Theor. Comput. Fluid Dyn. DOI 10.1007/s00162-012-0261-2 ORIGINAL ARTICLE Bérengère Podvin · Patrick Le Quéré Nonlinear dynamics between two differentially heated vertical plates in the presence of stratification Received: 17 October 2011 / Accepted: 21 February 2012 © Springer-Verlag 2012 Abstract We consider the numerical simulation of the flow between infinite, differentially heated vertical plates with positive stratification. We use a two-dimensional Boussinesq approximation, with periodic bound- ary conditions in the vertical direction. The relative stratification parameter γ = ( 1 4 RaS) 1/4 , where Ra is the Rayleigh number and S the adimensional stratification, is kept constant and equal to 8. The Prandtl num- ber is 0.71. We derive a complex Ginzburg-Landau equation from the equations of motion. Coefficients are computed analytically, but we find that the domain of validity of these coefficients is small and rely on the numerical simulation to adjust the coefficients over a wider range of Rayleigh numbers. We show that the Ginzburg-Landau equation is able to accurately predict the characteristics of the periodic solution at moderate Rayleigh numbers. Above the primary bifurcation at Ra = 1.63 × 10 5 , the Ginzburg-Landau model is found to be Benjamin-Feir unstable and to be characterized by modulated traveling waves and phase-defect chaos, which is supported by evidence from the DNS. As the Rayleigh number is increased beyond Ra = 2.7 × 10 5 , nonlinearities become strong and the flow is characterized by cnoidal waves. Keywords Natural convection · Instabilities 1 Introduction We consider the dynamics of a differentially heated vertical channel in the presence of stable stratification. This configuration is of interest for several reasons. Firstly, differentially heated cavities are important from an industrial point of view, from building applications to cooling of silicon chips. Secondly, the analogy between rotating and free thermal convection flows [39] ensures that insight from the simulations studied here has some relevance for rotating flows such as the Ekman layer, which is essential for meteorology and geophysical studies. A large number of studies focus on differentially heated cavities with isothermal walls, where the prescribed temperature is constant. Elder’s experimental results [12, 13] indicated the presence of streamwise traveling waves along the walls. Armfield and Patterson [2] showed that the development of the flow on an isolated isothermal plate was similar to that of a cavity flow. Nagata and Busse [27] studied the three-dimensional development of instabilities of a plane-parallel shear flow in the limit of zero Prandtl number. Chait and Korpela [7] examined the stability of the secondary flow between isothermal plates and found that the flow became rapidly three-dimensional. However, Henkes and Le Quéré [15] compared the transition to turbulence for two-dimensional and three-dimensional cavities and found that three-dimensional effects only affected the wall-heat transfer by about 15 %. Communicated by Eldredge. B. Podvin (B ) · P. Le Quéré LIMSI-CNRS, Orsay, France E-mail: podvin@limsi.fr