0. Introduction

1. Trying to bend the Burgers cylindrical vortex into a torus

2. Matching principle and anomalies in the Euler Hamiltonian

3. Topological Euler Flow

3.2. Boundary conditions at the surface and its edge

This surface is bounded by a singular line C where the angular velocity in the transverse plane diverges as $1/r$, preserving a finite circulation for an infinitesimal dual loop $\delta C$ encircling C (Figure 3).

There is only a tangent discontinuity of velocity at the surface $S_C\setminus C$, coming from the delta function in vorticity. The boundary values are

$$\begin{array}{ccc}& & \overrightarrow{\omega }\to 2\pi n\delta \left(z\right)\overrightarrow{\sigma }\times \overrightarrow{\nabla }{\varphi }_1;\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill & & {\overrightarrow{v}}^{+}-{\overrightarrow{v}}^{-}=2\pi n\overrightarrow{\nabla }{\varphi }_1;\hfill \end{array}$$

(43)

$$\begin{array}{ccc}& & S_3^{+}=S_3^{-};\hfill \end{array}$$

(44)

$$\begin{array}{ccc}& & {\varphi }_2^{+}={\varphi }_2^{-}-2\pi n;\hfill \end{array}$$

(45)

$$\begin{array}{ccc}& & {\omega }_n^{+}={\omega }_n^{-}=-\overrightarrow{\sigma }·\overrightarrow{\nabla }{\varphi }_2^{\pm }\times \overrightarrow{\nabla }{\varphi }_1^{\pm }\hfill \end{array}$$

(46)

Here $\overrightarrow{\sigma }$ is the local normal vector to the surface, and z is the normal coordinate.

The velocity gap must vanish at the edge of the surface, which requires the boundary condition

$$\begin{array}{c}\hfill \overrightarrow{\nabla }{S}_3\left(C\right)=0;\end{array}$$

(47)

On a simple surface with a boundary $\partial S=C$, as a consequence of incompressibility, the velocity would be directed along the edge, i.e., the velocity $v_n$ along the inner normal to the loop will vanish

$$\begin{array}{c}\hfill v_n=\overrightarrow{v}\left(\overrightarrow{C}\left(l\right)\right)·{\overrightarrow{C}}^{\prime \prime }\left(l\right)/\left|{\overrightarrow{C}}^{\prime \prime }\left(l\right)\right|\stackrel{?}{=}0\end{array}$$

(48)

However, our velocity is singular at the edge, and the Euler solution only applies outside a thin tube surrounding C. We must match this velocity with the Burgers solution at the surface of this tube (see Figure 1, Figure 2).

It remains a computational problem to build a full velocity field, at least for the simple example of a flat circle C; here, we just sketched the flow using its general properties, which follow from our equations. These parametric 3D plots were created in [13] with Mathematica^® by solving the model particle motion equation in a velocity flow with the Kelvinon topology and geometry.

The singular rotational part of Euler velocity ${\overrightarrow{v}}_{\theta }=\frac{{\Gamma }_B}{2\pi r}$ (matching the Burgers vortex inside) will contribute to the "normal inner" velocity $v_n$, at the junction with the tube; the result will depend on the angle $\beta$ in this plane normal to ${\overrightarrow{C}}^{\prime }$ and diverge in the Euler limit.

Together, the rotational and axial flow inside the tube and inside the vortex sheet results in a spiral motion, as shown in Figure 1. In Figure 1, the thin spirals inside the green vortex sheet are tangent to both sides. Each spiral’s big (alpha) cycle leads to vorticity normal to the vortex sheet, responsible for the circulation ${\Gamma }_C$. The small (beta) cycles of each spiral correspond to rotation around the local direction of the spiral, which corresponds to large tangent vorticity inside the vortex sheet.

In addition, there is an Euler flow outside the tube, which follows the Burgers spiral, but cannot penetrate the vortex sheet, as the normal velocity vanishes at both sides. Therefore, the rotational motion around the small cross-section of the torus does not go through the vortex sheet.

