Wind work at the air-sea interface: A Modeling Study in Anticipation of Future Space Missions

. Wind work at the air-sea interface is the transfer of kinetic energy between the ocean and the atmosphere and, as such, is an important part of the atmosphere-ocean coupled system. Wind work is deﬁned as the scalar product of ocean wind stress and surface current, with each of these two variables spanning, in this study, a broad range of spatial and temporal scales, from 10 km to more than 3000 km and hours to months. These characteristics emphasize wind work’s multi-scale nature. In the absence of appropriate global observations, our study makes use of a new, global, coupled ocean-atmosphere simulation, 5 with horizontal grid spacing of 2–5 km for the ocean and 7 km for the atmosphere, analyzed for 12 months. We develop a methodology, both in physical and spectral spaces, to diagnose three different components of wind work that force distinct classes of ocean motions, including high-frequency internal gravity waves such as near-inertial oscillations, low-frequency currents such as those associated with eddies, and seasonally averaged currents, such as zonal tropical and equatorial jets. The total wind work, integrated globally,


Introduction
Wind work is known to drive a large part of ocean dynamics (Ferrari and Wunsch, 2009), and is defined in this study as the scalar product of the wind stress and surface ocean current vectors (Renault et al., 2016;Yu et al., 2018). Wind work forces zonal jets (spatial scales of ∼1000 km and time scales of days to months), in particular at equatorial and tropical latitudes, where they are key players in the El Niño Southern Oscillation (ENSO) (Maximenko et al., 2005). Wind work also forces or 25 damps mid-latitude currents (with spatial scales from 10 km to more than 500 km and time scales of days to months), such as those associated with submesoscale, and mesoscale eddies, which are critical players in the horizontal and vertical transport of heat at these latitudes (Eden and Dietze, 2009;Zhai et al., 2012;Zhai, 2013;Klein et al., 2019;Rai et al., 2021). Additionally, wind work generates near-inertial oscillations and internal gravity waves (spatial scales of 10 to 1000 km and time scales of hours), which impact mixing in the ocean interior and therefore contribute to setting the structure and strength of the  ional Overturning Circulation (MOC) (Komori et al., 2008;Polzin and Lvov, 2011;Nikurashin et al., 2013;Alford et al., 2016).
To better identify the nonlinearities and spatial and temporal scales that characterize wind work, let us examine the dynamical variables involved. The wind stress vector, τ , can be written as (Large and Yeager, 2004) 35 where ρ air is the air density, and C d a drag coefficient that is a function of the wind field and stability of the atmospheric boundary layer (see next section). U a is the vector wind usually taken at an altitude of 10 m, and u o the ocean-surface current vector. Then the wind work, F s , is Eq. 2 highlights that wind work is nonlinearly related to wind stress and ocean current.

40
Referring to Eq.2, some examples of the multiscale issues we have to address are the following. Wind fluctuations with time scales of one hour impact the wind stress at these short time scales. The resulting wind work, in regions of atmospheric storm tracks, generates near-inertial motions and internal gravity waves (with time scales less than a day) whose kinetic energy can be up to twice larger than when only wind fluctuations with time scales longer than six hours are considered (Klein et al., 2004; 45 Rimac et al., 2013). However, wind fluctuations at short time scales also impact the weekly-averaged and monthly-averaged wind stress and therefore the wind work at these longer time scales. This is due to the quadratic relationship between winds and wind stress (Eq. 1). For example, in regions of atmospheric storm tracks, the resulting monthly-averaged wind work is larger by a factor four when wind fluctuations at short time scales are taken into account than when only weekly or monthly winds are used (Zhai et al., 2012;Zhai, 2017). Thus, high-frequency winds can lead to a larger forcing of low-frequency ocean 50 currents. This example and the more detailed arguments developed in Section 3 emphasize the need to have observations of winds and currents over a broad range of temporal and spatial scales in order to diagnose the different wind work components that energize or damp oceanic motions.
To assess the broad range of scales that influence wind work and impact ocean currents, we make use of model outputs of 55 wind stresses and ocean currents from a new, global, coupled ocean-atmosphere model that includes tidal forcing in the ocean and has horizontal grid spacing of 2-5 km in the ocean and 7 km in the atmosphere. This model has been integrated for more than a year. The resulting numerical simulation produces wind and current fluctuations at very short time scales (45-s oceanatmosphere coupling time step), enables spatial collocation and contemporaneity of atmospheric winds and ocean currents, and takes into account the impacts of winds and ocean currents on wind stresses. Our study focuses on the impact of wind work 60 on ocean currents including near-inertial oscillations, mesoscale eddies, large-scale currents and gyres, but does not account for high frequency motions such as surface gravity waves, Langmuir circulation, and mixed layer turbulence. The next section describes the global numerical model used. Section 3 describes the methodology employed in physical and spectral spaces, and discusses the multiscale issues we have to address. An analysis of the wind work components that force different classes of motion is presented in Section 4. Conclusions follow in Section 5.

