Influence of initial conditions on the large-scale dynamo growth rate
aa r X i v : . [ a s t r o - ph . E P ] A ug Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 2 March 2018 (MN L A TEX style file v2.2)
Influence of initial conditions on the large-scale dynamogrowth rate
Kiwan Park ⋆ Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA
ABSTRACT
To investigate the effect of energy and helicity on the growth of magnetic field, helicalkinetic forcing was applied to the magnetohydrodynamic(MHD) system that had aspecific distribution of energy and helicity as initial conditions. Simulation resultsshow the saturation of a system is not influenced by the initial conditions, but thegrowth rate of large scale magnetic field is proportionally dependent on the initial largescale magnetic energy and helicity. It is already known that the helical component ofsmall scale magnetic field(i.e., current helicity h j · b i ) quenches the growth of largescale magnetic field. However, h j · b i can also boost the growth of large scale magneticfield by changing its sign and magnitude. In addition, simulation shows the nonhelicalmagnetic field can suppress the velocity field through Lorentz force. Comparison of theprofiles of evolving magnetic and kinetic energy indicates that kinetic energy migratesbackward when the external energy flows into the three dimensional MHD system,which means the velocity field may play a preceding role in the very early MHDdynamo stage. The generation and amplification of magnetic field inastrophysical systems are ubiquitous phenomena. Theorigin and exact mechanism of growth of magnetic fieldsin stars or galaxies have been long standing problems. Ithas been thought that helical kinetic motion or turbulenceamplifies the magnetic field( B field). However, the helicalcomponent does not seem to be an absolute necessity forthe amplification of large scale magnetic field. In astro-physical dynamos, for instance, the kinetic energy of somecelestial objects like supernovae or galaxy clusters has lowor practically zero level of helical component. The evolutionof B fields in these objects is thought to be dominatedby small scale dynamo( SSD ): the amplification of fieldsbelow the large scale eddy without helicity(Kazantsev(1968), Kulsrud & Anderson (1992), Meneguzzi et al.(1981), Haugen et al. (2003), Schekochihin et al. (2004)),Mininni et al. (2005)). So, it is important to understand thedetailed mechanism of dynamo in MHD equations whetheror not the driving force is helical.As of yet some problems in the MHD dynamo pro-cess are not completely understood: the role of he-lical or nonhelical kinetic(magnetic) field, the effectsof initial conditions( IC s) such as kinetic(magnetic) en-ergy and helicity. There were trials to see the effectsof IC s on the dynamo(Haugen & Brandenburg (2004),Maron et al. (2004)). However, the trials are not yetenough; moreover, there are few analytic studies to ex-plain the effects of initial conditions. Some statisticalmethods like Eddy Damped Quasi Normal Markovian approximation( EDQNM , Pouquet et al. (1976)) can beused to explain the influences of IC s on the profile of grow-ing B field qualitatively. However, it is partial and incom-plete. Development and verification of the theoretical resultswith more detailed simulation data are necessary. Nonethe-less, related simulation results still provide us many detailedphenomena that are helpful to understanding the MHD tur-bulence. In this paper the effects of initial magnetic energyand helicity on the large scale dynamo were investigatedusing simulation data and analytic methods. The main aim of this paper is to find out the effectof initial conditions( IC s) on the growth and saturationof magnetic helicity( H M = 1 / h A · B i , B = ∇ × A )and magnetic energy( E M ). For this, the combinations ofthree simulations were carried out: Non Helical MagneticForcing( NHMF ), Helical Magnetic Forcing(
HMF ), andHelical Kinetic Forcing(
HKF ). To explain simulationresults, the equations derived from
EDQNM and two scalemean field dynamo theory(Field & Blackman (2002)) wereused.For the simulation code, high order finite difference PencilCode(Brandenburg (2001)) and the message passing inter-face(MPI) were used. The equations solved for
HKF in the c (cid:13) Kiwan Park E-mail: [email protected] Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA code are,
DρDt = − ρ ∇ · v (1) D v Dt = − c s ∇ ln ρ + J × B ρ + ν (cid:0) ∇ v + 13 ∇∇ · v (cid:1) + f (2) ∂ A ∂t = v × B − η J (3) ρ : density; v : velocity; B : magnetic field; A : vector poten-tial; J : current density; D/Dt (= ∂/∂t + v · ∇ ): advectivederivative; η : magnetic diffusivity; ν (= µ/ρ , µ : viscosity, ρ :density): kinematic viscosity; c s : sound speed; f : forcingfunction(helical or nonhelical). The unit used in the code is‘ cgs ’. Velocities are expressed in units of c s , and magneticfields in units of ( ρ µ ) / c s (i.e., B = √ ρ µ v , µ ismagnetic permeability and ρ is the initial density). Notethat ρ ∼ ρ in the weakly compressible simulations. Theseconstants c s , µ , and ρ are set to be ‘1’. In the simulations η (= c / πσ , σ : conductivity) and ν are 0.006.In case of the magnetically driven simulation(magneticforcing, MF ), forcing function f is located in the magneticinduction equation( ∂ A /∂t = v × B − η J + f ) instead ofthe momentum equation. As Ohm’s law( η J = E + v × B )implies, f symbolizes sort of the external electromagneticforce that drives the magnetic eddy(Einaudi & Velli (1999),Park & Blackman (2012b)).We employ a cube like periodic box of spatial volume(2 π ) with mesh size of 256 for runs. The forcing function f (http://pencil-code.nordita.org) used in the simulations iseither fully helical(in fourier space, ∇ × f = k f , k : wavenumber) or non-helical( ∇ × f = k f ). f ( x, t ) is representedby N f k ( t ) exp [ i k f ( t ) · x + iφ ( t )]( N : normalization factor, k f ( t ): forcing wave number). And to prevent the shock phe-nomenon, forcing magnitude f k is 0 .
