Period ratios for standing kink and sausage modes in magnetized structures with siphon flow on the Sun

1.6 MB·

by  Hui Yu

ResearchinAstron.Astrophys. 2012Vol.XNo.XX,000–000 R
esearchin
http://www.raa-journal.org http://www.iop.org/journals/raa A
stronomyand
A
strophysics
Period ratios for standing kink and sausage modes in magnetized
6
1 structures with siphon flow on the Sun
0
2

n HuiYu1,Shao-XiaChen1,BoLi1,&Li-DongXia1
a
J ShandongProvincialKeyLaboratoryofOpticalAstronomyandSolar-TerrestrialEnvironment,

8 InstituteofSpaceSciences,ShandongUniversity,Weihai,264209,China;[email protected]

]
R
Abstract Standing oscillations with multiple periods were found in a number of atmo-
S
sphericstructuresontheSun.Theratiooftheperiodofthefundamentaltotwicetheone
.
h of its first overtone,P /2P , is importantin applications of solar magneto-seismology.
1 2
p
We examine how field-aligned flows impact P /2P of standing modes in solar mag-
- 1 2
o netic cylinders.For coronalloops, the flow effectsare significantfor bothfast kinkand
r sausagemodes.Forkinkones,theyreduceP /2P byupto17%relativetothestaticcase
1 2
t
s evenwhenthedensitycontrastbetweentheloopanditssurroundingsapproachesinfinity.
a Forsausage modes,the reductionin P /2P duetoflow istypically. 5.5%compared
[ 1 2
with the static case. However,the threshold aspect ratio, only abovewhich can trapped

1 sausage modes be supported, may increase dramatically with the flow magnitude. For
v photospherictubes, the flow effect on P /2P is not as strong. However,when applied
1 2
3 tosausagemodes,introducingfield-alignedflowsoffersmorepossibilitiesininterpreting
0
the multiple periodsrecently measured. We concludethat field-alignedflows should be
8
taken into accountto help better understandwhat causesthe departureof P /2P from
1 1 2
0 unity.
.
1
Key words: magnetohydrodynamics (MHD) – Sun: corona – Sun: magnetic fields –
0
waves
6
1
:
v 1 INTRODUCTION
i
X
The frequently measured waves and oscillations can be exploited to deduce the physical param-
r
eters of the structured solar atmosphere that are otherwise difficult to yield, thanks to the di-
a
agnostic power of the solar magneto-seismology (SMS) (see e.g., the reviews by Roberts 2000;
Aschwanden 2004; Nakariakov&Verwichte 2005; Nakariakov&Erde´lyi 2009; Erde´lyi&Goossens
2011; DeMoortel&Nakariakov 2012). In the context of SMS, multiple periodicities inter-
preted as a fundamental standing mode and its overtones detected in a substantial number
of oscillating structures are playing an increasingly important role (see Andriesetal. 2009;
Ruderman&Erde´lyi2009,forrecentreviews).Inthecaseofstandingkinkoscillations,bothtwo(e.g.,
Verwichteetal.2004;VanDoorsselaereetal.2007)andthreeperiodicities(DeMoortel&Brady2007;
vanDoorsselaereetal. 2009; Inglis&Nakariakov 2009; Kupriyanovaetal. 2013) have been found.
Moreover, the ratio between the period of the fundamentaland twice the period of its first overtone,
P /2P ,deviatesingeneralfromunity.ThiswasfirstfoundbyVerwichteetal.(2004)intwoloopsina
1 2
post-flarearcadeobservedbytheTransitionRegionandCoronaExplorer(TRACE)inits171A˚ pass-
bandon2001April15,wherevaluesof0.91and0.82weremeasuredforP /2P .Asimilarvalue(0.9)
1 2
2 HuiYuetal.
wasfoundforTRACE 171A˚ loopson1998November23(VanDoorsselaereetal.2007),andalsoin
flaringloopsasmeasuredwiththeNobeyamaRadioheliograph(NoRH)on2002July3whereP /2P
1 2
isdeducedtobe0.83(Kupriyanovaetal.2013).InthislatterstudythedeviationP /2P fromunityis
1 2
likelytobeassociatedwithwavedispersionatafiniteaspectratiooftheflaringloop.However,loops
seemthininEUVimages,therebypromptingAndriesetal.(2005)toattributethefinite1 P /2P to
1 2
−
thelongitudinalstructuringinloopdensities,giventhatwavedispersionisexpectedtobeminimalfor
kinkmodessupportedbyastaticlongitudinallyuniformloopwithtinyaspectratios.Whenitcomesto
standing sausage modes, fundamentalor globalmodestogetherwith their first overtoneswere identi-
fied.Inflare-associatedquasi-periodicpulsationsmeasuredwithNoRH on2000January12,theglobal
(fundamental)sausage mode was foundto have a P of 14 17 s, and its first overtonecorresponds
1
−
to a P of 8 11s (Nakariakovetal. 2003;Melnikovetal. 2005). Interestingly,sausagemodeswere
2
−
alsoseenincoolpost-flareloopsinhighspatialresolutionH imagesandcorrespondtoaP 587s
α 1
≈
andaP 349s(Srivastavaetal.2008).Actually,sausagemodeshavebeendirectlyimagedinmag-
2
≈
neticpores(Mortonetal. 2011)andchromosphere(Mortonetal. 2012).UsingtheRapid Oscillations
in the Solar Atmosphere (ROSA) instrument situated at the Dunn Solar Telescope, the former study
employedanEmpiricalModeDecomposition(EMD)analysisto revealanumberofperiods,someof
whichseemtocorrespondtoafundamentalmodeanditshigherovertoneswiththestandingmodeset
upbyreflectionsbetweenthephotosphereandtransitionregion.
