PowerPoint Presentation - A liquid metal Taylor-Couette experiment

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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
                      What is an accretion disk?

•   Gas, dust, and plasma                  HH30
    accumulated by a strong point-        By HST
    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
     – 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)
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 instability…
  two most likely causes are either MRI or subcritical hydrodynamic
• We wish to demonstrate the MRI in the laboratory
                         The MRI mechanism

Accretion disk flow follows
Keplerian orbits:
• Ω(r) = (GM)1/2 r3/2
• Anti-cyclonic since dΩ/dr < 0
• Rayleigh stable since d(r2 Ω)/dr > 0
• Re > 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)
              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
    – Plausible given high Re for proto-planetary disks (Re ~ 1012)
    – Evidence for subcritical instability claimed in observations of onset
      to turbulence in Taylor and Wendt’s 1930s (and also more recent)
      hydrodynamic experiments
    – Doesn’t account for stabilization by high rates of rotation in
      Keplerian disks
                       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
                     Linear stability analysis

     • Assume ideal Couette profile:

                     b                     2
                                       2r2  1r12          2
                                                         r12r2 (1  2 )
          (r)  a  2              a    2     2
                                                    ;b 
                    r                    r2  r1               2
                                                              r2  r12

     • MHD equations:
                  V              (B  )B        B 2 
                       (V  )V            p        2 V
                    t                 0        20 
                  B                
                        (V  B)   2B
                  t                0

Taylor-Couette experiment well suited to study the MRI

Re based on inner cylinder

                                But can be             Keplerian
                                                       accretion disks
                                stabilized             destabilized
                                by field               by magnetic field

                                           solid-body rotation is
                                           stable: minimum energy

                                      Re based on outer cylinder

                             Ji, Goodman, and Kageyama, MNRAS (2001)
                   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)
    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
     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
        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
• Profiles measured with LDV
• After several iterations, ring speeds   Guess #0
  were found that produce velocity
  profiles indistinguishable from ideal

               Ekman flow                 Guess #6
     Proposed solution: segmented end caps

• New scheme for driving end caps scales with rotation speed
  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
                       Reynolds stress measurements
     •   Richard & Zahn (1999) proposed a
         turbulent viscosity based on Wendt’s
         cyclonic data:
                                 ˜ ˜
                     3        VrV
                                        (1 2) 105
          turb  R         2     2
                       R      q V
     •   which cannot be excluded by
                         1 2%
     •   Simultaneous measurement of Vr and
         V by a dual synchronized LDV to
         determine Reynolds stress
                          ˜ ˜
                       ˆ    VrV
                             2                               Proposed level of
                                                                Turbulent stress
         with accuracy achieved by large
     •   Results show that the turbulent
         viscosity is far too weak to explain
         accretion disk transport
                                                        Ji et al., Nature, 2006
Stability diagram for new apparatus

Rayleigh unstable           n=4
                    MRI     n=3

         Always stable
     Predictions by 2D simulations of experiment

100% speeds, B2.5kG                45% speeds, B1.9kG
      23.6/s                            1.3/s
                   So far: 65% in water
                   30% in liquid metal
                             Liquid metal experiments

•   Modified Taylor-Couette
     –   Inner cylinder: R1=7cm, 1 < 4000rpm
     –   Outer cylinder: R2=21cm, 2 < 500rpm
     –   Chamber height: H=28cm
     – 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
            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
                   Initial magnetic data

• Initial data is contaminated with noise from the AC motor and
  port plug magnetization
Fluctuations vary with flow shear
               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
               Qualitative results from data

• Fluctuation amplitudes vary in time for marginally stable and
  centrifugally unstable flow profiles at 3.8 kG applied field
    – Can’t 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)
 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
    – 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
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

• 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
• The experiment should be able to produce the MRI, though
  other instability due to boundary layer effects may also be
• 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
                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

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