JPRS ID: 9650 USSR REPORT METEORLOGY AND HYDROLOGY

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• Content Type: REPORTS

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JPRS L/9650
7 April 19~ 1 -
_ ~ -
USSR Re ort
- p
= METEOROLOGY AND HYDROLOGY
No. 11, November 1980
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~ JPRS L/9650
7 April 1981
USSR REPORT
METEOROLOGY AND HYDROLOGY
No. 11, November 1980
Translation of the Russian-langua.ge monthly journal METEOROLOGIY~a I
GIDROLOGIYA publisned in Moscow by Gidrometeoizdat.
CONTENTS
Wave Triplets in the Atmospheric ~quatorial Zone 1
Model of General Circulation of the Atmosphere Used at the USSR
Hydrometeorological Center 13 -
Use of Nested Grids P�iethod in Three-Dimensional Atmospherie Model 25
Medium-Range Prediction of H500 by a Physicostatistical Method 34 -
Simple Climatic Model of the Latitudinal Distribution of Atmospheric
Precipitation 43
Experimental Investigations of Orog~aphic Waves and Vertical Movements in the -
Neighborhood of Krasnovodsk Airport S1
On the Problem of the Movement of Continental Ice 57
Bispectral Analysis of Sea Level Fluctuations 67
, Prediction of the Channel Process and Solutions of the Problem 77
Reliability of Choice of Parameters of a Gamma Distribution in the Processing
of I:unoff Data 87
liydrological Basis of a Plan for Protecting Leningrad From Floods 97
Method for ~valuating the Admissible Contamination of Water by Sma12. Ships With
~ngines 103
_ Evaluation of Accuracy in Computing Mean Soil Moisture Reserves in Rainy
- Periods .................................e............................o....... 112
a - a- [III - USSR - 33 S&T F4U0]
~no nr~r~r ~ r r tcF nt~~ v
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,
Stochastic Dynamic Programming Method f.or Computing the Most Advantageous
Ship Navigation Routes 121
More on Evaluating the Accuracy in Measuring Water Dis~harges 12b F
Investigation of the Structure of a Fluid Flow in a Channel Using an Optical
- Dopp~er Hydrometer....~ 131 ~
Forming and Development of the Theoreticsl Investigations of the Atmosphere in
the Studies of V. A. Khanevskiy (On the Hundredth Anniversary of His Birth).. 137 -
Review of Monograpli by A. I. Lazarev, A. G. Nikolayev and Ye. V. Khrunov: i
`Optical Investigations in Space' ('OPTICHESKIYE ISSLEDOVANIYA V KnSMOSE~), ,
Leningrad, Gidrometeoizdat, 1979, 256 Pages 144
Eightieth Birthday of Nikolay Fedorovich Gel'mgol'ts 147 -
. Seventieth Birthday of Anatoliy Ivanovich Karakash 149
Conferences, Me~.tings, Seminars 151 ~
s
Notes From Abroad ......................e............. 154 _
- h -
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- UDC 551.511.32(-062.4)
_ WAVE TRIPLETS IN THE ATMOSPItERIC EQUATORIAL ZONE
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 80 pp 5-15
[Article by Ye. M. Dobryshman, profesaor, Institute of Atmospheric Physics, manu-
- script submitted 22 May 80] -
- [TextJ Abstract: The existence of a triplet of waves,
which are in resonance interaction, is estab- -
lished for a system of two nonlinea~ equations
relating the stream function and tempsrature in
a very simple model of circulation in a narr~w
equatorial zone. The article gives an analysi~ -
of interacting wave vectors and graphs of the
geometrical locus of poiflts representing the ~
ends of pairs of vectors interacting with a -
third. The solution for ~he amplitudQs o� in-
teracting vectors in the wind and temperature
~ields is expressed through el],iptica'1 Jacobi ~
functions. A compar~.son is given with a simpler
case Rossby waves, being a solution of one .
- nonlinear equation.
More and more attention is being devoted to an investigation of nonlinear process-
es in the atmosphere. This is entirely understandable since it is primarily non-
_ linear processes which are responsit~le for the restructuring of circulation mech-
anisms ot the most different scale~. Tl~e ~eneral difficulties i~i study of nonlin- -
ear processes are well known and therefore it need only be mentioned that virtual-
ly every nonlinear prc,blem requires its own approach and the formulation of its own
solution method. True, the general nature of the nonlinearity of the equations of -
hydrodynamics also makes it possib?e to eatploy some general procedures for solviug ~
individual classes of problems. Among such classes is the problem of �;aave triplets,
that is, three waves with different compor~ents of the wave vectors which are in
resonance interaction there is an exchange of energies (the total energy of the
triplet is conserved). It can be noted at once that the solutions found are usual-
ly not.absolutely precise; they are obtained using different approximations. The
most us~d variant is the weak interaction approximation, which will be discussed
below.
- Within the framework of linear theory, or better said, linear approxtmation theory,
the amplitude and phase of each wave are independent characteristics of the pro- _
cess and thus the waves do not interact with one another. Rossby waves are the most
important macroscale waves in the atmosphere. ~ single Rossby wave is a precise
1 -
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solution of the nonlinear vorticity equation (to ba more precise, the projection ~
of the vorticity vector onto the vertical axis z) within the framework of quasi-
- horizontality of motion, characteristic for processes of a synoptic scale every- -
- where except for a narrow equatorial zone [5]. Source [2] gave a general proced- -
- ure for seeking solutions of the nonlinear vorticity equation in the form of the
sum of noninteracting waves. This method was developed further in different as-
pects (see ~3, 7]). The nonlinearity of Che corresponding operator indicates the
existence of interaction between waves and in the 1960's studies appeared with
- individual special solutions in the form of sums of specially "selected" Rossby _
waves in one way or another interacting with one another. An important step in _
its theoretical aspects was [6], which gave a method for analysis of a resonance -
triplet; the nonlinear part of the triplet is described by ~~erator -
which is natural for hydrodynamics vorticity advection. Tn numerous subsequent -
s~udies the method was refined and developed in the most different directions,
the most important of which, indeed, is a study of the interaction of two waves
with the zonal flow the "third wave." _
Tt1E objective of the a-rticle is to demonstrate that in a more complex case, spe-
cifically for a system of two nonlinear equations describing the simplest models
of circiilation ir a narrow equatorial zone, resonance triplets can exist. -
_ 1. Formulation of problem. The simplest zonal models of small-scale processes,
that is, not dependent on the x-cooxdinate along the equator, are described by a
system of equations correct for a narrow equatorial zone [4];
dG' (v U) - 0
dt + ~
~ darJ + i~~ ~'k) = x dy
ar , _ _ a~~ ~ ~1~ -
- -dt + T) - r a~; ,
Here t is time, y is the horizontal ~:oordinate, reckoned from the equator to the
nor.th, z is the vertical coordinat~,o~~ 1/30 m�sec'2��C'1 is the buoyancy para-
metex,2(-N3�10~'3 �C�m 1 is the i~ :.:ameter of vertical atmospheric stability, U=
u-~y / r0 + Z~ z is zonal angular moment, u is tlie zonal velocity component,
cJ = 7.29�10~5 is the angular velocity of the earth's rotation, r~ = 6.37�10� m
is the earrh's mean radius, ~ is the atream function in the meridional plane,
so that v= - d~/ d z; w= d~'/ d y are the velocity components along the y and z
axes respectively, T is the deviation of temper:~ture from a linear profile along ~
z; ~
dA dB dA ~dc3
- l�'~ ~ B) � d~, d~ - as ' is a Jacobian,
, _ d= 0=' -
ey= ~ dt= '
It; can be seen from system (1) that first of all, the movement is three-dimension- -
a1 a11 three velocity comp~~~ents are different from zero, and second, U, aud
this means, well, are determined after finding the ~ and T functions.
System (1), even in more general for~r,, has a number of solutions of the type of
7~aves superposed on some basic state characterized by constant values of the mer-
- idional. and vertical wind veloclty components f5].
2 -
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We will attempt to find a solution of the system of equations -
' dvr~ + = z dT ; ~ + (L, T) - - I' d--L
Y Y ~~f~
in the form
N _
Lr Tk t ~~kY'~"nk=-~kt~
,~j ' - Vz ~ e
. ~T ~ ~J(1
k-1 ~
~ T_ rTket(/!!ky-~-R~Z-zRl) _
~all the parameters are constants).
After the substitution of (2) into the first equation of system (1') and shortening -
by the factor i we obtain , =
~ �k ?~'~k et ~ )k + ~ ~ m~ ,~k et i )k � ~ ~ - n~ oj ~+j e` ( )1~ _ _
. k= I k=i j_I .
" N N
- - v [ ~ /lh ~jk e~ l )k ~ m/�j ,~/el ~ _
R_ ~ , ~3)
i=~
N
= Z ~ //ik Tk t'~ ~ )k
Rsl
where P k= mk + nk is the square of the wave vector modulus,
( )k = ~mr. Y nk a - t). _
- Equating the coefficients on ei~ ~k, we obtain N expressions -
(~k - mk V~ ~k ~k = a mk ~k ~k = N)� Z4) -
When multiplying the nonlinear terms it is necessary to examine two cases: when j= -
k and j~ k. In the first case the coefficients on the doubled frequencies become
equal to zero. For j=~ ic it is possible to avoid combination frequencies by re- -
_ quiring that the coefficients on ~'k, W~ become equal to zero. These, it is easy
to see, will be as follows:
~Pk - Pj1~~k - m~ na)�
~5)
The first possibilifiy t
/i2 - - r, = CnnSt ?
,R P~-O-. k - -
is of no interest (the same as with Rossby waves [3]). It leads to the trivial re-
sult
3
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mk _V�~1F ~
~ (6)
- and thus the linear velocity of all the waves Q'k/mk is one and the same. -
,
The 3 econd possibility is: ,
mk n~ - mj rtk = 0.
The number of expressions (7) will be C2 = N(N + 1)/2; however, only N- 1 will be
indep endent. With N= 3 the number of i~idependent expressions will be equal to 2;
this circumstance will be used in the future.
= Subs tituting (2} into the second equation of system (1'), a~'ter similar calcula- -
tions, we first obtain N e;cpressions
(~k-mhV) Ta=['"zk'~r. ~8~
and second, N(N - 1)/2 expressions t~kT~ (mkn� - m~nk) = 0, that is, the same ex- -
pressions (7), ~-hich can be represented in t~e form
mk; nk = q=> pk - mk ~ 1 t 9'')� _
Solving (7) and (S) jointly, we find
- mh V)= = xr mk _,~r ok = mk V� 1% i i~ (9)
'k ' q -
and
~
- I mk )~1 t q=, . (10)
i
Thus, there are waves moving along ~ d against the flow; their linear velocity
relative i:o the flow dk/u~ - V=;; �Y f(1 + q2)/m~, all other conditions being
equal, is the greater the greater tr~e dimension of the wave alon g the meridian
_ (that- is, the lesser the ~ vali~?`. The amplitude of the wave in the ~ field,
_ firs t of aII, is proportional to the amplitude of the wave in the temperature
field, and second, is the greater the lesser the q value, that is, the lesser the
- ratio of the wave length vertically to the wave length along the meridian. The
superposing of such waves can give a rather mottled pattern.
We emphasize that these wave solutions are precise solutions of a nonlinear system ~
o� equations. However, the described waves do not 3.nteract with one another (there
is no energy ^~cchange) and this means that one of the important characteristics of
, virtually a~., nor7.inear process has not found its reflection in solutions deter-
mined by formulas ~'Z)-(10) .
It is possible to find the int:Practing waves if it is noted that with N= 3 it is
_ not mandatory to require annu~meiit of a fluctuation with the combination fxequency
()k In this case there should be terms compensating the combination
frequencies. We introduce the hypothesis of weak interaction, namely tha*_ the a~r
_ plitudes of the triplet ~k, Tk; k= 1, 2, 3 are not great and slowly change with
- ~ ,
_ 4 ~
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- time so that the order of magnitude of the paraneters d~k/dt, dTk/dt and the
_ quadratic term~ y~k ,~kT~ is idantical.
We will seek the solutien of (1' ) in tihe form -
J - ,Ja _ VZ ~ ~ ,yh (t) ej l )k ~
a-i I
(11)
T=~ Tk(t) e` ~ )k
k-1 -
Substituting (11) into (1'), we obtain (sum~nation occurs everywhere from 1 to 3;
therefore, in the future the su~ation limits will not be indicated) -
U~ d ef ( )h, ~ ~ 1 ~Gk (~~'~k Er ( ~h L (T'. Lii1k Qk'Jk~X
; T ~
~12i
~ in, ~5~ e` t ( V - ~ irik ~Lk e~ ( )k~ . ~(-im1 Pi ~~I e~ ~ ~1)~ _ ' _
- i a~. Tk m,~ e' t)k
r ddt e~ l )k _ ~ c�~k Tkel ( )k ~ I~ ~h y~te~ l )r, . ~ ~t~ T~e~ ( _ . (13)
- V-{-iEitkuke'( )k~ . ~ lmj7'~e't 1'= -~~inr~;~~~e' ( )a -
The linear (final) terms give the expressions (4) and (8); we will not repeat them.
They determine the phase velocities of the triplet
~ ok ~ mk V� m j`a['~ok ~14~
and the relationship between the amplitudes of tYie waves `
~~k = � V~�%r ra;Pk~ cl5) -
In order to compensate the small terms it is necessary to require that each combin-
ation frequency, that is, the sum of two interacting frequencies ()k be
equal to the third, for example,
(~n~J+~i,~.-~s, t) + (msJ+n.~~-~~t) _ (nt3J+ns~ -7yt) (1C) _
and similarly ( )2 + ( )3 = ( )1; ( )3 + ( )1 - ( )2. Since the components of -
. the wave vectors mk, nk can be of different signs, the same as dk - u~V, without
loss of universality it is possible to write expression (16) and two expressions
similar to it in the simple form
[(irr,-}-ms+ms) Y~` ~ni-~ns+n3)~- l,, ~ t~ = 0.
Since this should be satisfied for any y, z and t, we obtain two expressions for
- the components of the wave vectors
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. m~-Lnt:~m~_01
n?+n,~-n3-0 ~ _
- ~1~~ -
arid o~e expression for the phase velocities of waves
- Q, ~Q~t?s=~� ~17~)
This synchronism condition, after the substitution (14) and allowance for the first
- expression from (17), assumes the form
- m,; P~ m., + m~;' p.~ _ (lg)
Thus, the triplet can be in resonance interaction only unde~ the condition that
their wave vectors and phase velocities satisfy expressions (17)-(18).
n
~
. a
~ -
,
- ~ i m
-
_ /
~
z
~ ~
Fig. 1. Locus of points of ends of pairs ~
of vectors w::~ch can interact with the
- vector (0, -1) (limiting case).
II. Analysis of interacting wav^ Jectors. On the plane of wave vectors (m, n) it
is necessary to find a trio of points which are related by formulas (17)-(18). We
note, first of all, that formulas (17)~(18) are symmetric relative to the sub-
- scripts and since mk and nk can have arbitrary signs it is s ufficient to li.mit
ourselves to an examiriation of one quadrant, for exa.mple IV: m~ 0; n< 0(see Fig.
_ 1). For m 5 0 the picture will be a mirro r reflection relative to the n-aYis, for
n> 0-- relative to the nraxis.
, .
One of the v,.:Cors, for example (m3, n3), should be stipulated and it is necessary -
to find the ~ocv~ of pairs of points (ml, nl) ;(m2, n2), which can interact with
= the stipulated vec~c~r. The answer is ambiguaus since there is not only one pair of
- numbers, but a lo cus, and therefore with stipulated (mg, n3) for the remaining `
- four numbers there are only r~~ree correlations. First we will examine a limiting
~~ase when mg = 0 and n3 is arbitrary, that is, one wave is "infinite" in length
':ilong y, jExcept f:or the case n3 = 0. With mg = n3 = 0 in actuality there are two
waves in the solution (11), which, a~ .I5 easy to see,, do not interact with one
another.] It follows from (17)-(18) that ml = m2 = 0 and the locus of the pairs ~
6 '
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of points is situated on the vertical axis (n); the pairs of points satisfy
, the expression nl + n2 =-n , or ml =-m2 ~ 0, and then
m~ +(-n3 - nl)~- m�+ n= 0~ nl 1/2n3,
that is, the pair of vectors interacting with the (0, -n) vector is situated sym-
~ metrically relative to the n-axis. Thus, the locus of the points in this case
involves two mutually perpendicular straight lines: one is the n-axis and the `
other is a straight line parallel to the m-axis and separated from it by the dis-
tance n=-1/2 n3. Figure 1 shows these straight lines and examples of pairs of
vectors interacting with the vector (0, n3) and with n3 =-la (If n3 ~-1, it
is necessary simply to change the scale by a factor of i n3I
_ In a general case it is convenient to make an analysis af the system of equations
(17)-(18) in polar coordinates m=~o cos Sp , n= j~ sin
Now (17)-(18) are rewritten in the form
P~ cos Y~ c~~s fi3 COS D~~ p, sin t P: sin ~:2 sin i
- (19)
- cos + co~ - - cos J.
The system (19) can be simplified by introducing the ratios /~1//~3 = X1~ ~1~~3 .
= X2. (In the sense ~1, Jr 2> 0) . We obtain -
CO$ c~~ ~OS 'y_, _ - l0S '.:3
s:n T ~n sin sin �:3 ~20~
cos cos c:~ cos _
The proble~m is reduced to the following: with a stipulated wave vector, lying on
a single circle, find the possible pairs of vectors 7~1, ~1~ x2~ ~2, forming a
resonance triplet.
We will note some corollaries which follow easily from system (20); ~ and ~ can-
not be of different orders of magnitude. Assuming the opposite (1C we see
that X2 cos ~2 N 0~ 5~2~ f n'/2 and x2 sin SP2z 0~ SP2~ 0; that is, we obcain
= a contradiction. If x1~ 1, then x2 cos 502 =-cos ~g and x2 sin c92 sin ~P2
- ~ X 22~1. This means that if one of the triplet vectors is sma11 the other two in
absoIute value will be close to unity. Then we will examine a case when 7c1, X2 ~
l. It is evident that this is possible only when cos ~lx -cos ~2 and sin ~lw -sin
So2. It therefore follows that ~2~n+5o1 and that cos ~3~0~~3~ f n'/2, that is, ~
a pair of "large" wave vectors can interact only with a vector for which ~ m3
- (n3 Then, by virtue of symmetry o� the system relative to the subscripts 1 and
2 the locus of the points will be a curve with two axes of symmetry. The curve
has four points in common with the circle P= 1. These points are arranged in th.e
following way: `P3 t2'rY/3 and +~+J[. With ~3 = 0(mg = 1; n3 = 0) the second
pair of points merges into one the curve touches the circle on the outside.
