Salmon's Hamiltonian approach to balanced flow applied to a one-layer isentropic model of the atmosphere

Salmon's Hamiltonian approach is applied to formulate a balanced approximation to a hydrostatic one-layer isentropic model of the atmosphere. The model, referred to as the parent model, describes an idealized atmosphere of which the dynamics is closely analogous to a one-layer shallow-water model on the sphere. The balance used as input in Salmon's approach is a simplified form of 'linear balance', in which the balanced velocity vb is given by v. = k x V j-1 (M - M). Here k is a vertical unit vector, f is
the Coriolis parameter, M is the Montgomery potential and M is the value of the Montgomery potential at the state of rest. This form of balance behaves acceptably on the whole sphere, in contrast with 'classic' geostrophic balance, vs = k x f-1V M, which forces the meridional wind velocity to be zero at the equator. Salmon's Hamiltonian approach is applied to obtain an equation for the time-change of the balanced velocity that guarantees both material conservation of potential vorticity as well as conservation of energy. New in this application of Salmon's approach is a nonlinear relation between Montgomery potential and surface pressure (characteristic for an isentropic ideal gas in hydrostatic equilibrium) in combination with spherical geometry and a variable Coriolis parameter. We will discuss how the unbalanced velocity a can be calculated in a practical way and how the model can be stepped forward in time by advecting the balanced potential vorticity with the sum of the balanced and unbalanced velocity. The balanced model is tested against a ten-day period from a long integratife with the parent model.