Vertical propagation of planetary-scale waves in variable background winds in the upper cloud region of Venus

Icarus 248 (2015) 560–568

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Vertical propagation of planetary-scale waves in variable background winds in the upper cloud region of Venus Toru Kouyama a,⇑, Takeshi Imamura b,c, Masato Nakamura b,c, Takehiko Satoh b,c, Yoshifumi Futaana d a

Information Technology Research Institute, National Institute of Advanced Industrial Science and Technology, Umezono 1-1-1, 305-8568 Tsukuba, Ibaraki, Japan Department of Basic Space Science, Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, 252-5210 Sagamihara, Kanagawa, Japan c Department of Solar System Science, Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Japan d Swedish Institute of Space Physics, Box 812, SE-981 28 Kiruna, Sweden b

a r t i c l e

i n f o

Article history: Received 29 January 2014 Revised 10 July 2014 Accepted 10 July 2014 Available online 18 July 2014 Keywords: Venus, atmosphere Atmospheres, dynamics Meteorology

a b s t r a c t Recently it was found that the low-latitude zonal wind and the amplitudes of Kelvin and Rossby waves at the cloud top of Venus show long-term variations in a synchronized manner. For the purpose of explaining this synchronization, we investigated the influence of the background zonal wind profile on the upward propagation of Kelvin and Rossby waves at altitudes 60–80 km. Results from a linearized primitive equation model suggests that Kelvin waves can reach the cloud top height when the background wind speed is slow, whereas Rossby waves can reach the cloud top when the background wind speed is fast. These features obtained from the model are consistent with the observations. Since the momentum deposition by these waves can accelerate or decelerate the mean flow, these waves may contribute to the variation of the background wind. The calculated spatial distributions of the momentum dissipation indicate that the Kelvin waves accelerate the low-latitude atmosphere, and thus they can act to induce transition from the slow wind period to the fast wind period. On the other hand, the Rossby waves decelerate mainly the mid-latitude atmosphere, so that additional mechanisms are required to decelerate the low-latitude atmosphere. A possible mechanism is momentum advection caused by the Rossby wave-induced meridional circulation. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In contrast to the small rotational velocity of Venus (1.6 m s1 on the equator), the westward zonal wind speed increases monotonically with height and reaches about 100 m s1 at the cloud top around 70 km altitude (Counselman et al., 1980; Schubert, 1983). To maintain this wind system, called the superrotation, the transport of westward momentum from the lower atmosphere to the cloud top is needed. The mechanism is still unclear, although various mechanisms have been proposed (Gierasch, 1975; Rossow and Williams, 1979; Fels and Lindzen, 1974; Takagi and Matsuda, 2006; Yamamoto and Tanaka, 1997). Planetary-scale atmospheric waves are embedded in this superrotational flow according to imaging observations. Wavenumber-1 traveling waves with periods of 4 d (d = Earth days) and 5 d have been observed as cloud brightness and wind velocity oscillations in the data taken by Orbiter Cloud Photo-Polarimeter onboard Pioneer Venus Orbiter (Del Genio and Rossow, 1990; ⇑ Corresponding author. E-mail address: [email protected] (T. Kouyama). http://dx.doi.org/10.1016/j.icarus.2014.07.011 0019-1035/Ó 2014 Elsevier Inc. All rights reserved.

Rossow et al., 1990). The 4-d wave is identified as a Kelvin wave based on the equatorial confinement of the zonal wind oscillation, absence of the meridional oscillation, and the prograde propagation relative to the background wind. The 5-d wave is considered as a symmetric Rossby wave having the meridional oscillation in the mid-latitude, the zonal oscillation in the high latitude and the equatorial region, and the retrograde propagation. These waves were also observed in the cloud-tracked velocity fields derived from images taken by Solid State Imaging system onboard Galileo (Belton et al., 1991; Toigo et al., 1994; Peralta et al., 2007; Kouyama et al., 2012) and by Venus Monitoring Camera (VMC) onboard Venus Express (Kouyama et al., 2013; Khatuntsev et al., 2013). Covey and Schubert (1982) studied the linear response of a model Venus atmosphere to external forcing over broad frequencies, and showed that Kelvin-like waves and Rossby-like waves are excited as preferred modes for cloud-level forcing. Del Genio and Rossow (1990) suggested that the observed Kelvin waves provide mean-wind acceleration of the order of 0.1 m s1 d1 at the equatorial cloud top. Imamura (2006) estimated the acceleration by Kelvin waves near the equator to be about 0.3 m s1 d1 and the deceleration by Rossby waves in

