We present a numerical analysis of an incompressible decaying
magnetohydrodynamic turbulence run on a grid of 1536^3 points. The Taylor
Reynolds number at the maximum of dissipation is ~1100, and the initial
condition is a superposition of large scale ABC flows and random noise at small
scales, with no uniform magnetic field. The initial kinetic and magnetic
energies are equal, with negligible correlation. The resulting energy spectrum
is a combination of two components, each moderately resolved. Isotropy obtains
in the large scales, with a spectral law compatible with the
Iroshnikov-Kraichnan theory stemming from the weakening of nonlinear
interactions due to Alfven waves; scaling of structure functions confirms the
non-Kolmogorovian nature of the flow in this range. At small scales, weak
turbulence emerges with a k_{\perp}^{-2} spectrum, the perpendicular direction
referring to the local quasi-uniform magnetic field.Comment: 4 pages, 4 figure

A. Pouquet

J. V. Shebalin

P. D. Mininni

P. S. Iroshnikov

English

arXiv

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Energy Spectra Stemming from Interactions of Alfvén Waves and Turbulent Eddies

We present a numerical analysis of an incompressible decaying magnetohydrodynamic turbulence run on a grid of 15363 points. The Taylor Reynolds number at the maximum of dissipation is 1100, and the initial condition is a superposition of large-scale Arn'old-Beltrami-Childress flows and random noise at small scales, with no uniform magnetic field. The initial kinetic and magnetic energies are equal, with negligible correlation. The resulting energy spectrum is a combination of two components, each moderately resolved. Isotropy obtains in the large scales, with a spectral law compatible with the Iroshnikov-Kraichnan theory stemming from the weakening of nonlinear interactions due to Alfvén waves; scaling of structure functions confirms the non-Kolmogorovian nature of the flow in this range. At small scales, weak turbulence emerges with a k-2 spectrum, the perpendicular direction referring to the local quasiuniform magnetic field. © 2007 The American Physical Society.Fil: Mininni, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Pouquet, A.. National Center for Atmospheric Research; Estados Unido

Mininni, Pablo Daniel

Pouquet, A.