Instead, the flow turns and rotates, sliding along each side of the vortex sheet, as shown in Figure 2.

The external flow is sketched in Figure 2. Vorticity points in a local direction of each spiral; the rotation velocity is orthogonal to vorticity, and the potential part of velocity is parallel to vorticity, providing helicity. The tornadoes attached to the vortex sheet are connected to the spiral vortexes circling the tube. We have not shown this connection here.

One can see that the circulations ${\Gamma }_C,{\Gamma }_B$ have opposite signs, in agreement with the Kelvinon theory. The tornadoes can only move towards the vortex sheet as the negative normal strain $S_{nn}<0$ leads to the exponential decay of the normal distance $z\propto exp\left(t{S}_{nn}\right)$. Therefore, the tornadoes rotate clockwise, corresponding to negative circulation ${\Gamma }_C$, screwing the flow onto the surface in Figure 2.

The velocity in the center of the spiral flow around the tube also goes clockwise to provide negative ${\Gamma }_C$, but this makes the circular motion in the cross-section plane go anticlockwise, meaning positive circulation ${\Gamma }_B$. Note that the trajectories are tangent to the vortex sheet at the surface. Vorticity points in a local direction of each spiral; the rotation velocity is orthogonal to vorticity, and the potential part of velocity is parallel to vorticity, providing helicity.

Figure 1. Inner Navier-Stokes flow. The idealized spiral motion of liquid particles around the axial vortex line inside the vortex tube (transparent yellow) and vortex sheet (transparent green). The thickness of the vortex sheet and the vortex tube is magnified compared to the turbulent regime we are studying. We trace 25 particles with different colors.

Figure 1. Inner Navier-Stokes flow. The idealized spiral motion of liquid particles around the axial vortex line inside the vortex tube (transparent yellow) and vortex sheet (transparent green). The thickness of the vortex sheet and the vortex tube is magnified compared to the turbulent regime we are studying. We trace 25 particles with different colors.

Figure 2. Outer Euler flow. The idealized spiral motion of liquid particles around the axial vortex line outside the vortex tube (solid yellow) with the tangent velocity boundary conditions at the vortex sheet (solid green). This flow goes around the tube, sliding along the surface on each side rather than passing through it. The thickness of the vortex sheet and the vortex tube is magnified compared to the turbulent regime we are studying. We trace 25 particles with different colors.

Figure 2. Outer Euler flow. The idealized spiral motion of liquid particles around the axial vortex line outside the vortex tube (solid yellow) with the tangent velocity boundary conditions at the vortex sheet (solid green). This flow goes around the tube, sliding along the surface on each side rather than passing through it. The thickness of the vortex sheet and the vortex tube is magnified compared to the turbulent regime we are studying. We trace 25 particles with different colors.

Let us note in passing that this singular tube and change of the Neumann boundary conditions at the edge invalidate the conditions of the de Lellis-Brue theorem in our case.

This theorem [9], in particular, claims that the surface Laplacian cannot be positive at the whole disk-like vortex sheet if the velocity is nonsingular and tangent to the edge.

We need this positivity for stability, as it was argued in [9] as part of the CVS conditions.

Let us come back to our boundary conditions.

The potential velocity part enters the strain near the loop and has no singularity.

The true boundary condition is an eigenvalue requirement (14).

The analogous CVS conditions [9] for the vortex sheet were equivalent to the velocity gap being a null vector of a boundary strain at the surface

$$\begin{array}{c}\hfill \widehat{S}·\Delta \overrightarrow{v}=0.\end{array}$$

(49)

The shape of the vortex sheet in [9] was fixed by this requirement, up to a few parameters.

We expect the same relation here to provide the same vortex sheet stability.

However, the velocity gap vanishes at the boundary, so this relation does not fix the boundary value of the strain. The eigenvalue requirement represents this missing condition at the edge of the discontinuity surface.

We conjecture that in the case of the Kelvinon, the shape of the discontinuity surface is determined by (49) inside, plus (14) at the edge.

Note that here we deviate from our old hypothesis [14,15,16,17] that the shape of the discontinuity surface minimizes its area.

