## 4 Analysis of KH Stability Near Its Threshold Around R \(\approx\) 1
### Dependance of the KH Front Inclination for a Given Incident Laser Power
* For P/P\({}_{t}\) > (P/P\({}_{t}\))\({}_{max}\) = A(1)/A\({}_{0}\)(\(\approx\)1.7), only one solution also exists for R such that R \(>\) R\({}_{max}\)\(\approx\) 2.6 (with R\({}_{max}\) being the solution of A\({}_{0}\)R\({}_{max}\)=A\({}_{G}\)(R\({}_{max}\))). * For (P/P\({}_{t}\))\({}_{min}\) < P/P\({}_{t}\) < (P/P\({}_{t}\))\({}_{max}\), two solutions, R\({}_{1}\) and R\({}_{2}\), are such that R\({}_{min}\) = A\({}_{0}\)/A\({}_{G}\)(1)(\(\approx\)0.5) \(<\) R\({}_{1}\), R\({}_{2}\)\(<\) R\({}_{max}\). In fact, it is tempting to consider that for this discontinuous variation of A(R), these two solutions for R, obtained in the power range \(\approx\)0.8 < P/P\({}_{t}\) < \(\approx\)1.7, are representative of some unstable regime that could occur on the KH front. Indeed, as these two solutions are physically possible, there is no reason that the KH is stabilized on only one solution, because the melt KH surface is not static; it is a moving fluid with ripples on its surface [34], so switching from one geometry to another is quite possible. Now, if one considers the effect of the second type of A(R) variation, which has a continuous but rather sharp transition at R = 1 (red curves in Figure 9), one can see that R/A(R) shows a decreasing part located between a maximum and a minimum located on either side of R = 1. But for the same reason as previously, this decreasing part is not physically possible because it would correspond to an increase of the aspect ratio when the incident power decreases. Therefore again, two solutions for R would be possible for a given range of normalized power P\({}_{n}\), but now inside a reduced range compared to the first case. Finally, for the third type of A(R) variation with a smoother transition around R = 1 (green curves, in Figure 9), one observes that R/A(R) is always a monotonous increasing function. Therefore, only one solution for R can be obtained, which means that a stable geometry of the KH front is obtained for any value of the incident power P\({}_{n}\). This model-based approach of the effect of the absorptivity variation around R = 1, seems to show that a stable KH front geometry is unlikely for these conditions. Due to the fact that the corresponding real variation of A(R) is rather unknown, in order to avoid this unstable zone, the use of incident powers P high enough in order to obtain aspect ratios R greater than about R\({}_{max}\) (\(\approx\)2 for steel) seems recommendable. One can also add that the range of P/P\({}_{t}\) that corresponds to the instable behavior is also dependent of the absorptivity A\({}_{0}\) of the used material. For example, by using the same discontinuous variation model of A(R), inside the \"unstable\" zone, one can compare the ranges of P/P\({}_{t}\) and the corresponding solutions R\({}_{1}\) and R\({}_{2}\) for copper (with A\({}_{0}\)\(\approx\) 0.1) and for steel (with A\({}_{0}\)\(\approx\) 0.31) (it should be recalled, however, that more realistic conditions for A(R) should of course show narrower ranges). For steel, we have seen that 0.8 < P/P\({}_{t}\) < 1.7 with R\({}_{1}\), R\({}_{2}\) varying from 0.5 to 2.6. For copper, (with A\({}_{0}\)\(\approx\) 0.10), one obtains 0.6 < P/P\({}_{t}\) < 1.9, but with R\({}_{1}\), R\({}_{2}\) varying from 0.35 to 6.5. Therefore, one sees that the resulting range for the variations of R\({}_{1}\) and R\({}_{2}\) is enlarged when the material absorptivity decreases, and higher laser power is required to enter in the stable KH regime. It is interesting to notice that [PERSON]'s experiments on copper welding [27], clearly show that only high incident laser powers, which generate deep KHs, produce high quality welds. A possible consequence of this result would be to consider that any increase of the absorptivity, either by using materials with higher absorptivities or by using shorter laser wavelengths, should improve the stability of the KH front for these operating conditions.