Examination of Scaling Relationships Involving Penetration Distance at the Bottom of a Stellar...

THE ASTROPHYSICAL JOURNAL, 529 :402È413, 2000 January 202000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

EXAMINATION OF SCALING RELATIONSHIPS INVOLVING PENETRATION DISTANCE AT THEBOTTOM OF A STELLAR CONVECTIVE ENVELOPE

E. SAIKIA,1,2 HARINDER P. SINGH,1 K. L. CHAN,3 I. W. ROXBURGH,4 AND M. P. SRIVASTAVA2Received 1999 March 12; accepted 1999 September 13

ABSTRACTA number of studies in the recent past have proposed a variety of scaling relationships among the

penetration depth at the bottom of a convective region, the vertical velocity of the Ñuid and the(*d) (V

z),

input Ñux While a relationship of the form has been proposed by Schmitt and coworkers(Fb). *

dPV

z3@2

on the basis of the equations of motion for buoyant plumes, Zahn proposed a similar relationship basedon scaling arguments. The relationships involving and the input Ñux are based on recent two-*

d, V

z,

dimensional numerical simulations by Hurlburt and coworkers. All these scalings were recently lookedinto by Singh, Roxburgh, & Chan, who performed full three-dimensional simulations of turbulent com-pressible convection for a stable-unstable-stable sandwich conÐguration. In the present study, we numeri-cally solve the full set of Navier-Stokes equations in three dimensions in order to study the behavior ofconvective motions penetrating into the bottom stable layer. We take up a series of models di†ering inresolution or mesh size and aspect ratio with a view to examine, in greater detail, the scaling relation-ships between the penetration distance and other Ñow parameters.Subject headings : convection È hydrodynamics È stars : interiors È turbulence

1. INTRODUCTION

The structure of a star is governed by the balance amongthe pressure due to the liberation of energy by nuclearfusion and self-gravity, the distribution of chemical ele-ments, and the radiative and convective transport of energy.Changing chemical composition due to the nuclear burningof elements in the core of a star and the redistribution ofthese products by mixing processes determine how the starwill evolve with age. Convection is the dominant mixingprocess, and the region of convective penetration below astellar convection zone is of particular interest (Roxburgh1996, 1998).

The mixing-length theory, which is normally used tomodel convection in stellar interiors, assumes a turbulenteddy to rise or sink under the action of buoyancy. The eddytravels a distance l, the mixing length, conserves entropy,and remains in pressure equilibrium with its surroundings.The mixing length l is taken as where is the pres-aH

p, H

psure scale height (PSH) and a is an adjustable parameter oforder unity such that a stellar model has the observedradius. In this model, the superadiabatic gradient *+> 1inside the convection zone. At the boundary, *+\ 0, v\ 0,and the penetration is absent. The model is not self-consistent as the eddy is accelerated in the convection zoneand thus should have a velocity at the bound-v0^ (F/o)1@3ary of the unstable-stable region. Furthermore, a self-consistent model should incorporate feedback of theovershooting on the energy transport in the penetrativeregion.

A number of nonlocal models that incorporate somemeasure of feedback have been studied by various authors

1 Department of Physics, Sri Venkateswara College, University ofDelhi, New Delhi 110 021, India.

2 Department of Physics and Astrophysics, University of Delhi, Delhi110 007, India.

3 Department of Mathematics, Hong Kong University of Science andTechnology, Kowloon, Hong Kong.

4 Astronomy Unit, Queen Mary and WestÐeld College, University ofLondon, Mile End Road, London E1 4NS, UK.

(Shaviv & Salpeter 1973 ; Maeder 1975 ; Roxburgh 1978,1985, 1997 ; van Ballegooijen 1982 ; Schmitt, Rosner, &Bohn 1984 ; Zahn 1991 ; Canuto & Dubovikov 1997, 1998).Schmitt et al. considered the equations of motion forbuoyant plumes and applied these to the convective under-shoot region starting at the location where and+ad \+radchoosing an initial (area-) Ðlling factor f, a characteristicvelocity and the number of plumes N present at anyVzo,given time. Using an entrainment parameterization thatdescribes a wide variety of laboratory and atmosphericmeasurements, they obtained a relationship *

dP f 1@2V zo3@2between the amount of undershoot and the vertical velocity

at the bottom of the convection zone. They suggested nodependence of the amount of penetration on the number ofplumes, N (i.e., only the plume density mattered in thestudied parameter domain). It was also predicted that therewould be an almost adiabatically stratiÐed region below theunstable layer provided we take the initial velocities as theyare given by mixing-length theory. It was also proved thatthe predicted overshoot was quite insensitive to the otherparameters such as entrainment rate or the number or thewidth of the plumes.

