Outline:

1
Momentum Transport
D. Craig General Meeting of the Center for
Magnetic Self-Organization In Laboratory and
Astrophysical Plasmas August 4-6, 2004 in
Madison, WI
Outline I. Introduction, background, and
examples of momentum transport II. Momentum
transport physics topics being addressed by
CMSO - Physics, Plans, and Progress
2
Why Study Momentum Transport?

• Momentum transport is an important issue in
• Accretion Disks
• Astrophysical Jets
• Solar Interior
• Laboratory Experiments
• Collisional viscosity fails to explain transport
  of
• momentum in all of the above cases
• Magnetic fluctuations can have a large, often
  dominant
• effect on the system in all of these situations
• A theme of Center research in this area is to
  significantly
• further our understanding of when and how
  magnetic
• fluctuations contribute to momentum transport

3
Accretion Disks

• Thin disk of material orbits a compact object and
  slowly falls onto it
• Angular momentum must be
• removed from accreting material
• Leading explanation for this
• is torque associated with magnetic
  fluctuations

Protostellar diskjet (Hubble Space Telescope)
4
Astrophysical Jets

• Associated with disks of protostars, Xray
  binaries, Active Galactic Nuclei.
• Synchrotron radiation reveals B field in AGN
  AGN jets
• Probably rotationally driven and magnetically
  confined
• Helical field pinch
• Axial flow decelerates by transfer of momentum
  toward edge of jet
• Analogous to lab?

Optical jet in galaxy M87 (NASA/HST)
Cartoon of magnetically collimated jet
5
Internal Rotation Profile of the Sun

• Helioseismology shows the
• internal structure of the Sun.
• Surface differential rotation
• is maintained throughout the
• convection zone
• Solid body rotation in the
• radiative interior
• Thin matching zone of shear
• known as the tachocline at
• the base of the solar convection
• zone
• How does this come about?
• Momentum sources transport

6
MST (Wisc) Experiment and Tools
n 1019 m-3 Te,i 0.1-1 keV b 10
R 1.5 m a 0.52 m B 0.2 T

• Tools
• FIR Interferometer / Polarimeter
• Doppler Spectroscopy
• - Passive - chord averaged flow
• - Active Charge Exchange Recombination
• Spectroscopy (CHERS) - 1 cm resolution (in
  development)
• Coil arrays - magnetic fluctuation spectrum
• Insertable probes - Langmuir, Mach, magnetic,
  spectroscopic
• Auxiliary flow drivers
• - biased probes in edge
• - neutral beam in core (in development)

7
Helical Flows Are Naturally Present in MST Plasmas

• In core, v mostly parallel to B
• In edge, have vparallel and vperp
• Origin of flows unclear

(sketches based on incomplete flow profile
measurements)
8
Plasma Momentum Changes Spontaneously in MST with
Bursts of Magnetic Activity

• Plasma rotation
• slows in 100 ms
• Not classical
• - 100 times too fast
• - n, T, ... do not
• change enough
• on this timescale
• Leading explanation
• involves coupled
• magnetic fluctutaions

vtoroidal (km/s)
9
Spontaneous Flows Also Measured in MRX

• Two kinds of flows
• 1. v associated with reconnection
• 2. toroidal (azimuthal) flows
• Momentum transport not examined
• yet

n 1-20 x 1019 m-3 T 4-30 eV B 0.05 T b
0.1-10
10
Momentum Transport Physics and Plans
We have chosen to focus our efforts on 5 physics
topics
1. Momentum transport by stochastic magnetic
fields 2. Momentum transport by Maxwell stress
from current-driven instabilities 3. Momentum
transport by Maxwell stress from
magnetorotational instability 4. Generation
and relaxation of momentum as part of a 2-fluid
form of magnetic relaxation 5. Momentum
transport in the sun
11
Transport by Stochastic Magnetic Fields

• Mechanism
• B field lines wander in space
• Particles or waves follow field lines
• ? Momentum carried in space
• Stochastic fields often found
• in lab and space
• - All Center devices other lab plasmas
• - Accretion disks (in MHD computation)
• - Likely in jets and in sun
• Stochastic fields NOT often invoked for
• momentum transport

