Numerical Calculation of Convection with Reduced Speed of Sound Technique (Research Note)

A&A 539, A30 (2012) Astronomy DOI: 10.1051/0004-6361/201118268 & c ESO 2012 Astrophysics

Numerical calculation of convection with reduced speed of sound technique (Research Note)

H. Hotta1,M.Rempel2, T. Yokoyama1, Y. Iida1,andY.Fan2

1 Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan e-mail: [email protected] 2 High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO, USA

Received 13 October 2011 / Accepted 31 December 2011

ABSTRACT

Context. The anelastic approximation is often adopted in numerical calculations with low Mach numbers, such as those including stellar internal convection. This approximation requires so-called frequent global communication, because of an elliptic partial differential equation. Frequent global communication is, however, negative factor for the parallel computing performed with a large number of CPUs. Aims. We test the validity of a method that artificially reduces the speed of sound for the compressible fluid equations in the context of stellar internal convection. This reduction in the speed of sound leads to longer time steps despite the low Mach number, while the numerical scheme remains fully explicit and the mathematical system is hyperbolic, thus does not require frequent global communication. Methods. Two- and three-dimensional compressible hydrodynamic equations are solved numerically. Some statistical quantities of solutions computed with different effective Mach numbers (owing to the reduction in the speed of sound) are compared to test the validity of our approach. Results. Numerical simulations with artificially reduced speed of sound are a valid approach as long as the effective Mach number (based on the lower speed of sound) remains less than 0.7. Key words. methods: numerical – stars: interiors – convection – Sun: interior

1. Introduction the parity of solar global field, the strength of polar field and so on (Hotta & Yokoyama 2010a,b). Turbulent thermal convection in the solar convection zone plays ff There have been numerous LES numerical simulations of a key role for the maintenance of large scale flows (di erential the solar and stellar convection (Gilman 1977; Gilman & Miller rotation, meridional flow) and solar magnetic activity. The an- 1981; Glatzmaier 1984; Miesch et al. 2000, 2006; Brown et al. gular momentum transport of convection maintains the global 2008) and their magnetic fields (Gilman & Miller 1981; Brun mean flows. Global flows relate to the generation of the global ff et al. 2004; Brown et al. 2010, 2011). In these studies, the anelas- magnetic field, i.e. the solar dynamo. Di erential rotation bends tic approximation is adopted to avoid any of the difficulties the pre-existing poloidal field and generates the strong toroidal Ω ff caused by the high speed of sound. At the base of the convec- field (the e ect), and mean meridional flow transports the tion zone, the speed of sound is about 200 km s−1. In contrast, magnetic flux equatorward at the base of the convection zone the speed of convection is thought to be 50 m s−1 (Stix 2004). (Choudhuri et al. 1995; Dikpati & Charbonneau 1999). The in- ff The time step must therefore be shorter owing to the CFL condi- ternal structures of the solar di erential rotation and the merid- tion in an explicit fully compressible method, even when we are ional flow are revealed by the helioseismology (see review by interested in phenomena related to convection. In the anelastic Thompson et al. 2003). Some mean field studies have repro- approximation, the equation of continuity is treated as duced these global flows (Kichatinov & Rüdiger 1993; Küker

