# Hydrodynamic Equations in Comoving Coordinates¶

## Conservative Form¶

We solve the equations of gas dynamics in a coordinate system that is comoving with the expanding universe, with expansion factor, $$a,$$ related to the redshift, $$z$$, by $$a = 1 / (1 + z).$$ The continuity equation is written,

$\begin{split}\label{eq:dens} \frac{\partial \rho_b}{\partial t} = - \frac{1}{a} \nabla \cdot (\rho_b {\bf U}) , \\\end{split}$

where $$\rho_b$$ is the comoving baryonic density, related to the proper density by $$\rho_b = a^3 \rho_{proper},$$ and $${\bf U}$$ is the proper peculiar baryonic velocity.

The momentum evolution equation can be expressed as

\begin{aligned} \frac{\partial (\rho_b {\bf U})}{\partial t} &=& \frac{1}{a} \left( - \nabla \cdot (\rho_b {\bf U} {\bf U}) - \nabla p + \rho_b {\bf g} + {\bf S}_{\rho {\bf U}} - \dot{a} \rho_b {\bf U} \right) , \end{aligned}

or equivalently,

\begin{aligned} \label{eq:momt} \frac{\partial (a \rho_b {\bf U})}{\partial t} &=& - \nabla \cdot (\rho_b {\bf U} {\bf U}) - \nabla p + \rho_b {\bf g} + {\bf S}_{\rho {\bf U}} , \end{aligned}

where the pressure, $$p$$, that appears in the evolution equations is related to the proper pressure, $$p_{proper},$$ by $$p = a^3 p_{proper}.$$ Here $${\bf g} = - \nabla \phi$$ is the gravitational acceleration vector, and $${\bf S}_{\rho {\bf U}}$$ represents any external forcing terms.

The energy equation can be written,

\begin{aligned} \frac{\partial (\rho_b E)}{\partial t} &=& \frac{1}{a} \left[ - \nabla \cdot (\rho_b {\bf U} E + p {\bf U}) + ( \rho_b {\bf U} \cdot {\bf g} + S_{\rho E} ) - \dot{a} ( 3 (\gamma - 1) \rho_b e + \rho_b ( {\bf U} \cdot {\bf U}) ) \right] . \end{aligned}

or equivalently,

\begin{aligned} \label{eq:energy} \frac{\partial (a^2 \rho_b E)}{\partial t} &=& a \left[ - \nabla \cdot (\rho_b {\bf U} E + p {\bf U}) + \rho_b {\bf U} \cdot {\bf g} + S_{\rho E} + \dot{a} ( \; ( 2 - 3 (\gamma - 1) ) \; \rho_b e ) \right] . \end{aligned}

Here $$E = e + {\bf U} \cdot {\bf U} / 2$$ is the total energy per unit mass, where $$e$$ is the specific internal energy. $$S_{\rho E} = S_{\rho e} + {\bf U} \cdot {\bf S}_{\rho {\bf U}}$$ where $$S_{\rho e} = \Lambda^H - \Lambda^C$$ represents the heating and cooling terms, respectively. Additionally, $${\bf S}_{\rho {E}}$$ may include any external forcing terms on the total energy, for example as in the stochastic forcing application. We can write the evolution equation for internal energy as

\begin{aligned} \frac{\partial (\rho_b e)}{\partial t} &=& \frac{1}{a} \left[ - \nabla \cdot (\rho_b {\bf U} e) - p \nabla \cdot {\bf U} - \dot{a} ( 3 (\gamma - 1) \rho_b e ) + S_{\rho e} \right] . \end{aligned}

or equivalently,

\begin{aligned} \frac{\partial (a^2 \rho_b e)}{\partial t} &=& a \left[ - \nabla \cdot (\rho_b {\bf U} e) - p \nabla \cdot {\bf U} + S_{\rho e} + \dot{a} ( \; ( 2 - 3 (\gamma - 1) ) \; \rho_b e ) \right] . \end{aligned}

Note that for a gamma-law gas with $$\gamma = 5/3,$$ we can write

\begin{aligned} \frac{\partial (a^2 \rho_b E)}{\partial t} &=& a \left[ -\nabla \cdot (\rho_b {\bf U} E + p {\bf U}) + \rho_b {\bf U} \cdot {\bf g} + S_{\rho e} \right] . \end{aligned}

and

\begin{aligned} \frac{\partial (a^2 \rho_b e)}{\partial t} &=& a \left[ - \nabla \cdot (\rho_b {\bf U} e) - p \nabla \cdot {\bf U} + S_{\rho e} \right] . \end{aligned}