- [dith ~3 n'/6 the points ~3 - 2Tr/3 and Ti+ ~P3 merge the curve touch~s the _
circle from the inside. Thus, with different SG3 angles the curves can pass
through one and the same points of the circle (two points), for example, for ~g =
- rt/18 and ~3 =-8/9 7T such points will be 10/9n and 17/ 18 TCo ~
The first two equations of system (20) are a system of linear equations relative
to xl and 'X2. Accordingly, the procedure for computing the locus of the points
of the ends of the pairs of vectors, interacting with given (~g) , is carried out -
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most simply in the following way. Stipulating sp3 and SA2, we will determine SOl
trom the third equation (there may be either two solutions or none). This means
tf~at the coefficients on and ,'~2 become known after X1 and ~2 (only posi-
tive solutions are suitable~ a~e found using the formulas ~
;c - sin (~s-4.) ~c. - _ sin (?a-4~)
a-
t -sin (~x--4i)' sin ("r_-,~)' '
. As we saw above, the denominator can be close to zero only for ~3 �7t'/2, and .
then ~2~ ~flx 0 or ~f'2 z~1 + n; this is the case whir_h is close to the ~;tmiting -
case represented in Fig. l.
R ~ Or .
~5 .
~ ~ ~
/ ~ ~ `
/ ~ 11
~ ~ ~ i+~ 1 .
Q _ S' I , ~ ,
� , i
~ ? ~ il % i
_ ~ y I I ~ _ L i /
~ .~`j ~ -
_ ~ �.~IF JI~ i 1 / -
% ~ i r; ~
~ S - -a.a _ _ - - - _ -
�l I g ' . 1 Jl _ r 15 ~ ~ 'J :S' " - - - ~ - - 'S m ~
�~,~Zi i f:...
, ~
~ 'I
~ . " ~ ; " ;~Si r -
_ i ~y`/ ' ~
- 'z' ( j~ ~
~ ~ / I
J.'Ol/ ~i~�l.i
l \ / �
Fig~ 2. Locus of pairs of ends vectors which can interact with a stipulated
_ ~ector. The numbers of the four curves correspond to the numbers of the stipulat-
ed vectors. a) 1) S~3 = 0�; 2) SPg =-30�; 3) f~3 =-60�; 4) ~3- 80�; 5) circle
of the radius r= 0.5 locus of centers of curves. The point A(~ =Tf ) is double,
where curve 1, corresponding to ~3 = 0, touches cir~les of a unit radius f rom the _
outside. The point B( 7/6 iT) is double, corresponding to ~g 1T/6, touching
circles oF a unit radius fxom within. b) 1) S~3 =-80� (that is, curve 4 in Fig.
2a); 2) So3 =-85.0�; 3) S~3 =-87.5�; 4) ~_-89.0�. Fur curve 4 the scales
are not ad}~e..;:ci to .Eor the length of the "lobes": the numbers at the ends of the
- maximum dia~~~~te. ;.ndicate the approximate X 1 and x2 value.s. The dashed line -
r.epresents the lin,~_ing straight line n= 0.5.
It is easy to establish that ~s in the case of Rossby waves [6], with ~3 = 0 the
locus of rhe points is an oval curve; its minimum diameter lies on the m-axis and
is equal to 1; the maximum diameter ~s parallel to the a-axis and is equal to '
Figur.e Za s}~ows the curves corresponc~in~ to four SP3 values. As indicated by com-
putatioris, the locus of the centers of the curves is a circle of the radius 0.5. -
8
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- With approach of ~3 to - Tf/2 the "lobes" are broadened and elongated, the "waist" _
- is narrowed, aud at the limit with ~3->-Tt/2 the cantours of the "lobes" tend to
occupy the values n-axis and the line n= 1/2, shown in Fig. 1, whereas the
'~waisC" is drawn to the point (0, 1/2). The tendency fio a limiting value is s1ow~ ~ -
as is illus~rated in Fig. 2b, which, at a scale reduced by a factor of 5 in com-
parison with Fig. 2as shows curves corresponding to ~3 values close to - lt'/2.
_ With movement of the end of the stipulated vector for the unit circle from 0 to
- r/2 the maximum diameter of the curve (the same as the minimum diameter) rotates
by a half.-angle and when ~3 =-~7'/2 attains values 7t'/4. Thus, for the stipulated ~
vector (m3, n3) (length of 1) the pair of vectors is situated at the ends o� tht~ -
- diameter of the corresponding curve. If the length of tlie stipulated vector (m3,.
= n3) is changed by a factor of ~3, all the scales also change by a factor of ~03.
_ (In this sense and also for Rossby waves the first three figures cited in [6] in
actuality should duplicate one another; there is only a change in scales along
both axes. The fourth figure and its interpretation are not entirely precise, _
_ which is associated primarily with the simplifications adopted in order to facil-
itate analysis of ''very long" waves). _
III. Analysis of amplitude functions. The "small" terms have remained in expres- '
sions (12) and (13). Equating of these terms gives a system of nonlinear equations
- for amplitude functions. From equation (12) we obtain
_ dt I ms it.; - m�, tt~](!�.; p~) ~L, l
. . - ' I
~
~r' = I m~ n, - nt, n,l (P~, - !.'1) ,5, t . (21)
: i'~~ di~ ~ m ~ n: - m: ra ~ 1(!~i - p. ) _
l
The expressions in brackets are equal to one another, as can be confirmed easily
by using expressions (17). (This is easily understood from "vector" considerations:
- the triplet of vecto rs forms a triangle and each b racket is a result of the vector
product of two sides of the triangle, that is, i5 equal to double the area of the
triangle). Therefore, we will ~enote [ J as a.
System (21) has two integrals which can contain arbitrary parameters. In order to
_ obtain one integral we~will multi ly the first equation by (A/p 2~2 + B+ C~Oi),
the second by (A/p ip 3+ B+ cp , the third by (A/p 2~ 2+ B~ C~o and
_ add. On the right-hand side we ob~ain zer.o and after in~egx~ation we find the "con- _
= servation law" in the general form -
_ ~ _
- ~ I � B C~,a 1 ~~k = const = M ~Po = P:~; p; - oi), (22)
l ~k_~ ?R_~ 1
where A, B, C, M are constants. The latter is determined from initial data, that =
_ is, with t= 0 ~ k~ sha;~ld be known, and thus M is equal to the value of the sum
in (22) with the replacement of ~ k by ~ k~. ~Jith A= C= 0 expression (22) can be
_ interpreted as conservation of the "density" energy of the triplet, that is, the
sum of the energy of the waves, multiplied by the "area" of the wave vector (/0 2),
- and with A= B= 0(by analogy with Rossby waves) as conservation of "vorticityk'r
iditti A~ 0 an interpretation is difficult. Such a great arbitrariness (three ~
free parameters: A, B, C) evidently is attributable entirely to the simplifica- ~
tion of the weak interaction, since for the initial system oL equations (1') it -
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is not possible to find the conservation law in such ~eneral form. Iii orde.r to
obtain another integral we multiply the first equation by ~i/ ( p2 -~G' 3), the _
second by ~2(P 3- P i) and subtract. After integration we ohtain
~1 ~~ji IR: - 03~ - ~i~'~.;. ~Pj - Pj ~ = N~ = COfISt. ~23~ -
Combining in completely the same way, for example, the second and third equations,
_ it is possible to obtain the expression
- ~,~,~~~i �i~ ~s'~:~ ~~'i - = N_ = con~t. (23r ~ _
tlowev~r, the integrals (22) and (23), (23') are linearly dependent, as is easily
confirmed, computing the rank of the matrix from the coefficients on j/1k. It is
equal to 2.
- Sin~e with equal nioduli of the wave vectors no energy exchange occurs, without lim-
_ iting universality it can be assumed that /01 >~o Z~~o3. (This means that Nl> 0,
- NZ ~ 0) . It follows Erom (21) that
,
~ ~ � ,
; i d ,1 _ ?Z _ F:i d ~~'3 = 2 ci �5.,
p; - dl - ~j dt aj - d~
It is clear that if the sign on a Jdl ~J2 S~3 is positive, then ~ ~ and yJ 3 increase
_ there is "suction" of energy from the middle to the outer wa~'es; if a i~ll 2~12 ~13 <
0, the pic~ure is the opposite there is an increase in the modulus of the ampli-
tude of the middle wave at the expense ~f the outer waves. There can be no other
~ituations. .
After this brief analysis it is possible to rewrite system (21) in one of two equiv-
alent forms (the parameters are dimensionless):
= ~~1 - - J~ 'J.. 'J.; _ 'J~ 'J.~ ~21~ ~ ?
or
, I,
~ - - ~JY i~:,+ ~~�i - i I 'J., ~ 'J,~ _ - 'l ~ 'J~.
~21~~~
Excluding ~1 and tjl3 in any of these systems using expres ions of the type (23), ,
which in dimensionless form are written c~l2 + cf1~ = pi; ~l3 + tf1~; pl~ p2 are con-
stants, we obtain an equation for ~12, lea~ding to an e11iptical integral of the
first kind [1] F(c~\pC). [P1 ~ P2; in the opposite case a divergent interval is ob-
tained.]
t-t~= I' d~, , = F~"~\x)_F~',u1x)~
,J `(Nl �.2) (/''-~11
.,,,ti -
where
_ ; =~~re sin P_: ~:o = ~ire sin P~: (P, > P,l: sin x - P: P,.
The s ubscript 0 denotes the value at tl~~ initial moment. _
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Now it is easy to compute ~1 and t~I using the formulas for P2 and P2. In order
to siinplify the computations and for3greater clarity we assumeifor the initial
moment t~ = 0 and ~20 = 0. Then the solution for y~ , y~2 and ~3 acquires an es- ~
pec3.ally simple form when using ellipti.~al Jacobi functions. Specifically -
J+~ ~ pti dp (t ~ a); ~z = P~ stl (t I a); - p2 c n(t ~ ac).
~ -
~
o,e ~
_ 0,4 ~ J
2 '
- ~ _
~
. te 4 6 8 t
_ -D,4 _
- -D,B
Fig. 3. Curves of change of amplitude functions with time (the parameters are di-
mensionless) . 1) y~l; 2) ~3; 3) ~2.
_ Figure 3 shows t~/ k curves for P2/Pl = 0.8. If at the initial moment tp ~20 ~ 0, -
in Fig. 3 the t-axis must be displaced to the right by the t~ vali~.e, and beginn-
ing the reading from the moment tp, the curves are continued to a complete doubled
period of the elliptical functions (with selected values of the parameters to 9.0
- + t~) . The vertical dashed line shows an example when with t= tp ~20x 0.6 and
d 4~2/dt < 0. _
It remained to analyze the behavior of the amplitude functions for temperature
waves. Using (13) we obtain a system of three linear differential equations for
three functions ~T
dr = m2 n;; ~Lz Ta - m, rt: T:
_ and two similar equations with a cyclic replacement of *_he subscripts. Although -
the system is linear, the coefficients on the right-hand sides are elliptical
_ functions and therefore it is difficult to integrate it as a linear system. It is
simpler to use the correlation between t~1k and Tk (15) and obtain a system of non-
linear equations which now will be easy to integrate:
d r, = 1~a
~
f n3 - m3 n~ T T_
= d~ ~ [ ~a 1 ~ ~ 24;
(and two similar systems) . Since mq~,/~O k= O"k, and ~k conform to the synchronism
condition (17'), it is easy to establish that one and the same nurnber will stand -
in the brackets for all three functions. Thus, with an accuracy to one and the same
factor for all three functions Tk the system of nonliriear equations for T is the
same as for ~k. The system (24) has an integral dependent on the parameters A
= and B. Multiplying each of the equations (24) by ATk + B~fk and adding, we obtain
1i
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(.q Ta BTk'~k~ = CO[lSt = E~ ~ 7k o+ BTk p'Lk p~,
" t~here the subscript 0 denotes the values at the initial moment.
After introducing the dimenaionless parameters, system (24) assumes precisely the
sa.me form as (21') or (21") with the replacement of Z~k by Tk, and this means that
its solution can be obtained simply from the solution fo?- ;j/k�
*~~t
- Since wave triplets have been theoretically substantiated fo~ waves of different
� nature, it is desirable to develop a common method for p rocessing the observation-
a 1 data on the wind and temperature fields so that, on the one hand, the theory is
reinforced, and on the other hand, so that the physical picture of nonlinear inter-
a ction of atmospheric processes~will be refined. Americ~~.n meteorologists progose
that correlation coefficients between the wave parameters be used for finding the
- interacting triplet of Rossby waves. Along these line.s the finding of a triplet in
= the equatorial atmosphere requires greater care ar.d involves more complexities than -
in the case of Zossby waves in accordance with the model in [6]. It can be shown
th at the trip'c'_ of Rossby waves can exist against a background of a zonal con-
- s tant f1ow.
Note. Equafiions (17) are universal for any model of weak interaction in hydrody- ~
namics since they are generated by the operator ('i~f,l1~?) . This determines the topo-
- logical nature of the locus of points the triplet of vecto rs forms a triangle;
two vectors (in absolute value) are clc se, because it follows from (20) that I,'~=2 -
- ~I~ 1. The specific form of the cur?e is determi.ned by the phase velocities, that
is, by the specific formulation of the problem.
~
BYBLIOGRAPHY ~
1. Ab ramovits, M., Stigan, I., SPrJ,VOCfINIK PO SPETSIAL'NYM FUNKTSIYAM (Manual on
Special Functions), Moscow, N~.uka, 1979.
2. Blinova, Ye. N., "Determination of the Velocity of Troughs and Ridges from a
- Nonlinear Vorticity Equation," PRIKLADNAYA MATEMATIKA I MEKHANIKA (Applieci
- Mathematics and Mechanics), No 5-6, 1946.
3. Dobryshman, Ye. M., "Examples of Precise Solutions of Nonlinear Prognostic
Equations," IZV. AN SSSR, SERIYA GEOFIZICH, (News of the USSR Academy of Sci-
ences; Geophysical Series), No 2, 1961. ~
4. DobrysY~..~3n, ~e. M., "Wave Movements in the Equatorial Zone," MET~OROLOGIYA I
GIDROLOGIYA (1�.~~eorology and Hydrology), No 1, 1977.
5. Dobryshman, Ye. M., DIDTAI~'KA EKVATORIAL~NOY ATMOSFERY (Dynamics of the Equator- ~
ial Atmosphere), Leningrad, Gidrometeoizdat, 1980.
6. Longuet-Higgins, M. S., Gill, A. E., "Resonance Interaction of Planetary
tiJaves," NELINEYNAYA TEORIYA RASPROSTRANENIYA VOLN (Nonlinear Theory of Wave
- Propagation), Moscow, Mir, I970.
7. Sharinova, S. M., "Computation and Use of Examples of Precise Solutions of Non-
linear Prognostic Equations," TRUDY MMTs (Transactions of the Moscow Meteoro-
_ logical Center), No 4, 1964. ~
12 ~
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UDC 551.(513:509.313) -
MODEL OF GENERAL CIRCULATION OF THE ATMOSPHERE USED AT TIiE USSR
HYDROMETEOROLOGICAL CENTER
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 80 pp 16-26 -
_ jArticle by T. V. Trosnikov, candidate of g::ysical and mathematical sciences, USSR
Hydrometeorological Scientific Research Center, manuscript submitted 28 May 80]
[Text] Abstract: The article describes a finite-differ- -
ence model of general circulation of the atmosphere
developed at the USSR Hydrometeorological Center.
Basic information is given concerning the model:
system of equations, spatial and temporal ~iscret-
- ization, interpolation method. The model takes in-
to account the principal physical processes forming
the circulation: radiat~.an, interaction with the -
underlying surface, turb~uler~t mixing. T!~e hydrolog-
- ical cycle, including the accumulation of water in ~
the soil and the formation of the snow ~over, is
taken into account quite fully. The results of a
numerical experiment for the modeling of January
~ circulation are given.
