Adiabatic Invariance and Applications: From Molecular Dynamics to Numerical Weather Prediction

A wide class of Hamiltonian systems exhibit a mixture of slow motion with superimposed fast oscillations. Under the assumption of scale separation, these systems can be investigated using the principle of adiabatic invariance. In this paper, we start with a review of some of the main theoretical and numerical findings. We then briefly summarize a few important implications for molecular dynamics (MD) before we provide a more extensive discussion of numerical weather prediction (NWP). In particular, the conservative Hamiltonian particle-mesh (HPM) method is extended to Euler's equation and the fundamental concepts of geostrophic and hydrostatic balance are illustrated on the level of fluid blobs. We also demonstrate numerically that symplectic time-stepping methods are able to maintain hydrostatic balance to high accuracy.