Changeset 1634 for palm/trunk/TUTORIAL


Ignore:
Timestamp:
Aug 31, 2015 1:29:42 PM (7 years ago)
Author:
hoffmann
Message:

updated tutorial on Lagrangian particles

Location:
palm/trunk/TUTORIAL/SOURCE
Files:
3 added
1 edited

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  • palm/trunk/TUTORIAL/SOURCE/particle_model_cloud_physics.tex

    r1532 r1634  
    443443% Folie 8
    444444\begin{frame}[t]
    445    \frametitle{Basic Particle Parameters (V)}
    446    \small
    447    Parameter that defines the mode of particle movement:
    448    \vspace{+2mm}
    449    
    450    The concept of LES ...\\
    451    \begin{center}
    452       \includegraphics[scale=0.1]{particle_model_figures/basic_particle_parameters_5.png}
    453    \end{center}
    454    \vspace{-5mm}
    455    \onslide<2->
    456    ... transferred to the embedded particle model leads to particle velocity:\\
    457    \begin{center}
    458          $\vec{V}_{{\text{particle}}} = \vec{V}_{{\text{resolved}}} + \vec{V}_{{\text{subgrid}}}$\\
    459     \end{center}
    460          \onslide<3->
    461          \scriptsize
    462          Accordingly, the particle movement is a result of:
    463          \begin{itemize}
    464             \scriptsize
    465             \item<3-> resolved flow $\vec{V}_{{\text{resolved}}}$
    466             \vspace{-2mm}
    467             \item<3-> and subgrid scale turbulence $\vec{V}_{{\text{subgrid}}}$
    468             \begin{itemize}
    469             \scriptsize
    470             \item<4-> $\vec{V}_{{\text{subgrid}}}=0$, if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .F.} (default value)
    471 \item<4-> $\vec{V}_{{\text{subgrid}}}\ne0$, if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .T. } determination of the subgrid part of the particle velocity as a solution of a stochastic differential equation (see Weil et al., 2004, JAS)
    472 
    473             \end{itemize}
    474          \end{itemize}
    475 
    476 
    477 
    478 \end{frame}
    479 
    480 % Folie 9
    481 \begin{frame}[t]
    482445   \frametitle{Basic Particle Parameters (VI)}
    483446   \small
     
    568531\end{frame}
    569532
    570 % Folie 10
     533% Folie 9
    571534\begin{frame}
    572535   \frametitle{Basic Particle Parameters (VIII)}
     
    596559\end{frame}
    597560
    598 % Folie 11
     561% Folie 10
    599562\begin{frame}[fragile]
    600563   \frametitle{An Example of a Particle NAMELIST}
     
    603566   \begin{tikzpicture}
    604567   \node [yellow] {\begin{lstlisting} 
    605  &particles_par  dt_dopts = 25.0,
    606                  bc_par_b = 'absorb',
     568 &particles_par  bc_par_b = 'absorb',
    607569                 density_ratio = 0.001,
    608570                 radius = 1.0E-6,
     
