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 Atmospheric waves and their interactions in the thermospheric neutral wind are studied based on Arecibo incoherent scatter radar observations. Our analysis suggests that the thermospheric atmosphere is usually disturbed by certain types of waves, including tides, gravity waves, and planetary waves, of which the diurnal tide is almost always the dominant disturbance. Strong interactions (defined as the coexistence of strong positive and negative correlations among the interacting waves) between the diurnal tide and gravity waves are frequently observed during the entire observation period. These strong interactions can persist for several days, although they are highly intermittent. Moreover, the sum and difference interactions between the diurnal tide and gravity waves always occur simultaneously and the energy exchange between the interacting waves is sometimes reversible. In addition to tide–gravity wave interactions, tide–planetary wave and tide–tide interactions are also found in our data. In tide–planetary wave interactions, the tidal oscillations are modulated at the interacting planetary wave periods. A combination of bispectral and correlation analyses verifies the occurrence of nonlinear interactions among different tidal components in the middle thermosphere. Moreover, during tide–tide interactions, the energy transfer trend changes very frequently, indicating that tidal energy is frequently redistributed among different tidal components. Generally, our study provides proof of strong tide–gravity wave, tide–planetary wave, and tide–tide interactions in the middle thermosphere, which has rarely been reported to date.
 Previous work on wave–wave interactions has mainly focused on the mesosphere and lower thermosphere. Possibly because of a lack of observational data, wave–wave interactions in the middle thermosphere, the subject of this paper, have received little attention. In past wave–wave-interaction research, observed amplitude anticorrelations of the interacting waves were usually taken as a proxy of the interaction, while insufficient attention was paid to positive correlations between the amplitudes of interacting waves. In this paper, we provide observational evidence of positive correlations between interacting waves and offer a possible explanation of this finding. Because GWs and PWs span a broad spectrum in a realistic atmosphere (due to their complex sources and propagating environments), if a tide with frequencyf1 and a GW or PW with frequency f2 have sufficiently large amplitudes, the sum and difference interactions, i.e., the interactions among the wave triplet (f1, f2, f1 + f2) and (f1, f2, f1 − f2/f2 − f1), respectively, would possibly occur simultaneously. This is indeed observed in our study but has rarely been reported in previous work.
 In the present paper, we focus primarily on atmospheric waves in the thermosphere with periods ranging from several to tens of hours and their interactions. The incoherent scatter radar (ISR) data adopted in this paper, and our derivation of the magnetic meridian component of the neutral wind, are introduced in the next section. All significant wave signatures in the thermospheric neutral wind derived from the ISR data, including the tides, PWs, and GWs, are presented in section 3. Our evidence of their mutual interactions is provided in section 4. In the final section, we briefly summarize our observations.
2. Data Description and Derivation of the Meridian Neutral Wind
 This paper considers four data sets from the 430 MHz ISR at Arecibo (18.34°N, 293.25°E) acquired between 1988 and 1993. They cover 144.6, 101.8, 145.0, and 232.4 h, beginning at local times of 10:56 on 5 December 1988, 10:55 on 30 March 1992, 15:34 on 30 July 1992, and 10:35 on 20 January 1993, respectively. Herein, we will refer to these data sets using the letters A, B, C, and D, respectively. The electron density, electron and ion temperatures, and the component of the ion bulk velocity along the radar's line of sight are directly measured. The ion drift antiparallel to the magnetic field lines, vap (in m s−1; positive is opposite to the magnetic field direction), is then inferred from the line-of-sight ion velocities measured at different antenna positions, assuming horizontally spatial and temporal homogeneity. The adopted data have uneven time resolution, varying from several minutes to several hours, while most time intervals are shorter than 35 min and rarely longer than 1.5 h. In the thermosphere, there are 15 measurement heights from 145 to 664 km, evenly spaced at an interval of 37.1 km. Because the data below 256 km are very sparse and the neutral wind characteristics deduced at altitudes above 367 km are affected by large uncertainties, our investigation only focuses on altitudes between 256 and 367 km; i.e., at four observation altitudes (256, 293, 330, and 367 km).
