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palm/trunk/TUTORIAL/SOURCE/particle_model_cloud_physics.tex
r1532 r1634 443 443 % Folie 8 444 444 \begin{frame}[t] 445 \frametitle{Basic Particle Parameters (V)}446 \small447 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 \scriptsize462 Accordingly, the particle movement is a result of:463 \begin{itemize}464 \scriptsize465 \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 \scriptsize470 \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 9481 \begin{frame}[t]482 445 \frametitle{Basic Particle Parameters (VI)} 483 446 \small … … 568 531 \end{frame} 569 532 570 % Folie 10533 % Folie 9 571 534 \begin{frame} 572 535 \frametitle{Basic Particle Parameters (VIII)} … … 596 559 \end{frame} 597 560 598 % Folie 1 1561 % Folie 10 599 562 \begin{frame}[fragile] 600 563 \frametitle{An Example of a Particle NAMELIST} … … 603 566 \begin{tikzpicture} 604 567 \node [yellow] {\begin{lstlisting} 605 &particles_par dt_dopts = 25.0, 606 bc_par_b = 'absorb', 568 &particles_par bc_par_b = 'absorb', 607 569 density_ratio = 0.001, 608 570 radius = 1.0E-6, … … 624 586 \subsection{Theory} 625 587 588 % 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} 621 622 626 623 % Folie 12 627 624 \begin{frame}[t] 628 \frametitle{Theory of the L agrangian Particle Model (I)}625 \frametitle{Theory of the LPM (II) -- Passive Advection (II)} 629 626 \footnotesize 630 \textbf{Advection of Passive Particles}\\631 627 \vspace{1mm} 632 628 \begin{itemize} 633 629 \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}:\\ 635 631 \begin{center} 636 $\vec{V}_{\text{particle}} = \ vec{V}_{\text{res}} (+ \vec{V}_{\text{sub}})$\\632 $\vec{V}_{\text{particle}} = \textcolor{red}{\vec{V}_{\text{res}}} (+ \textcolor{blue}{\vec{V}_{\text{sub}}})$\\ 637 633 \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: 647 635 \begin{tikzpicture}[remember picture, overlay] 648 636 \node [shift={(1.7 cm,2.8cm)}] at (current page.south west) … … 718 706 \end{tikzpicture} 719 707 720 \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). 722 710 \end{itemize} 723 711 \end{frame} … … 725 713 % Folie 13 726 714 \begin{frame}[t] 727 \frametitle{Theory of the L agrangian Particle Model (II)}715 \frametitle{Theory of the LPM (III) -- Non-Passive Advection} 728 716 \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:\\ 732 719 \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} 734 723 \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} 738 730 \begin{tabular}{llll} 739 731 $g$ & = gravitational acceleration & $\rho_l$ & = density of water\\ … … 743 735 \end{tabular} 744 736 737 745 738 \end{frame} 746 739 747 740 % Folie 14 748 741 \begin{frame}[t] 749 \frametitle{Theory of the L agrangian Particle Model (III)}742 \frametitle{Theory of the LPM (IV) -- Cloud Droplets (I)} 750 743 \footnotesize 751 \textbf{Using Particles as Cloud Droplets}\\752 \small753 744 \begin{itemize} 754 745 \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 e very 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). 757 748 \end{itemize} 758 749 \end{frame} … … 760 751 % Folie 15 761 752 \begin{frame}[t] 762 \frametitle{Theory of the L agrangian Particle Model (IV)}753 \frametitle{Theory of the LPM (V) -- Cloud Droplets (II)} 763 754 \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):\\ 774 760 \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 $A_i = \textit{number of droplets represented by one simulated particle}$ 764 777 765 \end{center} 778 \vspace{- 2mm}766 \vspace{-1mm} 779 767 \item Initial weighting factor can be assigned with the parameter \texttt{initial\underline{ }weighting\underline{ }factor} 780 \end{itemize}781 768 \end{itemize} 782 769 783 784 770 \end{frame} 785 771 786 772 % Folie 16 787 773 \begin{frame}[t] 788 \frametitle{Theory of the L agrangian Particle Model (V)}774 \frametitle{Theory of the LPM (VI) -- Diffusional Growth} 789 775 \footnotesize 790 \textbf{Simulation of Cloud Droplets (II)}\\ 791 \scriptsize 792 \begin{itemize} 776 % \scriptsize 777 \begin{itemize} 793 778 \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*} 780 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}} 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} 803 787 \hspace{0.5cm} 788 \footnotesize 804 789 \begin{tabular}{llll} 805 790 $r$ & = Droplet radius & $S$ & = Supersaturation\\ 806 791 $a$ & = Curvature effect & $b$ & = Solution effect\\ 807 $e$ & = Vapor pressure & $e_\text{s}$ & = Saturation vapor pressure\\808 792 $F_\text{k}$ & = Effect of heat conduction & $F_\text{d}$ & = Effect of vapor diffusion 809 793 \end{tabular} … … 813 797 % Folie 17 814 798 \begin{frame}[t] 815 \frametitle{Theory of the Lagrangian Particle Model (VI)} 799 \frametitle{Theory of the LPM (VII) -- Collisions (I)} 800 \vspace{-2mm} 816 801 \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} 826 812 \end{frame} 827 813 828 814 % Folie 18 829 815 \begin{frame}[t] 830 \frametitle{Theory of the L agrangian Particle Model (VII)}816 \frametitle{Theory of the LPM (VIII) -- Collisions (II)} 831 817 \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 $r_1<r_2<...