use of ineasurements of soil surface
temperature. At meteorological stations they are carried out on bare sec-
tors of the soil. Therefore, their use for the purposes mentioned above
in principle is possible only when the measurements of the radiation bal-
ance are carried out over a sector with bare soil. On the basis of these "
measurements the Q Iu value can be computed using the expression
~/i,=4vS(273+Tz)3(T�-T2). _ (3)
The second possible method for indirect but independent determination of
the difference Tu - T2 involves use of the formula
P=a p ~pD(Tn-T2), ~4)
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_ in which P is turbulent heat exchange, oC is a conversion factor, is air
density, cp is the specific heat capacity of air at constant pressure, D
is a coefficient having the dimensionality of velocity. It characterizes
the turbulent conductivity of the air layer bet~een the soil surface and
any fixed altitude (in this case 2 m). Henceforth we will call it the ex-
change coefficient.
The turbulent heat exchange value enters into the heat balance equation
R=P+B~-LE, (5)
in which R is the radiation balance, B is heat exchange in the soil, L is
the latent heat of evaporation, E is the turbulent flux of moisture or
evaporation.
The day-to-day changes in heat exchange in the soil are usually small and
rarely exceed 10% of the radiation balance. Therefore, in an approximate
estimate of turbulent heat exchange this heat balance component can be
omitted or estimated approximately using data from the literature [3, 5].
- Under extremely dry conditions, especially in the arid zone during the -
summer months, evaporation from the surface of the bare soil can be assum-
~d equal to the precipitation (H). Accordingly, assuming E~ H and limiting
ourselves to the above-mentioned anproximate estimate of heat exchange in
the soil, it is possible to obtain turbulent exchange values which are
employable for practical purposes. In their subsequent use for computing
Che differences Tu - T2, and then L~1 it is necessary to know the values
of the exchange coefficient. The literature gives estimates of this coef-
ficient used in climatological computations. However, a high percentage of
these coefficients were obtained on the basis of observations over the
plant cover. Therefore, we used additional observational data given in
[4]. They include information on all the components of the heat balance
and meteorological elements, including on the temperature of the soil
surface, measured using scattered temp2rature sensors [6], giving the spa-~
tially averaged temperature and having a small radiation error. The ob-
' servations were made in the Fergana valley in cotton.fields in 1956 be-
_ ginning on 7 May. In the study observational data were used from early in
- May through the first 10-day period of 3une inclusive, when the cotton
" plants are extremely small and do not Exert a significant influence on for-
mation of soil surface temperature.
Computations of the exchange coefficient D were made using expression (4)
a~d on their basis it was possible to construct the dependence D= f(U),
where U is wind velocity at a height of 2 m(see Fig. 1). We note in pass-
ing that an attempt to construct a curve of the dependence of the exchange
coefficient on ~ind velocity, taking into account the influence of temper-
ature stratification, did not give significant advantages. ~
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Dca/ceK cm/sec
- 1,3
0 0 0 ~
0 0
1,0 0
0 0
o Q
0 0 J
4s o o ~
i
i
0 Z U~r mf sec
Fig. 1. Curve of dependence of exchange coefficient on wind velocity. Along
y-axis: D cm/sec; along x-axis: U m/sec
~I? dl, ~ d J~ .
0
O ~ G~Q ~ uC0
O
C ~O
~ ~A G~~ ~ ~ ^ ~ ~~O O
0 0 CA O ~ J c` `~j9 C
O
Op~ eS ~ O .
C~ yCJ ~ . O O~ g C,
~ O O ~ O J
2 O ~ 00 O C
� O n
� O
p � comp
a u i
J 2 !t ~.D ~ Z 9 C _ din
r'ig. 2. Curves of correlation between Q Ii values computed by different
methods. -
In order to estimate the error in the dependence D= f(U) on the basis of
data on the diurnal swns of turbulent heat exchange,by means of inverse
computations we computed the temperature difference Tu - T2 and compared
this with the corresponding measured values. Then after scaling the com-
puted and measured Tu - T2 values into radiation units we determined the
standard deviation characterizing the error in determining Q I, asso-
ciated with the use oi the mentioned dependence. According to computation
data, c7 = 0.46 Cal/(cm2�month).
