[MUSIC] Hello, the knowledge of the vertical velocity profiles is often crucial for a correct assessment of the kinetic energy available for wind or marine turbines. But in several cases, the velocity measurements are restricted in the vertical. For instance, if you use a 40 meter [INAUDIBLE] to measure the wind speed, you need to extrapolate these measurements at the hub height, which could be around 100 meters or even higher. It is therefore tempting to assume the universal law for the boundary layer. And use some generic profiles to extrapolate the vertical distribution of the velocity. The logarithmic law is one of the most famous relation used to describe the mean velocity distribution in a turbulent layer. It works quite well for our flood bottom, when the surface layer is naturally stable, and not too strongly stratified. This turbulent profile is then controlled by a single parameter, the bottom of this z0. This logarithmic law is derived from the dynamical properties of turbulent flow and it takes into account some physical assumptions. The main assumption is that the turbulence intensity is controlled only by the amplitude of the shear flow and the bottom roughness. This roughness coefficient is then a constant value, which depends only on the local characteristic of the ground or the sea floor. These curves correspond to various tidal flow profiles. Measured at the same location, but at different times. All these profiles are rescaled by the vertically average velocity. We can see that for this [INAUDIBLE], Ebb in blue or Flood in red. And values flow intensities, Spring or Neap tides, all the velocity profiles collapse on the same logarithmic feet having a constant bottom roughness. This example shows that logarithmic law could be appropriate for a marine or a river flow. It works quite well especially when the surrounding sea floor is relatively flat and the bottom roughness is uniform. However, if we consider an atmospheric boundary layer, the situation could be much more complex. Indeed, the convection induced by the solar heating brings another source of turbulent fluctuations that may affect the velocity profile close to the surface. We should then take into account the development of the convective boundary layer during daytime. The heat source at the ground triggers a convective instability, which leads to a well mixed layer. Small and patchy cumulus clouds are the usual signature of this convective motion. Corrective terms could be added to the logarithmic law in order to take into account the turbulent convection. Monin-Obukov introduces a typical length scale, which is the ratio of the cube of the friction velocity u* divided by the surface buoyancy flex B0. This characteristic length correspond to the height at which turbulence is generated more by buoyancy than by the wind shear. Then, a similarity theory combined to institute that asset provides the shape of corrective function psi. And generally, we get quite complex calculations. Facing this increasing complexity, an empirical power law was suggested to fit the vertical distribution of the mean horizontal velocity. The Power Law relationship is often used as a substitute for logarithmic law when surface softness or stability information is not available. It gives a simple relationship between the wind speed at one height and those at another. We generally use the ten meter wind speed as the reference level. The standard height prescribed by the World Meteorological Organization, the WMO, for measuring open wind speed above a flat count is indeed ten meters. Such power law is control by a single parameter. The wind shear exponent a. Engineers do like this power law because it is very simple and convenient to use. However, you should keep in mind that it is an empirical relation, which have no physical basis. Nevertheless, the power fit may works as you can see on this mean wind profile, they can, in Summer, at the middle of the day. The wind shear exponent is quite small and equal to 0.12. But later on in the early evening, we need to fix the wind shear exponent at 0.5 to fit approximately the wind data. Hence, the wind shear exponent is expected to vary with the atmospheric stability. And therefore, the time of the day or the seasons for conductivity in stable configuration, the vertical shear of the horizontal velocity should be weak. And typical value of wind shear exponent are close to 0, generally, below 0.15. Around the neutral stability, the standard value of 1 over 7 provides a correct feet. While for a stable and stratified atmospheric boundary layer, the wind shear exponent exceed 0.2 and could reach as value up to 0.6. The bottom roughness will also impact the value of the wind shear exponent. And we can found typical values of 0.1 for very smooth surface. And 0.25 for a high forest or a large urban area. The power law is often used to extrapolate the wind speed from the reference level to a higher altitude. Moreover, Justus and Mikhail have demonstrated that the Power Law relationship also provides a direct relation for the height variation of the Weibull wind speed distribution. If we assume a constant value of the wind shear coefficient, the shape parameter k, and the scale parameter C, of the Weibull distribution could be simply transposed from the height z1 to the height z2 according to the following formulas. To sum up, various correlations are used to describe the velocity profile within the atmospheric or the oceanic boundary layers. For marine or [INAUDIBLE] layers, the bottom roughness have a major impact on the turbulence intensity and the velocity profiles could be correctly fitted with a logarithmic law. Why, for at mostly boundary layers, both the bottom roughness and the solar heating should be taken into account. Hence, there is a wider diversity of velocity profiles. So generalized logarithmic relations or [INAUDIBLE] law are the two most commonly used analytical models for extrapolating wind speed to higher heights. Although there is no universal analytic expression varied for all stability conditions. And these generic velocity profiles should be taken with care to extrapolate the wind resource from one height to another. Thank you.
