Computing Method Applied for Simulation and Forecast of
Hydrophysical Fields in the Southeastern Black Sea
DEMURI DEMETRASHVILI
M. Nodia Institute of Geophysics,
I. Javakhishvili Tbilisi State University,
M. Aleksidze str., 1, Tbilisi,
GEORGIA
also with
Institute of Hydrometeorology,
Georgian Technical University,
David Agmashenebeli str., 150, Tbilisi,
GEORGIA
Abstract: - In this paper, a numerical regional model of the Black Sea hydrodynamics with a spatial resolution
of 1 km is applied to simulate and forecast the mesoscale circulation and thermohaline structure in the
southeastern part of the Black Sea. The regional model is based on a nonlinear nonstationary system of
differential equations in z coordinates, describing the evolution of three-dimensional fields of currents,
temperature, salinity, and density. The solution of the equation system is based on the use of finite-difference
methods, in particular, on the method of multicomponent splitting of the differential operator of the original
problem into simpler operators. To illustrate the implementation of the numerical model, the paper presents the
calculated fields of flow, temperature, and salinity for the summer season of 2020 under real atmospheric
forcing derived from the atmospheric model SCIRON. The influence of basin-scale processes on regional
processes was taken into account by boundary conditions on the liquid boundary separating the regional area
from the open part of the sea basin. A comparison of the forecast results with available satellite data shows that
the model reproduces well the basic peculiarities of hydrophysical fields.
Key-Words: - Black Sea, circulation, system of equations, boundary conditions, splitting method, atmospheric
forcing, mesoscale eddies.
Received: May 13, 2024. Revised: September 25, 2024. Accepted: October 27, 2024. Published: November 26, 2024.
1 Introduction
In recent decades, mathematical modeling methods
have become one of the main research tools in
physical oceanography. The Study of the
hydrothermodynamics of the Black Sea using
mathematical models began in the 1970s of the last
century. A review and analysis of numerical models
of this period are given in monographs, [1], [2].
Numerical models are based on systems of
differential equations describing the hydrophysical
processes, whose solution is carried out by finite-
difference (numerical) methods. In early studies, two
types of models were clearly distinguished:
diagnostic and prognostic. In diagnostic models, the
density field was determined based on observational
data (and not during the solution process). This
makes it possible to simplify the system of
equations, as it eliminates the need to consider the
heat and salinity transfer equations, [3]. Prognostic
models based on a complete system of ocean hydro-
and thermodynamic equations provide a more
adequate reproduction of the processes of sea
dynamics, [3], [4], [5]. The complete system
includes the equations of motion projected on
horizontal coordinate axes, the equation of
hydrostatics, the continuity equation for an
incompressible fluid, heat and salt advection-
diffusion equations, and the empirical equation of
state of seawater. The pioneering work in this
direction was [4], where a prognostic model was
developed for simulating basin-scale wind and
thermohaline circulation in the Black Sea. To solve
the problem, a numerical scheme was used based on
the two-cycle method of splitting the problem both
into physical processes and spatial variables. This
method, which has found wide application in
problems of geophysical fluid dynamics and
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ecology, significantly simplifies the solution of
complex spatial problems and reduces them to the
solution of one-dimensional and two-dimensional
problems, [6].
Calculations based on diagnostic and prognostic
models have made a significant contribution to the
study of the dynamic processes of the Black Sea.
Analysis of numerical experiments showed that the
calculated fields well reflect the basic peculiarities of
the hydrological regime, identified from
observational data: the main cyclonic circulation,
including two internal cyclonic circulations in the
western and eastern parts of the sea, the rise of
waters in the central parts of the cyclonic
circulations, maximum salinity and density in the
central part of the cyclonic circulations, which
testifies to the upwelling of salty deep waters in the
central areas of the cyclonic circulations. In [4] the
significant role of the bottom relief in the northern
and western parts of the sea basin was shown.
