Starting from a set of fully nonlinear equations, the whole process of a gravity wave excited through resonant interaction under sum resonant conditions is clearly exhibited. In the whole interaction, the wavelength and frequency of the excited wave are in agreement with the values derived from the sum resonant conditions. And the energy growth of the excited wave is mainly from the primary wave. Moreover, strong energy exchange in the interaction shows that the nonlinear interaction may play a significant role in determining the atmospheric wave spectrum and transporting the momentum to higher altitudes. A discussion on the resonant and nonresonant interactions is also presented, and the detuning degree of interaction is proposed, which may be applied to measure whether or not the effective energy exchange occurs in the nonlinear interactions of gravity waves.
 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].
 Resonant interactions were discovered in the fluid dynamical context by Phillips . He pointed out that for prescribed wave vectors and frequencies ω, two waves with phases · − ω1t and · − ω2t can force a third wave with the sum or difference phase. If their wave vectors and frequencies satisfy the resonant conditions,
where ωj = 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.
 Nonlinear interactions among oceanic internal waves can take place, which were reviewed by Müller et al. . 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  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  numerically investigated the degree and characteristic time of resonant interaction in a compressible atmosphere, and Huang et al.  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 ω1 − ω2 = ω3. 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 ω1 + ω2 = ω3 has received little attention.
 In this letter, we will confirm that the resonant interaction of three waves satisfying the sum resonant conditions of + = and ω1 + ω2 = ω3 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.
 The simulations are carried out by using a set of primitive hydrodynamic equations in an adiabatic, inviscid and two-dimensional compressible atmosphere, of which the difference scheme is described in detail by Huang et al. . In this numerical computation, the horizontal and vertical domains are chosen to be 0 km ≤ x ≤ 2600 km and 0 km ≤ z ≤ 179.8 km, and the horizontal and vertical step lengths are set to be Δx = 4.0 km and Δz = 0.2 km, respectively.
 Aiming at the excitation process of gravity wave by the resonant interaction alone, the effects of mean wind and inhomogeneous temperature fields are excluded by assuming a windless and isothermal background atmosphere with initial constant temperature of T0 = 290 K and initial density profile of ρ0 = where g and R are the gravitational acceleration and universal gas constant, and ρc = 1.2 kg m−3 is the density on the ground. Two initial Gaussian packets which are introduced as the primary and secondary waves have the following horizontal perturbation velocities
where x and z are the horizontal (positive eastward) and vertical (positive upward) coordinates, respectively; ucj (j = 1, 2 and 3) is the maximum horizontal amplitude; xcj and zcj are the center positions of the wave packets in x and z directions, respectively; the horizontal and vertical half-widths of the wave packets, i.e. σxj and σzj, are chosen to be σxj = λxj and σzj = λzj, where λxj and λzj are the wavelengths of the waves in x and z directions, respectively.
 Previous studies have shown that the body force due to local instability and breaking of waves generally tends to create high-frequency gravity waves [Fritts et al., 2002; Vadas and Fritts, 2002; Vadas et al., 2003], and the local body force and wave-wave nonlinear interaction [Zhang and Yi, 2004; Huang et al., 2007] are likely the important source of gravity waves in the middle and upper atmosphere [Fritts et al., 2006]. Thus these increasingly important sources at higher altitudes may generate many downward propagating gravity waves, moreover, significant (more than 20%) gravity waves in the stratosphere and the upper mesosphere are observed propagating downward [Lintelman and Gardner, 1994; Zhang and Yi, 2007]. Hence, we can specify a downward propagating primary wave and an upgoing secondary wave, of which parameters are listed in Table 1, in which the negative sign of the vertical wavelength and wave number represents downward phase propagation. Next, we want to give a simple analysis. If an excited wave obeys the sum matching condition of + = , according to the parameters listed in Table 1, its wavelengths are calculated to be λx3 = 36.09 km and λz3 = −4.26 km, and then its frequency is ω3 = 2.12 × 10−3 Rad s−1 derived from the dispersion relation of gravity waves, which satisfies exactly the sum matching condition of ω1 + ω2 = ω3. This means that such selected three waves satisfy the sum resonant conditions in both wave numbers and frequencies, as expressed in equations (1) and (2), and the third wave would be excited through the resonant interaction if it arose.