Inthesolar atmosphere,flowsseemubiquitous(e.g.,Aschwanden2004), andhavebeenfoundin
oscillating structures in particular (e.g., Ofman&Wang 2008; Srivastavaetal. 2008). In the coronal
case,wheretheflowspeedstendtobe.100 kms−1 andthereforewellbelowtheAlfve´nspeed,they
arenotnecessarilyalwayssmall.Asamatteroffact,speedsreachingtheAlfve´nicrange( 103kms−1)
∼
havebeenseenassociatedwithexplosiveevents(e.g.,Innesetal.2003;Harraetal.2005).Inthecontext
of standing modes supported by loops, a siphon flow causes their phases to depend on the locations,
which is true even for the fundamental mode where only two permanent nodes are present and are
locatedatloopfootpoints.Actuallythislocation-dependentphasedistributionwasseenforthestanding
kinkmodemeasuredwith TRACE andSOHO on2001September15 (Verwichteetal. 2010), and
yields a flow speed indeed in the Alfve´nic regime (Terradasetal. 2011). The authorswent on to find
thatneglectingtheflowsleadstoanunderestimationoftheloopmagneticfieldstrengthbyafactorof
three.
Giventhatmultipleperiodicitieshavereceivedconsiderableinterest,andthatasignificantflowmay
play an important role as far as the applications of the solar magneto-seismologyare concerned,one
naturallyasks:howdotheflowsaffectmultipleperiodicitiesfroma theoreticalperspective?andwhat
wouldbetheobservationalimplications?Inaslabgeometry,thesequestionswereaddressedbyLietal.
(2013) (hereafter paper I) where a rather comprehensive analytical and numerical examination was
conducted. In cylindrical geometry, the flow effect on the period ratio for standing kink modes was
assessed by Ruderman (2010) for thin coronal loops. The present work extends both paper I and the
one by Ruderman (2010) by examining how the flows affect the dispersion properties and hence the
periodratiosofbothstandingkinkandsausagemodessupportedbyamagnetizedcylinderofarbitrary
aspectratio.Besides,inadditiontoacoronalenvironment,aphotosphericonewillalsobeexaminedin
detailtodemonstratehowintroducingaflowhelpsoffermorepossibilitiesininterpretingtherecently
measuredmultipleperiodsinoscillatingphotosphericstructures.
Thispaperisorganizedasfollows.Section2presentsabriefdescriptionofthecylinderdispersion
relation.Section3isconcernedwithcoronalcylinders,wherewefirstgiveanoverviewofthedispersion
diagrams,briefly describe a graphicalmeansto computethe period ratios, and examinehow the flow
affects the period ratios for standing kink and sausage modes. Likewise, section 4 examinesin detail
isolatedphotosphericcylinders.Finally,section5summarizestheresults,endingwithsomeconcluding
remarks.
Periodratiosforstandingkinkandsausagemodes 3
Fig.1 Schematic diagram illustrating the magnetic cylinder (denoted by subscript 0) and
its environment(subscripte). The variablesρ ,c ,v and U (i = 0,e) representthe mass
i i Ai i
density,adiabaticsoundspeed,Alfve´nspeed,andtheflowspeed,respectively.
2 CYLINDERDISPERSIONRELATION
Consider a cylinderof radiusa with time-independentfield-alignedflows. As illustrated in Fig.1, the
cylinderisinfiniteinthez-direction,andisborderedbytheinterfacer = ainacylindricalcoordinate
system(r,θ,z).Thephysicalparameterstaketheformofastepfunction,characterizedbytheirvalues
externalto (denotedby a subscripte) and inside (subscript0) the cylinder.The backgroundmagnetic
fields(B andB ), togetherwith theflow velocities(U andU ),are inthe z-direction.Letρ andp
0 e 0 e
denote the mass density and thermal pressure. It follows from the force balance condition across the
interfacethat
ρ 2c2+γv2
e = 0 A0, (1)
ρ 2c2+γv2
0 e Ae
whereγ = 5/3istheadiabaticindex,c = γp/ρistheadiabaticsoundandv = B2/4πρisthe
A
Alfve´nspeed.Itisalsonecessarytointroducethetubespeeds,c (i=0,e),
Ti
p p
c2v2
c2 = i Ai , (2)
Ti c2+v2
i Ai
4 HuiYuetal.
andthekinkspeedc ,
k
c2 =ρˆ v2 +ρˆ v2 , (3)
k 0 A0 e Ae
whereρˆ =ρ /(ρ +ρ )isthefractionaldensitywithi=0,e.
