The Astrophysical Journal, 734:118 (17pp), 2011 June 20 doi:10.1088/0004-637X/734/2/118C© 2011. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

THREE-DIMENSIONAL SPECTRAL SIMULATIONS OF ANELASTIC TURBULENT CONVECTION

Kaloyan Penev1, Joseph Barranco

2, and Dimitar Sasselov

31 Astronomy Department, Harvard University, 60 Garden St., M.S. 10, Cambridge, MA 02138, USA

2 Department of Physics and Astronomy, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132-4163, USA3 Astronomy Department, Harvard University, 60 Garden St., M.S. 16, Cambridge, MA 02138, USA

Received 2009 July 13; accepted 2011 April 11; published 2011 June 6

ABSTRACT

We have adapted the anelastic spectral code of Barranco & Marcus to simulate a turbulent convective layer withthe intention of studying the effectiveness of turbulent eddies in dissipating external shear (e.g., tides). We derivethe anelastic equations, show the time integration scheme we use to evolve these equations, and present the testswe ran to confirm that our code does what we expect. Further, we apply a perturbative approach to find anapproximate scaling of the effective eddy viscosity with frequency and find that it is in general agreement with anestimate obtained by applying the same procedure to a realistic simulation of the upper layers of the solar convectivezone.

Key words: convection – hydrodynamics – methods: analytical – turbulence

Online-only material: color figures

1. INTRODUCTION

Dissipation of stellar tides and oscillations is often consideredto be mainly due to the turbulent flow in their convective zones.Usually, the effects of the turbulent flow are parameterizedby some sort of effective viscosity coefficient. Clearly, thesituation is not as simple as that and the usual “fix” is to allowthis viscosity coefficient to depend on the perturbation beingdissipated, most notably its frequency and perhaps the directionof the shear it creates.

Completely analytical treatments start by assuming aKolmogorov spectrum for the turbulent flow and combine itwith some prescription for the effectiveness of eddies in dis-sipating perturbations of the given period. Since Kolmogorovturbulence is isotropic, the direction of shear is unimportant insuch prescriptions. Two such prescriptions have been used.

The first, proposed by Zahn (1966, 1989), states that when theperiod of the perturbation (T) is shorter than the turnover times ofsome eddies, their dissipation efficiency should be proportionalto the fraction of a churn they manage to complete in half aperturbation period. In this case the dissipation is dominated bythe largest local eddies, and the effective viscosity coefficientscales like

ν = νmax min

[(T

2τ

), 1

], (1)

where τ is the turnover time of the eddies with the largest size.The second prescription is due to Goldreich & Nicholson

(1977) and Goldreich & Keeley (1977). They argue that eddieswith turnover times much greater than the period of the per-turbation will not contribute appreciably to the dissipation, andhence the effective viscosity should be dominated by the largesteddies with turnover times shorter than T/2π . Then, the Kol-mogorov prescription of turbulence predicts that the effectiveviscosity will scale as

ν = νmax min

[(T

2πτ

)2

, 1

]. (2)

Zahn’s prescription has been tested against tidal circulariza-tion times for binaries containing a giant star (Verbunt & Phin-ney 1995) and is in general agreement with observations. Also

Zahn’s prescription is in better agreement with observed tidaldissipation of binary stars in clusters (Meibom & Mathieu 2005)and with the location of the red edge of the Cepheid instabilitystrip (Gonczi 1982).

The less efficient prescription has been used successfully byGoldreich & Keeley (1977), Goldreich & Kumar (1988), andGoldreich et al. (1994) to develop a theory for the damping ofthe solar p-modes. In this case, the more effective dissipationwould require dramatic changes in the excitation mechanism inorder to explain the observed amplitudes.

Finally, Goodman & Oh (1997) developed a perturbativederivation of the convective viscosity, which for a Kolmogorovscaling gives a result that is closer to the less efficient Goldreich& Nicholson viscosity than it is to Zahn’s. While providing afirmer theoretical basis for the former scaling, this does not re-solve the observational problem of insufficient tidal dissipation.

The development of two-dimensional (2D) and three-dimensional (3D) simulations of solar convection hints at apossible resolution of this problem. The convective flow thatthese simulations predict is fundamentally very different fromthe assumed Kolmogorov turbulence (Sofia & Chan 1984; Stein& Nordlund 1989; Malagoli et al. 1990). The first major differ-ence is that the frequency power spectrum of the velocity fieldis much flatter than the Kolmogorov power spectrum, and henceone might expect that the dissipation will decrease significantlyslower as frequency increases relative to the Kolmogorov case.Another major difference is that the velocity field is no longerisotropic, and hence one would expect it to react differently toshear in different directions. That is, if we would use an effec-tive viscosity coefficient, it should be a tensor and not a scalarquantity.

As a first step in investigating that possibility (Penev et al.2009b), we applied the perturbative approach developed byGoodman & Oh (1997) to numerical models (Robinson et al.2003) of small portions of the stellar convection of low-massstars to find the scaling of the components of the effectiveviscosity tensor with frequency. Somewhat unexpectedly, wefound that the scaling closely follows Zahn’s prescription, eventhough when the same approach is applied to Kolmogorovturbulence it gives results closer to those of Goldreich andcollaborators.

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That left a lot of questions unanswered. For example: arethe effects of turbulence anything at all like that of molecularviscosity? Is taking only the lowest order term in a powerseries expansion of the energy dissipation rate an acceptableapproximation? Is the turbulent dissipation dominated by thelargest eddies as Zahn (1966) claims or are the eddies withturnover times closest to the external period the most importantones, like in the Goldreich and collaborators picture?

The perturbative approach also assumes that the spatial scaleof the external shear being dissipated is large compared to allconvective scales and hence can be assumed to be a linearfunction of position, but it would be interesting to see howquickly dissipation efficiency is lost as the spatial scale of theexternal shear decreases.

To tackle these questions, we adapted the anelastic spectralcode of Barranco & Marcus (2006) to simulate a convectivelyunstable box in which we are able to place external, time-dependent shear as part of the evolution equations and observethe effects of the turbulent convective flow directly.

In order to make the code suitable to simulate convection,it was necessary to provide a mechanism that would supplyenergy at the bottom of the box and remove it at the top.To accommodate this, heat diffusion was added that allowsheat to enter/leave through the bottom/top wall, and drive theconvective motions. This, in turn, necessitated that we allow fora more general background state, slightly modified treatment ofthe pressure, and the introduction of boundary conditions on thetemperature. In addition, we added an external forcing term andremoved the Keplerian shear and the accompanying it shearingcoordinates.

The resulting code is ideal for our purposes for the followingreasons.

1. The anelastic approximation means no shock or soundwaves can exist, and hence the time steps can be muchlarger than for a fully compressible code, and yet buoyancyphenomena, such as convection, can occur, unlike in com-pletely incompressible codes. This allows us to get muchcloser to the actual timescales of interest (usually of orderhours or days), than a fully compressible code would. TheBoussinesq approximation has those same properties, how-ever it is not suitable for simulating stratified convection,where the buoyancy does not depend only on the tempera-ture.

2. Spectral codes are more efficient at simulating turbulentflows than finite difference codes, because their spatialaccuracy is exponential rather than a power law, so theyare capable of reliably reproducing the turbulent flow, evenat modest resolutions.

3. The grid happens to have higher density near the top andbottom of the box, where it is needed in order to resolvethe boundary layers that develop there due to the boundaryconditions we impose.

In this paper, we present the details of the modified codeas well as a perturbative estimate of the efficiency of turbu-lent dissipation. In Section 2, we derive the anelastic approxi-mation to the Navier–Stokes equations, modified to include atime-dependent shear, in Section 3 we present the numericalalgorithm, in Section 4 we show the tests we ran to confirm thatour implementation actually evolves the desired equations andverify that the numerical scheme is indeed second-order accu-rate in time, and finally, in Section 5 we use the perturbativecalculation to get an estimate of the effective turbulent viscosity

in our simulated box and compare it with the results of Penevet al. (2009b).

