Yona Dvorkin

Chaotic Hamiltonian dynamics of particle's horizontal motion in the atmosphere

Abstract A non-separable, near-integrable, two degrees-of-freedom (d.o.f.) Hamiltonian system arising in the context of particles dynamics on a geopotential of the atmosphere is studied. In the unperturbed system, homoclinic orbits to periodic motion in the longitude angle exist, giving rise to several types of homoclinic chaos in the perturbed system; a new type of homoclinic chaos is found, exhibiting chaos which is non-uniformly distributed in the longitude angle along the homoclinic loop while it is topologically uniformly distributed in this angle in a neighborhood of the hyperbolic periodic orbit, i.e. the return map to different values of the longitude angle are topologically conjugate only near the hyperbolic periodic orbit. The concept of colored energy surfaces and its corresponding energy-momentum map are developed as analytical tools for delineating the regions in the four-dimensional phase space where the various types of homoclinic chaos prevail. Geometrical interpretation of the Melnikov analysis is offered for detecting this non-uniformity in angle. Applying the results of the analysis to the planetary atmosphere with an infinite wavelength perturbation of amplitude ϵ, we find that for eastward going particles with initial velocities {(ifu0, v0) = (ū, 0)} the chaotic band occupies two narrow (O(ϵ)) bands on both sides of the equator nea latitudes φ = ± arccos (1 − 2 u ) . By contrast, for westward going particles the chaotic zone is thicker ( O ( ϵ; ) ) and is centered on the equator. Moreover, the dependence of the chaotic zones thickness on the perturbation frequency is much more sensitive to the value of the initial speed for eastward going particles than for those going westward. Qualitative and quantitative differences between the distribution of the homoclinic chaotic motion in the longitude angle for westward and eastward travelling wave perturbations are predicted and numerically confirmed.

Analytical Considerations of Lagrangian Cross-Equatorial Flow

The Lagrangian description of cross-equatorial flow under any steady, strictly meridional, pressure gradient forcing is shown to comprise an integrable two-degree-of-freedom Hamiltonian system. The system undergoes a pitchfork bifurcation when the angular momentum passes through a critical value. It is shown that, even when the full variation of the Coriolis parameter is taken into account, the dynamical system is fully integrable, which implies that its evolution from any initial state can be calculated with sufficient accuracy (depending on the accuracy of the initial state) to any desired time. The role of the driving pressure gradient is merely to shift the latitude of the fixed points from their location in the inertial case. When zonal variation or time dependence of the pressure field is allowed, the system becomes nonintegrable and chaotic bands appear where nearby trajectories diverge exponentially.

A Practical, Hybrid Model for Predicting the Trajectories of Near-Surface Ocean Drifters

A hybrid Lagrangian‐Eulerian model for calculating the trajectories of near-surface drifters in the ocean is developed in this study. The model employs climatological, near-surface currents computed from a spline fit of all available drifter velocities observed in the Pacific Ocean between 1988 and 1996. It also incorporates contemporaneous wind fields calculated by either the U.S. Navy [the Navy Operational Global Atmospheric Prediction System (NOGAPS)] or the European Centre for Medium-Range Weather Forecasts (ECMWF). The model was applied to 30 drifters launched in the tropical Pacific Ocean in three clusters during 1990, 1993, and 1994. For 10-day-long trajectories the forecasts computed by the hybrid model are up to 164% closer to the observed trajectories compared to the trajectories obtained by advecting the drifters with the climatological currents only. The best-fitting trajectories are computed with ECMWF fields that have a temporal resolution of 6 h. The average improvement over all 30 drifters of the hybrid model trajectories relative to advection by the climatological currents is 21%, but in the open-ocean clusters (1990 and 1993) the improvement is 42% with ECMWF winds (34% with NOGAPS winds). This difference between the open-ocean and coastal clusters is due to the fact that the model does not presently include the effect of horizontal boundaries (coastlines). For zero initial velocities the trajectories generated by the hybrid model are significantly more accurate than advection by the mean currents on time scales of 5‐15 days. For 3-day-long trajectories significant improvement is achieved if the drifter’s initial velocity is known, in which case the model-generated trajectories are about 2 times closer to observations than persistence. The model’s success in providing more accurate trajectories indicates that drifters’ motion can deviate significantly from the climatological current and that the instantaneous winds are more relevant to their trajectories than the mean surface currents. It also demonstrates the importance of an accurate initial velocity, especially for short trajectories on the order of 1‐3 days. A possible interpretation of these results is that winds affect drifter motion more than the water velocity since drifters do not obey continuity.

