A statistical mechanics framework for the large-scale structure of turbulent von Kármán flows

The VK experimental set-up used for the present study has been thoroughly described in [24, 31, 32]. The fluid is confined inside a cylinder of radius R = 100 mm, and forced through two rotating impellers of radius R_t—see figure 1. All the lengths will now be expressed in units of the cylinder radius R. The aspect ratio of our experiment is defined as the distance between the inner faces of the two opposite impellers $2H=1.8$. Impellers are driven by two independent motors, whose frequencies f₁ and f₂ can either be set equal in order to get an exact counter-rotating regime, or set to different values ${{f}_{1}}\ne {{f}_{2}}$. To change the viscosity, mixtures of water or glycerol with different dilution rates were used.

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In this paper, three main global parameters are used to characterize VK turbulence. (i) The Reynolds number $\operatorname{Re}=\pi ({{f}_{1}}+{{f}_{2}}){{R}^{2}}{{\nu }^{-1}}$—with ν the fluid kinematic viscosity—ranges from 10² to $5\times {{10}^{5}}$ so that a full range of regimes can be spanned, from a purely laminar to a fully turbulent one. (ii) The rotation number $\theta =({{f}_{1}}-{{f}_{2}})/({{f}_{1}}+{{f}_{2}})$, measures the relative influence of global rotation over a typical turbulent shear frequency. It can be varied from −1 to +1, hereby exploring a regime of relatively weak rotation to shear ratio. (iii) Finally, the torque asymmetry $\gamma =({{C}_{1}}-{{C}_{2}})/({{C}_{1}}+{{C}_{2}})$ measures the difference between the torques C₁ and C₂ applied to each of the propellers. It is crucial to note that in the VK2 experiment, turbulence can be either generated by maintaining constant the frequencies or the torques applied to each of the propeller—please see [24, 25] for more details.

At a finer level of description, it has been shown that the turbulence properties (anisotropy, fluctuations, dissipation) are influenced by the geometry of the propellers, namely, their non-dimensional radius R_t, the oriented angle α between the blades and the rotation direction (see figure 1), the heights h_b and the number n of blades [24]. In the present paper, we consider only propellers with h_b = 0.2 and focus on changes induced by variations of α. Those propellers are the so-called 'TM60', 'TM87' and 'TM73' propellers, whose characteristics are summarized in table 1. A single propeller can be used to propel the fluid in two opposite directions, respectively associated to the concave or convex face of the blades going forward. This can be accounted by a change of sign of the parameter α. In the sequel, we denote $(-)$ (resp. $(+)$) a propeller used with the concave (resp. convex) face of its blades going forward.

Table 1.  Parameter space explored in our set-up.

Propellers	Number of blades	α (in degrees)	Re	θ
All	8 and 16	$[-90,90]$	105	0
TM60(+)	16	72	$[{{10}^{2}},{{10}^{6}}]$	$[-1,1]$
TM60(−)	16	−72	105	$[-1,1]$
TM87(+)	8	72	105	$[-1,1]$
TM87(−)	8	−72	105	$[-1,1]$
TM73(+)	8	+24	105	$[-1,1]$
TM73(−)	8	−24	105	$[-1,1]$

Table 1 summarizes the parameter space that was explored in our system. Schematically, the influence of the propeller geometry has been explored at $\operatorname{Re}={{10}^{5}}$, $\theta =0$. The Reynolds variation has been explored at $\theta =0$ using the TM60 propellers (±). The rotation variation has been explored at $\operatorname{Re}={{10}^{5}}$ using TM73(±), TM87(±) and TM60(±). The influence of the forcing type ('constant velocity' against 'constant torque' forcing) has been studied with the TM60(−) and TM87(−) at $\operatorname{Re}={{10}^{5}}$.

2.2.1. A qualititative description.

To analyze the topology of the averaged states, stereoscopic particle image velocimetry (SPIV) measurements were mostly used. The system provides the three components of the velocity field on a 95 × 66 points grid, which covers a whole meridian plane of the flow through a time series of about 600 to 5000 regularly sampled values at a 10 Hz frequency. We also performed a few laser Doppler velocimetry (LDV) measurements providing mean velocities over an 11 × 13 points grid covering a half meridian plane of the flow. To deal with non-dimensional velocity fields, they are divided by a typical 'forcing velocity' defined as ${{V}_{0}}=2\pi R({{f}_{1}}+{{f}_{2}})/2$. We write $(r,\phi ,z),$ the standard cylindrical coordinates, and denote ${{\langle .\rangle }_{\circ }}$ as an average over a time series of SPIV measurements. We also use the short-hand notation ${{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}f=(1/2\;H)\int _{0}^{1}rdr\int _{-H}^{H}(f(r,0,z)+f(r,\pi ,z))$ to denote the spatial average of any quantity $f=f(r,\phi ,z)$ over the PIV window. As an example, figure 2 shows the (r, z) dependence of the three components of a 3D velocity field $({{u}_{r}},{{u}_{\phi }},{{u}_{z}})$ reconstructed from a PIV measurement at Reynolds number approximately equal to $3\times {{10}^{5}}$. Although the SPIV system does not allow us to analyze in detail the azimuthal dependence of the velocity fields, it is clear from figure 2 that the instantaneous velocity field is not axially symmetric, as it is not symmetric with respect to the transformation $r\to -r$. Axial symmetry exists though at the level of the time averaged velocity field (and more generally for any quantity derived from it). At a coarse level of description, the topology of the average velocity fields is simple and appears to bear some kind of universality. It either consists of a two-cell state⁵ that is symmetric with respect to the equatorial axis, a two-cell state with broken symmetry or a one-cell state.

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2.2.2. A more quantitative description.

As first observed in [30], the axial symmetry of the averaged states is not just an artifact of the averaging procedure; it can also be used as a natural guideline to describe the topology of the steady states. As the average flow inside the tank is divergence free and axially symmetric (namely, symmetric with respect to any azimuthal change), a Helmholtz decomposion can be used to write the averaged velocity field as

$$\begin{eqnarray}&&{{\langle {\bf u}\rangle }_{\circ }}={{\langle {{u}_{\phi }}\rangle }_{\circ }}{{{\bf e}}_{\phi }}+\frac{1}{r}\nabla (r\Psi )\times {{{\bf e}}_{\phi }}.\end{eqnarray} \tag{ 1 }$$

Independently of the underlying dynamics, the azimutal component of the vorticity ${{\langle {{\omega }_{\phi }}\rangle }_{\circ }}=(\nabla \times {{\langle {\bf u}\rangle }_{\circ }})\cdot {{{\bf e}}_{\phi }}$ is then related to the stream function through ${{\langle {{\omega }_{\phi }}\rangle }_{\circ }}=-{{\partial }_{zz}}\Psi -{{\partial }_{r}}{{r}^{-1}}{{\partial }_{r}}r\Psi =-{\mathcal{L}}(\Psi )$. The knowledge of $({{\langle {{u}_{\phi }}\rangle }_{\circ }},{{\langle {{\omega }_{\phi }}\rangle }_{\circ }})$ is then sufficient to reconstruct the 3D averaged velocity field ${{\langle {\bf u}\rangle }_{\circ }}$. Using such a decomposition of the velocity field, Monchaux et al [30] evidenced that the axially symmetric averaged velocity fields observed in VK set-ups were peculiar steady solutions of the Euler axially symmetric equations, at least in a region far from the propellers and the boundaries. The Euler axially symmetric equations are derived from the 3D (incompressible) Euler equations by considering the dynamics of a 3D velocity field whose cylindrical components do not depend on the azimuthal coordinate, and depend on r and z only—see for example [27, 33] and section 3 below. Steady states of the axially symmetric Euler equations are obtained whenever the toroidal field $r{{u}_{\phi }}$, the poloidal field ${{\omega }_{\phi }}/r$ and the reduced stream function $\psi =r\Psi$ are related through relations of the kind [27, 33]:

$$\begin{eqnarray}&&r{{u}_{\phi }}=F(\psi )\quad {\rm and}\quad {{\omega }_{\phi }}/r-F{{F}^{\prime }}(\psi )/{{r}^{2}}=G(\psi )\quad {\rm for}\;{\rm any}\;{\rm function}\;{\rm F}\;{\rm and}\;{\rm G}.\quad \end{eqnarray} \tag{ 2 }$$

