Finite-amplitude, neutral baroclinic eddies and mean flows in an internally heated rotating fluid: 1. Numerical simulations and quasi-geostrophic ‘free modes’

A series of numerical simulations of steady wave flows in a rotating fluid annulus, subject to internal heating and various thermal boundary conditions, is examined to characterise their structures, energetics and potential vorticity transport properties. The last of these characteristics, together with more conventional scaling considerations, indicate the possibility of applying quasi-geostrophic theory to the interior flow in a formulation similar to the inviscid, adiabatic models of Kuo and White.The analytical model of White, describing finite amplitude, neutral baroclinic eddies and mean flows as illustrations of the Charney-Drazin non-acceleration theorem, is then extended to include uniform diabatic heating and the effects of different forms of lateral shear in the background mean zonal flow. Like the solutions discussed by White, those obtained in the present paper consist of steady, internal jet, mean zonal flows, and baroclinic and barotropic Rossby wave components, all having the same three-dimensional wavenumber. Provided the diabatic heating is proportional to the stratification of the background flow, measured by the square of the Brunt-Vaisälä frequency N, the potential vorticity equation remains homogeneous. All the solutions are then characterised by zero net transfer of potential vorticity despite the possibility of non-zero eddy fluxes of heat or momentum and non-trivial Lorenz energy cycles.A series of particular three-component solutions (which, like some of the solutions discussed by White, do not obey conventional lateral boundary conditions) is examined as possible theoretical analogues of the steady waves observed in the numerical simulations of the laboratory flows, and is found to agree encouragingly well in the spatial variations of their mean flows, eddy stream function (pressure) and eddy fluxes of heat and momentum. Potential vorticity fluxes in the numerical simulations are relatively small (though crucially non-zero), supporting the possible analogy with the analytical model and exposing some limitations of the latter in not accounting for weak dissipation and forcing processes present in the laboratory flows.Further implications of the results are discussed, including possible analogies between the laboratory experiments and certain features in planetary atmospheres and oceans.