Dissipative Structures in Supersonic Turbulence

0.1 MB·English

by  Liubin Pan

DissipativeStructures inSupersonic Turbulence
Liubin Pan,1 Paolo Padoan,1 and Alexei G. Kritsuk1
1Department of Physics, University of California, San Diego,
CASS/UCSD 0424, 9500 Gilman Drive, La Jolla, CA 92093
Weshow that density-weighted moments of the dissipation rate, ǫl, averaged over ascale l, insupersonic
turbulence can be successfully explained by the She and Le´veˆque model [Phys. Rev. Lett. 72, 336 (1994)].
Ageneral methodisdeveloped tomeasurethetwoparametersofthemodel, γ andd,baseddirectlyontheir
physical interpretations as the scaling exponent of the dissipation ratein the most intermittent structures (γ)
andthedimensionofthestructures(d). Wefindthatthebest-fitparameters(γ = 0.71andd = 1.90)derived
fromtheǫl scalingsinasimulationofsupersonicturbulenceatMach6agreewiththeirdirectmeasurements,
confirmingthevalidityofthemodelinsupersonicturbulence.
9
0 PACSnumbers:
0
2
Supersonicturbulenceisubiquitousinthecoldinterstellar plainedlater. Thismodelisverysuccessfulinpredictingζpin
n
medium [1] and is believedto play a crucialrole in the pro- incompressibleturbulencewithhighaccuracy.
a
J cess of star formation [2, 3]. If supersonic turbulence were In this Letter, we study the fluctuations of the dissipation
characterizedbyuniversalstatistics, asoftenassumedforin- rate in supersonic hydrodynamic(HD) turbulence using nu-
7
1 compressibleturbulence,theuniversalitycouldconstitutethe mericalsimulations. We showthatthe simulationresultsfor
foundationsfor a statistical theory of star formation. In this the scaling exponents, τ , are well represented by eq. (1),
p
] Letterwefocusonthestatisticalpropertiesofthemostinter- suggestingtheSLformulationforthescalingbehaviorofthe
h
p mittentstructures(MISs)ofsupersonicturbulence. dissipation rate, originally proposed for incompressible tur-
- Thetheoryoffully developedturbulenceassumesthatthe bulence,maybeappliedtosupersonicturbulenceaswell. We
o
scaling behavior of small-scale fluctuations in the inertial present a method to directly measure the parameters γ and
r
t range is flow-independent,e.g., the momentsof the velocity d accordingto their physicalinterpretation,which is general
s
a difference,hδv(l)pi =h(v(x+l)−v(x))pi ∝lζp,haveuni- andnotlimitedtothesupersonicregimeofinteresthere. This
[ versalscalingexponents,ζ . Theuniversalstateforfullyde- methodcan be used to test the validity of the physicalinter-
p
3 velopedincompressibleturbulenceproposedbyKolmogorov pretationoftheSLmodelinanyofitsapplications.Forsuper-
v in1941(K41)[4],withζp = p/3,hasbeenshowntodeviate sonicturbulence,wefindthattheparametersderivedwiththis
0 significantly from ζ measured in both experiments and nu- methodareinexcellentagreementwiththevaluesthatbestfit
p
3 mericalsimulations,atp>3. Thisdiscrepancyisduetospa- τ ,whichconfirmsthephysicalinterpretationofthemodel.
3 p
tial fluctuations in the dissipation rate, neglected in the K41 Insteadofdirectlyinvestigatingthestatisticsofthedissipa-
1
. theory[6]. Thescalingexponentsoftheaverageenergydis- tionrate, moststudiesofthis modelare primarilyconcerned
8
sipation,ǫ ,overascalel(seeeq. (3))givecorrectionstothe withthescalingexponent,ζ ,ofthevelocitydifference,δv(l).
