Quantum Turbulence: Vinen Vortex-Line Densities in Helium and Condensates Against the Shear Bound

A dense, random tangle of quantised vortices — quantum turbulence — is the most strongly sheared state a superfluid ordinarily reaches. This paper assembles representative vortex-line densities for helium-4 counterflow turbulence and for turbulent condensates, converts them to intervortex spacings and gradient scales, and compares the strongest case with the torsion shear ceiling. Strong helium counterflow reaches a line density of 1e+11 per square metre, corresponding to a spacing of 3.16 micrometres and a gradient ratio of 2.34e-11 to the ceiling; moderate counterflow gives a spacing of 31.6 micrometres. Even the turbulent state, in which the flow is nothing but vortex cores and their induced velocities, remains orders of magnitude below the fundamental shear scale, and its dynamics are governed by reconnections and Kelvin waves within classical-like cascade physics. The line densities are taken from the published turbulence record and converted here, not measured.