MHD Mode Identification by Higher Order Singular Value Decomposition in C-2W

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Abstract
TAE Technologies RF code, RF-Pisa, has been upgraded to study the transfer of momentum and angular momentum from RF waves to plasma species in an FRC configuration.
Thanks to this upgrade RF_Pisa has achieved the capability to fully address the efficiency of the wave-plasma coupling in standard configurations as well as in advanced plasma schemes which allow to control the wave polarization and select narrow resonance regions.
The frequency range of interest is mostly the ICRH regime but the code has the capability to address different regimes also.
We present here the new tools through the equations that have been implemented and through examples of single mode analyses and 3D reconstructions. The latter ones are strictly necessary to study (i) the effects due to realistic and finite size antennas and (ii) the role of the straps phase in defining the resonance volumes and efficiency.
RF-Pisa Code
The cylindrical FLR plasma wave equations are obtained including the density current J computed for the different FLR orders and different plasma species1 (CGS units used throughout the poster) .
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Angular Momentum Conservation Law
A conservation law for angular momentum density can be derived from the time derivative of the mechanical angular momentum density2,3
Momentum Conservation Law
n Similarly to the angular momentum case, the conservation law of the momentum density can be derived from the time derivatives of the mechanical momentum density and of the electromagnetic momentum density
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3D Results: Reconstructions
Differentmodesexhibitverydifferent1Dprofiles.Herebelowprofilesofplasma current density induced by different modes.
RF Transfer of Momentum and Angular Momentum in FRC plasmas
Francesco Ceccherini, Laura Galeotti, Dan Barnes, Sean Dettrick, Kevin Hubbard, Xiaokang Yang and the TAE team TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610
Mode mθ=1 nz=1
Mode mθ=1 nz=2
Mode mθ=2 nz=1
Mode mθ=3 nz=3
total current density
total charge density
total magnetic field and electric field of the system
using Maxwell’s equations
Cylindrical coordinates
Substituting in this equation the appropriate Maxwell’s equations, with the help of the time derivative of the electromagnetic angular momentum density,
an integral and a differential formulation of the conservation law for angular momentum density can be derived2,3, namely
Pseudo tensor of rank 2
Maxwell’s stress tensor
As we are interested in the local transfer of angular momentum density from waves to plasma we will use the differential formulation for our studies and utilize the time-averaged conservation law which is given by
Maxwell’s stress tensor
The 3D profile of each quantity strongly depend on the the antenna configuration. Here below we show we show the induced electron current Jz (statAmp/cm2) and the ion angular momentum Lz for different phases of a four strap antenna.
n Averaging over time we obtain
where
statAmp/cm2
g cm2/s2
statAmp/cm2
FLR approximation
n Using quasi neutrality the previous expression reads in cylindrical components as
where
Cold plasma
Ion FLR current
Landau FLR current
Electron FLR current
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1D Results: Single Modes
g cm2/s2
Conclusions and Future Work
Because the time dependence is of type
average of ⟨Lfield⟩ is computed we obtain a quantity with no explicit time dependence and therefore 𝜕⟨Lfield⟩/𝜕t = 0. Hence the conservation law for angular momentum density reduces to
Cylindrical coordinates
where
The system we consider is quasi neutral and we do not take into account any density feedback from the plasma. Therefore the previous equation in cylindrical components becomes
plasma
exp(−iωt) form, when the time
A fast wave with a frequency of 1.05Mhz is excited by the antenna
Plasma density profile
1st harmonic resonance
2nd harmonic resonance
rwall r
Up to 1e4 modes are usually computed. Example of single mode result:
Mode mθ=0 nz=1
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Perturbation fields are FFT decomposed as
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RF-pisa has been upgraded to include momentum and angular momentum transfer.
Reconstruction tools have been also upgraded making Tae’s RF codes a comprehensive and versatile suite to possibly address RF coupling and current drive4 in FRC configurations.
Planned applications include:
n Benchmark against measuments of RF wave coupling and propagation5 on LAPD
n Compare with results from Petra-M full wave code (2019 INFUSE program award)
n RF wave field profile and antenna loading for Norman (2020 INFUSE program award)
References
The problem is reduced to the solution of a system of complex equations The size of the system scales as six times the number of gridpoints
Equations are solved for each single partial wave with wave numbers kθ =m/r and kz
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1M.Brambilla, Plasma Phys. Control. Fusion 31, 723 (1989)
2J.D. Jackson, Classical Elctrodynamics, Wiley, 1998
3J.R. Myra, D.A. D’Ippolito, L.A. Berry, E.F. Jaeger and D.B. Batchelor, 15° Topical Conf. on Radio Frequency Powwer in Plasmas (2003) 4A.G. Livtak (Editor), High-Frequency Plasma Heating, AIP New York, 1992
5X.Yang et al., “Overview of TAE Technologies’ HHFW project on LAPD”, 23° Topical Conf. on Radio Frequency Power in Plasmas (RFPPC 2019)
wall antenna