Solutions are obtained for convective regions in a continuously stratified, linearized primitive equation model using a smoothly posed moist convective adjustment parameterization of cumulus convection. In the approximation in which the convective adjustment time is fast compared to other processes, the vertical structure of the temperature field is constrained to be close to the quasi-equilibrium structure determined by the convective scheme. This in turn constrains the vertical structure of the baroclinic pressure gradients and velocity field. Analytic solutions result for vertical structures, while the horizontal and time dependence is governed by equations akin to shallow water equations. These consist of equations linking baroclinic velocities and pressure gradients, plus a moist static energy equation governing thermodynamics. This system holds for basic states that are slowly varying in space, for regions where deep convection happens frequently enough to constrain the temperature field. An effective static stability for these convectively constrained motions, the gross moist stability M, is defined in terms of thermodynamic variables. In time-dependent solutions, M determines phase speeds in deep convective regions. In solutions forced by sea surface temperature, M determines the work that must be done by vertical motion, which must in turn be balanced by surface fluxes. Surface fluxes tend to draw boundary layer temperature and moisture toward values determined by SST, while the convection translates these into deep baroclinic temperature and pressure gradients. The balance between surface fluxes and the effect of the gross moist stability on vertical motion determines how closely boundary layer enthalpy can follow SST. This picture combines modified versions of mechanisms proposed in simple models by Lindzen and Nigam, and Neelin and Held within a thermodynamically consistent framework. It also helps interpret models with convergence feedback schemes and the Gill model, and allows free parameters in these models to be related to basic thermodynamic quantities.