& Stix 2001; Rempel 2005; Hotta & Yokoyama 2011). These ∇·(ρ0u) = 0, (1) studies, however, used some kinds of parameterization of the turbulent convection, i.e. turbulent viscosity and turbulent an- where ρ0 is the stratified background density and u denotes the gular momentum transport. Thus, a self-consistent thorough un- velocity. The anelastic approximation assumes that the speed of derstanding of global structure requires the detailed investigation sound is essentially infinite, resulting in an instantaneous adjust- of the turbulent thermal convection. Turbulent convection is also ment of pressure to flow changes. This is achieved by solving important when describing the magnetic field itself. The strength an elliptic equation for the pressure, which filters out the prop- of the next solar maximum in the predictions of the mean field agation of sound waves. As a result, the time step is only lim- model depends significantly on the turbulent diffusivity (Dikpati ited by the much lower flow velocity. However, owing to the & Gilman 2006; Choudhuri et al. 2007; Yeates et al. 2008). In existence of an elliptic the anelastic approximation has a weak addition, the turbulent diffusion has an important influence on point. When performing the numerical calculation using parallel Article published by EDP Sciences A30, page 1 of 7 A&A 539, A30 (2012) computing, we must employ the frequent global communication. The formulations are almost identical to those of Fan et al. At the present time, the efficiency of scaling in parallel comput- (2003), and given by ing is saturated with about 2000–3000 CPUs in solar global simulations with the pseudo-spectral method (Miesch, priv. comm.). ∂ρ1 1 = − ∇·(ρ0u), (3) Higher resolution, however, is needed to understand the precise ∂t ξ2 mechanism of the angular momentum and energy transport by ∂u ∇p ρ 1 = −(u ·∇)u − 1 − 1 ge + ∇·Π, (4) the turbulent convection and the behavior of magnetic field es- ∂t ρ ρ z ρ pecially in thin magnetic flux tube. 0 0 0 ∂s1 1 In this paper, we test the validity of a different approach to = −(u ·∇)(s + s ) + ∇·(Kρ T ∇s ) ∂t 0 1 ρ T 0 0 1 circumvent the severe numerical time step constraints in low 0 0 γ − 1 Mach number flows. We use a method in which the speed of + (Π ·∇) · u, (5) sound is reduced artificially by transforming the equation of con- p0 tinuity to (see, e.g., Rempel 2005) ρ1 p1 = p0 γ + s1 , (6) ρ0 ∂ρ 1 = − ∇·(ρu), (2) ∂t ξ2 where T0(z), and s0(z) denote the reference temperature and entropy, respectively, and ez denotes the unit vector along the z- direction, γ is the ratio of specific heats, with the value for an where t denotes the time. Using this equation, the effective speed ideal gas being γ = 5/3, and s1 denotes the fluctuation of entropy of sound becomes ξ times smaller, but the dispersion relationship from reference atmosphere. We note that the entropy is normal- for sound waves remains otherwise unchanged (wave speed de- ized by a value of the specific heat capacity at constant volume creases equally for all wavelengths). Since this technique does cv. The quantity g is the gravitational acceleration, which is as- not change the hyperbolic character of the underlying equations, sumed to be constant. The quantity Π denotes the viscous stress the numerical treatment can remain fully explicit, thus does not tensor require global communication in parallel computing. We refer to the technique as the Reduced Speed of Sound Technique ∂vi ∂v j 2 Πij = ρ0ν + − (∇·u)δij , (7) (RSST). This technique was used previously by Rempel (2005, ∂x j ∂xi 3 2006) in mean field models of solar differential rotation and non- kinematic dynamos, which essentially solve the full set of time- and ν and K denote the viscosity and thermal diffusivity, respec- dependent axisymmetric MHD equations. Those solutions were tively, and ν and K are assumed to be constant throughout the however restricted to the relaxation toward a stationary state or simulation domain. We assume for the reference atmosphere a very slowly varying problems on the timescale of the solar cy- weakly superadiabatically stratified polytrope cle. Here we apply this approach to thermal convection, where m the intrinsic timescales are substantially shorter. To this end, we z ρ0(z) = ρr 1 − , (8) study two- and three-dimensional (3D) convection, in particular (m + 1)Hr + the latter will be non-stationary and turbulent. z m 1 p (z) = p 1 − , (9) The detailed setting of test calculation is given in Sect. 2.The 0 r (m + 1)H results of our calculations are given in Sect. 3. We summarize r our paper and give discussion of the RSST in Sect. 4. z T0(z) = Tr 1 − , (10) (m + 1)Hr p H (z) = 0 , (11) p ρ g 2. Model 0 ds γδ(z) 0 = − , (12) In this section, we describe the detailed settting of this study. dz Hp(z) ρ δ(z) = δ r , (13) r ρ (z) 2.1. Equations 0 where ρr, pr, Tr, Hr,andδr denote the values of ρ0, p0, T0, The two- or three-dimensional equation of continuity, equation H0 (the pressure scale height), and δ (the non-dimensional su- of motion, equation of energy, equation of state, are solved in peradiabaticity) at the bottom boundary z = 0. Since |δ|1, Cartesian coordinate (x, z)or(x,y,z), where x and y denote the the value m is nearly equal to the adiabatic value, meaning horizontal direction and z denotes the vertical direction. The ba- that m = 1/(γ − 1). The strength of the diffusive parameter sic assumptions underlying this study are as: ν and K is expressed in terms of the non-dimensional parameters: the Reynolds number Re ≡ vcHr/ν and the Prandtl number ≡ ≡ 1/2 1. the time-independent hydrostatic reference state; Pr ν/K, where the velocity unit vc (8δrgHr) . We note that 2. that the perturbations caused by thermal convection are in this paper the unit of time is Hr/vc. In all calculations, we set = small, i.e. ρ1 ρ0 and p1 p0,hereρ0 and p0 denote Pr 1. the reference state values, whereas ρ1 and p1 are the fluctua- tions of density and pressure, respectively. Thus a linearized 2.2. Boundary conditions and numerical method equation of state is used as Eq. (6); 3. that the profile of the reference entropy s0(z) is a steady state We solve Eqs. (3)–(6) numerically. At the horizontal boundaries solution of the thermal diffusion equation ∇·(Kρ0T0∇s0) = 0 (x = 0, Lx and y = 0, Ly), periodic boundary conditions are with constant K. adopted for all variables. At the top and the bottom boundaries,

