Tidal inertial waves in the differentially rotating convective envelopes of low-mass stars - I. Free oscillation modes

8 MB·

by  M. Guenel

Astronomy&Astrophysicsmanuscriptno.GBMR2015-p (cid:13)cESO2016
January19,2016
Tidal inertial waves in the differentially rotating convective
envelopes of low-mass stars
I. Free oscillation modes
M.Guenel1,C.Baruteau2,S.Mathis1,3,andM.Rieutord2
1 LaboratoireAIMParis-Saclay,CEA/DSM-CNRS-UniversitéParisDiderot,IRFU/SApCentredeSaclay,F-91191Gif-sur-Yvette
Cedex,France
2 InstitutdeRechercheenAstrophysiqueetPlanétologie,ObservatoireMidi-Pyrénées,UniversitédeToulouse,14avenueEdouard
Belin,31400Toulouse,France
6 3 LESIA,ObservatoiredeParis,PSLResearchUniversity,CNRS,SorbonneUniversités,UPMCUniv.Paris6,Univ.Paris-Diderot,
1
SorbonneParisCité,5placeJulesJanssen,92195Meudon,France
0
e-mail:[mathieu.guenel,stephane.mathis]@cea.fr;[clement.baruteau,michel.rieutord]@irap.omp.eu
2
Received.../accepted...
n
a
J
ABSTRACT

8
1 Context.Star-planettidalinteractionsmayresultintheexcitationofinertialwavesintheconvectiveregionofstars.Inlow-massstars,
theirdissipationplaysaprominentroleinthelong-termorbitalevolutionofshort-periodplanets.Turbulentconvectioncansustain

] differentialrotationintheirenvelope,withanequatorialacceleration(asintheSun)ordeceleration,whichcanmodifythewaves’
R
propagationproperties.
S Aims.Weexploreinthisfirstpaperthegeneralpropagationpropertiesoffreelinearinertialwavesinadifferentiallyrotatinghomoge-
. neousfluidinsideasphericalshell.Weassumethattheangularvelocitybackgroundflowdependsonthelatitudinalcoordinateonly,
h
closetowhatisexpectedintheexternalconvectiveenvelopeoflow-massstars.
p
Methods.Weusei)ananalyticalapproachintheinviscidcasetogetthedispersionrelation,fromwhichwecomputethecharacteristic
o- trajectoriesalongwhichenergypropagates.Thisallowsustostudytheexistenceofattractorcyclesandinferthedifferentfamiliesof
r inertialmodes;ii)high-resolutionnumericalcalculationsbasedonaspectralmethodfortheviscousproblem.
t Results.Wefindthatmodesthatpropagateinthewholeshell(Dmodes)behavethesamewayaswithsolid-bodyrotation.However,
s
a anotherfamilyofinertialmodesexists(DTmodes),whichcanpropagateonlyinarestrictedpartoftheconvectivezone.Ourstudy
[ shows that they are less common than D modes and that the characteristic rays and shear layers often focus towards a wedge –
orpoint-likeattractor.Moreimportantly,wefindthatfornon-axisymmetricoscillationmodes,shearlayersmaycrossacorotation

1 resonancewithalocalaccumulationofkineticenergy.TheirdampingratescalesverydifferentlyfromwhatweobtainforstandardD
v modesandweshowanexamplewhereitisindependentofviscosity(Ekmannumber)intheastrophysicalregimeinwhichitissmall.
7
1 Keywords. hydrodynamics–waves–planet-starinteractions–stars:rotation
6
4
01. Introduction Verbunt&Phinney(1995)andmorerecently,Meibom&Math-
.
1 ieu (2005) who showed that the oldest systems have circular-
0The tidal interaction between a star and an orbiting companion izedorbits,thoughthespin-orbitsynchronizationisstillunclear
6results from the differential force exerted by each body on the (Meibometal.2006).Observationsofextrasolarplanetsbythe
1other.Indeed,thegravitationalpotentialcreatedbythecompan- radial-velocityandtransitsmethodshavedevelopedrapidlyover
v:ion is not uniform inside the star since it is an extended body the past decade and stimulated interest in looking for signa-
iand not a point-like mass. If the orbit is close enough, the star tures of tidal interactions in star-planet systems : for instance,
Xexperiencesanon-negligibleperiodicalforcingwhichgenerates Pont (2009) looked for an excess rotation in stars with plan-
rtidal flows. Their dissipation into heat through various friction ets (due to substantial inward migration) while Husnoo et al.
a
mechanisms (see Zahn 2013; Mathis & Remus 2013; Ogilvie (2012) used observed eccentricities to conclude that tidal inter-
2014)takesenergyawayfromthesystemandallowsforaredis- actions play a prominent role in the orbital evolution and sur-
tributionofangularmomentum,whichleadstotheevolutionof vival of hot Jupiters (see also Lai 2012; Guillot et al. 2014).
the spin and orbital parameters on secular timescales. Depend- Moreover, Jackson et al. (2008) consistently checked that the
ingontheinitialdistributionofangularmomentumbetweenthe observedloweccentricitiesofclose-inplanetscanbeexplained
individualspinsandtheorbit,thesystemmayevolvetowardsan by tidal evolution processes, and calibrated the required tidal
equilibriumstate—wheretheorbitiscircularandthespinsof quality factors. Hansen (2010) and Penev et al. (2012) used
thebodiesaresynchronizedandalignedwiththeorbit—orlead observed populations of exoplanets and tidal evolution models
totheinspiralorejectionofthecompanion(Hut1980). to constrain tidal dissipation in stars, and both studies seem to
agree that their results are inconsistent with the dissipation in-
Observational evidence for ongoing tidal interactions was
ferredfromthecircularizationofbinarystars,suggestingthata
foundinclosebinarystars,forinstancebyGiuricinetal.(1984),
Articlenumber,page1of17
A&Aproofs:manuscriptno.GBMR2015-p
different mechanism is at play in this case. Winn et al. (2010) the effects of resonant excitation of gravito-inertial modes and
andAlbrechtetal.(2012)alsoproposedthattidaldissipationin possibleresonancelocking,whichcanspeedupsignificantlythe
convective regions could be responsible for the low obliquities secularevolution.
