JPRS ID: 8483 METEOROLOGY AND HYDROLOGY

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JPRr L/84$3
29 May 1979
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METEOROLOGY AND HYDROLOGY
No. 3, MARCH 1.979
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U. S. JOINT PUBLIC~eTIONS RESEARCH SERVICE
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J'PRS L/8483
29 May 1979
METEOROLOGY AND HYDROLOGY
No. 3, March Z979
- . �
Se].ected arti.cl.es from the Russian-].anguage journal
, METEOROLOGIYA ~ GIDROLOGIYA, Moscow. -
CONTENTS PAGE
Si~ificeace of Different Types of Iaformation in Long-Range
Forecasting
(M. I. Yudin) 1
Simple Statistical Model of Nbdern Clima,te
~ (V. M. Voloehchuk) 17
~lnpirica]. Nbdel of Modern Climatic Cb~..;,~~
(K. Ya. Vinnikdv, P. Ya. Graysman) 31
Study oP Formation oP Tropospheric Temperature Field by the
Con~ugate Equations Method ~
(D. B. Shteynbok) ~+7
, Computation of Characteriettcs of the Atmospheric Boundary
Layer Using Data Obtained Using a Meteorological Me,st
(N. L. Byzova~ V. A. Shnaydma,n) 54
~ Use of Data on Atmospheric Mnisture Content for Predicting the
Altitude of the Lower Boundary? of Frontal Cloud Cover
(N. I. Sholokhova) 64 -
Tropospheric Nbisture Content in the Typhoon Zone
(N. I. Pavlov) ?3
Turbidity Factor at Moscaw fo~ DifPerent Wind Directions and ~
Velocities
(G. M. Abakumova, T. V. Yevnevich) 80
Contamination of S~rface Waters of the Cerrtral Regi~ of the
North Atla.atic by the Atmospheric Fh]1.out oP Strontium-9o
(K. P. Me.khonfko) 87
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COI~ENTS (Continued) ~e
Evaluation and Predictiion of Change in Water Mineralization in
LL~rrge Rl,v~rs of the E1~ropee~n USSR With Allowance Eor the
ZnfJ.uence ~f Economic Activity
(Ye. A. Leonov) 95
Computation of Voluraea of Wind Transport of :9edimen~s in Channel
Fora~tions in ~he Nadr~^n River ~
(A. A. Levash~v, 0. N. Baryshnikova) 109 ~
Computation of He~.t Supply for the Development oP 3om~e Agricu].tural _
Crops During Sp~ing '
(Yu. I. Chirkov, L. G. Lerin) 115
Influence of the Beta EfPect in the Friction Field on Vertical
Nbvements in a Cyclone
(V. I. Ponomarev) 125
Model Ap~atus for Simulating Atmoapheric Condi~ione ~
M. B. Fridzon) 132 ~
Ozone ~n3 Globa]. Contamination oP the Atmosphere
(2. T,. Karol') 1!+1 -
Ozonometric Network in the USSR
(G. P. Gushchin) 1.57
Twentieth Anniversary of Discovery oP the Pole of Relative
Inaccessibility in Antarctica
(Yu. N. Avsyuls~ 168 ~
Review of Mono~aph by F. M. Ku~erman and V. A. Moisey~hik: _
Vyprevaniye Oziaprkh Ku1'tur (UamPing oY Wi~er Crops),
~ I,eningrad, Gidrometeoizd.at, 1977, 168 Pages
(A. I. Korovin, Y. A. Korneyev) 171
Review of Monngraph by G. T. Gutiyev and A. S. Moiyash: Ifl.imat
i Nbrozostoykost' Subtropicheskikh Rasteniy (Clime~te and F~ost
Resistance oP S'~:btropical Plants), ?~eningrad, Gidrometeoizdat, -
1977, 280 P~.ges ~ i
~
(S. A. Gol'tsberg) 173 :
Sixtieth Birthday of Vladimir Nikolayevich Parshin 175 '
Axard of A. A. Fridman Prize 178
At the USSR State Coimmittee on ~drometeorology and Environmental
Monitoring
(V. N. 2,akharov) 180
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CONL'~u ( Continued) Page
Conferences, Meetings snd Seminsre
(L. P. Yermakova, et al.) 181
Compa,rison of Dobson Spectrophotometers
(Yu. Ye. Kazakov) 190
Notes From Abroad ' .
- (8. I. Sil:rin) 192
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PUBLICATION DATA
f
English title ; NiETE0ROL0(~Y AND HYDROLO(iY No 3,
Me,r ?9 ;
Russian title : METEOROLO(~IYA I GIDROL'O(~IYA
f
Autihnr (s) . i
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Editor (s) ; Ye. I. Tolstokov . ~
Publishing House : GIIaROMETEOIZDAT .
;
Place of Publication ; Moscow
,
Date of Publication : 1979
Signed to press : 22 Feb 1979
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Copies . 4030 ;
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COPYRIGHT , "Meteorologiya i gidrologiya", 1979 ~
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i~'Uk UP'~I.CiAI. US~ ONLY ~
UDC 551.509.33
SIGNIFICANCE OF DIFFERENT TYPES OF INF'ORMATION IN LONG-itANGE FORECASTING
� Moscow METEOROLOGYYA I GIDROLOGIYA in Russian No 3, Mar 79 pp 5-14
~ (Article by Professor M. I. Yudin, Main Geophysical Observatory and Lenin-
grad Hydremeteorological Institute, aubmitted for publication 22 August
1978]
- AbstracC: The auChor gives a criCical analysis
of Che range of information drawn upon for the
purpose of making long-range hydrometeorological
forecasts9 from the poi~nt of view of the theory
of fundamental and con~ugate equations of dynamics
o� the atmosphere and ocean developed by G. I. -
Marchuk. Some concepts are based on the results
obtained by the use of the asympCotic methods
of nonlinear mechanics used in atmospheric dy-
namics. The desire is expressed that large-scale
indirect computations be made and that the archives �
be bolster~d with data for use in hydrodynamic and
physical-statistical long-range forecasting methods.
(Text~ Until recently the problem of collecting Che meteorological and hy-
- drometeorological informa~~.on necessary for predicting for a month and a
season in advance was so,lv~d purely intuitively by each researcher. Hocoever,
in the 1960's-1970's a number of researchers began to define the require-
' ments on the information to be drawn upon for long-ranp,e forecasta, linking ~
Chis to general considerations on the predictability of atmospheric process- '
es and the regularities in Che dynamics of the atmosphere-ocean-soil active
- layer system. Attention was given to a number of elements which earlier had
not been regarded at a11 as characteristics of long-period processes (cloud
cover, precipitation, boundary of_the snuw cover, etc.). Empirical and logical
bases were defined for the need to employ new characteristics.
During these years another direction arose in research. It was based on
variation of the initial conditions in the numerical forecasting problem.
[60J. It was discovered thaC a change in the initial values of a number
of ineteorological elements exerts relatively litrle influence on the results
of short-range nwnerical forecasting. This result means that such
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metenrological elemenCs chnnge rapidly, adapting tio the "fundameneal" in-
i~ial conditions.
The Cnnclusion was rherefore drawn rhat in short-range, and especially in
lnng-rnnge forecasCi~g, it is posaible to not collect information on Ch~
- init~.A1 values o� a number of ineteorological elements. The real condi- ;
rions in this case are replaced by some standard conditions.
