Feedback on vertical velocity . Rotation , convection , self - sustaining process

We have shown in the previous lectures that the −vU ′ term in the u equation, that is ∂tu = −vU ′ + · · · , is the key term leading to momentum transport −uv ∼ vvU ′t and perturbation energy production −uv U ′ ∼ vv(U ′)2t > 0. This term is the redistribution of streamwise velocity that releases energy from the background shear to enable bifurcation to turbulent flow. However, in shear flows we have not yet identified a mechanism that can feedback from the u fluctuation to v, thus v creates just the right u through the −vU ′ advection term, but how is v sustained? In this lecture, we first review two linear mechanisms of feedback on v involving extra physics, (1) through the Coriolis term in rotating shear flow, (2) through buoyancy in Rayleigh-Bénard convection. We derive the famous Lorenz model for convection, then consider a similar model for shear flows that illustrates the mechanisms involved in the nonlinear feedback from u to v, yielding a self-sustaining process for shear flows v → u → · · · → v. This is the model that was already discussed in lecture 1.