Zahn (1991) studied the problem analytically, basing it onscaling arguments. Under the assumption that the radiativeconductivity varies smoothly from the unstable to the stablelayer, he could explain the reason for scaling the nearlyadiabatic penetration as The Peclet number was takenV [email protected] larger than unity so that the convectivemotions become efficient enough to establish an adiabaticstratiÐcation and transport most of the heat produced. Heassumed that the downward-directed Ñows transport mostof the convective Ñux and that they occupy only a smallfraction of the area (as observed in computer simulationsand laboratory experiments). Using this model he derivedthe expression for the extent of subadiabatic penetration as

Lp

Hp\ V zo3@2(cf )1@2

A32

gQKsp+adB~1@2

, (1)

where is the extent of subadiabatic penetration ; is theLp

HpPSH; is the root mean square (rms) velocity of the ÑowVzo

402

PENETRATION AT BOTTOM OF CONVECTIVE ENVELOPE 403

TABLE 1

NUMERICAL PARAMETERS

AspectModel N

xN

yN

zRatio (A) t dt d NCFL Ck

M1 . . . . . . 35 35 51 1.5 1224 0.0073 0.0094 1.0 0.2M2 . . . . . . 35 35 51 1.5 2622 0.0073 0.0094 1.0 0.2M3 . . . . . . 64 64 51 1.5 1093 0.0036 0.0094 0.5 0.2M4 . . . . . . 20 20 37 1.5 1614 0.0154 0.0132 1.5 0.2M5 . . . . . . 20 20 64 1.5 1608 0.0057 0.0074 1.0 0.2M6 . . . . . . 28 28 46 1.5 1608 0.0081 0.0105 1.0 0.2M7 . . . . . . 28 28 46 3.0 1608 0.0081 0.0105 1.0 0.2M8 . . . . . . 35 35 64 1.5 1608 0.0057 0.0074 1.0 0.2

at the bottom of the convective region ; c is a coefficient thatmeasures the degree of asymmetry of the Ñow; g is the localgravity ; is the expansion coeffi-Q\[(L log o/L log T )

pcient at constant pressure, where o is the density ; K denotesthe thermal di†usivity ; is the radiative conductivity ats

pconstant pressure ; and is the+ad\ (L log T /L log P)adadiabatic gradient. This proposed relationship is similar inits scaling behavior to that put forward by Schmitt et al.The factor cf is kept constant over the whole penetrationregion so that the calculations become simpler. As a resultof this, the entrainment of matter and the spread of thedowndrafts are not permitted as they reach the bottom ofthe penetration region. The results obtained from this exer-cise came very close to those found by Bressan, Bertelli, &Chiosi (1981) using a nonlocal mixing-length formalism.

There have been some attempts recently to study theovershoot by directly solving the governing Ñuid equationsin two and three dimensions (Roxburgh & Simmons 1993 ;Hurlburt et al. 1994 ; Singh, Roxburgh, & Chan 1994, 1995,1996, 1998a, 1998b ; Muthsam et al. 1995 ; Freytag, Ludwig,& Ste†en 1996 ; Nordlund & Stein 1996). The numericalsimulations are constrained by the inability of the presentgeneration of computers to resolve Ñows at all scalespresent in the stellar convection zones. An alternative, thus,is to look for some scaling relationships between physicalquantities of interest based on the numerical simulations(Chan & SoÐa 1989, 1996 ; Lydon, Fox, & SoÐa 1992, 1993 ;Singh & Chan 1993 ; Hurlburt et al. 1994, Singh et al. 1994 ;Singh et al. 1995, 1998a, 1998b, hereafter SRC95, SRC98a,SRC98b, respectively). Hurlburt et al. gave scaling lawsrelating the penetration distance with the relative stabilityof the unstable zone to the stable zone. They studied theproblem in two dimensions and computed the physicalparameters by mainly changing the polytropic index andhence the relative stability of the lower stable layer in com-parison with the middle unstable layer in their three-layerconÐguration. It was calculated analytically and conÐrmedby numerical simulations that for nearly adiabatic penetrat-ion while for nonadiabatic penetration*

dP V zo3 /F

b, *

dP

where K is the piecewise radiative conductivityK1@2 P Fb1@2,

and is the input Ñux.Fb

Singh et al. (1994) performed three-dimensional simula-tions of turbulent convection with a view to study penetrat-ion above a convection zone. It was found that thepenetration distance above a convection zone was pro-portional to the maximum vertical velocity of the convec-tive motions, and this distance increased with the factor

where denotes the input Ñux and is theFb/oczt, F

bocztdensity at the top of the convective region. The behavior of

convective penetration below a convection zone wasstudied by systematically increasing the polytropic indexand hence the stability of the lower stable layer (SRC95). Itwas observed that the extent of downward penetrationdecreased as the stability of the lower stable layer wasincreased, a result that was also reported by Hurlburt et al.(1994) from their two-dimensional simulations.

