Weather Prediction by Numerical Process

Abstract

FINITE arithmetical differences have proved remarkably successful in dealing with differential equations ; for instance, approximate particular solutions of the equation for the diffusion of heat crO/dx" = dd/dt can be obtained quite simply and without any need to bring in Fourier analysis. An example is worked out in a paper published in Phil. Trans. A, Vol. 210*. In this book it is shown that similar methods can be extended to the very complicated system of differential equations, which expresses the changes in the weather. The fundamental idea is that atmospheric pressures, velocities, etc. should be expressed as numbers, and should be tabulated at certain latitudes, longitudes and heights, so as to give a general account of the state of the atmosphere at any instant, over an extended region, up to a height of say 20 kilometres. The numbers in this table are supposed to be given, at a certain initial instant, by means of observations.

Statistical properties of turbulence: An overview

“…Examples of such structures plumes in Rayleigh-Bérnard convection [72], structures behind a splitter plate [73], and large vortical structures in two-dimensional or stratified flows [1,35,36]. In three-dimensional flows, as we will see in greater detail below, energy that is pumped into the flow at the injection scale L cascades, as first suggested by Richardson [55], from large-scale eddies to small-scale ones till it is eventually dissipated around and beyond the dissipation scale η d . By contrast, two-dimensional turbulence [35,36,74,75] displays a dual cascade: there is an inverse cascade of energy from the scale at which it is pumped into the system to large length scales and a direct cascade of enstrophy Ω = 1 2 ω 2 to small length scales.…”

“…In Section 3 we introduce the equations that we consider. Section 4 is devoted to a summary of phenomenological approaches that have been developed, since the pioneering studies of Richardson [55] and Kolmogorov [56], in 1941 (K41), to understand the behaviour of velocity and other structure functions in inertial ranges. Section 5 introduces the ideas of multiscaling that have been developed to understand deviations from the predictions of K41-type phenomenology.…”

Large-eddy simulation of turbulent flows in a heated streamwise rotating channel

“…In 1922, Richardson [4] introduced the seminal idea of "energy cascade" to describe the scales of turbulence. Kolmogorov [5] Eventually, this energy will be dissipated by viscosity.…”

Hydrodynamic vortices and their gravity duals

“…There are on the other hand many phenomenological models of incompressible turbulence which can be dualized straightforwardly. We will consider the model of Richardson [6] which was made by precise by Kolmogorov [7]. This model correctly predicts the 3-point correlation function of the fluid velocities, although it fails to reproduce the higher correlators.…”