A method for reconstructing the variance of a 3D physical field from 2D observations: Application to turbulence in the ISM

0.32 MB·English

by  C. M. Brunt

Mon.Not.R.Astron.Soc.000,1–10(2009) Printed7January2010 (MNLATEXstylefilev2.2)
A method for reconstructing the variance of a 3D physical
field from 2D observations: Application to turbulence in
the ISM
0
1
0 C. M. Brunt1⋆, C. Federrath2,3, & D. J. Price4
2
1School of Physics, Universityof Exeter, StockerRoad, Exeter, UK
n 2Zentrumfu¨r Astronomie der Universita¨tHeidelberg, Institut fu¨r Theoretische Astrophysik, Albert-Ueberle-Str.2,
a D-69120 Heidelberg, Germany
J 3Max-Planck-Institute for Astronomy, K¨onigstuhl 17, D-69117 Heidelberg, Germany

7 4Centre for Stellar and Planetary Astrophysics, School of Mathematical Sciences, Monash University,Clayton Vic 3168, Australia

]
A
Accepted ;Received;inoriginalform
G
.
h
ABSTRACT
p
We introduce and test an expression for calculating the variance of a physical field in
-
o three dimensions using only information contained in the two-dimensional projection
r ofthefield.Themethodisgeneralbutassumesstatisticalisotropy.Totestthemethod
t
s weapplyittonumericalsimulationsofhydrodynamicandmagnetohydrodynamictur-
a bulence in molecular clouds, and demonstrate that it can recover the 3D normalised
[ density variance with ∼ 10%accuracyif the assumptionofisotropyis valid.We show

1 that the assumption of isotropy breaks down at low sonic Mach number if the tur-
v bulence is sub-Alfv´enic. Theoretical predictions suggest that the 3D density variance
6 should increase proportionally to the square of the Mach number of the turbulence.
4 Application of our method will allow this prediction to be tested observationally and
0
therefore constrain a large body of analytic models of star formation that rely on it.
1
. Key words: ISM:clouds – ISM: kinematics and dynamics – turbulence – magneto-
1
hydrodynamics – methods: statistical.
0
0
1
:
v
i 1 INTRODUCTION diction plays a key role in analytic models of star forma-
X
tion (Padoan & Nordlund 2002; Krumholz & McKee 2005;
r It is a fundamental problem in astrophysics that we typ- Elmegreen 2008; Hennebelle & Chabrier 2008; Padoan &
a ically only have access to two-dimensional (2D) physical
Nordlund2009; Hennebelle& Chabrier 2009).
fields that have been integrated or averaged over the line-
Ifthedensity,ρ,isexpressed in unitsof themean den-
of-sight,whilethephysicalfieldsofinterest areintrinsically
sity, ρ , the theoretical predictions for the relationship be-
three-dimensional (3D). Constraining 3D properties is very 0
tweenthedensityvarianceandrootmeansquare(rms)Mach
difficult,andislimitedtostructureswithafairlysimplege-
number,M, may be simply written as:
ometry(e.g.Lucy1974–seeReblinsky2000foranapplica-
tiontogalaxyclusters).Molecularcloudshaveverycomplex σ2 = b2M2, (1)
ρ/ρ0
structurethatisnotsuitablefordirectinversionfrom2Dto
3D, but there exist methods for inferring statistical infor- where b is a constant of proportionality. In the case of
mation on the scaling behaviour of the 3D structure of the isothermal gas, which is applicable to a good approxima-
density and velocity fields (e.g. Stutzki et al 1998). Molec- tion to molecular clouds, the probability density function
ular cloud evolution is driven in large part by the action (PDF) of the density field is thought to be lognormal in
ofturbulence,andrecently,muchinteresthasbeendirected form(V´azquez-Semadeni1994).Usingalognormal formfor
towardstheactionofturbulenceinshapingthedensityfield the density PDF, Padoan & Nordlund (2002) derive a re-
in molecular clouds, in particular towards the theoretically lation between the variance in the logarithm of the density
predicted increase in the variance of the density field with field,σl2n(ρ/ρ0), and the rms Mach number,M:
the Mach number of the turbulence (Padoan, Nordlund, &
σ2 =ln(1+b2M2), (2)
Jones 1997; Passot & V´azquez-Semadeni 1998). This pre- ln(ρ/ρ0)
which is equivalent to equation (1). A range of predictions
for b have been proposed, which were recently synthesized
⋆ [email protected] intoa unified model by Federrath et al (2008) who propose
2 C. M. Brunt, C. Federrath, & Price, D. J.
that b = 1/3 for solenoidal (divergence-free) forcing and Insertingequation (5) intoequation (3),
b=1 for compressive (curl-free) forcing in 3D.
