# Equations

On this page we give a small overview of the system of equations solved by Legolas. We use the full set of MHD equations, linearised around a dynamic background (that is, including flow). Physical effects include flow, (external) gravity, resistivity, optically thin radiative losses, anisotropic thermal conduction, viscosity and Hall effects.

## System of equations

\newcommand{\vbf}{\mathbf{v}} \newcommand{\gbf}{\mathbf{g}} \newcommand{\bbf}{\mathbf{B}} \newcommand{\HL}{\mathscr{L}} \newcommand{\HH}{\mathcal{H}} \newcommand{\kappabf}{\boldsymbol{\kappa}} \newcommand{\unit}{\mathbf{e}_{#1}} \begin{gather} \frac{\partial \rho}{\partial t} = -\nabla \cdot (\rho \vbf), \\ \rho\frac{\partial \vbf}{\partial t} = -\nabla p - \rho \vbf \cdot \nabla \vbf + (\nabla \times \bbf) \times \bbf + \rho\gbf + \mu\left[\nabla^2\vbf + \frac{1}{3}\nabla(\nabla \cdot \vbf)\right], \\ \begin{aligned} \rho\frac{\partial T}{\partial t} = &-\rho \vbf\cdot\nabla T - (\gamma - 1)p\nabla \cdot \vbf- (\gamma - 1)\rho\HL \\ &+ (\gamma - 1)\nabla \cdot (\kappabf \cdot \nabla T) + (\gamma - 1)\eta(\nabla \times \bbf)^2 + \mu\left|\nabla \vbf \right|^2, \end{aligned} \\ \begin{aligned} \frac{\partial \bbf}{\partial t} = ~&\nabla \times (\vbf \times \bbf) - \nabla \times (\eta\nabla \times \bbf) \\ &-\nabla\times \left[ \frac{\eta_H}{\rho_e}(\nabla \times \bbf) \times \bbf - \frac{\eta_H}{\rho_e}\nabla p + \frac{\eta_e}{\rho_e}\frac{\partial (\nabla \times \bbf)}{\partial t} \right]. \end{aligned} \end{gather}

## Physical effects

Below we briefly describe the various physical effects that can be added. For more information on how to add these in your particular setup take a look at the user submodule page.

### External gravity

An external gravitational field can be added, which can either be constant or dependent on position. In Cartesian geometry we assume this to be $x$-dependent and aligned with height:

$\gbf = -g(x)\unit{x}.$

A similar profile can be used in cylindrical geometries, e.g. for accretion disk setups.

### Resistivity

Resistivity can either be constant over the entire grid, or a more general temperature-dependent profile given by the Spitzer resistivity:

$\eta(T) = \frac{4\sqrt{2\pi}}{3}\frac{Z_\text{ion}e^2\sqrt{m_e}\ln(\lambda)}{(4\pi\epsilon_0)^2(k_B T)^{3/2}},$

with the Coulomb logarithm $\ln(\lambda) \approx 22$. The function $\eta(T)$ can be user-specified, in which case its temperature derivative should be provided as well. If the resistivity profile explicitly depends on position as well, providing the derivative of $\eta(x, T)$ with respect to $x$ is an additional requirement.

### Heating and optically thin radiative losses

Radiative cooling is governed by the heat-loss function, specified as the difference between energy gains and energy losses

$\HL = \rho\Lambda(T) - \HH(\rho, T),$

The function $\Lambda(T)$ here is called the cooling curve, which is a tabulated set of values resulting from detailed molecular calculations. Legolas has multiple cooling curves from existing literature implemented which are interpolated at high resolution and sampled on the given grid. We have also included analytical, piecewise prescriptions (e.g. the rosner curve). See the physicslist for an overview of the different options. The function $\Lambda(T)$ can be user-specified, in which case its temperature derivative should be provided as well.

The function $\HH(\rho, T)$ specifies the heating function. If thermal balance is forced we assume that this term only depends on the equilibrium background, meaning that it is constant in time but possibly varying in space and as such that it balances out the cooling contribution to ensure thermal equilibrium. $\HH(\rho, T)$ can be user-specified, in which case its density and temperature derivatives should be provided as well.

#### Cooling curves

At the moment 8 tabulated cooling curves are implemented, along with 1 analytical cooling curve. The list below gives an overview of the cooling curves that are available, along with their references.

