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Register a new userNonlinear stability of the ideal magnetohydrodynamic interchange mode at marginal conditions in a transverse magnetic field
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The stability of the ideal magnetohydrodynamic (MHD) interchange mode at marginal conditions is studied. A sufficiently strong constant magnetic field component transverse to the direction of mode symmetry provides the marginality conditions. A systematic perturbation analysis in the smallness parameter, $|b_2/B_c|^{1/2}$, is carried out, where $B_c$ is the critical transverse magnetic field for the zero-frequency ideal mode, and $b_2$ is the deviation from $B_c$. The calculation is carried out to third order including nonlinear terms. It is shown that the system is nonlinearly unstable in the short wavelength limit, i.e., a large enough perturbation results in instability even if $b_2/B_c>0$ (linearly stable). The normalized amplitude for instability is shown to scale as $|b_2/B_c|^{1/2}$. A nonlinear, compressible, MHD simulation is done to check the analytic result. Good agreement is found, including the critical amplitude scaling.
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The rapid deposition of energy by Edge Localised Modes (ELMs) onto plasma facing components, is a potentially serious issue for large Tokamaks such as ITER and DEMO. The trigger for ELMs is believed to be the ideal Magnetohydrodynamic Peeling-Ballooning instability, but recent numerical calculations have suggested that a plasma equilibrium with an X-point - as is found in all ITER-like Tokamaks, is stable to the Peeling mode. This contrasts with analytical calculations (G. Laval, R. Pellat, J. S. Soule, Phys Fluids, {bf 17}, 835, (1974)), that found the Peeling mode to be unstable in cylindrical plasmas with arbitrary cross-sectional shape. However the analytical calculation only applies to a Tokamak plasma in a cylindrical approximation. Here, we re-examine the assumptions made in cylindrical geometry calculations, and generalise the calculation to an arbitrary Tokamak geometry at marginal stability. The resulting equations solely describe the Peeling mode, and are not complicated by coupling to the ballooning mode, for example. We find that stability is determined by the value of a single parameter $Delta$ that is the poloidal average of the normalised jump in the radial derivative of the perturbed magnetic fields normal component. We also find that near a separatrix it is possible for the energy principles $delta W$ to be negative (that is usually taken to indicate that the mode is unstable, as in the cylindrical theory), but the growth rate to be arbitrarily small.
In the first part to this papercite{part1} it was shown how a simple Magnetohydrodynamic model could be used to determine the stability of a Tokamak plasmas edge to a Peeling (External Kink) mode. Stability was found to be determined by the value of $Delta$, a normalised measure of the discontinuity in the radial derivative of the radial perturbation to the magnetic field at the plasma-vacuum interface. Here we calculate $Delta$, but in a way that avoids the numerical divergences that can arise near a separatrices X-point. This is accomplished by showing how the method of conformal transformations may be generalised to allow their application to systems with a non-zero boundary condition, and using the technique to obtain analytic expressions for both the vacuum energy and $Delta$. A conformal transformation is used again to obtain an equilibrium vacuum field surrounding a plasma with a separatrix. This allows the subsequent evaluation of the vacuum energy and $Delta$. For a plasma-vacuum boundary that approximates a separatrix, the growth rate $gamma$ normalised by the Aflven frequency $gamma_A$ is then found to have $ln(gamma/gamma_A)=-{1/2} ln (q/q)$. Consequences for Peeling mode stability are discussed.
The CFETR baseline scenario is based on a H-mode equilibrium with high pedestal and highly peaked edge bootstrap current, along with strong reverse shear in safety factor profile. The stability of ideal MHD modes for the CFETR baseline scenario has been evaluated using NIMROD and AEGIS codes. The toroidal mode numbers (n=1-10) are considered in this analysis for different positions of perfectly conducting wall in order to estimate the ideal wall effect on the stability of ideal MHD modes for physics and engineering designs of CFETR. Although, the modes (n=1-10) are found to be unstable in ideal MHD, the structure of all modes is edge localized. Growth rates of all modes are found to be increasing initially with wall position before they reach ideal wall saturation limit (no wall limit). No global core modes are found to be dominantly unstable in our analysis. The design of $q_{min}>2$ and strong reverse shear in $q$ profile is expected to prevent the excitation of global modes. Therefore, this baseline scenario is considered to be suitable for supporting long time steady state discharge in context of ideal MHD physics, if ELMs could be controlled.
We find and investigate via numerical simulations self-sustained two-dimensional turbulence in a magnetohydrodynamic flow with a maximally simple configuration: plane, noninflectional (with a constant shear of velocity) and threaded by a parallel uniform background magnetic field. This flow is spectrally stable, so the turbulence is subcritical by nature and hence it can be energetically supported just by transient growth mechanism due to shear flow nonnormality. This mechanism appears to be essentially anisotropic in spectral (wavenumber) plane and operates mainly for spatial Fourier harmonics with streamwise wavenumbers less than a ratio of flow shear to the Alfv{e}n speed, $k_y < S/u_A$ (i.e., the Alfv{e}n frequency is lower than the shear rate). We focused on the analysis of the character of nonlinear processes and underlying self-sustaining scheme of the turbulence, i.e., on the interplay between linear transient growth and nonlinear processes, in spectral plane. Our study, being concerned with a new type of the energy-injecting process for turbulence -- the transient growth, represents an alternative to the main trends of MHD turbulence research. We find similarity of the nonlinear dynamics to the related dynamics in hydrodynamic flows -- to the emph{bypass} concept of subcritical turbulence. The essence of the analyzed nonlinear MHD processes appears to be a transverse redistribution of kinetic and magnetic spectral energies in wavenumber plane [as occurs in the related hydrodynamic flow, see Horton et al., Phys. Rev. E {bf 81}, 066304 (2010)] and differs fundamentally from the existing concepts of (anisotropic direct and inverse) cascade processes in MHD shear flows.
We study the linear and nonlinear evolution of the tearing instability on thin current sheets by means of two-dimensional numerical simulations, within the framework of compressible, resistive magnetohydrodynamics. In particular we analyze the behavior of current sheets whose inverse aspect ratio scales with the Lundquist number $S$ as $S^{-1/3}$. This scaling has been recently recognized to yield the threshold separating fast, ideal reconnection, with an evolution and growth which are independent of $S$ provided this is high enough, as it should be natural having the ideal case as a limit for $Stoinfty$. Our simulations confirm that the tearing instability growth rate can be as fast as $gammaapprox 0.6,{tau_A}^{-1}$, where $tau_A$ is the ideal Alfvenic time set by the macroscopic scales, for our least diffusive case with $S=10^7$. The expected instability dispersion relation and eigenmodes are also retrieved in the linear regime, for the values of $S$ explored here. Moreover, in the nonlinear stage of the simulations we observe secondary events obeying the same critical scaling with $S$, here calculated on the emph{local}, much smaller lengths, leading to increasingly faster reconnection. These findings strongly support the idea that in a fully dynamic regime, as soon as current sheets develop, thin and reach this critical threshold in their aspect ratio, the tearing mode is able to trigger plasmoid formation and reconnection on the local (ideal) Alfvenic timescales, as required to explain the explosive flaring activity often observed in solar and astrophysical plasmas.
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