Turbulent Compressible Convection with Rotation - Penetration above a Convection Zone
Partha S. Pal, Harinder P. Singh, Kwing L. Chan, M. P. Srivastava
aa r X i v : . [ a s t r o - ph ] F e b Astrophysics and Space Science manuscript No. (will be inserted by the editor)
Partha S. Pal · Harinder P. Singh · Kwing L. Chan · M. P. Srivastava
Turbulent Compressible Convection with Rotation -Penetration above a Convection Zone
Received: date / Accepted: date
Abstract
We perform Large eddy simulations of turbulent compressible convection in stellar-typeconvection zones by solving the Navi´er-Stokes equations in three dimensions. We estimate the extentof penetration into the stable layer above a stellar-type convection zone by varying the rotation rate( Ω ), the inclination of the rotation vector ( θ ) and the relative stability ( S ) of the upper stable layerThe computational domain is a rectangular box in an f-plane configuration and is divided into tworegions of unstable and stable stratification with the stable layer placed above the convectively unstablelayer. Several models have been computed and the penetration distance into the stable layer above theconvection zone is estimated by determining the position where time averaged kinetic energy flux hasthe first zero in the upper stable layer. The vertical grid spacing in all the model is non-uniform, andis less in the upper region so that the flows are better resolved in the region of interest. We find thatthe penetration distance increases as the rotation rate increases for the case when the rotation vectoris aligned with the vertical axis. However, with the increase in the stability of the upper stable layer,the upward penetration distance decreases. Since we are not able to afford computations with finerresolution for all the models, we compute a number of models to see the effect of increased resolution onthe upward penetration. In addition, we estimate the upper limit on the upward convective penetrationfrom stellar convective cores. Keywords convection – stars · interior – Sun · interior – rotation. PACS
First · Second · More
Simulations of penetrative convection above a stellar-type convection zone have been performed intwo-dimensions by Hurlburt et al. (1986), and in three-dimensions by Singh et al. (1994, 2001) and
Partha S. PalDepartment of Physics & Astrophysics, University of Delhi, Delhi - 110 007, IndiaHarinder P. SinghDepartment of Physics & Astrophysics, University of Delhi, Delhi - 110 007, IndiaE-mail: [email protected] L. ChanDepartment of Mathematics, Hong Kong University of Science & Technology, Hong Kong, ChinaM. P. SrivastavaDepartment of Physics & Astrophysics, University of Delhi, Delhi - 110 007, India Partha S. Pal et al.
Robinson et al. (2004). In the studies by Hurlburt et al. and Singh et al., several models were computedto study the overshoot of convective motions from unstable into the upper stable layer in a non-rotatingconfiguration. It was found that the motions from the convective region penetrate a significant fractionof pressure scale height into the stable layer above. Singh et al. (1994) also found that the penetrationdistance above the convection zone ( ∆ u ) scales as ∆ u ∼ ( F b /ρ ctop ) / , where F b is the input fluxand ρ ctop is the density at the top of the convection zone. Robinson et al. (2004) performed three-dimensional simulations of the upper radiation-convection transition layer with more realistic modelingfor three subgiant stars. For the present Sun they found the overshoot to be 0 . H p , for the 11 . . H p , while for a 11 . H p , H p being the localpressure scale height.Simulations of the effects of rotation on convective penetration or overshooting have been studiedby Brummel et al. (2002), Ziegler & R¨udiger (2003), Browning et al. (2004), K¨apyl¨a et al. (2004), andPal et al. (2007). Brummel et al. (2002) and Pal et al. (2007) examined the behaviour of penetrativeconvection below a convection zone under the influence of rotation by means of three-dimensionalsimulations. They found that with an increase in rotational velocity, the downward penetration de-creased. A similar behaviour was observed when the stability of the lower stable layer was increasedin a rotating configuration. Furthermore, the relative stability parameter S showed an S − / depen-dence on the penetration distance implying the existence of a thermal adjustment region in the lowerstable layer rather than a nearly adiabatic penetration zone. Ziegler & R¨udiger (2003) and K¨apyl¨a etal. (2004), using their 3D MHD codes also found that, as a general feature, the overshooting at thebottom decreases as a function of increasing rotation at a given latitude.Browning et al. (2004) performed three-dimensional simulations of core convection within A-typestars at a range of rotation rates. They found that as convective motions enter the stable stratificationof the radiative envelope, increasing the rotation rate enhances the overshooting.In this paper, we perform large eddy simulations of turbulent convection in a f -plane configurationto study the effect of rotation on the penetration distance above the convection zone. A total of 17two-layer (lower unstable – upper stable) models have been set up to examine the effect of rotationrate ( Ω ) and colatitude ( θ ) on the penetration distance. In some models, we systematically vary thestability of the upper stable layer to study its effect on the penetration distance. In one model, weincrease the horizontal resolution to see if the penetration distance is altered.In the next section, we give the essential ingredients of the simulations and the parameters of thecomputed models. In Section 3, we provide the results and their discussion. Important conclusions ofthe study are listed in Section 4. The general behaviour of convective transport in a stellar-type setting has been studied using LargeEddy Simulation (LES) approach by several groups (Chan & Sofia 1986, Hossain & Mullan 1991, 1993,Muthsam et al. 1995, Singh & Chan 1993, Singh et al. 1994, 1995, 1996, 1998a,b, 2001 Saikia et al.2000, Chan 2001, Pal et al. 2007). LES, being less demanding on speed and memory as it can have acoarser grid, allows large scale flows to be modeled explicitly while the smaller scales are modeled bysome sort of sub-grid scale formulation (Smagorinsky 1963).Pal et al. (2007) numerically solved the Navi´er-Stokes equations and incorporated the rotationaleffects by considering an f -plane configuration. A plane parallel layer of perfect gas was considered ina rectangular box which can be viewed as a small portion of a spherical shell. We use a configurationthat is similar to that of Pal et al. (2007) except that the computational domain has two layers (lowerunstable – upper stable) rather than their three layer sandwich (stable-unstable-stable) configuration.The choice is justified as we are interested in studying the behaviour of penetrative convection abovea convection zone rather than below as was the aim of Pal et al. (2007). The spherical shell rotatesaround the polar axis from west to east and the angular velocity vector Ω points toward the northpole. We use a right handed cartesian coordinate system in which X and Y denote the horizontaldirection and Z denotes the upward vertical direction. The gravity vector is denoted by g and is alongthe negative Z direction. The angle between Ω and the Z axis is denoted by θ and the rotation vector Ω lies in the XZ plane. The tilted rotation vector is kept constant implying a uniform angular velocityof the sphere. enetration above rotating convection zones 3 Table 1
Physical Parameters for Models R1 to R17Polytropic Indices Layer Thickness(PSH)Models Grid F b Top Bottom P b T b ρ b Top BottomR1-R13 35 × ×
96 0.125 2.0 1.5 4655 19.5 238 6.04 2.40R14 35 × ×
96 0.125 3.0 1.5 19767 17.8 1106 7.12 2.77R15 35 × ×
96 0.125 4.0 1.5 65092 16.8 3862 8.01 3.07R16 35 × ×
96 0.125 5.0 1.5 178290 16.1 11024 8.76 3.33R17 46 × ×
96 0.125 2.0 1.5 4655 19.5 238 6.04 2.40
The rectangular computational domain has an aspect ratio of 1.5 and a mesh of 35 × ×
96 points.The top and bottom boundaries are kept impenetrable and stress-free while the side boundaries aremade periodic. The domain is divided into two layers with the convectively unstable layer positionedbelow the convectively stable one. A constant flux F b is fed from the bottom and all the thermodynamicvariables are expressed in units which set the total depth and initial density, pressure and temperatureat the top to unity.We use the Large Eddy Simulations (LES) approach to solve the Navi´er-Stokes equation for anideal gas having the ratio of specific heats ( γ ) as 5 /
3. The equations of the problem are given in Pal etal. (2007). We use the Smagorinsky coefficient of viscosity to represent the sub-grid-scale eddy viscosityin the form: µ = ρ ( c µ ∆ ) (2 σ : σ ) / , (1)where c µ is the Deardorff coefficient (Deardorff 1971), σ is the strain rate tensor and the colon signinside the bracket denotes tensor contraction and ∆ = (∆ x ∆ y ) / ∆ z . The SGS turbulent diffusivityis computed from this viscosity by assuming a constant Prandtl number Pr = 1 / N CF L = C s ∆t/∆ min , (2)where ∆ min is the minimum grid size in any direction and C s = γ / is the dimensionless sound speedat the top. The boundary conditions imposed are : F b = constant = 0 .