07 for KF and 0.01 for MF (note that ∇ × f = k f f for the helical forcing). Thismakes mach number(= v/c s ) less than 0.3. Fig.1 includes the early time profiles of large scale | H M | (solidline) and E M (dotted line). In Fig.1(a), top line group in-cludes | H M | and E M for the case of NHMF → HKF : afterNon Helical Magnetic Forcing(as a precursor simulation, k f =30 t ≤ .
6) Helical Kinetic Forcing( k f =5, t> .
6) wasdone over this preliminary simulation. The middle linesare | H M | and E M for HMF → HKF : Helical MagneticForcing( k =30, t ≤ . HKF ( k f =5, t> . | H M | and E M for HKF system at k f =5 as areference simulation. The spectra show NHMF → HKF isthe most efficient in the growth of large scale H M and E M in the early time regime. HKF of which energy transferchiefly depends on α effect appears to be the least efficientin energy transfer. In case of HMF → HKF , the efficiencyis between ‘
NHMF → HKF ’ and ‘
HKF ’. And during
HMF , the features of
NHMF and
HKF due to the helicaland nonhelical field are observed.The spectrum of magnetic energy E M is always positive,but the sign of H M is influenced by the external driving (a) | H M | and E M (Logarithmic scale)(b) H M and E M (Linear scale)(c) | H M | and E M (Linear scale) Figure 1.
Seed energy and helicity in each case are the same.Precursor simulation ( N ) HMF changes the given seed field intothe specific energy distribution, which is used as new initial con-ditions for the consecutive main simulation
HKF . (a) Prelim-inary simulation
NHMF ( | f k | = 0 .
01 at k f = 30) finishes at t = 10 .
6. During this time regime, H M is negative. In contrast, HMF ( | f k | = 0 .
01 at k f = 30 for t ≤ .
0) generates positive H M . HKF ( | f k | = 0 .
07 at k f = 5) follows these preliminary sim-ulations. And HKF without a precursor simulation was doneseparately as a reference.(b) Except
NHMF , the magnetic fieldsin the other cases are indistinguishably small. (c) The left linegroup shows the influence of IC s. The difference in the onset po-sitions is mainly decided by large scale E M and H M generatedby the precursor simulations. And right line group includes theshifted E M and H M of each case for the comparison.c (cid:13) , 000–000 nfluence of initial conditions on the large-scale dynamo growth rate (a) E kin ( NHMF → HKF )(b) E mag ( NHMF → HKF )(c) E kin and E mag Figure 2.
NHMF ( fh m = 0, f = 0 . k f = 30) finishes at t = 10 .
6, and then
HKF ( fh k = 1, f = 0 . k f = 5) begins.Initially, only tiny E M is given( E kin is zero ). But E kin growsquickly, catches up with E M till t ∼ .
2, and outweighs it. (a) E kin which is transferred from magnetic eddy through Lorentzforce migrates backward and forward. (b) The diffusion of energyamong magnetic eddies without α effect in NHMF is tiny. Exceptthe forced eddy, the energy in magnetic eddies seems to be mostlyfrom kinetic eddies. After the precursor simulation, the peak of E M (nonhelical) at k = 30 disappears within a few time steps. (c)Comparison of E kin and E M . (a) E kin ( HMF → HKF )(b) E mag ( HMF → HKF )(c) E kin and E mag Figure 3.
HMF ( fh m = 1, f = 0 . k f = 30) finishes at t = 13 .
