  Kelvin's Theorem on conservation of circulations is an essential ingredient
of G. I. Taylor's theory of turbulent energy dissipation by the process of
vortex-line stretching. In previous work, we have proposed a nonlinear
mechanism for the breakdown of Kelvin's Theorem in ideal turbulence at infinite
Reynolds number. We develop here a detailed physical theory of this ``cascade
of circulations''. Our analysis is based upon an effective equation for
large-scale ``coarse-grained'' velocity, which contains a turbulent-induced
``vortex-force'' that can violate Kelvin's Theorem. We show that singularities
of sufficient strength, which are observed to exist in turbulent flow, can lead
to non-vanishing dissipation of circulation for an arbitrarily small filtering
length in the effective equations. This result is an analogue for circulation
of Onsager's theorem on energy dissipation for singular Euler solutions. The
physical mechanism of the breakdown of Kelvin's Theorem is diffusion of lines
of large-scale vorticity out of the advected loop. This phenomenon can be
viewed as a classical analogue of the Josephson-Anderson phase-slip phenomenon
in superfluids due to quantized vortex lines. We show that the circulation
cascade is local in scale and use this locality to develop concrete expressions
for the turbulent vortex-force by a multi-scale gradient-expansion. We discuss
implications for Taylor's theory of turbulent dissipation and we point out some
related cascade phenomena, in particular for magnetic-flux in
magnetohydrodynamic (MHD) turbulence.
