CC BY
0ISSN 1573-160X, PHYSICAL OCEANOGRAPHY, Vol. 31, Iss. 2, pp. 161-177 (2024)
Original Russian Text O A. A. Slepyshev, N. O. Ankudmov, 2024, published inMorskoy Gidrofizicheskiy Zhurnal, 2024, Vol. 40, Iss. 2, pp. 180-197
Original article
Generation of Vertical Fine Structure by Internal Waves
on a Shear Flow
A. A. Slepyshev N. O. Ankudinov
Marine Hydrophysical Institute of RAS, Sevastopol, Russian Federation H slep55@mail.ru
Abstract
Purpose. The work is purposed at studying the wave mechanism of fine structure generation, as well as determining the vertical wave fluxes of mass.
Methods and Results. In contrast to the previously used mechanism of forming a fine structure by internal waves due to breaking, a new approach based on determining the vertical wave fluxes of mass in the field of an inertia-gravity internal wave without breaking is proposed. The inertia-gravity internal waves on the Black Sea northwestern shelf are considered on a flow with a vertical velocity shift. The flow is assumed to be geostrophically balanced with the vertical velocity shifts compensated by a horizontal density gradient. The ./-plane approximation is used. Thus, the classical scheme for describing a wave field by the hydrodynamic equations is applicable with regard to the nonlinear effects. A weakly nonlinear approach is used. In the linear approximation, the eigenfunction and the dispersion relation are found by solving numerically the boundary value problem which determines the vertical structure of a mode in the presence of an average flow. In this case, the wave frequency is of a complex character since the coefficients in the differential equation of the specified boundary value problem are complex. Depending on the wave period and the mode number, either weak attenuation of a wave or its weak amplification is possible. The eigenfunction of internal waves is also complex. Therefore, the vertical wave fluxes of mass and the vertical component of the Stokes drift velocity are nonzero and lead to the generation of vertical fine structure which is irreversible. Conclusions. In the presence of a two-dimensional shear flow, taking into account the horizontal inhomogeneity of the average density field enhances the effect of generation of a vertical fine structure by the inertia-gravity internal waves. The vertical wave fluxes of mass also increase. The indicated fluxes and generated fine structure for the waves of different frequencies are close and the effect is enhanced in the presence of the waves of different frequencies.
Keywords: internal waves, fine structure, Stokes drift
Acknowledgments: The study was carried out within the framework of the theme of state assignment FNNN-2021-0004.
For citation: Slepyshev, A.A. and Ankudinov, N.O., 2024. Generation of Vertical Fine Structure by Internal Waves on a Shear Flow. Physical Oceanography, 31(1), pp. 161-177.
O 2024, A. A. Slepyshev, N. O. Ankudinov
O 2024, Physical Oceanography
Introduction
Fine structure of a liquid medium is a very interesting object studied both theoretically and experimentally under laboratory conditions. Structures amenable to detailed experimental study and theoretical analysis [1, 2] arise when solids move in a liquid. No less interesting structures [3] appear when a liquid drop falls onto a liquid. Such structures can explain the processes in the sea surface layer during precipitation.
The fine structure of hydrophysical fields was discovered almost half a century ago owing to the creation of high-resolution sounding equipment. Indeed, no one had
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imagined before its discovery that temperature and salinity vertical profiles on small spatial scales were very variable. This variability was initially thought 1 [4] to be due to small-scale turbulence, also not fully understood object. But then it turned out that not only turbulence was responsible for the fine structure generation as the kinematic effect of internal waves also gave a fine structure in measurements [5, 6]. However, this was not a long-lived structure since the kinematic effect disappeared after the wave train passage.
Double diffusion provided a mechanism for vertical stratification of temperature and salinity profiles in the ocean 2 [7-10]. The formation of convective cells in the form of salt fingers [11-14] is possible if temperature and salinity decrease with depth. They differ markedly from the surrounding liquid layers in terms of characteristic gradients, i.e. a structure appears with alternating thin stratified layers and quasi-homogeneous ones. This is actually the fine structure. Such conditions are very typical for the World Ocean and approximately 70% of its column is subject to such stratification 2. Double diffusion takes place when temperature and salinity increase with depth [15, 16]. Geothermal heat sources at the bottom contribute to this process. Double diffusion can lead to the formation of step structures which have been repeatedly observed in field experiments [17, 18]. Intrusive layering with mutual penetration of waters with different T, S-characteristics [19-21] is possible in the area of fronts. Intrusions are possible due to internal wave breaking at the shelf edge in the continental slope area [22].
