JPRS ID: 8727 USSR REPORT METEORLOGY AND HYDROLOGY NO.8, AUGUST 1979

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~ ANO . .

23 OCT06ER i9T9 N0. 8. AUt3UST i9T9 i OF 2
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1~()It ()H'1~1('IAI, l~til~: ()NI,Y
~ ~PRS ~~s727
23 October 1979
U~SSR Re ort ~
p
~ METEOROLOGY AND HYDROLOGY
No. 8, August ~ 979 _
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JPRS L/8727
23 October 1979
USSR REPORT
METEOROLOGY AND HYDROLOGY
- No. 8, August 1979
Selected articles from the Russian-language journal METEOROLOGIYA
I GIDROLOGIYA, Moscow.
. ~ CONT~NT~ PAGE
= N~umerical Modeling of Urban Microclimate
(G. I. Marchuk, Pt al.) 1
Numerical Prediction of the Pressure and Geopotential Fields for the
Northern Hemisphere With Alloi,rance for the Barotropic Boundary
I~ayer
(L. V. Berkovich, V. A. Shnaydman) 15
Modeling of a Cloud Ensemble
- ~A. I. Fal'kovich) 24
- About O~~e Method for. Increasing the Accuracy of Difference Solutions
in Forecasting With the Use of Nested Grids
(Ye. Ye. Kalenkovich) 37
D,ynamic qssimilation of Potential and Wind Fields in the Low Latitudes
~A. F. Kivganov, U. Ch. Mokhanti) 1~1~
Two Noniinear Problems in Dynamics of the Equatorial Atmosphere
~S. Evtimov, @fi 'c11~~ �~~ea~~~~~~~~~~~~~~~.~~~~~a~~~~~~.~~~~~~~~� 7~
Vertica7_ Distribution of Absolute Humidity of Air and Moisture
~ Content in the Atmosphere Over tne Oceans
(N. A. Timofeyev) 51
Computation of Available Potential Energy in the Northwestern
Pacific Ocean
(V. F. Kozlov, et al.) ..~o 72
~ - a - [III - USSR - 33 S& T FOUO]
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CONTENTS (Continued) Pa~e
Use of the E;rpcnential Smocthing Method for Predicting the
Long-Term Variation of Salinity in the Sea of Azav
(V. N. Bortnik, A. N. Qvsyannikov) 7~i
- Prediction ot i:he Quality of River Water During the Period of
Spring High Water
(F. Ya. Rovinskiy, Z. I,. Sinitsyna) 85
Investigations of the Water, Heat and Salt Balances in
Ameliorated I,ands _
(A. A. Sokolov, S. I. Kharchenko) 91
Method fo1~ Determinin~ the Runoff of F~trained Sediment:; in
_ Rivers Using Data on Their Accumulation in Backwaters
(V. S. Laps~enko~, T. A. Boguslavskaya) 100
Method for Predicting the Mean Oblast Yield of Rice in the
Ukraine
(V. M. Prosunko, Yu. I. Chirkov) lp9
Descending Flu~c of Long-Wave Ra.diation at the 50-mb Level
(L, R. Ilmitriyeva-Arrago, L. V. Samoylova) 115
RolF~ of Radiati~n in the Formation of Stratiform Clouds
(Ye. M. Feygel'son) 121
Helical Clouds in the El'brus Region
(T. N. Bibikova) 124
Use of the Higr-Altitude Pressure Field for Increasing the
A~vance Time for 5hort-Range Forecasts of Sea Level
(V. L. Andryushchenko) 130
Law of Distribu~tion o.f Sum of Random Values With Type-III
Pearson Probability Densities in Hydrology
(I. V. Busalayev) 133
I~idar Measurements of Atmospheric Humidity
(V. M. Zakharov, et al.) 139
Method and Apparatus for Measuring Water Velocity or Disch~rge
in Open Shallow-Depth Flows
(M. I. Biritskiy) 149
Disc;ussion of the Carbon Dioxide Proble~n
(K. Ya. Vinnikov)..e 15~
- b -
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FOR OFF] .
CONTENTS (Continued) Page
Review of Monograph by A. V. Karaushev: Teoriya i Metody Rascheta -
Rechnykh Nanosov (Theory and Methods for Computing River
Sedimen ts') ax~d Book Edited by A. V. Karaushev: Stok Nanosov,
Yego Izucheniye i Geograficheskoye Raspredeleniye (Runoff of
Sediments, Its Study and Geographic I}istribution), Leningrad,
Gidrometeoizdat, 1977
(N. A. NLikhaylova) 162
Reti~iew of Mono~raph by Georgi Markov: Prognozira,ne Na Niizhdite
ot Voda Za Na.~oyav~ane V Meliorativnite Rayoni (Px~ediction of' -
the Needs for Water for Irrigation in Meliorated Regions),
Sofia, Institute of Water Problems, Publishing House of the
~ulga(A MA~Alpat'yev)ciences:.1978..128�Pages
166
Six.tieth Birthday of Vera Aleksandrovna Moiseychik 168
Sixtieth Birthday of Arkadiy Ivanovich Korovin 170
Conferences, Meetings and Seminars _
_ (Yu. G. Slatinskiy, K. P. Vasil'yev) 172
' Notes F`rom Abroad
(B. I. Silkin) l76
-c-
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I' Vlt UP C 11.1N1, UJL' UN1,Y
PUBLICATION DATA
English ritle : METEOROLOGY AND HYDROLOGY
_ No, 8, Aug 79 -
Russian title : METEOROLOGIYA I GIDROLOGIYA
Author .
Ed:ttor (s) : Ye. I. Tolstikov
Pul~lishing House , GIDROMETEOIZDAT
Plac~~ of Publication ; Moscow
Dat_e of Publication . 1979
Si.gned to press , 23 Jul 79 -
c~pies . 3870
COPYRIGNT , ~~Meteorologiya i gidrolo~iya,"
1979
_d_
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UDC 551.(509.313:584.5)
" NLTMERICAL MO1)ELING OF URBAN M:LCROCLIMATE
Moscow METEOROLOGIYA I GIDROLUGIYA in Russian No 8, Aug 79 pp 5-15
[Article by Academician USSR Academy of Sciences G. I. Marchuk, Doctor of
Physical and Mathematical Sciences V. V. Penenko and Candidate of Physical
and Mathematical Sciences A. Ye. Aloyan, G. L. Lazriyev, Computation Center
Siberian Department USSR Academy of Sciences and Transcaucasian Scientific
Research Hydrometeorological Institute, submitted for publication 1 December
1978]
Abstract: The article gives a numerical model of the
microclimate of large cities, based on a model of
atmospheric processes. The authors give an example
of the modeling of the microclimate of a city typic-
al for the mid~'te latitudes and situated in lowland
- terrain for t.e summer season. The contribution of
different fa tors to the formation of a heat island
over a city ~s analyzed.
[Text] It is well known that the climate of a city differs from the climate -
of the surrounding area [9, 10, 12J. Numerous measurements of air tempera-
ture in a citv and in its neighborhood show that a city is almost always
warmer. The temperature difference T between a city and its neighborhood
sometimes attains rather high values (more than 10�C). In the literature
this phenomenon is called an urban heat island (HI). It was found that a
HI is expressed particularly well at night, in windless and cloudless weath-
er. The intensity of a HI is also dependent on the season of the year, on
geographic latitude, on the popula.tion of the city, etc.
Urban built-up areas exert a strong influence on the external wind. Accord-
ing to [9], the wind velocity in the lower 500-m layer over Moscow is 1-3
m/sec less than in the outskirts of Moscow. On the average the wind velo- -
city in the city is 20-35% less than in the outskirts [10]. The city also
changes other meteorological characteristics, such as relative humidity,
the frequency of recurrence of fogs, the number of cases of falling of pre-
cipitation and its intensity, the intensity of solar radiation, etc.
1
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An imuortant. role in the formation of urban microclimate i.s played by ar-
~ tificial he~it i~lows, .forming due to the operation of industrial enter-
prises, heating systems, auto transportation, etc. With time their influ-
ence will be intensified, since according to the data of the WMO world -
energy consumption is increasing on the average by 6% yearly. In some
large cities, especially those situated in the high lar_itudes, which are
reached by little solar energy, the quantity of heat released during the
year is comparable with the quantity of solar radiation absorbed by the
earth. For middle-latitude cities this factor is especiall.y important in
- winter. For example, in Moscow the winter temperature is higher than in
the suburbs, despite the fact that the radiation balance in Moscow is
less than in the area surrounding Moscow.
In the opinion of many researchers [9, 12, 17], the f.ollowing fundamental
~:ictors exert an influence on the forn~ation of urban climate: -
the urban built-up area, whose influence on the external wind is mani-
fested through an increase in the roughness of the underlying surface; ~
ttie diPference in the thermophysical properties of tlie underlying sur-
face. in the city and in the neighborhood;
- artificial heat flows; -
mechanical with.drawal of precipitation and a decrease in the freely
evaporating surface in the city;
air contamination.
In addition to these factors, we should also mention albedo, which for a �
city during the course of the entire year is less than for a rural area.
_ ':'his is evidently attributable to the presence in the city of dark construc-
~ion materials, with the contamination of snow and with its removal from
the streets in winter, etc.
During recent years, together with experimental investigations, much at-
reiltj011 has been devoted to the mathematical modeling of the microclimate
of large industrial centers [11-14, 16-19;, and this is natural because -
the concentration of industrial plants and transportation facilities in
r_ities and densely populated regions has the result.that cities more and
more ar.e Feeling the negative influence of industrialization. The authors 4
of j13, 16, 18-19] describe the results of numerical modeling of the micro-
climate of cities such as Montreal [13], Saint Louis [18], Tokio [16] and
ottiers. In these studies for the most part there was an investigation of
the inEluence of some ciefinite factor on the climate of a city. For ex-
ample, in [13], on the basis of a two-dimensional model, a study was made _
aE the influence of a HI on variations of the roughness parameter and an
lrtificial heat flow at nighttime under winter conditions. In [1.9], also
us~ng a two-dimensi.onal model, a study was made of the dependence of a
nighttime HI on atmospheric stability and velocity of the geostrophic
w~nd. We sh~uld mention [14], in which a study was made of a on~-dimen-
sional model, but for the first time use was made there of the heat bal-
ance equation for dztermining the temperature of the underlying surface
in a city and in a rural area. Three-dimensional models were described
- in [11, 16, 18]. In [18] a study was made of the dependence of the
2
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microclimate of Saint Louis on the velocity and direction of thf~ external
wind with stipulated ;~T. Refereiice [16], also with stipulated ~aT, gives
' a study of the dependence of the wind field on the HI in Tokio, taking
into account th~ real shoreline in this region.
In this paper we describe a numerical model for study of the mir.roclimate .
of large cities. First of all, we will examine the hydrodynamic aspects
of the microclimate problem the interaction of an ~ir mass with the
underlying surface, the formation of a heat island ~.~nd local. circulations
agaiiisr_ the background of an external flow. The splitting method [SJ is
- the methodological basis for constructing a num~rical model.
l. Ttie system of model equations has the following form (8]:
' ~ d du
u-I- c1 i v urt d"- >.+1 '
~~i ax a~ -1v' + Uz dz o�, (1)
d~~' ' ~
dt + div uv aa~, ~.8' dy - lu' -f- dZ Y~~ d~ + 0 2~, ~ 2>
ad~ + div u S~ta,' + u' ax + v~ ay aZ YU d
d +'a~
- -u'Nx-v'8y + c,~. ~+Qr ~3)
n '
aar rt div uq'= - dQ (w' vz + v' dy ) d ds + ~
~ (4)
- u'Qs - ~u'Qy - (5)
~ d" _ ~8'
ds '
du' dv' drv' (6)
dx dv + dz -
u= C~ u', v- V-}- v', ~~v = W-T- w', 9-;- 4= Q~- Q~,
r=n-}-''s~.
= a E~~ a+ d~z a~~
_ oX ax ay a,~
Here t is time, x, y, z are curvilinear coordinates; x, y are mutually ortho-
gonal and are directed along the relief, z= zl -~(x, y), zl is elevation
- above sea level, the equation zl = a'(x, q) describes relief, u is the wind
~ ve:locity vector, u, v, w are its components in the directions x, y, z re-
spectively, ,.1 ,,Q N S are the convection, Coriolis and stratification
parameters, �l, �2 are the horizontal turbulence coefficients, yu,
_ y~y are the vertical turbulence coefficients for mom~ntum and heat, ~ is
3
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- run Vrr1~1NL UJ~ U1VLY
potent:ial temperature, q is specific humidity, 3Z is a val!ie proportional
to }~re~ssure, U, V, W, 0, Q, 1T are the background values c~f the meteor-
ologic~al Fic~lds, u', v', w', q', T1' are deviatiuns from the background,
e x, ~~y and QX, (ly are the horizontal gradients of background potential
temperature and background specific humidity, I,W is the latent hE:at of evap-
_ oration, cp is the specifi.c heat capacity of air at constant p~essure, ~ is
the rate of formation of the liquid phase, Qr is the radiation component
of the heat influx.
In order to describe the surface layer we use the fol.lowing system of equa-
- ttons:
"Z d ~a~ I - u~~ ' az = P;.. ~e ~F~ = 9)~ (7)
x I'' i= t~;:f� ~u), P- Po = P;~ f~ C= L,
Z� Z~~
~~r = L ~ ,p - L ~
~8~
u, x z u,;_ x h h r~=
~ Yi ~ ~f~/~ 'ti (~h) ~l _ [l, i)~~ ~h = L ~ L %.R=J, ~
(9)
whF.re ~ V I = u2 + v2 is the wind velocity modulus, u,~ is friction velo-
city, ,g q* are the scales of potential temperature and specific humid-
ity, ~ is the Karman constant, zu, z$ are the roughness parameters for
wi.nd and temperature, h is the height of the surface layer, the subscripts
0 ancl h denote tt:e meteorological fields when z= z.g and z= h respective-
? y) , L, is sc~~.le length, ~ is dimensionless length, ~pi, fi are c:ontinuous
unjversal functions [4J. The initial conditions are stipulated in the fol-
EoLlo~~ing form [8]:
u'=v'=0, q'=0, t`}'=0 When t= 0. (10)
'I'h~~ system is solved with the following boundary. conditions [8]: ~
vr~' `~v' = 0, d s' _ ~ ~q~ _ O when x= fX, (11)
d.r d,~ rlx - ' dx
- du' _ l) dv' _ Q J;F � d9' when y= fY , (12 )
ny dy ~ d~ - dy = O
~
rr.' = 0, v' = 0, = p, q' = p, ~m' - 0 when z= H, (13)
= U. h d IL _ `Fu ~~h) ll dP
aZ { v - .fu c~.h, ;u~ {v ~ h az =
_ ~a ~;h~ -
`Y~I (=h� :n) -Pn) when z = h. (14)
- G.
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fiere II is the hei~;ht of the boundary layer o.f the atmosphere, X, Y~ire
_ the lateral bound~iries of the region.
Over t:he water ~ 0 and qp are considered stipulated:
t9o=0o, ~lo=9H ~eo). (15)
Over Y_he land the qp function is compute3 using the formula
90 = ~10 9H ~ (16 )
where ~7~ is relative humidity, assumed to be a known function of x, y and
- t, qH is saturating specific humidity, which is computed using the Mageius
formuia [ 6 ] .
For dEtermining temperature at the ground surface we write the heat balance
equat3.on
,
G~ - P Cp dz lo -[7~ p La, ( va dz lo /a ( l- rlsl + IS - Fs, 17
~ i \ 1 ( )
where G= i1s ( d T/ a z)S is heat transfer through the surface z= 0 (the sub-
script.s denotes values when z= 0), i1 S is ~he heat conductivity coeffic-
ient, :AS is albedo, T is absolute soil temperature, is air density, FS
i.s effective long-�wave radiation [6], a ~ is a dimensionless coefficient,
makin~; it possible to take into account the fact that at different points
on the underlying surfac:e an unlike quantity of heat is expended on evap-
oratic~n (condensation) as a result of its inhomogeneity, Ip is short-wave
solar radiation, which is determined using the Albrecht formula [2], IS(x,
y, t) is a�unction describing the flow of heat which is released in the
process of production and consumption of energy in a city.
The pi�oblem of taking into account artificial heat flows in models of urban
microclimate for the time being remains open [13-14, 17]. A release of
thermal energy into the atmosphere is possible either in the form of real
_ or ~Ln the form of latent heat, and depending on this, different methods
for irs parameterization are possible. In this model the artificial heat
flow, ~y analogy with [14], is taken into account as an increment to the
radiation heat influx.
Under calm conditions the temperature drop between the levels z= 0 and
z= z;~ ca:1 attain rather large values. Accordingly, in solving equation
(17) it is desirable to use a known semi-empirical parameterizati.on for- -
mula for a viscous sublayer [3, 15] -
i}S - = 0,09Ei2 a . (u� z;f;,,~n.as~ ~18~
where is the kinematic viscosity coefficient for air.
Then the solution of equation (17) is written in tr?e following form [4]:
S
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~
,4~ = c~o ~,~h~,~~ t+ o?, v=_ cF,
(ly) -
where
C_ AcP IB 4(FS/TS)~-'J O,u962 ce ( ~u
I_~ Ih
=5 l~,as .
x l J ,
- N
i?o =~o-~ F Jo (1 -A S) - F1-~- B 0,414 ~1-~ /(n Ts-rt -f-
s
nv2
~9h - qo-i~ AFLw -I- AFLw !a l$u~t -`~s-1)~ ~4 - Pc~c~ ~ V ~h+
B = 2 ~Si(KS r,~ t)1j2~ Ko = 1, KR = -f- 1)~12 - 2 ia~f= (n - ~ ~il~
when n > 1.
The e}:pressions for � and F were given in [4J. The remaining notations are:
KS ~s ttie soil therrnal conductivity coefficient, cu, c,y are the friction and
heat transier coefficients, t is the time interval, ~GS-II =(7'~'n - T)
is the_temperature deviation of the underlying surface from its mean daily
value T at the time t= (j - n)L1 t.
On the basis of �sp, from (17) it is possible to determine Z~S.
Now we will enumerate the principal input data necessary for modeling the
microclimate of specific cities:
geographic coordinates and plan of the city;
local relief;
characteristics of the underlying surface: roughness parameter, relative
humidity, al.bedo, mean daily temperature, heat and thermal conductivity -
- c:~efficients for the soil; distribution and intensity of artificial heat
sources; ttiermal stratificatiori of the background atmosphere and the values
oE tlie bac!cground fields of ineteorological elements: temperature, specific
liumtdi.ty and components of the velocity vector.
~ ' .
All ttie values stipulated at the input information level are functions of
space coordinates. The background values of the meteorological elements
Ecr prognostic purposes are obtained from models of large-scale atmospheric
processes and in solution of problems in a diagnostic regime are stipulated
~.ising the results of processing measurement data for the real atmosphere.
2. As an i_llustration we will consider the example of modeling of the micro-
climare of a cir.y for summer. Figure 1 schematically illustrates the plan
of a city whose boundaries are encompassed by a dashed line whereas the
most densely built-up areas are surrounded by a thick line and shaded. We
will assume ~hat the city is situated in the middle latitudes in a lowland
area. A river divides the city into two parts and th~ saa (lake, reservoir) .
6
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is sit:uated alongside it. A considerable part of the territory is occupied
by green plantings, for the most part pine forest typical f.or the middle
lar.itudes (grid puints of intersection with circlesj, wherc~~as the remaining
area can be regarded as rural, characterized by scattered ane-story build-
ings, law shrubs, fields, etc.
y HM
36 � �
. . .g~. o�^a., . o . . . . .
27 a~ ~ o~~~,.. . ?
. . ,
, o[''~i~ . .
_ . . ,,~n.~~;\ 1c;.7ro~
_ . . . . . . .
;,g o o � . . . .
o o ~ . . . . . .
~ o.\ , o a . . . ~ . . .
~ � '
0 o`~.:J � , ' / . . . . . .
g � .
000
_ . . , o . . . . . . . .D~ \ . .
~ 0 9 1B 21 3ti x KM
Fig. 1. Schematic plan of city.