i i 0 e
The dispersion relation (DR) for linear waves trappedin a cylinder with flow has been examined
by a number of authors(e.g., Narayanan1991; Somasundarametal. 1999; Terra-Homemetal. 2003;
VasheghaniFarahanietal.2009;Zhelyazkov2009,2012).Itsderivationstartswiththeansatzthatany
perturbationδf(r,θ,z;t)totheequilibriumf(r)takestheform
δf(r,θ,z;t)=Re f˜(r)exp[i(kz+nθ ωt)] , (4)
−
n o
where Re( ) means taking the real part of the function. Besides, k and n are the longitudinal and
···
azimuthalwavenumbers,respectively.Thephasespeedv isdefinedasv = ω/k.Oneproceedsby
ph ph
defining
c2 (v U )2 v2 (v U )2
m2 =k2 i − ph− i Ai− ph− i , (5)
i h (c2+v2 ) c2ih (v U )2 i
i Ai Ti− ph− i
h i
wherei=0,e.Toensurethewavesaretrapped,m2hastobepositive,meaningthat
e
c2 <(v U )2 <c2 or(v U )2 <c2 , (6)
m,e ph− e M,e ph− e Te
where c = min(c ,v ) and c = max(c ,v ). On the other hand, the spatial profiles of the
m,e e Ae M,e e Ae
perturbationsinther directionaredeterminedbythesignofm2.Whenm2 <0(m2 >0),thewaves
− 0 0 0
arebody(surface)ones,correspondingtoanoscillatory(aspatiallydecaying)r-dependenceinsidethe
cylinder.
With Eq.(4) inserted into the linearized, ideal MHD equations, the DR follows from the require-
mentsthattheradialcomponentoftheLagrangiandisplacementξ andthetotalpressureδp becon-
r T
tinuousatr =a.TheDRreads
ρe m0 vA2e−(vph−Ue)2 In′(m0a) = Kn′(|me|a) (7)
ρ0 me hv2 (v U )2iIn(m0a) Kn(me a)
| | A0− ph− 0 | |
h i
forsurfacewaves,and
ρe n0 vA2e−(vph−Ue)2 Jn′(n0a) = Kn′(|me|a) (8)
ρ0 me hv2 (v U )2iJn(n0a) Kn(me a)
| | A0− ph− 0 | |
h i
for body waves (n2 = m2 > 0). Furthermore,kink and sausage waves correspondto the solutions
0 − 0
totheDRwithnbeing1and0,respectively.TheprimedenotesthederivativeofBesselfunctionwith
respecttoitsargument,e.g.,J′(n a) dJ (x)/dxwithx=n a.Onemaynotethatm appearsonly
n 0 ≡ n 0 e
asabsolutevaluestoensurethatthewavesexternaltothecylinderareevanescent.
Itprovesnecessarytoexaminetheimportanceofdensityfluctuationrelativetothetransversedis-
placement.ThisisreadilydonebyevaluatingX (ρ˜/ρ )/(ξ˜/a) ,
0 r
≡ r=a
(cid:12)
(cid:12)
X = (m20a)(ω−kU0)2 p˜T (cid:12) . (9)
[k2c2 (ω kU )2]dp˜ /dr
0− − 0 T (cid:12)r=a
(cid:12)
(cid:12)
(cid:12)
Periodratiosforstandingkinkandsausagemodes 5
Forbodywaves,p˜ insidethecylinderisproportionaltoJ (n r)forakinkwave,andtoJ (n r)fora
T 1 0 0 0
sausageone,resultingin
(n a)J (n a)/J′(n a)
0 1 0 1 0
X = (vph−U0)2 kink, (10)
(vph−U0)2−c20 (n0saa)uJs0a(gne0.a)/J0′(n0a)
Likewise,forsurfacewaves,insidethecylinderp˜ I (m r)forakinkwave,and I (m r)fora
T 1 0 0 0
∝ ∝
sausageone,leadingto
(m a)I (m a)/I′(m a)
0 1 0 1 0
X = (vph−U0)2 kink, (11)
c20−(vph−U0)2 (m0saau)Is0a(gme.0a)/I0′(m0a)
The dispersion relations (7) and (8) possess three symmetric properties that allow us to simplify
ourexaminationofthestandingmodes.Thefirsttwodictatethatif[v ,k;U ,U ]representsasolution
ph 0 e
to the DR, then so does [v , k;U ,U ]; if [v ,k;U ,0] is a solution, then so is [ v ,k; U ,0]
ph 0 e ph 0 ph 0
− − −
(seeEq.(5)withU = 0).TheyweredetailedintheappendixofpaperIwhichadoptsaslabgeometry,
e
and can be readily shown to hold in the cylindrical case if one recognizes that xZ′(x)/Z (x) is an
n n
evenfunctionforBesselfunctionsZ ofintegerordern,whereZ isJ orI .Theyaresummarized
n n n n
here for one to realize that as long as the external medium is at rest (U = 0), as will be assumed
e
throughoutthisstudy,thenforthe purposeof examininghowthe periodratio dependson the internal
flowU ,oneneedsonlytoconsiderpositiveU .ThethirdsymmetrypropertyreflectssimplyaGalilean
0 0
transformation,which relates the phase speed v (k;U ,U ) in one frame,where the speeds read U
ph 0 e 0
andU ,tov (k;U†,U†)inadifferentonewherethespeedsreadU† andU†.Certainlyonerequires
e ph 0 e 0 e
thatU† U† = U U . One thensees that v (k;U†,U†) = v (k;U ,U )+(U† U ), and in
e − 0 e − 0 ph 0 e ph 0 e 0 − 0
particular,v (k;U U ,0) = v (k;U ,U ) U . Whatthismeansisthat, eventhoughthe wave
ph 0 e ph 0 e e
− −
dispersionpropertiesexpressedasaseriesofanalyticalexpressionsinanumberofphysicallyinteresting
limitsinbothcoronalandphotosphericenvironmentsaretobederivedinaframewhereU = 0,they
e
canbeeasilyextendedtoanarbitraryframeofreference.
