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Strong Anisotropic MHD Turbulence with Cross Helicity Benjamin D. G.Chandran [email protected] SpaceScience Center andDepartmentofPhysics,Universityof NewHampshire 8 0 0 2 ABSTRACT n a J This paper proposes a new phenomenology for strong incompressible MHD tur- 1 bulence with nonzero cross helicity. This phenomenology is then developed into a 3 quantitativeFokker-Planck model that describes the time evolution of the anisotropic ] h power spectra of the fluctuations propagating parallel and anti-parallel to the back- p - ground magnetic field B0. It is found that in steady state the power spectra of the o 5/3 r magnetic field and total energy are steeper than k− and become increasingly steep st asC/E increases, whereC = d3xv B is the cr⊥oss helicity, E is the fluctuation en- a Z · [ ergy, and k is the wavevector component perpendicular to B0. Increasing C with 1 E ⊥ fixed increases the time required for energy to cascade to smaller scales, reduces v 3 the cascade power, and increases the anisotropy of the small-scale fluctuations. The 0 implicationsoftheseresultsforthesolarwindandsolarcoronaarediscussedinsome 9 4 detail. . 1 0 8 Subject headings: turbulence — magnetic fields — magnetohydrodynamics — solar 0 wind— solarcorona— solarflares : v i X r a 1. Introduction Much of our current understanding of incompressible magnetohydrodynamic (MHD) turbulence has its roots in the pioneering work of Iroshnikov (1963) and Kraichnan (1965). These studies emphasized the important fact that Alfveń waves travelling in the same direction along a backgroundmagneticfielddonotinteractwithoneanotherandexplainedhowonecanthinkofthe cascade of energy to small scales as resulting from collisions between oppositely directed Alfveń wavepackets. Theyalsoargued thatintheabsenceofameanmagneticfield,themagneticfieldof the energy-containing eddies at scale L affects fluctuations on scales L much in the same way ≪ as wouldatrulyuniformmean magneticfield. – 2 – AnotherfoundationofourcurrentunderstandingisthefindingthatMHDturbulenceisinher- entlyanisotropic. Montgomery&Turner(1981)and Shebalin,Matthaeus,& Montgomery(1983) showedthatastronguniformmeanmagneticfieldB inhibitsthecascadeofenergytosmallscales 0 measured in the direction parallel to B . This early work was substantially elaborated upon by 0 Higdon(1984),Goldreich&Sridhar(1995,1997),Montgomery&Matthaeus(1995),Ng&Bhat- tacharjee (1996, 1997), Galtier et al (2000), Cho & Lazarian (2003), Oughton et al (2006), and manyothers. Forexample, Cho & Vishniac(2000) used numerical simulationsto showthat when thefluctuating magneticfield d B is&B thesmall-scaleturbulenteddies becomeelongated along 0 thelocal magneticfield direction. Goldreich &Sridhar (1995)introduced theimportantand influ- ential idea of “critical balance,” which holds that at each scale the linear wave period for the bulk ofthefluctuationenergy is comparabletothetimeforthefluctuationenergy tocascade to smaller scales. Goldreich & Sridhar (1995, 1997), Maron & Goldreich (2001), and Lithwick & Goldre- ich (2001)clarified a number of importantphysical processes in anisotropicMHD turbulenceand used the concept of critical balance to determinetheratio ofthe dimensionsof turbulenteddies in thedirectionsparallel and perpendiculartothelocalmagneticfield. Over the last several years, research on MHD turbulence has been proceeding along several