Resilience of helical fields to turbulent diffusion II: direct numerical simulations
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Resilience of helical fields to turbulent diffusion II: directnumerical simulations
Pallavi Bhat ⋆ , Eric G. Blackman † and Kandaswamy Subramanian ‡ IUCAA, Post Bag 4, Ganeshkhind, Pune 411007, India. Department of Physics and Astronomy, University of Rochester, Rochester, NY14618, USA
16 April 2018
ABSTRACT
The recent study of Blackman and Subramanian (Paper I) indicates that large scalehelical magnetic fields are resilient to turbulent diffusion in the sense that helical fieldsstronger than a critical value, decay on slow (resistively mediated), rather than fast ( ∼ turbulent) time scales. This gives more credence to potential fossil field origin modelsof the magnetic fields in stars, galaxies and compact objects. Here we analyze a suiteof direct numerical simulations (DNS) of decaying large scale helical magnetic fields inthe presence of non-helical turbulence to further study the physics of helical field decay.We study two separate cases: (1) the initial field is large enough to decay resistively, istracked until it transitions to decay fast, and the critical large scale helical field at thattransition is sought; (2) the case of Paper I, wherein there is a critical initial helicalfield strength below which the field undergoes fast decay right from the beginning. Forcase (1), the initial decay rate in the slow regime is on an average about twice thatof a purely resistive decay and both simulations and solutions of the two scale model(from Paper 1), reveal that the transition energy, E c , is independent of the scale ofthe turbulent forcing, within a small range of R M . We also find that the kinetic alpha, α K , is subdominant to magnetic alpha, α M , in the DNS, justifying an assumption inthe two scale model. For case (2), we show more comprehensively than in Paper I,how the two scale theory predicts that large scale helical energy at the transition is E c = ( k /k f ) M eq , where k and k f are the large scale and small turbulent forcingscale respectively and M eq is the equipartition magnetic energy. The DNS in thiscase agree qualitatively with the two scale model but the R M currently achievable, istoo small to satisfy a condition 3 /R M << ( k /k f ) , necessary to robustly reveal thetransition, E c . The fact that two scale theory and DNS agree wherever they can becompared and also the two scale theory predicts the transition of case (1) gives ussome confidence that E c of Paper I should be identifiable at higher R M in DNS aswell. Key words: dynamo–(magnetohydrodynamics) MHD–turbulence–galaxies:magneticfields–stars:magnet
Astrophysical systems, such as stars, galaxies and evengalaxy clusters, are observed to host coherent largescale magnetic fields (Clarke et al. 2001; Clarke 2004;Govoni & Feretti 2004; Brandenburg & Subramanian 2005;Vogt & Enßlin 2005; Fletcher 2010; Beck 2012). The originof such cosmic magnetic fields has been a long standing openquestion. A popular paradigm is that coherent large scale ⋆ [email protected] † [email protected] ‡ [email protected] magnetic fields arise due to dynamo amplification of smallseed fields. An interesting alternative would be if the fieldfrom a previous evolutionary phase has simply been fluxfrozen when a star, galaxy or a galaxy cluster was formed.Astrophysical systems are generally turbulent and such ini-tial fields could then in principle decay due to turbulentdiffusion. Indeed due to the above reason, the continued ex-istence of primordial large scale fields in galaxies has beenmostly considered to be untenable (Ruzmaikin et al. 1988).However, if coherent magnetic fields in these astrophysi-cal systems were initially of helical nature, and sufficientlystrong, Blackman & Subramanian (2013) (henceforth PaperI) argued on the basis of magnetic helicity conservation, that c (cid:13) they would be resilient to turbulent diffusion and hence, sur-vive up to the present epoch.If sub-equipartition helical fields can avoid turbulentdecay, then another practical implication is that if helicalfields are observed in a system–such as astrophysical jets–the observed helical fields would not necessarily be indicativeof magnetic energy domination in the system.Magnetic helicity is a nearly conserved quantityin general astrophysical context and has been usefulin understanding of dynamo saturation, by leading tothe development of the dynamical quenching formal-ism (Kleeorin et al. (2000); Field & Blackman (2002);Blackman & Brandenburg (2002); Blackman & Field(2002); Subramanian (2002); Brandenburg & Subramanian(2005) and references therein). Paper I used the largeand small scale magnetic helicity evolution equations ina two scale model, along with the mean field inductionequation and the minimal τ -approximation, to understandthe decay of helical large scale fields. An intriguing resultof their work is that even fields which are initially of fairlysub-equipartition strength, would undergo a slow resistivedecay if they are helical. It is important to check thevalidity of this simple two scale model and the results ofPaper I, by comparing with results from direct numericalsimulations (DNS) of decaying large scale helical magneticfields in presence of non-helical turbulence. This is the mainmotivation of the current work.There have been previous DNS studies of decayinghelical fields by Yousef et al. (2003) motivated by tryingto understand the quenching of turbulent diffusion. Also,Kemel et al. (2011) discuss simulations of decaying helicalfields in non-helical turbulence, applied to the cylindricalgeometry. These simulations emphasize the decay of ini-tially strong fields of order equipartition value. On the otherhand, Paper I focused on the situation where the initialfield strength is lowered to smaller and smaller values anda threshold energy, E c = ( k /k f ) M eq was shown to setthe transition from slow to fast decay. Here k and k f arethe wave numbers associated with the large scale field andthe small turbulent forcing scale, respectively and M eq isthe equipartition energy. Such a threshold was not evidentin earlier work. We wish to examine here through DNS, thedecay of helical field in more generality and with differentsets of initial strength and k f . One motivation is also toexamine if there is indeed a k f dependent threshold energy.We limit our present study to initially fully helical fields,where the measure of helicity is defined as the ratio of thehelical magnetic energy to the the total magnetic energy.As will be evident, there is enough richness and subtlety tobe understood here, even without considering fractionallyhelical cases.In the next section, we discuss the setup for the simu-lations and the quantities to be estimated. We find that thelarge scale helical magnetic field decays in two stages. Thefirst phase is of a slow decay, due to only microscopic resis-tivity. We discuss the slow regime in detail in Section 2.1.The second phase comprises of fast decay of the large scalemagnetic field and is discussed in Section 2.2. We also esti-mate the transition point which marks the transition fromslow to fast decay in section 3. There are two kinds of tran-sition points, arising in two different contexts. One is iden-tified in simulations of decaying field which start with the Fully helical large scale field decay in non-helical turbulence B r m s k f =3, 256 k f =5, 256 k f =7, 256 k f =10, 256 k f =7, 512 k f =10, 512 Figure 1.
