A weakly nonlinear theory, based on the combined amplitude–multiple timescale expansion, is developed for the flow of an arbitrary fluid governed by the low-Mach-number equations. The approach is shown to be different from the one conventionally used for Boussinesq flows. The range of validity of the applied analysis is discussed and shown to be sufficiently large. Results are presented for the natural convection flow of air inside a closed differentially heated tall vertical cavity for a range of temperature differences far beyond the region of validity of the Boussinesq approximation. The issue of possible resonances of two different types is noted. The character of bifurcations for the shear- and buoyancy-driven instabilities and their interaction is investigated in detail. Lastly, the energy transfer mechanisms are analysed in supercritical regimes.