## 4 Results and Discussion
### Critical Parameters via the Neutral Stability Method
\begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{K}\) & \(\mathbf{Pr}\) & \(\mathbf{G}\) & \(\mathbf{k_{crit}}\) & \(\mathbf{s_{crit}}\) & \(\mathbf{M_{crit}}\) \\ \hline \(1.0\times 10^{-2}\) & 1.001 & 1.000 & 0.2820 & 6.664 & \(1.032\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 1.833 & 0.2821 & 6.674 & \(1.034\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 3.360 & 0.2823 & 6.692 & \(1.036\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 6.160 & 0.2826 & 6.724 & \(1.041\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 1.129 \(\times 10^{1}\) & 0.2833 & 6.783 & \(1.049\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 2.069 \(\times 10^{1}\) & 0.2847 & 6.892 & \(1.064\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 3.793 \(\times 10^{1}\) & 0.2877 & 7.095 & \(1.087\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 6.952 \(\times 10^{1}\) & 0.2949 & 7.479 & \(1.120\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 1.274 \(\times 10^{2}\) & 0.3140 & 8.264 & \(1.152\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 2.336 \(\times 10^{2}\) & 0.3736 & 10.16 & \(1.130\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 4.281 \(\times 10^{2}\) & 0.5052 & 14.17 & \(1.008\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 7.848 \(\times 10^{2}\) & 0.5703 & 18.22 & \(1.042\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 1.438 \(\times 10^{3}\) & 0.5501 & 21.89 & \(1.334\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 2.637 \(\times 10^{3}\) & 0.4826 & 25.08 & \(1.979\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 4.833 \(\times 10^{3}\) & 0.3936 & 27.34 & \(3.224\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 8.859 \(\times 10^{3}\) & 0.3048 & 28.57 & \(5.548\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 1.624 \(\times 10^{4}\) & 0.2296 & 29.13 & \(9.835\times 10^{3}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 2.976 \(\times 10^{4}\) & 0.1710 & 29.38 & \(1.771\times 10^{4}\) \\ \(1.0\times 10^{-2}\) & 1.001 & 1.0 \(\times 10^{4}\) & 0.2105 & 210.3 & \(1.693\times 10^{5}\) \\ \(8.859\times 10^{-4}\) & 10. & 1.0 \(\times 10^{4}\) & 0.2102 & 210.0 & \(1.694\times 10^{5}\) \\ \(7.848\times 10^{-5}\) & 10. & 1.0 \(\times 10^{4}\) & 0.2069 & 207.4 & \(1.702\times 10^{5}\) \\ \(1.0\times 10^{-6}\) & 10. & 1.0 & 0.1563 & 89.24 & \(6.994\times 10^{4}\) \\ \(1.0\times 10^{-6}\) & 10. & 1.0 \(\times 10^{4}\) & 0.1403 & 156.2 & \(2.096\times 10^{5}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Critical values \(\{M_{crit},k_{crit},s_{crit}\}\) for fixed \(\{\Pr,G,K\}\) found by the neutral stability/minimization algorithm and parameter space continuation within the database. Samples were tested against the dispersion relation method and were found to be in agreement and global maximum of growth rate versus \(k\). The table shows only selected values; the database is \(\{\Pr\in[1,10]\}\times\{K\in[10^{-6},10^{-2}]\}\times\{G\in[10^{0},10^{5}]\}\). There are twenty values for each parameter, spaced exponentially. Figure 5: Magnitude of velocity field on \(z=0\) mm plane: (**a**) crossover wavenumber \(k\) for the \(M\)-\(k\) neutral stability curve (\(Re\left\{\omega\right\}=0\)) for the parameters \(\Pr=1.001,K=0.0001,G=1\). \(M_{crit}\) is found from the global minimum of the neutral curve and \(k_{crit}\) is the associated wavenumber; (**b**) frequency \(s=Im\left\{\omega\right\}\). velocity, \(c_{0}=\sqrt{(G+M)Pr}\), is plotted against the longwave phase velocity, \(s_{crit}/k_{crit}\), for each of the 8000 critical parameter sets considered. The plot suggests that \(c_{0}\) is the correct order of magnitude for the group velocity of the critical mode regardless of the parameter set in the range mapped. Furthermore, it is observed that the magnitude of the group velocity is constrained to be within \(\frac{1}{2}\) to \(\frac{2}{3}\) of the longwave phase velocity.