JPRS ID: 9499 USSR REPORT METEORLOGY AND HYDROLOGY

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JPRS L/9499 ~
22 January 1981
- USSR Re ort _
p
~ METEOROLOGY AND HYDROLOGY
~ No. August 1980
_ Fg~~ FOREIGN BR~ADCAST INFORMATION SERVICE -
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JPRS L/9499
22 January 1981
USSR REPORT ~
METEOROLOGY AND HYDftOLOGY
No. 8, August 1980 ~
Translation of the Russian-languaqe monthly journal METEOROLOGIYA I
GIDROLOGIYA put~lished in Moscow by Gidrometeoizdat.
r
CONTENTS
Some Problems in the Theory of Cloud Condensation Nuclei 1
- Computation of Wave Fluctuations of Atm~spheric Pressure .......................10
Some Methods for Evaluating the Parameters of Correlation Functions of =
- Meteorological Fields .....................o...................~......~o..... 18
Mean Annual Zonal Atmospheric Circulation ........................a............ 30
Determination of Glind Vel6city and Direction and Te~perature in tlte
Atmospheric Surface Layer by the Radioacoustic Sounding Method 37
' Organochlorine Pest:icides in Precipitation 48
Natural Components of the Field of Total Ozone Content in the Northern
Hemisphere 56
Parameterization of Heat and Moisture Exchange in Storms Applicable to
Problems in Interaction Between the Atmosphere and Ocean 63
Variability of Active Layer Characteristics in the Northwestern Pacific Ocean
- During Passage of a Storm ............................o...................... 72
Meandering and Eddy Formation in Zonal Ocean Curreuts 77
Com~utation of Current Velocity in the Quasi-Isothermal Layer of the
- Equatorial Zone in the ~Jcean ..............~......,..,.e....o................ 88
- Computation of *he Cancentration of Global Atm~ospheric Impurities in Rivers
and Closed Watez Bodies...~ ........................................o........ 93
- a- [III p USSR - 33 S&T FOUO]
.~nn ~r~rw~ � ? � Tc~r+ ~i?n a~
I . . . . . . . . . . . .
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Weather and Optimum Times foi the Sawing of Winter Cxops 102
Some Possibilities for Simplification of Adaptive Algorithms in Prognostic
5chemes .......................................a.............................. lI3
Pressure Gradients in Narrow Zones of Cold Fronts....e 120
Instrumentation, Methods and Results of Investigation of Atmospheric Ice-
Forming Nuclei 125
Remote Activation and Control of Instrument Operation 135
History of Meteorological Observations in Turkmenistan 139
'CLOUD ATLAS' (ATLAS OBI.~'1KOV), Edited bq A. Kh. Khrgian and N. I. I~ovozhilov,
Leningrad, Gidrometeoizdat, 1978, 267 pages ...................e,a............. 143
Review of Monograph by S. S. Levkovskiy:'Water Resources of the Ukraine. Use
and Consexvation'(VODNYYE RESURSY UKRAINY. ISPOL'ZOVANIYE I OKHRANA), Kiev,
"Vishcha Shkola," 1979, 200 pages 146
- High Award to Yuriy Antoniyevich Zzrael 148
S~xtieth Birthday of Leonid Tikhonovich Matveyev 149
Seventieth Birthday of David Yakovlevich Surazhskiy 152
Sixtieth Birthday of Vasiliy Mikhaylovich Pasetskiy ..............a............. 155
. Sixtieth Birthday of Lev Grigor'yevich Kachuriii 157
Conference~;, Meetings and Seminars 159
Notes from Abroad 166
Obi�uary of Viktor Mironovich Sklyarov (1916-1980) 169
~b-~
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UDC 551.574.11
SOME PROBLEMS IN THE THEOAY OF CLOUD CONDENSATION NUCLE7
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 5-12
[Article by Doctor of Physical and Mathematical Sciences I. P. Mazin, Central Aero-
logical Observatory, submitted for publication 18 Feb 80] ~
[Text] Abstract: Some problems in the theory of cloud .:on-
densation nuclei {CIN) are defined. It is demonstrat- _
ed that the effectiveness of C~TT is fully detezmined -
by their soluble part, which is unambiguously describ-
_ ed by the introduced effective radius of the condensa-
tion nuclei. The exp~rimental data indicate unsuit-
- ability of a power law approximation for describing
the real spectrum of CCN. This imposes restrictions -
on the applicability of the well-known Ttaomey formula
_ relating the maximum supersaturation and concentra-
tion of cloud droplets and the parameters of C~."~I. .
Introduction. A physically clear dnd graphic exposition of the theory of cloud con- _
densation nuclei (CCN) can be found in a study by L. M. Levin and Yu. S. S?dunov
' [2, 4]. We will briefly repeat the fundamental points in the theory, introducing
, some refinements in passing. In contrast to the authors of j2j, we will use a
natural definition of supersaturation
- i So = e e F11.--- - (1>
In [2, 4] it is assumed that - ~
e- E~ e-ES
$o= ~ , S= e .
, Here e is the actual elasticity of water vapor, ED is the saturating elasticity of. -
water vapor over a plane surface. -
As in (2J, we introduce the value x= 1/r. Then
E` = 1 ~ Bx - Cx�, ~2~ ~
Eo -
where ES is the saturating elasticity of vapor over a droplet of solution.
- [Here and in the text which follows use is made of the following nota-
tions: v-- coefficient of surface tension (75.6 g/sec2), W-- 1 g/cm3 is water~
density, R= 8.314�].0~ (erg/{IDOl�degree) is the unive;csal gas constant, T is abso-
lute temperacure (0�C = 273.15 K).] `1'he B coefficient c~iarar_.terizes the increase in _
_ the saturating elasticity of vapor over a droplet with a 3ecrease in its radiu~. _
1 -
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.
B_ 2 ~~tt~; 1,19~5 ~ 10-.~ � m. -
~ k T
(with T = 273.15 K). ,
2'he coefficient C, in j2] called nucleus activity, characterizes the decrease tn
the saturating elasticity of vapor over th.e solution. We will call the parawete~
~ 3~)
rn = C' -
the effective radius of the condensation nucleus. (Usually rn differ~ from the ra-
dius of the dry hygroscopic nucleus rdry by not more than tens of percent).
Supersaturation over droplets of solution S= e- ES/ES can be related to Sp, and
_ with an accuracy to the second order of magnitude relative to Sp and N we find
S _ t S~ - 1 S~ - { 1 '+)(Bx - r'R x'). (4 )
1 - 1 Bx - / Rfi:S~
where ?_-?(x) = Sp -(Bx - rn x3j. In clouds ~J~ l. [If S is determined as ~
- e - ES/Ep, then 'Y = 0. ]
- ~ ,
J
' J, -
o,
~j _
~1~ G~
_
"7 Y -
~ J.
I ` ~
~ I
~ I }
~ X1 i XI Xn X, ..i
} ~ _
r ,rm,. � ~:v Xlim
, 1
s,
_ _
s
0
- Se ?'e.t '
~
~ 'y
- w X.~,-r
Fig. 1.
a Now we will examine the family of curves S(x) for different Sp an.d rn (see Fig. 1). _
_ We wil.l a~sum2 that equation (4~ is correct in the entire range of sizes x< x3 =
_ rnl. When the radius of the CCN becomes equal to the effective radius (x = rnl�),
equation (4) is transformed into a straight-Iine equation (5), limiting the -
S(x) curves to the right,
1
2 -
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J-So--(1+~�)(Bxa-]j. (5) -
With S~) 0 there is suc.h ,3 limiting value rn = rnl~ (SD) at which the S(x) curve
touches the x-axis at some point xli~ = r'1 . All the droplets fc~rming on CCN, whose _
effective radius is greater than rnlim~ a~~min air which is supersaturated rel.ative `
to them (S(x)~ 0 with any x~x3) and grow witiwut limit. Whe.~ rn~rnlim the S(x) _
curve twice intersects the x-axis. At the points xl and x2 S(x) = 0 and the droplets
are in equilibriu:n with tt~e medium. However, at the point xl = rll the equilibrium
is unstable, whereas at the point x2 = r ?1 the equilibrium is st.able. This means
that with SQ ~ 0 tha droplets forming on the CCN either increase in size without 11.m-
it (r ~ rl) or are in a stable state, having the radius r= r2. If the supersatura-
tion SD increases, then the points rl, rlim~ r. approach one ar~other and are shifted
_ to the right, whereas rnlim decreases. The rl, rlim~ r2~ rnlim ?~alues are dependent
only on SQ. And this means that with a known supersaturation Sp the stable water- ~
enveloped nuclei have the radius r< rlim and they were formed on CCN whose effective ~
, radius is rn < rnlim� This made it possible to draw a distinct boundary between cloud
droplets and water-enveloped condensation nuclei. Droplets whose radius is r~rl~
with a given Sp must be classified as water-enveloped condensation nuclei, whereas
- those with r> rlim must be classified as cloud droplets.
With S~ < 0 with any rn there is only one point of intersection of the S(x) curve _
_ with the x-axis corresponding to s~able equilibriinn. With S~ ~ 0 the two points (xl, ~
X2) appear, which with an increase in S~ be~in to be drawn closer to one another. -
Finally, with a definite S~ value for stipulated rn these points merge into one,
- that is, xl = x2 = xlim - rlim'
Thus, water-enveloped nuclei are formed on condensation nuclei whose effective radi-
us is rn ~ rnlirn� However, nu~lei with r~l? rnlim are activated and give rise to cloud
droplets.
, Correlation Between Characteristic Radii of Nuclei and Supersaturation
~ The correlation between rnlim and rlim With S~ is found f.rom the conditions
P= lim) dS (X) I = 0
~ .c o r~P ~6~
S ~X~p) = U.
- Rejecting sma11 terms of the order of V, from (4) and (6) we find
:3r3 `~l'- _ . .
RrP ) ~7/
~~P V xTPI , B '
A rn~P j }'f3 6~.3~~~ ' 1~ " 5p ~ 3--`~. ~~013685 Ss~ `8~ -
` 0~ '
~The :~eaying of terms of the order y in similar expressions in [2, 4] is not legit-
imate. The ,:elationship rlim = 2B/3S0 was obtained earlier by Mason [3].]
i
3
.r~ - - rR - 1,2599 r� S,~'~3 : ~,8479 r� S'~~3,
_ P ~ So ~P ~P � ~9~ _
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- where r is in ~-m, S* is supersaturation in percent and ~p is in fractions of
~ unity. For example, with S:~ = 1% rn l. 37� 10'2 ~(m, and rli~ N 5.85rnlim ^ 0.08
� m. Since with the rising (cooling~ of the air the increase in S~ takes place
gradually, in the first approximation it can be assimmed that at any moment in
time the condensation nuclei are in a water-enveloped equilibrium state. And this
- means that all droplets with the radius r~ rl~ are water-enveloped nuclei and all
droplets with r) rli~ are growing cloud droplets [2,4]. As a result, in clouds
a minimum in the curve of the size distribution of cloud droplets is formed in the
region r~ rl~.
Still another useful expression can be derived from equation (4). It was proposed
by A. G. Laktionov and used as a basis in his development of an isothermal chamber
- - a prornising instrument for measuring the distribution of atmospheric condensa-
tion nuclei by supersaturations. In these chambers SQ = 0 and the equilibrium radii
r2 are measured. It follows from (4) with S= S~ = 0 that
_ - r3
r; _ ~ . (10)
- 'Taking (8) into account, we find that an equiiibrium droplet of the radius r2 grew
' on a nucleus rn, active with a supersaturation
_2co a (11)
S: - t00 S~ _ - _~,6 � 10-' r,-`,
3y3r.:
where r2 is in~.tm. Expression (11) somewhat refines the formula used by Laktionov,
i for which S* = 4�10'2 r21 (6].
_ In conclLeding this section we will compare the critical radius of an insoluble nuc-
leus r~r, active at a given supersaturation, with ~:he critical radius of a soluble
nucleus rn cr, active at this same supersaturation.
- a---- a tay a (12)
_ r"p - I n ( e ~ ) ~ So - S, .
It follows f rom (12) and (8) that
rKP 3
[kp = cr] = = rn = q~,3 ~~3 ~ l ~89~ 1~~ ~ 8~%j% ~~3. (1,3)
rp
With S* = 0.1% ~~19, whereas with S* = 1~6 E~9, that is, in order for the ac-
tivity of insoluble nuclei to be comparable to the activity of hygroscopic nuclei
- their radius must be greater by an order uf magnitude (and the mass by three orders
of magnitude). Thus, if in an aerosol particle the soluble part is more than 0.1%,
it is more important for the formation of ~roplets than the remaining part. Ac-
cording to data from a number of auth4rs, the mass of hygroscopic particles in the
atmosphere is several percent. (From 5 eo 30%, according to [lJ)o Therefore, with
assurance we can ssq g'~ cl.oud condensation nuclei that they constitute soluble or.
- nixed nuclei and that a factor of real importance is the distribution of CGT1 by
supersaturations, or what is the same, by their effecti~~e radii, which are close
t~ the radius of the soluble part of the nuclei. However, the formation of droplets
on insoluble (both wettable and nonwettable) particles plays vit~tually no signif-
icant role in the physics of droplet clouds. (At negative tempE~ratures this process
can be important from the point of view of foxmat,ion of ice crystals.J
4
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Spectrum of Distribution of Cloud Co .densati~n Nuclei by Supersaturations 4
Using (8), we will find the relationship betarPen the distribution of CCN by super-
saturations f(S) and the distrihution by effective radii cP(rn): -
~r,~) = f~S) I d
~ I= f~>>641 � 10-3 rn 3~Z~ ' 2~401,~ � 10-~ ~n s/z~ (14) -
where r is in ~,t, m, ~ is in cm'3� ~ m l. [Here and in the text which follows we will
dispense with the subscri.pt * on S* and by the letter S we will denate supersatura-
tion over a plane surface of pure water in
_ :~ow we will turn to the experimental data. It is known that aerosol particles (AP) -
with radii R from 0.1 to tens of ~,tm are distrib~xted in the atmosphere, in general,
in confoxmity to the Junge law: .
_ . (R) = R �
1 (15)
If the radius of the soluble part rs is related to the radius R of the particle
by the expression [4]
' rS = bR P > (16) -
- then, since th~ fraction of the soluble fraction in the atmosphe re decreas~:s with
an increase in R, l. Using (15) and (16), we find that the distribution of AP
by the radii of the soluble part will have the form
[c = s ] ~~~1 = ll ti l l' J dR ~ r~ (17)
~
- where
~ 1 } z-1 1 ~
~ �(g -1~=-~i1-~ a -111. -
~ , L ~ l1
S:tnce GCi 1. (usually 4) and 1/~8 ~ 0, then x> ac x 4. Thus, with a distribu-
tion of atmospheric aerosols in conformity to the Junge law and correctness of the
. correlation 'oetween rs and R of the type (16) the AP are distributed by the radii
_ of their soluble particles in conformity to a power law with the exponent 4.
Now we will turn to the distribution of aerosol particles by supersaturations. Ac-
cording to numerous data [8],
f 1S~ = CfZ Jk-~, ~~.8~ -
_ where k is usually less than 1. According to (14),
3 4 _ 1
' (rrt) ~ ~A ~ _ ~rs ca+~~. ~19) =
However, the integral distribution has the form -
_ ~v ~r~) _ ~V~ ~ ~=1
rn ~2~~
where = Z k, ~~'1 0,050Ei3k c. (21) _
5
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Table 1
Characteristic Distributions of CCN by Masses of Soluhle Part and Effective
Condensation Radii*
c.,t-a
_ 17! 1 I �'t MA.11 .VA.C
I rs �.m ~.,m backgronnd~ continent. -
;,QO� 10-16 t 3,38� ]0-~ 1.90� 10-= 42,0 1246
_'.37 3.1'� 2.98 40,7 I 996
4.63 ~ 3.9 i 3,16 38,5 831
c.UO ' -1.76 3;79 36,3 701 -
;.?0 � I 0- ' F.3i 5.06 33,4 ~O 1
~.70 ~,93 6,32 28,6 365
n.40 ~ 9.5? 7.59 21,9 270
1.^c,� i0-t< i 1.19� 10-' 9.48 12.4 173
^.y6 ~ I.c,9 1.26 � 10-' 4.7R 85,8
= ~ y ~ I.98 1.58 2,78 42,1
_~.n0 � I 0- :;.00 2,39 1.68 22,3
'ti,15 3.60 2.79 1,06 1 I,6
a.74 ! ~.00 3,19 0,710 5,82
9,2ti ~ ~.00 3,98 0,370 1,67
1,60� 10-'~ i 6.00 4,78 0,200 0,63
'?.54 ' 7,00 5,58 0.113 0,29
3,79 ' 8.00 6,37 0,07! 0,14
~,40 I 9,0(1 ".17 0,046 0.077 '
7,~ 1 1.00 � 10� 7.96 0,039 0.039
= 1,28�10" ~ l~0 9,56 0,011 0,011
'~,03 ; 1.40 1.11 � 10� 0.0035 0.0032
*Data on the concentration N were taken from [7]; the r~ and rn col~ns i~3e_com- ~
puted from th~ conditions rn = 0.7967 r8i p= 1.77 ;/cm , rs =(3m/4~p)
0.512837 m1~3. The background distribution is recommended for the modeling of sea _
clouds.
Taking into account that usually k 4. (We recall that rn ~ rs) .
- However, this contradiction is only apparent. The fact is that formula (15), from
which (17) follows, describes aerosols with a radius R> 0.1�m; however, formula
(20) is correct with S~ 0.1%, that is, rn~G 0.064~..t,mo The conclusion can therefore
- be drawn that either in particles with R~ O.l~,c m there is a soluble part whose
radius rn emerge~ beyond this limit, that is, greater than 0.064~.t.m, and then for- _
mula (17) describes another part of the spectrum, or in the considered region the =
relationship between rn and R is not described by a dependence of the type (16).
It is possible that both to some degree are c~rrect and thiLS the real spectrum of
eff~ctive canden~atian radii in the ran~e �r.c~m O,Q~. ~o l~,c m and above is evidently
not described by a power law.
Up to Che present time the overwhelming ma3ority of the measurements of the distrih- _
- ution of CCN hy supersaturations have heen carried out using diffusion chambers in _
the region S~ Q.1%, that is, in the region rn < 0.06f.,cm. In this field the measure-
- ments were satisfactorily described by empirical expression (18). However, the very
first experiments of Laktionov j6], who carried measurements into the region of low
supersaturations (to S N 0.025%), in actuality revealed that formula (18} with k=
- const no longer is suitable for describing the spectrum of CCN in the entire meas-
ured range. According to the data in [4], in the region 0.025 < S~ 0.16 k on the
6
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average is approxisnately equal to 1.6, that is, twice as great as the k value for
the range 0.16 ~ 5 tG 1%.
" Meszaros [7J, who was greatly involved in measurements of the cancentration of CCN _
in the atmosphere, generalized the accumulated experimental data and cited a table
- of the mean distribution of CCN with resp ect to the mass of their hygroscopic part,
assuming that the solid part of the nucleus is amm~nium sulfate (NH)2SO4. In repro-
_ ducing the I~eszaros table, we added columns for rs and rn, from which it can be
seen that the distribution which he proposed covers the range rn f~oai 0.019 to 1.1
� m and that it is essentially not a power law distribution. T"i~e value d lg N/d lg
rn changes in this distribution by almos t an order of magnitude with a change in rn
from 0.02 to l�m.
Applicability of Power Law Approximation of Spectrum of CCN With Respect to Super- -
saturations _
- A representation of the CCN spectrum in the form
:ti' = csk~ ~22~ ~
wltere N is the total concentration of nuclei active with the supersaturation S~has _
come a.nto wide use ~n cloud physics. Tfte well-known expressions derived by Ztaomey -
[9) for determini~~ the maximum 5upersat uration in a portion of air rising with
the stipulated v~'~~i~y u are based specifically on this distribution. And knowing
SmaY' using (22' also possible to deteimine the concentration of forming
- droplets Nk = cS~{ . Taking into account the importance of expression (22), Braham
[S] proposed that ~at.� on the CCN spectra be represented in (c, k}-space and, using
the ~aomey expressior.s, a diagram be constructed making it possible to fin~? S~X
and P~k easily from known c and k. -
However, we should note that expression (22), as emphasized above, approximates the
experimental data in a rather narrow ran ge of sizes. If it is taken into account
that in ~louds S< 1-2%, this range of change in S from 0.1% to approximately 2% is
equivalent to a change in rn from 0.01 to 0.06 N m. At the same time the Twomey for-
mulas and the Braham diagram were constructed on the assumption that (22) is correct
in the entire ran~;e of sizes, and what is especially important, in the field of
smaller S, up to S= 0, that is, to rrz The question arises: possibly can such
a substantial increase in the concentrat ion of large nuclei with rn~ O.Ob � m given _
by extrapolation of formula (22) into re gions of small S distort the results of de-
termi_nation of Smax and N? Are the Twomey expressions and the Braham diagram applic-
a~+le? _
It is evident that if the flux of water vapor onto droplets growing on nuclei with
rn > 0.06� m is small in comparison ~ith the flux of vapor onto droplets forming on
smaller nuclei the Twomey expressions are applicable. If the flux is not small, the .
expressions derived in such a way must, as a minimum, be examined attentively.
- In order to solve this problem we recall that the equilibrium radius requil of the
droplets at 100o humidity is related to rn by the expression
7
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requil = B-~!' rRl ~ (23)
-3/2k-1
And if initially the CCN were distrihuted in con~ormity to the la~ ~(rn)~rn ~
then, upon attaining a IQOI humidity, the water-enveloped nuclei will be distrib-
uted in conformity to the law -
(3 `
� - l 2 k Y~l ~ dr~ T-'-?. 24) -
jP = equil] ~rP) -"ra drp P ~
In the first approx.imation it can be assumed that the vapor fl~ax on a grcwi:zg drop-
. let is proportional to izs radius. Thus, in the estimates it can be assumed that
the total flux of vapor on droplets of a radius greater than r~ is proportional to ~
�
_ rmaz - rma
TP ~rP~ drP ~V #11 25~ _
- jp = equil] r:, i~ t
According to experimental data, k varies from 0.3 to 1.2 with a mean value of about _
- 0.5 (5J. Accordingly, with a distribution of droplets by radii of the effective
- condensation nuclei described by (19) the flux of vapor, in a~iy case during the
_ initial period of cloud formation, proceeds primarily on large droplets. It can
be postulated that if the distribution ll9), obtained from the distriUution (22),
- substantially exceeds the concentration of large condensation nuclei, this can
- lead to an appreciable exag~eration of the computed fluxes of vapor and the under- 6
stating of S~X and Nk associated with this. With an increase in the size o~ the
� droplets (due to rapid growth of the small droplets) the noted effect will be
weakened. How should one regard the estimates of Smax and Nk made using the 'I`womey
= formulas or the Braham diagram? It appears that the probl~m is not that difficult.
A change in k in a relatively broad range, such as from 0.3 to 1.6, changes the Nk
concentration by a fact~r not greater than 1.5-2 and S~X even less. Since usual-
- ly k does not vary so greatly, the real deviations of S~X and Nk from the esti-
mates obtained using the ~aomey formula must not exceed tens of percent. Thus, de-
spite the sseming unsatisfactory approximation of the CCN distribution by formula
~ (22), the estimates of S~X and Nk bas ed on it are entirely acceptable. ~
It appears that (Nl,~l)-space, transformation to which from (c, k)-space is readily ` -
accomplished using formulas (21), can beco~e equally convenient for a practical
comparison of the CCN spectra with one another and can be even more graphic. Tt
seems that the N1 value, equal to the ~oncentration of CCN, whose e~fective radius
is greater than O.l~tm, is more giaphic in comparison with the c value.
S uimnary ~
The numerical modeling of the processes transpiring in clouds is becoming an in- _
creasingly effective means for investigating cloud physics. The results in some
cases can be essentially dependent on the initial spectrum of CCN introduced into
the model. In order to he able to understand correctly the reasons for the dis-
crepancy in the results_ohtained in different models (for example, one-, two-,
- three-dimensional, etc.) it is necessary that each model be tested for a standard,
= oenerally accepted distribution of CCN. The distributions of C Qd proposed by E.
Meszaros and cited in the table for sea and continental clouds are entirely ac-
ceptable for this purpose.
8
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Taking advantage of the opportunity, I express appreciation to the participants in _
the seminar on cloud physics at Che Central Aerclogical Observatory for useful dis-
cussions.
~ BIBLIOGRAPHY
- 1. Aleksandr~v, E. L., Yasevich, N. P., "Determination of the Weight Concentra-
- tion of Atmospheric Aerosols and Their Content of Soluble Substances," PRO- -
l CEEDINGS OF THE VIII INTERN. CONF. ON NtTCLEA~ION, Gidrometeoizdat, Moscow, -
1975. -
~ 2. Levin, L. M. , Sedunov, Yu. S. ,"Some Probl~s in the Theory of Atmospheric -
Condensation Nuclei," DOKLADY AN SSSR (Repurts of the USSR Academy of Sci-
~ ences), Voi 170, No l, 1966.
- 3. Mason, B. J., FIZIiCA OBLAKOV (Cloud Physics), tr~nslated from English, Lenin- _
- grad, Gidrometeoizdat, 1961. _
4. Sedunov, Yu. S., FIZIKA OBRAZOVANIYA ZHIDKOKAPEL'NOY FAZY V ATMOSFERE (Physics
of Formation of the Liquid-Drop Phase in the Atmosphere), Leningrad, Gidro-
meteoizdat, 1972. `
_ 5. Braham, R. R., "CCN Spectra in c-k Space," J. ATMOS. SCI., Vol 33, 1976. _
6. Laktionov, A.G., "Spectra of Cloud Condensation Nuclei in the Supersaturation
- Range 0.02-1%," PROC. OF Tl~ VIII INTERN. CONFER. ON NUCLEATION, Gi~lrometeo- -
izdat, Moscow, 1975. -
7. Meszaros, E., "Present Status of Our ICuowledge on the Atmospheric Condensation
Nuclei," VOPROSY FIZIKI OBLAKOV (Problems in Cloud Physics), Leningrad, Gidro-
_ meteoizdat, 1978.
8. Pruppacher, H. R., Klett, J. D., MICROPHYSICS OF CLOUDS AND PRECIPITATION, ~
D. Rudel Publishing Company, Dordrecht, Holla.~d, 1978. _
9. Ttaomey, S., "The Nuclei cf Natural Clo ud Formation. Part 1. The Chemical Dif-
~ fusion Method and its Application to Atmospheric Nuclei," GEOFIS. PLTRA APPL., _
-I 43, 1959.
'i ,
-
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UDC 551.54 -
COMPUTATION OF WAVE FLUCTUATIONS OF ATMOSPHERIC PItESSURE
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 13-19
[Article by V. V. Simonov, Main Geophysical Observatory, submitted for publication
4 Feb 80 ]
[Text] Abstract: The author clarifies the possibil-
- ity of computing pressure on an undulating
boundary directly from the first and third
equations of motion. An analysis of numerical
- experiments is given for the purpose of find-
ing a preferable formula.
Sources [2,3] give a formulation of the problem and some results of computations
of the structure of a two-dimensional stratified turbulent flow of fluid over a
chain of r~nochromatic waves with the amplitude a, length L, velocity cf~ and re- -
lief ~(xj t). A fourth-degree equation for the stream function ~ and a second-
_ degree equation for turbulent energy b are considered. The system is closed by the -
generalized Karman hypothesis for the turbulence scale. The problem is solved in
curvilinear coordinates m~ving with the velocity cfl, related to s(x) and
_ the Cartesian coordinates x, z by the expressions
z - C (zl ~l)
E = x; ~ = 1 _ { ~X~.