F (x,y)=
Giventheimportanceofequation(1)foranalyticmod- 2
els of star formation, it is essential to test it with observa- 1 L/2 dz ∞ ∞ ∞ F˜(k ,k ,k )e2πik·r/L(6)
tionaldata.InitialobservationaltestsbyGoodman,Pineda, L4 Z 3 x y z
& Schnee (2009) did not find any obvious support for the −L/2 kxX=−∞kyX=−∞kzX=−∞
theoretical predictions. Federrath et al (2009) have sug- and computing thez integral first, we find that:
gested a reason for this lack of agreement, citing variations
∞ ∞
in b caused by different turbulent forcing mechanisms. Per- F (x,y)= 1 F˜(k ,k ,k =0)e2πik·r/L,
hapsmoreimportantly,theseobservationsreliedonmeasur- 2 L3 X X 3 x y z
kx=−∞ky=−∞
ing the variance in the projected 2D column density rather
(7)
than in the 3D density field that appears in the theoretical
predictions.Fromobservations,wedonothaveaccesstothe since
3Ddensityandvelocityfieldstodirectlytestthisprediction,
1 L/2
andevaluateitsapplicabilitytomolecularcloudsandtothe dze2πikzz/L=1 for k =0, (8)
LZ z
predictivestar formation models. −L/2
Inthispaper,wederiveandtest anexpression that re-
latestheobservablevarianceinthe2Dcolumndensityfield 1 L/2
dze2πikzz/L=0 for k =0. (9)
to the true 3D variance of the density field. The method is LZ z 6
−L/2
completely general, albeit with the assumption of statisti-
cal isotropy, and may also be applied to the projection of Sincethe2D Fourierseries, overtheinterval[ L/2,+L/2],
−
otherphysicalfields.InSection 2,wepresenttheanalytical of F2 is:
expressions needed to convert the 2D variance into the 3D L/2 L/2
variance. In Section 3, we demonstrate the application of F˜2(k2)=Z Z d2r2 F2(r2)e−2πik2·r2/L (10)
the method to numerical simulations of hydrodynamic and −L/2 −L/2
magnetohydrodynamicturbulence.Oursummaryisgivenin where r =(xxˆ,yyˆ) and k =(k xˆ,k yˆ), and:
2 2 x y
Section 4.
∞ ∞
1
F (r )= F˜(k ,k )e2πik2·r2/L, (11)
2 2 L2 2 x y
X X
kx=−∞ky=−∞
we see that, comparing equation (7) and equation (11):
2 THE STATISTICS OF PROJECTION FROM
1
3D TO 2D F˜(k ,k )= F˜(k ,k ,k =0). (12)
2 x y L 3 x y z
2.1 Development of the Method
In other words: the 2D Fourier series of F is propor-
2
tional to the k = 0 cut through F˜. Previous studies have
Given a physical field in 3D, we wish to examine how the z 3
made use of this result (e.g. Stutzki et al. 1998; Lazar-
statisticalpropertiesofa2Dprojectedfieldarerelatedtothe
ian & Pogosyan 2000; Brunt & Mac Low 2004; Miville-
intrinsicproperties.Bydoingthis,wecantheninferintrinsic
Deschenes&Martin(2007)).Asthepowerspectrum,P (k),
quantities from the measureable quantities. We focus here 3
is the squared modulus of the Fourier transform (P (k) =
onthevariance,measuredin3Dandinprojected2D.Inour 3
F˜(k)F˜∗(k)), 2D power spectra of projected fields have
initial derivation, we consider a 3D field, F (x,y,z), which 3 3
3
been used to infer the 3D power spectrum, under the as-
is averaged along the z axis to produce a projected field
sumption of isotropy, i.e. that the k = 0 cut through the
F (x,y),via: z
2
powerspectrumisstatisticallyrepresentiveofthefullpower
1 L/2 spectrum.Inouranalysisbelow,wewillalsomakeuseofthe
F (x,y)= dzF (x,y,z) (3)
2 LZ 3 power spectrum and theassumption of isotropy.
−L/2
Nowwecomputethevariancesinthe3Dand2Dfields.
where L is the physical size of the region, assumed cubical, From the mean valueof F3:
that contains F .
3 1 L/2 L/2 L/2
Wewillmakeuseofthe3DFourierseriesofF ,obtained F = d3r F (r), (13)
3 h 3i L3 Z Z Z 3
overthe interval[ L/2,+L/2], which is: −L/2 −L/2 −L/2
−
and themean square valueof F :
L/2 L/2 L/2 3
F˜3(k)=Z−L/2Z−L/2Z−L/2 d3r F3(r)e−2πik·r/L (4) F2 = 1 L/2 L/2 L/2 d3r F2(r), (14)
h 3i L3 Z Z Z 3
−L/2 −L/2 −L/2
where r =(xxˆ,yyˆ,zzˆ) and the spatial frequencies are k =
we findthe variance of F as:
(k xˆ,k yˆ,k zˆ), for integerk ,k ,and k .The field F can 3
x y z x y z 3
then bewritten as: σ2= F2 F 2. (15)
3 h 3i−h 3i
F (r)= 1 ∞ ∞ ∞ F˜(k ,k ,k )e2πik·r/L. Wealso make useof theFourier transform of F3, bynoting
3 L3 3 x y z that:
X X X
kx=−∞ky=−∞kz=−∞ 1
(5) hF3i= L3F˜3(0,0,0) (16)
Variance of a 3D physical field from 2D observations 3
and, through Parseval’s Theorem: (23)
F2 = 1 ∞ ∞ ∞ F˜F˜∗, (17) The3Dvariancecanthenbecalculatedasσ32 =σ22/R.Note
h 3i L6 X X X 3 3 however that since λ is necessarily finite, the observed 2D
kx=−∞ky=−∞kz=−∞
variance and the calculated 3D variance are lower limits to
allowing usto rewrite equation (15) as: thetruevariancesthatwouldbeobtainedinthelimitλ
−→
.This is discussed further below.