Barotropic instability of a zonal jet : From nondivergent perturbations on the β plane to divergent perturbations on a sphere

The linear instability of divergent perturbations that evolve on a cos 2 mean steady zonal jet embedded in a zonal channel on the plane and on a rotating sphere is studied for zonally propagating wavelike perturbations of the shallow-water equations. The complex phase speeds result from the imposition of the no-flow boundary conditions at the channel walls on the numerical solutions of the linear differential equations for the wave latitude-dependent amplitude. In addition, the same numerical method is applied to the traditional problem of linear instability of nondivergent perturbations on the plane where results reaffirm the classical, analytically derived, features. For these nondivergent perturbations, the present study shows that the growth rate increases monotonically with the jet maximal speed and that the classical result of a local maximum at some finite westward-directed speed results from scaling the growth rates on the jet’s speed. In contrast to nondivergent perturbations, divergent perturbations on the plane have no shortwave cutoff, and so the nondivergent solution does not provide an estimate for the divergent solution, even when the ocean is 1000 km deep (i.e., when the speed of gravity waves exceeds 10 Mach). For realistic values of the ocean depth, the growth rates of divergent perturbations are smaller than those of nondivergent perturbations, but with the increase in the ocean depth they become larger than those of nondivergent perturbations. For both perturbations, a slight asymmetry exists between eastward- and westward-flowing jets. The growth rates of divergent perturbations on a sphere are similar to those on the plane for the same values of the model parameters, but the asymmetry between eastward and westward jets is more conspicuous on a sphere. The value of gH (g is the reduced gravity; H is the equivalent mean layer thickness), which is filtered out in nondivergent theory, determines for divergent perturbations the relative magnitude of zonal velocity, meridional velocity, and height but has little effect on the growth rates.

Improving the calculation of particle trajectories in the extra-tropical troposphere using standard NCEP fields

Abstract The calculation of particle trajectories in the extra-tropical troposphere is improved by a hybrid model that employs the temperature and geopotential fields to supplement the velocity field. The hybrid model uses the temperature and geopotential fields to construct the Montgomery Stream function, which, together with a Rayleigh friction force and the Coriolis force determine the evolution of a “correctional velocity” based on Newtons 2nd law of motion. This velocity, however, is decoupled from the continuity equation so its horizontal divergence does not affect the pressure. The improvement of the trajectory calculation is obtained by integrating a linear combination of National Centers for Environmental Predictions’ (NCEP) velocity field and the “correctional velocity” computed from NCEPs temperature and geopotential fields. The improvement of the model-generated trajectories over those obtained from a straightforward advection by the velocity field is verified by comparing the calculated trajectories to the observed trajectories of 379 constant-level balloons launched in 1971 as part of the EOLE experiment. For flight times between 2 and 10 weeks the new algorithm generates trajectories that are statistically closer to the observed EOLE trajectories than those obtained from advection by the velocity field only. There are, however, several balloon flights where for certain, isolated, values of its parameters the hybrid model generates trajectories that are actually less accurate than those of straightforward advection. For balloon flights between 2 and 9 weeks the worst hybrid model trajectories are only about 4% less accurate than those of straightforward advection. The models improvement over advection by the velocity field reaches a maximum value of over 15% for 4–7 week long trajectories while for 1 week long, or longer than 10 week long trajectories the model offers no improvement over straightforward advection by the velocity field.