Here and in the text which follows we use a dimensionless form of writing for _
which, for the sake of simplicity, no special notations are introduced. As the
velocity scale we used u~ some characteristic dynamic velocity equal in this
case to the dynamic velocity at the upper boundary of the wave sublayer h; the
horizontal coordinate is normalized to L; in all the remaining cases the length
- scale is h. ,
The transformation. _ . '
~=z---~(x). (2)
is simpler and more ~onvenient than (1}, used, for example, in [1). However, the
transformation (1) more completely reflects the considered formulation of the
_ problem. The thickness of the wave sublayer, at whose upper boundary the induced
oscillations attenuate, is assumed to be limited and does not exceed the limits =
of the air surface layer. The stipulation of boundary conditions at a relatively
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= low height does not make the use of (2) very constructive. In "straightening"
, the lower boundary, this replacement "curves" the upper boundary near which the
vertical gradients of different characteristics are still quite great. The trans-
formation (2) is eff~ctive in analytical methods for solution and formulation
of boundary conditions with z-y o0 or in an examination of the entire atmospheric
boundary layer, in whose upper part there is attenuation of all the perturbations
caused by the underlying surface. In computing the prQSSUre distribution in the `
wave, which is closely associated with vorticity and its gradients, in the for- _
_ mulation of the problem there is a need for greater accuracy and a minimtun of =
different kinds of simplifications. OthenTise such a cIiaracte~c~tic important in
the problem of wave generation as the averaged flux of momentum at the discontin-
uity Fdis caused by p ressure forces will be cumputed with a substantial error. Even -
small changes in the relationship of phases between pressure and the wave surf ace
_ can lead to a change in the sign of Fdis, not to mention the quantitative resultso ~
But the physically obvious and simply reatized boundary conditions are more easily -
formulated for ~ than for c~ , Accordingly, the formulation of the prob~.em does not
contain equations for vorticity, but only the stream function is considered, de-
spite the fourth degree of the corresponding equationo
The system of finite- difference equations is solved by the method of successive
. Gauss-Seidel displacements. The formula for successive displacements for computing
the stream function cor.tains a whole series of coefficients at the internal points
of grid int~rsection which are computed in advance in [2, 3] in the form of two-
dimensional masses of data. For example, one of the intermediate factors in this -
formula has the �orm - - -
M=2 1+r-=, i t ai ~b~ V~+i. b: VT-~, -
An error was introduced ix? [2, 3], naraely the coefficients Vl and V2 were taken _
at a point of grid intersection with the indices i, j. -
In contrast to [Z, 3] we changed the grid in the upper part of the wave sublayer, ~
which in the last five intervals now became uniform with the interval 0.1~ Some
- approximation expressions were refined. In particular, the derivatives dY~ldyj and _
~y~/~~d~j , necessary f or computing the transformation term in the equation for the
balance of turbulent energ,~, are now determined using fo rmulas with a second degree
of accuracy, and not the first, as in [2, 3].
It is not the distrib ution of total pressure whicn is of fundamer.tal interest, but
its deviation from values at some boundary or from some point. We will examine the
devi.ation of pressure from the vertical = 0, that is, Li p(~,~) = p(~,rf -
p(0, In particular, with ?'j = 0, which will be indicated by the subscript "n' ,'r
~ we wil.l be concerned with the values -
~p~`s, 0)=dPn=P~~, 0)-P~~~ ~3)
Henceforth, a similar notation Q f T.i. will also be used for other functions,
By definition Fdis -~
a: - - ; o o,; � -
- Using (11) we find ~
E
Op�� - -~~J 7 d~ dE. ~12> -
Computations on the basis of (12) give for Q p~ values which are substantially
~ different from those which could be expected, although with 0( S is wave
steepness) entirely reasonable results are obtained. For example, with S= 0.08 -
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and c fl = 0 we obtained dp~� 1) _-3.16, whereas with 0 computations using
(12) gave -0.078. Since the real Q p~ values are small in comparison with other w
terms in fozmula (9), except in the neighborhood of Lhe point 1, where 0 pn-
_ ~ p~, computations of pressure using the third equation of motion are carried =
out without allowance for ~p~ . We feel that this cauaes a lesser error than al-
lowance for Qp~with the use of formula (12).
U~ing the same simplifications as in the derivation of (7), from ($)-(9) wP find
- ~pa=- rr -I"~N-DC'zp-~s~bo; (13) .
N(E)=D 74~~1�
0
3 It is aot very clear for what reason there was an appreciable deterioration of con-
vergence of the iteration process. Approxi.mately 20y of thettotal time expendi-
tures are on computation of the term _ . _
- d= :
_ 2no v- E
d
r, _
= in the equation for But allowance for it, together with the other above-mer.- -
tioned refinements and corrections, introduces significant changes into the be-
:iavior of many characteristics. As an illustration of this we can use the results
of computations of shearing stress with 0 shown in Fig. 1 for the case -
0.08 and c f 1 = 0 obtained in [3] and in this study. At the same time, the nature
of pressure distribution in the wave remained as before, although the quantita-
- tive changes are extremely significant. These results are also shown in Fig. 1.
As a comparison, p 13 gives a line denated Fa which contains the values of the
mean flux of momentum to the waves obtained ~~s[3] f rom the third equation of
motion.
dv; � ~0" Z; r~
_ I . 7F,� ?p-z '
~ ~ -
Z y~ ~ _
~ ~ ~
_ ~ ~ _
2 ~ ~ -
~ ~
- r ~
~
l' 2
- ~ ~ - ~
i
1 ~ ~ 1 ~ e ~
- i i ~ ~
~ ~ ~ ~ �J ~
~
~ ~ Y~~ J~~~ / / ~ 1
/ I
- \ / ~ `,\1
- ~ i
~ ~ ~i~ ' ~ ~ \
p ' O,S ~ 0 QS ~
Fig. 1. Distribution of ,~p~ (1, 2) and Fig. 2. D~stribution of dpn. (1, 2) and -
'Cn (3, 4) according to the results in ~ p~(3, 4) with cfl = 0 and b= 0.04
this study (1, 3) and from [3J (2, 4). (1, 3) and 0.12 (2, 4), -
- 0.08; cf1 = 0. _
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Expressions (7) and (13) are formally equally ~ustified. Figure 2 shows L~p~ and
, Qp.r for cfl = 0~ind~= 0.04 and 0.12, and on p 13 the corresponding Fdis val- _
ues. Available e~erimental data do not make it possible without reservation to
give preference to any of the formulas. The solution of this prablem is related
to a great extent to :he accuracy of computation of the fundamental terms in -
these formulas. The last term in (7) is decisive, to be more precise, the inte-
gral r,f a~G/d 7 with 0, ~'.nce yN. fl 1. But the results of computatian of the
- deriv.ative~ of the flux of momentum with 0 are highly dependent Qn the method
for camputing the flux itself at this boundary~ The table, for S= 0.08 and cfl -
- 0, gives the values i.fR.�r, computed using a 4-point formula, and -G fotntd by _
extrapolation. The table also gives the values of uniformly obtained a-G/ar~ with -
r~ = 0 with the use of 2 fT�r and ~,~X. It can be seen that not only the value, but =
also the sign of the derivative are dependent on the method for computing 2~~ Ac-
. cordingly, both the pressure distribution in the wave (fifth and sixth lines in
the table) and the resistance will be substantially different.
Our numerical experiments indicated that the values of the derivatives at the _
boundary, obtained by use of the formula and by extrapolation, differ little from
one another. The choice of the turbulence coefficient in the approximation formula -
~ for computing the fluxes is important. It is common to use the turbulence coef-
- ficient obtained as the sum clkl + c2k2, where the subscript on k indicates the
number of the grid point of intersection in the vertical direction. Assigning ci
different va1_ues from 0 to 1, the value of the flux can change substantially. The
_ ~or values were determined with cl = 1 and c2 = 0. On the basis of a compari.son _
of the results of computations, and not only those cited in the table, in this
study all the turbulent fluxes at the boundaries were determined by extrapolatian -
= using the three grid points of intersection adjacent to the boundary. _
- In formula (13) the main term is d N, which is considerably less sensitive to the
- numerical values 'G,~. The values Qpn computed from (13) with 'G ~�r and 'G ~ coin-
ci.ded with an accuracy to the third place. Therefore, the table gives only one
- line p'~..
- The computed value of the pressure decraase in the wave using formula (7), with =
0.08 and cfI = O,was equal. to -9.82. After comparing the figures -9.82, -3.16 -
and -0.078, it must be concluded that different types of approximation and comput-
ation e;.rors lead to greater errors in computing pressure on the basis of the
_ first equation of irrotion than when apoo in formula (13) is not taken into ac- _
count. The sources and magnitude of the computation errors remain the very same
if formula (4) instead of (3) is taken as a bas:is.
Still another advantage of formula (13) is that it was obtained without use of
the first equacion of motion, if the term Q p~ is not taken into account, which
- in an extreme case can be taken into account, assuming Q p~ _-q where q is
evaluated from the thickness of the wave sublayer and has a value of about 0.1.
The only thing which does not support (13) is the presence in the main term L~N
of a vertical integral whose upper limit, that is, the thickness of the wave sub-
layer, is stipulated quite subjectively.
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~ The following circ~nstance is of more than a little importance~ In order to ob-
tain reliable Qpn vaZues it is nacessary to have a lesser number of iterations
than for dpn. This is particularly noticeable wi~h respect to the Fdis valueso For
example, for the case 0.12 and cfl = 0 after 5,000, 10,000 and 15,000 itera-
- tions the ~'d values were equal to -2.95, -0.33, 0,37 reapectivelyo At the same _
~ ti~:a the si~~ar Fdis Were 2.13, 2.35, 2.38.
'raking into account everything stated above, the following conclusions can be -
drawn. First, with the inclusion of pressure among the sought-for characteristics -
it is desirable to use one fourth-degree equation for the stre~am function instead
of a system of two equations in the variables w and especially a system of
~ three equations ir. the variables u, p. Second, when it is necessary t,o comgute
the pressure distribution the use of the third equation of motion is more justif-
ied.
E ~
- I 0 0,1 f 0,2 0,3 0,4 ~ U,5 0,6 U,7 0,3 0,9 I 1,0
~ I I l I ~ I I I ( i
0,88j 0,52 0,23 0.12 O.lOI 0,14 0,24 0,4G ~J,65I 0,91 0,88
t9 2.40~ 1.38 0,58 0.29 0.71~ 0,40 0,69 l.l~ 1.88i ?.SE 2,40
i
d~�t0 -3 8.25I 4,97~ i,24 1,16~ 1,03~ 1.46 2,40 3,551 6,30~ 9.63j 3,25
d r~ I I i
I
tlr3 '10-1 1-0,441 0,27~ 0,32 U.22 O.l3I O,O~I-0,06 - 0,24I-U.~ -0.36(-0,04
~ !
-1p~'~�10-= j 0 I 6.93~ 10,4 11,9 12.9 14.1 15.9 18,5 i3,6I 31,0 ~ 39,9 -
~po 3� 10-~ I 0 U, l3 l 0,:.2 O,SO 0.98 1.07 l,OS 0,5;; 0,"4 ~ 0,1 I I- 0,10
]p~ �10-'- ~ 0 O,U9I 0,34 0,7~) I,IS 1,42 1,30 O,~J~ U.4hl U,13i 0
; ~ ~
[Tr = 'ft ; ~ = for (mula) ; 3 = extrap (olatic~n) ]
BIBLIOGRAPHY
1. Makin, V. K., Chalikov, D. V., "Numerical Modeling of Wind Waves," METEOROL- '
OGIYA I GIDROLOGIYA (Meteorology and Hydrology), No 10, 1979.
2. Simonov, V. V., "Turbulent Flaw Over an Undulating Boundary," TRUDY GGO (Trans-
actions of the Main Geophysical Observatory), No 423, 1979.
3. Simonov, V. V., "Some Results of Computation of the Structure of a Turbulent
Flaw Over an Undulating Boundary," TRUDY GGO, No 423, 1979.
4. Phillips, 0. M., DINAMIKA VERKHIv:?GO SLOYA OKEANA (Dynamics of the Upper Layer
of the Ocean), translated from English, Moscow, Mi.r, 1969.
5. Gos~men, A. D., Pan, V. M., et al., CHISLENNYYE METODY ISSLEDOVANIYA TECHENIY -
VYAZKOY ZHIDKOSTI (Numerical Methods for Investigation of Flows of a Viscous
Fluid), Moscow, Mir, 1972.
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= 6. Banner, M. L., Phillips, 0. M., "On the Incipent Breaking of Small-Scale
Waves," J. FLUID MECH., Vol 65, Part 4, 1974.
7. Barnett, T. P., Kenyon, K. E., "Rscent Advances in the Study of Wind Waves,"
REPORTS ON PROGRESS IN PHYSICG, Vol 38, No 6, 1975.
8. Ben~amin, T. B., "Shearing Flow aver a Wavy Boundary," J. FLUID MECH., Vol 6, -
- Pt 2, 1959.
9. Gent, P. R., Taylor, P. A., "A Note on ~Separarion~ Qver Short Wtnd Waves," _
BOUND.-LAYER METEOROL., 'Vol 11, No 1, 1977.
10. Gent, P. R., Taylor, P. A., "A Numerical Model of the Air Flow Above Water -
Waves," J. FLUID MECH., Vol 77, Pt 1, 1976.
- 11. Taylor, P. A., "Some N~erical Studies of Surface Boundary-Layer Flow Above -
Gentle Topography," BOUND.-LAYER METEOROL., Vol 11, No 4, 1977.
17 '
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UDC 551.501:519.272
SOME METHODS FOR EVALUATING THE PAP.AMETERS OF CORRELATION FUNCTIONS OF ~
METEOROLOGIGAL FIELDS
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 20-29
[Article by Candidate of Technical Sciences V. L. Savanov and T. A. Yarygina, Mos-
cow Power Institute, submi.tted for publication 18 Jan 80]
= [Text] Ahstract: The article gi~es a validation of
one of the possible approaches to solution
of the problem of evaluating the para~eters
of the correlation function of a random meteori
ological field which essentially involves find-
ing evaluations of the coefficients of a re-
gression model when there are observation cor- '
- related errors. The authors propose several
methods far evaZuating the carrelation func-
tion of a random (in a general case noniso- "
tropic and nonuniform) field. The choice of _
the method is dependent on whether the correl-
ation function is linear or nonlinear with re-
spect to the parameters and on the availability
of a priori information concerning the distribu- -
tion law or on the central moment function of
the fourth order of the investigated field. Ex-
pressions are derived making it possible to
evaluate the accuracy of the proposed methods.
Introduction. A factor of great importance for solving p roblems in meteorolo gy and
_ oceanography is finding methods for describing the spatial-temporal variability
of p arameters characterizing the different properties of the atmosphere and ocean. _
These parameters form fields (temperature, pressure, humidity, etc.) which, as is
- well known, in a general case are nonstationary relative to the time coordinate, _
nonuniform and nonisotropic relative to the space coordinates. One of the principal
prob lems arising in study of random meteorological and oceanographic fields is an
experimental determination of their statistical characteristics. For a complete
- statistical description of spatial-temporal random fields it is necessary to know
the multidiinensional distribution functions, but in actual practice it is common
to 1 imit the examination to simpler characteristics of the random fields mat~ie-
matical expeczations and cavariation or correlation functions.
A co rrelation analysis is one of the most important methods for the study of random ~
fields. A knowledge of the covariation function (CF) mak~s it possible to form a
general idea concerning the structure and properties o~ meteorological fields,
- 18
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so1~?g many applied prohlems, including the abjective analysis of the investigat-
ed fields [2, 5, 6], solve problems involved in rationalization of the network of
measuring stations in space and the optimum discretization of ineasurements in time
[3, 4, 10].
An experimental determi.nation of the CF of the field~ of ineteorological elements
usually essentially involves computations of evaluations of the parameters of a
CF of a stipulated forin f rom the readings of one or mpre independent field series.
The methods for determinin~ the CF of a random field used at the present time as-
_ sume satisfaction of a series of conditions which in practical situations frequent- ~
- l~r are violated. Thus, in the stipulation of the readings of several field series
it is usually required that these readings for all series correspond to one and
the same set of field arguments [6]. There are methods applicable in cases when _
- there are gaps in the series of observations [7], but even they can be employed
eff ectively only when there are relatively few gaps and the arguments coincide -
_ fo r most of the readings of different series.
However, if the readings for only one field series are stipulated, the limitations
become still more rigorous. In this case the developed methods are a~plicable only
- to isotropic random fields and are rigorously valid with stipulation of the readings
a t the points of intersection of a uniform grid of the values of the argument. Mod-
ifications of these methods with a nonuniform network essentially involving averag-
ing of the initial readings witnin the limits ~f some regions (gradati,ons) of the
values of the arg~nnent [2, 6] have been inadequately formalized and lead to the
_ ap pearance of systematic errors whose value usually cannot be evaluated. The stud- _
ies of R. L. Kagan [8] are devoted to investigation of a number of problems involv-
e d in evaluation of the accuracy in computing the CF by the gradations method. -
Be low we present one approach to determination of the CF of a random field making
it possible to propose a number of inethods for its evaluation free of the above-
mentioned shortcomings. We will examin: the problem of evaluating the CF with
s tipulation of the readings of only one series for the random fieldo Such a formul-
ation of the problem corresponds well to the practical situations arisitig in an
investigation of ineteorological and oceanographic fields. In addition, the results
are generai.ized for the case of several series [11].
Fo rmulation of problem. Assume that F(u), where u=(ul,...,uk) is a k-di.mensional
ar gument, denotes the investigated scalar random f ield. For problems in meteorology
and oceanography in a genexal case u=(x, y, z, t), where x, y, z are the space
coordinates, t is time. It is assinned that in a closed li~ited region of UE Rk
_ values of the argument the mathematical expectation (ME) of the field is equal to _
ze ro, that is, E[F(u)] = 0 with u~ U(this means that the determined field compon-
- en t is absent or first excluded), whereas the CF KF(u, s) for the field is known
w ith an accuracy to m unknown parameters: -
K~, (U, S) = E~F (u) F(Sl~ = k lu. S, bl~ u. S C U. (1)
where b=(b ...,bm)T is the vector of the unkno~n parameters (b ~ Rn), k(u,s,b) is ~ -
a function o~ a known type.
y
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Such a formulation is entirely justified since for most meteorological elements
the type of CF is kr..own approxi.mately on the basis of dzta accwnulated in the
course of observations over a series of years. In a special case when the field
i.s uniform k(u,s,b) is dependent on the difference of the arguments r= u, s,
ehat is~ k(u, s, b) = k(r, b). If~ in addition, the field is isotropic, then the CF
is dependent only on Che modulus of the difference in the arguments: k(u~ s, b) ~
k(r, b), where r _ ~ r ~ _ ~ u - s ~ = Y(u, - s, -t- . . . ~llk - Sk~=. -
. It should be noted that the methods for determining the CF used at the present ~
time, although to some degree they can be modified for computing the character-
istics of nonisotropic and nonunifozm fields, nevertheless are oriented primarily
on isotropic and uniform fields. However, the condition of unifoxmity and isotrop-
icity of ineteorological and oceanographic fields is very rigorous, and only for
extremely limited regions is it possible to speak of its approximate satisfaction.
Accordingly, henceforth we will examine the general case of a nonuniform and non-
~ isotropic field. -
Assume that as a result of the experiment, whose purpose is determination of the
- CF parameters, N values are obtained _
fk - 1 k~ uk~+ Uk C U~ k~ ~ ~2~ -
where fk denotes the result of ineasurement (reading) of the series f(u) of the F(u) ~
field at the point uk.
~ Conversion to regression model. Using the available readings (2), we form L values
~kr r ~4lr =I ~uk~ J ~ V~~r -
k, r-l, N, k_r, L= 1 N(;V-~-1)
and introduce a vector of the ~imensionality L
w = l~ftr . . . t ~1 Nr ~Y2, Zc'1g~~ . . . . ZC2 N. . . . , ~(N-11 N~ `LFINN~T� ~3~ -
It is easy to see that the wkr values can be interpreted as readings of the series
w(u ; s) of the random field W(u ; s~ = F(u) F(sl at the point u;s = uk ; ur. Here
u;s =(ul...,uk, sl...,sk). For the mathematical expectati~n (ME) and CF of the `
W(u~s) field we accordingly have
F( ti% (u ; s1J = E IF (u? F(s)~ = k(u~ S~ b), ~q) -
K~. (u ; S, P 1 91 = E I~F ~u~ F+s) k(u~ s, b)~ X (5)
(F(P) F~9) - k (p, q, b)~I, .
where p, q ~ U.
Therefore the field W(u~ s) can be represented in the form
- W(u ; s j= k(u, s, b) I' (u j s), _
- where the field r(u;s) has a zero ME and a CF K~ (u;s, p;q) = ICW(u; s, p;q) and
can be considered as the field of errors in measuring the nonrandom function k
(u,s,b).
2Q '
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_ By analogy with the W vector we in.troduce the vectors K and(~ with the components
kkr = k(uk, ur, b) and y~r =)'(uk~ ur) respectively under the condition k ti r,
where y(uk~ ur) denotes the reading of the field r(u s) at the point u;s = uk;
ur; we write
w = K (b) + r. ~6~
The W and vectors can be interpreted as sample values of the vectoral random
values for which there is satisfaction of tlie conditions E(r 0, E(W K(b),
� and their covariation matrices D((-) = D(W) havP tfie order L.
Thus, the problem of finding the parameters b of the CF k(u, s, b) can be formulated
as the problem of evaluating the parameters of a linear or nonlinear (depending on
tfie type of function k(u, s, b)) regression model with correlated observation =
errors (13].
A sufficiently general approach to formulation of evaluations of the parameters of
_ the model (6~ is a~ follows. We u~ill use ~ to denote so~ evaluation of the b vec- _
= tor and use W= K(b) to denote the ME evaluation of the W vec~or. We inLroduce the -
vector of t!?p residues - . _ .
~ ~
V(b~=V~-W =W-K(b). (7)
Then a broad class of evaluations of the b vecto r can be obtained by minimizing ~
some appropriately selected evaluation function P[V(b)J, that is
` ^
b - argminp ~V (b)J - argmin p (W - K (b)J. ~8~ _
b ,b
Henceforth we will limit ourselvea to the special case of auch an evaluation which
corresponds to the use in (8) of the function.___ _
p (V (b)]~ VT (b) CV (b?,
where C is an appropriately selected positively determined matrix of the order L.
Then we come to the evaluation -
^ N
b = arg min (W - K (b)]T C [W - K (b)J, ~9~ _
b
minimizing the generalized sum of the squares of the resi~ues, th.e so-called eval-
- uation of the generalized least squares method [13].
Evaluation of parameters of a linear mc~del. In a linear parameterization the CF
of the random field F(u) is stipulated in the fo rm
k(u, s, b) = bT m(u, s), u, s E U, _
where the vector ~(u, s) 1(u,s),..., ~ m(u,s))T; ~ i(u, s) are known func-
tions. Then
- W =~b+r~
where ~ is a matrix (measuring L x m) of the fo rm
~ _ ~`~ll~ S71?~ . . . ~ N~ ~14r ~2Sr . . . ~ N~ . . . ~ ~~N_~~ Nr -
_ ~NN~T� ~kr = ~ (Uk, l1!)~ k < r.
21 =
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In this case (9) wtll be equivalent to the expression
~ b~ b~'~ _~~T C~)-' ~t C W,
(10)
in this case the evaluations are not biased (E[bjl}] = b), and for the covariation '
matrix of the evaluations
D ~bl?1~ -E [ib~'~ - b)(b(~~ _ b)T~ ~
- we have
p ~bct~~ _ (~*~~A)-' ~T CD (W~ C~ (~TC~)-1. (11)
This formula makes it possible for any selected weighted matrix C and the known co-
variation matrix D(W) (for this it is sufficient to know the CF of the field W(u;s))
- to evaluate the statistical accuracy (c~variations) of the evaluations
If D(W) is known, then for attaining the maximum accuracy of the evaluations as the
_ weighted matrix it is necessary to select C= D-1(W) jl, 13]. In this case instead
of (10) we obtain
- b~z~ - p-' ~~y~ ~T D-' (W) W, (12) -
and the covariation matrix of the evaluations
p (br~i) = D-` (W ) (13)
We^note that with a stipulated CF of the field W(u;s) the matrices D(b~~~) and
D(b~2~), characterizing the st~tistical accuracy of the evaluations (10) and (12),
are dependent on the position of the points uk at which the f(uk) measurements were
made since the elements uf the matrices5~? and D(W) are dependent on the uk values.
Accordingly, in this case it is possible to formulate the problem of a priori for-
mulation of optimum ex~erimental pla.ns, that is, rational distribution of the net-
work of observation stations. However, if the D(W) matrix is unknown, as is usual-
ly the case in actual practice, as the weighted matrix C in (10) it is necessary
_ (depending on the available a priori information) to take some syuanetric positive-
ly determined matrix (for example, of the diagonal type). 'I'he simplest variant
of such, a matrix is a unit matrix. In this case we come to the usual evaluations
= of the least squares method. -
Evaluation of parameters of nonlinear model. Now we will examine a case when
k(u, s, b) is a f unction nonlinear relative to the evaluated parameters b, contin-
uous with respect to b. If the function k(u, s, b) is smooth with respect to b in
the neighborhood of the true values~of the h parameters and if equation (9) has a `
~:nique solution (we will denote it b~3~), then with sufficiently great dimensions
of the U region in which the measurements (Z) were made and with a Iarge number of `
measurem~nts N it can be ass~ed approximately that b~3~ is an unbiased evaluation
of the b vector and it can be assumed that
~ . _ - ~
k(u. s, b~3~)'-== k(u, s, bl (b?3~ - b)T g(u, s),
~ahere in contrast to the linear case ~(u, s) are dependent on the true values of
the parameters
t~
= ZL
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c(a, s) - m(u, s, b) k(u, s~ b), (14)
~ CT=(
b~~. . ~
Accordingly, the covariation matrix of the evaluations b~3~ is also dependent on
- b, although it is also determined, as for the linear case, by the expression
G ( b~3~) = U (b~3~, b) (~T C ~T CD (W ) C ~ (~7 C ~ ~~5~ -
However, here the ~ matrix is dependent on b, that is, ~~~(b), since in its
- forming use is made of elements formed by the values of the function (14). If the
D(W) matrix is known, by selecting C= D-1(W) (as in the linear case); we obtain -
- evaluations with the best statist~cal properties. The dispersion matrix of these
evaluations (we will denote them b~4~) has the form -
D (bi'~) = D ~bi'~, b) (~T D-' (W ) (16)
: With a CF quite smooth relative to b, instead of the values of the function (14}
in computing the matrix ~_~(b) it is possible to use the values of the functions
~ - n n
c (u, S1 = ~ lu, s, b~ _ `k ~u. s, b~'~),
therefore, it can be assimmed approximately that ~(b~ ' With a known CF of
the fi.eld W(,u; s), as in a linear case, it is possible to formulate the problem of
, optimum planning of an experiment, but a priori planning here is already impossible
although it is possible to use sequential planning, minimax or Bayes approaches _
(12].
_ Evaluation of CF of a Gaussian random field. Now we will ex,amine an important spec-
ial case of practical importance evaluation of the parameters of the CF of a
Gaussian (normal) field. As is known [9], for such a field F(u) the CF of the field
W(u; s) = F(u) F(s) is expressed in the following way through the CF of the initial
field: -
%~Q, (u ~ S, P~ 9) = k(u, P, b} k(s, q, b) + k(U, q, bl k(S, P, b).
Thus, the CF of the field W(u;s), like the field F(u), and accordingly, also the
covariation matrix D(W), is dependent in a known manner on the values of the eval-
uated b parameters, that is D(W) = D(W, b). In this case, regardless of the nature
of the function k(u, s, b) (linear or nonZinear relative to b) it is possible to
~ propose evaluations of three types. -
Evaluations of the first type are computed in accordance with (9) (for the case of
linear parameterization - in accordance with (10)). The covariation matrix of these
- evaluations, dependent (through D(W)) on the unknown b values, can be computed ap-
_ p roximately using formula (11) or (15) by a replacement of b by their determined
evaluations.
- An evaluation of the second type differs from the first in that in order to increase _
the accuracy of the evaluation use is made of ah iteration procedure
23 r
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n
bi =argmin ~w - K ~b))T D-'(W , bt-,)(W -K (bi), l- 1, 2, . . . (17)
b
n
wher~Ab,L is the evaluation of r?~e b parameters in the ~.-th step of the proced- _
ure; b~ is the appropriately selected initial evaluation of the parameters;
D(W, b~) is a matrix of a stipulated type or a matrix selected on the ba^is of -
a priori information. Together with the values of the evaluation of the ~ vector,
in each step of the procedure the evaluation of the matrix D(W,b) is refined. Ac-
cordingly, such a procedure can also be used in the case of a vi.rtually complete
~ a priori uncertainty concerning the form D(W,b); in this case it is possible to
stipulate D(W, bp), for example, in the form of a unit ma.trix. The covariation
matrix of the evaluations can be computed approxi.mately using the formula
n ~ ~ ~
D(W, bi) =(~T D-' (W~ b~-i) ~)-i ~T p-~ ~yy~ bi-~) U lW, bi) i:
- ~ ~ -
f D-i ( W, b~-~ i~T D-i ( W, b~-i 1~)'' �
With adherence to the necessary conditions the procedure (17) converges
~ . . -
( I tm b~ = bm
ro
in this case
D cb.~ }=(~r D-' cw, b~ -
The procedure can be sto ped under the cundition of attaining a stipulated close-
ness of the evaluations and _1 or their covariation matrices.
- Finally, the third method for obtaining the evaluations is a direct minimizing of
the generalized sum of the squares of the residues with use of a weighted matrix
dependent on the evaluated parameters, that is, it essentially involves solution
of the equation � ~ _ ~ - ~ ~ -
b=argmin (W - K ~b))T D-' (W, b)(W -K (b1)�
(18)
b
However, even with linear parameterization of the function k(u,s,b) computation of
- the evaluation (18) involves satisf action of the nonlinear operations for the com-
po~ents of the W vector. It is extremely difficult to obtain an expression for -
D(b) in this case but it is clear intuitively that the evaluations (18) and (17)
must not differ significantly in their statistical proPerties. We note that with
- use of evaluations of the f3rst two types considered above there is a possibility
for t.he optimum planning of an experiment (both with the sequential and with the
- minimax or Bayes approaches)a
~ Some models of the CF for a nonuniform field. At the present time in meteorolog-
ical and oceanographic computations use is made pri.marily of models of an iso-
tropic and uniform field, although in the literature there is constant mention of
the limitations on their use for the description of macroscale processes. In part
such a situation is evidently attributable to the lack of sufficiently effective .
algorithms for the processing of data for general cases of nonuniform and noniso- -
tropic fields.
