On stochastic evolution equations for chaos and turbulence

✍ Scribed by Hazime Mori

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 122 KB
• Volume: 205
• Category: Article
• ISSN: 0167-2789
• DOI: 10.1016/j.physd.2005.01.005

✦ Synopsis

The chaotic orbits of dynamical systems have positive Liapunov exponents, and become stochastic and random on long timescales due to the orbital instability, leading to various remarkable phenomena, such as (1) the loss of memory with respect to the initial states, (2) the dissipation of the kinetic energy into random fluctuations, (3) turbulent transport phenomena (e.g., turbulent viscosity, turbulent thermal diffusivity).

In order to obtain a statistical-mechanical approach to these phenomena, we have formulated the randomization of chaotic orbits by deriving a non-Markovian stochastic evolution equation in terms of a nonlinear fluctuating force and a memory function.

In the following, we outline its derivation and its application to the Boussinesq equations of turbulent Bénard convection. Then we find that turbulence produces an interference between the velocity flux and the heat flux which is similar to the interference between the electric current and the heat flux in the thermoelectric phenomena of metals.