where Vnis the neutral-wind vector, which is characterized by zonal (u), meridian (v), and vertical (w) components. Here we assume that w ≪ u, v.B is the magnetic field vector; B is the corresponding magnitude; D and I are the magnetic declination and inclination, respectively; Us is the southward neutral wind in the magnetic meridian plane; and vd represents the antiparallel O+ diffusion velocity, which can be expressed as
where Ti, Te, and Tn are the ion, electron, and neutral temperatures, respectively; Tp = and Tr = ; ne is the electron density; z is the altitude; mi is the ion mass; k is Boltzmann's constant; g is gravitational acceleration; and Dinis the ion-neutral diffusion coefficient given by
where υiis the ion-neutral collision the mass of an O+ ion, and and are the collision frequencies between the dominant O+ ion and the major neutral constituents, O, N2, and O2. If we use the formulae for ion-neutral collision frequencies fromBuonsanto and Witasse , Din can be expressed as
where the neutral densities of O, N2, and O2 (i.e., [O], [N2], and [O2]) are expressed in m−3. We obtain ne, Ti, and Te from the ISR data, and Tn, [O], [N2], and [O2] from the MSIS-86 model [Hedin, 1987]. A more detailed analysis of the derivation errors can be found in Buonsanto and Witasse .
3. Atmospheric Waves
 The adopted ISR raw data sets are sampled unevenly at intervals of several minutes to several hours, for time spans of over 100 h. This allows us to study a variety of atmospheric oscillations, including tides, GWs, and PWs, in the temporal domain. First, we performed Lomb–Scargle periodogram analysis [Scargle, 1982] on the time series of the derived southward wind in the magnetic meridian, Us. Figure 1 shows the normalized Lomb–Scargle periodograms at four chosen measurement heights for our four data sets. It can be clearly seen that statistically significant tidal oscillations exist (e.g., diurnal, semidiurnal, terdiurnal, and quarterdiurnal tides), indicating that the thermospheric atmosphere is usually perturbed by intensive tides, which has also been shown by previous authors [Duboin and Lafeuille, 1992; Hagan, 1993; Canziani, 1994a, 1994b; Hocke, 1996; Gong and Zhou, 2011]. The tidal amplitudes observed in data sets A, B, C, and D display visible dissimilarities. The maximum tidal amplitudes in data sets A/B/C/D are 18.3/20.6/28.9/32.1 m s−1. Among these tidal oscillations, the diurnal tide is the most dominant component in most cases. This is consistent with the results of Duboin and Lafeuille  and Hagan  at an altitude of 300 km. Its amplitude shows distinct altitude alteration, which has rarely been reported. For data sets A and C, a minimum magnitude is seen at a height of 330 km, while for data set B the minimum occurs at 367 km. For data set D, the diurnal tidal amplitude increases with height, and evident sidelobes with periods of 28 and 21 h can be found at all heights. Magnitude maxima occur at 367 km for data sets A, C, and D and at 330 km for data set B. The complex altitude variation of the diurnal tidal amplitude suggests that atmospheric tides in the middle thermosphere may be affected by complex dynamical processes. We also detect obvious signatures of semidiurnal, terdiurnal, and even quarterdiurnal tides in these data. The prominent semidiurnal tide is present only in data sets A, B, and C, while a distinct terdiurnal tide is observed in all four data sets. A significant quarterdiurnal tide only occurs in data sets A and B, at a level that is significantly weaker than that of the other three tidal components. In data set A, the terdiurnal tide is almost the most significant tidal component. Its amplitude is greater than that of the diurnal tide for the height range of 293–367 km. On the basis of the terdiurnal amplitudes in these four data sets, we come to the same conclusion as Duboin and Lafeuille ; i.e., the terdiurnal tide is the most variable of the first four harmonic tidal components.
 In addition to tides, some significant oscillations with periods greater than 36 h can also be identified in Figure 1. It is usually assumed that those waves observed in the wind field with periods longer than approximately 1.5 days are PWs [Beard et al., 1999]. In data sets A/B/C, 3.4/2.7/3.0-day waves appear with maximum amplitudes of 9.9/12.0/13.3 m s−1. This is clearly weaker than the corresponding tidal signatures. In data set D, two PW components are evident, characterized by 2.6- and 1.6-day oscillations with maximum amplitudes of 11.0 and 10.5 m s−1, respectively. Additionally, we also observe some distinct oscillations with periods shorter than 24 h, which are likely GWs (we will refer to them as such). In data set B, a 7.1 h wave is clearly present with a maximum amplitude of 14.0 m s−1, while a 5.5 h wave appears with a maximum amplitude of 9.1 m s−1. In data set D, 17.4, 10.1, and 7.1 h waves occur at the two lowest altitudes, all with amplitudes of approximately 10 m s−1, slightly above the 90% confidence level (not shown).