of hot Jupiters orbiting around cool stars, though Mazeh et al. Unlikestablystratifiedradiativezones,stellarconvectivere-
(2015)foundthatthispropertyextendstoKeplerObjectsofIn- gionsareveryslightlyunstablystratifiedandgravitymodescan-
terestwithorbitalperiodsuptoatleast50days,forwhichtidal not propagate inside them. For more than thirty years, the pos-
interactions should be negligible. These results have to be put sible effects of a dynamical tide in convective zones have been
into perspective since many other processes are responsible for overlookedinfavourofZahn’sequilibriumtidetheory.However,
shapingthearchitectureofthesystems—suchasmigrationin asimpleapproachtomodelthisdynamicaltideistosimplyig-
theprotoplanetarydisk(e.g.Baruteauetal.2014),planet-planet noretheconvectivemotionsandtoconsiderlineardisturbances
scattering (e.g. Chatterjee et al. 2008), Kozai oscillations (e.g. toaneutrallystratifiedfluid.Inthiscontext,thedynamicaltide
Naoz et al. 2011), star-planet magnetic interactions (see Stru- consistsoflow-frequencyoscillationsprimarilyrestoredbythe
garek et al. 2014), magnetic spin-down of the star (see Barker Coriolis acceleration — otherwise known as inertial waves —
&Ogilvie2009;Damiani&Lanza2015),etc.—andtheglobal that propagate in a sphere (if the body is fully convective) or a
pictureremainsquiteuncertain. sphericalshell(forsun-likestarswithanouterconvectiveenve-
Onthetheoreticalside,thetidalresponseofastarhasbeen lope). These waves have been studied long ago in a uniformly
investigated in the past, starting with the theory of the equilib- rotating,incompressibleandinviscidfluidbyThomson(1880);
rium tide, which is the flow induced by the quasi-hydrostatic Poincaré (1885); Bryan (1889); Cartan (1922) (also Greenspan
adjustment to the tidal potential, and its dissipation by turbu- (1968))andtheyexhibitremarkableproperties:theirfrequency
lentfrictionintheconvectiveenvelopeoflow-massstars(Zahn ω˜ in the fluid frame (rotating with angular velocity Ω) is re-
1966a,b). This work was followed by the study of the radiative strictedto[−2Ω,2Ω]whiletheirspatialstructureisgovernedby
damping of tidal gravity modes (Zahn 1975) in more massive ahyperbolicsecond-orderpartialdifferentialequationnamedaf-
starswithanouterradiativeregion.Zahn(1977)thenestimated terPoincaré.Theypropagatealongraysthatareinclinedwitha
thestrengthsofthesemechanismsinordertoderivethecharac- constant angle arcsin(ω˜/2Ω). Such hyperbolic equations along
teristictimescalesforcircularization,synchronizationandalign- with boundary conditions in a closed container generally yield
ment for binary stars. Since then, most works about tidal inter- an ill-posed problem whose solutions may be singular. The be-
actions were based on Zahn’s approach and focused on giving haviorofinertialwavesisdrivenbytheraysproperties:inafull
morerealisticdescriptionsofthemechanismsthoughttobere- sphere,theyfillthewholedomainforallfrequenciesandasetof
sponsiblefortidaldissipation—especiallythedynamicaltidein globalnormalmodeswithfrequenciesthatisdensein[−2Ω,2Ω]
radiativeregionsinearly-typestars.Forinstance,Savonije&Pa- hasbeenfoundinthecaseofafullsphere(Bryan1889).Thisis
paloizou(1983,1984)performednumericalsimulationsofgrav- not true in the case of a spherical shell (Rieutord & Valdettaro
itymodesexcitedbyaperiodictidalpotentialinanon-rotating 1997),forwhichraysoftenfocustowardslimitcyclescalledat-
massive star and derived the associated timescales for the or- tractors(seeMaas&Lam1995;Rieutordetal.2001)thatexist
bital evolution of massive binaries. The effects of rotation on in narrow frequency bands and depend on the size of the inner
thesegravitymodesandtheassociatedtidaldissipation,ignored core.Theinclusionofviscosity—asasimplificationofturbu-
byZahn,weretheninvestigatedbyRocca(1987,1989)whoin- lence—givesrisetopeculiarmodesthatpossessshearlayerslo-
cludedtheeffectsoftheCoriolisacceleration,andGoldreich& calizedaroundtheaforementionedattractors(e.g.Rieutordetal.
Nicholson(1989)whotookintoaccountpossibleinternaldiffer- 2001).
entialrotation. The tidal force exerted by a close planetary or stellar com-
After the discovery of the first exoplanetary systems in the paniononitshoststarmaythereforeexciteinertialwavesinthe
mid1990’s,thestudyofstar-planettidalinteractionregainedin- external convective envelope of low-mass stars, if one or more
terestandstudiesshiftedtowardslower-massstarswithradiative components of the tidal potential has a frequency that fall into
coresandconvectiveenvelopes,forwhichtheacceptedtidaldis- the inertial range. The energy dissipated and the angular mo-
sipationmechanismsofarwastheviscousdissipationofZahn’s mentum carried by these waves may play an important role on
equilibrium tide in the convective zone. Terquem et al. (1998), theevolutionoftheorbitalarchitectureofclose-inplanetarysys-
Goodman & Dickson (1998) and Witte & Savonije (2002) per- temsandontherotationoftheircomponents,yetithasonlyre-
formed numerical calculations of the dynamical tide in the ra- centlystartedtobeinvestigated(seeOgilvie&Lin2004,2007;
diativecoreofnon-rotatingsolar-likestarsandstudiedtheeffect Ogilvie 2009; Goodman & Lackner 2009; Rieutord & Valdet-
ofresonancesinthecoreofasolar-typestar,andfoundthatthe taro 2010; Le Bars et al. 2015) in uniformly-rotating fluids.
associatedtidaldissipationcouldbemoreefficientthanthevis- Thetidaldissipationinducedbyforcedinertialmodeshasbeen
cous dissipation of the equilibrium tide in the convective enve- found to be a very erratic function of the excitation frequency
lope.Barker&Ogilvie(2010,2011);Barker(2011)alsostudied andvariesoverseveralordersofmagnitude,thoughfrequency-
thepossiblenon-linearbreakingoftidalgravitywavesnearthe averaged estimates can be obtained analytically (Ogilvie 2013)
centre of a solar-type star and its influence on tidal dissipation and evaluated along stellar evolution (Mathis 2015), showing
and on the survival of hot Jupiters and angular momentum de- strongvariationswithstellarmass,ageandrotation.Ithasbeen
position. Progressively, the low-frequency regime — where ro- shown by Baruteau & Rieutord (2013) that differential rotation
tation is likely to have the most important effects — of stellar may strongly affect the propagation properties of linear inertial
oscillations regained interest with the works by Savonije et al. modesofoscillations.Theirstudywasrestrictedtoshellularand
(1995); Savonije & Papaloizou (1997); Papaloizou & Savonije cylindricalrotationprofiles,butturbulentconvectioncanalsoes-
(1997) who simulated the full tidal response of a uniformly- tablishvariousdifferentialrotationprofiles(seeBrun&Toomre
rotatingmassivestarandfoundsignaturesoftidally-excitedin- 2002;Brownetal.2008;Mattetal.2011;Gastineetal.2014),
ertial waves in the convective core, as well as gravito-inertial including conical rotation as observed in the Sun (Schou et al.
wavesintheradiativeenvelope.Witte&Savonije(1999,2001) 1998;Garcíaetal.2007).Inthiswork,weexplorethepropaga-
studied the tidal evolution of massive binary systems including tion and dissipation properties of inertial modes in stellar con-
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Gueneletal.:Tidalinertialwavesinthedifferentiallyrotatingconvectiveenvelopesoflow-massstars-I
vective envelopes (for low-mass stars only) with conical differ- perioddetermineswhatwecallthe“Rossbyscale"belowwhich
entialrotation,namelydifferentialrotationthatonlydependson turbulenceislittleinfluencedbyrotationandisfastcomparedto
latitude. thewaveperiod.Ourmodelassumesthateddiessmallerthanthe
In Sect. 2, we expose our physical model and the associ- Rossbyscalecanberepresentedbyaturbulentviscosity.Tofix
atedhypothesesaswellasthedifferentialrotationprofileweuse. ideas,letusconsidertheexampleoftheSun.Theboundingfre-
TheninSect.3,wecarryoutananalyticalanalysiswhichyields quency2Ω correspondstowaveswithaperiodof12.5days.