It should be noted that such conditions are not always realized and this phenomenon is more exotic than widespread in inland seas far from fronts. The fine structure still exists and can be explained by other mechanisms which, of course, also work in the open ocean. The internal waves breaking and hydrodynamic instability of flows is the mechanism that has been proposed to explain the fine structure formation [23-26]. However, it turned out that internal waves do not necessarily have to break in order to generate a fine structure. The wave field is intermittent, and internal waves often propagate as wave trains [27]. Due to nonlinearity, trains of internal waves produce small corrections to the flow density and velocity [28] with a horizontal scale equal to the train envelope scale, and the vertical scale is determined by relation cg / N, where cg is group velocity of
train and N is Brunt-Vaisala frequency. However, these corrections are proportional to the square of the current wave amplitude and disappear after the wave train passage.
The waves attenuate and form boundary layers in the vicinity of the boundaries [29, 30] in a dissipative medium when viscosity and diffusion are taken into account. A beam of three-dimensional internal waves is reflected from the layer where wave and buoyancy frequencies coincide and the reflection occurs with energy losses [31] which can amount to several percentage points. Internal
1 Gibson, C.H., 1980. Fossil Temperature, Salinity, and Vorticity Turbulence in the Ocean. In: J. C. J. Nihoul, ed., 1980. Marine Turbulence. Amsterdam: Elsevier Publising Co., pp. 221-257. https://doi.org/10.1016/S0422-9894(08)71223-6
2 Fedorov, K.N., 1978. The Thermohaline Finestructure of the Ocean. Oxford; New York: Pergamon Press, 170 p.
boundary layers are formed at discontinuities in the buoyancy frequency or its derivatives in the beam of internal waves with a reflected beam also present [31]. Boundary layers have split density and velocity scales. These boundary flows generate fine structure of the periodic flow.
Internal waves attenuate [32, 33] when turbulent viscosity and diffusion are taken into account. Vertical wave fluxes of heat, salt and momentum are nonzero and lead to the formation of an already irreversible fine structure [34-38]. However, not only consideration of turbulent viscosity and diffusion leads to the vertical fine structure generation. When taking into account the Earth rotation and inhomogeneous average flows, vertical wave fluxes of heat, salt and mass are nonzero even when turbulent viscosity and diffusion are not taken into account and result in formation of a vertical fine structure [39]. Vertical wave momentum fluxes are also nonzero [40-44]. However, the horizontal average density gradient which is always present in a geostrophically balanced flow was not considered in [39]. It was assumed that the horizontal scale of wave is much smaller than the scale of the average density field inhomogeneity. We take into account the horizontal gradient of average density and perform a comparison with the results for a homogeneous case in the present work. In [39], the boundary value problem for the vertical velocity amplitude is solved by the perturbation method with the introduction of small parameter e = V0 / (H • m), where V0 is characteristic flow
velocity, H is depth and m is wave frequency. However, this parameter is not always small and in this work the boundary value problem for the amplitude of internal wave vertical velocity is solved numerically using the implicit Adams scheme of the third order accuracy [40, 42].
Problem statement
Free internal waves are considered in the Boussinesq approximation with regard to the Earth rotation and the average two-dimensional vertically inhomogeneous flow in a horizontal infinite basin of constant depth [34-44]. The /-plane approximation is used. Horizontal gradients of average density are found from geostrophic relations 3 and expressed through vertical shifts of flow velocity components, as in [45]. The system of hydrodynamic equations for wave disturbances in the Boussinesq approximation is as follows:
Du
Dt
ß , x u
dU0 1 P
-u3 — = -—VP + g±-, (1)
dX3 P0 P0
D + (uV)P0 = 0, (2)
divu = 0 , (3)
3 Kamenkovich, V.M., 1977. Fundamentals of Ocean Dynamics. Amsterdam; New York: Elsevier Scientific Pub. Co., 249 p.
where u(u1,u2,u3) is vector of flow velocity wave disturbances; p, P are density and pressure wave disturbances; p0 is unperturbed average density; x3 axis is directed opposite to the gravitational acceleration vector g; is vector of projection of the Earth rotation angular velocity onto the axis x3; U0
(U°(x3), U20(x3),0) is vector of average flow velocity; operator D has form
D = d + ((u + U0)V) . Dt dt v '
From the "thermal wind" relations 3 [45], the components of average density
horizontal gradient are expressed through vertical shifts in the flow velocity as
follows:
dpp =_ pf dUl (4)
dx1 g dx3
dp0 = pf dUL (5)
dx2 g dx3
where f = 2() = 2QE sin^ is Coriolis parameter, e3 is unit vector of x3 axis, QE
is angular velocity of the Earth rotation, ^ is latitude.
