This paper numerically investigates the interaction between the Marangoni and the double diffusive convection in the Maxworthy [Journal of Fluid Mechanics, 1983, 128, 259-282] set-up. The model consists of a two-cavity rectangular system in which the smaller cavity is located at the top left comer of the larger one. The larger cavity is filled with hot salty fluid while the smaller one contains cold fresh fluid. This problem is solved for two different cases : the first one contains a rigid upper cavity, thereby, eliminating the Marangoni effect, and the second one contains a free-surface upper cavity. For the latter case, the interaction between the Marangoni (thermal and solutal) and the double diffusive regimes is investigated in detail. Finite element modelling results indicate that salinity induces stronger convection than the thermal ones. For the case for which a free surface exists, thermal and solutal Marangoni convection further enhances the strength of the cellular flow. Numerical results agree qualitatively with the Maxworthy experiment. It is shown that the governing equations should be solved in the complete form and that the surface tension should be taken into consideration during the numerical analysis.