Quasi-geostrophic Convection Research Articles

Quasi-geostrophic convection-driven dynamos in a thick spherical shell

SUMMARY We present dynamos computed using a hybrid QG-3D numerical scheme in a thick spherical shell geometry. Our model is based on a quasi-geostrophic convection code extended with a 3-D treatment of heat transport and magnetic induction. We find a collection of self-sustained, multipolar, weak field dynamos with magnetic energy one or two orders of magnitude lower than the kinetic energy. The poloidal magnetic energy is weak and, by construction, there is a lack of equatorially antisymmetric components in the Buoyancy and Lorentz forces. This leads to configurations where the velocity field is only weakly impacted by the magnetic field, similar to dynamos found in 3-D simulations where zonal flows and the Ω-effect dominate. The time-dependence of these dynamos is characterized by quasi-periodic oscillations that we attribute to dynamo waves. The QG-3D dynamos found so far are not Earth-like. The inability of our setup to produce strong, dipole-dominated, magnetic fields likely points to a missing ingredient in our QG flows, and a related lack of helicity and α-effect. The models presented here may be more relevant for studying stellar dynamos where zonal flows are known to dominate. This study was carried out at modest control parameters, however moving to lower Ekman numbers, when smaller values of both the magnetic and hydrodynamic Prandtl numbers can be of interest, our approach will be able to gain in efficiency by using relatively coarse grids for the 3-D magnetic and temperature fields and a finer grid for the QG velocity field.

Pizza: an open-source pseudo-spectral code for spherical quasi-geostrophic convection

We present a new pseudo-spectral open-source code nicknamed pizza. It is dedicated to the study of rapidly-rotating Boussinesq convection under the 2-D spherical quasi-geostrophic approximation, a physical hypothesis that is appropriate to model the turbulent convection that develops in planetary interiors. The code uses a Fourier decomposition in the azimuthal direction and supports both a Chebyshev collocation method and a sparse Chebyshev integration formulation in the cylindrically-radial direction. It supports several temporal discretisation schemes encompassing multi-step time steppers as well as diagonally-implicit Runge-Kutta schemes. The code has been tested and validated by comparing weakly-nonlinear convection with the eigenmodes from a linear solver. The comparison of the two radial discretisation schemes has revealed the superiority of the Chebyshev integration method over the classical collocation approach both in terms of memory requirements and operation counts. The good parallelisation efficiency enables the computation of large problem sizes with $\mathcal{O}(10^4\times 10^4)$ grid points using several thousands of ranks. This allows the computation of numerical models in the turbulent regime of quasi-geostrophic convection characterised by large Reynolds $Re$ and yet small Rossby numbers $Ro$. A preliminary result obtained for a strongly supercritical numerical model with a small Ekman number of $10^{-9}$ and a Prandtl number of unity yields $Re\simeq 10^5$ and $Ro \simeq 10^{-4}$. pizza is hence an efficient tool to study spherical quasi-geostrophic convection in a parameter regime inaccessible to current global 3-D spherical shell models.

Turbulent geodynamo simulations: a leap towards Earth’s core

We present an attempt to reach realistic turbulent regime in direct numerical simulations of the geodynamo. We rely on a sequence of three convection-driven simulations in a rapidly rotating spherical shell. The most extreme case reaches towards the Earth's core regime by lowering viscosity (magnetic Prandtl number Pm=0.1) while maintaining vigorous convection (magnetic Reynolds number Rm>500) and rapid rotation (Ekman number E=1e-7), at the limit of what is feasible on today's supercomputers. A detailed and comprehensive analysis highlights several key features matching geomagnetic observations or dynamo theory predictions -- all present together in the same simulation -- but it also unveils interesting insights relevant for Earth's core dynamics.In this strong-field, dipole-dominated dynamo simulation, the magnetic energy is one order of magnitude larger than the kinetic energy. The spatial distribution of magnetic intensity is highly heterogeneous, and a stark dynamical contrast exists between the interior and the exterior of the tangent cylinder (the cylinder parallel to the axis of rotation that circumscribes the inner core).In the interior, the magnetic field is strongest, and is associated with a vigorous twisted polar vortex, whose dynamics may occasionally lead to the formation of a reverse polar flux patch at the surface of the shell. Furthermore, the strong magnetic field also allows accumulation of light material within the tangent cylinder, leading to stable stratification there. Torsional Alfv{\'e}n waves are frequently triggered in the vicinity of the tangent cylinder and propagate towards the equator.Outside the tangent cylinder, the magnetic field inhibits the growth of zonal winds and the kinetic energy is mostly non-zonal. Spatio-temporal analysis indicates that the low-frequency, non-zonal flow is quite geostrophic (columnar) and predominantly large-scale: an m=1 eddy spontaneously emerges in our most extreme simulations, without any heterogeneous boundary forcing.Our spatio-temporal analysis further reveals that (i) the low-frequency, large-scale flow is governed by a balance between Coriolis and buoyancy forces -- magnetic field and flow tend to align, minimizing the Lorentz force; (ii) the high-frequency flow obeys a balance between magnetic and Coriolis forces; (iii) the convective plumes mostly live at an intermediate scale, whose dynamics is driven by a 3-term 1 MAC balance -- involving Coriolis, Lorentz and buoyancy forces. However, small-scale (E^{1/3}) quasi-geostrophic convection is still observed in the regions of low magnetic intensity.

