Turbulent characteristics in the intensity fluctuations of a solar quiescent prominence observed by the \textit{Hinode} Solar Optical Telescope

0.46 MB·English

by  Ersilia Leonardis

THEASTROPHYSICALJOURNAL,PAGES,DATE
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
(cid:13)c 2011.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedintheU.S.A.
TURBULENTCHARACTERISTICSINTHEINTENSITYFLUCTUATIONSOFASOLARQUIESCENTPROMINENCE
OBSERVEDBYTHEHINODESOLAROPTICALTELESCOPE
E.LEONARDIS,S.C.CHAPMAN,ANDC.FOULLON
CentreforFusionSpaceandAstrophysics,DepartmentofPhysics,UniversityofWarwick,Coventry,CV47AL,UnitedKingdom
Submitted2011October13;accepted2011November24;published2011January17
2
1 ABSTRACT
0
We focus on Hinode Solar Optical Telescope (SOT) calcium II H-line observations of a solar quiescent
2
prominence (QP) that exhibits highly variable dynamics suggestive of turbulence. These images capture a

n sufficient range of scales spatially (∼0.1-100 arc seconds) and temporally (∼16.8 s - 4.5 hrs) to allow the
a application of statistical methods used to quantify finite range fluid turbulence. We present the first such
J application of these techniquesto the spatial intensity field of a long lived solar prominence. Fully evolved

3 inertial range turbulence in an infinite medium exhibits multifractal scale invariance in the statistics of its
2 fluctuations,seenaspowerlawpowerspectraandasscalingofthehigherordermoments(structurefunctions)
offluctuationswhichhavenon-Gaussianstatistics;fluctuationsδI(r,L)=I(r+L)- I(r)onlengthscaleLalong

] agivendirectioninobservedspatialfieldI havemomentsthatscaleas<δI(r,L)p>∼Lζ(p). Forturbulencein
R
a system thatisof finite size, orthatis notfullydeveloped,oneanticipatesa generalizedscale invarianceor
S
extendedself-similarity(ESS)<δI(r,L)p>∼G(L)ζ(p). FortheseQPintensitymeasurementswefindscaling
.
h inthepowerspectraandESS.Wefindthatthefluctuationstatisticsarenon-GaussianandweuseESStoobtain
p ratiosofthescalingexponentsζ(p): theseareconsistentwithamultifractalfieldandshowdistinctvaluesfor
- directions longitudinal and transverse to the bulk (driving) flow. Thus, the intensity fluctuations of the QP
o
exhibitstatisticalpropertiesconsistentwithanunderlyingturbulentflow.
r
t Subjectheadings:Sun:corona–Sun:prominences–magnetohydrodynamics(MHD)–plasmas–turbulence
s
a
[
1. INTRODUCTION personic. Indeed, evidence of bow-shock compressions are

2 seen in Bergeretal. (2010) and the correspondingReynolds
Solar prominences or filaments in the lower solar corona
v arerelativelycool,denseplasmastructureswithtemperatures numberisestimatedas∼105.Thequestionthenimmediately
9
5 of about 104 K. Solar filaments can be seen on the disk, arisesastowhethertheobservedfluctuationsdoinfactcorre-
spondtoaturbulentflow.