 To sum up, we find a variety of oscillation signatures, including tides, PWs, and GWs in the thermospheric neutral wind. Tidal oscillations are the most dominant, while PWs and GWs are not as strong but still statistically significant.
4. Wave–Wave Interactions
Figure 1shows that a variety of wave components are present in the thermospheric neutral wind that display complicated variations with altitude. This implies the potential presence of nonlinear wave–wave interactions. To reveal interactions among tides, GWs, and PWs, we performed bispectral analysis, a quadratic phase-coupling detector, on the evenly sampled data (the observational data were processed using linear interpolation so that the resulting neutral windUs has an even time resolution of 30 min). This technique is extensively used to provide evidence of the presence of nonlinear wave–wave interactions [Beard et al., 1999; Pancheva, 2000; Pancheva et al., 2000; Pancheva and Mukhtarov, 2000; Suresh Babu et al., 2011]. In this paper, we use the conventional “Fourier type” methods to obtain the bispectral estimates by ensemble averaging. All the bispectral plots are normalized by dividing by the maximum value, leading to a relative amplitude between 0 and 1.
4.1. Tide–GW Interactions
 Except for data set C, the bispectra of the other three data sets exhibit notable peaks. However, the peaks arising at tidal–GW frequencies are only present in data sets B and D. Therefore we started from these two data sets to explore tidal–GW interactions. Figure 2 shows the bispectra of data set B at an altitude of 330 km (left) and data set D at 293 km (right).
Figure 2 (left) exhibits strong peaks near the bifrequencies (0.099, 0.0416) and (0.141, 0.0416) cycles h−1. The other three peaks occur near tidal frequencies and cannot be simply regarded as evidence of second-order nonlinear tide–tide interactions because the phase consistency among tides may, to some extent, be a direct result of their common force. In turn, this could lead to a nonzero bispectrum in the absence of nonlinear interactions. The bifrequencies (0.141, 0.0416) and (0.099, 0.0416) cycles h−1 correspond to the sum and difference triplets for the interaction between the diurnal tide and the 7.1 h GW. On the basis of the Lomb–Scargle periodograms for data set B in Figure 1, we find that the 24 and 7.1 h waves are very prominent. If a nonlinear interaction occurred between the diurnal tide and the 7.1 h GW, 5.5 h (sum frequency) and 10.1 h (difference frequency) waves would be produced. Interestingly, we observe these two weak but distinct spectral peaks in Figure 1, which provides additional support for the existence of this tide–GW interaction. The bispectrum (not shown) and Lomb–Scargle periodogram for data set B at an altitude of 367 km also clearly imply the occurrence of such an interaction. For simplicity, we will refer to the 24, 7.1, 10.1, and 5.5 h waves as DT, GW1, GW2, and GW3, respectively.
 To further explore these sum and difference interactions, we harmonically fit the DT, GW1, GW2, and GW3 oscillations over a sliding 24 h window with increments of 30 min from the evenly interpolated neutral wind at each height. The fit formula we used was
where f(t) is the evenly interpolated neutral wind, f0(t) is the background neutral wind (the arithmetic average of f(t) in each window), T is the wave period, and A(t) and ϕ(t) are the wave amplitude and phase, respectively. Here we choose T = 24/7.1/10.1/5.5 h. Since there exist time gaps up to several hours in the utilized data sets, we would like to provide a list of longer time gaps here. The largest time gap in data set C is 92.76 min, and no wave-wave interaction was found in it. So, we only discuss the longer time gaps in data sets A, B, and D in detail. For these three data sets, we only presented the measurement gaps larger than one quarter of the shortest period of the concerned waves, i.e., 5.5/4 = 1.375 h for data sets B and D and 6/4 = 1.5 h for data set A. In data set A, there are two gaps: 18.49–20.25 h and 72.36–79.67 h. In data set B, there is one gap: 31.95–37.37. In data set D, there are nine gaps: 12.43–14.08 h, 14.08–15.72 h, 69.26–75.37 h, 92.37–97.40 h, 117.56–121.13 h, 140.84–145.10 h, 156.67–158.90 h, 212.87–216.67 h, and 236.05–240.23 h. In the harmonical fit, the interpolated points corresponding to the time gaps larger than 1.375 h (for data sets B and D) or 1.5 h (for data set A) are ignored, and no fit was attempted if less than 16 h of data are present in any 24-h window, i.e., 2/3 of the window length [Beard et al., 1999; Pancheva and Mukhtarov, 2000]. Fortunately, in any 24-h window of the data sets utilized in our study, there are enough data points to accomplish the fit, producing continuous wave amplitudes and phases at evenly spaced intervals of 30 min.