After substituting relations (4), (5) into equation (2), the latter becomes as follows:
Dp p0f dU2 p0f dU1 3p0
— _u—--2 + u2 —--1 + u3-i-° = 0. (6)
Dt g dx3 g dx3 cx3
The boundary condition on the sea surface is the "rigid lid" condition which filters internal waves from surface ones and on the bottom - the impermeability condition 4:
u3 = 0 at x3 = 0 , (7)
u3 = 0 at x3 = _H , (8)
where H is sea depth.
Linear approximation. We look for solutions to equations (1), (3), (6) in a linear approximation under the condition of horizontal homogeneity of Brunt-Vaisala frequency on the wave scale in the following form [34-45]:
u, = ui0( x3) Ae'0 + c.c., i = 1,2,3, (9)
P = P( x3) Aei0 + c.c., p = p1( x3) Aei0 + c.c., (10)
4 Miropolsky, Yu.Z., 1981. Dynamics of Internal Gravitational Waves in the Ocean. Leningrad:
Gidrometeoizdat, 302 p. (in Russian).
where c.c. are complex-conjugate terms; A is amplitude multiplier; 0 = kx1 — mi is
wave phase, k is horizontal wavenumber, m is wave frequency. We assume that the wave propagates along x1 axis.
Expressions for amplitude functions u10, u20,p1, P1 through u30 follow from system (1), (3), (6):
i du-s
k dx-^
Q = ro - k ■ U0
(11)
P _ i
Q du30 + dU1
k dx dx.
0 f, dU.
30 + 77 (i ■
Q dx.
_)
30 7 J u30)
k dx
1 ( f du3,
Q
dU.
oA
k dx->
■ _ iu,,
dx.
3 y
pi = 0
i Pof dU
dxo d*x*
30 _ u «Pû. - Pof dUi
u30
dx3 gQ dx3
f du30
y k dx3
dU
0 Y
dx
3 y
(12) (13)
where Q = m — k • U0 is frequency with Doppler shift.
The amplitude function of the vertical velocity component u30 satisfies the equation
d 2u3, dx,2
+ku,,
2 ifdfa^
2 f
2 dU^
Q2 _f2 Q(Q2 _ f2)
du
dx
d 2U0
k(W2 _Q2) + Qa-Ur
dx3 Q2 _ f2
if
d 2U 20 dx 2
2 ifk
dU0 dU20
d^3 d^3
Q( Q 2_ ff
= 0,
(14)
where N2 = _—
is squared Brunt-Vâisâlâ frequency.
p0 ^3
Boundary conditions for u30 are as follows:
u30(0) = u30 (_H ) = 0.
(15)
Equation (14) has complex coefficients, therefore, generally speaking, its solution is a complex function and boundary value problem (14)-(15) must also have complex values of natural frequencies mat a fixed wave number [34-43]. Indeed, subsequent calculations reveal that the wave frequency has a small imaginary part. Boundary value problem (14)-(15) was solved analytically in [46, 47] without taking into account the Earth rotation at / = 0 on a flow with a constant vertical velocity PHYSICAL OCEANOGRAPHY VOL. 31 ISS. 2 (2024) 165
gradient at constant Brunt-Vaisala frequency. A solution is obtained in the form of imaginary order modified Bessel functions. The dispersion equation was solved using both asymptotic and numerical methods. It is shown that the flow leads to a noticeable anisotropy of the dispersion curves; in particular, frequency isolines in the plane of horizontal wave numbers can open up. Equation (14) has real coefficients at f = 0 , so the wave frequency and the eigenfunction of internal
waves are real when the Richardson number is larger than 1/4 [46, 47].
