Competition of rotation and stratification in flux concentrations
aa r X i v : . [ a s t r o - ph . S R ] M a y Astronomy&Astrophysicsmanuscript no. paper c (cid:13)
ESO 2018October 20, 2018
Competition of rotation and stratification in flux concentrations
I. R. Losada , , , , A. Brandenburg , , N. Kleeorin , , , I. Rogachevskii , , Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden Department of Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm, Sweden Department of Astrophysics, Universidad de La Laguna, 38206 La Laguna (Tenerife), Spain Instituto de Astrof´ısica de Canarias, C/ V´ıa L´actea, s/n, La Laguna, Tenerife, Spain Department of Mechanical Engineering, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel Department of Radio Physics, N. I. Lobachevsky State University of Nizhny Novgorod, RussiaOctober 20, 2018, Revision: 1.142
ABSTRACT
Context.
In a strongly stratified turbulent layer, a uniform horizontal magnetic field can become unstable to spontaneously form localflux concentrations due to a negative contribution of turbulence to the large-scale (mean-field) magnetic pressure. This mechanism,which is called the negative effective magnetic pressure instability (NEMPI), is of interest in connection with dynamo scenarios inwhich most of the magnetic field resides in the bulk of the convection zone, and not at the bottom, as is often assumed. Recent workusing the mean-field hydromagnetic equations has shown that NEMPI becomes suppressed at rather low rotation rates with Coriolisnumbers as low as 0.1.
Aims.
Here we extend these earlier investigations by studying the effects of rotation both on the development of NEMPI and on theeffective magnetic pressure. We also quantify the kinetic helicity resulting from direct numerical simulations (DNS) with Coriolisnumbers and strengths of stratification comparable to values near the solar surface, and compare with earlier work at smaller scale-separation ratios. Further, we estimate the expected observable signals of magnetic helicity at the solar surface.
Methods.
To calculate the rotational effect on the effective magnetic pressure we consider both DNS and analytical studies usingthe τ approach. To study the effects of rotation on the development of NEMPI we use both DNS and mean-field calculations of thethree-dimensional hydromagnetic equations in a Cartesian domain. Results.
We find that the growth rates of NEMPI from earlier mean-field calculations are well reproduced with DNS, provided theCoriolis number is below about 0.06. In that case, kinetic and magnetic helicities are found to be weak and the rotational effect onthe effective magnetic pressure is negligible as long as the production of flux concentrations is not inhibited by rotation. For fasterrotation, dynamo action becomes possible. However, there is an intermediate range of rotation rates where dynamo action on its ownis not yet possible, but the rotational suppression of NEMPI is being alleviated.
Conclusions.
Production of magnetic flux concentrations through the suppression of turbulent pressure appears to be possible onlyin the upper-most layers of the Sun, where the convective turnover time is less than 2 hours.
Key words. magnetohydrodynamics (MHD) – hydrodynamics – turbulence – Sun: dynamo
1. Introduction
In the Sun, magnetic fields are produced by a large-scale dy-namo (see, e.g., Moffatt, 1978; Parker, 1979; Krause & R¨adler,1980; Zeldovich et al., 1983; Ossendrijver, 2003; Brandenburg& Subramanian, 2005a). Although many details of this processremain subject to debate, it seems relatively clear that rotationenhances the efficiency of the dynamo if the Coriolis parame-ter is not very large. In the absence of rotation and shear, onlysmall-scale magnetic fields are generated by what is often re-ferred to as small-scale dynamo action (see, e.g., Zeldovich etal., 1990; Brandenburg & Subramanian, 2005a). Rotation leadsto an α effect (Steenbeck et al., 1966) if there is also stratifica-tion in density or turbulent intensity. The α effect can producemean magnetic field and net magnetic flux.Stratification leads to yet another effect which does notproduce magnetic flux, but merely concentrates it locally bywhat is now referred to as negative effective magnetic pressureinstability (NEMPI). Direct numerical simulations (DNS) ofBrandenburg et al. (2011a) have shown in surprising detail manyaspects of NEMPI that were previously seen in mean-field sim-ulations (MFS) of Brandenburg et al. (2010) and that have been anticipated based on analytical studies for some time (Kleeorinet al., 1989, 1990, 1993, 1996; Kleeorin & Rogachevskii, 1994;Rogachevskii & Kleeorin, 2007).The main physics of this effect is connected with the sup-pression of turbulent pressure by a weak mean magnetic fieldthat is less than the equipartition field. At large Reynolds num-bers, the resulting reduction of the turbulent pressure is largerthan the added magnetic pressure from the mean magnetic fielditself, so that the effective magnetic pressure that accounts forturbulent and non-turbulent contributions, becomes negative. Ina strongly stratified layer, i.e., a layer in which the density variesmuch more rapidly with height than the magnetic field, thisleads to an instability that is analogous to Parker’s magneticbuoyancy instability, except that there the magnetic field variesmore rapidly with height than the density. Because the effectivemagnetic pressure is negative, magnetic structures are negativelybuoyant and sink, which has been seen in DNS of Brandenburget al. (2011a).One of the main successes of recent comparative work be-tween DNS and MFS is the demonstration of a high degree ofpredictive power of MFS. The examples include details regard-ing the shape and evolution of structures, the dependence of their
1. R. Losada et al.: Competing effects in concentrating magnetic flux depth on the magnetic field strength, and the dependence of thegrowth rate on the scale separation ratio. Recent MFS of Losadaet al. (2012) (hereafter LBKMR) have shown that in the pres-ence of even just weak rotation, the growth rate of NEMPI issignificantly reduced. Expressed in terms of the Coriolis num-ber, Co = 2Ω /u rms k f , where Ω is the angular velocity, u rms isthe rms velocity of the turbulence, and k f is the wavenumber ofthe energy-carrying eddies, the critical value of Co was predictedto be as low as 0.03. Although this value does not preclude theoperation of NEMPI in the upper parts of the Sun, where Co isindeed small (about − at the surface), it does seem surpris-ingly low, which raises questions regarding the accuracy of MFSin this case. The purpose of the present paper is therefore to com-pare MFS of LBKMR with DNS of the same setup. It turns outthat, while we do confirm the basic prediction of LBKMR, wealso resolve an earlier noticed discrepancy in the growth ratesbetween DNS and MFS in the absence of rotation (see the ap-pendix of Kemel et al., 2012a). Indeed, in the particular case ofa magnetic Reynolds number of 18 and a scale separation ratio of30, the formation of structures is unusually strong and the aver-aged stratification changes significantly to affect the determina-tion of