< r_{N_\text{p}-1} <r_{N_\text{p}}$: 844 832 \scriptsize 845 833 \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}834 A_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} 849 837 \end{alignat*} 850 838 \end{itemize} … … 853 841 % Folie 19 854 842 \begin{frame}[t] 855 \frametitle{Theory of the L agrangian Particle Model (VIII)}843 \frametitle{Theory of the LPM (IX) -- Collisions III} 856 844 \footnotesize 857 \textbf{Simulation of Cloud Droplets (V)}\\858 845 \begin{itemize} 859 846 \item \textbf{Collision kernel without turbulence effects}:\\ … … 865 852 Wang and Grabowski, 2009, ASL):\\ 866 853 \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}} $\\ 868 855 \vspace{+1mm} 869 856 all \textcolor{red}{red} variables parameterize effects of turbulence … … 941 928 \begin{itemize} 942 929 \item Handling hundreds of millions of particles, efficient storing is essential for a good performance 943 \item The easiest method for storing Lagrangian particles is an one-dimensional array 944 \item Most applications demand particles located at a certain location (e.\,g., collision process is computed for all particles located in a certain grid box) 945 \item Finding this particles demands \texttt{N}$^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} 945 961 946 962 947 % 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 \scriptsize968 \begin{lstlisting}969 DO n = n_start(k,j,i), n_end(k,j,i)970 ...971 ENDDO972 \end{lstlisting}973 \normalsize974 \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 sorting976 \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 23985 948 \begin{frame}[t] 986 949 \frametitle{Storing Lagrangian particles (III)} … … 1066 1029 \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\\ 1067 1030 \ \\ 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 };} 1071 1035 \end{tikzpicture} 1072 1036 1073 1037 \end{frame} 1074 1038 1075 % Folie 2 41039 % Folie 23 1076 1040 \begin{frame}[fragile] 1077 1041 \frametitle{Storing Lagrangian particles (IV)} 1078 1042 \scriptsize 1079 1043 \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 stored1044 \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 1082 1046 \end{itemize} 1083 1047 \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} 1050 TYPE grid_particle_def 1051 TYPE(particle_type), DIMENSION(:), ALLOCATABLE :: particles 1052 END TYPE grid_particle_def 1053 1054 TYPE(grid_particle_def), DIMENSION(:,:,:), ALLOCATABLE :: & 1055 grid_particles 1056 \end{lstlisting}}; 1057 \vspace{-0.2mm} 1093 1058 \end{tikzpicture} 1094 1059 \vspace{-4mm} … … 1104 1069 IF ( n_par <= 0 ) CYCLE 1105 1070 particles(1:n_par) = & 1106 grid_particles(k p,jp,ip)%particles(1:n_par)1071 grid_particles(k,j,i)%particles(1:n_par) 1107 1072 DO n = 1, n_par 1108 1073 particles(n)%radius = 1.0E-6 … … 1115 1080 %\end{lstlisting} 1116 1081 \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} 1117 1105 1118 1106 \end{frame} … … 1372 1360 \begin{frame}[t] 1373 1361 \frametitle{Application Examples of the LCM (I)} 1374 \footnotesize1375 1362 \textbf{The Lagrangian Cloud Model has many advantages:} 1376 1363 \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 \\ 1379 1365 $\Rightarrow$ (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, ... 1381 1368 \end{itemize} 1382 1369 \textbf{How to use these advantages?} 1383 1370 \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} 1389 1373 \vspace{1.5mm} 1390 1374 \end{frame} … … 1392 1376 % Folie 31 1393 1377 \begin{frame}[t] 1394 \frametitle{Application Examples of the LCM (I I)}1378 \frametitle{Application Examples of the LCM (I)} 1395 1379 \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 1411 1385 \end{frame} 1412 1386 1413 1387 % Folie 32 1414 1388 \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)} 1427 1390 \footnotesize 1428 1391 \textbf{From Hoffmann et al. (2015, AR):} … … 1433 1396 \end{frame} 1434 1397 1435 % Folie 3 41398 % Folie 33 1436 1399 \begin{frame} 1437 1400 \frametitle{General Warning}
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