arXiv subjects

Michael A. Calkins

Publications and source records attributed to Michael A. Calkins.

At least 19 recordsLinked to original sources

arXiv2026

Internal heating in rapidly rotating convection is not a shortcut to geostrophic turbulence

Convective turbulence in planets and stars is often driven by internal heating. This forcing mechanism has also been proposed as a means of accessing the diffusion-free scalings of the geostrophic turbulence (GT) regime at modest forcing. We test this with the asymptotically reduced quasi-geostrophic model, which is formally valid in the limit of vanishing Ekman number, $Ek \rightarrow 0$, and retains the Prandtl number as an independent parameter. We find no such shortcut: the Nusselt and Reynolds numbers are no closer to their diffusion-free predictions than in the boundary-heated case, and at $Pr = 7$ they are further from them; the predicted $Pr^{-1/2}$ collapse fails; and the prefactor depends on the form of the heating. Our no-slip/stress-free cases track the radiatively driven simulations of Hadjerci et al. (2024) case by case, yet a $25\%$ variation in $Nu$ persists between sweeps that share the same reduced Rayleigh and Prandtl numbers but differ in Ekman number. Those data span only the narrow range of forcing over which the compensated Nusselt number is stationary; across the wider range accessible to the reduced model it rises to a maximum and then falls. We argue that bulk transport scalings are incomplete diagnostics of GT, whereas the saturation of the interior mean temperature gradient and of the vertical velocity kurtosis remain reliable indicators.

physics.flu-dyn↗

arXiv2026

Quasi-geostrophic Rayleigh-Bénard convection on the tilted $f$-plane

Rapidly rotating Rayleigh-Bénard convection on a $f$-plane at colatitude $\vartheta_f$ is investigated numerically using an asymptotically reduced equation set valid in the limit of very rapid rotation. The equations provide a non-hydrostatic but quasi-geostrophic description in a non-orthogonal coordinate system. The tilt changes the structure of the large-scale barotropic condensate from large-scale vortices to zonal flows as the colatitude of the $f$-plane increases, with bistable states present for certain parameter ranges, extending prior work to a geophysically significant parameter regime. This behaviour is understood through the impact of broken rotation symmetry on the barotropic source terms resulting from baroclinic vortical stresses and baroclinic torque. As the tilt angle $\vartheta_f$ increases, global heat and momentum transport is reduced relative to upright-polar convection, a result that is explained through linear theory and nonlinear power maps both of which demonstrate increased attenuation of the domain of dynamically active spatial scales as the convective modes depart from a North-South alignment in the horizontal plane. A key finding is that the predominance of lateral thermal mixing allows for the maintenance of a persistent unstable mean temperature gradient that saturates at increasing forcing levels and remains insensitive to the colatitude.

physics.flu-dyn↗

arXiv2026

Effects of global core-mantle boundary topography on outer-core convection and topographic torques

Topography at the core-mantle boundary (CMB) couples the outer core to the mantle and likely generates observable variations in the length of day ($Δ$LOD) and the geomagnetic field, though these effects remain poorly understood. We use direct numerical simulations of rotating shell convection with finite-amplitude CMB topography to investigate dynamical effects on the outer core. A range of topographic shapes is used, including individual spherical harmonics and a model representing seismically inferred heterogeneities in the deep mantle. As predicted by prior linear theory in the rotating annulus model, a new instability arises for Rayleigh numbers below the onset of convection; we confirm its existence in a global geometry, though the predicted scalings are quantitatively modified. The shape of the geostrophic contours -- lines of constant axial height -- plays a central role: deformed contours allow buoyancy to do work on the time-averaged flow, driving increases in Reynolds and Nusselt numbers of up to $\sim$100\% relative to a spherical boundary. Previous work showed that topographic torques scale linearly with topographic amplitude and quadratically with flow speeds; we confirm this scaling and extend it with new theory that estimates the torques for global, spectrally broad topography. When extrapolated to core conditions, the predicted torques are consistent with the magnitude required to drive observed decadal and subdecadal $Δ$LOD variations.

