http://journals.cambridge.org Downloaded: 02 Jun 2009 IP address: 170.215.39.246 J. Fluid Mech. (2009), vol. 628, pp. 269–297. c  2009 Cambridge University Press doi:10.1017/S0022112009006193 Printed in the United Kingdom 269 Centrifugal effects in rotating convection: nonlinear dynamics J. M. LOPEZ 1 † AND F. MARQUES 2 1 Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA 2 Departament de F´ ısica Aplicada, Universitat Polit` ecnica de Catalunya, Barcelona 08034, Spain (Received 8 July 2008 and in revised form 18 January 2009) Rotating convection in cylindrical containers is a canonical problem in fluid dynamics, in which a variety of simplifying assumptions have been used in order to allow for low-dimensional models or linear stability analysis from trivial basic states. An aspect of the problem that has received only limited attention is the influence of the centrifugal force, because it makes it difficult or even impossible to implement the aforementioned approaches. In this study, the mutual interplay between the three forces of the problem, Coriolis, gravitational and centrifugal buoyancy, is examined via direct numerical simulation of the Navier–Stokes equations in a parameter regime where the three forces are of comparable strengths in a cylindrical container with the radius equal to the depth so that wall effects are also of order one. Two steady axisymmetric basic states exist in this regime, and the nonlinear dynamics of the solutions bifurcating from them is explored in detail. A variety of bifurcated solutions and several codimension-two bifurcation points acting as organizing centres for the dynamics have been found. A main result is that the flow has simple dynamics for either weak heating or large centrifugal buoyancy. Reducing the strength of centrifugal buoyancy leads to subcritical bifurcations, and as a result linear stability is of limited utility, and direct numerical simulations or laboratory experiments are the only way to establish the connections between the different solutions and their organizing centres, which result from the competition between the three forces. Centrifugal effects primarily lead to the axisymmetrization of the flow and a reduction in the heat flux. 1. Introduction One of the most fascinating aspects of rotating convection is the observation of spatio-temporal chaos essentially at the onset of convection as the Rayleigh number (non-dimensional imposed vertical temperature gradient) is increased (Krishnamurti 1971; Busse & Heikes 1980; Niemela & Donnelly 1986; Hu, Ecke & Ahlers 1997). This experimentally observed spatio-temporal chaos has been associated with the K¨ uppers–Lortz (KL) instability (K¨ uppers & Lortz 1969; K¨ uppers 1970; Clever & Busse 1979). The KL instability occurs when the system is rotating faster than a critical level; convection rolls in a horizontally unbounded layer are unstable to rolls oriented at about 60 ◦ . The KL instability is formally found in an unbounded rotating system in the limit of zero centrifugal force. Experiments, of course, are conducted in bounded containers and most employ large horizontal-to-vertical aspect ratios. † Email address for correspondence: lopez@math.asu.edu