σ2= 1  ∞ ∞ ∞ F˜F˜∗ F˜2(0,0,0). ∞ Amorecompactformofequation(23)canbeobtained
3 L6 X X X 3 3 − 3 bydefining:
kx=−∞ky=−∞kz=−∞  
(18) λ/2 λ/2
2D,λ
P (k)= P (k) P (0) (24)
By similar analysis, we find thevariance of F2 is: X 2 X X 2 − 2
k6=0 kx=−λ/2+1ky=−λ/2+1 
∞ ∞
σ2= 1  F˜F˜∗ F˜2(0,0), (19) and:
2 L4 kxX=−∞kyX=−∞ 2 2 − 2  3D,λP2(k)=
or, applying equation (12): Xk6=0
1 λ/2 λ/2 λ/2
σ2=
2 L6×  P2(k) P2(0)
X X X −
∞ ∞ kx=−λ/2+1ky=−λ/2+1kz=−λ/2+1 
 F˜3(kz =0)F˜3∗(kz =0) F˜32(0,0,0). (25)
X X −
kx=−∞ky=−∞   so that:
(20)
2D,λ
P (k)
The ratio, R, of the variance of F to the variance of F is 2
2 3 X
therefore: R= k6=0 (26)
3D,λ
σ2 P2(k)
2 =R= X
σ2 k6=0
3
For this method to work, it is essential that the pro-
∞ ∞
 F˜3(kz =0)F˜3∗(kz =0) F˜32(0,0,0) jectedfield,F2,istheline-of-sightaverageofthe3Dfield,F3
X X − (seeequation(3)).Inmanyinstances,theprojected2Dfield
.kx=−∞ky=−∞  istheline-of-sightintregralofthe3Dfield(e.g.columnden-
∞ ∞ ∞
 F˜F˜∗ F˜2(0,0,0) sity versus density). A simple solution for integrated fields
3 3 3
X X X − that ensures the above requirements are satisfied is to ex-
kx=−∞ky=−∞kz=−∞  pressthefieldF innormalisedunits–i.e.bydividingF by
2 2
(21)
its mean value. In this case, the variance of the normalised
In practice, the measureable physical fields, either 3D density field can be calculated from the variance of the
through observations or numerical simulations, will consist normalised 2D column density field, as discussed below in
of a discrete number of measurements at a fixed set of grid Section 2.6.
points.Wedefinethescaleratio,λ,astheratiooftheimage
(orcube)sizetothepixelsize.Takingthegrid tobeofsize
2.2 Approximations
λ λ λorλ λpixelsin3Dand2Drespectively,theFourier
× × ×
transformsarecarriedoutatadiscretesetofspatialfrequen-
The power spectra of many fields of interest are power-law
cies,k= λ/2+1, λ/2+2,..., 2, 1,0,1,2...,λ/2 1,λ/2 in form: P k−α. If the power spectrum is steep (large
along eac−h axis. Th−e quantity F˜−F˜−∗ is the spectral−power, 3 ∝
3 3 α) then the variance is sensitively dependent on the power
P , which can be observationally derived (up to an unim-
3 at low spatial frequencies. Because of the quantisation in
portant constant of proportionality) as follows. If we had a
equation(26)andthesmallamountofinformationavailable
projectedmean2Dfield,F ,andwantedtousethistoinfer
2 at low k, the above procedure may be inaccurate for large
the3Dvariance,thenwewouldcomputeitspowerspectrum,
α.Brunt&MacLow(2004)examined3Dandprojected2D
P (k ,k ), and from this produce an azimuthally-averaged
2 x y standard deviations in the normalised density field (ρ/ρ )
0
powerspectrum P (k),which dependsonlyon themodulus
2 and normalised column density field (N/N ) obtained from
0
ofthespatialfrequency,k.Thekeyideabehindthemethod
numerical simulations, with λ=128. The (column) density
is that we can, assuming isotropy, take: power spectra were reasonably well fitted by P (k) k−3,
3
∝
P3(k)∝P2(k), (22) albeitwithsomecurvature.TheyfoundthatσN/N0/σρ/ρ0 ≈
0.34 0.04. Evaluating equation (26), using λ = 128 and
to obtain: ±
P (k) k−3, to compute the ratio of standard deviations
3
∝
λ/2 λ/2 (i.e. √R), we predict σ2/σ3 = 0.39, which is in acceptable
 P2(k) P2(0) agreement with the Brunt & Mac Low (2004) experimental
R= kx=X−λ/2+1ky=X−λ/2+1 − , result, given thespectral curvature.
λ/2 λ/2 λ/2 With a power law form for the power spectrum, it is
 P (k) P (0) temptingtoapproximateequation(26)withsimpleintegral-
2 2
X X X − based expressions for R. If the spectral slope is α we could
kx=−λ/2+1ky=−λ/2+1kz=−λ/2+1 
4 C. M. Brunt, C. Federrath, & Price, D. J.
write: resolution limit. These, naturally, are additional sources of
variance that we cannot measure.