physics.geo-ph↗

arXiv2026

Asymptotics of spherical dynamos exhibiting a small-scale MAC balance

Understanding the asymptotic behaviour of numerical dynamo models is critical for extrapolating results to the physical conditions that characterise terrestrial planetary cores. Here we investigate the behaviour of convection-driven dynamos reaching a MAC (magnetic-Archimedes-Coriolis) balance on the convective length scale and compare the results with non-magnetic convection cases. In particular, the dependence of physical quantities on the Ekman number, $Ek$, is studied in detail. The scaling of velocity dependent quantities is observed to be independent of the force balance and in agreement with quasi-geostrophic theory. The primary difference between dynamo and non-magnetic cases is that the fluctuating temperature is order unity in the former such that the buoyancy force scales with the Coriolis force. The MAC state yields a scaling for the flow speeds that is identical to the so-called CIA (Coriolis-inertia-Archimedes) scaling. There is an $O(Ek^{1/3})$ length scale present within the velocity field irrespective of the leading order force balance. This length scale is consistent with the asymptotic scaling of the terms of the governing equations and is not an indication that viscosity plays a dominant role. The peak of the kinetic energy spectrum and the ohmic dissipation length scale both exhibit an Ekman number dependence of approximately $Ek^{1/6}$, which is consistent with a scaling of $Rm^{-1/2}$, where $Rm$ is the magnetic Reynolds number. For the dynamos, advection remains comparable to, and scales similarly with, both inertia and viscosity, implying that nonlinear convective Rossby waves play an important role in the dynamics even in a MAC regime.

physics.geo-ph↗

arXiv2025

Boundary layers in quasi-static magnetoconvection with a vertical field and their implications for heat transport

Heat transport in quasi-static magnetoconvection with a vertical magnetic field in a plane layer geometry is investigated with direct numerical simulations and asymptotic theory in the limit of large Chandrasekhar number ($Q$). It is shown that thermal and magnetic boundary layers with thickness $O(Q^{-1/6})$, of the same order as the horizontal scale of the convection, are persistent over the range of investigated parameters. Thermal boundary layer control of the heat transport indicates that the Nusselt number depends on $Q$, suggesting that no asymptotic state of heat transport independent of $Q$ occurs in this system. The magnetic boundary layers necessitate leading order modifications to the dominant vertical force balance as well as the amplitude of the horizontal induced magnetic field near the boundaries. The scaling behavior of the thermal boundary layer and resulting heat transport is found to be independent of the choice of mechanical and electromagnetic boundary conditions.

physics.flu-dyn↗

arXiv2025

Turbulence in Earth's core generates large topographic torques on the mantle

Seismic and geodynamic studies indicate that the boundary between the Earth's liquid outer core and solid mantle is not spherical, but is likely characterized by topography in the form of inverted mountains and valleys that have typical amplitudes of several kilometers. One of the dynamical consequences of these deformations is that turbulent flow in the core can exert pressure torques on the mantle, thereby resulting in a transfer of angular momentum between the outer core and the mantle. Understanding this transfer of angular momentum is important for explaining variations in the Earth's rotation rate, or length of day. Whether kilometer-sized topography can explain observed variations in length of day is a longstanding question in geophysics. Here we use a suite of state-of-the-art numerical simulations of hydrodynamic convection in a rotating spherical shell with boundary topography to show that topographic torques exhibit a linear dependence on topographic amplitude and approach a quadratic dependence on flow speeds. This observation is explained with the asymptotic theory of rapidly rotating convection. These results imply that topographic torques are of sufficient magnitude to explain length of day variations.

physics.geo-ph↗

arXiv2023

A numerical investigation of quasi-static magnetoconvection with an imposed horizontal magnetic field

Quasi-static Rayleigh-Bénard convection with an imposed horizontal magnetic field is investigated numerically for Chandrasekhar numbers up to $Q=10^6$ with stress free boundary conditions. Both $Q$ and the Rayleigh number ($Ra$) are varied to identify the various dynamical regimes that are present in this system. We find three primary regimes: (I) a two-dimensional (2D) regime in which the axes of the convection rolls are oriented parallel to the imposed magnetic field; (II) an anisotropic three-dimensional (3D) regime; and (III) a mean flow regime characterized by a large scale horizontal flow directed transverse to the imposed magnetic field. The transition to 3D dynamics is preceded by a series of 2D transitions in which the number of convective rolls decreases as $Ra$ is increased. For sufficiently large $Q$, there is an eventual transition to two rolls just prior to the 2D/3D transition. The 2D/3D transition occurs when inertial forces become comparable to the Lorentz force, i.e. when $\sqrt{Q}/Re = O(1)$; 2D, magnetically constrained states persist when $\sqrt{Q}/Re \gtrsim O(1)$. Within the 2D regime we find heat and momentum transport scalings that are consistent with the hydrodynamic asymptotic predictions of Chini and Cox [Phys. Fluids \textbf{21}, 083603 (2009)]: the Nusselt number ($Nu$) and Reynolds number ($Re$) scale as $Nu \sim Ra^{1/3}$ and $Re \sim Ra^{2/3}$, respectively. For $Q=10^6$, we find that the scaling behavior of $Nu$ and $Re$ breaks down at large values of $Ra$ due to a sequence of bifurcations and the eventual manifestation of mean flows.

