Study of Low-Frequency Core-Edge Coupling in a Tokamak: II. Spatial Channeling & Focusing In Antenna-Driven MHD

Abstract

Motivated by evidence for core-edge coupling in the form of double-peaked fishbone-like low-frequency modes ($\lesssim 20\,{\rm kHz}$) in KSTAR, which exhibit synchronized Alfvénic activity both in the central core and near the plasma edge [1], we study the nonlocal response of a tokamak plasma in a visco-resistive full MHD simulation model using the code MEGA. The waves are driven by an internal "antenna" that is localized both radially and azimuthally in the poloidal $(R,z)$ plane and has a sinusoidal form $\exp(inζ- iωt)$ with Fourier mode number $n=\pm 1$ in the toroidal angle $ζ$ and fixed angular frequency $ω$ in time $t$. By flattening the safety factor profile $q(r)$ at suitable locations in the minor radius $r$, we created plateaus in the low-frequency Alfvén continua that act as wave "receivers". First, we confirm that such continuum plateaus respond with a coherent quasi-mode even when the driving antenna is located at a distant radius. Second, by varying the antenna location, we confirm the expectation of inward drive being more efficient than outward drive, which we attribute to volumetric focusing. Third, we find that the central core also responds well at frequencies below the central Alfvénic continuum plateau, which could facilitate chirping. Our results show that a core-localized low-frequency response does not necessarily require core-localized drive nor an exactly matching continuum, but may be driven from the edge and sub-resonantly. It remains to be seen to what extent the examined effects play a role in double-peaked fishbone-like activity. Other possible contributing mechanisms are discussed to motivate further study. Our analyses also elucidate the mode structure formation process, from transients to quasi- or eigenmodes, here in the realm of MHD, and to be followed by a verification study against kinetic models.

Figures (30)

