Unit normalisations

All equations in Legolas are in dimensionless form, as is common practice when dealing with (M)HD. As usual we have three degrees of freedom.

Mean molecular weightPermalink

Unit normalisations depend on the molecular weight $\bar{\mu}$, and in Legolas we usually distinguish between two cases:

• Electron-proton plasma: $\bar{\mu} = 0.5$, this is the default case.

  $$\bar{\mu} = \dfrac{m_e n_e + m_i n_i}{n_e + n_i} \simeq \dfrac{m_i n_i}{n_e + n_i} = \dfrac{1}{2}m_i \rightarrow \bar{\mu} = \dfrac{1}{2}$$

• Pure proton plasma: $\bar{\mu} = 1$, in this case the molecular weight should be explicitly set.

  $$\bar{\mu} = \dfrac{m_i n_i}{n_i} = m_i \rightarrow \bar{\mu} = 1$$

NormalisationsPermalink

Legolas has three options to specify units, all in cgs. In what follows $m_p$ denotes the proton mass, $k_B$ the Boltzmann constant, and $\mu_0 = 4\pi$ the magnetic constant.

1. Reference unit density, unit magnetic field and unit length $(\rho_u, B_u, L_u)$, then

   $$p_u = \frac{B_u^2}{\mu_0}, \quad T_u = \frac{\bar{\mu} p_u m_p}{k_B \rho_u}, \quad n_u = \frac{\rho_u}{m_p}, \quad v_u = \frac{B_u}{\sqrt{\mu_0 \rho_u}}.$$

2. Reference unit temperature, unit magnetic field and unit length $(T_u, B_u, L_u)$, then

   $$p_u = \frac{B_u^2}{\mu_0}, \quad \rho_u = \frac{\bar{\mu} p_u m_p}{k_B T_u}, \quad n_u = \frac{\rho_u}{m_p}, \quad v_u = \frac{B_u}{\sqrt{\mu_0 \rho_u}}.$$

3. Reference unit numberdensity, unit temperature and unit length $(n_u, T_u, L_u)$, then

   $$p_u = \bar{\mu} n_u k_B T_u, \quad \rho_u = m_p n_u, \quad v_u = \sqrt{\frac{p_u}{\rho_u}}, \quad B_u = \sqrt{\mu_0 p_u}.$$

All other normalisations follow from those above and are given by

• unit mass: $M_u = \rho_u L_u^3$
• unit time: $t_u = \dfrac{L_u}{v_u}$
• unit resistivity: $\eta_u = \dfrac{L_u^2}{t_u}$
• unit cooling curve: $\Lambda_u = \dfrac{p_u}{t_u n_u^2}$
• unit conduction: $\kappa_u = \dfrac{\rho_u L_u v_u^3}{T_u}$

Note: the unit normalisations are only relevant when radiative cooling, thermal conduction or temperature-dependent resistivity is included. We always set base values though (as one should), which are set using option 2. with default values $B_u = 10$ G, $L_u = 10^9$ cm and $T_u = 10^6$ K.