Using TM60(±) propellers for a wide range of rotation numbers, scatter plots of both the toroidal field ${{\langle r{{u}_{\phi }}\rangle }_{\circ }}$ and the poloidal field ${{\langle {{\omega }_{\phi }}/r\rangle }_{\circ }}$ against the reduced stream function $\psi =r\Psi$ showed a clear functional relationship between those quantities [30, 31]. To understand the general trends of the topologies, it is enough to choose linear functions for F and G. In this linear approximation, the VK topologies are characterized by four constant numbers, say A, B, C and D, defined by⁶ :

$$\begin{eqnarray}&&{{\langle {{u}_{\phi }}\rangle }_{\circ }}=B\Psi +A/r\quad {\rm and}\quad {{\langle {{\omega }_{\phi }}\rangle }_{\circ }}=(B+D{{r}^{2}}){{u}_{\phi }}+Cr.\end{eqnarray} \tag{ 3 }$$

A, B, C, and D are computed using the least square fits from the scatter plots of ${{\langle {{u}_{\phi }}\rangle }_{\circ }}$ and ${{\langle {{\omega }_{\phi }}\rangle }_{\circ }}$ against Ψ. An example is shown in figure 3, obtained for a large Reynolds number at $(\theta ,\gamma )=(0,0)$. In such a case, the fit is rather good. As already seen in [30], the fit deteriorates at lower Reynolds numbers and when the rotation number is too high (say $|\theta |\gtrsim 0.3$). Still, equation (3) provides a general framework for the interpretation of the data. The fitting procedure was carried out with various propellers at large Reynolds number in the symmetric state ($\gamma =0$ and $\theta =0$) with LDV data, that provide lower resolution representations of the mean flow. The resulting values for A, B, C and D as a function of the propeller's radius and angle are provided in figure 4. Because of the measurement technique, these fits are less accurate than with the PIV data. The LDV-measured values of A, B, C and D should therefore be used here to observe trends rather than to provide quantitative values. We observe that in all cases, both A and C are vanishing. The fact that A = 0 is compatible with ${{u}_{\phi }}$ being finite at r = 0, while the fact that C = 0 is a consequence of the symmetry of the basic state at $\theta =0$. The value of B depends mostly on the propeller's angle α, being positive when the angle is negative, and negative otherwise. The absolute value of B remains fairly constant in between 3 and 4, regardless of the angle—except for the $\alpha =0$ case. At negative α, the value of D is rather low, and close to 0. Increasing $\alpha \gt 0$ yields a linear decrease for D, from 0 to −20 (at $\alpha =72{}^\circ$). At this value of α though, there are some indications that a second branch of solutions can be found with D = 0. This has been confirmed by a study of the variations of our parameters with the Reynolds number, using PIV data, at $\theta =0$ in the symmetric state. With these better resolved data (not shown), we found that the coefficient D displays a clear bi-modal behavior, with two branches of solution: one extends around D = 0, and the other decreases linearly with ${\rm log} \operatorname{Re}$. A closer look at the values of D for the TM60/87(+) propellers indicates that the negative branch of solutions corresponds to a branch that connects continuously from a two-cell to a one-cell solution. As for the other coefficients, we found that B is rather insensitive to the Reynolds number while both the A and C coefficients remain zero, at any Reynolds number, in agreement with the previously described regularity and symmetry properties.

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When the forcing is fully symmetric ($\theta =0$ and $\gamma =0$), all the impellers that we have tested yield a symmetric two-cell state similar to the one depicted in figure 2. Similarly, when the forcing is clearly non-symmetric ($|\gamma |\simeq 1$ or $|\theta |\simeq 1$), the fluid is globally in rotation and the average state is one-cell. Yet, the nature of the transition between the symmetric and the non-symmetric states does strongly depend on the geometry of the impellers. On the one hand, the use of low curvature impellers ($\alpha \gtrsim -30{}^\circ$) yields a continuous transition that occurs via a sequence of increasingly non-symmetric two-cell states, with one cell becoming larger at the expense of the other. The sharpness of the transition can be characterized throughout the use of susceptibility coefficients (see section 4), analogous to the magnetic susceptibilities in the theory of ferro-magnetism. Cortet et al [34, 35] observed that those susceptibilities diverge at a finite turbulent Reynolds number $(\operatorname{Re}\simeq 4\times {{10}^{4}})$, a feature clearly reminiscent of second-order phase transition in statistical physics. On the other hand, the use of high curvature impellers ($\alpha \lesssim -30{}^\circ$) yields an abrupt change, which gives rise to multi-stability between the two-cell symmetric state, and one of the two one-cell states (symmetric to each other with respect to the equatorial axis) [24]. As a result, a hysteresis cycle for γ is described when the rotation number θ is used as a control parameter, and cycled from 1 to −1 and back, over a given time scale. Increasing the curvature of the blades increases both the width and the height of the hysteresis cycle that the system describes in the $\theta -\gamma$ plane [24]. When the torque number γ is used as a control parameter (i.e. when the system is forced at constant torque rather than at constant velocity),the hysteresis cycle is regularized [25, 36]—see figure 5. Because of the apparent simplicity and universality of the steady states, there is hope that a global understanding of the steady states can be provided through general arguments based on symmetries and conservation laws. This is precisely the outcome of statistical physics. In the sequel, we try to explain the topology of the axially symmetric mean velocity field and explain their stability as a function of the control parameters $\operatorname{Re},\theta ,\gamma ,\alpha$, using some tools borrowed from statistical physics.

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Inside an axially symmetric domain ${\mathcal{D}}$, an axially symmetric fluid is described in terms of a 3D velocity field ${\bf u}$, whose components in cylindrical coordinates $({{u}_{r}},{{u}_{\phi }},{{u}_{z}})$ do not depend on the azimuthal coordinate ϕ but on r and z only. The evolution equation of a perfect axially symmetric fluid is therefore obtained by setting ${{\partial }_{\phi }}=0$ in the incompressible Euler equations

$$\begin{eqnarray}&&{{\partial }_{t}}{\bf u}+{\bf u}\cdot \nabla {\bf u}+\nabla p=0\quad {\rm and}\quad \nabla \cdot {\bf u}=0.\end{eqnarray} \tag{ 4 }$$

Rather than prescribing the 3D velocity field, a convenient description of an axially symmetric flow can be achieved in terms of only two fields: (i) the azimuthal velocity ${{u}_{\phi }}$ (later named 'toroidal field'), and (ii) the azimuthal vorticity field ${{\omega }_{\phi }}=(\nabla \times {\bf u})\cdot {{{\bf e}}_{\phi }}$ (alternatively named 'poloidal field'). The entire 3D velocity field can then be reconstructed by using the incompressibility condition ($\nabla \cdot {\bf u}=0$), which allows the following Helmholtz decomposition for ${\bf u}$:

$$\begin{eqnarray}&&{\bf u}={{u}_{\phi }}{{{\bf e}}_{\phi }}+\frac{1}{r}\nabla (r\Psi )\times {{{\bf e}}_{\phi }},\end{eqnarray} \tag{ 5 }$$

where Ψ is the stream function, deduced from the azimuthal vorticity by the relation:

$$\begin{eqnarray}&&{{\omega }_{\phi }}=-\left( {{\partial }_{zz}}+{{\partial }_{r}}\frac{1}{r}{{\partial }_{r}}r \right)\Psi =-{\mathcal{L}}(\Psi ).\end{eqnarray} \tag{ 6 }$$

We also define $\psi =r\Psi$ the reduced stream function. To invert the differential operator ${\mathcal{L}}$, one needs to work with specified boundary conditions. We will assume here that ψ is vanishing at the boundaries, a condition that comes for the impenetrability condition for the velocity field on the outer walls. Finally, for a cylindrical domain ${\mathcal{D}}$ with height $2h$ and radius R, we will later use the short-hand notation ${{\int }_{{\mathcal{D}}}}\equiv (1/(h{{R}^{2}}))\int _{0}^{R}r{\rm dr}\int _{-h}^{h}{\rm dz}$. This simple geometry will serve as a guideline for the statistical theories described in the remainder of the section.

