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We propose new implicit constitutive relations for the heat fluxes of a two-temperature mixture of fluids. These relations are frame-indifferent forms. However, classical explicit forms of the stress tensors and the interaction forces (specified as explicit forms of constitutive relations) as given in mixture theory are used. The focus here is to establish constraints imposed on the implicit terms in the heat fluxes due to the Second Law of Thermodynamics. Our analysis establishes that the magnitude of the explicit entropy production is equal to or greater than that of the implicit entropy production. [ABSTRACT FROM AUTHOR]

Surface convergence in the ocean is associated with accumulation of buoyant pollutants as well as with vertical transport that is important to biological activity. Such surface convergence regions are marked by a high dilation rate, i.e., the finite time Lagrangian average divergence. Dilation-rate observations are most easily derived from the change of the area encompassed by a drifter swarm over time. The technological advances that have enabled the deployment of large numbers of drifters in a single experiment have raised new questions about optimal deployment strategies for extracting dilation-rate information with acceptable accuracy and as much spatial coverage as possible. Using a submesoscale-resolving operational model of the Mediterranean Sea, we analyze synthetic trajectories of drifter polygons to evaluate the impact of the number of drifters and their initial separation on the accuracy of the resulting dilation-rate estimates. The results confirm that estimates improve as the circumradius of the polygon decreases and as more drifters are added, but with only a marginal improvement for drifter polygons containing more than four drifters. Moreover, GPS positions obtained from drifters in the ocean are subject to uncertainty on the order of 2–50 m, and when this uncertainty is taken into account, an optimal circumradius can be identified that balances uncertainty from position measurements with that from the area approximations. Significance Statement: Locating regions of convergence over a finite time interval on the ocean surface can help in pollution mitigation, locating biological hotspots, and even search-and-rescue operations. Finite time convergence can be quantified using the dilation rate, but it is hard to measure in the ocean. Hence, we present a method to estimate the dilation rate using trajectories of drifters, which are instruments widely used by oceanographers during field experiments to understand the local flow features. We show that even though the drifter-based dilation rates are prone to error as a result of a finite number of drifters and limited GPS accuracy, the estimates locate around 90% of the strongest convergent features in our model. [ABSTRACT FROM AUTHOR]

Loop Current rings (LCRs) are among the largest mesoscale eddies in the world ocean. They arise when bulges formed by the Loop Current in the Gulf of Mexico close off. The LCR formation process may take several weeks, and there may be several separations and reattachments before final separation occurs. It is well established that this period is characterized by a persistent saddle point in the sea surface height field, as seen in both model and satellite data. We present here a detailed study of this saddle region during the formation of Eddy Franklin in 2010, over multiple days and at several depths. Using a data-assimilating Gulf of Mexico implementation of the HYbrid Coordinate Ocean Model (HYCOM), we compare the vertical structure of the currents and temperature fields on 5 and 10 June 2010. Finite-time Lyapunov exponents (FTLE) are computed from the surface down to 200 m to estimate the location of relevant transport barriers. Several new features of the saddle region associated with LCR formation are revealed: the ridges in the FTLE fields are shown to be excellent surrogates for the manifolds delineating the material flow structures with only slight degradation at depth. The intersection of the ridges representing stable and unstable manifolds drops nearly vertically through the water column at both times; remarkably, the material boundary shapes are maintained even as they are advected. Moreover, velocity stagnation points and saddle points in the temperature field are consistently found near the intersections at all depths, and their geographic positions are also nearly constant with depth.

Simple Summary: The second law of thermodynamics has a mystical appeal in disciplines with tenuous connections to its origins. We hypothesize that many of these appeals instead should be to another principle heretofore unrecognized: the law of mixed-up-ness (LOM). Instead of using a number such as entropy to characterize randomness, non-thermodynamic systems can be arranged in simple diagrams according to their degree of mixed-up-ness. Curiously, the evolution of such systems from degrees of low to high mixed-up-ness is both consistent with and richer than the principle of increasing entropy. Mixed-up-ness can be traced to unpublished notes by Josiah Gibbs. Subsequently, the concept was developed independently, and under somewhat different names, by other investigators. The central idea of mixed-up-ness is that systems states can be organized in a hierarchy by their degree of mixed-up-ness. In its purest form, the organizing principle is independent of thermodynamic and statistical mechanics principles, nor does it imply irreversibility. Yet, Gibbs and subsequent investigators kept entropy as the essential concept in determining system evolution, thus retaining the notion that systems evolve from states of perfect "order" to states of total "disorder". Nevertheless, increasing mixed-up-ness is consistent with increasing entropy; however, there is no unique one-to-one connection between the two. We illustrate the notion of mixed-up-ness with an application to the permutation function of integer partitions and then formalize the notion of mixed-up-ness as a fundamental hierarchal principle, the law of mixed-up-ness (LOM), for non-thermodynamic systems. [ABSTRACT FROM AUTHOR]

Application of the Brown-Samelson theorem, which shows that particle motion is integrable in a class of vorticity-conserving, two-dimensional incompressible flows, is extended here to a class of explicit time dependent dynamically balanced flows in multilayered systems. Particle motion for nonsteady two-dimensional flows with discontinuities in the vorticity or potential vorticity fields (modon solutions) is shown to be integrable. An example of a two-layer modon solution constrained by observations of a Gulf Stream ring system is discussed.

