1. The example of Rayleigh-Benard convection

2. Pattern-forming instabilities: The example of Rayleigh-Benard convection

3. Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

4. Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

5. Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

6. Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

7. Rayleigh-Benard convection: static state

8. Rayleigh-Benard convection

9. Rayleigh-Benard convection: static state with pure conduction Fixed temperature b.c.

10. Rayleigh-Benard convection. Boussinesq approximation, incompressible flow

11. Rayleigh-Benard convection: non dimensional formulation

12. Rayleigh-Benard convection Important parameters: R = g D 3 T 2 -T 1 ) / = a = L D

13. Rayleigh-Benard convection. If R < R crit conduction T(x,y,z,t)=T cond (z)=T 2 - z (u,v,w)=(0,0,0) If R > R crit convection T= T cond + (u,v,w) non zero

14. Rayleigh-Benard convection. Linear stability analysis

15. 2D Rayleigh-Benard convection (non dimensional formulation)

16. 2D Rayleigh-Benard convection

17. 2D Rayleigh-Benard convection: Linearization around the static state

18. 2D Rayleigh-Benard convection: Linearization around the static state

19. 2D Rayleigh-Benard convection: Linearization around the static state

20. 2D Rayleigh-Benard convection: Linearization around the static state

21. 2D Rayleigh-Benard convection: Linearization around the static state

22. Rayleigh-Benard convection. Linear stability analysis

23. 2D Rayleigh-Benard convection: Threshold to convection

24. Rayleigh-Benard convection: above R crit, convective motion occurs. This takes the form of parallel rolls

25. The convective rolls saturate the instability

26. Amplitude expansion

28. Amplitude expansion: first order

29. Amplitude expansion: second order

30. Amplitude expansion: third order

32. Amplitude expansion: third order The Fredhom alternative: eliminate the secular term and get a solvability condition: The Landau equation

33. Amplitude expansion: third order The Landau equation with real coefficients

34. Pitchfork bifurcation at R 2 =0

35. Stability of the rolls: Busse balloon

36. Stability of the rolls