ISSN 0097-8078, Water Resources, 2021, Vol. 48, No. 6, pp. 854–863. © Pleiades Publishing, Ltd., 2021.
Russian Text © The Author(s), 2021, published in Vodnye Resursy, 2021, Vol. 48, No. 6, pp. 633–642.
PALEOHYDROLOGY OF THE CASPIAN SEA
Dynamic-Stochastic Modeling of the Paleo-Caspian Sea Long-Term
Level Variations (14–4 Thousand Years BC)
A. V. Frolov*
Water Problems Institute, Russian Academy of Sciences, Moscow, 119333 Russia
*e-mail: anatolyfrolov@yandex.ru
Received April 1, 2021; revised May 28, 2021; accepted May 31, 2021
Abstract—The article considers an estimate of the effect of positive feedback in the mechanism of level oscillations in the Caspian Sea on the long-term level regime. The study was based on a modified dynamic-stochastic model of level variations, taking into account the spatial heterogeneity of evaporation from sea water
area. The evaporation from Caspian shallows is considered as the sum of a deterministic and a stochastic component. The probability density of Caspian level was obtained for supposed paleotime conditions as a solution
of the steady-state Fokker–Planck–Kolmogorov equation. In addition, Caspian level variations were simulated by Monte-Carlo method, the results of which confirmed the analytical calculations. Under some realistic assumptions accepted in the modeling, long-term variations of the Caspian level can have a nonstationary character under steady-state climate. The main result of the study is the conclusion that the nonlinear
dependence of evaporation on the Caspian level is to be taken into account not only in paleoclimatic reconstructions, but also in the estimates of the future sea level regime.
Keywords: Caspian Sea level variations, Fokker–Planck–Kolmogorov equation
DOI: 10.1134/S0097807821060051
INTRODUCTION
Data on lake level regimes, either direct or indirect,
are an important source of data when studying the
problem of climate change at scales from tens to thousands years. The climate conditions, which form water
balance of lakes at relatively long time scales from tens
to thousands years, have their effect on level variations, which can be identified by the composition of
bottom and coastal deposits (the presence of pollen
and shells), geomorphological characteristics of lake
beds, archeological monuments on the coast, and
other characteristics. The physically obvious (though
not always unambiguous) correlation between the climate conditions on lake basin, water balance, and
level variations in lakes can be used to retrodict the climatic characteristics in paleotime.
The dependence of lake water balance on climate
makes it possible to describe the climate conditions in
its basin based on water level and lake surface area.
Possible variants of water balance of Lake Chad and
the dependence of lake water area on its level were
used to evaluate the average long-term precipitation
onto lake basin in paleotime [32]. Studies of level variations in lakes can also be used to evaluate the character of interannual “intraclimatic” and long-term climatic variations, thus contributing to the determining
the cause of the past level variations in lakes [31].
Studying the regularities of level variations in lakes is
used to retrodict the global atmospheric circulation in
paleotime and to forecast lake level regimes in the
future [4, 6, 29, 31, 37–39].
Many researchers of paleoclimate based on data on
level variations in lakes implicitly assumed that the
natural level variations in natural water bodies are
always due to climate changes. When there were signs
indicating to changes in lake water levels, this assumption suggested conclusions regarding climate changes.
The nonstationarity of lake level regimes was assumed
to be directly related with the nonstationarity of climate, a conclusion that in some cases was quite reasonable. However, a question arises: is it possible that
under stationary climate, level variations form a nonstationary process because of the peculiarties of the
mechanism of lake water level variations? In a more
general formulation, it looks reasonable to assess the
contribution to considerable lake level variations not
only due to climate impact, but also the specifics of the
formation mechanism of water body level regime.
PROBLEM FORMULATION
The objective of this study was to assess the possible
contribution of the mechanism of Caspian level variations to the considerable (tens of meters) sea level variations in paleotime. The author tried to answer the
question: were the climate changes of the sea water
balance the only cause of the wide level variations in
854
DYNAMIC-STOCHASTIC MODELING
paleotime, or there existed some other way for the
internal mechanism of sea level regime formation to
contribute to this process? In this study, the regime of
Caspian level variations in the interval 14–4 thousand
years BC was studied.
