Models of turbulent dissipation regions in the diffuse interstellar medium

0.65 MB·English

by  Benjamin Godard

Astronomy&Astrophysicsmanuscriptno.0803prin (cid:13)c ESO2009
January23,2009
Models of turbulent dissipation regions in the diffuse interstellar
medium
B.Godard1,E.Falgarone1 andG.PineaudesForeˆts2,1
9
0
0
2

n
a
J 1 LRA/LERMA,CNRSUMR8112,E´coleNormaleSupe´rieure&ObservatoiredeParis,Paris
3 2 Institutd’AstrophysiqueSpatiale,CNRSUMR8617,Universite´Paris-Sud,Orsay
2
Received14august2008/Accepted11december2008

]
A Abstract
G
Aims.Supersonic turbulence is a large reservoir of suprathermal energy in the interstellar medium. Its dissipation, because it is
. intermittentinspaceandtime,candeeplymodifythechemistryofthegas.Thisisclearlyseenintheframeworkofshockchemistry.
h
Intenseturbulentdissipationalsooccursinregionsoflargevelocityshears,sharingwithshocksthepropertyofintermittency.Whether
p
theseburstsofdissipation,short-livedandlocalized,haveameasurableimpactonmolecularabundancesinthediffusemedium,and
-
o howthechemicalenrichmenttheydrivecomparestoobservations,arethequestionsweaddresshere.
r Methods.Wefurtherexploreahybridmethodtocomputethechemicalandthermalevolutionofamagnetizeddissipativestructure,
t undertheenergeticconstraintsprovidedbytheobservedpropertiesofturbulenceinthecoldneutralmedium.Forthefirsttime,we
s
a model a random line of sight by taking into account the relative duration of the bursts with respect to the thermal and chemical
[ relaxationtimescalesofthegas.Thekeyparameteristheturbulentrateofstrainaduetotheambientturbulence.Withthegasdensity,

it controls the size of the dissipative structures, therefore the strength of the burst. It also sets the relative importance of viscous
1
dissipationandion-neutralfrictioninthegasheatingandchemicalenrichment.
v Results.Foralargerangeofratesofstrainanddensities,themodelsofturbulentdissipationregions(TDR)reproducetheCH+column
2 densitiesobservedinthediffusemediumandtheircorrelationwithhighlyexcitedH .TheydosowithoutproducinganexcessofCH.
2
1 Asanaturalconsequence,theyreproducetheabundanceratiosofHCO+/OHandHCO+/H O,andtheirdynamicrangeofaboutone
2
7 orderofmagnitudeobservedindiffusegas.LargeC HandCOabundances,alsorelatedtothoseofHCO+,areanotheroutcomeofthe
2
3 TDRmodelsthatcomparewellwithobservedvalues.Neutralcarbonexceedstheabundanceexpectedationizationequilibrium,in
1. agreementwithfine-structurelineobservations.TheabundancesandcolumndensitiescomputedforCN,HCNandHNCareoneorder
0 ofmagnitudeabovePDRmodelpredictions,althoughstillsignificantlysmallerthanobservedvalues.Thedependenceofourresults
9 ontherateofstrainanddensityrevealsthatthechemicalenhancementsareinbetteragreementwithobservationsifthedissipationis
0 dominatedbyion-neutralfriction,involvingshearstructuresofthickness∼100AU.
:
v Keywords.Astrochemistry-Turbulence-ISM:molecules-ISM:kinematicsanddynamics-ISM:structure-ISM:clouds
i
X
r 1. Introduction (Miville-Descheˆnesetal.2003),
a
(3) its spatial and velocitystructure is even a greaterchallenge
Thediffusemediumhasamajorcontributiontothemassofthe since it has to reconcile the existence of structuresobservedat
interstellar medium (ISM) in galaxies like the Milky Way and allscalesinemissionandaremarkablesimilarityoflineprofiles
as such is a key player in the star formationprocess. Although observedinabsorption(seethediscussioninLiszt&Lucas1998
it is the first component of the ISM to have been discovered, whoquestiontheelusivedimensionalityofthediffusemedium).
and later on extensively analyzed throughabsorption measure- Sincemanytracersofthekinematicsandsmall-scalestructureof
mentsofatoms,ionsandmolecules(seethereviewofSnow& thediffuseISMaremolecularlines,cluesarelikelytobefound
McCall,2006),itsstructureandpropertiesremainachallengein initssurprisinglyrich,butpoorlyunderstood,chemistry.
manyrespects:
(1)longthoughttoconsistoftwostablephases-thewarmand Onemajorpuzzleinthischemistry,raisedbythedetectionof
cold neutral medium (WNM at temperatures T ∼ 8000 K and CH+ inalmosteverylineofsightsamplingtheCNM,persisted
CNM at T ∼ 100 K) in thermal pressure equilibrium - a sig- fordecadesbecausenoformationpathwasfoundtobeefficient
nificantfractionofitsemissionisnowdetectedattemperatures enough in the diffuse ISM (Black, Dalgarno & Oppenheimer
coveringthewholerangebetweenthoseoftheCNMandWNM 1975, Black & Dalgarno 1977). It is rooted in the fact that in
(Heiles&Troland2003), suchdiffusegas,CH+ formsviaahighlyendoenergeticreaction
(2) the CNM is turbulent with supersonic velocities, yet the C+ + H (∆E/k = 4640K) unlikelyto proceedatthe lowtem-
2
velocity and density power spectra carry the signature of the peraturesoftheCNM.Similarly,onewaytoactivatetheoxygen
Kolmogorov power spectrum for incompressible turbulence chemistryinthediffusemedium,andthereforetheformationof
2 B.Godardetal.:Modelsofturbulentdissipationregions
OHandH O,involvesthereactionofO+H whichhasanen- In their long-lasting effort dedicated to unravelling molecular
2 2
ergybarrierof2980K. abundancesin the diffuseISM, Liszt& Lucas(2002andrefer-
A possibly related issue is the existence of rotationally ex- encestherein)haveprovideduswithinvaluableconstraints.Not
cited H in the diffuse gas. FUSE observations have revealed onlydidtheyshowthattheabundancesofseveralmoleculesstay
2
large populations in the J > 2 levels of H , inconsistent with proportionaltoeachother,withverywelldefinedabundancera-
2
fluorescencegeneratedbytheambientUVfield(Sonnentrucker tios,buttheyfoundthatthecolumndensitiesofthesemolecules
etal.2003;Gryetal.2002;Nehme´etal.2008a,b;Martin-Za¨ıdi varybymorethanoneorderofmagnitudeacrosscloudsthatall
etal.2005;Lacouretal.2005).ThisexcitedH couldbelocal- haveaboutthesametotalhydrogencolumndensity,correspond-
2
ized in circumstellar material1, but it also has been detected in ingtodiffuseandtranslucentclouds.Theyalsorevealedthere-
thedirectionoflateBstars,devoidofcircumstellarmatter,asin markablesimilarityofHCO+ andOHlineprofiles,allthemore
the data of Gry et al. (2002). Stellar UV photons are therefore surprising for an ion and a neutral species, differently coupled
unlikelytocontributesignificantlytotheUV-pumpingofH .In tothe magneticfield.On thecontrary,thehigh-spectralresolu-
2
particular Lacour et al. (2005) find an increase of the Doppler tionspectraofCrane,LambertandSheffer(1995)convincingly
parameterof the H lines with J, supportingthe existence of a showedthattheCH+ lineprofilesaredefinitelybroaderandless
2
warm component that cannot be heated by UV photons. They GaussianthanthoseofCH,alongthesamelinesofsight,while
arguethatthiswarmcomponentcannotbeduetoH formation Lambert,Sheffer&Crane(1990)foundthat,inthedirectionof
2
pumping,asproposedbySternberg&Dalgarno(1995)indense ζ Oph,theCHlineprofilescouldbeseenasthesumofabroad
PDRs,becauseitwouldrequireanH formationratelargerthan componentsimilar to the CH+ profile, and a narrowone, close
2
thatinferredfromobservations,andwouldnotreproducetheob- tothatoftheCNline.AsimilarresultwasobtainedbyPanetal.