- 24
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. As one of the possible models of a CF of a nonisotropic and nonuniform field it
- is possible to propose a function in the form
KF (u, s) (u, s, bl ~ (r, b1, (19)
where r = Y(u, - s, -I- . . . (!!k - Skl~r
~(u,s,b) and ~(r,b) are appropriately selected functions. In special cases only
- one of the functions ~ and ~ may be dependent on b.
' We note that CF in the form
KF(u, s) _ (u~ bj (s, b) ~ (r, b),
~ obtained fram (19) with the choice ~(u,s,b} _~1(u,b)~1(s,b), corresponds to
- the CF of a nonuniform and nonisotropic field represented in the form of the
product of a uniform and isotropic field Q(u) and a CF of the form ~(r, b) and a
determined function ~1(u, b).
Example. We will examine computation of the CF parameter dependent.on two argiunents
and being a special case of (19),
' Kf (u, s) - b~F (u, s) v(r}, n, s~ U,
a ~ahere U is a part of a plane,
r = (tt, - s~ )1-F- (u= - s~
i
~ ~(r) and ~(u,s) are known functions, b is the sought-for parameter.
I
I Assume that in the U region there are 2 n readings of the centered Gaussian ran-
dom field. With a li.mited +i~ (u, s) and with a function I~(r)~ attenuating with an
increase in r it is always possible to indicate such an rp for which ~KF(u,s)~
< S, where ~ is a prestipulated small parameter. Assume that the existing 2n
readings fall at points forming n pairs of points ui, vi, i= l,n with the dis-
tance ri (rr = ~u~ i - vr 1 -F' (urz --vl2)' )
I
' between the points of the i-th pair (see Fig. 1); the correlation between the read-
- ings at the points of each pair is significant, whereas between the readings of
adj acent pairs it is negligible (that is, the distance between the points of any
- different pairs is greater than rp).
~~.,._,.y ~I
r~
u
~_y v~
~
I r' � un
I I
~ ~
i g~g. 1
~
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As a simplif ication we will f irst examine one pzi.r of points ui and vi, situated
_ at the distance ri. In this case, writing the vectors
~r = (ut, ut) ~ ~~)i ~ ~u1, 2'r) ~ (Ti~~ ~ lvtr vt) ~ (0))T,
W t = (f' (ut); t (u~) f (v;); f (~i)), -
fo~ning the D(Wi) matrix and making computations in accordance with (12) and (13),
- we obtain ~ - - - - - - _ . _
_ b1= 1W r) ~Pi D-~ t W;` W; = 2 (vt~ vr) 4(~) f' (ut) - .
-2 ~F (u;~ ~~1 ? ~r~) I ~u~) f (v;) tF (u,, url ~ ~ ~vi)~~ 1~ (b,)- b=.
Thus, in this case the accuracy of the evaluation is dependent onYy on the value ~
- of the evaluated parameter b itself and is not dependent on the distance ri be-
- tween the points at which the measurements were nade.
Proceeding now to the case of n pairs of points ui, vi, i= l,n, forming the vec-
tors ~ and W of the di.mensionality 2n(2~+-1)/2 and the D(W) matrix, making computa-
tions using formula (13), it is easy to show that D(~) = b~i. It therefore fol-
lows that the accuracy of the evaluation is dependent for the particular example
only on the value of the parameter to be evaluated and the number of pairs of
points and is not dependent on the distances between them. It is easy to show
that the evaluation of the b parameter, together with (13), can be computed uaing
the formula ^ 1 " `
b = rs ~ bj.
This is attributable to the fact that the dimensionality of the ~ and W vectors in
this case without a loss in accuracy can be reduced to 3n, excluding from consider-
ation the noninformative elements of the ~ vector corresponding to a zero correla-
tion between the readings at the points of different pairs and the corresponding
components of the W vector, in this case also changing the D(W) matrix. (In this
case the order of D(W) is equal to 3n).
Concluding remarks. The examined methods are quite universal and are applicable
for evaluating the parameters of the CF of arbitrary random fields (including non-
uniform and nonisotropic fields) with stipulation of one field series.
The results can also be generalized for the case of stipulation of the readings
of several field series at arbit?-?_ry (not necessarily c.oinciding for alI series)
points of a finite set of values of the arg~mment. In this case for each i-th ser-
ies we will have a vector Wi which is represented in the form
W~ = K; (b) + ~ i, i- l, n.
~here the vectors Wi, Ki, r i have the same sense as the corresponding W, K, ~ vec-
tors in fo~ula (6).
' 1,...,Wn) , which can be represented in the form
We introduce the vector W(W T
w=K cb?T r, _ -
, where K ~b~ - ~K, (b)~ . . . , Krt tbl)' and .
= c r� , r~~T
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It is easy to see that the covariation matrices of the W and r vectors have a
block structure
D~1 , . . U~ rt
f~= U~ 1'1' Oit u4Y Q': n
_ ~
On I ~n 2� D~~ ~
- where Dii = D((- i) = D(Wi), and 0i are zero matrices. The principal formulas for
finding the b evaluations and thei~ covariation matric~s in this case do not
change if W, K(b) in them is replaced by W and K(b).
However, in the case of stipulation of readings of field series at points coinciding
for all the series it is necessary to average the readings corresponding to differ-
ent series. Thus, there is a conversion to the readings of one series of another
~ field differing from the initial field in that the values of its CF are n times _
less than the corresponding values of the CF of the initial field and further pro-
cessing is carried out the same as for one series.
Finally, in a case when the initial field is not centered and its ME is a function
of a known form, dependent on several unknowa para~eters, for application of the
proposed methods for evaluating the CF it is first necessary to f ind evaluations
of the ME parameters (for example, using the least squares method) and correspond-
ing evaluations of the ME itself at the points of stipulation of the readings of
- series for the initial field and proceed to field readings which can be considered
approximately as already centered. Since the accuracy in evaluating the field ME
is usually greater than the accuracy in evaluating its CF, such a conversion does -
not lea.d to appreciable errors. _
' The proposed approach can be used not only for finding the covariation function of
the field, but also for determining the parameters of the cross-variation function
of the fields of two meteorological elements. Assimme that F1(u) and F2(u) are ran-
dom fields whose cross-covariation function k(u,s,b) is known with an accuracy to .
the unknown vector of the b parameters, and as a result of the experiment Nl read-
ings of the F1(u) field are obtained, _ _
J I k- f~uk~� uk E Ur k-~ r N~
and N2 readings of the F2(u) field,
/ Z r= f? ( ur~+ Ur E v~ r=-~ ~�`~S
We will fo mn L values
- ~kr =f~ kfz, =f, (u~1 f. (u,), k=- 1. N,, r= 1, N_, L= N~ .N~� -
We introduce the dimensionality vector L:
~ - f~11~ `1PJ12r . . . ~ ':.~1 ~ ~=ti 7F1?~~ � . . ~ 4C~.~ N~ ~ . . . ~
. ?DN2 i, 2FJn~` , . . . . ?,vN: ,v,), L = N, N2-
T'hen reasoning the same as in the case of the co~ariation function, it can be con-
firmed that all the formulas for computing the b vector will also be correct for
this case.
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It should be noted that the proposed procedures for computing the CF with their
use with an electronic computer can require a greater computer memory vol~e in
comparison with those used at the present time, but also have a number of advan-
tages, the most important of which is their applicability for nonuniform and non--
iso tropic fields. T'hese procedures are particularly effective in the case of a
small nimmber of initial data (thin network of ineasuring stations in the consider-
ed region and an infrequent reading of data at an individual station or research
ship). However, if the masses of data are large and the direct realization of the
proposed procedure is diff icult due to limitations on the volume of the computer
. mesnory (this is c:~pendent on the specific type of contputer used for the proc:ess-
ing) it is possible to propose a number of simplified, decomposition or iteration
pro cedures for the processing of data.
- BIBLIOGRAPHY
1. Anderson, T., STATISTICHESKIY ANALIZ VREMENNYKH RYADOV (Statistical Analy- -
sis of Time Series), Moscow, Mir, 1976.
2. Belyayev, V. I., OBRABOTKA I TEORETICHESKIY ANALIZ OKEANQGRAFICF~SKIKH NABLYUD-
ENIY (Processing and Theoretical Analysis of Oceanographic Observations), Kiev,
Naukova Dumka, 1973.
3. Brimkulov, U. N., Krug, G. K., Savanov, V. L., PLANIROVANIYE REGRESSIONNYICH
EKSPERIMENTOV PRI ISSLEDOVANII SLUCHAYNYIQi POLEY (Planning of Regression Ex- -
periments in an Investigation of Random Fields), Preprint USSR Academy of Sci-
ences, Scientific Council on the Complex Problem "Cybernetics," Moscow,
VINITI, 1978.
4. Brimkulov, U. N., Krug, G. K., Savanov, V. L., "Rationalization of the Measur-
- ing Network U~;*!o the Cr.iteTi,~n ~f .".~c::~~~; 3f rhe Mathemati.^a.?. nPscrintion of
the Field of Norms," METEOROLOGIYA I GIDROIAGIYA (Meteorology and Hydrology), _
No 7, 1978.
5. Gandin, L. S., OB"YEKTIVNYY ANALIZ METEOROLOGICHESKIKH pOLEY (Ob3ective Anal-
ysis of Meteorological Fields), Leningrad, Gidrometeoizdat, 1963.
6. Gandin, L. S., Kagan, R. L., STATISTICHESKIYE METODY INTERPRETATSII METEORO- -
LOGICHESKIKH DANNYKH (Statistical Methods for the Interpretation of Meteor-
ological Data), Leningrad, Gidrometeoizdat, 1976.
n -
7. Zhuravleva, Ye. B., Kagan, R. L., Polyak, I. I., Computation of Autocorrela-
tion and Cross-Correlation Functions from Several Records of a Random Pro-
cess," TRUDY GGO (Transactions of the Main Geophysical Observatory), No 289,
1972.
8. Kagan, R. L., On the Accuracy in Computing Space Correlation Functions,"
TRUDY GGO, No 308, 1973; No 336, 1974.
~ 9. Levin, B. R., TE~RETICHESKIYE OSNOVY STATISTICHESKOY RA.DIOTEIQ~NIKI (Theoret-
ical Principles of Statistical Radio Engineering), Moscow, Sovetskoye Radio, -
1974.
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10. Mashkovich, S. A., "Increasing the Quality of Ob~ective Analysis of the
Pressure Field Qver Regions With a Thin Network of Aerological Stations,"
TRUDY MMTs (Transactions of the World Meteorological Center), No 10, 1965.
11. Savanov, V. L., Yary~ina, T. A., "Some Methods for Determining the Correla-
tion Function of a Random Field," TRUDY MEI (Transactions of the Moscow
~ Power Institute), No 399, 1979.
12. Fedorov, V. V., TEORIYA OPTIMAL'NOGO EKSPERIMENTA (Theory of an Optimum Ex- -
periment)~ Moscow, 1971. _
~ 13. Kfiudson, D., STATISTIKA DLYA FIZIKOV (Statistics for ~'hysicists), Moscow, Mir,
- 1970.
29 ~
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UDC 551.513.1
MEAN ANNUAL ZONAL ATMOSPHERIC CIRCULATION
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 30-35
~ [Article by Candidate of Physical and Mathematical Sciences N. S. Sidorenkov, USSR
Hydrometeorological Scientific Research Center, submitted for publ3.~cation 9 Jan 80j
[TextJ Abstract: A semi-empirical differential equa-
tion is derived for the distribution of the ab-
_ solute moment of momentimn. It is shown that the
change in the moment of m~ment~m in the atmo-
sphere is determined by the balance of two flux- .
- es of the moment of momentum: horizontal, a~ij~
ifi~ as a result of macroturbulent exchange and
vertical, associated with microturbulent viscos-
ity. Its analytical solution is obtained in the
case of a very simple model of change in the co- .
efficients of turbulent exchange. It is shown
that the mean annual zonal circulation arises as
a result of the redistribution of the absolute
moment of momentimm of air between the low and
- high latitudes under the ir.f~upnce of macrotur- -
bulent mixing. Macroturbulent vl.scosity weakens
zonal circulation. Its stationary state is at-
tained only after accumulation of a definite
nositive moment of momentum in the atmosphere,
which is transferred from the earth to the at-
mosphere.
It is well known that as an average for the year in the low latitudes there is a
- prevalence of winds having a velocity component from east to west, whereas in the
- temperate and high latitudes there is a prevalence of west-east transpo rt. A
change in the sign of the zonal wind velocity component is observed at the ear th's
surface in the so-called "horse latitudes" (near f35�). Zones 6f calms are situat-
ed here. During recent decades it has become clear that variations in the intensity
of zonal circulation of the atmosphere are associated with changes in the earth's
rate of rotation [5, 6]. An intensification of zonal circulation of the atmosphere
_ occurs as a resu~t of influx of the moment of momentwn from the earth, whereas
weakening occurs as 3 result of its transfer toward the earth. It was a lso estab-
lished that the moment of moment~mn of zonal winds is not equal to zero and as an
average for the year is ab~ut 13�1025 kg�m2�sec'1 [6]. `
- 3~
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Modern numerical models of zonal circulation of the atmosphere, applied on an
- electronic computer, reproduce the pattern of zonal circulation of the atmosphere
well [3]. However, they did not lead to a significant deepening of our ideas con- .
- cerning the physics of the processes forming and supporting zonal circulation. The
results of the numerical models are sets of numbers which differ little from the
data from meteorological observations. We feel that the simplest qualitative theory
_ makes possible a deeper understanding of the physical essence of the phenomena
than even the most inventive of the existing numericai models. The purpose of this
article is the proposal of such a theory.
We will select a fixed Cartesian coordinate system with its origin at the center
of the earth's mass and with axes oriented relative to "fixed" stars. We will di-
rect one of the axes along the earth's axis of rotation and the other two will be
laid out ir~ the equatorial plane.
We will write the equation of motion of a unit volume of air for this inertial ref-
erence svstem: ~
. fM-~ P
~ dd ~ - - ~ P - R R F, - ~1~
where Va x R] + V~ is the velocity of absolute motion, Vp is wind velocity,
: ~is the angular velocity of the earth's rotation, R is the geocentric radius-
= vector of the considered volume,f~is air density, f is the gravitational constant,
M~ is the earth's mass, a P is the gradierit of atmospheric pressure P, F is
- frictional force, related to a unit volume, t is time.
We will multiply each term in equation (1) vectorally from the left by the geocen-
tric radius-vector R and we will take into account that in the free atmosphere the
frictional force is negligible. Then we obtain an equation for the absolute moment
of momentum of a unit volume of air in the atmosphere
J
RXada' =0 di [R,'~~s1=[cPXR1� ~2~
The moment of terrestrial gravitation is equal to zero since this force is direct-
ed along the radius R. Projecting the vector equation (2) onto the earth's axis of
rotation, we obtain
� d! - dP
p ~r - a
~ ~ ~3~
where S~R2 sin2 e+ uR sin ( SZ+oc)R2 sin2 ~ is the projection of the abso-
lute moment of momentum of a unit air mass onto the axis of the earth's rotation,
- u= OCR sin e is the velocity of the zonal wind (positive direction to the east),
p[. is the angular velocity of zonal air motion relative to the earth's surface,
- ~ is the polar angle oz� colatitude ~ t~ Tt/2, a is langitude, reckoned to the east.
We note that the projections of the absolute moment of momentum onto the axes situ-
_ ated in the equatorial plane are negligible in comparison ~vith the ~ value since
the deviation of the instantaneous vector ~ from the mean long-term value does not
exceed 10-6 radian [7], and VD ~ p~.
The left-hand side of equation (3) by means of the continuity equation can be rep-
resented in the form
31
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d! dpl dp' dp! da
_ P ar - dt -1 at - dt P 1 V, - l~ at -i-
a?t (4)
-~Pv'V.)= dc 1'a'P1V,.
_ We will represent the divergence ~'�P~-Va of the flux of the projection of the ab- -
solute moment of momentum in a spherical coordiriate system and substitute (4) into
equation (3). Then we obtain
adP ~ lRzolz~R)- Rs~ne ae (sind;olve)- (5)
_ 1 d ~P IvA _ dP
k sla 9 d'A ~ d;~ '
Since we are interested in the zonal circulation of air in the atmosphere, ::e =.~ill
zverage equation (S) in longitude -~re will integrate all its terms for a from 0
to 2?~. We take into account that the integ~als of the last two terms are equal to
zero, since it is possible to neglect the jumps of pressure and velocity u at moun-
tain ranges. Introducing the traditional notations _
X - xd;., -
~
we obtain d o l_ 1 d _
ar - R= aR [R2 ~P 1vR -E- P ~~2~R) , -
(6)
R sin ci e(sin 9(P[ve + p 1've)~�
In equation (6) terms of the type pr[vi and P~,'v'i reflect transfer of the projec-
tion of the absolute moment of momentinn by ordered circulation and eddies (tur-
bulence) respectively. Investigations of the components of the balance of moment of
momentum in the atmosphere, whose results are s~nnmarized in the monographs [4, 8],
indicated that the transfer of the mAment of mamentum is accomplished for the most
part by turbulence. The role of ordered circulation in the atmosphere is relative-
ly small. On the basis of these empirical data terms of the type P~,vican be
' neglected in comparison with the terms ~~,'v' i. -
As is well known, in physics extensive use is made of intuitive phenomenological
- expressions relating the rluxes of different physical parameters and the gradients
- of these parameters. In particular, in the semi-empirical theory of turbulence the
fluxes of momentum described by Reynolds stresses are assumed to be proportional
to the gradient of inean wind velocity. Adh~ring to these hypotheses, the fluxes of _
- the moment of momentum in the vertical and meridional directions can be expressed
by the following approximate equations:
~.!'vRZ-AR_aR plve-.,-AeRd~d~ _
where AR and A e are the coefficients of vertical and meridional exchange respec-
t ively . -
32
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Substituting (7) into (6), we obtain a second-degree differential equation in par-
tial der~vatives for the projection of the absolute moment of momentum onto the
earth's axis of rotation
d c l ! d d[ 1 d d!
dr - R= dR (f~ AR ok R-~ sin v
M(sin 9 Ad (8) -
Equation (8) shows that the rate of change in the projection of the absolute moment
of momentum in a unit volume is determined by the sum of the ~onvergences of the _
~ moment of momentum, or, w~at is the same, the moments of the forces arising due to
microturbulent (first term) and macroturbulent (second term on the right-hand side)
- transfers of the moment of momentum. It makes it possible to compute the zonal cir- -
culation of the atmosphere because the distri~ution of the ~ value in an atmo-
sphere rotating together with the earth is known.
Now we will attempt to find a solution of equation (8). For this we will select
the boundary conditions and make simplifications. An obvious boundary condition is -
the condition of "attachment" of the air to the earth`s surface
a- 0, 1= s Ro sin= 9 when R= R0. (9)
Here R~ is the earth's radius. A less obvious condition is the condition at the
upper boundary of the atmosphere. We will assiune that there (when R= R~) the veri
tical flux of the moment of moment ~n tends to zero and therefore
aa - U when (10)
u
x~' uR - R= R~ �
We will.:examine a stationary (not dependent or. tjme) zonal circulation of the at-
- mosphere. In this case ap~~d t N o an.d equation (8) can be written in the form of -
an equality of the moments of forces operative in a unit volume due to the vertical -
and meridional transfers of the moment of momentum in the atmosphere respectively:
d I A R= ar \ ~ d (i sin H a` ~
aR R oR ~ sinb dH l,' a dH,~ (11)
The coefficients of iner.idional and vertical exchange are essentially dependent on
latitude and altitude. In the first very rough approximation it can be assumed
that
. A9 = const, (12)
AR = AZ sin 9, -
(r3)
where AZ = const. It is impossible to assume the coefficient AR to be independent
- of latitude because, as indicated by the theory of the atmospheric boundary layer,
it is proportional to wind velocity u= c~ R sin Q.
Substituting into (11) the exchange coefficients r~e and AR from (12) and (13) and
taking into account that
R= dR R~ sinz 9-}- 2(Q + a} R sin'- A.; dR R= sin' 9,
_ d
e=(~' + 7) R' sin^~ H-}- ~e K~ sin= 9;. ~ R~ sin 2 A,
33 u
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we obtain _ _ .
d ~ d s_- 6 4 '~e 213 -~in= A Rz� (14)
dR d1t ~ AZ sin b -
We will denote the constant value
~ Q as - G~ -
and the latitude function - -
2(3 - sin2 H
sia~H - ~ (fi).
Integrating equation (14), we have
a _ - G ~ la) In R-C, ~g~ .3 C..~y~ (15) _
The unknown functions of latitude C1(~) and C2(~) are easily found from condi- -
tions (12) and (13) respectively:
- _ - (16) -
C~ ~e) = G ~ (e) RB~,, -
_ C, (6) - G ~ (9) (tn Ro + ~Ro 1. (17~
`
After the substitution of these functions into expression (15) we obtain a final
expression for the relative angular velocity of rotation of the atmosphere
z=G~(H?flR)-2 ~ A:. 2_~-sin=H -.t _In Rl ~18~
AZ ~in~ H ~ 3\ Rp k~ ~ ku J�
Formula (18) shows that as an average for the year the relative angular velocity
in both hemispheres is negative at latitudes below 35� and positive at latitudes
above 35�. A change in sign occurs at latitudes 35�N and 35�S. With altitude the
velocity increases in conformity to a complex law.
The negativeOC values indicate that the atmosphere lags in its rotation behind the
ear tn, whereas positive oC values indicate that it outpaces the earth. The earth
rot ates from west to east. Accordingly, in the first case (oC< 0) there is an
= easterl~ wiand, whereas in the second (oC~ 0) there is a westerly wind.
Thus, the resulting solution (18), despite the exceeding roughness of the assump-
tions (12)-(13), satisfactorily describ~es the latitudinal distribution of zonal _
winds in the atmospheric surface layer. The circumstance that the angular velocity
OG at the poles tends to infinity is ~aithout fundamental significance since there
- the R sin B value tends to zero.
Plow we will check to see whether the derived solution (18) satisfies the observed
fact that the moment of moment~rm of the zonal winds is not equal to zero but as
an average for tne year is about 13�1025 kg�m2�sec-1. For this purpose we will es-
timate the moment of momentum of the determined zonal circulation of the atm~-
sph ere (18). In the first approximation we will consider a homogeneous atmosphere
with a constant density P= P p= 1.29 kg�m 3, so that its height is equal to
~ ~ R~- RQ N 8 km. -
34
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R~ r.
h= ~ xR'sin=9~d~-�?-GPo k k'f(R> dRf ~(6)sin�bdd~
Q' V "
V
Ar r0 "V~~~U~' -1~ ~ llJ-~ Ap kg~TQ~�sec 1�
Here it is assumed that R= 6.37�106 m; 7.29�10-5 sec'1; W is the volume of ~
the atmosphere. According to empirical dat4, ~he values of the coefficients of
horizontal macroturbulent exchange fall in the range 106-10~ kg�.m 1�sec 1, whereas
the vaiue~ the coefficients of microturbulent exchange fall in the range 1-10
kg�m 1�sec-1 [1]. Therefore the ratio A e/AZ ~106, and accordingly, the moment of
momentum is h~ 17�102~ kg�m2�sec'1 or approximately 140 times greater than the ob-
served value. Evidently, this noncorrespondence is obtained due to the fact that
_ the values of the coefficients of microturbulent exchange AZ used in meteorology
do not coincide with their values for the entire atmosphere (because they are
computed, as a rule, on the basis of local, rather than planetary data). We note
- that a similar conclusion was drawn by N. Ye. Kochin and his students. In comput-
ing zonal circulation on the basis of the stipulated temperature distribution in -
the atmosphere they obtained plausible results only with exchange coefficients AZ
of the order of magnitude 103 kg�~n 1�sec 1[2J.
' Thus, the above-cited rough se~i-empirical theory describes well the principal ~
peculiarities of the mean annual zonal circulation of the atmosphere: presence
of zones of easterly winds in the low latitudes, westerly winds in the temperate
_ and high latitudes, zones of calms near latitu des f3.5�, and presence in the atmo-
sphere of a constant positive value of the unoment of momentum of zonal windso The
= physical essence of the processes transpiring in the atmosphere is also clearo
Zonal circulation of the atmosphere arises and is maintained under the influence of
meridional and vertical fluxes of the absolute moment of moment~, which are cau~ed -
by macroturbule:.t anc~ microturbulent mixing of the atmosphere respectively.
Macroturbulence, caused first and foremost by the equator-pole temperature con- -
- trast, transfers the moment of momentum in the direction of a decrease in the
value, that is, from the earth's equator to the poles. As a result, in the low lat-
itudes the moment of momentum decreases (easterly winds develop), whereas in the
- temperate latitudes the moment of momentum inc reases (westerly winds appear). At
_ latitudes t35� the k, value is equal to the average value for the entire atmosphere.
AccordinglyD here at the time of mixing the moment of momentum does not change
and no wind arises. -
With appearance of winds the forces of microturbulent friction of the air against
the earth's surface arise, that is, vertical fluxes of the moment of moment~
arise. In zones of easterly winds the moment of momentum flows from the earth to
the atmosphere, whereas in zones of westerly winds, vice versa, from the atmosphere
- to the earth. The vertical flux of the moment of momentum weakens the wind. The
greater the wind velocity, the greater is the vertical flux and the greater is
the weakening of the wind. The weakening of wind velocity continues until the ver-
tical microturbulent flux of the m:,ment of momentum comes into equilibrium with
= the horizontal macroturbulent flux in the direction of the meridian. Only then
does a stationary state set in and will the wind velocity not change with time.
The geome~ry of the zones of influx and loss of the moment of momentum in the atmo-
- sphere is different. As a result, the statianary state of zonal circulation is
35
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= attained only after accumulation of a definite positive moraent of momentum in -
the atmosphere, which is transferred from the earth to the dtmosphere~ -
BIBLIOGRAPHY -
1. Gandin, L. S., Laykhtman, D. L., Matveyev, L. T., Yudin, M. I., OSNOVY DINAM-
ICHESKOY METEOROLOGII (Principles of Dynamic Meteorology), Leningrad, Gidro-
meteoizdat, 1955.
2. DINAMICHESKAYA METEOROLOGIYA. CHAST' II (Dynamic Meteorology. Part II), edited
by B. I. Izvekov and N. Ye. Kochin, Moscow-Leningrad, Gidrometeoizdat, 1937. -
3. D_ymnikov, V. P., Perov, V. L., Lykosov, V. N., "Hydrodynamic Zonal Model of
General Circulation of the Atmosphere,'~ IZV. AN SSSR. FIZIKA ATMOSFERY I
OKEANA (News of the USSR Academy of Sciences, Physics of the Atmosphere and
Ocean), Vol 15, No S, 1979.
4. Lorentz; E. N., PRIRODA I TEORIYA OBSHCH~Y TSIRKULYATSII ATMOSFERY (Nature and
~ the Theory of General Circulation of the Atmosphere), Leningrad, Gidrometeo- _
izdat, 1970.
W 5. Munk, W., MacDonald, G., VRASHCHENIYE ZEMLI (The Earth's Rotation), Moscow,
_ Mir, 1964. -
6. Sidorenkov, N. S., "Investigation of the Atmospheric Moment of Momentum," IZV. -
AN SSSR, FIZIKA ATMOSFERY I OKEANA, Vol 12, No 6, 1976.
7. Sidorenkov, N. S., "Tensor of Atmospheric Inertia, Annual Changes in its Com-
ponents and Variations of the Earth's Rotation," IZV. AN SSSR, FIZIKA ATMOSFERY
I OKEANA, Vol 9, No 4, 1973.
= 8. Starr, V., FIZIKA YAVLENIY S OTRITSATEL~NOY VYAZKOST'YU (Physics of Phenomena
With Negative Viscosity), Moscow, Mir, 1971.
