We present a mechanism for finite time blow-up in the
two-dimensional incompressible Boussinesq
equations, achieved through a multi-layer degenerate pendula dynamics with a uniform $C^{1, \alpha}$ forcing term.
The construction gives rise to a vortex layer cascade in which
vorticity and temperature gradients concentrate successively, leading to singularity formation in finite
time. We will outline the key ideas behind the cascade mechanism and discuss its implications for singularity formation in incompressible fluid equations. These results are joint works with Luis Martínez-Zoroa, Andrés Laín Sanclemente and Fan Zheng.
