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transverse instabilities with non-linear, non-rigid calcula- tions. At the moment, there is no simulation code available that could be used for this purpose (the Q2D code is restricted to axisymmetric systems), but this may change as FPIC37 development progresses.
VII. SUMMARY
The FRC concept offers several attractive features for a fusion reactor. This includes engineering simplicity, high b, and reduced risk of dangerous disruptions. FRCs have histor- ically been very unstable, but recent advances with beam- driven FRCs have achieved stabilization over all dynamic plasma lifetimes and re-invigorated interest in the FRC as a potential fusion device. To further explore the feasibility of an FRC reactor, the study of the scaling to higher tempera- tures and magnetic fields is required. This requires a feed- back control of the positional instability.
We have presented a linear model for this instability. The model has a single free parameter, which is the (shape of the) volume that is considered rigid. We find that, in the absence of superconductors, any axisymmetric system is unstable to either a radial or an axial displacement (but not both). In the presence of a conducting wall, the growth rate of the instability becomes proportional to the wall resistivity. As wall resistivity decreases, plasma inertia becomes insig- nificant and the feedback control becomes feasible.
We have simulated the unperturbed evolution of an axi- symmetric, beam-driven FRC over a relatively quiescent period using the Q2D code. At three different points in time, we perturbed the plasma and compared the non-linear evolu- tion of the instability with the linear predictions. We found that predictions had a strong dependence on the choice of rigid volume and used this to fix the free parameter. There does not seem to be a way to determine this parameter with- out running non-linear simulations, and so the primary use of the linear model is not the determination of uncontrolled growth rates but the design of control algorithms. We also found differences between non-linear growth rates observed for perturbations at different times even when the macro- scopic plasma parameters were approximately constant throughout the unperturbed evolution. While the underlying physics that cause this difference are not clear, they are apparently well captured in the chosen free parameter.
When used for the control algorithm design, we find that the (now parameter-free) linear model performs satisfacto- rily. The stability of the controlled system and its global characteristics are predicted correctly in almost all cases over a wide range of control parameters. Linear and non- linear results also indicate the same potential improvements to the control algorithm (taking into account the resistivity of the wall to avoid a breakdown of performance at very fast cycle times and low latencies) and similar design goals for an experimental control system (favoring longer latencies over longer cycle times).
Analysis with the linear model also allowed us to simu- late the instability evolution in the presence of a non- axisymmetric wall and revealed the importance of 3-D effects on the instability growth rate. Growth rates were
found to be increased by a factor of 2 or more if the wall had cutouts in the regions that best couple to the plasma. This is particularly important because such cutouts may not signifi- cantly affect the characteristic L/R time of the wall.
We are confident that this work will enable us to suc- cessfully control the positional instability in the C-2W device and thereby contributes a crucial ingredient for the validation of FRCs as another potential way towards fusion energy.
Furthermore, it is encouraging that despite the additional complexity that the large particle orbits impose on FRC plasma physics (which has repeatedly foiled attempts to cal- culate FRC stability), the FRC remains amenable to simple models if the regimes are chosen carefully. This gives hope that in addition to the advances made by further development of 3-D simulation codes, some further insights into FRCs might still be gained with less computationally intensive methods.
ACKNOWLEDGMENTS
The authors would like to thank Jim Bialek for his assistance with the VALEN code and Richard Milroy for his assistance with the NIMROD code. Data analysis for this work was done with Python in Jupyter, using the SciPy, Matplotlib, and Pandas modules. The authors also gratefully acknowledge the support of Tri Alpha Energy’s investors and team members.
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