PATTERN FORMATION IN RAYLEIGH-B ´ ENARD CONVECTION IN A RAPIDLY ROTATING CYLINDER Michael Sprague, Keith Julien Department of Applied Mathematics, University of Colorado UCB 526, Boulder, CO 80309-0526, USA Michael.Sprague@Colorado.edu, Keith.Julien@Colorado.edu Eric Serre MSNM-GP UMR6181 CNRS Universit´ es d’Aix-Marseille I, II & III, La Jet´ ee Technopˆ ole de Chˆ ateau-Gombert, 38 rue Joliot-Curie, 13451 Marseille cedex 20, France serre1@l3m.univ-mrs.fr J.J. S´ anchez- ´ Alvarez, E. Crespo del Arco Departmento F´ ısica Fundamental, U.N.E.D., Apdo. 60.141,28080 Madrid, Spain jsanchez@bec.uned.es, emi@fisfun.uned.es To appear in Proceedings of the Fourth International Symposium on Turbulence and Shear Flow Phenomena, 2005. ABSTRACT Pattern formation in a rotating Rayleigh-B´ enard convec- tion configuration is investigated for moderate and rapid rota- tion in moderate aspect-ratio cavities. While the existence of K¨ uppers-Lortz rolls is predicted by theory at the onset of convection (K¨ uppers and Lortz, 1969; Busse and Clever, 1979), square patterns have been observed in physical (Ba- jaj et al., 1998) and numerical experiments (S´ anchez- ´ Alvarez et al., 2005) at relatively high rotation rates. Direct numer- ical simulation (DNS) of the Boussinesq equations becomes progressively more difficult as the rotation rate is increased due the presence of increasingly thin Ekman boundary layers and fast inertial waves. In addition to presenting numerical results produced from DNS of the full Boussinesq equations, we derive a reduced system of nonlinear PDEs valid for con- vection in a cylinder in the rapidly rotating limit. Reduced equations have been of great utility in the investigation of rapidly rotating convection on the infinite plane (Julien et al., 1998, 2005; Sprague et al., 2005) INTRODUCTION Rotating Rayleigh-B´ enard (RB) convection has impor- tant applications in geophysical and astrophysical flows as well as industrial applications (Boubnov and Golitsyn, 1995; Knobloch, 1998; Bodenschatz et al., 2000). In this paper, we consider Rayleigh-B´ enard convection in a closed cylinder with radius R and height H that is under uniform rotation (with an- gular velocity Ω) about the vertical axis and has a temperature difference ΔT imposed between the top an bottom surfaces. We focus on pattern formation near the onset of convection and under rapid rotation for ε = (ΔT/ΔTc − 1) << 1 where ΔTc is the temperature difference at which convective motion sets in. We are primarily interested in the formation of square patterns and K¨ uppers-Lortz (KL) rolls (K¨ uppers and Lortz, 1969; Ecke and Liu, 1998; Ning and Ecke, 1993). KL rolls is a time-dependent state where rolls lose stability to rolls of an- other orientation. While the existence of KL rolls is predicted by theory (K¨ uppers and Lortz, 1969; Busse and Clever, 1979), precessing square patterns have been observed in physical (Ba- jaj et al., 1998) and numerical experiments (S´ anchez- ´ Alvarez et al., 2005) carried out for cavities with moderate rotation rates and aspect ratios; the formation mechanism is not well understood. Flow in this configuration is well described by the Boussi- nesq equations (Chandrasekhar, 1961) in a cylindrical coordi- nate system (r,θ,z): Dt u + E −1 b z × u = − P ∇p + b z Ra σ T + ∇ 2 u + C ∇· u = 1 r ∂r (ru)+ 1 r ∂ θ v + ∂z w =0 (1) DtT = σ −1 ∇ 2 T where u =(u,v,w) T , Dt ≡ ∂t + u ·∇, p is pressure, P is the Euler number, σ = ν/κ is the Prandtl number, E = ν/(2ΩH 2 ) is the Ekman number, where ν is the kinematic viscosity, κ is the thermal diffusivity, Ra = gαΔTH 3 /(νκ), where g is gravity and α is the coefficient of thermal expansion, T is temperature, ∇ 2 := 1 r ∂r (r∂r )+ 1 r 2 ∂ 2 θ + ∂ 2 z