A comparison of the Graffi and Kazhikhov–Smagulov models for top heavy pollution instability

A model to describe convective overturning of a fluid layer due to density differences is derived based on equations of Kazhikhov & Smagulov. This is related to an analogous model of a reduced system based on equations of Dario Graffi. It is shown how the Graffi equations are recovered from the Kazhikhov–Smagulov equations as a non-dimensional parameter G, the Graffi number, tends to zero. The model is analysed numerically and instability thresholds are derived. It is seen that the results are realistic for small diffusion but for relatively large diffusion the approximation of Kazhikhov and Smagulov may have to be replaced by the full non-linear version. The question of spurious eigenvalues is addressed in two versions of the Chebyshev tau method employed in the numerical solution of the instability problem. It is seen that for the Kazhikhov–Smagulov theory the question of spurious eigenvalues is a non-trivial one.