Is the Alfven-wave propagation effect important forenergy decay in homogeneous MHD turbulence?

Murshed Hossain, Perry C. Gray, Duane H. Pontius, Jr., William H. MatthaeusBartol Research Institute, University of Delaware, Newark

Sean OughtonDept of Mathematica, University College London

Abstract. We investigate the role of three-pointdecorrelation due to Alfven wave propagation in three-dimensional incompressible homogeneous MHD turbu-lence. By comparing numerical simulations with the-oretical expectations, we have studied how this effectinfluences the decay of turbulent energy caused by bothan external mean magnetic field and the fluctuating tur-bulent field. Decay is initially suppressed by a meanmagnetic field, as expected, but the effect soon satu-rates. The decay rate does not scale with mean mag-netic field for higher values. The disagreement with the-oretical predictions can be accounted for by anisotropicspectral transfer. Thus, phenomenological models forenergy decay that include decorrelation due to Alfvenicpropagation are not substantiated. This work comple-ments our detailed study of various models of energydecay in homogeneous MHD [Hollain et al., 1995].

Decay Rates and Propagation Effects

For homogeneous MHD turbulence we are particu-larly interested in the dynamical behavior of the fluc-tuation energy per unit mass, E = (lul2 + IbI2}/2 =(Iz+ 12 + Iz-12) /4, and the fluctuation cross helicity den-sity, He = (u' b). Angle brackets represent volume aver-ages. These quantities are rugged invariants of the idealMHD equations and particularly relevant to the theoryof turbulent spectral transfer [Kraichnan, 1973;Frilch etal., 1975; Stribling and MatthaeUl, 1991]. The amountof E or He present is not modified by the nonlinearterms that mediate spectral transfer among eddies ofdifferent wavenumbers. In the high Reynolds numberlimit, these quantities can change only when excitationsreach the dissipation region at very small scales. The socalled Elsasser energies z~ = (lz+12) and Z: = (lz-12)are equivalent invariants.

Kraichnan [1965] and Dobrowolny et al. [1980b] ap-plied the approach of Karman [1938] and Kolmogorov[1941a,b] for hydrodynamic models to the turbulent de-cay of the Elsasser energies zl in MHD. For the presentpurposes, the first step is to write the decay of Zl interms of spectral transfer times T,%,

dZl Zl-d = -a% -::f""' (2)t T,

Introduction

The parameters ax are phenomenological constants oforder unity; the degree to which they do remain con-stant in the simulations indicates the validity of a con-jectured T,:f=. The spectral transfer times are estimatedfrom several characteristic time scales arising from thethe dynamical equations (1). The first term on the rightyields T~I the characteristic nonlinear time generalizedfor MHD in Elsasser representation [Dobrowolny et al.,

1980],:t: ,}.:t:7"n! = z. (3)

Of

This time scale represents the approximate rate at whichan eddy at the energy-containing length scale ,}.:t: de-cays due to correlations with turbulent structures ofthe other sense. Consequently, the typical lifetime ofthese correlations 7"f (called triple correlations, afterthe Fourier representation of the term) is also impor-tant. The spectral, nonlinear, and triple-decorrelationtimescales are related by [MatthaeU8 and Zhou, 1989]:

7"1' =~. (4)7"3

The pressure term acts on very short times cales to en-sure solenoidal vector fields and will be ignored. Dissi-pation due to viscosity and resistivity, important mainlyat small scales, is represented by S%.

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MHD turbulence is recognized as being of centralimportance to numerous subjects of ongoing research,e.g., the sun, the heliosphere, space plasmas, and cos-mic rays. While these applications may eventually re-quire precise accounts of the dynamical evolution ofturbulence, an immediate need is to include reasonablequantitative estimates ofMHD turbulence in dynamicalmodels appropriate to the specific application. The sim-plest such models involve phenomenological treatmentsof the energy decay, which can be used to estimate heat-ing rates in solar wind transport models.

We start with the incompressible MHD equations in-volving the solenoidal fluid velocity u and the magneticfield B. The magnetic field is written as the sum of alocally uniform mean value Bo and a fluctuating partb. Because mass density is constant, the equations arecast in Alfven speed units [Fyle and Montgomery, 1976],where magnetic field variables are equivalent to their as-sociated Alfven velocity, i.e., (47rp)-1/2Bo -Bo = VA.The dynamical equations are conveniently written usingEl.!Ii.!.!er [1950] variables z% = u :I: b as

HOSSAIN ET AL.: ALFVEN PROPAGATION EFFECT ON ENERGY DECAY

Figure 1. Evolution of total energy for three runs withvarying mean magnetic field Bo. Note the suppressionof decay as Bo increases from 0 to 3 and the compara-tively negligible change when Bo is increased to 8.

the Alfven wave decorrelation hypothesis is incorrect inthe energy-containing range.