The evolution of B rms in helical magnetic field de-cay simulations starting with a fully helical field of strength0.2 (superequipartition). Here, we show the evolution curves for k f = 3 , , ,
10 from runs of resolution 256 and also from tworuns with higher resolution of 512 at k f = 7 and 10 same initial field strength (of equipartition value) and resis-tivity, but different k f (forcing or the turbulent scale). Theseshow a transition of the evolving field from a slow to fastdecay regime after decaying to a critical energy threshold.The other kind could arise in simulations of decaying field,where the initial field strength is decreased until a criticalvalue is reached, below which the field decays at the fast rateright from the beginning. The second kind has been empha-sized in Paper I and is discussed in Section 3.2. In general,throughout the paper, we have juxtaposed the results fromthe simulations with the numerical solutions of the corre-sponding two scale model from Paper I. A discussion of ourresults and the conclusions are given in Section 4. One of the primary aims of our work is to determine how fasta helical large scale field decays when subject to turbulentdiffusion by small scale forcing. We use the
Pencil Code to simulate the decay of helical large scale fields in the pres-ence of non-helical turbulence. The fluid is assumed to beisothermal, viscous, electrically conducting and compress-ible. We solve the continuity, Navier-Stokes and inductionequations given by, D lnρDt = −∇ · u , (1) D u Dt = − c s ∇ lnρ + J × B ρ + F visc + f, (2) ∂ A ∂t = u × B + η ∇ A . (3) http://pencil-code.googlecode.com (Brandenburg 2003)c (cid:13) , 1–15 umerical simulations of decaying helical fields Here ρ is the density related to the pressure by P = ρc s ,where c s is speed of sound. The operator D/Dt = ∂/∂t + u ·∇ is the lagrangian derivative, where u is fluid velocity field.The induction equation is being expressed in terms of thevector potential, A and B = ∇ × A , is the magnetic field J = ∇ × B /µ is the current density and µ is the vacuumpermeability ( µ = 1 in the DNS). The viscous force is givenby, F visc = ν (cid:20) ∇ u + 13 ∇ · ∇ u + 2 S · ∇ lnρ (cid:21) (4)where, S = 12 (cid:18) ∂u i ∂x j + ∂u j ∂x i − δ ij ∇ · u (cid:19) , (5)is the traceless rate of strain tensor. The term f = f ( x , t ) isresponsible for turbulent forcing localised in k-space in mag-nitude and randomly changing phase at every time step (seeHaugen et al. (2004) for more details). These equations aresolved in a Cartesian box of a size l = 2 π on a cubic grid with N mesh points, adopting periodic boundary conditions.The initial magnetic field is a Beltrami field, B =B(sinkz , coskz , k = (2 π ) /l . In each run, the he-lical magnetic field is allowed to decay under the influenceof a non-helical turbulent forcing. We generate the turbu-lent flow in the box by randomly forcing the fluid aboutan average wavenumber k f , which is much larger than thewavenumber at which the large scale magnetic field is placed.The initial velocity field is zero in all the simulations. Wehave run a suite of simulations with varying k f (from 3 to10), and initial field strength. Most of the simulations havea ‘resolution’ of 256 , with 2 higher resolution, 512 , runs.The magnetic and fluid Reynolds numbers throughout thispaper are defined as R M = u rms /ηk f and Re = u rms /νk f ,respectively, where η and ν are the resistivity and viscosity ofthe fluid and are taken to be equal here and hence, P M = 1.Table 1 gives a list of all the simulations run towards thestudy.Starting with a helical magnetic field, the rms magneticfield B rms , decays exponentially as shown in Fig. 1, in ba-sically two stages. The field decays at a slow rate first andthen transitions to a much faster rate before finally reach-ing saturation due to the floor provided by the fluctuationdynamo (given that all the simulations have an R M whichis supercritical enabling the fluctuation dynamo to operate(Kazantsev 1967; Haugen et al. 2004; Schekochihin et al.2004; Bhat & Subramanian 2013)). In the top panel ofFig. 2, we show the evolution of u rms for run B with k f = 5(considering this to be the fiducial case). The kinetic en-ergy decays by less than 10% along with magnetic field inthe first stage. After transition, the magnetic energy decaysat a fast rate and as a result, the effect of Lorentz forceson the velocity field is reduced, thus increasing the u rms .In the bottom panel of Fig. 2, the corresponding kinetic en-ergy spectral evolution has been shown at times, t=10, 1000,2200 and 2800, with decreasing line thickness. The peak at k = 5 corresponds to the constant forcing.The corresponding evolution of the magnetic energyspectrum, M ( k ) for run B with k f = 5, is shown in the toppanel of Fig. 3. The top spectrum is at t=100 and evolves to Evolution of kinetic energy, Run B : k f =5
0 1500 3000time0.100.110.120.130.140.15 u r m s -8 -7 -6 -5 -4 -3 -2 K i n e t i c e n e r g y s p ec t r a Figure 2.
The top panel shows the evolution of u rms with time.It grows to an average initial value of 0.12. In the bottom panel,we show the spectra of the kinetic energy at times t=10, 100, 2200and 2800 for curves with decreasing line thickness respectively. the bottom at t=2700, with an interval of △ t=200 betweensuccessive spectra. Initially, the total helicity and the asso-ciated helical energy is on k = 1, which is then transferredto smaller scales, on time scale of few eddy turn over times.In the bottom panel of Fig. 3, we show the fractional helic-ity spectrum, defined as the ratio of helical energy, kH ( k ) / M ( k ), where H(k) and M(k) are themagnetic helicity and energy spectra respectively. We findthat the helicity on the small scales is of the same sign asthat on the large scale, as can be seen in the bottom panelof Fig. 3. The upper three curves corresponding to times,t=100, 1100, 2100 show that in the large scales, the energyis almost fully helical. And the fractional helicity in smallscales is <
1, due to the non-helical energy being constantlypumped at k = k f (where k f > k ), due to the non heli-cal forcing. By t=2700, corresponding to the bottom mostspectrum of highest thickness, the large scale field has al-most decayed completely. And hence, the sign of helicity isfluctuating across all scales.The constant non-helical forcing at k f , generates tur-bulence and subsequently facilitates the transfer of helicityand energy from to k to smaller scales. And then it be-comes imperative to identify the ‘large’ scale field, to be ableto analyse the simulation results. We consider contributionsfrom k = 1 to k = 2, to form the large scale field. This seemsan appropriate choice given that the power spectra of var-ious quantities like magnetic energy and magnetic helicity,have a minimum at k=2 and peak again at k = k f as canbe seen from Fig. 3. Consequently, for k >
2, the spectralenergy has been considered to be a part of the small scalefield. Thus each variable is split into a large scale (mean)and small scale (fluctuating) quantity, with an overbar de- It is difficult to decide an unambiguous scale separation betweenthe large and small in general. But in cases such as the α dy-namo, the scale separation can be decided based on the oppositesign of helicity on the two scales (Brandenburg & Subramanianc (cid:13) , 1–15 Table 1.
All the simulations have u rms = 0 .
12 and start withthe fully helical magnetic field of strength B rms = 0 .
2. The runsare at P M = 1. Also γ is the average initial decay rate.Run Resolution k f η × R M M γ A 256
10 2.0 60 1 0.0009E 512
10 1.5 80 1 0.0007G 256 k Power spectra with turbulent forcing at k f =5 -5 -4 -3 -2 M ( k ) -3 -2 -1 k H ( k ) / ( * M ( k )) Figure 3.
