Numerical heating of electrons in particle-in-cell simulations of fully magnetized plasmas

The role of spatial resolution of the electron gyroradius in electrostatic particle-in-cell (PIC) simulations is studied. It is demonstrated that resolving the gyroradius is crucial for simulations of strongly magnetized plasmas and that nonresolving it results in substantial anisotropic heating of electrons. The numerical heating can, in some cases, be suppressed by the higher-order weighting to the grid, but it cannot be avoided. Possible mechanisms behind this numerical heating are discussed. The study is carried out with a fully three dimensional electrostatic PIC code with an external magnetic and electric fields.

Figure 1

The electron temperature evolution for all simulation runs. The first-order weighting is given in the left column, while the second-order weighting is in the right column. Magnetic field is oriented in the $x$ direction. The scale on right side corresponds to the $r_{\mathrm{L}}/\mathrm{\Delta }x$ ratio.

Figure 2

Temporal evolution of RMS values of normalized potential fluctuations. In each figure the simulation with the first-order weighting is depicted by solid black line and the case with the second-order weighting by dashed blue line. Note that the vertical scale in panel (c) is different than in other panels.

Figure 3

Cuts in the $y\text{−}z$ plane for the potential density showing time evolution of electric potential in the plane perpendicular to the magnetic field in the case A1. Panel (a) corresponds to the initial state of potential, panel (b) shows the cut for the time $t=90/{\omega }_{\mathrm{p}}$, and panel (c) shows potential for the time $t=270/{\omega }_{\mathrm{p}}$. The evolution of filamentary structures along magnetic field is clearly visible. Both axes are normalized to the Debye length ${\lambda }_{\mathrm{D}}$.

Figure 4

$\omega \text{−}{k}_z$ amplitude spectra of electric potential for simulations resulting in temperature and potential growth. Top panels depict spectra for A1 electron gyro/upper hybrid frequency range (left), and A1 ion gyrofrequency range (right), while bottom panels depict spectra for B1 electron gyro/upper hybrid frequency range (left), and B1 ion gyrofrequency range (right). The frequency axis is normalized to ion gyrofrequency and wave number axis is normalized to the Debye length. We have ${\mathrm{\Omega }}_{\mathrm{e}}=500{\mathrm{\Omega }}_{\mathrm{i}}$ and ${\omega }_{\mathrm{uh}}\approx 504.5{\mathrm{\Omega }}_{\mathrm{i}}$ and thus we can see activity in the upper hybrid frequency range which is typical for perpendicular direction in magnetized plasmas.

Figure 5

The electron temperature evolution for the addtional simulations with drifting plasma. The left panel shows temperature evolution for the first-order weighting while the right panel shows temperature for the second-order weighting. The magnetic field is oriented in the $\widehat{\mathbf{x}}$ direction, and the electric field in $\widehat{\mathbf{y}}$ direction. It is clear that in these simulations, the second-order weighting only reduces the growth of temperature.