# Filling a flux surface in a tokamak or stellarator

Assume you have a magnetic confinement device like a tokamak or a stellarator. Let's assume further a configuration with closed magnetic field lines resulting in typical flux surfaces.

If you now apply energy to the electrons or ions on one particular flux surface, how long does it take for these particles to completely fill the flux surfaces (i.e. how long does it take to spread the energy across that particular flux surface)?

• Your main question and your parenthetical sound like different processes to me. Do you want to know about energy transport from surface to surface or transport from one part of a surface to the rest of that surface? – Endulum May 18 '18 at 14:12

Particles will move parallel to the magnetic field at the sound speed.

$v_s = \sqrt{\frac{Z T_e + 3 T_i}{M}}$

For a typical tokamak, the plasma is usually mostly deuterium, so $M=2$ u and $Z=1$. Let's say $T_e\sim500$ eV and $T_i\approx T_e$. This gets us $v_s = 3.1\times10^5$ m/s.

The safety factor $q = \frac{d\phi}{d\theta}$ measures how the magnetic field is twisted. For a quick estimate, let's pick a typical value of $q=3$. So for every poloidal turn, there will be three toroidal turns. Taking the circular approximation and $R=2$ m, $r=0.5$ m, an ion has to go a distance of roughly $2\pi R q + 2\pi r = 41$ m to get to around the poloidal circumference once.

Traveling 41 m at $v_s$ would take about 130 $\mathrm{\mu}$s. During this time, particles would have some opportunity to mix energy with neighboring field lines. If you only want one turn toroidally, then it's just one third the distance (since we assumed $q=3$), or 43 $\mathrm{\mu}$s.

So, spreading of heat and particles across a flux surface generally happens on the scale of microseconds.

• I also had the sound speed in mind, but then there is the question how these particles "spread" their energy. My guess is now that there are collisions involved: is the parallel collision time a good guess for the time it takes to spread the energy? – Alf Jul 31 '18 at 7:16