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 Dar io Graffi. It is shown how the Graffi equations are recovered from the Kazh ikhov-Smagulov equations as a non-dimensional parameter G, the Graffi numbe r, tends to zero. The model is analysed numerically and instability thresho lds are derived. It is seen that the results are realistic for small diffus ion but for relatively large diffusion the approximation of Kazhikhov and S magulov may have to be replaced by the full non-linear version. The questio n of spurious eigenvalues is addressed in two versions of the Chebyshev tau method employed in the numerical solution of the instability problem. It i s seen that for the Kazhikhov-Smagulov theory the question of spurious eige nvalues is a non-trivial one. (C) 2001 Elsevier Science Ltd. All rights res erved.