Spatiotemporal mode structure of nonlinearly coupled drift wave modes

This paper presents full cross-section measurements of drift waves in the linear magnetized plasma of the Mirabelle device. Drift wave modes are studied in regimes of weakly developed turbulence. The drift wave modes develop azimuthal space-time structures of plasma density, plasma potential, and visible light fluctuations. A fast camera diagnostic is used to record visible light fluctuations of the plasma column in an azimuthal cross section with a temporal resolution of $10\mu \mathrm{s}$ corresponding approximately to $10\%$ of the typical drift wave period. Mode coupling and drift wave dispersion are studied by spatiotemporal Fourier decomposition of the camera frames. The observed coupling between modes is compared to calculations of nonlinearly coupled oscillators described by the Kuramoto model.

Figure 1

Thermionic plasma device Mirabelle: (a) three-dimensional drawing and (b) schematic.

Figure 2

Localization of the recorded emitted light. The dash-dotted (black) line shows the radius of the circle of confusion $r_{\mathrm{cc}}$ and the dashed (blue) line is the energy of a light-emitting object arriving at the camera lens. The energy per pixel is a combination of both curves (red), i.e., $[E\left(z\right)/E_0]{\left(\pi r_{\mathrm{cc}}^2\right)}^{-1}$.

Figure 3

Spectra of (a) ion saturation current fluctuations and (b) visible light fluctuations of a coherent $m=3$ drift wave mode. (c) Mode structure of light fluctuations in the azimuthal cross section. (d) Space-time diagram of an azimuthal camera pixel array at $r=3\mathrm{cm}$.

Figure 4

Fourier decomposition of one camera frame recording weakly developed drift wave turbulence. The first row shows the light fluctuations in the azimuthal cross section and the second row shows the corresponding radial fluctuation amplitude profiles for (a) the camera measurement and (b)–(g) the decomposed modes $m=1-6$. The propagation direction of the mode structures is counterclockwise.

Figure 5

Time evolution of the decomposed modes (a)–(d) $m=1$, (e)–(h) $m=2$, and (i)–(l) $m=3$. Each mode starts from the same camera frame.

Figure 6

Average mode-number spectrum of weakly developed drift wave turbulence in a helium plasma.

Figure 7

Measured radial evolution of the frequency for the mode numbers (a) $m=1$, (b) $m=2$, (c) $m=3$, and (d) $m=4$. The vertical lines indicate the error bars obtained from the temporal average of the frequency.

Figure 8

Measured and fitted radial evolution of the frequency: (a) measurement and (b) fits according to Eq. (2). (c) Profiles of the fits and the measurement for the plasma density (solid line and circles), electron temperature (dash-dotted line and crosses), and plasma potential (dashed line and squares).

Figure 9

Phase differences of decomposed modes over the time of the transient.

Figure 10

(a) Trajectories of the two-oscillator Kuramoto model with fixed points [$\Delta \omega ={\omega }_2-{\omega }_1=0$, $\sum K=K_{1,2}+K_{2,1}=1$ (solid line)] and without fixed points [$\Delta \omega =\{1.1,-1.1\}$, $\sum K=1$ (dashed and dotted lines)]. Evolution of the phase differences $\Psi$ between two oscillators and the coupled signal $S$ for (b) and (c) the phase locked case and (d) and (e) the quasiperiodic case (periodic pulling).

Figure 11

Calculated phase differences due to the Kuramoto model for five interacting oscillators with arbitrary coupling constants derived from (a) a random coupling matrix and (b) a coupling matrix obtained from the measurements.