the subcritical baroclinic instability in local accretion disc models · physical process....

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0Astronomy & Astrophysicsmanuscript no. baro7 c© ESO 2018November 1, 2018

The subcritical baroclinic instability in local accretion disc modelsGeoffroy Lesur1 and John C. B. Papaloizou1

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences,Wilberforce Road, Cambridge CB3 0WA, UKe-mail:[email protected]

Received date/ Accepted 13 January 2010

ABSTRACT

Context. The presence of vortices in accretion discs has been debatedfor more than a decade. Baroclinic instabilities might be a wayto generate these vortices in the presence of a radial entropy gradient. However, the nature of these instabilities is still unclear and 3Dparametric instabilities can lead to the rapid destructionof these vortices.Aims. We present new results exhibiting a subcritical baroclinicinstability (SBI) in local shearing box models. We describethe 2Dand 3D behaviour of this instability using numerical simulations and we present a simple analytical model describing the underlyingphysical process.Methods. We investigate the SBI in local shearing boxes, using eitherthe incompressible Boussinesq approximation or a fully com-pressible model. We explore the parameter space varying several local dimensionless parameters and we isolate the regime relevantfor the SBI. 3D shearing boxes are also investigated using high resolution spectral methods to resolve both the SBI and 3Dparametricinstabilities.Results. A subcritical baroclinic instability is observed in flows stable for the Solberg-Hoıland criterion using local simulations. Thisinstability is found to be a nonlinear (or subcritical) instability, which cannot be described by ordinary linear approaches. It requires aradial entropy gradient weakly unstable for the Schwartzchild criterion and a strong thermal diffusivity (or equivalently a short coolingtime). In compressible simulations, the instability produces density waves which transport angular momentum outwardwith typicallyα . 3 × 10−3, the exact value depending on the background temperature profile. Finally, the instability survives in 3D, vortex coresbecoming turbulent due to parametric instabilities.Conclusions. The subcritical baroclinic instability is a robust phenomenon, which can be captured using local simulations. The in-stability survives in 3D thanks to a balance between the 2D SBI and 3D parametric instabilities. Finally, this instability can lead to aweak outward transport of angular momentum, due to the generation of density waves by the vortices.

Key words. accretion, accretion disks – instabilities – planet formation – turbulence

1. Introduction

The existence of long-lived vortices in accretion discs wasfirst proposed by von Weizsacker (1944) in a model of planetformation. This idea was reintroduced by Barge & Sommeria(1995) to accelerate the formation of planetesimals througha dust trapping process. Moreover, large scale vortices canlead to a significant transport of angular momentum thanks tothe production of density waves (Johnson & Gammie 2005b).Many physical processes have been introduced in the lit-erature to justify the existence of these vortices such asRossby wave instabilities (Lovelace et al. 1999), planetarygap instabilities (de Val-Borro et al. 2006, 2007), 3D circu-lation models (Barranco & Marcus 2005), MHD turbulence(Fromang & Nelson 2005) and the global baroclinic instability(Klahr & Bodenheimer 2003).

Baroclinic instabilities in accretion discs has regained in-terest in the past few years. A first version of these instabil-ities (the global baroclinic instability or GBI) was mentionedby Klahr & Bodenheimer (2003) using a purely numerical ap-proach and considering a disc with a radial entropy gradient.However, the presence of this instability was affected by chang-ing the numerical scheme used in the simulations, casting strongdoubts on this instability as a real physical processes. Thelinearproperties of the GBI were then investigated by Klahr (2004)andJohnson & Gammie (2005a). However, only transient growths

were found in this context. Klahr (2004) speculated that thesetransient growths could lead to a nonlinear instability explain-ing the result of Klahr & Bodenheimer (2003), but it is still un-clear whether transient growths are relevant for nonlinearinsta-bilities (see e.g. Lesur & Longaretti 2005, and reference thereinfor a complete discussion of this point). The nonlinear problemwas then studied in local shearing boxes by Johnson & Gammie(2006) using fully compressible numerical methods. These au-thors found no instability in the Keplerian disc regime and con-cluded that the GBI was “either global or nonexistent”.

Nevertheless, baroclinic processes were studied again byPetersen et al. (2007a) using anelastic global simulationsandincluding new physical processes. As in Klahr & Bodenheimer(2003), a weak radial entropy gradient was imposed in the sim-ulation. However, a cooling function was also included in theirmodel, in order to force the system to relax to the imposed radialtemperature profile. In these simulations, Petersen et al. (2007a)observed spontaneous formation of vortices with radial entropygradients compatible with accretion disc thermodynamics.Ina subsequent paper (Petersen et al. 2007b), these vortices werefound to survive for several hundreds orbits. According to theauthors, their disagreement with Johnson & Gammie (2006) wasdue to a larger Reynolds number and the use of spectral methods,a possibility already mentioned by Johnson & Gammie (2006).

In this paper, we revisit baroclinic instabilities in a localsetup (namely the shearing box) both in the incompressible-

http://arxiv.org/abs/0911.0663v2

2 Geoffroy Lesur and John C. B. Papaloizou: The subcritical baroclinic instability in local accretion disc models

Boussinesq approximation and in fully compressible simula-tions. The aim of this paper is to clarify several of the pointswhich have been discussed in the literature for the past 6 yearsabout the existence, the nature and the properties of baroclinicinstabilities. We show that a subcritical baroclinic instability (orshortly SBI)does exist in shearing boxes. This instability isnon-linear (or subcritical) andstrongly linked to thermal diffusion,a point already mentioned by Petersen et al. (2007b). We alsopresent results concerning the behaviour of the SBI in 3 dimen-sions, as vortices are known to be unstable because of paramet-ric instabilities (Lesur & Papaloizou 2009). We begin in§2 bypresenting the equations, introducing the dimensionless num-bers and describing the numerical methods used in the rest ofthe paper.§3 is dedicated to 2D incompressible-Boussinesq sim-ulations and a qualitative understanding of the instability. Wepresent in§4 our results in compressible shearing boxes and in§5 the 3D behaviour of the instability. Conclusions and implica-tions of our findings are discussed in§6.

