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Due to their limited resolution, numerical ocean models need to be interpreted as representing filtered or averaged equations. How to interpret models in terms of formally averaged equations, however, is not always clear, particularly in the case of hybrid or generalized vertical coordinate models, which limits our ability to interpret the model results and to develop parameterizations for the unresolved eddy contributions. We here derive the averaged hydrostatic Boussinesq equations in generalized vertical coordinates for an arbitrary thickness‐weighted average. We then consider various special cases and discuss the extent to which the averaged equations are consistent with existing ocean model formulations. As previously discussed, the momentum equations in existing depth‐coordinate models are best interpreted as representing Eulerian averages (i.e., averages taken at fixed depth), while the tracer equations can be interpreted as either Eulerian or thickness‐weighted isopycnal averages. Instead we find that no averaging is fully consistent with existing formulations of the parameterizations in semi‐Lagrangian discretizations of generalized vertical coordinate ocean models such as MOM6. A coordinate‐following average would require “coordinate‐aware” parameterizations that can account for the changing nature of the eddy terms as the coordinate changes. Alternatively, the model variables can be interpreted as representing either Eulerian or (thickness‐weighted) isopycnal averages, independent of the model coordinate that is being used for the numerical discretization. Existing parameterizations in generalized vertical coordinate models, however, are not always consistent with either of these interpretations, which, respectively, would require a three‐dimensional divergence‐free eddy tracer advection or a form‐stress parameterization in the momentum equations. 

{"Abstract":["Released to complement a revised version of Haine et al. 2024.\n\nJulia package to complement "A Review of Green's Function Methods in Ocean Circulation Models," by Haine et al. (2024). See https://essopenarchive.org/users/528978/articles/1215807-a-review-of-green-s-function-methods-for-tracer-timescales-and-pathways-in-ocean-models. This package has two goals.\n\n1. One of the stated goals of the manuscript is to make Green's Function methods accessible for learning purposes. A computational notebook is provided with this package and detailed below.\n\n2. Here, we also aim to make a Julia package that is useful and computationally efficient for research purposes. The codes are contained in the source directory (`src`) and can be used and imported by other Julia projects. \n\n "]} 

Abstract Water Mass Transformation (WMT) theory provides conceptual tools that in principle enable innovative analyses of numerical ocean models; in practice, however, these methods can be challenging to implement and interpret, and therefore remain under‐utilized. Our aim is to demonstrate the feasibility of diagnosing all terms in the water mass budget and to exemplify their usefulness for scientific inquiry and model development by quantitatively relating water mass changes, overturning circulations, boundary fluxes, and interior mixing. We begin with a pedagogical derivation of key results of classical WMT theory. We then describe best practices for diagnosing each of the water mass budget terms from the output of Finite‐Volume Generalized Vertical Coordinate (FV‐GVC) ocean models, including the identification of a non‐negligible remainder term as the spurious numerical mixing due to advection scheme discretization errors. We illustrate key aspects of the methodology through the analysis of a polygonal region of the Greater Baltic Sea in a regional demonstration simulation using the Modular Ocean Model v6 (MOM6). We verify the convergence of our WMT diagnostics by brute‐force, comparing time‐averaged (“offline”) diagnostics on various vertical grids to timestep‐averaged (“online”) diagnostics on the native model grid. Finally, we briefly describe a stack of xarray‐enabled Python packages for evaluating WMT budgets in FV‐GVC models (culminating in the newxwmbpackage), which is intended to be model‐agnostic and available for community use and development. 

Abstract Density-driven steric seawater changes are a leading-order contributor to global mean sea level rise. However, intermodel differences in the magnitude and spatial patterns of steric sea level rise exist at regional scales and often emerge during the spinup and preindustrial control integrations of climate models. Steric sea level results from an eddy-permitting climate model, GFDL CM4, are compared with a lower-resolution counterpart, GFDL-ESM4. The results from both models are examined through basin-scale heat budgets and watermass analysis, and we compare the patterns of ocean heat uptake, redistribution, and sea level differ in ocean-only [i.e., Ocean Model Intercomparison Project (OMIP)] and coupled climate configurations. After correcting for model drift, both GFDL CM4 and GFDL-ESM4 simulate nearly equivalent ocean heat content change and global sea level rise during the historical period. However, the GFDL CM4 model exhibits as much as a 40% increase in surface ocean heat uptake in the Southern Ocean and subsequent increases in horizontal export to other ocean basins after bias correction. The results suggest regional differences in the processes governing Southern Ocean heat export, such as the formation of Antarctic Intermediate Water (AAIW), Subpolar Mode Water (SPMW), and gyre transport between the two models, and that sea level changes in these models cannot be fully bias-corrected. Since the process-level differences between the two models are evident in the preindustrial control simulations of both models, these results suggest that the control simulations are important for identifying and correcting sea level–related model biases. 