65
2 Numerical simulation of the coupled ocean-atmosphere system  Strobach et al. (2020Strobach et al. ( , 2022 except that the ocean model includes tidal forcing, which triggers the generation of internal tides and promotes a more realistic internal gravity wave continuum.
The GEOS model was configured to use the C1440 cubed-sphere grid, which has nominal horizontal grid spacing of 6.9 km.
The vertical grid type is hybrid sigma-pressure with 72 levels. A detailed description of the GEOS atmospheric model configu-75 ration used in COAS is found in Molod et al. (2015) and Strobach et al. (2020). The surface layer parameterisation of turbulent fluxes is a modified version of the parameterisation documented in Helfand and Schubert (1995), with a wind stress and surface roughness model modified by the updates of Garfinkel et al. (2011) for a mid-range of wind speeds, and further modified by the updates of Molod et al. (2013) for high winds.  Su et al. (2018), and many others. The LLC2160 solves the hydrostatic primitive equations for velocity, potential temperature, and salinity with a seawater equation of state. The finite volume method is used to discretize the equations in space. The LLC2160 configuration uses an implicit free surface, real freshwater surface forcing, and the K-Profile Parameterization (KPP) vertical mixing scheme of Large et al. (1994)  The formalism of the coupling between the atmosphere and the ocean is classical and can be explained as follows. The Monin-Obukhov similarity theory-based parameterization of surface layer turbulence used to compute air/sea fluxes of heat, moisture and momentum is described in Helfand and Schubert (1995). This parameterization includes the effects of a viscous sublayer over oceans based on Yaglom and Kader (1974), which describes a resistance to enthalpy transfer that increases with surface 95 roughness. The stability functions for unstable surface layers are the KEYPS equation of Panofsky et al. (1977) for momentum and its generalization for scalar quantities. For stable surface layers the stability functions are those of Clarke (1970) for momentum and heat. The ocean roughness is determined by a polynomial which is a blend of the algorithms of Large and Pond (1981) and Kondo (1975) for low wind speeds, modified in the mid-range wind regime based on recent observations in the Southern Ocean according to Garfinkel et al. (2011) and in the high wind regime according to Molod et al. (2013). Note 100 that the ocean and atmosphere exchange momentum, heat, and fresh water through a "skin layer" interface which includes a parameterization of the diurnal cycle (Price et al., 1978). For the high-resolution simulation discussed here, the inertia of the skin layer is small. Finally, computations of momentum and heat fluxes at the air-sea interface take into account the differences between ocean and atmosphere resolutions. This is done using an exchange grid, created by the intersection of the ocean and atmospheric grids, which ensures complete conservation of momentum, heat, and freshwater flux across the air-sea interface.

105
The COAS simulation was initialized on January 20 using 2012 ocean initial conditions from the forced LLC2160 MITgcm simulation and 2020 atmospheric initial conditions from the Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2) interpolated to the C1440 GEOS grid. The reason for using 2012 ocean initial conditions is that there was no other spun-up MITgcm simulation of sufficient resolution available at the time the coupled simulation began.

