# Changeset 1634 for palm/trunk/TUTORIAL

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Timestamp:
Aug 31, 2015 1:29:42 PM (7 years ago)
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updated tutorial on Lagrangian particles

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

 r1532 % Folie 8 \begin{frame}[t] \frametitle{Basic Particle Parameters (V)} \small Parameter that defines the mode of particle movement: \vspace{+2mm} The concept of LES ...\\ \begin{center} \includegraphics[scale=0.1]{particle_model_figures/basic_particle_parameters_5.png} \end{center} \vspace{-5mm} \onslide<2-> ... transferred to the embedded particle model leads to particle velocity:\\ \begin{center} $\vec{V}_{{\text{particle}}} = \vec{V}_{{\text{resolved}}} + \vec{V}_{{\text{subgrid}}}$\\ \end{center} \onslide<3-> \scriptsize Accordingly, the particle movement is a result of: \begin{itemize} \scriptsize \item<3-> resolved flow $\vec{V}_{{\text{resolved}}}$ \vspace{-2mm} \item<3-> and subgrid scale turbulence $\vec{V}_{{\text{subgrid}}}$ \begin{itemize} \scriptsize \item<4-> $\vec{V}_{{\text{subgrid}}}=0$, if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .F.} (default value) \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) \end{itemize} \end{itemize} \end{frame} % Folie 9 \begin{frame}[t] \frametitle{Basic Particle Parameters (VI)} \small \end{frame} % Folie 10 % Folie 9 \begin{frame} \frametitle{Basic Particle Parameters (VIII)} \end{frame} % Folie 11 % Folie 10 \begin{frame}[fragile] \frametitle{An Example of a Particle NAMELIST} \begin{tikzpicture} \node [yellow] {\begin{lstlisting} &particles_par  dt_dopts = 25.0, bc_par_b = 'absorb', &particles_par  bc_par_b = 'absorb', density_ratio = 0.001, radius = 1.0E-6, \subsection{Theory} % Folie 11 \begin{frame}[t] \frametitle{Theory of the LPM (I) -- Passive Advection (I)} \small Parameter that defines the mode of particle movement: \vspace{+2mm} The concept of LES ...\\ \begin{center} \includegraphics[scale=0.1]{particle_model_figures/basic_particle_parameters_5.png} \end{center} \vspace{-5mm} \onslide<2-> ... transferred to the embedded particle model leads to particle velocity:\\ \begin{center} $\vec{V}_{{\text{particle}}} = \vec{V}_{{\text{resolved}}} + \vec{V}_{{\text{subgrid}}}$\\ \end{center} \onslide<3-> \scriptsize Accordingly, the particle movement is a result of: \begin{itemize} \scriptsize \item<3-> resolved flow $\vec{V}_{{\text{resolved}}}$ \vspace{-2mm} \item<3-> and subgrid scale turbulence $\vec{V}_{{\text{subgrid}}}$ \begin{itemize} \scriptsize \item<4-> $\vec{V}_{{\text{subgrid}}}=0$, if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .F.} (default value) \item<4-> $\vec{V}_{{\text{subgrid}}}\ne0$, if \texttt{use\underline{ }sgs\underline{ }for\underline{ }particles = .T. } \end{itemize} \end{itemize} \end{frame} % Folie 12 \begin{frame}[t] \frametitle{Theory of the Lagrangian Particle Model (I)} \frametitle{Theory of the LPM (II) -- Passive Advection (II)} \footnotesize \textbf{Advection of Passive Particles}\\ \vspace{1mm} \begin{itemize} \item The position of the particle is found by integrating $\dfrac{d \vec{X}_{\text{particle}}}{dt} = \vec{V}_{\text{particle}}$ \item The particle's velocity consist of a resolved and a subgrid part:\\ \item The particle's velocity consist of a \textcolor{red}{resolved} and an optional \textcolor{blue}{subgrid part}:\\ \begin{center} $\vec{V}_{\text{particle}} = \vec{V}_{\text{res}} (+ \vec{V}_{\text{sub}})$\\ $\vec{V}_{\text{particle}} = \textcolor{red}{\vec{V}_{\text{res}}} (+ \textcolor{blue}{\vec{V}_{\text{sub}}})$\\ \end{center} %   \vspace{2mm} %   \onslide<2->Transferring the LES concept ... %   \includegraphics[scale=0.1]{particle_model_figures/basic_particle_parameters_5.png}\\ %   \vspace{-2mm} %   \scriptsize \hspace{2em}total energy\hspace{2.25em}=\hspace{3em}resolved part\hspace{2.5em}+\hspace{3em}modelled part\\ %   \vspace{1mm} %   \small ... to the embedded particle model leads to: $\vec{V}_{\text{particle}} = \vec{V}_{\text{res}} (+ \vec{V}_{\text{sub}})$\\ %   \vspace{2mm} \item<2-> The resolved part $\vec{V}_{\text{res}}$ is derived by a tri-linear interpolation: \item<1-> The resolved part $\textcolor{red}{\vec{V}_{\text{res}}}$ is derived by a tri-linear interpolation: \begin{tikzpicture}[remember picture, overlay] \node [shift={(1.7 cm,2.8cm)}]  at (current page.south west) \end{tikzpicture} \vspace{+2.5cm} \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). \vspace{+3.2cm} \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). \end{itemize} \end{frame} % Folie 13 \begin{frame}[t] \frametitle{Theory of the Lagrangian Particle Model (II)} \frametitle{Theory of the LPM (III) -- Non-Passive Advection} \footnotesize \textbf{Advection of Non-passive Particles}\\ \ \\ Newton's second law for a droplet (considering Stoke's drag, gravity and buoyancy):\\ % \footnotesize 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:\\ \vspace{4mm} $\dfrac{dV_i}{dt} = \dfrac{1}{\tau_p} (u_i - V_i) - \delta_{i3}( 1 - \rho_0 / \rho_l ) g$\\ \begin{center} $\dfrac{dV_i}{dt} = \textcolor{red}{\dfrac{1}{\tau_p} (u_i - V_i)} - \delta_{i3}( 1 - \textcolor{orange}{\rho_0 / \rho_l} ) \cdot \textcolor{blue}{g},$\\ \end{center} \vspace{2mm} 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)\\ \vspace{10mm} \onslide<1-> with the inertial response time \begin{align*} \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)}, \end{align*} including a \textcolor{magenta}{correction term} for high Reynolds numbers (see Clift et al., 1978)\\ \vspace{1mm} \begin{tabular}{llll} $g$   & = gravitational acceleration & $\rho_l$ & = density of water\\ \end{tabular} \end{frame} % Folie 14 \begin{frame}[t] \frametitle{Theory of the Lagrangian Particle Model (III)} \frametitle{Theory of the LPM (IV) -- Cloud Droplets (I)} \footnotesize \textbf{Using Particles as Cloud Droplets}\\ \small \begin{itemize} \item This feature is switched on by setting the initial parameter \texttt{cloud\_droplets = .TRUE.} \item In this case, the change in particle radius by condensation/evaporation and collision/coalescence is calculated for every timestep. \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. \item In this case, the change in particle radius by condensation/evaporation and collision/coalescence is calculated for each time step. \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). \end{itemize} \end{frame} % Folie 15 \begin{frame}[t] \frametitle{Theory of the Lagrangian Particle Model (IV)} \frametitle{Theory of the LPM (V) -- Cloud Droplets (II)} \footnotesize \textbf{Simulation of Cloud Droplets (I)}\\ \begin{itemize} \item Simulation of enormous particle numbers like in real clouds\\ is impossible \begin{itemize} \footnotesize \item Ensembles of water droplets are simulated \vspace{1mm} \item Every simulated droplet stands for a very high number\\ of real droplets \vspace{1mm} \item Concept of weighting factor (Shima et al., 2009, QJRMS):\\ \begin{itemize} \item Simulation of realistic particle numbers (as found in clouds) is impossible \item Ensembles of water droplets are simulated instead \item Each simulated particle represents a very high number\\ of real droplets \item Concept of super-droplets (Shima et al., 2009, QJRMS):\\ \begin{center} $A_i = \textit{real number of droplets represented by one simulated droplet}$ \includegraphics[scale=1.0]{particle_model_figures/super.jpg} \includegraphics[scale=1.0]{particle_model_figures/super.jpg} $A_i = \textit{number of droplets represented by one simulated particle}$ \end{center} \vspace{-2mm} \vspace{-1mm} \item Initial weighting factor can be assigned with the parameter \texttt{initial\underline{ }weighting\underline{ }factor} \end{itemize} \end{itemize} \end{frame} % Folie 16 \begin{frame}[t] \frametitle{Theory of the Lagrangian Particle Model (V)} \frametitle{Theory of the LPM (VI) -- Diffusional Growth} \footnotesize \textbf{Simulation of Cloud Droplets (II)}\\ \scriptsize \begin{itemize} % \scriptsize \begin{itemize} \item The growth of the radius of single droplet by condensation/evaporation:\\ \vspace{1mm} $r_i \dfrac{dr_i}{dt} = \dfrac{(S - a\,r^{-1} + b\,r^{-3})}{F_\text{k} + F_\text{d}}$\\ \vspace{1mm} 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 $b$) \vspace{1.5mm} \item<2-> For $r > 1\,\text{\textmu m}$ solution and curvature effects are neglected $\Rightarrow$ an analytic solution is possible: \\ $r_i \dfrac{dr_i}{dt} = \dfrac{S}{F_k + F_d} \Rightarrow r_i(t) = \sqrt{ r_{i,0}^2 + 2 \cdot \Delta t \cdot \left( \dfrac{S}{F_\text{k} + F_\text{d}} \right)}$ \end{itemize} \vspace{1.0cm} \begin{align*} r \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}} \end{align*} 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 ($b$)} \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.) \end{itemize} \hspace{0.5cm} \footnotesize \begin{tabular}{llll} $r$   & = Droplet radius & $S$   & = Supersaturation\\ $a$ & = Curvature effect    & $b$  & = Solution effect\\ $e$ & = Vapor pressure  &  $e_\text{s}$ & = Saturation vapor pressure\\ $F_\text{k}$ & = Effect of heat conduction & $F_\text{d}$ & = Effect of vapor diffusion \end{tabular} % Folie 17 \begin{frame}[t] \frametitle{Theory of the Lagrangian Particle Model (VI)} \frametitle{Theory of the LPM (VII) -- Collisions (I)} \vspace{-2mm} \footnotesize \textbf{Simulation of Cloud Droplets (III)} -- Idealized concept of droplet collisions\\ \begin{tikzpicture}[remember picture, overlay] \node [shift={(6.0cm,4.0cm)}]  at (current page.south west) {% \begin{tikzpicture}[remember picture, overlay] \node at (0.0,0.0) {\includegraphics[scale=0.25]{particle_model_figures/collision1.png}}; \end{tikzpicture} }; \end{tikzpicture} \begin{itemize} \item Two prognostic quantities: $\ \ \$ (i) \textbf{weighting factor} $A$ and (ii) \textbf{total mass} of super-droplet $m$ \item total mass: mass of all droplets represented by one super-droplet \end{itemize} \vspace{-3mm} \begin{center} \includegraphics[scale=0.5]{particle_model_figures/coll.pdf} \end{center} \end{frame} % Folie 18 \begin{frame}[t] \frametitle{Theory of the Lagrangian Particle Model (VII)} \frametitle{Theory of the LPM (VIII) -- Collisions (II)} \footnotesize \textbf{Simulation of Cloud Droplets (IV)}\\ \scriptsize \begin{itemize} \item Calculation of droplet growth due to collisions considers three types of collisions: \vspace{-0.4cm} \begin{enumerate} \scriptsize \item collisions with smaller droplets $\Rightarrow$ increase the radius \item collisions with larger droplets $\Rightarrow$ decrease the weighting factor \item internal collisions $\Rightarrow$ decrease the weighting factor and increase the radius \end{enumerate} \item Two prognostic quantities: Weighting factor $A_n$ and mass of super-droplet expressed as volume averaged droplet radius $r_n=(m_n / (4/3 \pi \rho_\text{l} A_n))^{1/3}$: \begin{itemize} \item Calculation of droplet growth due to collisions considers three types of collisions (for all droplets located in one grid box): % \vspace{-0.4cm} \begin{itemize} \footnotesize \item \textcolor{red}{collisions