- 1. Introduction. At the present time the numerical modeling of macroscale atmospher-
ic processes is one of tl~e principal methods for investigating general circulation -
of the atmosphere and also its evolution in the past and future. Using models of =
oeneral circulation of the atmosphere it is possible to study the interaction of
processes at different scales, synthesize into a unified picture the diverse and
far from complete observations and using thESe data predict the future state of
the atmosphere. Such a broad circle of problems, solvable by means of numerical -
models, determines their great diversity because a new problem can require modi- -
fication of the entire model. At the same time, models. of general circulation of 4
the atmosphere have common structuxal and organizational characteristics. This
makes it possible to have a common program "core" on the basis of which different
_ variants of the model can be formulated. At the USSR Hydrometeorological Center -
specialists have developed such a program "core," including packets of control and
- servicing programs. It has served as a basis for creating several variants of a
~ model of general circulation of the atmosphere. The following sections give a de-
scription of one of them, applied using the CDCr172 computer, and the results ob-
- tained in the reproduction of January circulation in the northern hemisphere. _
. 13
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2. Description of Model
2.1. Notations and System of Model Equations
2.1.1. List of notations
p is pressure, t is time, ~ is longitude, S~ is latitude; ~ is the rate of condensa-
tion. The quantity of condensing moisture per unit mass of dry air per unit time; _
cD is the drag coefficient; D~ are the influxes of momentum (x = V), heat (x = T) _
and moisture (x~= q) to a unit air ma~s per unit time caused by horizontal turbu-
~ lence (y = H) and vertical turbulence (y = V); E is the evaporation rate. The quan- :
tity of water evaporating from a unit surface in a unit time; f= 2,(Lsin~P is the
Coriolis parameter; F~, is the flux of outgoing long-wave radiation at the upper
boundary of the atmosphere; FS is ~he flux of short-wave solar radiation at the
upper boundary of the atmosphere; Fi is the effective long-wave radiation of the
earth's surface; FS is the total short-wave radiation at the earth's surface; fi _
- is the turbulent heat flux at the earth's surface; k is a unit vector normal to
the earth's surface; kM, k'M, k"M are horizontal turbulence coefficients; kV is
the coefficient uf vertical turbulenee; ~h is the moisture content of the soil. The -
- qua.ntity of wat~r in the meter layer of soil of a unit area; m is the rate of snow _
_ tha.wing. The quantity of thawing snow in a unit area in a unit time; n is the
- tenths of cloud cover; q is the mixin~ ratio; qa is the mixing ratia for surface ,
_ air; ~R~ is the energy influx to a unit air mass, governed by radiation and con- ~
vective processes; p is precipitation. The quantity of water falling on the earth s
- surface per unit area and unit time; ps is pressure at the earth's surf ace; R= FS
(1 S) = F~ is the radiation balance at the earth's surface; S is the depth of
the snow cov~r. The quantity of snow per unit surface, determined as the equiva-
lent quantity of water, multiplied by 10; ~ is the intensity of falling of the ;
snow. The quantity of snow falling on a unit area in a unit time; T is air temper-
ature; Ta 3s temperature of the surf~~e air; TS is temperature of the ground sur- ,
face; tr is the relaxation time of radiation processes; V is horizontal wind velo-
_ city; u is the V component directed along v is the V~component directed along _
, VS is the velocity of surface air;
Vlp)- ps ~ Vdp =
is integral velocity; z is the altitude of the isobaric~surface; zT is altitude of _
the tropopause;oCis planetary albedo; �S is the albedo of the underlying surface;
b a b b P, ~ t are the discretization intervals in space and time; V(x)
is a~ ep function equal to zero when x~ 0, equal to unity when x~ O;~G is the vector
of frictional stress; ta,'~ ~ are the components of the t vector directed along ~
and B= T(pp/p)~` is potential temperature; dp/dt is vertical velocity in a _
p-coordinate Gystem; a= 6.37�106 m is the earth's radius; cp = 1005 J/(kg~K) is
- the specifi lieat capacity of dry air at a constant pressure; c~ = 718 J/(kg�K)
is the specific tie~+: capacity of dry air at a constant volume; g= 9.81 m/sec2 -
is the acceleration of free falling; L= 2.501�106 J/kg is the latent heat of con-
- densation of water vapor; p~ = 100~ mb ie standard pressure; R= 287 J/(kg�K) is ~
the gas constant of dry air; ~~y = 461.5 J/(kg�K) is the gas canstant of water
+ vapor; Ym = 6.5�C/km is the ~tandard vertical temperature gradient; 7~= R/cp =
~.288; ~,Y = 5.67�10-8 W/(m2�K ) is the Stefan-Boltzmann constant;,SZ = 7.29�10-5
sec`1 is the angular velocity of ~he ear~h's rotation.
14 ~
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- 2.1.2~ System of Equations
aX = F~ (X, Y) + F~ (X, 1') + Q, -
~t~ _GK~
dP
- -c~`'~~-dapV -fkX~-'gct
RT
_ G.~,, T_ o apT p~p .
F,= -c' `,9 ~dpQ , F~= ~ ?
Q 0
_ 0 0
0 - c ~ PS (P~)
DH D~ -
,
- D" DT + (QRC + 1_C)iCP
-p 0 0 0 0 U
DQ -r D9 - C ^ n n
Q- � � ~ v= lC~gp ~ l' ~ ~ r
_ (1 -v(s'})(P-E)-0,1 m 1 0 0 C 0 0
~ . . .
_ 5-10~-m
_ ~ ~
V
T
DH�
~ _ 9 1. _ Z ~ pH. v = " ~
- ~ , DH. I
PS
~ '
- 1 rduX dvX cos 9 -
~ �VX=acosp1 dn + d~p
~Y - t dx t dX 1
-(acos~ di.' a dml'
, 1 ( 1 d= .Y ~3 dX
, X- a'- cos p l cos ~ v A~ + a~ cos ~ j~~.
2.2. Vertical Structure of Model
_ . Figure la shows the vertical structure of the model and the method for stipulating `
_ the sought-for variables. In those cases when it is necessary to predetermine the
function at the level where it has not been determined this is done by linear in-
terpolation.
1S ~
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- 1lonwc .Pole -
~
. 6~ , , ~ -
_ , p.Q _ w~or-Q z__xaj ~ _
a) 1~ '
_ _ U_ T 4_j ! .
- 2 ~
~ 1'z ' -
_ _ U,
T~,? . _
- ~ T - y~, ~ _
_ 1'S , mU T, 4e' p' ~ ,
. E9uator
~ao aa. ~80eW 3KOarnp 1B0�e.i 180�E
rig. l. Grid region. a) vertical structure of model; b) latitude-longitude grid of
model. -
2.3. Integration Itegioii
The integratio~~ of the model equations is carried out in a latitude-longitude grid
determined by the parameters: NF = 18 is the number of latitudes, ML = 32 is the
number of longitudes (Fig. lb). The grid points whose indexing is determined by I
the number of the latitude i and the number of the longitude j, have the coordin- -
ates:
- , - ~)0� + (1;"' i) i = 1, . . . , NF�
~ i
~--180�-}-lj-1)%:~., j=l,. . .,114L. ~
~
- where Sy0 = 5�, ,3?1= 11.5�. '
2.4. Boundary Conditions
At the upper. boundary of the atmosp~�~re with p= 0 it is assumed that vertical vel- I
ocity W, integral velocity V and the fluxes of momentum, heat and moisture become '
- equal to zero. The fluxes of moi.~Cntum, heat and moisture are stipulated at the un- ;
derlying surface with p= ps. ~
_ ~
At the ].ateral boundaries along ~ natural periodicity conditions are assumed and
along ~ are assumed in Che zone d z/~cp = 0, together with limitation of wind velo-
city; at the equator d z/~c~~ = 0 and v= 0. ;
;
2.5. Spatial Approximation
In approxima~ing ,fie model system of equations use is made of finite-difference
formulas making it ~ossibi_e, in the case of absence af fluxes and internal sources ;
at the lateral boundaries, to "co.iserve" the quadratic iiitegrals [9, 11]. In the ~
notations generally employed the meteorological literature ~
n''= Z(a;+t~. a;_,~~ ~x a= a;+~~~ - p;_,~~~ ~ ~
~
these expressions can be written as follrws:
i
~
16 ~
i
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HU^n UFFICIAL USE ONGY
~ aX _ ~GfX,. + ~ dX~�~~ _ L~ ~X~Qcos 4) + c ~s~z~,
_ dXYcosy ~F (X~ Y` cos ~p) ~ O( Z) dXY _ ba (X~` Y") + O o~~
_ a ~ = o� S~? ~ d a~ '1,
d u~ X ~~p I,w Xp)
d p = Z~ + O~o p-).
� 1
2.6. Time Integration -
For time integration of the model equations use is made of a modified Euler itera-
tion scheme proposed in f101.
XY=Xk ~ot(F; +Fll, _
Xk+~ =~k ~ ~ t~~ 1- r~i ) F~ + w, F;l -f- ~ t(( l- wz) Fk -i- w, F_,)~
where k is the number of the time interval,~ t is the time interval, equal to 15 -
minutes. [The author expresses appreciation to Doctor I. Kurihara, who directed
his attention to this integration method.] ~
The weigh~s w1 and w2 are equal to 0.506 and 3.0 respectively. _
2.7. Filtering -
Fourier fi].tering of the V, T and ps fields is used in suppressing the parasitic
modes arising in the integration of the model..equations, especiallq in the polar
regions, which is associated with the closer spacing of the grid points and with
- a given time interval impairment of the computational stability condition. In the
filtering of wind velocity first there is transfoYmation from the spherical co~
ponents of the velocity vector to the velocity components in the stereographic ~
plane
ca--usin).-vcosi., v=ucos~-vsin}�.
After filtering there is reverse transformation to spherical components. The fil--
tering parameters are similar to those used in [6].
2.8. Physical Processes
2.8.1. Macroturbulence
The macroturbulent exchange of momentum, heat and moisture by small eddies, which
cannot be describea by the finite-difference scheme in the model, is taken into ac-
count on the Uasis of very simple hypotheses concerning the nature of the macro-
turbulent regime of the atmosphere and its inclusion in the model was selected in
the form described in [8],
H_ u H r~ H T
D� - k,~ ~ tt, Dt~ = k.tii nT = ksi D,~ = k i~ ~ q, -
k`~r=k~~~tt~�� kii =k:,~~~v~, kt.._knl~Tl, kq,=a,~~~9~,
where k~ = 1015 m3, and k'~ and k"~ are selected at each latitude from the condi-
tion that the mean value of the corresponding'coefficient of macroturbulent ex-
change at this latitude is equal to the mean value (1cM + 1cM)/2.
17
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2.Fi.2. Vertical Ditfusion
The change in momentum, heat and moisture caused by vertical turbulent mixing are
deCermined by the following formulas:
- D~_ 1 d ~k au i d dv
u o vz ~ ~ r1z t' ? dz ' V dz '
- DT = P d= p k~, d~ , U~~ n ~ k~� d`~ , '
where the coefficient of vertical turbulence kv 10 m2/sec in the lower tropo- ~
sphere and zero above.
2.8.3. Fluxes at the Underlying Surface . ;
Frictional stress, the fluxes of heat and moisture at the underlying surface are ;
- determined using the formulas ,
: ~ - GCDNI vsI Vs~ -
_ H -o cocP~VsI l~s- Ta)~
i
[B = airJ
E_? c~ ~ V~ ~~9.s ( Ts~ - 9a)
. ~
For determining evaporation the surfa~e mixing ratio of air qs is determined as: _
y,. ( T,.) y'~ ( ps, T,) over the ocean
* L
l 7,s-. Te ) ` -
(B = air ] y`~ r' 9 ~ P~' r"~ ~ R~ T A- I over the land
and the mixing ratio with a saturation corresponding to a particular pressure and
temperature is determined using the Magnus formula
- ;.s:i r - r~~ri_,
(l,h~~p~ T~_ 3.;9~12 ~I) r-3i
P
for pressure, expressed 3n millibars and temperature, expressed in �K. The deter-
mination cF r~e ~ coefficient is ginen in the next section. The drag coefficients _
cDH and cn ar.e u.:r.crmined using a scheme proposed in [5].
- c~y-1),002 ~ 1,2 � l0-e z~ over the land
c~,H = Q,001 ( l-I- 0,0? ~ VS over the ocean
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- CU _ CDH~~~ 7 ~ T if ~ T= Ts - Tair
CD = COH (1 +)~o T' ~~~5;)~ if d T= Ts - Tair ~ t~.
In these formiil.as zp is the elevation of the terrain above sea level in meters.
Since the drag coefficients are dependent on surface wind velocity and the tem-
perature ~ump, these values in the computations are taken from the preceding time -
' interval. -
The wind velocity, temperature and humidity in the surface layer are determined on
the assumption that in the lower layer of the model there is no accumulation of
momentum, heat and moisture (see [5]). -
2.8.4. Hydrology of the Land
- A scheme proposed by M. I. Budyko jl] is used in determining evaporation from the
land surface. With a soil moisture content m greater than half the'critical value
n1c = 150 kg/m2 the maximum quantity of water which can be stored in the meter
soil layer, evaporation does not differ from the evaporation from the water sur-
~ace. With lesser m values the evaporation rate decreases, which is determined by
= the ~ coefficient:
1 with m~ m~/2, 2 m/m~ with m< m~~2.
The moisture content of the snow-free soil is determined by.integration of the
equation for m in the main system of the model. In a case when the soil is covered
with snaw the change in soil moisture content is determined only by the melting of
the snow,
dm '
di -
and the coefficient ~ = 1/2.
The melting of the snow is dependent on the temperature of the surface air: m= 0
with Tair~ ~~C and ~5 = 5�10'~ (Tair - 273) with Tair>
The surface of an ice-covered ocean is dealt with in the same way. In this case `
~ = 0.25.
2.8.5. Determination of Temperature of the Underlying S~nrface
- The temperature of the ocean surface is assumed to be stipulated and the tempera-
ture of the underlying surface of the land and sea ice is determined from the heat .
balance condition: LE. Since $ and E are lYnearly dependent on TS! then
- R-N(T;)-LE (Ts)
, ,
R - ~ T L'
_ 7~ JS TB+F~P~~~s~~p(1 + 9 ( a) ~
\ ~p~'rfe ~
where k is the number of the time interval. '
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2.8.6. Macroscale Condensation ~ '
Macroscale condensation processes are included in tfie f~rm of a modification of
Che followi.ng algoritfim: condensation occurs in tfiose sc}uares of a spatial grid
in which the value of the mixing ratio e~tceeds some fraction E(in the experiments
, 85%) of the mixing ratio at a saturation corresponding to the pressure p and the -
temperature T~ in the grid square q* (pp, T~). Assuming that the energy released -
in this case is expended only on a change in air temperai:ure, tlien the process -
~ itself transpires up to the time when the new temperature TN and humidity qN
' va~.ues sat~isfy the relationship qv =~q*(p~, Ti1) . Thus, in the course of the pro- -
cess the energy released will be equal to cP(TN T~). Equating this expression to the ~
energy equivalent of the released moisture, we obtain the equation _
= c~,(TN-To)=L~9u-~9ss~TN))� .
For its sulution in each time inrerval use is made of one iteration by the Newton-
Rafson method
L (E 9" ~~o) - 9olr2k
cp
~ ~ N= T0 - 1-~ 4251,6;5 E 9~ ( To) , -
~ (T�-3?)~
where k is minimum, making it possible to satisfy the condition _
C8t=9o-.:q*(T~,)~0.
All the condensing moisture falls in the form of liquid precipitation or snow if
the mean ~emperature in the lower layPr of the atmosphere with a thickness of 340
m is below 0�C. This temperature is determined by linear interpolation using the
temperature values at the lower levet of the model and in the surface layex�.
' 2..8.7. Radiation
The heat influr.es to a unit air mass,as a result of radiation and convective pro-
cesses are computed by the metho~ described in j4]. At the p level these influxes
are determined using the formula
( P`,r
Q~t~: = tP I 7'* - T- PS .I, ~T" - T) dP -I- p~FS ~1 - a) - F~ - R)~
\ ~
_ where T~ = Tair - ymz with z~ zT and T* = Tair - rm zT with z>zT. The altitude of
the tropopause is stipulated as dependent on latitude. The relaxation time tr of
the radiati~-~ nrocesses is assumed equal to two weeks. The radiation fluxes are ~
computed u. 'r~t the following formulas:
the flux of long-wave rad~ation passing through ~he upper boundary of the atmo-
sphere j3J
IB = air] F! =a-}-bTn-~a~+b~T~)?t, -
wiiere a=-800, b= 4.667, al = 81Q, bl = 3.333 are empirical coefficients (the
dimensionality of the flux is cal/(cm2�day));
2Q
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the total short-wave radiation at the underlying surface is
Fs - 0,7 FS (0,29 0,71 (1 - n));
the effective long-wave radiation of the earth's surface is
[B = air] F~ - a Te (4 ( TS - TB) Te ((0,25 - 0,004 p,.l (48 0,623))(1 - cnz),
= where c are empirical coefficients dependent on latitude [1]. -
. ZDDm/Ka ,Q~r~r/Kt J/k$ .
Q~ f00 6)
140
70
BD
~ ~ ~ ~
10 30 SO 70 90 JO SO 70 c~m days
Fig. 2. Temporal change of the mean (for the entire atmosphere) zonal (a) and eddy
- (b) components of specific kinetic energy.
-~soo
. ^n5~- ~ b) `J7 `i i
~TS q) f~ _ 20 3 i i'~ ~300
1l' Jp 0 ~'~b~ ~p ~0 ; ~ ~SDO
_ 61 a ~ ~'=5 ; S,. S.~ ; -Y ~ 700 -
e7 1`~ 2o i ~~T ti9oo
100
O) ~ ^ ~ L) 0 , '300
i
_ SoOI-4 ~~1 I + -y 0 ; ~`~i 0 Z 2 ~ ` ~60D
75C ~ 4~ ~ 6
i~ b~ I ~ ~ 700 -
� "4 900
~ZS -~6 - 0 e~ -sa -~~r"~I~ao -
) J75 J6 -14 _ j'-"'~s~1/G 3~. 300
/ 120 - I h'' Z -1~SCC
615 - ,10""_'..- 7p9
9P~
Z~"'~~ 900
' 90' B2 54 J6 1d 0 70' 60 ~i0 20 `i -
latitude
_ Fig. 3. January mean latitudinal distributions of ineteorological elements. a) zon- -
al velocity (m/sec) in model and b) according to [12], c) vertical velocity in
model (dimensionless) and d) according to [12] (10'lmb/sec~, e) temperature (�C)
in model and f) according to [12].
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- 3. Numerical Experiment
Below we give some results of modeling of January circulation of the atmosphere in
- the northern hemisphere. As the initial conditions in the experiment we selecC~d
the fields of ineteorological elements for 4 Novemb er 1969 [7]. The "boundary" con-
ditions were fixed and corresponded to January: temperature of the ocean surface, -
zonal flux of solar radiation at the upper boundary of th~ atmosphere, surf ace al- _
bedo and Cenths of cloud cover. Thus, the formation of January circulation proceed-
ed from a state of the atmosphere corresponding to autumn conditions and transition
to wlnter required about 1 1/2 months of model time. The presence of two circula-
tion regimes transitional and quasistationary can be seen in Fig. 2a. It
shows the temporal variation of the specific zonal kinetic energy averaged for the ,
- entire atmosphere. Its rapid increase during the first 45 days was associated with
� the transition from autumn to wl.nter circulation..The further small temporal trend
:i..s associated with the slow cooling of the upper layers af the polar atmpsphere _
continuing in the experiment. Figure 2b shows the behavior of the eddy part of the _
kinetic energy. Here one should note the first 20 days, when adaptation to winter ;
conditions took p1ac~, after which a cyclogenesis-anticyclogenesis regime was estab-
lished in the m~;el. One should also note the relatively high level of the ratio of ~
the zona~. part ~..i the kinetic energy to the eddy component. i
~ /la/c m ~a/cym Pa~day I_
, 900 y p) a~ s 300 f b> i
100 � ' -
0 j
-100 � ;
- - j
J00
-Z00 '
zJ d) ~ . ~ i~
.
~
100 . � �
- -400 ~
- ~00~ f00 ~ �
B~ , e . . ~
~ C ` .