A new approach to the problem of ~he infarmation necessary for hydrodynamic
lon~-range weather forecasting was developed in the investigaCiona of G. I. "
Mnrchuk (32, 33, 35, 57 and othera]. The ~nomaliea of ineeeorological ele-
ments are regarded as perturba~iona relative to the mean climatic values. ~
~or their determinaCion it is po~asible Co write a system of fundamental �
~ and con~ugate equaCions for dynamics of Che atmosphere and ocean. IC is ;
shown that solution of the con~ugate problem determines Che significance
of different Cypes of inetedrological and hydrologl.cal informntion in depend-
ence on the advance time of tihe forecast and Che intervals for averaging � i
the sought-for function in time and space. ~
The application of the theory of conjugate equations affords a possibility
for ~ more rigorous validation of a considerable parC of Che conclusions
concerning the information necessary for forecasts for a month and a sea-
son~ earlier having only empirical confirmations. AnoCher part of the con-
~ clusions is sub~ect to refinement. In this light a reatudy has been made of ~
the problem of the makeup of Che informaCion whic?~ is used in predicting
temperature and precipitation for the principal agriculturul regiona of the
USSR by the physical-statistical method. Earlier this prob],em was analyZed ~
in [38, 49, 51, and others]. In t'iie examination of some of the disputable
problems we will rely on the concept of the atmosphere-active layer of the ~
ocean and land system as a multifrequency system in which rapid and slow ;
movements in the first approximation can be separated [36, 55].
Tnformation on the state of the upper layer of the ocean (waCer tempera-
ture, ice content). This source of information has long attracted the aC-
tention of researchers investigating the problems relating to large-scale
interaction between the atmosphere and ocean. In particular, in the well-
known studie~ of V. Yu. Vize [15, 16] a detailed study was made of the ice "
content of the Barents Sea as a predictor in the long-range forecasting of ,
_ temperature and precipitation. The correlation established by Meinhardus ~
[58] between the temperature of the Gulf Stream in November-January and
air temperature in Central Europe in February-April was checked and re-
- evaluated several times. There were found to be considerable variations
in the sample values of the correlation coefficient up to a change in sign i
of the correlation during individual periods. In a number of studies it ~
was therefore concluded that data on water temperature in the ocean are
of liCtle practical importance for the purpc'ee of making forecasts for a
month and a season. A useful review of the discussion of this problem and
clarification of a number of disputable points is contained in a book by
T. V. Pokrovskaya [45).
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The ehought has been exprer~sed eh~t the sta~e of the upper layer of ~he
ocenn is a result of atmospheric circulation conditiona. On Chis basis
it is postulatad that models of Che developmenC of macroproceseea can be .
fnrmulnted independently of ~nformation on the ocean.
Scimetlme~ nne he~rs the contention th,il wuter temperaCure h~s 11tC1e varia-
b11:iCy. The conclusion is therefore drawn rt~aC heat exch~nge between the
- ocean and ehe atmosphere is dependent to a higl?er degre~ nn the air ~emper-
~ture unomalies th~n on waCer temperature anomalies. However, an analysis
- of data on Che atandard deviations of waCer temperature and air tem-
perature ( O'~a) in the North Atlantic [26, 27, 29] carr3ed ouC during recent
years does not confirm these statements. Comparison of the O'~ and aTg val-
ues cited in these studies shows that these ar.e comparable valuea. We car=
ried out preliminary computations for checking the naCural hypothesis that
- the raeio of the standard deviation of the mean water temperature during
some time interval rela~ive to this same air Cemperature characteristic
increases wiCh an increase 3n the av~raging interval. (The aense of ~he hy- -
pothesis is that water temperature is more conservative than air tempera- _
ture.) Some tendency to ~n increase in the coneidered ratio with a trans-
ition from monthly t~ seasonal meana is actunlly d~tected. However, it is
weaker than can be expected a priori. EvidenCly, the intensity of sir
transformatioti over the ocean is so great Chat the air temp~erature over
the sea surface also acquires the property of conservatism. This problem
- requires further analysis. -
A solid basis for use of data on temperaCures of the upper layer of the
ocean was the theory of fundamental and conjugate equations~ mentioned
above. The theory makes it possible to represent the air temperature anom-
aly during some time interval in the form of the sum of Cwo tertns, one of
which characterizes the influence of initial disturbances in the atmosphere,
whereas the other characterizes the influence of initial disturbances in
the active layer of the ocean and land. It has been demonstrated that with
an increase in the averaging interval the characteristic ~�alue of the first
term decreases rapidly snd the second becomes decisive. It is understandable
ehat this conclusion does not relate only Co the significance of watex tem-
perature in the ocean. In principle it can also serve as a basis for use of
- data on soil temperature and moisture content, ice content, times of dis-
appearance of the snow cover in different regions, etc.
- It would be of great interest Co make a theoretical evaluaCion of the prog-
nostic significance of the enumerated factors by means of solution of the
fundamental and conjugate equations of the problem of disturbances of the
thermal or precipitation regime. It should be noted that in principle the
incorporation of data on ice content and on the dynamics of forming and~
disappearance of the snow cover and a number of factors important for
monthly and seasonal forecasts do not require validation by means of com-
putations using complex models of atmospheric circulation. An adequate
basis is the conclusions drawn with the application of important simplif-
. ications, such as those obtained by M. I. Budyko [11, 12, 14, and oChers]
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~vnl.unt~nns o~ Chc inEluence Gxc~r~~~cl l~y bueti ~li~ HI1111J ~~nd .[r.e auver
- on the Chermal r~g~.me oF the atmosphere. However, eppllcaCion o� the theory
of perturbations affords a posaibility of localizing the expected effeces
in rime nnd in s~~nce. :
InCormurion on ?�he disrribution of cloud cnver. There are contradictory =
opinions on the problem of the significance of data on cloud cover for
forecasCing purposes. As is we11 knnwn, the cloud cover field has long
been considered sma11-scale (spotry). Moreover, an analysis of the rela-
tive significance of different initial data employed in numerical fore-
casCing [28, 60] indicated that a change in init~.~l data on the humidity
field has relaCively little inf luence on rhe forecasting resulCa. On tihis
basis V. M. Kadyshnikov ~.n a numerical model of computaCion of sCeady pre- -
cipieation adopted Ch~ iniCial condition for saturation of the atmod;+here
by wnter vapar in the ~n'cire space, and cloud cover and precipitiaCion at
subsequenti moments in ti.ne are functions of the compuCed verCical velociC-
ies. '
In [46] an attempt was mac'e Co refine Che initial condition for Che humid-
ity field. Humidity was determ~ned as a function of the ob~~erved valuea ;
~ of nonconvective cloud ccrver and the computed verCical veloctty values
at the time used as the initial value. Using preliminary, stiil exL�remely
limited data, such a formulation of the problem led to a satisfactory pre-
diction of cloud cover for a t~ne of 24 hours. With a more precise meChod ,
for introducing data on initial cloud cover there is also some increase _
in the success of the forecast.
We feel that these results must be taken into accounC in solving the problem
of raCional~methods for stipulaCing the initial humidity field for short-
range forecasting purposes. However, for validaCion of the significance
oE data on cloud cover in long-range forecasting it is necessary to have
~ demonstrat3on o~ a different kind.
I1e can mention two bases for inclusion of cloud cover among Che predictors:
the concepC of a regulating role of cloud cover in p:~ocesses ot thermaL .
interaction between the atmosphere and the active layer;
the concept of presence of a low-frequency global component in Che cloud
cover field. ~
i I%
Now we will examine these bases in ~omewhat greaCer deCail.
Ideas concerning the important role of cloud cover as a regulator with a
feedback in Che processes of thermal interaction between the ocean and Che
atmosphere were expressed in 1963 by A. S. Monin [39]. A simple model
for illustraCing this role was formulaCed in [17]. It was demonstrated
that variations of the type .
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warm sea
~
development of convection, increase in cloud cover and cooling
of the aea
cold ~ea
stable stratiEication, decrease in cloud cover and heatiing
- of th~ sea
c~n hnve periods of abouC a month.