2π λ/2 dkkk−α
R= 4π Rλ1/2 dkk2 k−α, (27) derivTedhebyfurlelmvaorviianngc(efroofmthtehefidelednoFm3isnhaotuorld)tihneperffinecctipolfetbhee
1
R beampattern,andsummingoverthespatialfrequencyrange
whichiseasilysolvedforaspecifiedα.However,thisisnota
16k6k . In general, we will havelittle or no information
terriblygoodapproximationastheintegralshavebeencom- c
aboutk ,unlessofcourseadistinctbreakinthepowerspec-
puted over circular and spherical regions of k-space rather c
trumisobserved.Apossible,theoretically-motivated choice
thanthesquareandcubicalregionswhichcontainthefields.
isthatk isthespatialfrequencycorrespondingtothesonic
At low αthis is not sufficiently accurate, and in general we c
scale (V´azquez-Semadeni et al 2003; Federrath et al 2009).
recommend that the direct summation method presented
We may try to account for this, by assuming a form for P
in equation (26) be used. As a trivial example, for a cube 3
in the inaccessible range k > k > λ/2, by extending an
of Gaussian noise (α= 0), equation (26) correctly predicts c
observed power-law for example. This is quite dangerous,
R = (λ2 1)/(λ3 1), which tends to 1/λ for large λ. In
− − particularly for α63 for which the summation diverges at
thesamelimit,equation(27)incorrectlypredictsR=3/2λ.
large spatial frequencies. The case α = 3 provides a con-
Equation (27) is useful however in gaining some intuitive
venient reference point, at which R is underestimated by a
understanding of how R depends on the form of the power
factor ln(λ/2)/ln(k ) – a logarithmic divergence. With-
spectrum: R decreases if proportionally more power resides ∼ c
outadirectconstraintonk itismorereasonabletorestrict
at large k because the denominator is weighted by k2 and c
our knowledge of the 3D variance to the spatial dynamic
the numerator by k. The physical effect underlying this is
rangeaffordedbytheobservations,andutilizeequation(28)
that,ifthevarianceismostlyathighk(smallspatialscales),
asstated,whilerecognizing that the3Dvariancemustbea
line-of-sight averaging suppresses more of thepower than if
lower limit to the true 3D variance. For numerical simula-
thevariance is mostly at low k (large spatial scales). Equa-
tions,thisisnotanissue,aswehaveaccesstothefullrange
tion(27)isalsousefulforexploringscale-dependenceeffects
of spatial frequencies at which structureis present.
onthecalculatedvariances.Notethatfor2<α63,the2D
varianceconvergesasλ butthe3Dvariancediverges!
−→∞
2.4 Accounting for Non-Periodic Fields
2.3 Application to Observational Data TheFourierseries utilised aboveimplicitly assume that the
field is periodic. While this is true of some numerical simu-
For fields acquired through observations, it is necessary to lations, it is obviously not trueof real physicalfields. If the
account for the telescope’s beam response (point spread field F has a significant amplitude at the field boundaries,
2
function) in the calculation of variances. The observational then elevated power is produced in the 2D power spectrum
version of equation (26) is: at some spatial frequencies, typically along the k =0 and
x
2D,λP (k)B˜2(k) ky =0axes,andthisisnotconsistentwithanisotropicform
2 for the 3D power spectrum. Firstly, we note that a signif-
X
R= k6=0 (28) cant amplitude of F2 at the field edges is almost certainly
3D,λP (k)B˜2(k) notconsistentwiththeassumptionofisotropy,asitimplies
2
X that F has been extracted from a larger field with signifi-
k6=0 2
cant power on larger scales. In this case, restriction of the
where λ is the number of pixels along each axis, B˜2(k) = fieldintheprojectedxandydirectionsisimposedbutthere
B˜(k)B˜∗(k) is the square of the Fourier space representa- has been no corresponding restriction on the extent of the
tion of the telescope beam pattern, and P2(k) is the power field in the z direction. Sensible definition of the field F2 is
spectrum of F2 in the absence of beam-smoothing and thereforerecommended,with,ideally,F2fallingtonearzero
instrumental noise. (Note that the observable quantity is amplitudeat the field boundaries.
P2(k)B˜2(k)+P2N(k)whereP2N(k)isthenoisepowerspec- It is possible to ameliorate the effect of edge disconti-
trum.) Equation (28) includes accounting for the fact that nuities by edge-tapering the field (e.g. Brunt & Mac Low
the observed variance in the 2D field has been suppressed 2004).Inthiscasewerecommendthatλissufficientlylarge
by the smoothing imposed by the telescope beam, and this thatthetaperingcausesanegligible modification ofthe2D
must be taken into account. In equation (28), we have also variance.Agood solutionis,inadditiontotapering,topad
applied the beam pattern to the denominator (represent- thefieldwithzerovalues.Ifanobservedfieldwithscaleratio
ing the 3D variance). This procedure may at first appear λ is zero-padded out to a larger size λ , then it is straight-
p
an odd choice, but it is motivated by the requirement that foward to show that the 3D variance should be calculated
welimit ourknowledgeof the3Dvariancetotheresolution via:
providedbythedata.In effect,equation (28) is an attempt
1
to construct a 3D beam that samples the 3D density field σ32= η3(((σ22+hF2i2)η2−hF2i2)/Rp+hF2i2)−hF2i2, (29)
and affords the same effective linear resolution as that pro-
vided by the telescope beam in projected 2D. The Fourier where η =λ /λ and R is the 2D-to-3D variance ratio cal-
p p
series F˜ extends to k = , or at least to a (potentially culated from the power spectrum of the padded field. The
3
verylarge) spatialfrequenc∞y,k ,beyondwhichthereareno quantities F , σ2, and σ2 apply to theunpaddedfield.