_ ~ Table 1
Parameters of Underlying Surface
- z. z;~ l0 ~'s 6 /S 7 8
~'~CCTHOCTb ~t M a~~ JCQIa KR.1 _ n~,
_ , C('ti 5 M� cerc ��C AI~ � CQK
2 ropo,q I I I 0,01 12,5 0.62 I 0,2 1(t) 0.25
3 Ceno 0,1 0,01 5 0,~ 0,3 0 I
- 4 6op I 0,5 I 0,01 5 0,26 ~ 0,4 0 1
KEY:
_ 1. Area (land use) 5. m2/sec
2. City 6. cal/m�sec��C
Cot~ntryside 7. cal/m2�sec
4. Forest g� a r
For tt~e computations we used a numerical scheme, constructed by analogy
with [7-8] with some modifications taking into account the specifics of
~he model. All the numerical experiments described below were carried out
_ with tlie following values of the input parameters:
X= Y= 21. 375 km; h= 50 m; Fi = 1650 m; L1 t= 1200 sec; G x= ~ y= 2250 m;
z= i 100 m when z~ 300 m, 150 m with 300 m< z
at ~"~''Sv~-�p41-QR~'L(c-e), ~2~
,
~9 v~ 9v dp = e-- c. ~ 3)
Elere the nota~ia:ns are those generally employed:
S= c~T + gz, h= cpT + gz + Lq is the static energy of dry and moist air,
Qg is tne rate of radiation heating, c is the rate of condensation of a
unit mass of air, e is the rate of evaporation of cloud particles. ;
_ The line at the top denotes averaging along a horizontal region, which must
be sufficiently large to contain a cloud ensemble but sufficiently small
in order to be only a part of a large-scale disturbance.
For convenience in investigating the cloud ensemble we will rewrite equa-
tions (2) and (3) in the form �
Q~ � ~i p�Sv d dp =QR +L ~c-e)- dp S'~', ~4)
26 ~
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L( d y+ C' 9z~ d
9`"1= L(c e) + L p q'
Q: d~ dP I ~ 5~
Here the primes dPnote deviations from mean values.
It follows from (4) that Q1 (it is called apparent heating due to large-scale
movements) is governed by radiation heating, the release of the latent heat
of condensation and vertical convergence of vertical turbulent transfer of
sensible heat. Equation (5) shows that Q(it is called the apparent loss
of moi.sture due to large-scale movements~j is goYerned by condensation and
vertical divergence of the vertical transfer of moisture.
From equatians (4)-(5) it is easy to derive the equation
Q~ - ~R- - dP ~S' + Lq') _ - ap h~,,,', (6)
where h'cJ' is the measure of the vertical turbulent transfer of energy.
This value can serve as a measure of activity of cumulus convecti.on. _
Using the conservation of mass, heat and water vapor equations in each cloud,
averaging them for the cloud ensemble, equations (4)-(S) can be reduced to
the form (see [9])
. Y QR _ m~ dP
- LDI, (7)
Q., - LM~ n~ - LD (qT - y+ l)~ ~g~
where M~ is the total vertical flux of mass in clouds, multiplied by g(its
dimensionality coincides with the dimensionality of uJ), D is the rate of ex-
pulsion (detrainment) of mass from the clouds for a unit pressure interval,
~ is the ratio of the mixture of liquid water in the clouds at the expul-
sion level.
In the derivation of these equations we made the following assumptions:
1. Active clouds occupy only an insignificant part of the cloud ensemble.
These are columns occupied by an ascending flow of a section constant in
height. Their bases lie at the level zg.
2. All the liquid water which is expelled from the cloud is also evaporated
here and the rate of evaporation is determined from the expression e= D~,.
_ 3. ~ach cloud produces an outsurge in a very thin layer where the cloud loses -
its buoyancy, that is, where h~ = h*.
n
Excluding ~ from (7) and (8), it is possible to obtain a fundamental equa-
tion for the cloud ensemble
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~ v~ ~ a.ya~~~ Vv~J Va\~.JL
Q~ Q: - Qa = U( h* -!t) - M~ dP .
(9)
[n a study by Yanai, et al. [9] equations (7)-(8) are used jointly with the
m~iss budget equati.on, equation for ttie static energy of dry air, wat~r vapur
and liquid water :quations for the entire cloud ensemble. In order to c]ose
the system of equations it is necessary to use a sufficiently artificial de-
pendence of the rate of falling of precipitation on the quantity of liquid
water. The derived nonlinear system of equations is solved by the iterations
meth~d. When Q1 - QR ~ 0, as is frequently observed in the Trades zone, the
scheme is not suitable. At the same time, the Yanai method does not give ,
- e:cplicit information on the spectr~ of distribution of different types of
clouds.
Arakawa and Shubert [7] postulated that the single positive parameter ~ can
completely characterize the type of cloud. Here ~1 is selected in such a way
that the level of the outsurge zD( a) decreases when ~l increases.
Then the total flux of mass in the clouds M~ can be expressed (7) as
ao(zl
~Yl: = J' ~n (z, i.) d a,, ~10~
0
where m~z~ a) cf a. -~,ti1:
?.i~f~., ~.�--da)
is t~~e mass flux into subensemble clouds for which ~i fal'ls in the interval
d~). The total outsurge at the z level will be
d i.o (z) ~11~
D 1 = - m (z. ~.o �
It is convenient to normalize m(z, T);
nt ( z, = InB ( i.) y, ( z, i.).
Iiere mB(~l) = m(zB,i1) is the density of the mass flux at the z-level for -
a cloud subensemble a . .
I t is easy to f ind, that E E; ( z) = nt (z, i.) u(z, i.) d i.,
at C( i,, i~ - d i. ~ .
where Ei(z) is entrainment in clouds of the type al at the z-level,
lz? = 1 f~ r. 1 z, 1, )
~ dz
is L-he fractional rate of entrainment of the subensemble
Substituting (10)-(11) into (9) and transforming to a p-coc~rdinat:e system,
we arrive at the following form of the fundamental equation for a cloud
ensemble
~
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d ( p ) _
Qi - Qs QR = lh^' - lt) mn (~.u (P)) yi (l', (P)) -
(P)
- ~h J� nle r~ (P, d i.. (12~ .
dp
n
Thus, for formulating a model of a cloud ensemble it is necessary to select
the ~ parameter and obtain mB( a) and n(p,~ 1n order to simplify solu-
tion of the problem substantially, Arakawa and Shubert [7] postulated that
the fractional velocity of entrainment of the f~ux of mass in a cloud, aver-
aged for the lifetime of the cloud, is constant with altitude. It is also
selected as the parameter assuming that ~,l (z, a. Iience we immediate-
ly derive an equation for ~(z,~~)
d r, ~p, al g
p ~~P. i.).
(13)
In this paper the physics of cloud clusters is investigated diagnostically
on the basis of observational data. Therefore, the R1 and Q2 values can be
consic;ered stipulated (they are determined on the basis of experimental
data). Thus, expression (12) can be considered as an equation for_ obtaining
mB(il) if the ~1 D distribution is known. (In forecasting problem.5, in order
to describe the parameterization scheme completely and derive an equation
for dFtermining mg(a one other additional condition is required. For this
purpose Arakawa and Shubert propose use of the hypothesis that a cloud en-
semble is in a state of quasi-equilibrium with a large-scale effect.)
In determining ~lD(p) we use the expression
- h~ (p~ _ ~ cP. a~ hB f ~7 (P', h (P') dz (P') , (14) .
Pe
which is derived by integration of the equation for conservation of the
static energy of moist air in the cloud subensemble. Here h~(p, 7l) and
hB(a) is the staric energy af moist air of the subensemble a at the
level~ p and pg. k'e note that the choice hg(a ) and pg for the time being
is arbitrary. We will find ~ D(p) from the condition that h~(p, at the
level of the outburst, where the cloud loses its buoyancy, coincides with
h* ~PD) �
A similar problem in investigation of the properties of the cloud ensemble
was solved in a study by Nitta [8J. That author used data from aerological
soundi.ng in the Trades zone, obtained during the BOMEX expedition. The sound-
ing was carried out to an altitude of 500 mb. Integration was carried out
with a 20-mb interval. In our case, since we will use observational data
in GA'I'E polygon AB at the principal isobaric surfaces, integration will be
carried out with an interval 50 mb up to the 100~mb surface. As hg we will
take hg = h*(pg), and pg = 950 mb. This assumption means that all the clouds
have ~.n identical energy at their bases. The same condition is also used in
8~�
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For numerical solution of the problem the entire troposphere from 950 to 100
mb is broken down into layers with a 50-mb interval. We will examine all
L-he clouds having an outsurge between pi and Pi+l and wi11. assigri them the
n?can ~~ci l ue ~i . Then ttie m.iss .flux ~it the base of tliese clr~uds wi.ll be
av (Pt)
Me _ ` mB(~) d'n. �
~p (Pt-f-i )
p M6
100 1
~ Q';~ B)
300
OR 400 i ~
1
/
S00 % ~
~ ~ I
_ 60U ~2 S j ;
700 ~ h ~ i n~
B00 \ I
~ ~ 1
950 '
1000 ~ ' ~
-2 0 5 �C/cym 75 Bo Kan/x 3 -
Fig. 1. Vertical profiles of the apparent heat source Q1, apparent loss of
moisture 42 and radiation cooling QR (a) and also profiles of static energy
of dry (S), moist (h) and saturated air (h*) (b)
KEY:
1. mb ~
�C/day
3. cal/g
It. is also convenient to introduce the mean outsurge and entrainment (pi,
Pi+l)� From the transcendental equation (14), by the iterations method, we
Find ~ i for each interval. Equation (12) is the Volterra integral equation
of the second kind (in it the upper integration limit is dependent on p).
Averaging this equation in the interval (pi, pi+l), it can be reduced to an
algebraic system of equations for the unknowns Mg( ~~i).
After finding MB, having the corresponding parameterization of precipitation,
it is already easy to compute all the parameters of the cloud ensemble. Now
- we will cite the results of computations on the basis of observational data
for polygon AB in the GATE program. The computations were made for mean
values in class "B" [4] (class "B" included situations when well-developed _
- ct~mulonimbus cloud cover was observed over polygon AB),
Figure la shows the vertical profiles of the apparent heat source 41 and the
apparent moisture loss 42 in units af heating rate. These values are means
for class "B" (24 observation times in the first phase of observations in
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polygan AB). The profiles shown here are very similar to ttie Ql and Q2 pro-
files obtained by Yanai, et al. [9] for the Marshall Islands. True, since
for us the averaging was carried out using cases of a strotigly developed
IC? (class "B"), the maximum Ql and 42 values in our case are almost twice
greater. The same as in [9], the Q1 maximum is situated appreciably higher
than the 42 maximum. In order to compute the vertical turbiilent energy flux
it is also necessary to know the profile of radiation cooling QR. Since di- -
rect measurements of QR were made in GATE only at nighttime, here we were
Eorced to use the climatic profile QR obtained by Doplik and already used
in a study by Yanai, et al. This profile is also shown in Fig. 'la.
Now we will proceed (Fig. lb) to an investigation of the static energy of -
dry (S), moist (h) and saturated (h*) a ir. In the upper troposphere (above
300 mb) these profiles were very close to the profiles obtained by Yanai,
et al. [9]; on the other hand, in the lower troposphere ttie values of all
three characteristics in our case were appreciably less.
In our opinion, this is attributable to the fact that the mean stratifica-
tion, obtained using data registered on island stations, characterizes re-
~ions with the strongest convection, since clouds, as indicated in [5], are
most frequently formed over the islands. The mean stratification for polygon
AB, even when ther~e is considerable cloud cover over the polygon, is far
from the str + y,",
n ~1~
where u is a precise solution of the differential problem, h is the grid
_ interval, i is the generalized number of the grid point of intersection, n
is the order of convergence, v~~~ are adequately smooth functions, not de-
penden.t on h, 1'jh is a grid function having the order 0(he
- Then, using the set (Q - n+ 1) of solutions with successively decreasing
steps, we find a linear combination of these solution~ in wfiich the second
term in the representation (1) is equal to zero. According~.y, the total solu-
_ tion will have the order of convergence h~ in the initial grid.~Th"us, this ~
method will make it possible, using simple schemes of the f-irst ar second
order of accuracy, to obtain higher-order solutions.
We wi].1 note some shortcomings of the extrapolation method. The equality
(1) in essence is another method for formulating convergenr_e of the scheme.
Naturally, proof of the correctness of expansion (1) can be accomplished .
only for a limited class of problems. Another shortcoming is rela.ted to
the asymptotic nature of expression (1). For example, in applying short-
range weather forecasting models the restrictions on the capabilities of
electronic computers do not make it possible to use very small intervals.
It tt?erefore follows that an increase in the order of the approximation wich
specific finite h values does not give a guarantee of an actual decrease in
ttie error of the corrected solution for any norm in comparison with the in-
itial solutions.
In this paper. we propose a ne~a method for the correction of difference solu-
tions i.n which the no nn of the error in the result in L2 is not greater
than the.errars in the initial solutions =n a sequence of ~rids. If expan-
7 sion (1) is correct, then at least in the case of a combination of two solu-
' tions it is possible to demonstrate an increase in accuracy by the order h.
for the norm C.
,lssume that we have two difference solutions ul and u2 for some problem with
the intervals h and mh, f~r which expansions of the type (1) are correct:
u~ = (u) + h� (vj + hR+~ ;i,
u = ~ ) Imh)" (v) (/r~h)"-~'1 Y~~� C2>
l~c~re m was selected in such a way as to avoid reinterpolation from one grid
to another (that is, m is either a whole number or a fraction with a numer-
ator ec~ual tc~ unity). The grid funetions ui, ~~l are taken in a set of
points of the coarser grid. The precise soluti~n u and the function v are
projected onto this same set. Thus, the dimensionality of the vectors in (2)
- is determined by the number of points in the coarse grid. It is easy to
show that the linear comoination
38
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a~u~--~b'us,
a' - "`n b~ - 1
- where ,n" - t ' m~~ - l
_ ~3~
approaches a precise solution with the order hn+l~
- We will demonstrate the following statement.
Theorem. The linear combination of solutions (2) ua
i+ bu2 with the coef-
ficien.ts a, b, minimizing the square of the error [a~i1 + Su
2 -~u~]2, con-
verges to a precise solution with the order hn+l~
Proof. We wi.ll assume that the vectors ui, u2 are linearly independent. We
will introduce notations for the scalar products:
~=((u), (u))~ d=((u), (v)), e=((U), (v))~
~
s( (4)
.f ~~u'~~ 711 r~ ~ - 11u~r yl^~~ S - llv~f `711~r ` - ~l ~~r Y~?~i
~ ~
k=(~I~, Y~a)~ 9= t''~i~ r=~rs~ Yi:),
~ > >
~ii=~(ui~ ui)~ ~sz=(us, u2), ria=(ui, u2)~
roi=((u), u~)~ ro2=~~~u), us)�
Then, in accordance with the least squares methud,
r�~ r~~ - r~,~ r�
_ Q = ~ ~
~~~a r, i- rui ~ ~ 5)
6- o , ~ _
~ - r~ ~ rzs - r12 .
We will write expressions for the scalar products, denoted by the letter r
with the subscripts, through the preceding values in (4):
r,, = c-}- 2 dh" 2 fhn+l eh-" 2 Shz n-Ft + 9Ji2 n-L2~ '
- ' r,= - c 2 ~t (mh)n 2g (mh1"+' e (mlz;=^ 2 t (mh)2
r (mhlz "~2, -
(6)
r~; - C -f- d (1 -L ntn ) h~ -f- ~ f g-"tR+' ) h�+1 em" hz "
-F- (s -f- tm) m~ fi" n~"~ "I" IzJ71"+~ l12
ro, = c d!t" -j- fh"+~, .
~ ~ r1z = c + d (~nh)� + g (rnh)R+~.
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We will denote the numerators in the expressions for a, b by ra, rb. Sub-
stituting (6) into (5), we obtain
o = (m" - 1)' (ce - d') ha R ~ ~ 1 - ~n" )~cs - ctm"+~ dg. - d
f ) l12
-I- Icq crm~ CjZl1i�-}'l f_' gmn-Fl~z~ h2 n-}-2 p tha R+ i).
rQ - (rn" - l ) m" (ce - d') h=" m^ [clm (2 m" - 1) cs -f- df -1-
_ dgrrt (1 - 2 m�J~ I12 n-11 ..F, mn+i [~rm"+I - ~k -f- g~f - gmn~-1)~ ji2 "~-s~-
~ (/13 "-~-1) ~
1'b = - ~ Illn 1; ~CB - l~ ~ Jl' n ~CS ~2 - JTL^~ - Ct/il"~'1 fi
- df (nc" 2) -i- daflln-~-1] h2 "+t (cq - Ckm"-~-1 ~ gfm"~'~
- f h2 n+= O (h3 "+1).
Hence it can be seen that a= a' + 0(h), b= b' + 0(h) and ra + rb =
0(h3n+l~~ that is, a+ b= 1+ 0(hn+l~~ The theorem is proved.
Remark. If the last terms in (2) have the order 0(hn+2), then the accuracy
uf. the total solution will have the order hn+2~
Assume now that there are - n+ 1) solutions in successively more "bunch-
ed" gi:ids. The proof of such a theorem for the general case of the combin-
ation n+ 1) of solutions is difficult since it is necessary to invest-
igate expansions in powers of h of determinants of the order ( Q- n+ 1).
But if we limit ourselves to the requirement of an increase in the accur-
acy of the solutions at the norm of the space L2, then as follows from the -
least squares method, the error in the total solution in a general case also
- will tiot be greater than the error in the corresponding solution obtained
by extrapolation.
A shortcoming of the method described in the paper is the necessity for a
precise solution. In prognostic models, even if there is convergence and it
is assiuned that there is a possibility of reckoning with grid intervals as
small as desired, with a sufficient accuracy we can obtain a solution of
the initial system of differential equations which is only som~ model of
ttie atmosphere. Therefore, in the specific use of the described algorithm
for a forecast it is necessary to check the stability of the coefficients
of linear combinations of solutions or investigate the nature of their be- _
havior with time. This will make it possible to carry out time extrapola-
tion of the coefficients.
The described method can also be used for an experimental determination of
convergence of the scheme and the order of the convergence. In actuality,
if, for example, we compute the coefficients a, b in two solutions and
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decrease the intervals, then with h-->0 a, b should tend to a', b'. Then from
expressions (3) it is possible to compute the order of accuracy n of the
initial solutions.
Tzble 1
~9
_ Comparison of Evaluations of Hemispherical and Regional For.ecasts and Their
Combination
I t% Q.s
Ypone}ib, - -
M6 ~ I ~ I 3 ~ I 2 I 3 i I I 2 3
Level, mb I
~
] 000 22 I t~ 1:3 27 . I 3 10 17 l0 3
850 1:~ I;t ti Iri 7 4 7 g q
700 l:i ~i ~3 17 ;3 fi 7 5
500 11 :3 p 8 'l !1 4 I
300 15 U 0 6 l l 13 1 I ~
200 17 1 U 12 0 1 14 1 0
Note. First columns comparison of hemispherical and regional forecasts;
second columns comparison of regional forecast and combined forecast
with coefficients from (5); third column same for combination of fields
with "prediction" of coefficients.
Now we wfll cite some preliminary results of computations using the proposed
method. jde used one of the variants of a"telescoped" system for prediction
with full equa.tions in an isobaric coordinate system [1]. The computations
were made for two regions: outer, covering the greater part of the northern
hemisphere (hemispherical model), and inner, adapted for short-range fore-
- casting in the West Siberian .region (regional model). The problems far both
models were formulated uniformly and differ only in the substitution of the
' lateral boundazy conditions: the condition of absence of flows through the
lateral boundariea is set in the hemispherical model, whereas the regional
model uses the results of the hemispherical forecast for correcting the val-
ues of the meteorological elements at the boundaries of the region. The tur- -
bulent terms in the equations of motion and the h~eat influx equation, and
also in the equation for predicting the geopotential of the 1000-mb surface,
being the lower boundary condition, were written in the form of the Laplacian
of the corresponding function with the constant coefficient 5�105 m2/sec.
Only horizontal turbulence was taken into account. The horizontal interval
- in the hemispherical model was 600 km, in the regional model 300 km; the
time intervals were 1 hour and 30 minutes respectively.
The coefficients a and b from (5) for the combination of the hemispherical
and regional predicted geopotential fields, and also the standard evalua-
tions: the relative error, r-- the correlation coefficient for the
geopotential trends and the mean square error o-for each of the six levels
(1a00, 850, 700, 500, 300, 200 mb) were computed for a limited territory.
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V..a va a iva~aL1 Vu1.1 Vl\LL .
Table 2
Comparison of ~valuations of Two Hemispherical Forecasts W:ith a Different
- Itesolution and Their Combinations
- f �k ~ �k a ~
~~pOBP.Hb,
_ M6 ~I ZI 3 I 4 I5 ll 2I 3 I 4 I.i ]I 2I 3 I 4 I r~ -
1000 11 22 1S -2i -4 13 21 14 -12 -3 4 14 10 -16 -4
850 12 S S-3:~ -S 8 7 4-I;, -G 3 7 5-lti -4
7U~ 1S 5 ~-3% - 6 3 2 1-10 -3 4 n 4-IEi -4
500 9 3 2-31 -7 5 1 0-14 -4 4 3 2-1!~ -4 '
300 7 `l p ---2~ -G .i 1 0 -11 -2 (i 1 0 -11) -4
200 14 0--4 I-21 -3 8 1 -2 -S -2 fi 1-2 -17 -3
Note, first-third columns same as in Table 1, but instead of ~i re-
gional forecast, a hemispherical forecasr in a fine grid; fourth columns
extrapolation of first-order schemes; f~fth columns extrapolation of
second-order schemes.