A basic input of the statistical physics of axially symmetric flows is the existence of conserved global quantities in the inviscid, unforced limit—at least provided that the perfect fluid can be considered to remain 'sufficiently regular'. For instance, it is well known that the Euler dynamics (4) preserve both the kinetic energy $E=1/2{{\int }_{{\mathcal{D}}}}{{{\bf u}}^{2}}$ and the Helicity $H={{\int }_{{\mathcal{D}}}}{\bf u}\cdot (\nabla \times {\bf u})$ [19]. In terms of ${{u}_{\phi }}$ and ${{\omega }_{\phi }}$ these quantities read:

$$\begin{eqnarray}&&E=\frac{1}{2}{{\int }_{{\mathcal{D}}}}u_{\phi }^{2}+{{\omega }_{\phi }}\Psi \quad {\rm and}\quad H=2{{\int }_{{\mathcal{D}}}}{{u}_{\phi }}{{\omega }_{\phi }}.\end{eqnarray} \tag{ 7 }$$

Axially symmetric velocity fields are 3D fields that are invariant with respect to the azimuthal coordinate. This spatial continuous symmetry around the axis makes the axially symmetric Euler dynamics further preserve two infinite families of global quantities, namely, the toroidal casimirs C_f and the generalized helicities H_g (see [27, 33]). This is a consequence of Noether's theorem and of the fact that the Euler dynamics is Hamiltonian [41, 42]. The additional invariants read

$$\begin{eqnarray}&&{{C}_{f}}={{\int }_{{\mathcal{D}}}}f\left( r{{u}_{\phi }} \right)\;\quad {\rm and}\quad \;{{H}_{g}}={{\int }_{{\mathcal{D}}}}g\left( r{{u}_{\phi }} \right){{\omega }_{\phi }}/r\;{\rm for}\ {\rm any}\ {\rm regular}\ {\rm enough}\;f\;{\rm and}\;g.\end{eqnarray} \tag{ 8 }$$

Those quantities allow us to interpret, thermodynamically, the linear description of the VK topologies provided by equations (2), (3) and (6). The idea is to use the invariants of equation (8) to build a 'free energy' functional, which can be thought of as an analog of a Landau–Ginzburg functional [27, 28, 30]. To stick to a linear level of description, one uses a quadratic form for the functions f and a linear form for g, namely, $f(x)=\alpha {{x}^{2}}-\mu x$ and $g(x)=hx-\gamma$, with α, μ, h, γ yet unprescribed scalars. The free energy then reads:

$$\begin{eqnarray}&&{\mathcal{F}}[{{u}_{\phi }},{{\omega }_{\phi }}]={{\int }_{{\mathcal{D}}}}(\beta /2+\alpha {{r}^{2}})u_{\phi }^{2}+h{{u}_{\phi }}{{\omega }_{\phi }}+\beta {{\omega }_{\phi }}\Psi /2-\gamma {{\omega }_{\phi }}/r-\mu r{{u}_{\phi }}.\end{eqnarray} \tag{ 9 }$$

If the fluid inside the VK set-up was both perfectly axially symmetric and inviscid, then we could expect the local minimizers of ${\mathcal{F}}$ to play a peculiar role. Indeed, the reader familiar with dynamical systems may have already recognized that the free energy (9) is an Arnold function relevant for axially symmetric perfect fluid, whose minima (if any) provide axially symmetric profiles that are formally stable—see [33, 42] for more details about the stability of infinite dimensional dynamical systems and the stability of axially symmetric perfect flows in particular⁷ . The critical points of ${\mathcal{F}}$ are determined by the following class of axially symmetric fields $(u_{\phi }^{\star },\omega _{\phi }^{\star })$:

$$\begin{eqnarray}&&(\beta +2\alpha {{r}^{2}})u_{\phi }^{\star }+h\omega _{\phi }^{\star }-\mu r=0\quad {\rm and}\quad hu_{\phi }^{\star }+\beta {{\Psi }^{\star }}-\gamma /r=0.\end{eqnarray} \tag{ 10 }$$

Setting $A=\gamma /h$, $B=-\beta /h$, $C=\mu /h$ and $D=-2\alpha /h$, we exactly retrieve equation (3), which we previously used to characterize the VK topologies. This provides a clear connection between the steady states inside the VK tank and the steady states of the axially symmetric perfect fluid. However, it is easily shown that the fields $(u_{\phi }^{\star },\omega _{\phi }^{\star })$ that satisfy (10) do not, in general, locally minimize the free energy ${\mathcal{F}}$, unless h = 0 (non-helical case). In this case, positive values for β and α ensure that the $(u_{\phi }^{\star },\omega _{\phi }^{\star })$'s are indeed minimizers of ${\mathcal{F}}$. If h is non-zero, the profiles that satisfy (10) are saddle points of the free energy and are therefore unstable with respect to any non-trivial perturbations. Were we dealing with an inviscid axially symmetric fluid, we would therefore conclude that such 'meta-stable' profiles could not be observed in the long term. It is, however, an experimental fact that those profiles (with non-zero 'h') are relevant to approximate those observed in a VK flow [30].

To connect the VK topologies with the meta-stable profiles of equation (10), some crude identifications need to be made: the equilibrium fields $(u_{\phi }^{\star },\omega _{\phi }^{\star })$ of the inviscid theory with the averaged PIV fields $({{\langle {{u}_{\phi }}\rangle }_{\circ }},{{\langle {{\omega }_{\phi }}\rangle }_{\circ }})$, and the axially symmetric domain ${\mathcal{D}}$ with the PIV measurement domain ${{{\mathcal{D}}}_{{\rm piv}}}$. The VK analogs of the axially symmetric casimirs are the angular momentum I, the circulation Γ, the helicity H, the toroidal energy T, and the poloidal energy P, defined as

$$\begin{eqnarray} I & = & {{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}{{\langle r{{u}_{\phi }}\rangle }_{\circ }},\quad \Gamma ={{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}{{\langle {{\omega }_{\phi }}\rangle }_{\circ }},\quad H={{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}{{\langle {{u}_{\phi }}\rangle }_{\circ }}{{\langle {{\omega }_{\phi }}\rangle }_{\circ }} \\ T & = & (1/2){{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}\langle {{u}_{\phi }}\rangle _{\circ }^{2}\quad {\rm and}\quad P=(1/2){{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}\langle {{u}_{r}}\rangle _{\circ }^{2}+\langle {{u}_{z}}\rangle _{\circ }^{2}. \\ \end{eqnarray} \tag{ 11 }$$

Equations (10) and (3) actually provide a good approximation of the steady states of the VK set-up. Figure 6 shows the comparison between steady states obtained using the TM73(+) propeller and solutions predicted by inviscid thermodynamics. To obtain it, we solved equation (3) with the SPIV-measured values of the constants A, B, C and D—and appropriate boundary conditions [36]. To make the computation easier, we approximated the quantity $B+D{{r}^{2}}$ by a constant ${{K}_{{\rm eff}}}$. We will later refer to this approximation as a 'Beltrami' approximation⁸ .

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3.3.1. Analogy with a lattice model: a qualitative description.