125 ( at the bottom ) ,T = T t = constant ( at the top ) ,v z = 0 ; ∂v x ∂z = ∂v y ∂z = 0 . We have computed 17 models and the physical parameters for the models are listed in Table 1.The numerical parameters of the simulations are given in Table 2. Columns (6) - (9) in Table 3 listthe values of Coriolis number ( Co ), Rossby number ( Ro ), effective Reynolds number ( Re ), and Taylornumber ( T a ) for the simulated cases. Following Chan (2001), they are defined as Co = Ω d/ h v ′′ i ,Ro = 1 /Co,Re = h v ′′ i d/ h ¯ µ/ ¯ ρ i ,T a = (2 Ω d / h ¯ µ/ ¯ ρ i ) , with v ′′ ≡ ( v x ′′ + v y ′′ + v z ′′ ) / where v ′′ x denotes the root mean square (rms) fluctuation of v x etc.The extent of domain is denoted by d . Partha S. Pal et al. F l u x e s F l u x e s Fig. 1
Profiles of the time- and horizontally-averaged energy fluxes for (a) case R1 ( Ω = 0 , θ = 0 ◦ ), (b) caseR7 ( Ω = 0 . , θ = 45 ◦ ). Models R1 to R13 are computed to examine the effect of rate of rotation ( Ω ) and the angle ( θ )between the rotation vector and the vertical axis on the extent of penetration below the convectivelystable layer. The polytropic index of the upper stable layer is taken to be 2.0 in all these models(R1-R13). In models R2 to R5, θ is varied from 22 . ◦ to 90 ◦ while Ω is fixed at 0.25. In the second setof models R0, R10, R11 and R13, Ω is varied from 0 to 1 in steps of 0.25, respectively, keeping θ fixedat 0 ◦ . In models R3, R7, and R12, Ω is 0.25, 0.5 and 1.0, respectively, while θ is fixed at 45 ◦ . In thelast set of models R14 to R16, polytropic index of the upper stable layer is varied from 3 to 5 whileall the other parameters are the same as model R7. Another model R17, having a grid of 46 × × Ω = 0 . , θ = 45 ◦ ) is computed to examine the effect ofincrease in the number of grid points (resolution) in the horizontal plane.As described , four models, namely, R7, R14, R15 and R16 have been set-up to examine the effectof stability of the upper stable layer on the penetration height of these rotating configurations. Eachof these four models have different polytropic indices in the upper stable layer, namely, 2, 3, 4 and 5for the cases R7, R14, R15 and R16, respectively. For all these four models, the values of Ω and θ arekept constant at 0.5 and 45 ◦ respectively. Similar to the relative stability parameter S for the lowerstable layer (Hurlburt et al. 1994, Singh et al. 1995, Pal et al. 2007), we can define a relative stabilityparameter for the upper stable layer for these four cases as: S = m i − m a m − m a ; m i = 2 , , , , (3)where m i denote the polytropic indices of the upper stable layer for various cases, m is the polytropicindex of the upper stable layer for our case R7 and m a = 1 / ( γ −
1) is the adiabatic index. Here, wehave taken γ = c p /c v as 5/3. Thus, for our reference case R7 the relative stability parameter S is equalto unity and for cases R14, R15 and R16 it comes out to be 3, 5 and 7, respectively. enetration above rotating convection zones 5 Table 2
Numerical Parameters for the 17 modelsModels g C µ ∆t t N CF L ∆ min R1-R15,R17 50 0.2 0.00054795 1698 0.1 0.007074R16 50 0.2 0.00027397 1698 0.05 0.007074
Table 3
Dynamical Parameters of the Computed ModelsModel Ω θ/π θ h v ′′ i Co Ro Re T a ∆ u ∆ p (PSH)R1 0 0 0 o . ∞ . o o . o o . o o . o o o o o o o o o o All 17 models computed in this study have a two layer configuration in which the stable region isplaced above the convectively unstable region. Model R1 is non-rotating while models R2 to R17 areall rotating, differentiated by a set of values of rate of rotation ( Ω ), the angle between the Z -axis andthe rotation vector ( θ ) and the polytropic index (or stability) of the upper stable layer. Tables 1 and2 list the physical parameters of all the models.The interface of the unstable-stable layer is located at a height of 0.6 from the bottom. This impliesthat the thickness of the lower stable layer is 0.6 or 60% of the total domain of computation while theupper stable layer has a thickness of 0.4 corresponding to 40% of the total domain. The total domainof computation contains about 8.44 pressure scale heights (PSH) for