1 whilst prominences are observed above the solar limb. In
Many models have been developed to describe possi-
3 practice, they are classified in three main categoriesaccord-
ble scenarios for the production of dynamical structures
. ing to their locationon the Sun, namelyactive, intermediate
0 in the corona. The local magnetic field is suggested
and quiescent. The latter usually occur on the quiet Sun at
1 to play a key role as it is thought to be the driver of
highlatitudesandasaconsequencearealsoknownas"polar
1 the prominence threads (e.g., Low&Hundhausen 1995;
crown"prominences,whileactiveandintermediatefilaments
1 Foullonetal. 2009; Hershawetal. 2011). Recently, in
areoftenobservedatlowlatitudesassociatedwith activere-
: stronglyinhomogeneouscoronalplasmastructures,processes
v gions (Engvold 1998). All prominencesoriginate from fila-
i mentchannelsanddevelopabovethepolarityinversionline. such as magneto-thermal convection in solar prominences
X (Bergeretal.2011)andKelvin-Helmholtzinstabilitiesinthe
They show many different morphologiesand dynamics (see
r Mackayetal.2010,forarecentreview). corona (Foullonetal. 2011) have been suggested as mech-
a anisms for the generation of dynamical structures. Fur-
The Hinode Solar Optical Telescope (SOT) provides ob-
thermore, the observed complexity of the coronal mag-
servations of solar prominences revealing detailed internal
netic field may be generated by photospheric turbulence
dynamics at unprecedented spatio-temporal resolution. In
(Abramenkoetal.2008;Dimitropoulouetal.2009). Intrigu-
particular, dynamics associated with quiescent prominences
ingly,correlationsbetweenoutercoronaandsolarwindhave
(QPs)areseentoexhibitspatio-temporalevolutioncharacter-
also been found in the statistics of large-scale density fluc-
izedbyhighvariability(Bergeretal.2008). MostoftheQPs
tuations (Tellonietal. 2009) suggestive that the signature of
intheSOTdatasetappearverticallystructuredanddominated
coronal turbulence is convected with the solar wind plasma
by upward and downward transport of matter; the ascend-
ing flows appear dark and are faster (25 kms- 1on average) (Matthaeus&Goldstein 1986). To distinguish these pro-
cesses from turbulenceevolving in-situ (locally) in the flow,
than the descending flows (about 10 kms- 1). The upflows
wewillapplyanalysismethodsthathavebeenspecificallyde-
have often been observed to ultimately evolve into vortices
velopedtoquantifyfiniterangefluidturbulence.
(Liggett&Zirin 1984) and are considered to be associated
The characteristic, reproducible properties of a turbulent
with smallscale turbulence(Bergeretal. 2010). Thepromi-
flow are statistical in nature. They characterize a scale in-
nence is a low-β plasma with electron density ∼1011cm- 3 variance of the statistical properties of fluctuations - that is,
and temperatures up to ∼104 K (Tandberg-Hanssen 1995); these properties are unchanged as we move from scale to
these typical parameters suggest that the upward flow is su- scalesubjecttoarescaling. Thusinafullyevolvedmagneto-
2 LEONARDISETAL.
hydrodynamic(MHD) turbulent flow in an infinite medium, erageataspatialresolutionof0.10896arcsecperpixel,that
onefindspowerlaw dependenceof the physicalobservables is,onepixelcorrespondsto∆r∼77.22kmonthesolarsur-
of the flow - e.g., the velocity and magnetic field fluctua- face;eachimageis800×420pixels. Theimageshavebeen
tions have power law power spectra over a range of scales, calibrated(normalizedtotheexposuretime)andalignedwith
which is identified as the inertial range of the turbulence. respecttothesolarlimb.Furthermore,thesespecificobserva-
As power law power spectra are not unique to turbulence tionsarealongalineofsightthatistoagoodapproximation
(Sornette2000) anddonotuniquelycharacterizethe scaling perpendiculartotheprominencesheet.
ofthefluctuations(Chapmanetal.2008),multifractalscaling Figure1showsthefirstframeofthedataset. Notethedif-
ofthehigherordermoments(structurefunctions)offluctua- ferentstructures: largescalestructuresappearbrighteratthe
tionsisalsoneededtoidentifyaturbulentflow(Frisch1995). edge of the prominence while at smaller scales, bright and
Thesestatisticalmethodshavebeenappliedextensivelytoin- dark threads alternate within the plasma sheet. We will ex-
situobservationsoftheoutgoingflowfromthesolarcorona, aminefluctuationsinspacebytakingdifferencesinintensity
namely the solar wind, and have established its turbulent along directions longitudinal (vertical) and transverse (hori-
character (e.g., Horbury&Balogh 1997; Sorriso-Valvoetal. zontal)tothedirectionofupward/downwardflow.Thisproce-
1999;Pagel&Balogh2003;Bruno&Carbone2005). How- dureisshownbytheoverlaidgridwhichismadeof10strips
ever,thesein-situobservationsaretypicallysingleorfroma labelledasstripsL1toL5alongthelongitudinaldirectionand
few pointin space so that Taylor’shypothesis(Taylor1938) stripsT1toT5alongthetransversedirection;eachstripis10
isusuallyevokedtocharacterizethescalingpropertiesofthe pixels wide. We will also examine fluctuationsin time, that
flow. Here,SOTobservationsofaQPprovideadirectobser- is,fromoneimagetothenext.Fivewhitesquares,labelledA
vationofthespatialfieldoffluctuations. toE,withsize21×21pixels,indicatetheregionsoverwhich
In this paper we present the first application of these sta- the respective intensity time series are formed across all the
tistical methods to the spatial intensity field of a long lived images.
