source: palm/tags/release-3.10/TUTORIAL/SOURCE/numerics_bc.tex @ 1291

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several updates in the tutorial

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1% $Id: numerics_bc.tex 1226 2013-09-18 13:19:19Z raasch $
2\input{header_tmp.tex}
3%\input{header_lectures.tex}
4
5\usepackage[utf8]{inputenc}
6\usepackage{ngerman}
7\usepackage{pgf}
8\usetheme{Dresden}
9\usepackage{subfigure}
10\usepackage{units}
11\usepackage{multimedia}
12\usepackage{hyperref}
13\newcommand{\event}[1]{\newcommand{\eventname}{#1}}
14\usepackage{xmpmulti}
15\usepackage{tikz}
16\usetikzlibrary{shapes,arrows,positioning}
17\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
18\usepackage{amsmath}
19\usepackage{amssymb}
20\usepackage{multicol}
21\usepackage{pdfcomment}
22
23\institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
24\date{last update: \today}
25\event{PALM Seminar}
26\setbeamertemplate{navigation symbols}{}
27
28\setbeamertemplate{footline}
29  {
30    \begin{beamercolorbox}[rightskip=-0.1cm]&
31     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
32    \end{beamercolorbox}
33    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,
34      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}
35      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}
36    \end{beamercolorbox}
37    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}
38    \end{beamercolorbox}
39  }
40%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.pdf}}
41
42\title[Numerics and Boundary Conditions]{Numerics and Boundary Conditions\\
43(used in PALM)
44}
45\author{Siegfried Raasch}
46
47\begin{document}
48
49% Folie 1
50\begin{frame}
51\titlepage
52\end{frame}
53
54% Folie 2
55\begin{frame}
56   \frametitle{Overview}
57   \scriptsize PALM is (almost) using simple, standard and fast numerical schemes:
58   \begin{itemize}
59      \scriptsize
60      \item<2-> \textbf{Spatial and temporal discretization by finite differences}\\
61      \item<3-> \textbf{Explicit timestep methods:}\\
62         - Euler\\
63         - \underline{Runge-Kutta}, second or \underline{third order}
64      \item<4-> \textbf{Advection method}\\
65         - Upstream\\
66         - Piacsek-Williams (second order central finite differences)\\
67         - Bott-Chlond-scheme (monotone, positiv definit, for scalars only)\\
68         - \underline{5th-order scheme of Wicker and Skamarock}, (as used in WRF model)
69      \item<5-> \textbf{Poisson-equation for pressure}\\
70         - \underline{Direct FFT-method}\\
71         - Multigrid-method
72      \item<6-> \textbf{Lagrangian particle model included}
73      \item<7-> \textbf{Boundary conditions:}\\
74         - \underline{Cyclic} and non-cyclic horizontal boundary conditions\\
75         - Surface layer with Monin-Obukhov similarity\\
76         - Topography\\
77         - Turbulent inflow (for non-cyclic horizontal boundary conditions)
78         
79   \end{itemize}
80\end{frame}
81
82
83\section{Numerics}
84\subsection{Numerics}
85
86% Folie 3
87\begin{frame}
88   \frametitle{Numerical Grid}
89   \footnotesize
90   \vspace{2mm}
91   \includegraphics[width=\textwidth]{numerics_bc_figures/numerical_grid.png}
92   \begin{itemize}
93      \item<1->Equations are spatially discretized on an Arakawa-C grid.
94      \item<2->All scalar variables s (e.g. , $p^*$, $e$, $K_{\mathrm{m}}$,
95               $K_{\mathrm{h}}$) are defined at the cell centers.
96      \item<3->Velocity components ($u$, $v$, $w$) are shifted by half of the grid spacing.
97      \item<4->Spacings are equidistant, stretching along $z$ is possible.