Since the topology of VK flows can be retrieved from a simple thermodynamic argument, one may wonder about the practical use of a full statistical theory. However, we also noticed that the VK steady profiles did not match any axially symmetric genuine equilibrium, but rather a class of axially symmetric 'meta-stable' equilibria. One aim of statistical mechanics is to understand this discrepancy, and to highlight important distinctions between VK flows and truly axially symmetric fluids. In particular, we will argue that although the VK profiles are in principle meta-equilibrium ones, they can still be interpreted as profiles that maximize a suitably defined configuration entropy.

The statistical origin of the coarse-grained steady states can be qualitatively intuited, by looking at the averaged signs of the azimuthal field for nearly symmetric forcing ($\gamma \simeq 0$ and $\theta \simeq 0$). In a simplified interpretation of the VK experiment, one may want to think about the signs of the instantaneous azimuthal velocity field (as measured on each position of the SPIV grid) as a 'spin' that could take either a + 1 or −1 value. In the sequel we drop the precautional commas '' and call it a spin. Then, each propeller could be thought of as a statistical reservoir of '+' and '−', ensuring the numbers of '+' and '−' to remain steady. The scatter plot of the average azimuthal velocity sign against the stream function makes a hyperbolic tangent law emerge whatever the Reynolds number—see figure 7. At a qualitative level, the tan h law is reminiscent of the sin h laws observed in decaying 2D turbulence [43], or to some particular 'two-level discrete case' found in the statistical theories developed for 2D inviscid flows in [6, 7, 37]. It is also reminiscent of the mean-field closure equations that appear in the study of long-range lattice models of ferromagnetism. Consider for example the Curie–Weiss model, one of the simplest lattice models of ferromagnets that can be treated analytically [44], and whose Hamiltonian reads:

$$\begin{eqnarray}&&{\mathcal{H}}=-\frac{1}{2N}\mathop{\sum }\limits_{i=1}^{N}\mathop{\sum }\limits_{j\ne i}{{s}_{i}}{{s}_{j}}-h\mathop{\sum }\limits_{i=1}^{N}{{s}_{i}}\quad {\rm with}\quad {{s}_{i}}=\pm 1.\end{eqnarray} \tag{ 12 }$$

In the thermodynamic limit ($N\gg 1$), it is well-known that the canonical free energy per site at temperature ${{\beta }^{-1}}$ of the Curie–Weiss model can be written as an infimum over all the possible values of the magnetization $\mu =(1/N)\sum _{i=1}^{N}{{s}_{i}}$, namely

$$\begin{eqnarray}&&f(\beta )=\mathop{{\rm inf} }\limits_{\mu }\left\{ \frac{\beta {{\mu }^{2}}}{2}-{\rm log} \;{\rm cosh} \;\beta (\mu +h) \right\}.\end{eqnarray} \tag{ 13 }$$

The infimum is reached for the value of the magnetization that satisfies a self-consistent tan h law, namely $\mu ={\rm tan} h\beta (\mu +h)$. In the Curie–Weiss model, the tan h law emerges as a consequence of the interactions being long-range and the mean-field approximation being exact. In fact, we show in the sequel that the tan h relation between the stream function and the averaged azimuthal velocity spins in the VK set-up has some origin in this analogy with the Curie–Weiss model.

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3.3.2. Analogy with a lattice model: the Euler perspective.

The statistical mechanics of inviscid fluids, as developed by Robert, Sommeria and Miller [6, 7] for the 2D case, aims to determine the coarse-grained configuration (if any) that a perfect fluid is the most likely to adopt, were it described in terms of an equilibrium ensemble, be it micro-canonical, canonical, grand-canonical. The coarse-grained description can usually be achieved in terms of a 'macro-state probability field' p. In the case of axially symmetric fluids, the latter is defined as

$$\begin{eqnarray}&&{{p}_{{\bf r}}}(\sigma ,\xi )={\rm Proba}(r{{u}_{\phi }}=\sigma \;{\rm and}\;{{r}^{-1}}{{\omega }_{\phi }}=\xi \;{\rm in}\ {\rm the}\ {\rm vicinity}\ {\rm of}\;{\bf r}).\end{eqnarray} \tag{ 14 }$$

As the number of degrees of freedom of the underlying axially symmetric Euler equations is formally infinite, the statistical ensembles and corresponding macro-state probability fields need to be defined through an appropriate discretization and coarse graining, whose details are here omitted—see [29] for the axially symmetric case and [37, 45] for the 2D case. This construction provides a useful analogy with standard lattice models of ferro-magnetism. Indeed, the statistical ensemble that we wish to compute can now be thought of as the thermodynamic spin-wave limit of a finite-size bi-dimensional lattice model of N² interacting spins ${\bf S}$, whose Hamiltonian is prescribed by the specific shape of the inviscid dynamical invariants given by equation (8). In the axially symmetric case, the spins ${{{\bf S}}_{ij}}=({{\sigma }_{ij}},{{\xi }_{ij}})$ are two-degrees-of-freedom objects that represent the micro-scale value of the pair $(r{{u}_{\phi }},{{\omega }_{\phi }}/r)$ at position ${{{\bf r}}_{ij}}$ ($1\leqslant i,j\leqslant N$). The analogy is clearer in the particular situation where the ${{\sigma }_{ij}}{\rm s}$ can take only two values, say ${{\sigma }_{\pm }}=\pm 1$ ⁹ , (which would be the natural assumption if we wanted, for example, to study the statistics of the signs of the ${{\sigma }_{ij}}{\rm s}$ only). In this case, the appropriate micro-canonical lattice model interaction Hamiltonian per spin is

$$\begin{eqnarray}&&{\mathcal{H}}/{{N}^{2}}=\frac{1}{{{N}^{4}}}\mathop{\sum }\limits_{i,j,k,l}J_{kl}^{ij}{{\xi }_{ij}}{{\xi }_{kl}}\;\;{\rm with}\ \ \ J\;{\rm a}\;{\rm discretization}\;{\rm of}-{{{\mathcal{L}}}^{-1}},\end{eqnarray} \tag{ 15 }$$

subject to the constraint that both the toroidal magnetization $I=(1/{{N}^{2}}){{\sum }_{i,j}}{{\sigma }_{ij}}$ and the poloidal conditional magnetizations ${{\Gamma }_{\pm }}=(1/{{N}^{2}}){{\sum }_{i,j:{{\sigma }_{ij}}=\pm }}{{\xi }_{ij}}$ remain constant. However, it turns out that the statistical ensembles related to this 'axially- symmetric lattice model' are ill-defined in the thermodynamic limit [27, 29]. The problem comes from the poloidal degrees of freedoms, namely the ${{\xi }_{ij}}{\rm s}$, which are not sufficiently constrained by the prescription that both the energy and the conditional magnetizations remain finite in the limit $N\to \infty$. As a consequence, an ultra-violet (UV) catastrophe occurs: unless the averaged poloidal field is uniformly zero, the most probable configurations are those for which most of the ${{\xi }_{ij}}{\rm s}$ are infinite and mask the coherent structures. Two problems are now apparent. The first one is theoretical. It relates to the existence or not of equilibrium measures for the axially symmetric Euler equations. The second problem is more practical: how can an ill-defined statistical theory possibly describe the topology of the flows observed inside the VK set-up? As previously emphasized, the present paper does not deal with the axially symmetric Euler equations but with VK turbulence. Therefore, the first question is clearly far beyond our present concerns. We now describe a strategy to deal with the second question.

