 boundary at 70$^{\circ}$C), we have checked that those top convective motions have no significant impact on the dynamics of the two-layer system. It is also important to say that despite the thick acrylic wall and the intermediate fluid layer between the cylinder and the tank, the working region is not fully thermally insulated on the sides. Nevertheless, our fully stratified, test experiment has shown no motion within the fluid driven by these lateral losses. 
\\

The equation of state of water is non-linear with a  maximum density close to 4$^{\circ}$C (International Equation of State of Seawater, 1980):
\begin{equation}
    \begin{split} 
    \rho(T)= 999.842594+6.793952.10^{-2}T-9.095290.10^{-3}T^2+1.001685.10^{-4}T^3 \\
    -1.120083.10^{-6}T^4+6.536332.10^{-9}T^5.\\
    \end{split}
    \label{eq:eos_eau}
\end{equation}
Thus, due to the imposed temperature at top and bottom boundaries, the bottom part of the tank, between 0$^{\circ}$C and 4$^{\circ}$C, is convectively unstable (see figure \ref{fig:setup}). Cold plumes detach from the bottom plate and rise in the tank due to buoyancy. Reciprocally, ``hot'' fluid sinks from the 4$^{\circ}$C  interface due to gravity. While convective motion takes place in the lower layer, the upper part of the tank, between 4$^{\circ}$C and 35$^{\circ}$C, is stably stratified, with an assumed linear temperature profile at equilibrium. The temperature is indeed linear for an ideal case without thermal losses, assuming that the stratified layer has a bottom boundary at fixed temperature $4^\circ$C and top boundary at $35^\circ$C (\textit{i.e.} constant diffusive flux through the whole layer). However, the density profile is not linear, due to the non-linear equation of state of water. Stratification is characterised by the Brunt-V\"ais\"al\"a frequency $N^* = \frac{1}{2 \pi} \sqrt{-\frac{g}{\rho_0} \frac{\partial \rho}{\partial z}}$. Because of the non-linear equation of state, $N^*$ is not constant with depth, as shown in figure \ref{fig:setup}. For simplicity, we also use below the global buoyancy frequency defined as $N = \frac{1}{2 \pi} \sqrt{-\frac{g}{\rho_0} \frac{\Delta \rho}{\Delta z}}$, where ${\Delta \rho}$ is the global density contrast within the stratified layer of depth ${\Delta z}$.
\\

\begin{figure}[h]

  \includegraphics[scale=.3]{schema_cuve_profilN_NB_v3.png}
  \caption{2D sketch of the tank. A cylinder (light grey shaded area) is placed in a larger cubic tank. The system is cooled down at the bottom at 0$^{\circ}$C and heated up at the top at 35$^{\circ}$C. The bottom half is convective with an almost constant density, apart from the bottom boundary layer. The upper half is stably stratified and waves are generated due to the fluid motions in the convective layer. The graph on the right shows the theoretical profile for the buoyancy frequency $N^*$. It is computed considering a linear temperature profile and the equation of state of water (\ref{eq:eos_eau}). The dashed line is the global buoyancy frequency $N$ calculated on the stratified layer. The various length scales are the cylinder diameter $D$, the vertical extent of the tank $H$ and the vertical extent of the convective layer $h$. $\delta$ is the minimal width between the outer square tank and the inner cylinder. The $x$, $y$ and $z$-velocity components are noted $u$, $v$ and $w$ respectively.}
  \label{fig:setup}
\end{figure}

Before starting the experiment, the bath and the heater are allowed to reach their assigned temperature. Then, the upper half of the tank is filled with water stratified in temperature from 4$^{\circ}$C to 35$^{\circ}$C, using the double bucket technique \cite{oster_density_1965}. The bottom half is filled with water with a temperature close to 4$^{\circ}$C. This filling process is used to avoid tremendously long transient before reaching steady state by thermal diffusion. Typically, we fill the tank at the end of a day and start the measurements the next day in order to let the system reach its equilibrium state over night. Each experiment then lasts four days, with no change in the location of the interface. Note that this steady interface position is the result of the heat budget equilibrium between the convective heat flux extracted from the bottom plate, the diffusive heat flux through the stratified layer, and the lateral heat losses.