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the mid-latitude to be 0.15 m s1 d1. The magnitudes of the acceleration and deceleration correspond to the change in the mean zonal velocity of 10–20 m s1 over 100 d. The latitudinal distribution of the cloud-tracked zonal velocity and the predominant wave modes at the cloud top change from epoch to epoch (Del Genio and Rossow, 1990). Kouyama et al. (2013) presented observational evidence of a quasi-periodical variation of the mean zonal velocity by 20 m s1 with a period of about 255 days in the southern low latitude region based on an analysis of VMC cloud images in 2006–2010. 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. The mechanism of this variation and its correlation with the background zonal wind speed has not been well understood. Khatuntsev et al. (2013) also detected long-term variations of the zonal wind speed at the cloud level, although their periodicities are less evident. In this study, we aim to understand the synchronous variation of the mean zonal velocity and the planetary-scale wave activity. We investigate how the variation of the background zonal wind speed influences the vertical propagation of Kelvin and Rossby waves at the cloud level of Venus using a linearized primitive equation model. Section 2 describes further analysis of the longterm variation originally made by Kouyama et al. (2013), but by adding newly obtained data. Section 3 introduces the basic concept of the influence of the background zonal wind speed on the vertical propagation of planetary-scale waves in the venusian atmosphere. Section 4 describes the setup of the numerical model, and Section 5 gives calculated wave structures. Section 6 evaluates the waveinduced acceleration and discusses possible roles of the acceleration in inducing the transition of the zonal wind state. Section 7 gives conclusions. 2. Long-term variation of the zonal wind and planetary-scale waves at the cloud top level In this section we show the long-term variation of the wind velocity and the dominant planetary-scale wave at the cloud top during the period from May 2006 to December 2011 using a cloud tracking method (Ogohara et al., 2012; Kouyama et al., 2012) applied to VMC ultraviolet (UV) images. VMC is a wide-angle camera having 17.5°  17.5° field of view with 512  512 pixels for each of the four wavelength channels. The UV filter is centered at 365 nm wavelength, which is the region of the specific spectral feature of the unknown absorber and the maximum contrast on the Venus disc (Markiewicz et al., 2007). 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. Following Kouyama et al. (2013), for each orbit we prepare a pair of images with a time interval of 1.0 ± 0.1 h when the sub-spacecraft latitude was 55–70°S; this criterion assures a wide coverage in the equatorial region. The typical spatial resolution at the sub-spacecraft position is 20 km. 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 from which we could deduce cloud motion vectors regularly an ‘‘epoch’’. Ten epochs are available as shown in Table 1. Here, epochs 6 and 7 are divided into two sub-epochs because of gaps of several tens of days, and epoch 9 is also divided into two sub-epochs because the characteristics of periodical perturbations are clearly different between the first half and the second half in this epoch. Epochs 1, 2 and 6-2 were excluded from the analysis because the number of sampling is not enough in these epochs.