servedcolumndensitiesofCH+ foundtocorrelatewithexcited (2004,2005)towardsstarsoftheCepOB2andCepOB3regions.
H (Spitzeretal. 1974;Snow1976,1977;Frisch& Jura1980; These sets of results suggest that the velocity field is involved
2
Lambert&Danks1986). in the originand the evolutionof these molecules, and doesso
ISO-SWS observations further support the possible exis- differentlyforeachspecies.
tence of a small fraction of warm gas in the Galactic diffuse Itisthereforechallengingtocomparetheseavailableobser-
mediumbyrevealingitspurerotationallineemission(Falgarone vations with models of a random line of sight across the dif-
et al. 2005).Interestingly,the ratio N(H ) /N ∼ 2×10−4, fuseISM,whereactivedissipationburstscoexistwithothersin
2 warm H
whereN(H ) istheH columndensityinlevelsJ ≥3,isthe theirrelaxationphase.Inparticular,thepossibilitythatanumber
2 warm 2
same across the Galactic diffuse medium as in the direction of of transient events may dominate the observed molecular col-
nearbylateBstars.RecentSpitzerobservationshaveconfirmed umndensitieshasneverbeenaddressed.Thisis whatwe do in
theISO-SWSlinefluxvalues(Verstraeteetal.inpreparation). the present paper. We restrict our study to densities lower than
Both the observed abundances of CH+ and column densi- nH =200 cm−3because,aswillbeseen,athigherdensitiesand
for the turbulent energy observed in the CNM, turbulentdissi-
ties of rotationally excited H suggest that large amounts of
2
pation does not heat the gas enoughto open the endoenergetic
suprathermalenergy are deposited in the cold diffuse medium.
barriersmentionedabove.Wecannotruleout,though,raredis-
One obviousreservoirof suprathermalenergyin the ISM is its
sipationburstsofexceptionalintensitythatwouldbeabletoheat
turbulent kinetic energy. Attempts at incorporating this energy
stilldensergastotherequiredtemperature.Weextendtheprevi-
in the chemical networks of magneto-hydrodynamical(MHD)
ousstudiesinawaythatallowsustoexploretheparameterdo-
shockshavebeenpartlysuccessfulatreproducingtheobserved
main,inparticularthosecharacterizingtheambientturbulence.
propertiesofthediffusemedium(PineaudesForeˆtsetal.1986,
Wealsoextendthechemicalnetwork.Wemodelarandomline
Draine&Katz 1986,Flower&PineaudesForeˆts1998).Other
of sight across the diffuse medium and compare the predicted
routes have been explored, involving dynamic interactions of
columndensitiesofavarietyofmolecularspeciestotheobser-
the gas and the star cluster in the Pleiades (White 1984, 2003;
vations.Thedynamicsteadystateiscomputedanddescribedin
Ritchey et al. 2006), turbulent transport between the WNM
Section2,thechemistryintheactivedissipationandrelaxation
and CNM (Lesaffre et al. 2007) and turbulent dissipation tak-
phasesispresentedinSections3and4.Themodellingofaline
ing place in regions of large velocity shears. Turbulence be-
ofsightisdiscussedinSection5.Comparisonsofcomputedcol-
ingintermittentin spaceandtime(seethe reviewofAnselmet,
umndensitieswithobservations,aswellasexcitationdiagrams
Antonia& Danaila 2001),velocityshears may locally be large
ofH ,areshowninSection6andtheseresultsarediscussedin
enoughtodrivelargelocalheatingratesandtriggertheendoen- 2
Sect.7.
