An update of results from the Princeton MRI Experiment Mark Nornberg Contributors: E. Schartman, H....

An update of results from the Princeton MRI Experiment

Mark Nornberg

Contributors: E. Schartman, H. Ji, A. Roach,W. Liu, and Jeremy Goodman

CMSO General Meeting6 April 2009

Princeton Plasma Physics Lab

Outline

• Motivation for studying magnetorotational instability (MRI)

• Experiment to study magnetically induced turbulence in rotating shear flow

• Theory of basic waves in a rotating magnetized fluid from linear stability analysis

• Observations of propagating waves in hydrodynamically turbulent liquid metal flow

• Conclusions

We would like to understand accretion disks

• An accretion disk forms from gas, dust, and plasma accumulated by massive stars and black holes

• Accumulation 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)

HH30By HST

Accretion Disk + Black Hole in the Core of Galaxy NGC 4261

What governs the accretion rate in 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

The MRI mechanism

Accretion disk flow follows Keplerian orbits:• Ω(r) = (GM)1/2 / r3/2

• dΩ/dr < 0 • Centrifugally stable d(r2 Ω)/dr > 0• Re > 1012

Balbus and Hawley, APJ (1991)Rev. Mod. Phys. (1998)

Magnetic tension can lead to a runaway instability creating effective radial flux of angular momentum.• Free energy flow shear• Only requires dΩ/dr < 0• Purely growing mode• Resistively limited (minimum Rm required)

Novel Couette-Taylor experiment

• High Reynolds number flows• Control secondary flow due to boundary layers Ji et al., Nature, 2006

Burin et al., Exp. Fluids, 2006

PROMISE

Helical MRI

Stefani, PRL, 2006

Sisan, PRL, 2004Maryland Spherical CouetteMaryland Spherical Couette

• Assume ideal Couette profile:

• MHD equations:

• Flow shear is quantified by vorticity parameter

• Dispersion relation from local (WKB) stability analysis:

Linear MHD stability analysis

€

Ω(r) = a+b

r2

€

a =Ω2r2

2 − Ω1r12

r22 − r1

2;b =

r12r2

2(Ω1 − Ω2)

r22 − r1

2

€

ρ∂V

∂t+ (V ⋅∇)V =

(B ⋅∇)B

μ0−∇ p+

B2

2μ0

⎛

⎝ ⎜

⎞

⎠ ⎟+ ν∇2V

∂B

∂t=∇ × (V × B) +

η

μ0∇2B

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)

• Magnetocoriolis waves result as a combination of

• Coriolis & Lorentz forces:– Together: Fast MC wave– Opposed: Slow MC wave– Split Alfvén frequency– Add resistivity– Add sufficient shear– Slow wave becomes

unstable (MRI)

Basic waves in rotating incompressible conducting fluids

Alfvén waves and Inertial waves

Couette-Taylor experiment well suited to study the MRI

Re based on outer cylinder (106)

Re

base

d on

inne

r cy

linde

r (1

06 )

Ji, Goodman, and Kageyama, MNRAS (2001)Ji, et al., Exp. & Models (2004)

Liquid metal experiments

• Inner cylinder: r1=7cm, Ω1 < 4000 rpm• Outer cylinder: r2=21cm, Ω2 < 500 rpm• Chamber height: H=28cm (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)

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

Non-axisymmetric modes observed

B0 = 3.30 kGBr (Gauss) measurements at surface show azimuthal (m=1) mode

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Z(cm)

Toroidal angle(radians)

Fourier decomposition of modes

• Two nonaxisymmetric modes with different phase velocity

(0,1) (1,1)(axial mode, azimuthal mode)

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Observation of rotating modes

• (0,1) and (1,1) are dominant nonaxisymmetric modes• Each show different phase velocity which changes with

field strength

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

Internal velocity measurements through ultrasound

• Velocity measurements in liquid metals are possible through Ultrasonic Doppler Velocimetry (UDV)– Natural oxides are

effective scattering targets

– Concentration important for effective measurement

• Initial results suggest unexplained hydrodynamic behavior as well as profile modification by MHD effects

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

• We will be using UDV to measure changes to the flow profile and local Reynolds stress