models R1-R13 and R17 withthe upper stable layer containing 6.04 PSH while the lower convective layer contains 2.40 PSH. Formodels R14, R15 and R16 the total domain has around 9.89, 11.08 and 12.09 PSH, respectively withthe upper stable layer containing around 7.12, 8.01 and 8.76 PSH, respectively.It may be noticed that although the upper stable layer occupies only 40% of the domain, it containsmore than 70% of the PSH. This is because the length of the PSH decreases as we go from bottom tothe top. The density contrast ( ρ bottom /ρ top ) of models R1-R13 is 238.0 while the temperature contrast( T bottom /T top ) is 19.5 (cf. Table 1). For model R16 the density contrast is the maximum (11024).After the fluid has thermally relaxed, it is further evolved for another 1 , ,
000 time steps andthe time and horizontal averages are taken for quantities of interest. Figure 1 shows the distributionof various energy fluxes with height for a number of models. The calculation of the extent of thepenetration into the upper stable layer has been based on time- and horizontally averaged kinetic flux(F k ). Such a choice is obvious as the kinetic flux is directly related with the motions and the profileis also convenient for estimation of the extent of penetration (Hurlburt et al. 1986, Hurlburt et al.1994, Singh et al. 1994,1995, Saikia et al. 2000, Pal et al. 2007). We illustrate this point by plottingthe distribution of kinetic energy flux for several sets of models in Fig. 2. The kinetic energy fluxis negative in most of the convectively unstable region. Near the interface in the unstable region itincreases and becomes positive and then falls to zero in the upper stable region. Ideally, the extent ofpenetration should correspond to the first zero of the kinetic energy flux in the stable layer above theunstable-stable interface. However, we define the penetration height (∆ u ) to be the distance from theinterface of the unstable-upper stable layer to where F k has fallen to a value of 0 . Partha S. Pal et al. K i ne t i c E ne r g y F l u x K i ne t i c E ne r g y F l u x K i ne t i c E ne r g y F l u x Fig. 2
Distribution of time- and horizontally averaged kinetic energy fluxes with height for models (a) R1,R10, R11 and R13. The corresponding values of angular rotational velocity ( Ω ) are 0, 0.25, 0.50 and 1.0,respectively. For these four cases the angle ( θ ) between the rotation vector and the vertical axis is 0 ◦ , (b)R10, R2, R3, R4 and R5. All these models have Ω fixed at 0.25 while θ changes from 0 ◦ to 90 ◦ in steps of22 . ◦ , (c) R7, R14, R15 and R16. The polytropic indices in the upper stable layer for these four models are,respectively, 2, 3, 4 and 5. The corresponding relative stability parameters are 1, 3, 5 and 7, respectively. Allthe four models have Ω = 0 . θ = 45 ◦ . stable layer. The extent of penetration distance (∆ u ) and penetration distance in PSH (∆ p ) for all the17 models are given in Table 3.In Fig. 3(a), we show the horizontally averaged kinetic energy flux as a function of depth and timefor model R1. We have plotted 500 profiles of F k which corresponds to every 200th time step. In Fig.3(b), we show a time series of penetration distance ∆ u calculated by using the criterion outlined above.The mean penetration is indicated by the horizontal dashed line.We now describe the effect of varying various parameters, e.g., Ω , θ , S , and the horizontal resolutionon the penetration distance above the convectively unstable region.3.1 Dependence of penetration height on Ω Two sets of models R1, R10, R11 and R13 and R3, R7 and R12 have been computed to examine theeffect of rate of rotation on the penetration distance. In models R1, R10, R11 and R13, the rotationrate ( Ω ) is systematically increased from 0 to 1.0 in steps of 0.25 (cf. Table 3). In all these four models,the rotation vector ( θ ) is kept at 0 ◦ , implying that the rotation vector coincides with the Z -axis orthe vertical direction. Figure 4(a) shows the dependence of penetration distance on Ω for these fourmodels. In model R1, which is a non-rotating case ( Ω = 0), the penetration distance ∆ u = 0 .
