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Jarzynski Equality and Crooks Relation for Local Models of Air–Sea Interaction

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Jarzynski Equality and Crooks Relation for Local Models of Air–Sea Interaction Earth Syst. Dynam., 12, 689–708, 2021 https://doi.org/10.5194/esd-12-689-2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. Jarzynski equality and Crooks relation for local models of air–sea interaction Achim Wirth1 and Florian Lemarié2 1LEGI, Univ. Grenoble Alpes, CNRS, Grenoble INP, 38000 Grenoble, France 2LJK, Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, 38000 Grenoble, France Correspondence: Achim Wirth ([email protected]) Received: 26 October 2020 – Discussion started: 15 December 2020 Revised: 30 March 2021 – Accepted: 23 April 2021 – Published: 4 June 2021 Abstract. We show that the most prominent of the work theorems, the Jarzynski equality and the Crooks rela- tion, 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 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 stochas- ticity 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 βD of the turbulent −1 fluctuations (the equivalent of the thermodynamic beta β D (kBT ) in thermal fluctuations). The Crooks re- lation gives the βD 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. 1 Introduction atmosphere mostly loses energy at the interface by friction and the ocean mostly gains energy (e.g., Wirth, 2019). The To better understand the interactions between different com- energy exchange is not conservative, and most of the energy ponents of the climate system is an important and difficult is dissipated (Duhaut and Straub, 2006; Wirth, 2018). task. The problem lies in the different science proper to each Since the work of Einstein(1906) (see also Einstein, 1956; component leading to disparate processes, evolving on dis- Perrin, 2014), fluctuations have been the focus of research similar scales in space and time. This heterogeneity complex- in statistical mechanics, which had traditionally been con- ifies the research, from an observational, theoretical and nu- cerned with averages. Fluctuations in a thermodynamic sys- merical perspective. Air–sea interaction is one example. The tem usually appear at spatial scales which are small enough exchange of heat, momentum and matter between the atmo- so that thermal, molecular motion leaves an imprint on the sphere and the ocean has a strong influence on our climate dynamics, as was first noted by Einstein(1906) (see also (Stocker et al., 2007). In the present work only the exchange Einstein, 1956; Perrin, 2014). The importance of fluctua- of momentum is considered. It is caused by the shear at the tions is, however, not restricted to small systems where ther- sea surface due to the difference between the atmospheric mal fluctuations are important, since they leave their imprint winds and the ocean currents in the corresponding planetary on the dynamics at all scales when (not necessarily ther- boundary layers. For a general discussion on air–sea inter- mal) fluctuations are strong enough. A typical example of action we refer to Csanady(2001). The atmospheric winds non-thermal fluctuations is fluctuating turbulent fluid motion are usually faster than the ocean currents, and therefore the Published by Copernicus Publications on behalf of the European Geosciences Union. 690 A. Wirth and F. Lemarié: Jarzynski equality and Crooks relation for local models of air–sea interaction (e.g., Frisch, 1995). The average motion of a turbulent fluid considered based on different approaches: Hamiltonian dy- can not be understood without some knowledge about the namics subject to an external forcing, Fokker–Planck equa- turbulent fluctuations. The importance of turbulent fluctua- tions and Langevin dynamics (see Seifert, 2012, for a re- tions is especially pronounced in geophysical flows, which view). Here only the last approach is used. are highly anisotropic due to the influence of gravity. This The concepts developed for micro-dynamics with fluctua- leads to a quasi-two-dimensional dynamics, an energy cas- tions due to thermal motion are here applied to macroscopic cade from small to large scales and strong fluctuations (see fluid dynamics, where an atmospheric planetary boundary Boffetta and Ecke, 2012, for a recent review on 2D turbu- layer interacts with an oceanic mixed layer. In this case the lence). Likewise, the air–sea interaction on hourly to climatic fluctuations are due to the smaller-scale turbulence in both timescales can not be understood without some knowledge of layers. The concepts of fluctuation theorems have been pre- the fluctuations at smaller and faster scales (see McWilliams viously applied to cases with turbulent rather than thermal and Huckle, 2006; Shrira and Almelah, 2020). Furthermore, fluctuations. Examples are the experimental data of the drag in many natural systems the focus is on the fluctuations rather force exerted by a turbulent flow (Ciliberto et al., 2004) and than on an average state. Examples are weather and climate the local entropy production in Rayleigh–Bénard convection dynamics, where we focus on the fluctuations in the same (Shang et al., 2005). system on different timescales. For the weather the timescale A system that is subject to an external forcing typically of interest is from roughly an hour to a week; for the cli- evolves in time; it is in a non-stationary state. If there is a bal- mate the focus is from tens to thousands of years. As pro- ance between external forcings and/or internal dissipation in cesses with very different timescales intervene, the system is such a way that ensemble averages do not evolve in time, the not in a stationary state at those timescales but is constantly system is in a non-equilibrium stationary state. In the here- evolving in time. The different components of the system ex- considered work theorems a dissipative system is subject to change energy; they do work on each other. The exchange of forcing and also the average large-scale quantities evolve in energy between fluctuating components is the subject of the time; the dynamics is in a non-stationary non-equilibrium present work. state. A recent concept, which is presently the subject of at- The concepts of non-equilibrium statistical mechanics tention when non-equilibrium thermal systems are consid- have been applied to momentum transfer between the at- ered, are work theorems. The most prominent ones are the mosphere and the ocean in a non-rotating frame in Wirth Jarzynski equality (Jarzynski, 1997) and the Crooks rela- (2018, 2019). This was done by adapting the mathematics de- tion (Crooks, 1998). Rather than looking at average values of veloped to study the movement of a Brownian particle. The the thermodynamic variables, they consider their probability present work prolongs this research by considering work re- density functions (pdf’s) which allow the replacement of in- lations and extending it to the dynamics in a rotating frame. equalities of equilibrium statistical mechanics by equalities. The motion of a particle in a rotating frame is similar to As an example, the second law of thermodynamics states that Brownian motion of a charged particle in a magnetic field, a the work W performed on a system is larger than or equal problem which has been studied since Taylor(1961) (see also to the increase 1G in its free energy: W ≥ 1G. When the Czopnik and Garbaczewski, 2001). The structure of the equa- work is seen as a fluctuating quantity w, which differs even tions is identical when the Larmor frequency of a charged when a specific process is repeated with the same determin- particle in a magnetic field is replaced by the Coriolis fre- istic forcing protocol but is subject to thermal fluctuations, quency. The passage from a non-rotating frame to a rotating the Jarzynski equality says that hexp(−βw)i D exp(−β1G), frame is, however, far from straightforward, for principally where the average hi is taken over the ensemble of thermal two reasons. First, the dynamics is no longer invariant by fluctuations. This not only includes the second law on av- time reversal, even in the non-dissipative limit. In the words erage but also says that individual exceptions have to exist of statistical mechanics, detailed balance, which is the basis (see Sect.2). When thermal fluctuations are considered, the of many analytical results, is lost. Secondly, it is not clear −1 (thermodynamic) β D (kBT ) is the inverse of the product that results from simple models that do not explicitly resolve of the Boltzmann constant and the temperature. In the case of the vertical structure in the atmospheric and oceanic bound- air–sea interaction, considered here, the dynamic β, (denoted ary layer are useful to investigate the situation in a rotating βD) is the inverse of an energy related to the macroscopic tur- frame with a Coriolis force (see McWilliams and Huckle, bulent fluctuations. It is the inverse of a “temperature”, that 2006). Indeed, the dynamics in the planetary boundary layer is, in the present context, of a turbulent kinetic energy.
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