Volume 25, Issue 6

This paper contains a review of the progress in mathematical study of the ocean dynamics equations and data assimilation problems for oceanic models. Solvability of boundary value and data assimilation problems for quasigeostrophic models and for general ocean equations are given. The existence of strong solutions for the simplified system of basic equations of the ocean is studied. In addition, the existence and uniqueness theorems for the model of ocean tides are considered.

In this paper we propose a finite difference approximation of the tidal wave system of equations in spherical coordinates on an unstructured grid, prove its stability and study the properties of the matrix of the finite difference scheme. In particular, we obtain the same results as for the case of the Cartesian coordinates. Namely, it is proved that the matrix after elimination of a part of the unknowns is an M -matrix, so it is possible to use the classical numerical methods for solving this system of linear algebraic equations.

Let an operator S act in a Banach space H and determine some evolutionary process S i ( u ), i = 0, 1, . . . , u ∈ H . The problem of construction of the set of initial conditions u 0 for a given point z 0 so that the trajectory { S i ( u 0 )} approaches the trajectory { S i ( z 0 )} for 0 < i ⩽ n is considered for the operator S . In this case the condition of closeness of z 0 and u 0 is not imposed. The original problem is reduced to the construction of a set of locally stable manifolds forming a subset of the globally stable manifold of the point z 0 . Based on this approach, the problem of nonlocal stabilization is solved, i.e., given a point a 0 , an ‘admissible’ replacement of the operator S by S ∗ is constructed so that the trajectory, approaches the trajectory { S i ( z 0 )} for 0 < i ⩽ n . An iterative method is proposed in the paper for the construction of an approximation of the global stable manifold for an abstract semidynamical system of the saddle type. Using this method, the problem of nonlocal stabilization is solved for the Lorenz equation and for an equation of the Navier–Stokes type.

An approach to estimation of the immune system efficiency is proposed in the paper. The sum of energy costs of the immune system support and the body losses caused by infectious diseases determines the energy value of the immune system protection. It is assumed that the state of the immune system providing the minimum energetic cost for the given environment is the norm, because it increases adaptability. The adaptation of the immunity to external conditions is realized through the minimization of the energy cost. The adaptation of the immune system may result in the maintenance of chronic infections and formation of immunodeficiency. Mathematical models of an immune system should be constructed as components of a model of the homeostatic maintenance supersystem.

The problem of mathematical modelling of the baroclinic dynamics of the World Ocean and its large areas is considered. The mathematical model has been developed at the Institute of Numerical Mathematics RAS (INMRAS) and is based on the equations of general circulation represented in the generalized σ-system of coordinates with a free surface in the hydrostatics and Boussinesq approximation. The computational solution technique for this problem is based on multicomponent splitting and has a flexible, hierarchically scalable modular structure. The main ‘cycle’ is the decomposition of a complicated system of equations into a series of large and energetically balanced subsystems called modules, according to physical processes. The complete numerical model is determined by the set of split subsystems and approximate the original model with some order of time accuracy. Along with the solution of direct prognostic problems, the method of multicomponent splitting can be efficiently used for solution of four-dimensional problems of observation data variational assimilation. The computing methodology is illustrated by several problems of large-scale ocean circulation, including the problem with four-dimensional variational initialization of hydrophysical fields.