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The quasi-geostrophic equations: Approximation, predictability and equilibrium spectra of solutions

✍ Scribed by A. F. Bennett; P. E. Kloeden

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
964 KB
Volume
107
Category
Article
ISSN
0035-9009

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✦ Synopsis

Abstract

Two well‐posed initial‐boundary value problems for the quasi‐geostrophic equations have been identified. the first is defined in a vertical annular cylinder with specified temperatures on the horizontal bottom and top surfaces. the second is defined in a box with periodic boundary conditions on the sides and with rigid bottom and top surfaces; bottom topography is included. Well‐posedness is established by theorems which determine conditions under which there exist unique solutions depending continuously on the initial and other data. the theorems are stated here, but proved elsewhere.

Completely rigidly contained quasi‐geostrophic motion is in general an ill‐posed problem, due to over‐specification of circulation integrals around edges where vertical and horizontal boundaries meet.

The theorems are used to infer (1) sufficient conditions for the convergence of finite‐element Galerkin approximations of Fix (1975) to exact solutions, (2) the predictability type of the solutions as defined by Lorenz (1969), and (3) that the long‐time high wave‐number spectra of the equations are steeper than the inertial ranges and the statistical equilibrium spectra of Kraichnan (1967). Implications for numerical circulation models are drawn.

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