A Heuristic Framework for Next-Generation Models of Geostrophic Convective Turbulence

Many geophysical and astrophysical phenomena are driven by turbulent fluid dynamics, containing behaviors separated by tens of orders of magnitude in scale. While direct simulations have made large strides toward understanding geophysical systems, such models still inhabit modest ranges of the governing parameters that are difficult to extrapolate to planetary settings. The canonical problem of rotating Rayleigh-B\'enard convection provides an alternate approach - isolating the fundamental physics in a reduced setting. Theoretical studies and asymptotically-reduced simulations in rotating convection have unveiled a variety of flow behaviors likely relevant to natural systems, but still inaccessible to direct simulation. In lieu of this, several new large-scale rotating convection devices have been designed to characterize such behaviors. It is essential to predict how this potential influx of new data will mesh with existing results. Surprisingly, a coherent framework of predictions for extreme rotating convection has not yet been elucidated. In this study, we combine asymptotic predictions, laboratory and numerical results, and experimental constraints to build a heuristic framework for cross-comparison between a broad range of rotating convection studies. We categorize the diverse field of existing predictions in the context of asymptotic flow regimes. We then consider the physical constraints that determine the points of intersection between flow behavior predictions and experimental accessibility. Applying this framework to several upcoming devices demonstrates that laboratory studies may soon be able to characterize geophysically-relevant flow regimes. These new data may transform our understanding of geophysical and astrophysical turbulence, and the conceptual framework developed herein should provide the theoretical infrastructure needed for meaningful discussion of these results.Comment: 36 pages, 8 figures. CHANGES: in revision at Geophysical and Astrophysical Fluid Dynamic

Jump Rope Vortex in Liquid Metal Convection

Understanding large scale circulations (LSCs) in turbulent convective systems is important for the study of stars, planets and in many industrial applications. The canonical model of the LSC is quasi-planar with additional horizontal sloshing and torsional modes [Brown E, Ahlers G (2009) J. Fluid Mech. 638:383--400; Funfschilling D, Ahlers G (2004) Phys. Rev. Lett. 92(19):194502; Xi HD et al. (2009) Phys. Rev. Lett. 102(4):044503; Zhou Q et al. (2009) J. Fluid Mech. 630:367--390]. Using liquid gallium as the working fluid, we show via coupled laboratory-numerical experiments that the LSC in a tank with aspect ratios greater than unity takes instead the form of a "jump rope vortex", a strongly three-dimensional mode that periodically orbits around the tank following a motion much like a jump rope on a playground. Further experiments show that this jump rope flow also exists in more viscous fluids such as water, albeit with a far smaller signal. Thus, this new jump rope mode is an essential component of the turbulent convection that underlies our observations of natural systems

Oscillatory thermal-inertial flows in liquid metal rotating convection

We present the first detailed thermal and velocity field characterization of convection in a rotating cylindrical tank of liquid gallium, which has thermophysical properties similar to those of planetary core fluids. Our laboratory experiments, and a closely associated direct numerical simulation, are all carried out in the regime prior to the onset of steady convective modes. This allows us to study the oscillatory convective modes, sidewall modes and broadband turbulent flow that develop in liquid metals before the advent of steady columnar modes. Our thermo-velocimetric measurements show that strongly inertial, thermal wind flows develop, with velocities reaching those of comparable non-rotating cases. Oscillatory bulk convection and wall modes coexist across a wide range of our experiments, along with strong zonal flows that peak in the Stewartson layer, but that extend deep into the fluid bulk in the higher supercriticality cases. The flows contain significant time-mean helicity that is anti-symmetric across the midplane, demonstrating that oscillatory liquid metal convection contains the kinematic components to sustain system-scale dynamo generation.Comment: 29 pages, 12 figure

Thermoelectric Precession in Turbulent Magnetoconvection

We present laboratory measurements of the interaction between thermoelectric currents and turbulent magnetoconvection. In a cylindrical volume of liquid gallium heated from below and cooled from above and subject to a vertical magnetic field, it is found that the large scale circulation (LSC) can undergo a slow axial precession. Our experiments demonstrate that this LSC precession occurs only when electrically conducting boundary conditions are employed, and that the precession direction reverses when the axial magnetic field direction is flipped. A thermoelectric magnetoconvection (TEMC) model is developed that successfully predicts the zeroth-order magnetoprecession dynamics. Our TEMC magnetoprecession model hinges on thermoelectric current loops at the top and bottom boundaries, which create Lorentz forces that generate horizontal torques on the overturning large-scale circulatory flow. The thermoelectric torques in our model act to drive a precessional motion of the LSC. This model yields precession frequency predictions that are in good agreement with the experimental observations. We postulate that thermoelectric effects in convective flows, long argued to be relevant in liquid metal heat transfer and mixing processes, may also have applications in planetary interior magnetohydrodynamics

Geomagnetic polar minima do not arise from steady meridional circulation

Observations of the Earth’s magnetic field have revealed locally pronounced field minima near each pole at the core–mantle boundary (CMB). The existence of the polar magnetic minima has long been attributed to the supposed large-scale overturning circulation of molten metal in the outer core: Fluid upwells within the inner core tangent cylinder toward the poles and then diverges toward lower latitudes when it reaches the CMB, where Coriolis effects sweep the fluid into anticyclonic vortical flows. The diverging near-surface meridional circulation is believed to advectively draw magnetic flux away from the poles, resulting in the low intensity or even reversed polar magnetic fields. However, the interconnections between polar magnetic minima and meridional circulations have not to date been ascertained quantitatively. Here, we quantify the magnetic effects of steady, axisymmetric meridional circulation via numerically solving the axisymmetric magnetohydrodynamic equations for Earth’s outer core under the magnetostrophic approximation. Extrapolated to core conditions, our results show that the change in polar magnetic field resulting from steady, large-scale meridional circulations in Earth’s outer core is less than 3% of the background field, significantly smaller than the ∼ 100% polar magnetic minima observed at the CMB. This suggests that the geomagnetic polar minima cannot be produced solely by axisymmetric, steady meridional circulations and must depend upon additional tangent cylinder dynamics, likely including nonaxisymmetric, time-varying processes

