Efficient Nonlinear Low-Order Models for Atmospheric and Climate Dynamics

Kevin A Grady, Purdue University

Abstract

The governing equations of atmospheric and climate dynamics present enormous mathematical challenges when studied analytically. Following the pioneering work of Kolmogorov, Lorenz, and Obukhov, a popular approach to handle these difficult partial differential equations (PDEs) is to approximate them with finite systems of ordinary differential equations (ODEs), called low-order models (LOMs). One such LOM is the celebrated Lorenz (1963) model of just three ODEs, but attempts to extend it to larger, more realistic models of atmospheric dynamics have sometimes led to LOMs exhibiting unphysical behavior, such as a lack of energy conservation in the dissipationless limit. These behaviors can be avoided by constructing LOMs using 3-mode nonlinear dynamical systems known in mechanics as Volterra gyrostats, the simplest one being equivalent to the Lorenz model. Gyrostatic LOMs guarantee energy conservation, suggesting they may offer a general framework for deriving efficient LOMs for atmospheric and climate dynamics. This study explores the use of gyrostatic LOMs in three important related problems of atmospheric dynamics. The first is 2D Rayleigh-Bénard convection (RBC), where an algorithm for studying gyrostatic LOMs was developed. Before now this had to be done manually, limiting the LOMs that could be studied as well as their size. This algorithm permits the study of LOMs larger than previously possible as well as their conservation properties. It was used here to demonstrate that all physically sound LOMs for this problem from recent publications have a gyrostatic form. The second problem is the interplay of buoyancy and shear in the formation of rolls versus cells in mesoscale shallow convection (MSC). A gyrostatic LOM for 3D RBC with the ability to parameterize buoyancy and shear was developed using an adopted version of the algorithm for 2D RBC. This model was run for hundreds of different combinations of buoyancy and shear, with the results generally matching those of other observational and modeling studies. The third problem is convection driven by internal heating, where the algorithm developed for 2D RBC was applied to derive several gyrostatic LOMs. In general these LOMs were shown to match reasonably well with the actual physics of this problem.

Degree

Ph.D.

Advisors

Gluhovsky, Purdue University.

Subject Area

Atmospheric sciences

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