ergeticreactionsofcarbonandoxygenchemistriesinthediffuse
ISM(Falgarone,PineaudesForeˆts&Roueff1995).Alongthese
lines,Joulainetal.(1998,hereafterJ98)haveexploredtherole
2. Steadystateofamagnetizedvortexina
ofion-neutraldecouplinginduced,intheweaklyionizeddiffuse
weakly-ionizeddiffusegas
medium,bythesharpgasaccelerationsintheregionsoflargest
velocity-shearanditsimpactonion-neutralchemistry,inpartic- 2.1.Theneutralflow
ular the formation of CH+ and HCO+. Falgarone et al. (2006)
haveanalysedthethermalandchemicalrelaxationphaseinthe Turbulence in the diffuse ISM is supersonic with respect to its
evolutionofagaschemicallyenrichedinadissipationburst. coldphase,theCNM.Supersonicturbulencedissipatesinshocks
and regions of large velocity shear (Kritsuk et al. 2007). Their
The observationaldata probingthe moleculardiversity and
richnessofthediffuseISMarenotrestrictedtoCH+andHCO+. respective importance has been studied in numerical simula-
tions. Porter et al. (1992; 1994) and Pavlovski, Smith & Mac
Low (2006) showed that most of the turbulent kinetic energy
1 Here,wedeliberatelyoverlookmostoftheearlyCopernicusresults
is rapidly transferred to high wavenumber non-compressible
on H high-J lines obtained in the direction of hot stars (e.g. Spitzer
2
et al. 1973; Savage et al. 1977; Shull & Beckwith 1982) that led to modes,oncetheshocksgeneratedbysupersonicturbulencehave
theconclusionthatexcitedH absorptionisoccurringincircumstellar started to interact, reducing the role of compressible modes
2
materialheated/shockedbythestaritself. (shocks)inturbulentdissipation.Intheso-calledquiescentISM
B.Godardetal.:Modelsofturbulentdissipationregions 3
(i.e. far from star forming regions), the smooth observed line- where π is the stress tensor. This rate is written in cylindrical
ij
shapessupporttheviewofaturbulencedevoidofstrongshocks coordinates:
(Falgarone et al. 1994) implying that the dissipation preferen-
tiallyoccursinshear-layers. Γnn = ∂uθ − uθ 2+ ∂uz 2+2 ∂ur 2+2 ur 2+2 ∂uz 2(9)
Dissipativestructuresaremodelledasshear-layersbelonging η " ∂r r # ∂r ! ∂r ! (cid:18) r (cid:19) ∂z !
to a solution of the Helmholtz equation for vorticity, close to
the Burgers vortex adopted in J98: this analytical solution has where η = ρν is the dynamicviscosity in a gasof density ρ. It
the merit that it has only two free parameters that describe the is computedforhydrogenatomsusing the Kay & Laby(1966)
balancebetweenthestretchingactionofthelargescalesandthe tablesofphysicalandchemicalconstants:η = 6×10−6T1/2 g
k
diffusion of vorticity across the vortex edge, at small scale. It cm−1s−1 whereT isthegaskinetictemperature.
k
providesananalyticalframeworkinwhichwecancomputethe In all the following,it is assumed that the fluid description
effectofpartialdecouplingbetweenionsandneutralsuponthe of the gasmotionsis justified becausethe mean freepath ofH
steadystateconfigurationofvelocityandmagneticfieldandthus atoms λ = 0.23(n /50 cm−3)−1 AU in the diffuse medium,
H−H H
explore the effect of both the ion-neutral friction and viscous for a H-H elastic collision cross section2 σ = 5.7×10−15
H−H
heatinguponthechemicalnetwork. cm2(Spitzer1978),issmallerthanallthelengthscalesinvolved
The modified Burgers vortex is an axisymmetric solution inthemodel.
elaborated in atmospheric sciences by Nolan & Farrell (1998).
ItisidenticaltotheBurgersvortexatsmallradiir,incylindrical
coordinates(r,θ,z),anddiffersfromitatlargeradiiinthesense 2.2.Interstellarconstraintsonthevortexparameters
thattheradialinflowvelocityisnotdivergentatinfinity:
As said above, each vortex is defined by a set of three inde-
ur(r)=−ar·e−βr2 (1) pneonidneflnutepnacreameitehteerrso,na,thβeadnydnaωm0.icTshoercount-tohffe cphaerammisettreyroβfhthaes
2
structureaslongasu issmallcomparedtou .Hereafter,β
r θ,max
where a is the turbulent rate of strain (in s−1) and β (in cm−2) is chosenin orderto satisfy this condition.In otherwords,tur-
describesthecut-offoftheradialvelocity.Theaxialvelocityu , bulentdissipationinthevortexhappensthroughradialvorticity
z
thevorticityω andtheorthoradialvelocityu areinferredfrom distribution,notthrougha radialfluxofmatter. Inthese condi-
z θ
theHelmholtzandcontinuityequations: tions,thedominantcontributiontotheviscousdissipationisthe
firstterminEq.(9).
u (r)=az·e−βr2 · 1−βr2 , (2) Numerical simulations of incompressible turbulence
z (Jimenez1997)andexperiments(Belinetal.1996)haveshown
(cid:16) (cid:17)
that the maximum tangential velocity of filaments of vorticity
ωz(r)=ω0·e−4aνβ(cid:20)1−e−βr2(cid:21), (3) uθ,max ∼ ω0r0 is of the order of the rms velocity dispersion of
the ambient turbulence σ . Since the equilibrium radius r
turb 0
1 r is set by the rate of strain a (Eq. 5), ω0 is also determined. In
uθ(r)= r′ωz(r′)dr′ (4) thecaseofinterstellarturbulence,itimpliesthattheorthoradial
r Z
0 velocityinthevortexissupersonicwithrespecttothecoldgas.
It is noteworthy that slightly supersonic Burgers vortices have
whereω isthepeakofvorticityandνisthekinematicviscosity.
0 been found in experiments of rotating magnetized plasmas by
Anyvortexis thereforeentirelydefinedbythreeparameters,a,
Nagaokaetal.(2002)inconjunctionwithaninnerdensityhole.
βandω .
0 Moreover, as will be seen later, the gas being violently and
Notethat,accordingtotheradialdependenceofthevorticity,
rapidly heated in the layers of largest orthoradial velocity, the
thesame equilibriumvortexradiusr asfortheBurgersvortex
0 Machnumberthereislikelytodropbelowunity.Theonlyfree
canbedefined,
parameter left in the vortex description is therefore the rate of
r2 =4ν/a (5) straina,althoughthegasdensityisinfactafreeparameterthat
0 determines the vortex size, through the kinematic viscosity ν
involvingthetwoquantitiesthatactonit,therateofstrainaand (Eq.5).
theviscosityν.Accordingly,thevortexcrossingtime
2.3.Magneticfieldconfigurationandionizedflow
kr dr′ 2
τ = = ln(1/k) (6)
c Zr ur(r′) a The configuration of the magnetic field and ions reached once
the above vortex has developed in the partially ionized gas is
foranyconstantk < 1dependsonlyontherateofstrain,while numericallycomputed.Theionsareinitiallyatrest,threadedby
thevortexperiod,definedas a uniform magnetic field parallel to the z axis, B = B k. The
0 0
ionsarepredominantlyC+,theneutralsaremainlycomposedof
r
P= 0 (7) HandH andtheionizationdegree, x = 2×10−4,isweak3.At
2
u (r )
θ 0 t = 0,theionsaresuddenlyputintomotionbyfrictionwiththe
vortexthatdevelopedintheneutralgasi.e.thethreecomponents
dependsonν,a,andω .