Nonlinear effects. The flow velocities in the Euler u and Lagrange uL representations are related by relation 5 [36, 43] up to second-order terms in wave steepness
ul = u+(Jul dxV)u . (16)
0
Solving this equation by iteration method up to terms quadratic in the wave amplitude, we obtain an expression for the average Lagrangian velocity after averaging over the wave period
uL = U +(j udxV)u , (17)
0
where vectorU(U0,V0) is average flow velocity; u is field of wave Euler velocities;
the overbar means averaging over the wave period. The second term in (17) is the Stokes drift velocity which is determined by the following formula [34, 36, 38, 39, 41-43, 48]:
u s = (J u dxV)u. (18)
0
Vertical component of the Stokes drift velocity is nonzero for the complex frequency [39, 42]:
u3S = iA1 A1\1 _-4 ]'d (u30u3*0) , (19)
^ a a ) dx3 v '
where A1 = A exp(Sa • t) , Sa = Im(a) is wave frequency imaginary part. Two horizontal components of the Stokes drift velocity are determined by the following formulas:
= A A*
U\S = ,
1 d
c dx3
( \
+ c.c.
du^0
U 30
V dx3 )
(20)
5 Dvoryaninov, G.S., 1982. Effects of Waves in the Boundary Layers of the Atmosphere and Ocean. Kiev: Naukova Dumka, 176 p. (in Russian).
u2S = A1A1
1 du,.
aQ k dx,
if
du
30
_ A, a,
c Q2 +c.c.
(
Q
v dx3
0
_ k
dU2 dx
0
+ c.c.
/
du30 dU2 + if d
v dx3 dx3 k dx3
30 d U2 ^
r+"30 d"2)
+k
dU
0
dx
dU20 + if du3
dx k dx
'3 )
(21)
We find the vertical wave mass flux taking into account formulas (10), (13) [37, 39]:
pu3 = № 1Q
i p0 f dU0 du30 dp0 p0 f dUj0 \ f du30 . dU2
u 30 I iu3
k g dx3 dx3
dx3 gQ dx3 1 k dx.
u30 + c.c.
.. (22)
The equation for a correction to average wave density p that does not oscillate on a time scale, accurate to terms quadratic in the wave amplitude, has the form:
dp +u dp i tt0 dp i dpu i dpu t dpu3
- + U.
+ u,
dp0
+ u.
dp0
+ u,
dp0
dt dx1 cx2 cxl cx2 dx3 cxl cx2 dx Considering relations (4), (5), equation (23) is transformed to the form:
= 0. (23)
dp + U dp + U .dp +5pu1 . 5Pu2 , 5Pu3
dt
dx,
dx2 dxl
dx, dx,
_ u,
M dU2
g dx,
+ u,
pf dU1
g dx
+ u,
= 0.(24)
dx.
In the horizontally homogeneous case equation (24) takes the form:
dp + Spu dt dx.
3 .. "puf dU2 _ p0f U _ 5p0=0
dx.
- + u,
- + u,
A-3 6 6
Integrating equation (25) over time, we obtain:
dx.
A p = _J
dpu3 „, po f dU2
dx.
v dx3
-_ u.
+ u
pf g dx,
+ u.
dx.
(25)
(26)
■3 )
Substituting pu3 from formula (22) and three components of the Stokes drift velocity from formulas (19)-(21) into expression (26), after integration we obtain:
Ap =
3pu
3 o p0f dU0 0 p0 f dU0
dx.
_ u.
dx
dx
dpc dx.
where
-0 I a\2 I i
pu3 =1 AMq
i p0 f dU° du30
u0 =
1S _
Ml2
dx^ dx^
du*
_ u.
p0f dU10 \ f du3(i k dx.
dx3 g Q dx3
dU
0
_ iu.
dx
■3 )
a dx.
dx,
c.c.
3 )
u3S =
ilAl21"° ^ |-T"(u30u3*0) =
1 a a ) ax3 7
= 1 A[ '
1 du30 œQ k dx3
if
du*
- k
dU 20
+ c.c.
- A"
œ Q0
(
Q
du30 dU0 + f d u30 +u
dxz dxz k dx3
d 0U dx
0
+ k
dU
0
dx
dU
0 + if du3l
dx k dx
3 y
+ c.c. !
Correction to density Ap determined by formula (27) is a vertical fine structure generated by an internal wave and irreversible [37, 39]. The specified correction is retained after the wave train passage.
u
Calculation results
We calculate the vertical fine structure generated by the internal wave using the data from the third stage of the 44th cruise of R/V Mikhail Lomonosov at the Black Sea northwestern shelf 6 [34, 35, 39-43].
Fig. 1 shows the time course of temperature isolines obtained by four gradient-distributed temperature sensors.
