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Register a new userHigh-m Kink/Tearing Modes in Cylindrical Geometry
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The global ideal kink equation, for cylindrical geometry and zero beta, is simplified in the high poloidal mode number limit and used to determine the tearing stability parameter, $Delta^prime$. In the presence of a steep monotonic current gradient, $Delta^prime$ becomes a function of a parameter, $sigma_0$, characterising the ratio of the maximum current gradient to magnetic shear, and $x_s$, characterising the separation of the resonant surface from the maximum of the current gradient. In equilibria containing a current spike, so that there is a non-monotonic current profile, $Delta^prime$ also depends on two parameters: $kappa$, related to the ratio of the curvature of the current density at its maximum to the magnetic shear, and $x_s$, which now represents the separation of the resonance from the point of maximum current density. The relation of our results to earlier studies of tearing modes and to recent gyro-kinetic calculations of current driven instabilities, is discussed, together with potential implications for the stability of the tokamak pedestal.
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Double-tearing modes (DTMs) have been proposed as a driver of `off-axis sawtooth crashes in reverse magnetic shear tokamak configurations. Recently differential rotation provided by equilibrium sheared flows has been shown capable of decoupling the two DTM resonant layers, slowing the growth the instability. In this work we instead supply this differential rotation using an electron diamagnetic drift, which emerges in the presence of an equilibrium pressure gradient and finite Larmor radius physics. Diamagnetic drifts have the additional benefit of stabilizing reconnection local to the two tearing layers. Conducting linear and nonlinear simulations with the extended MHD code MRC-3d, we consider an m=2, n=1 cylindrical double-tearing mode. We show that asymmetries between the resonant layers and the emergence of an ideal MHD instability cause the DTM evolution to be highly dependent on the location of the pressure gradient. By locating a strong drift near the outer, dominant resonant surface are we able to saturate the mode and preserve the annular current ring, suggesting that the appearance of DTM activity in advanced tokamaks depends strongly on the details of the plasma pressure profile.
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 external forcing or a sheared velocity profile are involved. The calculation provides a new relationship between the saturated island width and the $Delta $ stability parameter that involves a $ln{w/w_{0}}$ term, where $w_{0}$ is a nonlinear scaling length that was missing in previous work. It also yields the modification of the equilibrium magnetic flux function.
Kinetic treatments of drift-tearing modes that match an inner resonant layer solution to an external MHD region solution, characterised by $Delta^{prime}$, fail to properly match the ideal MHD boundary condition on the parallel electric field, $E_{parallel}.$ In this paper we demonstrate how consideration of ion sound and ion Landau damping effects achieves this and place the theory on a firm footing. As a consequence, these effects contribute quite significantly to the critical value of $Delta^{prime}$ for instability of drift-tearing modes and play a key role in determining the minimum value for this threshold.
The nonlinear dynamo effect of tearing modes is derived with the resistive MHD equations. The dynamo effect is divided into two parts, parallel and perpendicular to the magnetic field. Firstly, the force-free plasma is considered. It is found that the parallel dynamo effect drives opposite current densities at the different sides of the rational surface, making the $lambda =boldsymbol{j}cdotboldsymbol{B}/|boldsymbol{B}|^2$ profile completely flattened near the rational surface. There are many rational surfaces for the turbulent plasma, which means the plasma is tending to relax into the Taylor state. In contrast, a bit far from the rational surface, the parallel dynamo effect is much smaller, and the nonlinear dynamo form approximates the quasilinear form. Secondly, the pressure gradient is included. It is found that rather than the $lambda$ profile, the $boldsymbol{j}cdotboldsymbol{B}$ profile is flattened by the parallel dynamo effect. Besides, the perpendicular dynamo effect of tearing modes is found to eliminate the pressure gradient near the rational surface. In addition, our result also provides another basis for the assumption that current density is flat in the magnetic island for the tearing modes theory.
The effects of line-tying on resistive tearing instability in slab geometry is studied within the framework of reduced magnetohydrodynamics (RMHD).citep{KadomtsevP1974,Strauss1976} It is found that line-tying has a stabilizing effect. The tearing mode is stabilized when the system length $L$ is shorter than a critical length $L_{c}$, which is independent of the resistivity $eta$. When $L$ is not too much longer than $L_{c}$, the growthrate $gamma$ is proportional to $eta$ . When $L$ is sufficiently long, the tearing mode scaling $gammasimeta^{3/5}$ is recovered. The transition from $gammasimeta$ to $gammasimeta^{3/5}$ occurs at a transition length $L_{t}simeta^{-2/5}$.
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