Dissipation mechanisms for convection in rapidly rotating spheres and the formation of banded structures

Abstract

We report banded zonal structures in numerical simulations of weakly nonlinear rapidly rotating convection. A quasigeostrophic model of convection is used to demonstrate how, in the presence of Ekman pumping, banded structures can develop immediately above the onset of convection, and in the absence of developed turbulence. We argue that these bands necessarily correspond to a regime in which both Ekman pumping and bulk viscosity equally affect the zonal flow and that their width scales with the Ekman number E as E 1/4 . We investigate the respective role of two dissipation mechanisms, bulk viscosity and Ekman friction, acting on weakly nonlinear rapidly rotating convection. To make the problem tractable in a parameter regime of small Ekman numbers ͑low viscosity͒, we use a simplified approach relying on a z-integrated set of equations ͑known as quasigeostrophic convection, or ␤ convection 1 ͒. In a previous investigation using a similar approach, 2 we have shown, in the presence of bulk viscosity only, that as the Ekman number is decreased toward realistic values ͑which are presently out of reach of fully three-dimensional modeling͒ the zonal flow becomes increasingly important near the onset of convection. We have shown that, from an asymptotic point of view, the zonal flow is strong enough to suppress convective motions on a quasiperiodic basis and to yield relaxation oscillations immediately above the onset. 2 In all studies of quasigeostrophic or fully threedimensional convection, when only bulk viscosity is retained, the zonal flow near onset takes the form of a large "S" structure in the radius, with two counter-rotating bands ͑retrograde near the axis and prograde near the equator͒. As the Rayleigh number is further increased above critical, the number of bands slowly increases. 3 Recently, Jones et al. 4 investigated the effect of dissipation in the viscous boundary layers. They used a twodimensional rotating annulus model with mixed mechanical boundary conditions. Their model is strongly nonlinear and relies on a prescribed radial variation of the stream function. Introducing the effect of Ekman pumping, they were able to achieve a significantly higher number of bands ͑up to six on the ␤ plan͒ for a given Rayleigh number. The role of Ekman pumping in producing these jets still needs to be clarified, as well as the resulting balance between dissipation mechanisms. This turbulent approach differs from the weakly nonlinear investigation presented here.The zonal flow is driven by nonlinearities. Independent of which dissipation mechanism affects the axisymmetric component of the flow, its radial profile exhibits prograde and retrograde velocities. It is the shearing action of the zonal flow on the convective columns which yields the kinetic energy saturation. The zonal flow has to produce a strong enough shear to saturate the growth of convective motions. If only bulk viscosity is retained in the zonal flow equation, this shear will be achieved while minimizing the second derivative of the zonal velocity in...

Reynolds stresses and mean fields generated by pure waves: applications to shear flows and convection in a rotating shell

“…exterior). Figure 7(b) shows that this is qualitatively correct; a mean flow of a similar form has been obtained with a simpler, Cartesian QG model by Busse & Hood (1982), and with cylindrical QG models by Aubert et al (2003), Gillet & Jones (2006), Morin & Dormy (2006). The fact that the formula (3.22) fails to describe the zonal flow amplitude quantitatively, since here for instance it would yield min(|A| 2 V 2 ) = −0.265 instead of min(|A| 2 V 2 ) = −0.150, underlines the importance of both dissipation mechanisms, bulk viscosity and Ekman pumping, as noticed by Gillet & Jones and Morin & Dormy.…”

The effects of boundary topography on convection in Earth’s core

“…We extend the QGCM used by previous authors (Cardin & Olson 1994; Aubert et al 2003; Morin & Dormy 2004, 2006; Gillet & Jones 2006; Gillet et al 2007) with the inclusion of non‐axisymmetric boundary topography. The QGCM is an extension of the original theory developed by Busse (1970) for convection in rapidly rotating spherical shells.…”

“…In the present work, we employ the 2‐D quasigeostrophic convection model (QGCM) (e.g. Cardin & Olson 1994; Aubert et al 2003; Morin & Dormy 2004, 2006; Gillet & Jones 2006; Gillet et al 2007) to examine the effects of CMB topography on the convective dynamics of the core. The use of the QGCM allows us to reach lower Ekman number and higher Reynolds number flows than is possible in most 3‐D models of core convection.…”

Three branches of dynamo action

“…Such zonal flows are very weakly damped. If their radial structure is larger than E 1/4 L they are dominated by boundary layers dissipation (Morin & Dormy 2006). It results that the large-scale zonal flows will behave in a different manner in the case of stress-free boundary conditions (for which only the bulk viscous effects will be relevant).…”