Direct Prediction of Nonlinear Tearing Mode Saturation Using a Variational Principle

Direct prediction of nonlinear tearing mode saturation using a variational principle

J. Loizu1, Y.-M. Huang2, S. R. Hudson2, A. Baillod1, A. Kumar3, and Z. Qu3 1 Ecole´ Polytechnique Fédérale de Lausanne, Swiss Plasma Center, CH-1015 Lausanne, Switzerland 2 Princeton Plasma Physics Laboratory, PO Box 451, Princeton NJ 08543, USA 3 Mathematical Sciences Institute, the Australian National University, Canberra ACT 2601, Australia∗ (Dated: June 8, 2020) It is shown that the variational principle of multi-region relaxed magnetohydrodynamics (MRxMHD) can be used to predict the stability and nonlinear saturation of tearing modes in strong guide field configurations without resolving the dynamics and without explicit dependence on the plasma resistivity. While the magnetic helicity is not a good invariant for tearing modes, we show that the saturated tearing mode can be obtained as an MRxMHD state of a priori unknown helicity by appropriately constraining the current profile. The predicted saturated island width in a tearing-unstable force-free slab equilibrium is shown to reproduce the theoretical scaling at small values of ∆0 and the scaling obtained from resistive MHD simulations at large ∆0.

Magnetic reconnection is a ubiquitous phenomenon in very fact that a saturated state can be reached proves the the Universe that exhibits fast conversion of tremendous existence of a nearby, generally three-dimensional MHD amounts of magnetic energy into kinetic energy. It is ob- equilibrium. These observations raise the question: is served in solar eruptive flares [1, 2] and in the interaction there a variational principle that would allow direct cal- between the solar wind and the Earth’s magnetic field [3]. culation of the saturated tearing modes, without resolv- Magnetic reconnection has an immediate practical im- ing the complex, resistivity-dependent dynamics? In this portance. In magnetic fusion devices, reconnection drives letter, we provide compelling analytical and numerical fast relaxation processes, such as sawtooth-like temper- evidence that this is possible. This opens the possibility ature crashes in tokamaks [4], reversed field pinches [5], of performing much simpler numerical calculations, but and stellarators [6], as well as other typically slower re- most importantly provides deep insight into the funda- laxation processes, such as neoclassical tearing modes in mental physical mechanisms that underpin tearing mode tokamaks [7], which degrade plasma confinement, slow formation and saturation, leading to new general under- plasma rotation, and can trigger plasma disruptions. standing of magnetic reconnection phenomena. In the presence of a strong guide field (e.g., the toroidal The theory of Taylor relaxation [15, 16], which does field in tokamaks), reconnection often occurs via the tear- not explicitly involve resistivity as a parameter, seems to ing mode instability. Within resistive magnetohydrody- indicate that it is indeed possible to find certain glob- namics (MHD), tearing modes appear when a certain ally relaxed, saturated states by minimizing the classical parameter, ∆0, which depends on the equilibrium pro- MHD potential energy [17], files, exceeds a critical value, ∆c, which is zero except for toroidal effects [8, 9]. A magnetic island then grows with Z p B2 W = + dV , (1) a time scale between the ideal (Alfvén)time scale τ and A Vp γ − 1 2µ0 the resistive (diffusion) time scale τη [10]. In tokamaks, nonlinear saturation of the tearing mode is typically ob- where Vp is the plasma volume, p is the plasma pressure, served, and the corresponding saturated island width, and γ is the adiabatic index. Taylor relaxed states satisfy wsat, is a critical parameter for machine operation [7]. a Beltrami equation, ∇ × B = µB, with µ being a con- Numerical simulations of the growth and saturation of stant. They are obtained by taking p = 0 in Eq. (1) and resistive tearing modes in realistic parameter regimes are extremizing W for arbitrary variations in the magnetic expensive, primarily because of the very large Lundquist field subject to the only constraints of conserved total number, S = τη/τA 1, which imposes resolving very magnetic fluxes and conserved global magnetic helicity, disparate time scales [11]. For example, the ITER toka- 9 Z mak [12] can easily reach S ∼ 10 in the core. K = A · B dV , (2) There are three crucial observations to make: (1) while Vp the linear growth rate of the tearing mode always depends on resistivity, e.g. γ ∼ η3/5 in a cylinder [8], where A is the magnetic vector potential. In an infinitely the instability parameter ∆0 does not, and the instability conducting, ideal plasma, K is an exact invariant in time, threshold is ∆c = 0 except for toroidal corrections [9]; (2) thereby reflecting the topological implications of Alfvén’s while the nonlinear growth rate of the island depends on theorem [18]. In a plasma with finite resistivity, K is dis- resistivity [13], the value of wsat does not [14]; and (3) the sipated in time, but often much slower than W . In fact, it can be argued that K is a good invariant during relaxation processes that involve small-scale magnetic fluc- tuations or fast reconnection time scales [16, 19]. Ex- ∗Electronic address: [email protected] periments have indeed confirmed the quasi-invariance of 2

