Development of a Pulsed ~100MW Rotating Magnetic Field Ionization System for C-2W
P. 1

Abstract
n  The TriAlpha Energy (TAE) code RF-Pisa is a Finite Larmor Radius (FLR) full wave code developed through the years to study RF heating in the Field Reversed Configuration (FRC) in both the ion and electron cyclotron regimes.
n  The FLR approximation is perfectly adequate to address RF propagation and absorption at the fundamental and second harmonic frequencies (as in the minority heating scheme), but it is not able to describe higher order processes such as high-harmonic fast waves (HHFW).
n  HHFW might be effective in FRCs, and would be a useful complement to NB injection because it may allow 1) different heat deposition profile compared to NBs, and 2) heating without fueling, as suggested by recent results obtained at NSTX [1].
n  A significant upgrade of RF-Pisa based on the so-called “quasi local approximation” [2] to include HHFW has been undertaken.
n  Here we present the first results of the application of the new code to FRC equilibria of interest for the TAE project.
Analytical Model
n  In order to derive the HHFW wave equations for FRC configurations we have revisited and adapted the theory developed by M. Brambilla for the toroidal geometry [2-3]. Here below we briefly summarize the very large amount of algebraic calculations involved and we defer to the cited literature for all details and complete derivations.
n  To compute the wave electric field E(r) from the wave equation
we need first the expression for the constitutive relation, i.e. j=j(E)
n  The starting point for the computation of j(r) is the linearized Vlasov equation as
n  The integrals along the unperturbed orbits can be solved thanks to some approximations:
1)  Using:
n  the drift approximation
where
Term due to the distribution function anisotropy
Term related to the drift velocity of charged particles
HHFW local kr vs r solutions: Bernstein and fast waves
the unperturbed orbits can be written as a slow guiding center motion separated from the fast gyration as
Plasma of deuterium and electrons
Quasi-neutrality nD = ne
gyro-center position gyration vector u n i t v e c t o r p a r a l l e l t o t h e equilibrium magnetic field
Solution of multi-mode E-field components
n  the three constant of motion: energy, magnetic moment and the canonical angular momentum for the guiding center
j(E) can be written as j = jM + jA +jD +jB
1 N. Bertelli et al., Nucl. Fusion 57, 056035 (2017)
2 M. Brambilla, Plasma Phys. Control. Fusion 44, 2423 (2002)
drift velocity
3)  Assuming that the radial dependence of the field can be written in Eikonal form as
parameter representing the ratio of the wavelength to the equilibrium gradient lenghts
allows to transform the wave equation in the algebraic expression
Ion O-point T=1000eV
Electron O-point T = 500eV
n 
n  Investigation of HHFW modes power absorption characteristics n  Exploration of parameters space
3 M. Brambilla, Plasma Phys. Control. Fusion 41, 775 (1999)
4 M. Brambilla, “Kinetic Theory of Plasma Waves”, Clarendon Press, Oxford (1999)
Numerical study of HHFW heating in FRC Plasmas 
Francesco Ceccherini, Laura Galeotti, Sean Dettrick, Xiaokang Yang , and the TAE team
TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610
n  Since we are interested in an FRC where all the equilibrium quantities depend on the radial coordinate only, we can assume no coupling between modes and therefore perturbation fields can be fft decomposed as
2)  Utilizing the quasi local approximation and considering a Maxwellian
cold
HHFW
equilibrium distribution function allow to write where
plasma species
Bessel function order
where D, S, L, R are the usual cold plasma element of the dielectric tensor and σ2, λ0 ,δ2 are the the temperature correction terms as described in Ref.4
HHFW local conductivity tensor components for fast wave
Dominant term, it represents the only that survives if the distribution function is isotropic
Term due to spatial gradients, referred to as diamagnetic current
n  Therefore local wave numbers k0 of any possible mode for each radial location can be found solving
where
n  Finally the wave number of the mode is used to calculate the local current as
and inserted in the wave equation to compute the wave electric field amplitude
Code
n  In order to study the HHFW regime the original TAE’s code RF_Pisa has been branched and a major new development is in progress. A merging to a unique broad RF code is planned as a near future project.
n  From a numerical point of view one of the main challenges introduced by the HHFW regime is represented by the need to obtain the modes wave vector at each radial location through the complex roots of the det(A) equation which eventually results in a significantly tangled complex function.
n  Because the spatial width of the high harmonics absorption region can be very narrow a very fine mesh is required, usually 5000 to 10000 radial points are used. Moreover, the presence of large numbers of Bernstein waves implies the need to have at each radial location hundreds or more of initial guesses in the {Re(Kr), Im(Kr)} space to make sure that all possible modes are properly detected.
n  The previous two constraints on the required quantities for mesh points and initial guesses for the root finder make the problem not suitable for table top machines. For such a reason an MPI parallelization in the {r, Re(Kr), Im(Kr)} space. This kind of parallelization implies a very limited quantity of communication between the head node and the slave nodes and no communication at all between the slaves. Therefore the scaling of the code performance versus number of cores is extremely efficient. A typical run requires 100 to 200 cores and a few hours.
Numerical Results
Characteristics of test equilibrium
Bernstein harmonic 13th
Fast wave
2) hot plasma FLR wkb
hot-FLR
complex Bessel function of order p
Together to the electric field amplitude, the conductivity tensor is the fundamental ingredient to calculate the absorbed power through the anti-hermitian part of the dielectric tensor.
Next Steps
Perpendicular injection
___ abs(Re(kr)) ___ abs(Im(kr))
1) cold plasma wkb
Bernstein harmonic 14
th
Injection at 73 degrees
Bernstein harmonic 20th
Fast wave local kr: HHFW vs cold and hot-FLR results
HHFW
___ Re(4 πi/ω σ)
___ Im(4 πi/ω σ)


































































































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