  In this paper, we study the 2D free boundary incompressible Euler equations
with surface tension, where the fluid domain is periodic in $x_1$, and has
finite depth. We construct initial data with a flat free boundary and
arbitrarily small velocity, such that the gradient of vorticity grows at least
double-exponentially for all times during the lifespan of the associated
solution. This work generalizes the celebrated result by
Kiselev--{\v{S}}ver{\'a}k to the free boundary setting. The free boundary
introduces some major challenges in the proof due to the deformation of the
fluid domain and the fact that the velocity field cannot be reconstructed from
the vorticity using the Biot-Savart law. We overcome these issues by deriving
uniform-in-time control on the free boundary and obtaining pointwise estimates
on an approximate Biot-Savart law.