110
The 2020 atmospheric initial conditions were imposed by the DYnamics of the Atmospheric general circulation Modeled On Non-hydrostatic Domains (DYAMOND; Stevens et al., 2019) Phase II protocol. The mismatch in ocean and atmospheric initial condition years is not ideal but given that this is an unconstrained coupled simulation, the simulation year is notional. The results shown in this study are based on a simulation period of 14 months (January 20, 2020 to March 25, 2021). We did not take into account the first two months that correspond to the spin-up period. Model outputs concern the last 12 months and include 115 hourly three-dimensional fields for all oceanic and atmospheric variables, some higher-frequency (15-minute) two-dimensional atmospheric fields, and many diagnostic variables, for a total storage requirement of ∼2 Petabytes. resulting from atmospheric weather patterns that propagate rapidly, for example, going from South Africa to South America within 6-10 days ( Figure 1a). Embedded within these large-scale patterns are smaller-scale patterns (as small as 100 km), some of them propagating with the large-scale ones, others being quasi-stationary (see Box 1 on Figure 1a and the movie). The latter are mostly the signature of ocean currents on the wind stress and can be identified from watching the movie. Impact of land topography on the wind stress is also noticeable in the movies, in particular close to the east coast of Asia at mid-latitudes and but containing less energy than small-scale currents, are also noticeable in the movie. These patterns propagate with large-scale atmospheric storms. They are the signature of near-inertial motions and internal waves driven by the large-scale winds stress.

Wind stresses and ocean currents in spectral space
The different temporal and spatial scales of wind stresses and ocean currents are further characterized in spectral space in 140 regional domains. Figure 2 shows spectra of wind stresses and currents in the Kuroshio Extension region in the North Pacific (see Appendix A for spectra calculation details). The frequency and wavenumber ranges span, respectively, periods between 2 hours and 40 days and length scales from 10 km to 1000 km. Wind stresses and currents occupy different regions in spectral space. On the one hand, wind stresses ( Figure 2a) are mostly characterized by high frequencies (< 2 days) comprised of large spatial scales (>500 km) for time scales larger than 12 hours and small spatial scales (20-500 km) for time scales smaller 145 than 12 hours. Such wind stresses are the signature of large-scale atmospheric storms and the associated small-scale patterns that propagate with them. Small-scale wind stress patterns associated with slowly-moving ocean eddies are weaker and occupy periods larger than 2 days. On the other hand, energetic ocean currents ( Figure 2b) are mostly characterized by smaller spatial scales (<500 km) and lower frequencies (periods >2 days). As sketched in Figure 2c, these currents are associated with mesoscale eddies known to be driven by the baroclinic instability of mean currents. However, ocean currents also have a large 150 magnitude in the near-inertial band (ω ≈ f ) with scales larger than 500 km. These currents are associated with near-inertial waves (see Figure 2c) forced by high-frequency winds.
Based on the properties discussed above, we analysed the wind work over a three-month period, during winter, spring, summer and fall to emphasize its seasonality. During each season, we consider the following decomposition for surface wind stress τ 155 and ocean-surface currents u o : where X represents either τ or u o , the overline operator represents a time-average over 3 months, also called time-mean or seasonal-mean, and the prime operator represents time fluctuations with periods smaller than 3 months. The time fluctuations are further decomposed into a high-frequency component (hf ) for periods smaller than 3 days and a low-frequency component 160 (lf ) for periods between 3 days and 3 months. Varying the 3-day threshold between low-frequency and high-frequency motions does not have a significant impact on the results of the present study. The hf component captures high-frequency contributions such as those at the inertial frequency. The low-frequency component is further decomposed into two contributions in terms of spatial scales: the large-scale contribution (lf >) for spatial scales larger than a critical length scale L c and the small-scale contribution (lf <) for scales smaller than L c . Following Rai et al. (2021), we define L c as the length scale for which the low-165 frequency component of wind work is negative for scales smaller than L c and positive for larger scales. Negative wind work at these scales has been referred to as "eddy killer" or "eddy damping", a mechanism that has been thoroughly investigated during the past 15 years (Eden and Dietze, 2009;Renault et al., 2016Renault et al., , 2018Rai et al., 2021). Using the same procedure as Rai et al. (2021), we found that L c ≈ 250 km (see Section 4.3 for more details).