Name Notes Approximate $\log_{10}(T)$ range Reference
rosner Analytical cooling curve $3.891 - 7.605$ Rosner et al. (1978)
jc_corona Cooling curve for coronal conditions $4.000 - 7.954$ Colgan et al. (2008)
dalgarno Cooling curve for low temperatures $2.000 - 9.000$ Dalgarno & McCray (1972)
dalgarno2 Cooling curve for very low temperatures $1.000 - 4.000$ Dalgarno & McCray (1972)
ml_solar Cooling curve for solar abundances $2.000 - 9.000$ Mellema & Lundqvist (2002)
spex Cooling curve for solar abundances $3.800 - 8.160$ Schure et al. (2009)
spex_dalgarno spex supplemented with dalgarno for lower temperatures $2.000 - 8.160$ -
colgan The original cooling curve $4.065 - 9.065$ Colgan & Feldman (2008)
colgan_dm colgan supplemented with dalgarno2 for lower temperatures $1.000 - 9.065$ -

All cooling curves are interpolated at high resolution using a local second order polynomial and then sampled on the grid, with the exception of the analytical curves. Note that all tabulated cooling curves are extended to higher temperatures (i.e. outside of their tabulated range) by assuming pure Brehmsstrahlung ($\Lambda(T) \sim \sqrt{T}$). For temperatures below the tabulated range the cooling is set to zero.

### Thermal conduction

The thermal conduction prescription relies on a tensor representation to model the anisotropy in MHD and is given by

$\kappabf = \kappa_\parallel\unit{B}\unit{B} + \kappa_\bot(\boldsymbol{I} - \unit{B}\unit{B}),$

with $\boldsymbol{I}$ the unit tensor and $\unit{B} = \bbf / B$ a unit vector along the magnetic field. For the actual coefficients the Spitzer conductivity is assumed:

\begin{align} \kappa_\parallel &\approx 8 \times 10^{-7}T^{5/2}~\text{erg cm^{-1}s^{-1}K^{-1}}, \\ \kappa_\bot &\approx 4 \times 10^{-10} n^2B^{-2}T^{-3}\kappa_\parallel, \end{align}

Alternatively, both of these can be independently set to constant values. In the absence of a magnetic field we set $\kappa_\bot = \kappa_\parallel$, such that the tensor representation reduces to $\kappabf = \kappa_\bot\boldsymbol{I}$.

When overriding parallel thermal conduction a function $\kappa_\parallel(T)$ can be given, as well as its temperature derivative. The derivative with respect to position (denoted with $’$) is automatically calculated as $\kappa_\parallel’ = \frac{\partial \kappa_\parallel}{\partial T}T_0’$.

For perpendicular thermal conduction a function $\kappa_\bot(\rho, T, B)$ can be given, as well as its derivatives

$\frac{\partial \kappa_\bot}{\partial \rho}, \qquad \frac{\partial \kappa_\bot}{\partial T}, \qquad \frac{\partial \kappa_\bot}{\partial B^2}.$

The derivative with respect to position is automatically calculated as

$\kappa_\bot' = \frac{\partial \kappa_\bot}{\partial T}T_0' + \frac{\partial \kappa_\bot}{\partial \rho}\rho_0' + \frac{\partial \kappa_\bot}{\partial B^2}2B_0B_0'.$

### Viscosity

The viscosity addition to the momentum equation is usually given in a full tensor form, but in Legolas this is simplified to good approximation to

$\mathbf{F}_{\mathrm{v}} = -\nabla\cdot\boldsymbol{\Pi} \simeq \mu \left[ \nabla^2\vbf + \frac{1}{3} \nabla\left( \nabla\cdot\vbf \right) \right],$

where the constant $\mu$ denotes the dynamic viscosity. The viscous heating term in the energy equation is expanded using a Frobenius norm and written as

$\left|\nabla \vbf\right|^2 = \sum_{i=1}^{3}\sum_{j=1}^{3}\left(\nabla \vbf\right)_{ij}^2$

The heating term can be toggled on or off independently of the main viscosity addition in the momentum equation, see the physicslist for details on how to do this.

### Hall MHD

Legolas allows for the addition of Hall effects, including electron contributions:

$\underbrace{\frac{\eta_H}{\rho}(\nabla \times \bbf) \times \bbf}_\text{main Hall term}, \qquad\qquad \underbrace{\frac{\eta_H}{\rho}\nabla p}_\text{electron pressure}, \qquad\qquad \underbrace{\frac{\eta_e}{\rho}\frac{\partial (\nabla \times \bbf)}{\partial t}}_\text{electron inertia}.$

The main Hall contribution, electron pressure term and electron inertia term are treated as separate entities, meaning that these terms can be toggled on or off independently of one another. See the physicslist for details.