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Wettersysteme-ZF.tex
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\documentclass[8pt,landscape]{extarticle}
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% To make this come out properly in landscape mode, do one of the following
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% 1.
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% pdflatex latexsheet.tex
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%
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% 2.
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% latex latexsheet.tex
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% dvips -P pdf -t landscape latexsheet.dvi
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% ps2pdf latexsheet.ps
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% If you're reading this, be prepared for confusion. Making this was
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% a learning experience for me, and it shows. Much of the placement
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% was hacked in; if you make it better, let me know...
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% 2008-04
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% Changed page margin code to use the geometry package. Also added code for
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% conditional page margins, depending on paper size. Thanks to Uwe Ziegenhagen
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% for the suggestions.
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% 2006-08
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% Made changes based on suggestions from Gene Cooperman. <gene at ccs.neu.edu>
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% -----------------------------------------------------------------------
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\begin{document}
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\raggedright
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\footnotesize
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\begin{multicols*}{4}
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\begin{center}
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\Large{ZF Wettersysteme asd} \\
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\small{701-0473-00L Wettersysteme, bei M. Sprenger \& F. Aemisegger} \\
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\small{Jannis Portmann \the\year}
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\end{center}
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\begin{center}
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\rule{\linewidth}{0.25pt}
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\end{center}
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\section{Thermodynamik}
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\subsection{Potentielle Temperatur}
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$$\theta = T \bigg(\frac{p_o}{p} \bigg)^\kappa$$
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Bsp.
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$$\frac{T_{Boden}}{T_{LCL}} = \bigg( \frac{p_{Boden}}{p_{LCL}} \bigg)^\kappa$$
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\subsection{Hydrostatische Grundgleichung}
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$$\frac{dp}{dz} = -\rho g$$
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integriert
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$$h = \frac{RT}{g}\ln \bigg(\frac{p_o}{p} \bigg)$$
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\subsection{Stabilität}
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\begin{figure}[H]
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\centering
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\includegraphics[width=3.5cm]{stability.png}
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\caption{Hydrostatische Stabilität}
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\label{fig:stability}
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\end{figure}
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\subsubsection{Brunt-Väisälla Frequenz}
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$$N^2 = \frac{g}{\theta}\frac{\partial \theta}{\partial z}$$
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$N^2 > 0: stabil$
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\section{Winde und Fronten}
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\subsection{Geostrophischer Wind}
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$$fu_G = -g \frac{\partial \phi}{\partial y}$$
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$$fv_G = g \frac{\partial \phi}{\partial x}$$
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wobei $f$ der Coriolis-Parameter ist.
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Geostrophische Näherung ist gültig, wenn der Rossby-Parameter $<1$.
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$$Ro = \frac{U}{fL}<1$$
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\subsection{Thermischer Wind}
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$$\frac{\partial v_g}{\partial z} = \frac{g}{fT} \vec{k} \times \nabla_hT$$
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integriert
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$$\vec{v_T}=\vec{v_g}(p_1)-\vec{v_g}(p_2) = \frac{R}{f}\ln \bigg(\frac{p_1}{p_2} \bigg)\vec{k} \times \nabla_h T$$
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wobei
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$\vec{k} \times \nabla_h T = \frac{\Delta T}{\Delta y}$
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\begin{figure}[H]
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\centering
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\includegraphics[width=5cm]{thermischer_wind.png}
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\caption{Thermischer Wind}
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\label{fig:therm-wind}
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\end{figure}
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\subsection{Temperaturadvektion}
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Verschiebung warmer oder kalter Luft (Norhemispäre: von S nach N Warmlufadvektion z.B. durch Barokline Welle (s. auch \ref{fig:energy-baroclinity}))
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$$F = -\vec{v}\cdot\vec{\nabla} T$$
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\subsection{Ageostrophischer Wind}
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Senkrecht auf den Wind (normal)
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$$V_{an} = \frac{1}{f}\frac{DV}{Dt}$$
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Entlang dem Wind (streamwise)
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$$V_{as} = \frac{1}{f}\frac{V^2}{R_t}$$
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wobei $V$ die horizontale Windgeschwindigkeit, $f$ der Coriolisparameter und $R_t$ die Krümmung der Trajektorie (zyklonal = positiv) ist.
