Long-term variation in the cloud-tracked zonal velocities at the cloud top of Venus deduced from Venus Express VMC images
Information Technology Research Institute, National Institute of Advanced Industrial Science and Technology, Ibaraki, Japan
Corresponding author: T. Kouyama, Information Technology Research Institute, National Institute of Advanced Industrial Science and Technology, 3-1-1 Umezono, 305-8568, Tsukuba, Ibaraki, Japan. (firstname.lastname@example.org)
 We present observational evidence of the variation of the cloud-tracked zonal velocity by ~20 m s–1 with a timescale of a few hundred days in the southern low latitude region based on an analysis of cloud images taken by the Venus Monitoring Camera on board Venus Express. A spectral analysis suggests that the variation has a periodicity with a period of about 255 days. Although cloud features are not always passive tracers, the periodical variation of the dynamical state is a robust feature. Superposed on this long-term variation of the zonal velocity, Kelvin wave-like disturbances tend to be observed in periods of relatively slow background velocity, while Rossby wave-like disturbances tend to be observed in periods of fast background velocity. Since the momentum deposition by these waves can accelerate and decelerate the mean flow, these waves may contribute to the suggested long-term oscillation.
 Periodical oscillations of the zonal wind speed with periods of the order of 1 year have been observed in planetary atmospheres. The terrestrial atmosphere exhibits the quasi-biennial oscillation (QBO) of the equatorial stratospheric wind with a period of 24–30 months [Andrews et al., 1987]. QBO is thought to be driven by the interaction of upwardly propagating eastward and westward waves, which are generated by convective systems, with the mean zonal flow [Plumb, 1977]. The Jovian atmosphere shows the quasi-quadrennial oscillation (QQO) of the equatorial wind speed [Leovy et al., 1991; Friedson, 1999]. QQO may also be caused by the interaction between waves and the mean zonal flow [Friedson, 1999; Li and Read, 2000]. Equatorial oscillations have also been suggested in Saturn atmosphere [Orton et al., 2008; Fouchet et al., 2008]. Such internal oscillations of the atmosphere may occur also on other planets with rotating stratified atmospheres and equatorial convection zones [Baldwin et al., 2001].
 Year-to-year variations of the zonal wind speed at the Venus’ cloud top have been reported. The cloud top level is considered to be 71–74 km altitude in latitudes lower than 50°, while it descends to ~65 km in the polar region [Kawabata et al., 1980; Titov et al. 2008; Ignatiev et al., 2009]. Cloud tracking using ultraviolet (UV) images taken by Pioneer Venus covering more than 7 years suggested that the zonal wind speed averaged over a few months decreased by about 8 m s–1 from 1979 to 1982 and then increased by about 10 m s–1 from 1982 to 1985 [Del Genio and Rossow, 1990]. However, these observations were conducted intermittently only once every 1 or 2 years, and thus the periodicity of the variation has been unclear. Another difficulty is that there are some inconsistency in the strength of the midlatitude jet between the cloud-tracked wind and the thermal wind calculated from the temperature data [Limaye, 1985]. Further observational clues are definitely needed to understand the variability.
 The Venus Monitoring Camera (VMC) on board ESA's Venus Express [Markiewicz et al., 2007] offers an opportunity to study long-term variations of Venus's cloud-top dynamics by providing cloud images almost continuously over 6 years since April 2006. In this study we investigate the variations of the cloud motion during the period from May 2006 to January 2010 by applying a newly developed cloud tracking method [Ogohara et al., 2012; Kouyama et al., 2012] to VMC UV images.
2 Data Set
 Venus Express follows an elongated polar orbit with the periapsis near the north pole, the orbital period of 24 h, and the apoapsis distance of 66,000 km. The UV imaging by VMC is conducted when the spacecraft is located around the periapsis and in the ascending (northwardly moving) portion of the orbit [Svedhem et al., 2009]. VMC took typically 10–100 UV images per orbit. For each orbit we chose a pair of images with a time interval of 1.0 ± 0.1 h taken when the subspacecraft latitude was 55°S–70°S; this criterion assures a wide coverage in the equatorial region. Since the orbital plane is nearly fixed in the inertial frame of reference, the local time of the observation changes periodically with a period of one Venus year; therefore, the data used in our study, chosen based on the criterion above, are not continuous. We call a cluster of observation days in which we could deduce cloud motion vectors regularly as an “epoch”, and we could use seven epochs in total as shown in Table 1.
