[MUSIC] Hello. It is well-known that atmospheric or oceanic motions are turbulent. However, turbulence is not easy to define. It relies on several statistical tools. We will not give here a basic lecture on free turbulence, it will take several hours. Nevertheless, we will try to give some general definitions and very simple concepts to start with. Turbulent flows are directly related to disorder or [INAUDIBLE]. Fascinating illustration of atmospheric turbulence is provided by the famous painting of van Gogh, The Starry Sky, which qualitatively illustrate how the turbulent swirling motion of the atmospheric boundary layer affects the light propagation. The flickering of the stars, amazingly reproduced by the artists, is indeed directly related to the atmospheric turbulence in the boundary layer. If now, we consider the precise measurements of the atmospheric wind velocity at a given location, we generally get this type of signal. This curve is probably less artistic than Van Gogh's painting, but we can also distinguish here rapid fluctuations of the wind's speed around the slower evolution of the mean wind. These high frequency fluctuations are measured by an ultrasonic anemometer, and it is quite simple to perform a temporal averaging to filter out the rapid fluctuations. The main hypothesis of turbulent flow analysis is that we are able to separate the non-turbulent flow from the turbulent fluctuations. If you use a temporal averaging to perform this separation, on which characteristic time scale averaging should be done? To answer this question, let's have a look at the wind power spectrum. Here is a typical curve for wind power density. There is a large peak around five to six days, which corresponds to large scale synoptic deprivations. A peak at 12 hours corresponding to the night and day variations, and a large spectral gap of flow energy between few hours and ten minutes. And the last peak at higher frequency is associated to the atmospheric turbulence. All the habit wind variations of the order of minutes of few seconds correspond to turbulent fluctuations in use by small scales eddys or waves. The spectral gap is crucial to validate the temporal separation between the metrological time scales and the atmospheric turbulence. We can then easily extract from any wind measurement the turbulent path of the signal. If the mean wind is expected to be predicted by the national weather forecasting system, the turbulent fluctuations are fully [INAUDIBLE], and we can only try to estimate some statistical properties. By definition, the mean of the turbulent velocity u hat is 0. But not the mean of the square. Hence, we use a root mean square to build a dimensionless number, which quantifies the quality intensity of turbulence. The turbulence intensity is the ratio of the root means squared of the turbulent wind over the mean wind. In the atmosphere, this turbulent intensity could reach quite large values. Typical values of 20 or 30% could be reached for low winds, and relative turbulence intensity decays when the mean speed increases. On these scales, we can see that the turbulence intensity is maximal near the ground within the boundary layer where the vertical velocity shear is strong. While at the high altitude outside the boundary layer, the velocity shear and the turbulence intensity are much weaker. The bottom roughness is a major source of small scale and turbulent motion. And therefore, it will have a strong impact on the turbulence intensity. For the same geostrophic wind forcing, a ten-meter square wind in this example, the wind intensity close to the ground could strongly differ. On one hand, if you consider a quite rough configuration, such as an urban area with tall buildings, the vertical extend of the bottom boundary layer will be large. The wind at 50 meter will be significantly reduced by the bottom friction. On the other hand, for a quite smooth bottom, such as the sea surface or fields, the bottom boundary layer will be confined in a thin layer close to the ground. And in such case, a 50-meter wind speed will be closer to the geostrophic winds and could reach a value up to eight meters per second. Hence, we've seen that the vertical profile of the average horizontal wind speed strongly depends on the bottom roughness, and we could wonder if some universal curb exists for the velocity profiles in the bottom boundary layer. We plot here the mean vertical profile of an oceanic boundary layer forced by the tides. As for the atmospheric boundary layer, the bottom roughness strongly impacts the velocity profiles. And quite surprisingly, when we re-scale the velocity with vertically average speed, most of the girls collapse on a single one for distinct directions have off load, and various flow intensity spring on the tides, it seems that we get a generic velocity profile for the bottom boundary layer. The main goal of the overall session will be to understand how we can derive such a universal velocity profile for a highly turbulent flow. And the Reynolds decomposition is the first step to achieve this goal. The underlying assumption is that if the instantaneous flow is unpredictable, there exists a mean flow. In other words, a mean velocity profile which is predictable. We showed you several distinct observations of the turbulent velocity profiles along a flat bottom. And here, if we merge all these observations, we are able to extract an average profile and separate the turbulent flow from this average. However, we consider here a statistical mean and not a temporal mean. You may ask, what is a statistical mean? The statistical mean corresponds to an average of a large number of realizations of the same repeated event. For instance, you restart several times the same experiment with the same initial conditions. And the properties of the statistical mean, the mean of the travel and velocity filtration is equal to zero. The mean of the product of two mean velocity is equal to the product of the means. The statistical average of a mean velocity times a turbulent velocity is equal to the product of the mean velocity times the mean of the turbulent filtration, and therefore equal to zero. The statistical mean of the project of two turbulent velocity is not equal to zero. And finally, the mean of any derivatives will be equal to the derivatives of the mean variables. To sum up, as far as the boundary layer is concerned, the Reynolds decomposition splits the instantaneous velocities in a mean path, the boundary layer profile, which mainly depends on the altitude and the turbulent velocity fluctuations. The spectral gap allows us to use the temporal average to perform the splitting. Thank you.