Download Turbulent convection in stellar interiors

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