2. THE EVOLUTION EQUATIONS

The fully compressible set of equations we would like toevolve, in a box-like domain, with the x- and y-axes perpendicu-lar to the acceleration of gravity and the z-axis pointing againstthe gravity, are

∂v∂t

= −v · ∇v − 1

ρ∇p − gz + f, (3)

∂ρ

∂t= −∇ · ρv, (4)

∂θ

∂t= −v · ∇θ +

θ

CpTρ∇ · (κ∇T ) , (5)

θ = T

(p0

p

) RCp

, (6)

p = ρRT, (7)

where p is the pressure, T is the temperature, ρ is the density, vis the velocity, θ is the potential temperature, Cp is the specificheat at constant pressure, R is the ideal gas constant, g is theacceleration due to gravity, κ is the heat diffusion coefficient,and f is some external forcing (e.g., tidal force).

As was said before, we do not actually evolve Equations (3)–(7), but rather their anelastic approximation. To develop theanelastic equations, each flow variable is split into a sum of atime-independent background component (denoted by an overbar) and a time-varying perturbation (denoted by tilde over itssymbol) superimposed on top of that.

The background variables satisfy the above equations with allx, y, and t derivatives set to zero and no velocity. The boundaryconditions that complete these equations are that we require thebackground temperature on the top and bottom walls to havesome fixed values Tlow and Thigh, respectively, and the pressureat the top wall to have some value, ptop.

Alternatively, we could fix the temperature gradients at theboundary, which would set the flux that the simulated convectivelayer must transport. Neither of those two options is exactlywhat the actual physical problem requires. We could base ourboundary conditions on reliable physics, if we were planning tosimulate the entire convective zone, however, this is impossibleto do at the resolution required to study the turbulent dissipation,so we will ultimately be interested only in the flow that developsin the interior of the box and plan to exclude the regions nearthe top and bottom boundaries from any analysis we do. For thisreason the choice of the exact thermal boundary conditions willhopefully have little impact on our results.

For convenience, we define the following quantities:

K (z) ≡∫ z

−Lz/2

dz

κ(z)(8)

and α ≡ Th − Tl

K(Lz/2), (9)

where the z coordinate has a value −Lz/2 at the bottom of thebox and Lz/2 at the top of the box.

The background variables that emerge as the solution toEquations (3)–(7) are

T = Th − αK, (10)

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

ρ = ptop

RTl

exp

[∫ Lz/2

z

1

T

( g

R− α

κ

)dz

], (11)

p = ρRT , (12)

θ = T

(p0

p

) RCp

. (13)

To get the anelastic equations, we write all the variables in theoriginal equations as a background and perturbation component:

p = p + p, (14)

ρ = ρ + ρ, (15)

T = T + T , (16)

θ = θ + θ . (17)

Inserting Equations (14)–(17) in the fully compressibleEquations (3)–(7) and dropping terms quadratic in the pertur-bation variables, we get the set of anelastic equations we wouldlike to evolve with time:

∇ · ρv = 0, (18)

∂v∂t

= v × ω − ∇h +θ

θgz + f, (19)

∂θ

∂t= −vz

dθ

dz− v · ∇ θ +

θ

CpT ρ∇ · (κ∇T ), (20)

h ≡ p

ρ+

v2

2, (21)

p

p= ρ

ρ+

T

T, (22)

θ

θ= p

ρgHρ

− ρ

ρ, (23)

where we have introduced the enthalpy h (defined byEquation (21)), the density scale height Hρ ≡ −( d ln ρ

dz)−1, and

the vorticity ω ≡ ∇ × v. Equation (23) is not the correct lin-earized equation for the potential temperature. The denominatorof the pressure term should have been γ p instead of ρgHρ . Thisreplacement was introduced by Bannon (1996). He showed thatit is required to ensure that the anelastic equations conserveenergy.

Note that in Equation (20) we do not include a heat diffusionterm proportional to the potential temperature gradient as issometimes done in simulations of the solar convection to allowfor some subgrid turbulent heat transport (cf. Clune et al. 1999).

To define the time evolution completely, we need to addboundary conditions to the above equations. The boundaryconditions on the four vertical walls of the domain are setby the fact that we use Fourier series expansion for thehorizontal dependence of all quantities, and hence we mustimpose periodicity in those directions. In addition to that, wewant the temperature at the top and bottom boundaries to havethe values we specified for T (Tl and Th, respectively), and hence

its perturbation should be zero. Also we require that the top andbottom walls are impermeable, but with no friction. That meansthat we set vz to zero at the top and bottom boundaries and donot require anything for vx and vy .

The anelastic equations above obey a set of energy conser-vation equations for the following definitions of the kinetic andthermal energies:

EK ≡∫

V

ρv2

2dV, (24)

ET ≡∫

V

CpρTθ

θdV . (25)

Using the anelastic evolution equations, the rates of change ofthese two energies can be shown to be

dEK

dt= E1, (26)

dET

dt= −E1 + E2, (27)

where the sinks/sources are defined as

E1 ≡ g

∫V

ρ

θvθdV , (28)

E2 ≡∫ (

κ∂T

∂z

∣∣∣∣Lz2

− κ∂T

∂z

∣∣∣∣− Lz

2

)dxdy. (29)

3. NUMERICAL TIME EVOLUTION

3.1. Spectral Method

The spectral method used in this code is exactly the same asin Barranco & Marcus (2006, Section 3.2). For completenesswe reproduce it below adopting their notation.

The spectral representation for a flow variable (q) used inthe code, at a resolution of Nx × Ny × Nz in the x-, y-, andz-directions, respectively, is given by

q(x, y, z, t) ≈Nx2∑

k=− Nx2 +1

Ny

2∑l=− Ny

2 +1

Nz∑m=0

qklm

× (t)ei2πkx/Lx ei2πly/Ly Tm(z), (30)

where the vertical basis functions, Tm(z) ≡ cos(m cos−1 zLz

), areChebyshev polynomials. The numerical method we use doesalmost all calculations in the wavenumber/Chebyshev space,except for taking products of variables, which are done byfirst transforming back to physical (x, y, z) space on a gridof collocation points, taking the product there and transformingback.

The grid of collocation points is defined by {xl = −Lx/2 +lΔx, ym = −Ly/2 + mΔy, zn}, where Δx = Lx/Nx andΔy = Ly/Ny , and the vertical collocation points are definedby

z = Lz

2cos (ξ ) , ξn ≡ nπ

Nz

. (31)

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

With this definition, the vertical Chebyshev basis functions of zare mapped to cosine basis functions in ξ :

Tm

(2z

Lz

)= cos

[m cos−1

(2z

Lz

)]= cos (mξ ) , (32)

so with this choice of unevenly spaced z collocation points, theChebyshev transform in the vertical direction can be computedas a discrete cosine transform in the evenly spaced ξ coordinate.

3.2. Time Integration

Time integration is performed using a fractional time stepmethod. Each fractional step consists of updating the flow vari-ables according to only one or several terms of the correspondingevolution equation, introducing as many steps as are necessaryto incorporate all terms. Special care must be taken to ensurethat the order of the fractional steps does not introduce spurioussecond-order terms that would render the time integration first-order accurate in time. Below we present the steps in the orderin which they are performed in the code. In what follows, wewill use superscripts to refer to the index of the time step, withfractional values corresponding to fractional steps.

3.2.1. Advection Step

The advection step is fully explicit. It uses values fromthe current and previous time steps to achieve second-orderaccuracy. It first calculates the quantities

M ≡ v × ω +θ

θgz (33)

and N ≡ −vz

dθ

dz− v · ∇ θ . (34)

Then, the velocity and the temperature are updated according to

vN+ 13 = vN +

Δt

2(3MN − MN−1) (35)

and θN+ 13 = θN +

Δt

2(3NN − NN−1). (36)

3.2.2. Hyperviscosity Step

The purpose of this step is to suppress the highest modesboth in the horizontal and vertical directions, in order toavoid the buildup of power there due to the truncation ofthe spectrum at some finite number of spectral coefficients.For finite difference codes, a step of this sort is unnecessary,because there is some finite “grid viscosity,” associated withthe numerical scheme, while for spectral codes, there is noequivalent effect. We implement the hyperviscosity step exactlyin the way described in Barranco & Marcus (2006), suppressingeach spectral coefficient by a factor as follows:

vN+ 2

3klm = v

N+ 13

klm exp[−Δt

(ν

hyp⊥ k

2p

⊥ + νhypz m2p

)], (37)

θN+ 2

3klm = θ

N+ 13

klm exp[−Δt

(ν

hyp⊥ k

2p

⊥ + νhypz m2p

)], (38)

where νhyp⊥ and ν

hypz are some hyperviscosity coefficients,

p is an integer between 1 and 6, k2⊥ ≡ k2

x + k2y is the

horizontal wavenumber, and m is the order of the Chebyshevpolynomial that this particular amplitude applies to.

