BAROCLINIC VORTICITY PRODUCTION IN PROTOPLANETARY … · BAROCLINIC VORTICITY PRODUCTION IN PROTOPLANETARY DISKS. II. VORTEX GROWTH AND LONGEVITY Mark R. Petersen Department of Applied

BAROCLINIC VORTICITY PRODUCTION IN PROTOPLANETARY DISKS. II.VORTEX GROWTH AND LONGEVITY

Mark R. Petersen

Department of Applied Mathematics, University of Colorado, Boulder, CO; and Computer and Computational Science Division

and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM; [email protected]

Glen R. Stewart

Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO

and

Keith Julien

Department of Applied Mathematics, University of Colorado, Boulder, CO

Received 2006 November 3; accepted 2006 December 5

ABSTRACT

The factors affecting vortex growth in convectively stable protoplanetary disks are explored using numerical sim-ulations of a two-dimensional anelastic-gas model that includes baroclinic vorticity production and radiative cooling.The baroclinic feedback, in which anomalous temperature gradients produce vorticity through the baroclinic termand vortices then reinforce these temperature gradients, is found to be an important process in the rate of growth ofvortices in the disk. Factors that strengthen the baroclinic feedback include fast radiative cooling, high thermal dif-fusion, and large radial temperature gradients in the background temperature. When the baroclinic feedback is sufficientlystrong, anticyclonic vortices form from initial random perturbations and maintain their strength for the duration of thesimulation, for over 600 orbital periods. Based on both simulations and a simple vortexmodel,we find that the local angularmomentum transport due to a single vortex may be inward or outward, depending on its orientation. The global angularmomentum transport is highly variable in time and is sometimes negative and sometimes positive. This result is for ananelastic-gas model and does not include shocks that could affect angular momentum transport in a compressible-gasdisk.

Subject headinggs: accretion, accretion disks — circumstellar matter — hydrodynamics — instabilities —methods: numerical — solar system: formation — turbulence

Online material: color figures

1. INTRODUCTION

The baroclinic term is a source of vorticity in the vorticity equa-tion and is derived by taking the curl of the pressure gradient inthe Navier-Stokes equation,

: < � 1

�:p

� �¼ 1

�2:� < :p; ð1Þ

where p is the pressure and � is the density. The baroclinic term isnonzero when pressure and density gradients are not aligned.

An intuitive example of baroclinicity is the land-sea breeze,which is initiated when air temperatures above the land rise morethan over the nearby ocean. The warm air over the land expands,isobars rise relative to those over the ocean, and consequently,the isobars tilt toward the ocean. At the same time, the colder airover the ocean has a higher density than over the land, so theisopycnals tilt toward the land. The tilting of isobars and iso-pycnals in opposite directions is a baroclinic source of vorticity,which causes a circulation in the vertical plane that blows fromthe ocean to the land near the surface. Thus, the potential energyof the tilted isopycnals is converted into the kinetic energy of theland-sea breeze, which dissipates through surface friction andreduces the land-sea temperature contrast through temperatureadvection (see, e.g., Holton 2004).

A related concept is the baroclinic instability, which is of cen-tral importance to the production of vortices and Rossby waves

at midlatitudes. Here the decrease in solar insolation from theequator to the pole causes colder temperatures, and consequentlyhigher density, at the surface at higher latitudes; thus, the isopycnalsurfaces are tilted toward the equator. A system with tilted iso-pycnals has more potential energy than one with level isopycnals,just like an inclined free surface has more potential energy than alevel one. This potential energy is available to processes that canflatten out the isopycnals. For example, vortices in the atmosphereand ocean convert the potential energy of the inclined isopycnalsto the kinetic energy of their meso-scale motion. Vortices flattenthe isopycnals by transferring heat poleward through their mixingaction.The baroclinic instability is so named because of the tilted

isopycnals, but the physics is fundamentally different from theland-sea breeze. In the land-sea breeze, the circulation is in thevertical plane and is caused directly by the baroclinic term, i.e., bynonaligned density and pressure gradients in the vertical. In thebaroclinic instability, the isopycnals are titled in the vertical, butthe vortices are in the horizontal plane, so they could not be pro-duced by the baroclinic term directly. Rather, the tilted isopycnalspresent an unstable configuration that is ripe for processes, thatcan convert the potential energy to kinetic energy, much like howan avalanche levels out a steep incline of snow.The baroclinic processes discussed in this paper for a proto-

planetary disk are similar to the land-sea breeze, but in radialgeometry. Due to the gravity and radiation of the central star, thedensity, temperature, and pressure of the disk’s gas all decrease

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# 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.

radially. Any azimuthal variations in temperature (and thus den-sity or pressure by the ideal gas law) would lead to an increase invertical vorticity due to the baroclinic term in equation (1). Thefocus of this work is the baroclinic feedback, in which a vortexenhances azimuthal temperature gradients to reinforce the vortexitself.Under the right conditions, the baroclinic feedback strengthensvortices, so that they can exist for long periods of time. Thesevortices could play a crucial role in planetary formation, as theyare efficient at collecting particles from the disk (Tanga et al.1996; Johansen et al. 2004; Barge & Sommeria 1995; Klahr &Bodenheimer 2006). The high density of solids in the vortexwould speed the formation by core accretion, which is so slow inthe rest of the disk that it may not be a feasible theory of plane-tary formation there (Wetherill 1990). Avortex that collects solidsis also a potential site of gravitational instability. In thismechanism,matter accumulates in the vortex until it is dense enough to collapseinto a planet through gravitational self-attraction (Boss 1997).

Our study was motivated by Klahr & Bodenheimer (2003),who investigated the effects of baroclinicity in a radially stratifieddisk using a finite difference model of the compressible Navier-Stokes equation combined with a radiative transfer model. Theyfound the baroclinic instability to be a source of vigorous turbu-lence, which leads to the formation of long-lasting vortices andpositive angular momentum transport. Barotropic simulations inwhich the entropy (temperature) is constant in the radial directiondid not develop turbulence, even with large initial perturbations.To explain these results, Klahr (2004) performed a local linear anal-ysis for a disk with constant surface density and found that modesdo not grow if the growth time of the instability is longer than theshear time.

