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Title: Observation of magnetocoriolis waves in the Princeton MRI Experiment


1
Observation of magnetocoriolis waves in the
Princeton MRI Experiment
  • Mark Nornberg

Princeton Plasma Physics Lab
Contributors E. Schartman, H. Ji, A. Roach, W.
Liu, and Jeremy GoodmanPST Seminar 18 December
2008
2
Outline
  • Motivation for studying magnetorotational
    instability (MRI)
  • Experiment to study magnetically induced
    turbulence in rotating shear flow
  • Theory from linear stability analysis
  • Observations of propagating waves in
    hydrodynamically turbulent liquid metal flow
  • Conclusions
  • Possible discussion of free surface MHD

3
What is an accretion disk?
  • Gas, dust, and plasma accumulated by massive
    stars and black holes
  • Accretion of material onto the central object
    releases energy which is radiated away producing
    the measured luminosity of the object
  • Accretion is responsible for many important
    astrophysical processes
  • Star and planet formation
  • Mass transfer in binary systems
  • Huge amounts of radiation from quasars and active
    galactic nuclei (1015 times luminosity of the sun)

4
What is the cause of turbulence in accretion
disks?
  • Accretion rate limited by angular momentum
    transport
  • Accretion disks have faster inflow than predicted
    by classical viscous transport, so flow is likely
    turbulent
  • Disk flows are linearly stable in hydrodynamics
  • Possibilities for instability leading to disk
    turbulence
  • Nonlinear hydrodynamic instability
  • MHD instability
  • The magnetorotational instability (MRI) is the
    destabilization of rotating shear flow by a
    magnetic field
  • We wish to demonstrate the MRI and study angular
    momentum transport in the laboratory

5
The MRI mechanism
  • Magnetic tension can lead to a runaway
    instability creating effective radial flux of
    angular momentum.
  • Free energy flow shear
  • Only requires dO/dr lt 0
  • Purely growing mode
  • Resistively limited (minimum Rm required)
  • Accretion disk flow follows
  • Keplerian orbits
  • O(r) (GM)1/2 / r3/2
  • dO/dr lt 0
  • Centrifugally stable d(r2 O)/dr gt 0
  • Re gt 1012

Balbus and Hawley, APJ (1991) Rev. Mod. Phys.
(1998)
6
Novel Couette-Taylor experiment
Maryland Spherical Couette
Sisan, PRL, 2004
Stefani, PRL, 2006
  • High Reynolds number flows
  • Control secondary flow due to boundary layers

Ji et al., Nature, 2006 Burin et al., Exp.
Fluids, 2006
7
Linear MHD stability analysis
  • Assume ideal Couette profile
  • MHD equations
  • Flow shear is quantified by vorticity parameter
  • Dispersion relation from local (WKB) stability
    analysis

8
Basic waves in rotating incompressible conducting
fluids
  • Dispersion relation for ideal fluid
  • Assume no rotation, recover shear Alfvén waves
  • Transverse polarization
  • Restoring force caused by Lorentz force
  • Flow becomes uniform along field
  • Assume no field and no flow shear, obtain
    inertial waves
  • Transverse polarization
  • Restoring force caused by Coriolis force
  • Generated by deviations from uniform rotation
    along rotation axis(Taylor-Proudman Theorem
    flow becomes uniform along axis of rotation)

9
Basic waves in rotating incompressible conducting
fluids
  • Magnetocoriolis waves result as a combination of
  • Coriolis Lorentz forces
  • In phase Fast MC wave
  • Out of phase Slow MC wave
  • Split Alfvén frequency
  • Add resistivity
  • Add sufficient shear
  • Slow wave becomesunstable (MRI)

Alfvén waves
and Inertial waves
10
Couette-Taylor experiment well suited to study
the MRI
Re based on inner cylinder
Re based on outer cylinder
Ji, Goodman, and Kageyama, MNRAS (2001) Ji, et
al., Exp. Models (2004)
11
Liquid metal experiments
  • Inner cylinder r17cm, ?1 lt 4000 rpm
  • Outer cylinder r221cm, ?2 lt 500 rpm
  • Chamber height H28cm (aspect ratio 2.1)
  • Re 107 and Rm 10
  • Liquid metal GaInSn eutectic (Pm 10-6)
  • Six coils provide 5 kG axial field (S 0.3 -
    3.0)
  • External magnetic field measured by array of
    pickup coils and Hall sensors

E. Schartman (thesis 2008)
12
Experimental procedure
  • Establish flow in liquid metal by starting motors
  • Flow develops over several Eckman times ?200 s
  • Apply axial magnetic field (up to 5 kG, ??10 ms)
  • Observe external magnetic fluctuations on array
    of radial Hall probes (1 Gauss resolution) and
    pickup coils (0.5 G/sec sensitivity)
  • Compare results for different rotation speeds,
    shear, and magnetic field strength

