Three-dimensional quasi-geostrophic convection in the rotating cylindrical annulus with steeply sloping endwalls

Abstract

The rotating cylindrical annulus geometry was first developed by Busse (J. Fluid Mech., vol. 44, 1970, pp. 441-460) as a simplified analogue for studying convection in rapidly rotating spherical geometries. Although it has provided a more tractable two-dimensional model than the sphere, it is formally limited to asymptotically small slopes and thus weak velocities in the direction parallel to the rotation axis. We present an asymptotically reduced three-dimensional equation set to model quasi-geostrophic convection in the annulus geometry where order-one slopes are permissible; this model provides a closer analogue to quasi-geostrophic convection in spheres and spherical shells where steeply sloping boundaries are present. A linear stability analysis of the reduced equations shows that a new class of three-dimensional, convectively driven Rossby waves is present in this system. The gravest modes exhibit strong axial variations as the slope of the boundaries becomes large. In addition, higher-order eigenmodes showing increasingly complex axial dependence are found that possess critical Rayleigh numbers close to that of the gravest mode.

The breakdown of the anelastic approximation in rotating compressible convection: implications for astrophysical systems

“…The critical parameters are those values associated with the smallest value of the Rayleigh number characterized by a zero growth rate; the resulting parameters are denoted by Ra c , k c and ω c . Further details of the numerical techniques employed can be found in [29]. Figure 2 shows the critical parameters as a function of the Taylor number for two different Prandtl numbers and three different values of N ρ for the NSE; the case N ρ = 10 −2 yields results close to those obtained from the OBE, whereas N ρ = 5 represents a relatively strong background stratification.…”

Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers

“…The three-dimensional cylindrical annulus model developed by Calkins et al [7] offers an intermediate stage between the development of a global spherical model and the plane layer model investigated in the present work. Christensen [16] argues that, providing the flow is sufficiently rotationally influenced, the large-scale dynamo properties in spherical shell dynamo calculations may be relatively insensitive to the hydrodynamic Reynolds number and the magnetic Prandtl number.…”

“…Moreover, reduced QG convection equations were recently developed for the three-dimensional cylindrical annulus geometry [7], extending the small-slope two dimensional model first developed by Busse [5]. Collectively, these previous investigations highlight the importance of developing and employing asymptotic models for the purpose of improving our understanding of flow regimes that are characteristic of planets.…”

Laboratory-numerical models of rapidly rotating convection in planetary cores

“…These lower latitude convection columns are generated by thermal Rossby waves and are not fully equivalent to our Cartesian cases, which better simulate convection at higher latitudes within the tangent cylinder (e.g. Busse & Cuong 1977;Sreenivasan & Jones 2006a;Takehiro 2008;Calkins et al 2013). Because the vorticity changes sign across the mid-layer only in high latitude columns, they have differing topologies (Chandrasekhar 1961), differing heat transfer behaviours (e.g.…”