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The `Alfv\'en Paradox' is that as resistivity decreases, the discrete eigenmodes do not converge to the generalized eigenmodes of the ideal Alfv\'en continuum. To resolve the paradox, the $\epsilon$-pseudospectrum of the RMHD operator is considered. It is proven that for any $\epsilon$, the $\epsilon$- pseudospectrum contains the Alfv\'en continuum for sufficiently small resistivity. Formal $\epsilon-pseudoeigenmodes$ are constructed using the formal Wentzel-Kramers-Brillouin-Jeffreys solutions, and it is shown that the entire stable half-annulus of complex frequencies with $\rho{|\omega|^2}=|\bf{v} \cdot \bf{B}(x)|^2$ is resonant to order $\epsilon$, i.e.~belongs to the $\epsilon-pseudospectrum$. The resistive eigenmodes are exponentially ill-conditioned as a basis and the condition number is proportional to $\exp(R_M^{1\over 2})$, where $R_M$ is the magnetic Reynolds number.

In this work, we address the question of the influence of magnetic islands on the perpendicular transport due to steady-state ITG turbulence on the energy transport time scale. We demonstrate that turbulence can cross the separatrix and enhance the perpendicular transport across magnetic islands. As the perpendicular transport in the interior of the island sets the critical island size needed for growth of neoclassical tearing modes, this increased transport leads to a critical island size larger than that predicted from considering collisional conductivities, but smaller than that using anomalous effective conductivities. We find that on Bohm time scales, the turbulence is able to re-establish the temperature gradient across the island for islands widths $w \lesssim \lambda_{turb}$, the turbulence correlation length. The reduction in the island flattening is estimated by comparison with simulations retaining only the perpendicular temperature and no turbulence. At intermediate island widths, comparable to $\lambda_{turb}$, turbulence is able to maintain finite temperature gradients across the island.

In this work, the Flux-Coordinate Independent (FCI) approach to plasma turbulence simulations is formulated for the case of generic, static magnetic fields, including those possessing stochastic field lines. It is then demonstrated that FCI is applicable to nonlinear turbulent problems with and without X-point geometry. In particular, by means of simulations with the FENICIA code, it is shown that the standard features of ITG modes are recovered with reduced toroidal resolution. Finally, ITG turbulence under the influence of a static island is studied on the transport timescale with ITER-like parameters, showing the wide range of applicability of the method.

A Flux Coordinate Independent (FCI) approach for anisotropic systems, not based on magnetic flux coordinates has been introduced in [F. Hariri and M. Ottaviani, Comput. Phys. Commun., 184,2419 (2013)]. In this paper, we show that the approach can tackle magnetic configurations including X-points. Using the code FENICIA, an equilibrium with a magnetic island has been used to show the robustness of the FCI approach to cases in which a magnetic separatrix is present in the system, either by design or as a consequence of instabilities. Numerical results are in good agreement with the analytic solutions of the sound-wave propagation problem. Conservation properties are verified. Finally, the critical gain of the FCI approach in situations including the magnetic separatrix with an X-point is demonstrated by a fast convergence of the code with the numerical resolution in the direction of symmetry. The results highlighted in this paper show that the FCI approach should be able to address turbulent transport problems in X-point geometries., Comment: figures have been removed on purpose. Article submitted for publication with full content

Unstable electrostatic resistive modes, driven by density gradients, are identified in a bounded sheared slab. The boundary conditions play a crucial role and are shown to change the nature of the problem, which is related to so called "universal" mode. The dispersion relation and the structure of the eigenmodes of the instability are derived and are shown to depend on a limited set of dimensionless parameters. The occurrence and possible impact of these modes on numerical simulations and actual plasmas are discussed., Comment: 27 pages, 10 figures. The following article has been submitted to Physics of Plasmas. After it is published it will be found at http://pop.aip.org/pop

Turbulence simulation codes can exploit the flute-like nature of plasma turbulence to reduce the effective number of degrees of freedom necessary to represent fluctuations. This can be achieved by employing magnetic coordinates of which one is aligned along the magnetic field. This work presents an approach in which the position along the field lines is identified by the toroidal angle, rather than the most commonly used poloidal angle. It will be shown that this approach has several advantages. Among these, periodicity in both angles is retained. This property allows moving to an equivalent representation in Fourier space with a reduced number of toroidal components. It will be shown how this duality can be exploited to transform conventional codes that use a spectral representation on the magnetic surface into codes with a field-aligned coordinate. It is also shown that the new approach can be generalised to get rid of magnetic coordinates in the poloidal plane altogether, for a large class of models. Tests are carried out by comparing the new approach with the conventional approach employing a uniform grid, for a basic ion temperature gradient (ITG) turbulence model implemented by the two corresponding versions of the ETAI3D code. These tests uncover an unexpected property of the model, that localized large parallel gradients can intermittently appear in the turbulent regime. This leaves open the question whether this is a general property of plasma turbulence, which may lead one to reconsider some of the usual assumptions on micro-turbulence dynamics., Comment: 19 pages (once in pdf format). 1 LaTeX file and 10 eps figures in the zip folder

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.

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., Comment: 4 pages, submitted to Physical Review Letters

During the 2003 experimental campaign, the aim of realizing reliable fully non inductive discharges has been successfully completed in Tore Supra. One the major difficulties in optimizing this scenario comes from MHD stability issues. Indeed, the current profile obtained with LH antennas in these experiments is prone to the triggering of single or double tearing modes. Under certain conditions, the discharges enters in a regime of permanent MHD activity. The present work investigates the MHD properties of these fully non-inductive discharges in the operational space defined by the toroidal magnetic field B, the total plasma current Ip, and the mean parallel index of LH waves . Linear MHD stability properties divide the operational space in regions separated by rational values of the minimum of the safety factor. We discuss in our work the localisation of the transition between benign MHD activity and the MHD regime., Comment: 12th International Congress on Plasma Physics, 25-29 October 2004, Nice (France)