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\section{Satellitenbilder}
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\subsection{Kanäle}
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\begin{itemize}
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\item \textbf{VIS}: Intensität abhängig von Albedo, hohe Intensität = hohereflektierende Fläche = weiss, Unterscheidung Wolken - Eisschwierig, nur am Tag VIS Bilder
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\item \textbf{WV}: durch Strahlungsmessung von obersterstark feuchter Schicht in Atmosphäre. Obere Troposphäreund tiefe Temperaturen $\Rightarrow$ geringe Intensitäten = weiss. Für Feuchteverhältnisse in oberer Troposphäre (300-600 hPA). Passiver Tracer der atmosphärischen Strömung
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\item \textbf{IR}: Temp. der abstrahlenden Oberfläche. Warm = hohe Intensität = schwarz. Hohe Wolken weiss, weil Oberfläche kalt.Hohe/tiefe Wolken lassen sich gut unterscheiden. Tiefe Wolken/Nebel kaum sichtbar, da $\Delta T$ zu gering. Misst $\lambda_{max} \Rightarrow T_{Wolke}$
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\end{itemize}
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\section{Dynamik}
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\subsection{Vorticity}
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$$\xi = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = \vec{k} \cdot \nabla \times \vec{v_h}$$
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$$\frac{d\xi}{dt} = -\vec{v}\cdot \vec{\nabla}(\xi + f) - (\xi + f)\bigg(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\bigg)$$
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\subsection{Potentielle Vorticity (PV)}
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$$Q = \frac{1}{\rho}(f+\xi)\frac{\partial \theta}{\partial z}$$
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für synoptische Skalen ($\xi \ll f$) vereinfacht sich der Ausdruck zu
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$$Q = \frac{1}{\rho}f\frac{\partial \theta}{\partial z}$$
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\begin{itemize}
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\item Grenze der Stratosphäre bei 2PVU
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\item Bleibt bei trockenadiabatioschen Prozessen erhalten
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\end{itemize}
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\subsubsection{Invertibilitätsprinzip}
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PV-Verteilung in Atmosphäre zusammen mit Verteilung derpotentiellen Temperatur am Boden legt die quasi- horizontaleStrömung (Druck-, Temperatur-, Windfeld) fest.
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\subsection{PV-Streamer}
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\begin{figure}[H]
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\centering
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\includegraphics[width=5cm]{pv-streamer.png}
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\caption{Wind entlang eines PV-Streamer}
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\label{fig:pv-streamer}
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\end{figure}
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\subsection{PV-Anomalien}
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\begin{figure}[H]
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\centering
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\includegraphics[width=5cm]{pv-anomaly.png}
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\caption{Schnitt eines PV-Streamer (positive Anomalie)}
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\label{fig:pv-anomaly}
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\end{figure}
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\subsubsection{Erzeugung und Vernichtung von PV}
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$$\frac{D}{Dt} Q = -g \vec{\eta_p} \cdot \vec{\nabla_p} \dot{\theta} - g\vec{\nabla_p} \theta \cdot (\vec{\nabla_p} \times \vec{F})$$
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Wobei $\dot{\theta} \space [\mathrm{Ks^{-1}}]$ die adiabatische Heizrate und $\vec{F}$ die Summe der nicht-konservativen Kräfte
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\section{Lagrange'sche- vs Euler'sche Perspektive}
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\subsection{Lagrange'sche Perspektive}
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Aus Sicht eines Partikels $\Rightarrow$ materielle Ableitung\\
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Z.B.
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$$\frac{D \theta}{Dt} = \frac{\partial \theta}{\partial t} + (v \cdot \nabla) \theta$$
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\subsection{Euler'sche Perspektive}
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Aus Sicht eines ortsfesten Punktes\\
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Z.B.
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$$\frac{\partial \theta}{\partial t}$$
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\section{Globale Zirkulation}
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\subsection{Antrieb}
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\begin{figure}[H]
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\centering
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\includegraphics[width=5cm]{rad_balance_ERBE_1987.jpg}
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\caption{Differentielle Erwärmung}
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\label{fig:radiation-balance}
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\end{figure}
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Zirkulation (Wärmefluss gegen Pole) wirkt Strahlungsunterschieden entgegen.
