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 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.

A submesoscale-permitting global ocean model is used to study the upper ocean turbulence. Submesoscale processes peak in winter and, consequently, geostrophic kinetic energy (KE) spectra tend to be relatively shallow in winter (∼k-2) with steeper spectra in summer (∼k-3). This seasonal transition from steep to shallow power-law in the KE spectra indicates a transition from quasi-geostrophic (QG) turbulence in summer to pronounced surface-QG-like turbulence in winter. It is shown that this transition in KE spectral scaling has two phases. In the first phase (late autumn), KE spectra show a presence of two spectral regimes: ∼k-3 scaling in mesoscales (100-300 km) and ∼k-2 scaling in submesoscales (<50 km), indicating the coexistence of QG, surface-QG, and frontal dynamics. In the second phase (late winter), mixed-layer instabilities convert available potential energy into KE, which cascades upscale leading to flattening of the KE spectra at larger scales, and k-2 power-law develops in mesoscales too.

We describe an idealized primitive equation model for studying mesoscale turbulence and leverage a hierarchy of grid resolutions to make eddy-resolving calculations on the finest grids more affordable. The model has intermediate complexity, incorporating basin-scale geometry with idealized Atlantic and Southern oceans, and with non-uniform ocean depth to allow for mesoscale eddy interactions with topography. The model is perfectly adiabatic and spans the equator, and thus fills a gap between quasi-geostrophic models, which cannot span two hemispheres, and idealized general circulation models, which generally have diabatic processes and buoyancy forcing. We show that the model solution is approaching convergence in mean kinetic energy for the ocean mesoscale processes of interest, and has a rich range of dynamics with circulation features that emerge only due to resolving mesoscale turbulence.