The issue of whether the baroclinic instability is a mechanismfor nonlinear growth and the formation of vortices has been arecent source of debate. Johnson and Gammie are critical of thefindings of Klahr &Bodenheimer (2003) and Klahr (2004). Theirlinear analysis found no exponentially growing instabilities, ex-cept for convective instabilities in the absence of shear (Johnson& Gammie 2005a). Furthermore, they use a shearing-sheet nu-merical model to show that disks with a nearly Keplerian rotationprofile and radial gradients on the order of the disk radius are stableto local nonaxisymmetric disturbances (Johnson&Gammie 2006).

The goal of this study is to understand the effects of baroclinicinstabilities and radiative cooling on the generation of turbulence,vortex formation, and vortex longevity in protoplanetary disks.One of our motivations is to shed light on the conflicting obser-vations of baroclinic instabilities byKlahr&Bodenheimer (2003)and Johnson & Gammie (2006).

Thiswork is presented in two parts. Part I (Petersen et al. 2007),which precedes this article, presents the equation set, details of thenumerical model, and results of the small-domain simulations,which are used to study the process of vortex formation. This paper,Part II, explores the parameters that affect the baroclinic feedbackduring the growth phase of the vortices; these simulations use thelarger quarter-annulus domain and are run for hundreds of orbitalperiods to observe the long-term behavior of the vortices. Webegin with a quick review of the equation set in x 2. The results inx 3 discuss the evolution of a typical simulation, the process of thebaroclinic feedback, the Richardson number as a diagnostic, andthe � -viscosity. In x 4 we discuss the angular momentum trans-port in our simulations, which is highly variable and depends onthe orientation of individual vortices. In x 5 we conclude that thebaroclinic instability is an important mechanism for vortex gener-ation and persistence, and we review the conditions that affect theinstability. For conciseness there is little repetition between Parts Iand II, so the reader is advised to read both together.

2. DESCRIPTION OF THE EQUATION SET

The model equations are described fully in Part I of this workand are only briefly reviewed here. They model an anelastic gas,which filters out pressure waves that restrict the time step of thenumerical model, but do not impact the physics of interest here.Our equation set is similar to those in Bannon (1996) and Scinocca& Shepherd (1992), which are anelastic models of the atmospherederived from the conservation of momentum, the conservation ofmass, the second law of thermodynamics, and the ideal gas law.Our equations use two-dimensional polar coordinates (r; �), wheretemperature and density are stratified in the radial direction. Var-iables, such as the vertical component of vorticity �, stream func-tion �, potential temperature �, thermal temperature T, surfacedensity �, and Exner pressure �, are written as the sum of abackground and perturbation term, e.g., � ¼ �0(r)þ �0(r; �; t),where the background functions only vary radially.

The model equations are

� 0 ¼ 1

r

@

@r

r

�0

@�0

@r

� �þ 1

r2�0

@ 2�0

@�2; ð2Þ

@� 0

@tþ @ �;

1

�0

�

� �¼ cp

r

@�0

@r

@�0

@�þ �e9

2� 0; ð3Þ

@�0

@tþ 1

�0

@ �; �ð Þ ¼ � �0

�þ e9

2�0: ð4Þ

The first is the relationship between the perturbation stream func-tion�0 and the perturbation vorticity � 0; the second and third areprognostic equations for perturbation vorticity � 0 and perturba-tion potential temperature �0, respectively. Radial and azimuthalvelocities u ¼ (u; v) are related to the stream function by �0u ¼�: < �z. Other variables include the radiative cooling time � ,specific heat at constant pressure cp, time t, viscosity �e, thermaldissipation e, vertical unit vector z, and the Jacobian @(a; b) ¼(@ra@�b� @�a@rb)/r.

The baroclinic term,

cp

r

@�0

@r

@�0

@�; ð5Þ

is a central focus of this paper. It is the only source term in vor-ticity equation (3), and it plays an important role in the baroclinicinstability, as one might expect. Most people are familiar withthe baroclinic term using density and pressure, shown in equa-tion (1). This operation in terms of our variables is

: < �cp�0:�0

� �¼ � 1

r

@

@��cp�

0 d�0

dr

� �¼ cp

r

@� 0

@�

d�0

dr: ð6Þ

The factor @r�0 indicates that the baroclinic feedback should bestrengthened if @r�0 is large, i.e., if radial pressure gradients arelarge. But

@p0@r

� @�0T0

@r¼ �0

@T0@r

þ T0@�0

@r; ð7Þ

so we expect large radial temperature or density gradients tostrengthen the baroclinic feedback.

The radiative cooling term �0/� diffuses the perturbation po-tential temperature equally at all scales with an e-folding time of� . The two Laplacian terms, �e9

2� 0 and e92�0, diffuse potential

vorticity and potential temperature at their fastest at the highestwavenumbers. They were added to the numerical model to dis-sipate energy for numerical stability. In the nondimensionalized

BAROCLINIC VORTICITY PRODUCTION. II. 1253

version of the equation set, �e ande are replacedwith theReynoldsand Peclet numbers,

Re ¼ L2sc�etsc

; Pe ¼ L2scetsc

;

where the length scale Lsc and timescale tsc are described in Part I.

3. RESULTS

The simulations discussed in this paper vary parameters suchas the radiative cooling rate, the background temperature andsurface density gradients, and the Peclet number (Table 1). Thesesimulations capture the salient features of the physics of the an-elastic equation set. The topic of Part I was vortex formation andthus used a smaller domain for only five orbital periods. Herewe are interested in vortex growth and longevity due to thebaroclinic feedback and have chosen a larger domain and du-rations of 300Y600 orbital periods. (This is 6200Y12,400 yrfor a 1M� star.) The simulations were performed on the quarterannulus with a radial extent from 5 to 10 AU and a resolution of256 ; 256 and 512 ; 512 grid points.

