## 1. Introduction

[2] Nonlinear interaction of gravity waves inspires many scientists' interest because it is one of the most important mechanisms in energy exchanges, amplitude constraints, and spectral evolution for gravity waves in the middle and upper atmosphere [*Hines*, 1991; *Fritts and Alexander*, 2003].

[3] Resonant interactions were discovered in the fluid dynamical context by *Phillips* [1960]. He pointed out that for prescribed wave vectors and frequencies *ω*, two waves with phases · − *ω*_{1}*t* and · − *ω*_{2}*t* can force a third wave with the sum or difference phase. If their wave vectors and frequencies satisfy the resonant conditions,

where *ω*_{j} = Ω(_{j}) is the dispersion relation, energy is systematically transferred from two initial waves into the third wave. Here in equations (1) and (2), the subscripts *j* = 1, 2 and 3 denote the initial wave with higher frequency, named as the primary wave; the other initial wave with lower frequency, regarded as the secondary wave; and the excited third wave, respectively.

[4] Nonlinear interactions among oceanic internal waves can take place, which were reviewed by *Müller et al.* [1986]. Subsequently, many scientists researched the characteristics of resonant interaction among atmospheric gravity waves based on weakly nonlinear interaction approximation [*Yeh and Liu*, 1981; *Klostermeyer*, 1984; *Dunkerton*, 1987; *Dong and Yeh*, 1988; *Fritts et al.*, 1992; *Yi and Xiao*, 1997], and these theoretical works provided us an insight into the essential property of resonant interaction of gravity waves in a linear regime. Adopting the linearized resonant interaction equations, *Yi* [1999] has numerically studied the temporal and spatial evolutions of gravity wave packets in resonant interactions. Recently, starting from a set of nonlinear equations, *Zhang and Yi* [2004] numerically investigated the degree and characteristic time of resonant interaction in a compressible atmosphere, and *Huang et al.* [2007] further simulated the nonresonant interaction of gravity waves and discussed the effect of viscous dissipation on the nonresonant interaction. However, in fact, most studies above focused on the interactions with respect to three waves matching or mismatching the difference resonant conditions of − = and *ω*_{1} − *ω*_{2} = *ω*_{3}. Moreover, most above cited numerical studies exhibited the interaction process of three wave packets by presenting the wave energy density. The resonant interaction of atmospheric gravity waves under the sum resonant conditions of + = and *ω*_{1} + *ω*_{2} = *ω*_{3} has received little attention.

[5] In this letter, we will confirm that the resonant interaction of three waves satisfying the sum resonant conditions of + = and *ω*_{1} + *ω*_{2} = *ω*_{3} can occur in the atmosphere, and the whole interaction process is clearly exhibited through numerical experiments for the first time. The evolution of spectrum and energy of the excited wave is quantitatively analyzed, and quite considerable energy transfer in the interaction demonstrates that the nonlinear interaction is an important mechanism in forming the atmospheric wave spectrum and transporting the energy and momentum to much higher altitudes. Finally, a discussion on the resonant and nonresonant interactions is presented.