98   \end{itemize}
99
100   \tikzstyle{plain} = [rectangle, draw, fill=white!20, text width=0.33\textwidth, font=\small]
101   \begin{tikzpicture}[remember picture, overlay]
102      \node at (current page.north west){%
103      \begin{tikzpicture}[overlay]
104         \node[plain, draw,anchor=west] at (88mm,-55mm) {
105         \noindent \scriptsize general definition (cylic):\\
106         $\Psi$(0:nz+1,-1:ny+1,-1:nx+1)\\
107         $\Psi$(:,-1,:) $=\Psi$(:,ny,:)\\
108         $\Psi$(:,ny+1,:) $=\Psi$(:,0,:)
109
110         };
111      \end{tikzpicture}
112      };
113   \end{tikzpicture}
114\end{frame}
115
116% Folie 4
117\begin{frame}
118   \frametitle{Timestep Methods (I)}
119   \footnotesize
120   \begin{itemize}
121      \item<1->\textbf{Euler}\\
122      \vspace{3mm}
123      $\dfrac{\partial \psi(t)}{\partial t} = F (\psi(t)) \rightarrow 
124      \dfrac{\psi(t + \Delta t) - \psi(t)}{\Delta t}
125      \approx F (\psi(t))$ \hspace{8mm} 
126      \onslide<2-> $u\dfrac{\Delta t}{\Delta x}=C<1$\\
127      \begin{flushright}
128         for stability
129      \end{flushright}
130
131      \onslide<1->$\psi (t+\Delta t) = \psi(t) 
132      + \Delta t \cdot F(\psi(t)) \hspace{28mm} \mathcal{O}(\Delta t)$\\
133      (used for SGS-TKE in special cases)
134         
135      \vspace{3mm}
136      \item<3-> \textbf{Runge-Kutta, third-order}\\
137      \vspace{2mm}
138      $k_1=F(\psi(t))$\\
139      \vspace{1mm}
140      $k_2=F \left( \psi(t) + \frac{1}{3} \Delta t \cdot k_1 \right)$\\
141      \vspace{1mm}
142      $k_3=F \left( \psi(t) - \frac{3}{16} \Delta t \cdot k_1 
143      + \frac{15}{16} \Delta t \cdot k_2 \right)$\\
144      \vspace{1mm}
145      $\psi(t + \Delta t) = \psi(t) + \frac{1}{30}\Delta t (5 k_1 + 9 k_2 + 16 k_3)$ 
146      \hspace{12mm} $\mathcal{O}(\Delta t^2)$ \hspace{3mm} $C \le 0.9$\\
147   \end{itemize}
148\end{frame}
149
150% Folie 5
151\begin{frame}
152   \frametitle{Timestep Methods (II)}
153   \footnotesize
154   \onslide<1->In the PALM code, the different timestep schemes are treated by one
155   equation using switches:
156   $\psi (t + \Delta t ) = (1 - c_1) \cdot \psi (t - \Delta t ) + c_1 \cdot \psi (t) 
157   + \Delta t \cdot \left[ c_2 \cdot F (\psi (t) ) + c_3 \cdot F (\psi (t - 
158   \Delta t ) ) \right]$
159   \vspace{1mm}
160
161   \onslide<2->
162   \begin{centering}
163      \begin{table}
164         \begin{tabular}{cccc}
165            \bf{Scheme} & \bf{c$_1$} & \bf{c$_2$} & \bf{c$_3$}\\
166            Euler & 1 & 1 & 0\\
167            RK (1st step) & 1 & 1/3 & 0\\
168            RK (2nd step) & 1 & 15/16 & -25/48\\
169            RK (3rd step) & 1 & 8/15 & 1/15\\
170         \end{tabular}
171      \end{table}
172   \end{centering}
173
174   \onslide<3->
175   \begin{align*}
176      \psi (t - \Delta t) &= \psi (t) \hspace{15mm} \textbf{after each RK substep}\\
177      \psi (t) &= \psi (t + \Delta t)
178   \end{align*}
179\end{frame}
180
181% Folie 6
182\begin{frame}
183   \frametitle{Advection Methods (I)}
184   \small
185   \begin{itemize}
186      \item<1-> Piacsek Williams C3 (1970, J. Comput. Phy., 6, 392).
187      \item<2-> Scheme of 2nd order accuracy.
188      \item<3-> Conserves integrals of linear and quadratic quantities.
