Local mass conservation and velocity splitting in PV-based balanced models. Part II: Numerical results

The effects of enforcing local mass conservation on the accuracy of non-Hamiltonian potential-vorticity-based balanced models (PBMs) are examined numerically for a set of chaotic shallow-water f-plane vortical flows in a doubly periodic square domain. The flows are spawned by an unstable jet and all have domain-maximum Froude and Rossby numbers Fr ~0.5 and Ro ~1, far from the usual asymptotic limits Ro [arrow right] 0, Fr [arrow right] 0, with Fr defined in the standard way as flow speed over gravity wave speed. The PBMs considered are the plain and hyperbalance PBMs defined in Part I. More precisely, they are the plain-, plain-, and plain- PBMs and the corresponding hyperbalance PBMs, of various orders, where "order" is related to the number of time derivatives of the divergence equation used in defining balance and potential-vorticity inversion. For brevity the corresponding hyperbalance PBMs are called the hyper- hyper-, and hyper- PBMs, respectively. As proved in Part I, except for the leading-order plain- each plain PBM violates local mass conservation. Each hyperbalance PBM results from enforcing local mass conservation on the corresponding plain PBM. The process of thus deriving a hyperbalance PBM from a plain PBM is referred to for brevity as plain-to-hyper conversion. The question is whether such conversion degrades the accuracy, as conjectured by McIntyre and Norton. Cumulative accuracy is tested by running each PBM alongside a suitably initialized primitive equation (PE) model for up to 30 days, corresponding to many vortex rotations. The accuracy is sensitively measured by the smallness of the ratio ... where Qsub PBM and Qsub PE denote the potential vorticity fields of the PBM and the PEs, respectively, and