K during sawtooth oscillations in tokamaks [20] and re- We start by considering a force-free, doubly periodic, versed field pinches [21]. Tearing modes, however, usu- plasma current slab described by the following equilib- ally go through a slow nonlinear phase [13] and hence rium magnetic field [27]: B = Bz ez + ez × ∇ψ, where z invariants other than K shall be sought to describe their is the guide-field direction and the flux function ψ(x) = 2 saturation in terms of a more general variational princi- ψ0/ cosh (x) determines the “poloidal” magnetic field, 0 ple. Moreover, Taylor’s theory cannot describe saturated By(x) = ψ (x), as well as the “toroidal” current density 00 states with non-trivial current profiles in strong guide jz(x) = ψ (x) (Figure 1). The toroidal field, Bz, has a field configurations. In fact, since the current density “radial” dependence such as to satisfy the force-balance 2 2 2 in a Taylor state satisfies j = µB, its spatial variation is condition Bz (x) + By (x) = B0 , where B0 is the magni- quite limited in systems with a magnetic field whose mag- tude of the magnetic√ field on the resonant surface (x = 0). nitude, B, is almost uniform. In addition, Taylor states By setting ψ0 = 3 3/4 we have that the maximum do not support pressure gradients, which are ubiquitous poloidal field is By,max = 1, and hence we can ensure to any relevant magnetic fusion experiment. that a strong guide field is present by choosing B0 1. Over a decade ago, the theory of multi-region relaxed This current sheet can be characterized by a radial√ equi- p 00 MHD (MRxMHD) was proposed by Dewar et al [22, 23] librium scale length a ≡ |jz(0)/jz (0)| = 1/ 8 ≈ 0.35, in an attempt to generalize Taylor’s relaxation theory and we choose a box size Lx = 2π a. The poloidal and allow one to explore energetically favourable recon- extent is what controls the stability of the system and nection events while maintaining some ability to con- thus we choose Ly = Lz = L as control parameter. strain the pressure and current profiles. MRxMHD starts with the same energy W , Eq. (1), but considers a con- 1.5 1 strained minimization with additional topological con- 1 straints. The plasma is partitioned into N nested vol- 0 umes separated by N −1 ideal interfaces that are assumed 0.5 y to remain magnetic surfaces during the minimization of z

