Demo

Onset of Tilt Instability in FRC Plasmas
Francesco Ceccherini, Laura Galeotti, Sean Dettrick, Dan Barnes, Kevin Hubbard and the TAE team TAE Technologies, Inc., 19631 Pauling, Foothill Ranch, CA 92610
Abstract
n Tilt instability in FRC plasmas1 is a well-known MHD phenomenon that has been extensively investigated over the years by different authors and an empirical scaling law of the numerical growth rate versus the plasma E/S* parameter has been discussed in the literature.
n FPIC, TAE Technologies’ proprietary hybrid code, was also recently utilized to study the n=1 toroidal modes as tilt, radial shift and interchange2,3.
n In spite of the investigations already carried out what is the exact onset mechanism and what are the plasma structures which characterize the very early phase of the tilt instability remain largely unknown and any possible investigation requires very large computational resources.
n Thanks to the recently approved INCITE project4 it has been possible to utilize HPC capabilities and apply FPIC at unprecedented levels of resolution. Detailed plasma structures at tilt onset and subsequent times are presented and discussed.
FPIC Code
n FPIC is an hybrid 3D code which includes the generalized Ohm’s law with finite electron mass.
n The electron mass can be set to zero, yielding the usual Ohm’s law. n Faraday’s law, and Ampere’s law neglecting displacement current,
are also used in the field solver.
n A Cartesian staggered Yee mesh is used and shaped boundaries can be straightforwardly included.
n The Dey-Mittra method for cut-cell boundaries on a Yee mesh is used to model any kind of boundary5
n The full Lorentz equation for ions is used in the particle advance.
n Esirkepov’s current deposition scheme is used to obtain Ji 6
n Finite resistivity is implemented for both the FRC inner “high density” and outer “low density” regions.
n The code is MPI parallelized using PETSc and domain decomposition.
Recent upgrades
n New diagnostic tools to “measure” the FRC geometric evolution are available.
n Standard Alfven waves benchmark tests have been carried out and passed.
n Periodic boundary conditions
n Initial tilt mode perturbation
n Multi species -> possibility to add neutral beams
Velocity structures for toroidal m=1 modes
n The most relevant MHD instabilities in FRCs for the toroidal mode number n=1, i.e., tilt, radial shift, ballooning and interchange7,8,9,10 have in common the formation of poloidal velocity structures during the early phase of the instability.
n Therefore these modes can be identified during the onset phase using the symmetric and anti-symmetric parts with respect to the midplane of the velocities vr and vz 11, i.e
vr- = vr(z) – vr(-z) vr+ = vr(z) + vr(-z) vz- = vz(z) – vz(-z) vz+ = vz(z) + vz(-z)
with the help of a full Fourier decomposition.
n For our analyses we select the toroidal n = 1 mode and the axial mode m = 1 for tilt, radial shift and ballooning and m = 0 for interchange instability. Radial components have been summed up in all cases.
Poloidal velocity and ion density structures in the early phase of the different modes
Case under study
n To study possible paths to the onset of the tilt instability we have
utilized the recent upgrades introduced in FPIC as periodic
boundary conditions, domain decomposition and the new virtual
CD Ion density
Jz
Ion density
Jz
particle technique for boundary current deposition
Ion density
Toroidal VS Rotational modes
Tilt instability
delay
ABCD
Delay between the onsets of tilt instability and “tilt-induced” rotational
instability
A
B Ion density C D
Conclusions
Ion density
Jθ
A significant rotational spin-up is associated to the start of the tilt
Jθ
Internal tilt mode n=1 m=1
Radial shift mode n=1 m=1
n The use of periodic boundaries seems to have a paramount role in terms of the tilt instability. In particular, with respect to previous results obtained with conducting boundaries we observe a significant slowing down of the mode overall growth (see stable phase between time “B” and time “C”).
n In addition, after an initial readjustment (time “A” to time “B”) a quite clear correlation between the onset of the tilt instability and the saturation of the different rotational modes (time “C”) seems to establish. In other words, the presence of continuously growing rotational modes (time “B” to time “C”) seems to prevent the tilt instability to start and become the leading player.
n Once the tilt instability starts and a steady growth rate is achieved, the amplitude of the rotational modes begins to increase again (time “D”). The observed delay between tilt and rotational modes is in the order of 20 μs.
References
gives rise to (vr-, vz+), with vz+ gives rise to vz- (quite small) and
predominant.
predominant vr+ Equilibrium ion density
Ions have no poloidal velocity at t = 0
O-point
Ls
rs
Interchange mode n=1 m=0
gives rise to (vr+, vz-), with vr+ predominant
Ballooning mode n=1 m=1
gives rise to (vr+, vz-).
modes is about 20 μs
AB Ion density
Ion density
1 M.W. Binderbauer et al., Phys. Plasmas 22, 056110 (2015)
2 F. Ceccherini et al. CP10.00087, APS-DPP 2016
3 S. Dettrick et al. PP11.00097, APS-DPP 2018
4 INCITE 2018 - Kinetic Simulation of FRC Stability and Transport
(P.I. Sean Dettrick, co-P.I. Toshi Tajima)
5 S.Dey et al., IEEE Microw. and Guided Wave Lett. 7, 273 (1997) 6 T. Z. Esirkepov, Comput. Phys. Commun. 135, 144 (2001)
7 E. Belova et al, Phys. Plasmas 8, 1267 (2001)
8 Y. A. Omelchenko, Phys. Rev. E 92, 023105 (2015) 9 E. Belova et al, Phys. Plasmas 10, 2361 (2003)
10 E. Belova et al, Phys. Plasmas 11, 2523 (2004)
11 N. Ohtani et al., Phys. Plasmas 10, 145 (2003)
J Jz z
Jθ Jθ
Acknowledgement
An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357.