The background surface density and background temperatureare constant in time and are power functions in the radial,

�0(r) ¼ ar

rin

� �b

; T0(r) ¼ cr

rin

� �d

; ð8Þ

where rin ¼ 5 AU is the inner radius of the annulus. The coef-ficients are a ¼ 1000 g cm�2, c ¼ 600 K for the quarter-annulusdomain, and b and d are varied and shown in Table 1. For ex-ample, for simulation A1 the background surface density variesfrom 1000 to 350 g cm�2, and the background temperature de-creases radially from 600 to 150 K. This range of temperaturescan only be achieved in a realistic disk when the radius rangesfrom 1 to 10 AU (Boss 1998). We have artificially enhanced the

radial temperature gradient in order to compensate for the lowerresolution of our global simulations. We have demonstrated inPart I, using a higher resolution local simulation, that more real-istic temperature gradients can still produce vortices. Most sim-ulations were run to 300 orbital periods, measured as a full (2�)orbit at rmid ¼ 7:5 AU. This is 6200 yr for a 1 M� star.Thermal temperature T and potential temperature � are related

by

� ¼ Tp0(rin)

p

� �R=cp¼ T

�; ð9Þ

where R is the gas constant and � is the Exner pressure. Allresults in this paper are expressed in terms of the thermal tem-perature T in order to compare to observations. The potentialtemperature is ameasure of entropy. If entropy increases radially(d�0/dr > 0), then the disk is convectively stable—this is theSchwarzschild criterion (Schwarzschild 1958). If the entropygradient is accompanied by differential rotation, the Solberg-Høiland criterion (Tassoul 2000; Rudiger et al. 2002) is used totest convective stability (see Part I, x 4). For the simulationspresented in this paper, the Solberg-Høiland value is positive(0.035Y0.299 yr�2), indicating that they are convectively stable.The initial condition for the perturbation temperature is shown

in Figure 1. It is created with a specified wavenumber distribu-tion in Fourier space, transformed to Cartesian coordinates, andinterpolated to the Fourier-Chebyshev annular grid (see Part I, x 3.2).The initial vorticity perturbation is created in a similar fashion.The magnitude of the initial conditions is 25% of the maximumof the background function.The small-domain simulations in Part I (r 2 ½9:5; 10�, � 2

½0; �/32�) required a much smaller initial perturbation to initiatevortices—a temperature perturbation of only 5% and an ini-tial vorticity perturbation of zero. This is possible because thesmall domain is of a higher resolution relative to the backgroundshear. The sensitivity analysis in Part I showed that smaller ini-tial perturbations are required to initiate vortices with progres-sively higher resolution and Reynolds number. The same is trueof the background temperature; the quarter-annulus domain useshigher temperatures (c ¼ 600 K) and steeper gradients (d ¼ �2)than the small-domain simulations. Again, the sensitivity anal-ysis in Part I showed that at higher resolutions, vortices can beformedwith progressively cooler disks and shallower backgroundgradients.The evolution of a typical simulation can be described as fol-

lows. The initial distribution of vorticity shears due to the differ-ential rotation of the nearly Keplerian rotational profile (Fig. 2).Even at these early times, the perturbation vorticity and perturbation

TABLE 1

Model Parameters for the Numerical Simulations

Discussed in This Paper

Name Grid � d b Re Pe End Time

A1................... 5122 1 �2 �1.5 4e7 4e7 100

A2................... 5122 1 �1 �1.5 4e7 4e7 60

A3................... 5122 1 �0.75 �1.5 4e7 4e7 60

B1................... 2562 1 �0.75 �1.5 2e7 2e7 300

Tau1................ 2562 1 �2 �1.5 2e7 2e7 300

Tau2................ 2562 3 �2 �1.5 2e7 2e7 300

Tau3................ 2562 10 �2 �1.5 2e7 2e7 300

Tau4................ 2562 100 �2 �1.5 2e7 2e7 300

T1 ................... 2562 1 �2 �1.5 2e7 2e7 300

T2 ................... 2562 1 �1 �1.5 2e7 2e7 300

T3 ................... 2562 1 �0.75 �1.5 2e7 2e7 600

T4 ................... 2562 1 �0.5 �1.5 2e7 2e7 300

T5 ................... 2562 1 �0.25 �1.5 2e7 2e7 300

D1................... 2562 1 �2 �1 2e7 2e7 300

D2................... 2562 1 �2 �1.5 2e7 2e7 300

D3................... 2562 1 �2 �2 2e7 2e7 300

Pe1.................. 2562 100 �2 �1.5 2e7 2e7 300

Pe2.................. 2562 100 �2 �1.5 2e7 1e6 300

Pe3.................. 2562 100 �2 �1.5 2e7 1e4 300

Notes.—Here � is the radiative cooling time in orbital periods, d and b are thepowers on the background temperature and surface density functions, Re and Peare the Reynolds and Peclet numbers, and end time is in orbital periods. In sim-ulationB1, the baroclinic term is turned off at various times during the simulation.

Fig. 1.—Initial temperature perturbation T 0. [See the electronic edition of theJournal for a color version of this figure.]

PETERSEN, STEWART, & JULIEN1254 Vol. 658

kinetic energy grow due to the baroclinic term (Fig. 5). Afterabout five orbital periods, anticyclonic vortices begin to form,and by 10 orbital periods, the domain is populated by numeroussmall anticyclones. Cyclonic (anticyclonic) fluid rotates in thesame (opposite) direction as the background fluid and is denotedby positive (negative) vorticity perturbation in the figures. It iswell known that anticyclones can be long lived in a Kepleriandisk, while cyclones shear out into thin filaments that eventuallydissipate away (Godon& Livio 1999; Marcus 1990; Marcus et al.2000). An anticyclonic vortex has a positive azimuthal velocity atsmall inner radii and a negative azimuthal velocity at large outerradii. This means that anticyclonic vortices can smoothly matchthe background shear flow and therefore extract energy from theKeplerian shear. Cyclonic vortices cannot smoothly match thebackground shear flow and are therefore sheared apart.

