  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.