0
of the neutral gas velocity u are those of the vortex given by
Because vorticity is radially non uniform there is a differ- n
entialrotation of the fluid within the structure which inducesa
2 For comparison the H -H elastic collision cross section is
viscousdissipationrate: 2 2
σ ∼3 10−15cm2(Monchicketal.1980).
H2−H2
3 Computedforadiffusemolecularcloudofdensityn ∼50cm−3,
∂u H
Γ = π i (8) temperatureT ∼100K,illuminatedbythestandardinterstellarradia-
nn Xi,j ij∂xj tionfield(ISRkF).
4 B.Godardetal.:Modelsofturbulentdissipationregions
Eqs. (1), (2) and (4). Boundaryconditions are provided by the
assumptionthatthevortexhasafinitelengthL ,apodisedover
V
alengthC .
V
Thealignmentofωwiththeambientmagneticfieldissup-
ported by the results of numericalsimulations. Brandenburget
al. (1996) showed that in MHD turbulence, magnetic field and
vorticity vectors tend to align with each other. More recently
Mininni et al. (2006a,b) observed a similar behaviour in their
15363numericalsimulations.
Undertheseassumptions,wecomputethetwo-dimensional
time-dependent evolution of the ion velocity u and the mag-
i
netic field B. We neglect the retro-action of the ions upon the
neutralmotionsbecause,fordensitiesintherange10-200cm−3
and an ion-neutral drift velocity comparable to u (see Fig.
n
1), the friction force F they exert on the neutrals is negli-
in
gible compared to the advection force in the vortical motion:
F ∼10−3(l/10AU)×ρ u .∇u ,lbeingthespatialscaleforthe
in n n n
variationofu ,intherange10to100AU.Theneutralsvelocity
n
componentsare thereforethose of the vortex(Eqs. (1), (2)and
(4))atanytime.
In the interstellar medium B is frozen in the charged fluid
(Spitzer1978)anditsevolutionissimplywritten:
∂B
+∇×(B×u)=0. (10)
∂t i
Neglecting the pressure gradientsin the evolution equation
oftheionizedflow(thisassumptionisjustifiedinSect.7)leads
to:
∂u hσvi 1
i +(u ·∇)u = in ρ (u −u)+ (∇×B)×B(11)
∂t i i (µ +µ) n n i 4πρ
n i i
whereµ andµ arethemeanmassperparticleoftheneutrals(H,
n i
H ) andions(mostlyC+)respectively.hσvi = 2.2×10−9cm3
2 in
s−1 is the momentum transfer rate coefficient between the ion-
Figure1.Magneticpropertiesofthevortex.Panel(a):evolution
izedandneutralfluidscalculatedbyFlower&PineaudesForeˆts
of|u | ,themaximumionvelocitycomponentperpendicular
(1995,AppendixA),andclosetotheLangevinrate. i⊥max
to the axis z, as a function of time. The different curvescorre-
spondtothemodels M , M , M and M ,(seeTable1).Panels
0 1 2 3
M M M M (b) and (c): orthoradial and axial components of the magnetic
0 1 2 3
Magneticfield B µG 10 5 10 10 fieldatt=100yrforthemodelM asfunctionsofrandz.
0
Vortexlength L AU 200 200 100 200
V
Apodisationlength C AU 100 100 50 50
V
Table 1. Parameters of the four models of Fig. 1. The density neutrals. After ∼ 100 yr, |u | < 0.4 km s−1 which is small
n andtheturbulentrateofstrainarefixed:n = 50cm−3,a = i⊥max
H H comparedto |u | in the vortex (see Fig. 2 in Sect. 3.2) for all
5×10−10s−1. n⊥
models. A steady state is reached in which the ions are almost
back to rest and the magnetic field slightly helical (Fig. 1b). A
large and steady state drift is set between the ion and neutral
orthoradialvelocitieswithanamplitudeclosetotheorthoradial
WeintegrateEq.(10)andEq.(11)bymeansofatwodimen- velocityoftheneutralsinthevortex.
sionalimplicit schemeusing the AlternatingDirectionImplicit Suchadrifthasadeepimpactonthechemistryofthegas,as
method(ADI).To validate ourapproachwe also use two other was shownby J98, andcontributesto the dissipationof its tur-
integrationschemes:anexplicitandanimplicitwithouttheADI bulentenergy.Theadditionalheatingtermduetotheion-neutral
method.Theresultsofour300×200pointsgridsimulationsare frictioniswritten:
mdiaspxliamyuemdinioFnigv.e1lo.cPiatyneclo(ma)psohnoewntsptheerpeevnodluictiuolnarotfo|uthi⊥e|maaxx,isthze, Γin = µρn+ρiµ hσviin µ µ+iµ (ui−un)2. (12)
asafunctionoftime.Thecurvescorrespondtothefourmodels n i n i
presentedinTable1.Panels(b)and(c)showtheorthoradialand Wealsofoundthatthereisonlyaverysmallion-neutraldrift
axialcomponentsofBforthemodelM att=100yr. inthezdirectionbecausethemagneticfieldisonlyslightlyhe-
0
Fig.1ashowsthattheions,initiallyatrest,areputintomo- lical.Sinceinadditionthecontributionsofthespatialvariations
tion by the friction force from the neutrals in the vortex. This ofu totheheatingtermΓ arenegligible(Eq.9),weconsider
z nn
motion (includingits boundaryconditions)generatesan ortho- thevortexasinvariantalongtheaxiszandrestrictourstudyof
radialcomponent B of B and a gradientof B . These terms in thespatialandtimedependencestothoseoccurringradially.