max
thegeneralpropagationpropertiesofinertialwavesintheinvis- FromMixing-LengthTheory,whichsaysthatthetypicalscaleof
cid case. These results are compared to the viscous problem in turbulenceisthemixing-length(abouttwicethepressurescale
Sect.4wherewecomputenumericallythevelocityfieldsofsuch height), we note that the largest eddies have a typical scale of
inertial modes, which sometimes have very different properties 100Mm,withatypicalvelocityof50m/s.Thiswouldbethetyp-
thaninthesolid-bodyrotationcase.Finally,weidentifyinSect. ical numbers for the so-called giant cells characterized by a 20
5 which of these properties are important for the forced prob- daysturnovertimescale(Mieschetal.2008).Withthesenum-
lemthatwewilltreatinasubsequentpaper,andinparticularthe bersweeasilyfindthattheRossbyscaleis50Mm.Thusallthe
evaluationofthetidaldissipationinducedbythesewaves. eddies of scale much smaller than 50Mm are assumed to influ-
encetheinertialmodesthroughaturbulentdiffusion.Theinflu-
enceofturbulentmotionswithlargerscalesisadmittedlymuch
2. Inertialmodesindifferentiallyrotating moredifficulttotakeintoaccount.However,wemaynotethatin
convectiveenvelopes thecaseoftheSunthedifferentialrotationflowismuchstronger
thanthelargescaleeddies(1km/scomparedto50m/s).Thus,as
2.1. Physicalmodel
afirststep,neglectingthedirectinteractionoftheseeddieswith
Sincewewanttostudythepropagationofinertialwavesinthe theinertialwavesisconsistentasfarasordersofmagnitudeare
differentially rotating convective envelopes of low-mass-stars, concerned.
wewillconsiderasimplifiedsetupwithanincompressible,vis-
cousfluidinsideasphericalshellofexternalradiusRandinter-
2.2. Mathematicalformulation
nalradiusηR(0<η<1)thataccountsfortheboundarybetween
theradiativecoreandtheconvectiveenvelope. Inaninertialframewiththeusualsphericalcoordinates(r,θ,ϕ),
AsmotivatedinSect.1,weuseaconicaldifferentialrotation andafterlinearizationoftheNavier-Stokesequationaroundthe
profile—dependingonlyonthecolatitudeθ—whichreads steady state (described by the “0” subscripts), we obtain (e.g.
Baruteau&Rieutord2013)
Ω(θ)=Ω0(θ)/Ωref =1+εsin2θ, (1) ∂u1+Ω ∂u1+2Ω e ×u +rsinθ(u ·∇Ω )e =−∇p +ν∇2u ,
∂t 0 ∂ϕ 0 z 1 1 0 ϕ 1 1
so that the dimensionless angular velocity of the background
flow is 1 at the poles and 1+ε at the equator. The quantity ε (2)
isaparameterthatdescribesthebehaviorofthedifferentialro-
where ∂/∂t denotes the partial time-derivative, Ω is the back-
tation: 0
ground angular velocity of the fluid, e = cosθe − sinθe is
z r θ
– ε > 0 is for solar differential rotation (equatorial accelera- the unit vector along the rotation axis, u1 is the wave velocity
tion), fieldand p1 denotesthereducedpressureperturbation,whichis
– ε < 0isforanti-solardifferentialrotation(equatorialdecel- the pressure perturbation divided by the reference density (ρ0).
eration). Notethatonlythelasttermoftheleft-handsideisproportional
totheshearofthedifferentialrotation,whiletheothertermsare
The value (cid:15) ≈ 0.3 approximates the differential rotation profile formallyidenticaltothecaseofsolid-bodyrotation.
intheSun’sconvectiveenvelope. Along with this equation, we use the linearized continuity
We point out that the above rotation profile does not sat- equation
isfy the Taylor-Proudman theorem, which states that the veloc-
∇·u =0 (3)
ityfieldofarotatingincompressiblehomogeneousinviscidfluid 1
mustbeconstantalongcolumnsparalleltotherotationaxis.As sincethefluidisalsoassumedtobeincompressible.
is now well-known (e.g. Brun & Toomre 2002; Brown et al. We look for velocity (u ) and reduced pressure (p ) per-
2008),thisdifferentialrotationprofileissustainedbyReynolds turbations of angular frequen1cy Ω (in the inertial fram1e) and
p
stressesduetoconvectiveturbulentmotions.Thismeanflowis azimuthal wavenumber m. This means they are proportional to
assumed to be dynamically stable. Our analysis will show that exp(cid:16)iΩ t+imϕ(cid:17).Eqs.(2)and(3)canberecastas
somemodesareunstableforsomeparametersofthemodel(see p
Eq.(19)).Theseinstabilities,interestingperse,shouldbeinter-
(cid:40)
pretedasgivingthelimitsoftheparametersrangeofthismodel, iΩ˜pu1+2Ω0ez×u1+rsinθ(u1·∇Ω0)eϕ =−∇p1+ν∇2u1,
especiallyinitsfutureusefortidallyforcedflows. ∇·u =0,
1
Thewavesthatwestudyinthefollowingarelow-frequency
(4)
waves: their frequency is less than 2Ω , where Ω is the
max max
maximum angular velocity of the fluid. Such waves propagate where Ω˜ = Ω +mΩ is the Doppler-shifted wave frequency
p p 0
onaturbulent backgroundandpresumablyinteractwith eddies (that is the local wave frequency in a frame rotating with the
that have similar time scale (just like the stochastically excited fluid).
acoustic modes of solar-type stars (Zahn 1966b; Goldreich & WenormalizeallfrequenciesbyΩ —whichwedefineas
ref
Keeley 1977)). In a Kolmogorov type turbulence, the turn-over the background angular velocity at the poles i.e. Ω (θ = 0) —
0
timescaleofeddiesdecreasesas(cid:96)2/3 if(cid:96) isthescaleoftheed- and distances by R. We introduce the associated dimensionless
dies. The equality between the turnover time scale and rotation quantities:theradius x = r/R,thevelocityfieldu = u /RΩ ,
1 ref
Articlenumber,page3of17
A&Aproofs:manuscriptno.GBMR2015-p
thereducedpressurep= p /R2Ω2 ,thedifferentialrotationpro- formercase,theequationgoverningthepathsofcharacteristics
1 ref
file Ω = Ω /Ω , the wave frequency ω = Ω /Ω in the in- inameridionalplanereads
0 ref p p ref
ertial frame (resp. ω˜p = Ω˜p/Ωref in the fluid’s frame) and the dz 1 (cid:16) (cid:17)
EkmannumberE =ν/R2Ωref.Fromnowon,weshalluseexclu- ds = 2ω˜2 Az±ξ1/2 , (9)
sively the dimensionless quantities defined above. This finally p
yieldsthedimensionlesssystemthatweshallsolvenumerically which shows that, contrary to the case of solid-body rotation
inSect.4: (Az = 0,ξconstant),differentialrotationtendstocurvepathsof
characteristicssincetheright-handsideofEq.(9)nowdepends
(cid:40)
iω˜pu+2Ωez×u+xsinθ(u·∇Ω)eϕ =−∇p+E∇2u, (5) both on s and z. On the other hand, no characteristics exist in
∇·u=0. theellipticdomain,thereforetherelationξ = 0defines“turning
surfaces”onwhichcharacteristicsreflect(seeFriedlander1982).
