1 | % $Id: cloud_physics.tex 1515 2015-01-02 11:35:51Z raasch $ |
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2 | \input{header_tmp.tex} |
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3 | |
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4 | \usepackage[utf8]{inputenc} |
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5 | \usepackage{ngerman} |
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6 | \usepackage{pgf} |
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7 | \usetheme{Dresden} |
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8 | \usepackage{subfigure} |
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9 | \usepackage{units} |
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10 | \usepackage{multimedia} |
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11 | \usepackage{hyperref} |
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12 | \newcommand{\event}[1]{\newcommand{\eventname}{#1}} |
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13 | \usepackage{xmpmulti} |
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14 | \usepackage{tikz} |
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15 | \usetikzlibrary{shapes,arrows,positioning,decorations.pathreplacing} |
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16 | \def\Tiny{\fontsize{4pt}{4pt}\selectfont} |
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17 | |
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18 | %---------- neue Pakete |
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19 | \usepackage{amsmath} |
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20 | \usepackage{amssymb} |
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21 | \usepackage{multicol} |
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22 | \usepackage{pdfcomment} |
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23 | \usepackage{xcolor} |
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24 | \usepackage{siunitx} |
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25 | \sisetup{% |
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26 | mode = math, detect-family, detect-weight, |
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27 | exponent-product = \cdot, |
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28 | number-unit-separator=\text{\,}, |
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29 | output-decimal-marker={,}, |
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30 | } |
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31 | |
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32 | \institute{Institute of Meteorology and Climatology, Leibniz UniversitÀt Hannover} |
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33 | \selectlanguage{english} |
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34 | \date{last update: \today} |
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35 | \event{PALM Seminar} |
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36 | \setbeamertemplate{navigation symbols}{} |
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37 | \setbeamersize{text margin left=.5cm,text margin right=.2cm} |
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38 | \setbeamertemplate{footline} |
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39 | {% |
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40 | \begin{beamercolorbox}[rightskip=-0.1cm]& |
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41 | {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}} |
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42 | \end{beamercolorbox} |
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43 | \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,% |
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44 | leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}% |
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45 | {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}% |
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46 | \end{beamercolorbox}% |
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47 | % \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}% |
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48 | % \end{beamercolorbox} |
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49 | }%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.eps}} |
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50 | |
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51 | \title[PALM - Cloud Physics]{PALM - Cloud Physics} |
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52 | \author{PALM group} |
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53 | |
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54 | % Notes: |
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55 | % jede subsection bekommt einen punkt im menu (vertikal ausgerichtet. |
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56 | % jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet) |
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57 | \begin{document} |
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58 | |
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59 | % Folie 1 |
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60 | \begin{frame} |
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61 | \titlepage |
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62 | \end{frame} |
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63 | |
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64 | % Folie 2 |
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65 | \begin{frame} |
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66 | \frametitle{Contents} |
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67 | |
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68 | \begin{itemize} |
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69 | \item<1->{Motivation} |
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70 | \item<1->{Approach} |
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71 | \item<1->{Extension if basic equations and SGS-model} |
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72 | \item<1->{Additional Sources / Sinks in prognostic equations} |
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73 | \item<1->{Control parameters} |
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74 | \item<1->{Example of shallow cumulus clouds} |
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75 | \end{itemize} |
