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SUMMARY:Dynamics of singularities and wavebreaking in 2D hydrodynamics wit
 h a free surface - Prof Pavel Lushnikov\, University of New Mexico
DTSTART:20170608T140000Z
DTEND:20170608T150000Z
UID:TALK72971@talks.cam.ac.uk
CONTACT:Prof Natalia Berloff
DESCRIPTION:2D hydrodynamics of ideal fluid with a free surface is conside
 red. A\ntime-dependent conformal transformation is used which maps a free\
 nfluid surface into the real line with fluid domain mapped into the\nlower
  complex half-plane. The fluid dynamics is fully characterized\nby the com
 plex singularities in the upper complex half-plane of the\nconformal map a
 nd the complex velocity. The initially flat surface\nwith the pole in the 
 complex velocity turns over arbitrary small\ntime into the branch cut conn
 ecting two square root branch points.\nWithout gravity one of these branch
  points approaches the fluid\nsurface with the approximate exponential law
  corresponding to the\nformation of the fluid jet. The addition of gravity
  results in\nwavebreaking in the form of plunging of the jet into the wate
 r\nsurface. The use of the additional conformal transformation to\nresolve
  the dynamics near branch points allows analyzing\nwavebreaking in details
 . The formation of multiple Crapper capillary\nsolutions is observed durin
 g overturning of the wave contributing to\nthe turbulence of surface wave.
  Another possible way for the\nwavebreaking is the slow increase of Stokes
  wave amplitude through\nnonlinear interactions until the limiting Stokes 
 wave forms with\nsubsequent wavebreaking. For non-limiting Stokes wave the
  only\nsingularity in the physical sheet of Riemann surface is the\nsquare
 -root branch point located. The corresponding branch cut\ndefines the seco
 nd sheet of the Riemann surface if one crosses the\nbranch cut. The infini
 te number of pairs of square root\nsingularities is found corresponding to
  an infinite number of\nnon-physical sheets of Riemann surface. Each pair 
 belongs to its own\nnon-physical sheet of Riemann surface. An increase of 
 the steepness of\nthe Stokes wave means that all these singularities simul
 taneously\napproach the real line from different sheets of Riemann surface
  and\nmerge  forming 2/3 power-law singularity of the limiting\nStokes wav
 e. It is conjectured that non-limiting Stokes wave at the\nleading order c
 onsists of the infinite product of nested square root\nsingularities which
  form the infinite number of sheets of Riemann\nsurface. The conjecture is
  also supported by high precision\nsimulations\, where a quad (32 digits) 
 and a variable precision (up\nto 200 digits) were used to reliably recover
  the structure of square\nroot branch cuts in multiple sheets of Riemann s
 urface.\n
LOCATION:MR11\, CMS
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