ANGULAR MOMENTUM TRANSPORT

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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
ACCRETION DISC
? thin
? thin
? differentially rotating
? differentially rotating
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COMPARISON
TACHOCLINE
ACCRETION DISC
? thin
? thin
? differentially rotating
? differentially rotating
? magnetized (probably)
? magnetized (probably)
? turbulent (probably)
? turbulent (probably)
? large-scale dynamo?
? large-scale dynamo?
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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?
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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
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ANGULAR MOMENTUM TRANSPORT
GENERAL
? anisotropic motion (Reynolds stress)
? anisotropic magnetic fields (Maxwell stress)
? non-axisymmetric gravitational fields
SMALL-SCALE FEATURES
LARGE-SCALE STRUCTURES
? spiral arms / shocks
? waves
? vortices
? turbulence
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SHEARING SHEET
? local model of a differentially rotating disc
? uniform rotation O ez plus uniform shear flow
2Ax ey
? appropriate for studies of thin discs
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MAGNETOROTATIONAL INSTABILITY
OPTIMAL MODE (channel flow)
? layer analysis (incompressible ideal fluid, ?
µ0 1)
u
b
? exact nonlinear solution but unstable (Goodman
Xu 1994)
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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
? viscosity
? alpha viscosity
? AM transport by convection
? nonlinear hydrodynamic shear instability
? 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 (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
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CONTINUOUS SPECTRUM
LINEAR THEORY IN IDEAL MHD
? arbitrary reference state
? Lagrangian displacement ?
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CONTINUOUS SPECTRUM
BASIC STATE
? steady and axisymmetric
? cylindrical polar coordinates (s,f,z)
? differential rotation
? toroidal magnetic field
SOLUTIONS
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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
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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
? Tayler (1973) stability criteria
? necessary and sufficient
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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
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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
? appropriate ordering schemes are unclear (to me)
? assume overwhelming stable stratification
? weak B MRI when
(NB no MRI in 2D)
? O 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
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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
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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)
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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