Table 1. Definition of Epochs
Number of Orbits
20 May 2006 to 3 Jul. 2006
15 Nov. 2006 to 13 Feb. 2007
30 Jun. 2007 to 17 Sep. 2007
28 Jan. 2008 to 1 May 2008
6 Sep. 2008 to 16 Nov. 2008
6 Apr. 2009 to 22 Jul. 2009
30 Nov. 2009 to 11 Mar. 2010
 The images were projected onto a local solar time (LST)-latitude coordinate system located at 70 km altitude from the surface, and were interpolated into regular grids with intervals of 0.2° in both latitude and longitude to benefit from the VMC's highest spatial resolution in the adopted images (about 20 km at the subspacecraft point, corresponding to 0.2°). The typical resolution in the original images used is around 30 km.
3 Analysis Procedure
3.1 Cross-Correlation Method
 An automated cloud tracking method [Ogohara et al., 2012; Kouyama et al., 2012] was applied to these images to derive motion vectors. Both the template and the target area have a size of 6° in latitude and longitude, and the template is placed every 3° in both latitude and longitude to obtain cloud motion vectors at this spatial resolution. Generally, each cloud motion vector is obtained by maximizing the cross-correlation between a template in the first image and a target area in the second image. However, it tends to be difficult to avoid creating many erroneous vectors through this process. We have developed a cloud tracking algorithm that can modify erroneous cloud motion vectors to more plausible ones using a method proposed by Wu  and Evans . The algorithm was validated by Kouyama et al.  by comparing cloud motion vectors obtained from Galileo/SSI Venus images with those obtained by Belton et al.  and Peralta et al. [2007b] from the same data set. In this algorithm, we consider all vectors that correspond to the positions of nonnegative local maxima of the correlation surface as candidate vectors. Then we compare the candidate vectors for the focused position with those for the neighboring positions, and then the most plausible vector is chosen from the candidates assuming that the flow at any point has a direction similar to the flows at neighboring points.
 The adopted template size of 6° × 6° (~640 × 640 km at the equator) is partly based on previous observations that small-scale cloud features exist down to scales of several tens of kilometers [Murray et al., 1974; Belton et al., 1976; Toigo et al., 1994]. Having not a few contrast features in each template enables a unique identification of the displaced air parcel. The sizes of tracked features range from the spatial resolution (typically 30 km) to the template size (640 km). Since cloud patterns are not well spatially resolved near the limb and have less contrast near the terminator, we restricted the analysis to regions where both the solar incident angle and the emission angle are less than 71°.
 The cloud-tracked zonal velocity as a function of the latitude averaged on the illuminated side obtained by our method is shown in Figure 1 and compared with that of Moissl et al. , who analyzed images from 40 orbits covering from May 2006 to October 2007. We used the same orbits except for the orbit number of 29 and 530, in which the cloud images do not have enough dayside coverage. Our result is mostly in agreement with both the visual and the automated tracking results of Moissl et al.  in the region equatorward of ~50°S, while it is in agreement only with the automated tracking result at higher latitudes.
Moissl et al.  implied that visual tracking is more reliable than automated tracking: they argued that the cloud morphology and low contrast at latitudes higher than 40°S make pattern matching by their cross-correlation method difficult. We can conclude that the motion vectors obtained by our method are reliable at least in the region from the equator to 50°S based on this comparison. The rather large standard deviation in Ogohara et al.  as compared to this study is attributed to the use of nonoptimized parameters in the cloud tracking algorism in Ogohara et al.
 Previous studies showed that LST-dependent structures exist in the cloud-top wind field [Limaye, 1988; Del Genio and Rossow, 1990; Limaye, 2007; Sánchez-Lavega et al., 2008; Moissl et al., 2009]. Cloud motion vectors can be obtained only on the dayside and thus the zonal-mean zonal velocity cannot be obtained from the data. Therefore, we focus on the relative variation in a limited local time range from 12:00 to 14:00 to minimize the influence of the LST-dependent structure in this study. The use of this afternoon section maximizes the number of cloud motion vectors in the low latitude. Cloud-tracked velocities in this local time range were averaged at each latitude and then smoothed in latitude with a 9°-width running average.