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Energy Spectra Stemming from Interactions of Alfvén Waves and Turbulent EddiesP. D. Mininni1,2 and A. Pouquet21Departamento de Fı́sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria, 1428 Buenos Aires, Argentina2NCAR, P.O. Box 3000, Boulder, Colorado 80307-3000, USA(Received 24 July 2007; published 21 December 2007)We present a numerical analysis of an incompressible decaying magnetohydrodynamic turbulence runon a grid of 15363 points. The Taylor Reynolds number at the maximum of dissipation is 1100, and theinitial condition is a superposition of large-scale Arn’old-Beltrami-Childress flows and random noise atsmall scales, with no uniform magnetic field. The initial kinetic and magnetic energies are equal, with negli-gible correlation. The resulting energy spectrum is a combination of two components, each moderately re-solved. Isotropy obtains in the large scales, with a spectral law compatible with the Iroshnikov-Kraichnantheory stemming from theweakening of nonlinear interactions due to Alfvénwaves; scaling of structure func-tions confirms the non-Kolmogorovian nature of the flow in this range. At small scales, weak turbulenceemerges with a k2? spectrum, the perpendicular direction referring to the local quasiuniform magnetic field.DOI: 10.1103/PhysRevLett.99.254502 PACS numbers: 47.65.d, 47.27.Jv, 94.05.Lk, 95.30.QdMagnetic fields permeate the Universe, and with theincreased resolving capabilities of instruments, a flurry ofdetails on highly complex flows emerge. The origin of suchmagnetic fields, the dynamo problem, is still not fullyunderstood, in particular, when the magnetic Prandtl num-ber PM—the ratio of kinematic viscosity  to magneticdiffusivity—differs substantially from unity, as it does inthe interstellar medium (PM  1) or in convective regionsof stars and liquid cores of planets (PM  1). Numeroustheoretical and numerical studies have been written (see[1] ), and recent laboratory experiments address this fun-damental problem as well. Here, we take a different ap-proach by assuming that a dynamo mechanism works,presumably leading to approximate equipartition of themagnetic and kinetic energies, as observed, for example,in the solar wind. We then ask questions about the nature ofthe nonlinear dynamics of such a flow. The completeproblem (a dynamo taken all the way to the nonlinear stageat a high Reynolds number as encountered in nature, andfor PM differing substantially from unity) is still out ofreach, but it might be among the key outcomes of petascalecomputing facilities and experimental devices currentlybeing developed. We specifically address in this Letterthe nature of the energy spectra that arise in such condi-tions. A more detailed account of the development andevolution (up to the peak of enstrophy) of structures andcorrelations of the flow will be presented elsewhere.The magnetohydrodynamic (MHD) equations read: @v@t v  rv  10rP  j b r2v; (1) @b@t r 	v b  r2b; (2)v is the velocity, b is the magnetic field, j  r b is thecurrent density, P is the pressure, 0  1 the density,    2 104, and r  v  r  b  0. One can also writethese equations in terms of the Elsässer variables z v b, of energy E and flux . We solve Eqs. (1) and (2)in a three-dimensional box of length L0  2 using peri-odic boundary conditions and a pseudospectral method,dealiased by the standard 2=3 rule; minimum and maxi-mum wave numbers are kmin  1 and kmax  N1=3=3, withN  15363 grid points. At all times kD=kmax < 1, wherekD is the dissipation wave number. The initial conditionsfor the velocity and magnetic fields are constructed from asuperposition of three Beltrami (helical) flows to whichsmaller-scale random fluctuations are added with initialkinetic and magnetic energy EV  EM  0:5 [with respec-tive spectra EV	k and EM	k]; magnetic helicity HM ha  bi  0:45 (b  r a, where a is the vector potential,and the brackets denote volume average), and cos	v;b hv  bihjvjjbji1  104 (see [2] for details). The compu-tation is stopped when the growth of the total dissipationsaturates (t  3:7), at which time the Reynolds numberbased on the mechanical integral scale is Re  ULV= 9200, and that based on the mechanical Taylor scale isR  UV=  1100; U is the rms velocity, the integralscales are defined as Li  2E1iRk1Ei	kdk, the Taylorscales are i  2Ei=Rk2Ei	kdk1=2, and i is either V orM. These scales at t  3:7 are LV  2:6, V  0:31,LM  3:1, and M  0:39. It was shown in [2] that at earlytimes, current and vorticity sheets form, and further roll up,fold, or pile up. Here we focus on the fully developedregime close to the peak of enstrophy.The total energy flux 	k as a function of wave numberat t  3:7 is shown in Fig. 1 (inset), together with the total,kinetic, and magnetic energy spectra [E	k  EV	k EM	k], compensated by either k5=3 (Kolmogorov spec-trum, hereafter K41) or k3=2 (Iroshnikov-Kraichnan, or IK[3] ); the latter takes into account the slowing down ofnonlinear transfer because of Alfvén waves. The scalingPRL 99, 254502 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 DECEMBER 20070031-9007=07=99(25)=254502(4) 254502-1 © 2007 The American Physical Societyof E	k and EM	k is close to k3=2 in a range of scalescovering almost a decade. Energy spectra in MHD havebeen reported in the literature, both for decaying andforced flows, for data stemming from either numericalsimulations or observations (mostly in the solar wind). Inmost cases, a spectrum close to K41 is found [4], althoughthere are times when the IK solution is preferred [5], and insome cases both scalings may be observed [6]. It is notclear whether universality obtains (in which case, for verylarge Reynolds numbers it might either be K41 or IK), orwhether there are different behaviors, the boundaries towhich are not necessarily known. Different spectra havealready been observed for different forcings in reducedMHD (RMHD, an approximation valid in the presence ofa strong magnetic field) [7], and recently reported for MHDruns with a strong imposed magnetic field [8]. Note that theflow in our run has the largest R obtained in a directnumerical simulation, at the expense, though, of not beingstationary.To clarify the issue of which spectra should arise inMHD turbulence, at least two things can be done. On onehand, one computes isotropic high-order structure func-tions, in which case the differentiation between K41 and IKis enhanced, as already found in [9]. On the other hand, onecan quantify the degree of anisotropy in the flow, some-thing that will be discussed later. We define the longitudi-nal (parallel to the displacement l) structure function oforder p for the quantity f as Sfp	‘  hfL	lpi hff	x  f	x l  l=‘gpi. If the field is self-similar weexpect a scaling Sfp	‘  ‘fp in the inertial range, where fpare the scaling exponents. In hydrodynamic turbulence,K41 theory predicts p  p=3, and in MHD the IK theoryleads to p  p=4. In practice, these relations are modifiedby intermittency corrections due to extreme events at thesmall scales. The exact scaling laws for MHD turbulencederived in [10], hzL 	ljz	lj2i  4‘=3, can beused to define the inertial range and to improve the esti-mate of the scaling exponents, as is often done in hydro-dynamics when the extended self-similarity (ESS) hy-pothesis is used [11]. These scaling laws, and structurefunctions up to order 8, were computed at t  3:7 forincrements in the x̂, ŷ, and ẑ directions, and averaged.Figure 2 shows the resulting scaling exponents p for theElsässer fields z, computed in the range in which theexact scaling laws hold.Consistent with the computation of the energy spectra,the scaling of z is closer to the IK prediction, with 3 0:831 0:006, 3  0:806 0:005, 4  0:9850:008, and 4  0:958 0:007 (for higher orders, devia-tions from K41 and IK are larger because of intermittencycorrections). The values of 3 are not consistent with theexact scaling law for hydrodynamic turbulence. As statedin [10], the exact scaling laws in MHD involve correlationsbetween the velocity and magnetic fields (or equivalentlybetween z), and such correlations play a role in thedynamics of the flow. In particular, they can induce a shiftfrom a standard K41 coupling; this effect has been mod-eled in [12] as a scale variation of the angle between thevelocity and magnetic field, i.e., in the relative cross he-licity cos	v;b. Also, from the values of 3 and 4 , thepossibility of a Kolmogorov scaling hidden by a bottleneck(as reported in [13] ) can be ruled out for this simulation.However, even when the scaling exponents in the inertialrange for p  3 and 4 are closer to the IK prediction (withstrong corrections due to intermittency), it is worth men-tioning that the structure functions after using ESS do notshow the same scaling at all scales. This is different fromthe hydrodynamic case where, after using the ESS hy-pothesis, a unique scaling at all scales is observed, evenin the dissipative range. This brings us to our second point,linked to anisotropy.Indeed, it is known that in MHD anisotropy develops atsmall scales under the influence of a large-scale magneticfield. Several models have been written to explain how thisFIG. 