The top panel shows the evolving magnetic powerspectra, M(k) for run B. Spectral evolution has been plotted fromtop to bottom, corresponding to times, t=100 to t=2700, with aninterval of △ t = 200, between subsequent spectra. The bottompanel shows the ratio of helical energy, kH ( k ) / M ( k ). The red ’plus’ symbol indicates a positive value,and the black star indicates a negative value. The four curvesfrom top to bottom are at times t=100, 1100, 2100 and 2700 noting the mean; for example the magnetic field B = B + b ,where B and b are respectively the large and small scalefields. Further, we will consider volume averages of severalquadratic quantities like magnetic energy density over thewhole simulation box, and denote such averages by angularbrackets hi .The three quantities we are mainly interested in arethe following. First, the total large scale magnetic energy, h B i /
2, defined as, h B i Z M ( k ) dk. (6) Second, large scale helical energy (LSHE), M H , M H = Z kH ( k )2 dk. (7)This is an important quantity because the two scale modelfrom Paper I is applied to study the evolution of LSHE.And then the behaviour of LSHE can be extended to thetotal large scale energy, h B i /
2, upto some time scale. Third,small scale helicity (SSH), h a · b i , h a · b i = Z ∞ H ( k ) dk (8)An important understanding derived from the two scalemodel is that the SSH remains in steady state for most ofthe slow decay phase. And the transition from slow to fastdecay is largely governed by the change in SSH. And hence itis critical to check the nature of SSH evolution in the DNS.We show the time evolution of h B i / M H and h a · b i in Fig. 4 for runs A, B, C and D with a resolution of 256 . Allthese quantities have been normalised by the equipartitionenergy M eq = ρu rms /
2, with ρ ≈ η , but different k f and hencedifferent R M (see Table 1). In Fig. 5, we show the time evo-lution of these three quantities in higher resolution (512 )runs E and F. In the upper panels of Fig. 4 and Fig. 5, theevolution of the total large scale energy, h B i /
2, is shown indot-dashed red line and the LSHE, M H , is shown in solidblack, along with the solution for M H from the two scalemodel in blue dashed line. (The evolution equations for thetwo scale model are given below). In the lower panels, thedotted red line shows evolution of SSH, h a · b i . One can ob-serve that the gap between the curve for the two quantities, h B i / M H increases once the helicity has decreasedsubstantially in the fast decay phase. Also, the gap becomesmore pronounced for smaller k f runs due to the smaller scaleseparation.The decay rate in a particular decay phase is calculatedby two methods. One is by simply fitting an exponentialform to the M H (t), given by, M H ( t ) = M H e − γt (9)where M H and γ are the free parameters. Note that weretain the code time scale, t, while plotting the decay curvesin the Fig. 4 and Fig. 5. Here, t = k c s , where k = 1.It can be seen from Fig. 4 and Fig. 5, that slope ofLSHE evolution curve is changing continuously. Thus, thedecay rate obtained by the first method, will be an averageestimate. In the second method, we fit for the entire LSHEevolution curve, using the function, M H ( t ) = exp (cid:18) A + Bt + 1 C + Dt (cid:19) (10)where A, B, C and D are free parameters. Then the loga-rithmic slope of LSHE is derived from the fit. This gives thedecay rate as a function of time.In both methods of estimating the decay rate, best fitswere decided by the calculation of least squares. We nowdiscuss the two phases of decay. Other fitting forms were tried, to fit the entire curve of LSHEevolution. This form provides the best fit by the method of leastsquares. c (cid:13) , 1–15 umerical simulations of decaying helical fields Run A : 256 , k f =3, R m =200 L a r ge s ca l e m ag n e t i c e n e r g y LSE from DNSLSHE from DNSLSHE from two scale modelAverage slow decay rate = 0.0009 0.0340 500 1000 1500 2000 2500 3000time0.0010.0100.100 SS H SSH from DNScurve fitSSH from two scale model
Run B : 256 , k f =5, R m =120 M ag n e t i c L S e n e r g y LSE from DNSLSHE from DNSLSHE from two scale modelAverage slow decay rate = 0.0009 0.0400 500 1000 1500 2000 2500 3000time0.0010.0100.100 SS H SSH from DNScurve fitSSH from two scale model
Run C : 256 , k f =7, R m =86 M ag n e t i c L S e n e r g y d eca y LSE from DNSLSHE from DNSLSHE from two scale modelAverage slow decay rate = 0.0009 0.0480 500 1000 1500 2000 2500 3000time0.0010.0100.100 SS H SSH from DNScurve fittwo scale model solution
Run D : 256 , k f =10, R m =60 M ag n e t i c L S e n e r g y LSE from DNSLSHE from DNSLSHE from two scale modelAverage slow decay rate = 0.0009 0.0460 500 1000 1500 2000 2500 3000time0.0010.0100.100 SS H SSH from DNScurve fitSSH from two scale model
Figure 4.
We show the evolution of h B i / M H and h a · b i for runs with 256 resolution at k f = 3 , , M eq . The thin vertical line marks the time by when the SSH decreases by 50% of its initial steady state value, in thebottom panel and intersects the LSHE curve at the transition energy indicated by the horizontal thin line. The thin blue line shows thefit using Eq. 9 to the slow decay phase. The purely resistive decay rate for the large scale magneticenergy k = 1 mode is given by 2 ηk = 4 × − (in dimen-sionless units) for runs A to D with η = 2 × − . The slowdecay regime for assessing the average decay rate is identi-fied from t = 0 to t = t slow , where t = t slow is chosen asan arbitrary time comfortably less than the time, the curveevolves towards the transition region.From the exponential fit to the initial slow decayregime, the average decay rate, γ S ∼ × − , is al-most twice the purely resistive decay rate for k = 1(where the large scale field resides). Nevertheless, this γ S ismuch smaller than the corresponding turbulent decay rate ∼ η t k = (2 / u rms /k f . For example, in the fiducial caseof run B, where k f = 5 and with u rms = 0 .
12, we have(2 / u rms /k f = 0 . ∼
18 times larger than the γ S obtained from DNS. Notice that the initial field is quiteclose to the equipartition strength. This goes to show that helical magnetic field of a sufficiently large inital strength,decays slowly at a rate which is of the order of the resistivetime-scale and does not decay turbulently as one may havenaively expected.In the top panel of Fig. 6, we show the fit to LSHEevolution curve for runs A-D, using the form in Eq. 10. Inthe bottom panel of Fig. 6, we show the logarithmic slope ofLSHE from the fit. On taking the mean of the logarithmicslope values in the resistive decay phase (from t = 0 to t = t slow ), we again obtain the average estimate of γ S ∼ × − . Hence matching with the γ S obtained from thefirst method. Nonetheless, as can be seen from Fig. 6, thedecay rate is continuously changing even in the slow decayphase. This slowly changing decay rate can be understoodby considering the following.The large scale field in the simulations is almost fullyhelical. Hence the large scale field is expected to decay ac-cording to the equations governing the evolution of magnetic c (cid:13)000
18 times larger than the γ S obtained from DNS. Notice that the initial field is quiteclose to the equipartition strength. This goes to show that helical magnetic field of a sufficiently large inital strength,decays slowly at a rate which is of the order of the resistivetime-scale and does not decay turbulently as one may havenaively expected.In the top panel of Fig. 6, we show the fit to LSHEevolution curve for runs A-D, using the form in Eq. 10. Inthe bottom panel of Fig. 6, we show the logarithmic slope ofLSHE from the fit. On taking the mean of the logarithmicslope values in the resistive decay phase (from t = 0 to t = t slow ), we again obtain the average estimate of γ S ∼ × − . Hence matching with the γ S obtained from thefirst method. Nonetheless, as can be seen from Fig. 6, thedecay rate is continuously changing even in the slow decayphase. This slowly changing decay rate can be understoodby considering the following.The large scale field in the simulations is almost fullyhelical. Hence the large scale field is expected to decay ac-cording to the equations governing the evolution of magnetic c (cid:13)000 , 1–15 Run E : 512 , k f =7, R m =120 M ag n e t i c L S e n e r g y LSE from DNSLSHE from DNSLSHE from two scale modelSlow decay slope = 0.0008 0.0410 500 1000 1500 2000 2500 3000time0.0010.0100.100 SS H SSH from DNScurve fitSSH from two scale model
Run F : 512 , k f =10, R m =80 M ag n e t i c L S e n e r g y LSE from DNSLSHE from DNSLSHE from two scale modelAverage slow decay rate = 0.0007 0.0360 500 1000 1500 2000 2500 3000time0.0010.0100.100 SS H SSH from DNScurve fitSSH from two scale model
Figure 5.
