Regimes of axisymmetric flow in an internally heated rotating fluid

Abstract

A boundary-layer scale analysis is presented for steady, zonally symmetric flow in a Cartesian channel of rectangular cross-section, subject to uniform internal heating, and cooling at the lateral boundaries, using an approach based on that of Hignett, Ibbetson & Killworth for a related system. Six main flow regimes are identified, depending chiefly upon the magnitude of the parameter ℙ defined as the square of the ratio of the (non-rotating) thermal-boundary-layer thickness scale to that of the Ekman layers adjacent to the horizontal boundaries. For , where [Aopf ] is the Rayleigh number and ε the channel aspect ratio), the flow consists of an advectively dominated interior, characterized by a balance between vertical advection and internal heat generation, diffusively dominated thermal boundary layers adjacent to the sidewalls, and horizontal, viscously dominated Ekman layers (for non-zero rotation rate). If ℙ [Lt ] 1, the flow is only weakly modified by rotation, but as ℙ increases through unity, rotation tends to inhibit heat transfer and thickens the thermal boundary layers. Provided ℙ [Gt ] ε2σ −2, (where σ is the Prandtl number), the zonal flow is predominantly geostrophic, though not given by the conventional thermal-wind scale (based on the total thermal contrast ΔT) unless ℙ [Gt ] 1.The results of the scale analysis are compared with laboratory measurements and numerical simulations of steady flow in a rotating, cylindrical annulus subject to (radially non-uniform) internal heating and sidewall cooling. Over the range of parameters accessible in the laboratory, the azimuthal velocity scale and thermal contrast were found to vary with rotation and heating rates in the way predicted from the scale analysis for the Cartesian system. Above a certain critical value of ℙ (for the geometry used here ℙt ≈ 1), the baroclinic wave regime was found to occur, corresponding to where rotational constraints first begin to influence significantly the heat transfer of the axisymmetric flow. The numerical simulations are compared with the laboratory measurements, and used to extend the ranges of rotation rate and aspect ratio over which the scale analysis could be verified. Good agreement was found for the dependence of globally averaged flow parameters on ℙ, and the dynamical characteristics of each regime were further verified using explicit calculations of the balance of terms in the basic equations from the numerical model.Further applications of the scaling technique to other, related systems are also discussed, together with a consideration of its generalization to systems of geophysical interest.