the effective magnetic pressure. However, by restrictingthe analysis to early times, we obtain coefficients that are notonly in better agreement with an earlier formula of Brandenburget al. (2012a) with a smaller scale separation ratio, but that alsogive MFS results that agree better with our new DNS.The DNS are used primarily to compute the growth ratesand magnetic field structures during the saturated state with-out invoking the mean-field concept at all. By contrast, the τ approach (Orszag, 1970; Pouquet et al., 1976; Kleeorin et al.,1990; Rogachevskii & Kleeorin, 2004) is used to determine thedependence of mean-field coefficients on the rotation rate. Thiscan also be done with DNS (Kemel et al., 2012a). Here we applythose calculations to the case with rotation.We recall that we adopt here an isothermal stratification andan isothermal equation of state. This is done because the ef-fect that we are interested in exists even in this simplest casewhere temperature and pressure scale height are constant. Non-isothermal setups have been studied at the mean-field level bothwith (K¨apyl¨a et al., 2012, 2013) and without (Brandenburg et al.,2010) entropy evolution included. In a stably stratified layer, en-tropy evolution leads to an additional restoring force and henceto internal gravity waves (Brunt-V¨ais¨al¨a oscillations) that stabi-lize NEMPI (K¨apyl¨a et al., 2012). Thus, by using both isother-mal stratification and an isothermal equation of state, we recovera situation that is similar to an adiabatic layer, except that thenthe temperature and hence the pressure scale height decreasewith height.The system we are thus dealing with is governed by the com-bined action of rotation and stratification. In principle, such sys-tems have been studied many times before, for example to de-termine the α effect in mean-field dynamo theory (Krause &R¨adler, 1980; Brandenburg & Subramanian, 2005a). The differ-ence to earlier work is the large scale separation ratio, where thedomain is up to 30 times larger than the scale of the energy-carrying eddies. As mentioned in the beginning, stratificationand rotation lead to kinetic helicity and an α effect. We there-fore also quantify here the amount of kinetic helicity producedand whether this leads to observable effects in the resulting mag-netic structures. We use here the opportunity to explore the fea-sibility of determining the magnetic helicity spectrum from mea-surements of the magnetic correlation tensor along a longitudinalstrip. We begin by discussing first the basic equations to determinethe effective magnetic pressure from DNS and the τ approach(Section 2), compare growth rates for MFS and DNS (Section 4),and turn then to the measurement of kinetic and magnetic helic-ity from surface measurements (Section 5), before concluding inSection 6.
2. The model
We consider DNS of an isothermally stratified layer(Brandenburg et al., 2011a; Kemel et al., 2012a) and solvethe equations for the velocity U , the magnetic vector potential A , and the density ρ , in the presence of rotation Ω , D U D t = − Ω × U − c ∇ ln ρ + 1 ρ J × B + f + g + F ν , (1) ∂ A ∂t = U × B + η ∇ A , (2) ∂ρ∂t = − ∇ · ρ U , (3)where D / D t = ∂/∂t + U · ∇ is the advective derivative, ν is the kinematic viscosity, η is the magnetic diffusivity due toSpitzer conductivity of the plasma, B = B + ∇ × A isthe magnetic field, B = (0 , B , is the imposed uniformfield, J = ∇ × B /µ is the current density, µ is the vac-uum permeability, F ν = ∇ · (2 νρ S ) is the viscous force, S ij = ( ∂ j U i + ∂ i U j ) − δ ij ∇ · U is the traceless rate-of-strain tensor.The angular velocity vector Ω is quantified by its scalar ampli-tude Ω and colatitude θ , such that Ω = Ω ( − sin θ, , cos θ ) . Asin LBKMR, z corresponds to radius, x to colatitude, and y to az-imuth. The forcing function f consists of random, white-in-time,plane, non-polarized waves with a certain average wavenumber k f . The turbulent rms velocity is approximately independent of z with u rms = h u i / ≈ . c s . The gravitational acceleration g = (0 , , − g ) is chosen such that k H ρ = 1 , so the densitycontrast between bottom and top is exp(2 π ) ≈ in a domain − π ≤ k z ≤ π . Here, H ρ = c /g is the density scale heightand k = 2 π/L is the smallest wavenumber that fits into the cu-bic domain of size L . In most of our calculations, structuresdevelop whose horizontal wavenumber k x is close to k . Weadopt Cartesian coordinates ( x, y, z ) , with periodic boundaryconditions in the x - and y -directions and stress-free, perfectlyconducting boundaries at the top and bottom ( z = ± L z / ). Inall cases, we use a scale separation ratio k f /k of 30, a fluidReynolds number Re ≡ u rms /νk f of 36, and a magnetic Prandtlnumber Pr M = ν/η of 0.5. The magnetic Reynolds number istherefore Re M = Pr M Re = 18 . The value of B is specified inunits of the volume-averaged value B eq0 = √ µ ρ u rms , where ρ = h ρ i is the volume-averaged density, which is constant intime. As in earlier work, we also define the local equipartitionfield strength B eq ( z ) = √ µ ρ u rms . In our units, k = c s = µ = ρ = 1 . In addition to visualizations of the actual mag-netic field, we also monitor B y , which is an average over y anda certain time interval ∆ t . Time is sometimes specified in termsof turbulent-diffusive times t η t0 k , where η t0 = u rms / k f is theestimated turbulent diffusivity.The simulations are performed with the P ENCIL C ODE http://pencil-code.googlecode.com which usessixth-order explicit finite differences in space and a third-orderaccurate time stepping method. We use a numerical resolutionof mesh points.We compare with and extend earlier MFS of LBKMR, wherewe solve the evolution equations for mean velocity U , mean
2. R. Losada et al.: Competing effects in concentrating magnetic flux density ρ , and mean vector potential A , in the form ∂ U ∂t = − U · ∇ U − Ω × U − c ∇ ln ρ + g + F MK , (4) ∂ A ∂t = U × B − ( η t + η ) J , (5) ∂ρ∂t = − U · ∇ ρ − ρ ∇ · U , (6)where F MK = F M + F K , with ρ F M = − ∇ [(1 − q p ) B ] (7)being the mean-field magnetic pressure force, and F K = ( ν t + ν ) (cid:0) ∇ U + ∇∇ · U + 2 S ∇ ln ρ (cid:1) (8)is the total (turbulent plus microscopic) viscous force. Here, S ij = ( U i,j + U j,i ) − δ ij ∇ · U is the traceless rate-of-straintensor of the mean flow and q p is approximated by (Kemel et al.,2012b) q p ( β ) = β ⋆ β + β , (9)which is only a function of the ratio β ≡ | B | /B eq ( z ) . Here, β ⋆ and β p are coefficients that have been determined from previousnumerical simulations in the absence of rotation (Brandenburg etal., 2012a). In Eq. (7) we have taken into account that the meanmagnetic field is independent of y , so the mean magnetic tensionvanishes.The strength of gravitational stratification is characterizedby the nondimensional parameter Gr = g/c k f ≡ ( H ρ k f ) − (Brandenburg et al., 2012b). Another important nondimen-sional parameter is the Coriolis number, Co = 2Ω /u rms k f .Alternatively, we normalize the growth rate of the instability bya quantity λ ∗ ≡ β ⋆ u rms /H ρ , (10)which is motivated by the analytic results of LBKMR and thefinding that NEMPI is suppressed when > ∼ λ ∗ .