- 100 � ~ BO E; 40 20 C+ i
� I
- ~ �
BD 60 ~r0 20 0
-fOP - 1 � 2 i
_ i
~
_ I
Fig. 4. .January c~.~n latitudinal fluxes. a) radiation bal,ance at the upper boundary '
of the atmospher_e ~c,:ording to model (1) and according to [13] (2), b) radiation ~
balance at the surface acco rding to model (1) and according to [1] (2), c) precip- I
itation according to model (1.` and according to [1] (2), d) evaporation according to ~
model (1) and according to [1J (2), e) thermal flux according to model (1) and ac- ~
cording to [1) (2). ~
i
The mean characteristics of Jai:uary circulation, reproduced by the model, were ob-
- tained by 30-day averaging, b eginning with the 60th day of model time. Figure 3 ~
shows the mean latitudinal values of zonal velocity ~a), vertical velocity (c) and ~
~
- 22 !
I
i=
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temperature (e).. As noted, in this experiment a somewhat reduced temperature was
~ obtained in the upper troposphere of the polar regions, with its exaggeration in
the tropics, which in the model 1ed to the formation of a strong jet in the upper
atimosphere (see Fig. 3a) . Another peculiarity of the zonal flow, distinguish3.ng ~.t
- from the observed pattern ('b), is the inadequate development of easterly winds in
_ the tropical region, which o ccupy only the lower half of the troposphere. This
peculiarity of the circulation is associated with the inadequate spatial resolu-
- tion of the model. The strueture of the circulation cella in the atmosphere (Had- -
ley and Ferrel cells) can be seen in Fig. 3c, which shows the mean latitudinal
_ vertical velocity cJ . A comparison with the observed pattern (d) shows a fairly
- good agreement of tfie maxima and minima, which indicates a quite precise reproduc-
- tion of the circulation cells by the model. -
Figure 4 shows the mean latitudinal values of the r~d3ation, moisture and heat
fluxes. On these graphs the dots represent the observed January values of the cor- ^
- responding parameters. The comparison indicates a quite precise model reproduction
of the peculiarities in the distribution of the radtation and moisture fluxes ob-
served in January. An except ion is the heat flux fram the surface. However, one
shuuld note the considerable uncertainty in the empirical data on the heat fluxes
from the land surface for January.
- In conclus3on the author considers it his duty to note the important role played
by the specialists of the Laboratory on Numerical Modeling of General Circulation
of the Atmosphere at the USSR Hydrometeorological Center in creating a programmed
' core of a model of general circulation of the atmosphere in the carrying out and '
analysis of numerical experiments.
BIBLIOGRAPHY
- 1. Budyko, M. I. r TEPLOVOY BALANS ZEMNOY POVERKHNOSTI (Heat Balance of the Earth's
- Surface), Leningrad, Gidrometeoizdat, 1956.
2. Budyko, M. I., ATLAS TEPLOVOGO BALANSA ZEMNOGO SHAR~'~ (Atlas of the Earth's Heat
Balance), Moscow, Mezhduvedomstvennyy Geofizicheskiy Komitet, 1963.
y 3. Budyko, M. I., IZMENENIYE KLIMATA (Climatic Change), Leningrad, Gidrometeoizdat,
1969, _
- 4. Trosnikov, I. V., Yegorova, Ye. N., "Use of Empirical Formulas for Computing
~ Radiative Energy Influxes in the Modelin~ of General Circulation of the Atmo-
sphere," TRUDY GIDROMETTS~NTRA SSSR (Transactions of the USSR Hydrometeorolog-
ical Center), No 160, 19 75. .
5. Arakawa, A., "Design of the UCLA General Circulation Model. Univ. of Califor-
nia, Los Angeles, Dep. of Meteorol.," TECHrTICAL R~;PORT, No 7, 1972. ~
- 6. Bengtsson, L., "ECMWF Global Fcrecasting System European Centre for Medium
Range Weather Forecasts," TECHNICAL REPORT, 1978.
7. Carson, D. J., "First Res ults from the GARP Basic Data Set Fro~ect," THE GARP
� PRJGRAI~~IE ON NUMERICAL EXPERIMENTATICN, Report No 17, 1978.
' 23 _
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8. Gilchrist, A., Corby, G. A., Rowntree, P. R., "United Kingdom P4eteorological
Office Five-Level General Circulation Model," METHODS IN CQ'~:JTATIONAL PHYS- -
TCS, Vc,l 17, 1977. -
9. Holloway, J., Leith, Jr., Spellman, P3. J., I~Ianabe, S., "Latitude-Longitude
Grid Suitable for Numerical Time Integration of a Global Atmospheric Model,"
MON. WEATHER REV., Vol 101, No J, 1973. _
10. Kurichara, Y., Tripoli, G. J., "An Iterative Time Integration Scheme Designed
to Preserve a Low-Frequency Wave," MON. WEATHER REV., Vol 104, No 6, 1976. -
11. Miyakoda, K., "Cumulative Results of Testing a Meteorological-Mathematical .
Model," PROCEEDINGS OF THE ROYAL IRISH ACADEMY, Vol LXXIII, Section A, No 9, -
1973.
- 12. Oort, A. H., Rasmusson, E. M., "Atmospheric Circulation Statistics," NOAA
PROFESSIONAL PAPER 5, 1971. _
13. Oort, A. H., Vonder Haar, T. H., "On the Observed Annual Cycle in the Ocean- ' _
Atmospher.e Heat Balance Over the Northern Hemisphere," J. OCEANOGR., Vol 106,
:~0 6, 197t;. -
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UDC 551.(509.313+513)
USE OF NESTED GRIDS I~THOD IN THREE-D IMENS IONAL ATMOSPHERIC MODEL
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 80 pp 27-33
_ [Article by A. I. Degtyarev, USSK Hydrometeorological Scientific Research Center,
- manuscript submitted 2$ May 80]
- [Text] Abstract~ A hemispherical finite-differ-
ence model of the atmosphere in full equa-
_ tions is considered iii a quasistatic approx-
imation. The model makes it possible t.o use
a fine nested grid for improving spatial
resolution in a stipulated region. The al-
gorithms~employed in carrying out joint
camputations for two grids are discussed
in the article. The author gives the re-
= sults of control computations based on real
data. The results of experiments with the
influence of a solution for a nested grid
on the solution for a thin grid and without ~
this influence are compared,
In solving problems in the numerical modeling of the atmosphere by the finite-dif- _
= ferences method the question inevitably arises of the spatial resolution of the
mode. The answer to this question is usually dictated by the capabilities of the
electronic computer used. Accordingly, computations in a region comparable to a
hemisphere (or sphere) must be made with great spatial intervals which cannot de-
scribe the development of atmospheric processes sufficiently well. The errors
caused by too thin a grid lead to a distortion of both the amplitudes and the
phase velocities of wave components of atmospheric disturbances. An improvement
in the spatial resolution model will make it possible to take shorter waves into _
account, and what is equally important, describe long waves with a greater degree =
of accuracy.
7n order to improve the spatial resolution of the model in some region, a method
based on the use of so-called nested grids (NG method) has now come into wide use.
_ The basic idea of the method is that the computations are mad~ using two (or more)
grids with a different spatial-temporal interval. The grid with the higher reso- ~
= lution (dense grid) in such a case is atipulated in a definite region, whereas
25
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Che grid with the poorer resolution (thin grid) covers a greater region, such as =
_ a hemisphere.
There are two methods ("two strategiesy" to use the terminology employed by Phil- _
lips and Shukla j12]) for solution of a problem when employing the NG method.
Strategy I assumes the stipulation of boundary conditions for the dense network
on the basis of a solution for the thin grid. Strategy II, in addition, affords
~ossibility for using the solution in the dense grid in integration for the thin
grid ("feedback"). A comparison of these strategies was presented in [12] for dif- -
ferent sets of m43e1 initial data. The mean square errors, computed over the re- ~ _
gion with the dense grid for one strate~y or the other relative to the control
computations for the dense grid in the entire regiony indicated the advantage of -
strategy II for all sets of initial data. However, this str~.tegy, in contrast to
the first, requires the carrying out of joint com~utations for both grids, which
leads to additional expenditures of computer memory. -
At the present time strategy I has come into wider use and is employed quite suc-
- cessfully in numerical modeling and hydrodynamic short-range we~ther forecasting -
[l, 2, 14]. Spertfic realizations of the NG method differ with r~~spect to the
methods for form,~l.atin~ the boundary and initial conditions for '.he dense grid,
methods for nliminating the computational gravitational waves arising in the re-
~ gion of the dense grid and some other details. From tiii~ point of view the meth-
_ odological studies of Chen and Miyakoda [5] and Miyakoda and Rosati [11] are of
interest. These studies give a comparison of different methods for formulating .
the boundary conditions for the dense grid. The results of the experiments indi-
cated that the so-called "sponge" metho~l, based on use of highly dissipative vis-
cous terms for suppressing parasitic w3ves in the boundary regions, gives excess-
ively smoothed fields of ineteorological elements. Then we find that the apglica-
tion of the "radiation" condition at the points of outflow from the region of the
dense grid does not completely eliminate the reflected high-frequency waves at the
boundaries of the dense grid and re~.sires additional boundary smoothing, as when
using the solution for the thin g~i~i at all the boundary points of the dense grid.
Tn addition, the method for stipulation of the boundary condition with "radiation" -
is by no means trivial with use j-~ a system of full equations.
Strategy II is presently used primarily in the modeling of the development of trop-
ical cyclones. In the studies of Jones [8], Ley and Elsberry [10], the authors
examine models consisting of two nested grids with a"feedback." These authors
feel that the presence of a feedback does not allow a significant discrepancy in
the solutions for the thin and dense grids, which can arise in computations when
- using strategy I. Therefore, the boundary conditions for the dense grid, obtained
- from the sole�~i.on for the thin grid, to a high degree correspond to the solution
= fo r the deu.. gr-t d. -
In this article we examine an application of the NG method in a;nodel in full equa-
- tions and also compare the re:;ults obtained when using strategies I and II. The
principal mathematical model ::ur the experiments with use of the NG method is a
model of general circulation of the ?tmosphere developed by I. V. Trosnikov [3].
The NG method was realized in a varianC ~f this model without allowance for radi-
_ ative and phase heat influxes. We will briefly discuss the problem of formulating
the problem in a thin grid (a more detailed description of the model is given in
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[3]), and then we will examine the principal algorithms for the organization of ~
the computational process in a dense grid.
- ~~te system of full equations in hydrothermodynamics is written in a hydrostati~C
appraxi.mation in a spherical coordinate system (SD is latitude, a is longitude).
- A p-coordinate system is used vertically. The state of the atmosphere in the
- model is determined by the zonal u- and meridional v-components of wind velocity,
the analogue of vertical velocity cJ, elevation above sea level z, temperature ~
- T and surface pressure PS. Then, using the generally employed notations, the sys-
tem of equations can be written in the form
dt a cos m d
n+(f ~ u) v-F- F~H FH, ~1~
dc a vp -(I +'Q4 u~ u-t-PH~-Fu,. (2)
dT RT~~~ ~3~
ar ' r FH -i- F .
l'p P
Ps ~s ~ .
~p.~ + t a udp + a(~ v cos dp 0, (4)
ar a cos q d), f
0 U �
a cos ( d~ dv~cosx 1-f dw =0 (5) _
~ , r J p '
~ ds RT ~6)
tlP 8P ' -
where f is the Coriolis parameter, a is the earth's mean radius, c is specific
heat capacity at a constant pressure, R is the spec3fic gas constant of air, Fu, ~
FH, FH are terms for horizontal turbulent diffusion of momentum and heat. H
_ In the model use was made of a simplified method for taking into account the non-
, linearity of the diffusion coefficient in dependence on flow deformation proposed
in [6]. This coefficient is assumed to be proportional to the modulus of the La-
placian of the corresponding value. The terms for vertical turbulent diffusion
P'W, FW, FW determine vertical turbulent movement.
The solution of system (1)-(6) will be sought in a hemisphere (region G with the
boundaries and in a region (region e with the boundaries d). We have e+ d E G.
- In the hemisphere the boundary conditions are stipulated similar to [3], with the
- single difference that the surface temperature at the present time is considered _
_ fixed. This is attributable to the fact that in the experiments carried out the
nonadiabatic in�luxes are not taken into account and the computations are made
for a time up to 3 days. The formulation of boundary conditions for a dense grid
will be discussed below.
- With conversion to the difference problem in the region G+(- there is stipulation
of a grid "1~tg with the interval H, and in the ~ region e+ d-- the grid 'lJ~dg with
- the interval h in such a way that H= Kh. In these experiments K= 2. Both grids
- are latitudinal-longitudinal. In writing the system of equations (1)-(6) in finite- -
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difference form use was made of the scheme et~loyed in j7]. As is we11 knoc~m, such
a scheme makes it possible to conserve the mass of the atmosphere and also the
total energy of atmospheric movements under the condition of adiabaticity and ab-
eenCe of friction.
When using a latitude-longi*_ude grid in the near-polar latitudes the horizontal
spatial interval becomes small ar~d therefore Fourier filtering is applied to the
main prognostic functions fo r maintaining computational sfiability in these lati-
tudes.
For time integration in the model use is ma.de of a modified Euler scheme with the
scaling proposed in [9],
,r" - X' , ~ ( f~ -'t- F; l,
(7)
- X=+' = Xt -4- ~z ((1 ~ w~) F~ w, F'~1 - ~ra~_) F~ + w, F~).
In (7), in the second half-interval, the terms on,the right-hand sides are divid-
- ed into advective terms F1 and a11 the remaining terms F2. The values of the
weights wL = w2 = 1 give the known Euler scheme with scaling. Scheme (7) with wl
= 0.506 and w2 = 3 differs from the Euler scheme by scaling with considerably
lesser dissipa`:..~ ty in the region of long waves.
- I1ow ~ae wi11 examine the principal algorithms determining the characteristics of
integration in the dense grid. As a convenience we will represent the set of
points of intersection of a thin grid
B ~
~Vt~ _ ~V t~ + Y~g + 4~tg,
~ where ~tg are the grid points of intersection within tl~ e region, y~g are the
gr.;.d po~.nts of interGP~rinn ]ving on the boundary d. SPtb are the orid noints of
intersection outside the region e+ d. ~
The dense grid y1d~ consists of the boundary points of intersection 'Ydg, a series
of points of intersection 'Y ldg :-.l~acent to the boundary points of intersection, and
the remaining internal points of intersection 't}fdg, that is
- ~dg = Ydg + '}/ldg + y~dg.
The solution of the problem in the dense grid is dependent on the solution in the
thin grid in the s ense that for each time interval in the dense grid it is neces-
sary to stipulate the boundary values at the grid points of intersection 'yd . The
seeking of *r,~ miss ing values at these points of intersection is accomplishe~ by
parabolic iiYrerp.tation. As the control points for the interpolation we use the
values at the poinc~ of intersection of the thin gr3.d 1~tg.
In the model we used an alg:~._hm for the partial change in the spatial computa-.
tion region with interpolation in tj.me in accordance with scheme (7) in the fi.rst
and second half-intervals, This algorithm was tested by the author using a one-
dimensional model. It is easy Co clarify the effect of this model, examining the
~ time integration scheme (1). The computai~ions in the first half-interval are made
28
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_ at the points of intersection 'Yldg +~dg, uaing the boundary points of intersec-
tion yd . Ttie revults of these computationa are the X* values at the points of
interaec~ion yldg + y~d . In order to obtain the X'G+1 values at these points of
inCersection it is nece~sary to have the X* values at the points of intersection
ydg. In general, the X* values have only an auxiliary character in the iteration
scheme (7). It is therefore not entirely correct to obtain them by interpolation
_ of the corresponding values from the points of intersection of a thin grid, or,
ris in the first half-interval, use the X values. Accordingly, the computation re-
- gion in the second half-interval is na.rrowed and the X t+1 values are computed only
at the points of intersection ~dg with the use of Y1dg as boundary points. The
X t+1 values at the points of intersection '~ldg are found using parabolic inter-
polation.
_ Since the solution for the dense grid in principle should give a lesser error, as-
sociated with the spatial resolution, in calculations for the thin grid at the
points of inrersection '}JB it is possible to use the results of computations in
the dense ~rid. This opera~ion is carried out using a nine-point Shuman operator
_ [13]. The Xp value at each of the points of intersection Bg in the thin grid is
- determined from the value for X~ at the corresp~nding point of intersection ?~dg
in the dense grid and the valuea at the 8 surrounding points of intersection.
Xo = Xu -i' 2 Y(1 - ti)~X, + Xa -f- X,; XR - 4,l'o)
. 9 v~1X1.'F'X3-~X3"~'X~-4X~~.
- The opera~or (8) with 'V= 1/2 best eliminates two-step perturbations.
~ For a better correspondence of the comp utational solution for the dense grid to
- the boundary conditions obtained from the solution for the thin grid and for
_ suppressing the high-frequency computational modes, in the model use is made of
boundary smoothing, which is emp~oyed for only one row of points of intersection
_ 'Y ldg along the boimdary of the dense grid. A three-point smoothing operator, fre-
� quently employed for these purposes, is used
X('Yldg) = 0.5 X('Yldg) + 0.25 (X('j'tg) + X(~ag)),
= where '4fag is the row of points of intersection in the region ~dg adjacent to
Yldg.
As the initial data for testing the model we used the GARP BDS set of data in [4].
These data represent an analysis of the wind velocity and temperature fields and
the pressure field at sea level.
- Tn these experiments the thin grid was determined j.n the northern hemisphere with
a meridional interval 5� and along the circles of latitude 11.25�. The
dense grid haa half the horizontal spatial interval and is supErposed on the e~ist-
ern hemisphere from 82.5� to 42.5�N. In the central part of this region, and spe-
cifically over the northeastern part of the European USSR, there is a cyclone ac-
- companied by a deep trough at H500 which moves in an easterly direction. During
the first two days f:he generation of this low takes place and then it is gradual-
ly filled. The nested grid method is employed in order to ascertain the movement
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of this low more precisely and also the movement of pressure formations associated
with it. _
- �ao a a sa o so �e. a. ~eo�~. a 90 ~ sa � a;. E
aa - Q, ~ eo 6~b G~.