More recently, SRC98a and SRC98b simulated a numberof cases to study the scaling relationship between the pen-etration distance and the rms vertical velocity by varyingthe input Ñux It was found that in agreementF

b. *

dP F

b1@2,

with the relationship suggested by Hurlburt et al. for non-adiabatic penetration and a piecewise conductivity proÐle.A relationship of the form was evident for*

dD (0.64)F

b1@2

the four cases studied. Furthermore, it was concluded thatthe relationship is a fairly good approximation to*

dPV zo3@2describe the relationship between the penetration distance

and the rms vertical velocity at the bottom of the(Vzo)convection zone, as suggested earlier by Zahn (1991) basedon scaling arguments. A relationship of the form *

dD

was noticed for the four cases.(7^ 0.3)V zo3@2In the present work, we study the behavior of penetrativeconvection below a deep stellar-type convective envelope bycomputing a series of models for various mesh sizes/resolutions and aspect ratios. We thus try to Ðnd the depen-dence of penetration distance on vertical and horizontal*

dresolution, aspect ratio, time evolution, and the input Ñux.The scaling relationships put forward by SRC98b are alsoexamined in the light of new calculations. In the nextsection, we describe the essential features of our models andthe simulation parameters. In ° 3, we give the detailedresults of our computation in terms of time- and space-averaged quantities and a discussion of the obtained results.

TABLE 2

INITIAL PHYSICAL PARAMETERS FOR ALL MODELS

Polytropic Indices Thickness of Layers (in PSH)

Pr Fb

Top Middle Bottom Tb/T

tob/o

tP

b/P

tTop Middle Bottom

1/3 . . . . . . 0.125 1.5 1.5 2.0 19.7 87.2 1716 0.84 5.57 1.04

404 SAIKIA ET AL. Vol. 529

FIG. 1.ÈTime- and space-averaged magnitudes of the kinetic energyÑux are plotted against the depth for (a) models M1, M2, and M3, (b)models M4, M5, and M8, and (c) models M6 and M7. The peak of isF

kmaximum for M2 (B0.08), reaching a magnitude that is about 64% of theinput Ñux. Note that the time of evolution for model M2 is 2622.

Important conclusions of the study constitute the lastsection.

2. NUMERICAL PARAMETERS

Our computational domain is a three-dimensional box ina Cartesian coordinate system (x, y, z) with z denoting thevertical direction and, thus, the depth of the box. The gov-erning equations representing the stratiÐed compressiblehydrodynamical Ñow and the simulation parameters arediscussed in detail in SRC95. A subgrid-scale (SGS) eddyviscosity is used and is represented by the equation

k \ o(Ck d)2(2r : r)1@2 , (2)

FIG. 2.ÈVariation of the enthalpy Ñux with depth for various models

where is the Deardor† coefficient (Smagorinsky 1963 ;CkDeardor† 1971), d is the minimum grid size, r is the rate-of-strain tensor, and k is the dynamic viscosity. The SGS di†u-sivity is computed from this eddy viscosity by assuming aconstant Prandtl number Pr.

The three-layer stable-unstable-stable conÐguration isgenerated by controlling the conductivities of di†usive Ñux,

which is given asFd,

Fd\ [K

T+T [ K

S+s , (3)

where the speciÐc entropy

s \ c ln T /(c[ 1)[ ln P , (4)

where c is the ratio of speciÐc heats. While describes theKSdi†usive e†ect of the SGS turbulence, describes the radi-K

Tative e†ect. We use a piecewise conductivity proÐle (seeChan & Gigas 1992) such that, in a stable layer, the coeffi-

No. 1, 2000 PENETRATION AT BOTTOM OF CONVECTIVE ENVELOPE 405

FIG. 3.ÈRoot mean square vertical velocity plotted against depth forall eight models

cient is zero whereas is given by the input Ñux (fromKS

KTthe lower boundary) divided by the subadiabatic tem-F

bperature gradient used for setting the initial polytropic dis-tribution of the layer. In the unstable layer, is set to aK

Tsmall value equal to 0.0004 times that of the upper stablelayer.