� Further inve~tigations of the role of cloud cover over the acean in heat ex-
change pY�,-~~;sses are relared to Che theory of conjugate equationa. In [34,
41-43] the influence of summer cloud cover anomalies over the North Atlan-
tic on autiumn and winter air temperature anomalies over Europe was examined
from the theoretical point of view. There wae also a statistical analysis -
of asynchronous carrelations. The principal statistical conclusiona of n~c-
essity are based on a~ime-limite4 aeries of ineasuremenC data on cloud
cover made from a satellite, so that Che investigaCion w~s far from cow-
pleted. Nevertheless, the results confirm the considered influence is im-
portant.
Now we will consider what bases there ure for asserting that there is a low-
frequet?c��; global ~omponenC in the cloud cover field. Here the reference is
not to the ~hermal effect mechanism, which was mentioned above, but the dy-
namic mec~~anism deCermining long-persisting extensive regions of ascending
movements`~, with which the formation and persistence of cloud cover are as-
sociaCed-,
The fi~r!at conclusions that in planetary movements, in contrast to strictly _
large-r?cale movements, the potential part of velocity is not very small
in cdtaparison with the solenoidal part, were drawn in [47, 56]. In partic-
ular, it is of interest to study a graph of the characteristic values of
thr: horizontal divergence of velocity and its component terms, obtained in
- ~+7] by means of "scale analysis" of the tercns in the equations of atmo-
' spheric dynamics. It follows from the graph that under mean conditions the
considered ratio attains a minimum fo~ a scale L1 ~800 km a~d increases ,
rapidly in the region of scales L y L1. Th3s conclusion gave hope for suc-
- cess in an attempt at empirical discrimination of the stable planetary com-
ponent in the cloud cover field. It was surmised that its values can be a
significant predictor in a long-range forecast. On a practical basis it
was possible to collect long-tQrm series of data on cloud cover in the
Atlantic-European secCor adequate for analysis. This material was processed
by the method of expansion in empirical orthQgonal components (EOC). The
expansion coefficients were regarded as predictors. In [52] there was check-
ing of the hypothesis of statistical significance of a set of asynchronous ~
correlations relative to different groups of predictors. It was shown that
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the liypotihesis of significance of data on cloud cover iy satiafactorily
confirmed on ~he basis of Ctie coneidered maCerial.
1~urCl~er results were nbtttined in a cycle of etudies [50, 54, 55] in formuL-
r~ting ~ hydrodynamic-stnCistical model of atmospheric clrculaCion. The
sCream func~ion and velocity potiential are among the principal variables.
Spectral transformations of the equations of atmospheric dynamics were
carried out; these lead in Che tirst approximation to the discrimination ~
of a 1lmited number of subsy~tems of ordinary diff erential equations. The
characteristic frequencies and eigenvectors of the �undfimental and con~ug-
ate equations [36] were compuCed in this same first approximaCion for a
number ~f subsys~ems. T~~e important conclusion was drawn thaC some plan-
etary movements are characterized by very small characC~ristic frequencies
(periods of several tens of days). It follows from the form of the eigenvec-
tors thaC for zonal wave numbers n= 1 and n= 2(f irst ewo harrnonics) the
contribution of the potential part of ~he vector is significant. This means
that Che planetary disturbances arising in the vertical velocity field,
and accordingly, also in the cloud cover field, are~ not propagated rapidly
f~om east to west (as follows frorn the Rossby theory), but are very s'lowly
moving waves. The nonlinear interactions of waves exert a sCrong influence
on the movements o~ waves and ridges from day to day. However, with con-
siderable time averaging the Cotal influence of th ese more or leas irreg-
ular inCeractions lessens relative to the systematic movement of the waves.
We can therefore assert that the global cloud cover field is characterized
by peculiarities which long retain their sign and therefore exert a signif-
icant influence on macroprocesses, whose prediction is rhe basic objective
of monthly and seasonal f orecasts. It is easy to see that both are basie
for regarding cloud cover as a very important element in the system for
long-range forecasting and supplement one anoCher. The concept of a regulat-
ing role of cloud cover in heat exchange of Che sea-atmosphere system rests -
on an analysis of forced oscillations of the system and the discrimination
of the low-frequency component rests on an analysis of free oscillations
in the atmosphere.
Now it is possible to understand c:hy in [28, 60) it was not possible to de- .
tect a significant influence of the initial humidity (and cloud cover)
field on the numerical forecasting result. The fact is that the variation
of the initial data was a small-scale and accordingly a rapidly oscillat-
ing disturbance. In order to clarify the principal effect it was necessary
to carry out variation of the planetary field component. i
Information on the pressure field. First we note that in the light of the '
taodern theory of predictability of individual synoptic processes Chere
is no basis for c~nsidering a11 data on the pressure field as a source of
useful information. In particular, at the present time it appears that the use-
fulness of the well-known similarity indices ,/~x for long-range fore- ;
casts requires careful validation. A shortcoming of these indices is that
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the U~~~iy for eh~ir d~t~rmingtinn i~ a~nmpr~zi~on di the ~ma11-ar~1e Chnr-
~i~trri~ticg of fieid~ ~r indtvidual pnint~.
Muge ~uthvr~ nf hydrodyn~niC, hydrodyn~mic-~CaCi~ttcal~ phyoic~l-~t~tiigricai
rind synoptic meChode of long-range forecaeting limit the use of dat~ on the
~reggur~ f i~ld ta a few cheracCertetiGa. For ex~mple, in th~ Ye. N. glinova
ln:~g-rang~ foreanating theory ~2-5, gnd othera~ the circulation iadex ie nf
p~rtleul~r importance. The impor~anc~ of th~ circuletion index aC tihe ini- -
t1c~1 mn~nen~ and computatinn of ite chenge~ ie an import~nt pare of the
` routin~ mcthod far predicting mean monthly air temperature anomaliee for
tt~e northern hemigphere. T11e modern stage in d~vplopment of the th~ory ie
relat~d to the p~rri~l line~rization mpehnd (discrimination of the mgin
� w~veg) developed by Ye. N. Blinova (6-8 and others). Ae th~ initi~l data
it i~ nece~~ary to ~xipulaCe a relgeiv~ly gmal~ number nf coe�ficiente for
~ph~ric~1 funcCinns.
In lcsng-rgnge forecaeCin~ methndg bgeed on ~ claggificaeion nf mpteorolog-
ic~l procegseg and fieldg the number of characCerietics used in forecaeting ,
~s nlsd reduced to u minin?um. But the information ueed ict the claseification
i'r:self can be extremely extensive, as can be eeen~ for example, from the
_ bookg [18, 19J. The eame can be said about the cnmplex phygical-gtatieticgl
m~thnd, in which Che gpproach of expansion of Che presaure field into seriee
in empirical orthogonal functions is ueed for the purpoge of discriminating
ttie mosr large-scale and long-period processes [31, 48, 53, and othere).
We note that with the eubatantial difference in the methads for proceasing
information in the macrocirculation and physical-sCatistical meChode the
initial information etill hae much in common. Thig ia attributable to the -
striving af researchers to g49p~10N adapt all available information to use.
Now we will discuss conclusions from the G. I. Marchuk perturbation theory.
Atnon~ the approximate formuLations of the long-range forecasting problem _
mentidned in [32j thpre is nne whicti does nor require the use of data on
pr~ssure and wind. It is assumed that the distribution of temperature in
the ~ctive layer of water and land at tf~e initial moment is everywhere
knnwn and the variations of the heat influx in the atmosphere were computed
' independently. An approximate solution of the forecasting prdblem with re-
placement of the values of the horizontal wind velocity component by the
r~e~n climatic values is obtained under these condirions. However, such
- a formulation of the problem is regarded only as inCernediate. A more
~ complete theory was based not only on informaCion on the state of the at-
mospherc in the time inCerval 0< t~ where the necessary information t~
restored by means of solution of the prognostic problesn and the correspond-
ing conjugate problem, but also precise information on the fields of ineteor-
ological elements in the time interval - 00 < t< 0, Which should be obtained
from observations.
