2 Model and control parameters

We consider the fixed-boundary problem of a finite $\unicode[STIX]{x1D6FD}$ equilibrium in a classical $l=2$ stellarator (Freidberg Reference Freidberg2014). Namely, we must provide (i) the geometry of the boundary, e.g. via the Fourier coefficients of the cylindrical coordinates defining the boundary surface, $\{R_{mn},Z_{mn}\}$ ; (ii) the pressure profile as a function of the enclosed toroidal magnetic flux, $p(\unicode[STIX]{x1D6F9})$ ; and (iii) an additional profile, e.g. the rotational transform, , or the net toroidal current, $I_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D6F9})$ . The total toroidal magnetic flux enclosed by the boundary, $\unicode[STIX]{x1D6F9}_{\text{edge}}$ , is also provided but its value is irrelevant and only acts as a global scale-factor for the magnetic field strength.

2.1 Boundary

The simplest boundary representation that can model an $l=2$ stellarator is that of a rotating ellipse with no toroidally averaged elongation. Namely,

(2.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}R(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})=R_{00}+R_{10}\cos \unicode[STIX]{x1D703}+R_{11}\cos (\unicode[STIX]{x1D703}-N_{p}\unicode[STIX]{x1D711})\\[5.0pt] Z(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})=Z_{00}+Z_{10}\sin \unicode[STIX]{x1D703}+Z_{11}\sin (\unicode[STIX]{x1D703}-N_{p}\unicode[STIX]{x1D711}),\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with $Z_{00}=0$ , $Z_{10}=-R_{10}$ and $Z_{11}=R_{11}$ . For our $\unicode[STIX]{x1D6FD}$ -limit study, the main parameters of interest in (2.1) are the major radius, $R_{00}$ , and the number of field periods, $N_{p}$ . In fact these can be used to vary independently the inverse aspect ratio, $\unicode[STIX]{x1D716}$ , and the vacuum rotational transform, , which are predicted to determine the ideal equilibrium $\unicode[STIX]{x1D6FD}$ -limit. We therefore choose to fix the other parameters to $R_{10}=1$ and $R_{11}=0.25$ . Two examples of such boundaries with different values of $N_{p}$ are shown in figure 1.

The inverse aspect ratio is

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D716}=\frac{r_{\text{eff}}}{R_{00}},\end{eqnarray}$$

where the effective minor radius is $r_{\text{eff}}=\sqrt{r_{\text{max}}r_{\text{min}}}$ , with $r_{\text{max}}=R_{10}+R_{11}=1.25$ and $r_{\text{min}}=R_{10}-R_{11}=0.75$ , respectively the major and minor axis of the rotating ellipse. The vacuum rotational transform can be estimated analytically (Helander Reference Helander2014) as

(2.3)

For example, for $N_{p}=5$ we get .

Figure 1. Boundary of a classical $l=2$ stellarator with $N_{p}=5$ (a) and $N_{p}=10$ (b) field periods. The inverse aspect ratio is $\unicode[STIX]{x1D716}=0.1$ and the colour represents the amplitude of the vacuum magnetic field on the boundary as computed from SPEC.

2.3 Zero-net-current versus fixed-iota

The SPEC code calculates MHD equilibria as extrema of the Multiregion, Relaxed MHD (MRxMHD) energy functional (Hole, Hudson & Dewar Reference Hole, Hudson and Dewar2007; Hudson, Hole & Dewar Reference Hudson, Hole and Dewar2007). In essence, the energy functional is the same as in conventional ideal MHD equilibrium theory (Kruskal & Kulsrud Reference Kruskal and Kulsrud1958), but the constraints under which the function is extremized are different. While in ideal-MHD the magnetic topology is continuously constrained, in MRxMHD the topology is only discretely constrained, thus allowing for partial relaxation. More precisely, the plasma is partitioned into a finite number, $N_{V}$ , of nested volumes, $V_{v}$ , that undergo Taylor relaxation. These volumes are separated by $N_{V}-1$ interfaces that are constrained to remain magnetic surfaces during the energy minimization process. For the $\unicode[STIX]{x1D6FD}$ -limit study at hand, we have $N_{V}=2$ volumes separated by one ideal-interface. The location and shape of this interface is unkown a priori and determined self-consistently by a force-balance condition. MRxMHD equilibrium states satisfy

