Important note on the pressure normalisation, conversion to temperature and thermal speed

We remind that a normalised value \(\tilde{a}\) is defined using a background value \(a_0\) such as

\[\tilde{a} = \frac{a}{a_0}\]

We choose the background pressure to be

\[p_0 := m_i n_0 v_A^2 = B_0^2/\mu_0 \ ,\]

corresponding to a normalised pressure

\[\tilde{p} = n k T/(B_0^2/\mu_0) \ ,\]

which for the background plasma simplifies to

\[\tilde{p} = \beta/2 \ .\]

The background temperature is

\[T_0 = \frac{p_0}{n_0 k_B}\]

and we have

\[\tilde{T} = \beta \frac{\tilde{B}^2}{2\tilde{n}}\]

also equal to \(\beta/2\) in the background plasma.\

The particles’ velocities are initialised using a Maxwellian distribution, i.e. a normal distribution with a standard deviation \(\sigma = \sqrt{k_B T/m_i}\), which is a speed. The normalising speed is \(v_0 = v_A\), the Alfven speed. We then get

\[\tilde{\sigma} = \sigma/\sigma_0 = \sqrt{\beta/2}\]

To conclude, the background thermal speed, if defined as

\[v_{th} = \sqrt{k T / m} \ ,\]

(note that a factor 2 is often present, depending on the context) is normalised to

\[\tilde{v_{th}} = \sqrt{\beta} \ .\]