Formation and Quasi‐Periodic Variation of Equatorial Jet Caused by Planetary‐Scale Waves in the Venusian Lower Cloud Layer

The equatorial jet in the Venusian lower cloud layer (47–55 km altitudes) and its quasi‐periodic variation are found in a general circulation model (GCM). The equatorial jet is produced by the 5.8‐day wave and destroyed by the 7‐day wave, and its quasi‐periodic variation with a timescale of about 280 days is caused by the alternating development of these waves in the GCM. The 5.8‐day wave, which is excited by the Rossby‐Kelvin instability, produces the equatorward angular momentum (AM) flux, and accelerates the zonal‐mean zonal wind in the equatorial region. The 7‐day wave, newly found in the present study, is a planetary‐scale wave antisymmetric about the equator, although it has not yet been observed. It is excited by the coupling among the lower‐altitude equatorial Rossby mode, the mid‐latitude Rossby mode, and the high‐latitude Rossby mode. In the growth period, the 7‐day wave produces the poleward (equatorward) AM flux around the equatorward (poleward) critical latitude. As a result, the zonal‐mean zonal wind is decelerated (accelerated) in the equatorial region and high latitudes (mid‐latitudes). In the regrowth period, the lower‐altitude equatorial Rossby mode disappears due to the disappearance of the equatorial jet, but the high‐latitude Rossby mode is still enhanced by the coupling with the mid‐latitude Rossby mode. These results could provide a possible explanation of the equatorial jet inferred from the recent Akatsuki observation, although it should be investigated by further observations.


10.1029/2023JE007922
2 of 23 equatorward of 40°-45°, and decreases with latitude toward the poles.It has been also shown that the zonal winds fluctuate by about 20 m s −1 with a period of 255 days (Kouyama et al., 2013).On the other hand, information on the atmospheric motions in the lower cloud layer has been derived from infrared (IR) observations, but quite limited so far (e.g., Belton et al., 1991;Carlson et al., 1991;Crisp et al., 1991;Hueso et al., 2015;Limaye et al., 2018;Sánchez-Lavega et al., 2008).The zonal wind velocity around 50-60 km altitudes is also almost constant with latitude at about 60-70 m s −1 in latitudes equatorward of 50°, and decreases with latitude toward the poles.Recently, observations by the infrared camera in the 2-μm region (IR2) onboard Akatsuki showed a possibility that a significant equatorial jet could exist in the lower cloud layer at 47-57 km (Horinouchi et al., 2017), which had not been observed in the earlier observations.Note that the equatorial jet is a narrow band of zonal winds with a maximum rotational speed (i.e., a maximum angular velocity) near the equator.In the Akatsuki IR2 observation, the equatorial jet was detected in July and August 2016, but was not in March 2016.It is inferred from these results that the equatorial jet could vary significantly in time, repeatedly appearing and disappearing.Although the producing mechanism of the equatorial jet is not explained, they pointed out that the meridional and/or vertical angular momentum (AM) transport by atmospheric waves should be important for its formation.
At the cloud top level, it has been suggested that planetary-scale waves with periods of 4 and 5 days (referred to as the 4-day and 5-day waves) exist (e.g., Del Genio & Rossow, 1990).Since the 4-day (5-day) wave propagates faster (slower) than the SR, it is called Kelvin (Rossby) wave.The horizontal wind distribution associated with the 5-day wave is dominated by mid-latitude vortices symmetric about the equator with a zonal wavenumber of 1 (Imai et al., 2019;Kouyama et al., 2013), while that with the 4-day wave has not yet been obtained.The horizontal wind and temperature distributions associated with the thermal tide has been also derived at the cloud top level (Akiba et al., 2021;Horinouchi et al., 2020;Kouyama et al., 2019).Horinouchi et al. (2020) showed that the thermal tide produces the equatorward AM flux in low latitudes, and it could contribute to the maintenance of SR.They also pointed out that the SR in low latitudes is hardly affected by the 5-day wave, at least, at the cloud top level, because the meridional AM flux produced by the 5-day wave is almost negligible.It is noted, however, that the vertical structures of these waves have not been clarified by the previous observations.It has been inferred from time variations of the cloud opacity in low latitudes that there exist Kelvin-like waves in the lower cloud layer (Carlson et al., 1991;Crisp et al., 1991;Peralta et al., 2020).Recently, observations by the Longwave Infrared (LIR) camera onboard Akatsuki showed that multiple planetary-scale waves with periods of 3.6, 5.0, 5.4, and 6.1 days exist simultaneously around 65 km altitude (Kajiwara et al., 2021).They suggested that the 3.6-day wave is a Kelvin wave and the other waves are Rossby waves symmetric about the equator; however, the wind distributions associated with these waves were not derived.Thus, it remains largely unknown what kind of waves exist in the cloud layer.To understand the atmospheric circulation on Venus, it is crucially important to investigate the zonal wind distribution and its time variation, and the waves and how they affect the zonal-mean zonal wind by their AM and heat transports in the cloud layer.
The equatorial jet has been reproduced in several numerical studies using Venus general circulation models (GCMs).However, the altitudes at which the equatorial jet is reproduced and the mechanisms by which it is generated are not consistent among them.Takagi and Matsuda (2007) showed that the SR could be generated and maintained by the thermal tide mechanism (Fels & Lindzen, 1974;Plumb, 1975) in their model.The reproduced SR is dominated by the equatorial jet with a peak at 65 km altitude in the upper cloud layer, which is higher than that inferred from the Akatsuki IR2 observation.This equatorial jet is formed by the vertical AM transport due to the thermal tide, which is in a quasi-steady state, and hardly fluctuates in time.Note that the atmospheric motions are driven by the solar heating without a zonal-mean component in their model, that is, the mean meridional circulation is not taken into account.M. Yamamoto and Takahashi (2003b) reproduced the SR using a GCM driven by the solar heating including the zonal-mean component only.In their model, the SR is generated and maintained by the meridional circulation mechanism (Gierasch, 1975;Matsuda, 1980).Their result showed that the SR is the fastest at 56-58 km altitudes, and the equatorial jet and the mid-latitude jets coexist around these altitudes, although the thermal tide is excluded in their model.They did not investigate the generating mechanism of the equatorial jet, but it might be related to Kelvin-like waves with periods of 3-5 days in their model, which were generated at 40 km altitude, propagated upward, and absorbed at the critical level near 60 km, at which the zonal-mean zonal wind velocity equals to the zonal phase velocity of the wave.It should be noted, however, that some studies using similar GCMs showed different results in which the equatorial jet was not reproduced (Lee & Richardson, 2010;Lee et al., 2007;M. Yamamoto & Takahashi, 2003a, 2004).Lebonnois et al. (2010) reproduced the SR using a GCM with a realistic radiative transfer model.In their result, the equatorial jet with 10.1029/2023JE007922 3 of 23 a peak around 65 km altitude appears, which is similar to that obtained by Takagi and Matsuda (2007).They also showed that the SR and the equatorial jet disappear if the diurnal component of the solar heating is turned off during the numerical integration.This result suggests that both the SR and the equatorial jet are significantly contributed by the thermal tide.Similar results have been also obtained in their more recent works (Garate-Lopez & Lebonnois, 2018;Lebonnois et al., 2016).However, those results differ from their previous one in that the mid-latitude jets, which are thought to be formed by the mean meridional circulation, coexist with the equatorial jet.It should be also noted that some studies did not reproduce the equatorial jet despite the use of GCMs with realistic radiative transfer models (e.g., Mendonça & Read, 2016;M. Yamamoto et al., 2021).The structure of SR is also quite different in these studies; that is, the SR in the cloud layer is close to a solid body rotation in the work of Mendonça and Read (2016), while it is accompanied by significant mid-latitude jets in that of M. Yamamoto et al. (2021).As briefly summarized above, the equatorial jet in the lower cloud layer has not been fully reproduced in the previous GCM studies, although it was seen in the result of M. Yamamoto and Takahashi (2003b).Its generating and time variation mechanisms have not been elucidated so far.
Recently, using a GCM, Takagi et al. (2022) found planetary-scale waves in the cloud layer with a zonal wavenumber of 1.Since these waves have periods of 3.3, 4.8, and 5.8 days, they are called 3.3-day, 4.8-day, and 5.8-day waves.The 3.3-day and 5.8-day waves respectively correspond to the 4-day and 5-day waves observed at the cloud top (Del Genio & Rossow, 1990;Imai et al., 2019;Kouyama et al., 2012Kouyama et al., , 2013)).These waves are excited by the Rossby-Kelvin instability (Iga & Matsuda, 2005), in which an equatorial Kelvin mode interacts with Rossby modes in mid-and/or high latitudes across a critical latitude, at which the zonal phase velocity of the wave equals to the zonal-mean zonal wind velocity.One of the most interesting characteristics of these waves is that they produce the equatorward AM transport and accelerate the zonal-mean zonal wind in the equatorial region.Takagi et al. (2022) also showed that the weak equatorial jet appeared in altitudes of 40-50 km in their result.They pointed out that it could be generated by the equatorward AM transport due to the 5.8-day wave, because the equatorial Kelvin mode of the 5.8-day wave exists in altitudes of 40-52 km in low latitudes, and the zonal AM is accumulated in this region from mid-and high latitudes during the growth of 5.8-day wave.It is noted, however, that the generating mechanism and time variation of the equatorial jet have not been investigated in their study.In addition, the peak of the equatorial jet was located around 42 km altitude in their result.This altitude could be far below the observed one, since the observed equatorial jet should exist in the lower cloud layer (47-57 km).
In the present study, we extend the work of Takagi et al. (2022) to show that the strong equatorial jet in the lower cloud layer (47-55 km) is produced by the 5.8-day wave and destroyed by the 7-day wave, and the quasi-periodic variation of the equatorial jet is caused by the alternating development of these waves in our GCM.Since this equatorial jet significantly varies in time, it is not clearly seen in the field averaged over long periods of time.The equatorial jet found in the GCM could be consistent with that inferred from the recent Akatsuki IR2 observation (Horinouchi et al., 2017), although its existence should be confirmed by further observations as discussed in Section 4. The 7-day wave is a new type of planetary-scale wave antisymmetric about the equator with a period of about 7 days, although it has not yet been observed.We also investigate the three-dimensional structures of the 7-day wave in detail and discuss its excitation mechanism.