Table 1 gives the characteristics of improvement (for � and cr decreases,
for r-- increases) in the probable success of the forecasts. In the first
columns, for each type of evaluation we give a comparison of the hemispher-
ical and regional models, in the second columns a comparison of the re-
gional model and a combination of models by the described method. These -
values were averaged for two 24-hour forecasts with real initial data for
4, S April 1965. The table shows that the regional model gives much better
evaluations than the hemispherical model. We will assume that this is primar- _
ily a resulr of better horizontal resolution. A total forecast, even with
the use of a relatively poor hemispherical forecast, at the lower levels -
has a considerable advantage over a regional forecast. At the upper levels
the improvement is insignificant, but it must be taken into account that
in the cited example very good evaluations were obtained above for the in-
itial forecast using the regional model: for example, at the level 300 mb -
~ = 48%, r = 88%.
Although in a small number of examples it was impossible to ascertain the
n_iture of behavior of the coefficients a, b with time, we nevertheless
attempted to use the si.mplest method for their "prediction": employ for -
the combination of forecasts the coefficients obtained using computations
for the preceding day. In the third columns in Table l we give the results -
of a comparison of these, alxeady in a literal sense, forecasts with a re-
- alOIl$1. model, showing that even with the simplest method of extrapolation
of the coefficients a considerable increase in probable success is obtained
at the Iower levels. .
.
We note that altnough optimization was carried out on the basis of o', the
~ and r evaluations are also improved.
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In the next group of experiments we carried out a comparison of two hemi-
spherical models witli horizontal intervals 600 and 300 km. In contrast to
the preceding computations, the time intervals were 2 hours and 1 hour and
convective terms were not taken into account. There was also a difference
in the method for comparing the forecasts. In the preceding experiments the
hemispherical forecast was interpolated using bicubic splines, which does
_ not impair the accuracy of solution for the dense network for which the com-
parison was also made. Here the initial data for a model with good resolu-
tion were interpreted bilinearly in a dense network and the forecasting re-
sults inversely in a fine second-order grid.
Table 2 gives the experimental results. The notations are the same as in
Zable 1, but in place of a regional model we took a hemispherical model in
a fine grid. Here a similar effect was ohtained: at the lower levels there
is a considerable improvement in the quality of the combined forecast (even
greater than in the preceding computations); deterioration at the 200-mb
level is extremely insignificant.
IJe also checked the combination of forecasts by the extrapolation method.
Taking into account that the model does not use two-cycle splitting [2],
and accordingly the schemes have a first order with respect to time, we
computed a linear combination with the coefficients a= 2, b=-1(m = 2, n
= 1). The results of the comparison with the fiemispherical model in a fine
grid are represented in the four columns in Table 2. It can be seen that
these forecasts with respect to many indices are even worse than forecasts
using a coarse grid (mesh). Since with respect to space variables the
schemes have a second order of approximation, we made a combination with
the coefficients a= 4/3, b=-1/3 (fifth columns). The results~show that
the evaluations deteriorate at all levels, but nevertheless the order of
the schemes was closer to the second, although it is not attained due to
- the coarse resolution. We note that the coefficients, with respect to for-
mulas (5), in all the computations were different for different levels, pos-
itive and less than unity.
Thus, the results of the described experiments show the great promise of us-
ing the methad described ir. the paper for a combination of forecasts. Fur-
ther investigations must be carried out using longer series of initial data
for the purpose of finding methods for predicting the coefficients.
BIBLIOGRAPHY
1. Kalenkovich, Ye. Ye., Novilcova, N. V., Cholakh, I. V., "The Forecasting
- Problem for the Northern Hemisphere and a Region," TRUDY ZAP.-SIB. RNIGMI
(Transactions of the West Siberian Regional Scientific Research Hydro-
meteorological Institute), No 41, 1978 (in press).
2. Marchuk, G. I., M~TODY VYCHISLITEL'NOY MATEMATIKI (Methods in Computa-
tional Mathematics), Moscow, Nauka, 1977.
43
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UDC 551.509.313(-062.5)
DYNAMIC ASSIMILATION OF POTENTIAL AND WIND FIELDS IN THE LOW LATITUDES
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 79 pp 40-48
- [Article by Candidate of Physical and Mathematical Sciences A. F. Kivganov
and U. Ch. Mokhanti, Odessa Hydrometeorological Institute, submitted for
publication 24 October 1978]
Abstract: The article describes the formula-
tion of experiments for the dynamic assimila-
tion of the geopotential and wind fields with- -
in the framework of a regional barotropic model
employing full equations with the use of some
variants of boundary conditions and different
initial data. The results of assimilation in
the low latitudes are given.
[Text] Prognostic models with the full equations of hydrothermodynamics im-
pose increased demands on the representativeness and assimtlation of ini-
tial fields. For the filtering of high-frequency noise in the initial
fields, and also their assimilation within the framework of a specific
prognostic model the dynamic assimilation method has come into extensive
use [7, 14, 25]. In the temperate latitudes, in addition to the geopoten-
tial field, the geostrophic wind is used [3, 7, 8]. In the low latitudes
the quasigeostrophic approximation loses sense and the use of a solenoidal
approximation involves serious difficulties in solving the wind balance
= equation. Therefore, it is unquestionably of interest to investigate the
possibility of the use of information on the real wind for the purposes
of a hydrodynamic forecast using full equations. The desirability of such
an approach was mentioned in the studies of Krishnamurti, Washington, Gor-
cion and others [11-13].
In this ~aper we discuss the results of experiments for the dynamic assim-
ilatior. of the wind and geopotential fields within the framework of a re-
gional (3-40�N; 60-120�E) barotropic model employing full equations with
- the use of different initial information. In addition, a study is made of
the influence of different boundary conditions on the assimilation process.
44
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The latter m:itter is acquirin}; special importance in the realization of _
regional for~casting schemes and contains a number of fundamental difficult-
ies.
_ The dynamic assimilation of the initial fields is accomplished within the
framework of a barotropic model using the full equatinns
d,t _ F. ( )
� d~ - 1
He re ri
q~
X = ,
~ '
~I - r~ dz v dy - a~ +!v
dv dv r) ~
f-
_ - fl dX ~ Z~ O Oy -~ll ~ .
y
' a ~Y ~ d du dc~ .
-u d~ -z ~y -~~ox dy j
where u, v are the horizontal components of the wind velocity vector, ~ is
geopotential, is the Coriolis parameter, X and F are the matrix colLUnns
of the dependent variabl.es and the right-hand side respectively.
- The system of equations (1) is integrated in time in the form of a four-ele-
ment Matsuno scheme, that is, the dynamic assimilation of the initial wind
and geototential fields is accomplished using the pseudoforecasting method
[7]. As is well known [6, 15], the Matsuno scheme leads to the appearance
of computation viscosity, favoring a considerable decrease in the amplitude
of high-frequency noise caused by errors in the measurement and analysis of
the initial fields and also generated in the process of a finite-difference
salution of system (1).
Ttius, the P4atsuno scheme is reduced to the four following elements:
forward forecast
Xn+~ _ �Yr, -F- t F' ( ~';,1.
~(n�~t _ Xn ~ F ~i~n+~~; .
backward forecast .
~1'n =11'n~ t - : F ~Xn-~1~' ~2)
1 ~(R+~ - T (Xn). -
Here Yn+l and Xn is the preliminary X value at the times n+ 1 and n respect-
ively, is the time interval, V is the number of the iteration.
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~
~ a c~ c~ c~ v~ ~~n o v~ o 00
. N' p ci ci cil~ -~lei ~ilcv ~ilci
_
~ ' _
~ ~p Ip ^Io ^I^ OlO ~I(~ OICO -
O O
(d ~ ~
H
'c - �
~y~ ~ Cf t` C~ G~
~ r.~ J O O O
~ _
O
ti-i
~ cYJ C rJ ~1 ~
b ~~i~.-.~ O O O O ; j
G, _
O
U F ~
~ W ro ~IM MIM ~IM N~M ~I~
cd - - ^ l
b .
~ _
~ ~ y ~.q I~..~ ~ I~ ry Iti ~ I~ ti I~
N ~
~ ~
w ~
Q
~
~ ~ ~
~ ~
q.~ ? ~ 0 0 0
v
L1 F -
cC~ u
~-I ca M er M M
~ ~ 17' C p O O ~ p .
O O v
FL1 ~ C
O . C
~ 00 O' W IC1 u'~ l~ ~ I~y vj
~ ~ ro ~ M Qi tA ~D N h M' ~OIM ~
~ ~i
a~
w r-~
~ ~ a M 00 c+~ N... u" I~ I
~ rc ~a N~.: f~j IGV Cy I..: Cy N C~J
~
O E '
~ ~.s `a oo cc cv cc
N ~ ~ ^ N
~i -
V7
I;~ ~ ~ ~ _ .
~v _
G
m
'c a
~ M Q~ Q~ .r t~ V' ~f7IQ~
~J CV M l~ Qi M I^ ~ IM M~
- -
~ _ ~i
4-1 I ~ I~+ ^ I M h I00 O I C+~ O pp .
n w O~ ~ M p I... IN fA
ai u ~
U ~
~ 1J ~
` `c
a ~r i. ~ ro y �
'L~ ro O C'j f.V Cy �rl 1-I
~
~ - v ~
y ,y' ~ ~n ~ ~ P cn
~l o 0 0 0 0
- ' r-I N
~ ,LIIBH~Bg ~ ~ ~W
~4
46
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The f3.nite-difference approx~.mation of the right-hand side:~ of s.ystem (1)
is ca~~ried out using the F. Shuman scheme (advective form with a filtering
factor [9]). The Yiorizontal grid interval is 210 km (21 x 31), the time in-
terva7. is 10 minutes. The parameter is assumed to be var:Lable.
_ l~i vartant 1 on the boundary uf the grid region ttie valu~s of tlie X functions
were kept constanton the boundary of the grid region. Such a boundary condi-
tion in combination with the filtering procedure was used in a regional prog-
nostic scheme for the low latitudes [lj.
In variant 2 there was stipulation of a boundary condition of the solid wall
type
V,~ I ~ = lj, ~3~ -
where Vn is the normal comgonent of the wind velocity vector on the contour
- I. 7'he u and ~ values at the boundary are found directly from the equa-
tions of motLon of the system (1). ~
The following boui~dary condition was set in variant 3:
.3V,. ~ dV� ~
U~. ( us ' dn )i.~ 0, ~4)
where D is horizontal divergence, VS is the tangential component of the wind
veloci.ty vector, d/a n is the derivative along the normal to the boundary,
a/a s is the derivative along the contour.
In variant 4 for the westerly and easterly boundaries of the grid region
there was stipulation of the splicing conditions for periodicity [13, 15J, .
for the northerly and southerly boundaries the solid wall conditions (3).
{uo ~o - - - - ,zo
I a1 r- - - - - q1
S~dr'~ J7Y~
~ ~
1.: SJE
~d~ J78-
I~p Sf9l _ 560-~ I
\ `.3_- 70i
i ~ - I
~
~ ~ m ~ ';~6- ;
wc ~ ~ ,
J _-ti., ~
;,,;q ~ Fig. 1. Initial (a) and assimilated
a ~
~,o se; ~ S%r;,"1~ (b) AT500 fields at 0300 hours on
~ se2 s~o ssa se2 _ 6 November 1973.
+;p ~ ~ ~ i4J
SBO ~ , i
~ Ol 3'`~ - ! i
iw SSf4~ �`':a ~TS~ ,Py~7C~
~
i I
_ ~ B SB4 '9b ~i
S00 i ~
SB4 ~ 318 ~
sez , sen~ , y SIZ sevy~
Eo~ eo so , no ,zo
r
_ 4 7
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35~~`~ 10S 6oMlceK ~ ~
,~oo - t
240
1B0 KEY : . -
1. dam/sec
~20 2. m/sec2
E~ ~ 3. years
,i
G ~
lu~l~lOs rleeKz
96 ~
1
72 .
' 48
24
Z 3
0 6 i2 J8 t y
Fig. 2. Changes in ~~t~ (a) and ~ ut~(b) in process of dynamic assimilation
and farecasting: 1) in forecast without assimilation; 2) in forecasting with
assimi.lation; 3) in the assimilation process.
A comparison of the results of experiments with diffecent formulation of
the boundary conditions is given for four subregions I, II, III, IV (Fig.
la), urhich were selected with circulation conditions taken into account. For
each subregion (25 grid points of intersection) we comguted the mean abso-
lute differences of the assimilated and initial field~ ~b X~ and the maximum
values ~ b x
The results of the experiments, presented in Table 1, were obtained for a
free assimilation regime when not one of the three fields u, v, is
regenerated either within or after completion of the Matsuno cycle. As the
initial data we use the real wind, from which the stream function ~ field [2]
is computed with subsequent inversion of the wind balance equation. As the
boundary conditions on the contour we stipulated the actual geopotential
values and a solution of the equation was obtained by the iteration method
for the internal region 5-38�N and 62-118�E. Thus, before the inttializa-
tion process proper there is preliminary assimilation of the wind and geo-
potential fields within the framework of solenoidal equilibrium.
Figure 2 shows the dependence on the number of cycles (n), averaged for the
- entire region of assimilation of the absolute values of the trends ~ut~ and
~c~ t~ for variant 4. It can be seen that the greatest variability in the
- trends is observed during the first 4-5 hours of the pseudoforecast, and
t.heir variation is quite smooth. Therefore, for detecting the effect of in-
fluence of the boundary conditions on the process of assimilation the re- .
sults of the experiments in Tab1e 1 were obtained for 25 cycles corre~ond-
ing to the characteristic adaptation,time. The numerator gives the ~b u~ and
u~m values; the denominator gives the ~ b v ~ and ~ b v values; the fig-
ures in parentheses are the corresponding characteristics for 72 assimila-
tion cycles.
48
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M analysis Table 1 shows that for the internal subregic,n III, and also
" tlu~ boundary subr.~~;ic~n IV, characterized by a low-~ra~lient nresgure Cteld,
~li~ r.~~;;u.lts ~~F ~i11 exp~rlments differ little Erom one anoLher, th~it is, tlic
_ ,issimS11t~011 is virtual.ly not dependent on the type of boundary conditions.
I~or subregion I-- a region wi~h high pressure gradients the solid wall
condition gives the poorest results in comparison with the other boundary
conditions. The changes in~~~X ~ in subregion II for all variants are much
greater than for the remaining subregions. This can be attributed to the
presence of a tropical cyclone, inadequate initial information and close-
ness to the boundary. A comparison of experiments 2 and 4 shows that the
splicing condition for periodicity for the eastern and western boundaries
leads to an appreciable decrease in the ~b X~ values.
Tlius, an analysis of the results shows that for regional prognostic schemes
in the low latitudes it is possible to use boundary conditions in the form ~
of variants 3 and 4 as being the most preferable. However, further experi- -
ments for developing a regional prognostic scheme indicated that the best
Porecasting results are obtained from the use of a boundary condition in the
Eorm of variant 4 in comparison with variant 3. Therefore, all th.e subsequent
results Prom this study were obtained using a boundary condition in the form -
of variant 4.
We also note that in the wind and geopotential fields, obtained 18 hours
after the pseudoforecast, high-frequency noise is virtually absent, and there-
fore no need arises for ac~ditional use of space filters.
Now we will discuss the content of experiments in the field of dynamic as-
similation with different initial data.
Scheme 1. As the initial field we will use the real wind field. The geopoten-
tial field is reconstructed on the basis of the wind field by means of com-
puting the stream function ~ with subsequent inversion of the balance equa-
tion. In the assimilation process the fields are not regenerated, that is,
we solve the rPCiprocal adaptation problem (so-called free assimilation re-
gime). This scheme was used above in obtaining the results represented in
Table 1.
Scheme 2. The znitial fields are the actual wind and geopotential fields, -
that is, independent fields. A free assimilation regime is carried out.
_ Sclieme 3. The geopotential and wind fields are stipulated the same as in
" scheme l, but in the assimilation process the wind field is regenerated be-
fore the onset of this cycle, that is there is solution of the problem with
_ reconstruction of the geopotential field on the basis of the wind field.
Scheme 4. In contrast to scheme 3, in this experiment there is introduction
of artificial "mismatching of the,fields," when the values of the wind velo-
city components are assumed equal to half the values of the real wind.
49
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Scheme 5. The initial fields are stipulated completel,y in c~ccordance with
scheme~ 1, but instead of the Matsuno procedure, in the fre~ assimilation
_ proce~s use is made of the Okamura scheme
X~,~,~ = XR T F~Xn~r X~ = Xn.~.~ T F~~'~~~~~ l5~
X,~ 3 X;~ - 2 X;,',
having, in comparison with the Matsuno cycle, a doubled damping character-
istic curve [4].
Table 2
X~ and~b X~~ Values for Different Assimilation Schemes
Cxenta ~b~V~ I ~~,`1'~rn I ~Lu~ I ~aulm I ~bv~ I ~cv~~
1 0.4 2,5 1. I 8,7 1,1 3,6
(0,7) (~~0) ~Z~3) ~9~6) ~~~3) ~4~1? -
2 0,7 3,3 1,1 8,8 1,1 3,6
(1~1) (4~2) ~1~4) ~9~6) ~1~4) ~4,1)
3 0,6 2,g - ~ _ _ _
4 l,2 4,1 - - _
_ 5 0,8 3,0 1,3 9,2 1,3 (4,1) -
The results of the experiments are given in Table 2, where the fb X) and
{b X~ m values relate to the entire assimilation region (the corresponding
values for 72 cycles are given in parentheses).
The results obtained using scheme 1 show that geopotential changes in the
assimilation process are s~all. This is possible due to two princ.ipal fac-
tors: as a result of the procedure of preliminary assimilation of the geo-
potential and wind fields within the framework o� a solenoidal approxima-
tion and as a result of the specifics of the adaptation process itself,
caused by the smallness of the Coriolis parameter and pressure gradients
in the low latitudes.
As an example, Fig. 1 shows the initial and assimilated geopotential fields
.l8 hours after the pseuduforecast. The comparison shows that the nature of
tlie fiel.d did not change in the assimilation process; the pressure gradients
in this case somewhat decreased, especially in the region of pressure forma-
tions. .
A comparison of the results obtained using schemes 1 and 2 shows that the -
~ b u I and Ib v{ values, and also ~ bu ~ m and f~v~m differ insignificantly
(0.1-0.2 m/sec), whereas the mean and maximum absolute differences of the
geopotential fields in scheme 2 are greater than in scheme 1 by a factor
of 1.5-2, although as before, they remain small, not exceeding 0.7 dam, and
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their maximwn values are 4 dam.
This c:omparison gives basis for concluding that in the low latitudes, the
same as in the temperate latitv.des, the geopotential field adjusts to the
wind field. This is also confirmed by a comparison of the i-esults obtained
using schemes 1 and 3. With an increase in the "mismatching" of the initia.l
fields (schemes 3 and 4) the change in the geopotential field occurs
more intensively in the adaptation process. This is also correct for the
f:~:~_ assimilation regime (schemes 1 and 2).
Ttie diFference in the results obtained using schemes 1 and 5 can be attrib-
uted to the :Eact that when using the Okamura method the assimilated fields
are obtained more averaged than in the Matsuno method as a result of an in-
crease in computation viscosity and in this case the assimilation process
transpires far more intensively.
Comparison of assimilation processes in the low and temperate lat:itudes
for similar experiments [7] indicated that the reaching of a stea.dy regime
in the low latitudes is attained somewhat earlier than in the temperate
latitudes.
The effectiveness of the pxocess of assimilation of initial fields was check-
ed in a series of experiments for developing a regional prognostic scheme.
It can be seen from an analysis of Fig. 2 that the prognostic trendsf Xt~
increase rapidly in experiments using nonassimilated initial fields in
the forecasting process, reflecting clear indications of computation in-
stability. In contrast, in a forecast using assimilated fields no signif-
icant fluctuations are noted in the course of the averaged trends during
_ the course of the entire time of the forecast.
Fiowever, in the predicted fields themselves, obtained using assimilated
data, especially near the boundaries, there is a localization of high-fre-
quency oscillations with a wavelength of about two grid intervals. This
noise is generated in a prognostic model, in particular, due to the boun-
dary conditions, and can be suppressed using spatial-temporal filters.
Thus, in contrast to the problem of assimilation in a prognostic regional
model it is necessary to have additional filtering of high-frequency noise.
- In this study the spatial filter, based on the proc.edure o:F "smoothing-
- desmoothing" of prognostic fields [5], was applied after each four time
intervals (ttie time interval in the forecasting scheme was assumed equal
to 5 minutes). This made possible a complete filtering of noise in the
entire forecasting region.