A30, page 2 of 7 H. Hotta et al.: Convection with RSST (RN)

Table 1. Parameters for numerical simulations.

Case Dimension Lx × Ly × Lz(Hr) Nx × Ny × Nz δr Re ξ 128.72 × 2.18 384 × 96 1 × 10−6 260 1 228.72 × 2.18 384 × 96 1 × 10−6 260 10 328.72 × 2.18 384 × 96 1 × 10−6 260 30 428.72 × 2.18 384 × 96 1 × 10−6 260 50 528.72 × 2.18 384 × 96 1 × 10−6 260 80 638.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−4 300 1 738.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−4 300 5 838.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−4 300 10 938.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−4 300 15 10 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−4 300 20 11 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−4 300 40 12 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−4 300 80 −4 1/2 13 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10 300 20/(δ/δr) 14 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−3 300 1 15 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−3 300 5 16 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−3 300 10 17 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−3 300 15 18 3 8.72 × 8.72 × 2.18 384 × 384 × 96 1 × 10−3 300 20

Notes. In case 13, the value of ξ is 20 at the bottom and 4.5 at the top boundary.

impenetrative and stress-free boundary conditions are adopted The longest allowed time step, Δtmax, is determined by the for the velocities and the entropy is fixed to CFL criterion. When both advection and diffusion terms are in- cluded in calculation, the time step reads vz = 0, (14) Δt = min(Δt , Δt ), (24) ∂v max ad v x = 0, (15) ∂z where ∂v y = min(Δx, Δy, Δz) 0, (16) Δtad = cad , (25) ∂z ctot s1 = 0. (17) and ctot is the total wave speed At both the top and bottom boundaries (z = 0andz = Lz), we = | | + ctot v cs, (26) set p1 in the ghost cells such that the right hand side of the z- component of Eq. (4) is zero at the boundary (which is between where the effective speed of sound is expressed as ghost cells and domain cells), where the ghost cells are the cells beyond the physical boundary. = 1 p0 · cs γ (27) We adopt the fourth-order space-centered difference for each ξ ρ0 derivative. The first spatial derivatives of quantity q is given by The time step determined by the diffusion term is ∂q 1 min(Δx2, Δy2, Δz2) = −q + + q + − q − + q − Δ = Δ ( i 2 8 i 1 8 i 1 i 2), (18) tv cv , (28) ∂x i 12 x max(K,ν) where i denotes the index of the grid position along a par- and cad and cv are safety factors of order unity. = × −4 ticular spatial direction. The numerical solution of the system Using δr 1 10 , the original speed of sound is about 35vc at the bottom and 12vc at the top boundary, respectively. In is advanced in time with an explicit fourth-order Runge-Kutta −2 all calculations, Δx ∼ 2.3 × 10 Hr. Thus, if we use cad = 1and scheme. The system of partial equations can be written as −4 −1 cv = 1, Δtad = 6.4 × 10 Hr/vc and Δtv = 1.5 × 10 Hr/vc.For ∂U all the ξ values considered in this paper, the time step remains = R(U) (19) restricted by the (reduced) speed of sound, thus the calculation ∂t is about ξ times more efficient with RSST. for Un+1, which is the value at tn+1 = (n + 1)Δt is calculated in four steps 3. Results Δt 3.1. Two-dimensional study U + 1 = Un + R(Un), (20) n 4 4 We carried out four two-dimensional (2D) calculations with the Δt = + RSST and one calculation without approximation (case 1–5). Un+ 1 Un R(Un+ 1 ), (21) 3 3 4 The values of the free parameters are given in Table 1.Thesu- Δ − − = + t peradiabaticity (δ)is1× 10 6 at the bottom and about 2 × 10 5 Un+ 1 Un R(Un+ 1 ), (22) 2 2 3 at the top boundary. It is almost the same superadiabaticity value Un+1 = Un +ΔtR(U + 1 ). (23) as at the base of the solar convection zone. Figure 1 shows the n 2 A30, page 3 of 7 A&A 539, A30 (2012)

δ Entropy ( r cv) (a) Maximum Mach number (b) Mach number 2.1686 = 0.50 2.0 1.000 20 0.1000 1.5 10 ξ ) =1 r 0.100 ξ =10 1.0 0.0100 z (H 0 ξ =20 ξ 0.5 =40 -10 0.010 0.0010 ξ =80

0.0114 0.0 0.0 8.7 0 2 4 6 8 t=20.00 0.001 0.0001 x (Hr) 0 100 200 300 400 0.0 0.5 1.0 1.5 2.0 δ time (Hr/vc) z (Hr) Entropy ( r cv) 2.1686 = 0.50 2.0 (c) Horizontal RMS velocity (d) Vertical RMS velocity (e) Relative variation 20 1.0 0.4 1.04 1.5 10 0.8 ) r 0.3 1.02 1.0 z (H ) 0 0.6 ) c c (v (v 0.2 1.00 z h v 0.5 v -10 0.4 0.98

0.0114 0.0 0.0 8.7 0.1 0 2 4 6 8 0.2 t=400.00 x (H ) 0.96 r 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 z (H ) z (H ) z (H ) Fig. 1. Time-development of entropy in a 2D calculation with parame- r r r = = ters of case 1. (Top panel) t 20. (Bottom panel) t 400. Fig. 2. Some quantities in 2D calculation. a) Maximum Mach number of each time step. b) Distribution of Mach number estimated with the rms velocity. c) Distribution of horizontal rms velocity vh. d) Distribution of time-development of entropy. In the beginning, non-linear time- vertical rms velocity. e) Ratio of the rms velocities for each ξ and ξ = 1. dependent convection occurred (top panel), which changed to a steady state at later times (bottom panel). For the steady state, we ξ compared the root mean squared (rms) velocity with diﬀerent ξ. Dependence of growth rate on The rms velocity was deﬁned as 2.0