 Note that for the following checks of the wave amplitude's correlations, which occur simultaneously with the wave–wave interactions, the wave amplitudes have been detrended. This was done to take into account that their linear trends, which are most likely caused by long-term variations, would possibly lead to high correlation coefficients [Xu et al., 2009], which may not be due to wave–wave interactions.
Figure 3 provides the time series of the original and reconstructed southward neutral wind (top), time series of the reconstructed waves (middle), and time variation of the detrended wave amplitudes (bottom) for Data B at 330 km. It can be observed from Figure 3 that the original southward neutral wind (the black curve in Figure 3, top) fluctuates dramatically, implying it might be enduring some complex and violent dynamic processes, such as strong waves, intensive interactions among waves, and external forcing. The reconstructed wind (the red curve in Figure 3, top) with the background wind (the blue curve in Figure 3, top) and the concerned four wave components, i.e., DT, GW1, GW2, and GW3, which are fitted from the original wind, is very close to the original wind, indicating that these four waves are the primary wave components of the original wind and validating our fitting process. The reconstructed DT (the red curve in Figure 3, middle) reflects well the 24-h oscillation in the original wind. Compared with the other three reconstructed wave components, we can conclude that DT is the most dominating perturbation of the original wind. Observing the time-dependent variations of the detrended wave amplitudes, we find that the amplitudes of all four waves exhibit evident transient variations. Moreover, in the period from 19:00 on 31 March to 8:00 on 1 April, the instantaneous amplitudes of GW1 and GW2 show the almost same time variation while those of DT and GW2 show the almost opposite time variation, implying that strong positive or negative correlations exist among these interacting waves.
 Therefore we use a sliding 12-h time segment with increments of 30 min to calculate their time-varying correlation coefficients.Figure 4 shows the correlation coefficients of the detrended instantaneous amplitudes of the difference (i.e., interactions of DT, GW1, and GW2) and sum (i.e., interactions of DT, GW1, and GW3) interaction triads as a function of time. Significant correlations between interacting waves are frequently present over the entire observation period. We define a significant correlation as one where the magnitude of the correlation coefficient is greater than the 90% confidence level of a test of statistical significance. For the difference interaction (Figure 4, top), we find that the correlation coefficients exhibit significantly negative values during several periods; see, for example, the correlation coefficient between DT and GW1 (red curve) from 18:30 on 31 March to 7:00 on 1 April, as well as the correlation coefficient between DT and GW2 (purple curve) from 19:00 on 31 March to 8:00 on 1 April. Obviously, these anticorrelations resulted from energy interchange between the interacting waves, as has been reported in many papers in the literature [e.g., Harris and Vincent, 1993; Gurubaran et al., 2001; Lima et al., 2004; Pancheva, 2006].
