From a direct numerical simulation of the MHD equations we show, for the
first time, that velocity and magnetic-field structure functions exhibit
multiscaling, extended self similarity (ESS), and generalized extended self
similarity (GESS). We also propose a new shell model for homogeneous and
isotropic MHD turbulence, which preserves all the invariants of ideal MHD,
reduces to a well-known shell model for fluid turbulence for zero magnetic
field, has no adjustable parameters apart from Reynolds numbers, and exhibits
the same multiscaling, ESS, and GESS as the MHD equations. We also study
dissipation-range asymptotics and the inertial- to dissipation-range crossover.Comment: 5 pages, REVTEX, 4 figures (eps

Basu, Abhik

Dhar, Sujan K.

Pandit, Rahul

Sain, Anirban

English

arXiv

From a numerical study of the magnetohydrodynamic (MHD) equations we show, for the first time in three dimensions (d=3), that velocity and magnetic-field structure functions exhibit multiscaling, extended self-similarity (ESS), and generalized extended self-similarity (GESS). We propose a new shell model for homogeneous and isotropic MHD turbulence, which preserves all the invariants of ideal MHD, reduces to a well-known shell model for fluid turbulence for zero magnetic field, has no adjustable parameters apart from Reynolds numbers, and exhibits the same multiscaling, ESS, and GESS as the MHD equations. We also study the inertial- to dissipation-range crossover