θ z
turninducemagnetictensionandpressuregradient(seeEq.10), Inthenextsectionswefocusontherapidthermalandchem-
two forces resisting the orthoradialentrainment exerted by the icalevolutionofthegastrappedinsuchasteadystatestructure,
B.Godardetal.:Modelsofturbulentdissipationregions 5
andfollowitsthermalandchemicalrelaxation,oncethevortex dUi = d 3nkT = B +B + µnΓ + µnΓ (14)
hasblown-up. dt dt 2 i i! i e µ in µ en
i e
where B , B and B are the sums of all the heating and cool-
3. Theactivephase n i e
ingratesoftheneutrals,theionsandtheelectronsrespectively,
3.1.Numericalmodelling not induced by turbulent dissipation, and Γ and Γ the heat-
nn in
ingratesinducedbyturbulentdissipationdefinedintheprevious
As in J98, we follow the Lagrangianevolution of a fluid parti-
section.Γ istheheatingrateduetotheelectron-neutraldrift,a
cletrappedinthesteadystatevortexconfiguration.Becausethe en
term takeninto accountin thecode butnegligiblecomparedto
vortexcrossingtimeτ (seeSect.2.1)iscomparabletothechem-
c Γ andΓ .Thecoolingratesincludetheradiativede-excitation
icaltimescales,wecomputenon-equilibriumchemistrycoupled nn in
ofthe ro-vibrationallevelsof H , ofthe fine structurelevelsof
tothetime-dependentthermalevolution.Theinitialgastemper- C+,CandOandoftherotationa2llevelsofH O,OHandCO.
aturesandmolecularabundancesarethoseofasteadystatedif- 2
Lastly,sinceionsandneutralsaredecoupled,weusetheap-
fusecloudofdensityn ,illuminatedbytheambientinterstellar
H proximation of Flower et al. (1985) for the calculation of the
radiationfield(ISRF)(Draine1978)scaledbythefactorχ,and
chemicalratecoefficients.Thecross-sectionofa2-speciesreac-
shieldedbytheextinction A .Thecosmicrayionizationrateζ
V tion is integrated over a Maxwellian velocity distribution at an
andtheelementalabundancesaregiveninTable2.
effectivetemperature
Theneutralsandtheionsaretreatedseparately4 andafluid
particleisdefinedateachtimebyitspositionr,theneutraland
m T +m T 1 m m
itohneizteemdpveerlaotcuirteysfioefldthseunneauntrdalusi,TtnheanmdaisosndseTnis,itaiensdρthneanadbuρni-, Teff = 1m21+m22 1 + 3km11+m22u2D (15)
dancesn(X)ofeachspecies.Thesystemthereforecomprises11
dynamic time-dependentvariables (r, ni, nn, ρi, ρn, Ti, Tn, unr, wherem1,m2,T1andT2arerespectivelythemassesandthetem-
unθ,uir,uiθ).OurchemicalnetworkoriginatesfromtheMeudon peraturesofthe2reactantsanduD theirrelativedriftvelocity.
PDRcode(LePetitetal.2006).Itincorporates90speciesinter-
acting through 1524 reactions. Those include the formation of
3.2.Thermalevolutionofthegas
H ondust,thephotoprocessesandtheprocessesinducedbythe
2
cosmic rays.We also computethe time-dependentevolutionof Fig. 2 displaysthe mainpropertiesofa referencemodelwhere
thepopulationsofthe18firstro-vibrationallevelsofH2 (corre- a= 3×10−11s−1,nH =30cm−3andAV = 0.4mag.Thevortex
spondingtoTex <104K). hasanequilibriumradiusr0 = 38AUandgeneratesanaverage
turbulentheatingrate,Γ = 3.4×10−23 ergcm−3 s−1 defined
turb
Density n cm−3 10-200 as:
H
Radiationfield χ 1
ECxotsimncitcioranyionizationrate ζAV sm−a1g 03.×1-101−17 a Γturb =2/(Kr0)2 Kr0[Γnn(r)+Γin(r)]rdr. (16)
Z
Elementalabundances [X]/[H] 0
Helium [He] 1.00×10−1
TheintegrationdomainextendstotheradiusKr wherethetur-
Carbon [C] 1.38×10−4 0
Oxygen [O] 3.02×10−4 bulentheating has no significant influence on the gas tempera-
Nitrogen [N] 7.94×10−5 tureandchemistry(K ∼5).
Sulfur [S] 1.86×10−5 In the model presented here, the heating rate is dominated
Iron [Fe] 1.50×10−8 everywhere by ion-neutral friction (Fig. 2a). For higher values
oftheturbulentrateofstraina,r decreases(seeSect.5.3)and
Table 2. Physical conditions and elemental abundances of the 0
the importanceofthe viscousdissipationincreasesbecauseu
gasintheTDRmodels.a-Dalgarno(2006). nθ
isfixedbytheambientturbulence.
Thegasinthevortexneverreachesthermalbalanceandthe
thermalinertiaisvisiblebycomparingFigs.2aand2cwherethe
peaktemperatureofthe fluid particleis reacheda fewhundred
The system of variables is therefore driven by a set of
years after the peak of the heating rate. Emission in the pure
119first-ordercoupleddifferentialequationsthatareintegrated
rotationallinesofH isbyfarthedominantcoolantinthelayers
2
alongthefluidparticletrajectory.Toensurethatthetimestepis where T & 200K (Fig. 2c) while the cooling rate due to the
n
consistentwiththevariationsofalldependentvariablesweuse ionizedcarbonC+ decreasesin thewarmestlayers.Thisis due
theDVODEdifferentialequationsolver(Brownetal.1989).
tothechemicalevolution(seeSect.3.3).
The evolution of the thermal energydensities U and U is
n i Last, some chemical clues are provided in Fig. 2c. The
givenby:
neutral-neutral reactions only depend on the temperature T
n
whiletheion-neutralreactionsdependontheion-neutraldrift.In
dU d 3
n = n kT = B +Γ +Γ +Γ (13) Eq.(15),thesecondtermoftherighthandsidereaches1000K
n n n nn in en
dt dt 2 ! asu ∼ 3kms−1,aneffectivetemperaturehigherthanthepeak
D
kinetictemperatureinthevortex.Acomparisonoftheshapesof
4 Theionsandtheelectronsaretreatedasauniquefluidatatemper-
theorthoradialvelocityu andtheneutraltemperatureT shows
atureT because:(1)theion-electronvelocitydriftinducedbythemag- nθ n
neticfieildfluctuationsinthemodelis∼1cms−1,negligiblecompared thattheendo-energeticion-neutralchemistryisactivatedearlier
tothethermalvelocities,(2)theion-electrontemperatureequipartition inthefluidparticleevolutionthantheneutral-neutralchemistry.
time is ∼ 0.1(T/1000K)3/2(n/310−3cm−3)−1 yr (Spitzer 1978), also For each type of vortex, the relative importance of those two
i
negligiblecomparedtothedynamictimescales. chemicalregimesisdifferent.