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\subsection{Jets}
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Hadley Cell (thermisch direkt), Ferrel Cell (thermisch indirekt) und polar Cell (thermisch direkt) führen zu Jets zwischen den einzelnen Zellen
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\subsubsection{Thermisch direkte Zirkulation}
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Aufsteigen in tieferen Breiten, absinken in höheren Breiten
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\subsection{Umwandlung der Energie}
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\begin{figure}[H]
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\centering
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\includegraphics[width=5cm]{energy.png}
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\caption{Umwandlung der Energieformen}
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\label{fig:energy-forms}
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\end{figure}
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\subsubsection{Baroklinität}
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\begin{figure}[H]
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\centering
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\includegraphics[width=6cm]{baroclinity.png}
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\caption{Energie aus Baroklinität}
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\label{fig:energy-baroclinity}
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\end{figure}
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\begin{itemize}
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\item Baroklinität führt zu kinetischer Energie (grösserer Gradient $\rightarrow$ höhere potentielle Energie)
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\item Die Baroklinität ist im Winter grösser als Sommer (v.a. weiter südlich)
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\end{itemize}
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\subsubsection{Barokline Welle}
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\begin{figure}[H]
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\centering
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\includegraphics[width=6cm]{barocline-wave.png}
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\caption{Barokline Welle mit Wellenachse}
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\label{fig:wave-baroclinity}
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\end{figure}
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Tiefdruckgebietsbildung an Trog-Vorderseite
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\subsection{Heiztank Beispiel}
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\begin{figure}[H]
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\centering
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\includegraphics[width=6cm]{heat-tank-example.png}
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\caption{Thermische Zirkulation}
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\label{fig:circulation-example}
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\end{figure}
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\section{Tropopause}
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Sprünge bei Jetstream-Einflusss
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\subsection{Definitionen}
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\subsubsection{Thermische Tropopause}
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$$-\frac{dT}{dz} < 2Kkm^{-1}$$
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für min. 2km
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\begin{itemize}
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\item basiert nicht auf einer Erhaltungsgrösse (willkürlich)
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\end{itemize}
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\subsubsection{Dynamische Tropopause}
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$$Q = 2\mathrm{pvu}$$
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\begin{itemize}
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\item basiert auf der adiabatischen Erhaltungsgrösse $Q$ (PV)
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\end{itemize}
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\subsubsection{Chemische Tropopause}
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Fläche bestimmter Ozonkonzentration
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\subsubsection{Tropische Tropopause}
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Da am Äquator $f=0$:
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$$Q \approx \frac{1}{\rho}f\frac{\partial \theta}{\partial z} = 0$$
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Darum Isentrope Fläche 380K für tropische Regionen
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\subsection{Stratosphere-Troposhphere Exchange (STE)}
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Im Winter am grössten
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\subsubsection{Tropo- to Stratoshpere Transport (TST)}
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\begin{itemize}
|
||||||
|
\item Maximum über Nordatlantik und Westamerika
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\subsubsection{Strato- to Troposhpere Transport (STT)}
|
||||||
|
\begin{itemize}
|
||||||
|
\item Maximum über Nordatlantik und -pazifik (Stormtracks)
|
||||||
|
\item meist shallow exchanges
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\subsection{Prozesse}
|
||||||
|
\begin{itemize}
|
||||||
|
\item Tropo- bzw. Stratosphärische Cutoffs
|
||||||
|
\item Streamer
|
||||||
|
\item Tropopausenfalten
|
||||||
|
\item brechende Schwerewellen
|
||||||
|
\item Kovektion
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\section{Isotopen-Meteorologie}
|
||||||
|
\subsection{Isotopenverhältnis}
|
||||||
|
$$\delta = \frac{R_\mathrm{sample}-R_\mathrm{std}}{R_\mathrm{std}}$$
|
||||||
|
$R_\mathrm{std}2H = 0.00015576$ \\
|
||||||
|
$R_\mathrm{std}18O = 0.00200520$ \\
|
||||||
|
|
||||||
|
\subsection{Fraktionierung}
|
||||||
|
\subsubsection{Gleichgewichts Fraktionierung}
|
||||||
|
Bei $RH=100\%$
|
||||||
|
\begin{itemize}
|
||||||
|
\item Bei Phasenübergängen werden Isotopen nicht gleich verteilt
|
||||||
|
\item Schwere Isotopen bevorzugen Phase mit stärkerer Bindung (da tieferer Sättigungsdampfdruck)
|
||||||
|
\item Grösser bei tiefen Temperaturen
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\subsubsection{Nicht-Gleichgewichts Fraktionierung}
|
||||||
|
Bei $RH<100\%$
|
||||||
|
\begin{itemize}
|
||||||
|
\item Bei Phasenübergängen werden Isotopen nicht gleich verteilt
|
||||||
|
\item Schwere Isotopen haben eine geringere Diffusivität
|
||||||
|
\item Grösser bei starker Untersättigung
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\begin{figure}[H]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=6cm]{isotopes.png}
|
||||||
|
\caption{Schematische Verteilung von Isotopen}
|
||||||
|
\label{fig:isotopes}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\section{Gebirgsmeteorologie}
|
||||||
|
\subsection{Um- oder Überstömung}
|
||||||
|
Möglicher Ablauf
|
||||||
|
\begin{itemize}
|
||||||
|
\item (a) Deformation der Kaltfront und Ausbildung von Südföhn
|
||||||
|
\item (b) Kaltluftausbruch ins westliche Mittelmeer (Mistral) und Bildung einer Lee-Zyklone
|
||||||
|
\item (c) Bewegung der Lee-Zyklone nach Osten und Einsetzen von Bora und Nordföhn
|
||||||
|
\end{itemize}
|
||||||
|
\begin{figure}[H]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=6cm]{alpenumströmung.png}
|
||||||
|
\caption{Wechselwirkung Kaltfront}
|
||||||
|
\label{fig:alps}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\subsubsection{Lee-Zyklogenese}
|
||||||
|
Durch Mistral entsteht PV-Anomalie am Westrand der Alpen. Diese schnürt sich eventuell ab und beginnt die Zyklogenese im Golf von Genua.
