Table Of ContentThe Decay of Magnetohydrodynamic Turbulence from Power-Law Initial Conditions
Chirag Kalelkar∗ and Rahul Pandit†
Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India.
4
We derive relations for the decay of the kinetic and magnetic energies and the growth of the
0
Taylorandintegralscalesinunforced,incompressible,homogeneousandisotropicthree-dimensional
0
magnetohydrodynamic(3DMHD) turbulencewith power-law initial energy spectra. Wealso derive
2
boundsforthedecayofthecross- andmagnetichelicities. Wethenpresentresultsfrom systematic
n numericalstudiesofsuchdecaybothwithinthecontextofanMHDshellmodelanddirectnumerical
a simulations(DNS)of3DMHD.Weshowexplicitlythatourresultsaboutthepower-lawdecayofthe
J
energies hold for times t<t∗, where t∗ is thetime at which theintegral scales become comparable
0 to the system size. For t < t∗, our numerical results are consistent with those predicted by the
3 principle of ‘permanence of large eddies’.
] PACSnumbers: 47.27.Gs,47.65.+a,05.45.-a
h
c
e
m I. INTRODUCTION tain the following interesting results: We show first how
to generalise the results of Ref. [7, 8] and derive, for
-
t unforced, incompressible,homogeneousand isotropic[10]
a The decay of homogeneous and isotropic, fluid turbu-
three-dimensional MHD (3DMHD) turbulence, expres-
t lence has been the subject of extensive theoretical[1, 2]
s sions for the decay of the kinetic and magnetic energies
. and experimental[3, 4, 5] studies. These include wind-
t and the growth of the Taylor and integral scales, when
a tunnel experiments downstreamof a grid[3] and flow be-
m hind a towed grid in a stationary channel filled with He- we start with power-law initial conditions Ea0(k) ∼ kq
(q > 1), where the subscript a is v for the kinetic en-
- lium II[4]. However, unambiguous statements can still −
d not be made about the dependence of the decay of fluid ergy and b for the magnetic energy. Such initial condi-
n tionsareofinterestintheastrophysicalcontextofthede-
turbulence on the initial conditions. In experiments,
o cay of power-law ‘primordial’ energy spectra[8]. We also
these can be changed to some extent by changing the
c derive bounds for the decay of the cross- and magnetic
[ grid geometry and choice of virtual origin[5] but caution
helicity.We then show by systematic numerical studies
is advised in interpreting the results since our ability to
2 of a shell model for MHD turbulence[11, 12] and an 803
choose initial conditions precisely is severely restricted.
v pseudospectralDNSofthe3DMHDequationsthat,given
In an earlier study[6], the energy-decay exponents were
3 power-lawinitialconditions,thekineticandmagneticen-
4 predicted to be initial-condition dependent, but the pre-
ergiesandtheTaylorandintegralscalesfollowthedecay
2 cisenatureofthisdependencewasnotelucidatedclearly.
expressions mentioned above within a regime governed
7 Direct numerical simulations (DNS) can control initial
by the temporal evolution of the integral scales.
0 conditions, but, to the best of our knowledge, have in-
3
vestigated a very limited class of initial conditions. A The unforced MHD equations are
0
recentexception[7]is a study ofthe decayoffluidturbu-
/
t lenceforthesetofinitialconditionswithapower-lawini-
ma tial energy spectrum E0(k) ∼ kq, where the superscript ∂v +(v )v = ▽p∗ + (b·▽)b +ν 2v,
0 denotes the choice of virtual origin of time t = t , k ∂t ·▽ − ρ 4πρ ▽
- 0
d is the magnitude of the wave vector,and the exponent q ∂b
= (v b)+η 2b, (1)
n distinguishes different initial conditions in this set. ∂t ▽× × ▽
o
Results, both experimental and numerical, for the de-
c
cayofmagnetohydrodynamic(MHD)turbulenceareeven
: where ν is the kinematic viscosity, η is the magnetic vis-
v morescarcethantheirfluid-turbulenceanalogs. Different
i cosity, p∗ = p +b2/8π is the effective pressure, and p
X power-lawdecayshavebeensuggestedinDNS[9]studies,
is the pressure. We enforce the incompressibility con-
but the sensitivity of these decays to the initial condi-
r dition v = 0 and eliminate p∗ in the usual manner.