3. Effective magnetic pressure
In this section we study the effect of rotation on the function q p ( β ) . We consider first the results of DNS and turn then to ananalytical treatment. In the MFS of LBKMR we assumed that P eff ( β ) does notchange significantly with Co in the range considered. With DNSwe can compute P eff ( β ) by calculating the combined Reynoldsand Maxwell stress for a run with and one without an imposedmagnetic field. This allows us to compute q p ( β ) using Eq. (17)of Brandenburg et al. (2012a): q p = − h ρ ( u x − u x ) + b − b x i. B , (11)where the subscripts 0 indicate values obtained from a referencerun with B = 0 . This expression does not take into accountsmall-scale dynamo action which can produce finite backgroundmagnetic fluctuations b . The effective magnetic pressure is thendetermined using the equation P eff ( β ) = [1 − q p ( β )] β . The Fig. 1.
Normalized effective magnetic pressure, P eff ( β ) , for three val-ues of Co, compared with Eq. (9) for different combinations of β ⋆ and β p , as discussed in the text. result is plotted in Fig. 1 for three values of Co during an earlytime interval when structure formation is still weak and the back-ground stratification remains unchanged so that the result is notyet affected. We note that even in the Co = 0 . case, in whichthe instability is no longer so prominent, we have to restrict our-selves to early times, since the negative effective magnetic pres-sure affects the background stratification and hence the pressurechanges at later times. The resulting profiles of P eff ( β ) are vir-tually the same for all three values of Co. We also comparewith Eq. (9) for different combinations of β ⋆ and β p . It turns outthat the curves for different values of Co are best reproduced for β ⋆ = 0 . and β p = 0 . . Let us now compare with theoretical predictions for q p ( β ) . Wetake into account the feedback of the magnetic field on the tur-bulent fluid flow. We use a mean-field approach whereby ve-locity, pressure and magnetic field are separated into mean andfluctuating parts. We also assume vanishing mean motion. Thestrategy of our analytic derivation is to determine the Ω depen-dencies of the second moments for the velocity u i ( t, x ) u j ( t, x ) ,the magnetic field b i ( t, x ) b j ( t, x ) , and the cross-helicity tensor b i ( t, x ) u j ( t, x ) , where b are fluctuations of magnetic field pro-duced by tangling of the large-scale field. To this end we use theequations for fluctuations of velocity and magnetic field in rotat-ing turbulence, which are obtained by subtracting equations forthe mean fields from the corresponding equations for the actual(mean plus fluctuating) fields. The equations for the fluctuations of velocity and magnetic fieldsare given by ∂ u ( x , t ) ∂t = 1 ρ (cid:0) B · ∇ b + b · ∇ B − ∇ p (cid:1) + 2 u × Ω + ˆ N u , (12) ∂ b ( x , t ) ∂t = B · ∇ u − u · ∇ B + ˆ N b , (13)
3. R. Losada et al.: Competing effects in concentrating magnetic flux where Eq. (12) is written in a reference frame rotating with con-stant angular velocity Ω , p = p ′ + ( B · b ) are the fluctuations oftotal pressure, p ′ are the fluctuations of fluid pressure, B is themean magnetic field, and ρ is the mean fluid density. For sim-plicity we neglect effects of compressibility. The terms ˆ N u and ˆ N b , which include nonlinear and molecular viscous and dissi-pative terms, are given by ˆ N u = u · ∇ u − u · ∇ u + 1 ρ (cid:0) j × b − j × b (cid:1) + f ν ( u ) , (14) ˆ N b = ∇ × (cid:0) u × b − u × b − η ∇ × b (cid:1) , (15)where ρ f ν ( u ) is the molecular viscous force and j = ∇ × b /µ is the fluctuating current density. To eliminate the pressure termfrom the equation of motion (12) we calculate ∇× ( ∇× u ) .Then we rewrite the obtained equation and Eq. (13) in Fourierspace. We apply the two-scale approach and express two-point correla-tion functions in the following form u i ( x ) u j ( y ) = Z d k d k u i ( k ) u j ( k ) exp { i ( k · x + k · y ) } = Z d k d K f ij ( k , K ) exp( i k · r + i K · R )= Z d k f ij ( k , R ) exp( i k · r ) (16)(see, e.g., Roberts & Soward, 1975). Here and elsewhere, wedrop the common argument t in the correlation functions, f ij ( k , R ) = ˆ L ( u i ; u j ) , where ˆ L ( a ; c ) = Z a ( k + K / c ( − k + K /
2) exp ( i K · R ) d K , with the new variables R = ( x + y ) / , r = x − y , K = k + k , k = ( k − k ) / . The variables R and K correspond to thelarge scales, while r and k correspond to the small scales. Thisimplies that we have assumed that there exists a separation ofscales, i.e., the turbulent forcing scale ℓ f is much smaller than thecharacteristic scale L B of inhomogeneity of the mean magneticfield. We derive equations for the following correlation functions: f ij ( k , R ) = ˆ L ( u i ; u j ) , h ij ( k , R ) = ρ − ˆ L ( b i ; b j ) and g ij ( k , R ) = ˆ L ( b i ; u j ) . The equations for these correlation func-tions are given by ∂f ij ( k ) ∂t = i ( k · B )Φ ij + L Ω ijmn f mn + I fij + ˆ N fij , (17) ∂h ij ( k ) ∂t = − i ( k · B )Φ ij + I hij + ˆ N hij , (18) ∂g ij ( k ) ∂t = i ( k · B )[ f ij ( k ) − h ij ( k ) − h ( H ) ij ]+ D Ω jm ( k ) g im ( k ) + I gij + ˆ N gij , (19)where Φ ij ( k ) = ρ − [ g ij ( k ) − g ji ( − k )] ,D Ω ij ( k ) = 2 ε ijm Ω n k mn ,L Ω ijmn = D Ω im ( k ) δ jn + D Ω jn ( k ) δ im . Hereafter we have omitted the R -argument in the correlationfunctions and neglected terms ∼ O ( ∇ R ) , and ε ijn is the fullyantisymmetric Levi-Civita tensor. In Eqs. (17)–(19), the terms ˆ N f , ˆ N h and ˆ N g are determined by the third moments appear-ing due to the nonlinear terms, the source terms I fij , I hij and I gij which contain the large-scale spatial derivatives of the meanmagnetic and velocity fields, are given by Eqs. (A3)–(A6) inRogachevskii & Kleeorin (2004). These terms determine turbu-lent magnetic diffusion and effects of nonuniform mean velocityon the mean electromotive force.For the derivation of Eqs. (17)–(19) we use an approach thatis similar to that applied in Rogachevskii & Kleeorin (2004). Wetake into account that in Eq. (19) the terms with tensors that aresymmetric in i and j do not contribute to the mean electromotiveforce because E m = ε mji g ij . We split all tensors into nonheli-cal, h ij , and helical, h ( H ) ij , parts. The helical part of the tensorof magnetic fluctuations h ( H ) ij depends on the magnetic helicity,and the equation for h ( H ) ij follows from magnetic helicity con-servation arguments (see, e.g., Kleeorin & Rogachevskii, 1999;Brandenburg & Subramanian, 2005a, and references therein). τ -approach The second-moment equations (17)–(19) include the first-orderspatial differential operators applied to the third-order moments M (III) . To close the system, we express the set of the third-orderterms ˆ N M ≡ ˆ N M (III) through the lower moments M (II) . Weuse the spectral τ approximation which postulates that the de-viations of the third-moment terms, ˆ N M (III) ( k ) , from the con-tributions to these terms afforded by the background turbulence, ˆ N M (III , ( k ) , are expressed through similar deviations of thesecond moments: ˆ N M (III) ( k ) − ˆ N M (III , ( k ) = − M (II) ( k ) − M (II , ( k ) τ ( k ) (20)(Orszag, 1970; Pouquet et al., 1976; Kleeorin et al., 1990;Rogachevskii & Kleeorin, 2004), where τ ( k ) is the scale-dependent relaxation time, which can be identified with the cor-relation time of the turbulent velocity field for large Reynoldsnumbers. The quantities with the superscript (0) correspond tothe background turbulence (see below). We apply the spectral τ approximation only for the nonhelical part h ij of the tensor ofmagnetic fluctuations. A justification for the τ approximation indifferent situations has been offered through numerical simula-tions and analytical studies (see, e.g., Brandenburg et al., 2004;Brandenburg & Subramanian, 2005b, 2007; Rogachevskii et al.,2011). We solve Eqs. (17)–(19) neglecting the sources I fij , I hij , I gij withthe large-scale spatial derivatives. The terms with the large-scalespatial derivatives which determine the turbulent magnetic dif-fusion, can be taken into account by perturbations. We subtractfrom Eqs. (17)–(19) the corresponding equations written for thebackground turbulence, use the spectral τ approximation. We as-sume that the characteristic time of variation of the second mo-ments is substantially larger than the correlation time τ ( k ) forall turbulence scales. This allows us to get a stationary solution
4. R. Losada et al.: Competing effects in concentrating magnetic flux for the equations for the second-order moments, M (II) . Thus, wearrive to the following steady-state solution of Eqs. (17)–(19): f ij ( k ) = L − ijmn h f (0) mn ( k ) + iτ ( k · B )Φ mn ( k ) i , (21) h ij ( k ) = − iτ ( k · B )Φ ij ( k ) , (22) g ij ( k ) = iτ ( k · B ) D − im [ f mj ( k ) − h mj ( k )] . (23)We have assumed that there is no small-scale dynamo in thebackground turbulence. Here the operator D − ij is the inverse ofthe operator δ ij − τ D Ω ij (R¨adler et al., 2003) and the operator L − ijmn is the inverse of the operator δ im δ jn − τ L Ω ijmn (Elperinet al., 2005). These operators are given by D − ij = χ ( ψ ) ( δ ij + ψ ε ijm ˆ k m + ψ k ij )= δ ij + ψ ε ijm ˆ k m − ψ P ij + O ( ψ ) , (24) L − ijmn ( Ω ) = [ B δ im δ jn + B k ijmn + B ( ε imp δ jn + ε jnp δ im )ˆ k p + B ( δ im k jn + δ jn k im )+ B ε ipm ε jqn k pq + B ( ε imp k jpn + ε jnp k ipm )]= δ im δ jn + ψ ( ε imp δ jn + ε jnp δ im )ˆ k p − ψ ( δ im P jn + δ jn P im − ε imp ε jqn k pq ) + O ( ψ ) , (25)where ˆ k i = k i /k , χ ( ψ ) = 1 / (1 + ψ ) , ψ = 2 τ ( k ) ( k · Ω ) /k , B = 1 + χ (2 ψ ) , B = B + 2 − χ ( ψ ) , B = 2 ψ χ (2 ψ ) ,B = 2 χ ( ψ ) − B , B = 2 − B and B = 2 ψ [ χ ( ψ ) − χ (2 ψ )] , P ij ( k ) = δ ij − k i k j /k , δ ij is the Kronecker tensor.We use the following model for the homogeneous andisotropic background turbulence: f (0) ij ( k ) = h u i P ij ( k ) W ( k ) ,where W ( k ) = E ( k ) / πk , the energy spectrum is E ( k ) =( q − k − ( k/k ) − q , k = 1 /ℓ f and the length ℓ f is the maxi-mum scale of turbulent motions. The turbulent correlation timeis τ ( k ) = C τ ( k/k ) − µ , where the coefficient C = ( q − µ ) / ( q − . This value of the coefficient C corresponds tothe standard form of the turbulent diffusion coefficient in theisotropic case, i.e., η T = h u i R τ ( k ) E ( k ) dk = τ h u i / .Here the time τ = ℓ f / p h u i and p h u i is the characteris-tic turbulent velocity in the scale ℓ f . For the Kolmogorov’s typebackground turbulence (i.e., for a turbulence with a constant en-ergy flux over the spectrum), the exponent µ = q − and thecoefficient C = 2 . In the case of a turbulence with a scale-independent correlation time, the exponent µ = 0 and the coeffi-cient C = 1 . Motions in the background turbulence are assumedto be non-helical.Equations (21)–(25) yield: f ij ( k ) = f (0) ij ( k ) − h ij ( k ) , (26) h ij ( k ) = Ψ1 + 2Ψ (cid:18) − ψ (cid:19) f (0) ij ( k ) , (27)where Ψ = 2 τ ( k · c A ) , c A = B / √ ρ , and we have taken intoaccount that L − ijmn P mn ( k ) = P ij ( k ) . After the integration in k space we obtain the magnetic tensor h ij in physical space: h ij ( β ) = q ( β ) δ ij + q ( β ) β ij , (28)where β = B/B eq , and the functions q ( β ) and q ( β ) are givenin Appendix A. We consider the case in which the angular veloc-ity is perpendicular to the mean magnetic field. The results caneasily be generalized to the case of the arbitrary angle betweenthe angular velocity and the mean magnetic field. Fig. 2.