6 B.Godardetal.:Modelsofturbulentdissipationregions
Cviathedissociativerecombinationwithelectrons,HCO+ and
COvia
CH++O→HCO++H (18)
3 2
andCN,HCNandHNCvia
CH++N→HCN++H . (19)
3 2
TheproductionofCH+isalsoattheoriginofC HandCSsince
3 2
thesemoleculesarebothproductsofCH(throughthereactions
CH+C+ →C+ +HandCH+S+ →CS+ +Hrespectively).
2
The second main reaction, absent in the ambient medium,
which plays an importantrole in the chemicalevolution of the
vortexis:
O+H →OH+H −∆E/k=2980K. (20)
2
BesidesthedirectproductionofOHittriggerstheproductionof
H Ovia
2
OH+H →H O+H −∆E/k =1490K (21)
2 2
andO via
2
OH+O→O +H. (22)
2
Fig.3 displays the evolution of several relative abundances
in the magnetized vortex. The impact of the turbulent heating
issuchthatmostspeciesabundancesrise from2to5 ordersof
magnitudewithinthestructure.Theformationofvorticesinthe
turbulentgasflowthereforehasspecificchemicalsignaturesthat
we expect to observe in the diffuse medium. One remarkable
exampleisHCO+.SincethismoleculeisadirectproductofCH+
3
Figure2. Vortex physical properties as functions of the radius (inthevortex)itbecomesrelatedtoalmostallthespecieswhose
(bottom axis) or time (top axis, arbitrary origin) for the refer- abundanceisenhancedinthevortex.
encemodel:a=3×10−11s−1,n =30cm−3andA =0.4mag. Time (and space) stratification is also visible in Fig.3:
H V
Panel(a):TheheatingtermsΓ duetotheviscousdissipation, the chemical enrichments occur successively, because the ion-
nn
Γ due to the ion-neutral friction and Γ due to the photo- neutral chemistry is triggered earlier than the neutral-neutral
in ph−e
electriceffect.Panel(b):Thedominantcoolingtermsduetothe chemistry (see previous section) and because the chemical in-
radiativedesexcitationofH (purerotationallines),C,OandC+ ertiaofeachspeciesisdifferent.
2
(fine-structurelines).Panel(c):Thetemperatureandorthoradial
velocityoftheneutrals.
4. Therelaxationphase
4.1.Numericalmodelling
3.3.Chemicalevolutionofthegas
Once the burst of turbulent dissipation is over, some chemical
Thechemicalevolutionofthegasduringthevortexactivephase signaturesimprintedinthegaspersistforseveralthousandyears
issimilartothatreportedinJ98,althoughthechemicalnetwork as shown by Falgarone et al. (2006).To compute the chemical
isupdatedandincludesnitrogen-andsulfur-bearingmolecules. andthermalrelaxationofthegas,weassumethat,oncethevor-
TheoutlineofthisnetworkisgiveninAppendixCwherewedis- tex has vanished, the gas is dynamically frozen: u = 0. The
playthemainproductionanddestructionroutesofthemolecules previous Lagrangian approach is switched to Eulerian, and we
ofinterest(1)intheambientdiffusemedium(nH =30cm−3and computethetime-dependentevolutionofeachcellinthevortex.
AV =0.4mag)and(2)inthevortexforthereferencemodelata Theinitialconditionsoftherelaxationaretheconditionsofthe
radiusr=r0. active stage at every position. The thermal equations (13) and
Themostimportantreactionrouteopenedbythedissipative (14)arestillvalidwithΓ =0andΓ =0.
nn in
structureistheendothermichydrogenationofC+: While the numerical code has been conceived to treat iso-
C++H →CH++H −∆E/k=4640K. (17) baricorisochoricrelaxation,alltheresultspresentedinthispa-
2
perwere obtainedassumingisochoricrelaxation,becauseit al-
BesidesthedirectproductionofCH+,thisreactionisrespon- lowsustobetterdisentanglewhatisduetothechemistryitself
sibleformostofthechemicalrichnessofthevortexasshownin fromwhatisduetothegasdensity.Inparticular,itshowsmore
Fig. C.2 (AppendixC): it enhancesthe productionof CH+ via clearlytheroleoftherelaxationtimescalesofthemolecules,i.e.
3
the successive hydrogenationby H of CH+ andCH+. CH+, in onlydrivenbythechemicalnetworkandthethermalevolution,
2 2 3
turn, enhances the production of molecules including CH and independentlyofthegasdensity.
B.Godardetal.:Modelsofturbulentdissipationregions 7
Figure3.Fractionalabundancesrelativeton = n(H)+2n(H ) Figure4. Vortex thermal features during the relaxation phase,
H 2
of selected species as functions of the radius (bottom axis) or as functions of the radius and time (after the vortex blow-up)
time (top axis, arbitrary origin) for the reference model: a = forthe referencemodel:a = 3×10−11 s−1, n = 30cm−3 and
H
3×10−11s−1,n =30cm−3andA =0.4mag. A = 0.4 mag. An isochoricrelaxation is assumed. Top panel:
H V V
themaincoolingterms,i.e.theradiativedesexcitationofH and
2
C+.Bottompanel:thetemperatureoftheneutrals.
4.2.Thermalevolutionofthegas
Fig. 4 displaysthe main coolingrates(top panel)andgastem- forCO).Theexistenceoftherelaxationphasesmodifiesthecor-
perature(bottompanel)asfunctionsoftimeandpositioninthe relations between molecular abundances. One example is pro-
vortex (after the vortex blow-up).It shows that as in the active videdbyHCNandC2Hthatfollowasimilarenhancementinthe
phase,thecoolingrateduringtherelaxationphaseisdominated activephase(seeFig.3)andhavemarkedlydifferentbehaviours
bytheemissioninthepurerotationallinesofH .Inthemodel intherelaxationphase.
2
presented here (n = 30 cm−3), other cooling agents (mainly The richest phases are not necessarily those contributing
H
C+)becomedominantatt∼104yr. the most to the observablecolumn densities because they have
shortlifetimes.Inthenextsection,wedetailhowwetaketime-
variability into account in our modelling of a random line of
4.3.Chemicalevolutionofthegas sightacrossthediffusemedium.