a tions or evolution of the relevant length scale has not ▽·
Theinvariantsofinviscid,unforced3DMHDarethetotal
been investigated in any detail.
energy (E = E +E 1 (v2 +b2) dV), the cross-
Inthispaper,weinitiatesuchaninvestigationandob- T v b ≡ 2
helicity (H v b dV), and the magnetic helicity
C ≡ · R
(H A b dV, where A is the vector potential).
M
We res≡trict ou·rseRlvesto the case with H0 close to zero
R | M|
andexcludeanyhelicalcontributions[9]tothedecaypro-
∗Electronicaddress: [email protected]
cessduetotheinversecascadeofH . Mostofourresults
†Electronicaddress: [email protected]; M
are obtained for initial equipartition of energy, namely,
Also at the Jawaharlal Nehru Centre for Advanced Scientific Re-
search,Bangalore,India. E0 =E0, but we alsopresenta few results for E0 =E0.
v b v 6 b
2
II. THEORETICAL RESULTS where q = 1 2h (q > 1)[13].
− − −
For q = 1, Eqs. (4) and (10) lead to especially
We observe that Eqs. (1) are invariant under the simple forms for the temporal decay of the kinetic-
rescaling and magnetic-energies. We are interested in the limits
2π/L 0 and K , of relevance to high-Reynolds
→ → ∞
x lx,t l1−ht,v lhv,b lhb, number turbulence, so
→ → → →
ν l1+hν,η l1+hη, (2)
→ → 1 ∞
E (t)= dy Φ (y,1/ν2,1/η2). (11)
a a
wherelisanarbitraryscalefactorandh<0ascalingex- 2t Z0
ponent. Thelocalkinetic-andmagnetic-energydensities
Thus both E and E decay as 1/t, which general-
in three dimensions are given by v b
izes the result of Ref. [7] to the case of decaying
(k,t,ν,η,K,2π/L)= MHD turbulence. For all q = 1, the simple power-
Ea law dependence of E (t) on t6 does not follow. The
Ωk2 L a
d3xd3y eik·(x−y) a(x,t) a(y,t) , (3) fluid and magnetic Taylor scales, λa(t,ν,η,2π/K,L)
(2π)3L3 Z2π/K h · i [ 2Kπ/Ldk Ea(k,t,ν,η)/ 2Kπ/Ldk k2Ea(k,t,ν,η)]1/2, als≡o
showasimple power-lawdependence ontonly forq =1.
whereLand2π/K arethelarge-andsmall-distancecut- R R
In the limits 2π/L 0, K ,
offs,thesubscriptaisvforthekineticenergydensityand
→ →∞
b for the magnetic energy density and Ω is the solid an-
gle. Angular brackets denote an average over the initial ∞dy Φ (y,1/ν2,1/η2) 1/2
λ (t)=t1/2 0 a . (12)
caonnddmitiaognnsebtiecinegnecrogniseisdienretdh(issereabngeleowof).leTnghtuhssthcaeleksinaerteic a " R0∞dy Φa(y,1/ν2,1/η2)y#
R
K Similarly, the integral scales La(t,ν,η,2π/K,L)
≡
Ea(t,ν,η,2π/K,L)=Z2π/Ldk Ea(k,t,ν,η,K,2π/L).(4) [th2Keπ/sLudbkscrEipat(ka,t,isν,vη)/fokr]/th2eKπ/flLudidk iEnat(ekg,rat,lνs,cηa)le(awnhderbe
R R
By combining Eqs. (2) and (3) we get for the magnetic integral scale), also grow as a power
of t with exponent equal to 0.5 for q = 1 in the limits
(k/l,l1−ht,l1+hν,l1+hη,K/l,2π/Ll)= 2π/L 0, K .
