Defining turbulence is not easy, although it is a phenomenon exhibited by solutions to particular partial differential equations, namely the Navier-Stokes equations, there is no precise mathematical definition.
On the other hand everyone has seen boiling water, or smoke twirling in the air. These are two examples of fluid flows that exhibit turbulence.
Turbulence theory is important for the parameterization of climate models, especially with regard to statistical properties and long range effects of both localized turbulent flows and turbulent flows spanning the whole atmosphere.
The butterfly effect? for example plays a crucial role in the question what effects are limiting the validity of long time numerical solutions of the Navier-Stokes equations and therefore climate models.
Low dimensional systems of ordinary differential equations (ODE) have been used as a model of turbulence. It has been questioned in the first half of the 20th century if the existence of turbulence in real flows invalidates the Navier-Stokes equations, because these equations are deterministic. Examples from chaos theory have since shown that simple nonlinear systems of ODE can exhibit seemingly chaotic, turbulent behaviour. This effect is sometimes called intrinsic randomness.
It is an open problem if any and which turbulent solutions of the Navier-Stokes equations are ergodic.
A numerical simulation of a turbulent flow that resolves all relevant length scales is called a direct numerical simulation.
If there is a cut and all effects below a certain length scale are ignored or simplified, the simulation is called a large eddy simulation. Large eddy simulations have been invented in meteorology. (Physicists may be reminded of an ultraviolett cutoff in quantum field theory.)
It is not trivial to visualize processes in 3 spatial dimensions. One possibility is a plot of the time dependent vorticity $\omega = rot(\vec v)$ of the velocity field $\vec v$. Some researchers have plotted iso-surfaces where the vorticity is above a certain threshold.
Example: In a finite volume we can compute the average value of the vorticity $\langle \omega \rangle$ and the variance $var(\omega)$. Then we can draw surfaces of regions where
for some value of k.
A friendly introduction with many pictures from experiments and an overview of the various theoretical approaches to turbulence is this:
P.A. Davidson: Turbulence. An introduction for scientists and engineers. (Oxford University Press 2004, ZMATH)
Also see Eddy Who? and Wave turbulence here on Azimuth.
Direct numerical simulation, Wikipedia
Y. Kaneda and T. Ishihara: High-resolution direct numerical simulation of turbulence, Journal of Turbulence, Volume 7, No. 20, 2006, online here