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76 | \end{frame} |
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77 | |
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78 | % Folie 3 |
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79 | \begin{frame} |
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80 | \frametitle{Why simulating clouds?} |
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81 | |
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82 | \begin{itemize} |
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83 | \item<2->{Atmospheric boundary layers are usually covered with shallow clouds like cumulus or stratocumulus which are the inherent characteristic of more realistic boundary layers.}\\ \par\medskip |
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84 | \item<3->{Optional feature to account for:}\\ \par\medskip |
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85 | \begin{itemize} |
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86 | \item<4->{Microphysical processes} |
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87 | \begin{itemize} |
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88 | \item<4->{Evaporation / condensation of cloud droplets} |
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89 | \item<4->{Precipitation} |
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90 | \item<4->{Transport of humidity and liquid water}\\ \par\medskip |
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91 | \end{itemize} |
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92 | \item<5->{Radiation processes} |
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93 | \begin{itemize} |
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94 | \item<5->{Short-wave radiation} |
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95 | \item<5->{Long-wave radiation} |
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96 | \end{itemize} |
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97 | \end{itemize} |
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98 | \end{itemize} |
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99 | \end{frame} |
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100 | |
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101 | % Folie 4 |
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102 | \begin{frame} |
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103 | \frametitle{Approach} |
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104 | |
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105 | \begin{itemize} |
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106 | \item<1->{One-moment bulk model $\Rightarrow$ in contrast to PALM's Lagrangian cloud model (LCM) (see also particle\_model\_cloud\_physics.pdf, Riechelmann et al., 2012)} |
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107 | \item<2->{Dynamics like advection and diffusion are covered by Navier-Stokes equations (see basic\_equations.pdf)} |
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108 | \item<3->{Thermodynamics are considered by parameterizations $\Rightarrow$ non explicit treatment of microphysical processes} |
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109 | \item<4->{Total water specific humidity $q$ is prognosed as an additional variable $\Rightarrow$ one-moment} |
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110 | \item<5->{Liquid water specific humidity $q_l$ is determined diagnostically} |
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111 | \end{itemize} |
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112 | \uncover<6->{PALM's basic equations are extended to account for cloud microphysics} |
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113 | \end{frame} |
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114 | |
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115 | % Folie 5 |
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116 | \begin{frame} |
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117 | \frametitle{Definitions (I)} |
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118 | |
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119 | \begin{itemize} |
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120 | \item<1->{Liquid water potential temperature $\theta_{l}$ (defined by Betts, 1973)\\ |
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121 | \begin{minipage}[c][1.5cm][c]{0.38\textwidth} |
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122 | \qquad$\theta_{l}=\theta -\frac{L_{v}}{c_{p}}\left( \frac{\theta}{T} \right) q_{l}$ |
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123 | \end{minipage} |
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124 | \begin{minipage}[c][1.5cm][c]{0.52\textwidth} |
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125 | {\scriptsize $L_{v}$: latent heat of vaporization; $L_{v}=\SI{2,5e6}{J/kg}$\\ |
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126 | $c_{p}$: specific heat of dry air; $c_{p}=\SI{1005}{J/kg K}$} |
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127 | \end{minipage}\\ |
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128 | is the potential temperature of an air parcel if all its liquid water evaporates due to an reversible moist adiabatic descent.} |
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129 | \item<2->{Total water specific humidity $q$\\ |
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130 | \begin{minipage}[c][1.5cm][c]{0.38\textwidth} |
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131 | \qquad$q = q_{v} + q_{l}$ |
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132 | \end{minipage} |
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133 | \begin{minipage}[c][1.5cm][c]{0.52\textwidth} |
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134 | {\scriptsize $q_{v}$: specific humidity\\ |
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135 | $q_{l}$: liquid water speciffic humidity} |
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136 | \end{minipage}\\ |
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137 | } |
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138 | \item<3->{$\theta_{l}$ and $q$ are the prognostic variables when using PALM's cloud physics model} |
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139 | \end{itemize} |
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140 | \end{frame} |
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141 | |