Geomagnetic polar minima do not arise from steady meridional circulation

Observations of the Earth’s magnetic field have revealed locally pronounced field minima near each pole at the core–mantle boundary (CMB). The existence of the polar magnetic minima has long been attributed to the supposed large-scale overturning circulation of molten metal in the outer core: Fluid upwells within the inner core tangent cylinder toward the poles and then diverges toward lower latitudes when it reaches the CMB, where Coriolis effects sweep the fluid into anticyclonic vortical flows. The diverging near-surface meridional circulation is believed to advectively draw magnetic flux away from the poles, resulting in the low intensity or even reversed polar magnetic fields. However, the interconnections between polar magnetic minima and meridional circulations have not to date been ascertained quantitatively. Here, we quantify the magnetic effects of steady, axisymmetric meridional circulation via numerically solving the axisymmetric magnetohydrodynamic equations for Earth’s outer core under the magnetostrophic approximation. Extrapolated to core conditions, our results show that the change in polar magnetic field resulting from steady, large-scale meridional circulations in Earth’s outer core is less than 3% of the background field, significantly smaller than the ∼ 100% polar magnetic minima observed at the CMB. This suggests that the geomagnetic polar minima cannot be produced solely by axisymmetric, steady meridional circulations and must depend upon additional tangent cylinder dynamics, likely including nonaxisymmetric, time-varying processes

A nonlinear model for rotationally constrained convection with Ekman pumping

It is a well established result of linear theory that the influence of differing mechanical boundary conditions, i.e., stress-free or no-slip, on the primary instability in rotating convection becomes asymptotically small in the limit of rapid rotation. This is accounted for by the diminishing impact of the viscous stresses exerted within Ekman boundary layers and the associated vertical momentum transport by Ekman pumping. By contrast, in the nonlinear regime recent experiments and supporting simulations are now providing evidence that the efficiency of heat transport remains strongly influenced by Ekman pumping in the rapidly rotating limit. In this paper, a reduced model is developed for the case of 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 predominately 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 for spatially resolving 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 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 a heat transport that diverges to infinity. This layer arrests this blowup resulting in finite heat transport at a significantly enhanced value.Comment: 38 pages, 14 figure

Experimental study of internal wave generation by convection in water

We experimentally investigate the dynamics of water cooled from below at 0^oC and heated from above. Taking advantage of the unusual property that water's density maximum is at about 4^oC, this set-up allows us to simulate in the laboratory a turbulent convective layer adjacent to a stably stratified layer, which is representative of atmospheric and stellar conditions. High precision temperature and velocity measurements are described, with a special focus on the convectively excited internal waves propagating in the stratified zone. Most of the convective energy is at low frequency, and corresponding waves are localized to the vicinity of the interface. However, we show that some energy radiates far from the interface, carried by shorter horizontal wavelength, higher frequency waves. Our data suggest that the internal wave field is passively excited by the convective fluctuations, and the wave propagation is correctly described by the dissipative linear wave theory

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

We present the first investigation that explores the effects of an isolated topographic ridge on thermal convection in a planetary core-like geometry and using core-like fluid properties (i.e. using a liquid metal-like low Prandtl number fluid). The model′s mean azimuthal flow resonates with the ridge and results in the excitation of a stationary topographic Rossby wave. This wave generates recirculating regions that remain fixed to the mantle reference frame. Associated with these regions is a strong longitudinally dependent heat flow along the inner core boundary; this effect may control the location of melting and solidification on the inner core boundary. Theoretical considerations and the results of our simulations suggest that the wavenumber of the resonant wave, LR, scales as Ro−1/2, where Ro is the Rossby number. This scaling indicates that small-scale flow structures [wavenumber ] in the core can be excited by a topographic feature on the core-mantle boundary. The effects of strong magnetic diffusion in the core must then be invoked to generate a stationary magnetic signature that is comparable to the scale of observed geomagnetic structures [

Magneto-Stokes Flow in a Shallow Free-Surface Annulus

We analyse a magnetohydrodynamic flow inspired by the kinematic reversibility of viscous Taylor-Couette flows. The system considered here shares the cylindrical-annular geometry of the Taylor-Couette cell, but uses applied electromagnetic forces to drive "magneto-Stokes" flow in a shallow, free-surface layer of electrolyte. An analytical solution is presented and validated with coupled laboratory and numerical experiments. The dominant balance of Lorentz forcing and basal viscous drag reproduces the kinematic reversibility observed by G.I. Taylor with precise electromagnetic control. Induced fluid deformation may be undone by simply reversing the polarity of electric current through the system. We illustrate this analogy with theory and experiment, and we draw a further connection to potential flow using the Hele-Shaw approximation. The stability and controllability of the magneto-Stokes system make it an attractive tool for investigating shear flows in a variety of settings from industrial to astrophysical. In addition, the set-up's simplicity and robustness make magneto-Stokes flow a good candidate for PIV calibration and for educational demonstrations of magnetohydrodynamics, boundary layers, and flow transition