) 1.8 1 Lx Ly r

= 2 /H vrms v dxdy. (29) c LxLy 0 0 1.6 Figure 2 shows our results where the value of ξ is from 1 to 80. ff = 1.4 The e ective Mach number is defined as MA vrms/cs.Ifwe have larger ξ value than 80, we cannot reach a stationary state, since there are some shocks generated by supersonic convection. growth rate (v 1.2 The discussion of unsteady convection is given in the next ses- sion of 3D calculations. Even though the Mach number reaches 1.0 0.6 using ξ = 80 (panel a), the horizontal and vertical rms veloc- 1 10 100 1000 ities are almost identical to those calculated with ξ = 1 (without ξ = RSST). The ratio of the rms velocities for each ξ and ξ 1are Fig. 3. Behavior in linear phase. Dependence of growth rate on ξ is ff −6 showninFig.2e. The di erence is always a few percent. This shown. Black and red lines show the results with δr = 1 × 10 and result is unsurprising since in the stationary state the equation of 1 × 10−4, respectively. continuity becomes 1 0 = ∇·(ρ0u), (30) ξ2 (downflow) generates a positive (negative) entropy perturbation whose solution must not depend on the value of ξ. We note that and then a negative (positive) density perturbation is generated the cell size is to some degree affected by the aspect ratio of the by the sound wave. If the speed of sound is fairly slow, the gen- ff eration mechanism of density perturbation is ineffective. In the domain. This does not a ect our conclusions since this influence −6 −4 calculation with δr = 1 × 10 (δr = 1 × 10 ), the growth rate is the same for all considered values of ξ. To confirm the ro- = = bustness of our conclusions, we repeated this experiment with a with ξ 200 (ξ 20), however, is almost same as that with ξ = ξ = ξ = wider domain (26.16H × 2.18H instead of 8.72H × 2.18H )in 1, even though flow with 200 ( 20) is expected to be r r r r Ma = . which we find four steady convection cells with about 10% dif- the Mach number 1 3, i.e. supersonic convection flow in ferent rms velocities. In addition we find a dependence on ξ that the saturated state. is similar to that shown in Fig. 2, i.e. the solutions show differ- ences of only a few percent as long as ξ<80. In this section, we 3.2. Three-dimensional study confirm that the RSST is valid for the 2D stationary convection when the effective Mach number is less than 1, i.e. ξ<80 with We now investigate the suitability of the RSST for 3D unsteady −6 −4 δr = 1 × 10 .Ifweuseδr = 1 × 10 (result is not shown), the thermal convections (case 6–13). The value of superadiabatic- criterion becomes ξ<8 for the 2D calculation. ity at the bottom boundary is 1 × 10−4 and at the top 2 × 10−3. As a next step, we investigate the dependence of the linear Although this value is relatively large compared to solar value, growth rate on ξ during the initial (time-dependent) relaxation the expected speed of convection is much smaller than the speed phase toward the final stationary state. Figure 3 shows the lin- of sound, hence is small enough to allow us to investigate the ear growth of maximum perturbation density ρ1 with different validity of the RSST. The entropy of 3D convections with ξ = 1, −6 ξ. Black and red lines show the results with δr = 1 × 10 and 20, and 80 at t = 100Hr/vc are shown in Fig. 4. The convection −4 δr = 1 × 10 , respectively. These calculation parameters are is completely unsteady and turbulent (animation is provided). not given in Table 1. In the calculation with δ = 1 × 10−6,the The appearance of convection with ξ = 20 is almost the same as growth rate decreases for values of ξ>300, whereas it occurs that with ξ = 1. We verify this using the Fourier transformation −4 at ξ>30 in the calculation with δr = 1 × 10 . The reason and auto-detection technique. However, the appearance of con- can be explained as follows. In the convective instability, upflow vection with ξ = 80 (bottom panel) differs completely from the

A30, page 4 of 7 H. Hotta et al.: Convection with RSST (RN)

(a) z=2.18H (b) z=1.09H (c) z=0 δ δ r r s1 (cv ) at z=2.18Hr s1 (cv ) at z=0 100 100 100 ξ =1 10 ξ =5 8 8 ξ =10 2 ξ =15 ) ) ) c 10-2 c 10-2 c 10-2 ξ =20 ξ =20/(δ/δ )1/2 5 r 6 6 1 10-4 10-4 10-4 ) ) r r amplitude (v amplitude (v amplitude (v 0 0 -6 -6 -6 y(H 4 y(H 4 10 10 10

-1 -8 -8 -8 -5 10 10 10 2 2 1 10 100 1 10 100 1 10 100 k = (k2 +k2 )1/2 k = (k2 +k2 )1/2 k = (k2 +k2 )1/2 -2 x y x y x y 0 -10 0 0 2 4 6 8 0 2 4 6 8 ﬀ ξ =1 x(Hr) x(Hr) Fig. 6. Comparison of horizontal velocity spectra with di erent ξ. a) = = = δ δ z 2.18Hr b) z 1.09Hr c) z 0. Velocities are averaged in time s1 (cv ) at z=2.18Hr s1 (cv ) at z=0 10 between t = 100 and 200H /v . 8 8 r c 2 5 6 6 1 ρ