0, and then
HKF ( fh k = 1, f = 0 . k f = 5) begins.(a) E kin of HMF is smaller than that of
NHMF . (b) E mag of HMF is also smaller than that of
NHMF . The second small peakaround k = 9 ,
10 is the inversely cascaded energy due to α effect.This peak moves backward to be merged into the new forcingpeak( k = 5) when HKF begins. The peak of E M at k f = 30 alsodisappears within a few time steps. function ‘ f ’ and forcing method. For example, in caseof HMF , H M and ‘ f ’ have the same sign(middle lines, t ≤ . HKF , H M has opposite sign of ‘ f ’.The cusp in this group( t ∼
13) is the rapid change of H M from positive( HMF , fh m = h k a · b i / h b i =+1, full helical)to negative( HKF , fh k = h v · ω i / h k v i =+1, full helical). The c (cid:13) , 000–000 Kiwan Park E-mail: [email protected] Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA (a)(b)(c)
Figure 4. (a), (b) are the preliminary simulation before
HKF .(c) This plot is the same as that of (a), but the forced eddy is k f =5, closer to the large scale. It shows basic profile of E kin does not so much depend on the position of forced eddy. Alsolinear and uniform kinetic energy distribution implies the energytransfer is more local and contiguous rather than nonlocal. reasons of reversed or equal sign of large scale magnetichelicity in HKF and
HMF were partially explained inPark & Blackman (2012a), Park & Blackman (2012b). Incontrast, the magnitude of | H M | tends to be continuous,which implies the relation between H M and E M . Thiswill be dicussed again. On the contrary, the direction andmagnitude of H M in NHMF are irregular because of thefluctuating ‘ f ’. During NHMF (t ≤ .
6) the actual sign of H M is negative. However, if NHMF keeps going on, H M will change the sign slowly and irregularly.Fig.1(b) includes the linearly scaled plots in Fig.1(a). Allsimulations started with the same seed field. However,as the plots show, the system driven by NHMF has thelargest and fastest growing E M and H M in the early timeregime. In addition, during HKF after the preliminary
NHMF , E M and H M still grow fastest, and maintain thelargest values until they get close to the saturation. Thisshows the effects of IC s and forcing method clearly.Fig.1(c) is to compare directly the onset and saturationof E M and | H M | . The group of three lines in the leftpart has the original data plots, and the lines in the rightgroup are their shifted plots for comparison. The onsetpoint of NHMF → HKF is the earliest, followed by
HMF → HKF , and then
HKF . The order of onsetposition is closely related to the different amount of mag-netic energy and helicity generated during the preliminarysimulation. However, as Table.1 shows, H M,L ( L : largescale), E M,L , E kin,L , and E kin,s ( s : small scale) in eachcase are of similar values at their onset positions in spiteof the different IC s. They are sort of critical values forthe onset, and the time to reach this critical point isinversely proportional to the magnitude of IC s. Since theforcing method after the preliminary simulation is the same( HKF ), IC s are determinants of the onset time. However,the saturation of turbulence dynamo is independent of IC s.Rather the saturation is decided by the external forcing ‘ f ’and intrinsic properties of system like viscosity ν ( ∼ Re − ,Reynolds number Re = V rms L/ν . L: characteristic lineardimension) or magnetic diffusivity η ( ∼ Re M − , MagneticReynolds number Re M = V rms L/η ). All three simulationshave the same saturated Re M ∼
30. Those features of criticalvalues and saturation imply there is no long lasting memoryeffect in turbulence. This validates Markovianization inMHD equations for closure.Fig.2(a) ∼ E kin and E M of NHMF → HKF . During the preliminary
NHMF at k f = 30, most E M is localized at the forced eddy(Fig.2(b)). But E kin which is larger than E M in mostrange spreads over from large to small scale(Fig.2(a), 2(c)).This relatively linear profile of E kin spectrum indicates thepressure that makes the system homogeneous transportingthe energy forward and backward is dominant in thevery early time regime. On the other hand, these figures,especially Fig.2(c), imply the relation between E kin and E M . Initially only tiny seed E M was given to the system.However, once NHMF started, E kin caught up with E M by t ∼ . E kin getsto induce E M which is the source of H M . That E kin isone of the sources of E M is coincident with the result ofEDQNM approximation(Pouquet et al. (1976)). On theother hand, the backward transfer of kinetic energy seemsto contradict the accepted theorem that inverse cascadein three dimensional magnetohydrodynamic turbulence isnot possible. However, when the energy or vorticity is notconserved, E kin can be inversely cascaded. We will comeback to this problem later. c (cid:13) , 000–000 nfluence of initial conditions on the large-scale dynamo growth rate Fig.3(a) ∼ E M and E kin for HMF → HKF . HMF shows two kinds of energy transports: nonlocaltransport of E M and local transport of E kin . The for-mer is caused by helical field( α effect), and the latter iscaused by the pressure. In Fig.3(a), backward migrationof E kin indicates the role of pressure. And the increaseof E M in large scale implies the direct energy transferfrom kinetic eddy(Fig.3(b), 3(c)). Furthermore, the helicaldriving source generates current helicity h j · b i (= h k a · b i )and kinetic helicity h v · ω i , which forms α effect in thesystem. This α effect generates the secondary peak of H M around k ∼ HKF begins at k f = 5.The difference between Fig.4(a)( NHMF ) andFig.4(b)(
HMF ) is just magnetic helicity ratio( fh m = 0, fh m = 1). Fig.4(a) and Fig.4(c)( NHMF ) have the samehelicity ratio fh m = 0 but different forced eddies( k f = 30, k f = 5). These clearly show kinetic energy migration andthe basic profile of field evolution do not depend on theposition of forced eddy so much.Fig.5(a) ∼ dH M /dt , dE M /dt , E M ( × . H M ( × . h v · ω i - h j · b i )/2( × . h v · ω i - h j · b i ∼− α ) shows the effects of ‘ f ’ and IC s clearly.With the smallest IC s in HKF (Table.1), its durationtime(0 < t < ∼ α coefficient islonger than that of other cases. In contrast the simulationof ‘ NHMF → HKF ’ has the shortest duration time ofconstant α coefficient( ∼ < t < ∼ dE M /dt is not always larger than d | H M | /dt .Fig.6(a) shows the profiles of d | H M | /dt in the early timeregime. And d | H M | /dt in Fig.6(b) are shifted plots for thecomparison. All d | H M | /dt converge to zero; but, the pro-file of HKF follows different paths. d | H M | /dt of HKF issmaller than that of the other cases until it reaches theonset position.