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UDC 534.221.1+551.(508.85+524+55)
DETERMINATION OF TiIIND VELOCITY AND DIRECTION AND TII~ERATURE IN THE ATMOSPHERIC
SURFACE LAYER BY THE RADIOACOUSTIC SOUNDING METHOD
~ Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 36-45
u [Article by S. I. Babkin, G. N. Miloserdova, Candidate of Physical and Mathematical
Sciences M. Yu. Orlov, Yu. I. Pakhomov, Candidate of Technical Sciences Ye. G. Pro-
shkin, Yu. N. U1'yanov and B. S. Yurchak, Institute of Experimental Meteorology,
submir.ted for publication 4 Mar 80]
[Ter,t] Abstract: A study was made of the influence ex-
- erted on the accuracy of ineasurements of temper-
ature and wind velocity in the atmospheric surface
layer which are sense3 by the radioacoustic sound-
_ ing (RAS) method, as well as the accuracy of the
speed of sound, angular coordinates and air humid-
- ity. A comparison is made of the results of ineas ure-
ments of the temperature and wind velocity pro-
files by the RAS method with similar data obtained
simultaneo~_~sly with sensors mounted on the high
meteorological mast of the Institute of Experimen-
tal Meteorology. The results obtained by the two
methods coincide within the limits of error. The
possibility of using a priori information on atmo-
spheric parameters in making measurements by the
RAS method is investigat~d. It ia demonstrated that
the use of such ir~formation makes possible an ap-
= preciable improvement in the accuracy of the re-
y sults. Recommendations are given on the choice of
- the directions for multiple sounding which are best
= from the point of view of the accuracy of the re-
sults.
- In~roduction. T'he radioacoustic sounding (RAS) method is based on measurement of
` the velocity of propagation of a sound pulse in the air by a Doppler radar [7].
The spee~i of sound in the air is
c - a Vr, ~1~
37
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where T is absolute temperature, Vr ie the pro~ection of the velocity of motion of
air (wind) on the direction of sound propagation, a is a coefficient dependent on
_ the gas composition. For dry air a= 20.0528 m�sec'1�K 1~2.
When measuring the speed of ~ound in the atmosphere in not less than four different
directions by means of the RAS method there is a possibility of simultaneous deter- _
mination of the pr.ofiles of temperature (T) and all wind components (Vx, Vy, VZ)
[4, 5, 8].
, The RAS method can be ~ised in the followin~ variants: -
- vertical sounding for measuring the temperature profile [7];
sounding in four directions for simultaneous determination of the profiles of
~ temperature and the wind components [8];
_ sounding in several different directions (n ~ 4) for determination of the profiles
of r.emperature and the wind components by the least squares method.
Vertical Sounding
In vertical sounding for determining the temperature profile the accuracy of the
_ temperature value will be determined using the formula
T ~~~i~ 1~~~= l v~T e vs1 =
~T='? a 1~ (~n)z ~ T , r (~0) � (2)
Here ~ a is the accuracy in determining the proportionality factor in the Laplace _
- formula (see above); L~ c is the accuracy in determining the speed of sound; L~vZ
is the accuracy in determining the vertical wind component; VX and ~ VX are the
horizontal wind velocity and the error in its determination.
The assumption that the vertical wind component is equal to zero is introduced in
the case of vertical sounding. Thus, the uncertainty in (~VZ is a measure of the
accuracy with which this assumption is satisfied. _
The fourth term in the radicand of formula (2) determines the contribution to the
accuracy in measuring temperature from the accuracy in determining the horizontal
- component of wind velocity and the accuracy in determining the angle between
the vertical and the direction of sounding. It will be asstuned that sin a S=~.
The coefficient a for dry air is known with a ve r~ high accuracy. If it is assumed
that relative humidity in the lower layer of the atm~sphere is equal to (75f25)% _
(as is satisfied in most cases in the middle zone of the USSR), the a value can be
assumed equal to 20.lOfG.02 [4].
The accuracy in measuring the speed of sound with a Doppler radar is assumed to be
equal to f0.1 m/sec.[8]. If the vertical wind component is known with an accuracy
to ~0.3 m/sec, the accuracy in determining temperature in a vertical sounding re-
- gime will be t0.8�C. If the uncertainty in the vertical wind component (according .
to data from independent meas urements) is equal to zero, the accuracy in measuring
temperature is f0.65�C. ~
- The presence of vertical or horizontal wind velocity at the time of sounding leads
to a systematic error in estimating temperature. Computations using formula (1)
- show that with a deviation of the direction of sounding from the vertical by fl�,
38
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= a= 20.10, T= 273 K
dT - 1.6 K d, -
dvj = , (m/sec) and o~s = - O,U�~8i K~ (m/sec), -
that is, the presence of ari ascending flow with a velocity of about 10 cm/sec or
a moderate horizontal wi.nd with a velocity 7 m/sec leads to an exaggeration o~ the
, measured temperature by approximately 0.2 K. The horizontal wind will exert an in- -
fluence on the measured temperature value only in a case when the sounding direc~
tion differs from the vertical.
Strictly speaking, such a situation exists only when the radar antenna has a very
narrow directional diagram. For real directional diagrams, even in the case of r~g-
orously vertical sounding, the presence of a horizontal wind will exert an influ- -
ence on the accuracy in measuring temperature. However, an examination of this fac-
tory dependent on design of the apparatus9 is beyond the scope of this article.
Thus, vertical sounding makes it possible to determine the temperature profile but
requires the use of additional (a pr3ori) information for increasing the accuracy of
the results or for introducing corrections.
An improvement in the accuracy of the temperature value can be achieved, in partic-
ular,with a decrease in the uncertainty of the a coefficient by means of use of ad-
ditional information on air humidity in the place where the temperature is measured. -
A decrease in the uncertainty of the a value makes possible a doubling of the ac-
_ curacy of the temperature value (to f0.45�C). Further improvement in accuracy re-
- quires an improvement in the accuracy in measuring the speed of sound and a de-
czease in the uncertainties of the a and Vx values.
Sounding in Four Directions
Formulation of problem. In more detailed fo nn formula (1) for the speed of sound
propagation in the direction with the ntmmber i will read as follows:
c; - cr i~-~ tt; VX vl V,, 2e'; Vz. ~ 1~)
where ui, vi, wi are the direction cosines of the vector of the direction i.
In carrying out sounding in directions indicate~ in Fig. 1 the system of four equa-
tions for determir~ing the four unknowns (x? ~ T; x2 = VX; x3 = V~; xi = V~ in
_ matrix form is written as follows:
C = ~s X. (3)
Here M is the matrix of the system of equations and
a sin cos a, sin sin a, cos
1I - a sin o, cos a: sin o2 sin a.. cos o= (4)
I a sin ~3 cos :~a sin o3 sin a, cos a,
I~ a sin cos a~ sin sin cos
i
39
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z "
~
~
_ ~ _
~ - -
.
o ~ /
` ~ o -
h ~ -
\ ~
_ ~ ~
6~ i~,
~ -
, `
~ d.s CZ t
~ Q - X
/
- Fi~. 1. Directions of sounding with simultaneous determination of temperature and
wind components (four soundings). -
The angles S i are the angles between the direction of sounding and the vertical
- (z-axis), the angles OCi are the angles between the pro~ections of the sounding
directions onto the (x, y) plane and the positive direction of the x-axis. -
- Similar expressions can be written as well for a case when the sound pulse irradi- -
ated by the radar is propagated vertically along the z-axis and the electromagnetic
radiation reflected from it is received by four separate antennas with axes inclin-
ed at some angles relative to the z-axis. When making soundings in several direc- _
tions the resulting values for temperature and wind direction are averaged for the
area of the base of a cone with the radius H t~ and the ~easurements in indi- -
- vidual directions are not made simultaneously. However, with the propagation of a
sound pulse in a vertical direction and reception of the reflected radi.ation in
four directions the me~~srements give the instantaneous values of temperature and
the wind components at one point at a given altitude.
The parameters whose errors in determination will exert an influence on the accur- _
acy of determination of temperature and the wind components will be
~J (1= j - ~;--2~ 3. 4) .
Sk (i - 5, 6, 7, 8; 2, 3, 4)
P~ ~ a~ = 9, 10, 1 l, 12,; 1- 1, 2, 3, 4)
a (i = 13)
The soZution of equations (3) is wri.tten in the form -
X~ X-~ C _ (5~
The accuracies of the values of components of the X vector can be determined from
the diagonal ter~s of the covariation matr__ix _
~ _
D t~ _ dX D ~p~ ~6)
- dF dP
Here dX/a p is the matrix of derivatives of the values of the components of ~the '
X vector for the p parameters; D(p) is the covariation matrix of the p parameters;
the symbol N denotes transposition.
40
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The matrix d X/a p was obtained numerically, by solution of system (3) with the
- introduction, in turn, of small increments Q,pi~cpi in the values of all the pi
- parameters. In this case no assumption is made conceming the smallness of the -
changes in elements of the inverse matrix I~1 with stoall changes of the a pi para-
metera, such as adopted in obtaining the resulta in [4].
The errors in determining the values for temperature and the wind components can
be computed using th2 formulas
~T-2x,~x,=2~Td�(X),
� -
~ ~
~ ~
~ ~X = v d_z t�t'): ~ V~ = d�>~ (X): ~ V: _~dss fX)�
In a case when in the measurements it is necessary to obtain the wind modulus and _
its dir.ection, other than the wind components,
I V~ z~ x3: 3= arctg z, ~g~ -
The e~rors in determining IVi and ~ are easily computed from the known values of
th-e~ errors in the wind components (with allowance for the nondiagonal terms of the
D(x1 matrix)�
� Choice of values of parameters and their uncertainty. Computations of the values
~ of the errors were made with some typical values of temperature and the wind com-
- ponents. It was assumed that the temperature at the sounding point was 273 K; the
wind components were Vx = Vy = 10 m/sec and VZ = Ool m/sec.
ev, � e~, m/sec
o:a ,
- , 5; b ~
' .dT'L dV= H/c , �
~S Q7~ zar
u~ 8~ ~ I t)
i
~
zc � ~ r GS
~ ' ~ _
~ i, i ~
4S0 n. ~ S , 0 4~s o. ec Mlc m/sec
Fig. 2. Dependence of accuracy in determini~g temperature (a), horizontal wind com-
ponents (b) and vertical wind component (cj on accurac}~ in measuring speed of sound
(four soundings).
On the basis of available data on uncertainties of the parameters used in the meas-
� urements (coefficient, speed of sound, angles in the horizon*al and vertical planes)
the principal computations were made with the following values of the uncertainties: _
Qa ==~.02 m�sec-1 K'1~~ [4); psi = a~'i = t0.001745 (1�); Bc = tO.I m/sec.
41
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'I'he value of the uncertainty ~ a was discusaed above. The accuracy in determin-
_ ing the angular coordinates is ehsured by modern instrumentation. The components -
of the uncertainty c are dependent on the instr~ments and apparatus used in the
method and therefore a detailed analysis of their values is not given here. We
will only point out that for a RAS system constructed at the Khar'kov Institute
of Radioelectronics [2], the principal components 3f uncertainty in the measured
- speed of sound value are caused by the following phenomena or errors:
_ a) the error with which the velocity of propagation of electromagnetic waves in
the atmosphere is known; _
b) the short- and long-term frequency stability of the Doppler radar transmitter;
c) the error in measuring the Doppler frequency, which in turn is dependent on the
frequency distortions of the echo signal in its processing by the receiver and -
tracking filter, the errors in measuring the Doppler frequency and the width of
the Doppler spectrum of echo signal frequencies.
Results of conpu~azions ar.d experiments. The contributions of individual components
of errors to the total uncertainties of the measured values were evaluated. _
The dependence of the total uncertainties of temperature and the wind components _
on the accuracy in measuring the speed of sound is given in Fig. 2. Our computa-
tions also made it possible to select the optimum (for obtaining the most precise
results) combinations of sounding directions. ?'he best accuracy for temperature _
T=�1.4 K) and the vertical wind componentc (~VZ =�0.35 m/sec) is obtained with
~ with a~combination of the angles b L= ~ g= 30�; b 2= ~ 4= 10�. The components
. VX and Vy are dete=mined with an accuracy to f0.3 m/sec. In this case the points
- for which the speeds of soimd are determined at a particular altitude are situated
on an ellipse with the ratio of the semimajor and semiminor axes equal to al/a2 =
2a8; the semimajor axis of this ellipse is oriented along the vector of the hori-
zontal wind.
With the combination of angles ~ 1= ~ 2= ~ 3= 30�; ~ 4= 0�, which in routine
_ work is ensured far more simply, the error in determining temperature in this case
increases by 6% (to f1.49 K) and the e nor L~VZ by 10~ (to f0.92 m/sec). There
is also a 50% improvement in the accuracy in determining the horizontai wind com- _
- ponents (to f0.2 m/sec).
Thus, it is possible to recommend the following sequence for carrying out simultan- .
eous measurements by the RAS method when using four soundings ensuring the best
accuracy of the results:
l. Preliminary measurements with an arbitrary orientation of the directions of sound-
- ing relative to the direction of the horizontal wind.
- 2. On the basis of the results of these measurements, the choice of th~e orientation
- of directions ensuring rhe best accuracy of the results. The principal measurements
are also made with this orientation. -
The dependence of the accuracy in determining temperature on the choice of saunding
angles is shown in Fig. 3.
- It should be noted that a case when the sounding directions coincide with the gen-
eratrices of a circular cone is exceedingly unfavorahle with respect to obtaining
a high accuracy of the results, specifically, when the axis of the cone is inclined
- by a small angle to the vertical. _
~ 42
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- e~t~- -
_ J
2

- ~ degrees ~
~ ?0 d zp~d
Fig. 3. Dependenee ~f the accuracy in determining temperature on the value of the
angle b 4 in the vertical plane (,0 c= t0.1 m/sec; a 1= S2 = a3 = 30�; four
- soundings).
- ~.4~ 1 ~ ~ ~
i~. ~ Il
~~~i !
/ 1
~s~ , , k ,
_ ~a , ~ ~ _
i: ~ ~
; i , ,
_ ~4 r ,
- ~u; , F ?
ee , ~ i -
~s j ~ ; _
~t '
_ ~ i ~
~ ~ ` ~
13 j V m/se~ ~
~ l ~
~03 S 7 r�n r c~c ea v
- Fig. 4. Comparison of results of determination of te~;y~ra~ure, modulus and direc-
tion of wind by the RAS method (1) and sensors on high meteorological mast (2).
- The results of ineasurements of temperature and the horizontal wind components in
the case of multiple sounding are given in Fig. 4. In this same figure, as a com- _
parison, we show the profiles of temperature and wiad obtained during measurements
by the RAS method using the set of sensors on the high meteorological mast (FIMM) of
the Institute of Experimental Meteorology. _
= The measurements were made using radioacoustic sounding apparatus developed at the -
- Khar'kov Institute of Radioelectronics. The operating principle for this instrumen-
tation and the principal technical specifications were described in [2].
- As already noted in [4J, the presence of correlations of errors of the parameters -
- exerts considerahle influence on the results. An analysis of the structure of ~
errors in measuring the speed of sound indicated that the correlation coefficients ~
- between the errors are not e~~ual to zero because the individual components of errors
= for measurements in different directions are the same. A similar situation also ex-
ists for error~ in deter~ining angles in the vertical plane and the angles in the
horizontal plane, that is
- v c; ~ c~ ! r > Qoj ~o~ ~ ; j ~ ~s~ ,z~ ~ ~ -i >
~ CJ = fl G~ 9a~ _~3~ ;~a1 =
~Cp ~Q ~~Gt Q~Q J~tcO.
The evaluations show that with correlation coefficients P t~i~~) = p( S S�) _
p to~i a~) = 0.5 the accuracy of the results is apprec~ably better (by a~ac~or of
approximately 1.5).
~ 43
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. Multiple Sounding
pne of the possible modifications of the multiple sounding method is.sounding in
~pVar~l (n> 4) directions. In this case the method for processing the results will
be similar to that which is used in the Manning and Greenhow quasi-multiray iue~hods
[lJ for determining the wind in the upper atmosphere from the movement of ineteor _
trails.
In this case the matrir, of the system of equations for multiple soundin~ has the
form
,
- i � R R
- ~ ira= a~ u; a~ v; a~ w; ~
;=t ~=t
n n ~ ~
~ 11i ~ il~ ll~ Sl~ ~ 11~7L'~ ~8~ ~
~i - ~ i-1 i-1 t=1 i~l
- - ~ n n A ~ I
a~ 2'i ~ vI ui ~ v? ~ v~ ~~i .
ia 1 icl i= l ial
n ll ~R+ ~
~ ~T I.r ~~i /r ~1 f Ir~ 2~~ `
I i~l i_1 1~1 i-1 ,
and the column of free terms will consist of the following values:
f `i ~ c; -
- ~
I ~i
~
~ ui I ~8~~ ~ -
S=I R
1 ` ct ~j -
F- 1
i
~ 1
~ ~ C[ ~l
I~ ~
= Use of A Priori Infor-u,a~ion on Temperature and Wind Values
The examination made above was made on the assumption that when making measuremerits
of temperature and wind there are no data concerning them, or at least, a priori
information was taken into account in implicit form, as in the case of vertical .
sounding for determining temperature.
However, the experience accumulated in meteorology makes it possible, prior to the _
onset of ineasurements, to obtair_ some information concerning the temperature and -
wind at different altitudes (for example, see [3]).
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~ Now we will examine the possibility of the use of a priori information in deter-
mining temperat~re and the wind components if the RAS method is used in measuring
k values ck(T, V) (speed of sound). One of the possibilities of such use is asso-
ciated with an approach based on the Bayes theorem and set forth in [6).
Assume that prior to the onset of ineasurements of temperature and wind velocity at
- some altitude we know their a priori values: tl t2, t3, t4 the wind com-
ponents, VX, Vy, VZ; t~(j = 5., 6, 7, 8) the angles of the directions of sounding
relative to the vertical; t~ 9, 10, 11, 12) the angles between the direc-
tion of sounding and the x-axis in the horizontal plane; t13 - the value of the a
coefficient. We will also assume that the a priori covariation nata-ix R of these
- values is known.We will make measurements of k values of the speed of sound ck and
determine their covariation matrix V.
Then, using the Bayes theorem, the assumption of a Gaussian distribution of a11 the
parameters and the measured values and a linear representation of changes in the
ck values in dependence on the parameters ti in some region of change in the para-
meters
. _ _ - _ . . _
~a(t,~=~ct1?~-E ga, ~r;-r;),
% (9)
we will find the corrections to the a priori values of the parameters and the a
posteriori matrix of errors of the parameters R' [6]:
. lR-' + gV-'g1(~~- ~ = gV-~ (C-C'). ~10~
R' = R- Rg (gRg V)-' gR. (11)
~ It follows from the positive determinancy of the R and V matrices that with satis-
faction of procedure (11) the diagonal elements of the a posteriori covariation
- matrix R' (a posteriori dispersions ti) will be less than the mentioned diagonal
elements of the a priori covariation matrix (a pr.iori dispersions ti). Thus, there
is a refinement of the a priori values of the ti parameters.
- The system of equations (10) makes it possible to determine the corrections to the
a pri_ori values of the parameters ti, taking into account the information obtained
in the experiment.
As an illustration of the possibilities of the RAS method with the use of a priori
information, Fig. 5 shows the t~mperature profile determined by such a method in
the case of vertical sounding (curve 3). As a comparison, in this same figure we
show the temperature profiles obtai.ned using I~i sensors (curve 1) and by means of
the traditional RAS method (curve 2). The estimated uncertainties in temperature
values are also cited.
In obtaining the temperature profile, with a priori information taken into account,
- it was assumed that prior to the onset of ineasurements we knew the relative humid-
ity value at tne ground level with an accuracy to f10%, the temperature at the
ground level with an accuracy to +3�C and the vertical wind component at the ground
level with an accuracy to f0.3 m/sec. It was also assumed that in the lower 300-m
layer of the atmosphere the relative humidiry changes with altitude by not more
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than 10%, the vertical wind com}~onent does not change with altitude, the temper- _
_ ature decreases linearly with altitude and its gradients can be taken from the -
the data in [3] for the corresponding meteorological conditions. The accuracies
in the a priori values of the a coefficient, temperature T at an altitude of 150
m and the vertical wind VZ are 0.005 m�sec'1/IC-1~2, 1�C and 0.3 m/sec. _
H.M
~
~
_ ~ -
/ q
1 \
\ o
~
~ o
20~ ; 1
~ ~
~ 1
_ 3 2~ ~
- I1 �
\
\
\
1017 \
\ �
\ -
~
o \ -
~ ~
0
- r_
- ~ . - -
- ~7 .E -S 7'G
Fig. S. Comparison of temperature profile measured with HhIIi [high meteorological
mast] sensors (1), by the RAS nethod in vertical sounding (2) and by RAS. method
with the use of a priori information on humidity, temperature and vertical wind
component (3). -
, The accuracy in measuring the speed of sound was assumed equal to 0.3 m/sec. The
data cited in Fig. 5 show that the temperature values obtained using the sensors
_ on the high meteorological mast, the RAS method and the RAS method with the use
of a priori information coincide with one another in the limits of their estimated
uncertainties. Computations on the basis of formula ~Z) on the assumption of ab-
sence of a priori information gives for the temperature values obtained by the RAS _
- method an uncertainty of about f0.9�C (see above). Thus, allowance for a priori in-
formation wi~h an insignificant complicating of the processing procedure makes pos-
sible an appreciabl.e improvement in the accuracy of the results without expending
_ efforts on the improvement of instrumentation for improving the accuracy in meas-
urement of the speed of sound.
~
BIBLIOGRAPHY -
- 1. Babadzhanov, P. B., et al., RADIOMETEORNYYE ISSLEDOVANIYA TSIRKULYATSII VERKH--
- NE~' ATMOSFERY (Radiometeor Znvestigations of Circulation of the Upper Atmo-
sphere), Dushanbe, Donish, 1974.
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~
2. Babkin, S. I., et al., "System for Radioacoustic Sounding of the Atmosphere
in the Centimeter Wavelength Range," TEZISY DOKLADOV NA V VSESOYUZNOM SIMPOZ-
IUME PO LAZERNOMU T AKUSTICHESKOMIJ ZONAIROVANi~I7 ATMOSFERY (Su~naries of Re-
ports at the Fifth All-Union Symposium on Laser and Acoustic Sounding of the
Atmosphere), Part 3, Tomsk, 1978.
- 3. Mashkova, G. B., "Pecul3arities of Stratification of Air Temperature in the Low-
er 300 m of the Atmosphere," TRUDY IEM (Transactions of the Institute of Exper-
imental Meteorology), No 10(53), 1975.
4. Orlov, M. Yu., Yurchak, B. S., "Accuracy in Measuring Temperature and the Wind
_ Co~onents by the Radioacoustic Method," rRUDY IEM, No 19(72), 1978.
_ 5. Stepanenko, V. D., RADIOLOKATSIYA V METEOROLOGII (Radar in Meteorology), Lenin-
grad, Gidrometeoizdat, 1973.
6. Dragt, J. B., "Statistical Considerations on Techniques for Adjustment of Dif-
ferential Cross Sections With Measured Integral Parameters," TRUDY TREKI~STORON-
' NEGO SOVETSKO-BEL'GIYSKO-GOLLANDSKOGO SIMPOZIUMA PO NEKOTORYM VOPROSAM FIZIKI
BYSTRYI~i REAKTOROV (Transactions of the Trilateral Soviet-Belgian-Dutch Sym-
posium on Some Problems in the Physics of Fast Reactors), Report R-28, Melekess,
Moscow, Atomizdat, 1570.
- 7. North, E. M., et al., "RASS, a Remote Sensing System for Measuring Low-Level
- Temperature Frofile," BULL. AMER. METEOROL. SOC., Vol 54, No 9, 1973.
8. Mazshall, J. M., et al., "Combined Radar-Acoustic Sounding System," APPL.
OPTICS, Vol 11, 1972.
,
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UDC 551.577.53(100) `
ORGANOCHLORINE PESTICIDES IN PRECIPITATION
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 46-51
[Article by Candidate of Technical Sciences Ts. I. Bobovnik,ova and A. V. Dibtseva, .
Institute of Experimental Meteorology, submitted for publication 2 Jan 80J
[Text] Abstract: Contamination of precipitation by
organochlorine pesticides (DDT, DDE, DDD, oC -
and 'y-BHC) is examined. The article gives
the results of investigations carried out
in the Moscow region in 1978-1979. It is
~ shown that despite the ban or sharp restric-
tion on the use of DDT in the well-develop- -
- ed countries in the early 1970's there has -
been no appreciable decrease of DDT in pre-
cipitation. The fallout of DDT with rain ex- _
ceeds the fallout of DDT w3.th snow by a fac-
tor of 5-6 and the fallout of '}~-xHC in ra.i.n
exceeds its fallout in snow by a factor~of
1Q-20. -
Considerable quantities of pesticides after their use for contending with agricul-
tural pests and weeds enter the atmosphere. This is attributable, first of all,
to the fact that the pressure of the vapors of poisonous chemicals is 'sufficiently
great for their evaporation from the soil surface, water and leaves [6]. In addi-
tion, a rather high percentage of the pesticides does not reach the surface to be -
protected even in the process of their application. For example, a ccording to data
_ published by Taylor [13~, in the application of dieldrin and heptachlor on fields _
covered with grass by means of a surface spraying apparatus 60~ of the dieldrin
and 42% of the heptachlor did not reach the surface but remained in the atmosphere.
Pesticides can be present in the atm~osphere in the form of vapor o r aerosols [5]. =
Their fate thereafter is dependent on the size of the particles and meteorological
conditions. Reference for the most part is to organochlorine pesticides and some _
herbicides which are rather resistant to photochemical reactions and which there-
fore can be present for a long time in the atmosphere. Most ~f *he organophosphorus
pesCicides, as a result of their small persistence, decompose rapidly and it is dif-
ficult to trace their further behavior. -
. The atmosphere is therefore a potential reservoir for the transport of pesticides.
Pesticides can be washed out from it by precipitation in the form of rain and snow -
and the soil and sLrface waters can be recontaminated.`According to [16], the
- 48
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principal mechanism hy ~hich pesticides are r~ashed from the at~osphere is rain. _
Many scientists in different countries bave determined the concentrations of
pesticides in samples of air and rain water taken both at the places ~here they
have been applied and in clean regions. The concentrations of pesticides in the
air vary in a wide range from 1�1U'15 to 1�10"3 g/m3, depending on the time
and place of the sampling. Higher c~r.centrations have been noted in the regions
of application of the poisonous chemicals and low concentrations are associated
with global residue [6J. The content of pesticides in precipitation was first
measured in the 19b0's, when their application, especially the use of DDT, at-
tained the maximtnn s cale [ 16 J.
In 1965 Weibel, et al. (United States) determined the content of organochlorine
pesticides (DDT, DDE, chlordan, dieldrin, etc.) and the herbicide 2,4,5-T in
samples of rain falling in a rural area 500 feet from the place of application
of poiso:ious chemicals [14]. The DDT concentration in this rain was Oa6~,~,g/liter,
DDE 0.2, chlordan 0.5, dieldrin 0.003, 2,4,5-T Oo04Ng/liter.
In Great Britain studies of the contamination of precipitation by o rganochlorine
pesticides (OCP) began in 1964. Wheatly and Hardman [15] in the spring of 1964
decided to clarify the possibility of detecting OCP in the atmosphere and there-
fore in rain water in order to explain their presence in uncultivated soils at
- Wellesbourne. The investigations were made at the aerometeorological station at
Wellesbourne. A copper rain gage with a diameter of 8" was set out in a sector
overgrown with grass. Rain water was collected in a glass container with a vol-
- ume of 2a5 litere. Each month the conCainer was changed. The observations were
- made from April 1964 through February 1965. Individual rains were analyzed some-
times. The determination was made by the methods of thin-layer and gas-fluid
chromatography. The concentrations of aldrin, dieldrin, ~'-BHC and DDT were at
the level of tenths and himdredths ~f ~.t.g/liter, that is, at the li.mit of response -
in determining pesticides. On the basis of data on the use of OCP and the rate
of their elimination from the soil the authors calculated the residues of aldrin,
dieldrin, y-BHC and DDT in the soil. These were 1, 8, 2 and 30 tons respectively, -
which probably were distributed in the upper S mm of all the occupied ag*_-~.cultural
areas in England and Wales .
Seven stations for the collection of precipitation have functioned in England -
since 1966> Rain water was collected in a glass container which was painted yel-
low in order to decrease photochemical decomposition. Samples of rain water col-
lected during August 1966-July 1967 were found to contain organochlorin.e pesticides
_ (dieldrin, p, p'-DDT, o,p'-DDT, p, p~-DDE, OC- and y-BHC) in concentrations equal
t~ tenths and hundredths of �-g/liter. Great differences in the content of OCP in _
dependence on the place of sampling were not discovered.
Systematic measurements of pesticides in the air and in rain water at five back- `
ground and two standard (for comparison) stations have been made in West German~~
since 1970 [7, 9]. Precipitation was collected in a 3.0-Iiter glass container .
_ through a glazed funnel 30.5 cm in diameter (intake area 0.730 m2). The samples
were taken monthly and were analyzed by the gas-fluid chromatography method [8].
49
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Aerosol particles of air ~ere ~ound to contain DDT, dieldrin and lindane in con-
centrations of about 1 ng/m3; ~ x~ ~ater it averages 10 ng/g or more. Despite
the banning of the use of DDT in West ~ermanp since 1470 tnere has heen no signif- ~
icant decrease in the insecticide i.n the air and precipitation. In 1971 the DDT
' content in rain was abvut 400 ng/liter and the content of r-BHC ~ras about 300 ng/
liter [9].