Table 1. Initial Parameters of Primary and Secondary Wave Packets
kx (Rad m−1)
kz (Rad m−1)
ω (Rad s−1)
1.26 × 10−4
1.26 × 10−3
1.80 × 10−3
0.48 × 10−4
−2.73 × 10−3
0.32 × 10−3
 Three cases with only different initial amplitudes of the primary and secondary waves, which are listed in Table 2, are performed. The propagation and interaction process of the wave triad in horizontal disturbance velocities for case 1 is shown in Figure 1. At the beginning time, the primary wave is closely above the secondary wave. After 3 h propagation, the primary and secondary waves meet with each other and overlap partly with evident phase staggering. At the time of 4 hours, an apparent fresh disturbance which staggers with the primary and secondary waves appears, and the tilt of its phase indicates that the excited wave propagates upward and eastward. From 5 h to 7 h, the primary and excited waves propagate farther downward and upward, respectively. After 9 h propagation, these three waves depart from each other in space by and large, and the maximum horizontal velocity of the excited wave grows to 2.6 m s−1 due to energy absorbing through the resonant interaction. Simultaneously, the maximum amplitude of the primary wave rapidly reduces from 6.0 m s−1 at the beginning time to about 0.3 m s−1 as a result of the energy transfer, the expanded wave coverage and the exponential increase of the background density with the decreasing height.
Table 2. Initial Amplitudes and Final Energies of Interacting Waves for Cases 1–3
Amplitude of Wave 1 (m s−1)
Amplitude of Wave 2 (m s−1)
Energy of Wave 1 (105 J)
Energy of Wave 2 (105 J)
Energy of Wave 3 (104 J)
Degree of Interaction (%)
 The comparison of the wave vectors and frequencies of the interacting wave triad is interesting. The wave number spectra of the wave triad are obtained by making a Discrete Fourier Transform (DFT) on the horizontal disturbance velocities over the whole computational domain. Because the spectra of these three waves are apart from each other, we can easily extract the spectra of each wave, respectively, and the normalized wave number spectra of the excited wave after 4 h are shown in Figure 2. When the apparent excited wave fluctuation begins to occur at the time of 4 hours, its dominant wavelengths are 36.17 and −4.28 km in the horizontal and vertical directions, which are rather close to the values of 36.09 and −4.26 km derived from the sum resonant condition, respectively. And according to the dispersion relation, the wave frequency can be calculated to be 2.12 × 10−3 Rads−1, which equals the sum of ω1 = 1.80 × 10−3 Rad s−1 and ω2 = 0.32 × 10−3 Rad s−1. This indicates that the sum resonant conditions are exactly satisfied and the new wave is excited through the sum resonant interaction. In addition, the wavelength of the excited wave shifts slightly after 6 h.
 We calculate the wave energies at each integral hour by integrating the wave energy density ɛw = ρ0 (u′2 + w′2) + + over the whole computational domain, where p′ is the perturbation pressure; va is the acoustic speed, and γ is the ratio of specific heats at constant pressure and volume. Here, the perturbation quantities of each wave are obtained from the corresponding total field by applying a wave number spectrum band-pass filter. The values of the energies of the primary and secondary waves at t = 0 h and the excited wave at t = 9 h is given in Table 2, and Figure 3 shows the evolution of the wave energies with time. After 9 h propagation, the energy of the primary wave is 7.90 × 105 J, which is 30.46% less than its initial value of 11.36 × 105 J, and the energy of the secondary wave increases slightly from 4.86 × 105 J to 5.28 × 105 J. A striking result shown in Figure 3 is that the excited wave obtains the energy of approximately 2.36 × 105 J through the sum resonant interaction, which is 20.78% of the initial energy of the primary wave. This percentage is regarded as the degree of interaction, which is a measure of the strength of interaction [Zhang and Yi, 2004; Huang et al., 2007]. The total energy of these three waves decreases about 4.19% after 9 h of interaction owing to a little energy conversion from the wave disturbance to the large-scale background movement and the energy loss in the filtering process, which are about 3.37% and 0.82% of the total energy, respectively.
