Within the traditional frame of reduced MHD, a new rigorous perturbation expansion provides the equation ruling the nonlinear growth and saturation of the tearing mode for any current gradient. The small parameter is the magnetic island width w. For the first time, the final equation displays at once terms of order w ln(1/w) and w which have the same magnitude for practical purposes; two new O(w) terms involve the current gradient. The technique is applicable to the case of an external forcing. The solution for a static forcing is computed explicitly and it exhibits three physical regimes.
The nonlinear saturation of the tearing mode is revisited in slab geometry by taking into account higher-order harmonics in the outer solution. The general formalism for tackling this problem in the case of a vanishing current gradient at the resonant surface is derived. It is shown that, although the higher-order harmonics lead to corrections in the final saturation equation, they are of higher order in the perturbation parameter, which provides a formal proof that the standard one-harmonic approach is asymptotically correct.
The saturation of the tearing mode instability is described within the standard framework of reduced magnetohydrodynamics (RMHD) in the case of an $r$-dependent or of a uniform resistivity profile. Using the technique of matched asymptotic expansions, where the perturbation parameter is the island width $w$, the problem can be solved in two ways: with the so-called flux coordinate method, which is based on the fact that the current profile is a flux function, and with a new perturbative method that does not use this property. The latter is applicable to more general situations where an externa…
We study the nonlinear evolution of the resistive tearing mode in slab geometry in two dimensions. We show that, in the strongly driven regime (large Delta'), a collapse of the X-point occurs once the island width exceeds a certain critical value ~1/Delta'. A current sheet is formed and the reconnection is exponential in time with a growth rate ~eta^1/2, where eta is the resistivity. If the aspect ratio of the current sheet is sufficiently large, the sheet can itself become tearing-mode unstable, giving rise to secondary islands, which then coalesce with the original island. The saturated stat…
Recent MHD simulations find that the reversal of the local resistive interchange parameter $D_R$ from negative to positive due to impurity radiation cooling is able to trigger the resistive tearing mode growth in a tokamak above a threshold in impurity level. A layer of perturbed Pfirsch-Schlüter current density and resistivity are also induced by the impurity radiation, which further govern the tearing mode growth and saturation in the nonlinear stage. The impurity threshold and the tearing mode growth strongly depend on the parallel thermal conductivity, and such a dependence derives from th…
In this study, the evolution of a highly unstable m = 1 resistive tearing mode, leading to plasmoid formation in a Harris sheet is studied in the framework of full MHD model using the NIMROD simulation. Following the initial nonlinear growth of the primary m = 1 island, the X-point develops into a secondary elongated current sheet that eventually breaks into plasmoids. Two distinctive viscous regimes are found for the plasmoid formation and saturation. In the low viscosity regime (i.e. P r . 1), the plasmoid width increases sharply with viscosity, whereas in the viscosity dominant regime (i.e.…
A model system of equations has been derived to describe a toroidally rotating tokamak plasma, unstable to Resistive Wall Modes (RWMs) and metastable to Neoclassical Tearing Modes (NTMs), using a linear RWM model and a nonlinear NTM model. If no wall is present, the NTM growth shows the typical threshold/saturation island widths, whereas a linearly unstable kink mode grows exponentially in this model plasma system. When a resistive wall is present, the growth of the linearly unstable RWM is accelerated by an unstable island: a form of coupled RWM-NTM mode. Crucially, this coupled system has no…
The equilibrium of a finite-beta cylindrical RFP plasma in the presence of saturated-amplitude tearing modes is investigated. The singularities arising from the perturbative analysis of the MHD force balance equation JxB=grad(p) are resolved through a proper regularization of the cylindrical symmetric component of the pressure and parallel current density profiles, by setting to zero their gradient at the modes rational surfaces. An equilibrium model, which satisfies the regularization rule at the various rational surfaces, is developed. The comparison with the experimental data from the Rever…
This is a brief account of our numerical study of the tearing mode reconnection. We demonstrate two main points. First, we show that, given sufficiently small resistivity, the Rutherford regime always exists; larger values of Delta' require smaller values of resistivity. Rutherford's negligible-inertia assumption is validated and the asymptotically linear dependence of the time derivative of the island width on the resistivity and Delta' is confirmed. Second, we find that, at large Delta', the Rutherford regime is followed by a nonlinear stage of fast growth linked to X-point collapse and form…
A detailed understanding of island seeding is crucial to avoid (N)TMs and their negative consequences like confinement degradation and disruptions. In the present work, we investigate the growth of 2/1 islands in response to magnetic perturbations. Although we use externally applied perturbations produced by resonant magnetic perturbation (RMP) coils for this study, results are directly transferable to island seeding by other MHD instabilities creating a resonant magnetic field component at the rational surface. Experimental results for 2/1 island penetration from ASDEX Upgrade are presented e…
We investigate the role of kinetic instabilities driven by a proton anisotropy on the onset of magnetic reconnection by means of 2-D hybrid simulations. The collisionless tearing of a current sheet is studied in the presence of a proton temperature anisotropy in the surrounding plasma. Our results confirm that anisotropic protons within the current sheet region can significantly enhance/stabilize the tearing instability of the current. Moreover, fluctuations associated to linear instabilities excited by large proton temperature anisotropies can significantly influence the stability of the plas…
Magnetic reconnection associated with the tearing instability occurring in double-current sheet systems is investigated within the framework of reduced resistive magnetohydrodynamics (MHD) in a two-dimensional Cartesian geometry. The explosive non linear phase is particularly explored using the adaptive finite-element FINMHD code. The critical aspect ratio, that is defined as the minimum $L/x_s$ ratio (with $L$ and $x_s$ being the periodic system length and half-distance between the two current layers respectively) necessary for non linear destabilization after the linear and early non linear …
We present nonlinear resistive MHD simulations of the response of a rotating plasma to an error field when the plasma has weakly damped linear tearing modes (TM's), stabilized by pressure gradient and favorable curvature. Favorable curvature leads to the Glasser effect: the occurrence of real frequencies and stabilization with positive stability index $Δ'$. A cylinder with hollow pressure is used to model the toroidal favorable curvature. Linear simulations with rotation and an error field $\tildeψ_w$ show, in agreement with analytic results, that the peak reconnected flux occurs for a rotatio…
Brée, Demircan, and Steinmeyer [Phys. Rev. Lett. 106, 183902 (2011)] recently calculated higher-order Kerr effect (HOKE) coefficients using the nonlinear Kramers-Kronig relations. We show that the saturated and negative nonlinear index of refraction obtained by Brée et al. is an artifact of their use of a perturbative theory of ionization in an intensity region where it breaks down.