Equation (37) is equivalent to adding a term to the right-handside of the momentum Equation (3) of the form(

∂v∂t

)hyp

= −{

νhyp⊥

[∂2

∂x2+

∂2

∂y2

]p

+ νhypz

[√1 − z2

∂

∂z

(√1 − z2

∂

∂z

)]p}v.

(39)

The complicated z operator has been chosen to have theChebyshev polynomials as its eigenfunctions. This replacementwas suggested by Boyd (1998), because having an actualdiffusion operator would raise the order of the differentialequations and require the introduction of additional nonphysicalboundary conditions at the top and bottom of the box.

In addition, choosing a large value for p (typically p = 6)means that only the high spatial frequency modes will beaffected, leaving the large-scale flow closer to inviscid, whilestill maintaining numerical stability.

This approach is different from what is often done in simula-tions of the solar convection where actual viscosity is introduced(cf. Chan & Sofia 1989; Cattaneo et al. 1991; Stein et al. 1989).This is sometimes necessary, especially if magnetic fields areconsidered, in order to have appropriate ratios between magneticand molecular viscosity, for example.

Similarly, Equation (38) is equivalent to adding a term to theright-hand side of the temperature Equation (5) of the form(

∂θ

∂t

)hyp

= −{

νhyp⊥

[∂2

∂x2+

∂2

∂y2

]p

+ νhypz

[√1 − z2

∂

∂z

(√1 − z2

∂

∂z

)]p}θ .

(40)

It is important to note that the introduction of this artificialhyperviscosity term breaks the conservation of energy in ourbox, since we do not add a corresponding heat source toreplenish the energy lost.

3.2.3. Pressure Step

The pressure step ensures that the velocity at the end of thetime step satisfies the anelastic constraint—Equation (18). In thenumerical scheme, we abandon the enthalpy at a specific time asa variable, and instead use its average between two consecutivetime steps:

ΠN+1 ≡ 1

2(hN+1 + hN ). (41)

We then achieve second-order accurate time evolution byupdating the velocity according to

vN+1 = vN+ 23 − Δt

2(∇hN+1 + ∇hN ) = vN+ 2

3 − Δt∇ΠN+1. (42)

Imposing the anelastic constraint, we get

0 = ∇ · vN+1 + vN+1z

d ln ρ

dz,

= ∇ · vN+ 23 − Δt∇2ΠN+1 + v

N+ 23

z

d ln ρ

dz− Δt

∂ΠN+1

∂z

d ln ρ

dz.

(43)

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

Regrouping and requiring vz|±Lz/2 = 0, we get the differentialequation for updating Π with its boundary conditions:[

∇2 +d ln ρ

dz

∂

∂z

]ΠN+1 = 1

Δt

(∇ · vN+ 2

3 + vN+ 2

3z

d ln ρ

dz

)(44)

∂ΠN+1

∂z

∣∣∣∣±Lz/2

= 1

Δtv

N+ 23

z

∣∣∣±Lz/2

. (45)

Equation (45) is imposed at the expense of the two highestChebyshev modes of the momentum Equation (19). For detailsof the implementation of this step in our code, see Appendix A.

3.2.4. Heat Diffusion Step

The heat diffusion step updates the potential temperatureaccording to

θN+1 = θN+ 23 +

Δt

2

θRκ

Cpp

[∇2 +

d ln κ

dz

∂

∂z

](T N + T N+1). (46)

When we express T in terms of v, h, and θ , this gives a second-order differential equation for θN+1. In terms of the followingdefinitions,

P ≡ − αθ

gκT

[∇2 − d ln κ

dz

∂

∂z+

κ ′2

κ2− κ ′′

κ

] (h − v2

2

), (47)

C1 ≡ d ln κ

dz− 2g

CpT, (48)

C2 ≡ g

CpT

(d ln θ

dz− d ln κ

dz

), (49)

C3 ≡ −2Cpρ

κΔt, (50)

C4 ≡ αθ

gκT, (51)

the equation we solve during this step is(∇2 + C1

∂

∂z+ C2 + C3

)θN+1 = ,

= (PN+1 + PN ) −(

∇2 + C1∂

∂z+ C2

)θN + C3θ

N+ 23 .

(52)

This is a second-order equation for θN+1, so we need twoboundary conditions to make the solution unique. These comefrom the requirement that the temperature perturbation on theboundary vanishes T

∣∣± Lz

2= 0. This requires

θN+1∣∣± Lz

2= C4

(hN+1 − v2

2

). (53)

We see that the differential Equation (52) uses only the valueof ΠN+1, but the boundary conditions need the value of theenthalpy at the updated time. There are two options of how toobtain this value. One is to keep track of the enthalpy from the

very beginning, and after each pressure step, update it accordingto

hN+1 = 2ΠN+1 − hN . (54)

However, if the initial value of h is not perfectly set thisprescription will lead to oscillations in the value of the enthalpyat the top and bottom boundaries. To illustrate this, assume thatinitially we set h0 to a value that is slightly higher than whatit should be. Since Π1 is calculated without reference to theinitial conditions, it will have the correct value. This will leadto h1 being slightly smaller, and so on. These oscillations arethen translated to the interior of the box, through their effect onthe potential temperature variable. Further, we found that theamplitude of these oscillations grows with time for all the caseswe ran.

In order to avoid this problem, instead of introducing the ad-ditional variable h, with the only purpose of getting temperatureboundary conditions, we use a second-order approximation toits value through Π:

hN+1 ≈ 1

2(3ΠN+1 − ΠN ). (55)

The boundary conditions (Equation (53)) are then imposedby ignoring the differential Equation (52) at the top andbottom walls (where it does not make sense anyway). For theimplementation details of this step, see Appendix B.

4. TESTS

Since all the Fourier transforms and differential operators arecomputed in exactly the same way as in Barranco & Marcus(2006), we refer to Section 4.1 of that paper for a discussion ofthe performance of the code. There are two main performancedifferences between our version of the code and Barranco &Marcus (2006). The first one is that the evaluation of the heatdiffusion step requires the evaluation of four additional 3Dfast Fourier transforms (FFTs), which are the slowest operationperformed by their code. The second difference is the fact thatwe have allowed for a more general background density and heatdiffusion profile. On one hand, this introduces several additionalFFTs along the vertical direction, and more importantly solvingan Nz × Nz matrix Nx × Ny times for every pressure, Greens,and heat diffusion time step, making these the slowest steps forus. However, none of these changes lead to a different scalingof the number of time steps calculated per unit time with thenumber of processors used, and the overall approximately linearscaling is preserved for computational grids with resolutions ofat least 1283 on up to 128 processors, which were the typicalresolution and number of processors used.

Below, we present various tests we ran to confirm that the codeis evolving the equations described above and that the numericalscheme employed is second-order accurate in time. We repeatall the tests that Barranco and Marcus performed to confirm ournumerical scheme and add a test of the Brunt–Vaisala frequencyin our box. Since this is the critical quantity that determines thetransition from convective stability to instability, it is an essentialtest for the intended use of the code.

4.1. The Background State

Most of the tests we performed share the same backgroundstate, which is also the background we used for the finalconvective box simulation. The only time we modified it was

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

in order to simulate a convectively stable box and observe thebuoyancy mode. Here we describe that background state.

We use a self-consistent non-dimensional set of units. Hence,the results apply for any choice of units.