0 j -1 the energy. In each volume, the plasma undergoes Taylor B -0.5 relaxation and the magnetic helicity, Eq. (2), is conserved -2 within the volume, along with the toroidal and poloidal -1 magnetic fluxes. While varying the plasma potential en- -1.5 -3 ergy, W , the ideal interfaces are allowed to undergo geo- -2 0 2 -2 0 2 x x metrical deformations. The resulting MRxMHD equilibrium states satisfy FIG. 1: Equilibrium radial profiles for the poloidal magnetic 0 field, By(x) = ψ (x), and the toroidal current density, jz(x) = 00 ∇ × Bl = µlBl (3) ψ (x). Red circles and red stairs describe the equilibrium 2 profiles obtained from the SPEC solution with N = 41, B = B 0 p + l = 0 (4) 10, L = 2π,Ψw = 0.1. 2µ0 where l = 1, ...N labels the volumes, µl is a constant char- While this equilibrium is ideally stable [28], it is unsta- acterizing the Taylor states that is related to the parallel ble to a tearing mode [29], which grows at the resonant surface x = 0. The instability parameter for the tearing current, and · l is the jump across the interface sep- arating volumes l and l + 1. mode is independent of resistivity and can be calculated A numerical solution to the MRxMHD equations can analytically [29], be found by using the Stepped-Pressure Equilibrium 2(5 − k2)(3 + k2) Code (SPEC), which can run in slab [24], cylindrical [25], ∆0 = √ , (5) or toroidal geometry [26]. Provided that the boundary k2 4 + k2 geometry and the number N of relaxation volumes are given, the solution is determined by a set of N quadru- with an instability occurring for ∆0 > 0. Here k is the plets, {p, Ψt, Ψp,K}l=1...N . Namely, in each volume one poloidal wavenumber whose smallest√ possible value is k = must provide: the pressure p, the enclosed toroidal flux 2π/L. Hence ∆0 > 0 for 2π/L < 5. The condition for Ψ , the enclosed poloidal flux Ψ , and the enclosed mag- instability thus becomes a condition for the current sheet t p p netic helicity K. Alternatively, an equilibrium solution aspect ratio, a/L < 5/(32π2) ≈ 0.126 (Figure 2). can be found by replacing, in each volume, the con- We now describe this current sheet equilibrium and straint on K with a constraint on the Beltrami constant its stability from the perspective of MRxMHD. In order µ = j·B/B2. This becomes particularly crucial if the cur- to accurately approximate the equilibrium profiles, we rent profile is to be constrained. In fact, while K is not a construct an MRxMHD equilibrium with a large num- good invariant for tearing modes, we will show that the ber of volumes N 1. In each volume, we provide a quasi-invariance of the current profile can be exploited quadruplet {p = 0, Ψt, Ψp, µ}l=1...N by proceeding as fol- by SPEC to describe the saturated tearing equilibrium lows. First, we select the positions, xl, of the interfaces as an MRxMHD state of a priori unknown helicity. defining the relaxation volumes, with x0 = −Lx/2 = −π 3

The eigenvalues of H are evaluated numerically using the SPEC code for diﬀerent values of the equilibrium current sheet aspect ratio a/L. We observe that the smallest eigenvalue, referred to as λ, corresponds to an m = 1 mode and becomes negative for suﬃciently small a/L (Figure 2). The value of a/L at which λ changes sign depends on the choice of Ψw and, for Ψw → 0, coincides with that at which ∆0 changes sign. Since λ < 0 implies that the mode is unstable, we have just shown that the MRxMHD stability boundary is in agreement with linear tearing mode theory. We now consider equilibria with ∆0 > 0, for which the tearing mode is expected to develop an island until saturation is reached. An exact nonlinear saturation theory, valid in the limit of small ∆0, was developed indepen-

0 dently by Escande and Ottaviani [31] and by Militello FIG. 2: Solid line: ∆ versus a/L obtained from Eq. (5) with and Porcelli [32]. In this theory, sometimes referred to as k = 2π/L. Dashed lines: smallest eigenvalue λ of H computed the POEM theory [27], the saturated island width is from SPEC. Diﬀerent curves are for diﬀerent values of Ψw, the smallest being Ψw = 0.01 (in blue). Here N = 121. 0 2 wsat = 2.44∆ a (7)