12
Plans Momentum Transport in Stochastic B
1. Measure in MST, a direct measure of
this effect Requires diagnostic development (
1-2 yrs) 2. Drive flows in MST, vary
fluctuations, measure momentum transport Requires
electrically biased probes and/or neutral beams
( 0.5-1 yr) 3. Measure mean flow profile and its
evolution in MST Requires diagnostic development
( 1-2 yrs) 4. Drive flows in MRX and measure
momentum transport Requires electrically biased
probes and/or neutral beams (mid-long term) 5.
Measure flows in SSX (diagnostic development,
near term) 6. Include momentum transport in
self-consistent theory for transport in
stochastic magnetic fields ( 1 yr) 7. Assess
relevance of self-consistent theory to
astrophysics ( 1 yr)
13
Flow Perturbation Experiments

• Insertable biased probes create pulse of edge
  flow in MST
• Core responds with some delay
• ? global momentum transport timescale 1 ms

• Neutral beam injection
• might be able to make
• pulsed core flows

14
Charge Exchange Recombination Spectroscopy
(CHERS) Basic principles
(1) Charge exchange
(2) Radiative decay
We observe this!
15
CHERS Profile measurement
Doppler shift (Dl) gives vimpurity
2000
Area gives nimpurity
Signal level (photons)
1000
Doppler width (s) gives Timpurity
0
3431
3435
Wavelength (Å)
Measure Doppler shifted and broadened line
emission profile Need accurate model for
profile shape Need accurate technique for
data fitting
16
Beam-driven CHERS emission is localized
Fiber bundle views of beam and background
Beam current monitor
Perpendicular viewing chords
30 keV H beam
MST vessel

• View emission resulting from charge exchange
  between beam
• neutrals (H) and background impurity ions
• Intersection volume between beam and fiber
  views is small
• localized measurement of impurity
  Ti, vi (and possibly ni)

17
Upgraded CHERS system installed on MST (April
2004)
Initial measurements made on CVI line emission
(344 nm) Data exhibit large signal, low
signal-to-noise Will allow impurity Ti, vi to
be resolved on fast time scale ( 100 ms) Atomic
modeling initial fitting of CVI line shape has
been done
Beam ON
Beam off
Beam off
Ti (eV)
time (ms)
18
Momentum Transport Physics and Plans
1. Momentum transport by stochastic magnetic
fields 2. Momentum transport by Maxwell stress
from current-driven instabilities 3. Momentum
transport by Maxwell stress from
magnetorotational instability 4. Generation
and relaxation of momentum as part of a 2-fluid
form of magnetic relaxation 5. Momentum
transport in the sun
19
Current-Driven Tearing Modes

• Perturbations with kB 0 do not bend B field
  lines
• Fluctuations with kB 0 somewhere are called
  resonant
• Position (surface) where kB 0 called resonant
  surface
• In MST, have helical B ? helical resonant
  perturbations
• Pitch of B field lines changes with radius
• Multiple resonances throughout plasma
• Tearing Modes
• One class of resonant perturbations
• Driven primarily by ? J(r)
• Tear magnetic field to form islands
• Typically see full spectrum of
• tearing modes in MST

20
Magnetic Maxwell Stress From Nonlinearly Coupled
Tearing Modes

• Fluctuating B can make net force, ltJk?Bkgt
• - Can rewrite as ?? (BkBk) ? magnetic analog of
  ?? (vkvk)
• Nonlinear mode coupling can give
• Force at resonant location for mode k has the
  form
• In MHD, forces localized to resonant positions
  of coupled modes
• Forces are differential (3 forces at 3 locations
  all add to 0)
• - Momentum transport, no net force

coupling coefficient
phases of modes
21
Coupled Tearing Modes Produce Strong Torques in
MST

• Maxwell stress in core estimated from edge
  measurements of B
• Mode amplitude and coupling increase during
  relaxation events
• Strong ltJ?Bgt forces result