 In case 2, two initial waves have the same amplitude of 1.0 m s−1 and the numerical result shows an interaction degree of 21.25%, which is considerably close to that in case 1. This indicates that the energy of the excited wave may be approximately proportional to that of the primary wave when the initial energy of the secondary wave remains a constant value. In case 3, the energy of the secondary wave with initial amplitude of 1.53 m s−1 is the same as that of the primary wave, and Figure 4 shows the variation of the wave energies with time. The energy of 45.72 × 104 J absorbed by the excited wave and the corresponding interaction degree of 40.25% are an indication that violent energy transfer happens in case 3. By comparing case 3 with case 1, one can note that the energy of the excited wave increases with the initial energy of the secondary wave on condition of the constant initial energy of the primary wave. Additionally, in these three cases, as shown in Figures 3 and 4, the energy tends to transfer mainly from the primary wave to the excited wave, which is consistent with the conclusion in the nonresonant interaction [Huang et al., 2007] though the excited wave has the highest and lowest frequencies in the resonant and nonresonant studies, respectively.
4. Discussion and Summary
 The whole interaction process of three wave packets is clearly exhibited. In the resonant interaction, the wave vectors and frequencies of three waves coincide with the sum resonant conditions of + = and ω1 + ω2 = ω3, and intense energy transfer from the primary wave to the excited wave occurs. For example, the degree of interaction arrives at 40.25% in case 3, and such a great value hasn't been reported before. The study presented also suggests that it is possible to create an excited wave which has larger energy than the final primary wave energy if the secondary wave has sufficient energy. Therefore, not only can the resonant interaction under the sum resonant conditions indeed happen in a compressible atmosphere, but also the surprisingly strong energy exchange in the interaction demonstrates that nonlinear interaction can be an important factor in determining the atmospheric wave spectrum. This investigation supports the conclusion proposed by Fritts and Alexander  that wave-wave interaction is an increasingly significant source at higher altitudes. More importantly, since the upgoing wave excited by the sum resonant interaction has large vertical wavelength and intrinsic phase speed, it can propagate to much higher altitudes, thus this nonlinear interaction may play a significant role in the momentum budget well into the thermosphere.
 A downward propagating gravity wave excited through the nonresonant interaction with three waves mismatching the difference resonant conditions of − = and ω1 − ω2 = ω3 was confirmed by Huang et al. . In weakly nonlinear interaction approximation, the nonlinear interaction occurs only for those wave triads satisfying the wavenumber matching condition, and the frequency mismatch (Δω) is allowable for the nonresonant interaction [Dong and Yeh, 1988; Yi and Xiao, 1997]. In this case, we introduce a detuning degree of interaction (δ), which is written as
In case 1 presented in the previous study [Huang et al., 2007], for the specified primary wave with λx1 = 45 km and λz1 = −4.5 km and secondary wave with λx2 = 120 km and λz2 = −2.3 km, the nonresonant interaction with effective energy exchange has a detuning degree of δ = 0.23. In case 8 in this previous study, δ is 0.66, and the degree of interaction decreases to 2.73%, which means that the effective energy exchange is rather weak. Thus, the detuning degree may be used to roughly estimate the strength of interaction. Here, for the given primary and secondary waves in this study, two detuning degrees are calculated to be δ1 = 0 and δ2 = 3.22. δ1 = 0 represents the resonant interaction corresponding to satisfying the sum resonant conditions, and δ2 = 3.22 represents the nonresonant interaction corresponding to the mismatching of the condition of ω1 − ω2 = ω3. So large detuning degree of δ2 = 3.22 indicates the corresponding nonresonant interaction without effective energy exchange, and the corresponding excited wave can't be found in our numerical result. Nevertheless, according to the analysis above, a more interacting phenomenon may be expected that two new gravity waves with considerable energies are simultaneously excited through the resonant interactions for some special gravity wave groups if the sum and difference resonant conditions of ± = and ω1 ± ω2 = ω3 can be satisfied simultaneously, or both δ1 and δ2 equal to 0 simultaneously, and we will further verify and investigate this phenomenon in the future work.
 The authors would like to thank the editor and anonymous reviewers for their comments on the manuscript. This work was jointly supported by the National Natural Science Foundation of China through grants 40825013, 40731055 and 40774085; the Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) and the Open Foundation of State Key Laboratory of Space Weather.