189      \item<4-> Requires incompressibility $\rightarrow$ flux form of advection term.
190                \onslide<4-> \includegraphics[width=0.8\textwidth]{numerics_bc_figures/advection_methods.png}
191   \end{itemize}
192   $$\left.\frac{\partial (u \psi)}{\partial x}\right\vert_i = \frac{1}{2 \Delta x}
193   \left( u_{i+\frac{1}{2}} \psi_{i+1} - u_{i-\frac{1}{2}} \psi_{i-1} \right)$$
194   \begin{itemize}
195      \item<5-> In case of momentum advection (e.g. $\psi=u$), $u_{i-1}$ and
196                $u_{i+1}$ have to be obtained by linear interpolation.
197      \item<5-> May cause $2 \Delta x$ wiggles in case of sharp gradients.
198   \end{itemize}
199\end{frame}
200
201% Folie 7
202\begin{frame}
203   \frametitle{Advection Methods (II)}
204   \begin{itemize}
205   
206      \item<1-> \small Bott-Chlond\\ \scriptsize
207         \onslide<1-> - Chlond (1994)\\
208         \onslide<2-> - Monotone, positive definit. Can only be used for scalars\\
209         \onslide<3-> - Conserves sharp gradients\\
210         \onslide<4-> - Numerically expensive\\
211         \onslide<5-> - Not optimized for use on cache-based machines.
212      \par\bigskip
213      \item<6-> \small Default: Wicker and Skamarock scheme (5th order)\\ \scriptsize
214         \onslide<6-> - Much better accuracy than Piacsek Williams\\
215         \onslide<7-> - Much simpler algorithm than Bott-Chlond\\
216         \onslide<8-> - Requires additional ghost layers\\
217         \onslide<9-> - Adds additional numerical dissipation
218         
219   \end{itemize}
220\end{frame}
221
222% Folie 8
223\begin{frame}
224   \frametitle{Advection Methods – Wicker/Skamarock (I)}
225   \footnotesize
226   \begin{itemize}
227      \item Wicker and Skamarock (2002, Mon. Wea. Rev. 130, 2088 – 2097).
228      \item Based on flux form of advection term
229      \item Difference of fluxes at the edge of the grid cell is used to
230      discretise advection term
231   \end{itemize}
232
233   \begin{columns}[T]
234      \begin{column}{0.55\textwidth}
235         \hspace{8mm}\includegraphics[width=0.8\textwidth]{numerics_bc_figures/numerical_grid_small.png}
236      \end{column}
237      \begin{column}{0.45\textwidth}
238         $\frac{ \partial \psi}{\partial t} = - \nabla (u_i \psi) \approx 
239         - \frac{F_{i+\frac{1}{2}} - F_{i-\frac{1}{2}}}{\Delta x}$
240      \end{column}
241   \end{columns}
242
243   \tikzstyle{plain} = [rectangle, text width=0.1\textwidth, font=\small]
244   \begin{tikzpicture}[remember picture, overlay]
245      \node at (current page.north west){%
246
247      \begin{tikzpicture}[overlay]
248         \node[plain, anchor=west] at (2mm,-68mm) {
249         \tikz 
250         {
251         \draw[blue, -latex', line width=5pt] (1,0) -- (2,0);
252         }
253
254         $F_{i-\frac{1}{2}}$
255         };
256      \end{tikzpicture}
257
258      \begin{tikzpicture}[overlay]
259         \node[plain, anchor=west] at (62mm,-68mm) {
260         \tikz 
261         {
262         \draw[blue, -latex', line width=5pt] (,0) -- (2,0);
263         }
264         $F_{i+\frac{1}{2}}$
265         };
266      \end{tikzpicture}
267
268      };
269   \end{tikzpicture}
270\end{frame}
271
272% Folie 9
273\begin{frame}
274   \frametitle{Advection Methods – Wicker/Skamarock (II)}
275   
276
277
278   \textbf{Finite difference approximation of 6$^{\text{th}}$ order}
279   \begin{tikzpicture}[scale=2]
280      \tikzstyle{ann} = [draw=none,fill=none,right]
281      \matrix[nodes={draw, thick, fill=blue!20}, row sep=0.3cm,column sep=0.5cm]{
282      \node[rectangle, rounded corners]{
