THE EADY PROBLEM WITH LINEAR HORIZONTAL SHEAR

We consider the quasi-geostrophic linear shear stability analysis of a n inviscid, channel flow with linear horizontal and vertical velocitie s. This modified Eady basic state yields a variety of flow behaviors a s the horizontal shear is increased. The presence of y-shear allows fo r unstable modes with the same growth rate, different phase speeds and different amplitude structures. We speculate that these modes may con tribute to amplitude and structural vaccilation. We find that the two- mode truncation in y provides both quantitative, for small horizontal shears, and qualitative, for larger y-shears, results which compare fa vorably with the 25 y-mode numerical results. The instability in this model is baroclinic modified by the projection of the horizontal basic flow onto the y-modes. The destabilization of relatively short waves, associated with the horizontal shear, is interpreted physically using the constraints provided by energy conservation. It is demonstrated t hat the presence of a horizontal shear may be interpreted as a reducti on in the 'effective' vertical channel depth in a baroclinic Eady mode l. In other words, waves with smaller vertical penetration depth, with wavenumbers larger than the wavenumber of the short-wave cut-off in t he Eady model, can maintain the necessary phase relationship to provid e instability.