055 or0.384 PSH. As the rotation rate increases to 0.25 in model R10, the distance ∆ u increases to 0.058 enetration above rotating convection zones 7 Fig. 3 (a) Horizontally averaged kinetic energy flux as a function of time. Total of 1,00,000 profiles werecollected out of which 500 (every 200th) are plotted for case R1. The height is denoted in the grid pointsstarting from one at the bottom (b) Penetration height (∆ u ) with time for model R1 for 1,00,000 time steps(computed at every 10th time step). which corresponds to 0.407 PSH. For models R11 and R13 having Ω = 0 . θ is changed to 45 ◦ . We find that unlike in the first set of models (with θ = 0 ◦ ), the penetrationdistance decreases with increase in the rotation rate. For example, the upward penetration decreasesfrom 0.399 PSH (Model R7, Ω = 0 .
25) to 0.363 PSH (Model R12, Ω = 1 . θ is 45 ◦ .The penetrative behaviour is markedly different from the case of downward penetration (Pal et al.2007), in which the penetration distance ( ∆ d ) decreased with increase in the rotation rate, for anyfixed value of colatitude θ .For the two sets of models with θ = 0 ◦ and θ = 45 ◦ , we examine the behaviour of pentration heightwith Rossby number. Figure 5(a) shows the variation of penetration height in PSH (∆ p ) with Rossbynumber ( Ro ) for three cases R10, R11 and R13 having θ = 0 ◦ and Ω = 0 . , .
50 and 1 .
0, respectively.A smaller Rossby number implies a larger rotation rate and the penetration distance increases withdecrease in the Rossby number from 0 .
407 PSH for model R10 to 0 .
446 PSH for model R13. For thisset of models with θ = 0 ◦ , we see a relationship ∆ p ∼ .
38 Ro − . . Partha S. Pal et al.
Fig. 4
Dependence of penetration height in PSH (∆ p ) with (a) Ω varying from 0, 0.25, 0.50 and 1.0 for themodels R1, R10, R11 and R13, (b) colatitude ( θ ) varying from 0 ◦ to 90 ◦ in steps of 22 . ◦ for models R10, R2,R3, R4 and R5 having Ω = 0 .
25. In this as well as in all the subsequent figures, error bars denote the standarddeviation in ∆ p computed for 1 , ,
000 time steps after the fluid has thermally relaxed.
Fig. 5(b) shows the variation of ∆ p with Ro for the second set of models R3, R7 and R12 having θ = 45 ◦ and Ω = 0 . , .
50 and 1 .
0, respectively. For this set, the penetration distance decreases withthe decrease in Rossby number. Model R3 with Ro = 0 .
255 has ∆ p = 0 .
399 PSH, while R12 with Ro = 0 .
069 has ∆ p = 0 .
363 PSH. This set of models shows a scaling relation of ∆ p ∼ .
44 Ro . .3.2 Dependence of penetration height on colatitude θ A set of five models R10, R2, R3, R4 and R5 has been examined to study the effect of co-latitude ( θ )on the penetration distance. All the models have Ω = 0 .
25 while θ is changed from 0 ◦ (Model R10) to90 ◦ (Model R5) in steps of 22 . ◦ . Figure 4(b) shows the plot of ∆ p against θ for these five models. At θ = 0 ◦ , the penetration distance is maximum 0 .
407 PSH. It decreases slightly for successive values of θ , and for θ = 90 ◦ (case R5) has the minimum value of 0 .
210 PSH.Singh et al. (1994) computed many non-rotating cases having different input fluxes and founda scaling relationship between the upward penetration and the input flux ( F b ) of the form ∆ p ∼ enetration above rotating convection zones 9 Fig. 5
Variation of penetration height with Rossby number for (a) three models R10, R11 and R13 having Ω = 0 . , .