and thus linearly increases with ∆0 (Figure 3). We can and xN = Lx/2 = π. In order to describe the equi- also obtain the saturated island width at arbitrarily large librium profiles efficiently we pack the interfaces where values of ∆0 by solving, e.g., the compressible, visco- the gradients are larger, namely, by selecting interfaces resistive, MHD equations in 2D [33], with equal ψ-spacing instead of equal x-spacing. Second, we evaluate the enclosed poloidal and toroidal fluxes, Ψp ∂tρ + ∇ · (ρv) = 0 and Ψt, in each volume by using the analytical solution B2 for B (x) and B (x). Finally, we evaluate the Beltrami ∂ (ρv) + ∇ · (ρvv) = −∇( ) − ∇ · (BB) + ν∇2(ρv) y z t 2 constant by performing a radial average within each vol- 2 ∂tB = ∇ × [v × B − η(j − jeq)] (8) ume, µ = h j · B/B ix. For the special case of the volume containing the resonant surface, x = 0, we al- where η and ν are the resistivity and viscosity, respec- low for an arbitrary radial width that we parametrize by tively, and jeq is the (initial) equilibrium current profile the enclosed toroidal flux normalized to the total toroidal and represents a source term in Faraday’s law that en- flux, 0 < Ψw < 1. In other words, the size of the reso- sures the stationarity of the plasma current on resistive nant volume is proportional to Ψw. With this procedure, time-scales. As in the POEM theory, the toroidal current the SPEC equilibrium solution is only determined by the density at saturation is a flux-function, number of volumes, N, and the resonant flux Ψw. Figure 1 shows an example of profiles obtained from the SPEC jz = hjz,eqiψ (9) solution for N = 41 and Ψw = 0.1. In the limit N → ∞ and Ψw → 0, the analytical profiles are retrieved exactly. where h·iψ is the average over a flux-surface and ψ(x, y) is The stability of this MRxMHD equilibrium can be as- the saturated poloidal flux function. Equation (9) implies sessed by studying the derivative of the interfaces force the quasi-invariance of the current profile, the invariance 2 balance, fl = p + B /2µ0 l , with respect to poloidal being exact in the limit of small island width. This prop- perturbations in the interface geometry, xl [30]. Writ- erty will be exploited by SPEC to replace the invariance P P ing fl = m fl,m cos (mθ) and xl = m xl,m cos (mθ), of K with that of jz in the nonlinear calculations. where θ = 2πy/L, we are interested in the matrix ele- The saturated island width, wsat, obtained by solving ments Eqs. (8) for different initial values of ∆0 is shown in Fig. 3. As reported by Loureiro et al [27], the predictions of the ∂fli,mi POEM theory are well recovered for a surprising range Hij = (6) 0 0 ∂xlj ,mj of values of ∆ a, up to about ∆ a ≈ 2. For larger values 0 of ∆ a, we also observe a curve-bending [27], with wsat where lq = b1 + (q − 1)/Mc and mq = q − 1 − (lq − 1)M, below the POEM prediction. Importantly, the results for q = 1...(N − 1)M, with N the number of relaxed vol- become independent of resistivity for sufficiently small umes and M the number of Fourier modes. When H is values of η (and similarly for ν), a property that can be evaluated at fixed magnetic helicity and fluxes, its eigen- anticipated by a mere rescaling of the velocity in Eqs. (8). values provide information about the stability of each More precisely, a transformation v → v/η leaves only 2 eigenmode m with the corresponding eigenfunction char- dependencies of order η and ην when considering ∂t = 0. acterized by the eigenvector {xl,m}. 4

width that is possibly achievable goes to zero. For in- POEM creasingly large values of Ψw, the island is allowed to resistive MHD 15 grow over a larger volume, although it may well only oc- SPEC fixed K cupy a fraction of it. However, if the value of Ψ exceeds SPEC fixed j w z ∗ a certain threshold, Ψw, the initial equilibrium becomes stable and no island grows at all. This is illustrated in 10 0

/ a Fig. 4 for two cases: one with ∆ a ≈ 1 and one with 0 0

sat ∆ a ≈ 2. Obviously, for each ∆ , the saturated island w width obtained with SPEC depends on Ψw. However, 5 there is a distinct point, which we identify as wsat, and that is the maximum achieved island width, which oc- ∗ curs at around Ψw ≈ Ψw (dashed lines in Fig. 4). We thus apply the following procedure: for each value of ∆0a, 0 a linear stability analysis is first performed with SPEC ∗ 02468and the value of Ψw is obtained by solving λ(Ψw) = 0. ∗ a Then, SPEC is run nonlinearly with Ψw . Ψw and the obtained value of wsat is extracted from the correspond- 0 FIG. 3: Saturated island width, wsat/a, as a function of ∆ a. ing saturated state. It can also be easily verified that Predictions shown for the POEM theory (solid line), the re- the result does not depend much on the exact choice of −2 sistive MHD simulations solving Eqs. (8) with η = 10 , Ψw, i.e., that the corresponding value of w is around its −4 ν = 10 , and conducting wall boundary conditions (blue maximum (Figure 4). stars), and the SPEC solution at fixed magnetic helicity (pur- ple diamonds) and at fixed toroidal current (red circles). The saturated island width, wsat, obtained from SPEC nonlinear calculations for a wide range of ∆0a is shown in Fig. 3 for two types of constraints. When the magnetic helicity and poloidal flux, K and Ψp, in each volume are constrained to be exactly conserved (diamonds w = w 0 max in Fig. 3), the value of wsat increases with ∆ a but 1.5 = * stays below the predictions of both POEM and resistive w w w = w sat MHD simulations. This discrepancy reflects the fact that the magnetic helicity is not well conserved during the a 2 1 slowly reconnecting tearing mode [19]. However, when

