Evidence of a fluctuation theorem for the input of mechanical power to the ocean at the air-sea interface from satellite data

The ocean dynamics is predominantly driven by the shear 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 5 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 length of temporal averaging, guides the temporal downand up-scaling and constrains the episodes of extreme events.


Introduction
The exchange of heat, momentum and matter between the atmosphere and the ocean has a strong influence on our climate (Stocker et al. (2013)). Recent advances in satellite and in-situ based global Earth Observation (EO) systems and platforms, have significantly improved our ability to monitor ocean-atmosphere interactions. In the present work the exchange of momentum is considered, which is described by the fluxes of mechanical power at the ocean surface. It is caused by the shear at turbulent dynamics in the atmosphere and the ocean the quantities are fluctuating over a large range of scales in time and space.
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 P(t) > 0 (where . represents an average over the observation period and several surface areas A i ) and (ii) the power input is fluctuating, in time and space, due to the turbulent motion in the atmosphere and the ocean and negative events, with P(t) < 0, occur. 10 Today, fluctuations are the focus of research in statistical mechanics, which was traditionally concerned with averages.
Fluctuations in a thermodynamic system usually appear at spacial scales which are small enough so that thermal, molecular, motion leaves an imprint on the dynamics as first noted by Einstein (1906) (see also Einstein (1956) and Perrin (2014)). The importance of fluctuations is, however, not restricted to small systems and can leave their imprint on the dynamics at all scales when (not necessarily thermal) fluctuations are strong enough. in geophysical flows, which are highly anisotropic due to the influence of gravity and rotation. This leads to a quasi twodimensional dynamics and an energy cascade from small to large scales and strong fluctuations (see i.e. Boffetta and Ecke 20 (2012) for a review on 2D turbulence). Likewise, the description of air-sea interactions on large time scales may not be understood without some knowledge of the fluctuations at smaller and faster scales. Furthermore, in many natural systems the focus is on the fluctuations rather than on an average state, weather and climate dynamics are examples where we focus on the fluctuations of the same system on different time scales. For the weather the time scale of interest is from roughly an hour to a week, for the climate the focus is from tenths to thousands of years and beyond.

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A recent concept which is presently subject of growing attention in non-equilibrium statistical mechanics are Fluctuation Theorems (FT) (see e.g. Evans et al. (1993), Gallavotti and Cohen (1995a), Gallavotti and Cohen (1995b), Ciliberto et al. (2004), Shang et al. (2005) and Seifert (2012)). Not only the average values of quantities like entropy, work, heat or other, are studied, but their fluctuating properties are scrutinised. There are different forms of FTs, reviewed in detail by Seifert (2012).
In the present paper we focus on the FT put forward in Gallavotti and Cohen (1995a), Gallavotti and Cohen (1995b) and 30 Gallavotti and Lucarini (2014), corresponding to the detailed fluctuation theorem in the limit of large averaging times. When the FT applies to a fluctuating quantity, as i.e. P(t) in the present study, it relates the probability to have a negative event, i.e. the ocean loses energy, to the probability of a positive event, i.e. the ocean gains energy, of the same magnitude. The FT is not only concerned with instantaneous values but considers the fluctuations of temporal averages over varying averaging time. The FT, which is stated precisely in the next section, thus puts a strong constraint on the fluctuations of the quantity considered and its temporal averages of varying length.
FTs have been established analytically for Langevin type problems with thermal fluctuations (Seifert (2012)). Most experimental data comes also from micro systems subject to thermal fluctuations. The thermodynamic frame of the quantities considered, as entropy, heat and work is not necessary to establish FTs. Examples of non-thermal fluctuations are the experimental 5 data of the drag-force exerted by a turbulent flow (Ciliberto et al. (2004)) and the local entropy production in Rayleigh-Bénard convection (Shang et al. (2005)). For these non-Gaussian quantities the existence of a FT was shown empirically. Our work is strongly inspired by these investigations of the FT in data from laboratory experiments of turbulent flows.
In Wirth (2019) the FT was investigated for three parameterizations of air-sea interaction and we refer the reader to this work for the theory and analytical solutions on fluctuating air-sea interaction in these idealised models. In that publication the concept 10 of FT is also placed in a broader context of fluctuating dynamics and the relation to the fluctuation-dissipation-relation and the fluctuation-dissipation-theorem is given (see also Seifert (2012) for a general discussion). Here we extrapolate the research of Wirth (2019) by applying the concept of FTs to data derived from satellite measurements and discuss their relevance. It is important to notice that even in the case of the idealised models the FT was not established by analytical calculation, but it was confirmed numerically that the FT is obtained asymptotically, in the long-time limit, when the averaging time is larger than the 15 characteristic time-scale of the slow ocean-dynamics.