The dimensions of the box are 4 × 4 × 4, with a resolution of128 collocation points per direction.

We need to choose the vertical profile of the heat diffusioncoefficient. On the one hand, because the vertical grid spacingis very small near the boundaries, having large vz in thoseregions would mean that the CFL stability condition wouldimpose an extremely small time step. In order to avoid this,we need to set up a background that is convectively stable nearthose boundaries. On the other hand, we would like to simulatetop-driven convection, which is typical for stars with surfaceconvection zones. To achieve that we need the most unstablestratification to occur near the top of the box. Because most ofthe driving appears near the top of the box and the density ismuch higher near the bottom of the box, we found that it wassufficient to have a convectively stable stratification only nearthe top wall. Further, since the heat diffusion step uses the heatdiffusion coefficient, and its first and second derivatives, weneed our expression for the heat diffusion coefficient to have acontinuous second derivative.

To achieve all these requirements, we construct the particularκ(z) profile used in this paper from six separate pieces asfollows:

κ(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

κ0, z < z0

κ1 + κ2 sin[k(z − z0)], z0 < z < z1

κ3 + κ4(z − z1)

[(z − z1

z2 − z1

)2

− 3

], z1 < z < z2

κ5 + κ6(z − z2)2 + κ7(z − z2)3 + κ8(z − z2)4, z2 < z < z3

κ9 + κ10 sin

(π

z − z3

z4 − z3

)+ κ11(z − z3), z3 < z < z4

κ12, z4 < z

,

(56)where the parameters κi , i = 1 . . . 12, zj , j = 1 . . . 4, and kare chosen to make κ , κ ′, and κ ′′ continuous and to select theshape of the curve. The shape of the curve used in this articlewas determined from the following constraints.

1. The values that κ(z) takes at the boundaries: κ0 = 20 andκ12 = 21.

2. The depth above which κ(z) remains at its maximum valueof κ12: z4 = 1.95.

3. The depth at which κ(z) has a minimum and the value atthat minimum: z2 = 1.4 and κ(z2) = 19.8.

4. The depth below which κ(z) is held constant at its bottomvalue of κ0: z0 = −1.8.

5. The locations of the two inflection points: z1 = 0.2 andz3 = 1.85.

A plot of the resulting depth dependence of κ(z) is presented inFigure 1(a).

Next, we need to choose values for the background temper-ature at the top (Tlow) and bottom (Thigh) of the box. Those aredictated by the requirement that the flow speeds should neverapproach the local speed of sound; otherwise, the anelastic ap-proximation is no longer acceptable, and the fully compressibleequations should be used.

Finally, we need to choose values for the pressure at the topof the box (ptop), the external gravity (g), the specific heat atconstant pressure (Cp), and the value of the ideal gas constant(R). These values together with Tlow determine the pressure

and density scale heights. Since we are interested in studyingturbulence in a stratified medium we need to choose these valuessuch that our box encompasses at least several pressure anddensity scale heights. The particular values we use are

ptop = 1.0 × 105, (57)

g = 2.74, (58)

Cp = 0.21, (59)

R = 8.317 × 10−2, (60)

Tlow = 10.0, (61)

Thigh = 62.37. (62)

We deliberately do not include any units in the above quantities,since the simulated flow is independent of the choice of units.

The resulting background potential temperature, pressure,and density are presented in Figures 1(b)–(d). One can see thatthe steepest negative slope of θ occurs for heights between 1and 1.5 units and that significant stratification is imposed. FromFigures 1(c) and (d), we see that the convective box encompasses2.8 density scale heights and 4.6 pressure scale heights.

4.2. Energy

We first verify that the energy-like conserved quantitiesdefined in Equations (24) and (25) evolve according toEquations (26)–(29). The initial conditions we used for thistest were that all components of the velocity were set to 0, andthe initial potential temperature contained random fluctuationsin the lower 10% of the spectral modes. We chose the timestep to be much smaller than the smallest absolute value of thebuoyancy period, which in this case is an imaginary quantity formost of the box. This way, the time step is short both comparedto the growth rate of the instability near the middle of the boxand the g-mode period near the boundaries.

The test was performed by running the code with no hypervis-cosity, which does not conserve energy. The kinetic and thermalenergies, as well as the rates E1 and E2, were saved at everytime step. We then compute the time integrals of E1 and E2 us-ing Simpsons’s method. This is sufficient, since the numericalevolution is only second-order accurate in time. In Figure 2, weshow that Equations (26)–(29) are indeed satisfied to one partin a million.

4.3. Normal Modes

A second test we performed was checking that the normalmodes of the linearized equations of motion evolve as expected.We look for normal modes of the form

q(x, y, z, t) = q(kx, ky, z)e−iωt+ikxx+ikyy . (63)

For convenience, define

τ ≡ gθ

θ, (64)

ω2B ≡ g

d ln θ

dz. (65)

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.196 0.198 0.2 0.202 0.204 0.206 0.208 0.21 0.212

Heig

ht

Heat Diffusion Coefficient

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

9.88 9.9 9.92 9.94 9.96 9.98 10 10.02

Heig

ht

Potential Temperature

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

11.5 12 12.5 13 13.5 14 14.5

Heig

ht

ln[density]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5

Heig

ht

ln[Pressure]

(a) (b)

(c) (d)

Figure 1. (a) Heat diffusion coefficient (κ), (b) the background potential temperature (θ), (c) the background density (ρ), and (d) the background pressure (p) profiles.

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

0 10 20 30 40 50

Ene

rgy

time

-ET

ET + EK − ε2dt 1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

0 10 20 30 40 50

Ene

rgy

time

EK

EK − ε1dt

Figure 2. Energy conservation. Left: negative thermal energy—solid line, the absolute difference between the total energy calculated directly and as the integral ofE2—dashed line. Right: kinetic energy (solid line), the absolute difference between the kinetic energy calculated directly and as the integral of E1—dashed line.

In terms of Equations (63)–(65), after dropping all nonlinearterms, the anelastic Equations (18)–(23) simplify to includeonly two variables vz and τ :

−iω

(I − DDA

k2⊥

)vz = τ (66)

−iω(τ + L1vz

) = − ω2Bvz + L2τ , (67)

where the operators D and DA are defined in Appendix A,Equations (A4) and (A5), and (with k2

⊥ ≡ k2x + k2

y) the operators

L1 and L2 are

L1 = αR

Cpp

[1

k2⊥

(κ ′2

κ2− κ ′′

κ

)DA − d ln κ

dz− d ln ρ

dz

], (68)

L2 = κ

Cpρ

[−k2

⊥ − α

κT

d ln κ

dz+

(d ln κ

dz− α

κT

)D + D2

].

(69)

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

−4 −2 0 2 4−1

−0.5

0

0.5

1

z

Vz

−4 −2 0 2 4−1

−0.5

0

0.5

1

z

θ

Figure 3. Example of the shape of the normal modes, we initialized the box with. On the left is the z component of velocity and on the right is the potential temperature.

10−3

10−2

10−1

10−8

10−6

10−4

Vx

Time Step

Fra

ctional E

rror

10−3

10−2

10−1

10−8

10−6

10−4

Vy

Time StepF

ractional E

rror

10−3

10−2

10−1

10−8

10−6

10−4

Vz

Time Step

Fra

ctional E

rror

10−3

10−2

10−1

10−8

10−6

10−4 θ

Time Step

Fra

ctional E

rror

Figure 4. Maximum error in normal mode evolution, after a fixed evolution time. The error is scaled by the maximum value of the quantity at time t = 0. The solidline corresponds to quadratic scaling of the error with the time step.

