Three-dimensional linear instability on a sphere: resolution experiments with a model using vertical orthogonal basis functions

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 by using a third vertical structure function. Spherical harmonics are used for the horizontal structure. The nonseparable eigenvalue problem is derived. Solutions are presented which use 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 unstable solution of a simpler problem. Another OBF (labeled 2-L) is intended to simulate a two-layer model. Increasing the horizontal resolution from R15 (rhomboidal truncation at zonal wavenumber 15) to R30, is found to decrease the growth rates in nearly all cases. (One exception is solid body rotation.) Surprisingly high resolution is needed to represent properly the instability of most of the basic flow jets studied. For some of these flows, R30 resolution may not be high enough. In none of the flows examined was R15 considered 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-L version of the model obtained nearly stationary, rapidly growing modes whose counterpart was not found in the MOBF model. Initially, the results suggested that much higher resolution may be needed than suggested by a previous researcher. This contradiction was seemingly resolved by the author's observation 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 circulation models are made.