physics.flu-dyn↗

arXiv2023

Small scale quasi-geostrophic convective turbulence at large Rayleigh number

A numerical investigation of an asymptotically reduced model for quasi-geostrophic Rayleigh-Bénard convection is conducted in which the depth-averaged flows are numerically suppressed by modifying the governing equations. The Reynolds number and Nusselt number show evidence of approaching the diffusion-free scalings of $Re \sim Ra E/Pr$ and $Nu \sim Pr^{-1/2} Ra^{3/2} E^2$, respectively, where $E$ is the Ekman number and $Pr$ is the Prandtl number. For large $Ra$, the presence of depth-invariant flows, such as large-scale vortices, yield heat and momentum transport scalings that exceed those of the diffusion-free scaling laws. The Taylor microscale does not vary significantly with increasing $Ra$, whereas the integral length scale grows weakly. The computed length scales remain $O(1)$ with respect to the linearly unstable critical wavenumber; we therefore conclude that these scales remain viscously controlled. We do not find a point-wise Coriolis-Inertia-Archimedean (CIA) force balance in the turbulent regime; interior dynamics are instead dominated by horizontal advection (inertia), vortex stretching (Coriolis) and the vertical pressure gradient. A secondary, sub-dominant balance between the buoyancy force and the viscous force occurs in the interior and the ratio of the rms of these two forces is found to approach unity with increasing $Ra$. This secondary balance is attributed to the turbulent fluid interior acting as the dominant control on the heat transport. These findings indicate that a pointwise CIA balance does not occur in the high Rayleigh number regime of quasi-geostrophic convection in the plane layer geometry. Instead, simulations are characterized by what may be termed a \textsl{non-local} CIA balance in which the buoyancy force is dominant within the thermal boundary layers and is spatially separated from the interior Coriolis and inertial forces.

physics.flu-dyn↗

arXiv2023

Asymptotic scaling relations for rotating spherical convection with strong zonal flows

We analyse the results of direct numerical simulations of rotating convection in spherical shell geometries with stress-free boundary conditions, which develop strong zonal flows. Both the Ekman number and the Rayleigh number are varied. We find that the asymptotic theory for rapidly rotating convection can be used to predict the Ekman number dependence of each term in the governing equations, along with the convective flow speeds and the dominant length scales. Using a balance between the Reynolds stress and the viscous stress, together with the asymptotic scaling for the convective velocity, we derive an asymptotic prediction for the scaling behaviour of the zonal flow with respect to the Ekman number, which is supported by the numerical simulations. We do not find evidence of distinct asymptotic scalings for the buoyancy and viscous forces and, in agreement with previous results from asymptotic plane layer models, we find that the ratio of the viscous force to the buoyancy force increases with Rayleigh number. Thus, viscosity remains non-negligible and we do not observe a trend towards a diffusion-free scaling behaviour within the rapidly rotating regime.

physics.flu-dyn↗

arXiv2023

The Puzzling Structure of Solar Convection: Window into the Dynamo

The operation of the solar dynamo, with all of its remarkable spatio-temporal ordering, remains an outstanding problem of modern solar physics. A number of mechanisms that might plausibly contribute to its operation have been proposed, but the relative role played by each remains unclear. This uncertainty stems from continuing questions concerning the speed and structure of deep-seated convective flows. Those flows are in-turn thought to sustain both the Sun's turbulent EMF and the large-scale flows of differential rotation and meridional circulation suspected of influencing the dynamo's organization and timing. Continued progress in this area is complicated by (i) inconsistencies between helioseismic measurements of convective and meridional flow made with different techniques and instruments, and (ii) a lack of high-latitude data for convection, differential rotation, and meridional flow. We suggest that the path forward to resolving these difficulties is twofold. First, the acquisition of long-term helioseismic and emissivity measurements obtained from a polar vantage point is vital to complete our picture of the Sun's outer convection zone. Second, sustained and expanded investment in theory-oriented and combined theory/observational research initiatives will be crucial to fully exploit these new observations and to resolve inconsistencies between existing measurements.

