JPRS ID: 9841 USSR REPORT METEORLOGY AND HYDROLOGY NO.2, FEBRUARY 1981

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JPRS L/9841
13 July 1981
USSR Report
METEOROLOGY AND HYDROLOGY
_ No. 2,- February 1981
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JPRS L/9841
13 July 1981
USSR REFORT
~
~
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METEOROLOGY AND HYDROLOGY
No. 2, February 1981
Translation of the Russian-language monthly journal METEOROLOGIYA I
GIDROLOGIYA published in Moscow by Gidrometeoizdat. ;
CONTENTS
Impact of Carbon Dioxide on Climate............................................... 1
Telescoped Scheme for Hydrodynamic Short-Range Weather Forecasting 16
Prediction of Mean Monthly Air Temperature Fields Over the Northern Hemisphere
Using an Automated Group Analogue Scheme 29
Model of Cloud Cover on a Stationary Front 43
Evaluation of the Effectiveness of Antihail Protection in Bulgaria 55
Variations in the Intensity of the Indian Summer NLonaoon According to Cloud Cover
Data From Satellites........................................................... 62
Kara-Bogaz-Gol Gulf and the Caspian Sea Prohlem.................................. 70
Variability of Ice Conditions on Ship Navigation Routes 79
Influence of Errors in Statistica,l Characteristics on the Accuracy of Optimum
Interpolation.......r��os��.s~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~00, gooses���000000 � 90
In�luence of Solid Particles on the Kinematics of the Transporting Fluid Flow... lOZ
Dynamic-Statistical Methods for Predicting the Yield of Agricultural Crops..... . 110
Organization at Camputerized Data Archives From Analyses of Meteorological
Fields Obtained Under the FGGE Program........................................ 124
Visibility of Light Signals in a Crystalline Fog 130
Measures of the Ozone Concentration in the Troposphere.......................... 134
MPteorological Support for the Taenty-Second Olympic Games in Moscow by the
USSR Hydrometeorological Center............................................... 139
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Activity of the International Data Center on Atmoapheric Electricity............ 148
Fiftieth Anniversary of the Main Aviation Meteorological Center................. 152
Review of Monograph by l. A. Shiklomanov: Antropogennyye Izmeneniya Vodnosti
i Rek (Anthropogenic Changes in the Water Volume in Rivers), Leningrad,
Gidrometeoizdat, 1979, 300 Pages 159
'Seventieth Birthday of Isay Grigar'yevich Guterman 162
At the USSR State Committee on Hydrometeorology and Environmental Monitoring.... 164
Notes From Abroad............................................................... 165
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UDC 551.588.7
IMPACT OF CARBON DIOXIDE ON CLIMATE
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 2, Feb 81 pp 5-17
[Article by M. I. Budyko, corresponding member USSR Acxdemy of Sciencea, and N. A.
Yefimova, doctor of gecgraphical sciences, State Hydrological Institute, manuscript
submitted 18 Aug 80]
[Text] Abstract: A study was made of the dependence
of climatic conditions on the concentration
of atmospheric carbon dioxide on the basis
of data on climates of the geological past.
The change in the mean annual air tempera-
ture at the earth's surface with a doubled
carbon dioxide concentration is estimated. Ap-
proximate data are given on the change in the
quantity of precipitation on the land in the
high and middle latitudes for these conditions.
The problem of the impact of atmoapheric carbon dioxide on climate was studied in
two cycles of investigations. The first of these was devoted to a clarification
of the relationship between climatic changes in the geological past and variations
in the C02 concentration in the atmosphere. Investigations in this direction were
initiated late in the 19th century in studies by Arrheniue [14] and Chamberlin '[22].
In the second cycle a study was made of the influence of an increase in the C02 con-
centration caused by man's economic activity on modern climate. These investiga-
tions, initiated in the 1930's by Callender [19], have now acquired great impor-
tance in relation to long-range anthropogenic changes in global climate.
~ Until recently the studies of
ited the possibilities for cl,
~ During recent years ways were
i
-i This paper gives materials on
ical past, whose use makes it
i
climate.
these cycles were poorly tied in together, which lim-
ixifying the influence of carbon dioxide on climate.
found to make a 3oint study af the two problems [3, 61.
the impact of carbon dioxide on climate of the geolog-
possible to clarify the patterns of modern changes in
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Sensitivity of climate. In a quantitative explanation of climatic changes it is
necessary to know its sensiti.vity to variations in external climate-forndng fac-
tore. In particular, for an explanation of the dependence of climate on the atmo-
spheric C02 concentration it is necessary to evaluate the sensitivity of climate
to changes in the heat influx caused by variations of atmospheric transparency for
long-wave radiation.
Until recently in determining the sensitivity of climate use was made primarily of
computations with models of the theory of climate. Since all existing climatic�
models contain different simplifications, the accuracy of such computations has
remained unclear, as a result of which the conclusion is sometimes drawn that
there are no adequately'reliable evaluations of the aensitivity of climate to
changes in the heat influx.
During recent years considerable progress has been attained in this field as a re-
sult of the use of a series of empirical methods for evaluating the sensitivity of
climate to changes in the heat influx, based on the use of data on changes in cli-
mate in the geolagical past, on modern changes in climate and on seasonal changes
in the meteorological regime. All these different and completely independent ap-
.proaches give close results in evaluating the parameters c'iaracterizing the sen-
sitivity of climate. These results agree well with data f-eom computations of sen-
sitivity using models of the theory of climate, including detailed models of gen-
eral circulation of the atmosphere and schematic energy models of the thermal re-
gime with determination of their parameters on the basis of sufficiently reliable
empirical materials. Among the general characteristics of the impact of carbon di-
oxide on climate it is common to use the parameter ,Q Tmean. the change in mean '
global air temperature at the earth's surface with doubling of the C02 concentra-
tion in comparison with its value for the end of the pre-industrial epoch. On the
basis of a comparison of the results of computations of thia parameter using dif-
ferent models of the theory of climate a comanission of the United States Climate
Council under the chairmanship of J. Charney coacliided that its value is 3t1.5�C
[20]. This conclusion coincides with the conclusion drawn in other studies made
recently, where in determining tre parameter ATmean use-was made of the results of
computations based on the application of both theories of climate and empirical
methods [5, 6]. Table 1
Change in Mean.Air Temperature With Doubling of the Atmoapheric C02 Concentration
Simplified models of ModPls of gen- Modern Climatic changea
climatic theory eral circulation climatic in geological
of atmosphere changes past
1. 2.4�C (1967) 5. 2.9�C (1975) 8. 3.3�C (1977) 9. 3.5�C (1979)
2. 2.5-3.5�C (1914) 6. 2.0�C (1979) 10. 3.4�C (1980)
3. 2.0-3.20C (1977)
4. 3.30C (1979)
_ The results of dTmean computations are given in Table 1, which for each evaluation
of the,6Tmean parameter gives the years of publication of the corresponding atudies.
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The first of these evaluations, ohtained in a study by Manabe and Wetherald [26],
pertains to conditions for the earth as a whole and includes nu allowance for the
feedback between the temperature field and the area of the polar snow-ice cover.
Since this feed6ack is poeitive, that is, ineensifying the sensitivity of the
thermal regime to changes in the heat influx, the corresponding value of the ATmean
parameter must be somewhat reduced. This feedback was also not taken into account
in the detailed computations of Augustsson and Ramanathan [15] (evaluation 3). Eval-
uation 2 was based on computations of the distribution of the mean latitudinal tem-
peratures made taking the mentioned feedback into account [2]. Evaluation 4 was ob-
tained by Ramanathan, et al. [29], also with allowance for this feedback.
Evaluations 5 and 7 were found in the studies of bLanabe and Wetherald [27, 281. In
the first of these studies computations were made of the changes in the mean lati-
tudinal values of elements of the meteorological regime with an increase in the at-
mospheric C02 concentration, relating to mean annual conditions. In the second study
a more general problem was solved, including computations of both latitudinal and
longitudinal changes in the meteorological elements for an idealized topography of
the continents and oceans. Evaluation 6 was obtained in an investigation by Manabe
and Stouff er, in which the authors took into account the real distribution of the
continents and oceans as well as the annual variation of ineteorological elementa [25].
In these studies use was made of a detailed model of general circulation of the atmo-
sphere, including allowance for the principal feedbacks between climatic elements,
including the feedback between the thermal regime and the snow-ice cover.
Evaluation 8 is based on an analysis of empirical data on modern climatic changes
[5]. Evaluations 9-10 were obtained using data on climatic change in the geological
past [5, 6], when these changes were dependent on additional feedbacks between the
albedo of the earth's surface and the thermal regime, which do not have great impor-
tance for the modern change in climate. Among these additional relationships are
the mutual relationships between the thermal regime and changes in the area of the
continental glaciations and albedo of the surface of the continents occupied by a
vegetation cover [11, 21], which increase the sensitivity of the thermal regime to
variations in C02 concentration. In [S, 6], on the basis of approximate computations,
it was ass uned that the influence of these ieedbacks increases the Wmean value by
approximately 1�C. This correction was taken into account in determining the values
of the LSTmean Parameter in the �ourth column of the table for ensuring comparability
of these evaluations with the evaluations relating to the conditions of modern
change of climate.
Table 1 shows that the mean value of the parameter ATmean, determined by different
~ methods, is close to 3�C; the maximum deviation of individual evaluations from this
I value does not exceed 1�C. Computations of the mean deviation of individual evalua-
' tions from the indicated ATmean value make it possible to assume that its probable
~ error is about 15%.
t;
; It is obvious that the problem of determining the change in the mean planetary. air
; temperature with an :tncrease in the C02 concentration is not the whole of the prob-
lem of the influence of carbon dioxide on climate. In order to answer practical
; questions concerning modern changes in climate it is necessary, in particular, to
~ have materials on change in the temperature of the lower air layer at different lat-
~ itudes and longitudes for definite seasons. It is also necessary to evaluate the
~ changes in the precipitation sums on the continents.
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In answering these questions it is possihle to use two methods: computations using
different models of the theory of climate and an analysis of data on the climates
of the past relating to warmer epochs. It has already been repeatedly noted that
it is extremely desirable to make simultaneous use of both these methods since the
accuracy of eac.h of these methods is limited and is usually inadequately known. With
matching of the results of the two independent approaches it can be hoped that these
results will be adequately reliable [4, 61.
Paleoclimatic data. Returning to the problem of the use of empirical data for study-
ing the influence of an increase in the concentration of C02 on climate, we should
note the possibility of the use for this purpose of materials on the,modern change
of climate, on the meteorological regime in the epoch of the climatic optimum of
the Holocene, during the interglacial epochs and during the Tertiary j6, 10, 241.
The importance of these materials for solving the in3lcated problem varies. The cli-
iaatic change during the last century was only partially dependent on the increase
in the C02 concentration; the warmings of the Holocene and interglacial epochs evi-
dently were determined for the most part by variations of elements of the earth's
orbit and the inclination of the earth's axis. Although it has been established
in studies of the theory of climate that changes in the heat influx to the earth-
atmosphere system caused by different factors lead to similar climatic changes [28],
the indicated difference can be a source of additional errors in a study of the de-
pendence of climate on the GOZ concentration.
Among t'ie already enumerated forms of cli.matic change in the past it is the Cenozoic
cooling which is most closely associated with variations in the COZ concentration
and data on this cooling are of apecial value in a study of the dependence of cli-
mate on the atmospheric carbon dioxide content. This value incieases considerably
due to the great interval of change in mean temperature at the earth's surface in
the course of the Cenozoi.c era, which -f3 about 10�C. Such a value, much greater
than the ranges of tempE_ature variations during the time of the other enumerated
warmings, corresponds to a change in the C02 concentration in an interval which is
approximately equal to the anticipated increase in the C02 concentration under the
influence of economic activity in the course of the coming centuries.
The use of data on climatic changes c'luring the Tertiary period for a study of the
dependence of climate on the C02 concentration involves a number of difficulties.
In particular, in the Paleogene the form of the oceana and continents differed ap-
preciably from.their modern form, which could lead to additional climatic changes
not associated with variations in the atmoapheric content of carbon dioxide. These
changes, however, evidently were'of a regional character and exerted little influ-
ence on variations in mean annual temperature [2]. In the Neogene the form of the
earth's sur.face was close to that of the present day, which facilitates the use
of paleoclimatic data for this,time for a atudy of the dependence of interest to
us .
Other difficulties in the use of paleoclimatic data are rel.ated to the limited de-
tail of materials on the concentration of carbon dioxide in the past, which are
available only for entire divisions of geological periods, the inadequa.tely clar-
ified accuracy of these data, the schematic nature and limited accuracy of paleo-
climatic materials. The existence of these difficulties gives the results of study
of the dependence of climate on the C02 concentration on the basis of paleoclimatic
data an approximate character and requires their detailed checking. One of the ways
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to carry out such checking is a comparison of available data on variations in the
cincentration of carbon dioxide with changes in the thermal regime in the Ter-
tiary. The results of such a comparison are given in the next aection.
In order to use paleoclimatic data in a study of the dependence of the thermal re-
gime on the carbon dioxide concentration in the atmosphere it is necessary to de-
termine as precisely as possible the changes in this concentration in the past in
comparison with the mDdern epoch.
