% $Id: numerics_bc.tex 915 2012-05-30 15:11:11Z fricke $ \input{header_tmp.tex} %\input{header_lectures.tex} \usepackage[utf8]{inputenc} \usepackage{ngerman} \usepackage{pgf} \usetheme{Dresden} \usepackage{subfigure} \usepackage{units} \usepackage{multimedia} \usepackage{hyperref} \newcommand{\event}[1]{\newcommand{\eventname}{#1}} \usepackage{xmpmulti} \usepackage{tikz} \usetikzlibrary{shapes,arrows,positioning} \def\Tiny{\fontsize{4pt}{4pt}\selectfont} \usepackage{amsmath} \usepackage{amssymb} \usepackage{multicol} \usepackage{pdfcomment} \institute{Institut für Meteorologie und Klimatologie, Leibniz Universität Hannover} \date{last update: \today} \event{PALM Seminar} \setbeamertemplate{navigation symbols}{} \setbeamertemplate{footline} { \begin{beamercolorbox}[rightskip=-0.1cm]& {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}} \end{beamercolorbox} \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex, leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot} {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber} \end{beamercolorbox} \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot} \end{beamercolorbox} } %\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.pdf}} \title[Numerics and Boundary Conditions]{Numerics and Boundary Conditions\\ (used in PALM) } \author{Siegfried Raasch} \begin{document} % Folie 1 \begin{frame} \titlepage \end{frame} % Folie 2 \begin{frame} \frametitle{Overview} \scriptsize PALM is (almost) using simple, standard and fast numerical schemes: \begin{itemize} \scriptsize \item<2-> \textbf{Spatial and temporal discretization by finite differences}\\ \item<3-> \textbf{Explicit timestep methods:}\\ - Euler\\ - Leapfrog\\ - \underline{Runge-Kutta}, second or \underline{third order} \item<4-> \textbf{Advection method}\\ - Upstream\\ - Piacsek-Williams (second order central finite differences)\\ - Upstream-spline\\ - Bott-Chlond-scheme (monotone, positiv definit, for scalars only)\\ - \underline{5th-order scheme of Wicker and Skamarock}, (as used in WRF model) \item<5-> \textbf{Poisson-equation for pressure}\\ - \underline{Direct FFT-method}\\ - Multigrid-method \item<6-> \textbf{Lagrangian particle model included} \item<7-> \textbf{Boundary conditions:}\\ - \underline{Cyclic} and non-cyclic horizontal boundary conditions\\ - Surface layer with Monin-Obukhov similarity\\ - Topography\\ - Turbulent inflow (for non-cyclic horizontal boundary conditions) \end{itemize} \end{frame} \section{Numerics} \subsection{Numerics} % Folie 3 \begin{frame} \frametitle{Numerical Grid} \footnotesize \vspace{2mm} \includegraphics[width=\textwidth]{numerics_bc_figures/numerical_grid.png} \begin{itemize} \item<1->Equations are spatially discretized on an Arakawa-C grid. \item<2->All scalar variables s (e.g. , $p^*$, $e$, $K_{\mathrm{m}}$, $K_{\mathrm{h}}$) are defined at the cell centers. \item<3->Velocity components ($u$, $v$, $w$) are shifted by half of the grid spacing. \item<4->Spacings are equidistant, stretching along $z$ is possible. \end{itemize} \tikzstyle{plain} = [rectangle, draw, fill=white!20, text width=0.33\textwidth, font=\small] \begin{tikzpicture}[remember picture, overlay] \node at (current page.north west){% \begin{tikzpicture}[overlay] \node[plain, draw,anchor=west] at (88mm,-55mm) { \noindent \scriptsize general definition (cylic):\\ $\Psi$(0:nz+1,-1:ny+1,-1:nx+1)\\ $\Psi$(:,-1,:) $=\Psi$(:,ny,:)\\ $\Psi$(:,ny+1,:) $=\Psi$(:,0,:) }; \end{tikzpicture} }; \end{tikzpicture} \end{frame} % Folie 4 \begin{frame} \frametitle{Timestep Methods (I)} \footnotesize \begin{itemize} \item<1->\textbf{Euler}\\ \vspace{1mm} $\dfrac{\partial \psi(t)}{\partial t} = F (\psi(t)) \rightarrow \dfrac{\psi(t + \Delta t) - \psi(t)}{\Delta t} \approx F (\psi(t))$ \hspace{8mm} \onslide<2-> $u\dfrac{\Delta t}{\Delta x}=C<1$\\ \begin{flushright} for stability \end{flushright} \onslide<1->$\psi (t+\Delta t) = \psi(t) + \Delta t \cdot F(\psi(t)) \hspace{28mm} \mathcal{O}(\Delta t)$\\ (used for SGS-TKE in special cases) \vspace{3mm} \item<3-> \textbf{Leapfrog}\\ $\psi(t + \Delta t) = \psi(t - \Delta t)+2 \Delta t \cdot F(\psi(t))$ \hspace{16mm} $ \mathcal{O}(\Delta t^2)$ \hspace{3mm} $C \le 0.1$\\ Time-splitting requires a weak time filter (Asselin filter) \vspace{3mm} \item<4-> \textbf{Runge-Kutta, third-order}\\ $k_1=F(\psi(t))$\\ \vspace{1mm} $k_2=F \left( \psi(t) + \frac{1}{3} \Delta t \cdot k_1 \right)$\\ \vspace{1mm} $k_3=F \left( \psi(t) - \frac{3}{16} \Delta t \cdot k_1 + \frac{15}{16} \Delta t \cdot k_2 \right)$\\ \vspace{1mm} $\psi(t + \Delta t) = \psi(t) + \frac{1}{30}\Delta t (5 k_1 + 9 k_2 + 16 k_3)$ \hspace{12mm} $\mathcal{O}(\Delta t^2)$ \hspace{3mm} $C \le 0.9$\\ \end{itemize} \end{frame} % Folie 5 \begin{frame} \frametitle{Timestep Methods (II)} \footnotesize \onslide<1->In the PALM code, the different timestep schemes are treated by one equation using switches: $\psi (t + \Delta t ) = (1 - c_1) \cdot \psi (t - \Delta t ) + c_1 \cdot \psi (t) + \Delta t \cdot \left[ c_2 \cdot F (\psi (t) ) + c_3 \cdot F (\psi (t - \Delta t ) ) \right]$ \vspace{1mm} \onslide<2-> \begin{centering} \begin{table} \begin{tabular}{cccc} \bf{Scheme} & \bf{c$_1$} & \bf{c$_2$} & \bf{c$_3$}\\ Euler & 1 & 1 & 0\\ Leapfrog & 0 & 2 & 0\\ RK (1st step) & 1 & 1/3 & 0\\ RK (2nd step) & 1 & 15/16 & -25/48\\ RK (3rd step) & 1 & 8/15 & 1/15\\ \end{tabular} \end{table} \end{centering} \onslide<3-> \begin{align*} \psi (t - \Delta t) &= \psi (t) \hspace{15mm} \textbf{after each RK substep}\\ \psi (t) &= \psi (t + \Delta t) \end{align*} \end{frame} % Folie 6 \begin{frame} \frametitle{Advection Methods (I)} \small \begin{itemize} \item<1-> Piacsek Williams C3 (1970, J. Comput. Phy., 6, 392). \item<2-> Scheme of 2nd order accuracy. \item<3-> Conserves integrals of linear and quadratic quantities. \item<4-> Requires incompressibility $\rightarrow$ flux form of advection term. \onslide<4-> \includegraphics[width=0.8\textwidth]{numerics_bc_figures/advection_methods.png} \end{itemize} $$\left.\frac{\partial (u \psi)}{\partial x}\right\vert_i = \frac{1}{2 \Delta x} \left( u_{i+\frac{1}{2}} \psi_{i+1} - u_{i-\frac{1}{2}} \psi_{i-1} \right)$$ \begin{itemize} \item<5-> In case of momentum advection (e.g. $\psi=u$), $u_{i-1}$ and $u_{i+1}$ have to be obtained by linear interpolation. \item<5-> May cause $2 \Delta x$ wiggles in case of sharp gradients. \end{itemize} \end{frame} % Folie 7 \begin{frame} \frametitle{Advection Methods (II)} \begin{itemize} \item<1-> \small Upstream-spline\\ \scriptsize \onslide<1-> - Requires Euler