structure of homogeneous nonhelical magnetohydrodynamic turbulence

Structure of homogeneous nonhelical magnetohydrodynamic turbulenceR. S. Miller,a) F. Mashayek, V. Adumitroaie, and P. GiviDepartment of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo,New York 14260-4400

~Received 10 July 1995; accepted 24 May 1996!

Results are presented for three-dimensional direct numerical simulations of nonhelicalmagnetohydrodynamic~MHD! turbulence for both stationary isotropic and homogeneous shear flowconfigurations with zero mean induction and unity magnetic Prandtl number. Small scale dynamoaction is observed in both flows, and stationary values for the ratio of magnetic to kinetic energy areshown to scale nearly linearly with the Taylor microscale Reynolds numbers above a critical valueof Rel'30. The presence of the magnetic field has the effect of decreasing the kinetic energy of theflow, while simultaneously increasing the Taylor microscale Reynolds number due to enlargementof the hydrodynamic length scales. For shear flows, both the velocity and the magnetic fieldsbecome increasingly anisotropic with increasing initial magnetic field strength. The kinetic energyspectra show a relative increase in high wave-number energy in the presence of a magnetic field.The magnetic field is found to portray an intermittent behavior, with peak values of the flatness nearthe critical Reynolds number. The magnetic field of both flows is organized in the form of ‘‘fluxtubes’’ and magnetic ‘‘sheets.’’ These regions of large magnetic field strength show a smallcorrelation with moderate vorticity regions, while the electric current structures are correlated withlarge amplitude strain regions of the turbulence. Some of the characteristics of small scale MHDturbulence are explained via the ‘‘structural’’ description of turbulence. ©1996 AmericanInstitute of Physics.@S1070-664X~96!01009-9#

I. INTRODUCTION

Fifty years of widespread scientific investigation havepassed since Kolmogorov’s original theory of incompress-ible hydrodynamic~HD! turbulence.1 Yet, due to its chaoticnature HD turbulence remains among the most complex un-solved problems in classical physics. Even more challengingis the problem of turbulence in electrically conducting fluidsobeying the magnetohydrodynamic~MHD! equations.2 Non-linear coupling between the velocity and the magnetic induc-tion fields produces many additional phenomena. In particu-lar, production due to stretching and folding of field lines canlead to growth or sustainment of the magnetic field~the‘‘MHD dynamo’’ effect3,4!. The production effect is a directresult of Maxwell’s equations, i.e., an electromotive force isproduced through Ohm’s law when a conductor movesacross magnetic field lines. Unbounded magnetic growth isprevented by the action of the Lorentz force on the velocityfield resulting in a so-called saturated field condition.

Some of the earliest theoretical considerations of the dy-namo problem suggest that the magnetic Reynolds number isone of the primary parameters in determining the strength~orlack! of field amplification and sustainment.3 Analytic stud-ies of this phenomenon are performed by modeling the non-linear terms,5,6 and by multiple scale analyses.7–9These stud-ies emphasize the importance of the magnetic helicity inamplifying the dynamo phenomenon. Other early studies en-list second order closure techniques to quantify this effect,and find that a magnetic steady state is achieved above acritical magnetic Reynolds number of order 30.10 The dy-

namo effect is difficult to observe in the laboratory due to thesmall values of the magnetic Prandtl number of availableconducting fluids; see, e.g., Roberts and Hensen.11 However,direct numerical simulation~DNS! offers a viable alternativeto conventional laboratory experiments. Although current su-percomputer technology limits DNS to moderate values offlow parameters~e.g., the Reynolds number! many importantfeatures of turbulence may be captured by suchsimulations.12,13

One of the first three-dimensional~3-D! DNS of MHDturbulence is conducted by Pouquet and Patterson.14 Theypropose that large scale kinetic energy is transferred to highwave numbers through the traditional cascade process, whereit is distributed between the velocity and the magnetic fieldsby Alfven waves. In the presence of magnetic helicity theenergy then follows an inverse cascade process from thesmall scales of the magnetic field to the large scales. Due tothe turbulence decay, no validation of field sustainment isgiven; however, rapid field growth during early times of thesimulation is observed. One of the first genuine nonlinearsimulated dynamo effects is a convection driven dynamo in aspherical shell.15 This configuration is chosen to study someissues of importance to the solar dynamo. Meneguzzi, Frisch,and Pouquet16 study stationary isotropic turbulence and ob-serve a small scale dynamo effect in the absence of helicityabove a critical magnetic Reynolds number~of order 40!.Several numerical simulations of 3-D MHD turbulence con-firm that the magnetic Reynolds number is the primary pa-rameter influencing the dynamo effect.10,15–17

The scope of research pertaining to turbulence modifica-tion, intermittency, and coherent structures in MHD turbu-lence is somewhat limited. Most previous numerical simula-tions are confined to two-dimensional~2-D! flows. Biskamp,

a!Present address: Jet Propulsion Laboratory, California Institute of Technol-ogy, Pasadena, California 91109-8099.

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Welter, and Walter18 employ high resolution simulations of2-D isotropic turbulence to address issues related to intermit-tency. By adapting theb model19 they conclude that whilesecond order correlation functions are relatively unaffectedby intermittency, the corrections necessary for fourth- andhigher-order coefficients become significant. In addition,they observe departures from lognormality for small valuesof the local energy dissipation, a result in agreement withHD studies ~e.g., Miller20!. Pouquet, Sulem, andMeneguzzi,21 incorporating both 2-D and 3-D simulations,illustrate how the correlations between the velocity and themagnetic fields can damp the energy exchange and reducethe amplitude and the intermittency of the derivative fields.Orszag and Tang22 find that the magnetic field is more inter-mittent than the velocity field in 2-D MHD turbulence. Thisobservation is verified in 3-D flows for the thermal convec-tion problem,17 and also in isotropic turbulence.16 Intermit-tent magnetic field regions organize into magnetic ‘‘fluxtube’’ structures similar to the vorticity tubes of hydrody-namic turbulence. These magnetic structures are observed in3-D MHD simulations of isotropic turbulence,16 high sym-metry turbulence employing hyperviscosity and hypermag-netic diffusivity,23 and in compressible stratified convectionabove a stable overshoot layer.24,25

Although a zero mean magnetic field can be sustainedthrough dynamo action, most MHD flows in nature occurwith a mean field. The mean field is typically ‘‘slowly’’varying with respect to the dynamic fluctuations. Thus, insome cases, it is possible to consider the mean field to belocally steady and uniform. The most obvious effect of amean magnetic field is to create an anisotropic turbulencestate for both the velocity and the magnetic fields. The Lor-entz force has zero component parallel to the induction field,and anisotropic states occur due to less restriction of fluidmotions parallel to the mean. Shebalin, Matthaeus, andMontgomery26 and Oughton, Priest, and Matthaeus27 inves-tigate the influence of a mean magnetic field in both 2-D and3-D homogeneous flow simulations. Their results indicatethat an externally applied dc field preferentially transfers en-ergy to modes with wave vectors perpendicular to the meanfield and also inhibits the development of turbulence. Theseobservations indicate that traditional ‘‘return to isotropy’’theories may not be applicable to the small scales of natu-rally occurring MHD turbulence, and that increased under-standing of anisotropic MHD turbulence is warranted.