2
ltJ?Bgt
22
Plans Momentum Transport by Maxwell Stresses
from Tearing Modes
1. Measure ltJ?Bgt directly in MST ( 1-2 yrs) 2.
Calculate ltJ?Bgt directly in MHD computation ( 1
yr) 3. Drive flows in MST, vary fluctuations,
measure momentum transport Requires electrically
biased probes and/or neutral beams ( 0.5-1
yr) 4. Measure mean flow profile and its
evolution in MST ( 1-2 yrs) Look for evidence of
localized forces near resonant surfaces 5.
Measure flows in SSX (near term) 6. 3D MHD
computation in SSX geometry with hybrid code
(near term) 7. Assess relevance for astrophysical
jet problem ( 1 yr)
23
Maxwell Stress in MHD Computation
(On behalf of F. Ebrahimi, by way of S. Prager)

• Using DEBS code (3D nonlinear resistive MHD in
  periodic cylinder)
• Generate saturated RFP state with many tearing
  modes
• Apply ad hoc uniform toroidal momentum force

24
Maxwell Stress in MHD Computation

• Will examine ltJ ? Bgt from tearing fluctuations
  and v(r) evolution
• First numerical runs now underway

25
Momentum Transport Physics and Plans
1. Momentum transport by stochastic magnetic
fields 2. Momentum transport by Maxwell stress
from current-driven instabilities 3. Momentum
transport by Maxwell stress from
magnetorotational instability 4. Generation
and relaxation of momentum as part of a 2-fluid
form of magnetic relaxation 5. Momentum
transport in the sun
26
Magnetorotational Instability (MRI)

• Believed to dominate angular momentum transport
  in disks
• Exists in ideal MHD for arbitrarily weak fields
  ? gtgt ?
• Feeds on differential rotation
• Converts toroidal kinetic energy to magnetic
  energy turbulence
• Growth rate ? shear rate
• Saturates at ? ? 10 ? 100 (?)
• Demonstrated in simulation, not yet in lab

Top view along rotation axis
Side view in poloidal plane
27
Outstanding Issues Concerning MRI

• How far from ideal can the plasma be?
• Some are quite resistive protostellar disks,
  quiescent cataclysmic variables, etc.
• Can AMT be explained by hydrodynamic
  instabilities?
• Can MRI exist only when ? gt 1 ?
• Do simulations get the transport rate right?
• Answer to latter two questions may be No if the
  scale height of the magnetic field is much larger
  than that of the plasma a magnetized corona.

28
Plans Momentum Transport by MRI
1. Calculate linear stability of MRI in lab,
apply to MST ( 1 yr) 2. Investigate MRI in
liquid metal Gallium experiment Operate
experiment (near term) Apply nonlinear MHD theory
to experiment (near term) Develop incompressible
MHD computation (near term) 3. Evaluate the role
of active disk coronae in angular momentum
transport in accretion disks Requires code
development (longer term)
29
The Princeton MRI Experiment

• Liquid gallium Couette flow
• Centrifugal force balanced by pressure force from
  the outer wall
• MRI destabilized with appropriate ?1, ?2 and Bz
  in a table-top size.
• Identical dispersion relation as in accretion
  disks in incompressible limit

Bzlt1T
30
Status

• Water experiments and hydrodynamic simulations
  revealed importance of Ekman effect due to end
  plates. Paper published.
• Optimized design includes 2 independently driven
  rings at each end
• Ekman effect minimized, and thus much wider
  operation regimes
• Much more complex apparatus
• Engineering design completed, reviewed, bid
  awarded, and the apparatus fabricated and
  assembled. Testing underway.
• Magnetic coils designed, fabricated. Other
  components completed or underway. Ready for
  gallium experiments later in the year.
• Modeling a new spectral-element code working
  (Fausto et al.) and the existing ZEUS code being
  adapted (Liu, Stone, Goodman).

