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Small-scale magnetic helicity losses from a mean-field dynamo

Small-scale magnetic helicity losses from a mean-field dynamo
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    a  r   X   i  v  :   0   9   0   5 .   0   2   4   2  v   1   [  a  s   t  r  o  -  p   h .   S   R   ]   3   M  a  y   2   0   0   9 Mon. Not. R. Astron. Soc.  000 , 000–000 (0000) Printed 3 May 2009 (MN L A TEX style file v2.2) Small-scale magnetic helicity losses from a mean-field dynamo Axel Brandenburg 1 , Simon Candelaresi 1 and Piyali Chatterjee 2 1  NORDITA, AlbaNova University Center, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden 2  Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India 3 May 2009, Revision: 1.36 ABSTRACT Using mean-field models with a dynamical quenching formalism we show that in finite do-mains magnetichelicity fluxes associatedwith small-scale magneticfields are able to alleviatecatastrophic quenching. We consider fluxes that result either from advection by a mean flow,the turbulent mixing down the gradient of mean small-scale magnetic helicity concentration,or the explicit removal which may be associated with the effects of coronal mass ejectionsin the Sun. In the absence of shear, all the small-scale magnetic helicity fluxes are found tobe equally strong both for large-scale and small-scale fields. In the presence of shear there isalso an additional magnetic helicity flux associated with the mean field, but this flux does notalleviate catastrophic quenching. Outside the dynamo-active region there are neither sourcesnor sinks of magnetic helicity, so in a steady state this flux must be constant. It is shownthat unphysical behavior emerges if the small-scale magnetic helicity flux is forced to vanishwithin the computational domain. Key words:  magnetic fields — MHD — hydrodynamics – turbulence 1 INTRODUCTION Both mean-field theories as well as direct simulations of the gen-eration of large-scale magnetic fields in astrophysical bodies suchas the Sun or the Galaxy invoke the effects of twist. Twist is typ-ically the result of the Coriolis force acting on ascending or de-scending magnetic field structures in a stratified medium. The neteffect of this systematic twisting motion on the magnetic fieldis called the  α  effect. In text books the  α  effect is normally in-troduced as a result of helical turbulence (Moffatt 1978; Parker 1979; Krause & R¨adler 1980), but it could also arise from mag- neticbuoyancy instabilities(Schmitt1987;Brandenburg & Schmitt 1998). The latter may also be at the heart of what is knownas the Babcock–Leighton mechanism that describes the net ef-fect of the tilt of decaying active regions. Mathematically, thismechanism can also be described by an  α  effect (Stix 1974).Regardless of all these details, any of these processes face aserious challenge connected with the conservation of magnetichelicity (Pouquet, Frisch & L´eorat 1976; Kleeorin & Ruzmaikin 1982; Kleeorin, Rogachevskii & Ruzmaikin 1995). The serious- ness of this is not generally appreciated, even though the conser-vation of magnetic helicity has long been associated with whatis called catastrophic  α  quenching (Gruzinov & Diamond 1994,1995, 1996). Catastrophic  α  quenching refers to the fact that the α  effect in helical turbulence in a periodic box decreases withincreasing magnetic Reynolds number for equipartition strengthmagnetic fields (Vainshtein & Cattaneo 1992; Cattaneo & Hughes 1996).Thiswouldbe ‘catastrophic’ because themagneticReynoldsnumber is large ( 10 9 in the Sun and  10 15 in the Galaxy).A promising theory for modeling catastrophic  α  quenchingin a mean-field simulation is the dynamical quenching formula,i.e. an evolution equation for the  α  effect that follows from mag-netic helicity conservation (Kleeorin & Ruzmaikin 1982). Later,Field & Blackman (2002) showed for the first time that this for-malism is also able to describe the slow saturation of a helical dy-namo in a triply-periodic domain (Brandenburg 2001a). As this dy-namo runs into saturation, a large-scale magnetic field builds up,but this field possesses magnetic helicity. Indeed, the eigenfunc-tion of a homogeneous  α 2 dynamo has magnetic and current he-licities proportional to  α . However, this concerns only the meanfield, and since the helicity of the total field is conserved, the small-scale or fluctuating field must have magnetic helicity of the oppo-site sign (Seehafer 1996). This leads to a reduction of the  α  effect(Pouquet, Frisch & L´eorat 1976). The dynamical quenching formalism is now frequently usedto model the nonlinear behavior of mean-field dynamos with andwithout shear (Blackman & Brandenburg 2002), open or closedboundaries (Brandenburg & Subramanian 2005), and sometimeseven without  α  effect (Yousef, Brandenburg & R¨udiger 2003; Brandenburg & Subramanian 2005). However, it became soonclear that the catastrophic quenching of the  α  effect can onlybe alleviated in the presence of magnetic helicity fluxes outof the domain (Blackman & Field 2000a,b; Kleeorin et al. 2000, 2002). There are various contributions to the magnetic helic-ity flux (Rogachevskii & Kleeorin 2000; Vishniac & Cho 2001; Subramanian & Brandenburg 2004, 2006), but one of the most ob- vious ones is that associated with advection. Shukurov et al. (2006) have implemented this effect in a mean-field model with dynamicalquenching in order to model the effects of a wind on the evolutionof the galactic magnetic field. One goal of the present paper is to c  0000 RAS  4 Figure 1.  