Magnetic Self-Organization in the RFP

Magnetic Self-Organization in the RFP Prof. John Sarff University of Wisconsin-Madison Joint ICTP-IAEA College on Plasma Physics ICTP, Trieste, Italy Nov 7-18, 2016

The RFP plasma exhibits a fascinating set of magnetic selforganization phenomena Magnetic Relaxation Event Cycle Magnetic Reconnection (Tearing Instability) V,ion (km/s) Dynamo Parallel Momentum Two-Fluid Momentum Relaxation (current and ion flow) Magnetic Turbulence & Transport 1.2 0.6 Ions Electrons Non-Collisional Particle Energization

Magnetic self-organization in natural plasmas Momentum Transport Ion Heating in the Solar Corona Sun Hydrogen Oxygen Solar/Geo Dynamo Cranmer et al., ApJ, 511, 481 (1998) Kuang & Bloxham, Nature, 97

The MST RFP at UW-Madison Magnetic induction is used to drive a large current in the plasma Plasma current, I p < 0.6 MA ; B < 0.5 T Externally applied inductive ohmic heating is 5-10 MW (input to electrons) T i ~ T e < 2 kev, despite weak i-e collisional coupling (n ~ 10 19 m 3 ) Minor radius, a = 50 cm ; ion gyroradius, r i 1 cm ; c/w pi 10 cm b < 25% ; Lundquist number S = 5 10 5-6

Reversed BT forms with sufficiently large plasma current, and persists as long as induction is maintained V pol (V) B tor (G) V tor (V) 80 40 0 1000 500 0-500 200 100 0 B tor B tor (a) Poloidal loop voltage Toroidal field Toroidal loop voltage I p (ka) 400 200 0-20 0 20 40 60 80 100 Time (ms) Toroidal plasma current

However, a reversed-bt should not be an equilibrium

An imbalance in Ohm s law yields a similar conclusion E + V B = J Ohm s law: and E = J V? = E B/B 2 There is less current in the core than could be driven by E, and more current in the edge than should be driven by E current profile is flatter than it should be 2.0 1.5 E 1.0 0.5 0 hj 0.5 0 0.2 0.4 0.6 0.8 1 r/a

The RFP as a minimum energy configuration Minimize magnetic energy, with constrained global magnetic helicity yields r B = B constant (J.B. Taylor, 1974) Solution in a cylinder: Bessel Function Model B z (r) =B 0 J 0 ( r) B (r) =B 0 J 1 ( r) B z a =2.8 J 0 ( a) < 0 for a>2.4 B q resembles an RFP equilibrium r/a

Current profile exhibits a cycle of slow peaking followed by an abrupt flattening during impulsive relaxation events discrete dynamo cycle magnetic relaxation cycles I φ B φ MST 5 ms B φ (a) Time before after λ = J / B peaking from: (1) E B maximum on axis (2) current diffusion to hot core radius, r Tendency toward a Taylor state, but not fully relaxed a

Relaxation cycles result from quasi-periodic impulsive magnetic reconnection events (a.k.a. sawteeth) Toroidicity allows distinct k = 0 resonant modes at many radii in the plasma: 0 = k B = m r B θ + n R B φ m = poloidal mode number n = toroidal mode number ~ 0.2 q(r) rb φ RB θ 1,6 m=1, n 6 resonances 1,7 1,8 m=0, all n conducting shell Linear Instability Nonlinear Excitation 0 minor radius, r a Minor Radius, r multiple magnetic islands

A dynamo-like emf arrests the peaking tendency of the current profile, i.e., this is how tearing instability saturates in the RFP With non-axisymmetric quantities, (i.e., tearing instability): B = B + B ì toroidal surface average ë spatial fluctuation B ~ b (r)e i(mθ nφ) B B <<1 Then mean-field parallel Ohm s law becomes: E η J =! V! B dynamo-like emf from tearing instability Correlated product of fluctuations " represents nonlinear saturation at equilibrium magnitude

Nonlinear, resistive MHD provides a base model for the origin of the dynamo E = ηj S V B ρ V t = S ρv V + SJ B+ P m 2 V S = τ R τ A = Lundquist number P m = ν /η = Magnetic Prandtl number Ohm s Law ηj E V B nonlinear dynamo from tearing fluctuations S = 6 10 3 Dynamo emf maintains the current profile close to marginal stability. V, B = fluctuations associated with tearing modes

Plasma (ion) flow also affected during relaxation events Implies coupled electron and ion momentum relaxation B (G) Tearing fluctuation V φ, ω /k φ (km/s) Plasma flow and mode rotation Typical relaxation event

Profile of the parallel flow also flattens during relaxation events (km/s) Time (ms)