 The standard deviation in the LST range of 12:00–14:00 for each latitude before averaging represents the sum of the contributions of the real spatial variation in this region and the random error of velocity determination. These two contributions lead to the error of the averaged velocity similarly, and thus they are treated together. The standard deviation is typically 7 m s–1 (corresponding to ~1.5 pixel displacement near the subspacecraft point) in the region equatorward of 45°S, and we expect that the error after averaging is a statistically times smaller value of ~2 m s–1 by virtue of the summation of 30 vectors (6° × 6° templates are placed every 3° in both latitude and longitude with an overlap of 50%, and thus there are 10 bins in LST and 3 bins in latitude). On the other hand, in the southern high latitude the low contrast of features makes cloud tracking difficult, and the error exceeds 10 m s–1 in the region poleward of 48°S; data in this region are not used in this study.
 The pointing error is estimated to be ~0.1 pixel in our analysis, in which the line of sight of VMC is determined from Venus's limb position in each image using a limb-fitting technique developed by Ogohara et al. . Since the projected Venus disk becomes ellipse on the CCD plane, an ellipse fitting method [Kanatani and Sugaya, 2007] is used for determining the center position of Venus. The influence of the pointing inaccuracy on the cloud-tracked velocity is estimated to be ~1 m s–1 around the subspacecraft point, provided that we use a pair of Venus images with the time interval of 1 h and the distance from Venus center of 40,000 km, which is typical of the images used in the analysis.
3.2 Sensitivity to the Template Size
 The sensitivity of the cloud-tracked velocity to the template size was examined for template sizes from 2° × 2° to 10° × 10°. An example using a pair of cloud images is shown in Figure 2, which compares the cloud-tracked zonal velocity averaged in a region 12–13 LST, 15°S–30°S for different template sizes. The result suggests that the cloud tracked velocity depends only weakly on the template size, although larger standard deviations are obtained for smaller template sizes. This tendency is common to other image pairs provided that the averaged correlation coefficient is greater than 0.6 for the template size of 2° × 2°.
 We have also repeated the whole analysis presented in this paper with template sizes of 4° × 4° and 8° × 8° (not shown here). These two cases yielded roughly the same result as the case with the template size of 6° × 6°. This is consistent with the conclusion of Kouyama et al. , who reported that the change of the template size from 3° × 3° to 6° × 6° did not alter the result significantly in the cloud tracking using Venus images taken during Galileo's Venus flyby.
3.3 Limitation of Cloud Tracking
 We consider that cloud features having spatial scales of several tens of kilometers or more are not always passive tracers. They can be meteorological phenomena that move at speeds different from the ambient wind. It is natural to assume that cloud-tracked velocities are affected by such processes because the typical image resolution we used is ~30 km. Candidates of such meteorological processes include upward penetration of convection and gravity waves that originate in the altitude region below the cloud top.
 Convection-like features have been observed at the cloud top [Belton et al., 1976; Rossow et al., 1980; Peralta et al., 2007a; Titov et al., 2012]. Although their origin is unclear, a possible mechanism is the upward penetration of convective motions that are thought to exist in the middle and lower part of the cloud [Baker et al., 1998]. If this is the case the phase speeds of convective patterns reflect the mean zonal wind speed at a certain level below in the cloud, which is somewhat slower than that at the cloud-top level.
 Gravity waves will also be generated below the cloud top. A possible source of gravity waves is convection in the middle and lower part of the cloud layer [Leroy and Ingersoll, 1996]. Although gravity waves generated by convection have various phase velocities, the mean phase velocity is expected to be similar to the background zonal wind speed at the level of excitation [e.g., Lane and Clark, 2002]. Winds blowing over topography will also excite gravity waves [Sagdeev et al., 1986; Young et al., 1987].
 Cloud features caused by the above-mentioned processes will, on average, tend to move slower than the true zonal wind at the cloud top level depending on the source altitude. For example, the mean zonal phase velocity of gravity waves generated at 55 km would be 10 m s–1 slower than the zonal wind velocity at the cloud top level (65–70 km) assuming the mean vertical shear of the zonal wind of 1 m s–1 km–1. Higher resolution images would allow discrimination of such propagating features from passive tracers, resulting in larger zonal velocities on average. The pixel resolution of ~30 km in our analysis does not allow strict discrimination. Implications of the results of cloud tracking under the influence of such processes are discussed in section 6.
