Unstable periodic trajectories of a barotropic model of the atmosphere

Abstract

Unstable periodic trajectories of a chaotic dissipative system belong to the attractor of the system and are its important characteristics. Many chaotic systems have an infinite number of periodic solutions forming the skeleton of the system attractor. This allows one to approximate the system trajectories and statistical characteristics by using periodic solutions. The least unstable orbits may generate local maxima of the system state distribution functions on the attractor. With respect to atmospheric systems this means that orbits may determine dynamic circulation regimes and typical variability modes of the system. In some cases, given a small number of periodic solutions, one can describe the dynamics on the attractor of the system and the basic statistics with sufficient precision. Thus, the information concerning periodic trajectories of a particular dynamic system may be very important for analysis of its behavior.

A search for periodic trajectories is reduced to the solution of a system of nonlinear equations with respect to the initial condition of an orbit and its period. The choice of a numerical solution method and an initial guess is an important aspect here. In this paper we consider the problem of the calculation of periodic trajectories for a barotropic model of the atmosphere. Several methods for determination of periodic orbits of the model are formulated and implemented, including the Newton method with step suppression, the Newton method with a second order tensor correction, the quasi-Newton method with step suppression, the quasi-Newton method with minimization of the error functional and approximate Hessian inversion by the LBFG scheme and the GMRES method. A comparison of the efficiency of these numerical methods and different choices of initial conditions is performed. Various factors influencing the rate of convergence of the methods are considered.