Taking into account what has been said above, on the basis of (4) and (5)
we can write
~ ~!o = 4 ~S (273 T,)3 R-B-LH
�pcoU ~6)
Since the errors in computing effective radiation, acr_ording to the com-
ment made above, are completely included in ~ Icomp. their evaluation in-
volves the paired comparison of the determined d Ii values.
It is possible to obtain some idea concerning the systematic differences
of the ~ Ii values computed by the three above-mentioned metliods by
using the expressions .
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~lP=Q,~jo,
~11P = a_ ~ /n,
~Ip =Q~~/~~
\
~~/P `'~lP
~1 = ~�S jD r - c, Ip ' n~ - E~1 .
IP = comp; Tf = u]
Graphs of the dependence (7) are shown in Fig. 2.
It is evident that the measure of the relative value of the systematic dif-
~ ferences between the L~Ii values, determined by the two independent meth-
ods, is the deviation of the corresponding ai values from 1.
The random deviations between the paired comparable L1I values, computed
by different methods, can be characterized by the values of the corres-
ponding dispersipns
~2= -a ~8~
~ ( ~ ~ o
02 = (t~ Ip - aZ /n)'-~
Q3 = (~fp- R3~Ia~=.
Here the horizontal line is the averaging symbol.
Since the L1I~omp ,[~ID and p Iu values were computed by independent meth-
- ods, the errors in these computations are also independent. In actuality,
the random errors in romputing Q I~o~P are related primarily to the devia-
tion of the real vertical distribution of. air temperature and humidity,
the deviation of clouds from their, "typical" values adopted in validation
of the parameters in the computation formulas,.and also their approximate
character. The error in computing ~ID is determined by the c~rrespond-
ing error in ~he dependence D= f(U), incomplete adherence to the hydro-
dynamic conditions for its use, and also the conditions necessary for ad-
equately reliable determination of the monthly sums of turbulent heat ex-
- change. Finally, the random error in computing Q Iu is related to the cor-
responding errors in measuring the soil surface temperature. Accordingly,
the va~ues of the dispersions of each of the two paired comparable values
- must be the sum of the two corresponding dispersions, to wit
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y"~o Tab 1
qpe~~ae Q~ Q_ ~y Qi ~1 Q3 Qp I QD Jn a~ = aP + Jp~
Nu ber o te in s~r s a -
- ie p cbmp n~ u
29 1.Oi 1.~~0 ' 1.49 1,10 1.26 0,9~ 0,84 O,G3 U,7-1 0! -'P + on+ -
~ n,`~~'~ I.~.; ! 1,43 1,03 t,33 0.88 0.36 0,54 U,77 ~9~
b8 I,i1'' I.:,_' ; 1,-~9 1,06 1,30 0,91 U,:,~ 0,58 O.iG �3-~Q+�Q'
(T~= u; p= rad] Here O"rad is used to denote the dispersion Q Ip, Ct D
denotes the dispersion Q ID and O~'u denotes the dispersion L~Iu.
1'I~e three equations of the very simple system (9) contain three unknown
parameters O"rad, v"D and ~u, which can be determined using the already
computed values cYi, O'2 and O'3.
In order to apply the described method to an evaluation of the errors in
computing effective radiation we used observational data for the actino-
metric station Takhiatash (lower course of the Amudar'ya River), in whose
description, given in [1] and in handbooks on climate, there is a clear
indication that the observations were made in a sector with a bare soil
surface, that is, there was satisfaction of the requirements necessary
for computing the L~ID ~and ~ Iu values. In addition, this region of Central
t\sia is characterized by very little precipitation (the annual precipita-
tion norm is 98 mm). Therefore, here, more frequently than in other re-
gions, for monthly time intervals th~re is satisfaction of the condition
E~ H necessary for computing Q ID.
In the computations we used the Brent formula, the values of whose coeffic-
ients were refined by M. Ye. Berlyand [2] on the basis of theoretical com-
putations. A linear dependence was used in taking into account the influ-
ence of cloud cover. The value of the parameter c entering into it was
also taken from M. Ye. Berlyand.