Ea → →∞
l1+2h (k,t,ν,η,K,2π/L). (5) A consequence of the positive-definiteness of
a
E the energy spectra is that the helicity spectra
HbeenscuepfporrtehssetdhefodrenpoetnadteionncaelocfonEvaenoinenKce.aIntdis2πus/eLfuwl tilol s[avti(skfy,t,νr,eηa)lizba(bki,litt,yν,ηc)o]1n/s2t,raianntsd,, |HCM((kk,,tt,,νν,,ηη))| ≤
E E |H | ≤
define the functions b(k,t,ν,η)/k, where C(k,t,ν,η) and M(k,t,ν,η)
E H H
are the cross- and magnetic helicity densities defined
Φ (k,t,ν,η) k1+2h (k,t,ν,η) (6) like the energy densities (Eqs. (3)). Hence, in the limits
a a
≡ E
2π/L 0, K (with variables other than k and t
→ → ∞
in terms of which Eq. (5) becomes suppressedfornotationalconvenience),thecross-helicity
(for q =1) satisfies
Φ (k/l,l1−ht,l1+hν,l1+hη)=Φ (k,t,ν,η); (7)
a a
∞
when differentiated with respect to l, this yields, in the HC(t) dk C(k,t)
| |≤ |H |
limit l 1, Z0
→ 1 ∞
dy [Φ (y)Φ (y)]1/2 (13)
v b
∂Φ ∂Φ ∂Φ ≤ 2t
a +(1 h) a +(1+h) a + Z0
−∂lnk − ∂lnt ∂lnν
∂Φ and the magnetic helicity (for q =1) satisfies
a
(1+h) =0. (8)
∂lnη
∞
H (t) dk (k,t)
M M
The general solution of the above equation is | |≤ |H |
Z0
1 ∞ Φ (y)
b
Φ (k,t,ν,η)=F ((1 h)lnk+lnt,(1+h)lnt+ dy . (14)
( a1+h)lnν,(1+a h)l−nt+( 1+h)lnη). (9) ≤ 2√tZ0 √y
− −
The results obtainedaboveapply fortimes t<t , where
∗
This solution corresponds to
t is the crossovertime at which L (t) becomes equal to
∗ a
thesizeofthesystem(orthelinearsizeofthesimulation
(k,t,ν,η)=kq Φ (k(3+q)/2t,t(1−q)/2ν−(3+q)/2,
Ea a box in a DNS). For t > t∗, finite-size effects, that might
t(1−q)/2η−(3+q)/2), (10) well be nonuniversal, modify the decay of E (t).
a
3
III. NUMERICAL RESULTS 10−10
n=9
n=12
10−15
We now report on the numerical studies we have car-
ried out to check the results given above. In most of our 10−20
runs, we concentrate on the region 0 < t < t . We use
∗
dseonutbaltei-vpereccaisseiosnthaartitohumrerteics,ulbtustarheavuenacffheecckteedd iifnwreepursee- s∈ /kbn10−25
quadruple-precision arithmetic. 10−30
Shell models comprise a set of ordinary differential
10−35
equations containing suitable non-linear coupling terms
that respect the analogs of the conservation laws of the (a)
10−40
3DMHD equations in the inviscid, unforced limit. They 101 102 103k2τs 104 105
n
exhibit an energy cascade in the presence of viscosities.
10−5
Weconfineourstudytoashellmodelproposedin[11,12] n=9
n=12
that enforces all the ideal 3DMHD invariants in the in-
viscid, unforced case, that reduces to the well-known 10−10
GOY[14]modelforfluidturbulencewhenmagneticterms
aresuppressed,has noadjustable parametersapartfrom
the fluid and magnetic Reynolds numbers, and exhibits s∈b10−15
thesamemultiscaling(withinerrorbars)asthe3DMHD
equations.