Dependence of λ/λ ∗ on /λ ∗ for DNS (red dashed line),compared with MFS (i) where q p0 = 20 and β p = 0 . (black solidline), as well as MFS (ii) where q p0 = 32 and β p = 0 . (blue dash-dotted line). In this case no growth was found for Co ≥ . . In allcases we have B /B eq0 = 0 . . The contribution of turbulence to the mean-field magneticpressure is given by the function q p ( β ) = [ q ( β ) − q ( β )] /β : q p ( β ) = 112 β h A (0)1 (0) − A (0)1 (4 β ) − A (0)2 (4 β ) − τ ) (cid:16) A (2)1 (0) − C (2)1 (0) − A (2)1 (4 β )+40 C (2)1 (4 β ) + 92 π (cid:2) ¯ A (16 β ) − C (16 β ) (cid:3) (cid:17)i , (29)where the functions A ( j ) i ( x ) , C ( j ) i ( x ) , ¯ A i ( y ) and ¯ C i ( y ) , andtheir asymptotics are given in Appendix A. Following earlierwork (Brandenburg et al., 2012a), we now define a magneticReynolds number based on the scale ℓ f = 2 π/k f , which isrelated to the Re M defined earlier via Rm = 2 π Re M . For B ≪ B eq / / , the function q p ( β ) is given by q p ( β ) = 45 ln Rm − Co , (30)and for B eq / / ≪ B ≪ B eq / the function q p ( β ) isgiven by q p ( β ) = 1625 (cid:0) | ln(4 β ) | + 32 β (cid:1) − Co , (31)where Co = 2Ω τ . This shows that for the values of Co of inter-est (Co ≤ . ), the correction to q p is negligible (below − ),which is in agreement with the numerical findings in Fig. 1.
4. Coriolis effects of NEMPI in DNS and MFS
We have performed DNS for different values of Co and calcu-lated the growth rate λ ; see Fig. 2. It turns out that λ shows adecline with increasing values of Co that is similar to that seenin the MFS of LBKMR, who used q p0 = 20 and β p = 0 . (corresponding to β ⋆ = 0 . ). However, for Co = 0 . and0.66, some growth is still possible, but the field begins to attainsystematic variations in the z direction which are more simi-lar to those in a dynamo. In that case, we would have to deal
5. R. Losada et al.: Competing effects in concentrating magnetic flux
Fig. 3. yt -averaged B y for Co = 0 . (left), 0.03 (middle), and 0.06 (right) at different times. with a coupled system and a direct comparison with the NEMPIgrowth rate would not be possible. We return to this issue laterin Sect. 5.1.In Fig. 2 we compare with the MFS of LBKMR, who used q p0 = 20 and β p = 0 . (corresponding to β ⋆ = 0 . ). Thisset of parameters is based on a fit by Kemel et al. (2012a) for k f /k = 30 and Re M = 18 . Note that the growth rates forthe MFS are about 3 times larger than those of the DNS. Asexplained in the introduction, this might be caused by an inac-curate estimate of the mean-field coefficients for these particu-lar values of k f /k and Re M . Indeed, according to Eq. (22) ofBrandenburg et al. (2012a), who used k f /k = 5 , these param-eters should be q p0 = 32 and β p = 0 . (corresponding to β ⋆ = 0 . ) for Re M = 18 . This assumes that these parametersare independent of the value of k f , which is not true either; seeKemel et al. (2012a). To clarify this question, we now perform3-D MFS with this new set of parameters. Those results are alsoshown in Fig. 2. It turns out that with these parameters the re-sulting growth rates are indeed much closer to those of the DNS,suggesting that the former set of mean-field coefficients mightindeed have been inaccurate. As alluded to in the introduction,a reason for this might be the fact that for k f /k = 30 NEMPIis very strong and leads to inhomogeneous magnetic fields forwhich the usual determination of mean-field coefficients, as used
Fig. 4.
Evolution of B /B eq for runs of which three are shown inFig. 3. The three horizontal lines correspond to the approximate valuesof B /B eq in the three rows of Fig. 3. by Brandenburg et al. (2012a), is no longer valid, because for in-homogeneous magnetic fields there would be additional terms
6. R. Losada et al.: Competing effects in concentrating magnetic flux
Fig. 5. B y at the periphery of the computational domain for Co = 0 . (left), 0.03 (middle), and 0.06 (right) at the same times as in Fig. 3. The x, y, z coordinates are indicated in the middle frame. Note the strong surface effect for Co = 0 . in the last time frame. in the expression for the mean Reynolds stress (cf. Kemel et al.,2012c).In Fig. 3 we show the yt -averaged B y for Co = 0 . , 0.03,and 0.06 at different times. When comparing results for differ-ent rotation rates, one should take into account that the growthrates become strongly reduced. Indeed, in the last row of Fig. 3we have chosen the times such that the amplitude of NEMPI iscomparable for Co = 0 . and 0.03, while for Co = 0 . wehave run much longer and the amplitude of NEMPI is here evenlarger; see Fig. 4, where we show B /B eq , which is the normal-ized magnetic field strength for horizontal wavenumber k = k in the top layers with ≤ k z ≤ . It is clear that the formationof structures through NEMPI remains more strongly confined tothe upper-most layers as we increase the value of Co. Even forCo = 0 . there is still noticeable growth of structures, which isdifferent from what is seen in MFS; see Fig. 2. Fig. 6.
Results of MFS of LBKMR showing B y at the periphery of thecomputational domain for Co = +0 . in the LBKMR case (left) andwith the new set of parameters (right) at the same time. (The range in B y /B shown here is larger than that shown in LBKMR.) 7. R. Losada et al.: Competing effects in concentrating magnetic flux Fig. 7. B y at the periphery of the computational domain for Co = 0 . and θ = 45 ◦ (upper row) and ◦ (lower row), at three different times(from left to right). The x, y, z coordinates are indicated in the middle frame. These figures show the generation of structures that beginto sink subsequently. However, for Co = 0 . and larger, thissinking is much less prominent. Instead, the structures remainconfined to the surface layers, which is seen more clearly in vi-sualizations of B y at the periphery of the computational domainfor Co = 0 . ; see Fig. 5, which is for approximately the sametimes as Fig. 3.To our surprise, the large-scale structures remain still inde-pendent of the y -direction, which is clearly at variance with re-sults of the corresponding MFS. In Fig. 6 we reproduce a resultsimilar to that of LBKMR for Co = ± . . Even at other anglessuch as θ = 45 ◦ and ◦ , no variation in the y -direction is seen;see Fig. 7. The reason for this discrepancy between DNS andthe corresponding MFS is not yet understood. Furthermore, theconfinement of structures to the surface layers, which is seen soclearly in DNS, seems to be absent in the corresponding MFS. The apparent lack of y dependence of the large-scale mag-netic field in the DNS shows that this contribution to the mag-netic field is essentially two-dimensional. In the lower panel ofFigure 5 of LBKMR, a comparison between 2-D and 3-D MFSwas shown for Co ≈ . as a function of latitude. At thepole, the normalized growth rates were λ/λ ∗ ≈ . and 0.14for 2-D and 3-D MFS, respectively. This difference is smallerfor smaller values of Co, but it increases with increasing val-ues of Co; see Fig. 8. We note in passing that the 2-D resultin this figure supersedes that of Figure 3 of LBKMR, were λ was determined from the amplification of the total magnetic field(which includes the imposed field), rather than the deviations of the magnetic field from the imposed one. This resulted in a 4times smaller estimate of λ . Furthermore, the critical value ofCo, above which NEMPI shuts off, is now delayed by a factor ofabout 2–3. ∗0 λ / λ ∗0 λ H ρ / η t Co
2D 3D
Fig. 8.