The polytropic index plays a vital role in the initial strati-Ðcation of the layers. For the present case of perfect gasunder the inÑuence of a constant gravitational acceleration,each layer has a stratiÐcation initially in which temperature,density, and pressure can be expressed in terms of the poly-tropic index in that layer. In all our models, the polytropicindices from the top down are 1.5, 1.5, and 2.0. In ourpresent exercise, we make use of the alternating directionimplicit on staggered mesh (ADISM) technique of Chan &Wol† (1982) to evolve the Ñuid with time to a statistically

stationary state.The time step for this implicit scheme ischaracterized by the CFL number NCFL :

NCFL\ vsdt/d , (5)

where d is the minimum grid size in any direction, at the topin our case ; is the dimensionless sound speed at thev

s\ c1@2

top ; and denotes the time step that can be taken inNCFLexcess of that for an explicit scheme. As far as the construc-tion of the mesh is concerned, the grid points in the x- andy-directions are uniformly spaced, while in the verticaldirection there are a roughly equal number of grid pointsper pressure scale height.

The Ñuid is an ideal gas with c\ 5/3 and is contained in arectangular box with a particular aspect ratio (width/depth)A with periodic side walls and impenetrable and stress-freetop and bottom walls. All the variables are made dimen-sionless by scaling so that the initial temperature, density,pressure, and the total depth of the domain all equal 1 at thetop. The velocities are scaled to while the scaling(P

t/o

t)1@2,

for various Ñuxes, including the input Ñux, is Various(PtVt).

energies are scaled to where the subscript t refers to thept,

initial values of the quantities at the top of the domain.Tables 1 and 2 list the numerical and initial physical

parameters of a total of eight cases that we have computedfor examining the e†ect of various parameters on the pen-etration distance into the bottom stable layer. It has beenreported by Hurlburt et al. 1994 and SRC95 that the pen-etration distance decreases if the stability of the lower stablelayer is increased. Therefore, to eliminate this e†ect, we havekept the polytropic index of the lower stable layer andhence the stability parameter unchanged for all eight cases.Instead, we see the variation in the penetrative behavior forvarious grid sizes and aspect ratios. The models M1 and M2are chosen to see the impact of the time for which the Ñuid isevolved after thermal relaxation has set in. Model M1 wascomputed earlier by SRC95. Both these models have a meshof 35] 35 ] 51 and an aspect ratio of 1.5 and are similar inall respects except that M1 was evolved until time 1224while M2 is integrated until time 2622 before the physicalquantities of interest are accumulated. Model M3 has amesh of 64 ] 64 ] 51 and has been incorporated to checkthe penetration behavior when the horizontal resolution isincreased from 35] 35 as in model M1 to 64] 64. Simi-larly, we have included models M4 and M5 with mesh sizes20 ] 20 ] 37 and 20 ] 20 ] 64, respectively, to Ðnd theinÑuence of the vertical resolution on the penetration dis-tance. Besides these Ðve models, there are two models, M6and M7, for determining the dependence of the penetrationon the aspect ratio of the box. The aspect ratio for therectangular box has been increased from 1.5 in model M6 to3.0 in model M7. An additional model M8 having a mesh of35 ] 35 ] 64 is computed, and the results are comparedwith model M1, which has the same horizontal resolution,and model M5, which has the same vertical resolution.

The total Ñux is made up of four Ñuxes : the enthalpy Ñuxthe kinetic energy Ñux the di†usive Ñux and(F

e), (F

k), (F

d),

the viscous Ñux see SRC95). The three-layer conÐgu-(Fv;

ration is evolved with time until the sum of these four Ñuxesbecomes almost equal to the input Ñux, at all depths.(F

b),

The required physical quantities are calculated in terms ofcombined temporal and spatial (horizontal) means andcomputed by integrating the equations for some more timeafter the Ñuid is thermally relaxed. The output Ñux at thetop ranges from 90% of the input Ñux for model M3 toF

b


FIG. 4a

FIG. 4b

FIG. 4.ÈPseudostreamlines for (a) model M2 and (b) model M3. A total of 1600 points (40 ] 40) are released from the y-z plane into the convecting Ñuidat a distance x \ 0.6. The rectangular box is seen from the front and the eye coordinates are (4.0, 0.75, 0.5). Streamlines for model M1 are given in SRC95 and,thus, are not shown here.

99% of for model M2. For model M1, the output Ñux isFb92%, while for models M4ÈM8 it is 94% of F

b.

3. RESULTS AND DISCUSSION

In all eight models, the three-layer conÐguration is suchthat the size of the upper stable layer is 2% whereas that ofthe lower stable layer is 40% of the whole vertical domain.The objective of placing the upper stable layer is to pad theconvective region from the upper solid boundary of the box.Although the upper stable layer has a small width (0.02), itinitially contains almost 1 PSH, as the length of the pressurescale height is minimum near the top.

