The second variation of the stream function energy of water waves with vorticity

We compute the second variation of the stream function energy of two-dimensional steady free surface gravity water waves with vorticity in the stream function formulation. We prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with Stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point. The particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy.