Rossby wave energy: a local Eulerian isotropic invariant

Conservation laws relate the local  time-rate-of-change of the spatial integral of a density function to the divergence of its  flux through the boundaries of the integration domain. These provide integral constraints on the spatio-temporal development  of a field. Here we show  that  a new type of conserved quantity exists, that does not require integration over a particular domain but which holds locally,  at any point in the field.  This is derived for the pseudo-energy density of  nondivergent Rossby waves where  local invariance is obtained for (1) a single plane wave, and (2) waves produced by an impulsive point-source of vorticity. 

The definition of pseudo-energy used here  consists of a conventional kinetic part, as well as an unconventional pseudo-potential part, proposed by  Buchwald (1973).  The anisotropic nature of the nondivergent energy flux that appears in response to the point source further clarifies the role of the beta plane in the  observed western intensification of ocean currents.