In [3,8] ttie values of the C02 concentration for different divisions of the Tertiary
times were found using materials published by A. B. Ronov an.the rate of deposition
of calcareous sediments. These materials do not include data for the modern epoch.
The corresponding value, which we will denote Sp, can be found by the extrapolation
method using data for divisions of the Tertiary and Cretaceous periods. Such comput-
ations made it possible to find a Sp value of 0.05�1015 g/year [3]. Recently�A. B.
Ronov somewhat refined the rates of carbonate formation for the Pliocene. Computa-
tions made with allowance for this refinement, $ccording to data for the Ter.tiary
and Cretaceous periods, give So = 0.07�1015 g/year. Extrapolation of the values
for the last two divisions of the Tertiary on the basis of the refined data publish-
ed by A. B. Ronov gives SO = 0.045�1015 g/year. The corresponding value, which can
be found using data published by Bowen [17], has this same order of magnitude.
On the basis of the evaluations cited here it can be postulated that the Sp value
falls in the interval 0.045-0.07�1015 g/year. This makes it posaible not to change
the earlier adopted evaluation of this value, equal to 0.05�1015 g/year.
Table 2 gives the carbon dioxide concentration values determined with allowance for
this value in comparison with its modern value in the Tertiary period and in the
Upper Cretaceous. It can be concluded from the data in Tab1Q 2 that in the Pliocene
the C02 concentration was greater than the modern level by almost twice and in the
Miocene by almost a factor of 4. It must be remembered that the changes in the car-
bon dioxide concentration during the last hundred million years had a more complex
structure, which is only partially characterized by the six mean values cited here,
relating to long time intervals. Evidently, in the.past there were repeated briefer
variations of the C02 concentration and the theraial regime.
! Thermal reaime. The most detailed investigations of the influence of carbon dioxide
! on the thermal regime Uy means of models of general circulation of the atmosphere
' were made in the studies of Manabe and Stouffer [25] and Manabe and Wetherald [28].
' 'lhe results of these studies do not completely coinc~de. Whereas the values of the
~ Tmean Parameter obtained in the second of them agree well with the conclusions of
, a nunber of theoretical and empirical studies, thet3Tmean value found by Manabe and
Stouffer is somewhat less than the results of other modern investigations. It is
worth noting, as indicated by Table 1, that this value is lower than the values of
the 6Tmean Parameter obtained in the absence of allowance for the feedback between
; the thermal regime and the snow-ice cover, which, as noted above, increases the
sensitivity of climate to the values af the C02 concentratian. From the preliminary
' communication on the investigation by Manabe and Stouffer [25] it is difficult to
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explain tle reason far such a discrepancy, which scarcely is dependent on the dif-
ferenc detail. of tlie used climat2c models and evidently is associated with a nonco-
incidence of the parameterization of the studied processes in [25] and [28]. Until
this problem is clarified we limit ourselves to a comparison of the results of a
recent study by Manabe and Wetherald [28] with paleoclimatic data.
Figure 1 gives the differences in the mean latitudinal temperature at different
latitudes, according to Manabe and Wetherald, with a doubling of the C02 concentra-
tion in comparison with its modern value (MW curve). This graph also shows the re-
sults of determination of temperature for the middl.e of the Pliocene st different
latitudes in the northern and southern hemispheres according to materials from empir-
ical investigations generalized by I. I. Borzenkova, M. V. Muratova and I. A. Suyet-
ova. The mean latitudinal temperature values for the temperate and high latitudes
for which tliere were groups of uniform data are represented in the form of two
segments of continuous lines. For the lower latitudes it was possible to use dsta
from temperature measurementa only in four regione. These data are represented in
che form of dots. 'Lhe dashed line corresponds to the mean temperature distribution
according to empirical data. d)
f6
e
B
4
?
d;
Fig. 1. Dependence of change in air temperature Fig. 2. Change in mean global
(L1'P) an latitude with doubling of C02 concentra- air temperature (Q Tp) with dif-
Lion. ferent C02 concentrations.
It follows from the indicated inaterials that the mean air temperature for the earth
:it the earth's surface in the t4iddle Pliocene was above the present-day 1eve1 by
1.9�C. Tliis value almost coincides with tlie &me8n value found by Manabe and Weth=r-
ald. We note that in the Pliocene the thermal regime was influenced by the feedback
between temperatures and reflectivity of the continents, dependent on the state of
the vegetation cover, which was not taken into account in the study by Manabe and
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~ Wetherald. Allowance for this feedback increases the value of the parameter aTmeBn
i by approximately 0.50C [llj, a& a zeault of which. the QTmean values in these two
~ computations are equal to 2.9 and 3.5�C. It is ofivioua that these values agree
! cluite well.
~ It is worth noting, as indicated in Fig. 1, that the dependence of the Q T value on
I latitude accarding to empirical data and according to the computations of Manabe
and jJetherald is very similar. It can be noted that the empirical. data indicate a
somewhat more considerable increase in the aT difference with latitude in compar-
i ison with the results of theoretical computations. ~ In order to clarify the dependence of the thermal regime on the chaiiges in the COZ
: concentration in a broad range of its values it is goasible to use data on paleo-
temperatures during the last hundred mitlion years, during which the C02 concentra-
tion decreased almost tenfold. In earlier studies [6, 9] for this purpose use was
~ made of materials from the investigations of V. M.'Si.nitsyzi (12, 131, which up to
-i the present time are the sole source providing information on the thermal regime
' of a considerable part of the earth's surface during the entiwe Phanerozoic. A com-
parison of the air temperature maps constructed by V. M. Sinitsyn with ttce mater-
j ials of later investigations makes it possible.to assume that for the Tertiary per-
i iod and the Mesozoic era V. M. Sinitsyn somewhat exaggerated the temperature dif-
~ ferences of the past in comparison with the modern epoch in the high and middle
latitudes and understated these differences in the low latitudes. However, the
~ mean air temperature differences for the northern hemisphere, computed using the
~ V. M. Sinitsyn maps, are quite reliable, as can be seen, in particular, from the
' data in Fig. 2.
Table 2
Relative Changes in C02 Concentration
Division
Absolute age of
Rate of deposition
Concentration
beginning and
of carbonates, 1015
of C02, %
end of di.vision,
g/year C02
millions of years
Pliocene
2-9
0.09
0.055
Miocene
9-25
0.18
0.110
Oligocene
25-37
0.08
0.050
Eocene
37-58
0.31
0.185
Paleocene
.58-66
0.20
0.120
Upper Cretaceous
66-101
0.44
0.265
This figure, in the form of dots, shows the values of the noted difference from the
maps constructed by V. M. Sinitsyn for the Pliocene, Miocene, Oligocene, Eocene-
Paleocene and Upper Cretaceous in dependence on the corresponding concentrations of
carbon dioxide. The circle represents the similar value for the Pliocene, computed
tising the data in Fig. 1. The empirical values for temperature change can be compared with the results of
theoretical computations. In these computations it was assumed, in accordance with
the results of several modern investigations, that with a constant albedo a doubling
of the C02 concentration leads to an increase in the mean air temperature at the
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earth.'s surface by 2.50C. Since over the course of a hundred million years the
earth's albedo changed, these changes must be taken into account in determining
air temperatures in the geological past.
One of the reasons for change in albedo was the increasing aridity of the continents,
which occurred at the end of the Pliocene and the Pleistocene. As noted above, this
led to a decrease in the mean air temperature by approximately 0.50C.
The second reason was the appearance and gradual increase in the area of the snow
and ice cavers in the high latitudes. Although the influence of these covers on the
thermal regime is taken into account in most modern theories ef climate, the accur-
acy of such computations is usually rather limited due to the schematic nature of
the parameterization of the corresponding processes.
The influence of snow and ice covers on the thenaal regime of the past can be evalu-
ated on the basis of the following empirical data.
It follows from materials from satellite dbservations [23] that in the high lati-
tudes of the northern hemisp;;ere in summer in the region free of ice the albedo of
the eazth-atmosphere system is about 0.40, whereas the mean albedo of the zone with
an ice cover is approximately 0.55. Taking into account the difference in these
values and the area of permanent and seasonal snow and ice covers, as well as the
ratio of the radiation values in the zone of the anow and ice cover and its mean
global value, on the basis of data on the sensitivity of inean temperature to change
in the heat influx it can be found that the now existing snow and ice cover decreases
the temperature of the lower air layer for the entire earth by appro ximately 2�C.
It is known that the snow and ice cover in the high latitudes already develoQed in
the Paleogene in the form of mountain glaciations which occupied'a relatively small
area and exerted no substantial influence on global climate. In the Miocene this
cover occupied a considerable part of the territory of Antarctica, which, as indi-
cated by computations, could reduce the mean global temperature by 0.2-0.3�C. A
considerable broadening of the snow and ice cover occurred in the Pliocene and es-
pecially at the end of the Pliocene, when exteneivs zonea of sea ice developed
and the seasonal snow cover on the continents expanded.
An increase in the area of the land as a result of a decrease in ocean levels due to
the formation of continental glaciers could also exert some influence on the earth's
albedo. Since the mean albedo of the surface of the continents was greater than the
albedo for the oceans, for this reason with the development of large glaciations
there was a decrease in mean temperature. An approximate evaluation shows that a11
other conditions being equal, such a decrease for the aodern epoch is about 0.3�C
in comparison with the Paleogene. This, value, however, can scarcely be included in
computations of changes in mean temperature because it is only part of the varia-
tions of temperature due to the advance and retreat of sea waters as a result of
rising and subsidence of the earth's surface, which it is difficult to estimate with
suff icient accuracy for the earth as a whole.
Limiting ourselves to allowance for the first two temperature differences, caused by
variations in albedo, using the data in Table 2 it is possible to compute the tem-
perature changes for all the divisione of the Tertiary and Upper Cretaceous.
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Fig. 3. Comparison of changes in mean glo-
Fig. 4.
Temperature
change.in the geo-
bal air temperatures found using empirical
logical
past in comparison with modern
data (L1TP) and results of computatians of
epoch.
these changes on the basis of the COZ con-
centration (6T'p).
KEY:
A)
Paleocene
D) Miocene
B)
Eocene
E) Pliocene
C)
Oligocene
F) millions of
years
Differences in Mean Temperature in Geological Past in Comparison With Present Epoch
Divisions
Temperature differ-
ences, �C
Pliocene Miocene 0ligocene Eocene Paleocene Upper
Cretaceous
3.7 6.9 4.3 9.1 7.5 10.4
The results of these computations are given below.
The represented temperature differences consist of a temperature increase as a re-
sult of changes in the greenhouse effect caused by variations in the C02 concentra-
tion and the temperature increase as a result of changes in albedo of the earth-atmo-
sphere system. In the computations of the second of these values it was assumed that
it is equal to 2.5�C for the Oligocene and earlier time intervals, 2.2�C for the
Miocene and 1.5�C for the Pliocene (the value for the Pliocene is the mean value for
that time interval during which the albedo appreciably changed). These temperature
differences are represented in Fig. 2 in the form of crosses.
The solid curve drawn on the basis of these data agrees well with empirical mater-
ials.
I A similar conclusion can be drawn f rom Fig. 3, which shows the relationship between
~ the temperature differences found on the basis of empirical data (L1TP) and the dif-
i-ereces obtained as a result of the mentioned (Q T'p) computations. Such an agree-
' ment is evidence, in particular, o� the reliability of the V. M. Sinitsyn materials
~ used in this case.
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The greatest discrepancy between the data published by V. M. Sinitsyn and the re-
sults of computations are for the Oligocene. This discrepancy is probably attrib-
utable to the fact that V. M. Sinitsyn, on the basis of the limited data available
at the time he made his investigations (1940's-1950's), could not discriminate re1-
atively brief climatic variations against the background of the general trend of
temperature change at the end of the Mesozoic era and the Tertiary period. In this
connection he did not detect a marked cooling in the Oligocene which could be ob-
s.erved using data on variations in the C02 concentration and which can be clearly
detected on the basis of the newer materials on the thermal regime of the Tertiary
cited below.
Figure 1 shows that the changes in air temperature in the low latitudes are rela-
tively small during climatic variations. Accordingly, in an empirical investigation
of climatic changes it is of great importance to have data on the increase or de-
creas.e in temperature in the middle and high latitudes, where the temperature vari-
ations are great in absolute value and they can be detected more easily.
m
Fig. 5. Change in air temperature (Q T) Fig. 6. Change in precipitation with
with different C02 concentrations. 1) ac- doubling of C02 concentration. MW
cording to data from Axelrod and Baily, data from Manabe and Wetherald, C ~
2) Shackleton and Kennett, 3) Buchart data from Sinitsyn, MC data from
Muratova and Suyetova.
Figure 4 gives data on the temperature differences in comparison with the modern
epoch, computed using materials f rom three investigations: Axelrod and Baily [16]
(computations of paleotemperatures on the basis of palynological data for the most
part for western North America, curve 1), Shackleton and Kennett [30] (isotopic an-
alysis of remnunts of marine organisms in the oceans of the southern hemisphere,
curve 2), Buchart [18] (isotopic analysis of remnants of marine organisms in the
southern part of the North Sea, curve 3).
By comparing the data in Fig. 4 witli the changes in the C02 concentration cited in
'l'able 2, one can note the good qualitative agreement of these independent materials.