Note that in the numerical simulations presented in Sect. 4 Notethatiftherotationprofileissymmetricbyz→−z,thenEq.
below,weactuallytakethecurlofthefirstequationinorderto (7)isalsosymmetricbyz → −z,meaningonlypositivevalues
getridofthe∇pterm: of z can be considered. We detail below the case of the conical
rotationprofiledefinedinEq.(1).
 (cid:16) (cid:17)
 ∇× iω˜pu+2Ωez×u+xsinθ(u·∇Ω)eϕ = E∇×∇2u,
 ∇·u=0. 3.2. Pathsofcharacteristics,turningsurfacesandclassesof
inertialmodes
(6)
3.2.1. Pathsofcharacteristics
In addition to Eqs. (6), we use stress-free boundary condi-
Fortherotationprofilethatweadopt,giveninEq.(1),wehave
tions (u·e = 0 and e ×[σ]e = 0, where [σ] is the viscous
r r r (cid:16) (cid:17)
stress tensor) at the inner and outer boundaries of the spherical A =4Ω(θ) Ω(θ)+εsin2θcos2θ (10)
s
shell.
and
In the following section, we define several useful quanti-
ties for the description of the propagation properties of inertial A =−4Ω(θ)εcosθsin3θ, (11)
z
waves, and Eq. (1) implies that they only depend on the colati-
sothat
tudeθandareindependentoftheaspectratioηofthespherical
shell. ξ =16Ω2(θ)ε2cos2θsin6θ
(cid:16) (cid:17)
+4ω˜2 4Ω2(θ)+4Ω(θ)εcos2θsin2θ−ω˜2 . (12)
p p
3. Inviscidanalysis:pathsofcharacteristics,
We assume that ε > −1 so that the Rayleigh stability criterion
existenceofturningsurfacesandcorotation
(A > 0 everywhere in the shell) is always satisfied. From Eq.
s
resonances
(9), the slope of the paths of characteristics in the hyperbolic
As a first step, we study the solutions to Eqs. (5) in the invis- domainreads
tceirdisltiimcsito(fEin=er0ti)a.lItwaalvloews,swuhsictohsatruedyatvheerydyunsaemfuilcstoooflcfohraruanc-- ddzs =−A(θ)cosθsin3θ+4ωξ˜41/2 (13)
derstanding the solutions to the full viscous problem for small p
viscosities. where
ξ 4Ω2(θ)−ω˜2
3.1. ThegeneralPoincaré-likeequation = A2(θ)cos2θsin6θ+2A(θ)cos2θsin2θ+ p,
4ω˜4 ω˜2
p p
FollowingBaruteau&Rieutord(2013),weadoptcylindricalco- (14)
ordinates (s,ϕ,z) and we eliminate the components of the ve-
locity perturbations, which yields a partial differential equation and
on p only. Focusing on the second-order terms, we obtain the Ω(θ)
A(θ)=2ε . (15)
followingmixed-typepartialdifferentialequation: ω˜2
p
 
∂∂2sp2 + ωA˜2z ∂∂s2∂pz +1− ωA˜2s∂∂2zp2 +···=0, (7) Achsaeraxcpteecritsetdic,saldletpheenadboonveθqounalnyt.itiesaswellasthedynamicsof
p p
wherezerothandfirstordertermshavebeenomitted–thatcor-
3.2.2. Turningsurfaces
respondstotheWKBJapproximation.InEq.(7),
Inthesetupdescribedin3.2.1,ξ issymmetricbytheequatorial
2Ω∂(s2Ω) 2Ω∂(s2Ω) planeoftheshellsoonlyvaluesθ ∈ [0,π/2]needtobeconsid-
A (s,z)= and A (s,z)= . (8)
s s ∂s z s ∂z ered.Asexplainedinthepreviousparagraph,equationξ(θ) = 0
definesturningsurfacesthatseparatethesphericalshellintoan
In the case of solid-body rotation (ε = 0), Eq. (7) reduces
hyperbolicdomain(ξ >0)andanellipticdomain(ξ <0).Turn-
to the well-known Poincaré equation for inertial waves (Cartan
ingsurfacesareportionsofconesofaxiszandaperture2θ.How-
1922;Greenspan1968).Itishyperbolicinthedomainwherethe
(cid:16) (cid:17) ever, their location cannot be determined analytically because
discriminantξ(s,z) = A2z +4ω˜2p As−ω˜2p ispositive(whichwe ξ(θ)=0isapolynomialequationofdegree6inX =sin2θ.Still
will refer to as the “hyperbolic domain”) and becomes elliptic theirexistenceleadstotwoclassesofinertialmodeswithconical
inthedomainwhereξisnegative(the“ellipticdomain”).Inthe rotation:
Articlenumber,page4of17
Gueneletal.:Tidalinertialwavesinthedifferentiallyrotatingconvectiveenvelopesoflow-massstars-I
1.0 1.0
0.5 0.5
(cid:182) 0.0 (cid:182) 0.0
(cid:45)0.5 (cid:45)0.5
m(cid:61)0 m(cid:61)1
(cid:45)1.0 (cid:45)1.0
(cid:45)6 (cid:45)4 (cid:45)2 0 2 4 6 (cid:45)6 (cid:45)4 (cid:45)2 0 2 4
Ω Ω
p p
1.0 1.0
0.5 0.5
(cid:182) 0.0 (cid:182) 0.0
(cid:45)0.5 (cid:45)0.5
m(cid:61)2 m(cid:61)4
(cid:45)1.0 (cid:45)1.0
(cid:45)8 (cid:45)6 (cid:45)4 (cid:45)2 0 2 4 (cid:45)10 (cid:45)8 (cid:45)6 (cid:45)4 (cid:45)2 0 2
Ω Ω
p p
Fig.1.IllustrationofthetwokindsofinertialmodespropagatinginafluidwithconicalrotationprofileΩ(θ) = Ω (θ)/Ω = 1+εsin2θ,for
0 ref
m={0,1,2,4}fromtop-lefttobottom-right.Theeigenfrequencyintheinertialframeω isshowninx-axisandthedifferentialrotationparameter
p
εisiny-axis.Blue:Dmodesthatexistinthewholeshell(ξ >0everywhere).White:DTmodesthatexhibitatleastoneturningsurfaceinside
theshell(thesignofξ changesintheshell).Red:Noinertialmodesmayexistintheshell(ξ < 0everywhere).Orange:Themodesinthe
regionbetweenthetwoorangelinesfeatureacorotationlayer(ω˜ =0)insidetheshell.Themeaningoftheblacksolid,dottedanddashedlinesis
p
explainedinparagraph3.2.2.
