POTENTIAL VORTICITY AND THE ELECTROSTATICS ANALOGY - ERTEL-ROSSBY FORMULATION

The isomorphism between the theory of electrostatics and the quasi-geo strophic 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 gi ven in terms of the divergence of a vector field. The vector is the su m of linear and non-linear contributions and can be written as a diele ctric tenser acting on the geopotential gradient. The linear component s of the tenser differ from those for the quasi-geostrophic potential vorticity only if there exists a vertical variation of background pote ntial vorticity, such as occurs at the tropopause, or if there is shea r of the assumed background flow. The non-linear components are absent in the quasi-geostrophic case. The forms of free, bound and total cha rge are defined for accurate non-linear forms of the potential vortici ty. The free-space Green's function for the operator defining the tota l charge is identical to that for quasi-geostrophic theory and provide s a scheme whereby the field attributed to each potential vorticity el ement is an invariant quantity. One of the most important results aris ing from this formulation is that the non-linearities in the definitio n of potential vorticity can be neglected when considering the far-fie ld effect of potential vorticity anomalies. An analytical example of t hese ideas is given for a uniform anomaly of semi-geostrophic potentia l vorticity embedded in an otherwise uniform background potential vort icity. The dielectric constant and bound charge are calculated and giv e a clear insight into the differences between this and the quasi-geos trophic solution.