In early studies, the low level of computing
resources did not allow the creation of models with
high spatial resolution. In [4], the spatial resolution
was 37 km, and in most calculations using diagnostic
and prognostic models, the spatial resolution was 40-
50 km, which was insufficient for a full study of the
features of hydrodynamic processes in the Black
Sea, especially in coastal/shelf areas, where eddy
structures of various sizes are often formed, [7].
The rapid progress of computing technology
over the past 30 years has given great impetus to the
creation of high-resolution numerical models, which
has made the modeling results more reliable and
adequate. Currently, a number of publications are
devoted to high-resolution modeling of the Black
Sea hydrothermodynamics, [8], [9], [10], [11], [12],
[13], [14], [15]. Almost all modern models are based
on solving systems of non-stationary, non-linear
differential equations using finite-difference
methods. It should be noted that in most publications
the main attention is paid to the study of the upper
layer of the sea, which plays an active role in the
interaction of the sea and the atmosphere.
The Black Sea hydrodynamic models have
found wide application in problems of modeling the
spread of oil pollution accidentally entering the sea,
[16], [17].
Modern numerical models of sea dynamics have
become not only a powerful research tool, but also
have created a solid basis for the development of
marine short-term forecasting systems for some
European seas, similar to weather forecasting
systems. Improvements in remote sensing (satellite)
methods and assimilation of in situ observation data
have also contributed to the development of marine
forecasting systems.
A great achievement of the Black Sea
operational oceanography is the development of the
basin-scale nowcasting/forecasting system at the
beginning of the XXI century, [18], [19]. One of the
components of the system became the Black Sea
regional forecasting system for the southeastern part
of the Black Sea, [20]. The regional system is based
on the regional model of the Black Sea dynamics
(RM-IG) with a spatial resolution of 1 km, which is
developed by adapting the basin-scale numerical
model with a spatial resolution of 5 km [12] to the
southeastern part of the sea basin. In turn, this basin-
scale model is an advanced version of [4]. Within
the framework of the EU projects, the computational
grid of the RM-IG with a spatial resolution of 1 km
was nested in the computational grid of the basin-
scale model of the Marine Hydrophysical Institute
(Sevastopol, Ukraine) with a resolution of 5 km,
which ensured the consideration of the impact of
hydrothermodynamic processes of the open part on
regional processes through the conditional liquid
boundary (one-way nesting method).
This paper presents a regional model of sea
hydrodynamics and the main aspects of the
numerical algorithm for solving the model equation
system based on the splitting method. The results of
computer implementation of the RM-IG are also
discussed.
2 Problem Formulation
To describe nonstationary hydrophysical processes
in the Baroclinic Sea, we consider the following
system of differential equations (the x coordinate is
directed to the east, y to the north, and z is vertically
downwards)
.
, (1)
(2)
(3)
(4)
(5)
=
1
+
0x
p
lvuudiv
t
u
z
u
xy
uDD
,D=
1
++v
v xy
v
0
z
v
D
y
p
luudiv
t
,
g
z
p
0,= udiv
,
1
.
0
T
z
T
xy
T
T
T
DD
z
I
cz
wTudiv
t
T
,
=+
SS
S
z
S
xy
SDD
z
wSudiv
t
S
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(6)
,
where
.
Here u, v and w represent the components of the
flow velocity; T’, S’, P’ and ρ’ are the deviations of
the physical quantities temperature, salinity,
pressure, and density from their vertically averaged
values and ; l = l0 + βy is the Coriolis
parameter; I0 is the total flux of solar radiation to the
sea surface determined by the Albrecht formula
[21], A is the albedo of the sea surface; α is the
absorption coefficient of solar radiation by seawater;
(6) is a linearized formula for the equation of state
of seawater ρ = f (T, S) applied in [12]; η is the
parameter describing the influence of cloudiness.
μ and ν are the factors of horizontal and vertical
eddy viscosities, and μ1 and ν1- are the factors of
horizontal and vertical diffusion for heat and salt.
Other designations are well known.
