A pretty good sponge: Dealing with open boundaries in limited-area ocean models

The problem of computing within a limited domain surrounded by open boundaries is discussed within the context of the shallow-water wave equations by comparing three different treatments, all of which surround the domain by absorbing zones intended to prevent reflections of outgoing waves. The first, which has attracted a lot of attention for use in electromagnetic and aeroacoustic applications, is intended to prevent all reflections. However, it has not yet been developed to handle the second important requirement of open boundaries, namely the ability to pass information about external conditions into the domain of interest. The other two treatments, which absorb differences from a specified external solution, allow information to pass through the open boundary in both directions. One, based on the flow relaxation scheme of [Martinsen, E.A., Engedahl, H., 1987. Implementation and testing of a lateral boundary scheme as an open-boundary condition in a barotropic ocean model. Coastal Eng. 11, 603–627] and termed here the “simple sponge,” relaxes all fields toward their external counterparts. The other, a simplification and generalization of the perfectly matched layer, referred to here as the “pretty good sponge,” avoids absorbing the component of momentum parallel to the open boundary. Comparisons for a case that is dominated by outgoing waves shows the pretty good sponge to perform essentially as well as the perfectly matched layer and better than the simple sponge. In comparisons for a geostrophically balanced eddy passing through open boundaries, the pretty good sponge out-performed the simple sponge when the only external information available was about the advecting flow, but when information about the nature of the eddy in the sponge zones was also available, the simple sponge performed better. For the case of an equatorial soliton passing through the boundary and no information provided about its nature outside the open domain, again the pretty good sponge performed better. Proving useful for situations governed by nonlinear equations forced by external conditions and being easy to implement, the pretty good sponge should be considered for use with existing limited-area ocean models.