  The interaction of an acoustic wave with a stratified fluid can drive strong
streaming flows owing to the baroclinic production of fluctuating vorticity, as
recently demonstrated by Chini et al. (J. Fluid Mech., 744, 2014, pp. 329). In
the present investigation, a set of wave/mean-flow interaction equations is
derived that governs the coupled dynamics of a standing acoustic wave mode of
characteristic (small) amplitude {\epsilon} and the streaming flow it drives in
a thin channel with walls maintained at differing temperatures. Unlike
classical Rayleigh streaming, the resulting mean flow arises at O({\epsilon})
rather than at O({\epsilon^2}). Consequently, fully two-way coupling between
the waves and the mean flow is possible: the streaming is sufficiently strong
to induce O(1) rearrangements of the imposed background temperature and density
fields, which modifies the spatial structure and frequency of the acoustic mode
on the streaming time scale. A novel Wentzel-Kramers-Brillouin-Jeffreys
analysis is developed to average over the fast wave dynamics, enabling the
coupled system to be integrated strictly on the slow time scale of the
streaming flow. Analytical solutions of the reduced system are derived for weak
wave forcing and are shown to reproduce results from prior direct numerical
simulations (DNS) of the compressible Navier Stokes and heat equations with
remarkable accuracy. Moreover, numerical simulations of the reduced system are
performed in the regime of strong wave mean flow coupling for a fraction of the
computational cost of the corresponding DNS. These simulations shed light on
the potential for baroclinic acoustic streaming to be used as an effective
means to enhance heat transfer.