 More interestingly, we also find periods when the correlation coefficients have significantly positive values; e.g., the period from 5:00 on 31 March to 16:30 on 1 April for the coefficient between GW1 and GW2. We will now attempt to explain this positive correlation. When a three-wave interaction takes place, there should be one wave (referred to as wave 1) that exchanges energy with the other two waves (waves 2 and 3). The correlation coefficients between waves 1 and 2, as well as between waves 1 and 3, would be significantly negative at that time. On the condition that these two negative correlation coefficients have very large magnitudes, which implies that wave 1 releases (or absorbs) energy almost simultaneously to (or from) both waves 2 and 3, wave 1 would display an amplitude variation that is almost opposite to those of waves 2 and 3. Waves 2 and 3 might then display somewhat consistent variations in amplitude. Thus a significant positive correlation coefficient between waves 2 and 3 would be expected. This conceptual explanation is clearly demonstrated inFigure 4. On the basis of the three correlation coefficients among the difference interaction triad in Figure 4 (top), we find three periods when two of the coefficients are significantly negative: one period runs from 19:00 on 31 March to 7:00 on 1 April (period A) and the other two last from 21:30 to 23:00 on 1 April (period B) and from 2:00 to 3:30 on 2 April (period C), respectively. During period A, GW1 and GW2 are highly anticorrelated with DT, and the correlation coefficients are both equally large at −0.92 (in magnitude). Apparently, such strong negative correlations indicate that the amplitudes of both GW1 and GW2 display almost entirely opposite variations with respect to that of DT. Therefore it is very well possible that GW1 is positively correlated with GW2. In fact, these two waves are indeed highly correlated, the correlation coefficient reaches 0.99, which is consistent with our conceptual explanation.
 Periods B and C are much shorter than period A. During the former two periods, the correlation coefficient between DT and GW1 (red curve) during period B and that between DT and GW2 (purple curve) during period C have lower values (in magnitude, approximately −0.78 and −0.64, respectively). This is despite that fact that one of the two negative correlation coefficients (i.e., the coefficient between GW1 and GW2; blue curve) is higher than −0.9 (in magnitude), which is close to the equivalent values during period A. The remaining correlation coefficients during both periods B and C have lower positive values, slightly below the significance level, which confirms that positive correlation is closely linked to the magnitude of the negative correlations. In other words, only two strong, negative correlations could lead to a significant positive correlation. Moreover, note that during the entire observation period, positive or negative correlations among the interacting-wave triad may change during different interaction periods, which can be regarded as different interaction stages. For instance, DT and GW1 are negatively correlated with each other during period A but positively correlated during period C. Similar variations can also be found for the correlation between DT and GW2, as well as for that between GW1 and GW2. The changes in sign of the correlation coefficients indicate changes in the direction of energy transfer and imply that the energy exchange is reversible. This has previously been reported in a numerical study on fully nonlinear interactions among GWs [Zhang and Yi, 2004]. In fact, the strong positive correlation between the interacting waves has also been revealed in both observational and numerical studies [Thayaparan, 1997; Yi and Xiao, 1997; McCormack et al., 2010]. However, these authors did not present detailed physical explanations.
 The characteristics of the correlation coefficients for the sum interaction (Figure 4, bottom) are very similar to those for the difference interaction. We find two periods, from 19:00 to 23:30 on 31 March (period D) and from 20:00 to 23:00 on 1 April (period E), when two significant anticorrelations and one significant positive correlation coexist. During these two periods, both DT and GW3 are highly anticorrelated with GW1, while they are highly correlated with each other, indicating that wave energy is transferred among the three waves and GW1 exhibits an opposite amplitude change compared to the other two waves. The energy-transferring pairs did not change during the entire observation period, as opposed to their behavior during the difference interaction. In periods D and E, the maxima of the correlation coefficients of the interacting waves are generally greater than 0.7 in magnitude, except for the correlation coefficient between GW3 and GW1 during period E, which is −0.69. Because GW3 is negatively correlated with GW1 and positively correlated with DT, we suggest that GW3's energy mainly originated from GW1 rather than from DT.
 Moreover, periods A and B (in which we observed the difference interaction) almost overlapped with periods D and E (characterized by the sum interaction), respectively, clearly illustrating that the difference and sum interactions occurred simultaneously. This phenomenon has rarely been reported to date.