~ a9s ~
r0 O Sy?~1B 60 ^ ~ ~
512
~a s~s
6D szs
'r0 O O ~ 34f
60 ~ �
!r~ ~ ~ ~s76 z0 O n t~s76
, 90 /~n ~o )i i:'.-~~~~ ~n
J 90' d. . 1''.r. 90 ?0' e d
~ ~o e~ ao'
,--~1
s~l ~ s~2 60 s12 '
544 ~ \ ~ 52d
S60 f0 / 560
~ ~ s~s ,,J ,s;s
�Q ~ ~ zo o~
_o� ' ~ ys 9a ~as�e.a. o� ~ .9
5 0 !Jf i ~
d) e)
' ia e 7d ~ ~
'~s~2
sn 0 f~1 60
~ sze _
so s: e so
L ~60 ~ ~i ~ I / 1 \ 544
~ Fig. Z. Fields of heights of isobaric surface H500� a) initial, b) actual, after
3 days, c) result of computations for 3 days in experiment B1 for thin grid, d)
- same, in experiment B1 for dense grid, e) in experiment B2 for thin grid, f) B2
- for dense grid.
' I Experiment B1 ~ Experiment B
D~ i thin ~rid dense Qrid ~ thin rid deizse grid
- ~ I k E I P � I k I E I P ~ I k I~ P a I k I E I P -
I ~ I ~
1 11,3 0,69 O,ii 0.4~I ~0,6 0,75I0,70 0,34 9,7 0,79 0,660,42 9,F U,83 O,F5 J,42
2 15,8 0,66 U,93 0,33 14,3 0,73 0,80 0,37 12,8 0,31 0,7i,0,33 t1,7 0.82 0,76 0,45
3 18.8 0.56 1.2d 0.2% 17,4 O.B:~ :,~2 0,45 14.9 0,71 ~~.990.27 14.9 0.700,960,36 '
3Q
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- ra~~~ W en c 9o'a~ E rE~AMiKa J/kg
" a; t~ ~ ~ ~ ~ a ~ �
/ \ ! \ ~ -00 ~,~,~.,k;. � .
/ \ i
- i ' I I ~ BO ~
\y` ~ ~ ~ ` ~ ~ \'~.:y:
' ' b~
61 t ~ ~ y0
, I I ~ ~ p~~
f; ~ ~ o
f ~ ~ ~
b-
.b~ ~ ~ I ~ n�Z ~,n /Ra J~kg ~ .
- Z I . ? - ~ ~ cP, since the heat introduced into such a system is e*pended not
anly on heating, but also on phase transitions. It is important that cp can be es-
sentially dependent on temperature. '
As an example, we will assume that
jH = sat ] 9_ k P~~ ( T)
N1 0 ,
where k is a dimensionless value less than unity, sat~T) is the density of the -
, sa~urated vapor at the temperatt~r~~ T, ~ is air density at the underlying surface.
~
Q9 K�O,o
~�O,g .
0, 7
. 0,5 -
N G, m, 60 40 10 3 ZO 40 60 b ar. S
Fig. 1. Change in p~ value with latitude.
This assumption makes it possible to compute the ol value at different latitudes
which correspond to the observed latitudinal. distribution of inean annual tempera~
tures. The results of such computations ior different k= 0.8-0.6 are represent-
ed in Fig. l. It can be seen that the effective heat capacity in Che southern ~
46 -
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latitudes can be two or three times greater than in the northern latitudes. It
can be concluded from this that fluctuations of the energy source E identical in
intensity will lead to fluctuations of temperature greater in value in the north-
ern and temperate latitudes in comparison with temperature fluctuations~in the
southern latitudes.
It follows from (9) that the intensity of precipitation is determined by local
evaporation E, the local rate of change in atmospheric temperature and the mer-
idional transfer of water vaFor, which is regulated by heat transfer and which -
- is described by the last term in (9).
It is important that the averaged intensity of precipitation was not sensitive to
different variations of cloud cover. This conclusion i~ a result of the fact that
- after spati.al averaging along the circles of latitude and for time intervals ex-
. ceeding the time of atmospheric relaxation that part of the condensed moisture
which is expended on an increase in the size or liquid-water content of clouds
_ is a small fraction in comparison with the total quantity of condensed moisture.
- It should also be noted that it was possible to match the heat and moisture ba1-
ances by the introduction of only one additional parameter dq/dT. Such an approach
made it possible to exclude precipitation and evaparation from the underlying sur- _
face in explicit form from the thermal regime equation (8) and make equation (8)
independent of (9). It goes without saying that (8) will nevertheless be dependent
on (9) if one takes into account the possible influence of precipitation on the
zonally averaged albedo and the intensity of long wave radiation.
- Variant of model formulated by M. I. Budyko. We will use the M. I. Budyko hypothesis
[1] in which
L ~r - r~, _ ; ~ T - T~,1, (10)
where T is the mean planetary temperature of the lower air layer. For modern con-
_ ditionspTP = 288 K. This relationship was confirmed by materials from satellite
observafiions for the mea~ annual conditions of the radiation balance of the earth-
atmosphere system [3]. The following is obtained
y ~ 2.784 Cal/(cm2�degree�year)
We will examine a case when r~ ~-(T - TP). Then expression (10) makes it possible
to represent the flux QT by a nanlocal expression through the temperature field
Qr ~ S~ ~ .I ~T - Tn~ sIn ' d (11) -
0
where y- y* determines the fracti.on of heat transported by ocean currents. The
physical sense of the nonlocal parameterization of the flux (11) is explained in
[4].
The representation of the flux in form (li) makes it possible to write equations
(8) and (9) in the form
47 -
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_ ;
~
i
~ tNco d T + _ (12) ~
~
a dr i lT - Tpl r,., ;
i
1=E- dq d~ -(1-a) ~ (T-T~)-i-
;
r~ '
~ d~ L sin ~ 1~T - Ta) Sin d~'. i
- ' � ' (13) ~
~
i
~ The semiempirical equation (12), except for the nonstationary term, retained the ;
same form which was proposed in the initial. M. I. Budyko va:~iant [1]. Using for-
mula (13) it is possible to compute the mean annual latitudinal distribution of
precipitation corresponding to the latitudinal distribut-ton ot mean temperature j
obtained from solution of equation (12) [13] (with a T/ a t~ O) and by means of !
measurement. As the mean latitudinal distribution of evaporation E use was made ~
' of the data published by Sellers [9, 13]. As an example it was assumed that q/M =
- k /~sat~T~~P � . i
~
~ - ~oo"~`~eod mm/year
. ~
~ ~ ~
J ~I1SG0 h�QP
~ ( : ~ ` ~ ~ ,
n�QB ~ ~ , ~�0.6
r�;!E', _ ' `
~
~
. ~ ~ ~ ~
~ ,
. ~ ~ S00
~!i
- ! -
N c.~ so fo zo .c Ma so b;u:S
i
Fig. 2. Experimental curves of the mean annual distribution of precipitation (1)
and evaporation (2) [9], and also c-~puted distributions of precipitation (3) '
'on earth.
The results of such computation~ ~rith k= 0.8-0.6 and y* are represented in
Fig. 2. Here also for comparison we give the experimental curve for the mean an- ~
nual distribution of precipitation on the earth [9]. ;
�
The computed curves of the distribution of precipitation virtually coincide (Figo ~
2) for cases of temperature distributions measured and obtained from solution of
equations (12). The dependence of the distribution of precipitation on the para-
meter k in the range of its values 1-0.6 is weak. There is a qualitative agreement _
of the.compu`ed and measured distributions of precipitation in the high and te~ -
perate lati~~ules. However, there is a considerable noncorrespondence between
theory and experin:~ntal data for the low latitudes. This discrepanGy is attrib-
utable, in particular, to the fact that the considered mechanism of ineridiunal
transport of hea t and moisture does not apply in the tropical zone uf the atmo- -
i sphere. Moreover, it follows xrom (13) that this mechanis~ gives an incorrect -
direction of Che transfer of moistur~ in the tropical zone from the equator to the -
po1e. ~
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In actuality, however, the transfer of moisture by the Trade winds occurs here in -
th~e opposite direction, which also leads to the appearance of heavy precipitation
in the ICZ. However, in this model heavy precipitation does not appear in the ICZ.
It can be concluded from this that the region of Hadley circulation is not satis-
factorily described within the ~ramework of one-dimensional models, which can be
used correctly only in those cases when movement in the troposphere has a quasi-
two-dimensional character and the transfer of heat and moisture along the meridian
is accomplished by macroscale atmospheric eddies and not by regular currents, as
- in the region of the Trades. ~
Next we note that on the basis of equations (12) and (13) it is possible to solve
_ self-consistent problems if the E value is determined. According to the schematiz-
ation cited in [2], E= ftw, where tW is the temgerature of the surface of ocean
- waters in degrees Celsius. Since the change in tW with latitude qualitatively coin- -
- cides with the variation E(Sp), using the correlation between E and tW and in a
definite way carefully selecting the proportionality factor f, we qualitatively ob-
tain the same results for the distribution of precipitation as shown in Fig. 2,
The correlation E= ftW was obtained with neglecting of the dependence of E on
~aind veloc3ty variations. Tl-.is is justified by the fact that in the computations
in [2] use was made only of the mean latitudinal and mean seasonal values of the ~
parameters for which different variations of wind velocity possibly are real but -
play a small role.
In a general case, however, it is usually assumed that E and rp are dependent on -
wind velocity. Then for the closing of equation (9) it is necessary to have dy- `
namic equations. Such equations within the framework of a one-layer model, with
allowance for the mechanism of interaction between moving atmospheric eddies and
the main flow of the atmosphere, were proposed in [8].
In conclusion the author expresses appreciation to N. S. Okhrimenko for assistance
in carry3ng out the compu~ations.
: BIBLIOGRAPHY
- 1. Budyko, M. I., "Origin of Glacial Epochs," METEOROLOGIYA I GIDROLOGIYA (Meteor-
ology and Hydrology), No 11, 1968. .
2. Budyko, M. I., IZMENENIYE KLIMATA (Climatic Change), Leningrad, Gidxometeoizdat,
1974.
3. Budyko, M. I., "Semiempirical Model of the Thermal Regime of the Atmosphere
and Real Climate," METEOROLOGIYA I GIDROLOGIYA (Meteorology and Hydrology), No
4, 1979.
- 4. Voloshchuk, V. M., Svirkunov, P. N., "On the Problem of Nonlocal Parameteriza-
tion of Turbulent Flows," METEORGLOGI~'A I GIDROLOGIYA, No 7, 1980.
5. Voloshchuk, V. M., Se,3unov, Yu. S., "Kinetic Equation for Evolution of the ~
Spectrum of Droplets in a Turbulent Medium in the Condensation Stage of Cloud
_ Development," METEOROLOGIYA I GIDROLOGIYA, No 3, 1977. -
- 49
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6. Gandin, L. S., Dubov, A. S., CHISLEI3NYYE METODY KRATKOSROCHNOGO PROGNOZA
POGUDY (Numerical Methods for Short-Range Weather Forecasting), Leningrad, ~
Gidrometeoizdat, 1968.
7. Kabanov, A. S., Sedunov, Yu. S., "On the Equations for the Transfer of Heat
and Moisture in Clouds," TRUDY IEM (Transactions.of the Institute of Experi-
mental Meteorology), No 30, 1972. �
8. Kabanov, A. S., Shmerlin, B. Ya., "On Allowance for the Influence of Rotation
- and Movement of Eddies in the Main Flow of a Fluid," METEOROLOGIYA I GIDRO-
LOGIYA, No 10, 1979.
~ 9. Pal'men, E., N'yuton, Ch., TSIRKULYATSIONNYYE SISTEMY ATMOSFERY (Circulation
- Systems in the Atmosphere), Leningrad, Gidrometeoizdat, 1973.
_ 10~ Stepanov, A. S., "Condensation Growth of Cloud Droplets in a Turbulent Atmo-
sphere," IZV. AN SSSR, FIZIKA ATMOSFERY I OKEANA (News of the USSR Academy
of Sciences: Physics of the Atmoaphere and Ocea.n), Vol 11, No 1, 1975. ,
11. Coakley, J. A., "Climate Modeling Radiative-Convective Mode1," REV, GEOPHYS.
SPAC~ PuY5., Vol 16, No 4, 1978. -
12. North, G., "Analytical Solution to a Simple Climate Model With Diffusive Heat
Transport," J. ATMOS. SCI., Vo1 32, No 6, 1975.
13. Sellers, W. D., PHYSICAL CLIMATOL0~1, Univ.. of Chicago Press, Chicago, Illi-
nois, 1965.
14. Sellers, W. D., "A G1oba1 Climatic Model Based on the Energy Balance of the
Earth-Atmosphere System," J. AP?L. METEOROL., No 8, 1969.
- 15. Sellers, W. D., "A New Globa:. C~.imatic Model," J. APPL. METEOROL., Vol 12,
_ No 2, 1973.
. -
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UDC 551.558(575.4)
EXPERIMr,NTAL INVESTIGATIONS OF OROGRAPxIC WAVES AND VERTICAL MOVEMENTS IN THE
_ NEIGHBORHOOD 0~' KRASNOVODSK AIRPORT
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 80 pp 49-53
_ [Article by 0. A. Lyapina and Ye. I. Sofiyev, candidate of physical and mathemat-
_ ical sciences, Central Asiatic Regional Sci~ntific Research Institute, manuscript
- submitted 7 Apr 80]
[Text] Abstract: Thie study was carried out in
- connection with the complex conditions _
involved in the landing of aircraft during
presence of strong northerly winds. Oro- -
graphic waves were investigated using con-
- stant-level pilot balloons tracked by
theodolite. Vertical. movements were ir.ves- -
tigated using conventional pilot balloonso
The parameters ~f waves are given, the con-
ditions for their development are described
and their influence on a landing aircraft
is discussedo
Krasz~ovodsk airport is situated on a plateau which drops off steeply (at an angle
of 45�) to Krasnovodskiy Gulf. The dxopcff extends latitudinally and at the edge -
- there are highlands with a sma11 (about 50 m) relatit?e relief.
_ . 4M ~~~}11~y~
u ~j
c7ol Bemen Wind , BT~~~+~ _
~
_ ~ %~7 Flight strip ~ - ~ ~ -
= enn ,--A . - -
- ~~.Ii~f.~ � 11� �frM�
C \
_ 2 ,3 4 3 6 7 L ~rM ' -
- Fig. 1. Orograph:ic waves near Krasnovodsk airport. -
_ Figure 1 schematically shows the vertical profil.e of the terrain ~long the landing
b:~=am when an approach is made from the north.
In an investigation of the orogranhic deformation of the flow in the case of ~,orth-
erly winds we released constant-level pilot balloons from point B, situated at the
edge of the dropoff. The tra~ e~tories of the pilot balloons were tracked by two op-
tical theodolites set up along thP shores of Krasnovodskiy Gu1f (point C, ~ig. 1)0
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HM
k nxxx
~400 x xx
. x X ~ xx"x ,uc
x r:X ~ , x "x ~w~xxx x
9,i`DO kk " xxx "x x xX
~x~a~, x
k
x
x
-
- +
_ 300 ~ o ' 2 0~0 ~~~oo
+ + o~�~
~o
+ �
200 + : � ~ ~
+ , ~
+
~ o +
~DO y � �
+ ~ a~ ~�c
cb
. oc
i i ~ i i i
D % ? ~ ~ S 6 7G~M
Fig. 2. Traject^~-~~s of constant-level pilot balloons for different temperature
gradientso Oc~~ber 1978. 1)~'= 0.6�C/100 m(24 Oct, 1130 hours), 21 = lo0�C/100 '
m(24 Oct, 1430 hours), s) y= i,l�C/100 m(14 Oct, 1130 hours)o .
In addition, at points A and C ordinary (non-con5tant level) pilot ba1].oons were
= launched for measuring the vertical wind profiles and these were tracked by theo- ~
_ dolite. All ~he pilot balloons (both confitant-level and ordinary) were released ~
- each houro The coordinates were reckonel with a discreteness of 15 sec with AShT -
~ theodolites (accuracy of readings O,OI�). The observations were processed on a -
- "Minsk-2L" electronic computer using a specially formulaCed algorithm.[1], making -
lt possible to compute zhe coordinates with any arrangement of the pilot balloons
relative to the base (in particular, ahen the pilot balloon is at the zenith, in -
the vertical plane of the base, an~ a:.so at the same level with or even below the =
observation points).
~ The ten~perature stratification was evaluated by making measurements of temperature -
at point C and on the landing strip (difference in elevations 120 m), and also us- ~
ing data from aircraft soundings and radiosonde measurements.
. Wave movements were observed when there were northerly winds exceeding 6 m/sec and
a stable thermal stratification. These conditions are evidently necessary for the
~ appea*_-ance of waves, that is, are always observed when waves are present, For the
time being it is not clear whether they are also adequate, that is, whether they -
iaevitably "le~d t~ waves. The vertical wind profile measured at point A, despite
expectations [3,4], ~;id not exhibit a significant correlation with the presence
of waves. It is easy to trace.the development and destruction of waves dur3ng the
course of one northerly intrus-~~n when they developed during a stable stratifica-
tion and were destroyed when Liie stratification was unstable (for example, on 17
= and 24 October 1978). The waves usualiy have a lifetime of 0.5-2 hours; their fre- -
quency of appearance is relatively sma1~: in 50 cases of observations of constant-
level balloons waves were noted in only I1.
.
$
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The vertical extent of the wave disturbances is dependent on the thickness of the
- inversion layer. According to data from research flights it usually is 800-1,000
~ m, that is, exceeds the height of the obstacle by a factor of 10. The horizontal
eXtent of the waves can exceed l0 km.
- Table 1
Characteristic Parameters of Lee Waves. October, 1978
- Number Time 'Y�C/100 m u m/sec H km ~ km Amax ~ Wmax I `
_ 2 17h27m 1.2 5.0 0.25 1.7 Si~O 2.7
8 14h10m 8.1 0.3 2.1 200 3.4
10 9h38m 5.2 0.6 1.9 100 1..3
10 1Oh30m 6.3 0.5 3.0 130 0.8
H-- mean wave height; A~X maxim~.im wave amplitude
Figure 2 shows the projections of the trajectories of constant-level pilot balloons
entering into waves ontc the vertical plane. The trajectories 1, 2, 3 relate to _
stable, neutral and unstable stratifications respectively. Trajectory 3 merits
special mention. 'I'he stratification in the lower 100-m layer at the time of ob-
~ servation was unstable and no waves were~observed here. The constant-level pilot
balloon rose monotonically. At an altitude of 1300 m it entered int~ a clearly ex-
pressed attenuating wave, probably arising on the boundary of an inversion.
