## Hamiltonian tools applied to nonhydrostatic modeling

#### Almut GassmannMax-Planck-Institut für Meteorologie

Before starting out developing an NWP model, one has to formulate the model equations
very carefully. Full moist turbulent flow equations require to consider the heterogene
composition of the air, which contains vapour, liquid and frozen particles at the same
time. This has severe implications on the Hamiltonian formulation of the dynamical
system. In fact, a Poisson bracket description is unique only in the dry limit case,
phase conversions and friction terms give addional source terms to the dry limit
Hamiltonian.

The formulation of model equations in Poisson bracket form allows us to choose prognostic
variables as desired. Thus we can avoid the need for an explicit energy budget equation.
In our case, we can formulate an entropy equation, represented by the potential
temperature, so that the entropy source terms are unveiled and attributable to a certain
physical process.

With focus on the numerical discretisation, it is straight-forward to discretize Poisson
brackets, which reflect the product rule for derivatives in space. A similar discrete
product rule might also be invoked for the temporal discretisation. Classical
forward-backward time integration can be interpreted in that way.

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