Gökce Tuba Masur

• Optimal balance is a numerical decomposition method of geophysical flows into a balanced and unbalanced components without any asymptotic analysis. It was introduced under optimal potential vorticity (PV) balance by Viúdez and Dritschel (2004) in a special semi-Lagrangian PV-based scheme. The method adiabatically deforms the nonlinear model into its linear form where mode decomposition is exact. It leads to a boundary value problem in time where gravity waves are removed at the linear end and a base-point coordinate is restored at the nonlinear end. This problem is solved by an iterative backward-forward nudging scheme. As global geophysical ocean models use primitive variables, we study optimal balance on an existing f-plane shallow water model in the primitive velocity-height variables. Our model, nevertheless, includes kinematic PV-inversion formulas if the PV is base point. We, here, systematically investigate our numerical model for several design parameters. We found that optimal balance works with PV-based projectors which are the most robust choice with primitive variable-based projectors which are useful for general domains and global models. The PV-based projectors are the linear oblique projector and the base point PV. The linear oblique projector can be reformulated as a PDE-based projector preserving linear PV. Besides, the height field as a prominent candidate of base point and a linear PDE-based projector support more general cases. The method returns high-quality balance with rapid convergence of the nudging scheme, but its convergence is, still, an open question. We proved the ''quasi-converge'' of the nudging iterates up to a small termination residual, and this residual is as small as the balance error which is of algebraic order in the time-separation parameter for a lower-dimensional system. Hence, optimal balance is an accurate diagnostic tool in primitive variables and can be implemented on complicated models without fundamental obstacles.