Model
The Venus GCM used in this study, AFES-Venus (Sugimoto et al., 2014a;Suzuki et al., 2022), is based on an Earth GCM, AGCM for the Earth Simulator (AFES) (Enomoto et al., 2008;Ohfuchi et al., 2004), which is a highly reliable GCM widely used in the Earth meteorology.The AFES-Venus has been successful in reproducing various meteorological phenomena on Venus, such as baroclinic waves in the cloud layer (Sugimoto et al., 2014b), the cold collar in the polar region (Ando et al., 2016(Ando et al., , 2017)), the thermal tide (Suzuki et al., 2022;Takagi et al., 2018), the streak structure in the lower cloud layer (Kashimura et al., 2019), the three-dimensional cloud distribution (Ando et al., 2021;Ando, Takagi, et al., 2020), the gravity waves generated by the thermal tide (Sugimoto et al., 2021), and the thermal structure in the polar region (Ando et al., 2021(Ando et al., , 2022)).The experimental conditions, which are the same as those used in the work of Takagi et al. (2022), are briefly described below.The horizontal resolution is a T42 spectral truncation, corresponding to 128 × 64 grid points in the zonal and meridional directions, respectively.The vertical domain of 0-120 km altitudes is divided into 120 layers by a constant thickness of 1 km.The vertical coordinate is σ, which is defined by p/p s , where p is pressure, and p s is surface pressure.The surface topography is neglected.Rayleigh friction with a relaxation time of 10 days is applied to the lowest layer only to represent the surface friction.The horizontal eddy diffusion is represented by second-order hyperviscosity with a relaxation time of about 1 day for the largest wavenumber components.See the work of Sugimoto et al. (2023) for a discussion of how the model depends on the horizontal hyperviscosity.The vertical eddy diffusion is also used in the model, whose coefficient is fixed to be 0.0015  m 2 s −1 (Sugimoto et al., 2019).
Convective adjustment is used to represent vertical heat transport by convection.The solar heating is prescribed based on the work of Haus et al. (2015), but it is neglected above 90 km altitude for numerical stability in the uppermost layers.The infrared radiative transfer process is simplified by a Newtonian cooling scheme: where κ(z) is the inverse of a relaxation time based on the work of Crisp (1989).T 0 (z) is a horizontally uniform reference temperature profile based on the Venus international reference atmosphere (VIRA) data (Seiff et al., 1985) and recent radio occultation observations (Ando, Imamura, et al., 2020;Tellmann et al., 2009).Note that the planetary rotation is assumed to be eastward, opposite to that of Venus; therefore, the SR is also eastward (positive) in the present model.See also the works of Takagi et al. (2022) and Suzuki et al. (2022) for further details of the model.
The initial state for the time integration is an idealized super-rotating one.The zonal wind is in a zonally uniform solid body rotation, whose velocity increases linearly with altitude and reaches 100 m s −1 at 70 km on the equator.Above 70 km, the angular velocity of the initial zonal wind is assumed to be constant.The initial temperature is in gradient wind balance with the initial zonal wind.The model atmosphere is integrated for 51 Earth years, and reaches a quasi-steady state even in low altitudes below the cloud layer.We analyze the data obtained in the last 585 days (5 Venus days).

Zonal-Mean Fields
Figure 1a shows a latitude-altitude distribution of the time-averaged (over 585 days) zonal-mean zonal wind (black contour) maintained in the quasi-equilibrium state.At 70 km altitude near the cloud top, the zonal-mean zonal wind velocity is almost constant at about 120 m s −1 in latitudes equatorward of 30°, and the weak mid-latitude jets with a peak velocity of 130 m s −1 are formed in latitudes of 30°-50°.These features agree well with the recent observations (e.g., Gonçalves et al., 2020;Horinouchi et al., 2018;Machado et al., 2017Machado et al., , 2021)).The rotation period of the zonal-mean zonal wind around the planet (blue contour) decreases with latitude around 70 km: it is about 4 days at the equator, 3 days at 30°N/S, and 2 days at 60°N/S.On the other hand, in 40-52 km altitudes, the rotation period of the zonal-mean zonal wind near the equator is shorter than that in mid-latitudes around 30°.In these altitudes, the rotation period of the zonal-mean zonal wind is the longest (about 7.5 days) in latitudes of about 15°-40°, while it decreases with latitudes in latitudes poleward of 40°, as in other altitudes.This fact means that a time-averaged equatorial jet is formed in 40-52 km altitudes.Note, however, that the equatorial jet inferred from the Akatsuki IR2 observation could exist in altitudes of 47-57 km (Horinouchi et al., 2017), which might be above the altitudes in which the time-averaged equatorial jet is formed in the present result.As pointed out by Takagi et al. (2022), this equatorial jet could be produced by the 5.8-day wave excited by the Rossby-Kelvin instability (Iga & Matsuda, 2005).The equatorward AM flux produced by the 5.8-day wave accelerates the zonalmean zonal wind in 42-53 km altitude in latitudes equatorward of 15° to which the Kelvin mode of 5.8-day wave is confined (see Figures 7, 8a, and 10d in Takagi et al. (2022)).This acceleration region coincides well with the region where the time-averaged equatorial jet is formed.
Figure 1a also shows a latitude-altitude distribution of the root-mean-square (RMS) of the zonal-mean zonal wind from its time average,  ū , which is calculated as follows: where (3) The result shows that the zonal-mean zonal wind below the cloud top level fluctuates in time mainly in the following regions: (a) an equatorial region in altitudes of 45-60 km in and above the time-averaged equatorial jet, (b) mid-latitude regions in latitudes of 30°-60° and altitudes of 60-73 km below the mid-latitude jets, and (c) high-latitude regions in latitudes of 60°-90° and altitudes of 45-70 km.
Figure 1b shows time variations of the zonal-mean zonal wind in altitudes of 30-80 km on the equator.The zonal-mean zonal wind fluctuates quasi-periodically with a period of about 280 days in the lower cloud layer (45-55 km altitudes).The peak magnitude of the fluctuation is about ±8 m s −1 at 53 km.These results indicate that the strong time-varying equatorial jet appears in the lower cloud layer above the time-averaged equatorial jet (see also Figure 6), which could be consistent with the Akatsuki IR2 observation (Horinouchi et al., 2017).Similar fluctuations of the zonal-mean zonal wind can be seen above 60 km, whose period is almost the same as that in the lower cloud layer, although its amplitude is not so large in altitudes of 60-70 km.It is interesting to point out that the phase of fluctuation in the upper cloud layer (60-70 km) propagates upward if it is observed in a reference frame fixed on the ground.As mentioned in Section 1, Kouyama et al. (2013) showed that the zonal wind velocity at the cloud top fluctuates by about 20 m s −1 with a period of 255 days in the southern low-latitude region.To explain the observed fluctuations, Kouyama et al. (2015) proposed a dynamical mechanism similar to that of the quasi-biennial oscillation (QBO) in the Earth's stratosphere.In their mechanism, the zonal-mean zonal wind near the cloud top is accelerated (decelerated) by the equatorial Kelvin wave (the mid-latitude Rossby wave) which is artificially forced and propagates upward from below when the zonal-mean zonal wind is relatively slow (fast) in their model.The present result seems to be qualitatively consistent with the observations, although the observed amplitude is larger than that obtained in the present model.We should investigate how the zonal-mean zonal wind fluctuates in the upper cloud layer and examine the validity of the mechanism proposed by Kouyama et al. (2015) in the future.For other details of the zonal-mean field, see also the work of Takagi et al. (2022).