In terms of vz and τ the remaining flow variables can beexpressed as

k2⊥h = iωDAvz, (70)

iωvx = ikxh, (71)

iωvy = ikyh. (72)

We calculated several numerical solutions to the eigenmodeequations (see Figure 3) and supplied them as initial conditionswith very low amplitude to the code. We expect that for latertimes the evolution is done simply by multiplying the initialamplitudes by e−iωt . We ran the code with the backgrounddiscussed above, and a time step that was no larger than π

250ωfor the mode in question. The comparison with the simulatedevolution of one of these eigenmodes is presented in Figure 4.As we can see, for large enough time steps, the error scales as thesquare of the time step, which confirms that the code is indeedsecond-order accurate in time at least as far as the linear termsare concerned. For very small time steps, the error deviates fromthat scaling due to numerical round off. The minimal fractionalerror is much larger than the numerical precision, because we

have chosen the heat diffusion to be as small as possible, whichcauses the matrix we invert during the heat diffusion step to havevalues along the diagonal that are many orders of magnitudelarger than the off-diagonal values, which causes the numericalround off to be amplified many times. This also explains whythe error in the potential temperature is the largest: the othervariables suffer from this only indirectly.

4.4. 2D Vortices

Another test we performed was initializing the box with apair of circular columns of vorticity running from the top tothe bottom boundary. In physical space, for each column thevorticity was constant and in the ±z direction. The two columnshad opposite signs of vorticity, in order to ensure that the totalvorticity in the box is zero, as required by the periodic boundaryconditions (see the left panel of Figure 5). Evolving the systemforward in time, one expects that the two vortices will remainstable, although distorted to a small extent, and move parallel toeach other at a constant rate determined by the distance betweenthem and the magnitude of their vorticity. In Figure 5, we showsnapshots of a horizontal cross section of the vorticity in our boxat different times. With tcross denoting the expected box crossingtime, the left panel shows the initial conditions, the middle

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

Figure 5. Snapshots at three different times of horizontal slices of the vorticity in the 2D vortices simulation. Red corresponds to positive vorticity, blue to negative,and green to zero. The left panel is at the start of the simulation, the middle at a little more than tcross/4, and the right at tcross/2.

(A color version of this figure is available in the online journal.)

panel corresponds to a little more than tcross/4, and the rightpanel corresponds to tcross/2. Because of the periodic boundaryconditions, having two vortices in our box is equivalent to havingan infinite number of vortices, copies of the two in the box. Thereis no analytical expression for the infinite series that determinesthe speed with which each column should move, but if thetwo vortices are far away from the walls as compared to thedistance between them we expect that their motion is at leastapproximately that of the situation with only two vortices. Thetime step we chose in this case was 0.005tcross.

The difference between the actual and expected position ofaverage absolute vorticity in the box is shown in Figure 6. Wesee that the average rate with which the vortices are moving isthe correct rate. The oscillations evident in the figure are dueto the deformation of the vortices from their initial cylindricalshapes. The drop at the end of the simulated time interval is dueto the fact that, at this time, the vortices are leaving the box fromthe right boundary and their periodic copies are entering fromthe left (the last panel of Figure 5).

4.4.1. Convectively Stable Box

The last test we performed was imposing a convectively stablestratification in the box and initializing with random temperatureperturbations. In this case, one expects to see buoyancy oscil-lations with a frequency given by the Brunt–Vaisala frequency:ω2

B ≡ g ddz

ln θ . So for this test, we needed the buoyancy fre-quency to be approximately constant throughout the box, andmuch larger than the time step, but small enough to allow usto simulate many buoyancy periods. So, we made the heat dif-fusion coefficient constant throughout the box and set the topand bottom temperatures to be the same. This way the buoyancyfrequency is independent of height. The particular values for theparameters of this run were

ptop = 1.0 × 105, (73)

g = 2.0, (74)

Cp = 0.2, (75)

R = 8.317 × 10−2, (76)

Tlow = 10, (77)

0 20 40 60 80 100−5

−4

−3

−2

−1

0

1

2 x 10−4

time

δ x /

Lx

Figure 6. Difference between expected vortex position and the average positionof vorticity. The period of the oscillations corresponds to the rotational periodof the vortices and the drop in the end is due to the fact that the vortices areexiting the box on the right and hence a bit of them is appearing from the left,causing the average position of vorticity to move toward zero.

Thigh = 10, (78)

δt = 2 × 10−3, (79)

giving a buoyancy frequency of ω2B = 2.0,

A plot of the kinetic energy for this run is presented inFigure 7. It can be seen that, as expected, the kinetic energygoes through two cycles every buoyancy period. This is becausethe velocity has to go through one cycle, and the energy hasa maximum every time the velocity has a maximum, or aminimum. The decay in the curve is caused mostly by theheat diffusion smoothing out the perturbations over time. Sincedifferent modes decay at different rates, and they are coupledthrough the nonlinear terms, we have not shown an expecteddecay curve.

5. ESTIMATING THE DISSIPATION IN AN UNSTABLYSTRATIFIED CONVECTIVE BOX

5.1. Steady-state Flow

Having confirmed that the code is solving the correct equa-tions, we simulated a box with the background state describedin Section 4.1 for long enough to reach a steady state. In orderto minimize the impact that the artificial hyperviscosity stephas on our flow, we chose the smallest values of ν

hyp⊥ and ν

hypz ,

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

0.00016

0 1 2 3 4 5 6 7 8 9

EK

ωBt/2π

Figure 7. Kinetic energy as a function of time for a convectively stable box. Thedistance between every two consecutive vertical lines is the buoyancy period.The horizontal axis is time in units where the buoyancy period is 1.

separately for the temperature and the velocity that result in anumerically stable solution for the particular choice of resolu-tion and temperature gradient, and we fix the hyperviscosity pparameter to 6, which means that only the highest few spectralcoefficients are significantly affected by the hyperviscosity step.This is sufficient, since as we discussed earlier that numericalinstability is due only to the two highest spectral modes grow-ing exponentially. We also verified that the global properties ofthe flow—the equilibrium kinetic and thermal energies as wellas the spatial power spectra of the velocity components andthe potential temperature do not change if we double the νhyp

values.We used two criteria for determining that a steady state has

been reached. First, the kinetic and thermal energies should stopdrifting systematically up or down, see Figure 8. The oscillationswe see in the kinetic energy have a period close to the convectiveturnover time of the box. Second, the spatial spectra of thevelocity and potential temperature must be steady to within afew percent.

The vertical spectra are not directly obtained from theoutput of the simulation since the collocation points of thecomputational grid are not evenly spaced, and the verticaldirection is not periodic. We first re-sample to an evenly spacedgrid and apply some window function, before performing thediscreet Fourier transform. Since in the vertical direction we

simulate the Chebyshev series of the quantities, the most naturalway to re-sample to an even grid is to evaluate this series foreach of the new collocation points.

Fourier power spectra of the three velocity components andthe potential temperature are presented in Figure 9, along withthe scaling that Kolmogorov statistics predict (spectral power∝ k−5/3).

The sharp cutoff at high wavenumbers is due to the finiteresolution of the box (using a box of half the resolution producesa cutoff at half the wavenumber we see in Figure 9). It is notdirectly due to the hyperviscosity, since reducing the value ofνhyp does not decrease its slope, but rather causes the highest fewspectral components to grow exponentially with time, and onlyafter they have grown to values higher than the spectrum beforethe cutoff, is power pumped to intermediate wave numbers,eliminating this cutoff.

Horizontal cross sections of vz and θ + θ at three differentheights of a typical steady-state frame of the flow are presentedin Figure 10. The three heights we chose were z = 1.98, z = 0,and z = −1.98 for a box in which the vertical coordinate runsfrom −2 to 2. Horizontal sections of the z component of thevorticity ωz as well as typical vertical sections of the abovequantities are presented in Figure 11.

Near the top boundary of the box (Figures 10(a) and (d)),the flow exhibits a cellular pattern of narrow down-flow lanes.Traces of the cells are visible only in the top 5% of the box, afterthat the flow organizes itself into a pair of perpendicular down-flow lanes that horizontally span the entire box. With time, thoselanes get distorted, break up, and reform, but are well definedfor at least half the time, for most of the upper half of the box.They are generally parallel to the grid x- and y-axes.

Similar patterns were first observed by Porter & Woodward(2000). Their tests show that the lanes tend to align themselveswith the periodic directions of the models and not with the gridaxis per se (rotating the periodic directions at 45◦ relative to thegrid axis caused the lanes to rotate as well). So they concludethis to be due to the small horizontal to vertical aspect ratio ofthe simulations.