differentlines. Forexample,onegroupofstudieshasattemptedtodeterminethepowerspectrum, intermittency, and anisotropy of strong incompressible MHD turbulence using direct numerical simulations. (See,e.g.,Cho&Vishniac2000,Mu¨ller&Biskamp2000,Maron&Goldreich2001, Cho et al 2002, Haugen et al 2004, Muller & Grappin 2005, Mininni & Pouquet 2007, Perez & Boldyrev 2008). Another series of papers has explored the properties of anisotropic turbulence in weakly collisional magnetized plasmas using gyrokinetics, a low-frequency expansion of the Vlasov equation that averages over the gyromotion of the particles. (Howes et al 2006, 2007a, 2007b; Schekochihin et al 2007). These authors investigated the transition between the Alfveń- wavecascadeandakinetic-Alfveń-wavecascadeatlengthscalesofordertheprotongyroradiusr , i aswellasthephysicsofenergydissipationandentropyproductioninthelow-collisionalityregime. Turbulence at scales . r has also been examined both numerically and analytically within the i framework of fluid models, in particular Hall MHD and electron MHD. (Biskamp, Schwarz, & Drake 1996, Biskamp et al 1999, Matthaeus et al 2003; Galtier & Bhattacharjee 2003, 2005; Cho & Lazarian 2004; Brodin et al 2006, Shukla et al 2006). Another group of studies has investigated the power spectrum, intermittency, and decay time of compressible MHD turbulence. (Oughton et al 1995, Stone et al 1998, Lithwick & Goldreich 2001, Boldyrev et al 2002, Padoan et al 2004, Elmegreen & Scalo 2004). Additional work by Kuznetsov (2001), Cho & Lazarian (2002,2003),Chandran(2005),andLuo&Melrose(2006)hasbeguntoaddressthewayinwhich Alfveń waves, fast magnetosonic waves, and slow magnetosonic waves interact in compressible weak MHD turbulence. Another recent development is the finding that strong incompressible MHD turbulence leads to alternating patches of alignment and anti-alignment between the veloc- – 3 – ity and magnetic-field fluctuations. (Boldyrev 2005, 2006; Beresnyak & Lazarian 2006; Mason, Cattaneo,&Boldyrev2006;Matthaeusetal2007)Thesestudiesexaminedhowthedegreeoflocal alignment(andanti-alignment)dependsuponlengthscale,aswellastheeffectsofalignmentupon theenergy cascadetimeandthepowerspectrumoftheturbulence. The topic addressed in this paper is the role of cross helicity in incompressible MHD turbulence. Thecrosshelicityis defined as C= d3xv B, (1) Z · where v is the velocity and B is the magnetic field. The cross helicity is conserved in the absence of dissipation and can be thought of as a measure of the linkages between lines of vorticity and magneticfieldlines,bothofwhicharefrozentothefluidflowintheabsenceofdissipation(Moffatt 1978). In the presence of a background magnetic field, B = B zˆ, the cross helicity is also a 0 0 measureofthedifferencebetweentheenergyoffluctuationstravellinginthe zand+zdirections. − Dobrowolny, Mangeney, & Veltri (1980) showed that MHD turbulence with cross helicity decays to a maximally aligned state, with d v= d B/√4pr , where d v and d B are the fluctuating velocity ± andmagneticfieldandr isthemassdensity. Differentdecayratesfortheenergyandcrosshelicity were also demonstrated by Matthaeus & Montgomery (1980). In another early study, Grappin, Pouquet, & Leórat (1983) used a statistical closure, the eddy-damped quasi-normal Markovian (EDQNM)approximation,tostudystrong3DincompressibleMHDturbulencewithcrosshelicity, assumingisotropicpowerspectra. TheyfoundthatwhenC=0,thetotalenergyspectrumissteeper 6 thantheisotropicIroshnikov-Kraichnank 