The values ~ I~o~P, L1Iu and L~1D were determined using expressions (2), (3)
and (6). In the computations we used observational data obtained during tt~e
warm half-year (April-September) during 1955-1959 and 1961-1968, given in
j2], in actinometric handbooks and in handbooks on USSR climate. Additional
data on temperature of the soil surface, temperature and air humidity, wind
velocity, total cloud cover and precipitation were taken from handbooks on
climate and archival data. In computati,ons we excluded cases~'when satisfac-
tion of the condition E~II caused well-founded doubt or when there were
gaps in the observations of one of the elements. The results of observa-
tions for a total of 58 months were used in the computations. In order to
form some idea concerning the statistical stability of the computed charac-
teristics the series was broken down into two equal parts.
The results of paired comparison of the computed values are given in Fig. 2.
- The computed ai and ~i values (in Cal/(cm2�month)) are given in Table 1.
We must note that the differences between the ai and Cl i values obtained
for the.first and second halves of the investigated geriod and also for
the period as a whole do not exceed the limits of the admissible error
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in estimating the sample values of the corresponding parameters, that is,
the ohtained ai and o'~ values have an adequate statistical stability. The
second important conclusion following from an examination of the computed
values is that the values of the coefficient al (the systematic discrepan-
cies between the compared L~1~o~ and ~ID values) do not differ aignif- ~
icantly from unity not only for the investigated period as a whole, but al-
so for it~ parts, that is, tfie two values virtually ~oincide. On the other
hand, the values of tfie coefficients a2 and a3 differ substantially from
unity, evidence of considerable systematic differences between the comput-
ed d Icomp andLlID values, on the one hand, and L~1u on the other. The rea-
son for this, as is clearly noted, is the systematic error in measuring
temperature of the soil surface, data for which are considerably too lowo
The latter for tfie conditions in Central Asia during the summer months is
entirely natural since the surface of even dry soil has a considerably less-
er reflectivity in comparison with the reflectivity of the thermometer res-
ervoir. As a result the mean value of the characteristic temperature of the
latter is considerably lielow the temperature of the dry soil surface. Thus,
the results, even with the most cautious approach, give basis for drawing
the conclusion that there are no significant systematic differences be-
_ tween the effective radiation parameters obtained by computations and their
measured values (under the conditton Tu = T2). _
In evaluating tfie determined values characteri~ing tlie rand4m error ~.I in
computations attention must be given tQ the fact that for all three co~
putation methods they differ little from one another and in general are not
great. A mare complete idea concerning the magnitude of the computation
error ~ Icomp can be obtained by comparing it with the computed values cf
effective radiation and the radiation balance.
The mean value of the computed values of the effective radiation for April-
September during the aFiove-mentioned time interval (1955-1959, 1961-1968)
is 3.2 Cal/(cm2�month). Accordingly, the relative probable error in such
computations is 18%, and the error with an 80% guaranteed probability -
34%.
The mean value of the radiatioi: ~~lance, obtained on the basis of the re-
sults of ineasurements,of absorbed short-wave radiation and the computed
values of effective radiation, is 10.8 Cal/(cm2�month). In such a case the
probable error and the error with 80% guaranteed probability, related to
the use of the computed values of effective radiation fox the warm half-
year, on the average will be 5.3 and 10.1% respectively, which does not ex-
ceed the errors in measuring the monthly sums of the radiation balance,
- which, in accordance with [7], are estimated at 10%.
Next we note that the Oro value, cited atiove in the text, and ~D, cited in
Table 1, determining the error in computing L11D, differ relatively little
from one another. However, insignificant differences between them are en-
tirely to be expected because cTO was ohtained on the basis of use of heat
balance ohservations and d D on the basis of mass materials. It is more ~
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important that ~D is. appreciahly less than ~~o~ and u. This conclu-
sion is important fo'r evaluating the error associated with the introduc-
tion of a correction to the radiation halance in computations of evapora- -
tion from the surface of the moistened soil. ~
l~in~i.lly, it should he noted that despite the wel..l.-known incorrectnese of
standard observations of temperature of the soil surface carried out in the
_ network of inetEOrological stations the 0'u value, dependent in the last
analysis on the random errors of these observations, is relatively small.
