The statistical equilibrium solution of a primitive-equation model

The statistical equilibrium solution of an f-plane, primitive-equation model with a single quadratic energy invariant is determined by numerical integration. The initial condition resembles the atmosphere in terms of the shape and magnitude of its energy spectrum. The equilibrium solution is one in which energy is equipartitioned among all the linearly independent modes of the system. This state is attained after two simulated years.

The approach to equilibrium is explored in detail. It is characterized by (at least) two stages. The first is dominated by quasi-geostrophic dynamics and nonlinear balances. The approximate conservation of quasi-geostrophic potential enstrophy is important during this stage, so that the solution initially tends to the equilibrium solution of a quasi-geostrophic form of the model. The second stage is characterized by a very slow transfer of energy from geostrophic modes to inertial-gravity waves. The rate of transfer of energy during this stage is shown to be very sensitive to initial conditions.