HKF starts with the smallest IC s, but allquantities except E M,s become the same as those of othercases by the onset position. When the field is about to arise, E M,s of HKF ( ∼ . × − ) is smaller than that of othercases( ∼ × − ). In theory, this term is discarded becauseof the seemingly little influence on the evolution of E M or | H M | (Pouquet et al. (1976), Field & Blackman (2002),Blackman & Field (2002)). However, E M,s is closely relatedto the conservation of magnetic helicity in the system andconstraining velocity field. We will discuss about this again.
There is no theoretical method that can completelyexplain the influence of IC s like E kin , E M , or H M onMHD dynamo yet. However, some approximation like EDQNM (Pouquet et al. (1976)), though limited, canbe used. The representations of H M and E M of thismethod are quite similar to those of two scale meanfield method(Blackman & Field (2004), Field & Blackman(2002)). The equations are composed of Alfv´ e n effectterm by the larger eddies, α effect term by the smaller eddies, and dissipation term. These approximate equationsassume the field is composed of helical and nonhelicalpart. If helical component in the field is zero or ignor-ably small( NHMF or NHKF ), these equations are notvalid. The system is divided into large( k = 1) and smallscale( k = 2 ∼ k max ), and this small scale can be subdividedinto the forcing( k = 2 ∼
6) and smaller scale( k = 7 ∼ k max ). ∂H M ∂t ∼ = (cid:0) Alfv ´ en effect z }| { (Γ /k )( H v − k H M ) + e Γ E v (cid:1) + α R E M − ν v k H M ∼ = α R E M − ν v k H M , (4) ∂E M ∂t ∼ = (cid:0) Alfv ´ en effect z }| { k Γ( E v − E M ) + e Γ H v (cid:1) + α R k H M − ν v k E M ∼ = α R k H M − ν v k E M ( E kin ≡ E v ) . (5)The coefficients are, α R = − (cid:2) Z ∞ k/a θ kpq ( t ) (cid:0) H v ( q ) − q H M ( q ) (cid:1) dq (cid:3) , ( a < ν v = 23 Z ∞ k/a θ kpq ( t ) E v ( q ) dq, θ kpq ( t ) = 1 − exp( − µ kpq t ) µ kpq ,µ k = C s (cid:2) Z k q ( E v ( q ) + E M ( q )) dq (cid:3) / +(1 / √ k (cid:2) Z k E M ( q ) dq (cid:3) / + ( ν + η ) k , Γ = 43 k Z ak θ kpq ( t ) E M ( q ) dq, e Γ = 43 Z ak θ kpq ( t ) q H M ( q ) dq. (6) H v is kinetic helicity(=1/2 h v · ω i ), H M is magnetichelicity(=1 / h A · B i , please note the coefficient), and ν v is kinetic eddy diffusivity. α R that transfers H M and E M to larger scale is composed of the residualhelicity( q H M ( q ) − H v ( q )) and triad relaxation time θ kpq . θ kpq is the function of eddy damping rate(see appendix) µ kpq (= µ k + µ p + µ q ), and connects smaller scale eddiesand larger scale eddies.The influence of Alfv´ e n terms( k = 0) on the large scale( k =1) is physically meaningless. Ignoring Alfv´ e n terms( k = 0),we find those coupled equations have two normal coordi-nates: ‘ E M + H M ’ and ‘ E M − H M ’. ∂ ( E M + H M ) ∂t = ( α R − ν v )( E M + H M ) , (7) ∂ ( E M − H M ) ∂t = − ( α R + 2 ν v )( E M − H M ) . (8)Assuming R t ( α R − ν v ) dt ≡ ( α − ν ) t , the solution is H M ( t ) = 12 (cid:2) H M (cid:0) e ( α − ν ) t + e − ( α +2 ν ) t (cid:1) + E M (cid:0) e ( α − ν ) t − e − ( α +2 ν ) t (cid:1)(cid:3) , (9) E M ( t ) = 12 (cid:2) E M (cid:0) e ( α − ν ) t + e − ( α +2 ν ) t (cid:1) + H M (cid:0) e ( α − ν ) t − e − ( α +2 ν ) t (cid:1)(cid:3) . (10) c (cid:13) , 000–000 Kiwan Park E-mail: [email protected] Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA
These solutions show how H M and E M are generated. Forexample, in case of H M ( t ) both H M and E M are sourcesof H M ( t ), but E M produces H M ( t ) like an auxiliarysource( ∼ Sinh). In the early time regime the effect of E M on H M ( t ) is tiny, but finally becomes on a level with H M . H M,sat converges to E M,sat (Fig.1(c)) as t →∞ . Thissolution shows that as long as α is larger than dissipation2¯ ν , large scale magnetic field eventually becomes fullyhelical by α effect.(Stribling & Matthaeus (1991), Biskamp(2008))On the other hand, one of our interests is how long theeffects of initial values last in turbulence. In Maron et al.(2004), the influence of imposed large scale magnetic energyon the system was tested(nonhelical kinetic forcing). Thestrong magnetic field in the large scale was expected tosuppress the formation of small scale fields. However, theeffect of imposed magnetic energy disappeared soon, andthe system eventually followed the external forcing source.Like an oscillator driven by an external driving source, theeffect of IC s exist only in the early transient mode. Inturbulence smaller eddy loses the information faster thanlarge eddy does.To see how long the effect of IC s lasts, the formal solutionof Eq.(4) may be useful: H M ( t n ) = e − R tn ν v ( τ ′′ ) dτ ′′ (cid:2) Z t n e R τ ν v ( τ ′ ) dτ ′ α R ( τ ) E M ( τ ) dτ + H M (0) (cid:3) . (11)Using the trapezoidal method for the integration part withthe assumption of R t n ν v ( τ ) dτ ≡ V ( t n ) and t n ≡ n ∆ t , wefind the approximate solution: H M ( t n ) ∼ (cid:2) e V (0) − V ( t n ) α R (0) E M (0)∆ t + e − V ( t n ) H M (0) (cid:3) ++ (cid:2) e V ( t ) − V ( t n ) α R ( t ) E M ( t ) + e V ( t ) − V ( t n ) α R ( t ) E M ( t )+ ... + e V ( t n − ) − V ( t n ) α R ( t n − ) E M ( t n − ) (cid:3) ∆ t. These show all previous results affect the current magnetichelicity in principle. However the influence decreases expo-nentially, which is coincident with the simulation results.The decaying speed depends on the several factors: energy,helicity, ν , and η . Of course the actual ν v varies with time.But, since ν v ( ∼ V ) changes rather smoothly and satu-rates to a constant, this inference is qualitatively reasonable. Table.1 provides information on the energy distributionsof the evolving variables at each stage. When E M or H M is about to rise, most E kin is located in the forcing scaleregime( k = 2 ∼
6) regardless of its initial distribution.The ratio of smaller scale E kin ( k = 7 ∼ k max ) to E kin ofthe whole small scale regime( k = 2 ∼ k max ) is about 1%.After the onset, as large scale E M or H M grows, kineticenergy migrates towards the smaller scale. At this timethe saturated ratio elevates up to 8 ∼ H M or E M (onset position), more(helical) kinetic energy is located in the forcing scale. Andif the inverse cascade of H M is less required, E kin in the (a)(b)(c) Figure 5.
During
HKF , ( h v · ω i - h j · b i )/2 in each case drops atdifferent time position. It depends on the energy and helicity( IC s)from the preliminary simulation. Also due to the different eddyturnover time between large and small scale, there is a phase dif-ference in the profile of growth rate, E M ( H M ), and α relatedterm. For the growth ratio, usually logarithmic growth ratio isused: d log | H M | /dt = − α R E M / | H M | − k ν v ( k = 1), but lineargrowth rate was used for the mathematical convenience and visi-bility. All quantities but E M and H M are the averages of 50 ∼ (cid:13) , 000–000 nfluence of initial conditions on the large-scale dynamo growth rate (a) (b) Figure 6. (a) Growth ratio proportionally depends on the IC s. The area between the line and time axis is H M . (b) The profile ofevolving growth ratio of HKF is slightly different from that of others. It seems to be caused by the turbulent effect. Each growth ratiois the average of 50 nearby points.(a) (b)
Figure 7.
