Internal Gravity Waves in a Stratified Medium with Model Shear Flow Distributions
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rnal Gravity Waves in a Stratified Medium with Model Shear Flow Distributions V. V. Bulatova,* and Yu. V. Vladimirova,** aIshlinsky
Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia * e-mail: [email protected] ** e-mail: [email protected] Received March 1, 2020; revised March 12, 2020; accepted March 12, 2020
Abstract—The problem of the field of internal gravity waves in a stratified medium of finite depth is considered for model distributions of background shear flows. The constant distribution of the buoyancy frequency and various linear dependences of background shear flow on the depth are used to solve the problem analytically. The dispersion relations expressed in terms of the modified Bessel function of imaginary index are obtained. In satisfying the Miles–Howard stability condition in the case of large Richardson numbers, the Debye asymptotics of the modified Bessel function of imaginary index are used to construct the analytical solutions. The properties of the dispersion relation are studied and the basic analytical characteristics of the dispersion curves are investigated as functions of the parameters of background shear flows. The phase patterns of the excited fields are calculated numerically for various wave generation models. Key words: stratified medium, internal gravity waves, buoyancy frequency, shear flows, modified Bessel function DOI: 10.1134/S0015462820050031
The important characteristic of natural stratified media (ocean and atmosphere) is the presence of flows with vertical velocity shear which depend only slightly on time and horizontal coordinates. In the Earth’s ocean such flows can manifest themselves in the region of seasonal thermocline and have an appreciable effect on internal gravity wave (IGW) dynamics. If the horizontal scale of variations in the flows is much greater than the internal gravity wave length and their scale of time variability is much greater than the internal wave periods, then such flows can be considered to be steady-state and horizontally homogeneous [1–7]. The aim of the present study is to construct the solutions that describe the dynamics of fields of internal gravity waves in a stratified medium of finite depth for model velocity distributions of background shear flows. 1. FORMULATION OF THE PROBLEM We will consider an ideal vertically stratified incompressible medium of finite depth H. Let (U (z) , V(z)) be the velocity vector of background shear flow at a depth z. In the Boussinesq approximation the equation for small perturbations of the vertical velocity component w takes the form [1–3, 6, 8]: 2 D Δw − D ∂ 2U ∂w + ∂ 2V ∂w + N 2(z)Δ w = 0, 2 Dt ∂z 2 ∂x ∂z 2 ∂y Dt 2 2 Δ = Δ2 + ∂ 2 , ∂z
2 2 Δ2 = ∂ 2 + ∂ 2 , ∂x ∂y
N 2(z) = −
g d ρ0(z) , ρ0(z) dz
(1.1)
D = ∂ + U (z) ∂ + V (z) ∂ , Dt ∂t ∂x ∂y
where N 2(z ) is the square of the Brunt–Väisälä frequency (buoyancy frequency), g is the free fall acceleration, and ρ0(z) is the undisturbed density of the medium. The boundary conditions are taken in the for
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