The widespread use of integration limits for t to t in the formulas
of the theory of perturbaCions is equivalent to Che followring: the iniCial
values of the fields of ineteorological elements (when t= 0), necessary for
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prrdirtirig weAeh~r uHing ehc? fnrmulud nf p~r~urbuCinn th~nry ar~ replgced
by tiGrur~1 intnrmntinn nn th4 ~rr~C~ nE ehp ntmo~ph~re ne nll the timee pre-
c�~ding Chp zprn mom~nt t< 0) at the level of the ocean and cnnCin-
entg. '~hey mu~t b~ kn~wn with rh~ a~c~ssary ~ccuracy (~37j, pp 205-206).
It fc~llnw~ from thi~ ref~rence to the literature thar information on the
pr~~sure fieid ~or the Cim~ t~ 0 ie neceesary to thaC degree to which ie
will mnk~ iti pogsible tn ~upply thp lacking data nn the Cemperature nf the
upper layer of tY~~ oc~an~ contin~neal temperature anomglies~ heat influx -
in th~ ntmosphere aC the iniCial momenC t! 0. Such gn underetanding of
tt~~ significance of thi~ information is fundamenCally differenC �rom Che
Kynoptic approach tn Che long-range forecaeting problem based on the clae~- .
ification and tr~cking of macroproceasea in Che aCmoaph~re ae independent
~bjpcCg of investig~tion.
Nnw w~ will pxgmine how to solve the problem of the eignificance o� dgta
on ehe preggufe field in a hydrodynamic-statiatical model. A solution can
b~ nbtgined by aeparating the eiger~v~ctors nf the fundamentel and con~ugate
equationg into "glow" variables correeponding to gmall characteristic
frequenciea and "fast" varigblee correapvnding to large frequencies. T'hen
the approach of asymptotic metf~ods of nonlinear mechenics is applied, mak- _
�ing it pogaible to obCain an approximate solution for the averaged valuea
6f ehe slow variables~ which is not dependent on the valuea of the "fast"
variables. The ov~rwhelming majority of eig~nvecCors relate to frequencies
much greater than 2 ~t/month.
'I'his means that for the purpoae of forecaeting for a month or a season "nl- .
most a11" the spectral componente of the pressure field must be regarded~
r~~ v~lues whnse initial phases exert no influence on the forecagting results. _
5uch a conclusion fully corregponds to the concepts follnwing from use of
perturbation theory.
Table 1
Dependence of Characteristic Frequencies of
Planetary Waves on CirculaCion Index -
~/~u '
n
O.O.i O.Q~ I 0.03 O,V: .
~
1 0,005 0,018 U.U1.i U,O~i ~
'l 0~008 ~~U28 U.WN ~,UG,~
3 0.003~3--8) o.~rY o,a~ u,os~
4 0,034(3-B) u.Op~ 0.0~~; u,04;
[3~c~; s=~~
}{owev~r~ the presence of even a very small number of slow variables shows
that from the initial data on Che pressure field it is possible to draw
some direct conclusions concerning the nature of development of atmospheric
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FOR O~FICIAL U5.~ Ot~t.Y
t~r~ced~~~ in the monehg ~tlttGEdiBtpLy gh~~d. True, thesp cnnclueion~ to a
~~c~nNldhrubl~ dp~ree nr~ nnnditinnnl, ~ince tt~e chnractrristic frequencie~
tlic?'Hinw vgriableg ar~ highly dependpnt nn thh futur~ v~lue of th~ rir-
rulution index. As a confirmazion we cite Tab1e 1 of the minimum charneter-
igtiic frequencies chnraCterizing pianetary baroclinic wav~e with different
vnlue~ of the circulation index o~/W . Z'h~ �requency values with w~ 0.
Cited tn (36j, ~ervp ee initial data. In c~ses when the deeignation (W-B)
gtandg nfter the frequency, the waves are di~plac~d from aest to eaet. In
the remnining casee the compuCaCions indicate an opnoeite mov~tnenC of thp
� wnveg.
T~ble 1 ehows that allowance for the initial phase of the planetary r+avee
� ie mo~t important in the caee of large values of the circulation index~
thnt i~, in the cold half of the year.
_ We nnt~ Chat if th~r~ is n~ gpectral expanaion of the fields in eigenvectore
of the fundgmental nnd conjugate problems, it ie difficulC to indicaCe an
~lternntive method for ieolating the information on the initial presaure
field impnrtant for long-r~nge forecasting. Z'herefore, it ie promieiag ta
develop a veriant of rhe theory of perturbaCione on the basis of a hydro-
dynamic-statietical model.
We do not have an opportunity of discussing the probl~n, very important for
n hydrodynamic-atatistical model, of deacribing the effecta aseociated with
"f~se" variables. We only note Chat in the prepared variant of the model
prcvision is made for a statiatical descriptioa of ehose effects uhich
are ussociated with atipulation of a sm~ll volume of datg concerning th~
pressure (and thermal) field.
Indirect computations and archivizing of data. It can be expected that an
approach based on perturbation theory leads to the establishing of a number
of phy~ically more significant predictors Chan those which long-range fore-
casters }iave at the present time. These predictors in essence will repre- ~
sent the substitut~s for the lacking characteristica of the state of the
.~ctive layer of water and land, and the total v~lues of the heat and mois-
ture fluxes. In addition, here it is necesaary to include an extremely
' limited number of characCeristics of the planetary pressure field.
In this connecCion it is very important to apply maximum effort�s for ob-
taining empirical data on the global characteristics of heat and moisture
exct~ange during individual monthe and seasons. Such a problem is real
since the methods for indirect computations developed by Soviet and foreign
researchers for indirect computations of the components of the heat and
water balance, and parametric description of the state of the active layer,
Ln principle can be used for large-scale computations. A description ~f a
number of modern computation methodR is conta~ned, for example, in the
books [13, 23-25, 40, 44, 59].
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FOIt OI~FICLAL US~ dNLY
r.r~n U~ ~linwn thul CI~~ rrnpnrn~inn df nrci~iv~g nf tht~ ~nrt over a long
~c~rie~ of yenr~ is of beeic int~reat fnr sp~cia].isCe develQping phy~ical-
KtntiKticnl fnr~aagtirr~ m~eth~ds, wh~re~e ~i~~cieligt~ in the field of hydro-
clynnmic m~thod~ fnr inng-r~~nge [nrecesting nre reletivcly littl~ inrereated
in yuc11 data. Howev~r, such nn idea ig erraneous. The development of hy-
drodynamic forecasCing methode for intermediete and long periods showe
that to a high degree tl~ey are dependent on tihe empirical dara necegegry
for analygig n� sy~tematic errors in tha model and ita refinemenC. If it
~sg wned~ on thp bagi~ nf th~ regults published by G. P. Kurbatkin ~30~ 31~ .
and otherg~~ that for the ~dju~tment and correction of en intermediate- ~
l~ngth forecast it ig nec~ssgry to have archival data for approximately
five years, the neceseaYy length o� the series for the purposes of fore-
casting �or a month and eea~on muat not be leas than 20-30 yeara.
Relying on ~n analy~ig of the stability o~ the correlations~ which is reg-
ularly accompli~hed in the practice of forecasts m~king uge nf the complex ;
physi~gl-statigtic~l method,it is nece.esary to deem preferable a aeries of
- 30-40 years. Z'hus, the requiremenCs on informaCion for the purpoaes of hy-
drodynamic or complex phyaical-gtatiaticgl methode agree entirely eatig-
f~ctnrily. It appears that the preparation of the above-mentioned data
will be of great imporCance for the development of a number of other meth-
ods having features in common in the approach to the initial information.
