Discrete and continuous spectra of the barotropic quasigeostrophic vorticity model. Part I

The evolution processes of small disturbances in an arbitrary basic flow can be expressed as a combination of spectral functions of the discrete spectra and continuous spectrum of a model that bear distinctly different evolutionary characteristics. Using the linearized barotropic quasigeostrophic vorticity model, this study formulates the discrete spectral solution into a form that is consistent with traditional normal modes in time and space, and the continuous spectral solution into a form with the continuum covering the range between minimum and maximum zonal angular velocities. An estimation of the bounds of the spectral points is derived to complement those derived from integral constraints. A theorem is given to describe the possible number of discrete spectral points away from the continuum. The theoretical analysis is then used to aid the numerical identification and interpretation of discrete and continuous spectra of the model with realistic atmospheric basic zonal flows. It is shown that neutral spectral points correspond to either ultralong waves with global meridional coverage or synoptic-scale waves in low latitudes. The unstable spectral points correspond to localized waves with developing or decaying timescales longer than 2 weeks. Structures of spectral function of the continuum are also presented and discussed. They are shown to restrict on one side to the critical latitude and on the other side to the jet core under certain conditions.