Publications of the IAS Fellows

Multiscaling in Models of Magnetohydrodynamic Turbulence
Abhik Basu,1 Anirban Sain,1 Sujan K. Dhar,2 and Rahul Pandit1,*
1Department of Physics, Indian Institute of Science, Bangalore 560012, India
2Supercomputer Education and Research Center, Indian Institute of Science, Bangalore 560012, India
From a numerical study of the magnetohydrodynamic (MHD) equations we show, for the first time
in three dimensions sd ­ 3d, that velocity and magnetic-field structure functions exhibit multiscaling,
extended self-similarity (ESS), and generalized extended self-similarity (GESS). We propose a new
shell model for homogeneous and isotropic MHD turbulence, which preserves all the invariants
of ideal MHD, reduces to a well-known shell model for fluid turbulence for zero magnetic field,
has no adjustable parameters apart from Reynolds numbers, and exhibits the same multiscaling,
ESS, and GESS as the MHD equations. We also study the inertial- to dissipation-range crossover.
[S0031-9007(98)07096-3]The extension of Kolmogorov’s work (K41) [1] on fluid
turbulence to magnetohydrodynamic (MHD) turbulence
yields [2] simple scaling for velocity v and magnetic-field
b structure functions, for distances r in the inertial range
between the forcing scale L and the dissipation scale hd .
Many studies have shown that there are multiscaling cor-
rections to K41 in fluid turbulence [3]. Solar-wind data
[4], numerical studies of two-dimensional MHD [5], and
recent shell-model studies [6,7] of MHD turbulence yield
similar multiscaling. We elucidate this for homogeneous,
isotropic MHD turbulence, in the absence of a mean mag-
netic field, by presenting the first evidence for such multi-
scaling in a numerical, pseudospectral study of the MHD
equations in three dimensions s3dMHDd. We propose a
shell model with no tunable parameters except Reynolds
numbers, study it by an Adams-Bashforth method, show
it has this multiscaling, and that it reduces to the Gledzer-
Ohkitani-Yamada (GOY) shell model [8,9] for 3d fluid
turbulence if b ­ 0. To extract multiscaling exponents
we develop the ideas of extended self-similarity (ESS)
[10,11] and generalized extended self-similarity (GESS)
[11,12] in both real and wave-vector skd spaces, used in
fluid turbulence [10–12].
We use the structure functions S ap ­ kjasx 1 rd 2
asxdjpl, where a can be v , b, or one of the Elsässer
variables Z6 ­ v 6 b, x and r are spatial coordinates,
and the angular brackets denote an average in the sta-
tistical steady state. S ap , rz
a
p at high fluid and mag-
netic Reynolds numbers Re and Reb , respectively, and for
the inertial range 20hd & r ¿ L. The extension [2] of
K41 to homogeneous, isotropic MHD turbulence with no
mean magnetic field yields z ap ­ py3. Shell models [6,7]
and solar-wind data [4] have obtained multiscaling in
MHD turbulence, i.e., z ap ­ py3 2 dzap , with dzap . 0
and z ap nonlinear, monotonically increasing functions of
p. Work on fluid turbulence shows [3] an extended in-
ertial range if we use ESS [10] and GESS [12]: Thus
with ESS, in which z ap yz
a
3 follows from S ap , fS a3 gz
a
p yz
a
3
,
we should expect by analogy that it extends down tor . 5hd (as exploited in some MHD shell models [6,7]).
In GESS, which employs Gap srd ; S ap srdyfS a3 srdgpy3 and
postulates a form Gap srd , fGaq srdgr
a
pq
, with rapq ­ fz ap 2
pz a3 y3gyfz aq 2 qz
a
3 y3g, it has been suggested [12] for
fluid turbulence that the apparent inertial range is ex-
tended to the lowest resolvable r; however, k-space GESS
[11] shows a crossover from inertial- to dissipation-range
asymptotic behaviors.
Our studies yield many interesting results: The mul-
tiscaling exponents we obtain from 3DMHD and our
shell model agree (Figs. 1a and 1b) and z bp . z Z
1
p *
z Z
2
p . z
y
p . z
b
p lie close to the She-Leveque (SL) pre-
diction [13] for fluids sz SLp ­ py9 1 2f1 2 s2y3dpy3gd,
but z yp lie below it (Fig. 1c) [14]. The probability dis-
tribution functions (Fig. 1d) for dyasrd ­ yasx 1 rd 2
yasxd and dbasrd ­ basx 1 rd 2 basxd are also differ-
ent. ESS works both with real- and k-space structure func-
tions (Fig. 2). To study the latter we postulate k-space
ESS [for real-space structure functions we use S and G
and for their k-space analogs (not Fourier transforms) S
and G]:
Sap ; kjaskdjpl ø AaIpsSa3 dz
0a
p , L21 ¿ k & 1.5kd ,
Sap ; kjaskdjpl ø AaDpsSa3 da
a
p , 1.5kd & k ¿ L ,
(1)
where askd is the Fourier transform of asrd, AaIp and AaDp
are, respectively, nonuniversal amplitudes for inertial and
dissipation ranges, kd , h21d , and L21 the (molecular)
length at which hydrodynamics breaks down (cf. [11] for
fluid turbulence). The exponents aap and z 0ap characterize
the asymptotic behaviors of the structure functions in
dissipation and inertial ranges. They are universal, but
aap Þ z
0a
p . In our shell model z 0ap ­ z ap , but our data for
3DMHD suggest z 0ap ­ 2sz ap 1 3py2dy11 (i.e., Sapskd ,
k2sz
a
p 13py2d in the inertial range [15]); the difference arises
because of phase-space factors [11]. z 0ap and aap seem
universal (the same for all our runs [Table I)]; aap is close
to, but systematically less than, py3. The k dependences
FIG. 1. (a)–(c) Inertial-range
exponents versus p from typi-
cal 3DMHD and shell-model
runs (Table I) and their com-
parison with the SL formula:
(a) z yp yz y3 , (b) z bp yz b3 , and
(c) z yp , z bp , z z
1
p , and z z
2
p from
SH2. (d) Semilog (base
10) plots of the probability
distributions Psdyasrdd and
Psdbasrdd with r in the
dissipation range [we average
over 6tey (Table I) and
suppress a since we average
over Cartesian components]; a
Gaussian distribution is shown
for comparison.of Sap follow from that of S
a
3 . We find
Sa3 ø B
a
I k
2z a3 29y2, L21 ¿ k & 1.5kd , (2)
Sa3 ø B