'Phis can be said, in particular, about the studies planned by G. V. Cruza
nn the applicaCion of the analogues method [20-22) to a monthly weather
forecast and also on the methods of Ye. P. Borisenkov [9, lOj, N. A. Bag- ,
rov [lj, and othere.
The great volume of work mak~s it desirable to combine the efforts of a nwn-
ber of groups, p~ssibly in the form of a special inCernational program (af-
ter ending of the primary processing of FGE data). It is desirable to dis-
cuss this problem in the 5cientific Council on the "Weather Forecasting
Problem" and f.n the 5oviet Ceophysical Com~nittee.
;
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~ox o~~tcini, us~ nxi.Y
BIBLIOCRAPHY
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11
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i
FOit O~FICIAL USE ONLY � :
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18. Girs, A. A., MAKROTSIttKULYAT810NNYY METOD DOLGOSROCNNYKH METEOROLOGICH-
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20. Gruza, G. V., Soldatkina, A. M., "Principles for Formulating a Method
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21. Gruza, G. V., Reytenbakh, R. G., "Application of the Analogue Principle
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22. Gruza, G. V., Ran'kova, E. Ya., "Model of Adaptive Statistical Forecast- ~
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23. Doronin, Yu. P., TEPLOVOYE VZAIMODEYSTVIYE ATMOSFERY I GIDROSFERY V
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28. Yadyshnikov~ V. M., "Pseudoadiabatic Model of Continuous~Precipitation
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pp 33-40~ 1968.
29. Krindin, A, N. (compiler), SREDNIY~ MESYACHNYYE TEMPEItATURY VODY, VOZ-
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TRUDY SII~OZIUMA "RAZNOSTNYYE I SPEKTRAL'NYYE METODY RESNENIYA ZADACH
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Leningrad, Gidrometeoizdat, 1974, 303 pages.
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the Total Flux of Radiant Energy for the Purpose of Long-Range Fore-
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IYA, No 8, pp 10-16, 1974.
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i'
I
35. Marchuk, G. I., Skiba, Yu. N., "Numerical Computation of the Con~ug-, ~
ate Problem for a Mode1 of Thet~al InCeraction AeCween tihe Atmosphere~
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i
36. MaCyugin, V. A., Yudin~ M. I.~ "On the Linear Theory of Baroclinic Plan- ~
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I
37. Meshcherskaya, A. V., Ruk~oveCs, L. Yudin, M. I., Yakovleva, N. I.,
YESTESTVENNYYE SOSTAVLYAYUSHCHIYE METEOROLOGICHESKZKH POLEY (Natiural � ~
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38. Meshcherskaya, A. V., Ledneva, K. V., Vlazhevich, V. G., "CharacCeris- ~
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39. Monin, A. S., "Physical Mechenism of Weather Changea," METEOROLOGIYA
I GIDROLOGIYA, No 8, pp 43-46, 1963. ~
40. More, L., RAZVITIYE IDEY I NABLYUDENIY, SVYAZANNYKH S IZUCHENIYEM MOREY I
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42. Musayelyan, Sh. A., Ugryumov, A. I., 2adorozhnaya, T. N., "On ttre Prob- ~
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46. Pyatygina, K. V., Yudin, M. I., Krnnorina, E. A., Polekhin, V. B., "Para- `
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, f
~
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47. Yudin, M. I., "Relationghips nf the Elemenes of Large-Scale ACmo-
spheric Movements~and Snme Pragnostic Corollc~ries~" MATERIALY SOVESHCH-
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Forecasting Methods), Moscow~ Gidrometeoizdat, pp 5-24, 1961.
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eral CirculaCion of the Atmosphere," TRUDY MEZHbUNAItODNOGO SIMPOZIUMA
PO DINAMIKE KRUPNOMASSHTABNYKH ATMOSF~RNYKH PROTSESSOV. MOSKVA, ~3-30
. IYUNYA (Transactions of the International Symposiwn on the Dynamice of
Large-Scale ACmospheric Processes. Moecow, 23-30 June 1965), Moscow,
"Nauka," pp 213-220, 1967.
~ 49. Yudin, M. I., FI2IK0-STATISTICHESKIY METOb DOLGOSROCHNYKH PROGNOZOV PO- ~
GODY (Physical-SCatiatical Method for Long-Range Weather For~casts),
Leningrad, GidromeCeoizdat, 1968, 28 pages.
50. Yudin, M. I., "DeCermination of Mean Motion in Long-Itange Forecaeting
Problems and the Theory of Climate," TRUDY GGO, No 272, pp 3-14, 1972.
51. Yudin, M. I.~ Meshcherskaya, A. V., Blazhevich, V. G., "CharncterisCica
of Hydrometeorological InformaCion Used in Che Long-Range Phyaical-
Statisti.cal Forecasting of Temperature and PrecipiCation for Regiona
of Inadequate Moistening," TRUDY GGO, No 236, pp 45-63, 1970.
52. Yudin, M. I., Meshcherskaya, A. V., "Results of Use of the Physical-
Stat3stical Method for Predicting PrecipitaCion and Temperature for
a Long Time in Advance," TRUDY V VSESOYUZNOGO METEOROLOGICHLSKOGO
S"YEZDA (Transactions of the Fif th All-Union MeCeorolugical Congress),
Vol II, Leningrad, GidrometeoizdaC, pp 83-84, 1972.
53. Yudin, M. I., Meshcherskaya, A. V., "Some Evaluations of the Natural
Components as PredicCors and Predictants," TRUDY GGO, No 273, pp 3-15,
1972.
54. Yudin, M. I., "Hydrodyni3mic-SCatisCical Model of ACmospheric Circula-
- tion. Integral CharacteY~istics of Atmospheric Circulation," DOKLADY
VSE50YUZNOGO SIMPOZIUMA FO PIGAP (Reports of the All-Union GARP Sym-
posium)(21-24 December 15~71), Moscow, Gidrometeoizdat, pp 202-206,
. 1973.
55: Yudin, M. I., "Principles for FormulaCing a Spectral Model of Atmospher-
ic Circulation With Allowar~ce for the Peculiarities of Movements of -
Different Spatial Scales," TRUDY SIMPOZIUMA "RAZNOSTNYYE I SPEKTRAL'NYYE
METODY RESHENIYA ZADACH DINA~MIKI ATMOSFERY I OKEANA" (Transactions of
the ~ymposium: Difference an~3 Spectral Methods for Solving Problems in
Dynamics of the Ocean and Atatosphere), Novosibirsk, pp 49-67, 1974.
56. Burger, A. P., "Sca1e Considex~ation of Planetary Motions of the Atmo-
sphere," TELLUS, Vol 10, No 2, pp 195-ZOS, 1958.
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. ,
57.' Marchuk, G. x. ~"Formulation of rhe 'Theory nf perCurbat~.ons for Com-
plicated Mod~ls. P. Ix: Weather PredicCion~" GEOFISICA INTERNACIONAL,
Vol 15, No 3, pp 169~].82, 1975.
!i8. Meinardus, W., "Uber meteorologische Bezei.chungen zwischen dem Nordae-
lantischen Ocean und Europa im WinCerhalb~ahr," M~T. ZEITSCHRIFT~ No
3~ PP 3-48, 1898. ;
59. PARAMETERIZATION OF SUB-GRID SCALE PROCESSES. (GARP Publicationa Ser-
ies, No 8), 1972, 121 pages.
60. Smagorinsky, J., Miyakoda, K.~ Strickler, R.~ "The Relative ImporCance 4
of Variables in Initial Conditions for Dynamical WeaCher Prediction," ;
,
T~LLUS, Vol 22, pp 141-157, 1970.
