Barotropic instability with downstream and asymmetric cross-stream variations: idealized calculations

The barotropic instability of basic states with downstream and asymmetric cross-stream variations is investigated using the linearized, non-divergent barotropic vorticity equation in a periodic beta channel. A single expression for streamfunction defines the basic state; the variations are introduced by parameter changes. Eigenvalue and time integration methods are employed to determine the instabilities. When downstream variation is present, the maximum amplitude of the unstable streamfunction is always located downstream from the location of maximum latitudinal shear. This occurs because the disturbance propagates through a range of longitudes where the flow is locally barotropically unstable. The instability is sensitive to the degree of downstream variation. When there is strong variation there is more concentration of streamfunction amplitude in the regions downstream of the jet maximum, smaller disturbance scales and smaller growth rates than when the flow is more parallel. The smaller growth rate for the downstream variation case results from the propagating disturbance being subjected to strong shear for only a finite time before moving into regions of lesser shear where it cannot grow as fast. Asymmetric cross-stream variation has little effect on the growth rate and frequency of the most unstable mode but significantly affects the structure of the instability. Larger amplitude and more pronounced tilt opposite to the shear occur on the side of the jet with the strongest shear. This can be expected from a theoretical consideration of the energetics. Instabilities in the presence of both downstream and asymmetric cross-stream variations combine the effects of each of those individual kinds of variations. The time integration and eigenvalue methods compare very well; in fact, they complement one another. Both may be needed for a thorough understanding of these types of instabilities. Preliminary nonlinear calculations show that as the disturbances grow to finite amplitude, they split, with high centers moving to the north and low centers to the south. The size of the disturbances increases as well. Downstream variation causes split centers of significant amplitude to be concentrated in the eastern part of the channel while cross-stream asymmetry introduces an asymmetry in the strength of the highs and lows. Additional studies are proposed to investigate the hypothesis that the development of blocking patterns may depend crucially on whether the larg