Torsional Alfv\'en resonances as an efficient damping mechanism for non-radial oscillations in red giant stars

1.2 MB·

by  Shyeh Tjing Loi

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2017 (MNLATEXstylefilev2.2)
Torsional Alfv´en resonances as an efficient damping
mechanism for non-radial oscillations in red giant stars
7
⋆
1 Shyeh Tjing Loi and John C. B. Papaloizou†
0
2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre forMathematical Sciences,
Wilberforce Road, Cambridge CB3 0WA, UK
n
a
J
0 Lastcompiled:1February2017
3

] ABSTRACT
R Stars are self-gravitating fluids in which pressure, buoyancy, rotation and magnetic
S fields provide the restoring forces for global modes of oscillation. Pressure and buoy-
. ancy energetically dominate, while rotation and magnetism are generally assumed to
h
be weakperturbations andoften ignored.However,observationsofanomalouslyweak
p
dipole mode amplitudes in red giant stars suggest that a substantial fraction of these
-
o are subject to an additional source of damping localised to their core region, with in-
r directevidence pointing to the role ofa deeply buriedmagnetic field. It is alsoknown
t
s thatinmanyinstancesthe gravity-modecharacterofaffectedmodesispreserved,but
a
so far no effective damping mechanism has been proposed that accommodates this
[
aspect. Here we present such a mechanism, which damps the oscillations of stars har-

1 bouringmagnetisedcoresviaresonantinteractionswithstandingAlfv´enmodesofhigh
v harmonic index. The damping rates produced by this mechanism are quantitatively
1 on par with those associated with turbulent convection, and in the range required
7 to explain observations, for realistic stellar models and magnetic field strengths. Our
7
results suggest that magnetic fields can provide an efficient means of damping stel-
8
lar oscillations without needing to disrupt the internal structure of the modes, and
0
lay the groundwork for an extension of the theory of global stellar oscillations that
.
1 incorporates these effects.
0
7 Key words: stars:oscillations—stars:magneticfield—stars:interiors—methods:
1 analytical — MHD
:
v
i
X
r
a 1 INTRODUCTION largeinteriordisplacements(Deubner& Gough1984).Inre-
ality,andparticularlyinthecaseofevolvedstars,modesare
Surface convection in many stars stochastically excites
not purely one typeor the other but havemixed character,
global oscillations (normal modes), which can be de-
exhibiting large fluid displacements both near the surface
tected through the intensity fluctuations associated with
and in thedeep interior (Osaki 1975).
temperature variations induced at the stellar surface
Thefluiddisplacementfieldξ(r,t)atthepointwithpo-
(Houdek& Dupret 2015). These normal modes can be re-
sition vectorr associated with anormal modeof oscillation
garded as standing superpositions of waves associated with
canbedescribedbyaspatialamplitudefunctionmodulated
restoringforcesproducedbypressure,buoyancy,theCoriolis
by a time-harmonic component exp( iωt). If rotation and
force (in the presence of rotation) and the Lorentz force (if −
magnetic fieldsare weak, which is thecase for thevast ma-
thestar harbours a magnetic field).Pressure and buoyancy
jority ofstars, thenthereisnegligible departureof thestel-
effects dominate energetically over those produced by rota-
lar background from spherical symmetry. This allows one
tionandmagneticfields,andsotoafirstapproximationone
to expand the spatial part in terms of vectorial spherical
identifies in the asymptotic limit of high and low frequen-
harmonics,i.e.theoverallfluiddisplacementcanbewritten
cies two types of modes: p-modes, restored mainly by pres-
sure and associated with large surface displacements; and ξ(r,t)=[ξrYℓmˆr+ξh∇Yℓm+ξTˆr×∇Yℓm]exp(−iωt), (1)
g-modes, restored mainly by buoyancy and associated with
where we adopt spherical polar coordinates (r,θ,φ) and
there is an implicit summation over spherical harmonics
Ym(θ,φ). Radial dependencies are captured solely by the
ℓ
⋆ E-mail:[email protected] scalar functions ξr(r), ξh(r) and ξT(r), which describe dis-
† E-mail:[email protected] placementsinthreemutuallyorthogonaldirectionsforgiven
2 S. T. Loi and J. C. B. Papaloizou
ℓandm.ThefirsttwotermsontheRHSofEq.(1)arecol- main sequence life the field should relax into a long-
lectively referred toas thespheroidal component,while the lived equilibrium state if sufficiently large-scale magnetic
third(involvingξ )isthetorsional component.Mathemat- structure can be retained during the cessation phase. Al-
T
ically, spheroidal motions are thosefor which ( ξ) =0, though there have been many previous works investigat-
r
∇×
the subscript r denoting the radial component. Physically, ing the effects of magnetic fields on stellar oscillations
these are motions that involve deformation but no twist. (e.g., Campbell & Papaloizou 1986; Cunha& Gough 2000;
Torsional motions have ξ = 0, and correspond to mo- Rincon & Rieutord2003;Reese et al.2004;Lee2007),these
∇·
tions that involve twist but nodeformation. are not directly applicable here as they have mainly been
In theabsence of rotation and magnetic fields, onecan concerned with cases where the magnetic field of interest
show from the fluid equations of motion that for ω = 0, extendsbeyondthestar andis dynamically significant only
6
ξ =0,implyingthatpressureandbuoyancyareonlycapa- in a thin layer near the surface. Prior to the discovery of
T
bleofrestoringspheroidalmotions.However,itispossibleto thedipoledichotomyproblem,theexistenceofcore-confined
accessthethirdspatialdegreeoffreedom(torsionalmotions) fields,thoughnot generally disputed,wasnotconsidered to
in the presence of rotation and/or magnetic fields. In this giverisetoobservableconsequences.Untilrecently,verylit-
work we ignore rotation. Our aim is to investigate the dy- tle attention has been paid to their possible influence on
namical consequences of interactions between torsional mo- stellar oscillations.
tionsrestoredbytheLorentzforcewiththeusualspheroidal
The link to main-sequence dynamo action led to sug-
(i.e., p- and g-) modes. We deal with the limit where the
gestions that the mechanism behind the dipole dichotomy
magnetic field is sufficiently weak that it does not disrupt
might involve a deeply buried magnetic field (Garc´ıa et al.
