a negative temperature vortex to its energy content. The argument is based on the argument of Chorin.

The well-known power law for dissipation of energy with scale, $\langle E\rangle(k)\sim k^{-5/3}$, where $k$ is the wave number, can be derived by a scaling argument due to Kolmogorov and is also supported by Chorin’s filament model using results from a Monte Carlo simulation [@chorin-akao91; @chorin].

Chorin’s filament model can be applied to analyze a vortex tube in a sparse, homogeneous suspension of tubes. Consider a narrow and straight enough vortex tube $\mathcal{T}$ that has a center line $\mathcal{C}$, parametrized by $s$, and cross sections through $\mathcal{C}(s)$, denoted by $\mathcal{S}(s)$, orthogonal to the center line and such that $\mathcal{S}(s_1)$ and $\mathcal{S}(s_2)$ do not intersect for $s_1\ne s_2$.

Given a point ${\bf x}$ in the vortex tube and $r>0$, we define the ball $B_r({\bf x})=\{{\bf x'}\colon|{\bf x'}-{\bf x}|<r\}$. We take $r$ small enough so that $B_r({\bf x})$ contains no points that belong to other vortex tubes in the suspension. We denote by $\Sigma(s)$ the part of the cross section $\mathcal{S}(s)$ inside $B_r({\bf x})$ and by $\mathcal{C}_r$ the part of the center line of the vortex tube for which $\Sigma(s)$ is non-empty, i.e., $$\Sigma(s)
  =
  