The direction of magnetic helicity is decided according to the conservation of total magnetic helicity in the system. Theminimum (t ∼ ∼ H M in Fig.5(c). forcing scale decreases and moves toward the smaller scalewhich has less helical effect but more dissipative effect. E kin,s , more exactly h v · ω i plays the role of balancingthe growth of large scale magnetic field. However, theevolution of E kin,L shows rather an irregular feature. In theearly time regime E kin,L leads the growth of E M,L whichgenerates H M (Fig.1, 2, and 3). According to EDQNMapproximation, the role of E kin,L with E M,L is related tothe self distortion effect(the eddy damping rate µ k , Eq.(6)).Smaller magnitude of E kin,L decreases µ k , which increases α effect and dissipation at the same time. However, moredetailed simulation is necessary to check these theoreticalinference.Analytic equation like EQDNM or mean field dynamotheory does not explicitly explain the role of magneticenergy in the small scale. But simulation results providesome clues to the influence of E M,s on the dynamo. Theratio of smaller scale E M to that of the whole smallscale is consistently regular, i.e., from onset: ∼
30% to saturation: ∼ E M,s , more exactly h j · b i , is related with the inverse cascade of H M (or E M )and balancing the growth rate of large scale magneticfield. h j · b i in α coefficient does not always quench thelarge scale magnetic field. As Fig.7(a), 7(b) show, whenthe necessity of inverse cascade of magnetic energy islarge, forcing scale H M is negative so that α effect is en-hanced(the kinetic helicity in α coefficient keeps positive).As the large scale field saturates, the sign of forcing scalemagnetic helicity grows to be positive, i.e., lowering α effect.With more detailed plots we can investigate the dynamicproperties of small scale regime with respect to the largescale B field growth.Fig.8(a) shows the evolution of small scale E M,s (= P k = k max k =2 E M ( k )) and nonhelical E M,s (= E M,s − kH M,s , thin line). To remove the preliminarysimulation,
NHMF → HKF ( ≡ . t ′ ≡ t − . HMF → HKF ( ≡ . c (cid:13) , 000–000 Kiwan Park E-mail: [email protected] Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA
NHMF → HKF HMF → HKF HKF
Init. Onset Sat. Init. Onset Sat. Init. Onset Sat.( t = 10 .
6) ( t ∼
70) ( t → ∞ ) ( t = 13 .
0) ( t ∼ t → ∞ ) ( t = 0) ( t ∼ t → ∞ ) H M,L − . × − − . × − − . × − . × − − . × − − . × − − . × − − . × − − . × − E M,L . × − . × − . × − . × − . × − . × − . × − . × − . × − E M,S . × − . × − . × − . × − . × − . × − . × − . × − . × − (5 . × − ) (2 . × − ) (1 . × − ) (5 . × − ) (2 . × − ) (1 . × − ) (1 . × − ) (1 . × − ) (1 . × − )(100%) (32%) (21%) (100%) (33%) (19%) (100%) ( ∼ E K,L . × − . × − . × − . × − . × − . × − . × − . × − . × − E K,S . × − . × − . × − . × − . × − . × − . × − . × − . × − (6 . × − ) (1 . × − ) (8 . × − ) (6 . × − ) (2 . × − ) (9 . × − ) (2 . × − ) (2 . × − ) (8 . × − )(97%) (1%) (9%) (88%) (1%) (10%) ( ∼ Table 1.
Large scale: k = 1, forcing scale: k = 2 ∼
6, smaller scale: k = 7 ∼ k max (quantity in parentheses). E M,s ( t ) = P k max k =2 E M,s ( k, t ).The percentage is the ratio of smaller scale to small scale: ∼ P k max k =7 / P k max k =2 . time unit using t ′ ≡ t − .
0. But
HKF ( ≡ h j · b i >
0) or suppress it( h j · b i < E M in the earlytime regime. E M of the 1st simulation is relatively largerthan that of other cases, and E M of the 3rd simulation isthe least. The initial small scale E M ,s (5 . × − of the1st simulation, 5 . × − of the 2nd simulation) due tothe preliminary simulation drops till t ∼
10 and begins togrow again. The different minimum value and evolution ofeach field profile imply some important clues to the relationbetween large scale magnetic field and small scale magneticfield. In addition, the origin of helical magnetic field can beinferred from Fig.8(a), 8(b).The magnetic helicity is the topological linking number ofmagnetic fields. But statistically it can be considered asthe correlation between different components of magneticfield(Yoshizawa (2011)). h B i ( k ) B j ( − k ) i = P ij ( k ) E M ( k )4 πk + i k l k ǫ ijl H M ( k ) (12) (cid:18) P ij ( k ) = δ ij − k i k j k (cid:19) . Since magnetic helicity h a · b i cannot be larger than 2 E M /k .Small scale magnetic energy has a lower bound proportionalto the small scale magnetic helicity. In addition, the growthof small scale magnetic helicity depends on that of largescale magnetic helicity in terms of the conservation ofmagnetic helicity in the system. All of these explain thereasons of quick drop of E M and different evolution of E M in the small scale. The role of nonhelical E M becomesclear with the comparison of E kin . We will discuss aboutthis again. The initial B-field plays the role of seed fieldin MHD dynamo, and at the same time the correlationbetween its different components constrains the growth oflarge scale magnetic field dynamically changing the signand magnitude.Fig.8(c), 8(d) show E kin spectra of the 1st and 2ndsimulation are very similar. Fig.8(b), 8(c), and 8(d) implythe profile of kinetic energy does not depend on the small scale E M much as long as E M is not too muchdifferent. Fig.8(e) and 8(f) also show large scale E kin is independent of E M when the magnetic energy is notsignificantly different. In addition, Fig.8(d), 8(e) clearlyshow that E kin drops when large scale E M begins to rise,i.e., onset position. At the onset point of the 1st simulation, t ∼ E kin of this simulation begins to drop. Andaround t ∼ E kin of this simulation also begins to drop. They meet againeach other when large E M of each simulation gets saturated.In Navier Stokes equation(Eq.2), Lorentz force( J × B )can be decomposed into magnetic tension( B ·∇ B ) andpressure( −∇ B / B /R c − ∇ ( B / n , R c : radius of curvature, Priest(2003)) exists, as the definition of Lorentz force implies.When the growth of E M accelerates near the onset point,the compressive force( −∇ B /
2) normal to B-field growsso that the net effect of Lorentz force becomes negative.This presses the plasma and causes the geometrical changesof magnetic fields. The kinetic motion of plasma slows down.Figure.8(c), 8(f) also show there is a time regime( t < ∼ E kin is independent of theinitial values. This occurs when Lorentz force is still weak,and looks like the corresponding concept of the kinematicregime.Fig.9(a) ∼ E M insmall scale be determined by the growth of large scalemagnetic energy(helicity). As mentioned, small scale E M of the 1st and 2nd simulation are almost the same. Weknow the different large scale E M (or H M ) causes thedifferent growth rates( ∂ B /∂t ∼ h v · ω i B ). The fast growthof negative H M in the large scale requires the fast growth ofpositive H M in small scale to conserve H M in the system.The evolution of small scale E M or H M is highly influencedby the initial large scale E M or H M . The quick change of c (cid:13) , 000–000 nfluence of initial conditions on the large-scale dynamo growth rate (a) (b)(c) (d)(e) (f) Figure 8.