) ) (a) at z=2.18H (b) Detection of convective cell

r r 1 r

0 0 #16

y(H 4 y(H 4 8 8 #13 #14 -1 2 -5 2 6 6 #11 -2 #15 0 -10 0

) ) #12 0 2 4 6 8 0 2 4 6 8 r r #09 #10 ξ x(Hr) x(Hr) =20 y (H 4 y (H 4 #07 #08 #04 δ δ #06 s1 (cv ) at z=2.18Hr s1 (cv ) at z=0 #05 10 8 8 2 2 #01 #03 2

5 #02 6 6 1 #17 0 0 ) ) r r 0 2 4 6 8 0 2 4 6 8 0 0 x (H ) x (H ) y(H 4 y(H 4 r r

-1 2 -5 2 Fig. 7. Detection of convective cell. a) Original contour of density per- -2 = 0 -10 0 turbation at z 2.18Hr. b) Distribution of detected convection cell. 0 2 4 6 8 0 2 4 6 8 Color and label (#n) show each convective cell. ξ =80 x(Hr) x(Hr) Fig. 4. Snapshots of entropy of the 3D convection. Top, middle, and bottom panels correspond to ξ = 1, 10, and 80, respectively. Left (right) panels show entropy at top (bottom) boundary. (Animation is provided, This supersonic downflow frequently generates shocks and pos- and the difference between ξ = 1 and 80 is most clearly visible in the itive entropy perturbation, thus downflow is decelerated. This animation of Fig. 4, which is provided with the online version.) is the reason why the rms velocities with large ξ(=40, 80) are small. When ξ = 5, 10, 15, and 20, however, the rms velocities = (a) Mach number (b) Vertical RMS velocity (c) Horizontal RMS velocity are in good agreement with that for ξ 1. We estimate the rms 0.6 0.6 power density of pressure, buoyancy, and inertia (Fig. 5). This 0.5 0.5 1.00 results display almost the same trend as for the rms velocities. 0.4 0.4 = 0.3 0.3 The power profiles ξ 5, 10, 15, 20 agree with each other and 0.10 0.2 0.2 those with ξ = 40, 80 disagree with those with ξ = 1. These re- 0.1 0.1 = 0.01 sults show that the RSST is a valid technique at least for ξ 20, 0.0 0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 at which the Mach number is around 0.7. We note that the con- Height (Hr) Height (Hr) Height (Hr) ⋅ ∇ ρ ρ ⋅ ⋅∇ vection pattern among our 2D and 3D cases varies substantially. (d) <-v ( p1+ 1g)> (e) <- 0v (v )v> 0.015 0.015 ξ =1 As a consequence, the ξ values for which the validity of RSST 0.010 0.010 ξ =5 ξ =10 breaks down are also different in the 2D and 3D setups. 0.005 0.005 ξ =15 ξ =20 ξ In Fig. 6, we compare the average spectral amplitudes for 0.000 0.000 =40 ξ =80 ξ δ δ 1/2 different values of ξ. There is no significant difference between =20/( / r) -0.005 -0.005 the spectral amplitudes for different values of ξ in the range -0.010 -0.010 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 from 1 to 20. Height (Hr) Height (Hr) We investigate the distribution of cell sizes at the top bound- = Fig. 5. Some quantities averaged in time between t 100 and 200Hr/vc ary. This value is strongly related to the turbulent diffusivity and in 3D calculations. a) Distribution of Mach number estimated with the transport of either angular momentum or energy. The method for rms velocity. b) Distribution of vertical rms velocity vh. c) Distribution detecting the cell is as follows. At the boundary of the cell i.e., of horizontal rms velocity. d) Distribution of rms power density of pres- the region of downflow, the perturbation of density is positive sure and buoyancy. e) Distribution of rms power density of inertia. and has a high value. When the density in a region exceeds a threshold, the region is regarded as the boundary of a convective cell. When a region is surrounded by one continuous boundary others. This difference is most clearly visible in the animation of region, the region is defined as one convective cell. Figure 7 Fig. 4 that is provided with the online version. shows the detected cells. Each color and each label (#n) corre- We estimate the rms velocities for different ξ by calculat- spond to each detected cell. We estimate the size of all cells and ing the average of values between t = 100 to 200Hr/vc (Fig. 5: compare the distribution of cell sizes for different ξ. The results panel b and c). Without the RSST, i.e. ξ = 1 and using δr = are shown in Fig. 8. The cell size distribution follows a power × −4 × −2 × −2 2 1 10 , the Mach number is 1 10 at the bottom and 4 10 at law from about 0.01 to 10 Hr and there is no dependence on ξ in the top boundary. The rms velocities at ξ = 40and80differ from the range from 1 to 20. Although when using this type of tech- those at the ξ = 1 by less than 15% and 30%, respectively. When nique the size of cells tends to be large when neglecting smaller we adopt ξ = 40 and 80, the Mach number estimated by rms ve- cells, our conclusion is not wrong, since all auto-detections are locity exceeds unity (Fig. 5: panel a), i.e. supersonic convection. equally affected.