This minimal area would be an asymptotic solution of the loop equation at large smooth loops [9], but not the exact shape of the discontinuity surface at an arbitrary loop.

Note also that as a consequence of (14), the eigenvalues of a strain at the surface have a form of

$$\begin{array}{c}\hfill \mathrm{diag}\left(-\lambda \left(\overrightarrow{r}\right),0,\lambda \left(\overrightarrow{r}\right)\right)\end{array}$$

(50)

The highest eigenvalue $\lambda \left(\overrightarrow{r}\right)>0$ has the eigenvector in the tangent plane of the surface, orthogonal to the velocity gap (43), as it follows from (49) and stability requirement (the normal component of strain at the surface must be negative).

Thus, if the velocity gap is orthogonal to the surface edge near the edge, the leading eigenvector of the strain would indeed be directed along the loop, as we required.

This condition is compatible with the fact that $\overrightarrow{\nabla }{S}_3$ vanishes linearly near the edge

$$\begin{array}{ccc}& & \overrightarrow{\nabla }{S}_3(\overrightarrow{r}\to \overrightarrow{C}\left(l_0\right))\propto (\overrightarrow{r}-\overrightarrow{C}\left(l_0\right));\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& & \Delta \overrightarrow{v}(\overrightarrow{r}\to \overrightarrow{C}\left(l_0\right))=2\pi nZ\overrightarrow{\nabla }{S}_3\propto (\overrightarrow{r}-\overrightarrow{C}\left(l_0\right))\perp {\overrightarrow{C}}^{\prime }\left(l_0\right);\hfill \end{array}$$

(52)

The last relation follows from the fact that $\overrightarrow{C}\left(l_0\right)$ is the nearest point at the edge to the point $\overrightarrow{r}\in S$, which leads to

$$\begin{array}{c}\hfill {\partial }_l{(\overrightarrow{r}-\overrightarrow{C}\left(l_0\right))}^2\propto {\overrightarrow{C}}^{\prime }\left(l_0\right)·(\overrightarrow{r}-\overrightarrow{C}\left(l_0\right))=0\end{array}$$

(53)

It would be very interesting to find the Kelvinon solution for the flat circular C, where the discontinuity surface is the unit disk $D^2$.

Figure 3. The dual loop (red) encircling the monopole ring (blue). The Burgers vortex resolves the singular vortex line.

Figure 3. The dual loop (red) encircling the monopole ring (blue). The Burgers vortex resolves the singular vortex line.

3.3. The mapping onto a disk on a sphere

This tube cross-section’s boundary $\delta C$ is mapped on the loop $\gamma$ on ${\mathbb{S}}^2$, covered n times. Topologically, a circle is mapped on a circle with homotopy ${\pi }_1\left({\mathbb{S}}_1\right)=\mathbb{Z}$.

The circulation ${\oint }_{\delta C}{v}_{\alpha }d{r}_{\alpha }$ is related to the vorticity flux through the tube’s cross-section, which equates it to the Burgers parameter ${\Gamma }_B$.

The circulation around a loop on a 2-sphere equals the area of one of two complementary spherical caps ${\Omega }_{\pm };\partial {\Omega }_{\pm }=\gamma$, depending upon the orientation of $\gamma$, i.e., the sign of the winding number n (see Figure 4).

Figure 4. The regions at ${\Omega }_{\pm }\in {\mathbb{S}}^2$, with opposite orientations of the boundary loop $\partial {\Omega }_{\pm }=\gamma$. The areas $|{\Omega }_{\pm }|$ add up to $4\pi$.

Figure 4. The regions at ${\Omega }_{\pm }\in {\mathbb{S}}^2$, with opposite orientations of the boundary loop $\partial {\Omega }_{\pm }=\gamma$. The areas $|{\Omega }_{\pm }|$ add up to $4\pi$.