The equations (1) - (6) are solved under the
following boundary and initial conditions:
On the sea surface z = 0, which is considered a rigid
surface, boundary conditions describing
atmospheric forcing are:
w = 0,
, ;
On the sea bottom z = H (x, y)
u = v = w = 0, ;
On the lateral solid surfaces Г0
0n/S , 0n/T 0,= v,0
u
.
On the lateral liquid surfaces Г1 two kinds of
boundary conditions are considered:
a) in the case of inflow into the sea
S
~
= ,
~
= , v
~
= v,
~
STTuu
,
b) in the case of outflow from the sea
Here n is the vector of outer normal to the
lateral surface,
zyzx
,
are wind stress components
along the axes x and y;
cQQ TT /
, where
T
Q
is
the turbulent heat flux between the sea and
atmosphere;
are the current velocity components
along the axes x and y and the deviations of
temperature and salinity at the liquid boundary,
respectively; PR and EV are the atmospheric
precipitation and evaporation.
00 0 0 = , = , v= v,S STTuu
at t = 0.
The theorem of uniqueness for the solution of
this problem has been proven [2].
The formulas for calculating the factors of
horizontal and vertical turbulent viscosity and
diffusion were the same as in [20].
The influence of basin-scale processes on
regional processes was taken into account by
boundary conditions on the liquid boundary
separating the regional area from the open part of the
sea basin. Values of on the liquid
boundary were derived from the basin-scale model of
the Black Sea dynamics of the Marine Hydrophysical
Institute.
3 Numerical Scheme
For the numerical implementation of the above
problem, the two-cycle splitting method was used
to split the model equation system both by physical
processes and by vertical coordinate planes and
lines. Note that in the case of non-commutative
operators, the two-cycle splitting method provides
second-order accuracy with respect to time, [6].
We will consider the basic stages of the
algorithm for solving the system of equations (1) –
(6). Before splitting with respect to physical
processes, Let's divide the entire time interval (0, T)
into equal intervals
11 jj ttt
(the time step τ
= tj-1- tj) and linearize the equations (1), (2), (4),
(5) on each such interval. On each extended interval
11 jj ttt
we apply the two-cycle method of
splitting with respect to physical processes. (for
, S + T= ST
, )1( 0
z
eIAI
,/ ,/ S zSzT
T
pzppz
SzSSTzTT
)( ,)( =
,)( ,)(
PST ,,
Sf
S /
,/ Tf
T
yyxx
Dxy
vu
,
,
,zz
Dzvu
,
11, yyxx
DxyST
,
1, zz
DzST
,,,, STvu
,
0
xz
z
u
,
v
0
y
z
z
T
TQ
z
T
,)( 0
SEVPR
z
S
S
S - z/S , z/T
T
,0/ nu
,0/ nv
,0/
nT
,0/
nS
STu ~
,
~
, v
~
,
~
STu ~
,
~
, v
~
,
~
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simplicity, we will omit the primes on the functions
T, S, and P).
Fig. 1: Block diagram of the numerical algorithm
In order to better understand the numerical
solution algorithm, it is shown in Figure 1 in a
schematic form.
From Figure 1 it is evident that as a result of
splitting the main differential operator of the 3D
problem into elementary operators the following
basic stages are identified: 1. advection-diffusion
stage of physical fields; 2. adaptation of physical
fields. At the advection-diffusion stage, the problem
is reduced to solving one-dimensional problems,
and at the adaptation stage we get two problems:
barotropic and baroclinic. As a splitting of
baroclinic problem with respect to vertical
coordinate planes, it is reduced to a set of two-
dimensional problems.
On the time interval,
jj ttt
1
we consider
the advection-diffusion stage of physical fields.
(7)
where components uj, vj, wj of the vector
j
u
satisfy the continuity equation:
.0
j
udiv
System of equations (7) is solved under the
above-mentioned boundary conditions and initial
data
In the second interval we have the
adaptation stage:
(8)
under boundary conditions:
w2 = 0 at z = 0, H
and initial data:
Using the equation of state, density is excluded
from the hydrostatic equation.