The effectiveness of the assimilation process and filtering can also be
judged using the data in Table 3, which gives the results of a spectral
breakdown of the kinetic energy K of the initial and predicted fields for
6-7 November 1973. Here use was made of the fast Fourier transform method
[10] and a filter with the parameter 0.07.
51
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~
. .
0
~
~ ~ ~ - ~ ~
o
M Y C ~ N `T N .
_ ~ ~ 3 ~
.r~ .
~ a�
H
a ~
r, , TJ .C
r-1 � ^ ~
~ 'U b0
- �i ~ o c~ o �rl ~ G
~ ~ ^ ~ ~ � ~ w O r-1
, ~ ~ ?G c~d
~ ~ _ a' c,~ 3
u cc
G d rA I
~ ^ ^ I
~ ~t co a
. ~ J
v F
~ : c~'. ~p � . `o .
CV ~ O N 'C~ d~ ~-1
F' u~ ~ rl ~7 C~
C~+ L
~ o u ^ W ~
W �c. ,G,~ q
U C .~u ~-~i ~
ci
~ u cd
~ v ~ ' .
~ ` ~ ~ 3
~4 ~ ~ ~ ~ c~ oo - ~ ~
N ~ ~ a ~
o ,r�, ~
~ M ~ ~
. w y�A
N -
~ ~~~z
~
~ ~
~ o ~n
N I ~ 4-I Ul N
I ~ ~
r,,,~ 0 ~ ~ ~ Il~ ~ N
O (U ,C
~ ~ I u ~ ~ e; 1~J ~rl GNl
~ p I u c ~ ~
V r C'r cd
~ cv H ~ '~3
a c t~
C/] _ _
~ ~ ~
p N � ^
0 ~ u ~ A O
~ ~ O
N ~ ~ ~ O ~ ~ ~ ~ N J V ~ ~
~ fD O �r~ ~ ~1
~ ~ O N N . v 1~1 rl r-I
N ~ ~ f1~ e'-~
~ �rl
a -
~ a � m a~i ~m cn
~ ~ a~
~ ~ ~ cud ~ c~U
~
. ~ c~U ~ 3
.C rl U
Y ' z c.~ 3
s ~
~ cv 3 $ o ~ ~-I cl,
x o
o ~ ~ ~ ~
ax ~ II �m� S~/ w U G cn
~ u S`9 �m'~ ~ ~ c 41 O
~ _ � ' u~.l v/ Y ~ ar C]. �rl ~
~C ' ~ Y W y ^ � ~n ~ ~ ~ (d ~
. ~C i � f' ~ x k ` E ^ ~ E d r~-i ~~rl
_ a, ~ ~ ~ C
o ��e o� jj o�.o p y
e a~ x~~ 3
a0 rn O ~-I N ,
ri r-I r-i
51a
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It can be seen from an analysis of Table 3 that the assimilation and filter-
ing regimes are expressed first and foremost in the short-wave range, to a
lesser degree in the synoptic and long-wave ranges, 3nd there is virtual-
ly no ~hange in the kinetic energy of zonal flow.
A comparison with the results of a forecast made using unassimilated fields
:indicated that the. assimilation and filtering regimes are indispensable parts
of a regional forecasting model using full equations in the low latitudes.
In conclusion, the authors consider it their pleasant duty to express apprec-
iation to M. S. Fuks-Rabinovich and L. V.Berkovich for useful comments and
advice expressed in the course of discussion of the results of th:is study.
BIBLIOGRAPHY
1. Bedi, Kh. S., Datta, R. K., Krichak, S. 0., "Numerical Forecast of Meteor-
ological Elements Under Summer Monsoon Conditions," METEOROLOGIYA I GIDRO-
LOGIYA (Meteorology and Hydrology), No 5, 1976.
2. Kivganov, A. F., Mokhanti, U. Ch., "Computation of the Stream Function
~dith Different Formulation of Boundary Conditions," METEOROLOGIYA, KLI-
MATOLOGIYA I GIDROLOGIYA (Meteorology, Climatology and Hydrology),
Kiev, No 14, 1978. _
3. Kluge, I., Esberg, E., "Effectiveness of Some Methods for Assimilation
of the Geopotential and Wind Fields," METEOROLOGIYA I GIDROLOGIYA, No
1, 1976.
4. Nitta, T., "Preparation of. Initial Data and Objective Analysis for a
Model of Primitive Equations," TRUDY VTOROGO TOKIYSKOGO SIMPOZIUMA PO
CHISLENNYM METODAM PROGNO'LA POGODY (Transactions of thF Second Tokio
Symposiwn on Numerical Weather Forecasting Models), Leningrad, 1971.
5. Spektorman, A. D., ruks-Rabinovich, M. S., "Baroclinic Prognostic Model
Using the rull Equations of Atmospheric Dynamics With Detailed Horizon-
tal Resolution," TRUDY GIDROMETTSENTRA SSSR (Transactions of the USSR
Iiydrometeorological Center), No 180, 1976.
6. Fal`kovich, A. I., Yurko, T. A., "Wind Reconstruction in the Low Lati-
tudes by the Dynamic Assimilation of Fields Method," TRUDY GIDROMET-
'?'S~NTRA SSSR (Transactions of the USSR Hydrometeorological Center), No
103, 1972.
7. Fedorova, N. G., Fuks-Rabinovich, M. S., "On Dynamic Assimilation of
Initial Fields for Models Using Full Equations in Hydrothermodynamics,"
METEOROLOGIYA I GIDROLOGIYA, No 5, 1972.
52
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t3. Fe~iorova, N. G., "Application of the Method of Dynarnic Assimi:lation of
Fields in Conputing Wind for a Prognostic Model in the Full Equations of
Hydrodynamics," TRUDY GIDROMETTSENTRA SSSR, No 103, 1972.
9. Shuman, F. G., Stekaul, Dzh. D., "On the Problem of Writing Finite-Dif- -
ference Equations With Allowance for Map Scale," LEKTSII PO CIiISLENNYM
ME'YODAM KRATKOSROCHNOGO PROGNOZA POGODY (Lectures on Numerical Methods
for Short-Range Weather Forecasting), T,eningrad, Gidrometeoizdat, 1969.
10. Cooley, J. W., Tukey, J. W., "An Algorithm for the Ma.chine Calculation
oE Complex Fourier Series," MATH. COMPUT., Vol 19, 1965. -
11. Gordon, C. G., Umscheid, L., Miyakoda, K., "Simulation Experiments for
Determining Wind Data Requirements in the Tropics," J. ATMOS. SCI., Vol
29, No 6, 1972.
12. Houghton, D. D., Washington, W. M., "On Global Initialization of the
Primitive Equations," Part I, J. APPL. METEOROL., Vol 8, No 5, 1969. -
13. Krishnamurti, T. N., "An Experiment in Numerical Weather Prediction in
Equatorial Latitudes," QUART. J. ROY. METEOROL. SOC., Vol 95, No 405,
19b9.
14. Miyakoda, K., ?Koyer, R. W., "A Method of Initialization of Dynamical
Weather Forecasting," TELLUS, Vol 20, No 1, 1968.
- 15. Nitta, T., Hovermale, B. J., "A Technique of Objective Analysis and In-
itialization for the Primitive Forecast Equations," MON. ~dEATHER REV.,
Vol 97, No 9, 1969. .
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. UDC 551.509.313(-062.4)
TWO NONLINEAR PROBLEMS IN DYNAMICS OF THE EQUATORIAL ATMOSPHERE
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 79 pp 49-54
[Article by S. Evtimov, Professor S. Panchev and R. Chaka~ov, Sofia Uni-
- versity, Bulgaria, submitted for publication 6 December 1978]
Abstract: The authors have derived precise solu-
tions for two stationary systems of equations
(1)-(3) and (17)-(20) modeling movements i.n
the equatorial atmosphere. The peculiarities of
these solutions are discussed. -
[Text] First Problem. Dobryshman [2~ proposed the following dimensionless
system of equations of motion for the equatorial atmosphere:
ou du v ^ w, '
vay+r~a1=Y ~1~
ydy+wax=-Yu-da ~ (2)
a`' + d~ =0, (3)
dy dz
where x, y, z are the axes of coordinates directed to',!the east, north and
zenith respectively; u, v, w are the velocity components and dS~/ ~ y is
_ the meridional pressure gradient. The latter is usually considered stip-
ulated, that is, system (1)-(3) is closed. It has been solved by many
authors and by different methods [2, 3, 5, 6, 9]. Here we propose still
another class of elementary precise solutions which can have importance
in the interpretation of observed atmospheric circulation in the equator-
ial region. In addition to (u, v, w), a solution is obtained for
This is achieved by the introduction of requirements on the separation
of variables, which is equivalent to the addition of a new equation to
_ (1)-(3).
In actuality, in accordance with (3) we write that
vly, z~ =f(y)h'(z), ~4~
~ ~y, z) _ -f'(~) h {z~,
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whe~re the prime denotes differentiation for the argument. Then equations
(1) and (2) assume the form
h' l ay y) f~ d~ -F~ 1~ _ ~ (5)
ff' (h'2 - hh") _ - yu - day . . (6) ' _
In order for the variables to be separated in equation (5) it must be as-
sumed that
ay - Y = ~i (z), ~s + 1 =13 (Y), (7)
where A(y) and. B(z) for the time being are arbitrary functions. However,
2u/ i3 y~ z= ~ 2u/ d z a y, hence we obtain A' (z) = B' (y) const,
which gives A(z) _-,~(z - zp)~and B(y) y. Substituting this into (7)
and integrating, we obtain an expression for zonal velocity
u (Y, z) = rto Y'' y -1)(z - za), (8)
where zp > O~is some fixed altitude. This is a new equation which closes
the system (1)-(3). .
Now equation (5) assumes the form ~
h (z ~ Zo) = ff ~ y - a, - const. ~9~
- After integration we find
f (v) =foya ,
h (z) = ho (z - zola , . (10)
where a= a 1/~, and f~, h~ are the integration constants. Finally, from
equation (6) we also obtain the pressurp field
dy Vu - v~ az y2 (2 - zo)' a-=, (11)
T~here v~ = fph~, and u(y, z) is given by expression (8). Finally, for
solving the initial system we obtain
' u cy, z) - u~ -F- 2 y= r Y- 1)(z - zol,
~y, Z~ _ ~.vo ya (Z - zo~a-1, . c12>
~ (y, z) - - a 2~o Ya-' ~z - xo)a , .
1 2 t 4 ~ a_ ~ z-z)-
~(y~ z) - cp~ - 2 uo y- g y-(~ y Z Y'" J~ n
- 2 vo y2 � (z - z~)'- ~a_'~�
This salution has the free parameter a. Varying this, it is possible to
derive formulas approximating the real wind and pressure fields in the
equatorial atmosphere. In order that there be no singularities in (12),
it is necessary to adhere to the conditions
55 ~
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a~l or a=0. (13)
We will examine some special cases. Assume that 0~' = 0. Then, ~_n accordance
with (12),
v=r~=0, _
1 z - x ) (14)
ta = uo 2 y= Y-)~ o~
a~/ay=-yu,
that is, we have obtained purely zonal geostrophic motion.
Suc}i a conclusion also follows directly from the initial systeta with v= '
w= 0, with the exclusion of an explicit dependence of u and ~ on y and
z in accordance with (14). As can be seen from (14), at the altitude z=
zp the F field is symmetric, and the equation d~/a y= 0 has the roots
yl = 0 and Y2~3 = f -2up. With up ~ 0(easterly wind at the equat~r it-
self`)
~ has a minimum at the equator and a maximum at latitudes Y2~g =
, t~-2up. With u~ > 0~' has one extremum (maximum) at the equator. At
other altitudes (z ~ z~) the number and form of the extrema wi11 be de-
termined by the relationship of the parameters up, f3 and z- zp. We ~
also note that a geostrophic solution in the case of a symmetric pressure _
- field was also obtained by Bulakh [1].
We will assume that oc.= 1. We obtain
v=voy, (15)
c~ =-vo (z-zoj ,
a~/ay=-yu-v~y.
In this example v~ has the sense of ineridional (dimensionless) velocity
at the boundary y= 1 of the equatorial atmosphere. If vp = 0 there, then,
- as r_an be seen from (15), in the entire region ly~~ 1 we will haue v= w
= 0 and a geostrophic balance between u and d~/a y. ~
Finally, if 4C = 2, then v=2voy2(z-ZO)~ '
r~ _ -2vo9 ~z-zo) 2,
a~/ay=-~u-4voy3(z-zo)2. ~ (16)
It can be seen that 1 du I
v�- 2 dz y=1,
that is, v~ is proportional to the vertical gradient of ineridional velo-
city at the discontinuity y= 1 of the equatorial atmosphere. If vp = 0,
_ we again arrive at the preceding derivation. -
The selection of a specific solution from all the possible solutions must
be made on the basis of ttieir comparison with observational data.
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_ ,
Sec~~nd Pr~blem
For seeking wave solutions the authors of [4] proposed a more complere sys-
tem of equations of motion, which in dimensional form reads
' dt -2 u~ v-wl, (17)
o ,
_ dv 1 dp - 2~ y it, ~18~
dt p dy ro
dw I dp +2 u~lt-g, (19)
dt - p dz
dv dw ~20~
ay a: - o,
where dt - dr + v y-'~ w dz , -
r~ is the earth's radius, c+~ is .the angular velocity of its rotation. On
the basis of this system we will investigate the influence of Coriolis
- force on stationary movements of the equatorial atmosphere within the
framework of its zonal model.
Introducing, as usual, the stream function v=- c3 ~.r/ a z, w= d 3~ /dy,
we obtain from (17) and (18)-(19) respectively
dr U ~ U~ _ ~21)
a ,~li~ o~Y} _ {U, Q}, (22) ,
where and ~ A, B~ are the Laplacian and the Jacobian for the variables ~
Y~ z, .
ro y2 2 w z, U= rc o� ( 23 )
With v/ a t= 0 equations (21) and (22) give
U=F(~), (24)
0~ -F' - 0, . (25)
where F' = dF/d and F(yI) is an arbitrary function.
Integrating (25), with (23) taken into account, we obtain the equation
~2 ~ Z- fo y=~ F~ c~~ = H(~), c26>
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whe::e H(~) is also an arbitrary function. The further solutioii of (26)
is ;~ossible only after the specific choice of F(~ and H(~). For this
we iise physical considerations.
Si~ice the initial system (17)-(20) does not contain derivatives oE x(zon-
ality), it will apply only near the equator, where this condition is ap-
proximately satisfied. Outside this region, due to the well-expressed
zonal transfer, meridional and vertical movements must be absent. Accord-
ingly, solutions of equation (26) must be localized near the equator,
that is, with increasing distance from the latter ~ and its derivatives
must tend rather rapidly to zeru. Then it follows from (26) that
F'(p) -H(0) =0 when y (27)
Limiting ourselves to the first nontrivia~ terms in the expansion of the -
F( ) and H( functions, we obtain
F _ o + 2 ~2, H _ (28)
where � , ~ are constants.
These expressions for F and H correspond to a regime of "weak nonlinear-
ity" and a"weak influence" of Coriolis force. It is a circumstance of
not a little importance that only in this case is it possible to obtain
a precise analytical solution of equation (26).
In actuality, substituting (28) into (26), we find the linear equation
~~Y'-(2 maz- ro Y') (29)
~
wtiich formally coincides with the stationary Shrodinger equati~~n.
We will solve (29) by the method of separation of variables:
~(y. 2) = Y(y~Z(z)�
Substituti.ng into (29), we obtain ~wo boundary problems: -
. Y" ( ro y'= - Y = 0, Y ( ~o ) = 0: ( 30 ~ -
Z"-(2 u~a2a-rj Z=O, Z (U)=Z(It)=Q, (31) -
wtiere and h is the altitude of the atmospheric layer where
movement occurs. In other words, we will assume that w(0) = w(h) = 0
("solid top" r_ondition).
The soluti~;n of equation (30) has the form
Y_e-~2I.~AnHnl~~, ~=1U~lal)'I~Sr~ (32) .
~ r~
n
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� v~? v~ � ~.~t~~u ~uL V~\LL �
whe:-e Hn ) are ~Iermite polynomials [ 7] and � and are re~.ated by the
exptession
~
r~�ra~l-'= �=-2 IZ--1, aCO, 33) -
l ~ 1 ~
n i:~~ a wliole number, An are constants.
Introducing into (31) the new variable
~1(z) _ (2 ~a)'~3 z v (2 wa)-'l~~
we write equation (31) in the form ~
Z"�--~Z-O. ~ (3t,)
A general solution of this equation is given by the Airy functions [8]
Z ~7i) = C, A! !Y~) -E- C~ Bl (~1)� (35) ;
Assume that ~1 ~ = r~ (0) and ~1 h = ~ (h) . Since ~ 0, then ~k < , The
functions Ai( and Bi(~ ) are monotonic when -1.02 and, oscillating,
- tend to zero when Y1--' - oo . Accordingly, the boundary conditions (31)
will be satisfied if ~ h:-1.02, and, as it is easy to see, if
Ai ~~lo) Bi ~~IB) - Ai ~rh~ B~ ~ lo) = 0. (36)
With fixed h this transcendental equation relates ~'-and . In addition,
C2 =-C1Ai~'~ p) I Bi( .
Thus, for the stream function we finally obtain
~Y, z) = P- [`q~ ~y~~ - ni Bi ~ ~,i1 N� (37 )
l' ~n ) ~
and the constant C1 is included in An. It can be seen that the solution
(37) contains tne free constants An, a' and h. They determine differ-
ent possible regimes of cellular circulatior..
In conclusion it can be stated that within the framework of a stationary
zonal model of the equatorial atmosphere the fundamental effect of Cor-
iolis force is a strong localization of vertical and meridional movements
near the equator and the setting in of a cellular circulation regime.
The authors express appreciation to Ye. M. Dobryshman for discussion of
the problems touched upon in this work.
BIBLIOGRAPHY
1. Bulakh, B. M., "Correlation Between Wind Velocity and Pressure at the
~quator," IZV. AN SSSR, FIZIKA ATMOSFERY I OKEANA (News of the USSR
Academy of Sciences, Physics of the Atmosphere and Ocean), Vol VII,
No 3, 1971. ~
- 59
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2. Dc~bryshman, Ye. M., "Some Peculiarities of Circul.ation in the Tropo-
:;~~li~~r~~ i.n llir I~,c~u;il.~~ri;~l I:i'~;it~it," P11~;'I'I~,l)I~l)I,O(~I1'~ L(~IUltl)I,I)(;Il'A (~~t~~l~~~~r-
ology and liydrology) , No 5, 1964. -
3. llobryshman, Ye. M., Fal'kovich, A. N., "Some Models of Stationary Move-
ment of Air in the Equatorial Zone," METEOROLOGIYA I GIDROLOGIYA, No
7, 1967.
4. Dobryshman, Ye. M., "Wave Movements in the Equatorial Zone (Zonal Model),"
METEOROLOGIYA I GIDROLOGIYA, No 1, 1977.
5. Ingel', L. Kh., "A Model of Circulation in the Troposphere in the Low
Latitudes," METEOROLOGIYA I GIDROLOGIYA, No 9, 1975.
6. Panchev, S., Chakalov, R., "One Class of Precise Solutions of Nonlinear _
Equations of Dynamics of the Equatorial Atmosphere," KHIDR~~LOGIYA I
METEOROLOGIYA (Hydrology and Meteorology), Sofia, Nc S, 1974.
7. Smirnov, V. I., KURS VYSSHEY MATEMATIKI (Course in Higher :'~athematics),
P art 2, Vol III, Moscow, Nauka, 1974.
8. Fok, V. A. , PROBLEMY DIFRAKTSII I RASPROSTRANEPIIYA ELL'hTROP4AGNITNYI:II
VOLN (Problems in Diffraction and Propagation of Electromagnetic Waves),
Moscow, Sovetskoye Radio, 1970.
9. Chakalov, R., "Some Special Solutions of Hydrodynamic Equations for the
Equatorial Atmosphere," KHIDROLOGIYA I METEOROLOGIYA, Sofia, No 5, 1975. _
~
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,
i?DC 55]..571
VERTI(.:E1L DISTRIBUTION OF ABSOLUTE HUMIDITY OF AIR AND MOIS'fURE CC?NTENT IN
THF. A7'MOSPHERE OVER THE OCEANS
Moscow MGTEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 79 pp 55-62
[Article by Candidate of Physical and iiathematical Sciences N. A. Timofeyev, -
Marine Hydrophysical Institute, submitted for publication 27 December 1978]
Abstract: On the basis of an analysis of radio-
sonde data from Soviet scientific research ships ~
the author has derived formulas describing in _
the annual variation the mean latitudinal ver-
tical profiies of the volume concentration of
water vapor aH and the moisture content of the
atmosphere w g(H altitude) as a function
of cloud cover and air humidity at the ice-free
ocean surface. An evaluation is given for the
errors in computations of aH and w H using these
formulas under conditions extremely different
from those in the middle latitudes. Data are giv-
en on the total atmospheric moisture content ~ ~
over the oceans, in arctic regions, over the land
and as a whole for the earth.