The Fluctuation Theorem
We are interested in the mechanical power, P(t), absorbed by the ocean over a given surface area, A, of the ocean surface and an observation period t obs . We suppose that P(t) is a statistically stationary random variable, meaning that its statistical properties (mean value, moments and temporal correlations) do not change when shifted in time. Its statistical properties, at 20 every instance of time, are completely described by its probability density function (pdf), p(z), which gives the probability that P(t) takes values between z 1 and z 2 by integration: Pr[z 1 < P(t) < z 2 ] = z2 z1 p(z)dz. The symmetry function is: It compares the occurrence of events when the ocean receives power of magnitude z to the occurrence when the ocean loses power of the same magnitude. We further denote the normalised energy received during an interval τ starting at time t 0 , by: where t obs is the total length of the available data record. The corresponding pdf is denoted by p(z, τ ) and the symmetry function by S(z, τ ). Note that the averaging starts at time t 0 and extends over the interval τ .
The Galavotti-Cohen fluctuation theorem (called FT in the sequel for brevity) holds for P if two conditions are satisfied: (i) the symmetry function depends linearly on the variable z, and (ii) on τ , for large averaging times (τ τ 0 ): appear explicitly and only the kinematic viscosity has a slight dependence on temperature. There is, therefore no reason why k B T is a governing parameter of the problem.
If the FT holds it is sufficient to know the probability for either z > 0 or z < 0 to obtain the whole pdf, when σ is known.
The FT therefore constraints "half" of the pdf, a strong constraint in the absence of an equivalent of the Boltzmann distribution.
This property also allows to calculate the probability of the rare events of z < 0 from frequent events z > 0.
10 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 Gallavotti and Cohen (1995a) and Gallavotti and Cohen (1995b) that the FT might only be valid for values z < z * , when the large deviation function (see i.e. Touchette (2009) More recently it was recognised that boundary conditions, that is the value P(t) at t = t 0 and t = t 0 + τ , can leave their signature in the symmetry function S(z, τ ), even when the limit of τ → ∞ is taken, whenever the pdf p(z) has tails which are exponential or less steep than exponential 15 (see Farago (2002), Van Zon and Cohen (2004) and Rákos and Harris (2008)). In such case an extended FT (EFT) is expected, which shows a linear scaling of the symmetry function near the origin with a transition to a flatter curve for larger values.
An analytic expression of the symmetry function, or the value of z * is obtained only for very idealised cases and the results presented here are empirical.
3 Power Input 20 The calculations of the power input to the ocean are based on the shear at the surface and the ocean velocity. The shear is usually evaluated, based on the difference between the horizontal wind velocity u s a , usually taken at 10m above the ocean surface and the horizontal ocean surface-current u s o , using the quadratic drag law (see i.e. Renault et al. (2017)): The drag coefficient C d depends on the sea-state and the stratification in the atmosphere and the ocean, it is obtained using 25 bulk formulas (Fairall et al. (1996)).
To obtain the power input, the vector product between the shear and the ocean current-velocity is taken: For the work done on the large-scale geostrophic-circulation, Wunsch (1998) and Zhai et al. (2012) used the surface goestrophic velocity estimates from altimetry. Using model data, Rimac et al. (2016) used the velocity at the surface to calculate the total 30 power input, to evaluate that only a fraction of this power is transmitted to the interior ocean at the base of the mixed layer. In the present work, largely building on 15-m drogued drifter velocities (Rio et al. (2014)), we use for u o the estimation of the current velocity at 15m depth.

Data
In this study, we build on the newly released GlobCurrent products, now available via the Copernicus Marine Environment Monitoring Service (CMEMS, http://marine.copernicus.eu/services-portfolio). Essentially building on the quantitative estima-

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Satellite winds are from the Copernicus project (http://marine.copernicus.eu/services-portfolio/access-to-products/). They are from scatterometer and radiometer wind observations. It is a blended product based on the different missions (ERS-1, ERS-2, QuikSCAT, and ASCAT) available at 1/4 o spatial resolution and every 6 hours and is described in Bentamy et al. (2017) and Desbiolles et al. (2017). The data record for which wind and current data is available extends over 24 years, 1993-2016, at a resolution of 6h in time and 1/4 o in space. 20 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 A i that obey similar The pdfs of E(t) τ are calculated for an interval that spans twice the mean value of each pdf from the origin. Note, that the average is unity by definition. The pdfs are calculated with three different resolutions (bin sizes). The interval is separated into 21, 31 and 41 bins of equal size and the pdfs are obtained by counting the number of occurrences for each bin. The symmetry function is only calculated when probabilities are lager than 10 −3 per bin, this led to an omission of bins in exp. 1, only.

