How strong are the Rossby vortices?

1.2 MB·English

by  H. Meheut

Mon.Not.R.Astron.Soc.000,1–7(2012) Printed11December 2013 (MNLATEXstylefilev2.2)
How strong are the Rossby vortices?
H. Meheut1,2⋆, R.V.E. Lovelace3 and D. Lai3
1Physikalisches Institut & Centerfor Space and Habitability, Universita¨tBern, 3012 Bern, Switzerland
2CEA, Irfu, SAp, Centre de Saclay, F-91191 Gif-sur-Yvette,France
3 3Department of Astronomy, Cornell University,Ithaca, NY 14853, USA
1
0
2 11December 2013

n
a
J ABSTRACT
The Rossby wave instability, associated with density bumps in differentially rotating
4
discs, may arise in several different astrophysical contexts, such as galactic or proto-

] planetary discs. While the linear phase of the instability has been well studied, the
R nonlinear evolutionand especially the saturationphase remain poorly understood. In
S this paper,we test the non-linearsaturationmechanismanalogousto thatderivedfor
. wave-particle interaction in plasma physics. To this end we perform global numerical
h
simulations of the evolution of the instability in a two-dimensional disc. We confirm
p
- the physical mechanism for the instability saturation and show that the maximum
o amplitudeofvorticitycanbe estimatedastwicethelineargrowthrateofthe instabil-
r ity.Weprovideanempiricalfittingformulaforthisgrowthrateforvariousparameters
t
s of the density bump. We also investigate the effects of the azimuthal mode number
a
of the instability and the energy leakage in the spiral density waves. Finally, we show
[
that our results can be extrapolated to 3D discs.

1
v Key words: accretion, accretion discs - hydrodynamics - instabilities.
9
8
6
0
1 INTRODUCTION can be concentrated and the size of the resulting planetes-
.
1 imals (Meheut et al. 2012). The Rossby vortices can also
0 Understanding the evolution of accretion discs is impor-
influence planet migration in the disc in the latter stage of
3 tant for several astrophysical applications. The Rossby
planet formation (Koller et al. 2003), and the amplitude of
1 wave instability (RWI) (Lovelace et al. 1999) and Rossby
migration is determined by the strength of the vortices. In
: vortices it form have been discussed in a large va-
v theRWImodelofQuasi-PeriodicOscillations(QPO)inmi-
riety of disc systems, from large-scale galactic discs
i croquasars,thesaturationamplitudeoftheRWIisofimpor-
X (Lovelace & Hohlfeld1978;Sellwood & Kahn1991)andthe
tancefortheunderstandingoftheobservedQPOamplitude.
r Galactic centre (Tagger & Melia 2006), to discs in mi-
Thesaturation mechanismoftheRWIwasalreadydis-
a croquasars (Lovelace et al. 2009) and protoplanetary discs
cussedinLi et al.(2001),butnocriterionforsaturationwas
(Varni`ere& Tagger 2006; Meheut et al. 2012). The latter
given. In this paper, we interpret the growth of the insta-
are of particular importance since Rossby vortices may ac-
bility in term of particle-wave interaction and use classical
celerate planetesimal formation. The RWI operates around
plasma physics results in this context, which give the non-
thedensitybumpsorvortensitybumps(seeSection2)inthe
linear saturation condition as proposed by Lovelace et al.
disc. In the linear theory, the growth rate of the instability
(2009). In section 2, we present the physical interpretation
can be computed as an eigenvalue problem. This was done
of thegrowth of the instability and the model for thesatu-
by Li et al. (2000) for 2D discs, by Umurhan (2010) in the
rationmechanism.Thenumericalsimulationsarepresented
shallow water approximation, and by Meheut et al. (2012)
and compared with the model in section 3. We then sum-
and Lin (2012) for 3D vertically stratified discs. After the
marise and conclude in thelast section.
linear phase, the instability saturates and the linear theory
breaks down. The saturation amplitude of the instability is
of importance in the several applications. For example, in
protoplanetary discs, the vortices formed by the RWI can
2 THE RWI
concentratethesolid particles andhenceaccelerate thefor-
mation of planetesimals. The amplitude of the vortices is
2.1 Linear growth
a key parameter that determines the amount of solids that
The RWI can be seen as an equivalent of the Kelvin-
Helmholtzinstabilityindifferentiallyrotatingdiscs,andhas
⋆ E-mail:[email protected] a similar instability criterion: an extremum in the quantity
(cid:13)c 2012RAS
2 H. Meheut, R.V.E. Lovelace, D. Lai
L
(i.e., increasing wave amplitude tends to to remove energy
fromthefluid).Ontheotherhand,thewaveinthenegative
gradientofLhasω/m>Ωandcarriespositiveenergy.The
growthoftheRWIisduetotheinteractionbetweenthetwo
Rossby waves with respectively positive and negative ener-
Rossby Rossby gies.Thetotalenergyisconservedwhilethewaveamplitude
Spiral density wave wave Spiral density increases.Themechanismforthegrowthoftheinstabilityis
wave E<0 E>0 wave thenlocalised inthecorotation region anddoesnotdepend
E<0 E>0
on the boundaryconditions.
As explained by Tagger (2001), in accretion discs the
differentialrotation couplesRossbywavestodensitywaves.
These density waves can only propagates outside the two
rILR rc rOLR r (innerand outer) Lindblad resonances radii rLR defined as
Figure 1.SchematicviewoftheRWIwiththetwopropagating ω−mΩ(rLR)=±κ(rLR). (5)
regions forthe Rossbywaves anddensity waves, and inbetween
Between the two propagation zones the wave is evanescent
theevanescentregions.Atthecorotationradiusrc thewavepat-
andtunnellingispossibleasplottedinFig.1.Asdetailedin
tern frequency matches the orbital frequency (Eq. 4). The inner
andouterLindbladradii,rILR andrOLR,aredefinedbyEq.(5) Tsang & Lai (2008) and Lai & Tsang (2009), these propa-
gationregionsclearlyappearwhenexpressingthelinearized
.
perturbationequationsintheformofSchr¨odingerequation,
with theeffective potential given by
L related to the vorticity of the equilibrium flow. In a non-
magnetised thin disc, this quantitycan bewritten as 2mΩ d ΩΣ m2 κ2−ω˜2
V = ln + + . (6)
eff rω˜ d (cid:16) κ2−ω˜2(cid:17) r2 c2
L= ΣΩ(pΣ−γ)2/γ = Σ (pΣ−γ)2/γ, (1) r s
κ2 2(∇~ ×~v) ThewavesarepropagatingintheregionwhereVeff(r)isneg-
z
ativeandevanescentwhereV (r)ispositive.Thiseffective
eff
where Σ is the surface density, ~v the velocity of the fluid, potential is plotted in Fig. 2 for the specific configuration
γ the adiabatic index, Ω the rotation frequency, and κ2 = considered below.
4Ω2+2rΩΩ′ thesquaredepicyclicfrequency(sothatκ2/2Ω
isthevorticity).Heretheprimedenotesaradial derivative.
For barotropic discs considered in this paper, the quantity 2.2 Non-linear saturation
L reduces to
To estimate the amplitude at which the linear calcula-
ΣΩ Σ
L= = , (2) tion stops to be valid, we use a standard plasma crite-
κ2 2(∇~ ×~v)z rion which gives, for instance, the breakdown of the Lan-
corresponding to1/2 of theinversevortensity1. dau damping theory due to particle trapping (O’Neil 1965;
Krall & Trivelpiece 1973). We consider a fluid particle in a
In a discwith such an extremum,twostanding Rossby
Rossby wave with the vorticity amplitude ω exponentially
wavescancoexist,oneoneachsideoftheextremum.Were- v
growing with time duringthelinear stage,
callherethatRossbywavesarevorticitywavespropagating
on the gradient of vortensity, and their dispersion relation
|ω |∝exp(γt). (7)
v
is given by
We here use the convention to express time in units of
ω˜ =−c2s(kr2+2mκ22c/2sr2)+κ2mrΣL′ (3) Ω−01.The estimation of therotation time of a fluid particule
where ω is the wave frequency, ω˜ ≡ ω −mΩ is the wave in a vortex is related to the vorticity. For instance for a
frequencyintherotatingframe,mandk itsazimuthaland circular vortex with constant vorticity, it can be estimated
r
radial wave numbers, and c is the sound speed. The wave as
s
corotation radius r is defined by:
c τ ∼4π/|ω |. (8)
T v
ω˜(r )≡ω−mΩ(r )=0. (4)
c c HoweverinthecaseofthevorticesformedbytheRWI,
Thus,withthenegativegradientofthediscrotationvelocity, the vortex doesn’t have an analytical expression and the
we have ω˜ < 0 and ω˜ > 0. From the dispersion turnover timescale can not be computed analytically. From
r>rc r<rc
relation, one can see that Rossby waves propagate only in thesimulations presented in thenext section, we estimated
theregionwhereL′ >0ifr<r andwhereL′<0ifr>r .
