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Results of high numerical accuracy are reported for the growth rate σ of axisymmetric disturbances to circular Couette flow between concentric cylinders with the inner one rotating and of two-dimensional disturbances of the pure conduction state of a Boussinesq fluid between horizontal parallel plates heated from below. When σ is positive, the former case leads to Couette–Taylor (CT) flow and the latter to Rayleigh–Bénard (RB) convection. The curve of vanishing growth rate (marginal stability curve) Tc(q) or Rc(q), where q is the wave vector of the disturbance and T and R are the Taylor and Rayleigh number, respectively, is obtained for both cases. The correlation length amplitude ξ0 is calculated by obtaining the curvature of the marginal stability curve. The amplitude σ0 of σ was determined from data with T>Tc(q) or R>Rc(q). In that range, the wave vector qm of maximum σ is also found. For the CT system, the calculations were performed systematically as a function of radius ratio η for 0.975≥η≥0.1. For the RB system, those parameters which depend on the Prandtl number P were obtained for 10−3≤P≤103. Some of the RB results could be compared with analytic calculations and served to test this numerical procedure. It is expected that these results will be useful in calculations using an amplitude equation for the weakly nonlinear state and in studies of wavenumber-selection processes.

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Experimental results are reported for the selection of axial wavelengths of Taylor vortices in a system where the Reynolds number, R 9 is "ramped" spatially from above to below the critical value, R c , for the onset of vortex flow. It is found that a sufficiently slow ramp connecting the supercritical region (containing vortices) to a subcritical region (of purely azimuthal flow) results in the selection of a unique wavelength. A more rapid ramp results in a small R-dependent band of allowed wavelengths which grows in width as R-^R C from above.PACS numbers: 47.20. + m, 05.70.LnIn nonlinear dissipative systems subjected to an external stress, R, a transition frequently occurs to a state of reduced symmetry having a spatial structure with a characteristic wavelength. Examples are Rayleigh-B6nard convection, 1 Taylor vortex flow, 2 " 4 certain chemical reactions, 5 flame-front propagation, 6 and crystal growth. 7 The equations governing systems of infinite spatial extent usually possess a continuum of linearly stable solutions , 8 corresponding to a band of wave numbers having a width which varies as e 1/2 (e = R/R C -1, and R c is the value of the stress parameter at the transition). Imposition of nonperiodic boundary conditions may reduce the width of the allowed band to order e, 9 * 10 but for the investigated cases a finite band remains. Such finite systems generally select discrete states from the band of stable solutions, with the selected wave numbers dependent upon the particular boundary conditions imposed.For Taylor vortex flow (TVF) between concentric rotating cylinders, the multiplicity of stable states was first explored in detail by Coles 3 and Snyder. 11 They observed TVF to be stable for a range of axial wavelengths when the Reynolds number, R, was above R c , the value for vortex formation, but below the value where azimuthal waves appear. The observable range of wavelengths 11 * 12 is roughly equal to that expected theoretically, 13 but the factors determining the wavelength selected in a particular experiment are not understood. It has been suggested quite recently, 14 » 15 however, that a gradual axial variation of e from positive to negative values (a spatial ramp) could permit a continuous wavelength adjustment mechanism to become effective, which might result in the selection of a unique although possibly e -dependent state. The question of the existence and nature of such selection processes is largely unexplored. One might expect that their nature would have a strong influence on the development of temporally nonperiodic (or turbulent) behavior in large real systems. 16 More specifically, we would expect a "weak" selection mechanism to render the primary modes of the system relatively sensitive to the influence of noise and other degrees of freedom of the system.To investigate wavelength selection experimentally, we used apparatus in which the gap between the cylinders tapered linearly from supercritical to subcritical values as shown in Fig. 1. The working fluid was a 30% solution of glyc...