## 5 Scaling Laws and Optimisation
### Non-Power Law Scalings and \(f_{ren}\) for Different Optimisation Levels of W7-AS  
The same analysis reported in the previous subsection has also been performed for the other device, W7-AS, whose entries in the database present a significantly different level of optimisation. In the bottom plots of Figure 3, the trend of the energy confinement time is plotted versus the ratio \(\text{R}_{\text{W-AS/ISS04}}\). For a \(t_{2/3}\) exponent of 0.3, the functional dependence of \(R_{\text{W-AS/ISS04}}\) is  
\[R_{W7-AS/ISS04}\approx 0.4168\cdot a^{-0.11}\cdot R^{-0.00}\cdot p^{-0.01} \cdot n_{e}^{0.20}\cdot B^{0.41}\cdot t_{\frac{3}{3}}^{-0.11}\cdot\frac{1}{ \left(1+1.44\cdot e^{-\frac{2R}{K_{\text{AP}}}}\right)} \tag{6}\]  
The dependency does not change dramatically for the case of \(t_{2/3}\) being elevated to 0.2. The ratio \(R_{\text{W-AS/ISS04}}\) is compared with \(f_{ren}\) in Figure 3. The part depending on the minor and major radii, \(R_{W7-AS/ISS04}\)(\(a\),\(R\)), is shown versus R in the plots of Figure 5 (while the other major dependencies are reported in Figures A3-A6). Again, for an easier visual comparison, the values of this ratio have been averaged for each level of optimisation.  
Figure 4: Analysis of LHD data for various optimisation levels vs. the major radius. (**Top left**) The part of the model Equation (2), which depends on major and major radius \(R_{\text{LHD/ISS04}}\)(\(a\),\(R\)). (**Top right**) the actual confinement time in LHD vs. the major radius. (**Bottom**) trend of \(f_{ren}\) in ISS04 vs. the major radius.  
In the case of the shearless device W7-AS, the ratio \(R_{W7-AS/ISS04}\) depends strongly on plasma density and magnetic field, whereas in the case of LHD, the most important factors are geometrical, the minor and major radii. This evidence needs to be further investigated, but some points can be already emphasised. First, the completely different trends of \(R_{LHD/ISS04}\) and \(R_{W7-AS/ISS04}\) stress again the need for particularising the scaling laws for the different magnetic configurations, with and without shear. Second, the dubious and confusing nature of \(f_{rm}\) becomes even more evident. This arbitrary factor tends to obscure more than clarify the underlying trends and the phenomenology. On the other hand, a significant effort should be devoted to understanding the causes of the different forms of Equations (5) and (6), for example, by studying whether they are due to dissimilar approaches to optimisation or to intrinsic specificities of the magnetic configuration physics.  
6 Comparison with the Tokamak in the L Mode of Confinement and Extrapolation to Stellarator Reactors  
Extrapolation is always a delicate matter. In principle, data driven equations, such as those derived in the present work, have very limited validity out of sample, where they can indeed become misleading. On the other hand, the trends of the derived scaling laws near the already explored operational space can provide useful insight. This is the objective of Section 6.1, in which the comparison with the Tokamak in L mode seems to indicate that, in optimised configurations, the dominant transport mechanism in Stellarators is probably turbulence. Larger extrapolation should be mainly aimed at checking the basis consistency of the scaling laws, meaning that they do not take absurd turns, when particularised for a parameter region of interest far from that already explored. This is the objective of  
Figure 5: Analysis of W7-AS data for various optimisation levels vs. the major radius. (**Top left**) The part of the model Equation (3), which depends on major and minor radius/_fa.R_). (**Top right**) the actual confinement time in W7-AS vs. the major radius. (**Bottom**) trend of \(f_{rm}\) in ISS04 vs. the major radius.  
Section 6.2, devoted to investigating the predictions of the various scalings to the region covered by the Stellarator main proto-reactor designs.