On Truncation Errors in Sigma‐System Models

Abstract

A detailed examination is given as to how temperature T ‐ and thus geopotential ‐ of a synoptic state appears in sigma(σ)‐coordinate systems, i.e., systems whose lowest coordinate surface coincides with the earth's surface. It is shown that the description of T along constant σ requires many more Fourier modes than would be expected by merely regarding the horizontal synoptic variation. This effect is related to the scale of the orographic field and to the vertical variation of T. With these results as a basis, possible truncation errors ‐ directly associated with the use of σ‐coordinates ‐ of horizontally discretized functions are discussed. It is found that if the resolution is inadequate, horizontal advection in the thermodynamic equation may become significantly distorted as a result of truncation. It is furthermore inferred that, with proper difference formulation of the terms of the pressure gradient force, it ought to be possible to limit ‐ though not to eliminate ‐ errors in this quantity.

To illustrate the theoretical/qualitative deductions, a few numerical applications are carried out. These also indicate that the maximum truncation error in the pressure force would be about 0.5 ms⁻¹, expressed as equivalent geostrophic wind at mid‐latitudes. This figure was obtained with centered difference (second order accuracy) approximation of derivatives, a grid distance of 370 km and a ground slope corresponding to that on the western side of the Rocky Mountains.

Notes

Contribution number 310.

Present address: Dept. of Met., Arrheniuslab., Fack, S‐104 05 Stockholm 50, Sweden.