3.3.3. A phenomenological treatment for the vorticity fluctuations.

The main idea. Over the past decade, many statistical theories have been brought up to describe the statistical equilibria of the ideal axially–symmetric perfect fluid [27–29, 40, 46, 47]. Those theories yield different outcomes. The reason stems from the existence of a so-called UV catastrophe in the ideal theory for the axisymmetric fluids. In other words: due to the special nature of the inviscid invariants, a full equilibrium statistical theory for the axially symmetric fluid cannot predict both a finite non-zero energy and a non-trivial large-scale flow. This degeneracy originates from the non-zero vorticity stretching term, which allows any initial vorticity field to grow unbounded in the limit of a vanishing viscosity. In order to write down a self-consistent and non-degenerate statistical theory, one therefore needs to model the statistics of the small scales. Different Ansätze lead to different outcomes that may or may not be relevant to describe VK flows. In order to select the suitable Ansatz, we first review the gist of the theories in the next paragraph. Their outcomes are summarized in table 2. We then discuss the relevance of those theories to VK flows and define two statistical inverse temperatures.

Table 2.  Equilibria obtained using different ansatz for the azimuthal vorticity ξ. Greek letters denote Lagrange multipliers, Latin letters combinations of those—that are different in each cell. In the Gaussian modeling only the quadratic invariants are taken into account, while in the two-level theory it is assumed that the toroidal field can only take two values ±1. More details can be found in the appendix.

Name	Macro state	Gaussian	Two-level
Frozen-${\bf r}$	$\delta (\xi -{{\xi }_{0}}({\bf r})){{p}_{{\bf r}}}(\sigma |\xi )$	$\langle \sigma \rangle =\left( \mu -h{{\xi }_{0}}({\bf r}) \right)/D({\bf r})$	$\langle \sigma \rangle ={\rm tan} h(A+B{{\xi }_{0}}({\bf r}))$
		$\langle \xi \rangle ={{\xi }_{0}}({\bf r})$	$\langle \xi \rangle ={{\xi }_{0}}({\bf r})$
Frozen-σ	$p_{\bf r}(\sigma )\underbrace{{p}_{\bf r}(\xi |\sigma )}_{\rm prescribed}$	$\langle \sigma \rangle =(C-\beta A({\bf r})\psi ({\bf r}))/D({\bf r})$	$\langle \sigma \rangle ={\rm tan} h({{A}_{1}}+{{B}_{1}}\psi ({\bf r}))$
		$\langle \xi \rangle =A({\bf r})\langle \sigma \rangle +B$	$\langle \xi \rangle =\frac{{{\xi }_{+}}{{{\rm e}}^{{{A}_{+}}+{{B}_{+}}\psi }}+{{\xi }_{-}}{{{\rm e}}^{{{A}_{-}}+{{B}_{-}}\psi }}}{{{{\rm e}}^{{{A}_{+}}+{{B}_{+}}\psi }}+{{{\rm e}}^{{{A}_{-}}+{{B}_{-}}\psi }}}$
Gaussian	${{p}_{{\bf r}}}(\sigma ,\xi )$	$\langle \sigma \rangle =(A({\bf r})+B({\bf r})\psi ({\bf r}))/D({\bf r})$	$\langle \sigma \rangle ={\rm tan} h({{A}_{1}}+{{B}_{1}}\psi ({\bf r}))$
		$\langle \xi \rangle =(C({\bf r})+F({\bf r})\psi ({\bf r}))/D({\bf r})$	$\langle \xi \rangle =-\beta \psi /(4\nu )$
	with $\langle {{\xi }^{2}}\rangle \lt +\infty$		$+\frac{1}{2\nu }\frac{{{g}_{+}}{{{\rm e}}^{{{A}_{+}}+{{B}_{+}}\psi }}+{{g}_{-}}{{{\rm e}}^{{{A}_{-}}+{{B}_{-}}\psi }}}{{{e}^{{{A}_{+}}+{{B}_{+}}\psi }}+{{e}^{{{A}_{-}}+{{B}_{-}}\psi }}}$
Micro canonical	${{p}_{{\bf r}}}(\sigma ,\xi )$	$\langle \sigma \rangle =A({\bf r})$	$\langle \sigma \rangle ={{A}_{1}}({\bf r})$
	with $\langle {{\xi }^{2}}\rangle \to +\infty$	$\langle \xi \rangle =B\psi ({\bf r})+C$	$\langle \xi \rangle ={{B}_{1}}\psi ({\bf r})+{{C}_{1}}$

The gist of the ideal theories. As just mentioned, common to all the works on axially symmetric fluid equilibria is the use of additional phenomenological assumptions to control the fluctuations of the poloidal degrees of freedom. Ensemble averages are usually computed in terms of a most probable macro-state probability field that maximizes a macro-state entropy, determined by an appropriate use of Laplace's theorem, and the saddle-point method. The idea is therefore to restrict the set of macro-state probability fields over which the maximization is performed. [46, 47] impose that the axially symmetric flow has a vanishing toroidal field, so that an extra-enstrophy constraint appears for the poloidal field; [27, 28, 40] assume that the poloidal field is non-fluctuating, namely, $\langle \omega _{\phi }^{2}\rangle ={{\langle {{\omega }_{\phi }}\rangle }^{2}}$ at any position ${\bf r}$; [48] suggests to freeze the poloidal degrees of freedom, namely, write the macro-state probability field as ${{p}_{{\bf r}}}(\sigma ,\xi )={{p}_{{\bf r}}}(\sigma ){{p}_{{\bf r}}}(\xi |\sigma )$ and prescribe the conditional probability distribution ${{p}_{{\bf r}}}(\xi |\sigma )$; it was also suggested to impose a cut-off on the poloidal degrees of freedom, either used as a physical parameter [27] or as an intermediate regularization constraint [29].

Table 2 presents the different outcomes that the various Ansätze lead to. Results are here summarized for two cases. In the 'Gaussian modeling', only the quadratic and linear invariants enter the theory. In the two-level modeling, it is assumed that the toroidal degrees of freedom can only take the two values±1. Both cases give consistent results and are easily extended to the general case (see the appendix). Table 2 shows that the different modelings yield different outcomes. This suggests that the poloidal fluctuations play a crucial role in VK mixing.

Discussion: temperatures and VK mixing. If one indeed takes for granted that the axially symmetric Euler equations can be used as a guideline to understand the VK topologies—which we recall is far from obvious but relies on experimental observations—then those statistical theories shed a qualitative light on the mixing processes at stake inside the VK tank. On the one hand, the assumption that the poloidal field is non-fluctuating and fixed in space (the 'frozen-${\bf r}$' model), so that only toroidal degrees of freedom do indeed mix, is not satisfying. This assumption yields only one of the two constitutive thermodynamic equations (10). It thus does not allow to self-consistently determine the averaged field without a further Ansatz about the poloidal profile. On the other hand, if one allows for too high a level of fluctuations in the theory (the 'micro-canonical theory'), one obtains a closed but very restricted class of profiles, viz., the '$(h=0)$' solutions of equation (10). Those solutions correspond to cases where the coarse-grained toroidal field is completely decoupled from the stream function. Those solutions may therefore be formally relevant for an axially symmetric perfect fluid, but are clearly not those observed inside the VK tank. Rather, the mixing inside the tank seems to lie in between those two extreme cases. The 'frozen-σ' theory provides such an example of mixing. The theory describes poloidal degrees of freedom that cannot mix independently from the toroidal ones. Observe from table 2 that this assumption leads to a tan h relation between the coarse-grained toroidal field and the stream-function, as is apparent in figure 7.