During relaxation, the cooling of the gas causes all the endo-
energeticreactionstriggeredduringthedissipativebursttoslow
5. Modellingofalineofsight
downandstop,oneaftertheother.Thegaslosesitschemicalen-
richmentataspeedthatdependsonthemolecularspecies.This Inturbulentflows,thespatialdistributionoftheregionsofhigh-
isillustratedinFig.5thatdisplaysthetime-dependentevolution estdissipationrate(extremaofvelocityshear,extremaofnega-
ofthecolumndensityNVR(X,t)ofselectedspeciesX integrated tivevelocitydivergencei.e.shocks)isfarfromspace-filling,one
acrossthevortex. oftheaspectsoftheintermittentnatureofturbulence.Thefilling
For most species (CO, C , OH, H O, C H...) the signature factoroftheseextremahasbeencomputedinnumericalsimula-
2 2 2
of the turbulentdissipationpersists overmore than103 yr. The tionsofmildlycompressible(Pety&Falgarone2000)orsuper-
characteristictimescales(e-foldingtimes)differbetweenspecies sonic MHD turbulence (Pavlovski et al. 2006, Pan & Padoan
bymorethanafactorof30(≈2×102yrforCH+and7×103yr 2008).Morerecently,Moisy&Jimenez(2004)haveshownthat
8 B.Godardetal.:Modelsofturbulentdissipationregions
results, namely because, as it will be seen, the results depend
weaklyona.
The contribution of one active phase of duration τ to the
V
totalcolumndensityofaspeciesX isN (X).Thecontribution
VA
oftherelaxationphaseofthatspeciesiscomputedbyassuming
thatthelongeritsrelaxationtimescale,thegreaterthecontribu-
tion of the relaxation phase in the observed column density so
thatthechemicalcompositionofalineofsightisentirelydeter-
mined by the number of active vortices N and their lifetime
VA
τ .TheresultingcolumndensityofaspeciesXis:
V
∞
N(X)=N (X) N (X)+1/τ N (X,t)dt +N (X)(23)
VA VA V VR M
" Z #
0
whereN (X)isthecontributionoftheambientmedium.
M
5.2.Constraintsprovidedbytheenergyavailableinthe
turbulentcascade
5.2.1. Thenumberofactivevortices
The number of active vortices N in a line of sight and their
VA
lifetimeτ ,areconstrainedbytheturbulentenergyavailablein
V
thecascadeanditstransferrate.Inturbulentflows,thetransfer
rateofkineticenergyatscalelofcharacteristicvelocityu is,on
l
average:
1 u3
ε = ρ l . (24)
l
2 l
The Kolmogorov scaling of turbulence, valid for incompress-
ible turbulence, postulates that this transfer rate is scale-
independent. In the highly compressible interstellar medium,
one would expect this quantity to differ from one scale to an-
other.Measurementsoftheinternalvelocitydispersionofclouds
Figure5.Columndensitiesofselectedspecies,integratedacross
of size l combined with their density provide an estimate of
the reference vortex, as functions of time during an isochoric
theturbulentenergytransferrateatthisscale.Acompilationof
relaxationphase.
CO(1-0)lineobservationsofinterstellarstructuresofsizerang-
ing between 10−2 and 103 pc shows that ε is remarkably in-
l
dependent of the scale in the Galaxy and that there is a large
scatter (by a factor of 100) about a well defined average value
theregionsofintensevorticitytendtoformfilaments,whilere- ε ∼2×10−25 ergcm−3s−1(Falgarone,1998;Hily-Blantetal.
obs
gionsofmostintensedissipationratherformsheetsorribbons,
2008).AsimilarvalueholdsforturbulenceintheHICNMand
allofthembeingorganizedinclusters,probingtheorganization
in non-star-formingdense cores (Falgarone 1999). The unifor-
ofsmall-scaleintermittentstructures.
mityofthisvalueacrossthelocalISMsuggeststhattheturbulent
Forourpurpose,weassumethatanylineofsightinterceptsa cascadeencompassesthedifferentregionsofthecoldmedium,
numberofvortices,eitheractiveorintheirrelaxationphase.The andthatthetransferisdrivenbyturbulenceatthesamerate,in
totalnumberofvorticesperlineofsightisconstrainedbytheav- allmedia,whateverthegasdensity.
eragetransferrateofturbulentenergyperunitvolumeavailable Weadopttheabovevalueofε inourmodelasrepresenta-
obs
inthecascade. tiveoftheturbulentenergytransferratethroughscales.Wethus
imposethat,atanytime,thedissipationrateinalltheactivevor-
ticesinalineofsightisequaltotheaverageenergytransferrate
5.1.Method
intheturbulentcascade,sothat:
Anylineofsightsamplesthreekindsofdiffusegas:(1)theambi-
entmediuminwhichthechemistryiscomputedassteadystate εobs =NVAΓturb2Kr0/L (25)
UV-dominated chemistry, (2) a number N of active vortices
VA where L is the depth of the line of sight, inferred from L =
and (3) a numberof relaxation phases related to N . We also
VA N /n . This fixesthe numberof active vorticesin a givenline
considerthatthelineofsightishomogeneousandcharacterized H H
ofsight5.Thisnumberandmanyoftheresultsarethereforepro-
byitsdensityn anduniformshieldingA fromtheISRF.Such
H V portionaltoε .
anapproximationisusefultotesttheimportanceofeachparam- obs
eter (a,n and A )on thefinalchemicalstate of thegas. Last,
H V 5 N alsodependsontheangleofinclinationofthevorticesalong
weassumethatallthevorticesinthelineofsightareidentical: VA
thelineof sight.Because thedynamics andthechemistryinavortex
theyallhavethesameturbulentrateofstraina.Inthefollowing donotdependontheaxialcoordinateszandbecausethemodelisax-
sectionweshowwhyamorerealisticdescription,withadistri- isymmetric,thisanglehasnoinfluenceonthefinalresults.N isthus
VA
butionofrateofstrainvalues,wouldnotprovideverydifferent definedforvorticesperpendiculartothelineofsight.
B.Godardetal.:Modelsofturbulentdissipationregions 9
Figure6.MainphysicalpropertiesoftheTDRmodelsasfunctionsoftheturbulentrateofstrainaforseveralvaluesofthedensity
n . Top panels: the vortex heating terms Γ due to the viscous dissipation (left) and Γ due to the ion-neutral friction (right).