c h 2i 2 3
variationsin thefield.Inotherwords,therewill befluctua- Abovewehaveassumedasquare(cubical)boxthatcon-
tions in the projected 2D field (and therefore the 3D field) tains the field F (F ). In practice, we may wish to apply
2 3
that we are not directly sensitive to, as they lie below our this method to fields that are not exactly square, although
Variance of a 3D physical field from 2D observations 5
this then obviously raises questions about the assumption Note that it is not necessary to assume a lognormal form
of isotropy. For a field of size λ λ , we recommend zero- for the column density PDF, although this indeed may be
x y
×
paddingtoproduceasquarefieldofsizeλ λ ,andusing true (Ostriker, Stone, & Gammie 2001; V´azquez-Semadeni
px py
equation (29) with η =(λpxλpy/λxλy)12. No×te that this as- & Garc´ıa 2001).
1 Considering that σ2 may be subject (for spectral
sumes that the line-of-sight extent of F3 is λz = (λxλy)2. ρ/ρ0
slope α 3) to a logarithmic divergence because of unre-
Clearly, one should try to ensure that λ λ to respect
x y ≈
≈ solveddensitystructure,asdiscussedintheprevioussection,
theassumption of isotropy.
equation(30)hasthefortunatepropertythatthecomputa-
tion of σ2 suppresses this divergence to a log(log) di-
ln(ρ/ρ0)
2.5 Summary of the Method vergence,sothat σl2n(ρ/ρ0) can bequitewell estimated even
from observations at finite resolution, provided the spatial
Tosummarizethemethod:toestimatethe3Dvariance,σ32, dynamic range is sufficiently large to measure σN2/N0 ac-
thefollowing procedure should befollowed: curately, and the field is of good linear spatial resolution
(1) measurethevariance,σ2,ofthenormalisedprojected such that λ/2 approaches reasonable expectations for kc.
2 We recommend that estimates of the unresolved variance
field F ,
2
be made, using equation (27) as a guide for suitable values
of k (e.g. the spatial frequency corresponding to the sonic
(2) measure the power spectrum, P (k), of F , and c
2 2
scale, or even k ) to better assess the utility of the
assume, through isotropy, that P3(k) P2(k), c −→ ∞
∝ observationalmeasurement(Brunt2009).Wenotealsothat
theabovemethod for estimating σ2 doesnot requirethe
(3) using P2(k), compute the 2D-to-3D variance ratio, ρ/ρ0
cloud distance to be known. However, observations at fixed
R,via equation (26) or equation (28),
angular resolution applied to clouds at different distances
will probe different physical scales, so the method is not
(4) computeσ2,accountingforanyzero-paddingusedto
3 entirely distance-independent.
computethe power spectrum (equation (29)).
The above prescription can, in principle, be applied to
other positive-definite fields, such as temperature. Estima-
tionofmeanprojectedtemperaturefieldsisusuallydoneby
quite a different procedure than that used for column den-
2.6 Density Fields
sityfields,however.Typically,weareabletoderivealine-of-
Sofar,otherthanisotropy,wehavenotimposedanypartic- sightaveragetemperatureforeachpixel,eitherthroughflux
ularstructuretothefieldF (e.g.bychoosingaformforthe ratios at far infrared wavelengths, or by excitation analysis
3
power spectrum and PDF). It is well established that pro- of millimetre-wave spectral lines, for example. These mea-
jectedcolumndensitypowerspectraofmolecularcloudsare surementsareoften not straightforward spatial averages, as
power-law in form (e.g. Stutzkiet al 1998; Bensch, Stutzki, assumedbyourmethod,butmaybe(e.g.)density-weighted
&Ossenkopf2001),implyingalsothatthe3Dpowerspectra averagesinstead.Somecautionmustbeappliedtothetreat-
are also of power-law form. Typically, the 2D power spec- ment of such fields.