This interpretation of statistical mechanics provides a zero order approximation to the actual nature of the VK stirring. The difference between the equilibrium steady states (the 'h = 0' case) and the non-equilibrium ones is here provided through a restriction of phase space. This idea is similar to the notion of restricted partition functions recently proposed by Herbert [49] to explore the nature of inverse cascades in helical turbulence. Finally, note that our discussion was here restricted to predictions for the averages of first-order quantities. Obviously, a statistical theory based on an 'axi-symmetric Ansatz' fails to give an accurate and quantitative view about the fluctuations measured in VK turbulence, as those are clearly observed not to be axially symmetric—see figure 2. It is, however, interesting to remark that the minimal 'frozen-σ' theory does not prescribe any shape for the poloidal fluctuations. In particular, those are not prescribed to be Gaussian, as one could in principle expect from a statistical mean field theory that uses quadratic invariants as inputs. Because of the seemingly crucial role played by the fluctuations, it is natural to investigate in more detail both the poloidal and the toroidal fluctuations inside the VK tank. To this end, we introduce two quantities, which can be thought of as two toroidal and poloidal inverse temperatures, namely, ${{\beta }_{{\rm tor}}}$ and ${{\beta }_{{\rm pol}}}$. Those are defined by

$$\begin{eqnarray}&&\beta _{{\rm tor}}^{-1}={{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}\left( {{\langle u_{\phi }^{2}\rangle }_{\circ }}-\langle {{u}_{\phi }}\rangle _{\circ }^{2} \right)\quad {\rm and}\quad \beta _{{\rm pol}}^{-1}={{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}\left( {{\langle \omega _{\phi }^{2}\rangle }_{\circ }}-\langle {{\omega }_{\phi }}\rangle _{\circ }^{2} \right).\end{eqnarray} \tag{ 16 }$$

These temperatures should depend on the experimental control parameters that can now be seen as analogous of 'thermostats'. In figure 8, we show, for example, the dependence of $1/{{\beta }_{{\rm tor}}}$ as a function of the angle α (left panel) and the Reynolds number Re (right panel). It can be seen that the toroidal temperature increases with decreasing α, and increases from zero past $\operatorname{Re}\sim {{10}^{3}}$ (the laminar/turbulent transition). The temperature peaks around $\operatorname{Re}=40000$ for TM60(+) impellers, and then saturates or slightly decreases. Both the toroidal and the poloidal temperature will allow us to describe VK topologies at any control parameter from a statistical perspective, which goes beyond the insight of the inviscid theories.

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A major drawback of the theories described in the previous section is their intrinsic rooting on an inviscid description (i.e. force-free, zero viscosity) of the VK flows. As such, they do not predict anything about finite Reynolds number effects. Besides, the presence of forcing and dissipation, as well as the lack of instantaneous axial-symmetry concur to destroy the conservation of the global invariants on which the statistical description relies. However, the broad description of the VK flow topologies in terms of an equilibrium statistical theory allows for a thermodynamical interpretation of the transitions, in the spirit of the Curie–Weiss theory of ferro-magnetism. Here, we shall not specify in as much detail as in the previous section the spin lattice model that we consider. What we retain from the analogy is the following. The VK flow can be seen as a lattice model, whose spins each have two components: one linked with the toroidal velocity ${{u}_{\phi }}$, the other one linked with the toroidal vorticity ${{\omega }_{\phi }}$. Those two-component spins evolve under the action of both a thermostat and a symmetry-breaking external field, which are provided by the four control parameters Re, γ, θ, α. The ability of the spin to orientate itself as a function of the forcing can be traced by the local 'magnetization vector' $(r{{u}_{\phi }}({\bf r}),{{\omega }_{\phi }}({\bf r}))$, whose (space-time) average reads:

$$\begin{eqnarray}&&{\bf M}=(I,\Gamma ).\end{eqnarray} \tag{ 17 }$$

As illustrated in figure 9, different shapes for the propellers give rise to different behavior for ${\bf M}$, implying different 'preferred orientations' for the spins. We think that the propensity of each spin to deviate from this orientation can be captured by the behavior of the two temperatures ${{\beta }_{{\rm pol}}}$ and ${{\beta }_{{\rm tor}}}$, based on toroidal vorticity and velocity fluctuations (see equation (16))—The more curved the propellers, the higher the temperatures. In the same way, increasing the Reynolds number for a given shape increases the temperatures. In the spirit of statistical physics, the transition from a two-cell state towards a one-cell state can be thought of as a symmetry-breaking transition. It is then natural to introduce a susceptibility vector ${\boldsymbol{ \chi }}$ as:

$$\begin{eqnarray}&&{\boldsymbol{ \chi }} =({{\chi }_{I}},{{\chi }_{\Gamma }})={{\left. \frac{\partial {\bf M}}{\partial \theta } \right|}_{\operatorname{Re},\alpha }}\end{eqnarray} \tag{ 18 }$$

The complete analogy is summarized in table 3. It allows for an interpretation of the salient hydrodynamical observations previously observed in the VK set-up in terms of phase transition and critical exponents. With this analogy in mind, a new light can be shed on the finite-Reynolds number 'phase transition' observed in the VK set-up by [34, 35]. The main signature of the phase transition is obtained by monitoring the behavior of the susceptibility ${{\chi }_{I}}(\theta =0)$, in order to characterize the situation of a weak symmetry breaking due to the external forcing. In [34, 35], the control parameter was taken to be the logarithm of the Reynolds number, in analogy with a definition formulated by Castaing in [50] for the temperature of a turbulent flow. Here, we show that the phase transition can be identified and further characterized by the fluctuations of ${{u}_{\phi }}$ and ${{\omega }_{\phi }}$, which play the role of temperatures, as suggested by the inviscid statistical theory.

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Table 3.  Analogy between a two components spin system and the VK flow.

'Ferro-magnetic' Quantity	Hydrodynamic Analog	Name
Spin	$(r{{u}_{\phi }},{{\omega }_{\phi }}/r)$	'Beltrami Spin'
Magnetization	${\bf M}=(I,\Gamma )$	Angular momentum and circulation
Thermostat		Forcing and dissipation
Temperature	$(1/{{\beta }_{{\rm tor}}},1/{{\beta }_{{\rm pol}}})$	Fluctuations
Symmetry breaking fields	$(\theta ,\gamma )$	Rotation and torque numbers
Susceptibility	$({{\chi }_{I}},{{\chi }_{\Gamma }})$

The phase transition is made apparent by the study of the behavior of the magnetization as a function of the temperature $1/{{\beta }_{{\rm tor}}}$. This is shown in figure 10, where the mean magnetizations at $\theta =0$ for all impellers (any α) and Reynolds numbers have been gathered. The data collapse nicely on a well-defined curve. Compared to a ferro-magnetic system, the behavior is here reversed. A non-zero magnetization occurs when the temperature is higher than a critical temperature $1/\beta _{{\rm tor}}^{\star }({\bf M})\simeq 0.044\pm 0.03$. The data then suggest that the magnetization grows as the square root of the distance to the critical temperature, namely, $|{\bf M}|\propto \sqrt{1/{{\beta }_{{\rm tor}}}-1/\beta _{{\rm tor}}^{\star }}$, a behavior reminiscent of standard lattice models with mean-field interactions between the spins. Similarly, the behavior of the susceptibility χ seems to exhibit a divergence around a critical temperature $1/\beta _{{\rm tor}}^{\star }(\chi )\simeq 0.044\pm 0.03$—see figure 11. Because of the crucial role played by the poloidal fluctuations in the statistical theories described in section 3, it is tempting to check whether some sort of fluctuation–dissipation relation holds, involving both the poloidal and the toroidal fluctuations. Under its simplest expression, a fluctuation relation involving the susceptibility ${{\chi }_{\Gamma }}$ can be written as

$$\begin{eqnarray}&&{{\chi }_{\Gamma }}={{\beta }_{{\rm tor}}}{{\int }_{{{{\mathcal{D}}}_{{\rm piv}}}}}\left( {{\langle \omega _{\phi }^{2}\rangle }_{\circ }}-\langle {{\omega }_{\phi }}\rangle _{\circ }^{2} \right)\propto {{\beta }_{tor}}/{{\beta }_{pol}}.\end{eqnarray} \tag{ 19 }$$

To check whether such a relation holds, in figure 12 we have plotted ${{\chi }_{\Gamma }}$ against the quantity ${{\beta }_{{\rm tor}}}/{{\beta }_{{\rm pol}}}$. Indication of a linear trend is apparent. This is compatible with the observation made in the previous section—that the poloidal and the toroidal degrees of freedom are far more correlated in a VK set-up than a purely axially symmetric interpretation of the dynamics would imply. This observation is in a sense compatible with the 'frozen-σ' scenario, which tells us that the poloidal degrees of freedom are somehow enslaved to the toroidal ones. Note that the linear trend also exists for ${{\chi }_{I}}$. In both cases, the prefactor is small (of the order of $1/1000$ to $1/20000$). In a Beltrami approximation, the ratio ${{\beta }_{{\rm tor}}}/{{\beta }_{{\rm pol}}}$ is inversely proportional to the mode numbers, i.e. to the number of degrees of freedom [28]. This remark would therefore provide an interpretation of the prefactor in the fluctuation–dissipation relation as the inverse number of degrees of freedom. This would give an estimate for the number of degrees of freedom of the order of $N\sim {{10}^{3}}$ to 10⁴—much smaller than the traditional estimate $N={{\operatorname{Re}}^{9/4}}$ (which would rather give a number of the order of 10¹³).