The instruments were hung from the board of the vessel and located at the following depths: 5-15 m - the first instrument; 15-25 m - the second instrument; 25-35 m - the third instrument; 35-60 m - the fourth instrument (indicated in Fig. 1 by Roman numerals I - IV, respectively).
14 15 IS 1.7 IS 19 20 Î1 22 t. tl
F i g. 1. Dependence of vertical displacements of temperature isolines on time
Figure 1 shows clearly a powerful wave train of 15-minute second-mode internal waves. The maximum amplitude at elevations is 0.5 m.
Figure 2, a shows Brunt-Vâisâlâ frequency dependence on the vertical coordinate, and Fig. 2, b - dependence on two components of flow velocity.
6 Panteleev, N.A., 1985. Report on the Work on the 44th Voyage (3rd Stage) of the NIS Mikhail Lomonosov August 7 - September 15, 1985. Sevastopol: MHI Academy of Sciences of the Ukrainian SSR. Vol. 1, 135 p. (in Russian).
F i g. 2. Dependence of Brunt-Vaisala frequency (a) and flow velocity components (b) on the vertical coordinate
xltf
0.025
« 0,2 a
b 0.015
w
O.Ol
0.0U5
fl >
/j?
•r JT /
a
0.05 0.1 0.15 0.2 k, rad/m
4 2
TZ u.
é -4
2
\ 1 b
0.05 0.1 0.15 0.2 k, rad'm
F i g. 3. Dispersion curves (a) and graphs of the frequency imaginary part (b) of the first (1) and second (2) modes
The boundary value problem (14), (15) is solved numerically using the implicit Adams scheme of the third order accuracy [34-43]. The shooting method is applied to determine the wave number and frequency imaginary part at a fixed wave period for a given mode of internal waves. The resulting solution is complex, as well as the wave frequency with a small imaginary part. Figure 3, a shows the dependence of frequency real part ro0 = Re(ro) on the wave number for the first two modes. Figure 3, b reveals the dependence of the frequency imaginary part on the wave number. The imaginary part of first-mode frequency 5® = Im(®)is negative, i.e. the wave is slightly attenuated.
For the second mode, the imaginary part of the frequency is negative when frequency a0 is less than 12 cph; if the frequency is higher, its imaginary part is
positive, i.e., weak attenuation in the low-frequency region is replaced by weak amplification in the high-frequency region. The first-mode damping decrement is greater in absolute value than the second-mode damping decrement for a fixed wave number. For 15-minute internal waves of the second mode, the attenuation decrement is equal to Sa = — 1,4 -10 _8 rad/s, the wavelength is 197 m.
We calculate the vertical component of the Stokes drift velocity using formula (19). Normalizing factor A1 is found by the known maximum amplitude of 0.5 m
vertical displacements. The dependence of the Stokes drift velocity vertical component on the vertical coordinate for the first two modes is demonstrated in Fig. 4, a, the wave frequency was 4 cph.
Fig. 4, b shows the average density vertical profile.
ui%, m/s p(), kg/m!
F i g. 4. Dependence of vertical component of the Stokes drift velocity on the vertical coordinate of internal waves of the first (1) and second (2) modes (a); average density profile (b)
Fig. 5 shows the profiles of vertical mass flow determined by formula (22) and non-oscillating correction to density (formula (27)) at t = 9 h for the first two
modes of internal waves with 4 cycle/h frequency at the same maximum amplitude of 0.5 m.
The vertical wave mass flux of the first mode predominates significantly in the upper 40-meter layer, while the wave fluxes of the first and second modes are comparable in absolute value at greater depths. The intensity of the generated vertical fine structure of the density field for the first mode is greater than for the second one at the same maximum wave amplitude.
F i g. 5. Profiles of the vertical wave mass flux (a) and fine structure of density field (b) for the first (1) and second (2) modes of internal waves
Further we compare the wave fluxes and the generated fine structure of inertia-gravity internal waves of different frequencies. To do this, we consider the second mode with frequencies of 4, 1 and 12 cph at the same maximum wave amplitude of 0.5 m.