Frequency Analysis
Several planetary-scale waves have been found by Takagi et al. (2022) in the present GCM simulation.Among those waves, they investigated 3.3-day, 4.8-day, and 5.8-day, and showed that the 3.3-day and 5.8-day waves corresponds to the Kelvin (4-day) and Rossby (5-day) waves observed at the cloud top, respectively, as mentioned in Section 1.They also pointed out that another wave with periods of 6.5-7.6 days (hereafter referred to as the 7-day wave) is identified both in the zonal and meridional winds at a wide range of latitude except in the zonal wind on the equator, although it may not have been found in previous observations.Because the amplitude of the 7-day wave is equal to or larger than that of the 5.8-day wave, it could play an important role on dynamics of the Venusian atmosphere, at least in the present model.In the following analysis, we investigate the structure of 7-day wave and show that the 7-day wave produces the poleward (equatorward) AM momentum transport in low (high) latitudes, and the zonal-mean zonal wind is decelerated in low and high latitudes and accelerated in mid-latitudes in the lower cloud layer.
As in the work of Takagi et al. (2022), we first analyze the meridional and vertical distributions of frequency spectrum of the meridional wind field; note that the meridional distribution was not shown by Takagi et al. (2022).The model outputs are sampled at every 6 hr at each grid point on the 0°E longitude.The fast Fourier transform is applied to the time series of meridional wind data over 585 days.Figure 2a shows amplitude of the Fourier component of the meridional wind as a function of period and latitude obtained at 53.5 km altitude.It is confirmed that the 7-day wave has a clear signal in the meridional wind over a wide range of latitude, including latitudes equatorward of 30°.This result suggests that the 7-day wave has a meridional structure antisymmetric about the equator, contrasting with the fact that the 3.3-day, 4.8-day, and 5.8-day waves have meridional structures symmetric about the equator, and consequently they have no signal in the meridional wind on the equator (see also Figure 4b in Takagi et al. (2022)).The amplitude of meridional wind associated with the 7-day wave is about 10 m s −1 in high latitudes at 53.5 km altitude, which is larger than those with the 4.8-day and 5.8-day waves.As the 7-day wave has a zonal wavenumber-1 structure as shown later, its zonal phase velocity is slower than the zonal-mean zonal wind at all latitudes at 53.5 km altitude.Note that the signal of 7-day wave spreads over a period range of 6.7-7.2 days in low latitudes and that of 6.7-7.6 days in high latitudes.As discussed in Section 4, the spread of periodicity of the 7-day wave could be closely related to its excitation mechanism.
Figure 2b shows amplitude of the Fourier component of the meridional wind as a function of period and altitude obtained at 75°N latitude.The 7-day wave has large amplitude in altitudes of 40-75 km.The zonal phase velocity of 7-day wave is slower than the zonal-mean zonal wind at 75°N, that is, the 7-day wave propagates westward in a reference frame moving with the zonal-mean zonal wind at this latitude, as the SR is assumed to be eastward in the present model.