Noise-Induced Interhemispheric Particle Transport—Stochastic Resonance in a Hamiltonian System

A Lagrangian model is employed to study the characteristics of a horizontal cross-equatorial flow. The Coriolis force and the mean meridional pressure field assumed here render the dynamics of particle flow across the equator a nonlinear Hamiltonian system of a bistable potential that has a local maximum at the equator. In the absence of any additional forces this local maximum at the equator prohibits particles from flowing from one hemisphere to the other. When all other (i.e., in addition to the mean meridional pressure gradient) forces are introduced into the system as stochastic forcing, modeled by Gaussian white noise, anomalous diffusion up the mean pressure gradient occurs and particles launched in one hemisphere can reach the other. Spectral estimations of equator crossing events show that at low noise intensity the spectral peak is low, narrow, and situated at low frequencies, and that as the amplitude of the noise increases, the peak becomes higher, wider, and shifts toward higher frequencies. At very large noise intensities the spectral peak flattens out, which implies that the process of equator crossing becomes noise dominated. The results demonstrate the existence of an optimal noise intensity where the signal-to-noise ratio of equator crossings exhibits a sharp maximum, and this optimal noise intensity is insensitive to the precise value of the mean meridional pressure gradient. These findings are applicable to the terrestrial atmosphere where the mean meridional geopotential height gradient and the (zonal and temporal) deviations from it are of the same order. These results demonstrate, for the first time, the occurrence of stochastic resonance in a Hamiltonian system.

Chaotic lagrangian motion on a rotating sphere

We study the motion of Lagrangian particles launched on a geopotential surface of a rotating sphere (e.g. floats in the deep ocean) where the latter is zonally perturbed by some travelling pressure wave (e.g. tidal waves). The motion of these particles is described by a near integrable, two-degrees-of-freedom Hamiltonian system. For some regions in parameter and phase space, the system may be reduced to a one-and-a-half-degrees-of-freedom Hamiltonian system and standard tools may be applied to prove the existence of chaotic homoclinic behavior, hence of the phenomena of anomalous transport associated with the homoclinic chaos. In other regions zonally localized structures and homoclinic tangles with back-flows, associated with the three dimensionality of the energy surfaces appear, as well as resonant behavior of unstable periodic orbits.

Linear instability of uniform shear zonal currents on the β-plane

A unified formulation of the instability of a mean zonal flow with uniform shear is proposed, which includes both the coupled density front and the coastal current. The unified formulation shows that the previously found instability of the coupled density front on the f -plane has natural extension to coastal currents, where the instability exists provided that the net transport of the current is sufficiently small. This extension of the coupled front instability to coastal currents implies that the instability originates from the interaction between Inertia-Gravity waves and a vorticity edge wave and not from the interaction of the two edge waves that exist at the two free streamlines due to the Potential Vorticity jump there. The present study also extends these instabilities to the β-plane and shows that β slightly destabilizes the currents by adding instabilities in wavelength ranges that are stable on the f -plane but has little effect on the growthrates in wavelength ranges that are unstable on the f -plane. An application of the β-plane instability theory to the generation of rings in the retroflection region of the Agulhas Current yields a very fast perturbation growth of the scale of 1 day and this fast growth rate is consistent with the observation that at any given time, as many as 10 Agulhas rings can exist in this region.

Divergent versus Nondivergent Instabilities of Piecewise Uniform Shear Flows on the f Plane

Abstract The linear instability of a piecewise uniform shear flow is classically formulated for nondivergent perturbations on a 2D barotropic mean flow with linear shear, bounded on both sides by semi-infinite half-planes where the mean flows are uniform. The problem remains unchanged on the f plane because for nondivergent perturbations the instability is driven by vorticity gradient at the edges of the inner, linear shear region, whereas the vorticity itself does not affect it. The instability of the unbounded case is recovered when the outer regions of uniform velocity are bounded, provided that these regions are at least twice as wide as the inner region of nonzero shear. The numerical calculations demonstrate that this simple scenario is greatly modified when the perturbations’ divergence and the variation of the mean height (which geostrophically balances the mean flow) are retained in the governing equations. Although a finite deformation radius exists on the shallow water f plane, the mean vortici...