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Hydrodynamic Helicity in Boussinesq-Type Models of the Geodynamo M ISSN 1069-3513, Izvestiya, Physics of the Solid Earth, 2006, Vol. 42, No. 6, pp. 449–459. © Pleiades Publishing, Inc., 2006. Original Russian Text © M.Yu. Reshetnyak, 2006, published in Fizika Zemli, 2006, No. 6, pp. 3–13. Hydrodynamic Helicity in Boussinesq-Type Models of the Geodynamo M. Yu. Reshetnyak Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Bol’shaya Gruzinskaya ul. 10, Moscow, 123995 Russia Received October 17, 2005 Abstract—The generation of hydrodynamic helicity is considered in models of thermal convection of a rapidly rotating incompressible liquid (small Rossby numbers). It is shown that a system of cyclonic vortices arising in geodynamo models generates a large-scale hydrodynamic helicity. The spatial distribution of the helicity is ana- lyzed as a function of the intensity of heat sources in models with various boundary conditions for the velocity field. PACS numbers: 91.25.Cw DOI: 10.1134/S1069351306060012 INTRODUCTION The violation itself of the reflection invariance in the rotating turbulence is usually associated with the Cori- The origination of a large-scale magnetic field in olis force action on eddies moving in a medium with a celestial bodies is related to magnetic convection pro- zero gradient of density [Parker, 1979]. Let the density cesses. It is known that the magnetic field can be gener- of the body decrease with its radius. Then, an ascending ated by both large-scale flows [Bullard and Gellman, eddy broadens, producing a velocity component 1954] and small-scale flows [Vainshtein, 1983]. In the directed along the eddy radius. Accordingly, the Corio- latter case, along with small-scale fluctuating magnetic lis force arises due to the radial velocity and general fields, large-scale fields arise [Vainshtein et al., 1980] (a rotation. This force makes the eddy rotate in the sense synergic process), which is closely connected with the opposite to the general rotation. In a similar way, a concept of the α-effect in the theory of mean fields descending eddy contracts, also giving rise to a radial [Krause and Rädler, 1980]. Averaging over one coordi- velocity, and the Coriolis force makes the eddy rotate in nate, e.g., over the longitude, and using an additional the same sense as the general rotation. This results in a generation term with α, the 3-D dynamo problem could nonzero correlation, be reduced in many cases to the 2-D problem, while χH 〈〉⋅ preserving the validity of the limitation imposed by the = v curlv , (1) Cowling theorem, which precludes the generation of an i.e., the average helicity of the flow (here, v is the tur- axisymmetric magnetic field by an axisymmetric flow. bulent velocity and the broken brackets mean averag- α ing). Correlation (1) determines the value of the The origin of the -effect consists in the fact that the α mirror symmetry is violated in a convective rotating -effect (the Moffat formula): body and, as a result, the number of, e.g., clockwise H H ατ= – χ /3, (2) rotating eddies in the northern hemisphere is systemat- ically greater than the number of anticlockwise rotating where τH is the characteristic time of correlation. Note ones [Krause and Rädler, 1980]. On the contrary, anti- that the helicity V · curlV is interesting in itself, without clockwise rotating eddies in the opposite hemisphere regard for the α-effect, because it is the integral of dominate over clockwise rotating ones. This violation motion, like the kinetic energy [Kurganskii, 1993; Frik, of the mirror symmetry, resulting from the averaging of 2003; Lesieur, 1997]. In mechanics, the original of χH the Maxwell equations over turbulent pulsations, gives is the squared angular momentum [Dolzhanskii, 2001]. rise to a component of the average magnetic field B par- It is remarkable that the pseudoscalar value χH can be allel to the average electric current J = αB (whereas the estimated from symmetry considerations because a magnetic field is usually perpendicular to the current). pseudoscalar can be constructed from the density gra- It is this generally small parallel component of the mag- dient and angular rotation velocity (the so-called netic field that is capable of closing the self-excitation Krause formula, see for details [Reshetnyak and circuit of the magnetic field in the Faraday phenome- Sokolov, 2003]). This traditional interpretation of the non of electromagnetic induction. α-effect is, however, of limited applicability. First of 449 450 RESHETNYAK all, the averaged flow V often cannot be separated from –1 ∂V EPr ------- + ()V ⋅ ∇ V the turbulent pulsations v in applied problems.1 The ∂t (3) α relationship between the helicity and -effect indicated = curlBB× – ∇P – 1 × V ++RaTr1 E∆V above has such a simple form only in the case of locally z r homogeneous and isotropic turbulence. Finally, the ∂ T ()⋅ ∇ ()∆ medium in which the magnetic field is generated is ------ + V TT+ 0 = T. ∂t often incompressible, so that the validity of the idea of expanding and compressing eddies is limited. It is The dimensionless numbers of Prandtl, Ekman, Ray- worth noting that, after the α-effect has been estab- ν ν lished, the variability of density of the medium is, as a leigh, and Roberts are written as Pr = --- , E = ------------- , κ Ω 2 rule, neglected in dynamo models. 