A maximum entropy approach to the parametrization of subgrid processes in two-dimensional flow

In numerical models of geophysical fluid systems parametrization schemes are needed to account for the effect of unresolved processes on processes that are resolved explicitly. Usually, these parametrization schemes require tuning of their parameters to achieve optimal performance. We propose a new type of parametrization that requires no tuning as it is based on an assumption that is not specific to any particular model. The assumption is that the unresolved processes can be represented by a probability density function that has maximum information entropy under the constraints of zero average time-derivatives of key integral quantities of the unresolved processes. In the context of a model of a simple fluid dynamical system it is shown that this approach leads to definite expressions of the mean effect that unresolved processes have on the processes that are resolved. The merits of the parametrization, regarding both short-range forecasting and long-term statistics, are demonstrated by numerical experiments in which a low-resolution version of the model is used to simulate the results of a high-resolution version of the model. For the fluid dynamical system that is studied, the proposed parametrization turns out to be related to the Anticipated Potential Vorticity Method with definite values of its parameters.