3 PERIODRATIOSFORSTANDINGMODESSUPPORTEDBYCORONALCYLINDERS
3.1 OverviewofCoronalCylinderDispersionDiagrams
Consider first the coronal case, where the ordering v > v > c > c holds. To be specific, we
Ae A0 0 e
choosev = 4c andc = 0.72c ,theobservationaljustificationofwhichwasgiveninpaperI.For
A0 0 e 0
theexternalAlfve´nspeed,unlessotherwisespecified,we willdiscussindetailareferencecasewhere
v =2v .Evidently,thelargertheratiov /v ,thestrongerthedensitycontrast.
Ae A0 Ae A0
Figure 2 presents the dependence on longitudinal wavenumber k of the phase speeds v for a
ph
seriesofU =M c ,wheretheinternalMachnumberM reads0,0.8,1.2and3.2,respectively.Kink
0 0 0 0
andsausagewavesareplottedwiththedashedandsolidcurves,respectively.AsshowninFig.2b,they
are labeled by combinationsof letters b/f+F/S+K/S, representingbackward or forward, Fast or Slow,
KinkorSausage.“Fast”or“Slow”isrelatedtothemagnitudeofthephasespeed,while“backward”or
“forward”derivesfromthesignofthephasespeedswhentheflowisabsent,andwastermed“originally
backward-(forward-)propagating”byAndriesetal.(2000)inthesamesense.Thenumberappendedto
thelettersdenotestheorderofoccurrence,meaningthatfFK1representsthefirstbranchofforwardFast
Kink wave. The characteristicspeedsexternal(interior)to the cylinderare givenon the left(right)of
eachpaneltoaidwavecategorization.InagreementwithTerra-Homemetal.(2003)(hereafterTEB03),
Fig.2indicatesthatallwavesinsuchacoronalenvironmentarebodywaves.
6 HuiYuetal.
AclearflowdependencecanbeseeninFig.2.Considerfirsttheslowwaves.Thepropagationwin-
dowsalwaysencompass( c +U , c +U )and(c +U ,c +U ),whichisreadilyunderstandable
0 0 T0 0 T0 0 0 0
− −
whenoneexaminestheslenderandthickcylinderlimits.Intheformerlimit(ka 1),
≪
c4 k2a2
v U c 1+ T0 , (12)
ph ≈ 0± T0s c20vA20 h2l,±
whereh hasaninfinitenumberofvalues.Forkinkwaves,h isanarbitraryrootofthetranscen-
l,± l,±
dentalequation
xJ′(x) ρ v2 c2
1 = 0 A0− T0 , (13)
J (x) −ρ v2 c2
1 e Ae− T0,±
wherec = c +U ,andxdenotestheunknown.Forsausagewaves,h canbeapproximated
T0,± T0 0 l,±
±
by
h j . (14)
l,± 1,l
≈
Whentheoppositelimitholds(ka 1),onefinds
≫
c2 g2
v U c 1 0 l , (15)
ph ≈ 0± 0s − vA20−c20k2a2
where
j kink
g = 1,l (16)
l j sausage
0,l
(cid:26)
in which l = 0,1,2, , and j denotes the l-th zero of J . The plus and minus signs in Eqs.(12)
n,l n
···
and(15)correspondtotheupperandlowerbands,respectively.However,theslowwavesinthecoronal
casearenotofinterestasfarastheperiodratioP /2P isconcerned,fortheyarenearlydispersionless
1 2
duetothenearlyindistinguishablevaluesofc andc ,andthedeviationofP /2P fromunityinthe
0 T0 1 2
presentstudyderivesentirelyfromwavedispersion.
Inviewoftheirstrongerdispersion,letuspayacloserlookatfastwaveswhosepropagationwin-
dowsencompass( v ,U v )and(U +v ,v ).Onemayreadilyunderstandthisbyexamining
Ae 0 A0 0 A0 Ae
− −
thethickcylinderlimit(ka 1),whereonefinds
≫
v2 h2
v U v 1+ A0 l,±. (17)
ph ≈ 0± A0s v2 c2k2a2
A0− 0
WiththeexceptionofbFK1andfFK1,thereexistwavenumbercutoffsforbothkinkandsausagewaves,
andthesearegivenby
(ka) =g Λ , (18)
c l ±
where
(c2+v2 )[(v U )2 c2 ]
Λ = 0 A0 Ae∓ 0 − T0 .