1
f
1
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UDC 551.583.15
SII~LE STATISTICAL MODEL OF MODERN CLIMAT~
Moacow METEOROLOGIYA I GIDROLO~IYA in Russian No 3, t4gr 79 pp i5-24
[Article by Doctor of Physical and Maehematical Sciences V. M. Voloshchuk,
Institute of Experimental Meteorology, submitted for pub~ication 30 May
19 78.]
Abstract: For an analysis of climatic procesa-
es wiCh characteristic times of change of tihe
order of tens of years the author has formul-
ated a simple statistical model whoae basic
difference from Che simple deterministic mod-
� els used earlier by different researchers is
allowance for weather fluctuations. The author
has derived the fundameneal equation for the
model, in which only surface temperature, aver-
aged for some region, is considered, on the
assumption that this parameter is staCistical-
ly independent of the other climatic parameCers
and modern climate has an isolated point of :
equilibrium. Ztao very simple variants of the
model are investiRated: Gaussian and power-lxw
four-parameCer.
[Text] Introduction. The problem of the possible influence exerted on the
earth's climate by human activiCy is acquiring ever-increasing timeliness.
A rather greaC number of hypotheses, validated to different degrees, con-
\ c~rning the physical mechanisms of such an influence, has now been advanc-
' ' ed (for example, see [1, 2, S]). However, despite the extremely broad
front of investigations carried out in this direction, for the time be-
ing reliable experimental resulCs have not been obtained; the rigorous as-
sumptions used in the theoretical investigations for the time being give to
the latter only the nature of frequently heur~stic reasonings and propo-
sitions.
Evidently, when reference is to investigation of a possible anChropogenic
change in modern climaCe, it is necessary, in particular, to study cli-
matic processes with characteristic times of change of the order t~f tens
of years. In the opinion of a rather large number of climatologists,
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prec~.gely during aunh ~ e~.me ~.1: is pogsible tn expect, Eor exnmple, a
c~~.maric eff~cti �rom Che influence exerted on the rgdiation balance o~
eh~ ~~rCh-atmogpher~ sysCem by the carbon d3oxide entering 3nto the ~C-
mosphere as a result o� the combustiion o� coal nnd peCroleum. :
liue during Che indicated cht~racteriat~.c times a substantiial influence on ;
the chgnge in some climatic characteris~ics wi11 be exerted by their
weather tl.uceuaCions. In tihis case we are dealing wi~h a phenomenon ~
which eo some degree is similar to the inf?uence of turbulence on the
courge of some atmospheric processes (�or example, on cloud formation and ~
cvolueion) or the influence of Brownian motion on the behavior of aero-
so1 parricles in the medium. Naturally, in an analysis of anthropogenic
change of modern climaCe ie is necessary somehow eo t~lce c~euther fluctun-
tions intio account. This article is devoted to the �ormulatiion oP one of
the possible models for tiaking them into account. We ttote Chat such c~n ~
artempC was undereaken recently in [6]. We assume t;haC we have been able ;
to generalize somewhat and carry out further development of the'ideas ;
set forrh in ~his study. ;
First of all, we will be more specific in what is meant by the terms "cli-
maCic change" and "weather fluctuations." We feel Chat it is necessary to
do this here due to the fact that for the time being there are ~eersingly ,
no definitions of~these concepCs which are generally accepted in climaCic
theory. .
For Che purpose of clarity we wi11 have in mind the surface temperaCure
Tsurf~ that is, the tiemperature of the air mass measured at different sur- ,
face (or shipboard) meteorological stations or posts. Similarly it is pos-
sible to examine other so-called weather characteristics, such as the pre-
cipit~tion total, humidity, tenths of cloud cover, etc.
A time series of data on surface temperature, if Che measuremenCs ha~~e been
made rather frequently, has an extremely irregular nature. If no aCtention i
- is given to the diurnal variation, for short times Che change in surface
temperature at a stipulated poinC will be purely random and related, in
parCicular, to the random movement of air masses in the atmosphere. We in- .
troduce Che following notations t-{-~n ~
~ ` dt' Tn (t'), T' = T~ - TK~ (1, ,
TK =
. ~ .
['R = surf] where "Lsurf is the charac~eristic correlaCion time T' (rate of ~
- change of fluctuations). [K = cli(matic)] I
It seems to us that '~surf is the most fitting and a compleCely natural ;
parameter making it possible Co separate the weather fluctuations of the i
value from its climatic change. In this case, for example, the weather
J characteristic Tsurf becomes a purely climatic parameter when it is aver-
aged in time 'Gsurf; the deviation of Tsurf from Tcli~ thaC is, T~ becomes ;
the weather fluctuation of surface temperature. The time ~G surf in order of ~
~ 18
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m~gniCude can scarcely be more than a few month~. Iti is avident that Che
cl~.mn~ic parameter T 1~ is ~ random value, and this meana ehat it makea
sense to speak nnt o~ any values of th3.s parameter a~ a atipulated mnment
in rime, and about the distributiian funcCion ~(Tc1~,, C) (reference is tio
Che probability dene~.ty funceion, norm~li.zed Co unitiy). Therefore, hence-
fdreh under ehe tierm "cllmatic change in surface Cemperature" wa will menn
th~ evolueion oE the introduced dietribu~ion funct~.on ~f (TCli, t)�
We noCe that in the invesCigation of possib~.e anthrnpogenic changes in cli-
. maCe the emphasia up to the present time has been devoted, as a rule, only
to n sCud;~ of the behavior of the first moment of Chia distrib stion ~unc-
tion, that is, when reference was Co surface temperature~ '
� ~TK~ ~dTKTKtO~lK~ ~2~
FrequenCly on1.y intuitive consideraCions are preaented with respect to Che
~~ossible behavior of Che variabillty of climatic parameters (climaCology
of second and higher momenta). And indeed, it is not impossible that the
vigoraus development of industry at the present stage can begin Co exert'
an influence (or has already influenced) not the climaCic trends of dif-
ferent climatic characteristics, but specifically ~heir variabilitiy, al-
so playin~z an extremely significant role in our 'life.
Despite the rather rough description of the real processes, simple climatic
models (Budyko, 5ellers, Zat'sman, Adam and oCher models [SJ) have an ex-
tremely high heuristic value. Therefore, it seemed desirable to us ~o for-
mulate, proceeding on the basis of the simplest physical considerations,
a similar model which would describe the evolution of Che~distribution
function of the climatic parameters (in the considered case surface
temperature) under the inf luence of the disturbing facCors.
Principal physical assumptions. We feel that the simplest variant of a
staristical model of climate can be formulat~d on the basis of a kinetic
equaCion of the diffusion type for the ~ distribution. However, for this
it is necessary that the correlation time for the rate of change in weather
fluctuations ~L surf could be considered negligible in comparison with the
- characteristic time of change in ~(we will denote it by 'G~li)� The two
characteristic times of change in ~ closest to 'G surf are detercnined by
the seasonal variation of temperature and the high-frequency variation of
_ climate with a characteristic time of 2-4 years; the next characteristic.
time evidently has the order of several tens of years (warming beginning
in the second half of the past century and ending in the 1940's of this
century, for example). A rather detailed analysis of the state of inves-
rigations of the time sCrucCure of processes in the earCh-atmosphere sys-
tem is given in [4]. Thus, if it is necessary to examine the seasonal var-
iation of ~ as well, then, generally speaking, it is possible only with a'
definite strain to consider the condition 'Gsurf Q ticli to be satisfi~d.
However, nevertheless even in this case in some firsC approximation it can
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be used, since ther~ scarcely should be any effort Co ob~nin more precige
quantitatiive relaCionships 3.n simple model~.