A multiscale dynamo model driven by quasi-geostrophic convection

A convection-driven multiscale dynamo model is developed in the limit of low Rossby number for the plane layer geometry in which the gravity and rotation vectors are aligned. The small-scale fluctuating dynamics are described by a magnetically modified quasi-geostrophic equation set, and the large-scale mean dynamics are governed by a diagnostic thermal wind balance. The model utilizes three time scales that respectively characterize the convective time scale, the large-scale magnetic evolution time scale and the large-scale thermal evolution time scale. Distinct equations are derived for the cases of order one and low magnetic Prandtl number. It is shown that the low magnetic Prandtl number model is characterized by a magnetic to kinetic energy ratio that is asymptotically large, with ohmic dissipation dominating viscous dissipation on the large scale. For the order one magnetic Prandtl number model, the magnetic and kinetic energies are equipartitioned and both ohmic and viscous dissipation are weak on the large scales; large-scale ohmic dissipation occurs in thin magnetic boundary layers adjacent to the horizontal boundaries. For both magnetic Prandtl number cases the Elsasser number is small since the Lorentz force does not enter the leading order force balance. The new models can be considered fully nonlinear, generalized versions of the dynamo model originally developed by Childress & Soward (Phys. Rev. Lett., vol. 29, 1972, pp. 837–839), and provide a new theoretical framework for understanding the dynamics of convection-driven dynamos in regimes that are only just becoming accessible to direct numerical simulations.

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

AbstractThe 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 influence of fluid properties on the morphology of core turbulence and the geomagnetic field

Here we investigate the effects of fluid properties on the morphology and dynamics of convection in the Earth's outer core. The results of two quasi-geostrophic convection simulations are carried out at comparable convective velocities for fluids in which the ratio between the kinematic viscosity and thermal diffusivity (the Prandtl number, Pr) is 0.1 and 10. The Pr=0.1 case is representative of thermal convection in a liquid metal, whereas the Pr=10 case is representative of chemical convection. We find the influence of the Prandtl number to be significant; low Prandtl number fluids have a propensity for large-scale coherent vortex formation and slowly varying dynamics. Conversely, the high Prandtl case is dominated by significantly smaller length scales and more rapidly varying dynamics. However, both cases have zonal flows with similar strength, demonstrating that Reynolds stresses in high Prandtl number convection can be large when the buoyancy forcing is strong. By using a simple kinematic magnetic induction model we show that the structure of the magnetic field is not a direct indication of the underlying convective morphology when the magnetic diffusivity is large, as in Earth's core. Thus, our simulation results imply that the convective turbulence differs between thermally and chemically dominated convection, but that it may be difficult to determine the dominant forcing from geomagnetic field structure alone.

Kinetic energy cascades in quasi-geostrophic convection in a spherical shell

We consider triadic nonlinear interaction in the Navier–Stokes equation for quasi-geostrophic convection in a spherical shell. This approach helps us understand the origin of kinetic energy transport in the system and the particular scheme of mode interaction, as well as the locality of energy transfer. The peculiarity of convection in the sphere, concerned with the excitation of Rossby waves, is considered. The obtained results are compared with the results of our previous study on Cartesian geometry.

A quasi-geostrophic convection model for planetary systems using a domain decomposition method

A numerical study is described of a quasi-geostrophic model for thermal convection in rapidly rotating planetary systems where the effect of Coriolis forces is dynamically predominant. In the quasi-geostrophic convection model, a spherical shell is divided into the three domains: a spherical annulus outside the tangent cylinder touching the equator of the inner sphere, the southern and northern polar regions inside the tangent cylinder. Convection in the rotating spherical annulus is investigated using a quasi-two-dimensional geostrophic approximation. Both linear stability analysis and fully nonlinear simulations are carried out. A new domain decomposition method suitable for massively parallel computers is employed in numerical simulations. It is found that, while the structure of mildly nonlinear flows at R = O ( 10 R c ) , where R denotes the Rayleigh number and R c is its critical value at the onset of convection, is spatially complicated and irregular, the structure of strongly nonlinear flows at R = O ( 100 R c ) is dominated by an axisymmetric zonal flow that is largely laminar, stable and nearly time-independent, similar to that on giant planets Jupiter and Saturn. We also model the atmospheric circulation of a 51 Pegasus b-type extrasolar planet which is synchronized with its parent star and receives an intensive radiation on its dayside.