Magnetic helicity in stellar dynamos: new numerical experiments

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<ul><li><p>Astron. Nachr./AN 323 (2002) 2, 99122</p><p>Magnetic helicity in stellar dynamos: new numerical experiments</p><p>A. BRANDENBURG , W. DOBLER , and K. SUBRAMANIAN </p><p> NORDITA, Blegdamsvej 17, DK-2100 Copenhagen , Denmark Department of Mathematics, University of Newcastle upon Tyne, NE1 7RU, UK National Centre for Radio Astrophysics - TIFR, Pune University Campus, Ganeshkhind, Pune 411 007, India</p><p>Received 2001 November 30; accepted 2002 February 5</p><p>Abstract. The theory of large scale dynamos is reviewed with particular emphasis on the magnetic helicity constraint inthe presence of closed and open boundaries. In the presence of closed or periodic boundaries, helical dynamos respond tothe helicity constraint by developing small scale separation in the kinematic regime, and by showing long time scales inthe nonlinear regime where the scale separation has grown to the maximum possible value. A resistively limited evolutiontowards saturation is also found at intermediate scales before the largest scale of the system is reached. Larger aspect ratioscan give rise to different structures of the mean field which are obtained at early times, but the final saturation field strengthis still decreasing with decreasing resistivity. In the presence of shear, cyclic magnetic fields are found whose period isincreasing with decreasing resistivity, but the saturation energy of the mean field is in strong super-equipartition with theturbulent energy. It is shown that artificially induced losses of small scale field of opposite sign of magnetic helicity as thelarge scale field can, at least in principle, accelerate the production of large scale (poloidal) field. Based on mean field modelswith an outer potential field boundary condition in spherical geometry, we verify that the sign of the magnetic helicity fluxfrom the large scale field agrees with the sign of . For solar parameters, typical magnetic helicity fluxes lie around </p><p>per cycle.</p><p>Key words: MHD turbulence</p><p>1. Introduction</p><p>The conversion of kinetic into magnetic energy, i.e. the dy-namo effect, plays an important role in many astrophysicalbodies (stars, planets, accretion discs, for example). The gen-eration of magnetic fields on scales similar to the scale ofthe turbulence is a rather generic phenomenon that occurs forsufficiently large magnetic Reynolds numbers unless certainantidynamo theorems apply, which exclude for example two-dimensional fields (e.g., Cowling 1934).</p><p>There are two well-known mechanisms, which allow thegeneration of magnetic fields on scales larger than the eddyscale of the turbulence: the alpha-effect (Steenbeck, Krause&amp; Radler 1966) and the inverse cascade of magnetic helic-</p><p>Correspondence to: Axel Brandenburg, brandenb@nordita.dk current address: Kiepenheuer Institute for Solar Physics,</p><p>Schoneckstr. 6, 79104 Freiburg, Germany current address: Inter University Centre for Astronomy and As-</p><p>trophysics, Post Bag 4, Pune University Campus, Ganeshkhind,Pune 411 007, India</p><p>ity in hydromagnetic turbulence (Frisch et al. 1975, Pouquet,Frisch, &amp; Leorat 1976). In a way the two may be viewed asthe same mechanism in that both are driven by helicity. The -effect is however clearly nonlocal in wavenumber space,whereas the inverse cascade of magnetic helicity is usuallyunderstood as a (spectrally) local transport of magnetic en-ergy to larger length scales. Both aspects are seen in simula-tions of isotropic, non-mirror symmetric turbulence: the non-local inverse cascade or -effect in the early kinematic stageand the local inverse cascade at later times, when the fieldhas reached saturation at small scales (Brandenburg 2001a,hereafter referred to as B01).</p><p>It is not clear whether either of these two mechanisms isactually involved in the generation of large scale magneticfields in astrophysical bodies. Although much of the largescale magnetic field in stars and discs is the result of shearand differential rotation, a mechanism to sustain a poloidal(cross-stream) magnetic field of sufficiently large scale isstill needed. The main problem that we shall be concernedwith in this paper is that of the associated magnetic helic-</p><p>c WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, Germany 2002 0044-6337/02/0207-0099 $ 17.50+.50/0</p></li><li><p>100 Astron. Nachr./AN 323 (2002) 2</p><p>ity production, which may prevent the large scale dynamoprocess from operating on a dynamical time scale. However,not many alternative and working dynamo mechanisms havebeen suggested so far. In the solar context, the BabcockLeighton mechanism is often discussed as a mechanism thatoperates preferentially in the strong field regime (e.g. Dik-pati &amp; Charbonneau 1999). However, Stix (1974) showedthat the model of Leighton (1969) is formally equivalentto the model of Steenbeck &amp; Krause (1969), except that inLeighton (1969) the -effect was nonlinear and mildly sin-gular at the poles. Nevertheless, a magnetically driven ef-fect has been suspected to operate in turbulent flows driven bythe magnetorotational instability (Brandenburg et al. 1995) orby a magnetic buoyancy instability (Ferriz-Mas, Schmitt, &amp;Schussler 1994, Brandenburg &amp; Schmitt 1998). In the frame-work of mean-field dynamo theory, magnetically driven dy-namo effects have also been invoked to explain the observedincrease of stellar cycle frequency with increased stellar ac-tivity (Brandenburg, Saar, &amp; Turpin 1998). However, as longas these mechanisms lead to an -effect, they also producemagnetic helicity and are hence subject to the same problemas before.