A maximum entropy approach to the interaction between small and large scales in two-dimensional turbulence

A numerical model of a fluid dynamical system is necessarily limited to scales of motion down to a given minimum length scale. All interactions with smaller scales have to be dealt with in some approximate way by methods that are called parametrizations. The problem can be studied particularly well in a system that is geophysically relevant yet relatively simple and therefore amenable to very high resolution modeling: two-dimensional turbulence. By using spectral models with high and low spatial resolutions and concentrating on the energy and enstrophy and their time-derivatives as functions of wavenumber we study the process of interaction between small and large scales. We show that this interaction can be modelled quite well by representing the small scales by means of a probability density function that is based on the principle of maximum entropy and on constraints on their energy and enstrophy as well as their time-derivatives. Two sets of constraints are discussed and assessed in terms of the performance of their corresponding parametrizations.