the structure of the spheroidal modes. We find that reso-
2014).Follow-uptheoreticalworkbyFuller et al.(2015)and
nant interactions between the two types of modes can pro-
Lecoanet et al. (2016) has established that if the magnetic
vide an efficient energy sink for spheroidal motions. This
field strength exceeds a critical threshold, complete con-
sourceofdampingmaybepotentiallyimportantforexplain-
version of gravity waves to magnetoacoustic waves occurs,
ing the anomalously low amplitudes of non-radial (particu-
whichthendissipatewithinthecore(dampingprocessesas-
larly dipole) modes observed in some evolved stars.
sociatedwithconversionbetweendifferentwavemodeshave
The existence of red giant stars exhibiting low ampli-
been previously been investigated mainly in the context of
tudes of their dipole (ℓ = 1) modes was first reported by
the solar atmosphere, e.g. Spruit& Bogdan (1992)). This
Mosser et al. (2012a), accounting for roughly 20% of their
actstoselectivelydampnon-radialmodes,whilealsoimply-
sample. For the remainder, the higher amplitudes of their
ingthatmodeshavingthecharacterofg-modescouldnotbe
ℓ=1modesareconsistentwiththeprimarysourceofdamp-
constructed (only purep-modescould exist). Anadditional
ing being convection alone. Given that the red giant popu-
prediction is that affected p-modes should be magnetically
lation appears to be divided into those with either high or
splittoanextentcomparabletog-modeperiodspacingsand
lowℓ=1modeamplitudes,withrelativelyfewintermediate
rotational splittings (Cantiello et al. 2016). However, more
cases,weshallrefertothisasthedipoledichotomy problem.
detailed analyses of the observational data performed by
Whilethefrequenciesofthelow-amplitudeℓ=1modesare
Mosser et al. (2016) indicate that (i) additional splitting of
close to those predicted by the usual asymptotic relation
thissortisnotseen,(ii)themeasuredmodeamplitudesare
(Tassoul1980;Gough1986)obeyedbytheremainderofthe
inconsistent with total energy conversion, and (iii) in many
sample(Mosser et al.2011,2012b),theirwidthsareconsid-
instancesthemixedcharacterofthemodesisretained.The
erably larger (Garc´ıa et al.2014),suggesting that theℓ=1
current consensus is that an independent mechanism is re-
modes of these stars are subject to an additional source of
quired to explain the existence of stars possessing mixed
damping. Follow-up analyses by Stello et al. (2016) estab-
modes with weak amplitudes, and where theamplitude de-
lished that a dichotomy also exists for the ℓ = 2 modes,
pression is only partial.
but to a lesser extent than ℓ = 1. Radial (ℓ = 0) modes
appear tobe unaffected.Asargued by Garc´ıa et al. (2014), In this work we present a new mechanism for damp-
sources of damping such as turbulent viscosity localised to ing spheroidal modes involving resonant interactions with
the convective envelope should affect all low-degree modes torsional Alfv´en modes localised to the magnetised core (a
to a similar extent, and so to selectively affect non-radial similarideawasbrieflyspeculatedonbyReeseet al.(2004)
modes theextra source of dampingneeds to belocalised to in the context of roAp stars, but thishas not been pursued
the core. A further piece of evidence is that the behaviour further in any context). No critical field strength is neces-
ismass-dependent(Mosser et al.2012a),beingrestrictedto sary: our mechanism is capable of operating in the regime
stars more massive than 1.1M⊙ (Stello et al. 2016). This is below the threshold required by Fuller et al. (2015). More-
roughly the threshold mass above which stars on the main over,nodisruptiontothestructureofthespheroidalmodes
sequencepossess convective ratherthan radiative cores. isrequired,implyingthatthemixedcharacterofmodescan
Convectiveregionsofstarsarestronglyassociated with beretained.Dampingratesareafunctionofseveralparam-
dynamo action and therefore the existence of a magnetic eters including the field strength, so in general one expects
field (Proctor & Gilbert 1994; Charbonneau & MacGregor only partial energy loss. In Section 2 we describe the back-
2001). Numerical simulations suggest that convective core groundstellar modelandmagnetic fieldconfiguration used.
dynamos in massive stars may generate field strengths of In Section 3 we explain the details of the damping mech-
10–100kG or more (Brun et al. 2005; Featherstone et al. anism, and present quantitative results for the application
2009). The long timescales of magnetic diffusion in stel- of this to a 2M⊙ red giant model in Section 4. We discuss
lar cores, which greatly exceed nuclear timescales, sug- observationalconsequencesandlimitationsinSection5.We
gest that after the dynamo ceases at the end of the star’s concludein Section 6.
Magnetic damping of stellar oscillations 3
×1030 ×107 ×1019 ×107
4 6 4 8
3 3 6
4 )
3
m (kg)2 5 ×1029 T (K)2 p (Pa)2 (kg/mρ4
1 0 1 2
0 5
×107
0 0 0 0
0 5 0 1 2 0 1 2 0 1 2
r (m) ×109 r (m) ×108 r (m) ×108 r (m) ×108
×105
1.8 0.1 12 0.1
l = 0
10
0.08 0.08 l = 1
1.6
8 l = 2
z)0.06 s) z) 0.06 l = 3
Γ11.4 N (H0.04 c (m/ 46 S (Hl0.04
1.2
0.02 0.02
2
1 0 0 0
0 5 0 1 2 0 5 0 1 2
r (m) ×109 r (m) ×108 r (m) ×109 r (m) ×108
Figure 1. From left to right, top to bottom: plots of the enclosed mass, temperature, pressure, density, adiabatic index, buoyancy
frequency,soundspeedandLambfrequencyprofiles,fora2M⊙redgiantstellarmodel(generatedbyCESAM).Notethatthetemperature,
pressure, density and buoyancy/Lamb frequency plots are onlyshown up to 5% of the stellar radius. Inthe plotof enclosed mass (top
left),aninsetplotzoominginto1%ofthestellarradiushasbeenincludedtoillustratethehighcentralmassconcentration.
