  The Kelvin-Helmholtz theorem on conservation of circulations is supposed to
hold for ideal inviscid fluids and is believed to be play a crucial role in
turbulent phenomena, such as production of dissipation by vortex
line-stretching. However, this expectation does not take into account
singularities in turbulent velocity fields at infinite Reynolds number. We
present evidence from numerical simulations for the breakdown of the classical
Kelvin theorem in the three-dimensional turbulent energy cascade. Although
violated in individual realizations, we find that circulations are still
conserved in some average sense. For comparison, we show that Kelvin's theorem
holds for individual realizations in the two-dimensional enstrophy cascade, in
agreement with theory. The turbulent ``cascade of circulations'' is shown to be
a classical analogue of phase-slip due to quantized vortices in superfluids and
various applications in geophysics and astrophysics are outlined.