In our coordinates, this loop $\gamma$ is some horizontal circle $S_3=cos\lambda =\mathit{const}$.

$$\begin{array}{ccc}\hfill & & {\Gamma }_B={\oint }_{\delta C}{v}_{\alpha }d{r}_{\alpha }={\int }_{{\Omega }_{\pm }}d{\varphi }_2\wedge d{\varphi }_1=2\pi nZ(1-signncos\lambda );\hfill \end{array}$$

(54)

The boundary value of $S_3\left(C\right)=cos\left(\lambda \right)$ remains as a constant parameter of our Clebsch field, to be determined later from the minimization of the Hamiltonian.

There is, of course, a possibility to get zero circulation (and, therefore, zero velocity, in case $Z=0$ or $cos\lambda =signn$).

Let us now turn to the circulation ${\Gamma }_C$ around the original loop. In the same way, as with the Burgers circulation ${\Gamma }_B$, this circulation can be written as a vorticity flux through some Stokes surface bounded by the loop and passing through the Euler region.

A small part of this surface will pass through the Burgers tube. As the vorticity inside the Burgers tube is directed towards its axis, we can choose this surface to pass this tube in the local tangent plane to the loop. The normal to this local tangent plane $\overrightarrow{n}\Vert {\overrightarrow{C}}^{\prime }\left(l\right)\times \overrightarrow{C}"\left(l\right)$ is orthogonal to the direction of the vorticity $\overrightarrow{\omega }\Vert {\overrightarrow{C}}^{\prime }\left(l\right)$.

Therefore, the flux is determined solely by the vorticity in the Euler region, which means that the singular-line solution (23) with potential flow outside the loop does not provide a finite circulation ${\Gamma }_C$.

This requirement is the ultimate reason for the topological solution.

There is no contradiction at this level with the Clebsch field, but there is an interesting relation based on the Stokes theorem.

Let us compute the Euler flux through the discontinuity surface on the upper side ${\mathcal{S}}_C^{+}$.

The flux through the surface is

$$\begin{array}{ccc}\hfill & & {\Gamma }_C={\int }_{S_C^{+}}d\overrightarrow{\sigma }·{\overrightarrow{\omega }}^{+}={\int }_{{\Omega }_{\pm }}d{\varphi }_1\wedge d{\varphi }_2=2\pi mZ(1-signmcos\lambda );\hfill \end{array}$$

(55)

where $m\in \mathbb{Z}$ is a winding number for ${\varphi }_2$ around the loop C.

These two winding numbers $n,m$ are consistent with the boundary condition [9] at the surface $\partial \mathcal{T}$ of the infinitesimal tube $\mathcal{T}$

$$\begin{array}{ccc}\hfill & & {\varphi }_2\left(\overrightarrow{C}\left(l\right)+\overrightarrow{\xi }\right)=m\alpha +n\beta ;\hfill \end{array}$$

(56a)

$$\begin{array}{ccc}& & \overrightarrow{\xi }=ϵ\left(\overrightarrow{n}cos\beta +\overrightarrow{\sigma }sin\beta \right);\hfill \end{array}$$

(56b)

$$\begin{array}{ccc}& & \overrightarrow{n}=\frac{{\overrightarrow{C}}^{\prime \prime }}{|{\overrightarrow{C}}^{\prime \prime }|};\hfill \end{array}$$

(56c)

$$\begin{array}{ccc}& & \overrightarrow{\sigma }={\overrightarrow{C}}^{\prime }\times \overrightarrow{n};\hfill \end{array}$$

(56d)

$$\begin{array}{ccc}& & \alpha =\frac{2\pi l}{\left|C\right|};\hfill \end{array}$$

(56e)

$$\begin{array}{ccc}& & \left|C\right|={\oint }_{C}dl;\hfill \end{array}$$

(56f)

Here, $ϵ\to 0$ is the radius of the tube.

4. Energy balance revisited

5. Asymptotic freedom revisited

Figure 5. The beta function $\beta \left(g\right)$ and the running coupling constant $g_R(logR)$

Figure 5. The beta function $\beta \left(g\right)$ and the running coupling constant $g_R(logR)$

6. Multifractals

Figure 6. DNS data [8] for classical circulation fractal dimensions $\lambda \left(p\right)$ fitted against our asymptotic formula (95) (blue) and K41 line $\frac{4p}{3}$ (green).

