1. Turbulence in Astrophysics (Theory)
Wolfram Schmidt
Institut für theoretische Physik und Astrophysik
Universität Würzburg

2. Astrophysical Turbulence
Stirring of Fluid
Mechanical force stirring fluid into rotational motion
Turn-over time T, wavelength L
What happens in the limit t → ∞?
It depends on the Reynolds number!
21 September 2004
Astrophysical Turbulence

3. Laminar vs. Turbulent Flow
Reynolds, 1883
If Re is relatively small, only eddies of size L are produced
For Re ~ 1000, the motion of adjacent fluid layers becomes unstable
21 September 2004
Astrophysical Turbulence

4. Astrophysical Turbulence
21 September 2004
Astrophysical Turbulence

5. Astrophysical Turbulence
Fluid motion forces vortices to stretch, and a stretching vortex must fold to accomodate an increasing length in a fixed volume. To the extent that the flow is scaling, I conjecture the vortex tends toward a fractal.
Mandelbrot,
The Fractal Geometry of Nature
21 September 2004
Astrophysical Turbulence

6. Astrophysical Turbulence
Vortices
Turbulent fluid motion is inherently rotational
21 September 2004
Astrophysical Turbulence

7. Astrophysical Turbulence
Strain and Vorticity
Symmetric derivative
Antisymmetric derivative
Rate of strain
Vorticity
Dilatation
21 September 2004
Astrophysical Turbulence

8. Astrophysical Turbulence
Vortex Formation
Porter et al.
ASCI, 1997
Vortices are streched and folded in three dimensions
21 September 2004
Astrophysical Turbulence

9. The Turbulence Cascade
Richardson, 1922; Onsager, 1945
Breaking up of laminar flow structure due to large |S| produces high vorticity ω
Force of wavelength L produces structure on scales much smaller than L for high Re
Small vortices are quasi random
Turbulence is a non-linear multi-scale phenomenon
21 September 2004
Astrophysical Turbulence

10. Astrophysical Turbulence
Isotropic Turbulence
Taylor, 1935
Statistically, there is no prefered direction (random orientation of vortices)
In nature, turbulence is never exactly isotropic on large scales (forcing, BCs)
However, turbulent flows tend to become asymptotically isotropic towards small scales (randomisation by non-linear energy transfer)
21 September 2004
Astrophysical Turbulence

11. Astrophysical Turbulence
The Kolmogorov Theory
Hypothesis of local isotropy: At sufficiently high Re, the dynamics on small scales tends to become statistically isotropic
First similarity hypothesis: The statistics of isotropic velocity fluctuations on sufficiently small scales are universal und uniquely determined by the viscosity and the rate of kintetic energy dissipation
Second similarity hypothesis: There is a subrange of scales for which the statistics of turbulent fluid motions are independent of the mechanism and the length scale of dissipation
21 September 2004
Astrophysical Turbulence

12. The 5/3 Power Law (K41) Rate of dissipation ε Wave number k = 2π/l
log E
k -5/3
transfer
Rate of dissipation ε
Wave number k = 2π/l
log k
L-1
ηK-1
Length scale of viscous dissipation
21 September 2004
Astrophysical Turbulence

13. Astrophysical Turbulence
But the hope that „homogeneous turbulence“ would be a sensible model was dashed by Landau & Lifschitz , which notes that some regions are marked by very high dissipation, while other regions seem by contrast nearly free of dissipation.
Mandelbrot,
The Fractal Geometry of Nature
21 September 2004
Astrophysical Turbulence

14. Astrophysical Turbulence
Realistic Turbulence
Convective boundary layers: Anisotropy in stratified medium (convection zones, atmospheres)
Turbulent combustion: Anisotropy across flame surface, transient flow (thermonuclear supernovae)
Gravoturbulence: Inhomogeneous and supersonic turbulence in self-gravitating fluids (star formation)
MHD turbulence: Instability of fluid motion due to interaction with magnetic field, multi-scale anisotropy (ionized gas in ISM, jets, accretion disks)
21 September 2004
Astrophysical Turbulence