1t ~s even somewhat less than the random error in computations of effec-
~ tive radiation characterized ~v the ~comp value.
The described method for estimating the errors in computations of effective _
- radiation can also be used for other points if there is satisfaction of
the above-mentioned conditions necessary for computing the Q Iu and ~ ID
~alues.
BIBLIOGRAPHY
1. Barashkova, Ye. P., Gayevskiy, V. L., D'yachenko, L. N., Luchina, K. M., -
Pivovarova~ Z. I., RADIATSIONNYY REZHIM TERRITOFII SSSR (Radiation Re-
gime of the USSR), Leningrad, Gidrometeoizdat, 1961.
2. Berlyand, M. Ye., Berlyand, T. G., "Determination of the Earth's Effec-
tive Radiation With Allowance for the Influence of Cloud Cover," IZV.
AN SSSR, SER. GEOFIZ. (News of. the USSR Academy of Sciences, Geophys-
ical Series), No 1, 1952.
3. Budagovskiy, A. I., ISPARENIYE POCHVENNOY VLAGI (Evaporation of Soil
Moisture), Moscow, Nauka, 1964.
4. Budagovskiy, A. I., Mii;~yeva, Ye. N., "Results of Investigation of Evap-
oration from Irrigated Fields in Central Asia," TRUDY GGI (Transactions
of the State Hydrological Institute), No 151, 1968.
5. Budyko, M. I., KLIMAT I ZHIZN' (Climate and Life), Leningrad, Gidro-
meteoizdat, 1971.
6. Kaganov, M. A., Ch~idnovskiy, A. F., "Instruments for Measuring Tempera-
ture of the Soil Surface," SBORNIK TRUDOV PO AGRONOMICHESKOY FIZIKE
(Collection of Yapers on Agronomic Physics), No 5, Moscow-Leningrad,
1952.
7. Lebedeva, K. D., Sivkov, S. I., "Accuracy in Measuring the Radiation
Balance by Thermoelectric Balance Meters," TRUDY GGO (Transactions of
the Main Geophysical Observatory), No 129, 1962.
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UDC 551.593.5(263)
OPTICAL CHARACTERISTICS OF THE ATMOSPHERE IN THE TROPICAL ZONE OF
THE ATLANTIC OCEAN
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 79 pp 62-69
[Article by V. N. Adnashkin, L. K. Veselova and Candidates of Physical and
Mathematical Sciences 0. D. Barteneva, A. G. Laktionov and N. I. Nikitin-
skaya, Main Geophysical Observatory, Leningrad State University and Insti-
tute of Applied Geophysics, submitted for publication 22 May 1979]
Abstract: This paper presents the results
of investigations of the spatial-temporal
variability of the aerosol-optical character-
istics of the near-water layer and atmospher-
ic layer of the tropical zone in the Atlantic
Ocean according to data from TROPEKS-72 and
GATE-74. Also considered is the latitude var-
iation of integral and spectral values in
the region 0.35-1.OOfcm of atmospheric trans-
parency, the selectivity index of aerosol at-
tenuation of solar radiation and moisture content -
of the atmospheric layer, and also a number of ,
characteristics of the near-water layer: meteor-
ological ranRe of visibility, concentrations of
large (d> 0.63�m) and giant (d > 10 �m) aerosol
particles. It is shown that the principal fac-
tor responsible for the inconstancy of optical
weather ir~ the tropical region of the North At-
lantic is the transport of dust from the des-
erts of the African continent.
[Text] Experimental investigations of the parameters of microstructure
and optical characteristics~of aerosol in the near-water layer and atmo-
spheric layer in the tropical latitudes of the Atlantic Ocean indicate
their considerable temporal variability and spatial nonuniformity [1, 3,
4, 6, 7, 12-14, 16]. It is shown in this study that the principal factor
responsible for the inconstancy of optical weather in the tropical region
of the North Atlantic is the transport of great quantities of dust into
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the ocean from the deserts of the African continent.