astro-ph.SR↗

arXiv2022

Asymptotic behaviour of rotating convection-driven dynamos in the plane layer geometry

Dynamos driven by rotating convection in the plane layer geometry are investigated numerically for a range of Ekman number ($E$), magnetic Prandtl number ($Pm$) and Rayleigh number ($Ra$). The primary purpose of the investigation is to compare results of the simulations with previously developed asymptotic theory that is applicable in the limit of rapid rotation. We find that all of the simulations are in the quasi-geostrophic regime in which the Coriolis and pressure gradient forces are approximately balanced at leading order, whereas all other forces, including the Lorentz force, act as perturbations. Agreement between simulation output and asymptotic scalings for the energetics, flow speeds, magnetic field amplitude and length scales is found. The transition from large scale dynamos to small scale dynamos is well described by the magnetic Reynolds number based on the small convective length scale, $\widetilde{Rm}$, with large scale dynamos preferred when $\widetilde{Rm} \lesssim O(1)$. The magnitude of the large scale magnetic field is observed to saturate and become approximately constant with increasing Rayleigh number. Energy spectra show that all length scales present in the flow field and the small-scale magnetic field are consistent with a scaling of $E^{1/3}$, even in the turbulent regime. For a fixed value of $E$, we find that the viscous dissipation length scale is approximately constant over a broad range of $Ra$; the ohmic dissipation length scale is approximately constant within the large scale dynamo regime, but transitions to a $\widetilde{Rm}^{-1/2}$ scaling in the small scale dynamo regime.

physics.flu-dyn↗

arXiv2022

Strong large scale magnetic fields in rotating convection-driven dynamos: the important role of magnetic diffusion

Natural dynamos such as planets and stars generate global scale magnetic field despite the inferred presence of small scale turbulence. Such systems are known as large scale dynamos and are typically driven by convection and influenced by rotation. Previous numerical studies of rotating dynamos generally find that the large scale magnetic field becomes weaker as the flow becomes more turbulent. The underlying physical processes necessary for sustaining so-called large scale dynamos is therefore still debated. Here we use a suite of numerical simulations to show that strong large scale magnetic fields can be generated in rotating convective turbulence provided that two conditions are satisfied: (1) the flow remains rotationally constrained; and (2) magnetic diffusion is important on the small convective length scale. These findings are in agreement with previous asymptotic predictions and suggest that natural dynamos might satisfy these two conditions.

physics.flu-dyn↗

arXiv2021

Quasi-static magnetoconvection with a tilted magnetic field

A numerical study of convection with stress-free boundary conditions in the presence of an imposed magnetic field that is tilted with respect to the direction of gravity is carried out in the limit of small magnetic Reynolds number. The dynamics are investigated over a range of Rayleigh number $Ra$ and Chandrasekhar numbers up to $Q = 2\times10^6$, with the tilt angle between the gravity vector and imposed magnetic field vector fixed at $45^{\circ}$. For a fixed value of $Q$ and increasing $Ra$, the convection dynamics can be broadly characterized by three primary flow regimes: (1) quasi-two-dimensional convection rolls near the onset of convection; (2) isolated convection columns aligned with the imposed magnetic field; and (3) unconstrained convection reminiscent of non-magnetic convection. The influence of varying $Q$ and $Ra$ on the various fields is analyzed. Heat and momentum transport, as characterized by the Nusselt and Reynolds numbers, are quantified and compared with the vertical field case. Ohmic dissipation dominates over viscous dissipation in all cases investigated. Various mean fields are investigated and their scaling behavior is analyzed. Provided $Ra$ is sufficiently large, all investigated values of $Q$ exhibit an inverse kinetic energy cascade that yields strong `zonal' flows. Relaxation oscillations, as characterized by a quasi-periodic shift in the predominance of either the zonal or non-zonal component of the mean flow, appear for sufficiently large $Ra$ and $Q$.

physics.flu-dyn↗

arXiv2019

Heat transfer and flow regimes in quasi-static magnetoconvection with a vertical magnetic field

