An Interesting Poster to look at from the Tri Alpha Energy Team in California

FRC as a Two-Fluid Flowing Relaxed State
B. S. Nicks, L. Steinhauer, and the TAE Team
TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610
Central Question and Introduction Relaxed FRC-like State with Prescribed Density Profile
Contribution to the Central Question
Central Question: Can FRCs (field-reversed configurations) be shown to be a preferred state of plasma under certain conditions?
Background: Taylor’s conjecture
 Real plasma might spontaneously find a preferred state1
 Global magnetic energy is minimized while magnetic
helicity constrained.
 Lagrangian minimization procedure gives Euler- Lagrange equations for relaxed state. For Taylor case, procedure results in force-free states
 “Relaxed” states burst upon the plasma world most prominently in the form of spheromaks and reversed- field pinches (RFP).
Our contribution: Generalized relaxation theory
 Multi-fluid theory: species allowed substantial (Alfvénic) flow2,3. Allows non-force-free states, finite-beta plasmas
 Flow is extremely important to FRC modeling
 Minimized quantity is now magneto-fluid energy 𝑊
while fluid species helicity and angular momentum are constrained (see Appendix)
 Lagrange multipliers corresponding to species helicities and angular momentum determine relaxed state
 Determining relaxed state requires solving three coupled PDEs. Analytical treatment available only for uniform density.
 Initial efforts used uniform density and imposed separatrix, resulting in spheromak-like states
 Current status: impose density model and solve for fields and flows. Adjust Lagrange parameters to find FRC-like state
𝑓𝑚
Separatrix
 2D (𝑟𝑧) cylindrical grid, mirror-like Dirichlet conditions for 𝑟, Neumann for 𝑧. Only axisymmetric solutions thus allowed
 Toroidal field is small compared to poloidal field except at null point and at large radius. Field at null point caused by strong poloidal current in core. Viscosity may prevent this from occurring in reality.
 Prescribed density profile: elongated Hills vortex inside presumed separatrix, uniform outside
Boat wakefield
 Lagrangian parameters chosen to demonstrate FRC-like solution for model density profile
 Relaxed state with assumed density shows region of closed flux (magnetic reversal) and nearly on-axis X points
 Magnetic reversal in core caused by strong toroidal current inside separatrix
 Plasma has net species helicity; ideally net helicity should be near zero
 Initial results are encouraging: FRC-like state shown to be preferred with imposed density profile.
 Relaxed states with flow show reversed core and nearly on-axis X points, but 𝐵 warrants closer examination
 Flow and density profiles can be compared with or drawn from experiment
𝜃
Appendix of Equations
 Each species 𝛼 has drift surfaces 𝑌 = 𝜓 + 𝑚 𝜙 Τ𝑞 𝑛 𝛼𝛼𝛼𝛼 𝛼
 Poloidal flux 𝜓 and toroidal flow 𝜙𝛼 Magneto-fluidenergy:𝑊 =1𝐵2 ׬ +σ 𝑚 𝑛 𝑢2 𝑑3𝑟
Ԧ Canonical momentum: 𝑃 = 𝑚 𝑢 +𝑞 𝐴
2 𝑞2
𝛼𝛼𝛼𝛼 𝑚𝑓2𝑉𝜇0
𝛼𝛼𝛼𝛼
𝑟𝑑3 𝑃×𝛻 ⋅ 𝑃 ׬= 𝐾:Specieshelicities 𝛼𝛼𝑉𝛼
𝑟𝑟𝑚𝛼𝑛𝛼𝑢𝜃𝛼𝑑3 ׬ = 𝜃𝐿 : Angular momentum 𝑉
 Minimization 𝑊 and hold 𝐾 and 𝐿 constant 𝜃 𝛼 𝑓𝑚
(Lagrange multipliers 𝜆𝛼 and Ω)
 Total enthalpy and entropy constant and 𝑚𝑒 = 0
 Euler-Lagrange and Bernoulli system for 𝜓, 𝑌 , 𝑛: 𝑖
∆∗𝜓 + 𝜆2 − 𝑙−2 𝑛 𝜓 + 𝜆 𝜆 + 𝑙−2 𝑛 𝑌 = 𝜇 𝑞 𝑛Ω𝑟2, 𝑒𝑖𝑖𝑒𝑖𝑖00
𝑙2𝜆2∆∗𝑌 = 𝜆2 −𝑙−2 𝑛 𝑌 +𝑙2𝜆 𝛻𝑛⋅𝛻𝑌 +𝜆 𝜆 𝜓+𝜇 𝑞 𝑛Ω𝑟2, 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑛𝑖𝑖𝑒00
𝐶′ 𝑛𝛾−1+ 0 𝜆2𝑙4 𝑛 𝛻𝑌 + 𝑌−𝜓2 −𝑞Ω𝑌−𝜓 =𝑐𝑜𝑛𝑠𝑡 𝑖0 𝑖𝑖𝑖𝑖𝑒2𝑚𝑖𝑟2+𝑖
Simpler Case: Uniform Density
Current Efforts
 Uniform density  𝜓 → 0 at all
boundaries
 Result is spheromak- like state, no separatrix
 Can be useful for modeling FRC core
0.100 0.075
0.050 0.025
0.000
0.75 0.50 0.25 0.00 0.25- 0.50- 0.75-
 Full PDE system must be solved simultaneously to provide self- consistent density without imposition of model
 Machine-learning PDE methods may allow for solution of full system
 Once full system solved, Lagrange parameters can be scanned to find true minimum magneto-fluid energy states
 Extension to 3D to allow for non-axisymmetric solutions (such as possible tilt modes)
References
1. J. B. Taylor, Phys. Rev. Lett. 33, 1139 (1974)
2. L.C. Steinhauer, Phys. Plasmas 6, 2734 (1999).
3. L.C. Steinhauer and A. Ishida, Phys. Plasmas 5, 2609 (1998).
𝜓 (Normalized) 𝐵 (Normalized) 𝑧