2 MODELS operation is strong enough to influence the velocity field)
(Braithwaite & Spruit 2015). The timescale of relaxation is
2.1 Red giant stellar model
theAlfv´entraveltimeacrossthecore,whichcanbeasshort
Wrede wgiiallntillmusotdraeltewohuorsembeacchkagnriosumndinptrhoefilceosnwteexrteogfenaer2aMte⊙d ρas11y0r0(gfocrma−c3o)r.eWdihaemreettehreoyfRexcis∼t i1n00nMatmur,eB, fi∼eld1s0kinGnaonnd-
∼
by the CESAM (Code d’Evolution Stellaire Adaptatif et convective regions should therefore obey the force-balance
Modulaire) stellar evolutionary code (Morel 1997). We ob- condition
tainedtheparametergridsfromanonlinesource1.Thevar- 1
p+ρ Φ= ( B) B, (2)
iousbackgroundquantitiesareshownplottedinFig.1.The ∇ ∇ µ0 ∇× ×
ageofthemodelis963Myr,whenthestarisatanintermedi-
wherep,ρ,Φand Bare thepressure,density,gravitational
ateposition initsascentalongtheredgiantbranch(RGB).
potential and magnetic field, respectively. If that were not
Atthisstagetheradiusofthestaris7.7R⊙andthedynam-
ical timescale (given by R∗3/GM∗, where R∗ and M∗ are thecasethentheywouldevolvetowardssuchastateonthe
Alfv´en timescale.
thestellarradiusandmass)is6.8hr.Althoughourproposed
p To model a physically realistic equilibrium field that
mechanism is quitegeneral, we havechosen todemonstrate
might be found in a red giant core, one seeks a solu-
itusingareasonablyrealistic stellar modeltoobtain mean-
tion to Eq. (2) that (i) is spatially confined, (ii) is fi-
ingful estimates of the damping rates for comparison with
nite throughout, (iii) is continuous at the boundary in-
observation.
terior to which the field is confined, and (iv) is stable.
The third condition is needed to avoid infinite current
2.2 Magnetic field configuration sheets on the boundary. It has been shown that purely
poloidalandpurelytoroidalfieldsareunstable(Tayler1973;
Following cessation of convective fluid motions and there-
Markey & Tayler 1973; Flowers & Ruderman 1977), and so
fore the dynamo at the end of the main sequence, the
for stability, magnetic equilibria necessarily involve a mix-
magnetic field is expected to relax rapidly into an equi-
ture of poloidal and toroidal components. Numerical stud-
librium state (note that the Lorentz force during dynamo
ies, which find that random initial fields tend to settle into
mixedpoloidal-toroidal configurationswith roughlycompa-
1 https://www.astro.up.pt/helas/stars/cesam/A/data/ rable strengths of the two components, support this no-
4 S. T. Loi and J. C. B. Papaloizou
tion (Braithwaite & Spruit 2004; Braithwaite & Nordlund theresult
2006). Notably, this means that a simple dipole field is in- βλr r
adequatefor ourpurposes, since it violates all four criteria. Ψ(r)= f(r,r1;λ) ρ(ξ)ξ3j1(λξ)dξ
More generally, it can be shown that magnetic fields which j1(λr1)(cid:20) Z0
r1
are force free throughout, spatially confined and continu- +j1(λr) ρ(ξ)ξ3f(ξ,r1;λ)dξ , (7)
ousontheboundarymustvanishidentically(Roberts1967; Zr (cid:21)
Braithwaite & Spruit 2015), and so we require a non-force
where
free configuration.
Early analyticwork byPrendergast (1956) successfully f(ξ1,ξ2;λ) j1(λξ2)y1(λξ1) j1(λξ1)y1(λξ2), (8)
obtained an axisymmetric, mixed poloidal-toroidal solution ≡ −
satisfying the first three criteria. Though originally derived j1 andy1 are spherical Bessel functions ofthefirstand sec-
forincompressiblestars,itsextensiontocompressibilitypro- ond kind,and λ is a root of
ducesaqualitativelysimilarresult(Braithwaite & Nordlund r1
2006; Duez& Mathis 2010). Although we do not check for ρ(ξ)ξ3j1(λξ)dξ=0. (9)
stability of this configuration for the red giant model con- Z0
sidered here, its stability for n = 3 polytropes has previ- Theoscillatory natureofj1 meansthatmorethanonepos-
sible value of λ may satisfy Eq. (9); we chose to use the
ously been verified numerically (Duezet al. 2010). In addi-
tionweexpectthatconfinedfieldswithpoloidalandtoroidal smallestone.Wehavesetr1 =0.005R∗ (theinferredbound-
ary of what used to be the convective core) on the basis of
components of comparable strength should in general be
inspectionofthestellarprofiles.Thiscorrespondstoamass
adequately characterised by the configuration we employ.
To obtain this field solution, one begins by introducing the coordinate of 0.11M∗. For the ρ(r) profile of the red giant,
weobtainλ=2643.2R−1,yieldingcomparablepoloidaland
poloidal flux function ψ(r,θ), in terms of which an axisym- ∗
metric magnetic field B=(B ,B ,B ) may bewritten toroidalfieldstrengths(themaximumvaluesofeachofthese
R φ z
differby less than a percent).
B= 1 ψ φˆ +B φˆ . (3) Thecomponentsof BintermsofΨaregivenin spher-
R∇ × φ ical polar coordinates by
Note that (R,φ,z) will be used to refer to cylindrical polar 2
B (r,θ)= Ψ(r)cosθ,
coordinates, while (r,θ,φ) are spherical polar coordinates r r2
(the φ coordinate is the same for each and refers to the B (r,θ)= 1Ψ′(r)sinθ, (10)
azimuthal direction). Physically, ψ is a flux surface label, θ −r
meaning that poloidal projections of the field lines are the λ
B (r,θ)= Ψ(r)sinθ.
level surfaces of ψ. Substituting Eq. (3) into Eq. (2) and φ −r
applyingtheappropriatevectoridentities,onecanshowthat Thecorrespondingfieldconfiguration,whichwewillreferto
thequantityF RBφ isinvariantonfluxsurfaces,i.e.F = as Prendergast’s solution, is displayed in Fig. 2.
≡
F(ψ). If in addition we assume a barotropic configuration, Prendergast’s solution qualitatively resembles a dipole
wearriveatanonlinearPDEknownastheGrad-Shafranov initsangulardependence,butunlikeadipolefieldpossesses
equation: no singularity, has a mixed poloidal-toroidal topology, and
dF vanishes smoothly at the spherical boundary r = r1 in all
∆∗ψ+F = µ0ρR2G, (4) three components of B. Beyond this radius we set B = 0,
dψ −
i.e.weneglect theenvelopefieldundertheassumptionthat
thisismuchweakerthanthecorefield.Theoverallstrength
where
ofthePrendergastfieldiscontrolledthroughtheparameter
∂2 ∂2 1 ∂ β, which sets the amplitude of the field but not its shape.