Around the middle of the box, the pair of perpendicular lanesare still sometimes visible, but they are a lot less persistent.Rather, at those depths, the flow consists of one or two down-flows, which take up less than a quarter of the cross-sectionalarea, the rest being occupied by a gentler up-flow (Figures 10(b)and (e)).

0

20000

40000

60000

80000

100000

120000

140000

160000

0 100 200 300 400 500 600 700

Kin

etic E

ne

rgy

Time

-350000

-300000

-250000

-200000

-150000

-100000

-50000

0

50000

0 100 200 300 400 500 600 700

Th

erm

al E

ne

rgy

Time

Figure 8. Kinetic (left) and thermal (right) energy content of the convective box used as one of the criteria for having reached a steady state. We decided that steadystate was reached at a time of 400.

10

The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

-2

0

2

4

6

0.5 1 1.5 2

log(

Spec

tral

Pow

er)

log(kx)

VX

VY

VZ

θk−5/3x

-2

0

2

4

6

0.5 1 1.5 2

log(

Spec

tral

Pow

er)

log(ky)

VX

VY

VZ

θk−5/3y

-1

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

log(

Spec

tral

Pow

er)

log(kz)

VX

VY

VZ

θk−5/3z

7

8

9

10

11

12

13

14

15

-2.5 -2 -1.5 -1 -0.5 0 0.5

log(

Spec

tral

Pow

er)

log Ω

VX

VY

VZ

θΩ−1.2

Figure 9. (x: top left; y: top right; z: bottom left; time: bottom right) Spectra of the three velocity components and the potential temperature. The thick line correspondsto Kolmogorov scaling (Ek ∝ k−5/3) in the first three plots and to the scaling we find for the effective viscosity (see the next section) in the last plot.

For the lower half of the box, the asymmetry between up- anddown-flows gradually decreases as we get further down. Nearthe bottom boundary the large scale of the flow disappears, untilonly small-scale structure is left (Figures 10(c) and (f)).

This behavior with depth of the convective structures seems tobe a general more or less scale independent property of stratifiedconvection. It is seen both in local simulations (e.g., Porter &Woodward 2000) and in global ones.

Since we ran our simulations with the smallest possible valueof the heat diffusion coefficient, we expect to have efficientconvection, hence we would expect the z gradient of thepotential temperature to be very close to zero except near theimpenetrable top and bottom boundaries. The averaged overtime and horizontal directions logarithmic gradient of θ + θcan be seen in Figure 12. As we can see, the scale height ofθ is indeed more than two orders of magnitude larger than thebox, as long as we are not very close to the boundaries. Thetop of the box has a local minimum of the entropy gradient anda significant convectively stable region, so as to keep the largemagnitude vertical flow from getting too close to the boundaryand causing numerical problems. We also see that another localminimum of the entropy gradient has developed near the bottomof the box. This is caused by the lower impenetrable wall. Nothaving a significantly thick convectively stable region near thebottom is acceptable, because the larger density of the fluidmeans that the flow velocities are much smaller than at the topof the box, and thus the CFL condition near the bottom wallgives a lot less stringent limitation on the time step.

5.2. Adding Units

If one wants to compare our values to physical conditions,in, for example, the solar convective zone, one needs to select a

Table 1Example Set of Units Applied to the Unstably Stratified Simulation

Quantity Unit

Length 108 cmTime 10 sMass 1014 gTemperature 1000 K

particular set of units in which all the variables will be expressed.Of course, the results are equally applicable to any choice,but here we present one, converting all quantities discussedabove for the unstably stratified case just as an illustration. Theparticular choice we make is shown in Table 1.

This would make our simulated box have dimension of4 Mm, densities ranging from 10−5 to 2 × 10−4 g cm−3, andpressures in the range of 10–103 bar. The gravity we imposethen would be 2.74 × 104 cm s−2 (exactly the solar surfacegravity), the typical convective velocities that develop in ourconvective zone are a few km s−1, the convective turnover timeis on the order of 1000 s, and the flux transported by our box,estimated as the diffusive heat flux through the bottom boundary:κ(dT /dz)−Lz/2 = 2.8 × 1011 erg s−1 cm−2 or about four timesthe solar surface flux. So in this set of units, the simulated flowlooks similar to the actual flow a bit under the solar photosphere,at least in gross terms.

5.3. Lowest Order Perturbative Expansion

Ignoring the part of the simulation before steady state wasreached, we applied the perturbative method of Goodman & Oh(1997) as described by Penev et al. (2009b) to find a lowestorder perturbative expansion estimate of the energy transfer rate

11

The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

(a) (d)

(b)

(c) (f)

(e)

Figure 10. Horizontal cross sections of the vertical velocity component (left) and the total potential temperature (right) at three different heights of the box: z = 1.98(a, d), z = 0 (b, e), and z = −1.98 (c, f). Darker areas correspond to lower values, with the gray scale in each figure adjusted to the range of values encountered in theslice.

between a small amplitude external forcing and the turbulentflow in our box, and, respectively, from that we can derive thecomponents of an effective viscosity tensor that reproduces theenergy dissipation rate due to the turbulence.

We assume forcing in the form of an external velocity field:

V = A(t) · x. (80)

Goodman & Oh (1997) define two dimensionless parameters:the tidal strain Ω−1 |A| and (Ωτc)−1, where Ω is the frequency ofthe perturbation and τc ≡ Lc/Vc. The characteristic convectivelength scale is Lc and Vc is the characteristic convective velocity.In the case of hierarchical eddy structured convection τc is theeddy turnover time.

We then take only the lowest nonzero term in the expansion ofthe energy transfer rate E with respect to those two parameters.The resulting expression is

E(Ω = 2πR/T ) = Re{SR,−R + SR,R}+

∑r =0

1

πrIm{Sr+R,r−R + Sr+R,r+R}, (81)

with Sρ,ρ ′ ≡ T

N 2Nz

∑λ,μ,ν,ν ′

ρ∗ν−ν ′v

1λ,μ,ν,ρPλ,μ,ν ′v2∗

λ,μ,ν ′,ρ ′ ,

where N ≡ NxNyNzNt , Pλ,μ,ν ≡ I−k′λ,μ,νk′

λ,μ,ν/k′2λ,μ,ν , with

k′λ,μ,ν ≡ kλ,μ,ν + z/Hρ , ρ(kz) is the Fourier transform of the

density averaged over x, y, t .

12

The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

(a) (d)

(b)

(c) (f)

(e)

Figure 11. Horizontal cross sections of the vertical vorticity component (left panels) at heights: z = 1.98 (a), z = 0 (b), and z = −1.98 (c). Vertical cross sections(right panels) of the vertical velocity (d), the total potential temperature (e), and the vertical vorticity (f). Darker areas correspond to lower values, with the gray scalein each figure adjusted to the range of values encountered in the slice.

In this case, we do not need to worry about the compressibilityof the flow, since the anelastic approximation does not allow forany. However, we do allow for density stratification, so we willneed to check that the logarithmic gradient of the density is muchsmaller than the logarithmic gradient of vz. A plot comparingthe two quantities is presented in Figure 13. We see that thedifference is more than an order of magnitude, enough to justifyignoring the density term for the perturbative calculation.

To perform the calculation, we use a discreet Fourier trans-form to estimate the spectra of the velocities and density neededin Equation (81), but before that, we again re-sample to a gridthat has its collocation points evenly spaced in the vertical direc-tion. We have to be careful when using discreet Fourier trans-forms to estimate spectra. In particular, we need to pay special

attention to the z and time directions, since they are not periodicand hence the discreet Fourier transforms suffer from window-ing effects. To confirm that our results are not significantlyaffected, we perform the same calculations using no windowingin those two directions, as well as Welch (parabolic) and Bartlett(triangular) windows. Also it is possible that the impenetrabletop and bottom walls might influence our result. So we alsotried Welch and Bartlett window functions that exclude the top12.5% and the bottom 5%. Finally, we performed the calcula-tion with two different density scale heights: infinity and thevolume-averaged Hρ . The resulting effective viscosity scalingsfrom all these tries were within 15% of each other.

