## 4 Results and Discussion
### Critical Parameters via the Neutral Stability Method  
Identifying the critical Marangoni number and associated wavenumber and frequency by continuation in \(k\) and \(M\) for fixed \(\left\{K,Pr,G\right\}\) is computationally expensive if parameter space is to be mapped. Once a single critical parameter set is known, continuation along the neutral stability surface should be possible. Ref. [14] recognized that the Marangoni number only arises linearly in Equation (20), so that it is possible to solve the tangential stress boundary condition for \(M\) and eliminate all quantities \(\alpha\), thus finding the neutral stability curve by the condition that at arbitrary \(k\) and with \(Re\left\{\omega\right\}=0\), \(Im\left\{M\right\}=0\) is imposed by adjusting \(s=Im\left\{\omega\right\}\). [PERSON] [15] also used this technique. Neither study, however, identified the long wave stationary branches put forth here. The critical parameters \(\left\{M_{crit},k_{crit},s_{crit}\right\}\) are then found by minimizing \(M\) over \(k\). In this paper, the relation \(\Delta=0\) was solved for \(M\), and neutral stability was found by using [PERSON]'s method to adjust \(s\) to achieve \(Im\left\{M\right\}=0\). Use of numerical root finding methods for \(s\) such that \(Im\left\{M\right\}=0\) are faster than those that seek values of \(s\) such that \(\Delta=0\). An algorithm for parameter space continuation in \(\left\{K,Pr,G\right\}\) for \(\left\{M_{crit},k_{crit},s_{crit}\right\}\) by traversing the neutral stability surface was developed. The essential steps are to find the \(M\)-\(k\) and \(s\)-\(k\) neutral curves by computing a neighborhood in \(k\) surrounding an estimated critical point \(\left\{M,k,s\right\}\) from adjacent values of \(\left\{K,Pr,G\right\}\). Subsequently, [PERSON]'s method is used to find the minimum of the \(M\)-\(k\) neutral curve. This is illustrated in Figure 5 for the parameters \(Pr=1.001\), \(K=0.0001\) and \(G=1\). The crossover wavenumber \(M-k\) neutral stability curve, i.e., that for which \(\Re(\omega)=0\), is plotted, along with a curve showing the functional dependence of the frequency \(s=Im(\omega)\) on the wavenumber \(k\). A space of twenty values of each of \(\left\{K,Pr,G\right\}\) was mapped by this method. Table 1 exhibits a sampling of the database built up from [PERSON] continuation in \(\left\{K,Pr,G\right\}\). The range of parameters was selected to represent some physically realizable liquids (small capillary number and O(1) Prandtl number) and layer depths (microgravity to terrestrial gravity). From Table 1, it is clear that values of \(M_{crit}\sim 10^{4}\)-\(10^{5}\) are achievable under combinations of small capillary number and thin layers in liquids. Critical wavenumbers \(k_{crit}\) are typically of intermediate scale, neither long (<0.1, say) nor short (>10, say), but usually about \(k_{crit}\sim 0.25\). Figure 4: (**a**) wavenumber \(k_{max}\); (**b**) growth rate \(Re\left\{\omega_{max}\right\}\); (**c**) group velocity \(Im\left\{\omega\right\}/k\) reduced by long wave phase velocity (26) for which \(\Delta=0\) and \(Re\left\{\omega_{max}\right\}>Re\left\{\omega\right\}\) for all \(k\) at the parametric values \(\left\{K=0.0001,Pr=1.001,G=1\right\}\) and \(M\in\left\{500,1250\right\}\). These curves identify the most dangerous mode (wavelength with most rapid growth or slowest decay). Clearly, there is neutral stability at \(M\sim 1100\). The jaggedness of (**a**) reflects the granularity of the discretization in \(k\)-\(M\)-continuation. Figure 4 is suggestive that the group velocity \(s/k\) of the critical mode might actually be well predicted by the long wave formula (20) for the phase velocity. The scatter plot in Figure 6 tests this hypothesis by crudely applying it to the entire mapped set of 8000 critical points. The critical group