2. Local model

2.1. Equations

In the following, we will assume a local model for the ac-cretion disc, following the shearing sheet approximation.Thereader may consult Hawley et al. (1995), Balbus (2003) andRegev & Umurhan (2008) for an extensive discussion of theproperties and limitations of this model. As a simplification,we will assume the flow is incompressible, consistently withthe small shearing box model (Regev & Umurhan 2008). Theshearing-box equations are found by considering a Cartesianbox centred atr = R0, rotating with the disc at angular velocityΩ = Ω(R0). We finally introduce in this box a radial stratificationusing the Boussinesq approximation (Spiegel & Veronis 1960).Defining r − R0 → x andR0φ → y as in Hawley et al. (1995),one eventually obtains the following set of equations:

∂tu + u · ∇u = −∇Π − 2Ω× u

2ΩS xex − ΛN2θex + ν∆u, (1)

∂tθ + u · ∇θ = ux/Λ + µ∆θ (2)

∇ · u = 0, (3)

whereu = uxex+uyey+uzez is the fluid velocity,θ the potentialtemperature deviation,ν the kinematic viscosity andµ the ther-mal diffusivity. In these equations, we have defined the meanshearS = −r∂rΩ, which is set toS = (3/2)Ω for a Kepleriandisc. One can check easily that the velocity fieldU = −S xeyis a steady solution of these equations. In the following we willconsider the evolutions of the perturbationsv (not necessarilysmall) of this profile defined byv = u −U .

The generalised pressureΠ is calculated solving a Poissonequation with the incompressibility condition (3). For homo-geneity and consistency with the traditional Boussinesq ap-proach, we have introduced a stratification lengthΛ. Note how-ever thatΛ disappears from the dynamical properties of theseequations as one can renormalise the variables definingθ′ ≡ Λθ.The stratification itself is controlled by the Brunt-Vais¨ala fre-quencyN, defined for a perfect gas by

N2 = − 1γρ

∂P∂R

∂

∂Rln( Pργ

)

, (4)

whereP andρ are assumed to be the background equilibriumprofile andγ is the adiabatic index. With these notations, onecan recover the ordinary thermodynamical variables asθ ≡ δρ/ρ,

whereδρ is the density perturbation andρ is the background den-sity. The stratification length is then defined byΛ ≡ −∂RP/ρN2.

2.2. Dimensionless numbers

The system described above involves several physical processes.To clarify the regime in which we are working and to make eas-ier comparisons with previous work, we define the following di-mensionless numbers:

– The Richardson numberRi = N2/S 2 compares the sheartimescale to the buoyancy timescale. This definition is equiv-alent to the definition of Johnson & Gammie (2005a).

– The Peclet numberPe = S L2/µ.– The Reynolds numberRe = S L2/ν.

L is a typical scale of the system, chosen to be the radial box size(Lx) in our notations.

The linear stability properties of this flow are quite well un-derstood. The flow is linearly stable for axisymmetric perturba-tions when the Solberg-Hoıland criterion is satisfied. This crite-rion may be written locally as:

2Ω(2Ω − S ) + N2 > 0 Stability, (5)

or equivalently in a Keplerian disc,Ri > −4/9. Another linearstability criterion, the Schwarzschild criterion, is often used forconvectively stable flowswithout rotation nor shear. This crite-rion reads

N2 > 0 Stability, (6)

or equivalentlyRi > 0. As we will see in the following, thiscriterion is the relevant one for the SBI.

When viscosity and thermal diffusivity are included,the Solberg-Hoıland criterion is modified, potentially lead-ing to the Goldreich-Schubert-Fricke (GSF) instability(Goldreich & Schubert 1967; Fricke 1968). The stabilitycriterion is in that case

µ2Ω(2Ω − S ) + νN2 > 0 Stability. (7)

This criterion is satisfied when both (5) is verified andν/µ < 1,which corresponds to the regime studied in this paper.

2.3. Numerical methods

We have used two different codes to study this instability.When using the Boussineq model (§3 and§5), we have usedSNOOPY, a 3D incompressible spectral code. This code usesan implicit scheme for thermal and viscous diffusion, allowingus to study simulations with smallPe without any strong con-strain on the CFL condition. The spectral algorithm of SNOOPYhas now been used in several hydrodynamic and magnetohy-drodynamic studies (e.g. Lesur & Longaretti 2005, 2007) andisfreely available on the author’s website. For compressiblesim-ulations (§4), we have used NIRVANA (Ziegler & Yorke 1997).NIRVANA has been used frequently in the past to study var-ious problems involving MHD turbulence in the shearing box(Fromang & Papaloizou 2006; Papaloizou et al. 2004).

In the following, we use the shearing sheet boundary condi-tions (Hawley et al. 1995) in the radial direction. This is madepossible by the use of the Boussinesq approximation in whichonly thegradients of the background profile appear explicitly,throughΛ andN. Therefore, whenΛ andN are constant throughthe box, one can assume that the thermodynamic fluctuationθ isshear-periodic, consistently with the shearing-sheet approxima-tion.

Geoffroy Lesur and John C. B. Papaloizou: The subcritical baroclinic instability in local accretion disc models 3

0 2000 4000 6000 8000 10000

10−3

10−2

10−1

t

<ω

2 / 2

>

Ap=0.2Ap=0.4Ap=0.6Ap=0.8Ap=1.0

Fig. 1. Volume averaged enstrophy for several initial amplitudeperturbation (Ap) in arbitrary units. A subcritical instability isobserved for finite amplitude perturbations betweenAp = 0.2andAp = 0.4.

3. 2D subcritical baroclinic instability inincompressible flows

To start our investigation, we consider the simplest case ofa2D (x, y) problem in an infinitely thin disc. This setup is the lo-cal equivalent of the 2D global anelastic setup of Petersen et al.(2007a), and one expects to find similar properties in both casesif the instability is local. In two dimensions, it is easier to con-sider the vorticity equation instead of (1). Defining the verticalvorticity of the pertubations byω = ∂xvy − ∂yvx, we have:

∂tω + v · ∇ω − S x∂yω = ΛN2∂yθ + ν∆ω (8)

In this formulation, the only source of enstrophy〈ω2/2〉 =∫

dxdyω2/2 is the baroclinic termΛN2∂yθ which will be shownto play an important role for the instability.

The simulations presented in this section are computed in asquare domainLx = Ly = 1 with a resolution of 512× 512 gridpoints using our spectral code. When not explicitly mentioned,the time unit is the shear timescaleS −1. We choose the fiducialparameters to beRe = 4× 105 ; Pe = 4 × 103 andRi = −0.01,in order to have a flow linearly stable for the Solberg-Hoılandcriterion (5). These parameters are close to the one used byPetersen et al. (2007a) and are compatible with disc thermody-namics.

