Ocean dynamics is predominantly driven by the shear stress between the atmospheric winds and ocean currents. The mechanical power input to the ocean is fluctuating in space and time and the atmospheric wind sometimes decelerates the ocean currents. Building on 24 years of global satellite observations, the input of mechanical power to the ocean is analysed. A fluctuation theorem (FT) holds when the logarithm of the ratio between the occurrence of positive and negative events, of a certain magnitude of the power input, is a linear function of this magnitude and the averaging period. The flux of mechanical power to the ocean shows evidence of a FT for regions within the recirculation area of the subtropical gyre but not over extensions of western boundary currents. A FT puts a strong constraint on the temporal distribution of fluctuations of power input, connects variables obtained with different lengths of temporal averaging, guides the temporal down- and up-scaling and constrains the episodes of improbable events.

The exchange of heat, momentum and matter between the atmosphere and the ocean has a strong influence on our climate

More precisely, we consider the mechanical energy exchange between the atmosphere and the ocean at a time,

We focus on two properties of the mechanical power input to the ocean at the surface: (i) on average the ocean gains energy at the interface

Today, fluctuations are the focus of research in statistical mechanics, which was traditionally concerned with averages. Fluctuations in a thermodynamic system usually appear at spatial scales which are small enough so that
thermal, molecular, motion leaves an imprint on the dynamics as first noted by

Turbulent fluid motions are typical examples (i.e.

A recent concept which is presently the subject of growing attention in non-equilibrium statistical mechanics is fluctuation theorems (FTs) (see e.g.

FTs have been established analytically for Langevin-type problems with thermal fluctuations

In

We are interested in the mechanical power,

The Galavotti–Cohen fluctuation theorem (called FT in the following for brevity) holds for

If the FT holds, it is sufficient to know the probability for either

For a dynamical system the FT may or may not hold, and it might only be valid for a range of values. It was already noted in

The calculations of the power input to the ocean are based on the shear stress at the surface and the ocean velocity. The shear stress is usually evaluated based on the difference between the horizontal wind velocity

To obtain the power input, the vector product between the shear stress and the ocean current velocity is taken:

In this study, we build on the newly released GlobCurrent products, now available via the Copernicus Marine Environment Monitoring Service (CMEMS,

Strongly based on altimeter data, this global ocean surface current product and also similar global observation-based products

Satellite winds are from the Copernicus project (

The FT is a property that concerns the tails of a pdf, and it is necessary to consider a large amount of data, as provided by the GlobCurrent products. Still, a time record of 24 years of data coverage at a single location
is too small for empirically suggesting or refuting the existence of a FT.
To increase the amount of data, we use different tiles

Four domains are considered: the first is in the recirculation area of the subtropical gyre (20–30

The pdfs of

The pdfs

With increasing averaging periods, the pdfs become more centred around unity, which is the average value; see Eq. (

The verification of the FT, that is of Eq. (

The pdf

The pdf

The pdf

The pdf

For the four domains, we observed a convergence of the normalized symmetry function with increasing averaging time. This indicates the existence of a large deviation principle (see e.g.

The contraction rate

Index

We did not attempt to present error bars in the figures and numbers in the tables, as uncertainties depend on the number of statistically independent events, that is the correlation time. In the case of air–sea interaction there are correlations due to the atmospheric dynamics (mostly synoptic), the ocean dynamics, the annual cycle, interannual variability and a climatic trend. How these processes contribute to the tails of the pdfs, to improbable events, is currently a hot topic in climate science (see e.g.

We obtain evidence that a FT applies to data within the recirculation area of the subtropical gyre in the Atlantic and Pacific oceans. In these cases the FT can be used to estimate the occurrence of rare negative events from frequent positive events of the same magnitude for all averaging periods

The FT does not seem to apply in the highly non-linear Gulf Stream extension for

During data analysis, we also found that a FT does not apply when islands or coastlines are present (not shown here). Departure from a FT for the power input to the ocean is found where horizontal dynamics dominates over the vertical ocean–atmosphere momentum exchanges. The influence of the horizontal transport of energy with respect to the injection of energy through the surface decreases with domain size considered, as the circumference of a domain grows linearly, whereas its surface growth is quadratic. Yet determining the existence of a FT for larger ocean domains requires more data, which are currently not available. Our results are purely empirical: a theory explaining why the power input follows a FT in some cases and not in others is still missing.

Finally, we put the theory of FTs in the more general context of climate dynamics. A measurement, especially when coming from satellites, always contains some averaging in space and time. A FT, when it applies, will help to relate averages over varying periods and is a powerful tool to guide the up- and down-scaling of observational data in time and obtain the statistical information on shorter and longer timescales, which are not explicitly observed. More precisely, when the pdf of the power supply and therefore also the symmetry function is known from observations for given averaging times, the symmetry function can be calculated for shorter and larger averaging times and constrains “half” of the pdf. This is useful in down-scaling and the construction of statistical parameterizations of not directly observed dynamics over shorter timescales. On the other hand, the information can be useful for developing models for the persistence of events over large timescales not yet observed. A FT can help to decide whether the persistence in time of a phenomenon is within the likeliness of the statistically stationary dynamics or due to external influences. Furthermore, when data from observations follow (or not) a FT, model data should do likewise. As such, the FT becomes a tool for investigating the fidelity of models.

Recently there has been increasing interest in the variability of atmosphere and ocean dynamics and in the exchange of the two, also at high frequency, and their contribution toward the lower-frequency air–sea momentum and energy fluxes (e.g.

We conclude by looking at our results from the standpoint of dynamical systems. Statistical mechanics of systems in equilibrium are described by the Boltzmann distribution, which is completely determined by the temperature.
In non-equilibrium statistical mechanics no such universal distribution is known (see e.g.

Data are available at

AW performed the coding. Writing was shared by both authors.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

These data were provided by the Centre de Recherche et d'Exploitation Satellitaire (CERSAT) at IFREMER, Plouzane (France), and CMEMS. Part of this work was performed when AW visited LOPS, Brest. We are grateful to Abderrahim Bentamy for explanations concerning the data and Mickael Accensi and Jean-Fancois Piolle for help with the data analysis. The authors were funded by the GlobCurrent Project, funded by the ESA Data User Element (DUE).

This research has been supported by Labex OASUG@2020 (Investissement d’avenir – ANR10 LABX56) and the GlobCurrent Project funded by the ESA Data User Element (DUE) (contract no. 4000109513/13/I-LG).

This paper was edited by Balasubramanya Nadiga and reviewed by four anonymous referees.