a
Dk
da exps2cakykdd, 1.5kd & k ¿ L , (3)
where BaI and BaD are nonuniversal amplitudes [Equa-
tion (2) holds [11] for 3DMHD; for our shell model the
factor 9y2 is absent]. Thus all Sap , ku
a
p exps2caaapkykdd
for 1.5kd & k ¿ L, with uap ­ aapda (cf. [11] for fluid
turbulence). In Eq. (3) da, ca, and kd are not univer-
sal; they depend on whether we use the 3DMHD equa-
tions or our shell model. We extract the universal part
of the inertial- to dissipation-range crossover via our k-
space GESS as follows: We first define Gap ; SapysS
a
3 dpy3;
log-log plots of Gap versus Gaq yield curves with universal,
but different, slopes for asymptotes in inertial and dissi-
pation ranges. The inertial-range asymptote has a slope
rap,q (as in real-space GESS); the dissipation-range one
TABLE I. The viscosities and hyperviscosities ny , nb , nyH , and ybH , the Taylor-microscale Reynolds numbers Rel and Rebl,
the box-size eddy-turnover times tey and teb , the averaging time tA, the time over which transients are allowed to decay tt ,
and kd (dissipation-scale wave number) for our 3DMHD runs (kmax ­ 32 for MHD1 and MHD2 and kmax ­ 40 for MHD3) and
shell-model runs SH1–4 skmax ­ 225k0d. The step size sdtd is 0.02 for MHD1–3, 2 3 1025 for SH1–2, and 1024 for SH3–4.
Note that tey . 8tI the integral time for our 3DMHD runs.
Run ny nyH nb nbH Rel Reml teyydt teyydt ttytey tAytey kmaxykd
MHD1 8 3 1024 7 3 1026 1023 8 3 1026 .24.8 .14.3 .8.8 3 103 .6 3 103 .2 .2.3 .1.83
MHD2 8 3 1024 9 3 1026 8 3 1024 9 3 1026 .24.1 .18.1 .8.8 3 103 .5.6 3 103 .2 .2.3 .1.83
MHD3 8 3 1024 9 3 1026 8 3 1024 9 3 1026 .26 .19.6 .7.9 3 103 .4.8 3 103 .1 .2.2 .2.22
SH1 1029 0 1029 0 .4.6 3 108 .7.8 3 108 .107 .6 3 106 .50 .450 .25
SH2 1028 0 1028 0 .4.3 3 107 .6.5 3 107 .107 .6 3 106 .50 .450 .28
SH3 1026 0 2 3 1026 0 .4 3 106 .3 3 106 .2 3 106 .106 .500 .2500 .210
SH4 4 3 1026 0 1026 0 .1.2 3 105 .1 3 106 .106 .1.7 3 106 .500 .3000 .211has a slope vasp, qd ; faap 2 py3gyfaaq 2 qy3g. These
slopes are universal, but not the points at which the curves
move away from the inertial-range asymptote. To obtain
a universal crossover scaling function [different for each
sp, qd pair because of multiscaling] we define logsHapqd ;
Dapq logsGapd and logsHaqpd ; Daqp logsGaq d; the scale fac-
tors Dapq ­ D
a
qp are nonuniversal, but plots of logsHapqd
versus logsHaqpd, for both 3DMHD and our shell model,
collapse onto a universal curve within our error bars for
all k, Rel, and Rebl (Fig. 3).
The MHD equations are [2]
›Z6
›t
1 sZ7 ? =dZ6 ­ n1=2Z6 1 n2=2Z7
2 =pp 1 f6, (4)
where n6 ; sny 6 nbdy2, ny and nb are, respectively,
fluid and magnetic viscosities, pp ; fp 1 sb2y8pdg,
FIG. 2. Log-log plots (base 10) of Sa10 versus Sa3 showing k
space ESS for 3DMHD with (a) a ­ v and (b) a ­ b. Insets
illustrate real-space ESS for 3DMHD and ESS for our shell
model; the lines show the inertial-range asymptotes (a few
points on the right correspond to forcing scales and are not
used for inertial-range fitting).
with p the pressure, the density r ­ 1, f6 ; sf 6 gdy2,
and f and g are the forcing terms in the equations for
›vy›t and ›by›t. We assume incompressibility and use
a pseudospectral method [11] to solve Eq. (4) numerically.
We force the first two k shells, use a cubical box with side
LB ­ 2p, periodic boundary conditions, and 643 modes
in runs MHD1 and MHD2 and 803 modes in run MHD3
(Table I). We include fluid and magnetic hyperviscosities
nyH and nbH [i.e., the term 2sny 1 nyHk2dk2 in the
equation for ›vskdy›t and the term 2snb 1 nbHk2dk2
in the equation for ›bskdy›t] [16]. For time inte-
gration we use an Adams-Bashforth scheme (step-
size dt). We use Rel ­ yrmslyny , Rebl ­FIG. 3. GESS log-log plots (base 10) of Hy9,6 versus Hy6,9 and
(inset) Hb9,6 versus Hb6,9 showing the inertial- to dissipation-
range crossover; lines are inertial-range asymptotes.
brmslynb , ly ­ f
R‘
o Eyskd dky
R‘
o k
2Eyskd dkg1y2, lb ­
f
R‘
o Ebskd dky
R‘
o k
2Ebskd dkg1y2, Eyskd , Sy2 skdk2, and
Ebskd , Sb2 skdk2. Parameters for runs MHD1–3 are
given in Table I, where tea ; LByarms is the box-size
eddy-turnover time for field a and tA the averaging time;
initial transients are allowed to decay over a period tt .
We use quadruple-precision arithmetic; results from our
643 and 803 runs are not significantly different.
The Richardson-cascade picture suggests that the mul-
tiscaling behavior in turbulence might arise in simplified
dynamical models with a reduced number degrees of free-
dom arranged hierarchically. Shell models of turbulence
[3,8], which cannot be derived from the Navier-Stokes
equation, but build in the cascade and all conservation
laws, achieve this reduction with complex scalar veloci-
ties in a logarithmically discretized k space; they obtain
large Rel and exponents in agreement with experiments.
Similar shell models for MHD turbulence have been pro-
posed earlier [6,7,17], but there is no MHD shell model
that enforces all ideal 3DMHD invariants and which re-
duces to the GOY shell model for fluid turbulence, when
magnetic-field terms are suppressed. We present such a
model and show that it yields z ap in agreement with those
we obtain for 3DMHD. Our shell-model equations
dz6n
dt
­ ic6n 2 n1k
2
nz
6
n 2 n2k
2
nz
6
n 1 f
6
n (5)
use the complex, scalar Elsässar variables z6n ;
syn 6 bnd, and discrete wave vectors kn ­ koqn,
for shells n; c6n ­ fa1knz
7
n11z
6
n12 1 a2knz
6
n11z
7
n12 1
a3kn21z
7
n21z
6
n11 1 a4kn21z
6
n21z
7
n11 1 a5kn22z
7
n21z
6
n22 1
a6kn22z
7
n21z
6
n22gp, which ensures z1n , z2n , k21y3 is
a stationary solution in the inviscid, unforced limit
[6–9] and preserves the n1, Z1 $ n2, Z2 symmetry
of 3DMHD. We fix five of the parameters, a1 2 a6,
by demanding that our shell-model analogs of the to-
tal energy f;
P
