ANGULAR MOMENTUM TRANSPORTBY MAGNETOHYDRODYNAMIC TURBULENCE

Gordon OgilvieUniversity of Cambridge

TACHOCLINE DYNAMICS11.11.04

INTRODUCTIONSOME 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

COMPARISONTACHOCLINE► thin

ACCRETION DISC► thin

► differentially rotating ► differentially rotating

COMPARISONTACHOCLINE► thin

ACCRETION DISC► thin

► differentially rotating► magnetized (probably)► turbulent (probably)► large-scale dynamo?

► differentially rotating► magnetized (probably)► turbulent (probably)► large-scale dynamo?

COMPARISONTACHOCLINE► 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?

COMPARISONTACHOCLINE► 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► difficult to simulate

ANGULAR MOMENTUM TRANSPORTGENERAL

► 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 ez plus uniform shear flow –2Ax ey

► appropriate for studies of thin discs

MAGNETOROTATIONAL INSTABILITYOPTIMAL MODE (‘channel flow’)► layer analysis (incompressible ideal fluid, ρ = μ0 = 1)

► exact nonlinear solution but unstable (Goodman & Xu 1994)

u

b

MAGNETOROTATIONAL INSTABILITYNONLINEAR DEVELOPMENT (A. Brandenburg)

MAGNETOROTATIONAL INSTABILITYNONLINEAR DEVELOPMENT

MAGNETOROTATIONAL INSTABILITYNONLINEAR DEVELOPMENT

ENERGY AND ANGULAR MOMENTUMENERGY 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 MODELSEDDY-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 SPECTRUMINTRODUCTION► 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

● contain many of the most important instabilities● derive sufficient conditions for instability

CONTINUOUS SPECTRUMLINEAR THEORY IN IDEAL MHD

► Lagrangian displacement ξ► arbitrary reference state

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

SOLUTIONS

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

CONTINUOUS SPECTRUMREQUIRED ORDERING

CONTINUOUS SPECTRUMLOCAL DISPERSION RELATION

CONTINUOUS SPECTRUMCASE OF ZERO MAGNETIC FIELD

► Høiland (1941) stability criteria► necessary and sufficient for axisymmetric disturbances

CONTINUOUS SPECTRUMLIMIT OF WEAK MAGNETIC FIELD

► Papaloizou & Szuszkiewicz (1992) stability criteria► necessary but not sufficient for stability

CONTINUOUS SPECTRUMCASE 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 instabilitiesNOTES 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