Figure 6. DNS data [8] for classical circulation fractal dimensions $\lambda \left(p\right)$ fitted against our asymptotic formula (95) (blue) and K41 line $\frac{4p}{3}$ (green).

Figure 7. DNS data [19] for quantum circulation fractal dimensions $\lambda \left(p\right)$ fitted against our asymptotic formula (95) (blue) and K41 line $\frac{4p}{3}$ (green).

Figure 7. DNS data [19] for quantum circulation fractal dimensions $\lambda \left(p\right)$ fitted against our asymptotic formula (95) (blue) and K41 line $\frac{4p}{3}$ (green).

7. Questions and answers

• Q:
  By differentiating your relation for the velocity at the center of the Burgers vortex, you get an identity

  $$\begin{array}{ccc}& & \frac{\partial v_t\left(l\right)}{\partial l}=\frac{\partial \left(\overrightarrow{v}\left(\overrightarrow{C}\left(l\right)\right)·{\overrightarrow{C}}^{\prime }\left(l\right)\right)}{\partial l}=\hfill \\ & & {\overrightarrow{C}}^{\prime }\left(l\right)·\widehat{S}\left(\overrightarrow{C}\left(l\right)\right)·{\overrightarrow{C}}^{\prime }\left(l\right)+\overrightarrow{v}\left(\overrightarrow{C}\left(l\right)\right)·{\overrightarrow{C}}^{\prime \prime }\left(l\right)=\hfill \\ & & c\left(l\right)+\overrightarrow{v}\left(\overrightarrow{C}\left(l\right)\right)·{\overrightarrow{C}}^{\prime \prime }\left(l\right)\hfill \end{array}$$

  (96)
  From that, we conclude that

  $$\begin{array}{c}\hfill v_t\left(l\right)=\int dlc\left(l\right)+\int dl{v}_n\left(l\right)\kappa \left(l\right)\end{array}$$

  (97)

  where $\kappa \left(l\right)$ is the local curvature of the loop, and $v_n\left(l\right)$ is the velocity projection on the inner normal of the loop.
  According to the Burgers solution, the second term vanishes at the center of the loop. Therefore, with the positive $c\left(l\right)$ you need for stability, this formula contradicts the periodicity of velocity at the loop.
  A: I am afraid I disagree with your formula for the tangent component of the velocity field. My formula, (16), is manifestly periodic (and also parametric invariant, as it should be). With explicit initial data, it reads

  $$\overrightarrow{v}\left(\overrightarrow{C}\left(l\right)\right)=\overrightarrow{v}\left(\overrightarrow{C}\left(0\right)\right)+{\int }_0^{l}d{l}^{\prime }\widehat{S}\left(C\left(l^{\prime }\right)\right)·{\overrightarrow{C}}^{\prime }\left(l^{\prime }\right)$$

  (98)
  Using the eigenvalue condition $\widehat{S}\left(C\left(l^{\prime }\right)\right)·{\overrightarrow{C}}^{\prime }\left(l^{\prime }\right)=c\left(l^{\prime }\right){\overrightarrow{C}}^{\prime }\left(l^{\prime }\right)$ it can be written as

  $$\overrightarrow{v}\left(\overrightarrow{C}\left(l\right)\right)=\overrightarrow{v}\left(\overrightarrow{C}\left(0\right)\right)+{\int }_0^{l}d{l}^{\prime }c\left(l^{\prime }\right){\overrightarrow{C}}^{\prime }\left(l^{\prime }\right)$$

  (99)
  Multiplying this equation by ${\overrightarrow{C}}^{\prime }\left(l\right)$, we find the correct relation for the tangent velocity

  $$\begin{array}{c}\hfill v_t\left(l\right)=\overrightarrow{v}\left(\overrightarrow{C}\left(0\right)\right)·{\overrightarrow{C}}^{\prime }\left(l\right)+{\int }_0^{l}d{l}^{\prime }c\left(l^{\prime }\right){\overrightarrow{C}}^{\prime }\left(l^{\prime }\right)·{\overrightarrow{C}}^{\prime }\left(l\right)\end{array}$$

