Published: April 10, 2018

New Integrals of Motion and Singularities in 2D Fluid Dynamics with Free Surface

We study the problem of 2D incompressible fluid dynamics with free surface, we assume  the the fluid is ideal and the flow is potential. Following the conformal mapping  technique we reformulate the problem to surface variables and demonstrate the existence  of previously undiscovered constants of motion associated with singularities in the  analytic continuation of conformal map and complex potential. In numerical simulations we recover the analytic structure of the surface shape and observe simple poles and  branch point singularities of the square-root type. We use the Alpert-Greengard-Hagstrom method to recover the location, type and magnitude of the singularities. We show how  the approach of square-root type singularities may be responsible for the breaking of waves in the ocean, following the nonlinear stage of modulational instability.