with smaller droplets $\Rightarrow$ increase  total mass} \item \textcolor{blue}{collisions with larger droplets \ \ $\Rightarrow$ decrease  weighting factor \hphantom{collisions with larger droplets \ \ $\Rightarrow$} and total mass} \item \textcolor{orange}{internal collisions \hphantom{ager droplets} $\Rightarrow$ decrease  weighting factor} \end{itemize} \item Total mass of super-droplet not useful $\Rightarrow$ volume averaged droplet radius $r_n=(m_n / (4/3 \pi \rho_\text{l} A_n))^{1/3}$ \item Droplets are sorted that $r_1 A new, efficient approach for storing particles is implemented in PALM: \\ \only<1>{\begin{center} \vphantom{\colorbox{red}{\textbf{a four-dimensional array}}} \end{center} } \only<2>{ \begin{center} \colorbox{red}{\textbf{a four-dimensional array}} \end{center} } \end{itemize} \end{frame} % Folie 22 \begin{frame}[fragile] \frametitle{Storing Lagrangian particles (I)} \begin{itemize} \item Sorting the particles by their respective grid-box reduces the operations to \texttt{N}: \scriptsize \begin{lstlisting} DO n = n_start(k,j,i), n_end(k,j,i) ... ENDDO \end{lstlisting} \normalsize \item This was done in the previous version of PALM, reducing CPU time of LPM by 9.6\,\% \item However, sorting increases CPU time and demands a second, one-dimensional array for efficient sorting \item<2-> To overcome these issues, a new approach has been developed for the current version of PALM: \\ \begin{center} \textbf{a four-dimensional array} \end{center} \end{itemize} \end{frame} % Folie 23 \begin{frame}[t] \frametitle{Storing Lagrangian particles (III)} \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\\ \ \\ - LPM CPU time decreases by 22\,\%\\ \ \\ - Available memory doubles, since no large additional arrays are needed for assigning the particles to their grid-box\\};} - LPM CPU time decreases by 22\,\% (in comparison to storing particles in a one-dimensional array)\\ % \ \\ % - Available memory doubles, since no large additional arrays are needed for assigning the particles to their grid-box\\ };} \end{tikzpicture} \end{frame} % Folie 24 % Folie 23 \begin{frame}[fragile] \frametitle{Storing Lagrangian particles (IV)} \scriptsize \begin{itemize} \item A new \textbf{3D-array} of another FORTRAN derived data type: \texttt{grid\_particle\_def} \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 \item A \textbf{3D-array} of another FORTRAN derived data type: \texttt{grid\_particle\_def} \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 \end{itemize} \tikzstyle{yellow} = [rectangle, draw, fill=yellow!30, text width=1.05\textwidth, font=\scriptsize,scale=0.95] \begin{tikzpicture} \node[yellow]{\begin{lstlisting} TYPE grid_particle_def TYPE(particle_type), DIMENSION(:) :: particles END TYPE grid_particle_def TYPE(grid_particle_def), DIMENSION(:,:,:) :: grid_particles \end{lstlisting} }; \vspace{-0.2mm} \begin{tikzpicture}\node[yellow]{\begin{lstlisting} TYPE grid_particle_def TYPE(particle_type), DIMENSION(:), ALLOCATABLE :: particles END TYPE grid_particle_def TYPE(grid_particle_def), DIMENSION(:,:,:), ALLOCATABLE :: & grid_particles \end{lstlisting}}; \vspace{-0.2mm} \end{tikzpicture} \vspace{-4mm} IF ( n_par <= 0 ) CYCLE particles(1:n_par) = & grid_particles(kp,jp,ip)%particles(1:n_par) grid_particles(k,j,i)%particles(1:n_par) DO n = 1, n_par particles(n)%radius = 1.0E-6 %\end{lstlisting} \end{itemize} \end{frame} % Folie 24 \begin{frame}[fragile] \frametitle{Storing Lagrangian Particles (V) -- Efficient interpolation} \begin{columns} \begin{column}{6cm} \footnotesize \begin{itemize} \item For interpolating any LES quantity on the location of a particle, the data from 8 grid points is needed \item