  (100)
  Take $l=L$ in (99); we have the periodicity condition

  $$\overrightarrow{0}={\int }_0^{L}d{l}^{\prime }c\left(l^{\prime }\right){\overrightarrow{C}}^{\prime }\left(l^{\prime }\right)$$

  (101)
  This expression does not change if you shift $c\left(l^{\prime }\right)$ by any constant, as the extra term will integrate into zero

  $${\int }_0^{L}d{l}^{\prime }{\overrightarrow{C}}^{\prime }\left(l^{\prime }\right)=\overrightarrow{C}\left(L\right)-\overrightarrow{C}\left(0\right)=\overrightarrow{0};$$

  (102)
  In particular, one may subtract from $c\left(l^{\prime }\right)$ its (positive) mean value on the loop. Thus positivity of $c\left(l\right)$ does not contradict the periodicity.
• Q: What is the boundary value of the Euler velocity at the boundary C of the vortex surface? The normal to the loop component of velocity must vanish for mass conservation (incompressibility), which condition directs the boundary velocity along the loop. Do you agree?
  A: There is no boundary value of the Euler velocity, as it diverges as $1/r$, where $r\to 0$ is the radius of the vortex tube.
  Velocity at the center of the Burgers vortex is directed along the loop, the same as vorticity, thus providing nontrivial helicity density.
  This velocity component normal to the loop $v_n$ is not zero for the following reason. This normal inner component of velocity is singular (diverges as $1/r$ near the edge), so one cannot neglect the existence of a thin Burgers tube (see Figure 1 ).
  The boundary conditions for the Euler flow are imposed at the surface of the tube, not at its center, and there is a match.
• Q:
  What prevents the Kelvinon from flying in the normal direction to the ring, as all vortex rings do? I also wanted to ask if the Kelvinon has a non-zero impulse $\propto {\int }_{allspace}\overrightarrow{x}\times \overrightarrow{\omega }{d}^{3}x$. The impulse is proportional to the net linear momentum. For the Burgers loop part, it is not difficult to compute the impulse as a line integral around the loop, which is non-zero.
  A: The Neumann boundary condition for the potential part in the velocity (with the rotational part being the Biot-Savart integral), by definition of the stationary state, cancels the sheet’s motion (normal velocity vanishes, and tangent one does not mean a moving surface). As for the Burgers tube attached to the surface at the edge, the same thing happens. The mean velocity in the normal plane to the ring vanishes at every point of the loop, so it does not move.
  The finite negative radial strain pushes flow inside the ring, moving along the axis in the rotating vortex with circulation ${\Gamma }_C$. In addition, there is a (much faster) circular motion in the normal plane, going around the small circle at each point of the loop. Together, these motions create a spiral, shown in Figure 1.
  That self-induced velocity $v=\Gamma logR/w$ comes from the region far from this surface, just like my anomalous part in the Hamiltonian. This part, indeed, cancels by the surface contribution by the Neumann boundary conditions, which are designed for a steady state.
  So, the vortex surface attached to the vortex ring acts like a parachute, or, better to say, as a wing, stopping the normal motion. The flux goes around the surface and the ring in circles, as shown in Figure 2.
  As for the net momentum, I cannot prove it is zero, even for a stationary velocity field tangent to the surface and the ring.
  All I need is a stationary distribution of circulation based on stationary flow. The loop and the surface must also be stationary, but the net momentum of the flow could be finite. The integral of the velocity field over volume is not restricted by its value at the surface. Take an example of the flow around the rigid body.
  I analyze this question in the Section 3.2.
  In that section, I also discuss the relation of the CVS conditions for the vortex sheet $\widehat{S}·\Delta \overrightarrow{v}=0$ and the new boundary condition $\widehat{S}·\overrightarrow{n}=c\overrightarrow{n}$. I argue that they are related, given that the velocity gap linearly vanishes at the edge of the surface.
• Q:
  Is the diffusion of the sheet also everywhere balanced on its surface by strain presumably induced by other portions of the sheet?