Dependence of λ/λ ∗ on /λ ∗ for θ = 0 ◦ in the 3-D and 2-Dcases for θ = 0 (corresponding to the pole). The plot in Fig. 8 has been done for the more optimistic setof mean-field parameters ( q p0 = 20 and β p = 0 . ), but theessential conclusions that the growth rates in 2-D and 3-D aresimilar should not depend on this. The remaining differences be-tween DNS and MFS regarding the lack of y dependence of the
8. R. Losada et al.: Competing effects in concentrating magnetic flux mean field and the confinement of structures to the surface layersmight be related to absence of mean-field transport coefficientsother than q p , η t , and ν t . By and large, however, the agreementbetween DNS and MFS is remarkably good in that the predicteddecline of NEMPI at rather modest rotation rates is fully con-firmed by DNS.
5. Kinetic and magnetic helicity
By adding rotation to our strongly stratified simulations, we au-tomatically also produce kinetic helicity. In this section we quan-tify this, compare with earlier work, and address the questionwhether this might lead to observable effects. All results pre-sented in this section are based on time series with error barsbeing estimated as the largest departure from any one third ofthe full time series.
In turbulence, the presence of rotation and stratification givesrise to kinetic helicity and an α effect (Krause & R¨adler, 1980;Brandenburg et al., 2013). As a measure of kinetic helicity, wedetermine the normalized helicity ǫ f = ω · u /k f u . (32)In Fig. 9 we compare our present runs at k f /k = 30 with thoseof Brandenburg et al. (2012b) at k f /k = 5 showing ǫ f versusGr Co. For our present runs (red filled symbols), kinetic helic-ity is clearly very small, which is a consequence of the smallvalue of Co. Compared with earlier runs at k f /k = 5 , whichgave ǫ f ≈ Gr Co, the present ones show about twice as muchhelicity. Interestingly, in the limit of rapid rotation the relativekinetic helicity declines again when the product Gr Co is largerthan about 0.5. The maximum value of ǫ f that can be reachedis about 0.3. In a fully periodic domain, dynamo action wouldbe possible when ǫ f > ( k f /k ) − , which is . in this case.However, because of stratification and boundaries, the onset isdelayed and no dynamo action has been found in the simula-tions of Brandenburg et al. (2012b). However, in the presentcase, dynamo action is possible for ǫ f > / which leads toa Beltrami-like magnetic field with variation in the z direction.Dynamo action is demonstrated in the absence of an imposedfield, which leads to slightly smaller values of ǫ f for the samevalue of Gr Co (see blue symbols in Fig. 9). The case Co = 0 . is close to marginal and the field is slowly decaying, which is inagreement with the expected position of the marginal line.In Fig. 10 we show visualizations of B x and B y for a runwith Co = 0 . showing dynamo action. Note the approximatephase shift of ◦ between B x and B y which has been seen inearlier simulations of α -type dynamo action from forced turbu-lence (Brandenburg, 2001). As alluded to in Sect. 4.1, the pos-sibility of dynamo action might be responsible for the continuedgrowth found in DNS for Co ≥ . . Visualizations of the yt -averaged B y for Co = 0 . and 0.31 show that structures withvariation in the x direction do still emerge in front of a new com-ponent that varies strongly in the z direction and that becomesstronger as the value of Co is increased.Our results for Co = 0 . and 0.31 might be examples of adynamo coupled to NEMPI. Such coupled systems are expectedto have an overall enhancement of the growth. This possibility,which is not included in the present mean-field model, has re-cently been demonstrated in spherical geometry (Jabbari et al.,2013) by coupling an α dynamo to NEMPI. Looking at Fig. 2, Fig. 9.
Relative kinetic helicity spectrum as a function of Gr Co forGr = 0 . with Co = 0 . , 0.06, 0.13, 0.49, and 0.66 (red andblue symbols) compared with results from earlier simulations ofBrandenburg et al. (2012b) for Gr = 0 . (small dots connected bya dotted line). The solid line corresponds to ǫ f = 2 Gr Co. The two hori-zontal dash-dotted lines indicate the values of ǫ ∗ f ≡ k /k f above whichdynamo action is possible for k f /k = 5 and , respectively. Runswithout imposed field (blue filled symbols) demonstrate dynamo actionin two cases. The blue open symbol denotes a case where the dynamois close to marginal. Fig. 10.
Visualization of B x and B y for the run with Co = 0 . showingdynamo action. Note the clear signature of a Beltrami field showingvariation in the z direction. Fig. 11.
Comparison of yt -averaged B y for Co = 0 . and 0.31. we are led to conclude that for Co ≥ . , the coupled systemwith NEMPI and dynamo instability is excited in a case wherethe dynamo alone would not be excited, and that the growth ratebegins to be larger than that of NEMPI alone. Obviously, morework in that direction is necessary.
9. R. Losada et al.: Competing effects in concentrating magnetic flux
As a consequence of the production of kinetic helicity, the mag-netic field should also be helical. However, since magnetic he-licity is conserved, and since it was zero initially, it should re-main zero – at least on a dynamical time scale (Berger, 1984).This condition can be obeyed if the magnetic field is bi-helical,i.e., with opposite signs of magnetic helicity at large and smallwavenumbers (Seehafer, 1996; Ji, 1999). We may now askwhether signatures of this could in principle be detected at thesolar surface. To address this question, we use our simulationat intermediate rotation speed with Co = 0 . , where magneticflux concentrations are well developed at the surface of the do-main, and compare with a larger value of 0.13.Measuring magnetic helicity is notoriously difficult, becauseit involves the magnetic vector potential which is gauge de-pendent. However, under the assumption of homogeneity andisotropy, the Fourier transform of the magnetic correlation ten-sor is M ij ( k ) = ( δ ij − ˆ k i ˆ k j ) µ E M ( k )4 πk − ǫ ijk i k k H M ( k )8 πk , (33)where ˆ k = k /k is the unit vector of k , and E M ( k ) and H M ( k ) are magnetic energy and magnetic helicity spectra, which obeythe realizability condition, µ E M ( k ) ≥ k | H M ( k ) | , where thefactor in front of E M ( k ) is just a consequence of the factor 1/2in the definition of energy. Matthaeus et al. (1982) used the solarwind data from Voyager II to determine H M ( k ) from the in situ measurements of B and Brandenburg et al. (2011b) applied it tomeasuring H M ( k ) at high heliographic latitudes where H M ( k ) is finite and turned out to be bi-helical. We now adopt the samemethod using Fourier transforms in the y direction. In the Sun,this corresponds to measuring B along a π ring at fixed polarlatitude, where one might have a chance in observing the fullcircumference at the same time. In Fig. 12 we show the resultfor three values of Co.It turns out that H M ( k ) is compatible with zero for our in-termediate value of Co. For faster rotation (Co = 0 . ), H M ( k ) is negative both at large wavenumbers ( k ≫ k f ), and positive(but still compatible with zero within error bars) at intermedi-ate wavenumbers ( . < k/k f < . ). For k/k f < . , themagnetic helicity is again negative. However, the error bars arelarge and rotation is already so fast that structure formation viaNEMPI is impossible. It is therefore unclear whether meaningfulconclusions can be drawn from our results.In the northern hemisphere of the Sun, a bi-helical spectrumis expected where magnetic helicity is negative on all scales ex-cept the largest ones where the α effect operates. In this con-nection we remind the reader that this sense is reversed in thesolar wind far from the Sun (Brandenburg et al., 2011b). Thishas also been seen in simulations of magnetic ejecta from adynamo-active sphere (Warnecke et al., 2011), which may beexplained by a diffusive magnetic helicity transport (Warneckeet al., 2012). To put the above considerations in relation to the actual helicitycontent, we now compare with the magnetic and kinetic energyand helicity spectra computed from the fully three-dimensionaldata set; see Fig. 13. The magnetic and kinetic helicity spectraare normalized by k/ and / k , respectively, so that one canestimate how much the absolute values of these spectra are be-low their maximum possible values given by the corresponding Fig. 12.