∆∗ + InspectionoftheCESAMgridofmodelsforthisstarshows
≡ ∂R2 ∂z2 − R∂R
that the core contracts by roughly a factor of 10 in radius
∂2 ∂2
+(1 µ2) , (5) from 403 Myr (on the main sequence) to 963 Myr. Under
≡ ∂r2 − ∂µ2
conservationofmagneticflux,centralfieldstrengthsshould
increasebyafactorofabout100.Ifmagneticfieldstrengths
µ cosθ, and G = G(ψ) is another flux surface invari-
≡ onthemainsequencewere10–100kG,thenoneexpectsfield
ant.Neitherthefunctionalforms of F or Gareconstrained
strengths in the red giant core to be of order several MG.
within ideal MHD (different choices correspond to different
Wehaveset β such that the central field strength is 4 MG.
equilibria),andsofollowing Prendergast (1956),asimplify-
ingchoiceistosetF(ψ)=λψ andG(ψ)= β/µ0,whereλ
and β are constants. Separating ψ(r,θ) = Ψ−(r)sin2θ turns
Eq. (4) into 3 ALFVE´N RESONANCE DAMPING
MECHANISM
2
Ψ′′− r2 −λ2 Ψ=βρr2, (6) In this section we present a mechanism for damping
(cid:18) (cid:19) spheroidal modes through interaction with an embedded
which is an inhomogeneous, second-order ODE that can be magnetic field, illustrating this for a red giant containing
solved usingthemethodofGreen’s functions.Applyingthe a Prendergast field in the core. This section is structured
boundary conditions Ψ(0) = 0, Ψ(r1) = 0 and Ψ′(r1) = 0, as follows. First, we isolate eigenmodes of oscillation that
where r1 is the boundary of the field region, this produces in the limit of a weak magnetic field are purely torsional
Magnetic damping of stellar oscillations 5
in nature (e.g. Mestel 2012) and correspond to standing
x 10−3 Alfv´en waves localised to the field region (Section 3.1). We

showthatthesecoupletothespheroidalmodesthroughthe
Lorentzforce(Section3.2).Wethenincorporateviscousand
4
Ohmicdissipationandshowthatthisproducesadampingof
thetorsional modes, implying that the torsional problem is
3 one of a driven-dampedmechanical oscillator (Section 3.3).
Under resonant conditions, the rates of driving and dissi-
2 pation for such a system exactly balance. In the context of
the stellar problem, this means that where resonances be-
1 tweenspheroidalandtorsionalmodesexist,theenergydissi-
patedequalstheworkdoneagainsttheLorentzforcebythe
R)* 0 spheroidalmotions.Integratingthisoverthestar,wearrive
z ( atananalyticalexpressionfortheoveralldampingrateγ of
a spheroidal mode (Section 3.4). Throughout this work we
−1
assume linearity of the fluid motions, and specialise to the
case of axisymmetric modes.
−2
3.1 Torsional Alfv´enic oscillations
−3
Theequationofmotionofadrivenoscillatorcanbewritten
−4
∂2ξ
+ [ξ]=S(r,t), (11)
∂t2 L

0 1 2 3 4 where ξ denotes mechanical displacement, is a linear op-
R (R*) x 10−3 eratorcontainingthespatialderivativesofξLandthesource
termSrepresentstheexternaldriving/forcing.HereSisre-
garded as being external because it does not depend on ξ,
5 although itmay beafunction ofposition rand timet.The
4 normal modes of the oscillator are thesolutions of Eq. (11)
with S = 0 (the homogeneous problem). Imposing a time-
2Nr/g 23 hthaermeiogneincpdroebpleenmden[cξe]ξ=∝ω2eξx,ps(a−tisiωfiet)d,ftohrisoncloyrrsepspecoinadlsvatlo-
ues of ω2 = ω2. TLhese are the natural frequencies of the
1 0
oscillator, and the associated forms of ξ are the eigenfunc-
0 tions of the system.
0 0.005 0.01 0.015 0.02
r (R*) Thefluidequation ofmotion intheabsenceofrotation
can bewritten
x 106
∂u 1
ρ +u u = p ρ Φ B2+(B )B,
10 ∂t ·∇ −∇ − ∇ − 2∇ ·∇
(cid:18) (cid:19)
(12)
ρ〈ρ〉/ 5 twhheeruesuua=l µD0ξf/aDcttorisstihnetofluthidevdeelfiocniittyioanndofwBe.hTavheealbassotrbtwedo
terms on the RHS of Eq. (12) correspond to the magnetic
pressureandtension,respectively.WehereadopttheCowl-
0
0 0.005 0.01 0.015 0.02 ing approximation under which the gravitational potential
r (R*) is fixed and depends only on r. Then upon linearising and
taking the curl of Eq. (12) the r-components of the first
threetermsontheRHSvanish,leavingmagnetictensionas
Figure 2. The Prendergast magnetic field solution (top) calcu-
theonlyforce capableofrestoring torsional motions. Inthe
lated over the assumed core region (0.005 of the stellar radius).
axisymmetriccase,whichwefocusonhere,thetorsionaldi-
Thisregionisshownshadedinthebottomtwopanels,whichplot
rectioncorrespondstotheφ(azimuthal)direction(wecom-
thedimensionlesssquaredbuoyancy frequencyandmass density
profiles near the centre of the star. In the top panel, a selection mentonthenon-axisymmetriccaseinSection5.2).Consider
ofmagneticfluxsurfaces(poloidalfieldloops)areshownasblack nowthetorsionalcomponentofEq.(12),whichlinearisesto
lines, while the underlying colour represents the strength of the give
toroidalcomponent.Theabsolutescalingofthefieldatthisstage
itshaersboilturatiroyn;iilsluasxtrisaytmedmheetrreicisajnudststoheonolvyeraalmlgeeroidmioentrayl.hNaolft-eptlahnaet ρ0∂∂2tξ2φ = BR0 ·∇(RBφ′)+ BR′ ·∇(RB0φ). (13)
needstobeshown. Subscript 0’s denote static background quantities, while
primes denote (small) time-dependent perturbations about
the background. Using the linearised induction equation
6 S. T. Loi and J. C. B. Papaloizou
B′= (ξ B0),thisallowsustoexpressEq.(13)interms
∇× ×
of just ξ and backgroundquantities.This can bewritten in
x 10−3 x 10−4
theform
2.5
∂2ξφ + [ξ ]= fTS , (14) 4
∂t2 LT φ ρ0
where 3 2
LT[ξφ]=−ρB0R0 ·∇(cid:20)R2B0·∇(cid:18)ξRφ(cid:19)(cid:21) , (15) 12 1.5
fTS =−BR0 ·∇(cid:20)R2ξ·(cid:18)BR0φ(cid:19)+RB0φ(∇·ξ)(cid:21) z (R)* 0
1
+ R[∇×(ξ×B0)]·∇(RB0φ). (16) −1 1
See that fTS depends only on the spheroidal displacement −2
ξ (ξ ,0,ξ ),notξ ,andcan thusberegarded asaforc-
inSg≡termRin Ezq. (14) pφrovided that the spheroidal displace- −3 0.5
ment is assumed known, which will effectively be the case
−4
whenthemagneticfieldisweak.Inthatcasethespheroidal
modes will be relatively unperturbed.

