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.