Dynamo Generated by the Centrifugal Instability
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Dynamo generated by the centrifugal instability Florence Marcotte1,2 and Christophe Gissinger3 1Laboratoire de RadioAstronomie, Ecole Normale Sup´erieure, CNRS, 24 rue Lhomond, 75005 Paris, France 2Institut de Physique du Globe de Paris, CNRS, 1 rue Jussieu, 75005 Paris, France 3Laboratoire de Physique Statistique, Ecole Normale Sup´erieure, CNRS, 24 rue Lhomond, 75005 Paris, France We present a new scenario for magnetic field amplification where an electrically conducting fluid is confined in a differentially rotating, spherical shell with thin aspect-ratio. When the angular momentum sufficiently decreases outwards, an hydrodynamic instability develops in the equatorial region, characterised by pairs of counter-rotating toroidal vortices similar to those observed in cylindrical Couette flow. These spherical Taylor- Couette vortices generate a subcritical dynamo magnetic field dominated by non-axisymmetric components. We show that the critical magnetic Reynolds number seems to reach a constant value at large Reynolds number and that the global rotation can strongly decrease the dynamo onset. Our numerical results are understood within the framework of a simple dynamical system, and we propose a low-dimensional model for subcritical dynamo bifurcations. Implications for both laboratory dynamos and astrophysical magnetic fields are finally discussed. PACS numbers: Introduction tary cores for instance, shear profiles may be produced by dif- ferential rotation between the inner core and outer core-mantle During the last decades, satellite observations have signif- boundaries, which explains why spherical Couette flow, i.e. icantly improved our knowledge of both the internal struc- the flow between two concentric rotating spheres, has been ex- ture and the magnetic field of planets in the solar system. tensively studied in both geophysical and astrophysical frame- Similarly, observations of stellar surface magnetic fields have work. made important progress in the past few years. For instance, Recently, the ability of spherical Couette flow to generate a while classical axial dipole magnetic fields are observed for dynamo magnetic field has been studied by [16]. They show the Earth, Jupiter or Saturn (e.g. [1–3]), data from the Voyager that in a large gap, the single pair of equatorially symmet- II mission have shown that Neptune’s and Uranus’ dynamo ric eddies due to secondary meridional circulation in such magnetic fields both exhibit strong non-dipolar components Couette flows fails to sustain dynamo action in Earth-like and inclined dipolar field [4, 5]. geometry (with aspect ratio χ = 0.35). Indeed, they find These observations have challenged the widely accepted within this geometry that dynamo instability can only build scenario for magnetic field generation in natural systems, i.e. on a first hydrodynamic instability, consisting in the desta- self-generation of the magnetic field by the turbulent motions bilization of the strong equatorial jet, or destabilisation of of an electrically conducting fluid due to thermo-chemical the quasi-geostrophic shear-layer encompassing the tangent 1/4 convection. For instance, Uranus and Neptune presents sim- cylinder (the Stewartson E layer, [17]) when overall rota- ilar internal structures, surface magnetic fields and zonal tion is added to the system. flows, but their measured outward heat flux are very differ- In smaller shell gap, Couette flow may be prone to the cen- ent. Although several studies focused on modifications of trifugal instability if the angular momentum of the fluid suf- convection-driven dynamo models (see for instance [6–9]), it ficiently decreases outward. There has been considerable in- is therefore tempting to find mechanisms that do not rely on terest in characterising such hydrodynamic Taylor-Couette in- thermal convectionbut rather depend on other properties, such stability of spherical Couette flow – experimentally [18, 19], as planetary internal structure or differential rotation. In this numerically [20–23], and analytically [24, 25]. Although a perpective, some authors have studied new types of dynamo few number of studies are devoted to the generation of mag- forcings likely to occur in planets, such as precession [10] or netic field by such flow in cylindrical geometry [26–29] , the arXiv:1606.04078v1 [physics.flu-dyn] 13 Jun 2016 tidal motions [11]. ability of centrifugally-unstable spherical Couette flow to am- On the other hand, due to the difficulty to generate vigorous plify and maintain a magnetic field has never been demon- convection in experiments, it is currently impossible to repro- strated yet. The aim of the present paper is to show that duce convection-driven dynamos in the laboratory. For this centrifugally-unstable flows are very efficient to produce dy- reason, successful dynamo experiments have been obtained namo magnetic fields in thin spherical shell, and may be rele- so far in setups significantly