An experimental demonstration of the Magnetorotational Instability

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An experimental demonstration of the
Magnetorotational Instability

• Mark Nornberg

Contributors E. Schartman, H. Ji, M. J. Burin,
W. Liu, and Jeremy Goodman
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What is an accretion disk?

• Gas, dust, and plasma accumulated by a strong
  point-like central object through gravitational
  attraction
• Accretion of material onto the central object
  releases energy which is radiated away results
  in the measured luminosity of the object
• Accretion is responsible for many important
  astrophysical processes
• Star and planet formation in proto-stellar disks
• Mass transfer in binary systems
• Huge amounts of radiation from quasars and active
  galactic nuclei (1015 times luminosity of the sun)

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What is the cause of turbulence in accretion
disks?

• Rates of material inflow in an astrophysical disk
  are limited by how quickly angular momentum can
  be transported
• Accretion disks show fast rates of material
  inflow compared with a centrifugally stable flow,
  so flow is likely turbulent
• Various possibilities for instability leading to
  disk turbulence hydrodynamic turbulence, density
  stratification, MHD instabilitytwo most likely
  causes are either MRI or subcritical hydrodynamic
  instability
• We wish to demonstrate the MRI in the laboratory

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The MRI mechanism

• Accretion disk flow follows
• Keplerian orbits
• O(r) (GM)1/2 r3/2
• Anti-cyclonic since dO/dr lt 0
• Rayleigh stable since d(r2 O)/dr gt 0
• Re gt 1012
• Magnetic tension can lead to a runaway
• instability creating effective radial flux
• of angular momentum.
• Instability is axisymetric
• Free energy flow shear
• Stabilized by strong magnetic field
• Resistively limited (minimum Rm required)

Balbus and Hawley, Rev. Mod. Phys. (1998)
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What is subcritical instability?

• It is proposed that cooler proto-planetary disks
  may be insufficiently conductive to sustain the
  MRI
• An alternative mechanism for the transition to
  turbulence relies on nonlinear instability as in
  the transition to turbulence in pipe flow
• Plausible given high Re for proto-planetary disks
  (Re 1012)
• Evidence for subcritical instability claimed in
  observations of onset to turbulence in Taylor and
  Wendts 1930s (and also more recent) hydrodynamic
  experiments
• Doesnt account for stabilization by high rates
  of rotation in Keplerian disks

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Proposed experiment

• Create hydrodynamically stable flow of a liquid
  metal between two rotating cylinders
  (Taylor-Couette experiment)
• Destabilize the flow through the MRI by applying
  an axial magnetic field
• Observe onset of fluid instability through either
  direct measurements of vr or indirect
  measurements of Br and change in torque on motors

Bzlt1T
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Linear stability analysis

• Assume ideal Couette profile
• MHD equations

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Taylor-Couette experiment well suited to study
the MRI
Centrifugally Unstable, But can be stabilized
by field
Keplerian accretion disks destabilized by
magnetic field
Re based on inner cylinder
solid-body rotation is stable minimum energy
Re based on outer cylinder
Ji, Goodman, and Kageyama, MNRAS (2001)
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Prototype experiment

• Initial experimental attempt revealed need to
  address effects of finite length cylinders
• Flow profiles were not ideal Couette due to
  strong secondary flow

Kageyama et al., JPSJ (2004)
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Finite height of cylinders leads to Ekman suction

• Endcaps of cylinders typically rotate with the
  outer cylinder causing a pressure imbalance
• Boundary layer develops to balance pressure
• Ideal Couette profile is modified by return flow
  which results in secondary radial circulation
• Secondary flow can be unstable

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Proposed solution segmented end caps

• Eliminate the boundary layer formation by
  matching the rotation of the end caps to the
  ideal Couette profile

Burin et al., Exp. Fluids, 2006
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Proposed solution segmented end caps

• Original idea was to reduce pressure gradient
  causing the secondary circulation by matching the
  speed of the end caps to the ideal Couette
  profile
• Profiles measured with LDV
• After several iterations, ring speeds were found
  that produce velocity profiles indistinguishable
  from ideal Couette

Guess 0
Ekman flow
Guess 6
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Proposed solution segmented end caps

• New scheme for driving end caps scales with
  rotation speed

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Fluctuations are extremely small for new flow

