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It is often quoted (see https://doi.org/10.1016/j.piutam.2013.09.008) that the separation $\delta$ between quantised vortices in a vortex tangle in a superfluid is approximately

$$\delta \approx L^{-1/2}$$

where $\delta$ is the separation between the quantised vortices and $L = \frac{\Lambda}{V}$ is the vortex line density.

Where does this approximation come from? How can it be derived?

My initial thoughts are to use the Onsager-Feynman rule $n = \frac{2\Omega}{\kappa}$ where $n$ is the number of vortex lines per unit area, $\Omega$ is the angular velocity and $\kappa$ is the quantum of circulation. But beyond that I am unsure.

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  • $\begingroup$ Dimensional analysis? What are the other scales in the problem? $\endgroup$ – Ryan Thorngren Apr 13 '18 at 17:02
  • $\begingroup$ I could use dimensional analysis, but I'd like to derive this algebraically e.g., by using the geometry of the vortex tangle. What do you mean by "other scales" in the problem? JG $\endgroup$ – Jack G Apr 13 '18 at 17:12
  • $\begingroup$ If you slice the tangle with a plane, the question is whether this plane intersects each tangle an O(1) number of times. If so, then the formula follows. The only way this could fail is if other length scales (maybe like typical extrinsic curvature of the vortex loops) enter the game. $\endgroup$ – Ryan Thorngren Apr 13 '18 at 17:18
  • $\begingroup$ Let's assume a basic model where there's no other length scales. What is O(1)? $\endgroup$ – Jack G Apr 13 '18 at 17:33
  • $\begingroup$ Order 1. That is, a constant that doesn't change much as we vary system parameters. Then the formula just follows from a scaling argument. The danger is that even in a simple model, length scales like typical extrinsic curvature can appear seeming out of nowhere (but are really reflecting some microscopic length scale). By the way above I mean the slice intersects a single loop an O(1) number of times. $\endgroup$ – Ryan Thorngren Apr 13 '18 at 17:55
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Q: "Where does this approximation come from?"

I couldn't find a free link for the originator's work, but here's the PayWall:

"Three-dimensional vortex dynamics in superfluid $^4$He: Homogeneous superfluid turbulence", by K. W. Schwarz, in Phys. Rev. B 38, 2398 – Published 1 August 1988

See also Andrew Baggaley's WikiDot page: "Quantum Turbulence (old)" and it's link: "Non-linear Dynamics Lab", and his new version of that webpage: "Quantum Turbulence".

Q: "How can it be derived?"

See: "Vortex density spectrum of quantum turbulence" (The .PDF at arXiv seems corrupted, try here.), by Roche, or "Velocity Statistics Distinguish Quantum Turbulence from Classical Turbulence" by Paoletti or "Coherent vortex structures in quantum turbulence", by Baggaley, page 2:

"The Biot–Savart interactions of vortex lines over length scales larger than the average inter-vortex distance $ℓ ≈ L^{−1/2}$ has induced the same Kolmogorov energy spectrum $E(k) ∼ k^{−5/3}$ (for $k ≪ k_ℓ = 2π/ℓ$) which is observed in ordinary turbulence. The Kolmogorov spectrum was also observed in experiments with turbulent super-fluid helium, and in calculations performed with both the vortex filament mode and the Gross–Pitaevskii equation".

Each of those works contain numerous formulas and references. Unfortunately I don't have time to MathJax all the details ATM. I can revisit this in a few hours if needed.

A further reference is the "Quantum Turbulence" section of "Vortex filament method as a tool for computational visualization of quantum turbulence", by Risto Hänninen and Andrew W. Baggaley, page 4:

"Counterflow Turbulence

The earliest experimental studies of QT were reported in a series of groundbreaking papers by Vinen in the 1950s. In these experiments, turbulence was generated by applying a thermal counterflow, in which the normal and superfluid components flow in opposite directions. This is easily created by applying a thermal gradient, e.g., by heating the fluid at one end. The most common diagnostic to measure is the vortex line density, $L=Λ/V$, where $Λ$ is the total length of the quantized vortices and $V$ is the volume of the system; from this, one can compute the typical separation between vortices, the inter vortex spacing $ℓ≈L^{−1/2}$. This can readily be measured experimentally using second sound, and higher harmonics can probe the structure of the tangle. Numerical simulations have played a crucial role in visualizing the structure of counterflow turbulence and probing the nature of this unique form of turbulence; indeed, it has no classical analog. Some of the very earliest studies using the VFM were performed by Schwarz; however, computational limitations forced him to perform an unphysical vortex mixing procedure. A more recent study by Adachi et al. made use of modern computational power and studied the dependence of the steady-state vortex line density on the heat flux of the counterflow. Within the parameter range of the study, there was good agreement with experimental results, vindicating the use of the VFM for counterflow turbulence."

Again, the quoted text contains references to papers where further information on the derivations can be found.

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  • $\begingroup$ Thank you very much for the additional references. They've helped a lot! $\endgroup$ – Jack G Apr 21 '18 at 10:48

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