Wakes in inhomogeneous plasmas

The Debye shielding of a charge immersed in a flowing plasma is an old classic problem. It has been given renewed attention in the last two decades in view of experiments with complex plasmas, where charged dust particles are often levitated in a region with strong ion flow. Efforts to describe the shielding of the dust particles in such conditions have been focused on the homogeneous plasma approximation, which ignores the substantial inhomogeneity of the levitation region. We address the role of the plasma inhomogeneity by rigorously calculating the point charge potential in the collisionless Bohm sheath. We demonstrate that the inhomogeneity can dramatically modify the wake, making it nonoscillatory and weaker.

Figure 1

Sketch of the problem. The solid curves illustrate the unperturbed sheath, showing the electric potential ${\phi }_{\mathrm{s}}\left(z\right)$, ion number density $n_{\mathrm{i}}\left(z\right)$, electron number density $n_{\mathrm{e}}\left(z\right)$, and ion flow velocity $v\left(z\right)$. The dashed line shows the potential perturbation ${\phi }_q$ (on the $z$ axis) due to the immersed charge $q<0$.

Figure 2

The velocity and field inhomogeneity lengths, $L_v$ and $L_E$ [Eq. (24)], as well as the electron Debye length ${\lambda }_{\mathrm{e}}$ and ion shielding length ${\lambda }_{\mathrm{i}}$ [Eq. (23)]. The lengths are normalized by ${\lambda }_{\mathrm{e}\infty }$ and shown as functions of the normalized sheath potential $\psi$.

Figure 3

Potential perturbation in the downstream direction, ${\phi }_q(r_{\perp }=0;z)$. The solid and dashed lines represent the results of the inhomogeneous and homogeneous models, respectively. The numbers near the curves indicate the values of ${\psi }_0$. The graph is obtained by truncating the integration at $k_{\perp }=20$ (see text).

Figure 4

Potential perturbation in the direction perpendicular to the flow, ${\phi }_q(r_{\perp };z=0)$, divided by the Coulomb potential and shown on a log-log graph. The line styles and numbers near the curves bear the same meanings as in Fig. 3.

Figure 5

The derivative $(\partial {\phi }_q/\partial z){|}_{z=0}$ as a function of $r_{\perp }$. Shown are the results for ${\psi }_0=1.5$; the line styles bear the same meanings as in Fig. 3.

Figure 6

Potential perturbation ${\phi }_q(r_{\perp },z)$ for (a) the inhomogeneous and (b) homogeneous models. Shown are lines of constant ${\phi }_q$ for ${\psi }_0=1.5$ (the potential step is not kept constant). The charge is located in the center of the left edge; the flow is directed to the right. In both cases, the dimensions are $15{\lambda }_{\mathrm{e}\infty }$ in the flow direction and $8{\lambda }_{\mathrm{e}\infty }$ in the perpendicular direction. Note that for the cold-ion approximation employed in this paper, in both cases the potential perturbation logarithmically diverges as one approaches the line $r_{\perp }=0$, $z>0$ (also drawn).