Structure of 7-Day Wave
To remove waves and/or disturbances other than the 7-day wave, we first apply a bandpass filter (Duchon, 1979) with a period range between 6.5 and 8.0 days.This period range is slightly broader than that of the 7-day wave.However, signals in a period range of 7.6-8.0days are so weak that the following results are hardly affected.
Figure 3 shows a latitude-altitude distribution of the zonally and time-averaged (over days 117-351) kinetic energy of horizontal winds (KE),  0 , where ρ 0 is the basic atmospheric density, u′ and v′ are the zonal and meridional wind deviations obtained by the bandpass filter, respectively.The KE is relatively large in the following four regions: an upper equatorial region (denoted by the red rectangle in Figure 3), an lower equatorial region (the black rectangle), northern mid-and high-latitude region in 45-65 km altitudes (the blue rectangle), and southern mid-and high-latitude region in 45-65 km altitudes (the violet rectangle).This distinctive distribution implies that the 7-day wave consists of several modes in low latitudes and mid-and high latitudes.
We next perform composite analysis in order to investigate the three-dimensional structure of 7-day wave.As shown in detail later (Section 3.4 and Figure 6a), time variations of the KE obtained in the four regions suggest that the time evolution of the 7-day wave can be devided into three periods: growth, regrowth, and decay periods.Each part of the wave grows coherently in the growth period.Therefore, to obtain a clear structure, the bandpass-filtered data are averaged over the growth period of 7-day wave (days 117-195) after shifting them zonally in such a way that the maximum of meridional wind at the equator and 53.5 km altitude is located at a fixed longitude of 0°E.Note that the almost same result is obtained if we use the meridional wind at 70°N or 70°S latitude for the composite analysis.
Figure 4a shows a horizontal structure of the 7-day wave obtained at 48 km altitude.The 7-day wave is dominated by a zonal wavenumber-1 component, and the meridional distribution of horizontal winds is antisymmetric about the equator, as inferred from Figure 2a.Interestingly, there exist four critical latitudes at an altitude between 41 and 51 km (represented by the green lines), at which the zonal-mean zonal wind velocity equals to the zonal phase velocity of 7-day wave, and they are located at about 10°N/S and 38°N/S at 48 km altitude (Figure 4a).This is because the critical surface, on which the zonal-mean zonal wind velocity equals to the zonal phase velocity of the wave, is depressed by about 10 km in the equatorial region due to the equatorial jet (Figures 3 and 5).The critical latitude on the lower (higher) latitude side is hereinafter referred to as the equatorward (poleward) critical latitude.
In mid-and high latitudes poleward of the poleward critical latitudes (about 38°), geostrophic vortices, in which the horizontal winds are almost parallel to the contour lines of the geopotential height deviations, are predominant (Figure 4a).Since the zonal phase velocity of 7-day wave is slower than the zonal-mean zonal wind velocity in this region, that is, the 7-day wave propagates westward in a reference frame moving with the zonal-mean zonal wind, we call these vortices the high-latitude Rossby mode.In subtropic latitudes between the equatorward critical latitude (about 10°) and the poleward one (about 38°), the horizontal winds are also almost in the geostrophic balance with the geopotential height deviations.As discussed later in Section 4.2.1, the quasi-geostrophic motion in the subtropic latitudes between the equatorward and poleward critical latitudes is associated with the mid-latitude Rossby mode.Note that the 7-day wave propagates eastward relative to the zonal-mean zonal wind in this region.In the equatorial region between the equatorward critical latitudes (10°S-10°N), the meridional winds with the maximum amplitude at the equator are predominant, although the amplitude is not so large (about 0.3 m s −1 ).As shown clearly in Figure 4b, these equatorial meridional winds are associated with vortices centered at the equator, which are confined to latitudes between the equatorward critical latitudes.The structure of these vortices is similar to that of the equatorial Rossby wave with n = 2 or the mixed Rossby-gravity wave with n = 0 (Matsuno, 1966), where n represents a meridional index of the equatorial waves, corresponding to the number of zeros of meridional wind associated with the mode.This is also consistent with the fact that the 7-day wave propagates westward relative to the zonal-mean zonal wind in this region, since the equatorial Rossby wave and the mixed Rossby-gravity wave propagate westward.Considering the meridional wind velocity associated with the equatorial vortices vanishes near the equatorward critical latitude, the equatorial vortices are thought to be associated with the equatorial Rossby wave with n = 2 rather than the mixed Rossby-gravity wave with n = 0. Therefore, we call these vortices the lower equatorial Rossby mode.Note that "lower" is added to distinguish it from the upper equatorial Rossby mode which exists above 52 km, as shown later (see also Figures 4c-4e).Near the poleward critical latitude at 38°N (38°S), the phase of geopotential height deviation is tilted westward (eastward), and the equatorward AM flux is produced by a negative (positive) correlation between the zonal and meridional winds across the poleward critical latitude (see also Figure 5a).On the other hand, the poleward AM flux is produced by a positive (negative) correlation between the zonal and meridional winds across the equatorward critical latitude at 10°N (10°S), although the correlation is rather weak near the equatorward critical latitude.
The AM is thus transported from the equatorial region and the high latitudes into the subtropic region between the equatorward and poleward critical latitudes.This means that the zonal-mean zonal wind is decelerated in latitudes equatorward of 10° and latitudes poleward of 38°, and accelerated in latitudes of 10°-38° (see also Figure 5a).As a result, the meridional shear measured by angular velocity of the zonal-mean zonal wind is decreased, and the zonal-mean zonal wind gets close to a solid body rotation, as shown later (Figures 6 and 7).Note also that the equatorward critical latitudes in both hemispheres vanish when the 7-day wave has finished growing, and the equatorial structure below about 52 km altitude significantly changes in time, as shown later in Figures 9 and 10a.
Figure 4c shows a horizontal structure of the 7-day wave obtained at 54 km altitude.There is no critical latitude at this level, since the zonal phase velocity of 7-day wave is slower than the zonal-mean zonal wind above the critical surface (Figure 5).The geostrophic vortices of the high-latitude Rossby mode are predominant in latitudes poleward of 30°, as at 48 km altitude, but the phase tilt seen in latitudes of 10°-45° at 48 km has disappeared.The equatorial region between 30°S and 30°N is dominated by vortical motions centered at the equator, which are similar to those observed at 48 km but much larger and stronger.The time-averaged maximum meridional wind velocity on the equator is about 0.3 m s −1 at 48 km and 1.5 m s −1 at 54 km (Figures 4a and 4c).We call these equatorial vortices the upper equatorial Rossby mode; note that "upper" is added to distinguish it from the lower one, as mentioned above.The geopotential height deviations antisymmetric about the equator are isolated in latitudes equatorward of 30°.As indicated by the horizontal wind vectors near 30° latitude that are tilted eastward (westward) in the northern (southern) hemisphere, the AM is transported poleward from low latitudes by the correlation between the zonal and meridional winds, suggesting that the zonal-mean zonal wind is decelerated in the equatorial region (see also Figures 5a and 5d).This upper equatorial Rossby mode extends poleward with altitude, but the local maximum of meridional wind at the equator disappears around 60 km altitude (Figure 4d).
As shown in Figure 4e, the meridional winds associated with the lower equatorial Rossby mode are confined to altitudes between the critical level (about 41 km) and about 50 km, whose phase tilts eastward with altitude.On the other hand, the upper equatorial Rossby mode extends above 52-80 km or higher, and its phase tilts westward with altitude.The KE of the lower and upper equatorial Rossby modes is separated vertically at about 52 km (Figure 3).These results suggest that the lower and upper equatorial Rossby modes are not directly connected.Since the zonal phase velocity of 7-day wave is slower than the zonal-mean zonal wind above the critical level located at 41 km in the equatorial region, it is also suggested that the vertical phase (group) velocity of the lower equatorial Rossby mode is positive (negative), and that of the upper equatorial Rossby mode is negative (positive), if it is observed in a reference frame moving with the zonal-mean zonal wind (e.g., Andrews et al., 1987).
Figure 4f shows that the vertical structure of the geostrophic vortices associated with the high-latitude Rossby mode is almost barotropic except below about 45 km altitude.At about 62 km, the meridional wind has the maximum amplitude of about 10 m s −1 , and the temperature deviation changes its sign.Below 45 km, the phase tilts eastward with increasing altitude, and a negative correlation occurs between the meridional wind and the temperature deviation, producing the equatorward heat transport (see also Figure 5c).
Next, we investigate how the 7-day wave affects the zonal-mean zonal wind using the transformed Eulerian-mean (TEM) equation in log-pressure coordinates (Andrews et al., 1987): where    is the zonal-mean zonal wind velocity,    * and  w * are the zonalmean meridional and vertical wind velocities of the residual mean meridional circulation, respectively,  is latitude, z* is log-pressure height, a is the planet radius, f is the Coriolis parameter, ρ 0 is the basic atmospheric density, and  X a zonal-mean zonal component of friction and/or viscosity.
) is the Eliassen-Palm (EP) flux defined as follows: where v′ and w′ are the meridional and vertical wind deviations, respectively, θ′ is the potential temperature devi- where T is the temperature, p is the pressure, p 00 is a reference pressure of 1,000 hPa, R is a gas constant of 191.4  J K −1 kg −1 for the CO 2 atmosphere, and C p is the specific heat.Note that C p is fixed at a constant value of 1,000 for simplicity in the present study, although it depends on temperature in the real Venusian atmosphere.Note also that, since the AM flux associated with waves is defined by ) in the present study, the signs (directions) of the AM and EP fluxes are opposite to each other if the meridional heat flux,  0 ′  ′ , is neglected.
Figure 5 shows the meridional and vertical AM fluxes, the meridional heat flux, and the EP flux produced by the 7-day wave, averaged over days 117-234, which correspond to the period between the beginning of the growth phase and the beginning of the regrowth phase of the 7-day wave (see the red and black lines in Figure 6a).In altitudes of 45-50 km, where the lower equatorial Rossby mode exists, the poleward (equatorward) AM flux is produced by the 7-day wave across the equatorward (poleward) critical latitude located at about 10° (38°) (Figure 5a), as suggested from Figure 4a.The poleward AM flux is confined to narrow latitudes near the equatorward critical latitude, while the equatorward AM flux extends over latitudes of 25°-80°.In altitudes of 52-64 km, where the upper equatorial Rossby mode and the high-latitude Rossby mode are predominant, the poleward AM flux is produced from the equator to 60° latitude.The weak poleward AM flux is also produced below 45 km in latitudes poleward of the poleward critical latitude.It is noteworthy that the direction of the meridional AM flux produced by the 7-day wave is opposite to that by the 5.8-day wave in latitudes of 20°S-20°N and altitudes of 45-55 km (see Figure 10a in Takagi et al. ( 2022)).The distribution of the vertical AM flux is similar to that of the meridional AM flux in mid-and high latitudes (Figure 5b).This is because the AM and EP fluxes produced by the 7-day wave approximately follow the isentropic surface, as shown in Figure 5d.The heat flux is poleward above 49 km altitude (Figure 5c).This poleward heat flux is one order of magnitude weaker than that produced by the 5.8-day wave (Takagi et al., 2022), although the meridional and vertical AM fluxes of the 7-day wave have magnitudes comparable to those of the 5.8-day wave.In the region below 49 km and above the critical surface in mid-and high latitudes, the upgradient equatorward heat transport is produced against the meridional temperature gradient of the basic state.Since the temperature field is in the gradient wind balance with the horizontal wind, and the zonal-mean zonal wind increases with altitude below the cloud top, the meridional temperature gradient is negative (positive) in the northern (southern) hemisphere.The equatorward heat transport could be caused by the downward propagation of the high-latitude Rossby mode, as discussed in Section 4. This result contrasts with those obtained for the 5.8-day and 4.8-day waves, which produce the large poleward heat flux and very little equatorward heat flux.
The distribution of the EP flux is rather complex (Figure 5d).In low latitudes at 48 km altitude, the EP flux originated near 20° latitude proceeds equatorward across the equatorward critical latitude at 10°, and converges near the equator, decelerating the zonal-mean zonal wind.It also proceeds poleward on the isentropic surface of 343 K with increasing its magnitude, and reaches around 60 km altitude in high latitudes.The zonal-mean zonal wind is accelerated by the EP flux divergence in a wide range of latitude from the equatorward critical latitude to 60°.Below the 343 K surface, the EP flux is directed downward in high latitudes, reaches about 40 km altitude, and changes its direction equatorward.On the isentropic surface of 355 K, the equatorward EP flux, which is originated from a region around 60° latitude and 60 km altitude, predominates in a wide range of latitude from the equator to 60°.This equatorward EP flux also decelerates the zonal-mean zonal wind in the equatorial region in 52-57 km altitudes.The EP flux along the isentropic surfaces could be mainly attributed to the meridional and vertical AM fluxes (Figures 5a  and 5b), while the downward EP flux in high latitudes below 49 km altitude could be mainly attributed to the equatorward heat flux, since the vertical AM flux is weak below 49 km (Figures 5b and 5c).Note that the equatorial region where the zonal-mean zonal wind is decelerated by the 7-day wave, which is located in latitudes of 10°S-10°N and altitudes of 45-55 km, almost coincides with that where the zonal-mean zonal wind is accelerated by the 5.8-day wave (see Figure 10d in Takagi et al. (2022)).These results suggest that the time-varying equatorial jet in the lower cloud layer could be produced by the 5.8-day wave and destroyed by the 7-day wave in the present model.

