Abstract
In Saltzman’s seminal paper from 1962, the author developed a framework based on the spectral method for the analysis of the solution to the classical Rayleigh–Bénard convection problem using low-order models (LOMs), LOM (n) with n ≤ 52. By way of illustrating the power of these models, he singled out an LOM (7) and presented a very preliminary account of its numerical solution starting from one initial condition and for two values of the Rayleigh number, λ = 2 and 5. This paper provides a complete mathematical characterization of the solution of this LOM (7), herein called the Saltzman LOM (7) [S-LOM (7)]. Historically, Saltzman’s examination of the numerical solution of this low-order model contained two salient characteristics: 1) the periodic solution (in the physical 3D space and time) that expand on Rayleigh’s classical study and 2) a nonperiodic solution (in the temporal space dealing with the evolution of Fourier amplitude) that served Lorenz in his fundamental study of chaos in the early 1960s. Interestingly, the presence of this nonperiodic solution was left unmentioned in Saltzman’s study in 1962 but explained in detail in Lorenz’s scientific biography in 1993. Both of these fundamental aspects of Saltzman’s study are fully explored in this paper and bring a sense of completeness to the work.
Saltzman’s Model. Part I: Complete Characterization of Solution Properties