    624586\subsection{Theory}
    625587
     588% Folie 11
     589\begin{frame}[t]
     590   \frametitle{Theory of the LPM (I) -- Passive Advection (I)}
     591   \small
     592   Parameter that defines the mode of particle movement:
     593   \vspace{+2mm}
     594   
     595   The concept of LES ...\\
     596   \begin{center}
     597      \includegraphics[scale=0.1]{particle_model_figures/basic_particle_parameters_5.png}
     598   \end{center}
     599   \vspace{-5mm}
     600   \onslide<2->
     601   ... transferred to the embedded particle model leads to particle velocity:\\
     602   \begin{center}
     603         $\vec{V}_{{\text{particle}}} = \vec{V}_{{\text{resolved}}} + \vec{V}_{{\text{subgrid}}}$\\
     604    \end{center}
     605         \onslide<3->
     606         \scriptsize
     607         Accordingly, the particle movement is a result of:
     608         \begin{itemize}
     609            \scriptsize
     610            \item<3-> resolved flow $\vec{V}_{{\text{resolved}}}$
     611            \vspace{-2mm}
     612            \item<3-> and subgrid scale turbulence $\vec{V}_{{\text{subgrid}}}$
     613            \begin{itemize}
     614            \scriptsize
     615            \item<4-> $\vec{V}_{{\text{subgrid}}}=0$, if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .F.} (default value)
     616\item<4-> $\vec{V}_{{\text{subgrid}}}\ne0$, if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .T. }
     617            \end{itemize}
     618         \end{itemize}
     619         
     620\end{frame}
     621
     622
    626623% Folie 12
    627624\begin{frame}[t]
    628    \frametitle{Theory of the Lagrangian Particle Model (I)}
     625   \frametitle{Theory of the LPM (II) -- Passive Advection (II)}
    629626   \footnotesize
    630    \textbf{Advection of Passive Particles}\\
    631627   \vspace{1mm}
    632628   \begin{itemize}
    633629   \item The position of the particle is found by integrating $ \dfrac{d \vec{X}_{\text{particle}}}{dt} = \vec{V}_{\text{particle}}$
    634    \item The particle's velocity consist of a resolved and a subgrid part:\\
     630   \item The particle's velocity consist of a \textcolor{red}{resolved} and an optional \textcolor{blue}{subgrid part}:\\
    635631   \begin{center}
    636     $\vec{V}_{\text{particle}} = \vec{V}_{\text{res}} (+ \vec{V}_{\text{sub}})$\\
     632    $\vec{V}_{\text{particle}} = \textcolor{red}{\vec{V}_{\text{res}}} (+ \textcolor{blue}{\vec{V}_{\text{sub}}})$\\
    637633    \end{center}
    638 %   \vspace{2mm}
    639 %   \onslide<2->Transferring the LES concept ...
    640 %   \includegraphics[scale=0.1]{particle_model_figures/basic_particle_parameters_5.png}\\
    641 %   \vspace{-2mm}
    642 %   \scriptsize \hspace{2em}total energy\hspace{2.25em}=\hspace{3em}resolved part\hspace{2.5em}+\hspace{3em}modelled part\\
    643 %   \vspace{1mm}
    644 %   \small ... to the embedded particle model leads to: $\vec{V}_{\text{particle}} = \vec{V}_{\text{res}} (+ \vec{V}_{\text{sub}})$\\
    645 %   \vspace{2mm}
    646    \item<2-> The resolved part $\vec{V}_{\text{res}}$ is derived by a tri-linear interpolation:
     634   \item<1-> The resolved part $\textcolor{red}{\vec{V}_{\text{res}}}$ is derived by a tri-linear interpolation:
    647635   \begin{tikzpicture}[remember picture, overlay]
    648636      \node [shift={(1.7 cm,2.8cm)}]  at (current page.south west)
     
    718706      \end{tikzpicture}
    719707     
    720       \vspace{+2.5cm}
    721       \item<3-> The $\vec{V}_{\text{sub}}$ is only computed if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .T.} (this also requires \texttt{use\underline{ }upstream\underline{ }for\underline{ }tke = .T.} in the \texttt{inipar} namelist). Then, a solution for $\vec{V}_{\text{sub}}$ is derived from a stochastic differential equation (see Weil et al., 2004, JAS).
     708      \vspace{+3.2cm}
     709      \item<1-> $\textcolor{blue}{\vec{V}_{\text{sub}}}$ is only computed if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .T.} Then, a solution for $\textcolor{blue}{\vec{V}_{\text{sub}}}$ is derived from a stochastic differential equation \\(see Weil et al., 2004, JAS).
    722710            \end{itemize}
    723711\end{frame}
     
