2D hydrodynamics of ideal fluid with free surface is addressed. A time-dependent conformal transformation is used which maps a
free fluid surface into the real line with fluid domain mapped into the lower complex half-plane. The fluid dynamics is fully
characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. The
initially flat surface with the pole in the complex velocity turns over arbitrary small time into the branch cut connecting two
square root branch points. Without gravity one of these branch points approaches the fluid surface with the approximate
exponential law corresponding to the formation of the fluid jet. The addition of gravity results in wavebreaking in the form of
plunging of the jet into the water surface. The use of the additional conformal transformation to resolve the dynamics near
branch points allows to analyze wavebreaking in details. The formation of multiple Crapper capillary solutions is observed during
overturning of the wave contributing to the turbulence of surface wave. Another possible way for the wavebreaking is the slow
increase of Stokes wave amplitude through nonlinear interactions until the limiting Stokes wave forms with subsequent
wavebreaking. For non-limiting Stokes wave the only singularity in the physical sheet of Riemann surface is the square-root
branch point located. The corresponding branch cut defines the second sheet of the Riemann surface if one crosses the branch cut.
The infinite number of pairs of square root singularities is found corresponding to infinite number of non-physical sheets of
Riemann surface. Each pair belongs to its own non-physical sheet of Riemann surface. Increase of the steepness of the Stokes wave
means that all these singularities simultaneously approach the real line from different sheets of Riemann surface and merge
together forming 2/3 power law singularity of the limiting Stokes wave. We found that that non-limiting Stokes wave at the
leading order consists of the infinite product of nested square root singularities which form the infinite number of sheets of
Riemann surface [1,2].

[1] P.M. Lushnikov. Structure and location of branch
point singularities for Stokes waves on deep water. Journal of
Fluid Mechanics, v. 800, 557-594 (2016).

[2] S.A. Dyachenko, P.M. Lushnikov, and A.O. Korotkevich.
Branch cuts of Stokes wave on deep water. Part I: Numerical
solution and Pade approximation. Studies in Applied Mathematics,
v. 137, 419-472 (2016).