c c τ ∼2/|ω |. (9)
T v
We consider here the case of a maximum of L as plot-
ted in Fig. 1. The wave located in the positive gradient of Thiscircularmotionaroundthevortexcentreisamajor
L (r < r ) has the pattern frequency smaller than the disc perturbationtothecircularorbitofthefluidparticlearound
c
rotation frequency (ω/m < Ω) and carries negative energy thedisccentreanditmodifiestheradialstructureofthedisc
in the region of the vortices on a time scale τ . The linear
T
calculation of the growth of the Rossby wave instability is
1 Vortensityisdefinedastheratioofverticalvorticitytosurface determinedbythisradialstructureofthediscandespecially
density:(∇~×~v)z bytheradialstructureofL.Therefore,thislinearapproach
Σ
(cid:13)c 2012RAS,MNRAS000,1–7
How strong are the Rossby vortices? 3
isonlyvalidwhenthegrowthtimescale(1/γ)issmallerthat m=2
1000
τ . This occurs when
T
500
|ω |
v ∼γ. (10) Veff 0
2
-500
In other words, the exponential growth of the wave am-
-1000
plitude necessarily ends when the fluid particle motion in
-1500
the vicinity of resonant radius r differs appreciably from
c
-2000
the strictly azimuthal motion in the equilibrium. Note that
thesaturationcriterion(10)isonlyorder-of-magnitude,and -2500
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
similar criterion was proposed by Lovelace et al. (2009) in r/r
0
a different context. Below we perform non-linear numerical m=5
1000
simulations to calibrate thiscriterion.
500
Veff 0
-500
3 NON-LINEAR SIMULATIONS
-1000
3.1 Methods -1500
-2000
Wehaveperformedtwodimensional(2D)non-linearsimula-
-2500
tionsofadifferentiallyrotatingdiscwithaconstantsurface 0,2 0,4 0,6 0,8 1 1,2 1, 4 1,6 1,8 2
r/r
densityexceptforaGaussianbump.Thisover-densitygives 0
an extremumof L in which theinstability can grow. Figure2.EffectivepotentialforadensityprofilegivenbyEq.13
We work in cylindrical coordinates (r,ϕ) with the 2D withχ=0.2for m=2(upper panel) and m=5 (lower panel).
Euler equations Waves canpropagateonlyintheregionswhereVeff(r)<0.
∂ Σ+∇~ ·(ρ~v)=0, (11)
t
a resolution of (1024,256) was needed for the highest az-
∂ (Σ~v)+∇~ ·(~vΣ~v)+∇~p=−Σ∇~Φ , (12)
t G imuthal modes number m = 5 and the simulations with
whereΣisthesurfacedensityofthefluid,~vitsvelocity,and white noise perturbations. There is a null radial velocity at
p its pressure. ΦG =−GM∗/r is thegravitational potential theinnerand outer boundaryof thegrid.
of the central object with G the gravitational constant and This numerical configuration is very similar to the one
M∗ the mass of the star. We consider a globally isothermal of Meheut et al. (2012) which has proven to accurately de-
flow, i.e. with a radially constant sound speed, and a linear scribe the instability, with a very good agreement with the
relation between pressure and density p=c2Σ. analytical approach in the linear phase. For all these simu-
s
Theinitialsurfacedensity,normalizedtoΣ isgivenby lations, thesquareof theepicyclic frequency,κ2, is positive
0
at all radii and the disc is stable against axisymmetric per-
(r−r )2
Σ/Σ =1+χexp − 0 . (13) turbations.