3/2spectrum. Pouquetetal(1988)foundasimilarresult − in direct numerical simulations of 2D incompressible MHD turbulence. More recently, Lithwick, Goldreich,&Sridhar(2007)andBeresnyak&Lazarian(2007)addressedtheroleofcrosshelicity instrongMHD turbulencetakingintoaccount theeffects ofanisotropy. This paper presents a new phenomenology for strong, anisotropic, incompressibleMHD tur- bulencewith nonzero cross helicity,and is organized as follows. Section 2 presents somerelevant theoretical background. Section 3 introduces the new phenomenology as well as two nonlinear advection-diffusion equations that model the time evolution of the power spectra. Analytic and numericalsolutionstothisequationintheweak-turbulenceandstrong-turbulenceregimesarepre- sented in Sections 4 and 5. Section 5 also presents a simple phenomenological derivation of the power spectra and anisotropy in strong MHD turbulence. Section 6 presents a numerical solution to the advection-diffusionequation that showsthe smoothtransitionbetween the weak and strong turbulenceregimes. Section7addressesthecaseinwhichtheparallelcorrelationlengthsofwaves propagating in opposite directions along the background magnetic field are unequal at the outer scale. InSection8,theproposedphenomenologyisappliedtoturbulenceinthesolarwindandso- larcorona,andinSection9theresultsofthisworkarecomparedtotherecentstudiesofLithwick, Goldreich,& Sridhar(2007)and Beresnyak &Lazarian (2007). – 4 – 2. Energy Cascadefrom Wave-PacketCollisions TheequationsofidealincompressibleMHD can bewritten ¶ w ± + w v zˆ (cid:209) w = P(cid:209) (2) ¶ t ∓∓ A · ± − (cid:0) (cid:1) where w =v (d B/√4pr ) are the Elsasser variables, v is the fluid velocity, d B is the magnetic ± ± field fluctuation, r is the mass density, which is taken to be uniform and constant, v =B /√4pr A 0 is the Alfveń speed, B = B zˆ is the mean magnetic field, and P = (p+B2/8p )/r , which is 0 0 determined by the incompressibility condition, (cid:209) w = 0. Throughout this paper it is assumed ± · thatd B.B and w .v . 0 ± A In thelimitofsmall-amplitudefluctuations(w v ), thenonlinearterm w (cid:209) w inequa- ± A ∓ ± ≪ · tion(2)can beneglectedto afirst approximation,andthecurl ofequation(2)becomes ¶ ¶ v (cid:209) w =0, (3) ¶ t ∓ A¶ z × ± (cid:18) (cid:19) which is solved by setting (cid:209) w equal to an arbitrary function of z v t. Thus, w represents ± A ± × ± fluctuations with v = b that propagate in z direction at speed v in the absence of nonlinear A ± ∓ interactions. Intheabsenceofanaveragevelocity,thecrosshelicitydefinedinequation(1)canbe rewrittenas pr √ C= d3x (w+)2 (w )2 . (4) − 2 Z − Thecrosshelicityisthusproportionaltothediffe(cid:2)renceinenergy(cid:3)betweenfluctuationspropagating inthe z and +z directions. − Equation (2) shows that the nonlinear term is nonzero only at those locations where both w+ and w are nonzero. Nonlinear interactions can thus be thought of as collisions between oppo- − sitely directed wave packets (Kraichnan 1965). When both w+ and w are nonzero, equation (2) − indicates that the w fluctuations are advected not at the uniform velocity v zˆ, but rather at ± A ∓ the non-uniform velocity v zˆ+w . Maron & Goldreich (2001) elaborated upon this idea by A ∓ ∓ showing that to lowest order in fluctuation amplitude, if one neglects the pressure term, then w+ wave packets are advected along the hypothetical magnetic field lines corresponding to the sum of B and the part of d B arising from the w