To achieve this goal, an improved dynamic-stochastic model of long-term level variations of the Caspian Sea was developed. In this model, the evaporation was considered separately for two parts of the sea,
i.e., the shallow part (mostly, the Northern Caspian
Sea) and the deep-water part (the Middle and Southern Caspian Sea). The data on the level regime and the
morphometry of the sea, required to use this model,
were taken from [1, 2, 7, 11, 15–20, 36].
THE FEATURES OF CASPIAN SEA
MORPHOMETRY TAKEN INTO ACCOUNT
IN THE SIMULATION OF ITS LEVEL REGIME
The morphometric characteristics of the Caspian
Sea, including schematic maps of Caspian Sea water
area in paleotime, are given in [8, 11, 17]. The Caspian
Sea has a unique morphometric feature—a considerable share of shallows in the entire water area. In this
study, shallows in the Caspian Sea will be assumed to
be water areas with depth up to 40–50 m; in the deep
part of the sea, such interval accounts for about a half
of the active layer, which is equal to 100 m [18].
According to data in [8], at an elevation of –28.0 m in
the Baltic System (BS), the area of the Northern Caspian Sea, at an average depth of 4.4 m, is ~90 thousand
km2, accounting for 24.3% of the total sea area. The
wind mixing at wind speed >7 m/s extends to a depth
of 10 m, i.e., embraces almost the entire Northern
Caspian. The deep-water part of the sea also has
coastal shallows, the area of which is estimated at
~30 thous. km2. The shallows of the sea also include
Kara-Bogaz-Gol Baty, which, at a sea level of
‒28.0 m BS, has a mean depth of 5–7 m and an area
of ~20 thousand km2.
Under some conditions, e.g., in the absence of positive feedback in the level oscillation mechanism and
at a steady increase in water area, the equilibrium area
is a unique solution.
If the dependence of sea water area is approximated
by a linear function
F (h) = a + bh,
(1)
the mean volume (mathematical expectation) of the
inflow and the evaporation depth are v + and e ,
respectively, then the equilibrium area F* of the water
body becomes:
+
(2)
F * = v = F (h*),
e
where h* is the equilibrium level. For relatively small
ranges of level variations, F ( h*) can be approximated
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855
by a power function of the 2nd–3rd order. The dependence (2) is often used for scenario estimates of the
mean values of the main water balance components of
the Caspian Sea—the total river inflow and effective
evaporation—in
the
form
of
equality
+
v = F (h*)e = F *e , in the cases were evaporation
does not depend on the level (e.g., in [1, 17, 20]).
Figure 1 shows the schematic position of shorelines
during the Late Khvalynian transgression and a space
photograph of a part of its basin.
The comparison of the possible boundaries of the
shoreline during the Late Khvalynian transgression of
the Caspian on the schematic map and the relief of the
sea coast on the space photograph of the sea and its
basin (Figs. 1a, 1b) clearly demonstrates the possible
increase in the area of the shallow Northern Caspian
at an increase of the level from –30 to 0…+10 m BS.
The flat plains of the Caspian Depression, covered
with water, considerably increase the area of shallows—the Northern Caspian increases threefold to
270 thousand km2 (with the incorporation of data
from [8, 11]).