561

Latitudinal structures of the periodical fluctuations are determined by applying the Lomb–Scargle periodogram method (Lomb, 1976; Scargle, 1982) to the velocity time series at latitudes 15–45°S. Fig. 1 shows the obtained spectra as functions of wave period and latitude from epoch 3 to epoch 10, with the circulation period of the mean zonal wind also plotted for comparison. Based on the latitudinal structure and the sign of the intrinsic phase velocity, which is defined as the phase velocity in the reference frame moving with the background flow, we classified the distinct wave modes seen in the spectra into Kelvin and Rossby waves (Table 1). The employed criteria are: waves propagating faster than the background wind with oscillations mainly in the zonal wind in low latitudes are Kelvin waves; and waves propagating slower than the background wind with fluctuations in both zonal and meridional winds in the mid-latitude are Rossby waves. Though dominant waves are unclear in epochs 5, 8, 9-2, and 10, the distinct wave in epoch 3 is Kelvin wave, and those in epochs 4, 6-1, 7-1, 7-2, and 9-1 are Rossby wave. Kelvin waves were detected only in epoch 3, in which the background wind velocity is the smallest in the whole observation period, while Rossby waves were frequently observed in relatively fast wind periods (more than 100 m s1 in epochs 4, 6-1, 7-1, 7-2 and 9-1). These two types of waves are not clearly identified at the same time. The reason of the observed correlation between the dominant wave mode and the background wind speed will be studied in the subsequent sections. Here, a caution should be exercised as the Kelvin wave dominated only in one epoch. This is because the phase relationship between the VMC observation and the long-term variation of the wind speed did not allow sampling of the slow-wind period more than once. Additionally, the mean zonal velocity at the cloud top shows an increasing trend since 2006 (Khatuntsev et al., 2013), further restricting the occasion of the slow-wind condition. We must admit that the statistical significance is not very high; more observations are required to confirm the correlation. Del Genio and Rossow (1990) reported a similar tendency based on a spectral analysis of the cloud brightness measured by Pioneer Venus OCPP: the 4-d (Kelvin) wave predominated in fall 1979, spring 1980 and spring 1985, while the 5-d (Rossby) wave predominated in spring 1982, fall 1982 and summer 1986, although both modes are seen in spring 1979 and spring 1983. On the other hand, Rossow et al. (1990) reported that the velocity amplitudes of the 4d wave and the 5-d wave show a common year-to-year variability based on the cloud-tracked velocities obtained from OCPP images. The cause of this discrepancy is unclear. Here we should note that these wave amplitudes were estimated by compositing the data into longitude systems that rotate with the periodicities of exactly 4 d and 5 d, and thus they are lower limits because, unless the actual wave periods are exactly 4 and 5 days, some smearing is unavoidable (Rossow et al., 1990). Although the magnitude of such an error is uncertain, direct comparison between the results obtained by different analysis methods is difficult.

3. Phase velocity dependence of the vertical group velocity of Kelvin and Rossby waves In the upper cloud region of Venus (altitude >60 km), radiative damping is expected to be the primary agent of dissipation of planetary-scale waves. For example, even a disturbance with an infinite vertical wavelength decays within 10 d at 70 km altitude (Crisp, 1989). Since a wave would be attenuated significantly when the propagation time of one scale height is longer than the radiative relaxation time scale, the vertical group velocity is the critical factor determining whether a wave can reach the cloud top from the lower atmosphere. Here we assume that the horizontal phase

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Table 1 Characteristics of the epochs. Epoch

Period

Orbit no.

Number of orbits

Dayside-mean zonal velocity at 18°S (m/s)

Dominant wave

1 2 3 4 5 6-1 6-2 7-1 7-2 8 9-1 9-2 10

May 20, 2006–July 3, 2006 November 15, 2006–February 13, 2007 June 30, 2007–September 17, 2007 January 28, 2008–May 1, 2008 September 6, 2008–November 16, 2008 April 6, 2009–May 27, 2009 July 4, 2009–July 22, 2009 November 30, 2009–December 28, 2009 January 23, 2010–March 11, 2010 August 27, 2010–October 15, 2010 February 24, 2011–April 6, 2011 April 8, 2011–May 20, 2011 November 11, 2011–December 21, 2011

30–74 208–298 436–515 648–741 869–940 1082–1133 1171–1189 1320–1347 1374–1420 1589–1638 1770–1812 1814–1856 2031–2080

13 30 62 77 74 44 17 26 35 37 27 26 35

93 ± 7 97 ± 9 89 ± 8 101 ± 9 101 ± 9 108 ± 9 107 ± 8 105 ± 11 117 ± 8 106 ± 8 112 ± 8 107 ± 9 108 ± 7

(Low statistics) (Low statistics) Kelvin Rossby (Unclear) Rossby (Low statistics) Rossby Rossby (Unclear) Rossby (Unclear) (Unclear)

