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

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

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

Astrophysical Turbulence 21 September 2004 Astrophysical Turbulence

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

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

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

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

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

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

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

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

Astrophysical Turbulence But the hope that „homogeneous turbulence“ would be a sensible model was dashed by Landau & Lifschitz 1953-1959, 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

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

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

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

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

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

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

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

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

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

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

Astrophysical Turbulence 21 September 2004 Astrophysical Turbulence

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

Astrophysical Turbulence 21 September 2004 Astrophysical Turbulence

Astrophysical Turbulence 21 September 2004 Astrophysical Turbulence

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