4 Temporal Variation of Cloud-Tracked Velocities
 Figures 3 and 4 show temporal variations of the cloud-tracked zonal velocity and meridional velocity, respectively, at 18°S, 24°S, 30°S, 36°S, and 42°S. We also plotted velocities smoothed by a Gaussian filter with an e-folding full width of 24 d (d = Earth days) to emphasize long-term variations. This filtering removes 4–6 d period oscillations, which prevail in all epochs and are attributed to planetary-scale waves [Del Genio and Rossow, 1990; Rossow et al., 1990; Belton et al., 1991]. Figure 5 shows the periodical features of the zonal velocity in epoch 4 as an example. In addition to the gradual decrease, we see short-timescale fluctuations with peak-to-peak amplitudes of 20–30 m s–1 with timescales of 4–6 d. We can see similar short-timescale fluctuations in all epochs. Since the magnitudes of the short-timescale fluctuations are much larger than the errors and these fluctuations showed clear periodicity in each epoch (see section 6), we conclude that the short-timescale fluctuations seen in Figures 3 and 4 are mostly attributed to waves rather than measurement errors.
 In Figure 3, the smoothed zonal velocities at 18°S and 24°S decreased by about 20 m s–1 during epochs 4 and 5. They increased by about 20 m s–1 during the data gaps between epochs 3 and 4, between epochs 4 and 5, and between epochs 5 and 6. The result suggests that oscillatory variations with peak-to-peak amplitudes of about 20 m s–1 and timescales of a few hundred days exist. There also exist temporal variations in the smoothed meridional velocity with peak-to-peak amplitudes of up to 8 m s–1 (Figure 4), with apparently no correlation with the zonal velocity.
5 Spectral Analysis
 The periodicity of the zonal and meridional velocities is further studied using Lomb-Scargle periodogram [Lomb, 1976; Scargle, 1982]. The analysis allows us to obtain power spectra from unevenly sampled data with data gaps like our cloud-tracked velocities (see Appendix A). This method is equivalent to least-squares fitting of sine and cosine functions to the data [Scargle, 1982].
 Figure 6 shows the power spectra of the observed zonal and meridional velocities without smoothing at 18°S, together with the level of 99% statistical significance. In Lomb-Scargle method the period interval can be chosen arbitrarily, and is 0.1 d in this study to resolve short-period oscillations. The zonal velocity spectrum exhibits a prominent peak at 255 d with the half-width at half-maximum of ~20 d, being clearly separated from the 225 d period Venus year cycle. Peaks around 100–150 d are attributed to an interference from the 225 d cycle because (1/255 + 1/225)–1 ~ 120 (see Appendix B). The absence of signatures of the 4–6 d oscillations is attributed to the variation of the periods from epoch to epoch based on the result using test data (see Appendix B); this point will be discussed more in detail in section 6. Figure 7 shows the Lomb-Scargle periodogram of the zonal velocity as a function of the latitude. The long-term oscillation having a period of 250–260 d is predominant on the low-latitude side of ~40°S.
 The meridional component also shows a peak around the period of ~255 d; however, the peak is below the confidence level and cannot be treated as a real feature in this study. Although several other peaks exceed the confidence level, their magnitudes are marginal compared to the confidence level, and thus we also hesitate to conclude that these peaks are real.
 Lomb-Scargle periodogram also yields the amplitude and the phase of each spectral component (see Appendix A). The 255 d-period component was reconstructed using the retrieved amplitude of A = 11 m s–1 and the phase at t = 0 of φ0 = –51°, and is compared with the observed cloud-tracked velocities in Figure 8. A constant offset was given to the retrieved sinusoidal function so that the mean-square difference between the function and the data is minimized. The retrieved 255 d period oscillation roughly follows the development of the smoothed zonal velocity. Similar results were obtained also for 24°S, 30°S, 36°S, and 42°S and summarized in Table 2. Long-term variations of the zonal velocity with periods 250–260 d seem to occur widely in the low latitude. The correlation coefficient between the zonal velocity at 18°S and that at 24°S is 0.94, and the coefficient between 18°S and 30°S is 0.80, suggesting that the variation is phase coherent in the low latitude. The correlation coefficient decreases at higher latitudes.