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; Thus, in particular, on all three curves one can note a warming cccurring in the Eo-
cene in comparison with the Paleocene. On all the curves there is a marked cooling
~ in the Oligocene, tempprature maximwn in the Miocene, a lesser maximum in the Eo-
! cene, a cooling in the Pliocene in comparison with the Miocene. Some differenced
in the position of the indicated temperature maxima and minima, established on the
basis of materials from different investigations, are attributable to the incomplete
- agreement of the time scales of each investigation, t�he accuracy of which is limited.
In addition to the principal patterns of the thermal regime of the Tertiary, from
the data presented in Fig. 4 it is possible to find more detailed features of cli-
matic variations. For example, on two curves after the temperature maximum in the Mi-
ocene one can note a relatively brief second temperature increase in the Neogene.
Since the time scale of this climatic variation is less than the duration of the
divisions of the Tertiary, it is impossible to compare this change in the thermal
regime with available data on the atmospheric content ot carbon dioxide.
Since data on paleotemperatures, from which Fig. 4 was constructed, in contrast to
the materials of V. M. Sinitsyn, pertain to individual limited regions, their depen=
dence on variations in the C02 concentration is complicated by the influence of a
nunber of additional factors. Nevertheless this dependence is adequately clear, as
is indicated in Fig. 5, where the mean temperature differences, determined for
each division of the Tertiary period from the three curves represented in Fig. 4,
are compared with the C02 concentration.
It follows from the data in this figure that with an appreciable scatter of indi-
vidual points there is a clear dependence between the considered values, which is
described satisfactorily by a line corresponding to the curve in Fig. 2 with an in-
crease in its ordinates by a factor of 1.7.
- Such an increase can be attributed to the fact that the data represented in Fig. 5
relate to the temperate latitudes., where the temperature changes are greater than
their mean global values, represented in Fig. 2.
The good qualitative and quantitative agreement in temperature changes determined
by empirical methods and found from materials on variations in-the C02 conceatra-
tion makes it possible to conclude that there is an adequate reliability of data
on the concentration of carbon dioxide, a satisfactory accuracy of materials on
paleatemperatures and correctness of the evaluations of the sensitivity of the
thermal regime to changes in the quantity of atmospheric C02 used here.
For a more detailed characterization of changes in the thermal regime caused by an
increase in the mass of carbon dioxide it is necessary to construct maps of air
temperature for different C02 concentrations. In studies on the theory of climate
such maps, based on allowance for the real form of the continenta and oceans, fur
the time being have not been constructed. In an earlier investigation [9] the V. M.
Sinitsyn air temperature maps., relating to the Early-Middle Pliocene, were used for
this purpose. It was assumed that these maps correspond to the condition of an in-
_ crease in the C02 concentration by a factor of approximately 2.
A detailed investigation of the studies of V. M. Sinitsyn shows that in the geo-
chronolog"ical scale which he used the Pliocene was more prolonged and also takes in
the latter part of the Miocene in the modern geochronological scale. Accordingly,
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- his climatic mape f.or the F.ar1y-Middle Pliocene relate to a higher level of C02
concentration falling in the interval of its increase by a factor of 2-4 in com-
parison with modern conditions. Evidently, the data of V. M. Sinitsyn for the
Late Pliocene, which corresponds to the chronology of the Pliocene in its modern
definition, are closest to a double increase in C02 concentration.
We note that the use of paleoclimatic data for reconstructiqn of the meteorolog-
ical regime in the case of a doubled C02 concentration involves apecial difficulties
because in the Pliocene there was a relatively rapid change in the albedo of the
earth's surface. Therefore, using data on climates of the past it is easier to
study the climatic conditions for higher COZ concentrations when the albedo of
the earth's surface was more constant. In particular, it is worth noting the pos-
- sibility of using for this purpose data for the Miocene, which correspond to an
increase in the concentration of carbon dioxide by a factor of approximately 4
in comparison with the present-day level.
A comparison of new paleoclimatic maps for the Miocene and their comparison with
similar maps constructed using models of general circulation of the atmosphere
can have great importance for clarifying the irifluence of the C02 concentration
on climate.
Precipitation. It follows from simple physical considerations that with an in-
crease in the C02 concentration the quanCity of precipitation falling on the
earth's surface increases. In particular, it was established long ago that evap-
oration from the surface of water bodies and from.the surface of the land under
. conditions of adequate moistening is proportional to the radiation balance value
and in the first approximation is equal to the value of this balance, divided by
_ the latent hearc of vaporization [1].
Since with an increase in the concentration of carbon dioxide rhe radiation balance
of the earth's surface increases, there is a corresponding increase in total evap-
oration and the sum of precipitation equal to it for the earth as a whole. This de-
pendence can be investigated approximately on the basis of use of the heat balance
equation for the earth's surface [7].
However, it is evident that in individual regions of the earth with an increase
in the C02 concentration the quantity of precipitation can both increase and de-
crease in accordance with changes in atmospheric circulation and other factora.
For practical purposes data on variations of the precipitation regime on the contin-
ents are most necessary.
In computations of the precipitation regime, as well as in determination of ctianges
in air temperature for conditions of a considerable increase in the quantity of at-
mosptieric carbon dioxide, it is possible to use two methods: detailed models of the
tlieory of climate and paleoclimatic data relating to epochs with a higher C02 con-
tent i;: atmospheric air.
Figure ti shows the results of determination of the differences in the mean latitud-
inal annual s wns of p'recipitation falling on the continents of the northern hemi-
sphere for a doubled C02 concentration and precipitation sums with a pre-induatrial
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level of the quantity of carbon dioxide. The MW curve is based on materials of
the latest investigation of Manabe and Wetherald [28]. The curve C was constructed
using the data published by V. M. Sinitsyn for Eurasia, which relate, according to
his nomenclature, to the Late Pliocene. The curve MC corresponds to the mean liitii.-
tudinal values found using the research materials of M. V. Muratova and I. A. Su-
yetova, who on the basis of paleobotanic data computed the mean annual sums of pre-
cipitation in ten regions of the temperate latitudes of Eurasia and North America
for the Middle Pliocene.
Figure 6 shows that the results of al.l three investigationa are in satisfactory
~ agreement. They all indicate an increase in the precipitation sums in the higher
~ latitudes and a decrease in precipitation ia the lower latitudes.
i
~ The great absolute values of the considered difference warrant attention. In most
cases this difference is 10-30% or more of the quantity of falling precipitation.
f
The difference between the curves in Fig. 6 can be attributed in part to a nonco-
j incidence of the regions to which the corresponding data pertain, and in part to
; the errors of each of the methods for determining the values of precipitation
i
, change.
It can be assumed that the agreement between these curves is adequate for deter-
mining the general pattern of change of the precipitation sums in the middle lati-
tudes of the continents with an increase in the C02 concentration. However, for
many practical purposes it is necessary to have more precise and detailed infor-
mation on the dependence of the precipitation regime pn the concentration of car-
bon dioxide, obtaining which is a task of iuture investigations.
The principal conclusion from the materials presented here is that it is now gos-
sible to study the influence of the carbon dioxide concentration on climate by
! two independent methods: theoretical and empirical.
-i The satisfactory agreement of the results of application of these methods for de-
termining the change in temperature and precipitation with an increase in the C02
~ concentration indicates a correspondence between theae evaluations and the condi-
~ tions of real climate.
The authors express appreciation to I. I. Borzenkov'a, V. A. Zubakov, M. V. Muratova
and I. A. Suyetova, who assisted considerably in carrying out Chis work.
BIIILIOGRAPHY
l. Budyko, M. I., KLIMAT I ZHIZN' (Climate and Life), Lpningrad, Gidrometeoizdat,
1971.
2. Budyko, M. I., IZMENENIYE KLIMATA (Climatic Change), Leningrad, Gidrometeoizdat,
1974.
3. Budyko, M. I., GLOBAL'NAYA EKOLOGIYA (Global Ecology), Moscow, Mysl', 1977.
`i
4. Budyko, M. I., "Investigation of Modern Climatic Changes," METEOROLOGIYA I
GIDROLOGIYA (Meteorology and Hydrology), No 11, 1977.
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5. Budyko, M. I., PROBLEMA UGLEKISLOGO GAZA (The Carbon Dioxide Problem),. Len-
ingrad, Gidrometeoizdat, 1979.
6. Budyko, M. I.,'KLIMAT V PF.OSHLOM I BUDUSHCHEM (Climate in the Past and Future),
Leningrad, Gidrometeoizdat, 1980.
7. Budyko, M. I., Drozdov, 0. A., "Reasons for Changes in the Moisture Gycle,"
VODNYYE RESURSY (Water Resources), No 6, 1976.
8. Budyko, M. I., Rnnov, A. B., "Evolution of the Atmosphere in the Phanerozoic,"
GEOKHIMIYA (Geochemistry), No 5, 1979.
9. Budyko, M. I., et al., "Impending Climatic Changes," IZV. AN.SSSR: SERIYA GEO-
GRAFICHESKAYA (News of the USSR Academy of Sciences: Geography Series), No 6,
1978.
10. Gerasimov, I. P., "Climates of Past Geological Epochs," METEOROLOGIYA I GIDRO-
LOGIYA, No 7, 1979.
11. Yefimova, N. A., "Influence of Change of Albedo of the Earth's Surface on the
Earth's Thermal Regime," METEOROLOGIYA I GIDROLOGIYA, No 7, 1980.
12. Sinitsyn, V. M., DREVNIYE KLIMATY YEVRAZII. CH. I. PALEOGEN I NEOGEN, 1965,
CH. II. MEZOZOY, 1966 (Ancient Climates of Eurasia, Part I. Paleogene and Neo-
gene, 1965, Part II. Mesazoic, 1966), Leningrad, Izd-vo LGU.
13. Sinitsyn, V. M., WEDENIYE V PALEOKLIMATOLOGIYU (Introduction to Paleoclimat-
ology), Leningrad, Nedra, 1967. ,
14. Arrhenius, S., "On the Influence of the Carbonic Acid in the Air Upon the
Temperature of the Ground," PHILOS. MAGAZ., Vol 41, 1896.
15. Augustsson, T., Ramanathan, V.,."A Radiative-Convective Model Studq of the
C02-Climate Problem," J. ATMOS. SCI., Vol 34, 1977.
16. Axelrod, D. I., Baily, H. P., "Palaeotemperature Analysis of Tertiary Floras,"
PALEOGEOGR. PALEOCLIMAT. PALEOECOL., Vol 6, 1969.
17. Bowen, H. J. M., TRACE ELM4ENTS IN BIOCHEMISTRY, Acad. Preas, N. Y., 1966.
18. Buchart, B., "Oxygen Isotope Paleotemperatures from the Tertiary Period.in
the North Sea," NATURE, Vol 275, 1978.
19. Callender, G. S., "The Artificial Production of Carbon Dioxide and its Influ-
ence on Temperature," QUART. J. ROY. METEOROL. SOC., Vol 64, 1938.
20. CARBON DIOXIDE AND CLIMATE: A SCIENTIFIC ASSESSMENT, National Academy of Sci-
ences, TJashington, D. C., 1979.
- 21. Cess, R. D., "Biosphere-Albedo Feedback and Climate Modeling," J. ATMOS. SCI.,
Vol 35, No 9, 1978.
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22. Chamberlin, T. C., "A Group of Hypotheses Bearing on Climatic Changes," J.
GEOLOGY, Vol 5, 1897.
23. Ellis, J. S., Vonder Haar, T. H., "Zonal Average Farth Radiation Budget Meas-
urements From Satellites for Climate Studies," ATMOS. SCI. PAPER, Vol 240,
Colorado State Univ., 1976.
24. Flohn, H. H., CLIMATE AND ENERGY. A SCENARIO TO THE 21st CENTURY PROBLEM.
CLIMATIC CHANGE, Vol 1, 1977.
25. Manabe,'S., Stouffer, R. J., "A C02 Climate Sensitivity St.udy With a Mathe-
matical Model of the Global Climate," NATURE, Vol 282 [year not given].
26. Manabe, S., Wetherald, R. T., "Thermal Equilibrium of the Atmosphere With a
Given Distribution of Relative Humidity," J. ATMOS. SCI., Vol 24, 1967.
27. Manabe, S.; Wetherald, R. T., "The Effects of Doubling the C02 Concentration
an the Climate of a General Circulation Model," J. ATMOS. SCI., Vol 32, 1975.
28, Manabe, S., Wetherald, R. T., "On the Horizontal Distribution of Climate
Change Resulting From an Increase in C02 Content of the Atmosphere," J. ATMOS.
SCI., Vol 37, 1980.
29. Ramanathan, V., Lian, M. S., Cess, R. D., "Increased Atmospheric C02: Zonal
. and Seasonal Estimates of the Effect on the Radiation Energy Balance and Sur-
face Temperature," J. GEOPHYS. RES., Vol 84, 1979.
30. Shackleton, N. J., Kennett, J. P., "Paleotemperature History of the Cenozoic
and the rnitiation of Antarctic G].aciation: Oxygen and Carbon Isotope Analyses
in DSDr Sites 277, 279, 281," INITIAL REPORTS OF DSDP, Vol 29, 1973.
