Volume 20, Issue 1

The inverse problem of the mathematical theory of tides is considered, i.e., a problem of functions defining boundary values on liquid parts of the boundary. A closure equation is based on the observation data on the function of the sea level (free surface elevation) on a part of the boundary. The existence and uniqueness of the solution is investigated. We formulate an iteration algorithm for solving the problem studied.

In this paper we consider two nonstationary heat convection–diffusion models. The first model describes convection–diffusion processes in the 'surface' ocean layer. The second one is a 3D model and describes heat propagation processes in the whole ocean. We pose identification problems for the models and propose methods for solving the problems posed. In the case of the 2D model, the methods of its solution are theoretically justified. Experimental results using real data for the water area of the Indian Ocean are given.

We consider a 'data assimilation problem' for nonlinear delay differential equations. Our problem is to find an initial function that gives rise to a solution of a given nonlinear delay differential equation, which is a close fit to observed data. A role for adjoint equations and fundamental solutions in the nonlinear case is established. A 'pseudo-Newton' method is presented. Our results extend those given by the authors in [(C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for delay differential equation: Part I: The continuous problem & an integral equation analysis. NA Report No. 431 , MCCM, Manchester, England, 2004.), (C. T. H. Baker and E. I. Parmuzin, Analysis via integral equations of an identification problem for delay differential equations. J. Int. Equations Appl. (2004) 16 , 111–135.)] for the case of linear delay differential equations.

In this paper, we study inviscid shallow-water equations in a domain on a rotating sphere, in which advection terms are omitted and the nonlinear force of bottom friction is taken into account. We prove the existence and uniqueness of the solution and the continuity of the resolving operator. Then we study the solvability of the basic and adjoint linear systems.

We consider the problem of variational assimilation of observational data with the aim to reconstruct the initial condition in a locally one-dimensional model of vertical heat exchange in the ocean, which is described by the nonstationary heat equation with a nonlinear diffusion coefficient. We investigate the solvability of the data assimilation problem, develop and justify algorithms for its numerical solution. The results of numerical experiments are given.

The aim of this paper is to formulate a mathematical σ-model of thermohaline sea dynamics and its numerical solution. The novelty of the work is taking account of the nonhydrostatic effect, establishing the law of conservation for a complete nonlinear problem, and the generalization (for the nonhydrostatic case) of the numerical algorithm for solving the problem. The algorithm is based on the method of splitting with respect to physical processes. We describe nonhydrostatic effects at a separate splitting stage and introduce a new function describing the deviation of pressure from the hydrostatic one, which is calculated at an additional splitting stage. The elimination of this stage from the chain of split systems automatically leads to a special model case describing hydrostatic dynamics. Further, main attention is given to the barotropic dynamics problem. We formulate two finite difference algorithms of its solution: the first one by solving a linear hyperbolic system in terms of ( u , v ,ζ ), the second algorithm by reducing it to the equation ζ .