After the initial period of vortex formation, the vortices mergeand grow in strength (Figs. 3 and 4). This merging behavior is

similar to the merging of like-signed vortices in two-dimensionalisotropic turbulence, which transfers energy from smaller to largerscales (the inverse cascade). However, in shearing flows vorticesdo not merge as readily and must be sufficiently close in the radialdirection. It is not at all clear that this merging of vortices canoccur in a fully three-dimensional disk if the initial radial vortexscale is small compared to the disk scale height. On the other hand,if vortices primarily form on the upper and lower surfaces of avertical stratified disk, as found by Barranco & Marcus (2005),then it may be possible for small-scale vortices to merge in thesesurface layers. Further discussion and images of vortex merger,longevity, and distribution can be found in Godon & Livio(1999) and Umurhan & Regev (2004).

There is a clear ‘‘sandwich’’ pattern of temperature perturba-tions around each vortex (Fig. 4); the vortex advects warmer fluidtoward the outside of the disk and cooler fluid toward the inside ofthe disk. In the sandwich analogy, the temperature perturbations are

Fig. 2.—Perturbation vorticity � 0 in the quarter-annular computational domain for simulation A1. The time t refers to the orbital period in the middle of the annulus. Atvery early time (left), the vorticity is simply sheared by the background differential rotation. By five orbital periods (middle), a few anticyclonic vortices begin to fold over,and by 10 orbital periods (right), numerous small anticyclonic vortices have formed. [See the electronic edition of the Journal for a color version of this figure.]

Fig. 3.—Perturbation vorticity � 0 (top row) and perturbation temperature T 0 (bottom row) for simulation A1, where the radiative cooling time is fast (� ¼ 1). In thisregime, the baroclinic feedback remains strong, and vortices remain strong for the full simulation. [See the electronic edition of the Journal for a color version of this figure.]

BAROCLINIC VORTICITY PRODUCTION. II. 1255No. 2, 2007

the bread, and the vortex is the meat between the bread. These per-turbations have azimuthal temperature gradients that play a role inthe baroclinic feedback.

3.1. Baroclinic Vorticity Production

The model equations for vorticity and temperature perturbationsare coupled by the baroclinic term in vorticity equation (3) and theadvection term in temperature equation (4). This coupling is re-quired to support long-lived vortices; without it , vorticity andtemperature perturbations simply decay to zero.

The baroclinic feedback operates as follows.

1. Azimuthal gradients in the perturbation temperature field,@�0/@�, make the baroclinic term in the vorticity equation nonzero.

2. The baroclinic term is a source of vorticity that strengthensanticyclonic vortices.

3. Vortices stir the fluid, moving warm fluid from the innerannulus outward and cool fluid from the outer annulus inward.

4. This local advective heat transport enhances azimuthal tem-perature gradients, @�0/@�, completing the feedback cycle.

In order to show that the vortex growth is indeed due to thisbaroclinic feedback, the baroclinic termwas turned off at varioustimes in simulation set B (Fig. 5). In all of these trials, perturba-tion vorticity and kinetic energy drop off immediately when thebaroclinic term is turned off. This is particularly striking at t ¼10 and 100, when vortex strength is growing quickly in the ref-erence simulation. The kinetic energy in these plots is computedfrom the perturbation velocity fields.

The rate of thermal dissipation, � , plays a crucial role in theformation and growth of vortices. Figure 6 shows that there are

two distinct stages in these simulations: vortex formation, fromt ¼ 0 to about 5 orbital periods, and vortex growth, which occursafter t ¼ 5. During vortex formation, small thermal dissipation( large �) allows the strongest vortices to form, because the initialtemperature perturbation dies off quickly when thermal dissipa-tion is large, so that azimuthal temperature gradients are smallerand the baroclinic term produces less vorticity. This is not yet thebaroclinic feedback, because steps 3 and 4 are missing—it is justbaroclinic vorticity production from the initial temperature gradi-ents, which are steps 1 and 2.Once vortices form, they advect fluid about them (step 3),

creating the distinctive ‘‘sandwich’’ pattern of cool (warm) tem-perature perturbations on the inside (outside) of the vortex, asshown in Figure 4. These temperature perturbations create localazimuthal temperature gradients (step 4), completing the cycleof the baroclinic feedback. Sometime after 5 orbital periods,the vortices have formed, and the simulation transitions from thevortex formation stage to the vortex growth stage. Now that thebaroclinic feedback is operating, thermal dissipation has the op-posite effect than it had at early times (Fig. 6). If the disk coolsquickly (small �), then the warm and cool temperature pertur-bations can remain tight about each vortex, so that @�0/@� in thebaroclinic term is large, and the baroclinic feedback is strong. Ifthe disk cools slowly ( large �), the perturbation temperature re-sponds sluggishly to mixing by vortices, @�0/@� is small, thebaroclinic feedback is weak, and vortices simply dissipate away(Fig. 7). Quantitative measures of disk activity, like kinetic en-ergy and maximum temperature and vorticity, clearly show thatthe strength of the feedback and rate of growth of vortices isstrongly dependent on � (Fig. 6). In simulationswhere the radiative

Fig. 4.—Perturbation vorticity and temperature for simulation T3, where d ¼ �0:75. Despite dissipation of vorticity from the numerical code, the vortices remain longlived, because baroclinic vorticity production reinforces the vortices and balances the dissipation. Here the ‘‘sandwich’’ pattern about each vortex is clearly seen;temperature perturbations track each vortexwith awarm band to the outside and a cool band to the inside. [See the electronic edition of the Journal for a color version of thisfigure.]

Fig. 5.—Comparison of maximum perturbation vorticity j� 0j (left), perturbation kinetic energy (middle), and maximum perturbation temperature jT 0j (right) forsimulation B1, where the baroclinic term is turned off at the times indicated. When this occurs, the vorticity and kinetic energy immediately drop off, indicating that vortexgrowth is due to the baroclinic term. [See the electronic edition of the Journal for a color version of this figure.]


cooling rate was sufficiently fast, the baroclinic feedback countersdissipation, and vortices remain strong and coherent for hundredsof orbital periods (Fig. 4). The longest running simulation, T3,where d ¼ �0:75, ended at 600 orbital periods, at which point allthe vortices had merged into a single anticyclonic vortex.

