Outline:

I. Introduction, background, and examples of momentum transport

II. Momentum transport physics topics being addressed by CMSO

- Physics, Plans, and Progress

Momentum Transport

D. CraigGeneral Meeting of the Center for Magnetic Self-Organization

In Laboratory and Astrophysical Plasmas

August 4-6, 2004 in Madison, WI

Why Study Momentum Transport?• Momentum transport is an important issue in:

Accretion DisksAstrophysical JetsSolar InteriorLaboratory 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

• 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

GMRMRTorque )(

Protostellar disk+jet (Hubble Space Telescope)

Accretion Disks

• 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 jetin galaxy M87(NASA/HST)

Cartoon ofmagneticallycollimated jet

Astrophysical Jets

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

MST (Wisc) Experiment and ToolsR = 1.5 ma = 0.52 mB ~ 0.2 T

n ~ 1019 m-3

Te,i ~ 0.1-1 keV ~ 10 %

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)

v toro

idal

r

vmax ~ 30 km/s

v polo

idal

r

vmax ~ 10 km/s

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 incompleteflow profile measurements)

Plasma Momentum Changes Spontaneously in MST with Bursts of Magnetic Activity

m=1,n=6m=1,n=7

m=0,n=1

20

30

40

10 15 2050

50B (

Gau

ss) 150

100

0

20B (

Gau

ss)

40

0

10v

(k

m/s

)n=

6

25 30Time (ms)

sawteethcore modes

edge mode

~~

• Plasma rotation slows in ~ 100 s

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

v toro

idal (

km/s

)

z

R

• Two kinds of flows:

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

Spontaneous Flows Also Measured in MRX

Toroidal (out of reconnection plane) flows

null helicity

separatrix

co-helicity(guiding B)

separatrix

v (

km/s

)

v (

km/s

)n = 1-20 x 1019 m-3

T = 4-30 eV

B = 0.05 T

= 0.1-10

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

We have chosen to focus our efforts on 5 physics topics:

Transport by Stochastic Magnetic Fields

• Mechanism:B field lines wander in spaceParticles 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

Tor

oida

l Ang

le /

r/a

Puncture Plot of B Field in MST

Plans: Momentum Transport in Stochastic B

1. Measure in MST, a direct measure of this effectRequires diagnostic development (~ 1-2 yrs)

2. Drive flows in MST, vary fluctuations, measure momentum transportRequires electrically biased probes and/or neutral beams (~ 0.5-1 yr)

3. Measure mean flow profile and its evolution in MSTRequires diagnostic development (~ 1-2 yrs)

4. Drive flows in MRX and measure momentum transportRequires 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)

˜ p i|| ˜ B r

Flow Perturbation Experiments

• Insertable biased probes create pulse of edge flow in MST• Core responds with some delay

global momentum transport timescale ~ 1 ms

Pulse

Time (ms)

2.5 ms

0

10

20

30

40

0 10 20 30 40

Bias

No Bias

Tor

oida

l ve

loci

ty (

km/s

)

Core ion flow (Cv)

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

Charge Exchange Recombination Spectroscopy (CHERS): Basic principles

)(AHAH 1)-(ZZ0 n,l (1) Charge exchange

hWe observe this!

hlnln ),(A),(A 1)-(Z1)-(Z

(2) Radiative decay

3431 3435Wavelength (Å)

Sign

al l e

vel (

phot

ons)

Doppler shift () gives vimpurity

Doppler width() gives Timpurity

Area givesnimpurity

Measure Doppler shifted and broadened line emission profile Need accurate model for profile shape Need accurate technique for data fitting

2000

0

CHERS: Profile measurement

1000

Beam-driven CHERS emission is localized

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)

30 keV H beam

MST vessel

Fiber bundle views of beam and background

Perpendicular viewing chords

Beam current monitor

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 s)Atomic modeling & initial fitting of CVI line shape has been done

12 14 16 18

200

400

600

800

1000

time (ms)

Ti (

eV)

Beam ONBeam off Beam off

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

Current-Driven Tearing Modes• Perturbations with k·B = 0 do not bend B field lines

Fluctuations with k·B = 0 somewhere are called “resonant”Position (surface) where k·B = 0 called “resonant surface”

• In MST, have helical B helical resonant perturbationsPitch of B field lines changes with radius

Multiple resonances throughout plasma

• Tearing ModesOne class of resonant perturbations

Driven primarily by J(r)

Tear magnetic field to form islands

• Typically see full spectrum of

tearing modes in MST toro

idal

dir

ectio

n

radius

Puncture plot for single mode

• Fluctuating B can make net force, <JkBk>- 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

k'-kk'k ~ B~B~J~

)sin(C~F k'-kk'kk'kk'kk'

k'-k,k'k,k BBB

phases ofmodes

Magnetic Maxwell Stress FromNonlinearly Coupled Tearing Modes

coupling coefficient

Coupled Tearing Modes ProduceStrong Torques in MST

<JB>

tvM

2

• Maxwell stress in core estimated from edge measurements of B

• Mode amplitude and coupling increase during relaxation events

• Strong <JB> forces result

Plans: Momentum Transport byMaxwell Stresses from Tearing Modes

1. Measure <JB> directly in MST (~ 1-2 yrs)

2. Calculate <JB> directly in MHD computation (~ 1 yr)

3. Drive flows in MST, vary fluctuations, measure momentum transportRequires 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)

Maxwell Stress in MHD Computation

• 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

(On behalf of F. Ebrahimi, by way of S. Prager)

Maxwell Stress in MHD Computation

• Will examine <J B> from tearing fluctuations and v(r) evolution

• First numerical runs now underway

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

• Believed to dominate angular momentum transport in disks

• Exists in ideal MHD for arbitrarily weak fields: >>

• 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

Topviewalongrotationaxis

Side viewin poloidalplane

Magnetorotational Instability (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 > 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.

Outstanding Issues Concerning MRI

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 experimentOperate 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 disksRequires code development (longer term)

• 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

Bz<1T

The Princeton MRI Experiment

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).

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

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!)

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

Initial Conditions

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

Parallel Momentum Relaxation• Taylor relaxation - single fluid MHD

Global helicity (AB dV) “conserved” Relax to minimum magnetic energy (via vB) Constant JB/B2 profile

• 2-fluid relaxation Generalized helicity for each species (AsBs dV) is “conserved”

where As = A + (ms/qs) vs and Bs = As

Relax to minimum magnetic + flow energy (via vB and JB)

Constant JB/B2 and nvB/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

Plans: Two Fluid Relaxation

1. Observe momentum profile relaxation in 2 fluid MHD computation

in MST geometryRequires 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

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

Plans: Momentum Transport in the Sun

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)

Note: Work to be done in conjunction with work on the solar dynamo problem

Observations and Opportunites inMomentum 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