An accurate hyperbolic system for approximately hydrostatic and incompressible oceanographic flows

Abstract

It is well known that severe restrictions on accuracy and stability arise when using a numerical model based on the Eulerian equations to compute the low frequency motions of physical oceanography. The loss of accuracy is due to the extreme skewness of the system. Although the system can be transformed to symmetric hyperbolic form, the diagonal transformation matrix contains factors that produce mathematical estimates indicating loss of accuracy and these estimates are verified in practice. The stability restriction is just a result of the multiple timescales present in the system, i.e. the presence of Rossby, gravity, and sound waves. Three alternatives to overcome these restrictions are discussed. The first two alternatives are the primitive and quasi-geostrophic equations. Although models based on these systems alleviate some of the restrictions of the Eulerian model, they operate under a new set of limitations. The primitive equation model requires more resolution than a quasi-geostrophic model to obtain the same degree of numerical accuracy and is ill-posed for the initial-boundary value problem. Although the quasi-geostrophic equations are accurate to the order of the Rossby number, there are cases when this error is of the order of 10%. Reducing the error in these cases, e.g. by using the balance equations, requires a considerable increase in the computational complexity of the system. The quasi-geostrophic equations also cannot be used in the equatorial region and any improvements which would allow them to be used there would result in a computationally inefficient system. The third alternative is to slow down the gravity and sound waves by decreasing the size of the appropriate terms in the equations. Although the resulting approximate system alleviates the severe accuracy requirement, the stability requirement may still be unacceptable. By using the reduced system derived from the approximate system, a system which has many desirable properties is obtained. The resolution requirements for a model based on the reduced system are the same as those of the quasi-geostrophic model, i.e. less than for a model based on the primitive equations. Because the reduced system is the proper mathematical limit of a hyperbolic system, a wide range of boundary conditions can be chosen so that the resulting initial-boundary value problem is well posed. The reduced system analytically describes the low-frequency solutions to two digits of accuracy (even when the Rossby number is O (0.1)), yet only requires the solution of a linear, constant coefficient, three-dimensional elliptic equation. Finally, the reduced system is applicable in the equatorial region.