Code Conversion and Mode Conversion

Using the spectral representation in Eq. (1b) and from the condition that $k_\bot\rho_\textrm{i} \simeq 1$ we estimate the maximum poloidal mode number needed for mode conversion studies. Taking $k_\bot \sim \frac{m}{r}$, the required resolution is approximately $M_\textrm{max} \equiv r/\rho_\textrm{i} \equiv \rho^*$. In Fig.1, two plots of the power spectrum of the Fourier transform of the right circularly polarized component of the electric field are shown for a specified set of flux surfaces. The two plots demonstrate that in the bulk of the plasma and especially at mode-conversion layers ($\rho=0.5$ for the cases shown) and at the outermost surfaces, several hundred poloidal modes are needed for convergence. Even at the outermost flux surfaces (around $r/a=0.9$) the amplitude is down to 1%. Simulations at higher numbers of poloidal modes show that the spectrum begins to fall off much more quickly just past this resolution and the solution doesn't change appreciably. The parts of the spectrum shown at $\vert M\vert > 64$ in the left panel and at $\vert M\vert > 256$ in the right are the part of the spectrum above the Nyquist frequency. The poloidal resolution is twice the spectral resolution to avoid aliasing.

Figure 1: The left panel is a D($^3$He) mode-conversion case in Alcator C-Mod with only 127 poloidal modes. The power-spectrum does not fall off at the largest modes and indicates insufficient resolution. Increasing to 511 modes in the right panel yields a well-converged spectrum, even at the largest radii.

We may also graphically see the non-physical effects of insufficient resolution in the left panel of Fig. 2. At this low resolution the loss of horizontal localization is exaggerated and electric field bleeds over along the flux surface into ion cyclotron resonances ($^3$He indicated by dashed red line), causing ``spurious'' ion power absorption at that location. In contrast, when the resolution is increased to the point where the spectrum was shown to converge, the mode conversion layer becomes well localized vertically and the spurious ion damping vanishes.

Figure 2: The left panel shows a blow-up of the mode conversion region for the same scenario as in Fig. 1 with only 15 poloidal modes used to resolve the layer. The dashed red line indicates the ($3$He) cyclotron resonance. With 255 modes on the right, the vertical layer is well localized and the multiple scales of waves are observed.

New physics regimes are available at these higher resolutions. In the typical Asdex-Upgrade MC discharge in Fig. 3, the layer is far from the center of the device and requires much more poloidal resolution than the C-Mod case. While the C-Mod simulation converged acceptably at $N_m$ = 255, Asdex-Upgrade requires at least $N_m$ = 511. The complete poloidal cross-section of Fig. 3 also shows the three waves (FW, ICW, and IBW) all present simultaneously. The large structures on the right side are the FW propagating in from the antenna. The midrange waves off axis at about -20 cm are the ICW traveling backward to the right and the smallest wavelength mode in the midplane to the left of -20cm is the IBW. These cases were not possible with the serial version of the code and represent the extension of the TORIC code to a new class of problems.

Figure 3: The simulation of a D($^3$He) mode conversion scenario in Asdex-Upgrade shown here requires twice the resolution of the more central case in C-Mod.

We may progress to even higher resolution and run simulations in the lower hybrid range of frequencies (LHRF). The case in Fig. 4 has the proper parallel wavelength and frequency for lower hybrid waves, but couples only to the fast wave due to the antenna current strap being oriented perpendicular to ${\mathbf{B}}$. Presently, a new antenna model is being developed in the code to couple properly to the lower hybrid slow wave polarization.

Figure 4: The electric field from a LHRF simulation using TORIC. The coupled wave corresponds to the fast wave polarization.