A particle-in-cell ansatz for solving the Euler equations in a rotating frame is described. The approach is ideally suited for 'layered' models of flows with sharp density and velocity fronts. The material and Coriolis accelerations in the Euler equations are solved at each particle while the gradient accelerations are evaluated on a grid and interpolated at each time step to the particles. The height of each particle is fixed and, depending on the application may be constant for all particles or may vary from particle to particle. The approach is used here to predict the evolution of a lens in a layered model with lower layer outcropping. The integral invariant of the volume is conserved exactly and total energy and total angular momentum are conserved to within 3% throughout a 30 day simulation. Exceptional resolution of the density and velocity fronts is maintained during the simulation without imposing any numerical viscosity. the model also reproduces essential characteristics of analytic solutions to a parabolic shaped lens. This algorithm is well suited to parallel implementation; all of the calculations reported here were done on an IBM SP2. Performance speedup and execution time as a function of the number of processors is given.

Assuming that the behaviour of a nonlinear stochastic system can be described by a Markovian diffusion approximation and that the evolution equations can be reduced to a system of ordinary differential equations, a method for the calculation of prediction time is developed. In this approach, the prediction time depends upon the accuracy of prediction, the intensity of turbulence, the accuracy of the initial conditions, the physics contained in the mathematical model, the measurement errors, and the number of prediction variables. A numerical application to zonal channel flow illustrates the theory. Some possible generalizations of the theory are also discussed.

A two dimensional system of autonomous nonlinear ordinary differential equations models glacier growth and temperature changes on an idealized planet. We apply standard perturbative techniques from dynamical systems theory to study small amplitude periodic orbits about a constant equilibrium. The equations are put in cononical form and the local phase space topology is examined. Maximum and minimum periods of oscillation are obtained and related to the radius of the orbit. An adjacent equilibrium is shown to have saddle character and the inflowing and outflowing manifolds of this saddle are studied using numerical integration. The inflowing manifolds show the region of attraction for the periodic orbit. As the frequency gets small, the adjacent (saddle) equilibrium approaches the radius of the periodic orbit. The bifurcation of the periodic orbit to a stable homoclinic orbit is observed when an inflowing manifold and an outflowing manifold of the adjacent equilibrium cross.

A unique set of coutemporaneous satellite-tracked drifters and five-day composite Advanced Very High Resolution Radionmeter (AVHRR) satellite imagery of the North Atlantic has been analyzed to examine the surface flow structure of the Gulf Stream. The study region was divided into two sections, greater than 37° N and less than 37° N, in order to answer the question of geographic variability. Fractal and spectral analyses methods were applied to the data. Fractal analysis of the Lagrangian trajectories showed a fractal dimension of 1.21 + 0.02 with a scaling range of 83 - 343 km. The fractal dimension of the temperature fronts of the composite imagery is similar for the two regions with D = 1.11 + 0.01 over a scaling range of 4 - 44 km. Spectral analysis also reports a fairly consistent value for the spectral slope and its scaling range. Therefore, we conclude there is no geographic variability in the data set. A suitable scaling range for this contemporaneous data set is 80 - 200 km which is consistent with the expected physical conditions in the region. Finally, we address the idea of using five-day composite imagery to infer the surface flow of the Gulf Stream. Close analyses of the composite thermal fronts and the Lagrangian drifter trajectories show that the former is not a good indicator of the latter.