solar prominence. The SOT images are of intensity, rather In order to improve statistics we will construct local spa-
than velocity, and so intrinsic to our analysis is the assump- tial averagesand will presentthe variation aboutthese aver-
tion that the moving structures in the images follow the ages (DudokdeWit 2004). The procedure used to analyse
flow, acting as markers or passive scalars for the plasma the intensity measurementsin the strips consists in calculat-
dynamics (e.g. the intensity measurements can be treated ingstatisticalquantitiesforsmallensemblesof10neighbour-
as proportional to the squared density of the plasma flow). ingrows(columns)foreachstripalongthehorizontal(verti-
This assumption is supported by the correspondence be- cal)directionandthenperforminganaverageacrossthestrip
tween traceable UV motions and true mass motions, found width.Forexample,themeanvalueoftheintensitiesforstrip
using combined imaging and Doppler data of prominences T1 will be the average over the mean values calculated for
(Kuceraetal.2003). Importantly,theQPisoffinitephysical eachofthe10rowswithinT1.Thesameprocedureisadopted
size, and the turbulence may not be fully developed. Under for the analysis in the time domain: the statistical quantities
thesecircumstances,weanticipateageneralizedformofscale calculatedforeachtimeseriesassociatedwiththepixelsthat
invariance, that is, generalized similarity (scale invariance) composethesquareareaveragedoverthe21x21pixels.
also known as Extended Self-Similarity (ESS) (Benzietal. Figure 2 plots the variation in intensity I(r) for strip T5
1993). Generalizedsimilarityhasbeenseen in thefast solar versus pixel position r (top-leftpanel) and I(t) for square D
wind(Carboneetal.1996;Hnatetal.2005;Nicoletal.2008; versus time (bottom-left panel). They fluctuate strongly in
Chapman&Nicol 2009), in laboratorysimulations of MHD theirfirstdifferences,whicharedefinedinspaceasδI(r,L)=
turbulence (Dudsonetal. 2005; Dendy&Chapman 2006) I(r+L)- I(r)withL=1pixel(top-rightpanel)andintimeas
and in hydrodynamics (Grossmannetal. 1994; Bershadskii δI(t,τ)=I(t+τ)- I(t) with τ =∆t = 16.8 sec (bottom-right
2007). WefindthattheintensityfluctuationsintheQPdoin- panel). Thisisatypicalaspectofastochasticprocessinclud-
deedexhibitquantitativefeaturesconsistentwithafinitesize ingturbulence(Kantz&Schreiber1997).
turbulentflow,namely,ESS,multifractalityandnon-Gaussian
statistics.
3. POWERSPECTRA
2. THEDATASET Fully evolvedMHD turbulenceis self-similar in character
The Hinode spacecraft was launched in September 2006 exhibitingpowerlawscalinginthepowerspectrum. Wethus
andmovesinasun-synchronousorbitovertheday/nightter- analyse the power spectral densities (PSDs) of the intensity
minator, allowing near-continuous observations of the Sun. measurementsbothinthetimeandspacedomains.
TheSOTonboardHinodeisa diffraction-limitedGregorian TheleftpanelinFigure3showsthePSDsoftheintensity
telescopewitha0.5maperture,whichisabletoprovideim- measurementsforstripsT4andT5alongthetransversedirec-
ages of the Sun with an unprecedented resolution up to 0.2 tion and L1 and L2 along the longitudinal one. All power
arcsecandcadencesbetween15and30sec. TheBroadband spectra are dominated by two main slopes: at small wave
Filter Imager (BFI), one of the four instruments of the Fo- numbersthespectrascale as∼k- 2 consistentwith aBrown-
cal Plane Package on the SOT, provides observations over ianprocess,thatis,additivenoise(Percival&Walden2000),
a rangeof wavelengths(380-670nm) which distinguishdif- whileatlargerwavenumbersthespectrascaleas∼k- α with
ferent coronal structures. We use the Ca II H spectral line spectralindexαreportedinTable1foreachstripandsugges-
(396.85 nm) images of a QP observed by the SOT on the tive of non-trivialdynamics. The α values are estimated by
north-west solar limb (90W 52N) on November 30th, 2006 extractingthegradientofthelinearfitstotheplotsinFigure
(see the time evolution of the QP of interest in animation 1 3 (left) within the wave number ranges 2.43-4.55Mm- 1 for
ofBergeretal. (2008)). Thetime intervalconsideredcovers thelongitudinalstripsand2.07-4.61Mm- 1 forthetransverse
∼4.5 hrs, from 01:00:00 UT to 05:30:00 UT corresponding strips. The α values found are distinct from -5/3, which is
to about 1000 images with a cadence ∆t =16.8 sec on av- thevalueexpectedfortheKolmogorovspectrumforanideal
TURBULENTCHARACTERISTICSINASOLARQUIESCENTPROMINENCE 3
turbulentflow. Thisisnotsurprisingsincetheseobservations in the right panel of Figure 5 confirms a distribution nearly
areintegrated,lineofsightintensitymeasurements.However, Gaussianforthetemporalfluctuations.Thequasi-normaldis-
we should still expectline of sight measurementsto capture tributionofthefluctuationsinthetimedomainmayagainbe
qualitativefeaturesofturbulence(suchasnon-Gaussianfluc- aconsequenceofthecadenceoftheobservationsasdiscussed
tuations, multifractal scaling and ESS) whilst not necessary intheprevioussection.
