Quantitative Diagnostics of Mixing in a Shallow Water Model of the Stratosphere

Two different approaches are applied to quantify mixing in a shallow water model of the stratosphere. These are modified Lagrangian mean (MLM) theory and a technique referred to as ‘‘reverse domain filling with local gradient reversal’’ (RDF-LGR). The latter is similar to a previously existing technique using contour advection and contour surgery. It is first proved that in an inviscid shallow water atmosphere subject to mass sources and sinks, if the mass enclosed by a potential vorticity (PV) contour is steady in time, then the integral of the mass source over the area enclosed by the contour must be zero. Next, the MLM and RDF-LGR approaches are used to diagnose the time-averaged transport across PV contours in the model simulations. The model includes a sixth-order hyperdiffusion on the vorticity field. Except in a thin outer ‘‘entrainment zone,’’ the hyperdiffusion term has only a very weak effect on the MLM mass budget of the polar vortex. In the entrainment zone, the hyperdiffusion term has a significant effect. The RDF-LGR results capture this behavior, providing good quantitative estimates of the hyperdiffusion term, which is equivalent to the degree of radiative disequilibrium at a PV contour. This agreement shows that the main role of the hyperdiffusion is to ‘‘mop up’’ the filaments that are produced by the essentially inviscid large-scale dynamics. All calculations are repeated for two values of the hyperdiffusion coefficient that differ by a factor of 50, with little difference in the results. This suggests that the amount of material entrained from the vortex edge into the surf zone does not depend on the details of the small-scale dissipation, as long as it is sufficiently weak and has some degree of scale selectivity.