283      $F^{\text{6th}}_{i-\frac{1}{2}} = \frac{1}{60} u_{i-\frac{1}{2}} \left( 37 
284      (\Psi_i + \Psi_{i-1}) - 8 (\Psi_{i+1} + \Psi_{i-2}) + (\Psi_{i+2}
285      + \Psi_{i-3}) \right)$ 
286       };\\
287       };
288   \end{tikzpicture}
289   
290   \vspace{5mm}
291   
292   \textbf{Artificially added numerical dissipation term}
293   \begin{tikzpicture}[scale=2]
294      \tikzstyle{ann} = [draw=none,fill=none,right]
295      \matrix[nodes={draw, thick, fill=blue!40}, row sep=0.3cm,column sep=0.5cm]{
296      \node[rectangle, rounded corners]{
297      $-\frac{1}{60} \left| u_{i-\frac{1}{2}} \right| \left( 10 (\Psi_i - 
298      \Psi_{i-1}) - 5 (\Psi_{i+1} - \Psi_{i-2}) + (\Psi_{i+2} - \Psi_{i-3}) \right)$ 
299       };\\
300       };
301   \end{tikzpicture}
302\end{frame}
303
304% Folie 10
305\begin{frame}
306   \frametitle{Advection Methods – Wicker/Skamarock (III)}
307   
308   \begin{tikzpicture}[scale=2]
309      \tikzstyle{ann} = [draw=none,fill=none,right]
310      \matrix[nodes={draw, thick, fill=blue!20}, row sep=0.3cm,column sep=0.5cm]{
311      \node[rectangle, rounded corners]{
312      $F^{\text{6th}}_{i-\frac{1}{2}} = \frac{1}{60} u_{i-\frac{1}{2}}
313      \left( 37 (\Psi_i + \Psi_{i-1}) - 8 (\Psi_{i+1} + \Psi_{i-2}) + (\Psi_{i+2} + 
314      \Psi_{i-3}) \right)$ 
315       };\\
316       };
317   \end{tikzpicture}
318   
319   \begin{columns}[T]
320      \begin{column}{0.7\textwidth}
321         \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_oscillations.png}
322      \end{column}
323      \begin{column}{0.3\textwidth}
324         Centered Finite Differences produces numerical oscillations (''wiggles'')
325         near sharp gradients.
326      \end{column}
327   \end{columns}
328\end{frame}
329
330% Folie 11
331\begin{frame}
332   \frametitle{Advection Methods – Wicker/Skamarock (IV)}
333   \footnotesize   
334
335   \begin{tikzpicture}[scale=2]
336      \tikzstyle{ann} = [draw=none,fill=none,right]
337      \matrix[nodes={draw, thick, fill=blue!40}, row sep=0.3cm,column sep=0.5cm]{
338      \node[rectangle, rounded corners]{
339      $F^{\text{5th}}_{i-\frac{1}{2}} = F^{\text{6th}}_{i-\frac{1}{2}}
340         - \frac{1}{60} \left| u_{i-\frac{1}{2}} \right| \left( 10 (\Psi_i - \Psi_{i-1}) - 
341            5 (\Psi_{i+1} - \Psi_{i-2}) + (\Psi_{i+2} - \Psi_{i-3}) \right)$ 
342       };\\
343       };
344   \end{tikzpicture}
345   
346   \begin{columns}[T]
347      \begin{column}{0.7\textwidth}
348         \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_oscillations_2.png}
349      \end{column}
350      \begin{column}{0.3\textwidth}
351         \vspace{3mm}
352         \underline{Advantage}\\
353         Numerical Dissipation damps small scale oscillations.\\
354         \vspace{3mm}
355         \underline{Disadvantage}\\
356         In a turbulent flow numerical dissipation removes energy from small scales.
357
358      \end{column}
359   \end{columns}
360\end{frame}
361
362% Folie 12
363\begin{frame}
364   \frametitle{Advection Methods – Wicker/Skamarock (V)}
365   
366    \begin{columns}[T]
367      \begin{column}{0.6\textwidth}
368         \includegraphics[width=1\textwidth]{numerics_bc_figures/numerical_properties.png}
369      \end{column}
370      \begin{column}{0.4\textwidth}
371         \includegraphics[width=1\textwidth]{numerics_bc_figures/pw_ws.png}
372      \end{column}
373   \end{columns}
374   
375   \begin{itemize}
376      \item Better resolution of larger scales $(> 8\,\Delta x)$ and hence less
377      numerical energy transfer from larger to smaller scales compared to lower
378      order schemes.
