Conservative Properties of the Variational Free-Lagrange Method for Shallow Water

The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the additional property that it preserves the variational structure of shallow water. The VFL method was first derived in this context by Augenbaum (1984) who discretised Hamilton’s action principle with a free-Lagrange data structure. The primary purpose of this article is to demonstrate, through the use of geometric integrators, that the VFL method exhibits no secular drift in the energy error over long-time shallow water simulations. We additionaly derive the semi-discrete divergence and potential vorticity equations in the Lagrangian frame, both of which augment the description of the discrete momentum equation by characterising the evolution of its respective irrotational and solenoidal components. Like the continuum equations, the former exhibits a divU term which indicates that the flow has a very strong tendency towards a purely rotational state. The latter equation provides crucial insight into the form of discrete curl operator required for conservation of discrete potential vorticity. Numerical results demonstrate the conservative properties of the VFL method and motivate the application of the VFL method to long-time climate simulations.