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142 | % Folie 6 |
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143 | \begin{frame} |
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144 | \frametitle{Definitions (II)} |
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145 | |
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146 | |
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147 | \begin{itemize} |
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148 | \item<1->{Why using $\theta_{l}$ and $q$?}\\ \par\medskip |
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149 | \begin{itemize} |
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150 | \item<2->{$\theta_{l}$ and $q$ are conservative quantities in the absence of precipitation, radiation and freezing processes.} |
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151 | \item<3->{Phase transitions do not have to be described explicitly in the prognostic equations.} |
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152 | \item<4->{In case of dry convection (no condensation): $\theta_{l} \rightarrow \theta$ and $q \rightarrow q_{v}$} |
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153 | \item<5->{Parameterizations of SGS-fluxes can be retained.} |
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154 | \item<6->{...$\rightarrow$ see also Deardorff, 1976}\\ \par\medskip |
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155 | \end{itemize} |
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156 | \item<7->{Virtual potential temperature $\theta_{l}$\\ |
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157 | \begin{minipage}[c][1.5cm][c]{0.65\textwidth} |
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158 | \qquad$\theta_{v}=\left[\theta_{l} +\frac{L_{v}}{c_{p}}\left( \frac{\theta}{T} \right) q_{l}\right] \left(1+0,61 q - 1,61q_{l}\right)$ |
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159 | \end{minipage} |
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160 | \begin{minipage}[c][1.5cm][c]{0.22\textwidth} |
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161 | \end{minipage}\\ |
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162 | } |
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163 | \end{itemize} |
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164 | \end{frame} |
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165 | |
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166 | % Folie 7 |
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167 | \begin{frame} |
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168 | \frametitle{Extension of basic equations (I)} |
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169 | |
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170 | \begin{itemize} |
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171 | \item<1->{First principle is solved for $\theta_{l}$ (instead of $\theta$)\\ |
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172 | \begin{minipage}[c][1.5cm][c]{0.46\textwidth} |
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173 | \qquad$\frac{\partial\bar{\theta}_{l}}{\partial t}= - \frac{\partial\bar{u_{k}} \bar{\theta_{l}}}{\partial x_{k}}- \frac{\partial H_{k}}{\partial x_{k}} + Q_{\theta}$ |
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174 | \end{minipage} |
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175 | \begin{minipage}[c][1.5cm][c]{0.46\textwidth} |
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176 | {\scriptsize SGS flux: $H_{k}=\overline{u_{k} \theta_{l}} - \bar{u}_{k}\bar{\theta}_{l}$} |
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177 | \end{minipage}\\ \par\medskip |
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178 | } |
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179 | \item<2->{Conservation equation for total water specific humidity $q$ (instead of $q_{v}$)\\ |
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180 | \begin{minipage}[c][1.5cm][c]{0.46\textwidth} |
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181 | \qquad$\frac{\partial\bar{q}}{\partial t}= - \frac{\partial\bar{u_{k}} \bar{q}}{\partial x_{k}}- \frac{\partial W_{k}}{\partial x_{k}} + Q_{\theta}$ |
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182 | \end{minipage} |
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183 | \begin{minipage}[c][1.5cm][c]{0.46\textwidth} |
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184 | {\scriptsize SGS flux: $W_{k}=\overline{u_{k} q} - \bar{u}_{k}\bar{q}$} |
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185 | \end{minipage}\\ |
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186 | } |
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187 | \end{itemize} |
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188 | \end{frame} |
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189 | |
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190 | % Folie 8 |
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191 | \begin{frame} |
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192 | \frametitle{Extension of basic equations (II)} |
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193 | |
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194 | \begin{itemize} |
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195 | \item<1->{Sources / Sinks due to radiation (RAD) and precipitation (PREC) |
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196 | \begin{minipage}[c][3.0cm][c]{0.46\textwidth} |
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197 | \begin{align*} |
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198 | Q_{\theta} &= \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{RAD}} + \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{PREC}}\\ |
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199 | Q_{W} &= \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}} |
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200 | \end{align*} |
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201 | \end{minipage}\\ \par\medskip |
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202 | } |
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203 | \item<2->{Diagnostic approach for $\bar{q}_{l}$ (all-or-nothing schema) |
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204 | \begin{minipage}[c][1.5cm][c]{0.44\textwidth} |
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205 | \begin{align*} |
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206 | \bar{q}_{l} = |