givingthesamenumericalvaluesofscalingexponentsasin-
situpointobservations. 5. STATISTICALSCALINGPROPERTIESOFFINITE
The time series associated with squaresA to E revealdif- RANGETURBULENCE
ferent dynamics: they have power law power spectra in the
Akeypropertyofturbulenceisthatitcanbecharacterized
frequency domain with fitted spectral indices α in the fre-
and quantified in a robust and reproducible way in terms of
quency range 1-20 mHz very close to -1 (see Table 1 and
the ensemble averaged statistical properties of fluctuations.
the right panel of Figure 3). This ∼ 1/f scaling may be
Wecanaccesstothestatisticalscalingpropertiesofaspatial
simply attributable to a "random telegraph" process, that
series, f(x),alongagivendirectionx,byconstructingdiffer-
is, how a series of uncorrelated pulses or features in the
encesδf withincrementL:
flow moving through the line of sight of the observations
(Kaulakys&MešKauskas 1998; Kaulakysetal. 2005). We
δf(x,L)= f(x+L)- f(x) (1)
estimatethe"maximumobservablespeed"ofstructuresmov-
ingpastalineofsightasu=∆r/∆t∼ 4.6kms- 1. Sincethe on the spatial field. Generalized structure functions (GSFs)
prominenceflowhasabulkvelocity(u ∼25kms- 1)larger areapowerfultooltotestforstatisticalscalingandaredefined
flow
thanuthen,atagivenpixel,intensityfluctuationsaremoving as:
∞
toofastforustoobservecorrelationsintime. Inotherwords, Sp(L)= |δf|p = |δf|pP(δf,L)d(δf), (2)
thetimeneededtocatchacoherentstructure(e.g.,up-flows), (cid:10) (cid:11) Z- ∞
at fixed space coordinatesacross two consecutive frames, is
wheretheangularbracketsindicateanensembleaverageover
muchshorterthanthecadence,therefore,allthemovingflows
x, implying an assumption of approximate statistical homo-
intheprominenceappeardecorrelatedintime.
geneity. Fully developed inertial range turbulence in an in-
4. PROBABILITYDISTRIBUTION finite mediumexhibitsthe followingscaling forthe pth mo-
mentoftheGSF:
We now investigate the statistics of the intensity fluctua-
tionsinthespacedomain,δI(r,L)=I(r+L)- I(r)withlength Sp(L)∼Lζ(p) , (3)
scaleL,andinthetimedomain,δI(t,τ)=I(t+τ)- I(t),with wheretheζ(p)arethescalingexponents,whicharegenerally
time scale τ. Turbulent fluctuations in the inertial range in- anonlinearfunctionof p.
variably possess a non-Gaussian "heavy tailed" probability
Forthe specialcase of statistical self-similar(fractal) pro-
density function (PDF) that arises from the intermittent na- cessesonefindsalinearformofζ(p)in p,suchthat:
ture of the energy cascade in the flow (Marsch&Tu 1997;
Sorriso-Valvoetal.1999;Hnatetal.2002). ζ(p)=pH , (4)
The left panel in Figure 4 shows the PDF of the intensity
whereH istheHurstexponent.
fluctuationsforstripT5,normalizedtothemeanvalueµand
Influidturbulence,weanticipateintermittency,thatisζ(p)
standard deviation σ, in order to allow comparisons with a
isquadraticin p(Frisch1995). Determiningthepreciseζ(p)
Gaussian distribution (solid red line). The PDF of the spa-
iscentraltotestingturbulencetheories. Sincewedonothave
tial variations appears to be more peaked compared to the
measurements in-situ here, we cannot directly compare our
Gaussian distribution. A measure of the "peakedness" of a
observed ζ(p) value with predictions of turbulence theories.
