[MUSIC] We have seen that the flow within the atmospheric or the oceanic boundary layer is highly turbulent. If we use an appropriate averaging, we are able to split the velocity in two parts. A mean or horizontal velocity that will mainly depends on the entity of z, and random turbulent fluctuations you had. The mean of these fluctuations vanishes but not the average of high such as the average of two distant components of the velocity fluctuations. Hence, a turbulent closure is a set of equations that express the i order moments of the turbulent fluctuations, as a function of the mean velocity field. We will see in what follows the simplest closure that we can use, which is often called eddy viscosity, or turbulent viscosity. To go further, I first recall the Reynolds-averaged Navier-Stokes equations. The left-hand side corresponds to the inviscid Euler equations applied to the mean flow. While the right-hand side describes the impact of turbulent velocity filtrations. The eddy viscosity concept is simple. It assumes that the turbulent dissipation can be described by a laplace operator, as for the molecular viscosity. We get here a set of equation formerly identical to the logistic equation. But, we have a new coefficient called eddy of turbulent viscosity. This formal equation implies that, the first order of the relative of the non-linear turbulent fluctuations are equal to the laplace operator. And we get the following equations. Let's try to find the closure of relation from a simple boundary layer. We consider here a typical boundary layer or profile where the horizontal velocity depends on the altitude deep. This profile was measured in a coastal channel, and the horizontal flow is forced here by the tides. We assume, that the flow is parallel and uniform along the x-axis, how are you going to write the closure of relations assuming a constant eddy viscosity coefficient? Take a paper, find the solution and write it down. Then answer the following quiz. When you use the eddy viscosity closure for uniform and parallel boundary layer, you get this simple relation. The statistical average of the product of the horizontal velocity times the vertical velocity fluctuations is proportional to the vertical shear of the mean velocity. What does it mean? Why there is a minus sign? To gain some physical insight into this closure relation, let's consider the mean shear flow of the boundary layer, plus a single eddy which induces small velocity filtrations. Then, we know that the momentum of any fluid parcel, will be conserved by the eddy advection. If the eddy induces a downward motion. The velocity fluctuation w is negative, and the horizontal velocity fluctuation will be positive. The difference between the momentum of the mean shear and the fluid parcel momentum is indeed positive. Therefore, the product of velocity fluctuations u times w will always be negative. Conversely, if the eddy induces an upward motion, the velocity fluctuation w, is now positive. And the horizontal velocity fluctuations, will then be negative. And here again, the product of velocity fluctuations u times w, will be negative. In both case, this term is negative, which means that the vertical flux of horizontal momentum induced by the turbulent filtration is negative. It makes sense because, the main source of the vertical shear of the horizontal velocity is the bottom friction, which is opposite to the main flow direction. The turbulent filtration induce a vertical aggression of this negative momentum flux. The intensity of this vertical eviction is quantified here by the eddy viscosity parameter. However, a question still ignites, how can we quantify this turbulent viscosity coefficient? Do we need to measure precisely the turbulent viscosity filtrations? In fact, no, we will see in what follows that local measurements are not needed, and an integral approach is sufficient. Let's create for a few minutes the atmospheric boundary layer and consider a simple laboratory experiment. Let me make a river or a tidal flow over a rough bottom. As far as the bottom roughness is the main source of turbulent filtrations, there is indeed a lot of similarity between the atmospheric and the marine monolayers. We will now apply the concept of turbulent eddy viscosity for a simple hydraulic configuration. Keep your concentration few more minutes. You can see on this picture, an open channel flow. The bright line corresponds to the free surface, and is slightly tilted over the flat, and rough bottom. If we assume the Hydrostatic balance, the slope of the free surface, is directly proportional to horizontal pressure radiant. We, then consider a steady, unidirectional, and parallel flow. The eddy viscosity closure with a constant coefficient, and the Reynolds-averaged Navier-Stokes equations. According to the available assumptions, the later leads to a simple linear equation. In order to solve this second order differential equation, we need two indefinite boundary conditions on the velocity profile. One standard condition is fixed by the no sleep condition at the bottom, the velocity should vanish at the bottom. The second condition is a bit more tricky. There is no force or any external stress on the free surface, and therefore, the vertical flux of the horizontal momentum is zero. In other words, the vertical derivatives of the mean horizontal velocity should vanish at the poiseuille phase. Hence, the solution of this simple ODE leads to a parabolic velocity profile, which corresponds to a half positive flow, we then integrate the velocity along the vertical to get the flow rate and with some basic algebra leads to a simple relation. That gives the Eddy viscosity coefficient as a function of global quantities, such as the flow rate, the water depth, H, and the free surface slope, s. We can easily quantify the value of this turbulent viscosity for a large scale guide on channel for instance. And we get an eddy viscosity, which is save all the magnitude larger than the molecular viscosity, which make sense, because the turbulent filtrations are much more efficient to transport the momentum rather than molecular chokes. Hence, the eddy viscosity closure is a very simple concept. You just have to replace the molecular viscosity by another value, the eddy viscosity. Which could be a estimated at a global scale. However, if now we look at the boundary layer profile, the eddy viscosity closure provides a poiseuille profile plotted here with a solid line, which differs significantly from the real instrument measurements, the small dots. The simple turbulent closure do not change the dissipation operator and therefore, the boundary layer profile is similar to viscous profile. The eddy viscosity concept, could be use as a first guess to estimate an equivalent turbulent viscosity. But, to get a realistic boundary layer, we should use a more elaborated turbulent closure. The next step will be to use the pond closure, which leads to a logarithmic boundary layer profile.