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207 | \begin{cases} |
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208 | \bar{q}-\bar{q}_{s} & \text{if } \bar{q} > \bar{q}_{s} \\ |
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209 | 0 & \text{if } otherwise |
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210 | \end{cases} |
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211 | \end{align*} |
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212 | \end{minipage}\\ \par\medskip |
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213 | $\bar{q}_{s}$ is the saturation value of the specific humidity which is determined based on Sommeria and Deardorff, 1977 and further described in cloud\_physics.pdf |
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214 | } |
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215 | \end{itemize} |
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216 | \end{frame} |
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217 | |
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218 | % Folie 9 |
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219 | \begin{frame} |
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220 | \frametitle{Extension of SGS model (I)} |
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221 | |
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222 | \begin{itemize} |
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223 | \item<1->{SGS fluxes are modelled by means of a down-gradient approximation |
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224 | \begin{minipage}[c][1.5cm][c]{0.6\textwidth} |
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225 | \begin{equation*} |
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226 | H_{k} = - K_{h} \frac{\partial\bar{\theta}_{l}}{\partial x_{k}} \qquad \text{;} \qquad W_{k} = - K_{h} \frac{\partial\bar{q}}{\partial x_{k}} |
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227 | \end{equation*} |
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228 | \end{minipage}\\ \par\medskip |
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229 | } |
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230 | \item<2->{SGS flux of potential temperature $\overline{u_{3}' \theta'}$ in prognostic equation of the SGS-TKE $\bar{e}$ is replaced by the flux of the virtual potential temperature $\overline{u_{3}' \theta_{v}'}$ which is modelled according to Deardorff, 1980 as: |
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231 | \begin{minipage}[c][1.2cm][c]{0.44\textwidth} |
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232 | \begin{equation*} |
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233 | \overline{u_{3}' \theta_{v}'} = K_{1} \cdot H_{3} + K_{2} \cdot W_{3} |
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234 | \end{equation*} |
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235 | \end{minipage}\\ \par\medskip |
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236 | } |
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237 | \end{itemize} |
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238 | \end{frame} |
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239 | |
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240 | % Folie 10 |
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241 | \begin{frame} |
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242 | \frametitle{Extension of SGS model (II)} |
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243 | |
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244 | \begin{itemize} |
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245 | \item<1->{The coefficients $K_{1}$ and $K_{2}$ depend on the saturation state of the grid volume (see also Cuijpers u. Duynkerke, 1993)\\ \par\medskip |
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246 | \begin{itemize} |
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247 | \item<2->{Unsaturated grid box ($\bar{q}_{l} = 0$)\\ |
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248 | \begin{minipage}[c][1.5cm][c]{0.35\textwidth} |
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249 | \begin{align*} |
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250 | K_{1} &= 1,0 + 0,61\cdot\bar{q}\\ |
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251 | K_{2} &= 0,61\cdot\bar{\theta} |
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252 | \end{align*} |
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253 | \end{minipage}\\ \par\medskip |
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254 | } |
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255 | \item<3->{Saturated grid box ($\bar{q}_{l} \neq 0$)\\ \par\medskip |
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256 | \begin{minipage}[c][2.2cm][c]{0.64\textwidth} |
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257 | \begin{align*} |
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258 | K_{1} &= \frac{1,0 - \bar{q} + 1,61\cdot\bar{q}_{s}\left(1,0 + 0,622\frac{L_{v}}{R T}\right)}{1,0 + 0,622\frac{L_{v}}{R T}\frac{L_{v}}{c_{p} T} \bar{q}_{s}}\\ |
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259 | K_{2} &= \theta \left(\frac{L_{v}}{c_{p} T}\cdot K_{1} -1,0\right) |
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260 | \end{align*} |
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261 | \end{minipage} |
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262 | } |
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263 | \end{itemize} |
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264 | } |
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265 | \end{itemize} |
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266 | \end{frame} |
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267 | |
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268 | % Folie 11 |
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269 | \begin{frame} |
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270 | \frametitle{Sources / Sinks (I)} |
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271 | |
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272 | \begin{itemize} |
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273 | \item<1->{Radiation model (based on Cox, 1976) $\Rightarrow$ scheme of effective emissivity\\ \par\medskip |
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274 | \begin{itemize} |
