Potential vorticity and the electrostatics analogy: Ertel—Rossby formulation

Abstract 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 tensor acting on the geopotential gradient. the linear components of the tensor 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.