POTENTIAL VORTICITY AND THE ELECTROSTATICS ANALOGY - ERTEL-ROSSBY FORMULATION

The isomorphism between the theory of electrostatics and the quasi-geostrophic potential vorticity is extended to the Ertel-Rossby potential vorticity. Anomalies of mass-weighted potential vorticity are defined relative to an arbitrary zonal-mean or horizontal-average flow and given in terms of the divergence of a vector field. The vector is the sum of linear and non-linear contributions and can be written as a dielectric tenser acting on the geopotential gradient. The linear components of the tenser differ from those for the quasi-geostrophic potential vorticity only if there exists a vertical variation of background potential vorticity, such as occurs at the tropopause, or if there is shear of the assumed background flow. The non-linear components are absent in the quasi-geostrophic case. The forms of free, bound and total charge are defined for accurate non-linear forms of the potential vorticity. The free-space Green's function for the operator defining the total charge is identical to that for quasi-geostrophic theory and provides a scheme whereby the field attributed to each potential vorticity element is an invariant quantity. One of the most important results arising from this formulation is that the non-linearities in the definition of potential vorticity can be neglected when considering the far-field effect of potential vorticity anomalies. An analytical example of these ideas is given for a uniform anomaly of semi-geostrophic potential vorticity embedded in an otherwise uniform background potential vorticity. The dielectric constant and bound charge are calculated and give a clear insight into the differences between this and the quasi-geostrophic solution.