We show the evolution of h B i / M H and h a · b i for runs with 512 resolution at k f = 7 and 10. All the quantities arenormalised by M eq . The thin vertical line marks the time by when the SSH decreases by 50% of its initial steady state value, andintersects the LSHE curve at the transition energy indicated by the horizontal thin line. The thin blue line shows the fit using Eq. 9 tothe slow decay phase. helicity (which is a conserved quantity in the limit of η → d h A · B i dt = − η h J · B i (11)All the quantities can be split into a mean (large scale) anda fluctuating (small scale) component. Accordingly, Eq. 11can be written as, d h A · B i dt + d h a · b i dt = − η (cid:0) h J · B i + h j · b i (cid:1) (12)where cross terms between the large and small scales vanish.Now, the small scale magnetic helicity is expected to reachsteady state much faster than the large scale magnetic he-licity. And hence, if d h a · b i /dt → h J · B i and h j · b i .To obtain an equation for large scale helicity, one canuse the mean field induction equation, given as, ∂ B ∂t = − c ∇ × E = ∇ × (cid:0) V × B + E − η ∇ × B (cid:1) (13)where, E = v × b is the electromotive force (or theEMF). Using the first order smoothing approxima-tion (FOSA) (Moffatt 1978; Krause & Raedler 1980)or a τ -approximation closure scheme (Pouquet et al.1976; Blackman & Field 2002; R¨adler et al. 2003;Brandenburg & Subramanian 2005), E can be shownto be given by, E = ( α K + α M ) B − η t J . (14)Here the kinetic alpha effect ( α K ), magnetic alpha effect( α M ) and the turbulent diffusivity ( η t ) are given by, α K ≃ − τ v · ω , α M ≃ τ ρ j · b , η t ≃ τ v (15)and τ is the correlation time which can be estimated to be of order the dynamical or eddy turn over time, t eddy =1 / ( u rms k f ). We uncurl Eq. 13 to obtain an equation for themean vector potential, A . This can be used to obtain thedynamical equation for large scale helicity,12 d h A · B i dt = h ( α K + α M ) · B i − η t h J · B i − η h J · B i (16)where we have put divergence terms to zero for periodicboundary conditions. The SSH evolution is obtained by sub-tracting Eq. 16 from Eq. 12,12 d h a · b i dt = −h ( α K + α M ) · B i + η t h J · B i − η h j · b i (17)From Eq. 16, we see that the large scale helical field woulddecay due to turbulent diffusion in the absence of the alphaeffect. In our context of forced non-helical turbulence, thekinetic alpha effect is expected to be negligible. We verifythis below, directly from DNS, in section 2.3. However, α M could be generated by the action of turbulent diffusion ona large scale helical field. This can be seen explicity in theterm, η t h J · B i in Eq. 17, which leads to the generation of h a · b i and hence h j · b i (even if were initially zero), hav-ing the same sign as h J · B i . Thus, the resulting α M can inprinciple, balance the turbulent diffusion leading to a slowresistive decay of the LSHE. This implicitly constitutes largescale dynamo action, driven by the small scale current he-licity.We also see from Eq. 17, that −h α M · B i ∝− k f h a · b ih B i , causes rapid damping of the SSH and leadsto a steady state. For such a steady small scale helicity,Eq. 17 can be used to derive a relation between, h J · B i and h j · b i . We have for d h a · b i /dt → − α M h B i + η t h J · B i − η h j · b i (18)where, we have dropped the kinetic alpha term followingPaper I. And by substituting the expression for α M into c (cid:13) , 1–15 umerical simulations of decaying helical fields time 0.0010.0100.1001.000 L S H E ( M H / M e q )
500 1000 1500 2000 2500 0.0010.010 E n e r g y d eca y r a t e , γ corresponding turbulent decay rate, 2 η t k k f =3k f =5k f =7k f =10 Figure 6.
Top panel shows the normalised LSHE evolution forruns A-D along with the fit using the function in Eq. 10. Bottompanel gives the logarithmic derivative using the fit, i.e.the decayrate evolution for all runs. The symbol, ∗ , in the bottom panel,marks the value of respective turbulent decay rates. Eq. 18, we have, h j · b i = h J · B i ηη t + h B i / M eq (19)where, M eq is equal to ρu rms /
2, in dimensionless units.In the denominator of the RHS of the Eq. 19, if oneconsiders that η << η t , and hence negligible, then with h B i / M eq = 1 (which is the case initially in the runs A-F),we have h j · b i ≃ h J · B i . This implies that from Eq. 12, with d h a · b i /dt →
0, the LSHE decays at twice the resistive de-cay rate. This is an average estimate as ( h B i /
2) is actuallydecaying slowly, and hence, h j · b i / h J · B i will increase overtime, increasing the decay rate as can be seen from the bot-tom panel of Fig. 6. So, in fact the decay rate of LSHE ischanging continuously even in the slow decay phase.To further corroborate this continuous change in de-cay rate from theory, consider the following: Using theEq. 19, we show in Fig. 7, an estimate of the quantity M = h B i / M eq = h j · b i / h J · B i − η/η t , for run B as a solidyellow line. While LSHE calculated directly using Eq. 7 isshown in solid black. Then, we use the function in Eq. 10to fit for both LSHE and M . Subsequently, we derive thelogarithmic slope of the evolution curves using the fit andhave shown them as black and yellow dash-dotted lines fordirect LSHE and M , respectively. While the amplitude ofthe curve for M is smaller than that of direct LSHE by ∼ M , matchesclosely with that of direct LSHE for most of the resistive de-cay phase. And we can see directly from Fig. 7, that decayrate is increasing constantly by a small amount for most ofthe slow decay phase. The match in the decay rate evolu-tion of M H with that from the model M , shows that the twoscale model is quite useful for understanding the simulationresults in the slow decay phase.Note that the Fig. 7 of Kemel et al. (2011), shows k f =5, Run B M ag n e t i c l a r ge s ca l e h e l i ca l e n e r g y , L S H E direct LSHEM = (< −J.−B>/
Figure 7.
The nomalised LSHE evolution curve is overlayed bythe predicted noisier LSHE from the two scale model, as accordingto the Eq. 19, in the fiducial case of run B. The dashed curvesare the corresponding logarithmic slopes evaluated using the fitfor each of the LSHE curves. the decay of a highly superequipartition field with time.They find the initial decay rate for the magnetic energyto be γ = − ηk (while we obtain the decay rate of 4 ηk for equipartition initial fields). For h B i / >> M eq , fromEq. 19, h j · b i << h J · B i and hence the large scale field isthen predicted to decay at a purely resistive rate, which isconsistent with their finding.In passing we also note that the correct prediction ofthe rate of slow decay, γ S by the two scale model, whichuses the closure relation for E in the Eq. 16, also lends somecredence to such mean field closures. We see from Fig. 4 that the LSHE decays at a much fasterrate after it drops below some critical energy threshold. Thefast decay phase is identified from some time after the curvedtransition region, to the time just before the field saturates,to a level determined by the tail of the fluctuation dynamoat large scales. (A more precise definition of the transitionto the fast decay regime follows in section 3). Here the ex-pected decay rate is the turbulent decay rate, γ F = 2 η t k ,where η t = u rms / k f . In the simulations, u rms = 0 .
12 and η t = 0 .
027 for run A, η t = 0 .
016 for run B, η t = 0 . η t = 0 .
008 for run D. The two scale modelsolutions match with expected decay rate of 2 η t k and areshown as the blue dashed lines in Fig. 4. It can be seen fromFig. 4, that in almost every DNS run, the slope of the LSHEcurve in fast decay phase, is steeper than that of the twoscale model solution. In fact, we find that the decay ratein the fast decay phase does not settle to a specific value,but keeps increasing with time, until the LS energy has de-creased sufficiently to be dominated by noise.The top panel of Fig. 6 shows that the fit for LSHEevolution curve does not reach an asymptotic slope at latetimes. The logarithmic slope of the large scale energy derived c (cid:13) , 1–15 time A l p h a E v o l u t i o n -6 -5 -4 -3 -2 -1 Positive Kinetic AlphaNegative Kinetic AlphaMagnetic Alpha0 500 1000 1500 2000 2500 300010 -7 -6 -5 -4 -3 -2 -1 Positive Kinetic AlphaNegative Kinetic AlphaMagnetic Alpha
Figure 8.
The top and bottom panels show α K and α M againsttime in the simulations, with k f =5 and k f =10 respectively. from the fit is shown in the bottom panel of Fig. 6 . Itcan be seen that for runs B-D, the logarithmic slope goesto values much larger than the turbulent decay rate. In allthese cases, the large scale and small scale is constituted bya sharp split at k = 2. However, this is an imperfect split andthe effective large scale wave number, k could increase to ahigher value, as the large scale field decays. Such an increasein the wavenumber for ’large scale’ would then increase theexpected turbulent rate.Fig. 4 also shows SSH evolution obtained in the DNSwith different k f . Initially SSH is zero in the DNS, but risesto a non-zero value due to transfer of helicity from large tosmall scales and then stays roughly constant before decayingat late times. Paper I predicts the initial value for the steadystate SSH, to be ( k /k f ) M eq . The corresponding two scalesolutions are shown as dashed lines in the Fig. 4, and we seethat the steady SSH come close to expected values, but arelarger in DNS runs B-F. This shows the limitations of thetwo scale model in capturing the whole spectral evolution ofthe DNS, nevertheless there is a reasonable agreement withexpectations of the two model. Also, we find that the secondslope of SSH, after the steady state phase, is steeper than thecorresponding two scale model slope. Here again we expectthat the effective wavenumber for small scale field increasesfrom k f to larger values, resulting in a faster decay of SSHin the DNS. Whereas such an increase would be restrictedin the two scale model, where the small scale is fixed at k f . Paper I has discussed at length, the contribution of kineticalpha, α K to E . In the derivation leading to the two scalemodel, it was assumed the contribution of α K to E is neg-ligible as compared to α M . The kinetic alpha, α K , couldbe generated due to the Lorentz force and then would op-pose the magnetic alpha, α M . If α K was significant, thenthe large scale helical field would decay much faster thanthe resistive decay rate. It was argued in Paper I that thegenerated α K is indeed small. Nonetheless, it is important to make an estimate of α K from the DNS and quantify itscontribution to the net E .In Fig. 8, we show both α K and α M estimated from theDNS using Eq. 15. It can be seen that α K fluctuates but ismostly negative and opposite in sign compared to α M in theslow decay phase. One also sees from the Fig. 8, that α K inthe slow decay phase is found to be a factor of 4-5 smallerthan α M in the case of k f = 5 (Run B) and a factor of ∼ k f = 10 (Run D). The contribution of α K to the EMF is thus considerably smaller than α M andhence subdominant as argued in Paper I. In the saturatedphase, when all the magnetic helicity (and hence, α M ) hasdecayed, the α K alternates equally between being positiveand negative values. It is important to identify the threshold below which theslow decay turns into a fast one, because smaller the tran-sition energy is with respect to equipartition value, longerwould be the timescale for which the helical large scale fieldremains resilient to turbulent diffusion.We identify two kinds of transition energy, E c and E c ,arising in two different contexts. One threshold E c , arisesin the context where as the field decays in time, it transitsfrom the slow decay phase to the fast decay phase after cross-ing the threshold energy, E c . This behaviour is what hasbeen examined so far, in runs A-F, where we started witha field of equipartition strength. The other context is whereone starts with different initial large scale field strengths.As initial magnetic energy is decreased, below a threshold, E c , the field ceases to start with resisitive decay phase andinstead decays at a much faster rate right from the begin-ning. Paper I argued on the basis of two scale model thatthe latter threshold or critical energy is k f dependant with E c = ( k /k f ) M eq . We will examine both types of transi-tion points. We first focus on E c in section 3.1 and later on E c in Section 3.2. To identify this transition, we first consider the two scalemodel and then turn to the DNS. We solve numericallyEq. 16 and Eq. 17 of the two scale model, for different k f ,and then plot the time evolution of the large scale magneticenergy. To explore the behaviour of the transition point un-der the variations in k f alone, we keep R M and the initialmagnetic energy, M fixed across different numerical solu-tions. In the left hand side panel of Fig. 9, we show the de-cay of a fully helical large scale field with time for different k f = 3 , , , ,
20. The initial magnetic energy, M = M eq and the R M is fixed to a value of 120 which is comparableto the value in the DNS. One immediately notices that allthe curves almost coincide. Note that in these solutions, the η is the same, which explains the same slope in the slow de-cay regime. And in order to keep the turbulent decay rate, ∼ u rms k / k f = 2 u rms / k f , the same, we compensate theincreasing k f , by increasing the u rms . This forms the idealexperiment to understand the behaviour of the transition c (cid:13) , 1–15 umerical simulations of decaying helical fields Two scale model solutions for varying k f with fixed R m =120 -5 -4 -3 -2 -1 N o r m a l i s e d L S m ag n e t i c e n e r g y kf=20kf=10kf=7kf=5kf=3 1600 1700 1800 1900 0.010.10 time Two scale model solutions for varying k f with fixed R m =120 N o r m a l i s e d SS H ( < a . b> k f / M e q ) d ( l n γ ) / d t Figure 9.
For panel in the left, two scale model solutions for normalised LSHE is given for varying kf, at 3,5,7,10,20. Panel in the rightshows the corresponding two scale model solutions for normalised SSH. R M is fixed at 120. Two scale model solutions for varying k f with fixed R m =12000 • • • time10 −5 −4 −3 −2 −1 N o r m a l i s e d L S m ag n e t i c e n e r g y kf=20kf=10kf=7kf=51.66 • • −8 −7 −6 −5 −4 −3 −2 −1 time Two scale model solutions for varying k f with fixed R m =12000 N o r m a l i s e d SS H ( < a . b> k f / M e q ) • • • • • d ( l n γ ) / d t Figure 10.