The vertical structure of ocean eddies is generally surface-intensified, commonly attributed to the dominant baroclinic modes arising from the boundary conditions (BCs). Conventional BC considerations mostly focus on either flat- or rough-bottom conditions. The impact of surface buoyancy anomalies—often represented by surface potential vorticity (PV) anomalies—has not been fully explored. Here, we study the role of the surface PV in setting the vertical distribution of eddy kinetic energy (EKE) in an idealized adiabatic ocean model driven by wind stress. The simulated EKE profile in the extratropical ocean tends to peak at the surface and have ane-folding depth typically smaller than half of the ocean depth. This vertical structure can be reasonably represented by a single surface quasigeostrophic (SQG) mode at the energy-containing scale resulting from the large-scale PV structure. Due to isopycnal outcropping and interior PV homogenization, the surface meridional PV gradient is substantially stronger than the interior PV gradient, yielding surface-trapped baroclinically unstable modes with horizontal scales comparable to or smaller than the deformation radius. These surface-trapped eddies then grow in size both horizontally and vertically through an inverse energy cascade up to the energy-containing scale, which dominates the vertical distribution of EKE. As for smaller horizontal scales, the EKE distribution decays faster with depth. Guided by this interpretation, an SQG-based scale-aware parameterization of the EKE profile is proposed. Preliminary offline diagnosis of a high-resolution simulation shows the proposed scheme successfully reproducing the dependence of the vertical structure of EKE on the horizontal grid resolution. 

Abstract West Antarctic Ice Sheet mass loss is a major source of uncertainty in sea level projections. The primary driver of this melting is oceanic heat from Circumpolar Deep Water originating offshore in the Antarctic Circumpolar Current. Yet, in assessing melt variability, open ocean processes have received considerably less attention than those governing cross-shelf exchange. Here, we use Lagrangian particle release experiments in an ocean model to investigate the pathways by which Circumpolar Deep Water moves toward the continental shelf across the Pacific sector of the Southern Ocean. We show that Ross Gyre expansion, linked to wind and sea ice variability, increases poleward heat transport along the gyre’s eastern limb and the relative fraction of transport toward the Amundsen Sea. Ross Gyre variability, therefore, influences oceanic heat supply toward the West Antarctic continental slope. Understanding remote controls on basal melt is necessary to predict the ice sheet response to anthropogenic forcing. 

Abstract The climatological mean barotropic vorticity budget is analyzed to investigate the relative importance of surface wind stress, topography, planetary vorticity advection, 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 coarse‐grained. At length‐scales greater than 1,000 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 km in size, and the associated bottom pressure anomalies in local vorticity balances at length‐scales smaller than 1,000 km. Overall, there is a transition from the Topographic‐Nonlinear regime at scales smaller than 1,000 km to the Topographic‐Sverdrup regime at length‐scales greater than 1,000 km. 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 viscous friction generally peak. 

Abstract Understanding advective‐diffusive dispersal of trace substances in environmental fluids like the global ocean is a ubiquitous challenge in geophysics. Since the turn of the millennium, substantial progress has been made in the theory, implementation in models, and application of such tracers in oceanography. For the first time, this progress is reviewed here in a synthetic way. We focus on tracer techniques in ocean models, including real and virtual tracers that diagnose timescale information, and we emphasize the connection to the Green's function that solves the advection‐diffusion equation. Implementation of these techniques in ocean models is explained in an accessible way. We present example applications of these techniques to questions concerning ocean circulation, transport of biogeochemicals, and paleoceanography, including future opportunities. 

Here, we present an estimate for the ocean's global scale transfer of kinetic energy (KE), across scales from 10 to 40,000 km. Oceanic KE transfer between gyre scales and mesoscales is induced by the atmosphere’s Hadley, Ferrel, and polar cells, and the intertropical convergence zone induces an intense downscale KE transfer. Upscale transfer peaks at 300 gigawatts across mesoscales of 120 km in size, roughly one-third the energy input by winds into the oceanic general circulation. Nearly three quarters of this “cascade” occurs south of 15°S and penetrates almost the entire water column. The mesoscale cascade has a self-similar seasonal cycle with characteristic lag time of ≈27 days per octave of length scales; transfer across 50 km peaks in spring, while transfer across 500 km peaks in summer. KE of those mesoscales follows the same cycle but peaks ≈40 days after the peak cascade, suggesting that energy transferred across a scale is primarily deposited at a scale four times larger. 

Abstract The use of coarse resolution and strong grid‐scale dissipation has prevented global ocean models from simulating the correct kinetic energy level. Recently parameterizing energy backscatter has been proposed to energize the model simulations. Parameterizing backscatter reduces long‐standing North Atlantic sea surface temperature (SST) and associated surface current biases, but the underlying mechanism remains unclear. Here, we apply backscatter in different geographic regions to distinguish the different physical processes at play. We show that an improved Gulf Stream path is due to backscatter acting north of the Grand Banks to maintain a strong deep western boundary current. An improved North Atlantic Current path is due to backscatter acting around the Flemish Cap, with likely an improved nearby topography‐flow interactions. These results suggest that the SST improvement with backscatter is partly due to the resulted strengthening of resolved currents, whereas the role of improved eddy physics requires further research.