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 similar to 0.5 and Ro similar to 1, far from the usual asymptotic limits Ro -> 0, Fr -> 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-delta delta , plain-gamma gamma , and plain-delta gamma 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-delta delta, hyper-gamma gamma, and hyper-delta gamma PBMs, respectively. As proved in Part I, except for the leading-order plain-gamma gamma 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 epsilon = parallel to Q(PBM) - Q(PE) parallel to (2)/ parallel to Q(PE) parallel to (2), where Q(PBM) and Q(PE) denote the potential vorticity fields of the PBM and the PEs, respectively, and parallel to parallel to (2) is the L-2 norm. At 30 days the most accurate PBMs have epsilon approximate to 10(-2) with PV fields hardly distinguishable visually from those of the PEs, even down to tiny details. Most accurate is defined by minimizing epsilon over all orders and truncation types delta delta, gamma gamma, and delta gamma. Contrary to McIntyre and Norton's conjecture, the minimal epsilon values did not differ systematically or significantly between plain and hyperbalance PBMs. The smallness of epsilon suggests that the slow manifolds defined by the balance relations of the most accurate PBMs, both plain and hyperbalance, are astonishingly close to being invariant manifolds of the PEs, at least throughout those parts of phase space for which Ro less than or similar to 1 and Fr less than or similar to 0.5.

As another way of quantifying the departures from such invariance, that is, of quantifying the fuzziness of the PEs' slow quasimanifold, initialization experiments starting at days 1, 2,... 10 were carried out in which attention was focused on the amplitudes of inertia - gravity waves representing the imbalance arising in 1-day PE runs. With balance defined by the most accurate PBMs, and imbalance by departures there-from, the results of the initialization experiments suggest a negative correlation between early imbalance and late cumulative error epsilon. In such near-optimal conditions the imbalance seems to be acting like weak background noise producing an effect analogous to so-called stochastic resonance, in that a slight increase in noise level brings PE behavior closer to the balanced behavior defined by the most accurate PBMs when measured cumulatively over 30 days.