In this article, results are presented of DNS of 3-D ho-mogeneous magnetohydrodynamic turbulence for both sta-tionary isotropic and homogeneous shear flow configura-tions. For simplicity, only incompressible nonhelical velocityfields and zero mean magnetic fields with magnetic Prandtlnumber equal to unity are considered. However, many im-portant features of MHD turbulence are captured, includingsmall scale dynamo action, equipartition of energies in thesmall scales, and effects of the Lorentz force on the velocityfield. All MHD simulations are repeated for the case of HDturbulence. The homogeneous shear flow is a relativelysimple configuration, and is convenient for the study of an-isotropic MHD turbulence in the absence of a mean magneticfield. The specific objectives of the study are to~1! assess

modifications to the turbulence due to the magnetic field,~2!quantify the dynamo effect in both isotropic and homoge-neous shear flows, and~3! investigate some issues of impor-tance in relation to coherency and intermittency in MHDturbulence.

II. FORMULATION AND PARAMETRIZATION

The incompressible form of the MHD equations are em-ployed to describe the turbulent transport in both isotropicand homogeneous shear ‘‘box’’ flows. The nondimensionalMHD equations for the fluctuation fields~zero mean! areconsidered in conservative form:

]uj]xj

50,]bj]xj

50, ~1!

]ui]t

1]

]xj~uiuj !52

]pt

]xi1

]

]xj~bibj !1

1

Re0

]2ui]xj]xj

1 f i ,

~2!

]bi]t

1]

]xj~biuj !5

]

]xj~uibj !1

1

Re0m

]2bi]xj]xj

. ~3!

Here ui ~i51,2,3! denotes the components of the velocityvectoru ~boldface indicates a vector!, bi represents the com-ponents of the magnetic induction vectorb, andpt denotesthe total pressure. The transport variables are normalizedwith respect to the reference length~L0!, velocity ~U0!, in-duction ~B0!, and density~r0! scales. The nondimensionaldiffusivity is the inverse of the ‘‘box Reynolds number,’’Re05U0L0/n, wheren is the kinematic viscosity. The boxmagnetic Reynolds number is defined asRe0m5U0L0/nm5Prm Re0, where the ratio of kinetic tomagnetic diffusivities is denoted by the magnetic Prandtlnumber, Prm5n/nm . In Eq. ~2!, f i is a forcing term to bediscussed below.

Simulations are conducted of both stationary isotropicand homogeneous shear turbulence within the domain0<xi<L52p ~x1[x, x2[y, x3[z!. A Fourier pseudospec-tral method with triply periodic boundary conditions is em-ployed for the spatial discretization of all transport variables.All calculations are performed in Fourier space with the ex-ception of the nonlinear terms, and time advancement is per-formed using an explicit second order accurate Adams–Bashforth technique. The computational routine is capable ofsimulating both stationary or decaying isotropic, and homo-geneous shear flows for either the HD~b[0! or the MHDequations. The computational requirement for the solution ofthe Navier–Stokes equations~on a Cray-C90 supercomputer!is approximately 0.18, 0.60, 1.25, and 4.80 s per iteration for323, 483, 643, and 963 collocation points, respectively. Forthe full MHD equations, 0.40, 1.35, 3.2, and 10.8 s per itera-tion are required for the same respective resolution.

To emulate the stationary isotropic turbulence field, alow wave-number forcing scheme is imposed. This is imple-mented by adding energy to the large scales of the turbulenceat a statistically constant rate, whereby an energy cascade isdeveloped for sufficiently large Reynolds numbers. The en-ergy is then dissipated at high wave numbers at the same rateand a statistically stationary state is achieved. Provided that

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there is sufficient separation between the low and high wavenumbers, the small scale turbulence is considered to be un-affected by the artificial forcing. The forcing algorithm em-ployed is based on a scheme developed by Eswaran andPope28 and is described in detail by Miller.20

In homogeneous shear turbulence simulation, a linearmean velocity profile is added to the fluctuating velocity.The primary effect of the mean shear is to provide a natural~albeit idealistic! homogeneous forcing. In contrast to theisotropic case, no stationary state is achieved and the Rey-nolds number grows until the turbulence structures outgrowthe box size. The magnitude of the imposed shear is givenwith the amplitude of the mean gradient, in this caseY5dU1/dx25const~whereU1 denotes the mean componentof the velocity in thei51 direction!. In order to use theFourier spectral method, the governing equations are solvedfor their fluctuating quantities on a grid which deforms withthe mean flow. The transformation is an extension of theprocedure employed by Rogallo29 for HD turbulence and de-tails of its implementation of MHD flows are provided byMiller.20 The mean shear imposed by the grid transforma-tions skews the grid in time. In order to allow the simulationto progress for a substantial time, it is necessary to remeshthe grid at regular intervals. Aliasing errors introduced by theremesh process are removed via a truncation of the variablesin Fourier space outside of the spherical wave-number shellof magnitude&N/3, whereN is the number of collocationpoints in any direction. This results in a slight loss of kineticand magnetic energy; however, if the simulation is well re-solved this truncation is considered to be negligible. Thesimulations are performed until the length scales of the tur-bulence become too large to be accurately resolved, at whichtime the simulation is terminated.

Both the velocity and magnetic fields are initialized asrandom Gaussian, isotropic, and solenoidal fields. Non-mirror-symmetricb fields with arbitrary magnetic helicitiesare generated by the method described in Pouquet andPatterson.14 The relative magnitude of the helicity is speci-fied through the correlation coefficient of theb field and itsvector potentiala ~b5“3a!: This coefficient is defined asC(a,b) 5 âibi&/Aâjaj&^bkbk&, where the angular brackets~^ &! indicate an ensemble average over all grid points. Theinitial energy for the velocity isEv5ûiui&53 in all cases.The forcing amplitude~AF51.25!, forcing radius~KF52&!,and the magnitude of the mean velocity gradient~Y52! arekept fixed for the respective flows. In the isotropic simula-tions, an initial kinetic energy spectrum havingEv8(k); k25/3 ~the prime indicates the energy spectra as a functionof the wave-number magnitude! is imposed. This choice issomewhat arbitrary since the asymptotic statistical state ofthe turbulence is independent of the initial conditions, and isprimarily dependent on the forcing parameters and the boxReynolds number. For the shear~SHR! flow simulationcases, the velocity field and all of the magnetic fields arespecified with a spectrum;k4 exp@22(k/ks)

2#, where theparameterks specifies the wave-number location for themaximum amplitude of the energy spectrum. The valueks57 is chosen for the initial velocity spectrum in all shearsimulations, whereasks is varied for the magnetic fields in

order to study its influence. Detailed studies of the effects ofboth the initial conditions and the rate of shear have beenconducted by Rogallo29 for HD flows.

A listing of all the simulation parameters are provided inTable I for the isotropic~ISO! flow simulations and in TableII for the SHR flow simulations. The labels in the first col-umn of each table are preceded by ‘‘I’’ and ‘‘S’’ for the ISOand the SHR flow simulations, respectively. The informationlisted for each simulation includes the box Reynolds number~Re0!, the ratio of initial magnetic and kinetic energies~Em/Ev , whereEm5^bibi&!, the wave number of maximalinitial magnetic energy (ks), the magnetic helicity, the gridresolution, the time step, the duration of simulation, the eddyturnover time~te!, and the resolution parameter~hkmax!. Theeddy turnover time is defined as the ratio of the integrallength scale to a characteristic velocity:

te5L

urms, L5

3p

2EvE0

kmax Ev8~k!

kdk, ~4!

where the subscript rms denotes the root mean square~urms iscalculated as the average of the rms of all three velocitycomponents! andL is the integral length scale. The resolu-tion parameter is the product of the Kolmogorov length scale@h5~n3/ê&!1/4, e52nsi j si j , where the symmetric rate ofstrain tensor issi j5(ui , j1uj ,i)/2 and the derivative notationis ui , j5]ui /]xj # and the maximum resolved wave number~kmax!. The box Reynolds number, the grid resolution, andthe time step are held constant for the SHR flow simulationsand take the values Re05200, N596, andDt52.531023.For all cases the magnetic Prandtl number is Prm51, inwhich caseh represents the smallest length scale of both theturbulence and the magnetic field. The values given for bothte and hkmax correspond to time averaged values for the

TABLE I. Conditions for the ISO simulations. All simulations are repeatedwith Em[0.

