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Vertical transport in the ocean plays a critical role in the exchange of freshwater, heat, nutrients, and other biogeochemical tracers. While there are situations where vertical fluxes are important, studying the vertical transport and displacement of material requires analysis over a finite interval of time. One such example is the subduction of fluid from the mixed layer into the pycnocline, which is known to occur near density fronts. Divergence has been used to estimate vertical velocities indicating that surface measurements, where observational data is most widely available, can be used to locate these vertical transport conduits. We evaluate the correlation between surface signatures derived from Eulerian (horizontal divergence, density gradient, and vertical velocity) and Lagrangian (dilation rate and finite time Lyapunov exponent) metrics and vertical displacement conduits. Two submesoscale resolving models of density fronts and a data-assimilative model of the western Mediterranean were analyzed. The Lagrangian surface signatures locate significantly more of the strongest displacement features and the difference in the expected displacements relative to Eulerian ones increases with the length of the time interval considered. Ensemble analysis of forecasts from the Mediterranean model demonstrates that the Lagrangian surface signatures can be used to identify regions of strongest downward vertical displacement even without knowledge of the true ocean state., Comment: 30 pages, 8 figures. Submitted to Ocean Modelling

Monami is the synchronous waving of a submerged seagrass bed in response to unidirectional fluid flow. Here we develop a multiphase model for the dynamical instabilities and flow-driven collective motions of buoyant, deformable seagrass. We show that the impedance to flow due to the seagrass results in an unstable velocity shear layer at the canopy interface, leading to a periodic array of vortices that propagate downstream. Each passing vortex locally weakens the along-stream velocity at the canopy top, reducing the drag and allowing the deformed grass to straighten up just beneath it. This causes the grass to oscillate periodically. Crucially, the maximal grass deflection is out of phase with the vortices. A phase diagram for the onset of instability shows its dependence on the fluid Reynolds number and an effective buoyancy parameter. Less buoyant grass is more easily deformed by the flow and forms a weaker shear layer, with smaller vortices and less material exchange across the canopy top. While higher Reynolds number leads to stronger vortices and larger waving amplitudes of the seagrass, waving is maximized at intermediate grass buoyancy. All together, our theory and computations correct some misconceptions in interpretation of the mechanism and provide a robust explanation consistent with a number of experimental observations., Comment: 16 pages, 9 figures, supplementary information at the end of the document

Partitioning ocean flows into regions dynamically distinct from their surroundings based on material transport can assist search-and-rescue planning by reducing the search domain. The spectral clustering method partitions the domain by identifying fluid particle trajectories that are similar. The partitioning validity depends on the accuracy of the ocean forecasting, which is subject to several sources of uncertainty: model initialization, limited knowledge of the physical processes, boundary conditions, and forcing terms. Instead of a single model output, multiple realizations are produced spanning a range of potential outcomes, and trajectory clustering is used to identify robust features and quantify the uncertainty of the ensemble-averaged results. First, ensemble statistics are used to investigate the cluster sensitivity to the spectral clustering method free-parameters and the forecast parameters for the analytic Bickley jet, a geostrophic flow model. Then, we analyze an operational coastal ocean ensemble forecast and compare the clustering results to drifter trajectories south of Martha's Vineyard. This approach accurately identifies regions of low uncertainty where drifters released within a cluster remain there throughout the window of analysis. Drifters released in regions of high uncertainty tend to either enter neighboring clusters or deviate from all predicted outcomes., Comment: 21 pages, 11 figures, Supplementary Material A and B at the end of the document

The growth rate of material interfaces is an important proxy for mixing and reaction rates in fluid dynamics, and can also be used to identify regions of coherence. Estimating such growth rates can be difficult, since they depend on detailed properties of the velocity field, such as its derivatives, that are hard to measure directly. When an experiment gives only sparse trajectory data, it is natural to encode planar trajectories as mathematical braids, which are topological objects that contain information on the mixing characteristics of the flow, in particular through their action on topological loops. We test such braid methods on an experimental system, the rotor-oscillator flow, which is well-described by a theoretical model. We conduct a series of laboratory experiments to collect particle tracking and particle image velocimetry data, and use the particle tracks to identify regions of coherence within the flow that match the results obtained from the model velocity field. We then use the data to estimate growth rates of material interface, using both the braid approach and numerical simulations. The interface growth rates follow similar qualitative trends in both the experiment and model, but have significant quantitative differences, suggesting that the two are not as similar as first seems. Our results shows that there are challenges in using the braid approach to analyze data, in particular the need for long trajectories, but that these are not insurmountable., Comment: 18 pages including Supplementary Material, 16 figures. RevTeX 4.1 document class