At the last step of splitting, we again have the
advection-diffusion stage
(9)
System of equations (9) is solved under the
same boundary conditions as (7) and initial data:
To approximate in time all nonstationary
relatively simple problems received after splitting
, =
xy
u1
1z
u
jDDuudiv
t
u
,D=v
v xy
v1
1z
v
jDudiv
t
,
z
c
1
-
0
1
1
I
z
DDTudiv
t
TTT
z
T
xy
T
j
,=
1
1z
DDSudiv
t
SSS
z
S
xy
S
j
,
222 o
z
w
y
v
x
u
,0 =
1
+
2
0
2
2x
p
lv
t
u
,0 =
y
1
+
v 2
0
2
2
p
lu
t
, 0
2
2 w
t
TT
, 0
2
2 w
t
SS
, )(
22
2STg
t
PST
,
11
1
jj uu
,
11
1
jj vv
,
11
1
jj TT
.
11
1
jj SS
11 jj ttt
,
1
1
2
j
juu
,vv 1
1
2
j
j
,
1
1
2
j
jTT
.
1
1
2
j
jSS
0)( 2
n
u
)( 1
jj ttt
, =
xy
u3
3z
u
jDDuudiv
t
u
,D=v
v xy
v3
3z
v
jDudiv
t
,
z
c
1
-
0
3
3
I
z
DDTudiv
t
TTT
z
T
xy
T
j
,=
3
3z
DDSudiv
t
SSS
z
S
xy
S
j
,
1
23
j
juu
,
1
23
j
jvv
.
1
23
j
jSS
,
1
23
j
jTT
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the main task, the Crank-Nicholson scheme is used
providing second-order accuracy in time.
Note that equations for u, v, T, S in (7) and (9)
on the advection-diffusion stage are independent of
each other. Using the two-cycle splitting method
with respect to geometrical coordinates x, y, z, and
after finite-difference approximation of equations,
we obtain a set of one-dimensional problems which
are solved effectively by the factorization method.
At the adaptation stage, the solution is divided
into barotropic and baroclinic components, i. e. we
obtain two problems on the extended time interval
. For this purpose we will consider the
average on-depth values:
, , .
We assume that:
,
2uuu
,
2vvv
,
2ppp
Then, from (8) we will receive the following
equation system for the barotropic problem:
(10)
with boundary condition on the lateral surface σ
(11)
The barotropic problem (10)-(11) is reduced to
solving a finite-difference system of algebraic
equations for integral stream function, which is
solved by iterative method.
The operator of the baroclinic problem is
preliminary splitting with separating the term of the
Coriolis force as an independent stage. Thus, the
system of equations is considered:
(12)
The analytical solution of (12) is:
u = u0coslt +v0sinlt, v = v0coslt – u0sinlt,
where initial conditions u0 and v0 are baroclinic
parts of the solution received after the adcection-
diffusion stage (9).
The remaining part of the differential operator
of the baroclinic problem (without the Coriolis
terms) splits into vertical coordinate planes zx and
zy. As a result, the adaptation problem is reduced to
solving a sequence of similar two-dimensional
problems for analogs of the stream function, which
are solved using an iterative algorithm.
During the elaboration of the numerical
algorithm and corresponding software on “Fortran”
the accuracy of the numerical solution of each sub-
problem, received as a result of splitting the 3D
equation system, was tested as follows: let’s in the
area
D
with lateral surface σ the problem is
considered, which in operator form can be written:
(13)
with boundary condition on σ:
gB
(14)
and initial condition:
0
at = 0 (15)
where, in general, A and B differential operators, g
is given function. Let's choose any analytical
function
, that satisfies the conditions (14), (15)
and substitute it into the equation (13). Then we get
(16)
The function is defined from equation (16).
It is easy to guess that the numerical solution of
the equation:
should coincide with the analytical solution
with a certain accuracy. In the conducted numerical
experiments the accuracy of the numerical solution
was evaluated using the formula:
,
where k, l are the numbers of grid nodes along
horizontal axes. Test numerical experiments showed
that the value of is in the range of 0.002 – 0.007.
