BISAC NAT010000 Ecology
BISAC NAT045050 Ecosystems & Habitats / Coastal Regions & Shorelines
BISAC NAT025000 Ecosystems & Habitats / Oceans & Seas
BISAC NAT045030 Ecosystems & Habitats / Polar Regions
BISAC SCI081000 Earth Sciences / Hydrology
BISAC SCI092000 Global Warming & Climate Change
BISAC SCI020000 Life Sciences / Ecology
BISAC SCI039000 Life Sciences / Marine Biology
BISAC SOC053000 Regional Studies
BISAC TEC060000 Marine & Naval
In this work, the waves and currents generated by prognostic wind in the Sea of Azov are investigated using a three-dimensional nonlinear sigma-coordinate model. The mathematical model was also used for studying the transformation of passive admixture in the Sea of Azov, caused by the spatiotemporal variations in the fields of wind and atmospheric pressure, obtained from the prediction SKIRON model. Comparison of the results of numerical calculations and the data of field observations, obtained during the action of the wind on a number of hydrological stations was carried out. The evolutions of storm surges, velocities of currents and the characteristics of the pollution region at different levels of intensity of prognostic wind and stationary currents were found. The results of a comprehensive study allow reliably estimate modern ecological condition of offshore zones, develop predictive models of catastrophic water events and make science-based solutions to minimize the possible damage.
mathematical modeling, Sea of Azov, storm, surge phenomena processes, surface currents, evolution of passive admixture, three-dimensional hydrodynamic model
I. Introduction
Currently steady growth of interest in the mathematical modeling of wave motions of various natural stratified environments is observed. This is due to problems of geophysics, oceanography, atmospheric physics protection and study of the environment, the operation of complex hydraulic structures, including offshore oil complexes and other. Industrial activity in the offshore including those related to mining is important task wave dynamics, and obtained characteristics are used to assess the environmental impact on marine technology design and development of effective methods of forecasting of extreme hydrological events. Numerical study of hydrodynamic processes in the Azov Sea, arising from the different types of atmospheric circulation will be performed using the three-dimensional nonlinear sigma coordinate model with high spatial resolution adapted to the peculiarities of this basin. New modules and routines were included in the model in order to obtain assessment of the response patterns of surface and bottom currents at strengthening wind stress. Based on the analysis of results of a series of numerical experiments are performed to identify extreme synoptic perturbation level and flow velocity of the Azov Sea.
Studies based on the results of numerical simulation of the dynamic structure of the Azov Sea is also relatively scarce. The geometrical characteristics of the shallow pools caused requirements to hydrodynamic models that use curvilinear coordinate system. Numerical simulation of the dynamics of the waters of the Azov Sea is devoted to a series of works for a long time by Belov [1], Phillipov [1, 2] and Ovsienko [3], for the first time on the basis of linear two-dimensional models of the main characteristics of wind currents, wind-surges processes for typical stationary wind fields have been studied. From recent publications on the modeling of circulation work of Chikin A.L. [4] is the most interesting, which used three-dimensional non-linear shallow water model for the study of stationary motions at a wind speed of 7 m/s. A number of works Fomin V.V. [5, 6] is devoted to studying of stationary currents in the Sea of Azov with the use of three-dimensional non-linear models of the dynamics of currents.
II. The three dimensional primitive equations. the Boundary and initial conditions
We shall consider a rectangular coordinate system in which the x-axis is directed to the east, the y-axis is directed to the north, and the z-axis is directed vertically upwards. The mathematical model is based on the equations of motion and continuity using the hydrostatic approximation [7, 8]. We denote that u, v, and w are velocity projections on the axes x, y, and z, respectively; t is the time; p is the pressure; ρ is the density; g is the acceleration due to gravity; and f is the Coriolis parameter
(1)
, (2)
, (3)
. (4)
The coefficient of the horizontal viscosity AM is calculated using the Smagorinskii model of subgrid viscosity [9] depending on the horizontal velocity gradients. The relations for the calculation of the vertical viscosity coefficients KM and the turbulent diffusion KH according to the semi-empirical model of Mellor-Yamada (level 2.5) [10] are written as follows:
, 
, (5)
where SM and SH in a neutrally stratified flow are equal to 0.30 and 0.49, respectively. This parameterization is based on the solution of two additional equations in partial derivatives to determine the turbulent kinetic energy (q2/2) and the turbulence macroscale (l).