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UDC 551.509.313
TELESCOPED SCHEME FOR HYDRODYNAMIC SHORT-RANGE WEATHER FORECASTING
Moscow METEOROLOGIYA I GIDROLOGIYA in Russian No 2,.Feb 81 pp 18-27
[Article by V. M. Kadyshnikov, candidate of physical and mathematical sciences,
USSR Hydrometeorological Scientific Research Center, manuscript received 27 May 801
[Text] Abstract: A scheme for short-range forecasting
of ineteorological elements is propoaed on the
basis of telescoping with a one-sided influence.
Some results of its operational testing are pre-
sented. The merits and shortcomings of different
methods for evaluating a hydrndynamic forecast
of press.ure fields. are discusaed.
An important and obvious reserve for increasing the quality of hydrodynamic ahort-
range weather forecasts is a reduction of the apatial (horizontal) steps in numer-
ical schemes. If in this case we do not wish to impose qualitatively new require-
ments on the speed of computers, the extent of the region of the forecast must be
reduced; indeed, even with retention of the number of computation points, the com-
putation tiaie with the use of explicit f inite-difference achemes increases by as
many times as the grid interval decreases. However, a decrease in the dimensions
of the region makes it necessary to desist from physically inadequate boundary con-
ditions at its lateral boundaries. In actuality, in the case of an ordinary re-.
gional forecast for a time of 24-36 houss for a region measuring about 20 x 20
points with a distance of 300 km between them it is poesible, for example, to con-
sider the boundary values of the meteorological.elements to be constant with.time.
The errors arising as a result of this (under the condition of absence of computa-
tional instability associated with incorrect stipulation of.the boundary conditions),
manifestPd in presence of rapid reflected waves of a great amplitude propagating
within the computation region with the veloci�ty, of synoptic formations, are not
_ ref lect.ed in the quality of the forecast in some internal region. With the same
number 6f points, but with a decrease in the distance between them by a factor of
2-3.the internal region, free af such errors, virtually disappears. This can be
avoided o�nly by stipulating physically adequate conditions on the boundaries, that
is, if at some boundary point using the computation algorithm it is necessary to
stipulate some function, it must be assigned a true value. But aince such are un-
known, they must first be found by solving first the problem of forecasting for a
larger region which includes the boundary points of our region with a small inter-
val as internal boundary conditions. This can be a regional or hemispherical fore-
cast. The resulting forecast in the fine grid can in turn be regarded as auxiliary,
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giving the boundary conditiona for computing the forecast in a atill finer grid,
being internal relative to that just uaed.
Such a problem of the successive precomputation of ineteorological elemeats with the
use oi embedded grids with decreasing intervals with the use as boundary conditions
of the values of the elements already computed using the coarser grid is called the
telescoping of the forecast. This idea and the term itself for the first time were
proposed in [21].
If reference is to the telescoping of a barotropic forecast, the problem of specif-
ically what functions must be stipulated at what boundary points (a11 the functions
are known to us at each boundary point from the forecast for a large territory) is
solved simply. For the first time it was investigated by Charny [18], who at the
points of inflow stipulated normal velocity and vorticity, and at the pointa of
outflow only the normal velocity. The correctness of the corresponding problem
follows [16] from the theory of characteristics. In a one-dimensional case there are
other formulations of the problem of a barotropic forecast for a limited territory.
In a two-dimensional problem this matter was studied in detail in [4].. However,
not all the theoretically possible formulat3ons are satisfactory on a practical
basis. In [7], for example, it was shown that the replacement of normal velocity
in the Charny problem by divergence can lead to negative reaults.
The situation is different in a haroclinic case. Tt was demonstrated in [6] that
depending on the vertical velocity profile at a boundary point, that is, on the
totality of its values at the levels of breakdown of the atmosphere into computa-
tion layers, at this point it is poasible to stipulate definite combinations of
values of the sought-for functions at different levels, determining the others
from solution of the equations (in this case, in particular, the entire profile of
any function cannot be stipulated at~the outflow points). But the problem of deter-
mining the correspoitding combinations is too unwieldy: at each point in each time
interval it is necessary to solve the-:gu1.1 problem of the eigenvalues far a matrix�
whose order is proportional to the number of levels in the model; its e:Lements are

dependent on the coefficients of; t t~:~and the normal velocity values at
different levels. A simtlar conclusion or a continuous model it is impossible
to stipulate certain meteorological elements (as a function of altitude) at the
points of inflow and certain others at the points of outflow was drawn still
earlier in [15].
It should not be thought that since computations for the internal region differ
only with respect to the dimensions of the grid used, solutions for both problems
(for the external and internal regions) virtually coincide at comnon points, so
Chat at all boundary points it is possible tc stipulate all functions. This is im-
possible even under the condition that the initial fields in the small grid are
obtained by a simple interpolation of the corresponding fields with the coarse
grid. In addition, one of the purposes of telescoping is allowance, already in the
initial data, for more detailed information on meteorological fields, including
microscale information; then, very frequently in the telescoping procedures the
scheme includes physical factors not at all taken into account in forecasts in a
coarser grid or taken into account less completely; finally, the forecast for the
large area, used as the auxiliary in the stipulation of physically adequate boun-
dary conditions for forecasting for the smaller, internal region, can be prepared
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s
tising a scheme differing considerably in computational respects (number of levels,
altitudinal boundary conditions, finite-difference algorithm, etc.) from the
scheme for making forecasts for the small territory.
Thus, it is impossible to stipulate everything on the boundary of the small grid
Uecause the data potentially intended for this purpose are not solutions of the
problem in the fine grid. At the same time it is virtually impossible to deter-
mine specifically what should be stipulated.
Accordingly, one can understand the interest in the literature [20, 21, 24, 25, 27,
281 with respect to the problem of "interaction" of the values of the meteorolog-
ical elements at the boundary points of the fine grid obtained from computations
made with the coarse grid with the values within the fine grid. In a number of
these studies the problem in a barotropic case is also formulated. And although,
as indicated above, in a barotropic scheme it is possible to get by without a
precise formulation of the boundary conditions, such investigations are considered
useful in the working out of empirical rules, which are possibly suited for a baro-
clinic case as well.
And nevertheless, apparently, all the practical results in this field are very
closely related to specific finite-difference schemes for the integration of dif-
ferential equations.
We will employ r.he Lax-Wendroff scheme [23], for the first time used in solving the
barotropic forecasting equations in [13, 221. A detailed description of the scheme
can be found in [13]. It can also be used directly in an integration of baroclinic
prognostic equations [14]. However, we will apply it only to "plane" nonstationary
equations derived already after vertical discretization of the equations. It is
lcnown that the Lax-Wendroff scheme has great merits [14]: being explicit, it is
stable and if it distinguishes disturbances, only in the most insignificant degree,
which enables it, in particular, to describe well cases with large gradients. This
is attributable to the fact, as demonstrated in the linear variant in [12], that
the difference system of equations, having a higher order than the differential
system, precisely describes the propagation of these same waves, 7.ike the differen-
tial sysCem,plus some additional, rapidly attenuating waves.
It goes without saying that all these properties appear only under the condition
that the vertical discretization of equations was selected in such a way that the
forecasting problem for the derived system of plane equations is correct, that is,
the solution exists uniquely and is continuously dependent on the input data. In
the literature proper attention has not yet been devoted to this problem: for solu-
tion of the baroclinic problem use is made of inethods whose merits were investigat-
ed in the example of the equation of advection, or in the best case, a system
of barotropic equations. Moreover, a quasilinear system of evolutionary equations
in first-degree partial derivatives, which is obtained after vertical discretiza-
tion and exclusion of some functions from it by means of diagnostic expressions,
may not be hyperbolic, that is, by means of canonization it will be impossible
to reduce it to a system consisting of individual systems of "barotropic" equa-
tions with different velocities of "gravitational" waves. Since the problem of fore-
casting for such a system of equations is not correct, there is no method by which
its solution is possible.
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' If the discretization of equationa has already been selected on the basis of one
consideration or another, it is exceedingly difficult to check whether the de-
! rived system of first-degree equations is hyperbolic, even with a fixed wind pro-
; file and temperature stratification: it is necessary to be convinced that in ths
~ already mentioned matrix (reference is to the formulation of boundary conditions)
~ all tfie eigenvalues are real and that it has a full set of eigenvectors. I't is all
~ the more impossible to check the selected discrett.zation from the point of view
~ of carrectness of the corresponding problem in the entire phase space of solutions,
t that is, for any profiles of the mentioned meteorological elements. It was demon-
~ strated in [8] how it is possible to formulate such discretizations which will
' knowingly yield a positive result. We will use precisely such a discretization
~ without pretense to the conservation of energy or any other integral characteris-
i tics present in an undiscretized system of equations because in our opinion their
i retention ie not of great practical value: integral properties by no means guaran-
~ tee differential properties, which in essence are the purpose of numerical model-
The initial system of equations is written in the form [9].
ut-+ uuX+vrt, -}-(o u.=-(bz+ lzr-{-KuVu'+vY,
z'r + uvx -I- vvy W TIC (Dy - tu ~ Kv Yu2 v2 ,
- R T = (N (E =1 n C),
T, -I-u7'x-I-vTr=I'w rr_RT(Bt--T
~ ~
- (llz py ) = mcr
(1)
(2)
(3)
(4)
(5)
where u, v, W are components of the velocity vectox along the x, y. C axes, is
geopotential, T is temperature, is the Coriolis parameter, R is the gas con-
stant, g is the acceleration of free falling, y and ya are the vertical and dry
adiabatic vertical temperature gradients, K is the drag coefficient, different
from zero only at the underlying surface (it was assumed that
K IS = 1= 1.71�10-4m 1/TIt= 1) .
The boundary conditions for 9 are
c0=0
with I = 0, (6)
(Pe + u(Dx + vO y = c' w-I- uLx + vLy W= Rr with 1, (7)
wtiere L is the geopotential of the underlying surface above sea level. The sense
uf tiie lower boundary condition (bearing in mind that it was written for I= 1) is
that we take the slope of relief into account, but not its elevation.
The vertical structure of a 5-level atmospheric model which we will use in a fin-
ite-difference solution of the problem�(1)-(7) ia as follows: at the maiii levels
300, 500, 700, 850 and 1000 mb (their numbers are 1, 3, 5, 7, 9) the geopotential
and horizontal velocity components are determined; at the intermediate levels 400,
600, 775 and 925 mb (their numbers are 2, 4, 6, 8) the vertical velocity and tem-
perature are determined. In addition, with Z= 1 in accordance with the boundary
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condition we also determine the vertical velocity. It is not necessary to deter-
mine TI 1. Otherwise ther.e would be 5 T values. After discretization of the
model in accordance with (:S) they would be expressed through 5(1> values using a
denerate matrix (otherwise all ~ could be expressed through T, that is, throiigh
C", which is impossible). Accordingly, T~~ = 1 is a linear combination of the al-
ready introduced four T values.
The convective derivatives in the equations of uoDtion (1), (2) are frequently
omitted in a forecast. In actuaZity they are usually small. Although we wi11 not
neglect the corresponding terms, taking into account their role in frontal zones,
we will carry out the theoretical analysis of vertical breakdown in the text which
follows withaut them. In the analysis we will also assume that the parameters and
c2 dependent on the sought-for functions are always positive..
Guided by.the deriyatives of I in the hydrostatic equation (3) and the continuity
equation (5), we will write the equations of motion (1) and (2) and the continuity
equation (5) at the main levels and the hydrostatic equation (3) and heat influx
equation (4) at intermediate levels. In accordance with [8] we will assume that
- RTs =(0z 1+1 -0s 1-1 )/(b i+i - cx i-i (1= 1, 2, 3, 4), (g)
'

- Dz 1-1 _ (WS i - Ws )I (~z , - r ~s i-z ) (D ~ uX + vy, i = i, 2, , ~
- D, = (ws - (ue)A'G's - CO�
(9)
To these nine equations it is necessary to add equations (1) and (2), written at
the levels 1, 3, 5, 7, 9, equation (4) written at the levels 2, 4, 6, 8, and also
the boundary conditione (6), (7). A system of 25 equations is derived for the 25
initial functions.
In [8] a model with such a vertical structure was not analyzed in detail from the
point of view of correctness of the forecasting problem for the corresponding
system of plane equations; only the final result was cited. The writing of the hy-
drostatic equation in a logarithmic coordinate system was also not proposed. Ac-
cordingly, we will now check to see if the formulated problem is actually correct.
Our purpose is to demonstrate that the evolutionary (soluble relat ve to the deriv-
atives of t) system of plane equations which is derived, if 4~1, 3, ~5 and C~7
are excluded by means of (8), and if (9), with (6) taken into account, is used
to exclude cJ2, `J4, w6, wg and W 9, that is, a system of 15 equations,, is hyper-
bolic. Since the hyperbolicity of our two-dimensional system of equations and the
corresponding one-dimensional system exiat simultaneously, we can limit ourselves
to a one-dimensional case and equation (2) can be excluded from consideration [8].
In addition, in epuations (1 ,(4) and (7), from which it is now necessary to ex-
clude all W and C2 , except q, we omit all the nnndifferential terms, bearing in
mind that the type of system of first-degree differential equations is determined
liy the matrix of coefficients with derivatives of x.