trum is of the form P k−α where α 3. Density fields
2
∝ ≈
arealwayspositive,anditisconvenienttoexpressboththe
density, ρ, and column density, N, in normalised units, by
2.7 Velocity Fields
dividing by their respective mean values, ρ , and N , re-
0 0
spectively. In this way, they will conform to the properties Applicationofthemethodtoprojectedmeanvelocityfields
of the fields F and F discussed above (i.e. that F is the is also possible. Obviously, since a velocity field is not
3 2 2
line of sight average of F ). Otherwise, the column density positive-definite, we do not normalize the field as was done
3
will bescaled bythephysicallength, L,as it is theintegral for the density field. In principle, we do have access to the
ofρalongtheline-of-sight,ratherthantheaverage.Inmost mean velocity field, which can be obtained through an op-
cases, our 2D observations will be of column density – e.g. tically thin spectral line. However, such a field is density-
through an optically thin spectral line, or through extinc- weighted, rather than a direct spatial average as assumed
tion mapping. The physical line-of-sight length, L, needed by equation (3), and significant problems in estimating the
to convert column density to projected mean density, may powerspectrum (Brunt& Mac Low2004), and presumably
beunknown.WhileLmaybeinferredthroughtheassump- therefore the 2D variance, can arise for supersonic turbu-
tion of isotropy, if the distance is known, it is better to use lence. There will be lines of sight with insufficient signal-
thenormalised density,ρ/ρ , and column density,N/N . to-noisetocalculateameanvelocity.Thereforethevariance
0 0
Using the observed variance, σ2 , in the normalised mustbecalculatedforthedetectedlinesofsight,withtheas-
N/N0
column density field, and the angular-averaged power spec- sumptionthatthesereliably represent theentirefield.Note
trum, P (k), we can then use equation (28) to obtain that the problem in estimating the mean of the field (as
N/N0
R. With appropriate treatment of any zero-padding (equa- required in the case of column density / density analysis)
tion (29) with F = 1, σ2 = σ2 , and σ2 = σ2 ), the does not apply here. We also have access to only one com-
h 2i 2 N/N0 3 ρ/ρ0
variance in normalised density,σ2 , can be calculated. ponentofthevelocityfield (theline-of-sight componentvz)
ρ/ρ0
IfalognormalPDFin3Disassumed,wecanthenderive and we must assume not only isotropy in this, but also ap-
σ2 via: ply the isotropic assumption to the entirely inaccessible vx
ln(ρ/ρ0)
andv components.Withtheseprovisos,themethodwould
y
σ2 =ln(1+σ2 ). (30) proceed as outlined at the end of Section 2.1.
ln(ρ/ρ0) ρ/ρ0
6 C. M. Brunt, C. Federrath, & Price, D. J.
Figure 1.Plotoftheestimated3Ddensityvariance,σ2/R,ver- Figure2.Plotoftheestimated3Ddensityvariance,σ2/R,ver-
2 2
sus theactual 3D density variance, σ2,obtained fromnumerical susthe actual 3D density variance, σ2,obtained fromnumerical
3 3
simulationsofhydrodynamicturbulence.Eachplottedpointrep- simulations of magnetohydrodynamic turbulence. Each plotted
resentsthemeanvalueofσ2/Robtainedfromanalysisofcolumn point represents the mean value of σ2/R obtained fromanalysis
2 2
densityfieldsprojectedalongthethreecardinaldirections,while ofcolumndensityfieldsprojectedalongthethreecardinaldirec-
the error bars represent the standard deviation in the values of tions,whiletheerrorbarsrepresentthestandarddeviationinthe
σ2/Rforthethreedirections. values of σ2/R forthe threedirections. The solidlineis theline
2 2
ofequality.
There is a rather more straightforward method of es-
timating the line-of-sight velocity dispersion, σ , which is
vz
simply to measure the dispersion of the mean line profile
obtained through imaging observations of an optically-thin
spectralline.(Thisisalsoadensity-weightedmeasure,how-
ever.) Ifwehaveanestimate ofthecloud temperature,and
an estimate of the mean molecular weight, we can then de-
rive the Mach number to construct the right-hand-side of
equation (1).
3 APPLICATION TO NUMERICAL
SIMULATIONS
Asaninitialtestofthemethod,wenowapplyittonumeri-
calsimulationsofhydrodynamicandmagnetohydrodynamic
turbulence.Giventhattheabovetheoryisderivedassuming
isotropic fields,we investigatetheeffect of anisotropy using
magnetohydrodynamicsimulations.Inallthestudiesbelow,
weusenormalisedcolumndensities(N/N )andnormalised
0
densities (ρ/ρ0) so that σ22 =σN2/N0 and σ32=σρ2/ρ0. Figure 3. Plot of the standard deviation in σ22/R expressed as
afraction of the mean σ2/R, versus the Alfv´enicMach number,
2
MA,fortheMHDsimulations.
3.1 Hydrodynamic Turbulence
First, we analyse hydrodynamic simulations at a range of weperformedasuiteoflowresolution1283 particlecalcula-
rmsMachnumbers:1.25,2.5,3.5, 5,7,10,and20.Multiple tions, interpolated to 2563 grids. Since the key idea here is
snapshotsareusedat eachMach numbersetting,separated thereconstructionofthedensityvariance,wefindthateven
by at least a crossing time. The hydrodynamic simulations low resolution calculations are sufficient as demonstrators
were run with the phantom Smoothed Particle Hydrody- ofthemethod.Foreach simulation, we takethe3D density
namicscode,withturbulencedrivenartificiallyover 5cross- field, of mean unity, and create three normalised projected
ing times using large-scale solenoidal Fourier driving (Price fields by averaging along each spatial axis. We compute σ2
3
& Federrath 2009; Federrath et al 2009; Brunt, Heyer, & in 3D and a value of σ2 for each projected 2D field.
