On the relationship between the multi-region relaxed variational principle and resistive inner-layer theory

Kumar, A.; Loizu, J.; Hole, M. J.; Qu, Z. S.; Hudson, S. R.; Dewar, R. L.

doi:10.1088/1361-6587/acc96e

On the relationship between the multi-region relaxed variational principle and resistive inner-layer theory

Journal Article · Tue May 23 00:00:00 EDT 2023 · Plasma Physics and Controlled Fusion

DOI:https://doi.org/10.1088/1361-6587/acc96e· OSTI ID:1986329

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Australian National Univ., Canberra, ACT (Australia)
Ecole Polytechnique Federale Lausanne (EPFL) (Switzerland)
Australian National Univ., Canberra, ACT (Australia); Australian Nuclear Science and Technology Organisation (ANSTO), NSW (Australia)
Nanyang Technological Univ. (Singapore)
Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)

We show that the variational energy principle of the multi-region relaxed magnetohydrodynamic (MRxMHD) model can be used to predict finite-pressure linear tearing instabilities. In this model, the plasma volume is sliced into sub-volumes separated by 'ideal interfaces', and in each volume the magnetic field relaxes to a Taylor state, where the pressure gradient $$\nabla p = 0$$. The MRxMHD model is implemented in the Stepped-Pressure Equilibrium Code (SPEC) so that the equilibrium solution in each region is computed while preserving the force balance across the interfaces. As SPEC computes the Hessian matrix (a discretized stability matrix), the stability of an MRxMHD equilibrium can also be computed with SPEC. In this article, using SPEC, we investigate the effect of local pressure gradients and the $$\nabla p = 0$$ in the vicinity of the resonant surface of a tearing mode. For low-beta plasma, we have been able to illustrate a relationship between the resistive singular-layer theory (Coppi et al 1966 Nucl. Fusion6 101; Glasser et al 1975 Phys. Fluids18 875–88) and the MRxMHD model. Within the singular layer, the volume-averaged magnetic helicity and the flux-averaged toroidal flux are shown to be the invariants for the linear tearing modes in SPEC simulations. Our technique to compute MRxMHD stability is first tested numerically in a cylindrical tokamak and its application in toroidal geometry is demonstrated. We demonstrate an agreement between the stability boundary obtained with SPEC simulation and the resistive inner-layer theories.

View Accepted Manuscript (DOE)

Research Organization:: Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)

Sponsoring Organization:: Australian Research Council; European Commission (EC); Simons Foundation; USDOE

Grant/Contract Number:: AC02-09CH11466

OSTI ID:: 1986329

Journal Information:: Plasma Physics and Controlled Fusion, Journal Name: Plasma Physics and Controlled Fusion Journal Issue: 7 Vol. 65; ISSN 0741-3335

Publisher:: IOP ScienceCopyright Statement

Country of Publication:: United States

Language:: English

References (37)

