The Downstream and Upstream Development of Unstable Baroclinic Waves

Abstract

The response of a baroclinically unstable atmosphere to a localized initial perturbation is examined. In a preliminary numerical experiment, using a nonlinear primitive-equation model with spherical geometry, new disturbances grow regularly downstream in broad agreement with a particular type of observed development. The initial disturbance moves downstream, while smaller scale disturbances develop upstream of it—each forming at about the same longitude. Later downstream disturbances have upper level amplitudes significantly larger than found in nonlinear integrations using normal-mode initial conditions. Similar development is found in a quasi-geostrophic model with spherical geometry, and the rate of spreading of the instability is insensitive to the amplitude of the initial perturbation. For the baroclinic instability model considered by Eady the downstream disturbances appear first at the upper surface at a position which moves with a speed close to that of the zonal-mean flow at this surface, while the upstream disturbances all form close to a fixed location it the surface mean flow is zero. New disturbances possess a short horizontal length scale which subsequently expands toward that of the most unstable normal mode, and their amplitude increases at a rate initially much in excess of normal-mode growth rates. Phase speed is close to that of normal modes. Diagnosis is performed both using a form of the omega equation and through an asymptotic analysis of the solution represented by a sum of normal modes of different wavenumbers. Further calculations have been performed for a linearized β-plant model. Inclusion of a mean density variation with height and the β-effect slows the downstream spread of the instability by up to 15%, and this downstream development occurs on a longer length scale. The upstream disturbances quite generally develop close to a fixed location when the mean surface flow is zero, and possess a relatively short zonal scale if the low-level mean flow has vertical shear. An additional primitive-equation integration confirms the results of the preliminary nonlinear experiment.