379      \item Less spectral energy at smaller scales.
380   \end{itemize}
381   
382\end{frame}
383
384% Folie 13
385\begin{frame}
386   \frametitle{Pressure Solver (I)}
387   \footnotesize
388   \begin{itemize}
389      \item<1-> Governing equations of PALM require incompressibility
390      \item<2-> Incompressibility is reached by a predictor-corrector method\\
391            \scriptsize
392            1. Momentum equations are solved without the pressure term giving a
393            provisional velocity field which is not free of divergence.\\
394            \vspace{2mm}
395            $\overline{u}^{t+\Delta t}_{i_{\mathrm{prov}}} = \overline{u}^t_i +
396            \Delta t \left( - \frac{\partial}{\partial x_k} \overline{u}^t_k
397            \overline{u}^t_i - (\varepsilon_{ijk} f_j \overline{u}^t_k
398            - \varepsilon_{i3k} f_3 u_{\mathrm{g}_k}) 
399            + g \frac{\overline{\theta^*}^t}{\theta_0} \delta_{i3}
400            - \frac{\partial}{\partial x_k} \overline{u'_k u'_i}^t \right)$\\
401            \vspace{2mm}
402            \onslide<3-> 2. Assign all remaining divergences to the (perturbation)
403            pressure $p^*$ so that the new corrected velocity field is the sum of the
404            provisional, divergent field and the perturbation pressure term.\\
405            \vspace{2mm}
406            $\overline{u}^{t+\Delta t}_{i} = 
407            \overline{u}^{t+\Delta t}_{i_{\mathrm{prov}}} + 
408            \Delta t \left(-\frac{1}{\rho_0} \frac{\partial \overline{p^*}^t}{\partial x_i} \right)$\\
409            \vspace{2mm}
410            \onslide<4-> 3. The divergence operator is applied to this equation.
411            Demanding a corrected velocity field free of divergence, this leads to a
412            Poisson equation for the perturbation pressure.\\
413            \vspace{2mm}
414            $\frac{\partial^2 \overline{p^*}^t}{\partial x_i^2} = \frac{\rho_0}
415            {\Delta t} \frac{\partial \overline{u}_{i_{\mathrm{prov}}}^{t + \Delta t}}
416            {\partial x_i}$\\
417            \vspace{2mm}
418            \onslide<5-> 4. After solving the Poisson equation, the final velocity field
419            is \\
420            calculated as given in step 2.\\
421
422   \end{itemize}
423\end{frame}
424
425% Folie 14
426\begin{frame}
427   \frametitle{Pressure Solver (II)}
428   \small
429   
430   \begin{itemize}
431      \item FFT-solver\\
432      \onslide<1-> 1. Discretization of the Poisson-equation by central differences\\
433      \onslide<2-> 2. 2D discrete FFT in both horizontal directions\\
434      \onslide<3-> 3. Solving the resulting tridiagonal set of linear equations\\
435      \onslide<4-> 4. Inverse 2D discrete FFT in both horizontal directions leading
436      to the perturbation pressure
437
438      \begin{itemize}
439         \item<5-> Very fast and accurate, $\mathcal{O}(n \log n)$, $n$:
440         number of gridpoints
441         \item<6-> CPU requirement $<$ 50\% of total CPU time
442         \item<7-> Due to non-locality of the FFT, transpositions are required
443         on parallel computers
444         \item<8-> Requires periodic boundary conditions and uniform grids
445         along $x$ and $y$
446      \end{itemize}
447   \end{itemize}
448\end{frame}
449
450% Folie 15
451\begin{frame}
452   \frametitle{Pressure Solver (III)}
453   \scriptsize   
454   \begin{columns}
455      \begin{column}{1.03\textwidth}
456         \begin{itemize}
457            \item<1-> Multigrid-method\\
458            \begin{itemize}\scriptsize 
459               \item Iterative solver\\
460               \scriptsize
461               basic idea: Poisson equation is transformed to a fixed point problem:\\
