Understanding planetary core convection dynamics requires the study of convective flows in which the Coriolis and Lorentz forces attain a leading-order, so-called magnetostrophic balance. Experimental investigations of rotating magnetoconvection (RMC) in the magnetostrophic regime are therefore essential to broadly characterize the properties of local-scale planetary core flow. Toward this end, we present here the first thermovelocimetric measurements of magnetostrophic, liquid metal convection, which are made using liquid gallium as the working fluid, at moderate rotation rates (Ekman numbers 10-4 <= Ek <= 10-5) and in the presence of dynamically strong magnetic fields (Elsasser number A = 1). Complementary rotating convection (RC) experiments are performed at the same rotation rates to serve as reference cases. Our RMC velocity measurements adequately follow a geostrophic turbulent scaling for cases in which local-scale convective inertial forces exceed the Lorentz forces in the fluid bulk. In cases where Lorentz forces exceed local-scale inertia (local or convective interaction parameter Nl or N 3), the root-mean-square RMC velocities are magnetically damped, yielding values below the geostrophic turbulent RC scaling prediction. An enhancement in heat transfer is observed, which we attribute to the increased coherence of vertically aligned magnetostrophic convective flow. Our results and comparisons to dynamo simulations suggest that the system is in the strong-field regime, where convection-scale flows are magnetically damped with N 3.

Accurate measurement of fluid velocity remains a significant challenge in experimental fluid dynamics, as it requires capturing detailed spatial and temporal information within flows that are frequently turbulent. Obtaining velocity fields requires tracking identifiable features within the flow, such as particles, tracer elements, or coherent flow structures, over time. This typically requires accurate optical visualization of the flow. The most prevalent and effective method for measuring fluid velocity in the laboratory is particle image velocimetry (PIV), which uses the tracking of suspended reflective or fluorescent particles to determine the fluid's velocity. The mathematical method commonly used to track particles is the cross-correlation between two images. PIV relies on a laser sheet to illuminate the particles in a plane and requires the fluid to be transparent to visible light. To overcome these optical constraints, we propose to use the surface temperature field, which is accurately measurable in the context of thermal convection experiments. This provides structures that can be tracked if the temperature field is mainly advected by the flow. We employ the Farneback dense optical flow algorithm [Farnebäck, Two-frame motion estimation based on polynomial expansion, in , edited by J. Bigun and T. Gustavsson (Springer, Berlin, 2003), pp. 363–370] to track the features of the temperature field. Similar methods have proven effective in tracking planetary flows, as demonstrated by measuring Jupiter's surface or Earth's ocean currents. We present here a benchmark study using a simple, synthetic two-dimensional advection-diffusion simulation, as well as a rapidly rotating quasigeostrophic simulation based on the “Coreaboloid” experimental setup [Lonner, Aggarwal, and Aurnou, Planetary core-style rotating convective flows in paraboloidal laboratory experiments, ]. We investigate the applicability of thermal image velocimetry to measuring the velocity field in rotating convection experiments. The reconstructed velocity field allows for a quantitative physical analysis of the flow dynamics.

This paper is associated with a poster winner of a 2025 American Physical Society's Division of Fluid Dynamics (DFD) Milton van Dyke Award for work presented at the DFD Gallery of Fluid Motion. The original poster is available online at the Gallery of Fluid Motion, at https://doi.org/10.1103/APS.DFD.2025.GFM.P012

Convection is ubiquitous in stellar and planetary interiors, where it likely plays an integral role in the generation of magnetic fields. As the interiors of these objects remain hidden from direct observation, numerical models of convection are an important tool in the study of astrophysical dynamos. In such models, unrealistically large values of the viscous (nu) and thermal (kappa) diffusivities are routinely used as an ad hoc representation of the effects of subgrid-scale turbulence, which is otherwise too small to resolve numerically. However, the functional forms of these diffusion coefficients can vary greatly between studies, complicating efforts to compare between results and against observations. We explore this issue by considering a series of nonrotating, nonmagnetic, solar-like convection models with varying radial functions for the diffusivities and differing boundary conditions. We find that the bulk kinetic energy scales similarly regardless of the diffusivity parameterization. This scaling is consistent with a freefall scaling, wherein viscosity plays a subdominant role in the force balance. We do not, however, observe such diffusion-free behavior in the convective heat transport. Our results also indicate that the functional form adopted for the diffusion coefficients can impact the distribution of turbulence within the convective shell. These results suggest that some care should be taken when comparing solar convection models directly against helioseismic observations.

