Nonmodal growth in spherical geometry

A great number of studies have examined nonmodal growth in the quasi-geostrophic (QG) framework, however very few of them have dealt with the subject in the context of spherical geometry. In this study, a QG spherical model is developed and results are compared to those obtained from models run in a Cartesian domain.

The linear QG spherical model developed for this study was derived following the steps outlined in Hollingsworth, et. al. (1976, QJRMS, 901-911). Eigenvalue and initial value versions of the model were developed. The model uses a parallelogrammic truncation scheme, keeping 15 zonal wavenumbers and 30 meridional nodes (P15/30 truncation). Eight model levels are kept. The basic state zonal wind used in the model is a 30 degree jet, described in Simmons and Hoskins (1976, JAS, 1454-1476) and Frederiksen (1978, JAS, 1816-1826). The model was tested against previously published results from similar models. Despite minor differences in the handling of the vertical structures of the models, the results obtained from the QG model described here were in close agreement with those published by Frederiksen. The fastest growing normal modes occurred at zonal wavenumbers 8 and 9, both of which showed growth rates very near 0.6/day.

The existence of growth rates in excess of the normal mode growth rate is evidence of nonmodal growth (NG). An examination of time series of growth rates for quantities such as energy, L2 norm, and potential enstrophy can then illustrate situations where NG is occurring. Two general types of initial conditions (IC) are used: 1. a “connected” IC, which has been found to develop large NG in Cartesian models; and 2. a “separated” IC, which is similar to observed atmospheric conditions prior to cyclogenesis. Previously reported results in Cartesian coordinates indicate that the connected IC has larger NG than the separated IC, and NG is much greater at higher wavenumbers in both initial conditions.

Preliminary results in spherical coordinates have broad similarities with results found in Cartesian coordinates. The connected IC again has much more NG than the separated IC. Again, NG is much bigger at higher wavenumbers than at the most unstable wave. For the connected IC, the greater the tilt, the longer it takes to reach the normal mode growth rate. However, the normal mode growth rate is reached quite quickly (within three days). These similarities and differences from Cartesian geometry results will be highlighted in our talk.