Re0 EM/EV ks C~a,b! N Dt~31023! tfinal te hkmax

IRun0 200 0.01 7 0 96 2.5 45 1.80 1.65IRun1 150 0.01 7 0 64 3.75 56 1.51 1.28IRun2 150 1.00 7 0 64 3.75 56 1.50 1.24IRun3 150 0.01 2 0 64 3.75 56 1.51 1.29IRun4 150 0.01 7 1 64 3.75 56 1.49 1.22IRun5 110 0.01 7 0 64 3.75 70 1.41 1.48IRun6 60 0.01 7 0 48 5.0 70 1.33 1.43IRun7 40 0.01 7 0 48 5.0 120 1.47 1.84IRun8 20 0.1 7 0 32 7.5 150 1.52 1.98IRun9 10 0.1 7 0 32 7.5 150 2.07 3.33

TABLE II. Conditions for the SHR simulations, Re05200, N596,Dt52.531023.

EM/EV ks C~a,b! tfinal te hkmax

SRun0 ••• ••• ••• 5 1.05 1.16SRun1 0.01 7 0 6 1.18 1.26SRun2 0.1 7 0 6 1.48 1.45SRun3 1.00 7 0 6 2.06 1.74SRun4 0.01 2 0 6 1.39 1.34SRun5 0.01 7 1 6 1.17 1.25

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stationary ISO runs in the presence of the magnetic field. Forthe SHR simulations, these parameters are given at the timewhen the relative energy of theu1 velocity componentreaches its maximal value~discussed below!. All of the ISOMHD cases corresponding to different Re0 values are re-peated with purely HD flow, and SRun0 represents a purelyHD case having the same parameters as cases SRun1–SRun5.

III. RESULTS

In each of the ISO simulations, the equations governingthe hydrodynamics are first integrated forward in time for aminimum of five eddy turnover times to reach a stationarystate. At this time, the magnetic field is added to begin thesimulation or, in the HD cases, the integration is continuedfor more than 10 eddy turnover times to calculate time aver-aged statistics. The MHD simulations are conducted for asufficient duration to determine the existence of stationarymagnetic states. Those simulations which attain stationarystates are continued for more than 10 additional eddy turn-over times so that time averaging can be performed on thedata. Time is normalized by the HD eddy turnover time(t/te) for the ISO flow simulations.

For the SHR flow cases, the mean shear is directly ap-plied to the initial isotropic fields. No stationary states existin this flow; however, energy component ratios~relative en-ergies! are known to attain approximate ‘‘asymptotic’’ val-ues as indicated in previous HD simulations~e.g., Rogersand Moin30!. This is observed to be the case in the MHDflow also. Figure 1 portrays an example of this effect for case

SRun1~time is normalized via the mean shear magnitude forthe SHR flow simulations!. The relative energy of a vectorcomponent is defined as the ratio of the variance of thatparticular component relative to the total vector energy, i.e.,^ui

2&/Ev and ^bi2&/Em . The time at which the ratiou1

2&/Eu

reaches a maximum value is denoted as the ‘‘peak time.’’Note that the relative energies of all components of both theu andb fields are initially about13, due to the initial isotropicconditions.

Comprehensive resolution studies have been conductedto assess the effects of grid resolution on both the time av-eraged ISO simulation statistics and on the temporal evolu-tion for the SHR flows. Two of the low Reynolds numbercases, IRun6 and IRun8, were repeated with 163, 243, 323,483, and 643 grid points and case IRun1 was repeated with323, 483, and 963 grid points. In addition, the SHR flow reso-lution was investigated by repeating case SRun1 with 483,643, and 1283 grid points. In all cases, the magnetic fieldmoments are more sensitive to resolution than the equivalentstatistics of either the velocity field or its gradients. Anevaluation of statistical moments as large as fourth orderreveals that the parameterhkmax is the primary parameter forthe resolution consideration. Results from the larger Rey-nolds number ISO and SHR cases indicate that adequateresolution is obtained forhkmax'1.2 and larger. This is inagreement with previous findings in the case of HD station-ary turbulence.28 However, for smaller ISO grid sizes~IRun6and IRun8! a larger resolution is necessary to yield accuratestatistics for the magnetic field;hkmax'1.4 and larger. Thisis a result of the relatively small separation between thesmallest scales of the turbulence and the forcing scales. Allof the results indicate that the simulations listed in the tablesare sufficiently resolved, and the results presented hereinafterare independent of the grid size.

A. Effects of initial conditions

For both the ISO and SHR simulations, a variety of ini-tial conditions are considered to study the effects of the ini-tial ratio of the magnetic and hydrodynamic energies, theinitial length scale of the magnetic field, and the initial mag-netic helicity on flow evolutions. This is particularly impor-tant in the ISO simulations, as it is important to establish thatstationary states are independent of initial conditions beforemaking quantitative comparisons. The SHR flow does notrelax to a steady state condition; therefore, it is expected thatthe effects of initial conditions will prevail throughout thesimulated times considered here.

Simulations IRun1 through IRun4 are considered to es-tablish the independence of ISO steady state results to initialconditions. Figure 2 presents the time evolution of both theenergy ratio (Em/Ev) and the magnetic enstrophy~Hm5^ j i j i&, where j5“3b is the electric current density!for a duration of approximately 37 eddy turnover times. Itappears that both quantities asymptotically approach thesteady state condition. Analysis of other quantities~notshown here! confirms that this is the case. However, the earlytemporal evolutions are highly dependent on the initialbfields. Figure 2 is also useful in assessing the effects of theinitial conditions at early times. Simulations IRun1 and

FIG. 1. Temporal evolution of the relative component energies for the ho-mogeneous shear simulation SRun1:~a! kinetic energy,~b! magnetic energy.


IRun3 contain initially relatively small scale~ks57! andlarge scale~ks52! magnetic fields, respectively, and other-wise have identical parameters@Em/Ev50.01 andC~a,b!50#. The early time behavior of both the energy ratio and themagnetic enstrophy exhibits substantially larger initialgrowth rates for the large scaleb field ~IRun3!. The expectedvalue of the magnetic dissipation is the primary parameteraffecting the early time evolutions of these fields. The rela-tively large energy at high wave numbers in simulationIRun1 results in a mean dissipation which is more than anorder of magnitude larger than that of IRun3 with an initiallylarge scale field ~the respective values areêm&5nm^bi , jbi , j&56.8131023 and 6.3831024!.

The early time effects of initial magnetic helicity arealso illustrated in Fig. 2 by the data of simulations IRun1 andIRun4. The temporal evolutions of both the energy ratio andthe magnetic helicity are nearly independent of the initialmagnetic helicity. This is due to the absence of any mecha-nism by which the magnetic helicity can be supported. Thecorrelation between theb field and its potential in simulationIRun4 decreases to less than 10% of its initial value in lessthan two eddy turnover times. Note that case IRun2 with aninitial unity energy ratio is observed to decay rapidly towardsthe same approximate stationary state as cases IRun1, IRun3,and IRun4. The long time fields are thus considered to beindependent of initial conditions and, of these runs, only thetime averaged data of simulation IRun1 are considered in theremainder of the article.

The above discussion of the early time effects of the

magnetic field initial conditions is also applicable to the SHRconfiguration. However, in these flows the hydrodynamicfield also experiences rapid modifications at early times dueto the action of the applied mean velocity gradient, and nostationary state exists. Due to the action of the Lorentz force,additional initial condition effects with regard to turbulencemodification are expected. As such, the effects of initial con-ditions are felt throughout the simulations.