timestep\\ \onslide<2-> - Nonlocal scheme: produces heavy load on communication network\\ \onslide<3-> - Numerically unstable under stable stratification \item<4-> \small Bott-Chlond\\ \scriptsize \onslide<4-> - Chlond (1994)\\ \onslide<5-> - Monotone, positive definit. Can only be used for scalars\\ \onslide<6-> - Conserves sharp gradients\\ \onslide<7-> - Numerically expensive\\ \onslide<8-> - Not optimized for use on cache-based machines. \item<9-> \small Default: Wicker and Skamarock scheme (5th order)\\ \scriptsize \onslide<9-> - Much better accuracy than Piacsek Williams\\ \onslide<10-> - Much simpler algorithm than Bott-Chlond\\ \onslide<11-> - Requires additional ghost layers\\ \onslide<12-> - Adds additional numerical dissipation \end{itemize} \end{frame} % Folie 8 \begin{frame} \frametitle{Advection Methods – Wicker/Skamarock (I)} \footnotesize \begin{itemize} \item Wicker and Skamarock (2002, Mon. Wea. Rev. 130, 2088 – 2097). \item Based on flux form of advection term \item Difference of fluxes at the edge of the grid cell is used to discretise advection term \end{itemize} \begin{columns}[T] \begin{column}{0.55\textwidth} \hspace{8mm}\includegraphics[width=0.8\textwidth]{numerics_bc_figures/numerical_grid_small.png} \end{column} \begin{column}{0.45\textwidth} $\frac{ \partial \psi}{\partial t} = - \nabla (u_i \psi) \approx - \frac{F_{i+\frac{1}{2}} - F_{i-\frac{1}{2}}}{\Delta x}$ \end{column} \end{columns} \tikzstyle{plain} = [rectangle, text width=0.1\textwidth, font=\small] \begin{tikzpicture}[remember picture, overlay] \node at (current page.north west){% \begin{tikzpicture}[overlay] \node[plain, anchor=west] at (2mm,-68mm) { \tikz { \draw[blue, -latex', line width=5pt] (1,0) -- (2,0); } $F_{i-\frac{1}{2}}$ }; \end{tikzpicture} \begin{tikzpicture}[overlay] \node[plain, anchor=west] at (62mm,-68mm) { \tikz { \draw[blue, -latex', line width=5pt] (,0) -- (2,0); } $F_{i+\frac{1}{2}}$ }; \end{tikzpicture} }; \end{tikzpicture} \end{frame} % Folie 9 \begin{frame} \frametitle{Advection Methods – Wicker/Skamarock (II)} \textbf{Finite difference approxiamtion of 6$^{\text{th}}$ order} \begin{tikzpicture}[scale=2] \tikzstyle{ann} = [draw=none,fill=none,right] \matrix[nodes={draw, thick, fill=blue!20}, row sep=0.3cm,column sep=0.5cm]{ \node[rectangle, rounded corners]{ $F^{\text{6th}}_{i-\frac{1}{2}} = \frac{1}{60} u_{i-\frac{1}{2}} \left( 37 (\Psi_i + \Psi_{i-1}) - 8 (\Psi_{i+1} + \Psi_{i-2}) + (\Psi_{i+2} + \Psi_{i-3}) \right)$ };\\ }; \end{tikzpicture} \vspace{5mm} \textbf{Artificially added numerical dissipation term} \begin{tikzpicture}[scale=2] \tikzstyle{ann} = [draw=none,fill=none,right] \matrix[nodes={draw, thick, fill=blue!40}, row sep=0.3cm,column sep=0.5cm]{ \node[rectangle, rounded corners]{ $\frac{-1}{60} \left| u_{i-\frac{1}{2}} \right| \left( 10 (\Psi_i - \Psi_{i-1}) - 5 (\Psi_{i+1} - \Psi_{i-2}) + (\Psi_{i+2} - \Psi_{i-3}) \right)$ };\\ }; \end{tikzpicture} \end{frame} % Folie 10 \begin{frame} \frametitle{Advection Methods – Wicker/Skamarock (III)} \begin{tikzpicture}[scale=2] \tikzstyle{ann} = [draw=none,fill=none,right] \matrix[nodes={draw, thick, fill=blue!20}, row sep=0.3cm,column sep=0.5cm]{ \node[rectangle, rounded corners]{ $F^{\text{6th}}_{i-\frac{1}{2}} = \frac{1}{60} u_{i-\frac{1}{2}} \left( 