± s[(vAe∓U0)2−c20][(vAe∓U0)2−vA20]
Ontheotherhand,forbFK1andfFK1intheslendercylinderlimitka 1,v maybeapproximated
ph
≪
by
ρˆ v2 (d U )2
v± d 1 0 A0− ±− 0 (λ ka)2K (λ k a) , (19)
ph ≈ ± ± h 2d d i ± 0 ±| | 
± k
 
 
Periodratiosforstandingkinkandsausagemodes 7
where
d =ρˆ U d , (20a)
± 0 0 k
±
d = c2 ρˆ ρˆ U2, (20b)
k k− 0 e 0
q
(d2 c2)(v2 d2)
λ = ±− e Ae− ± . (20c)
± s(c2e +vA2e)(d2±−c2Te)
Moreover,v+ andv− representtheupperandlowerbranches,respectively.
ph ph
Compared with available ones, our study offers some new analytical expressions for the phase
speed v in a numberof physicallyinteresting limits. Equations(12) and(15) offerthe approximate
ph
expressionsofv forslowwavesintheslenderandthickcylinderlimits,respectively.Forthefastones,
ph
Eq.(17) presentsan explicitexpressionforv in the limitof ka 1, therebyextendingthe original
ph
≫
discussionof static cylindersin thissituation byEdwin&Roberts(1983) (hereafterER83)where the
authorsemphasizedtheanalogywiththeLovewavesofseismologyandPekeriswavesofoceanography
(seeEq.(13)inER83).Moreover,Eq.(19)examinesfastkinkwavesintheslendercylinderlimitka 1,
≪
andextendsavailableresultsinthreeways.First,neglectingthefirstordercorrection,Eq.(19)reduces
to d , which agrees with Eq.(70) in Goossensetal. (1992). Second, taking U = 0, we recover the
± 0
expression for a static cylinder, namely Eq.(15) in ER83. Our expression also shows that Eq.(15) as
giveninER83isinfactnotrestrictedtothecoldplasmalimit(c =c =0),butvalidforarathergeneral
e 0
coronalenvironmentaslongasλisgeneralizedtoincorporatec andc ,asgivenbyourEq.(20).Third,
e Te
theplusversionv+ reducestoEq.(5)inVasheghaniFarahanietal.(2009)wherethetransversewaves
ph
propagatinginsoftX-raycoronaljetsareexamined,whenonenotesthat(d2 c2)/(d2 c2 ) 1,
+ − e + − Te ≈
andv2 c2.However,itturnsoutthatexceptforextremelysmallka,retainingtheoriginalformin
Ae ≫ e
termsof the modifiedBessel functionK is more accuratethanthe logarithmicformgivenbyEq.(5)
0
in VasheghaniFarahanietal. (2009). Furthermore, the expression v− gives the phase speed for the
ph
wavesthatarebackwardpropagatingintheabsenceofflow.
3.2 ProceduresforComputingStandingModes
By“standing”,werequirethattheradialLagrangiandisplacementξ (r,θ,z;t)iszeroattheinterface
r
r = aatbothendsofthecylinderz = 0,L,irrespectiveofθandt.Onerequirementforthistobetrue
forarbitraryθ isthatonlypropagatingwaveswithidenticalazimuthalwavenumbersncancombineto
formstandingmodes.Apairofpropagatingwavescharacterizedbyacommonangularfrequencyωbut
differentlongitudinalwavenumbersk andk thenleadtothat
r l
ξ (r,θ,z;t)= Re ξ˜ (r)exp[i(k z+nθ ωt)]
r r,l l
−
n o
+ Re ξ˜ (r)exp[i(k z+nθ ωt)] . (21)
r,r r
−
n o
Specializing to (r,z) = (a,0), one finds ξ˜ (a) = ξ˜ (a), meaning that one is allowed to choose
r,l r,r
−
ξ˜ (a)=A tobereal.Itthenfollowsthat
r,l ξ
ξ (a,θ,z;t)
r
= A [cos(ωt k z nθ) cos(ωt k z nθ)]
ξ l r
− − − − −
k k k +k
r l l r
= 2A sin − z sin ωt z nθ . (22)
ξ
− 2 − 2 −
(cid:18) (cid:19) (cid:18) (cid:19)
Forξ (a,θ,L;t)tobezeroatarbitraryt,thisrequires
r
2πm
k k = ,m=1,2, (23)
r l
− L ···
8 HuiYuetal.
Byconvention,m=1correspondstothefundamentalmode,andm=2toitsfirstovertone.
Atthispoint,itsufficestosaythattheprocedureforcomputingtheperiodratiosofstandingmodes
isidenticaltotheslabcase,whichwasdetailedinpaperI.Basicallyitinvolvesconstructinganω k
−
diagram where each propagating wave in a pair to form standing modes corresponds to a particular
curve, meaning that a horizontal cut with a constant ω would intersect with the two resulting curves
attwopoints.Iftheseparationbetweenthetwopointsis2π/L,thenonefindsthefundamentalmode.
If it is twice that, one finds the first overtone. Let the angular frequency of the fundamental mode
(firstovertone)bedenotedbyω (ω ),theperiodratioissimplyP /2P = ω /2ω .Theexistenceof
1 2 1 2 2 1
cutoffwavenumbersforsausagewavestobetrappedtranslatesintotheexistenceofcutoffaspectratios
(a/L) forstandingsausagemodestobenon-leaky.AsemphasizedbypaperI(seeFig.3therein),
cutoff
this (a/L) is not determinedby the differencebetween the two cutoffsof the two ω k curves
cutoff
−
dividedby2π,butlargerthanthat.