1. modes for which at least one turning surface exists within
theshell,referredtoasDTmodesfollowingtheterminology
introducedinBaruteau&Rieutord(2013)(Dfordifferential ωp =−m±2, (16)
rotation,Tforturningsurface),
andisshownbytheverticaldottedlinesinFig.1,
2. modesthatpropagateinthewholeshell,whichisentirelyhy-
perbolicbecausethereisnoturningsurfacewithinit,named – ξ(θ=π/2)=0,whentheturningsurfacereachestheequator,
Dmodes. whichyields
The occurrence of D and DT modes is depicted in Fig. 1 ωp =(−m±2)(1+ε), (17)
fortheaxisymmetriccase(m = 0)andafewnon-axisymmetric
andisshownbythesolidblacklinesinFig.1.
cases(m>0).WhiteareasrepresentDTmodeswhichhaveξ <
0 at least once within the shell, while blue regions represent D We point out that there may exist two turning surfaces in a
modes for which ξ > 0 everywhere within the shell. The red meridional plane for a given set of parameters. For instance, in
areasillustratethecasewhereξ <0everywhereintheshell,for the bottom-left panel of Fig. 1 (where m = 2), the transition
whichtheshellhostsnoinertialmodes.Thecurvesthatseparate fromthecentralblueregiontothewhiteregionaroundω = 0
p
DandDTmodessatisfy: showsthattwoturningsurfacesoccuratθ=π/4beforesplitting
towardsthepoleandtheequator.Ourinvestigationsshowedthat
– ξ(θ = 0) = 0,whichcorrespondstothecasewheretheturn- thisonlyoccursform = ±2aroundω = 0,whichwechecked
p
ingsurfaceisontherotationaxis,andwhichcanberecastas bysolvingnumericallytheequationξ(θ=π/4)=0foreverym,
Articlenumber,page5of17
A&Aproofs:manuscriptno.GBMR2015-p
whosesolutionsaredepictedbytheblackdashedlinesinFig.1. (23)
Anexampleisgiveninthefollowingsubsection(seethebottom-
rightpanelofFig.5). AssumingthefluidisstableagainsttheGSFinstability,wede-
We point out that for D modes — for which ∀θ, ξ(θ) > 0 fine
— we have |ω˜p(θ)| < 2Ω(θ), which means that for D modes, (cid:34)(cid:32) εsin2θcos2θ(cid:33) k εcosθsin3θ(cid:35)1/2
the standard criterion for propagation of inertial waves is met B= 1+ + s , (24)
everywherelocally. 1+εsin2θ kz 1+εsin2θ
sothatEq.(23)becomes
3.2.3. Criticallayers
|k |
For non-axisymmetric modes (m (cid:44) 0), ω˜p may vanish in the ω˜p =±2Ω(θ)B(cid:107)kz(cid:107). (25)
shell, which corresponds to so-called corotation resonances or
critical layers (Baruteau & Rieutord 2013). Since ω˜ is a func- Forsolid-bodyrotation,B = 1andEq.(25)istheusualdisper-
p
tionofθonly,acriticallayeristheintersectionofaconeofaxis sionrelationforinertialwaves.Consequently,thephasevelocity
zwiththesphericalshell.AscanbeseenfromEq.(13),pathsof intheframerotatingwiththefluidis
characteristicsmaybecomelocallyverticalatcorotationlayers.
k
Theirlocationisgivenby: v =±2Ω(θ)B z k, (26)
p (cid:107)k(cid:107)3
(cid:114)
(cid:18) ω (cid:19)
sinθ= ε−1 −1− p (18) whilethegroupvelocityisgivenby
m
and they exist only when ωp ∈ [−m,−m(1+ε)] (if mε < 0) or v =±2Ω(θ) ks (−k e +k e )
ω ∈ [−m(1+ε),−m] (if mε > 0). Critical layers exist for the g (cid:107)k(cid:107)3 z s s z
p
D and DT modes located between the two orange lines in the (cid:34) 2Ω(θ)εcosθsin3θ (cid:35)
panelsofFig.1. × B− (cid:107)k(cid:107)2B−1 . (27)
k k
s z
FromEq.(25),weseethatacriticallayermayformintheshell
3.3. Dispersionrelation,phaseandgroupvelocities
(ω˜ =0)inthefollowingformalcases:
p
Weexpandinthisparagraphthestudyofthepropagationofin-
ertial waves in the inviscid limit by considering the wave dis- – |ks| → ∞atfinitekz,sothatvp = 0andvg = 0atcorotation
persionrelation.Thelatterisobtainedbyassumingsolutionsto :inertialwavescannotcrosscriticallayers.
Eq.(7)intheform p∝exp[i(kss+kzz)]andadoptingtheshort- – |(cid:12)kz| →(cid:12) 0 at finite ks, so that vp = 0, vg · es = 0 and
wavelengthapproximation: (cid:12)(cid:12)vg·ez(cid:12)(cid:12) → ∞, indicating that inertial waves may propagate
acrossacriticallayerwithlocallyverticalpathsofcharacter-
(cid:34) (cid:35)
ω˜2 = kz2 A − ksA , (19) istics. (cid:12) (cid:12)
p (cid:107)k(cid:107)2 s kz z – B → 0(cid:12), whic(cid:12)h implies again that vp = 0 while (cid:12)(cid:12)vg·es(cid:12)(cid:12) →
(cid:113) ∞ and (cid:12)(cid:12)vg·ez(cid:12)(cid:12) → ∞, which means that in this case as well
where (cid:107)k(cid:107) = k2+k2. Assuming the bracket-term is positive, inertialwavesmaypropagateacrossthecorotationresonance
s z
which is equivalent to assuming that the background rotation withafiniteslopeforthepathsofcharacteristics,sinceB=0
profileisdynamicallystabletoinertialwaves(seethediscussion implies
inSec.2.1),wecandefine: (cid:16) (cid:17)
dz k 1+εsin2θ 1+cos2θ
(cid:34) k (cid:35)1/2 =− s =− . (28)
B˜ = A − sA (20) ds kz εcosθsin3θ
s k z
z
Thisresultshowsthatwhenitcomestocriticallayers,con-
sothatthewavedispersionbecomesω˜ =±B˜|k |/(cid:107)k(cid:107).Thephase ical and shellular rotation profiles behave similarly — in the
p z
and group velocities in a frame rotating with the fluid are then sensethatthethreeaforementionedcasesarepossible—while
givenby: cylindrical rotation stands out as the case where characteristics
of inertial waves can formally never cross critical layers (see
k
v =±B˜ z k (21) Baruteau&Rieutord2013).Itisimportanttokeepinmindthat
p (cid:107)k(cid:107)3 this analysis only reflects the hyperbolic part of the problem
while the solutions to the full viscous problem (Sect. 4) may
and behaveabitdifferentlythanthecharacteristicsofthepurelyin-
(cid:34) (cid:35)
k A (cid:107)k(cid:107)2 viscidproblem(asshownforinstancebyFig.13ofBaruteau&
vg =±(cid:107)k(cid:107)s3 (−kzes+ksez) B˜+ 2z kskzB˜−1 , (22) Rieutord2013).
whichcorrespondstoEqs.(A9)and(A10)inBaruteau&Rieu-
tord(2013).Notethatk·v =0asforpureinertialwavesprop- 3.4. Dynamicsofthecharacteristicsofinertialwaves:
g attractorcycles,focusingpointsandLyapunov
agatinginasolid-bodyrotatingfluid.
exponents
Applying these formulae to our conical rotation profile
Ω(θ)=1+εsin2θ,thedispersionrelationisgivenby Inthecaseofsolid-bodyrotation,thepathsofcharacteristicsina
meridionalplanefollowstraightlines,whichformaconstantan-
(cid:34)(cid:32) (cid:33) (cid:35)
k2 εsin2θcos2θ k εcosθsin3θ glewiththerotationaxis—i.e.thez-axis—,thatonlydepends
ω˜2p =4Ω2(θ)(cid:107)kz(cid:107)2 1+ 1+εsin2θ + kzs 1+εsin2θ . on the (uniform) Doppler-shifted wavefrequency ω˜p = ωp +m
Articlenumber,page6of17
Gueneletal.:Tidalinertialwavesinthedifferentiallyrotatingconvectiveenvelopesoflow-massstars-I
1.0
ε= 0.00
ω=-3.7500
p
m= 2
η= 0.71
0.8
0.6
z
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
s
Fig.2.Pathsofcharacteristicsforsolid-bodyrotation(ε=0).Left:Pathsofcharacteristicsdonotconvergetowardsanyperiodicorbit(m=0,
ω =1.305,η=0.35andsolidbody-rotationε=0).Right:Exampleofanattractorcycleofthepathofcharacteristicsform=2,ω =−3.75,
p p
solid-bodyrotation(ε=0)andthesolaraspectratioη=0.71.