The boundary conditions at the free surface (z = ζ(x, y, t)) for the equations of motion are written as follows:
, 
, (6)
We note that 
and 
are projections of the tangential wind stress [8]; W = (Wx, Wy) is the wind velocity vector at a height of 10 m over the sea level; and Ca is an empirical coefficient of the surface friction [11], which varies depending on the wind velocity:
. (7)
The normal component of the velocity at the bottom is zero (z = ‒H(x, y)); the bottom tangential stresses are related to the velocity by the quadratic equation [8]
, 
, (8)
where 
, 
is the bottom friction coefficient, which is found from the formula 
, where zb is the vertical step in the bottom layer; and z0 = 0.03 mm is the roughness parameter that characterizes the hydrodynamic properties of the underlying bottom surface. The non-slip conditions are specified at the lateral boundaries.
Zero fluid motion and zero fluctuations of the free surface before the atmospheric forcing are specified as the initial conditions (t = 0):
, 
, 
, 
. (9)
Let use the equation of transport and diffusion to calculate the spreading of the admixture with the concentration C(x, y, z, t) [8]:
, (10)
Here, AH = 10 m2/s [3, 5] is the coefficient of horizontal turbulent diffusion and KH is the vertical diffusion; KH is found from equation (5). The conditions of zero admixture fluxes through the free surface, lateral walls (S), and bottom of the basin are added to the dynamic boundary conditions at the free surface and bottom layer [8]:
, 
, 
. (11)
The initial pollution region for all the atmospheric perturbations considered below is located in the surface layer:
, (12)
where z1 is the thickness of the pollution region, R is its radius, and 
is the distance from the center of this region (x0, y0) to the point at which the concentration is calculated. We select the time of the dispelling of the admixture (td) and the coefficient of the maximum square of its spreading at different levels (Kmax) as examples of the parameters characterizing the evolution of the passive admixture. Then, Kmax = Smax/S0, where S0 is the square of the initial region of pollution in the surface layer, and Smax is the maximum square of the pollution at the considered depth during the transformation of the admixture. The condition of complete pollution dispelling is for a concentration that does not exceed 2.5 × 10‒2 over the entire basin of the sea (Cd = 2.5 × 10‒2).
The transition from the z-coordinate to the sigma-coordinate is made to perform the numerical realization [5, 8]. In this case, the solution algorithm is based on the application of two-layer differential schemes. The transport operators are approximated [12] using a TVD scheme (a linear combination of the scheme of directed differences and the Lax-Wendroff scheme); the spatial digitization of the equations is performed using a C grid. Uniform steps over the horizontal coordinates ∆x and ∆y and the σ-coordinate are used.
The resolutions of the model by latitude and longitude are (1/59)° × (1/84)°, at which the linear sizes of the cell are ∆x = ∆y = 1.4 km; the number of the horizontal grid nodes is 276 × 176. The number of σ-sigma levels by the vertical is 11. The equations were integrated with a step of ∆t = 18 s to determine the average two-dimensional velocity components and the sea level and with a step of ∆tA = 10∆t = 3 min to calculate the deviations from the found mean values and vertical velocity component. The choice of the integration steps over the temporal and spatial coordinates was performed according to the stability criterion for barotropic waves [13].
The bottom topography is interpolated to the model grid using the depths given in the navigation charts. The deviations of the Azov Sea level were analyzed at nine stations located near large towns (Fig. 1).
Fig. 1. The bottom relief (m) the Sea of Azov, the position of coastal stations
III. Analysis of numerical experiments
Stationary motions in the Sea of Azov are generated by the western wind field, whose velocity at the sea surface (
= 5, 10 m/s) does not depend on x and y; in the first three hours (0 < t < 3 h), it increases with time according to a linear law (
= 0 at t = 0) and reaches the maximum value; later (t ≤ 3 h) it does not change. The moment when the currents reach the stable regime (t = t0) is determined by the fact that no notable deviations of the level and current velocities (the variations do not exceed 3%) occur between two neighboring time moments (T > t0 and T+ ∆t). From this we find the time when the fluid motion reaches the stable regime (t = t01, 
; t = t02, 
).