11 jj ttt
fA
t
f
ffA
t
ffA
t
lklk ,,
max
,0 =
lv
t
u
.0 =
lu
t
v
Hudz
H
u0
1
Hvdz
H
v0
1
Hpdz
H
p0
1
,
2TT
.
2SS
,0 =
1
+
0x
Hp
Hvl
t
Hu
,0 =
y
1
+
0
Hp
Hul
t
Hv
.0 =
y
Hv
x
Hu
0)(
n
u
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4 Model Implementation
The model described above is implemented in the
southeastern part of the Black Sea with dimensions
of approximately 215x340 km. The maximum sea
depth in the model is assumed to be 2006 m. The
considered water area covers the Georgian sector of
the Black Sea and adjacent area which is allocated
from the open sea by meridian 39.080E. The
southeastern part of the basin is covered with a grid
of 215x347 having a horizontal resolution of 1 km.
Vertically, 30 calculated horizons were used with a
minimum grid step of 2 m near the sea surface and a
maximum of 100 m below a depth of 200 m. The
time step was 30 minutes. The other parameters had
the following values: g = 980 cm/s2, ρ0 = 1g/cm3, l0
= 0.95.10-4sec-1, β = 10-13cm-1sec-1, c = 4.09Jg-1K-1 ,
α = 0.0023m-1, As boundary conditions at the open
boundary of the region, we used the fields of
currents, temperature and salinity, calculated using
the model of the Marine Hydrophysical Institute for
the entire sea with a step of 5 km.
Eddy viscosity factor on a vertical:
The calculations took into account the inflow of
6 Georgian rivers. atmospheric forcing was derived
from the atmospheric model SCIRON.
Calculations based on the RM-IG under real
nonstationary atmospheric wind and thermohaline
forcing show that the considered water area is
characterized by significant seasonal and
interannual variability of circulation processes. In
the cold season, cyclonic character predominates in
the regional circulation, although not infrequently
anticyclonic gyre structures can also be observed
against the general cyclonic background. In many
cases, the summer circulation is distinguished by its
anticyclonic character, including the formation of
the well-known Batumi eddy in the southeastern
part of the sea.
It should be noted that as the calculations
carried out under the conditions of real atmospheric
forcing show, the Batumi anticyclonic eddy is
characterized by different intensities in different
years. For example, the Batumi eddy was a very
stable formation in 2010, occupying an area
approximately 150-200 km in diameter throughout
the summer. The circulation regime essentially
determines the structure of the salinity field, to
which many marine organisms are highly sensitive.
For demonstration purposes, Figure 2 shows the
surface current, salinity on z=50 m horizon and the
sea surface temperature (SST) corresponding to
August 12, 2020 calculated using the RM-IG. The
forecasting period was 9-13 August 2020, 00:00
GMT.
(a)
(b)
(c)
Fig. 2: Simulated surface current (a), salinity (b) at z
= 50 m and SST (c) corresponding to 12 August
2020
mzscm
mzscm
55,/10
55,/50
2
2
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Figure 2(a) clearly shows the Batumi
anticyclonic eddy, which is the main element of the
regional circulation for the mentioned day. The
formation of small vortex structures is also
observed. Comparison of the salinity field (Figure 2
(b)) with the velocity field (Figure 2(a)) shows a
good correlation between these fields. The relatively
low salinity in the central part of the anticyclone is
due to the downward current, which transports low-
salinity waters to the deep layers.
The SST correlates relatively weakly with the
circulation structure, and the formation of the
temperature field is largely determined by the
exchange of heat between the sea and the
atmosphere (Figure 2(c)).
Fig. 3: Geostrophic surface current on 12 August,
2020, 00:00 GMT, which is created using satellite
altimeter data. By rectangle, the forecasting area is
marked
Fig. 4: Satellite SST corresponding to 15:12 GMT,
12 August, 2020. By rectangle, the forecasting area
is marked
With the purpose of comparing forecasted fields
with observational data, in Figure 3 the geostrophic
current created using satellite altimeter data is
presented for 12 August, 2020 and in Figure 4 –
satellite SST derived from NOAA for 15:12 GMT,
12 August, 2020. Comparison of predicted surface
circulation (Figure 2(a)) with the geostrophic
current (Figure 3) shows that an anticyclonic eddy
was actually observed for this day, as was obtained
as a result of calculations.
