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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. Craig General Meeting of the Center for Magnetic Self-Organization In Laboratory and Astrophysical Plasmas August 4-6, 2004 in Madison, WI

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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

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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 disk+jet (Hubble Space Telescope) Accretion Disks

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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 Astrophysical Jets

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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

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MST (Wisc) Experiment and Tools R = 1.5 m a = 0.52 m B ~ 0.2 T n ~ m -3 T e,i ~ 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)

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v toroidal r v max ~ 30 km/s v poloidal r v max ~ 10 km/s Helical Flows Are Naturally Present in MST Plasmas In core, v mostly parallel to B In edge, have v parallel and v perp Origin of flows unclear (sketches based on incomplete flow profile measurements)

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Plasma Momentum Changes Spontaneously in MST with Bursts of Magnetic Activity Plasma rotation slows in ~ 100 s Not classical times too fast - n, T,... do not change enough on this timescale Leading explanation involves coupled magnetic fluctutaions v toroidal (km/s)

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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) n = 1-20 x m -3 T = 4-30 eV B = 0.05 T =

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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:

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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 Toroidal Angle / r/a Puncture Plot of B Field in MST

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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 (~ 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)

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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

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Charge Exchange Recombination Spectroscopy (CHERS): Basic principles (1) Charge exchange We observe this! (2) Radiative decay

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Wavelength (Å) Signal level (photons) Doppler shift ( ) gives v impurity Doppler width ( ) gives T impurity Area gives n impurity Measure Doppler shifted and broadened line emission profile Need accurate model for profile shape Need accurate technique for data fitting CHERS: Profile measurement 1000

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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 T i, v i (and possibly n i ) 30 keV H beam MST vessel Fiber bundle views of beam and background Perpendicular viewing chords Beam current monitor

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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 T i, v i to be resolved on fast time scale (~ 100 s) Atomic modeling & initial fitting of CVI line shape has been done time (ms) T i (eV) Beam ON Beam off

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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

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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 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 toroidal direction radius Puncture plot for single mode

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Fluctuating B can make net force, - Can rewrite as (B k B k ) magnetic analog of (v k v k ) 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 phases of modes Magnetic Maxwell Stress From Nonlinearly Coupled Tearing Modes coupling coefficient

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Coupled Tearing Modes Produce Strong Torques in MST 2 Maxwell stress in core estimated from edge measurements of B Mode amplitude and coupling increase during relaxation events Strong forces result

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Plans: Momentum Transport by Maxwell Stresses from Tearing Modes 1. Measure directly in MST (~ 1-2 yrs) 2. Calculate directly in MHD computation (~ 1 yr) 3. Drive flows in MST, vary fluctuations, measure momentum transport Requires electrically biased probes and/or neutral beams (~ 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)

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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)

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Maxwell Stress in MHD Computation Will examine from tearing fluctuations and v(r) evolution First numerical runs now underway

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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

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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 (?) Demonstrated in simulation, not yet in lab Top view along rotation axis Side view in poloidal plane Magnetorotational Instability (MRI)

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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

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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)

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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

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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).

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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

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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!)

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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

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Initial Conditions

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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

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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/B 2 profile 2-fluid relaxation Generalized helicity for each species ( A s B s dV) is conserved where A s = A + (m s /q s ) v s and B s = A s Relax to minimum magnetic + flow energy (via v B and J B) Constant J B/B 2 and nv B/B 2 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

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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

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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

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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

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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

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