At the moment (t = t0) when the fluid motion stably adjusts to the stationary wind (, a temporally and spatially inhomogeneous wind field obtained from the reanalysis data (WSKIRON) is added to the stationary wind. At t > t0, the wind maintains the stationary motion, and WSKIRON adds a non-stationary component to the motion. We study the joint influence of the stationary (and variable WSKIRON) wind on the maximum velocities of the currents and the extreme values of the offshore and onshore transport using a series of numerical experiments. These results are compared with the data obtained from the calculation of the winds and currents caused only by constant wind forcing or only by wind determined from the reanalysis.
The investigation of the stationary currents is performed for the constant western wind, whose velocity is 5 and 10 m/s. The surface wind field from the SKIRON model [14] in the period from September 8 to 18, 2007, is used as the variable wind in time and space. Its forcing in the existence of the stationary currents in the Sea of Azov occurs during 10 days starting from September 8 at 00:00 h.
Table 1 presents the maximum wind velocities and wind directions (WSKIRON) over the Sea of Azov from September 11 to 18, 2007, as functions of time. The wind velocity changes monotonously between two neighboring times. The deviations of the wind velocity vector from the x-axis (the x-axis is directed to the east at an angle of 50° to the latitude) are given in degrees. It is seen from this that, during the studied period (192 h), the maximum and minimum velocities were 12.7 and 1.6 m/s, respectively. The dominating directions of the wind are northeastern and northwestern. We note that the results of the long-term observations of the atmospheric perturbations in the region of the Sea of Azov [6] in this period (September) agree well with the data of the SKIRON model given in this table.
Table 1. The maximum wind velocities versus the time obtained from the SKIRON model data from September 11, 2007 at 00:00 to September 18, 2007 at 24:00
|
Time, h |
Wind velocity, m/s |
Wind direction, degrees |
Time, h |
Wind velocity, m/s |
Wind direction, degrees |
|
2 |
6.8 |
107 |
76 |
9.4 |
354 |
|
14 |
2.6 |
344 |
92 |
12.7 |
350 |
|
28 |
5.8 |
9 |
104 |
9.6 |
100 |
|
30 |
3.0 |
10 |
106 |
7.5 |
210 |
|
32 |
5.2 |
107 |
108 |
5.7 |
200 |
|
44 |
5.8 |
344 |
112 |
3.5 |
110 |
|
48 |
8.1 |
100 |
124 |
5.2 |
344 |
|
50 |
4.2 |
213 |
130 |
3.3 |
354 |
|
54 |
5.8 |
195 |
132 |
1.6 |
347 |
|
56 |
7.9 |
192 |
140 |
2.1 |
10 |
|
58 |
9.6 |
108 |
152 |
4.9 |
106 |
|
62 |
11.6 |
350 |
192 |
5.8 |
200 |
Numerical experiments were realized for two velocities of the stationary western wind to study the influence of the induced currents on the sea level fluctuations and the variations in the velocity field of the non-stationary currents caused by the WSKIRON wind.
Table 2 gives the maximum sea level deviations caused by the stationary wind (, only by the wind based on the reanalysis data (WSKIRON), and by their joint forcing ( + WSKIRON) at the coastal stations of the Sea of Azov. The sea level variations caused by onshore winds are given in the upper part of the table, and the offshore values are given in the lower part of the table. It follows from the analysis of these results that the maximum onshore sea level changes generated by the system of stationary and variable wind were recorded at the following stations: 20.7 cm () and 62.4 см () in Taganrog, 57.1 cm (WSKIRON) and 80.4 cm ( + WSKIRON) in Primosko--Akhtarsk, and 102.2 cm ( + WSKIRON) in Eisk. It is seen from here that the maximum sea level change caused by onshore winds in the case of ( + WSKIRON) (102.2 cm) is 1.27 times greater than in the case of ( + WSKIRON) (80.4 cm). The minimum sea level changes caused by onshore winds appear in Mysovoye (7.5 cm for , 13.9 cm for ) and in Opasnoye (9.4 cm for WSKIRON, 16.1 cm for + WSKIRON, and 24.8 cm for ( + WSKIRON).
