Jarzynski equality and Crooks relation for local models of air-sea interaction

 We show that the most prominent of the work theorems, the Jarzynski equality and the Crooks relation, can be applied to the momentum transfer at the air-sea interface using a hierarchy of local models. In the more idealized models, with and without a Coriolis force, the variability is provided from a Gaussian white-noise which modifies the shear between the atmosphere and the ocean. The dynamics is Gaussian and the Jarzynski equality and Crooks relation can be obtained analytically solving stochastic differential equations. The more involved model consists of interacting atmospheric and oceanic boundary-layers, where only the dependence on the vertical direction is resolved, the turbulence is modeled through standard turbulent models and the stochasticity comes from a randomized drag coefficient. It is integrated  numerically and can give rise to a non-Gaussian dynamics. Also in this case the Jarzynski equality allows for calculating a dynamic-beta ß_D of the turbulent fluctuations (the equivalent of the thermodynamic-beta  ß=(k_B T)^-1 in thermal fluctuations). The Crooks relation gives the ß_D as a function of the magnitude of the work fluctuations. It is well defined (constant) in the Gaussian models and can show a slight variation in the  more involved models. This demonstrates that recent concepts of stochastic thermodynamics used to study micro-systems subject to thermal fluctuations can further the understanding of geophysical fluid dynamics with turbulent fluctuations.