The characteristic parameters of some waves are given in the ;able.
- The length of the waves in kilometers can be roughly est3tnated using the tormula
~ N~ where u is the wind velocity at the level of the waves in m/sec, that is,
half as great as computPd using the approxia~ate Dorodnitsyn formula [2]. The mod-
ulus of the vertical velocity can be from 0.1 to 0.3 of the harizontal velocity at
the level of the waves or from 0.2 to 0.4 of the wind velocity at the level of the
- landing strip.
Thus, with great wind velocities (15-20 m/sec) the vertical velocity components in
the waves can attain 5-7 m/sec, which is not safe for an aircraft coming in for a -
landing. _
- Another danger of the waves is as follows. An aircraft on the descent path a?ter-
nately enters into the ascending and descending branches of the wave. Upon enter-
ing into the ascending branch the pilot parries its effect using the elevators and =
as a result (after 8-12 sec) the aircraft can be in the descending branch with a
negative pitching angle. Such a situation is especially dangerous directly near
elevations where the descending flows are most intensive and overcoming them may
- be beyond the technical capabilities of the aircraft. .
The descending flows on the lee side of a steep dropoff are well known to the ex- -
perienced pilots of Krasnovodsk airport. A spontaneous loss of altitude, which
here is called an "indraft," appears at a distance of 1-2 km from the edge of the -
st~ep scarp, which agrees well with the lengths of the waves cited in the tdbleo
The "indraft" is observed both against the background of calm flight and when
there i~ bumping of different intensity.
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- According to our observations, a stable stratification, favoring the formation of
waves at the time of northerly and northwesterly intrusions, does not exist for a
long time, 0.5-2 hours, and is replaced by an equally brief period of instability.
'~his circumstance, evidently, imposes definite limitations on the maximum poss3ble
advance time for predicting waves, limiting it to 1-2 hours.
Vertical fluctuations of the wind were computed using data from base pilot balloon
observations made on the windward (point A, Fig. 1) and leeward (point C) sides
_ using the formula
Wj-W-W;.
- ~ahere W is the vertical velocity of the pilot balloon, averaged for the entire as- -
- cent, ~di is the vertical velocity of the pilot balloon in the i-th layer. The
thickness of the i-th layer, with a discreteness of observations of 15 sec, was
30-100 m. _
44i
J6
_ ,rl
;G'f E: J M
-
- !v, -
1 '
- 1 EGJ,v , ~
G ~ ~
~
au~� ~
- f~!. 400M
y1 ~
~v~ ,
~v: ~ -
� JG ZOOn
0
~ t~6 l~6 1,4 11.~P- 4�0,6 Q6 1,f ?,7 3.0JB
A -
_ .-~'rs... � ~ -
Fig. 3. Frequency of recurrence of verti.cal gusts at different altitudes over wind-
ward and leeward sides of obstacles (with winds of northerly directions).
Figure 3 sl~: s the frequency of recurrence of vertical fluctuations of the wind
over the windward ~~rd leeward sides of topographic rises when there are strong
winds of northerly directions. The figure shows that over the leeward side the
vertical fluctuations are ~wo or three times stronger than over the windward side.
We note a considerable distc..~lon of the histogram of the frequency of recurrence
after the passage of an air flow over. topographic rises. On the leeward side there
- is a sharp increase in the fraction of high values of the fluctuations (the histo- _
gram is "blurred"). In addition, the nea~ly normal distribution of frequency of
recurrences on the windward sides becomes bimodal on the leeward side. All these
differences are smoothed with an increase in alt-.itude.
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The appreciable asymmetry of the histogram at an altitude of 200 m over the wind-
ward side is evidently associated with the ordered rising of air along the slope.
- As m:Cght be expected, bumping is much more probable and intense on the leeward
side of the topographic rises. Instrumental measurements af vertical accelerati~ons
_ of the AN-24 aircraft in the case of northerly winds registered accelerations up
to 1 g on the leeward side. At the same time, on the windward side the accelera-
tions did not exceed 0.1 g.
~umping has a sharply defined limit: regardless of intensity, when coming in for a
. landing it ceases 1-0.5 km from the edge of the topographic rise.
_ In all probability, on the leeward side considerable vertical gusts are attribut-
able either to waves or to eddies with a horizontal axis which become detached
from the edge of the steep rise. Judging from the oscillogram of accelerations,
the horizontal dimensions of eddies are about 500 m. Flight on the leeward side is
usually accompanied by considerable fluctuations of air velocity. According to
communications of pilots, there were cases when the amplitude of fluctuations of
air velocity attained up to 200 km/hour. This is possible when the aircraft passes
through eddies in which the linear velocity is close to 30 m/sec and seems entire-
ly real, since at point B(Fig. 1) velocities of 22-25 m/sec were repeatedly ob-
_ served.
It is usua.lly assumed that the reason for turbulence on the leeward side of the
topographic rise is the hills situated at its very edge (Figo 1). Our observations
- for the time being have not confirmed this. The flights made over a part of the
scarp where there were no rises revealed that here bumping of the same intensity is
- encountered.
Conclusions
_ In the region of southern rises near Krasnovodsk airport, when there are strong
- winds of northerly directions, waves create the "indraft" effect a spontan-
eous dropping of the aircraft when coming in for a landing. The waves, experiencing
breakdown, lead to bumping. Orographic bumping on the leeward side has a limited
- localization: when coming in for a landing it ceases 0,5-1 km in front of the edge
of the topographic rise.
Vertical wind fluctuations on the leeward side of the topographic rise can attain
0.4 of the velocity measured on the landing stripo
In order to predict waves and aircraft bumping it is necessary that thermal strat-
ification be taken into accounto The advance time of a forecast cannot exce~d 1-2
- hours due to the great time variability of the phenomenon.
BIBLIOGRAPHY ,
1. Denisov, Yu. M., Sofiyev, Ye, Io, "New Algorithm for Computing the Coordinates
- of a Pilot in Base Observations," TRUDY SARNIGMI (Transactions of the Central
Asian Regional Scientific Research Hydrometeorological Institute), No 38(119),
1976.
- SS
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2. Dorodnitsyn, A. A., "Some Problems in the F1ow of Air Currents Around Irreg-
u.larities of the Earth's Surface," TRUDY GGO (Transactions of the Main Geo-
physical Observatory), No 31, 1940.
3. Musayelyan, Sh,, VOLNY PREPYATSTVIY V ATMOSFERE.(Obstacle Waves in the Atmo-
sphere), Leningrad, Gidrometeoizdat, 1962.
4. Scarer, R. S., "Airflow Over Mountains: II The Flow Over a Ridge," QUART, J.
POY. METEOROL. SOC., Vol 79, 1953. -
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- UDC 551.324.51
_ ON THE PROBLEM OF THE MOVEMENT OF CONTINENTAL ICE
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 80 pp 54-62
jArticle fiy N. A. Bagrov, professor, USSR Hydrometeorological Scientific Research -
Center, manuscript suhmitted 28 May 80] -
jText] Abstract: A study was made of the problem
of movement of continental ice on the as- -
simmption of linearity in the resistance to
movement. In this case the rate of movement
is proportional to the surface slope of the -
glacier. A model of a circular glacier was
formulated applicable to the Antarctic glacier,
~ The results of the computations agree rather
c~ell with availafile data with stipulation of -
one constant the resistance coefficient.
_ At the present time most scientists acknowledge that during the T~rtiary the earth'3
temperature ciecreased almost monotonically. At the end of this period and during
the entire period which followed, the Quaternary, this weak decrease began to be .
accompanied by great variations with an alternation of cold glacial and relativa-
ly warm interglacial epochs. The last glacial epoch, evidently, was one of the
longest and mast severe [5, 11~.
A. I. Voyeykov was one of the first to bring attention to the fact that the snow ~ '
and ice covers are not only a result of cooling, but themselves are responsible
for their own existence because of their cooling effect. This point of view was
further developed in the studies of Brooks [2].
In his studies on the theory of climatp Budyko [4] and other scientists made a de-
tailed examination of the influence of ice on climateo It was gradually clarified -
that ice is the principal factor in the instability of Quaternary climateo Cli- -
matic variations were abaent in the course of the Tertiary and a la.rge part of the
Cenozoic era because a warm climate, especially warm polar regions, do not arise
in the presence of an ice cover. But by no means everything is clear here. One of
the unsolved problems concerning climatic variations is that more or less signif-
icant coolings or warmings are evidently associat~d with some important restructur-
ings of the entire atmospheric circulation, which are accompanied by changes in
the properties of macroturbulence.
. S~ .
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In actuality, for example, cooling is reflected primarily j.n the polar regions,
which causes an increase in the equator-pole gradiento It would seem that in this
case there should be a substantial increase in the influx of advective hea~ from
the equator. But this for some reason or another does not occur, but there is a
further cooling, the formation of continental ice, a decrease of the ocean level,
and accorcli.ngly, a decrease in oceanic advection.
Evidently, with an increase in the equator-pole thermal gradient there is an in-
_ crease in the intensity of zonal circulation and circulation itself becomes more
= stable. The latter can be attributed to tha fact that with a decrease in tempera-
ture and the ocean level there is an incre~se in dryness of the air; this xesults
in a decrease in cyclonic activity and the latter in turn reduces meridional ex-
change.
Finally, the propagation of ice is stopped at some latitu.des (prowided it does not -
- cross a critical latitude). But this position wi11 also not be stable, and the -
less stable the closer the ice approaches the critical latitude. Should there be
a random variation in the direction of a warming whj.ch is quite significant, the _
entire process c7n go in the opposite direction, that is, in the direction of a
_ further warmin~. :Iowever, for such a reverse process there must be some exceed-
. ing of. the tea~perature conditions in comparison with the conditions for advance
of the ice.
In any case, according to the computations made by Budyko [4], whereas a decrease
in the mean temperature by several tenths of a degree makes it possible for thE
- ice to propagate southward by 10 or more degrees of latitude, the reverse temper-
- ature increase to the initial value st~ll does not ensure retreat of the glacier, -
The ice advances easily but retreats with difficult~.
What about the ice itself? The ice has many properties of a solid brittle body rel-
ative to rapidly changing stresses. .Lt is capable of retaining its form for a long
time almost unchanged and is capablr:.of withstanding some dilatation. The elastic-
_ ity of ice is very sma11. Under the influence of relatively small long-persisting
_ and slowly changing stresses the '.ce is capable of being slowly deformed, flow and
change its form. In this property it resembles a very viscous fluid. However, the
tensor of stresses in the ice mass is related very complexly to the tensor of de-
; formation and this relationship is very highly dependent on temperature and the
fiype of ice and the degree of nonlinearity increases with an increase in the -
stresses. More or less rigorous solutions of the equations for the flow of ice
have been derived only for extremely schematic conditions and they are ill-suited
for practical computations of the regime of glaciers. Fo r this reason it makes -
sense to use `:ia simplest assumptions for studying the movement of glaciers. TEze
simplest ass~?~npt~~n here will b~ the hypoth~esis of linearity of the correlation
between resistance u.id the rate of movement. Using the equation for movement of
a fluid in which all the teru~s on the left-hand side are dropped, we write equa- -
tions for the movement of ic^ rectangular coordinates:
- p ~x k u+X' ~1)
_ P dy - k 1J ~f,
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= ~-_p a:_k`r~'+~, -
~ ~y + dz = (1) _
Here k is some coeff icient having the dimenaionality of velocity; X, Y, Z are the
projections of mass forces (gravitational forces) onto the coordinate axes, ~O is
- density, p is pressure.
` The use of these equations for ice seems extremely unrealistic. There is a definite
difficulty in their interpretation. But their application to definite situations
is easily interpreted and gives acceptable results.
Now we will examine a v~ry broad glacier on a bed having a common slope with the _
nonunifoxmities (see Figo 1). By aelecting the x-axia along the slope and the y- _
~ axis in an upward sloping direction, we obtain _
u_ P
g az k 5tn
p
g ay - k sin ~2~
Obviously, the rate of movement along the y-axis can be neglected if the nonuni-
formities of the bed are small and the glacier surface is slightly slanting along
the x-axis.
- y
. Qdx
_ , ,
~
trH
_ 0 -?uH*d.dr(uN)dx _
- dx
x
Fig. 1. Diagram of movement of glacier along sloping surface. ~ is the slope of the
' glacier bed to the horizon, H is glacier thickness, u is the rate of movement along
the x-axis, Q(x,t) is the rate of growth of ice on the surface due to the precipita-
tion - evaporation difference.
By int~grating tl:e continuity equation along y we obtain: -
ax (uH) vH = 0, ( 3)
where u is the mean rate of movement of the ~lacier in the cross section, vH is
the rate of movement at the glacier surface~.
For the rate vg we have the obvious exp~cession -
~'H - a + u ax - Q (x, t)� (4)
Here Q(x,t) is the layer of growth of the glacier from above as a result of the
falling of precipitation. 7'he second term for vg can be dropped as a result of -
its smallness.
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Integrating the second equation (2), we find that -
k p-kcossH. _
~g _
Introducing this result into the first equation, we wi11 have
u-- k~ dz - sln e1. ~5~ `
It therefore can be seen that the rate u is not de endent on
p y, that is, it.remains
- constant for the entire cross section. 1'his means that the glacier seemingly
slides along its bed, being drawn out and becoming slightly deformed. .
Collecting the results of (4) and (5), and introducing them into (3), we obtain
finally an equation of motion in the following torm:
dH d(kf~ aN 1 d kH sin fi~ Q(x, t)�
dr dx)-~ (6)
This equation _~_.L~ be derived by another, more graphic method. According to the fig-~
ure shown above, a mass of ice flows into a defined elementary volume with a mean -
rate of movement u, equal to uH, whereas the outflow is uH a x(uH)dx. The
difference, equal to - a~a X(uH)dx, arises due to the increase of the surface
d Hfa t dx and falling preciFitation Qdx. Comparing tb.ese values, we obtain the
continuity E:quation in the following form: ~ -
adt + az (uH) = Q (x, t)� _
The principal hypothesis now is that there is assumed to be a proportionality be-
tween the rate of movement u and tne slope of the glacier surface, that is, it is
assumed that ~
u = -k( dH -sine~,
~ _
_ where ~ is the slope of the x-a.:~.s to the horizon. Introducing this expression into
~ the continuity equation, we again obtain the equation (6). _
We will examine the simplest case of a plane horizontal bed. AssLm?e that the ac-
c~ulation of precipitation Q= const. Then equation (5) in the case of steady
movement a~sumes the form ~
Q
a ~NdX~+k-~�
Integratin~ ::~is expression once, we obtain
N dx -I- k x= const.
,.~aa~~ -
If with x= 0 d H/ d x= 0, t' cons~ant will be equal to zero.
. Integrating the derived equation once aqain, we obtain
yz.~ k x'=Ho� (7)
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~ The surface of such a glacier is an ellipse whose minor axis is b= Hp and whose
major axis is k/Q times greater.
The oint x = H ~
p p Q/k is the runoff point. This is a singularity: here H(x) = 0.
In its neighbor'hood our ass~ption of smallness of the derivative d H/ ~ x is sub-
stantially impaired. However, this condition comes close to the conditions when
the glacier descends from the land to the sea and its end is constantly broken off
and seemingly becomes vertical. It is easy to confirm that at the end of the
_ glacier 9.ts "discharge" of ice is equal precisely to the entire quantity of ac-
cumulati~~n of precipitation on the glacier surface.
In the case of steady movement and a sloping bed ~quati~n (6) allows one integral
- in the form
H aX -@ H~ k J~ k~ dx - const.
We will examine the simples t case Q= const and write the preceding equation in the
form
~
- Yx=C-~-aH-Nh". ~8~
Here Y= Q/k, and H' is used in denoting the derivative dH/dx. -
- We will differentiate this equation for H. We wi11 have
7~ =E-N'-HdH~ _
dH ' _
This equation allows the separation of variables (if H' is assumed to be some new
variable). Obviously, we will have
- dH H' dH' H' dH'
~ H Hiz t H' -~'T ~H~ _ 1 ~ (4 .
The integral of this equation in the case 4'y'- E 2> 0 can be written in the form
InH= 2 tn (Hi2-eH'-}-7)~
, ~9~ _
� + 4` ~l arctg 2~4
�
E~ const.
- ~ 1 Y -
In the case 4Y- E2=
=Bffz, -f~-f~)=B(-f~-f2, f~)� (4>
These relationships show that the bispectrum need only be determined for fl and
f2 values satisfying the condition 0~ fl C F2 ~ oo .
i]
FJe will examine the Kramer represent~ation for a stationary random process
~
x (t) - ~ e~f~ dz ( f
where dz(f) is an uncorrelated rand~..n~easure.
Then it is possible to write the following expressions: < dz(f)> =0,
_ =s(f,)b (f~+t2)df~,
- =B (i~, f2) a (f~+f~+f3) ~f~df2,
where b(f) is the delta fuizction.
Thus, if the spectrum represents the contribution to the mean square, that is, the
dispersion of the process due to the two Fourier components, whos~ frequencies
together equal zero, the bispectrum gives the contribution to the mean cube, that
is, the asyicmetry due to the product of the components whose frequencies together .
are equa.l to zero .
As is well known, for a Gaussian process the third moment (asymmetry) is equal to
zero. In this case
~xg (t) > - R (0,0) =0
(5)
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and it follows from (1) that
~ ~f~, fs)'=0.
7'hus, the bispectrum shows how the cieviations of the process from a Gaussian prv-~
- cess are "expanded" ia frequency, that is, arise due to some frequencies which
= exist under synchronism conditions,
' ti'~'~i'~'fa~~� ( 6 )
~ On the other hand, if it is assumed that some investigated process x(t) is Gauss-
ian at the initial moment or that it was formed due to the passage of some Gauss-
ian process through a nonlinear system, then the bispectrum becomes a character-
istic of the nonlinearity of this system. [Between the linearity of the wave field
_ and the Gaussian distribution there is some dependence established by a theorem
(Hasselman [1]) which asserts that in a linear approximation ~he random homogen-
= eous field of waves with dispersion asymptotically tends to a Gaussian stare,
- even if at the initial moment it was not Gaussian.] In this case it is possible
_ to speak of the interaction between some frequencies.
Now we will examine two examples illustrating the zeed to use bispectral character-
istics.
