Three-Dimensional Linear Instability on a Sphere: Resolution Experiments with a Model Using Vertical Orthogonal Basis Functions

Abstract

A new form of the linear, quasi-geostrophic model is derived on a sphere. The new feature is the use of empirically defined orthogonal basis functions (OBFs) to represent the vertical structure of the perturbation solutions. The prescribed basic state is expressed using a third vertical structure function. Spherical harmonics are used for the horizontal structure. The nonseparablc eigenvalue problem is derived. Solutions are presented using various vertical OBFs, basic flows (both zonally varying and zonally uniform) and horizontal truncations(both rhomboidal and triangular). One OBF (labeled "MOBF') is patterned after the structure found in the most unusable solution of a simpler problem. Another OBF (labeled "2-L") is intended to simulate a two-layer model. Increasing the horizontal resolution from RI5 (rhomboidal truncation at zonal wavenumber 15) to R30 is found to decrease the growth rates in nearly all cas. (One exception is solid body rotation.) Surprisingly high resolution is needed to properly represent the instability of most of the basic flow jets studied herein. For some of these flows, R30 may not be high-enough resolution. In none of the flows examined did we conclude that RI 5 was adequate. The phase speeds in the "MOBF" cases are frequently much faster.than the most unstable modes in the "2-L" cases. In a few instances, the "2-U" version of the model obtains nearly stationary, rapidlygrowing modes whose counterpart is not found in the "MOBF" model. Initially, our results suggested that much higher resolution may be needed than suggested by a previous researcher. This contradiction was seemingly resolved by our obse~vatlon that the convergence to the correct solution was faster when the basic jet was centered at a lower latitude. Some implications for low-resolution general circulaton models are made.