 Similarly, we also found some evidence for tide–GW interactions in data set D. The bispectrum for data set D at a height of 293 km exhibits strong peaks near the bifrequencies (0.0576, 0.0416), (0.0992, 0.0416), and (0.1408, 0.0416) cycles h−1. In addition, a peak near the bifrequencies (0.0576, 0.125) is obvious but not as strong. The presence of the former three peaks implies the occurrence of nonlinear interactions between the diurnal tide and GWs, and the latter's presence implies nonlinear interactions between the terdiurnal tide and GWs. Here we only focus on interactions between the diurnal tide and GWs because they are much more significant. Those three strong bispectrum peaks correspond to the triplet (24, 17.4, 10.1), (24, 10.1, 7.1), and (24, 7.1, 5.5) h. Figure 1 clearly shows that these GW spectral components involved in the interactions. Except for the 5.5 h GW, which is weak but distinct, the other three GW components (i.e., the 17.4, 10.1, and 7.1 h GWs) are prominent. Because the 10.1 and 7.1 h waves are both involved in two interaction triads, the corresponding wave–wave interactions are very complicated. Despite this complexity, we still checked the relevance of the wave amplitudes of the interaction triads. Again, we refer to the 24/7.1/10.1/5.5 h waves as DT/GW1/GW2/GW3, while we call the 17.4 h wave GW4 for simplicity. We apply identical methods as before to harmonically fit the wave amplitudes and calculate the correlation coefficients of the detrended wave amplitudes. Figure 5 shows the correlation coefficients of the interaction triads (waves 1, 2, and 3), i.e., (DT, GW4, GW2), (DT, GW2, GW1), and (DT, GW1, GW3) in Figure 5 top, middle, and bottom, respectively, for data set D at an altitude of 293 km. The red, purple, and blue curves are the correlation coefficients between waves 1 and 2, 1 and 3, and 2 and 3, respectively, in each triad. Note that the first and second triads can be regarded as the difference and sum interactions between DT and GW2, while the second and third triads are the equivalent interactions between DT and GW1. We first explore the interactions between DT and GW2. It is very clear that strong (anti)correlations occur frequently. In addition, for some periods two correlation coefficients are highly negative while the third is significantly positive. This further confirms the conclusion drawn above. These periods, labeled F–M, occur occasionally across the entire observation period of data set D. This indicates that interactions between DT and GW2 could persist for a long time (8 days), although they are highly intermittent. Moreover, a strong interaction could consecutively persist for more than 24 h, as exemplified by periods H (26 h) and L (27 h), which are much longer than the relevant periods in data set B.
 During period H, DT and GW4 are highly anticorrelated with GW2, and the correlation coefficients are as high as −0.90 and −0.98 (in magnitude), respectively, while DT is positively correlated with GW4 (its correlation coefficient reaches 0.93). In the difference interaction between DT and GW2, the energy-exchange wave pairs remain unchanged, and DT and GW4 keep absorbing (or releasing) energy from (or to) GW2, similarly as inFigure 4 (bottom).
 In period L, both DT and GW1 are highly anticorrelated with GW2, and the correlation coefficients are as high as −0.90 and −0.97 (in magnitude), respectively, while DT is positively correlated with GW1 (its correlation coefficient reaches 0.98). However, in the sum interaction between DT and GW2, the energy-exchange wave pairs may change. For example, in period M, DT and GW2 are highly anticorrelated with GW1, while DT is positively correlated with GW2. DT exchanged energy with GW2 in period L, but with GW1 in period M, a situation that is similar to that inFigure 4 (top). We also find periods F and H in Figure 5 (top), almost overlapping with periods J and L, respectively, in Figure 5 (middle). This further confirms that strong difference and sum interactions would occur simultaneously.
 On the basis of the difference and sum interactions between DT and GW1, similar conclusions could also be drawn: (1) strong correlations and anticorrelations occur frequently; (2) interactions may occur intermittently throughout the entire observation period (8 days); (3) strong interactions could consecutively persist for more than a day; (4) strong difference and sum interactions could occur simultaneously, as shown by the overlapping periods K and N, L and O, and M and P; and (5) energy-exchange wave pairs may change in the interaction triad.
4.2. Tide–PW Interactions
 We also can find some evidence for tide–PW interactions in data set D. In addition to the tide–GW interactions, Figure 2 (right) also exhibits strong peaks near the bifrequencies (0.0258, 0.0158) and (0.0416, 0.0158) cycles h−1 and distinct but not strong peaks near the bifrequencies (0.0992, 0.0258), and (0.125, 0.0158) cycles h−1. The former two peaks imply nonlinear interactions between the diurnal tide and PWs, while the latter two imply such interactions between the terdiurnal tide and PWs. Again, we only focus on interactions between the diurnal tide and PWs because they are much stronger than those between the terdiurnal tide and PWs.
