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Dynamo Effects in Laboratory Plasmas S.C. Prager University of Wisconsin October, 2003

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The lab plasma dynamo does Generate current locally Increase toroidal magnetic flux Conserve magnetic helicity Act through alpha and other effects Arise from fluctuations superposed on the mean field Achieve a nonlinearly saturated steady state (with full backreaction) The lab plasma dynamo does NOT Generate magnetic field from a small seed field Increase magnetic energy (it redistributes magnetic field)

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The toroidal magnetic field is measured by the safety factor q weak field, large fluctuations self-organized strong field, small fluctuations externally controlled Dynamo and self-organization occurs in laboratory plasmas with weak toroidal magnetic field

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Examples: reversed field pinch (RFP) spheromak The RFP: toroidal plasma with helical magnetic field apply toroidal electric field E T --> j T --> B P --> J P

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The RFP Today, approximate as cylinder

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The MST Experiment (Madison Symmetric Torus) T ~ 1 keV; n ~10 13 cm -3 ; I ~ 0.5 MA, S ~ 10 6

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The Spheromak a compact torus

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Outline Evidence for field generation The standard MHD model The backreaction Measurements of the MHD dynamo Dynamo effects beyond MHD (measurements) Open issues and relation to astrophysics

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Evidence of field generation Cowlings Theorem Toroidal flux generation Ohms law

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Cowlings theorem applied to the RFP A time-independent, cylindrically symmetric plasma cannot contain a reversed magnetic field Proof: assume B z is reversed. at the radius where B z = 0 Thus, magnetic flux decays within reversal surface, in constrast to experiment BzBz r

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in experiment E || j || radius additional current drive mechanism (dynamo)

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The Standard MHD model Mean field ohms law dynamo effect For high conductivity, Lab: from tearing instability (reconnection) Astrophysics: from convection, rotation…

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The nonlinear dynamo energy source instability dynamo Quasilinear theory: current diffusion Nonlinear MHD computation: a complete description (Bhattacharjee, Hamieri; Strauss;Boozer…..)

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yields a collection of spatial Fourier modes (~R/a) z r Flow vectors In poloidal plane: 2 counter-rotating vortices, in toroidal plane: more complicated magnetic field: stochastic

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Nonlinear MHD Computation radius

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The Lab Dynamo and the Backreaction The lab dynamo is strong, with the backreaction, self-induced Compare with backreaction theories predicting dynamo suppression (Cattaneo/Vainshtein, Kulsrud/Hahm, Gruzinov/Diamond, Bhattacharjee) From Pouquet et al., for isotropic, homogenous turbulence backreaction

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Combining two equations, large resistivity -suppression with small resistivity No obvious suppression, laboratory regime, Astrophysical regime???

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Measurements of MHD dynamo Measure each term in Ohms law In the hot core passive spectroscopy, active spectroscopy (under development) (charge exchange recombination spectroscopy) (den Hartog, Craig, Ennis) Laser Faraday rotation (Ding, Brower, UCLA) Motional Stark effect (Craig, den Hartog, under development In the cool edge Insertable probes: magnetic, Langmuir (E), spectroscopic

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Active Spectroscopy

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3-Wave Polarimeter-Interferometer System MST R 0 = 1.50 m a = 0.52 m I p = 400 kA n e ~ 10 19 m -3 B 0 ~ 4 kG Faraday rotation/interferometer system

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Spectroscopic probe

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Measure quantities during discrete dynamo event Toroidal Magnetic Flux (Wb) MST time (ms)

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Flow velocity fluctuations

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time (ms) r/a = 0.9 MHD dynamo dominant at some radii, not everywhere r/a = 0.8 Measurement of MHD dynamo 0 -10 -20 0 -10 Volts m Volts m -0.500.5 time (ms) r/a = 0.9 r/a = 0.8

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Dynamo Effects Beyond MHD Hall dynamo Diamagnetic dynamo Kinetic dynamo (current transport)

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Hall dynamo: a two-fluid effect MHD dynamo Hall dynamo Two fluid effects also alter the dynamo

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Quasilinear Theory of Hall Dynamo Three layer analysis Ideal MHD v e ~ v i Ideal two-fluid v e ~ v i distance from reconnection layer 0 d R, d e s Resistive two- fluid v i ~ 0 V. Mirnov

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For experimental parameters distance from resonant surface dede s

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Faraday rotation angle time (ms) 2426 Magnetic fluctuations Measuring fluctuations with Faraday rotation

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Time Evolution of Current Density Fluctuation

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m = 1, n = 6 The reconnection current sheet

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Hall Dynamo Measurements W. Ding et al

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Hall dynamo localized in radius

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The diamagnetic dynamo parallel component of mean-fields, or, writing yields MHD dynamo diamagnetic dynamo

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Measurement of diamagnetic dynamo Ji et al TPE-1RM20 RFP Different dynamo mechanisms dominate in different parameter regimes

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Kinetic Dynamo Radial transport of parallel current (electron momentum) by particle motion along stochastic magnetic field Can show, radial flux of parallel current ~ not yet measured

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Open questions ( and relation to astrophysics) Nonlinear aspects of MHD dynamo Is nonlinear physics of growing field similar to that of steady state dynamo Does strong dynamo effect in lab have implication for astrophysical dynamo saturation? What is the role of reconnection in astrophysical dynamos? Does current (magnetic field) transport play a role in astrophysics? What is role of nonlinear coupling in altering wave functions near reconnection surface?(Need a nonlinear theory)

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Non-MHD effects What are the relative contributions of the various mechanisms? Dependence on parameters? Does the detailed mechanism matter? Are non-MHD mechanisms active in astrophysics?

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