  A: This balance leads to the CVS condition, which I analyzed in my previous work. In a stationary solution, the diffusion term exactly cancels the advection term, which happens with both Burgers and Townsend solutions in the vorticity region. For the flat surface, the normal velocity vanishes, and the negative normal strain makes the solution stable.
  For a curved vortex sheet, this is provided by the CVS equations. So, it does not happen for arbitrary sheets. The sheet’s shape must adjust for this cancellation, which is what happens in my exact CVS solution with the hyperbolic sheet ${x\left|y\right|}^{\mu }=const$.
  As for decay, the decay time (and reconnection time) are not infinite, but they grow with the Reynolds number. Thus my "stationary" solution only holds for this growing decay time.
• Q:
  When you compute the energy, for example, the equation with Euler’s constant, do you use an expression for the energy in terms of the vorticity? I think that the expression to use is $\propto {\int }_{allspace}\overrightarrow{B}·\overrightarrow{\omega }{d}^{3}x$, where $\overrightarrow{B}$ is the vector potential. Is this the expression you used?
  A: No, why? I used ordinary ${\int }_V{d}^3\overrightarrow{r}{\overrightarrow{v}}^2/2$ and substituted the Burgers solution. BTW note that the Burgers vortex may generally have a constant velocity in the z direction.
  Look at the computation in the appendix in my Phys Reports [9].
• Q:
  Did you prove the positivity of $c\left(l\right)$?
  A:
  I could not prove it in the general case, but I have presented arguments in favor in Section 3.2. It is left as an open problem to prove it.
• Q:
  I wanted to mention the Kelvin-Benjamin maximum energy principle: A steady inviscid vortex has maximum energy with respect to isovortical perturbations (i.e., imagine perturbing the vortex by an arbitrary incompressible velocity field acting for a short pseudo-time). Given this, shouldn’t the relaxation procedure maximize the Hamiltonian? Perhaps, this principle does not apply because the Hamiltonian is infinite in an inviscid limit?
  A:
  Mathematically, I am investigating the turbulent (not inviscid) limit where the Reynolds number is arbitrarily large but not infinite.
  The Hamiltonian minimization is a tricky part, indeed. Consider the Euler flow outside the Burger’s tube and the attached sheet. The parameters of this flow (including the normalization of the Clebsch field Z) minimize the Euler Hamiltonian ( the integral of the space outside the tube and the sheet), with the boundary conditions at the surface of the tune and the sheet. As I have shown, these boundary conditions involve the outside flow parameters (the boundary value of $S_3$, the shape of the vortex sheet, and the Z factor).
  Now, the Euler part of the Hamiltonian (integral over the outside region) is not conserved. Its time derivative equals minus anomalous dissipation. However, the total Hamiltonian, which includes the integral inside the vortex tube and sheet) is conserved. Its time derivative vanishes with a proper value of the normalization factor Z, an Euler integral of motion.
  For this particular value of Z, which provides the energy balance, the total Hamiltonian is conserved in the NS (dissipation is exactly canceled by energy pumping).
  Our Matching Principle at work here: all the variables and parameters of the Euler flow are to be defined as those of the NS flow with an infinitesimal viscosity. Logarithmic divergences prevent from setting viscosity to zero, but I generalize the Euler flow (call it turbulent flow) as the one minimizing the conserved Hamiltonian.
  So, this is a definition of the inviscid limit of the NS in the presence of anomalies; this definition formally becomes the Euler theory with anomalous terms in the Hamiltonian. There is no inviscid limit, but there is an intermediate logarithmic regime where all the negative powers of the Reynolds number are neglected, but the logarithmic terms remain. These terms are summed up by the renormalization group equation, leading to asymptotic freedom.
  Our work belongs to theoretical physics, not applied math. Rather than solving established equations, we postulate new ones to be verified in experiments and DNS. Our new matching principle removes ambiguities from the weak Euler solutions, leading to asymptotic freedom.
  Remains to be seen whether it applies to real turbulence.