There are two dissipative terms in temperature equation (4):the Laplacian terme9

2�0, which dissipatesmost quickly at smallscales, and the radiative cooling term ��0/� , which dissipatesequally at all scales. Can the Laplacian term play the same role asthe radiative cooling term in the baroclinic feedback? SimulationsPe1YPe3, in which Pe ranges from 104 to 2 ; 107, show that theLaplacian term can indeed play that role (Fig. 8); higher thermaldiffusion (smaller Peclet number) produces a stronger baroclinicfeedback. Higher diffusion produces warm and cool areas aroundeach vortex that aremore localized azimuthally and therefore havelarger azimuthal temperature gradients (Fig. 9). The azimuthal

temperature gradients then produce more vorticity through thebaroclinic term (step 1 of the baroclinic feedback).

Other simulations explore the role of background temperature(T1YT5) and background surface density (D1YD3). Largerbackground temperature gradients in simulations T1YT5 resultin larger and stronger vortices (Fig. 10). Quantitative measures,such as the kinetic energy, maximum vorticity, and maximumtemperature, all grow faster with larger temperature gradients(Fig. 11). The evolution of these quantities does not change asthe background density gradient is varied (Fig. 12). It is some-what surprising that the baroclinic feedback responds strongly tothe background temperature gradient but not the background den-sity gradient when these gradients seem to be on equal footing inthe baroclinic term (see eq. [7]). The background temperaturegradient is a source of available potential energy that can be trans-formed into the kinetic energy of vortex motion, as the vortices

Fig. 6.—Data from simulations Tau1 through Tau4, where � , the radiative cooling time, varies between 1 and 100. Two distinct stages can be seen: during vortexformation—at early times—the disk cools rapidly and cools fastest with small � ; once vortices have formed, the baroclinic feedback takes effect, and vortices grow fastestwith small � . [See the electronic edition of the Journal for a color version of this figure.]

Fig. 7.—Perturbation vorticity and temperature at 85 orbital periods from simulations Tau1, Tau2, and Tau3, where the radiative cooling time � is varied from 1 to 10.When the radiative cooling time is slow (large � , right), the temperature responds sluggishly to vortices, making the baroclinic feedback weak. [See the electronic edition ofthe Journal for a color version of this figure.]


transport heat from the hot inner disk to the cold outer disk. Thisnonlinear heat advection cannot be captured in a linear stabilityanalysis. Since the surface density is time independent in ouranelastic model, the background surface density gradient cannotprovide a source of potential energy for vortex formation.

3.2. Richardson Number

Several previous studies have used the Richardson numberto characterize instabilities in protoplaneary disks (Johnson &Gammie 2005a, 2006), and we compute the Richardson numberhere for comparison. We believe that the Solberg-Høiland crite-rion (Tassoul 2000; Rudiger et al. 2002; also see Part I, x 4), whichwas specifically created for differentially rotating astrophysicalfluids, is the best way to judge whether a disk is convectivelyunstable. For the simulations presented in this paper, the Solberg-Høiland values are positive (0.035Y0.299 yr�2), indicating thatthey are all convectively stable. However, the Richardson number

also provides useful information about the instability. We foundthat the baroclinic feedback is stronger (i.e., vortex growth isfaster) in simulations with more negative Richardson numbers.The Richardson number is often evoked in geophysical tur-

bulence to quantify the relationship between stratification andshear. For the atmosphere this dimensionless ratio is typically

Ri(z)¼ N 2(z)

@u=@zð Þ2¼ � g=�ð Þ d�=dzð Þ

@u=@zð Þ2; ð10Þ

whereN (z) is the local Brunt-Vaisala buoyancy frequency, u is thehorizontal velocity, z is the vertical coordinate, � is the density, andg is the gravitational force (Turner 1973). The numeratorN 2 givesthe strength of the stratification, where N 2 is negative for a con-vectively unstable fluid, positive and small for weakly stablestratification, and positive and large for strongly stable stratifica-tion. The denominator gives the strength of the shear.

Fig. 8.—Data from simulations Pe1YPe3, which compare the effects of varying Peclet number Pe. High Peclet number indicates low thermal diffusion. Increasingthermal diffusion (decreasing Pe) strengthens the baroclinic feedback, as exemplified by the slopes of the kinetic energy after t ¼ 10. This is similar to increasing theradiative cooling rate. [See the electronic edition of the Journal for a color version of this figure.]

Fig. 9.—Perturbation vorticity and temperature after 50 orbital periods for simulations Pe2 and Pe3. Simulation Pe3 (right column) has higher thermal diffusion (lowerPe), resulting in temperature perturbations that are larger in size (bottom right) and a stronger baroclinic feedback. [See the electronic edition of the Journal for a colorversion of this figure.]


In our equation set, the Richardson number is

Ri ¼ N 2

r @�0=@rð Þ½ �2¼ �cp d�0=drð Þ d�0=drð Þ

r @�0=@rð Þ½ �2: ð11Þ

By comparing the Richardson number (Fig. 13) with kineticenergy or maximum vorticity (Fig. 11) for simulations T1YT5, itis clear that the Richardson number is an excellent way to predictthe strength of the baroclinic feedback. When Ri � 0 (T4 andT5, where d ¼ �0:5 and �0.25, respectively), kinetic energyand vorticity simply decay away. When Ri < 0 (T1YT3, whered ¼ �2 to �0.75), the baroclinic feedback operates and kineticenergy and vorticity grow. In fact, the simulation with the mostnegative Richardson number (T1, where d ¼ �2) also has thefastest vortex growth.