0 (cid:16) 2σ2 (cid:17)
Ourcanonical parameters are r =1., the amplitude of the
0 3.2 Results
bump χ is chosen in the range [0.15,0.3], and its width is
given by σ/r0 =0.05. Although we will also present results For each of the 20 simulations, the linear growth rate of
for different valuesof σ/r0. Thedisc scale height is fixedto the instability is obtained by fitting the amplitude of the
h=cs/Ω0 =0.1r0,Ω0beingtheKeplerianorbitalfrequency vorticity in the Rossby waves region as a function of time
at r0. The initial azimuthal velocity is chosen to achieve (see Fig. 3). We have previously shown that this method
radial equilibrium. A low amplitude perturbation is added givesgrowthrateverysimilartotheoneobtainedbysolving
to this equilibrium: the linear equations (Meheut et al. 2012). The amplitude
of the waves initially grows exponentially until saturation.
(r−r )2
v =ǫsin(mφ)exp − 0 (14) As explained in section 2.2, we estimate saturation to arise
r (cid:16) 2σ2 (cid:17)
when the waves reach a vorticity of the order of 2γ. This
with ǫ = 10−4 and the azimuthal mode number m ∈ [2,5]. amplitudeis plotted as a dashed line in Fig. 3.
Wehavealsoperformedsimulationswithinitialrandomper- Foreachsimulation,theratioofthemaximumofvortic-
turbations of thesame amplitude. itytotheexpected value,ωmax/2γ, isgiven.Thesaturation
v
We use the Message Passing Interface-Adaptive Mesh amplitude reached in the non-linear numerical simulations
Refinement Versatile Advection Code (MPI-AMRVAC) for m = 4 and m = 5. Similar to the non-linear evolution
(Keppenset al. 2012), with a uniform grid. The numeri- of the Landau damping, we also obtain oscillations in the
calschemeistheTotalVariationDiminishingLax-Friedrich waves amplitude.
scheme (see T´oth 1996) with a third order accurate Koren For low mode numbers, the maximum vorticity is
limiter (Koren 1993) on the primitive variables. We use a slightly below the estimation. This is due to the shape of
uniform cylindrical grid with r/r ∈ [0.2,2.4], and the full thevortices.Indeedthevorticeswith lowmhaveelongated
0
azimuthal direction ϕ∈ [0,2π]. Numerical convergence has shapesascanbeseeninFig.4wherevorticesstreamlinesin
been tested and two different resolutions have been used: the(r,ϕ)planeareplotted.Thesestreamlinesarecomputed
(1024,128) for most of the simulations (m = 1 to 4) but with the perturbed velocity during the linear stage of the
(cid:13)c 2012RAS,MNRAS000,1–7
4 H. Meheut, R.V.E. Lovelace, D. Lai
Figure3.AmplitudeoftheRossbywaves(vorticity)onlogarithmicscaleasafunctionoftime(inunitsofΩ−1).Afitoftheexponential
0
growth(solidline)givesthegrowthrateγ.Thedashedlinecorrespondstothesaturationamplitudeestimatedbythemodel.
instability. To allow for a comparison between different m density waves propagating outside the Lindblad resonances
cases,thetimeslicesfortheseplotsarechosensuchthatthe that were not considered in the local mechanism proposed
perturbations have about the same amplitudes. This shape for growth and saturation in section 2.2. The amplitude of
was to be expected as the width of the Rossby wave prop- the density waves and the energy loss is the highest when
agation region is fixed by the width of the initial density the Lindblad resonances are close to the corotation radius
bump and the length of the vortices is directly related to and the width of the evanescent wave region is small. This
the azimuthal mode number m. Assuming a doubled cir- regions correspond to a positive effective potential V in
eff
culation time for the elongated m = 2 vortices gives the Fig. 2. Since the Lindblad resonances are closer to corota-
correct saturation amplitude, as one can see in Fig. 3 for tion for higher azimuthal mode number, the evanescent re-
the[m=2,χ=0.30] simulation. gionisnarrower.Energytransmissionthroughthisregionis
thencarriedawaybydensitywaves.Thesedensitywavescan
On the other hand, the vorticity maximum is overes-
then become non-linear when they reach high amplitudes.
