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Coupled atmosphere–ocean models are widely used for climate change modelling. However, there is now more and more evidence on necessity to use such models in numerical weather prediction at different time scales. A coupled model is developed at the Institute of Numerical Mathematics, Shirshov Institute of Oceanology (Russian Academy of Sciences), and Hydrometeorological Research Centre of Russia. Particularities of program implementation for this model are discussed. The atmosphere model SLAV and the World Ocean model INMIO are coupled using the original program for models coupling. The results of numerical experiments with the coupled model demonstrate an agreement with observation data and show a possibility to use this model for probabilistic weather forecasts at time scales from weeks to year.

The fluctuation dissipation theorem (FDT), a classical result coming from statistical mechanics, suggests that, under certain conditions, the system response to external forcing can be obtained using the statistics of natural fluctuation of the system. The application of the FDT to the most sophisticated climate models and the real climate system represents a difficult problem due to the huge dimensionality of these systems and the lack of the data available for proper sampling of the system natural variability. As a consequence, one has to use some regularization procedures constraining the form of permitted perturbations. Naturally, the skill of the FDT depends on the type and parameters of the regularization procedure. In the present paper we apply FDT to predict the response of a recent version of the NCAR climate system model (CCSM4) to salinity and temperature forcing anomalies in the North Atlantic. We study the sensitivity of our results to the amount of available data and to key parameters used in our numerical algorithm.

A class of two-point compact difference schemes of the second–third orders of accuracy on a twopoint coordinate stencil is considered for a one-dimensional transfer equation. All difference schemes are based on interpolation polynomials constructed on a given stencil. Based on the behaviour of the solution and the character of interpolation polynomials, we propose hybrid compact difference schemes of 2–3rd orders of accuracy on smooth functions producing solutions weakly smoothing the front of discontinuities. The study of grid convergence for the constructed difference scheme is carried out and the propagation of a pulse of complex form is simulated numerically for studying the behaviour of the difference scheme on discontinuous solutions.

Some problems of the theory of electron transfer in gases under the action of a strong external electric field is considered in the paper. Based on the three-dimensional ELSHOW algorithm, samples of states of particles in an electron avalanche are obtained for a given time moment in order to calculate the corresponding ‘diffusion radii’ and diffusion coefficients. Randomized projection estimators and kernel estimators (for test purpose) are constructed with the use of grouped samples for evaluation of the distribution density of particles in an avalanche. Test computations demonstrate a high efficiency of projection estimators for calculation of diffusive characteristics.

The weighted method of similar trajectories (MST) allows one to construct estimators of functionals on a single Markov chain simultaneously for a given range of parameters of the problem. Choosing an appropriate Markov chain, we take into account additional conditions providing the finiteness of MST variance. A modification of the weighted MST with branching of chain trajectory is constructed in the paper according to the parametric maximum of the auxiliary weight. It is proved that the computational cost of this algorithm is bounded if the basis functionals are also bounded. Numerical study of the efficiency of the modified MST in comparison with analog modelling was carried out on the example of the standard problem of transfer theory on estimation of the probability of albedo and transmission of a particle.