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= Governing equations =
By default, PALM has six prognostic quantities: the velocity components ''u'', ''v'', ''w'' on a Cartesian grid, the potential temperature ''θ'', specific humidity ''q'',,v,, or a passive scalar ''s'', and the SGS turbulent kinetic energy (SGS-TKE) ''e''. The separation of resolved scales and SGS is implicitly achieved by averaging the governing equations over discrete Cartesian grid volumes as proposed by [#schumann1975 Schumann (1975)]. Moreover, it is possible to run PALM in a direct numerical simulation mode by switching off the prognostic equation for the SGS-TKE and setting a constant eddy diffusivity. For a list of all symbols and parameters, that we will introduce in this section, see the following Table 1 and 2.


[[Image(Table1.png,600px,border=1)]]
[[Image(Table2.png,600px),border=1)]]


The model is based on the non-hydrostatic, filtered, incompressible Navier-Stokes equations in Boussinesq-approximated form. In the following set of equations, angle brackets denote a horizontal domain average. A subscript 0 indicates a surface value. Note that the variables in the equations are implicitly filtered by the discretization (see above), but that the continuous form of the equations is used here for convenience. A double prime indicates SGS variables. The overbar indicating filtered quantities is omitted for readability, except for the SGS flux terms. The equations for the conservation of mass, energy and moisture, filtered over a grid volume on a Cartesian grid, then read as
{{{
#!Latex
\begin{align*}
 \frac{\partial u_i}{\partial t} &= - \frac{\partial u_i u_j}{\partial x_j} -\varepsilon_{ijk}f_j u_k + \varepsilon_{i3j}f_3 {u_{\mathrm{g},j}} - \frac{1}{\rho_0} \frac{\partial \pi^\ast}{\partial x_i} + g \frac{\theta_\mathrm{v} - \langle\theta_{\mathrm{v}}\rangle}{\langle\theta_{\mathrm{v}}\rangle}\delta_{i3}-\frac{\partial}{\partial x_j} \left(\overline{u_i^{\prime\prime} u_j^{\prime\prime}} - \frac{2}{3}e\delta_{ij}\right), \\
 \frac{\partial u_j}{\partial x_j}&=0, \\
 \frac{\partial \theta}{\partial t} &= - \frac{\partial u_j \theta}{\partial x_j} -\frac{\partial}{\partial x_j}\left(\overline{u_j^{\prime\prime}\theta^{\prime\prime}}\right) - \frac{L_\mathrm{V}}{c_p \Pi} \Psi_{q_\mathrm{v}}, \\
 \frac{\partial q_\mathrm{v}}{\partial t} &= - \frac{\partial u_j q_\mathrm{v}}{\partial x_j} - \frac{\partial}{\partial x_j}\left(\overline{u_j^{\prime\prime}q^{\prime\prime}_\mathrm{v}}\right) + \Psi_{q_\mathrm{v}},\\
 \frac{\partial s}{\partial t} &= - \frac{\partial u_j s}{\partial x_j} - \frac{\partial}{\partial x_j}\left(\overline{u_j^{\prime\prime}s^{\prime\prime}}\right) + \Psi_s.
\end{align*}
}}}
Here, ''i'', ''j'', ''k'' ∈ {1, 2, 3}. ''u'',,i,, are the velocity components (''u,,1,, = u, u,,2,, = v, u,,3,, = w'') with location ''x'',,i,, (''x,,1,, = x, x,,2,, = y, x,,3,, = z''), ''t'' is time, ''f,,i,, = (0, 2 Ω cos(φ), 2 Ω
sin(φ))'' is the Coriolis parameter with ''Ω'' being the Earth's angular velocity and ''φ'' being the geographical latitude. ''u'',,g,k,, are the geostrophic wind speed components, ''ρ'',,0,, is the density of dry air, ''π^∗^ = p^∗^ + 2/3 ρ,,0,, e '' is the modified perturbation pressure with ''p^∗^'' being the perturbation pressure and the SGS-TKE 
{{{
#!Latex
\begin{align*}
e = \frac{1}{2} \overline{u_i^{\prime\prime} u_i^{\prime\prime}},
\end{align*}
}}}
and ''g'' is the gravitational acceleration. The potential temperature is defined as
{{{
#!Latex
\begin{align*}
 \theta = T/\,\Pi,
\end{align*}
}}}
with the current absolute temperature ''T'' and the Exner function
{{{
#!Latex
\begin{align*}
  & \Pi = \left(\frac{p}{p_0}\right)^{R_\mathrm{d}/c_p},
\end{align*}
}}}
with ''p'' being the hydrostatic air pressure, ''p'',,0,, = 1000 hPa a reference pressure, ''R'',,d,, the specific gas constant for dry
air, and ''c'',,p,, the specific heat of dry air at constant pressure. The virtual potential temperature is defined as
{{{
#!Latex
\begin{align*}
  & \theta_\mathrm{v} = \theta \left[1+\left(\frac{R_\mathrm{v}}{R_\mathrm{d}}-1\right) q_\mathrm{v} - q_\mathrm{l}\right],
\end{align*}
}}}
with the specific gas constant for water vapor ''R'',,v,,, and the liquid water specific humidity ''q'',,l,,. For the computation of ''q'',,l,,, see the descriptions of the embedded cloud microphysical models in Sects. ~\ref{sec:micro} and \ref{sec:lcm}. Furthermore,
''L'',,V,, is the latent heat of vaporization, and ''Ψ'',,q,,v,,,, and ''Ψ'',,s,, are source/sink terms of ''q'',,v,, and ''s'', respectively.

== References ==
* [=#schumann1975]'''Schumann U.''' 1975. Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comput. Phys. 18: 376–404.