Subcritical convection of liquid metals in a rotating sphere using a quasi-geostrophic model

We study nonlinear convection in a rapidly rotating sphere with internal heating for values of the Prandtl number relevant for liquid metals ($Pr\in[10^{-2},10^{-1}]$). We use a numerical model based on the quasi-geostrophic approximation, in which variations of the axial vorticity along the rotation axis are neglected, whereas the temperature field is fully three-dimensional. We identify two separate branches of convection close to onset: (i) a well-known weak branch for Ekman numbers greater than $10^{-6}$, which is continuous at the onset (supercritical bifurcation) and consists of thermal Rossby waves, and (ii) a novel strong branch at lower Ekman numbers, which is discontinuous at the onset. The strong branch becomes subcritical for Ekman numbers of the order of $10^{-8}$. On the strong branch, the Reynolds number of the flow is greater than $10^3$, and a strong zonal flow with multiple jets develops, even close to the nonlinear onset of convection. We find that the subcriticality is amplified by decreasing the Prandtl number. The two branches can co-exist for intermediate Ekman numbers, leading to hysteresis ($Ek=10^{-6}$, $Pr=10^{-2}$). Nonlinear oscillations are observed near the onset of convection for $Ek=10^{-7}$ and $Pr=10^{-1}$.Comment: 30 pages, 16 figures, published in JF

Effect of metallic walls on dynamos generated by laminar boundary-driven flow in a spherical domain

We present a numerical study of dynamo action in a conducting fluid encased in a metallic spherical shell. Motions in the fluid are driven by differential rotation of the outer metallic shell, which we refer to as "the wall". The two hemispheres of the wall are held in counter-rotation, producing a steady, axisymmetric interior flow consisting of differential rotation and a two-cell meridional circulation with radial inflow in the equatorial plane. From previous studies, this type of flow is known to maintain a stationary equatorial dipole by dynamo action if the magnetic Reynolds number is larger than about 300 and if the outer boundary is electrically insulating. We vary independently the thickness, electrical conductivity, and magnetic permeability of the wall to determine their effect on the dynamo action. The main results are: (a) Increasing the conductivity of the wall hinders the dynamo by allowing eddy currents within the wall, which are induced by the relative motion of the equatorial dipole field and the wall. This processes can be viewed as a skin effect or, equivalently, as the tearing apart of the dipole by the differential rotation of the wall, to which the field lines are anchored by high conductivity. (b) Increasing the magnetic permeability of the wall favors dynamo action by constraining the magnetic field lines in the fluid to be normal to the wall, thereby decoupling the fluid from any induction in the wall. (c) Decreasing the wall thickness limits the amplitude of the eddy currents, and is therefore favorable for dynamo action, provided that the wall is thinner than the skin depth. We explicitly demonstrate these effects of the wall properties on the dynamo field by deriving an effective boundary condition in the limit of vanishing wall thickness.Comment: accepted for publication in Physical Review

Jets and large-scale vortices in rotating Rayleigh-Bénard convection

One of the most prominent dynamical features of turbulent, rapidly rotating convection is the formation of large-scale coherent structures, driven by Reynolds stresses resulting from the small-scale convective flows. In spherical geometry, such structures consist of intense zonal flows that are invariant along the rotation axis. In planar geometry, long-lived, depth-invariant structures also form at large scales, but, in the absence of horizontal anisotropy, they consist of vortices that grow to the domain size. In this work, through the introduction of horizontal anisotropy into a numerical model of planar rotating convection by the adoption of unequal horizontal box sizes (i.e., Lx≤Ly, where the xy plane is horizontal), we investigate whether unidirectional flows and large-scale vortices can coexist. We find that only a small degree of anisotropy is required to bring about a transition from dynamics dominated by persistent large-scale vortices to dynamics dominated by persistent unidirectional flows parallel to the shortest horizontal direction. When the anisotropy is sufficiently large, the unidirectional flow consists of multiple jets, generated on a time scale smaller than a global viscous time scale, thus signifying that the upscale energy transfer does not spontaneously feed the largest available mode in the system. That said, the multiple jets merge on much longer time scales. Large-scale vortices of size comparable with Lx systematically form in the flanks of the jets and can be persistent or intermittent. This indicates that large-scale vortices, either coexisting with jets or not, are a robust dynamical feature of planar rotating convection