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\times \boldsymbol{B}=\unicode[STIX]{x1D707}_{v}\boldsymbol{B}\quad \text{in the volumes} & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\hspace{-2.0pt}\left[p+\frac{B^{2}}{2\unicode[STIX]{x1D707}_{0}}\right]\hspace{-2.0pt}\right]=0\quad \text{on the interface} & \displaystyle\end{eqnarray}$$

for $v=1,2$ . In addition to providing the enclosed toroidal fluxes in each volume ( $\unicode[STIX]{x1D6F9}_{a}$ and $\unicode[STIX]{x1D6F9}_{\text{edge}}$ ), the solution to equation (2.4) requires one more parameter if the volume is a topological torus (the innermost volume) and two more parameters if the volume is an annulus (the outer volume). Hence we must provide a total of 3 parameters to determine the equilibrium solution at a given value of $\unicode[STIX]{x1D6FD}$ .

If we want to enforce a zero net-toroidal-current, $I_{\unicode[STIX]{x1D711}}=0$ , we can impose $\unicode[STIX]{x1D707}_{1}=\unicode[STIX]{x1D707}_{2}=0$ and then iterate on the total enclosed poloidal flux, $\unicode[STIX]{x1D713}_{p}$ , until the surface current has no net toroidal component. At each iteration step, the net toroidal surface current can be easily calculated as

(2.6) $$\begin{eqnarray}I_{\unicode[STIX]{x1D711}}^{\text{CS}}=\int _{0}^{2\unicode[STIX]{x03C0}}\unicode[STIX]{x27E6}\boldsymbol{B}\unicode[STIX]{x27E7}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\,\text{d}\unicode[STIX]{x1D703}\end{eqnarray}$$

by virtue of Ampère’s law. Here $\boldsymbol{e}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}\boldsymbol{x}$ and $\boldsymbol{x}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ parametrizes the surface. The iterative procedure can be implemented via a Newton method and brings $I_{\unicode[STIX]{x1D711}}^{\text{CS}}$ down to machine precision in a few steps. We refer to this mode of operation as zero-net-current.

If we want to constrain the rotational transform, , we can force it to remain constant on both sides of the ideal-interface,  and at the edge, . Once again, this can be achieved by iterating on the values of $\unicode[STIX]{x1D707}_{1,2}$ and $\unicode[STIX]{x1D713}_{p}$ . We refer to this mode of operation as fixed-iota.

We would like to remark that while the zero-net-current mode guarantees $I_{\unicode[STIX]{x1D711}}=0$ , it does not guarantee that the rotational transform remains constant and in particular we expect . Conversely, the fixed-iota mode guarantees that  remains constant on certain surfaces but in general we expect $I_{\unicode[STIX]{x1D711}}\neq 0$ , in particular at the location of the pressure-gradient.

3 High- $\unicode[STIX]{x1D6FD}$ equilibria and Shafranov shift

Figure 2 shows Poincaré plots of the equilibrium magnetic field at different values of $\unicode[STIX]{x1D6FD}$ for both the zero-net-current stellarator and the fixed-iota stellarator. In both cases there is a Shafranov shift that increases with $\unicode[STIX]{x1D6FD}$ . However, the Shafranov shift of the axis, $\unicode[STIX]{x1D6E5}_{\text{ax}}$ , increases with $\unicode[STIX]{x1D6FD}$ much faster in the zero-net-current stellarator. This can be explained as follows. For small $\unicode[STIX]{x1D6FD}$ , one expects  (Miyamoto Reference Miyamoto2005). As shown in § 4,  decreases with $\unicode[STIX]{x1D6FD}$ in the zero-net-current stellarator, thus amplifying the effect.

Figure 2. Poincaré section ( $\unicode[STIX]{x1D711}=0^{\circ }$ ) of the equilibrium magnetic field at different values of $\unicode[STIX]{x1D6FD}$ . (a) Zero-net-current stellarator. (b) Fixed-iota stellarator. Here $N_{p}=5$ and $\unicode[STIX]{x1D716}=0.1$ . Indicated in red are the boundary surface and inner interface supporting the pressure pedestal.