Analysis of the wind work 170
The wind work depends not only on the amplitudes of time-mean and fluctuating surface wind stress and currents but also on their cross-correlation. We apply the Reynolds decomposition to Eq. 2 using Eq. 3. The resulting wind work at each grid point averaged over 3 months includes a time-mean component (τ .u o ) and a total time-dependent component (τ .
H is a mixed-layer depth assumed to be constant. From Eq. 5, surface currents, u o lf < , are directly forced by τ lf < .u o lf < .
However, these currents are also affected by the first RHS term in Eq. 5 that involves u o lf > . If currents, u o lf > , are unstable, production of surface currents at smaller spatial scales (u o lf < ) can occur through this first RHS term. From Eq. 6, u o lf > is directly forced by τ lf > .u o lf > . The consequence is that, through the first RHS term in Eq.5, u o lf < is indirectly forced by Of course, scale interactions are more complex and involve more frequencies and spatial scales as discussed at the end of Appendix B (see Eq. B14). Such nonlinear interactions enable the kinetic energy transfer between scales (inverse and direct kinetic energy cascades) as well as current instabilities. However, the present example illustrates that, in order to understand the wind impact on the ocean dynamics, we need to consider the different components of the wind work displayed in Eq. 4 altogether, and not just focusing on one or two components. The present study analyzes all wind work components.

200
A more thorough future study should be dedicated to the kinetic energy budget in the upper oceanic layers that involves both wind work forcing and nonlinear advection of momentum.

Multiscale decomposition of wind work
In this section we analyse the time-mean component of the wind work (first term on the RHS of Eq. 4) as well as the total 205 time-dependent components (the last three terms on the RHS of Eq. 4). The total time-dependent components include the high-frequency component and low-frequency component (see Eq. 4). From Table 1, time-mean (COAS TM) and total timedependent (COAS TD) components represent respectively ∼44% and ∼56% of the total wind work. Their relative contributions as well as the total wind work (COAS total) weakly varying with seasons. The total wind work is larger than 5 TW, a value close to recent estimations (Yu et al., 2018;Yu and Metzger, 2019). However, the spatial distribution of the different wind work 210 components vary with seasons, as discussed in the following subsections.  summer. Wind work in tropical and equatorial regions has the same order of magnitude as the wind work at mid-latitudes.
In all seasons, wind work in tropical and equatorial latitudes, i.e. between 30 • N and 30 • S, displays several zonal patterns elongated over ∼ 3000 km mostly across the Indian, Pacific and Atlantic oceans with an intensification across the equator 220 (Figures 4a-d). Such wind work, known to be associated with westward trade winds, impacts tropical and equatorial zonal jets (Maximenko et al., 2008;Chelton et al., 2011;Laurindo et al., 2017). The negative zonal band around 6 • N is associated with the well-known eastward equatorial jet (Qiu et al., 2017). At these latitudes, wind work experiences a strong seasonality (Figures 4a-d)  confirmed by the zonally averaged wind work (panel e). This seasonality at mid-latitudes is explained by synoptic atmospheric storms (with time scales of a few days) that are intensified in winter. These characteristics are consistent with previous studies (Watanabe and Hibiya, 2002;Zhai, 2013;Yu et al., 2018). Also, the emergence of hurricanes in summer and fall, in particular in the northern hemisphere, have a strong signature on the wind work (see for example Figure 5c, and also movies in Supporting Information).

245
Spectra of ocean current and wind stress for different seasons (respectively first and second columns of Figure 6a) in the Kuroshio Extension reveal how the different time and spatial scales associated with the wind work components vary seasonally. The effective spatial scales considered in these spectra are smaller than 1000 km and time scales smaller than three months.
Within this time and spatial domain, ocean motions are dominated by near-inertial and higher frequency motions as well as by 250 lower frequency mesoscale eddies (<500 km). Currents with larger scales are weakly energetic in this region. The wind stress is found to be much weaker in spring and summer than in fall and winter. During fall and winter the wind stress has greater energy at higher frequencies (Figure 6a, second column). High-frequency ocean currents, such as near-inertial oscillations (NIO) forced by the wind stress, exhibit a seasonality different from the wind stress: NIOs are energetic in summer and fall (third and fourth rows on Figure 6a, first column) and weaker during winter and spring (first and second rows on Figure 6a, first column).

