Volume 17, Issue 5

- In this paper we present a method for constructing an approximate response operator of the average state of an atmospheric general circulation model to small external forcing. The method is based on Kraichnan’s fluctuation-dissipation relationship under the assumption of ‘quasiregularity’ of the systems studied. Its advantage is that only the statistical characteristics of an unperturbed system are needed to calculate the approximate response operator. This method is implemented for CCM0, an atmospheric general circulation model of the National Center for Atmospheric Research, USA. It is shown with the aid of a number of numerical experiments that the constructed approximate response operator allows us to reproduce both the amplitude and the spatial structure of the linear part of the CCM0 model response with high accuracy. Thus, it approximates the real response operator of the model with good accuracy. Therefore using this method we can solve the problem of determining the model sensitivity to small external forcing and the problem of reconstructing the forcing that generates a given response.

We propose the approximation technique of the nonstationary Stokes problem by a system of the Canchy-Kovalevskaya type equations with a symmetric space operator that contains a small parameter ε. We prove the convergence of the solution v of a modified problem to the solution u of the original problem with rate O(ε) and convergence q to p with rate O(√ε) on an infinite time interval as ε → 0.

-The problem of describing the whole dynamics on the globall attractor of a semidynamical system is studied. The solution is reduced to finding the function of rate of attraction to an attractor. The method of determing numerically an estimate for this function is presented

- When cyclic iterative methods (with cycle of length N) are used to solve ill-conditioned equations, instability can occur in the iterative process. There are two reasons for this instability: (a) large nonns r i N of intermediate (transition operators for 1 ≤ i < N, (b) large norms q i N of residual operators for 1 < i ≤ N. In the first case the solution at intermediate iterations may have a large norm. In the second case the round-off errors arising at intermediate iterations can increase dramatically. The described instability phenomena occur when the sequence of parameters is not chosen properly. The ‘stable’ sequences were first proposed by the authors in [1-5]; there the norms r i N could be larger than unity. In [6, 7] the stable algorithms for ordering the iterative parameters for which r i N , ≤ i = 1.... ,N. are presented. In the present paper, for brevity, we derive estimates for q i N only for some values of i and for all i in the limiting case. These algorithms allow us to construct Chebyshev stable infinitely extendable iterative methods [5-7] as well as explicit stable methods for solving stiff systems of differential equations [8, 9]

- We investigate the properties of a control operator in the initial condition reconstruction problem for a viscous barotropic fluid dynamic equation on a sphere. We prove the theorems on the reconstruction problem solvability in special function spaces.