(see the dispersion relation given in Eq. 25 for ε = 0). For a m=0 η=0.100 ε=0.300
givenaspectratiooftheshellη,certainvaluesofthefrequency 0.0
ω cause the paths of characteristics to converge towards peri- -0.5
p
Λ -1.0
odicorbitscalled“attractors”,whileforsomeotherfrequencies,
-1.5
paths of characteristics nearly occupy the whole shell with no -2.0
obviousperiodicpattern.Thesetwodifferentbehaviorsareillus- 0.5 1.0 1.5 2.0
trated in Fig. 2. The existence of one or several attractor(s) for ω
p
a given set of parameters may be investigated by studying the
exponential convergence rate of two close characteristics along m=0 η=0.350 ε=0.300
multiplereflexions.ThisisquantifiedbytheLyapunovexponent 0.0
Λ(formoredetails,seeRieutordetal.2001;Baruteau&Rieu- -0.5
Λ -1.0
tord2013).TheLyapunovexponentcanbedefinedas -1.5
-2.0
Λ= lim 1 (cid:88)N ln(cid:12)(cid:12)(cid:12)(cid:12)dxk+1(cid:12)(cid:12)(cid:12)(cid:12), (29) 0.5 1.0ω 1.5 2.0
N→∞ N k=1 (cid:12) dxk (cid:12) p
m=0 η=0.710 ε=0.300
where dx is the separation between two characteristics after k
k 0.0
reflections.Inourcase,thisquantitycanbeevaluatedusingei- -0.5
therthereflexionsontheequator(ds ),ontherotationaxis(dz ) Λ -1.0
k k
or on the inner (resp. outer) boundary of the shell dθin (resp. -1.5
k -2.0
dθout). Λ ≈ 0 corresponds to the case of “space-filling” paths
k 0.5 1.0 1.5 2.0
of characteristics, whereas Λ < 0 identifies the existence of an
ω
attractor(seeleftandrightpanelofFig.2respectively). p
Inthecaseofsolid-bodyrotation,thelocationswherechar- m=0 η=0.900 ε=0.300
acteristics bounce off the shell boundaries, the equator and the 0.0
rotationaxiscanbedeterminedanalytically,thusallowingfora -0.5
semi-analyticcalculationofΛ(Rieutordetal.2001).However, Λ -1.0
-1.5
differentialrotationrequiresthenumericalintegrationofthepath
-2.0
ofcharacteristics.Forthatpurpose,wechooseastartingpointin
0.5 1.0 1.5 2.0
theshellandwefollowthepropagationbynumericallyintegrat- ω
p
ingEq.(13).Sincecharacteristicscannotpropagateintheelliptic
domain,itisnecessarytoimposereflexionatturningsurfaces. Fig. 3. Numerical spectrum of the Lyapunov exponent for m = 0,
Forillustrationpurposes,weevaluatednumericallytheLya- ε = 0.30 and ω ∈ [0.2,2.0]. From the top to bottom panels, η =
punovexponentsasafunctionoftheeigenfrequencyform = 0 {0.10,0.35,0.71,0p.90}.Theredopensquarescorrespondtothemodes
fordifferentvaluesofηandε,moreprecisely: showninFigs.6to8.
– η={0.10,0.35,0.71,0.90}alongwithafixedsolar-likevalue
ofthedifferentialrotationparameterε=0.30forFig.3,
– ε = {−0.50,−0.25,0.30,0.60} along with a fixed solar-like Weused800frequenciesperunitintervaloffrequenciesand
valueoftheconvectiveshellaspectratioη=0.71forFig.4. tenpairsofcharacteristicsforeachdatapointbeforeaveraging
Articlenumber,page7of17
A&Aproofs:manuscriptno.GBMR2015-p
m=0 η=0.710 ε=-0.500 entialrotationparameterε.Weseethattheoccurrenceofshort-
0.0 period attractors is not sensitive to ε but some features of the
-0.5 Lyapunov spectra can be tracked as they are progressively al-
Λ -1.0
teredandshiftedtohigherfrequencieswithincreasingε—for
-1.5
-2.0 instance, the deep double-peak starting from ωp ∈ [0.3,0.5] in
0.5 1.0 1.5 2.0 thetop-leftpanelandshiftedtoωp ∈ [1.25,1.5]inthebottom-
ω rightpanel.
p
m=0 η=0.710 ε=-0.250 Whenthefluidisdifferentiallyrotatingwithangularvelocity
Ω(θ),pathsofcharacteristicsmaystillconvergetowardsattrac-
0.0
-0.5 tors — just like in the case of solid-body rotation — but with
Λ -1.0 curved characteristics as illustrated in the top-left panel of Fig.
-1.5
5. They also depend on the azimuthal wavenumber m through
-2.0
thenon-uniformDoppler-shiftedfrequencyω˜ .Wecanalsofind
0.5 1.0 1.5 2.0 p
attractorsofcharacteristicswhenaturningsurfaceexistsinthe
ω
p shell(top-rightpanel).Sometimes,characteristicstendtofocus
m=0 η=0.710 ε=0.300 at the intersection of a turning surface with the boundaries of
0.0 the shell as illustrated by the bottom-left plot of Fig. 5, show-
-0.5 ingabehaviorthatissimilartotheonefoundbyDintransetal.
Λ -1.0 (1999)forgravito-inertialwaves.Finally,weemphasize thatin
-1.5
rare cases, two turning surfaces can exist within the shell, af-
-2.0
fectingthewaves’propagationdomainaccordingly:itiseither
0.5 1.0 1.5 2.0
restrictedtothepolarandequatorialregions(asillustratedinthe
ω
p
bottom-right panel of Fig. 5) or restricted to mid-latitudinal re-
m=0 η=0.710 ε=0.600 gions.
