I will discuss amplification mechanisms in incompressible flows generated by geophysical forcing (gravity) and by nonlinear effects (vortex stretching). 

In the first part, I will introduce Elgindi’s decomposition of the Biot-Savart law in C^α (Elgindi, Annals of Math. 2021; Kiselev & Svérak, Annals of Math. 2014) and show how it can be used to determine the precise pointwise growth of solutions of the 2D Boussinesq equations for short times. This is based in part on joint work with Hientzsch and Iandoli.

In the second part, I will present a proof of finite-time blow-up for C^∞ solutions of a class of stretching equations. This proof introduces a new virial-type method that is, in a certain sense, inspired by the same decomposition. It is based on a joint work with Elgindi.