  This paper presents calculations of the Castaing (Physica D, 46, 177, 1990)
cascade kernels for five well-known models of the turbulent cascade and
demonstrates that these kernels provide conceptually simple and direct
descriptions of turbulence models equivalent to their multifractal spectra. The
problem with a log-normal model seems to be that its Castaing kernel predicts a
small but non-zero probability that finite-amplitude turbulence will decay to
arbitrarily large turbulent amplitudes at smaller scales. Kernels for
log-Poisson models do not have this pathology. It is shown how kernels evolve
when the cascade model itself varies with scale. Future studies of other
kernels compatible with and even dictated by detailed fluid physics may assist
understanding of the turbulent cascade.
