\mathcal{S}(s)\cap B_r({\bf x})
  \quad\text{and}\quad
  \mathcal{C}_r
  =
  \left\{\mathcal{C}(s):\ \Sigma(s)\ne\emptyset\right\}.$$

In order to compute the energy spectrum, Chorin defines, for $r>0$, the vorticity correlation integral $$S_r
  =
  \left\langle
    \int_{\mathcal{T}\cap B_r({\bf x})}\boldsymbol\omega({\bf x})\cdot\boldsymbol\omega({\bf x'})\,d\mathcal{H}_\mathcal{T}
  \right\rangle,$$ where $d\mathcal{H}_\mathcal{T}$ denotes the appropriate Hausdorff measure, related to the set capacity on $\mathcal{T}$, and the average is taken over the ensemble of all possible configurations. Then, using disintegration of measure [@schwartz76], we have $$S_r
  =
  \left\langle
    \boldsymbol\omega({\bf x })\cdot\int_{\mathcal{C}_r}\,ds\int_{\Sigma(s)}\boldsymbol\omega({\bf x'})\,d\mathcal{H}_\Sigma
  \right\rangle.$$ If vorticity is roughly uniform throughout the cross section $\Sigma(s)$ so that $\boldsymbol\omega({\bf x'})\approx\boldsymbol\omega(s)$, and if $|\Sigma|(s)=\mathcal{H}_\Sigma(\Sigma(s))$ denotes the Hausdorff measure of $\Sigma(s)$, then $$S_r
  \approx
  \left\langle
    \boldsymbol\omega({\bf x})\cdot\int_{\mathcal{C}_r}|\Sigma|(s)\boldsymbol\omega(s)\,ds
  \right\rangle.$$ Assuming that the Hausdorff dimension of $\Sigma(s)$ remains constant throughout $\mathcal{T}\cap B_r({\bf x})$ and denoting it by $D_{\Sigma}$, we obtain $$S_r
  =
  \mathcal{O}(r^{D_{\Sigma}+1}).$$ To obtain the vorticity spectrum, $Z(k)$, we integrate the Fourier transform of $S_r$ over a sphere of radius $k=|{\bf k}|$. This gives $Z(k)=\mathcal{O}(k^{-D_{\Sigma}+1})$, and, consequently, the energy spectrum satisfies $$\label{eq:e-k-ksq}
  \langle E\rangle(k)
  =
  Z(k)/k^2
  =
  \mathcal{O}(k^{-D_{\Sigma}-1}).$$

Let $D_c$ be the dimension of the center line $\mathcal{C}$ of the vortex, $D_\Sigma$ the dimension of the vortex cross section, and $D$ the dimension of the support of the vorticity in the vortex filament $\mathcal{T}$. We 