On shallow-water potential-vorticity inversion by Rossby-number expansions

For the f-plane shallow water (SW) equations, the basic properties of formal Rossby-number expansions and potential vorticity (PV) inversion by the implied asymptotic balance conditions are re-examined. An indeterminacy inherent in conventional expansions beyond leading-order geostrophic balance is highlighted. The indeterminacy gives us the freedom to define, in principle, infinite sets of 'balance conditions' relating inertia-gravity modes to Rossby modes. Examples discussed include balance conditions that are obtained when one of the variables, linearized PV, vorticity, or perturbation height is regarded purely as a leading-order geostrophic quantity.

For the sake of conservation of global mass and boundary circulation, SW PV inversion is defined as one that employs the PV anomaly, i.e. the deviation of PV from its domain-area average. SW PV inversion by means of various Rossby-number balance conditions and the modified expansion procedure for the PV anomaly are presented. In the latter, called the 'WBSV procedure' in this paper, the PV anomaly is regarded purely as the leading-order geostrophic quantity and the Rossby formula for SW PV anomalies is satisfied asymptotically. A modification of the WBSV procedure satisfying the exact SW equation for the PV anomaly is also presented.

Numerical results for these PV inversion procedures, accurate up to second order in Rossby number, are presented for two SW simulations involving weakly and strongly ageostrophic flows. Also presented are results from PV inversion using the delta-gamma hierarchy of balance conditions, which sets the Nth-order time derivatives of divergence and ageostrophic vorticity to zero. It is shown that the WBSV procedure incurs a significant loss of accuracy relative to both the delta-gamma hierarchy and other asymptotic procedures like the one that regards the linearized PV as a leading-order geostrophic quantity. The modified WBSV procedure shows little improvement. The inaccuracy of this procedure stems from a poor leading-order solution for the balanced fields.