255
One explanation is that wind stress forces NIO kinetic energy integrated over the mixed-layer depth. Since the mixed-layer depth is smaller in summer than in winter, the velocities associated with the total mixed layer NIO kinetic energy are larger in summer. This is confirmed when the surface NIO kinetic energy is multiplied by the mixed-layer depth: we recover a seasonality close to that of wind field and therefore of the wind work, as displayed on Figure 6b. Ocean currents with low-frequency, such as mesoscale eddies (with sizes larger than 100 km) and submesoscale structures, are less energetic in summer and fall 260 compared with winter and spring (see Figure 6b). Such seasonality of low-frequency ocean motions is consistent with previous studies Qiu et al., 2018;Callies et al., 2015;Rocha et al., 2016). Indeed, submesoscales (<50 km) become energetic in winter when the mixed-layer depth is large, with the kinetic energy of these scales being transferred to mesoscale eddies through the so-called inverse kinetic energy cascade, leading to higher mesoscale kinetic energy in spring Lawrence and Callies, 2022).
From these wind work co-spectra, the total wind work is larger in fall and winter and smaller in spring and summer ( We now examine the low-frequency component of the wind work corresponding to wind stresses and ocean currents with periods larger than 3 days. Low-frequency ocean currents at mid-latitudes are often referred to as currents in geostrophic balance 275 (balance between Coriolis and pressure gradients terms in the momentum equations), usually diagnosed from satellite altimetry (Chelton et al., 2011). The present numerical study includes, in addition, ageostrophic low-frequency currents (departing from geostrophy) that comprise ageostrophic eddy currents and surface wind-driven Ekman flow. These ageostrophic currents can explain 30 to 50 % of the total low-frequency currents in energetic areas Chassignet and Xu, 2017). However, we expect the correlation between wind stress and wind-driven Ekman flow to be larger than the correlation between wind 280 stress and total eddy currents since the latter are mostly driven by the ocean interior dynamics.
Referring to Eq. 4, the low-frequency component is decomposed into two parts using a critical length scale, L c : the first part, corresponding to wind stresses and ocean currents with scales smaller than L c . The value, L c , is defined such that the low-285 frequency component to the wind work is negative for scales smaller than L c and positive for larger scales. The methodology to determine L c follows Rai et al. (2021). The low-frequency fields (X lf = τ lf or u o lf ) are convolved with a window function G > (Top-Hat kernel to define the low-frequency component with spatial scales larger than L c : X lf > = G > * X lf , where * is a convolution on a sphere as described in Aluie (2019). Then, the low-frequency component to the wind work of all spatial scales smaller than a given scale, L, has been estimated for the global ocean using 12 month outputs of wind stresses and ocean 290 currents and is shown on Figure 7a. As expected, the wind work is negative for small scales and reaches a minimum at L = L c equal to 250 km for COAS simulation. This means that wind work for L < L c is negative and becomes postive for L > L c .
From Figures 8a-d, τ lf < .u o lf < (the wind work corresponding to scales < L c ) is negative in all seasons in most oceanic regions. This points to the eddy damping effect explained by many studies (Eden and Dietze, 2009;Renault et al., 2016Renault et al., , 2018295 Rai et al., 2021). An heuristic argument is that winds usually have scales larger than 500 km and therefore an approximation of the wind stress and wind work at small scales (L < L c ) is given by (using Eqs. 1 and 2): More detailed arguments are found in Renault et al. (2017) and Rai et al. (2021). From Figures 8a-d, the negative con- from τ lf < .u o lf < , not only in terms of sign but also in terms in terms of magnitude and seasonality. These figures indicate that magnitude of τ lf > u o lf > is 10 times larger than τ lf < .u o lf < and is closer to, although smaller by a factor 2 than, the total timedependent wind work (Figures 5e and 9e). The strong seasonality of τ lf > u o lf > resembles the total time-dependent wind work.
These results indicate that the total time-dependent wind work at mid-latitudes splits almost equally into high-frequency and low-frequency components. In tropical and equatorial regions, a comparison between Figures 5e and 9e reveals that patterns 315 have a comparable magnitude, indicating that the high-frequency component of the wind work is small at these latitudes. The positive contribution of τ lf > u o lf > , at mid-latitudes, should strengthen large-scale low-frequency ocean currents. As emphasized by Chen et al. (2014) and Yang et al. (2021), shear and baroclinic instabilities of these large-scale currents may generate smaller-scale eddy currents. This suggests that production of smaller-scale eddies (L < L c ) by the positive part, τ lf > .u o lf > , through instabilities, may be larger than the eddy damping as discussed in Section 3.2. This points to the importance of ac-320 counting for all wind work components to better infer the wind forcing of the ocean dynamics. The result also emphasizes that a full kinetic energy budget should account for all the wind work components as well as the nonlinear advection terms in the momentum equations, since the instabilities mentioned before are explained by these terms. To further understand the impact of high-frequency wind stress on the wind work, we have tested this phase relationship in spectral space, focusing on the Kuroshio Extension region. Figure 10a shows the wind work co-spectrum estimated from the coupled simulation. As expected, the co-spectrum reveals a positive maximum around the inertial and higher frequencies, which 340 corresponds to the forcing of near-inertial and higher frequency motions (Klein et al., 2004;Alford et al., 2016). The magnitude of the wind work for these high frequencies (in terms of period, T <3 days) is 52 mW/m 2 . For time scales larger than 3 days, the wind work is negative (Figure 10a). This is related to the mesoscale eddy damping mentioned before, since low-frequency ocean motions in the Kuroshio Extension are mostly associated with mesoscale eddies and not large-scale currents. In terms of negative wind work, the magnitude is -16 mW/m 2 . Considering positive and negative wind work, the net wind-work is positive 345 and equal to 36 mW/m 2 . Similar results have been found in other areas of the world ocean, such as the ACC, the Gulf Stream and tropical regions (see Figure 11). We repeated this spectral calculation by applying a phase lag of 12 hours between wind stresses and ocean currents. Results (see the comparison of Figures 10a and b) indicate that the 12-hour offset impacts the wind work in the high-frequency band. In this spectral band, the wind work now displays alternating positive and negative values. The wind work that impacts low frequency motions is almost unchanged by the offset. As a result the total wind work spatial collocation of wind stress and currents, but also their contemporaneity, which has an impact on the integrated wind work.