50 and 1 .
0, respectively, and θ = 0 ◦ , and (b) three models R3, R7 and R12 having Ω = 0 . , . .
0, respectively, and θ = 45 ◦ The dashed lines represent scaling laws: (a) Ro − . and (b) Ro . . (F b /ρ ctop ) / , where ρ ctop is the density at the top of the convection zone. We attempt to find asimilar relationship which includes the effects of rotation.In Fig. 6(a), we have plotted (F b /ρ ctop ) / / ∆ p for Ω = 0 .
25, 0 .
5, and 1 . ∆ p ∼ (F b /ρ ctop ) / / . − . forthe three cases having θ = 0 ◦ . For cases R3, R7, R12 having θ = 45 ◦ (Fig. 6(b)), we have a relationshipof the form ∆ p ∼ (F b /ρ ctop ) / / . . .For a set of models with a particular value of θ , we have ∆ p = c Ro x and ∆ p = (F b /ρ ctop ) / c Ω x , (4)where c and c are constants and x and x are the scaling powers for Ro and 1 /Ω .One can use relations (4) to compute back the power laws of Fig. 6 by writing Ω x = ( F b /ρ ctop ) / c c Ro x , (5) x = ln ( F b /ρ ctop ) / c c Ro x ln Ω . (6)
Fig. 6
Plot of (F b /ρ ctop ) / / ∆ p for different Ω for (a) cases R10, R11 and R13 having θ = 0 ◦ and (b) casesR3, R7, R12 having θ = 45 ◦ . The dashed line shows the scaling relationship (a) 1 /Ω − . and (b) 1 /Ω . for the two sets of models. Substituting the value of
Ro, c , c , x , and Ω in Eqn.(6), we find that for cases with θ = 0 ◦ , x to be − .
063 and for cases with θ = 45 ◦ it is 0 .
062 which are close to the values − .
065 and 0 . S We have constructed four models with different stability parameter ( S ) defined in Eqn. (3) by varyingthe polytropic index of the upper stable layer. We have S = 1 , , Ω ) and angle between the rotation vector andvertical axis ( θ ) are fixed at 0 . ◦ , respectively, for these models. The corresponding penetrationheights ∆ u and ∆ p (in PSH) are given in the last two columns of Table 3. Figure 2(c) clearly showsthe differences in the kinetic flux profile with S especially in the region near the stable-unstable layerinterface. We find a decrease in penetration distance from 0.385 PSH in model R7 ( S = 1) to 0.058PSH in model R16 ( S = 7). Figure 7 shows the instantaneous vertical velocities at the horizontalinterface of the unstable-stable interface for models R7 and R16. The vertical velocities for model R16with stiffer stable layer (S=7) are much lower compared to that of model R7 (S=1).Fig. 8 shows the variation of penetration distance with stability factor S for models R7, R14, R15,& R16. These models show a scaling relationship ∆ p ∼ S − . This means that the convection is efficientenough to establish a nearly adiabatic stratification in the upper stable layer and the penetration isnearly adiabatic (Hurlburt et al. 1994, Zahn 2002). Brummel et al. (2002) and Pal et al. (2007) found a enetration above rotating convection zones 11 Fig. 7
Instantaneous vertical velocity at a horizontal surface at unstable-upper stable layer boundary locatedat a height of 0.6 from the bottom for two cases R7 ( Ω = 0 . , θ = 45 ◦ ) and R16 ( Ω = 0 . , θ = 45 ◦ ). Polytropicindices of the upper stable layer are 2 (Model R7) and 5 (Model R16). Fig. 8
Variation of penetration distance ∆ p with stability parameter S for models R7, R14, R15 and R16.Here, Ω = 0 . θ = 45 ◦ . The dashed line represents scaling law, S − . scaling relationship of ∆ p ∼ S − / for downward penetration for rotating convection zones associatedwith a thermal adjustment region at the bottom.3.4 Dependence of penetration height on resolutionTable 4 lists additional models computed to see the effect of resolution on the upward penetration.Model R1A is same as model R1 except that it has a total of 524 ,
288 (64 × × ,
600 (35 × ×
96) grid points, having better horizontal as well as vertical resolution.