For panel in the left, two scale model solutions for normalised LSHE is given for varying kf, at 5,7,10,20. Panel in the rightshows the corresponding two scale model solutions for normalised SSH. R M is fixed at 12000. point as k f is varied. From the left panel of Fig. 9, the be-haviour of the transition point is seen to be independent ofthe changing turbulent forcing scale, k f .To determine the transition point, we have adopted thefollowing method. The evolving decay rate of the large scalehelicity, from the two scale model is given by, γ = 12 h A · B i d h A · B i dt = − η t k (cid:18) − k f h a · b i k M eq (cid:19) − ηk (20)where, for fully helical fields, h J · B i ∼ k h A · B i and h j · b i ∼ k f h a · b i . Also, h B i ∼ k h A · B i and therefore,Eq. 20 also describes the evolving decay rate of the largescale energy. The decay rate is expected to be fairly con-stant in the slow decay phase and sharply increases duringthe transition region and then settles to the turbulent de- cay value. Thus, the logarithmic slope of γ will go througha maximum, when the decay rate changes the fastest. Thepoint in time when the maximum occurs, can be then de-fined as the point of transition and the corresponding largescale energy is defined to be the transition energy. In thebottom right panel of Fig. 9, we show the d ( lnγ ) /dt curves,while the top right panel shows the evolution of SSH. Notealso that the maximum of d ( lnγ ) /dt coincides with the pointat which the SSH changes slope, i.e. SSH goes from nearlysteady state value to decaying resistively at k f . We thus findthe transition energy to be E c /M eq =0.031, 0.029, 0.026,0.025, 0.025 for k f =3, 5, 7, 10, 20 respectively. For com-pleteness, along with the Eq. 20, we give here the corre- c (cid:13) , 1–15 sponding equation for the small scale helical field,12 d ln( h a · b i ) dt = − η t k f ( k h A · B i ) M eq (cid:18) − k M eq k f h a · b i (cid:19) − ηk (21)In Fig. 10, we show similar plots of two scale solutionsat a much larger R M = 12000, to also test the sensitivity ofthe results with respect to changes in R M . The right panelof the Fig. 10, shows at the top, evolution of SSH whilethe bottom panel shows the evolution of d ( lnγ ) /dt . It canbe seen from such plots for both the cases of R M = 120and 12000, that as the k f increases, the point at which SSHchanges slope, occurs later in time. And the correspondinglarge scale energy curve also transitions later in time. Asa result, the transition energy would be similar across dif-ferent k f . The transition energy estimated in this case is E c /M eq = 0.0011, 0.0009, 0.0009, 0.0008 for k f = 5, 7, 10,20 respectively. We find that the change in the transitionenergy from R M = 120 to R M = 12000 is by a factor of ∼ −
30. Thus interestingly, E c seems to scale as R − / M .Now we turn to the DNS and determine the transitionenergy of type E c for the various runs in Table 1. In the caseof simulations, we find that the decay rate in fast regime, ischanging with time and and does not settle to a final valueas is the case in the two scale model. Therefore, we do notfind a maximum in the evolving logarithmic slope of LSHEto be able to determine the transition point. Instead, weadopt a slightly different method of estimating the transitionenergy in the case of DNS. From the right panels of Fig. 9and Fig. 10, we pointed out that d ( lnγ ) /dt is maximumwhen the SSH begins to decrease. Thus it seems plausibleto define the transition point as the time when the SSHdecreases from its initial steady state value by say, 50%.From Eq. 20, it can be seen that the first term on RHS goesto 0 when h a · b i = ( k /k f ) M eq is in steady state, and bythe time SSH, h a · b i decreases by 50%, the large scale fieldis expected to decay at a rate, of the order of the turbulentdecay rate.We use again the form in Eq. 10 to determine a fitfor the SSH evolution. Then we estimate the point in timeby when SSH decreases by 50% of its initial steady statevalue. The corresponding value of large scale energy is thetransition energy. This method of determining the transitionenergy is conceptually similar to the one used for two scalemodel as it determines the point at which there is a changein the SSH evolution from steady state (or nearly zero decayrate) to a non-zero decay rate. This method of determiningthe transition energy is illustrated in the lower panels ofFig. 4 and Fig. 5. The vertical lines in these figures give thetime when SSH has decreased by 50% of its initial steadystate value. This line intersects the LSHE curve at a tran-sition energy value indicated by the horizontal line in eachupper panel of Fig. 4 and Fig. 5. The transition energy thusdetermined, gives the critical energy as E c ∼ . M eq for run B. This is similar to the transition point we obtainfrom the corresponding two scale model solution of E c ∼ . M eq . For the other runs A, C, D, E and F we find thetransition energy to be E c /M eq = 0 . , . , . , . .
034 respectively, as can be read from the Fig. 4 andFig. 5. This indicates the near universality of the transitionpoint, E c , with respect to k f in a small range of R M .One can also determine the transition energy from the point of intersection between the slopes fit to the two decayregimes, slow and fast. The slope intersection method de-pends on accurate fits for the two decay regimes and henceis subject to uncertainity. Also, as we see in Fig. 6 that theslopes are not constant in any of the decay regimes, and arecontinually changing, and therefore, the determined slope isan approximate average estimate. The fit in especially thefast decay regime seems highly uncertain, depending on thewindow of time chosen. Hence we do not pursue this methodfor determining the transition energy. Now we will examine the second kind of transition point, E c . This occurs when the initial magnetic field strength islowered to a critical point below which the the field decaysat a fast decay rate right from the beginning (i.e. the initialslow decay phase is now absent). We consider the evolutionEq. 16 for the large scale helicity and substitute h j · b i in α M (in the emf E term) and h J · B i , with the corresponding twoscale approximation of k f h a · b i and k h A · B i , respectively.We then get,12 ∂ h A · B i ∂t = 13 ρ k f h a · b i τ h B i− u rms k f k h A · B i− ηk h A · B i (22)Let us focus on the ideal limit of η →
0, for which the totalmagnetic helicity is conserved at all times. Then, one cansubstitute for the small scale helicity in Eq. 22, h a · b i = (cid:0) h A · B i − h A · B i (cid:1) , where the total helicity (cid:0) h A · B i (cid:1) isconserved. Converting all the quantities to a dimensionlessform, we get, dM d ˜ t + 23 M − M (cid:0) M − ( k /k f ) (cid:1) = 0 (23)where M = h B i / (2 M eq ) = k h A · B i / (2 M eq ), t/τ = ˜ t and M = M (˜ t = 0) is the normalised initial energy of the largescale helical field. Eq. 23 can now be solved to give, M = M − ( k /k f ) − (cid:16)(cid:16) ( k /k f ) e ( − t ( M − ( k /k f ) ) / (cid:17) /M (cid:17) (24)When M > ( k /k f ) , at late times ˜ t → ∞ , we have, M → M − ( k /k f ) (25)indicating that, the reduction in the field strength is bya finite amount. When M = ( k /k f ) , Eq. 23 becomes dM /dt = − (2 / M and hence M → M < ( k /k f ) , we obtain at late times when˜ t → ∞ , M = M (cid:0) − ( k f /k ) M (cid:1) e ( − t (( k /k f ) − M ) / ) (26)which implies that the large scale field undergoes a rapiddecay. Hence, ( k /k f ) forms a natural transition point, inthe case of large R M (or here in the ideal limit of R M → ∞ ),which determines when the LSHE will directly transit torapid decay. This was emphasized in Paper I but withoutgiving the above argument. The question arises how well thisthreshold, which holds in the ideal limit, obtains for morerealistic R M , both in the two scale model and the DNS. Wefirst reconsider the two scale model. c (cid:13) , 1–15 umerical simulations of decaying helical fields Time-shifed two scale solutions for varying initial energy, kf=5, Rm=12000 time10 -8 -6 -4 -2 N o r m a l i s e d L S m ag n e t i c e n e r g y f r o m t w o s ca l e m o d e l /k f ) Unshifted
Time shifted two scale solutions for varying initial energy, kf=5, Rm=12000 time 0.0010.010.11 N o r m a l i s e d SS h e l i c i t y ( < a . b> k f / M e q ) /k f ) Figure 11.
