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.

Normalisations

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. $a$ and $b$ are constants determined by the plasma composition. By default they depend only on the He abundance $f_\mathrm{He}$ as

\[a = 1 + 4 f_\mathrm{He}, \quad b = 2 + 3 f_\mathrm{He},\]

such that $f_\mathrm{He} = 0$ corresponds to a fully ionised hydrogen plasma. The He abundance is set to $f_\mathrm{He} = 0$ if not specified.

Note: for alternative plasma compositions, users can define $a$ and $b$ in the user module and update the units from there, see e.g. the pre-implemented equilibrium smod_equil_magnetothermal_instabilities.f08.

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

    \[p_u = \frac{B_u^2}{\mu_0}, \quad T_u = \frac{p_u}{b n_u k_B}, \quad n_u = \frac{\rho_u}{a m_p}.\]

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

    \[p_u = \frac{B_u^2}{\mu_0}, \quad n_u = \frac{p_u}{b k_B T_u}, \quad \rho_u = a m_p n_u.\]

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

    \[p_u = b n_u k_B T_u, \quad \rho_u = a m_p n_u, \quad B_u = \sqrt{\mu_0 p_u}.\]

All other normalisations follow from those above and are given by

  • unit velocity: $v_u = \sqrt{\frac{p_u}{\rho_u}}$
  • 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.