Discussion and Conclusion
The scalar product of wind stress and ocean surface current, called wind work, is the kinetic energy transfer between ocean 355 and atmosphere. Our study has examined the impact of the wind work on the forcing of ocean currents using outputs of wind stresses and currents from a new coupled ocean-atmosphere simulation with high spatial resolution. Resulting wind stresses and surface ocean currents involve a broad range of time and space scales, from one hour to one year and 10 km to more than 3000 km. Our examination makes use of a simple method that splits the wind work into three components. (i) The highfrequency component, corresponding to wind stress and ocean current with time scales less than 3 days, (ii) the low-frequency 360 component, corresponding to wind stress and ocean current with time scales between 3 days and 3 months, and (iii) the timemean, or seasonal-mean, component diagnosed from wind stress and ocean current averaged over 3 months. Each of these three components, when integrated over the world ocean, does not vary much with seasons and explains 28% of the total wind work for the first two and 44% for the third one. This leads to a total wind work larger than 5 TW, a value close to recent estimations Figure 11. Co-spectrum of wind stresses and surface ocean currents in several regions from the COAS simulation (see square boxes in the middle panel). Note the color range is different in each panel. (Yu et al., 2018;Yu and Metzger, 2019). However, the analysis in physical and spectral spaces of each of these components 365 reveals a strong diversity their characteristics.
The high-frequency component of the wind work (time scales smaller than 3 days) dominates in regions of mid-latitude atmospheric storm tracks where it directly forces internal gravity waves, mostly near-inertial oscillations with large spatial scales.
One important characteristic of this component is its sensitivity to the phase relationship between wind stresses and ocean 370 currents. Thus, a spectral analysis shows that a phase shift of 3 to 12 hours between wind stress and ocean current, reduces this component by a factor up to 10. The high frequency component also has a strong seasonality because of winter atmospheric storms .
The low-frequency component (corresponding to wind stress and ocean current with time scales between 3 days and 3 months) 375 has been analyzed following Rai et al. (2021)'s approach which consists of defining a critical length scale that splits this component into a negative small-scale part and a positive large-scale part. The positive part exhibits a strong seasonality at mid-latitude, not present in the negative part. In addition, magnitude of the positive part is 10 times larger than the negative part. These characteristics of the wind work, not found when using geostrophic currents diagnosed from altimetry datasets, point to the importance of low-frequency ageostrophic current contribution to the wind work (Rai et al., 2021). The large 380 magnitude of the positive part may be explained by the strong correlation between wind stress and wind-driven Ekman flow.
Confirmation of this explanation will be the focus of a future study. The small-scale part acts as an eddy damping for mesoscale eddies as discussed before (Eden and Dietze, 2009;Renault et al., 2016Renault et al., , 2018. This eddy damping effect may be counterbalanced by the positive large-scale part of the wind work. Indeed, instabilities of larger-scale currents may energize currents associated with smaller eddies (L < 250 km). Reciprocally, smaller eddies can energize larger eddies through an inverse ki-385 netic energy cascade. This emphasizes that the two low-frequency parts of the wind work cannot be examined separately, but need to be examined through a full kinetic energy budget, a focus of a future work.
The time-mean, or seasonal-mean, component is significant in equatorial/tropical regions as well as at mid-latitudes. This component forces equatorial and tropical zonal jets as well as stationary and propagating large eddies. It may also force tropical 390 wave instabilities, through nonlinear advection terms, but this has still to be assessed using a full kinetic budget as mentioned before.
Our results emphasize the need to have satellite observations of wind stresses and currents, that are collocated and contemporaneous, with a resolution of at least 10 km and a temporal resolution less than 12 hours. The present ASCAT wind observations 395 and ocean currents diagnosed from conventional altimeters do not meet these requirements. However, several future projects, such as the wind and current mission (Odysea) (Rodríguez et al., 2019) and the Ocean Surface Current Multiscale Observation Mission (OSCOM) (Du et al., 2021), intend to address the limitations of existing wind stress and ocean current products.
Odysea mission aims to measure wind stresses and ocean surface currents (including both, geostrophic and ageostrophic currents) with sensors on-board a single satellite, with a spatial and temporal resolution of ∼ 10 km and twice a day. In addition to 400 being collocated and contemporaneous, these global measurements of wind stress and ocean current will have wide-swaths, as large as ∼ 1, 800 km. Observations from the upcoming Surface Water and Ocean Topography mission (SWOT) should allow for the diagnosis of geostrophic currents with a resolution of ∼15 km over a wide-swath of 120 km (Fu and Ferrari, 2008;Wang et al., 2019). Then, combining Odysea and SWOT observations should permit to assess the relative contributions of geostrophic and ageostrophic currents to the wind work. We envision that these future observations, exploited in combination 405 with SST observations from Advanced Microwave Scanning Radiometer (AMSR-E), will permit estimates of not only the kinetic energy budget (including the wind work and nonlinear advection of momentum), as suggested by the present study, but also the heat budget in the upper ocean . Modelling studies like the present one should help to better assess the potential of these missions.

