The Mathematical Representation of Turbulent Heat Fluxes Using Reynold’s Decomposition

From a mathematical perspective, it is fundamental to develop a rigorous background upon which to study the physical quantities of a turbulent flow. The first problem in the mathematical theory is related to the deterministic nature of chaotic systems assumed in dynamical system theory and believed to hold inturbulence. This has actually not been proved for the Navier-Stokes equations. It is in fact one of the most outstanding open problems in mathematics to determine whether given an initial condition for the velocity field there exists, in some sense, a unique solution of the Navier-Stokes equations starting with this initial condition and valid for all later times. In a turbulent atmosphere, a turbulent stress term, the Reynolds stress, must be applied. All the terms in the horizontal motion equations are the order of 10^﹣4 - 10^﹣3 m·s². Under certain condition, some terms are very small and can be neglected for example, the rotational term is insignificant in the equation of vertical motion and has been omitted, instead gravitational acceleration term appears in the equation for vertical motion [1]; for steady flow, the tendency can be neglected; in the centre of high and low pressure, gradient force can be neglected; at the equator of for small scale processes, the Coriolis force can be neglected, and above the atmospheric boundary layer the stress terms can be neglected.