In Japan special attention has been devoted to the study of hexachloran in t~.~ em-
vironment, especially in rain. This was attributed to the fact that at the end
of the 1960's BHC was used in great quantities. The precipitation co llector was
situated on the roof o.f a building at Tokio University, in a city area with no
nearby fields. The samples were taken in dark five-liter bottles, from December
- 1968 to November 1969 [10].
The concentration of OC -BHC was fram 45 to 830 ng/liter and the concentration of 'r
BHC was from 29 to 398 ng/liter. Seasonal variations in the concentrations ofcY - -
and y-BHC were observed. The maxi~m values were discovered in the sum~er months
and the minimum values were observed in winter, which was associated with the use _
of pesi.icide in ~immer. However, the contamination of rain water with BHC was ob-
' served throughout the entire year. This fact gives basis for assuming that the
principal portions of BHC entering into the a~osghere fall onto the earth in
the course of a short time; the remaining quantity remains in the atmosphere and
gradually falls out over the course of a prolonged period.
During 1973-1974 Gitsova in Bulgaria [3] investigated the content of organochlor-
ine pesticides in natural waters, including in rain water. The precipitation
samples were collected at the hygiene center in Sofia over a period of 13 months
(October 1973 - November 1974). The average concentrations of ~-BHC were 35 ng/
liter, and for p,p'-DDT - 68 ng/liter. -
~
- According to the data in [11J, the quantity of DDT used on the earth was about _
100,000 tons, of which up to 40% of the ~7DT was used for contending with malaria
and the remaining b0% for contending with domestic insects and agricultural pests.
DDT is used for the most part in the developing countries (India, Republic of Chad, _
and elsewhere) to which 90% of the DDT produced in the well-developed countries _
(United States, West Germany, Japan, France and others) is ~xported. _
In 1974 specialists at the Institute of Experimental Meteorology began work on
study of contamination of the soils and rivers by residual quantities of organo-
chlorine pesticides. In earlier studies we de~onstrated that at the present time
- organochlorine pesticides are discovered in all objects of the biosphere soil,
_ surface waters, fish and precipitation [2]. The discovery of DDT, its metabolites,
pC - and Y-BHC in the soils of preserves and in rivers flowing f ar from agricul-
- tural regions led to the idea of initiating systematic observations of the levels
of contami.nation of precipitation,. For this purpose we used an automatic collector
of dry and wet fallout developed at the Institute of Experimental Meteorology by _
G. G. Belov [lJ. The instrument consists of a precipitation indicator, actuating
mechanism and two enameled receiving containers. The sensitivity of the precipita-
tion indicator makes it possible to activate the ac~uating mechanism with the en-
- try of rai.ndrops or snowflakes with a diameter of 200 t,l mo From the moment of -
5Q -
" F~R OFFICIAL USE ONLY
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_ onsee of the falling of precipitation the receiver of dry fallout (atmospheric
uust) is tightly covered by a lid. Upon the ending of the rain or snow the lid
covers the tank with precipitation and the tanlc for the collection of dry fall- _
_ out is opened. The precipitation for the month is poured into a dark glass bottle
an1 analyzed. With the falling of more than 2 mm of precipitation each rain is
analyzed. _
The precipitation is doubly extracted with distilled n-hexane; after purification
_ and concentration of the extract the analysis is made with a gas-fluid chromato- ,
graph. The analysis error is 15Y.
Snow falling on a backing over a month period is analyzed. For comparative purposes
the snow is sampled at several points around the backing. After the snow is thaw-
e~i in an enameled container the volume of the sample is determined and then it is
analyzed using the method described in [4]. In determining organochlorine pesti- _
cid2s in precipitation an effort is made to reduce to a minimum the contaruination
rrom reagents and other sources within the laboratory. The container is carefully
was~ed ~nd all the reagents are purified and checked for chromatographic purity.
In addition, blank experiments are made in each case.
TaLle 1 gives the m,e3a monthly concentrations of organochlorine pesticides and
~ _heir fal~out with precipitation during 1978 and the nine months af 1979. The
table shows that DDT and }'-BHC are discover.ed in all seasons of the year. On the
average, the observed concentrations are at the level of tenths of ng/liter, spe-
_ cifically: DDT from 15 to 120 ng/liter, ~/-BHC from 13 to 100 ng/liter. The
conte:�-lt of the metabolite DDE in precipitation exceeded the quantity of DDT or was -
equal to it. It is known that DDE is a product of the transformation of DDT caused _
to a considerable degree by photochemical processes transpiring in the atmosphere.
During su~umer the transforniation of DUt into DDE is accelerated appreciably. A con-
firmation of this fact is comparative data on the ratio of the concentrations of _
DDE/DDT in separately collected rains and precipitation collected duri~g the month. _
Such an e:cperiment with the use of tWO separate p recipitation collectors was carried
o ur in June and July 1979. Individual rains were collected in one collector for an-
" alysis and the monthly precipitation was collected in the other. At the end of the
month the precipitation was poured out and analyze~ ior its OCP content. In addi-
tion, dry faliout was collected in the dry fallout collector. At the end o� the
month this collector was washed out with acetone and carefully wiped with gauze _
wetted with acetone. The pesticides were extracted from the wash fluid and marl by
hexane. -
~ The results obtained in this experiment are given in Table 2. The DDE/ DDT ratio for
individual cases of precipitation varied near 1 and did not exceed 1.5; for precip-
itation collected during the month it was 3.2-3.5 (Table 2). The transformation
o� DDT into DDE was still more appreciable in the sample of dry fallout in July when
_ there were few days with precipitation (four days) . The remaining time the collector
of dzy fall~ut was ~pen; the mean air temperature was 17.6�C, which favored the
- photoctiemical transformat~_on of DDT in contrast to a closed collector. In July the
_ quaDtity of. precipitation was maximtinn (147 the number of days with precipita-
tion was ;.7, and ~Pspite such a large number of days with precipitation the DAE/DDT
- 51
~
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V ~ r~ ~n L~ ~h c~
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C) I N~�-~���NNOO ~ O:V `~C'~CVM
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CVN CV.f'~O ~ ~N~flaq
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~ x OGVN-~~N~~^O O ON NM^N
~ _
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^
O~ C~. _
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GL a0int~:Jc:J~:aO-00 M'00 O~O~
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3 a' ~ VJUUcis~=~'.G1V G ~oo - . . . . . . . c
o V C: cv ~r ~ r-i r~-1 r-I r-i r~l r~-I rl
(A
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a ~ ~ ~
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~
- y.~ -
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p . ~
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a -
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~ S aq~ n:ootiviac~^:CCfl nci.rQao~rit~ciir p v-1 p.
F" y Q~ - GO Qf ~j i!: :C ~ N iL] N'C M a c..'1 C: V U f3
~ Yu~ r ~
U ~ t3~ _
~ a o~ ~
ao ~ ~ ~
~ r~ ~ ~ D, ~ t
~ ~ ~ O
- O ~ ~ ~ ~c ; ~ c ~ ' ~ c~d ~ ~ c~d .-i ~
. c.~i ..b a~o _m ~ Y , � N C+ f~ ~ ~rl N D, ~
m F ~ x a~. a u G>, ~ ~ ct1 cE ~ 3-+ 5' ~
I S G~ � g m L Y J~ , a+ m 4~ m J ^ O'~".. ~'7 d' ~~'7 ~7 C'i -
GQ~ :.+.4U~~.-a .+.E~C .i.: -
1 k / -
a =
P= 4, ,
k, n are dimensionless constants, selected on the basis of observational data, D is
droplet diameter, Dm is the so-called modal diameter a~ the droplet for which the
f function attains a maximum value, (~(x) is the gamma f unction, PW is water den-
sity, is the liquid water content of the spray cloud,
64 -
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218 u~ _ _ u ~
_ o� - 0,='~ 63 u,~ u= 100 c.K c, [cm/sec]
~ f - ?~2 + ~6 ~
, , ~4~
[6 = sprayj u is wind velocity, wp is the initial vertical component of droplet
- velocity, _
J `~o = ~m ~o rn~ ~5~ _
w0 m is its value with D= Dm~ Hdrop s~ Edrop s are the total quantities of heat
_ anc~ moisture respectively released by ~iroplets of the diameter D during the time
('L) of their presence in the air,
_ . -
HKS=?-Di.kH~ :~ta'"_tp~'t' tIDa lp ~eXp(-~r:)- 1]}~ ~6)
t
E15= 2 -DpdkE ~(~~tp---t�~t) --1- (t~-tn~ (~-exp (-a,.)]}, (7~ ~
ar
" = Ao ~D, ~o exp ao (D. ~o m~ u~~
r,
[p = equil]
�`~o (D~ ~'om) and ao ~D~ ~om)
~ are approximation coefficients,
. . _ ~g~
. a~dpkEL+ky1~
a=12 ,
t ~m ,Q, ps
f 10 3.s air density, cW is the heat capacity of water., -
- 4?:~0 qs ~ - - - -
~ - ~141.9 - u.~ ~ra + r~, (10)
(11)
,`.i~ _ - a~ ta,
ta is air temperature, tW is the temperature of the water surface, qs i_s saturating -
,,peci_fic humldity,
- - _ . _
3.8 e I 17,57 rQ
9s = p XP l 24i ,9 to ~12~
j~, i.s rhe deficit of specific humidity in the near-water layer,
~ - 9S - Qa ~ (1.3) -
qa is air specific humidity, p is pressure, teq~l is the so-called equilibrium tem-
_ pera*ure o` the droplet, y _
kh,atQ-j~dpkEL (14)
tequil - kH a alapk� ~ +
kt~, kF are the ventilation coefficients, taking into account the influence of
wind velocity on the intensity of heat an~ moisture exchange between a droplet and
the sc~r.;ounding me3ium j10],
65
FOR OF'FICIAL USE ONLY ~
~
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EOR OFFICIAL USE ONLY
kH - t-~- 0,23 Pr,'~~~ Re'/', kE - 1+ 0,23 Scy~~~ Re'!~, (15)
Re, PrM, ScM are the Reynolds, Prandtl and Schmidt numbers respectively,
_ Re - D~ P , pr� - ~ , Sc�= -P'd , (16) -
~ r.
a,~C., Yj are coefficients characterizing the molecular properties of the air: its
thermal conductivity, thermal diffusivity and dynamic viscosity respectively, d is
the coefficient of molecular diffusion of water vapor, L is *he latent heat of
evaporation. In accordance with [10],
_ _ . . - - - _ .
10~=5.77+0,16 ta, r~� 106=172+0.~ iQ, _ (17) -
d= 0,22 + 0,0015 t Q, L= 597,26-0,647 t Q. ' ~
However, it seems entirely natura~ that ir. the first approximation these values can
be considered constant and equal to their values with ta = 10�C. With respect to
the values cW, PW, they a1so, without substantial error, can evidently be assumed -
to be as follows:
_ cW = 1 cal/~g'~~), P w= 1 g/~m3� -
For the practical use of (3) and (5), the following values of the parameters are
_ recommended in [3]:
_ D~ = 0.015 cm, k= 1, n= 2, w~ m~ 300 cm/sec. (18) _
' In this connection it must be noted that qualitative and statistically support-
ed experimental data on change in the f~ction f and the parameters D~, k, n asso-
ciated with it, and also on the liquid water content during strong winds are lack- _
ing. What has been said also applies to w~ m, Accordingly, adopting nevertheless
~ for further practical use the values for the parameters (1�3), it must be remembered
that they can be replaced by others corresponding to more reliable measurements or =
numerical experiments. For this same reason we considered some values (as in [3]) _
with allowance for a possible range of changes wp m= 50-300 cm/sec. _
Derivation of Economical Parametric Formulas _
It follows from the form of formulas Z3), (5)-(7) that after their substitution into `
(1), (2) the greatest difficulties for analytical integration are the expressions
(6) and (7). Therefore, ~ae will discuss in detail their transformation so as to
facilitate the determination of (1), (2). An analysis of (6), (7) shows that the
parameters cx i, tequil az?d ~ are functions of droplet diameter. Taking into account
the relative smallne:~s of the difference in the ventilation coefficients, for the
_ heat and moisture exchange of a droplet ~rith the surrounding air with the usually -
- employed values PrM = 0.71 and Sc~,1 = 0.62, we introduce some mean ventilation coef-
ficient
k- 1~- 0,2 ti' Re (~9)
and we will transform expressions (9) and (14) to the form
66
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FOR OFFICIAL USE ONLY =
~ ~
- : . _C~~~k... _ .
= eff] ar= t2 cmP~, . ~20~
~ [p = equil] tP = ta A, (21)
where -
~
_ _ ;
d -F- a, L.
(22)
In order to explain the nature of the dependence of droplet flight time on its dia-
' meter we used expression (8) and a table of the coefficients A~ and c?~.0 cited in
_ [3]. An analysis of the resulting graphic material in~dicated that there is a pos-
sibility for approximating this dependence by a formula in the form
' ~D (23)
where _
f ^
. _ .o - ta (u - uo) �
. (24)
~
In finding the proportionality factor and the exponent y we used as a point of de-
parture the fact that the relative error in.computing 'C from (23) in comparison -
with its values computed from (8) must not exceed 10% and insofar as possible must
be minimum near D= Dm. In addition; since the influence of spray in the general
balance of heat and moisture exchange becomes appreciable at wind velocities ~ 15 -
m/sec, we limited the possible range of u changes downward to the value up = 15 m/ '
sec. The values of the parameters 'C0, ~G'u and computed in accordance with the =
conditions stipulated above are given in Table 1. We note, however, that since ex- -
pression (23) is an approximation of formula (8), w}iich in turn is an approximate
result of nimmerical experiments, the resulting values of the parameters are also
approximate and in the future can be made more precise.
Table 1
Dependence of the y, Zp and Z'u Parameters on the Modal Value of the Vertical
_ Velocity of Departure of Droplets f rom Water Surface
~n ~ CN;C (~~geC
_ 7~ I ~ ~ ~ `l.~l ~ ~ -
- 0,89 ~ 0,74 0,65 d,6S `
' 10= i,13~J7~ y,3221 14,044 16,8516
:,,-10~ 0,9831~ 1,0415 1,5239 1,829
We introduce the notation -
y _ 1 - exp a;
~ a~.~ ~25~
67
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and we will transform expressions (6), (7) with the introduced simplifications (19)-
- (24) taken into accou~t. Then we have
-D- T . _ . L
HKS=`~'~i.k:(~m) [a, -(tu,-tQ-}-~1 y,, ~Zb~
E~S=2 -Dpdk~(u ~T[A (1 -a, ~~)+a,y~t~,-tQ-f-T1. ~
~11' (27>
_ [K = drop, = eff]
Table 2
Dependence of Integrals Il, I2 on 2iodal Value of Vertical Velocity of Departure of
_ Droplets from Water Surface -
~nrn ~yl~ (~~SeC
50 I ]00 ( 200 ~ 300
~
I U.35309I0,3271410,31791` 0.3179(
0.4?691 0.4327 {~~.41687~o,aiss~
- Now substituting (3), (5), (26) and (27) into (1), (2), as a result of integration -
we obtain the following parametric formulas:
_ _ - .
_ HK - C ~ f p, - (ta - t~,l (28) -
l
= LE,~ = LCd ~ ~a. I ni - a~ ~4~ - mz)1 at (t9 _ ta~ zzJ ~ (29)
J
- where
- ' n~ ~ -
_ . C_ kl R ~~mo~z (30~
= rln-f-~ 1 D '
l k 1
Q~ ~,2 I: l~Re,~, ~ 1, + ~~2 ll Rem , ( 31) -
I, x;+" exp (-Nxk) dx, 1:= X~5+7+n exp (~--~.x'~) ax, ~32~
a' u
- +rs o.s + , R k ( 33 )
- f x' y exp ;1 x'~} dz, I~ x y exp ~-LL x) dx,
o ~
_ u P ( 34 )
Rem - r r
~ (35)
D -
- x Um ~
The values of the I1, I2 integrals were found analytically in [8] and are given in
Table 2. With respect to computation of the integrals I3, I4, for their determina- -
tion we used the Lobatto nimnerical method with automati.c choice of the integration
interval j7]. In this. connection, for evaluating the exror arising due to tl-,~
� 68
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' ~ tl) O~p tt~p N t~ GV
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a' o0 00 00 00
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r0 �r o0 00 00 00
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~
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ha0 O p~~ti ~a~0
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.J
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rl W ' 0 a~DO Oa0 cMpa~O
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~
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q t~
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1-~ ~ ~ N~ ~D ~M GV OD t0
O C O O O O O C O
N
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~
t~
ro
a~ _
a
E
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C O O O O O O O
Qi
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w
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~ I NM NN ~N ~N
~ O O O O O O O O
~
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A $ $ _
69
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introduction of the finite droplet spectrum we first carried out computations us- -
ing formulas (32) for comparison of the numerical and analytical integration meth- -
ods. This comparison indicated that when using the selected droplet spectrum (D =
103-2�10-1 cm) the relative error in nimmerical integration was ~ 10-6%. Table 3
gives the values of the integrals I3, 14 obtained by averaging of their extremal
~alues with possible changes in pressure p= 970-1030 mb and wind velocity u= 15-
40 m/sec for each stipulated air temperature value ta.
Now we will note another important property of the derived parametric formulas (28)
and (29). This is the relative influence of the temperature and humidity fields on
- the heat and moisture exchange of a spray cloud. Thus, the apparent heat flux is de- -
pendent not only on the diffusion of heat, but also on the moisture deficit in the _
near-water layer. And evaporation, in turn, is determined, in addition to the
deficit of saturation in the surrounding air, by the conditions of thermal strat-
ification in the near-water layer.
The algorithm for computing the fluxes of apparent and latent heat due to spray
clouds during storms cited above was applied in the form of subprograms in FORTRAN
for the BESM-6 electronic computer. The time for computing one variant with the
input data ta, tW, qa, p and u does not exceed 0.01 sec. At the present time this
- method is being tested in a model of general circulation of the atmosphere and
- ocean formulated by the Computation Center Siberian Department USSR Academy of Sci-
ences.
In conclusion the authors consider it their pleasant duty to express their appreci-
ation to Ye. P. Borisenkov, R. S. Bortkovskiy, M. A. Kuznetsov and~A. G. Yantsen
for useful discussion of the article.
BIBLIOGRAPHY
1. Ariel', N. Z., Bortkovskiy, R. S., "Refined Model of Energy and Mass Exchange -
_ of Spray Over the Ocean Surf ace During a Storm,".TAYFUN-75 (Typhoon-75), Vol 2,
Leningrad, Gidrometeoizdat, 1978.
2. Borisenko, Ye. P., "Some Mechanisms of Interaction Between the Atmosphere and
Ocean During Stormy Weather Conditions," PROBLEMY ARKTIKI I ANTARKTIKI (Prob-
- lems of the Arctic and Antarctic), No 43-44, 1974.
3. Borisenkov, Ye. P., Kuznetsov, M. A., "Parameterization of Interaction Between
the Atmosphere and Ocean Under Stormy Weather Conditions Applicable to Models
of General Circulation of the Atmosphere," IZV. AN SSSR, FIZIKA ATMOSFERY I
OKEANA (News of the USSR Academy of Sciences, Physics of the Atmosphere and
Ocean), Vo1 14, No 5[year not given]. -
4. Bortkovskiy, R. S., "Mechanism of Interaction Between the Ocean and the Atmo-
sphere During a Stoxm," TRUDY GGO (Transactions of the Main Geophysical Ob-
servatory), No 282, 1972.
5. Bortkovskiy, R. S., "Refinement of Estimates of Heat and M~oisture Exchange Be- -
tween the Ocean and the Atmosphere," TRUDY GGO, No 326, 1975.
~ 70
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6. Bortkovskiy, R. S., Kuznetsov, M. A., "Some Results of Investigation of the
State of the Sea Surface," TAY~UN-75, Vol l~ Leningrad, Gidrometeoizdat,
1977.
- 7. Brushlinskaya, 0. V., Vasil'yeva, L. G., "Set of Standard Programs for Approx-
imate Computation of Simple Integrals With Automatic Choice of Interval,"
CHISLENNYY ANALIZ NA FORTRANE (Numerical Analysis in FORTRAN), Moscow, MGU,
No 8, 1974.
8. Gradshteyn, I. S., Ryzhik, N. M., TABLITSY INTEGRALOV, SLTMM, RYADOV I PROIZ-
VEDENIY (Tables of Integrals, Sums, Series and Products), Moscow, Nauka, 1971.
9. Kraus, Ye. B., VZAIMODEYSTVIYE ATMOSFERY I OKF,ANA (Interaction Between the At-
mosphere and the Ocean), Leningrad, Gidrometeoizdat, 1976. ~
= 10. Matveyev, L. T., KURS OBSHCHEY METEOROLOGII (Course in General Meteorology),
Leningrad, Gidrometeoizdat, 1976.
~ 11. PreoFarazhenskiy, L. Yu., "Evaluation of the Content of Spray Droplets in the
Near-Water Layer of the Atmosphere," TRUDY GGO, No 282, 1972.
= 12. Bortkovskiy, R. S., Byutner, E. K., Malevskiy-Malevich, S. P., Preobrazhenskiy,
L. Yu., PROTSESSY PERENOSA VBLIZI POVERI~iNOSTI RAZDELA OKEAN-ATMOSFERA (Trans-
~ port Processes Near the Ocean-Atmosphere Discontinu3ty), Leningrad, Gidrometeo-
izdat, 1974.
- 13. Wu, J., "Spray in the Atmospheric Surface Layer: a Review and Analysis of Lab-
- oratory and Oceanic Results," JGR, Vol 84, No C4, 1979. -
71 -
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UDC 551.(465.7+553.8)(265) -
VARIABILITY OF ACTIVE LAYER CHA.RACTERISTICS IN THE NORTHWESTERN 2ACIFIC OCEAN
DURING PASSAGE OF A STORM
Moscow METEOROIAGIYA I GIDROLOGIYA in 8ussian No 8, Aug 80 pp 65-68
[Article by K. A. Rogachev, Pacific Ocean Oceanological Institute, submitted for pub-
lication 20 Aug 79)
[Text] Abs tract: The interaction between the boundary
_ layers of the atmosphere and the ocean is ~del- -
ed at a synoptic time scale. The wind mixing of
different thermal structures in the northwestern
_ part of the Pacific Ocean is examined in a differ-
ential model of the active layer. A strong dependence
of the change in active layer characteristics during
a storm on the vertical thermal structure is demon-
strated.
Among the most important disturbances occurring in the active layer (AL) o~ the
- ocean are synoptic and seasonal disturbances. Un~er the term "synoptic variability" -
we will understand processes with a characteristic time scale of several days, which
corresponds to the synoptic maximwn of atmospheric variability. Such a variability _
- is associated with the p assage of individual pressure systems, which account for
80% of the kinetic energy in the atmosphere in the middle latitudes [2]. Elsberry
- and Camp [5J, who examined the characteristics of strong atmospheric disturbances,
pointed out that their main part is concentrated in short periods of time with a
duration of about two days asso ciated with the passage of cyclones. Accordingly, in
the study of thermal structure of the AL it is the reaction of the AI, ta the Fas-
- sage of powerful atmospheric formations which is of the greatest interest in study
of tYse ~~thermal structure of the AL.
We can discriminate three physical processes responsible for restructuring of the
thermal structure of the AI,during a storm: cooling of the ocean surface as a re- _
, sult of evaporation and heat exchange with the atmosphere, including nonturbulent
mechanisms, wind mixing of the AL and vertical circulation of waters in the zone v
affected by the storm.
In this article we will examine changes in the structure of the AL as a result of
evaporation and turbulent heat exchange and wind mixing. A one-dimensional differ-
ential model of the AL is used for this purpose. The role of one-dimensional pro-
cesses in the reaction of the AL to strong atmospheric d~sturbances was examined
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by Camp and Elsherry. In j4] they dre~ the conclusion that they play a decisive
role at a synoptic time scale.

~ Using a one-dimensional model of a nonstationary Ekman friction layer for the AL,
we wi1Z have the following sysCem of equations: -
- o" - !v - a K au - .
- ar a: d~
dr' lu = K ~v
dt ds da ~1~
' dr a dT
ac = a: K aZ
K= Ko (u, v, T, z)
- with the boundary conditions
da ~ dv _ _ _ ~ dT
~ K az - - ~o, x ~Z - �y~Po~ K or - - Q~ Z = 0� ~
u-v=0, dz -0, z=H~, (2>
where u, v are components of velocity of the drift current, ~ is the Coriolis para-
meter, t is time, z is the vertical coordinate. The origin of coordinates is placed
- at the ocean surface with an axis directed vertically downward, T(z, t) is water
tem~erature, K is the coefficient o� turbulent diffusion, assumed equal to the co- -
_ efficient of turbulent viscosity along the vertical, 'G X, 'C y are the components of -
the shearing str.ess of the wind at the ocean s4rface, P~ is water density, Q is
! the heat flux into the AL.
~ The function Kp is determined from the balance equation for turbulent energy, which
' includes the following components: generation of turbulence by velocity shear, dis-
sipation of turbulent energy and work against buoyancy forces. The solution of this
equation has the form [1] -
_ _ - -
~
Ko = (~h)Z v ~ az )2 + ~ d )Z + g ~ as ,
= where h is the depth of the well-mixed layer, g, ~ are the acceleration of free fall-
ing and the coefficient of thermal expansion of water, c is some constant.
By the term "depth of the well-mixed layer" is meant the depth at which the expres-
sion under the radical sign becomes equal to zero. Deeper than the we11-mixed layer
the value of the K coefficient was ass~ed to be constant and equal to the minimum
_ value 1 cm2/sec. This value is two orders of magnitude less than the value of the
~ coefficient in the well-mixed layer and two orders of magnitude greater than the
molecular viscosit~ coefficient in watPr. Such a determination of the diffusion
_ coefficient deeper than the mixed layer is associated with the paucity of ideas
concerning mixing processes in the thermoclinee
_ If one uses the scale [t] = 105 sec (day) and a scale of changes in depth of the
mixed layer [H] = 20 m, then from (1) [KJ =[H]2/[t] = 40 cm2/sec. If [Q] = 800 W/
m2, then (T] = 1�. If the velocity scale [u] = 10 cm/sec, then [ t] = Oo2 dyne/cm2.
The thermal balance of the ocean surface Q is dependent on many factors: wind velo-
city, water-air temperature difference, air humidity, c~oud cover, intensity of
shor_t-wave raaiation. In this study the heat flux was represented in the form
73
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a
- Q_~ H,.
r~, -
Here the following components are considered:
Hl =~.l R is short-wave rad~ation, orhere R is the f lux of short-wave radiation inci-
dent on the upper boundary of tfie atmosphere, � is the a~tenuation coefficient, de-
- pendent on surface albedo, air humidity and atmospheric albedo. _
H2 = cl(T + 273�)4 is the Iong-wave radiation of the surface, where cl is some con-
stant, H3 =-c2(T - TA)cD w is turbulent heat exchange between the ocean surface
and the atmosphere, where cD is the friction coefficient, w is wind velocity, TA
- is air temperature, c2 is a constant, _
H4 = BoH3 is heat exchange as a result of evaporation, where Bo is the Bowen ratio.
r'C - _
10
' .
i
F
~
~
L
y J TO 'r0 60 DO 1QB
'S
6~
;o
s
~
~y~ t days
JO 10 f0 :0 80 sx
Fig. 1. Reaction of different thermal structures to passage of a storm. a) 41�N, 170�
b) 46�N, 170�E.
System (1) with the boundary conditions (2) was solved numerically with stipulated
TA, X~y values with different initial conditions for u(z,0), v(z,0), T(z,0), K(z,
0). The thermal conductivity equation was solved using an implicit triangle scheme
by the factorization method and the equation of m~tion was solved by the transverse
fractional steps method [3] with a time interval of 5�103 sec and with an interval
along the vertical coordinate 2 m with Hp = 100 m.
- In order to investigate the reaction of the active layer to wind intensifications
_ and changes in air temperature we selected the mean monthly long-term tempera-
ture profiles in the northwestern part of the Pacific Ocean. Data on water temper-
ature were taken f rom the archives of the Pacific Ocean Oceanological Institute of
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:
the Far Eastern Scientific Center USSR Academy of Sciences, data on air tempera-
ture from the ATLAS OF THE OCEANS (Pacific Ocean), and the',t and ci parameters _
- from j6].
Thus, different vertical thermal structures were disturbed by an identical wind. We
will examine the reaction of the AL to identical disturbances by the wind in the -
northwestern part of the Pacific Ocean. Figure 1 shows the chang~ in the tempera-
ture profiles with ti~e for August with an intensification of the wind in conform-
ity to a linear law from 3 to 20 m/sec and attenuation to the initial value: _
~ O'2 . t 3,
_ where time is in days. The air temperature decrea~ed by 5�C relative to the mean _
- monthly value during wind intensification. The figure shows that despite identical
disturbance by the wind, different thermal structures in the AL react differently.
