### Theory Weekly Highlights for December 2005

##### December 23, 2005

Dr. Klaus Hallatschek has joined the GA theory group for a one-year visit as the 2005 Rosenbluth Award recipient. Dr. Hallatschek received his doctoral degree form the Technical University-Munchen in 1998 and subsequently took a position as a research scientist at the Max-Planck Institute of Plasma Physics in Garching, Germany. At GA he will be working closely with GA scientists in furthering his research in core and edge plasma turbulence studies.

Dr. S. Guo from the RFP Group in Frascati has completed a four week visit at GA during which she worked with Dr. Ming Chu on the physics aspects of resistive wall modes common to RFPs and Tokamaks and on related MHD phenomena.

##### December 16, 2005

The MHD and two fluid growth rates of a low beta m=2/n=1 tearing mode in the presence of well-separated central sawtooth oscillations were examined using reconstructions of experimental DIII-D equilibria. The outer ideal MHD solutions between the rational q=1, 2, and 3 surfaces were determined using the PEST3 code for a low beta equilibrium and the outer region solution was matched asymptotically to the Glasser, Greene and Johnson resistive MHD inner layer solutions. This yields a dispersion relation for the linear growth rate in the form of a matrix equation for the matching conditions. The most important effects in the dispersion relation are found to be the resistive interchange parameter D_R and the coupling to the 1/1 surface, both of which were stabilizing in the case considered. Two-fluid diamagnetic effects reduce the growth rates significantly, while inducing a mode rotation near the electron diamagnetic frequency. It is expected that this will be important for large diamagnetic frequencies since the rotation shear between surfaces will then significantly affect the coupling between them through the ideal matching data. This will be studied in future work.

##### December 09, 2005

It has been shown that the implementation of the nonconforming and conforming Finite Hybrid element approximations in GATO are equivalent if evaluated at the half node points. The Finite Elements represent the small Frobenius MHD solutions. This implies that the traditional procedure implemented in the code, of reconstructing the final small solution from the conforming (and therefore continuous) elements is valid, despite the discontinuity of the basis (nonconforming) finite elements used to describe the actual displacements. This reconstruction scheme had previously been chosen only heuristically. The nonconforming elements also then provide a straightforward way to allow displacement jumps at rational surfaces generated by the presence of a large Frobenius solution component. This allows a relatively easy extension of the code to treat limiting marginal, but otherwise physically valid, eigenmodes manifested as tearing modes.

##### December 02, 2005

Several key results were obtained in the theoretical formalism of the plasma response to external nonaxisymmetric perturbations. For the linearized response, the conditions for completeness of the 2D eigenfunctions of the ideal MHD operator, as a basis for the plasma response, were obtained. Completeness can be lost following projection on the plasma boundary and then inversion back to the full plasma domain. In addition to the obvious case where internal modes exist, the conditions can be violated in certain well-defined situations that can be easily monitored and possibly avoided. In addition, the Nuhrenberg-Boozer application for determining the resonant displacement jumps and associated island sizes was generalized to any (including nonresonant) response feature. The generalization can then be applied to determine the specific externally applied perturbations needed to induce nonresonant perturbations, for independent control of MHD modes and plasma rotation.

**Disclaimer**

These highlights are reports of research work in progress and are accordingly subject to change or modification