Mathematical Description of the Flows near the Bottom of the Ocean

Abstract—We construct an explicit solution for a boundary value problem for a system of partial differential equations which describes small linearized motions of three-dimensional stratified flows in the half-space. For large values of t, we obtain uniform asymptotical decompositions of the solutions on an arbitrary compact in the half-space. In the vicinity of the boundary plane, we establish the asymptotical properties of the boundary layer type: we can observe a worsening of the decay in the approximation to the bottom. The results can be used in the meteorological modelling of water flows near the bottom of the Ocean, as well as Atmosphere flows near the Earth surface.


I. INTRODUCTION
We consider a system of equations of the form  The equations (1) are deduced in [1], [2] under the assumption that the function of stationary distribution of density is performed by the exponentially decreasing function 3 * Nx e   , which corresponds to the Boltzmann distribution. The system (1) can be considered as modelling Manuscript received April 20, 2019; revised July 25, 2019. This work was supported by "Programa de Investigaciones 2018-2020", Facultad de Ciencias, Universidad de los Andes.
A. Giniatoulline is with the Department of Mathematics, Los Andes University, Cra. 1 Este No 18A-10, Bogota D. C., Colombia, South America (e-mail: aginiato@uniandes.edu.co). the linearized movement of three-dimensional fluid in a homogeneous gravitational field (Ocean or Atmosphere), as it can be visualized in the following Fig. 1: Fig. 1. Initial distribution of density as a function of 3 x .
Some physical properties of the flows described by (1) can be found, for example, in [3]- [5].
The Cauchy problem for (1) was considered in [6], where the explicit solution was constructed and p L -estimates were obtained. The half-space domain for (1) was considered in [7], where the solution was constructed using unilateral Fourier transform; the uniqueness of the solutions was proved in a class of growing functions, and it was established that the solution decays for large t as 1 t . The spectral properties of the differential operator of (1) for compressible viscous case were studied in [8]. In this work, we consider the system (1) in the half-space and the boundary condition Without loss of generality, we may assume velocity and density and use the same notation for modified functions: In that way, instead of system (1), we will consider the system:

II. CONSTRUCTION OF THE SOLUTION
To construct the solution of (4) in the half-space 3 R  , we will solve an ordinary differential equation with respect to 3 x . Here we proceed in a different way if we compare it with the solution from [7], where "unilateral" sine and cosine Fourier transforms were used with respect to 3 x . Supposing that the initial data  (4). In this way, we obtain the following system: In particular, for the function P from (5) we have the following ordinary differential equation: In terms of Fourier-Laplace transform, the boundary condition (3) will take the form We denote the characteristic roots of the equation (6) as In what follows, we will consider 0 Re 0 The general solution of (6) contains two arbitrary constants which depend on  and   .
One of them can be defined from the condition (7), and the other -from the condition 3 lim 0 x P   . In this way, we obtain If we substitute the representation (8) in (5), we obtain a linear algebraic system for the functions . After solving that system, we have the following: ; where 0 J and 1 J are Bessel functions. We use (9) and apply the inverse Fourier and Laplace transforms to (8 (10) Analogously, for the rest of the components of the solution for the problem (2)-(4), we obtain the representations:

III. ASYMPTOTICAL DECOMPOSITION ON A COMPACT SET FOR LARGE VALUES OF TIME AND BOUNDARY LAYER PROPERTIES AT THE BOTTOM
It is easy to see that the obtained representations (10)-(11), essentially, have the same qualitative form. Thus, without loss of generality, we will investigate in detail the asymptotical behavior of one of the components of the solution. In particular, let us consider the function   To obtain the estimate (17), the expression of the main term of the asymptotic expansion was used ( [10]):