Here we use a new analysis schema, the Freshening Length, to study the transport in the Irminger Current on the east and west sides of Greenland. The Freshening Length schema relates the transports on either side of Greenland to the corresponding surface salinity gradients by analyzing climatological data from a data assimilating global ocean model. Surprisingly, the warm and salty waters of the Current are clearly identified by a salinity maximum that varies nearly linearly with distance along the Current's axis. Our analysis of the climatological salinity data based on the Freshening Length schema shows that only about 20 % of the transport east of Greenland navigates the southern tip of Greenland to enter the Labrador Sea in the west. The other 80 % disperses into the ambient ocean. This independent quantitative estimate based on a 37-year long record complements seasonal to annual field campaigns that studied the connection between the seas east and west of Greenland more synoptically. A temperature-salinity analysis shows that the Irminger Current east of Greenland is characterized by a compensating isopycnal exchange of temperature and salinity, while west of Greenland the horizontal convergence of less dense surface water is accompanied by downwelling/subduction. [ABSTRACT FROM AUTHOR]

Surface drifter observations from the LAgrangian Submesoscale ExpeRiment (LASER) campaign in the Gulf of Mexico are paired with Eulerian (ship-borne X-band radar) data to demonstrate that velocity structure functions from drifters differ systematically from Eulerian structure functions over scales from 0.4 to 7 km. These differences result from drifters oversampling surface convergences and regions of intense vorticity. The first-, second-, and third-order structure functions are calculated using quasi-Lagrangian (drifter) and Eulerian data from approximately the same location and time. Differences between quasi-Lagrangian and Eulerian structure functions are attributed to two forms of bias. The first bias results from the mean divergence or vorticity of the background flow creating nonzero first-order structure functions. This background bias affects both quasi-Lagrangian and Eulerian data when insufficiently time-averaged. It severely biases the drifter third-order structure functions but is smaller in Eulerian structure functions at both second and third order. This bias can be corrected for using lower-order structure functions. The second form of bias results from drifters accumulating in regions with flow statistics that differ from undersampled regions. This accumulation bias is diagnosed by identifying the dependence of the Eulerian structure functions on divergence and vorticity as well as scale. Together, both biases suggest that caution is needed when interpreting second-order drifter statistics and that linking raw third-order drifter statistics to energy fluxes is often erroneous in ocean data: Even with background correction and sufficient time-averaging, drifters overestimate the Eulerian estimate of the third-order structure function by up to a factor of 5 when signs are consistent. [ABSTRACT FROM AUTHOR]

We present an analysis of ocean surface dispersion characteristics, on 1-100-m scales, obtained by optically tracking a release of ... (600) bamboo plates for 2 h in the northern Gulf of Mexico. Under sustained 5-6m s-1 winds, energetic Langmuir cells are clearly delineated in the spatially dense plate observations. Within 10min of release, the plates collect in windrows with 15-m spacing aligned with the wind. Windrow spacing grows, through windrow merger, to 40m after 20min and then expands at a slower rate to 50m. The presence of Langmuir cells produces strong horizontal anisotropy and scale dependence in all surface dispersion statistics computed from the plate observations. Relative dispersion in the crosswind direction initially dominates but eventually saturates, while downwind dispersion exhibits continual growth consistent with contributions from both turbulent fluctuations and organized mean shear. Longitudinal velocity differences in the crosswind direction indicate mean convergence at scales below the Langmuir cell diameter and mean divergence at larger scales. Although the second-order structure function measured by contemporaneous GPS-tracked surface drifters drogued at ~0.5m shows persistent r2/3 power law scaling down to 100-200-m separation scales, the second-order structure function for the very near surface plates observations has considerably higher energy and significantly shallower slope at scales below 100m. This is consistent with contemporaneous data from undrogued surface drifters and previously published model results indicating shallowing spectra in the presence of direct wind-wave forcing mechanisms. [ABSTRACT FROM AUTHOR]

Floating oil, plastics and marine organisms are continually redistributed by ocean surface currents. Prediction of their resulting distribution on the surface is a fundamental, long-standing, and practically important problem. The dominant paradigm is dispersion within the dynamical context of a nondivergent flow: objects initially close together will on average spread apart but the area of surface patches of material does not change. Although this paradigm is likely valid at mesoscales, larger than 100 km in horizontal scale, recent theoretical studies of submesoscales (less than ∼10 km) predict strong surface convergences and downwelling associated with horizontal density fronts and cyclonic vortices. Here we show that such structures can dramatically concentrate floating material. More than half of an array of ∼200 surface drifters covering ∼20 x 20 km2 converged into a 60 x 60 m region within a week, a factor of more than 105 decrease in area, before slowly dispersing. As predicted, the convergence occurred at density fronts and with cyclonic vorticity. A zipperlike structure may play an important role. Cyclonic vorticity and vertical velocity reached 0.001 s-1 and 0.01 ms-1, respectively, which is much larger than usually inferred. This suggests a paradigm in which nearby objects form submesoscale clusters and these clusters then spread apart. Together, these effects set both the overall extent and the finescale texture of a patch of floating material. Material concentrated at submesoscale convergences can create unique communities of organisms, amplify impacts of toxic material and create opportunities to more efficiently recover such material. [ABSTRACT FROM AUTHOR]