Comparison of predicted (Figure 2(c)) and
satellite SST (Figure 4) shows a good agreement
with each other. According to both modeling results
and satellite observations for 12 August 2020 water
temperature on the most territory of the southeastern
water area was 27-280C. Only on small territory in
the southwestern part the temperature was about
260C.
With the purpose of a more quantitative
comparison of the predicted and satellite SST fields,
we calculated a root-mean-square error (RMSE)
using the formula:
RMSE =
where Ts is satellite SST and Tp – predicted SST, N
–the number of grid nodes. It was found that for the
considered day RMSE = 0,720c.
As can be seen from Figure 2(b) and Figure
2(c), the salinity and temperature fields in the upper
layer are characterized by significant horizontal
heterogeneity. Our calculations show that in the
deep layers, below a depth of about 300 m, their
horizontal gradients decrease significantly, which is
in good agreement with the results of instrumental
observations and numerical modeling, [1], [2]. The
horizontal uniformity of thermohaline fields
provides the barotropic nature of circulation in the
deep layers of the Black Sea.
In Figure 5 velocity vector field is shown on
horizons of 200, 500 and 1000 m for the same day.
It is clear from Figure 5 that the general
anticyclonic nature of the mesoscale circulation is
practically preserved in the 1000 m thick layer.
Depending on the depth, the influence of the
configuration of the sea basin increases and the
circulation undergoes a certain transformation. On
the horizon of 200 m, the Batumi eddy is well
expressed, but its radius decreases. At the same
time, the formation of small unstable eddies is
observed. Due to the weakening of atmospheric
forcing with depth, the current undergoes sharp
quantitative changes: maximum speed on the sea
surface 24 cm/s decreases to 12 cm/s on z = 200 m
and to 4 cm/s on z= 1000 m. These speed values are
N
TT p
lk
lk
slk 2
,
,
,)(
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in good agreement with the profiling drifters ARGO
data, [22].
(a)
(b)
(c)
Fig. 5: Simulated current field on horizons of 200 m
(a), 500 m (b) and 1000 m (c)
5 Conclusion
The paper presents a mathematical modeling
method applied to simulate and forecast the
hydrological regime of the southeastern part of the
Black Sea. The effectiveness of the method, which
is based on splitting the main differential operator
into elementary operators, is demonstrated by
modeling and forecasting of main hydrophysical
fields – the current, temperature, and salinity - with
1 km spatial resolution for 12 August of 2020.
Forecast outputs are compared with satellite data
showing the reality of predicted fields.
In perspective, we plan to develop a high-
resolution regional modeling system that will make
it possible to simulate and forecast coastal dynamic
processes with very high resolution in the Georgian
nearshore zone with sizes of about 50-120 km,
which is distinguished by high economic and
recreational activity. With this purpose, a very high-
resolution version of the RM-IG will be developed
(with a spatial resolution 200 m). This advanced
version, which will be nested in the RM-IG, will
allow the simulation and forecast of nearshore
processes with a higher accuracy.
Acknowledgment:
This work was supported by Shota Rustaveli
National Science Foundation of Georgia (SRNSFG)
[grant number FR-22-365].
References:
[1] Stanev E. V., Truhchev D. I., Roussenov V.
M. The Black Sea water circulation and
numerical modeling of currents, Kliment
Ohridski University Press, Sofia, 1988,
pp.222
[2] Kordzadze A. A. Mathematical modeling of
the current dynamics (theory, algorithms,
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Demuri Demetrashvili developed software for a
numerical solution algorithm in Fortran; conducted
computational experiments, graphical interpretation
and analysis of the obtained results and prepared a
scientific article.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The work was supported by Shota Rustaveli
National Science Foundation of Georgia (SRNSFG)
[Grant number FR-22-365).
Conflict of Interest
The author has no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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