The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection

The influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is investigated for the case of stress-free mechanical boundary conditions. It is shown that whereas the leading order system satisfies fixed temperature boundary conditions implicitly, a double boundary layer structure is necessary to satisfy the fixed heat flux thermal boundary conditions. The boundary layers consist of a classical Ekman layer adjacent to the solid boundaries that adjust viscous stresses to zero, and a layer in thermal wind balance just outside the Ekman layers adjusts the temperature such that the fixed heat flux thermal boundary conditions are satisfied. The influence of these boundary layers on the interior geostrophically balanced convection is shown to be asymptotically weak, however. Upon defining a simple rescaling of the thermal variables, the leading order reduced system of governing equations are therefore equivalent for both boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading order influence on the interior convection

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

Spin orbit alignment for KELT-7b and HAT-P-56b via Doppler tomography with TRES

We present Doppler tomographic analyses for the spectroscopic transits of KELT-7b and HAT-P-56b, two hot-Jupiters orbiting rapidly rotating F-dwarf host stars. These include analyses of archival TRES observations for KELT-7b, and a new TRES transit observation of HAT-P-56b. We report spin-orbit aligned geometries for KELT-7b (2.7 +/- 0.6 deg) and HAT-P-56b (8 +/- 2 deg). The host stars KELT-7 and HAT-P-56 are among some of the most rapidly rotating planet-hosting stars known. We examine the tidal re-alignment model for the evolution of the spin-orbit angle in the context of the spin rates of these stars. We find no evidence that the rotation rates of KELT-7 and HAT-P-56 have been modified by star-planet tidal interactions, suggesting that the spin-orbit angle of systems around these hot stars may represent their primordial configuration. In fact, KELT-7 and HAT-P-56 are two of three systems in super-synchronous, spin-orbit aligned states, where the rotation periods of the host stars are faster than the orbital periods of the planets.Comment: 9 pages, accepted for publication in MNRA

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 [

Small scale quasi-geostrophic convective turbulence at large Rayleigh number

A numerical investigation of an asymptotically reduced model for quasi-geostrophic Rayleigh-B\'enard 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.Comment: 32 pages, 11 figure

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

Quasi-static Rayleigh-B\'enard 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.Comment: 31 pages, 9 figure

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