Low wavenumber instability on the equatorial beta-plane

Unstable waves on the equatorial beta‐plane are studied when the mean flow U(y) is a function of latitude only, which permits separation of variables. The additional restrictions of (i) a linear shear U(y) = Sy where S is a constant and y is non‐dimensional latitude and (ii) small zonal wavenumber k permit simple approximations that allow identification of two distinct modes of instability. The "mixed Kelvin‐inertial" mode is an unstable Kelvin wave for |S| < 2 but an inertial instability for larger shear. The "mixed gravity‐Kelvin" mode is the neutral, n=0 positive frequency gravity wave for small shear but an unstable Kelvin wave for |S| > 2. In contrast to normal hydrodynamics problems, there is no minimum shear for instability but instead Im(c) is proportional to exp[−5.34/|S|] for small S. Although the inertial instability was known previously, the two branches of unstable Kelvin waves and the connection between them and the neutral and inertially unstable gravity waves are original discoveries. For weak shear, only the Kelvin wave is unstable.