[[NoteBox(warn,This site is currently under construction!)]]

= Lagrangian particle model (LPM) =

The embedded LPM allows for studying transport and dispersion processes within turbulent flows. In the following we will describe
the general modeling of particles, including passive particles that do not show any feedback on the turbulent flow. In Sect. [wiki:/doc/tec/lcm Lagrangian cloud model] we will describe the use of Lagrangian particles as explicit cloud droplets.

== Formulation of the LPM ==

Lagrangian particles can be released in prescribed source volumes at
different points in time. The particles then obey
{{{
#!Latex
\begin{align*}
  & \frac{\mathrm{d} x_{\mathrm{p}, i}}{\mathrm{d}t} = u_{\mathrm{p},i}(t)
\end{align*}
}}}
where ''x'',,p,i,, describes the particle location in ''x'',,i,, direction (''i ∈ {1, 2, 3}'') and ''u'',,p,i,, is the
respective velocity component of the particle. Particle trajectories are calculated by means of the turbulent flow fields provided by PALM
for each time step. The location of a certain particle at time ''t + Δ t,,L,,'' is calculated by
{{{
#!Latex
\begin{align*}
  &
  x_{\mathrm{p},i}(x_{\text{ps},i},t + \Delta t_\mathrm{L}) =
x_{\mathrm{p},i}(x_{\text{ps},i},t) + \int \limits_t^{t + \Delta
t_\mathrm{L}} u_{\mathrm{p},i}(\hat{t}) \mathrm{d}\hat{t}\,,
\end{align*}
}}}
where ''x'',,ps,i,, is the spatial coordinate of the particle source point and ''Δ t'',,L,, is the applied time step in the Lagrangian particle model. Note that the latter is not necessarily equal to the time step of the LES model. The integral in the Eq. above is evaluated using either a Runge-Kutta (2nd- or 3rd-order) or the (1st-order) Euler time-stepping scheme.

The velocity of a weightless particle that is transported passively by the flow is determined by
{{{
#!Latex
\begin{align*}
  & u_{\mathrm{p},i} = u_i(x_{\mathrm{p},i})\;,
\end{align*}
}}}
and for non-passive particles (e.g., cloud droplets) by
{{{
#!Latex
\begin{align*}
  & \frac{\mathrm{d}u_{\mathrm{p},i}}{\mathrm{d}t} =
  \frac{1}{\tau_\mathrm{p}} \left(u_i(x_{\mathrm{p},i}) - u_{\mathrm{p},i}\right)
  -\delta_{i3} \left(1 - \frac{\rho_0}{\rho_{\mathrm{l},0}} \right) g,
\end{align*}
}}}
considering Stoke's drag, gravity and buoyancy (on the right-hand side, from left to right). Note that this Eq. is solved analytically assuming all variables but ''u'',,p,i,, as constants for one time step. Here, ''u'',,i,,''(x'',,p,i,,'')'' is the velocity of air at the particles location gathered from the eight adjacent grid points of the LES by tri-linear interpolation (see
Sect. [wiki:/doc/tec/particle particle code structure]). Since Stoke's drag is only valid for radii ''≤ 30 ''μm (e.g., [#rogers1989 Rogers and Yau, 1989]), a nonlinear correction is applied to the Stokes's drag relaxation time scale:
{{{
#!Latex
\begin{align*}
  & \tau_\mathrm{p}^{-1} =
  \frac{9\,\nu\,\rho_0}{2\,r^2\,\rho_{\mathrm{p},0}}\,\cdot\,\left(1 +
    0.15 \cdot {Re}_\mathrm{p}^{0.687} \right).
  \label{eq:lpm2}
\end{align*}
}}}
Here, ''r'' is the radius of the particle, ''ν = 1.461 x 10^-5^'' m^2^ s the molecular viscosity of air, and ''ρ'',,p,0,, the density of the particle. The particle Reynolds number is given by
{{{
#!Latex
\begin{align*}
  & {Re}_\mathrm{p}=\frac{2\,r\,\left|\,u_i(x_{\mathrm{p},i}) -
      u_{\mathrm{p},i}\,\right|}{\nu}.
\end{align*}
}}}
Following [#lamb1978 Lamb (1978)] and the concept of LES modeling, the Lagrangian velocity of a weightless particle can be split into
a resolved-scale contribution ''u'',,p,,^res^ and an SGS contribution ''u'',,p,,^sgs^:
{{{
#!Latex
\begin{align*}
 u_{\mathrm{p},i} = u_{\mathrm{p},i}^{\text{res}} +
 u_{\mathrm{p},i}^{\text{sgs}}\,.
\end{align*}
}}}
''u'',,p,i,,^res^ is determined by interpolation of the respective LES velocity components ''u'',,i,, to the position of the particle. The SGS part of the particle velocity at time ''t'' is given by
{{{
#!Latex
\begin{align*}
 u_{\mathrm{p},i}^{\text{sgs}}(t) = u_{\mathrm{p},i}^{\text{sgs}}(t -
 \Delta t_\mathrm{L}) + \mathrm{d} u_{\mathrm{p},i}^{\text{sgs}}\,,
\end{align*}
}}}
where d''u'',,p,i,,^sgs^  describes the temporal change of the SGS particle velocity during a time step of the LPM based on [#thomson1987 Thomson (1987)]. Note that the SGS part of ''u'',,p,i,, in the second equation of this section is always computed using the (1st-order) Euler
time-stepping scheme. [#weil2004 Weil et al. (2004)] developed a formulation of the Langevin equation under assumption of isotropic Gaussian turbulence in order to treat the SGS particle dispersion in terms of a stochastic differential equation. This equation reads as
{{{
#!Latex
\begin{align*}
  \mathrm{d}u_{\mathrm{p},i}^{\text{sgs}} = &-\frac{3 c_{\text{sgs}}
    C_\mathrm{L}\epsilon}{4}\frac{u_{\mathrm{p},i}^{\text{sgs}}}{e}
    \Delta t_\mathrm{L} + \frac{1}{2} \left(\frac{1}{e} \frac{\mathrm{d}
        e}{\Delta t_\mathrm{L}} u_{\mathrm{p},i}^{\text{sgs}} +
      \frac{2}{3}\frac{\partial e}{\partial x_i} \right) \Delta
    t_\mathrm{L} + \left(c_{\text{sgs}} C_\mathrm{L} \epsilon
    \right)^{\frac{1}{2}} \mathrm{d}\zeta_i\,
\end{align*}
}}}
and is used in PALM for the determination of the change in SGS particle velocities. Here, ''C'',,L,,'' = 3'' is a universal constant (''C'',,L,,'' = 4 ± 2'', see [#thomson1987 Thomson (1987)]). ''ζ'' is a vector composed of Gaussian-shaped random numbers, with each
component neither spatially nor temporally correlated. The factor
{{{
#!Latex
\begin{align*}
 c_\text{sgs} =
 \frac{\langle\,e\,\rangle}{\langle\,e_\text{res}\,\rangle +
   \langle\,e\,\rangle}\,,\quad 0 \leq c_\text{sgs} \leq 1\,,
\end{align*}
}}}
where ''e'',,res,, is the resolved-scale TKE as resolved by the numerical grid, assures that the temporal change of the modeled SGS
particle velocities is, on average (horizontal mean), smaller than the change of the resolved-scale particle velocities ([#weil2004 Weil et al., 2004]). Values of ''e'' and ''ε'' are provided by the SGS model described in Sect. [wiki:/doc/tec/sgs turbulence closure] (see Eqs. for ''∂e/∂t'' and for the dissipation rate ''ε'',respectively). The first term on the right-hand side of the Eq. for d''u'',,p,i,,^sgs^ represents the influence of the SGS particle velocity from the previous time step (i.e., inertial "memory"). This effect is considered by the Lagrangian time scale after [#weil2004 Weil et al. (2004)]:
{{{
#!Latex
\begin{align*}
  &
  \label{eq:LS5a}
  \tau_\mathrm{L} = \frac{4}{3}\frac{e}{c_{\text{sgs}}
    C_\mathrm{L}\epsilon}\,,
\end{align*}
}}}
which describes the time span during which ''u'',,p,,^sgs^''(t - Δt'',,L,,'')'' is correlated to ''u'',,p,,^sgs^''(t)''. The applied time step of the particle model hence must not be larger than τ,,L,,. In PALM, the particle time step is set to be smaller than ''τ'',,L,,'' / 40''. The second term on the right-hand side of the Eq. for d''u'',,p,i,,^sgs^ ensures that the assumption of well-mixed conditions by [#thomson1987 Thomson (1987)] is fulfilled on the subgrid scales. This term can be considered as drift correction, which shall prevent an over-proportional accumulation of particles in regions of weak turbulence (#rodean1996 Rodean, 1996). The third term on the right-hand side is of stochastic nature and describes the SGS diffusion of particles by a Gaussian random process. For a detailed derivation and discussion of
the Eq. for d''u'',,p,i,,^sgs^ see [#thomson1987 Thomson (1987)], [#rodean1996 Rodean (1996)] and [#weil2004 Weil et al. (2004)}.