0.0
-0.5
Λ -1.0
-1.5
-2.0
0.5 1.0 1.5 2.0 4. Viscousproblem:shearlayers,comparisonto
ω
p inviscidanalysis,behaviouratcorotation
resonances
Fig. 4. Numerical spectrum of the Lyapunov exponent for m = 0,
η = 0.71 and ω ∈ [0.2,2.0]. From the top to bottom panels, ε =
p
{−0.50,−0.25,0.30,0.60}. The blue solid line marks the frequency of Our study of the propagation properties of inertial waves in a
thetransitionbetweenDandDTmodes.Theredopensquarescorre- differentially rotating inviscid fluid with a conical rotation pro-
spondtothemodesshowninFigs.6to8. file(seesection3)showsthattwofamiliesofinertialmodesof
oscillationexist:Dmodesthatmaypropagateintheentireshell,
andDTmodesthataretrappedinpartoftheshellbecauseofthe
theresults.TheresultsshowninFigs.3and4shouldtherefore
existence of a turning surface. We find that paths of character-
beconsideredratherasaqualitativeindicatorofthepresenceof
isticsdependonmthroughtheDoppler-shiftedwavefrequency,
short-periodattractorcyclesthanaquantitativeonebecauseour whichisnotthecaseforsolid-bodyrotation,butthemaindiffer-
method probably yields substantial uncertainties on the values encebetweenm = 0andm (cid:44) 0modesisthepossibleexistence
of Λ. Note also that the focusing of characteristics towards a
ofcorotationresonancesinthelattercase.
wedgeinthefrequencyrangeofDTmodes(seethebottom-left
panel of Fig. 5 below) prevented us from computing values of InSect.4.1,webrieflypresentthenumericalmethodweused
ΛfortheDTmodes.Asexpected,attractorsofvariousstrengths inordertosolvetheviscousproblemexposedinSect.2.1.Then
existinsmallintervalsoffrequencieswhereΛisstrictlynegative we study in Sect. 4.2 (respectively Sect. 4.3) the general prop-
whereas intervals where Λ ≈ 0 are associated with very long- ertiesofweakly-dampedviscousDmodesthatpropagateinthe
periodattractorsorquasi-periodicorbitsofcharacteristics. wholesphericalshell(resp.DTmodesthatpropagateinpartof
Fig.3illustrateshowtheexistenceofshort-periodattractors theshell)vianumericalsimulations.Weshowinparticularhow
in the frequency range of inertial waves can be affected by the the shear layer structure can be related to the inviscid analysis
size of the inner core. It seems that when the core is small, at- detailed above in Sect. 3. As explained in the introduction of
tractorsofcharacteristicsareveryrarebutstillexistasillustrated this work, we will mostly focus on singular modes for which a
bytheregionaroundω ≈ 1.66inthetop-leftpanelofFig.3. short-periodattractorofcharacteristicsexistssincetheymaybe
p
Wecheckedthatthisattractorexistsregardlessofthesizeofthe related with strong viscous dissipation (Ogilvie 2005). Finally,
innercoreaslongasthelatterissmallenough,andwecomputed we examine m (cid:44) 0 modes with corotation resonances in Sect.
thecorrespondingeigenmode(seeFig.8,andFig.8inBaruteau 4.4.
&Rieutord(2013)).However,theotherpanelsofFig.3indicate
thatmoreandmoreattractorsexistwithincreasingthevalueofη,
duetoincreasinglynumerousreflectionsontheinnercore.The 4.1. Numericalmethod
casewhereη = 0.90showsthatalargenumberofshortperiod
attractorsexistoverthefrequencyrangeofDmodes. ThelinearizeddimensionlesssystemofEqs.(6)andtheassoci-
Fig. 4 shows the evolution of the Lyapunov exponent spec- ated stress-free boundary conditions are solved using a unique
trum for a given core size but for various values of the differ- decomposition of the unknown velocity field u onto vectorial
Articlenumber,page8of17
Gueneletal.:Tidalinertialwavesinthedifferentiallyrotatingconvectiveenvelopesoflow-massstars-I
1.0 1.0
ε= 0.30 ε= -0.30
ω=-3.7500 ω= 1.6800
p p
m= 2 m= 0
η= 0.71 η= 0.71
0.8 0.8
0.6 0.6
z z
0.4 0.4
0.2 0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
s s
1.0 11..00
ε= 0.50 εε== --00..9900
ωp= 2.4000 ωωpp==--00..11550000
m= 0 mm== 22
η= 0.71 ηη== 00..7711
0.8 00..88
0.6 00..66
z zz
0.4 00..44
0.2 00..22
0.0 00..00
0.0 0.2 0.4 0.6 0.8 1.0 00..00 00..22 00..44 00..66 00..88 11..00
s ss
Fig.5.Upperleft:ExampleofattractorcycleforaDmodewithm = 2,ω = −3.75inasolar-likeconvectiveenvelope(ε = 0.3,η = 0.71).
p
Upperright:SameforaDTmodewithm=0,frequencyω =1.68andanti-solarconicalrotation(ε=−0.3),thewhitedashedlineshowsthe
p
turningsurface.Bottomleft:Illustrationofthefocusingofthepathsofcharacteristicsattheintersectionoftheturningsurface(dashedline)and
theouterboundaryoftheshell(m=0,ω =2.4andε=0.5).Bottomright:Form=2,ω =−0.126andε=−0.90,twoturningsurfacesexist
p p
withintheshell,allowingthepropagationofcharacteristicsneartheequatororthepolesbutnotinbetween.
sphericalharmonics(Rieutord1987): 2. the right-hand side of the fourth-order differential equation
satisfied by ul is a linear combination of wl±3, dwl±3/dx,
(cid:88)∞ (cid:88)l (cid:110) wl±1, dwl±1/dmx, ul±2, dul±2/dx and d2ul±2/dmx2 (themse last
u(x,θ,ϕ)= ulm(x)Rml (θ,ϕ)+vlm(x)Sml (θ,ϕ) thmreetermmsvanishmform=m0). m
l=0 m=−l
(cid:111)
+wl (x)Tm(θ,ϕ) (30) Thefactthattheradialfunctionsul ,vl andwl withdifferent
m l m m m
with Rm = Ym(θ,ϕ)e , Sm = ∇ Ym and Tm = ∇ ×Rm where azimuthal wavenumbers m are notcoupled is aconsequence of
Ym(θ,ϕl)isthel usualsrphelricalhaHrmlonicofldegreeHlandlorderm theaxisymmetryofthebackgroundflow.Thecouplingbetween
l differentlatitudinaldegrees(froml−3tol+3)resultsfromthe
normalizedontheunitsphere,and∇ =e ∂ +e (sinθ)−1∂ is
H θ θ ϕ ϕ choiceofourspecificconicalrotationprofilethroughtheCorio-
thehorizontalgradient.
lisaccelerationterms.
TheprojectionofthecontinuityequationonYm givesvl as
l m In order to solve these equations numerically, they are dis-
a simple function of ul and dul /dx. The momentum equation
m m cretizedintheradialdirectionontheGauss-Lobattocollocation
isthenprojectedonRm andTm,whichyieldstwolongordinary
l l nodesassociatedwithChebyshevpolynomials.Eachsetofequa-
differentialequationsinvolvingonlytheradialfunctionsul and
m tions is truncated at order L for the spherical harmonics basis,
wl (sincevl canbeeliminated):
m m and at order Nr for the Chebyshev basis. This yields a finite
1. therighthandsideofthesecond-orderequationsatisfiedby eigenvalue problem of order N × L with a block banded ma-
r
wl is a linear combination of ul±3, dul±3/dx, ul±1, dul±1/dx trix composed of (up to) seven block bands. Then we use the
m m m m m
andwl±2(thistermvanishesform=0); linearsolverbasedontheincompleteArnoldi-Chebyshevalgo-
m
Articlenumber,page9of17
A&Aproofs:manuscriptno.GBMR2015-p
rithm(seedetailsinValdettaroetal.2007)tocomputepairsof C as a function of the spherical harmonic degree : for a given
l
eigenvalues(equaltoiω )andeigenvectors(valuesoful ,vl and l, we take the maximum value among all the Chebyshev coef-
p m m
wl oneachpointoftheradialgrid),givenaninitialvalueguess ficients.Thereforewearecertainthatnumericalconvergenceis
m
foriω .SincethevaluesofN andLrequiredtoachievespectral achievedforthismode.