Fig. 1. Time series power spectra of the zonal and meridional cloud-tracked velocities as functions of period and latitude obtained by Lomb–Scargle periodogram method. Colors represent statistical significance levels and dotted contours indicate 90% significance level. The periods corresponding to the dayside-average zonal velocities are also plotted (solid curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

velocity is determined by the dynamics in the lower atmosphere where the wave is excited and that it is almost invariant. Then, the change in the mean zonal velocity near the cloud top would modify the vertical group velocity through the change in the intrinsic phase velocity. In this section we review the dependence of the

group velocity on the intrinsic phase velocity for Kelvin and Rossby waves. Considering a planetary-scale Kelvin wave, whose vertical structure is identical to those of internal gravity waves, the vertical group velocity is given approximately by

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C gðzÞ 

k  j2 ; jc  u N

ð1Þ

where N denotes the Brunt–Vaisala frequency, k is the horizontal  ðzÞ is the height-dependent background zonal wind wavenumber, u velocity, and c is the horizontal phase velocity. Eq. (1) indicates that the vertical propagation speed increases with the absolute value of  j. For example, considering a Kelvin the intrinsic phase speed jc  u wave with the zonal wavenumber of unity propagating around 70 km altitude where the scale height is H  5 km and N  0.02 s1,  j  10 m s1 , it would take 70 d to travel a scale height when jc  u  j  30 m s1 . while it would be less than 10 d when jc  u Although the planetary-scale symmetric Rossby waves observed in the venusian atmosphere have wide structures in latitude, the basic characteristics can be discussed in terms of the equatorial wave theory (Imamura, 2006). From the dispersion relationship of an equatorial Rossby wave (cf. Chapter 4 in Andrews et al., 1987), the vertical group velocity is given by

C gðzÞ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 8 !9 u 2 2 =  Þ3 k3 <3 u   ðc  u 9 ðc  ðc  u Þk u Þk 42 ¼ t þ 1þ :2 ; 4 Nb b b   31  Þk2 ðcu 1þ2 b 7 1 7 þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 ; 2 2 9 ðcuÞk2   Þk ðcu þ b 1þ b 4  Þk2 ðcu b

Fig. 2. Vertical group velocity of an equatorial Rossby wave as a function of the absolute value of the intrinsic phase speed. Horizontal axis indicates the normalized intrinsic phase velocity and vertical is the normalized vertical group velocity of the wave.

ð2Þ

where b = 2X/a is the meridional derivative of the Coliolis parameter on the equator in the coordinate system moving with the superrotation, X is the angular velocity of the super-rotation, and a is the planetary radius. The vertical group velocity above also increases  j (Fig. 2). with jc  u 4. Model description To investigate the upward propagation of Kelvin and Rossby waves around the cloud top region, the waves are forced at the bottom of a linearized primitive equation model that extends from 60 to 100 km altitude. The governing equations are written in a nonrotating frame of reference as follows:



 d uv tan / 1 @ U þ rR u  ¼ ; dt a a cos / @k   d u2 tan / 1 @U þ rR v þ ¼ ; dt a a @/   @ u @ v @ @U þ rN þ þ þ Sw ¼ 0; @t a cos / @k a @/ @z u @u 1 @ @w þ ðv cos /Þ þ  w ¼ 0; a cos / @k a cos / @/ @z

@T R2 T þ ; @z cp

ð10Þ

is taken from Covey and Schubert (1982). The model altitude domain is divided into 400 grids for integration. Following Covey and Schubert (1982), Eqs. (3)–(6) are linearized about a basic state with the zonal wind distribution given by

 ðz; /Þ ¼ XðzÞa cos /; u

ð11Þ

where X is the angular velocity of atmospheric rotation. This form of the mean zonal wind imposes a constant angular velocity at each altitude and simplifies the numerical technique. The real zonal wind distribution shows excess angular velocities in the mid-latitude both in the fast and slow background period. Imamura (2006) showed that mid-latitude jets do not influence the structure of Kelvin waves significantly but cause a poleward expansion of Rossby waves. Writing

ð/; zÞ þ u0 ð/; zÞ exp½iðsk  xtÞ; uðt; k; /; zÞ ¼ u

ð12Þ

ð5Þ

with similar expressions for v, w and U, we obtain four perturbation equations of motion from (3)–(6). The horizontal momentum equations are replaced by the vorticity and divergence equations in which the horizontal velocity components are replaced by the stream-function w0 and velocity potential v0 , then the linearized equations are written as