different from natural systems, in vant for non-convective magnetized systems. Moreover, such which a liquid metal is pumped inside a constrained pipe ge- spherical Taylor-Couette dynamos display strong analogies ometry [12, 13] or driven by mechanical impellers [14]. More with dynamos generated by thermo-chemical convection, al- recently, there have been efforts to develop new dynamo ex- though much easier to generate in a laboratory. Their experi- periments based on more realistic flows, such as precession- mental or theoretical study may lead to a better understanding driven dynamos [15]. In this perspective, differential rotation of magnetic field generation in planetary interiors. may play an important role in astrophysical objects. In plane- The outline of the present article is as follows: we first Typeset by REVTEX 2 present our numerical model (section 1) and some hydrody- in the covered parameters range by convergence tests (up to namic results on the centrifugal instability in a spherical shell lmax = 256, mmax = 128, and 96 radial gridpoints). (section 2). We then study the Taylor-Vortex dynamo with both non-rotating (section 3) and rotating (section 4) outer sphere. Finally, we derive in section 5 a simple model based CENTRIFUGAL INSTABILITY IN SPHERICAL SHELL on symmetry arguments which describes the subcritical nature of the dynamo transition. In the idealized situation where the spheres are replaced by We conclude with a few remarks on the relevance of this two infinitely long rotating cylinders, the ideal laminar Cou- flow for astrophysical and laboratory dynamos. ette solution for the rotation Ω is given by : A MODEL Ω(r)= A + 2 , (3) 1 r2 We consider an incompressible, electrically conducting Ω 2 Ω 2 2 2 2 2 Ω in which A1 = ( 2r2 1r1)/(r2 r1) and A2 = r1r2( 1 fluid confined between two co-rotating spheres. r and r Ω 2 2 − − − 1 2 2)/(r2 r1). In this case, the Rayleigh criterion predicts are respectively the radius of the inner and outer spheres, and linear stability− if the angular momentum increases outwards Ω , Ω are respectively the angular speed of the inner and the Ω 2 Ω 2 1 2 ( 1r1 2r2). Otherwise, the centrifugal instability is ob- outer spheres. The governing equations are the magnetohy- served≤ when the Reynolds number is large enough, and takes drodynamic (MHD) equations, i.e., the Navier-Stokes equa- the form (at onset) of axisymmetric Taylor-Couette vortices tion coupled to the induction equation. The problem is made contained in the poloidal plane. A similar behavior occurs dimensionless following [16]: this approach allows us to com- in a spherical gap: if the angular momentum decreases out- pare simulations with and without global rotation for a fixed wards along a cylindrical radius, Taylor vortices can be gener- Reynolds number. ated near the equator if the gap is small enough. Note that in ∂ this case, the bifurcation is strongly imperfect, since it appears u 1 Re˜ + (u ∇)u + 2E− ez u = from a background flow already involving a poloidal recircu- ∂t · × (1) lation. Nevertheless, for a sufficiently thin spherical shell, one 1 1 ∇Π + ∆u + E− + Re˜ χ− (∇ B) B, can expect Rayleigh’s criterion to remain valid at least close to − × × the equatorial plane, which translates with our dimensionless ∂B 1 numbers to the condition: ∇ ˜ − ∆ ∂ = (u B)+ RePm B. (2) t × × 1 1 E χ− χ δRe− . (4) The dimensionless parameters are the Ekman number E = ≥ − ∆Ω ν ˜ r2r1 Ω 2 , the Reynolds number Re = ν , the magnetic Prandtl 2r2 number Pm = ν/η and the aspect ratio of the spherical shell Let us first study the case with no global rotation, i.e. when χ ν η = r1/r2, where and are respectively the kinematic Ω2 = 0. At low Reynolds number, the mean flow is equato- viscosity and the magnetic diffusivity. The inner and outer rially symmetric and essentially toroidal, except for a large- spheres are electrical insulators, and no-slip boundary con- scale meridional recirculation between polar and equatorial ditions are used on these boundaries. In all the results re- regions (Fig.1-a ). However, characterisation of the equato- ported in this paper, a thin gap χ = 0.9 is used. Note that rial flow in the narrow-gap spherical shell indicates the exis- the Reynolds number used here as a control parameter (Re˜ ) is tence of a Rayleigh-like stability threshold, above which sev- based on the outer shell radius. In the following, our results eral pairs of counter-rotating, toroidal and axisymmetric Tay- will be presented in terms of a more conventional Reynolds lor vortices (TV) build up on the primary base flow. The new number (Re) based on the gap width δ = (r2 r1)/r2) to al- base state above this threshold corresponds to the saturated − δ r1∆Ω low easier comparison with previous works: Re = ν . The TV concentrated in the vicinity of the equator, as shown in equations (1)-(2) are integrated using the PARODY code [30]. Fig.1 for several Reynolds numbers. This bifurcation has been This code relies on a toroidal/poloidal decomposition of the widely studied in different aspect-ratios by [20–23].