Fluctuating Field Effect

To examine possible effects of wave propagation dueto the fluctuating magnetic field, we treat exampleswith no mean magnetic field. In that case .,.1 = A%/band

~ = -~~ (8)dt A% ZT + b

We will evaluate this equation by testing the constancyof Q%. First we consider the case when the cross helicityand energy difference D are zero (i.e., equipartition inkinetic and magnetic energies), so that Z+ = Z- == Zand b2 = Zi/2 = Z~/2 = Z2/2. We also now have onlyone length scale )., so that .,.1 = ..;2)./Z. Therefore,the decay model with wave propagation effect due tothe fluctuating magnetic field is:

dZ2 1 Z3=- Q-(1 + ..;2) ).

In addition, energy decay has been argued to dependupon wave-propagation effects arising from the secondterm and represented by a characteristic Alfvenic timescale T;. The superscript :i: on each of these variablesadmits the possibility that they are generally differentfor the two Elsasser fields.

Each of these quantities has been discussed previously[Kroichnan, 1965; Groppin et al., 1982, 1983; Groppinet al., 1983; MatthaeU8 and Zhou, 1989], but mainly inthe context of inertial-range phenomenology, and in es-sentially an approximation of isotropic turbulence. Itis not immediately clear how these inertial-range argu-ments should be applied in the energy-containing rangefor phenomenological models of MHD energy decay.

For steady inertial-range energy transfer, Kroichnan[1965] argued that the triple correlation lifetime TJshould be dominated by Alfvenic decorrelation whenpropagation is sufficiently strong that TA < Tn'. Hesuggested that TA should be the period of Alfven wavesof the appropriate scale in the mean magnetic field.Pouquet et al. [1976] suggested that inertial-range triplecorrelations also decay due to propagation in the fluc-tuating magnetic field. Under the isotropy assumption,the wave period at wavenumber k is of order (kV A)-l.These ideas can be directly carried over to the energy-containing range by using ),% in place of k-l. This leadsto a characteristic Alfven time

% ),%T A = ITT2 I 1.2' (5)

YVA +b2

where b is the rms magnitude of b.In general, triple correlations also decay due to the

nonlinear process characterized by Tn', but the K roich-nan [1965] phenomenology neglects this effect becausethe mean magnetic field strength is assumed to be large.For cases where both effects are important, the totalrate of triple decorrelation is plausibly given by the sumof the two rates [MatthaeU8 and Zhou, 1989]:

1 1 1~ = ~+~. (6)TJ Tn' TA (9)dt

This suggests that the effect of wave propagation wouldbe to scale the decay rate by a simple numerical con-stant of order unity. Given the inherent uncertainty inthe correct value of Q%, there does not appear to bea practical way to distinguish the two theories in thissituation.

To demonstrate the level at which the models aresuccessful, we begin by computing the right-hand sideof the decay equation (8) with the propagation term bfirst included then omitted. These values are comparedto the decay rate on the left-hand side determined bytaking finite differences in the simulations. The ratioof the model value for dZi/dt to that determined fromthe simulation can be thought of as a time-dependentvalue of Q+, which is supposed constant in the models.Figure 2 shows two values of computed Q+ (with andwithout the propagation term b) for a run started fromD = 0 and Z+ = Z-. After an initial transient period,

Mean Field Effect

For strong mean magnetic field with Z=*, « Bo, theextension of Kraichnan's [1965] model into the energy-containing range predicts an energy decay rate of

dZ2 Z2 Z2~=-Q=*,~. (7)

This suggests that decay should be suppressed by in-creasing the mean magnetic field. To evaluate this pre-diction, we plot the time histories of energy for threecomparable simulations with varied mean fields in Fig-ure 1. The two runs with Bo = 3 and 8 clearly decayslower than the Bo = 0 run, but the suppression seemsto have saturated with Bo = 3. If (7) were true, thenthe decay rate in the Bo = 8 case would be almost threetimes smaller than in the Bo = 3 case. Obviously, thisdoes not happen in the simulations (at least for the rel-atively low Reynolds number cases reported here), so

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HOSSAIN ET AL.: ALFVEN PROPAGATION EFFECT ON ENERGY DECAY

Figure 2. Computed Q+ from simulation data for zeroinitial cross helicity and energy difference.

both curves approach level asymptotes, verifying thateither model reasonably represents the dynamic evolu-tion of the turbulence.

