Michael A. Calkins

Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry

We report on axisymmetric numerical simulations of rapidly rotating spherical shells in which the axial rotation rate of the outer shell is modulated in time. This allows us to model planetary bodies undergoing forced longitudinal libration. In this study we systematically vary the Ekman number, 10−7≤E≲10−4, which characterizes the ratio of viscous to Coriolis forces in the fluid, and the libration amplitude, Δϕ. For libration amplitudes above a certain threshold, Taylor–Gortler vortices form near the outer librating boundary, in agreement with the previous laboratory experiments of Noir et al. [Phys. Earth Planet. Inter. 173, 141 (2009)]. At the lowest Ekman numbers investigated, we find that the instabilities remain spatially localized at onset in the equatorial region. In addition, nonzero time-averaged azimuthal (zonal) velocities are observed for all parameters studied. The zonal flow is characterized by predominantly retrograde flow in the interior, with a stronger prograde jet in the outer equatori...

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 timescales that respectively characterize the convective timescale, the large-scale magnetic evolution timescale, and the large-scale thermal evolution timescale. 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-scales. 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 solid 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 and Soward [Phys. Rev. Lett., \textbf{29}, p.837, 1972]. These models may be useful for understanding the dynamics of convection-driven dynamos in regimes that are only just becoming accessible to direct numerical simulations.

A nonlinear model for rotationally constrained convection with Ekman pumping

A reduced model is developed for low-Rossby-number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominantly aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need to resolve the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping (which correlates positively with interior convection) allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer, the nonlinear feedback from Ekman pumping would be able to generate heat transport that diverges to infinity at finite Rayleigh number. This layer arrests this blowup, resulting in finite heat transport at a significantly enhanced value. With increasing buoyancy forcing, the heat transport transitions to a more efficient regime, a transition that is always achieved within the regime of asymptotic validity of the theory, suggesting that this behaviour may be prevalent in geophysical and astrophysical settings. As the rotation rate increases, the slope of the heat transport curve below this transition steepens, a result that is in agreement with observations from laboratory experiments and direct numerical simulations.

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

The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e. incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilize the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.

Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

A linear stability analysis for compressible convection in a plane layer geometry both with and without the influence of rotation is presented. For the rotating cases we employ the tilted -plane geometry that allows for varying angles between the rotation and gravity vectors. The stability criteria for compressible and anelastic ideal gases is compared. As expected, the critical parameters for the compressible equations approach those of the anelastic equations as the background stratification approaches the adiabatic (anelastic) limit. For the rotating cases, we observe asymptotic scaling behavior in the critical parameters in both compressible and anelastic fluids as the Taylor number becomes large. In contrast to the incompressible limit, finite tilt angles between the gravity and rotation vectors result in propagating compressible Rossby waves as the most unstable eigenmode and the critical parameters are established for a range of stratification levels and Taylor numbers; all wave orientations are found to propagate in prograde and equatorward directions for non-isothermal background states. We also compare the linear stability of the thermodynamically rigorous anelastic equations with an anelastic model that replaces thermal diffusion with an entropy diffusion-like term in the energy equation; it is shown that the linear stability of the entropy diffusion model yields qualitatively similar results for the critical parameters in comparison to the full anelastic set. We show that a thermodynamically rigorous alternative to the entropy diffusion model is the isothermal adiabatic background state in which temperature and entropy become equivalent thermodynamic quantities and viscous heating becomes subdominant in the energy equation; the stability characteristics of this model are also presented.

Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers: Single mode solutions

The onset of dynamo action is investigated within the context of a newly developed low Rossby, low magnetic Prandtl number, convection-driven dynamo model. This multiscale model represents an asymptotically exact form of an α^{2} mean field dynamo model in which the small-scale convection is represented explicitly by finite amplitude, single mode solutions. Both steady and oscillatory convection are considered for a variety of horizontal planforms. The kinetic helicity is observed to be a monotonically increasing function of the Rayleigh number. As a result, very small magnetic Prandtl number dynamos can be found for sufficiently large Rayleigh numbers. All dynamos are found to be oscillatory with an oscillation frequency that increases as the strength of the convection is increased and the magnetic Prandtl number is reduced. Kinematic dynamo action is strongly controlled by the profile of the helicity; single mode solutions which exhibit boundary layer behavior in the helicity show a decrease in the efficiency of dynamo action due to the enhancement of magnetic diffusion in the boundary layer regions. For a given value of the Rayleigh number, lower magnetic Prandtl number dynamos are excited for the case of oscillatory convection in comparison to steady convection. With regard to planetary dynamos, these results suggest that the low magnetic Prandtl number dynamos typical of liquid metals are more easily driven by thermal convection than by compositional convection.

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

Most large-scale planetary magnetic fields are thought to be driven by low Rossby number convection of a low magnetic Prandtl number fluid. Here kinematic dynamo action is investigated with an asymptotic, rapidly rotating dynamo model for the plane layer geometry that is intrinsically low magnetic Prandtl number. The thermal Prandtl number and Rayleigh number are varied to illustrate fundamental changes in flow regime, ranging from laminar cellular convection to geostrophic turbulence in which an inverse energy cascade is present. A decrease in the efficiency of the convection to generate a dynamo, as determined by an increase in the critical magnetic Reynolds number, is observed as the buoyancy forcing is increased. This decreased efficiency may result from both the loss of correlations associated with the increasingly disordered states of flow that are generated, and boundary layer behavior that enhances magnetic diffusion locally. We find that the spatial characteristics of

A computationally efficient spectral method for modeling core dynamics

\alpha

Prandtl-number Effects in High-Rayleigh-number Spherical Convection

, and thus the large-scale magnetic field, is dependent only weakly on changes in flow behavior. However, our results are limited to the linear, kinematic dynamo regime, and future simulations including the Lorentz force are therefore necessary to assess the robustness of this result. In contrast to the large-scale magnetic field, the behavior of the small-scale magnetic field is directly dependent on, and therefore shows significant variations with, the small-scale convective flow field.

Rotating convective turbulence in Earth and planetary cores

An efficient, spectral numerical method is presented for solving problems in a spherical shell geometry that employs spherical harmonics in the angular dimensions and Chebyshev polynomials in the radial direction. We exploit the three-term recurrence relation for Chebyshev polynomials that renders all matrices sparse in spectral space. This approach is significantly more efficient than the collocation approach and is generalizable to both the Galerkin and tau methodologies for enforcing boundary conditions. The sparsity of the matrices reduces the computational complexity of the linear solution of implicit-explicit timestepping schemes to