A General Theorem on Angular-Momentum Changes due to Potential Vorticity Mixing and on Potential-Energy Changes due to Buoyancy Mixing

Abstract

An initial zonally symmetric quasigeostrophic potential vorticity (PV) distribution q i ( y) is subjected to complete or partial mixing within some finite zone jyj , L, where y is latitude. The change in M, the total absolute angular momentum, between the initial and any later time is considered. For standard quasigeostrophic shallow-water beta-channel dynamics it is proved that, for any q i ( y) such that dq i /dy . 0 throughout jyj , L, the change in M is always negative. This theorem holds even when ''mixing'' is understood in the most general possible sense. Arbitrary stirring or advective rearrangement is included, combined to an arbitrary extent with spatially inhomogeneous diffusion. The theorem holds whether or not the PV distribution is zonally symmetric at the later time. The same theorem governs Boussinesq potential-energy changes due to buoyancy mixing in the vertical. For the standard quasigeostrophic beta-channel dynamics to be valid the Rossby deformation length L D ) L where is the Rossby number; when L D 5 ' the theorem applies not only to the beta channel but also to a single barotropic layer on the full sphere, as considered in the recent work of Dunkerton and Scott on ''PV staircases.'' It follows that the M-conserving PV reconfigurations studied by those authors must involve processes describable as PV unmixing, or antidiffusion, in the sense of time-reversed diffusion. Ordinary jet self-sharpening and jet-core acceleration do not, by contrast, require unmixing, as is shown here by detailed analysis. Mixing in the jet flanks suffices. The theorem extends to multiple layers and continuous stratification. A least upper bound and greatest lower bound for the change in M is obtained for cases in which q i is neither monotonic nor zonally symmetric. A corollary is a new nonlinear stability theorem for shear flows.

How mesoscopic staircases condense to macroscopic barriers in confined plasma turbulence

“…Patterns of mixed layers, observed in the ocean [1], emerge from double-diffusive convective instability and salt fingering [2][3][4]. In the turbulent gas of planetary atmospheres, pattern formation manifests itself by the formation of quasi-periodic flow patterns, such as the lateral belts in the Jovian atmosphere [5,6]. In magnetized plasmas, E×B zonal flow (ZF) shear patterns develop from drift-wave (DW) turbulence.…”

“…Away from x = 0, ∂ x n 0 is uniform and close to −g i . The density equation including the flux drive is given by: (6) where Γ(x, t) = − l 2 ε α + D c ∂ x n is the sum of the turbulent and the collisional diffusion fluxes. As the solutions evolve to their final steady-state, Γ tot saturates and becomes uniform.…”

Friction and mixing effects on potential vorticity for bottom current crossing a marine strait: an application to the Sicily Channel (central Mediterranean Sea)

“…This interesting result points out that an increase in violates the all those assumptions for flow stability theorems (see, for instance, Wood and McIntyre, 2010).…”

The Atmospheric Wave–Turbulence Jigsaw

“…In the atmosphere, at least, there is no doubt that the breaking of Rossby waves is Nature's principal way of causing PV mixing and its typical consequences, anti-frictional jet sharpening and eddytransport-barrier reinforcement. In many cases the associated radiation stress or wave-induced momentum transport is an essential part of how the wave-turbulence jigsaw fits together [19], [10], [36]. One way of seeing more precisely how it fits together is through what is called the Taylor identity; see (1.9) and (1.10)ff.…”