Johnson & Gammie (2006) found that disks with a nearlyKeplerian rotation profile and radial gradients on the order of thedisk radius have Ri � �0:01 and are stable to local nonaxi-symmetric disturbances. Our simulations are not restricted to thisRi � �0:01 criterion, as simulations T1YT3 have quickly grow-ing instabilities, but have Richardson numbers in the range of�5 ; 10�5 to �5 ; 10�4.

The most likely difference between the two models that ac-counts for this disagreement is that our simulation allows smallinitial temperature perturbations to evolve into strong local vor-ticity perturbations that can produce stable vortices. This initialevolution can only occur if the viscous dissipation is sufficientlylow (high Reynolds number).

3.3. � -Viscosity

Protoplanetary disks are often described by the dimensionlessnumber �, which is used to parameterize an effective viscosity

� ¼ �csHp, where Hp is the vertical pressure scale height of thedisk and cs is the local sound speed. This simple description wasused to calculate the density structure, temperature structure, andmean components of laminar and turbulent gas flow in a disk(Shakura & Sunyaev 1973; Lynden-Bell & Pringle 1974; Lin &Papaloizou 1980).

The � -viscosity, rather than Reynolds number, is commonlyreported in the astrophysical literature to characterize the dissi-pation of energy in the disk. If the pressure scale height is scaledas Hp ¼ cs/�0, where �0 is the background angular velocity,the � -viscosity can be calculated as � (r) ¼ �e�0(r)/c

2s . In our

anelastic model, this measure cannot be used directly, becausecs 3 ju0j and pressure waves are temporally constrained to ad-just instantaneously. In order to compare the� -viscositywith otherprotoplanetary disk models, we report the ratio of the � -viscosityto the Mach number squared,

�

M 2¼ �e�0

u0j j2¼ �0

Re u0j j2; ð12Þ

whereM ¼ ju0j/cs and tildes indicate nondimensionalized variables.Azimuthal averages of � /M 2 for all simulations are between

10�5 and 10�2 (Fig. 14). Mach numbers of 0.01 or 0.1 wouldproduce corresponding � -viscosity ranges of 10�6 to 10�9 and10�4 to 10�7, respectively. Klahr & Bodenheimer (2003) reportMach numbers of 0.03 to 0.3 and � ¼ 10�2 to 10�4 for theirtwo-dimensional simulations with radial temperature gradients,which have resolutions of 622 and 1282, respectively. Godon &Livio (1999) report a viscosity parameter� ¼ 10�4 and 10�5 fortheir 1282 and 2562 simulations, respectively. In general, higherReynolds numbers (and thus smaller�-viscosity) can be achievedwith higher resolution. Our simulations have slightly higher res-olution (2562 and 5122) than other studies, and the effective

Fig. 11.—Comparison of data for simulations T1YT5, where the background temperature T0 � r d and d ranges from �0.25 to �2. Simulations in which thebackground temperature gradient is larger in magnitude increase the strength of the baroclinic feedback, resulting in increases in all three measures. [See the electronicedition of the Journal for a color version of this figure.]

Fig. 10.—Perturbation vorticity for simulations T1, T2, and T3, where the background temperature T0 � r d and d ¼ �2,�1, and�0.75, respectively. Larger backgroundtemperature gradients produce stronger baroclinic instabilities, so that vortices grow faster. [See the electronic edition of the Journal for a color version of this figure.]


Reynolds number is also higher (�107) due to the use of hy-perviscosity (see Part I, x 3). These characteristics contribute toa lower effective� -viscosity, so that our results include numerousfine, small-scale structures, such as layers of filaments, swirlingaround the vortices.

4. ANGULAR MOMENTUM TRANSPORT

The transport of angular momentum is of critical interest inthe study of protoplanetary disks. The traditional view of disk evo-lution is that angular momentum is transported outward as mass istransported inward. In Keplerian motion, gas near the star has afaster angular velocity than the gas further out. Turbulence in thegas creates an effective viscosity, so that faster moving gas in theinner disk will speed up slower gas in the outer disk, and the outerfast gas will tend to slow down the inner gas. Thus, angular mo-mentum is transported outward. As the inner gas slows down, it isno longer rotationally supported at that orbit and falls toward thestar to gain speed. Thus, mass is transported inward. Similar argu-ments can be made for particle collisions, which would enhancethis process.

The theory of outward angular momentum transport is based onazimuthally uniform dynamics in a viscous disk. Turbulence andcoherent structures may have radically different effects and are cur-rently a topic of intense scientific interest. Klahr &Bodenheimer

(2003) report that, just like in laminar flow, turbulence in baroclinicdisks transports angular momentum outward and mass inward,while releasing potential energy. Li et al. (2001) used afinite volumemodel of the compressible Euler equations to model Rossby wavesand vortex generation and found that individual vortices trans-port angular momentum outward. Johnson & Gammie (2005b)also found positive angular momentum flux in their compress-ible shearing-sheet model when they used strong initial vorticityperturbations to trigger vortex formation.As a simple example, consider locally Cartesian coordinates

in the radial and azimuthal direction. A slanted vortex of this localcoordinate system could have the stream function

�0 ¼ A exp � �

�0

� �2

þr�þ r

r0

� �2" #( )

: ð13Þ

Each streamline is a rotated ellipse centered at the origin withradial extent r0, azimuthal extent �0, and amplitude A. The angleof the ellipse is only affected by . This vortex is superposed onsome background flow, so the perturbation velocities in locallyCartesian coordinates are u0 ¼ �@��

0 and v0 ¼ @r�0. The angular

momentum transport of this vortex is

�0

Z 1

�1u0v0 d� ¼ � A2�0�0

ffiffiffiffiffiffi2�

p

4

; exp1

2

r

r0

� �2

�4þ 2�20r

20

� �" #: ð14Þ

Fig. 13.—Richardson number Ri(r) for simulations T1 through T5, where thebackground temperature T0 � r d . The Richardson number depends only onbackground functions, so it is constant in time. By comparing to Fig. 11, one cansee that a more negative Ri (for example, T1, d ¼ �2) indicates a stronger in-stability, and less negative Ri (for example, T3, d ¼ �0:75) indicates a weakerinstability. For example, for T4 (d ¼ �0:5) and T5 (d ¼ 0:25) Ri is zero and po-sitive, respectively, and the baroclinic instability is not active in either case. [Seethe electronic edition of the Journal for a color version of this figure.]