    725713% Folie 13
    726714\begin{frame}[t]
    727    \frametitle{Theory of the Lagrangian Particle Model (II)}
     715   \frametitle{Theory of the LPM (III) -- Non-Passive Advection}
    728716   \footnotesize
    729    \textbf{Advection of Non-passive Particles}\\
    730    \ \\
    731    Newton's second law for a droplet (considering Stoke's drag, gravity and buoyancy):\\
     717  % \footnotesize
     718   Newton's second law of motion for a (spherical!) particle, for which \textcolor{red}{Stoke's drag}, \textcolor{blue}{gravity} and \textcolor{orange}{buoyancy} are considered:\\
    732719   \vspace{4mm}
    733    $\dfrac{dV_i}{dt} = \dfrac{1}{\tau_p} (u_i - V_i) - \delta_{i3}( 1 - \rho_0 / \rho_l ) g$\\
     720   \begin{center}
     721   $\dfrac{dV_i}{dt} = \textcolor{red}{\dfrac{1}{\tau_p} (u_i - V_i)} - \delta_{i3}( 1 - \textcolor{orange}{\rho_0 / \rho_l} ) \cdot \textcolor{blue}{g},$\\
     722      \end{center}
    734723      \vspace{2mm}
    735   with the inertial response time $\tau_p^{-1} = \dfrac{9 \nu \rho_0}{2 r^2 \rho_l} \left( 1 + 0.15 \text{Re}^{0.687} \right)$ including a correction term (in parenthesis) for high Reynolds numbers (see Clift et al., 1978)\\%based on $\text{Re} = \dfrac{2r \left| \vec{u}_i - \vec{V}_i \right|}{\nu} $ (see Clift et al., 1978)\\
    736    \vspace{10mm}
    737    \onslide<1->
     724  with the inertial response time
     725  \begin{align*}
     726  \tau_p^{-1} = \frac{9 \nu \rho_0}{2 r^2 \rho_l} \textcolor{magenta}{\left( 1 + 0.15 \cdot \text{Re}^{0.687} \right)},
     727    \end{align*}
     728    including a \textcolor{magenta}{correction term} for high Reynolds numbers (see Clift et al., 1978)\\
     729   \vspace{1mm}
    738730   \begin{tabular}{llll}
    739731           $g$   & = gravitational acceleration & $\rho_l$ & = density of water\\
     
    743735   \end{tabular}
    744736
     737
    745738\end{frame}
    746739
    747740% Folie 14
    748741\begin{frame}[t]
    749    \frametitle{Theory of the Lagrangian Particle Model (III)}
     742   \frametitle{Theory of the LPM (IV) -- Cloud Droplets (I)}
    750743   \footnotesize
    751    \textbf{Using Particles as Cloud Droplets}\\
    752    \small
    753744   \begin{itemize}
    754745      \item This feature is switched on by setting the initial parameter \texttt{cloud\_droplets = .TRUE.}
    755       \item In this case, the change in particle radius by condensation/evaporation and collision/coalescence is calculated for every timestep.
    756       \item In case of condensation or evaporation, the potential temperature and the specific humidity (computed by the LES) have to be adjusted. This is done within the subroutine \texttt{interaction\_droplets\_ptq}, which is therefore the major coupling between the LES and the LPM.
     746      \item In this case, the change in particle radius by condensation/evaporation and collision/coalescence is calculated for each time step.
     747      \item In case of condensation or evaporation, the LES variables potential temperature and the specific humidity have to be adjusted. This is done within the subroutine \texttt{interaction\_droplets\_ptq} (which is the major coupling between LES and LPM).
    757748   \end{itemize}
    758749\end{frame}
     
