INSTITUTE OF GEOPHYSICS, TBILISI, GEORGIA
INSTITUTE OF NUMERICAL MATHEMATICS, RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF GEOGRAPHY OF THE ACADEMY OF SCIENCES OF REPUBLIC OF MOLDOVA
BSEC PROJECT
Hydro and Thermodynamic Processes in the “Black Sea –
Land – Atmosphere” System and Regional Climate.
Development of Fundamentals of Monitoring and Forecasting
System
Final Report
Head of the Project
Professor A. Kordzadze
Tbilisi, 2007
The participating organizations
M. Nodia Institute of Geophysics
1, M. Alexidze str., 0193 Tbilisi, Georgia
Contact person: Head of the Department of Mathematical
Modeling of Geophysical and Ecological Processes in the Sea and Atmosphere,
Professor Avtandil Kordzadze
Tel.: (+995 32) 33-38-14; Fax: (+995 32) 332867; e-mail: avtokor@ig.acnet.ge
Institute of Numerical Mathematics of Russian Academy of Sciences
8, Gybkina str., GSP-1, 119991 Moscow, Russia
Contact person: Professor Vladimir Zalesny
Tel.: (+7 095) 938-39-07; Fax: (+7095) 938-18-21; e-mail: zalesny@inm.ras.ru
Institute of Geography of the Academy of Sciences of Republic of Moldova
MD – 2028. Chishinau 1, Academiei str., Republic of Moldova
Contact person: Professor Tatiana Constantinova
Tel/Fax: (+37322) 73-98-38; e-mail: const@asm.md
2
Contents
A. TECHNICAL PART
.................................................4
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Modern State of a Problem of the Sea-Atmosphere Interaction . . . . . . . . . . . . . . . . . . . . . . .6
3. General Structure of the Coupled Regional Numerical Model
of the Black Sea – Land – Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4. The Black Sea Dynamics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
4.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Numerical Scheme of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 Simulation of the Black Sea Circulation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5. Atmospheric Boundary Layer – Soil Quasi-one-dimensional
Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1 Equations, Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Defenation of a Turbulence Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 Numerical Scheme of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.4 Results of Test Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6. Numerical Model of Large-Scale Atmospheric Processes for a Limited Area . . . . . . . . . 18
6.1 Statement of the Atmospheric Dynamics Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.2 Numerical Scheme of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.3 Realization of the Atmospheric Dynamics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
7. A Coupled Regional Hydrodynamic Model “Sea – Land –Atmosphere” . . . . . . . . . . . . . 25
7.1 Model Equations, Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.2 Some Problems of Realization of the Coupled Regional Model . . . . . . . . . . . . . . . . .26
7.3 About Software of the Coupled Regional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
7.4 Preliminary Results from the Coupled Regional Model . . . . . . . . . . . . . . . . . . . . . . . 28
8. Semi-Lagrangian Model of the Atmosphere for Numerical Weather Prediction . . . . . . . .38
9. Estimation of Climatic and Agroclimatic Potential of Territories with a Mountain .
Relief in Conditions of a Varying Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
10. Conclusions. Perspectives of Works on Creation of Monitoring and Forecasting
System for the Black Sea Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
11. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51
B. FINANSIAL PART . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3
1. Introduction
For the last decades the natural environment undergoes significant modifications. These changes
are caused by the increased economic activity of humanity and are connected with its intensive
anthropogenic action on the environment. People have always acted on the environment, however
while the scales of these actions were small, the nature always had time to regenerate itself. From the
middle of 20th century the intensity of anthropogenic action on the nature has increased so, that it has
violated the ecological equilibrium, which was kept within many centuries [1-5]. The increased
anthropogenic influence on the natural environment had in view well-known geophysics R. Revel
and G. Suess, when they noticed else in 1957 that humanity carries out ,,the large-scale geophysical
experiment » not in the laboratory or on the computer, but on the own planet [3].
From the point of view of deterioration of an ecological situation the Black Sea and its
adjoining region is not exception. The Black Sea is an estuary basin, facing rapid environmental
degradation. Scientific results clearly show that profound changes have occurred in the Black Sea
ecosystem during the last two decades, as a consequence of a negative anthropogenic impact in the
region.
The Black Sea region is located in the central zone of the transport corridor connecting Europe
with Asia. In conditions of intensive transportation of power resources danger of occurrence of mancaused failures is considerably increased. Such failures may become the reason of significant
environmental contamination, and as a whole of ecological accident.
The Black Sea is the richest source of natural resources, its influence on a social-economic
state of Black Sea riparian countries is very large. Therefore the ecological state of the Black Sea is
very important for these countries.
The intensification of human activity on exploitation of the sea considerably raises importance
of skill and ability to supervise and predict a state of the sea environment and resources to avoid
large financial expenses, and also negative critical consequences of economic activity.
Commercial exploitation of a shelf of the sea and, in particular, its uses for extraction and
transportation of petroleum and gas inevitability results its uses in increase of probability of large
