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ANGULAR MOMENTUM TRANSPORT BY MAGNETOHYDRODYNAMIC TURBULENCE Gordon Ogilvie University of Cambridge TACHOCLINE DYNAMICS 11.11.04
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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
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COMPARISON TACHOCLINE ► thin ACCRETION DISC ► thin ► differentially rotating
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COMPARISON TACHOCLINE ► thin ACCRETION DISC ► thin ► differentially rotating ► magnetized (probably) ► turbulent (probably) ► large-scale dynamo? ► differentially rotating ► magnetized (probably) ► turbulent (probably) ► large-scale dynamo?
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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?
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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
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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
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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
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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
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MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT (A. Brandenburg)
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MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT
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MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT
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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
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TURBULENCE MODELS EDDY-VISCOSITY MODEL (von Weizsäcker 1948) VISCOELASTIC MODEL (O 2001; O & Proctor 2003) REYNOLDS-MAXWELL STRESS MODELS (Kato; O 2003)
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SOME CONTROVERSIES ► nonlinear hydrodynamic shear instability ► ‘viscosity’ ► ‘alpha viscosity’ ► AM transport by convection ► baroclinic / Rossby-wave instability
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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
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CONTINUOUS SPECTRUM LINEAR THEORY IN IDEAL MHD ► Lagrangian displacement ξ ► arbitrary reference state
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CONTINUOUS SPECTRUM BASIC STATE ► steady and axisymmetric ► cylindrical polar coordinates (s,φ,z) ► differential rotation ► toroidal magnetic field SOLUTIONS
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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’
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CONTINUOUS SPECTRUM REQUIRED ORDERING
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CONTINUOUS SPECTRUM LOCAL DISPERSION RELATION
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CONTINUOUS SPECTRUM CASE OF ZERO MAGNETIC FIELD ► Høiland (1941) stability criteria ► necessary and sufficient for axisymmetric disturbances
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CONTINUOUS SPECTRUM LIMIT OF WEAK MAGNETIC FIELD ► Papaloizou & Szuszkiewicz (1992) stability criteria ► necessary but not sufficient for stability
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CONTINUOUS SPECTRUM CASE OF ZERO ANGULAR VELOCITY ► necessary and sufficient ► Tayler (1973) stability criteria
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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
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APPLICATION TO THE TACHOCLINE ► appropriate ordering schemes are unclear (to me) ► assume overwhelming stable stratification
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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)
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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
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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)
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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
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