3.1. Influence of initial perturbations

In the first numerical experiment, we choose to test the ef-fect of the initial conditions, keeping constant the dimension-less parameters. In our initial conditions, we excite randomlythe largest wavelengths of the vorticity field with an amplitudeAp. This initial condition is slightly different from the one usedby Petersen et al. (2007a,b) who introduced perturbations in thetemperature field. In each experiment we modify the amplitudeof the initial perturbationAp, and follow the time evolution ofthe total enstrophy (Fig. 1)

The numerical results indicates clearly the presence of anonlinear or subcritical transition in the flow. Indeed, we findthe “instability” for large enough initial perturbations.This alsoconfirms the result of Johnson & Gammie (2006): there is no

linear instability in the presence of a weak radial stratification.Moreover, one of the reasons that Johnson & Gammie (2006)did not observe any transition could be because of the weakinitial perturbations they used (between 10−12 and 10−4). Usingsimilar initial conditions, we did not observe any transition ei-ther. The existence of the instability in our simulations waschecked by doubling the resolution (1024× 1024), keeping con-stant all the dimensionless numbers. We found no significantdif-ference between high and low resolution results, showing thatour simulations were converged for this problem.

When the system undergoes a subcritical transition, it de-velops long-livedself-sustained vortices, as shown forAp = 1in Fig. 2 (top). For such a large Reynolds number, it is knownthat vortices survive for long times (see e.g. Umurhan & Regev2004), even without any baroclinic effect. To check that the ob-served vortices were really due to the baroclinic term, we havecarried out the exact same simulation but without stratification(Fig. 2 bottom). This simulation also shows the formation ofvor-tices but these are lately dissipated on a few hundred shear times.At t = 500S −1 the difference between the simulation with andwithout baroclinicity becomes obvious.

The vortices observed in these incompressible simulationslead to a very weak and strongly oscillating turbulent transport ofangular momentum. One finds typically aninward transport withα = 〈vxvy〉/S L2 ≃ −3× 10−5. This is consistent with the resultspublished by Petersen et al. (2007b) in global simulations.

In several other numerical experiments (not shown here), wehave noticed that the amplitude threshold required to get the in-stability depends onRe andPe, a larger Reynolds number beingassociated with a weaker initial perturbation. This dependencywas pointed out by Petersen et al. (2007a) and it indicates thatthe amplitude threshold in a realistic disc could be very small(i.e. much smaller than the sound speed). Note however that thisthreshold might also be scale dependent, a problem which hasnot been addressed here.

3.2. Influence of the Richardson number

To understand how the instability depends on the amplitudeof the baroclinic term, we have carried out several runs withRe = 4 × 105 and Pe = 4 × 103, varying Ri from −0.02 to−0.16 and from 0.02 to 0.016. We compare the resulting shell-integrated enstrophy spectra ( ˆω2

k/2) on Fig. 4. WhenRi < 0 and|Ri| becomes larger, the instability tends to amplify smaller andsmaller scales. In particular forRi = −0.02, the dominant scaleis clearly the box scale, as already observed for the fiducialrun.This trend is also observed looking directly at vorticity snapshots(Fig. 3). The sign ofRi is also of importance for the onset of theSBI. To demonstrate this effect, we show the time history of thetotal enstrophy for positive and negativeRi on figure 5. In thecases of positiveRi, the total enstrophy decays until the flow be-comes axisymmetric. Perturbations are then damped very slowlyon a viscous timescale. We conclude from these results that anecessary condition for the SBI to appear is a flow which do notsatisfy the Schwarzschild criterion (6).

3.3. Influence of thermal diffusion

The importance of thermal cooling and thermal diffusion wasalready pointed out by Petersen et al. (2007b). To check thisde-pendancy in our local model, we have considered several simula-tions withRi = −0.01, varyingPe from Pe = 20 toPe = 16000.The resulting enstrophy evolution is presented for severalof


Fig. 2.Evolution of the vorticity in the fiducial case. Top row has a baroclinic term withRi = −0.01. Bottom row has no baroclinicity.We showt = 10 (left), t = 100 (middle),t = 500 (right).

Fig. 3. Vorticity map for Ri = −0.02 (left) andRi = −0.16 (right) att = 500. As already shown by the spectra, small scales aredominant for larger|Ri|.

these runs on Fig. 6. Looking at the snapshots of the vorticityfield for these simulations, we find the SBI approximatively for50 ≤ Pe ≤ 8000. Assuming a typical vortex sizel ∼ 0.25 (seee.g. Fig.2), these Peclet numbers correspond to thermal diffusiontimes 3S −1 ≤ τdiff ≤ 500S −1 over the vortex size, with an opti-mum found forτdiff ≃ 10S −1 (Pe = 250). Note that the coolingtime used by Petersen et al. (2007b) lies typically in this rangeof values.

3.4. Instability mechanism

Since the flow is subcritically unstable, no linear anal-ysis can capture this instability entirely. As shown byJohnson & Gammie (2005a), an ensemble of shearing waves ina baroclinic flow is subject to a transient growth with an am-

plification going asymptotically like1 |Ri|t1−4Ri for |Ri| ≪ 1whenµ = ν = 0, the waves being ultimately decaying whenµ > 0 or ν > 0. However, this tells us little to nothing aboutthe nonlinear behaviour of such a flow. Indeed, it is known thatbarotropic Keplerian flows undergo arbitrarily large transientamplifications (see e.g. Chagelishvili et al. 2003), but subcrit-ical transitions are yet to be found in that case (Hawley et al.1999; Lesur & Longaretti 2005; Ji et al. 2006). In the SBI case,the subcritical transition happens only for negativeRi as shownabove, in flagrant contradiction with the linear amplification de-scribed by Johnson & Gammie (2005a). This is a strong indica-tion that linear theory is of little use for describing this instabil-ity.

To isolate a mechanism for a subcritical instability, oneshould start from a non-linear structure observed in the simu-

1 Note that the amplification occurs independently of the signof Ri,as already mentioned by Johnson & Gammie (2005a).


101

102

10−5

10−4

10−3

10−2

10−1

100

101

102

k/2π

k5/3 ω

2

Ri=−0.02Ri=−0.04Ri=−0.08Ri=−0.16Ri=−0.32

Fig. 4.Enstrophy spectrum as a function of the Richardson num-ber Ri. As |Ri| becomes larger, the instability moves to smallerscales.

