Avalanches triggered by Kelvin-Helmholtz instability in a cylindrical plasma device

A profile-evolving simulation of the Controlled Shear Decorrelation Experiment (CSDX) linear device is performed with our newly developed code. The simulation result shows an excellent agreement with the experimental observations of profiles and fluctuations of plasma density and electric potential in the $B=1000$ G standard discharges, suggesting the mechanism of their evolutions. According to our simulation, an avalanche of plasma density, featuring a rapid destruction of particle profile, is triggered every time the dominant instability transits from near adiabatic collisional drift wave to non-adiabatic Kelvin-Helmholtz instability. The avalanches always start at the point where the local vorticity is the maximum among the whole device. A critical vorticity is found for any avalanche to happen. The avalanches always lead to intermittent particle and heat convective structures outside the main plasma column, and these structures are ejected out as avaloids when zonal flow intensity at birth time is weak.

Figure 1

Diagram of the simulation domain. Equations (1) to (4) are advanced in the $r=10$ cm cylinder. The view angle of an operational fast camera is used [16].

Figure 2

Simulation result of the evolution of (a) zonal flow $\lg 〈v_{E,\theta }/\left(\mathrm{cm}/\mathrm{s}\right)〉$, (b) $m=3$ perturbation autopower $\lg 〈{\left({\tilde{n}}_3/{\mathrm{cm}}^{-3}\right)}^2〉$, (c) $m=1$ perturbation autopower $\lg 〈{\left({\tilde{n}}_1/{\mathrm{cm}}^{-3}\right)}^2〉$, (d) particle count radial distribution $〈2\pi rn/{\mathrm{cm}}^{-2}〉$, (e) outward radial particle convection $\lg 〈2\pi rn{v}_{E,r}/\left({\mathrm{cm}}^{-2}\mathrm{cm}/\mathrm{s}\right)〉$ (left white for inward particle flux regions). We use same color bar for different azimuthal mode numbers of perturbations for comparison. We use the same dark blue color for regions where $〈{\tilde{n}}_m^2〉\le {10}^{16}{\mathrm{cm}}^{-6}$ and $0<2\pi r〈n{v}_{E,r}〉\le {10}^{15}{\mathrm{cm}}^{-2}\mathrm{cm}/\mathrm{s}$, in case of miscalibration of the color bars due to some trivial low values. Vertical white and black lines show the times when zonal flow at $r=3$ cm starts to decrease and increase respectively, labeling phase transition times.

Figure 3

Simulation result of early shear flow growth phase features a linear growth of a CDW. (a) Zonal-averaged auto-spectrum of electric potential perturbation $\lg 〈{\tilde{\varphi }}_m^2/{\mathrm{V}}^2〉$ at $t=25.19$ ms. (b) Evolution of perturbed electric potential at a certain position at $r=3.6$ cm. The green line connects the beginning and ending of two adjacent CDW periods, whose incline is used to estimate linear growth rate $\gamma =3.34{\mathrm{ms}}^{-1}$ and interval is used to estimate real frequency $f={\omega }_r/2\pi \approx 5.86\mathrm{kHz}$.

Figure 4

Late shear flow growth phase features a significant growth of both (a) zonal flow $〈v_{E,\theta }〉$ and (b) radially sheared parallel flow $〈v_{\parallel i}〉$.

Figure 5

Evolution of key scalars in the whole case. (a) is the zonal-averaged azimuthal $\mathit{E}\times \mathit{B}$ velocity at $r=3.6$ cm and the global maximum, indicating the time of transitions between two phases. (b) is the global maximum vorticity contribution by the radial shear of $\mathit{E}\times \mathit{B}$ velocity $\widehat{\mathit{r}}\times {\partial }_r{\mathit{v}}_E$, compared to maximum total vorticity. (c) is the maximum vorticity contribution by the curl of azimuthal $\mathit{E}\times \mathit{B}$ velocity $\mathbf{\nabla }\times \left(v_{E,\theta }\widehat{\mathit{\theta }}\right)$, compared to maximum total vorticity. The global maximum of a scalar $f, \mathrm{Max}\left\{f\right\}$, is the maximum $f$ evaluated in the region $0.5\mathrm{cm}\le r\le 9.5\mathrm{cm}$.

Figure 6

Illustration of a typical avalanche with the ejection of an avaloid. With sequential numbers on the left top labeling time, the three groups of pictures show the evolution of density, electric potential, and vorticity at the entire center $x-y$ plane of the vacuum vessel during $t=15.6347–16.3240$ ms. The time interval between adjacent pictures is $\mathrm{\Delta }t=0.0313\mathrm{ms}$. The view angle of Fig. 1 is used.

Figure 7

Illustration of a typical avalanche without an avaloid during $t=25.6296–26.3189$ ms.

Figure 8

3D structures of (a) density $n$, (b) potential $\varphi$, and (c) vorticity $w$ in the whole device when the CDW and K-H instability coexist. Contours of the fields are drawn in the $x-y$ planes with equal intervals. The time of the snapshot is $t=15.8540$ ms, which is the same time labeled number 8 in Fig. 6. The magnetic axis $\left(x,y\right)=\left(0,0\right)$ is shown by the white line.

Figure 9

3D structure of (a) density, (b) potential, and (c) vorticity of an avaloid at $t=16.1046$ ms, which is the same time labeled number 16 in Fig. 6. The smaller density peak in each $x-y$ plane is the avaloid.

Figure 10

Radial particle flux at $r=1.5$ cm (a) for the snapshots in Fig. 6 (with avaloid ejection) and (b) for snapshots in Fig. 7 (without avaloid ejection).

Figure 11

Global maximum amplitude of magnetic perturbations perpendicular to the dominant magnetic field direction.