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Turbulent convection in stellar interiors A numerical approach using time-implicit simulations Maxime Viallet University of Exeter (UK) Collaborators: Casey Meakin (Los Alamos, US), David Arnett (Tucson, US) Isabelle Baraffe (Exeter), Rolf Walder (ENS Lyon) IPAG - Grenoble April 26th 2012 Outline • Introduction • The MUSIC code • 2D/3D stellar convection • Conclusion Outline • Introduction • The MUSIC code • 2D/3D stellar convection • Conclusion Stellar convection Why are star convective ? Sun Structure Radiative zone • Star: self-gravitating gas (~75 % H + ~25 % He + metals) in hydrostatic and thermal equilibrium • Huge release of energy by nuclear burning reactions in the core of the star • Energy transport processes: • Photons diffusion (radiative zone) 3 4acT � kr = F�r = −kr ∇T 3κρ • Convection Core 4 H → He kr ∝ T 5−7 Convective zone Stellar convection Where are convective zones ? For main sequence stars Convection: Gas becomes opaque to radiation Radiative from “Stellar Structure and Evolution” R. Kippenhahn, A. Weigert Convection: Energy release is very important (CNO cycle) Convection in stars is ubiquitous ! Stellar convection Properties of convection in stellar envelopes (Sun) • Source for microscopic viscosity: • Due to momentum transport by photons (collisions between ions negligible) ! ν~ 1 - 104 cm2/s => Re > 1010 => Highly turbulent flow VL Pe = Pr ≾ ≫1 • Ra ~ χr • Large stratification: density decreases by orders of magnitude throughout the convective zone 1015, 10-6, • Low-Mach flow (10-4-10-3) in the interior, but transonic (~1) transition toward the surface • Interplay with rotation and magnetic field • Flow regime that is not possible to model numerically ! Computing stellar structures • Stellar evolution models: • 1D approach: important physical processes have to be parametrised ! ex: convection (MLT, Böhm-Vitense 1958), rotation, magnetic field, ... Many successes of 1D models (calibration of free parameters from observations) • Improvement of observational technics • Helioseismology/asteroseismology - interferometry • Accurate observational data have to be met by sophisticated theoretical models ! • Stellar evolution calculations remain central, but a better description of the physics is required ! • Ex: convection beyond mixing-length parametrisation: • Convective boundary mixing, waves generation, ... " Multi-D computations are the way to make progress ! Multi-D # 1D strategy dr 1 = dm 4πr2 ρ dp Gm =− dm 4πr4 dL = �nuc dm ∇rad d ln T = or d ln P ∇mlt { Multi-D computation of stellar structure Difficulties • We have: ∆x � ∆tacc (N.B. in stars sound speed increases with depth!) • “Stiff” hydro: min cs kr 4acT 3 • Stiff diffusion: χr = ρc = 3κc ρ2 can become very large near the surface p p ! Severely restrict time-step when using explicit methods (CFL condition) un+1 = un + ∆tf ∆t (un ) ! Motivation for a fully implicit method du(t) • = f (u(t)) dt un+1 = un + ∆tf (un+1 ) unconditionally stable (A-stability) • Time step choice is driven by physical and accuracy considerations ! • Advantage: no approximation ! MUSIC: MUltidimensional Stellar Implicit Code Outline • Motivation • The MUSIC code • 2D/3D stellar convection • Conclusion MUSIC M. Viallet, I. Baraffe, R. Walder, A&A, 2011 • Gas dynamics equations • No explicit viscosity: “Implicit Large Eddy Simulation” paradigm (see e.g. Boris 2007) � v) ∂t ρ = −∇.(ρ� � v − ∇P � + ρ�g ρ∂t�v = −ρ�v .∇� � � v − ∇. � F�rad ∂t (ρe) = −∇.(ρe� v ) − P ∇.� • Radiation: diffusion approximation • Realistic microphysics (EOS, opacities) • Spatial method: finite volume on 2D/3D spherical wedges � F�rad = −kr ∇T P = P (ρ, e) T = T (ρ, e) dU • Semi-discretised system: = R(U ) dt dU = R(U ) dt U n+1 − U n − ∆t < R(U ) >= 0 where < R(U ) >= βR(U n+1 ) + (1 − β)R(U n ) with β = 0.5 ! Non-linear system: F (U n+1 ) = 0 MUSIC Implicit time-stepping strategy • Fully-implicit scheme (used in 2D only) • F(Un+1)=0 is solved by Newton-Raphson iterations • F � (U (k) )δU (k) = −F (U (k) ) is solved by preconditioned GMRES iterations • Jacobian matrix computed explicitly, needed for algebraic preconditioner ILU(2) • Pro: unconditionally stable, contra: expensive ! Forming Jacobians burden the computation • Minimum Residual Approximate Implicit scheme (c.f. Botchev et al. 1997) • Perform an explicit step with a large (unstable!) time step (15-20x Δtstab) • Stabilise the scheme by performing a few GMRES iterations on the implicit residual • Pro: Jacobian-free GMRES (i.e. tractable in 3D!), contra: conditionally stable • Parallel computing: both methods were used successfully on up to 128 cores Outline • Motivation • The MUSIC code • 2D/3D stellar convection • Conclusion Application Stellar convection • Cold giant star Numerical domain • M=5 Msun, Teff=4500 K , R=60 Rsun, L = 103 Lsun Last 50% in radius are convective -> deep envelope • Good lab for studying stratification effects and the physics of the convective/radiative boundary • Numerical