accidents with irreparable damage to recreational and biological resources of the sea. Therefore it is
clear, that simultaneously with an intensification of industrial development of the sea the monitoring
and forecasting system of changes of the sea environment capable to give a solid data for acceptance
of administrative decisions, updating working and substantiation of the future economic projects
should develop.
There is also the other problem, which is connected to change of a regional climate. In the last
years problems of anthropogenic change of a climate and adaptation of human activity to new
climatic conditions have become one of the most paramount problems of a modern civilization. The
Black Sea region in this respect deserves the significant attention. According to many experts’
estimations on a background of the global warming of a climate, the cooling of the Black Sea surface
and its adjoining territory is observed [6, 7].
Real opportunity of essential changes of a climate have strengthened anxiety on consequences
of these changes for an agriculture of the Black Sea region countries. It is obvious that influence of
climatic changes on this branch of a national economy is shown through change of various
agroclimatic characteristics - durations of the vegetative period, the sums of temperatures, deposits
etc. Set of estimations of changes of various agroclimatic parameters may form a basis for
construction of the general picture of change of agroclimatic potential of the Black Sea region
countries.
The Black Sea plays an important role in formation of weather and regional climate [8]. As well
as ocean and atmosphere, the Black Sea and the atmosphere form uniform hydrothermodynamic
system, where exchangeable processes of energies and different substances on the interface sea4
atmosphere take place continuously. Therefore, for the successful solution of problems of forecast of
weather, the Black Sea state and regional climate changes the Black Sea and atmosphere must be
considered as an entire hydrothermodynamic system.
Thus, in the beginning of the XXI century protection of the environment, restoration and
preservation of ecological equilibrium, forecasting of anthropogenous change of a global and
regional climate, ecologically safe and rational assimilation of natural resources became the
necessary condition for the sustainable development of humanity. Management policies need answers
to concrete questions concerning the response of nature to both natural and man-made changes in
environmental forcing factors and loading. Numerical modelling is an important and necessary tool
for a better understanding of the relations between the processes, and for forecasting these response.
The rational use of natural recourses and optimal organization of the economic activity of
humans significantly depends on operational reception of information about state and changes of the
natural environment. Therefore the creation of a monitoring and forecasting system for the sea and
atmosphere is a very relevant problem regarding the Black Sea region. The realization of this system
will enable to observe continuously the temperature, salinity of the sea, currents, zones of
contamination, ets, that describe current and future states of the Black Sea and atmosphere.
The principal goal of the suggested project was preparation of the scientific base for
implementation of the operational monitoring and forecasting system of hydrothermodynamic fields
in the Black Sea and the atmosphere. With this purpose the coupled regional model of the system
“the Black Sea – atmosphere – soil” is developed within the framework of the given project, test
numerical experiments are carried out for the extended limited area with the centre – the Black Sea
region. It is necessary to note, that presented in the framework of BSEC project work is the first
attempt of unification of models of dynamics of the sea and the atmosphere for the Black Sea region.
Besides, it is estimated climatic and agro-climatic potential of territories with a mountain relief (on
an example of a Republic Moldova), The tendency of climatic characteristics is revealed.
In the present scientific Report the detailed description of separate modules of the coupled
regional model and results of their realization for the Black Sea region are consistently given. There
is also description of a global semi-Lagrangian finite difference atmospheric general circulation
model of the Institute of Numerical Mathematics of Russian Academy of Sciences (Moscow, Russia).
Outcomes of realization of this global model will used in the coupled regional model for simulation
of regional climate fields in perspective. There is description of the coupled regional hydrodynamic
model of the system “The Black Sea-land-atmosphere” and some results of its realization for the
expanded territory with the center - the Black Sea region are given.
The basic modules of the coupled model represent models of dynamics of the atmosphere and
ocean which are based on full systems of hydro and thermodynamic equations of the ocean and the
atmosphere. For solution of the tasks included in the coupled model, well-known splitting methods
are used, which were suggested by G. I. Marchuk for the first time for solution of problems of
dynamics of the atmosphere and ocean.[9, 10].
It is necessary to note, that the works carried out within the framework of the given project
cannot be consider as final. They have preliminary character and there is preparatory stage for the
further works on improvement of the coupled model “the Black Sea-land-atmosphere”.
5
2. Modern State of a Problem of the Sea-Atmosphere Interaction
Nowadays methods of mathematical modelling are widely used in studying of hydro and
thermodynamic, and ecological processes including the problem of ocean-atmosphere interaction.
Interest to the problem of global interaction between the ocean and atmosphere is caused first of all