Ttius, we derive a system of ten evolutionary one-dimensional equations (the prime
denotes differentiation for x)
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F'UR l)FM'It'IAl. lhh: l)Nl.l'
du. ,
or ~ uu's 1-i - R j: b: k Ts ,r - 4)9 (j = 1� 1,3, 4, 5),
R-1
/
da1.' ur r- T=azx_~

5
aa y u~9 - CZ
Ir QZ R-1 u2 A-1'
4=1
b.' k = E1 Rt 1 - t21F-1. a2 k-1 = Cy k - C2 R-R �
where
(10)
Its hyperbolicity is equivalent to the correctness of the Cauchy problem for it
and accordingly to the forecasting problem if the boundary conditions are proper-
ly formulated.
This system of equations is hyperbolic at any point in the phase space of solutions
if in the matrix of coef ficients on the right-hand side with any values of the cor-
responding parameters all the eigenvalues are real and it has a full set of eigen-
vectors. It might be added that this cannot be checked directly. But it follows
from the definition of hyperbolieity that such a system can be canonized (at each
point in its special way), introducing new fk functions auah that for each of
them the following equation will apply
.kf + kt, ?k, = 0 (k - 1, 2, . . . , 10),
and aZl Ak are real.
It therefore follows that if a problem periodic with respect to x is formulated,
at each point in phase space there is a positively determined quadratic form whose
integral does not change with time. It was demonstrated in [8] that if, on the
other hand, a system of type .(10) is not hyperbolic, no positively determined
quadratic form is retained in it. Thus, to demonstrate the hyperbolicity of sys-
tem (10) means to indicate for it some positively determined quadratic form re- '
maining in the periodic problem.
We will multiply the equations (10) (respectively) by . �
( aul_ ( l= t, 2, 3, 4, 51; ( R bT) (l 1' 3' 4)' �
si
We add them and integrate the sum for the periodicity region. At any fixed point
in the phase space of solutions, that is, with "frozen" values of the coefficients,
the integral of the advective derivatives becomes equal to zero. The remaining
terms on the right-hand side with an accuracy to the factors have the form fx50
(f and T are the sought-for functions). It is easy to confirm that the tertn f X
corresponds accurately to each such term with the same coefficient. Accordingly,
the entire right-hand side disappears after integration. At the left we obtain
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. s ' ~2
J ~ ~ iau2 ,~.,-i + ~ (-'j~ r! ~ + dx.
21
Thus, for system (10) we found a positively determined quadratic form whose inte-
gral is retained. It is easy to understand its sense: beneath tlie integral is the
difference analogue of energy for a simplified (without convective derivatives,
mountains and friction) syatem of equations before its vertical diacretization
under the coadition of "freezing" of the coefficients:
o ~uz+ t c~~ )2 d=~.. f r r
1 S g1 C=1
o -00
This follows from (8) and (9) if it is required that the Newton-Leibnitz theorem
on the detertnined integral be satisfied. �
We note that the result obtained in no way is related to.the specific number of
levels in the model. It is a corollary of the discretization formulas (8),.(9).
We also note that the considered "energy" has no relationship to the true energy
of the problem (1)-(7), also examined.without allowance for convective deriva-
tives, mountains and friction. In order that there be a quadratic integral for
the corresponding nonlinear problem, it is necessary to take the convective de-
rivatives into account, but at the same time it is necessary to replace the boun-
dary condition (7) by the condition 0 and consider the 4r parameter to be
constant with height. For example, an energy conservation law is derived in [10].
4Je note that the velocity values in the advective derivatives remained arbitrary,
that is, they can be selected in any way, using only approximation considerations
as a point of departure. The convective derivatives in equationa (1), (2) were
also not determined. With respect to vertical velocity, it can be assumed, for
example, that w 2 1_1 = 1/2(w 2 1-2 '1(J2 i) (i m 1, 2, 3, 4), and the derivatives
of g with i= 1, 5 are replaced by one-sided differences, but with i= 2, 3, 4--
by central differences.
As already mentioned, we will solve the system of plane equations by the Lax-Wen-
droff inethod. It goes without saying that it need not be reduced to the form (10),
which we require for demonstrating corxectness. We will take equations (1), (2)
and (4) directly and supplemen�t them in each time interval by the expressions (8)
and (9).
The telescoping of the forecast was accompliahed in the following way. First for
a territory measuring 20 x 24 points with a horizontal interval 300 km (taking in
approximately Europe; ttie corresponding L(x,y) was taken from [19]) and a time
interval of 12 minutes a forecast is computed 24 hours in advance. In this case
the time-constant tendencies of geopotential at.the lateral rows of points and
the geostrophic wind at them, computed using data on geopotential in the two 1at-
eral rows, are stipulated from the known geopotential values at the final time of
the forecast. The initial wind is stipulated geostrophic. Thus, informattion is re-
quired on geopotential at the initial moment in time everywhere and in the two
lateral rowa after a day. The "matching" of the stipulated bourtdary values with the
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! MOR OHF'ICIA1. IItiM; qNI.Y
i
i
; �
computed internal values was accomplished as in [20]: in each time interval the
values of all the "evolutionary" variables at the points of the first internal
row are corrected using the formula i
j fa = Z f: -F 4(ft + fa)
(point 1 is a boundary point, reckoning is inward from the boundary along the nor-
rual).
; During the computations each four hours the values of all the meteorological ele-
. ments at points which are boundary points for the internal region subject to tele-
; scoping (it is situated at the center of the first region) are stored. Upon com-
~ pletion of the computations a forecast begins for the internal region for this
~ same time. The number of points in it is the same, but the interval in horizontal
coordinates and time is half as great. In such new computations the tendencies at
the boundaries are "piecewise-constant" in time: they are changed each 4 hours.
; The initial and boundary values in the deneer grid of points are obtained from the
~ corresponding.values in the coarse grid by simple linear interpolation. The match-
ing is accomplished the same as before. .
~ By way of trial runs we computed 12 examples with the use of the actual (including
; the future values at the boundaries) initial data. The results were good. Without
, discussing tfiese in detail, we will examine a table in whose upper part there are
' some results yielded by the scheme in comparative tests under operational condi-
tions in July-November 1979 carried out by the test laboxatory of the USSR Hydro-
meteorological Center. Ob3ective analysis [1] was used, and the results of 36-hour
forecasts in accordance with the scheme in [3] were taken as the future values at
the boundaries of the first region with a 300-km interval.
i
i Unfortunately, the results are not entirely comparable: E, R and S1 for the tele-
scoped scheme were computed using 37 points, whereas fo r the other schemes 50
i points were used, including these 37; in addi.tion, S1 for these three achemes was
~ determined using a greater ninnber of cases 67.
~ With respect to the relative error F_ the telescoped scheme is appreciably poorer
than the others. In our opinion, however, an exaggerated importance is attached
to this evaluation, highly dependent on the quality of the forecast of the back-
� ground, which is of little importance. For the weatherman-forecaster the gradients
of the isohypses (wind) and the position of the pressure centers (the vertical
gradients of pressure fields-temperature should also be evaluated) are far more
, important. S1 is widely (and frequently by itself) used abroad as a statistical
evaluation of a forecast of horizontal gradients. We note in passing that the other evaluations used abroad are not dependent on the background. The data 3:n
this part of the table show that S1 is related most closely to the synoptic eval-
uation Q r(the schemes are arranged in the order of an increase in 6r; what
place they would occupy with reapect to E,, R and S1 is shown; the N value char-
acterizes the degree of difference of the corresponding arrangement of places from
the initial place): the arrangement of the schemes with respect to S1 and 6r
! is identical. '
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Distribution of Pr.ognostic Schemes by Places in Dependence on the Quality Criterion
(Schemes: T-- Telescoped, SHD Synoptic-Hydrodynamic [11], S-- Synoptic Op�ra-
tional Forecasts, R-- Regional Variant of Scheme [3], SIB Computation Center Of
the Siberian Department of the USSR Academy of Sciences [5], AMER United Stiates
National Meteorological Center [26], Q-- Quasigeostrophic [21)
bJr e R S, % of predicted
Scheme I'new formations
For 33 forecasts of surface pressure for 24 hours in July-
November with 44 pressure centers
T .
SHD
220
276
6
0,85 4 3
0,68 2 0
0,68 4 3
0,76 2 0
49, t 1 0
50,8 2 0
58
47
S
293
309
17
16
0,66 1 2
0,74 3 l
0,81 1 2
0,72 3 l
51,7 3 0
51,8 4 0
53
37
R
i iv
6
6 o I
For 34 forecasts of surface pressure
for 24 hours
in March-May�
1976 with
40 pressure centers
S
292
322
I 30 I
0,75 3
93 5
0
2 I 0,72
69
30
3
4
2
2
49,3
52,9
2
4
! I
2
59
50
SIB
~
R
324
2
,
0,64 I
2
,
0,80
1
2
45,6
l
2
59
41
g~
I 330
3ti~~
I 35 I 0,70 2
85 4
0
2 I
1
0,73
0,65
2
5
2
0
50,0
54,8
3
5
I I
0
59
n
~
.
I
l
8~
6~
1
N
10
Same, but
for
forecasts for 36 ho
urs (the
quasigeostrophic
scheme was not
370
evaluated)
12 4
1
68
3 0
2�
1 58,7
2
1
55
gIg
~R
398
405
2~
,
0,65 1
83 2
0
1
1
,
0,71
60
0
l
4
1
I
51
62,2
1
3
1
0
61
57
g~
~
433
28
,
0,88 3
1
,
0,62
3
1
62,7
4
0
71
N1 gl
Notations. r-- error (in km) in forecast of position of pressure centers, bdr
difference in L r between adjacent schemes; the first number in the F_ , R, S1
coliunns denotes the relative error, correlation coefficient and,gradient error re-
spectively (their determination was given in [17]), the second number is the place for the corresponding index, the third number is the deviation from the initial
place with respect to 6r; N is the sum of the deviations.
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' In order to confirm the presence of a relationship between these two.highly im-
~ portant characteristics, in this same table we giv:! data obtained using mater-
ialg f rom other tests [17]. Although the distribution of schemes with respect Co
~.1 a r and S1 now does not coincide (a precise coincidence in all cases, to be sure,
is impossible: S1 characterizes the structure of the entire field, and A r charac-
`tErizes only the position of singularities), the N value shows that here they are
closest to one another. From this point of view the telescoped scheme yielded fair resulte. It is consid-
erably superior to other schemes with respect to Ar. The synoptic scheme is in-
' ferior to it even with respect to the prediction of new formations (as can be
seen from the middle and lower parts of the table, in the tests of 1976 forecasts
using other schemes did not surpass it in this respect).
We note that despite the prevailing opinion that.the forecasts of the United States
National Meteorological Center are the best with respect to all indices, in ac-
tuality this is not the case. The acheme developed by the Computation Center of
the Siberian Department USSR Academy of Sciences gives better results with re-
spect to the highly important synoptic characteristic Ar. Whereas in a forecast
for 24 hours it surpasses the two subsequent schemes insignificantly, evtdently
within the limits of accuracy of the evaluations (and with respect to S1 is ap-
preciably inferior to Chem), in a forecast for 36 hours it is superior to all the
schemes with a greater interval, although, to be sure, it occupies second place
w3th respect to S1. T1-.e poor F, value for this scheme is attributable simply to an
uns.uccessful forecast of the background.

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BIBLIOGRAPHY
, 1. Bagrov, A. N.,*"Operational Scfieme for tfie Ofi3ective Analys3s of Aerological
Information for the Northern Hemispfiere," TRUDY GIDROHETTSENTRA SSSR (Trans-
actions of tfie USSR 'Hydrometeorological Center), No 196, 1978.
2. Belousov, S. L., MNOGOUROVENNY'YE KVAZIGEOSTROFICHESKIYE MODELI PROGNOZA: LEK-
TSII PO CHISLENNYM KRATKOSROCHNYM PROGNOZAM POGODY (Mnlttlevel Quasigeostrophic
Forecasting Models.: Lectures on Numerical Short-Range Weatfier Forecasting),
Leningrad, Gidrometeoizdat, 1969.
3. Belousov, S. L., et al., "Qperational Model for the Numerical Forecasting of
Meteorological Elements in the Idorthern Hemisphere," TRUDY GIDROMETTSENTRA
SSSR, No 212, 1978.
4. Gordin, V. A., "Mixed Boundary-Value Prohlem for a Darotropic Model of the
Atmosphere," TRUDY GIDROMETTSENTRA SSSR, No 196, 1978.
5. Dymnikov, V. P., IContarev, G. R., Guseva, N. V., Kolotovkich, I. V., Kulinych,
A. G., Gazetova, N. P., Shemetova, G. V., Kaminskaya, L. Ye., Torbina, Z. V.,
"Prediction of tieteorological Elements in a Limited Territory Using Full Equa-
,tions," METEOROLOGIYA I GIDROLOGIYA (Meteorology and Hydrology), No 9, 1975.
6. Kadyslinikov, V. M., "Boundary Conditions: in tlie Problem of Short-Range Weather
Forecasting Using a Baroclinic Model of the Atmosphere," IZV. AN SSSR: FIZIKA
ATMOSFERY I OKEANA (News of the USSR Academy of Sciences:, Physics of the Atmo-
sphere and Ocean), No 1, 1973.