Fig. 12.—Comparison of data for simulations D1YD3, where the background surface density�0 � rb and b ¼ �1,�3/2, and�2, respectively. The data show that vary-ing the gradient of surface density results in little difference in the strength of the baroclinic feedback. [See the electronic edition of the Journal for a color version of this figure.]

Fig. 14.—Azimuthal averages of � /M 2 for a typical simulation (A1) for varioustimes, in orbital periods. This quotient ranges between 10�2 and 10�5 for all simu-lations. [See the electronic edition of the Journal for a color version of this figure.]


The sign of this quantity depends only on , the angle of thevortex. Positive and negative vortices with the same orientationhave the same angular momentum transport, as the sign of thevortex only affects A, a squared quantity in equation (14). Thisindicates that it is only the orientation of the vortex within theflow that affects whether momentum travels toward the insideor outside of the disk; the direction of rotation of the vortex isinconsequential.

Our simple analytic example is shown in Figure 15 for ¼�0:5 (top left) and 0.5 (top right), where the other constants are�0 ¼ 1, r0 ¼ 2, and A ¼ 1. The bottom panels show vorticeswith similar orientations in the full numerical model. Clearly,the direction of the angular momentum transport only dependson the angle of the vortex, as in the analytic example. These vor-tices are not from the simulations in Table 1, but are from shortsimulations that were specifically designed to produce theseorientations.

What is the effect of vortices on angular momentum transportwhen they are imbedded in a turbulent flow populated with fila-ments and other interacting vortices? To investigate this, the an-gular momentum transport,�0u

0v0, was recorded using azimuthaland global averages in the numerical model. In typical simula-tions, like A1, the angular momentum transport is highly variablein space and time (Fig. 16). Specifically, the global angular mo-mentum transport cycles chaotically between positive and nega-tive periods as the vorticity field evolves. The angular momentumtransport in these simulations is influenced by the interaction ofnumerous vortices and vorticity filaments, which is much morecomplicated than the single vortex case. We would expect thatindividual vortices within this flow would contribute angular mo-mentum based on their orientation and that these individual con-tributions could be summed tofind the angularmomentum transport.However, a separate study of a small number of vortices andfilaments in the flow would be required to say conclusively.

Fig. 15.—Vorticity from the analytic example (top row) and numerical model (bottom row) in which vortices are radially leaning out (left column) or leaning in (rightcolumn). The inset shows the angular momentum flux, which is positive for outward-leaning vortices and negative for inward-leaning vortices. For these pedagogicalexamples, the scale for the vorticity and angular momentum flux is arbitrary. [See the electronic edition of the Journal for a color version of this figure.]

Fig. 16.—Global angular momentum transport (left) and angular momentum transport as a function of radius (right) for simulation A1, where each curve is an averageof 10measurements taken over one-half of an orbital period and times correspond to the images shown in Fig. 3. As described in the text, angular momentum transport dueto a particular vortex depends on the orientation of the vortices. The spatial and temporal variability in angular momentum transport shown here is due to variability in theorientation of the vortices. [See the electronic edition of the Journal for a color version of this figure.]


Based on the simulation results in Figure 16, we conclude thatthe total angular momentum transport in an anelastic-gas tur-bulent disk with vortices and vortex filaments may be inward oroutward and can vary locally in the disk depending on the ori-entation of the vortices. In contrast, studies of compressible-gasdisks have all found that vortices transport angular momentumoutward (Klahr & Bodenheimer 2003; Li et al. 2001; Johnson &Gammie 2005b). Compressible-gas models include acousticwaves, which are filtered out of our anelastic model. Shocks pro-duced by acoustic waves in these studies may orient the vorticesuniformly, so that angular momentum is transported outward,or transport angular momentum by other means.

5. CONCLUSIONS

In this study we are interested in exploring the necessary con-ditions for vortex formation in an anelastic protoplanetary diskmodel that includes baroclinicity and radiative cooling. We haveshown that long-lived vortices can be formed by initial randomtemperature perturbations through themechanism of the baroclinicinstability. Vortex production must compete with the strong in-hibiting effects of Keplerian shear, an effect observed by otherauthors (Bracco et al. 1999; Godon & Livio 1999). Only anti-cyclones survive in Keplerian disks, while cyclones shear outand diffuse away. Many previous studies do not include barocliniceffects due to a lack of thermodynamics (Bracco et al. 1999;Umurhan & Regev 2004; Johnson & Gammie 2005b) or an as-sumed polytropic relation (Godon & Livio 1999) and, therefore,onlymodel decaying turbulence froman initial vorticity distribution.

In the baroclinic feedback, local azimuthal temperature gradientsproduce vorticity through the baroclinic term in the vorticity equa-tion. This strengthens vortices, which advect the surrounding ther-mally stratified gas, producing stronger local temperature gradients.

The baroclinic feedback can only operate once vortices haveformed, as a coherent vortex is required to produce the ‘‘sandwich’’pattern of warm and cold gas about each vortex. In our simulations,two distinct stages can be seen: vortex formation, in which theinitial temperature perturbation rapidly decays; and vortex growth,in which the baroclinic feedback takes effect and both perturbationvorticity and perturbation temperature grow for the rest of thesimulation.

The conditions required for the baroclinic feedback are: (1) asufficiently large radial temperature gradient in the backgroundstratification and either (2) a fast radiative cooling time or (3) highthermal dissipation (i.e., small Peclet number). If the backgroundradial temperature gradient (condition 1) is too small, advectionby the vortices does not strengthen local azimuthal temperaturegradients. Both conditions 2 and 3 allow temperature perturba-tions to track vortices, so that the structure of the vorticity andtemperature fields are strongly coupled. The difference betweenthe two mechanisms is that thermal dissipation smooths outsmall-scale features, resulting in large-scale thermal perturba-tions, while radiative cooling affects all scales equally and pro-duces smaller thermal perturbations. Varying the backgroundsurface density gradient had no effect on the strength of thebaroclinic feedback.