by necessity of solution of problems of long-term weather forecast and the forecast of change of a
global climate.
The problem of global change of a climate, widely discussed nowadays, finds the reflection in
changes at a regional level also and interest grows to the simulation of a regional climate.
Studying of a global climate occurs on the basis of hydrodynamic models of global circulation
which intensively develop. In hierarchy of such models from the point of view of perfection and
complexities at the supreme step are three-dimensional coupled models of the system “ocean –
atmosphere” (for example, [11-14]). In the majority of these models the equation systems are written
in spherical coordinate system, and on a vertical the  coordinate system is used. These models
describe well the basic features of climatic system “ocean – atmosphere”, but at the same time some
results are in the unsatisfied consent with the observational data.
Except of the three-dimensional models of ocean - atmosphere interaction by some authors it
was considered also rather simplified problems in which separate aspects of interaction was studied
[15-18].
Object of ours research is small-scale sea – atmosphere interaction. Quantitative characteristics
of this interaction represent turbulent heat, moisture and movement fluxes, with the help of which
interaction between atmospheric and sea surface layers are carried out. In [19] the comparative
analysis of calculation methods of turbulent fluxes between the ocean and atmosphere was given.
Some Questions of small-scale interaction are considered in well-known monographies [20-22]. In
[23] it is presented the methodology to develop a coupled modeling system between atmosphere and
ocean. Meso-scale Modeling System and Princeton Ocean Model (POM) have been used as the basic
tools for the proposed methodology. Computational, and some physical aspects, of the coupled
system are investigated for the Southwest Atlantic Bight.
The analysis of the literature accessible to us shows, that the coupled regional model “seaatmosphere” for the Black Sea region till now is not realized.
3. General Structure of the Coupled Regional Numerical Model
of the Black Sea-Land-Atmosphere
The coupled model consists of separate blocks, each of them represents the mathematical
model describing hydro and thermodynamic processes in separate objects of the environment (the
sea, the atmosphere, active layer of the soil). The model is based on full systems of ocean and
atmosphere hydro and thermodynamic equations, equation of heat conductivity in the soil and heat
balance equations on the underground surface (land, water).
In Fig.1 the vertical structure of the model is schematically showed. The vertical structure of
the model comprises the following layers: 1. Troposphere; 2. Atmospheric surface layer; 3. Active
layer of the soil; 4. Active layer of the sea; 5. Deep layer of the sea.
In each layer for description of physical processes the following differential equation systems
are considered:
I.
In the troposphere
6
the full system of atmospheric hydrothermodynamic equations in hydrostatic
approximation;
II.
In the lower turbulent layer of the atmosphere
simplified one-dimensional equation system of atmospheric boundary layer;
III.
In the active and deep layers of the sea
the full system of ocean hydrothermodynamic equations
IV.
In the active layer of the soil
the equation of heat conductivity.
These equations are connected with one another with boundary conditions on a vertical, which
basically express continuity of solutions and their first derivatives at transition from one layer to
another. As one of boundary conditions on the underground surface (water, land) the equation of heat
balance is considered.
In the following sections there are description of separate modules of the coupled models.
HT  10  12 km
Troposphere
HT   x, y   ha 
 x, y   ha
ha  50  80 m surface layer
surface layer
above a sea
upper layer
of a sea
h  1m
hm
lower layer of a sea
 x, y 
Z
Z
H M x, y 
ZM
Fig.1. The schematic image of vertical structure of the coupled regional model.
7
hm
4. The Black Sea Dynamics Model
4.1 Model Description
The goal of the sea dynamics model is to describe temporal-space evolution of currents,
temperature, density, and salinity in all basin of the Black Sea from the sea surface to the sea bottom
on a basis of numerical integration of full system of ocean hydrothermodynamic equations. Besides,
the results of realization of the task will be applied as lower boundary conditions for the task of the
upper active layer of the sea. For this purposes baroclinic prognostic sea dynamics model [24, 25] is
used [24, 25]. The offered model is an improved version of before developed models of the Black
Sea dynamics [26-29].
In the reporting period within the framework of the given BSEC Project space resolution of the
model has been increased (from 10 km to 5 km spacing). With this purpose for the repoting period
realization of the following works was required:
- Elaboration of a designed program on the algorithmic language “Fortran” intended for translation
of input data from the grid with 10 km spacing to the grid with 5 km spacing ;
- Preparation of input data on the grid with 5 km spacing;
- Corresponding modification of software of the Black Sea dynamics model in view of the new
grid;
- Performance of test numerical experiments using horizontal grid step 5 km.
The model takes into account water exchange with the Mediterranean Sea, Danube river
inflow, atmospheric forcing, quasi-realistic bottom relief, the absorption of short-wave radiation by
surface layer of the sea, space-temporal variability of horizontal and vertical exchange.
The model equations have been written down for deviations of thermodynamic values from the
appropriate standard vertical distributions. The equation system of the model in the Cartesian
coordinate system (the axis x is directed eastward, y - northward, and z - from a sea surface vertically
downwards) has the following form:

 u
u
1  p
,

 divuu  lv +
= u 
z z
t
0  x

v
1  p
 v
 divuv + lu +
= v 

,
t
0  y
 z z

div u = 0,
 p / z  g ,

 T

 T    T T
1  I T
 div uT    T .w  T T  
T



,
t
z
z
z
c  z  t

 S

 S  S  S  S
 div uS +  S w =  S S  
S


,
t
 z
 z
 z
 t
 =  T T  +  S S  ,
T 
 T
,
 z
S 
(4.1.1)
S
,
 z
T  T ( z, t )  T , S  S ( z, t )  S ,  =  ( z, t )   , p  p( z, t )  p ,
8
 =




,



 x  x  y  y
I 0  a sinh 0  b sinh 0 ,
I   (1  A) I 0 e z ,
sinh 0  sin  sin   cos  cos cos

12
t,
~
  1  (a~  b n~)n~ .
From the Mamaev’s empirical formula of equation of state for marine water   f (T , S ) [30] are
defined
 T   f / T  10 3 (0.0035  0.00938T  0.0025 S ),  S   f /  s  10 3.(0.802  0.002T ) .

Here u, v, and w are the components of the current velocity vector u along axes x, y, z ,
respectively; T , S , P ,   - the deviations of temperature, salinity, pressure and density from their
standard vertical distributions T , S, P,  ; l  l 0   . y - the Coriolis parameter; g , c  0 - the
gravitational acceleration, the specific heat capacity and the average density of seawater;
 , T ,S , , T ,S - the horizontal and vertical eddy viscosity, heat and salt diffusion coefficients,
respectively; I 0 - the total radiation flux determined by the Albrecht formula [31] at z = 0, A -albedo
of a sea surface, h0 - the zenithal angle of the Sun;  - the geographical latitude,  - the parameter of
declination of the Sun,  - the factor which takes into account influence
of a cloudiness on a total
~
radiation and depends upon ball of cloudiness n~ [32]; a, b, a~, b - the empirical factors;  - the
parameter of absorption of short-wave radiation by seawater.
The equation system (4.1.1) was solved at the following boundary and initial conditions:

 u
 zx ,
 z
0 
zy
 v

, w  0,
 z
 0
T = T *  T (0, t ) , S = S*  S (0, t )
at z  0 ,
u  0, v = 0,  T / n  0 ,  S/ n  0
on
Г0,
~
~
u  u~ , v = ~
v , T = T , S = S
оn
Г1 ,
u  0 , v = 0 , w = 0,  T / z   T  S/ z,   S
аt z  H ,
u  u 0 , v = v 0 , T = T 0 , S = S 0
at t = 0,
(4.1.2)
(4.1.3)
where H describes the bottom relief of the sea basin;  zx , zy , T * , S * are the wind stress components
along axes x and y , temperature and salinity on a sea surface z = 0, respectively; Г0 – the rigid lateral
boundary, and Г1 - the liquid boundary separating the sea basin from other water area (in our case –
the boundary between the Black Sea basin and the Bosphorus strait or the Danube River); n - the
~ ~
external normal to surface Г0; u~ , ~v , T , S - the velocity components, deviations of temperature and
salinity on liquid boundaries, respectively.
Factors of turbulence  ,  T, S , and  T , S were calculated by the formulas presented in [33,34] :
9
  x.y 2  u  +   u   v
  x    y  x
2
2



 , T 
, S  S ,

cS
cT

2
2

g  
 v

 T , S  (0.05h)
    z     z ,
0

where  x and  y are horizontal grid steps along x and y , respectively; cT and c S are some
constants; h is the depth of the turbulent surface layer, which is defined by the first point z m , in
2
 u

 z

2

 v
 + 2

  y


which following condition is satisfied:
g  
 u 
 v
(0.05 z m ) 2 
  T0 , S .
 
  z
  z  0  z
2
2
In case of unstable stratification, which may be appear during integration of the equations
 
(
 0 ), the realization of this instability in the model was taken into account by increase of
 z
factor of vertical turbulent diffusion  T ,S 20 times in appropriate columns from a surface to the
bottom.
4.2 Numerical Scheme of Solution
The problem (4.1.1) - (4.1.3) is solved numerically on the basis of a two-cycle splitting
method by physical processes, coordinate planes and lines described in details in [10, 29]. With this
purpose the entire time interval (0,T) is broken up into equal broadened intervals tj-1  t  tj+1 and on
each such interval is made linearization of the advective members. Obtained quasi-linear equation
system on each time interval tj-1  t  tj+1 is splitted by physical processes, as a result of which
following problems are allocated:
1. The transfer of the physical fields taking into account eddy viscosity and diffusion;
2. The adaptation of the physical fields with division of the solution into barotropic and baroclinic
components.
At the transfer stage on time interval tj-1  t  tj the following equations are considered:
j
 u1  u1  u1
 u1
 t  divu u1  x  x  y  y  z  z ,

