Conservative parameterization schemes

TL;DR

This paper introduces methods to develop physical parameterization schemes in weather and climate models that preserve conservation laws and symmetries, ensuring physically consistent simulations.

Contribution

It presents direct and inverse classification methods for constructing conservative parameterization schemes based on differential equation analysis.

Findings

Methods successfully applied to shallow-water equations

Conservative schemes preserve physical invariants

Invariant schemes also retain symmetries

Abstract

Parameterization (closure) schemes in numerical weather and climate prediction models account for the effects of physical processes that cannot be resolved explicitly by these models. Methods for finding physical parameterization schemes that preserve conservation laws of systems of differential equations are introduced. These methods rest on the possibility to regard the problem of finding conservative parameterization schemes as a conservation law classification problem for classes of differential equations. The relevant classification problems can be solved using the direct or inverse classification procedures. In the direct approach, one starts with a general functional form of the parameterization scheme. Specific forms are then found so that corresponding closed equations admit conservation laws. In the inverse approach, one seeks parameterization schemes that preserve one or more pre-selected conservation laws of the initial model. The physical interpretation of both classification approaches is discussed. Special attention is paid to the problem of finding parameterization schemes that preserve both conservation laws and symmetries. All methods are illustrated by finding conservative and conservative invariant parameterization schemes for systems of one-dimensional shallow-water equations.