We examine the role discrete symmetries, time-reversal, and mirror symmetries play in the context of geophysical waves and instabilities. By looking at three special cases from the two-layer quasi-geostrophic model as well as developing a general framework for translating real-space transformations to Fourier space, we are able to (1) show that baroclinic instability is an example of spontaneous parity-time symmetry breaking; (2) show that pure parity symmetry for a fluid system is exactly analogous to charge-conjugation-parity symmetry in a condensed matter system; and (3) show that when a pure parity symmetry is broken, this is associated with the suppression of wave propagation. Furthermore, in the latter case, instability can arise without a corresponding symmetry breaking. This study highlights the role of symmetry breaking behind the dynamics of geophysical waves and instabilities.

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