w the toroidal current in each volume is constrained to be * 0 = fixed (circles in Fig. 3), the scaling of wsat with ∆ a is well w = w w w sat reproduced for the entire range of ∆0a. The constraint 0.5 on the current in SPEC is in fact an approximation of a 1 Eq. (9) and can be achieved by following the approach described in Ref. [34], i.e. by constraining µ to the ini- 0 tial value and iterating on Ψp in each volume in order to 0 0.05 0.1 0.15 avoid the formation of sheet currents on the interfaces, thereby ensuring that the current profile remains smooth. w We would like to emphasize that, regardless of the ap- plied constraint and for all ∆0, the obtained equilibria (1) FIG. 4: Black circles: island width, w, obtained from SPEC 0 0 have lower energy than the initial state, (2) satisfy the as a function of Ψw, for two cases: ∆ a ≈ 1 and ∆ a ≈ 2. ∗ MRxMHD equilibrium equations, and (3) correspond to Vertical blue lines: location of the marginal point, Ψw = Ψw, above which the SPEC equilibrium is stable. Oblique red line: minima of the MRxMHD energy functional [26]. It is also maximum allowed size of the island wmax = 2ΨwLx = 4πΨw. important to notice a limitation of our results. For suffi- 1/4 Dashed lines: saturated island width obtained from SPEC, ciently large islands, w/a (κ⊥/κk) , the flattening of wsat, as given by the maximum of the achieved island size. the temperature modifies the plasma resistivity and thus jz(ψ) must be modified accordingly in order to balance the constant induced electric field [35]. Here κk and κ⊥ SPEC is run nonlinearly in order to find saturated are the parallel and perpendicular heat conductivities. equilibria by starting from initially unstable equilibria, The prediction of wsat with SPEC in such regime [35] ∆0 > 0, and perturbing the geometry of the ideal inter- is beyond the scope of this paper and is left for future investigations. In general, our procedure should be able faces, xl(θ), according to the unstable eigenmode. For each value of ∆0, the current sheet aspect ratio a/L is to directly predict the saturation of tearing modes so as determined (Figure 2), and a choice must be made for long as a constraint akin to Eq. (9) is obtained from the considered model equations. Ψw. In the limit Ψw → 0, the volume available for reconnection vanishes and therefore the maximum island We would like to remark that the SPEC results shown 5 in Fig. 3 are numerically converged in the sense that in- maks and arbitrarily shaped stellarators. To that aim, creasing further the Fourier resolution (here m = 4 modes future investigations will start by extending these calcu- are considered) or the number of volumes (here N = 121) lations to cylindrical geometry, the results of which can does not significantly change the values of wsat. Finally, be verified against generalizations of the POEM theory we would like to emphasize that each resistive MHD cal- to asymmetric equilibrium current profiles [37]. culation (each blue star in Fig. 3) takes about 1 CPU hour, while each corresponding SPEC run (red circles in Fig. 3) takes about 5 seconds. In this letter, we have demonstrated, for the first time, that it is possible to use an equilibrium code to directly We acknowledge very helpful discussions with Ami- predict the nonlinear saturation of tearing modes, with- tava Bhattacharjee, Per Helander, Robert Dewar, Adelle out resolving the complex resistive-dependent dynamics Wright, Alessandro Zocco and Jonathan Graves. This and without free parameters. These saturated states work has been carried out within the framework of the are lower energy MRxMHD equilibria of a priori un- EUROfusion Consortium and has received funding from known helicity and are obtained by exploiting the quasi- the Euratom research and training programme 2014 - invariance of the current profile. These findings provide a 2018 and 2019 - 2020 under grant agreement No 633053. precise answer to the recent suggestion that tearing sat- The views and opinions expressed herein do not necessar- uration occurs “when reaching some topologically con- ily reflect those of the European Commission. This work strained minimal energy state” [14]. In practice, nu- was also supported by a grant from the Simons Foun- merical codes like SPEC or BIEST [36], which calcu- dation/SFARI (560651, AB). The data that support the late MRxMHD equilibria in toroidal geometry, may be findings of this study are available from the correspond- used to quickly predict saturated island widths in toka- ing author upon reasonable request.

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