nsjynj2 1 jbnj2dy2g, the cross helicity
f; 1y2
P
nsynbpn 1 ypnbndg, and the magnetic helicity
f;
P
ns21dnjbnj2ykng be conserved if n6 ­ 0 and
f6n ­ 0; while enforcing the conservation of energy,
we also demand [18] that the cancellation of terms
occurs as in 3DMHD. We fix the last parameter by
demanding that, if bn ­ 0 for all n, our model reduces
to the GOY model, with the standard parameters [9]
that enforce conservation laws. Finally a1 ­ 7y12,
a2 ­ 5y12, a3 ­ 21y12, a4 ­ 25y12, a5 ­ 27y12,
a6 ­ 1y12, and q ­ 2. We solve Eq. (5) numerically
by an Adams-Bashforth scheme (step size dt), use
25 shells, force the first k shell [11], set ko ­ 224 ­
1ys2Lsd, where Ls is the box size, and use Ey ­
Sy2 skndykn, ly ­ s2pykod f
P
n S
y
2 skndy
P
n k
2
nS
y
2 skndg1y2,
lb ­ s2pykod f
P
n S
b
2 skndy
P
n k
2
nS
b
2 skndg1y2, yrms ­
fko
P
n S
y
2 skndypg1y2, and brms ­ fko
P
n S
b
2 skndypg1y2.
Parameters for our four runs SH1–SH4 are given in
Table I. These use double-precision arithmetic, but we
have checked in representative cases that our results are
not affected if we use quadruple-precision arithmetic.
As in the GOY model the structure functions Spsknd
oscillate weakly with kn because of an underlying three
cycle [9,18]. These oscillations can be removed either
(a) by using ESS or (b) by using the structure func-
tions
Pa
n,p ­ kImfanan11an12 1 an21anan11y4gpy3l [9].
Method (a) yields z ap yz a3 , which we find are universal.
Method (b) gives exponents z ap . These have a mild
dependence on Rel and Rebl but this goes away if we
consider the ratios z ap yz
a
3 , as in the GOY model [11];
thus the asymptotes in our ESS and GESS plots have
universal slopes.
The Navier Stokes equation (3DNS) follows from
3DMHD if b ­ 0 or, equivalently, Rebl ­ 0. However,
if we start with Rebl . 0, the steady state is characterized
by the MHD exponents and RelyRebl . Os1d (i.e., an
equipartition regime) [19]. Since our MHD shell model
reduces to the GOY model as Rebl ! 0, we use it
(and not costly 3DMHD) to study the fluid turbulence to
MHD turbulence crossover: A small initial value of Rebl
yields a transient with GOY-model exponents, but finally
the system crosses over to the MHD turbulence steady
state [18].
In summary, we have shown that structure functions in
3DMHD turbulence display multiscaling, ESS, and GESS,
with exponents and probability distributions different
from those in fluid turbulence. Our new shell model (a)
gives the same exponents as 3DMHD and (b) reduces
to the GOY model as Rebl ! 0. Our ESS and GESS
uncover a universal crossover from inertial- to dissipation-
range asymptotics. It would be interesting to compare
our results with experiments, but with caution: (i) solar-
wind data might yield exponents different from ours
because of the presence of a mean magnetic field andcompressive effects [20]; (ii) the inertial- to dissipation-
range crossover might not apply to the solar wind because
a hydrodynamic description might break down in the
dissipation range [20]. However, our results should apply
to MHD systems with an equipartition regime [2]. The
agreement of z bp with the SL formula is interesting but,
we believe, fortuitous since vorticity organizes itself into
filamentary structures [13] in fluid turbulence but into
sheetlike structures in 3DMHD (we have checked this in
our study).
We thank J. K. Bhattacharjee and S. Ramaswamy for
discussions, CSIR (India) for support, and SERC (IISc,
Bangalore) for computational resources.
*Also at Jawaharlal Nehru Centre for Advanced Scientific
Research, Bangalore, India.
[1] A.N. Kolmogorov, C. R. Acad. Sci. USSR 30, 301 (1941).
[2] D. Montgomery, in Lecture Notes on Turbulence, edited
by J. R. Herring and J. C. McWilliam (World Scientific,
Singapore, 1989); D. Biskamp, in Nonlinear Magnetohy-
drodynamics, edited by W. Grossman et al. (Cambridge
University Press, Cambridge, England, 1993).
[3] For recent reviews, see K. R. Sreenivasan and R.A.
Antonia, Annu. Rev. Fluid Mech. 29, 435 (1997); S. K.
Dhar et al., Pramana J. Phys. 48, 325 (1997).
[4] R. Grauer, J. Krug, and C. Marliani, Phys. Lett. 195, 335
(1994); L. F. Burlaga, J. Geophys. Res. 96, 5847 (1991);
Horbury et al., Adv. Space Res. 19, 847 (1996).
[5] R. Grauer et al., Phys. Plasmas 2, 41 (1995).
[6] D. Biskamp, Phys. Rev. E 50, 2702 (1994) also finds
z bp . z
y
p , for p ­ 2, in a shell model.
[7] V. Carbone, Phys. Rev. E 50, R671 (1994).
[8] E. B. Gledzer, Sov. Phys. Dokl. 18, 216 (1973); K.
Ohkitani and M. Yamada, Prog. Theor. Phys. 81, 329
(1989).
[9] L. Kadanoff, D. Lohse, and J. Wang, Phys. Fluids 7, 517
(1995).
[10] R. Benzi et al., Phys. Rev. E 48, R29 (1993).
[11] S.K. Dhar, A. Sain, and R. Pandit, Phys. Rev. Lett. 78,
2964 (1997).
[12] R. Benzi et al., Europhys. Lett. 32, 709 (1995); for GESS
in solar-wind data V. Carbone, Geophys. Res. Lett. 23,
121 (1996); R. Grauer, Phys. Plasmas 2, 41 (1995).
[13] Z. S. She and E. Leveque, Phys. Rev. Lett. 72, 336 (1994).
[14] We use the SL formula as a convenient parametrization
for the multiscaling exponents in fluid turbulence.
[15] For a heuristic justification for fluid turbulence, see [11].
[16] Hyperviscosities do not influence multiscaling (Ref. [11]
for fluids) in the inertial range, where conventional
viscosities dominate. Even if the viscosity vanishes,
hyperviscosity does not affect zpyz3 in fluid turbulence [E.
Leveque and Z. S. She, Phys. Rev. Lett. 75, 2690 (1995)].
[17] C. Gloaguen et al., Physica (Amsterdam) 17D, 154
(1985).
[18] A. Basu and R. Pandit (unpublished).
[19] Thus in a renormalization-group calculation Rebl should
appear as a relevant operator.
[20] E. Marsch, in Reviews in Modern Astronomy Vol. 4, edited
by G. Klare (Springer-Verlag, Berlin, 1991); C.-Y. Tu and
E. Marsch, Space Sci. Rev. 73, 1 (1995).