Thus, the change in surface temperature at the~point 46�N, 170�E attains 6�C, and at
- the point 41�N, 170�E 2�C. -
The value of the turbulence coefficient is different for different structures, but
its variability is far more dependent on the wind stress. The values of the coef-
f icient increase by a factor of 20 during the wind stress peak and attain 600 cm2/
sec. A cemparison of the temperature gradient values in the AL with the changes in
surf ace temperature indicates their full correspondence. Such a correspondence in-
dicates that for an adequate description of changes in surface temperature during
the passage of a storm Tt is necessary to use a differential model which makes it
possible to describe the continuous [hermal structure of the AL.
- _
~ Q1 .`~y v 6) .`'~151'y1~2
. , ) ~
_ a ~ . .
~ � ~ / ( ~
'~--3 J O ~yt'~~ 1,3~ ? 1,6 ?
~ ~i~~~~ / ~
.
~ L~-}~0 ~
Z ~ ~z
I ~~.1 I ~ 3~ i"~'~~ 12 ~ .
_ 1
_ Fig, 2.
Figure 2a shows the decrease in surface temperature (�C) in the northwestern part
of the Pacific Ocean, computed for 30 profiles for August and Fig. 2b shows the
change in depth (x 20 m) of the well-mixed layer with the passage of a two-day
storm. The considered region is chara~rexized by a complex hydrological structure.
This is associated primarily with the existence here of the subarctic front bet~~een
subarctic and subtropical waters. The maximum change in temperature attains 6�C _
and the maximum increase in depth of the well-mixed layer is 30 m.
JS
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Computations of the variahility of ciiaracteristics of the AL during passage of a -
model storm, carried out for the entire northern part of the Pacific Ocean, indi- ~
_ cate that the maximum variability is ohserved in the northwestern part of the
Pac3.fiC Ocean.
Thus, after examining the reaction of different theYmal structures to the passage
of a model storm it is possible to draw tfie following conclusion: the change in
surface temperature is determined by th,e thermal structure of the AL. The maximum
changes must be expected in an AL with large vertical temperature gradients. This -
indicates that capture of cold water from the deep layers is the dominant ~rocess
determi.ning the change in surface temperature. -
The author e~resses appreciation to N. I. Dyul'dina for assistance in the comput-
ations.
BIBLIOGRAPHY
1. Kalatskiy, V. I., MODELIROVANIYE VERTIKAL'NOY TERMICHESKOY STRUKTURY DEYATEL'-
NOGO SLOYA OKEANA (Modeling of the Vertical Thermal Structure of the Active
Layer in the Ocean), Leningrad, Gi3rometeoiadat, 1978.
2. Lappo, S. S., SREDNEMASSHTABNYYE DINAt-IICHESKIYE PROTSESSY OKEANA, VOZBUZHDA-
YEMYYE ATMOSF'EROY (Mesoscale Dynamic Processes in the Ocean Excited by the
Atmosphere), Moscow, Nauka, 1979.
~ 3. Yanenko, N. N., METOD DROBNYKH SHA~GOV RESHENIYA MONOGOMERNYKH ZADACH MATEMAT-
ICHESKOY FIZIKI (Fractional Steps Method for Solution of Multidimensional
Problems in Mathematical Physics), Moscow, Plauka, 1967.
4. Camp, N. T., Elsberry, R. L., "Oceanic Thermal Res~onse to Strong Atmospheric
Forcing: The Role of One-Dimensional Process," J. PHYS. OCEANOGR., Vol 8, No =
3, 1978.
5. Elsberry, R. L. , Camp, N. T. ,"Oceanic Thennal Response to Strong Atmospheric
- Forcing: Characteristics of Forcing Events," J. PHYS. OCEANOGR., Vol 8, No 3,
- 1978.
6. Saltzman, B., Ashe, S., "The Variance of Surface Temperature Due to Diurnal
and Cyclone-Scale Forcing," TELLUS, Vol 2$, No 4, 1976.
76 -
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UDC 551.465.558
MEANDERING AND EDDY FORMATION IN ZONAL OCEAN CUR.RENTS
- Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug $0 pp 69- 77
[Article by Candidate of Physical and Mathematical Sciences V. F. Kozlov and Ye. V. ~
Ivanchenko, Far Eastern State Uttiversity and Pacific Ocean Oceanological Institute,
_ submitted for publication 28 Sep 79]
- [Text] Abstract: A etudy was made of the reaction
of a stationary poorly stratified flow with -
a transverse velocity shear in an infinite -
channel whose width in some section changes
in a.jump. It was possible to discriminate
a class of profiles of undisturbed flow lead-
ing to a linear equation of conservation
. of potential vorticity. The meandering and
eddy formation conditions were investigated. .
The results of numerical computations of the
streamlines are given for different syffinetric
profiles. The conclusion is drawn that the -
curvature of the velocity profile of the on-
comir.g flow pI.ays a decisive i�ole.
The meandering and eddy formation phenomena in regions of strong currents of the -
Gulf Stream or Kuroshio type have become the subject of theoretical investigations
- only in recent decades [4, 8, 10, 12y 14, 18, 20, 22, 24-26). As a model it is cus-
tomary to examine limiting cases either of thin inertial jets [2, 6, 16, 19, 21],
or, on the other hand, kinematically homogeneous (without a transverse velocity
shear) flows [Z, 3, 7, 13, 15, 17]. A qualitative ?xplanation of the observed ef-
- fects can be given using a fundamental relationship in geophysical hydrodynamics
the law of conservation of potential vorticity, which in the simplest station-
ary barotropic case has the form r+ 4 ~
N - CI
Here ~ is relative vorticity, ~ is the Corioli~ parameter, H is the total height
_ of the column of fluid, ~~J is the integral stream function.
It can be seen from this expression that the relative vorticity closely associated '
with the curvature of the streamlines changes every time vrhen along the tra~ectory �
there are changes in ~(meridional displacements of the streamlines) and/or H
(change in bottom relief). An extreme form of manifestation of bottom relief is
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the presence in the flow of lateral solid boundaries, including islands. It is of
interest to study the influen~e of disturFiance of the ~orm of the lateral houndary
on a flow with a transverse mean (vertically) velocity shear.
A prcblem of a similar type is examined in this article. Within the framework of
an inertial quasigeostrophic model, on the assumption of weak stratification, a `
study is made of the behavior of a flow with a transverse velocity shear in a zonal
- channel whose width in some section changes in a jump. In the first approximation
such a schematic representation can be used for the Kuroshio region to the south
o� Japan.
Now we will formulate the principal assumptions of the model. We will introduce
the characteristic scales of a horizontal extent L*, horizontal velocity U~, -
depth H*, Coriolis parameter rise of bottom relief h*, buoyancy frequency N*;
assume that dSL/dy is the constant Rossby parameter. The fluid is assumed to
be incompressible and nonviscous and motion is assumed to be steady; in addition,
it is assumed that
_ - - - ' = t)
= C"
0) o Thus,
for the appearance of wave mAdes it is necessary that the profile be of the trigo-
nometric type; on the other hand, an adequate condition for the absence of wave
~ modes is that the profile belong to the parabolic or hyperbolic type. If wave
_ modes are absent, in place of (29) we have for large x
(z, y) ~Y) = ~ ~Y) (1) - ~ (l)~ Y (y)~ -
that is, purely zonal flow. It may be fe~.~ d that in some part of the channP?
~~(y)~ 0; this carresponds to the appearance of return flows. In a more general
case (29) with M~ 1 the pattern will be complicated by the presence of individual
eddies forming a quasiperiodic structure. -
On the basis of the described model we carried out several tens of numerical ex-
periments with different values of the determining parameters C, Q and The ~
computed fields ~(x,y) were printed out in the form of isolines on an automatic
- digital printout unit in zhe region -0.5 < x< 2.0. Here we will illustrate only one
series of experiments carried out with a fixed discharge Q and width for several
values of the C parameter. !'liis makes it possible to trace the influence of curva-
- ture (or shear) of the undisturbed flow on the currents in the transition region.
_ F igure 2 shows the computed streamlir,.cs ~/b = const fo r the case Q/b = 1/~1 2 and
= 1.7 with C/~12 = 4 (a), 0 (b), -0.5 {c)s -1 (d), -1.4 (e), -1.8 (f), In all
- the figures the corresponding velocity profile of the undisturbed flow is shown
at the left.
Figure 2a,b shows hyperbolic and parabolic cases with concave profiles of the velo- _
: city o� the undisturbed fJ_ow; here M= N= 0, the wave modes are absent. Return
- currents can be seen in the expanding part of the channel. In the first case they -
occupy only the central part, and in the second case the lower half of the re-
gion.
Figure c-f shows the patterns of currents for trigonometric profiles varying from _
convex to concave. In these cases N= 0, M= 1(c), N= M= 1(d), N= 1, M= 2
(e,f) respectively. The appearance of wave modes makes the field of currents more
_ complex and leads to the appearance of eddy formations.
Theoretical conclusions and numerical experiments show thnt in the considered mod-
f el eddy fo?-mation and meandering of~the flow in a widening channel are essential-
ly dependent on the C parameter. The condition tor the appearance of wave modes
- (30) can assume a different form. Using (11), by differentiation we find the _
identity ~ U" ( y) b CU , -
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by means of which in place of (30) we ohtain
u� ~ n ~ / ; l~ ~31~
l 1
This expression can be interpreted as thP instability condition [1], as a re-
sult of which a process of ineandering and eddy formation arises. It is of interest
tn apply this criterion to real observation data on the transverse structure of cur-
rents.
_ By multiplying (31) by tT(y) and integrating for the width of the channel upstream, '
we obtain a"softened" condition for the appearance of wave modes
U' (1)-U't0)_ Q+b. (32) ~
Conditions (31) and (32) are equivalent for shearless currents, for which they are
always satisfied.
What has been said above leads to the conclusion that any attempts at prediction
of the behavior of currents cf the Gulf Stream, Kuroshio or Circumpolar Current _
_ type must be based on a study not only of changes in their discharge, but (which
- is evidently more important) also on an analysis of the transverse sCructure up-
stream from the considered region.
BIBLZOGRAPHY
1. Arnol'd, V. I., "Conditions of Nonlinear Stability of Plane Stationary Curvi- ~
linear Currents of an Ideal Fluid," DOKLADY AN SSSR (Reports of the USSR Acad- -
_ emy of Sciences), Vol 162, No S, 1965.
2. Kozlov, V. F., "Model of Meandering of Inertial Currents in a Baroclinic _
Ocean," IZV. AN SSSR, FIZIKA ATMOSFERY I OKEANA (News of the USSR Academy ~f
Sciences, Physics of the Atmosphere and Ocean), Vol 6, No 9, 19700
3. Kozlov, V. F., Ten, Ye. V., "One Stationary Problem in MESOOCeanology," METEOR-
OLOGIYA I GIDROLOGIYA (Meteorology and Hydrology), No l, 1978.
4. Arnason, G., Welsh, J. G., "Numerical Prediction of the Gulf Stream by Means
o� the Equivalent Barotropic Model," MITT. ZNST. MEERESKUNDE, Univ. Hamburg,
No 10, 1968. -
5. Bretherton, F. P., Haidvogel, D. B., "i~o-Dimensional Turbulence Above Topo-
- graphy," J. FLUID MECH., Vol 78, No l, 1976.
6. Gadgil, S., "Structure of Jets in Rotating Systems," J. FLUID. MECH., Vol 47, ~
No 3, 1971.
7. Gadgil, S., "Time-Dependent Topographic Meandering of a$aroclini.:: Current,"
DYN. ATMOS. AND OCEANS, Vol 1, No 2, 1976.
8. Godf rey, J. S., Robinson, A. R., "The East Australian Current as a Free Iner-
" tial Jet," J. MA.RINE RES., Vol 29, PTo 3, I971.
85
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9. Jones, 0. K., "The Flow of a Stratified Fluid Over a Vertical Step," TELLUS, _
Vol 22, No 5, 1970.
10. Luyten, J. R., Robinson, A. R., "Transient Gulf Stream Meandering. Part II:
- Analysis Via a Quasigeostrophic Time-Dependent Model," J. PHYS. OCEANOGR.,
Vol 4, No 2, 1974.
_ 11. McCartney, M. S., "The Interaction of Zonal Currents With Topography With
Applications to the Southern Ocean," DEEP SEA AES., Vol 23, No 5, 19760
12. McCreary, J. P., White, W. B., "On a Theory of the Kuroshio Meander," DEEP
SEA RES., Vol 26A, No 3, 1979. -
13. Mclntyre, M. E., On Stationary Topography-Induced Rossby Wave Patterns in a
Barotropic Zonal Current," DTSCH. HYDR. Z., Vol 21, No 5, 1978.
14. Niiler, P. P., Fobinson, A. R., "The Theo ry of Free Inertial Jets. II. A _
Numerical Experim,ent for the Path of the Gulf Stream," TELLUS, VoI 19, No 4,
- 1967.
15. Porter, G. H., Rattray, M., Jr., The Influence of Variable Depth on Steady
Zonal Barotropic Flow," DTSCH. HYDR. Z., Vol 17, No 4, 19640
it
lb. Robinson, A. R., Niiler, P. P., The Theory of Free Inertial Currents. I,
Pach and Structure," TELLUS, Vol 19, No 2, 1967.
17. Robinson, A. R., Gadgil, 5., "Time-Dependznt Topographic Meandering," GEO-
- PHYS. FLUID DYN., Vol 1, No 4, 1970. -
18. Robinson, A. R., Taft, B. A., "A N~erical Experiment for the Path of the
Kuroshio," J. MARINE RES., Vol 30, No 1, 1972.
19. Robinson, A. R., Luyten, J. R., Flierl, G., "On the Theory of Thin Rotating
Jets: A Quasigeostrophic Time Dependent Model," GEOFHYS. FLUID DYN., Vol 6,
GEOPHYS. FLUID DYN., Vol 5, No 3, 1975. _
20. Robinson, A, R., "Dynamics of the Kuroshio Current: Experimental and Theor-
~ etical 5tudies from South of Kyushu to the Izu-Ogasawara Ridge," BRUUN MEM.
LECT., Paris, 1976.
21. Sain*_-Guily, B., "Les Meanders des VeinES de Gourant dans les Ocean," BULL.
INS~. OCEANOGR., No 1108, 1957.
22. Solomon, H., "Co~nent on a Theory of the Kuros~:{o Meander," DEEP SEA RES.,
- voi z5, xo io, 19~s. _
23. Stern, M. E., "Minimal Properties of Planetary Eddies," J. MARINE RES., Vol
33, No 1, 1975.
24. Thacker, W. C., "Spatial Graw*h of Gulf Stream Meanders," GEOPHYS. FLUID
DYN., Vol 7, No 3-4, 1976.
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25. Warren, B. A., "Topographic Influences on the Path of the Gulf Stream," _
TELLUS, Vol 15, No 2, 1963.
26. White, Wo B., McCreary, J. P,, "On the Formation of the Kuroshio Meander and
its Relationship to the Large-Scale Ocean Circulation," DEEP SEA RES., Vol =
23, No 1, 1976.
~
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UDC 551.465(26) (-062.4)
COMPUTATION OF CURRENT VELOCITY IN THE QUASI-ISOTHERM~.L LAYER OF THE EQUATORIAL
- ZONE IN THE OCEAN
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 up 78-81
- [Article by A. B. Polonskiy, Marine Hydrophysical Institute, submitted for publica-
tion 29 Aug 79J
[Text] Abstract: Within the framework of the model
proposed in [5} it was possible to obtain
the velocity profile in the quasi-isothermal
layer at the equator with different coefficients
of turbulent exchange. The balance of turbulent
energy is analyzed.
In [5] the author proposed a two-layer stationary model of a baroclinic ocean, in-
cludin~ the equator, and carried out a number of numerical experiments for com-
puting the characteristics of the quasi-isothermal layer (QL). Use was made of
depth-integrated equations of motion, continuity and heat balance equations. No as-
sumptions were made concerning the mechanism of vertical turbulent exchange of heat ,
and momentum in the QL. Such an approach does not make it possible to determine the
current velocity profile in the QL. Using the hypothesis of proportionality of the
stress and strain tensors adopted in the theory of currents and stipulating the
coefficient of turbulent exchange it is easy to obtain a solution of the equations
of mAtion in a linear approxima4ion and find the current velocity profile in the
QL. However, the following question remains open: to what degree is the solution
dependent on the b~havior of the coefficient of turbulent exch3nge. Evidently, in
the case of a strong dependence it is desirable to confine oneself to computations
of the total fluxes within the limits of the QL, as was done in [5], or consider
the coefficient of turbulent exchange to be an unknown value and find it from addi-
tional considerations.
In [2] a similar problem was solved within the framework of the classical Ekman
problem for purely drift currents in the middle latitudes. ~~e will examine a baro-
clinic ocean, including the equator, with a stipulated temperature model [5]. We
will assume that the baroclinic pressure gradients have already been determ:ined
from solution of the boundary value problem, and we must find the velocity profile
on a stipulated vertic3l.
The equation of motion and the boundary conditions have the following fo rm:
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j(ku')' + fv - ru - G,~ - g a T�s z, ~
~ (kv'j' - fu - rv - G~ - ga ~ z. ~1~
z-0: ku'~- kv'-- Ty, -
P p
~ z=h: ku'-kv'=0. (2)
- Here k is the coefficient of turbulent exchange,
_ Gs = g z f 7x ( h-~- c)-r T' hs~; _
Gv = g a[T~ (h ; c) + T� (hr t c~)],
g a_ 0'25 �C c~ ~ r= 10-6 c-',
T� and h are temperature and ~hickness of the QL, c is the~thickness of the thermo-
cline, considered to be a known function of y, the prime denotes differentiation
for z, the oz-axis is directed downward.
~ 67e recall that in writing (1) use was made of the following model for t:~mperature:
_ ~ _ - - - when z = h (3) -
T 7� eXp h ~ z when z~ h� _
We introduce the characteristic scales: length L(horizontally) and h(vertically),
' velocity U�, temperature T� and limit ourselves to an examination ~f a narrow zone
; near the equator within which f/r < 1. Then for k it is natural to introduce the
scale h2r, and for 'C/f~ the scale U�hr. We note that it is possible to select the
_ characteristic scales somewhat difterently, but for the purposes of this a~ticle
this choice is not fundamental. Multiplying the second equation of sys tem (lj by
~ i and adding it to the first, we ~btain one equation for complex values. Now we
will proceed to a dimensionless form. Asstuni.n~ t~13t k= kp = const, we obtain
. - _ ~ _ . (4)
m"-~~in-G-Tz. _
_ .
, Here m = u + iv, j"- - ` ~ 1 ~ lo _ f ~ _ -
G= (Gs + tCiy), 7'= (TX-}- iTy),
RaT~~(c+h) gaT�h
~ (5)
- ' ko LrUo ' 1"- Ro LrUo '
- z=0: m'=-:, z=1: m'=0,
k~ is the dimensionless value of the coeff~c.ient of turbulent exchange.
The solution (4), with the boundary conditions (5) taken into account, has the form
_ _ . . .
T,~'~) ch (j (1-s)~ G-Tz T ch (jz)
m- j sh j j'~ sh j' ~6)
- Now we will assimme that the coefficient~.of turbulent exchauge decreases with depth _
in conformity to the following law:
,
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k = koe-=.- - (7)
The dependence (7) models the characteristic behavior af the coefficien~ of tur- ~
bulent exchange with depth [3]. In this case system (1) is reduced to the equa- _
tion
m." - m' - mes = (G - Ti) eZ. ~8~
Solving (8), with allowance for the boundary conditions (5), using power series and
limiting oursel~es to terms of the order z3, we obtain the following approximate
~ _
so ution:
m=-(t 1) s(1 2-}- 1 b~' z'1-}- z-}-
~ ~ -
+ G-I 2_+ :(~'~+5)-4G;-T (.Is)= i'z3 1~ ' ~9)
2 4 J7 2 + ~
+ 2 G+j+-}-1-T ~ -
6
In expressions (5), (61, (8) t is norma.lized to kp, that is, in actuality is the .
ratio of the dimensionless wind s~ress ('G = 2 X+ i'C Y) to the dimensionless coef-
ficient of turbulent exchange (kp). Separating the real and fictitious parts in -
(6) and (9), we obtain an expression fo r the velocity components.
Now we will exami:ie a current at the equator. As a simplification we will assume
that the meridional velocity component is absent (the motion is symmetric relative _
to the equatorial plane). We note that this limitation is not fundamental. From (6)
and (9) we find f ; _ T1 ch (1 --s~ _ . _ . _ -
u= -(G- Ta)--T snl
sh 1 .
- ~6')
ua.~l,~-z~-- e~~.. 4(1-~.. 1'- 2"1-G. i
~ ~ (9' )
We then assume L= 5�10~ cm, T= 20�C, c= 2�104 cm, -
U�=5~ c.x/c, h=3� 103 c~c, rX ~-0,5 dyne/csn2. (10)
In accordance with the observational data and the results of computations [5], [6] _
T� and h decrease from west to east. This means tha* G and T are negative values.
Now it is possible to detexmine the velocity pr~files from equations (6') and (9`),
- which, with (10) taken into account, are rewritten in the form
_ .
u- sn 1~ ch (1 - z) -(0,6 z- 4,6) -F. sh l' ~ 6~~~
.
- u=- 3,3 (1,5 - z-~- 4-~- e 0,15 ~ 1-I- 2- 3~'' 4,6.
~ (9 )
In (6'), (6"), (4'), (9") we assumed kp = 1. The velocity profiles aotained using
(6") and (9") are cited in the figure. ~
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ss o o. ~ zn tr2;
, ` ---~--r--T---
.
1
- Q5 1
.
.
, ~
~
~
i
i
_ jrh ~ ~ !
Fig. 1. Current velocity profiles at equator with different coefficients of turbu-
lent exchange. k k -
~)k=-� ~ sfh. s~ A==.
h!~ h=r
The quantitative discrepancies in the distribution of velocity with depth in the
two considered cases are extremely great. We note that some difference in velo-
city at the lower boundar~ of the QL is associated for the most part with the ~
error in the approximate solution (9), which here is maximtmm.
Using the derived solutions we will evaluate the principal terms in the equation
for the budget of turbulent energy within the limits of the QL production due
to velocity shear and dissip~:.ion. ,
Determining ~
_ ` k(u'}~ dz
U ~
and proceeding to dimensional values, we obta:tn the integral production of t::rbulent
energy in the QL. It appears that the principal contribution to the pro duction of
, t urbulent energy will be from drift velo city she ar. The contribution of the velo-
city shear arising due to baroclinic effects is 1 ess than 10%. The characteristic
value for the production of turbulent energy in the QL with the adopted val.ues of
the coefficient of turbulent exchange is 10'2 cm 2/sec3, which coincides with the ex-
perimental data in [7].
Now we will detezmi.e integral dissipation within the limits of the QL, using the
known expression [4]
k3 .
E=~o ,
. ~i1>
where ~ is dissipation, Q is the turbulence scale, c~ = 0.0625.
Integrating (11) within the limits of the QL, we obtain the integral dissipation.
We will assume for the evaluation that the turbulence scale is proportional to the
thickness of the QL [l, 5]. Tt~en :it appears that integral dissipation with a con-
stant coefficient of turbulent exchange is approximately three times greater than
when using dependence (7)o The characteristic dissipation value iz this case is
too 1ow by several orders of magnitude in comparison with the experimental data in
(7]. In order to obtain the dissipation of turbul ent energy in the QL of the ord^r
of magnitude 1o-2 cm2/sec3 it is necessary to increase the coefficient of turbulent
exchange to a value of the order of magnitude lOZ cm2/sec. (We recall that o~e as-
sumed k= h~r = 9 cm2/sec}.
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In conclusion will formulate the principal conclusions:
~1. The veloci~y profile in the QL is determined to a considerable degree by the co-
efficient of turbulent exchange. Accordingl~, it is desirable, using integral ~odels
similar to those proposed in j5], to limit ourselves to a determination of the total -
flu~:es into the QL.
2. It is necessary to make precise use of the equation for the budget of turbulent .
energy in camputations of t;~.e characteristics of the QL with stipulated coef-
ficients of turbulent exchange, since an insignif icant change in these coefficients
_ can lead to an inco rrect description of the balance of turbulent energy in the QL.
BIBLIOGRAPHY
1. Kitaygorodski}-, S. A., "Dynamics of the Upper Thermocline in the Ocear: Results
of Science and Technology," OKEANOLOGIYA (Oceanology), VINITI, Vol 4, Moscow,
1977. -
2. Koziov, V. F., "Influence of Change in the Coefficient of Vertical Exchange on '
Drift Currents," IZV. AI3 SSSR, SERIYA GEOFIZICH. (News of the USSR Academy of
Sciences, Geophysical Series), No 7, 1963. ~
3. Marchuk, G. I., Kochergin, V. P., Klimok, V. I., Sukharukov, V. A., "Mathemat-
ical Modeling of Seasonal Variability of the Surface Turbiilent Layer in the
Ocean," IZV. 4N SSSR, FIZIKA ATMOSFERY I OKEANA (News of the USSR Academy of
Sciences, Physics of the Atmflsphere and Ocean), Vol 14, No 9, 1978.
4. Monin, A. S., Yaglom, A. M., STATISTICHESKAYA GIDROMEKHANIKA (Statistical Hy- ~
dromechanics), Part I, Moscow, Nauka, 1965.
5. Polonskiy, A. B., "Computation of the Characteristics of the Quasi-Isothermal
Layer in the Equatorial Zone of the Ocean," METEOROLOGIYA I GIDROLOGIYA _
(Meteorology and Hydrology), No 3, 1980. _
G. "The Lomonosov Current," CB. STATEY MGI (Collection of Articles of the Marine
Hydrophysical Institute Ukrainian Academy of Sciences), Vol 34, 1963. _
7. Jones, J. H., "Vertical Mixing in the Equatorial UndPrcurrent," J. PHYS. U~r'.,ANO-
GR. , Vol 3, No 3, 1973.
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~ -

~ UDC 556 (531. 4+535. 8)
COMPUTATION OF THE CON~ENTRATION OF GLOBAL ATMOSPHERIC IMPURITIES IN RIVERS AND
- CLOSF.D WATER BODIES
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 82-89
[Article by Candidate of Pnysical and Mathematical Sciences K. P. Makhon'ko, Insti-
tute of Experimental Meteorology, submitted for publication 14 Mar 80]
- [Text] Abstract: A study was made of the process of =
contamination of rivers and closed water bodies
by an impurity entering from an atmospheric source =
_ of a global scale. The contribution of atmospheric -
_ fallout of impurity to the contamination of water
- in the territory of a basin and ocean area, the -
- erosion from the soil surface of the basin and the _
- role of bottom deposits are taken into account. In
the case of rivers an allowance is made for the in-
= fluenc:e of runof f and the dilution of contaminated =
waters by pure ground water~ A comparison of the
- results of theoretical computations of the change
in the concentrations of Sr90 in the rivers of the
Moscow area and the USSR during 1954-1975 with data -
- from direct observations indicated good agreement.
- T'iie conta~ination of rivers and water bodies occurs as a result of local effluent
- and the action of sources of a global scale. An example of this type of source is
- t~ie atmosphere the carriPr of atmospheric dust, cosmogenic isotopes, radioac-
tive products of powerful nuclear explos ions injected earlier into the strato-
spheres etc. All these substances have a com-non criterion the global scale of -
their entry from the a~mosphere onto the imderlying surf~ce; henceforth for brev-
ity they will be called global atmospheric impurities.
~Je will examine the entry of global atmospheric i~purities into rivers and water
J bodies.
The contamination of a closed water body is caused by the following factors: a)
the entry into it oi atmospheric precipitat~~i: flowing from the surface of the
soil-vegetation cover and carrying a contan~inating impurity falling from the atmo-
= sphere in the course of the year; h) fal.'..ing of the impurity directly on the sur-
= face of the water body; c) partial washing-out from the soil of an impurity ac- -
- cumulating in it during all the preceding years due to the falling of precipita- _
tion [9]; d) so~ption interaction between an impurity present in the water and an _
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impurity in the bottom deposits of a water hody j10]; e) biogenous migration of
an impurity. Biogenous migration can usually be neglected [I1]. Then, summing the -
- concentrations created in the water body by the factors enumerated above, the
concentration of impurity in the water is represente~l in the form
C= Cp T CM + C, CE � (1)
Now we will examine the individual terms in (1). The concentration created by the
runoff of atmospheric fallout of an impurity from the area of a drainage basin
into a water body in the current year is represented in the form
P (t) ~2~
Cp = hP ~t~ kp + -
where P is the annual atmospheric fallout of the impurity, hp is the r_hickness of
the annual layer of atmospheric precipitation which in the form of surface runoff
~ carries the impurity falling in its basin to the water body, t is time, kp is the
- coefficient of dilution of runoff water contaminated by atmosph?ric fallout af
_ the impurity in the waters of the water body.
