Collisionless Plasma Shocks in Striated Electron Temperatures

The existence of low frequency waveguide modes of ion acoustic waves is demonstrated in magnetized plasmas for electron temperatures striated along the magnetic field lines. At higher frequencies, in a band between the ion cyclotron and the ion plasma frequency, radiative modes develop and propagate obliquely to the field away from the striation. Arguments for the subsequent formation and propagation of electrostatic shock are presented and demonstrated numerically. For such plasma conditions, the dissipation mechanism is the “leakage” of the harmonics generated by the wave steepening.

Figure 1

Illustrative examples of numerical solutions of (2) for the electron temperature profile shown in the upper panel, and defined by $T_e(\xi )/T_0=1-\frac{1}{2}\mathcal{D}+\mathcal{D}\exp (-{\xi }^2/{\mathcal{W}}^2)$, where $\mathcal{W}=4$ and $\mathcal{D}=24/23$. Waveguide mode solutions for $\Omega =0$, $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$ are shown in the middle panel. Free modes solutions obtained for $\Omega =\frac{5}{4}$, $\frac{3}{2}$, $\frac{7}{4}$, 2 and $\gamma =2$ are shown in the lower panel.

Figure 2

Examples of numerical simulations showing the variations of the normalized potential $\Phi =e\psi /T_i$ as a function of position at a fixed time. The electron temperature enhancement is localized as a Gaussian in the $x$ direction. We have $\omega ={\Omega }_{\mathrm{pi}}\pi /5$ in both cases while ${\Omega }_{\mathrm{ci}}/{\Omega }_{\mathrm{pi}}=0.05$ and ${\Omega }_{\mathrm{ci}}/{\Omega }_{\mathrm{pi}}=1$ in the left and right panels, respectively. We have here $T_e/T_i=50$. For clarity, only a part of the simulation domain is shown.

Figure 3

Numerical simulation showing normalized potential $\Phi =e\psi /T_i$ and relative density $n_i/n_0$ during the formation and propagation of a shock under the conditions mentioned before. We have $T_e/T_i=25$, ${\Omega }_{\mathrm{ci}}/{\Omega }_{\mathrm{pi}}=\frac{1}{2}$ and $\delta n_i/n_0\approx 0.24$, where $\delta n_i$ is the actual detected density perturbation at the time where the shock is fully formed. The background density $n_0$ is normalized to unity. Lower frame shows sample of shock fitting for the ion density.

Figure 4

Shock velocity $U$ in units of $C_{s0}$ (upper panel) and saturated shock thickness $\Delta$ in units of $C_{s0}/{\Omega }_{\mathrm{ci}}$ (lower panel) as function of the fully formed shock relative height $\delta n_i/n_0$, for different combinations of the parameters $T_e/T_i=25$ and 75, and ${\Omega }_{\mathrm{ci}}/{\Omega }_{\mathrm{pi}}=\frac{1}{4}$ and $\frac{1}{2}$.