• Figure 1: Example of a moderate-strength double-peaked fishbone-like mode in KSTAR similar to those analyzed in Ref. Lee23 (adapted from Fig. 5 of Ref. Lee26). Panels (a) and (c) show time traces of the magnetic fluctuation signal $\dot{B}_\vartheta \equiv {\rm d}B_\vartheta/{\rm d}t$ measured by Mirnov coils. The vertical red lines indicate the time at which electron cyclotron emission imaging (ECEI) snapshots (b) and (d) were taken. For orientation, the ECEI data are overlaid with contours of EFIT-reconstructed Lao85 magnetic flux surfaces (gray dashed). Rational magnetic surfaces with safety factor values $q=2,3,4$ are indicated by solid colored lines. The last closed flux surface is drawn black. The inner and outer portions of ECEI were taken by different detectors that, at the time of these experiments (post-2018), had a toroidal angular distance of $\Delta\zeta = 18.5^o$. To be precise, panels (b) and (d) show $\delta Y/\overline{Y} \equiv (Y-\overline{Y})/\overline{Y}$, where $Y(R,z,t)$ is the measured signal subject to $25$-$35\,{\rm kHz}$ band-pass filtering, and $\overline{Y}(R,z)$ is its time-average. In an optically thick plasma, one can assume that $Y \approx T_{\rm e}\times C(R,z)$, with a scaling factor $C(R,z)$ that differs between detector channels. Since $\delta T_{\rm e}/\overline{T}_{\rm e}$ is a relative amplitude with a radially nonuniform reference profile satisfying $\overline{T}_{\rm e}^{\rm core} \gg \overline{T}_{\rm e}^{\rm edge}$, it is clear that, in absolute terms, we have $\delta T_{\rm e}^{\rm edge} \ll \delta T_{\rm e}^{\rm core}$. It would have been interesting to estimate also the radial displacement $\delta\xi_r \approx \delta T_{\rm e}/T_{\rm e}'$ (or the displacement in flux space), but unfortunately the electron temperature profile measurement (Thomson scattering) was not accurate enough to yield reliable values for its radial gradient $T_{\rm e}' \equiv {\rm d}T_{\rm e}/{\rm d}r$, especially near the edge.
• Figure 2: Volumetric focusing of material or energy fluxes along the yellow arrows towards the poles of a rotating sphere (a) or the axis of a cylinder (b). The cylindrical case also represents the situation in the poloidal cross-section of a torus, where the singular axis is located at $(R,z) = (R_0,z_0)$ of the coordinate system $(R,z,\zeta)$ drawn magenta. Within a poloidal cross-section of the cylinder or torus, we also use the poloidal coordinates $(r,\vartheta)$ drawn blue, where $0 \leq r \leq a$ ranges from the axis $0$ to the boundary radius $a$.
• Figure 3: Simulation setup based on KSTAR shot 18567 @ $8.85\,{\rm s}$Lee23. Panel (a) shows the plasma's poloidal $(R,z)$ cross-section, with the toroidal beta $\beta$ (color contours), magnetic flux surfaces with $q=1,2,3,4,5$ (white), last closed flux surface (dashed black), magnetic midplane (magenta), and wall (green). The coordinates of the magnetic axis $(R_0,z_0)$, the on-axis magnetic field strength $B_0$, and the mean minor radius $a$ are also shown. Panel (b) shows the safety factor profiles $q(r)$ as functions of the normalized minor radius $\hat{r} = r/a$ for the original reference case (gray) and for the two model cases (blue, orange) that we consider here. Panel (c) shows the bulk ion density profiles $N_{\rm i(r)}$ that are used in these two models.
• Figure 4: MHD wave continua for the two model cases in Fig. \ref{['fig:03_kstar_profiles']}, computed by solving the equations in Appendix A of Ref. Deng12 for toroidal mode number $|n|=1$. Arrows indicate flat portions of Alfvénic low-frequency continua with dominant poloidal mode numbers $m=1$ and $m=2$, whose frequencies (a) $9.8...10.9\,{\rm kHz}$ and (b) $8.5...9.5\,{\rm kHz}$ are close to the plasma-frame frequency of the core-localized component of the double-peaked fishbone modes studied in Ref. Lee23.
• Figure 5: Illustration of our antenna model and its effect on the plasma in the poloidal cross-section of our KSTAR Model 1. The four bright spots in panel (a) represent the antenna envelopes $\delta\hat{E}_{\rm ant} \propto \cos^3\chi$ of Eq. (\ref{['eq:setup_ant']}) at the four radii $R_{\rm ant}$ listed in Table \ref{['tab:setup_ant']}. In the background, we show a snapshot $\delta\hat{\Phi}_{|n|=1}(R,z)$ of the decaying electrostatic potential fluctuations after initializing a MEGA simulation with an arbitrary perturbation and running it for $\hat{t}\equiv t\omega_{\rm A0} = 350$ Alfvén times ($0.15\,{\rm ms}$) without antenna. After activating the innermost antenna ($R_{\rm ant} = 1.98\,{\rm m}$, $\hat{r}_{\rm ant} \approx 0.13$), and running the simulation for the same duration of $350$ Alfvén times, we obtained the wave field $\delta\hat{\Phi}_{|n|=1}(R,z)$ shown in panel (e): a coherent core-localized response with dominant Fourier component $|m/n|=1/1$, which rotates counter-clockwise as indicated by the three violet arrows. As described in the text, this antenna-driven quasi-mode forms in four stages, which we label 0, 1, 2 and 3. The main features of Stages 0--2 are shown in panel groups (b)--(d) in terms of snapshots of the fluctuating component $\delta\hat{B}_{\rm tor}(R,z)$ of the toroidal magnetic field. The arrows in (b-1) and (c-1) indicate wave front propagation directions. The dotted curves indicate the plasma boundary, which can be seen to act as a reflector in (b). Note that the colorbar in snapshot set (b) has been scaled and clipped at a low amplitude in order to reveal the fine structure of the earliest transients in $\delta\hat{B}_{\rm tor}$. In contrast, the color scale in (c) and (d) is chosen to display the full range of $\delta\hat{B}_{\rm tor}$. The final Stage 3 is shown in Fig. \ref{['fig:06_kstar-1.051_decay-ant_spec']}(c,d).