probabilitydistributionisgivenbythekurtosisparameter,k,
However we can test whether the ζ(p) that we observe are
defined as k=<δI>4/σ4, where <δI>4 is the fourth mo-
non-linearwith p,consistentwithamultifractal,intermittent
ment probability distribution. Since Gaussian distributions
flowandwediscussthisinthenextsection.
havek=3,thentheexcesskurtosiskiscommonlyused,which First we will focus upon the direct observationsof fluctu-
isdefinedask=k- 3. ationsinthespatialfieldasthesecapturenon-trivialcorrela-
The excess kurtosis k calculated for the PDF of strip T5 tionsinthefluctuationsintheflow. TheleftpanelsofFigure
is 2.44 ± 0.17 indicating a non-Gaussian distribution. Fur- 6showlog-logplotsoftheaveraged3rdmomentoftheGSF,
ther evidence of non-Gaussianstatistics is given by the nor- <S >,versusL (toppanel)forstripsT1toT5andversus
3 trans
mal probability plot of the cumulative distribution function L (bottompanel)forstripsL1toL6,whereL andL
long trans long
(CDF). This is a quantile-quantile (Q-Q) plot where quanti- identify the pixel increments of the fluctuations δI(L) along
ties of the observed CDF (y-axis) are plotted against that of thetransverseandlongitudinaldirectionrespectively. Recall
a normal or Gaussian CDF (x-axis). If the data are normal that<S >referstotheaverageoverthestructurefunctions
3
distributed then the normalprobability plot of the CDF will calculatedforeachofthe10rows(orcolumns)formingasin-
belinear,whileotherdistributiontypeswillintroducecurva- glestrip. Thestructurefunctionanalysisprovidesameasure-
tureintheplot. TheleftpanelofFigure5showstheCDFof mentofthecorrelationofthefluctuationswithlengthscaleL.
strip T5, which does not follow the theoretical function ex- TheincreaseoftheGSFswithLintheleftplotsofFigure6
pectedforaGaussiandistribution(reddot-dashedline). The thussuggeststhatthe spatialintensityfluctuationsofthe QP
temporalfluctuationsareofdifferentcharacter:theyaremore arehighlycorrelated.Thisisasignatureofthepresenceofco-
closely described by a Gaussian distribution function. The herentstructuresintheflow.Inparticular,theintensityfluctu-
right panel of Figure 4 shows the PDF of the intensity fluc- ationsinthelongitudinaldirection(bottom-leftpanel)reveala
tuations for square D which has k =0.78±0.16, indicating correlationoverabroaderrangeofspatialscalesasthecoher-
statistics very close to Gaussian. Furthermore, the Q-Q plot entstructuredetectedareassociatedtotheupanddownflows
4 LEONARDISETAL.
oftheQP,whichmovealongtheverticaldirection;thisisthe tions of strip T5. In Figure 7 we plot the 3rd moment of
longitudinaldirectioninwhichweexpecttoseethestrongest the structure function, S , normalized to a value L ∼0.54
3 0
correlationinaturbulentflow(Frisch1995).Alongthetrans- MmagainstL/L (inlogarithmicaxes)for7consecutivetime
0
verse direction (top panel of Figure 6), the curves exhibit a intervals separated by ∆T = 1.12 min and starting at t =
0
knee within a range of length scales of 0.9-2 Mm (dashed 01:10:31 UT. The choice of the value for the parameter L
0
black lines). These "break points" delimit the crossover be- arises from the characteristic width (on average) of the up-
tweenthesmall-scaleturbulenceandthelarge-scalecoherent flows. The collapse of all the GSFs onto each other within
structures;theabovelengthscaleshavebeenattributedtothe the inertial range indicates the existence of a single scaling
typical distances between the dark up-flows and seem to be function G(L/L ). The overlapping of the various GSFs in
0
ingoodagreementwith themulti-moderegimeofRayleigh- Figure7breakswheretheeffectsoflarge-scalestructuresbe-
Taylorinstability(Ryutovaetal.2010). comeimportant. ThisbreakpointforthestripT5occursata
lengthscaleof∼2.8Mm,whichcorrespondstothewidthof
6. EVIDENCEFORESSANDMULTIFRACTAL thelargebrightstructureshowninFigure1ontherightofthe
SCALING prominence and crossed by strip T5 and the length scale of
transitiontocoherentstructuresinFigure6(top-leftpanel).
