The stability properties of m≥2 tearing instabilities in tokamak plasmas are analyzed. A boundary layer theory is used to find asymptotic solutions to the ideal external kink equation, which are used to obtain a simple analytic expression for the tearing instability parameter Δ’. This calculation generalizes previous work on this topic by considering more general toroidal equilibria (however, toroidal coupling effects are ignored.) Constructions of Δ’ are obtained for plasmas with finite beta and for islands that have nonzero width. A simple heuristic estimate is given for the value of the saturated island width when the instability criterion is violated. A connection is made between the calculation of the asymptotic matching parameter in the finite beta and island width case to the nonlinear analog of the Glasser effect [Phys. Fluids 18, 875 (1975)].

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Stability of tearing modes in tokamak plasmas

Numerical studies show that the m=1 tearing mode continues to grow exponentially well into the nonlinear regime, in contrast with the slow, ‘‘Rutherford,’’ growth of m>1 modes. A single helicity calculation is presented which generalizes that of Rutherford [Phys. Fluids 16, 1903 (1973)] to the case when the constant‐ψ approximation is invalid. As in that theory, the parallel current becomes an approximate flux function when the island size W exceeds the linear tearing layer width. However, for the m=1 mode, W becomes proportional to δB, rather than (δB)1/2 above this critical amplitude. This implies that the convective nonlinearity in Ohm’s law, which couples the m=0 component to the m=1 component, dominates the resistive diffusion term. The balance between the inductive electric field and this convective nonlinearity results in exponential growth. Assuming the form of the perturbed fields to be like that of the linear mode, we find that growth occurs at 71% of the linear rate.

Analytic theory of the nonlinear m=1 tearing mode

The saturation of the tearing mode instability is described within the standard framework of reduced magnetohydrodynamics in the case of an r-dependent or 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 external forcing or a sheared velocity profile are involved. The calculation provides a new relationship between the saturated island width and the Δ′ stability parameter that involves a lnw∕w0 term, where w0 is a nonlinear scaling length that was missing in previous work. It also yields the modification of the equilibrium magnetic-flux function.

Rigorous approach to the nonlinear saturation of the tearing mode in cylindrical and slab geometry

Higher order asymptotic matching procedure is developed to derive the nonlinear equation for saturated magnetic island with the symmetric profile of the equilibrium current. The theory extends the previous results to include higher order effects such as nonlinear modification of the equilibrium current and asymmetry in the boundary conditions in the outer region. It is shown that due to a finite width of the nonlinear region, the magnitude of the magnetic flux at the rational surface is different from the asymptotic value found from the outer solution, resulting in the suppression of the island growth for higher values of Δ′ parameter.

Higher order and asymmetry effects on saturation of magnetic islands

The growth of a tearing instability in Rutherford's non-linear regime is investigated. Using a singular perturbations technique, we recover to lowest order Rutherford's result. To the following order we show that the mode generates a quasi-linear deformation of the equilibrium flux profile, whose resistive diffusion slows down the growth and shows the possibility of a saturation of the instability Etude de la croissance d'une instabilite de dechirure dans le regime non lineaire de Rutherford. L'usage des techniques de perturbations singulieres permet de trouver a l'ordre le plus bas, les resultats de Rutherford. A l'ordre suivant on montre que le mode engendre une deformation quasi lineaire du profil d'equilibre du flux magnetique; la diffusion resistive de cette deformation ralentit la croissance du mode et indique la possibilite d'une saturation de l'instabilite

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M. Tagger

Papers

Quasi-linear evolution of tearing modes