Rigorous characterization of boundary layer separations

This chapter focuses on a rigorous characterization of boundary layer separations of 2-D incompressible fluid flows and its applications to geophysical fluid dynamics. In the boundary layer, the shear flow can detach/separate from the boundary that generates bubbles and leads to more complicated turbulent behavior. It is important to characterize the separation. It is observed experimentally that the points where the normal derivative of the velocity field vanishes may just be the separation point. No known theorem is available that can be applied to determine the separation until the recent work that provides a first rigorous characterization. The chapter also emphasizes on the main results and tools used for this characterization. The characterization is based on the structural bifurcation theory for a parameter family of divergence-free vector fields. From the mathematical point of view, the structural bifurcation theory provides a natural tool for characterizing the boundary layer separations of incompressible flows.