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We present numerical simulations of steady, laminar, axisymmetric convection of a Boussinesq fluid in a shallow, rotating, cylindrical domain. The flow is driven by an imposed vertical heat flux and shaped by the background rotation of the domain. The geometry is inspired by that of tropical cyclones and the global flow pattern consists of a shallow, swirling vortex combined with a poloidal flow in the r-z plane which is predominantly inward near the bottom boundary and outward along the upper surface. Our numerical experiments confirm that, as suggested by Oruba et al 2017, an eye forms at the centre of the vortex which is reminiscent of that seen in a tropical cyclone and is characterised by a local reversal in the direction of the poloidal flow. We establish scaling laws for the flow and map out the conditions under which an eye will, or will not, form. We show that, to leading order, the velocity scales with V=(\alpha g \beta)^{1/2}H, where g is gravity, \alpha the expansion coefficient, \beta the background temperature gradient, and H is the depth of the domain. We also show that the two most important parameters controlling the flow are Re=VH/\nu and Ro=V/\Omega H, where \Omega is the background rotation rate and \nu the viscosity. The Prandtl number and aspect ratio also play an important, if secondary, role. Finally, and most importantly, we establish the criteria required for eye formation. These consist of a lower bound on Re, upper and lower bounds on Ro, and an upper bound on Ekman number., Comment: 18 pages, 14 figures, 1 table

In dynamo power-based scaling laws, the power $P$ injected by buoyancy forces is measured by a so-called flux-based Rayleigh number, denoted as ${\rm Ra}_Q^*$ (see Christensen and Aubert, 2006). Whereas it is widely accepted that this parameter is measured (as opposite to controlled) in dynamos driven by differential heating, the literature is much less clear concerning its nature in the case of imposed heat flux. We clarify this issue by highlighting that in that case, the ${\rm Ra}_{Q}^*$ parameter becomes controlled only in the limit of large Nusselt numbers (${\rm Nu} \gg 1$). We then address the issue of the robustness of the original relation between $P$ and ${\rm Ra}_Q^*$ with the geometry and the thermal boundary conditions. We show that in the cartesian geometry, as in the spherical geometry with a central mass distribution, this relation is purely linear, in both differential and fixed-flux heating. However, we show that in the geometry commonly studied by geophysicists (spherical with uniform mass distribution), its validity places an upper-bound on the strength of the driving which can be envisaged in a fixed Ekman number simulation. An increase of the Rayleigh number indeed yields deviations (in terms of absolute correction) from the linear relation between $P$ and ${\rm Ra}_Q^*$. We conclude that in such configurations, the parameter range for which $P$ is controlled is limited., Comment: 14 pages, 3 figures

A comprehensive study of the classical linear spin-down of a constant density viscous fluid (kinematic viscosity \nu) rotating rapidly (angular velocity \Omega) inside an axisymmetric cylindrical container (radius L, height H) with rigid boundaries, that follows the instantaneous small change in the boundary angular velocity at small Ekman number $E=\nu/H^2\Omega \ll 1$, was provided by Greenspan & Howard (1963). $E^{1/2}$-Ekman layers form quickly triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the $O(E^{-1/2}\Omega^{-1})$ time-scale. On the longer lateral viscous diffusion time-scale $O(L^2/\nu)$, the QG-flow responds to the $E^{1/3}$-side-wall shear-layers. In our variant the side-wall and top boundaries are stress-free; a setup motivated by the study of isolated atmospheric structures, such as tropical cyclones, or tornadoes. Relative to the unbounded plane layer case, spin-down is reduced (enhanced) by the presence of a slippery (rigid) side-wall. This is evinced by the QG-angular velocity, \omega*, evolution on the O(L^2/\nu) time-scale: Spatially, \omega* increases (decreases) outwards from the axis for a slippery (rigid) side-wall; temporally, the long-time ($\gg L^2/\nu)$ behaviour is dominated by an eigensolution with a decay rate slightly slower (faster) than that for an unbounded layer. In our slippery side-wall case, the $E^{1/2} \times E^{1/2}$ corner region that forms at the side-wall intersection with the rigid base is responsible for a $\ln E$ singularity within the $E^{1/3}$-layer causing our asymptotics to apply only at values of E far smaller than can be reached by our Direct Numerical Simulation (DNS) of the entire spin-down process. Instead, we solve the $E^{1/3}$-boundary-layer equations for given E numerically. Our hybrid asymptotic-numerical approach yields results in excellent agreement with our DNS., Comment: 33 pages, 10 figures

We investigate the transition from steady dipolar to reversing multipolar dynamos. The Earth has been argued to lie close to this transition, which could offer a scenario for geomagnetic reversals. We show that the transition between dipolar and multipolar dynamos is characterized by a three terms balance (as opposed to the usually assumed two terms balance), which involves the non-gradient parts of inertial, viscous and Coriolis forces. We introduce from this equilibrium the sole parameter ${{\rm Ro}}\,{{\rm E}}^{-1/3} \equiv {{\rm Re}}\,{{\rm E}}^{2/3}$, which accurately describes the transition for a wide database of 132 fully three dimensional direct numerical simulations of spherical rotating dynamos (courtesy of U. Christensen). This resolves earlier contradictions in the literature on the relevant two,terms balance at the transition. Considering only a two terms balance between the non-gradient part of the Coriolis force and of inertial forces, provides the classical ${{\rm Ro}}/{\ell_u}$ (Christensen and Aubert, 2006). This transition can be equivalently described by ${{\rm Re}} \, {\ell^{2}_u}$, which corresponds to the two terms balance between the non-gradient part of inertial forces and viscous forces (Soderlund {\it et al.}, 2012)., Comment: 14 pages, 4 figures

State of the art numerical models of the Geodynamo are still performed in a parameter regime extremely remote from the values relevant to the physics of the Earth's core. In order to establish a connection between dynamo modeling and the geophysical motivation, {it is necessary to use} scaling laws. Such scaling laws establish the dependence of essential quantities (such as the magnetic field strength) on measured or controlled quantities. They allow for a direct confrontation of advanced models with geophysical {constraints}. (...) We show that previous empirical scaling laws for the magnetic field strength essentially reflect the statistical balance between energy production and dissipation for saturated dynamos. Such power based scaling laws are thus necessarily valid for any dynamo in statistical equilibrium and applicable to any numerical model, irrespectively of the dynamo mechanism. We show that direct numerical fits can provide contradictory results owing to biases in the parameters space covered in the numerics and to the role of a priori hypothesis on the fraction of ohmic dissipation. We introduce predictive scaling laws, i.e. relations involving input parameters of the governing equations only. We guide our reasoning on physical considerations. We show that our predictive scaling laws can properly describe the numerical database and reflect the dominant forces balance at work in these numerical simulations. We highlight the dependence of the magnetic field strength on the rotation rate. Finally, our results stress that available numerical models operate in a viscous dynamical regime, which is not relevant to the Earth's core.