Optimal balance for rotating shallow water in primitive variables

ABSTRACT Optimal balance is a near-optimal computational algorithm for nonlinear mode decomposition of geophysical flows into balanced and unbalanced components. It was first proposed as “optimal potential vorticity balance” by Viúdez and Dritschel [J. Fluid Mech., 2004, 521, 343] in the specific setting of semi-Lagrangian potential vorticity-based numerical codes. Later, it was recognised as an instance of the more general principle of adiabatic invariance of fast degrees of motion under slow perturbations. From this point of view, the system is slowly deformed from a linearised configuration to the full nonlinear dynamics. In the former, linear analysis yields an exact separation of balanced and unbalanced flow. In the latter, a given base-point coordinate, e.g. the height or potential vorticity field, can be matched. This formulation leads to a boundary value problem in time. In this paper, we show that this more general viewpoint leads to practical implementations of optimal balance on top of a primitive variables (here, velocity-height variables) numerical code. We identify preferred choices for several design parameters. The most critical choices concern the linear projector onto the slow modes at the linear-end boundary and the choice of base-point coordinate at the nonlinear end. We find that, even though the evolutionary model is formulated in primitive variables, potential vorticity based end-point conditions are advantageous. In particular, the only universally robust linear projector is the oblique projector onto the Rossby modes along the gravity-wave modes, which can be interpreted as the distinct non-orthogonal projector onto the Rossby modes that preserves the linear potential vorticity. Hence, the projector can be formulated as an elliptic partial differential equation which holds promise for using the method to produce an accurate nonlinear mode decomposition for more general models without the need to resort to asymptotic analysis.