To perform PIV measurements, particles are  chosen as small as possible and with a density as close as possible to water density in order to avoid their sedimentation in the stratified layer over the long duration of the experiment. We use fluorescent orange polyethylene micro-spheres whose size ranges from $10 \, \mathrm{\mu m}$ to $20 \, \mathrm{\mu m}$ and density is $1.00 \pm 0.01$~g/cc. The fluorescent property allows us, with a high pass filter on the camera, to remove any laser reflection, significantly enhancing the images quality. The tank is illuminated with a green laser $532$~nm. Power is set at $1$~W. 
We perform side view PIV to measure convection and IGWs spectral characteristics, and top view PIV to observe the large scale flow and its fluctuations over time. The camera used for the side view PIV is a HiSense Zyla $2560 \times 2160$ pixels recorded on 12 bits. Acquisition rate is $2$~Hz with $100$~ms exposure time. Typical acquisition time for spectral characteristics is $50$~min.
For the top view, we use a Point grey camera $1920\times1080$ pixels on 8~bits. Exposure time is $1$~s, acquisition rate $0.1$~Hz and acquisition time $8$~hours. Captured movies are processed either by the DantecDynamics software DynamicStudio for the side view or by DPIVSoft \cite{meunier2003analysis} for the top view. Both are resolved into $32\times32$ pixels boxes with 50\% overlapping. 

Side view PIVs are performed in the middle of the tank at $y=16$~cm in a laser sheet crossing the cylinder along its diameter. This is the case for all figures shown in the $(x,z)$ plane and thus not mentioned in the results section. The vertical fields (see an example in figure \ref{fig:LSCexpe}) do not show the whole $(x,z)$ plane (where the origin $(0,0)$ is located in the bottom left corner of the cubic tank): it was indeed chosen to zoom in, in order to have the best resolution for the very weak motions in the stratified layer. The interface between the layers is localised between $11 \mathrm{~cm} \leqslant z \leqslant 12$~cm. Typical Rayleigh number for the convection based on this depth is $Ra = 7 \times 10^6$, and the global Brunt-V\"ais\"al\"a frequency is $N = 1.35 \times 10^{-1}$~Hz.

\begin{figure*}[h]
    \centering
    \includegraphics[scale =.45]{champ_int_730_colored.pdf}
    \hspace{.1cm}
    \includegraphics[scale =.45]{LSC_0510_1800_colored.pdf}
    \caption{(Left) Instantaneous velocity field. An ascending plume is visible at $x=150$ mm, and transported by the large-scale circulation. (Right) Large-scale circulation in the convective layer obtained by time-averaging velocities over a 50 minutes signal. The large-scale circulation is a counter clockwise cell. No inversion of the circulation has been seen in our experiment. The velocities under $z=10$ mm are noisy and thus not shown here. Maximum instantaneous velocities are $2.5$ times bigger than the maximum averaged velocities. The left edge of the cylinder is located at $x=19$~mm and the centre of the cylinder is at $x=160$~mm. Approximately $40$~mm of the right side of the cylinder is not captured with our PIV measurments.}
    \label{fig:LSCexpe}
\end{figure*}


To observe the large-scale flow, horizontal views at different heights are performed. A linear translation axis platform from Igus, driven by homemade software, is used to automate the process. With two mirrors, one fixed at the exit of the laser head and the other fixed on the translating platform, it is possible to sweep along one direction with the laser sheet. We typically make measurements during 15~s in each of 11 successive planes separated by 0.5~cm. The full scan duration is about 3~min, and is repeated during at least 8~hours.
\\

The cylindrical geometry described here differs from the cylindrical shell