With water level in the sea varying within
‒28.0…‒24.0 m BS, a change in the level by 1 m
causes a change in the total area of deep-water parts of
the Caspian Sea—the Middle and Southern—on the
average by 1.5 thousand km2 [8], which is almost 10
times less than the same characteristic for the Northern Caspian: bNC = 12.5 thous. km2/m. In other words,
the damping of level variations in the drainless sea is
almost completely due to the effect of the variable area
of the Northern Caspian. Therefore, as a first approximation, we can admit that the area of the deep-water
Caspian Sea at level variations above –30.0 m remains
constant; we will include the value of bMSC in bNC, thus
accounting for, though minor, but real damping effect
of the variability of the total area of the Middle and
Southern Caspian on sea level variations. The value of
bNC within the range of interest (–30.0…+10.0 m BS)
was evaluated with the use of the hypsographic relationship given in [17]. In accordance with these data,
bNC = 9.6 ≈ 10 thous. km2/m. Note that the inclusion
of Kara Bogaz Gol in shallows of the sea increases
their area; however, at a level above ~ –26 m BS, it has
almost no effect on the increment of the area of shallows in the entire Caspian Sea accompanying level
rise, because the shore of the bay is abrupt (approximately vertical).
In the coordinate system with zero at –30. 0 m BS,
a linear approximation of the dependence of the total
Caspian area on its level can be written as
F ( h) = Fd + Fs ( h) = ( 410 + 10 h) thous. km 2, (3)
the area of shallows is Fs ( h) = (120 + 10 h) thous. km2,
and the area of deep-water part of the sea is
Fd = const = 290 thous. km2.
856
FROLOV
PR
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Fig. 1. (a) Schematic map of the occurrence of the Late Khvalynian transgression according to [11]; (b) space photograph of the
Caspian Sea. Shorelines at (a): (1) the maximal stage 0–2 m BS, (2) Sartasskaya stage 10–12 m, (3) at elevations of –16…–17 m,
(4) abrasion shores, (5) coastal accumulative forms, (6) Late Khvalynian deltas and coastal-alluvial plains, (7) incised deltas,
(8) riss shores. Photos from the site Instagram:com/p/CAZ5SGtAn56.
Plots in Fig. 2 show the dependence of Caspian
area on its level, based on modern data [17], and a linear approximation of this dependence for the range 0–
40 m (or –30…+10 m BS). Note that the small discrepancy between Caspian area at a level of –30.0 m
BS according to data [17] and its area by (30) is of no
significance for this study, because variations of the
sea level are much higher than this level.
As follows from (1), under this approximation, the
dependences of the area of the entire Caspian on the
level at levels 10 and 20 m in the new coordinate system accepted by the author (–20.0 and –10.0 m BS,
respectively)
are
as
follows:
F(h)
=
2 and F(h) =
(120 + 10 × 10) + 290 = 510 thous. km
(120 + 10 × 20) + 290 = 610 thous. km2, which is close
to the areas of 511.7 and 621.0 thous. km2, respectively,
by [17].
At an increase in the total area of the Caspian Sea,
the role of shallows of the Northern Caspian (along
with other shallows) in the mechanism of level variations increases because of the greater evaporation. At a
level of –10.0 m BS, the area of Caspian shallows is
320 thousand km2, i. e. ~52% of the total sea area.
Such proportion between the total Caspian area and
the area of its shallows is unique among large natural
water bodies and has a considerable effect on the formation of the level regime of the sea.
THE INPUT COMPONENT OF CASPIAN
WATER BUDGET
Studying regularities in long-term water level variations in the drainless Paleocaspian requires the use of
adequate models of total river inflow into the sea, as
well as evaporation and precipitation over its water
area. The assumptions regarding Caspian water budget
in the paleotime, accepted in this study, are based on
the current concepts of the relative role of each budget
component in the formation of sea level variations.
The groundwater inflow into the Caspian can be
Caspian area, thous. кm2
600
400
200
10
20
30
40
Caspian level, m, above the mark –30.0 m BS
Fig. 2. Dependence of Caspian area on its water level: dots
are data of [17], the line is approximation F = 410 + 10h
(thous. km2).