In order to wriCe a kinetic equatian for only one randam value Tcli ~r ~g
necessary co saCisfy the condition of ita sCatistical nondependence on
other random c].imatic parametera. It seems Cha~ it is scnrcely poasible to
indicaee situaeions when this condition would be satisfied cnmpletely for ;
ehe earCh's climate. Therefore, we will use it ~s a simplification nec~g- :
sary for conaideration. If Chere were no nthers, then already due to this ,
simplificat~on a11 thQ subsequenC description would be Cransformed into
modeling. '
We note that here reference is to possible TcZi values at a sCipulated ~ E
point. BuC for d3.fferent pointa on Che earth the Tc1~, disCribution will
knowingly be different, and in general, even in Che simplest model iC will
be inadmissible to examine the evoluCion of ~ at some painC independenCly ;
of its evoluCion at other points on the earth. However, it seems to ua
that it is possible to discriminate individual regions which in energy re-
spects are more or less closed, so that iti is possible to neglect the
statistical dependence of the surface temperatures averaged for these re-
fions in some first approximation. The choice of the exCent of the region ,
mu~t be made on the basis of additional physicoclimaCic data. Henceforth
by T we will mean the surface temperature Tsurf for which there was aver-
aging for such a type of region.
�
_ ~
~
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'1't~~ rnn~id~rgtivn~ Ciepd ab~v~ m~k~ ir po~~ibi~ Cn rgpr~sent the rate of
T~hgng~ in eh~ fnrm ~f r~gul~r ~nd fluctu~Cing pgrC~
r~arr, r~, r~~~(T, r). (3~
whprh ~ ig ehe delr~-cnrr~lated ~andom �unction with a mean equni to zerot
~ro. ~t~ ~l~-t)a2arr, r~ a~~c). c4~
~ Integrnting (3) in th~ timp int~rvgl (e~ t+dt), wg nrriv~ aC gnnCh~r fnrm
~f ehig ~quation:
dr~ a c r, r~ dc + 1~~~ ~s~
where ~ ie ~ rnndom vglue, with ~ normgl diatributton~ with n maan equal
t~ zero gnrl a diapor~ion equsl te unity: ~y � 0, 1.
in cnntrggt tin th~ nrdingry diffprential equ~tione~ which for etipulated
valueg nf the function aC th~ tim~ tp deC�rmin~ ite b~havior at an arbi-
tr~ry momenC in time t> Cp, equgtion (5) mnkeg it poegible tn computp not
some T vnlue~ but itg di~tributioc~ at tihe moment in time t> t(~ on the
basig of the sripulated digtribution at rhe time Cp. Such a claes of
equatinn is usunlly call~d stoChastic differentinl equationg nf the ito
ryp~ [3J.
1'h~ procedure fnr solving equatione of the type (S) is ~xtretn~ly complex
~nd unwieldy. Therefore, u~ually there ie transformation from these equa-
tiona tn the equivalent ordinary differential equationg in partial deriv-
~tives for the distribution function kinetic equatione. The derive-
tion of the kinetic equations from (5) was obtained and valideted by A.
N. Kolmagorov and is cited in many monographs on the theory of rendom pro-
cesses; therefore~ here it makes no sense to discuss thi~ in detail.
The kinetic equation ~COllowing from (5) for describing the evolution of the
distribution funcCion ~ with time has the form
d~ r, I~ o D(T, c) c. (6)
~ This is the ordinary equation af convectiv~ diffusi~n with the drift co-
efficient a ana ~he diffusion coefficient D.
If D= 0, we arrive at a simple deterministic climatic cnodel such as wae :
mentioned in the introduction. Thus, a new element of the proposed climatic
model is the inclusion of a dif.fusion term, which makes it possible to take
into account the influence of weather fluctuations (to be sure, within the
frnmework of the adopted assumptions and simplifications).
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t~~k na~rtrrn~, us~ oN~,Y
Gnu~~i~n vari~n~ df ~nodpl. W~ wili dpnnt~ th~ ClimneiC ~urf~c~ t~mp~r.e- `
tur~, av~r~g~d nv~r ~ p~riod df gbout 100 y~a~rg, by TN, and eh~ devi~tion
~f ~ g~nm ~N>>y SZ . W~ wili can~id~r th~ pnint TN t~ be nn isol~ted atablp
pnlnt in mad~rn eiima~c~, tihuC ig, a pnint tu which, firgt nf gli~ th~ Cpm-
~~~rn~ur~ atwny~ r~~urn~ nfe~r n~ma11 di~turbanC~ nnd in whose immedinte
n~ikl~burh~~d, ~~ec~nd ~f a11, eh~r~! are n~ eimilnr poine~. Mc ..over~ v~
wi1] n~~wee ehdt ~eh~~ ~in~iiar pnint~ nf equilibriwfl of modern rlimat~
~r~ ut ~uch ~rpat digt~nchg ehnt eh~ir infiuence i~ not r~fl~ated ~n th~
ev~~utinn of thp SZ d~viaeic~ne c~u~~d by either regulnr n~ rendom fgctor~.
Unf~rtun~C~1y, invemCi~ati~n nf th~ factuai mgt~rinl ~n th~ b~hevinr nf '
m~d~rn ~licnat~ mgkp~ it paegibl~ tn dr~w a Conelu~idn only about the pog-
~ibiiity of ~ hypoth~tirgl ~itu~~ion ~nd doeg not makp it pogsible tn ~~y
nnyehing nbout hos~r cld~e it is tn re~lity, ~bouC how far the other pointg ~
nf equilibrium are fr~m TN.
Th~ cirpd hypoeheei~ n~k~~ it poes3bl~ tn ~xpand ehe drifC and diffueinn
r.o~ffieir~rr~ in ~qu~tion (6) into a T~y1or s~ri~s in gnm~ neighborhood
~f ~N:
~ ( T, t 1 ~ a ~ T~., t ) ~g . . .
(7)
' d D w ~ ~ ,
o~r, t~=ocrN,
r ~ rh
'1'h~ n(TN, t) vnlue characteriz~e the factors leading to n regular change
in SZ� In this connectian it can be conveniently broken down into two parta
~ o(T~, t)=A(!)-}-R,
whprp A(t) conteing irifarmation on the mechanisms leading to a seasonal
rh~nge in surface temperature and its variations with a period of 2-4
ye~rs if the latter actually ~xiet and R gives inforeoation on the mechan-
igms exerting an influence on its change With a greater characteristic
time. We can introduce inCo R information on the regular anthropogenic in-
fluence an surface temper~ture, fnr example, by virtue of an increase in
the concpntration of gtmospheric C02, a change in albedo of the underlying
surface~ etc. if We $ubgtitute the cited t~+a terms of the Tsylor series fnr .
a('~,t) into (4) it immedigtely ig clear that the ~ parameter characterizes
the resistance of the climatic system relative to different regular or ran-
dom disturbancee gnd equal in order of magnitude to Che inverse time of ex- .
ponential relgxation of surface temperature to the TN level after sudden ;
~egsation of expogure to the distiurbing factor. The ~ parameter cannoC be ;
equal to zpro since othernrise the dispersion of the random value without i
influence of external regular facCorg aould increase With time Without lim-~
it. The stability of modern climate indicates thaC this does not occur. We
note thaC similar considerations on the need for existence of a finite value
for p are also 3iven in (6J. Moreove~, ~~annot assume negative values,
since the TN point Was determined as a point of stable equilibrium. The
dependence of ~ on t can be negleated for the considered ticies, since its
vulue muat be determined by thn~~ same processes in the Environment as the
TN value. To be sure, tttis assumption requires additional checking, but
sp~ms very probable to us. M4reover, in the climatic statistical model var-
iants examined here allrn+ancp for a possible dependence of ~ an t leads to
nothing ne~+ from the physical point of vie~+.
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~ir~t w~ wi11 gggwne that D, thgt is~ th~ int~neity of ehe rate of change
~f wc~aeh~r fiuctu~tion~ with tim~, ie nnti d~p~ndent on 5~,. On~ of the po~-
~ib1h Cnrmg of th~ dppendence ef D on S'Z ai11 be examin@d in det8il in th~
i~~xt ~~c:Cion of thie articl~.