|
||||||
|
Höhen-PV-Streamer unterstütz dieses Vorgehen mit Cut-Off. (Zusammenspiel von Höhen- und Boden-PV-Anomalien)
|
||||||
|
|
||||||
|
\subsubsection{Inverse Froude-number}
|
||||||
|
Zum Abschätzen ob die Luft ein Gebirge Um- oder Überströmt (kleine $Fr \rightarrow$ wahrscheinlichere Überströmung).
|
||||||
|
$$Fr = \frac{NH}{U}$$
|
||||||
|
Wobei $N$ die Schichtung (Brunt-Väisälla), $H$ die Gebirgshöhe und $U$ die Anströmgeschwindigkeit ist.
|
||||||
|
|
||||||
|
\subsection{Schwerewellen}
|
||||||
|
\subsection{Entstehung}
|
||||||
|
Störung in der Druckverteilung durch auf- und absteigende Bewegungen, die sich vertikal ausbreitet.
|
||||||
|
\begin{figure}[H]
|
||||||
|
\centering
|
||||||
|
\includegraphics[width=5cm]{gravity-waves.png}
|
||||||
|
\caption{Schwerewellen bei Überströmung eines Gebirges}
|
||||||
|
\label{fig:gravity-waves}
|
||||||
|
\end{figure}
|
||||||
|
Verantwortlich für die Bildung von Lenticularis \\
|
||||||
|
|
||||||
|
\subsection{Brechende Schwerewellen}
|
||||||
|
Verändert das Windfeld (vertikal und horizontal) stark, kann zu starken Turbulenzen führen.
|
||||||
|
|
||||||
|
\section{Planetare Grenzschicht}
|
||||||
|
\begin{itemize}
|
||||||
|
\item Bis ca. 1km (Höhe der tieffsten Inversion)
|
||||||
|
\item Geostrophisches GGW gilt hier nicht
|
||||||
|
\item Hohe Aerosolkonzentration
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\subsection{Turbulente kinetische Energie (TKE)}
|
||||||
|
$$TKE = \frac{1}{2}(\bar{u'}^2+\bar{v'}^2+\bar{w'}^2)$$
|
||||||
|
|
||||||
|
$$\frac{\partial}{\partial t}(TKE) = -\overline{u'w'}\cdot \frac{\partial \bar{u}}{\partial z} - \overline{v'w'}\cdot \frac{\partial \bar{v}}{\partial z} + \frac{g}{\bar{\theta_v}} \cdot \overline{w'\theta'_v}$$
|
||||||
|
$$-\frac{\partial}{\partial z}(\overline{w'TKE}+\frac{\overline{w'p'}}{\rho})-\epsilon$$
|
||||||
|
|
||||||
|
\subsubsection{Richardson Zahl}
|
||||||
|
$$Rf = \frac{g}{\bar{\theta_v}} \cdot \overline{w'\theta_v'} \cdot (\overline{u'w'}\frac{\partial \bar{u}}{\partial z} + \overline{v'w'}\frac{\partial \bar{v}}{\partial z})$$
|
||||||
|
$Rf < 1$: Turbulenz, $Rf > 1$: keine Turbulenz
|
||||||
|
|
||||||
|
|
||||||
|
\section{Konstanten}
|
||||||
|
\begin{itemize}
|
||||||
|
\item $R_\mathrm{s, dry-air} = 287.058 \space \mathrm{J}\mathrm{kg}^{-1}\mathrm{K}^{-1}$
|
||||||
|
\item $c_\mathrm{p, dry-air} = 1005 \mathrm{J}\mathrm{kg}^{-1}\mathrm{K}^{-1}$
|
||||||
|
\item $\kappa = \frac{R_\mathrm{s, dry-air}}{c_{p,\mathrm{dry-air}}} = 0.28$
|
||||||
|
\item $1 \mathrm{pvu} = 1 \times 10^{-6}\mathrm{m}^2\mathrm{s}^{-1}\mathrm{K}\mathrm{kg}^{-1}$
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\scriptsize
|
||||||
|
|
||||||
|
\section{Copyleft}
|
||||||
|
|
||||||
|
\doclicenseImage \\
|
||||||
|
Dieses Dokument ist unter (CC BY-SA 3.0) freigegeben \\
|
||||||
|
\faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp} \\
|
||||||
|
\faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/wettersysteme-zf} \\
|
||||||
|
Jannis Portmann, HS20
|
||||||
|
|
||||||
|
\end{multicols*}
|
||||||
|
\end{document}
|
||||||
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