Profiles of  α and U   for w α k 1  = 0 . 2  and w U  k 1  = 1 . the domain size,  H   =  L . However, in order to separate boundaryeffects from effects of the dynamo we also consider the case whereweextend thedomain inthe z  direction and choose L  = 4 H   and let α K   go smoothly to zero at z   =  H   and U  z  either goes to a constantfor  z > H   or it also goes smoothly to zero. Thus, we choose α  =  α 0 z H  Θ( z  ; H,w α ) ,  (22)where we have defined the profile function Θ( z  ; H,w ) =  12  1 − tanh  z  − H w  ,  (23)which is unity for z  ≪ H   and zero otherwise, and w  quantifies thewidth of this transition. For the wind we choose the function U  z  =  U  0 z H   [1 + ( z/H  ) n ] − 1 /n Θ( z  ; H  U  ,w U  ) ,  (24)with  n  = 20 . Both profiles are shown in Fig. 1. The strictly linearprofiles of  Shukurov et al. (2006) can be recovered by taking  L  = H  , w α → 0 , and  n →∞ .As length unit we take  k 1  =  π/ 2 H  , and as time unit we take ( η t k 21 ) − 1 . This deviates from Shukurov et al. (2006), who used π/H   as their basic wavenumber. Our motivation for this changeis that now the turbulent decay rate is equal to η t k 21 , without an ex-tra 1/4 factor. We adopt nondimensional measures for  α 0 ,  U  0 , and S  , by defining C  α  =  α 0 η t k 1 , C  U   =  U  0 η t k 1 ,  and  C  S  =  S η t k 21 .  (25)To match the parameters of  Shukurov et al. (2006), we note that C  U   = 0 . 6  corresponds to their value of 0.3, and the value k f  /k 1  =10  corresponds to their value of 5.We obtain solutions numerically using two different codes.One code uses an explicit third-order Runge-Kutta time steppingscheme and the other one a semi-implicit scheme. Both schemesemploy a second order finite differences. We begin by reporting re-sults for the srcinal profile of  Shukurov et al. (2006) with L  =  H  . Figure 2.  Space-time diagrams for B x  and B y  for the marginal values of  C  α  for  L  =  H   with  C  U   = 0  and either the symmetric solution (S) witha vacuum boundary condition on z  =  H   or the antisymmetric solution (A)with the perfect conductor boundary condition. In both cases the criticalvalue C  α  = 5 . 13  is applied. Light (yellow) shades indicate positive valuesand dark (blue) shades indicate negative values. 3 RESULTS3.1 Kinematic behavior of the solutions When the magnetic field is weak, the backreaction via the Lorentzforce and hence the  α M   term are negligible. The value of   R m does then not enter into the theory. The effects of magnetic he-licity fluxes are therefore not important, so we begin by neglectingthe wind or other transporters of magnetic helicity. For the linear α profile we find that the critical value of   C  α  for dynamo action tooccur is about 5.13. These solutions are oscillatory with a dimen-sionless frequency  ˜ ω ≡ ω/η t k 21  = 1 . 64 . The oscillations are asso-ciated with a migration in the positive z   direction. This is shown inFig. 2 where we compare with the case of a perfectly conductingboundary condition at  z   =  H   for which we find  C  crit α  = 7 . 12  and ˜ ω  = 2 . 28 .The fact that there are oscillatory solutions to the  α 2 dynamois perhaps somewhat unusual, but it is here related to the factthat  α  changes sign about the equator. Similar behavior has beenseen in some other  α 2 dynamos where  α  changes sign with depth(Stefani & Gerbeth 2003; R¨udiger, Elstner & Ossendrijver 2004; R¨udiger & Hollerbach 2004; Giesecke, Ziegler & R¨udiger 2005) c  0000 RAS, MNRAS  000 , 000–000   Magnetic helicity losses from a mean-field dynamo  5 and in simulations of helically forced turbulence with a change of sign about the equator (Mitra et al. 2009). In the latter case, how-ever, the outer boundaries were perfectly conducting. In our mean-field model such a case is also oscillatory, as will be discussed be-low.Note that we have made here the assumption that the solutionsare symmetric about the midplane, i.e.  B i ( z,t ) =  B i ( − z,t )  for i  =  x  or  y . For the application to real systems such a symmetrycondition can only be justified if the symmetric solution is moreeasily excited than the antisymmetric one for which  B i ( z,t ) = − B i ( − z,t )  for  i  =  x  or  y . This is indeed the case when we adoptthe vacuum condition at  z   =  H  , because the antisymmetric solu-tion has  C  crit α  = 7 . 