• Fluctuation levels were indistinguishable from
  solid body rotation for new ring speeds, even at
  Re 106
• Casts doubt on ability of nonlinear instability
  to occur in quasi-keplerian flow

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Reynolds stress measurements

• Richard Zahn (1999) proposed a turbulent
  viscosity based on Wendts cyclonic data
  
• which cannot be excluded by

• Simultaneous measurement of Vr and V? by a dual
  synchronized LDV to determine Reynolds stress
  with accuracy achieved by large
  statistics
• Results show that the turbulent viscosity is far
  too weak to explain accretion disk transport

Proposed level of Turbulent stress
Ji et al., Nature, 2006
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Stability diagram for new apparatus
n5
Rayleigh unstable
n4
n3
MRI
n1,2
Multi-modes unstable
Always stable
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Predictions by 2D simulations of experiment
100 speeds, B?2.5kG ??23.6/s
45 speeds, B?1.9kG ??1.3/s
So far 65 in water 30 in liquid metal
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Liquid metal experiments

• Modified Taylor-Couette
• Inner cylinder R17cm, ?1 lt 4000rpm
• Outer cylinder R221cm, ?2 lt 500rpm
• Chamber height H28cm
• Liquid metal GaInSn eutectic
• Six coils provide 5 kG axial field
• External magnetic field measured by 36 pickup
  coils and 8 Hall effect sensors
• Internal pressure and magnetic sensors will be
  introduced in a fin probe
• Exploring possibility of direct velocity
  measurements through ultrasound Doppler
  velocimetry

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Experimental Demonstration of MRI

• Establish hydrodynamically quiescent flow in
  liquid metal (segmented rings demonstrated to
  work with water)
• Flow becomes quiescent over several Eckman times
  ?200 s
• Destabilize the MRI with axial magnetic field (up
  to 5 kG, ??10 ms)
• Observe growing external magnetic fluctuations on
  array of radial Hall probes (1 Gauss resolution)
  and pickup coils (0.5 G/Hz sensitivity)
• Compare results for several different velocity
  profiles with different levels of shear
  centrifugally unstable, marginally stable, low
  shear, and solid body

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Initial magnetic data

• Initial data is contaminated with noise from the
  AC motor and port plug magnetization

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Fluctuations vary with flow shear
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Qualitative results from data

• For the marginally stable and unstable flows, the
  pickup coils exhibit changes in the fluctuation
  levels
• Fluctuations are constant for applied fields of
  2.5 kG and 5.0 kG
• Larger for profiles with larger flow shear and
  higher speed

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Qualitative results from data

• Fluctuation amplitudes vary in time for
  marginally stable and centrifugally unstable flow
  profiles at 3.8 kG applied field
• Cant be due only to hydrodynamic turbulence
  since character changes with strength of magnetic
  field
• Either due to global instability (MRI) or
  instability caused by boundary layer flow
  (magnetic Ekman effect)

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Ekman-Hartmann layer develops with magnetic field

• The boundary layer is modified by the applied
  magnetic field
• Thickness transitions from Ekman layer (viscous)
  to Hartmann layer (resistive)
• Changes secondary circulation
• Can lead to another instability unrelated to the
  MRI
• Even if the boundary layer is disrupted in the
  experiment, the magnetic field may reestablish it
  and generate a secondary flow
• We need to distinguish between instability caused
  by the boundary flow from the MRI which is an
  instability of the bulk flow

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Possible mechanism for boundary layer disruption

• Summer research on prototype experiment to
  explore possible causes of boundary layer
  disruption
• Wobbling of inner cylinder
• Gap along end caps near inner cylinder

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Conclusions

• We have a unique experiment which has
  demonstrated the possibility of hydrodynamically
  quiescent flows at high Reynolds number (Re
  106)
• Nonlinear instability is not a likely mechanism
  for accretion disk turbulence
• The experiment should be able to produce the MRI,
  though other instability due to boundary layer
  effects may also be present
• We observe an MHD instability caused by the
  application of an axial magnetic field
• We will distinguish the MRI from magnetic Ekman
  effects by comparing measurements from profiles
  that are MRI stable and unstable

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What is laser Doppler velocimetry?
Interference patterns form when two coherent
laser beams intersect. Frequency of
backscattered light from reflective tracer
particles is proportional to velocity. Unevenly
sampled at a point as a function of time. Scan in
r, z.
Image courtesy of Dantec Dynamics