Table 2. Maximum sea level displacements caused by onshore and offshore winds (cm) at coastal stations of the Sea of Azov under a stationary regime and caused by a prognostic wind in the presence of stationary currents
|
Coastal stations |
|
|
WSKIRON |
|
|
|
Genichesk |
- |
- |
25.4 |
32.2 |
62.3 |
|
Berdyansk |
- |
- |
9.6 |
16.9 |
44.3 |
|
Mariupol |
9.8 |
37.3 |
29.3 |
46.4 |
80.4 |
|
Taganrog |
20.7 |
62.4 |
50.6 |
63.1 |
89.5 |
|
Eisk |
13.8 |
52.2 |
38.1 |
76.0 |
102.2 |
|
P. Akhnarsk |
8.1 |
43.2 |
57.1 |
80.4 |
91.1 |
|
Temryuk |
10.2 |
26.9 |
24.5 |
29.7 |
49.9 |
|
Opasnoye |
- |
- |
9.4 |
16.1 |
24.8 |
|
Mysovoye |
7.5 |
13.9 |
12.1 |
19.6 |
34.2 |
|
Genichesk |
12.2 |
51.7 |
42.5 |
76.5 |
87.0 |
|
Berdyansk |
4.0 |
17.6 |
17.3 |
30.9 |
62.1 |
|
Mariupol |
- |
- |
18.2 |
26.0 |
39.7 |
|
Taganrog |
- |
- |
29.0 |
42.4 |
72.9 |
|
Eisk |
- |
- |
18.9 |
41.1 |
45.3 |
|
P. Akhnarsk |
- |
- |
14.1 |
23.8 |
35.5 |
|
Temryuk |
- |
- |
8.7 |
15.2 |
23.0 |
|
Opasnoye |
3.3 |
11.1 |
10.6 |
20.5 |
34.2 |
|
Mysovoye |
- |
- |
22.3 |
39.4 |
63.9 |
The maximum sea level changes become greater under the joint influence of variable and constant offshore winds than in the stable regime and in the case of zero stationary currents. Among all the types of winds, the maximum sea level changes under offshore winds occur in Genichesk: 12.2 cm (), 51.7 cm (WSKIRON), 76.5 cm ( + WSKIRON), and 87.0 cm ( + WSKIRON). The minimum sea level changes under offshore winds occur in Opasnoye (3.3 cm (), 11.1 cm () and in Temryuk (8.7 cm (WSKIRON), 15.2 cm ( + WSKIRON), and 23.0 cm ( + WSKIRON)).
Let us compare the modeling results and the data of the field measurements presented in the tables of the hourly sea level values of the State Meteorological Service of Ukraine in the period from September 8 to 18, 2007. Let us estimate numerically the obtained extreme values of the sea level changes under onshore winds caused by the WSKIRON wind with the hourly data from these tables. The simulated maximum of the onshore sea level change in Genichesk is 25.4 cm, which is 4.7 cm (16%) smaller than from the data of the observations. It follows from this that the amplitudes of the sea level fluctuations obtained from the field data and from the numerical calculations agree quite well. The indicated differences are likely to be caused by the errors of the measurements and mathematical modeling.
Variations in the profile of the sea surface caused by wind forcing are shown in Fig. 2. It is seen from the figure that, in the stable regime (Fig. 2a), the sea level decreases along the western coast and increases near the eastern coast. The node line (dashed line) crosses the central part of the sea normal to the wind velocity. The regions in which the maximum and minimum sea level deviations appear three change during the wind forcing (Figs. 2b-2d).
Fig. 2. Sea level fields of the Sea of Azov at different time moments: (a) stationary regime (,t = 48 h); non-stationary regime ( + WSKIRON) (b) 68 h; (c) 90 h; (d) 140 h.
The objective of the following numerical experiments is to estimate the influence of the wind fields and generated currents on the spreading of the passive admixture transported to the central region of the sea. The initial position of the center of the region where the admixture was released is located at a point with the coordinates x0 = 180 km, y0 = 120 km; the depth of the sea at this point is 12 m. The region where the admixture was released is a cylinder with radius R = 9 km and depth h1 (0 > z ³ h1), where h1 (1 m) is the vertical step in the surface layer. The initial concentration is constant in this region and equal to unity (C0(x, y, z, 0) = 1).