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275 | \item<2->Very simple, accounts only for absorbtion and emission of long-wave radiation due to water vapour and cloud droplets and neglects horizontal divergences of radiation\\ |
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276 | \begin{minipage}[c][1.5cm][c]{0.35\textwidth} |
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277 | \begin{equation*} |
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278 | \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{RAD}} = \left(\frac{\theta}{T}\right) \frac{1}{\varrho c_{p} \Delta z}\left[ \Delta F(z^{+}) - \Delta F(z^{-}) \right] |
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279 | \end{equation*} |
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280 | \end{minipage}\\ \par\medskip |
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281 | \begin{tabbing} |
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282 | $\Delta F$: \qquad \=Difference between upward and downward irradiance at\\ |
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283 | \>grid points above ($z^{+}$) and below ($z^{-}$) the level in\\ |
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284 | \>which $\bar{\theta}_{l}$ is defined. |
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285 | \end{tabbing} |
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286 | Further information: cloud\_physics.pdf |
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287 | |
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288 | \end{itemize} |
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289 | } |
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290 | \end{itemize} |
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291 | \end{frame} |
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292 | |
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293 | % Folie 12 |
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294 | \begin{frame} |
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295 | \frametitle{Sources / Sinks (II)} |
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296 | |
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297 | \begin{itemize} |
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298 | \item<1->{Precipitation model (based on Kessler, 1969)\\ \par\medskip |
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299 | \begin{itemize} |
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300 | \item<2->{Simplified scheme which accounts only for the process of autoconversion for the formation of rain water.\\ |
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301 | \begin{minipage}[c][1.5cm][c]{0.44\textwidth} |
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302 | \begin{align*} |
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303 | \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}} = |
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304 | \begin{cases} |
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305 | (\bar{q}_{l}-\bar{q}_{l_{\text{crit}}})/\tau & \text{if } \bar{q}_{l} > \bar{q}_{l_{\text{crit}}} \\ |
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306 | 0 & \text{if } \bar{q}_{l} \leq \bar{q}_{l_{\text{crit}}} |
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307 | \end{cases} |
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308 | \end{align*} |
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309 | \end{minipage}\\ \par\medskip |
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310 | } |
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311 | \item<3->{precipitation leaves grid box immediately if the threshold $\bar{q}_{l_{\text{crit}}} = \SI{0,5}{g/kg}$ is exceeded.}\\ \par\medskip |
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312 | \item<4->{Timescale $\tau = \SI{1000}{s}$.} |
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313 | \item<5->{ |
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314 | \begin{minipage}[c][1.5cm][c]{0.35\textwidth} |
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315 | \begin{equation*} |
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316 | \left(\frac{\partial\bar{\theta}_{l}}{\partial t}\right)_{\text{PREC}} = \frac{L_{v}}{c_{p}}\left(\frac{\theta}{T}\right) \left(\frac{\partial\bar{q}}{\partial t}\right)_{\text{PREC}} |
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317 | \end{equation*} |
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318 | \end{minipage} |
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319 | } |
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320 | \end{itemize} |
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321 | } |
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322 | \end{itemize} |
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323 | \end{frame} |
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324 | |
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325 | % Folie 13 |
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326 | \begin{frame} |
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327 | \frametitle{Control parameters} |
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328 | |
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329 | \begin{itemize} |
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330 | \item<1->{The following settings in the parameter file enable the use of the bulk cloud model:}\\ \par\medskip |
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331 | \scriptsize |
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332 | \begin{itemize}\scriptsize |
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333 | \item<2->{ |
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334 | $\left. |
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335 | \begin{array}{ll} % fÌr mehrzeiligen Text nötig |
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336 | \text{humidity = .TRUE.}\qquad\qquad\qquad |
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337 | \end{array} |
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338 | \right\}: |
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339 | \begin{array}{ll} % fÌr mehrzeiligen Text nötig |
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340 | \text{prognostic equations for specific} \\ \text{specific humidity } \bar{q} \text{ is solved} |
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341 | \end{array} |
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342 | $ |
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343 | }\\ \par\medskip |