The GSFs shown in the left panels of Figure 6 clearly do
notfollowthepower-lawscalingofEquation(3). Corrections
7. CONCLUSIONS
tothisequationindeedhavetobetakenintoaccountforreal
Weperformedthefirstqualitativetestforin-situturbulence
turbulentflowsforwhichfiniterangeturbulenceeffectsmay
inaQPobservedbyHinode/SOT.Weanalysedthestatistical
arise (Dubrulle2000; Bershadskii 2007); either when turbu-
propertiesofthespatio-temporalintensityfluctuationsassoci-
lence is notcompletelyevolved(low Reynoldsnumber),the
atedwiththeimagedQPfromtheprospectiveofafinitesized
data sets are of finite size (realistic cases) or the system is
turbulentsystem. Wefoundthefollowing:
bounded,thensymmetriesintheflowarebroken,andthesim-
Inspace:
ilarity is lost. Nevertheless, a generalized similarity or ESS
1.ThePSDsoftheintensitymeasurementsinthespacedo-
has been observed, which suggests a generalized scaling for
mainexhibitpowerlaw scaling suggestiveof non-trivialdy-
the pthmomentoftheGSFbyreplacingLinEquation(3)by
namics.
aninitiallyunknownfunctionG(L),suchthat
2. The PDFsofthe intensityfluctuationsare describedby
Sp(L)∼G(L)ζ(p). (5) non-Gaussianstatisticsconsistentwithsmall-scaleMHDtur-
bulence.
This arises directly from ESS (Benzietal. 1993; 3. The GSFs of the intensityfluctuationssuggesta gener-
Carboneetal. 1996). Comparing structure functions of alized scaling for the structure functions with a dependence
differentorder pandq,wecanthenwrite: on a function G(L). They also reveal a high degree of cor-
relationespeciallyalongthelongitudinaldirectiontothebulk
Sp(L)= Sq(L) ζ(p)/ζ(q) . (6) (driven)flow.Characteristiclengthscalesinthetransversedi-
(cid:2) (cid:3) rectionhavebeendetectedandassociatedtothecharacteristic
Alog-logplotofSp versusSq willthereforegivetheratioof
distancesbetweentheup-flows.
therespectivescalingexponents,ζ(p)/ζ(q).
4.ESSholdsforallthestripsconsideredanditisconsistent
ThemiddlepanelsinFigure6showtheESSinlogarithmic
for each direction transverse and longitudinal to the flow as
scale of the GSF for p=2 and q=3. These are straight lines
a signature of the generalized similarity expected for finite
on the log-logof Figure6 thusconfirmingthatEquation(6)
rangeturbulentsystems.
holds. The gradient of such plots in the inertial range pro-
5. Theratio ofthe scaling exponentsζ(2)/ζ(3)is roughly
videsameasurementoftheratioζ(2)/ζ(3). Departuresofthe
constant for all the strips along each direction and its value
curvesfromalinearbehaviouroccurforlengthscalesoutside
is distinct from 0.66, that is the value expected for a frac-
the inertial range and are associated with large-scale coher-
tal system. The prominenceflow is thereforemultifractalin
ent structuresin the flow. Finally, the right panelsin Figure
character,againconsistentwithin-situturbulence.
6showζ(2)/ζ(3)forstripsT1toT5alongthetransversedi-
6.Theintensityfluctuationsinthespacedomainsatisfythe
rection(toppanel)andforstripsL1toL5inthelongitudinal
generalizedscalinganticipatedbyESSandascalingfunction
direction(bottompanel). Theerrorbarsprovideanestimate
G(L/L )isobservedfordifferentsuccessivetimeintervals.
0
of the uncertainty in the gradients of the fitted lines in the
Intime:
inertialrange.Theratioζ(2)/ζ(3)appearstoberoughlycon-
7. ThePSDsshow∼1/f scaling,consistentwithuncorre-
stantacrossallthestripsand,moreinterestingly,differsfrom
latedpulsesmovingpastthelineofsightoftheobservations;
the value that one would expect if ζ(p) was linear in p, i.e.
TheintensityfluctuationsareclosetoGaussiandistributed.