fluctuations. This result can be used to construct a 0 − geometrical picture for how wave-packet collisions cause energy to cascade to smaller scales, as depictedinFigure1. Inthisfigure,twooppositelydirectedwavepacketsofdimension l inthe plane perpendicular to B and length l along B pass through one another and get sh∼eare⊥d. Col- 0 0 lisions between wavepackets of similarkl are usually the dominant mechanism for transferring ⊥ energy from large scales to small scales. The duration of the collision illustrated in the figure is approximatelyD t l /v . Thefluctuatingvelocityandmagneticfieldaretakentobeintheplane A ∼ k – 5 – perpendicular to B , as is the case for linear shear Alfveń waves. The magnitude of the nonlinear 0 terminequation(2)isthen w+ w /l ,wherew isthermsamplitudeofthew wavepacket. l −l ±l ± ∼ ⊥ The fractional change in the v an⊥d b⊥fields of the w ⊥wave packet induced by the collision is then ∓ roughly w+l w−l D t w±l l = k. (5) l⊥ ⊥ × w v ⊥l ! ∓l ! A ⊥ ⊥ ⊥ BEFORE COLLISION: l field lines l B 0 group w − wave packet velocity w+ wave packet AFTER COLLISION: B 0 w+ wave packet group w − wave packet group velocity velocity Fig. 1.—Whentwowavepacketscollide,eachwavepacketfollowsthefieldlinesoftheotherwavepacketandgets sheared. If this fractional change is 1 for both w+ and w fluctuations then neither wave packet − ≪ is altered significantly by a single collision, and the turbulence is weak. Wave packets travel a distance l before being significantly distorted, and the fluctuations can thus be viewed as ≫ k linear waves that are only weakly perturbed by nonlinear interactions with other waves. In the wave-packetcollisiondepictedinFigure1,theright-handsideofthew wavepacketisalteredby − thecollisioninalmostthesamewayastheleft-handside,sincebothsidesencounteressentiallythe same w+ wave packet, sincethe w+ packet is changed only slightlyduring thecollision. Changes totheprofileofawavepacketalongthemagneticfield arethusweakerthanchangesintheprofile of a wave packet in the plane perpendicular to B (Shebalin et al 1983, Ng & Bhattacharjee 1997, 0 Goldreich &Sridhar 1997,Bhattacharjee & Ng 2001, Perez & Boldyrev2008). Asa result, inthe weak-turbulencelimit,thecascade ofenergy tosmalll is muchless efficient than thecascade of energy to smalll (Galtieret al 2000). k ⊥ On the other hand, if the fractional change in equation (5) is of order unity then a w wave ∓ packet isdistortedsubstantiallyduring asinglecollision,andtheturbulenceis saidto be“strong.” In thecase that thefractional change in equation (5) is 1 for one fluctuation type, (e.g., w ) but − ∼ 1 for the other (w+), the turbulence is still referred to as strong. It should be noted that strong ≪ turbulencecanarisewhenw v ,providedthatl l . Instrongturbulenceenergycascades ±l A ≪ ⊥≪ k ⊥ – 6 – to smallerl to a greater extent than in weak turbulence, but the primary direction of energy flow k ink-space isstillto largerk , asdiscussedin thenextsection. ⊥ 3. Anisotropic MHDTurbulence withCross Helicity In order to develop an analytical model, it is convenient to work in terms of the Fourier transformsofthefluctuatingw fields, givenby ± 1 w˜ (k)= d3xw(x)e ikx. (6) ± (2p )3Z − · Thethree-dimensionalpowerspectrumA (k) isdefined by theequation ± w˜ (k) w˜ (k ) =A (k)d (k+k ), (7) ± ± 1 ± 1 h · i where ... denotes an ensemble average. Cylindrical symmetry about B is assumed, so that 0 h i A (k) = A (k ,k ), where k and k are the components of k perpendicular and