Table 2. Parameters of the Most Significant Periodical Variation of the Zonal Wind Obtained by Lomb-Scargle Periodogram
Amplitude (m s–1)
Correlation With Winds at 18°S
6 Relations to Waves
 Variations of the zonal velocity and the cloud brightness with periods of 4–5 d were observed in the Pioneer Venus and Galileo missions and were attributed to equatorial Kelvin waves and Rossby waves [Del Genio and Rossow, 1982; Rossow et al., 1990; Del Genio and Rossow, 1990; Kouyama et al., 2012]. Venusian general circulation models involve various planetary-scale waves [e.g., Yamamoto and Takahashi, 2003; 2004; Lebonnois et al., 2010], and some of them may correspond to the observed cloud-top waves. Del Genio and Rossow  suggested that Kelvin waves provide mean-wind acceleration of the order of 0.1 m s–1 d–1 at the equatorial cloud top. Imamura  calculated, based on modeled vertical structures of the waves, the acceleration by Kelvin waves near the equator to be about 0.3 m s–1 d–1 and the deceleration by Rossby waves in the midlatitude to be 0.15 m s–1 d–1. The magnitudes of the acceleration and deceleration are consistent with the change in the mean zonal velocity of 20 m s–1 over 100 d suggested in this study.
 Although velocities obtained by cloud tracking do not necessarily represent ambient winds, in the following we tentatively assume that the cloud-tracked velocities represent ambient winds and discuss the dependence of the periods and the latitudinal structures of the observed 4–6 d period oscillations on the background wind speed. Figures 9 and 10 show the spectra of the zonal and meridional velocities as functions of the latitude. Here we applied Lomb-Scargle method independently for each epoch (3, 4, 5, 6, and 7). Because cloud-tracked velocities were not sampled regularly in time in each epoch, we adopted Lomb-Scargle periodogram rather than standard Fourier analysis. Epochs 1 and 2 were excluded from the analysis since the number of sampling is not enough in these epochs. Since periods of short period oscillations varied from epoch to epoch, 4–6 d period oscillations are not identified in the spectrum of the whole time series (Figure 3; see Appendix B for details). Also plotted in these figures are the periods corresponding to the dayside-mean zonal velocities averaged in each epoch as functions of the latitude. The mean zonal velocities are slower in epoch 3 than other four epochs, and thus we classified epoch 3 as a slow background wind period and other epochs as fast background wind periods.
 It is clear from Figures 9 and 10 that Kelvin wave-like oscillations prevail in slow background wind periods and Rossby wave-like oscillations prevail in fast background wind periods. In epoch 3, which is classified into periods of slow background wind, there exists a 4.2 d period zonal oscillation in the low-latitude with the zonal propagation faster than the background wind provided that the zonal wave number is 1, and there is no significant meridional oscillation. These characteristics are consistent with a Kelvin wave, which propagates faster than the background wind and yields fluctuations mainly in the zonal wind in low latitudes.
 On the other hand, in epoch 4, which is classified into periods of fast background wind, there exists a 4.8 d period zonal oscillation in the middle and low-latitude with the zonal propagation slower than the background wind, and a 4.8 d period meridional oscillation also exists in the midlatitude. These characteristics are consistent with a Rossby wave. In addition, we draw a vortex pattern (Figure 11) from spectral analysis of cloud-tracked velocities in the period from 660 to 710 days, where perturbations of the cloud-tracked velocity were relatively large. The phase difference between the zonal and meridional perturbations is about π/2, which is also consistent with a Rossby wave. Power spectra of epochs 6 and 7 (Figures 9 and 10), which are also considered as periods of fast background wind, gives results similar to that of epoch 4. Although epoch 5 is also considered as a fast background wind period, no prominent oscillation is identified both in the zonal and meridional velocities.
 Since Kelvin waves transport positive angular momentum from lower to higher altitudes and Rossby waves work in an opposite way, momentum deposition by these waves can cause acceleration and deceleration of the mean zonal flow at the cloud top level, similarly to the classical QBO theory. However, on Venus, how and where these waves are excited are unknown. Moreover, the latitudinal distributions of momentum deposition by Kelvin and Rossby waves are different from each other, with the former concentrated in the equatorial region and the latter in the midlatitude [Imamura, 2006]. It is also possible that waves that are not resolved in the present analysis, such as gravity waves, are driving the oscillation.
 Recently, Hueso et al.  examined the variability of the cloud-level wind field based on a cloud tracking analysis of Venus Express VIRTIS images. Their result from nightside 1.74 µm images suggests that there is no notable long-term variability near the cloud base at ~45 km altitude. Based on this result the long-term oscillation observed in our analysis will not have deep structures extending to the subcloud region, and furthermore, waves that contribute to the oscillation will be excited somewhere between the cloud base and the cloud-top height. This situation may correspond to the “cloud-level forcing” case of Covey and Schubert ; they studied the linear response of a model Venus atmosphere to external forcing over broad frequencies, and showed that a Kelvin-like wave and a Rossby-like wave similar to the observed ones are excited as a preferred mode for both cloud-level and surface-level forcing.