- [Text; Water is present in the atmosphere in the form of vapor, clroplets and
ice crystals. The ratio of the droplet and crystalline moisture present in
~ the clouds to the mass of water vapor as a whole for the earth is less than
1% [3]. Therefore, the total atmospheric moisture content ~p with great ac-
curacy can be identified with its vapor content in an equivalent layer of
precipitable water which could be formed as a result of water vapor conden-
sation. The ~p value is determined most accurately using radiosonde data
or data obtained by the optical j4], IR [3] or laser [11] nethods. However,
_ for the time being these. data are still available in inadequate volume for
climatic genera"lizations directly for oceanic conditions. Climatic maps of
rhe value for the oceans (for example, in [2, 6]) were compiled for the
most part using radiosonde data from island and coastal stations. However,
inFormation on atmospheric water vapor, especially on its vertical distrib-
utj.on, is necessary for solving a broad range of problems relating to the
r.adiation regime, moisture cycle, climate and conditions for the p ropaga-
tion of radio waves. This circumstance motivates a search for empirical
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dependences 1~etween the moisture content ~H in the layer from the ~ieight
H ~o the uppcar :timit of the armosphere and different chara~teristics uE
_ air lium.idil-y (absolute humtdity ap, water vapor elzstt.city ep, et.c.), mea5-
ured at the level of standard shipboard observations.
71
In the author's paper [20], on the basis of an analysis of data fcom 1,187
shipboard radiosonde observations it was demonstrated that
l 1.55 + 0,0~6 rt ) eo ~~s ~ ~ 1~
where and e~ are expressed in millimeters and the total cloud coverage
n in tenths.
[~ith fixed gradations of cloud cover the correlation coefficient between
lg ~ ~ and lg ep in the entire range of their values characteristic for
the ice-frPe surface of the world ocean changes in the range 0.90-0.94.
Tt~e computation errors (~p)com using formula (1) had a normal distribu-
_ tion. The probability of the fa~lling of ( wp)act values determined from cne
radiosonde observation into the interval ( wp)compf 0 is approximately -
70%. The mean square error v'~lp is 14%. The v'~0 value, in addition to
- the real var:iability of the moisture content with fixed ep and ny also in-
cludes the computation error ( w 0)act based on radiosonde data. The latter
is equal to 5-6% [7, IOJ.
W o MM
60 ~
/
/ ~
/
/
~
-
i/ -
4G �
y
:
~
v
z ~
i.
- Y~ Z
X ~i _ _ 3 .
;~x ~ x 9
o ~0 2o eo r+M
[~L~;. 1. Dependence of total atmospheric moisture content ~ on water vapor
elasticity ep at ocean surface. 1) computations made using formula (1) with
n= 0(lower curve) and n= 10/10; 2 and 3) on the basis of [22] and [19]
respec:tively; 4) observations of drifting stations "SP-4, -5 and -6" during
_ tlie period Erom April through September [15, 17J.
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. HHH � ( r � ~ ~ _
? 2 ,f 4 S 6 7 B~9
o x o _
n) 6! -
o . > .
- 5 �
c ~ .
x ~ ~1'~
~
x
0 `
0,6 ~7,8 1, 0 1,2 D,6 0,8 0 1, 2 1,Y ~
~QH ~ ~0~ I~ QN I Op ~ tP .
mean
Fig. 2. Normalized relative verti.cal profiles of absolute air humidity aH
with its different gradations at the o;,ean surface (a) and with different -
gradations o f total cloud cover (b) . 1) ap ~10; 2) a~ = 10.1-15; 3) a~ _
15.1-20 and 4) a~~ 20.1 g/m3; 5) n= 0-2; 6) n= 3-4, 7) n= 5-6, 8) n=
7-8 and 9) n= 9-10/10.
Table 1
Absolute Air Humidity ag(g/m3) at Different Altitudes and Total Atmospheric
Moisture Content WD (mm) Over the Oceans for Different Gradations a0 and.
Cloud Cover n
_ ao ~lat' ~ n 6aa~b~
H~ 20,1 0=2 3=4 ~=G ;+8 9=10 0-+-l0
7,2 :5,3 4,6 6,0 1,1 3,4 5,4 7,6 9,3 5,7*,
~
0 7,24 13,23 1~4.64 22,00 17,46 13,87 17,77 I18,16 16,52 17,61)
0,5 5,83 10,62 15,01 17,;3 14,00 15,OU 14,2;~ 1�i,SG 13,G0 l4.'3l
I 4.63 S,32 12,01 14,~5 1U,98 11,93 Il,t1t1 11,g5 I1,UU 11,41
2 2,56 ~,22 7,G3 9,64 6,5U 7,4G 7,41 7,77 7,53 7,3(i
4 1,06 1,94 2,97 3,99 2,36 2,53 2,S'' :;,1S 3.15 '~,92
6 0,40 0,71 1,13 1,:~5 0,82 1,1Q 1,10 1,23 1,27 1,12
~ 8 0,13 0,26 0,4U O,;i5 0,28 U,37 O,;i9 O,44 0,~l4 0,3h
l0 0.03 U.03 0,13 U,19 O,O~J 0,11 i),13 U,16 0,15 U,13
N(~;~icno P3) 3 179 16:i 344 4~J~! 25G 263 1;,2 151 3G5 1187
panEtosot~,~},.
poAat+ttA 4 l:i.l 27,ti 40,2 n(1,4 3~i.~ 39,2 41,1 3~1,1 ~3R,fi
(I) iFi.S ?~.7 4'l,~ .;;.1 ;(i,0 42.7 yl.-~ ~a.},$ -}2.~i 41,2
(4) 15,2 '17.~J ~l0,~ :iU.2' 3~,7 39~`3 35~7 lal,5 ;39,0 38,1i
5` CpenEiE?e aEravcimn ofiu[cii o6~~a~ixocrtr. '
k F: Y :
1. g/m3 -
2. n tenths
3. Nuinber of radiosonde observations
4. Radiosonde observations
5. Mean total cloud cover values
a 63 -
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Table 2
Atmospheric Moisture Content ~(mm) Ove.r Oce~ns
1 �0 2~"`'
I ~ ~ I U ~ 1 J ~ 2U ~ 25
H x.ir -
2 n 6a.an~t
0 I 10 I U I 10 I 0 I 10 I 0 I 10 ~ 0 I l9 I 0 I lOr .
1,72 2,Q7 8,$4 10,67 18,30 2'L,24 2~9,47 3d,S0I39,40 4S.~i2 :~1,21 G3,5:~
(1.`, ],2J 1,~~:~ G.6~~ 5.~15 1~,0t) 17.77 21,95 2S,OS ~i0,67 39,i~1 40,25 52,23
1 0,~17 1,2i ;~,U.`~ G,G6 10,6G la,l4 1f,y0 2'l,5ti 23,5�1 32,U3 31,~~5 42,76
2 U.:~3 0,7~; ?.S3 4,06 6,U~ 5,30 9,S1; 14,31 1~,20 2U,73 19,20 28,22
~ 0.29 O,~.'". 1,~~7 2,43 3,~1i, ~,,3S 5,i0 8,92 8,36 13,1J 11.~i4 18,32
�1 (1,16 p,2G 0,8G 1,43 1.9;.f 3.2~3 3,2~i 5,47 4,SS 3,2G li,g7 11.72
0,(Iti3 0,1`'~ 0,46 O,S~f 1,07 1,91 1,~33 3,31 'l,Sl :i,lU 4,05 7,38
_ ti Q,I)41 Q.(,1,~s'; ~,2i~ 0,47 O,~$ l,ll 1,02 1,97 1,61 3,1(1 2,37 4,59
~ 0,(12:3 0,04i.~ f~,l~ 0,`l7 O.;il (1,(i4 0,57 1,16 O,~l 1,3f~ 1,37 2,77
ti iO,tJl~ 0,0'l; U.U~;ft p,l:i 0.17 ~~,;.f6 0.31 O,Gi U,ril 1.1U U,79 1,70
~ O,Oh~; 0,(~l�1 ~~,~?~.fi U,U~I O,O;i3 0.?(1 U.17 (~.35 0,2C 0,1;-~ 0,4,~ 1,02
KEY: I0,003 O,OU7 O,U1,~ O,U43 U,0-16 0,11 0.090 U,2? 0,1:~ 0,3i U,1~ O.GO
1. g/m3
2. n tenths
Table 3 �
Atmospheric Moisture Content Over Yacific Ocean in Layer 0-5 km on
- the Basis of Radiosonde Data {1) and Computations (2)
~ 2 ~eepanb 3 May '
llI}~pora, apad i I (2) ~1) I ~2) -
1
50-60 c. 6,3 7,? 10,6 11,0
40-SQ 4 11,0 12,5 14,8 15,5 -
30-40 16,0 I8,0 21,6 23,0
20-30 ?3,0 25,4 30,4 3I,9
10-20 32,5 34,3 38,8 41,2
0-10 38,3 40,6 42,2 44,5
0-10 Fo. 5 42,1 43,0 40,3 42,4
10-2U 40,3 40,8 34, I 36,6
20-30 34.1 34.� 26,6 28,0
30-40 24,8 ?6,3 18,3 21,2
4 0-50 17,8 19, 3 12,9 16,2
5~-60 12,G 13,1 8.6 I l.l �
Kl?.Y :
1.. I,atitude, degrees 4. N
2. �February 5. S
3. May
f:xpression (1) was checked in definite regions of the Pacific and Indian
Oceans with the most diverse weather types on the basis of independent
r;idiosonde data. It was found that the maximum computation error (`~p)comp
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rvn urrt~lnl, uJn ulvl.,t
w.[th the avera~;ing of ohservations over a period of 1-3 weeks does nc~t ex-
c~+ed 10% ['L~J. 'T'he s;ime conclusion wa5 drawn by 13. N. Yef;c~rov ~ R~ u~t tlic~
basis oE an analysis of radiosonde data from scientific research ships
sailing in the North Atlantic. For other regions of the oceans the results
of computations of the total moisture content of the atmosphere on the basis
oF expression (1) in the case of average cloud cover condil_ions are in sat- -
isfact.ory agreement with the values (``~0)comp~ computed on the b~.sis of the -
empirical formulas published by I. A. Shatunov [22] and V. G. Snopkov [19]
_ (Fig. 1). However, these dependences cannot be reco~ended for use in the
high ].atitudes of oceans covered with ice. For example, in the Central Arc-
_ ~ic Basin the p)act values, obtained from radiosonde ob:,ervations, in the
warm half of the year exceed the computed values 0)comp by approximately
a factor of 1.5-2.
6 r. . . -
90
� , .
~ ';,7 200 /y .
Fig. 3, llependence of inean square error in computation of (~p)comp value
_ using formula (4) on number N of radiosonde observations. The solid curve -
corresponds to the case a"~~ = 12/ the dots correspond to the experi-
nental. data. HXM
- i ~
i .
~ ~
~
~
.
~
\`~bl ++b~ Q~
5 ' i i '
~ I
1 i '
? i
i - ~
~ --2
\ j
0
0, 5 1, 0 1, 5
~QNJQI~QHJp ~ ~L1M~4If LJN~p CO$P .
Fig. 4. Comparison of ineasured and computed vertical profiles of absolute
air hu.midity (1) and atmospheric moisture content (2) in ICZ (a), outside
it (b) and for entire GATE-74 (c) act; p= compJ
It Eollows from expression (1) that the w p value with fixed ep increases
w.ttli an increase in cloud cover. An increase in the number of clouds is ,
usually lssociated with an increase in the intensity of vertical movements
in the atmosphere and with an increase in the level of transfer of water
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_ vapor upward Er.om tl?e underlying surface. A sin~ular valve effect is the re-
= s~lt [6]: air currents, rich in water vapor, I'ise upward, lea.ve a part oE
tlicir m~isturc there and impoverished of moisture, descend toward the un-
der.lying surf.ace, where they again become saturated. Then the pracess is re-
~~cateci again and again. The presence of a positive correlation between cloud -
cover and humidity, s wnmed vertically, was confirmed by satellite observa-
tions [14]. In the case of a fixed cloud cover an increase in wr/, with an
increase in e~ occurs more rapidly than in accordance with the 13.near law.
The physical nature of this effect is also associated with an intensifica-
tion of vertical air movements as a result of an increase in fluctuations
of its density with an increase in moisture content [12]. ~
Tlie noted regularities must also be reflected in the vertical distributfon
of absolute air humidity ag. For this purpose the entire mass of data from
shipboard radiosonde observations was broken d~wn into a number of submasses
with different gradations of ap and n(Table 1). For each submass we con-
_ structed the vertical profiles of the dimensionless ratios ag/ap, which
tlien were normalized by dividing by the values (aH~aO~mean~ found from the
mean (Eor the entire mass of data) ag profile. In Table 1 this profile cor-
responds to values ap = 17.60 g/m3 and n= 5.7 tenths. The experi.mental pro-
- files (at~/a~) /(a~~/a~)me~ln are shown by different notations in Fig. 2. The
solid curves represent the results of computations (with correspanding ap
and n values) using the expression
a = a~~ (1 -~-0.012 nf/) ~2~
f' ~a(U,'ll~/-/-{-U,00'?4H=)(I,I~I-U,UU~n�) '
, which describes the vertical distribution of the volume concentration of
water vapor aH in the atmosphere in dependence on absolute air t~umidity at ~
the ic.e-free surface of the oceans and cloud cover. A reconstruct:ion of the
vertical profiles ag using the ordinarily employed formula aH = a.0�10-bH~
where b= 0.434 a~/ w is carried out with an error especially appreciable
at grE~at altitudes [19]. A decrease in the ag value with a.ltitude H, in ac-
cordar..ce witl.i (2), occurs the more slowly the greater the cloud c:over and
ap. By~ knowing the law of change in ag as a function of H it is possible
to find the quantity of precipitable water W H as the integral
= cuH aH dH ct� f (l + 0,012 nH) dN. ~3~
y . H 10~0,11~1 H-1-U,UO'l4 1/-')(1,14--U,00$ an)
The ``~H values, obtai.ned by numerical integration with a vertical interval
/-~li = lU0 m, are shown in Table 2. In turn, these data with a very small error
are approximated using the following formulas:
~u =(1~72 -f- ~,04 ~t) ao~01 0,0018 ao~ ~4~
~H_ ~o .l~-[(0,243-0,0034 n-0,0017 a�) H-{-0,0038 H~] . ~5~
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4ere al~ and .i~ have the dimen:~ionality g/m3, W~~ and ~Q are in ;nm, H is
iii km, n is in tentlis.
I?m~,~uy(~ia; tli~~ i~x~~r~~:;5tc~nti (~i) ;i~id (5) witlt t~rhiLr~ir.y .i~~ ;ind Il (l: LS ~~:~tiy t~~
Eind tlie quautity of water vapor present in a column of the atmos2here be-
tween the levels Hl and H2 (H2 > Hl) , as ~gl ~ 2~~ gl g2�
Expressions (1) and (4) are virtually identical for the case n= 0. In the
case of a continuous cloud cover and its mean conditions the (``~p)comp val-
ues, computed using formula (1), are exaggerated by 6-7% (Table 1). The mean
square error U"wp for formula (4) with N= 1(N is the number of radiosonde
observations) is 12% and decreases inversely proportionally to ~(Fig. 3).
Thus, with N= 151-499 (Table 1) the value Q"u,~ = 0.8%.
In the new edition of the PiARINE ATLAS OF THE PACIFIC OCEAN [2] there are
maps of moisture content of the atmosphere in the layer~0-5 km for Febru-
ary, riay, July and November, constructed using data from measurements of
air huinidity at 160 radiosonde stations during a period from one to eighteen
.;~ears. The layer 0-5 km contains approximately 90% of all atmosphexic mois-
ture. Source [2] also gives the initial information necessary for computing
atmospheric moisture content using formulas (4) and (S). In order to exclude
c}ie random errors related to a different period for the averaging of radio- ~
sonde data we examined the mean latitudinal characteristics of moisture
:ontent ('Table 3). The cumputed values were exaggerated by an average of
6.6%. Tliis can be a result of systematic discrepancies in humidity measure=
ments when using radiosondes of different designs. Formulas (4) and.(5)�were -
obtained from an analysis of Soviet data and moisture content maps in [2],
compilpd for the most part on the basis of data from foreign radiosonde ab- -
servations'. The latter, as indicated by comparisons made during the GATE
period j10], usually give lower air humidity values. With the exclusion of -
r.tie systematic discrepancy, formulas (4) and (5) describe the spatial-tempor-
al distributions of the mean-latitudinal values of atmosplieric r~oisture con-
~ent in the layer 0-5 km over the oceans with a mean square error f4%.
It is natural to raise the question of the accuracy in obtaining mean ag
and ~.J1~ estimates in fix~d regions. For this purpose~we used radiosonde
dn5 prevailing in tl~e Northern Caucasus [4, 7], Volga region [2J, Far
l;ast [13J and Central Asia [1]. ,
In this arl-.icle we present the results of investigations which make it pos-
sible to develop a method for p`redicting the mean oblast yield of rice cul-
tivated in the southern Ukraine. The initial data used were materials from
thc Central Statistical Administration on the mean oblast yield for 1951-
l~)79 and data from agrometeorological observations during this period at
hydrometeorological stations situated in rice-growing regions of the Ukrain-
i:.in S5R.
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Vari:iLionS t~i Lhe yield of agricultural crops in each geog~-apliLc rcgLon
in whici~ the agric.ul.tural techniques and equipment used in cultivation and
thc seed v~ir:ieties are the same are a result of varlation c~F weather con- -
di tions [ 3, h, t0:~ .
The cultivatl.on of rice in our country is carried out under. conditions of
optimum moistening. This is achieved by the flooding of ric:e paddies with
water during the entire growing cycle. As indicated by the studies of dif-
ferent researchers [1, 2, 4, 7, 13], the limiting climatic factor exerting
a dominating influence on the formation of the rice yield i.s the thermal
regime.
'It~e integral and most stable biological index of rice productivity, as f.or
other agricultural crops [12, 14 and others],is the densit~~ of the prod~ac-
tive stand.
The mentioned points served as a basis for developing a method for predict-
ing the mean oblast rice yield. As the multiple correlation predictors in-
cLuded in the program for computations on an electronic cornputer we select-
ed the. following productivity indices for rice: density of stems with ears,
characterizing the potential possibilities of the crop, and the mean air
Cempezature for the interphase periods of development "sprouting - leaf tube
Eormation" and "leaf tube formation - heading of panicles," whicli reflects
the conditioiis for growth and development. The basis for selecting the
en.umerated independent variables in the pr~gnostic model oE the yield was
their information content and significance, determined in pair correlaLi.on
with tt~e mean oblast rice yield. The correlation coefficients between the
yield and the mean air temperature values during the mentioned pc~riods are
0.52 and 0.64 respectively. The closeness of the correlation between the
rice yield and the density of the productive stand is characterized by a
correlation ratio 0.70.
As ttie dei~endent variable in the rice yield model we used the rat:io of the
absoliite value of the mean oblast yield (y) to its statistical maximum
(YmzY), which was computed for each oblast with a probability 99.9% by the
Gumbel' metl~od [5]. As indicated in [3, 6], the replacement of ttie yield
val.ue hy tlie ratio y/Ymax makes it possible to take into account such
E;ictors constantly acting on the yield, characterizing the regional pecul-
iariti.es oE individual regions of crop cultivation as soil fertility, agri-
cultura.l tec}iniques and equipment, seed varieties and productivity of cli-
mate. Tliis also makes it possible to increase the volume of the sample
Ec~r tlie statistical processing of data by means of combining data for dif-
E~rent. oblasts [11], which is extremely important in the case of short ob-
5ervcition seri.es.
~5 ci r.esu:l.t of multtple correlation of the dependent variable y/Ym~x and the �
enumera~ed predictors we obtained the following linear regression equation:
~=Y,,,~, (0,32~x,+0,O1Sx2+0,023x3-0,250)
~ R=0,703-!-0,003 S~, =�4,6 centners/hectare, ~1~
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wliere y i.s tlie mE:~in c~bla5t y ield (limits oE variati.on Erum 20 L-~> GO cenln~~r~/
h~~:~t:rire); Y~n,lx is thc~ statistical maximum of the rice yield, equal. tu h5.0
c~c~ntners/hectare Ec~r Krymskaya Oblast and 60.7 centners/hectare for hher5on-
skay~i Ublast; xl is the mean (oblast) number of stems wiCh panicles, ciivided
by 1f1(l~ (]imits of ch~nge from 20Q to 900 stems per 1 m2); xZ, ;n eCFcact of SL-Sc-/1c cl.oud5 ~is much grenter tl~,in l-lie similar
e~facc c~f NS--l1s cl,ouds bccause, ~~tecording to the d~t~t publ.i.shed by L. T.