Multiscaling in models of magnetohydrodynamic turbulence

From a numerical study of the magnetohydrodynamic (MHD) equations we show, for the first time in three dimensions (d = 3), that velocity and magnetic-field structure functions exhibit multiscaling, extended self-similarity (ESS), and generalized extended self-similarity (GESS). We propose a new shell model for homogeneous and isotropic MHD turbulence, which preserves all the invariants of ideal MHD, reduces to a well-known shell model for fluid turbulence for zero magnetic field, has no adjustable parameters apart from Reynolds numbers, and exhibits the same multiscaling, ESS, and GESS as the MHD equations. We also study the inertial- to dissipation-range crossover

Dhar, Sujan K

Name not available

Multiscaling in Models of Magnetohydrodynamic Turbulence

Indian Academy of Sciences

, India
time
ng,
a new
riants
eld,
scaling,
ssover.Multiscaling in Models of Magnetohydrodynamic Turbulence
Abhik Basu,1 Anirban Sain,1 Sujan K. Dhar,2 and Rahul Pandit1,*
1Department of Physics, Indian Institute of Science, Bangalore 560012, India
2Supercomputer Education and Research Center, Indian Institute of Science, Bangalore 560012
From a numerical study of the magnetohydrodynamic (MHD) equations we show, for the first
in three dimensionssd ­ 3d, that velocity and magnetic-field structure functions exhibit multiscali
extended self-similarity (ESS), and generalized extended self-similarity (GESS). We propose
shell model for homogeneous and isotropic MHD turbulence, which preserves all the inva
of ideal MHD, reduces to a well-known shell model for fluid turbulence for zero magnetic fi
has no adjustable parameters apart from Reynolds numbers, and exhibits the same multi
ESS, and GESS as the MHD equations. We also study the inertial- to dissipation-range cro
[S0031-9007(98)07096-3]d
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The extension of Kolmogorov’s work (K41) [1] on flui
turbulence to magnetohydrodynamic (MHD) turbulen
yields [2] simple scaling for velocityv and magnetic-field
b structure functions, for distancesr in the inertial range
between the forcing scaleL and the dissipation scalehd .
Many studies have shown that there are multiscaling
rections to K41 in fluid turbulence [3]. Solar-wind da
[4], numerical studies of two-dimensional MHD [5], an
recent shell-model studies [6,7] of MHD turbulence yie
similar multiscaling. We elucidate this for homogeneo
isotropic MHD turbulence, in the absence of a mean m
netic field, by presenting the first evidence for such mu
scaling in a numerical, pseudospectral study of the M
equations inthree dimensions3dMHDd. We propose a
shell model with no tunable parameters except Reyn
numbers, study it by an Adams-Bashforth method, sh
it has this multiscaling, and that it reduces to the Gledz
Ohkitani-Yamada (GOY) shell model [8,9] for3d fluid
turbulence ifb ­ 0. To extract multiscaling exponen
we develop the ideas of extended self-similarity (ES
[10,11] and generalized extended self-similarity (GES
[11,12] in both real and wave-vectorskd spaces, used in
fluid turbulence [10–12].
We use the structure functionsS ap ­ kjasx 1 rd 2
asxdjpl, where a can bev, b, or one of the Elsässe
variablesZ6 ­ v 6 b, x and r are spatial coordinates
and the angular brackets denote an average in the
tistical steady state.S ap , r
z ap at high fluid and mag-
netic Reynolds numbers Re and Reb , respectively, and fo
the inertial range20hd & r ø L. The extension [2] of
K41 to homogeneous, isotropicMHD turbulencewith no
mean magnetic fieldyieldsz ap ­ py3. Shell models [6,7]
and solar-wind data [4] have obtained multiscaling
MHD turbulence, i.e.,z ap ­ py3 2 dzap , with dzap . 0
and z ap nonlinear, monotonically increasing functions
p. Work on fluid turbulence shows [3] an extended
ertial range if we use ESS [10] and GESS [12]: Th
with ESS, in whichz ap yz
a
3 follows from S ap , fS
a
3 gz
a
p yz
a
3 ,
we should expect by analogy that it extends downe
r-
s,
g-
i-
D
ds
w
r-
)
)
ta-
n
f
-
s
to
r . 5hd (as exploited in some MHD shell models [6,7]
In GESS, which employsGap srd ; S ap srdyfS
a
3 srdgpy3 and
postulates a formGap srd , fGaq srdg
rapq , with rapq ­ fz ap 2
pz a3 y3gyfz aq 2 qz
a
3 y3g, it has been suggested [12] fo
fluid turbulence that the apparent inertial range is
tended to the lowest resolvabler; however,k-space GESS
[11] shows a crossover from inertial- to dissipation-ran
asymptotic behaviors.
Our studies yield many interesting results: The m
tiscaling exponents we obtain from 3DMHD and o
shell model agree (Figs. 1a and 1b) andz bp . z
Z1
p *
z Z
2
p . z
y
p . z
b
p lie close to the She-Leveque (SL) pr
diction [13] for fluids sz SLp ­ py9 1 2f1 2 s2y3dpy3gd,
but z yp lie below it (Fig. 1c) [14]. The probability dis-
tribution functions (Fig. 1d) fordyasrd ­ yasx 1 rd 2
yasxd anddbasrd ­ basx 1 rd 2 basxd are also differ-
ent. ESS works both with real- andk-space structure func
tions (Fig. 2). To study the latter we postulatek-space
ESS [for real-space structure functions we useS and G
and for theirk-space analogs (notFourier transforms)S
andG]:
Sap ; kjaskdj
pl ø AaIpsS
a
3 d
z 0ap , L21 ø k & 1.5kd ,
Sap ; kjaskdj
pl ø AaDpsS
a
3 d
aap , 1.5kd & k ø L ,
(1)
whereaskd is the Fourier transform ofasrd, AaIp andA
a
Dp
are, respectively, nonuniversal amplitudes for inertial a
dissipation ranges,kd , h21d , and L21 the (molecular)
length at which hydrodynamics breaks down (cf. [11] f
fluid turbulence). The exponentsaap andz
0a
p characterize
the asymptotic behaviors of the structure functions
dissipation and inertial ranges. They are universal,
aap fi z
0a
p . In our shell modelz
0a
p ­ z
a
p , but our data for
3DMHD suggestz 0ap ­ 2sz ap 1 3py2dy11 (i.e., Sapskd ,
k2sz
a
p 13py2d in the inertial range [15]); the difference arise
because of phase-space factors [11].z 0ap and a
a
p seem
universal (the same for all our runs [Table I)];aap is close
to, butsystematically lessthan,py3. Thek dependences