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DYNAMIC-STOCHASTIC MODELING
857
Table 1. Evaporation from Caspian Sea water area [16] and the morphometric characteristics of the sea [7] (at level mark
of –28.0 m BS)
Characteristic
Evaporation, сm/year
Mean depth, m
Area, thous. km2
Northern Caspian
Middle Caspian
Southern Caspian
The sea as a whole
101
4.4
90.1
81
192
137.8
103
345
148.5
101
208
376.3
assumed zero considering its present-day estimate of
as little as 1–1.5% of the total river inflow. The Volga
runoff accounts for 80–85% of the current total river
inflow into the sea; the proportion in the past is
assumed to be about the same [17, 20, 36]. The author
of this article used the representation of Caspian water
balance with the input component as the sum v + of the
volumes of river inflow q and precipitation p onto the
water area, v + = q + p .
The output part in this case is the net physical
evaporation. Such approach allows the dependence of
evaporation on water level in the sea to be taken into
account more correctly.
In this study, Caspian level variations are simulated
for the interval of 14–4 thousand years BC. The reason for such choice is the considerable change of the
equilibrium levels, around which level variations were
taking place in (about) the first and second parts of the
above interval. The relatively stable regime of Caspian
level variations around –10 m BS in the first part of
the period (14–9 thousand year BC) changed into
nearly the same regime but about –20 m BS (9–
4 thousand year BC).
Therefore, in accordance with [1], the mean level
has shown a considerable and almost instant (by
~10 m) drop of the mean level, supposedly because of
climate changes. This study discusses the question:
could such changes in the level took place under
unchanging climate because of the specifics of the formation mechanism of level variations?
Because of the lack of data, we have to assume the
statistical parameters of precipitation onto the Caspian water area in paleotime to be similar to the modern estimates. The current mean precipitation depth is
evaluated at 0.2 m/year [2] with a coefficient of variation of ~0.2; the sea water area in paleotime is assumed
to be ~510 thous. km2 [17]. Therefore, the average volume of precipitation is ~100 km3/year, and the variance of this volume is evaluated at ~400 (km3/year)2.
The mean value and the coefficient of variation CV of
river inflow are taken equal to 410 km3/year and 0.2,
respectively, which is close to the characteristics in [2,
20]. Therefore, the variance of river inflow is q ~
6700 (km3/year)2. The mean annual water inflow
v + = q + p into the Caspian Sea through rivers q and
with precipitation p onto sea water area is evaluated at
~510 km3/year. Assuming that river water inflow and
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precipitation onto the water area are mutually independent, we obtain that the variance σ2v + of the process
v + ~ 7100 (km3/year)2. The assumption regarding the
independence is based on the fact that the areas of
river runoff formation in the Caspian drainage basin
are far from the water area onto which precipitation
falls; in addition, no plausible reasons are known to
explain the physical mechanism of such dependence.
The accepted mean values of the total water inflow
into the Caspian and its water area correspond to an
evaporation layer of ~1.0 m, as follows from somewhat
extended interpretation of the results given in [17].
THE EVAPORATION FROM THE CASPIAN SEA
WATER AREA
The possible dependence between evaporation and
sea level was mentioned in [9]. The effect of sea level
(depth) on evaporation from Caspian water area was
convincingly proved by the results of study [16] given
in Table 1.
The depth of evaporation from the Northern Caspian is 20% greater than that from the Middle Caspian, which lies south from the former, and it is almost
equal to the evaporation depth from the Southern
Caspian.
According to modern data, which practically coincide with data in [16], the long term mean of the depth
of annual evaporation from the entire sea is 0.97 m, the
coefficient of variation can be taken equal to 0.2,
hence the variance is ~0.038 (m/year)2 [2]. Note that
this estimate of the variance of evaporation all over the
water area takes into account the effect of both evaporation components—stochastic and deterministic; in
this case, the stochastic component refers to the entire
Caspian water area, i.e., both to its shallow and deepwater parts.