In Figures 1a, 1b, and 1c we have shown the distributionof the average Ñux of kinetic energy withF

k\ (1/2)oV 2V

z,

depth, while the distribution of the enthalpy Ñux Fe\

is shown in Figures 2a, 2b, and 2c, for all eight(e] p)Vzmodels. Figure 2b shows a bump in the enthalpy Ñux proÐle

at the boundary of the unstableÈlower stable layers formodel M4. This sudden jump in was also observed byF

eSingh et al. (1994) and was attributed to low resolution inthat region. Here, we see that model M4 has indeed thepoorest vertical resolution with 37 grid points in the vertical

direction. The bump in goes away when the number ofFegrid points in the z-direction is increased to 64 in models

M5 and M8.The rms vertical velocities of various models are depicted

in Figures 3a, 3b, and 3c. In all the Ðgures drawn here, zerodepth means top of the domain while the bottom of thedomain is at depth 1. The unstable middle region extendsfrom a depth of 0.02È0.6.

We use the proÐle of the kinetic energy Ñux to estimatethe extent of penetration into the stable layer at the bottomof the convectively unstable region. We deÐne the penetrat-ion distance to be the length of the region between the*

dinterface of the unstableÈlower stable layer and the depthwhere the kinetic energy Ñux, reaches close to zeroF

k,

([10~3). Besides this deÐnition of the penetration distance,we employ one more criterion that has been used in someearlier studies (Hurlburt et al. 1994 ; SRC95 ; SRC98a ;SRC98b) and is also based on the proÐle of kinetic energyÑux, According to this, the penetration extends to aF

k.

depth where is 5% of its value at the interface of theFkunstableÈlower stable layers. In order to compute we*

d,

perform polynomial interpolation to obtain the values of Fk


FIG. 5a

FIG. 5b

FIG. 5.ÈTop views of pseudostreamlines for the models M2 and M3. The points are released on the x-y plane at a depth (z) of (a) 0.5 from the top formodel M2, (b) 0.5 from the top for model M3, and (c) 0.7 from the top for model M3. A total of 3600 (60 ] 60) points are released, and the box is viewed fromthe top at an eye location of (0.75, 0.75, 3.5) and hence appears square, as the extent in both x- and y- directions is 1.5.

between the available grid points. The values of penetrationdistance for all eight models together with the penetrat-*

dion in pressure scale heights are given in Table 3.(*p)

The more vigorous the downward plumes, the greater isthe penetration, which can be seen by comparing the pen-etration distances and the rms vertical velocities at the(Vzo)

base of the convection zone for the eight models given inTable 3. Models M1 and M2 are similar except that thetime of evolution for M2 is longer, at 2622, while that formodel M1 (see SRC95) is 1224. This means that, in the twomodels, time- and space-averaged quantities were collectedat di†erent times. It may be noticed that for M1 and M2Vzo


FIG. 5c

are 0.103 and 0.111, and the corresponding penetration dis-tances are 0.233 and 0.254, respectively. Considering thatwe are dealing with turbulent, time-dependent Ñows, suchbehavior is not unexpected.

Models M1 and M3 have been set up to test the e†ect ofhorizontal resolution on the penetration depth. M1 has35 ] 35 mesh points, while M3 has 64 ] 64 points in thetwo horizontal directions. The two models are evolved toroughly the same time. The penetration distances in the twomodels are quite close to each other as are the values of Vzo.The near doubling of the horizontal resolution does notseem to have any a†ect on the average velocity of thedescending plumes and hence on the penetration *

d.

To have a more complete picture of the overall phenome-non of penetration for various models, we also study theÑow patterns and draw pseudostreamlines by taking snap-shots of the velocity vectors after the Ñow has reached astatistically steady state. We release particles into the Ñuidin the rectangular box from a horizontal plane and followthem in three dimensions along the Ñow vectors. The pro-jections of the three-dimensional Ñow onto a two-dimensional plane are then viewed from di†erent ““ eye ÏÏ

locations. In Figures 4È8, we have shown the instantaneousÑow patterns for di†erent models corresponding to thetimes given in Table 1.

