**Theorical Presentation**

Rayleigh Benard instability is one of the most famous
thermal instability. Innumerable number of publications shows great interest
of scientific community. If this phenomenon take part in elaboration of
recent theories as such dynamical system, the fact remains that the first
study dates back beginning of the century. Indeed, Benard suggested experimental
study :

This study showed clearly the existence of a critical threshold between two mode of heat transferts:

- __a mode of pure thermal conduction:__

So, it represents the solution of heat equation, consequently
the profile of temperature is linear ( in this case, heat transfert is

consequence of molecular phenomenon )

( temperature )

- __a mode of convection heat transfert :__

This mode is characterized by forming of stationnaries
contra-rotative rolls ( here, heat transfert is macroscopic phenomenon)

As difference of temperature increase, there
is, first, forming secondaries instabilities which are characterized by

oscillating motions of contra-rotative rolls and
finally, as T1 -T2 becomes bigger and bigger, more and more "unstructured'
behaviour appear ( until appearence of turbulence ).

( temperature )

Lord Rayleigh established by using theorical analysis
concordance with Benard 's experience. More particularly, he showed existence
of Ra the characteristic number of this instability :

for which :

- if Ra < Rac, heat tranfert 's mode is purely conductive

- if Ra > Rac, convection is established

where Rac is the critical Rayleigh number. Rayleigh showed
Rac = 1709.

The Modelisation of Rayleigh Benard relies on Navier Stokes
system with Boussinesq assumption : in other words, density is supposed
constant except that it feignes only quantities of motion by means of only
buyoncy forces (generator of the instability). Thus, variation of density
destabilizes fluid while drag force and thermal force stabilize it. Ra
repesents ratio between these two quantities.

A study of linear stability allows to obtain curve of
stabilty representing Ra in function of wave number

For Ra < 1708, all perturbations are absorbed

For Ra > 1708, a certain number of perturbations grows up.

This curve will be very useful in searching of initials
conditions ( see chapter III )

Finally, we give a figure showing differents states of
th instability :

with

In order to study stationnary rolls, our study will be
limited to epsilon < 30.