B. Integral flow statistics

For the ISO cases, the statistical analyses of the data aremost often conducted in terms of both spatial and temporalaveraging~when stationary states exist!. Statistics gatheredfrom several simulations can be used to study the effects ofthe flow parameters. There are a number of choices by whichto parametrize the problem, including the box size, the inte-gral length scale, and the Taylor microscale Reynolds num-bers~Rel!, etc. Analysis of the data indicates that Rel yieldsthe most insight into the problem. For SHR flows Rel isbased on the Taylor microscale~l1! in the x1 direction~la

25ûa2&/^(]ua/]xa)

2&! ~no summation over repeatedGreek indices hereinafter! and the turbulent velocity scale(AEv). For ISO flows the Reynolds number is defined as theaverage over all three directionsxj with correspondinglength and velocity scaleslj andAûj

2&, respectively. Re-sults for the SHR runs are presented as both temporal evo-lutions and occasionally in the form of tabular data at thepeak times of the simulations.

Leorat, Pouquet, and Frisch10 find a bifurcation to amagnetic state in their eddy damped quasinormal Markovian~EDQNM! approximation study of nonhelical MHD turbu-lence above a critical magnetic Reynolds number of a fewtens. Figure 3 presents both the Reynolds number~in ISO!and temporal~in SHR! dependence of the kinetic, the mag-netic and the total (Et5Ev1Em) energies. The data in bothfigures are normalized by the kinetic energy of the purelyHD flow ~SRun0 for the SHR flow!. The ISO results are ingood qualitative agreement with the EDQNM predictions10

and show a transition to ‘‘magnetic states’’ occurring at acritical Reynolds number of Rel,c'30. Above this criticalvalue, sustainedb fields are observed, and both the kineticand total energies decrease with the Reynolds number. Thetransition is more gradual in the present results than thatpredicted by the EDQNM approximation. The SHR data cor-respond to case SRun4 having a large scale initialb fieldwith 1% energy ratio. In this simulation, the initial seed mag-netic field is rapidly amplified in the early stages of devel-opment and then gradually decreases. Note that no conclu-sions can be made from this figure in regard to a sustaineddynamo effect for the SHR simulation. This is because theenergies are normalized by the kinetic energy of the HDsimulation. However, these results do illustrate several ef-fects of theb field. In both the ISO and the SHR cases thepresence of the magnetic field acts to damp the turbulenceenergy~also observed in all the other SHR simulations!. Inaddition, the normalized kinetic energy for the SHR simula-tion reaches a nearly constant value afterYt'2.

Time averaged values of the steady state ISO energy

FIG. 2. Effect of initial conditions on the stationary isotropic simulations asshown by the temporal evolutions of~a! the energy ratio and~b! the mag-netic enstrophy.


ratios in MHD turbulence are listed in Table III as a functionof various Reynolds numbers~the normalization is with theMHD Ev!. The ratios for the higher Reynolds number cases~Em/Ev;0.1 for ReL'100! are in accord with the results ofMeneguzzi, Frisch, and Pouquet.16 When presented graphi-cally ~not shown!, the energy ratios portray a nearly lineardependence on the Taylor microscale Reynolds number forthe range of parameters considered. A very weak magneticfield is sustained for case IRun7 with Rel530.3 ~for morethan 100 HD eddy turnover times!, while the two simulationswith lower Reynolds numbers do not support a magneticfield. The magnetic energy in simulation IRun8 was allowedto decay to a value ofEm/Ev;10225 and no evidence of amagnetic steady state was observed. Based on these observa-tions, the critical value of the Reynolds number for smallscale dynamo action is concluded to be Rel,c'30 ~orReL,c'55!.

The transition to magnetic states is characterized by sev-eral changes in both the hydrodynamic and the electromag-netic structure of the flow field. In particular, the kinetic andmagnetic enstrophies provide a measure of the respectiverotational and electric current density energies of the velocityand the magnetic fields~the kinetic enstrophy is defined asHv5^v iv i&, v is the vorticity vector!. Figure 4 depicts thesequantities as a function of Rel for the steady state values ofthe ISO simulations. This figure indicates a significant dropin Hv in the presence of a sustained dynamo effect. The HDdata show a near linear rise inHv over the range of Reynoldsnumbers simulated. However, the MHD flow shows an ap-parent plateau value of the kinetic enstrophy for Reynoldsnumbers above the critical value. As the ratio of the mag-netic to kinetic energy grows, the Lorentz force becomesincreasingly significant to small scale flow dynamics. As willbe shown later, this force tends to align perpendicular to thevorticity vector and acts to disperse high vorticity regions ofthe turbulence, hence decreasing the kinetic enstrophy. Fig-ure 4 also shows that the magnetic enstrophy increasessteadily for Rel.Rel,c and eventually becomes larger thanHv . This trend is observed in spite of the fact that the largestenergy ratio observed in the ISO simulations is ‘‘only’’about 15%~see Table III!. The larger magnetic enstrophy incomparison to the kinetic enstrophy is due to a more inter-mittent behavior of the magnetic field in comparison to thevelocity field.

Significant insight into MHD small scale dynamics andintermittency is gained by the examination of the higher or-der statistics of the field variables and their derivatives.These statistics are important in characterizing the transitionto magnetic states. The skewness and flatness of a zero meanrandom variablevj are defined asm3(vj)5^vj

3&/^vj2&3/2 and

m4(vj)5^vj4&/^vj

2&2, respectively. In agreement with previ-ous HD and MHD studies,31,32 the velocity field is found tobe nearly Gaussian for all flow fields considered, while thestatistics of both the velocity and magnetic fields and theirlongitudinal derivatives are observed to be symmetricallydistributed about the mean~i.e., zero skewness!. The timeaveraged flatness values of the magnetic field are presentedin Fig. 5 for the ISO simulations~presented as the average ofthe three coordinate components!. Below the critical Rey-nolds number~Rel,c'30! the dynamo effect is not present,

FIG. 3. Development of the kinetic, magnetic, and total energies as normal-ized by the purely hydrodynamic kinetic energyEv ~HD!, ~a! ISO simula-tion time averaged results as a function of the Reynolds number for casesIRun0, 1, 5, 6, 7, 8, and 9~right to left! and~b! temporal evolution of SHRsimulation, case SRun4.

TABLE III. Time averaged values for the ratio of magnetic to kinetic en-ergy for the ISO flow simulations.

Re0 Rel ReL Em/Ev

IRun9 10 9.66 13.8 •••IRun8 20 18.4 31.2 •••IRun7 40 30.3 55.6 2.031024

IRun6 60 41.2 79.9 2.331022

IRun5 110 55.0 118 6.531022

IRun1 150 68.5 150 1.231021

IRun0 200 90.0 205 1.631021

FIG. 4. Reynolds number dependence of the kinetic and the magnetic en-strophy for the ISO simulations IRun0, 1, 5, 6, 7, 8, and 9~right to left!.


but the magnetic field maintains an approximate ‘‘self-similar’’ decay and time averaged flatness values are calcu-lated. In these low Reynolds number cases, the flatness fac-tors for theb field are observed to be considerably largerthan that of Gaussian. The flatness for theb field exhibits amaximum value at Rel'40 and then decreases. It is sug-gested that this can be explained in terms of the energy ratio.In the next section it is shown that the crossing point~equi-partition! of the kinetic and the magnetic energy spectramoves towards lower wave numbers as the Reynolds numberand the energy ratios are increased. The magnetic stretchingterm in the transport equation for theb field is the innerproduct of a large scale field~b! with a hydrodynamic dissi-pation scale gradient term. Therefore, as the relative strengthof theb field is increased and the corresponding equipartitionwave number migrates away from the dissipation scales, themagnetic field stretching term in Eq.~3! appears more‘‘space filling’’ ~less intermittent!. For Reynolds numbersbelow the critical value, the primary scales for interactionsbetween the magnetic field and the velocity field are the dis-sipation scales. In this case, the magnetic field intermittencytherefore increases with the magnetic field strength. Addi-tional larger Reynolds number simulations are needed to in-vestigate the fate of the magnetic flatness at large Reynoldsnumbers.

