On nonresonant interactions of atmospheric waves in a rotating Earth

We investigate the effects of the Earth's rotation upon nonresonant interactions of atmospheric waves. The wave perturbations in the atmosphere are represented by eigen modes which, when formulated in terms of a Hermitian operator by a judicious choice of state variables, called the F-system in this paper, becomes orthogonal to each other. The effects of geostrophic terms on the eigen vectors are simply represented by the parameter δ, also introduced in this paper. They are important when the wavenumber vector is nearly vertical. The general theory of nonresonant wave-wave interactions is then formulated. Two cases of nonresonant interactions where the Earth's rotation plays an important role are investigated in detail: the case of a gravity wave interacting with two vortical modes (GVI) and the case of elastic scattering (ES). It is found that the previously found threshold condition on GVI in the absence of rotation is weakened or disappears, depending on the scale of vortical modes, when atmospheric rotation is taken into account. As a consequence, GVI may be an important process in extracting energy and momentum from gravity waves. The Earth's rotation is also found to influence ES in important ways. Through nonresonant interactions with a vertical wind shear of circular polarization, the exchange of energy and momentum between a large scale, horizontally propagating gravity wave and a mesoscale, nearly vertical propagating gravity wave is found to proceed at a rapid rate.