Magnetorotational Instability in Protostellar Discs

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Magnetorotational Instability in Protostellar
Discs

• Raquel Salmeron
• University of Sydney
• Supervisor
• Mark Wardle, Macquarie University

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Introduction
The Orion Nebula

• Low mass YSOs
• Accretion discs lt 10 Mstar
• Size up to 1000 AU
• Temperature from 10K to a few 103 K
• Weakly ionised
• Magnetically driven accretion

Hubble Space Telescope Credit C.R. O'Dell (Rice
University), and NASA
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Accretion processes

• Angular momentum transport
• Molecular viscosity (Pringle 1981)
• Convective turbulence (Lin Papaloizou 1980, Ryu
  Goodman 1992, Stone Balbus 1996)
• Outflows (Wardle Königl 1993)
• Tidal effects (Vishniac Diamond 1989)
• Hydromagnetic instabilities
• Magnetorotational Instability (Balbus Hawley
  1991)
• Non-ideal MHD conditions
• Resistive approximation
• Ambipolar diffusion
• Hall effect

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How the MRI transfers angular momentum outwards

• Two fluid elements, in the same orbit, are joined
  by a field line (Bo). The tension in the line is
  negligible.
• If they are perturbed, the line is stretched and
  develops tension.

• The tension acts to reduce the angular momentum
  of m1 and increase that of m2. This further
  increases the tension and the process runs
  away.

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Ideal MRI - An overview

• Requirements
• Angular velocity decreasing with radius
• Subthermal B with a poloidal component
• ? Satisfied in astrophysical discs

• Fastest growing modes
• growth rate
• independent of B
• wavelength proportional to B
• horizontal displacements

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The present study
Objective To model the linear growth of MRI
perturbations at different radial locations.

• Model includes
• vertical stratification
• conductivity as a tensor

• Growth depends on
• density,
• temperature,
• ionisation rate, and
• charged species
• ? conductivity
• These factors vary radially and vertically in the
  disc

• Applications
• Explore unstable zones
• Analyse properties of the MRI with a z-dependent
  conductivity tensor
• Motivate further work

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Governing Equations
Amperes Law
Conservation of mass
Ohms Law
Conservation of momentum
Perturbations
Induction Equation
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Disc Model

• Vertical Stratification
• Density distribution given by

• Evaluation of the conductivity tensor
• Method of the present work
• Allows to study the structure and growth of MRI
  with model parameters

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Linearisation

• Conductivity tensor not affected by perturbations
  
• Equations decouple into two subsystems
• Sound waves in the vertical direction
• MHD perturbations in the plane of the disc

• Initial state

• Assumptions

Terms lt H/r neglected
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Model Parameters

• Magnetic coupling at the midplane

• Ratio of the midplane Alfven to sound speed ?
  Strength of the magnetic field at the midplane

• Ratio of the conductivity terms perpendicular to
  the field ? Conductivity regime of the fluid

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Boundary Conditions

• At the midplane

• At the surface

System solved by shooting from the midplane to
the surface to determine ? and ?E?
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Test Models
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Comparison with local analysis
Growth Rate (?)
Hall regime (?1Bzgt0)
Ambipolar diffusion
Good coupling
Number of Nodes
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Typical results - structure
Mid plane
Surface
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Conductivity Regime
?1 ?2
Hall ?1Bz gt 0
Ambipolar diffusion
?1 -?2
Hall ?1Bz lt 0
Height above mid plane (z/H)
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Conductivity Regime
Hall regime ?1Bzgt0
Ambipolar diffusion
Radial
Radial
Amplitude of perturbations of B
Azimuthal
Azimuthal
Height above midplane
Height above midplane
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Conductivity Regime
?ovA/cs
Height above mid plane
Log(Magnetic coupling (?o))
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Magnetic Field Strength
Ambipolar diffusion
Radial
Amplitude of perturbations of B
vA/cs0.005 ? 0.7000
Good coupling
Height above midplane
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Magnetic Field Strength
Hall ?1Bz gt 0
Radial
Amplitude of perturbations of B
vA/cs0.005 ? 0.7499
Good coupling
Height above midplane
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Parameter space - Coupling
Log(Maximum growth rate ?max)
vA/cs0.1
Log(Coupling ?o)
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Parameter space - Magnetic field
Log(Maximum growth rate ?max)
Good coupling
Log(vA/cs)
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Parameter space - Magnetic field
Log(Maximum growth rate ?max)
Poor coupling (?02)
Log(vA/cs)
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Implications
Full conductivity
Ambipolar diffusion
?o 0.01
? 0.2913
? 0.3728
vA/cs0.01
vA/cs0.01
Height above midplane
Height above midplane
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Implications

• Allowing for different conductivity regimes to
  dominate at different heights, it is found that
• Hall effect modifies both the structure and
  growth of unstable modes
• The extent of the dead zone is reduced
• The growth rate is increased
• Configuration of active zones and angular
  momentum transport is significantly modified by
  Hall conductivity

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Main Findings - Summary

• Global modes are a discrete subset of possible
  local modes
• The most unstable MRI perturbations peak at
  higher vertical locations when ambipolar
  diffusion dominates
• At good coupling, structure and growth rate are
  determined by ambipolar diffusion
• At poor coupling, the Hall effect modifies both
  the structure and growth rate of the
  perturbations
• Perturbations grow at about the ideal rate for
• Hall limit ? gt vA2/cs2
• Ambipolar Diffusion ? gt vA/cs(Wardle 1999)

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Main Findings - Summary

• Unstable modes exist until a certain (vA/cs)max
  is reached
• At good coupling, (vA/cs)max ? 1 for all regimes
• At poor coupling, (vA/cs)max ? 2.9 for the Hall
  (?1Bzlt0) case
• At very weak fields (vA/cs lt 0.01), global
  effects are less relevant and MRI growth rates
  tend to local values
• Including the Hall regime in the study of
  dynamical processes in low conductivity discs is
  essential for the understanding of accretion.

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Work in progress

• Include a z-dependent conductivity tensor, which
  takes into account cosmic x-ray ionisation.
• Conduct modelling of structure and linear growth
  of MRI perturbations for a range of radial
  positions in low conductivity discs.