ANGULAR MOMENTUM TRANSPORT BY MAGNETOHYDRODYNAMIC TURBULENCE Gordon Ogilvie University of Cambridge TACHOCLINE DYNAMICS

INTRODUCTION SOME TACHOCLINE ISSUES (Tobias 2004) ► sources of instability : HD and MHD ► nonlinear development ► turbulence and turbulent transport : HD and MHD SOME ACCRETION DISC ISSUES ► differential rotation and AM transport ► HD and MHD instabilities ► turbulence and turbulent transport : HD and MHD

COMPARISON TACHOCLINE ► thin ACCRETION DISC ► thin ► differentially rotating

COMPARISON TACHOCLINE ► thin ACCRETION DISC ► thin ► differentially rotating ► magnetized (probably) ► turbulent (probably) ► large-scale dynamo? ► differentially rotating ► magnetized (probably) ► turbulent (probably) ► large-scale dynamo?

COMPARISON TACHOCLINE ► thin ACCRETION DISC ► thin ► differentially rotating ► magnetized (probably) ► turbulent (probably) ► large-scale dynamo? ► highly subsonic ► differentially rotating ► magnetized (probably) ► turbulent (probably) ► large-scale dynamo? ► highly supersonic ► strong stable stratification? ► weak or no stratification?

COMPARISON TACHOCLINE ► thin ACCRETION DISC ► thin ► differentially rotating ► magnetized (probably) ► turbulent (probably) ► large-scale dynamo? ► highly subsonic ► differentially rotating ► magnetized (probably) ► turbulent (probably) ► large-scale dynamo? ► highly supersonic ► strong stable stratification? ► difficult to resolve ► weak or no stratification? ► difficult to resolve ► difficult to simulate

ANGULAR MOMENTUM TRANSPORT GENERAL ► spiral arms / shocks ► vortices SMALL-SCALE FEATURES ► waves ► turbulence LARGE-SCALE STRUCTURES ► anisotropic magnetic fields (Maxwell stress) ► anisotropic motion (Reynolds stress) ► non-axisymmetric gravitational fields

SHEARING SHEET ► local model of a differentially rotating disc ► uniform rotation Ω e z plus uniform shear flow –2Ax e y ► appropriate for studies of thin discs

MAGNETOROTATIONAL INSTABILITY OPTIMAL MODE (‘channel flow’) ► layer analysis (incompressible ideal fluid, ρ = μ 0 = 1 ) ► exact nonlinear solution but unstable (Goodman & Xu 1994) u b

MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT (A. Brandenburg)

MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT

MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT

ENERGY AND ANGULAR MOMENTUM ENERGY EQUATION (shearing sheet) ► in either growing instability or saturated turbulence, ► AM transport down the gradient of angular velocity ► very natural outcome of MHD instabilities ► contrast (e.g.) convective instability or forced turbulence

TURBULENCE MODELS EDDY-VISCOSITY MODEL (von Weizsäcker 1948) VISCOELASTIC MODEL (O 2001; O & Proctor 2003) REYNOLDS-MAXWELL STRESS MODELS (Kato; O 2003)

SOME CONTROVERSIES ► nonlinear hydrodynamic shear instability ► ‘viscosity’ ► ‘alpha viscosity’ ► AM transport by convection ► baroclinic / Rossby-wave instability

CONTINUOUS SPECTRUM INTRODUCTION ► cf. Friedlander & Vishik (1995); Terquem & Papaloizou (1996) ► problems with a normal-mode approach in shearing media ● modes may require confining boundaries ● entirely absent ( k y ≠ 0 ) in the shearing sheet ● do not describe parallel shear flow instability ► continuous spectrum and non-modal localized approaches ● contain many of the most important instabilities ● derive sufficient conditions for instability

CONTINUOUS SPECTRUM LINEAR THEORY IN IDEAL MHD ► Lagrangian displacement ξ ► arbitrary reference state

CONTINUOUS SPECTRUM BASIC STATE ► steady and axisymmetric ► cylindrical polar coordinates (s,φ,z) ► differential rotation ► toroidal magnetic field SOLUTIONS

CONTINUOUS SPECTRUM ASYMPTOTIC LOCALIZED SOLUTIONS ► envelope localized near a point (s 0,z 0 ) ► plane-wave form with many wavefronts ► finite frequency and vanishing group velocity ► ‘frozen wavepacket’

CONTINUOUS SPECTRUM REQUIRED ORDERING

CONTINUOUS SPECTRUM LOCAL DISPERSION RELATION

CONTINUOUS SPECTRUM CASE OF ZERO MAGNETIC FIELD ► Høiland (1941) stability criteria ► necessary and sufficient for axisymmetric disturbances

CONTINUOUS SPECTRUM LIMIT OF WEAK MAGNETIC FIELD ► Papaloizou & Szuszkiewicz (1992) stability criteria ► necessary but not sufficient for stability

CONTINUOUS SPECTRUM CASE OF ZERO ANGULAR VELOCITY ► necessary and sufficient ► Tayler (1973) stability criteria

APPLICATION TO ACCRETION DISCS ► allows an understanding of the nonlinear state? ► appropriate ordering scheme for a thin disc reveals ● MRI (unavoidable) ● magnetic buoyancy instability (possible) differential rotation MRI

APPLICATION TO THE TACHOCLINE ► appropriate ordering schemes are unclear (to me) ► assume overwhelming stable stratification

APPLICATION TO THE TACHOCLINE ► conclusions change under weaker stratification ► appropriate ordering schemes are unclear (to me) ► assume overwhelming stable stratification ● weak B : MRI when ● Ω = 0 : Tayler ( m = 1 ) when ● suppressed at the poles if ● cf. Cally (2003) (but not requiring mode confinement) ● sensitivity to radial gradients; magnetic buoyancy (NB: no MRI in 2D)

REMARKS PROPER JUSTIFICATION ► prove existence of continuous spectrum ► asymptotic treatment of non-modal disturbances ► justifies ‘local analysis’ for a restricted class of disturbances ADVANTAGES ► algebraic character of eigenvalues and eigenvectors ► strictly local character, independent of BCs ► deals easily with complicated 2D basic states

REMARKS ► neglects the role of turbulent stresses in the basic state ► misses truly global instabilities NOTES OF CAUTION ► neglects diffusion (double / triple) in the perturbations ● Acheson (1978); Spruit (1999); Menou et al. (2004)

SUMMARY ► methods for analysing linear instabilities ► angular momentum transport and energy arguments ► MRI optimized for AM transport down the gradient of ► differences between HD and MHD systems ► analogies are imperfect but of some value ► methods for understanding and modelling turbulent states angular velocity but of limited applicability in the Sun ► continuous spectrum contains many of the important ones