The concentration forming as a result of fallout of an impurity from the atmosphere
directly onCo the surface of a deep water body, as was demonstrated in [4], can be
- represented in the form - - - - -
e P(e) _dM~t-e)
C~t -.1 H~e~ e d 6, (3)
0
where H is the thickness of the quasihomogeneous layer of water i.n the water body,
1~-M is the constant elimination of impurity from a layer with the thickness H into -
- the deeper layers of the water body. If the impurity decays with time (for example, _
there is radioactive decay of the isotope), then J~ M=/t'M that is, also in- _
cludes the constant of this decay
For a shallow water body, mixed tc the bottom, the H value acquires the sense of
_ its mean depth and 1~.M is the constant elimination of the impurity from the water
into the bed of the water body.
The C a. value is the concentration of impurity in the water body, forming as a re-
= sult of removal of that part of the impurity accumulating on the soil surface due -
_ to the falling of precipitation: -
_ (~1 ~4)
C,=a~~~ ,~~t~ k,, _
where c7 is the active store of im~urity in the surface soil layer with the thick- -
ness b~ which is subj ect to washing out, ~t/ is the water volume which is capable -
_ of being replaced in a unit volume of soil r~hen the impurity is washed out by -
atmospheric precipitation, k o- is the coefficient of dilution of runoff water con-
taminated by the impurity washed out of the soil in the waters of the water body. -
_ The value is dependent on the moisture content and moi.sture capacity of soils in
- the basin, the dissection of relief (in particular, due to slope steepness) and -
~ ~ similar factors. Some of the impurity in the course of migration gradually leaves
the soil layer subject to washing out or passes into fixed form, being held firmly _
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in the soil layer. The te~poral change in the aetive supply of impurity in the ~ur-
_ face layer of soil with the thickness b can be represented in the following way: -
dt -p~t~-'>>?~ C5) -
whoxe ./l. ~./1Q ie the constant eliminaCion of the mobile part of the ~mpurity
= from the surface layer of the soil, also taking its decay into account.
Assuming Or= 0 with t= 0, we obtain a solution of equation (5) in the form
_ _ _ _ _ .

- r
3_ ` p ~9~ e-s; ~r-e) d8. ~6~ -
J
0
The concentration of impurity forming as a result of exchange processes with bottom -
deposit~ in the water of a shallow water body mixed to the bottom, CE,is determined
- by the dynatnics of sorption-des~rption processes. If the concentration of the im- -
purity in the water is maintained at the same level (d~/dt = 0), a mobile equil-
_ ibrium is established relatively rapidly between the content of impurity in the =
_ water and in the bottom deposits; this equilibriinn is determined by the distribu-
tion ~oefficient Kd. We will neglect the slow process of diffusion of the impurity
into the depth of the layer of bottom deposits [10]. With an increase in the con-
centration in the water (dC/~dt > O) the equilibrium is impaired and a flux of im-
- purity into the bottom deposits arises~ with a decrease in the concentration in
the water (dC/dt < 0) a flux of impurity arises fxom the bottom deposits into the -
- water. Thus, the CE value can be assumed to be proportional to the rate of change .
, in the concentration of impurity in the river water and be written in the form
_ C 1 dC
E aE ar ~
where /tE _/t,'E + ~1 is the constant of exchange of the impurity between the active
' layer of bottom deposits and water. This value, evidently, is determined by the
rate and character of the bottom currents, the type of bottom deposits and simil ar -
- factors. Now substituting (2)-(4), (6) and (7) into (1), we obtain the equation
_ _ . . .
1 dC (t) ~ r P idl - -1M (t-A~_. +
~E dt + C~t~ kP hP + 1 H~H~ e d 9 _
- 0 C8~
+ k ~ P(Al e-.~, ~r-e) d d.
~f o~e, ~~M, .
0
- Assuming C= 0 with t= 0, we obtain the general solutior_ (8) in the form of quad-
ratures - - - - _ - . . _
- r
- C=~Lp1~E `h~ el~ e-~F (t-e) dfi-{-
J
_ (9)
- 1 E(`' -11� (t~{~ Ep(8 ~ dM (E-A) -
11 o e ~ H~g ~
I e-d, - e~ -
- + a, a~8~ Y~ ~ dE dfi. -
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For water bodies which are great in area but ahallow, which are mixed to the bot-
tom, the H value, as already noted above, acquires the sense of the mean depth of
the water body and in (9) can be removed from the integral sign. In this case for
~ some climatic zones, where evaparation of moistu~e from the surface of th~ water
- body is great, the right-'land side of formula (9) must be additionally multiplied
by the concentration coefficient kf.
- The expressions for the concentration coefficient and the dilution coefficients
in (`L) and (4) can be ~aritten in the form
~ V -1 hP SP. h~ SP v v
; k!-~1- v) ~ kP- (H-1-hp)J'~ v' k~- v+Y'`~v~ (10)
- where Q V is the quantity of w~ter evaporating from a water body with the volume V,
Sp and S are the areas of the drainage basin and water surface respectively.
If the quantity of fallout of the impurity from the atmosphere does not change with
- time, then assuming the mean values for the parameters hP, S and v varying little
from year to year, the solution (9) is easily represented in analytical form
(11)
- C= aP -(be-.~, t; ce~ t,{- de- 11E r~ p~
where _
- k
f k p k~ k, k~ k~ k, :1~.
a hP + 3~~1J + H ~M ~ b " a~: ~ (JE - .1J ) ' .
k~ .~E kJ kP k~ k' -
- ~ - d = - - (12 )
HAN(-lE -JM)' hp a�~ (.~y -_1E }
- kf
H~.1N_ ~F~.
In the case of small, shallow water b~dies the fallout of impurity onto the water
surface can be neglected. Then in (1) C~,~ = 0, the second term on the right-hand
side of (8) becomes equal to zero and the concentration of impurity in such a
water body will be described by the formula
I ` -.1~. (t-E)X
_ C k~kP ~E.lr p((a)' e- .~E ~~-N~ dH ; kfka :~E I ~ - -~~.,i _
- b ~1~~
~ 1` P (h? - ~ (c - ?3) ~
X i a~g~ �,~Aj e ' ~c'.
0
With P= const (13) is transformed into (14):
C= uP- (be-"i,r de-.~fr ~ P~ ~14~
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- where -
(I khpP d~,~ ' [t khpP a- a�~ (.1' k' _1E ) � ~15~ .
and the b parameter retains its former value (12). _
For large deep water bodies it is possible to neglect the role of bottom deposits -
~ in the contamination of water. Then in (1) CE _ ~ and the first ter~n on the left-
- hand side of (8) becomes equal to zero. The concentration of impurity in such a ~
water body is - - - . ~
ptr~ ; e--1,K(t-d) e-.lz (t-9)
- C = aP + ~ p (9) -H ~A~ - k. ~H~ ~e~ d 6. (16) -
. a ~
With P= const (16) can be represented in the form (14) with replacement of.JI.E by _
M in (14) and the following values of the parameters a, b, d: -
' Q_,~p _ i k, b_ k, d_ 1 (17)
hP H.\,K t dv:17 , ~7 . - N.l,u
Now we will examine contamination of the river system by an impurity. In this case
it is possible to neglecti the atmospheric fallout of impurity onto the surface of
the river water (C~~ 0); however, additionally it is necessary to take into ac- _
count the runoff of the impurity together with the river water and the dilution
of the contaminated river water h;- Fure ground water. If we limit ourselves to an
~ examination of the mean annual water runoff and the water suppln of the river and
. in the first approximation assume these values to be constant, the change in the _
~ content of the impurity in the river as a result of runoff, on the one hand, can be
represented in the form of the product of water volume V and the change in the con- -
~ centration of impurity in the river dC. On the other hand, with an intensity of _
water runoff Q along the river during the time dt there will be a runoff CQdt of _
the itnpurity. In a general case it is also necessary to take into account a dacay -
_ of the impurity during this time of ~ Cdt. Equating these expressions, we obtain -
an equation describing the change in the contamination of river water as a result
of the diiution of contaminated water by pure ground water, compensating the run-
off. -
d~ _ - ~ C - ~ C = - AQC. (18)
Here ~ is the dilution constant, characterizing the multiplicity of replacement of
the wa ers in the river during a unit time (with decay taken into account).
Solving (13) and (18) jointly, we obtain an expression for the concentration of im-
purity in river water: _
kl kP AF P~r~ k!'~a ~F ~ P(9)
C - ~E - ~Q hP l11 _~E -bQ ~ a ~d~ ~y~ X (19) -
= x e-- i r- e) d E~ + Co,
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where C~ is the "background" concentratitn in the river from sources of contamin-
ation not taken into account, with water ~vaporation taken inta account.
If P= const, with constant values hp, ~ and v E~.9) can be represented in ana-
1y tical form
- C=aP-be-"~,rP+Co, (20)
where kP k, k~ d f k~ k, ~.F
�f - ( hP + a�.a, ~ ~E - ~Q ~ b - atie, ~~E - ~Q ~ ' ~21~
Now we will compare the computed concentrations of impurity in the water with data
from direct measurements. At the present time the greatest amount of material has
been accumulated on the Sr90 concentration in the rivers of the Moscow area [1, 2]
and in the rivers of the USSR as a whole [3]. Source [2J gives an empirical formula
fo r the Sr90 concentration in the rivers of the Moscow region
C= a p~- b h-}- c, (22)
wtaere h is the annual quantity of precipitation, A is the reserve of Sr90 in the
- sail.
- TtLZ empirical coefficients a, b and c, refined in [1], are equal to 0.031, O.OI2
_ year 1 and 3�10'13 Ki/liter. By comparing formulas (22) and (19) it is possible to
_ no te a certain simi.larity between Chem. However, the hp value is dependent on the
state of the soil surface and therefore is unrelated linearly to h, and also to
b v. In addition, (22) does not take into account the temporal decrease in the
- quanr_ity of Sr90 subject to wa~hing out, although this effect is observed in ac-
tual practice [5]. All this explains why in Fig. la the concentration curve C(t),
_ computed using (22), after 1965 deviated from the experimental data. This figure
shows tl~.e results [ 6] of the computations which we made using foYronula (19) , which
_ is conveniently written in the form
. . _ _ . . _ _
C ~t) = aP (t) -t- b c (t) Co. (23)
The values of the parameters in (23) are as follows:
_ kjkP a~ 4 . k,ks e�
_ ~~E _ 1Q ~ hP = S+1 � 10 year/cm, b _ -dQ ) z�,
- 3,2 � l0-; c,+t-', ~1, - 0,14 Year 1, Co = 2.1p-13 Ki/liter. (24)
Since hp, ~ and v were assumed to be constant in the first approximation, the errors
arising ss a result of this are included in the C~ valuA, which in a case if th~
value of the errors is significant, to some degree acquires the sense of a fitting
parameter. In computations of C(t) the experimental values P(t) for the Moscow re-
gion in 1962-1967 were taken from [2]; data for th~ remaining years were found for
- the Mosc~w region on the basis of the intensity o~ tY1e fallout onto the territory
of the USSR as a whole [3, 7J (the fallout ontca the territory of the Moscow region
was 10% less).
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~
,
' FOR OFFICIAL USE ONLY
c,~o~'tKu/n Ki/liter aPe6,~o~'~xu~n Ki/liter
r
o, ai
-
,
~o ~ ; ~ 1
_ ` , i i: Q,S
- ~ .
- 0 2 ;
- 2
~ ,
r:3JJ 9960 i96S 1970 1915 ~ 1953 1960 1965 1910 19bf
Fig. l. Concentration of Sr90, averaged for Moscow region. a) in rivers, computed
using formula (22) (1) and using formula (23) (2); in surface waters contaminated
_ by runoff of isotope from soil surface, flowing in1:o rivers b 6(1) and washing
away of atmospheric fallout aP (2).
- In Fig. la the dots represent data from observations of the concentration of Sr9d
in the rivers of the Moscow region. It follows from Fig. la that the C(t) curve,
computed using formula (23), coincides well with the data from direct measure-
ments, This coincidence makes it possible to use formula (23) for computing the
S~9U concentration in the rivers of the Moscow region during the preceding years,
begiruiing in 1954, when the radioactive fallout fram the atmosphere increased
~ sharply. Figure 1a shows that in 1959 a peak in the concentration of Sr90, ap-
proximately corresponding to the 1967 level, should be observed in the rivers of
the Moscow region. This peak was caused by the entry of Sr90 into the river after
falling from_ Lhe atmosphere in this same year~ as can be seen clearly from Fig.
- lb, which shows the temporal change in the terms aP and b O' in formula (23). It
also follows fram this fi~ure that the constribution of the washing away of atmo-
spheric fallout aP by precipitation and the contribution of washing away of the
isotope b d, accumulating in the soil, to the contamination of river water, are
quite close in value during the periods of increase in atmospheric fallou,;, but
sharply diverge during periods of a decrease of fallout. In particular, after
1963 washing away from the soil exceeded the entry of Sr90 into the rivers with
fallout by a factor of 1.5 to 17.
It should be noted that the assumption of a constancy of the Cp value in (23) which
proved to be valid for the C(t) curve segments for 1961-1972 (Fig. la) cannot be
_ made for the preceding years, which makes the corresponding segment of the C(t)
curve less reliable.
We will make an estimate of the possible changes in C(t), for which we will com-
pute this value under the extreme ass~ption that duizng the period 1954-1961
the Cp parameter increased linearly from zero. In Fig. la this part of the curve
- has been plotted as a thin line. It can be seen that the general character of the
curve did not change under these circumstances.
Now we will examine the ~emporal changF in the mean Sr90 concentration in the riv-
- ers of the USSR.
94
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aP, d6,10-':A'u/n Ki/liter
c,~o'"Ku/a Ki/li.ter �
Z ~0 6)
Q)
m -
1 qs
_ , o
I oe
I / e ?
~ / !
~ 1953 1960 1965 1910 1975 ~ '95S 1960 796.i 1S7G 1975
Fig. 2. Sr90 concentration averaged for the C~rritory of the USSR. a) in rivers,
computed using formula (25); b) in surface waters contaminated by runoff of the -
isotope from the soil surface b O' flowing into rivers (1) and washing away of at-
mospheric fallout aP (2).
_ It was demanstrated in [3] th~~ with averaging for the entire territory of the
country the max~m~ of the Sr concentration in river water is observed in the
year following the maximimm of fallout of this isotope fram the atmosphere. This is ~
attributable to the effect of the lag of Sr�`--' migrating along the system of tribu- -
_ ta~ies into large rivers and along the way in a state of sorption interaction with
bottom deposits in the tributaries [8]. Taking the mentioned ti.me shift into ac- -
count, formula (23) for the rivers of ~he USSR is represeuted in the form ~
_ -
C(tj-aP(t- 1)+ba(t-1)-~-Co, (25) -
where a= 1�10-3 year/cm, b= 1.4�10-4 cm 1,.~Q = 0.16 year 1, C~ = 4.4�10'13 Ki/
liter. The corresponding C(t) curve is shown in Fig. 2a; the dots in this figure
represent ob~ervational data taken from [3]; data on the fallout of Sr90 were tak-
en from [3, 7]. As earlier, for 1954-1961 the C(t) curve was computed in two ex-
treme variants: with Cp = const and with Cp increasing linearly from zero (thin
line).
An examination of Fig. 2a makes it possible to conclude that despite the combining
- of data for extremely diverse rivers flowing through the territory of the USSR,
the coincidence of the theoretical curve with the observational data can be deemed ~
entirely satisfactory. The assumption made concerning an increase of the C~ para-
meter during the period up to 1961 did not substantially change the nature of the C
(t) curve; in particular, a clearly expressed maximum of the Sr90 concentration
in the rivers of the USSR in 1960 was maintained. It follows from Fig. 2b that
this maximum was caused by intensive fallout of the isotope from the atmosphere
in 1959. After powerful injections of Sr90 into the planetary atmosphere there was
also an increase in the intensity of its expulsion from the atmo5phere. During
these periods the contribution of washing ost of fallout of the isotope to the
contamination of river water predominated over the contribution from washing out
from the soil. After 1965 the picture was the reverse :zd the contribution of the
washing out of Sr90 from the soil up to six times axceeded the contribution from _
washing out of its fallout. -
100
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- ~ BIBLIOGRAPHY
1. Bobovnikova, Ts. I., Makhon'ko, K. P., "On the Problem of Migration of Sr90 =
in Fresh Waters of the Land," RADIOEKOLOGIYA VODNYKH ORGANIZMOV (Radioecology �
of Wat~r Organisms), No 2, 1973.
2. Bobovnikova, Ts. I., Sereda, G. A., Shulepko, Z. S., "Relationship Between
- the Sr90 Conteat in Fallout, Soil and Rivers, According to Measurements Dur- -
ing 1961-I967 in the Moscow Region," TRUDY IEM (Transactions of the Institute
of Experimenta?. Meteorology), No 5, 1970.
~ 3. Bobovnilcova, Ts. I., Avramenko, A. S., Makhon'ko, K. P., D3.btseva, A. V., Volo-
- kitin, A. A., Chumichev, V. B., "Sr90 Concentration in Surf ace Waters o� the
' Land in the Territory of the USSR," METEOROLOGIYA I GIDROLOGIYA (Meteorology
~ and Hydrology), No 9, 1977.
4. Makhon'ko, K. P., "Cem~utation of Contamination of Surface Waters of the Cen-
tral Region of the North Atlantic by Atmospheric Fallout of Sr90," METEOROLOG- -
IYA I GIDROLOGIYA, No 3, 1979. _
i -
5. Makhon'ko, K. P., Avramenko, A. S. Bobovnikova, Ts. I., Chunqichev, V. B., _
"Runoff Coefficient of Sr50 and Cs~37 from the Soil Surface of a River Basin,"
METEOROLOGIYA I GIDROLOGIYA, No 10, 1977. _
6. Makhon'ko, K. P., Rabotnova, F. A., "Rnle of Radioactive Contamination of the
Soil .and Atmospheric Fallout in the Entry of Sr90 into River Water," MATERIALY
VI VSESaYUZNOGO SIMPOZIUMA PO SOVREMENNYM PROBLEMAM SAMOOCHISHCHENIYA VODOYEM-
- Oil I REGULIROVANIYA KACHESTVA VODY. II SEKTSIYA. GIDROKHIMICHESKIYE I SANITARNO-
BIOLOGICHESKIYE ASPEKTY SAMOOCHISHCH~tIYA,, CH. I(Materials of the VI�All-Union -
Symposium on Modern Problems in the Self-Purification of Water Bodies and Regu- _
lation of Water Quality. Section II. Hydrochemical and Sanitary-Biological As- -
pects of Self-Purification, Part I), Tallin, 1979.
7. Makhon'ko, K. P., Avramenko, A. S., Silant'yev, A. N., Polyakova, T. V., Chum- _
, ichev, V. B., Malakhov S. G., "Soil Accumulation of Sr90~ ~S137~ Ce144 and
Zirconium With Niobium~s as an Average for the USSR," TRUDY IEMi, No 6(64), 1977. _
8. Makhon'ko, K. P., Bobovnikova, Ts. I., Avramenko, A. S., Dibtseva, A. Vo, Vo1-
oktina, A. A., "Vertical Distribution of Sr90 and Cs137 in the Bottom Deposits
of Rivers and Lakes," EKOLOGIYA (Ecology), No 3, 1975. _
- 9. Makhon'ko, K. P., Volokitin, A. A., Avramenko, A. S., Bobovnikova, Ts. I., "Mi-
gration of Radioisotopes in the Soils of the Coastal Sectors of Rivers," TRUDY
IEM, No 6(64), 1977.
10. Makhon'ko, K. P., "Migration of Radioisotopes of Global Origin in Some Natural
Media," RADIOEKOLOGICKA KONFERENCIA, 1 diel, Stary Smokovec, 1972.
- 11. Rovinskiy, r. Ya., Makhon'ko, K. P., "On the Problem of Migration of a Radioac-
tive Impurity in the Bottom Materials of Water Bodies Without Through Flow,"
TRUDY IPG (Transactions of the Institute of Applied Geophysics), No 8, 1967.
_ 101
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WEATHER AND OPTIMUM TIMES FOR THE SOWING OF WINTER CROPS
Moscow T.i??TEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 90-97
[Article by Professor A. P. Fedoseyev, All-Union Scientific Research Institute of
Agricultural Meteorology, submitted for publication 3 Apr 80]
[Text] Abstract: A study was made of the dynamics
of the decrease in the yie:.d of winter
wheat and rye, the change in structure of
th~ yieid and the effectiveness of mineral
fertilizers in connection with deviation of
the sowing times from the optimum dates. A
tendenGq to a shifting of sowing times to
later dates with an increase in the sophistic-
- ation of agricultural techniques is demon-
strated.
Sowing times exert a great influence on the quality and magnitude of the yield. Sow-
- ing dates are related to the conditions of growth and development of plants, rasis-
tance to unf avora~le meteorological phenomena, harvesting conditions and effective-
ness of agroengineering measures. By choosing different sowing times there is a def-
inite possibility for exposure of plants to conditions of a different length of day-
time, temperature and mo isture content of the air and soil.
Tt~e dependence of the yield of grain of winter crops on the sowing times has the
' character of a~single-peaked curve. Figure 1 shows a series~ of such curves for
- different natural.-economic regions of the E~iropean USSR constructed by thE author
in collaboration with A. I. Snetkova on the basis of a generalization of long-term
field data for experiment al agricultural stations and scientific institutions (a
tofal of 2,640 experimental years was analyzed). Most o� the experiments were
carried out during the period 1967-1976 and were carried out uising new regionaliz-
ed varieties of winter wheat and rye. The maxim~mm yield in the case of an optimum
sowing time in each case was assigned the value 100~.
The relative curves of the yield of winter crops by sowing times characteristic for
the regions of the Nonchernozem zone and the Middle V~lga hace a sharper peak. Un-
der these conditions the mean long-term range for the optimum period of sowing of
winter crops (wi~h a smal 1 underharvest of grain in the range 5% of the maximum)
is 12-14 days (Fig. 1, curves 1-4). Beyond the limits of this mean interval there
is a marked dropoff of the crop yield.
102
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-
i �~1G0"i ~
^ 90 r ~ � -
"V0~ BD o% ~
� 9G~ 70 ~ ~
90~ f00~
70~ 90 Z
100r DO -
90~ 7p ~ ~ -
_ 6D ;OJ
- 10~ 30
6~7~ 90 l00 ~
10 9Q y s
'00
90 ~ 6 -
- IOC ; BO
- ?Gr~ 70 ~ -
NOF 60 100 ~ � ~ ~
7C~ SO 90 -
700 g B
90
�30 BO ~
90 70 � r -
_ BO 70
~ ~ ;t
'
20 91 : J; 10 20 30 10 ZO J 1
uiDAe OC.1C' GCn/p6pn ,OKTA6da
Jul Aug Sep Oct
Fig. 1. Yield of winter wheat ar?d winter rye (in the northern regions) in dependence
on sowing ti.mes fox the European USSR. Regions: 1) Northeastern Nonchernozem; 2)
Northwestern Nonchernozem; 3) Middle Volga; 4) Central Nonchernuzem; 5) Wooded Steppe
Zone; 6) Lower Volga; 7) Steppe Zone; 8) Belorussia; 9) Baltic republics; 10) North-
, ern Caucasus; 11) Dry Steppe Zone �
SP nn ~o
~ 3Q n
-i 9
I
e
i 20 5 -
7
-i 6 0 0 ]
4
_ ~0 y
~ 10D 600 ~D00 1900 'C =
_ Fig. 2. Correlation between mean duration of period of sowing of winter crops (SP; -
in days) and sum of negative temperatures during winter (�C). Notations of the re-
gions (figures next to the circles) are the same as in Fig. 1.
The next group of more flattened curves refle~ts the conditions prevailing in the
wooded steppe and steppe zones, Belorussia and the Lower Volga (Fig. 1, curves 5-8),
Here the mean range of the sowing period increases to 20 days. _
The curves for losses of the grain y ield with deviation from the optimum sowing
times iu the southern ~egions with waYm winters and in the Baltic republics have -
a still smoother shape (Fig. 1, curves 8-11). The mean xange of the period of
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; sc~wing of winter crops with losses not greater than 5% of the yield atta~ns 27-32 -
days.
In regions with a prolonged gentle autu~a period (Baltic republics, Belorussia,
- Northwestern Nonchernozem, Northern Caucasus) a de+riation in the optimum sowing
times by 20 days both in the direction of earlier and later times exerts an effect
in an underharvest of grain yields to an approximately equal degree in the aver- -
age range 15-20%. In the remaining regions of the European USSR with a more contin-
- ental c'limate the greatest underharvest occurs c~rith late sowing times (on the aver-
age by 22-35% with a lag of 20 days) in comparison with early sowing times (15-25%).
Accardingly, the greatest loss is from a lag in the sowing of winter crops, which
most frequently is actually observed. '
It is a legitimate assumption that the duration of the possible sowing period for
different regions is related to the severity of winter [11]. In regions with gentle
warm winters there can be a periodic winter growing season for plants favoring an
intensification of shoot formation in incomplete~y developed crops. Also of impor- _
tance for the additional bushing-out of winter wheat is the duration of the early
spring period. For example, in the Baltic republics the transition of air tempera-
ture from 0 to 10�C occurs on the average in 42 days; in the central regions of the _
Nonchernozem zone this period is ?4 days. Milder conditions of winter and spring
to some degree favor a smoothing of the differences in the development of plants
sown at different times.
The dependence of the mean duration of the sowing period of winter crops on the sum ~
of negative temperatures in winter is clearly confirmed by Fig. 2. Such a correla-
tion is also traced with other indices of winter severity.
It should be noted that the mean long-term duration of the period of optimiun sowing
cannot be used for specific years. -
Plants sown in the optimum sowing periods ~re more winter-resi.stant and productive. _
~ With an increase in the deviations of the sowing times from the optim~ times the
density of the productive etem stand is reduced by the time o� the harvest. This is -
particularly conspicuous at later times, when with a lag of 30 days the density of
the stem stand is 409~ less than the mean density for optimum sowing times (Table 1).
One of the rea.sons for the thin~ning-out of the stem stand is an appreciable decrease ~
in productive bushiness in the case of late sowing times, whereas the bushiness for
early-sowed plants remains virtually unchanged. Plantings in late and to a greater
degree in early sowing seasons are less resistant to unfavorable wintering condi-
tio~s. Their survival during the winter on the average is 7-10% less for late and
15-22% less for early sowing times.
- The absolute mass of 1,000 grains, according to averaged data, also decreases and
this is particularly conspicuous for late times (on the average by 14%). The number -
of grains in an ear on the average is somewhat greater for plants sown at early
times and is somewhat less for plants sown at late times, although under the specific _
conditions of indiyidual years the inf ruct escence of the grains in the flowers can =
be more intensive in plants sown at late sowiug times.
104
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Sowings at early times are considerably more sub j ect to damage by f rit and Hessian
flies, are populated by greenbugs, are damaged by blight, brown rust and root rot. -
In the case of late sowing times there is an increase in the damage inflicted by
soil click beetles, surface caterpillars and also inflicted by covered smut. Plant~ _
with late sowing times are damaged more intensively by rust during earing. -
- In addition, with late sowing ti.mes the forming of the grair~ in the southern regions
falls in the period of highest temperatures, and in the northern regions, on the
other hand, transpires more slowly when there is cold, frequently rainy weather.
The change in the effectiveness of mineral fertilizers in dependence on the times
for the sowing of winter crops is of special interest. The displacement of response
, of winter crops to fertilizers in the direction of later sowing times observed in
some experiments is evidently attributable to a slowing of the pr~cesses of mineral-
_ ization cf organic substances in the soil with a decrease in air temperature. How-
- ever, other factors also become operative which give the reverse effect: a slowing
of the entry of nutrients into the plants with a definite minimum of temperatures
and a shortening of the period of the active growing season for late-sown crops. _
It is known that with a temperature of 8-10�C there is an appreciable decrease in
the entry of nitrogen into the roots and moveMent from the roots into ttie plant or- `
_ gans above the ground and its use in the formati.on of organic nitrogen compounds is -
weakened. At still lower temperatures (5-6�C and below) the root absorption of nitro- -
gen and phosphorus is sharply reduced.
On the average, in the Ivonchernozem zone the air temperature drops from 14�C in the
10-day period of optimum sowing times to 12�C by the next 10-day period. With such =
' a sowing time, as a result of weakening of the processes of mineralization of matter,
it is possible to expect some increase in the response of winter crops to fertilizer.
Sowings in the next (second after the optimum time) 10-day period occur under condi- _
tions of a reduced temperature of 10�C, already causing a slowing of the ~ovement of
. nutrients. In the third and fourth 10-day periods there is a further temperature de- -
crease to 8-6�C. Under these conditions there is a marked impairment in the entry of
_ mineral s ubstances into the roots and the growing season for plants is reduced by -
3-4 10-day periods. W ith these sowing times it is necessary to expect a considerabl e
decrease in the effectiveness of fertilizers.