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The pdfs p(z, τ ) for the four domains and for different values of averaging times τ are presented in the left panels of figs. 1, 2, 3 and 4. All clearly display non-Gaussianity. With increasing averaging period, the pdfs become more centred around unity (which is the average value, see eq. (2)), a consequence of the central limit theorem and occurrences of negative values become less likely. In the right panels of figs. 1, 2, 3 and 4 the symmetry function divided by the averaging period is plotted. These plots are similar to those in Gallavotti and Lucarini (2014), who verified the FT in an idealised numerical model. 10 The verification of the FT, that is of eq. (3), is two-fold. First, we verify the linear dependence of the symmetry function on z for different averaging periods τ and determine the slope. Second, we verify that the slope is a linear function of τ for times larger than the characteristic time, τ > τ 0 , of the system. This is demanding, and a large amount of data is necessary. For the first point, we have to consider the pdf for an extended range in z, including the tails, asking for ensemble sizes (number of intervals of length τ ) large enough so that we can observe a clear scaling behaviour. For the second point, we have to increase τ 15 to verify convergence. Furthermore, for larger τ the pdfs are more and more peaked around unity and negative events become less and less likely.
For the four domains, we observed a convergence of the normalised symmetry function with increasing averaging time. This indicates the existence of a large deviation principle (see i.e. Touchette (2009)   The contraction rate σ is the slope of the curves in the right panels of figs. 1, 2, 3 and 4. To estimate the alignment of the points for τ = 1250 days, we constructed an index γ: the slope of the normalised symmetry function from the origin to the first bin divided by the slope from the origin to the last bin. A value γ = 1 indicates a perfect alignment of the first bin with the last.
The index is presented in table 1 for the four different domains and three different resolutions of the pdf. For the recirculation area of the subtropical gyre cases, the index varies around unity for the different bin sizes. It is significantly greater than unity 5 in the Gulf Stream and the Kuroshio extension for all bin sizes considered.
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 pdf's, to extreme events, is a currently a hot topic in climate science (see i.e. Ragone et al. (2018)).

Discussion
We obtain clear evidence that a FT applies to data within the recirculation area of the subtropical gyre in the Atlantic and the Pacific Ocean. In this cases the FT can be used to estimate the occurrence of rare negative events from frequent positive events 5 of the same magnitude for all averaging periods τ (measured in days). If the FT applies, the probability of the rare extreme negative events can be calculated from frequent positive events. Extreme events are often key for the system in a variety of applications and are the focus of recent research in climate science (Ragone et al. (2018), Seneviratne et al. (2012)). As an example: in the Atlantic subtropical gyre case the slops of the symmetry function is S(z, τ ) = 2. · 10 −2 τ z, this means that an event of the magnitude z = −1 is p = exp(−2. · 10 −2 τ ) less likely than an event having the average value (z = 1) and an 10 event of the magnitude z = −2 is p = exp(−4. · 10 −2 τ ) less likely than an event having twice the average value (z = 2). A FT represents a tool to obtain the rare negative events from frequent positive events for all averaging times and demonstrates that, to leading order, the probability of negative events vanishes exponentially with the averaging time.
The FT does not seem to apply in the highly non-linear Gulf Stream extension for z 0.3 and Kuroshio extension z 0.5.
For these regions, the symmetry function follows a FT for small values of z, before the curve flattens. This resembles the behaviour found in the EFT (see section 2). Indeed, in these two cases (exp2 & 4) the tails of the pdf of P show pronounced super exponential tails and boundary values might be important leading to a behaviour predicted by an EFT. Nevertheless, a similar change of slope was also found using highly idealised models of air-sea interactions (discussed in Wirth (2019)), to 5 which a friction term was added to the ocean. This suggests that an increased energy cascade, in the extension of boundary currents, might be responsible for the departure from a FT. When the scaling of the symmetry function flattens for higher power-input, the manifestation of a negative extreme event, versus a positive event of the same magnitude, becomes more likely.
During data analysis, we also found that a FT does not apply when islands or coastlines are present (not shown here). 10 Departure from a FT for the power input to the ocean is found where horizontal dynamics dominates over the vertical oceanatmosphere 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 grows is quadratic. Yet, determining the existence of a FT for larger ocean domains asks for more data, which is currently not available. Our results are purely empirical, a theory explaining why the power input follows a FT in some cases and not in 15 others, is still missing.
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. When data from observations follow (or not) a FT, model data should do likewise. As such, the FT becomes a tool of investigating the fidelity of models. 20 Statistical mechanics of systems in equilibrium are described by the Boltzmann distribution and the other properties can be derived from it. In non-equilibrium statistical mechanics no such universal distribution is known (see i.e. Derrida (2007), Touchette (2009) andFrisch (1995)), but some quantities in some processes seem to follow a FT which constraints the pdf and might indicate some universality. The mechanical power-input to the ocean by air-sea interactions, as a forced and dissipative dynamical system, may thus belong to a class of particular non-equilibrium systems exhibiting a FT symmetry property and 25 offer guidance for climate studies.
Author contributions. AW has performed the coding, writing was shared by both authors