2 L α δ η g0 TL ν The limitations noted above are particularly charac- Ra = -------------------Ωκ , and q = ---κ , where is the kinematic vis- teristic of two very important cases of planetary and 2 cosity coefficient, α is the coefficient of volume expan- laboratory dynamos. In the laboratory experiment car- δ ried out by Frik [2003], liquid sodium used as a work- sion, g0 is the gravity acceleration, T is the unit pertur- bation of the temperature T relative to the equilibrium ing medium can be considered as completely incom- η pressible. The dynamo mechanism operates in the profile T0, and is the magnetic diffusion coefficient. Earth’s outer core, consisting mostly of liquid iron and The Rossby number is defined as Ro = EPr–1. having a density changing, according to current ideas, Problem (3) becomes fully defined by introducing by 15–20% along the radius [Geomagnetism, 1988]. boundary conditions at r = ri, r0. Zero boundary condi- The situation with compressibility is similar in planets tions are used for the temperature perturbations T, and, of the solar system. Moreover, the coefficient of volume in conjunction with the profile T0 specified above, this expansion in the Earth’s core is too small (~2 × 10–5 K–1) corresponds to fixed values of the total temperature to generate heating-induced helicity in terms of T0 + T: (1, 0). Note that variation in spherically sym- Parker’s model. metric boundary conditions for T (e.g., fixed tempera- ture or heat flux at boundaries or various forms of heat However, even the first 3-D models of the geody- source distribution in the core) affects weakly the gen- namo in the Boussinesq approximation had regimes erated magnetic field beginning from the onset of with nonzero large-scale helicity [Glatzmaier and Rob- developed convection [Sarson et al., 1997; Reshetnyak erts, 1995]. Below, we consider possible mechanisms and Steffen, 2005] but can dramatically change the of the helicity generation in such systems and compare regime of geomagnetic field reversals in the presence of our results with known models of the solar dynamo, in inhomogeneities of the heat flux at the core–mantle which the generation mechanisms of the α-effect have boundary ranging within a few tens of percent [Glatz- been discussed for many years [Krivodubskii, 1984; maier et al., 1999]. Therefore, without loss of general- Rüdiger and Kitchatinov, 1993]. ity in what follows, we accept the condition of the first ri/r – 1 kind with fixed temperatures: T0 = ----------------- at r0 = 1. EQUATIONS OF THE GEODYNAMO 1 – ri The impermeability condition Vr = 0 is accepted for Formulation of the Problem the velocity field. The tangential velocity components Vθ and Vϕ are subjected to the no-slip conditions Vθ = 0 We consider the geodynamo equations for an and Vϕ = 0 at the boundaries r r and r (the rigid core incompressible liquid (∇ · V = 0) in a spherical layer = i 0 ≤ ≤ in some models can rotate about the z axis) [Glatzmaier (ri r r0) rotating about the z axis at an angular veloc- Ω θ ϕ and Roberts, 1995], which is typical of rigid bound- ity in a spherical coordinate system (r, , ). We aries; an alternative to these boundary conditions for introduce the following units of measurement for the the tangential velocity components is the vanishing of V t P B κ L ↔ velocity , time , pressure , and magnetic field : / , τ τ τ 2 κ ρκ2 2 Ωρκµ tangential stresses at these boundaries: rθ, rϕ = 0 L / , /L , and 2 0, where L is the length unit, κ is the molecular heat conductivity coefficient, ρ is the [Kuang and Bloxham, 1997; Tilgner and Busse, 1997]. density, and µ is the magnetic constant; then, we have Boundary conditions of these types are used in models 0 of convection in stars with a free boundary [Chan- τ↔ ∂B –1 drasekhar, 1981] (exact expressions for in various -------= curl()VB× + q ∆B ∂t coordinate systems can be found in [Landau and Lif- shitz, 1975]). Such a type of boundary conditions, 1 Many geo- and astrophysical problems are characterized by con- somewhat strange for the Earth’s core, is related to the tinuous velocity field spectra and do not make the separation into following. It is known that, in the first case, the gener- large and small scales used in the dynamics of average fields. ated magnetic field concentrates near rigid boundaries IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol.
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