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I'm trying to make a numerical simulation of pulsar magnetosphere using FDTD on a log-spherical Yee lattice for fields and PIC for plasma particles. Field part is working like charm but issue arises when adding the plasma.

To simulate the dynamics of charged plasma particles I use Newton-Lorenz equation:

$\frac{d\gamma\vec{v}}{dt} = \frac{e}{m}\left( \vec{E} + \frac{\vec{v}}{c} \times \vec{B} \right)$

Adopting CGS system of units, characteristic values for $\vec{E}$ and $\vec{B}$ are of order of magnitule $10^{12}$, charge to mass ratio for electron is of order of magnitude $10^{17}$, so the value for acceleration is ~$10^{29}$.

In the code itself I normalized all variables in the following manner:

$ \vec{u} = \frac{\gamma\vec{v}}{c}\\ \vec{E}^{*} = \vec{E}/B_0\\ \vec{B}^{*} = \vec{B}/B_0\\ \tau = \frac{ct}{R}\\ $

Where $B_0=10^{12}$, $R=10^6$ (radius of the pulsar) and $c$ is the speed of light. As a result, my equation looks as follows:

$\frac{d\vec{u}}{dt} = \frac{eRB_0}{mc^2}\left(\vec{E}^{*}+\frac{\vec{u}\times\vec{B}^{*}}{\gamma}\right)$

Normalization factor $\frac{eRB_0}{mc^2}\sim 10^{15}$, acceleration is still huge.

I may choose different normalization for time as follows:

$t = t_0\tau\\ \frac{eB_0}{mc}t_0 = 1$

But with $\tau$ chosen in that way it takes literally forever for fields to get to stationary configuration.

What can I do?

Tajimura
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  • With fields that strong, you can't use the non-relativistic Newton-Lorenz equation. You must treat things relativistically in order to avoid diverging speeds. In addition, the field itself varies based on reference frame, and therefore varies with the speed of the particles. –  Feb 20 '18 at 03:46
  • @probably_someone, I do use relativistic equation in the code, so Lorenz factor is there in my equations (and also lapse function due to curvature of space). I just omitted them here in the question so they don't distract people from the essence of the problem. Will it be better if I edit them in? –  Feb 20 '18 at 04:11
  • If I recall correctly, there's a stack exchange just on scientific computing. This question might work better there? – Chris Feb 20 '18 at 04:26
  • @Chris, I tried posting the question there, but was not able to do it, 'cause SE refuses to add tags in computational physics site. I dunno why it's so, never had that happen in Physics SE and Math SE. Any way to move the question there? –  Feb 20 '18 at 04:46
  • @Tajimura I believe you can flag your post and ask a moderator to do it. – Chris Feb 20 '18 at 04:48
  • Meanwhile, edited the question to include more info. –  Feb 20 '18 at 04:50
  • In a pulsar magnetic field, are not the gyroradii quantized? I ask because one of the ways you may simplify this is to use a gyro-kinetic approach, i.e., follow the particles' guiding centers and avoid tracking their gyro-orbits, which should have huge accelerations in such strong magnetic fields. The non-relativistic gyrofrequency of an electron in a $10^{8}$ nT field is in the GHz range, i.e., the particle orbits the field billions of times per second. The gyroradii would need to be extremely small to make the acceleration small. –  Feb 20 '18 at 12:03
  • @honeste_vivere, unfortunately it doesn't apply here. First, there's no quantization, second, after a short transition phase, electron rotation synchronizes with the pulsar's rotation. My trouble is that I can't "catch" the transition due to timescales of fields and particles being drastically different. –  Feb 20 '18 at 12:13
  • It seems that you need to apply a method able of capture multiple scales. – nicoguaro Feb 22 '18 at 02:13

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