    760751% Folie 15
    761752\begin{frame}[t]
    762    \frametitle{Theory of the Lagrangian Particle Model (IV)}
     753   \frametitle{Theory of the LPM (V) -- Cloud Droplets (II)}
    763754   \footnotesize
    764    \textbf{Simulation of Cloud Droplets (I)}\\
    765    \begin{itemize}
    766       \item Simulation of enormous particle numbers like in real clouds\\ is impossible
    767       \begin{itemize}
    768          \footnotesize
    769          \item Ensembles of water droplets are simulated
    770          \vspace{1mm}
    771          \item Every simulated droplet stands for a very high number\\ of real droplets
    772          \vspace{1mm}
    773          \item Concept of weighting factor (Shima et al., 2009, QJRMS):\\
     755   \begin{itemize}
     756      \item Simulation of realistic particle numbers (as found in clouds) is impossible
     757         \item Ensembles of water droplets are simulated instead
     758         \item Each simulated particle represents a very high number\\ of real droplets
     759         \item Concept of super-droplets (Shima et al., 2009, QJRMS):\\
    774760         \begin{center}
    775           $A_i = \textit{real number of droplets represented by one simulated droplet}$
    776               \includegraphics[scale=1.0]{particle_model_figures/super.jpg}
     761                       \includegraphics[scale=1.0]{particle_model_figures/super.jpg}
     762                       
     763                        $A_i = \textit{number of droplets represented by one simulated particle}$
     764
    777765          \end{center}
    778          \vspace{-2mm}
     766         \vspace{-1mm}
    779767         \item Initial weighting factor can be assigned with the parameter \texttt{initial\underline{ }weighting\underline{ }factor}
    780       \end{itemize}
    781768   \end{itemize}
    782769   
    783 
    784770\end{frame}
    785771
    786772% Folie 16
    787773\begin{frame}[t]
    788    \frametitle{Theory of the Lagrangian Particle Model (V)}
     774   \frametitle{Theory of the LPM (VI) -- Diffusional Growth}
    789775   \footnotesize
    790    \textbf{Simulation of Cloud Droplets (II)}\\
    791    \scriptsize
    792    \begin{itemize}
     776  % \scriptsize
     777    \begin{itemize}
    793778      \item The growth of the radius of single droplet by condensation/evaporation:\\
    794       \vspace{1mm}
    795       $ r_i \dfrac{dr_i}{dt} = \dfrac{(S - a\,r^{-1} + b\,r^{-3})}{F_\text{k} + F_\text{d}}$\\
    796       \vspace{1mm}
    797       primarily depending on the supersaturation $S = e / e_\text{s}-1$, including the effects of the particle's curvature (parameter $a$) and amount of solute aerosol (parameter $b$)
    798       \vspace{1.5mm}
    799       \item<2-> For $r > 1\,\text{\textmu m}$ solution and curvature effects are neglected $\Rightarrow$ an analytic solution is possible: \\
    800       $ r_i \dfrac{dr_i}{dt} = \dfrac{S}{F_k + F_d} \Rightarrow r_i(t) = \sqrt{ r_{i,0}^2 + 2 \cdot \Delta t \cdot \left( \dfrac{S}{F_\text{k} + F_\text{d}} \right)}$
    801    \end{itemize}
    802    \vspace{1.0cm}
     779\begin{align*}
     780r \dfrac{\text{d}r}{\text{d}t} = \dfrac{(\textcolor{red}{S} - \textcolor{blue}{a}\,r^{-1} + \textcolor{orange}{b}\,r^{-3})}{F_\text{k} + F_\text{d}}
     781\end{align*}
     782      primarily depending on the \textcolor{red}{relative water supersaturation $S$}, and the effects of the \textcolor{blue}{particle's curvature ($a$)} and \textcolor{orange}{physical and chemical properties of aerosol ($b$)}
     783
     784\item Stiff differential equation: Numerical integration with a 4th-order Rosenbrock method, which adapts its internal time step for an accurate and computationally efficient solution (Grabowski et al., 2011, Atmos. Res.)
     785
     786   \end{itemize}
    803787   \hspace{0.5cm}
     788   \footnotesize
    804789   \begin{tabular}{llll}
    805790           $r$   & = Droplet radius & $S$   & = Supersaturation\\
    806791           $a$ & = Curvature effect    & $b$  & = Solution effect\\
    807            $e$ & = Vapor pressure  &  $e_\text{s}$ & = Saturation vapor pressure\\ 
    808792           $F_\text{k}$ & = Effect of heat conduction & $F_\text{d}$ & = Effect of vapor diffusion
    809793   \end{tabular}
     