15. The Navier-Stokes Equation
Conservation of momentum
Mechanical, magnetic, gravitational forces
Lagrangian
time derivative
Viscous dissipation tensor
Non-linear advection
21 September 2004
Astrophysical Turbulence

16. Astrophysical Turbulence
Further Equations
Conservation of energy
Mass conservation
Poisson equation
Maxwell equations
in the case of MHD
21 September 2004
Astrophysical Turbulence

17. Astrophysical Turbulence
Statistical Theories
Mixing length theory: one characteristic length scale lM =αHP (Kolmogorv spectrum → δ-peak)
ODT models: one-dimensional stochastic process for eddy size (reproduces Kolmogorv spectrum)
PDF models: determine probability distributions for velocity fluctuations etc.
Reynolds stress models: dynamical equations for moments of fluctuating fields
21 September 2004
Astrophysical Turbulence

18. Astrophysical Turbulence
Stellar Convection
Full Reynolds stress model for compressible turbulence (Canuto, 1997): multitude of coupled, non-linear PDEs → hopeless
Feasible model: reduced set of eqns. for mean K , Fc , ε and average squared fluctuations of temperature and horizontal velocity (Kupka, 1999)
Closure relations for higher order moments
Non-local & anisotropic
21 September 2004
Astrophysical Turbulence

19. Stellar Convection: Convective Flux
MPA, 2004
Kupka
21 September 2004
Astrophysical Turbulence

20. Stellar Convection: Vertical RMS Velocity
MPA, 2004
Kupka
21 September 2004
Astrophysical Turbulence

21. Numerical Simulations
Direct numerical simulation (DNS): Static grid, NSE or numerical viscosity
Large Eddy Simulation (LES): Subgrid scale model for unresolved turbulence
Smooth particle hydrodynamics (SPH): Particle ensemble represents the flow
Adaptive mesh refinement (AMR): Hierarchy of dynamically generated grid patches
21 September 2004
Astrophysical Turbulence

22. Thermonuclear Supernovae
Runaway turbulent deflagration of C+O in a Chandrasekhar-mass white dwarf
PPM for hydrodynamics (Fryxell et al., 1989)
Subgrid scale model for turbulent flame speed (Niemeyer & Hillebrandt, 1995)
Level set method for flame surface tracking (Reinecke et al., 1999 )
Homologous grid expansion to follow the explosion (Röpke, 2004)
21 September 2004
Astrophysical Turbulence

23. History of a SN Ia Explosion
t = 0 s
t = 0.3 s
t = 0.6 s
Röpke et al.
MPA, 2004
t = 2 s
21 September 2004
Astrophysical Turbulence

24. Astrophysical Turbulence
21 September 2004
Astrophysical Turbulence

25. Astrophysical Turbulence
Turbulence in the ISM
Supersonic turbulence in self-gravitating gas
Thermal processes, magnetic fields
Paradigm of turbulent star formation: Turbulence can induce local gravitational collapse, albeit it provides global support
SPH treatment: e.g. Klessen, 2001
AMR treatment with PPM/ZEUS: e.g. Kritsuk & Norman, 2002; Abel et al., 2002
21 September 2004
Astrophysical Turbulence

26. Astrophysical Turbulence
21 September 2004
Astrophysical Turbulence

27. Astrophysical Turbulence
21 September 2004
Astrophysical Turbulence

28. Astrophysical Turbulence
Résumé
Significant developments in the treatment of turbulent convection via statistical models
Three-dimensional simulations with sophisticated codes running on extremely powerful computers offer exciting insights
However, most simulations are ignorant of small-scale turbulence (SGS models!)
AMR is excellent for inhomogeneous and transient and astrophysical flows, but is it appropriate for turbulence?
21 September 2004
Astrophysical Turbulence