Abstract As an extension of CCI (conditional convective instability) theory, three-dimensional, deep and steady convective waves in a barotropic basic flow with an embedded moist zone were studied via perturbation methods. The basic state is assumed to be slightly supercritical to the gravest two-dimensional CCI mode which gives the cross-sectional structure for the leading order solution. Along the basic flow the inviscid solution is a solitary wave, manifesting an energy conservation of O(ϵ3). Far upstream (downstream), the exponential growth (decay) of the perturbation is similar to the linear, growing (decaying) CCI mode, except that the time variation now is related to the spatial variation through the basic flow advection. At the wave peak, the vertical displacement is “overshot” in that the buoyancy is maximally decreased (increased) in the moist ascent (dry descents) of the deep convection due to the strong warming effect and expansion of the moist zone in the upper levels, which tends to reverse ...