On spontaneous imbalance and ocean turbulence: generalizations of the Paparella–Young epsilon theorem

Recent progress in understanding the balance–imbalance problem is highlighted, with emphasis on spontaneous-imbalance phenomena associated with the exponentially fast “wave capture” of inertia–gravity waves. These phenomena are excluded from shallow-water models and are outside the scope of the classical Lighthill theory. Also discussed is progress on a different topic, an effort to extend the Paparella–Young epsilon theorem to realistic ocean models. The theorem constrains turbulent dissipation rates ε in horizontal-convection thought-experiments, in which mechanically-driven stirring is switched off. The theorem bears on the so-called “ocean heat engine” and “ocean desert” controversies. The original theorem (2002) applied only to very idealized ocean models. Several restrictions on the original proof can now be lifted including the restriction to a linear, thermalonly equation of state. The theorem can now be proved for fairly realistic equations of state that include thermobaric effects, and nonlinearity in both temperature and salinity. The restriction to Boussinesq flow can also be lifted. The increased realism comes at some cost in terms of weakening the bound on ε. The bound is further weakened if one allows for the finite depth of penetration of solar radiation. This is collaborative work with Francesco Paparella and William Young. To appear in Turbulence in the Atmosphere and Oceans (Proc. International IUTAM/Newton Inst. Workshop held 8–12 December 2008), ed. D. G. Dritschel, Springer-Verlag. Final version in press