When a dynamical system admits both fast and slow timescale dynamics, a simplified model can be constructed by systematically filtering out the fast dynamics. In the context of atmospheric dynamics, these models are called balance models, with the quasi-geostrophic (QG) model being the most notable example. Not only is the QG model used widely for advancing our theoretical understanding of atmospheric and oceanic dynamics, it is also crucial for practical applications such as model initialization and data assimilation . The reason is that since large scale dynamics are mostly balanced (i.e. devoid of fast motions), using unfiltered observations in numerical models can impart an unrealistically large component of fast motion due to observational errors. These errors can be reduced with the aid of balance models.
Although the QG model is immensely useful for the midlatitudes, it is not applicable to the tropics as a singularity develops. Previous attempts at deriving equatorial balance models have been unsuccessful as it filters out Kelvin waves, which play a dominate role in equatorial dynamics. In this talk I will give an introduction to balance models with the aid of a simple dynamical system. This will be followed with a discussion on how a modified asymptotic approach can be used to systematically derive equatorial balance models in the planetary scale regime, which captures slow Rossby and Kelvin wave modes while filtering out fast gravity wave motions.