462               $\vec{p}^{k+1} = T \cdot \vec{p}^k + \vec{c}^k$\\
463               \vspace{1mm}
464               \onslide<2-> starting from a first guess, the solution will be
465               improved by repeated execution of the fixed point problem:\\
466               $\begin{array}{rcl}
467               \vec{p}^{1} &=&T \cdot \vec{p}^0 + \vec{c}^0\\
468               \vec{p}^{2} &=&T \cdot \vec{p}^1 + \vec{c}^1
469               \vspace{-2mm}\\
470               
471               &\vdots&
472               \vspace{-1.5mm}\\
473               
474               \vec{p}^{k} &=&T \cdot \vec{p}^{k-1} + \vec{c}^{k-1}\\
475               \vec{p}^{k+1} &=&T \cdot \vec{p}^k + \vec{c}^k\\
476               \end{array}$\\
477               \vspace{1mm}
478               \onslide<3-> Depending on the structure of the matrix $T$ and vector
479               $c$ different iterative solvers can be defined, e.g.: Jacobi-scheme
480               (here on 2D-uniform grid):\\
481               $p^{k+1}_{i,j} = \frac{1}{4} \cdot \left( p^k_{i-1,j} + p^k_{i+1,j}
482               + p^k_{i,j-1} + p^k_{i,j+1} - \Delta x^2 f(i,j,k) \right)$\\
483               \scriptsize \vspace{2mm}
484               \item<4-> With each iteration step $k$ the improved solution converges towards
485               the exact solution.
486               \item<5-> Iterative schemes are 'local schemes' $\rightarrow$ information
487               is needed \\ only from neighboring grid-points.
488               \item<6-> Very low convergence: $\mathcal{O}(n^2)$.
489           \end{itemize}
490         \end{itemize}
491      \end{column}
492   \end{columns}
493\end{frame}
494   
495% Folie 16
496\begin{frame}
497   \frametitle{Pressure Solver (IV)} 
498   \begin{itemize}
499      \item<1-> Multigrid-method\\
500      \begin{itemize}
501         \item Due to their locality, iterative solvers show a frequency-dependent
502            reduction of the residual: low frequencies are reduced slower than high
503            frequencies.
504         \item<2-> The main idea of the multigrid method is to reduce errors of different
505         frequencies on grids with different grid spacing:
506         \begin{itemize}
507            \item errors of high frequency are reduced on fine grids
508            \item errors of low frequency are reduced on coarse grids.
509         \end{itemize}
510      \end{itemize}
511   \end{itemize}
512   \onslide<2->
513   \begin{figure}[htp]
514      \centering
515      \includegraphics[scale=0.35]{numerics_bc_figures/errors.png}
516   \end{figure}
517\end{frame}
518
519% Folie 17
520\begin{frame}
521   \frametitle{Pressure Solver (V)} 
522   
523   \begin{columns}[T]
524      \begin{column}{0.65\textwidth}
525         \begin{itemize}
526            \item<1-> Multigrid-method\\
527            \begin{itemize}
528               \footnotesize
529               \item On each grid-level an approximate solution of the fixed point
530                  equation is obtained performing a few iterations.
531               \item<2-> The solution is transmitted to the next coarser grid-level
532                  where it is used as the first guess to solve the fixed point problem.
533               \item<3-> This procedure is performed up to the coarsest grid-level
534                  containing two grid-points in each direction.
535               \item<4-> From the coarsest grid-level the procedure is passed in
536                  backward order to get the final solution.
537               \item<5-> For large grids faster than FFT method.
538               \item<6-> V- and W-cycles are implemented.