    813797% Folie 17
    814798\begin{frame}[t]
    815    \frametitle{Theory of the Lagrangian Particle Model (VI)}
     799   \frametitle{Theory of the LPM (VII) -- Collisions (I)}
     800   \vspace{-2mm}
    816801   \footnotesize
    817    \textbf{Simulation of Cloud Droplets (III)} -- Idealized concept of droplet collisions\\
    818    \begin{tikzpicture}[remember picture, overlay]
    819       \node [shift={(6.0cm,4.0cm)}]  at (current page.south west)
    820          {%
    821          \begin{tikzpicture}[remember picture, overlay]
    822             \node at (0.0,0.0) {\includegraphics[scale=0.25]{particle_model_figures/collision1.png}};
    823          \end{tikzpicture}
    824          };
    825    \end{tikzpicture}     
     802    \begin{itemize}
     803   \item Two prognostic quantities:
     804   
     805$\ \ \ $ (i) \textbf{weighting factor} $A$ and (ii) \textbf{total mass} of super-droplet $m$
     806\item total mass: mass of all droplets represented by one super-droplet
     807\end{itemize}
     808   \vspace{-3mm}
     809\begin{center}
     810\includegraphics[scale=0.5]{particle_model_figures/coll.pdf}
     811\end{center}
    826812\end{frame}
    827813
    828814% Folie 18
    829815\begin{frame}[t]
    830    \frametitle{Theory of the Lagrangian Particle Model (VII)}
     816   \frametitle{Theory of the LPM (VIII) -- Collisions (II)}
    831817   \footnotesize
    832    \textbf{Simulation of Cloud Droplets (IV)}\\
    833    \scriptsize   
    834    \begin{itemize}
    835       \item Calculation of droplet growth due to collisions considers three types of collisions:
    836       \vspace{-0.4cm}
    837       \begin{enumerate}
    838       \scriptsize
    839           \item collisions with smaller droplets $\Rightarrow$ increase the radius
    840           \item collisions with larger droplets $\Rightarrow$ decrease the weighting factor
    841           \item internal collisions $\Rightarrow$ decrease the weighting factor and increase the radius
    842       \end{enumerate}
    843       \item Two prognostic quantities: Weighting factor $A_n$ and mass of super-droplet expressed as volume averaged droplet radius $r_n=(m_n / (4/3 \pi \rho_\text{l} A_n))^{1/3}$:
     818     \begin{itemize}
     819         \item Calculation of droplet growth due to collisions considers three types of collisions (for all droplets located in one grid box):
     820     % \vspace{-0.4cm}
     821      \begin{itemize}
     822      \footnotesize
     823          \item \textcolor{red}{collisions with smaller droplets $\Rightarrow$ increase  total mass}
     824          \item \textcolor{blue}{collisions with larger droplets \ \ $\Rightarrow$ decrease  weighting factor \hphantom{collisions with larger droplets \ \ $\Rightarrow$} and total mass}
     825          \item \textcolor{orange}{internal collisions \hphantom{ager droplets} $\Rightarrow$ decrease  weighting factor}
     826      \end{itemize}
     827       \item Total mass of super-droplet not useful
     828       
     829       $\Rightarrow$ volume averaged droplet radius $r_n=(m_n / (4/3 \pi \rho_\text{l} A_n))^{1/3}$
     830
     831      \item Droplets are sorted that $r_1<r_2<...< r_{N_\text{p}-1} <r_{N_\text{p}}$:
    844832         \scriptsize
    845833\begin{alignat*}{3}
    846 A_n^{\ast}&=A_n &&-  K(r_n,\, r_n) \, \frac{1}{2} \, \frac{A_n (A_n - 1)}{\Delta V} \, \Delta t - \sum \limits_{m=n+1}^{N_\text{p}} K(r_m,\, r_n) \frac{A_n \, A_m}{\Delta V} \, \Delta t\\
    847  r_n^{\ast}&=\left(\vphantom{\sum \limits_{m=1}^{n-1}} \right. &&\left[ \vphantom{\sum \limits_{m=1}^{n-1}}\right. r_n^3 + \sum \limits_{m=1}^{n-1} K(r_\text{n}, \, r_\text{m}) \frac{A_m}{\Delta V} \, r_m^3 \, \Delta t -\sum \limits_{m=n+1}^{N_\text{p}} K(r_\text{m}, \, r_\text{n}) \frac{A_m}{\Delta V} \, r_n^3 \, \Delta t \left.\vphantom{\sum \limits_{m=1}^{n-1}}\right] \left/\vphantom{\sum \limits_{m=1}^{n-1}}\right. \notag \\
    848 &  && \left[\vphantom{\sum \limits_{m=1}^{n-1}}\right.\ 1 -  K(r_n,\, r_n) \, \frac{1}{2} \, \frac{A_n - 1}{\Delta V} \, \Delta t -\left. \left. \sum \limits_{m=n+1}^{N_\text{p}} K(r_m,\, r_n) \frac{A_m}{\Delta V} \, \Delta t \right] \right)^{1/3}
     834A_n^{\ast}&=A_n &&\textcolor{orange}{-  K(r_n,\, r_n) \, \frac{1}{2} \, \frac{A_n (A_n - 1)}{\Delta V} \, \Delta t} \ \textcolor{blue}{- \sum \limits_{m=n+1}^{N_\text{p}} K(r_m,\, r_n) \frac{A_n \, A_m}{\Delta V} \, \Delta t}\\
     835 r_n^{\ast}&=\left(\vphantom{\sum \limits_{m=1}^{n-1}} \right. &&\left[ \vphantom{\sum \limits_{m=1}^{n-1}}\right. r_n^3 + \textcolor{red}{\sum \limits_{m=1}^{n-1} K(r_\text{n}, \, r_\text{m}) \frac{A_m}{\Delta V} \, r_m^3 \, \Delta t } \ \textcolor{blue}{-\sum \limits_{m=n+1}^{N_\text{p}} K(r_\text{m}, \, r_\text{n}) \frac{A_m}{\Delta V} \, r_n^3 \, \Delta t} \left.\vphantom{\sum \limits_{m=1}^{n-1}}\right] \left/\vphantom{\sum \limits_{m=1}^{n-1}}\right. \notag \\
     836&  && \left[\vphantom{\sum \limits_{m=1}^{n-1}}\right.\ 1 \textcolor{orange}{-  K(r_n,\, r_n) \, \frac{1}{2} \, \frac{A_n - 1}{\Delta V} \, \Delta t} \ \textcolor{blue}{-}\left. \left. \textcolor{blue}{\sum \limits_{m=n+1}^{N_\text{p}} K(r_m,\, r_n) \frac{A_m}{\Delta V} \, \Delta t} \right] \right)^{1/3}
    849837\end{alignat*}
    850838\end{itemize}
     