The required values of the resolved-scale particle velocity components, ''e'', and ''ε'' are obtained from the respective LES fields using the eight adjacent grid points of the LES and tri-linear interpolation on the current particle location (see Sect. [wiki:/doc/tec/particle particle code structure]). An exception is made in case of no-slip boundary conditions set for the resolved-scale horizontal wind components below the first vertical grid level above the surface. Here, the resolved-scale particle velocities are determined from MOST (see
Sect. [wiki:/doc/tec/bc boundary conditions]) in order to capture the logarithmic wind profile within the height interval of ''z'',,0,, to ''z'',,MO,,. The available values of ''u'',,∗,,,
{{{
#!Latex 
$\overline{w^{\prime\prime}u^{\prime\prime}}_0$,
}}}
and 
{{{
#!Latex
$\overline{w^{\prime\prime}v^{\prime\prime}}_0$
}}}
are first bi-linearly interpolated to the horizontal location of the particle. In a second step the velocities are determined using the Eqs. for ''u'',,∗,,, ''∂u/∂z'' and ''∂v/∂z'' (\ref{eq:most:begin})--(\ref{eq:most:end}). Resolved-scale horizontal
velocities of particles residing at height levels below $z_0$ are set
to zero. The LPM allows to switch off the transport by the SGS
velocities.

== Boundary conditions and release of particles ==

== Recent applications ==

== References ==
* [=#rogers1989] '''Rogers RR, Yau MK.''' 1989. A short course in cloud physics. Pergamon Press. New York.

* [=#lamb1978] '''Lamp RG.''' 1978. A numerical simulation of dispersion from an elevated point source in the convective planetary boundary layer. Atmos. Environ. 12: 1297–1304.

* [=#thomson1987] '''Thomson DJ.''' 1987. Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J. Fluid Mech. 180: 529–556.

* [=#weil2004] '''Weil JC, Sullivan PP, Moeng C-H.''' 2004. The use of large-eddy simulations in Lagrangian particle dispersion models. J. Atmos. Sci. 61: 2877–2887.

* [=#rodean1996] '''Rodean HC.''' 1996. Stochastic Lagrangian models of turbulent diffusion. Meteor. Mon. 26: 1–84. [http://dx.doi.org/10.1175/0065-9401-26.48.1 doi].