parallel to B . ± ± 0 ⊥ k ⊥ k Themean-squarevelocityassociatedwith w fluctuationsisthen ± 1 (d v )2 = d3kA (k ,k ). (8) ± ± 4Z ⊥ k It is assumed that at each value of k there is a parallel wave number k±(k ) such that (1) ⊥ k ⊥ the bulk of thew± fluctuation energy is at k <k±(k ) and (2) A±(k ,k ) depends only weakly | k| k ⊥ ⊥ k onk for k <k±(k ). Aw± wavepacketatperpendicularscalek−1 thenhasacorrelationlength k | k| k ⊥ 1 ⊥ in the direction of the mean field of k± − . The rms amplitude of the fluctuating Elsasser ∼ k fields at aperpendicularscalek 1, denot(cid:16)edw(cid:17) , isgivenby − ±k ⊥ ⊥ (w±k⊥)2 ∼A±(k⊥,0)k⊥2kk±. (9) Asdescribedinthesection2,whenaw wavepacket atscalek 1 collideswithaw wavepacket ∓ − ± at scalek 1, thefractional changeinthew packet resultingfro⊥mthecollisionisapproximately − ∓ ⊥ k w c = ⊥ ±k . (10) ±k ⊥ ⊥ k±vA k Thewavenumberkc± isdefined to bethevalueofk± forwhichc ±k =1. Thus, k ⊥ k4A (k ,0) ± kc± = ⊥ v2 ⊥ . (11) A – 7 – 3.1. The energy cascadetime withWa whe−nwkka−v≫e pkac−ck,etht.eEvaalcuhesoufcch−k⊥coilsli≪sio1narnedquairwe+s aistoimnley(wke−avkAl)y−a1f.feTchteedebffyecatssinogflseuccoclelsissiiovne collisionsadd incoherently, and thus (c ) 2 collisionsare requkired for the w+ wavepacket to be −k − stronglydistorted,andforitsenergytop⊥asstosmallerscales. Thecascadetimet + foraw+ wave k packet at perpendicularscalek 1 isthusroughly ⊥ − ⊥ 1 t +k ∼(k−vA)−1(c −k )−2 ∼ k v (weak turbulence). (12) ⊥ k ⊥ c− A Similarly,ifc + 1, thent (k+v ) 1. k ≪ −k ∼ c A − ⊥ ⊥ Whenk−∼kc−,thevalueofc −k is∼1,aw+ isstronglydistortedduringasinglewavepacket collision,andktheturbulenceisstron⊥g. Each suchcollisiontakesatime(k−vA)−1. Sincek− kc−, k k ∼ 1 t +k⊥ ∼(kk−vA)−1 ∼ kc−vA (strongturbulence). (13) Similarly,ifc + 1, thent (k+v ) 1. k ∼ −k ∼ c A − ⊥ ⊥ Thecasekk±≪kc± (i.e.,c ±k⊥ ≫1)isexplicitlyexcludedfromthediscussion. Initialconditions could in principle be set up in which k± kc±. However, the cascade mechanisms described in ≪ k section 3.2 will not produce the condition k± kc± if it is not initially present. It should be ≪ k emphasized that in both weak turbulence and strong turbulence, the cascade time is given by the sameformula,t (k v ) 1, whichinvolvestheA spectrumevaluatedonlyat k =0. ±k ∼ c∓ A − ∓ k ⊥ 3.2. The Cascade ofEnergy to Larger k k The two basic mechanisms for transferring fluctuation energy to larger k were identified k by Lithwick, Goldreich, & Sridhar (2007). The first of these can be called “propagation with distortion.” Suppose a w+ wave packet of perpendicular scale k 1 and arbitrarily large initial − parallel correlation length begins colliding at t = 0 with a stream o⊥f w wave packets of similar − perpendicular scale. At time t =t + , the leading edge of the w+ wave packet has been distorted k substantially by the stream of w w⊥ave packets, but the trailing portion of the w+ wave packet − at distances & 2v t + behind the leading edge has not yet encountered the stream of w wave A k − packets. If the paralle⊥l correlation length of the w+ wave packet is initially>2v t + , then during A k a time t + the w+ wave packet acquires a spatial variation in the direction of th⊥e background k magnetic⊥field of length scale 2v t + 2(k ) 1. This process is modeled as diffusion of w ∼ A k ∼ c− − ± – 8 – fluctuation energy in the k direction with diffusion coefficient D (D k )2/D t, where D k =k ± c∓ and D t =t . “Propagationkwithdistortion”then leads toavalueokf∼D ofk (k )3v . k ±k ± ∼ c∓ A