    853841% Folie 19
    854842\begin{frame}[t]
    855    \frametitle{Theory of the Lagrangian Particle Model (VIII)}
     843   \frametitle{Theory of the LPM (IX) -- Collisions III}
    856844   \footnotesize
    857    \textbf{Simulation of Cloud Droplets (V)}\\
    858845   \begin{itemize}
    859846      \item \textbf{Collision kernel without turbulence effects}:\\
     
    865852      Wang and Grabowski, 2009, ASL):\\
    866853      \vspace{1mm}
    867       $K(r_\text{n}, r_\text{m}) = 2 \pi (r_\text{n} + r_\text{m})^2 \cdot \textcolor{red}{\eta_E} E(r_\text{n}, r_\text{m}) \cdot \textcolor{red}{\langle| w_r |\rangle} \textcolor{red}{g_\text{RDF}} $\\
     854      $K(r_\text{n}, r_\text{m}) = 2 \pi (r_\text{n} + r_\text{m})^2 \cdot \textcolor{red}{\eta_E} \cdot E(r_\text{n}, r_\text{m}) \cdot \textcolor{red}{\langle| w_r |\rangle \cdot} \textcolor{red}{g_\text{RDF}} $\\
    868855      \vspace{+1mm}
    869856      all \textcolor{red}{red} variables parameterize effects of turbulence
     
    941928   \begin{itemize}
    942929   \item Handling hundreds of millions of particles, efficient storing is essential for a good performance
    943    \item The easiest method for storing Lagrangian particles is an one-dimensional array
    944    \item Most applications demand particles located at a certain location (e.\,g., collision process is computed for all particles located in a certain grid box)
    945       \item Finding this particles demands \texttt{N}$^2$ operations:
    946       \scriptsize
    947 \begin{lstlisting}
    948 DO  n = 1, N
    949    DO  k = 1, N
    950       IF ( k /= n )  THEN
    951          IF ( ABS( particles(k)%x - particles(n)%x )   &
    952               < threshold )  THEN
    953             ...
    954          ENDIF
    955       ENDIF
    956    ENDDO
    957 ENDDO
    958 \end{lstlisting}
    959    \end{itemize}
    960 \end{frame}
     930    \item Most applications demand particles located at a certain location (e.\,g., collision process is computed for all particles located in a certain grid box)
     931      \item Sorting the particles by their respective grid-box increases the computability of the code, but needs time for the sorting itself
     932\item<2-> A new, efficient approach for storing particles is implemented in PALM: \\
     933\only<1>{\begin{center}
     934\vphantom{\colorbox{red}{\textbf{a four-dimensional array}}}
     935\end{center}
     936}
     937\only<2>{
     938\begin{center}
     939\colorbox{red}{\textbf{a four-dimensional array}}
     940\end{center}
     941}
     942
     943   \end{itemize}
     944\end{frame}
     945
    961946
    962947% Folie 22
    963 \begin{frame}[fragile]
    964    \frametitle{Storing Lagrangian particles (I)}
    965    \begin{itemize}
    966       \item Sorting the particles by their respective grid-box reduces the operations to \texttt{N}:
    967          \scriptsize
    968 \begin{lstlisting}
    969 DO  n = n_start(k,j,i), n_end(k,j,i)
    970    ...
    971 ENDDO
    972 \end{lstlisting}
    973 \normalsize
    974 \item This was done in the previous version of PALM, reducing CPU time of LPM by 9.6\,\%
    975 \item However, sorting increases CPU time and demands a second, one-dimensional array for efficient sorting
    976 \item<2-> To overcome these issues, a new approach has been developed for the current version of PALM: \\
    977 \begin{center}
    978 \textbf{a four-dimensional array}
    979 \end{center}
    980    \end{itemize}
    981 \end{frame}
    982 
    983 
    984 % Folie 23
    985948\begin{frame}[t]
    986949   \frametitle{Storing Lagrangian particles (III)}
     
