We study the Boussinesq system, which is the simplest fluid model that admits internal gravity waves. These dispersive waves are ubiquitous in the atmosphere and ocean and they owe their restoring mechanism to the stable stratification in these environments, i.e., to the fact that density decreases with altitude. Internal gravity waves are typically far too small in scale (especially vertical scale) to be resolvable within global numerical models and therefore their dynamics and their interactions with the mean flow must be parametrized (i.e., put in by hand) based on a combination of observations and theory. Consequently, the classic wave–mean interaction theory for internal gravity waves has been extensively developed and this provides a convenient starting point for us.

Boussinesq system and stable stratification

The simplest fluid model that captures the effect of stable internal stratification is the Boussinesq model, which can be derived from the Euler equations under the assumption of a small density contrast across the fluid together with ∇·u = 0. Importantly, the latter constraint filters sound waves and reduces the degrees of freedom compared to the Euler system.

Before writing down the governing equations we note that as a realistic model for atmospheric and oceanic flows the Boussinesq system is mostly limited by its global restriction to small density contrasts across the fluid, and that these limitations are much more severe in the atmosphere than in the ocean. Specifically, in the ocean the global density contrast is only about 3%, whilst in the atmosphere density changes by 100% in the vertical.