410
Appendix A: Frequency-wavenumber spectrum and co-spectrum The ω-κ spectrum of a given variable φ(x, y, t) is computed in a domain 1000 km-size and over 90 days. We refer the reader to Torres et al. (2018) for the full methodology. Briefly, before computing the ω-k spectrum of a φ(x, y, t), its linear trend is removed and a 3-D Hanning window is subsequently applied to the de-trended φ(x, y, t) . A discrete 3-D Fourier transform is then computed to retrieveφ(k, l, ω) the Fourier coefficients, where. is the Fourier transform, k the zonal 415 wavenumber, l the meridional wavenumber, and ω the frequency. Finally, the 3-D Fourier transform is used to compute a 2-D spectral density, |φ| 2 (κ, ω) where κ is the isotropic wavenumber defined as κ = √ k 2 + l 2 . The transformation from an anisotropic spectrum to an isotropic spectrum is performed following the methodology described by Torres et al. (2018).
The co-spectrum of the wind work is computed similarly to the ω-κ spectrum, following the methodology described in Flexas et al. (2019). First, the Fourier transforms of the wind stress τ (k, l, ω) and ocean current u o (k, l, ω) are calculated. The cospectrum of the wind work is then given by where Re is the real part of the complex quantity, and asterisk (*) the complex conjugate. The 2-D co-spectrum, τ .u o (κ, ω), 420 is retrieved using the procedure described in the first paragraph of this appendix.
The ω-κ spectrum and co-spectrum are presented in a variance preserving form for easier comparison across the frequencywavenumber domain.
Time-averaging Eq. B2 and multiplying the resulting equation by u o yields After multiplying Eq. B2 by u o + u o and taking the time average and using Eq. B3, we get The same operation can now be done using u o = u o hf + u o lf , which leads to The same arguments, but now in spatial space, can be applied using u o lf = u o lf > + u o lf > leading to with . the Fourier transform, * the conjugate, ω the frequency, k the wavenumber and R the real part. So, each frequency and each wavenumber of the wind stress forces surface currents with the same frequency and wavenumber.

