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In established three-dimensional turbulence, mean vortex stretching is positive: $⟨{\omega }_i{\omega }_j{S}_{ij}⟩>0$. The Betchov relation then gives $⟨abc⟩<0$. Bi-axial strain has two positive principal strain rates and one negative rate, hence $abc<0$; extension along one axis with compression along the other two instead gives $abc>0$.

Thus the cubic strain statistic favors bi-axial strain. Strictly, this is an amplitude-weighted preference: a negative average alone cannot prove that there are more events of one type, since a few strong events could dominate it. Positive mean vortex stretching is additional physical information, not a consequence of homogeneity alone.

Bi-axial strain spreads a nearly incompressible material volume along two directions while thinning it along the third, favoring sheet-like concentrations of vorticity. A sheet can then become unstable and roll up into vortex tubes. The Townsend-Betchov cascade cartoon uses this sequence to reconcile strain-favored sheets with the intense tubes observed at dissipative scales. Alignment of vorticity with the principal strain rates also affects the stretching; the determinant statistic does not specify the entire geometry.

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.

The Townsend-Betchov cascade cartoon organizes inertial range vorticity into sheets by bi-axial strain, then uses sheet instability and roll-up to generate finer vortex tubes. Dissipative tube cores balance stretching with kinematic viscosity. Such differing mechanisms across scales challenge literal identical-step, memoryless cascade models without contradicting a mean energy flux.