Numerical simulations of quasi-static magnetoconvection with a vertical magnetic field are carried out up to a Chandrasekhar number of $Q=10^8$ over a broad range of Rayleigh numbers $Ra$. Three magnetoconvection regimes are identified: two of the regimes are magnetically-constrained in the sense that a leading-order balance exists between the Lorentz and buoyancy forces, whereas the third regime is characterized by unbalanced dynamics that is similar to non-magnetic convection. Each regime is distinguished by flow morphology, momentum and heat equation balances, and heat transport behavior. One of the magnetically-constrained regimes appears to represent an `ultimate' magnetoconvection regime in the dual limit of asymptotically-large buoyancy forcing and magnetic field strength; this regime is characterized by an interconnected network of anisotropic, spatially-localized fluid columns aligned with the direction of the imposed magnetic field that remain quasi-laminar despite having large flow speeds. As for non-magnetic convection, heat transport is controlled primarily by the thermal boundary layer. Empirically, the scaling of the heat transport and flow speeds with $Ra$ appear to be independent of the thermal Prandtl number within the magnetically-constrained, high-$Q$ regimes.

physics.flu-dyn↗

arXiv2019

Magnetic quenching of the inverse cascade in rapidly rotating convective turbulence

We present results from an asymptotic magnetohydrodynamic model that is suited for studying the rapidly rotating, low viscosity regime typical of the electrically conducting fluid interiors of planets and stars. We show that the presence of sufficiently strong magnetic fields prevents the formation of large-scale vortices and saturates the inverse cascade at a finite length-scale. This saturation corresponds to an equilibrated state in which the energetics of the depth-averaged flows are characterized by a balance of convective power input and ohmic dissipation. A quantitative criteria delineating the transition between finite-size flows and domain-filling (large-scale) vortices in electrically conducting fluids is found. By making use of the inferred and observed properties of planetary interiors, our results suggest that convection-driven large-scale vortices do not form in the electrically conducting regions of many bodies.

physics.geo-ph↗

arXiv2018

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

Convection is the predominant mechanism by which energy and angular momentum are transported in the outer portion of the Sun. The resulting overturning motions are also the primary energy source for the solar magnetic field. An accurate solar dynamo model therefore requires a complete description of the convective motions, but these motions remain poorly understood. Studying stellar convection numerically remains challenging; it occurs within a parameter regime that is extreme by computational standards. The fluid properties of the convection zone are characterized in part by the Prandtl number $\mathrm{Pr}=ν/κ$, where $ν$ is the kinematic viscosity and $κ$ is the thermal diffusion; in stars, $\mathrm{Pr}$ is extremely low, $\mathrm{Pr}\approx 10^{-7}$. The influence of $\mathrm{Pr}$ on the convective motions at the heart of the dynamo is not well understood since most numerical studies are limited to using $\mathrm{Pr}\approx1$. We systematically vary $\mathrm{Pr}$ and the degree of thermal forcing, characterized through a Rayleigh number, to explore its influence on the convective dynamics. For sufficiently large thermal driving, the simulations reach a so-called convective free-fall state where diffusion no longer plays an important role in the interior dynamics. Simulations with a lower $\mathrm{Pr}$ generate faster convective flows and broader ranges of scales for equivalent levels of thermal forcing. Characteristics of the spectral distribution of the velocity remain largely insensitive to changes in $\mathrm{Pr}$. Importantly, we find that $\mathrm{Pr}$ plays a key role in determining when the free-fall regime is reached by controlling the thickness of the thermal boundary layer.

astro-ph.SR↗

arXiv2017

Inertia-less convectively-driven dynamo models in the limit of low Rossby number and large Prandtl number

Compositional convection is thought to be an important energy source for magnetic field generation within planetary interiors. The Prandtl number, $Pr$, characterizing compositional convection is significantly larger than unity, suggesting that the inertial force may not be important on the small scales of convection as long as the buoyancy force is not too strong. We develop asymptotic dynamo models for the case of small Rossby number and large Prandtl number in which inertia is absent on the convective scale. The relevant diffusivity parameter for this limit is the compositional Roberts number, $q = D/η$, which is the ratio of compositional and magnetic diffusivities. Dynamo models are developed for both order one $q$ and the more geophysically relevant low $q$ limit. For both cases the ratio of magnetic to kinetic energy densities, $M$, is asymptotically large and reflects the fact that Alfvén waves have been filtered from the dynamics. Along with previous investigations of asymptotic dynamo models for $Pr=O(1)$, our results show that the ratio $M$ is not a useful indicator of dominant force balances in the momentum equation since many different asymptotic limits of $M$ can be obtained without changing the leading order geostrophic balance. Furthermore, the present models show that inertia is not a requirement for driving low $q$, large-scale dynamos.

physics.geo-ph↗

arXiv2016

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 $α$, 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.

physics.geo-ph↗