</p><p>Although kinetic helicity is crucial in the usual expla-nation of the effect, it is not a necessary requirement formagnetic field generation, and lack of parity invariance is al-ready sufficient (Gilbert, Frisch, &amp; Pouquet 1988). However,as in every effect dynamo, magnetic helicity is produced atlarge scales, which is necessarily a slow process. A mecha-nism similar to the ordinary effect is the incoherent ef-fect (Vishniac &amp; Brandenburg 1997), which works in spiteof constantly changing sign of kinetic helicity (in space andtime) provided there is systematic shear and sufficient turbu-lent diffusion. This mechanism has been invoked to explainthe large scale magnetic field found in simulations of accre-tion disc turbulence (Brandenburg et al. 1995, Hawley, Gam-mie, &amp; Balbus 1996, Stone et al. 1996). In addition, how-ever, the large scale field of Brandenburg et al. (1995) showsspatio-temporal coherence with field migration away fromthe midplane. This has so far only been possible to explainin terms of an dynamo with a magnetically driven ef-fect (Brandenburg 1998). In any case, the incoherent effect,which has so far only been verified in one-dimensional mod-els, would not lead to the production of net magnetic helicityin each hemisphere.</p><p>Yet another mechanism was suggested recently by Vish-niac &amp; Cho (2001), which yields a mean electromotive forcethat does not lead to the production of net magnetic helicity,but only to a transport of preexisting magnetic helicity. Thisis why this mechanism can, at least in principle, work on afast time scale even when the magnetic Reynolds number islarge. Finally, we mention a totally different mechanism thatworks on the basis of negative turbulent diffusion (Zheligov-sky, Podvigina, &amp; Frisch 2001). This is what produces turbu-lence in the KuramotoSivashinsky Equation, which is thenstabilised by hyperdiffusion (i.e. a fourth derivative term).Among all these different dynamo mechanisms, the conven-tional effect and the inverse magnetic cascade are the onlymechanisms that have been shown numerically to produce</p><p>Large scale dynamos(e.g. in the sun)</p><p>Is significant net magnetic helicityproduced in each hemisphere?</p><p>NOfast effect:</p><p>(like in kinematiccase,</p><p>and large scaleseparation)</p><p>NOnon- effect:(e.g., negative</p><p>magn. diffusion,incoherent ,</p><p>Vishniac &amp; Choeffect)</p><p>YES</p><p>standard effect: what happens</p><p>to magnetichelicity?</p><p>Alternative Afast effect:</p><p>transferfrom small</p><p>to largescales, no</p><p>net helicity</p><p>Alternative Bnon- effect</p><p>dynamos:no</p><p>numericalevidence</p><p>yet</p><p>Alternative C(i)</p><p>dissipation(ii) recon-</p><p>nection(near</p><p>surface?)</p><p>Alternative Dloss</p><p>throughboundaries:(i) equator?</p><p>(ii) outersurface</p><p>Fig. 1. Sketch illustrating the different ways astrophysical dynamosmay be able to circumvent the magnetic helicity problem.</p><p>strong large scale fields under turbulent conditions. These arehowever exactly the mechanisms that suffer from the helicityconstraint.</p><p>The purpose of the present paper is to assess the signifi-cance of the helicity constraint and to present some new nu-merical experiments that help understanding how the con-straint operates and to discuss possible ways out of thedilemma. For orientation and later reference we summarise inFig. 1 various models and the possible involvement of mag-netic helicity in them. Dynamos based on the usual effectare listed to the right. They all produce large scale magneticfields that are helical. However, in a periodic or an infinitedomain, or in the presence of perfectly conducting bound-aries, the magnetic helicity is conserved and can only changethrough microscopic resistivity. This would therefore seem tobe too slow for explaining variations of the mean field on thetime scale of the 11 year solar cycle. Open boundary condi-tions may help, although this was not yet possible to demon-strate, as will be explained in Sect. 4 of this paper.</p><p>Given that the issue of magnetic helicity is central to ev-erything that follows, we give in Sect. 2 a brief review ofmagnetic helicity conservation, the connection between theinverse cascade and the realisability condition, and the sig-nificance of gauge-invariant forms of magnetic helicity andthe surface integrated magnetic helicity flux. Readers famil-iar with this may jump directly to Sect. 3, where we discussthe issue of magnetic helicity cancellation in kinematic and</p></li><li><p>A. Brandenburg et al.: Magnetic helicity in stellar dynamos 101</p><p>non-kinematic dynamos, or to Sect. 4, where we begin witha discussion of the results of B01 and Brandenburg &amp; Dobler(2001, hereafter referred to as BD01), or to Sect. 4.3 wherenew results are presented.