0 l p
K41theory,referredtoasintermittencycorrections[5,7]. A Themodelpredictsζ =(1−γ)p/3+γ(1−βp/3)/(1−β),
8 p
0 carefulstudyoffluctuationsintheenergydissipationisessen- whichfollowsfromtherefinedsimilarityhypothesis,δv(l)∼
: tialforunderstandingintermittencyinturbulence. ǫ1/3l1/3, and eq (1). Assuming that the largest available ki-
v l
i TheintermittencymodelbySheandLe´veˆque[7](hereafter netic energyin the strongest structures is ∼ U2, with U be-
X the SL model) is based on a hierarchy of dissipative struc- ing the rms velocity, and that the timescale in these struc-
r turesofdifferentintensitylevels, characterizedbytheratios, tures follows the usual Kolmogorov scaling, t ∝ l2/3, She
a l
ǫl(p) = hǫpl+1i/hǫpli, of successive moments of ǫl. With in- and Le´veˆque argued that ǫ(∞) ∼ U2/tl ∝ l−2/3, i.e., γ =
creasing order, p, ǫ(p) represents structures of increasing in- 2/3[21]. With this γ and with d = 1, correspondingto fil-
l
tensity and ǫ(∞) corresponds to the MISs. By invoking a amentarydissipativestructures,thevaluesofζp predictedby
l thismodelagreewithexperimentalresultsofincompressible
hypothetical“hiddensymmetry”thatrelatesthishierarchyof
turbulencewithanaccuracyofabout1%[8].
structures to the most intermittent ones, the model predicts
Although not directly measured from the MISs, (cf. [9]),
thescalingexponents,τ ,oftheenergydissipationmoments,
p
hǫpi∝lτp,ofallorders,p, γ =2/3hasbeenadoptedinmostapplicationsofthemodelto
l incompressible[10]andsupersonicMHDturbulence[11,12].
τ =−γp+γ(1−βp)/(1−β). (1) Thedimensiondwasobtainedeitherfromtheassumedgeom-
p
etryoftheMISs(d = 2forcurrentsheetsorshocksinMHD
Theparameterγisthescalingexponentofthedissipationrate orsupersonicturbulence),orfromthebestfittothenumerical
in the MISs, ǫ(l∞) ∝ limp→∞lτp+1−τp ∝ l−γ, and β is re- velocitystructurefunctions[12,13].Theseworkshaveshown
latedtoγ andtotheHausdorffdimension,d,oftheMISsby thattheSLmodelwith2DMISsisgenerallyconsistentwith
γ/(1−β)=D−d,whereD =3forthree-dimensional(3D) simulationsofMHDandhighlycompressibleturbulence.
turbulence. Thephysicalmeaningofthisrelationwillbeex- However, a strict verification of the validity of this model
2
requires a demonstration that the parameters that fit τ also
p
havethedeclaredphysicalmeaning,otherwisetheagreement γ = 0.71, d = 1.90
0
betweenthe modeland the simulationsmay be a mere coin-
cidence. Inthepresentworkwethusobtainγ anddbothby
adirectmeasurementfromtheirinterpretations,andbyfitting -0.5
τ . We are also interested in deriving the dimension of the
p
MISs in supersonic HD turbulence where about 1/3 kinetic τp
-1
energydissipates in dilatationalmodesand 2/3 in solenoidal
modes, with strongest shocks generally coinciding with the
locationsofstrongestshearandvortices. -1.5
Measuringτ .–Wetakethe10243simulationofsupersonic
p
HDturbulenceforisothermalidealgaswitharmsMachnum-
-2
ber of 6 from reference [14]. The simulation employs the 0 1 2 3 4
p
piecewiseparabolicmethodtosolvetheEulerequation[17].
We focusonthestatistics ofǫ . Thedissipationrateperunit
l FIG.1: Scalingexponents, τp, of thedissipationrate, ǫl, averaged
massatagivenpositionandtimeiscalculatedby[15],
over 9 snapshots. Error bars indicate snapshot-to-snapshot varia-
tions.Thebestfitgivesγ =0.71andd=1.90.
