## 1. Introduction

A thick CO_{2} gas covers Venus’ surface, maintaining a surface pressure of 92 (Earth) atmospheres, or 9.2 × 10^{4} hPa, and a temperature of 730 K. The greenhouse effect of CO_{2} determines the thermal structure in the Venus atmosphere. The atmospheric angular momentum is supplied from the surface, and the maximum is located in the dense lower atmosphere (Schubert et al. 1980). The zonal wind speed increases with height from the surface and reaches about 100 m s^{−1} at the cloud top (65–70 km).

Sulfuric acid clouds globally cover the planet in the height range between 47 and 70 km. In the cloud layer, 70% of incoming solar energy is depleted by the extinction and heats the atmosphere near the cloud top. The solar heating is important in the dynamics of the Venus middle atmosphere.

Planetary-scale near-infrared (NIR) markings, which rotate with the 5.5-day period, are observed in the low-latitudinal region (e.g., Crisp et al. 1991). The NIR opacity source is optically thick middle and lower clouds. The migrating velocities of the planetary-scale markings are faster than the zonal mean velocities of the small-scale markings (<2000 km). If the mean velocity of the small-scale markings corresponds to the longitudinally averaged velocity of zonal flows, the planetary-scale 5.5-day wave has phase velocity faster than that of the mean flow at the cloud base (47–48 km). Yamamoto (2001) suggested that the 5.5-day NIR markings are caused by a vertically propagating equatorial gravity wave (such as Kelvin wave).

Planetary-scale ultraviolet (UV) markings rotate with periods of 4 days at low latitudes and 5 days at midlatitudes (e.g., Del Genio and Rossow 1990; Rossow et al. 1990). The UV markings are associated with planetary-scale waves near the cloud top. The equatorial 4-day wave, such as a Kelvin wave with an *e*-folding latitude of ∼23° (Del Genio and Rossow 1990), corresponds to the dark equatorial band of Y-shaped cloud pattern (e.g., Belton et al. 1976; Rossow et al. 1980) in UV observation. Covey and Schubert (1981, 1982), Smith et al. (1992, 1993), and Yamamoto (2001) suggested that the equatorial cloud-top 4-day wave is identified as a vertically propagating Kelvin or gravity wave. Yamamoto and Takahashi (2003b) briefly reported a vertically propagating Kelvin wave with periods of 3–5 days in a T21 Venus-like AGCM. The midlatitude 5-day Rossby wave detected by UV observation is generated in the cloud layer (Covey and Schubert 1981, 1982; Smith et al. 1992, 1993; Yamamoto and Tanaka 1997).

Thermal tides are detected in the orbiter infrared radiometer (OIR) observation (e.g., Elson 1983; Schofield and Taylor 1983). The dynamical structures were theoretically and numerically investigated (e.g., Elson 1983; Pechmann and Ingersoll 1984; Covey et al. 1986; Takagi and Matsuda 2005). For the diurnal tide (Elson 1983), two peaks of the temperature component are observed at the 65° and 85° latitudes in the cloud-top region, and the phase difference between the two peaks is ∼180°.

Barotropic instability assuming nondivergent eddies has been applied to the dynamical phenomena in the middle atmosphere (e.g., Elson 1978; Rossow and Williams 1979). However, it is difficult that the nondivergent eddies, such as geostrophic flow, are realized in the lower atmosphere rotating slowly with horizontally differential angular velocity (Yamamoto and Tanaka 2006). As a matter of fact, in our preliminary AGCM experiments (Yamamoto and Takahashi 2003a, hereafter YT03; Yamamoto and Takahashi 2003b), the horizontal divergence cannot be neglected in the superrotation mechanism. Thus, we should consider an instability including horizontal divergence. Satomura (1981) suggested that shear instability contributes to the superrotation dynamics. Recently, Iga and Matsuda (2005) examined shear instability in the spherical shallow water system. In addition, inertial instability (Joshi and Young 2002) should be also considered in the lower atmosphere.