 The bifrequencies (0.0258, 0.0158) and (0.0416, 0.0158) cycles h−1, which correspond to the difference interaction triplet (24, 63.3, 38.8) h and the sum interaction triplet (24, 63.3, 17.4) h, confirm the existence of a quadratic nonlinear coupling between the diurnal tide and the 2.6 day PW. The secondary waves produced by this tide–PW interaction should have periods of 17.4 and 38.8 h. Figure 1 (bottom) displays two such predicted frequency peaks with significant amplitudes.
 In addition to the direct energy exchange between tides and PWs and excitation of secondary waves, the tidal-amplitude variability at the periods of PWs (i.e., tidal modulation by PWs) is taken as another symptom of the coupling between tides and PWs [Canziani, 1994b; Beard et al., 1999]. Here we provide the periodograms of diurnal tidal amplitude (Figure 6) for data set D at a height of 293 km. We used the same method as before to harmonically fit the diurnal oscillations. Next, we carried out a Lomb–Scargle periodogram analysis of the time-dependent variations of the diurnal tidal amplitudes.Figure 6shows the 2.6-day PW modulations, which further confirms the occurrence of interactions between the diurnal tide and the 2.6-day PW.Figure 6also clearly shows the 7-day modulation. Additionally, inFigure 1two notable sidelobes of the diurnal tide, with periods of 28 and 21 h, exactly corresponding to the frequency-matching conditions of wave–wave interactions between the DT and 7-day wave, further confirmed the modulation of the DT by the 7-day wave. However, there is no appreciable 7-day oscillation inFigure 1, which may be due to the short temporal coverage of the observation. An alternative explanation is that the interaction between the DT and 7-day wave may occur below the altitudes of observation. Although the 7-day wave itself may not have propagated into the observational altitudes, the interaction excited wave components (i.e., two sidelobes of the DT) can propagate upward into the observational altitudes because of their higher frequencies and associated higher vertical propagation velocities. Similar results have been reported byBeard et al. .
 As mentioned in section 3, we also observe evident PW signatures in the other three data sets, (A, B, and C). For data sets B and C, there are no discernable peaks in their bispectra corresponding to PW periods, indicating that no discernable tide–PW interactions occurred. However, for data set A, we find a weak peak at (82.6, 34.0, 24) h in the bispectum at an altitude of 293 km (Figure 7) and a strong 82.6 h modulation of the diurnal tide (not shown). This verifies our interpretation of an interaction between the diurnal tide and the 3.4-day PW, although it might be weaker than those between the different tidal components discussed in the next section.
4.3. Tide–Tide Interactions
 We find strong peaks near the tidal frequencies in the bispectra of both data sets A and B. Because there are no strong peaks corresponding to tide–GW or tide–PW interactions in the bispectrum of data set A, verification of tide–tide interactions in this data set should be much easier. Therefore we performed a detailed analysis of data set A. Figure 7 exhibits strong peaks near the tidal frequencies (0.0833, 0.0416) and (0.125, 0.0416) cycles h−1. These bispectrum peaks cannot be simply regarded as a sufficient indicator of nonlinear interactions between 24/12 and 24/8 h tides because of their likely common forcing. However, the primary diurnal, semidiurnal, and terdiurnal tides in the middle thermosphere is believed to be produced by different mechanisms: the diurnal tide is believed to be mainly in situ generated and the semidiurnal one is mainly generated in the lower atmosphere, while the terdiural one is a result of a complex interaction between the semidiurnal tide and ion-neutral momentum coupling associated with the diurnal variation of the ion density. Since these tides do not potentially come from the common forcing, these bispectrum peaks around tidal frequencies are more likely resulted from tide–tide interactions.