450
However, the full momentum equations, including nonlinear advection terms, highlight that fluctuating wind stresses indirectly force mean surface currents and, similarly, mean wind stresses force fluctuating surface currents. These nonlinear effects are illustrated below.
Let us start with the full momentum equations Applying for example the decomposition u o = u o + u o and τ = τ + τ to Eq. B11 leads, after some calculations, to From eqs. B12 and B13, the time-mean and fluctuations surface currents, resulting directly from the wind work forcing, 460 subsequently interact through the nonlinear advection terms in the momentum equations. For example, fluctuating surface currents forced by wind stress fluctuations (Eq. B13) impact mean current through the second RHS term in Eq. B12. Similarly, mean current forced by mean wind stress (Eq.B12) impact current fluctuations through the first RHS term in Eq. B13 since these mean currents can be unstable. This example emphasizes that mean wind stress can force indirectly fluctuating currents and fluctuating wind stress can force indirectly mean currents. It confirms that the different components of the wind work 465 displayed in Eq. 4 need to be considered altogether and not separately. Nonlinear interactions are more complex than shown in eqs. B12 and B13. This can be understood when moving again to the spectral space. Indeed, the generalization of Eq. B10 leads to where u o in the term u o .∇u o (ω, k) involve frequencies (ω 1 and ω 2 ) and wavenumbers (k 1 and k 2 ) such that ω 1 + ω 2 = ω and 470 k 1 + k 2 = k.
Code availability. The exact version of the model used to produce the results used in this paper is archived on Zenodo (https://doi.org/10. 5281/zenodo.6686083), as are input data and scripts to run the model and produce the plots.
Data availability. The coupled ocean-atmosphere simulation can be found at: https://portal.nccs.nasa.gov/datashare/G5NR/DYAMONDv2/ GEOS_6km_Atmosphere-MITgcm_4km_Ocean-Coupled/GEOSgcm_output/. In particular, the dataset contained in the folder geosgcm_surf/ were used in this study. The variables used in this study are U (east-west velocity component),V (north-south velocity component), oceTAUX (east-west wind stress component), and oceTAUY (north-south wind stress component).
Author contributions. HT and PK led the data analysis and data interpretation and drafted the manuscript. DM, CNH, AM, and ES helped develop and integrate the coupled ocean-atmosphere simulation. All authors contributed to the scientific interpretation of the results and reviewed the manuscript.
DM, HZ, and DPM were supported by the NASA Physical Oceanography (PO) and Modeling, Analysis, and Prediction (MAP) programs. PK acknowledges support from the SWOT Science Team, the NASA S-Mode project and the QuikSCAT mission. AFT and MF were supported by the NASA S-MODE project and PDRDF funding from NASA's Jet Propulsion Laboratory BQ acknowledges support from NASA OSTST project. High-end computing was provided by the NASA Advanced Supercomputing (NAS) Division at the Ames Research Center.