Magnetic field generation in giant planets and rapidly rotating stars produces a diverse range of field geometries, from large-scale dipole-dominated configurations to complex, small-scale multipolar structures. Earlier dynamo studies have suggested that multipolar solutions tend to arise when rotational effects become less dominant. We investigate the strength of non-dipolar magnetic fields generated in systems dominated by rotation. 40 three-dimensional, spherical-shell dynamo simulations were carried out using the MagIC code, primarily made up of bistable pairs - simulations with the same control parameters that can settle in both a dipolar and non-dipolar steady-state regime. We use this suite of models to test how their magnetic field strength scales with heat flux and velocity. Our dynamo simulations produce magnetic fields with morphologies that fall on the two distinct branches, dipolar or non-dipolar, yet have very similar convective velocities. The strength of the dipole component differs by an order of magnitude between the two regimes, when scaled as a function of driving power. However, their non-dipolar magnetic field strengths are very similar. We conclude that when attempting to predict the magnetic field strength of rapidly rotating planets and stars, one cannot assume that it will have a dipole-dominated geometry. In particular, the amplitude of the dipole component is expected to be an order of magnitude smaller in the non-dipolar regime.

We investigate how the strength of the Lorentz force alters stellar convection zone dynamics in a suite of buoyancy-dominated, 3D, spherical shell convective dynamo models. This is done by varying only the fluid's electrical conductivity via the nondimensional magnetic Prandtl number, Pm. Because the strength of the dynamo magnetic field and the Lorentz force scale with Pm, it is found that the fluid motions and mode of dynamo generation differ across the 0 . 25 <= Pm <= 10 range investigated here. For example, we show that strong magnetohydrodynamic effects cause a fundamental change in the surface zonal flows: differential rotation switches from solar-like with prograde equatorial zonal flow for larger electrical conductivities (i.e. stronger dynamo magnetic field) to an anti-solar differential rotation with retrograde equatorial zonal flow at lower electrical conductivities (i.e. weaker magnetic field). This study shows that the value of electrical conductivity is important not only for sustaining dynamo action, but can also drive first-order changes in the characteristics of the magnetic and velocity fields. It is further associated with the ratio of inertial and Lorentz forces, estimated by the local magnetic Rossby number, Ro(M,l). We show in our models that Ro(M,l) sets the characteristics of the large-scale convection regime that generates the dynamo fields, with Ro(M,l )less than or similar to 1 (Lorentz dominated) corresponding to solar-like differential rotation and Ro(M,l )greater than or similar to 1 (inertia dominated) corresponding to anti-solar-like differential rotation.

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.

Current structure models of Jupiter and Saturn suggest that helium becomes immiscible in hydrogen in the outer part of the planets’ electrically conducting regions. This likely leads to a layer in which overturning convection is inhibited due to a stabilizing compositional gradient. The presence of such a stably stratified layer impacts the location and mechanism of convectively driven dynamo action. Juno’s measurements of Jupiter’s magnetic field enabled an estimate of its dynamo radius based on the magnetic Lowes spectrum. A depth of ∼0.8 R _J is obtained, where 1 R _J is Jupiter’s radius. This is rather deep, considering that the electrical conductivity inside Jupiter is expected to reach significant values at ∼0.9 R _J . Here, we use three-dimensional numerical dynamo simulations to explore the effects of the existence and location of a stably stratified helium rain layer on both the inferred Lowes radius and the location of the radial extent of dynamo action. We focus on a Jupiter-like internal structure and electrical conductivity profile. We find that for shallower stable layers, there is no magnetic field generation occurring above the stable layer, and the effective dynamo radius and the inferred Lowes radius are at the base of the layer. For deeper stable layers, Lowes radii of ∼0.87 R _J are inferred as a shallow secondary dynamo operates above the stable layer. Our results strongly suggest the existence of a stable layer extending from ∼0.8 R _J up to at least ∼0.9 R _J inside Jupiter. The physical origin of this extended stable layer and its connection to helium rain remain to be elucidated.