FIG. 1. (a)–(c) Inertial-range
exponents versusp from typi-
cal 3DMHD and shell-model
runs (Table I) and their com-
parison with the SL formula:
(a) z yp yz
y
3 , (b) z bp yz
b
3 , and
(c) z yp , z
b
p , z
z1
p , and z
z2
p from
SH2. (d) Semilog (base
10) plots of the probability
distributions Psdyasrdd and
Psdbasrdd with r in the
dissipation range [we average
over 6tey (Table I) and
suppressa since we average
over Cartesian components]; a
Gaussian distribution is shown
for comparison.u
i
lo
o
ves
ainof Sap follow from that ofS
a
3 . We find
Sa3 ø B
a
I k
2z
a
3 29y2, L21 ø k & 1.5kd , (2)
Sa3 ø B
a
Dk
da exps2cakykdd, 1.5kd & k ø L , (3)
where BaI and B
a
D are nonuniversal amplitudes [Equa
tion (2) holds [11] for 3DMHD; for our shell model the
factor9y2 is absent]. Thusall Sap , k
uap exps2caaapkykdd
for 1.5kd & k ø L, with uap ­ a
a
pd
a (cf. [11] for fluid
turbulence). In Eq. (3)da, ca, and kd are not univer-
sal; they depend on whether we use the 3DMHD eq
tions or our shell model. We extract theuniversal part
of the inertial- to dissipation-range crossover via ourk-
space GESS as follows: We first defineGap ; SapysS
a
3 dpy3;
log-log plots ofGap versusG
a
q yield curves withuniversal,
but different,slopes for asymptotes in inertial and diss
pation ranges. The inertial-range asymptote has a s
rap,q (as in real-space GESS); the dissipation-rangeTABLE I. The viscosities and hyperviscositiesny , nb , nyH , and ybH , the Taylor-microscale Reynolds numbers Rel and Rebl,
the box-size eddy-turnover timestey and teb , the averaging timetA, the time over which transients are allowed to decaytt ,
and kd (dissipation-scale wave number) for our 3DMHD runs (kmax ­ 32 for MHD1 and MHD2 andkmax ­ 40 for MHD3) and
shell-model runs SH1–4skmax ­ 225k0d. The step sizesdtd is 0.02 for MHD1–3,2 3 1025 for SH1–2, and1024 for SH3–4.
Note thattey . 8tI the integral time for our 3DMHD runs.
Run ny nyH nb nbH Rel Reml teyydt teyydt ttytey tAytey kmaxykd
MHD1 8 3 1024 7 3 1026 1023 8 3 1026 .24.8 .14.3 .8.8 3 103 .6 3 103 .2 .2.3 .1.83
MHD2 8 3 1024 9 3 1026 8 3 1024 9 3 1026 .24.1 .18.1 .8.8 3 103 .5.6 3 103 .2 .2.3 .1.83
MHD3 8 3 1024 9 3 1026 8 3 1024 9 3 1026 .26 .19.6 .7.9 3 103 .4.8 3 103 .1 .2.2 .2.22
SH1 1029 0 1029 0 .4.6 3 108 .7.8 3 108 .107 .6 3 106 .50 .450 .25
SH2 1028 0 1028 0 .4.3 3 107 .6.5 3 107 .107 .6 3 106 .50 .450 .28
SH3 1026 0 2 3 1026 0 .4 3 106 .3 3 106 .2 3 106 .106 .500 .2500 .210
SH4 4 3 1026 0 1026 0 .1.2 3 105 .1 3 106 .106 .1.7 3 106 .500 .3000 .211-
a-
-
pe
ne
has a slopevasp, qd ; faap 2 py3gyfaaq 2 qy3g. These
slopes are universal, but not the points at which the cur
move away from the inertial-range asymptote. To obt
a universal crossover scaling function[different for each
sp, qd pair because of multiscaling] we define logsHapqd ;
Dapq logsGapd and logsHaqpd ; Daqp logsGaq d; the scale fac-
tors Dapq ­ D
a
qp are nonuniversal,but plots of logsHapqd
versus logsHaqpd, for both 3DMHD and our shell model,
collapse onto auniversal curvewithin our error bars for
all k, Rel, and Rebl (Fig. 3).
The MHD equations are [2]
≠Z6
≠t
1 sZ7 ? =dZ6 ­ n1=2Z6 1 n2=2Z7
2 =pp 1 f6, (4)
where n6 ; sny 6 nbdy2, ny and nb are, respectively,
fluid and magnetic viscosities,pp ; fp 1 sb2y8pdg,
e
e
n
r
e
ll
-
e
ur
ul-
d
ee-
ce
es
ion
ci-
ts.
o-
en
ch a
itFIG. 2. Log-log plots (base 10) ofSa10 versusS
a
3 showing k
space ESS for 3DMHD with (a)a ­ v and (b)a ­ b. Insets
illustrate real-space ESS for 3DMHD and ESS for our sh
model; the lines show the inertial-range asymptotes (a f
points on the right correspond to forcing scales and are
used for inertial-range fitting).
with p the pressure, the densityr ­ 1, f6 ; sf 6 gdy2,
and f and g are the forcing terms in the equations fo
≠vy≠t and≠by≠t. We assume incompressibility and us
a pseudospectral method [11] to solve Eq. (4) numerica
We force the first twok shells, use a cubical box with side
LB ­ 2p, periodic boundary conditions, and643 modes
in runs MHD1 and MHD2 and803 modes in run MHD3
(Table I). We include fluid and magnetic hyperviscositie
nyH and nbH [i.e., the term2sny 1 nyHk2dk2 in the