Since the turbulent flow is anisotropic we expect that theeffective viscosity is anisotropic. This means that the viscosity

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

-0.004

-0.002

0

0.002

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

dlo

gθ/dz

Z

Figure 12. Average logarithmic gradient of the potential temperature.

-2-1.5

-1-0.5

00.5

11.5

2

0.1 1 10 100

Dep

th

d ln ρdz 1

v2z

1/2∂vz∂z

2 1/2

Figure 13. Comparison between the logarithmic derivative of the verticalvelocity (dashed curve) and the background density (solid curve) for the steady-state flow simulated with our anelastic code.

for us is a rank-four tensor–Kijmn. This tensor must obey a set ofsymmetries to ensure that the torque on fluid elements is finite.Further, since there is no physical difference between the twohorizontal directions, we expect that Kijmn will by symmetricunder rotation/reflection around the vertical axis. After allthese symmetries are taken into account, many components ofKijmn are shown to be zero and the rest can be expressed interms of five independent quantities (see Penev et al. 2009b,Section 2.2)—K0, K ′

0, K00′ , K1, and K2—as follows:

K0 = 1

2(K1111 + K1122)

K0′ = K3333

K00′ = K1133 (82)

K1 = K1313

K2 = 1

2(K1111 − K1122) .

We can extract the Km quantities by comparing the energydissipation rate of Equation (81) to the dissipation that wouldoccur in the presence of actual anisotropic viscosity:

Evisc(Ω) = 1

2

∫ Lz

0dz[4K1|A1|2 + K2|A2|2 + K0A2

0

+ K0′A20′ + 2K00′A0A0′]. (83)

Since in Equation (83) we allow the viscosity to depend ondepth, and there is no way to constrain this dependence, we haveto choose some radial profile a priori. Our choice is motivatedby the mixing length theory:

Km(z) = K0m(Ω)ρ〈v2〉1/2Hp, (84)

10−4

0.01

1

K0 m

K0 m

1 3 10 30 100ΩτcΩτc

K00 K0

0 K000 K0

1 K02

Figure 14. Five viscosity parameters as estimated from the steady-state unforcedconvective flow. The vertical axis is the effective viscosity in units of 〈v2

z 〉1/2Hp

(Hp: pressure scale height) and the horizontal axis is the perturbation frequencyin units of inverse convective turnover times (τc).

(A color version of this figure is available in the online journal.)

where K0m are dimensionless constants, that depend on the

frequency of the external shear (Ω), and Hp is the local pressurescale height. This is a sensible choice, since the turbulentviscosity should scale as the length scale and the velocity scaleof the problem. Clearly, the velocity scale is that of convection,and in the spirit of mixing length theory, we use the pressurescale height as the length scale. If the mixing length paradigmis relevant, we expect that the value of K0

m should scale as themixing length parameter for the particular simulation.

The resulting frequency dependencies of the five viscosityparameters are presented in Figure 14. As we can see, all termshave similar scaling with frequency, and the relative magnitudesof the distinct components obey the same relations as the Penevet al. (2009b) viscosity calculated for realistic models of thesurface convection of low-mass stars.

The approximate period dependence of the viscosity compo-nents we find is

Km = K0mρ〈v2〉1/2Hp

(T

τc

)λ

,m ∈ {0, 0′, 00′, 1, 2}λ = 1.05 ± 0.1,

K0 = 0.068 ± 0.01,

K0′ = 0.18 ± 0.02, (85)

K00′ = 0.019 ± 0.005,

K1 = 0.066 ± 0.01,

K2 = 0.05 ± 0.01,

where T ≡ 2π/Ω and the errors quoted correspond to the rangeof values encountered for the different window functions anddensity scale heights.

6. CONCLUSION

The above result points to the possibility that viscosityin turbulent convection zones loses efficiency significantlyslower than what Kolmogorov scaling predicts at least on largetimescales. This seems to be due to the fact that the turbulenteddies with turnover times similar to these large timescalesare not in the inertial subrange and hence, the velocity powerspectrum is much shallower than the Kolmogorov 5/3 law.This result is not conclusive, since the possibility remains thatdropping higher order terms in the above expansion is not agood approximation.

In Penev et al. (2009a), we introduce horizontal depth depen-dent forcing into the flow equations and obtain the dissipation

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

properties of the turbulent convective flow directly. We werethen able to compare the average rate of work done on the flowby the external forcing to the expected rate of energy transportand dissipation by an assumed effective viscosity. We foundthat with sufficiently long-time average these two quantitieshave the same depth dependence, thus verifying the validity ofthe assumption that an effective viscosity coefficient is suffi-cient to parameterize the average dissipation and momentumtransport properties of the turbulent convective flow for largespatial scales. Further, by repeating the above procedure a num-ber of times we were able to derive the scaling of this effectiveviscosity coefficient with period and confirm that it is linear.

In this paper, we presented a code that is well suited for thepurpose of studying turbulent dissipation in convective zones.First, it is a spectral code, which means that the spatial accuracyis exponential, and hence the code is able to reliably simulateturbulent flows at modest resolutions. Further, by using theanelastic approximation, we do not need to deal with soundwaves and shocks. This allows us to take much larger timesteps (by more than an order of magnitude) than with a fullycompressible code, and hence we can have runs that cover amuch longer physical time than fully compressible simulations.This is important, since the external forcing regimes we areinterested in are tides with orbital periods of several days, whichis not achievable in reasonable time with fully compressiblecodes.

The price we pay of course is that the flow we simulateis not a good approximation to the flow in any star’s surfaceconvection zone. First, we cannot accommodate the regionwhere the driving of the convection occurs, because this region ischaracterized by strongly supersonic flows and shocks. Also thephoton mean free path in that region is not small so the radiativeeffects can no longer be captured by a heat diffusion coefficient.Second, our code uses the ideal gas equation of state which isa poor approximation to the upper layers of stars. Finally, wedo not consider either rotation or magnetic fields, both of whichmight have an important effect on the dissipative efficiency ofthe convective zone.

However, the scaling of the effective viscosity with frequencywe obtained (Equation (85)) for our box is linear just as thescaling obtained with the same perturbative approach appliedto realistic simulations of the upper layers of the convectionzones of the Sun and two smaller stars (Penev et al. 2009b).The effective viscosity of Equation (85) is a bit more than twotimes larger than the effective viscosities from the realistic stellarmodels, but this is expected, since the mixing length parameterfor our box is about two times larger than for those simulations,and as we noted before, we expect that the effective viscosityshould be proportional to the mixing length parameter. Thisgives us confidence that our results are applicable to the systemswe are interested in studying.

6.1. Comparing to Other Codes

There are a large number of other anelastic numerical codesaiming to study convection in stars or giant planets (Lantz &Fan 1999; Glatzmaier 1984; Jones & Kuzanyan 2009, etc.),the ASH code (Miesch et al. 2000; Elliott et al. 2000; Tobiaset al. 2001; Brun & Toomre 2002; Miesch et al. 2008; Brownet al. 2010; Featherstone et al. 2009; among many others).Each of these groups has made different design decisions forthe numerical scheme they implement, mostly dictated by theproblems the particular numerical code was designed to study.

Here we compare the design decisions we have made to thoseof the above authors.

Two major differences between the code presented in thispaper and (some) of these other codes are that they considerspherical shells encompassing the entire star/planet in someradius range, while we consider only a small region in whichwe ignore the spherical geometry and approximate it as plane-parallel, and that we ignore magnetic fields. Thus we applythe fast, albeit less detailed evolution typically used to studyconvective zones globally to a small localized rectangular box,usually treated fully compressibly (e.g., Chan & Sofia 1989,1996; Cattaneo et al. 1991; Kim et al. 1995; Stein et al. 1989;Robinson et al. 2003, etc.), allowing us to simulate a well-resolved local flow over a long-time period, which is necessaryfor our purposes.

In addition, as stated in Section 2, instead of using theappropriate expansion in perturbed quantities for the ideal gasequation of state we modify the expression in order to obtain aset of equations which preserve energy exactly instead of onlyto lowest order in perturbed quantities.

