1 / 30

ANGULAR MOMENTUM TRANSPORT

ANGULAR MOMENTUM TRANSPORT. BY MAGNETOHYDRODYNAMIC TURBULENCE. Gordon Ogilvie. University of Cambridge. TACHOCLINE DYNAMICS. 11.11.04. INTRODUCTION. SOME TACHOCLINE ISSUES (Tobias 2004). ► sources of instability : HD and MHD. ► nonlinear development.

rkellar
Download Presentation

ANGULAR MOMENTUM TRANSPORT

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ANGULAR MOMENTUM TRANSPORT BY MAGNETOHYDRODYNAMIC TURBULENCE Gordon Ogilvie University of Cambridge TACHOCLINE DYNAMICS 11.11.04

  2. 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

  3. COMPARISON TACHOCLINE ACCRETION DISC ► thin ► thin ► differentially rotating ► differentially rotating

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

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

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

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

  8. SHEARING SHEET ► local model of a differentially rotating disc ► uniform rotation Ωez plus uniform shear flow –2Axey ► appropriate for studies of thin discs

  9. MAGNETOROTATIONAL INSTABILITY OPTIMAL MODE (‘channel flow’) ► layer analysis (incompressible ideal fluid, ρ= μ0 = 1) u b ► exact nonlinear solution but unstable (Goodman & Xu 1994)

  10. MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT (A. Brandenburg)

  11. MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT

  12. MAGNETOROTATIONAL INSTABILITY NONLINEAR DEVELOPMENT

  13. 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

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

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

  16. 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 (ky≠0) in the shearing sheet ●do not describe parallel shear flow instability ► continuous spectrum and non-modal localized approaches ●derive sufficient conditions for instability ●contain many of the most important instabilities

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

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

  19. CONTINUOUS SPECTRUM ASYMPTOTIC LOCALIZED SOLUTIONS ► envelope localized near a point (s0,z0) ► plane-wave form with many wavefronts ► finite frequency and vanishing group velocity ► ‘frozen wavepacket’

  20. CONTINUOUS SPECTRUM REQUIRED ORDERING

  21. CONTINUOUS SPECTRUM LOCAL DISPERSION RELATION

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

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

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

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

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

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

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

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

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

More Related