37 (\Psi_i + \Psi_{i-1}) - 8 (\Psi_{i+1} + \Psi_{i-2}) + (\Psi_{i+2} + \Psi_{i-3}) \right)$ };\\ }; \end{tikzpicture} \begin{columns}[T] \begin{column}{0.7\textwidth} \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_oscillations.png} \end{column} \begin{column}{0.3\textwidth} Centered Finite Differences produces numerical oscillations (''wiggles'') near sharp gradients. \end{column} \end{columns} \end{frame} % Folie 11 \begin{frame} \frametitle{Advection Methods – Wicker/Skamarock (IV)} \footnotesize \begin{columns}[T] \begin{column}{0.1\textwidth} \begin{tikzpicture}[scale=2] \tikzstyle{ann} = [draw=none,fill=none,right] \matrix[nodes={draw, thick, fill=blue!40}, row sep=0.3cm,column sep=0.5cm]{ \node[draw=none,fill=none]{ $F^{\text{5th}}_{i-\frac{1}{2}} = F^{\text{6th}}_{i-\frac{1}{2}}$ };\\ }; \end{tikzpicture} \end{column} \begin{column}{0.8\textwidth} \begin{tikzpicture}[scale=2] \tikzstyle{ann} = [draw=none,fill=none,right] \matrix[nodes={draw, thick, fill=blue!40}, row sep=0.3cm,column sep=0.5cm]{ \node[rectangle, rounded corners]{ $- \frac{1}{60} \left| u_{i-\frac{1}{2}} \right| \left( 10 (\Psi_i - \Psi_{i-1}) - 5 (\Psi_{i+1} - \Psi_{i-2}) + (\Psi_{i+2} - \Psi_{i-3}) \right)$ };\\ }; \end{tikzpicture} \end{column} \end{columns} \begin{columns}[T] \begin{column}{0.7\textwidth} \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_oscillations_2.png} \end{column} \begin{column}{0.3\textwidth} \vspace{3mm} \underline{Advantage}\\ Numerical Dissipation damps small scale oscillations.\\ \vspace{3mm} \underline{Disadvantage}\\ In a turbulent flow numerical dissipation removes energy from small scales. \end{column} \end{columns} \end{frame} % Folie 12 \begin{frame} \frametitle{Advection Methods – Wicker/Skamarock (V)} \begin{columns}[T] \begin{column}{0.6\textwidth} \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_properties.png} \end{column} \begin{column}{0.4\textwidth} \includegraphics[width=1\textwidth]{numerics_bc_figures/pw_ws.png} \end{column} \end{columns} \begin{itemize} \item Better resolution of larger scales $(> 8\,\Delta x)$ and hence less numerical energy transfer from larger to smaller scales compared to lower order schemes. \item Less spectral energy at smaller scales. \end{itemize} \end{frame} % Folie 13 \begin{frame} \frametitle{Pressure Solver (I)} \footnotesize \begin{itemize} \item<1-> Governing equations of PALM require incompressibility \item<2-> Incompressibility is reached by a predictor-corrector method\\ \scriptsize 1. Momentum equations are solved without the pressure term giving a provisional velocity field which is not free of divergence.\\ $\overline{u}^{t+\Delta t}_{i_{\mathrm{prov}}} = \overline{u}^t_i + \Delta t \left( - \frac{\partial}{\partial x_k} \overline{u}^t_k \overline{u}^t_i - (\varepsilon_{ijk} f_j \overline{u}^t_k - \varepsilon_{i3k} f_3 u_{\mathrm{g}_k}) + g \frac{\overline{\theta^*}^t}{\theta_0} \delta_{i3} - \frac{\partial}{\partial x_k} \overline{u'_k u'_i}^t \right)$\\ \vspace{3mm} \onslide<3-> 2. Assign all remaining divergences to the (perturbation) pressure $p^*$ so that the new corrected velocity field is the sum of the provisional, divergent field and the perturbation pressure term.