Physical interpretation of unstable modes of a linear shear flow in shallow water on an equatorial beta-plane

Unstable modes of a linear shear flow in shallow water on an equatorial β-plane are obtained over a wide range of values of a non-dimensional parameter and are interpreted in terms of resonance between neutral waves. The non-dimensional parameter in the system is E ≡γ4/(gHβ2), where γ, g, H and β are the meridional shear of basic zonal flow, gravitational constant, equivalent depth and the north-south gradient of the Coriolis parameter, respectively. The value of E is varied within the range -2.50 < log E < 7.50. The problem is solved numerically in a channel of width γ/β. The structures of the most unstable modes, and the combinations of resonating neutral waves that cause the instability, change according to the value of E as follows. For log E < 2.00, the most unstable modes have zonally non-symmetric structures; the most unstable modes for log E < 1.00 are caused by resonance between equatorial Kelvin modes and continuous modes, and those for 1.00 < log E < 2.00 are caused by resonance between equatorial Kelvin modes and westward mixed Rossby-gravity modes. The most unstable modes for log E < 2.00 have symmetric structures and are identical with inertially unstable modes. Examinations of dispersion curves suggest that nonsymmetric unstable modes for 1.00 < log E < 2.00 and inertially unstable modes for log E < 2.00 are the same kind of instability.