2
Mac Low 2009; Brunt 2003). For the analysis in this paper The power spectrum for each column density field is
Variance of a 3D physical field from 2D observations 7
Figure4. Relationsbetweenσ22/Randthetrueσ32 foreachofthethreeaxes.RunswithMA61areshownasdots;runswithMA>1
areshownasopensquares.Thesolidlineisthelineofequalityandthedottedlinesmarkunderestimationandoverestimationofσ2 by
3
afactorof2.
then measured (specifically, we compute the average power
in bins of unit width in k.) Following this, equation (26) is
usedtocomputeavalueofRforeachprojectedfield.Todo
this, we interpolate the binned column density power spec-
trumtotheappropriatepowerforeachk resultingfromthe
nested sums. An extrapolation is required (using a power
law fit to the power spectrum in the high k region) for
√2λ/2<k6√3λ/2 in the denominator. In theabove pro-
cedure, we emphasize that only information present in the
column density field is used in the calculation of R. Values
ofRcomputedforthesefieldsvarybetweenabout0.03 and
0.15.Notethatingeneral,Rwilldependontheformofthe
power spectrum and thescale ratio of the field.
In figure 1 we plot the estimated 3D density variance,
σ2/R, versus the actual 3D density variance, σ2. For each
2 3
snapshot,werepresentthemeasurementsasamean(plotted
point)andstandarddeviation(errorbar)obtainedfromthe
three different projections. The method can predict the 3D
density variance to about 10% accuracy.
Figure 5.Plotoftheanisotropyindicator, σ2 /σ2 ,versusthe
2⊥ 2||
3.2 Magnetohydrodynamic Turbulence Alfv´enic Mach number, MA, for the MHD simulations. For ref-
erence, the horizontal dashed line marks the expected value of
Wenowanalysemagnetohydrodynamic(MHD)simulations σ2 /σ2 for an isotropic field; the vertical dashed line marks
2⊥ 2||
ofturbulence.TheMHDsimulationswererunwiththegrid- MA=1,belowwhichanisotropyisnotable.
based code flash (Fryxell et al. 2000). We used a new ap-
proximate Riemann solver for ideal-MHD (Bouchut, Klin-
genberg, & Waagan 2007, 2009), which preserves positive strengths (lower β) significant anisotropy can arise in the
states in highly supersonic MHD turbulence. This solver density and velocity fields (Mac Low 1999; Vestuto, Os-
was recently developed for flash by Waagan (2009). The triker, & Stone 2003; Heyer et al. 2008), and thus the as-
corresponding scheme for preserving positive states in hy- sumptionsofourmethodcouldbreakdown.UsingtheMHD
drodynamical studies has been successfully tested and ap- densityfields,werepeatthesameproceduredescribedabove
plied in Klingenberg, Schmidt & Waagan (2007). We used tocalculateR,andhencetheestimated3Dvariance,σ2/R;
2
the same solenoidal forcing scheme as used in Federrath et the comparison of estimated to true 3D variance is shown
al. (2009) and Price & Federrath (2009). All MHD models in Figure 2. We find that the method is still accurate in
were evolved on a fixed grid with 2563 grid zones. predicting the mean (plotted points) but the variation be-
In addition to varying the Mach number, in the MHD tween the different directions (error bars) is now signifi-
simulations we now also vary the ratio of thermal to mag- cantly larger for some parameterizations. Investigation of
netic pressure, β = P/(B02). For large magnetic field this shows that significant variations in σ2/R between the
2µ0 2
8 C. M. Brunt, C. Federrath, & Price, D. J.
Figure 6. Representative 2D slices through a sample of density fields created by sub-Alfv´enic turbulence. The mean magnetic field
directionisillustratedbyanarrow,labelledB0.Thevalues ofthe sonicMachnumber, M,the plasmabeta, β,and theAlfv´enicMach
number,MA aregivenaboveeachpanel.
Figure 7. Power spectra of the 2D density field slices from Figure 6. Greyscales are represented in logarithmic units, with contours,
eachseparatedbyadecadeinpower,overlayed;thecontoursbeensmoothedforclarity.
differentprojection axesareseeniftheAlfv´enicMachnum- tureinthedensityfieldthatisproducedwhenM <1.Asa
A
ber,M =M (β/2),issmallerthan unity.Toshowthis, simplemeasure of thisanisotropy, wedefinean “anisotropy
A
∼
in Figure3 wepplot thefractional errorin theestimated 3D indicator” using the 2D fields that are produced by averag-
densityvariance(i.ethestandarddeviationinσ2/Rdivided ingoverthethreecardinalaxes.Theanisotropyindicatoris
2
by the mean σ2/R obtained from the three different pro- definedasσ2 /σ2 =(σ2 +σ2 )/2σ2 ,whereσ2 ,σ2 ,and
2 2⊥ 2|| 2x 2y 2z 2x 2y
jection directions) versus MA. The fractional error in the σ22z arethevariancesinthe2Dfieldsproducedbyaveraging
estimated 3D density variance is around 10% for MA > 1, overthex,y, and z axes respectively. Themeasured values
while it increases significantly for MA <1. ofσ22⊥/σ22|| areplottedinFigure5versusMA.ForMA>1,
differAesntthdeirreecitsionnosw,itnoistawbolerthvaprilaottitoinnginthσe22i/nRdivbiedtuwaeleσn22t/hRe eσa22c⊥h/σo2f2||thiesp∼roujencitteyd, i2nDdicfiaetldinsg. FroorugMhAly<eq1uathlevavrairainancecseiinn
versusσ32 relationsobtainedfrom eachaxis.Theserelations the 2D field averaged over the z direction (parallel to the
areshowninFigure4,wherewehavedistinguishedtheplot- mean B field) is larger, by as much as a factor of 2, than
ted points with different symbols for MA 61 and MA >1. that of the2D fields produced by averaging over thex or y
The 3D variance in sub-Alfv´enicrunsis overestimated (un- directions (perpendicular to themean B field).