equation for≠vskdy≠t and the term2snb 1 nbHk2dk2
in the equation for≠bskdy≠t] [16]. For time inte-
gration we use an Adams-Bashforth scheme (ste
size dt). We use Rel ­ yrmslyny, Rebl ­ll
w
ot
y.
s
p-
FIG. 3. GESS log-log plots (base 10) ofHy9,6 versusH
y
6,9 and
(inset) Hb9,6 versus H
b
6,9 showing the inertial- to dissipation
range crossover; lines are inertial-range asymptotes.
brmslynb , ly ­ f
R`
o Eyskd dky
R`
o k
2Eyskd dkg1y2, lb ­
f
R
`
o Ebskd dky
R
`
o k
2Ebskd dkg1y2, Eyskd , Sy2 skdk2, and
Ebskd , Sb2 skdk2. Parameters for runs MHD1–3 ar
given in Table I, wheretea ; LByarms is the box-size
eddy-turnover time for fielda andtA the averaging time;
initial transients are allowed to decay over a periodtt.
We use quadruple-precision arithmetic; results from o
643 and803 runs are not significantly different.
The Richardson-cascade picture suggests that the m
tiscaling behavior in turbulence might arise in simplifie
dynamical models with a reduced number degrees of fr
dom arranged hierarchically. Shell models of turbulen
[3,8], which cannot be derived from the Navier-Stok
equation, but build in the cascade and all conservat
laws, achieve this reduction with complex scalar velo
ties in a logarithmically discretizedk space; they obtain
large Rel and exponents in agreement with experimen
Similar shell models for MHD turbulence have been pr
posed earlier [6,7,17], but there isno MHD shell model
that enforcesall ideal 3DMHD invariantsand which re-
duces to the GOY shell model for fluid turbulence, wh
magnetic-field terms are suppressed. We present su
model and show that it yieldsz ap in agreement with those
we obtain for 3DMHD. Our shell-model equations
dz6n
dt
­ ic6n 2 n1k
2
nz
6
n 2 n2k
2
nz
6
n 1 f
6
n (5)
use the complex, scalar Elsässar variablesz6n ;
syn 6 bnd, and discrete wave vectorskn ­ koqn,
for shells n; c6n ­ fa1knz
7
n11z
6
n12 1 a2knz
6
n11z
7
n12 1
a3kn21z
7
n21z
6
n11 1 a4kn21z
6
n21z
7
n11 1 a5kn22z
7
n21z
6
n22 1
a6kn22z
7
n21z
6
n22gp, which ensuresz1n , z2n , k21y3 is
a stationary solution in the inviscid, unforced lim
to
y,
ms
by
[9
in
w
a
tic
e
he
c-
.
av
m
ed
el
to
lly
ad
in
S
nt
a)
ce
n
are
r-
rs
an
n-
use
he
ply
e
t,
to
to
in
for
Sc,
ific
,
. A.
s
tion
].
1]
al
es,
d
[6–9] and preserves then1, Z1 $ n2, Z2 symmetry
of 3DMHD. We fix five of the parameters,a1 2 a6,
by demanding that our shell-model analogs of the
tal energy f;
P
nsjynj2 1 jbnj2dy2g, the cross helicity
f; 1y2
P
nsynbpn 1 ypnbndg, and the magnetic helicity
f;
P
ns21dnjbnj2ykng be conserved if n6 ­ 0 and
f6n ­ 0; while enforcing the conservation of energ
we also demand [18] that the cancellation of ter
occurs as in 3DMHD. We fix the last parameter
demanding that, ifbn ­ 0 for all n, our model reduces
to the GOY model, with the standard parameters
that enforce conservation laws. Finallya1 ­ 7y12,
a2 ­ 5y12, a3 ­ 21y12, a4 ­ 25y12, a5 ­ 27y12,
a6 ­ 1y12, and q ­ 2. We solve Eq. (5) numerically
by an Adams-Bashforth scheme (step sizedt), use
25 shells, force the firstk shell [11], setko ­ 224 ­
1ys2Lsd, where Ls is the box size, and useEy ­
Sy2 skndykn, ly ­ s2pykod f
P
n S
y
2 skndy
P
n k
2
nS
y
2 skndg1y2,
lb ­ s2pykod f
P
n S
b
2 skndy
P
n k
2
nS
b
2 skndg1y2, yrms ­
fko
P
n S
y
2 skndypg1y2, and brms ­ fko
P
n S
b
2 skndypg1y2.
Parameters for our four runs SH1–SH4 are given
Table I. These use double-precision arithmetic, but
have checked in representative cases that our results
not affected if we use quadruple-precision arithme
As in the GOY model the structure functionsSpsknd
oscillate weakly withkn because of an underlying thre
cycle [9,18]. These oscillations can be removed eit
(a) by using ESS or (b) by using the structure fun
tions
Pa
n,p ­ kImfanan11an12 1 an21anan11y4gpy3l [9].
Method (a) yieldsz ap yz
a
3 , which we find are universal
Method (b) gives exponentsz ap . These have a mild
dependence on Rel and Rebl but this goes away if we
consider the ratiosz ap yz
a
3 , as in the GOY model [11];
thus the asymptotes in our ESS and GESS plots h
universal slopes.
The Navier Stokes equation (3DNS) follows fro
3DMHD if b ­ 0 or, equivalently, Rebl ­ 0. However,