Some insight into the existence of a dynamo effect forSHR simulations is gained by examining the evolutions ofthe energy ratio and the magnetic energy~normalized by itsinitial value!, as shown in Fig. 6. Allb fields except the unityenergy ratio case~SRun3! experience an initial rise in energydue to the abrupt application of the mean velocity gradient attimeYt50. This causes strong stretching and bending effectson the initial random magnetic fields, and hence amplifica-tion of the magnetic energy. Simulation SRun3 has a suffi-ciently large energy to resist these effects and actually dampsthe u field to a large enough extent to display no initialmagnetic energy growth@Fig. 6~b!#. The sharpest relativeincrease in magnetic energy occurs for the large scale initialfield of SRun4 with nearly an order of magnitude increasewithin the first large scale turnover time. The positive growthrates of the magnetic energies observed in the latest stages ofthe simulations~except SRun3! in Fig. 6~b! suggest the pres-ence of a dynamo effect. The rates of growth of the magnetic

fields are slower than those of the velocity field, as evidencedby the monotonically decreasing energy ratios at later timesin Fig. 6~a!.

Some results portraying the anisotropy of the SHR floware presented in Table IV. This table lists various statistics ofboth the HD and MHD flows. Previous DNS data of Rogersand Moin30 ~RM! are also included for comparison. The cor-relation coefficient in the table is defined asC(va ,vb)5 ^vavb&/A^va

2&^vb2& for the vectorv. The addition of a

magnetic field delays the turbulence development and resultsin the increased peak times used for comparisons. The tablesuggests that an important parameter influencing the anisot-ropy and the Reynolds stress development is the instanta-neous energy ratio at the peak times. NotwithstandingSRun4, as the instantaneous energy ratio is increased, theprimary effects on the turbulence are to increase the anisot-ropy of both the velocity and the magnetic fields and also todecrease the magnitude of the Reynolds number. The caseSRun4 is initialized with a relatively large scale~‘‘force-free’’! magnetic field and is able to develop the largest valueof the Reynolds number despite having a moderate energyratio. The reason for the increase in anisotropy with the en-ergy ratio is that the Lorentz force exhibits preferred orien-tations in thex2-x3 plane. Therefore, this force causes therelatively largest resistance to fluid motions in directions per-pendicular to the streamwise direction, thus increasing thefluid anisotropy. This preferential orientation of the Lorentzforce is discussed in Sec. III D. Table IV also illustrates thatthe correlationC(u1 ,u2) decreases as the energy ratio is

FIG. 5. Reynolds number dependence of the flatness factor for the magneticfield for the ISO simulations IRun0, 1, 5, 6, 7, 8, and 9~right to left!.

FIG. 6. Temporal development of the energies for the SHR simulations:~a!the energy ratio, and~b! the magnetic energy normalized by the initial value.The legends are the same for both figures.


increased, while the correlationC(b1 ,b2) displays the oppo-site behavior. The production terms in the mean kinetic andmagnetic energy equations are related to these correlationsand are proportional tou1u2& and ^b1b2&, respectively.Hence, as the correlationC(u1 ,u2) is decreased, the relativeproduction of kinetic energy is also decreased and the turbu-lence grows at a slower rate. On the other hand, the relativeincrease of the magnetic Reynolds~Maxwell! stress compo-nent ^b1b2& results in an increased relative production ofmagnetic energy. Note also that the correlationsC(u1 ,u2)and C(b1 ,b2) have opposite signs. This is a result of theopposite sign of the source terms due to the mean velocitygradient in the transport equations of the fluctuatingu andbfields.

Based on the previous discussions, it may be expectedthat the magnetic field, by damping the turbulence energy,should yield a lower Reynolds number. This, however, is notnecessarily the case. Figure 7 illustrates this effect for boththe ISO and the SHR cases. The ISO data show a transitionat the critical box Reynolds number into distinct HD andMHD states. Above the critical value, a larger Rel is ob-served for the magnetic turbulence. The difference betweenthese states increases as Re0 increases. A similar trend isobserved in the SHR flows. The Reynolds number is in-creased over its HD values~at the sameYt values! for all ofthe three simulations which are initialized with a 1% energyratio ~SRun1, SRun4, and SRun5!. The obvious explanationfor these trends is that the magnetic field, while decreasingthe kinetic energy, also causes an increase in the lengthscales of the turbulence. However, if the initial magneticfield strength is large~SRun2 and SRun3!, the kinetic energycan be damped sufficiently to result in a net decrease of theReynolds number. The results for cases SRun5 and SRun1again show similar trends; in the discussions below, caseSRun5 is no longer considered.

The variation of the Taylor length scale is presented inFig. 8 for both the ISO and the SHR flows. The Taylor scaleis presented as the average over its value along all threedirections. This average value is used in the SHR results as arepresentative length scale. In both flows, the length scale isobserved to increase in the presence of a magnetic field. Asimilar behavior is observed for both the Kolmogorov andthe integral length scales~not shown!. In nonhelical MHDturbulence, a steepening of the kinetic energy spectrum in

the inertial range has been observed in EDQNMpredictions.10 The integral length scales would be increasedby this steepening. However, the enlargement of the Taylorscale is best explained in a structural sense as discussed be-low. The Taylor length scale in the MHD flows remain con-sistently larger than the HD case for the SHR flows exceptfor the case with a 100% initial energy ratio~SRun3!. In thiscase, the magnitude of the Lorentz force is of the same orderas the pressure gradient. The initial magnetic field is uncor-related with the velocity field, so that the Lorentz force actsas though it is a random forcing term applied as a step func-tion at timeYt50. This random force has the effect of ini-tially increasing the relative amount of high wave-numberenergy, thus decreasing the length scales at early times.

TABLE IV. Comparisons of instantaneous parameters for the SHR simulations at the peak times.

RM SRun0 SRun1 SRun2 SRun3 SRun4 SRun5

Ytpeak 8.0 8.0 8.0 10.0 12.0 10.0 8.0Rel 72.6 101 112 107 88.0 132 110û1

2&/Ev 0.53 0.50 0.53 0.59 0.65 0.58 0.54û2

2&/Ev 0.16 0.20 0.18 0.16 0.11 0.17 0.18û3

2&/Ev 0.31 0.30 0.29 0.25 0.24 0.25 0.28C(u1 ,u2) 20.57 20.54 20.56 20.51 20.36 20.55 20.55Em/Ev ••• ••• 0.031 0.128 0.747 0.089 0.030^b1

2&/Em ••• ••• 0.48 0.65 0.84 0.58 0.47^b2

2&/Em ••• ••• 0.23 0.15 0.06 0.23 0.23^b3

2&/Em ••• ••• 0.29 0.20 0.10 0.19 0.30C(b1 ,b2) ••• ••• 0.21 0.39 0.47 0.30 0.22

FIG. 7. Development of the Reynolds number based on the Taylor micro-scale~a! vs the box Reynolds number for the ISO simulations IRun0, 1, 5, 6,7, 8, and 9~right to left!, ~b! vs time for the SHR simulations.


