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 self- organization phenomena Magnetic Relaxation Event Cycle Magnetic Reconnection (Tearing Instability) Dynamo Two-Fluid Momentum Relaxation (current and ion flow) V || , ion Parallel Momentum (km/s) Magnetic Turbulence & Transport 1.2 Non-Collisional Ions 0.6 Particle Energization Electrons
Magnetic self-organization in natural plasmas Momentum Transport Ion Heating in the Solar Corona Sun Oxygen Hydrogen 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 80 V pol Poloidal loop voltage 40 (V) 0 1000 〈 B tor 〉 B tor 500 Toroidal field 0 (G) B tor ( a ) -500 200 V tor Toroidal loop voltage 100 (V) 0 I p 400 Toroidal plasma current (kA) 200 0 80 100 20 40 60 -20 0 Time (ms)
However, a reversed-BT should not be an equilibrium
An imbalance in Ohm’s law yields a similar conclusion V ⊥ = E × B /B 2 ⇒ E || = η J || • Ohm’s law: E + V × B = η J and 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 E || 1.5 1.0 0.5 h J || 0 –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 = λ B (J.B. Taylor, 1974) constant Solution in a cylinder: “Bessel Function Model” B z ( r ) = B 0 J 0 ( λ r ) λ a = 2 . 8 B z B θ ( r ) = B 0 J 1 ( λ r ) B q J 0 ( λ a ) < 0 λ a > 2 . 4 for 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 Time B φ ( a ) after before Tendency toward a Taylor state, but not fully relaxed λ = J || / B peaking from: E ⋅ B (1) maximum on axis (2) current diffusion to hot core a radius, r
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 m = poloidal mode number R B φ n = toroidal mode number m=1, n ≥ 6 Linear Instability ~ 0.2 resonances conducting 1,6 q(r) rB φ 1,7 1,8 RB θ shell m=0, Nonlinear Excitation all n a minor radius , r 0 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 ( r ) e i ( m θ − n φ ) B ~ ˜ ˜ B = 〈 B 〉 + ˜ • With non-axisymmetric quantities, b B (i.e., tearing instability): ë ì ˜ B 〈 B 〉 << 1 spatial toroidal fluctuation surface average • Then mean-field parallel Ohm’s law becomes: 〈 E 〉 || − η 〈 J 〉 || = 〈 ! V × ! Correlated product of fluctuations B 〉 || � � represents nonlinear saturation at equilibrium magnitude dynamo-like emf from tearing instability
Nonlinear, resistive MHD provides a base model for the origin of the dynamo S = τ R = Lundquist number E = η J − S V × B τ A ρ∂ V m ∇ 2 V ∂ t = − S ρ V ⋅∇ V + S J × B + P P m = ν / η = Magnetic Prandtl number E || S = 6 ´ 10 3 η J || Dynamo emf maintains the Ohm ’ s Law current profile −〈 ˜ V × ˜ B 〉 || close to marginal stability. ⬆ nonlinear dynamo from tearing fluctuations ˜ , ˜ V B = fluctuations associated with tearing modes
Plasma (ion) flow also affected during relaxation events • Implies coupled electron and ion momentum relaxation Tearing ˜ B fluctuation (G) Plasma flow and V φ , ω / k φ mode rotation (km/s) 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 ∂ J ne ∇ p e + η J + m e E = − V × B + 1 ne J × B − 1 Ohm’s law: ne 2 ∂ t d V nm i dt = J × B − ∇ p − ∇⋅Π gyro − ∇⋅ ν nm i W Momentum: 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 〈 ˜ j × ˜ b 〉 || ne v × ˜ b 〈 ˜ 〉 || 10 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 〈 E 〉 || − η 〈 J 〉 || = −〈 ! V e × ! B 〉 || ≈ 〈 ! E ⋅ ! B 〉 || / B ⬆ “total” dynamo emf Measured Balance of Parallel Ohm’s Law Comparison of Ohm’s Law Terms 15 Total Dynamo - Total Dynamo 5 E - eta*J 0 10 (V/m) 〈 ! E ⋅ ! -5 B 〉 || (V/m) -10 5 B -15 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 E_parallel 0 10 8 〈 E 〉 || 6 (V/m) 4 -5 2 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0 Time From Sawtooth (ms) -2 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 Time From Sawtooth (ms)
The Reynolds stress bursts in opposition to Hall emf, which is the Maxwell stress in parallel momentum balance Probe measurements r/a=0.85 N/m 3 (edge region)
Relaxation events similar to those in MST are seen in NIMROD extended MHD simulations Toroidal field “reversal parameter” use right axes à
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 20 20 “Deep insertion” magnetic probe 0 0 V/m à R − 20 − 20 E || − � J || <j × b>/n o e − 40 − 40 60 simultaneous measurements 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1.0 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 Z Z ∂ 1 B 2 dV = I ϕ V ϕ + I θ V θ η J 2 dV } − ∂ t 2 µ 0 Simulated OFCD | {z Ç prescribe AC loop voltages in global power balance • Evolve 1D equilibrium: fixed shape (marginal tearing-stable)
OFCD on MST produces 10% increase in plasma current, as much as expected V tor V dc V loop 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 gyro- scale turbulence Power Spectrum of Magnetic Fluctuations 10 –4 tearing 10 –8 P ( f ) instability (T 2 /Hz) 10 –12 1 10 100 1000 ⬆ Frequency (kHz) ≈ w ci / 2 π
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 Wavenumber Power Spectrum 10 –4 Tearing Range 10 –6 B 2 ( k ) 10 –8 (T 2 -cm) 10 –10 k ||–5.4 10 –12 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 Heating is Anisotropic 1.5 C +6 T ⊥ 1.0 T (keV) 0.5 T || 0.0 2 -2 -1 0 1 2 -2 Time (ms) time (ms) Time (ms)
Recommend
More recommend
