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 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 and Soward [Phys. Rev. Lett., 29, p.837, 1972], 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

Libration-driven flows in ellipsoidal shells

Planets and satellites can undergo physical librations, which consist of forced periodic variations in their rotation rate induced by gravitational interactions with nearby bodies. This mechanical forcing may drive turbulence in interior fluid layers such as subsurface oceans and metallic liquid cores through a libration‐driven elliptical instability (LDEI) that refers to the resonance of two inertial modes with the libration‐induced base flow. LDEI has been studied in the case of a full ellipsoid. Here we address for the first time the question of the persistence of LDEI in the more geophysically relevant ellipsoidal shell geometries. In the experimental setup, an ellipsoidal container with spherical inner cores of different sizes is filled with water. Direct side view flow visualizations are made in the librating frame using Kalliroscope particles. A Fourier analysis of the light intensity fluctuations extracted from recorded movies shows that the presence of an inner core leads to spatial heterogeneities but does not prevent LDEI. Particle image velocimetry and direct numerical simulations are performed on selected cases to confirm our results. Additionally, our survey at a fixed forcing frequency and variable rotation period (i.e., variable Ekman number, E) shows that the libration amplitude at the instability threshold varies as ∼E⁰·⁶⁵. This scaling is explained by a competition between surface and bulk dissipation. When extrapolating to planetary interior conditions, this leads to the E1/2 scaling commonly considered. We argue that Enceladus' subsurface ocean and the core of the exoplanet 55 CnC e should both be unstable to LDEI

Performance benchmarks for a next generation numerical dynamo model

Numerical simulations of the geodynamo have successfully represented many observable characteristics of the geomagnetic field, yielding insight into the fundamental processes that generate magnetic fields in the Earth's core. Because of limited spatial resolution, however, the diffusivities in numerical dynamo models are much larger than those in the Earth's core, and consequently, questions remain about how realistic these models are. The typical strategy used to address this issue has been to continue to increase the resolution of these quasi-laminar models with increasing computational resources, thus pushing them toward more realistic parameter regimes. We assess which methods are most promising for the next generation of supercomputers, which will offer access to O(106) processor cores for large problems. Here we report performance and accuracy benchmarks from 15 dynamo codes that employ a range of numerical and parallelization methods. Computational performance is assessed on the basis of weak and strong scaling behavior up to 16,384 processor cores. Extrapolations of our weak-scaling results indicate that dynamo codes that employ two-dimensional or three-dimensional domain decompositions can perform efficiently on up to ∼106 processor cores, paving the way for more realistic simulations in the next model generation

Effect of width, amplitude, and position of a core mantle boundary hot spot on core convection and dynamo action

Within the fluid iron cores of terrestrial planets, convection and the resulting generation of global magnetic fields are controlled by the overlying rocky mantle. The thermal structure of the lower mantle determines how much heat is allowed to escape the core. Hot lower mantle features, such as the thermal footprint of a giant impact or hot mantle plumes, will locally reduce the heat flux through the core mantle boundary (CMB), thereby weakening core convection and affecting the magnetic field generation process. In this study, we numerically investigate how parametrised hot spots at the CMB with arbitrary sizes, amplitudes, and positions affect core convection and hence the dynamo. The effect of the heat flux anomaly is quantified by changes in global flow symmetry properties, such as the emergence of equatorial antisymmetric, axisymmetric (EAA) zonal flows. For purely hydrodynamic models, the EAA symmetry scales almost linearly with the CMB amplitude and size, whereas self-consistent dynamo simulations typically reveal either suppressed or drastically enhanced EAA symmetry depending mainly on the horizontal extent of the heat flux anomaly. Our results suggest that the length scale of the anomaly should be on the same order as the outer core radius to significantly affect flow and field symmetries. As an implication to Mars and in the range of our model, the study concludes that an ancient core field modified by a CMB heat flux anomaly is not able to heterogeneously magnetise the crust to the present-day level of north–south asymmetry on Mars. The resulting magnetic fields obtained using our model either are not asymmetric enough or, when they are asymmetric enough, show rapid polarity inversions, which are incompatible with thick unidirectional magnetisation