A30, page 5 of 7 A&A 539, A30 (2012) Cell Size Distribution 0.06 106 ξ =1 ξ =5 ξ 5 =10 10 0.05 ξ =15 ξ

) =20

-2 r ξ =20/(δ/δ ) 104 r 0.04 103

102 0.03 ξ =1 1 ξ =5 10 ξ =10 Dist. Dens. (number H ξ =15 0.02 100 ξ =20 ξ δ δ 1/2 =20/( / r) -1 10 0.01 0.001 0.010 0.100 1.000 10.000 100.000 2 Cell Size (Hr ) 0.00 Fig. 8. Distribution of convective cell size with diﬀerent ξ is shown. The cell size distribution is averaged in time between t = 100 and 200Hr/vc. 0.5 1.0 1.5 2.0 Height (Hr) (a) Mach number (b) Vertical RMS velocity (c) Horizontal RMS velocity 0.6 0.6 Fig. 10. Dependence of [∇·(ρ0u)]rms on ξ. In case 13, the value of ξ is 1.00 0.5 0.5 20 at the bottom and 4.5 at the top boundary. 0.4 0.4

0.10 0.3 0.3 0.2 0.2 If the equation of continuity is satisﬁed, i.e. (∂ρ/∂t = −∇ · (ρu)), 0.1 0.1 0.01 the primitive form is obtained as 0.0 0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 u Height (Hr) Height (Hr) Height (Hr) ∂ ⋅ ∇ ρ ρ ⋅ ⋅∇ ρ = ρ(u ·∇)u + F. (33) (d) <-v ( p1+ 1g)> (e) <- 0v (v )v> 0.015 0.015 ∂t 0.010 0.010 However, using the RSST these two forms are no longer equiv- 0.005 0.005 ξ =1 alent. We used the primitive formulation at the expense that ξ =5 ξ 0.000 0.000 =10 energy and momentum are not strictly conserved; however, the ξ =15 ξ =20 ﬀ -0.005 -0.005 consistency of our results for di erent values of ξ strongly

-0.010 -0.010 indicates that this is not a serious problem for the setup we 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0