During the period of GATE-74, from 27 June through 10 September, on the
basis of data from Lhe geostationary satellite GMS-1, "dust cyclones"
nine times crossed the Atlantic Ocean from east to west in the latitude
zone from 10 to 20�N [2J. The eastern part of this region of the Atlantic
Ocean, the so-called "sea of gloom," in which the transport of dust from
the African continent by the NE Trades is systematically observed [10], -
can also be seen clearly in the averaged latitude variation of optical
and aerosol characteristics of the atmosphere from 50�N to the equator,
represented in Fig. 1 on the basis of data from TROPEKS-72 [1, 6]. Spe-
cifically in this region of the ocean all the aerosol-optical charactexL
istics considered below attain their extremal values. For example, in the
"sea of gloom" the following are noted: the lowest integral transparency
PZ [9] and the highest value of the aerosol component of the optical lay- -
er of the atmosphere 'G j~ for 1.OO�m; the maximum N concentrations
of large (d~0.63~m) and giant (d>l0�m) particles and the minimtun val-
ues of rhe meteorological range of visibility S in the near-water layer
of the atmosphere. The n parameter, characterizing the degree of selec-
_ tivity of aerosol attenuation of solar radiation in the well-known Ang-
strom farmula 'Cj~= /3~-n has a value on the order of 0.2, that is, the
spectral variation of the aerosol component of optical thickness of the
atuasphere in the spectral range 0.35-1.00 ~ m is extremely close to
neutral.
Figure 1 shows that the air masses in the temperate latitudes are charac-
terized by appreciably higher values of the transparency characteristics
- and the selectivity of aerosol attenuation, and also a relatively small
moisture content W of the atmospheric layer, which with the degree of ad-
vance into the tropics increases and attains maximum values in the ICZ
intertropical convergence zone [8]. _
The equatorial region from 10�N to the equator, within which the ICZ is
situated, is also subject to the influence of the transport of contin- -
ental dust from the central and southwestern regions of Africa. However,
the effect of the "cloud filter" in the ICZ region and the arrival of
pure oceanic air masses in the lower layers of the atmosnhere with south-
erly and Anti-Trades westerly winds favor this picture: with movement
from 10�N toward the equator there is a tendency to an increase in trans-
parency of the entire layer of the atmosphere and especially its near-
water layer [6]. Thus, Fig. 1 shows how nonuniform this region of the
Atlantic Oceara is with respect to its optical properties.
In Fig. 2, where the days of observations have ~een plotted along the x-
axis, we show the temporal variability of the parameters of microstruc-
ture and optical characteristics of aerosol considered above for the
period of the three phases of GATE-74 from 28 June throuQh 19 September
durin� which the "Passat" scientific research weather ship was situated
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at a point on the equator 10�W at a distance of about 500 km from Africa.
WcH
S
. '
J�
>
~ r'_.~.
x `h
~B
~
~6' s n
, , ~
O,G~ * ~
ta
~2 x
R
~r~ % x ~`t ' x A 3
0,0 "
Q `-x'x # x .
I
QS ~ ~
r
SKM
6
x x
f0 ~ ~~S
20 a \ x
Q 'A~ ,
Ncn'
a-J i,~~~~ s
1~ x~~
- ~oZ
1 ~ '�~l'\.,~7
10'
SO y0 JO ?0 90 G y GrrG N
Fig. l. Latitude variation of aerosol-optical characteristics of atmospher-
ic layer and near-water layer over the Atlantic Ocean (Northern Hemisphere).
Figure 2 shows that the amplitude of variation of aerosol-optica~. charac-
teristics of the atmosphere during this period is extremely significant.
The integral coefficient of atmospheric transparency P2 varied from 0.59
to 0.75. The concentration of large (d ~ 0.63 ~tm) aerosol particles in the
near-water layer varied from 2 to 30 cm 3 and the concentration of par-
ticles d~ 2 N.m was from C~.1 to 1.3 cm 3. The meteorological range of vis-
ibility S varied in the range 10-100 km.
