Stochastic geometry and topology of non-Gaussian fields

Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.Comment: 8 pages, 4 figure

On the wave-induced difference in mean sea level between the two sides of a submerged breakwater

Very simple formulae are derived for the difference in mean level between the two sides of a submerged breakwater when waves are incident on it at an arbitrary angle. The formulae apply also to waves undergoing refraction due to changes in depth and to waves in open channel transitions

On the statistical distribution of the heights of sea waves

The statistical distribution of wave-heights is derived theoretically on the assumptions (a) that the wave spectrum contains a single narrow band of frequencies, and (b) that the wave energy is being received from a large number of different sources whose phases are random. Theoretical relations are found between the root-meansquare wave-height, the mean height of the highest one-third (or highest one-tenth) waves and the most probable height of the largest wave in a given interval of time. There is close agreement with observation

The distribution of extremal points of Gaussian scalar fields

We consider the signed density of the extremal points of (two-dimensional) scalar fields with a Gaussian distribution. We assign a positive unit charge to the maxima and minima of the function and a negative one to its saddles. At first, we compute the average density for a field in half-space with Dirichlet boundary conditions. Then we calculate the charge-charge correlation function (without boundary). We apply the general results to random waves and random surfaces. Furthermore, we find a generating functional for the two-point function. Its Legendre transform is the integral over the scalar curvature of a 4-dimensional Riemannian manifold.Comment: 22 pages, 8 figures, corrected published versio

Quasi-exact-solution of the Generalized Exe Jahn-Teller Hamiltonian

We consider the solution of a generalized Exe Jahn-Teller Hamiltonian in the context of quasi-exactly solvable spectral problems. This Hamiltonian is expressed in terms of the generators of the osp(2,2) Lie algebra. Analytical expressions are obtained for eigenstates and eigenvalues. The solutions lead to a number of earlier results discussed in the literature. However, our approach renders a new understanding of ``exact isolated'' solutions

A note on wave set-up

Seaward of the breaker zone, the observations of Saville are in good qualitative agreement with the prediction that the mean surface level is increasingly depressed towards the shoreline

Signed zeros of Gaussian vector fields-density, correlation functions and curvature

We calculate correlation functions of the (signed) density of zeros of Gaussian distributed vector fields. We are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract Riemann-Cartan or Riemannian manifold. As an application, we discuss one- and two-point functions. The zeros of a two-dimensional Gaussian vector field model the distribution of topological defects in the high-temperature phase of two-dimensional systems with orientational degrees of freedom, such as superfluid films, thin superconductors and liquid crystals.Comment: 14 pages, 1 figure, uses iopart.cls, improved presentation, to appear in J. Phys.

The electrical field induced by ocean currents and waves, with applications to the method of towed electrodes

The purpose of this paper is to discuss the nature of the electrical field induced in the ocean by particular types of velocity distribution. It is believed that these examples will be helpful in the interpretation of measurements by towed electrodes in the sea. The electrical field induced by waves and tidal streams, originally predicted by Faraday (1832), was first measured experimentally by Young, Gerrard and Jevons (1920), who used both moored and towed electrodes in their observations. Recently, the technique of towed electrodes has been developed by von Arx (1950, 1951) and others into a useful means of detecting water movements in the deep ocean. While the method has been increasingly used, the problem of interpreting the measurements in terms of water movements has become of great importance. Two of the present authors have made theoretical studies (Longuet-Higgins 1949, Stommel 1948) dealing with certain cases of velocity fields, and Malkus and Stern (1952) have proved some important integral theorems. There seems, however, to be a need for a more extended discussion of the principles underlying the method, and for the computation of additional illustrative examples. This is all the more desirable since some of the theoretical discussions published previously have been misleading

Some model experiments on continental shelf waves

This paper describes some model experiments that verify the theoretical form of continental shelf waves. Both the dispersion relationship and the positions of the orbital gyres are confirmed. The existence of a maximum frequency for each mode, with a corresponding zero group velocity, may be of significance for field observations