About

This setup models the dynamics of a collisionless shock transition layer with a three-component homogeneous plasma with the periodic boundary condition in all three directions. The system consists of the reflected ions, the incoming core (upstream) ions, and the background electrons, all represented by isotropic Maxwellian distributions in the rest frame of each component. The simulation frame corresponds to the rest frame of electrons.

Physical Parameters

The following parameters should be defined in the configuration file:

• cc : speed of light $c$
• wp : electron plasma frequency $\omega_{pe}$
• mime : ion-to-electron mass ratio $m_i/m_e$
• ush : Four-shock speed in units of the speed of light $U_{sh}/c$
• theta : polar angle of the ambient magnetic field with respect to the x-axis $\theta$
• phi : azimuthal angle of the ambient magnetic field with respect to the x-axis $\phi$
• sigma : electron cyclotron-to-plasma frequency squared $\sigma = \Omega_{ce}^2/\omega_{pe}^2$
• alpha : density of the reflected ion beam normalized to the total density $\alpha = n_r/n_0$
• betae : electron plasma beta $\beta_e = 2 v_{th,e}^2/V_{A,e}^2$
• betai : core ion plasma beta $\beta_i = 2 v_{th,i}^2/V_{A,i}^2$
• betar : reflected ion plasma beta $\beta_r = 2 v_{th,r}^2/V_{A,i}^2$

The electron charge-to-mass ratio is assumed to be unity $|e|/m_e = 1$, which gives the ambient magnetic field $B_0 = c \omega_{pe} \sqrt{\sigma}$.
The three components of the ambient magnetic field are then given by

$$\begin{aligned} B_{0,x} &= B_0 \cos \theta \\\ B_{0,y} &= B_0 \sin \theta \cos \phi \\\ B_{0,z} &= B_0 \sin \theta \sin \phi \end{aligned}$$

Note that the electron velocity in the shock rest frame is written as $V_e = -(1 - 2\alpha) V_{sh}$ where $V_{sh} = U_{sh} / (1 + U_{sh}^2/c^2)^{1/2}$ is the three shock speed calculated from the four shock speed. The density ratio $\alpha$ is defined in the shock rest frame, which we need to convert to the simulation frame to yield

$$\alpha' = \alpha \frac{1 + (1 - 2\alpha) V_{sh}^2}{1 - (1 - 2\alpha)^2 V_{sh}^2}$$

Similarly, the core and reflected ion drift velocities are given by

$$\begin{aligned} V_i' = \frac{-2 \alpha V_{sh}}{1 - (1 - 2\alpha) V_{sh}^2}\\\ V_r' = \frac{+2 (1-\alpha) V_{sh}}{1 + (1 - 2\alpha) V_{sh}^2} \end{aligned}$$

These drift velocities are always parallel to the x-axis. It is easy to check that the above quantities satisfy the charge and current neutrality conditions.
Note that the electron and ion Alfven speeds are defined by $V_{A,e} = B_0 / \sqrt{n_0 m_e}$ and $V_{A,i} = B_0 / \sqrt{n_0 m_i}$, respectively.

Compiling and Executing the Code

The procedure is very similar to the description available here.
Note that the pic-nix repository will automatically be cloned in the build directory using the following procedure.
Alternatively, if you want to use an existing pic-nix source directory in the local filesystem, you can do so by specifying -DPICNIX_DIR=/path/to/pic-nix/dir in the cmake command line.

Clone

Clone the repository via:

$ git clone git@github.com:amanotk/pic-nix-foot.git

Compile

Compile the code to obtain an executable main.out as follows:

$ cd pic-nix-foot
$ cmake -S . -B build \
	-DCMAKE_CXX_COMPILER=mpicxx \
	-DCMAKE_CXX_FLAGS="-O3 -fopenmp"
$ cmake --build build

The executable file will be found in the build directory.

Run

Example 1: buneman

This example provides a setup for the Buneman instability, an electrostatic ion-electron beam instability.

Go to buneman directory:

$ cd buneman

and run the code, for instance, via:

$ export OMP_NUM_THREADS=2
$ mpiexec -n 8 ../main.out -e 86400 -t 200 -c config.json

The data files are written to data directory by default.

Set the environment variable PICNIX_DIR to the path to the pic-nix directory. Then, run the following command to examine the simulation results:

$ python batch.py data/profile.msgpack

You will find image files generated in the same directory.

Example 2: aflven

This example provides a setup for an instability where an ion beam propagating parallel to the ambient magnetic field generates Alfven waves via the cyclotron resonance.

Run with the same procedure with buneman, but this time with the following command to run up to $\omega_{pe} t = 5000$

$ mpiexec -n 16 ../main.out -e 86400 -t 5000 -c config.json

The rest is the same as buneman. For details of the problem, see Hoshino & Terasawa (1985).

References

• Hoshino, M., & Terasawa, T. (1985). Numerical Study of the Upstream Wave Excitation Mechanism, 1. Nonlinear Phase Bunching of Beam Ions. Journal of Geophysical Research, 90(A1), 57–64. https://doi.org/10.1029/JA090iA01p00057
• Matsukiyo, S., & Scholer, M. (2003). Modified two-stream instability in the foot of high Mach number quasi-perpendicular shocks. Journal of Geophysical Research: Space Physics, 108(A12), 1459. https://doi.org/10.1029/2003JA010080
• Matsukiyo, S., & Scholer, M. (2006). On microinstabilities in the foot of high Mach number perpendicular shocks. Journal of Geophysical Research: Space Physics, 111(6), 1–10. https://doi.org/10.1029/2005JA011409
• Amano, T., & Hoshino, M. (2009). Nonlinear evolution of Buneman instability and its implication for electron acceleration in high Mach number collisionless perpendicular shocks. Physics of Plasmas, 16(10), 102901. https://doi.org/10.1063/1.3240336
• Muschietti, L., & Lembège, B. (2013). Microturbulence in the electron cyclotron frequency range at perpendicular supercritical shocks. Journal of Geophysical Research: Space Physics, 118(5), 2267–2285. https://doi.org/10.1002/jgra.50224