C. Energy spectra

Of primary concern in this section are the modificationsof the HD spectra as caused by the magnetic field, and therelationship of the kinetic and magnetic spectra includingequipartition. Equipartition refers to the tendency of MHDturbulence to develop nearly equal values for both the kineticand magnetic energy at high wave numbers. This occurs dueto the small scale exchange of energy between the two fieldsthrough the action of Alfve´n waves. Figure 9 presents thenormalized kinetic energy spectra for both the ISO and theSHR flows. The time averaged ISO spectra are taken fromthe high Reynolds number case IRun0. The SHR spectra arepresented for cases SRun0–SRun3. These spectra are actu-ally functions of the wave-number vector component for theanisotropic flow fields. However, in order to simplify theanalysis, they are presented versus the wave-number magni-tude. The SHR energy spectra are given for instantaneousflow conditions corresponding to the peak times. The pri-mary effects of the magnetic field on the energy spectrum forthe ISO data are to steepen the slope in the moderate wave-number region and to increase the relative energy in the highwave numbers@Fig. 9~a!#. There is no distinct inertial rangeat these Reynolds numbers. However, the steepening is inagreement with the EDQNM predictions.10 The Lorentzforce creates a relative increase in the small scale energy ofthe turbulence. Of course, this is only a relative effect as thetotal energy is decreased. The SHR flows display this in-crease most significantly for case SRun3 which displays thelargest instantaneous energy ratio@Fig. 9~b!#. The dissipationspectra~not shown! for both flows display a mild flattening

in the moderate wave-number regions. The wave-number lo-cation of the maximum dissipation does not appear tochange. The broadening effect implies that the transfer of thekinetic energy to the magnetic energy occurs over a rela-tively large range of scales and is centered around the scalecorresponding to maximal viscous dissipation. The maximalmagnetic and kinetic dissipation scales are approximately thesame due to the unit magnetic Prandtl number consideredhere.

Examination of the equipartition effect is facilitated bythe normalized kinetic and magnetic spectra in Fig. 10. Bothenergies are normalized by the kinetic energy to retain therelative crossing points. These results represent both rela-tively large and smallEm/Ev ratios for both flows. Parts~a!and ~b! of Fig. 10 depict results of IRun0 and IRun6 withtime averaged energy ratios of 0.16 and 0.023, respectively.The SHR spectra are shown in Figs. 10~c! and 10~d! for theinstantaneous peak time data of simulations SRun2 andSRun1 with respective energy ratios of 0.128 and 0.031. Theforms of both the kinetic and magnetic spectra remain ap-proximately self-similar. However, the equipartition wavenumber is observed to migrate towards smaller wave num-bers for larger energy ratios~i.e., from case IRun6 to IRun0and from case SRun1 to SRun2!. This migration also resultsin a magnetic field with a larger length scale. This wasbriefly discussed above in relation to the decreasing trendsobserved for the magnetic field intermittency with increasingenergy ratios for Rel.Rel,c . Turbulence fields with largerlength scales generally portray less intermittent effects. For

FIG. 8. Development of the Taylor length scale~a! for the ISO simulationsIRun0, 1, 5, 6, 7, 8, and 9~right to left! and ~b! for the SHR simulations.

FIG. 9. Normalized energy spectra:~a! kinetic energy spectra for IRun0,~b!kinetic energy spectra for SHR flows. The ISO results are time averaged andthe SHR results are for the peak times.


example, passive scalar fields generally display asymptoticGaussian statistics, but derivatives of the scalar are charac-terized by smaller length scales and portray significant de-partures from Gaussian.20,33 It is unknown whether the mag-netic field intermittency will continue to decrease at largerReynolds numbers~Fig. 5!, and future simulations and/orexperiments are needed to address this issue.

Another feature pertaining to energy spectra is found forISO flows near the critical Reynolds number. The presentresults suggest that sustained dynamo action can occur in theabsence of equipartition of high wave-number kinetic andmagnetic energies. This is illustrated in Fig. 11 which depictsthe normalized kinetic and magnetic energy spectra for thecase IRun7. This is the lowest Reynolds number ISO simu-

lation for which a sustained dynamo action is observed. No-tice that there is no equipartition of energy in this case; themagnetic energy spectrum appears to be ‘‘floating’’ nearlytwo decades beneath the kinetic spectrum. Any exchange ofenergy must be occurring through only very weak Alfve´nwaves. The simulation was conducted for more than 100 HDeddy turnover times and confirmed that theb field is sus-tained. It may be possible for the equipartition to occur atvery low energy values~at very high wave numbers!. Tocapture the crossing point of the two spectra would requiresimulations with much higher resolution.

D. Structure and organization

This section is devoted to physical descriptions of coher-ency in 3-D nonhelical MHD turbulence. For this purpose,instantaneous fields from each of the cases IRun0 and SRun2are considered for detailed investigation. The results are con-sidered at timet/te525 for the ISO data and at timeYt510for the SHR data. Other simulated results from each configu-ration are confirmed to be in accord with the results pre-sented for these two cases. Figure 12 illustrates the smallscale structure of the ISO turbulence field. In Fig. 12~a! thevorticity field is highlighted with vorticity vectors in only theregions where the local vorticity magnitude is within 10% ofits maximum value. As in the HD flow field, the high ampli-tude vorticity regions tend to organize into tube structureshaving thickness of the order of the Kolmogorov scale andlength of the order of the integral scale. Figure 12~b! depictsmagnetic field vectors in only the regions where the localmagnetic field amplitude is within 10% of its maximum

FIG. 10. Kinetic and magnetic energy spectra normalized by the mean turbulence energy:~a! IRun0, ~b! IRun6, ~c! SRun2, and~d! SRun1. The ISO resultsare time averaged and the SHR results are for the peak times.

FIG. 11. Kinetic and magnetic energy spectra normalized by the mean tur-bulence energy for case IRun7.


value. The magnetic field also displays evidence of ‘‘tube-like’’ structures, hereinafter referred to as magnetic fluxtubes. In contrast to the vorticity field, the magnetic field alsoshows a substantial tendency to align in ‘‘sheet-like’’ struc-tures@the high amplitude induction regions at the upper rightcorner of Fig. 12~b! are confirmed with flow visualization asbeing sections of magnetic sheets#. Flow visualization showsthat the regions of maximal current density are related moreto the sheet structures than to the flux tubes. Both the vortic-ity and the induction fields are observed to be strongly inter-mittent. That is, the fields are concentrated in only a smallfraction of the volume in physical space. No obvious corre-lation is observed between the two types of structures.

The vortical structure of homogeneous shear flows isknown to be dominated by vortex tubes~aligned with theprincipal axis of the flow! which fold into the shape of a‘‘horseshoe.’’ These structures are referred to as ‘‘hairpin’’vortices.30 Figure 13 shows the interaction of a typical hair-pin vortex with a magnetic flux tube in the SHR flow. Thestructures are depicted with lines which are everywhere tan-gent to the vector of interest and which are concentrated only

in regions of relatively high amplitude vorticity or magneticfield, respectively. Flow visualizations provide no evidenceof magnetic hairpins. The magnetic flux tubes in the SHRflow show a tendency to align along directions close to theprincipal axis~45° in thex1-x2 plane!. Both the vorticity andthe magnetic tube structures are of approximately the samelength and width as those observed in the ISO flow.

The preferential alignment of the vorticity and the mag-netic field vectors in the SHR flow is quantified with theinclination angle @u5tan21(v2/v1) for any vector v i# ofthese vectors in thex1-x2 plane. Figure 14 shows the distri-bution of this angle~the PDF ofu multiplied by 360°! for thevorticity field and also for several of the electromagneticfields. The angleu50 indicates the direction of positivex1~streamwise!, with positive angles representing counter-clockwise rotation about thex3 axis. The HD vorticity resultsare in agreement with the previous analysis of Rogers andMoin.30 That is, there is peak alignment of the vorticity withthe principal axis~45°! with equal relative weights for bothpositive and negative vorticity. The effect of the magneticfield is to skew the vorticity alignment towards thex1 axis.This is explained by the alignment of the Lorentz forcewhich acts perpendicularly to theb field. The magnetic fieldand the Lorentz force show preferred angles approximatelyequal to 20° and280°, respectively. The net effect is toforce the vorticity vectors towards thex1 axis.

