Nonlinear effects in the bounded dust-vortex flow in plasma

The vortex structures in a cloud of electrically suspended dust in a streaming plasma constitutes a driven system with a rich nonlinear flow regime. Experimentally recovered toroidal formations of this system have motivated study of its volumetrically driven–dissipative vortex flow dynamics using two-dimensional hydrodynamics in the incompressible Navier-Stokes regime. Nonlinear equilibrium solutions are obtained for this system where a nonuniformly driven two-dimensional dust flow exhibits distinct regions of localized accelerations and strong friction caused by stationary fluids at the confining boundaries resisting the dust flow. In agreement with observations in experiments, it is demonstrated that the nonlinear effects appear in the limit of small viscosity, where the primary vortices form scaling with the most dominant spatial scales of the domain topology and develop separated virtual boundaries along their periphery. This separation is triggered beyond a critical dust viscosity that signifies a structural bifurcation. Emergence of uniform vorticity core and secondary vortices with a newer level of identical dynamics highlights the applicability of the studied dynamics to gigantic vortex flows, such as the Jovian great red spot, to microscopic biophysical intracellular activity.

Figure 1

Toroidal dust cloud with poloidal circulation in the laboratory dusty plasma experiment.

Figure 2

Schematic representation of the toroidal dust cloud with vortex motion and mapping of the boundary of its cross section to a toroidal domain of rectangular cross section accommodated in the cylindrical geometry of the present nonlinear solutions. Segments $AB, BC, CD$, and $DA$ of the cloud boundary map to the corresponding sides of the rectangle $ABCD$.

Figure 3

Profiles of the (a) driver velocity $v$ and (b) corresponding driver vorticity ${\omega }_s$ which are uniform along $\widehat{z}$ and $\widehat{\varphi }$.

Figure 4

(a) Dust flow streamlines, strengths of (b) the diffusive term and (c) the nonlinear term plotted from top to bottom for the values of dust viscosity $\mu =1\times {10}^{-3}, 1\times {10}^{-4}, 1\times {10}^{-5}$, and $8\times {10}^{-7}{U}_{0}L$, respectively, where the values $\nu =0.01U_0/L$ and $\xi =0.001U_0/L$ are used.

Figure 5

(a) Dust flow streamlines superimposed by straight lines and contours used to draw profiles of $\omega$ as function of $r\psi$, (b) vorticity $\omega$ as function of $r\psi$ along the straight line segments ($L1,L2,L3,L4,\mathrm{and}L5$) joining the vortex center to the domain vortices, and (c) that along the indicated contours ($C1,C2,C3,C4,\mathrm{and}C5$). The profiles correspond to $\mu ={10}^{-3}{U}_{0}L, \xi =0.001U_0/L$, and $\nu =0.01U_0/L$.

Figure 6

(a) Dust flow streamlines superimposed by straight lines and contours used to draw profiles of $\omega$ as a function of $r\psi$, (b) vorticity $\omega$ as function of $r\psi$ along the straight line segments ($L1,L2,L3,L4,\mathrm{and}L5$) joining the vortex center to the domain vortices, and (c) that along the indicated contours ($C1,C2,C3,C4,\mathrm{and}C5$). The profiles correspond to $\mu =8\times {10}^{-7}{U}_{0}L, \xi =0.001U_0/L$, and $\nu =0.01U_0/L$.

Figure 7

(a) Magnitude of the diffusion term and (b) magnitude of the nonlinear term plotted for $\mu =8\times {10}^{-7}{U}_{0}L, \xi =0.001U_0/L$, and $\nu =0.01U_0/L$ in a domain confinement limited to larger $r$ values.

Figure 8

The vorticity surface plots for values of parameter (a) $\mu ={10}^{-3}$, (b) ${10}^{-4}$, (c) ${10}^{-5}$, and (d) $8\times {10}^{-7}{U}_{0}L$. The developing points of separation are indicated by arrows on segments $AB$ and $BC$.

Figure 9

The normalized vorticity profile along the no-slip boundary $BC$ for various values of parameter $\mu$. The point of separation and associated bifurcation is indicated by an arrow where the profile corresponding to the critical value ${\mu }^{*}\approx {10}^{-5}{U}_{0}L$ has a single degenerate singular point.

Figure 10

The nonlinear boundary layer scaling with dust viscosity $\mu$. The profile shows a change of regime at ${\mu }^{*}\approx 1\times {10}^{-5}{U}_{0}L$.