ð6Þ

 ixD Dw0 þ 2isXw0 þ 2X sin /Dv0 þ 2X cos /

ð3Þ ð4Þ

 Xz cos /w0

where

d @ u @ v @ @ ¼ þ þ þw ; dt @t a cos / @k a @/ @z

SR

ð7Þ

and the log-pressure vertical coordinate is given by

z  lnðsurface pressure=pressureÞ;

ð8Þ

w  dz=dt:

ð9Þ

Here U is the geopotential, t the time, k the west longitude, / the south latitude, u and v the westward and southward components of wind velocity, respectively, T the temperature, R the gas constant for Venus atmosphere (191 J K1 kg1), cp  4.5R the specific heat at constant pressure, rR the coefficient of Rayleigh friction, and rN the radiative relaxation rate. In this coordinate system the observed velocity of the mean zonal wind is positive. The vertical profile of the static stability

@w0 ¼ 0; @/

@ v0 þ 2Xz sin /w0 @/ ð13Þ

@w0 þ isXz w0 ¼ a2 DU0 ; @/ ð14Þ   0 0 0 @U isw @v þ Sw0 ¼ 0; ixD  2XXz a2 sin / cos / þ ð15Þ @/ cos / @/ ixD Dv0 þ 2isXv0  2X sin /Dw0  2X cos /

Dv0 þ

@w0  w0 ¼ 0; @z

ð16Þ

where

xD  x  sX þ ir;

  s 1 @ @ : D þ cos / @/ cos2 / cos / @/

ð17Þ

2

ð18Þ

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Solutions are obtained by converting these equations to a set of ordinary differential equations in z by expanding the dependent variables in associated Legendre functions (Covey and Schubert, 1982). The analytical solution of a Kelvin or Rossby wave on the equatorial b-plane (Matsuno, 1966; Andrews et al., 1987) is imposed at the lower boundary. The geopotential field at the bottom is given by

b the ampliwhere, y is the poleward distance from the equator, U tude scale, lE the equatorial deformation radius, and x the frequency. The lE is a measure of the latitudinal pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi scale of the equatorially trapped wave defined by lE ¼ x=2kb. We set b to be 4.3  1012 m1 s1, corresponding to the equatorial wind speed of 85 m s1 which is typical of 60 km altitude. The disturbance winds at the bottom are determined by using (19) or (20) and (13)–(16). Radiation condition is imposed at the upper boundary. We assume two types of the background zonal wind profile to represent the fast and the slow wind period (Fig. 3). In the fast wind profile (Model F), which represents the fast wind period, the wind speed increases with altitude from 60 to 70 km and decreases above 70 km, which is based on direct measurements (Schubert, 1983) and estimations from thermal fields (Newman and Leovy, 1992; Zasova et al., 2007). In the slow profile (Model S), which represents the slow wind period, the wind speed is independent of the altitude. Although the exact wind profiles in these different background wind periods are unknown, experiments with such simplified conditions can still qualitatively reproduce the variation of the wave amplitude with the background wind speed. What is necessary for the simulation is that the Model S winds are mostly slower than the Model F winds by typically 20 m s1 around 70 km altitudes. In both Models F and S the phase speeds of the imposed Kelvin and Rossby waves are assumed to be 111 m s1 (period of 4.0 d) and 78 m s1 (period of 5.7 d), respectively. These speeds are chosen based on the observations of Kelvin waves in the slow wind period and Rossby waves in the fast wind period. This choice leads