models: 2D and 3D spherical wedge • 80% of the star in radius • Stellar surface has complex physics that we cannot resolve ! External isothermal zone drives convection Sun Fully implicit Δtimpl ~ 1000 Δtexpl (x5) CFLhydro ~ 100 2D (r,θ) 216x256 Time: 5.5 turnover times (~ 800 days) Stratification: Domain: 12 Hp CZ : 6 Hp Mach in CZ: 10-2 - 10-1 days • 2D turbulent flow: large-scale, long-lived vortices interacting • Strong down-flows, broader up-flows • Radiative zone filled with g-modes excited at the boundary layer c.f. Viallet et al. 2011 3D (r,θ,φ) 216x1282 Same effective resolution as 2D Approximate implicit Δtimpl ~ 15 Δtexpl (x2) CFLhydro ~ 1.5 Time: ~ 1 turnover time (~ 285 days) Colours: radial velocity Stratification: Domain: 12 Hp CZ : 6 Hp Mach in CZ: 10-2 - ≲10-1 7000 h CPU time on 96 cores (3 days wall-time) • 3D turbulent flow: smaller scales, short-lived structures • Again: fast downward plumes, slower/broader up-flows + g-modes in RZ 150 x more expensive than 2D ! Stellar convection Thermal balance τKH Eint = ∼ 2000 years L L/L� (%) • Thermal relaxation not reached Radiative • Overshooting/penetration region 2D Enthalpy Kinetic energy • Stronger in 2D • Bump in radiative luminosity (see e.g Dieng and Xiong, MNRAS, 2008) • Astrophysical relevance: convective boundary mixing ! L/L� ∇ ∇rad ∇ad 3D Stellar convection Convective velocities • Convective velocities • 2D: Ekin ~ 2.6x1043 erg/s • 3D: Ekin ~ 4x1042 erg/s • Stem from different dissipation properties: • 3D: vortices are rapidly destroyed by instabilities • 2D: vortices are more stable => flow is less prone to dissipation • 2D: stronger penetration, more kinetic energy is transferred to g-modes MLT: solid-wall boundary Stellar convection Kinetic energy equation • Euler equation: � u) = −∇P � + ρ�g ρ(∂t �u + �u.∇� � 0 + ρ0�g �0 = −∇P � u) = −∇p � � + ρ��g ρ(∂t �u + �u.∇� • Kinetic energy equation: � �� � � ∂t ρEk + ∇.(ρE � u + p � u ) = p ∇.� u + ρ �u.�g − �k k Wp = pressure dilatation Wb=buoyancy work sink: numerical dissipation INFERRED • Steady-state: !Wp + !Wb = !ɛk " driving is balanced by damping • MLT: convective velocity is obtained by integrating Wb over one mixinglength, all other terms are neglected Stellar convection Local kinetic energy balance • Horizontal integration + time-average (several turnover times) � F� = Wp + Wb − �k ∂t ρEk + ∇. 2D • Pressure-dilatation (Wp) • Overestimated in 2D, still not negligible in 3D • Transport term (-∇.F): • Important: moves KE from the top to the bottom • Total dissipation: • !ɛk ~ 0.5 L in 2D and ~ L in 3D => large dissipation ! • τdiss = Ekin/!ɛk ~ 50 days in 3D, ~ 0.25 turnover time ! • N.B. 2D and 3D have the same effective resolution => enhanced dissipation has a physical origin 3D Stellar convection Is the numerical dissipation physical ? • 3D: turbulent cascade of KE toward viscous scale (cf. Kolmogorov 1941, 1962) Wp + Wb • Dissipation rate is set by forcing, not viscosity ! • Numerical simulation: cascade stops at the grid scale • Kolmogorov estimate for isotropic turbulence • lD ~ 5.9x1011 cm, Hp ~ 1-4 x 1011 cm • Consistent with previous works: lD ~ min ( 4 Hp, lcz ) (see e.g. Meakin & Arnett 2009) • Justification of the “Implicit Large Eddy Simulation” paradigm • Numerical dissipation does a physical job in 3D ! ɛk 3D Outline • Motivation • The MUSIC code • 2D/3D stellar convection • Conclusion Conclusion and perspectives • MUSIC: new hydro code devoted to the modelling of stellar interiors • Based on implicit time stepping to relax numerical stability limits inherent to stellar hydrodynamical problems • Results in modest, but already useful, speed-up N.B.: the code beats its own explicit version Future improvement: Jacobian-free version of the fully implicit method ! • 2D/3D simulations of a stratified convective envelope: • Dynamics results from the balance between driving and small-scale dissipation Pressure dilatation and transport are essential in getting overshooting right ! Differences between 2D and 3D due to different dissipation properties ! • Turbulent convective zones are highly dissipative system Can we neglect turbulent heating any longer ? Conclusion and perspectives • Successor of MLT ? Work in progress since 1961 ! (3 years after MLT) e.g. Kuhfuss 1986, Canuto 1993, Deng 2006, among many others !!! • Best theoretical framework: Reynolds Average Navier-Stokes equations • But: turbulence closure models => more free parameters ? • Use Multi-D to understand: • Mixing processes, important for stellar evolution ! • Convective boundary mixing: ubiquitous ! e.g. lithium depletion in low-mass stars, hydrogen ingestion during He-flash, convectivecore overshooting in massive stars ... • Mixing in stably stratified layers, induced by waves and/or rotation (work in progress with C. Georgy & R. Walder in Lyon) e.g. surface enrichment of massive stars, light-element depletion in low-mass stars • Time-dependent background: stellar pulsations e.g. coupling the ϰ-mechanism and turbulent convection in Cepheids END