timated for the highest azimuthal mode number when the
This may explain the lower amplitude at saturation. More-
growth rate is low. See for instance the [m = 5,χ = 0.15]
overthesedensitywavesarealsoresponsiblefortheangular
simulation. We have checked that this is not related to nu-
momentumtransferthroughthediscandasaresultforthe
merical dissipation by doubling the resolution and obtain-
ingnomodification ofωmax.Indeedtheenergyloss respon- evolution of the radial structure of the disc from which the
v linear growth is computed. The distance between the two
sible for the low saturation amplitude may be due to the
(cid:13)c 2012RAS,MNRAS000,1–7
How strong are the Rossby vortices? 5
m=2 m=3 m=4 m=5 m=2 m=5
Figure 4.Streamlines showingtheshape ofthe vortices forthe Figure 5. Vorticity (Ω0) of the flow near corotation for m2χ25
four azimuthal mode numbers considered here. For the lowest and m5χ25. The white lines show the position of the Lindblad
modenumberthevorticesareelongated,forthehighestonesthe resonances.Thewidthoftheevanescent regionbetweenthevor-
vorticesarecircular. ticitywaveandtheLindbladresonanceislargerforlowerm.
propagationregionappearsclearlyonFig.5whereboththe
vorticity waves in the region of corotation and the position
oftheLindbladresonancesarevisible.Onecanalsoidentify
the spiral density waves propagating inwards and outwards
from the Lindblad resonances in the m = 5 plot. To fur-
ther check the importance of the Lindblad resonances, we
haveperformednumericalsimulationswithashallowerden-
sity bump (σ/r =0.4) but with the same azimuthal mode
0
numberm=5.Thedensitybumpbeingshallower, theLin-
blad resonances are situated furtheraway from thevortices
andlessenergyislostinthespiralswaves.Fig.6showsthat
inthiscasethesaturation amplitudeiscorrectlyestimated.
Theamplitudeofsaturation inthesimulationswithan
initial white noise perturbation is more complex. Multiple
modesgrowsimultaneouslyinthelinearphasewithdifferent
growth rates, as can be seen on Fig.7. However we still ob-
Figure6. AmplitudeoftheRossbywaves(vorticity)onlogarith-
tain an exponential growth of thetotal perturbation as the micscaleasafunctionoftime(inunitsofΩ−1)forthesimulation
growth rates of thedifferent modesare similar. Thegrowth 0
withσ/r0=0.4andm=5.Afitoftheexponentialgrowth(solid
rate of the total perturbation (0.21Ω0) is slightly smaller line)givesthegrowthrateγ.Thedashedlinecorrespondstothe
thanthehighestgrowthrateofthedifferentmodes(0.22Ω0 saturationamplitudeestimatedbythemodel.
for m = 4) due to this coexistence of multiple modes with
(cid:13)c 2012RAS,MNRAS000,1–7
6 H. Meheut, R.V.E. Lovelace, D. Lai
Figure7. AmplitudesoftheRossbywaveswithdifferentm’sas Figure 8.Amplitudeof the Rossbywaves (mid-planevorticity)
afunctionoftime.Thesimulationstartsoutwitharandomwhite onlogarithmicscaleasafunctionoftimeforthe3D simulation.
noise,withtheparametersχ=0.3,σ/r0=0.05,cs/(r0Ω0)=0.1. Afitoftheexponentialgrowthgivesthegrowthrate(solidline).