Normalized magnetic helicity spectra for different values of theCoriolis number, Co. In all panels, the same range is shown, but forCo = 0 . the normalized helicity exceeds to this range and reaches − . . Fig. 13.
Kinetic and magnetic energy and helicity spectra computedfrom the full three-dimensional data set for Co = 0 . . Positive (nega-tive) values of spectral helicity are indicated with filled (open) symbols.Note the enhancement of spectral power at the smallest wavenumber ofthe domain, k . realizability conditions, | H M | k/ ≤ E M and | H K | / k ≤ E K ,respectively.The spectra show that only at the largest scale the veloc-ity and magnetic fields have significant helicity, but they re-main clearly below their maximum possible values. At largescales (small k ), both helicities are negative (indicated by open
10. R. Losada et al.: Competing effects in concentrating magnetic flux symbols), but the magnetic helicity is predominantly positive atwavenumbers slightly below k f . This is consistent with Fig. 12,which also shows positive values, although only in the case offaster rotation (Co = 0 . ). Below the forcing scale, both en-ergy spectra show a k / spectrum, which is shallower than thewhite noise spectrum ( k ) and similar to what has been seen inhelically driven dynamos (Brandenburg, 2001). Note also the up-rise of magnetic and kinetic power at the smallest wavenumber( k = k ), which is again similar to helically driven dynamos, butit is here not as strong as in the dynamo case.
6. Conclusions and discussion
The present work has confirmed the rather stringent restrictionsof LBKMR showing that NEMPI is suppressed already for ratherweak rotation (Co > ∼ . ). This demonstrates the predictivepower of those earlier mean-field simulations (MFS). On theother hand, it also shows that the consideration of the mere ex-istence of a negative pressure is not sufficient. We knew alreadythat sufficiently strong stratification and scale separation are twoimportant necessary conditions. In this sense, the existence ofNEMPI might be a more fragile phenomenon that the existenceof a negative magnetic pressure, which is rather robust and canbe verified even in absence of stratification (Brandenburg et al.,2010). For the rather small Coriolis numbers considered here, nomeasurable change of q p was seen in the simulations, which isin agreement with our theory which predicts that the change isof the order Co .Applied to the Sun with Ω = 2 × − s − , the strong sensi-tivity of the instability to weak rotation implies that NEMPI canonly play a role in the upper-most layers where the correlationtime is shorter than Co / ≈ hours. Although this value mightchange with changing degree of stratification, it is surprising asit would exclude even the lower parts of the supergranulationlayer where τ is of the order of 1 day. On the other hand, wehave to keep in mind that our conclusions based on isothermalmodels, should be taken with care. It would therefore be usefulto extend the present studies to polytropic layers where the scaleheight varies with depth. It should also be noted that weak rota-tion (Co = 0 . ) enhances the surface appearance. At the sametime, as we have argued in Sect. 4.1, the sinking of structuresbecomes less prominent, which suggests that they might remainconfined to the surface layers. However, preliminary MFS do notindicate a significant dependence of the eigenfunction on Co forvalues below 0.1. Our interpretation, if correct, would thereforeneed to be a result of nonlinearity.If we were to apply NEMPI to the formation of active re-gions in the Sun, we should keep in mind that the scale of struc-tures would be 6–8 pressure scale heights (Kemel et al., 2012a).At the depth where the turnover time is about 2 hours, we es-timate the rms velocity to be about 500 m/s, so the scale heightwould be about 3 Mm, corresponding to a NEMPI scale of atleast 20 Mm. This might still be of interest for explaining plageregions in the Sun. Clearly, more work using realistic modelswould be required for making more conclusive statements.Regarding the production of kinetic helicity and the possi-ble detection of a magnetic helicity spectrum, our results sug-gest that the relative magnetic helicity cannot be expected tobe more than about 0.01. This is a consequence of correspond-ingly low values of kinetic helicity. We find that the normalizedkinetic helicity is given by ǫ f ≈ Gr Co. For the Sun, we ex-pect Gr = ( k f H ρ ) − ≈ . , which agrees with what is usedin our simulations, leaving therefore not much room for more optimistic estimates. In this connection we should note that inKemel et al. (2012a) the value of k f H ρ ( = Gr − ) was estimatedbased on stellar mixing length theory, using ℓ mix = α mix H p forthe mixing length with α mix ≈ . being an empirical parame-ter. For isentropic stratification, the pressure scale height H p isrelated to H ρ via γH p ≈ H ρ . With k f = 2 π/ℓ mix we obtain k f H ρ = 2 π γ/α mix ≈ π , so Gr = ( k f H ρ ) − ≈ . . We notehere that, owing to a mistake, we underestimated the value of k f H ρ by a factor of 2.6. This factor also has an enhancing ef-fect on the growth rate of NEMPI. The correct value should thenbe larger and would now be clearly faster than the turbulent–diffusive rate. Furthermore, as we have shown here, at the pointwhere NEMPI begins to be suppressed by rotation, effects re-lated to dynamo action reinforce the concentration of flux, eventhough the dynamo alone would not yet be excited. In this sense,the stringent restrictions of LBKMR from MFS appear now lessstringent in DNS. It might be hoped that this new feature caneventually be reproduced by MFS such as those of Jabbari et al.(2013) that take the α effect into account. Acknowledgements.
