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

Abstract

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 vb = k×δf−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 is used in preference to standard geostrophic balance, vb = k × f−1δM, which forces the meridional wind velocity to be zero at the equator. Salmon's Hamiltonian technique is applied to obtain an equation for the time rate of change of the balanced velocity that guarantees both the 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 discuss how the unbalanced velocity va 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 total velocity v = vb + va. The balanced model is tested against a ten-day integration of the parent model.