ǫ(x,t)=(2Re)−1(∂ v +∂ v −(2/3)δ ∂ v )2 (2)
i j j i ij k k
whereReistheeffectiveReynoldsnumbercontrolledbynu- Wecalculatemomentsofǫ fromeq. (4)for9snapshotsof
l
mericaldissipation. Wecomputethevelocitygradientsatthe oursimulation,coveringmorethan5dynamicaltimes.Weob-
resolutionscaleandassumeReisconstant. tainτ fromleast-squarefitstothelog (hǫpi)-log (l)curves
Wecalculatetheaveragedissipationrate,ǫl(x,t),atascale ineacphsnapshot. Theresultsaresho1w0ninl Fig. 11,0wherethe
l around x, from the definition given in [5] (generalized to data points and the error bars are, respectively, the average
accountfordensityfluctuations), exponents and the standard deviations over the 9 snapshots.
The error bars are negligible for p < 2, meaning that there
1
ǫ (x,t)= ρ(x+x′,t)ǫ(x+x′,t)dx′ (3) arelittle snapshot-to-snapshotvariationsfortheexponentsat
l ρ (x,t)V(l) Z
l loworders. Thescatterincreaseswiththeorder,andtheerror
|x′|<l
bar at p = 4 is alreadysignificant(7%). We find that, start-
whereV(l) = 4πl3/3is thevolumeofa sphericalregionof ingfromp=4,thelog10(hǫpli)-log10(l)curvesarenolonger
size l, and ρ (x,t) = 1/V(l) ρ(x + x′,t)dx′ is the wellfitbystraightlinesandthuswe onlyshow resultsup to
l |x′|<l the4thorder. Note,however,thatthe4thordermomentofǫ
average density of that region.RFor convenience, we divide l
correspondsto12thordermomentofδv(l).
thesimulationboxintocubes(insteadofspheres)ofdifferent
Comparing with eq. (1), we find that the SL model with
sizesinourcomputations.
Themoments,hǫpi,ofǫ ,canbeevaluatedby γ = 0.71and β = 0.35 (d = 1.90)givesan excellentfit to
l l the numericaldata. The fit shows that the SL model can be
1 successfully applied to the density-weighted statistics of the
hǫpli= ρ¯V Z ǫpl(x,t)ρl(x,t)dx (4) dissipation rate in supersonicturbulence. As mentionedear-
lier,ademonstrationthatthebest-fitparametersindeedcarry
whereV isthetotalvolumeofthesystemandρ¯istheoverall theirphysicalmeaningisneededto verifythevalidityof the
averagedensity. WehaveusedtheFavre[16]densityweight- model. Tothisend,wedirectlymeasuretheparametersfrom
ingfactor,ρ /ρ¯,toaccountforthedensityvariationsincom- the simulation data. A fairly large range of parameter pairs,
l
pressibleturbulence. Withthisdensityweighting,thefirstor- (0.67-078)forγ,andacorrespondingrangeof(2.04-1.60)for
der moment, i.e., hǫ i, is independent of scale l, as follows d, can give acceptable (but poorer) fits to the numerical re-
l
fromeqs. (3)and(4),resultingin τ = 0 (see Fig. 1). This sultsforτ withinthe2σ errorbars. IftheSLmodelworks,
1 p
suggeststhat,iftherefinedsimilarityhypothesisappliestosu- adirectmeasurementwouldfixtheseparameters.
personic turbulence, the density-weighted third-order veloc- Measuringγ.–Weobtainγ directlybymeasuringtheaver-
ity structurefunctionin compressibleturbulencewouldhave age dissipation rate profile aroundthe MISs. We first locate
ζ = 1,anexactresultforincompressibleturbulence,known theMISsinthecomputationaldomain,byselectingcellswith
3
as Kolmogorov’s 4/5 law [14]. Since the SL formula, eq. dissipation rate larger than a given threshold, ǫ , set to be
th
(1), gives τ = 0, it is appropriate to compute the density- close to the maximum dissipation rate over the domain, ǫ .