While a front view of the box (Fig. 4) shows us the behav-ior of downÑows, a top view depicts the projections of thehorizontal Ñows (Fig. 5). One can check the e†ect of hori-zontal resolution by comparing the streamline plots ofmodels M2 and M3. In Figure 5a we have a top view of thebox with streamlines projected onto the x-y plane at adepth of 0.5 from the top boundary for the model M2,which has a mesh of 35] 35 ] 51 points. In Figures 5b and5c, we have a similar view for model M3, which has the x-yplane located at two di†erent depths, 0.5 and 0.7 from thetop, respectively, and has a better horizontal resolution witha mesh of 64] 64 ] 51 points. It may be noticed (Figs. 5band 5c) that we are able to resolve several horizontal rollsfor model M3, which has a Ðner horizontal mesh comparedto model M2 (Fig. 5a).

Models M4 and M5 have been set up to look into thee†ect of vertical resolution on the penetration distance. Weobserve (see Table 3) that there is a remarkable increase inthe undershoot from 0.185 (0.63 PSH) to 0.365 (1.11 PSH)

TABLE 3

PENETRATION AND ITS SCALING RELATIONSHIPS

Model *d

*p

Vzo *d/V zo3 *

d/F

b1@2 *

d/V zo3@2

M1 . . . . . . 0.233 (0.208) 0.77 (0.70) 0.103 213.6 (190.4) 0.66 (0.59) 7.05 (6.29)M2 . . . . . . 0.254 (0.221) 0.83 (0.73) 0.111 185.6 (161.6) 0.72 (0.63) 6.87 (5.98)M3 . . . . . . 0.225 (0.201) 0.75 (0.67) 0.104 200.0 (178.4) 0.64 (0.57) 6.71 (5.99)M4 . . . . . . 0.185 (0.167) 0.63 (0.57) 0.108 147.2 (132.8) 0.52 (0.47) 5.21 (4.71)M5 . . . . . . 0.365 (0.320) 1.11 (1.00) 0.116 233.6 (204.8) 1.03 (0.90) 9.24 (8.10)M6 . . . . . . 0.226 (0.200) 0.75 (0.67) 0.101 219.2 (194.4) 0.62 (0.57) 7.04 (6.23)M7 . . . . . . 0.299 (0.267) 0.95 (0.86) 0.115 196.0 (175.2) 0.84 (0.75) 7.67 (6.85)M8 . . . . . . 0.290 (0.257) 0.92 (0.83) 0.108 230.2 (204.0) 0.82 (0.73) 8.17 (7.24)


FIG. 6a

FIG. 6b

FIG. 6.ÈFlow patterns projected on the x-z plane at y \ 0 for models (a) M4 and (b) M5. The number of particles released is 900 (30 each in the x- andz-directions) for both models, and the boxes are viewed from the positive x-axis. Prominent downÑows are seen for model M5. The periodicity of the sidewalls can be veriÐed by looking at the velocity vector proÐle on the left- and right-hand edges of the box.

as we increase the number of vertical grid points from 37(model M4) to 64 (model M5). The rms vertical velocity atthe bottom of the unstable zone is 0.108 for M4 and 0.116for model M5. Since the estimation of the penetration dis-tance is based on the Ñux of kinetic energy, it would beF

k,

interesting to examine Figure 1b in which we have plottedagainst depth for these models. This Ñux is negative inF

kthe whole of the unstable region, but the magnitude of thisÑux at the bottom of this region is much larger for modelM5. Furthermore, it falls to zero much more slowly, almostat the bottom of the box, in contrast to model M4 where thedrop is much steeper and we have an impression that thebox is too small for model M5 in which the vertical motionsare much stronger. To examine this issue in more detail, wehave computed an additional model M8 in which we havekept the vertical resolution the same as M4 and M5 whilethe horizontal resolution is increased to 35 ] 35. We seethat (see Fig. 1b) the kinetic energy Ñux now drops to zerowell inside the lower boundary of the box and the penetrat-ion depth is 0.290, corresponding to 0.92 PSH.This meansthat the horizontal resolution of models M4 and M5 maybe insufficient and the Ñows tend to be stronger in the verti-

cal direction where there are more grid points available. Acomparison of models M1 (35] 35 ] 51) and M8(35] 35 ] 64) also indicates a larger penetration for modelM8 where there are more grid points in the vertical.

Furthermore, the Ðnal density contrasts for models(ob/o

t)

M4 and M5 are 102.1 and 99.75, respectively. One could,therefore, attribute the larger vertical velocity (and hencepenetration) for model M5 to the lower density contrast.Another reason for the large di†erence in for the two*

dmodels is believed to be the di†erence in thickness of theunstable layers of the two models after they are thermallyrelaxed. The extent of the middle unstable region became5.48 PSH in M5 compared with 5.34 PSH in model M4.