\\ $\overline{u}^{t+\Delta t}_{i} = \overline{u}^{t+\Delta t}_{i_{\mathrm{prov}}} - \frac{\Delta t}{\rho_0} \frac{\partial \overline{p^*}^t}{\partial x_i}$\\ \vspace{3mm} \onslide<4-> 3. The divergence operator is applied to this equation. Demanding a corrected velocity field free of divergence, this leads to a Poisson equation for the perturbation pressure.\\ $\frac{\partial^2 \overline{p^*}^t}{\partial x_i^2} = \frac{\rho_0} {\Delta t} \frac{\partial \overline{u}_{i_{\mathrm{prov}}}^{t + \Delta t}} {\partial x_i}$\\ \vspace{3mm} \onslide<5-> 4. After solving the Poisson equation, the final velocity field is \\ calculated as given in step 2.\\ \end{itemize} \end{frame} % Folie 14 \begin{frame} \frametitle{Pressure Solver (II)} \small \begin{itemize} \item FFT-solver\\ \onslide<1-> 1. Discretization of the Poisson-equation by central differences\\ \onslide<2-> 2. 2D discrete FFT in both horizontal directions\\ \onslide<3-> 3. Solving the resulting tridiagonal set of linear equations\\ \onslide<4-> 4. Inverse 2D discrete FFT in both horizontal directions leading to the perturbation pressure \begin{itemize} \item<5-> Very fast and accurate, $\mathcal{O}(n \log n)$, $n$: number of gridpoints \item<6-> CPU requirement $<$ 50\% of total CPU time \item<7-> Due to non-locality of the FFT, transpositions are required on parallel computers \item<8-> Usage requires periodic boundary conditions and uniform grids along $x$ and $y$ \end{itemize} \end{itemize} \end{frame} % Folie 15 \begin{frame} \frametitle{Pressure Solver (III)} \scriptsize \begin{columns} \begin{column}{1.03\textwidth} \begin{itemize} \item<1-> Multigrid-method\\ \begin{itemize}\scriptsize \item Iterative solver\\ \scriptsize basic idea: Poisson equation is transformed to a fixed point problem:\\ $\vec{p}^{k+1} = T \cdot \vec{p}^k + \vec{c}^k$\\ \vspace{1mm} \onslide<2-> starting from a first guess, the solution will be improved by repeated execution of the fixed point problem:\\ $\begin{array}{rcl} \vec{p}^{1} &=&T \cdot \vec{p}^0 + \vec{c}^0\\ \vec{p}^{2} &=&T \cdot \vec{p}^1 + \vec{c}^1 \vspace{-2mm}\\ &\vdots& \vspace{-1.5mm}\\ \vec{p}^{k} &=&T \cdot \vec{p}^{k-1} + \vec{c}^{k-1}\\ \vec{p}^{k+1} &=&T \cdot \vec{p}^k + \vec{c}^k\\ \end{array}$\\ \vspace{1mm} \onslide<3-> Depending on the structure of the matrix $T$ and vector $c$ different iterative solvers can be defined, e.g.: Jacobi-scheme (here on 2D-uniform grid):\\ $p^{k+1}_{i,j} = \frac{1}{4} \cdot \left( p^k_{i-1,j} + p^k_{i+1,j} + p^k_{i,j-1} + p^k_{i,j+1} - \Delta x^2 f(i,j,k) \right)$\\ \scriptsize \vspace{2mm} \item<4-> With each iteration step $k$ the improved solution converges towards the exact solution. \item<5-> Iterative schemes are 'local schemes' $\rightarrow$ information is needed \\ only from neighboring grid-points. \item<6-> Very low convergence: $\mathcal{O}(n^2)$. \end{itemize} \end{itemize} \end{column} \end{columns} \end{frame} % Folie 16 \begin{frame} \frametitle{Pressure Solver (IV)} \begin{itemize} \item<1-> Multigrid-method\\ \begin{itemize} \item Due to their locality, iterative solvers show a frequency-dependent reduction of the residual: low frequencies are reduced slower than high frequencies. \item<2-> The main idea of the multigrid method is to reduce errors of different frequencies on grids with