Hydromagnetic stability of a diffuse linear pinch Newcomb, William A. Annals of Physics, Vol. 10, Issue 2 https://doi.org/10.1016/0003-4916(60)90023-3	journal	June 1960
Linear stability of periodic orbits in lagrangian systems Mackay, R. S.; Meiss, J. D. Physics Letters A, Vol. 98, Issue 3 https://doi.org/10.1016/0375-9601(83)90735-1	journal	October 1983
Stepped pressure profile equilibria in cylindrical plasmas via partial Taylor relaxation Hole, M. J.; Hudson, S. R.; Dewar, R. L. Journal of Plasma Physics, Vol. 72, Issue 06 https://doi.org/10.1017/S0022377806005861	journal	December 2006
Three-dimensional magnetohydrodynamic equilibria with continuous magnetic fields Hudson, S. R.; Kraus, B. F. Journal of Plasma Physics, Vol. 83, Issue 4 https://doi.org/10.1017/S0022377817000538	journal	July 2017
Combined plasma–coil optimization algorithms Henneberg, S. A.; Hudson, S. R.; Pfefferlé, D. Journal of Plasma Physics, Vol. 87, Issue 2 https://doi.org/10.1017/S0022377821000271	journal	April 2021
Curvature effects on the dynamics of tearing modes in tokamaks Lütjens, Hinrich; Luciani, Jean-François; Garbet, Xavier Physics of Plasmas, Vol. 8, Issue 10 https://doi.org/10.1063/1.1399056	journal	October 2001
Observation and analysis of a resistive mode with interchange parity in negative central shear plasmas in the DIII-D Tokamak Jayakumar, R.; Luce, T. C.; Taylor, T. S. Physics of Plasmas, Vol. 9, Issue 12 https://doi.org/10.1063/1.1522381	journal	December 2002
Thermal Equilibrium and Stability of Tokamak Discharges Furth, H. P. Physics of Fluids, Vol. 13, Issue 12 https://doi.org/10.1063/1.1692896	journal	January 1970
Finite-Resistivity Instabilities of a Sheet Pinch Furth, Harold P.; Killeen, John; Rosenbluth, Marshall N. Physics of Fluids, Vol. 6, Issue 4 https://doi.org/10.1063/1.1706761	journal	January 1963
Surface currents on ideal plasmas Webster, Anthony J. Physics of Plasmas, Vol. 17, Issue 11 https://doi.org/10.1063/1.3517096	journal	November 2010
Computation of multi-region relaxed magnetohydrodynamic equilibria Hudson, S. R.; Dewar, R. L.; Dennis, G. Physics of Plasmas, Vol. 19, Issue 11 https://doi.org/10.1063/1.4765691	journal	November 2012
Formation of current sheets in magnetic reconnection Boozer, Allen H. Physics of Plasmas, Vol. 21, Issue 7 https://doi.org/10.1063/1.4890491	journal	July 2014
Computation of resistive instabilities by matched asymptotic expansions Glasser, A. H.; Wang, Z. R.; Park, J. -K. Physics of Plasmas, Vol. 23, Issue 11 https://doi.org/10.1063/1.4967862	journal	November 2016
Multi-region relaxed magnetohydrodynamic stability of a current sheet Loizu, J.; Hudson, S. R. Physics of Plasmas, Vol. 26, Issue 3 https://doi.org/10.1063/1.5091765	journal	March 2019
A method for determining a stochastic transition Greene, John M. Journal of Mathematical Physics, Vol. 20, Issue 6 https://doi.org/10.1063/1.524170	journal	June 1979
Resistive instabilities in general toroidal plasma configurations Glasser, A. H.; Greene, J. M.; Johnson, J. L. Physics of Fluids, Vol. 18, Issue 7 https://doi.org/10.1063/1.861224	journal	January 1975
Resistive instabilities in a tokamak Glasser, A. H.; Greene, J. M.; Johnson, J. L. Physics of Fluids, Vol. 19, Issue 4 https://doi.org/10.1063/1.861490	journal	January 1976
Reduced equations for finite beta tearing modes in tokamaks Izzo, R.; Monticello, D. A.; DeLucia, J. Physics of Fluids, Vol. 28, Issue 3 https://doi.org/10.1063/1.865061	journal	January 1985
Magnetic reconnection via current sheets Biskamp, D. Physics of Fluids, Vol. 29, Issue 5 https://doi.org/10.1063/1.865670	journal	January 1986
Stability of tearing modes in finite‐beta plasmas Iacono, R.; Bhattacharjee, A.; Ronchi, C. Physics of Plasmas, Vol. 1, Issue 8 https://doi.org/10.1063/1.870592	journal	August 1994