0 200 400 600

10−2

10−1

100

t

<ω

2 / 2

>

Ri=0.02Ri=0.16Ri=−0.02Ri=−0.16

Fig. 5. Time history of the total enstrophy for positive and neg-ative Richardson number. The instability is observed only fornegativeRi.

lations. For the SBI, these structures are self-sustained vortices.In order to understand how baroclinic effects can feedback onthe vortex structure, we initialised our fiducial simulation with aKida vortex of aspect ratio 4 (Kida 1981). Although this vortex isknown to be an exact nonlinear solution of the inviscid equationof motions, it is slowly modified by the explicit viscosity and thebaroclinic term, leading to a slow growth of the vortex. We showon Fig. 7 the resulting vortex structure (left) and the associatedbaroclinic termΛN2∂yθ (right) at t = 100S −1. Note that thesestructures are quasi steady and evolve on timescales much longerthan the shear timescale. It is clear from these snapshots that thebaroclinic feedback tends to amplify the vorticity locatedinsidethe vortex, leading to the growth of the vortex itself.

To understand the origin of the baroclinic feedback, let’s as-sume the physical quantities are evolving slowly in time, asisobserved in the simulations, so we may write

ω = ω(ǫt) ≡ ω(τ), (9)

θ = θ(ǫt) ≡ θ(τ), (10)

0 1000 2000 3000 4000 5000 6000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t

<ω

2 / 2

>

Pe=25Pe=250Pe=500Pe=2000Pe=16000

Fig. 6. Time history of the total enstrophy for several Pecletnumber (Pe). The instability is observed for 50≤ Pe ≤ 8000.

whereǫ is a small parameter and it is assumed that|θτ| is of thesame order as|v · ∇θ|, and similarly forω. We therefore have tosolve:

ǫ∂τω + (v · ∇ − S x∂y)ω = ΛN2∂yθ + ν∆ω, (11)

ǫ∂τθ + (v · ∇ − S x∂y)θ = vx/Λ + µ∆θ. (12)

Since the Richardson number is assumed to be small, the baro-clinic feedback of (12) has to be small. As only this term canlead to an instability, we can assume it scales likeǫ. The role ofthe viscosity is to prevent the instability from happening,damp-ing vorticity fluctuations. Assuming we are in a regime in whichthe instability appears, the viscosity has to be of the orderof thebaroclinic term, scaling likeǫ. At zeroth order inǫ, we are leftwith:

(v · ∇ − S x∂y)ω = 0, (13)

(v · ∇ − S x∂y)θ = vx/Λ + µ∆θ (14)

This system of equations describes a quite simple physical sys-tem: the vorticity field is a steady solution of the inviscid vor-ticity equation with a constant shear and without any baro-clinic feedback, whereas the potential temperature results fromadvection-diffusion of the background entropy profile due to theflow structure. A family of steady solutions of the inviscid vor-ticity equation with a constant shear are known to be steadyvortices, like the Kida (1981) vortex. More generally, it can beshown that anyω of the form:

ω = F(ψ) (15)

whereψ is the stream function defined byu = ez × ∇ψ issolution of (13).

In the following, we assume (13) is satisfied byω and welook for a solution to the entropy equation (14). As a furthersim-plification, we neglect the advection of the perturbed entropy θby v, assuming this effect is small compared to the advection bythe mean shear, although this does not affect the argument lead-ing to the possibility of instability (see below). Using a Fourierdecompositionθ(x) =

∫

dk θ(k) exp(ik · x), one gets

S kydθdkx+ µk2θ = vx/Λ. (16)


Fig. 7.Vortex structure obtained starting from a Kida vortex. (left) vorticity map. (right) baroclinic term (ΛN2∂yθ) map.

A solution to this equation can easily be found using standardtechniques for solving first order ordinary differential equations.Assuming|vx| decays to zero when|k| → ∞, one may write theresult in the form

θ(kx, ky) =1

Λ|S ky|

∫ ∞

−∞dk′xvx(k′x, ky)G(kx, k

′x, ky), (17)

where we have defined

G(kx, k′x, ky) = H

(

(kx − k′x)S ky/|S ky|)

×

exp[

µ

S ky

(k′3x − k3x

3+ k2

y [k′x − kx])]

, (18)

H being the Heaviside function. From this expression, it is for-mally possible to derive the time evolution of the vorticityandlook for an instability using the first order term in (12). However,this requires an explicit expression for the vorticity, which is apriori unknown. Instead, we have choosen to compute the evolu-tion of the total enstrophy of the system〈ω2/2〉 =

∫

dxdyω2/2:

∂t〈ω2/2〉 = ΛN2〈ω∂yθ〉 − ν〈|∇ω|2〉 (19)

Evidently, an instability can occur only if the first term on theright hand side is positive. Using (17), is is possible to derive ananalytical expression for this term:

ΛN2〈ω∂yθ〉 = −4π2ℜ

N2∫

dkxdkydk′x

|ky||S ||k′|2 ω

∗(k)G(kx, k′x, ky)ω(k′)

, (20)

wherek′ = (k′x, ky). In this expression, the existence of an in-stability is controlled by the Green’s functionG(kx, k′x, ky) andthe shape ofω(k). To our knowledge, it is not possible to de-rive any general criterion for the instability, as such a criterionwould depend on the background vortex considered. It is how-ever possible to derive a criterion in the limit of a large thermaldiffusivity µ. In this limit, G is strongly peaked aroundk′x = kx,and one can assume in first approximation thatG is proportionalto a Dirac distribution. More formally, one can approximateinthe limit of large thermal diffusivity:

G(kx, k′x, ky) ≃ H

(

(kx − k′x)S ky/|S ky|)

exp[

µ

S kyk2(k′x − kx)

]

, (21)

whenkx ≃ k′x. It is then possible to derive an expression for thebaroclinic feedback as a series expansion in the small parameterµ−1:

LN2〈ω∂yθ〉 ≃ −4π2ℜ[

N2∫

dk( k2

y

µk4|ω(k)|2 −

S k3y

µ2k4ω∗(k)∂kx

[

ω(k)/k2]

+ . . .

