% !TeX encoding = UTF-8 % !TeX spellcheck = en_US \documentclass[11pt, aspectratio=169]{beamer} \usepackage{templates/beamerthemekit} \usepackage{makecell} \usepackage[export]{adjustbox} \usepackage{booktabs} \usepackage{siunitx} \usepackage{amssymb} \usepackage{amsmath, nccmath} \usepackage{xmpmulti} \usepackage{pdfpages} \usepackage{siunitx} \usepackage[absolute,overlay]{textpos} \usepackage[absolute,overlay]{textpos} \usepackage{multimedia} \usepackage{graphicx} \usepackage{xcolor} \usepackage{colortbl} \usepackage{adjustbox} \usepackage{tikz} \usepackage{ulem} \usepackage{bm} \graphicspath{{figures/}} \usepackage{enumitem} % Make links show up as coloured \hypersetup{ colorlinks = true, allcolors = black, urlcolor = blue, } %\newcommand{\sectionW} [1] {\section {\color{brown} {#1}}} % commande de Patrick \newcommand{\sectionW} [1] {\section [\color{white} #1] {#1}} \newcommand{\fullframeimage}[1]{ \begin{center}% \includegraphics[max width = \textwidth, max height=0.8\textheight]{#1}% \end{center} } \let\emph\relax % there's no \RedeclareTextFontCommand \DeclareTextFontCommand{\emph}{\color{slide-color}} \colorlet{slide-color}{indigo} \title{ Turbulent boundary layer measurements near the surface below the jet maximum in a katabatic wind along alpine slope. } %couleur du titre %\subtitle{ %Implications for numerical modelling and the definition of laws of the wall. %} \author{\underline{Christophe Brun}, Wilfred Bessem, Ambre Dublanche, \\ Muriel Lagauzère, Stéphane Pioz, } \institute{LEGI, UGA, Grenoble, France} \date{Barcelona, 3 September 2024} % You can add any logos you like by uploading them into the logos folder and replacing the blank image here \titleimage{belle2_wide_169} \titlelogoa{belle2-logo} \titlelogob{bmbf} \titlelogoc{blank} %\titlelogoa{belle2-logo} %\titlelogob{bmbf} %\titlelogoc{blank} \begin{document} \begin{frame} \titlepage \end{frame} \section*{\color{white} Introduction} \begin{frame}{Physical background } \begin{textblock}{15}(0.7,1.7) Process of katabatic wind formation \end{textblock} \begin{textblock}{15}(0.5,2.7) \only<1-2>{\includegraphics[width=.63\textwidth]{figures/Cata_nuit_complet1.PNG}} \end{textblock} \begin{textblock}{10}(7.9,3) \centering \only<1>{\textbf{Negative radiative budget}\\ \hspace{.25cm} $R_n^{night} = LW_{received} - LW_{emitted}<0 $\\ $\big\Downarrow$\\ \textbf{Surface temperature cooling}\\ \hspace{.25cm} $\Rightarrow$ Temperature gradient\\ \hspace{.8cm} $\Rightarrow$ Air cooling / densification\\} \only<1>{$\big\Downarrow$\\ \textbf{Downslope flow} \\ \hspace{1.15cm} $\Rightarrow$ Turbulent mixing\\} \end{textblock} \only<2>{ \begin{textblock}{20}(11,2) {\small \textbf{Night Anticyclonic conditions}} \\ {\small \textbf{Strong radiative cooling:}} \\ {\small $H_S \in [-80; -100] \ W/m^2$} \\ \end{textblock} \begin{textblock}{20}(11,5) {\small \textbf{background stratification:} } \\ {\small $N = \sqrt{\dfrac{g}{\theta_0} \dfrac{\partial \theta_v}{\partial z}} \approx 0.01-0.02$~Hz} \\ \end{textblock} \begin{textblock}{20}(10.9,9) {\small \textbf{Turbulent flow regime:} }\\ {\small $Re = \dfrac{2 g }{\nu \theta_s \sin{\alpha}} \dfrac{ H_s}{ N^2 }\approx 10^5-10^6$} \\ {\small \textbf{Shapiro \& Fedorovich BLM 2014} } \\ {\small \textbf{Xiao \& Senocak JFM 2019} } \end{textblock} } \end{frame} \sectionW{2. Theory } \begin{frame}{ RANS budget for katabatic jet } \begin{textblock}{15}(10.5,8) \only<1-3>{\includegraphics[width=.33\textwidth]{figures/Cata_nuit_complet1.PNG}} \end{textblock} \only<1-3> {\begin{textblock}{8}(.7,2.5) \only<1>{\begin{equation} \underbrace{\frac{\partial \overline{u}}{\partial t}}_{\substack{\text{Inertia}}} + \underbrace{\frac{\partial \overline{u'w'}}{\partial z}}_{\substack{\text{Turbulent} \\ \text{momentum flux} }} \approx \begingroup\color{lightgray} \underbrace{\overline {w}\frac{\partial \overline {u}}{\partial z}}_{\substack{\text{Advection}}} - \underbrace{g \frac{\overline {\theta}-\theta_a}{\theta_a}\sin \alpha}_{\substack{\text{Katabatic forcing}}} \endgroup =0 \nonumber \end{equation}} \only<2>{\begin{equation} \underbrace{\frac{\partial \overline{u}}{\partial t}}_{\substack{\text{Inertia}}} + \underbrace{\frac{\partial \overline{u'w'}}{\partial z}}_{\substack{\text{Turbulent} \\ \text{momentum flux} }} \approx \begingroup\color{lightgray} \underbrace{\overline {w}\frac{\partial \overline {u}}{\partial z}}_{\substack{\text{Advection}}} \endgroup - \underbrace{g \frac{\overline {\theta}-\theta_a}{\theta_a}\sin \alpha}_{\substack{\text{Katabatic forcing}}} \nonumber \end{equation}} \only<3>{\begin{equation} \underbrace{\frac{\partial \overline{u}}{\partial t}}_{\substack{\text{Inertia}}} + \underbrace{\frac{\partial \overline{u'w'}}{\partial z}}_{\substack{\text{Turbulent} \\ \text{momentum flux} }} \approx \underbrace{\overline {w}\frac{\partial \overline {u}}{\partial z}}_{\substack{\text{Advection}}} - \underbrace{g \frac{\overline {\theta}-\theta_a}{\theta_a}\sin \alpha}_{\substack{\text{Katabatic forcing}}} \nonumber \end{equation}} \end{textblock}} \only<1> {\begin{textblock}{8}(.7,5.5) \begin{fleqn}\begin{equation} \underbrace{\frac{\partial \tilde{\theta}}{\partial t}}_{\substack{\textnormal{Inertia}}} + \underbrace{\frac{\partial \overline{w'\theta'}}{\partial z}}_{\substack{\textnormal{Turbulent sensible } \\ \text{heat flux}}} \approx \begingroup\color{lightgray} - \underbrace{\overline w\frac{\partial \tilde{\theta}}{\partial z}}_{\substack{\textnormal{Advection}}} -\underbrace{\overline u \frac{\partial \theta_a}{\partial z} \sin{\alpha} -\overline w \frac{\partial \theta_a}{\partial z} \cos{\alpha} }_{\substack{\textnormal{Ambient stratification }}} \endgroup =0 \nonumber \end{equation} \end{fleqn} \end{textblock}} \only<2> {\begin{textblock}{8}(.7,5.5) \begin{fleqn}\begin{equation} \underbrace{\frac{\partial \tilde{\theta}}{\partial