Nonlinear dynamics of sand banks and sand waves

Sand banks and sand waves are two types of sand structures that are commonly observed on an off-shore sea bed. We describe the formation of these features using the equations of the fluid motion coupled with the mass conservation law for the sediment transport. The bottom features are a result of an instability due to tide–bottom interactions. There are at least two mechanisms responsible for the growth of sand banks and sand waves. One is linear instability, and the other is nonlinear coupling between long sand banks and short sand waves. One novel feature of this work is the suggestion that the latter is more important for the generation of sand banks. We derive nonlinear amplitude equations governing the coupled dynamics of sand waves and sand banks. Based on these equations, we estimate characteristic features for sand banks and find that the estimates are consistent with measurements

Non-stationary Spectra of Local Wave Turbulence

The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is studied in the limit of strongly local interactions where the usual kinetic equation, describing the time evolution of the spectral wave-action density, can be approximated by a PDE. If the wave action is initially compactly supported in frequency space, it is then redistributed by resonant interactions producing the usual direct and inverse cascades, leading to the formation of the K-Z spectra. The emphasis here is on the direct cascade. The evolution proceeds by the formation of a self-similar front which propagates to the right leaving a quasi-stationary state in its wake. This front is sharp in the sense that the solution remains compactly supported until it reaches infinity. If the energy spectrum has infinite capacity, the front takes infinite time to reach infinite frequency and leaves the K-Z spectrum in its wake. On the other hand, if the energy spectrum has finite capacity, the front reaches infinity within a finite time, t*, and the wake is steeper than the K-Z spectrum. For this case, the K-Z spectrum is set up from the right after the front reaches infinity. The slope of the solution in the wake can be related to the speed of propagation of the front. It is shown that the anomalous slope in the finite capacity case corresponds to the unique front speed which ensures that the front tip contains a finite amount of energy as the connection to infinity is made. We also introduce, for the first time, the notion of entropy production in wave turbulence and show how it evolves as the system approaches the stationary K-Z spectrum.Comment: revtex4, 19 pages, 10 figure

Dimensional Analysis and Weak Turbulence

In the study of weakly turbulent wave systems possessing incomplete self-similarity it is possible to use dimensional arguments to derive the scaling exponents of the Kolmogorov-Zakharov spectra, provided the order of the resonant wave interactions responsible for nonlinear energy transfer is known. Furthermore one can easily derive conditions for the breakdown of the weak turbulence approximation. It is found that for incompletely self-similar systems dominated by three wave interactions, the weak turbulence approximation usually cannot break down at small scales. It follows that such systems cannot exhibit small scale intermittency. For systems dominated by four wave interactions, the incomplete self-similarity property implies that the scaling of the interaction coefficient depends only on the physical dimension of the system. These results are used to build a complete picture of the scaling properties of the surface wave problem where both gravity and surface tension play a role. We argue that, for large values of the energy flux, there should be two weakly turbulent scaling regions matched together via a region of strongly nonlinear turbulence.Comment: revtex4, 10 pages, 1 figur

A weak turbulence theory for incompressible magnetohydrodynamics

We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0ê[parallel R: parallel]. Numerically and analytically, we find energy spectra E± [similar] kn±[bot bottom], such that n+ + n− = −4, where E± are the spectra of the Elsässer variables z± = v ± b in the two-dimensional case (k[parallel R: parallel] = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made

Norman Julius Zabusky OBITUARY

Norman Julius Zabusky, who laid the foundations for several critical advancements in nonlinear science and experimental mathematics, died of idiopathic pulmonary fibrosis on 5 February 2018 in Beersheba, Israel. He also made fundamental contributions to computational fluid dynamics and advocated the importance of visualization in science.Published versio

Pattern Universes

In this essay we explore analogies between macroscopic patterns, which result from a sequence of phase transitions/instabilities starting from a homogeneous state, and similar phenomena in cosmology, where a sequence of phase transitions in the early universe is believed to have separated the fundamental forces from each other, and also shaped the structure and distribution of matter in the universe. We discuss three distinct aspects of this analogy: (i) Defects and topological charges in macroscopic patterns are analogous to spins and charges of quarks and leptons; (ii) Generic (3+1) stripe patterns carry an (energy) density that accounts for phenomena that are currently attributed to dark matter; (iii) Space-time patterns of interacting nonlinear waves display behaviors reminiscent of quantum phenomena including inflation, entanglement and dark energy.Comment: 15 Pages, Essay with 3 technical appendice

Localization and Coherence in Nonintegrable Systems

We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian oscillator chains approaching their statistical asympotic states. In systems constrained by more than one conserved quantity, the partitioning of the conserved quantities leads naturally to localized and coherent structures. If the phase space is compact, the final equilibrium state is governed by entropy maximization and the final coherent structures are stable lumps. In systems where the phase space is not compact, the coherent structures can be collapses represented in phase space by a heteroclinic connection to infinity.Comment: 41 pages, 15 figure

Transport of sulfur dioxide from the Asian Pacific Rim to the North Pacific troposphere

The NASA Pacific Exploratory Mission over the Western Pacific Ocean (PEM-West B) field experiment provided an opportunity to study sulfur dioxide (SO2) in the troposphere over the western Pacific Ocean from the tropics to 60°N during February–March 1993. The large suite of chemical and physical measurements yielded a complex matrix in which to understand the distribution of sulfur dioxide over the western Pacific region. In contrast to the late summer period of Pacific Exploratory Mission-West A (PEM-West A) (1991) over this same area, SO2showed little increase with altitude, and concentrations were much lower in the free troposphere than during the PEM-West B period. Volcanic impacts on the upper troposphere were again found as a result of deep convection in the tropics. Extensive emission of SO2 from the Pacific Rim land masses were primarily observed in the lower well-mixed part of the boundary layer but also in the upper part of the boundary layer. Analyses of the SO2 data with aerosol sulfate, beryllium-7, and lead-210 indicated that SO2 contributed to half or more of the observed total oxidized sulfur (SO2 plus aerosol sulfate) in free tropospheric air. The combined data set suggests that SO2 above 8.5 km is transported from the surface but with aerosol sulfate being removed more effectively than SO2. Cloud processing and rain appeared to be responsible for lower SO2 levels between 3 and 8.5 km than above or below this region