Computational model for tearing-relaxation recently extended to include two-fluid effects Nonlinear multi-mode evolution solved using NIMROD Ohm s law: Momentum: E = V B + 1 ne J B 1 ne p e +ηj + m e ne 2 J t nm i dv dt = J B p Π gyro νnm i W Relaxation process couples electron and ion momentum balance

Generalized Ohm s law permits several possible mechanisms for dynamo action The MHD and Hall mechanisms are measured to be significant, summing together in a way that has not been completely diagnosed MHD Hall ~ Diamagnetic ( p e ) There s also a kinetic dynamo, i.e., stochastic transport of current

Probe measurements in the edge region show that both the MHD and Hall dynamo emf terms are important 20 10 j b ne v b V/m 0 q = 0-10 0.75 0.80 0.85 0.90 0.95 r/a

Measurements of the total dynamo emf show a balance in Ohm s law 85 re negligible. The dynamo is multiplied by 1 and plotted with E J in Fig.! B! E! B! /B E η J = V e.7. A balance in Ohm s law is observed to within a standard deviation whenever 77 he error bars of the two measurements overlap. Therefore, it is observed that Ohm s Ep wall Et wall aw balances to within one standard deviation over the entire sawtooth cycle. The 2 tself. The dynamo and Ek -4 k Jk far from the sawtooth are similar to the periods 0.6ms < t < -6 0.3ms and 0.2ms < t < 0.6ms. -8-10 -0.6-1.8-2.0-2.2-2.4-0.6 Measured Balance ofohm s Parallel Ohm s Law Comparison -0.4 of -0.2-0.0Law 0.2Terms 0.4 0.6 15-1.6 Time From Sawtooth (ms) 5 0.000-0.002 0-0.004! B! -0.008 E magnetic relaxation. -0.010-0.012 B -0.2outer -0.0 boron nitride mo -0.6 In MST, -0.4 probe 0.2 shields 0.4show substantially 0.6-5 -0.006-10 -0.4-0.2-0.0 0.2 0.4 Time From Sawtooth (ms) 0.6-15 -0.6 (V/m) 6 4 2-0.2-0.0 0.2 0.4-2 Time From Sawtooth (ms) -0.6-0.4-0.2-0.0 0.2 0.4 Time From Sawtooth (ms) - -0.4-0.2 0.2 Time From-0.0 Sawtooth (ms) 0.4 Time (ms) side onfrom whichsawtooth the fast electrons are 0.6 incident. During <Ep~ Bp~> /B E_parallel 10 8-0.40 - - wear on the Correction to E_parallel 0-5 -0.6 5 E - eta*j (Tesla) (V/m) -0.076-0.078-0.080-0.082-0.084-0.086-0.088-0.090-0.6 (V/m) (V/m) (V/m) 10 (Tesla) Total Dynamo Bp wall (blue) and Bp- probe (red) -0.4-0.2-0.0 0.2 0.4 0.6 Time From Sawtooth Figure 4.6: (left) Cartoon of the(ms) Dynamo Probe Bdot coil and Total Dynamo locations. The top four measurements are centered 1.5 cm from Bt wall (blue) and Bt probe (red) of each stalk. The second set is centered 2 cm below the top se Cross section view of a capacitive probe in one of the stalks of th Probe. The grey circle shows the capacitor electrode surround boron nitride dielectric shield. 0.6 0.6 Figure 5.7: The dynamo, E, and J are compared to see if parallel Ohm s law is a good model for the edge of the plasma (neglecting other Ohm s law 1 MST, the Dynamo Probe is rotated 90 degrees every 20 shots in 10 0 8 distribute this wear between the four stalks of the probe and extend 6-1 four boron nitride shields. It is also good to distribute this wear beca 4-2 the capacitor measurements increase as the thickness of the boron nitr 2 E 0 away. -3-2 -0.6-4 -0.6 4.2.2 (V/m) (V/m) -2 (V/m) total dynamo emf awtooth cycle is 3ms long but only 1.2ms is plotted to focus on the sawtooth (V/m) 0-1.2-1.4 - - -0.4-0.2-0.0 0.2 0.4-0.4 Time -0.2 0.2 Helicity Probe From-0.0 Sawtooth (ms) 0.4 0.6-1 0.6 Time From Sawtooth (ms) The Helicity Probe contains a Bdot triplet as well as a triple L

The Reynolds stress bursts in opposition to Hall emf, which is the Maxwell stress in parallel momentum balance N/m 3 Probe measurements r/a=0.85 (edge region)

Relaxation events similar to those in MST are seen in NIMROD extended MHD simulations Toroidal field reversal parameter use right axes à Figure 4.10: Field reversal parameter and magnetic and kinetic energy in dominant mode

NIMROD simulations reveal the same tendencies as observed in MST plasmas

NIMROD simulations motivated probe measurements of the Hall dynamo over a larger portion of the plasma A deep-insertion capacitive probe for the total dynamo is in development Deep insertion magnetic probe 20 0 à R V/m 20 40 E J <j b>/n o e 60 simultaneous measurements 0.6 0.7 0.8 0.9 1.0 Normalized Radius

