Balance, Potential-vorticity Inversion, Lighthill Radiation, and the Slow Quasimanifold

Practically our entire understanding of large-scale atmosphere–ocean dynamics depends on the notions of balance and potential-vorticity inversion. These are essential, for instance, for a clear understanding of the basic Rossby-wave propagation mechanism, or quasi-elasticity, that underlies almost every large-scale fluid-dynamical phenomenon of meteorological and oceanographical interest, from the global-scale transport of terrestrial greenhouse gases (and similar problems in the solar interior) to Rossby-wave-mediated global teleconnection, baroclinic and barotropic shear instability, vortex coherence, and vortex-core isolation. The ideas involved in understanding balance and inversion continue to hold special fascination because of their central importance both for theory and for applications, such as data assimilation, and the fact that complete mathematical understanding is still elusive. The importance for applications was adumbrated by Richardson in his pioneering study of numerical weather prediction. The importance for theory — and the exquisite subtlety involved — was adumbrated by Poincaré in his discovery of the homoclinic tangle, and by Lighthill in his discovery of the quadrupole nature of acoustic radiation by unsteady vortical motion.