Large-scale-vortex dynamos in planar rotating convection

Several recent studies have demonstrated how large-scale vortices may arise spontaneously in rotating planar convection. Here, we examine the dynamo properties of such flows in rotating Boussinesq convection. For moderate values of the magnetic Reynolds number (100≲Rm≲550, with Rm based on the box depth and the convective velocity), a large-scale (i.e. system-size) magnetic field is generated. The amplitude of the magnetic energy oscillates in time, nearly out of phase with the oscillating amplitude of the large-scale vortex. The large-scale vortex is disrupted once the magnetic field reaches a critical strength, showing that these oscillations are of magnetic origin. The dynamo mechanism relies on those components of the flow that have length scales lying between that of the large-scale vortex and the typical convective cell size; smaller-scale flows are not required. The large-scale vortex plays a crucial role in the magnetic induction despite being essentially two-dimensional; we thus refer to this dynamo as a large-scale-vortex dynamo. For larger magnetic Reynolds numbers, the dynamo is small scale, with a magnetic energy spectrum that peaks at the scale of the convective cells. In this case, the small-scale magnetic field continuously suppresses the large-scale vortex by disrupting the correlations between the convective velocities that allow it to form. The suppression of the large-scale vortex at high Rm therefore probably limits the relevance of the large-scale-vortex dynamo to astrophysical objects with moderate values of Rm, such as planets. In this context, the ability of the large-scale-vortex dynamo to operate at low magnetic Prandtl numbers is of great interest

Subcritical turbulent condensate in rapidly rotating Rayleigh-B\'enard convection

The possibility of subcritical behaviour in the geostrophic turbulence regime of rapidly rotating thermally driven convection is explored. In this regime a non-local inverse energy transfer may compete with the more traditional and local direct cascade. We show that, even for control parameters for which no inverse cascade has previously been observed, a subcritical transition towards a large-scale vortex state can occur when the system is initialized with a vortex dipole of finite amplitude. This new example of bistability in a turbulent flow, which may not be specific to rotating convection, opens up new avenues for studying energy transfer in strongly anisotropic three-dimensional flows.Comment: 12 pages, 6 figure

Self-consistent simulations of a von K\'arm\'an type dynamo in a spherical domain with metallic walls

We have performed numerical simulations of boundary-driven dynamos using a three-dimensional non-linear magnetohydrodynamical model in a spherical shell geometry. A conducting fluid of magnetic Prandtl number Pm=0.01 is driven into motion by the counter-rotation of the two hemispheric walls. The resulting flow is of von K\'arm\'an type, consisting of a layer of zonal velocity close to the outer wall and a secondary meridional circulation. Above a certain forcing threshold, the mean flow is unstable to non-axisymmetric motions within an equatorial belt. For fixed forcing above this threshold, we have studied the dynamo properties of this flow. The presence of a conducting outer wall is essential to the existence of a dynamo at these parameters. We have therefore studied the effect of changing the material parameters of the wall (magnetic permeability, electrical conductivity, and thickness) on the dynamo. In common with previous studies, we find that dynamos are obtained only when either the conductivity or the permeability is sufficiently large. However, we find that the effect of these two parameters on the dynamo process are different and can even compete to the detriment of the dynamo. Our self-consistent approach allow us to analyze in detail the dynamo feedback loop. The dynamos we obtain are typically dominated by an axisymmetric toroidal magnetic field and an axial dipole component. We show that the ability of the outer shear layer to produce a strong toroidal field depends critically on the presence of a conducting outer wall, which shields the fluid from the vacuum outside. The generation of the axisymmetric poloidal field, on the other hand, occurs in the equatorial belt and does not depend on the wall properties.Comment: accepted for publication in Physical Review

Numerical Simulations of Dynamos Generated in Spherical Couette Flows

We numerically investigate the efficiency of a spherical Couette flow at generating a self-sustained magnetic field. No dynamo action occurs for axisymmetric flow while we always found a dynamo when non-axisymmetric hydrodynamical instabilities are excited. Without rotation of the outer sphere, typical critical magnetic Reynolds numbers $Rm_c$ are of the order of a few thousands. They increase as the mechanical forcing imposed by the inner core on the flow increases (Reynolds number $Re$). Namely, no dynamo is found if the magnetic Prandtl number $Pm=Rm/Re$ is less than a critical value $Pm_c\sim 1$. Oscillating quadrupolar dynamos are present in the vicinity of the dynamo onset. Saturated magnetic fields obtained in supercritical regimes (either $Re>2 Re_c$ or $Pm>2Pm_c$) correspond to the equipartition between magnetic and kinetic energies. A global rotation of the system (Ekman numbers $E=10^{-3}, 10^{-4}$) yields to a slight decrease (factor 2) of the critical magnetic Prandtl number, but we find a peculiar regime where dynamo action may be obtained for relatively low magnetic Reynolds numbers ($Rm_c\sim 300$). In this dynamical regime (Rossby number $Ro\sim -1$, spheres in opposite direction) at a moderate Ekman number ($E=10^{-3}$), a enhanced shear layer around the inner core might explain the decrease of the dynamo threshold. For lower $E$ ($E=10^{-4}$) this internal shear layer becomes unstable, leading to small scales fluctuations, and the favorable dynamo regime is lost. We also model the effect of ferromagnetic boundary conditions. Their presence have only a small impact on the dynamo onset but clearly enhance the saturated magnetic field in the ferromagnetic parts. Implications for experimental studies are discussed