10−20
The unforced shell-model equations are[15]
dz± (b)
dtn =ic±n −ν+kn2zn±−ν−kn2zn∓, (15) 10−21500 102 k3/2τs 104 106
n
with the complex, scalar Els¨asser variables z± (v
bn), i = √ 1, and the discrete wavenumbersnkn≡= kn02±n FIG. 1: (a) Plot of Ebs/kn versus kn2τs for shells n = 9 and
− n = 12 with q = 1, illustrating the data-collapse implied by
(k sets the scale for wave-numbers), for shell index n
0
thescaling form of Eq. (10).
(n=1...N, for N shells) with
(b)Theanalogforq=0illustratingthelackofdatacollapse.
c± = [a k z∓ z± +a k z± z∓
n 1 n n+1 n+2 2 n n+1 n+2
+ a k z∓ z± +a k z± z∓
3 n−1 n−1 n+1 4 n−1 n−1 n+1 and Re0,s equal 34246 for q = 1 and 81920 for q = 0).
+ a5kn−2zn±−1zn∓−2+a6kn−2zn∓−1zn±−2]∗. The initbial velocity and magnetic fields are taken to be
Here a1 = 7/12, a2 = 5/12, a3 = 1/12, a4 = vn0 = kn(1+q)/2eiθn and b0n = kn(1+q)/2eiφn, with θn and
− φ being independent random variables distributed uni-
5/12, a = 7/12, a = 1/12 and ν = ν n
5 6 ±
−η. Shell-model−analogs of the total energy (Es =± formly between 0 and 2π (our shell-model energy densi-
a((EHHnvsdMCss+i≡n≡E(tbs1h/e≡n2()f−o(Pl11l/o)n2wn()|vibnPnngb|,n∗n2/(+t|khvvnenn∗)|bs2anur+)pe)e,|crbaosnncn|rds2ie)ptr)ht,veestdmhseiatfagcνnnr±deotss=iscfo0hhr.eelTlHsiiccheiiettrlyyel kt|HineFsM0o,Φsbr|ae/(iHqnkng2Mm=τdasxe,,fi11s/n,.νedE21,q01a.−/sη6(E,21as)0w.)≡hehIrhnae|asHnFt|IMm2hG/eak.xn,fs1oi()≃ra.m)EInwTE0,easas/l(slkkh1ono.u,wrτsro)unn=sa,
model. PWe solve Eqs. (15) numerically by an Adams- log-log plot that a data-collapse occurs for representa-
Bashforth scheme[16] (step size δts = 10−2) and use tive shells, here n = 9 and n = 12, when bs(kn,τs)/kn
N =22 shells with k =1/16 and ν =η =10−4. is plotted against k2τs. In FIG. 1(b) we Eshow the ana-
0 n
In decaying turbulence, it is convenient to measure log for q =0 illustrating the lack of data collapse. Note,
time in units of the initial large eddy-turnover times. however,thatthecollapseimprovesask 0;asweshow
→
For our shell model these are τ 1/(a0,s k ) with below this leads to a power-law decay of Es over a lim-
a0,s [ a0 2 ]1/2, the root-ame≡an-squarrmesva1lues of ited rangeof t andis relatedto the ‘permanbence of large
rms ≡ h n| n| i
the initial velocities and magnetic fields. Since τ are eddies’.
a
P
calculated at the start of the simulation runs, when the In FIG. 2(a) we show the decay (for q = 0 and 1)
velocity and magnetic fields differ only in phase (see be- of the magnetic energy Es with an observed slope of
b
low), here τ =τ equal 18.7 for q =1 and 7.8 for q =0. 1.000 0.001 (with errors from least-square fits) for
v b
Our runs are reported in terms of τs t/τ (t is the −q = 1. T±he final point on this graph for q = 1 exhibits
a
product of the number of iterations an≡d δts), and are the beginnings ofthe crossoverat t=t where Ls(t) be-
∗ a
ensemble averaged over 100 independent initial condi- comes comparable to k−1, the analog of the system-size
1
tionswithvaryingphases. WedefineRe0,s v0,s /(k ν) in the shell model.