When computingthe coronalstanding modes, we consideronly bFK1 and fFK1 for kink modes,
andbFS1andfFS1forsausagemodes.BrancheswithlargermodenumberslikebFK2orbFS2would
formstandingmodesonlyforrelativelythickcylinderswherea/Lis oftheorderunity.Forthe same
reason, we discard the combinationsbetween slow and fast sausage propagatingwaves. On the other
hand,combinationsof slow with fast kinkwave, such as bFK1 plus fSK, turn outextremelyunlikely
as well. Thisis because, while slow kinkwaves are dominatedby the intensityoscillationsinstead of
transversedisplacements(X 1),theoppositeholdsforfastones(X 1).Theendresultisthatif
| |≫ | |≪
afastkinkwavedoescombinewithaslowonetoformastandingmode,atransverseloopdisplacement
on the orderof the cylinderradiuswill lead to a relative intensity variationthat exceedsunity.To see
this,considerslendercylinderssuchthatka 0,andconsiderthecasewherethecomponentstoform
→
standingmodesarebFK1andanybranchoffSK.ForbFK1,oneseesthatv d andn a 0,and
ph − 0
henceX (d−−U0)2 (n a)2.Becaused U d ρˆ U isoftheorde≈rofv ,andv→2 c2,
≈ (d−−U0)2−c20 0 −− 0 ≈− k− e 0 A0 A0 ≫ 0
Xwouldberoughly(n a)2andhenceapproacheszeroaswell.However,forslowkinkwaves,bynoting
0
that(v U )2 c2 whenka 0,onefindsthatX (n a)2(1 v2 /v2 )(ρ v2 )/(ρ c2 ),which
ph− 0 → T0 → ≈ 0 − ph Ae e Ae 0 T0
is approximately(n a)2v2 /c2 since ρ v2 ρ v2 and c c . Note thatin coronalconditions,
0 A0 0 e Ae ≈ 0 A0 T0 ≈ 0
h as given by Eq.(13) can be approximatedby (l+3/4)π. When ka 0, with n a approaching
l,± 0
→
h ,X willbelarge.
l,±
3.3 PeriodRatiosforStandingKinkModes
Figure3presentsthedependenceontheaspectratioa/LoftheperiodratioP /2P pertinenttostand-
1 2
ing fast kink modes. Here the results for a number of different U are shown with different colors,
0
withU representedbytheinternalAlfve´nMachnumberM = U /v .Onecanseethatallcurves
0 A 0 A0
decreasefromunityatzeroa/L,attainaminimum,andthenincreasetowardsunity.IncreasingU sub-
0
stantially strengthensthe deviationof P /2P from unity relative to the static case (the black curve).
1 2
Take the minima for instance. While in the static case it reads 0.938,attained at a/L = 0.405,when
M = 0.8 it is significantly reducedto 0.778 attained at a/L = 0.267. At smaller aspect ratios, the
A
dispersionintroducedbytheflow,andhencethedeviationfromunityoftheperiodratioP /2P ,isnot
1 2
asstrong.However,atanaspectratioofa/L=0.19,onefindsthatP /2P decreasessignificantlyfrom
1 2
0.953inthestaticcaseto0.79whenM = 0.8.ActuallythisaspectratiocorrespondstotheNoRH
A
loop that experienced standing kink oscillations on 2002 July 3 with multiple periodicities that yield
P /2P = 0.82(Kupriyanovaetal. 2013).Onefindsthatwhilethewavedispersionduetotransverse
1 2
density structuring alone cannot account for this measured value of P /2P , it may be attained with
1 2
theaidoftheadditionalwavedispersionduetoflowshear.Inthisregard,weagreewithAndriesetal.
(2009)inthesensethatthecontributionofthedensitycontrastalonetothedeviationofP /2P from
1 2
unityseemstobemarginalforextremelythincylinders.However,wenotethatwhenasubstantialflow
shearexistsbetweenthecylinderanditssurroundings,theshear-associatedwavedispersionmaynotbe
neglectedfor loopswith finite aspectratios. As a matter of fact, for loopswith a/L as small as 0.05,
thefloweffectisstillsubstantialenoughtobeofobservationalsignificance:whileP /2P reads0.989
1 2
Periodratiosforstandingkinkandsausagemodes 9
inthestaticcase,whenM = 0.8itis0.934,whichisalreadybelowtheminimumP /2P canreach
A 1 2
whenthe flow is absent. We notethatthis a/Lis notunrealisticbutlies within the rangeofthe mea-
sured values of oscillating EUV loops examined in Ofman&Aschwanden (2002)(see their Table 1).
Thepointwe wanttomake hereisthatthewave dispersionassociated withthe transversestructuring
needstobeconsideredforatheoreticalunderstandingoftheperiodratiosofstandingkinkmodes,and
thisisparticularlynecessaryinthepresenceofastrongflowshearandwhentheloopaspectratiosare
notextremelysmall.
