Effects of equatorially-confined shear flow on MRG and Rossby waves

Linear modal stability analysis of a mean zonal shear flow is carried out in the framework of rotating shallow water equations (RSWE), both under the $\beta$-plane approximation and in the full spherical coordinate system. Two base flows -- equatorial easterly (EE) and westerly (EW) -- with Gaussian profiles highly confined to small latitudes are analyzed. At low Froude number, mixed Rossby-gravity (MRG) and Rossby waves are found to be particularly affected by shear, with prominent changes at higher wavenumbers. These waves become practically non-dispersive at large wavenumbers in EE. The perturbations are found to be more confined equatorially in EE than in EW with the degree of confinement being more pronounced in the $\beta$-plane system compared to the full spherical system. At high Froude number, the phase speeds are significantly larger in the $\beta$-plane system for all families of waves. Under the $\beta$-plane approximation, exponentially unstable modes can be excited, having negative (positive) phase speed in EE (EW). Strikingly, this flow is always neutrally stable with the full spherical system. This speaks for the importance of studying the whole spherical system even for equatorially confined shear.