v ≡ rms 1
and Re0,s b0,s /(k η) to be the values of the ini- Forq =0,anominalslopeof 0.66 0.01maybefitted
tial fluidb an≡d mrmagsneti1c Reynolds numbers (here Re0,s overaportionofthecurvebetw−eenτs±=1.28 10−2and
v ×
4
100 102
q=0 q=0
q=1 q=1
10−2
10−4
s0,sE/Ebb10−6 s0,sL /Lbb101
10−8
(a)
10−10 100
10−5 100 105 1010 10−6 10−4 10−2 100 102
τs τs
100
a=v FIG. 3: Log-log plots of the growth of the magnetic inte-
a=b
gral scale Ls (for q = 0 and 1) as a function of time with
b
E0,s = E0,s. The observed slope is 0.500±0.002 for q = 1
v b
in agreement with the magnetic integral-scale analog of Eq.
0,sEa10−1 (12).
sE/a
100
(a)
10−10
(b)
10−2
10−5 100 105 10−20
τs
s∈b
FIG.2: (a)Log-logplotsofthedecayofthemagneticenergy 10−30
Ebs as a function of time τs. The observed slope is −1.000± 06.92x10−2
0.001 for q = 1 in agreement with Eq. (11). A nominal 10−40 3.82
slope of −0.66±0.01 can be fit to the curve for q =0 in the 2.09x102
range τs = 1.28×10−2 and τs = 7.70×102. (b) Log-log 10−50 1.14x104
plots of the decay of the kinetic and magnetic energies (Es 10−2 100 102 104 106
v k
and Es respectively), without initial equipartition of energy n
(E0,s =b 103E0,s) for q=0. 100
v b (b)
10−10
τs = 7.70 102. This slope is in agreement with the
law obtaine×d from a hypothesis of ‘permanence of large 10−20
eddies’, whereinthe viscoustermsarenegligible atsmall s∈b
k in Eq. (10) leading to a decay law of 0.667. The 10−30
−
hypothesisof‘permanenceoflargeeddies’implies thatif 0
E0(k) kq, for k 0, then the total energy decays as 10−40 2.92x10−2
ETa (∼t t0)−2(13++qq→) as discussedinthe fluid turbulence 184.7.57.905×103
cont∼ext i−n Ref. [17]. 10−1500−2 100 102 104 106
k
So far we have used initial conditions with initial n
equipartition,i.e.,E0,s =E0,s. Wefindthatthisismain-
v b FIG. 4: Log-log plots of the MHD shell model magnetic
tainedduringthedecaywhichiswhywehaveshownplots energydensityEs as afunctionof wavenumber(temporalse-
obfotohnlEyvsthaenmdaEgbsnedteiccaeynearsgty−.1Foforrq0=<1ta<ndtE∗ v0a,nsd6=mEab0i,ns-, qanuden(cbe)wqit=h1τ.s vbalues indicated in the legend) for (a) q = 0
tain their initial ratio. In FIG. 2(b) we plot the tempo-
ral evolution of the kinetic and magnetic energies with-
oRuet0v,isni=tia1l0e3qRuei0bp,as)rtfiotironq,=by0.seUttninligkeEtv0h,se=ca1s0e3qEb=0,s1(,htehree [ohb(sernve|bdn|s2lo/pkne)i/s 0.5n0|0bn|20i].0a0s2aforfuqnc=tio1n. IonfFtIimGes.. 4T(hae)
kineticandmagneticpartsexchangeenergyhere,butthe anPd (b), we plotPa tem±poral sequence (with seperation
dominantone(thekineticenergyinourplot)decayswith in units of τs indicated in the legend) of the magnetic-
a slope 0.490 0.002. energy density s as a function of wavenumberfor q =0
− ± Eb
In FIG. 3, we plot (for q = 0 and 1) the shell and 1. The kinetic-energy density evolves in a similar
model analog of the magnetic integral scale Ls manner for the case of initial equipartition. Note that
b ≡
5
100 q=0
q=0
q=1
q=1
10−1
0E/Ebb 0L/Lbb
10−2
10−3 100
10−3 10−2 10−1 100 10−3 10−2 10−1 100
τ τ
FIG.5: Log-logplotsofthemagneticenergyEbasafunction FIG. 6: Log-log plots of the magnetic integral scale Lb as a
of time for q=0 and 1 for 3DMHD from our 803 DNS. function of time for q = 0 and 1 for 3DMHD from our 803
DNS.
