Parallel Computational Fluid
Implementations and Results Using Parallel Computers
A. Ecer, J. Periaux, N. Satofuka and S.
Taylor (Editors)
9 1995 Elsevier Science B.V. All rights reserved.
Application of a Parallel Navier-Stokes Model to Ocean Circulation
Chris Hill and John Marshall
Department of Earth, Atmospheric and Planetary Sciences,
Massachusetts Institute of Technology
A prognostic ocean circulation model based on the incompressible Navier-Stokes equations is
described. The model is designed for the study of both non-hydrostatic and hydrostatic scales;
it employs the "pressure method" and finite volume techniques. To find the pressure in the
non-hydrostatic limit, a three-dimensional Poisson equation must be inverted. We employ a
single-pass, two-grid method where the thin three-dimensional domain of the ocean is first
projected on to a two-dimensional grid using vertical integration as a smoother. The two-grid
method exploits the anisotropic aspect ratio of the computational domain and dramatically
reduces the computational effort required to invert the three-dimensional elliptic problem. The
resulting model is competitive with hydrostatic models in that limit and scales efficiently on
parallel computers.
The ocean is a stratified fluid on a rotating body driven from its upper surface by
patterns of momentum and buoyancy flux. The detailed dynamics are very accurately
described by the Navier-Stokes equations. These equations admit, and the ocean con-
tains, a wide variety of phenomenon on a plethora of space and time scales. Modeling of
the ocean is thus a formidable challenge; it is a turbulent fluid containing energetically
active scales ranging from the global down to order 1 - 10 km horizontally and some
10's of meters vertically- see Fig.1. Important scale interactions occur over the entire
Numerical models of the ocean circulation, and the ocean models used in climate
research, employ approximate forms of the Navier-Stokes equations. Most are based on
the "hydrostatic primitive equations" (HPE) in which the vertical momentum equation
is reduced to a statement of hydrostatic balance and the "traditional approximation" is
made in which the Coriolis force is treated approximately and the shallow atmosphere
approximation is made. On the large-scale the terms omitted in the HPE are generally
thought to be small, but on small scales the scaling assumptions implicit in them become
increasingly problematic.
In this paper we report on an ocean model firmly rooted in the incompressible
Navier-Stokes (INS) equations. The novelty of our approach is that it is sensitive to
the transition from non-hydrostatic to hydrostatic behavior and performs competitively
with hydrostatic models in that limit- more details can be found in Marshall [10],
[9] and arvind [1].

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