Figure 6, a shows vertical wave mass fluxes for waves of the indicated frequencies, and Fig. 6, b - profiles of the generated fine structure of the density field.
pi/,, kg/m3s Ap, kgW
F i g. 6. Profiles of the vertical wave mass flux (a) and fine structure of density field (b) of the second mode for 15-minute internal waves (red curve), one-hour internal waves (blue curve) and 5-minute internal waves (green curve)
The vertical wave mass fluxes of the second mode of 15-minute, one-hour and 5-minute internal waves are co-directed and decrease in absolute value with the wave period decreasing. Fine-structure corrections to density for waves with 15-minute
and one-hour periods are close and only enhance the effect when summed up. The mentioned corrections do not introduce inversions into the vertical density distribution. A similar calculation was carried out for the first mode of internal waves. Fig. 7 shows the graphs of the vertical wave mass flux and generated fine density structure of the first-mode internal waves for 4, 1, and 12 cph frequencies.
pu,, kg/iivs Ap. kg/mJ
F i g. 7. Profiles of the vertical wave flux of mass (a) and fine structure of density field (b) of the first mode for 15-minute internal waves (red curve), one-hour internal waves (blue curve) and 5-minute internal waves (green curve)
In the first mode, the vertical wave mass fluxes and fine-structure corrections are almost the same for waves with 1 and 4 cph frequencies. They differ most noticeably in the pycnocline for waves with 12 cph frequency.
It is of interest to study the impact of horizontal inhomogeneity of the average density field on vertical wave mass fluxes and generated fine structure. In the horizontally homogeneous case, the terms containing horizontal gradients of dp do
the average density —0 and —0 are neglected in equation (4). Fig. 8 shows
dx dy
the dependence of the vertical wave mass fluxes and generated fine structure on the vertical coordinate for the second-mode internal waves with 4 cph frequency in the horizontally homogeneous and inhomogeneous cases with the same maximum wave amplitude of 0.5 m.
Thus, the amplitude of fine-structure oscillations and vertical wave mass fluxes in the horizontally inhomogeneous case are higher than in the homogeneous one.
Similar calculations were also performed for the first mode of internal waves. Fig. 9 shows the profiles of the vertical mass flux and generated fine structure for the first mode of internal waves with 4 cph frequency in the horizontally inhomogeneous and homogeneous cases.
F i g. 8. Vertical distribution of the wave fluxes of mass (a) and generated fine structure (b) for the internal wave of the second mode in the horizontally inhomogeneous (red curve) and homogeneous (blue curve) cases
-2 -] 0 *10"7 0 2 4 *10'J
pTiit kg/nrs Ap, kg/mJ
F i g. 9. Dependence on the vertical coordinate of the wave mass flux (a) and generated fine structure (b) for the internal wave of the second mode in the horizontally inhomogeneous (red curve) and homogeneous (blue curve) cases upon the vertical coordinate
Similarly, the first mode vertical wave mass fluxes and fine-structure correction to the density are higher in the case that is horizontally inhomogeneous in average density.
Conclusion
The paper presents a mechanism for generating vertical fine structure due to vertical wave mass fluxes. These fluxes are nonzero for the inertia-gravity internal waves in the presence of a flow in which the velocity component normal to the direction of wave propagation depends on the vertical coordinate. Unlike
previous works, we do not assume horizontal homogeneity of average density field. In contrast, the flow is assumed to be geostrophically balanced, i.e. vertical velocity shifts are balanced by the horizontal gradient of the average density field. This gradient is found from the "thermal wind" relations. It is shown that consideration of horizontal inhomogeneity of average density field increases both vertical wave mass fluxes and generated fine structure, the amplitude of fine-structure oscillations. The wave mass fluxes and fine-structure corrections to density are very close for waves of 1 and 4 cph frequencies, i.e. the overall effect only intensifies for waves of different frequencies.
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Submitted 05.09.2023; approved after review 25.12.2023;
accepted for publication 18.01.2024.
About the authors:
Aleksandr A. Slepyshev, Leading Research Associate, Marine Hydrophysical Institute of RAS (2 Kapitanskaya Str., Sevastopol, 299011, Russian Federation), DSc (Phys.-Math.), ResearcherlD: V-6948-2017, ORCID ID: 0000-0002-9259-7558, slep55@mail.ru
Nikita O. Ankudinov, Leading Engineer-Researcher, Marine Hydrophysical Institute of RAS (2 Kapitanskaya Str., Sevastopol, 299011, Russian Federation), ankudinff@gmail.ru
Contribution of the co-authors:
Aleksandr A. Slepyshev - problem statement, theoretical part of the work: mathematical formulation of the model, derivation of all formulas
Nikita O. Ankudinov - numerical calculations using the proposed model, graphical presentation of the results
The authors have read and approved the final manuscript.
The authors declare that they have no conflict of interest.
