[MUSIC] Hello. The impact of the turbulent velocity fluctuations on the mean velocity profile is quantified by the turbulent closure relations. These mathematical relations express the statistical mean of the velocity fluctuations as a function of the mean velocity and, or. We have seen that the Eddy Viscosity closure leads to a simple relation. Where? The turbulent viscosity fluctuations are directly proportional to the sheer of the main velocity profile. The Eddy Viscosity closure is based on the simple concept. The turbulent velocity fluctuations will do that same job as the molecular diffusion. And therefore, we could replace the molecular viscosity by another coefficient, the Eddy Viscosity which is several order of magnitude larger. However, the Eddy Viscosity closure will always provide a Poiseuille profile plotted here with a solid line. And this profile differs significantly from the real institute measurements. Here, the small dots. Hence, other closure relation should be used to get more realistic profiles of the atmospheric or the oceanic [INAUDIBLE] layer. To suggest a better closure scheme, let's come back to the dynamical origin of turbulent velocity fluctuations. These fluctuations are due to the presence of small coherence structure such as vortices within the flow. But where do these structures, these vortices come from? They are produced by the instability or various cascading of instabilities of the large scale flow. And to better understand the phenomena, let's have a look to a classical shear instability. We plotted on this graph the special evolution along the horizontal axis of the vorticity plotted here with blue contours, associated to a non-stable shear flow. So upstream flow correspond to a simple velocity sheer localized arpund Z equals zero. We can see that if the vorticty is parallel at the entrance, downstream, this vorticity sheet starts to meander and splits in several entities. And four, the simplified two dimensional flow. The vorticity is a scalar which is equal to the curl of the velocity. Besides, if you consider an inviscid and purely two-dimensional flow, the vorticity companion satisfy the Lagrangian conservation. In other words, if you follow any free vessel, it's vorticity will stay constant along its trajectory. It means that the vorticity in the core of the small scale comes from the vorticity of the upstream shear flow. This results could be generalized to a more complex three dimensional flow if the mean flow remains uniform along the horizontal axis. On average, the vorticity of the small scale turbulent Eddy's will be equal to the mean flow vorticity. Hence, the vorticity of the mean boundary layer will drive the vorticity of the small scale turbulent fluctuations. If we consider a parallel flow for the boundary layer profile, the mean flow vorticity will be equal to the vertical derivative of the mean horizontal velocity. On the other hand, if we assume that the turbulent fluctuations correspond to small scales eddies, the vorticity of these Eddies will correspond to the curl of the velocity fluctuations. And now if we consider that these eddies have a typical length L, the vorticity of the turbulent fluctuations will scale as the ratio of the velocity fluctuations divided by L. And when we apply the vorticity conservation. We found that the vertical shear of the mean flow should be proportional to the amplitude of the velocity fluctuations. And we can then suggest the closure relations, where the statistical average of the product of the velocity fluctuations Is proportional to the square of the vertical velocity shear. This closer relation was put forward by a German physicist Ludwig Prandtl at the beginning of the 20th century. However, there is still an unknown value there, the typical length of the turbulent eddies. These characteristic scale is also called the mixing length. Then, Von Karman makes a very intuitive hypothesis, and suggested that the size of the turbulent eddies should depends on the distance to the boundary. Close to the bottom, the eddies are small and they could expand in size only if they are far away from the boundary. He assumes that the length of the turbulent eddies increases linearly with z, the distance to the boundary. Hence, we get a simple closure equation which depends on z and the vertical derivatives of the mean horizontal velocity. The second underlying hypothesis of the Prandtl closure is that the vertical flux of the horizontal momentum remains constant above the boundary. The addition of these two relations leads to a first order differential equation that have for general solution, a logarithmic velocity profile. Such logarithmic profiles were confirmed within institute measurements for many geophysical boundary layers. Here is an example for a marine boundary layer. Nevertheless the specificity of the logarithmic function is that it diverges towards minus infinity when z tends to zero. Hence, to get a physical solution, the bottom boundary condition Vx equals 0 should be reached at a given height, Z0, slightly above the bottom. For atmospheric or oceanic boundary layers, this characteristic height is related to the bottom roughness Indeed. Close to the boundary, the minimal size of the turbulent fluctuation is bounded by the bottom roughness. And Z0 increases with the bottom roughness. The smallest value below the centimeter scale are found for smooth and flat surfaces. While the largest value above 1 meter are obtained, for instance over urban area. To sum up, the dynamic of the vorticity was used to build a new turbulent closure. Which assumed that the mixing lens associated to small scale turbulent eddies increases with the distance to the boundary. This specific closure leads to logarithmic profiles for the mean flow velocity within the boundary layer. These profiles are controlled by a single a parameter, the bottom roughness. Such logarithmic profiles are often observed in the atmosphere or the ocean above a rough bottom. However, at the same location, various profiles could be observed. Here is an example of two wind profiles taken at different times. The black dot corresponds here to an intense wind flow and is in correct alignment with a logarithmic load. On the other hand the white dots correspond to a weaker flow and the log linear graph shows that we can hardly fit this profile with a logarithmic load. We will see that for the atmospheric boundary layer, the logarithmic flow is not a universal paradigm. And that another physical phenomena should be taken into account to describe more accurately the wind profiles. Thank you.
