A Multiscale Camassa--Holm Equation

A system of equations for Multiscale Geodesic Flow (MGF) is introduced whose solutions illustrate the paradigm of whorls within whorls within whorls, introduced by L. F. Richardson in 1922 to describe the cascade of energy in fluid turbulence. Numerical simulations are given for MGF on $S^1$, where the MGF equation comprises a multiscale generalisation of the Camassa-Holm (CH) equation whose emergent singular solutions generalise the peakon solutions of the CH.