derestimated) from 2D fields produced by averaging over
an axis parallel (perpendicular) to the mean magnetic field Interestingly,thedegreeofanisotropyappearstobere-
direction, by as much as a factor of 2. lated to the sonic Mach number, M, as is evident in both
∼
Thevariationsinσ2/Rarecausedbyanisotropicstruc- Figure3andFigure5.Indeed,forM >10,theestimationof
2
Variance of a 3D physical field from 2D observations 9
Figure 8. Columndensityplots(logarithmicgreyscale)ofthenon-periodicSPHsimulation.
the3Dvarianceisnotnoticeablyworseforsub-Alfv´enictur-
bulence than for super-Alfv´enic turbulence, and the largest
deviationsfrom isotropicbehaviourarefoundin theM =2
runs.Adirectvisualizationoftheanisotropiespresentinthe
densityfieldsisgiveninFigure6,whichshowsrepresentative
2Dslicesthroughasampleofdensityfieldsproducedbysub-
Alfv´enic turbulence. At low sonic Mach number (M = 2)
the density field is characterized by many long filamentary
structuresorientedparalleltothemeanmagneticfielddirec-
tion.AsthesonicMachnumberisincreased,thisdirectional
order in the density field is systematically reduced,until at
M = 10 it is barely noticeable. The power spectra of these
2D density field slices are shown in Figure 7. These con-
firm our visual assessment of the density field slices. The
high frequency spectral power along themagnetic field axis
isnotablylowerinthelowsonicMachnumbercalculations.
The tests on the MHD simulations provide a valuable
baseline for establishing the physical regimes in which our
method can beapplied. In most circumstances encountered
in molecular clouds with sizes of a few parsecs, sonic Mach Figure9. Comparisonofcalculatedtotrue3Dvariancesforthe
numbersarelargeenoughtoenforcesufficientisotropy that non-periodicSPHdensityfields.
the method can work with around 10% accuracy, even for
sub-Alfv´enic turbulence. However, these are rather ideal-
ized conditions, as the models do not include other rele- times (at t = 0; after one free fall time; and at the end of
vant physics such as self-gravity. Gravitational collapse in thecalculation where t = 1.274 tff) are shown in Figure 8.
thepresenceofamagneticfieldislikelytobeorientedalong (The column density fields are represented as logarithmic
themagneticfieldlinesifthefieldissufficientlystrong.Note greyscales as the variances are extremely large.) We zero-
that the combination of magnetic fields and gravity, in this padthesefieldstoλp =512from theoriginal λ=256,then
instance, can induce anisotropic structure in the orthogo- computethepowerspectraandfromthisderiveRp.Wethen
nal direction to that seen in the sub-Alfv´enic simulations calculate the3D variance byuse of equation (29).
described above. In Figure 9 we compare the 3D variances derived from
ourmethodwith theactual3D variancesmeasureddirectly
from the density fields. We find that the method is com-
parably accurate for these non-periodic fields, albeit with
3.3 Non-Periodic Fields
a rather large variation ( 40%) between different projec-
∼
In the foregoing analysis, we have used periodic fields for tions in the latest snapshot. In this field, the variance is
whichtherearenoedgediscontinuitiesandconsequentlyno very large indeed (7.7 106) and is dominated by a small
×
zero-paddingisrequired.Amorephysicallyrealisticscenario numberofverylargedensityvaluescausedbylocalisedgrav-
isprovidedbySPHsimulationsthatdonotemployperiodic itational collapse. The variances in the initial density field
boundaryconditions.Forouranalysis,wetakedensityfields (a spherical blob), in the t = t density field, and in the
ff
producedbythehydrodynamicsimulationsofPrice&Bate t = 1.274 t density field are recovered by the mean cal-
ff
(2009). Column density fields extracted at three different culated variances to within 0.3%, 20% and 20% accuracy
10 C. M. Brunt, C. Federrath, & Price, D. J.
respectively.Toevaluatethesuccessofthesemeasurements, Astrophysical Thermonuclear Flashes at the University of
we calculate an anisotropy indicator as was done in Sec- Chicago. We thank the anonymous referee for useful and
tion 3.2, by taking the ratio of the variance in one pro- interesting suggestions.
jected field to the mean of the variances in the other two
projected fields. Since the simulations contain no magnetic
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