In Figures 6a and 6b, we have projected the Ñow stream-lines on the x-z plane placed at y \ 0.0 for the two modelsM4 and M5, respectively. The front view of the box showsstrong downÑows for model M5 as compared to M4 andextends all the way to the bottom of the box. Figures 7a and7b provide another perspective of the Ñows as seen from thetop of the box for the two models (M4 and M5, respectively)in which we show the horizontal Ñows at the bottom solidboundary of the box in the x-y plane. We notice that for the


FIG. 7a

FIG. 7b

FIG. 7.ÈTop views of the Ñow patterns projected on the x-y plane at z\ 0.0 for models (a) M4 and (b) M5. A total number of 2500 (50 each in the x- andy-directions) particles were released. It may be seen that for the model with a coarser mesh (M4), the Ñows are weaker at the bottom of the box as comparedwith model M5, with 64 mesh points in the vertical.

model with a Ðner mesh in the vertical (M5), the Ñows arestronger as compared with model M4 with only 37 points inthe vertical.

To see the e†ect of aspect ratio, we have calculated for*dmodels M6 and M7, with aspect ratios (A) of 1.5 and 3.0,

respectively. The penetration distance increases from 0.226in model M6 to 0.299 in model M7. Hurlburt, Toomre, &Massaguer (1986) found only a slight increase in from*

dtheir two-dimensional simulations when the aspect ratiowas changed from 4 to 8. They attributed this to the time-


FIG. 8a

FIG. 8b

FIG. 8.ÈStreamlines for models (a) M6 with an aspect ratio 1.5 and (b) M7 with an aspect ratio 3.0

averaging interval, which was di†erent for the two aspectratios. In our model, we have kept the time-averaging inter-val the same for both our models M6 and M7.

We have plotted the pseudostreamlines for models M6and M7 in Figures 8a and 8b, respectively. The velocityvectors are projected on the y-z plane at x \ 0.6 for modelM6 (A\ 1.5) and x \ 1.2 for model M7 (A\ 3.0). Tocompare the behavior of the Ñows, the snapshots are takenat the same time (t \ 1608) in both the models. We Ðnd thatthere are several more downward-directed plumes formodel M7 (Fig. 8b), with the aspect ratio 3.0, comparedwith model M6 (Fig. 8a). The Ðnal density contrast is(o

b/o

t)

lower for model M7, and consequently the average rmsvertical velocity at the bottom of the unstable layer is higher(0.115) for M7 as is the penetration distance.

According to the helioseismic inversion of the recentsolar oscillation data, an upper limit of on the0.05H

pdownward penetration has been suggested. It is clear fromour results discussed above that the simulations yield ratherhigh estimates for the penetration. This is expected, as thedensity and pressure contrasts of all simulation models arefar lower than those prevailing in actual solar and stellarconvection zones. Since we are, at present, limited by theavailable computational power, a way around this problemis to Ðnd some scaling relationships relating the penetrationdistance to other physical quantities such as the verticalvelocity and the Ñuxes, which can then be applied to thesolar case as suggested by Schmitt et al. (1984), Zahn (1991),and Hurlburt et al. (1994).

Schmitt et al. (1984) and Zahn (1991) proposed *dP V zo3@2for a conductivity proÐle that is continuous across the

unstable and stable zones. Hurlburt et al. (1994) examinedthe behavior of the penetrative Ñows by simulating in twodimensions for a conductivity proÐle that varied piecewisein the three-layer conÐguration. For regions of nearly adia-batic penetration they found while for non-*

dP V zo3 /F

b,

adiabatic penetration a relationship was*dPF

b1@2

suggested, being the input Ñux fed at the bottom.FbSRC98b examined these relationships by simulating in

three dimensions. In both these studies, only a few caseswere considered. While Hurlburt et al. varied the aspectratio, SRC98b varied only the input Ñux. It is, therefore, ofconsiderable interest to study the relationships involvingthe penetration distance for our eight models as given inTable 3.

In all the models presently studied, we have kept theinput Ñux Ðxed at 0.125, the conductivities are unal-(F

b)

tered, and so also the boundary as well as inital conditions.Besides the changes in mesh size and aspect ratio, there is aslight variation in the time for which the Ñuid was allowedto relax before collecting the Ðnal averaged data. In Table 3,we have examined three relationships involving thepenetration distance, namely, (which is same as*

dP V zo3since is same for all the models),*

dPV zo3 /F

b, F

b*dPF

b1@2,

and *dPV [email protected] the input Ñux is Ðxed at 0.125 for all the models,F

bfor to be proportional to the penetration distance*d

Fb1@2,

for all the models should be the same. As we have already


FIG. 9.ÈSuperadiabatic gradient vs. depth for models (a) M1 to M3,(b) M4, M5, and M8, and (c) M6 and M7.