Strong zonal winds from thermal convection in a rotating spherical shell

Zonal wind (ZW) generation by thermal convection in rotating spherical shells is studied using numerical calculations. Strong ZW accompany quasi-geostrophic, high Rayleigh number convection in shells with stress-free boundaries. In a thin shell (radius ratio 0.75) with stress-free boundaries, nearly 90% of the total kinetic energy is contained in the ZW at Rayleigh number 10G and Taylor number 4.4x107. The same parameters in a thicker shell produce weaker convection and weaker ZW. Rigid boundaries reduce the kinetic energy in the ZW to less than 20% of the total. The ZW are eastward (prograde) in the equatorial region and westward at higher latitudes, and are driven by Reynolds stresses associated with the convection. Episodes with strong ZW alternate with episodes of strong convection. Although far from the dynamical regime of Jupiter and Saturn, our results support the interpretation that the prograde equatorial jets on these planets originate from deep convection

The cross-over to magnetostrophic convection in planetary dynamo systems.

Global scale magnetostrophic balance, in which Lorentz and Coriolis forces comprise the leading-order force balance, has long been thought to describe the natural state of planetary dynamo systems. This argument arises from consideration of the linear theory of rotating magnetoconvection. Here we test this long-held tenet by directly comparing linear predictions against dynamo modelling results. This comparison shows that dynamo modelling results are not typically in the global magnetostrophic state predicted by linear theory. Then, in order to estimate at what scale (if any) magnetostrophic balance will arise in nonlinear dynamo systems, we carry out a simple scaling analysis of the Elsasser number Λ, yielding an improved estimate of the ratio of Lorentz and Coriolis forces. From this, we deduce that there is a magnetostrophic cross-over length scale, [Formula: see text], where Λo is the linear (or traditional) Elsasser number, Rmo is the system scale magnetic Reynolds number and D is the length scale of the system. On scales well above [Formula: see text], magnetostrophic convection dynamics should not be possible. Only on scales smaller than [Formula: see text] should it be possible for the convective behaviours to follow the predictions for the magnetostrophic branch of convection. Because [Formula: see text] is significantly smaller than the system scale in most dynamo models, their large-scale flows should be quasi-geostrophic, as is confirmed in many dynamo simulations. Estimating Λo ≃1 and Rmo ≃103 in Earth's core, the cross-over scale is approximately 1/1000 that of the system scale, suggesting that magnetostrophic convection dynamics exists in the core only on small scales below those that can be characterized by geomagnetic observations

On the theory of core-mantle coupling

This article commences by surveying the basic dynamics of Earth's core and their impact on various mechanisms of core-mantle coupling. The physics governing core convection and magnetic field production in the Earth is briefly reviewed. Convection is taken to be a small perturbation from a hydrostatic, "adiabatic reference state" of uniform composition and specific entropy, in which thermodynamic variables depend only on the gravitational potential. The four principal processes coupling the rotation of the mantle to the rotations of the inner and outer cores are analyzed: viscosity, topography, gravity and magnetic field. The gravitational potential of density anomalies in the mantle and inner core creates density differences in the fluid core that greatly exceed those associated with convection. The implications of the resulting "adiabatic torques" on topographic and gravitational coupling are considered. A new approach to the gravitational interaction between the inner core and the mantle, and the associated gravitational oscillations, is presented. Magnetic coupling through torsional waves is studied. A fresh analysis of torsional waves identifies new terms previously overlooked. The magnetic boundary layer on the core-mantle boundary is studied and shown to attenuate the waves significantly. It also hosts relatively high speed flows that influence the angular momentum budget. The magnetic coupling of the solid core to fluid in the tangent cylinder is investigated. Four technical appendices derive, and present solutions of, the torsional wave equation, analyze the associated magnetic boundary layers at the top and bottom of the fluid core, and consider gravitational and magnetic coupling from a more general standpoint. A fifth presents a simple model of the adiabatic reference state. © 2012 Copyright Taylor and Francis Group, LLC