Table 4
Parameters of models computed to examine the effect of resolution on the upward penetrationModels Grid Ω θ/π θ t ∆ p (PSH)R1A 64 × ×
128 0 0 0 o × ×
128 0.50 1/4 45 o
907 0.333R7B 96 × ×
96 0.50 1/4 45 o
550 0.268
Table 5
Penetration as a percentage of the size of the Convection ZoneModel R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13% ∆ p ∆ p Penetration height for non-rotating model R1 is 0 .
384 PSH (cf. Table 3), while for the non-rotatingmodel R1A with increased resolution, it is 0 .
303 PSH.Two rotating models R7 and R7A examine the effect of change in both horizontal and verticalresolution on the penetration distance. We notice a decrease in penetration depth from 0 .
385 PSHin model R7 (35 × ×
96) to 0 .
333 PSH in model R7A (64 × × ∆ p with increasedresolution is smaller for model R7A which has a higher rotation rate compared to model R1A whichis non-rotating. It seems that the effect of increase in resolution on penetration is different for modelswith different rotation rates.Three models R7, R17, and R7B examine the effect of horizontal resolution on the penetrationdistance above the convection zone. While all the models have 96 grid points in the vertical, they differin their horizontal resolution. Model R7 has 35 ×
35 grid points in the horizontal direction, modelR17 has 46 ×
46 and model R7B has 96 ×
96 grid points. The penetration distance decreases with theincrease in horizontal resolution from 0 .
385 PSH (Model R7) to 0 .
340 PSH (Model R17) to 0 .
268 PSH(Model R7B).
We have computed 17 models and presented the results of our three-dimensional numerical simula-tions of turbulent compressible convection penetrating into a radiative envelope under the influence ofrotation. We find that the penetration distances for these models lie in the range 0 .
058 PSH ≤ ∆ p ≤ .
446 PSH.Recently, Woo & Demarque (2001) put an empirical constraint on the convective core overshoot forintermediate-to-low mass stars by using Roxburgh’s integral constraint. They found that the properlimit of core overshoot for these stars would be 15% of the core radius. In Table 5, we have shown thiscalculation for our 17 models. We have computed the penetration distance in PSH as a percentage ofthe total size of the convection zone. The size of the convection zone is taken from the last columnof Table 1 while the penetration height ( ∆ p ) has been taken from the last column of Table 3. As canbe seen from Table 5, We find an upper limit on the penetration into the upper radiative layer to bearound 18 . θ = 0 ◦ ), the penetration into the upper stable region increasesas the angular rotational velocity ( Ω ) increases or the Rossby number ( Ro ) decreases. However, thistrend is reversed for rotation around an inclined axis. When the angle ( θ = 45 ◦ ), the penetrationdistance into the radiative envelope decreases with increasing Ω owing to horizontal mixing. To seethe effect of change of angle of inclination on penetration, angle θ is systematically varied from 0 ◦ to90 ◦ in steps of 22 . ◦ for five models with a fixed angular velocity Ω = 0 .
25. We again find that thepenetration distance decreases as the colatitude θ is increased. We do not see this behaviour changingeven when the resolution is increased.We also find that the penetration distance above the convection zone obeys a scaling relation ofthe form ∆ p ∼ S − appropriate for nearly adiabatic penetration even in the presence of rotation. Thepresent simulations need to be extended to include more realistic input physics and a higher resolutionto enable us to get a better insight into the dynamics of rotating convection near the convective- enetration above rotating convection zones 13 radiative interface. Since the motivation of the study was to look into the effect of varying rotationrates on the penetration distance, we needed to compute several models. Due to the restrictions inspeed and memory, we were unable to increase the resolution of all the models. We hope to achievethis in future studies. Acknowledgements
PSP is grateful to University Grants Commission, India for a Senior Research Fellow-ship. HPS was supported by grants from Indian Space Research Organization and Deutscher AkademisherAustauschdienst (DAAD), Germany. KLC was partly supported by grants from Hong Kong Research GrantCouncil. Authors acknowledge the use of High Performance Computing facilities at IUCAA, Pune, India.
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