### Theory Weekly Highlights for June 2011

##### June 24, 2011

The physical accuracy and limitations of commonly used reduced collision operators were compared to the full Fokker-Planck operator recently implemented in the NEO code (see June 03 Highlight). The zeroth order Hirshman-Sigmar operator most closely followed the trend of the full Fokker-Planck operator for pure and impure plasmas, except in the highly collisional regime. This is due to missing the energy exchange in the zeroth order operator. While the full Hirshman-Sigmar operator, which also includes heating friction and energy diffusion, is most accurate for impure plasmas, it is not ideal for experimental analysis since it does not conserve momentum for species with unequal temperatures. A model with the full test particle component, but an ad hoc field particle component with simple momentum and energy conserving terms, commonly used in some PIC codes, does well in the Pfirsch-Schluter regime but is less accurate than the zeroth-order Hirshman-Sigmar operator in the banana regime. The Connor model is not accurate for impure plasmas due to lack of modeling the deceleration effect.

##### June 17, 2011

An analysis of the symmetry properties of the time-independent extended MHD equations shows that solutions of the, axisymmetric, ideal MHD equations remain solutions under reversals of the toroidal field, current density, rotation, or any combination thereof. Introducing non-axisymmetry, resistivity, and two-fluid effects each break different symmetries. In particular, both resistivity and two-fluid effects break the invariance of solutions under reversal of the toroidal rotation. Because the symmetry groups of ideal, resistive, and two-fluid MHD are distinct, it should be possible to ascertain the dominant physical mechanism of various phenomena through a series of experiments in which at least two of the toroidal field, current, and rotation are independently reversed. These results, which hold for nonlinear solutions in arbitrary geometry, should also be of use in testing numerical codes.

##### June 10, 2011

Initial tests of toroidal momentum transport with the Trapped Gyro-Landau Fluid model TGLF show good agreement between the predicted rotation and a limited set of DIII-D data. The first multi-species (electrons, deuterium, carbon), multi-channel (density, electron and ion temperature, toroidal rotation) predictions with TGLF have been performed using the XPTOR parallel transport code. The high accuracy neoclassical code NEO was used in these simulations.

##### June 03, 2011

The NEO code has been upgraded to include the full linearized Fokker-Planck collision operator. With this operator, NEO represents an exact calculation of the local neoclassical transport for multi-species plasmas. Highly accurate and fast numerical algorithms for treatment of the field particle operator, including the multiple scales that arise for multi-species plasmas, were explored. The method is Eulerian-based and uses an expansion in the cosine of the pitch angle of Associated Legendre polynomials. For the energy dimension, spectral methods, such as the commonly used Laguerre-based Sonine method, were found to yield rapid numerical precision loss, particularly for the cross-species field particle matrix elements for ion-electron plasmas. Finite element methods were also very inaccurate at large collision frequency, due to difficulty in obtaining exact conservation properties of the collision operator. The final optimal method was found to use zeroth order Laguerre polynomials of v^{2}, L_{m}^{0}(v^{2}), for the zeroth degree Legendre elements and first order Associated Laguerre polynomials, L_{m}^{1}(v^{2}), weighted with an additional factor v, otherwise.

**Disclaimer**

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