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A simple but extreme way to further constrain the optimization problem (A.4) is to assume that the poloidal field at position ${\bf r}$ is prescribed and non-fluctuating. We therefore solve (A.4) among the macro states of the kind:

$$\begin{eqnarray}&&{{p}_{{\bf r}}}(\sigma ,\xi )={{p}_{{\bf r}}}(\sigma |\xi )\delta (\xi -{{\xi }_{0}}({\bf r})).\end{eqnarray} \tag{ A.8 }$$

Whatever the field ${{\xi }_{0}}$, the macro-state entropy (A.3) now reads ${\mathcal{S}}[p]=-\int {{p}_{{\bf r}}}(\sigma |{{\xi }_{0}}){\rm log} {{p}_{{\bf r}}}(\sigma |{{\xi }_{0}})$. The extremal points of ${\mathcal{S}}[p]$ are achieved for the now correctly defined macro states:

$$\begin{eqnarray} p_{{\bf r}}^{\star }(\sigma ,\xi ) & = & \frac{1}{{\mathcal{Z}}_{{\bf r}}^{\star }}{\rm exp} \left\{ -\beta \frac{{{\sigma }^{2}}}{2{{r}^{2}}}+f(\sigma )+{{\xi }_{0}}({\bf r})g(\sigma ) \right\}\delta (\xi -{{\xi }_{0}}({\bf r})), \\ \quad {\rm where}\quad {\mathcal{Z}}_{{\bf r}}^{\star } & = & {{\int }_{\sigma }}{\rm exp} \left\{ -\beta \frac{{{\sigma }^{2}}}{2{{r}^{2}}}+f(\sigma )+{{\xi }_{0}}g(\sigma ) \right\}\delta (\xi -{{\xi }_{0}}({\bf r})). \\ \end{eqnarray} \tag{ A.9 }$$

An easy calculation then yields:

$$\begin{eqnarray} \langle \sigma \rangle & = & {{\int }_{\sigma ,\xi }}{{p}_{{\bf r}}}(\sigma ,\xi )\sigma =(\mu -h{{\xi }_{0}})/(2\kappa +2\beta /{{r}^{2}})\quad {\rm in}\;{\rm the}\;{\rm Gaussian}\;{\rm case},\quad \\ {} & = & {\rm tan} h(A+B{{\xi }_{0}}({\bf r}))\quad {\rm in}\;{\rm the}\;{\rm two}-{\rm level}\;{\rm case}.\quad \\ \end{eqnarray} \tag{ A.10 }$$

We write $D({\bf r})=2\kappa +2\beta /{{r}^{2}})$, $A=({{f}_{+}}-{{f}_{-}})/2$ and $B=({{g}_{+}}-{{g}_{-}})/2$. In both cases, we also obtain $\langle \xi \rangle ={{\xi }_{0}}({\bf r})$, as indicated in table 2.

Another way to remove the UV catastrophe is to prescribe that not only the poloidal averages conditioned on the toroidal areas but the whole conditional distributions $p(\xi |\sigma )$ are prescribed. We then choose to solve the optimization problem (A.4) over macro states of the kind

$$\begin{eqnarray}&&{{p}_{{\bf r}}}(\sigma ,\xi )={{p}_{{\bf r}}}(\sigma ){{p}_{{\bf r}}}(\xi |\sigma ),\quad {\rm with}\quad {{p}_{{\bf r}}}(\xi |\sigma )\quad {\rm being}\;{\rm prescribed}\quad \;a\;priori.\end{eqnarray} \tag{ A.11 }$$

The macro-state entropy (A.3) can now be written in terms of the (prescribed) entropy ${{s}_{{\bf r}}}(\sigma )$ of the conditional distributions as

$$\begin{eqnarray}&&{\mathcal{S}}[p]=-{{\int }_{{\mathcal{D}}}}{{\int }_{\sigma }}{{p}_{{\bf r}}}(\sigma )({\rm log} {{p}_{{\bf r}}}(\sigma )-{{s}_{{\bf r}}}(\sigma ))\quad {\rm with}\quad {{s}_{{\bf r}}}(\sigma )=-{{\int }_{\xi }}{{p}_{{\bf r}}}(\xi |\sigma ){\rm log} {{p}_{{\bf r}}}(\xi |\sigma ).\end{eqnarray} \tag{ A.12 }$$

Defining $\xi (\sigma )={{\int }_{\sigma }}\xi {{p}_{{\bf r}}}(\xi |\sigma )$, we find that the macro states (A.11) that solve the optimization problem and extremize ${\mathcal{S}}[p]$ are those satisfying

$$\begin{eqnarray} p_{{\bf r}}^{\star }(\sigma ) & = & \frac{1}{{\mathcal{Z}}_{{\bf r}}^{\star }}{\rm exp} \left\{ {{s}_{{\bf r}}}(\sigma )+\xi (\sigma )g(\sigma )+f(\sigma )-\beta {{\sigma }^{2}}/2{{r}^{2}}-\beta \xi (\sigma )\psi /2 \right\} \\ {\rm with}\quad {\mathcal{Z}}_{{\bf r}}^{\star } & = & {{\int }_{\sigma }}{\rm exp} \left\{ {{s}_{{\bf r}}}(\sigma )+\xi (\sigma )g(\sigma )+f(\sigma )-\beta {{\sigma }^{2}}/2{{r}^{2}}-\beta \xi (\sigma )\psi /2 \right\}. \\ \end{eqnarray} \tag{ A.13 }$$

In the two-level case, we write ${{\xi }_{\pm }}={{\int }_{\sigma }}\xi {{p}_{{\bf r}}}(\xi |\sigma =\pm )$, and easily obtain

$$\begin{eqnarray}&&\langle \sigma \rangle ={\rm tan} h({{A}_{1}}+{{B}_{1}}\psi )\quad {\rm and}\quad \langle \xi \rangle =\frac{\mathop{\sum }\limits_{\pm }{{\xi }_{\pm }}{\rm exp} \left\{ {{s}_{{\bf r},\pm }}+{{\xi }_{\pm }}{{g}_{\pm }}+{{f}_{\pm }}-\beta {{\xi }_{\pm }}\psi /2 \right\}}{\mathop{\sum }\limits_{\pm }{\rm exp} \left\{ {{s}_{{\bf r},\pm }}+{{\xi }_{\pm }}{{g}_{\pm }}+{{f}_{\pm }}-\beta {{\xi }_{\pm }}\psi /2 \right\}}\end{eqnarray} \tag{ A.14 }$$

with ${{A}_{1}}=(1/2)({{s}_{{\bf r},+}}+{{\xi }_{+}}{{g}_{+}}+{{f}_{+}}-{{s}_{{\bf r},-}}+{{\xi }_{-}}{{g}_{-}}+{{f}_{-}})$ and ${{B}_{1}}=-\beta ({{\xi }_{+}}-{{\xi }_{-}})/2$.