7. Kadyshnikov, V. M., "Formulation of Bouudary-Value Problems for the Equations
of a Barotropic Atmosphere in a Restricted Region," IZV. AN SSSR: FIZIKA ATMO-
SFERY I OKEANA, No 3, 1977.
8. Kadyshnikov, V. M., "Altitudinal Difference Differentiation and.Correctness of
the Weather Forecasting Problem," IZV. AN SSSR: FIZIKA ATMOSFERY I OKZANA, No
6, 1980. 1
9. Kibel', I. A., WEDENIYE V GIDRODINAMICHESKIYE METODY KRATKOSROCHNOGO PROGNOZA
POGODY (Introduction into Hydrodynamic Methods for Short-Range Weather Fore-
casting), Moscow, Gostekhizdat, 1957.
10. Marchuk, G. I., CHISLENNYYE METODY V PROGNOZE POGODY (Numerical Methods in
Weather Forecasting), Leningrad, Gidrometeoizdat, 1967.
11. Mertsalov, A. N., "Numerical Synoptic-Hydrodynamic Forecasts of the Surface
Pressure Field With an Advance Time of 24 Hours Using a Simplified Scheme,"
- TRUDY GIDROMETTSENTRA SSSR, No 129, 1974. 12. Pekelis, Ye. M., "On Solution of the Cauchy Problem in Finite Differences,"
TRUDY GIDROMETTSENTRA SSSR, No 151, 1974.
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13. Pressman, D. Ya., "One Difference Scheme for Short-Range Weather Forecast-
ing," TRUDY GIDROMETTSENTRA SSSR, No 6, 1965.
14. Pressman, D. Ya., "Difference Scheme for Short-Range Weather Forecasting Using
Full Equations (Spatial Problem)," TRUDY GIDROMTTSENTRA SSSR, No 14, 1966.
15. Pressman, D. Ya., "Formulation of Boundary Conditions in Horizontal Coordin-
ates in the Problem of Short-Range Weather Forecasting in Full Equations for
a Region Across Whose Boundary Flow Occurs," IZV. AN SSSR: FIZIKA ATMOSFERY I
OKEANA, No 9, 1969.
16. Rozhdestvenskiy, B. L., Yanenko, N. N., SISTEMY KVAZILINEYNYKH URAVNENIY
(Systems of Quasilinear Equations), Moscow, Nauka, 1968.
17. Ugryumov, A. I., Chernova, V. F., Ageyeva, A. K., Bukreyeva, L. A., "Compar-
ative Evaluation of Regional Schemes for Numerical Prediction of the Pressure
Field for 24 and 36 Hours," INFORMATSIONNYY SBOItNIK GIDROMETTSENTRA SSSR (In-
formative Collection of Articles of the USSR Hydrometeorological Center), No
6, 1978.
18. Charni, Dzh., "Integration of Primitive Equations and Balance Equations,"
TRUDY TOKIYSKOGO SWOZIUMA PO CHISLENNYM METODAM PROGNOZA POGODY (Transac-
tions of the Tokio Symposium on Numerical Weather Forecasting)(1960), Lenin-
grad, Gidrometeoizdat, 1967.
19., Berkovsky, L., Bertoni, R. A., "Mean Topographic Charts for the Entire Earth,"
BULL. AMER. METEOROL. SOC., Vol 36, No 7, 1955.
20. Chen, J. H., Miyakoda, K., "A Nested Grid Computation for the Barotropic Free
Surface Atmosphere," MON. WEATHER REV., No 2, 1974.
21. Hill, G. E., "Grid Telescoping in Numerical Weather Prediction," J. APPL.
METEOROL., No l, 1968.
22. Houghton, D., Kasahara, A., Washington, W., "Long-Term Integration of the
Barotropic Equations by the Lax-Wendroff Method," MON. WEATHER REV., No 3,
1966.
I 23. Lax, P. D., Wendroff, B., "Systems of Conservation Lawa," COMMtJNICATIONS ON
' PURE AND APPL. MATH., Vol 13, 1960. .
~ 24. Miyakoda, K., Rosati, A., "One-Way Nested Grid Models: the Interface Condi-
tions and the Numerical Accuracy," MON. LJEATHER RLV., No 9, 1977.
~ 25. Shapiro, M. A., 0'Brien, J. J., "Boundary Conditions for Fine-Mesh Litnited
i Area Forecasts," J. APPL. METEOROL., No 3, 1970.
i
' 26. Shuman, F. G., Hovermale, J. B., "An Operational Six-Layer Primitive Equation
~ Model," J. APPL. METEOROL., No 4, 1968. 27. Wang, H., Halpern, P., "Experiments With a Regional Fine-Mesh Prediction Mod-
el," J. APPL. METEdROL., No 4, 1970.
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28. Williamson, D., Browaing, G., "Formulation of the Lateral Boundary Condi-
tions for the NCAR Limited Area Model," J. APPL. METEOROL.,'No 1, 1974.
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UDC 551.509.(314+323+335)(215-17)
PREDICTION OF MEAN rIONTHLY AIR TEtIPERATURE FIELDS OVER THE NORTHERN HEMISPHERE
USING AN AUTOMATED GROUP ANALOGUE SCHEME
14oscow METEOROLOGIYA I GIDROLOGIYA in Russian. No 2, Feb 81 pp 28-39
[Article by G. V. Gruza, professor, L. K. Kleshchenko, candidate of geographical
sciences and E. Ya. Ran'kova, candidate of physical and mathematical sciences,
All-Union Scientific Research Inatitute of Hydrometeorological Information-l�7orld
Data Center, manuscript received 17 Jun 80] ,
[Text] Abstract: The article describes an automated scheme
for adaptive atatistical forecasting making use of
the group analogues method. The authors describe
the results of its application in numerical exper-
iments for the long-range prediction of air temper-
ature over the northern hemisphere. The authors
discuss the form of representation of forecasts,
including probabilistic, and give some evaluations
of the success of experimental forecasts based on
operational data.
Long-range weather forecasting ia one of the principal aspects of modern meteorol-
ogy [13], being of great importance for the national economy. Scientists in dif=
ferent fields of specialization are working on this problem, including thoae bas-
ing their work on the similarity principle [1, 16, 17). Since with the passage'of
time the technical possibilities of automatic numerical schemes are broadening
and the volume of archival data is increasing, the analogues method is still re-
taining its timeliness. Objective methods for the selection of analogues are be-
ing developed to an increasing degree, are being based on the methods of mathe-
matical statistics and are being applied with the use of electronic computers [3].
One of the realizations of such an approach is the GRAN (group analogues) approach,
described in the publications [4, S]. The scheme developed for the M-222 computer
is completely automated and standardized, as a result of which it can be used, in
~ particular, not only in a forecasting regime, but also in a diagnosis regime, for
example, in the meteorological interpretation of numerical (hydrodynamic) fore-
~ casts [9]. The scheme has found extensive application in solution of problems of
a research nature. For example, on its basis it has been possible to obtain evalu-
~ ations of p redictability (within the framework of the analogues method) of some
' meteorological features [6, 10]; the prognostic information yield of individual
~ predictors and their systems has been evaluated [8]; studies were made of a number
of problems related to optimization of parameters of the scheme [7]. As a prognos-
i tic system the GRAN scheme is also suitable for practical forecasting.
1 29
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Without repeating the detailed description of the scheme [4], we will only men-
tion that the forecast is constructed here as a statistical description of a
sample (group) of analogues, making it possible to formulate the forecast in a
categorical and in a prognostic form. The sample of analogues is formed from ar-
chival data on the basis of the stipulated criterion of aimilarity of the char-
acteristics of atmospheric processes designated at the input into the scheme. It
goes without saying that the choice of the indicated criterion, like the content
of the information on processes which is taken into account, to a considerable
degree predetermine the success in solution of the formulated problem.
= It should be noted that this scheme in all stages of preparation of the forecast
affords the researcher information on the prognostic significance of the predic-
tors used [11]. It is important that such information was provided for when us-
ing the scheme not only in a retrospective regime (for example, evaluations of
the success of experimental forecasts on the basis of archival material), but also
on a real time scale [12]. In obtaining and analyzing this sort of evaluations use
was made of one of the problems in the experiments described below, directed to
the development of a method for the long-range forecasting of air temperature
over the northern hemisphere.
Since a decisive role in the formation of the large-scale fields of ineteorological
elements is played by therroal and circulatory factors, in the investigation use was
made of data on the hemispherical fields of air temperature Tp and pressure Pp at
sea level (1891-1976), and also on the fields of geopotential H500 at the 500-mb
level (1949-1976). It should be especially emphaeized that this information does not seem to us to be
adequate for successful solution of the forecasting problem, but only reflects our
real possibilities at this stage in the research. '
The initial information on the meteorological fields was represented in such a way
as to ensure the possibility of a quantitative comparison of the processes (includ-
ing with respect to their inttasity) at different spatial-temporal scales. For this
purpose we carried out averaging (denoted below by the operator E) of data along
the circles of latitude of the Atlantic-European sector 40�W-100�E (conventional
notation E a, AES) and over the area of large regions of the northern hemisphere
(ER, NH). Such averaging was carried out for the Tp and PO fields stipulated by the
values at the points of intersection of a regular grid, but also the fields of the
zonal ((-A ) and mer.idional (r'y) components of their horizontal gradients. In ad-
dition, a study was made of the generalized parameters, to wit: the Ye. N. Blinova
circulatory indices at the ground level (IP) and at the 500-mb level (IH), the char-
acteristies of the high-level frontal zones in the northern (FZn) and southern (FZS)
hemispheres, computed on an electronic computer on the basis of H500 diurnal data
12]. .
An analysis of the information content of the characteristics of the temperature
and circulation regimes for the forecasting of temperature was accompliahed in
the process of two experiments (we will call them "experiment 1" and "experiment 2"
respectively). In experiment 1 we used surface data on air temperature and pressure
at sea level since 1891. Experiment 2 was broadened by the inclusi,on of data on
30
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i H500, but the duration of the analyzed period was only 28 years,(1949-1976). In
i accordance with the real possibilities (in particular, the speed of the M-222
, electronic computer and computer time availability) for numerical experiments, a
' limited set of systems to be taken into account was stipulated on an a priori
basis (Table 1).
! .
The table shows that among the predictors we included the diurnal and mean month-
ly data (the latter are indicated in the table by the upper horizontal line over
the designation of the element). The diurnal data were introduced for describing
the initial conditions (state of current atmospheric processes at the ground level
i and at the 500-mb level on the last day of the compared months). In addition, the
i diurnal data in the form of 30-day segments of time series of circulation indices
j (IH, IP) and temperature at Moscow (TM), in our opinion, must characterize the ten-
1 dency in the development of processes in the considered month (in the table they
; are supplied with the subscript "d").
All the data are represented with a time interval of 5 days (72 observations for
each year). In the case of inean monthly data use was made of 30-day moving means.
For temperature (due to the lack of diurnal data for the northern hemisphere for
a long period) such means were ob.tained by linear interpolation of the mean month-
ly values.
! The values of the parameters of the scheme (maximum number of analogues used,
~ "threshold" of similarity, type of weighting function), which to a considerable de-
i gree determine the effectiveness of its use, were selected as a result of several
~ specially undertaken studies of�a methodological character [11]. With respect to
the similarity indices, for the considered meteorological features we made use of
Euclidean distance [8], since the experiments included only vectors with uniform
' components (horizontal field or segment of a time series of one meteorological el-
i ement).
~ The testing period included the last 10 years (1967-1976). Zn experiment 1 the ana-
~ logues were selected from the remaining part of the archives (1891-1966), whereas
in experiment 2, due to the substantially lesser volume-of observations in the ar-
' chives the analogues in each case were selected from the entire series (1949-1976)
' with the exception of the tested year. In accordance with the local stationarity
i
hypothesis [7], the analogues in the archives were selected with displacement in
the limits of a month from calendar dates corresponding to the initial situation.
This made it possible to increase the volume of the accessible observations and im-
, prove the quality of the analogues by taking into account processes occurring in
, adjacent calendar periods.
The experimental forecasts were compared for the winter and summer seasons. Ac-
cording to evaluations made earlier [8, 9], allowance for the long-term prehistory
of development of atmospheric processes (within the framework of the information
used) exerts an insignificant influence on the success of the forecasts. There-
fore, in this investigation the choice of analogues was made on the basis of the
~ characteristics of only one initial month (the data cutoff time was the last day
of this month). As such a month for winter we chose December, and for summer
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June. The prediction was made with zero (in winter, for January, and in aummer,
for July) and one-month (for Fehruary and August respectively) advance times.
The prediction method was tested for three different territories: the extratropical
~ zone of the northern hemisphere (72 points of a regular geographic grid in the
zone 35-70�N), the lowland territory of the USSR (26 points of grid intersection)
and seven administrative regions of the USSP. [12]. A prediction was made for the
normalized air temperature anomalies, the deviations fram the mean long-term val-
ues, expressed in fractions of the standard deviation d; the mean and d were com-
puted for ttie 30-year period (1931-1960) recommended in [18]. The total number of
tested variants was 120 and the total number of evaluated forecasts was 1,080
(600 for winter and 480 for sur.uner). Success evaluations were made for each fore-
cast and as a whole for the entire period of the tests both for individual compon-
ents of the predictants and as an average for the predicted vector.