if we start with Rebl . 0, the steady state is characteriz
by the MHD exponents and RelyRebl . Os1d (i.e., an
equipartition regime) [19]. Since our MHD shell mod
reduces to the GOY model as Rebl ! 0, we use it
(and not costly 3DMHD) to study the fluid turbulence
MHD turbulence crossover: A small initial value of Rebl
yields a transient with GOY-model exponents, but fina
the system crosses over to the MHD turbulence ste
state [18].
In summary, we have shown that structure functions
3DMHD turbulence display multiscaling, ESS, and GES
with exponents and probability distributions differe
from those in fluid turbulence. Our new shell model (
gives the same exponents as 3DMHD and (b) redu
to the GOY model as Rebl ! 0. Our ESS and GESS
uncover a universal crossover from inertial- to dissipatio
range asymptotics. It would be interesting to comp
our results with experiments, but with caution: (i) sola
wind data might yield exponents different from ou
because of the presence of a mean magnetic field-
]
e
re
.
r
e
y
,
s
-
d
compressive effects [20]; (ii) the inertial- to dissipatio
range crossover might not apply to the solar wind beca
a hydrodynamic description might break down in t
dissipation range [20]. However, our results should ap
to MHD systems with an equipartition regime [2]. Th
agreement ofz bp with the SL formula is interesting bu
we believe, fortuitous since vorticity organizes itself in
filamentary structures [13] in fluid turbulence but in
sheetlike structures in 3DMHD (we have checked this
our study).
We thank J. K. Bhattacharjee and S. Ramaswamy
discussions, CSIR (India) for support, and SERC (II
Bangalore) for computational resources.
*Also at Jawaharlal Nehru Centre for Advanced Scient
Research, Bangalore, India.
[1] A. N. Kolmogorov, C. R. Acad. Sci. USSR30, 301 (1941).
[2] D. Montgomery, inLecture Notes on Turbulence,dited
by J. R. Herring and J. C. McWilliam (World Scientific
Singapore, 1989); D. Biskamp, inNonlinear Magnetohy-
drodynamics,edited by W. Grossmanet al. (Cambridge
University Press, Cambridge, England, 1993).
[3] For recent reviews, see K. R. Sreenivasan and R
Antonia, Annu. Rev. Fluid Mech.29, 435 (1997); S. K.
Dhar et al., Pramana J. Phys.48, 325 (1997).
[4] R. Grauer, J. Krug, and C. Marliani, Phys. Lett.195, 335
(1994); L. F. Burlaga, J. Geophys. Res.96, 5847 (1991);
Horbury et al., Adv. Space Res.19, 847 (1996).
[5] R. Graueret al., Phys. Plasmas2, 41 (1995).
[6] D. Biskamp, Phys. Rev. E50, 2702 (1994) also find
z bp . z
y
p , for p ­ 2, in a shell model.
[7] V. Carbone, Phys. Rev. E50, R671 (1994).
[8] E. B. Gledzer, Sov. Phys. Dokl.18, 216 (1973); K.
Ohkitani and M. Yamada, Prog. Theor. Phys.81, 329
(1989).
[9] L. Kadanoff, D. Lohse, and J. Wang, Phys. Fluids7, 517
(1995).
[10] R. Benziet al., Phys. Rev. E48, R29 (1993).
[11] S. K. Dhar, A. Sain, and R. Pandit, Phys. Rev. Lett.78,
2964 (1997).
[12] R. Benziet al., Europhys. Lett.32, 709 (1995); for GESS
in solar-wind data V. Carbone, Geophys. Res. Lett.23,
121 (1996); R. Grauer, Phys. Plasmas2, 41 (1995).
[13] Z. S. She and E. Leveque, Phys. Rev. Lett.72, 336 (1994).
[14] We use the SL formula as a convenient parametriza
for the multiscaling exponents in fluid turbulence.
[15] For a heuristic justification for fluid turbulence, see [11
[16] Hyperviscosities do not influence multiscaling (Ref. [1
for fluids) in the inertial range, where convention
viscosities dominate. Even if the viscosity vanish
hyperviscosity does not affectzpyz3 in fluid turbulence [E.
Leveque and Z. S. She, Phys. Rev. Lett.75, 2690 (1995)].
[17] C. Gloaguen et al., Physica (Amsterdam)17D, 154
(1985).
[18] A. Basu and R. Pandit (unpublished).
[19] Thus in a renormalization-group calculation Rebl should
appear as a relevant operator.
[20] E. Marsch, inReviews in Modern Astronomy Vol. 4,edited
by G. Klare (Springer-Verlag, Berlin, 1991); C.-Y. Tu an
E. Marsch, Space Sci. Rev.73, 1 (1995).


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Multiscaling in Models of Magnetohydrodynamic Turbulence

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