</p><p>2. The magnetic helicity constraint</p><p>Magnetic helicity evolution does not give any additional in-formation beyond that already contained in the inductionequation governing the evolution of the magnetic field it-self. Nevertheless, the concept of magnetic helicity provesextremely useful because magnetic helicity is a conservedquantity in ideal MHD, and almost conserved even in slightlynon-ideal conditions. The evolution equation for the mag-netic helicity can then be used to extract information thatcan be easily understood. At a first glance, however, mag-netic helicity appears counterintuitive, because it involves themagnetic vector potential, , which is not itself a physicalquantity, as it is not invariant under the gauge transformation . Only the magnetic field, , whichlacks the irrotational information from , is independentof and hence physically meaningful. Therefore isnot a physically meaningful quantity either, because a gaugetransformation, changes . For periodicor perfectly conducting boundary conditions, or in infinitedomains, the integral </p><p> is however gauge-invariant, because in</p><p> (1)</p><p>the surface integral on the right hand side vanishes, and thelast term vanishes as well, because .</p><p>The evolution of the magnetic field is governed by theinduction equation</p><p> (2)</p><p>where is the electric field. This equation can be integratedto yield</p><p> (3)</p><p>where is the scalar (electrostatic) potential, which is alsoreferred to as the gauge potential, because it can be chosenarbitrarily without affecting . Common choices are , which preserves the Coulomb gauge , and , which is convenient in MHD for numerical purposes.If const, another convenient gauge is ,which results in</p><p> ( const) (4)</p><p>Using Eqs (2) and (3) one can derive an equation for thegauge-dependent magnetic helicity density ,</p><p> (5)</p><p>Using Ohms law, </p><p> , we have </p><p> , where is the microscopic magnetic diffusivity,</p><p>the magnetic permeability in vacuum, and </p><p>the electric current density.</p><p>2.1. Magnetic helicity conservation with periodicboundaries</p><p>Consider first the case of a periodic domain, so the divergenceterm in Eq. (5) vanishes after integration over the full volume,and thus</p><p> (6)</p><p>where angular brackets denote volume averages, so</p><p> , where is the (constant) volume of thedomain under consideration. Note that the terms in Eq. (6)are gauge-independent [see Eq. (1)]. More important, how-ever, is the fact that the rate of change of magnetic helicity isproportional to the microscopic magnetic diffusivity . Thisalone is not sufficient to conclude that the magnetic helic-ity will not change in the limit , because the currenthelicity, , may still become large. A similar effect isencountered in the case of ohmic dissipation of magnetic en-ergy which proceeds at the rate </p><p> . In thestatistically steady state (or on time averaging), the rate ofohmic dissipation must be balanced by the work done againstthe Lorentz force. Assuming that this work term is indepen-dent of the value of (e.g. Galsgaard &amp; Nordlund 1996) weconclude that the root-mean-square current density increaseswith decreasing like</p><p> as (7)</p><p>whilst the rms magnetic field strength, </p><p>, is essentially in-dependent of . This, however, implies that the rate of mag-netic helicity dissipation decreases with like</p><p> as (8)</p><p>Thus, under many astrophysical conditions where the mag-netic Reynolds number is large ( small), the magnetic helic-ity , as governed by Eq. (6), is almost independent of time.This motivates the search for dynamo mechanisms indepen-dent of . On the other hand, it is conceivable that magnetichelicity of one sign can change on a dynamical time scale,i.e. faster than in Eq. (8), if magnetic helicity of the other signis removed locally by advection, for example. This is essen-tially what the mechanism of Vishniac &amp; Cho (2001) is basedupon. So far, however, numerical attempts to demonstrate theoperation of this mechanism have failed (Arlt &amp; Branden-burg 2001). Yet another possibility is that magnetic helicityof one sign is generated at large scales, and is compensatedby magnetic helicity of the other sign at small scales, as inthe kinematic effect. However, as we shall see in Sect. 4,this does not happen efficiently when nonlinear effects of theLorentz force come into play.</p><p>2.2. Realisability condition and connection with inversecascade</p><p>Magnetic helicity is important for large scale field genera-tion because it is a conserved quantity which cannot easilycascade forward to smaller scale. We shall demonstrate thishere for the case where the magnetic field is fully helical.The existence of an upper bound for the magnetic helicityis easily seen by decomposing the Fourier transformed mag-netic vector potential, </p><p>, into a longitudinal component,</p></li><li><p>102...</p></li></ul>

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