2. Scaling exponents p for z  v b. The K41(dotted line) and IK (dashed line) predictions are indicated. Asa reference, the scaling exponents in a 10243 hydrodynamic(HD) simulation are also shown [21]. Note 4  1, while forHD 3  1.FIG. 1. Energy spectra E	k (solid line), EM	k (dashed line),and EV	k (dotted line) compensated by k3=2; E	k compen-sated by k5=3 is also shown (dash-dotted line). In the inset is thetotal energy flux 	k, indicating the extent of the inertial range.PRL 99, 254502 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 DECEMBER 2007254502-2might affect the small-scale dynamics. The weak turbu-lence (WT) theory [14] leads to an exact spectrumk2? f	kk, where kk and k? are the amplitudes of thewave vectors in the directions parallel and perpendicular toa uniform magnetic field B0. It has long been argued that athigh Reynolds number, leading to a large-scale separation,the large-scale magnetic field BLS plays the same dynami-cal role as B0. As a result, we can expect the small scales tobe anisotropic with respect to BLS. However, since BLSevolves in time with a characteristic correlation time 	LSand has a curvature proportional to the Taylor scale M, inthe absence of B0 and for isotropic initial conditions, onemay assume that isotropy is still preserved in the large andintermediate scales (‘ * M).To illustrate this, we follow the procedure given in [15].A local mean field BLS	x is computed as the average ofthe magnetic field b in a box of size LM with center on x,such as BLS  hbibox	LM;x. Then, unit vectors lk and l?(k and ? with respect to BLS) are defined, and the twofollowing longitudinal magnetic structure functions Sb2?	‘  Sb2 	‘l?; Sb2k	‘  Sb2 	‘lk; (3)are computed for all points in the vicinity of x and for ‘ 22k1max; LM=2. The process is repeated for several valuesof x, and Sb2? and Sb2k are averaged over all the boxes (theresults shown are the average over 107 points). Figure 3gives the resulting Sb2? and Sb2k, as well as their ratio. In thelimit ‘! 0, this ratio can be associated with one of the so-called Shebalin angles [16] tan 2	  2lim‘!0Sb2?	‘Sb2k	‘; (4)which measures the anisotropy of b with respect to BLS.We see that scales smaller than ‘  0:4 are anisotropic,in agreement with [15]; Sb2?	‘>Sb2k	‘ in this range. How-ever, isotropy obtains in an intermediate range of scalescorresponding roughly to the 3=2 range in Fig. 1. Forvery small increments (‘  0:02  2k1D ), both structurefunctions follow‘2 scaling, as expected for smooth fieldsin the dissipative range. At scales larger than the dissipa-tion scale, but smaller than ‘  0:4  M, a range withSb2?  ‘ is observed, consistent with a k2? scaling in thespectrum as predicted by WT theory [14]. In the range ‘ 0:4; 1:2, where fluctuations are roughly isotropic, Sb2? Sb2k  ‘1=2. Figure 4 shows the structure functions compen-sated by ‘1=2 (IK scaling) and ‘2=3 (K41). The ‘1=2scaling is again consistent with the isotropic energy spec-trum and scaling of the isotropic (p  3 and 4) structurefunctions discussed previously.The recovery of isotropy observed at intermediate scalesshould not be associated with the small-scale fluctuationsbut rather with the fluctuations of the large-scale magneticfield, and thus may not always take place. Note thattan2	> 1, and for very small ‘, we have Sb2?  Sb2k ‘2. If BLS is strong, and there is enough scale separation forthe local mean field to look as a uniform field for the smallscales, WT theory is consistent with S2k	‘  ‘1=2. ForS2?	‘  ‘ with   1=2, the structure functions satisfyS2?=S2k  1 at all scales.Let M be the largest scale for the fluctuations to see themean magnetic field as a uniform field; for ‘  M, thecurvature of the large-scale magnetic field cannot be ne-glected, and fluctuations become more isotropic. At scaleslarger than M, we thus have isotropic fluctuations with alocal mean magnetic field BLS. Assuming for simplicityequipartition (see, e.g., [17] for extensions of the phenome-nology to the nonequipartition case), and constant energyflux " u2‘	A;‘=	2NL;‘ [where 	NL;‘  ‘=v‘ is the nonlinearturnover time, and 	A;‘  ‘=BLS is the Alfvén crossingtime (see [18] for a review of the relevant time scales inMHD)], we obtain b‘  v‘  ‘1=2 and the IK energy spec-trum E	k  k3=2. At scales smaller than M, the fluctua-tions are anisotropic and several phenomenologies havebeen put forward [12,14,19]. Assuming vl and bl aremostly in the direction perpendicular to BLS, 	NL;l ‘?=vl. The anisotropic extension of IK takes the Alfvéntime as 	A;l  ‘k=BLS, which results in vl  bl  ‘?‘1=2kFIG. 3. Second order structure functions Sb2?	‘ (solid line)and Sb2k	‘ (dotted line). Several slopes are given as a reference.The inset shows the ratio Sb2?	‘=Sb2k	‘ in linear coordinates.FIG. 4. Sb2? (solid line) and Sb2k (dotted line) compensated by‘1=2. The inset shows the same quantities compensated by ‘2=3.PRL 99, 254502 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 DECEMBER 2007254502-3and the spectrum E	k  k2? k1=2k. In the context of this run,where the correlation length of the large-scale magneticfield is LM, we can assume 	A;l  LM=BLS, which leads tothe same ‘? and k2? scaling. Note that this ? spec-trum is what WT theory predicts, and that for the isotropiccase (k?  kk), E	k  k2? k1=2kturns into the IK spectrum.In this light, it is not surprising that the IK spectrum at largescales can be followed, in the same dynamical vein, by itswave turbulence anisotropic counterpart at small scales.A different MHD spectrum has been advocated in [19],whereby the anisotropy of the flow induces a Kolmogorovspectrum. In the anisotropic range, this implies Sb2?  ‘2=3.Another anisotropic model based on dynamic alignment ofthe velocity and magnetic fields [12] implies Sb2?  ‘1=2.Such scalings are not observed in our computation, and theresults in Fig. 4 suggest Sb2?  ‘ is a better fit in theanisotropic range of our run.In conclusion, we presented evidence of energy scalingcompatible with IK phenomenology in data stemmingfrom a high resolution simulation of a freely decayingMHD flow. This scaling is observed at intermediate scaleswhere the turbulent fluctuations are approximately iso-tropic. By decomposing the fields into their componentsparallel and perpendicular to the local mean magnetic field,we confirmed the scaling and showed the emergence of ak2? spectrum at smaller scales. This is consistent withanisotropic extensions of the IK spectrum and with pre-dictions from weak turbulence theory. The behavior differsfrom scaling laws observed in previous numerical simula-tions or predicted by some theories of MHD turbulence.However, other scaling laws cannot be ruled out com-pletely for the following reasons. The various scaling ex-ponents discussed result from the relevant time scalesassociated with the energy transfer [18]. MHD turbulencetransfer is nonlocal [20], with time scales associated withthe large and the small scales, and a breakdown of univer-sality can be a candidate to explain the variety of solutionsreported in the literature. As an example, it is useful tocompare the present results with simulations of forcedMHD turbulence. In our simulations we can associate thescale where the transition from the isotropic to anisotropicfield takes place with M, and the correlation time of thelocal mean field 	LS with LM=BLS. In a forced simulation,the correlation time of the large-scale magnetic field isproportional to the correlation of the external forcing 	F. If	F  	A;‘, the local mean field changes faster than theAlfvén wave crossing time at ‘, and we should not expectanisotropy to develop or wave interactions to be relevant.The scale where the transition takes place would changeaccordingly. Such a dependence would be consistentwith results from forced RMHD [7] and MHD simula-tions [8].We acknowledge discussions with D. C. Montgomery;P. D. M. acknowledges discussions with D. O. Gómez. TheNSF-CMG Grant No. 0327888 was instrumental in ourwork. Computer time was provided by the NCAR BTSprogram. P. D. M. is a member of the Carrera delInvestigador Cientı́fico of CONICET.[1] A. Brandenburg and K. Subramanian, Phys. Rep. 417, 1(2005).[2] P. D. Mininni, A. G. Pouquet, and D. C. Montgomery,Phys. Rev. Lett. 97, 244503 (2006).[3] P. S. Iroshnikov, Sov. Astron. 7, 566 (1963); R. H.Kraichnan, Phys. Fluids 8, 1385 (1965).[4] M. K. Verma, D. A. Roberts, and M. L. Goldstein,J. Geophys. Res. 100, 19839 (1995); M. K. Verma et al.,J. Geophys. Res. 101, 21619 (1996); D. Biskamp andW.-C. Müller, Phys. Plasmas 7, 4889 (2000).[5] W.-C. Müller and R. Grappin, Phys. Rev. Lett. 95, 114502(2005).[6] J. J. Podesta, D. A. Roberts, and M. L. Goldstein,Astrophys. J. 664, 543 (2007).[7] P. Dmitruk, D. O. Gómez, and W. H. Matthaeus, Phys.Plasmas 10, 3584 (2003).[8] J. Mason, F. Cattaneo, and S. Boldyrev, arXiv:0706.2003.[9] H. Politano, A. Pouquet, and V. Carbone, Europhys. Lett.43, 516 (1998); D. Biskamp and W.-C. Müller, Phys.Plasmas 7, 4889 (2000).[10] H. Politano and A. Pouquet, Phys. Rev. E 57, R21 (1998);Geophys. Res. Lett. 25, 273 (1998).[11] R. Benzi et al., Europhys. Lett. 24, 275 (1993); Phys.Rev. E 48, R29 (1993).[12] S. Boldyrev, Phys. Rev. Lett. 96, 115002 (2006).[13] N. E. L. Haugen, A. Brandenburg, and W. Dobler, Phys.Rev. E 70, 016308 (2004).[14] S. Galtier, S. V. Nazarenko, A. C. Newell, and A. Pouquet,J. Plasma Phys. 63, 447 (2000).[15] L. J. Milano, W. H. Matthaeus, P. Dmitruk, and D. C.Montgomery, Phys. Plasmas 8, 2673 (2001).[16] J. V. Shebalin, W. H. Matthaeus, and D. Montgomery,J. Plasma Phys. 29, 525 (1983).[17] Zhou and W. H. Matthaeus, Phys. Plasmas 12, 056503(2005).[18] A. Pouquet, U. Frisch, and J. Leorat, J. Fluid Mech. 77,321 (1976); Y. Zhou, W. H. Matthaeus, and P. Dmitruk,Rev. Mod. Phys. 76, 1015 (2004).[19] P. Goldreich and S. Sridhar, Astrophys. J. 438, 763 (1995).[20] A. Alexakis, P. D. Mininni, and A. Pouquet, Phys. Rev. E72, 046301 (2005); P. D. Mininni, A. Alexakis, andA. Pouquet, Phys. Rev. E 72, 046302 (2005).[21] A. Alexakis, P. D. Mininni, and A. Pouquet, Phys. Rev.Lett. 95, 264503 (2005).PRL 99, 254502 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 DECEMBER 2007254502-4

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Energy spectra stemming from interactions of Alfvén waves and turbulent eddies

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