ζ(2)/ζ(3)=2H/3H∼0.66(seeEquation(4)). Theratiosof
thescalingexponentsfoundforallthestripsarethereforecon-
Theprincipalaimofthispaperhasbeentoexplore,forthe
sistent with a non-linear form of the scaling exponent ζ(p).
firsttime,thepossibilityofdiscerningthequantitativesigna-
This is a signature of the multifractal nature of this system
turesofturbulence,namelymultifractalorintermittentstatis-
whichindicatesintermittencywithintheQPflow.
tical scaling, within the flows of a long-lived QP. We have
Thegeneralizedsimilarityhasbeentestedexplicitlyinthe
shownhowtestsfornon-Gaussianity,multifractality,scaling
inertialrangeofsolarwindturbulencebye.g. Chapmanetal.
and ESS can be applied in order to fully identify and quan-
(2009)whoformalizedEquation(5)asfollows:
tify statistical properties of turbulent fluctuations. For these
Sp(L)=[Sp(L )]G(L/L )ζ(p) , (7) specificintensitymeasurementswearerestrictedtoaqualita-
0 0 tivecharacterizationofthefluctuationssincetheobservations
whereL is somecharacteristiclength-scaleofthe flow. We areintegratedalongthelineofsightratherthanin-situinthe
0
finally test this generalized scaling for the intensity fluctua- flow. Despitethisconstraint,thestatisticalmethodsusedare
TURBULENTCHARACTERISTICSINASOLARQUIESCENTPROMINENCE 5
Table1
Spectralindices
Domain Data α±∆α
T4 3.17±0.15
Wavenumber T5 2.93±0.19
L1 2.73±0.29
L2 2.74±0.37
A 1.21±0.04
B 1.17±0.04
Frequency C 1.29±0.04
D 1.27±0.04
E 1.20±0.04
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Fusion,48,B313
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ofinterestareconsistentwithaMHDturbulentflowforsys- Strintzi,D.,&Moussas,X.2009,A&A,505,1245
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6 LEONARDISETAL.
Figure1. QPobservedbySOTintheCaIIlineonNovember30th,2006at01:10:31UT.Imageresolution:1pixel∼77.22kmonthesolarsurface.Theimage
hasbeenrotatedtothehorizontalpositionwithrespecttothesolarlimb.Intensitylevelsincreasefrombluetowhite.Thewhitegridand5squaresareshownas
referencefortheanalysisinthespacedomain(transversestripsT1toT5andlongitudinalstripsL1toL5)andinthetimedomain(squaresA,B,C,DandE).
4
60
s) s) 2
N/50 N/
D D 0
I(r) (40 r,L) ( −2
30 T5 I( T5
0 200 400 600 800 d −40 200 400 600 800
r (pixel) r (pixel)
55 5
D D
)
s)50 N/s 2.5
N/ D
D ( 0
( )
I(t) 45 tI(t,−2.5
d
40 −5
0 100 200 300 0 100 200 300
Time (min) Time (min)
Figure2. Toppanels:IntensityseriesI(r)inthespacedomainforstripT5(left)andthecorrespondingfirstdifferencesδI=I(r+L)- I(r)withL=1pixel(right).
Bottompanels:IntensityseriesI(t)inthetimedomainforsquareD(left)andthecorrespondingfirstdifferencesδI=I(t+τ)- I(t)withτ=∆t=16.8sec(right).
TURBULENTCHARACTERISTICSINASOLARQUIESCENTPROMINENCE 7
T4 T5 L1 L2 A B C D E

m) 18 −1)
M z
H
2s) 14 2s) 10
N/ N/
D D
> (( 10 > ((
D D 5
S S
P 6 P
< <
g 10 "~k−2" g 10
lo 2 "k−a " lo 0 "f−a "

−1.5 −1 −0.5 0 0.5 1 −4 −3.5 −3 −2.5 −2 −1.5
log k (Mm−1) log f (Hz)
10 10
Figure3. Log-logplotsoftheintensityspectraforstripsT4,T5,L1andL2(left)andforsquaresA,B,C,D,andE(right). Allthespectraareshiftedinthe
y-directionforclarity. PSDsinthewavenumberdomain(left)revealtworegionswithdifferentscalingexponents: k- 2(solidline)andk- α(dashedline),while
inthefrequencydomain(right)thePSDsshowasinglescaling, f- α,withspectralindexα(dashedline).AlltheαvaluesaregiveninTable1.