The time of the release of the admixture in different experiments is not the same and depends on the characteristics of the wind. In the case of a nonstationary wind, the moments of the admixture release (t = t01, ; t = t02, ) and the stabilization of the fluid motion (t = t11, ; t = t12, ) coincide: t01 = t11 = 38 and t02 = t12 = 43 h. If only the wind WSKIRON forms the forcing, the time moment of release is September 11, 2007, at 00:00 (t03 = 72 h). If we consider the joint forcing of the stationary and non-stationary wind, the time of release is determined as follows: t04 = t01 + t03 ( + WSKIRON) and t05= t02 + t03 ( + WSKIRON). For the convenience of the analysis of the results, we assume that, in all the cases, the time of the release of the admixture is zero (t0 = 0).
Table 3 presents the coefficients of the maximum spreading of the passive admixture (Kmax), the time when it occurred (tmax, h), and the time of the complete dispelling of the admixture (td, h) at three depths for constant wind () and three variables (WSKIRON and + WSKIRON). One can see from the data analysis that the maximum square of the pollution at each depth depends on the wind velocity that leads to the stable motion.
Table 3. Parameters (Kmax; tmax, h; td, h) of the spreading evolution of the admixture at different depths of the Sea of Azov
|
Depth, m |
Max. values (K, t), and td |
|
W2st |
WSKIRON |
|
|
|
z = 0 |
Kmax |
1,14 |
1,18 |
1,25 |
1,30 |
1,32 |
|
tmax, h |
5,7 |
4,9 |
31 |
40 |
40 |
|
|
td, h |
17,3 |
18,2 |
57,1 |
84,5 |
86,5 |
|
|
z = -H/2 |
Kmax |
1,16 |
1,18 |
1,27 |
1,33 |
1,35 |
|
tmax, h |
14,3 |
14,7 |
34 |
42 |
42 |
|
|
td, h |
36,7 |
37,6 |
104 |
106 |
110 |
|
|
z = -H+hb |
Kmax |
1,16 |
1,19 |
1,33 |
1,37 |
1,38 |
|
tmax, h |
26,9 |
25,5 |
55 |
58 |
59 |
|
|
td, h |
53,4 |
55,2 |
108 |
110 |
115 |
In the case of greater velocity of the constant wind ( > ), the velocities of the currents increase, the square of the admixture region spreading increases (Kmax), and time of its complete dispelling also increases (td). The joint forcing of the stationary and WSKIRON wind causes an increase in the pollution region.
In this case, the maximum pollution square occurs if the velocity of the forcing wind is maximal ( + WSKIRON). In this case, Kmax at the free surface is 1.32 40 h after the release of the admixture, and the time of its complete dispelling () is 86.5 h. The maximum square of the pollution region at the depth z = ‒H/2 is gained 42 h after the release of the admixture (Kmax = 1.35), and the complete dispelling of the pollution occurs in 110 h. In the bottom layer (z = H + hb), the coefficient of the maximum spreading of the admixture is 1.38 (tmax = 59 h). We note that, in the case considered here ( + WSKIRON), the concentration of the admixture in the entire basin 115 h after its release does not exceed 2.5% of its initial value (Cd = 2.5 × 10‒2).
IV. Conclusion
In this work, we present the results of investigations of the phenomena caused by onshore--offshore winds and the evolution of a passive admixture by the current system generated by the constant and variable winds in the Sea of Azov. The reliability of these results is confirmed by the comparison of the simulated values of the extreme sea level changes caused by onshore and offshore winds with the field data obtained during wind forcing by the surface wind obtained from the SKIRON model at coastal hydrometeorological stations. The obtained results are presented in the table of the sea level changes caused by the onshore and offshore winds and the current velocities for different characteristics of constant and variable wind. We also performed the analysis of the influence of the wind velocity and the generated currents on the characteristics of the transformation of the passive admixture.
The analysis of the modeling results and the dynamic processes in the Sea of Azov allowed us to reach the following conclusions:
(1) It is found from the analysis of the stationary motions that, under constant wind forcing with a two-fold increase in the velocity (5 and 10 m), the maximum deviations of the sea level increase by a factor of 3.45 (0.2 and 0.69 cm), the minimum deviations increase by a factor of 3.9 (0.1 and 0.39 cm), and the maximum velocities of the stable currents increase by 12 times (0.16 and 1.17 m/s);
(2) An increase in the maximum wind velocity leads to an increase in the volume of the pollution region; the minimum pollution square appears in the absence of wind.
(3) The time needed for the pollution region to reach the maximum volume decreases when the wind velocity increases.
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