It should be emphasized that no preliminary sorting-out of the predictors was
made and therefore all the evaluations can be cansidered as obtained in an inde-
pendent sample. Evaluations of the quality.of the probabilistic forecasts are com-
puted outside the GRAN scheme [14, 15] and will not be considered here.
It was found that the mean success of the methodological forecasts for different
makeup of the information us.ed was very stable. In Table 2, for some systems of
predictors, we give one of the evaluations of quality of the categorical forecasts
the mean square error S(in the prediction of the normalized anomalies it was
expressed in fractions of O' and was adequate at each field point to the relative
error and as an average for the field to the widely used criterion Q employed
by long-range forecasters [1]). As a comparison, here we have also given the cor-
responding evaluations of climatic and random forecasts, as well as evaluations
of the "optimum" analogues, which are selected a posteriori directly prior to the
observed state of the forecasted phenomenon and characterize the "special predict-
ability" of the temperature fields considered in this case. The methods for obtain-
ing them are described in greater detail in [12]. The table shows that the errors
in the methodological forecasts, averaged for the test period, for all the predic-
tants are less than the random errors, do not attain the levels of "optimum" errors
and as a whole are comparaUle to the evaluations of climatic forecasts.
The geogranhical distribution of evaluations of the quality of experimental fore-
casts for tlte territory of the northern hemiaphere can be judged from maps of val-
ues of the mean square error S and the success in forecasting the sign of the
p anomalies (Fig. 1). Here we have defined (shading) regions of "successful" fore-
casts, corresponding to the arbitrary criteria j < 1 and J>> 0. The figure includes
the results tor tlie system of predictors X1 (experiment 1) for a forecast for a
month in advance fo;c two seasons.
Similar maps for the system of predictors X4 and a zero advance time reveal a def-
inite similarity in the geographical distribution of regions of "successful" fore-
casts, which can�be regarded as indirect evidence of their stability. In this case
the main territory of the Soviet Union does not fall in these regions and this is
evidently attributable to the relatively great difficulty in making a forecast for
,these regions.
32
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MOIt OFMI('IA1, IISI: QNLY
~
I:D

N
/
.
;
y
0
0
,
0
/
~
JO
60
Fig. 1. Region of successful forecasts of inean monthly sir temperature over north-
ern hemisphere (35-70�N). a) winter (February); b) summer (August). 1) P= 0; 2)
b= i; s) p>o, J< i.
Table 1
Group of Predictors for Numerical Experiments for Predicting Air Temperature
Mete
al i
tion
Air
tu
Circ,
gri
Circi
501
3rologic-
File �
xperiment 1
Expe im n 2 �
~
V
iforma-
cha
(1891-1966)
97b)
(~y4
.
ra
teris-
~ ~
~ o
terri
_
0
X1I
X2
X3I
X4
X5I
X6I
X7
X8I
X9
tic
b~
tory
'-empera-
ER1'
'
27
NH
+
,
1
y
+
+
+
+
+
re
E" ra
r
14
AI.: S
+
E" ('y T
14
AE$
-
-F
_
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TMJ
'o
Mosc
+
I1xfii.oa a
ERP
27
NH
-
+
+

>und leve
E" f', P
I I
AF.S -
l

.
Ex r_ P
ii.
AEs
~

I P,r
::0
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T
+
t
ilation a
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27
NH
+
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F X I I
27
NH

03,
:6
36
~I
T
y
,
+
1 H,l
30
Nn
I
'
y
-
~
C = N; ~0a $ '
ln evaluating the results of these numerical experiments the conclusion can be
clrawn that the group analogues method when tlhere is an informative system of pre-
dictors has a definite advantage in comparison with a climatic and especially a
random forecast. Accordingly, there is a need for further investigations for
seek�Lng and ttnalyzing the characteristics of atmospheric processes which must be
taken into account in the evaluation of similarity, that is, for improvement in
the system of predictors. Nevertheless, taking into account the real readiness
of the GRAN scheme for use in routine work and for the purpose of accumulating
prognostic experience, since 1979 attempts have been made to prepare forecaste on
the basis of group analogues in a routine regime.
33
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Fl)It OFM1('IAt. iISh' ()N1.1'
P ~
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Forecasting time
Fig. 2. Success of experimental categorical forecasts of sign of anomaly of inean
monthly air temperature over northern hemisphere. 1) evalustions of forecasts bas-
ed on routine data with month advance period (complex of predictors RO), 2) eval-
uations of individual predictability. Since the effectiveness of different systems of predictors (within the frameworkof the used information) is approximately identical, for oFerational tests we
selected the simplest variants in which the mean monthly amoothed fields (for 27
regions in the northern hemisphere) are used as initial information: only for sir
temperature (XO) and in combination with the surface pxessure field (X1' in cotr-
trast to system X1, where the Pp field was used for the last day of the month).
As the predictants we also included the field of normalized air temperature anom-
alies, smoothed over an area of 42 approximately equal-area squares in the zone 30=
80�N. The evaluations of success of forecasts obtained for this variant for the
main test period (1967-1976) are given in Table 3. We note that at the present
time the preparation of forecasts by the.proposed method under operational condi-
tions is technically feasible only for a month in advance. Accordingly, evalua-
tions of forecasts for a zero advance time are not given in this case.
The table shows that the mean evaluat3ons for the test period differ aubstantially
from the results in Table 2. We note that in individual, evidently extremal years
the relative errors E, of both methodological and climatic forecasts increase
sharply, indicating an increased difficulty in predicting such processes. Judg-
ing from the cited evaluations, some preference must be given to the system XO
in predictions for the territory of the northern hemisphere. Figure 2 shows the
success of the forecasts using this system prepared on the basis of operational
data for 1979.
A comparison of these evaluations with the evaluations for "optiminn." analogues in-
dicates that in the second half-year the forecasts to all intents and purposes at-
tained a success "ceiling" which was the maximian possible within the framework of
the existing archives. It must be remembered that analogues to the predictant were
not selected using the P criterion, but instead the Euclidean distance S , so that
in the sense of the evaluation P curve 2 cannot be optimum, which is indicated,
in particular, by the results for June 1979.
The content of the prognostic information issued using the GRAN scheme is shown in
Table 4 in the example of a forecast of the norntalized air temperature anomalies
for seven regions of the USSR (a predictant of minimum dimensionality was select-
ed). This information cantains a climatic forecast (evaluations of "unconditional
climatology," obtained for the entire period from which the selection of
35
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38
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Tig. 3. (see top of next page)
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Fig. 3. (see preceding page) Probabilistic forecast of normalized air temperature
anomalies over the northern hemisphere based on three equiprobable gradations
(forecast for January 1980 for a month in advance using the set of predictors X1').
analogues is allowed, in this case-1891-1976) and an adaptive forecast on the ba-
sis of a sample of analogues (in this case with use of the system of predictors
X1').
These two parts of the forecast are similar in content and are represented in
terms of a full detailed statistical description of the corresponding samples
through the statistical characteriatics and distribution functions. The latter are
given in the form of quantiles for stipulated probability levels and probability
densities for the introduced gradients of the predictants. Here a categorical an-
alysis corresponds to the one column "Mean" in the adaptive forecast.
Since probabilistic forecasts for the time being are not yet generally accepted, it
is of interest to examine possible forms of their graphic representation. Figure 3
shows two variants of graphic representation of a probabilistic forecast of the
distribution of normalized air temperature anomalies for the northern hemisphere.
The basis for this table was a forecast for three equiprobable gradations, which
can be interpreted as the gradations "below the norm," "near the norm" and "above
the norm" respectively. The limits of the gradations were determined individually
for each predictant as 33.3% quantiles of the unconditional distribution (with
five gradations these are 20% quantiles, as, for example, in Table 4).
The histograms (Fig. 3a) show the precomputed probability distributions of the con-
sidered gradations at each field point. The numerical values corresponding to them
are given in Fig. 3b. It is also useful to remember that for a climatic forecast
all the probabilities would be equal to 1/3 and the histograms would have the
form of a rectangle with the height 1/3. Figure 3b also shows regions where the
predicted probability of the gradations "below the norm" and "above the norm" is
greater than the climatic.probability. We note that there are regions where the
formed sample of analogues indicates an increased probability of occurrence of
anomalies of both signs (corresponding to the area of crosa-hatching). It is un-
derstandable that here the uncertainty of the categorical forecast (which would
give a value close to the norm) is substantially higher. This can be attributed to
the inadequately high quality of the selected analogues (in particular, due to the
incomplete description of the initial state). It is possible that the forecast for
these regions requires additional refinement (for example, by using analogues with
a different system of predictors).
We emphasize in conclusion that these results reflect only the current level of use
of the GRAN scheme for long-range forecasting. Further prospects may be afforded
by the discrimination of regions responsible for the formation of weather condi-
tions in specific regions and with the formation of group analogues with "regional"
systems of predictors. As a result, a forecast for global territories (northern
hemisphere or the USSR) will be formed by combining regional forecasts. Clearly
such an approach involves solution of optimization problems in each current situa-
tion, which is possible with sufficiently high-capacity computers. In addition,
later, in seeking analogues, allowance must be made for data on the underlying sur-
face (ocean temperature and ice content, snow cover, etc.), atmosphere-ocean inter-
action, global cloud cover, etc.
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BIBLIOGRAPHY
1. Bagrov, N. A., Vasyukov, K. A., Zverev, N. I., Ped', D. A., "The Similarity
Principle and its Use in Practical Work," TRUDY TsIP (Transactions of the
Central Institute of Forecasts), No 132, 1964. 2. Glyz', G. A., "Analysis of Position of the Planetary High-Altitude Frontal
Zone," TRUDY VNIIGMI-MTsD (Transactions of the A11-Union Scientific Research
Institute of Hydrometeorological Information-World Data Center), No 58, 1979.
3. Gruza, G. V., Soldatkina, A. M., "Principles for Developing a Method for Pre-
dicting Weather by the Analogue Method," TRUDY SANIGMI (Transactions of the
Central Asian Scientific Research Hydrometeorological Institute), No 29(44),
1967.
4. Gruza, G. V., Ran'kova, E. Ya., Esterle, G. R., "Scheme for Adaptive Statis-
tical Forecasting Using a Group of Analogues," TRUDY VNIIGMI-MTsD, No 13,
1976.
5. Gruza, G. V., Ran'kova, E. Ya., "Variant of a Scheme for Selecting and Eval-
uating Group Analogues Using an M-222 Electronic Computer," TRUDY VNIIGMI-
MTsD, No 35, 1977.
6. Gruza, G. V., Ran'kova, E. Ya., "Evaluation of the Difference of Some Meteor-
ological Ob3ects and Their Predictability by the Analogues Method," TRUDY
VNIIGMI-MTsD, No 53, 1977.
7. Gruza, G. V., Ran'kova, E. Ya., "Statistical Forecasting Using A Group of
Analogues," PRIMENENIYE STATISTICHESKIKH METODOV V METEOROLOGII. TRUDY II
VSESOYUZNOGO SIMPOZIUMA PO PRIMENENIYU STATISTICHESKIKH METODOV V METEOR-
OLOGII (Use of Statistical Methods in Meteorology. Transactions of the Sec-
ond All-Union Symposium on the Application of Statistical Methods in Meteor-
ology), Leningrad, Gidrometeoizdat, ly;?.
e
8. Gruza, G. V., Ran'kova, E. Ya., "Method for Statistical Weather Forecasting
on the Basis of Dynamic Climatology," PRIMENEIdIYE STATISTICHESKIKH METODOV
V METEOROLOGII. TRUDY III VSESOYUZNOGO SIMPOZIUMA PO PRIMENENIYU STATIS-
TICHESKIKH METODOV V METEOROLOGII (Use of Statistical Methods in Meteorol-
ogy. Transactions of the Third All-Union Symposium on the Application of
Statistical Methods in Meteorology), Leningrad, Gidrometeoizdat, 1978.
9. Cruza, G. V., Kleshchenko, L. K., Ran'kova, E. Ya., "A Method for the Meteor-
ological Interpretation of Numerical Long-Range Forecasting," TRUDY VNIIGMI-
MTsD, No 58, 1979.
10. Gruza, G. V., Ran'kova, E. Ya., "Long-Range Meteorological Forecasts With the
Use of a Group of Analogues and Evaluation of the Predictability of Meteor-
ological Processes," TRUDY VNIIGMI-MTsD, No 77, 1980.
41
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_ 11. Gruza, G. V., Kleshchenko, L. K., Ran'kova, E. Ya., "Results of Numerical
Experiments in Prediction of Mean Monthly Air Temperature Fields Over the
Northern Hemisphere hy the Group Analogues Method," TRUDY VNIIGMI-MTsD, No
77, 1980.
12. Gruza, G. V., Ran'kova, E. Ya., "Use of Analogues for Evaluating Predict-
ability and Long-Range Prediction of the Mean Monthly Air Temperature
Field," METEOROLOGIYA I GIDROLOGIYA (Meteorology and Hydrology) (in press).
13. Monin, A. S., PROGNOZ POGODY KAK ZADACHA FIZIKI (Weather Forecasting as a
Problem in Physics), Moscow, Nauka, 1970.
14. Radyukhin, V. T., "Checking the Hypothesis of Randomness of Forecasts Given
in Stochastic Form," TRUDY VNIIGMI-MTsD, No 77, 1980.
