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Existence of Three-Dimensional Ideal-Magnetohydrodynamic Equilibria with Current Sheets J Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets J. Loizu, S. R. Hudson, A. Bhattacharjee, S. Lazerson, and P. Helander Citation: Physics of Plasmas 22, 090704 (2015); doi: 10.1063/1.4931094 View online: http://dx.doi.org/10.1063/1.4931094 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/22/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The effect of three-dimensional fields on bounce averaged particle drifts in a tokamak Phys. Plasmas 22, 072510 (2015); 10.1063/1.4926818 Three-dimensional linear peeling-ballooning theory in magnetic fusion devices Phys. Plasmas 21, 042507 (2014); 10.1063/1.4871859 Two-dimensional magnetohydrodynamic simulations of poloidal flows in tokamaks and MHD pedestal Phys. Plasmas 18, 092509 (2011); 10.1063/1.3640809 Validation of the linear ideal magnetohydrodynamic model of three-dimensional tokamak equilibria Phys. Plasmas 17, 030701 (2010); 10.1063/1.3335237 Minimizing the magnetohydrodynamic potential energy for the current hole region in tokamaks Phys. Plasmas 11, 4859 (2004); 10.1063/1.1792284 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 198.35.1.205 On: Mon, 14 Sep 2015 20:21:40 PHYSICS OF PLASMAS 22, 090704 (2015) Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets J. Loizu,1,2 S. R. Hudson,2 A. Bhattacharjee,2 S. Lazerson,2 and P. Helander1 1Max-Planck-Institut fur€ Plasmaphysik, D-17491 Greifswald, Germany 2Princeton Plasma Physics Laboratory, PO Box 451, Princeton, New Jersey 08543, USA (Received 19 June 2015; accepted 4 September 2015; published online 14 September 2015) We consider the linear and nonlinear ideal plasma response to a boundary perturbation in a screw pinch. We demonstrate that three-dimensional, ideal-MHD equilibria with continuously nested flux-surfaces and with discontinuous rotational-transform across the resonant rational-surfaces are well defined and can be computed both perturbatively and using fully nonlinear equilibrium calcu- lations. This rescues the possibility of constructing MHD equilibria with current sheets and contin- uous, smooth pressure profiles. The results predict that, even if the plasma acts as a perfectly conducting fluid, a resonant magnetic perturbation can penetrate all the way into the center of a tokamak without being shielded at the resonant surface. [http://dx.doi.org/10.1063/1.4931094] The properties and numerical computation of three- crucial observation is explored in this letter, and it leads to dimensional, ideal-MHD equilibria are of fundamental impor- some remarkable conclusions. tance for understanding the behaviour of both magnetically Singularities in the current density areÐ allowed in the 1–3 confined fusion and astrophysical plasmas. In particular, ideal-MHD model, but the total current, Sj Á ds, passing singular current densities at rational surfaces are predicted through any surface, S, must remain finite for any physically in equilibria with continuously nested flux-surfaces, with acceptable equilibrium. While the integral of a d-current and without pressure.1,4–7 These singular currents play a density is always finite, surfaces may be constructed through crucial role in describing (i) the plasma response to non- which the Pfirsch-Schluter€ current diverges logarithmically. axisymmetric boundary perturbations,8–10 (ii) ideal and resis- This is not physical: it would seem that the ideal-MHD equi- tive stability,11,12 and (iii) the dynamics of reconnection librium model has the fatal flaw of not allowing for pressure. 1 phenomena, such as sawteeth.13 However, as noted by Grad, equilibrium solutions with- The singularities arise from requiring charge conserva- out infinite currents may be constructed by considering pres- tion, $ Á j ¼ 0, which gives rise to a magnetic differential sure profiles that are flat in a small neighbourhood of each rational surface. In order to construct non-trivial, continuous equation for the parallel current, B Á $u ¼$ Á j?, where pressure profiles, the pressure-gradient must be finite on a set j uB þ j?. Magnetic differential equations are densely sin- gular.14 Their singular nature is exposed in straight-field-line of finite measure, e.g., the irrationals that are sufficiently far coordinates, (w, h, f), which may be constructed on each from low order rationals, i.e., those that satisfy a Diophantine condition.18 In that case, p is continuous but its derivative is flux surface and imply J B Á $ ¼ i @h þ @f, where J is the not. The pressure profile must be fractal, with the pressure- Jacobian of the coordinates, i ðwÞ is the rotational-transform on a given flux surface, which is labeled by the enclosed to- gradient being discontinuous on a fractal set of finite measure. Grad1 described such equilibria as pathological. roidal flux, wP, and h and f are poloidal and toroidal angles. Alternatively, Bruno and Laurence19 showed that equili- Writing u ¼ umnðwÞ exp½iðmh À nfÞ, we have m;n bria with discontinuous pressure profiles are also possible umnðxÞ¼hmnðxÞ=x þ DmndðxÞ; (1) solutions, with the finite set of discontinuities in p occurring on surfaces with strongly irrational transform. These where x i m À n; hmn iðJ $ Á j?