t}}_{\substack{\textnormal{Inertia}}} + \underbrace{\frac{\partial \overline{w'\theta'}}{\partial z}}_{\substack{\textnormal{Turbulent sensible } \\ \text{heat flux}}} \approx \begingroup\color{lightgray} - \underbrace{\overline w\frac{\partial \tilde{\theta}}{\partial z}}_{\substack{\textnormal{Advection}}} \endgroup -\underbrace{\overline u \frac{\partial \theta_a}{\partial z} \sin \alpha }_{\substack{\textnormal{Ambient stratification }}} \begingroup\color{lightgray} -\overline w \frac{\partial \theta_a}{\partial z} \cos{\alpha} \endgroup \nonumber \end{equation} \end{fleqn} \end{textblock}} \only<3> {\begin{textblock}{8}(.7,5.5) \begin{fleqn}\begin{equation} \underbrace{\frac{\partial \tilde{\theta}}{\partial t}}_{\substack{\textnormal{Inertia}}} + \underbrace{\frac{\partial \overline{w'\theta'}}{\partial z}}_{\substack{\textnormal{Turbulent sensible } \\ \text{heat flux}}} \approx - \underbrace{\overline w\frac{\partial \tilde{\theta}}{\partial z}}_{\substack{\textnormal{Advection}}} \begingroup\color{lightgray} -\underbrace{\overline u \frac{\partial \theta_a}{\partial z} \sin \alpha -\overline w \frac{\partial \theta_a}{\partial z} \cos{\alpha} }_{\substack{\textnormal{Ambient stratification }}} \endgroup \nonumber \end{equation} \end{fleqn} \end{textblock}} \only<3> {\begin{textblock}{8}(.7,8.5) \begin{fleqn}\begin{equation} \underbrace{\frac{\partial \overline{w}}{\partial t}}_{\substack{\text{Inertia}}} +\underbrace{\frac{\partial\overline{w'^2}}{\partial z}}_{\substack{\text{Turbulent} \\ \text{velocity variance}}} %-\underbrace{\nu \dfrac{\partial^2 \overline{w}}{\partial z^2}}_{\substack{\text{Diffusion}}} \approx \underbrace{\overline{w}\frac{\partial \overline{w}}{\partial z}}_{\substack{\text{Advection}}} + \underbrace{g \frac{\overline{\theta} -\theta_a}{\theta_a}\cos \alpha}_{\substack{\text{Katabatic forcing}}} \nonumber \end{equation} \end{fleqn} \end{textblock}} \only<1>{\begin{textblock}{8}(10.0,3.3) boundary layer on a flat surface \\ \textcolor{indigo}{\scriptsize{Wyngaard (2010)}} \end{textblock}} \only<2>{\begin{textblock}{8}(10.0,3.3) Prandtl model for Katabatic jet \\ \textcolor{indigo}{\scriptsize{Prandtl (1942)}} \end{textblock}} \only<3>{\begin{textblock}{8}(9.0,3.3) Katabatic boundary layer along a slope \\ \textcolor{indigo}{\scriptsize{Charrondière BLM 2022}} \end{textblock}} %En modifiant W, on modifie aussi wT par advection (on voit un profil pas constant) et sûrement aussi T donc. Comme rétroaction, pas modélisable facilement Apport du modèle intégral : wTs \end{frame} \sectionW{Wall model for Katabatic flow} \begin{frame}{ Analytical model for turbulent katabatic flow } \begin{textblock}{15}(10.5,5) \only<1-4>{\includegraphics[width=.33\textwidth]{figures/Cata_nuit_complet1.PNG}} \end{textblock} \begin{textblock}{8}(7.8,2.5) \only<2-4>{ \begin{equation} {\color{red} \frac{1}{L_{Kat}} } u_*^2 =- g \frac{{\color{red} \overline {\theta_S}}-\theta_a}{\theta_a}\sin \alpha \nonumber \end{equation}} \end{textblock} \begin{textblock}{8}(.8,2.5) \only<1>{\begin{equation} \begingroup\color{lightgray}\underbrace{\frac{\partial \overline{u}}{\partial t}}_{\substack{\text{Inertia}}} + \underbrace{\overline {w}\frac{\partial \overline {u}}{\partial z}}_{\substack{\text{Advection}}} + \endgroup \underbrace{\frac{\partial \overline{u'w'}}{\partial z}}_{\substack{\text{Turbulent} \\ \text{momentum flux}}} \approx - \underbrace{g \frac{{\color{red} \overline {\theta_S}}-\theta_a}{\theta_a}\sin \alpha}_{\substack{\text{Katabatic forcing}}}= \frac{u_*^2}{ {\color{red}L_{Kat}} } \color{red}>0 \nonumber \end{equation}} \only<2-4>{ \begin{fleqn}\begin{equation} -\overline{u'w'} = u_*^ 2 \begingroup\color{red} (1- \frac{z}{L_{Kat}}) \endgroup \nonumber \end{equation} \end{fleqn} } \end{textblock} \begin{textblock}{8}(0.8,6.3) \only<1-3>{\begin{fleqn}\begin{equation} \begingroup\color{lightgray} \underbrace{\frac{\partial \overline{\theta}}{\partial t}}_{\substack{\textnormal{Inertia}}} + \endgroup \underbrace{\frac{\partial \overline{w'\theta'}}{\partial z}}_{\substack{\textnormal{Turbulent}\\ \text{heat flux}} } \approx - \underbrace{\overline w\frac{\partial \overline{\theta}}{\partial z}}_{\substack{\textnormal{Advection}}} \begingroup\color{lightgray} -\underbrace{\overline u \frac{\partial \theta_a}{\partial z} \sin \alpha }_{\substack{\textnormal{Ambient stratification}}} \endgroup {\color{blue} >0} \nonumber \end{equation} \end{fleqn}} \only<4>{\begin{fleqn}\begin{equation} \underbrace{\frac{\partial \overline{w'\theta'}}{\partial z}}_{\substack{\textnormal{Turbulent}\\ \text{heat flux}} } \approx - \underbrace{\overline w\frac{\partial \overline{\theta}}{\partial z}}_{\substack{\textnormal{Advection}}} \nonumber \end{equation} \end{fleqn}} \end{textblock} \begin{textblock}{7}(0.8,10.5) \only<1-2>{ \begin{equation} \begingroup\color{lightgray}\underbrace{\frac{\partial \overline{w}}{\partial t}}_{\substack{\text{Inertia}}} +\endgroup \underbrace{1/2 \frac{\partial \overline{w}^2}{\partial z}}_{\substack{\text{Advection}}} \begingroup\color{lightgray} +\underbrace{\frac{\partial\overline{w'^2}}{\partial z}}_{\substack{\text{Turbulent} \\ \text{velocity variance}}} \endgroup \approx \underbrace{g \frac{{\color{red}\overline{\theta_S}} -\theta_a}{\theta_a}\cos \alpha}_{\substack{\text{Katabatic forcing}}} = - %\frac{{\color{red} a} u_*^2}{ \tan \alpha} \frac{u_*^2}{ {\color{red} L_{Kat} \tan \alpha } } \color{red}<0 \nonumber \end{equation}} \only<3-4>{ \begin{fleqn}\begin{equation} \overline{w}^2 \tan \alpha \approx-2 \overline{u'w'} = 2 u_*^2 \begingroup\color{red} (1- \frac{z}{L_{Kat}}) \endgroup \nonumber \end{equation} \end{fleqn} } \end{textblock} \end{frame} \sectionW{1. Campagne de mesure in situ} %%%%%%%%%%%%%%%%%%%%% % PART2 %%%%%%%%%%%%%%% \begin{frame}{In Situ Measurements in the Alps} \only<1>{ \begin{textblock}{20}(1,2) \begin{overpic}[height=.85\textheight]{figures/mats2.JPG} \put (190, 186) {\small \textbf{November 2012}} \put (190, 176) {\small Blein phD 2016} \put (190, 166) {\small Brun et al. JAS 2017} \put (190, 156) {\small Charrondière et al. BLM 2020} \put (190, 137) {\small \textbf{April 2015} } \put (190, 127) {\small Unpublished results} \put (190, 108) {\small \textbf{February 2019} } \put (190, 98) {\small Charrondière et al. BLM 2022} \put (190, 88) {\small Charrondière et al. JFM 2022} \put (190, 78) {\small Charrondière et al. POF 2024} \put (190, 68) {\small Zenodo repository 2022 DOI: 10.5281/zenodo.6546702} \put (190, 49) {\small \textbf{February 2023} } \put (190, 39) {\small French Alps, Grenoble } \put (190, 20) {\small \textbf{February 2024-2025} } \put (190, 10) {\small Austrian Alps, Innsbruck (TeamX project)} \end{overpic} \end{textblock}} \only<2>{ \begin{textblock}{20}(1.1,1.8) \begin{overpic}[height=.85\textheight]{figures/mats.png} \put (294, 215) {\small \textbf{Winter 2023, Grenoble }} \put (294, 205) {\small \textbf{3-15 February}} %\put (2, 40) {\tiny 3D cobra probe} \put (173, 185) {\small \textbf{2D sonic anemometer (Waissala) }} \put (173, 175) {\small $z=3.5m$ } \put (173, 156) {\small \textbf{LW radiation sensor (IR120) $-> H_S, \ T_S$ }} \put (173, 127) {\small \textbf{3D sonic anemometer (CSAT3B) }} \put (173, 117) {\small $z=1m$ } \put (173, 98) {\small \textbf{4 Thermocouples (FW3) }} \put (173, 88) {\small $z=0.7m$, $z=1.2m$, $z=2.0m$, $z=2.9m$ } \put (173, 66) {\small \textbf{3D Pitot sensor} } \put (173, 56) {\small $z=2mm-900mm$} \put (173, 38) {\small \textbf{Micrometric }} \put (173, 28) {\small \textbf{ displacement }} \put (173, 18) {\small \textbf{ system }} \put (173, 08) {\small \textbf{(Rosier) }} \end{overpic} \end{textblock}} \only<2>{ \begin{textblock}{20}(10.1,11.5) \begin{overpic}[height=.22\textheight]{figures/Cobra_schema.pdf} \put (10, 55) {\small pressure transducers: $f=1250$~Hz} \put (10, 65) {\small 'Cobra' 3D pitot tube \textit{TFI}} \end{overpic} %\begin{textblock}{20}(13.28,11.57) %\includegraphics[width=.05\linewidth]{figures/Tete_cobra_photo.jpg} %\end{textblock} \end{textblock}} \only<3>{ \begin{textblock}{20}(1.1,1.8) \begin{overpic}[height=.86\textheight]{figures/night_mast.png} \put (294, 220) {\small \textbf{Winter 2024-2025, Innsbruck }} \put (294, 210) {\small \textbf{TeamX project}} \end{overpic} \end{textblock} \begin{textblock}{20}(8.1,1.9) \begin{overpic}[height=.85\textheight]{figures/mast_schema.png} \end{overpic} \end{textblock}} \end{frame} \begin{frame}{23 katabatic profiles (Grenoble 2023)} \begin{textblock}{20}(9.5,.2) \only<1>{\includegraphics[width = .2\linewidth]{figures/Extraction_katabatic_event_3.pdf}} \end{textblock} \begin{textblock}{20}(.5,2.4) \begin{itemize} \item Wind in the downslope direction \only<1>{\item Positive temperature gradient} \only<1>{\item Velocity profiles $\sim$ wall