Nonlinear interaction between transverse disturbances and longitudinal rolls has been investigated for flow in an inclined slot with a heated lower wall when both modes of instability occur at nearly the same value of the control parameter. This condition is shown to be possible for a fluid with Prandtl number greater than 0.263 897, For slightly supercritical values of the Rayleigh number (R) when the critical Rayleigh number for longitudinal rolls RLC is somewhat less than that for transverse stationary rolls, RSC, and for transverse travelling waves, RTC, longitudinal rolls occur first and then remain stable as R is increased beyond RSC or RTC; no mixed mode state occurs. In contrast, if RSc or RTc is slightly below RLC, pure transverse modes exist for only a relatively small range of R beyond RSC or RTC. Thereafter, a three-dimensional mixed mode state occurs well before RLC is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches RLC the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than RLC. Mixed mode convection is also investigated for a special choice of parameters when three modes, namely transverse stationary rolls, transverse travelling waves and longitudinal rolls, become unstable simultaneously. Longitudinal rolls again emerge as the stable supercritical state.

Convection patterns in a flow through a horizontal channel that is heated from below are predicted on the basis of a weakly nonlinear theory. At a certain value of the Reynolds number and the Rayleigh number, the conduction state with steady shear flow becomes linearly unstable to both longitudinal rolls and transverse modes, simultaneously. The longitudinal rolls align along the streamwise direction whereas the transverse modes are periodic in the streamwise direction. Amplitude equations for the interaction between the longitudinal rolls and the transverse modes are derived in a consistent manner. Coefficients in the equations are determined numerically for a wide range of parameters. The longitudinal rolls are found to bifurcate supercritically. On the other hand, the transverse modes bifurcate subcritically or supercritically, depending on the Prandtl number, the aspect ratio of the channel, and the boundary conditions on the sidewalls. Stable convection patterns are classified in a parameter space. A mixed mode pattern, which is a mixture of the components of the longitudinal rolls and the transverse modes, is found to be stable for some sets of parameters.

Two perturbation methods used in weakly nonlinear stability theory, namely, the method of multiple scales and the amplitude expansion method, are examined for their equivalence through formal analyses and numerical calculation of the Landau constants. The method of multiple scales is shown to give results equivalent to those obtained from the amplitude expansion formulation for slightly supercritical states if a normalization condition is applied to the fundamental mode. The convergence of the expansion in the method of multiple scales is also discussed.

The centre manifold reduction to derive the Stuart-Landau equation is examined. A double expansion in terms of the Fourier series and linear eigenfunctions is introduced in hydrodynamic equations. A centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the Stuart-Landau equation. Through a formal expansion in linear eigenfunctions, the latter equation is shown to be equivalent with the one derived by the method of multiple scales. Numerical coefficients involved in the quintic Stuart-Landau equation are evaluated for plane Poiseuille flow, convection in a vertical slot, and Rayleigh-Bénard convection. In all the cases, the coefficients converge as a dimension of the ODE system increases and approach the numerical values obtained by the method of multiple scales.

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The linear stability of unstably stratified shear flows between two horizontal parallel plates has been investigated. The eigenvalue problem was solved numerically by making use of the expansion method in Chebyshev polynomials, and critical Rayleigh numbers were obtained accurately in the Reynolds number range of [0.01, 100]. It was found that the critical Rayleigh number for two-dimensional disturbances increases with an increase of the Reynolds number. The result strongly supports previous stability analyses except for the analysis by Makino and Ishikawa (1985) in which a decrease of the critical Rayleigh number was obtained. For some cases. a discontinuity in the critical wavenumber occurs, due to the development of two extrema in the neutral stability boundary.

A higher harmonic resonance with wavenumber ratio of 1:3 is found to take place in Rayleigh-Bénard convection under rigid-rigid boundary conditions. Bifurcation diagrams for two-dimensional motion are obtained for various values of the Prandtl number P. It is found that a pure mode and mixed mode solutions exist as nonlinear equilibrium states of primary roll solutions for relatively high-Prandtl-number fluids (P ≥ 0.13) while the pure mode, mixed modes, travelling wave and modulated wave solutions exist for relatively low-Prandtl-number fluids (P ≤ 0.12).

In the numerical simulation of certain field theoretical models, the complex Langevin simulation has been believed to fail due to the violation of ergodicity. We give a detailed analysis of this problem based on a toy model with one degree of freedom (S = −β cos θ). We find that the failure is not due to the defect of complex Langevin simulation itself, but rather to the way how one treats the singularity appearing in the drift force. An effective algorithm is proposed by which one can simulate the 1/β behaviour of the expectation value < cos θ > in the small β limit.