k The second mechanism identified by Lithwick, Goldreich, & Sridhar (2007) can be called “uncorrelated cascade.” Consider a w+ wave packet of perpendicular scale k 1 and arbitrarily − large parallel correlation length, and consider two points within the wave packe⊥t, P and P , that 1 2 movewiththewavepacket at velocity v zând areseparated byadistancealongB of2v t − A 0 A −k ∼ 2(k+) 1. The w wave packets at perpendicular scale k 1 encountered by the portions of⊥the c − − − w+ wave packet at P and P are then uncorrelated, beca⊥use w wave packets are substantially 1 2 − distorted while propagating between P and P . Thus, the way in which the w+ wave packet 1 2 cascades at location P is not correlated with the way in which the w+ wave packet cascades at 1 location P . If the parallel correlation length of the w+ wave packet is initially > 2v t , then 2 A −k wave-packet collisions introduce a spatial variation along B into the w+ wave packet of⊥length 0 scale 2v t 2(k+) 1 during a timet + . Again, we model this as diffusion of w fluctuation ∼ A −k ∼ c − k ± energy in the⊥k direction with D (D k )⊥2/D t and D t =t , but now D k =k . “Uncorrelated cascade” thuslekadsto ak -diffusi±kon∼coeffikcient of (k )2k±k⊥v . k c± c± c∓ A k ∼ Accountingforbothmechanisms,onecan write D (k )2k v , (14) ± c,max c∓ A ∼ k where k (k ) is the larger of k+(k ) and k (k ). If k+ > k , then w+ energy diffuses in k c,max c c− c c− ⊥ ⊥ ⊥ k primarilythroughthe“uncorrelatedcascade”mechanism,whilew energydiffusesink primarily − k throughthe“propagationwithdistortioncascade” mechanism. 3.3. Advection-Diffusion Model forthePowerSpectra The phenomenology described in the preceding sections is encapsulated by the following nonlinearadvection-diffusionequation, ¶ A 1 ¶ c k2A h ¶ 2A ±k = 1 ±k ±k +c (k )2k v ±k +S g A , (15) ¶ t −k ¶ k t ⊥ 2 c,max c∓ A ¶ k2 k±− ±k ±k ±eff,k ! ⊥ ⊥ ⊥ k where A is shorthand for A (k ,k ), c and c are dimensionless constants of order unity, and ±k ± 1 2 S and g A are forcing and d⊥ampking terms, respectively. The first term on the right-hand side k± − ±k ±k of equation (15) represents advection of fluctuation energy to larger k , while the second term represents diffusionoffluctuationenergy to larger k . The quantityt ⊥ is an effectivecascade | k| ±eff,k time at perpendicular scale k 1. Usually, the transfer of energy to small⊥scales is dominated by − local interactions in k-space,⊥and the cascade time for a w+ wave packet is (k v ) 1. In some c∓ A − ∼ – 9 – cases, however, the shearing of small-scale wave packets by much larger-scale wave packets can becomeimportant. To accountforsuch cases, theeffectivecascadetimeis takento be q4A (q ,0) (t ±eff,k⊥)−1 =max(cid:20) ⊥ ∓vA ⊥ (cid:21) for0<q⊥ <k⊥, (16) i.e., (t ) 1 is the maximum value of k v for all perpendicular wave numbers between zero ±eff,k − c∓ A and k . Th⊥eflux ofw energy to largerk is ± ⊥ ⊥ ¥ c k2A h e (k )=2p dk 1 ±k ±k . (17) ± ⊥ Z ¥ k t ⊥±eff,k − ⊥ Thetermh is givenby ±k 1 ¶ h = k A (k ,0) , (18) ±k −A±(k ,0)¶ k ⊥ ± ⊥ ⊥ ⊥ andisincludedsothate increasesastheA spectru(cid:2)mbecomesa(cid:3)moresteeplydecliningfunction ± ± ofk ,inaccordancewithweakturbulencetheory(Galtieretal2000,Lithwick&Goldreich2003). ⊥ To match the energy flux in weak turbulence theory in the limit of zero cross helicity, one must set1 p J c = , (19) 1 − 2 where ¥ 1 2[(x2 1)(1 y2)]1/2(1+xy)2[8 (x+y)3]ln[(x+y)/2] J = dx dy − − − 1.87. (20) Z1 Z 1 (x2 y2)4 ≃− − − Forsimplicity, c =1. (21) 2 