1.5 1
T5 D
Gaussian Gaussian
) 0.8
)
s>)/ 1 s)/
I I> 0.6
d<
I − d I − < 0.4
dF((0.5 d((
D F
D
P P 0.2
0 0
−4 −2 0 2 4 −4 −2 0 2 4
(d I − <d I>)/ s (d I − <d I>)/ s
Figure4. PDFsoftheintensityfluctuationsinspaceδI=I(r+L)- I(r)forstripT5withL=15pixel∼1.16Mm(left)andintimeδI=I(t+τ)- I(t)forsquare
Dwithτ =1.12min(right).BothPDFsarenormalizedtothemean<δI>andthevarianceσoftheintensity. RedsolidlinesareGaussianPDFswithµ=0and
σ=1.
8 LEONARDISETAL.

0.999 0.999
T5 D
0.99 Gaussian 0.99 Gaussian
0.95 0.95
y y
bilit 0.75 bilit0.75
a a
b b
o 0.25 o0.25
Pr Pr
0.05 0.05
0.01 0.01
0.001 0.001

−20 −10 0 10 20 3 −2 −1 0 1 2 3
d I = I(r+L) − I(r) d I = I(t+t ) − I(t)
Figure5. Normalprobabilityplots(Q-QplotsagainstaGaussian)oftheCDFsoftheintensityfluctuationsforstripT5inspace(left)andsquareDintime
(right).LengthandtimescalesarethoseinFigure4.DashedredlinesrefertotheprobabilityexpectedforaGaussiandistribution.
TURBULENTCHARACTERISTICSINASOLARQUIESCENTPROMINENCE 9
T1 T2 T3 T4 T5

8
6
>] >] 0.8
)s6 )s
Ltran Ltran4 30.75
S(34 S(2 z/20.7
< <
[0 [02 z 0.66
12 1
g g
o o
l l 0 0.6
0.9 − 2 Mm
0
0.55
0 1 2 3 0 1 2 3 4 T1 T2 T3 T4 T5
log [L (pixel)]
10 trans log [<S (L )>] Strips
10 3 trans
L1 L2 L3 L4 L5
14
12 12 0.8
>] >]
)ng10 )ng10 0.75
o o
Ll 8 Ll 3
S(3 6 S(2 8 z/20.7
< <
[0 4 [0 z0.66
1 1
g g 6
o 2 o
l l 0.6
0
0 1 2 3 4 0 2 4 6 L1 L2 L3 L4 L5
log [L (pixel)] log [<S (L )>] Strips
10 long 10 3 long
Figure6. Leftpanels: Log-logplotof<S3>versusLtransfortheintensityfluctuationsalongthetransversedirectionforstripsT1toT5(top). Dashedlines
delimittherangeofscaleswherethecurveexhibitaknee.Thebottompanelshowsthelog-logplotof<S3>vs.LlongforstripsL1toL5alongthelongitudinal
directiontotheflow.Allthecurvesareshiftedinthey-directionforclarity. Middlepanels:Log-logplotsofS2againstS3forallthestripsconsideredshowing
evidenceofESS.DashedlinescorrespondtothelinearregressionfitsacrosstherangeLtrans=0.2-5Mmforthetransversestrips(top)andLlong=0.2-10Mm
forstripsL1toL5(bottom). Allthecurvesareshiftedinthey-directionforclarity. Rightpanels: Gradientsofthelinearfitsshownintherespectivemiddle
panels,whichcorrespondtotheratioofthescalingexponentsζ(2)/ζ(3). Noticethattheratiosareroughlyconstantforeachdirectionanddifferentfromthe
value0.66forbothtransverse(top)andlongitudinal(bottom)strips.Valuesofζ(2)/ζ(3)6=0.66indicatethatζ(p)isnon-linearinpandthereforetheprominence
flowshowsamultifractalcharacter.
10 LEONARDISETAL.
1.5
T5
] 1
>
)
0
L
( 0.5
3
S
<
>/ 0 L ~ 2.8 Mm
)
s
an−0.5
Ltr
(3 −1
S L = 7 pixel ~ 0.54 Mm
< 0
[0−1.5 D T ~ 1.12 min
1
g
o −2
l t = 01:10:31 UT
0
−2.5
−1 0 1 2
log [L /L ] (pixel)
10 trans 0
Figure7. Log-logplotof<S3(Ltrans)>/<S3(L0)>versusLtrans/L0withL0=0.54MmforstripT5forsevendifferentframesofthedatasetseparatedby∆T
=1.12min.t0isthetimeoftheobservationcorrespondingtothefirstframeofthedataset.NoticethatthevariousstructurefunctionssuperimposeuptoscaleL
∼2.8Mm.