Time Variations of 7-Day Wave and Zonal-Mean Zonal Wind
Figure 6a shows time variations of the normalized KE of the 7-day and 5.8-day waves and the zonal-mean zonal wind on the equator averaged over the lower cloud layer (48-54 km).The KE of 7-day wave is calculated for the four regions denoted by the rectangles shown in Figure 3, which correspond to the upper (the black line) and lower (the red line) equatorial Rossby modes and the high-latitude Rossby modes in the northern (the blue line) and southern (the violet line) hemispheres, while that of the 5.8-day wave (the green line) is calculated for a region in latitudes of 90°S-90°N and altitudes of 30-70 km.During days 117-195 (referred to as the growth period), all the modes of 7-day wave grow in the same way, and decay after day 195.However, the upper equatorial Rossby mode and the northern and southern high-latitude Rossby modes regrow during days 234-286 (referred to as the regrowth period), while the lower equatorial Rossby mode continues to decay.After day 286, all the modes of 7-day wave decay during days 286-351 (referred to as the decay period), and they start to grow again after day 380.The total period from growth to decay of the 7-day wave is about 280 days.It should be noted that the northern and southern high-latitude Rossby modes vary in time in different ways in the regrowth (days 234-286) and decay (days 286-351) periods, although they grow in the same way in the growth period.This is because the northern and southern high-latitude Rossby modes could be synchronized via the lower equatorial Rossby mode in the growth period, while they are not synchronized and could grow independently in the regrowth period, as discussed in Section 4. Figure 6a also shows that the 5.8-day wave grows prior to the 7-day wave during days 55-155, and decays during days 155-260 as the 7-day wave grows.Also, the zonal-mean zonal wind on the equator averaged over the lower cloud layer (48-54 km) increases and decreases in almost the same way as the 5.8-day wave.The timescale of these variations is about 280 days, the same as that of the 7-day wave.
Figure 6b shows time variations of the zonal-mean zonal wind at 54 km altitude.The zonal-mean zonal wind varies quasi-periodically not only in the equatorial region but also in a wide range of latitude with amplitude of 15 m s −1 on the equator and 12 m s −1 at 30°N/S.Before the 5.8-day wave begins to grow (around days 55 and 286), the equatorial jet, which is confined to latitudes of 15°S-15°N, has almost completely disappeared.It becomes stronger as the 5.8-day wave grows, increasing the difference in angular velocity of the zonal-mean zonal winds between the equator and 30°N/S (i.e., the meridional shear measured by angular velocity of the zonal-mean zonal wind in low latitudes).
It seems that the 7-day wave begins to grow when the equatorial jet matures (around days 117 and 380).As the 7-day wave grows during days 117-195 and 380-468, the 5.8-day wave decays and the equatorial jet decelerates in the lower cloud layer.At the same time, the zonal-mean zonal wind gradually accelerates in latitudes of 25°-50°, decreasing the meridional shear.Around days 234 and 515 before the regrowth period, the equatorial jet almost disappears, and the zonal-mean zonal wind gets close to a constant velocity distribution in latitudes of 60°S-60°N.In the regrowth period of 7-day wave (days 234-286 and 515-560), the zonal-mean zonal wind significantly decelerates in mid-latitudes (25°-45°).In the decay period of 7-day wave (days 23-94 and 286-351), the 5.8-day wave begins to grow again.The equatorial jet also accelerates, increasing the meridional shear of the zonal-mean zonal wind in low latitudes.
Figure 7 shows difference in the zonal-mean zonal wind before and after the growth period of 7-day wave.The equatorial jet in 45-55 km altitude strongly decelerates by about 11 m s −1 in the growth period, while the zonalmean zonal wind in latitudes of 20°-60° (60°-90°) accelerates (decelerates) in altitudes of 40-70 km.It could be confirmed that the equatorial jet in 45-55 km altitudes, which matures in the beginning of the growth period of 7-day wave, almost disappears before the regrowth period (Figure 6b).These results suggest that the meridional shear measured by angular velocity of the zonal-mean zonal wind is decreased in both the low and high latitudes as the 7-day grows, and the zonal-mean zonal wind with the equatorial jet changes to a nearly constant velocity distribution in a wide range of latitude.The change in the equatorial region is quite consistent with the EP flux produced by the 7-day wave and the acceleration rate of the zonal-mean zonal wind due to the EP flux convergence (Figure 5d).The changes in mid-and high latitudes are also consistent with the meridional and vertical AM fluxes produced by the 7-day wave (Figures 5a and 5b), although the distribution of acceleration rate of the zonalmean zonal wind estimated from the EP flux is considerably more complex (Figure 5d).It is suggested from these results that the change of the zonal-mean zonal wind in the equatorial region could be attributed mainly to the 7-day wave, while those in mid-and high latitudes could be contributed not only to the 7-day wave, but also to other waves such as the 5.8-day and 4.8-day waves (Takagi et al., 2022) and the meridional circulation.Note that the equatorial zonal-mean zonal wind near 54 km altitude is further decelerated by about 4 m s −1 in the regrowth period (days 234-286), as shown in Figure 6b, because the upper equatorial and high-latitude Rossby modes grow again after day 234, producing the poleward AM flux (the equatorward EP flux) which decelerates the equatorial zonal-mean zonal wind just above the critical surface (see also Figure 10).

Quasi-Periodic Variation of the Equatorial Jet
As shown in Section 3, the equatorial jet, which varies quasi-periodically with a period of about 280 days, is reproduced in the lower cloud layer in the present result.As it accelerates by 13-15 m s −1 during days 23-120 and 286-410 (i.e., over 97-124 days) as the 5-day wave grows (Figure 6), its acceleration rate could be estimated to be 0.1-0.15 m s −1 day −1 . Takagi et al. (2022) showed that the time-averaged acceleration rate of the zonal-mean zonal wind induced by the 5.8-day wave is about 0.1  m s −1 day −1 in the equatorial region at 54 km altitude, which agrees well with the above estimation.The 7-day wave begins to grow as the equatorial jet matures, and the equatorial jet decelerates as the 7-day wave grows (Figure 6).The deceleration rate of the equatorial jet at 54 km could be estimated to be 0.1-0.13 m s −1 day −1 , which agrees well with that induced by the 7-day wave (about 0.15 ), as shown in Figure 5d.These results strongly suggest that the equatorial jet in the lower cloud layer is produced (destroyed) by the equatorward (poleward) AM transport produced by the 5.8-day (7-day) wave, and its quasi-periodic variation is caused by the alternating development of these waves.Horinouchi et al. (2017) showed a possibility that the time-varying equatorial jet could exist in the lower cloud layer using the nightside images taken by the Akatsuki IR2 camera, as mentioned in Section 1.They detected the equatorial jet in 11-12 July 2016 and 13-26 August 2016, but did not in 25 March 2016.The period between 25 March 2016 and 11 July 2016 is 108 days, and that between 11 July 2016 and 20 August 2016 is 40 days.It is inferred from this fact that the equatorial jet could last for tens of days.The equatorial jet observed in July and August 2016 was confined to latitudes equatorward of 20°-30°, and its magnitude was 10-15 m s −1 in the difference with a solid body rotation derived from the zonal wind distribution in mid-latitudes, while the zonal wind distribution obtained on 25 March 2016 seemed to be an intermediate one between a solid body rotation and a constant velocity distribution.These observational results suggest that the equatorial jet could vary in time significantly with a timescale of tens to hundreds of days, and agree well with those obtained in the present study, as shown in Figures 6 and 7.
It should be pointed out that a possibility of concomitance of the zonal wind component of waves and/or disturbances in the zonal wind velocity obtained from the Akatsuki IR2 nightside images cannot be excluded in the work of Horinouchi et al. (2017), namely the observed zonal wind velocity could be different from the zonal-mean zonal wind velocity.This is because the zonal ranges covered by their observations were only about 60°-68° in longitude.In fact, the zonal wind distribution in latitudes of 0°-30°N obtained on 26 August 2016 was quite different by 10-15 m s −1 from that obtained on the day before.This rapid change might be caused by the superposition of some waves and/or disturbances on the zonal-mean field.In order to investigate the influence of waves and disturbances on the zonal wind in the lower cloud layer, we examine the meridional profiles of the zonal-mean zonal wind and the zonally averaged RMS of zonal wind deviation from its zonal average, σ u′ , at 54 km altitude in the first half of the growth period (days 117-147) and the regrowth period (days 234-286) of 7-day wave.Note that the zonal-mean zonal wind on the equator averaged over the lower cloud layer takes its maximum and minimum value in the former and latter period, respectively (Figure 6).σ u′ is defined as follows: where u is the zonal wind which is not bandpass-filtered,    is the zonalmean zonal wind, λ is longitude, and t 1 and t 2 are start and end times of the analysis period.Figure 8 shows that, in the first half of the growth period (days 117-147), the equatorial jet is formed at 54 km altitude (the red solid line).However, if the zonal observation range is restricted as in the Akatsuki IR2 nightside observation (Horinouchi et al., 2017), the observed zonal wind distribution could largely vary within the range indicated by the red translucent shade due to waves and disturbances.In the most extreme case, the observed meridional profile of the zonal wind could be almost a solid body rotation with a rotation period of 6.5 days, despite of the presence of the equatorial jet.The apparent change of the observed zonal wind is significantly larger in low latitudes than in mid-latitudes, as indicated by the red translucent shade in Figure 8.This is because the 5.8-day wave, which is the predominant wave in low latitudes in the lower cloud layer, matures in Figure 8. Meridional profiles of the zonal-mean zonal wind at 54 km altitude in latitudes between 40°S and 40°N averaged over the first half of the growth (days 117-147, red solid line) and regrowth (days 234-286, blue solid line) periods, in which the zonal-mean zonal wind on the equator averaged over the lower cloud layer is at its maximum and minimum, respectively (Figure 6a).Each translucent color shade indicates the range of zonal wind variation estimated by    ±   ′ , where    is the zonal-mean zonal wind and σ u′ is the zonally averaged root-mean-square (RMS) of the zonal wind deviation from its zonal average (Equation 8) obtained in each period.Dashed lines denote solid body rotations with periods of 4.5, 5.5, and 6.5 days.Note that the zonal wind is not bandpass-filtered in this analysis.), the zonal-mean zonal wind (black contour, m s −1 ), the rotation period of zonal-mean zonal wind (violet contour, days), and the isentropic surfaces at 343 and 355 K (red line) averaged over days 117-147 in the first half of the growth period.(b) The same as in panel (a) but averaged over days 234-286 in the regrowth period.
this period (Figure 6), and the equatorial Kelvin mode of 5.8-day wave is dominated by the zonal winds of about 7-9 m s −1 confined to latitudes of 20°S-20°N in the matured phase (see Figures 7 and 8 in Takagi et al. (2022)).As the zonal winds of the equatorial Kelvin mode of 5.8-day wave are superposed on the zonal-mean zonal wind, the apparent equatorial jet is enhanced and reduced periodically with a period of 5.8 days.On the other hand, in the regrowth period (days 234-286), the equatorial jet disappears, and the zonal-mean zonal wind velocity is almost constant with latitude in a wide range of latitude (the blue solid line).Since the 5.8-day wave is not active in this period, the apparent equatorial jet does not appear (the blue translucent shade).These results, which might be also consistent with the Akatsuki IR2 nightside observation (Horinouchi et al., 2017), strongly suggest that we need more observations in a wider range of longitude to confirm the existence and time variation of the equatorial jet in the lower cloud layer.It is also noted that, in the decay period of 7-day wave, the zonal-mean zonal wind distribution at 54 km altitude is close to a solid body rotation (not shown).Since the 5.8-day wave is growing in this period, the apparent equatorial jet could be observed due to the superposition of the zonal wind component of its equatorial Kelvin mode if the zonal observation range is not sufficient.

