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Register a new userMagnetohydrodynamic Stability at a Separatrix: Part II
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Added by
Anthony James Webster
Publication date
2009
fields
Physics
and research's language is
English
Authors
A.J. Webster
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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.
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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.
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.
The SOL width is a parameter of paramount importance in modern tokamaks as it controls the power density deposited at the divertor plates, critical for plasma-facing material survivability. An understanding of the parameters controlling it has consequently long been sought (Connor et al. 1999 NF 39 2). Prior to Chang et al.(2017 NF 57 11), studies of the tokamak edge have been mostly confined to reduced fluid models and simplified geometries, leaving out important pieces of physics. Here, we analyze the results of a DIII-D simulation performed with the full-f gyrokinetic code XGC1 which includes both turbulence and neoclassical effects in realistic divertor geometry. More specifically, we calculate the particle and heat ExB fluxes along the separatrix, discriminating between equilibrium and turbulent contributions. We find that the density SOL width is impacted almost exclusively by the turbulent electron flux. In this simulation, the level of edge turbulence is regulated by a mechanism we are only beginning to understand: $ abla B$-drifts and ion X-point losses at the top and bottom of the machine, along with ion banana orbits at the low field side (LFS), result in a complex poloidal potential structure at the separatrix which is the cause of the ExB drift pattern that we observe. Turbulence is being suppressed by the shear flows that this potential generates. At the same time, turbulence, along with increased edge collisionality and electron inertia, can influence the shape of the potential structure by making the electrons non-adiabatic. Moreover, being the only means through which the electrons can lose confinement, it needs to be in a balance with the original direct ion orbit losses to maintain charge neutrality.
In this work, we present the results of simulations carried out for N2-H2 capacitively coupled radio-frequency discharges, running at low pressure (0.3-0.9 mbar), low power (5-20 W), and for amounts of H2 up to 5 pct. Simulations are performed using a hybrid code that couples a two-dimensional time-dependent fluid module, describing the dynamics of the charged particles in the discharge, to a zero-dimensional kinetic module, that solves the Boltzmann equation and describes the production and destruction of neutral species. The model accounts for the production of several vibrationally and electronic excited states, and contains a detailed surface chemistry that includes recombination processes and the production of NHx molecules. The results obtained highlight the relevance of the interactions between plasma and surface, given the role of the secondary electron emission in the electrical parameters of the discharge and the critical importance of the surface production of ammonia to the neutral and ionic chemistry of the discharge.
We determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel-Korteweg-de Vries-Zakharov-Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio $b$ increases as the electron distribution becomes increasingly flat-topped. As $b$ and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle $theta$ at which the perturbation is applied. Solutions whose minimum value is zero and travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of $theta$ for which the first-order growth rate is not zero.
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