Letusexaminetheoperator moreclosely.Although 0 1 2 3 4
T
at first glance this appears to deLpend on three spatial di- R (R*) x 10−3
mensions, notice that B0 = Bp∂/∂s, where Bp is the
·∇
magnitude of the poloidal component of B, and s is arc
length(i.e.physicaldistance)alongthepoloidalprojections
of the field lines. Hence the problem is intrinsically one- ×10-3
dimensional.Rescalingtoanewdistancecoordinateσobey- 2
ing dσ/ds=1/(R2B ), theeigenproblem reduces to
p
∂2η ∂2η 1
φ =v2 φ , (17)
∂t2 A ∂σ2
)
where η ξ /R is a new scaled fluid displacement R* 0
and vA2 ≡φ 1≡/(ρ0φR4). One recognises Eq. (17) as the one- z (
dimensionalwaveequationwith spatially varyingadvection -1
speed v = v (σ), which can be identified as the Alfv´en
A A
speedwithrespecttothenewcoordinates. Fromhereitbe- -2
comes more convenient to work in terms of η rather than 2 4
φ
ξφ. The particular form of Eq. (17) allows the solutions to ×10-3 0 2 ×10-3
be understood intuitively as standing waves on stretched η -2 0 R (R )
1D loops. These are quantised vibrations whose frequencies φ *
increase as thespatial scale decreases.
Figure 3.SpatialdistributionoftheAlfv´enspeedoverlaidwith
WesolvedEq.(17)asamatrixeigenvalueproblemona
anarbitraryfieldloop (top), andthe amplitudefunction forthe
discrete 1D grid for each flux surface, with periodic bound-
j =7 eigenmode on that loop (bottom). Colour bar units arein
aryconditionsandspatialderivativesapproximatedbycen-
terms of the dynamical speed pGM∗/R∗. In the bottom panel,
treddifferences.WecalculatedtheeigenmodesXj(σ,ψ)and the equilibrium position of the field line is shown in black and
eigenfrequencies ω02,j(ψ) on 1000 evenly-spaced (in ψ) flux thedisplacedpositioninred.Arrowsareanaidtovisualisingthe
surfaces with 5000 uniformly-spaced (in σ) points on each directionofthedisplacement.
surface. Here j Z+ is the harmonic index. Although the
∈
total number of eigenfunctions obtainable by this method
equals the number of grid points, the accuracy of the so- a circle about the axis of symmetry (here the z-axis). The
lutions is expected to degrade for larger ω2 where spatial motion can be envisaged as segments of each flux surface
0,j
scales of the associated eigenfunctions become too small to (which are tori in 3D) twisting with respect to others. For
be adequately resolved. On each flux surface we restricted a given wave speed vA, one expects ω0,j for fixed j to in-
theeigenfunctions used in furtheranalysis to the1000 hav- crease as thelength of thefield loop shrinks.This is indeed
ing thelowest eigenfrequencies. observed in our model: the spatial distribution of ω0,j for
Theparameterβwaschosensoastoproducethedistri- j =300 is shown in Fig. 4.
butionoftheAlfv´enspeed,vA,shownintheupperpanelof Foragivenfluxsurface,onealsoexpectsω0,j toincrease
Fig.3.Aselectedeigenmodeisillustratedinthelowerpanel with j. Only an even number of nodes is allowed for vibra-
ofFig.3.Sincetheproblemisaxisymmetric,thespatialam- tions on a loop, so j =1 has zero nodes, j =2,3 have two,
plitude function need only be displayed on a poloidal field j =4,5havefour,andsoon.Hencejisroughlyproportional
loop, corresponding to a longitudinal slice of the flux sur- to thenumberof wavelengths around theloop, but there is
face. The full solution is obtained by sweeping each loop in apaired structureto thespectrum. This can be seen in the
Magnetic damping of stellar oscillations 7
30
8.4
8.2
8
25 7.8
7.6
7.4
7.2
20 190 200 210
0,j 15
ω
10
5
0
0 200 400 600 800 1000
j
Figure5.Thetorsionalspectrumcalculatedfor10evenly-spaced
(in ψ) flux surfaces. Each track corresponds to one flux surface,
and flux surfaces of lower tracks enclose those of higher ones.
Although apparently continuous, the tracks are infact made up
of discrete points, since j ∈Z+. The discreteness can beseen in
the inset plot, which zooms in to a small portion of the overall
spectrum.Thepairedstructurereflectsapproximatelydegenerate
modes,whichhaveequalnumbersofnodesbutareoddandeven
versions of one another. Frequencies are given in units of the
dynamicalfrequency,pGM∗/R3∗.
