Navier-Stokes model

The viscous incompressible fluid flow can be described by the Navier-Stokes equations as follows

where r is the fluid density, u=(u,v,w) is the flow velocity vector, n is the kinematics viscosity, p is the pressure.

The Navier-Stokes model is the base of theory of hydrodynamic turbulence. These equations can be solved in common case only with modern computers using the numerical algorithms. Utilized the direct numerical simulation (DNS) it has been shown that the hydrodynamic chaos and turbulence can be described with the Navier-Stokes model. There are many simplified models to understand how the chaotic motion is produced in the natural systems (Lorenz-like chaos and other). See for instance

http://hypertextbook.com/Chaos/

The Navier-Stokes model has been developed for various scientific and engineering researches including the airflows in the atmosphere. In this case air is assumed as a viscous, heat-conducting, incompressible gas in a rather slow turbulent motion. Thus the model of the turbulent flow in the atmospheric surface layer can be written as follows:

Here p is the pressure with the exception of the hydrostatic atmospheric pressure, is the gravity acceleration vector, is the equilibrium density, is the temperature, is the Prandtl number. The new term in the right part of the second equation is the buoyancy force, which produces the vertical movement (convection) in the atmosphere. This model can be closed with the hydrostatic equation and the standard Boussinesq approximation for the density fluctuations

,

where is the coefficient of expansion, for the perfect gases.