On the Pressure Gradient Force over Steep Topography in Sigma Coordinate Ocean Models

TLDR

The study investigates pressure‑gradient force errors near steep topography in sigma‑coordinate ocean models using Arakawa–Suarez vertical differencing schemes. Truncation error is quantified by inserting known buoyancy profiles into finite‑difference hydrostatic and pressure‑gradient terms, and hydrostatic inconsistency is also examined. Errors are distributed throughout the water column, increase with finer vertical resolution and grid placement relative to buoyancy structure, and removing a reference state cuts the lowest‑mode truncation error by 2–3×, yielding a 10–12 cm s⁻¹ false geostrophic current in a 10‑level model versus an order‑of‑magnitude smaller error in a 30‑level hydrostatically consistent model.

Abstract

The error in computing the pressure gradient force near steep topography using terms following (σ) coordinates is investigated in an ocean model using the family of vertical differencing schemes proposed by Arakawa and Suarez. The truncation error is estimated by substituting known buoyancy profiles into the finite difference hydrostatic and pressure gradient terms. The error due to "hydrostatic inconsistency," which is not simply a space truncation error, is also documented. The results show that the pressure gradient error is spread throughout the water column, and it is sensitive to the vertical resolution and to the placement of the grid points relative to the vertical structure of the buoyancy field being modeled. Removing a reference state, as suggested for the atmosphere by Gary, reduces the truncation error associated with the two lowest vertical modes by a factor of 2 to 3. As an example, the error in computing the pressure gradient using a standard 10-level primitive equation model applied to buoyancy profiles and topographic slopes typical of the California Current region corresponds to a false geostrophic current of the order of 10–12 cm s−1. The analogous error in a hydrostatically consistent 30-level model with the reference state removed is about an order of magnitude smaller.