Energy equation

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The energy equation implemented in FARGO reads:

$\frac{\partial e}{\partial t} + \vec{\nabla}\cdot(e\vec{v}) = -p\vec{\nabla}\cdot\vec{v} + Q_{+}-Q_{-},$

where

is the thermal energy density (thermal energy per unit area), $\vec{v}$ denotes the flow velocity,

is the vertically integrated pressure, and $Q_{+}$ ( $Q_{-}$ ) denote heating (cooling) source terms, assumed to be positive quantities. In this beta version of Fargo adiabatic, there is no cooling source term: $Q_{-}=0$ . A heating source term due to the disk viscosity is however implemented, that reads ([[D'Angelo, G., Henning, T., & Kley, W. 2003, ApJ, 599, 548]]):

$Q_{+} = \frac{1}{2\nu\Sigma}\left( \tau^2_{r,r} + \tau^2_{\varphi,\varphi} + \tau^2_{r,\varphi} \right) + \frac{2\nu\Sigma}{9}(\vec{\nabla}\cdot\vec{v})^2,$

where $\nu$ denotes the cinematic viscosity (calculated with function FViscosity in Viscosity.c), $\tau_{r,r}$ , $\tau_{r,\varphi}$ and $\tau_{\varphi,\varphi}$ are the components of the viscous stress tensor (calculated in function ComputeViscousTerms, as well as $\vec{\nabla}\cdot\vec{v}$ , in Viscosity.c). An ideal equation of state is used to close the hydrodynamics equations:

$p = \Sigma \frac{RT}{\mu},$

where

denotes the universal gas constant, $\mu$ the mean molecular weight and

the gas temperature.

and $\mu$ are fixed to unity (see fondam.h). Furthermore,

and

are connected with:

$e = \frac{\Sigma R T}{\mu (\gamma - 1)},$

where $\gamma$ denotes the adiabatic index. Its value is given by the ADIABATICINDEX parameter (default value: 1.4). Our equation of state therefore reads:

$p = (\gamma-1)e,$

and can also be recast as:

$p = \frac{\Sigma c_s^2}{\gamma},$

where

denotes the adiabatic sound speed. We comment that

is connected with the isothermal sound speed $c_{s,\rm iso}$ with $c_s = \sqrt{\gamma} c_{s,\rm iso}$ .

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