~tatveyev, dw/dz near the upper boundary in the first case is much grenter
tlian in the second [ 3] .
In our opinion, there is no need to.ask whether or not radi.ant heat e:cchange
exerts an influence on the cloud-formation process. Other questions are more
logic~l.l: what physical mechanisms arise as a result of the Eormation oE 5ucli
a powerFul "ti,ermal. pit," how do they lead to its smoothi~i~; r.incl wliat is tl~r.
resultant ef.fect?
It is possible that these questions at least in part have been answered by
L. T. Matveyev, who demonstrated that in clouds turbulent exchange is intens-
ified in comparison with that in the surrounding space [3].
Now we will further discuss the direct argumentation of article [2].
1. T}ie "results of computation of temperature change (L~T) after 10 liours,
cr:iused by absorption and emission of infrared radiation," cited on page 25,
in actuality represent the difference T' - T, where T is the temperature
computed in the L. Matveyev cloud-formation model [3] wi.thout allowance
for radiatioii, and T' is temperature in the same model witti addition oF a
radiation "bZock."
'Phe cited numbers (v T=+~.3�C near the lower boundary of the cloud and ~-1T
. _=0.2�C near its upper boundary) do not at all mean a real increase or de-
crease in the temperature at the boundaries after 10 hours~ Ttie radiation
only c}~anged the reaczion of temperature to other forms of heat e.xchange by
thc indicated not small values.
2. The rad~iaeion he~ting at the lower bound'ary of the clouci IayeY� was almost
an order.of magnitude less than the cooling at the upper boundary. There�ore,
the first, against the background of o,.her forms of heat exchange, may not
be manifested and will not lead to an anticorrelation of temperatures at
~~I rhe boundaries.
3. The aircraft sounding, whose data were used in [2], was carried out during
the daytime when the "therma.l pit" effect, if not completely, was,half reduced
by lieating of the cloud by solar radiation [5).
4~ If ~i comp~irison was made of temperature changes at the cloud boundaries
after 12 hours from the onset of cloud formation, the sought-for anticorrel-
;.ition of temperatures would probably be discovered under ttie condition that i~
~~t}~er. forms of heat exchange during this time retained their intensity. How-
ever, in the forming cloud layers considered in [2] the changes after 12
hours in liquid-water content and other factors mentioned above can lead to
bath an intens~fication and to ar. attenuation of radiation effects, and in-
de~~endently in the neighborhood of each boundary.
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~'liis ~hjectic7n against ti~e arbumer~tation in [2] is basic; a similr~r discus-
i.on is ~,iven on p 12 in [4
IJ~~ ~u~inl nut in c�c~nc�I�sinn that the author c~F this note dc~c5 n~~t Ansi:;t ~ind
ii:~v~~i li:i:; i n;~ i:;l ~~~I ~,u :i I i r~cl ;in~l I'~~r~~in~~~~l r~~ I~~ �f r:~~l innl li~~:il ~~xrli:ii?a~,~~ f n
~li~~ ~�I~~~i~l luriu.il i~~n ~~r~~~�~.~5:;, Wlinl w~, ~irc I.~iliclnY, ~~h~rul i.5 lli~~ ii~~~~~1 I~~r ~�r~~;it-
in~ c.l~sed models in whicti all forms of moisture and heat exchang~~ ~oould' in-
teract. Models of this kind are heing created and as a rule talce radiation
:into account [1].
BIBLIOGRAPHY
1.. I3uykov, M. V. , CIIISLENNOYB MODELIROVANIYE OBLAKOV SLOISTYKH FORM. OBZOR
(Nurnerical Modeling of Stratiform C1ouils. Review), Obninsk, 1978.
Matveyev, L. T., "Reasons for Cloud Formation," MET~OROLOGIYA I GIDRO-
LOCIYA (Meteorology and Hydrology), No 8, 1978. ,
'3. Pla~veyev, L. T., KU[tS OBSHCHEY METEOROLOG.II. FIZIKA ATMOST'ERY (Course
in C~eneral Pleteoro.Logy. Physics of the Atmospl~ere), Leningrad, Gidro-
me~eoizdat, 1976.
Feyhel'son, Ye. M., LUCHISTYY TEPLOOBM~N I OBLAKA (Radi.ant lleat ~xchange
and Clouds), Lenirigrad, Gidrometeoizdat, 1970.
Fe.~~el'son, Ye. Ht. , Krasnukutskaya, L. D. , POTOKI~ SOLN~CtINOGO IZLUCIIf,N CYA
I OIiLAKA (Fluxes of Solar Radiation and Clouds), Leningrad, Gidrometeo-
i�r.dat, 1978.
,
.
. 123 .
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UDC 551.576.1(234.9)
HELICAL CLOUDS IN THE EL'BRUS REGION
_ Koscow METEOROLOGTYA"I GIDROLOGIYA in Russian No 8, Aug 79 pp 99-102
[Article by '1'. N~ Bibikova, Moscow State University, submitted for publica-
tion 4 December 1978]
Abstract: The article gives an example of a
rare form of helical clouds in the mountains
of the Central Caucasus, in the E1'brus region. ~
~ The helical clouds developed during the flow-- `
ing of a WSW flow around E1'brus in the presence
of a~et stream in the upper troposphere.
[Text] Tn mountainous regions circulations are manifested in the form of
:~~ccial types of local weather with the development of definite cloud sys-
tem~, in par~icular, wave orographic systems consisting uf lenticular clouds
oC the type Ac lent.
Numerous observations carried out in different regions of the earth made it
possibte to prepare a detailed description of many varieties of orographic
clouds [2, S). However, in nature it is possible to encounter such forms
:.is are not contained in special atlases of mountainous cloud types.
I~car. ex~~mple, on 3 August 1964, in the mountains of tt~e CenL-ral Caucasus, in
the neighborhood of E1'brus, there were clotids of a~_r~;;re helical form. i'ig-
ur~ 1 shows three successive photographs of these clouds for 1200, 1210 and
1.230 h.ours respectively. The photographs were taken from Epchik :'ass (2296 m
.ibove sea level). Henceforth all elevations will be given from se.a level.
1)~ie tc.~ the exceptiunal form of the clouds we will briefly discuss the con-
ciitions Eor their genesis and development.
O~i 3 August the weather in the Caucasus was governed by a low-gradient pres-
sur.e field and rherefore during the first half of the day it was cloudless
in the mountains of the Central Caucasus. At 1200 hours the first cloud
appeared -[.n the I:1'brss region; it rapidly began to change its configura-
tion and a spiral twist was ~ormed (Fig. la). ~lt 1210 hours it developed
,i dist~inct helical form (Fig. lb~). Then along the line of sight a similar
cloud developed which seemed to be considerabZy ~smaller on the photograph
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due t~~ i.ts greater distance. The clouds retained their configuration for
].7 minutes. At 1220 hours similar helical clouds hegr.in to Eorm tii grea~cr
number. and by 1'~30 hours a whole series of helical clouds was fo nned (Fig.
:Lc).
',8
~ .
F
~ - .s
t;~, r;j~'
~ . ~ . ,,p a.~s.~.~:;r~.
, ~ . y~~a''~ 'l
~ .
- . . . N
6
.
f
~
~.'r*, 'N~~~ y ~
~~~~y~~e
r�r , ,n* s.. Y
. j r P�..~
r v'.,:'` �
~ ..?~o f
~ Xrq",~.p y~t . ~ ti.
�s,~ ~ } ~ . ~d ta'i ' y '
~ Pi 'S 1 yF {'y~y"f � H . ~ ~ ; M`
~f ~
~~Y r~q:.:'~~ r.,: g~ r ;;.~.~,.~,~P ~
~y;}.
nY
i Y!
.
. ~ ? �tA~. . u i^A':~ .
r� ~~~Firt~?~'.'~ . ~ ~ -~,a.:.r
y~~7y ~'14. G. i . y ~~,~'u .
5 a y. s f r'v .
}~k'.~ n tp i~.,r.:r~-~x~,t;~'~" ~ ..St3~'^a~~.
r;~c; ~a~m . r S ~r~~��~,
~ s ~
w :
, ,~4y;' r..5'Ad~+~+~R~ ~ . iF,
, ~ . - ~"~'.Y4~, ,.N ; '
~y. t~~: M . q'. G ?1 .
t
4:~
Fig. 1. Helical clouds i_1 the El'brus region. Photographs taken from Epchik
Pass on 3 August 1964. 1) 1200 hours; b) 1210 hours, c) 1230 hours.
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An analys.[s ~f the photographs revealed that the clouds had a dlsttnctly oro-
}.;r:ii~hic natur~ since their windw.ird part remained in place (desPite a con-
f cl~~ rrih I~~ w i ncl v~~ I oc. f ty I n the c I oucl -L~iyc~r) r~n~l 5~~e.~m i n}~, I y ~dn5 "t i~~~1" t u;i
duFinlCe lu~~~i.l. r.el[eE. Only l'hc leew.~rcl parL uE the c:loud w,ts r.~p[dly dr;iwn
uut and tw:LsCed. Unfortunately, the photographs were taken from one puinC
and the position of the clouds and their dimensions can be obtain~d hy stip-
ulating the corresponding altitude of the lower base. For this purpose we
used data from radiosonde observations at the nearest points: Sukhumi, Min-
eral'nyye Vody, Tbilisi and data from pilot balloon observations at Nal'chik
station. By comparing these data with the orography, it was possible to ad-
vance some well-founded h;~potheses. �
Ttie central part oF the Main Ca~r.asus Range is oriented from NW t~ SE and has
~ mean elevations of 3500-4000 m. Ircdividual peaks attain 4500-5500 m. The
highest point is El`brus, in whose region the helical clouds developed. E1'-
brus rises to 5633 m.
Figure 2a shows the vertical profiles of temperature and wind from the wind-
ward (Sukhumi station) and leeward (Mineral'nyy Vody station) sides of the
range on the basis of radiosonde data for 0900 hours. Figure 2a shows that
ti~e wind direction in the oncoming flow had a stable WSW direction (232-
253�) in the layer from 4000 m to the tropopause. If it is talcen into accoun~
tt~:it the direction perpendicular to the range averages 230�, it becomes clear
th~it the wind direction in the oncoming flow was deflected from the perpen-
dicular direction not more than 25�. Thus, the wind shear was caused for the
most part ~nly by a change in wind velocity with altitude. The greatest ver-
tical gradie~tt of wind velocity was observed at altitudes from 9000 to 10 000
m(6 m/sPC per 100 m) and from 10 000 to 11 000 m(12 m/sec per 100 m). In ad-
dttion, in the layer 11 000-14 000 m there was a jet stream wtiose axis pass-
ed through ttie E1'brus region (the wind velocity on the jet- stream axis at-
tained 45 m/sec). The tropopause was situated at an altitude of 1.4 270 m,
tt~:1t is, the upper boundary of the jet seream was situated near the lower
boundary of the tropopause.
Tlie variltion of temperature with altitude in the oncoming flow was as fol-
l.ows: in the layer 1000-1500 m the vertical temperature gradient was small
0.3�C/100 m. At altitudes 2600-2800 m there was an isothermic layer, and
above, in the layer from 3000 to 3840 m, there was again a reduced vertical
temperature gradient y= 0.33�C/100 m. Thus, in the lower layers of the at-
m~~sphere there was increased stability, which aloft was replaced by strong
instability. In the layer from 4000 to 9000 m the vertical temperature gradi-
ent was close to a dry adiabatic gradient ( Y= 0.85�C/100 m). Again, still .
higher, there was a layer with a small temperature gradient, that is, from
9000 to 11 000 m Y= 0.25�C/100 m. In general, the temperature variation
witti altitude in the windward flow (Mineral'nyye Vody stat:ton) was similar
to the temperature variation in the leeward flow.
'1'he relative humidity in the oncoming flow changed little with altitude and
in the l.ayer from 3000 m to the tropopause itself averaged 20%. In the lee-
w;ird flow the relative humidity values were somewhat greater (by not more
th:in ].5%), nevertheless remaining small.
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Ki~uwink e:he ~i_l.l- oE ll?e ~~hotograph, the Fc~cril length oC thc~ objec~:iv~~ ~.uid
the enlargement oE the photograph, it is possible to construct the plane
positi.on oF the clouds using the laws of perspective geometry. The results
of sucli a construction gave the following. [Construction from a photograph
taken at 12.1.0 liours. ] The clouds were formed bPhind ~1'brus, on i.ts l.ee-
ward side (on the northern side). If 9000 m is used as the altitude of the
lower base, t.he length of the first (closest to Epchik Pass) heli.cal cloud
was ].Fr ~QO m, and l-he second 9000 m. The distance betwec~n the hel ic~~frames o.E the motion picture
survey of these clouds. -
; -
I�1e a'%iso stipulated the altitude of the iower base of these helical clouds
at 11 OOQ m. The results of the corresponding computations are presented
below,
1)imensi.ons (lcm) of Helical Clouds for Altitudes of the Lower Base 9 and 11 km
Altitude of lower cloud.base 9 11
Length of first cloud ......................1(i 19
Length of second cloud 9 11
Distance between clouds (perpendicular to
flow) 7 8.5
Pitch of "screw" 6 7 ~
~i
ComptlrinY the computations of the corresponding parameters for two different
riltirudes, we note that they vary on the average by 20%.
1'hus, ~zn exainination of the photographs of helical clouds togethe.r with an
an;ilysis o� data from vertical c~unding of the atmosphere make it: possible
to draw some canclusions:
1. The helical clouds were formed during the flow Qf a perpendicular S[J wind
around L1'brus.
Thc~ presence of a jet stream at altitudes 11 000-13 000 m, in all prob- ~
.ibi1 ity, ~is char.acteristic for helical air movements in the uppei- tropo-
sphere a?Zd h~~lical. clouds are indicators of these movements.
3. Cl.auds developed in a flow having thick layers with small vertical tem-
~~erature gradients. , .
in conclusion it should be said that in the recent literature it is more and ~
more common to find articles indicatin~ a helical nature c;f air movement in
tt~e mountains. Thus, investigations carried out at the Central Aerological
Observatory in the mountains of Central Asia`in the neighborhood of Tashkent
using equal-altitude balloons made it possible to detect a helical structure
oE the flow at altitudes 10 000-12 000 m[3J.
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Summ~.~r. i�r.inf; everythin~ which has been said, and al.so proceedin~ on the hasls
uC Lhe concl.u~'ions which we drew earli.er in study of oru};r.apliic~ cl.ouds i.ii
tl~e Cr�imea [1] and the experimental results obtained ~y Shmeter and~Pinus
[4J, c,~e made the assumption that the helical clouds develo~~ed near a layer
with the maximum wind shear under an inversion. In our opinion, t.he most
probable altitude can be considered the a].titude 9U00 m(if it i~; also taken
into account that the elevation of the underlying surface in the cloud re-
gion is more than 5000 m).
H KH '
'S 0) 157' ?4e'
256 IS7
?53J 'F2::P .
?51 \ ?~Z
24E
yt 176
ZSy ' 7BJ
' 1J6 pgg
s 142 1-,
2?2 p cs
17B ?S'[
. ~~J 2d3
tQ ZO 40 1~4~ fD 60 VM./CCf( 1
-6D -40 -20 0 20 -y0 -20 0 20 t'C
BunmoG6le
oa` Q a
2 ~Zb`~~ a 5 o6no~ 6 ~
? 3
re6e,:,io~ ~
~ ;e,~.3r+vux
~ .3na6pyc 'j
4 S ~ 56JJ
4U40 2816
~f'`~x' ~ ~ � .~t ~7~: u~'~': ' ~A(fll
Qorrffau Hag.cOpu-~� ~-~3198 ~~--_,11
g 9 rooNdp~
F~ig. 2. Results of observations of orographic helical clouds on 3 August
1964. a) vertical profiles of temperature and wind according to radiosonde ~
data Eor stations Sukhum~ (at left) and Mineral'nyye Vody (at right).(the
wind direction is given in degrees reckoned clockwise from the di.rection to '
the north. The direction of the perpendicular to the main line oi: the range
is about 230�); b~~, horizontal position of clouds under the condirion that the
Lower Uase of the~~i~elical clouds is at an altitude 9000 m.
Ki:Y :
1.. V m/sec I~I 6. Helical clouds
2. Teberda River j~ 7. El'brus
3. Teberda 8. Dombay
4. Epchik Pass ~ 9. Klukhori ~i
5. Kuban.' River 10. Gvandra
11. Dongus-orun Pass
" 128
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~ ~ V..~ va ~ yVtlau V~L Vl\LL
Tliis Pact is al.so confirmed by the studies of An~ell, et al., who discc~ver-
~d sectors ol= spiral circulation in the Los Angeles regian over the southern
s~~urs oE the Sierra Nevada. The vertical and horizontal dimensions nf 5ectors
wi.tli spiral circu.lat:ion were G00-1000 m[6].
BIBLIOGRAPHY
1. Dyubyuk, A. F., Bibikova, T. N., "Conditions for the Formation of Cloud
Cover in Dependence on Orography," TRUDY GGO (Transactions of the Main
Ceophysical. Observatory), No 171, 1965.
2. Musayelyan, Sh. A., VOLNY PREPYATSTVIY V ATMOSFERE (Obstacle Waves in the
Atmo~phere), Leningrad, Gidrometeoizdat, 1962.
3. Patsayeva, V. A., "Investigation of Orographic Disturbances in the Atmo-
sphere Using ~qual-Altitude Balloons," TRUDY TsAO (Transactions of the
Central Aerological Observatory), No 59, 1964.
4. Pinus, N. Z., Shmeter, S. M., ATMOSFERNAYA TURBULENTNUST', VYZVAYUSHCHAYA
ROLTANKU SAMOLETOV (Atmospheric Turbulence Causing Aircraft Bumping),
Moscow, Gidrometeoizdat, 1962.
S. A1aka, M., "The Airflow Over Mountains," WMO, TECHNICAI, NOTE, No~ 34,
196~.
6. Angell, J. R., Pack, D. H., Holxworth, G. G., Dickson, G. R., "Tetroon
Tr.ajectories in an Urban Atmosphere," J. APPL. METEOROL., Vol 5, No 5,
1966.
. ~
jl ~ . ~
i,
. 129
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, _
~
< .
_ . . . ,ya_;.
, _ . ~ , . . : . . . . , .
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~
UDC 551.(461.25+465.755)
USC OF THE HIGH-ALTITUDE PRESSURE FIELD FOR INCREASING THE ADVANCE TIME FOR
SIIOKT-12ANGE FORCCASTS OF S~A LEVEL
PtoSCOw MGTEOROLOGIYA I GIDROLOGIYA in Rus~,ian No 8, Aug 79 pp 102-104 ~
[Article by V. I. Andryushchenko, Arctic and Antarctic Scien.*_,ific Researcli
Institute, submitted for publication 1 November 1978] ~ ~
Abstract: The author proposes a procedure fo-r.
~ using steering current rules for increasing the
advance time of predictions of surge variations
in sea level.
['i~xt] Predictions~of aperiodic variations of sea leyel are an important
component part of the system for hydrometeorological support of marine op-
erations. The effectiveness of planning of the latter is essentially depend-
ent on the advance time of the forecasts. However, the advance time of most
, prognostic methods deLeloped both in our country and abroad does not sur-
pass the time interval separating thF cause f.':ni the effect. The length of
this int~rval is different for different points, but its optimum value does
not eviden tly excPed 27 hours, .It is possible to achieve an objective in-
crease in the advance time of level fore:.asts without making a synoptic
forecast for this ~purpose by meaas of use of additional in:Eorniation in the
form of actual AT5(10 charts related to the time of preparation of: the fore-
cast. ~ �
Tn routine synoptic practice in the analysis and prediction of weather ex-
tensive use is made of the steering current rule, according to which the
m~vement of pressure systems and fronts occurs along the isohypes at ATS00
in the direction of the high-level wind. The rate of movem~nt of pressure
systems is proportional to the velocity of the high-level wind or the den-
sity of the :isohypses at AT500' Since the variability of high-pressure
Eields is considerably less than the variability of the surface pressure
fields, th is circumstance~is effectively used for prediction of movement of
surface ~ressure formations.
Applicable to level predictions, the steering current rule can be used for
transforming the coordinates of points at which atmospheric pressure is
read from the surface weather chart, The sense of this transformation is
to read pressure at that part of the surface chart from which with the
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' greatE~st probabtlity it :Ls possible to expect movements of any pi�essure Eor-
mation. Thus, if in the prognostic regression equation as the prF~.dlcturs we
use the pressure differences or the coefficients oF expans.ion of the pres-
sure f:f.eld, in addition to that advance time for w}iicli the p~nr~:Lcular equ:~-
tlon w~is computed, the transformation of~ coordin.ites oC t;iu~ pof.nts mc~k~~s it
~~ussible to ubt.iin an additional advance rime. '.Cl~e len~;Ch ~~f IIiI:: ~iddlC(un,~l.
advanc:e time is naturally determined as the minimum time iiitervcil 'T during
which the moving pressure system (such as a cyclone) can overcomE: the dis-
tance between the centers o~ action of the surface pressure field. The cen-
ters of action (there are usually two) are those field regions, being situ-
ated in which a medium-scale cyclone causes the greatest rise or fall at the
investigated point. They are easily determined by simple averaging of the
surface pressure fields relating to the times of extremal 1eve1 values.