)]

(22)

In this approximation, we find two competing effects. The firstterm on the right hand side appears when the thermal diffusioncompletely dominates over the shear in (16). It has a positivefeedback on the total enstrophy whenN2 < 0 and can be seen asa source term for the SBI. The second term involves a competi-tion between shear and diffusion. It can be seen in first approx-imation as a phase mixing term, and the resulting can be eitherpositive or negative, depending on the vorticity background oneconsiders. Physically, this term shears out the entropy structurecreated by the vortex and if it is too strong, it kills the positivefeedback of the first term. One should note however that thisterm will notreverse the sign of the baroclinic feedback but willjust weaken it by randomising the phase coherence betweenωandθ.

To understand the “growth” described by (22), on can de-rive an order of magnitude expression for the growth rateγ =d ln 〈ω2〉/dt combining (19) with (22):

γ ∼ (−N2)σ2

µφω(Sσ2/µ) − ν

σ2. (23)

In this expression,σ is the typical vortex size, with the assump-tion σ ∼ 1/k. We have included a phase mixing term with thefunctionφω, as an extension of (22). This function depends onthe background vortex solutionω(x, y) one chooses and cannotbe written explicitly in general. As explained above, this phasemixing weaken the baroclinic feedback whenPe is large but itdoes not change its intrinsic nature. According to these proper-ties, one expectφω(0) = 1 andφω(x)→ 0 whenx→ ∞.

Although φω can only be determined for a specific vortexsolution, one can still deduce a few general properties for theSBI. For very large thermal diffusivity (very smallPe) one willexpect the SBI when

(−N2)σ4

νµ> 1, (24)

sinceφω ∼ 1 in this limit. Although not quantitatively equiva-lent, this first limit corresponds to thePe > 25 threshold of our


A

B C

DA

B C

D

y

x (~r)

Fig. 8. Streamline in a vortex undergoing the SBI. A fluid parti-cle is accelerated by buoyancy effects on A-B and C-D branches.Cooling occurs on B-C and D-A branches (see text).

simulations. It is moreover similar to the criterion for theonsetof convection (in the absence of shear) based on the RayleighnumberRa = −N2σ4/νµ. One finds another instability condi-tion due to phase mixing whenµ → 0. Assumingφω(x) decaysfaster than 1/x whenx → ∞, one gets a minimum value forµsolving

φω(Sσ2/µ)µ

>ν

(−N2)σ4(25)

which can be calculated onceφω is explicitly provided. In oursimulations, this second limit corresponds to thePe < 16000threshold.

Although not entirely conclusive, this short analytical anal-ysis tends to explain why a relatively strong thermal diffusion isrequired to get the SBI. This dependancy was also pointed outby Petersen et al. (2007b) who used both a thermal diffusion anda cooling relaxation time in the entropy equation. Such a cool-ing time can also be introduced in our analysis in place of thethermal diffusion but this does not change the underlying physi-cal process. We also note that in this analysis, we have assumedthat a finite amplitude background vorticity satisfying (13) ex-isted in the first place, from which we have derived the baro-clinic feedback on the same vorticity field. In this sense, thisanalysis describes a non-linear instability and is different fromthe linear approach of Johnson & Gammie (2005a). In addition,we recall that we made the approximation of only including thebackground shear in the advection term on the left hand side ofequation (14). We remark that the dominant driving term in thelimit of large µ in (22) does not depend on this assumption be-cause the heat diffusion term dominates in this limit.

3.5. Phenomenological picture

Knowing the basic mechanisms underlying the SBI, one can ten-tatively draw a phenomenology of this instability. For thisex-ercise, let us consider a single streamline in a vortex subjectto the SBI (Fig. 8 left). For simplicity, we reduce the trajec-tory followed by a fluid particle in the vortex to a rectangle(Fig. 8 right). Let us start with a fluid particle on A, movingradially inward toward B. This fluid particle is initially inther-mal equilibrium, and we assume the motion from A to B is fastenough to be considered approximately adiabatic. As the par-ticle moves inward, its temperature and density slowly deviatefrom the background values. If the background entropy profileis convectively unstable (condition 6), the fluid particle movinginward is cooler and heavier than the surrounding, and is con-sequently subject to an inward acceleration due to gravity (theparticle “falls”). Once the particle reaches B, it drifts azimuthallytoward C. On this trajectory, the background density and temper-ature is constant. Therefore, the fluid particle gets thermalised

with the background and, if the cooling is fast enough, it reachesC in thermal equilibrium. Between C and D, the same buoyancyeffect as the one between A and B is observed: as the particlemoves from C to D, it is always hotter and lighter than the sur-rounding, creating a buoyancy force directed radially outwardon the particle. Finally, a thermalisation episode happensagainbetween D and A, closing the loop.

From this picture, it is evident that fluid particles get accel-erated by buoyancy forces on A-B and C-D branches. In theend, considering many particles undergoing this buoyancy cycle,the vortex structure itself gets amplified, explaining our results.Moreover, the role played by cooling or thermal diffusion is ev-ident. It the cooling is too fast, fluid particles tend to get ther-malised on the A-B and C-D branches, reducing the buoyancyforces and the efficiency of the cycle. On the other hand, if nocooling is introduced, the particles cannot get thermalised on theB-C and D-A branches. In this case, the work due to buoyancyforces on the B-A trajectory will be exactly opposite to the workdone on the C-D trajectory, neutralising the effect on average.

4. 2D SBI in compressible flows

In this situation we consider the 2D subcritical baroclinicinsta-bility within the framework of a compressible model. We ac-cordingy adopt the compressible counterparts of equations(1)-(3) for an ideal gas with constant ratio of specific heatsγ in theform

Du

Dt= −1

ρ∇P −∇Φ − 2Ω× u

−2ΩS xex + ∇ · Tv, (26)

Dc2

Dt=

(γ − 1)c2

ρ

DρDt+

1ρ

(

H +K∆c2)

(27)

∂tρ = −∇ · ρu. (28)

Herec is the isothermal sound speed which is proportional tothe square root of the temperature,Tv is a viscous stress tensor,H/(γ − 1) is the rate of energy production per unit volume,Φ isa general external potential, and the constantK = µρ where asfor the incompressible caseµ is a thermal diffusivity.