Example 1. We will assume that the record (for example, level) x(t) is known. This
was formed as a result of transformation of some input real stationary function
x0(t) in the case of passage of an inertialess system with quadratic nonlinearity:
- x (t) = axo (t) E bxo (t), (7)
where a, b, E are real constants.
In a real case this can correspond to the process of transformation of a tidal
wave as a result of friction against the bottom.
We will raise the question as to whether it 3s possible to determine the values of
the coefficients a and ~ b or the relative contribution of a nonlinear element rel-
ative to the 13near element on the basis of the characteristics of the input and
~ output of the system (a simple nonlinear "black box" variant). This problem is
_ broken down into two variants: a) the input function x~ is not known; b) it is
known. ~ .
Here we will be interested in the first variant, when the function at the input is
not known (as a rule, we do not know the characteristics of the t:Ldal wave at the
� input, that is, in deep water). However, if it is assumed that:
the input function xp(t) is a real xigorously stationary Gaussian random pro-
_ cess with a zero mean (a Gaussian state is essential)o ~
the nonlinearity ef the system is small ~ 1), then in this case it is pos-
sible to obtain some approxima*e solution of the problem, to be more precise, de-
termine the relative contribution of the nonlinear texm. Taking into account that
the third moment of the Gaussian process is equal to zero, from (7) we obtain
- 69
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< x~ (t) > _ ~Z < xo ~t) > -I- e= b2 < zo (tl
t$)
=~6.
, (9)
But
x (t) = axo -4- O that is, xn ~t) = a x (t) -f- ~
a, < x3 (t) > O (E) =1Q ~ a~ ~ x~ (t) > < x+ (t) > < x'= (l) > ~11~
Thus, in the case of an input Gaussian process and weak nonlinearity it is pos-
sible, using one output function, to determine (approxima.tely) the relative con-
tribution of the nonlinear element (but aot the value of the coefficients a and '
= E b, because a fu11 solution can be obtained with known input and output process-
es). -
~ Example 2. We wi11 examine the system ~
X~t~-, R~`~ XO ~t-~~ dS-~-E f(, ~~t~~ T,~ X~ (t-Y~ ~'C
(12)
X xo - ~i) tl", dts. ;
We will assume, first, that xp(t) is a real rigorously stationary random Gaussian
- process with a zero mean, and second, that the nonlinearity of the system is weak
~ 1). We know only the process at the system output x(t). As in the preceding
example, it 3s possible to solve two problems: _
a) determination of the ratio of the contributions to the spectrum of the process
- X(t) from the r~onlinear and linear parts of the system for each frequency f on
the basis c` the statistics oF the output function , that is -
a~f)~~); ~
b) determination of the Z(f, f') value through the statistics of the function at
the output x(t)}> (known; .~id the coefficients a(f) (unknown). Here Z(f, f') -
is the double Fourier transform of the transfer function K(~G l, 2 2); the Z(f, f')
value is usually called the interaction or biadmittance coefficient. [The term
"biadmittance," having the sense of a t~:~-frequency transfer �unction, was taken
from a study by Cartwright ~2].J
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For brevity we will cite the final expressions. The solution of the second prob-
- lem is as follows:
- EZ ( f, f~) _~~f) a~f') 2 S Bf) x)U~)' t~3)
where a(f) is the Fourier transform of a(t), SX(f) is the spectrum of the process -
x(t), B.P, f') is the bispectrum of the process x(t).
The first problem has the following solution:
I ~ ( J'. .f' -f) I' d
Q~f~ = J Sx lf) Sx lf') Sx lf' -f) f ~14~ ~ _
Thus, with the assumptions made above it is possible to determine (approximately) -
- the relative contribution of nonlinear interactions to the spectrum of level fluc- -
_ tuations on the basis of only one characteristic of the output function.
Beloc~r, when we sgeak of interaction between individual frequencies, it will always .
~ be understood that the investigated system conforms to the assumptions made above:
a Gaussian state of the input process and the weak nonlinearity of the system. -
Otherwise the reasonings concerning the interaction of frequencies can lose sense.
For example, when a non-Gaussian process is fed to the input of the linear sys-
~ tem an investigation of the bispectral characteristics of the output process does
- nflt give any information concerning the nonlinearity of the system but detennines -
the internal structure of the.output process, distorted by the linear system.~
~
_ It fallows from what has been stated above that the characteristics of nonlinear- -
ity of geophysical processes can be obtained on the basis of a bispectral analysis
of the corresponding time series. In an evaluation of the interaction between the
frequencies use is made of the real part of the bispeetrum Re[B(fl, f2], which is
_ a complex function, and the bicoherence function:
~ ~B (fi~ f~) .
~ 7~ ~ft, f~~ = SX (fi1 Sx (Isl Sx ~f~ +.fz) ' (15)
At the present time in solving practical problems use is made of the following
method for computing the bispectrum [2, 4]. The initial time series x(t) is divid-
ed intc n segments, each with the length N. The Fourier transfarm is computed for
each segment.
N ,
- ~ X~ ~f) _ ~ ~ e- `ft x~ (t') W� (t');
~ (2 aN) r,_1
_ ~ j-1, 2,. . rt; t'=1, 2,. . N, (16) .
~here Wn(t') is the weighting function~
Then the bispectrum is computed using the ~ormula
rs
B ~~i~ fz~ = n ~ X~ ~fi) X~ ~f9~ X~~ft -~-f2~� ~1~~ -
i=~
The bicoherence is computed using the formula -
71 -
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n~ I B(f~. f:l
~
7~ i, f~~ = n n n �
- ~ IX; (f?)I= ~Ix~(f~)I' ~ i�ri(I~ +f.)1=
1=~ (18)
It is us.ually assumed that if the bicoherence is equal to 1, a nonlinear interac-
~ tion exists between the frequencies, whereas if the Tiicoherence is equal to zero,
- interaction is absent j8]. Usual?'~ the bispectrum and bicoherence are represented
in th~ plane of the frequencies fl and f2 in the form of a triangle whose vertices
j have the coordinates -
- (0, 0) (fN, 0) and (1/2 fN, 1/2 fN),
where fN = 1/2Lat is the Nyqu3st frequency, Q t is the discreteness of observa-
tions. ,
_ Now, adhering to Cartwright [ 2], we will deteraiine the numerical algorithm for cour-
puting the interaction coefficient Z(fZ, f2). Assume that Xrl, Xr2, Xr3 are evalu-
= ations of the level spectrum for three frequencies. The sea 1eve1 spectrum Xr is
computed using the f~rmula 1 N _
_ X, = N ~ (1 cos 'N ~ exp (2 ir j/N) x ( j ~ t - to),
/ _ - N~ (19)
rr = l , `l, . . , ~l N 1 l.
~ i
In most cases in an analysis of the level of tidal seas N ~ t= 29.5 days (in order _
thatlaf be less than 1 cycle/month). Here rl corresponds to the arbitrary nontidal
f requency 2~1 r~ ~ f, r2 corresponds to the frequency of the main s emidiurnal tidal
wave 2 cycles%lunar day, r3 = r2 - rl f.~~r�convenience will be assumed positive.
. Xr3 contains the element hg, which reflects the interaction between Xrl and Xr , ~
and the residual element h'3, which is unrelated to hg. Similarly, Xr~ = h+ h~, '
where h reflects the interaction between Xr2 and Xr3. Thus; it can be written that
h~ Z ~-f>> f=) ~'r, Xr:'
- ~ (20) I
h, ~ Z(- f~, fs) -~r3 Xr.. ~ ~
where fk are frequencies proporti ~,nal to rk,. ~
We will examine the mean product for the set
' (,Y'r1 ,l'~_ ) .Yr~ _ (J?'r~ Xr_) X'~~ = X"r~ A'r, -
Assuming that the product of the incoherent components is equal to zero,
_ Xr.. Xr., ~Z ~-fi, f:) h, h," Z~-.f~~ ~a) h3 h3~].
_ we obtain
7~-fi,fs)^ 6 in formula (6) in place of 8 3 it is necessary to substitute the
value 6. This dependence makes it possible to compute the value of the general and ~
' optimum ~C2 tests with a relative error not exceeding 10-15%.
In actual hydrological computations it is customary to use only the general x2 test
and in order to avoid minimizing the statistical test in formula (1) it is neces- _
N
- sary to compute the t statistics: k . x�
` l'�t - Pr (Hi, 8.,, A.3 )
t- n~
P; (A~+ A?~ A3) ' (7)
where ei and e2 are evuluations of the parameters Bl = X~ a~nd e2 = C~ by the
moments method. I� the hypothesis Hp: 83 = e~ is true, the t statistics have a
central x2 distril~utio~ )C~2~with degrees of freedom, k- 35~,~ k- 1. If the _
- alternative H~:( el, e2, ~ 3) is correct, the t statistics have a noncentral dis-
tribution n S , where
~
~ 89
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k(p~ ce;. e;, e3~ -p~ ce;, e.;, e,~i~
.
P< c (z)- r'' ( )
- 1~ yl-�'~"`-"1~ ? _
The relative error in deternLining the value using formu.la (17) does not exceed
15%.
_ Similarly, the value of the approximate optimum x 2 test can be evaluated using the .
formula: m .
~ ~ O,Ua n; H:~ (H,; - P3~-,
(~1 - P ~ c : ix) - ' _ (18>
~ . ~ 1m )~1 . p=(m-~)
It is possib:i.e ;:o determiiie the critical value E~ in formula (13) with which the
p robability of refuting the Hp hypothesis when it is not true is assumed to
be inadec~uately large in comparison with the probability oC of refuting the Hp hy- _
po~hesis when it is true. For example, with E< 1 in formula (13) �(10%) < 40%, -
that is, the probability of adopting an untrue HD hypo~hesis is 60% and is only
- 1 1/2 times less than zhe probability of adopting a true H~ hypothesis.
This makes it possible to find the mean probable deviations of the true CS/Cv val- -
ue from the hypothetical value for each specific drainage basin, still not checked -
using the Pearson test. For example, when using the approximate optimum ~L2 test
with a stipulated critical ~ p the: ,m ar, maximum deviation Q 2=( eg i-~~) z
- for the i-th drainage basin, whicY; ~ ra~ist be expected in the case of adoption of
the hypo thes is Hp : Cs/ Cv = a 3, i:; determined as
i~~ ^ 1~1m ~rl=p'Im-11
m O,U4 n; H~;~ (19)
Similarly, the maximum deviatior~s Q 2, not checked bv the optimum x 2 test, are de- ~
termined: ,
. . . . 1~1m 1 I -�.~=~.~n-I.i.:: . .
rn lU"~ni~~k;-3~l?,,"0-}-0,2~Ayi(G-Pz~,)f~. (20) .
Formula (20) determines considerably greater values of the maximum deviations of
the hypothe~~cal Cs/Cv value from the true values and these deviations correspond
to the real passib~l.ities of statistical checking of the correctness of the choice
of the type of gamma distribution, taking into account, in contrast to formula
(19), the influence of the errors in determining the norm and variation coefficient. _
As an illustration we will examine the problem of checking the hypothesis of a
common CS/Cv value for the probability curves for minimum 30-day discharges of
the rivers of Transbaykalia, adopted on ~.:~e basis of data f~r 89 drainage basins. -
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~
- a~ a~
~ ~
_ .a ~ _
c~
- H 4-+
O
~J
_ ~
~
� c~
D
v
~ -
a-i '
~ Q ~ a v ~
tiy
_ I ~ ~
� ~ - - ~ ~
II ~ ~
p c
~ ~ ~ H ~~c~
- OpONMQ~ ~
v~n~.oo N
~ t ~ �rl
~Q, u.~
~
ri Ci tid.. ~
:+9 rt C ~t' a 3 v
a y- C`I ~r u'~ O d~ O ~n
~ ~~ro~~~
~ ~ ~
~v~ II
~ Gl ~ N V GV..~~Mc~D
- v ~ O
~ H ~ ~ ~~ci~~ ~~H ~ d. -
~N 1 ~ N rl
iH x ~ ~c u ~ ^ ~ N ~ ~ ~
~ v cU ~~~~oo -
r-1 .u - o o -r r. ~ 2 -
~ ~ c+ c: ~~7 i~ N ~
y.
U Ol v ~2 .-+~f?O~n~
~ 'k ~ ~ tl c~ -
a~ O " v -
- N 1.+ m F `n c+ 1~-r
- ~ ~ 1 ~ ~ ,s
~ ~ oo ....~r~dN~ �Q ~ -
~ s~ a
_ ~ ~ ao -
r, a~ d .d ~
_ 2~ ~ 4., ~
- v O
~ ~
~ v
c~i ~ j'
-
N
~ ~
LJ
fb _
~
_ " e
93
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0
M r-I
~O M ~t 01 O~ u'1
. . . . .
N 11 �
r-I (L ~t d' M ~Y' .T u'1
~
- H ~r1
~ ~
M U ~7 U1 u1 O~
II . . . � . .
~ r1 c~'1 N ~ M M
O
~ U1 ~t rl ~G ~D 00
~ ~n r-I rl e~-1 ~ r-I r-I
~ ~
~ ~
c~d ~-~I
,a O~ ' N U1 00 r-I n
- tA II � � � � � �
C1 ~ O rl r-I rl M c'~1
fd
~
H �
~
_ 'r'~ ~ V ~ t~ O~ O tr1 N ~O
tA II � . . � �
_ ~ n O r-I ri rl N N
~
~
U
' h-I ~ r1 ~t u'1 ~O O c~l '
A il � � � � � �
- O O O O rl r-I
~ ~
N
D
_ ~
- cD,O
A
= d~
.
- ~ V ~t c*1 N d' ri (`1
_ ~ `
~rl U
~ v
N
,L, M u1 M QC ~D ~0
~ ~ rl O~ ~O ~ c+1 M
� . . . �
O U rl O O O O O
~?~1
~ p rI N M N N ~-I
~
~d
D
_ ~ y ~
c~d ~ ~ t~A ~.~i ~ .C N
f
~ �r~l ~rI 00 tA ,'.~li
cA 6 c~ ~ A Dd~
~ cd
~r~i �r~l u ~d ~
~ r~i b0 O F+' 01
O d U ~ H
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I'or each of the 89 points we com~uted the mean maximum uncontrollable deviations
( e3 - e 3) of the true values e 3i of CS/Cv from the regional hy~othetical
valu~ e~, corresponding to the approximate optimum Pearson test (Q ) and the op-
_ timum Pear~on test w it h a brea
k down o f eac
h o f t he samp l e s i n 1 0 e q u i p r o b a
b l e i t
i-
tervals ) for ED = 1. The Q and ~ values were computed for different values
of the mean correlation coefficient/~ between the minimum 30-day discharges of
' the rivers of Transbaykalia. The case P= 1 corresponds to checking of the Hp hy-
pothesis: CS/CV = e 3 for ea~h individual series~
~ Table 3, listing six drainage basins, gives the durations of the series of observ-
� ations n, evaluations of the variation coefficient Cv and the ratio (CS/~~)N and
for/0 = 0; 0.5; 1 gives the maximum statistical uncontrollable deviations Q and
~ from the adopted hypothesis for the approximate and ;~recise optimum Pearson
tests .
~ The data cited in Table 3 show that if one does not take into account the influ-
- ence of the errors in determining Xp and Cv, the approximate optimum Pearson
test is not capable of checking even insignificant deviations of the true dis-
- tribution from the hypothetical distribution, even when using an individual ser-
ies (P = 1). The correlation of the series considerably lowers the value of the
statistical~tests, and accordingly, the effectiveness of regional generaliza-
tions.
BIBLIOGRAPHY
l. Blokhinov, Ye, G., RASPREDELENIYE VEROYATNOSTEY ~'ELICHIN RECHNOGO STOKA (Dis-
tribution of the Probabilities of River Runoff Values), Moscow, Nauka, 1974.
_ 2. Zhuk, V. A., Yevstigneyev, V. M., Chutkina, L. P., "Characteristics of Use of
Ma.tching Tests in Checking Hypotheses Cpncerning the Laws of Distribution of
Characteristic Runoff Values," PROBLEMY GIDROLOGII (Problems in Hydrology),
Moscow, Nauka, 1978.
_ 3. Kendall, M. Dzh., St'yuart, A., STATISTICHESKIYE VYVODY I SVYAZI (Statistical
~ Conclusions and Correlations), Moscow, Nauka, 1973.
4. Kritskiy, S. N., Menkel', M. R., "On Evuluating the Probabilities of Frequency
of Recurrence of Rarely Observed Hydrological Phenomena," PROBLEMY REGULIRO-
~ VAPdIYA RECHNOGO STOKA (Problems in Regulation of River Runoff), No 6, Moscow,
Izd-vo AN SSSR, 1956. ,
5. Rozhdestvenskiy, A. V., OTSENKA TOCHNOSTI KRIVYKH RASPREDELENIYA GIDROLOGI-
CHESKIKH KHARAKTERISTIK (Evalu~tion of the Accuracy of the Distributi:on Curves
of Hydrological Character.istics), Leningrad, Gidrometeoizdat, 1977.
6. Leman, E., PROVERKA STATISTICI~SKIKH GIPOTEZ (Checking of Statistical Hypothes-
es) , i~loscow, Nauka, 1964.
7. UKAZANIYA PO OPREDELENIYU RASCHETNYKH GIDROLOGICHESKIKH KHARAKTERISTIK. SN 435-
- 72 (Instructions on Determining Computed Hydrological Characteristics. SN [Con-
struction Norms]435-72), Leningrad, Gidrometeoizdat, 1972.
95
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- 8. Khristoforov, A. V., "Checki;~g 5tatistical Hypotheses in Computations of the
Maximum Water Discharges With a Lc:a Guaranteed Probability," METEOROLOGIYA
I GIDROLOGIYA (Meteoroiugy and Hydrolopsy), No 9, 1977. ,
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UDC 627.51(470.23)
HYDROLOGICAL BASIS OF A PLAN FOR PROTECTING LENINGRAD FROM FLOODS
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 80 pp 87-91
[Article by V. A. Znamenskiy, State Hydrological Institute, manuscript submitted
2 Jun 80]
. [Text] Abstract: The article gives formulation
of the problem, method and range of hy-
- drological, hydrochemical and hydrobiolog-
- ical investigations of the water system
� Lake Ladoga-Neva River-Neva Inlet and also
- a method for making predictions of changes
of the regime of this system as a basis
~ for protecting Leningrad from floods.