Preliminary simulation effect is removed by shifting the time unit(1st simulation: -10.6, 2nd simulation: -13.0). (a) E M ,s :5 . × − (1st), 5 . × − (2nd), 1 . × − (3rd). (c), (f) In the very early time regime, all E kin s evolve in the same way being independentof the initial values. (d) E kin of smaller scale eddy branches off earlier. sign with the sequent fast growth of small scale h j · b i ofthe 1st simulation and the fast decay of small scale E M ofthe second simulation support this fact very well.Fig.9(e), 9(f) show how residual helicity evolves. In the veryearly time regime the residual helicity does not depend onthe initial conditions. This phenomenon is an inevitableconsequence for the kinetically driven MHD dynamo. In Fig.10(a) ∼ t → t − (120 −
70) and t → t − (186 −
70) to compare the behaviors of field profilesafter the onset. Except some minor differences due tothe turbulene, all three simulations have the same fieldprofiles. This indicates the saturation of MHD dynamo isindependent of the initial conditions.On the other hand in Fig.4(a) ∼ c (cid:13) , 000–000 Kiwan Park E-mail: [email protected] Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA (a) (b)(c) (d)(e) (f)
Figure 9.
Small box includes h v · ω i and residual helicity for 3rd simulation. Their profiles are influenced by the large scale field of the3rd simulation and take different routes unlike 1st and 2nd simulation. (d) The evolution of h j · b i needs to be compared with that of E M,s in Fig.8(b). cascade of kinetic energy toward larger scale. This looks tocontradict the established theory that the inverse cascadeoccurs with the strong rotation effect or in the ideal (quasi)two dimensional hydrodynamic system. This conclusion isbased on the conservation of physical quantities like energyand enstrophy. The modified expression of h k i (Davidson (2004)) is, c (cid:13) , 000–000 nfluence of initial conditions on the large-scale dynamo growth rate (a) (b)(c) (d)(e) (f) Figure 10.
The plots of 2nd and 3rd are shifted by -50 and -116 for the comparison with 1st simulation after the onset. d h k i dt = ddt (cid:18) R ∞ k E v ( k, t ) dk R ∞ E v ( k, t ) dk (cid:19) = − k c ddt (cid:2) R ∞ ( k − k c ) E v ( k, t ) dk ] R ∞ E v ( k, t ) dk + k c ddt (cid:2) R ∞ ( k + k f ) E v ( k, t ) dk (cid:3)R ∞ E v ( k, t ) dk − ( R ∞ k E v ( k, t ) dk ) ddt R ∞ E v ( k, t ) dk ( R ∞ E v ( k, t ) dk ) . ( k c = R kE ( k, t ) dk/ R E ( k, t ) dk )If total energy R ∞ E v ( k, t ) dk and enstropy R ∞ k E v ( k, t ) dk are conserved, the first term in theright hand side determines h k i . Since the usual spreading E v ( k, t ) in turbulent flow makes R ∞ ( k − k c ) E v ( k, t ) dk grow, h k i decreases(inverse cascade). However, if enstro-phy or energy is not conserved(by the external source), h k i can grow or decrease according to ˙ E v and h ˙ ω / i .Biferale et al. (2012) showed that the reverse cascade of c (cid:13) , 000–000 Kiwan Park E-mail: [email protected] Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA hydrodynamic energy occurs when the mirror symme-try is broken(helicity). In addition, there were anothertrials to explain the forward cascade of E kin using thecanonical ensemble average(Biskamp (2008), Frisch et al.(1975)). If E tot , H M , and H c (= h u · b i ) are conservedquantities(ideal three dimensional MHD system), theform of E kin ( ∼ h u i u i i ) calculated using the canonicalpartition function Z − exp ( − αE tot − βH M − γH C ) ( Z : anormalization factor) shows forward cascade. However, if E tot (= E M + E kin ) is not conserved because of the externalforcing or some other sources, another term should beadded to E tot in the partition function; this changes theaveraged E kin into a new form that allows the backwardcascade. Based on simulation and theory, we have investigated theinfluence of IC s, the role of (non)helical field with thepressure in the energy transfer.The growth rate of large scale magnetic field is chieflyproportional to the large