A new nested boundary condition for a primitive equation ocean model

Abstract

Abstract. Nested boundary techniques are developed based on the results of Perkins [1993] and Blake [1991]. We focus on the numerical and physical consistency needs across the nested boundary. These techniques replace the transition zone used by other researchers with a numerically and a physically based correction step. We demonstrate our method using a nested, reduced gravity version of the Naval Research Laboratory (NRL) primitive equation ocean model and a two-layer hydrodynamic finite-depth version of the same model. Numerical experiments are performed using integrations of both an idealized double gyre and a realistic Greenland Iceland Norwegian (GIN) Sea configuration. To illustrate the need for improved boundary treatment, we present a numerical study of boundary errors. The study illustrates the fragile nature of nested boundary conditions. With even small errors, a dramatic impact is observed on the formation (or lack thereof) of the Atlantic-Norwegian Current, which is responsible for transporting North Atlantic water to the Arctic Ocean, in the GIN Sea configuration. In this paper we propose an alternative to transition zones based on a numerically and a physically based correction that utilizes the two-sided information available along the internal boundary, providing some support from the entire domain of dependence. Perkins [1993] stressed the two-sided nature of nested domains by viewing the nested model as one overall or aggregate solution. She showed that the "usual" one-sided nesting methodology did not lead, in general, to an aggregate solution that was continuous across the internal boundary. A boundary method that results in values at coincident coarsefine mesh locations being continuously "locked" together as we integrate in time seems attainable, but it cannot be achieved unless we utilize values on both sides of the internal boundary.Ookochi [1972] tried to use information from both sides of the internal boundary to find a nesting scheme that was both continuous (or single valued) and conservative. He concluded that he had to either ignore continuity or conservation; he sacrificed conservation. Among other methods that have utilized two-sided information, Kurihara et al.'s [1979] study enforced energy conservation across the internal boundaries, but they conclude that conservation did not work well for them. Supporting their conclusions, the work of Peggion [1994] indicates that conservation of energy is not appropriate on a nested mesh unless the numerical artifacts introduced where the grid is refined are taken into account. The application of her results would more appropriately be termed energy management, rather than conservation.These numerical artifacts do not effect conservation of mass, however.In this paper we demonstrate numerically some of the errors that can occur when continuity is not maintained. In order to 3483

Two-way embedding algorithms: a review

“…The advantage of this formulation is also that the diffusion term naturally cancels if there is no refinement. Using differences between coarse-and fine-grid variables can also be at the basis of the derivation of open boundary schemes suitable for embedding methods (e.g., Perkins et al 1997;Oddo and Pinardi 2008). Additionally, in a flux correction approach of conservation, it is easy to maintain conservation just by adding the diffusive flux of Eq.…”

Multiscale two-way embedding schemes for free-surface primitive equations in the “Multidisciplinary Simulation, Estimation and Assimilation System”

“…48-50 directly or to correct them to allow the outward radiation of scales unrepresented in the coarse domain. The radiation scheme is an extension of Perkins et al (1997) and updates our previous radiation schemes (Lermusiaux 2007;Haley et al 2009). Some more promising, recent boundary conditions that we have derived and that improve the continuity of horizontal fluxes and reduce jumps in vertical fluxes across the fine domain boundaries are presented in "Appendix 1.3".…”

“…For initialization, this superposition is done over the entire area before the final vertical interpolation of tracers and u . For lateral forcing, this is done at run time in the PE model at the exterior boundaries (and also along two bands inside these boundaries for Perkins et al (1997) boundary conditions). The resulting boundary values are used to generate Dirichlet boundary conditions for Eq.…”

“…Our options (see Haley et al 2009;Lermusiaux 1997) include using values based on data, applying radiation conditions (Orlanski 1976;Robinson 1989), or, following Perkins et al (1997), using radiation conditions to correct the provided values. For nested subdomains, we have first used the interpolated values directly or with Perkins et al (1997) corrections. Some other promising options we have explored with nested subdomains include using the coarse grid values in a narrow buffer zone around the fine domain, which reduces discontinuities.…”

“…boundary conditions: Following the algorithm ofPerkins et al (1997), corrections to the provided values (and nesting interpolation values) are obtained by applying the Orlanski radiation algorithm to the difference between the PE model values and these provided values and using these differences to correct the boundary values.For the barotropic transport, however, this is only done for the tangential component to the boundary. The correction to the normal component is derived from the correction obtained for the surface elevation, η, and the barotropic continuity equation∂ η ∂t + ∇ · [(H + η) U] = 0 .Appendix 1.4: Maintaining the vertically integrated conservation of massTo see how the free-surface algorithm maintains the vertically integrated conservation of mass, start from Eq.…”