For simplicity we adopt a model that may be either regardedas assuming that there is a prescribed energy production rateHandγ is close to unity in (27) so that the compressional heatingterm ∝ (γ − 1) can be dropped, or replacing the combinationof the compressional heating term andH by a new specifiedenergy production/ cooling rate. In either view, as the energyproduction rate is specified, the model is simplified as equation(27) becomes an equation forc2 alone.

We consider shearing box models of extentLx in the radialdirection andLy in the azimuthal direction. A characteristic con-stant sound speed,c0, defines a natural scale heightH = c0/Ω.We present simulations using NIRVANA (see section 2.3) fora small box withLx = Ly/2 = 0.6H and a large box withLx = Ly/2 = 1.2H. Thus in terms of the characteristic soundspeed, when Keplerian shear is the only motion, relative mo-tions in the small box are subsonic, whereas supersonic relativevelocities occur in the big box. We have also considered the tworesolutions (Nx,Ny) = (144, 288) and (Nx,Ny) = (288, 576) andtests have shown that these give the same results. We have alsoperformed simulations with no applied viscosity and with anap-plied viscosity corresponding to a Reynolds numberL2

xΩ/ν =12500. This was applied as a constant kinematic Navier Stokesviscosity but with stress tensor acting on the velocityv beingthe deviation from the background shear rather thanu. This is


Fig. 9. Time history of the evolution of the enstrophy corre-sponding to the perturbations for the small box. The uppermostcurve is forA = 0.25 with no applied viscosity. The next upper-most is the corresponding case with an imposed viscosity. Thelowermost curve is forA = 0.025 with applied viscosity andthe next lowermost curve is for the corresponding case with noviscosity

Fig. 10.Vorticity map for the small box simulation with appliedviscosity andA = 0.25 att = 2000.

not significant for an incompressible simulation but ensures thatunwanted phenomena such as viscous overstabilty produced byperturbations of the background viscous stress (see Kley etal.1993) are absent. The diffusivity corresponding to the viscositywe used is the same magnitude as the magnetic diffusivity used

Fig. 11. Time history of the evolution of the enstrophy corre-sponding to the perturbations for the large box. The uppermostcurve is forA = 0.5 with no applied viscosity. The lower curveis for the corresponding case with an imposed viscosity.

in Fromang et al. (2007) and as in their case we find that it isadequately resolved.

In order to perform simulations corresponding to the incom-pressible ones we need to impose gradients in the state vari-ables that produce a non zero Richardson number (appropriatefor γ = 1.)

Ri = − 1S 2ρ

∂P∂R

∂

∂Rln(Pρ

)

. (29)

Because of the shearing box boundary conditions, imposedbackground gradients have to be periodic. Thus we choose aninitially uniform densityρ = ρ0 and initial profiles forP andc2 of the formP = ρ0c2

0(1 − Cp sin(2πx/Lx)) andc2 = c20(1 −

Cp sin(2πx/Lx)),whereCp is a constant. For a given valueH/Lx,Ri depends only onCp which was chosen to give a maximumvalue for this quantity of−0.07. Note that in our case, asCp

is small, we have approximatelyRi ∝ cos2(2πx/Lx) in contrastto the incompressible simulations where it is constant. Once theinitial value ofµ has been chosen to correspond to a specifieduniform Peclet number, the external potentialΦ and the energysource termH were chosen such that the choice of state vari-ables with Keplerian shear resulted in an equilibrium stateandthen held fixed. For comparison with the incompressible simula-tions we adoptedPe = 379 for the simulations reported here. Atthis point we stress that in view of the rather artificial set up, thismodel is clearly far from definitive. However, it can be used toshow that the phenomena found in the incompressible case arealso seen in a compressible model but with the addition of theeffects of density waves in the case of the large box.

As for the incompressible case, solutions with sustained vor-tices were only found for sufficiently large initial velocity pertur-bations of the equilibrium state. We adopted an incompressiblevelocity perturbation of the form

vx = (3LyΩ/(8π))A sin(2πy/Ly) cos(2πx/Lx) (30)

and

vy = −(3L2yΩ/(8πLx))A cos(2πy/Ly) sin(2πx/Lx), (31)

whereA is specified amplitude factor.


Fig. 12. Vorticity map (left) and surface density map (right) for thelarge box simulation with applied viscosity andA = 0.5 att=190.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

50 100 150 200

< α

>

t

Fig. 13. Time history of the running means of the the spatiallyaveraged value ofα for the small box (lower curve) and largebox (upper curve) simulations with applied viscosity and withA = 0.5 andA = 0.25 respectively.

4.1. Simulation results

In Fig. 9 we show the the evolution of the enstrophy associatedwith the perturbations for the small box. Cases withA = 0.25with and without applied viscosity are shown. These lead to so-lutions for which anticyclonic vortices are sustained for longtimes. The case with no viscosity is more active as expected butotherwise looks similar to the case with applied viscosity.Bycontrast when the amplitude of the initial perturbation is reducedtoA = 0.025 no sustained vortices are seen. This demonstrates

that our numerical setup is not subject to any linear instability,such as Rossby wave instabilities (Lovelace et al. 1999). The en-strophy attains a low level in the case with no applied viscositywhich is a consequence of a long wavelength linear axisymmet-ric disturbance that shows only very weak decay provided bynumerical viscosity in this run. Fig. 10 shows a vorticity map forthe small box simulation with applied viscosity andA = 0.25.Anticylconic vortices are clearly seen in this case supporting thefinding from the incompressible runs that a finite amplitude ini-tial kick is required to generate them.

The time history of the evolution of the enstrophy for largebox simulations withA = 0.5 with and without applied vis-cosity is shown in Fig. 11. Corresponding vorticity and surfacedensity maps for the case with applied viscosity are shown inFig. 12. Again the inviscid case is more active but nonethelessthe corresponding maps look very similar. In these cases thean-ticyclonic vortices are present as in the small box case but thereis increased activity from density waves as seen in the surfacedensity maps. These waves could be generated by a processsimilar to the swing amplifier with vorticity source describedby Heinemann & Papaloizou (2009a,b), although the structureswe observe do not strictly correspond to a small scale turbulentflow. The density waves are associated with some outward an-gular momentum transport. However, the value ofα measuredfrom the volume average of the Reynolds stress is always highlyfluctuating. Accordingly we plot running means as a functionoftime for the small box and large box with applied viscosity inFig. 13. In the small box case there is a small residual time av-erage∼ 10−4 but in the large box this increases to∼ 3 × 10−3.This is clearly a consequence of the fact that the small box isclose to the incompressible regime, whereas the large box beingeffectively larger than a scale height in radial width allows the


0 100 200 300 400 500 600 700 800 900 1000−0.4−0.2

00.20.4

v x

0 100 200 300 400 500 600 700 800 900 1000−0.4−0.2

00.20.4

v y

0 100 200 300 400 500 600 700 800 900 1000−0.4−0.2

00.20.4

t

v z

Fig. 14.Time history of the maxima of the 3 components of thevelocity field when starting from 2D+3D noise. Once the SBIhas formed vortices, 3D motions due to parametric instabilitiesappears and balance the 2D instability.

vortices to become large enough to become significantly moreeffective at exciting density waves.