different grid spacing: \begin{itemize} \item errors of high frequency are reduced on fine grids \item errors of low frequency are reduced on coarse grids. \end{itemize} \end{itemize} \end{itemize} \onslide<2-> \begin{figure}[htp] \centering \includegraphics[scale=0.35]{numerics_bc_figures/errors.png} \end{figure} \end{frame} % Folie 17 \begin{frame} \frametitle{Pressure Solver (V)} \begin{columns}[T] \begin{column}{0.65\textwidth} \begin{itemize} \item<1-> Multigrid-method\\ \begin{itemize} \footnotesize \item On each grid-level an approximate solution of the fixed point equation is obtained performing a few iterations. \item<2-> The solution is transmitted to the next coarser grid-level where it is used as the first guess to solve the fixed point problem. \item<3-> This procedure is performed up to the coarsest grid-level containing two grid-points in each direction. \item<4-> From the coarsest grid-level the procedure is passed in backward order to get the final solution. \item<5-> For large grids faster than FFT method. \item<6-> V- and W-cycles are implemented. \end{itemize} \end{itemize} \end{column} \begin{column}{0.5\textwidth} \onslide<2-> \includegraphics[width=1\textwidth]{numerics_bc_figures/multigrid.png} \end{column} \end{columns} \end{frame} \section{Boundary Conditions} \subsection{Boundary Conditions} % Folie 18 \begin{frame} \frametitle{Boundary Conditions (I)} \begin{itemize} \item<1-> Lateral $(xy)$ boundary conditions:\\ \begin{itemize} \item Cyclic by default, allowing undisturbed evolution / advection of turbulence. \begin{columns}[T] \begin{column}{0.2\textwidth} %leer \end{column} \begin{column}{0.5\textwidth} \includegraphics[width=1\textwidth]{numerics_bc_figures/lateral_bc.png}\\ \vspace{2mm} \end{column} \begin{column}{0.4\textwidth} $\begin{array}{rcl} \Psi(-1) &=& \Psi(n)\\ \Psi(n+1) &=& \Psi(0)\\ \end{array}$ \end{column} \end{columns} \item<2-> Dirichlet (inflow) and radiation (outflow) conditions are allowed along either $x$- or $y$-direction. \item<3-> In case of a Dirichlet condition, the inflow is laminar (by default) and the domain has to be extended to allow for the development of a turbulent state, if neccessary. \item<4-> Non-cyclic lateral conditions require the use of the multigrid-method for solving the Poisson-equation. \end{itemize} \end{itemize} \end{frame} % Folie 19 \begin{frame} \frametitle{Boundary Conditions (II)} \begin{columns}[T] \begin{column}{0.7\textwidth} \scriptsize \begin{itemize} \item<1-> Surface boundary condition: \begin{itemize} \scriptsize \item<1-> Monin-Obukhov-similarity is used by default, i.e. a Prandtl-layer is assumed between the surface and the first grid layer.\\ $\frac{\partial \overline{u}}{\partial z} = \frac{u_{*}}{\kappa z} \Phi_{\mathrm{m}}$; \hspace{3mm} $u_{*} = \sqrt{- \overline{w' u'_0}} = \sqrt{\frac{\tau_0}{\overline{\rho}}}$\\ $\frac{\partial \overline{\theta}}{\partial z} = \frac{\vartheta_{*}} {\kappa z} \Phi_{\mathrm{h}}$; \hspace{3mm} $\vartheta_{*} = \frac{\overline{w' \theta'_0}}{u_{*}}$\\ \vspace{2mm} \item<2-> Integration between $z=z_0$ (roughness height) and $z=z_{\mathrm{p}}$ (top of Prandtl-layer, $k=1$) gives the only unknowns $u_{*}$ and $\theta_{*}$ which then define the surface momentum and heat flux, used as the real boundary conditions.