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A relatively simple way to evaluate bootstrap current in stellarators involves the use of the long mean free path asymptotic formula by Shaing and Callen [1]. This formula contains all the information about device geometry in a geometrical factor, independent of plasma parameters. This method is particularly suited for stellarator optimization, where multiple quick estimates of bootstrap current are needed.
Interestingly, even though there have been a few different derivations of this formula [1,2,3], all leading to the same or almost the same result, these are never reproduced by the numerical drift kinetic equation (DKE) solvers in the $1/\nu$ regime. This regime is relevant for the electron component in fusion reactors [4]. Conversely, DKE solvers show the trend of bootstrap current to converge towards the Shaing-Callen limit in the presence of a radial electric field [4], which is not explicitly accounted for in the existing derivations of this formula.
Qualitatively, the reasons for such anomalous behavior have been recently identified in Ref.[5] using an adjoint approach, where Onsager symmetry between bootstrap coefficient and Ware-pinch coefficient has been applied. In the present report, the convergence of bootstrap current to the Shaing-Callen limit is studied both analytically and semi-analytically, as well as numerically with the aid of the NEO-2 code [6]. The analysis remains in agreement with the qualitative observations of Ref.[5]. In addition to this, special cases where bootstrap current converges to this limit in the $1/\nu$ regime are demonstrated. The analysis shows that in realistic toroidal stellarator geometry, bootstrap current diverges in the $1/\nu$ regime with decreasing collisionality as $ν^{-1/5}$, and, conversely, it converges to the Shaing-Callen limit in the presence of banana orbit precession (particularly due to the radial electric field) as $\nu^{3/5}$.
A method for evaluating bootstrap current in the presence of a radial electric field is also discussed, which entails the use of rapid computations of this current by NEO-2 in the $1/\nu$ regime.
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 -- EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
References:
[1] K. C. Shaing and J.D.Callen, Phys. Fluids 26 (1983) 3315
[2] A. H. Boozer and H.J.Gardner, Phys. Fluids B 2 (1990) 2408
[3] P. Helander, J. Geiger, H.Maassberg, Phys. Plasmas 18 (2011) 092505
[4] C. D. Beidler et al, Nucl. Fusion 51 (2011) 076001)
[5] C. D. Beidler, "Final Report on Deliverable S2-WP19.1-T005-D005 Neoclassical and fast-ion transport" (2020) (MPG)
[6] W.Kernbichler et al, Plasma Phys. Control. Fusion 58 (2016) 104001
