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