\\ \vspace{2mm} \item<3-> $\Phi_{\mathrm{m}}$, $\Phi_{\mathrm{h}}$: Dyer-Businger functions\\ \onslide<4->$\Phi_{\mathrm{m}} = \left\{ \begin{array}{cc} 1+5\,\mathrm{Rif} & \text{stable}\\ 1 & \text{neutral}\\ (1-16\,\mathrm{Rif})^{-1/4} & \text{unstable}\\ \end{array} \right. $ \end{itemize} \end{itemize} \end{column} \begin{column}{0.3\textwidth} \onslide<1-> \includegraphics[width=1\textwidth]{numerics_bc_figures/surface_bc.png}\\ \vspace{-2mm} Prandtl-layer\\ \vspace{8mm} \onslide<5-> $\mathrm{Rif} = \frac{\frac{g}{\tilde{\theta}} \overline{w' \theta'_0}}{\overline{w' u'} \frac{\partial \overline{u}}{\partial z}}$\\ \scriptsize Richardson flux number \end{column} \end{columns} \end{frame} % Folie 20 \begin{frame} \frametitle{Boundary Conditions (III)} \begin{itemize} \item<1-> Surface boundary condition: \begin{itemize} \footnotesize \item<1-> Monin-Obukhov-similarity is only valid for a horizontal surface with homogeneous conditions. \item<2-> The surface temperature has to be prescribed. Alternatively, the surface heat flux can be prescribed. \item<3-> Instead of MO-similarity, no-slip conditions or free-slip conditions can be used \begin{equation*} u(z=0) = 0, \quad v(z=0)=0 \hspace{15mm} \frac{\partial u}{\partial z} = 0, \quad \frac{\partial v}{\partial z} = 0 \qquad \end{equation*} realized by \begin{flalign*} u(k=0)=-u(k=1) \hspace{18mm} u(k=0)=u(k=1)\\ v(k=0)=-v(k=1) \hspace{18mm} v(k=0)=v(k=1) \end{flalign*} \item<4-> Pressure boundary condition: $\dfrac{\partial p}{\partial z} = 0$ in order to guarantee $w(z=0)=0$ \item<5-> SGS-TKE condition $\dfrac{\partial e}{\partial z}=0$ \end{itemize} \end{itemize} \end{frame} % Folie 21 \begin{frame} \frametitle{Boundary Conditions (IV)} \begin{itemize} \item<1-> Boundary conditions at the top (default) \begin{itemize} \item<1-> Dirichlet conditions for velocities: $u=u_{\mathrm{g}}, \quad v=v_{\mathrm{g}}, \quad w=0$ \item<2-> Neumann conditions (temporal constant gradients) for scalars: $$\frac{\partial \theta}{\partial z} = \left. \frac{\partial \theta}{\partial z} \right\vert_{t=0}$$ \item<3-> Pressure: Dirichlet $p=0$ or Neumann $\dfrac{\partial p}{\partial z} = 0$ \item<4-> SGS-TKE: Neumann $\dfrac{\partial e}{\partial z} = 0$ \item<5-> A damping layer can be switched on in order to absorb gravity waves. \end{itemize} \end{itemize} \end{frame} % Folie 22 \begin{frame} \frametitle{Initial Conditions} All 3D-arrays are initialized with vertical profiles (horizontally homogeneous).\\ \quad \\ Two different profiles can be chosen: \begin{itemize} \item<2-> \textbf{constant (piecewise linear) profiles} \begin{itemize} \footnotesize \item \textbf{e.g.} $u=0, v=0, \dfrac{\partial \theta}{\partial z}=0$ \textbf{up to} $z=\unit[1000]{m}$, $\dfrac{\partial \theta}{\partial z}=+1.0$ \textbf{up to top} \end{itemize} \item<3-> \textbf{velocity profiles calculated by a 1D-model (which is a part of PALM)} \begin{itemize} \footnotesize \item \textbf{constant (piecewise linear) temperature profile is used for the 1D-model} \end{itemize} \end{itemize} \onslide<4-> \underline{Under horizontally homogeneous initial conditions, random}\\ \underline{fluctuations have to be added in order to generate turbulence!} \end{frame} \end{document}