Height (Hr) Height (Hr) considered. Alternatively, we could also implement our modified equation of continuity in a conservative formulation. = × −3 Fig. 9. The results with δr 1 10 . The format is the same as Fig. 5. This would ensure that density, momentum, and energy are strictly conserved at the expense of a modified set of primi- For the above two investigations, i.e. the Fourier analysis and tive equations. Figure 10 shows the dependence of [∇·(ρ0u)]rms study of cell detection, we conclude that the statistical features = 2 = ff ff ( [ξ ∂ρ1/∂t]rms)onξ.Usingξ 5, this term is almost the same are una ected by the RSST method as long as the e ective Mach as that with ξ = 1. Although the deviation becomes large as ξ number (computed with the reduced speed of sound) does not 2 = increases, it is not proportional to ξ . exceed values of about 0.7, which corresponds to ξ 20 in our In the previous discussion, we kept ξ constant in the entire setup. computational domain. In the solar convection zone, the Mach To confirm our criterion that the RSST is valid if the Mach number varies however substantially with depth, from ∼1inthe number is smaller than 0.7, we conduct calculations with larger photosphere to <10−7 at the base of the convection zone, the = × −3 superadiabaticity, i.e. δr 1 10 (cases 14–18). If our crite- speed of sound itself varies from about 7 km s−1 in the photo- rion is valid, required ξ with larger superadiabaticity must de- −1 −3 sphere to 200 km s at the base of the convection zone. A re- crease. The results are shown in Fig. 9.Usingδr = 1 × 10 , = duction in the speed of sound is therefore most significant in the the calculations with ξ 10, 15, and 20 generate a supersonic deep convection zone, but not in the near surface layers. This convection flow near the surface (Fig. 9a). It is clear that the re- = ff = could be achieved with a depth dependent ξ.Evenifweusea sults for ξ 10, 15, and 20 di er from those with ξ 1 and 5. conservative form of the equation of continuity, The calculation with ξ = 5 corresponds to a flow whose Mach number is around 0.6. Thus, our criterion is not violated even for ∂ρ 1 1 = −∇ · ρ u , (34) different values of the superadiabaticity. ∂t ξ2 0 Owing to our altered equation of continuity the primitive and conservative formulation of Eqs. (3)to(5) are no longer equiv- the result must not be same as the result without any approxi- alent. For example, the equation of motion in conservative form mation of the statistical steady state. When the value averaged is expressed as during a statistical steady state is expressed as a ,wherea is a physical value, the equation of continuity becomes ∂ u = −∇ · uu + (ρ ) (ρ ) F, (31) 1 ∂t = ∇· ρ u , 0 2 0 (35) where F denotes the pressure gradient, gravity, and Lorentz ξ force. With some transformations, we can obtain with inhomogeneous ξ. The solution of Eq. (35)isdifferent from ∂ρ ∂u the solution of the original equation of continuity in the sta- u + ρ = −u∇·(ρu) − ρ(u ·∇)u + F. (32) = ∇· u ∂t ∂t tistical steady state, i.e. 0 (ρ0 ). Thus, the statistical A30, page 6 of 7 H. Hotta et al.: Convection with RSST (RN) features such as rms velocity are not reproduced with an inho- 3. The base of the convection zone and the surface of the sun mogeneous ξ. If we use the non-conservative form of the equa- where the anelastic approximation is broken can be con- tion of motion given in Eq. (2), this problem does not occur. nected, using a space-dependent ξ. Thus, we investigate the nonuniformity of ξ in case 13 using the non-conservative form, i.e., Eq. (3). To keep the Mach number Overall we find that RSST is a very useful technique for studying 1/2 low Mach number flows in stellar convection zones as it sub- uniform for all heights, we use ξ = 20/(δ(z)/δr) ,i.e.ξ = 20 at the bottom and ξ = 4.5 at the top boundary. We note that the stantially alleviates any stringent time-step constraints without adding any computational overhead. ratio of the speed√ of convection to the speed of√ sound is roughly estimated as δ. In this setting, the value of δ is 1 × 10−2 at the bottom and 4 × 10−2 at the top boundary. The same analysis Acknowledgements. The authors want to thank Dr. M. Miesch for his help- as that for uniform ξ is done (see Figs. 5, 6,and10). There is ful comments. Numerical computations were in part carried out on Cray XT4 ff = 1/2 = at Center for Computational Astrophysics, CfCA, of National Astronomical no significant di erence between ξ 20/(δ(z)/δr) and ξ 1. Observatory of Japan. We would like to acknowledge high-performance com- Although mass is not conserved locally for a non-homogeneous puting support provided by NCAR’s Computational and Information Systems ξ using a non-conservativeform of the equation of continuity, the Laboratory, sponsored by the National Science Foundation. The National Center horizontally averaged vertical mass flux is approximately zero for Astmospheric Reseach (NCAR) is sponsored by the National Science Foundation. This work was supported by the JSPS Institutional Program for in the statistically steady convection and the conservation total Young Researcher Overseas Visits and the Research Fellowship from the JSPS mass is not broken significantly. We conclude that an inhomoge- for Young Scientists. We have greatly benefited from the proofreading/editing as- neous ξ is valid under the previously obtained condition, i.e. the sistance from the GCOE program. We thank the referee for helpful suggestions Mach number is less than 0.7. that helped to improve our paper.

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