To distinguish between the two models, we turn toa case with large cross helicity. Consider a situationwith D = 0 again, but let cross helicity be so large thatz+ < Z- and b ~ Z+/2. In that case 7"1 ~ 2A.i;/Z+,so the decay equation becomes:

Figure 3. Similar to Figure 2, only a significant andvariable croBB helicity is present in these simulationsto distinguish between the model predictions. TheQ+ curves have qualitatively different behaviors thatdemonstrate the advantage of the model without prop-agation (dashed curve) compared to that with it (solid).

dZ2 Z2 Z2~ ~ -Q% A%(Z:%;j+/2). (10)

By putting almost all the fluctuation energy into theZ+ field, the symmetry between the :f: equations hasbeen broken. For the Z+ field,

Summary and Discussion

~

dZ; Z;Z- [ 2Z_ ] ( )-=-a+- -, 11

dt A+ Z+

where the term in brackets appears only when the prop-agation term is included. This extreme example sug-gests that simulations with variable cross helicity shouldyield profiles of a+ that plateau better for one modelthan for the other. No pronounced difference is ex-pected for the minority fluctuations, apart from a con-stant of order unity between the a- values as in theprevious test.

We now evaluate the models against a simulationwith nonzero initial cross helicity. The ratio Z+ / Z-is initially 1.5 and rises to 4.5 by t = 8. Figure 3 showsthe values of ax computed from equation (8) with andwithout the propagation term, as before. Between t = 1and 8, the case with propagation rises by a factor of4, while the other remains within a factor of about 2of unity. While neither asymptotes as definitively asthe zero helicity case, it is clear that the model withoutpropagation is significantly better. There remains roomfor improvement, but including propagation does notappear to be a step in the right direction. The a- pro-files generally support either model to the same degreeof confidence and confirm that the two behave similarlyto each other, as suggested by our simple analysis.

These simulations suggest that the Alfven decorre-lation effect associated with the propagation of small-scale structures along the large-scale fluctuating mag-netic field does not contribute to the overall decay ofthe energy. This implies that the mechanism proposedby Pouquet et al. [1976] for the inertial range does notapply for the energy-containing range. Dropping thiseffect returns to a fluid-like model for the decay of theenergy-containing eddies in MHD turbulence. A classof such models and their properties have been examinedin a recent paper [Ho88ain et al., 1995].

Regarding the Alfven wave propagation effect dueto an external mean magnetic field, we showed thatthe decay rate does not scale with the strength of themean field as predicted by adapting Kraichnan's [1965]model to the energy-containing range. However, forfinite mean field the decay is indeed diminished fromthat when it is zero, an effect that has been attributedto anisotropic spectral transfer [Ho88ain et al., 1995].Numerical simulations of both freely decaying [Shebalinet al., 1983; Carbone and Veltri, 1990; Oughton et al.,1994] and driven [Ho88ain et al., 1985] MHD turbulencehave shown that wavenumber spectra become distinctlyanisotropic in the presence of a uniform mean magnetic

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HOSSAIN ET AL.: ALFVEN PROPAGATION EFFECT ON ENERGY DECAY

field of sufficient strength. This anisotropy is consistent Acknowledgments. This resea.rch was supported bywith relatively rapid spectral transfer of energy to wave NASA SPTP grant NAG5-157 and ATP grant NAGW-2456,vectors that are nearly perpendicular to the mean field and by NSF grant ATM 93-18728. Computations performedwith accordingly slower transfer to those aligned with at the San Diego Supercomputer Center.

the mean field. This phenomenon has been described interms of a resonance argument [Shebalin et al., 1983]. References

Models that try to include the effect of Alfvenic prop-agation assume that all wave-vectors experience the full Carbone, V., and P. Veltri, A shell model for anisotropicdecorrelation effect of the mean field Bo, as though each magnetohydrodynamic turbulence, GeophYI. AltrophYI.is aligned with the mean field. The increased anisotropy Fluid Dyn., 52,153, 1990.due to enhanced perpendicular transfer evidently tends Dobrowolny, M., A. Mangeney, and P. Veltri, Propertiesto suppress Alfvenic decorrelation effects. In the iner- of magnetohydrodynamic turbulence in the solar wind,t. I th '.t1' t d.fi h a . Alf '. Altron. AltrophYI., 83,26, 1980a.Ia range IS ellec -n:° 1 es t e ellective. ven tIme Dobrowolny, M., A. Mangeney, and P. Veltri, Fully devel-

from 7'A(k) = (koBo) ,the value approprIate for slab oped anisotropic hydromagnetic turbulence in interplane-

turbulence, to Ik. Bol-l, so that the orientation of k tary space, Phy'. Rev. Lett., 45,144, 1980b.enters explicitly. An anisotropic version of 7', is also ap- Elsasser, W.M., The hydromagnetic equations, PhYI. Rev.,propriate in the energy-containing range. One simple .79,183, 1950.approach would be to modif the s ectral transfer rate Fnach, 1!., A. Pouquet, J. Leorat, a~d A.. ~az';lre, Possibility

..Y P of an mverse cascade of magnetIc heliClty m magnetohy-(cf. (4)) for anIsotropIc turbulence to read drodynamic turbulence, J. Fluid Mech., 68,769, 1975.

