Cyclic reformation of subcritical perpendicular fast magnetosonic shocks due to oblique Whistler waves

Abstract

The stability of subcritical perpendicular fast magnetosonic shocks, which are propagating at 1.7 times the fast magnetosonic speed, is investigated using two-dimensional PIC simulations. The plasma, composed of electrons and fully ionized nitrogen, is permeated by a uniform magnetic field oriented at 45 degrees to the simulation plane normal. This configuration results in a diamagnetic current that sustains the shocks magnetic ramp and is partially resolved within the simulation plane. The diamagnetic current drives an oblique lower-hybrid gradient drift instability within the ramp. This instability has been observed in magnetic reconnection experiments and studied in the framework of a Harris-type sheath in previous studies. It arises from a reactive coupling between the oblique Whistler wave, which is propagating backward in the electron rest frame, and the forward-propagating ion acoustic wave. Our simulations show that the magnetic component of this wave modulates the shocks magnetic field, while the electrostatic ion density modulation forces the shock to collapse into a magnetic piston and then reform. The reformation is not forced by an external perturbation as in previous simulations but by the oblique Whistler wave.

Figures (11)

• Figure 1: Setup of the main simulation with the box size size $L_x \times L_y$ and periodic boundary conditions. Positions are expressed in units of the electron skin depth $\lambda_e$. The box is split in two. Each half has its own right-handed coordinate system with the z axis pointing up. A slab of dense plasma with thickness $6\lambda_e$ is centered on $x=0$ and surrounded by ambient plasma. The plasma conditions are uniform within the dense slab and within the ambient plasma outside the perturbation layer. A spatially uniform magnetic field $\mathbf{B}_0$ forms a right angle with x and is rotated relative to z by the angle $\theta=45^\circ$. Its $B_y$ component has opposite signs in the red and blue coordinate systems because their y-axes point in opposite directions. The dense plasma expands into the directions marked by the gray arrows. It expands into uniform plasma to the left and encounters a perturbation layer to the right.
• Figure 2: The power spectra of $\langle E_x^2 \rangle$ and $\langle B_z^2 \rangle$ for wave propagation perpendicular to the background magnetic $\mathbf{B}_0$ (simulation 1) are shown in (a) and (b), respectively. The power in both panels is normalized to the values of $c^2\langle B_z^2\rangle$ and $\langle B_z^2 \rangle$ at $k,\omega=0$, respectively. The dashed straight line, which is the electromagnetic approximation of the FMS/lower-hybrid wave branch, is $\omega_{fms}=v_{fms}k$. The dashed curve is the solution of Eqn. \ref{['Lower-Hybrid']}, which is the electrostatic approximation of the FMS/lower-hybrid wave branch. The fluctuation spectrum demonstrates a gradual change from the electromagnetic approximation to the electrostatic one near $\omega_{lh}$
• Figure 3: Waves propagating at the angle $\theta= 45^\circ$ relative to $\mathbf{B}_0$ (Simulation 2). Panels (a, b) show $\langle B_z^2 \rangle$ and $\langle E_x^2 \rangle$, respectively. The power spectra are normalized to $\langle B_z^2 \rangle$ and $c^2\langle B_z^2 \rangle$ at $k,\omega=0$. The dashed red curves show the solution of Eqn. \ref{['whistler']} and the red dashed line in (b) shows $\omega = c_s k$ of the ion acoustic wave. Panel (c) shows the dispersion relation extending to large $k$ and $\omega$. The red circle corresponds to $k=2\pi / L_y$ and $\omega = 1.28\omega_{lh}$. Panel (d) compares the phase speed $v_{ph}=\omega / k$ of the oblique Whistler wave with the speed $5.3 \times 10^6$ m/s (horizontal line). The vertical lines show $k = 2\pi/L_y$ and $4\pi/L_y$.
• Figure 4: The profile of a perpendicular subcritical shock at the time $\omega_{lh}t=12$. Panel (a) shows the ion density in units of $n_{i0}$ (black) and $B_z/B_0$; the background magnetic field pointed along the z axis in simulation 3. Panel (b) shows the electric field amplitudes $E_x$, $E_y$, and $E_z$ in units of $cB_0$. Panel (c) shows the electron mean velocity $\langle v_i \rangle$ along the directions $i=x, y, z$ expressed in units of $v_{th,e}$.
• Figure 5: Ion and magnetic field distributions at $\omega_{lh}t=2.9$ (upper row) and $\omega_{lh}t=5.8$ (lower row): Panels (a, d) show the square root of the phase space density, which was normalized to the peak value upstream. The dashed line in (a) follows the profile of the rarefaction wave and those in (d) show $x/\lambda_e = 5$ and 15. The ions above 1 correspond to shock-reflected ions and those above 2 to rarefaction wave ions. Panels (b, e) show the ion density normalized to the upstream value $n_{i0}$ and clamped to the value 3. Panels (c, f) show the modulus $| \mathbf{B}| /B_0$ of the magnetic field. The vertical dashed lines in (e, f) show $x/\lambda_e = 15$.