Home > Legacy archive > Specific versions > FARGO-AD > Energy equation

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 e is the thermal energy density (thermal energy per unit area), \vec{v} denotes the flow velocity, p 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 R denotes the universal gas constant, \mu the mean molecular weight and T the gas temperature. R and \mu are fixed to unity (see fondam.h). Furthermore, e and T 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 c_s denotes the adiabatic sound speed. We comment that c_s is connected with the isothermal sound speed c_{s,\rm iso} with c_s = \sqrt{\gamma} c_{s,\rm iso}.

Site Map | COAST COAST | Contact | RSS RSS 2.0