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Direct, simple and efficient computation of all components of the virtual-casing magnetic field in axisymmetric geometries with Kapur–Rokhlin quadrature

Published online by Cambridge University Press:  06 May 2024

Evan Toler Open the ORCID record for Evan Toler [Opens in a new window] ,
A.J. Cerfon Open the ORCID record for A.J. Cerfon [Opens in a new window]  and
D. Malhotra Open the ORCID record for D. Malhotra [Opens in a new window]
Evan Toler*
Affiliation:
Argonne National Laboratory, Lemont, IL 60439, USA
A.J. Cerfon
Affiliation:
Type One Energy Group – Canada Inc., Vancouver, Canada BC V6E 0A2
D. Malhotra
Affiliation:
Flatiron Institute, New York, NY 10012, USA
*
†Email address for correspondence: etoler@anl.gov
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Abstract

In a recent publication (Toler et al., J. Plasma Phys., vol. 89, issue 2, 2023, p. 905890210), we demonstrated that for axisymmetric geometries, the Kapur–Rokhlin quadrature rule provided an efficient and high-order accurate method for computing the normal component, on the plasma surface, of the magnetic field due to the toroidal current flowing in the plasma, via the virtual-casing principle. The calculation was indirect, as it required the prior computation of the magnetic vector potential from the virtual-casing principle, followed by the computation of its tangential derivative by Fourier differentiation, to obtain the normal component of the magnetic field. Our approach did not provide the other components of the virtual-casing magnetic field. In this letter, we show that a more direct and more general approach is available for the computation of the virtual-casing magnetic field. The Kapur–Rokhlin quadrature rule accurately calculates the principal value integrals in the expression for all the components of the magnetic field on the plasma boundary, and the numerical error converges at a rate nearly as high as the indirect method we presented previously.

Keywords

fusion plasmaplasma confinementplasma devices
Type
Letter
Information
Journal of Plasma Physics , Volume 90 , Issue 2 , April 2024 , 175900202
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press

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References

Blum, J., Boulbe, C. & Faugeras, B. 2012 Reconstruction of the equilibrium of the plasma in a tokamak and identification of the current density profile in real time. J. Comput. Phys. 231 (3), 960–980.CrossRefGoogle Scholar
Drevlak, M., Beidler, C.D., Geiger, J., Helander, P. & Turkin, Y. 2018 Optimisation of stellarator equilibria with ROSE. Nucl. Fusion 59 (1), 016010.CrossRefGoogle Scholar
Freidberg, J.P. 2014 Ideal MHD. Cambridge University Press.CrossRefGoogle Scholar
Gradshteyn, I.S. & Ryzhik, I.M. 2014 Table of Integrals, Series, and Products. Academic Press.Google Scholar
Hanson, J.D. 2015 The virtual-casing principle and Helmholtz's theorem. Plasma Phys. Control. Fusion 57 (11), 115006.CrossRefGoogle Scholar
Kappel, J., Landreman, M. & Malhotra, D. 2023 The magnetic gradient scale length explains why certain plasmas require close external magnetic coils. arXiv:2309.11342.CrossRefGoogle Scholar
Kapur, S. & Rokhlin, V. 1997 High-order corrected trapezoidal quadrature rules for singular functions. SIAM J. Numer. Anal. 34 (4), 1331–1356.CrossRefGoogle Scholar
Kress, R. & Martensen, E. 1970 Anwendung der rechteckregel auf die reelle hilberttransformation mit unendlichem intervall. Z. Angew. Math. Mech. 50 (1–4), 61–64.CrossRefGoogle Scholar
Landreman, M. 2017 An improved current potential method for fast computation of stellarator coil shapes. Nucl. Fusion 57 (4), 046003.CrossRefGoogle Scholar
Landreman, M. & Boozer, A.H. 2016 Efficient magnetic fields for supporting toroidal plasmas. Phys. Plasmas 23 (3), 032506.CrossRefGoogle Scholar
Lao, L.L., St. John, H.E., Peng, Q., Ferron, J.R., Strait, E.J., Taylor, T.S., Meyer, W.H., Zhang, C. & You, K.I. 2005 MHD equilibrium reconstruction in the diii-d tokamak. Fusion Sci. Technol. 48 (2), 968–977.CrossRefGoogle Scholar
Lazerson, S.A., Sakakibara, S. & Suzuki, Y. 2013 A magnetic diagnostic code for 3d fusion equilibria. Plasma Phys. Control. Fusion 55 (2), 025014.CrossRefGoogle Scholar
Lee, J. & Cerfon, A. 2015 ECOM: a fast and accurate solver for toroidal axisymmetric MHD equilibria. Comput. Phys. Commun. 190, 72–88.CrossRefGoogle Scholar
Lütjens, H., Bondeson, A. & Sauter, O. 1996 The chease code for toroidal mhd equilibria. Comput. Phys. Commun. 97 (3), 219–260.CrossRefGoogle Scholar
Malhotra, D., Cerfon, A.J., O'Neil, M. & Toler, E. 2019 Efficient high-order singular quadrature schemes in magnetic fusion. Plasma Phys. Control. Fusion 62 (2), 024004.CrossRefGoogle Scholar
Martinsson, P.-G. 2019 Fast Direct Solvers for Elliptic PDEs. SIAM.CrossRefGoogle Scholar
Marx, A. & Lütjens, H. 2017 Free-boundary simulations with the XTOR-2f code. Plasma Phys. Control. Fusion 59 (6), 064009.CrossRefGoogle Scholar
Pustovitov, V.D. & Chukashev, N.V. 2021 Analytical solution to external equilibrium problem for plasma with elliptic cross section in a tokamak. Plasma Phys. Rep. 47 (10), 956–966.CrossRefGoogle Scholar
Ricketson, L.F., Cerfon, A.J., Rachh, M. & Freidberg, J.P. 2016 Accurate derivative evaluation for any Grad–Shafranov solver. J. Comput. Phys. 305, 744–757.CrossRefGoogle Scholar
Shafranov, V.D. & Zakharov, L.E. 1972 Use of the virtual-casing principle in calculating the containing magnetic field in toroidal plasma systems. Nucl. Fusion 12 (5), 599–601.CrossRefGoogle Scholar
Sidi, A. & Israeli, M. 1988 Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J. Sci. Comput. 3 (2), 201–231.CrossRefGoogle Scholar
Toler, E., Cerfon, A.J. & Malhotra, D. 2023 A fast, accurate and easy to implement Kapur–Rokhlin quadrature scheme for singular integrals in axisymmetric geometries. J. Plasma Phys. 89 (2), 905890210.CrossRefGoogle Scholar
Trefethen, L.N. & Weideman, J.A.C. 2014 The exponentially convergent trapezoidal rule. SIAM Rev. 56 (3), 385–458.CrossRefGoogle Scholar
Zaitsev, F.S., Kostomarov, D.P., Suchkov, E.P., Drozdov, V.V., Solano, E.R., Murari, A., Matejcik, S., Hawkes, N.C. & Contributors, J.E.T-E.F.D.A. 2011 Analyses of substantially different plasma current densities and safety factors reconstructed from magnetic diagnostics data. Nucl. Fusion 51 (10), 103044.CrossRefGoogle Scholar
Zakharov, L.E. 1973 Numerical methods for solving some problems of the theory of plasma equilibrium in toroidal configurations. Nucl. Fusion 13 (4), 595–602.CrossRefGoogle Scholar
Zakharov, L.E. & Pletzer, A. 1999 Theory of perturbed equilibria for solving the Grad–Shafranov equation. Phys. Plasmas 6 (12), 4693–4704.CrossRefGoogle Scholar