Theory of Very Long Waves in a Zonal Atmospheric Flow

A derivation is given of the general wave equation for small quasi-static adiabatic perturbations in a zonal atmospheric flow. This equation is then specialized to the case of very long waver (wave-length 10,000 km or more). The proper modelling is obtained by assuming the (non-dimensional) wave number to be of order unity and by the assumptions of strong rotational constraint (Ro ⩽ 1), strong dynamic stability (Ri ⩾ 1) but weak stratification (ΔΘ/Θ ⩽ 1). The long waves obtained by the present model move at a speed that, to the first order, is independent of the wave number but dependent on the product Ro · Ri. A solution is worked out in the β-plane approximation, assuming a basic wind and potential temperature that vary linearly with pressure. The existence of stationary waves in this case is demonstrated. The effect of different boundary conditions is discussed. Some consequences of the theory of interest in connection with numerical weather prediction are pointed out.