White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation for Wave Beams in Turbulent Media

Starting with the Wigner distribution formulation for beam wave propagation in H\"{o}lder continuous non-Gaussian random refractive index fields we show that the wave beam regime naturally leads to the white-noise scaling limit and converges to a Gaussian white-noise model which is characterized by the martingale problem associated to a stochastic differential-integral equation of the It\^o type. In the simultaneous geometrical optics the convergence to the Gaussian white-noise model for the Liouville equation is also established if the ultraviolet cutoff or the Fresnel number vanishes sufficiently slowly. The advantage of the Gaussian white-noise model is that its $n$-point correlation functions are governed by closed form equations

Convergence of Passive Scalars in Ornstein-Uhlenbeck Flows to Kraichnan's Model

We prove that the passive scalar field in the Ornstein-Uhlenbeck velocity field with wave-number dependent correlation times converges, in the white-noise limit, to that of Kraichnan's model with higher spatial regularity

Richardson's Laws for Relative Dispersion in Colored-Noise Flows with Kolmogorov-type Spectra

We prove limit theorems for small-scale pair dispersion in velocity fields with power-law spatial spectra and wave-number dependent correlation times. This result establishes rigorously a family of generalized Richardson's laws with a limiting case corresponding to Richardson's $t^3$ and 4/3-laws

Self-Averaged Scaling Limits for Random Parabolic Waves

We consider 6 types of scaling limits for the Wigner-Moyal equation of the parabolic waves in random media, the limiting cases of which include the radiative transfer limit, the diffusion limit and the white-noise limit. We show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity (thus excluding the white-noise limit) leads to statistical stability or self-averaging in the sense that the limiting law is deterministic and is governed by various transport equations depending on the specific scaling involved. We obtain 6 different radiative transfer equations as limits

Taylor-Kubo Formula for Turbulent Diffusion in a Non-Mixing Flow with Long-Range Correlation

We prove the Taylor-Kubo formula for a class of isotropic, non-mixing flows with long-range correlation. For the proof, we develop the method of high order correctors expansion

Description of the All-Sky camera, its Method of Operation; An Instrument (Ascagraph) for Measuring the Film

The earlier orbits and ephemerides for the Soviet satellites were not sufficiently accurate to be very useful in making observations in Alaska. Extrapolations from our own observations gave better predictions. This merely pointed out the fact that rough observations of meridian transits at high latitudes will give better values of the inclination of the orbit than precision observations at low latitudes. Hence, it was decided to observe visually the meridian transits estimating the altitude by noting the position with respect to the stars or using crude alidade measurements. The times of the earlier observations were observed on a watch or clock and the clock correction obtained from WWV. Later the times were determined with the aid of stop watches, taking time intervals from WWV signals. This rather meager program of optical observations of the Soviet satellites was undertaken to give supplementary data for use of the radio observations, and particularly to assist in the prediction of position of the satellite so that the 61-foot radar of Stanford Research Institute could be set accurately enough to observe it (the beam width at the half-power points is about 3°). This report contains primarily the visual observations made at the Geophysical Institute by various members of the staff, and a series of observations by Olaf Halverson at Nome, Alaska. In addition there is a short discussion of the geometry of the trajectory, the illumination of a circumpolar satellite, and a note on the evaluation of Brouwer's moment factors.IGY Project Number 1.1 ; NSF Grant Number Y/1.1/44LIST OF FIGURES AND ILLUSTRATIONS -- I Introduction -- II Description of the All-Sky Camera -- III Installation and Operation -- IV Development of the Film , Inspection, and Preliminary Assessment -- V Preparation of Synoptic Maps -- VI Auroral Plotter (Ascagraph) for Reduction of All-Sky Camera PhotographsYe