To construct the model of evaporation from Caspian water area in paleotime, we make the following
assumptions. First, the evaporation from the deepwater part will be assumed to comprise two components, i.e., the main (constant) value and a stochastic
component, which reflects the stochastic character of
variations of evaporation conditions—variations of air
and water temperature, wind speed, etc. It is assumed
also that the evaporation from the shallow part of the
Caspian also contains a stochastic component, which
can be combined with the other component to form a
858
FROLOV
single one, referring to the entire water area. In such
case, the evaporation depth for the entire water area
can be written as
e (h)Fs (h) + Fd e
e(h, t ) = det
+ estoch (t ),
F (h)
Evaporation depth edet(h), m/year
1.1
1.0
(4)
0.9
where estoch (t ) is a stochastic component; edet ( h) is the
evaporation, depending on the level (depth) of the
shallow part of the Caspian (a deterministic component); e is the depth of evaporation from the deepwater part of the sea; F (h) and Fs ( h) are the dependences of the entire water area and the area of shallows
on water level in the sea, respectively; Fd = const is
the surface area of the deep-water sea, assumed constant; t is the time, years.
0.8
The evaporation e from the deep-water part of the
Caspian is taken equal to the modern estimate of the
mean evaporation from the entire water area [16] with
rounding: e ∼ 1.0 m/year. It is natural to take the stochastic component equal to zero, and its variance was
assumed equal to ~5.6 × 10–3 (m/year)2, considering
that the variance of evaporation from the entire sea
water area, derived by the authors’ model should be
close to the appropriate modern estimate. Note that
edet ( h) is a function of a random variable, i.e., level h;
therefore, it is also a random variable.
The dependence of evaporation on sea level
(depth) edet was chosen based on the following.
First, this function is positive, steadily decreasing,
maximal at small sea depths and minimal and large
depths.
Second, it is desirable that the function edet ( h) has
a form making it possible to obtain an analytical solution of the Fokker–Planck–Kolmogorov equation.
For this, the nonlinear relationship
edet ( h) = −marctan [n ( h − C )] + D,
(5)
was used, where m = 0.147 m/year, n = 0.4 m–1, C =
14.65 m, and D = 0.84 m/year are numerical coefficients (Fig. 3).
At low water level, the northern boundary of the
Northern Caspian water area shifts southward, the depth
of evaporation from the shallows with a depth of up to
~10 m reaches a considerable value of 1.04 m/year,
which is close to the average evaporation from the Southern Caspian—1.03 m/year (Table 1). The evaporation
from shallows with depths >20 m decreases to
0.63 m/year. At the accepted relationship (5), the gradients of the decrease of evaporation with increasing depth
of the Northern Caspian are maximal in the elevation
interval of 10–20 m (–20…–10 m BS).
0.7
0
8
10
15
20
25
30
Caspian level, m, above –30.0 m BS
Fig. 3. Dependence of the evaporation depth edet ( h) from
the shallow part of the Caspian on sea depth (level).
SIMULATING LONG-TERM LEVEL
VARIATIONS IN THE CASPIAN
AS A DYNAMIC-STOCHASTIC SYSTEM
Such approach to the simulation of level variations
in the drainless Caspian was first proposed by
S.N. Kritskii and M.F. Menkel [9] and later developed
by them in several studies. The further progress in the
notions of Caspian level variations as an output process of a dynamic-stochastic system was made by studies of S.V. Muzylev [14, 15], V.E. Prival’skii [25],
M.G. Khublaryan and V.I. Naidenov [26], V.N. Malinin [12, 13], and A.V. Frolov [22–24, 27]. In other
countries, the level regime of drainless lakes was studied with the use of the principles of dynamic-stochastic modeling in [28, 33, 35] and others. The level
regime of lakes in paleotime are commonly based on
deterministic models in the form of water balance
equations, not containing a stochastic component
([31, 34] etc.).
The present-day ideas regarding the mechanism of
variations of Caspian Sea level are presented in [23,
24]. Such variations are simulated with the use of
methods of nonequilibrium statistical mechanics ([3,
5, 7, 14, 21, 25] etc.).