Figure 4 further examines the flow effect by showing (a) the minimal period ratio, (P /2P )
1 2 min
and(b)itslocation,(a/L) ,asafunctionoftheinternalAlfve´nicMachnumberM .Inadditionto
min A
thereferencecase wherev /v = 2,Fig.4alsoexaminesotherratiosof3,4,10,and20,shownin
Ae A0
differentcolors.RegardingFig.4a,oneseesthatthefloweffectontheperiodratiosissignificantforall
theconsideredv /v ,orequivalently,thedensitycontrast.Asamatteroffact,atagivenM ,even
Ae A0 A
though (P /2P ) tends to decrease with increasing density contrast, this tendencyis rather weak,
1 2 min
and seems to saturate when v /v exceeds, say, 10, as evidenced by the fact that the two curves
Ae A0
correspondingtov /v being10and20canhardlybedistinguished.Consequently,whenv /v
Ae A0 Ae A0
is as large as 20, (P /2P ) decreases from 0.914 in the static case to 0.768 when M = 0.8,
1 2 min A
amountingto a relative difference of 16%, which is almost the same as in the case when v /v is
Ae A0
2wherethisfractionaldifferencereads17.1%.LookingatFig.4b,onenoticesthatforagivendensity
contrast,theaspectratioatwhichtheminimumperiodratioisattainedtendstodecreasewithincreasing
flow,andthis tendencyis clearerforweakerdensitycontrasts. Whenv /v is atthe two extremes,
Ae A0
(a/L) reads0.405and0.31inthestaticcase,andgoesdownto0.267and0.248foranM being
min A
0.8,respectively.Thefractionalchangeduetotheflowintheformerreads34%,while20%inthelatter.
Itisinterestingtocontrastthecylindercasewiththeslabone.Inbothcasestheminimalperiodratio
(P /2P ) andtheaspectratio(a/L) havebeenexaminedanalytically.Notethatintheslabcase,
1 2 min min
a refers to the half-width of the slab. For cold static slabs, Macnamara&Roberts (2011) established
that (P /2P ) can never drop below √2/2, which is attained for the infinite density contrast at
1 2 min
a zero aspect ratio. While this was established by employing the Epstein profile to connect the slab
densityandthedensityofitssurroundings,thenumericalresultsinbothMacnamara&Roberts(2011)
and paper I demonstrate that this lower limit for P /2P is also valid when the density profile is in
1 2
the form of a step function. When a flow U is introduced, paper I shows that P /2P is no longer
0 1 2
subject to this lower limit and the change in P /2P relative to the static case is typically 20%.
1 2
∼
Besides,(a/L) tendstoincreasewithincreasingU .Forcoldstaticcylinders,McEwanetal.(2006)
min 0
(hereafterM06)andalsoAndriesetal.(2009)establishedthat(P /2P ) also suffersfromalower
1 2 min
limitof 0.92whenthedensitycontrastapproachesinfinity,andtheaspectratiowherethislowerlimit
∼
is attained is 0.3 (see Figure 2 in M06, and note that the symbolL therein is the loop half-length,
∼
and hence their a/L corresponds to twice the value of a/L in the present study). The static case in
Fig.3 agreesremarkablywellwith Fig.2 in M06 despite thatthe soundspeeds are allowedto be non-
zeronow,whichisnotsurprisinggiventhatthesoundspeedsaresignificantlysmallerthantheAlfve´n
speeds. However,Fig.3 offersthe new result that in the cylinder case, the introductionof the internal
flow provides significant revision to the period ratio, making it no longer suffer from the lower limit
establishedforstaticcylinders.Thisistrueevenwhenthedensitycontrastapproachesinfinity,andthe
revisiontotheperiodratioistypically 16 17%,similartotheslab case.Ata givenv /v , the
Ae A0
∼ −
tendencyfor(a/L) todecreasewithincreasingU inthecylindercaseisoppositetowhathappens
min 0
forslabswithflows.
3.4 PeriodRatiosforStandingSausageModes
Figure 5 presents the period ratio P /2P as a function of aspect ratio a/L for a series of v /v ,
1 2 Ae A0
pertinenttostandingsausagemodes.Thesolid,dotted,anddashedcurvescorrespondtov /v being
Ae A0
2,3,and20,respectively.Asindicatedbythedifferentcolors,asetofU isinvestigatedandmeasuredin
0
unitsoftheinternalAlfve´nspeedU =M v .ItisclearfromFig.5thattheeffectofflowontheperiod
0 A A0
ratioP /2P isnotasstrongasforthekinkmodes.Sincethiseffectincreaseswithincreasingv /v ,
1 2 Ae A0
10 HuiYuetal.
onemayexaminetheextremewherev /v =20,inwhichcaseonefindsthatata/L=0.4,P /2P
Ae A0 1 2
reads 0.611 when M = 0.5, which is 5.4% lower than the value 0.646 obtained in the static case.
A
ThisfractionalchangeinP /2P istypicalinthiscaseatagivenaspectratio.However,thefloweffect
1 2
is much stronger when it comes to the cutoff aspect ratio (a/L) only above which can standing
cutoff
sausage modes be supported. This effect is substantial even when it is the weakest among the three
v /v considered:whenv /v = 2,(a/L) increasesfrom0.456to0.53to0.651withM
Ae A0 Ae A0 cutoff A
increasing from 0 to 0.1 to 0.2. Regardingthe other extreme v /v = 20, while (a/L) reads
Ae A0 cutoff
0.04forthestaticcase,itreads0.083whenM =0.2and0.197whenM =0.4.Thismeansthatata
A A
givenv /v ,relativetothestaticcase,cylinderswithflowcansupportstandingsausagemodesonly
Ae A0
whentheyaresufficientlythickerifthecylinderlengthisfixed.
