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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.

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.