    10661029      \uncover<1->{\node[text width=10em] at (10,0) {\scriptsize - All particles located in a certain grid-box are stored in a \textit{small} one-dimensional particle array permanently assigned to their grid-box\\
    10671030      \ \\
    1068       - LPM CPU time decreases by 22\,\%\\
    1069       \ \\
    1070       - Available memory doubles, since no large additional arrays are needed for assigning the particles to their grid-box\\};}
     1031      - LPM CPU time decreases by 22\,\% (in comparison to storing particles in a one-dimensional array)\\
     1032%      \ \\
     1033%      - Available memory doubles, since no large additional arrays are needed for assigning the particles to their grid-box\\
     1034};}
    10711035   \end{tikzpicture}
    10721036
    10731037\end{frame}
    10741038
    1075 % Folie 24
     1039% Folie 23
    10761040\begin{frame}[fragile]
    10771041   \frametitle{Storing Lagrangian particles (IV)}
    10781042   \scriptsize
    10791043   \begin{itemize}
    1080    \item A new \textbf{3D-array} of another FORTRAN derived data type: \texttt{grid\_particle\_def}
    1081    \item This type contains, as an element, a \textbf{1D-array} of the FORTRAN derived data type \texttt{particle\_type}, in which the particles, located at that grid box, are stored
     1044   \item A \textbf{3D-array} of another FORTRAN derived data type: \texttt{grid\_particle\_def}
     1045   \item This type contains, as an element, a \textbf{1D-array} of the FORTRAN derived data type \texttt{particle\_type}, in which the particles located at that grid box are stored
    10821046   \end{itemize}
    10831047   \tikzstyle{yellow} = [rectangle, draw, fill=yellow!30, text width=1.05\textwidth, font=\scriptsize,scale=0.95]
    1084    \begin{tikzpicture}
    1085    \node[yellow]{\begin{lstlisting} 
    1086  TYPE  grid_particle_def
    1087      TYPE(particle_type), DIMENSION(:) ::  particles
    1088  END TYPE grid_particle_def
    1089 
    1090  TYPE(grid_particle_def), DIMENSION(:,:,:) ::  grid_particles
    1091 \end{lstlisting}
    1092    };
     1048   \vspace{-0.2mm}
     1049   \begin{tikzpicture}\node[yellow]{\begin{lstlisting} 
     1050TYPE  grid_particle_def
     1051   TYPE(particle_type), DIMENSION(:), ALLOCATABLE ::  particles
     1052END TYPE grid_particle_def
     1053
     1054TYPE(grid_particle_def), DIMENSION(:,:,:), ALLOCATABLE ::     &
     1055                                                 grid_particles
     1056   \end{lstlisting}};
     1057   \vspace{-0.2mm}
    10931058   \end{tikzpicture}
    10941059   \vspace{-4mm}
     
    11041069      IF ( n_par <= 0 )  CYCLE
    11051070      particles(1:n_par) = &
    1106         grid_particles(kp,jp,ip)%particles(1:n_par)
     1071        grid_particles(k,j,i)%particles(1:n_par)
    11071072      DO  n = 1, n_par
    11081073        particles(n)%radius = 1.0E-6
     
    11151080%\end{lstlisting}
    11161081      \end{itemize}
     1082   
     1083\end{frame}
     1084
     1085% Folie 24
     1086\begin{frame}[fragile]
     1087   \frametitle{Storing Lagrangian Particles (V) -- Efficient interpolation}
     1088 \begin{columns}
     1089 \begin{column}{6cm}
     1090 \footnotesize
     1091 \begin{itemize}
     1092 \item For interpolating any LES quantity on the location of a particle, the data from 8 grid points is needed
     1093 \item The indices of these grid points have to be determined for each particle
     1094 \item Depending on the particle's location within the grid box, the same set of indices is needed for all particles in the same sub-grid box
     1095 \item Sorting the particles by their sub-grid box makes the determination of the indices for each particle unnecessary
     1096 \item Sorting increases CPU time by 3\,\%, but efficient interpolation \textbf{speeds up the model by 22\,\%}
     1097 \end{itemize}
     1098 \end{column}
     1099  \begin{column}{6cm}
     1100    \begin{center}
     1101   \includegraphics[scale=0.47]{particle_model_figures/interpolation_neu.pdf}
     1102   \end{center}
     1103 \end{column}
     1104 \end{columns}
    11171105   
    11181106\end{frame}
     