to lE = 2400 km for the Kelvin wave and lE = 2200 km for the Rossby wave at the bottom boundary. Once these phase speeds are given, the assumption that the waves originate from altitudes lower than 60 km requires that the background wind speed at this height should be in the range 78–111 m s1 to allow the vertical propagation; the wind speed of 85 m s1 (Fig. 3) is well within this range. The amplitude u0 of the Kelvin wave is set to be 5 m s1 at the bottom boundary for the u0 amplitude being ~5 m s1 at 70 km altitude in Model S, and the u0 of the Rossby wave is set to be 10 m s1 at the bottom so that the u0 amplitude is 10 m s1 at 70 km altitude in Model F, being consistent with the observations (Rossow et al., 1990; Kouyama et al., 2013). The use of a common phase speed in Model F and Model S for each wave needs discussion. Hueso et al. (2012) examined the variability of the cloud-level wind field based on a cloud tracking analysis of Venus Express VIRTIS images, and showed that there is no notable long-term variability near the cloud base at 45 km altitude although day-to-day variations are significant. The variable wind speeds at this height suggested by direct measurements (Schubert, 1983) are considered to reflect such short-term variations. Based on this observation, the observed long-term variation of the mean zonal wind at the cloud top (Kouyama et al., 2013; Khatuntsev et al., 2013) will not have a deep structure extending to the cloud base, and the characteristics of the dynamics near or below the cloud base will be rather stable. Although the wave source region remains unknown, the phase speeds of the waves will be largely unchanged from epoch to epoch if the waves are excited in the lower atmosphere. Fig. 4 shows the vertical profiles of the radiative relaxation time given in the model; the relaxation rate rN in (5) is the inverse of the radiative relaxation time. The relaxation time depends on the vertical wavelength. The model result given in the next section shows that the Kelvin and Rossby waves have vertical wavelengths of 9 km and 30 km, respectively, in Model F, and have vertical wavelengths of 15 km and 20 km, respectively, in Model S. Based on these values, the relaxation times for the Kelvin and Rossby waves in Model F are taken from the Crisp’s (1989) theoretical estimates for vertical wavelengths of 7 km and 30 km, respectively. The relaxation time in Model S is taken to be the middle of these for both waves. All relaxation time profiles are represented by smooth functions. In addition to the radiative relaxation, weak Rayleigh friction with a coefficient of rR = 1.2  107 s1 is also introduced into the model for the whole altitude range. The assumed coefficient corresponds to a relaxation time of 100 d. One may in principle

Fig. 3. Background zonal wind speeds as functions of the altitude in Model F (solid) and Model S (dashed). Phase speeds of the imposed Kelvin and Rossby waves are also indicated by allows.

Fig. 4. Radiative relaxation times given in the model for Kelvin (solid) and Rossby (dashed) wave in Model F, and for both waves in Model S (dotted), based on Crisp (1989).

! y2 b U ¼ U exp  2 expfiðkx  xtÞg for Kelvin wave; 2lE 0

" b U0 ¼ U

x  ku 2

blE k

1

y2 2

lE

for Rossby wave;

! 

y2 2

lE

# exp 

y2 2

2lE

ð19Þ

! expfiðkx  xtÞg ð20Þ

T. Kouyama et al. / Icarus 248 (2015) 560–568

disregard the friction, but we kept this process in the model, because it also helps to stabilize the calculation. On the other hand, dissipation by turbulent viscosity, which is not considered in this model, needs discussion. The timescale of the dissipation by viscosity is given by s = (kz/2p)2/D, where kz is the vertical wavelength and D is the turbulent diffusion coefficient. Letting D = 4 m2 s1 in the upper cloud region (Woo and Ishimaru, 1981), the timescale is s  6 d for kz = 9 km, s  16 d for kz = 15 km, s  29 d for kz = 20 km, and s  66 d for kz = 30 km. The radiative relaxation times for the waves given in Fig. 4 are all much shorter than their respective turbulent dissipation times near the cloud top, suggesting that turbulent viscosity is less important than radiative damping. Therefore, we expect that the inclusion of turbulent viscosity would not change the conclusions significantly, although its influence needs to be studied more in detail in future. Qualitatively, turbulent viscosity should enhance the dependence of wave dissipation on the intrinsic phase speed, because a smaller intrinsic phase speed means a shorter vertical wavelength, which leads to faster viscos damping. 5. Three-dimensional structures Fig. 5 shows the calculated vertical structures of the zonal wind in Models F and S for the Kelvin and the Rossby waves. The Kelvin wave in Model F shows significant attenuation with altitude and almost disappears at 70 km altitude. In Model S the Kelvin wave retains a significant amplitude even above 80 km. The vertical wavelength of the Kelvin wave in Model F is about 60% of that in Model S (Fig. 5b) due to the smaller intrinsic frequency. This indicates that the Kelvin wave in Model F propagates upward more slowly and is more effectively attenuated by radiative damping.

