 Solution

In the Townsend-Betchov cascade cartoon, bi-axial strain sweeps distributed vorticity into extended sheets at the upper part of the inertial range. These are predominantly inviscid structures: their shape and instability arise from advection and strain rather than local viscous smoothing. Strong velocity variation across a thin sheet permits Kelvin-Helmholtz instability, roll-up and three-dimensional deformation. The resulting small vortex tubes are stretched into still finer cores.

At the dissipative end, stretching and kinematic viscosity balance in Burgers-like cores. A core of strain rate $s$ has radius of order $(\nu /s{)}^{1/2}$; taking $s\sim (ϵ/\nu {)}^{1/2}$ gives the Kolmogorov length $({\nu }^3/ϵ{)}^{1/4}$. Circulation and shrinking cross-section allow finite integrated tube viscous dissipation. Thus sheet formation and tube formation play different roles, rather than the entire cascade being a succession of identical geometrical eddies.

The cartoon challenges a literal, uniformly self-similar, step-by-step cascade with independent interactions only between neighboring scales. Large-scale sheet formation can generate much smaller scales through an instability, and the resulting structures retain coherent spatial organization. A branching model that assumes every scale-transfer step has the same geometry and loses information about the parent flow can miss this mechanism. Closures that erase the relevant phase correlations also have difficulty representing its concentrated gradients and intermittency. The objection is to these dynamical assumptions, not to conservation of energy or the existence of a mean energy flux through scales. The cartoon alone is not a proof that all spectral transfer is nonlocal.