scale initial values. In contrast, itssaturation depends on the external driving source and theintrinsic properties like η or ν instead of IC s.Comparing the simulation results, we have seen how thehelical and nonhelical magnetic field constrain MHD dy-namo. The helical magnetic field in the small scale has beenthought to quench the growth of large scale magnetic field.As h A · B i increases the opposite sign of small scale h a · b i also grows. So the amplification of large scale magneticfield slows down and saturates. However, in the early timeregime h a · b i and large scale H M have the same sign; thus, h a · b i boosts the growth of large scale magnetic field.On the other hand, growing nonhelical magnetic fieldpresses the plasma through Lorentz force(magnetic pres-sure) and slows down the motion, which constrains themagnetic fields eventually. Of course, the evolution of smallscale fields is also influenced by the large scale IC s and theevolving large scale field.Besides, it is observed kinetic energy migrates backwardwhen the external energy flows into the three dimensionalMHD system. And the velocity field in the early time regimeseems to play a preceding role in the MHD dynamo. Kiwan Park acknowledges support from US NSF grantsPHY0903797, AST1109285, and a Horton Fellowship fromthe Laboratory for Laser Energetics at the University ofRochester.
APPENDIX A: EDDY DAMPED QUASINORMAL MARKOVIANIZATIONA1 Two point closure
Navier Stokes equation for the incompressible fluid is (cid:18) ∂∂t + νk (cid:19) u i ( k ) = − ik m P ij ( k ) X p + q = k u j ( p ) u m ( q ) ∼ h uu i . (A1)This equation requires information on the second order cor-relation equation. Then we need to solve another differentialequation: (cid:18) ∂∂t + ν ( k + k ′ ) (cid:19) h u ( k ) u ( k ′ ) i = h uuu i . (A2)We can derive the third order correlation term, which needsthe fourth order correlation: (cid:18) ∂∂t + ν ( k + p + q ) (cid:19) h u ( k ) u ( p ) u ( q ) i = h uuuu i . (A3)It is known that the probability distribution of turbulentvelocity is not far from the normal distribution. Then, thefourth order correlation term can be decomposed into thecombination of the second order correlation terms(QuasiNormal approximation, Proudman & Reid (1954), Tatsumi(1957)). h u ( k ) u ( p ) u ( q ) u ( r ) i ∼ X h uu ih uu i . (A4) A2 Eddy Damping coefficient
However, Ogura (1963) pointed out that Quasi Normalapproximation could make the energy spectrum negative.Later Orszag (1970) found that the decomposed value be-came too large when the fourth correlation was decomposedof the combination of second correlation terms. Orszag in-troduced eddy damping coefficient µ kpq ( ∼ /t ). (cid:18) ∂∂t + ν ( k + p + q ) + µ kpq (cid:19) h u ( k ) u ( p ) u ( q ) i = X h uu ih uu i . (A5)Orszag suggested µ kpq = µ k + µ p + µ q , µ k ∼ [ k E ( k )] / . (A6)( µ kpq used in Eq.6 is a little different from Orszag’s one.)However, if energy drops faster than k − , eddy dampingterm( ∼ t − ) decreases with ‘ k ’. This means the dampingtime of a smaller eddy can be larger than that of a largereddy. To solve this problem, another modified representationwas suggested by Lesieur & Schertzer (1978): µ k ∼ (cid:20) Z k E ( p, t ) dp (cid:21) / . (A7)This is ‘ Eddy Damped Quasi Normal ’ approximation.Then we have, (cid:18) ∂∂t + 2 νk (cid:19) h u i u j i k, t = Z t dτ Z k + p + q =0 e − [ µ kpq + ν ( k + p + q )]( t − τ ) X h uu ih uu i dp dq. (A8) c (cid:13) , 000–000 nfluence of initial conditions on the large-scale dynamo growth rate If time scale of P h uu ih uu i is much larger than [ µ kpq + ν ( k + p + q ] − , markovianization makes the equation muchsimpler. (cid:18) ∂∂t + 2 νk (cid:19) h u i u j i k, t = Z k + p + q =0 θ kpq X h uu ih uu i dp dq. (A9) θ kpq = Z t dτ e − [ µ kpq + ν ( k + p + q )]( t − τ ) dτ (A10)This is called ‘ Eddy Damped Quasi Normal Markovian approximation’(EDQNM, Pouquet et al. (1976), Davidson(2001), Lesieur (2008)).
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