Also, unlike the ASH code, for example, we consider ourbackground variables constant in time, which simplifies theevolution equations considerably, and unlike many anelasticcodes we evolve the actual velocity and potential temperaturerather than conserved quantities such as vorticity, momentum,energy, and poloidal/toroidal velocity fields.

And finally, our treatment of the subgrid dissipation ismarkedly different from all other anelastic codes in that we donot introduce actual viscosity (assumed to be due to unresolvedturbulent cascade), but rather keep the numerical integrationstable by the hyperviscosity discussed in Section 3.2.2, whichonly affects the simulated flow on scales a few times larger thanthe grid resolution.

We acknowledge the contribution of Philip Marcus to thedevelopment of the original code (Barranco & Marcus 2006).

APPENDIX A

IMPLEMENTING THE PRESSURE STEP

As discussed in Section 3.2.3 in order to advance the enthalpyby one time step we need to solve the following differentialequation/boundary conditions problem:[

∇2 +d log ρ

dz

∂

∂z

]ΠN+1 = 1

Δt

(∇ · vN+ 2

3 + vN+ 2

3z

d log ρ

dz

)∂ΠN+1

∂z

∣∣∣∣±Lz/2

= 1

Δtv

N+ 23

z

∣∣∣±Lz/2

. (A1)

The solution is obtained in two steps: first, we impose theboundary conditions by ignoring the differential equation forthe two highest Chebyshev modes and replacing it with theequations for the boundary conditions; then we use a Green’sstep (see Appendix A.1) to fix the equation for those twohighest modes and instead impose the boundary conditions atthe expense of ignoring the momentum Equation (19) at the twohighest modes.

We need to solve Equation (A1) as efficiently as possible.Clearly, we can make this a trivial matrix multiplication opera-tion if we were to store the inverse of the left-hand side operatorfor each horizontal mode at the start, and at each time step wemultiply every vertical slice of the transformed right-hand side

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

by the corresponding inverse to get the value of ΠN+1. How-ever, this would require Nx × Ny matrices of N2

z elements to bestored (where Nx, Ny, and Nz are the resolutions in the x, y, andz directions, respectively), which for large resolutions is likelyto exceed the amount of memory available on each node of thecluster where the code is to run.

To avoid this, we need to find the most efficient approachthat only stores things common to all horizontal modes. We seethat the pressure equation is almost identical for all horizontalmodes, except for the horizontal component of the ∇2 operator,which in Fourier space means simply multiplying by k2

⊥ ≡k2x + k2

y , where kx and ky are the corresponding x and ywavenumbers for each mode. So what we can reasonably do isto pre-compute the common part of the left-hand side operator,and then for each horizontal mode add k2

⊥ along the diagonal.We then overwrite the last two rows of the resulting matrix(Mi,j ) with

MNz,p = p2, (A2)

MNz−1,p = (−1)pp2, (A3)

in order to impose the boundary conditions, decompose Minto upper and lower triangular parts, and solve by backwardsubstitution. The right-hand side vector also needs to have itshighest two entries overwritten with the boundary conditions atthe top and bottom of the box, respectively.

A.1. Green’s Step

In this step, we fix the anelastic constraint even for thetwo highest Chebyshev modes, and instead break the twohighest modes of the momentum equation (19) to satisfy theboundary conditions which we break in the process. To makethe expressions shorter, define the following operators:

D ≡ d

dz, (A4)

DA ≡ d

dz+

d ln ρ

dz, (A5)

∇ ≡ ikx x + iky y + Dz, (A6)

∇A ≡ ikx x + iky y + DAz, (A7)

ΔA ≡ ∇A · ∇. (A8)

We would like to modify the pressure step in a way that willinclude two new degrees of freedom, which we can then useto fix the anelastic constraint for the two highest Chebyshevmodes. The particular modification useful in this case is

vN+1 = vN+ 23 − Δt∇ΠN+1 + z

(τN+1

1 TNz−1 + τN+12 TNz

), (A9)

where TN denotes a Chebyshev polynomial of order N, τ1 andτ2 are arbitrary coefficients to be chosen later, Nz is the orderof the highest Chebyshev coefficient we are simulating, andthe superscripts refer to the time index, with non-integer valuescorresponding to fractional steps.

Requiring the anelastic constraint and velocity boundaryconditions gives

ΔAΠN+1 = 1

Δt∇A · vN+ 2

3 + τN+11 DATNz−1 + τN+1

2 DATNz,

(A10)

DΠN+1|±Lz/2 = 1

Δtv

N+ 23

z + τN+11 TNz−1 + τN+1

2 TNz

∣∣∣±Lz/2

.

(A11)

To proceed we break up ΠN+1 into three pieces:

ΠN+1 = ΠN+10 +

1

Δt

(τN+1

1 ΓN+11 + τN+1

2 ΓN+12

). (A12)

This allows us to split the above equation into three separateequations with corresponding boundary conditions, the first ofwhich is the already calculated pressure step:

ΔAΠN+10 = 1

Δt∇A · vN+ 2

3 , (A13)

DΠN+1|±Lz/2 = 1

Δtv

N+ 23

z

∣∣∣±Lz/2

, (A14)

ΔAΓN+11 = DATNz−1, (A15)

DΓN+11 |±Lz/2 = TNz−1

∣∣±Lz/2 , (A16)

ΔAΓN+12 = DATNz

, (A17)

DΓN+12

∣∣±Lz/2 = TNz

|±Lz/2. (A18)

We solve the two new equations in the same way we solve thefirst one. This makes as before the anelastic constraint satisfiedfor all but the two highest Chebyshev modes, but this time wehave two arbitrary constants τ1 and τ2 which we can then set tovalues that will make the anelastic constraint hold for those twomodes as well.

APPENDIX B

IMPLEMENTING THE HEAT DIFFUSION STEP

The same considerations as the pressure step apply to thisstep. We again decide against making a table of pre-invertedmatrices for each horizontal mode, and instead we only pre-compute the part of the matrix that is common for all horizontalmodes. To make the description of the numerical procedurefollowed clearer, we define the following matrices:

CP : transforms a vector from Chebyshev to physical spacePC = CP−1: transforms a vector from physical to Chebyshev spaceD : differentiates a vector in Chebyshev space.

From those we construct a derivative operator and the commonpart of the left-hand side operator, both in physical space:

DP ≡ CP · D · PC, (B1)

MP ≡ DP · DP + F · DP + G, (B2)

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The Astrophysical Journal, 734:118 (17pp), 2011 June 20 Penev, Barranco, & Sasselov

where F and G are diagonal matrices defined by

Fij ≡ C1(zi)δij , (B3)

Gij ≡ (C2(zi) + C3(zi)) δij , (B4)

with zi being the ith vertical collocation point as defined inEquation (31), and C1 and C2 are given by Equations (48)and (49), respectively.

What we pre-compute and store is a matrix M, which weobtain by overwriting the first and last rows of MP with

MP 1,1 = 1 MP 1,j = 0, j = 2 . . . Nz, (B5)

MP Nz,j = 0, j = 1 . . . Nz − 1 MP Nz,Nz= 1, (B6)

to allow for imposing the boundary conditions, then sandwich-ing the resulting matrix between PC and CP:

M ≡ PC · MP · CP. (B7)

For each time step, in order to solve Equations (52) and (53) foreach horizontal mode we construct a new matrix M′(k⊥) fromM as follows:

M′i,j (k⊥) ≡ Mi,j − k2

⊥δi,j + k2⊥(PCi,0CP0,j

+ PCi,Nz−1CPNz−1,j ). (B8)

The first new term adds the horizontal part of the ∇2 operatorfor the given mode, however, this breaks the requirements forthe boundary conditions, so we repair them with the other twoterms. This way, the boundary conditions are directly imposedby breaking Equation (52) at the physical top and bottom of thebox, instead of ignoring it for the two highest Chebyshev modes.That means we do not need to perform extra Green’s steps likewe did for the pressure equation. We obtain the solution toEquation (52) by decomposing M′(k⊥) into upper and lowertriangular matrices and using back-substitution.

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