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The familiar divergence and Kelvin–Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the tensor product of the vector field with the exterior normal. The importance of this long-established yet relatively little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the two-dimensional divergence and Kelvin–Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given. The potential extension of the generalized two-dimensional identity to curved surfaces is considered and is shown not to hold. Finally, the paper includes a substantial background section on tensor analysis, and presents results in both symbolic notation and index notation in order to emphasize the correspondence between these two notational systems. 

Abstract The Minimet is a Lagrangian surface drifter measuring near-surface winds in situ. Ten Minimets were deployed in the Iceland Basin over the course of two field seasons in 2018 and 2019. We compared Minimet wind measurements to coincident ship winds from the R/V Armstrong meteorology package and to hourly ERA5 reanalysis winds and found that the Minimets accurately captured wind variability across a variety of time scales. Comparisons between the ship, Minimets, and ERA5 winds point to significant discrepancies between the in situ wind measurements and ERA5, with the most reasonable explanation being related to spatial offsets of small-scale storm structures in the reanalysis model. After a general assessment of the Minimet performance, we compare estimates of wind power input in the near-inertial band using the Minimet winds and their measured drift to those using ERA5 winds and the Minimet drift. Minimet-derived near-inertial wind power estimates exceed those from Minimet drift combined with ERA5 winds by about 42%. The results highlight the importance of accurately capturing small-scale, high-frequency wind events and suggest that in situ Minimet measurements are beneficial for accurately quantifying near-inertial wind work on the ocean. Significance Statement In this study we introduce a novel, freely drifting wind measurement platform, the Minimet. After an initial validation of Minimet sea surface wind measurements against independent wind measurements from a nearby research vessel, we investigate their utility in context of the near-inertial work done by the wind on the ocean, which is important for the ocean’s energy budget. We find Minimet near-inertial wind work estimates exceed those estimated using winds from a state-of-the-art wind product by 42%. Our results indicate that capturing storm events happening on time scales less than 12 h is crucial for accurately quantifying near-inertial wind work on the ocean, making wind measurements from platforms such as the Minimet invaluable for these analyses. 

The unsteady Ekman problem involves finding the response of the near-surface currents to wind stress forcing under linear dynamics. Its solution can be conveniently framed in the frequency domain in terms of a quantity that is known as the transfer function, the Fourier transform of the impulse response function. In this paper, a theoretical investigation of a fairly general transfer function form is undertaken with the goal of paving the way for future observational studies. Building on earlier work, we consider in detail the transfer function arising from a linearly-varying profile of the vertical eddy viscosity, subject to a no-slip lower boundary condition at a finite depth. The horizontal momentum equations, rendered linear by the assumption of horizontally uniform motion, are shown to transform to a modified Bessel’s equation for the transfer function. Two self-similarities, or rescalings that each effectively eliminate one independent variable, are identified, enabling the dependence of the transfer function on its parameters to be more readily assessed. A systematic investigation of asymptotic behaviors of the transfer function is then undertaken, yielding expressions appropriate for eighteen different regimes, and unifying the results from numerous earlier studies. A solution to a numerical overflow problem that arises in the computation of the transfer function is also found. All numerical code associated with this paper is distributed freely for use by the community. 

Abstract. A method for objectively extracting the displacement signals associated with coherent eddies from Lagrangian trajectories is presented, refined, and applied to a large dataset of 3770 surface drifters from the Gulf of Mexico. The method, wavelet ridge analysis, is a general method for the analysis of modulated oscillations, here modified to be more suitable to the eddy-detection problem. A means for formally assessing statistical significance is introduced, addressing the issue of false positives arising by chance from an unstructured turbulent background and opening the door to confident application of the method to very large datasets. Significance is measured through a frequency-dependent comparison with a stochastic dataset having statistical and spectral properties that match the original, but lacking organized oscillations due to eddies or waves. The application to the Gulf of Mexico reveals major asymmetries between cyclones and anticyclones, with anticyclones dominating at radii larger than about 50 km, but an unexpectedly rich population of highly nonlinear cyclones dominating at smaller radii. Both the method and the Gulf of Mexico eddy dataset are made freely available to the community for noncommercial use in future research. 

Abstract Observations from the past decades have promoted the idea of a long-lived anticyclonic vortex residing in the Lofoten Basin. Despite repeatedly recorded intense anticyclones, the observations cannot firmly decide whether the signature is of a single vortex or a succession of ephemeral vortices. A vortex persisting for decades requires some reinvigoration mechanism. Wintertime convection and vortex merging have been proposed candidates. We examine Lofoten Basin vortex dynamics using a high-resolution regional ocean model. The model is initialized from a coarser state with a weak eddy field. The slope current intensifies and sheds anticyclonic eddies that drift into the basin. After half a year, an anticyclone arrives at the center, providing the nucleus for a vortex that remains distinct throughout the simulation. Analyses show that this vortex is regenerated by repeated absorption and vertical stacking of lighter anticyclones. This compresses and—in concert with potential vorticity conservation—intensifies the combined vortex, which becomes more vertically stratified and also expels some fluid in the process. Wintertime convection serves mainly to vertically homogenize and densify the vortex, rather than intensifying it. Further, topographic guiding of anticyclones shed from the continental slope is vital for the existence and reinvigoration of the Lofoten vortex. These results offer a new perspective on the regeneration of oceanic anticyclones. In this scenario the Lofoten vortex is maintained through repeated merging events. Fluid remains gradually exchanged, although the vortex is identifiable as a persistent extremum in potential vorticity. 

Abstract The Lofoten Basin is the largest oceanic reservoir of heat in the Nordic Seas, and the site of important heat fluxes to the atmosphere. An intense permanent anticyclone in the basin impacts the regional hydrography, energetics, and ecosystem. Repeated sampling of this Lofoten Basin Eddy from dedicated cruises, autonomous profiling gliders, and acoustically-tracked subsurface floats enables the documentation of its dynamics and energetics over the course of 15 months. The eddy core, in nearly solid-body rotation, exhibits an unusually low vertical vorticity close to the local inertial frequency and important strain rates at the periphery. Subsurface floats as deep as 800 m are trapped within the core for their entire deployment duration (up to 15 months). The potential vorticity is reduced in the core by two orders of magnitude relative to the surroundings, creating a barrier. In the winter, this barrier weakens and lateral exchanges and heat flux between the eddy and the surroundings increase, apparently the result of dynamical instabilities and a possible eddy merger. Based on a simple energy budget, the dissipation timescale for the eddy energy is three years, during which wintertime convection seasonally modulates potential and kinetic energy. 

Stochastic processes exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm), with the spectral slope at high frequencies being associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matérn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matérn process and its relationship to fBm. An algorithm for the simulation of the Matérn process in O(NlogN) operations is given. Unlike fBm, the Matérn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.