14  in that case. However, this is not the casefor the perfect conductor boundary condition for which the anti-symmetric solution has  C  crit α  = 5 . 12 . We remark that there is astriking correspondence in the critical  C  α  values between the anti-symmetric solution with perfect conductor boundary condition andthesymmetric solutionwithvacuum condition on theone hand, andthe symmetric solution with perfect conductor condition and theantisymmetric solution with vacuum condition on the other hand.In the following we consider both symmetric solutions usingthe vacuum boundary conditions, as well as antisymmetric ones us-ing the perfect conductor boundary condition, which correspondsin each case to the most easily excited mode. In the cases wherewe use a vacuum condition we shall sometimes also apply a wind.This makes the dynamo somewhat harder to excite and raises  C  crit α from 5.12 to 5.60 for  C  U   = 0 . 6 , but the associated magnetic he-licity flux alleviates catastrophic quenching in the nonlinear case.Alternatively, we consider an explicit removal of magnetic helicityto alleviate catastrophic quenching. In cases with perfect conductorboundary conditions the most easily excited mode is antisymmetricabout the equator, which corresponds to a boundary condition thatpermits a magnetic helicity flux through the equator. This wouldnot be the case for the symmetric solutions. 3.2 Saturation behavior for different values of   R m We now consider the saturated state for a value of   C  α  that is su-percritical for dynamo action. In the following we choose the value C  α  = 8 . The saturation behavior is governed by equation (17).Throughout this paper we assume  k f  /k 1  = 10  for the scale separa-tionratio. Thiscorresponds tothevalue 5inShukurov et al. (2006), where  k 1  was defined differently. The dynamo saturates by build-ing up negative  α M   when  α K   is positive. This diminishes the total α  in equation (13) and saturates the dynamo. The strength of thisquenching can be alleviated by magnetic helicity fluxes that lowerthe negative value of   α M  .We plot in Fig. 3 the dependence of the saturation fieldstrength  B sat , defined here as the maximum of  | B ( z  ) | at the timeof saturation. To monitor the degree of quenching we also plot inFig. 3 the  R m  dependence of the maximum of the negative valueof   α M   at the time when the dynamo has saturated and reached asteady state. The maximum value of  − α M   is lowered by about 5%from 1.8 to 1.7 in units of   η t k 1  (see Fig. 3). Finally, we recall thatfor the  α 2 dynamos considered here both  B x  and  B y  oscillate, buttheir relativephase shift issuch that B 2 is non-oscillatory. Thenor-malized cycle frequency,  ˜ ω  ≡ ω/η t k 1 , is also plotted in Fig. 3 as afunction of   R m . It is somewhat surprising that  ω  does not stronglydepend on  R m . One may have expected that the cycle frequencycould scale with the inverse resistive time  ηk 21 . On the other hand,for oscillatory  α Ω  dynamos the cycle frequency is known to scalewith  η t k 21  Blackman & Brandenburg (2002), although that value Figure 3.  Scaling of the extremal value of  α M  , the saturation field strength B sat , and the cycle frequency  ω  with  R m  and either  C  U   = 0 . 6  (solidlines) or C  U   = 0  (dashed lines). could decrease if   η t ( B )  is strongly quenched. However, simula-tions only give evidence for mild quenching (Brandenburg et al.2008; K¨apyl¨a & Brandenburg 2009). There is a dramatic difference between the cases with andwithout magnetic helicity fluxes. For  C  U   = 0 . 6  the dynamoreaches asymptotic behavior for large values of   R m , while for C  U   = 0  the saturation field strength goes to zero and  max( − α M  ) reaches quickly an asymptotic value corresponding to a level of quenching that makes the dynamo marginally excited. 3.3 Helicity fluxes through the equator We have seen in Sect. 3.1 that in the perfect conductor case the an-tisymmetric solutions are the most easily excited ones. The bound-ary conditions for antisymmetric solutions permit magnetic helic-ity transfer through the equator. However, this alone does not suf-fice to alleviate catastrophic quenching unless a sufficiently strongflux is driven through the equator. A possible candidate for drivingsuch a flux would be a diffusive flux driven by the ∇ α M   term. InFig. 4 we plot the  R m  dependence of   max( − α M  ) ,  B sat , and  ˜ ω  for ˜ κ α  = 0 . 05  and 0. Again, catastrophic  α  quenching is alleviatedby the action of a magnetic helicity flux, but this time it is throughthe equator. The maximum value of  − α M   is lowered by 15% from c  0000 RAS, MNRAS  000 , 000–000
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