565

The Rossby wave shows an opposite signature. The Rossby wave in Model S is attenuated significantly with altitude and almost disappears at 70 km altitude. In Model F it does not decay above 62 km and retains a significant amplitude even above 70 km. The difference with the signatures of Kelvin wave is also attributed to the difference in the efficiency of radiative damping; the Rossby wave in Model S propagates upward more slowly due to the smaller intrinsic frequency, and thus more susceptible to radiative damping. This is also consistent with the emergence of Rossby wave-like oscillations only in the fast wind period (Kouyama et al., 2013 and Section 2). Fig. 6 displays the horizontal structures of the Kelvin and Rossby waves in both wind models at 70 km altitude, i.e. the cloud top level. In Model S for the Kelvin wave, the zonal component of the velocity disturbance is enhanced in latitudes 30°S– 30°N and the meridional component is virtually absent. In Model F for the Rossby wave, both zonal and meridional components are enhanced from low to high latitude region and the wind vectors draw vortex patterns in the mid-latitude. These latitudinal structures of the Kelvin and Rossby waves are in agreement with the Galileo SSI and Venus Express VMC results, respectively (Kouyama et al., 2012, 2013). The modeled temperature disturbances have latitudinal scales similar to those of the velocity disturbances. The temperature maximum is shifted eastward from the zonal velocity maximum in the low latitude for both waves: a feature characteristic of vertically propagating waves. The Kelvin and Rossby waves at the cloud top have never been detected by temperature measurements; simultaneous observations of the velocity and temperature disturbances associated with these waves would better constrain the characteristics of the waves.

Fig. 5. Vertical profiles of the zonal wind disturbance on the equator in Models F (solid) and S (dashed). (a) Amplitude and (b) phase of the Kelvin wave, and (c) amplitude and (d) phase of the Rossby wave. The direction of the atmospheric rotation is from east to west.

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Fig. 6. Horizontal structures of the wind velocity and temperature disturbances of the Kelvin wave in (a) Model F and (b) Model S, and the Rossby wave in (c) Model F and (d) Model S at 70 km altitude. Arrows represent wind velocity vectors. The direction of planetary rotation is from right to left. Color code displays the temperature disturbance. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6. Wave-induced zonal acceleration Wave-induced mean wind accelerations may play key roles in the transition between the fast and the slow wind period. For examining this possibility, the Eliassen–Palm (EP) fluxes (Andrews and McIntyre, 1976; Andrews et al., 1987) associated with the Kelvin and Rossby waves are evaluated. The latitudinal and vertical components of the EP flux vector F are defined by

F

ð/Þ

F

ðzÞ

v 0 h0

z ¼ q0 a cos / u hz  v

! 0 u0

;

( )   cos /Þ/ v 0 h0 ðu 0 0 ¼ Hq0 a cos / f  hz  w u ; a cos /

ð21Þ

ð22Þ

where q0(z) is the basic density, H is the basic scale height and h is the potential temperature. The prime (0 ) and overbar (-) denote disturbance and zonal-mean quantities, respectively, and subscripts denote partial derivative. The magnitude of the wave-induced acceleration of the background zonal wind speed is estimated by (q0a cos /)1r  F, where r  F is the EP flux divergence

rF¼

1 @ ð/Þ 1 @F ðzÞ ðF cos /Þ þ : a cos / @/ H @z

ð23Þ

Fig. 7 displays the meridional distributions of the acceleration by the Kelvin and Rossby waves. Since the Kelvin wave in Model F decays rapidly with altitude, the resultant acceleration is almost confined to the region below 65 km (Fig. 7a). On the other hand, in Model S, weaker radiative damping allows the Kelvin wave to propagate to even higher altitudes, and the acceleration is centered around 65–70 km altitude

(Fig. 7b). The acceleration of 0.1 m s1 d1 corresponds to a 10 m s1 increase of the zonal wind in 100 d. Such an equatorial acceleration by the Kelvin wave may contribute to the transition from the slow wind period to the fast wind period seen in the observations. The Rossby wave reaches high altitudes to induce significant deceleration only in Model F. Since the Rossby wave influences the zonal wind mainly in the mid-latitude in contrast to the Kelvin wave, the wave cannot contribute directly to the transition from the fast wind period to the slow wind period, which occurs mostly in the low latitude (Kouyama et al., 2013). A possible mechanism to decelerate the zonal wind is the vertical advection of the mean zonal momentum by the wave-induced meridional circulation. Since the latitudinal shear is small in the low latitude (e.g., Limaye, 2007; Moissl et al., 2009), the mean wind acceleration by the meridional circulation is given by