Figure 4. Spatial distribution of the j = 300 eigenfrequen-
cies. Colour bar units are in terms of the dynamical frequency
structureof the equations of motion:
pGM∗/R3∗.Sinceeigenmodesarelocalisedtoindividualfluxsur-
faces,ω0,j isconstantonanyfluxsurfaceforgivenj.Smallerflux ∂2ξS + [ξ ]= fST(ξφ) (19)
surfacestendtohavehigherω0,j forfixedj. ∂t2 LS S ρ0
∂2ξφ + [ξ ]= fTS(ξS) , (20)
insettoFig.5,whichplotsω0,j versusjforaselectionofflux ∂t2 LT φ ρ0
surfaces. Note that despite having equal numbers of nodes, where
eachpairstillcorrespondtodistincteigenmodes,thesebeing
1
othdedyawndouelvdenbevetrhseiosnisneofaonndecaonsointheesro(lue.tgio.nfosr),awciotnhstsalingthtvlAy LS[ξS]= ρ0 ∇p′+ρ′∇Φ0−fSS(ξS) , (21)
differenteigenfrequencies.Notethatsimilarbehaviourofthe [ξ ]= f(cid:2)TT(ξφ) . (cid:3) (22)
periodicsolutionsoftheMathieuequationoccurs.Theover- LT φ − ρ0
all slope of ω0,j versus j should also be proportional to the We see that the coupling between spheroidal and tor-
fundamentalfrequency(largerforsmallerloops,inlinewith sional motions is provided bytheLorentz force. To first or-
the picture of a vibrating string). As can be seen in Fig. 5, der, given that the Lorentz force is much smaller than the
this is indeed thecase. forces of pressure and buoyancy, the fSS term in Eq. (21)
can be neglected. This is akin to assuming that the mag-
netic field has negligible effect on the spheroidal eigensolu-
3.2 Coupling with spheroidal motions tion (i.e. we still get the usual p- and g-modes). An impor-
tant term not included explicitly in Eq. (19) is the forcing
The perturbation to the Lorentz force can be subdivided
associated with (purely spheroidal) convectivemotions, the
into terms that depend on the spheroidal displacement ξ
S source of energy for the whole system. Given that the cou-
but not the torsional displacement ξ , and the terms that
φ pling from spheroidal motions into torsional motions and
dependonξ butnotξ (thisispossiblebecauseonlyterms
φ S back into spheroidal motions is a second-order process, the
linearinξareretained).Writingthisoutincomponents,we
can expressthis separation as direct contribution of convection should dominate over fST
in the spheroidal equation of motion when magnetic fields
fS(ξ)=fSS(ξS)+fST(ξφ), are weak (the coupling strength scales like the magnetic
pressureB2,whichisfarsmallerthanthegaspressure).We
fT(ξ)=fTS(ξS)+fTT(ξφ), (18) shallthusneglectallmagnetictermsinEq.(19).Incontrast,
where fS and fT and the spheroidal and torsional compo- the Lorentz force has first-order significance in providing
nentstotheLorentzforceperturbation,fSS arethetermsin both the driving and the restoration of torsional motions,
fS that depend only on ξS, fST are those that depend only since there are no pressure or buoyancy forces to compete
on ξ , etc. This allows us to observe the following coupled with, and cannot be neglected in Eq. (20).
φ
8 S. T. Loi and J. C. B. Papaloizou
With magnetic terms neglected, finding the spheroidal
eigenmodesreducestothestandardhydrodynamicproblem 2 2x 10−6
oflinearadiabaticstellaroscillations.IntheCowlingapprox-
1
imation(i.e.neglectingtheperturbationtothegravitational 1
potential), this involves solving the following second-order 0
system of ODEs: ξr n = 1 0
−1
n = 2
dξr = 2 1 ξ + 1 Sℓ2 1 p′ −2 n = 3 −1
dr −(cid:18)r − Γ1Hp(cid:19) r ρ0c2s (cid:18)ω2 − (cid:19) n = 4
ddpr′ =ρ0(ω2−N2)ξr− Γ11Hpp′, (23) −30 r 0(R.5*) 1 −20 r 0(R.5*) 1
x 10−6
whichcanbeachievedbystandardnumericaltechniquesfor 20 2
ODEeigenvalueproblems.Wedidthisbyfirstinterpolating l = 0
the CESAM profiles onto a finer grid (to capture the small 15 l = 1 1
l = 2
spatial scales of g-mode oscillations) and then solving Eqs
(23) using the shooting method. The quantities Γ1, Hp, cs, ωS10 l = 3 0
S andNcharacterisethestellarbackgroundandcorrespond
ℓ 5 −1
to the adiabatic index, pressure scale height, sound speed,
Lamb frequency and buoyancy frequency, respectively. The
0 −2
horizontalcomponentξ ofthefluiddisplacementisrelated −1000 −500 0 0 0.5 1
to p′ through h n r (R*) x 10−3
p′
ξ = . (24) Figure6.Aselectionofeigenmodesandeigenfrequenciesforthe
h rω2ρ0 stellarmodelexaminedhere,showingthefourlowest-orderradial
We conducted a near-exhaustive search for all modes (top left), and an ℓ = 1 mixed mode near ω = 10 (top
spheroidal eigenmodes between ω =1 and 20 (expressed as right) with the central regions shown enlarged on the bottom
a multiple of the dynamical frequency GM∗/R∗3), refin- right so that the g-type oscillations can be seen. Red and black
in the two righthand plots correspond to horizontal and radial
ing this near the locations of p-dominated modes (of which
one exists per radial order), for sphericapl harmonic degrees fluiddisplacementsξh andξr,respectively.Notethatthescaling
of ξ isarbitrary. Thefrequencies (expressed as a multipleof the
ℓ = 0, 1, 2 and 3. The objective was to selectively extract
dynamicalfrequency)ofthefirstfourlowestsphericaldegreesare
themodesthatareobservableexperimentally,whicharere- plottedversusradialorderonthebottom left.
stricted to those with low ℓ (due to geometric cancellation
effects),strongp-modecharacter(associatedwithlargersur-
facemotionsandthereforeintensityvariations) andlocated ξS producecorresponding fine-scale oscillations of fTS(ξS).
within several radial orders of ω = 10 (typical frequency Thisillustrateshowtheexcitationofhighharmonicsbylow-
of maximum excitation for solar-like oscillators). Figure 6 degreemodescanoccur:ingeneralthefieldlinescutacross
shows several ℓ = 0 modes and one ℓ = 1 mode found by many radial shells, enabling large n tomap to large j.
