The ‘eddying’ ocean, recognized for several decades, has been the focus of much observational and theoretical research. We here describe a generalization for the analysis of eddy energy, based on the use of ensembles, that addresses two key related issues: the definition of an ‘eddy’ and the general computation of energy spectra. An ensemble identifies eddies as the unpredictable component of the flow, and permits the scale decomposition of their energy in inhomogeneous and non-stationary settings. We present two distinct, but equally valid, spectral estimates: one is similar to classical Fourier spectra, the other reminiscent of classical EOF analysis. Both satisfy Parseval’s equality and thus can be interpreted as length-scale dependent energy decompositions. The issue of ‘tapering’ or ‘windowing’ of the data, used in traditional approaches, is also discussed. We apply the analyses to a mesoscale ‘resolving’ (1/12$^\circ$) ensemble of the separated North Atlantic Gulf Stream. Our results reveal highly anisotropic spectra in the Gulf Stream and zones of both agreement and disagreement with theoretically expected spectral shapes. In general, we find spectral slopes that fall off faster than the steepest slope expected from quasi-geostrophic theory.

The thickness-weighted average (TWA) framework, which treats the residual-mean flow as the prognostic variable, provides a clear theoretical formulation of the eddy feedback onto the residual-mean flow. The averaging operator involved in the TWA framework, although in theory being an ensemble mean, in practice has often been approximated by a temporal mean. Here, we analyze an ensemble of North Atlantic simulations at mesoscale-permitting resolution (1/12$^\circ$). We therefore recognize means and eddies in terms of ensemble means and fluctuations about those means. The ensemble dimension being orthogonal to the temporal and spatial dimensions negates the necessity for an arbitrary temporal or spatial scale in defining the eddies. Eddy-mean flow feedbacks are encapsulated in the Eliassen-Palm (E-P) flux tensor and its convergence indicates that eddy momentum fluxes dominate in the separated Gulf Stream. The eddies contribute to the zonal meandering of the Gulf Stream and a northward migration of it in the meridional direction. Downstream of the separated Gulf Stream in the North Atlantic Current region, the interfacial form stress convergence becomes leading order in the E-P flux convergence.

Fine-scale motions ($<$100 km) contribute significantly to the exchanges and dissipation of kinetic energy in the upper ocean. However, knowledge of ocean kinetic energy at fine-scales (in terms of density and transfers) is currently limited due to the lack of sufficient observational datasets at these scales. The sea-surface height measurements of the upcoming SWOT altimeter mission should provide information on kinetic energy exchanges in the upper ocean down to 10-15 km. Numerical ocean models, able to describe ocean dynamics down to $\sim$10 km, have been developed in anticipation of the SWOT mission. In this study, we use two state-of-the-art, realistic, North Atlantic simulations, with horizontal resolutions $ \sim $ 1.5 km, to investigate the distribution and exchanges of kinetic energy at fine-scales in the open ocean. Our results show that the distribution of kinetic energy at fine-scales approximately follows the predictions of quasi-geostrophic dynamics in summertime but is somewhat consistent with submesoscale fronts-dominated regimes in wintertime. The kinetic energy spectral fluxes are found to exhibit both inverse and forward cascade over the top 1000 m, with a maximum inverse cascade close to the average energy-containing scale. The forward cascade is confined to the ocean surface and shows a strong seasonality, both in magnitude and range of scales affected. Our analysis further indicates that high-frequency motions ($<$1day) play a key role in the forward cascade and that the estimates of the spectral fluxes based on geostrophic velocities fail to capture some quantitative aspects of kinetic energy exchanges across scales.

An important characteristic of geophysically turbulent flows is the transfer of energy between scales. It is expected that balanced flows pass energy from smaller to larger scales as part of the well-known upscale cascade while submesoscale and smaller scale flows can transfer energy eventually to smaller, dissipative scales. Much effort has been put into quantifying these transfers, but a complicating factor in realistic settings is that the underlying flows are often strongly spatially heterogeneous and anisotropic. Furthermore, the flows may be embedded in irregularly shaped domains that can be multiply connected. As a result, straightforward approaches like computing Fourier spatial spectra of nonlinear terms suffer from a number of conceptual issues. In this paper, we endeavor to compute cross-scale energy transfers in general settings, allowing for arbitrary flow structure, anisotropy and inhomogeneity. We employ a Green's function approach to the kinetic energy equation to relate kinetic energy at a point to its Lagrangian history. A spatial filtering of the resulting equation naturally decomposes kinetic energy into length scale dependent contributions and describes how the transfer of energy between those scales takes place. The method is applied to a numerical simulation of vortex merger, resulting in the demonstration of the expected upscale energy cascade. Somewhat novel results are that the energy transfers are dominated by pressure work, rather than kinetic energy exchange, and dissipation is a noticeable influence on the larger scale energy budgets.

Understanding processes associated with eddy-mean flow interactions helps our interpretation of ocean energetics, and guides the development of parameterizations. Here, we focus on the non-local nature of Kinetic Energy (KE) transfers between mean and turbulent reservoirs. Transfers are interpreted as non-local when the energy extraction from the mean flow does not locally sustain energy production of the turbulent flow, or vice versa. The novelty of our approach is to use ensemble statistics to define the mean and the turbulent flow. Based on KE budget considerations, we first rationalize the eddy-mean separation in the ensemble framework, and discuss the interpretation of a mean flow driven by the prescribed (surface and boundary) forcing and a turbulent flow u’ driven by non-linear dynamics sensitive to initial conditions. We then analyze 120-day long, 20-member ensemble simulations of the Western Mediterranean basin run at 1/60 resolution. Our main contribution is to recognize the prominent contribution of the cross energy term .u_h’ to explain non-local energy transfers. This provides a strong constraint on the horizontal organization of eddy-mean flow KE transfers since this term vanishes identically for perturbations (u_h’) orthogonal to the mean flow (). We also highlight the prominent contribution of vertical turbulent fluxes for energy transfers within the surface mixed layer. Analyzing the scale dependence of these non-local energy transfers supports the local approximation usually made in the development of meso-scale, energy-aware parameterizations for non-eddying models, but points out to the necessity of accounting for these non-local effects in the meso-to-submeso scale range.

Circulation in the Southern Ocean is unique. The strong wind stress forcing and buoyancy fluxes, in concert with the lack of continental boundaries, conspire to drive the Antarctic Circumpolar Current replete with an intense eddy field. The effect of Southern Ocean eddies on the ocean circulation is significant – they modulate the momentum balance of the zonal flow, and the meridional transport of tracers and mass. The strength of the eddy field is controlled by a combination of forcing (primarily thought to be wind stress) and intrinsic, chaotic, variability associated with the turbulent flow field itself. Here, we present results from an eddy-permitting ensemble of ocean model simulations to investigate the relative contribution of forced and intrinsic processes in governing the variability of Southern Ocean eddy kinetic energy. We find that variations of the eddy field are mostly random, even on longer (interannual) timescales. Where correlations between the wind stress forcing and the eddy field exist, these interactions are dominated by two distinct timescales – a fast baroclinic instability response; and a multi-year process owing to feedback between bathymetry and the mean flow. These results suggest that understanding Southern Ocean eddy dynamics and its larger-scale impacts requires an ensemble approach to eliminate intrinsic variability, and therefore may not yield robust conclusions from observations alone.