Numerical simulation of harmonic, and trapped, Rossby waves in a channel on the midlatitude β-plane

The applicability of two competing wave theories of the rotating shallow-water equations, the harmonic planetary waves and the trapped planetary waves, is examined by solving the equations numerically in a zonal channel on the midlatitude β-plane. The examination is carried out by initializing the numerical shallow-water solver by harmonic and trapped waves and comparing the temporal evolution to that expected from the corresponding wave theory. The simulations confirm that harmonic waves provide accurate approximations for the temporal evolution in narrow channels while trapped waves do so in wide channels. The transition from a narrow channel to a wide channel occurs when the non-dimensional number (Formula presented.) (where L is the channel width, R_d is the radius of deformation, a is Earth's radius and ϕ₀ is the latitude of the centre of the channel) is increased above a threshold value that larger than 4 and that depends on the meridional mode number of the trapped wave. Numerical solutions of the nonlinear equations show that, for the same wave amplitude, trapped waves approximate the nonlinear solutions in a wide channel for much longer times than harmonic waves in a narrow channel.