 An indication of a periodical long-term oscillation of the Venus's atmospheric dynamics was reported for the first time. An analysis of the cloud-tracked dayside velocities derived from Venus Express VMC images showed that the mean zonal velocity varied repeatedly by more than 20 m s–1 with a timescale of a few hundred days. Although cloud-tracking does not necessarily provide exact ambient wind velocities (section 3.3), the periodicity of the cloud-tracked velocity at least suggests some variation of the dynamical state of the atmosphere.
 Although the data set contains significant lengths of data gaps due to observational constraint, Lomb-Scargle periodogram analysis allowed us to estimate the period of this oscillation. The estimated period of about 255 d is longer than one Venus year, and thus the oscillation will not be a seasonal cycle, although more continuous data are needed to draw a definitive conclusion. Combined with the absence of long-term variation near the cloud base as suggested by Hueso, et al. , the observed long-term variation does not seem to have a deep structure extending to the cloud base.
 The absence of a notable periodical variation in the meridional component is attributed the weakness of the mean meridional circulation as compared to the zonal circulation. Although cloud tracking yields dayside-mean meridional velocities of up to ~10 m s–1, such velocities are mostly attributed to thermal tides [e.g., Newman and Leovy, 1974; Takagi and Matsuda, 2006]. The zonally-averaged meridional velocities will be on the order of 1 m s–1 or less according to a diagnostic analysis of the zonally-averaged temperature field [e.g., Imamura, 1997]. The periodically-varying component, which should be even weaker, would not be clearly seen in our analysis. Detailed studies on the variation in the meridional flow will be a future work.
 On the assumption that the cloud-tracked velocities represent ambient winds, roles of shorter-period oscillations seen in the data were discussed. A Kelvin wave-like perturbation was observed when the background wind was relatively slow, and a Rossby wave-like perturbation was observed when the background wind was relatively fast. Since the momentum deposition by these waves can accelerate and decelerate the mean flow, these waves will contribute, at least partly, to the periodical oscillation of the super-rotation.
Appendix A: Calculation of Amplitude and Phase Using Lomb-Scargle Periodgram
 Lomb-Scargle method calculates the spectral density P at an angular frequency ω from evenly or nonevenly spaced data points hi = h(ti), i = 1,…, N, as [Scargle, 1982]
where ti is time. Here τ is defined by the relation
 This procedure is equivalent to a linear least-squares fitting of the function
to the data, where a and b are obtained by
 Then, the amplitude A is obtained by
 Since (A.4) can be rewritten as
 Thus the phase at t = 0 is given by
Appendix B: Lomb-Scargle Periodgram for Data With Large Data Gaps
 Before applying Lomb-Scargle periodogram to the observed wind speeds, we investigated the characteristics of the method by applying it to test data, which include data gaps similar to those of VMC data. First we tested a combination of sinusoidal functions
as the test data, where t is time in d. The data set has seven “observation” epochs of 70 d length separated by data gaps of 154 d length (Figure A1); the 224 d Venus year cycle is the same as the VMC observation. Figure A1b shows the spectral density calculated with the Lomb-Scargle periodogram, exhibiting distinct peaks at 4.2 d and 260.8 d, and several erroneous peaks. The third peak at 121 d and the fourth peak at 1571 d are attributed to the interference from the 224 d observation cycle because the frequencies are close to (1/224 + 1/260)–1 d and (1/224 – 1/260) –1 d.
 Next we tested a combination of another set of sinusoidal functions
where i denotes the i-th “observation” epoch (i = 1, 2,…, 7) and the period Ti varies from epoch to epoch as T1 = 4.0, T2 = 4.2, T3 = 4.4, T4 = 4.6, T5 = 4.8, T6 = 5.0 and T7 = 5.2 d (Figure A1c). In this case the spectral peaks of the short period oscillations are much suppressed (Figure A1) as compared to the constant-period case (Figure A1). The result suggests that short-period oscillations are difficult to identify in the spectrum of the whole time series when their periods vary even slightly from epoch to epoch.
 This study has been done using the Venus Express/VMC data distributed from ESA. The authors greatly appreciate the open data policy of the project. The authors also thank the anonymous reviewers for providing valuable comments on the draft of the paper.