Local mass conservation and velocity splitting in PV-based balanced models. Part I: the hyperbalance equations

This paper considers stratified and shallow water non-Hamiltonian potential-vorticity-based balanced models (PBMs). These are constructed using the exact (Rossby or Rossby-Ertel) potential vorticity (PV). The most accurate known PBMs are those studied by McIntyre and Norton and by Mohebalhojeh and Dritschel. It is proved that, despite their astonishing accuracy, these PBMs all fail to conserve mass locally. Specifically, they exhibit velocity splitting in the sense of having two velocity fields, v and v(m), the first to advect PV and the second to advect mass. The difference v - v(m) is nonzero in general, even if tiny. Unlike the different velocity splitting found in all Hamiltonian balanced models, the present splitting can be healed. The result is a previously unknown class of balanced models, here called "hyperbalance equations," whose formal orders of accuracy can be made as high as those of any other PBM. The hyperbalance equations use a single velocity field v to advect mass as well as to advect and evaluate the exact PV.