Turbulent Compressible Convection with Rotation - Penetration above a Convection Zone

a r X i v :0802.1824v 1 [a s t r o -p h ] 13 F e

b 2008

Astrophysics and Space Science manuscript No.

(will be inserted by the editor)

Partha S.Pal Department of Physics &Astrophysics,University of Delhi,Delhi -110007,India

Harinder P.Singh Department of Physics &Astrophysics,University of Delhi,Delhi -110007,India E-mail:hpsingh@physics.du.ac.in

Kwing L.Chan Department of Mathematics,Hong Kong University of Science &Technology,Hong Kong,China M.P.Srivastava Department of Physics &Astrophysics,University of Delhi,Delhi -110007,India

R1-R1335×35×960.125 2.0 1.5465519.5238 6.04 2.40 R1435×35×960.125 3.0 1.51976717.811067.12 2.77 R1535×35×960.125 4.0 1.56509216.838628.01 3.07 R1635×35×960.125 5.0 1.517829016.1110248.76 3.33 R1746×46×960.125 2.0 1.5465519.5238 6.04 2.40

?z =

?v y

;m i=2,3,4,5,(3)

m2?m a

where m i denote the polytropic indices of the upper stable layer for various cases,m2is the polytropic index of the upper stable layer for our case R7and m a=1/(γ?1)is the adiabatic index.Here,we have takenγ=c p/c v as5/3.Thus,for our reference case R7the relative stability parameter S is equal to unity and for cases R14,R15and R16it comes out to be3,5and7,respectively.

R1-R15,R17500.20.0005479516980.10.007074

R16500.20.0002739716980.050.007074

R1000o0.0740.000∞1088.8830.00E+00.0550.384 R20.251/822.5o0.059 4.2310.236894.394 5.728E+70.0570.399 R30.251/445o0.064 3.9140.255947.021 5.502E+70.0570.399 R40.253/867.5o0.070 3.5790.2791022.810 5.360E+70.0510.354 R50.251/290o0.066 3.7760.265957.403 5.227E+70.0310.210 R60.501/422.5o0.0568.9800.111840.540 2.279E+80.0600.422 R70.501/445o0.0598.4320.119879.430 2.200E+80.0550.385 R80.501/467.5o0.0677.4110.135975.684 2.091E+80.0520.362 R90.501/290o0.0647.7950.128875.589 1.864E+80.0220.147 R100.2500o0.058 4.2780.234889.006 5.787E+70.0580.407 R110.5000o0.0549.2830.108822.673 2.331E+80.0630.445 R12 1.01/445o0.06914.4740.0691011.6238.576E+80.0520.363 R13 1.000o0.05020.0040.050768.3119.448E+80.0630.446 R140.501/445o0.03613.7140.073876.052 5.774E+80.0130.111 R150.501/445o0.02619.0130.053895.473 1.160E+90.0060.061 R160.501/445o0.01925.4690.039799.152 1.659E+80.0050.058 R170.501/445o0.0627.9710.1261107.676 3.118E+80.0490.340

,(4)

c2?x2

where c1and c2are constants and x1and x2are the scaling powers for Ro and1/?.

One can use relations(4)to compute back the power laws of Fig.6by writing

(F b/ρctop)1/3

?x2=

c1c2Ro x1

R1A64×64×128000o10560.303

R7A64×64×1280.501/445o9070.333

R7B96×96×960.501/445o5500.268

Penetration height for non-rotating model R1is0.384PSH(cf.Table3),while for the non-rotating model R1A with increased resolution,it is0.303PSH.

Two rotating models R7and R7A examine the e?ect of change in both horizontal and vertical resolution on the penetration distance.We notice a decrease in penetration depth from0.385PSH in model R7(35×35×96)to0.333PSH in model R7A(64×64×128).It may be noticed that while models R1and R7had almost same penetration distance,the decrease in?p with increased resolution is smaller for model R7A which has a higher rotation rate compared to model R1A which is non-rotating.It seems that the e?ect of increase in resolution on penetration is di?erent for models with di?erent rotation rates.

Three models R7,R17,and R7B examine the e?ect of horizontal resolution on the penetration distance above the convection zone.While all the models have96grid points in the vertical,they di?er in their horizontal resolution.Model R7has35×35grid points in the horizontal direction,model R17has46×46and model R7B has96×96grid points.The penetration distance decreases with the increase in horizontal resolution from0.385PSH(Model R7)to0.340PSH(Model R17)to0.268PSH (Model R7B).

4Conclusions

We have computed17models and presented the results of our three-dimensional numerical simula-tions of turbulent compressible convection penetrating into a radiative envelope under the in?uence of rotation.We?nd that the penetration distances for these models lie in the range0.058PSH≤?p≤0.446PSH.

Recently,Woo&Demarque(2001)put an empirical constraint on the convective core overshoot for intermediate-to-low mass stars by using Roxburgh’s integral constraint.They found that the proper limit of core overshoot for these stars would be15%of the core radius.In Table5,we have shown this calculation for our17models.We have computed the penetration distance in PSH as a percentage of the total size of the convection zone.The size of the convection zone is taken from the last column of Table1while the penetration height(?p)has been taken from the last column of Table3.As can be seen from Table5,We?nd an upper limit on the penetration into the upper radiative layer to be around18.5%.

For rotation about a vertical axis(θ=0?),the penetration into the upper stable region increases as the angular rotational velocity(?)increases or the Rossby number(Ro)decreases.However,this trend is reversed for rotation around an inclined axis.When the angle(θ=45?),the penetration distance into the radiative envelope decreases with increasing?owing to horizontal mixing.To see the e?ect of change of angle of inclination on penetration,angleθis systematically varied from0?to 90?in steps of22.5?for?ve models with a?xed angular velocity?=0.25.We again?nd that the penetration distance decreases as the colatitudeθis increased.We do not see this behaviour changing even when the resolution is increased.

We also?nd that the penetration distance above the convection zone obeys a scaling relation of the form?p～S?1appropriate for nearly adiabatic penetration even in the presence of rotation.The present simulations need to be extended to include more realistic input physics and a higher resolution to enable us to get a better insight into the dynamics of rotating convection near the convective-