Conceptual study on using Doppler backscattering to measure magnetic pitch angle in tokamak plasmas

TL;DR

This work reframes Doppler backscattering (DBS) as a local diagnostic for magnetic pitch angle in tokamak plasmas by exploiting mismatch attenuation at the DBS cutoff. Using a beam-tracing (Scotty)

Abstract

We introduce a new approach to measure the magnetic pitch angle profile in tokamak plasmas with Doppler backscattering (DBS), a technique traditionally used for measuring flows and density fluctuations. The DBS signal is maximised when its probe beam's wavevector is perpendicular to the magnetic field at the cutoff location, independent of the density fluctuations. Hence, if one could isolate this effect, DBS would then yield information about the magnetic pitch angle. By varying the toroidal launch angle, the DBS beam reaches cutoff with different angles with respect to the magnetic field, but with other properties remaining similar. Hence, the toroidal launch angle which gives maximum backscattered power is thus that which is matched to the pitch angle at the cutoff location, enabling inference of the magnetic pitch angle. We performed systematic scans of the DBS toroidal launch angle for repeated DIII-D tokamak discharges. Experimental DBS data from this scan were analysed and combined with Gaussian beam-tracing simulations using the Scotty code. The pitch-angle inferred from DBS is consistent with that from magnetics-only and motional-Stark-effect-constrained (MSE) equilibrium reconstruction in the edge. In the core, the pitch angles from DBS and magnetics-only reconstructions differ by one to two degrees, while simultaneous MSE measurements were not available. The uncertainty in these measurements was under a degree; we show that this uncertainty is primarily due to the error in toroidal steering, the number of toroidally separated measurements, and shot-to-shot repeatability. We find that the error of pitch-angle measurements can be reduced by optimising the poloidal launch angle and initial beam properties.

Figures (16)

• Figure 1: Schematic to illustrate the toroidal response, and its geometrical intuition, for two different magnetic fields, $\mathbf{B}_{\rm 1}$ and $\mathbf{B}_{\rm 2}$. The green and blue solid lines, are the toroidal response to $\mathbf{B}_{\rm 1}$ and $\mathbf{B}_{\rm 2}$ respectively. The associated magnetic pitch angles, $\gamma_{\rm 1}$ and $\gamma_{\rm 2}$, are negative in our sign convention. The toroidal launch angle, $\varphi_{\rm t}$, which maximises the backscattered power is referred to as the optimal toroidal launch angle, $\varphi_{\rm t,opt}$. It depends on the angle between the probe beam's wavevector and magnetic field at cutoff Hillesheim:DBS_MAST:2015Hall-Chen:beam_model_DBS:2022Hall-Chen:mismatch:2022Damba:mismatch:2022Hall-Chen:mismatch:2024. Depending on whether the pitch angle is $\gamma_1$ or $\gamma_2$ the optimal toroidal angle will differ. As such, by measuring the dependence of backscattered power on toroidal launch angle, one should be able to infer the magnetic pitch angle at the cutoff location. All other plasma properties are kept the same. Note that $\varphi_{\rm t}$ lies in the midplane and the angle shown in the figure corresponds to $\varphi_{\rm t} > 0$. Also note that the difference between $\mathbf{B}_{\rm 1}$ and $\mathbf{B}_{\rm 2}$, and therefore $\varphi_{\rm t,opt,1}$ and $\varphi_{\rm t,opt,2}$, have been exaggerated for the purpose of illustration.
• Figure 2: Plasma properties of the five repeated shots as a function of time: (a) line integrated electron density, (b) safety factor $q$ at radial coordinate $\rho = 0.95$, (c) neutral-beam power $P_{\rm NBI}$, and (d) plasma current $I_{\rm p}$. We see that these shots are indeed nearly identical. We study DBS data during plasma current ramp up, at 500 ms, and at flat top, 1510 ms; these two times are marked with dotted lines. While eleven repeated shots were analysed in this paper, here we only show the five with significant DBS signal; while the other six shots had nearly identical plasma properties, the DBS signals across all frequency channels were negligible due to prohibitively high mismatch attenuation Hall-Chen:mismatch:2022Damba:mismatch:2022.
• Figure 3: Properties of the plasma at midplane for the time slices 500 ms (left) and 1510 ms (right). The experiments were carried out in X-mode polarisation for the set of frequencies: 55.0 GHz, 57.5 GHz, 60.0 GHz, 62.5 GHz, 67.5 GHz, 70.0 GHz, 72.5 GHz, and 75.0 GHz, as indicated by gray horizontal lines. Points with white centers are the cutoff positions calculated for launch at normal incidence on the midplane. Characteristic frequencies at midplane (top): $f_{\rm R}$ is the right-hand cutoff frequency, $f_{\rm PE}$ is the plasma frequency and the cutoff frequency for O-mode, $f_{\rm CE}$ is the cyclotron frequency, $2 f_{\rm CE}$ is the second harmonic of the cyclotron frequency. Pitch angle $\gamma$ at the midplane (bottom). We chose 1510 ms as the main time slice of study because it is the only time when MSE measurements are available. At this time, X-mode measurements were concentrated at the edge. We also analyse data from 500 ms, when the cutoff positions of the frequencies were deeper in core, probing a wider range of pitch angles.
• Figure 4: Workflow of inferring to magnetic pitch angle from DBS measurements, as used in this paper. We show experimental measurements (blue boxes), codes (green boxes), Scotty's input parameters (orange boxes), intermediate parameters (red boxes), and the output parameter (purple box).
• Figure 5: Radial location of cutoff locations, $\rho_{\rm c}$, (top) and wavenumbers at cutoff, $K_{\rm c}$, (bottom) at 500 ms as a function of toroidal launch angle, $\varphi_{\rm t}$, (left) and as a function of pitch angle scaling, $S$, (right). The latter is discussed in detail in subsection \ref{['subsec:DbsMeasurementsPitchAngleScaling']}. The equilibrium from shot 188839 was used for these beam-tracing simulations. The DBS was set to X-mode and poloidal launch angle $\varphi_{\rm p} = -11.4^\circ$. The sign convention for toroidal and poloidal angles is given in equation (\ref{['eq:launchK']}). Here $\rho$ is the square root of normalised toroidal flux. The three dots on each line (c) demarcate the following: the center dot is the optimal toroidal angle at which we get maximum backscattered power, the leftmost and rightmost dots are at toroidal angles such that the backscattered power is $1/\rme^2$ of the maximum backscattered power. $K_{\rm c}$ is approximately constant in this range of toroidal launch angles (c) and does not depend on pitch angle scaling (d). The radial coordinate $\rho_{\rm c}$ of the cutoff location has little dependence on toroidal launch angle (a) and no dependence on pitch angle scaling (b). At 1510ms, the dependence of $K_{\rm c}$ and $\rho_{\rm c}$ on $\varphi_{\rm t}$ and $S$ weakens further (not shown).