It is useful to define the quantity

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{0.5}\equiv \unicode[STIX]{x1D6FD}\left(\unicode[STIX]{x1D6E5}_{\text{ax}}=\frac{r_{\text{eff}}}{2}\right),\end{eqnarray}$$

namely, the value of $\unicode[STIX]{x1D6FD}$ at which the Shafranov shift of the axis reaches half of the minor radius. According to ideal-MHD equilibrium theory (Miyamoto Reference Miyamoto2005), $\unicode[STIX]{x1D6FD}_{0.5}$ is predicted to scale as

(3.2)

for large aspect ratios, $\unicode[STIX]{x1D716}\ll 1$ , and slowly varying , which is true for $N_{p}\sim 1$ . A scan in both $R_{00}$ and $N_{p}$ has been carried out in order to assess how $\unicode[STIX]{x1D6FD}_{0.5}$ scales in the numerical MHD calculations. Figure 3 shows the result of this scan. Despite the fact that SPEC allows for plasma relaxation, the scaling law (3.2) is very well reproduced in both modes of operation; except at high values of $\unicode[STIX]{x1D716}\sim 1/R_{00}$ and $N_{p}$ , for which the scaling law ceases to be valid. Moreover, the values of $\unicode[STIX]{x1D6FD}_{0.5}$ are much higher in the fixed-iota stellarator, by a factor of about 6. This reflects, once again, the fact that the Shafranov shift increases faster in the zero-net-current stellarator.

In § 4, the fundamental, macroscopic differences between the two modes of operation are explained in terms of the High-Beta-Stellarator (HBS) model developed in Freidberg (Reference Freidberg2014). In § 5, we attempt to describe and predict the $\unicode[STIX]{x1D6FD}$ -induced generation of islands and chaotic field-lines in the fixed-iota stellarator (figure 2).

Figure 3. Scaling of $\unicode[STIX]{x1D6FD}_{0.5}$ with the inverse aspect ratio, $\unicode[STIX]{x1D716}\sim 1/R_{00}$ (a), and with the vacuum iota, $\,\unicode[STIX]{x1D704}\text{-}_{v}\sim N_{p}$ (b). Black circles are for the fixed-iota stellarator. Magenta stars are for the zero-net-current stellarator. The dashed lines have slope 1 (a) and 2 (b).

Figure 4. Rotational transform as a function of toroidal magnetic flux from both SPEC (black stars) and VMEC (solid blue line) at $\unicode[STIX]{x1D6FD}=0.15\,\%$ . For comparison, the vacuum transform is also shown (dashed magenta line). Here $N_{p}=5$ , $\unicode[STIX]{x1D716}=0.1$ and the vertical dashed line indicates the location of the pressure pedestal.

4 Ideal $\unicode[STIX]{x1D6FD}$ -limit and the HBS theory

The HBS model for a classical stellarator developed in Freidberg (Reference Freidberg2014) and briefly summarized in appendix A predicts that the rotational transform at the plasma edge, , evolves with $\unicode[STIX]{x1D6FD}$ and plasma current as

(4.1)

where  is the transform produced by the net toroidal current,

(4.2)

and

(4.3)

where $a$ is the effective minor radius of the plasma edge and $\unicode[STIX]{x1D716}_{a}=a/R$ . For our system, we have $a=\sqrt{\unicode[STIX]{x1D6F9}_{a}/\unicode[STIX]{x1D6F9}_{\text{edge}}}r_{\text{eff}}$ and thus $\unicode[STIX]{x1D716}_{a}=\sqrt{\unicode[STIX]{x1D6F9}_{a}/\unicode[STIX]{x1D6F9}_{\text{edge}}}\unicode[STIX]{x1D716}$ .