539            \end{itemize}
540      \end{itemize}
541      \end{column}
542      \begin{column}{0.5\textwidth}
543         \onslide<2->
544         \includegraphics[width=1\textwidth]{numerics_bc_figures/multigrid.png}
545      \end{column}
546   \end{columns}
547\end{frame}
548
549
550
551\section{Boundary Conditions}
552\subsection{Boundary Conditions}
553
554% Folie 18
555\begin{frame}
556   \frametitle{Boundary Conditions (I)}
557   
558   \begin{itemize}
559      \item<1-> Lateral $(xy)$ boundary conditions:\\
560      \begin{itemize}
561         \item Cyclic by default, allowing undisturbed evolution / advection of turbulence.
562            \begin{columns}[T]
563               \begin{column}{0.2\textwidth}
564                  %leer
565               \end{column}
566               \begin{column}{0.5\textwidth}
567                  \includegraphics[width=1\textwidth]{numerics_bc_figures/lateral_bc.png}\\
568                  \vspace{2mm}
569               \end{column}
570               \begin{column}{0.4\textwidth}
571                  $\begin{array}{rcl}
572                     \Psi(-1) &=& \Psi(n)\\
573                     \Psi(n+1) &=& \Psi(0)\\
574                  \end{array}$
575               \end{column}
576            \end{columns}
577   
578         \item<2-> Dirichlet (inflow) and radiation (outflow) conditions are allowed along
579            either $x$- or $y$-direction.
580
581         \item<3-> In case of a Dirichlet condition, the inflow is laminar (by default) and
582            the domain has to be extended to allow for the development of a turbulent
583            state, if neccessary.
584         \item<4->  Non-cyclic lateral conditions require the use of the multigrid-method
585            for solving the Poisson-equation.
586           
587      \end{itemize}
588   \end{itemize}
589\end{frame}
590
591% Folie 19
592\begin{frame}
593   \frametitle{Boundary Conditions (II)}
594   
595   \begin{columns}[T]
596      \begin{column}{0.7\textwidth}
597         \scriptsize
598         \begin{itemize}
599            \item<1-> Surface boundary condition:
600            \begin{itemize}
601               \scriptsize
602               \item<1-> Monin-Obukhov-similarity is used by default, i.e. a Prandtl-layer is
603                  assumed between the surface and the first grid layer.\\
604                  $\frac{\partial \overline{u}}{\partial z} = \frac{u_{*}}{\kappa z}
605                  \Phi_{\mathrm{m}}$; \hspace{3mm} $u_{*} = \sqrt{- \overline{w' u'_0}}
606                  = \sqrt{\frac{\tau_0}{\overline{\rho}}}$\\
607                  $\frac{\partial \overline{\theta}}{\partial z} = \frac{\vartheta_{*}}
608                  {\kappa z} \Phi_{\mathrm{h}}$; \hspace{3mm} $\vartheta_{*} = 
609                  \frac{\overline{w' \theta'_0}}{u_{*}}$\\
610               \vspace{2mm}
611               \item<2-> Integration between $z=z_0$ (roughness height) and $z=z_{\mathrm{p}}$
612                  (top of Prandtl-layer, $k=1$) gives the only unknowns $u_{*}$ and
613                  $\theta_{*}$ which then define the surface momentum and heat flux,
614                  used as the real boundary conditions.\\
615               \vspace{2mm}
616               \item<3-> $\Phi_{\mathrm{m}}$, $\Phi_{\mathrm{h}}$: Dyer-Businger functions\\
617               \onslide<4->$\Phi_{\mathrm{m}} = \left\{ \begin{array}{cc}
618                     1+5\,\mathrm{Rif} & \text{stable}\\
619                     1 & \text{neutral}\\
620                     (1-16\,\mathrm{Rif})^{-1/4} & \text{unstable}\\
621                  \end{array} \right. $
622
623            \end{itemize}
624         \end{itemize}
625      \end{column}
626      \begin{column}{0.3\textwidth}
627         \onslide<1->
628         \includegraphics[width=1\textwidth]{numerics_bc_figures/surface_bc.png}\\
629         \vspace{-2mm}
630         Prandtl-layer\\
631         \vspace{8mm}
632         \onslide<5->
633         $\mathrm{Rif} = \frac{\frac{g}{\tilde{\theta}}
634         \overline{w' \theta'_0}}{\overline{w' u'}
635         \frac{\partial \overline{u}}{\partial z}}$\\
636         \scriptsize Richardson flux number
637      \end{column}
638   \end{columns}
639\end{frame}
640
641% Folie 20
642\begin{frame}
643   \frametitle{Boundary Conditions (III)}
644
645   \begin{itemize}
646      \item<1-> Surface boundary condition:
647      \begin{itemize}
648         \footnotesize
649         \item<1-> Monin-Obukhov-similarity is only valid for a
650            horizontal surface with homogeneous conditions.