Depending on the range of variations of the Caspian level, two main types of dynamic-stochastic
models can be used, differing in the feedbacks considered in the mechanism of sea level variations. At level
variations below ~–30 m BS (implying no seawater
outflow into Kara-Bogaz-Gol bay and the almost
complete disappearance of the shallow Northern Caspian), the only active feedback is a negative one, determined by the dependence F(h) of the surface area of
the sea on its level. The number of feedbacks is maximal within the range of level variations –31…–26 m
BS: two negative feedbacks are due to the morphometric
dependence F(h) and the hydraulic dependence v–(h) of
the outflow v– from the sea into Kara-Bogaz-Gol on
sea level; and a positive feedback is due to the dependence of the evaporation depth e(h) on the level (sea
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DYNAMIC-STOCHASTIC MODELING
depth in the shallow zone). The negative feedbacks
damp level variations and reduce their magnitude,
while the positive feedback has an inverse effect,
destabilizing the level regime. Note that at Caspian
level > –26.0 m BS, when the dependence of seawater
outflow into Kara-Bogaz-Gol ceases to damp level
variations, the bay becomes a shallow part of the sea,
which contributes to the mechanism of level variations
through the dependence of evaporation on the depth
of the bay. In this study, variations of Caspian level
above this mark are considered; therefore, the sea is
regarded as a drainless water body. In addition to the
commonly used system of marks BS, we use a reference
system with zero level corresponding to –30.0 m BS.
THE MAIN EQUATIONS
AND RELATIONSHIPS
The long-term level variations in the drainless Caspian Sea are described by the equation of sea water
balance:
+
+
dh (t ) v (t ) edet ( h) Fs ( h) + Fd e v (t )
=
−
+
− estoch , (6)
dt
F ( h)
F ( h)
F ( h)
where h is water level in the Caspian Sea,
v + (t ) = v + + v + (t ) is the water-budget input, equal to
the sum of river inflow and precipitation onto sea
water surface; v + is the average inflow, v + (t ) are fluc-
tuations of the inflow relative to its mean v + ; edet ( h)
and estoch are the evaporation, depending on the level,
and the stochastic component in evaporation from the
entire water area, respectively; e is the mean evaporation from the deep-water part of the sea; F (h), Fs ( h) ,
Fd = const and t have been determined earlier at (3).
The models for v + (t ) and es(t) will be represented
by autoregression processes (e.g., [6, 14]):
d v (t )
= −γ v v + (t ) + w1 (t ) ,
dt
+
(7)
des (t )
(8)
= −γ s es (t ) + w2 (t ) ,
dt
where γv = –lnrv, γe = –lnγe, rv and re are autocorrelation coefficients for processes v+(t) and es(t), respectively; w(i) (i = 1, 2) are white noises with known mathematical expectations w(i) and covariation functions
R(i)(τ) = D2(i )δ(τ), i = 1, 2; D2(i ) are the intensity coefficients of the appropriate white noises w(i), δ(τ) is Dirac
delta function.
The system of stochastic differential equations (6)–
(8) is a mathematical model of long-term level variations in the Caspian, allowing one to obtain a key
characteristic of level regime, i.e., the probability denWATER RESOURCES
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859
sity function (PDF) of the sea level as a solution of
Fokker–Planck–Kolmogorov equation (FPK)
Equation (6) can be written as
dh (t )
= f ( h) + g ( t ) ,
dt
(9)
v (t ) edet ( h) Fs ( h) + Fd e
, g(t) =
−
F ( h)
F ( h)
+
+
v (t )
v (t )
− estoch ≅
− estoch (t ), F * is the area, which
F ( h)
F*
in the model with bimodal PDF of the level is close to
some area A, equal, for example, to the half sum of
equilibrium areas of the sea.
where
f (t ) =
+
The processes v + (t ) and estoch (t ) are independent
and have equal autocorrelation coefficients; therefore,
g(t) is regarded as white noise. The stationary density
of the distribution of level p(h), corresponding to the
dynamic equation (9), can be found as the solution of
FPK equation with boundary conditions of zero flow
of probability [5, 21, 25] in the form:
 2 h f ( x) 
C
p ( h) =
exp 
dx  ,
2
g ( h)
N 0 h' g ( x ) 

where N0 is the coefficient of intensity of white noise
∞
g(t); N 0 = 4 k ( τ) d τ, where k ( τ) is the covariation
0
function of the process g(t); С is a normalization fac-

tor, determined by the condition

+∞
0
p ( h) dh = 1.