In the context of the HBS theory, the zero-net-current stellarator can be analyzed by taking . Equation (4.1) then implies that  decreases with increasing $\unicode[STIX]{x1D6FD}$ . This is visible in figure 4, where the profile  obtained from SPEC at finite $\unicode[STIX]{x1D6FD}$ is shown and compared to the vacuum transform. A jump in the rotational transform self-consistently develops on the ideal interface supporting the pressure gradient, namely at $\unicode[STIX]{x1D6F9}_{a}=0.3\unicode[STIX]{x1D6F9}_{\text{edge}}$ . The ideal MHD, variational moments equilibrium code, or VMEC (Hirshman & Whitson Reference Hirshman and Whitson1983), was also run for this case with a pressure pedestal of small but finite width (the calculation requires a rather high radial resolution, with about 3000 flux surfaces) and shown to produce essentially the same transform profile.

Figure 5. Rotational transform at the plasma edge, $\,\unicode[STIX]{x1D704}\text{-}_{a}^{+}=\,\unicode[STIX]{x1D704}\text{-}(\unicode[STIX]{x1D6F9}_{a}^{+})$ , as a function of $\unicode[STIX]{x1D6FD}$ , from SPEC calculations (stars), from the HBS prediction with Solov’ev pressure profile, equation (4.1) (dashed line) and from the HBS prediction adapted to a stepped-pressure profile, equation (A 10) (dash-dotted line). Two cases are shown: $N_{p}=3$ and $N_{p}=5$ at fixed $\unicode[STIX]{x1D716}=0.1$ .

In figure 5, the value of  is shown as a function of $\unicode[STIX]{x1D6FD}$ and compared to the HBS prediction, equation (4.1), showing fairly good agreement (notice that there are no free parameters). The agreement is even more remarkable if we notice that the HBS model assumes a circular cross-section and uses Solov’ev pressure profiles. In appendix A, we show that adapting the theory to the case of a stepped-pressure profile produces very similar predictions (see figure 5). This reflects the fact that the macroscopic equilibrium depends on integral quantities (e.g., the total plasma pressure) and not so much on the details of the profiles.

The point where , which from (4.1) happens when $\unicode[STIX]{x1D708}=1$ , corresponds to the emergence of a separatrix (see, e.g., figure 2) and this defines the ideal $\unicode[STIX]{x1D6FD}$ -limit, namely

(4.4)

For example, for the $N_{p}=5$ case depicted in figure 5 we have $\unicode[STIX]{x1D716}_{a}\approx 0.05$ and , thus (4.4) gives $\unicode[STIX]{x1D6FD}_{\text{lim}}\approx 0.4\,\%$ , in good agreement with the SPEC calculations. We note that the separatrix forming at $\unicode[STIX]{x1D6FD}>\unicode[STIX]{x1D6FD}_{\text{lim}}$ seems to possess two X-points that connect to the current sheet established on the ideal-surface (see middle-left panel in figure 2); a situation that is reminiscent of the Waelbroeck ribbon forming in the reconnecting kink-tearing mode in tokamaks (Waelbroeck Reference Waelbroeck1989; Zakharov, Rogers & Migliuolo Reference Zakharov, Rogers and Migliuolo1993).

For the fixed-iota stellarator, we can impose  in the HBS model. This leads to an expression for the value of the plasma current that is necessary to keep  constant. One obtains (Freidberg Reference Freidberg2014)

(4.5)

where

(4.6) $$\begin{eqnarray}H=\frac{\unicode[STIX]{x1D6FD}}{\unicode[STIX]{x1D6FD}_{\text{lim}}}.\end{eqnarray}$$

Figure 6 shows the net toroidal surface-current, $I_{\unicode[STIX]{x1D711}}$ , self-consistently generated in SPEC equilibria as a function of $\unicode[STIX]{x1D6FD}$ and compares it to the HBS prediction, equation (4.5), showing good agreement (again, there are no free parameters). Since the predicted current is entirely pressure-driven, we do not expect currents to develop in the relaxed volumes and we have checked that for these SPEC calculations the values of $\unicode[STIX]{x1D707}_{v}$ remain small $({<}10^{-2})$ , such that the corresponding volume currents are more than 10 times smaller than the pressure-driven surface-current. For large $H\gg 1$ , one has $I_{\unicode[STIX]{x1D711}}\sim \sqrt{\unicode[STIX]{x1D6FD}}$ and the HBS model predicts that no $\unicode[STIX]{x1D6FD}$ -limit is reached because the plasma current keeps rising and preventing the separatrix to form. From SPEC equilibrium calculations, however, where plasma relaxation is allowed, we observe that magnetic islands and chaotic field-lines emerge at sufficiently high $\unicode[STIX]{x1D6FD}$ , thereby providing a ‘non-ideal $\unicode[STIX]{x1D6FD}$ -limit’.

