The modified geostrophic Eady problem revisited

The Eady problem is one of the canonical examples for baroclinic instability and finds many references to it in astrophysical and geophysical fluid dynamics. Here we revisit the problem of the Eady linear instability problem in the geostrophic limit in the presence of a small slope, motivated by oceanic problems. We present three short but non-trivially related stories about the Eady problem: (i) the linear Eady problem is Parity-Time (PT)symmetric, a concept originating in quantum physics, and places a constraint on the solutions and bifurcations the governing system is allowed to have; (ii) exploring the mechanism leading to Eady instability and the exact role of the slope, and some subtleties relating to interpretation in terms of edge-wave phase-shifts versus eigenfunction phase-tilts; (iii) analysis of eigenfunctions within the GEOMETRIC framework, expressing eddy fluxes in terms of geometric quantities related to eddy covariance ellipses. Some implications in terms of eddy parameterisations and ocean modelling are discussed if time.