In the Gaussian modeling, one needs to specify a shape for $\xi (\sigma )$ and the entropies ${{s}_{{\bf r}}}(\sigma )$ in order to carry the calculation further. If one assumes for simplicity $\xi (\sigma )=A({\bf r})\sigma +B$, and ${{s}_{{\bf r}}}(\sigma )=const.$, one readily obtains

$$\begin{eqnarray}&&\langle \sigma \rangle =\frac{\gamma A+\mu -hB-A\beta \psi /2}{2\left( \kappa +\beta /2{{r}^{2}}+Ah \right)}\quad {\rm and}\quad \langle \xi \rangle =A\langle \sigma \rangle +B.\end{eqnarray} \tag{ A.15 }$$

We therefore put $C=2(\gamma A+\mu -hB)$, $D=4\left( \kappa +\beta /2{{r}^{2}}+Ah \right)$ to obtain the result of table 2.

An alternative way to give some sense to the problem (A.4) is to restrict the maximization procedure to the macro-state fields, ps whose local poloidal fluctuations are everywhere bounded, namely the ones for which a prescribed $\lambda \lt \infty$ exists such that ${{\int }_{\sigma ,\xi }}{{\xi }^{2}}{{p}_{{\bf r}}}(\sigma ,\xi )\lt \lambda$. One could rather wish to plug into the integral a r² coefficient so as to make this additional macro-state constraint the exact counterpart of the poloidal viscous dissipative term, but this subtlety goes beyond the present point. In the Gaussian modeling, the partition function logarithm of equation (A.5) can be computed as

$$\begin{eqnarray}&&{\rm log} {{{\mathcal{Z}}}_{{\bf r}}}={\rm log} \pi -\frac{1}{2}{\rm log} \delta ({\bf r})+\frac{\nu {{\mu }^{2}}-h\mu (\gamma -\beta \psi /2)+(\kappa +\beta /2{{r}^{2}}){{(\gamma -\beta \psi /2)}^{2}}}{4\delta ({\bf r})},\end{eqnarray} \tag{ A.16 }$$

where the Gaussian integration requires both ν and $\delta ({\bf r})=(\kappa +\beta /2{{r}^{2}})\nu -{{h}^{2}}/4$ to be positive. We deduce

$$\begin{eqnarray}&&\langle \sigma \rangle =\frac{2\mu \nu -h\gamma +h\beta \psi /2}{4\delta ({\bf r})}\quad {\rm and}\quad \langle \xi \rangle =\frac{2\gamma (\kappa +\beta /2{{r}^{2}})-h\mu -(\kappa +\beta /2{{r}^{2}})\beta \psi }{4\delta ({\bf r})}.\end{eqnarray} \tag{ A.17 }$$

Hence we retrieve the expression of table 2 by putting $A=4\mu \nu -2h\gamma$, $B=h\beta$, $D=8\delta$, $C=4\gamma (\kappa +\beta /2{{r}^{2}})-2h\mu$ and $F=-2(\kappa +\beta /2{{r}^{2}})\beta$. In the two-level case, the partition function reads

$$\begin{eqnarray}&&{\rm log} {{{\mathcal{Z}}}_{{\bf r}}}=\frac{1}{2}{\rm log} \pi -\frac{1}{2}{\rm log} \nu +{\rm log} \mathop{\sum }\limits_{\pm }{{{\rm e}}^{{{f}_{\pm }}+{{({{g}_{\pm }}-\beta \psi /2)}^{2}}/(4\nu )}}.\end{eqnarray} \tag{ A.18 }$$

We deduce

$$\begin{eqnarray}&&\langle \sigma \rangle ={\rm tan} h\left( {{A}_{1}}+{{B}_{1}}\psi \right)\quad {\rm and}\quad \langle \xi \rangle =-\frac{\beta \psi }{4\nu }+\frac{1}{2\nu }\frac{\mathop{\sum }\limits_{\pm }{{g}_{\pm }}{{{\rm e}}^{{{f}_{\pm }}-{{({{g}_{\pm }}-\beta \psi /2)}^{2}}/2\nu }}}{\mathop{\sum }\limits_{\pm }{{e}^{{{f}_{\pm }}-{{({{g}_{\pm }}-\beta \psi /2)}^{2}}/2\nu }}},\end{eqnarray} \tag{ A.19 }$$

with ${{A}_{1}}=(1/2)\left( {{f}_{+}}-{{f}_{-}}+(g_{-}^{2}-g_{+}^{2})/(2\nu ) \right)$, ${{B}_{1}}=-\beta ({{g}_{-}}-{{g}_{+}})/(4\nu )$. To obtain the expression of table 2, one needs to develop the squares in the exponentials in both the numerator and the denominator of the expression for $\langle \xi \rangle$, to obtain ${{A}_{\pm }}={{f}_{\pm }}-g_{\pm }^{2}/(2\nu )$ and ${{B}_{\pm }}={{g}_{\pm }}\beta$.

Finally, in order to define a micro-canonical measure for the axially symmetric Euler equations, the following strategy was proposed in [29]: (i) regularize the optimization problem (A.4) by using a temporary cutoff. (ii) Solve the cutoff dependent optimization problem. (iii) Relax the cutoff dependency by letting the cutoff go to $\infty$ while using an appropriate scaling for the Lagrange multipliers. The result of [29] can be retrieved from the Gaussian scenario just described and equations (A.17) and (A.19). From equation (A.17), one prescribes $\beta \sim {{\beta }_{0}}\nu$, $h\sim {{h}_{0}}\nu$, $\gamma ={{\gamma }_{0}}\nu$ as $\nu \to 0$ to obtain

$$\begin{eqnarray}&&\langle \sigma \rangle =\frac{\mu }{2\kappa }\quad {\rm and}\quad \langle \xi \rangle =\frac{{{\gamma }_{0}}}{2}-\frac{{{h}_{0}}\mu }{4\kappa }-\frac{{{\beta }_{0}}\psi }{4}.\end{eqnarray} \tag{ A.20 }$$

From equation (A.17), one prescribes $\beta \sim {{\beta }_{0}}\nu$, ${{g}_{\pm }}\sim {{g}_{\pm ,0}}\nu$, ${{f}_{\pm }}\sim {{f}_{\pm ,0}}$ as $\nu \to 0$, to obtain

$$\begin{eqnarray}&&\langle \sigma \rangle ={\rm tan} h\left( \frac{{{f}_{+,0}}-{{f}_{-,0}}}{2} \right)\quad {\rm and}\quad \langle \xi \rangle =-\frac{{{\beta }_{0}}\psi }{4}+\frac{1}{2}\frac{\mathop{\sum }\limits_{\pm }{{g}_{\pm ,0}}{{{\rm e}}^{{{f}_{{{\pm }_{0}}}}}}}{\mathop{\sum }\limits_{\pm }{{e}^{{{f}_{\pm ,0}}}}}.\end{eqnarray} \tag{ A.21 }$$

In both cases, the average toroidal field is constant and the average poloidal field is linear with ψ. Hence, we retrieve the expression of table 2 by putting $A=\mu /2\kappa$, $C={{\gamma }_{0}}/2-{{h}_{0}}\mu /4\kappa$, $B={{B}_{1}}=-{{\beta }_{0}}\psi /4$, ${{A}_{1}}={\rm tan} h\left( ({{f}_{+,0}}-{{f}_{-,0}})/2 \right)$ and ${{C}_{1}}=(1/2)\left( {{\sum }_{\pm }}{{g}_{\pm ,0}}{{{\rm e}}^{{{f}_{{{\pm }_{0}}}}}} \right)/\left( {{\sum }_{\pm }}{{{\rm e}}^{{{f}_{\pm ,0}}}} \right)$.