H nn in
Bottompanel:thevortexlifetimeτ .
V
5.2.2. Thevelocitydispersionoftheambientturbulence 5.2.3. Thelifetimeoftheactivephase
The lifetime of an active vortex,τ , is controlledby the large-
V
scale motions of the ambient turbulence that feed energy into
the small-scale structures (Moffatt, Kida & Ohkitani 1994). It
The energy transfer rate depends on the density, velocity dis- mayexceedtheperiodofthevortexassuggestedbyavarietyof
persion and timescale. We thus need to know the amount of experimentsinincompressibleturbulence(e.g.Douady,Couder,
turbulent energy available in the CNM component of the dif- Brachet1991).This lifetime is uncertain,though,andits value
fuse medium, or the rms turbulent velocity in the CNM. It is inourmodelisconstrainedbyenergeticconsiderations.
this quantity that sets the angular velocity of the vortex (see We assume that, for simplicity, all the vortices explored in
Section2.2).Thisquantityis difficulttodetermineon observa- themodelsdissipatethesametotalenergyovertheirlifetimeτ
V
tional grounds, because of the mixture of WNM and CNM in
the HI emission spectra and because of the lack of spatial in- E =π(Kr )2Γ L τ (26)
formationfor the absorptionspectra (dominatedby the CNM). 0 turb V V
Weadoptedarmsvelocitydispersionσ =3.5kms−1 forthe
turb Thisconstraintfixesthelifetimeτ ofthevortexi.e.thetime
CNMturbulencederivedfromtheHImapsofahighlatitudecir- V
duringwhich turbulentdissipation is active. In order to stay in
rus in the Polaris Flare (Joncas et al. 1992; Miville-Descheˆnes
thevortexframeworkweimposethatτ islargerthanthevortex
et al. 2003) in which HI emission is well correlated with the V
period P, which sets a lower limit to E. The influence of this
100µmemissionofdust,probingcolumndensitiesofgasrepre-
parameterontheresultsisdiscussedinAppendixB.
sentativeoftheCNM.Thisvalueisconsistentwiththosequoted
in Crovisier (1981) for the CNM. It is comparable to the geo-
metricmeanofthetwosmallestvalues,FWHM =4.9and12.0
5.3.Theroleoftherateofstrainanddensityunderthe
kms−1 inferredbyHaud& Kalberla(2007)fromtheGaussian
energeticconstraints
decompositionoftheHIprofilesoftheLeiden/Argentine/Bonn
surveyofgalacticHI(Hartmann&Burton1997).Last,thisrms Onceεobs,σturb and E aregiven,alineofsightisthereforede-
velocitydispersionisconsistentwiththeapproximateequiparti- finedbyonlythreeindependentparameters:theturbulentrateof
tion between magnetic and turbulentenergyinferred by Heiles straina,thegasdensitynHandtheshieldingAV.
& Troland(2005) from the median value of the magnetic field Tohelpthereaderunderstandthechemicalresultspresented
estimatedinthediffusemediumB=6µG. inthenextsection,wediscusstherolesoftherateofstrainand
10 B.Godardetal.:Modelsofturbulentdissipationregions
Figure7.MainphysicalpropertiesoftheTDRmodelsasfunctionsoftheturbulentrateofstrainaandofthedensityn ofthegas.
H
Toppanels:thevortexturbulentheatingrateΓ (left)andthevortexmaximumtemperatureT (right).Bottompanels:thevortex
turb max
radiusr (left)andthenumberofactivevorticesN alongalineofsightsamplingonemagnitudeofgas(right).
0 VA
gasdensitybecausetheaboveconstraintsontheenergydissipa- dissipation dominates because the orthoradial velocity is fixed
tionrateactuallycoupleaandn thatshouldbeindependentpa- andthinvorticesinducelargevelocityshears,thuslargeviscous
H
rameters.Thisissobecausethetwoenergyconstraints(transfer heating.However,thethermalinertiaofthegaspreventsitfrom
rateandenergy)involvetheequilibriumradiusr thatprimarily reaching much higher temperatures, because the most efficient
0
dependsonabutalsoonn ,viathedensitydependenceofthe vortices(largea)areshort-lived.Therangeofaandn explored
H H
kinematicviscosity(seeSection2.1)Thesetrendsareillustrated inourstudycoversthesetworegimesandwequantifythechem-
inFigs.6and7thatalsodisplaythedependenceofseveralkey icaleffectsofturbulentdissipationasitchangesfromdominated
quantitiesonaandn . byion-neutralfrictiontodominatedbyviscousdissipation.
H
Fig.6showsthat,asexpected,Γnn ∼ anH isalmostpropor- Last, we find thatthe numberof activevorticesNVA (com-
tionalto a becausethe highertherate of strain,the smaller the puted for a line of sight sampling one magnitude of gas) is
equilibrium radius and the larger the velocity shear. Γin is al- roughlyindependentofaanddecreasesalmostasn−2astheden-
mostindependentofaandincreaseswithdensityasΓin ∝uDn2H sityincreases.Thisisbecauseεobsisfixed(seeprevHioussection),
so that, depending on a and n , two regimes exist: one at low
H andbecauseofthecombineddependencesofr ,Γ and Lon
density and high rate of strain where the turbulent heating is 0 turb
aandn (seeEq.25).WenotethatN reacheslargevaluesat
dominated by viscous dissipation, and the other (small a, high H VA
lowdensity(upto severalhundredsalonglinesofsightofsev-
density)wherethisheatingisdominatedbytheion-neutralfric-
eraltensofparsecs).Howeverthefillingfactorofthevortices
tion.Therateofstrainathereforeplaysanimportantroleinthe
natureofthewarmchemistrytriggeredinthevortex.Thisfigure
also displays the run of τ with the rate of strain for different
V
densities,asaresultoftheconstraintonthetotalenergyE dis- 2Kr
f =N 0 (27)
sipatedineachvortex. v VA L
Fig.7showsthatr issmallintheformerregimeandreaches
0
values of the order of 100 AU in the regime where ion-neutral
frictiondominates.Thepeakgastemperaturereachedinthevor- neverexceeds f =4×10−2,itslowestvaluesbeing f ∼10−4at
v v
tex,T ,isalsoshown:itishigherintheregimewhereviscous highdensitiesandratesofstrain.
max