the evolution of these energy spectra illustrate quanti-
tatively, the qualitative notion of ‘permanence of large
eddies’: As time progresses,the large-k part of the spec- 10−5
tra get modified by viscous effects; however, the power- q=0
law part at small k maintains its initial form for t . t .
∗ 10−6
Wehavealsoexplicitlycheckedthatthebounds(13)and
(14) are respected in our shell-model simulations.
10−7
ForourDNSstudyofthe3DMHDequations,weusea
pseudospectral method[18] to solve Eqs. (1) in a cubical ∈b
box of side 2π with periodic boundary conditions and 10−8
803 Fourier modes. For the temporal evolution, we use
an Adams-Bashforth scheme (step size δt = 0.02) and 10−9
ν = η = 10−2 (we exclude any hyperviscosities). We
define the initial fluid and magnetic Reynolds numbers 10−10
100 101 102
to be Re0 2πv0 /ν, Re0 2πb0 /η (here Re0 and k
Re0 equavl≡34 forrmqs= 1 andb ≡17 forrqms= 0) with a0v 10−5
b rms ≡ q=1
[ a(k)2 ]1/2 the root-mean-squares of the velocities
h k| | i
andmagnetic fieldsandinitiallarge-eddyturnovertimes
toPbe τ 2π/a0 . As in the shell-model case, τ =τ 10−6
a ≡ rms v b
(herethey equal115.4forq =1and228.2forq =0)and
τ ≡t/τa (tistheproductofthenumberofiterationsand ∈b10−7
δt). The initial velocity and magnetic fields are taken to
bev(k) kq/2eiθk,b(k) kq/2eiφk withθ andφ being
k k
random∼variables distrib∼uted uniformly between 0 and 10−8
2π. OurDNSresultsarerestrictedtoinitialequipartition
of energy between the velocity and the magnetic field.
10−9
InFIG.5weplotthemagneticenergyEb asafunction 100 101 102
k
of time for q = 0 and 1. A slope of 1.057 0.004 is
o1b0−se2rvinedrefoarsoqn=ab1lebaegtwreeeemneτnt=w8i.t7h×E1q0−.−(31a1n).d±Iτn=FI8G.7.×6 aFIfGun.c7t:ionLoogf-lwoagvpenloutmsboferthke(mteamgnpeotriaclesneeqrugeyncdeeninsitsyteEpbsaosf
we plot the magnetic integral scale Lb as a function of 0.02τ for q = 0 and 0.05τ for q = 1) for 3DMHD from our
timeforq =0and1. Aslopeof0.401 0.003isobserved 803 DNS.
forq =1betweenτ =8.7 10−3andτ±=8.7 10−2. We
× ×
believetheslightdiscrepancywiththemagneticintegral-
scale analog of Eq. (12) arises because of the limited
spatialresolutionoftheDNS. InFIGs. 7(a)and(b), we
plot a temporal sequence (in steps of 0.02τ (q = 0) and FIG. 4. We believe this is because of the low spatial
0.05τ (q = 1)) of the magnetic energy density as a resolutionof our DNS for 3DMHD. Our DNS results are
b
E
function of wavenumber k for q = 0 and 1. The small-k presentedheretocomplementourshell-modelresultsand
part of the energy spectra in FIGs. 7(a) and (b) change to show, in particular, that our general conclusions are
somewhat more than their shell-model counterparts in not shell-model artifacts.