) r / ~ ~
~ ~ i2 ~
;~1, j ' . / B ~ ~ ~
1 ~ ~S~ ~ H
~ \
_ ~ 6 m ' ~ .
.:3, � ~~s~:.� � H
~ :
i ~
~
~ ~
B)> o / ~
> !1 ~
~ � ~2 ~ .
i
7
~ _ ~9~
~.,~.5
r y ~s
_ i ~ � \
-
I~I?;. l:xample of transformation of coordinates at which surface atmospheric
E>ressure is read. a) initial grid of points, b) AT500 chart 19 August 1972,
c) tr.r,nsEormation of grid points.
1'l~e pt�inciple for the transformation of coordinate points can be represent-
c~d in L-he fo.llowing way~ Each point of the working grid is matched with the
m:iddle oF a unit segment oriented along the noxmal to the isohypses at ~T500�
'1'}ien tt is displaced along the isohypses in the direction opposite the
steering current (high-altitude wind) at a distance proportional to the num-
I~er. oE isohypses intersecting the unit segment. As a unit segment it is
131
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i~ossible to take a r,~rt of the distance between the centers of action of
the surfar.e presst.~re fiel.d, for example, a half.
Si.nce the gre~itest path which a cyclone can overcome durin}; thc~ t.ime T i5
~,c~ u;~ l I u l-lic~ d i s~r~nee h~~ l-we~n the cen~ers o f r~r. t i ui~ i n Ch~~ ~~retisi~rc~ f i~~ l cl,
th~~ rntio oF tliis dis.tance to the maximum number of isohyp;;es inCersecting
it at ATSpp gives a distance unit corresponding to one isotiypse. In order
tt~at the obtaine.d distance unit correspond to the length of a unit segment,
i~ must be increased by as many times as the unit segment is les:; than the
distance between the centers of action of the pressure field. For.� routine
work it is mast convenient that the distance be exprAss~d in degrees of lati-
tude. -
An example of transformation of coordinates of the points is illustrated in
I~ig. 1, in whic}~ the initial grid of points after being superposed on AT500
;ind transEormation of the coordinates of points acquires a new form (Fig.
].c). 'I'fie operation of transformation of the coordinates of points makes pos-
sible an indirect allowance for the probable movement of the pressure forma-
tions acting at the level, but their possible evolution is not t~ken into ac-
count. The influence of this shortcoming can be somewhat lessened by adding
~o thc: pressure read from the surface chart by means of transforr~~ation of
the grid of points the value of its tendency, performing this addition al-
gebraically.
'I'esting of the proposed scheme for increasing the advance time of level fore-
casts at the bar of the Kolyma River gave fairly good results and at th~
present time this scheme is being used successfully in routine practical
work in the eastern regions of the Arctic.
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i
i
!
UDC 556.013 '
L~:W OF DISTRIBUTION OF SUM OF RANDOM VALUES WITH TYPE-III PEARSON
PR033ABILITY DENSITICS IN HYDROLOGY
Moscow M~TEOROLOGIY.A I GIDROLUGIYA in Russian No 8, Aug 79 pp 104-107
[Article by Candidate of. Technical Sciences I. V. Busalayev, Kazakh Sci- ~
entific R~search Institute of Power, submitted for publication 12 ~'ebruary ' ~
1979J
Abstracti The author fias derived a general, com-
�~pletely rigorous formula for the composition of ~
;'-distributed hydrological random values suit-
- able For any number af components. This formula ~
can be used in deriving different approximate or � ~
asymptotic expressions convenient for hydrolagi.cal
computations. On its basis (with specific values '
of the parameters) it is easy to determine the ~
approximation error. The formula is useful for hy- .
drological and water management applications. ,
[TextJ Despite a half-century of experience of use of the type-TTI Pearson
distributior. in hydrology, some of its important properties have still not
been clarified. In particular, a formula for the composition of distribu-
t:ions of this type in a general case remains unknown. [For the fi.rst time
thE type-III Pearson curve was used in hydrology by Foster in 1924 and
_ later in the 1930's by D. L. Sokolovskiy, S. N. Kritskiy and M.. F. Menkel'.]
~In our article [3] an attempt was made to answer this question. We derived
(lIl approximate formul.a, convenient for use and generalization, but not en-
~i.r.e.ly rigorous and precise. , ~
Ln this brief communication, which can be regarded as a continuation of the
mentioned article, we derive and demonstrate a composition formula entirely ~
correctly and examine some corollaries from it useful for hydrol~gical com- '
putations. , ~ '
Assume thar the random values xi, distributed in conformity to the law
a._
Bi (x;) = Ce-Yi.rlx! i ~ ;
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are ir;dependent and in addi.tion,
Ti ~ Ya < . . . < Yn < � , n),
iC bein~ required that the distribution of their sum be determincd
n
Y ~ X~�
~_t ,
For the mentioned densities we find their Lar.iace transform '
ai
r T~
~ IBt lX): P} � \P -F~ T~ ~
wliich, with some assumptions (positiveness of xi and monotony of L) unambig-
uousty determines the distribution (7].
By virtue of the multiplicity property, the Laplace transform of the sum of
tl~e independent random values is equal to the product of the corresponding
transEoXms of the terms [6]
7~ a~ Y" `a" ( )
L {P ~Y)~ P) _ ( p + Yi) . . . ~p~_Yn I .
~ i
The probability density function of the indicated sum of random values can
now he determined using an inverse transform of expression (1). In order to
~ acr_omplish this we will rewrite formula (1) in the f~llowing equivalent
form: ~ al r 1 a~
L ~~'~Y); P)=Ta~(P+Tn-l7n-7~)) 7a? 1P~7n-~7n-7x)) . . . (2)
~ `ar~
a
. . . 7R" (p + 7n - ~in - Tn)) .
Ttien, using the known pair of Laplace transforms [1, 5] .
t~-1 ~ ri. . . . ~n; 7~ t, . . . ,]~n lj = ~ ~7) ~
. pY `
. - p 1-?~ . � rl _ ~ ~-Pn
~ i
where 2 is a degenerate hypergeometric function of n variables [2, 4J.
We w:i1 I. multip.ly the original by e- ~nt, in connection with which in the `
[ransform we will have p+~ n instead of p. After some simple transforma-
tic~ns, also taki..ng into account that in our case f3i ~3n = r, we will
have
_ ~nt 7-~ ~ ~~i~ . ~
e ~ ~ . . . ~ ~n~ 7; t+ � .An l~_
- . , . . ~3~
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. . i
~ ~ ~ ~ . ~
;
_ ~~r)l~~ + a~~1' i ~ i an
_ r .
� ~(p+xre)Y 1 P'~'A6-~~ ~ . . . rn+%ii-l,h ~3~
~
Comparing the derived expression (3) with formula (2), using the inverse I
transform we find the density distrfbution function for the sum of gamma- ' '
distrihuted independent random values in the most general f:orm: ~
o. a ~
, 71 ~ . . . YR ~ - rn Y y a~~- . . . ~-rtn-1..
p ~ (a~-I- . . . +anl e ~4 j .
X ~ai~ . . . , an; at-I- . . . -~-an~ ~7n-7~) Y~ . . . ~ (7n-Tn) Y~� .
[de wi].1 show that the necessary normalization conditions a~.re alsc~ satisfied:
~ . Q~ an ~ ;
~ p ~Y) dY = - T + . . . �(n e-7n Y y n~-}~ . . . -ran-1 1Qt, . . . , a~; a~
I~ ~ ~ ~a~+ � � � -~-an) ~ � ;i
~ .
,I
i~ + . . . -i-an; ~7n-ii) Y~ . . . ~ ~7n-Tn) Y~ dY ~ ~ '
. ~
Yi i . . . 7n n f (ai-1- . . . ~-an 1
- X
- I' (a~+ . . . -4-an) 7 r~~t- . . . -ra~
, n . .
I,
� -n n
f- Tn-7~~ . . ~I-~in"7n i~
~ Tn ) l. Yn ) .
i,
T}ius, formula (4) in actuality represents the sought-for density of the sum ~
of y= distributed random values. ~
Now we will examin.e some special cases:
:i) ldith ~l =)~2 n= y all the variables in the hynergeon~etric func-
~ tion become equal to zero and 5~~2 =.1 and we obtain the kn~~wn result
. . . ~.a ~
P( Y)-.. i~ n T Y y a~ +...-!-aR-1
I (a~-~- . . . -f-an) ~
b) ~lssume that n= 2; then, since ( y2 - Y2~y = o, in the numerator tlie para-
meter a? disappears and from the general formula we have ~
P(Y) - T~al T2 ~ e 7,Y ~~+n�-I
. I' lu y iF~ ~ai� Qi �i- a_: (7==7i) ~5~ ~ -
wt~ere 1F1 is a degenerate hypergeometric Kummer function.
The mentioned distribution of the sum y= xl + x2 can be obtained in direct ~
computations [3J. . ;
For ctiis same reason, with n= 3, we obtain the probability density function, ~
expressed through a hypergeometric function of two variables
~ 135 ~ ~ l
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. ~ ~ ' ~ ; .
.
. . . ~
.
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. u ~i_ u.
( T; ~ 7:~ i,i ` _T.~ y n~ -i- n2 -t- o~ ~ X
_ ~ + a, a:i l .
%!~?s~ai~ az~ ~i+ax-F�as~ ~Ya-y~)y, (Ya-ys)yI�
c) Yroceeding from the general formula, it is also possibl~~ to obtain tiome
other equival~nt expressions for density. For example, substitutj.ng into
the exponent '}~min or the sum of all ~'i, we will have
~a, an -I~,7t~Y F+al-1 .
, ~ . . . 7~ \ ~ ;
f' lY1 - f� (ui;- . . . -I-an) ~ y ~z (a~, . . . . a~;
a~.~- . . . Tan; (~r 7t-Tt)Y, . . . , ~i-Tn) Y]�
~ i
This expression is convenient in that with sutiiciently large n(n~-~~-~ ) it
can be expressed through one variable.
ln actuali.ty, if it is assumed that
lim 7f - p, i= I,..., n
n-.oo r 7t
Lr
r
(any of yl at the ],i.mit is quite small in comparison with the entire sum),
L-hen, using a formula from [4, 8)
4's ~pi, . . . , a~~~ ai . . . an; x, . . . , x~ = ~s ~ai + . . . + a,,;
a~ + . . . + a,,; xJ,
we obtain
~~La~~ . . . , an: n~ . . . +a~r; (~7i-71~Y. . . . Yi-Yn~yl=
1
I !r ii - Tma:~ ) �
9 t . . . t al } . . aR V = C ` i
- ~ a 7~ - 7t~ . ~ ' �
L ;
Ilence., substituting the Pxponent .
j ~ 7i - 7maxl y
. e~ ; 1
in~o formula (6), we find that at the limit with n->~~ there is satisfaction
of th~ equation ~ ~
P lY 1= Ce- 7maxy~ a~} ...+un -1.
'I'I~e gener.al Eormulas (4) and (6) can be easily generalized for describing
tl~e l.inear form of the random values conforming to a gamma-distribution
V = h~ .~�i . . . b~ ,t'n, bj > U. ~ ~
Ln accordance with the similarity rule the Laplace transform of the probabil-
i.ty density function of the i-th term from (7) is equal to the expression
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t, 1"~ (7t b~l�r I u� i
L( b'i (b~ Xil: P i= 3i ~ Plbi -r' 7~ l - bt ( P-~ 7i bt
i
Since tlt.e probability density function of a linear for.m (7) (in the case of !
a nondependence oE the random values, which, is assumed) coi�responds to the ;
product of tlie Laplace transforms, we will have, after scuall transforma-
tions ,
~ L ~P lY)~ p; - (b~ ~~ila~ � (bn 7n~Q" r ~ la~ . . .
b;. . .bn `P+7n-~7a-1~bi1J
( I aR
. . . ` P � ln - lTn - 7n bn1)~~ ~
~
Ilence, similar to the preceding, we obtain i
(bt 7i)a~ � � � ~bn ~t)a" ,-Tn Y ai-}-. . . ~-art-1 ~
1' ~Y) C c y k: ~8~
~ (o~ -i- . . . an) bl . . . b~~
. X ~s ~Ri, . . . , an; a; -I- . . . r an~ ~Yn - 7t bt) Y. . . . , ~Tn - 7n bn) 9I ,
where C is a normalization factor.
. ~
A concl.usion of practical importance for hydrological applications follows
from this formula: if the coefficients.bi are assumed to be equal to Yn/ Yi, ~ ~
we obtain (since in this case in formula (8) ~ 2= 1) the ;'-distribution. ,
The established fact affords the possibility for programmed modeling of ~
distributed hydrological series with the stipulated parameters y' and a on
an electronic computer. In this respect it is particularly convenient to
use a formula of the type (6) with an exponent expressed through ~
n '
~ i.
i=1 ~
i
It: is suf.ficient.to multiply the initial series by the coefficients '
~ �
Z Y~~ Yi, ~
. i
in or~er to construct from them a Pearson series with stipulated character-
7.St1C~.
Thus, we confirmed that the derived formulas (4), (6), (8) are extremely
usefu.l for both theoretical examinations and for practical computations.
Un their basis it is possible to construct different types of approximate
or asymptotic expressions and also investigate the limiting properties of
d~nsity when n-> . In specific computations, using it one can then esti-'
mate the approximation error for a definite combination of parameters.
137
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. ~
~ ;
~ .
, . , , , , . . , _
. �
_ . . , . , _ - _ _
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, ~
BIBLIOGRAPHY
1. Beytmen, G., ~rdeyn, A., TABLITSY INTEGRAL'NYKil PREOBREIZOVANIY ('1'aUles
of Integral Transforms), Vol I, Moscow, Nauka, 1969.
2. Beytman, G., Erdeyn, A., VYSSHIYE TRANSTSENDENTtJYYE FUNKTSII (GIPER-
GEOMETRICHESKIYE FUNKTSII, FUNKTSII LEZHANDRA) (Higher Transcendental ~
' Functions (Hypergeometric Functions, Legendre Functions)), Moscow,
Nauka, 1965.
3. Busalayev, I. V., "Composition Formula for Distribution Curves oE Hydro-
logical Parameters (Type-III Pearson)," MET~OROLO(~IYA 3: GIDRGLOGIYA
(Meteorology and Hydrology), No 8, 1977.
4. Gradshteyn, I. S., Ryzhik, I. M., TABLITSY INTEGRALOV, SUI~'Il~i, RYADOV I
PROIZVEDENIY (Tables of Integrals, Sums, Series and Products), Moscow,
Fizmatizdat, 1962.
5. Ditkin, V. A., Prudnikov, A. P., SPRAVOCHNIK PO OPERATSIONNOMU ISCHISL~N-
IYU (l~andbook on Operational Calculus), Moscow, Vysshaya Shkola, 1965.
6o Prokhorov, Yu. V., Rozanov, Yu. A.,~TEORIYA VEROYATNOSTEY (Theory of
Probabilities), Moscow, Nauka, 1973.
7. SPRAVOCHNIK PO TEORII VEROYATNUSTEY I MATEMATICHESKOY STATISTIKE (Hand-
book on the Theory of Probabilities and Mathematical Statistics), Kiev,
Naukova Dumka, 1978.
8. Appell, P., Kampe de Feriet M., FONCTIONS ITYPERGEUN~TRIQUES ET HYPER-
SPHERIQUES, POLYNOMES d~HERMITE, Gauthier-Villars, 1926.
. ~
138 ~
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. ?v.i~ VL'1'iV1LaL UJL' VL\LL
' 1
~
~
UDC 551.~508.769:571)
LID'AR I~;ASUREMENTS OF ATMOSPHERIC HUMIDITY '
Pt~~scow M~TEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 79 pp 108-114
[Article by lloctor of Physical and Ma.thematical Sciences V. M. Zakharov,
S. F. Kalachinskiy, Candidate of Physical and Mathematical Scienr.es 0. K. . '
Kostke, G. A. Krikunov and I. S. Zhiguleva, Central Aerological ()bserv- ~
atory, submitted for publication 2 January 1979]
Abstract: The article gives a~description of
laser ranging apparatus developed by the auth-
ors for determining humidity of the lower tro- ~
posphere. The authors analyze the results of ~ ~
measurements of humidity of the lower tropo-
sphere obtained during 1976-1977 at the laser
sounding station at the Central Aerological
Observatory in Dolgoprudnyy.
[Text) Introduction. The development of remote methods for determining the
humidit~ field in the lower troposphere is necessary for a.nwnber of rea-
sons. As is well known, gas and aerosol contaminants, propaga~ting~from
~he sources of effluent, interact with the humidity field, which to a con-
sidArable degree can_determine the distribution of contaminating substances
tn the atmosphere and as a result, lead to the formation later of chemical
rerictions of the new componerits. These processes are of the greatest inter- ~
est in the atr basin of industrial centers, where regular radiosonde meas- ~
urements most frequently are,impossible. The results of humidity measure- ;
ments in the lower troposphere, obtained by remote metho.ds, can also be
used for the purposes of weather forecasting, since these measurement methods
cc~n supply a great mass of statistical data. Remote measurement method$ make
it Fossible to determine humidity on any sounding path, which is necessary ~
for solving some practica;, problems. Finally, joint measurements of the dis-
tr:ibution of humidity and aerosols im the atmosphere make it possible to in-
vestigate their joint in.fluence on atmospheric transparency, and according-
ly, on the radiative transfer of heat in the atmosphere. The enumerated
~roblems stimulated the development of the remote laser sounding method and
' the creation of apparatus for determining the humidity profile in the lower
troposphere.
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. Measurement Mettiod
As tfie remote method for determinin~ atmospheric humidity we selected the
"spontaneous comUination scattering method" (SCS) for the scattering of laser
radiation on water vapor. In using tlie SCS method the backscattering signal
from scattering on water vapor is compared with a signal caused b;y SCS on
molecular nitrogen. In the processing of sounding data this makes it possible
to exclude some instrument coefficients, and with simultaneous reception
at two SCS wavelengths, exclude instability of laser radiation. The distribution
of. tf~e water vapor concentration ~g2p with altitude H is determined by tl~e
E~~l.lowing expression from (4]:
H
Nct:P, i(H) Qnchp, a K~zK:~~l~ ~XP - f (H') dH'~ ~ ~1)
CKP = SCS . ~ o I
I ~ PH-o (i~) = Pn,~ cH) -
N C K P, 2~ H~ �,c C K P. i K u K s ~ ~
I~ eX P `t~ (H') QH' ~ ~
U
where NSCS~H) is the backscattering signal caused by SCS, K1 is the transmis-
sion coefficient for the optical receiving system and the :interference filter,
K2 is ttie transmission coefficient for the selecting filter, 1'~ i.s the quantum
efficiency of the photomultiplier, ~ is the attenuation index, U~SCS is the
SCS backscattering cross section, J� N2 is the concentration of molecular nitro-
gen, t.he subscripts 1 and 2 denote the wavelengths of SCS on water vapor and
molecu.lar nitrogen respectively. Since sounding in most cases was carried out
with high values of the meteorological range of visibility, then it was as-
sumed that
H
. qa~~.~ _ ~ ~ 9 = ex p 2 ~ e ! H' ) dH' .
9397,5 `
The maximum error in the estimates due to this assumption on the entire sound-
ing pa.th does not exceed 5% (see end of article).
Thus, expression (1) can be ~aritten in the form
Pti c~ ( H J~- pN. (H) NCKP, 1~N~ Q~ CKP. s K~z K~: ~1~ . ( 2>
~ NCKP, 2~N~ Qr CKP, t Ki~ K~~ ~I~
Formuta (2) includes the ratio of the SCS cross sections for Ii20 and N2. An
analysis of data in the literature [4) indicated that as the SCS cross sec-
tions it is possible to use the mean values ~~SCS 2- 3.05�10-3~ cm2/sr,
~~~SCS 1- ~�62'10-30 cm.2/sr. The quantum efficiency of the photomultiplier
f.or wavelengths 377.7 (N2) nm and 397.5 nm~(H20) nm was assumed to be iden-
tical = n2; the remaining coefficients entering into expression (2) were
determined experimentally.
In t}~e processing of the experimental lidar and radiosonde data we also used
Llie known formulas from [8]
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I
PH~o (Zl~~) = A 10~ P~i, u(c.~-3) _ .
s ~
, = 2~~�1~-V pH O ~~M-3>� .
