Evolution of a Vortex in a Strain Flow

Experiments and vortex-in-cell simulations are used to study an initially axisymmetric, spatially distributed vortex subject to an externally imposed strain flow. The experiments use a magnetized pure electron plasma to model an inviscid two-dimensional fluid. The results are compared to a theory assuming an elliptical region of constant vorticity. For relatively flat vorticity profiles, the dynamics and stability threshold are in close quantitative agreement with the theory. Physics beyond the constant-vorticity model, such as vortex stripping, is investigated by studying the behavior of nonflat vorticity profiles.

Figure 1

Measured vorticity field (colormap, vorticity out of the page) and stream function (black lines) at $t=0$, 40, and $80\mu \mathrm{s}$: below threshold, $\epsilon /{\omega }_0=0.116$, (a)–(c), and above threshold, $\epsilon /{\omega }_0=0.13$, (d)–(f), where ${\omega }_0$ is the peak vorticity. The initial vorticity profile is approximately flat. The separatrix is shown (thick black line) with saddle (center) points marked $X$ ($O$). Panels (g) and (h) show elliptical fits to the half-maximum vorticity contours at $20\mu \mathrm{s}$ intervals starting at $t=0$ for $\epsilon /\omega =0.116$ and 0.13, respectively.

Figure 2

(a) Experimental arrangement with electron source (I), confinement electrodes (II, IV, and VI), 8-sector electrode (III), 4-sector electrode (V), phosphor screen (VII), CCD diagnostic (VIII); (b) streamlines (black) of external flow created by electrode III voltages $+V_s$ (red), $-V_s$ (blue), $V=0$ (black); dashed rectangle is the region shown in Figs. 1; $\mathbf{B}$ out of the page; and (c) the initial experimental vorticity profiles: green, smoothly decreasing ($n=3$), and magenta, approximately flat top ($n=6$, cf. Fig. 1). Dashed lines are fits to Eq. (2).

Figure 3

Evolution of (a) total normalized circulation, (b) that inside the separatrix, and (c) total squared strain at the origin, normalized to ${{\omega }_0}^2$ for $\epsilon /{\omega }_0=0.087$ (black), 0.116 (blue), 0.130 (green), and 0.152 (magenta), where ${\omega }_0=220\text{krad}/\mathrm{s}$ and $n=6$. The $\epsilon /{\omega }_0=0.116$ and 0.130 data correspond to Fig. 1. In (c), the red dashed line is the Okubo-Weiss local stability boundary. Solid lines are guides to the eye.

Figure 4

Polar plot in ($\lambda -1$, $\xi$) space: (a) periodic orbit for $\epsilon =0.087$, direction of time indicated by the arrow; and (b) periodic and unstable orbits for $\epsilon =0.087$ (black), 0.116 (blue), 0.13 (green) and 0.152 (magenta); dashed line at $\lambda -1=4.4$ marks the instability threshold. Colored lines are the predictions of the Kida model with no fitted parameters. The $\epsilon /{\omega }_0=0.116$ and 0.130 data correspond to Fig. 1.

Figure 5

(a) Theoretical destruction thresholds for constant strain (dashed line) and equilibrium (dotted line), compared with experiment (error bars), with green (magenta) corresponding to $n=2–3$ ($n=5–7$) initial vorticity profiles; and simulation results with shaded bars for $n=3$ (green) and $n=7$ (magenta); and (b) normalized circulation remaining after a strain event of duration $t_f$. Magenta (green) circles correspond to simulations with $n=7$ ($n=3$) and $t_f$ up to $90/{\omega }_0$, while magneta (green) triangles are from experiment with $n=6$ ($n=2$) and $t_f=25/{\omega }_0$. Solid and dashed lines in (b) are a guide to the eye, vertical dashed line shows the theoretical destruction threshold.