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In this paper, we consider the classical solutions to a model of two-dimensional incompressible inviscid Boussinesq–Bénard equations. Notice that, in the case when the source term of temperature equation in this model is the second component of velocity u 2 or no source term, there is no global-in-time existence result for the general initial data. Here, if the source term is only chosen as Δ u 2 , then we can obtain the global well-posedness, inviscid limit and some exponential decay estimates. Our key observation is the nice symmetrical structure hidden in the corresponding system, which plays an extremely important role in the global well-posedness studied here. [ABSTRACT FROM AUTHOR]