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The magnetostrophic dynamo hypothesis has greatly influenced planetary dynamo research. Many magnetostrophic dynamo theories are founded upon the linear stability analysis by Chandrasekhar and Elbert, and by the canonical laboratory photographs taken by Nakagawa that show a significant enlargement of the convective flow scales in the magnetostrophic regime of liquid metal rotating magnetoconvection (RMC). We test whether these linear predictions are relevant for the nonlinear RMC system by exploring the five possible regimes using direct numerical simulations of RMC in the low magnetic Reynolds number quasi-static approximation. We map out the heat and momentum transport in these regimes, look at the flow structures and focus especially on the length scales. We have also included numerical counterparts of Nakagawa’s experiments and our results show an excellent agreement with three of these cases and linear theory. However, agreement with Nakagawa is not found in the magnetostrophic case: no enlargement of scales is observed, but still in good agreement with linear theory. Oscillatory bulk modes dominate all the RMC cases in which they exist, thus, suggesting that oscillatory convective flows may dominate all the other convective modes in planetary cores and may provide the motions that primarily generate planetary dynamo action. 

In this study, we analyse ‘magneto-Stokes’ flow, a fundamental magnetohydrodynamic (MHD) flow that shares the cylindrical-annular geometry of the Taylor–Couette cell but uses applied electromagnetic forces to circulate a free-surface layer of electrolyte at low Reynolds numbers. The first complete, analytical solution for time-dependent magneto-Stokes flow is presented and validated with coupled laboratory and numerical experiments. Three regimes are distinguished (shallow-layer, transitional and deep-layer flow regimes), and their influence on the efficiency of microscale mixing is clarified. The solution in the shallow-layer limit belongs to a newly identified class of MHD potential flows, and thus induces mixing without the aid of axial vorticity. We show that these shallow-layer magneto-Stokes flows can still augment mixing in distinct Taylor dispersion and advection-dominated mixing regimes. The existence of enhanced mixing across all three distinguished flow regimes is predicted by asymptotic scaling laws and supported by three-dimensional numerical simulations. Mixing enhancement is initiated with the least electromagnetic forcing in channels with order-unity depth-to-gap-width ratios. If the strength of the electromagnetic forcing is not a constraint, then shallow-layer flows can still yield the shortest mixing times in the advection-dominated limit. Our robust description of momentum evolution and mixing of passive tracers makes the annular magneto-Stokes system fit for use as an MHD reference flow. 

Rotating convective turbulence is ubiquitously found across geo- physical settings, such as surface and subsurface oceans, plane- tary atmospheres, molten metal planetary cores, magma chambers, magma oceans, and basal magma oceans. Depending on the thermal and material properties of the system, buoyant convection can be driven thermally or compositionally, where a Prandtl number (Pr = ν/κi) defines the characteristic diffusion properties of the system, with κi = κT representing thermal diffusion and κi = κC representing chemical diffusion. These numbers vary widely for geophysical sys- tems; for example, the liquid iron undergoing thermal-compositional convection in Earth’s core is defined by PrT ≈ 0.1 and PrC ≈ 100, while a thermally-driven liquid silicate magma ocean is defined by PrT ≈ 100. Currently, most numerical and laboratory data for rotat- ing convective turbulent flows exists at Pr = O(1); high Pr rotating convection relevant to compositionally-driven core flow and other systems is less commonly studied. Here, we address this deficit by carrying out a broad suite of rotating convection experiments made over a range of Pr values, employing water and three different sil- icone oils as our working fluids (Pr = 6, 41, 206, and 993). Using measurements of flow velocities (Reynolds, Re) and heat transfer effi- ciency (Nusselt, Nu), a baroclinic torque balance is found to describe the turbulence regardless of Prandtl number so long as Re is suf- ficiently large (Re 10). Estimated turbulent scales are found to remain close to onset scales in all experiments, a result that may extrapolate to planetary settings. Lastly, we use our data to build Pr-dependent predictive nondimensional and dimensional scaling relations for rotating convective velocities that can be applied across a broad range of geophysical fluid dynamical settings. 