1 m
weightedmomentsfromthesimulationdataandcomparewith Wewillcallthesecellsthedissipationpeaks.Wethenusecu-
the model. The Hausdorffdimensionof the MISs we obtain bicboxesofdifferentsizes,l,tocovereachpeak,andevaluate
fromfittingτ isthusdensity-weightedinthesenseofdensity- theaveragedissipationrateineachbox,ǫ (l),througheq.(3)
p p
weightingineq. (4). Foravalidcomparison,wewillinclude (with x at the peak). Taking the average over all peaks, we
density-weightinginourdirectmeasurementofd. obtainanaveragedissipationrate,hǫ (l)i,asafunctionofthe
p
3
0
d=1.92
0.710 -1 d=1.90
d=1.90
0.705
-0.5 γ
〉]m 0.700 -2
ε〉〈ε(l)/p -1 0.3 εt h0/.ε5m 0.7 [P(l)]10 -3 2.00
〈log[10 -1.5 11γ=//320 .mm70aa5xx log -4 d11..9905
γ=0.710
2/3 max 0.1 0.2 0.3
γ=0.710 -5 εth/εm
-2
0 0.5 1 1.5 2 0 0.5 1 1.5 2
log (l) log (l)
10 10
FIG. 2: Average dissipation rate, hǫp(l)i, (normalized to average FIG.3:Probabilityoffindingcubesofsizelwithaveragedissipation
maximum, hǫmi) around dissipation peaks selected by thresholds, ratelarger than 1/6, 1/4 and 1/3 ǫml−0.71. Scaling exponents of
1/3,1/2and2/3ǫm. Best-fitlinesgiveγ =0.705,0.710and0.710, P(l)correspondtod=1.92,1.90and1.90,respectively. Theinset
respectively.Theinsetshowsconvergenceofmeasuredγwithǫth. showsconvergenceofmeasureddwithǫth.
cubesize,l. Theslopeoftheprofileisexpectedtobetheex- e.g., using a box-countingmethod, is challenging. Here we
ponent, γ. We calculated the average ǫ (l) in two different take a simpler approach: we compute the probability, P(l),
p
ways: withandwithouttheaveragedensityρp(l)inaboxof of finding an MIS in a cube of size l, and derive d from the
sizelasaweightingfactor. Wefindlittledifferencebetween scalingofP(l)withl,basedonthephysicalargumentgiven
the slopesobtainedin the two ways, implyinga weak corre- above We need a criterion to judge whether a cube of size l
lationbetweenthedissipationrateandthedensityaroundthe in the simulation box contains an MIS at that scale. Based
peaks[22]. Weincreasethethresholdandcheckwhetherthe on the log-Poisson version of the SL model [8, 19], we find
slopeofǫ (l)converges. Theconvergedslopeistheparame- that the appropriatecriterionis thatthe cube in questionhas
p
terγ thatwepursue. anaveragedissipationratelargerthanathresholdthatscales
OurresultisshowninFig. 2forthreethresholds,1/3,1/2 likeǫ l−γ withl.Thefactorl−γ accountsforthedecreaseof
th
and2/3ǫ .Thethreecurvesaretheaverageoverallthepeaks the average dissipation rate in the MISs with scale. We will
m
inthesame9snapshotsusedtocalculateτ . Theprofilesare letǫ approachǫ inthesimulationbox.