The decay curves from two scale model for fully helical large scale magnetic field for different initial strengths for k f = 5and R M = 12000 are shown. The left panel of Fig. 11, and Fig. 12, show the de-cay of fully helical large scale magnetic field for a set ofdecreasing initial field strengths for the two scale model.We have adopted k f = 5 (Fig. 11) and k f = 20 (Fig. 12),both with R M = 12000. While the inset in the left panels ofFig. 11 and Fig. 12 show the evolution of large scale energywith decreasing initial strengths, the main plots show thecurves starting with subequipartition strength, time-shiftedto maximally coincide the M = 1 evolution. The labels inthe plot indicate the value of M . The thick black curveis the case of M = 1 (we will refer to this as the fiducialcurve). In the curves beneath that of M = 1, M has beendecreased to smaller and smaller values.The right panel of Fig. 11 shows the evolution of theSSH. For stronger initial fields, SSH achieves a steady valueand the resulting α M is large enough to offset the turbulentdiffusion. Then the large scale field decays initially at rate ofthe order resistive rate. When M is below a critical value,the initial helicity in large scales is insufficient to generatea large enough small scale helicity, and α M , by turbulentdiffusion. In this case the SSH decays, and the LSHE decaysfast due to turbulent diffusion (uncompensated by the α M effect).It can be seen from the right panel of Fig. 11 that thedash-dotted blue line starting with M = 0 .
05 is the lastto reach a steady state indicating the presence of resistivedecay regime initially. For such smaller M ∼ .
04 and be-low, the small scale helicity fails to rise to a steady stateand subsequently decays indicating the absence of slow de-cay regime, which means that the large scale field directlystarts decaying at a faster rate. This can be seen from theleft panel of Fig. 11. Note that curves with M above thethreshold have sharp initial drop (as SSH builds up), but donot decrease in their energy significantly before joining thefiducial curve. On the other hand, the time-shifted purpledotted curve with M = 0 .
02, which is below the threshold,drops by several orders of magnitude before joining the fidu-cial curve. Hence, we find that the transition point is close
Time shifted two scale solutions for varying initial energy, k f =20, R m =12000 time10 -8 -7 -6 -5 -4 -3 -2 N o r m a l i s e d L S m ag n e t i c e n e r g y f r o m t w o s ca l e m o d e l Unshifted -6 -4 -2 Figure 12.
The decay curves from two scale model for fully he-lical large scale magnetic field with different initial strengths for k f = 20 and R M = 12000 are shown. to the value of ( E c /M eq ) = ( k /k f ) = 0 .
04 as expectedfrom the work of Paper I and the Eq. 23 above.Again the same exercise is repeated at k f = 20. In theFig. 12, the solid green curve starting with M = 0 . k /k f ) ,is seen to drop in energy by few orders of magnitude be-fore joining the fiducial curve. Hence in this case, we findthe transition energy, E c ∼ . M eq , as is seen fromthe Fig. 12. This value of E c for the is again close to( k /k f ) M eq . Such large R M = 12000 is however beyondthe scope of current DNS. Thus in the DNS studies below,where R M has a more modest value of ∼ E c . Nevertheless, wedo expect to check the consistency with the two scale modeland hence indirectly substantiate the results of Paper I. c (cid:13) , 1–15 k f =5, R m =120, 256 DNS N o r m a l i s e d L S h e l i ca l e n e r g y ( M H / M e q ) k f =5 two scale model solutions k f =5, R m =120, 256 DNS SS h e l i c i t y ( < a . b>/ M e q ) k f =5 two scale model solutions Figure 13.
The decay curves for fully helical large scale magneticfield with different M for k f = 5 and R M = 120, from DNS, areshown. We have shown in Fig. 13, the results from the DNSruns G-K where the initial magnetic energy is lowered tosmaller and smaller values compared to M eq . In these setof simulations, we have fixed k f = 5, u rms ∼ .
12 and η =2 × − , and thus R M = 120, while varying M . The topleft panel of Fig. 13 shows the time evolution of large scalemagnetic energy in the DNS, while the top right panel theresults from the corresponding two scale model. The timeevolution of the SSH in the DNS and corresponding twoscale model are respectively shown in the bottom left andright panels of Fig. 13.A comparison beween the DNS (left panel) and the 2-scale model (right panel) in Fig. 13, shows that there is aqualitative agreement between the two. For example, theslow decay rate of large scale energy of both are compara-ble, and so also are the amplitudes of the steady state smallscale helicity. We also find that the average initial slopes(evaluated for the time period of t=0 to t=100) for all theruns G-K, listed in the Table 1, match closely with the es- Time shifted LSHE, k f =5, 256 , R m =120 N o r m a li se d L S h e li ca l e n e r g y ( M H / M e q ) Figure 14.
The time-shifted decay curve for fully helical largescale magnetic field for different sub-equipartition M at R M ≃ R M does not meet the condition 3 /R M ≪ ( k /k f ) needed to identify the predicted transition (see text). timate using Eq. 19, where the term in the denominatior, h B i / M eq is replaced with the value of M .In Fig. 14, we show the fiducial curve where M = 1 isthe solid black line. We also show the evolution curves start-ing with sub-equipartition energies, time-shifted to lie on thefiducial curve. As pointed out earlier, a clear energy tran-sition value would be revealed if curves with M below thethreshold decrease in their energy significantly before join-ing the fiducial curve. The graph shows all the curves nearlyfalling together without any such drop in the initial energywhich at face value means no clear identification of E c inthe energy scale. However, from Eq. 19, it can be seen thatthe term η/η t = 3 / R M has to be sufficiently small comparedto M , to be able to discern the E c dependence on k f , forthe subsequent evolution. If we compare the two terms in thedenominator of Eq. 19, then we require, 3 / R M << ( k /k f ) .Otherwise, η/η t would become important before M is low-ered to ( k /k f ) , and one cannot discern the influence of k f on E c . With k f = 5 and R M = 120, 3 / R M = 0 . k /k f ) = 0 .
04, thus the two terms are comparable.Hence, we seemingly need much higher R M runs to beable to properly check the more conservative threshold of E c /M eq = ( k /k f ) .Note that in Fig. 13, as M is decreased, approaching E c from above, the subsequent decay seems to be at an in-creasingly higher rate. This can be understood from Eq. 19(which applies only for M > E c since only in that regimedoes h j · b i reach the steady state assumed by that equa-tion.) For the curves with lower initial energy, the ratio, On the other hand, in the context of E c , it is the term η/η t which is responsible for the transition to faster decay phase. Andwe have seen that the transition energy E c depends on R M ,scaling as R − / M from the two scale model solutions. Thus DNSwith even a modest R M can enable us to discern the transitionenergy, E c , which morover agrees reasonably with that predictedby the corresponding two scale model solutions.c (cid:13) , 1–15 umerical simulations of decaying helical fields h j · b i / h J · B i will be larger, thus leading to a higher multi-ple of the resistive decay rate. This can be understood fromexamining Fig. 14. Since the curves with lower M , fall onthe curve with M = 1, later in time within the slow decayphase, they are expected to decay at an increasingly largerrate.Overall we see a qualitative agreement between the DNSand the 2-scale model. This again indicates that the re-sult of Paper I on the slow decay of helical magnetic fieldswhich have M > ( k /k f ) M eq , seems reasonably consistentwith the simulations that we have performed so far. Thoughhigher resolution simulations with high R M and small sub-equipartition initial fields are required to substantiate theabove results. E c = E c In the distinct contexts of the previous two sections wherewe have identified the transition energies E c and E c westarted with the SSH initially equal to zero. Here we discusshow the distinction between E c and E c can be traced tothe distinct levels to which h a · b i builds up in the two cases.The first context (section 3.1) in which we have identi-fied the the transition energy is the fiducial case of M = 1,for which we found a k f independent transition point E c .In this case, h a · b i builds up to a nearly steady state valueduring much of the resistive slow decay phase of h A · B i be-fore the fast decay occurs. In contrast, for the second context(section 3.2) of varying M , we found that when fast decayof the large scale field occurs right from the beginning, thehelicity transferred from large scale to small scales neverattains the aforementioned steady state value. The fast de-cay happens below a critical initial large scale helical energyvalue E c . If there is not enough initial helical large scale en-ergy to supply the needed SSH, the large scale field decaysfast. A source of SSH is crucial to explain the threshold of E c on k f (Paper I).To quantitatively study the importance of the role ofSSH source in distinguishing E c and E c , we can ask ifsetting the initial SSH equal to the maximum steady statevalue of the case of section 3.1 (rather than allow it to growfrom zero) and then vary the initial M (as in the case ofsection 3.2) do we recover E c ?Indeed, it is seen from Fig. 15, in the case of k f = 5,R M = 12000 and with the initial h a · b i = ( k /k f ) M eq ,the two scale model solutions for varying M fall togetheron the fiducial evolution curve for the context in section3.1 For evolution starting at lower and lower energy val-ues, the LSHE starts off initially with a flatter slope, buteventually joins the fiducial curve, which at that point intime is decaying at a much larger rate. The correspondingSSH quickly decays from the initial value of ( k /k f ) M eq tothe value on the fiducial curve at that point in time. Addi-tionally, it can be seen from Fig. 16 that if SSH is set to avalue, 0 < h a · b i < ( k /k f ) M eq , then the two scale modelsolutions are similar to the second context, where SSH ini-tially rises to attain the steady state value, and fails to do sowhen M is at the threshold or below. This shows that thedifference between the two contexts that led to E c = E c obtains due to the difference in the value of SSH attained inthe early transient phase of the evolution. Time-shifed two scale solutions for varying initial energy, kf=5, Rm=12000 -6 -5 -4 -3 -2 -1 N o r m a l i s e d L S m ag n e t i c e n e r g y time0.1 SS H ( < a . b> k f / M e q ) Figure 15.