One of the goals of this study was to see if the baroclinicinstabilities found byKlahr &Bodenheimer (2003) can be repro-duced in an anelastic equation set with simplified dynamics.They found that if the background radial entropy gradient is zero—this turns off all baroclinic effects—then the initial vorticity per-turbation just decays away (their model 2). When entropy variesradially so that the temperature T � r�1, the flow becomes turbu-lentwithin a feworbits, and vortices are formed (theirmodels 3Y6).Our results (condition 1, above) agree with this result and further-

more show that vorticity grows faster with steeper backgroundradial temperature profiles.Our conditions 2 and 3 state that thermal dissipation is required

for the baroclinic instability. Indeed, in our simulations when bothforms of thermal dissipation were sufficiently slow, the vorticitydied off after the initial vortex formation. This requirement is indisagreement with the findings of Klahr & Bodenheimer (2003),as they ‘‘got rid of radiation transport’’ (i.e., there is no radiativecooling) for their two-dimensional simulations. Because we usea simplified one-parameter radiative cooling model, we can seethat the baroclinic feedback strongly depends on the coolingtime � . It is not clear to us whyKlahr &Bodenheimer (2003) seevortex growth when radiative cooling is missing. Either there issome implicit thermal diffusion in their code, or their vortices aregrowing through a different mechanism than ours.The range of Richardson numbers at which we form vortices

differs from Johnson & Gammie (2006), who find that simu-lations with Ri � �0:01 are stable to local nonaxisymmetricdisturbances. Our simulations with �0:01 � Ri < 0 form tur-bulent instabilities and vortices quite easily. A likely explanationfor this difference is that our simulations have the large Reynoldsnumbers required to permit small initial temperature perturba-tions to evolve into strong local vorticity perturbations beforethey are viscously damped.The results of a model must be understood within the as-

ymptotic regime in which it is valid. Our model assumes that thedisk is thin and hydrostatically balanced in the vertical, so thatonly large-scale horizontal motions are considered. The verticaldynamics, which we do not consider, can affect vortex stabilityas well. Knobloch & Spruit (1986) argue that height variationsmust be included when discussing shear instabilities in the disk,because vertical gradients of azimuthal velocity are not small.Barranco & Marcus (2005) found that columnar vortices areunstable to small perturbations, but that internal gravity wavesnaturally create robust off-midplane vortices (see discussion inPart I, x 5).Indeed, the most serious limitation of this study is our as-

sumption of a two-dimensional disk. If the initial baroclinic in-stabilities have radial scales that are small compared to the diskscale height, as is suggested by our local simulations presentedin Part I, then the vertical stratification of the disk will likelyplay a major role in the nonlinear development and the longevityof vortices. Nevertheless, we believe that our two-dimensionalsimulations have served to identify physical processes that willlikely play a role in a fully three-dimensional simulation. Forexample, if vortices primarily form on the upper and lower sur-faces of disks, then the radiative cooling rates will likely be morerapid than if the vortices were buried in the optically thick mid-plane of the disk.We therefore expect that vortices confined to thesurface of a disk could have longer lifetimes due to their ability toefficiently transport heat radially outward.Our model is based on an anelastic mass conservation equation,

which filters out acoustic waves. Thus, shock waves and their po-tential interactions with vortices and angular momentum trans-port do not appear in our study, as they have in compressible-gasmodels (Li et al. 2001; Klahr & Bodenheimer 2003; Johnson &Gammie 2005b). These studies all found that angular momentumis always transported outward, while we found that it may be in-ward or outward and is highly variable in space and time. Theobvious difference in the dynamics is the lack of shocks in ouranelastic-gas model. This suggests that shocks play an importantrole in the transport of angularmomentum in protoplanetary disks.The baroclinic feedback enhances vortices so that they can be

long lived. In our simulations, they survived for the duration of


the longest numerical simulation—for over 600 orbital periods(12,400 yr)—and showed no signs of decaying. This studyshows that the baroclinic feedback is a viable mechanism for thegeneration and persistence of vortices in protoplanetary disks. Inthe baroclinic feedback, the background temperature gradient pro-vides a source of available potential energy that can drive thevortices indefinitely, even in the presence of a finite rate of vis-cous dissipation. As vortices are efficient at collecting particlesfrom the surrounding gas, they are a natural place for planets toform in the disk. If vortices are long-lived coherent structures inprotoplanetary disks, as suggested by this work, they offer a wayto overcome the difficulties presented by current planetary for-mation theories. The high particle concentrations in vortices speedup the core accretion process; likewise, this high particle concen-tration could lead to gravitational instability. Both core accretionand gravitational instability are hindered in the majority of thedisk, where particle concentrations are low. Strong concentrationof particlesmay require the vortices to grow large compared to the

scale height of the disk, so that they can extend through the mid-plane of the disk, where most particles will reside. Fully three-dimensional simulations are therefore required to establish therelevance of vortices to planet formation.

We thank P. Marcus for insightful feedback and practicaladvice, A. P. Boss for useful discussions, and an anonymousreferee for criticism that significantly improved the final version.M. R. P. has been supported by an NSF Vigre grant, DMS-9810751, awarded to the Applied Mathematics Department atthe University of Colorado at Boulder.M. R. P.’s workwas in partcarried out under the auspices of the National Nuclear SecurityAdministration of the US Department of Energy at Los AlamosNational Laboratory under contract DE-AC52-06NA25396. K. J.has been supported by NSF grant OCE-0137347 as well as theUniversity of Colorado Faculty Fellowship. G. R. S. was sup-ported by NASA’s Origins of Solar Systems research program.

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BAROCLINIC VORTICITY PRODUCTION IN PROTOPLANETARY … · BAROCLINIC VORTICITY PRODUCTION IN PROTOPLANETARY DISKS. II. VORTEX GROWTH AND LONGEVITY Mark R. Petersen Department of Applied