p r
convergencevaryalotfromonemodetoanother,especiallyfor Finally,theaxisymmetricmodeoffrequencyω ≈1.66pre-
p
theverydemandingsmallvaluesof E,somefigureswillbeac- sentedintheleftpanelofFig.8displaysashearlayerthatfol-
companiedbythespectralcontentofthevelocityfield.Inallthe lowsanattractorofcharacteristicsthatexistsforarbitrarilysmall
casespresentedinthisworkweassumedsymmetrywithrespect valuesoftheaspectratioη.Thismeansthatinconicaldifferen-
totheequatorialplanewhichexplainsthatourresultsareshown tialrotation,attractorsofcharacteristicsmayexistindependently
onlyforpositivevaluesofz,butanti-symmetryisalsopossible. oftheexistenceofaninnercore,whichwasalsofoundforcylin-
drical and shellular differential rotation profiles by Baruteau &
Rieutord (2013). We checked that this mode does exist for ar-
4.2. Axisymmetricandnon-axisymmetricDmodeswithno bitrarily small cores, which contrasts with the fact that inertial
corotationlayer modesinafullsphereareregularinthecaseofsolid-bodyrota-
tion(Greenspan1968;Zhangetal.2001).Ourresultshowsthat
Wepresentinthissectionournumericalresultsforafewrepre-
this is probably no longer the case when differential rotation is
sentativeDmodeswithabackgroundconicalrotationprofileand
taken into account. The case where characteristics do not con-
Ekman numbers between 10−7 and 10−8, and we compare the
vergetowardsanylimitcycleisshownintheright-panelofFig.
structural properties of the shear layers with the inviscid anal-
8,whichdisplaysamodeoffrequencyω ≈1.23withη=0.71
ysis detailed in Sect. 3. Note that in the rest of this paper, we p
and ε = −0.25. This set of parameters correspond to the open
alwaysshowthekineticenergydistributionofthecomputedve-
square in Fig. 4 for which Λ ≈ 0. As expected, the shear layer
locity fields, but that the viscous dissipation looks qualitatively
patternsvisibleherefollowthepropagationofcharacteristicsso
verysimilar(e.g.Fig.13inBaruteau&Rieutord2013).
that the kinetic energy is more smoothly distributed than in the
Fig. 6 displays the result of one of our numerical calcula- previousattractorcases.
tionsforanaxisymmetric(m = 0)Dmodewitheigenfrequency
ThefourmodesshowninFigs.6to8allcorrespondtoone
ω ≈ 0.91 obtained in a spherical shell of solar aspect ratio
p oftheopensquaresinFigs.3and4.Thishighlightstheinterest
η = 0.71withsolar-likeconicaldifferentialrotation(ε = 0.30).
of the inviscid approach undertaken in Sect. 3 since the kinetic
The left panel displays the spatial distribution of the mode’s
energyofagivenmodeisindeedmorelocalizedalongthinshear
kinetic energy in a meridional quarter-plane. The amplitude of layers when Λ < 0 and more uniformly-distributed in the shell
the mode is maximum near the rotation axis which is a feature when Λ ≈ 0. As a conclusion, the inviscid analysis allows us
sharedbyinertialmodesinauniformlyrotatingfluidshell.This tounderstandhowdifferentialrotationaffectsinertialmodes,al-
has been demonstrated in detail in the appendix A of Rieutord
though we restricted ourselves to D modes that can propagate
& Valdettaro (1997) who showed that the kinetic energy along
inthewholesphericalshell.WeseethatDmodesbehavequite
characteristictrajectoriesgrowsass−1/2whens→0.Thestruc-
similarly to inertial modes for solid-body rotation. In the next
ture of this specific mode mostly consists of a shear layer fol- section we turn to DT modes, which are specific to differential
lowingashort-periodattractorformedby(slightly)curvedlines,
rotation.
asexpectedinthecaseofdifferentialrotation(seeSect.3).This
attractor is overplotted with a thick white curve, which is the
predictionforthepropagationofcharacteristicsundertheshort- 4.3. Axisymmetricandnon-axisymmetricDTmodes
wavelengthapproximation.Thepatternsformedbytheattractor
and by the shear layers of the viscous mode are in very good In this section, we focus on modes for which a turning surface
agreement.ThemodeshowninFig.6wasextractedfromase- exists in the shell. The frequency range in which these modes
quence of calculations in which we followed a particular mode may exist is indicated by the white areas in Fig. 1. However,
whileprogressivelydecreasingtheEkmannumberfrom10−6 to notallsetsofparametersfallinginthesewhiteareaseffectively
10−9. The damping rate of this mode is displayed as a function correspond to an eigenmode. The inviscid analysis exposed in
of E in the right-panel of Fig. 6 where the dashed line clearly Sect. 3 reveals that characteristics cannot propagate in the el-
showsthatitisproportionalto E1/3,whichisthescalingthatis liptic domain of the shell, i.e. where ξ < 0 or equivalently
(cid:12) (cid:12)
expectedintheasymptoticregimeforsolid-bodyrotation(Rieu- (cid:12)(cid:12)ω˜p(θ)(cid:12)(cid:12) > 2ω(θ).Asaconsequence—forsufficientlysmallEk-
tord&Valdettaro1997),meaningthatthiskindofDmodeisnot mannumbers—DTmodesareexpectedtobetrappedinthehy-
deeplyaffectedbydifferentialrotation. perbolic domain out of which characteristics cannot propagate,
apropertywhichhasalwaysbeenverifiedbyournumericalcal-
We show in Fig. 7 a qualitatively similar axisymmetric D
mode of eigenfrequency ω ≈ 1.35 that was obtained with the culations.Wealsopresentresultsincaseswherecharacteristics
p
same aspect ratio but slightly lower Ekman number E = 10−8 converge towards the intersection of a turning surface with the
andanti-solardifferentialrotationε = −0.25.Thistimetheam- inneroroutershellboundaries(seeFig.5).
plitude of the mode is still maximum at the rotation axis but is
also quite large near the inner critical latitude (where the char-
4.3.1. ExistenceofDTmodes
acteristicsaretangenttotheinnercore),whichisreminiscentof
the solid-body rotation case where shear layers are sometimes In order to find DT modes, as a first step we checked the dis-
emittedatthecriticallatitude.TherightpanelofFig.7depicts tribution of the eigenvalues of our linear problem for a given
the spectral content of u and w for this mode : the top panel setofparameters,usingaQZfactorisationmethod.Thismethod
showsthemaximumChebyshevcoefficientsC asafunctionof rapidlybecomesverycostlyasthespectralresolution(andthus
k
theChebyshevorderk,takingthemaximumvalueamongallthe thedimensionsofthelinearproblem)increasesandthereforecan
sphericalharmonicscoefficientsforagivenk;similarly,thebot- only be performed at a low spectral resolution, effectively lim-
tompanelshowsthemaximumsphericalharmonicscoefficients itingustomoderatevaluesof E.Despitethislimitation,trends
Articlenumber,page10of17