THE RESULTS OF SIMULATING CASPIAN
LEVEL VARIATIONS
The dependence of the total volume of evaporation
E ( h) = edet Fs ( h) + eFd from the shallow and deepwater parts of the Caspian water area on sea level h is
given in Fig. 4.
The abscissas of the points of intersection of the
plot of E(h) and the straight line 2, i.e., the mean volume of inflow into the sea, are the values of the equilibrium levels: stable S1 and S2, close to 10 and 20 m
(–20 and –10 m BS), and unstable U, ~15 m (–15 m
BS). Figure 4 suggests the possible bimodality of level
PDF. However, the existence of two stable levels is a
necessary but not sufficient condition for the PDF of
Caspian level to be bimodal [23]. The existence of
PDF bimodality depends, in particular, on the mean
inflow, the variances of inflow and the stochastic
component of evaporation, and the gradient of evaporation depth decrease with increasing depth. In the
proposed model of Caspian level variations, all parameters listed above and having quite plausible values,
admit the existence of a bimodal PDF for sea level.
860
FROLOV
Volume of evaporation E(h)
from Caspian water area
and the mean volume of inflow v+,
km3/year
S2
520
U
S1
500
480
460
440
1
2
5
10
15
20
25
30
Caspian level, m above –30.0 m BS
Fig. 4. (1) Dependence of the total volume of water
E ( h) = edet ( h) Fs ( h) + Fd e evaporated from the water
area of the entire Caspian on sea level, (2) mean volume of
water inflow into the sea, S1 and S2 are stable equilibrium
level, U is unstable equilibrium level of the Caspian.
The stationary PDF of sea level, obtained with the
use of FPK equation, is given in Fig. 5а. PDF has a
distinct bimodal character, suggesting tendencies in
level variations to concentrate in the vicinities of stable
levels ~10 and ~22 m (~ –20 and ~ –8 m BS).
In addition to the construction of PDF of Caspian
level using FPK equation, simulation calculations
were carried out. This was made with the use of a discrete (finite-difference) analogue of the stochastic differential equation (6). The inflow into the sea and the
evaporation from the deep-water part of the Caspian
were simulated by 1st-order autoregression processes
with parameter 0.3; the length of realizations was 105,
i.e., enough to obtain sufficiently accurate estimates of
level parameters.
The histogram derived from the simulated series of
Caspian level marks (Fig. 5b), is in agreement with the
PDF of the sea level obtained as a solution of FPK
Probability
distribution
dencity
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0
(а)
5
10 15 20 25 30
Caspian level above –30.0 BS
equation (Fig. 5а). This agreement increases the reliability of modeling results.
Figure 6a gives a fragment of simulated Caspian
level curve and a plot of sea level variations in paleotime according to [1]. The plots of Caspian level in
Figs. 6a, 6b are in qualitative agreement; both plots are
fragments of realizations of nonstationary processes.
However, the causes of the nonstationary character of
the Caspian level in Figs. 6a, 6b are different. The
nonstationarity of level variations given in Fig. 6a is
due to the specifics of the mechanism of level formation in the form of positive feedback, formed by the
nonlinear dependence of evaporation on sea level as
an internal cause. The climatic conditions in this case
were assumed constant throughout the simulation
time interval; this was implemented in the simulation
of inflow, precipitation, and the stochastic component
of evaporation in the form of stationary autoregression
processes. The nonstationarity of level variations in
Fig. 6b is determined by the nonstationariy of the climate, which changes sea water balance. The mechanism of level variations in this case contains only one,
negative, feedback, caused by the variability of water
area as a function of the level. In the absence of positive feedback, only one stable equilibrium level is possible, which varies depending on climatically changed
water balance of the sea, i.e., an external cause. Therefore, the nonstationarity of level variations in Fig. 6 are
fundamentally different—in Fig. 6a, the cause is of
internal character, while that in Fig. 6b is external.