The electromagnetic structure of several flux tubes wasinvestigated via flow visualization. In general, these tubeswere observed to have width on the order of the Kolmogorovscale and length on the order of the integral scale. The flux

FIG. 12. Illustration of coherent structures in case IRun0 at timet/te525 asdepicted by vectors located in high amplitude regions:~a! vorticity vectorsand ~b! magnetic induction vectors.

FIG. 13. Interaction of a vorticity hairpin vortex and a magnetic flux tubefor case SRun1 at timeYt58: ~a! spanwise view, and~b! streamwise view.


tubes tend to repulse the local fluid by exerting a relativelystrong Lorentz force on the fluid element radially away fromthemselves. This offers a possible explanation for the resultsof Fig. 4 which displays a large reduction in kinetic enstro-phy for MHD turbulence, and may also explain the increasein turbulence length scales. Flux tubes forming in or nearregions of large velocity gradients can act to ‘‘break up’’these regions through the repulsive Lorentz action. Thiswould have the effect of increasing the Kolmogorov scaleand also decreasing the total kinetic enstrophy.

A more elaborate description of the small scale turbu-lence dynamics is achieved with the examination of theeigenvectors of the symmetric rate of strain tensor. Follow-ing the traditional notation, the eigenvalues are labeled asa>b>g, and the strain magnitude is denoted byS. In in-compressible flows,a1b1g50, a is extensive,g is com-pressive, andb has been shown to be typically extensive inHD turbulence ~i.e., positive ^b& values!.34,35 The corre-sponding eigenvectors are denoted byea , eb , andeg , respec-tively. The typically extensive nature of the second eigen-value is confirmed by the present MHD results. Table V liststhe mean, the skewness, and the flatness forb/brms. Thevalues correspond to instantaneous data of ISO simulationIRun0, and to the peak time data of SHR simulations SRun0and SRun2. The middle eigenvalue of the magnetic fieldstrain tensor [si j

(m)5(bi , j1bj ,i)/2] is also included, and isdenoted bybm . The mean value forb remains positive; how-ever, the magnitude is reduced for both of the MHD flows.This indicates a slight change in the small scale hydrody-namic structure of the MHD turbulence. In contrast to the

hydrodynamic field,bm remains symmetrically distributedwith near zero mean in both flows, and hence shows notendency for either compressive or extensive character. Thismay be related to the lack of skewness for the longitudinalderivatives of the magnetic induction mentioned in Sec.III B, and also provides insight into the existence of bothmagnetic tube and sheet structures. Magnetic tube structuresdisplay two compressive magnetic strain eigenvectors~bm,0!, whereas magnetic sheet structures are characterizedby two extensive eigenvectors~bm.0!.

Hereinafter, the analyses are limited to the IRun0 results,as the SHR results were observed to show similar trends. ThePDF of the alignment of the vorticity vector with the elec-tromagnetic fields is presented in Fig. 15. The magnetic fieldshows a strong tendency to align somewhat parallel to thevorticity vector. The vortex/flux tube interaction in Fig. 13illustrates this effect, as the tails of the hairpin vortex are inthe vicinity of the flux tube, where they run parallel to eachother. The electric current running helically around the fluxtubes would therefore be expected to show a preferred per-pendicular alignment with the vorticity vector. However,Fig. 15 shows that no significant preferred alignment existsfor these vectors. This can be explained in terms of the lo-cation of the strongest current magnitude within the flowfield. As noted above, the strongest current regions are asso-ciated with the magnetic sheet structures which are corre-lated with the high strain regions of the flow, not the vortic-ity regions. This effect can be quantified by examining thestatistics of the electromagnetic field conditioned on the sec-ond invariant of the deformation tensor:

FIG. 14. Distribution of the inclination angle of the projection of vectorfields in thex1-x2 plane for case SRun1 at timeYt58: ~a! vorticity vector,~b! electromagnetic vectors.

TABLE V. Statistics of the middle eigenvalue of the rate of strain tensors ofboth the velocity field~b! and the magnetic field~bm!, normalized by theirrespective standard deviations. IHD and SHD correspond, respectively, toHD results of IRun0 and SRun0 cases. IMHD and SMHD correspond, re-spectively, to MHD results of IRun0 and SRun2 cases.

b~IHD! b~SHD! b~IMHD ! b~SMHD! bm~ISO! bm~SHR!

Mean 0.750 0.547 0.685 0.443 0.013 0.001m3 1.17 0.838 1.31 1.41 0.030 0.046m4 4.34 4.60 4.12 5.78 4.77 4.52

FIG. 15. PDF of the angle of alignment between the vorticity vector and theelectromagnetic vectors. Case IRun0 at timet/te525.


II v521

2

]ui]xj

]uj]xi

521

2 SS221

4v iv i D .

Therefore, the positive and negative values ofII v correspondto regions of high vorticity and high strain, respectively. Fig-ure 16 presents the expectations of the squared magnitudesof the magnetic field, electric current, and the Lorentz force~j3b!, conditioned on the magnitude of the second invariant.The squared vector magnitudes are normalized by their meanvalues, and the second invariant is normalized by its standarddeviation. Notice that there is a small tendency for the mag-netic high amplitude regions to be located in regions of mod-erate to strong vorticity. However, these may only be rela-tively strong rotational fluid regions outside of the vortextubes, as the flow visualization here~and that by Meneguzzi,Frisch, and Pouquet16! show no indication that these struc-tures are correlated. Of more importance is the stronger cor-relation of the high amplitude current regions with the highstrain regions of the flow. The current shows only moderateamplitudes in high vorticity regions, and hence no significantpreferred alignment with the vorticity vector. The maximalregions of the Lorentz force are observed to be correlatedwith both high amplitude vorticity and high amplitude strainregions. Flow visualization also shows that regions of strongLorentz force are associated with both the magnetic sheetstructures~large strain! and the magnetic flux tubes~largevorticity!.

The effects of the magnetic field on the hydrodynamicscan be illustrated with the PDF of the alignment angle be-tween the Lorentz force and various hydrodynamic vectorsas presented in Fig. 17. In part~a! of the figure the entirefield is considered, but in part~b! the angles are conditionedon only regions of the flow having a positive second invari-ant greater than three times its standard deviation. In thismanner, the regions of predominantly vorticity tube struc-tures are isolated. Note the change in the heights for theprobabilities in the high vorticity regions. In agreement withthe previously described parallel alignment of theb andvi

vectors, the Lorentz force is typically perpendicular to thevorticity and parallel to the kinetic pressure gradient. Whenconditioned on the high vorticity regions, the probability for

these alignments becomes largely amplified. Also note thatthe PDF of the pressure gradient alignment in Fig. 17~b! isskewed towards antiparallel alignment. A phenomenologicaldescription of the vorticity tube structures can be made inwhich the low pressure vortical tubes are being broken up bythe outward radial Lorentz force acting against the naturalpressure gradient.

IV. SUMMARY AND CONCLUSIONS

Direct numerical simulations are conducted to study thesmall scale structure and dynamics of homogeneous nonhe-lical magnetohydrodynamic turbulence with unity magneticPrandtl number. Both stationary isotropic and homogeneousshear flows are considered in both the presence and the ab-sence of the magnetic field. The effects of the initial mag-netic field conditions are examined and found to be some-what insignificant for the stationary states of the isotropicflow. However, the homogeneous shear flow is significantlydependent on the initial conditions over the entire duration ofthe simulations. In particular, the initial length scale of themagnetic field is observed to influence the early time ampli-fication of the field. This is due to a larger mean magneticdissipation for initially small scale magnetic fields, and mayresult in an order of magnitude change in the ratio of mag-netic to kinetic energy over the entire duration of the simu-lations.

