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S. Lakshmivarahan
,
John M. Lewis
, and
Junjun Hu

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.

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S. Lakshmivarahan
,
John M. Lewis
, and
Junjun Hu

Abstract

Over the decades the role of observations in building and/or improving the fidelity of a model to a phenomenon is well documented in the meteorological literature. More recently adaptive/targeted observations have been routinely used to improve the quality of the analysis resulting from the fusion of data with models in a data assimilation scheme and the subsequent forecast. In this paper our goal is to develop an offline (preprocessing) diagnostic strategy for placing observations with a singular view to reduce the forecast error/innovation in the context of the classical 4D-Var. It is well known that the shape of the cost functional as measured by its gradient (also called adjoint gradient or sensitivity) in the control (initial condition and model parameters) space determines the marching of the control iterates toward a local minimum. These iterates can become marooned in regions of control space where the gradient is small. An open question is how to avoid these “flat” regions by bounding the norm of the gradient away from zero. We answer this question in two steps. We, for the first time, derive a linear transformation defined by a symmetric positive semidefinite (SPSD) Gramian G = F ¯ T F ¯ that directly relates the control error to the adjoint gradient. It is then shown that by placing observations where the square of the Frobenius norm of F ¯ (which is also the sum of the eigenvalues of G ) is a maximum, we can indeed bound the norm of the adjoint gradient away from zero.

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