13
Non-axisymmetric modes observed
Z (cm)
Toroidal angle (radians)
Br (Gauss) measurements at surface show azimuthal
(m1) mode
B0 3.30 kG
14
Fourier decomposition of modes
(0,1)
(1,1)
(axial mode, azimuthal mode)
  • Two nonaxisymmetric modes with different phase
    velocity

15
Observation of rotating modes
  • (0,1) and (1,1) are dominant nonaxisymmetric
    modes
  • Each show different phase velocity which changes
    with field strength

16
Observed rotation rates match fast/slow MC-waves
  • Phase speeds similar to Alfvén wave
  • Match behavior of MC-waves with B0
  • Should be damped when ??? so they must be driven
  • Positive growth for doubling the rotation speed

17
Beginning analysis of rotating shear flow
  • Similar separation in phase velocities observed
  • Speeds too low to observe significant growth
    rates
  • Excitingly close to marginal stability

18
3D simulations of experiment
  • Collaboration with U. Chicago to simulate
    experiment
  • Nek5000 spectral element code
  • Quantitative agreement for quiescent flows
  • Examining propagation of nonaxisymmetric modes

Obabko, Cattaneo, Fischer, Phys. Scr., 2008
19
Conclusions
  • Contributing to understanding of astrophysical
    MHD processes through laboratory experiments
  • In magnetized hydrodynamically unstable flows,
    we observe several nonaxisymmetric modes
  • The rotation rates of the two largest
    nonaxisymmetric modes match the dispersion
    relation for fast and slow magnetocoriolis waves
  • By mapping the field dependence of the
    magnetocoriolis waves we should be able to detect
    the threshold for the MRI
  • Experiments to observe destabilization of a
    quiescent flow by an applied magnetic field and
    further investigation of magnetized turbulent
    flows, both through experiments and simulations,
    are ongoing

20
2D simulation results
  • Ekman-Hartman layer development (boundary layer
    transitions to Hartmann layer
  • Steep velocity gradient (Stewartson layer)
    becomes unstable to MRI

Liu, PRE (2008)
21
Measured profiles for unstable flow
22
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23
Study of Free Surface MHD in the Liquid Metal
Experiment
  • Mark Nornberg
  • Contributors Hantao Ji, John Rhoads, Luc Peterson

24
Motivation Astrophysical free surface MHD
  • Important in describing dredge up mechanism in
    classical novae
  • Possible explanation for quasi-periodic x-ray
    emission from binary star systems

Lattimer Prakash, Science 304, 536 (2005)
25
Motivation Liquid Metal PFCs
  • Application of liquid metal walls in fusion
    reactors
  • Effective heat transfer
  • Protect solid wall
  • Reduce recycling
  • Possibly improve plasma stability
  • Not much is known about the stability of
    free-surface liquid metal flow in strong magnetic
    fields with electric current

26
Design of channel
  • Hydrodynamic flows studied to determine best way
    to reduce development length
  • Surface waves due to entrance effects suppressed
    using wave damper
  • Channel enclosed and filled with argon to prevent
    oxidization

27
Design of magnet
  • Up to 3 kG magnetic field provided by cooled
    copper wrapped around steel flux core
  • 2.5 kA from DC Rectifier

28
Laser reflection surface wave diagnostic
  • Laser beam reflections off surface onto screen
    tracked with CCD camera
  • Wavelength measured from relative phase between
    beams

29
Determination of dispersion from laser diagnostic
  • Laser positions determined by IDL image
    recognition routine for each frame
  • Time series of laser spot positions fitted to
    sinusoid packet
  • Phase determined from fit
  • Wave number calculated from linear fit

30
Surfactant effect of oxides
  • Exposure to oxygen creates oxide film which acts
    as surfactant, effectively reducing surface
    tension and eliminating capillary waves

Nornberg, Rev. Sci. Instr. (2008)
31
Modification of turbulence by magnetic field
  • Cross-channel magnetic field has no damping
    effect on driven streamwise waves
  • Yet natural surface fluctuations from turbulent
    flow are significantly reduced

Nornberg, Rev. Sci. Instr. (2008)
32
2D Turbulence
Position Sensitive Device
  • We are measuring the power spectrum of surface
    fluctuations to study the two-dimensionalization
    of surface wave turbulence
  • Observations indicate scale dependence in damping
  • Two sensors are used to obtain k-space spectrum
    from cross-correlation analysis

33
Measurements of flow profile
  • Simulations indicate that Hartmann boundary layer
    effect lead to peaking of flow profile
  • Support from preliminary measurements using
    potential probe velocimetry

34
Summary
  • We have developed a liquid metal channel flow
    experiment to study the basic physics of free
    surface MHD
  • We are developing diagnostics to study changes to
    both the flow profile and the surface turbulence
    due to magnetic fields
  • Further investigation of stability, heat
    transport, and the development of a weak
    turbulence theory is underway
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