Internal waves shoaling on the continental slope can break and form materially coherent vortices called boluses. These boluses are able to trap and transport material up the continental slope, yet the global extent of bolus transport is unknown. Previous studies of bolus formation primarily focused on systems consisting of two layers of uniform density, which do not account for the presence of ocean pycnoclines of finite thickness. We use hyperbolic tangent profiles to model the density stratification in our simulations and demonstrate the impact of the pycnocline on the bolus. A spectral clustering method is used to objectively identify the bolus as a Lagrangian coherent structure that contains the material advected upslope. The bolus size and displacement upslope are examined as a function of the pycnocline thickness, incoming wave energy, density change across the pycnocline, and topographic slope. The dependence of bolus transport on the pycnocline thickness demonstrates that boluses in continuous stratifications tend to be larger and transport material further than in corresponding two-layer stratifications., Comment: 29 pages, 14 figures, 2 tables, 4 appendices

Internal gravity wave energy contributes significantly to the energy budget of the oceans, affecting mixing and the thermohaline circulation. Hence it is important to determine the internal wave energy flux $\mathbf{J} = p \, \mathbf{v}$, where $p$ is the pressure perturbation field and $\mathbf{v}$ is the velocity perturbation field. However, the pressure perturbation field is not directly accessible in laboratory or field observations. Previously, a Green's function based method was developed to calculate the instantaneous energy flux field from a measured density perturbation field $\rho(x,z,t)$, given a $constant$ buoyancy frequency $N$. Here we present methods for computing the instantaneous energy flux $\mathbf{J}(x,z,t)$ for a spatially varying $N(z)$, as in the oceans where $N(z)$ typically decreases by two orders of magnitude from shallow water to the deep ocean. Analytic methods are presented for computing $\mathbf{J}(x,z,t)$ from a density perturbation field for $N(z)$ varying linearly with $z$ and for $N^2(z)$ varying as $\tanh (z)$. To generalize this approach to $arbitrary$ $N(z)$, we present a computational method for obtaining $\mathbf{J}(x,z,t)$. The results for $\mathbf{J}(x,z,t)$ for the different cases agree well with results from direct numerical simulations of the Navier-Stokes equations. Our computational method can be applied to any density perturbation data using the MATLAB graphical user interface EnergyFlux.

Determination of energy transport is crucial for understanding the energy budget and fluid circulation in density varying fluids such as the ocean and the atmosphere. However, it is rarely possible to determine the energy flux field $\mathbf{J} = p \mathbf{u}$, which requires simultaneous measurements of the pressure and velocity perturbation fields, $p$ and $\mathbf{u}$. We present a method for obtaining the instantaneous $\mathbf{J}(x,z,t)$ from density perturbations alone: a Green's function-based calculation yields $p$, and $\mathbf{u}$ is obtained by integrating the continuity equation and the incompressibility condition. We validate our method with results from Navier-Stokes simulations: the Green's function method is applied to the density perturbation field from the simulations, and the result for $\mathbf{J}$ is found to agree typically to within $1\%$ with $\mathbf{J}$ computed directly using $p$ and $ \mathbf{u}$ from the Navier-Stokes simulation. We also apply the Green's function method to density perturbation data from laboratory schlieren measurements of internal waves in a stratified fluid, and the result for $\mathbf{J}$ agrees to within $6\%$ with results from Navier-Stokes simulations. Our method for determining the instantaneous velocity, pressure, and energy flux fields applies to any system described by a linear approximation of the density perturbation field, e.g., to small amplitude lee waves and propagating vertical modes. The method can be applied using our Matlab graphical user interface EnergyFlux.

While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of an FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by an FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model., Comment: 32 pages, 10 figures

The detection of coherent structures is an important problem in fluid dynamics, particularly in geophysical applications. For instance, knowledge of how regions of fluid are isolated from each other allows prediction of the ultimate fate of oil spills. Existing methods detect Lagrangian coherent structures, which are barriers to transport, by examining the stretching field as given by finite-time Lyapunov exponents. These methods are very effective when the velocity field is well-determined, but in many applications only a small number of flow trajectories are known, for example when dealing with oceanic float data. We introduce a topological method for detecting invariant regions based on a small set of trajectories. In the method we regard the two-dimensional trajectory data as a braid in three dimensions, with time being the third coordinate. Invariant regions then correspond to trajectories that travel together and do not entangle other trajectories. We detect these regions by examining the growth of hypothetical loops surrounding sets of trajectories, and searching for loops that show negligible growth., Comment: 23 pages, 22 figures. PDFLaTeX with RevTeX4-1 format. Minor corrections to the text