Figure 6. Net toroidal plasma current as a function of $\unicode[STIX]{x1D6FD}$ , from SPEC calculations (stars) and from (4.5) for both Solov’ev profiles (dashed line) and stepped-pressure profile (dash-dotted line). Two cases are shown: $\unicode[STIX]{x1D716}=1/10$ and $\unicode[STIX]{x1D716}=1/20$ at fixed $N_{p}=5$ .

5 Non-ideal $\unicode[STIX]{x1D6FD}$ -limit and emergence of chaos

We can quantify the emergence of chaos by calculating the fractal dimension of the field-lines on the Poincaré section as a function of $\unicode[STIX]{x1D6FD}$ (Meiss Reference Meiss1992). More precisely, we can evaluate the so-called box-counting dimension, or Hausdorff dimension,

(5.1) $$\begin{eqnarray}D=\lim _{L\rightarrow 0}\left|\frac{\log (N)}{\log (L)}\right|,\end{eqnarray}$$

where $L$ is the size of the boxes and $N$ is the number of boxes containing at least one point of the magnetic field-line on the Poincaré section. If the field-line traces a magnetic surface, or even a magnetic island, one expects $D=1$ . If the magnetic field-line trajectory is chaotic, however, it fills up a certain ‘area’ in the Poincaré section and $D>1$ is expected. We remark that the accurate evaluation of $D$ requires a large number of toroidal transits, $N_{\text{trans}}$ , when generating the Poincaré section via field-line-tracing. Satisfactory convergence was found at values of about $N_{\text{trans}}>2\times 10^{4}$ . The value of $D$ for each field-line $i=1,\ldots ,n_{\text{lines}}$ can be plotted against an effective toroidal flux, $\unicode[STIX]{x1D6F9}_{i}\sim \unicode[STIX]{x1D70C}_{i}^{2}$ , which is proportional to the square of the radial coordinate, $\unicode[STIX]{x1D70C}_{i}$ , obtained from the interpolation of the inner and outer ideal surfaces.

Figure 7 shows the calculated fractal dimension as a function of the toroidal flux in equilibria of increasing $\unicode[STIX]{x1D6FD}$ . First, we observe that for sufficiently low $\unicode[STIX]{x1D6FD}$ we obtain $D(\unicode[STIX]{x1D6F9})=1$ , as expected, because magnetic surfaces are preserved in the entire volume. Second, we notice that for sufficiently high $\unicode[STIX]{x1D6FD}$ there are regions in which $D(\unicode[STIX]{x1D6F9})>1$ for $\unicode[STIX]{x1D6F9}>\unicode[STIX]{x1D6F9}_{a}$ . Third, the value of $D$ seems to be almost-binary, taking values at either $D\approx 1$ or $D\approx 1.6$ . Fourth, the regions with $D\approx 1.6$ correspond to what appears to be stochastic regions in the corresponding Poincaré section. Finally, the volume occupied by these regions increases with $\unicode[STIX]{x1D6FD}$ . These observations suggest that $D$ is a good proxy for the emergence of chaos, which greatly simplifies the task of probing the ‘non-ideal $\unicode[STIX]{x1D6FD}$ -limit’. In fact, we can now define the volume of chaos, $V_{\text{chaos}}$ , in the system as