the search. The ℓ = 0 modes (top left) are oscillatory only Thetorsional equationofmotionintermsofthescaled
near the surface, while the ℓ = 1 mode (top right and bot- fluid displacement ηφ can bewritten
tomright)hassignificantmixedcharacterandisoscillatory
∂2η 1 ∂2η
both near the surface and near the centre. Mixing occurs φ φ =F , (25)
also for ℓ = 2 and 3, but due to the weaker coupling for ∂t2 − ρ0R4 ∂σ2 φ
higher ℓ, the modes have purer p- or g-like character com- where Fφ =fTS/ρ0R. Wenow wish to derive an expression
paredtoℓ=1.Radialordersn(bottomleft)werecomputed for the expansion coefficients of F (the spheroidal forcing)
φ
usingtheEckartscheme(Eckart1960;Scuflaire1974;Osaki with respect to the torsional eigenmodes X identified in
j
1975), where the convention is to count p-type(g-type) ra- Section 3.1, i.e. the quantitiesa in
j
dialcrossingspositively(negatively).Thelargenegativeval-
ues for the ℓ > 0 modes indicate that these have a large Fφ(σ,ψ)= aj(ψ)Xj(σ,ψ). (26)
number(hundreds)of oscillations in theg-mode cavity. Xj
Recallthatfromthepointofviewofthetorsionalequa- First,weneedtoestablishtheorthogonalityrelationfor
tion of motion, spheroidal fluid motions act as a forcing X . Substituting the eigensolution ω2 ,X into the homo-
j 0,j j
function through their associated Lorentz force. In general, geneous form of Eq. (25), we have
if the forcing applied to a mechanical oscillator contains
∂2X
othneenorremsoonraentfreexqcuietnactiioesntohcacturms.atTchheitsstrnenatguthraolffrtehqeueexncciiteas-, −ρ0R4ω02,jXj = ∂σ2j . (27)
tiondependsonthegeometricsimilaritybetweentheforcing Integrating twice by parts and applying periodic boundary
function at the resonant frequencies and the corresponding conditions, one can show that
eigenmodes, which can be quantified as the coefficients of
∂2X ∂2X∗
the eigenfunction expansion of the forcing function. Figure X∗ j dσ= X k dσ, (28)
k ∂σ2 j ∂σ2
7showsthespatial distribution offTS nearthecorefor the I I
mode shown on the right of Fig. 6. Overlaid in black is the wheretheintegralisaroundaclosed fieldloop.Multiplying
samefieldloopshowninFig.3.Thefine-scaleoscillationsof Eq.(27)byX∗ andintegrating,andusingEq.(28),wefind
k
Magnetic damping of stellar oscillations 9
Figure8.Thetorsionalspectrumcomputedfor500fluxsurfaces.
Points are coloured according to the strength of coupling with
the spheroidal mode whose Lorentz force distribution is shown
in Fig. 7. The coupling strength is quantified as |aj|, the abso-
lute value of the coefficient of the eigenfunction expansion (see
Eq.(30)).
yieldsaphysicalupperboundona oftheorder104.Ascan
Figure 7. Spatial distribution of fTS (the torsional component beseenfromFig.8,theactualvalujesobtainedarea 100.
of the Lorentz force associated with spheroidal motions) for the j ∼
AlsoapparentfromFig.8isthatthetorsionalspectrum
mixed mode shown on the right of Fig. 6, overlaid withthe flux
is very dense. In fact, for every value of j Z+ there is a
surfaceshowninFig.3.Onlytheregionnearthecentreisshown.
∈
Fluid displacements have been normalised so that the total en- continuumofω0,j values,reflectingtheexistenceofacontin-
ergyofthemodeisunity.Oneseesthatthereisacross-cutofthe uum of flux surfaces. However, the numberof resonances is
fieldloopacrossmanyradialnodes(signchanges)offTS,suggest- finiteduetothediscretenatureofj,andatagivenωequals
ingthatthisshouldpreferentiallyexcitehigh-indexharmonicson the j range intersected by a horizontal line at that value of
thatloop. ω. For the spheroidal mode shown in Fig. 7, which is near
ω = 10, we identify around 900 resonances, i.e. there exist
thisnumberofmagneticfluxsurfaceswhichhaveatorsional
that
mode with thiseigenfrequency.
ω02,j−ω02,k ρ0R4XjXk∗dσ=0, (29)
I
(cid:0) (cid:1) 3.3 Dissipative effects
which implies that unless ω2 = ω2 , it must be that
0,j 0,k
ρ0R4XjXk∗dσ = 0. It is possible to normalise the Xj so We shall now incorporate dissipative effects, with the goal
Hthat ρ0R4XjXj∗dσ=1.Doingso,wearriveatthedesired ofeluciatingtheirroleincontributingtothedampingofthe
expression torsional oscillations. In the following derivation it will be
H
assumed that the damping coefficients in units of the dy-
aj = R3fTSXj∗dσ. (30) namical frequency are much less than unity (highly under-
I damped oscillator), so that the eigenfrequencies and eigen-
Thevaluesofa forthespheroidalmodeshowninFig.7 modesderivedaboveremainvalid.Attheendofthissection
j
and each of the torsional modes are plotted in Fig. 8. The we evaluate the expression for the damping coefficient ob-
spheroidal displacements have been normalised such that tained and verify that it is indeed small.
thetotal energy E (Unnoet al. 1989) equals unity,i.e. With dissipative terms included, the momentum and
induction equations are
E =ω2 ρ0r2 ξr2(r)+ℓ(ℓ+1)ξh2(r) dr=1. (31) ∂2ξ ∂ξ
Z + [ξ] ν 2 =0 (32)
(cid:2) (cid:3) ∂t2 L − ∇ ∂t
As a comment, the values obtained for a for the current
j ∂B′ ∂ξ
modelaresubstantiallylowerthanthemaximumphysically ∂t −νm∇2B′ =∇× ∂t ×B0 , (33)
allowed values,whichwouldoccurunderconditionsofcom- (cid:18) (cid:19)
plete geometric overlap (perfect constructive interference). where ν and ν are the viscous and Ohmic dissipation co-
m
As an order-of-magnitude estimate, the upper bound on aj efficients, and [ξ] refers to the linearised form of the RHS
is given by R1/2B3/2ξL−2, where R is the size of thecore, ofEq.(12).AnLticipatingsmall scales wehaveretained only
c c