In order to protect Leningrad against floods, prior to 1990 plans call for creat-
- ing a complex of structures which at the time of floods will block the path df
waters from the Gulf of Finland into the Neva Inlet and the Neva delta. In the
protective complex situated along the line Lom~onosov-Kotlin Island-Gorskaya vil-
lage and having an extent of 24.5 km there will be stone-earth dikes, two struc-
- tures through which ships can pass and six structures through which water can pass.
The construction of protective structures will fully solve the problem of protec-
_ tion of the city against floods and is clear evidence of the concern of the Party
and government for the city of Lenin and its four-million people.
At the same t~ime, ~luring 1980-1985 plans call for the carrying out of complex
natural and model hydro~ogical an~ ecological investigations of the water system
including Lake Ladoga, the Neva River and Neva Inlet. These investigations become
the scientific basis of ineasures for protecting Leningrad against floods and for
preservation and sanitizing Lake Ladoga, the Neva Ri~rer and the Neva Inlet.
The formulated scientific problem is extremely complex due to the need for taking
into account the complex regime of individual elements of the hydrographic system
and the influence of economic activity on it.
The upper link in this water system is Lake Ladoga, the largest lake in Europe,
whose regime, despite extensive investigations carried out in the past, has still .
heen inadequately studied.
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a
lluring recent year.s the water quality in the lake has heen influenced to a consid-
- erable degree by economic activity, in particular, the development of industry and
the implementation of agricultural and meliorative measures on its sh~res and in
a the territory of the basin.
All the mentioned fac tors act on the ecosystem of this large fresh-water body.
~he waters of Lake Ladoga, entering the Ne'va River and then into the Gulf of Fin-
land, for a distance of 32 km in the lower course flow through the territory of
= Leningrad. The h.ydro logical regime of the Neva, as one of the segments of the
" water system, differs ,,harply from the regime of a lake. In its upper part the
- water quality, level regime, current velocity, ice-therc~al conditions, etc. are .
determined by the re gime of Lake Ladoga. The lower part is under the influence
uf the Gulf of Finland. In addition, the conditions for tha forming of water _
quality and the ecosy stem of the Neva River is influenczd to a great degree by
the cities and indus t ry on its shores, including the econon~ic activity of Lenin- ~
grad. ^the plan for the development of Leningrad provides for a comp].ex system of -
measures for the sani tizing of water bodies within the city. It includes the _
_ clearing of riv4rs and channels, prevention of the dumping of waste water and
ground, and m~.~~L else. This has already brought substzntial results and during _
- the last decade the q uality of the water in the Neva River and in the arms of its
delta has substantially improved. ~ ~
The delta part of the Neva River is in the backwater of the Neva Inlet c~ the Gulf i
_ of rinland, whose influence is reflected not unly during the period of floods, but -
also in the day-to-day regime. Each da;~ t;he N~eva Inlet experiences the influence
. of the Gulf of Finland, whose waters periodically enter it under the influence =
- of atmospheric proces ses during the passage of cyclones over the Baltic Sea. As a -
_ result of penetration of long waves and the wind effect surges develop in the Neva _
Inlet which together with the runoff of the Neva are the basic distinguishing char-
acteristics of its hy drological reglme. These regime characteristics are manifested
most clearly during periods of floc~r;s. The combination of these factors determines
the physical backgro und against which the elements of the ecosystem of the natural
regime of this water body are f~..med. -
In addition, the reg ime of the Neva Inlet is presently influenced to a great de-
- gree (and will be so influenced in the future) by the economic activity: the ex-
isting and proposed discharge of waste water of the Leningrad sewer system, the .
construction of structures for the protection of the city froru floods and the pro-
= posed shifting of part of the rut~off from the Lake Ladoga basin. An evaluation of
its influence on the ecosystem of ~aater bodies is the objective of the planned in-
vestigatio~~~ .
Thus, the object or the investigation ts a complex water system with different
~haracteristics of the water, chemical and biolo gical regimes and diverse types -
of the effect exerted on it elements of man's economic activity. In order to
evaluate its changes it is necessary to take into acc~unt the conditions of in-
teraction among individual parts of the system and the interrelationship of the
elements of the hy dro logical, hydrochemi:al and hydrobiological regimes. In de-
veloping the programs for complex investigations it is evidently necessary to
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1 -
take into account the following principal problems:
_ study of the principal lau~s of vital functioning of the ecos.ystem Lake Ladoga-
Neva River-Neva Inlet and es~ablishing interrelationsliips among its abiotic and
biotic pa`rts; _
development of the methodological principles for the mathematical modeling of
_ multicomponent ecological systems;
~ development of a mathematical model of the ecosystem Lake Ladoga-Neva River-
Neva Inlet; ~
preparation of a forecast of the ecosystem Lake Ladoga-Neva River-Neva Inlet -
= under the influence of economic a~tivity; :
- evaluation flf the possibility of automated monitoring and optimum control of -
the ecosystem of Neva Inlet with respect to the conditions for the formation of ~
_ water quality. `
The research methods must provide for the carryin~ out of an in-situ study of water -
bodies for all elements of the regime and the carrying out of physical, physico-
biological and mathematical modeling. `
_ The program developed by the State Hydrological Insti_tute includes: _
~ 1. A complex in-situ.study of the conditions of the present-day status of hydro-
= logical, hydrochemical and hydrobiological regimes of the water system for obtain-
ing initial data for predicting changes in the regime under the influence of econ-
omic activity. The in situ investigations must give information on the present sta-
tus of the ~cosys tem of water boclies and the river and serve as a basis for a com-
parative evaluation of different elements of the regim~ in individual parts of the
system. On their basis it will be possible to develop the methodological aspects
of prediction of the change in the regime under the influence of plannad economic -
~ measures. These investigations will also be used in developing mathema.tical models
_ of individual regime e?ements for different parts of the water system and also in
_ a study of the regime in places where economic measures are being directly imple-
mented (protective structures and waste water outlets).
In situ investigations must be carried out for the entire complex of the inv~st- ~
' igated regime simultaneously at prestipulated times and under a unified program.
_ Only in this case will there be assurance of the necessary comparability of re-
.~1 sults for all parts of the water system. The program of :i:t situ investigations
_ takes into account the need for materials necessary for developing models and
- forecasting.
2. De~velopment of a method of physicochemical principles for prediction, specific-
ally prediction of changes in the hydrological, hydrodynamic and hydro:chemical
regimes of the water system. On the basis of in situ investigations there should
- ue refinement of the principal patterns of behavior of the hydrolog~.cal, hydrody- '
- namic and hydrochemical regimes of the water system, including determination of
the balanca of chemical substances for all parts of the water system.
The basis for prediction should be a mathematical model reproducing the hydrolog-
_ ical, hydrodynamic and hydrochemical regimes of both individual parts and th~ -
entire water sys~em. Such a model will take into account r_he processes of trans-
port, dilution and chemical transformatior. of matter. It is checked using materials
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from field invesL�igations and will serve as the mathematical basis for predicting
the quality of water with further allowance for changes in the vol.umes of river
runoff, systems of currents under the influence of protective structures and vol- -
~ umes of centralized discharge of waste water.
The results of the hydrological and hydrochemical forecasts will serve as a basis
for evaluating the influence of economic measures on the ecosystem and determina-
tion of zones and sectors dangerous with respect to eutrophication and also for
obtaining recommendations on the sanitizing of water bodies.
3. The development of forecasts of change in the ecosystem of water bodies under
the influence of planned economic measures, which is the main objective of the
formulated pr.oblem. Its implementation is based on a preceding field study of the _
energy characteristics and functioning of the ecosystems of coater bodies in their
present state in combination with ~tudy of abiotic factors and the hydrochemical _
- regime, and also on the results of proposed investigations using physical-hydro-
biological models having the ~urpose of evaluating the influence of discharge of
waste water on the ecosystem of Neva Inlet. For this purpose on the basis of pur-
ification struct~-res it is necessary to create apparatus reproducing the physical
conditions of ~a:,*^r exchange in combination with the entry of waste water and the
influenc:e ~f ~tie principal climatic fact~rs, which will make it possible to madel
the biological processes at the sites of discharge of urbzn waste waters in the Neva
Inlet.
Probably one of the important aspects of the investigation will be a study of the
- interaction of the water and bottom ecos~i5tems in Neva Inlet and evaluation of
' its possible change under the influence of economic activity. As a result, inform- -
. ation should be obtained on exchange processes in water bodies and the stability
of their ecosystems, modeled functions and parameters will be selected and expres-
s.ions will be derived wl~ich will formalize the cause-and-effect relationships in
the ecosystem for their further appl:.:ation in the form of a mathematical model. -
- On the basis of such a model it wiil '~e possible to develop a quantitative fore-
- cast taking into account the possible changes in the ecosystem as a result of the
planned economic measures and pr~r~sals on the sanitizing of the water system.
4. Tlie use of results of field and laboratory investigations and prediction of -
changes in the hydrological, hydrodynamic, hydrochemical and hydrobiological re-
gimes for the scientific validation of ineasures for preserving and sanitizing of
the Lake Ladoga-Neva River-Neva Inlet water systam.
- For this purpose, in addition to the studies pravided for in points 1-3, plans
call for the :+rrying out of investigations using a large-scale hydraulic model
of Neva Inle~, in ~ahich studies should be ma.de of the processes of water exchange
between Neva Inlet ,_nd the Gulf of Finl~nd under existing conditions and after the
- construction of protective structures and an evaluation should be made of the -
- chan.ge in the structure of flo~��s in a confined water body. Particular attention
- must be given to an investiga~lon of the mechanism of surge phenomena and their in-
~ fluence on the stiucture of currents, taking into account the influence of protec-
tive structures. The siting and size of water f1~~w-through structures must ensure -
the creation of an optimum Flow system in rleva Inlet for dilution and transport
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of waste water from existing and planned outlets of purification structures serv-
ing the city and its subu~bs. On the basis of an investigation of water exchange
processes and the processes of dilution of waste water a study will be made of the
possibility of control of the regime of flows and the processes of dilution of
waste *-�~iter, measures will be developed for the organization of optimum structure
of flows in Neva Inlet, stagnant and eddy zones will be detected and requirements -
wi11 be established on the norms for discharged waste water.
All the mentioned investigations must take into account the conditions related to
the ;r~sible extraction of part of the runoff from the basin of Lake Ladoga.
_ As a re5ult of comparison of information on the existing and predicted hydrochem- _
ical and k?ydrobiological regimes with materials from investigations on a hydraulic
model it should be possible to develop a scientific basis of ineasures for the san-
itizing of Neva Inlet, including the problem of contending with eutrophication,
contamination of bottom sediments and secondary contamination of the water mass.
- The system of ineasures for the sanitizing of Neva Inlet developed on this basis
should include:
- recommendations on the degree of purification and prepurification.of waste _
- waters;
recommendations on the siting of planned outlets of purification structures;
proposals on the creation of optimum regimes in Neva Inlet in different sea-
sons of the year;
- recommendations on control of the system of currents (by construction of per-
manent and temporary flow-controlling structures in delta distributaries and in
the ocean and also by maneuvering the gates of water flow openings in protective
- stri.~.ctures);
proposals for creating an automated system for control and monitoring of the
regime of Neva Inlet. `
The results of investigations carried out in Lake Ladoga and Neva River wi11 be -
used in developing a system of ineasures for sanitizing these water bodies.
For this purpose a scientific basis must be obtained for measures in the basin,
= on the shores and over the area of Lake Ladoga, directed, in particular, to con- _
tending with eutrophication and contamination of both individual regions and the .
entire lake. Similar recommendations on the sanitizing of the Neva River must
also be prepared for measures on its shores and in its drainage basin. _
As important results of the investigations recommendations must be developed on
the creation of an automated system for monitoring the quality of water and con-
trol of protective and purification structures in Neva Inlet, taking into ac-
count measures for the preservation and sanitizing of the water system. Provision
must be made for the automation of control of the technological process of puri-
fication of waste water in sewage purification structures (including the formula-
tion of algorithms and control models) and the gates of openings in protective
structures for the passage of water in dependence on the information obtained
by an automated system for monitoring water quality in different parts of the
water system.
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In the investigations of 1980-1985 provision is made for the participation of in-
stitutes of the USSR Academy of Sciences and agencies of the ministries of elec-
- tric power, melioration and water management, fishing, geology, health, higher _
and intermediate special education. The functi~ns of the key department are as~
signed to the USSR State Co~ittee on Hydrometeorology and Monitoring of the En-
vironment, and the functions of the key agency are assigned to the State Hydro-
logical Institute.
t
~ ioz
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UDC 556.531.4 -
METHOD FOR EVALiTATING THE ADMISSIBL~ CONTANILNATION OF WATER BY SMALL SHIPS
- WITH ENGINES ~
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 11, Nov 80 pp 92-99 -
jArticle by V. K. Plotnikov, candidate of physical and mathemati cal sciences, and
S. K. Revina, candidate of chemical sciences, Institute of Theoretical and Experi-
mental Physics and Institute of Applied Geophysics, manuscript s ubmitted 13 May
80]
[Text] Abstract: An atteffipt was made to ascertain the
maximum admissible disch~.rge of contaminating
substances into water bodies and watercourses
from a nonstationary source of contamination -
ships with small engines. Formulas are de-
, rived for estimating the admissible number of
ships with small engines for a particular water
body. ~
Small ships with engines must be regarded as a source of water contamination on
' a par with other water users, which makes it possible to raise the question of u
. tfie maximum admissible discharge of contaminating substances by these ships for
each specific water body rivers, lakes, reservoirs. The specif ic characteris-
- tics of small ships with engines as a source of contamination are the following:
their number in each water body is great, their tracks of movement quite imiform-
ly cover the surface of the water body (since the draft of such ships is small,
: they also move outside the channel), and the concentration of co ntaminating sub-
stances in the wake of a mov~.ng ship does not differ greatly from the mean for -
the water body (the latter will be demonstrated below). Everything said makes it ~
possible to assume that ships with small engines are continuously distributed
over the surface of a water body and a constantly operative source of contamina-
tion. In principle the chemical composition of the contamination, the mean quan-
= tity of each of the substances entering into the water in a unit time from one
_ ship and the mean time of o,peration of one ship are known, that is, the mean
discharge of each substancp by one ship during.a definite time, for exampl~, dur-
ing the navigation season, is known. Accordingly, the concept of the maximum ad-
missible discharge applicable to small ships must be replaced by the concept of -
the maximum admissible number of ships for each water body. The mathematica~ for-
mulatian of the problem in this case is as follows: assume that Mi is the admis-
sible discharge of the i-th substance into the water of a particular water body dur-
ing tfie time T, and qi is the mean discharge of the i-th substance during this same
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time by one ship. Then the number of ships for which the discharge of the i-th _
substance will be less than the maximum admissible is -
N _ Nt = , (1) -
qi
and the maximum adtai.ssible [ma] number of ships N~ for a particular water body
is equal to the minimum of the Ni values, that is, N~ = Ni~n'
In order to determine the maximum admissible discharge (MAD) of any substance it
is necessary to determine the correlation between the MAD and the maximum admis- -
sible concentration (MAC) of th3s substance. We will take into account the process
of self-purificat9.on of water and we will assume that the source of contamination
is distributed over the surface of the water body; the length of the sector of
change in the source distribution density is everywhere much greater than the
distance over which the contaminating substance diffuses in the water during the
- characteristic time of the self-purification process. Then, taking into account
the exponental nature of the self-purification process [6], we will write an _
_ equation for the mass of the contaminating substance per unit area of the water
body at the time T:
dm T d u( t 1 - r ~ rfnt� - r
~i~ - cts e ~ d t + e ~ ~ ( 2 )
0
where �(t) = dM/dt is the rate of discharge of the contaminating substance into
the water, dm0/ dS is the mass of the contaminating substance per unit area at
the initial moment in time, ti is the self-purification time constant. -
- The discharge of matter in a unit area in the time T is
e~t T ~~1 ~t~ dt. (3~
dS - dS `
c
- If the discharge is distributed uni~~rmly over the area of the water body and the
rate of the discharge is constant a*ic: equal to
� r mo' C4> -
then the mass of contaminating substance in the water at any moment in time is -
_ equal to m0 and the mass of the effluent is proportional to time and at the time
T is
M - ~0 T (5> -
The concentr~!~ion af contaminating substance in the water in this case is
lft~
� ~ - V~' ~6)
where V is the part of the w~...r volume in the water body in which the contamin-
~ ating substance is mixed.
In accordance with the assumptiuns made chove the contamination occurs unii'ormly
over the surface of the water body S~ Therefore
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I
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V = S/t,
- (7)
where h is the thickness of the water layer mixed with the substance. Wave action
is evidently the most effective process ensuring mixing. The amplitude of the
~vater fluctuations in a water body with a depth much greater than the wave length
decreases with an increase in the distance from the water surface exponentially
, with the exponent -2Tf/,1 z(here ~ is the wave length and z is the distance from
_ the surface). The distribution of the contaminating substance in the water upon
the ending of mixing will also be exponential with the same exponent. With an
identical concentration of substance at the surface equal quantities of matter
are contained in the entire water volume with an exponential density distribution
_ and in a layer with a thickness ~1/2Tt with a uniform density distribution. Ac-
cordingly, the thickness of the layer of mixed water can be assumed equal to
- . . . , h _ 1,._ . . . ~8~
We note that this assumption exaggerates the mean concentration of contamination.
Waves can be caused either by the ships themselves or by the wind. Since the
first factor constantly accompanies the process of contamination of water by ef-
fluent from ships, in estimating the thickness h of the layer it is necessary to
take into account the length of the ship's wave specifically. The mixing of the
impurity discharged by one ship is produced by all the ships passing along this
path after it. The length of the ship's wave is determined by the ship's speed
vship~
. 2 r. 2
i. = 'v .
[c = ship] B ` (9)
Here g is the acceleration of free falling.
With a wind velocity greater than the ship's speed the length of the wind wave,
and accordingly, the thickness of the mixing layer will be greater than that
caused by the movement of the ship. However, in order to simplify the computations
(with some understatement of the volume of mixed water) we will assume that the
thickness of the layer is determined from the ship's wave:
h = g . (lp)
In this expression vshi is the mean speed of ship movement. As a result, the
water volume into whichPcontaminating substances enter from ships with engines
is
z
- V = Sg ~ (11~
if the mean depth of the water body is H>h. If H I)(Fig. 3d). The computation formula for determining the mean moisture
reser.ves has the form
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