6
IV. DISCUSSION AND CONCLUSION equal initial kinetic and magnetic energies in the first
two shells, the MHD-shell-model simulations exhibit an
To summarize,we have derivedrelations for the decay energy cascade to higher shells. Once this cascade pro-
of the kinetic and magnetic energy, the growth of the cess is complete, the energy densities display a part that
Taylorandintegralscales,andbounds for the cross-and canbefittedtoak−5/3 formandtheintegralscalemoves
magnetic helicity in unforced, incompressible, homoge- towards small wavenumbers. The point we would like to
neous and isotropic 3DMHD turbulence. We have con- highlight here is that the evolution to a k−5/3-type en-
firmed our results numerically via shell-model and DNS ergy spectrum does not occur for the power-law initial
studies. We show explicitly that our results about the conditions we have concentrated on in this paper. This
power-law decay of the energies hold for times t < t , isespeciallyclearinthe MHD-shell-modelstudies shown
∗
where t is the time at which the integral scales become in FIG. 4. The reason for this is the ‘permanence of
∗
comparableto the systemsize. Fort<t ,ournumerical large eddies’: For small k, the energy spectrum retains
∗
results are consistent with those predicted by the princi- thepower-lawdependenceoftheinitialenergyspectrum.
ple of ‘permanence of large eddies’. Our study also has Deviationsfromthisinitialpowerlawbecomesignificant
some implications for suggestionsof ‘strong universality’ only for times t > t∗, at which the time-dependent in-
in decaying turbulence as we discuss below. tegral scale La(t) becomes comparable to the size of the
In an early study[19], an expression for the kinetic- system. Given the parameters, such as ν and η, that
energy density was hypothesized by assuming the exis- we have used, by time t∗, the Reynolds numbers is suffi-
tence of an ‘eddy-viscosity’. An energy decay similar to ciently low that a fresh energy-cascade to large k is not
whatwegetforq =1wasthenconsideredinthe context established again. Thus, suggestions of ‘strong univer-
of decaying fluid turbulence. However, this study did sality’ should be made with caution, for they apply only
not consider the initial-condition dependence of the de- to a class of initial conditions that does not include the
caylawsortheeffectofthegrowthoftherelevantlength power-law initial conditions used here for the decay of
scales. 3DMHD turbulence.
In a recent study[20], arguments have been given for The initial conditions used inRef. [9]are alsoqualita-
‘strong universality’ in forced and decaying fluid turbu- tively different from those that we use here in so far as
lence in a shell model. For decaying turbulence, this they begin with a large energy in a few Fourier modes.
means that scaling exponents of the n-th order velocity Thus they develop energy cascades and do not explore
structure functions and their coefficients in the isotropic the types of decay that are associated with the ‘perma-
sector, normalised by the mean energy flux, are univer- nence of large eddies’ we concentrate on here.
sal and the same as those for structure functions that
are obtained for forced turbulence. For the purpose of
our discussion here, this would imply that, irrespective Acknowledgments
of the initial condition used, after an initial period of
decay, the energy spectrum should evolve towards one C.K. thanks Tejas Kalelkar for useful discussions,
with a part that goes as k−5/3 (aside from multiscaling CSIR(India)forfinancialsupport,SERC(IISc)forcom-
corrections which we are not concerned with here). We putational resources, and R.P., the Indo-French Centre
havebeen able to getresults similar to those of Ref. [20] forPromotionofAdvancedScientificResearch(IFCPAR
by using the MHD shell model (Eqs. (15)) and initial ProjectNo. 2402-2)for support. We thank A. Basuand
conditions as in Ref. [20]. In particular, if we start with D. Mitra for discussions.
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