 v1
j
 v1  v1  v1
 t  divu v1  x  x  y  y  z  z ,


 T1  divu j T    T1    T1    T1   T  T ,
1
T
T
T
 t
x
x y
y z
z
z

 S1  divu j S    S1    S1    S1   S  S
1
S
S
S
 t
x
x y
y z
z
z
(4.2.1)
with (4.1.2) boundary conditions and initial conditions:
u1j-1 = uj-1, v1j-1 = vj-1, T1j-1 = Tj-1, S1j-1 = Sj-1,
10
For solving of equations (4.2.1) the two-cycle splitting method by coordinates is used, as a
result of which the problem is reduced to set of one-dimensional linear equation system, that are
effectively solved by the factorization method.
At the second stage (adaptation stage) on time interval tj-1  t  tj+1 the following equation
system is considered:
u 2
t
v 2
t
p 2
t
u 2
x
T2
t
S 2
t
1 p 2
 0,
 x
1 p 2
 lu 2 
 0,
 y
 lv 2 
 g  T T2   S S 2 ,
(4.2.2)
v
w
 2  2  o,
y
z
  T w2  0,
  S w 2  0,
with following boundary and initial conditions:
w 2  0,
u 2 n  0,
at
on
z  0, H
Г
u2j-1 = u1j, v2j-1 = v1j, T2j-1 = T1j, S2j-1 = S1j,
At the stage of adaptation the solution of the system (4.2.2) is divided into barotropic and
baroclinic components. The barotropic task is reduced to the solution of two-dimensional equation
for integral stream function.
The operator of the baroclinic task received after allocation of the Coriolis force, in addition is
splitted by vertical planes. As a result the adaptation problem is reduced to the solution of a set of the
same type two-dimensional tasks for analogues of stream function. Received two-dimensional
equations are effectively solved within the framework of uniform iterative algorithm [10, 29].
At the third stage (transfer stage) on the time interval tj  t  tj+1 the same equations (4.2.1)
are solved.
On each stage for approximation of tasks on a time the Krank-Nickolson scheme is used.
4.3 Simulation of the Black Sea Circulation Processes
With the purpose of numerical realization of the problem (4.1.1) - (4.1.3) the surface of the
Black Sea was covered with a grid with the constant steps equal to 5 km. The quantity of points on
axes x and y was 223 and 109, correspondingly. On a vertical the non-uniform grid with 32
11
calculated levels on depths: 1, 3, 5, 7, 11, 15, 25, 35, 55, 85, 135, 205, 305,..., 2205 m were
considered.
To determine values of parameters connected with absorption of short-wave radiation, some
works have been used [31, 32, 35, 36]. The dependence of the sea surface albedo upon zenithal angle
had basically such character, which is specified in the table [36], corresponding to excitement of the
sea surface with balls 1-3 and lower cloudiness with ball 0.2. According to this table the albedo
undergoes significant daily changes between 0.02 - 0.39. A seasonal course of common cloudiness
above the Black Sea basin was reproduced by linear interpolation on mean seasonal values of ball of
cloudiness given in [35]. The parameter of absorption of radiation was accepted equal to  = 0.0023
m-1, which corresponds to an usual ocean water in which about 10% of a radiation reaches a depth 10
~
m [37]. Empirical factors accepted values [31,32]: a=1.54 kW/m2, b=0.22 kW/m2, ~
a  b  0.38 .
Specific heat capacity c = 4.09 Jg-1K-1, which corresponds to a seawater with salinity about 18 %o.
The other parameters had the following values:
g = 980 cm2/s,  0 =1g/cm3, l = 0,95.10-4s-1,  = 10-13cm-1s-1,  t  1h , c T  cS  10 ,
(a)
(c)
(b)
(d)
Fig. 2. Computed current fields on horizons z = 3 m in August at different time moments:
(a) - 5690 h, (b) – 5734h, (c) – 5756h, (d) – 5798h (time is counted since 1 th January
of the 3 modeling year).
12
(a)
(c)
(b)
(d)
Fig. 3. Same as in Fig. 2, but on horizons z = 15m..
Figs. 2 and 3 illustrate calculated sea currents on depth of 3 m and 15m in August, when above
the Black Sea alternation of different types of atmospheric circulation, characteristic for the Black
Sea basin, took place. These wind types were taken from [38] Integration started on the 1 st of
January by 5 km horisontals spacing and as initial conditions annual mean climatic fields of current,
temperature, and salinity were used, which were obtained by the same model on annual mean input
climatic data. Results of numerical experiment were analyzed on the third model year when quasiperiodical regime was established.
In Figures basic features of the Black Sea circulation known from observations [39-43], such as
the general cyclonic character of current, cyclonic circulations in east and western parts of the basin,
anticyclones vortex in the southeast part (Batumi anticyclone) and some coastal anticyclonic vortexes
are well visible.
13
5. Atmospheric Boundary Layer – Soil Quasi – one-dimensional
Numerical Model
5.1. Equations, Boundary Conditions
Correct description of interaction of the atmosphere with the underlying surface (land, water) of
the Earth essentially depends on that fact as far as vertical distribution of meteorological values near
the surface is precisely described. The goal of the task is to obtain vertical distribution of wind,
temperature and moisture with high resolution in the atmospheric surface layer on the basis of onedimensional system of boundary layer equations (with taken in consideration of advective processes
parametrically). Besides that this model has independent scientific and practical value, in the coupled
regional model it will be play a role of the separate block with help of which will be describe
interaction with underlying Erth’s Surface.
For the reporting period atmospheric boundary layer – soil non-stationary quasi-onedimensional model (functions depend on time and vertical coordinates) with parametrical
consideration of advective processes and relief is developed. The model is based on simplified
system of planetary boundary equations, heat balance equation of the underlying surface and
equation of molecular heat conductivity in the active layer of a soil. Turbulent viscosity and diffusion
coefficients in the atmospheric surface layer are calculated during integration of model equations on
the basis of the similarity theory of Monin-Obukhov.
Model equation system has the following form [44]:
In the atmospheric boundary layer
u