Excitation Mechanism of 7-Day Wave
In order to investigate the excitation mechanism of 7-day wave, we calculate the potential vorticity (PV), P, which is defined as follows (Andrews et al., 1987): Figure 9 shows latitude-altitude distributions of the meridional gradient of the zonally averaged PV obtained in the growth and regrowth periods.The basic state of the atmosphere significantly changes in time in latitudes equatorward of 45° and altitudes of 45-65 km, with the disappearance of the equatorial jet in the lower cloud layer.As shown in Figure 6a, the development of 7-day wave could be divided into two stages: all the modes (the upper and lower equatorial Rossby modes and the high-latitude Rossby modes in both hemispheres) grow in the same way in the growth period, while the lower equatorial Rossby mode continues to decay and the others regrow in the regrowth period.These results suggest that the excitation mechanisms which work in the growth and regrowth periods could be different.Therefore, we investigate the excitation mechanisms of 7-day wave separately in the growth and regrowth periods in the following.

Excitation Mechanism in the Growth Period
In the growth period (days 117-195), the strong equatorial jet is formed in 40-56 km altitude, and the zonal-mean zonal wind is decelerated in mid-latitudes (Figure 9a).As a result, regions with the negative meridional PV gradient are formed from low to mid-latitudes in 43-65 km altitude (Figure 9a), so that baroclinic and/or barotropic instability could occur around these regions.The critical surface of the 7-day wave intersects the negative PV gradient regions, that is, the PV gradient changes its sign on the critical surface at about 10° and 40° latitudes.The altitude at which the PV gradient changes its sign on the critical surface is approximately located on the isentropic surface of 343 K. On this surface, the PV gradient is positive in latitudes between the equatorward critical latitudes (10°S-10°N) and latitudes poleward of the poleward critical latitude (40°-90°).The zonal phase velocity of 7-day wave is slower than the zonal-mean zonal wind velocity in these regions with the positive PV gradient, that is, the 7-day wave propagates westward relative to the zonal-mean zonal wind.This fact supports the interpretations that the vortices in latitudes between the equatorward critical latitudes at 48 km altitude are associated with the lower equatorial Rossby mode and those in latitudes poleward of the poleward critical latitude are associated with the high-latitude Rossby mode (Figures 4a and 9), and these Rossby modes propagate westward relative to the zonalmean zonal wind, as discussed in Section 3. On the other hand, in latitudes between the equatorward and poleward critical latitudes (10°-40°), the PV gradient is negative and the zonal phase velocity of 7-day wave is faster than the zonal-mean zonal wind velocity, that is, the 7-day wave propagates eastward relative to the zonal-mean zonal wind in this region.This fact suggests that the eastward propagating Rossby mode (hereinafter referred to as the mid-latitude Rossby mode) exists in the region with the negative PV gradient, and it makes the phase tilt of the 7-day wave which produces the meridional AM flux (Figures 4a and 5a).Note also that the equatorward (poleward) EP flux near the equatorward (poleward) critical latitude in altitudes of 46-50 km is distributed almost along the 343 K isentropic surface, suggesting that the interaction among the westward lower equatorial Rossby mode, the eastward mid-latitude Rossby mode, and the westward high-latitude Rossby mode could occur on this isentropic surface.
Using a shallow water model on a sphere, Iga (2002) investigated stability of the zonal-mean zonal wind with an equatorial jet.Because Iga (2002) mimicked the equatorial jet in the solar tachocline, as also done by Dikpati and Gilman (2001), the equatorial jet extends over a wider meridional range in the model of Iga (2002) than in the present result, and the meridional PV gradient changes its sign at 58° latitude, which is positive (negative) in latitudes equatorward (poleward) of this latitude.The results obtained by Iga (2002) can be summarized as follows.Unstable modes exist only for zonal wavenumbers of 1-3.These unstable modes are excited by the shear instability due to the equatorial jet because all the modes produce the poleward AM flux and decelerate the equatorial jet.The most preferred mode is a destabilized equatorial Rossby wave with a zonal wavenumber of 1 and n = 2, that is, the meridional structure antisymmetric about the equator, where n represents a meridional mode of the equatorial waves (Matsuno, 1966).The zonal phase velocity of the most preferred mode is slower (faster) than the zonal-mean zonal wind in latitudes equatorward (poleward) of the critical latitude.One of the most important features of this mode is that the critical latitude coincides with the latitude of 58° at which the meridional PV gradient changes its sign.Therefore, the most preferred mode propagates westward (eastward) relative to the zonal-mean zonal wind in latitudes equatorward (poleward) of the critical latitude where the meridional PV gradient is positive (negative).From these results, Iga (2002) concluded that the most preferred mode is excited by the coupling of the westward equatorial Rossby mode and the eastward high-latitude Rossby mode.
As shown in Figures 4a and 4b, the horizontal structure of the 7-day wave obtained in the present study is quite similar to that of the most preferred mode obtained by Iga (2002) in latitudes between the poleward critical latitudes (38°S-38°N) at 48 km altitude in the growth period, suggesting that the westward lower equatorial Rossby mode is coupled with the eastward mid-latitude Rossby mode through the equatorward critical latitude.At the same time, the eastward mid-latitude Rossby mode is also coupled with the westward high-latitude Rossby mode through the poleward critical latitude in the present result.Since these couplings occur simultaneously via the mid-latitude Rossby modes in both hemispheres, that is, the lower equatorial Rossby mode and the high-latitude Rossby modes in both hemispheres are connected by the mid-latitude Rossby modes, the zonal phases of the high-latitude Rossby modes in the northern and southern hemispheres are firmly aligned, that is, the centers of the cyclonic and anticyclonic vortices associated with the high-latitude Rossby modes in the northern and southern hemispheres, respectively, are located at the same longitude.Thus, the horizontal structure antisymmetric about the equator is maintained in the growth period (Figure 4).It is noted that the present result is similar to those obtained for the 5.8-day and 4.8-day waves (Takagi et al., 2022).In those results, the mid-and high-latitude Rossby modes in the northern and southern hemispheres are synchronized via the equatorial Kelvin mode, and this synchronization maintains their horizontal structures symmetric about the equator.

Excitation Mechanism in the Regrowth Period
In the regrowth period (days 234-286), as the equatorial jet disappears and the zonal-mean zonal wind in mid-latitudes is accelerated by the 7-day wave (Figure 7), the lower parts of the negative PV gradient regions disappear, and the critical surface of the 7-day wave splits into to two (Figure 9b).One is located near 53 km altitude in low latitudes, descends with latitude toward the poles, and reaches about 35 km in high latitudes.The other is located in the equatorial region in 40-46 km altitudes, forming a closed surface.The upper critical surface intersects the negative PV gradient regions at their lower latitude side (about 20°) at 52 km.However, the coupling between the westward equatorial Rossby mode and the eastward mid-latitude Rossby mode cannot occur in the regrowth period.The reason is as follows.The zonal-mean zonal wind distribution in low latitudes is intermediate between a constant velocity distribution and a solid body rotation in altitudes of 45-55 km (Figures 6 and 8), so that the zonal angular velocity of the zonal-mean zonal wind in the equatorial region is slower than that in mid-latitudes.As the result, the zonal angular phase velocity of the westward equatorial Rossby mode, if it exists, should be slower than and cannot be equal to that of the eastward mid-latitude Rossby mode.This fact means the impossibility of the coupling between the two Rossby modes.This consideration is also supported by the present results that, in the regrowth period, the lower equatorial Rossby mode has disappeared in altitudes of 48-52 km below the upper critical surface (Figures 10a and 10b), and the equatorward EP flux, which decelerates the zonal-mean zonal wind in the equatorial region in altitudes of 45-50 km in the growth period, has also disappeared (Figure 10f).On the other hand, it is inferred from the PV gradient distribution that the coupling of the eastward mid-latitude Rossby mode in the negative PV gradient region around 30° latitude and the westward high-latitude Rossby mode in the positive PV gradient region could be still possible in altitudes around 50 km, as in the growth period.In fact, the westward high-latitude Rossby mode grows again (Figure 6a), and the poleward EP flux along the 343 K isentropic surface is produced in mid-and high latitudes of the northern hemisphere (Figure 10f).These results suggest that the 7-day wave could be enhanced by the coupling of the eastward mid-latitude Rossby mode and the westward high-latitude Rossby mode in the regrowth period.
It is noted that the high-latitude Rossby mode in the southern hemisphere regrows faster than that in the northern hemisphere during days 234-286 (Figure 6a).The equatorward AM flux and the poleward EP flux across the critical surface along the 343 K surface disappeared in the southern hemisphere, and the poleward AM flux and the equatorward EP flux along the 355 K surface are also greatly reduced in latitudes of 30°S-70°S, compared to those in the northern hemisphere (Figures 10e and 10f).The zonal phases of the northern and southern high-latitude Rossby modes are not as strictly aligned as in the growth period, that is, the center of anticyclonic (cyclonic) vortex in the southern hemisphere is located slightly east of the center of cyclonic (anticyclonic) vortex in the northern hemisphere (Figures 10a-10d).It could be argued from these results that the high-latitude Rossby modes in both hemispheres are not synchronized in the regrowth period, and the coupling of the eastward mid-latitude Rossby mode and the westward high-latitude Rossby mode could occur independently in the northern and southern hemispheres.Furthermore, if it occurs independently in both hemispheres, it is not necessary that the periods of the northern and southern Rossby modes coincide with each other.Since the coupling occurs on an isentropic surface, as in the case of the 5.8-day wave (Takagi et al., 2022), it seems that a possible period of the unstable mode could be between 6 and 7 days (Figure 9b), although it also depends on possible meridional and vertical wavenumbers of the unstable mode.This consideration is qualitatively consistent with the result that the period of 7-day wave extends over a wider period range at higher latitudes than at lower latitudes (Figure 2).
The equatorial vortices in 53-57 km altitudes associated with the upper equatorial Rossby mode are also enhanced again in the regrowth period (Figure 6a).Their meridional winds at 54 km are significantly stronger than those in the growth period (Figures 4c and 10c).As suggested by the horizontal wind distribution in the equatorial vortices, the upper equatorial Rossby mode plays an important role in producing the equatorward AM flux in latitudes equatorward of 30° above the critical surface (Figure 10e).The zonal-mean zonal wind in the equatorial region is still decelerated in these altitudes by the convergence of the EP flux coming from the northern mid-latitudes around 60 km approximately along the 355 K isentropic surface (Figure 10f).However, it is difficult to explain why the zonal-mean zonal wind is still decelerated in the equatorial region, while the equatorial jet has been almost completely disappeared.The present analysis suggests that the upper equatorial Rossby mode could not be excited directly by the instability which excites the high-latitude Rossby modes.It remains unclear how it is excited and enhanced in the growth and regrowth periods.We should investigate its excitation mechanism in the future work.