The different m component is obtained by a Fourier analysis in
the azimuthal direction. The grow rates of the dominant modes
arealsogiven. growth rate is only slightly modified (decreased) in a fully
stratifieddisc.Moreovertheverticalvelocityinthevortices
is of low amplitude compared to the radial and azimuthal
similar growth rate. These different modes then interact in
perturbed velocity. As a result, we do not expect the cir-
thenon-linearphase,butonecanseethateq.(10)stillgives
culation time to be significantly modified by this vertical
a good approximation of the saturation amplitude of the
circulation. To confirm the relevance of our model for 3D
instability in thismore realistic case.
stratified discs, we perform a numerical simulation in such
a configuration. We choose m = 4 and χ = 0.25, and the
3.3 Fitting formula volumedensityischosentohaveasurfacedensitysimilarto
the2D case (see Meheut et al. 2012). The resulting growth
In order to give an estimation of the growth rate of the
and saturation of the instability is plotted in Fig. 8. The
instability based on the parameters of the density bump,
predicted maximum amplitude of the instability is correct,
weperformedsimulationswithinitialrandomperturbations
meaning that the model can be extended to 3D discs. This
and with χ ∈ [0.1,0.3] and σ/h ∈ [0.45,0.6]. Fitting sepa-
canthenbealsoappliedtothesaturationphaseobtainedin
ratelythedependenceofthegrowthrateonχandσ/h,and
Meheut et al.(2012)wherethelong-termstabilityofthe3D
then combining, we find that γ is approximately given by
vortices was studied. After the saturation the vortices con-
γ k χ 2/3 tinuetoevolve,theytendtomerge asinan inversecascade
≃ 1 −k , (15)
Ω (σ/h)4/3(cid:16)σ/h 2(cid:17) and eventually to decay.
0
with k = 0.142 and k = 0.138. Here Ω the Keplerian
1 2 0
frequency at the location of the density bump, σ/h and χ
specify the width and amplitude of the bump (see eq. 13), 4 SUMMARY
and h = c /Ω is the disc scale height. Note that this fit-
s 0
We have tested numerically the non-linear saturation
ting formula should be applied only in the range of param-
mechanism for the Rossby wave instability proposed by
eters specified above and require that the disc be stable
Lovelace et al. (2009). This is based on the classical results
against axisymmetric perturbations (ı.e., the Rayleigh cri-
terium κ2 > 0, is satisfied). Eq. 15 also corresponds to the ofparticle-waveinteractionwellknownforunstableelectron
plasmawaves. Wehaveshown thatnon-linearsaturation of
growthrateofthefastestgrowingmode(m∼4)inthelinear
the RWI occurs when the trapping frequency of the parti-
theory. The amplitude of maximum vorticity at saturation
clesintheRossbyvorticesequalstheinstabilitygrowthrate.
is given by
We have estimated the trapping frequency by the vorticity
|ω |∼ 2Ω0k1 χ −k 2/3. (16) oftheRossbywave;thisisaccurateforcircularvorticesbut
v (σ/h)4/3(cid:16)σ/h 2(cid:17) isonlyalowerlimitforelongatedvortices.Wealsodiscussed
thelinear coupling between the Rossby waves and the den-
sity waves. This has the effect of decreasing the amplitude
3.4 3D simulation
ofthevorticitywave.Finally,wegavean estimation for the
In the above we have considered 2D discs, neglecting their amplitudeofthevorticesatsaturationbasedonthecharac-
vertical structure. Recent study of the RWI in 3D verti- teristics of thedensity bump.
cally stratified discs has revealed unexpected vertical dis- Our paper focused on the non-linear saturation of the
placements inside the vortices (Meheut et al. 2010) that linearinstabilityinaninitially steadydisc.Wedidnotcon-
could be relevant for the saturation mechanism. However sidered the mechanism for the formation of the pressure
Meheut et al. (2012) and Lin (2012) have shown that the bump causing the instability. In a disc where the bump is
(cid:13)c 2012RAS,MNRAS000,1–7
How strong are the Rossby vortices? 7
continuouslysustainedbysomeaccretionprocesses,thedisc
is not steady and a future step is to take into account the
evolution of the shape of the bump in the study of the in-
stability.
ACKNOWLEDGMENTS
We thanks the referee G. Lesur for his useful comments.
This work has been supported in part by NSFgrants AST-
1008245 and AST-1211061 and the Swiss National Science
Foundation. RVEL was partly supported by NASA grant
NNX11AF33G.
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