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Abstract Mixing along isopycnals plays an important role in the transport and uptake of oceanic tracers. Isopycnal mixing is commonly quantified by a tracer diffusivity. Previous studies have estimated the tracer diffusivity using the rate of dispersion of surface drifters, subsurface floats, or numerical particles advected by satellite‐derived velocity fields. This study shows that the diffusivity can be more efficiently estimated from the dispersion of coherent mesoscale eddies. Coherent eddies are identified and tracked as the persistent sea surface height extrema in both a two‐layer quasigeostrophic (QG) model and an idealized primitive equation (PE) model. The Lagrangian diffusivity is estimated using the tracks of these coherent eddies and compared to the diagnosed Eulerian diffusivity. It is found that the meridional coherent eddy diffusivity approaches a stable value within about 20–40 days in both models. In the QG model, the coherent eddy diffusivity is a good approximation to the upper‐layer tracer diffusivity in a broad range of flow regimes, except for small values of bottom friction or planetary vorticity gradient, where the motions of same‐sign eddies are correlated over long distances. In the PE model, the tracer diffusivity has a complicated vertical structure and the coherent eddy diffusivity is correlated with the tracer diffusivity at the e‐folding depth of the energy‐containing eddies where the intrinsic speed of the coherent eddies matches the rms eddy velocity. These results suggest that the oceanic tracer diffusivity at depth can be estimated from the movements of coherent mesoscale eddies, which are routinely tracked from satellite observations. 

Abstract 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 Isopycnal mixing of tracers is important for ocean dynamics and biogeochemistry. Previous studies have primarily focused on the horizontal structure of mixing, but what controls its vertical structure is still unclear. This study investigates the vertical structure of the isopycnal tracer diffusivity diagnosed by a multiple‐tracer inversion method in an idealized basin circulation model. The first two eigenvalues of the symmetric part of the 3D diffusivity tensor are approximately tangent to isopycnal surfaces. The isopycnal mixing is anisotropic, with principal directions of the large and small diffusivities generally oriented along and across the mean flow direction. The cross‐stream diffusivity can be reconstructed from the along‐stream diffusivity after accounting for suppression of mixing by the mean flow. In the circumpolar channel and the upper ocean in the gyres, the vertical structure of the along‐stream diffusivity follows that of the rms eddy velocity times a depth‐independent local energy‐containing scale estimated from the sea surface height. The diffusivity in the deep ocean in the gyres instead follows the profile of the eddy kinetic energy times a depth‐independent mixing time scale. The transition between the two mixing regimes is attributed to the dominance of nonlinear interactions and linear waves in the upper and deep ocean, respectively, distinguished by a nonlinearity parameter. A formula is proposed that accounts for both regimes and captures the vertical variation of diffusivities better than extant theories. These results inform efforts to parameterize the vertical structure of isopycnal mixing in coarse‐resolution ocean models. 

In a recent paper [Chu (2023; Chu23)], the author formulated the equations governing atmospheric motion in a spheroidal coordinate system. Since the mass distribution of the Earth is not exactly spheroidal, the true gravity is not vertical in that coordinate system. Chu23 compared the magnitude of the static horizontal component of gravity in that system to those of the dynamically active forces and concluded that the horizontal components of gravity should not be neglected. In recent papers by the authors [Chang and Wolfe (2022; CW22) and Stewart and McWilliams (2022; CW22)], we explained that the actual interpretation of the approximation made in atmospheric and oceanic modeling is not neglecting the horizontal component of the true gravity, but is a geometrical approximation, approximating nearly spheroidal geopotential surfaces with bumps on which the true gravity is vertical by exactly spheroidal surfaces. We showed that under such an interpretation, the errors due to the geometrical approximation are small. Chu23 claimed that CW22 and SM22 erroneously neglected the gravity perturbations in their analyses. Here, we explain further the differences between these approaches, in the process showing that the criticisms of Chu23 on CW22 and SM22 are invalid, further supporting our conclusion that the horizontal component of the true gravity is not relevant in ocean and atmospheric dynamics. Physically, the reason why horizontal gravity is irrelevant in the coordinate system used by Chu23 is that it is balanced by a static horizontal pressure gradient force.