ξ is the characteristic fluid displacement and L is the char- the highest order derivatives in the viscous force and rate
acteristic length scale of variation in ξ. For our red giant of Ohmic diffusion. Invoking a time-harmonic separation,
model,substitutingappropriatevaluesfortheseparameters Eq. (33) becomes an inhomogeneous Helmholtz equation
10 S. T. Loi and J. C. B. Papaloizou
which can be solved by means of an integration kernel to the damping contribution iΓω (cf. the Landau prescription
yield B′(r) (ξ¯ B0), where from plasma physics).The final solution for bj is then
≈∇× ×
ξ¯(r)= K(r r′)ξ(r′)d3r′, (34) b (ψ)= aj(ψ0) . (40)
− j ω2 (ψ) ω2 iΓω
ZZZ 0,j − −
iω ω
K(r)= exp (1 i) r . (35) SincetheRHSofEq.(38)isapproximatelyconstant,wein-
4πνm|r| (cid:20)− − r2νm| |(cid:21) fer thecharacteristic scale to be x ν1/3. The local damp-
∼ tot
Herewehaveassumedthatξvariesmuchmorerapidlyover ing rate can be estimated from the first-order term of the
space than B0, which itself varies on a scale much larger Taylor expansion of ω02,j(ψ), and has the approximate ex-
telahfftaeencdtptohfνrdmoiu/sgsωihp.aξAtis=onfaξ¯risatsh(νutmhse/toLiωor)reepnl2atξ¯zc.efSoξurcbbesytiiξs¯tu,ctwoinnhcgicehrtnhaeisrdeintrhteoe- ufponrreitsdysi,eocΓnortΓruer∼lnast[i2ooωun0′t,tjto(oψbo0ec)cRtuhBrepoi]n2v/ve3erνrts1toe/ht3eo.fUwthpidetthtoimaoeffastcchatoelerrroeefsqoounriadrenedrt
− ∇
Eq. (32), neglecting products of ν and νm (given that both layer.Furtherinspection revealsthatthiswidth isprecisely
are small), defining η¯φ ξ¯φ/R and retaining only diffu- that for which the timescales of decorrelation and dissipa-
≡
sive terms involving second order spatial derivatives, the tion are equal, providing thephysical interpretation for the
torsional component can bewritten associated lossprocessasbeingcloselylinkedtophasemix-
∂2η¯ 1 ∂2η¯ ∂2 ∂η¯ ing. Given the small magnetic Prandtl numbers in stellar
∂t2φ − ρ0R4 ∂σ2φ −νtot∂n2 ∂tφ =Fφ, (36) cinietnertioνrs, νto1t0i9sTd−o3m/2inma2tesd−1by(StphietzOerhm19i6c2d)i.ssFiopratoiounr mcooedffie-l
m
where νtot = ν+νm is the total dissipation coefficient and (T 107K≈)wefindthatΓ 10−3 inversedynamicaltimes,
n is the direction normal to the flux surfaces. The reason and∼that the widths of the∼resonant layers are 10−7R∗.
for retaining only this spatial part of the Laplacian is that ∼
the finest-scale structure is likely to develop in the direc-
tion perpendicular rather than parallel to the flux surfaces, 3.4 Overall damping rates
as a result of phase mixing. This refers to thedecorrelation
A major objective of this work is to estimate the overall
of oscillations on adjacent surfaces having slightly different
damping rate γ of spheroidal modes due to the resonant
eigenfrequencies, which occurs on a timescale correspond-
couplingwithtorsionalmodes.Wewillnowcombineresults
ing to the inverse of their frequency difference (cf. the beat
from preceding sections to arrive at an expression for γ.
phenomenon). In the case of our model, the decorrelation
Thetotalrateofworkdonebythetorsionalcomponent
timescaleassociatedwiththespatialscalealongfluxsurfaces
( 10−5R∗)isonly 10dynamicaltimes,muchshorterthan of theLorentz force associated with spheroidal motions is
∼ ∼
the dissipation timescale associated with the same length dE ∂ξ∗ ∂ξ
scale ( 106 dynamical times). Decorrelation will therefore dt = fTS ∂tφ +fT∗S ∂tφ d3r
proceed∼downtomuchsmallerscalesbeforeitsdevelopment ZZZ (cid:18) ∂η∗ (cid:19)
is halted byviscous/resistive effects. =4πRe ρ0R4Fφ ∂tφ dσdψ . (41)
Weseekasolutiontothecoefficientsbj oftheeigenfunc- (cid:20)ZZ (cid:21)
tionexpansionη¯φ(t,σ,ψ)= jbj(ψ)Xj(σ,ψ)e−iωt.Noting Invoking the eigenfunction expansions for Fφ and ηφ, elim-
that ∂/∂n=RBp∂/∂ψ, we gPet inating bj in favour of aj using Eq. (40), making use of the
orthonormalityrelationforX andaveragingoveroneoscil-
∂2b (ψ) j
ω02,j−ω2 bj(ψ)+ iνtotωR2Bp2 ∂ψj2 =aj(ψ). (37) lation period, we obtain thetime-averaged rate of work
Let(cid:2)us focus(cid:3)on a small region in ψ near a resonant sur- ddEt ≈4π |aj(ψ0)|2 h(ψ)Γ+Γ2 dψ, (42)
face ψ0 whose j-th harmonic is of frequency ω0,j(ψ0) = ω. (cid:28) (cid:29) Xj Z
Locally, we adopt the Taylor expansion ω2 (ψ) ω2 +
2ωω0′,j(ψ0)[ψ −ψ0]. Consider the change o0f,jvariab≈le x = where
ECq[ψ. (−37ψ)0in]twohere C = (2ω0′,j(ψ0)/R2Bp2)1/3, which turns h(ψ) ω02,j(ψ) ω 2 . (43)
≡ ω −
(cid:18) (cid:19)
bjx+ iνtot∂∂2xb2j = 2ωCωa0′j,(jψ(ψ0)0) . (38) rTehsoisnaanssturmegeisonthaantdacjanvabreieasppslroowxliymaotveedrbtyheitswvidaltuheoafttthhee
The RHS,which we will call A,can beregarded as roughly resonantsurface ψ=ψ0 whereω0,j(ψ0)=ω.Inthelimit of
constantovertheresonantlayer.Equation(38)canbesolved smallΓtheresonantregionisspatiallynarrow,andsowecan
using Fourier transforms (x k and b (x) ˜b (k)), yield- approximate h(ψ) by the first term of its Taylor expansion
j j
ing → → about ψ0. This allows us to straightforwardly evaluate the
ψ-integral in Eq. (42). Wearrive at thefinal expression
0 k>0
˜b (k)= (39) −1
In the limit νtjot 0,(˜bijA/ieAxpt[ekn3dνstott/o31] kH<(k0) where H(k) (cid:28)ddEt (cid:29)=2π2Xj |aj(ψ0)|2 (cid:12)(cid:12)ddωψ0,j(cid:12)(cid:12)ψ0! , (44)
→ − (cid:12) (cid:12)
istheHeavisidestepfunction.Thissatisfies i [1 H(k)]= whichweseeisindependentofthe(cid:12)locald(cid:12)ampingcoefficient
F −
1/(x i0). Here denotes a Fourier transform and i0 is Γ. This reflects a basic property of driven-damped oscilla-
− F
aninfinitesimalimaginary componentthatweidentifywith tors that near a resonance, in the limit of weak dissipation