    13721360\begin{frame}[t]
    13731361   \frametitle{Application Examples of the LCM (I)}
    1374    \footnotesize
    13751362   \textbf{The Lagrangian Cloud Model has many advantages:}
    13761363   \begin{itemize}
    1377       \item Dynamics and microphysics of the cloud are directly related to physical processes of the individual droplets
    1378       \item Many microphysical processes are modeled by first principles \\
     1364         \item Many microphysical processes are modeled by first principles \\
    13791365      $\Rightarrow$ (almost) no parameterizations
    1380       \item The LCM provides detailed information, e.\,g., spatial and temporal evolution of the droplet spectrum, spatial distribution of the droplet concentration, droplet trajectories, ...
     1366         \item We are able to simulate cloud microphysics on a very accurate level, but we are also able to cope the macro-scale, i.\,e., a whole cloud or cloud ensemble by LES\\
     1367      \item The LCM provides detailed information, e.\,g., spatial and temporal evolution of the droplet spectrum, droplet trajectories, ...
    13811368   \end{itemize}
    13821369      \textbf{How to use these advantages?}
    13831370   \begin{itemize}
    1384       \item Many cloud microphysical processes are still not sufficiently understood, but have a large impact on macroscopic cloud properties
    1385       \item Open Issues: production of rain, interaction of clouds and aerosols, ...
    1386       \item Using the LCM, we are able to simulate cloud microphysics on a very accurate level, but are also able to cope the macroscale, i.\,e., the whole cloud or cloud ensemble\\
    1387       $\Rightarrow$ \textbf{new insights on clouds and their physics}
    1388    \end{itemize}
     1371   \item Some application examples will show!
     1372        \end{itemize}
    13891373   \vspace{1.5mm}
    13901374\end{frame}
     
    13921376% Folie 31
    13931377\begin{frame}[t]
    1394    \frametitle{Application Examples of the LCM (II)}
     1378   \frametitle{Application Examples of the LCM (I)}
    13951379   \footnotesize
    1396    \textbf{From Riechelmann et al. (2012, NJP):}
    1397       \begin{tikzpicture}[remember picture, overlay]
    1398       \node [shift={(6.3 cm, 4.2 cm)}]  at (current page.south west)
    1399          {%
    1400          \begin{tikzpicture}[remember picture, overlay]
    1401             \node at (0.0,0.0) {\includegraphics[scale=0.4]{particle_model_figures/turbulence_effects_2.png}};
    1402             \node at (-3.0,2.6) {Kernel without turbulence effects};
    1403             \node at (3,2.6) {Kernel with turbulence effects};
    1404 
    1405          \end{tikzpicture}
    1406          };
    1407    \end{tikzpicture}
    1408    \ \\ 
    1409    \vspace{52mm}
    1410    $\rightarrow$ Turbulence effects enhance droplet growth and lead to more realistic mass distribution function\\
     1380      \textbf{How to Track Particles:}
     1381   \begin{center}
     1382   \includegraphics[scale=0.28]{particle_model_figures/traj.jpg}
     1383   \end{center}
     1384   $\rightarrow$ Find out what a droplet is experiencing during its life time
    14111385\end{frame}
    14121386
    14131387% Folie 32
    14141388\begin{frame}[t]
    1415    \frametitle{Application Examples of the LCM (III)}
    1416    \footnotesize
    1417       \textbf{From Lee et al. (2014, MAP):}
    1418    \begin{center}
    1419    \includegraphics[scale=0.25]{particle_model_figures/lee.jpg}
    1420    \end{center}
    1421    $\rightarrow$ Confirm the importance of the cloud top and the affiliated mixing processes for the initiation of rain
    1422 \end{frame}
    1423 
    1424 % Folie 33
    1425 \begin{frame}[t]
    1426    \frametitle{Application Examples of the LCM (IV)}
     1389   \frametitle{Application Examples of the LCM (II)}
    14271390   \footnotesize
    14281391      \textbf{From Hoffmann et al. (2015, AR):}
     
    14331396\end{frame}
    14341397
    1435 % Folie 34
     1398% Folie 33
    14361399\begin{frame}
    14371400   \frametitle{General Warning}
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