CONCLUSIONS
A dynamic-stochastic model of long-term variations of Caspian Sea level is proposed; the model is
first to take into account the spatial heterogeneity of
evaporation from sea water surface.
In the model, the evaporation from Caspian Sea
water surface is simulated by the sum of two components—a deterministic function of the level (the depth
Frequency
(b)
1×104
5×103
0
10
20
30
Caspian level above –30.0 BS
Fig. 5. (a) Probability density distribution for Caspian level in accordance with Fokker–Plank–Kolmogorov equation; (b) histogram of Caspian level, obtained by simulation based on a discrete analog of the differential equation (6).
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DYNAMIC-STOCHASTIC MODELING
(а)
Caspian level above –30.0 BS
861
(b)
Caspian level above –30.0 BS
30
20
20
10
10
0
0
2 × 104
4 × 104
6 × 104 Years
140 120 100 80 60 40
Years ×100, BC
Fig. 6. (a) A fragment of a simulated series of Caspian levels; (b) Caspian level variations in paleotime (by data from [1]).
of shallows) and a stochastic component, determined
by evaporation from the shallow and deep-water parts
of the water area. These features of evaporation are
significant. Physically, the introduction of a stochastic
component means that a stochastic perturbation is
imposed on the functional dependence of evaporation
on the level; this perturbation reflects the year-to-year
variations of air temperature, wind fields, and other
processes above the sea area. Therefore, the deterministic component of evaporation can be masked by the
effect of the stochastic component. The neglect of this
feature of evaporation from the Caspian water area can
lead to errors in calculations of the sea level regime.
The model was used to obtain proofs of the possible
effect of positive feedback, reflecting the nonlinear
dependence of evaporation on sea depth, on Caspian
level variations. It is shown that an increase in the
mean level by ~10 m (from –20.0 to –10.0 m BS) 14–
4 thousand years ago for two successive time intervals
theoretically can be caused by a specific feature of the
mechanism of level variations in the Caspian, i.e.,
nonlinear dependence edet ( h) at invariable statistical
characteristics of the inflow. Under some quite realistic assumptions, accepted by the author in simulation,
the long-term variations of the Caspian level can be
nonstationary at stationary climate.
The nonlinearity of the function edet ( h) leads to
the bimodality of PDF of the Caspian level. However,
in this case as well, the nonlinear dependence of evaporation on the depth of the Caspian can affect the sea
level variations, increasing its variance.
The result of this study suggests the possible joint
effect on the level regime caused by the nonlinear
dependence of evaporation on the Caspian level and
the climatic changes of sea water budget. However, it
would be incorrect to interpret the obtained result
about possible considerable variations of the Caspian
level only under the effect of the nonlinearity of the
mechanism of sea level regime as an alternative to the
climatic explanation of such changes. No doubt, the
natural changes of Caspian level on time scales of tens,
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hundreds, and thousands of years depend on the climate on the sea drainage basin.
The main result of this study is the conclusion
regarding the need to take into account the role of the
nonlinear dependence of evaporation on sea depth in
the formation of Caspian level variations not only in
paleoclimatic reconstructions, but also in the forecast
estimates of the future regime of the sea level.
FUNDING
This study was carried out under Governmental Order to
Water Problems Institute, Russian Academy of Sciences in
what regards the simulation of the Caspian Sea level regime,
subject 0147-2019-0001, State Registration AAAA-A18118022090056-0, and was supported by the Russian Science
Foundation in what regards the construction of an
improved nonlinear dynamic-stochastic model of Caspian
level variations in paleotime, project 19-17-00215.
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Translated by G. Krichevets