  @u v @ u  @ u @u  ¼ w  w : a @/ @t @z @z

ð24Þ

The Rossby-wave induced meridional circulation at the cloud level has been studied by Imamura (1997), who estimated the meridional turnover time (timescale of advective exchange) to be sturnober  90 d. This corresponds to the vertical velocity of w = H/ sturnober  0.8 mm s1, and the substitution of this w to (24) gives  =@t  0:14 m s1 d1 when @ u  =@z  2 m s1 km1 is adopted @u based on previous observations. The magnitude of the suggested deceleration is similar to that of the Kelvin wave-induced acceleration. Then a possible scenario would be: Rossby waves deposit momentum in the mid-latitude; the wave-driven meridional circulation is intensified; the vertical advection in the low latitude is intensified; and the equatorial zonal wind is decelerated.

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Fig. 7. Meridional distributions of the zonal acceleration by the Kelvin wave in (a) Model F and (b) Model S, and by the Rossby wave in (c) Model F and (d) Model S.

7. Conclusions We investigated the influence of the variation of the background zonal wind speed on the vertical propagation of planetary-scale Kelvin and Rossby waves using a linearized primitive equation model. We imposed waves at 60 km altitude, and calculated the propagation and attenuation of them up to 100 km altitude. We set two kinds of vertical profiles of the background zonal wind speed: Model F represents the fast wind period which has a peak velocity at 70 km altitude, and Model S represents the slow wind period which has an altitude-independent wind velocity. The results suggest that the Kelvin wave can reach the cloud top level without significant attenuation by radiative damping only in Model S, and that the Rossby wave can reach the cloud top only in Model F. This is consistent with the observations, in which Kelvin wave-like perturbations were observed only in the slow wind period and Rossby wave-like perturbations were observed only in the fast wind period. The horizontal wave structures of the Kelvin wave in Model S and the Rossby wave in Model F are also in agreement with the observations. The calculated acceleration by Kelvin waves can explain the increase of the zonal wind speed from the slow wind period to the fast wind period. Rossby waves, on the other hand, decelerate the mean flow mainly in the mid-latitude, and thus other mechanisms are required to decelerate to the zonal wind speed in the low latitude; the vertical advection of zonalmean momentum by meridional circulation is one possibility. Even if the Kelvin and Rossby waves contribute to the transition between the fast and slow wind period, it is unclear why a steady state in which the waves’ opposite effects cancel each other out is not achieved. Contributions of gravity waves propagating at cloud heights (e.g., Peralta et al., 2008), which might be generated by convection in the lower part of the cloud (Baker et al., 2000; Imamura et al., 2014), are also expected. Observations of the phase relation-

ship among the long-term variations of the zonal wind, the meridional circulation, and various wave activities may provide clues to the interaction between the mean flow and the waves and the mechanism of the variation of the atmospheric state. Applications of the cloud-tracking method to future Venus image sets such as those obtained by the VMC onboard Venus Express and those from Akatsuki (Nakamura et al., 2014) could identify different types of waves. The stability of the dynamical state might be closely related to the mechanism of the maintenance of the super-rotation; close investigation of temporal variations in realistic Venus GCMs (e.g., Lebonnois et al., 2010) will also be helpful in understanding the super-rotation and the long-term variation in a consistent manner. Quasi-periodical oscillations of the zonal wind speed with periods of the order of years are observed also in other planetary atmospheres: the quasi-biennial oscillation (QBO) in the Earth’s equatorial stratosphere, (e.g., Andrews et al., 1987) and the quasi-quadrennial oscillation (QQO) in the jovian atmosphere (Leovy et al., 1991; Friedson, 1999) are well known. These oscillations are 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; Friedson, 1999; Li and Read, 2000). Comparative studies of the mean wind variations in the venusian and other planetary atmospheres might provide a new perspective on the wave-mean flow interaction in planetary atmospheres in general. Acknowledgments 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 reviewer for providing valuable comments on the draft of the paper.

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