jet} \end{itemize} \end{textblock} \begin{textblock}{20}(.7,8.2) \begin{overpic}[width=.70\textwidth]{figures/T_GC_fw.pdf} \put (86, 113) {\vector(0,-1){13.2}} \put (113, 113) {\vector(0,-1){13.2}} \put (118, 113) {\vector(0,-1){13.2}} \put (160, 113) {\vector(0,-1){13.2}} \put (190, 113) {\vector(0,-1){13.2}} \put (212, 113) {\vector(0,-1){13.2}} \put (218.5, 113) {\vector(0,-1){13.2}} \put (232, 113) {\vector(0,-1){13.2}} \put (238.5, 113) {\vector(0,-1){13.2}} \put (261.5, 113) {\vector(0,-1){13.2}} \put (264, 113) {\vector(0,-1){13.2}} \put (286, 113) {\vector(0,-1){13.2}} \put (291, 113) {\vector(0,-1){13.2}} \put (292, 113) {\vector(0,-1){13.2}} \put (312, 113) {\vector(0,-1){13.2}} \put (316, 113) {\vector(0,-1){13.2}} \put (338, 113) {\vector(0,-1){13.2}} \put (341, 113) {\vector(0,-1){13.2}} \put (370, 113) {\vector(0,-1){13.2}} \put (365, 113) {\vector(0,-1){13.2}} \put (386.5, 113) {\vector(0,-1){13.2}} \end{overpic} %13 eventsfrom 30 min to 3h15 %Rotation de l'inclinaison du capteur so that W est la normale à la pente pour la suite. \end{textblock} \end{frame} \begin{frame}{Normal to the slope velocity $\overline{w}$} \only<1>{\begin{textblock}{14}(1.5,1.9) {\small February 11, 2023 \\ (9h37-10h06) } \end{textblock}} \only<1>{\begin{textblock}{14}(6.2,1.9) {\small February 12, 2023 \\ (19h18-19h52) } \end{textblock}} \only<1>{\begin{textblock}{14}(11.2,1.9) {\small February 13, 2023 \\ (7h13-7h46) } \end{textblock}} \only<1>{\begin{textblock}{7}(4.8,10.5) \begin{equation} \overline{w}^2 = w_o \left( 1-\frac{z}{L_{Kat}} \right) \dfrac{2}{\tan \alpha} = u_*^2 \left( 1-\frac{z}{L_{Kat}} \right) \nonumber \end{equation} \end{textblock}} \only<1>{\begin{textblock}{15}(0.2,3.1)\begin{overpic}[width = 0.3\textwidth]{figures/s11_09h37W_mean.pdf} \end{overpic} \end{textblock}} \only<1>{\begin{textblock}{15}(5.2,3.2)\begin{overpic}[width = 0.3\textwidth]{figures/d12_19h18W_mean.pdf} \end{overpic} \end{textblock}} \only<1>{\begin{textblock}{15}(10.2,3.2)\begin{overpic}[width = 0.3\textwidth]{figures/l13_07h13W_mean.pdf} \end{overpic} \end{textblock}} \end{frame} \sectionW{Conclusions} \begin{frame}{Conclusion \& Outlook} \only<1>{\begin{textblock}{24}(.5,2.6) \hspace{.25cm} \bf Katabatic jet along a steep slope \vspace{0.5cm} \hspace{.25cm} $\Rightarrow$ Turbulent boundary layer in the inner layer \hspace{.25cm} $-$ non constant turbulent fluxes (gravity effect) \hspace{.25cm} $-$ log-lin law for the along the slope velocity \hspace{.25cm} $-$ normal to the slope velocity contribution $10\% Uj$ \hspace{.25cm} $-$ gravity lengthscale $ L_{Kat}= \dfrac {\theta_a}{\overline{\theta_S}-\theta_a} \dfrac{u_*^2}{g \sin \alpha}$ \vspace{0.5cm} \hspace{.25cm} $\Rightarrow$ TeamX project : Innsbruck valley \hspace{.25cm} $-$ log law for the temperature deficit? \hspace{.25cm} $-$ Analysis of the roughness regime near the surface \hspace{.25cm} $-$ Turbulence properties at the surface \end{textblock}} \end{frame} \end{document}