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344 | \item<3->{ |
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345 | $\left. |
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346 | \begin{array}{ll} % fÌr mehrzeiligen Text nötig |
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347 | \text{humidity = .TRUE.}\qquad\qquad\qquad \\ \text{cloud\_physics = .TRUE.} |
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348 | \end{array} |
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349 | \right\}: |
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350 | \begin{array}{ll} % fÌr mehrzeiligen Text nötig |
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351 | \text{prognostic equations for liquid water} \\ \text{potential temperature } \bar{\theta}_{l} \text{ and total water} \\ \text{specific humidity } \bar{q} \text{ are solved} |
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352 | \end{array} |
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353 | $ |
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354 | }\\ \par\medskip |
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355 | \item<4->{ |
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356 | $\left. |
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357 | \begin{array}{ll} % fÌr mehrzeiligen Text nötig |
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358 | \text{humidity = .TRUE.}\qquad\qquad\qquad \\ \text{cloud\_physics = .TRUE.} \\ \text{precipitation = .TRUE.} \\ \text{radiation = .TRUE.} |
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359 | \end{array} |
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360 | \right\}: |
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361 | \begin{array}{ll} % fÌr mehrzeiligen Text nötig |
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362 | \text{Kessler precipitation scheme and} \\ \text{radiation model are solved} |
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363 | \end{array} |
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364 | $ |
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365 | } |
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366 | \end{itemize} |
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367 | \end{itemize} |
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368 | \end{frame} |
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369 | |
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370 | % Folie 12 |
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371 | \begin{frame} |
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372 | \frametitle{Example - Setup for a cloudy boundary layer} |
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373 | |
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374 | \begin{figure}[H] |
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375 | \begin{minipage}[c][6.5cm][c]{.50\linewidth} |
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376 | \centering |
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377 | CBL with shallow cumulus clouds:\\ \par\bigskip |
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378 | \includegraphics[width=0.95\linewidth]{cloud_physics_figures/cbl5_preview.png} |
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379 | \end{minipage} |
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380 | \begin{minipage}[c][6.5cm][t]{.40\linewidth} |
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381 | \centering |
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382 | \includegraphics[width=0.9\linewidth]{cloud_physics_figures/param_clouds.png} |
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383 | \end{minipage} |
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384 | \end{figure} |
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385 | \end{frame} |
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386 | |
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387 | % Folie 13 |
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388 | \begin{frame} |
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389 | \frametitle{Example - Model output} |
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390 | |
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391 | \begin{figure}[H] |
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392 | \begin{minipage}[c][6cm][c]{.45\linewidth} |
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393 | \centering |
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394 | \includegraphics[width=0.95\linewidth]{cloud_physics_figures/profiles_cbl_cloud.png} |
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395 | \end{minipage} |
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396 | \begin{minipage}[c][6cm][c]{.45\linewidth} |
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397 | \centering |
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398 | \includegraphics[width=0.95\linewidth]{cloud_physics_figures/ql_xy_cbl_cloud.png} |
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399 | \end{minipage} |
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400 | \end{figure} |
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401 | \end{frame} |
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402 | |
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403 | % Folie 1$ |
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404 | \begin{frame} |
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405 | \frametitle{Bibliography} |
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406 | |
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407 | \tiny |
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408 | \begin{thebibliography}{} |
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409 | \bibitem[1]{betts1973} |
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433 | \bibitem[9]{cloudphys} |
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434 | \textsc{cloud\_physics.pdf:} \emph{Introduction to the cloud physics model of PALM.} |
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435 | \newblock {\tt trunk/DOC/tec/methods/cloud\_physics/cloud\_physics.pdf}. |
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436 | \end{thebibliography} |
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437 | \end{frame} |
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438 | |
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439 | \end{document} |
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