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\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}

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\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}