1
216 7 ~3~
PH~o (zl~~) = T' e~
~,s~zs t
E (.u6) = 6,107�1024~~9+t,
p(c.u 3) = 7~~� 1~18 T~
where M is molecular wei.ght, A is the Avogadro number, T is temperature in
�K, t is temperature in �C, E is the elasticity of saturating vapor, e is
tlie partial pressure of water vapor in mb, p is pressure in mb. ~
Instrumentation ~
In the. meas.urements we used a lidar with the following principal parameters: '
Laser emission power at a wavelength 347.2 nm 0.06 J~ '
Diameter of r.eceiving antenna 0.5 m~
Angle of field of view of receiving system 28 min ~
~ Transmission coefficient of optical system 0.7
Transmission coefficients of filters: '
interference at 397.5 nm 0.23 ;
interference at 377.7 nm 0.1 .
selecting at 397.5 nm 0.56
selecting at 377.7 nm ~ 0.26
Coefficient of suppression of signal at 347.2 nm by 377.7 and
397.5 mm ~ot less than 108. ~
Using a system for the electronic processing'of laser information it is pos-
sible to: ~
measure the SCS of laser radiation by atmospheric components at different
al.titudes;
measure the total energy of laser radiation during a stipulatc~d operating
time; ' ,
coiint the number of laser pulses.
The structural diagram of the electronic processing system is shown in Fig. ~
1. Measurement of laser radiation energy is accomplished using the FEK-22 ;
(3), an energy recorder (9) and an "accumulating counting rule" (15). The ;
F~K-09 (2) performs the function of a source of starting current pulses
controliing operation of the entire system. ~
i
. ~
~
141 ~
FOR OFFICIAL USE ONLY '
I
. ~
�
, ~
s;~...~.....:~ , . . _ . , . . . . . ..a.: . _ : , . . . _ . . ~ . , ,
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? 2 ~-~.J
1
~
4 5 6 7 8 9
10 11 12 1J 14 i5
,
16 11 18 99 ?1 21
Eig, l. S~ructural diagram of system for the electronic processir~g of laser
information. 1) FEU-71; 2) FEK-09; 3) FEK-2; 4) low-noise emitter.� follower;
5) prE~amplifier; 6) divider; 7) multiphase generator of strobe pulses; 8)
stand-by generator of nanosecond pulses; 9) recorder of energy of sounding
pulse; 10) limiter; ll) differentiating circuit; 12) emitter follower; 13)
dual trace, four-input storage oscillograph; 14) counter of number of sound-
ing pulses; 15) "accumulating counting rule"; 16) wide-band measuring am-
plifiE~r; 17) threshold device; 18) stand-by generator of nanosecand pulses;
19) di.vider; 20) pulse time analyzer; 21) recorder.
f(g6
~
. 5n - �
60 4
'0
/
~
1 2 3 � 5 GgN
Fig. 2. Integral counting curve for FEU-71. 1) input noise of electronic
channel; 2) FEU dark noise; 3) signal + noise with threshold levels of il-
lumination of photocathode. '
The f~-equency c1lar.acteristic of the FEU-71 output in a dynamic regime falls
in the range 0-250 MHz. The low-noise emitter follower~(4) is intended for
matching the FEU output with the input resistance of the preamplifier, hav- ~
ing a transmission band 400 MHz and an input noise temperature 450�K. The
signals are fed through the divider (6) and the emitter follower (12) to a
142
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stora~e oscil.lograph (13), functioning in a stand-by regime. The signals
:~rc~ r~J ~u Fed frum ~ divider (fi) to a l.imiter (10) , then to d~ifferentiat-
in~; circuit f11), wliGre ttie pulsc~d signals are shaped in duratiun and am-
plitude and i.n such a form are fed to the input of a wide-band measuring
amplifier (16), where they are amplified to the amplitude necessa.ry for
~the threshold device (17),.which is intended for the cutoff of diode noise
of the photomultiplier and the radionoise penetrating to the input of the
electronic channel.
'Che stand-by generator of nanosecond pulses (18) is triggered by pulses
passing through the threshold device (17) and sends to the output standard ~
pu.lses with a duration of S nsec, which with respect to time of receipt cor-
: respond to the photoelectronic pulses of the FEU-71. These pulsed signals
,~rc fed through the divider (19) to the oscillograph (13), where they can
be compared in time with the pulses from the emitter'follower (12) and also
~o the input of the pulse time analyzer (20). The pulse time analyzer (20)
is controlled by a multiphase generator of strobe pulses (7) and distributes
- the pulsed signals arriving at its input by channels.
`I'he width of each channel corresponds to the thickness of t:he analyzed layer
r~f the atmosphere and is 150 m. The total number of channels in the analyzer
is 16. The state oE the analyzer channels is monitored by rhe oscillograph
(13) and the information accumulated in the channels is registered on punch
tape using a recorder (21). The information loss in the pulse analyzer (20)
is not more than 2%. The main loss of information occurs in the threshold
device (17) and is dependent on the transfer coefficient o� the amplifica-
tion channel to the threshold device. The wide-band measuring amplifier (16)
ttas a fixed Ievel of the transfer coefficient which can vary in the limits
70 db with ari accuracy to f0.5 db. The presence.of such an amplifier in the .
system makes it possible to determirie the integral counting.curve of the
photomultiplier and select a working point on the plateau.of, this character-
istic curve. ~ .
Fioure 2 shoo~s the integral co unting characteristic curve iised in the FEU-71
syGtem.
The curve 1 determines. t.he dependence of the intensity of noise and radio
inrerference of the input.of the amplification channel at the output of the
threshald device (17) on the amplification factor of the measuring amplifier ~ ~
(16), wi.th current fed to the photomultiplier, curve 2-- dark current of
tlze FEU-71, curqes 3 and 4-- intensity of the signals and the dark current
witli threshold values of photocathode illumination. The threshold.value of
i.llumination was attained using neutral filters. As the light source use ~
was made af ~in~AL103V photodiode, placed in the plane of ttie telescope
.:iperture opening. The strength of the photodiode working current was 2 ma.
'Che plateau of the integral counting characteristic curve for the photo-
multiplier is found with values of the amplification factor 58-70 db. The
crror in measurements on the plateau of the integral counting characteristic
cur.ve with instability of the total transfer coefficient 10% is 6%. The
143
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~.T. . < .......r. , ..:a . . _ . _ . . . . . . , . , . . _ _ . . _ , . . .
, : _ , . . .
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:icCuul inti~.ctf~il.iL~,r uf LI�~ transEer coefficient Eor the ampl.iCic:atlun chnn-
ne]. i.r~ Lhe used system is not more than 7%.
L'xperi.mental Results
A determination of humidity by lidar was made beginning in 1974 at Dolgo-
prudnyy. The results of laser sounding were compared with radiosonde data.
The registry and processing of data are automated and the resu?ts are fed
out by an electronic computer in the form of tables of the profile of abso-
lute atmospheric humidity measured by the lidar, radiosonde data and the
elasticity of saturating vapor E. With simultaneous registry of the aerosol
scattering signal at a wavelength 347.2 nm or 694.3 nm we compute the cor-
relation coefficient R and the error in the correlation coefficient ~g
using the formulas
n
~ l �ti,o ~H~ - ~H~o ~N~ ~~Na~ ~f~) - N,~ (t111 ~4)
R = ~
n PHyo J`va
Y , .
where zR - ~-R'. N~ (H) _ (H) .
~ ~ ~+c ~f'1max) '
n = 51)- G0;
n is t:he n umber of records of the profiles of aerosol scattering v'7 and SCS
(H20), b'yH2~ and c7 2NA are the dispersions for P H2p and NA.
Among the total number of laser sounding sessions we selected and analyzed
96 sessions carried out under conditions of high atmospheric transparency. In
this case the errars caused by different atmospheric transparency at the SCS
wavelengths were minimum.
In 46 sounding sessions simultaneously at wavelengths 694.3 nm (16 sessions)
and 347.2 nm (30 sessions) we registered the backscattering signal and using
Eormula (4) we computed the R and ~R values. In 38 cases there wa~ a cor-
relation (R% 0.5), of which for 31 sounding sessions the correlation was
cl.ose (R=� 0.7). There is no substantial difference between sounding at
694.3 nm and 347.2 nm, which once again is evidence of absence of para-
si.tic optical interference, which could lead to the ohserved structure of
atmospheric humidi.ty.
For cc~mparison of the results of lidar sounding and radiosonde data, for each
of the sessions we computed the mean square differences
;~I+_O = ~ ri ~ ; F'` (H) - f?` ( t1) i . .
t
in the entire range of altitudes and computed the differential curves of the
statistical distribution of differences
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~~,H O l/I) = P~~.V..i ~f/) PII_O, s~fl),
whece ~H2O, lid and f'H.~p~ rad are th~ absolute humidities ~iccorclin~; to
.Cidar rind r~zdiosonde measur.ements. .
- ~ N~o ~ ',~r ~m3 _
- _ _ t,~ - -
t ( Z7J0%
4 = t(10J0j -8
~
^ � -y .
. o -~o
2 '
r ~
x 22~�
' e ~ ~ ~ - /
ti 1 _ - d0
' l22f~ ! C - ~ 20
0
~5~ 9350 1950 2550 y M
Fig. 3. Resu"lts ~of lidar and radioson~e measurements ~f humidity and aero-
sol scattering at wavele~ngth 347.2 nm. 14 December 1977, rc,lgoprudnyy.
In 96 sounding sessions : in 13 sessions d/U H2O, lid < 0. 5 g/m3, in 46
sessions 0.5 g/m3~ /~'H20 ~ 1.0 g/m3 and in 37 sessions ,~pg2p ? 1.0 g/m3.
Most frequently high a`pg2p values are characteristic for high values of
measured atmospheri;. humidity. For the purpose of determini.ng the vertical
scatter of Tidar and radiosonde data we constructed differential curves of
the statistical distribution a,~g2p(H) in the altitude ra*ge up t~ 2 km with
an interval :~50 m. At low altitudes about 75% of the i~ y H2~ values .fall in
the limits of errors in lidar measurements. With an increase in a.ltitude
~he scatter of lidar and radiosonde data increases. However, for all alti-
' tudes the lidar and radiosonde data on humidity measurement coinr.ide within
the limits of the total erxors in the results obtained by different methods.
As an example of joint measurements, Fig. 3 shows three humidity and aero-
sol backscattering profiles obtained after two hours. Figure 3 illustrates
the correlation ~f profiles.and characterizes typical radiasonde and lidar
_ m~asurement profiles.
~lmong the 96 analyzed sounding sessions 38 were carried out at a negative
t~mperature at the earth's surface (tmin =-16.8�C), 58 at a positive
temperature (tmax = 30.8�C). The atmospheric temperature for all sounding
sessions varied in a wide range; in many cases, according to radi.osonde
data, the temperature changed sign with altitude. High correlation coef-
ficients are characteristic for both positive and negative temperatures.
The laser sounding sessions carried out show that with a.ny atmospheric tem-
perature the humidity profile, determined by the lidar, as a rule has ex-
~r.ema. The humidity profile obtained using radiosonde data in most cases
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:~~,..:e: .-..:6t, , .i.r;...:_ _ . . . . . _ . , _ . . . . ~ . . .
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i~; i~u~itu ~un i~~, wli i~:li i ti ;i l L r:ibtiLrib 1~.~ Cu Lnc~rliu c~ C~h~~ r.id iusun.. ,i s~~n5o r, n~~ C
c~,i~~~~b1~~ c~f ru~LSCC'1'lllfy Shar.p gr.adients of water vapor concenL-ration. I~ :is
cfiaracterist:ic th~~t the profiles of atmospheric humidity measured with a
lidar with an interval 1-2 hours do vary but the peculiarities of the ver-
tical distribution of the concentration of water vapor remain as beFore.
It should be noted that in lidar experiments there is measiirement of abso-
lute humidity in the atmosphere. The generally accepted point of view is
that there is a correlation between relative humidity and aerosols in the
armosphere [3]. Computations of the relative humidity profile on the basis
of data from lidar measurements pg2p and radiosonde measurements of tempera-
ture indicated that there is a good correlation between the relative humid-
ity profile and the p g2p profile.
hieasurement Errors
In writing expression (2) we assumed that the ratio of the exponential terms
in (1) is equal to unity. The computations made by the authors for different
mereorologic~il ranges of visibility in the surface layer o~ the atmosphere
in specific ~zerosol models indicated that under conditions of tiigh atmospher-
i.c transpareticy (meteorological range of visibility not less than 20 km)
t}ie error due to the different transparency does not exceed S%. For lesser
altitudes the error varies in the range 2-4%, which agrees with similar
_ estimates by the authors in [1], using somewhat differing aerosol models.
In passing, we note that the computations which we made indicate that there
is no justification for using a laser with a radiation wavelength of 266 nm
for a rigorous qi.ianLitative determination of the contaminants (also includ-
ing For water vapor determination) by the SCS method. In the spectral region
2GU-300 nm there is much absorption of radiation.by such components as 03,
N02, HZO, SO~, HN03. The concentration of these components can vary in a
wide range. Computations indicate that only as a result of different ab-
sorption by ozone within the limits of the first kilometer the error due
to different transparency can be up to 20%. An increase in total absorption
on the path due to other components considerably increases the errors in
determining concentrations by the SCS method.
`Ct~e second sources of errors is governed by the errors in determining the
cueEEicient
[CKl' = SCS] K= K12K2 Y~2 c7
~ SCS, 2/K11K21 ~1 J~ SCS,1
'1'he minimum error in determining the SCS values for strong lines is not
more than 10%. The minimum accuracy in determining the
. K12~`22 ~ 2~K11K21 ~1
ratio during laboratory calibration does not exceed 20%. The maximum system-
atic error, determined by the accuracy in K, is 25%. This error is constant
with aTtitude and introduces a shift in the atmospheric humidity profile.
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Iil IiumicliCy measu~-enients Uy the SC:5 meCliod in a signal caused by SCS c~n
water v~~por there wi11 be partial pr~sence of a signal caused by SCS un
~h~ liqui.d phase of 1i20. For the exciting line 347.2 nm thc~ center of tlie
:1ine :Eor the liquid phase is situated at 394.6 nm, the spectral half-width
of the line is equal to 64� [7], and the SCS cross section for the liquid
phase exceeds by a factor of 7 the cross section for the vapor phase [6].
An analysis of the spectral contour of the filter and the SCS line (394.6
nm) indicated that for the used filters during sounding of the transparent
atmosphere the contribution of the signal from the liquid phase did not ex-
ceed 1.% in the SCS signal from H20 vapor. '
Finally, the random errors in measuring humidity are determined t+y L�he errors
in measuring signals caused by SCS. Using the criteria for select.ing the dis-
tribut~.on [2J, it was demonstrated that with the used instriunent parameters
the di~tribution of the photoreadings conforms to Poisson statisrics. Using ~
the standard formulas of the theory of errors, in each soutlding session we
computed the dispersion of photoelectrons.
Taking into account the comments made above, we determined the total error^
d'~1120/t'H20� ~dith minimum values for humidity and the meteorolof;ical. range
oF. visibility, equal to 20 km, the values dPH2O~PH2O F~~r. 1 km are 12%,
for 2 km 18%, for 3 km 33%. This same error decreases with an increase
in akmospheri.c hilmidity., and, for example,~for a humidity of 3 g/m3 for 3
km is 24%. rlaturally, the cited values characterize the mean errc>rs, which
vary i.n each experiment in dependence on atmospheric conditions and the
number of sounding pulses. ~
At po~;it:ive temperatures the mean square error in determining relative humid-
ity o1~ the lower troposphere by radiosondes is approximately 4-5% [5]. The ~
estiiu~~tes wt~ich were made indicate that relative humidity can be computed
with approximately the same error making lidar measurements of ~~g2p and
rzdio:;onde temperature measurements. . �
BIBLIOGRAPHY
1. Arshinov, Yu. l~e, Danichkin, S. A., "Influence of Atmospheric Transpar-
cncy on the [lccuracy of Humidity Measurements Using Combinat.ion Scat-
, terinK Spectra," IZVESTIYA AN SSSR, FIZIKA ATMOSFERY I OKEANA (News of
tl~e U5SR Academy of Sciences. Physics of the Atmosphere and Ocean), Vol
l..t, No 4, 1975.
2. Astaf.urov, V. G., G].azov, G. N., "Statistics of Photoreadings and Regimes
for rF~e Regisl~ry of an Atmospheric Lidar Signal," TEZISY DOKLADOV III
V:>1;SOYUZNOGO SIMPOZ]:UMA PO RASPROSTRANENIYU LAZERNOGO IZLUCHENIYA V ATMO-
SFf?RJ~ (Summaries of Reports at the Third Al1-Union Symposium on the Prop-
agation of Laser Radiation in the Atmosphere), Tomsk, I~A SO AN SSSR,
1975.
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3. Georgiyevskiy, Yu. S., Rozenberg, G. V., "Humidity as ~i Factor in Aero-
sol Variability," I7.VESTIYA AN SSSR, FIZIKA ATMOSFERY I OKF,ANA, Vol 9,
No 2, 1.973.
4. 7.~ikti~ir~v, V. M. , Kostko, 0. K. , M~TEOROLOGICHESKAYA LAi:LRNAYA T,OKATSIYA
(Meteorological Laser Sounding), Leningrad, Gidrometeoizdat, 1977.
5. NA.R?YUDENIYA NA GIDROMETEOROLOGICHESKOY SETI SSSR (Observations in the
US~F: Hydrometeorological Network), editeii by 0. A. Gorodetskiy, Lenin-
grad, Gidrometeoizdat, 1970.
6. Romanov, N. P., Shuklin, V. S., "Specrrum, Indicatrix and Section of Com-
bination Scattering of Liquid Water," TEZISY DOKLADOV :3-go VSESOYUZNOGO
SIMPOZIUMA PO LAZERNOMU ZONDIROVANIYU ATMOSFERY (Summa~~ies of` Reports at
the Thirci All-Union Symposium on Laser Sounding of the Atmosphere),
Tomsk, IOA SO AN SSSR, 1974.
7. Sushctiinskiy, A. M., SPEKTRY KOMBINATSIONNOGO RASS~YANtYA MOLEKUL I
KRISTALLOV (Spectra of Combination Scattering of Molecules and Crystals),
Moscow, Nauka, 1969.
8. Khrgian, A. Kh., FIZIKA. ATMOSFERY (Atmospheric Physics), Leningrad,
' Gidrometeoizdat, 1969.
148
FOR OFFICIAL USE ONLY
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APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000100100036-0
ru~c urrlultu. U~~ UNLY
UDC 556.08
METHOD AND APPARATUS FOR MEASURING WATER VELOCITY OR DISCHARGE IN OPEN
SHALLOW-DEPTH FLOWS ~
� ;
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 79 pp 114-116 ~
[Article by M. I. Biritskiy, Central Scientific Research Institute of Com-
plex Use of Water Resources, submitted for publication 4 December 1978]
Abstract: The article describes apparatus in-
troducing changes into measurements by micro-
current meters by the "five-point" method.when
determining velocity or discharge in open fl~~ws ~
with a depth up to 3 m and reducing the amount
of time spent on measurements. It consists o:~E
five measurement converters, rods and a pulse-
counting unit. The measurement converters are
rigi.dly attached on the racks of a rod which
mesh with a multioutput reducer. The pulse- ~
counting unit consists of five amplifiers, fre-
quency diwiders (division by 2), MES-54 counters,
and also a timer, a synchronizer for switching
the counters and timer on and off.
[Tet;t] Hydrometric current meters and microcurrent meters have come into '
wide use for measuring averaged velocities in open flows. The mean velocity�
on the vertical in the cross section of a watex flow is determined at one, ~
two, ttiree or five points. The error in the method when making measurements ~
with a hydrometric current meter for deter.mining averaged velocity on the ~ ~
vertical at one point is not lower than t5%, at two and three points ~
t(2-4%), and in th.e case of five points f2% [5], The last of the employ-
ed methods i.s the most precise, but requires the most time. At the Ceiltrcil
5r.ientitic Research Institute of Complex Use of Water Resources USSR Water
' Management Ministry specialists have developed apparatus [1, 2] introducing
changes into measurements by the five-point method and reducing time expend-
itures on measurements. ~ ,
It consists of ineasurement converters, rods and a pulse-counting unit.
149 ~
FOR OFFICIAL USE ONLY
. ' . . . ' .
. ~ . . . ~ . . ~ . . . ~ ` . . . . ~ i -
~ ' , t. ~ . . . . . . . ~ . . . .
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000100100036-0

APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000100100036-0
FOR OFFICIAL USE ONLY
5~~2 k s e 7
~ . ~ ,
1 .
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9 9
~ Fig. 1. Measurement converter.
A-A
. noOepHymo 4
I 2 '
,a ~ ' ` ;r ~ 3 , ~yf, ~
~ XCye.mvt:ry --4
. 1
,aL - 2 ~ . S ,
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KEY:
1. To counter
2. At surface
~ 3. At bottom
4. A-A/ turned
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