Convection occurs ubiquitously on and in rotating geophysical and astrophysical bodies. Prior spherical shell studies have shown that the convection dynamics in polar regions can differ significantly from the lower latitude, equatorial dynamics. Yet most spherical shell convective scaling laws use globally-averaged quantities that erase latitudinal differences in the physics. Here we quantify those latitudinal differences by analysing spherical shell simulations in terms of their regionalized convective heat-transfer properties. This is done by measuring local Nusselt numbers in two specific, latitudinally separate, portions of the shell, the polar and the equatorial regions, $$Nu_p$$ and $$Nu_e$$ , respectively. In rotating spherical shells, convection first sets in outside the tangent cylinder such that equatorial heat transfer dominates at small and moderate supercriticalities. We show that the buoyancy forcing, parameterized by the Rayleigh number $Ra$ , must exceed the critical equatorial forcing by a factor of $${\approx }20$$ to trigger polar convection within the tangent cylinder. Once triggered, $$Nu_p$$ increases with $Ra$ much faster than does $$Nu_e$$ . The equatorial and polar heat fluxes then tend to become comparable at sufficiently high $Ra$ . Comparisons between the polar convection data and Cartesian numerical simulations reveal quantitative agreement between the two geometries in terms of heat transfer and averaged bulk temperature gradient. This agreement indicates that rotating spherical shell convection dynamics is accessible both through spherical simulations and via reduced investigatory pathways, be they theoretical, numerical or experimental. 

Multi-scale instabilities are ubiquitous in atmospheric and oceanic flows and are essential topics in teaching geophysical fluid dynamics. Yet these topics are often difficult to teach and counter-intuitive to new learners. In this paper, we introduce our state-of-the-art Do-It Yourself Dynamics (DIYnamics) LEGO robotics kit that allows users to create table-top models of geophysical flows. Deep ocean convection processes are simulated via three experiments – upright convection, thermal wind flows, and baroclinic instability – in order to demonstrate the robust multi-scale modeling capabilities of our kit. Detailed recipes are provided to allow users to reproduce these experiments. Further, dye-visualization measurements show that the table-top experimental results adequately agree with theory. In sum, our DIYnamics setup provides students and educators with an accessible table-top framework by which to model the multi-scale behaviors, inherent in canonical geophysical flows, such as deep ocean convection. 

In magnetostrophic rotating magnetoconvection, a fluid layer heated from below and cooled from above is equidominantly influenced by the Lorentz and the Coriolis forces. Strong rotation and magnetism each act separately to suppress thermal convective instability. However, when they act in concert and are near in strength, convective onset occurs at less extreme Rayleigh numbers ( R a , thermal forcing) in the form of a stationary, large-scale, inertia-less, inviscid magnetostrophic mode. Estimates suggest that planetary interiors are in magnetostrophic balance, fostering the idea that magnetostrophic flow optimizes dynamo generation. However, it is unclear if such a mono-modal theory is realistic in turbulent geophysical settings. Donna Elbert first discovered that there is a range of Ekman ( E k , rotation) and Chandrasekhar ( C h , magnetism) numbers, in which stationary large-scale magnetostrophic and small-scale geostrophic modes coexist. We extend her work by differentiating five regimes of linear stationary rotating magnetoconvection and by deriving asymptotic solutions for the critical wavenumbers and Rayleigh numbers. Coexistence is permitted if E k < 16 / ( 27 π ) 2 and C h ≥ 27 π 2 . The most geophysically relevant regime, the Elbert range , is bounded by the Elsasser numbers 4 3 ( 4 4 π 2   E k ) 1 / 3 ≤ Λ ≤ 1 2 ( 3 4 π 2 E k ) − 1 / 3 . Laboratory and Earth’s core predictions both exhibit stationary, oscillatory, and wall-attached multi-modality within the Elbert range. 

The connection between the heat transfer and characteristic flow velocities of planetary core-style convection remains poorly understood. To address this, we present novel laboratory models of rotating Rayleigh–Bénard convection in which heat and momentum transfer are simultaneously measured. Using water (Prandtl number, Pr≃6) and cylindrical containers of diameter-to-height aspect ratios of Γ≃3,1.5,0.75, the non-dimensional rotation period (Ekman number, E) is varied between 10−7≲E≲3×10−5 and the non-dimensional convective forcing (Rayleigh number, Ra) ranges from 107≲Ra≲1012. Our heat transfer data agree with those of previous studies and are largely controlled by boundary layer dynamics. We utilize laser Doppler velocimetry (LDV) to obtain experimental point measurements of bulk axial velocities, resulting in estimates of the non-dimensional momentum transfer (Reynolds number, Re) with values between 4×102≲Re≲5×104. Behavioral transitions in the velocity data do not exist where transitions in heat transfer behaviors occur, indicating that bulk dynamics are not controlled by the boundary layers of the system. Instead, the LDV data agree well with the diffusion-free Coriolis–Inertia–Archimedian (CIA) scaling over the range of Ra explored. Furthermore, the CIA scaling approximately co-scales with the Viscous–Archimedian–Coriolis (VAC) scaling over the parameter space studied. We explain this observation by demonstrating that the VAC and CIA relations will co-scale when the local Reynolds number in the fluid bulk is of order unity. We conclude that in our experiments and similar laboratory and numerical investigations with E≳10−7, Ra≲1012, Pr≃7, heat transfer is controlled by boundary layer physics while quasi-geostrophically turbulent dynamics relevant to core flows robustly exist in the fluid bulk.