= 1-D model for precursor runs =

The initial profiles of the horizontal wind components in PALM can be prescribed by the user by piecewise linear gradients or by directly
using observational data. Alternatively, a 1-D model can be employed to calculate stationary boundary-layer wind profiles. This is particularly useful in neutral stratification, where inertial oscillations can persist for several days in case that non-balanced profiles are used for initialization. By employing the embedded computationally inexpensive 1-D model with a Reynolds-average based turbulence parametrization, these oscillations can be significantly damped. A stationary state of the wind profiles can thus be provided much faster in the 3-D model. The arrays of the 3-D variables are then initialized with the (stationary) solution of the 1-D model. These variables are ''u'',,i,, where ''i ∈ {1, 2}'', ''e'', ''K'',,h,,, ''K'',,m,, and, with MOST applied between the surface and the first vertical grid level, also ''L'', ''u'',,∗,, as well as
{{{
#!Latex
$\overline{u_i^{\prime\prime} u_3^{\prime\prime}}$
}}} 
(where ''i ∈ {1, 2}'').

The 1-D model assumes the profiles of ''θ'' and ''q'',,v,,, as prescribed by the user, to be constant in time. The model solves the prognostic equations for ''u'',,i,, and ''e'':
{{{
#!Latex
\begin{align*}
  & \frac{\partial u_i}{\partial t} = -\varepsilon_{i3j}f_3 u_j +
  \varepsilon_{i3j}f_3 {u_{\mathrm{g},j}} - \frac{\partial
    \overline{u_i^{\prime\prime}u_3^{\prime\prime}}}{\partial x_3}
\end{align*}
}}}
and
{{{
#!Latex
\begin{align*}
  & \frac{\partial e}{\partial t} = - \overline{u^{\prime\prime}w^{\prime\prime}} \frac{\partial u}{\partial z}
                                    - \overline{v^{\prime\prime}w^{\prime\prime}} \frac{\partial v}{\partial z}
                                    + \frac{g}{\theta} \overline{w^{\prime\prime}\theta^{\prime\prime}}
                                    - \frac{\partial \overline{w^{\prime\prime}e^{\prime\prime}}}{\partial z}
                                    - \epsilon\;.
\end{align*}
}}}
The dissipation rate is parametrized by
{{{
#!Latex
\begin{align*}
  & \epsilon = 0.064 \frac{e^{\frac{3}{2}}}{l}
\end{align*}
}}}
after [#detering1985 Detering and Etling (1985)]. The mixing length is calculated after [#blackadar1997 Blackadar (1997)] as
{{{
#!Latex
\begin{align*}
  & l = \frac{\kappa z}{1 + \frac{\kappa
      z}{l_{\text{Bl}}}}\,\;\text{with}\,\;l_{\text{Bl}} = \frac{2.7\times10^{-4}}{f}\sqrt{u_{\mathrm{g}}^2 + v_{\mathrm{g}}^2}\;.
\end{align*}
}}}
The turbulent fluxes are calculated using a 1st-order closure:
{{{
#!Latex
\begin{align*}
  & \overline{u_i^{\prime\prime}u_3^{\prime\prime}} = - K_\mathrm{m}
  \frac{\partial u_i}{\partial
    x_3}\;,\;\overline{w^{\prime\prime}\theta^{\prime\prime}} = -
  K_\mathrm{h} \frac{\partial \theta}{\partial z}
  \;,\,\overline{w^{\prime\prime}e^{\prime\prime}} = - K_\mathrm{m}
  \frac{\partial e}{\partial z}\;,
\end{align*}
}}}
where ''K'',,m,, and ''K'',,h,, are calculated as
{{{
#!Latex
\begin{align*}
 K_\mathrm{m} = c_\mathrm{m}\;\sqrt{e}\
  \begin{cases}         l & \text{for~}  \textit{Ri} \geq 0\\
    l_{\text{Bl}} & \text{for~}  \textit{Ri} < 0\\
  \end{cases}\;,\\
  &K_\mathrm{h} = \frac{\Phi_\mathrm{h}}{\Phi_\mathrm{m}} K_\mathrm{m}
\end{align*}
}}}
with the similarity functions ''Φ'',,h,, and ''Φ'',,m,, (see Eqs. in Sect [wiki:/doc/tec/bc boundary conditions]), using the gradient Richardson number:
{{{
#!Latex
\begin{align*}
  & \textit{Ri} =
  \frac{\frac{g}{\theta_\mathrm{v}}\frac{\partial\theta}{\partial
      z}}{\left[\left(\frac{\partial u}{\partial z}\right)^2 +
      \left(\frac{\partial v}{\partial z}\right)^2 \right]} \cdot
  \begin{cases}
    1\,\;&\text{for~}\,\;\textit{Ri} \geq 0\;,\\
    (1 - 16 \cdot
    \textit{Ri})^{\frac{1}{4}}\,\;&\text{for~}\,\;\textit{Ri} < 0\;.
  \end{cases}
\end{align*}
}}}
Note that the distinction of cases in the Eq. above is done with the value of ''Ri'' from the previous time step.

Moreover, a Rayleigh damping can be switched on to speed up the damping of inertial oscillations. The 1-D model is discretized in space using finite differences. Discretization in time is achieved using the 3rd-order Runge--Kutta time-stepping scheme ([#williamson1980 Williamson, 1980]). Dirichlet boundary conditions are used at the top and bottom boundaries of the model, except for ''e'', for which Neumann conditions are set at the surface (see also Sect. [wiki:/doc/tec/bc boundary conditions]).



== References ==
* [=#detering1985] '''Detering HW, Etling D.''' 1985. Application of the ''E''-''ε'' turbulence model to the atmospheric boundary layer. Bound.-Lay. Meteorol. 33: 113–133.

* [=#blackadar1997] '''Blackadar AK.''' 1997. Turbulence and Diffusion in the Atmosphere. Springer. Berlin. Heidelberg. NewYork. 185 pp.

* [=#williamson1980]'''Williamson JH.''' 1980. Low-storage Runge–Kutta schemes. J. Comput. Phys. 35: 48–56.