Atthispoint,acomparisonwithstudiesofsausagemodessupportedbymagnetizedslabsisinfor-
mative.AsdemonstratednumericallybyInglisetal.(2009)andanalyticallybyMacnamara&Roberts
(2011),forstatic coronalslabsP /2P may reachaslow as1/2withthe lowerlimitattainablewhen
1 2
thedensitycontrastisinfinite.Besides,thecutoffaspectratiolowerswithincreasingdensitycontrast.
WhileananalyticalexpectationofthelowerlimitofP /2P isnotavailableforcylinders,ourstudyof
1 2
an extremelylargedensitycontrast(v /v = 20)representedby the dashed curvesin Fig.5 shows
Ae A0
thatthe sausagemodesin a cylindricalgeometryfollowa similar pattern:P /2P is also subjectto a
1 2
lowerlimitof1/2,thecutoffaspectratiodecreaseswithv /v .Likewise,theinfluenceofflowonthe
Ae A0
standingmodesisqualitativelysimilarinbothgeometries:introducingaflowinthestructurehasamore
prominenteffectindeterminingthecutoffaspectratiothanonthevalueoftheperiodratio.WithM in
A
theexaminedrange[0,0.6],inbothgeometriesa flowmayalter(a/L) in anorder-of-magnitude
cutoff
senseandthefractionalchangeismorepronouncedathigherdensitycontrasts;whereasthefractional
changeinP /2P withrespecttothestaticcaseis.5%.
1 2
Figure5alsoallowsustopayacloserinspectionoftheobservedperiodratiosP /2P ofstanding
1 2
sausagemodes.While1 P /2P ofstandingkinkmodeshasbeenexaminedinconsiderabledetail(see
1 2
−
e.g.,the introductionin Macnamara&Roberts2011, andreferencestherein)andputin seismological
applications(e.g.,Andriesetal.2005,2009),theuseof1 P /2P ofstandingsausagemodesseemsnot
1 2
−
aspopular(seee.g.,Inglisetal.2009).Beforemakingitsserioususe,onemayfirstaskthequestionthat
whatleadstothedepartureofP /2P from1inthefirstplace.TheavailabledataforNoRH flareloops
1 2
yieldavalueofP /2P 15.5s/(2 9.5s)= 0.82atanaspectratioa/L = 0.12(Nakariakovetal.
1 2
≈ ×
2003;Melnikovetal. 2005), whilethose forcoolH alphapost-flareloopsyielda valueofP /2P
1 2
≈
587s/(2 349s) = 0.84atana/L = 0.03(Srivastavaetal.2008).InviewofFig.5whichaddresses
×
trappedmodes,thetwovaluesofP /2P aredifficulttoexplain:whichevervaluea/Ltakes,P /2P
1 2 1 2
isfarfromthemeasuredvalues,whichareactuallyoutsidetherangeoftheverticalextentofthisfigure.
Introducingaflowshearmakesthecomparisonofthetheoreticallyexpectedvalueswiththemeasured
onesevenmoreundesirable:atagivena/L,P /2P intheflowingcaseisactuallyevensmallerthanin
1 2
thestatic case. Adoptinga slabdescriptionforcoronalloopsaswasdoneinInglisetal.(2009, Fig.6)
andinpaperI(Figure6therein)doesnothelp,varyingtheparametersoftheequilibriumdoesnoteither,
fortheperiodsofstandingmodesaremostlydeterminedbythedensitycontrast(Inglisetal.2009).On
theotherhand,intheleakyregime,theperiods(andhencetheirratios)subtlydependontheparameter
range of the problem (Nakariakovetal. 2012). We conclude that for sausage modes, what causes the
deviationofP /2P fromunityneedsadedicateddetailedinvestigation.
1 2
Figure6extendsourexaminationontheeffectsofflowspeedonthecutoffaspectratio(a/L)
cutoff
pertinenttothestandingsausagemodesbyshowingthedistributionof(a/L) withvaryingAlfve´n
cutoff
speedratiosv /v andAlfve´nMachnumbersM .Thecontoursof(a/L) areequallyspacedby
Ae A0 A cutoff
0.02.Itcanbeseenthat(a/L) decreasesmonotonicallywithincreasingv /v atagivenM ,
cutoff Ae A0 A
but increases rather dramatically with increasing M at a given v /v . What is more importantin
A Ae A0
thecontextofSMSisthatFig.6helpsconstrainthecombinationsofdensitycontrastandinternalflow,
whentrappedstandingsausagemodesareobservedinacoronalcylinderwithknownaspectratio.This
pointcanbe illustratedif oneexaminesthe flaringloopreportedinNakariakovetal.(2003),whichis
25Mmlongand6Mmindiameter,resultinginanaspectratioofa/L=0.12.Nowthatthefundamental
sausagemodeoccurredinthisloop,itsaspectratiohastobelargerthanthecutoffvalue,meaningthat