(5.2) $$\begin{eqnarray}V_{\text{chaos}}=V_{\text{tot}}\mathop{\sum }_{i=1}^{n_{\text{lines}}}\frac{(\unicode[STIX]{x1D6F9}_{i}-\unicode[STIX]{x1D6F9}_{i-1})}{\unicode[STIX]{x1D6F9}_{\text{edge}}}{\mathcal{H}}(D(\unicode[STIX]{x1D6F9}_{i})-D_{\text{crit}}),\end{eqnarray}$$

where $V_{\text{tot}}$ is the total volume defined by the fixed-boundary, $n_{\text{lines}}$ is the number of traced field-lines, $\unicode[STIX]{x1D6F9}_{i}-\unicode[STIX]{x1D6F9}_{i-1}$ measures the enclosed toroidal flux between two neighbouring field lines, and ${\mathcal{H}}$ is the Heaviside function, with ${\mathcal{H}}=0$ for $D<D_{\text{crit}}=1.5$ and ${\mathcal{H}}=1$ otherwise. Figure 8 shows the profile of $V_{\text{chaos}}(\unicode[STIX]{x1D6FD})$ calculated for three different values of $N_{p}$ . Clearly, the emergence of chaos occurs at some critical value of $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\text{chaos}}$ , which we define as the non-ideal equilibrium $\unicode[STIX]{x1D6FD}$ -limit. The question remains: can we theoretically predict the value of $\unicode[STIX]{x1D6FD}_{\text{chaos}}$ ?

Figure 7. Fractal dimension of the magnetic field lines in a Poincaré section as a function of toroidal flux. The pressure pedestal is at $\unicode[STIX]{x1D6F9}/\unicode[STIX]{x1D6F9}_{\text{edge}}=0.3$ . Different curves are for different values of $\unicode[STIX]{x1D6FD}$ . All equilibria have $N_{p}=5$ and $\unicode[STIX]{x1D716}=0.1$ .

Figure 8. Volume of chaos as a function of $\unicode[STIX]{x1D6FD}$ for $N_{p}=3$ , $N_{p}=5$ and $N_{p}=10$ , at fixed $\unicode[STIX]{x1D716}=0.1$ . Vertical dashed lines indicate the predicted transition to chaos at $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\text{chaos}}$ given by (5.4).

We expect that chaos emerges due to the overlapping of magnetic islands according to the Chirikov criterion (Chirikov Reference Chirikov1979). That requires, however, predicting which resonances appear first and how does the width of the corresponding islands grow with $\unicode[STIX]{x1D6FD}$ . While we intend to investigate this question in more detail in the future, we derive here a criterion based on the following general idea. The expected width of an island generated by a resonance at  is expected to scale as  (Boozer Reference Boozer2004). Thus, new resonances can appear at finite $\unicode[STIX]{x1D6FD}$ due to (i) a change in the rotational transform, or (ii) a change in the harmonic content of $B$ inevitably caused by the Shafranov shift. As $\unicode[STIX]{x1D6FD}$ increases, the net toroidal current increases and modifies the rotational transform by increasing its value in the region $\unicode[STIX]{x1D6F9}_{a}<\unicode[STIX]{x1D6F9}<\unicode[STIX]{x1D6F9}_{\text{edge}}$ (data not shown). The rising of  allows new resonances to appear (and with lower values of $m$ ). At this point we make the following hypothesis: the emergence of chaos may occur when the perturbations in the poloidal field due to finite toroidal current are comparable to the vacuum poloidal field. Namely, chaos may emerge when . From the HBS theory we know that  increases with $\unicode[STIX]{x1D6FD}$ according to (4.5), hence applying our constraint we have

(5.3)

and hence

(5.4)

which gives $\unicode[STIX]{x1D6FD}_{\text{chaos}}\approx 0.5\,\%$ for $N_{p}=3$ , $\unicode[STIX]{x1D6FD}_{\text{chaos}}\approx 1.4\,\%$ for $N_{p}=5$ , and $\unicode[STIX]{x1D6FD}_{\text{chaos}}\approx 6.5\,\%$ for $N_{p}=10$ , thus in excellent agreement with the observed transition to chaos (see figure 8). More importantly, equation (5.4) predicts that the non-ideal equilibrium $\unicode[STIX]{x1D6FD}$ -limit scales exactly as the ideal equilibrium $\unicode[STIX]{x1D6FD}$ -limit but with a larger factor in front, of the order of $\sqrt{12}\approx 3.5$ .