651         \item<2-> The surface temperature has to be prescribed.
652            Alternatively, the surface heat flux can be prescribed.
653         \item<3-> Instead of MO-similarity, no-slip conditions or
654            free-slip conditions can be used
655            \begin{equation*}
656               u(z=0) = 0, \quad v(z=0)=0 \hspace{15mm}
657               \frac{\partial u}{\partial z} = 0, \quad 
658               \frac{\partial v}{\partial z} = 0 \qquad
659            \end{equation*}
660            realized by
661            \begin{flalign*}
662               u(k=0)=-u(k=1) \hspace{18mm} u(k=0)=u(k=1)\\
663               v(k=0)=-v(k=1) \hspace{18mm} v(k=0)=v(k=1)
664            \end{flalign*}
665         \item<4-> Pressure boundary condition:
666            $\dfrac{\partial p}{\partial z} = 0$ 
667            in order to guarantee $w(z=0)=0$
668         \item<5-> SGS-TKE condition $\dfrac{\partial e}{\partial z}=0$
669      \end{itemize}
670   \end{itemize}
671\end{frame}
672
673% Folie 21
674\begin{frame}
675   \frametitle{Boundary Conditions (IV)}
676
677   \begin{itemize}
678      \item<1-> Boundary conditions at the top (default)
679      \begin{itemize}
680         \item<1-> Dirichlet conditions for velocities: $u=u_{\mathrm{g}},
681            \quad v=v_{\mathrm{g}}, \quad w=0$
682         \item<2-> Neumann conditions (temporal constant gradients) for scalars:
683            $$\frac{\partial \theta}{\partial z} = 
684            \left. \frac{\partial \theta}{\partial z} \right\vert_{t=0}$$
685         \item<3-> Pressure: Dirichlet $p=0$ 
686            or Neumann $\dfrac{\partial p}{\partial z} = 0$
687         \item<4-> SGS-TKE: Neumann $\dfrac{\partial e}{\partial z} = 0$
688         \item<5-> A damping layer can be switched on in order to absorb
689            gravity waves.
690      \end{itemize}
691   \end{itemize}
692\end{frame}
693
694% Folie 22
695\begin{frame}
696   \frametitle{Initial Conditions}
697
698   All 3D-arrays are initialized with vertical profiles (horizontally homogeneous).\\
699   \quad \\
700   Two different profiles can be chosen:
701   \begin{itemize}
702      \item<2-> \textbf{constant (piecewise linear) profiles}
703         \begin{itemize}
704            \footnotesize
705            \item \textbf{e.g.} $u=0, v=0, \dfrac{\partial \theta}{\partial z}=0$ 
706               \textbf{up to} $z=\unit[1000]{m}$,
707               $\dfrac{\partial \theta}{\partial z}=+1.0$ \textbf{up to top}
708         \end{itemize}
709      \item<3-> \textbf{velocity profiles calculated by a 1D-model (which is a part of PALM)}
710         \begin{itemize}
711            \footnotesize
712            \item  \textbf{constant (piecewise linear) temperature profile is used
713               for the 1D-model}
714         \end{itemize}
715   \end{itemize}
716   \onslide<4->
717   \underline{Under horizontally homogeneous initial conditions, random}\\ 
718   \underline{fluctuations have to be added in order to generate turbulence!}
719\end{frame}
720
721\end{document}
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