On the Solvability of Inverse Problems Arising From the Two-Layer Lorenz '96 System
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Authors
Smith, Beau James
Issue Date
2022
Type
Dissertation
Language
Keywords
data assimilation , dynamical system , inverse , Lorenz , model , weather
Alternative Title
Abstract
The two-layer Lorenz ’96 model consists of two linearly coupled systems of ODEs with two distinct time scales. This simple model was designed to reflect the patterns of local instability and growth represented by the interaction of planetary and synoptic dynamics with mesoscale motions and convective clouds. Under the assumption that the large-amplitude variables in the first layer are fully observed, we consider two inverse problems. The first is to estimate the unobserved values of the second layer in the case where the dynamics are known; the second is to solve for both the unobserved small scales and the unknown dynamics governing them. For simplicity, we assume that the dynamics governing the small scales take on a parameterized form with a single unknown parameter. In this case, our goal is to simultaneously estimate that parameter and the unobserved small scales.Our study begins with a verification that the dynamics in the two-layer Lorenz ’96 model are dynamically interesting enough to merit further investigation. We then develop algorithms to solve the two types of inverse problems mentioned above and find theoretical conditions under which those algorithms allow us to estimate the unobserved small scales and, optionally, the unknown parameter. We begin by proving that directly inserting the observations into the model as it is being integrated in time results in synchronization that allows recovery of the unobserved small scales over time. We then make a novel use of derivative information�"i.e., the rate at which the observations change over time�"to obtain new forms of data assimilation that allow solving the inverse problem faster, under less stringent conditions, and when the parameter governing the small scales is unknown.Throughout we confirm our theoretical results with numerical experiments and remark that solving the inverse problem numerically turns out to be possible even when the system does not satisfy the hypotheses required by our theory.