Predictive Scaling Laws for Spherical Rotating Dynamos

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

Transition between viscous dipolar and inertial multipolar dynamos

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 figure

Spin-down in a rapidly rotating cylinder container with mixed rigid and stress-free boundary conditions

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 figure

Rapid Oceanic Response to Tropical Cyclone Oli (2010) over the South Pacific

International audienceThe effect of Tropical Cyclone Oli (2010) on the ocean is investigated using a variety of measurements. In situ temperature measurements on the cyclone track are available via the Centre de Recherches Insulaires et Observatoire de l'Environnement (CRIOBE) array of probes. This reflects an extreme fluctuation of the temperature some 18 h after the cyclone, lasting only 12 h and exceeding 38C in amplitude. This study interprets this extreme fluctuation in terms of enhanced mixing associated with the time-dependent inertial currents due to the cyclonic winds. The authors show, using Lagrangian simulations, that this rapid event is compatible with the severe length-scale shortening observed in Lagrangian simulations

On the role of thermal boundary conditions in dynamo scaling laws

International audienceIn dynamo power-based scaling laws, the power ? injected by buoyancy forces is measured by a so-called flux-based Rayleigh number, denoted as ? (see Christensen and Aubert, Geophys. J. Int. 2006, vol. 166, pp. 97-114). 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 ? parameter becomes controlled only in the limit of large Nusselt numbers (?). We then address the issue of the robustness of the original relation between ? and ? 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 ? and ?. We conclude that in such configurations, the parameter range for which ? is controlled is limited

Rôle de l'environnement grande échelle dans la canalisation et l'intensification des tempêtes

This thesis aims to a better understanding of the crossing of the jet-stream from its warm side to its cold side by a number of mid-latitude winter storms. Indeed, these storms were observed to experiment an explosive growth phase just after the crossing, which justifies the importance of the crossing issue. We investigate how the inhomogeneous spatial structure of the large-scale jet-stream affects, beyond baroclinic instability, the trajectory and the deepening of surface depressions during the jet crossing. First we study, in a barotropic numerical idealized context, how the large-scale deformation effects modulate the meridional displacement of a cyclonic eddy. This displacement is primarily due to the nonlinear effect of the meridional gradient of the large-scale potential vorticity gradient (called beta-drift, known in the context of tropical cyclones and ocean eddies). It is shown that the deformation effects reinforce the anticyclone created by the Rossby wave radiation due to the potential vorticity gradient, and with which the cyclonic eddy interacts. Then this mechanism is generalized to a baroclinic atmosphere by studying the crossing by a cyclonic surface eddy of a meandering and baroclinically unstable jet-stream, within a two-layer model. It is shown that a positive barotropic potential vorticity gradient induces a strong altitude anticyclone which is responsible for the crossing of the jet by the surface eddy with which it interacts. In addition, the energetic life cycle of an idealized eddy undergoing the deformation effects appears to be similar to those of some real storms, including intensification just after jet crossing.L'objectif de cette thèse est de mieux comprendre la traversée du courant-jet de son côté chaud vers son côté froid par un certain nombre de tempêtes des moyennes latitudes. En effet, on a observé que ces tempêtes croissent de manière explosive juste après cette traversée, d'où l'intérêt porté à la question de la traversée. On se demande par quel mécanisme la structure spatialement inhomogène du courant-jet influence, au-delà de l'instabilité barocline, la trajectoire et le creusement des dépressions de surface pendant la traversée du jet. On étudie d'abord, dans un cadre numérique barotrope idéalisé, comment les effets de déformation grande échelle modulent le déplacement méridien d'un tourbillon cyclonique. Ce déplacement est, en premier lieu, dû à l'effet non linéaire du gradient méridien de la vorticité potentielle grande échelle (concept de beta-drift, connu dans le contexte des cyclones tropicaux et des tourbillons océaniques). On montre que les effets de déformation renforcent l'anticyclone qui est créé par la génération d'ondes de Rossby due à la présence du gradient de vorticité potentielle, et avec lequel le tourbillon cyclonique interagit. Puis on généralise ce mécanisme à une atmosphère barocline en étudiant la traversée par un tourbillon cyclonique de surface d'un courant-jet avec méandres et instable barocliniquement, dans un modèle à deux couches. On montre qu'un gradient de vorticité potentielle barotrope positif induit un fort anticyclone d'altitude, responsable de la traversée du jet par le tourbillon de surface avec lequel il interagit. En outre, le cycle de vie énergétique d'un tourbillon idéalisé subissant les effets de la déformation est similaire à celui de certaines tempêtes réelles, avec notamment une intensification juste après la traversée du jet

Numerical models of astrophysical dynamos

International audienceThe parameters regime relevant to dynamo action in astrophysical objects is out of reach of present day numerical models because of computational limitations. It is thus necessary to derive scaling laws to extend numerical results to real world dynamos.We show that traditional power based scaling laws for the magnetic field strength are too general, since they mainly traduce the statistical balance between the energy production and dissipation, and are thus satisfied by any dynamo in statistical equilibrium. We introduce a predictive scaling law (i.e. depending on input parameters only) for the magnetic field strength in numerical dynamos, by guiding our reasoning on physical arguments. We thus show that dipolar dynamos operate in a viscous dynamical regime, which is not relevant to astrophysical objects. Finally, we show that the dipolar-multipolar transition occurring in numerical models can be described by a sole non-dimensional parameter corresponding to a three-terms balance

Bénard convection in a slowly rotating penny shaped cylinder subject to constant heat flux boundary conditions

International audienceWe consider axisymmetric Boussinesq convection in a shallow cylinder radius, L, and depth, H (<< L), which rotates with angular velocity Ω about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that Ω is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at sufficiently large Prandtl number, σ. As σ is reduced, the finite amplitude range of applicability of the asymptotics reduces in concert. Though the large meridional convective cell, predicted by the DNS, is approximated adequately by the asymptotics, the azimuthal flow fails almost catastrophically, because of significant angular momentum transport at small σ, exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction ψ derived from the asymptotics. With ψ given, we solve the now linear azimuthal equation of motion for the azimuthal velocity v by DNS. Our "hybrid" methods enable us to explain features of the flow at large Rayleigh number, found previously by Oruba, Davidson & Dormy (J. Fluid Mech.,vol. 812, 2017, pp. 890-904)