A Barotropic Model of the Angular Momentum–Conserving Potential Vorticity Staircase in Spherical Geometry

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Abstract
An idealized analytical model of the barotropic potential vorticity (PV) staircase is constructed, constrained by global conservation of absolute angular momentum, perfect homogenization of PV in mixing zones between (prograde) westerly jets, and the requirement of barotropic stability. An imposed functional relationship is also assumed between jet speed and latitudinal separation using a multiple of the “dynamical Rossby wave” Rhines scale inferred from the strength of westerly jets. The relative simplicity of the barotropic system provides a simple relation between absolute angular momentum and PV (or absolute vorticity). A family of solutions comprising an arbitrary number of jets is constructed and is used to illustrate the restriction of jet spacing and strength imposed by the constraints of global conservation of angular momentum and barotropic stability. Asymptotic analysis of the theoretical solution indicates a limiting ratio of jet spacing to the dynamical Rhines scale equal to the square root of 6, meaning that westerly jets are spaced farther apart than predicted by the dynamical Rhines scale. It is inferred that an alternative “geometrical” Rhines scale for jet spacing can be obtained from conservation of absolute angular momentum on the sphere if the strength of zonal jets is known from other considerations. Numerical simulations of the full (nonaxisymmetric) equations reveal a pattern of zonal jet evolution that is consistent with our construction of ideal PV staircases in spherical geometry (which can be considered as limiting cases), as well as with the asymptotic analysis of a geometrical Rhines scale. The evolution of the PV staircase originating from an upscale cascade of energy in the barotropic model is therefore seen to depend on conservation of energy (for the strength of jets) and conservation of absolute angular momentum (for the spacing and number of jets). Further analysis of the numerical results confirms a “Taylor identity” relating the flux of eddy potential vorticity to mean-flow acceleration. Eddy fluxes are responsible for the occasional transitions between mode number as well as for maintaining the sharp westerly jets against small-scale dissipation. Suggestions are made for extending the theoretical model to PV staircases that are asymmetric between hemispheres or with latitudinal variation of amplitude, as modeled in the shallow-water system.

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