We investigate how the strength of the Lorentz force alters stellar convection zone dynamics in a suite of buoyancy-dominated, three-dimensional, spherical shell convective dynamo models. This is done by varying only the magnetic Prandtl number, Pm, the non-dimensional form of the fluid's electrical conductivity σ. Because the strength of the dynamo magnetic field and the Lorentz force scale with Pm, it is found that the fluid motions, the pattern of convective heat transfer, and the mode of dynamo generation all differ across the 0.25 ≤ Pm ≤ 10 range investigated here. For example, we show that strong magnetohydrodynamic effects cause a fundamental change in the surface zonal flows: differential rotation switches from solar-like (prograde equatorial zonal flow) for larger electrical conductivities to an anti-solar differential rotation (retrograde equatorial zonal flow) at lower electrical conductivities. This study shows that the value of the bulk electrical conductivity is important not only for sustaining dynamo action, but can also drive first-order changes in the characteristics of the magnetic, velocity, and temperature fields. It is also associated with the relative strength of the Lorentz force in the system as measured by the local magnetic Rossby number, Ro_ℓ^M, which we show is crucial in setting the characteristics of the large-scale convection regime that generates those dynamo fields.

Previous studies focusing on the electrical conductivity and thermal evolution of an early magma ocean at the base of Earth's mantle have found that basal magma ocean (BMO) convection could have produced the ancient geomagnetic field. By advances in high-resolution dynamo modeling, we find that convection in a thin BMO-like spherical shell is able to sustain strong magnetic fields, including axial dipolar fields similar to current day field structure. However, integrating our dynamo results with improved thermal evolution models and taking the planet's rapid rotation into account using rotating convective turbulence models implies that an Earth-like magnetic field was unlikely to have been generated in the BMO, a finding relevant to the interpretation of ancient paleomagnetic signatures, Earth's global-scale dynamics, and long-term planetary evolution. Large uncertainties still remain, calling for refined models of deep Earth thermal and mineralogical processes, accurate determination of prefactors in convective scaling laws, and fully coupled core-BMO dynamo simulations. Nonetheless, our work highlights that BMO-type dynamos intrinsically require a larger product of electrical conductivity and velocity than core-type dynamos, and that they are similarly rotationally constrained, so that velocities are significantly reduced compared to nonrotating estimates.

Convection in planets and stars is predicted to occur in the "ultimate regime” of diffusivity-free, rapidly rotating turbulence, in which flows are characteristically unaffected by viscous and thermal diffusion. Boundary layer diffusion, however, has historically hindered experimental study of this regime. Here, we utilize the boundary-independent oscillatory thermal-inertial mode of rotating convection to realize the diffusivity-free scaling in liquid metal laboratory experiments. This oscillatory style of convection arises in rotating liquid metals (low Prandtl number fluids) and is driven by the temperature gradient in the fluid bulk, thus remaining independent of diffusive boundary dynamics. We triply verify the existence of the diffusivity-free regime via measurements of heat transfer efficiency Nu, dimensionless flow velocities Re, and internal temperature anomalies θ, all of which are in quantitative agreement with planar asymptotically-reduced models. Achieving the theoretical diffusivity-free scalings in desktop-sized laboratory experiments provides the validation necessary to extrapolate and predict the convective flows in remote geophysical and astrophysical systems.

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.

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.