Small scale dynamo action is observed in both flows. Forthe isotropic flow a critical Reynolds number of Rel'30 is

FIG. 16. Expected value of the squared magnitudes of the magnetic field,electric current, and the Lorentz force conditioned on the second invariant ofthe deformation tensor normalized by its standard deviation for simulationIRun0 at timet/te525. The squared magnitudes are normalized by theirexpected values.

FIG. 17. PDF of the angle of alignment between the Lorentz force and thevelocity, vorticity and negative of the kinetic pressure gradient:~a! the entireflow volume,~b! conditioned on regions where the second invariant is posi-tive and larger than three times its standard deviation. Case IRun0 at timet/te525.


found above which a stationary state is reached with a sus-tained magnetic field. For the range of parameters consid-ered, the time averaged ratio of magnetic to kinetic energiesscales nearly linearly with the Reynolds number. The homo-geneous shear flow does not attain stationary states. How-ever, monotonically increasing magnetic energies at latetimes indicate the presence of a dynamo effect. Although themagnetic energy is increasing, its ratio with the kinetic en-ergy shows a decreasing trend. This is due to a larger timescale for the magnetic field response, as compared with thegrowth rate of the turbulence.

The simulation results reveal several modifications to theturbulence structure due to the presence of the magneticfield. In particular, in both flows the turbulence kinetic en-ergy is decreased by the Lorentz force. However, except forvery high amplitude initial magnetic fields in the shear flowsimulations, the Taylor microscale Reynolds number is in-creased by the magnetic field. This occurs due to the enlarge-ment of the hydrodynamic length scales. Also due to themagnetic field, the turbulence intermittency and the kineticenstrophy are both observed to decrease, but the anisotropyis increased in shear flows. The induction is itself highlyanisotropic in shear flows, and displays an off-diagonal mag-netic Reynolds stress of opposite sign to the correspondingkinetic Reynolds stress.

Although the Reynolds numbers considered in thepresent simulations are not sufficiently large to produce dis-cernible inertial ranges, many interesting features of the ki-netic and magnetic energy spectra are portrayed. The kineticenergy spectra are shown to display increased relative highwave-number energy when a magnetic field is present. Thecrossing point of the normalized kinetic and magnetic energyspectra is observed to migrate towards the low wave num-bers as the ratio of the magnetic energy to the kinetic energyis increased. Additionally, the wave number of the maximumkinetic energy increases with the increase of energy ratio.

The magnetic field is found to organize into both sheet-and tube-like structures. The regions of large magnetic fieldmagnitude show a small correlation with the regions of mod-erate to large vorticity magnitude, and electric current struc-tures are found to be correlated with the regions of largestrain amplitude. The Lorentz force shows a strong tendencyto align antiparallel with the pressure gradient force in theregions of strong vorticity. This has the effect of ‘‘breakingup’’ the turbulence structures and results in a decreased ki-netic enstrophy. This structural description provides a physi-cal interpretation of some of the above-mentioned phenom-ena.

ACKNOWLEDGMENTS

We are indebted to Dr. John Shebalin for many usefulcomments.

This work is part of an effort sponsored by the NationalScience Foundation under Grant No. CTS-9253488 and bythe U.S. Office of Naval Research under Grant No. N00014-94-1-0667. Computational resources are provided by theSchool of Engineering and Applied Science Computing Cen-ter at SUNY—Buffalo and by the National Center for Super-computing Applications at the University of Illinois at Ur-bana.

1A. N. Kolmogorov, Dokl. Akad. Nauk SSSR,30 ~11!, 301 ~1941!.2D. Biskamp,Nonlinear Magnetohydrodynamics~Cambridge UniversityPress, New York, 1993!.

3G. K. Batchelor, Proc. R. Soc. London201, 405 ~1950!.4J. H. Thomas, Phys. Fluids11, 1245~1968!.5H. K. Moffatt, J. Fluid Mech.53, 385 ~1972!.6A. Pouquet, U. Frisch, and J. Leorat, J. Fluid Mech.77, 321 ~1976!.7D. Montgomery and T. Hatori, Plasma Phys. Controlled Fusion26, 717~1984!.

8W. H. Matthaeus, M. L. Goldstein, and S. R. Lantz, Phys. Fluids29, 1504~1986!.

9H. Chen and D. Montgomery, Plasma Phys. Controlled Fusion29, 205~1987!.

10J. Leorat, A. Pouquet, and U. Frisch, J. Fluid Mech.104, 419 ~1981!.11P. H. Roberts and T. H. Jensen, Phys Fluids B5, 2657~1993!.12P. Givi, Prog. Energy Combust. Sci.15, 1 ~1989!.13P. Givi, in Turbulent Reacting Flows, edited by P. A. Libby and F. A.Williams ~Academic, London, 1994!, Chap. 8, pp. 475–572.

14A. Pouquet and G. S. Patterson, J. Fluid Mech.85, 305 ~1978!.15P. A. Gillman and J. Miller, Astrophys. J. Suppl.46, 211 ~1981!.16M. Meneguzzi, U. Frisch, and A. Pouquet, Phys. Rev. Lett.47, 1060

~1981!.17M. Meneguzzi and A. Pouquet, J. Fluid Mech.205, 297 ~1989!.18D. Biskamp, H. Welter, and M. Walter, Phys. Fluids B2, 3024~1990!.19U. Frisch, P. L. Sulem, and M. Nelkin, J. Fluid Mech.87, 719 ~1978!.20R. S. Miller, Ph.D. thesis, State University of New York at Buffalo, NY,1995.

21A. Pouquet, P. L. Sulem, and M. Meneguzzi, Phys. Fluids31, 2635~1988!.

22A. Orszag and C. M. Tang, J. Fluid Mech.90, 120 ~1979!.23S. Kida, S. Yanase, and J. Mizushima, Phys. Fluids A3, 457 ~1991!.24A. Nordlund, A. Brandenburg, R. L. Jennings, M. Rieutord, J.Ruokolainen, R. F. Stein, and I. Tuominen, Astrophys. J.392, 647~1992!.

25A. Brandenburg, I. Procaccia, and D. Segel, Phys. Plasmas2, 1148~1995!.26J. V. Shebalin, W. H. Matthaeus, and D. Montgomery, J. Plasma Phys.29,525 ~1983!.

27S. Oughton, E. R. Priest, and W. H. Matthaeus, J. Fluid Mech.280, 95~1994!.

28V. Eswaran, and S. B. Pope, Comput. Fluids16, 257 ~1988!.29R. S. Rogallo,NASA Technical Memorandum 81315~NASA Ames Re-search Center, Moffat Field, CA, 1981!.

30M. M. Rogers, and P. Moin, J. Fluid Mech.176, 33 ~1987!.31Z. S. She, E. Jackson, and S. A. Orszag, Proc. R. Soc. London Ser. A434,101 ~1991!.

32D. Biskamp and U. Bremer, Phys. Rev. Lett.72, 3819~1993!.33F. Jaberi, R. Miller, C. Madnia, and P. Givi, J. Fluid Mech.313, 241

~1996!.34R. M. Kerr, Phys. Rev. Lett.59, 783 ~1987!.35W. T. Ashurst, A. R. Kerstein, R. M. Kerr, and C. H. Gibson, Phys. Fluids30, 2343~1987!.


structure of homogeneous nonhelical magnetohydrodynamic turbulence

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