The values of the aerosol component of the optical layer of the atmosphere
'~~j, for a= 1.O�.m were in the ran~e 0.10-0.40. A fact of importance is
that the nature of the spectral variation of aerogol attPnuation varied _
from close to neutral n= 0.1 to extremely selective n= 1.3. The water
vapor content in the atmospheric layer was 2.5-4.5 cm. The concentration
of giant particles with d> ZO ~.m during the entire period was very low
and did not exceed 10-3 cm 3.
74 -
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rrc~~ r pa3u d q~a~a ID~a3a
r ~1~ ; --;-T r-r-r~~~.,,~T_ -`1a_r~n-rt-r-~--rT-r~T-- -~-T--, . T
_ 4 ~ . , _ r _ F~ ,
. , TTrr~-i
r (
. aJ 1~~~ ~ i ~ ~ ~ TTl- .Tr. ~ ~ ~ !2
I~ II~~ I~~~ ' i
~ ~
i
i' I ~ ~
. 0;1 '
ae l T; ; ~ - , ! ;-~T~ ~1 r - ~ -
- ~ t - - ~ _ i ~
Qo
- -
0,7 ~ , i
rr1 ~I ~ i .
sr,r~
6 -
T--- ~
~ ~ ,
~ i- ^ ~
:IG1 ~ ~ I i I
~1I`~~_`-J .
' `
l.: ~'T~ 1 ~ i I
90' . _ ~ ~
2~ ~ _ ~ .
10 _ - _ _ , � - - ~ ~
S _ . - - ' ' - - ' - ~ ~,_..-.-...:y_:-: L
J . � L.1.,. _ ! ~ '
18 .JOI 3 S".'' 15 ll T? ~3 J1' 3 S 7 9 1t 13 95 97~9 d1 ; 3 S 7 9 11 13 73 fl 19
f~yn 2 3 aery'rm 4 CeHma6pe 5
P
Fig. 2. Temporal variatic,n of aerosol-optical characteristics of atmosphere .
- during GATE-74 period ("Passat" scientific research weather ship, 0� lati-
tude, 10�W). 1) moisture content of atmospheric layer; 2) aerosol compon-
ent af atmospheric layer for 1.0 �m; 3) index of selectivity of aero- -
sol attenuation of solar radiation; 4) integral coefficient of atmospheric
attenuation; S) meteorological range of visibility in near-water layer; 6)
concentration of giant particles (d> 2~m); 7) concentration of large par-
ticles (d~0.63~m).
KEY:
1. Phase ~
2. June
3. July
4. Augus t
5. September
We should note the presence of a hi;ii correlation between the values char-
acterizing atmospheric turbidity P2 and 'G;~, and also between the concen-
tration of large (d> 0. G3 �m) particles and the meteorological range of
visibility; in ttie latter case the correlation coefficient was 0.94. In
the considered region of the ocean the concentration of large particles
in the near water layer is essentially dependent on the direction of the
flows in the free atmosphere. The increase in the concentration of aerosol
particles is associated, in particular, with advection of northeasterly ~
flows in the layer 2-3 km, which carry dust-laden air from Africa, wher~as
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its decrease i~ associated with an intensification of monsoonal cirrula-
tion in the layer 1-1.5 km [7]. These results agree with the data obtain-
ed in TROPEKS-i2, on the basis of which the conclusion was drawn that the
basic mass of large aeresol particles is of continental origin [6].
Table 1
Aerosol-Optical Characteristics of Atmosphere During Du::t Intrusion Periods
. ~
_ _ ~
= U Q Q
MeCio ~ ~ ~ x x
_ I~aTa P.: S Ax ~ a_ o. s.,
xaxepex~ta ~ ~ 3 7~~ ~ F~ 5
1 'Z ~.a pS~ o=~
ii ~ e c~ x c. t-
r
6 HNCII RIlaccaT~, 15 e~ons 1974 r. 11 (~.61 17 0,9 0,2J 16 0,5
~ 0� m., 10' s.