5. The SBI in 3D

The results presented previously were obtained in a 2D setup.It is however known that vortices like the one observed in thesesimulations are linearly unstable to 3D perturbations due to para-metric instabilities (Lesur & Papaloizou 2009). The question ofthe survival of these vortices in 3D is therefore of great impor-tance, as their destruction would lead to the disappearanceof theSBI. As shown by Lesur & Papaloizou (2009), 3D instabilitiesinvolve relatively small scales compared to the vortex sizewhenthe aspect ratio2 χ of the vortex is large. This leads to strong nu-merical constraints on the resolution one has to use to resolveboth the 2D SBI and 3D instabilities. To optimise the computa-tional power, we have carried out all the 3D simulations usingour spectral code in the Boussinesq approximation, with a setupsimilar to§3 and constant stratification.

5.1. Fiducial simulation

We first consider a box extended horizontally to mimic a thindisc with L ≡ Lx = Ly = 8 and Lz = 1. We setRe =1.02× 106, Pe = 6400 andRi = −0.01 with a numerical res-olution Nx × Ny × Nz = 1024× 512× 128 in order to resolvethe small radial structures due to the 3D instabilities. Thehori-zontal boundary conditions are identical to the one used in 2Dand periodic in the vertical direction. Note that we don’t includeany stratification effect in the vertical direction to reduce com-putational costs. Initial conditions are to be chosen with care as3D random perturbations will generally decay rapidly due tothegeneration of 3D turbulence everywhere in the box. To avoid thiseffect, we choose to start with a large amplitude 2D white noise(〈√

v2〉 ∼ 0.1) to which we add small 3D perturbations with anamplitude set to 1% of the 2D perturbation amplitude.

We show on Fig.14 the time history of the extrema of thevelocity field in such a simulation. As expected, vertical motionstriggered by 3D instabilities appear att = 200. However, these

2 The aspect ratio of a vortex is defined by the ratio of the azimuthalsize to the radial size of the vorticity patch generated by the vortex

0 500 1000 1500

−0.2

0

0.2

v z

0 500 1000 1500

2

4

6

8

t

χ

Fig. 16.Time history ofvz extrema (top) and of the vortex aspectratioχ (bottom). Once the SBI has formed vortices, 3D motionsdue to parametric instabilities appears and balance the 2D insta-bility.

“secondary” instabilitiesdo not have a destructive effect on theSBI. Instead, an almost steady state is reached att ≃ 600.

Looking at the snapshots in the “saturated state” att = 800(Fig. 15) we observe the production of large scale anticyclonicvortices, as in the 2D case. However, in the core of these vortices(regions of negative vorticity), we also observe the appearanceof vertical structures. Looking at the vertical component of thevelocity field, one finds clearly that these vertical structures andmotions are localisedinside the vortex core. Although it is likelythat the 3D instability observed in this simulation is related to theelliptical instability described by Lesur & Papaloizou (2009), wecan’t formally prove this point as the vortices found in the simu-lation are different from the one studied by Lesur & Papaloizou(2009).

Despite the presence of 3D turbulence in these vortices, theturbulent transport measured in this simulation is directed in-ward, withα ≃ −3×10−5. This is to be expected as the 3D insta-bilities extract primarily their energy from the vortex structureandnot from the mean shear. However, allowing for compress-ibility will certainly change the turbulent transport as vorticeswill then also produce density waves (see section 4).

5.2. Evolution of a single vortex

To isolate the interplay between the SBI and 3D elliptical insta-bilities, we have considered the case of an isolated 2D vortex ina 3D setup. We take the same parameters as in the fiducial case,but we consider this time a Kida vortex of aspect ratioχ = 6 asan initial condition. Thanks to these initial conditions, it is possi-ble to follow the evolution of this single vortex, and in particularmeasure its aspect ratio as a function of time. This is done witha post-processing script, assuming that the vortex core boundaryis located atωb = 0.5[min(ω) + max(ω)]. We then measure thevortex aspect ratio as being the ratioly/lx of the boundary de-fined above. One should note however that this procedure doesnot check that the vortex is still an ellipse. Moreover, it giveswrong values forχ if 3D perturbations create strong fluctuationsof ω.


Fig. 15.3D structure obtained from our fudicial 3D simulation att = 800. Left: vertical component of the vorticity. Right: verticalcomponent of the velocity field. We find that 3D instabilitiesinvolving vertical motions and vertical structures are localised insidevortex cores.

Fig. 17.Vorticity map from a simulation starting with a 2D Kida vortex plus a weak 3D noise. We show a snapshot att = 660 (left),t = 750 (middle) andt = 830. After the 3D instability burst, a weak vortex survives and is amplified by the SBI.

We show on Fig. 16 the result of such a procedure as wellas the extrema on the vertical velocity for comparison. Startingfrom χ = 6, the first effect of the SBI is to reduce the vortexaspect ratio. This is consistent with the Kida vortex solution forwhich

ω

S= −1

χ

(

χ + 1χ − 1

)

. (32)

Therefore, the vorticity amplification in the vortex core due tothe baroclinic feedback leads to a reduction of the aspect ra-tio. Whenχ ≃ 4, we note the appearance of strong verticalmotions. During these “bursts” of 3D turbulence, the aspectra-tio measurement is not reliable anymore. However, the verticalmotions become ultimately weak, and a weaker vortex appearswith χ ∼ 10. The baroclinic feedback then amplifies the vortexand the cycle starts again. Interestingly, theχ < 4 Kida vor-tices described by Lesur & Papaloizou (2009) were shown to bestrongly unstable due to a “horizontal” instability. The behaviourobserved in these simulations tends to indicate that the paramet-ric modes unstable forχ > 4 are not fast enough to take over theSBI. However, asχ passes through the critical valueχ = 4, thehorizontal modes become unstable and strongly damp the vor-tex in a few orbits. This interpretation is somewhat confirmedby the snapshots of the velocity field (Fig. 17), where growingperturbations are localised inside vortex cores as observed byLesur & Papaloizou (2009).