Þmn, and Dmn is an as-yet “stepped-pressure” states are extrema of the multi-region, undetermined constant. The force-balance equation in its relaxed energy functional20 and resolve the singularities by 2 simplest form, j  B ¼ $p, implies j? ¼ B rp/B ; and allowing local relaxation, and thus are not globally ideal. 0 thus hmn is proportional to the pressure-gradient, hmn p ; The question remains: are there well-defined, non-patho- although it could potentially vanish if the Jacobian, J ,is logical, globally ideal, MHD equilibrium solutions in arbi- infinitely constrained.15 trary, three-dimensional geometry, with continuously nested The singularities consist of a pressure-driven, Pfirsch- surfaces and with arbitrary, smooth, continuous pressure pro- Schluter€ 1/x current density that arises around rational surfa- files? In this letter, we will suggest a new class of solution ces, and a Dirac d-function current density that develops at that satisfies each of these conditions. rational surfaces as a necessary mechanism to prevent the Historically, the cause of pathologies in MHD equilibria formation of islands that would otherwise develop in a non- with nested flux-surfaces has been attributed to the class of ideal plasma.16 Only recently have these singular current possible pressure profiles. However, the form of the pressure densities been computed numerically.17 In doing so, it was profile—whether it be smooth, continuous, or pathological in realized that infinite shear at the rational surfaces was some sense—is not the cause of the problem. The problem is required in order to have well-defined solutions.17 This the existence of singularities in the magnetic differential 1070-664X/2015/22(9)/090704/5/$30.00 22, 090704-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 198.35.1.205 On: Mon, 14 Sep 2015 20:21:40 090704-2 Loizu et al. Phys. Plasmas 22, 090704 (2015) equation, and these singularities are associated with the exis- tence of flux-surfaces with rational rotational-transform. In fact, the singularities remain even for zero pressure. As we shall see, ideal-MHD equilibria with rational surfaces are not analytic functions of the boundary, and it is only by remov- ing the rational surfaces that the singularities can be eliminated. Our approach is valid in arbitrary geometry, but for a transparent presentation and to enable verification calcula- tions we consider the linear and nonlinear, ideal plasma response to a non-axisymmetric boundary perturbation in a screw pinch with zero pressure and no flow. The linear r h z plasma displacement, n ¼ n er þ n eh þ n ez, induced by a FIG. 1. Solutions of Eq. (2) for an m ¼ 2, n ¼ 1 boundary perturbation and À2 non-axisymmetric, radial perturbation with a single Fourier for different values of Di (solid lines), ranging from Di ¼ 4  10 to Di ¼ 10À3. Also the singular case Di ¼ 0 is shown (discontinuous curve). harmonic, na cosðmh þ kzÞ, to the boundary satisfies the The corresponding SPEC linear calculations are also shown (squares). linearized force-balance equation, L0½n¼0, where L0½n dj½nÂB þ j  dB½n, where the “ideal” linear perturba- linear operator, L n , is singular; and higher order terms in tion to the magnetic field is dB½nrÂðn  BÞ, and 0½ 21 the perturbation expansion successively diverge. Writing the dj½nrÂdB½n. This reduces to Newcomb’s equation 2 perturbation in the geometry as n n1 þ n2 þÁÁÁ, the d dn equation for the second order term is L0½n2¼dj½n1 f À gn ¼ 0; (2) dr dr  dB½n1, and n2 is even more singular than n1. That perturbation theory is not a valid approach for where nr nðrÞ cosðmh þ kzÞ. The functions f(r) and g(r) treating ideal-MHD equilibria has long been known: are determined by the equilibrium Rosenbluth et al. arrived at a similar conclusion when study- ing the ideal internal kink,11 stating that “we must abandon 3 2 2 r the perturbation theory approach and go instead to a bound- f ¼ Bz ðÞi À i s 2 2 2 ; R þ r i s ary layer theory.” Rosenbluth et al. also realized that “all 2 harmonics are excited to comparable amplitude” by a small g ¼ B2½ i À i 2ðÞk2r2 þ m2 À 1 kþði 2 À i 2Þ2k r; z ðÞ s s boundary deformation.
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