The measurements in MST are qualitatively similar to NIMROD predictions

Relaxation of parallel flow is also in good qualitative agreement

Magnetic self-organization creates the possibility to sustain a steady-state fusion plasma using induction Magnetic helicity balance motivated by success of Taylor relaxation Conventional induction maintains helicity balance with constant V f & F Oscillating field current drive (OFCD) generates DC helicity injection using purely AC loop voltages apply oscillating V f & F : &

Energy balance with relaxed current profile for modeling OFCD @ @t Z 1 B 2 dv = I ' V ' + I V 2µ 0 {z } Z Ç prescribe AC loop voltages in global power balance J 2 dv Simulated OFCD Evolve 1D equilibrium: fixed shape (marginal tearing-stable)

OFCD on MST produces 10% increase in plasma current, as much as expected V tor V loop V dc V pol OFCD On t (ms) OFCD current drive efficiency measured the same as for steady induction ( 0.1 A/W)

Tearing instability at the global scale drives a cascade to gyroscale turbulence 10 4 Power Spectrum of Magnetic Fluctuations P( f ) (T 2 /Hz) 10 8 tearing instability 10 12 1 10 100 1000 Frequency (khz) w ci / 2π

The cascade is anisotropic and hints at a non-classical dissipation mechanism The k spectrum is well-fit by a dissipative cascade model (P. Terry, PoP 2009) Onset of exponential decay occurs at a smaller k than expected for classical dissipation 10 4 Wavenumber Power Spectrum B 2 (k) (T 2 -cm) 10 6 10 8 10 10 10 12 Tearing Range k 5.4 10 14 0 0.2 0.4 0.6 0.8 1.0 1.2 k (cm 1 )

Powerful ion energization occurs during the impulsive magnetic reconnection events Instantaneous heating rate can be as large as 10 MeV/s (50 MW!) (large-scale B) Time (ms) Relative to Reconnection Event

Heating is anisotropic and species dependent MST is equipped with several ion temperature diagnostics: Rutherford scattering for majority ion temperature Charge-exchange recombination spectroscopy (CHERS) for minority ions Neutral particle energy analyzers (energetic neutral loss from plasma) Minority Ions Hotter than Majority Ions 1.5 Heating is Anisotropic 1.0 x 10 cm C +6 T (kev) 1.0 0.5 T Time (ms) T 0.0-2 - -1 0 1 2 time Time (ms)

Heating depends on mass and charge Majority Ions Minority Ions ΔE ion He ++ ΔE mag H + D+ (varied fueling gas) Z/µ

An energetic ion tail is generated and reinforced at each reconnection event Distribution is well-fit by a Maxwellian plus a power-law tail Reminiscent of power laws observed for astrophysical energetic particles f D+ (E) = A e E/kT + B E γ Before Event After Event

Proposed Ion Heating Mechanisms

Existing models for ion heating in the RFP are based on distinct mechanisms Cyclotron-resonant heating: Feeds off the turbulent cascade to gyro-scale Preferential perpendicular heating, but with collisional relaxation!b 2 (ω ci ) Preferential minority ion heating, since is larger where w ci is smaller Mass scaling is predicted with dominant minority heating and collisional relaxation T α (ev) 400 200 (b) Fraction= 0.5 0 30 32 34 36 38 40 (c) Fraction= 1 Time (ms) T [D] T [C] T [D] T [C] Tangri et al., PoP 15 (2008) (similar to Cranmer et al)

Existing models for ion heating in the RFP are based on distinct mechanisms Stochastic heating: Feeds off large electrostatic electric field fluctuations and the distinct stochastic magnetic diffusion process Monte Carlo modeling yields MST-like heating rates (Fiksel et al, PRL 2009) Predicts mass scaling close to that observed (V/m) 1000 800 600 400 RMS!E r 200-0.6-0.4-0.2-0.0 0.2 0.4 0.6 Time From Sawtooth (ms) J/m 3 -Hz 10 2 10-2 10-6 Non-Alfvenic Cascade!B 2 / 2µ 0 1 2 m i n i! V 2!E B 0 10 3 10 4 10 5 10 6 10 7 Frequency (Hz) Emerging story: measurements not shown here suggest the electrostatic fluctuations for f 100 khz are drift waves excited in the turbulent cascade Importance of non-uniformity and gradients at the system scale and coupling of different types of modes/waves

Existing models for ion heating in the RFP are based on distinct mechanisms Viscous heating: No clear experimental evidence for the required large sheared flow Perpendicular flow is dominant for tearing modes for which the classical viscosity is small A reliable dissipation mechanism, but difficult to achieve the large heating rates seen in MST plasmas See, e.g., Svidzinski et al, PoP 15 (2009)