4. Steady-State WeakTurbulence + This section addresses weak turbulence in which k k− at the outer scale. The weak- k ∼ k turbulence condition, c ±k⊥ ≪ 1, is equivalent to the condition kc± ≪ kk±. Because kk-diffusion 1The value of c in equation (19) is a factor of 2 larger than the value that follows from the results of Galtier 1 et al (2000). It appears that this discrepancy results from the omission of a factor of 2 in equation (54) of Galtier et al (2000). This can be seen by starting from equation (46) of Galtier et al (2000) and using the expression on page 1045 of Leith & Kraichnan (1972) to simplify polar integrals of the form d2pd2qd (k p q)F(k,p,q) for − − two-dimensionalwavevectorsk, p,andq,whereF isafunctiononlyofthewaveR-vectormagnitudesandtheintegral isoverallvaluesof pandq. – 10 – involvesaD k k duringatimet ,thek -incrementoverwhichenergydiffuseswhilecascading k∼ c± ±k k ⊥ to larger k is much less than the breadth of the spectrum in the k direction ( k±), so the k - ⊥ k ∼ k k diffusion terms can be ignored to a good approximation. In this case, equation (15) possesses a steady-statesolutionin whiche + and e areconstant,and inwhich − A =g (k )k n±, (22) ±k ± − k ⊥ whereg+ and g arearbitrary functionsofk , andwhere − k n++n =6, (23) − with 2 < n < 4. Equations (22) and (23) match the results of weak turbulence theory for in- ± compressibleMHD turbulenceif oneallowsonly for three-waveinteractions among shearAlfveń waves (Galtier & Chandran 2006), or if one considers only the limit that k k (Galtier et al 2002). Ifonewrites n =3 a with a <1 andsets g+(k )=g (k ),then⊥eq≫uatikon(17)gives ± − ± | | k k e + 2+a = . (24) e 2 a − − In the limit a 1, e +/e = 1+a , in agreement with the weak-turbulence-theory result for − ≪ k k (Lithwick & Goldreich 2003), as in the weak-turbulence advection-diffusion model ⊥ ≫ | k| ofLithwick&Goldreich(2003). In steady state,A+(k ,0)and A (k ,0)are forced to beequal atthedissipationscaleso that − t + =t . Thisphenomeno⊥nof“pinning”⊥was discoveredby Grappinet al (1983)forstrong MHD k −k turbulence, and further elaborated upon by Lithwick & Goldreich (2003) for the case of weak turbulence. The dominant fluctuation type then has the steeper spectrum. If e +/e is fixed, then − theratiow+/w ofthermsamplitudesofthetwofluctuationtypesattheouterscalek 1 increases k −k −f f f as k /k increases, where k is the dissipation wave number. Alternatively, if w+/w is fixed, d f d k −k f f thene +/e approaches unityas k /k ¥ . − d f → Several of these results are illustrated by the numerical solution to equation (15) shown in Figure2. Thissolutionisobtainedusingalogarithmicgridfork ,withk =k 2i/n for0<i<N. ,i 0 Similarly, k = k 2j/n for 1 < j < M, but k = 0 for j = 0.⊥A is a⊥dvanced forward in time ,j 0 ,j ±k using a semki-implicit algorithm, in which thekterms h , t , and k on the right-hand side of ±k ±eff,k c± equation (15) are evaluated at the beginning of the time step,⊥and the A terms on the right-hand ± side of equation (15) are evaluated at the end of the time step. The algorithm employs operator splitting, treating the k -advection, forcing, and damping in one stage, and the k -diffusion is a ⊥ k second stage. In this approach, the matrix that has to be inverted to execute each semi-implicit time step is tri-diagonal. An advantage of this procedure over a fully explicit method is that the timestepisnotlimitedbythek -diffusiontimeatlargek andsmallk . Thediscretizedequations k ⊥ k