Mechanical coupling of the atmosphere to the ocean surface in general circulation models is represented using bulk wind stress formulations. The stress is often based on either absolute wind velocity, τa, or the more correct wind velocity relative to the ocean surface currents, τr. Here, we use coarse-graining to disentangle wind work by these formulations at different length-scales. We show that both can be reasonably accurate in forcing the ocean at length-scales larger than the mesoscales, with τa overestimating wind work by 10%. However, τa and τr show stark and opposing systematic biases in how they drive the mesoscales; τa does negligible (albeit positive) work on the mesoscales, while τr yields eddy-killing (negative work) that is artificially exaggerated by a factor of ≈4. We derive an analytical criterion for eddy-killing to occur, which shows that exaggerated eddy killing is due to resolution mismatch between the atmosphere and ocean. Our criterion highlights the disproportionate effect small-scale winds Ο(100)km can have on the dynamics of mesoscale ocean eddies, despite the dominant atmospheric motions being at length-scales larger than Ο(103) km. The eddy-killing criterion shows that large-scale winds do not necessarily cause eddy-killing but are merely an amplification factor for wind work on the mesoscales, which can be either positive or negative depending on the local alignment of small-scale winds with the ocean eddies. We propose a simple reformulation of τr, without introducing tuning parameters, to remove spurious eddy-killing from air-sea resolution mismatch that is often present in climate models.

We expand on a recent determination of the first global energy spectrum of the ocean’s surface geostrophic circulation \cite{Storer2022} using a coarse-graining (CG) method. We compare spectra from CG to those from spherical harmonics by treating land in a manner consistent with the boundary conditions. While the two methods yield qualitatively consistent domain-averaged results, spherical harmonics spectra are too noisy at gyre-scales ($>1000 $km). More importantly, spherical harmonics are inherently global and cannot provide local information connecting scales with currents geographically. CG shows that the extra-tropics mesoscales (100–500 km) have a root-mean-square (rms) velocity of $\sim15 $cm/s, which increases to $\sim30$–40 cm/s locally in the Gulf Stream and Kuroshio and to $\sim16$–28 cm/s in the ACC. There is notable hemispheric asymmetry in mesoscale energy-per-area, which is higher in the north due to continental boundaries. We estimate that $\approx25$–50\% of total geostrophic energy is at scales smaller than 100 km, and is un(der)-resolved by pre-SWOT satellite products. Spectra of the time-mean component show that most of its energy (up to $70\%$) resides in stationary mesoscales ($<500 $km), highlighting the preponderance of ‘standing’ small-scale structures in the global ocean. By coarse-graining in space and time, we compute the first spatio-temporal global spectrum of geostrophic circulation from AVISO and NEMO. These spectra show that every length-scale evolves over a wide range of time-scales with a consistent peak at $\approx200$ km and $\approx2$–3 weeks.

The climatological mean barotropic vorticity budget is analyzed to investigate the relative importance of surface wind stress, topography and nonlinear advection in dynamical balances in a global ocean simulation. In addition to a pronounced regional variability in vorticity balances, the relative magnitudes of vorticity budget terms strongly depend on the length-scale of interest. To carry out a length-scale dependent vorticity analysis in different ocean basins, vorticity budget terms are spatially filtered by employing the coarse-graining technique. At length-scales greater than 10o (or roughly 1000 km), the dynamics closely follow the Topographic-Sverdrup balance in which bottom pressure torque, surface wind stress curl and planetary vorticity advection terms are in balance. In contrast, when including all length-scales resolved by the model, bottom pressure torque and nonlinear advection terms dominate the vorticity budget (Topographic-Nonlinear balance), which suggests a prominent role of oceanic eddies, which are of Ο(10-100) km in size, and the associated bottom pressure anomalies in local vorticity balances at length-scales smaller than 1000 km. Overall, there is a transition from the Topographic-Nonlinear regime at scales smaller than 10o to the Topographic-Sverdrup regime at length-scales greater than 10o. These dynamical balances hold across all ocean basins; however, interpretations of the dominant vorticity balances depend on the level of spatial filtering or the effective model resolution. On the other hand, the contribution of bottom and lateral friction terms in the barotropic vorticity budget remains small and is significant only near sea-land boundaries, where bottom stress and horizontal friction generally peak.

We apply a coarse-grained decomposition of the ocean’s surface geostrophic flow derived from satellite and numerical model products. In the extra-tropics we find that roughly 60\% of the global surface geostrophic kinetic energy is at scales between 100 km and 500 km, peaking at ~300 km. Our analysis also reveals a clear seasonality in the kinetic energy with a spring peak. We show that traditional mean-fluctuation (or Reynolds) decomposition is unable to robustly disentangle length-scales since the time mean flow consists of a significant contribution (greater than 50%) from scales <500 km. By coarse-graining in both space and time, we find that every length-scale evolves over a wide range of time-scales. Consequently, a running time-average of any duration reduces the energy content of all length-scales, including those larger than 1000 km, and is not effective at removing length-scales smaller than 300 km. By contrasting our spatio-temporal analysis of numerical model and satellite products, we show that the AVISO gridded product suppresses temporal variations less than 10 days for all length-scales, especially between 100 km and 500 km.