15. Radyukhin, V. T., "Comparison of Effectiveneas of Some Quality Criteria of
Stochastic Forecasts," TRUDY VNIIGMI-MTsD, No 77, 1980.
16. RUKOVODSTVO PO MESYACHNYM PROGNOZAM POGODY (Manual on Monthly Weather Fore-
casts), Leningrad, Gidrometeoizdat, 1972.
17. Sonechkin, D. M., "Formulation of the Problem of Dynamic-Stochastic Weather
Forecasting 'by Analoguea'," TRUDY GIDROMETTSENI'RA SSSR (Transactions of the
USSR Hydrometeorological Center), No 181, 1976. ,
18. CLIMATIC NORMALS (CLINO) FOR CLIMATE AND CLIMATE SHIP STATIONS FOR THE
PERIOD 1931-1960, WMd/ONM, No 117, TP 52, 1962.
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UDC 551.576.11
MODEL OF CL01JDi COVER ON A STATIONARY FRONT
Moscow METEOROLOGIYA I GIDROLOGIYA in Rusaian No 2, Feb 81 pp 40-48
[Article by Yu. L. Matveyev and S. A. Soldatenko, Leningrad flydrometeorological
Institute, manuscript received 20 Jun 801
i [Text] Abstract: Using numerical methods applied on an
i electronic computer it was possible to formulate
~ a model of cloud'cover formation on a stationary
~ front. An evaluation was made of the influence of
; the temperature difference between warm and cold
; air, relative humidity of warm air, depth of the
' trough in which the front is situated, on the
~ �characteristics of frontal cloud cover: the alti-
; tude of its boundaries, horizontal extent, vertical
liquid-water content profile, quantity of precipit-
ation falling from a cloud, etc. It is shown that
' the model explains the principal features of the
i distribution of temperature with altitude and hor-
Iizontally in frontal zones (in particular, the for-
; mation of a frontal temperature inversion).
It follows from earlier investigations [2, 4, 7] that the processea of redistrib-
ution of heat and moisture, as well as the formation of cloud cover, are influ-
enced to the greatest degree by vertical movements, turbulent exchaage and advec-
tive receipts of heat and moisture. As is well known, fronts are situated in
troughs and therefore a convergence of air currents generated by the ascending
movement of air, caused by turbulent friction, is assaciated with them. This, in
turn, is accompanied by f rontal cloud cover. The role of turbulence is great and
Renerally recognized in the atmospheric boundary layer. However, in frontal zonea
under the influence of a horizontal temperature gradient there is a substantial
wind shear and the generation of the energy of turbulent fluctuations is observed
not only in the boundary layer, but also in the entire troposphere. However, if it
is taken into account that the cloud formation process also favors an intensifica-
tion of turbulent exchange (according to the experimental data in [8], the tur-
Uulence coefficient in clouds is greater than outside them and has an.order of
magnitude of several tens of ineters per second), it becomes obvious that turbulence
must be taken into account in formulating models of frontal cloud cover within the
limits of the entire troposphere.
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Although during recent decades less attention has been devoted to the study of
fronts than during the first decades after the discovery of this phenomenon,
nevertheleas studies [1, 3, 6, 9-21] have been made along these lines. Theee
liave developed and deepened our concepts concerning the mechanisms of formation
and structuring of frontal zones and the cloud s;azems associated with them.
This article is a development of [1, 2, 9]. In this article numerical methods are
used in modeling the steady fields of frontal cloud cover, vertical movements and
temperature under the influence of advective and turbulent influxes of heat, mois-
ture and momentum, as well as the phase traneitions of water vapor and the heat
of condensation. In contrast to the mentioned studies, we do not stipulate the
characteristics of turbulent exchange, these being extracted from the equation
for the balance of turbulent energy and expressions established in similarity
theory. In order to determine the wind velocity components use was also made of
more complete equations (with conservation of inertial terms) than ia studies made
up to the preaent time.
Initial Equations and Expressions
Assuming that4all the meteorological parameters and other eharacteristics do not
change in the direction parallel to the front (plane problem) and in time (sta-
tionary front), we write the initial system of equations in the following form:
(1)
rt dX -r- 2a~ ds = d k ds '
_
(2)
II = T ;a z + Lqlcp,
(3)
as
rt d
as a k as i
+ w d: d - p
aQ*
W '
(4)
S_Q+a,
.
11 d~ u~
d w: - vx) + d k
vs ' .
(5)
u d" w
- dx
2 ar(u uA)  �a o k 8u~i '
ds
(6)
dpu dom n
+ _ '
. .
(7)
dx
� .
db du
u 3X +~r~~ a
d db cG2
a: k o: - k
l
ds
k
g a e - ~(8)
s " a, O,G,, _ as J.
+ (I'-v)'
� � (9)
,
as - e as +
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tiere u, v, w are the projecCions of wind velocity onto the x-, y- and z-axes (the
x-axis is directed along the normal to the front, in the direction of the cold
air and the z-axis is directed vertically upward); T is air temperature; Yoc is
the dry adiabatic gradient; L, cP are the specific heat of condensation and heat
capacity of air; Qk is the flow of water droplets and ice crystals under the influ-
ence of gravity; q,a' are specific humidity of air and the liquid-water content of
the cloud; s is the specific moisture content of cloud air; ug, vg are the compon-
ents of the geostrophic wind along the x- and y-axes; 2 uJZ is the Coriolis para-
meter; P is air density; b, k are energy and the turbulence coefficient; g is the
acceleration of free falling; 8 is potential temperature; 26- 0.40, abO 0.73, c
N 0.046 are constants.
The equations for the inflow of heat (1) and moisture transfer (3) were written
taking into account [7, 81. Ttie form of equation (3) indicates that in ad-
dition to advective, convective and turbulent moisture influxes we have taken in-
to account the falling of droplets under the influence of gravity. The expression
for the flow of water droplets and ice cryatals has the form [8]
N '
Qk ~ - o (s - 9m) 'U~ (10)
where v is the mean weighted (hy mass) velocity of falling of cloud elements, qm
is the specific saturation hwnidity.
For v we used an expression cited in [5],
s - =x
[B = upper; H = lower ] y = vm eXp ~ ~ (11)
where vm is the maximum v value, which can be attained at the lower boundary
of the cloud zloWt zup, is the upper cloud boundary, p is a parameter character-
izing the decrease of 6 with altitude. �
In a cloud system (1)-(9) is supplemented by the expression
4 = 4m = 0,622 � (f�T)
'
(12)
where E(T) is the saturating pressure of water vapor, P is air pressure, which is
computed using the principal equation of statica.
The sought-for functions satisfy the following boundary conditions:
a) at the earth's surface (z ffi 0) ' u= v- as c 0, '
. . (13)
T( x, 0) - l -~2T th pr ,
(14)
f'x, 0) = f-- ~ th f- (15)
where T1, T2 are the air temperatures near the earth's surface in the warm (x-)-
-00) and cold (x-.-,+oo) air masses; fl, f2 are the relative air humidities with
these same values x; DT, Df are parametexs with the dimensionality of length,
characterizing the rate of change in temperature and relative humidity along the
45
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x-axis in the frontal zone,
r~ e T=T.-T:, f- r-' 2 t'
a e0n /1~ 1, = us o = C011St,
ev x
. r.o
n=,l~s o'~ 2 th Dv,
! (16)
(17)
where g(o), vgl0 are the velocity components of the geostrophic wind along the x
and y axes with x.->-- C)O; u(2b, vg24 are the same wind velocity components with x-s
8
+0o ; Dv is a parameter with the dimensionality of length, characterizing the rate
of change of the component vg p; _ v0
+ yr1, yto I Avso vi V(1)
;
=
b) at the level of the tropopause z= z op
T (x, z,, J"- _T (z.,) =const, (18) q (x, z�) = q (z ) = const, (19)
(20)
b(x, z.)=6(za� ) =0,
u (x, z.) ~ LiIz. ) = ug o, . (21)
~ r(s"~
v V. Za ~ _ yB (x, zw ) = vg o (x) T- +
g ( z� ) s" l d T '
+ T
dz;
1 W_ (22)
u
c) at a sufficiently great (theoretically with x-y -00) distance from the �ront in
a warm air mass
T(-oo, z) =T (z )=T t- i t z,
9(-oo, z)=q.x(z)=fi9, (Z),
(23)
(24)
where Y1, fl is the vertical temperature gradient and the relative hwnidity in
the warm air.
The expressions (14), (15) and (17) for a change in temperature, relative humidity
and the component of the geostrophic wind near the earth's surface in dependence
on x are in satisfactory agreement with the experimental data. These expressions
ensure a continuous conversion from the T1, fl and v(lg)p in warm air to the T2,
f2 and v(2) values of the correaponding meteorological parameters in the cold air.
0
The rategof change of the meteorological parameters near a surface front (that is,
near x= 0) is determined by the parameters DT, Df and Dv: the lesser their values,
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the more rapid is the change in the corresponding meteorological parameter in
the frontal zone and the lesser is the width of the frontal zone itself. With
the symbols and notations adopted in (14) and (17) the differences Q T= T~ - T2
atid A vgp = v(~) - v~~ are always positive, since Tl> T2, and v(~)> v~~~ in $
9 9 9 clearly expressed, unmasked trough vg~>> 0 and v~gb < 0). The* trough in which the
front is situated is the deeper the greater the difference Av p. [In accordance
with the adopted terminology the term "front" will be applied �o the line of in-
tersection of a frontal surface with the horizontal plane.]
Results of computations. In carrying out most of the computations we assumed T1 =
20�C, 'y 1= 0.7�C/100 m, ug p= 0.24)z = 1.12�10-4 sec71, the coefficient~8 = 3,
the mean weighted rate of falling of cloud droplets W= 13 cm/sec (such a rate cor-
responds to the modal radius of droplets rm = 3 m), v~~) _-v(24, a roughness par-
ameter zp = 1 cm (this parameter appeara with stipulat$on of tge turbulence coef-
ficient at the ground: ko = azp). .
The rate of vertical movements and the field of liquid-water content of the cloud
' cover forming in the frontal zone are dependent on such parameters as,6T, 6vg 00
f l, Dy.
The results of the computations of vertical velocity at diffsrent distances x
from a surface front and altitudes z with different Q T values are given ia Table
1. According to these data, with fixed x and 6T the vertical velocity increases
; with an increase in z(for example, with x= 100 km and 0 T=3�C from 0.09 cm/sec
I at an altitude of 0.3 km to 2.75 cm/sec at 5 km). In warm air (x < 0) and over a
' surface front (x = 0) vertical velocity is virtually not dependent on A T. With
Q T= 0�C at a fixed altitude the w field is symmetric relative to the front and
j the frontal surface in this case coincides with the vertical plane and at all alti-
' tudes the w maximum is attained with x= 0(at the front). With an increase in
contrast (,8 T) in temperatures the slope of the frontal surface decreases, with an
increase in altitude the front is more and more displaced in the direction of the
cold air and at the same time there is impairment of symmetry of the w f ield rela-
tive to the plane x= 0. For example, with,6T m 5�C the vertical velocity at all
; altitudes with x= 100 km is grgater than with x= -100 km (at an altitude of 1 km
it is equal to 0.27 and 0.54 cm/sec, at an altitude of 3 1m 0.34 and 2.65 cm/
sec respectively). With L1T = 10�C the w field is displaced sti.l more in the direc-
tion of the cold air (because of this the vertical velocity at a distance x= 300
; km with 0 T= 10�C at all altitudes is greater than with Q,T = 5�C). However, also
in the presence of a temperature contrast (d T.> 0) the w maximum is attained over
a surface front (with x= 0). Qua.litatively this is attributable to the fact that
the most substantial convergence of wind velocity is observed in the lower layer
near a surface front (x = 0); the vertical velocity generated by this convergence
exceeds ttie velocity arising under the influertce of wind convergence at higher
levels (where a front with high x values are found). This also explains the weak
, dependence of w on L1 T with x
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For deteimining OG in place of P* and 6TE we substituted the values from [5] and
minimized :he sum of the squares of the deviations of the real value from the pre-
dicted value:
-IIPjTA'=min. ,
i (2)
By equating the derivative dA/da to zero, we find
. a= K-1 T+ P a p/oa T_, '
(3)
Here K is the correlation coefficient between x and y; a'p, 47Q TE are the stan-
dard deviations. The line denotes averaging.
In the first variant of the.camputations the cvefficients of equation (1) were de-
termined directly from data on the energies and areas of hail falls during some
months and the prognoetic equations derived in this way were checked using the
materials for the same month or other periods. The results of the computations are
given in Tables 1 and 2. We should note the rather high correlation between P and
~ TZ . Table 2 gives the results of computations of the predicted area of damaged
crops using the derived prognostic relationships (1).
In general, the use of the prognostic relationships for a teaching sample (for
these same periods) gives a lesser relative mean square error than for control
samples (for other periods). The only exception is the prediction for July using
combined data for May and June. The use of the progaoatic relationship for May in
the prediction for June, July and August leads to a relative mean square error of
- more than 100%.
Table 2 also ~ives the mean values of the abeolute positive and negative errors
Q P} _(P - P)p* > p and the frequenciea of the positive and negative differences
p*< P
P- P*(nk), and also n