seen, the penetration distance is certainly a†ected by thevertical resolution and the aspect ratio. SRC98b found

and for four models in*dD (0.64)F

b1@2 *

dD (7 ^ 0.3)V zo3@2which only the input Ñux was varied, from 0.03125 to

0.1875. The penetration was found to proceed sub-adiabatically for a mesh of 35] 35 ] 51.

For our models M1, M2, and M3 the same law seems tohold even when the horizontal resolution was increasedfrom 35 ] 35 (models M1 and M2) to 64] 64 (for modelM3) or when the physical parameters were collected at dif-ferent times (1224 for M1 and 2622 for M2). Model M6, inwhich only the resolution is slightly poorer (28 ] 28 ] 46),also follows the two relationships. As may be seen fromTable 1, our models M4 (low overall resolution), M5(highest vertical resolution), and M7 (larger aspect ratio) do

not follow the above relationship. Model M4, with a meshof 20 ] 20 ] 37, yields a rather low penetration comparedto other models and hence does not seem to follow any ofthe three scaling relationships mentioned above (see Table3). Hence, this model with the poorest resolution may notbe suitable for any meaningful interpretation of the results.

In Figures 9a, 9b, and 9c, we have plotted the run ofsuperadiabatic gradient with depth for the eight models.Models M5 (20] 20 ] 64) and M8 (35] 35 ] 64) have thehighest vertical resolution. While the penetration seems toproceed nearly adiabatically for model M5, it becomesslightly subadiabatic for model M8 (see Fig. 9b). It may beseen from Table 3 that model M5 does not follow either ofthe two relationships given for the models M1, M2, M3, andM6. Model M7 with A\ 3 is slightly subadiabatic likemodel M8 (Fig. 9c) and seems to lie in between models M5(adiabatic) and M6 (nonadiabatic).

4. CONCLUSION

We have performed full three-dimensional simulations ofturbulent compressible convection for a three-layer conÐgu-ration in order to study the behavior of penetrative convec-tion at the bottom of a stellar-type convective envelope.Several models have been computed, and the e†ect of hori-zontal resolution, vertical resolution, aspect ratio, and thetime statistics on the extent of penetration is viewed. Sincewe are able to simulate only a small fraction of a stellarradius, it is more meaningful to examine some scalingrelationships of penetration distance with some dynamicalvariables like the vertical velocity and the input Ñux, whichcan then be applied to a realistic stellar model.

We have reported the values of penetration distance (*d),

based on the Ñux of kinetic energy Computations of(Fk). *

dwere also done using the enthalpy Ñux and the e-folding(Fe)

distance of vertical velocity. Since the variation of with*dvarious input parameters showed a similar trend, numbers

based on these two criteria were not shown in Table 3.We Ðnd that while the vertical resolution and the aspect

ratio of the box inÑuence the penetration distance con-(*d)

siderably, there is some e†ect of increasing the horizontalresolution on for the range of parameters presently con-*

dsidered. A horizontal resolution of 35 ] 35 may be sufficientfor a vertical grid count of 64. We expect that a furtherincrease in the vertical resolution will require a correspond-ing increase in the horizontal resolution. However, there islittle e†ect of the time statistics on once the Ñuid is*

d,

thermally relaxed.We have been able to examine and verify the scaling

relationship for the four models (M1, M2, M3,*dP F

b1@2

and M6) in which the penetration proceeds non-adiabatically. Model M4, with a 20] 20 ] 37 mesh, haspoor resolution and cannot be relied upon as far as inter-pretation of scaling relationships is concerned. We haveonly one deÐnite model (M5) where the penetration isnearly adiabatic, and, therefore, we cannot comment on thescaling law on the basis of this single model.*

dP F

b/V zo3Model M7, with a larger aspect ratio, and model M8 are

slightly subadiabatic ; it is difficult to say to which scalinglaw they are closer.

We Ðnd that more models need to be studied, especiallywith a continuous (instead of piecewise, as in the presentpaper) conductivity proÐle and by varying the stability ofthe lower stable layer. We hope to take up such studies infuture.


Parts of this work were supported by research grantsfrom the Particle Physics and Astronomy ResearchCouncil, UK, the Hong Kong Research Grant Council, andthe Indian Space Research Organization (ISRO). E. S. is

grateful to ISRO for a Junior Research Fellowship fromgrant 10/2/247 to H. P. S. Thanks are due to an anonymousreferee, whose comments improved the presentation of thepaper.

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