Vertical Propagation of the High-Latitude Rossby Mode
As shown in Figure 5c, the upgradient equatorward heat transport is produced against the meridional temperature gradient by the high-latitude Rossby mode.It would be difficult to explain such meridional heat transport by a kind of shear instability, but it is well known in the meteorology on Earth that the poleward (equatorward) meridional heat transport could be produced by a upward (downward) propagating Rossby wave (e.g., Andrews et al., 1987).In order to investigate this possibility, that is, the vertical propagation of the high-latitude Rossby mode, we estimate its vertical wavelength, λ z , using the dispersion relation of Rossby wave (Andrews et al., 1987): where c x is the zonal phase velocity, k, l, and m = 2π/λ z are the zonal, meridional, and vertical wavenumbers,  =  2  ∕ 2 , N is the Brunt-Väisälä frequency, and H s is the scale height.f e and β e are respectively defined as follows: where f is the Coriolis parameter.The zonal wavenumber, k, is 1/(a cos ϕ) for the 7-day wave.Since the high-latitude Rossby mode is a geostrophic vortex with a radius of about 6,000 km centered on the pole (Figure 4), the meridional wavelength, 2π/l, could be estimated to be 12,000 km. Figure 11 shows that the estimated vertical wavelength, λ z , of the high-latitude Rossby mode increases rapidly with altitude above the critical level and diverges at 45-50 km altitude, depending on latitude.This result suggests that the high-latitude Rossby mode could propagate vertically in altitudes between the critical level and about 50 km, while it could be vertically trapped above 50 km.From the dispersion relation (Equation 10), the vertical group velocity of the high-latitude Rossby mode, c gz , is obtained as follows: As shown in Figure 4f, the phase of the high-latitude Rossby mode tilts eastward with increasing altitude below 50 km.Therefore, the c gz is negative, that is, the energy propagation of high-latitude Rossby mode is downward.This result is also consistent with the downward EP flux in high latitudes below about 48 km altitude (Figure 5d), because the EP flux associated with the Rossby wave is in the same direction as its energy propagation (e.g., Andrews et al., 1987).These facts suggest that the high-latitude Rossby mode propagates downward in those regions, and the equatorward heat transport against the meridional temperature gradient can be explained by its downward propagation.Recently, Ando et al. (2022) pointed out that the equatorward heat transport produced by waves is important for the formation of deep low-stability layer in high latitudes.As the magnitude of the equatorward heat transport obtained in their study is comparable to that produced by the 7-day wave, the 7-day wave could substantially contribute to this process.

Meridional Winds at the Cloud Top
The kinetic energy of horizontal winds associated with the 7-day wave is confined to altitudes of 40-70 km (Figure 3).However, the amplitude of the upper equatorial Rossby mode significantly increases between the late growth period and the regrowth period, so that the meridional wind velocity associated with the 7-day wave reaches about 4-5 m s −1 in low latitudes at the cloud top level in these periods.Figure 12 shows a snapshot of the horizontal winds associated with waves and disturbances with periods less than 10 days (vectors) and the meridional wind velocity associated with the 7-day wave (color shade) obtained at 70 km altitude at day 234.Note that the horizontal winds represented by the vectors contain not only the 7-day wave component but also those of other waves except the thermal tide.At the cloud top level, the meridional winds associated with the short-period waves and disturbances are not symmetric about the equator, and the meridional winds across the equator are also present.The meridional winds in low latitudes and over the equator are dominated by a zonal wavenumber-1 component with amplitude of 7-8 m s −1 .The meridional wind associated with the 7-day wave is in phase with that of the waves and disturbances with periods less than 10 days, and its amplitude is about 4-5 m s −1 in low latitudes.This result suggests that more than half of the meridional wind velocity in low latitudes at the cloud top could be attributed to that associated with the 7-day wave.Using the UV cloud images taken by the Akatsuki UVI, Horinouchi et al. ( 2020) derived horizontal winds associated with transient waves and disturbances at the cloud top, excluding the thermal tide component.They showed that there exist the horizontal winds from one hemisphere to another, and their distribution is dominated by a zonal wavenumber-1 component.The amplitude of the observed meridional winds is about 5 m s −1 on the equator.Their result could be consistent with that obtained in the present study, suggesting a possibility that the observed meridional wind across the equator could be contributed by the 7-day wave which has meridional structures antisymmetric about the equator.

