Gradual instability of relaxation-extrapolation schemes

Many geophysical problems involve solving for the time derivative of the Laplacian of a streamfunction, particularly three-dimensional ocean circulation studies. This paper demonstrates the existence of two instabilities when relaxation methods are used to solve the resulting Poisson equation. One occurs as a result of over-relaxation. The other occurs even if under-relaxation is used, provided that there are variations in Coriolis parameter and that extrapolation provides an initial guess for the relaxation process. Both instabilities equilibrate with (erroneous) amplitudes determined by the degree of dissipation present. In the case of the latter instability, the amplitude can be six orders of magnitude larger than the error criterion for the relaxation; its structure resembles basic modes, which persist indefinitely. Some ways of circumventing the problem are discussed.