The two scale solutions for fully helical large scalemagnetic field (in top panel) and SSH (in bottom panel) withdifferent M for k f = 5 and R M = 12000, where the initial SSH = 0, but is set to the value of ( k /k f ) M eq Time-shifed two scale solutions for varying initial energy, kf=5, Rm=12000 -8 -7 -6 -5 -4 -3 -2 -1 N o r m a l i s e d L S m ag n e t i c e n e r g y /k f ) time0.010.10 SS H ( < a . b> k f / M e q ) Figure 16.
The two scale solutions for fully helical large scalemagnetic field (in top panel) and SSH (in bottom panel) withdifferent M for k f = 5 and R M = 12000, where the initial SSH = 0, but is set to the value of h a · b i k f /M eq = 0 . The extent to which large scale fields survive turbulent dif-fusion in the absence of dynamo action via kinetic helicityor shear is important in assessing the plausibility of dynamoversus fossil field origin of large scale fields in astrophysi-cal objects. Non-helical fields decay at the turbulent diffu-sion rate in the presence of non-helical turbulence, but largescale helical fields do not (Paper I). Here we have examinedthe survival of initially helical fields via direct numericalsimulations (DNS), and compared the results with the ba-sic two scale model of Paper I. Previous simulations havebeen done by Yousef et al. (2003) and Kemel et al. (2011). c (cid:13)000
The two scale solutions for fully helical large scalemagnetic field (in top panel) and SSH (in bottom panel) withdifferent M for k f = 5 and R M = 12000, where the initial SSH = 0, but is set to the value of h a · b i k f /M eq = 0 . The extent to which large scale fields survive turbulent dif-fusion in the absence of dynamo action via kinetic helicityor shear is important in assessing the plausibility of dynamoversus fossil field origin of large scale fields in astrophysi-cal objects. Non-helical fields decay at the turbulent diffu-sion rate in the presence of non-helical turbulence, but largescale helical fields do not (Paper I). Here we have examinedthe survival of initially helical fields via direct numericalsimulations (DNS), and compared the results with the ba-sic two scale model of Paper I. Previous simulations havebeen done by Yousef et al. (2003) and Kemel et al. (2011). c (cid:13)000 , 1–15 In particular, we have examined the decay of large scale he-lical fields in more general settings by studying in detail thedependence on initial field strength, forcing wavenumber ofthe turbulence and also R M . DNS takes into account thefull set of MHD equations but is limited by computationalpower to modest Rm of order 100. On the other hand, thetwo scale model involves simplifying assumptions, of havingonly two scales and also invokes a closure approximationfor the turbulent emf, but allows large Rm to be explored.Thus comparison will allow one to evaluate to what extentthe two scale model can be trusted, and also then the pos-sibility of extrapolating DNS to larger Rm. Overall, we findthat there is good qualitative agreement between the pre-dictions of two-scale theory and DNS.For the case in which we start with an initial helicalmagnetic energy of sufficient strength, of order the equipar-tition value ( M ∼ α K is much smaller than α M , which was made in Paper I, holds true. This indicatesthat the basic picture of Paper I, particularly regarding theimportance of a magnetic alpha, α M , being generated byturbulent diffusion of the large scale helical field, and thenitself acting to prevent the turbulent decay of the large scalefield, is reasonably robust. This also indirectly supports theideas behind the closure approximations used to derive themagnetic alpha.Subsequently, there is a transition to fast decay phase,which can exceed even the turbulent decay rate predictedby a two scale theory. For this fiducial case of startingwith equipartition energy, the threshold energy at whichthe transition occurs, E c , is independent of the forcingwavenumber k f and R M for the range of R M ∼
100 ex-plored. Meanwhile, the 2-scale model solutions at a muchhigher Rm=12000, do indicate a possible scaling of the tran-sition energy E c ∝ R − / M .A different transition energy threshold E c arises forthe case in which we seek the transition threshold at t = 0below which the field decays at a fast rate right from the be-ginning. This scenario is more astrophysically relevant, sincethe feasibility of the existence of subequipartition strengthinitial fields is higher as compared to the fiducial case. Insuch a case, Paper I argues that when the large scale he-lical field energy is below a critical initial magnetic energy E c ∼ ( k /k f ) M eq , it decays rapidly at the turbulent diffu-sion rate. We have reconfirmed this estimate by solving thetwo scale model exactly in the ideal limit, and also solving itnumerically for finite but very large R M , much larger thanpossible by DNS. For the moderate R M ∼
100 achievable byDNS (for DNS with large k f ), we have shown again that theDNS results are consistent with the 2-scale model. However,robustly identifying the transition energy E c predicted bythe 2-scale model, requires the condition 3 /R M << ( k /k f ) to be satisfied, and R M is not sufficiently large in the sim-ulations. Much higher R M simulations would be required.At present we can only say that, from the overall consis-tency between DNS and 2-scale model even for case 1, we do expect this later type of transition to obtain for high R M cases.Eventually it would be desirable to assess how the prin-ciples identified herein apply to more realistic conditions ofastrophysical rotators to assess whether large scale fieldsin astrophysical rotators such as galaxies could result frompost-processing of fossil helical fields without requiring atraditional in situ kinetic helicity. Real systems have shear,differential rotation, and stratification, all of which we havenot considered here. We also considered all large scale quan-tities to be averaged over closed volumes in the present work,thereby eliminating helicity fluxes. It would be of interest forfuture work to consider the influence of helicity fluxes on therelative decay of helical and non-helical large scale fields. Fi-nally we note that in real systems, there would in generalbe a combination of helical and non-helical fields and the re-sults herein would apply to the helical fraction of the largescale magnetic energy. ACKNOWLEDGMENTS
We thank Axel Brandenburg for useful discussions whichhelped to sharpen the arguments of this paper. We acknowl-edge the use of the high performance computing facility atIUCAA.
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