Energy‐preserving integrations of the primitive equations on the sphere

Long‐term global integrations of the primitive equations for the free‐surface model are performed using spherical polar co‐ordinates. Two methods are used to overcome the computational drawback of excessive resolution near the poles. In the first, time‐steps vary latitudinally on a grid with constant angular increments. Stable integrations result, in which the evolutions closely match (save for phase‐truncation due to divergence) those of the analytic solution of the non‐divergent vorticity equation for initial data in which non‐linear interactions vanish. In the second set of integrations zonal resolution varies latitudinally, and the resulting differential phase‐truncation causes spurious interactions which lead to rapid departure from the analytic solution. In both cases finite difference approximations are used which preserve domain sums of mass and of total energy (kinetic plus potential), in analogy with properties of the continuous equations. The space‐differencing method, together with the time‐smoothing inherent in the variable time‐step scheme, may have helped to suppress non‐linear instability in the first set of integrations.