Keywords: Boussinesq equations, wave-vortical interactions, quasi-geostrophic approximation, intermediate models Abstract: Starting from the rotating Boussinesq equations, we show how to derive a hierarchy of new models intermediate between the quasi-geostrophic (QG) approximation and the full equations. The new PDEs progressively include more effects of inertia-gravity waves. We explain how to derive the new models in two ways: (i) by eigenmode projection, and (ii) directly in physical space. We illustrate how the new reduced PDEs can be used to identify the nonlinear interactions primarily responsible for observed non-QG phenomena, such as cyclone/anticyclone asymmetry in geophysical flows. A waves-only model gives insight into the growth of horizontal shear flows. Simulations of the models highlight the practical implications of under-resolving wave-mode interactions in numerical calculations.