Cpe~aee 3a 1 c~asy 12 0,71 43 0,2 ~~,1~ 3,1 0,;~
29 xwax 1974 r. 11 u,~9 15 1,3 O,Y~ 9 ~).4
30 a 0,58 iG 0,7 U.dU 2:i 1,~~
11 aerycra 1974 r. 13 U,:'iU 8 - - 30 0,7
13 s G,59 1,0 (~,23 16 U,6
14 a 6.:~9 0.9 U.1ti. 14 U,8
Cpe~xee aa II ~asy 14 0,63 4~~ 0,5 O,Iti 8,5 0,6
g H~iC cAr.aaeMqK 1 aeryrra 1972 r. O,:i8 12 0,2 u,3�1
KypvaTOe 13
9 18,8� c. r .
is,s 3. k.
io
KEY:
1. Place of ineasurement 11....July
2~. Date ~2. tiean for phase I
3. �m 13. ..August ~
Concentration of large particles 14. Mean for phase II
5. Concentration of giant particles
6. "Passat" scientific research
weaLher ship
7. 0� latitude, 10�W -
8. "Akademik Kurchatov"
scientific research ship
A comparison of curves 2-7 in Fig. 2 shows that the greatest~variations
in aerosol-optical characteristics of the atmosphere occurred on 14-15
and 29-30 July and 9-14 August, when dus*-laden air masses arrived in
this regi~n of the Atlantic, some of which, moving from the arid zones
of North and Southwest Africa in a southwesterly direction, under def in-
ite synoptic conditions reached the equator. A~oint analysis of TV and
IR images for this region of the earth from the American meteorological
satellite GMS-1 indicated, for example, that on 30 July 1974 the centers
of the dust storms were situated in regions with the coordinates SP = 18�
N, ~l= 8�W and 31�N, ~l= 0�W at a distance of aBout 2000 km or more
from the measurement site j15].
76
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.
_ N CH'J
102
90
1
9d'
~d2 3 ~
- .
To~ ~
_ ~ 2 ,
4 ~
i0 S ~
� p~y 1 2 4 10 20rMHM N`
- Fig. 3. Spectra of size of aernsol particles in near-water layer obtainel
- in different regions of tropical zone of Atlantic Ocean. 1) "sea of gloom,"
- ~14�N, 19�W); 2) "sea of gloom" (18.8�N, 16.5�W), 1 August 1972; 3) Equa-
toriaJ. Atlantic (0� latitude, 10�W), 30 July 1974; 4) 1ow-dust region in
ocea >30 > 10 ~7 2-2
17 Hoeo�A.~e~:cauipouch 2 It 19i;t > >(0 >10 `2 J-~ ~
18 MNxafinoei:a 3 111 19iti ~ >30 >10 10:1 26-77
19 ~epThouo l X 197; c:?aGoe2 ~10 AHeM 2~ 2-2
I no~Nee ~
20 ~4so6irnbxo~ X 197G ro xce 2 Tana~ 7- ~7 i-7
21 Kpacnoraap~eNCKOC 2 `CI 19F~~ ~ To Hce - 6i 12 -!3
22 ~'Iaoli~:?b}ioe 2 ~CI I~~a cyxas2 0,5
4 ~ -0,2 77,0 E Srp e VI-VII 75 40 14 16
5 a -5,4 -77,0 ~ Cp. t Q$ I-III 75 5'J 1 16
6 s -5,2 I,0 1 Cp. tQ sa npeAwecxeyw- 31 9 16
u~a~ roA
7 asrycr ~-6,9 98,0 . ~tas VI-VIII 73 :~3 1 l 15
8 To kce 6-7.8 72,0. Cp. t� a ~'I11 80 f7 13 15
9 , -1,23 118,0 Cp. S~P s X-IX 100 6i 14 15
f' 12 15
10 ~ -0,33 G5,9 S S~P a VI-VII
fii ~:i 14 . l5
11 a -I,0 115,0 Qp, S~P s X-VI � -
12 s -5,0 31,0 Cp. taH I-VI 30 :,3 11 l5
13 ~ 0,5 63~U A~ p aall~ VI &.-A. 16 87 50 7 15
SAAt V11~7O
0,6 46,0 SA~~ ~ 0 �
� 2,1 80,0 ~ np~t ~A p a I-VI
K:
A.