31
(No Transcript)
32
(No Transcript)
33
Angular momentum transport in thin disks and
coronae

• Schnack Mikic visited Princeton Jan 04
• Met with Goodman, Yamada, Ji, Kulsrud
• Thin disk tutorial
• Formulated computational plan
• Summary notes written by Goodman

34
Status

• Princeton to hire post-doc (status?)
• Spend fraction of time at SAIC/San Diego to work
  on simulations (Schnack Mikic)
• Codes exist, but need modification of BCs
  (Goodman notes)
• Similar to coronal disruption/flare/CME problem
• Model problems (disk flares) done 10 years ago at
  SAIC (NASA proposal, not funded!)

35
Problem Formulation

• Magnetic loops in disk coronae are stressed by
  differential rotation of disk (similar to solar
  corona evolution)
• Two consequences
• Disruptions (disk flares)
• Non-local angular momentum transport between
  footpoints of loops (feedback on disk rotation)
• Modify existing code (MAC) to include Goodman
  model for disk dynamics (thin disk approximation)
• MAC developed and extensively used to study
  formation and disruption of solar coronal loops
• Initialize with potential field in corona
  (specified normal field distribution on disk
  surface)
• Apply differential rotationto boundary with
  feedback BC
• Analyze ensuing dynamics

36
Initial Conditions
37
(No Transcript)
38
Momentum Transport Physics and Plans
1. Momentum transport by stochastic magnetic
fields 2. Momentum transport by Maxwell stress
from current-driven instabilities 3. Momentum
transport by Maxwell stress from
magnetorotational instability 4. Generation
and relaxation of momentum as part of a 2-fluid
form of magnetic relaxation 5. Momentum
transport in the sun
39
Parallel Momentum Relaxation

• Taylor relaxation - single fluid MHD
• ? Global helicity (?A?B dV) conserved
• ? Relax to minimum magnetic energy (via v?B)
• ? Constant J?B/B2 profile
• 2-fluid relaxation
• ? Generalized helicity for each species (?As?Bs
  dV) is conserved
• where As A (ms/qs) vs and Bs ??As
• ? Relax to minimum magnetic flow energy (via
  v?B and J?B)
• ? Constant J?B/B2 and nv?B/B2 profiles
• ? Parallel current and parallel momentum profiles
  get coupled
• (alternatively dynamo and momentum
  transport coupled)
• Open question whether this actually happens in
  lab or space

40
Plans Two Fluid Relaxation
1. Observe momentum profile relaxation in 2 fluid
MHD computation in MST geometry Requires code
development ( 1 yr) 2. Measure parallel momentum
profile relaxation in any or all Center
devices (MST, MRX, SSX, SSPX) ( 2-3 yrs) Develop
diagnostics for v(r) Perform flow perturbation
and merging experiments Evaluate changes in
magnetic and kinetic helicity 3. 3D MHD
computation in SSX geometry with hybrid code
(near term) 4. Evaluate 2-fluid relaxation theory
for lab (near term) 5. Assess relevance of theory
for astrophysical cases
41
Momentum Transport Physics and Plans
1. Momentum transport by stochastic magnetic
fields 2. Momentum transport by Maxwell stress
from current-driven instabilities 3. Momentum
transport by Maxwell stress from
magnetorotational instability 4. Generation
and relaxation of momentum as part of a 2-fluid
form of magnetic relaxation 5. Momentum
transport in the sun
42
Plans Momentum Transport in the Sun
Note Work to be done in conjunction with work
on the solar dynamo problem
1. Develop incompressible/anelastic MHD spectral
element code ( 2 yrs) 2. Develop
sub-grid-scale models and compare to direct
numerical simulation ( 2 yrs) 3. Incorporate
sub-grid-scale models into spectral element
code ( 3 yrs) 4. Investigate physics of
integrated solar dynamo model ( 4 yrs)
43
Observations and Opportunites in Momentum
Transport
1. Opportunities for lab - astro coupling Coronal
MRI simulation - good start, waiting for
postdoc Liquid Gallium experiment - good
start MRI calculation for lab - will begin
soon Astrophysical jet ? lab connection - need
more effort Astrophysical applications for
stochastic B transport - need more 2.
Experimental and computational components are
strong Would benefit from increased theory effort
for several topics