The local scale of rotating convection, & ell;, is a fundamental parameter in many turbulent geophysical and astrophysical fluid systems, yet it is often poorly constrained. Here we conduct rotating convection laboratory experiments analogous to convecting flows in planetary cores and subsurface oceans to obtain measurements of the local scales of motion. Utilizing silicone oil as the working fluid, we employ shadowgraph imagery to visualize the flow, from which we extract values of the characteristic cross-axial scale of convective columns and plumes. These measurements are compared to the theoretical values of the critical onset length scale, & ell;crit, and the turbulent length scale, & ell;turb. Our experimentally obtained length scale measurements simultaneously agree with both the onset and turbulent scale predictions across three orders of magnitude in convective supercriticality (102 less than or similar to Ra similar to less than or similar to 105) $(1{0}<^>{2}\lesssim \tilde{Ra}\lesssim 1{0}<^>{5})$, a correlation that is consistent with inferences made in prior studies. We further explore the nature of this correlation and its implications for geophysical and astrophysical systems. Turbulent convection occurs within the liquid metal, water, and gaseous fluid layers of planetary interiors such as Earth's molten iron outer core, the subsurface oceans of icy moons, and the deep atmospheres of gas planets, respectively. The flow in each of these systems is strongly affected by the rotation of the planetary body. This rotation organizes the convecting flow into columnar structures elongated in the direction of the rotation axis. The horizontal width of the columnar flows is known as the local length scale of rotating convection, and is crucially the scale at which important planetary phenomena are driven, such as the induction of Earth's magnetic field. However, this quantity is not well known for geophysical systems. Here we conduct rotating convection laboratory experiments analogous to the convecting flows in planetary interiors, in which the local length scale is measured using a visualization technique called shadowgraph imaging. We compare our measurements to theoretical scaling arguments for laminar and turbulent rotating flows, and find a simultaneous agreement with both. This heretofore unappreciated correlation with both theoretical scales presents difficulties when interpreting laboratory and numerical experimental results in the context of more extreme geophysical flows, a challenge we address in the discussion. Laboratory convection experiments simulate turbulent flows within the polar regions of planetary cores and subsurface oceans The convective flow scale, measured via shadowgraph imaging, agrees with both the critical and turbulent scale predictions This agreement cannot be explained without advancement in convection theory, and implies mean field dynamo action occurs in Earth's core

Abstract Convection in rotating spherical layers of fluid is ubiquitous in spherical astrophysical objects like planets and stars. A complete understanding of the magnetohydrodynamics requires understanding of the linear problem—when convection onsets in these systems. This is a fluid dynamics problem that has been studied since the early 1900s. Theoretical scaling laws exist for the variation of critical quantities—the Rayleigh number Rac, the azimuthal wavenumber mc, and the angular drift frequency ωc—with respect to the Ekman number E. However, their variation with the radius ratio χ of the spherical shell is still poorly studied. To address this, we use an open source eigenvalue code Kore to compute these critical quantities over an extensive range of parameters spanning four decades in Ekman number and a dense grid of radius ratio from very thick to very thin shells, focusing on no‐slip and fixed temperature boundary conditions. We find that these variations are explained well by the theoretical scaling laws, especially at low E, but variations in radius ratio also exist. We obtain scaling laws of boundary layer thicknesses and spatial extent of onset modes with respect to the Ekman number which differ only slightly from theoretical scalings. We show that our data set can be used to obtain good estimates of critical quantities in the moderate E range, where the vast majority of current geophysical and astrophysical fluid dynamics simulations are performed, yet where asymptotic theory is only moderately accurate. We further verify asymptotic predictions and determine best‐fit asymptotic model coefficients.

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.

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.

Rotating convective turbulence is ubiquitously found across geophysical settings, such as surface and subsurface oceans, planetary 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 = \nu /\kappa _i $ Pr=nu/kappa i) defines the characteristic diffusion properties of the system, with $ \kappa _i = \kappa _T $ kappa i=kappa T representing thermal diffusion and $ \kappa _i = \kappa _C $ kappa i=kappa C representing chemical diffusion. These numbers vary widely for geophysical systems; for example, the liquid iron undergoing thermal-compositional convection in Earth's core is defined by $ Pr_{T} \approx 0.1 $ PrT approximate to 0.1 and $ Pr_{C} \approx 100 $ PrC approximate to 100, while a thermally-driven liquid silicate magma ocean is defined by $ Pr_{T} \approx 100 $ PrT approximate to 100. Currently, most numerical and laboratory data for rotating convective turbulent flows exists at $ Pr = O(1) $ 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 silicone oils as our working fluids ( $ Pr = $ Pr= 6, 41, 206, and 993). Using measurements of flow velocities (Reynolds, Re) and heat transfer efficiency (Nusselt, Nu), a baroclinic torque balance is found to describe the turbulence regardless of Prandtl number so long as Re is sufficiently large ( $ Re \gtrsim 10 $ Re greater than or similar to 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.