Fyfe, D., and D. Montgomery, High beta turbulence in two-dimensional magnetohydrodynamics, J. Plalma PhYI.,16,181, 1976.

Grappin, R., U. Frisch, J. Leorat, and A. Pouquet, Al!venicfluctuations as asymptotic states of MHD turbulence, A,-tron. A I trophy I. , 102,6, 1982.

Grappin, R., A. Pouquet, and J. Leorat, Dependence ofMHD turbulence spectra on the velocity field-magneticfield correlation, Altron. AltrophYI., 126,51, 1983.

Hossain, M., P. C. Gray, D. H. Pontius, Jr., W. H. Matt-haeus, and S. Oughton, Phenomenology for the decayof energy-containing eddies in homogeneous MHD turbu-lence, PhYI. Fluidl, 7,2886, 1995.

Hossain, M., G. Vahala, and D. Montgomery, Forced mag-netohydrodynamic turbulence in a uniform external mag-netic field, PhYI. Fluidl, 28,3074, 1985.

Karman, T., and L. Howa.rth, On the statistical theory ofisotropic turbulence, Proc. Roy. Soc. Lon., 164,192, 1938.

Kolmogoroff, A. N., On degeneration of isotropic turbulencein an incompressible viscous liquid, Dokl. Akad. NaukSSSR, 31,538, 1941a.

Kolmogoroff, A. N., Dissipation of energy in locally isotropicturbulence, Dokl. Akad. Nauk SSSR, 32,16, 1941b.

Kraichnan, R. H., Inertial-range spectrum of hydromagneticturbulence, Phy'. Fluidl, 8,1385, 1965.

Kraichnan, R. H., Helical turbulence and absolute equilib-rium, J. Fluid Mech., 59,745, 1973.

Matthaeus, W. H., and Y. Zhou, Extended inertialrange phenomenology of magnetohydrodynamic turbu-lence, Phy'. Fluidl B, 1,1929, 1989.

Oughton, S., E. Priest, and W. Matthaeus, The influence ofa mean magnetic field on 3d MHD turbulence, J. FluidMech., 280,95, 1994.

Pouquet, A., U. Frisch, and J. Leorat, Strong MHD heli-cal turbulence and the nonlinear dynamo effect, J. FluidMech., 77,321, 1976.

She balin, J. V., W. H. Matthaeus, and D. Montgomery,Anisotropy in MHD turbulence due to a mean magneticfield, J. Plalma PhYI., 29,525, 1983.

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1(12)

1 Z2=:t:"=::f:.-T. A:t:(Z'f+Bocos8:t:)'

where cos8:i: measures the degree of anisotropy [She-balin et al., 1983] of the Z:i: spectrum in the energy-containing range. Accounting for anisotropy by usingthe spectral transfer rate (12) to modify the decay equa-tions (2) has been successful [Ho.f.fain et al., 1995].

The straightforward extension of Kraichnan's [1965]inertial-range phenomenology to the energy-containingrange does not work as well as a simple energy de-cay model of the Karman-Kolmogoroff type. That is,there appears to be no explicit dependence of the mod-eled spectral transfer or the decay rates upon either thelarge-scale Alfven speed (for large enough Bo) or thatassociated with the rms value of b. The mean magneticfield does influence the simulation results, however, anda better phenomenological model can be obtained bytaking spectral anisotropy into account [Ho.f..ain et al.,1995]. This model has the energy decay equations

~ = _.!.~ (13)dt A A:i: '

where A ~ 1 when Bo ~ 0, and A ~ 2 when Bo ~ 1.Under suitable conditions this parameter emulates theeffects of spectral anisotropy.

In closing, we note that one might expect Alfven wavepropagation effects to remain valid for inertial-range dy-namics. But if spectral transfer through the inertialrange into the dissipation range is reduced by whatevermechanism, then that effect should slow the decay of thetotal turbulent energy and be reflected in the evolutionof the energy-containing eddies. Because no such effectis evident, the conclusions of this study for the energy-containing eddies suggest that the wave-propagation ef-fect may not be operative in the inertial range. Al-though propagation effects may strongly influence theangular distribution of energy flux in the inertial range,our results imply they do so without changing the totalenergy flux.

P. C. Gray, M. Hossain, W. H. Matthaeus,D. H. Pontius, Jr., Bartol Research Institute, University ofDelaware, Newark, DE 19716.

S. Oughton, Department of Mathematics, UniversityCollege London, WC1E 6BT, UK.

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