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 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

To better understand the interactions between different components of the climate system is an important and difficult task.
The problem lies in the different science proper to each component leading to disparate processes, evolving on dissimilar scales in space and time.
This heterogeneity complexifies the research, from an observational, theoretical and numerical perspective.
Air–sea interaction is one example.
The exchange of heat, momentum and matter between the atmosphere and the ocean
has a strong influence on our climate

Since the work of

A recent concept, which is presently the subject of attention when non-equilibrium thermal systems
are considered, are work theorems. The most prominent ones are the
Jarzynski equality

The here discussed work theorems are different but related to fluctuation theorems considered
in

The concepts developed for micro-dynamics with fluctuations due to thermal motion are
here applied to macroscopic fluid dynamics, where an atmospheric planetary boundary layer
interacts with an oceanic mixed layer. In this case the fluctuations are due to the
smaller-scale turbulence in both layers. The concepts of fluctuation theorems have been
previously applied to cases with turbulent rather than thermal fluctuations. Examples are
the experimental data of the drag force exerted by a turbulent flow

A system that is subject to an external forcing typically evolves in time; it is in a non-stationary state. If there is a balance between external forcings and/or internal dissipation in such a way that ensemble averages do not evolve in time, the system is in a non-equilibrium stationary state. In the here-considered work theorems a dissipative system is subject to forcing and also the average large-scale quantities evolve in time; the dynamics is in a non-stationary non-equilibrium state.

The concepts of non-equilibrium statistical mechanics have been applied to momentum
transfer between the atmosphere and the ocean in a non-rotating frame in

In the next section we introduce the theory of stochastic thermodynamics and work relations
applied to air–sea interaction. The models are introduced and solved, using stochastic calculus,
in Sect.

We consider the turbulent momentum transfer between the atmospheric and the
oceanic planetary boundary layer, which are coupled by a frictional force.
The atmospheric layer is also subject to a deterministic forcing imposed from
the exterior through a pressure gradient. The dynamics in the boundary layers
is investigated using Reynolds decomposition, in which the fast fluctuations
in the three-dimensional velocity are separated from the slowly evolving component
of the horizontal velocity field (called “velocity field” in the sequel). The
horizontal variations in the velocity field are neglected. This is justified in
a local model by the disparity of the vertical and horizontal scales. The
atmospheric planetary boundary layer is a few hundreds of meters thick. The oceanic
planetary boundary layer spans a few tens of meters in the vertical. The velocity
field in both layers varies considerably over the thickness of the corresponding
boundary layer. Horizontal variations are imposed by the weather systems that force
the dynamics and typically extend 1000 km in the horizontal. This leads to a classical
model of the planetary boundary layers

The superscript

The 0D version of the 1D2C model (Eq.

Let us also introduce the integrated mode, which gives the momentum integrated
over

The shear and the turbulence in the atmosphere and the ocean do
not affect the integrated momentum

The departures from the vertical average in the atmosphere and the ocean are given by

Schematic of the models considered: the integrated mode and the shear mode are forced.
The integrated mode is decoupled from the rest of the dynamics. The shear mode is coupled
to the internal modes of the atmosphere and the ocean by the surface stress.
The internal modes in the atmosphere and the ocean depend on the eddy viscosity in each layer
and the surface stress.
The randomness arises through the surface friction

The interaction between the different components is schematized in Fig.

The concept of stochastic thermodynamics was introduced by

We here apply these concepts to air–sea interaction; the “heat”, the source of the fluctuation, in our approach is (small-scale) turbulent motion,
all that is represented in magenta and red in Fig.

It is important to note that the Coriolis parameter does not explicitly appear in the equation of the work or the heat
as the Coriolis force is orthogonal to the local velocity.
However, the Coriolis parameter strongly influences the dynamics, that is

The forcing protocol on the time interval

If we neglect the turbulence in both layers, which is modeled by a stochastic term, the dynamics is deterministic.
During the forward process, starting from rest and applying the protocol

The reverse process starts from the converged state, is forced by

The turbulent motion within the system is due to internal dynamics and is modeled by stochastic terms.
When noise is added in the linear model, it does not interfere with the deterministic dynamics but simply adds to it.
Furthermore, the force is deterministic, so the randomness in the work is provided solely by the fluctuations in

In the Gaussian case the pdf's for the forward and reverse processes are

We denote the averaging with respect to the forward process by

Experiments can also be performed for different values of

Furthermore, neither the work nor the free energy depends on the relaxation process
and in an experiment it is not necessary to wait for the relaxation to the stationary
state to obtain the free energy. It is only necessary to repeat the experiment
sufficiently many times to obtain statistically significant results and use
the Jarzynski equality to obtain the free energy. The work does, however, depend
on

The pdf's of the forward (pdf

The Jarzynski equality (JE) considers an average with respect to the forward process,
whereas the Crooks relation (CR) compares the pdf's of the forward and reverse
process, without any averaging; it states

In the Gaussian case described above the CR is obtained by a straightforward calculation:

Note also that when the CR holds,

In cyclic or stationary processes the free energy gain vanishes. When a force is applied, it typically drives the system and does work, which is dissipated to heat. Rare events when the work is negative and heat does work must exist, following the derived results above.

The work relations are investigated for a hierarchy of models of air–sea interaction.
This not only favors a pedagogical discussion of the subject but also
helps to emphasize critical points in the application of the theory exposed above.
The simpler models, which are given by Eq. (

We then discuss in Sect.

The solution of the 0D1C model introduced in Sect.