 We will refer to the 24/12/8/6 h waves as DT/ST/TT/QT for simplicity. To explore the tides in detail, we harmonically fit the amplitudes of these tides and calculated the correlation coefficients of the detrended tidal amplitudes. Figure 8 shows the correlation coefficients among the interacting triads; i.e., (DT, ST, TT) in Figure 8 (top) and (DT, TT, QT) in Figure 8 (bottom), for data set A at an altitude of 293 km. If the first four tidal harmonics are all excited by the same forcing and no nonlinear interaction occurs, their amplitudes should exhibit exactly consistent variation trends. The correlation coefficients of the different tidal components would then all be positive. However, we found frequent negative correlation coefficients during the entire observation period. Similarly, during certain periods two correlation coefficients are often highly negative while the third is significantly positive. We label these periods Q–X. They also occur throughout the entire observation period. This indicates that interactions between tides could persist for more than 4 days, although they are highly intermittent. Moreover, strong interactions could consecutively persist for more than 10 h, as shown, for example, by periods R (11.5 h) and X (15.5 h).
 Note that the correlation coefficients exhibit rapid temporal variations. Large positive and negative values always supersede each other. Energy-exchange pairs change very frequently, indicating that tidal energy is constantly redistributed among the different tidal components. Compared with tide–GW energy exchanges, tide–tide exchanges seem much easier to reverse. Obviously, the frequent and significant variations of the correlation coefficients also verify the presence of nonlinear interactions among different tidal components.
 We conclude that in the middle thermosphere, in addition to interactions between the diurnal tide and GWs, interactions between tides and PWs or among different tidal components are found in our data, but generally they are not as frequent or as strong as those between the diurnal tide and GWs.
 In the thermospheric neutral wind derived from four Arecibo ISR data sets acquired between 1988 and 1993, we observe significant tides, GWs, and PWs in the altitude range of 256–367 km, indicating that the thermospheric atmosphere is usually perturbed by different kinds of atmospheric waves. Of these, the diurnal tide is almost always the strongest. Our bispectrum analysis indicates the occurrence of extensive interactions among different waves (tides, GWs, and PWs). Among these interactions, those between the diurnal tide and GWs are almost always the most prominent.
 We explored diurnal tidal–GW interactions by checking the relevance of the detrended instantaneous amplitude of the interacting-wave triplet. The three correlation coefficients between any two of the interacting waves are discussed in detail. We find that when the three waves are interacting intensely, two of the correlation coefficients are significantly negative, which may lead to a significant positive value of the remaining correlation coefficient. This phenomenon has not been revealed by previous observational studies. During the entire observation period of the four data sets adopted (the longest spans 8 days), strong correlations and anticorrelations are frequently present. Intense energy-exchange periods (when two significant negative correlation coefficients and one significant positive correlation coefficient coexist) could persist consecutively for more than a day. These occur throughout the entire observation period, which indicates that intense interactions could last for a very long time (although they may be highly intermittent).
 During the different interaction periods, the energy-transfer trend may change. This indicates that strong interactions can be reversible, although we also observed the situation that the energy transfer trend remains unchanged during the entire observation period. Moreover, note that our analysis reveals that strong difference and sum interactions between the diurnal tide and GWs usually take place simultaneously, which has rarely been reported to date. Generally, our study strongly suggests the presence of strong interactions between tides and GWs in the middle thermosphere.
 In addition to tide–GW interactions, tide–PW and tide–tide interactions are also observed in our data. During tide–PW interactions, the tidal oscillations are modulated at the interacting PW periods. Combining the bispectral and correlation analyses verifies the occurrence of nonlinear interactions among different tidal components in the middle thermosphere. Moreover, during tide–tide interactions, the energy-transfer trend changes very frequently, indicating that tidal energy is frequently redistributed among different tidal components.
 We thank the Arecibo Observatory staff for providing the observation data and the anonymous reviewers for their comments on the manuscript. The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST-1100968). This work was jointly supported by the National Basic Research Program of China (grant 2012CB825605), the National Natural Science Foundation of China (grants 41174126, 40825013, 41174127, 40974082, and 40890165), the National Science Foundation (grant AGS-1042223), the Specialized Research Fund for the Doctoral Program of Higher Education of China (grant 20100141110020), the Ocean Public Welfare Scientific Research Project of the State Oceanic Administration of the People's Republic of China (grant 201005017), a China Meteorological Administration Grant (GYHY201106011), the Open Programs of State Key Laboratory of Space Weather, and the Fundamental Research Funds for the Central Universities.