 u
  
lv 
 Fu ,
t
x
z z
v

 v
  
 lu  
 Fv ,
t
y
z z
 





 
 u (
S
)  v(
S
)  
,
t
x
x
y
y
z
z
 q1
q 
Q

Q


 u (
 q
)  v(
 q
) q 1 .
t
x
x
y
y
z
z
 l V

,
x  z
Fu  lV  U / t ,
(5.1.1)

l U

.
y
 z
Fv  lU  V /  t ,
S   / z ,   g /  0 ,
u  U  u, v  V  v, w  W  w,      , q1  Q  q1 .
In the active layer of a soil
 Tп
 2 Tп
 Kп
,
0  zп  h п .
(5.1.2)
t
 z п2
Here z = z1 -  (x, y), where z1 is the vertical coordinate from the bottom of the relief and  (x,y) the function describing the relief; u and v represent horizontal components of wind velocity vector;
14
U and V - - the background velocity components in the Cartesian coordinate system;   - deviation
of the potential temperature from the background value  ; g – the gravitational acceleration;
 0 =const – the average potential temperature of the atmosphere; l – the Coriolis parameter; S,  the parameters of thermal stability and buoyancy; q1 - the deviation of the moisture from the
background value Q. z п - the vertical coordinate downword from the surface to the soil; K п - the
factor of temperature conductivity; h п -the thickness of the active payer of a soil; T  п - deviation of
the soil temperature from daily averaged value. Fu and Fv parametrically describe the influence of
large scale processes on processes in the boundary layer.
For the equations (5.1.1) we shall consider the following boundary conditions:
on the top of the planetary boundary layer Ha
u  U , v  V ,    0, q1  0
At a level of a roughness z = z0
u  v  0,
- c p 
~
q

~ T
 L 1  c n K п  c1R ,
z
z
 zn
R  (1  A) I  F ,
 T,
q  rq H (T0 ) (0  r  1)
On the lower boundary of the active layer of a soil z п  hп
~
Tп  0.
Here c п is the volumetric heat capacity; R - the radiation balance of the underlying surface; A Albedo of a ground surface; I - the total short-wave radiation of the sun; F - the Earth’s surface
effective long-wave radiation; r - the relative humidity at a ground surface. q H - the specific
humidity in a saturated state; T0 - the temperature of the ground surface.
For solving of (5.1.1) and (5.1.2) the following hypothetical initial conditions are used:
u( z)  U( H)(1  e az cos(az)  V( H)e az sin az
v( z )  V( H )(1  e  az cos(az )  U( H )e  az sin az
~
   q  T  0 , a =
l / 2
Expressions for u and v represent known solutions of Ekman’s boundary layer [45, 46].
For calculation of short-wave radiation of the Sun the albrekht’s formula is accepted [31]
t
I  a1 sinh   b1 sinh  , sinh   sin  sin  cos  cos cos
12
where  is the declination of the Sun,  - the geographical latitude
Effective radiation of the Earth is defined by the Brent formula [46]
F  T04 a2  b2 e ,
where  is Stephan – Bolcman constant, e – the partial pressure of water vapor. The factors c1 is
defined by empirical formula [45].
Sating specific humidity was defined under Magnus’s formula [47].


15