The onset of thermal convection in a rapidly rotating sphere

Abstract

The linear stability of convection in a rapidly rotating sphere studied here builds on well established relationships between local and global theories appropriate to the small Ekman number limit. Soward (1977) showed that a disturbance marginal on local theory necessarily decays with time due to the process of phase mixing (where the spatial gradient of the frequency is non-zero). By implication, the local critical Rayleigh number is smaller than the true global value by an O(1) amount. The complementary view that the local marginal mode cannot be embedded in a consistent spatial WKBJ solution was expressed by Yano (1992). He explained that the criterion for the onset of global instability is found by extending the solution onto the complex s-plane, where s is the distance from the rotation axis, and locating the double turning point at which phase mixing occurs. He implemented the global criterion on a related two-parameter family of models, which includes the spherical convection problem for particular O(1) values of his parameters. Since he used one of them as the basis of a small-parameter expansion, his results are necessarily approximate for our problem.Here the asymptotic theory for the sphere is developed along lines parallel to Yano and hinges on the construction of a dispersion relation. Whereas Yano's relation is algebraic as a consequence of his approximations, ours is given by the solution of a second-order ODE, in which the axial coordinate z is the independent variable. Our main goal is the determination of the leading-order value of the critical Rayleigh number together with its first-order correction for various values of the Prandtl number.Numerical solutions of the relevant PDEs have also been found, for values of the Ekman number down to 10−6; these are in good agreement with the asymptotic theory. The results are also compared with those of Yano, which are surprisingly good in view of their approximate nature.

Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers

“…3a and 3b). As already explained in the case of thermal convection in a rapidly rotating sphere, the prograde spiralisation of the Rossby wave is a consequence of the decrease of β with increasing r (Zhang 1992;Cardin & Olson 1994;Jones et al 2000).…”

“…It corresponds to the spiraling of the Rossby wave observed for variable β (see fig. 3bc) Using the scalings for the Stewartson layer instability k ∼ E −1/4 and ω ∼ E 1/4 , relation B 8 leads to δ ∼ (2 |β|) −1/2 α −1 E 0 (B 9) which implies that the radial extension of the perturbation is independent of E. This is a fundamental difference with the thermal convection case investigated by Yano (1992) and Jones et al (2000) where the scaling k ∼ E −1/3 and ω ∼ E 1/3 implies δ ∼ (2 |β|) −1/2 α −1 E 1/6 (B 10) which corresponds to a slowly decaying radial extension when decreasing E.…”

“…In the flat case (figure 3c), the radial extent of the shear instability decreases as E 1/4 , following the width of the Stewartson layer. In the thermal convective case, the critical thermal Rossby wave is localised in a radial domain of width E 1/6 (Yano 1992;Jones et al 2000;Dormy et al 2004). On the contrary, the prograde Stewartson instabilities spread all the way to the outer equator.…”

Scale disparities and magnetohydrodynamics in the Earth's core

“…We shall focus on the second form of convection (Roberts 1968;Busse 1970; see also Jones et al . 2000), associated with large Prandtl number.…”

“…c are the corresponding wavenumber and frequency of convection (see also Soward 1977;Zhang 1991Zhang , 1992Jones et al . 2000).…”