~ The ideas expressed here are confirmed well by the actual data (Table 2). ~n the av-
- erage, some increase in the response of winter plants to fertilizers is observed ~
only with a small deviation of the sowing times from the optimum. With a lag of sow-
ing by 20 days the effectiveness of the fertilizers is already reduced by 20%. How-
ever, some shifting of the maximum effectiveness of fertiliz~rs to sowing times
somewhat later than the optimum times does not at all mean a real advantage. This
small effect is offset by a far more considerable underharvest associated with devia-
tion from the optimum sowing times (see Table 2). For example, against a fertilized
background with a lag in sowing by 14-20 days there is a Ioss of 10-35% of the crop .
yield. It is characteristic that on crop areas without fertilizers these losses are
somewhat greater 15-40X,
- Thus, rigorous adherence to the optimum sowing times for winter crops, dependent for
the most part on weather conditions, is in essence a considerable reserve for in-
- creasing the effectiveness of mineral fertilizers. For this reason it is of great
- practical importance to develop reliable methods for agxometeorological prediction
of sowing times. 105 .
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- The mean long-term times for the sowing of winter crops are rather con~ervative.
One of the founders of Soviet agroclimatology, A. I. Voyeykov, as early as 1884,
- on the basis of a generalization af the many years of experience of agricultural- _
ists, established the mean aowing times for the territories of Russian provin.ces
~1]-
N. N. Yakovlev compared these data with the long-term dates of sawing computed us-
ing climatic data on the duration of the period of autumn deve~opment of winter
crops of 52 days, reckoning from the date of transition of air temperature through
S�C in autumn [17]. The discrepancies between the mean sowing times according to
A. I. Voyeykov an3 the computed dates in most cases were insignificant and with
few exceptions fell in the range of four days.
A coc~arison of the mean sowing ti.mes for winter crops by provinces, established
in the 1870's-1880's, and the corresponding mean oblast sowing times for a 15-year
, periad taken from agroclimatic reference books (1940-1955),indicates their consis-
tent shift by 2-5 days later and in individual cases a few more days. In turn, a -
comparison of the mean sowing times for winter crops for the period 1940-1955 _
(according to data from agroclimatic reference books) with the mean statistical
values (50% of the sown area) for 1964-1976 for the most part confirms their -
shift by 2-9 days later.
The shift of the sowing times for winter crops tu a later time must be attributed
- to the introduction of new varieties of winter wheat of the intensive type and a _
general increase in soil fertility as a result of the application of fertilizers _
during recent years. _
It is known [9] that intensive varieties of winter wheat form higher yields in the _
case of sowing at later than the usual ti.mes. It is also known that the greater
the soil fertility, the more intensive is the growth of plants and the shorter is _
the time period which is required for the completion of autumn development. For
example, in the central regions of the Nonchernozem zone the optiminn period for
- the sowing of intensive varieties of winter wheat without the use of fertilizers
- is 2a-30 August, whereas with the application of fertilizers it is displaced to
the third ten-day period in August - first ten-day period in September [5]. With
a high background of fertilization and with the foxvierly reeo~ende3 earlier sow-
ing times th~ere is a nonproductive large autu~ loss of nutrients, an intensiried
- damping of plants and a decrease in their productivity.

_ With the variable meteorological conditions of specific years the optiminn sowing
times for winter crops can deviate considerably from those mean times established
for each zone. For e~mple, according to experimental data the sowing times for -
which the maximtmm grain yield is formed in the northern half of the European USSR =
in 70-80% of the years fit in the range of folir five-day periods. In the southern
_ regions the optimum sowing times are lengthier and in most (80%) cases fit in an _
interva]. of five or six five-day periods. In the remaining years (20-30% of the ~ -
cases) the sowing times can deviate by two or three additional five-day periods.
In agricultural practice recommendations are given for adherence to the mean opti-
mum sowing times established on the basis of experimental data for each soil-cli- _
matic zone or its individual parts. The duration of th~ mean sowing time period
106 -
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is u~ual~y determined for winter wheat at 10-15 days and for winter rye at 10-20
days.
Table 1 -
ChBnge in Yield Structure and Other Winter Wheat Indices in Dependence on Sowing _
li.mes (Mean of 200 Experiment-Years)
_ ~ Time -
~ Index ~ before opta.tu~n apti ~ter optimum, days~ `
da s -
= I 31 ~ 2a ~ 10 1 U ~ 20 I 3l? ~ 40
~
~HG'10 npos~'KTNBHNXI ~
1 cTe6nei+, .~i~ 36~1 36a 35;, 420 ~ 3fi0 I 31i~ 255 250
2 ~xcno sepeH I 29 'l6 49 26 24 2~ 25 -
3 Macca 1000 aepex, z I 3.3,3 3fi,S 37,:, 37,3 36,0 35,7 3'3,5 32,0
- q TIpoay~:r?+exa~ xycrxc� i -
rocrb 1,9 2,0 1,9 2.0 ~ I,i 1,6 1.6 1,4 -
5 FonnyecTao coxpat~xe� ' I
illNXCA pBCTCHH?1 38 3N�
MY. ~IO I 62 ~ ~0 ~ 8~ ~2 ~8 %J ~i -
5 floepe~+cltexNe cxpdro� .
cre6enbHU+xn Nyza~aH, I 43 , 22 12 5 1 3 U -
~ Tlopaxcetttie p~caeqN� i
- HON. �,o ~ 42 34 2' 23 ~4 41
$ I7opaKCeHHe no~iocaroH
M038NK0?(, qp ' ~1 4J ~7 4 Q ~ U - -
9 Co~epHCaxESe ~:IKB 8
3epHe. ;o 12.8 12.9 12.9 l:i.l 13.2 13.8 -
KEX: ~
- 1. Number of productive stems, m2
2. Number of grains
3. Mass of 1,000 grains, g
4. Productive bushiness
5. Number of plants survi.ving during winter, % '
6. Damage by stem flies, %
7. Damage by rust, %
8. Damage by streak mosaic, %
9. Protein content in grain, %
A comparison of the actual annual optimum sowing times established on the basis of
_ large amounts of experimental data with the mean recommended sowing t3.mes indicat- -
ed that their coincidence was observed in 60% of the years; in 20% of the years
- they differed by 10-20 days. Such deviations lead to considerable underharvests of
grain, in individual cases 5-10 centners/hectare or more.
- It is entirely obvious that the standard times for the sowing of winter crops recom-
= raencled by agricul~liral agencies are in need of annual correction from the point of
- view of both agroengi.neering and weather conditions.
Attempts of researchers to give an agrometeorological validation of the sowing times
for the most part essentially involved computation (prediction) of the optimum times
- on the basis of the sum of temperatures necessary for the formation of the optimum
107
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N
O
- ~ O~ N r-i
.t1 1 ~ -
cd ~ N ~D u'1 N
H N ~ ~
~ r~l fn
- W _
b ~D QO ~t
~ ~ -
v ~ v1 � � �
~I w ~ M ?~l O~
69 S7 N N
~
N ~ ~+-I
~rl ~ 1~ � .
,-i a~ a
~ ap u~ M
_ ~ W ia ~ r1 ~o ~O
d ~ N
- ~O 1~
ri W
~p N DO
c0 ~ ~ -
Sa W � � '
~ ~ ~ ~ N ^ M
~ ~ ~
'-1 -
~ 1~ V1 N
~ v
~ d ~ ~ 1~ ~D ~Y
4-I ~ ~rl rl N C~'1
O rl H ~
H ~
- V1 O
y 00
Cl ~
~ O
~ ~ O~ O ~
U i.l ~ � �
- y~,1 ~ ~ N ~ M
- 4~ Sa
W d
w
b ~
~ A cv o0 0 -
~ i~+ ~ ~ ~i' U1 ~ -
Cl O ~i N
4-1 ~
_ cd a
t~ ~ O
U N
' Gl U ~ ~ O ~7 ~ -
. . ~
~ ~ O rl ~t ~7
V1 lJ W N N
N tA C7 -
~ ~ A
~ ~ ~
~ ~
GJ ~ O . � �
U ' N N ~'"1 ~O
v .r{ N N
J~
v~ u
- o~ ~
- H W ~
U p
W
~ 3~+ 1
~ ~ ~ ~ H ~ r~i ~ c~d
~r~l V c~ N N ~ ~ ~
- O 00 cd $ rl ~+~-I d O N~ -
x > u w r+ ~
~ ~ a~i ~ o ~ ~
.y ~ ~ � ~ o0
~ ~ ~ ~
107a
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number of sprouts of winter crops by the end of the growing season (usually 2-4-6
sprouts, depending on the capacity of the plants for tillering and soil moisture
. content). The plants of winter wheat and rye attain such a bushiness when a sum
of effective (above 5�C) temperatures of 200-300�C is acci.mmulated during the peri -
iod from sowing to the cessation of the growing season in the autumn [16]. Con-
siderable corrections to derermination of the optimum sowing times are introduced
by the level of the mAisture reserves in the soil, which is taken into account in
the computations by different methods [2, 6, 7, 11, 15 and others]. The principle
of allowance for definite heat stiuns necessary for the development of plants also
served as a basis for constructing agroclimatic maps of the mean times for the
sowing of winter crops [3, 6, 8, 12, 16, 17 and others].
.
i'C~
i ~e �
i
r , � . r .
.
. , , r
r -
s~ � ~ � .
_ �a . .
~ .
. . . ~ _
- �
~
_ 5~
_ 1yr
~ ~
- ' 6~ t
~~L ~
,LI � i I
.
_ . .
"C
.
. ~
~ �
_ ~ . _
6
. ,
6 , ~ . .
15 d1 S 10 15 TO 15 J" ~"�0 15 1:' '
aAzycm ::~.-A~/70
Aug Sep
- Fig. 3, Optimum times for the sowing of winter crops in dependence on mean air tem- -
perature during period from 25 August through 20 October. Nonchernozem zone. a) un-
- der conditions of adequate moist~ning; b) under conditions of inadequate moisten- -
ing (precipitation during August-September less than 80
Table 3 -
Underharvest of Grain Yield for Different Winter 47heat Var~eties with Deviation
. ~of Sowing Ti~es f rom Optim~ (72 Experiment-Years) -
_ Variety Time
before optimtmm, days after optimum, days
LO lO 2O lO
Mironovskaya-808 23 8 7 13 ~
Odesskaya-51 14 6 8 18
Kavkaz 20 10 8 22
- Khar'kovskaya-63 18 12 12
- Bezostaya-1 32 14 12 24
I1'ichevka 17 14 20
108
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- An analysis of the mass e~eriments led to the conclusion that the optimality of
sowing times is determined not only by the bushiness index at the end of autumn.
T`he advantage of different sowing times in each specific case ia dependent on a -

~ whole series of factors: agricultural engineering techniques, variety, type of _
_ predecessor, multip~ication of pests and diseases, wintering conditions, condi-
tions of the spring and summer growing season, including even the resistance of

_ sowings at different times to bea.ting down. _
The processing of experimental data shows that the times of sowing of winter crops
in the Nonchernozem zone, determined using the criterion of 200-300�C sum of ef-
~ fective temperatures (in tre computations this limit was broadened by f25�C) in
60% of the cases actually corresponded to the maximum yields.
' Unfortunately, inadequate methodological possibilities do not make it possible in
_ the determination of the optim~n sowing timPS to introduce corrections for the
conditions of the future winter and spring-summer periods. Therefore, for the
time being agromet eorologists, as before, have at their disposal only a possibil- -
ity for taking into account the meteorological conditions developing prior to sc~:~-
ing or for evalvating the subsequent conditions of autumn with the use of synoptic
farecasts.
- Generally available recommendations on the use of agrometeorological data in the
choice of the optimimm times for the sowing of winter wheat for the territory af
the Ukraine were proposed by I. G. Grushko and V. P. Dmit~enko [3].
For the conditions of the Nonchernozem zone (Belorussia) L. K. Pyatovskaya [10]
proposed a method for computing the optim~an times for the sowing of winter crops
on the basis of inean air temperature for the period from 25 August through 20 Oc- -
tober.
P checking of the L. K. Pyatovskaya index on the basis of reliable experimental
data for the territory of the Nonchernozem zone indicated a possibility of its '
use for the choice of optimum sowing times. Under the conditions of an adequate
- moisture supply the optimality of sowing times was dependent for the most part on
the temperature level for the period (Fig. 3a). With a shortage of moisture (pre-
- cipitation quantity during August-September less than 80 the scatter of points
increases and the sowing times as a rule were displaced to earlier times (Fig. 3b). _
Different varieties of winter wheat in one and the same cli.matic rPgion have dif-
ferent optimtunsowing times. This is attributable to their photoperiodic reaction.
- Varieties with a well-expressed photoperiodic ~Leaction must be sown at earlier _
times. For example, under conditions of a gradually shortening day frost-resistant
varieties react mo re strongly than those with weak resistance to a short day with
a lag in their development and should be sown earl~ero Wintering Mexican short-stem
varieties with a weak photoperiodic reactipn can be sown latest [14].
It is known that intensive varieties of winter wheat (Avrora, Kavkaz, Bezostaya-2, _
Rostovchanka and others) form higher yields when sowa 10-12 days later than the
= usual times [9]. The opti.muun period of sowing is shortest for thema -
109 -
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The capacity for winter wheat of the Mironov~kaya-808 type on a good agricultur-
al background to bush out in autumn and spring broadens the intexvals of opti- _
mum sowing times. The variety Odesskaya-51 also is characterized t~y a high plas-
tfcity. W~.th a deviation of the sowing ti.mes from the optim~ its praductivity
- and res~stance to winter conditions decrease to a lesser degree than for the Var~- _
- ieties Kavkaz, Khar'kovskaya-63, I1'ichevka, Bezostaya-1 (Table 3)~ Odesskaya-16
and Novomichurinka are plastic varieties. The varieties Odesskaya-26 and Stepov-
, aya occupy an intermediate positinno
- The optimum times for the sewing of winter crops are dependent on the agroengineer-
ing background. For fertilized clean fallow the optimum sowing times are 10-12
days later than for predecess~rs not on fallow. On the other hand, with an inad-
equacy of precipitation it is desirable that the sowing begin with clean fallow.
The sowing of winter crops in dry or semidry ~oil cannot be allowedo According
- to data from the Zherebkovskaya Experimental Station, with a content of not more
than 3 mm of moisture avai.lable to p3ants in the 10-cro soil layer the wheat seeds _
~ may not lose their germinating power for a relatively lon~ time. With moisture
- resexves of 5-6 mm the seeds swell, become moldy and lose their germinating power
[13]. When the reserves of productive moisture are less than 15 ffin in the cultivat-
ed soil layer (0-20 cm) winter crops should not be sown prior to the falling of =
considerable precipitation because the plantings will be sparse [6]. Accordingly,
with a moisture shortage in the soil it is desirable that the sowing be postponed
to a later time or that the fields be left in spring cropso
It is possible to await precipitation in dry autumn only within the litnits of the
admissible times during which the sown winter ~rops could attain the tillering
phase by the end of the autumn growing s easono
P. G. Kabanov, for the conditions prevailing in the Vo1ga region, examines the pos- .
sibility of the sawing of winter rye at early times in a case of falling of rains
~ in lare July-early August and dry weather thereafter [4
As a result, it must be admitted that at the present time the most promising agro-
meteorological method for computing the optimum times of sowing of winter crops re-
- mains the method of computation on the bas is of the fndex of bushiness of winter
crops by the end of the growing season. The degree of bu~hiness characterizes not
. only the morphological. state of the plants but also the size of the apical cones ~
foz the autuinn sprouts. In its modern modification applicable to the state of the _
sprouts over great areas this method is b eing successfully developed at the USSR
Hydrometeorological Center [6, 7, 15J. -
BIBLIOGRAPHY
- :J~ ~
~ l~ Voyeykov, A. I., "On the Times of Sowing and Harvesting of Field Plants and
- the Mawing of Hay Fields in European Russia," SEL'SKOYE KHOZYAYSTVO I LESOVOD-
5TV0 (Agriculture and Forestry), January, 1884. _
110 -
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1. Grudeva, A. Ya., METODICHESKOYE POSOBIYE PO PROGNOZIROVANIYU SOSTOYANIYU
- SOSTOYANIYA OZIMOY PSHENITSY I OZIMOY RZHI KO VREMENI PREKRASHC~NIYA OSEN-
NEY VEGETATSII V CHERNOZENAIOY ZONE YeTS (Methodological Aid for Predicting
the State of Winter Wheat and Winter Rye by the Time of Cessation of the
- Autumn Growing Season in the Chernozem Zone of the European USSR), Moscow, -
- Gidrometeoizdat, 1974.
- 3. Grushka, I. G., Dmitrenko, Vo P., "Computation of the Anticipated Times of
Sowing of Winter Wheat and Evaluation of its Effectiveness," TRUDY U1r.rNIGMI,
{Transactions of the Ukrainian Scientific Research Hydrometeorological In-
stitute), No 84, 1969.
4. Kabanov, P. G., DIFFERENTSIROVANNOYE PRIl~[JENIYE AGROTEKHNIKI V POVOLZH~YE
(Differential Use of .Agricultural Techniques in the Volga Region), Saratov, _
Privolzhskoye Knizhnoye Izd-vo, 1968.
5. Kozlov, I. V., "Fertilizer, Sowing Times and Yie1d of Winter Wheat," KHIM-
IYA V SEL'SKOM KHOZYAYSTVE (Chemistry in Agriculture), No 2, 1978.
6. Maksimenkova, T. A., "A Methad for Predicting t:~e State of Winter Grain Crops _
by the Time of Cessation of Their Growing Season," METEOROLOGIYA I GIDROLOG- _
_ IYA (Meteorology and Hydrology), No 2r 1979.
, 7. Moiseychik, V. A., AGROMETEOROLOGICHESKIYE USLOVIYA I PEREZIMOVKA OZIMYKH
_ KUL'TUR (Agrometeorological Conditions and Wintering of Winter Crops), Len- _
ingrad, Gidrometeoizdat, 1975.
8. Nosatovskiy, A. Io, "Theoretical Validation of the Optimum Time of Sowing of
Winter Wheat," DOKI,ADY VASKhNIL (Reports of the All-Union Academy of Agri-
cultural Sciences), No 11-12, 1946.
9. Prutskov, F. M., POVYSHENIYE UROZAAYNOSTI ZERNOVYKH KUL'TUR (Increasing the
Yie1d of Grain Crops), Mosc~w, Rossel'khozi.zdat, 1977.
10. Pyatovskaya, L. K., AGROMETEOROLCIGICHESKOYE OBOSNOVANIYE SROKOV SEVA (Agro-
meteorological Validation of Sowing Times), Minsk, Uradzhay, 1977.
11. Svisyuk, I. V., "Sowing of ~Tinter Crops at Qptimum Times One of the Con-
ditions for Their Favorable Wintering," AGROMETEOROLOGICHESKIYE ASPEKTY PERE- �
ZIMOVKI RASTENIY (Agrometeorological Aspects of Wintering of Plants), Lenin- -
_ grad, Gidro~eteoizdat, 1977.
12. Stepanov, V. N., "Optimum and Limiting Times of Sowing of Winter Crops in the
USSR," DOKLADY NAUCHNOY KONFERENTSII TSKhA (Reports of the Scientific Confer-
ence of the Timiryazev Agricultural Academy), No 5, 1947. -
~
. 13. Tretyuk, A. I., "Times for the Sowing of Winter Wheat," SELEKTSIYA I TEIQiNO-
LOGIYA VOZDELYVANIYA KUKURUZY i NEKOTORYYE VOPROSY AGROTEKHNIKI DRUGIKH KUL'-
TUR V SEVERO-ZAPADNOY CHASTI STEPI UkrSSR (Selection and Technology for Corn
Cultivation and Some Problems in Agricultural Techniques for Other Crops in
the Northwestern Part of the Steppe UkrSSR), Dnepropetrovsk, 1976. =
111
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n
- 14. Turbin, N. V., Fedorov, A. K., Biological Validation of the Optim~ Time for
the SoGZing of Winter Wheat," DOKLADY VASKhNIL, No 3, 1976. -
15. Ulanova, Ye. S., AGROMETEOROLOGIL'HESKIYE USLOVIYA I UROZHAYNOST' OZIMOX
PSHENITSY (Agrometeorological Conditions and Yield of Winter Wheat;, Len-
ingrad, Gidrometeoizdat, 1.975. =
= 16. Shigolev, A. A., "Method for Preparing Phenological Forecasts," SBORNIK METOD-
ICHESKIKH UKAZANIY PO ANALIZU I OTSENKE SLOZHIVSHIKHSYA I OZHIDAYEMYKH AGRO-
METUSLOVIY (Collection of Methodological Instructions on Analysis an~ Evalu- �
ation of Prevailing and Anticipated Agrometeorological Conditions), Leningrad,
- Gidrometeoizdat, 1957. _
17. Yakovlev, N. N., KLIMAT I ZIMOSTOYKOST' OZIMOY PSHENITSY (Climate and Winter
Resistance of Winter Wheat), Leningrad, Gidrometeoizdat, 1966. -
_ ~
112 .
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~ UDC 551.509.314 -
SOM~ POSSIBILITIES FOR SIMPLIFICATION OF ADAPTIVE ALGORITI~SS IN PROGNOSTIC SCHEMES -
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 8, Aug 80 pp 98-102
- [Article by V. V. Plotnikov, Far Eastern Regional Scientific Research Institute,
= submitted for publication 11 Dec 79]
[Text] Abstract: In this article, on the basis of
an analysis of the variability of elements _
of a covariation matrix, the conclusion is
_ drawn that so-called homogeneity intervals
exist. It is shown that with a cha.nge in
_ the volume of the initial samples the stat-
. ic characteristics of the process virtually
- do not change in this interval. The applic-
ation of this fact makes posaible a substan-
tial decrease in the volinne of computations
= in prognostic problema making use of adaptive
~ algorithms. In addition, the auChor examines _
some assumptions relating to the optimum ad-
vance period of the prediction and the neces-
sary volume of the sample for statistical ex-
periments.
_ Adaptive forecasting methods have recently come into broad use in prognostic prac-
= Cice [3-5, 7]. Interest in these methods is attributable primarily to the fact that -
using them it is possible to solve problems of forecasting under nonstationary con-
_ ditions. This effect is achieved by constant rescaling of the prognostic operators
in each prediction interval. It is necessary to pay for such universality by a con- -
siderable complication of the algorithms, and as a result, an increase in the time
expended on the forecast. ~
When using adaptive algorithms in physicostatistical schemes, considering the coef-
ficients of expansion of the initial fields in natural orthogonal components (NOC)
as predictors and predictanCs [1, 7], a considerable part of the time is expended
on the scaling of natural components and their coefficients in each individual
forecast. -
The purpose of this study was an analysis of one of the possible ways to simplify
adaptive algorithms for physicostatistical prediction of hydrometeorological ob-
jects.
113
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The proposed appraach makes it possible to economize the time expended on con-

- stant scaling of the natural components~as a result of multiple use of the earlier
computed values. This is possible in connection writh the presence of sectors in the _
general nonstationary process corresponding to the requirements of statistical
homogeneity, with subsequent transfer of the properties of the preceding set to _
these sectors. As is well known, the determination of the field EOC is a process of
finding the eigenvalues and eigenvectors of the covariation (correlation) matrix
of this field. In actual practice we almost always deal with sample values of the
field elements and accordingly, we can obtain only an evaluation of the covariation _
matrix; this evaluation will be essentially dependent on the length of the sample.
_ A sample of the field values ~ F} i, ~=1 (we will call it a sample matrix) can be
- represented as follows:
Ft~Fsa....F.in
F=1 F~ . . . . Fs rc -
- F= . . . . . . . . . (1)
F~e~ Fm ~ . . . Finn _
Here m is the dimensionality of the vector P(the number of its components), n is
the volume of the sample (the number of observed values of the F vector included
~ in the computations).
The evaluation of the covariation m.atrix ~ M} i ~1 is obtained in the form of the
averaged product of the matrix (1) and its tra~s~o~ed value. _
_ - - M _ ~ F ET' _ `2>
n
where T is the transposition symbol. -
It is assumed here that the mean value of each component is already excluded.
Assuming the covariation matrix (2) to be quite well stipulated, we will determine
whether the coefficients of expansion of the field in EOC will change with a change
in the volume of the sample and if they do, how. In addition, we will attempt to .
find the maximum possible interval the increment of the sample volume with -
which tk~ese changes are statistically still not significant.
It is evident that the variability of the expansion coefficients is determined, in -
particular, by the variability of the elements of the covariation matrix. It is _
necessary to evaluate m(m+l)/2 parameters, of which m are diagonal and m(m-1)/2 are
independent nondiagonal values.
Now we will examine one of the elements of the covariation matrix as a function of
sample volume.
Without losing universality in our reasoning, in order to simplify the calculations
and for greater clarity we witl ass~e that this will be one of the diagonal ele-
ments of the matrix
114 ~
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, ~
R
- ~ ~Ft/ - F (n)~)- ~
M (n~11= 7_1 n . ~3) '
where F(n)~ is the mean value of the characteristic with a sam~le volume n.
n
~ F?i
- ~4~ -
F ~R~i ' n '
With a change in the volume of the sample the e~cpressions (3), (4) are writtea as ~
f ol lows : -
rt+-
~ (Ft/- F (n s)~~- ~5~ _
,4! (n + = 1= t
n : ~
n+~ -
~ Fr~
F (R _ ~ ~6)
n*s '
- where 'G is the number of observations added to the sample.
Now we will,examine an extremal variant when any subsequent observation of the F
v~ector, added to a sample of n terms, deviates from the mean value by a value which
is limiting for the particular process; we will denote it by KA~. Here A~ is the
- maximum possible amplitude of the process in the time interval n+'L and K is a co-
- efficient changing in dependence on the specific field from 0 to 1 and cha~acteriz- ~
ing the degree of nonstationarity of the particular field. Then equations (6), (5)
are represented in the following form:
. . .
' F + _ F ~~~1 + t KA~ . ~7)
rc+-
t KA~ ~ c'
\F't/- F (n)/- n + L (KA~~=
~R ~ �~l/ = la} + fo 1 /8\
\ /
l! t :
After obvious transformations we obtain
L _ ,N (n)1I�rt n-~ (KA~)= L S (KA~)~
M (n ')11= R~�~ (n-i-t)3 n+: ' ~9)
- The transformation M~~ with a change in the volume of the sample in a general case
is related to the influence of nonstationarity of the analyzed processes.
115
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~n order to find tha shift ~rith ~hich the level of variab3lity of the hroadened -
- sample in the interval n+'G with a stipulated confidence coefficient P does not
exceed (is co~ensurable with) the level of variability of the set of the volume n,
, ute will consider the following hypothesis.
' We will assume that M(n j~ _ 1.
M ~n~ll ~10)
In this case ~rith an increase (decrease) in the volume of the sample by 'C the ele-
ments of the covariation matrix do not change statistically. Then the random value
_ .
- M (n + T)i!
M (n)~~
has a F~~, y2 Fisher distribution with vl = n+2- 1 and v2 = n- 1 degrees of free- _
dom (4].
Hence with a stipulated confidence coefficient P it is possible to determine that
, increment in the volume of the sample (so-called homogeneity interval or possible
number of prediction intervals without scaling of the EOC) at which the change in
the covariation coefficients is staCistically not significant.
- For this we examine the inequality -
_ - -
M (n + ~ F,o r ' ~11~
- M ~n~ll ~ r
- where the significance level is
a-1-P.
' (12)
With (9) taken into account we obtain
, n + nt~K~Aj _ + SK=A~ ~P (13) -
n ~ ~n'�' T~3 ~ ~n~l1 ~n '~'t ~R~l! �i~ Dz,
In turn it is possible to determine the upper boundary of am~plitude of the process
(A~) by ha.ving the values of the diagonal eZements of the covariation matrix j,6]
A/�C M(nt':)11 ~ _ (14) -
- where C, using the results presented in [6], can be approximated by the following
expression:
C-1a(n+S-l}-0,069 n+t-;-l,ll. (15)
_ Accordingly, we finally write: _
_ _ . _
. .
` M(n -r T)11 n M(n +:;11f n Z7 f{'~ G'~ T 1C~ C'~ ' -
M ~n~11 ~ n + ~ + M (R)Il L � ~n tl3 + n +t (16)
Now the insquality (11) is transformed to the form
_ - . .
- n
n 1.~ n t= /C~ _ t!C1 C~ ~ Fo'' Dr Q. ~ I
(n + T)1
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t ~
f
2
1 ~
_
_
0 ' '
10 JD SO 10 90
Fig. 1. Dependence of ths upper boundary of the region of admissible values t on -
~ length of series n with confidence coefficient 95X for K= 0.5 (1), K= 0.75 (2)
- and K = 1.0 (3).
It goes without saying that all the proposed computations are correct under the con-
dition n 9 T,. Otherwise, with -?n there is a redistribution of the contributions
of the dispersion of the main (n) and additional (t ) samples, leading to a discon-
tinuity of the function ._-R---_-.._._.._. .
, � _
n+'~- ~R~~~ -cK~C=
Subsequent increases in the interval: lead to a dominant influence of the additional
sample and the relative fraction of dispersion of the principal sample decreases.
But from the point of view of the formulated problems this case is not of interest
and is not considered in the study. . ~ -
With n~j neglecting the second term in~ the denominator on the left-hand side of
inequality (17), by virtue of its smallness we obtain
" -F