The “burst” of turbulence observed in the Kida vortex casemight be a specific property of this vortex solution. Indeed,onlonger timescales, the spatial distribution of vorticity might bemodified, leading to a more progressive appearance of 3D insta-bilities and a state more similar to the one observed in 5.1 could

be achieved. However, this extreme example clearly demonstratethat 3D parametric instabilities do not lead to a total destructionof the vortices produced by the SBI.

6. Conclusions

In this paper, we have shown that a subcritical baroclinic insta-bility was found in shearing boxes. Our simulations producevor-tices with a dynamic similar to the description of Petersen et al.(2007a) and Petersen et al. (2007b). We find that the conditionsrequired for the SBI are (1) a negative Richardson number (orequivalently a disc unstable for the radial Schwarzschild crite-rion) (2) a non negligible thermal diffusion or a cooling functionand (3) a finite amplitude initial perturbation, as the instability issubcritical3.

When computed in compressible shearing boxes, the vorticessustained by the SBI produce density waves which could leadto a weak outward angular momentum transport (α ∼ 10−3), aprocess suggested by Johnson & Gammie (2005b). However, wewould like to stress that this transport cannot be approximatedby a turbulent viscosity, as the transport due to these wavesiscertainly a non local process. Several uncertainties remain in thecompressible stratified shearing box model. In particular,the im-posed temperature profile that is subsequently maintained by aheating source is somewhat artificial. This occurs while angu-lar momentum transport causes a significant density redistribu-tion causing the local model to deviate from the original form.Clearly one should therefore consider global compressiblesim-ulations with a realistic thermodynamic treatment to draw any

3 Note that point (1) and (2) were already mentioned byPetersen et al. (2007b), although in a somewhat different form.


firm conclusion on the long term evolution and the angular mo-mentum transport aspect of the SBI.

Although the vortices produced by the SBI are anticyclones,the precise pressure structure inside these vortices is notas sim-ple as the description given by Barge & Sommeria (1995). Inparticular, since the vorticity is of the order of the local ro-tation rate, vortices are not necessarily in geostrophic equilib-rium and pressure minima can be found in the centre of strongenough vortices. Moreover, these vortices interact with eachother, which leads to complex streamline structures and vortexmerging episodes. This leads us to conclude that the particle con-centration mechanism proposed by Barge & Sommeria (1995)might not work in its present form for SBI vortices.

In 3D, vortices are found to be unstable to parametric insta-bilities. These instabilities generates random gas motions (“tur-bulence”) in vortex cores, but they generally don’t lead to aproper destruction of the vortex structure itself. More impor-tantly, the presence of a weak hydrodynamic turbulence in vortexcores will lead to a diffusion of dust and boulders, balancing theconcentration effect described above. In the end, dust concentra-tion inside these vortices might not be large enough to trigger agravitational instability and collapse to form planetesimals.

Given the instability criterion detailed above, one can ten-tatively explain why Johnson & Gammie (2006) failed to findthe SBI in their simulation. First, we would like to stressthat our compressible simulations are not more resolved thantheirs. Since these simulations use similar numerical schemes,the argument of a too small Reynolds number proposed byPetersen et al. (2007b) seems dubious. We note however thatJohnson & Gammie (2006) did not include two other importantingredients: a thermal diffusivity and finite amplitude initial per-turbations. As shown in this paper, if one of these ingredient ismissing, the flow never exhibits the SBI and it gets back to alaminar state rapidly. Note that even if this argument explainsthe negative result of Johnson & Gammie (2006), it also indi-cates that the SBI should be absent from the simulations pre-sented by Klahr & Bodenheimer (2003), as no thermal diffu-sion nor cooling function was used (the initial conditions werenot explicitly mentioned). This suggests that either the globalbaroclinic instability described by Klahr & Bodenheimer (2003)and the SBI are two separate instabilities, or strong numericalartefacts have produced an artificially large thermal diffusion inKlahr & Bodenheimer (2003) simulations.

In the limit of a very small kinematic viscosity, the scale atwhich the instability appears has to be related to the diffusionlength scale (µ/S )1/2. In a disc, one would therefore expect theinstability around that scale, provided that the entropy profilehas the right slope (condition 1). However, as the instability pro-duces enstrophy, vortices are expected to grow, until they reach asize of the order of the disc thickness where compressible effectsbalance the SBI source term. We conclude from this argumentthat even if the SBI itself is found at small scale (e.g. because ofa small thermal diffusivity), the end products of this instabilityare large scale vortices, with a size of the order of a few discscale heights.

We would like to stress that the present work constitutesonly a proof of concept for the subcritical baroclinic instabil-ity. To claim that this instability actually exists in discs, one hasto study the disc thermodynamic in a self consistent manner,atask which is well beyond the scope of this paper. Moreover, itis not possible at this stage to state that the SBI is a solutionto the angular momentum transport problem in discs. Althoughthis instability creates some transport through the generation ofwaves, this transport is weak and large uncertainties remain on

its exact value. Finally, the coexistence of the SBI and the MRI(Balbus & Hawley 1991) is for the moment unclear. In the pres-ence of magnetic fields, vortices produced by the SBI mightbecome strongly unstable because of magnetoelliptic instabili-ties (Mizerski & Bajer 2009). Although the nonlinear outcomeof these instabilities is unknown, one can already suspect thatvortices produced by the SBI will be weaken in the presence ofmagnetic fields.

Acknowledgements. GL thanks Charles Gammie, Hubert Klahr, Pierre-YvesLongaretti and Wladimir Lyra for useful and stimulating discussions dur-ing the Newton Institute program on the dynamics of discs andplanets.We thank our referee G. Stewart for his valuable suggestionson this work.The simulations presented in this paper were performed using the DarwinSupercomputer of the University of Cambridge High Performance ComputingService (http://www.hpc.cam.ac.uk/), provided by Dell Inc. using StrategicResearch Infrastructure Funding from the Higher EducationFunding Councilfor England. GL acknowledges support by STFC .

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