Conclusions
Using a Venusian GCM, we reproduced the equatorial jet and its quasi-periodic variation in the lower cloud layer (47-57 km) which could be consistent with the recent Akatsuki IR2 observation (Horinouchi et al., 2017).This equatorial jet is produced by the equatorward AM flux due to the 5.8-day wave   and 6).As the zonal-mean zonal wind distribution approaches a constant velocity distribution, the 5.8-day wave begins to grow again, producing the equatorial jet.Thus, the quasi-periodic variation of the equatorial jet is caused by the alternating development of the 5.8-day and 7-day waves in the GCM.The timescale of this variation is about 280 days.These results could provide a possible explanation of the equatorial jet in the lower cloud layer inferred from the Akatsuki IR2 observation.
The development of 7-day wave could be divided into three periods: the growth period, the regrowth period, and the decay period (Figure 6a).In the growth period, the 7-day wave is excited by the coupling among the lower westward equatorial Rossby mode, the eastward mid-latitude Rossby mode, and the westward high-latitude Rossby mode (Figure 4).This mechanism is similar to the shear instability mechanism for the equatorial jet shown by Iga (2002), in which the westward equatorial Rossby mode is coupled with the eastward high-latitude Rossby mode, and the poleward AM flux is produced across the critical latitude.Since the lower equatorial Rossby mode and the high-latitude Rossby modes in both hemispheres are connected by the mid-latitude Rossby modes in the growth period, the horizontal structure antisymmetric about the equator is firmly maintained.Around the end of growth period and just before the regrowth period, the equatorial jet disappears due to the AM transport produced by the 7-day wave, and the zonal-mean zonal wind distribution becomes intermediate between a constant velocity distribution and a solid body rotation (Figures 6b and 7).In the regrowth period, the lower equatorial Rossby mode disappears due to the disappearance of the equatorial jet, but the high-latitude Rossby mode and the upper equatorial Rossby mode are still enhanced by the coupling of the mid-latitude and high-latitude Rossby modes (Figures 9 and 10).Since this coupling can occur independently in both hemispheres, that is, the high-latitude Rossby modes are not synchronized via the lower equatorial Rossby mode, the distributions of the AM and EP fluxes produced by the 7-day wave could be asymmetric about the equator in the regrowth period (Figure 10).
We examined the influence of waves and disturbances on the zonal wind observed in the lower cloud layer.In the first half of the growth period, the apparent equatorial jet could be largely enhanced or reduced by waves and/ or disturbances if the zonal observation range is restricted as in the Akatsuki IR2 observation (Figure 8).In the most extreme case, the observed meridional profile of the zonal wind could be almost a solid body rotation with a rotation period of 6.5 days, despite of the presence of the equatorial jet.This is because the 5.8-day wave, which is the predominant wave in low latitudes in the lower cloud layer (Takagi et al., 2022), matures in this period (Figure 6).The superposition of the zonal wind associated with the equatorial Kelvin mode of 5.8-day wave causes the apparent change of the equatorial jet with a period of 5.8 days.In the regrowth period, the equatorial jet disappears, and the zonal-mean zonal wind velocity is almost constant with latitude in a wide range of latitude (Figures 6 and 8).Since the 5.8-day wave is weak in this period, the apparent equatorial jet does not appear.In the decay period, the zonal-mean zonal wind distribution is close to a solid body rotation (Figure 6).Since the 5.8-day wave is growing in this period, the apparent equatorial jet could be observed if the zonal observation range is restricted (not shown).These results strongly suggests that we need more observations in a wider range of longitude to confirm the existence and time variations of the equatorial jet in the lower cloud layer.
The present result also shows that the significant upgradient equatorward heat transport is produced by the high-latitude Rossby mode against the negative (positive) meridional temperature gradient of the basic temperature field in the northern (southern) hemisphere (Figure 5c).The latitude-altitude distribution of the vertical wavelength of the high-latitude Rossby mode estimated from the dispersion relation of the Rossby wave shows that the high-latitude Rossby mode could propagate vertically in altitudes between the critical level and about 50 km in latitudes of 30°-90° (Figure 11).This result is also supported by the eastward phase tilt of the high-latitude Rossby mode below 45 km (Figure 4f) and the downward EP (energy) flux in high-latitudes below 50 km (Figure 5d).The equatorward heat transport produced by the downward propagation of the 7-day wave could affect the thermal structure in the polar region and play an important role in generating the deep low-stability layer extending in altitudes of 40-60 km, as pointed out by Ando, Imamura, et al. (2020).
We also examined the influence of 7-day wave on the meridional winds observed at the cloud top level.The recent observations show that there exist the meridional winds near the equator, whose distribution is dominated by a zonal wavenumber-1 component, and their amplitude at the equator is about 5 m s −1 (Horinouchi et al., 2020).The present result suggests a possibility that more than half of the meridional wind velocity could be attributed to the meridional winds associated with the 7-day wave in low latitudes at the cloud top level, and the observed meridional winds across the equator could be explained by the 7-day wave (Figure 12).
It should be emphasized again that the 7-day wave has not yet been observed.As the 7-day wave is small in amplitude in low latitudes and is not accompanied by strong vertical winds, it may be difficult to detect it in observations of the lower clouds.The existence of the equatorial jet and the waves in the lower cloud layer must be confirmed by further observations.Currently, observations on the both are overwhelmingly lacking.Large scale vortices were discovered in the lower cloud layer just recently (Horinouchi et al., 2023).Further observations are expected to discover new types of waves, including the 7-day wave.A future Venus mission, the Crosslink Radio Occultation measurements of the Venus Atmosphere (CROVA), is under consideration in Japan, in which three-dimensional structures of waves in 40-90 km altitudes could be derived by the cross-link radio occultation observations among the multiple spacecrafts (T.Yamamoto et al., 2021).Detailed observations of the atmospheric motions in the cloud layer are required to advance our understanding of the atmospheric circulation of Venus.

Figure 1 .
Figure1.(a) Latitude-altitude distribution of the zonal-mean zonal wind time-averaged over 585 days (black contour, the unit is m s −1 ), its rotation period (blue contour, days), and the root-mean-square (RMS) of the zonal-mean zonal wind calculated by Equation2(color shade, m s −1 ).(b) Time-altitude plot of deviations of the zonal-mean zonal wind from its time average over 585 days (color shade, m s −1 ) and the rotation period of the zonal-mean zonal wind (black contour, days) on the equator.

Figure 2 .
Figure 2. Amplitude of the Fourier component of the meridional winds obtained at 53.5 km altitude (a) and 75°N latitude (b) as a function of period.The unit is m s −1 .The rotation period of the zonal-mean zonal wind around the planet is shown by the white line in each panel.

Figure 3 .
Figure 3. Latitude-altitude distribution of the zonally integrated and time-averaged (over day 117-351) kinetic energy of horizontal winds associated with the 7-day wave extracted by the bandpass filter with a period range of 6.5-8.0 days.The unit is  kg m −1 s −2 .Four rectangles represent regions where the wave activity is high: 20°S-20°N latitude and 53-60 km altitude (red), 20°S-20°N latitude and 45-50 km altitude (black), 30°-90°N latitude and 45-65 km altitude (blue), and 90°S-30°S latitude and 45-65 km altitude (violet).The green line indicates the critical surface of 7-day wave, on which the rotation period of zonal-mean zonal wind is 7 days and the zonal-mean zonal wind velocity equals to the zonal phase velocity of 7-day wave.

Figure 4 .
Figure 4. Structure of the 7-day wave averaged over the growth period (days 117-195).(a) Longitude-latitude distributions of geopotential height deviation (color shade, the unit is m) and horizontal wind (vector, m s −1 ) obtained at 48 km altitude.The green lines indicate the critical latitudes where the zonal-mean zonal wind velocity equals to the zonal phase velocity of 7-day wave.In latitudes equatorward of 30°, the meridional wind is also shown by the black contour (the unit is m s −1 ).(b) The same as (a) but in latitudes of 25°S-25°N.(c) The same as (a) but at 54 km.(d) The same as (a) but at 60 km.(e) Longitude-altitude distributions of geopotential height deviation (color shade, m) and meridional wind (black contour, m s −1 ) on the equator.(f) Longitude-altitude distributions of temperature deviation (color shade, K) and meridional wind (black contour, m s −1 ) at 70°N latitude.The green lines in the panels (e) and (f) indicate the critical levels.

Figure 5 .
Figure 5. Latitude-altitude cross-sections of (a) the meridional angular momentum (AM) flux,   ′  ′ cos  (color shade, the unit is  kg m −1 s −2 ), (b) the vertical AM flux,   ′  ′ cos  (color shade,  kg m −1 s −2 ), (c) the meridional heat flux,   ′  ′ (color shade,  K kg m −2 s −1 ), and (d) the Eliassen-Palm (EP) flux (vector, kg s −2 ) and acceleration rate of the zonal-mean zonal wind due to the EP flux divergence (color shade,  m s −1 day −1 ), obtained for the 7-day wave, where ρ is basic atmospheric density,  is latitude, and  ( ) represents the zonal average.The green line indicates the critical surface of the 7-day wave.The magenta lines in panel (d) are the isentropic surfaces of 343 K (lower) and 355 K (upper).The violet lines in panel (d) indicate latitudes and altitudes where the meridional gradient of the potential vorticity (PV) changes its sign.The zonal-mean zonal wind distribution is shown by the black contours.All the quantities are time-averaged over days 117-234.

Figure 6 .
Figure 6.(a) Time variations of the zonally and vertically averaged kinetic energy density averaged over the upper equatorial region (red line), that over the lower equatorial region (black line), that over the northern mid-and high latitude region (blue line), and that over the southern mid-and high latitude region (violet line).Each region used for the average is indicated by a corresponding colored rectangle in Figure 3.The magenta and green lines show time variations of the zonal-mean zonal wind on the equator averaged over altitudes of 48-54 km (right axis, m s −1 ) and the kinetic energy density of 5.8-day wave taken from the work of Takagi et al. (2022), respectively.The kinetic energy densities are normalized by their maximum values.(b) Time-latitude plot of the zonal-mean zonal wind obtained at 54 km altitude.The unit is m s −1 .The black contour shows the rotation period (days) of the zonal-mean zonal wind.The days referred to in the text are denoted on the top of panels.

Figure 7 .
Figure7.Latitude-altitude cross-section of difference in the zonal-mean zonal wind between days 117 and 234, before and after the growth period (color shade, the unit is m s −1 ), and the zonal-mean zonal wind averaged over days 112-122 (black contour, m s −1 ).The zonal-mean zonal wind was averaged over 10 days in each period before taking the difference.The green line indicates the critical surface of the 7-day wave.

Figure 9 .
Figure 9. (a) Latitude-altitude distribution of the meridional gradient of zonal-mean potential vorticity (color shade, the unit is  K m kg −1 s −1

Figure 11 .
Figure11.Latitude-altitude distributions of the estimated vertical wavelength (color shade, the unit is km), λ z , the amplitude of the geopotential height deviations (black contour, m) of the mid-and high-latitude Rossby modes of 7-day wave, and the zonal-mean zonal wind (grey contour, m s −1 ) averaged over days 117-234.In the region without color shade, λ z is imaginary, that is, the high-latitude Rossby mode is vertically trapped.

Figure 12 .
Figure 12.Longitude-latitude distribution of the horizontal winds associated with waves and disturbances with periods less than 10 days (vectors) and the meridional wind velocity associated with the 7-day wave (color shade, the unit is m s −1 ) obtained at 70 km altitude at day 234.
1 and t 2 are start and end times of the analysis period (t 2 − t 1 is 585 days in the present study),