'Th~n rhp equarion for p ae~ua~es the following form:
~~-{=-~(~~w~A+R)p=D~ (8)
' 'I'hia ~quation d~termines the eimpleet poegible variant of the etatisticel
modpl df ciimate.
� Fnr ~quation (8) we wi~i ex~mine the problem vithout iniCial conditions. We
will ~~ek g solution in the form
In ~=bo-I-6~ (n~-1~, (9)
wh~re bp, bl and I~) are aome functions of t.
5ubstituting (9) inCo pquation (8)~ we arrive at the follor+ing expreseions
for these functions:
bo a-- In o~}- C, b~ e-~,
d ~ ~10~
ar - - ~ A -f- I~~
1 d~~ _ r ~s ~
ac � -
Itere C is soene constant which must be determined from normalization con-
dirians. It is easy to see that C=-1/2 ln 2 n.
Thus, solution of the problem without initial conditior~s for equation (8)
has the form of a normal distribution
. ~s-->= ~11~
I '
r - C ~
r. z
~ ~2
' where the mean value and the dispersion 0'2 are solutions of the dif-
ferential equations (10).
Thprefore~ the considered variant of the statistical model is natwrally
called "Gaussian."
We will denote the parameter characterizing the anthropogenic inf].uence
on climate by In the Gaussian variant of the model the evolution of
dispersion vith time, caused by the perturbing effect of h, can be ac-
complished only through a change in the parameters ~ and D. Assuming
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~ tc~ b~e u~me~il vnlu~, it c~n b~ ~sgumpd eha~ :
n-n�
R ^r
vhpr~ ~n ~nd D~ ar~ the vAlues of ehe parameeerg ~ end D when r]� 0.
~t~u~, th~ anthropageni~ chgnge in the m~an surface temperature can be, in
g~nern].~ compiet~ly ~igni�icnnr. However~ Ch~ change in diepereion in ehie
cee~ can be fuily ignnred. Thig meons Chat in the coneidered ~ituation Che
appe8r~nce o� a climaeic tr~nd etill sayg nothing wtwt~oever abour eignif- "
ie~ne changes in th~ disppreion of surfg~p temperatiure, about which some
climgtologigts speak very cat~gorically, aasuming it to be poasible~ proceed-
tng on the basie of close conbiderations. If the Caussian variant of the
model correctly raflects the r~al behavior of modarn cliroate, ehe posaible
gignificant evolution of its variability under the influence of man's in-
dugtrial activiry muet be investigated independently of an analysis of cli-
mgtic trende by m~ans of g search for Che phyaical mechanisms af dirpct sub-
gtantial disturbance of the ~ and D parametere. Insofar a~ we know, for the
time being only a hypotheais is being formulaeed concerning the possible
physical mechaniBmg of the anLhropogenic influence on climatic trende. Thi~
problem i~ extremely interesting and therefore beloar ae will examine a very
simple model for evaluating the posaible influence of climatic trends on
thp variabtlity of climate (with poatulatifln of exietence of such a correl- ,
gtinn).
Power-lgw four-parameter varianC of model. Thus, according to t;he common,
although, to be sure, not generally accepted opinion, the appearance of g
climatic trend in the behavior of surface temperature should lead to an
inten~ification of its variability. This assumption can be taken inro ac-
coune most simply, asguming ~
r~r, r~~Ftr~~+;~'(r-r~), c12~
where is a delta-correlated random function with a mean equal to zero.
Strictly speaking, such a representaCion of the random function F is en- � .
tirely legitimate since it is known that by virtue of the extremely com-
plex intertwining of a great number of direct and inverse correlations
in the climatic system there can be development of self-intensifying or
(vice versa) self-sustaining extremal situations. But (12) includea Che ~
new parameter T* characterizing the level with deviation frum which there ~
is an intensification of weather fluctuations of the rate of change in tem- ~
perature and the entire problem involves specifically a determination of
the Tk value. If it is assumed that T* = TN we do not obtain a de- ,
pendence of variability on trend. In Chis case the model is s3milar to the
Gaussian variant, but the distribution form is different. If we simply as-
sume T* = TN, then (12) will, in essence, in simplified tnathematical form
express the hypothesis of intensification of weather fluctuations with the
appearance for any reason of a climatic trend in aurface temperature. In
general, TN for this purpose can replace any other parameter not dependenC
on the parameters controlling the change in surface temperature for the
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Cim~ seale~. Nothing chang~a from thi~~ but F(T*) will be loaded witih addi-
tional information, but we do not need en explicit form of rhie function in
th~ future.
W~ notc~ that there can b~ a eomewh8t different inrerpYetation of expreseion
(12). In acCual3CV, if a~ eomp~r~ this ~xpre~eion with ~he firet expr~~-
~ion in ~7~, it can be seen that the eecond tarm in (12) is an exprees3on
t~king into accounC th~ weather �luctuatione of resistiviry in the climaeic.
~yg tem.
25
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;
~o~ o~~ict~t. usE orn.Y
Subetituting (].2) ~nto (4), we hove
~~r, r~-l~o+afl+YS3~,
(13)
~ D~ o (t) = N ) P ~v a ~ (t) ~ h 2 't~ ( ~ '
whare 'y ~ 0, and ot can assume an arbitrary ~ign~ but aithout fail thmre must ,
~be satiafaction of the cond3Cion D> 0. ~
;
Using for a the expression (7), and for D the expreasion (13)~ we arriva at ,
the following kineric equation: ;
'~~i'' a~ ~n.-1- A t R) p= du (~o aQ Y~') ~p. . (14)
Multiplying this equation by Qn, where n! 0~ 1,..., and integrating the
regult for ~!L in the 3nCerval + ao), after simple transformaeions
we arrive at tiie following recurrent expressions for the distribution mo-
mentg ~ : .
A ~t ~Q~~ = " ~3 - ~it - Y~ ~An~
(15~
-I- (A R -h (rc - t ) x~ �h -.1) D, = I.C n- a'~/ :~""1~ * . (17)
il~re the a~teriak denores the mean annual values of the corresponding
coeffici~nts~ divided by ,8 n> are tha mean annual valuee of the die-
. tribution moment~ E 3s the mean annual poseible rate of change of sur-
face Cemperature, divided by ~(we recall that vith the adopted aeeumption
E ! const).
it ig e~sy to s~e thet we arrive at a similar type of recurrent equationa
nlen for the extremal mean monthly values of the distribution moments
but in thig case 'y*, oC* and D* will represent Che mean monthly valuea of
the corresponding coefficiente, divided by
i:xpressions (17) cannot be used for computing moments whose numbere eatis-
fy the condition n> 1+ y-1. With satisfaction of this condition there
are no moments for Ch~ eCationgry probl~m. Thig is related to the pover-lav
of behavior of the function Sp ahen ~ fLI in the considered model. In ac-
tuality, with the same asaumptions which were used in deriving (17), equa-
tion (14) is tranaformed to the form
(18)
~ ~ ~ ' ou ~~r 'f' ~r.A ~ i+?tts~ ~
The asymptotic solution of this equation in the case of large~,~,~ has the
form
~...~Q~-~-~J:., ~~I-.x~
(19)
We will assume that < 1/3. Then the first four disCribution moments
. ~ are known to exist. Asaume that for some sufficiently greaC time interval
t* the parameter can be assumed equal to zero (unperturbed stationary
problem). In this case from expressiona (17), after simple transformations,
. for the mean value of the deviation of surface Cemperature from Tp, Che
dispersion d' 2, asymmetry S = < - / Cr 3 and excess E _ < ( .Q -
~ 4~~~ 4_ 3 we f ind