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Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) $A_t\\ll \\mathcal{O}(1)$ or (ii) $Fr_c^2 \\ll \\mathcal{O}(1)$, where $A_t$ is the ratio of density variation to the mean density and $Fr_c$ is the ratio of the phase speed to the long wave speed. The second clause implies, contrary to the commonly accepted notion, that a flow with large density variations can also be Boussinesq. 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