The indices of these grid points have to be determined for each particle \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 \item Sorting the particles by their sub-grid box makes the determination of the indices for each particle unnecessary \item Sorting increases CPU time by 3\,\%, but efficient interpolation \textbf{speeds up the model by 22\,\%} \end{itemize} \end{column} \begin{column}{6cm} \begin{center} \includegraphics[scale=0.47]{particle_model_figures/interpolation_neu.pdf} \end{center} \end{column} \end{columns} \end{frame} \begin{frame}[t] \frametitle{Application Examples of the LCM (I)} \footnotesize \textbf{The Lagrangian Cloud Model has many advantages:} \begin{itemize} \item Dynamics and microphysics of the cloud are directly related to physical processes of the individual droplets \item Many microphysical processes are modeled by first principles \\ \item Many microphysical processes are modeled by first principles \\$\Rightarrow$(almost) no parameterizations \item The LCM provides detailed information, e.\,g., spatial and temporal evolution of the droplet spectrum, spatial distribution of the droplet concentration, droplet trajectories, ... \item We are able to simulate cloud microphysics on a very accurate level, but we are also able to cope the macro-scale, i.\,e., a whole cloud or cloud ensemble by LES\\ \item The LCM provides detailed information, e.\,g., spatial and temporal evolution of the droplet spectrum, droplet trajectories, ... \end{itemize} \textbf{How to use these advantages?} \begin{itemize} \item Many cloud microphysical processes are still not sufficiently understood, but have a large impact on macroscopic cloud properties \item Open Issues: production of rain, interaction of clouds and aerosols, ... \item Using the LCM, we are able to simulate cloud microphysics on a very accurate level, but are also able to cope the macroscale, i.\,e., the whole cloud or cloud ensemble\\$\Rightarrow$\textbf{new insights on clouds and their physics} \end{itemize} \item Some application examples will show! \end{itemize} \vspace{1.5mm} \end{frame} % Folie 31 \begin{frame}[t] \frametitle{Application Examples of the LCM (II)} \frametitle{Application Examples of the LCM (I)} \footnotesize \textbf{From Riechelmann et al. (2012, NJP):} \begin{tikzpicture}[remember picture, overlay] \node [shift={(6.3 cm, 4.2 cm)}] at (current page.south west) {% \begin{tikzpicture}[remember picture, overlay] \node at (0.0,0.0) {\includegraphics[scale=0.4]{particle_model_figures/turbulence_effects_2.png}}; \node at (-3.0,2.6) {Kernel without turbulence effects}; \node at (3,2.6) {Kernel with turbulence effects}; \end{tikzpicture} }; \end{tikzpicture} \ \\ \vspace{52mm}$\rightarrow$Turbulence effects enhance droplet growth and lead to more realistic mass distribution function\\ \textbf{How to Track Particles:} \begin{center} \includegraphics[scale=0.28]{particle_model_figures/traj.jpg} \end{center}$\rightarrow$Find out what a droplet is experiencing during its life time \end{frame} % Folie 32 \begin{frame}[t] \frametitle{Application Examples of the LCM (III)} \footnotesize \textbf{From Lee et al. (2014, MAP):} \begin{center} \includegraphics[scale=0.25]{particle_model_figures/lee.jpg} \end{center}$\rightarrow\$ Confirm the importance of the cloud top and the affiliated mixing processes for the initiation of rain \end{frame} % Folie 33 \begin{frame}[t] \frametitle{Application Examples of the LCM (IV)} \frametitle{Application Examples of the LCM (II)} \footnotesize \textbf{From Hoffmann et al. (2015, AR):} \end{frame} % Folie 34 % Folie 33 \begin{frame} \frametitle{General Warning}
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