Parallel Computational Fluid Dynamics '95

Parallel Computational Fluid

Dynamics:

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.

545

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

Abstract

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.

Introduction

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

spectrum.

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 et.al. [10],

[9] and arvind et.al [1].