Pseudo-spectral approximations of Rossby and gravity waves in a two-Layer fluid


Graduation date: 2012The complexity of numerical ocean circulation models requires careful checking with\ud a variety of test problems. The purpose of this paper is to develop a test problem involving\ud Rossby and gravity waves in a two-layer \ud fluid in a channel. The goal is to compute very\ud accurate solutions to this test problem. These solutions can then be used as a part of the\ud checking process for numerical ocean circulation models.\ud Here, Chebychev pseudo-spectral methods are used to solve the governing equations\ud with a high degree of accuracy. Chebychev pseudo-spectral methods can be described in\ud the following way: For a given function, find the polynomial interpolant at a particular\ud non-uniform grid. The derivative of this polynomial serves as an approximation to the\ud derivative of the original function. This approximation can then be inserted to differential\ud equations to solve for approximate solutions. Here, the governing equations reduce to\ud an eigenvalue problem with eigenvectors and eigenvalues corresponding to the spatial\ud dependences of modal solutions and the frequencies of those solutions, respectively.\ud The results of this method are checked in two ways. First, the solutions using the\ud Chebychev pseudo-spectral methods are analyzed and are found to exhibit the properties\ud known to belong to physical Rossby and gravity waves. Second, in the special case\ud where the two-layer model degenerates to a one-layer system, some analytic solutions are\ud known. When the numerical solutions are compared to the analytic solutions, they show\ud an exponential rate of convergence.\ud The conclusion is that the solutions computed using the Chebychev pseudo-spectral\ud methods are highly accurate and could be used as a test problem to partially check numerical\ud ocean circulation models

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This paper was published in ScholarsArchive@OSU.

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