p th m
approximatedwellbypowerlaws(exceptatl =1,i.e.,atthe The chosen threshold is justified as follows. The SL
resolutionscale)andwefindγ =0.705,0.710and0.710,re- model is equivalent to a log-Poisson distribution for ǫ , i.e.,
l
spectively, for the three thresholds. This value is very close P(ǫ )dǫ = (l/L)(D−d)Σ∞ λnP (ln(ǫ lγ) − ln(ǫ¯Lγ) −
l l n=0n! L l
to2/3proposedbySheandLe´veˆque,suggestingthattheKol- nln(β))dln(ǫ ) where L is the integral scale, ǫ¯the overall
l
mogorov scaling for the timescale in the MISs, tl ∝ l2/3, dissipation rate, and λ = (D −d)ln(L/l)[3, 20]. The dis-
applies also to supersonic turbulence. The measured γ con- tribution,P (ln(ǫ /ǫ¯)), of the dissipationrate, ǫ , atL, de-
L L L
vergesto0.71atthethresholdof1/2ǫ ,whichconcideswith pendson the drivingforce and is thus non-universal. At the
m
the value obtained from the best fit to τp. Besides showing integral scale, ǫL is approximatelyequal to ǫ¯, thus PL(x) is
the applicability of the SL model to supersonic turbulence, supposed to be narrow and decrease very rapidly with in-
wehavethusverifiedthattheparameterγ carriestheprecise creasing x. If the SL model is valid for supersonic turbu-
physicalmeaninginthemodel. lence,theprobability,P(l),offindingaregionofsizel with
Measuring d.–The Hausdorff dimension, d, enters the SL ǫ > ǫ l−γ is given by the cumulative probability P(l) =
l th
model through the following argument, which also provides (l/L)(D−d)Σ∞ λn P (x)dxwhere
n=0n! ln(ǫth/ǫ¯)−γln(L)−nln(β) L
an explanation of the relation γ/(1− β) = D −d. In the landLareinunitsofRtheresolutionscale.Duetotherapidde-
limit p → ∞, the contribution to hǫpi would be primarily creaseofP (x)withx, foralargeǫ thecontributionfrom
l L th
from the MISs at scale l. Since the average dissipation rate the n-th term to P(l) decreases quickly with n because the
in regions of size l containing MISs goes like l−γ, we have lowerintegrallimitincreaseswithn(sinceβ <1).Asǫ in-
th
hǫpi ∝ l−γpP(l), in the limit p → ∞, where P(l) is the creases,thecontributionwouldbemoreandmoredominated
l
probabilityoffindingaregionofsizelthathostsadissipative by the n = 0 term, which is ∝ l(D−d). Therefore, in our
structureof the highestlevelat scale l. A geometricconsid- measurement,weincreaseǫ andcheckwhetherthescaling
th
eration suggests that P(l) ∝ lD−d if the dimension of the exponentof P(l) converges. If the SL model is correct, the
MISsis d [18]. This giveshǫpi ∝ l−γp+D−d as p → ∞. It convergedexponentisexpectedtobethecodimensionD−d
l
immediatelyfollowsfromeq. (1)thatγ/(1−β)=D−d. oftheMISsandtoagreewiththatderivedfromτ . Tobecon-
p
Directly measuring the Hausdorffdimension of the MISs, sistentwiththedensity-weightingin thedissipationratemo-
4
ments,eachcubethatsatisfiesthecriterionisgivenaweight- plicationsandbyNASAHighEndComputingProgram.
ingfactorproportionaltotheaveragedensityinthecube.
Noteaddedinproof.–TheauthorsaregratefultoDr. Chris
Fig.3showsthescalingofP(l)withl(averagedfromthe9
McKeeforpointingoutanerrorinEq.(2)inanearlydraftof
snapshots),for3differentǫ :1/6,1/4,and1/3ǫ .Forthe3
th m
thisLetter.
choicesofǫ ,thescalingexponentsofP(l)are,respectively,
th
1.08,1.10,and1.10(meaningd = 1.92,1.90and1.90). As
ǫ increases,dconvergesto1.90,whichagainagreesexactly
th
withthebest-fitvaluefromthescalingofthedissipationrate
moments. Togetherwiththemeasurementofγ,thisestimate
[1] R.B.Larson,Mon.Not.Roy.Astron.Soc.,194,809,1981;M.
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l incorporatedinthefirstedition(inRussian)ofthebook: L.D.
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1
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This research was partially supported by a NASA ATP
[21] A direct check for theaccuracy of thisargument can bedone
grant NNG056601G, by NSF grants AST-0507768, AST-
withourmethodformeasuringγ.
0607675 and NRAC allocation MCA07S014. We utilized [22] Wedonotusedensityweightingforhǫp(l)ishowninFig.2.It
computing resources provided by the San Diego Supercom- ismoreappropriatetouseitintheprobabilityoffindingMISs
puterCenter,bytheNationalCenterforSupercomputingAp- ateachscalewhenmeasuringd.