The simplest case is a constant force

The energy difference is

The free energy starting from rest is

The reverse process starts from the converged state is forced with amplitude

When noise is added in the linear model it does not interfere with the deterministic dynamics but just adds to it.
Furthermore, the force is prescribed (therefore deterministic) and the randomness in the work is provided solely by the fluctuations in

In order to apply the JE to the present problem,
we identify the heat by

Note that in the above all dependence is on the product

The CR is obtained by a straightforward calculation introducing

The calculations performed for the one-component model will now be extended to the two-component model where the two components interact through the Coriolis force (see Appendix

We start from a system at rest and apply the force constant

As the model is linear, all statistics are Gaussian and the statistical properties are completely described by the first-order moments,
which are described by the deterministic equations and the second-order moments.
Assuming the noise to be isotropic in the horizontal (

Note that for the work fluctuations only the

In this model we resolve part of the dynamics in the interior of the atmospheric and the oceanic layer explicitly.
The model consists of Eq. (

Evolution of the free energy (black) and the work performed on the integrated mode (red) from the numerical integration is shown. The evolution is deterministic and periodic and results agree with the analytic solution.

The forward pdf (pdf

For this model the free energy is still given by the kinetic energy of the integrated mode,
as all other motion decays when forcing subsides.
It is governed by the same equation as in the linear 0D Coriolis model;
that is, its dynamics is independent of the shear and the internal modes in the atmosphere and the ocean.
We call

The dynamics of the shear mode is governed by the same equations as in the linear 0D Coriolis model with a deterministic forcing and friction at the air–sea interface. The difference to the 0D model is that the dynamics of the internal modes within the atmosphere and the ocean are explicitly resolved and they influence the shear force that acts on the shear mode. That is, the stochastic term in the 0D models mimics the influence of the internal modes in the atmosphere and the ocean. The 1D model also resolves the shear modes, not only between the atmosphere and the ocean but also within them. These modes interact in a non-linear way and exchange energy, which is ultimately dissipated when the external forcing subsides. In the 1D model the internal modes within the atmosphere and within the ocean interact through the surface friction term and the internal eddy viscosities. In more involved 2D or 3D models, not studied here, they also interact through non-linear horizontal advection.

The numerical model to solve the above-discussed equations is a variation on the one used in

The work performed on the atmosphere is now a random process.
The numerical results show that the average work performed on the atmosphere in the forward process in the two experiments
is

We numerically found the JE

Figure gives

For evaluating the CR we plotted

We started by introducing the concept of work theorems into a simple model of air–sea interaction,
in which the atmosphere and ocean were represented by their corresponding mixed layer.
In this case the JE and the CR can be obtained analytically.
We then performed the same calculations on a model including a Coriolis force.
In that case the time reversibility is broken and the dynamics lags detailed balance,
which is at the basis of the original proofs of the JE and the CR
in the Hamiltonian system.
Analytical integrations of the stochastic differential equations governing the dynamics of the system
prove the existence of the JE and the CR.
They furthermore show that the limit of

In the applications of work theorems where fluctuations arise from thermal dynamics,
the thermodynamic

The physical interpretation of the dynamic

We have shown that the modern concepts of non-equilibrium statistical mechanics can be applied to large-scale environmental fluid dynamics, where fluctuations are not thermal but come from the turbulent fluid motion. We have demonstrated that the concepts of the dynamic beta, that is the equivalent of temperature in dynamical systems, can be extended to the momentum transfer at the air–sea interface using the formalism developed by Jarzynski and Crooks. It is important to note that work theorems are valid for forces of arbitrary amplitude; they are not a perturbative theory. This is, to the best of our knowledge, the first time that the concepts of work relations are investigated in geophysics and climate science. We successfully adapted the work theorems to the subject of air–sea momentum transfer, but they can, in the same way, be applied to other components of the climate system.

Work theorems also have important practical applications.
When the work pdf's of the forward and backward process are obtained,
the free energy of the system and the dissipated energy can be obtained
and a mechanical efficiency of the air–sea momentum transfer calculated.
This is key in understanding the energetics of the climate system.
For a discussion of the ocean circulation kinetic energy, we refer the reader to

The mechanics of air–sea momentum transfer has advanced considerably since the pioneering work of

The difficulty in performing simulations in air–sea interaction is the large difference in the characteristic timescales of the fast atmosphere and the slow ocean, the stiffness of the problem. Therefore integrations of the fast atmospheric dynamics are necessary with a long spin-up, as the ocean has to be in a statistically stationary state followed by a long integration to obtain a statistical significant ensemble of ocean states. When observations are considered, the stiffness asks for observations over extended periods of time which are just becoming available.

Similar problems appear when the interaction of other components of the climate system are considered. The momentum transfer at the air–sea interface is just one example where work relations between fluctuating components of the climate system increase our understanding. Their extension to other components is straightforward.

We consider a state vector given by the atmospheric and oceanic velocity:

The first step to solve the system of ordinary differential equations (ODEs) (Eq.

The deterministic and the stochastic dynamics are statistically independent,
so when calculating statistical moments we can ignore the deterministic
one (i.e.,

For the work theorems the focus is on the statistics of averages over a time interval

The calculations performed for the one-component model in the previous section will now be extended to the two-component model where the two components interact through the Coriolis force.

To simplify the algebra we temporarily manipulate complex quantities in
this subsection. For the 0D2C model given in Eq. (

For the sake of clarity we use the notations

The Fortran code corresponding to the 1D2C model described in Sect. 2.1
and used in Sect. 3.3 is available at

No data sets were used in this article.

AW and FL contributed to the theoretical calculations and the writing of the paper. AW designed and carried out the numerical experiments and their analysis. FL developed the 1D2C model.

The authors declare that they have no conflict of interest.

This work was funded by the French LEFE (Les Enveloppes Fluides et l'Environnement) MANU (méthodes MAthématiques et NUmériques) program through project FASIL.

This paper was edited by Rui A. P. Perdigão and reviewed by two anonymous referees.