We thank Koen Kemel for helpful comments concerningthe influence of magnetic structure formation on the measurement of q p0 and β p in DNS and an anonymous referee for useful suggestions that have ledto improvements in the presentation and a more thorough analysis. Illa R.Losada was supported by PhD Grant ‘Beca de Investigaci´on CajaCanarias paraPostgraduados 2011’. This work was supported in part by the European ResearchCouncil under the AstroDyn Research Project No. 227952, by the SwedishResearch Council under the project grants 621-2011-5076 and 2012-5797 (IRL,AB), by EU COST Action MP0806, by the European Research Council under theAtmospheric Research Project No. 227915, and by a grant from the Governmentof the Russian Federation under contract No. 11.G34.31.0048 (NK, IR). Weacknowledge the allocation of computing resources provided by the SwedishNational Allocations Committee at the Center for Parallel Computers at theRoyal Institute of Technology in Stockholm and the National SupercomputerCenters in Link¨oping, the High Performance Computing Center North inUme˚aand the Nordic High Performance Computing Center in Reykjavik. Appendix A: The identities used in Sect. 3.2 forthe integration in k –space To integrate over the angles in k –space in Sect. 3.2 we used thefollowing identities (Rogachevskii & Kleeorin, 2004, 2007): ¯ K ij = Z k ij sin θ a cos θ dθ dϕ = ¯ A δ ij + ¯ A β ij , (A.1) ¯ K ijmn = Z k ijmn sin θ a cos θ dθ dϕ = ¯ C ( δ ij δ mn + δ im δ jn + δ in δ jm ) + ¯ C β ijmn + ¯ C ( δ ij β mn + δ im β jn + δ in β jm + δ jm β in + δ jn β im + δ mn β ij ) , (A.2) ¯ H ijmn ( a ) = Z k ijmn sin θ (1 + a cos θ ) dθ dϕ = − (cid:18) ∂∂b Z k ijmn sin θb + a cos θ dθ dϕ (cid:19) b =1 = ¯ K ijmn ( a ) + a ∂∂a ¯ K ijmn ( a ) , (A.3)where β = B/B eq , ˆ β i = β i /β , β ij = ˆ β i ˆ β j , and ¯ A = 2 πa (cid:20) ( a + 1) arctan( √ a ) √ a − (cid:21) , ¯ A = − πa (cid:20) ( a + 3) arctan( √ a ) √ a − (cid:21) , ¯ C = π a (cid:20) ( a + 1) arctan( √ a ) √ a − a − (cid:21) ,
11. R. Losada et al.: Competing effects in concentrating magnetic flux ¯ C = ¯ A − A + 35 ¯ C , ¯ C = ¯ A − C . (A.4)In the case of a ≪ these functions are given by ¯ A ( a ) ∼ π (cid:18) − a (cid:19) , ¯ A ( a ) ∼ − π a , ¯ C ( a ) ∼ π (cid:18) − a (cid:19) . In the case of a ≫ these functions are given by ¯ A ( a ) ∼ π √ a − πa , ¯ A ( a ) ∼ − π √ a + 8 πa , ¯ C ( a ) ∼ π √ a − π a . The functions A ( m ) n ( ˜ β ) are given by A (0) n ( ˜ β ) = 3 ˜ β π Z ˜ β Rm / ˜ β ¯ A n ( X ) X dX , (A.5) A (2) n ( ˜ β ) = 3 ˜ β π Z ˜ β Rm / ˜ β ¯ A n ( X ) X dX, (A.6) Z ¯ A n ( a (¯ τ ))¯ τ m d ¯ τ = 2 π A ( m ) n ( ˜ β ) , (A.7)and similarly for C ( m ) n ( ˜ β ) , where a = [ ˜ βu kτ ( k ) / , ˜ β = √ B/B eq , and X = ˜ β ( k/k ) / = ˜ β / ¯ τ = a . The ex-plicit form of the functions A ( m ) n ( ˜ β ) and C ( m ) n ( ˜ β ) for m = 0; 2 are given by A (0)1 ( ˜ β ) = 15 (cid:20) β ˜ β (3 + 5 ˜ β ) − β − ˜ β ln Rm − β ln (cid:18) β β √ Rm (cid:19)(cid:21) , (A.8) A (0)2 ( ˜ β ) = 25 (cid:20) − arctan ˜ β ˜ β (9 + 5 ˜ β ) + 9˜ β − ˜ β ln Rm − β ln (cid:18) β β √ Rm (cid:19)(cid:21) , (A.9) A (2)1 ( ˜ β ) = 263 (cid:20) β ˜ β (7 + 9 ˜ β ) − β − β M ( ˜ β ) (cid:21) , (A.10) C (2)1 ( ˜ β ) = 133 (cid:20)
221 + arctan ˜ β ˜ β (cid:18) β + 11 ˜ β + 92 (cid:19) −
192 ˜ β −
92 ˜ β − ˜ β M ( ˜ β ) (cid:21) , (A.11)where M ( ˜ β ) = 1 − β + 2 ˜ β ln(1 + ˜ β − ) . Here we have takeninto account that Rm ≫ . For B ≪ B eq / / these func-tions are given by A (0)1 ( ˜ β ) ∼ −
15 ˜ β ln Rm ,A (0)2 ( ˜ β ) ∼ −
25 ˜ β (cid:20) ln Rm + 215 (cid:21) ,A (2)1 ( ˜ β ) ∼ (cid:18) −
310 ˜ β (cid:19) , A (2)2 ( ˜ β ) ∼ −
25 ˜ β , C (2)1 ( ˜ β ) ∼ (cid:18) −
314 ˜ β (cid:19) . For B eq / / ≪ B ≪ B eq / these functions are given by A (0)1 ( ˜ β ) ∼ β (cid:20) β − β (cid:21) ,A (0)2 ( ˜ β ) ∼
25 ˜ β (cid:20) β − − β (cid:21) . Other functions in this case have the same asymptotics as in thecase of B ≪ B eq / / . For B ≫ B eq / these functions aregiven by A (0)1 ( ˜ β ) ∼ π ˜ β − β , A (0)2 ( ˜ β ) ∼ − π ˜ β + 6˜ β ,A (2)1 ( ˜ β ) ∼ π β −
32 ˜ β , A (2)2 ( ˜ β ) ∼ − π β + 3˜ β ,C (2)1 ( ˜ β ) ∼ π
28 ˜ β −
12 ˜ β . The functions q ( β ) and q ( β ) are given by q ( β ) = 112 h A (0)1 (0) − A (0)1 (4 β ) − A (0)2 (4 β ) − (Ω τ ) (cid:16) A (2)1 (0) − C (2)1 (0) − A (2)1 (4 β )+20 C (2)1 (4 β ) + 92 π (cid:2) ¯ A (16 β ) − C (16 β ) (cid:3) (cid:17)i , (A.12) q ( β ) = 112 h A (0)2 (4 β ) + (Ω τ ) (cid:16) A (2)1 (0) − C (2)1 (0) − A (2)1 (4 β ) + 60 C (2)1 (4 β ) + 92 π (cid:2) ¯ A (16 β ) − C (16 β ) (cid:3)(cid:17)i . (A.13) References
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12. R. Losada et al.: Competing effects in concentrating magnetic flux