The TORIC Code and Physics Model

In this paper we discuss the FLR approach and the MPP modifications that have been made to increase the speed and resolution of the TORIC finite Larmor radius (FLR) full wave code. ``Full wave'', means that it solves Maxwell's equations in the presence of a plasma and wave antenna and so includes the effects of dispersion and mode-conversion, while FLR refers to the approximation that the Larmor radius is smaller than the perpendicular wave length and results in a finite order system, instead of an integral one. The system is solved for a fixed frequency with a linear plasma response [Eq. (1a)] in a mixed spectral-finite element basis [Eq. (1b)], where ${\mathbf{J}}^A$ and ${\mathbf{J}}^P$ are the antenna and plasma currents, $\sigma$ is the plasma conductivity, $m$ is the poloidal mode number, and $n_\phi$ is the toroidal mode number. The mixed Fourier-finite element basis introduces an algebraic parallel wave-number, $k_\Vert$, that depends on the local metric coordinates, $N_\tau$ and $R$, and $\Theta=\mathrm{atan} B_\tau/B_\phi$. $N_\tau$ reduces to the minor radius, $r$, in the limit of an orthogonal toroidal system and $\Theta$ is the angle between the poloidal and toroidal fields. The parallel wavenumber provides a connection to homogeneous theory that is useful in comparisons to the dispersion relations from plane-stratified theory and is exploited to modify the anti-Hermitian part of $\sigma$ in cases where the FLR approximation breaks down.

$$\begin{eqnarray}{\mathbf{\nabla}}\times {\mathbf{\nabla}}\times {... ...igma\left( k_\Vert^m, N_\tau \right) \cdot{\mathbf{E}}_m(N_\tau) \end{eqnarray}$$ (1a)

In TORIC the FLR approximation retains the second harmonic wave frequency and the second order ion gyro-radius effects, $\left(\rho_\textrm{i}= v_\textrm{ti}/\Omega_\textrm{ci}\right)$, for plasma interactions with the wave. This contributes terms from the dielectric that are of the same order or less as those that come from Maxwell's equations. It also retains the physics of the three ICRF waves: ion cyclotron waves (ICW)[8], ion Bernstein waves (IBW), and fast waves (FW). Near the mode conversion region, the FLR approximation breaks down and $k_\bot\rho_\textrm{i} \sim 1$, where $k_\bot$ is the perpendicular wave number. In these regions, damping from a WKB approximation is used to modify the anti-Hermitian part of the conductivity operator in TORIC to capture the proper damping, while the real part describing propagation remains unchanged[1]. This approximation holds if the particles experience phase decorrelation within a few gyro-periods of their wave interactions.