Stability of tearing modes in tokamak plasmas Hegna, C. C.; Callen, J. D. Physics of Plasmas, Vol. 1, Issue 7 https://doi.org/10.1063/1.870628	journal	July 1994
Helical temperature perturbations associated with tearing modes in tokamak plasmas Fitzpatrick, Richard Physics of Plasmas, Vol. 2, Issue 3 https://doi.org/10.1063/1.871434	journal	March 1995
Direct prediction of nonlinear tearing mode saturation using a variational principle Loizu, J.; Huang, Y. -M.; Hudson, S. R. Physics of Plasmas, Vol. 27, Issue 7 https://doi.org/10.1063/5.0009110	journal	July 2020
Computation of Equilibrium Foam Structures Using the Surface Evolver Phelan, Robert; Weaire, Denis; Brakke, Kenneth Experimental Mathematics, Vol. 4, Issue 3 https://doi.org/10.1080/10586458.1995.10504320	journal	January 1995
Stability of coupled tearing modes in tokamaks Fitzpatrick, R.; Hastie, R. J.; Martin, T. J. Nuclear Fusion, Vol. 33, Issue 10 https://doi.org/10.1088/0029-5515/33/10/I11	journal	October 1993
Relaxed MHD states of a multiple region plasma Hole, M. J.; Mills, R.; Hudson, S. R. Nuclear Fusion, Vol. 49, Issue 6 https://doi.org/10.1088/0029-5515/49/6/065019	journal	May 2009
Resistive instabilities in a diffuse linear pinch Coppi, Bruno; Greene, John M.; Johnson, John L. Nuclear Fusion, Vol. 6, Issue 2 https://doi.org/10.1088/0029-5515/6/2/003	journal	June 1966
On the difficulty of determining tearing mode stability Bishop, C. M.; Connor, J. W.; Hastie, R. J. Plasma Physics and Controlled Fusion, Vol. 33, Issue 4 https://doi.org/10.1088/0741-3335/33/4/009	journal	April 1991
Nonlinear three-dimensional MHD simulations of tearing modes in tokamak plasmas Lütjens, H.; Luciani, J. F.; Garbet, X. Plasma Physics and Controlled Fusion, Vol. 43, Issue 12A https://doi.org/10.1088/0741-3335/43/12A/326	journal	November 2001
Controlling sawtooth oscillations in tokamak plasmas Chapman, I. T. Plasma Physics and Controlled Fusion, Vol. 53, Issue 1 https://doi.org/10.1088/0741-3335/53/1/013001	journal	November 2010
Tearing stability in toroidal plasmas with shaped cross section Ham, C. J.; Liu, Y. Q.; Connor, J. W. Plasma Physics and Controlled Fusion, Vol. 54, Issue 10 https://doi.org/10.1088/0741-3335/54/10/105014	journal	September 2012
Greene's residue criterion MacKay, R. S. Nonlinearity, Vol. 5, Issue 1 https://doi.org/10.1088/0951-7715/5/1/007	journal	January 1992
Free-boundary MRxMHD equilibrium calculations using the stepped-pressure equilibrium code Hudson, S. R.; Loizu, J.; Zhu, C. Plasma Physics and Controlled Fusion, Vol. 62, Issue 8 https://doi.org/10.1088/1361-6587/ab9a61	journal	July 2020
Computation of linear MHD instabilities with the multi-region relaxed MHD energy principle Kumar, A.; Qu, Z.; Hole, M. J. Plasma Physics and Controlled Fusion, Vol. 63, Issue 4 https://doi.org/10.1088/1361-6587/abdbd0	journal	February 2021
Nature of ideal MHD instabilities as described by multi-region relaxed MHD Kumar, A.; Nührenberg, C.; Qu, Z. Plasma Physics and Controlled Fusion, Vol. 64, Issue 6 https://doi.org/10.1088/1361-6587/ac53ee	journal	April 2022
Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields Taylor, J. B. Physical Review Letters, Vol. 33, Issue 19 https://doi.org/10.1103/PhysRevLett.33.1139	journal	November 1974
Resistive Interchange Modes in Negative Central Shear Tokamaks with Peaked Pressure Profiles Chu, M. S.; Greene, J. M.; Lao, L. L. Physical Review Letters, Vol. 77, Issue 13 https://doi.org/10.1103/PhysRevLett.77.2710	journal	September 1996

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70 PLASMA PHYSICS AND FUSION TECHNOLOGY
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linear tearing mode theory
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resistive instabilities

On the relationship between the multi-region relaxed variational principle and resistive inner-layer theory

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