Superrotational flows of ∼100 m s^{−1} are observed at the cloud top (e.g., Rossow et al. 1990). Many scenarios of the superrotation have been proposed in previous studies from simple mechanistic models (e.g., Schubert and Whitehead 1969; Fels and Lindzen 1974; Gierasch 1975) to Venus-like AGCMs (e.g., Young and Pollack 1977; Rossow 1983; Del Genio et al. 1993; Del Genio and Zhou 1996). AGCMs are expected to be a powerful tool to elucidate the dynamical processes of Venus’ superrotation. The fully developed superrotations are reproduced under the condition of the surface pressure of 92 atm and the planetary rotation of 243 days (YT03; Yamamoto and Takahashi 2003b). In their experiments using a zonally uniform solar heating, the meridional circulation drives the superrotation with the help of the equatorward eddy momentum flux, instead of large eddy diffusion in the Gierasch mechanism (Gierasch 1975; Matsuda 1980, 1982). The meridional circulation is a primary angular momentum transporter supplying the angular momentum to the superrotation, since the upward eddy momentum transport is much smaller than that of the meridional circulation. The simulation using a three-dimensional (3D) solar heating is briefly reported in Yamamoto and Takahashi (2004). In this paper, the maintenance mechanism of the Venusian superrotation is investigated and discussed further on the basis of the wave analysis. The sensitivities to the surface boundary conditions are also examined since the unknown surface physical processes are important in the superrotation dynamics.

## 2. Model

The model used is version 5.6 of AGCM developed at the Center for Climate System Research/National Institute for Environmental Study (Numaguti et al. 1995) and is the same as that used by Yamamoto and Takahashi (2004). The horizontal wavenumbers are truncated at 21 for the dynamical processes, and latitude–longitude plane is divided by grids of 64 × 32 for the physical processes. The model atmosphere ranging from the surface to ∼90 km is divided by 52 layers, and the sigma coordinate system is utilized. The thickness of the layer is ∼2 km in the region between 10 and 80 km, and ∼1 km in the region between 1 and 10 km. Three layers of *σ* = 0.995, 0.980, and 0.953 are given near the surface (below 1 km).

The planetary rotation period is set at 243 days (Earth day), a Venusian day is 117 days, the planetary radius *a* is 6050 km, the gravity acceleration *g* is 8.87 m s^{−2}, and the standard surface pressure *P _{s}* is 9.2 × 10

^{4}hPa. Thermodynamic constants of CO

_{2}are assumed in the model. The gas constant

*R*is 191.4 J kg

^{−1}K

^{−1}, and the specific heat at constant pressure Cp is assumed as a constant value of 8.2 × 10

^{2}J kg

^{−1}K

^{−1}. The temperature and pressure dependences of thermodynamic constants are neglected, since thermodynamics under the extreme condition is not our main research purpose.

The surface topography and the seasonal variation are neglected in our simplified model. The radiative processes are simplified by the 3D solar heating and the thermal damping for longwave radiation using the Newtonian cooling parameterization. Figure 1 shows the vertical profiles of the global mean solar heating rate and the Newtonian cooling time scale. The heating at the subsolar point has a rate 4.81 times higher than the global-mean profile shown in Fig. 1a. The maximum heating rate given in this model is 30 K day^{−1} at the subsolar point and the 65-km altitude. The zenith angle (*λ*) dependence of the solar heating is cos^{1.4}*λ* (Hou and Goody 1989). Although the heating distribution is improved in comparison with YT03 and Yamamoto and Takahashi (2003b), the heating rate is larger than that in the real atmosphere below 55 km. The radiative processes are not included on the surface and in the upper atmosphere. The Newtonian cooling rate of YT03 is based on the parameterization for waves in Hou and Farrell (1987). According to Crisp (1989), radiative relaxation time at 70 km is ∼2 days for a vertical wavelength of 7 km and ∼10 days for infinity. In the present study, since small-scale gravity waves seen in YT03 and Yamamoto and Takahashi (2003b) are not predominant, the cooling rate is changed to that with the relaxation time of ∼10 days at the cloud top. The changed value is similar to the cooling rate of zonal mean temperature. The time constant above the cloud top is ∼5 times longer than that of YT03. Convective adjustment is applied each time step.

The fourth-order horizontal diffusion of the *e*-folding time of 4 days at the maximum wavenumber and the vertical diffusion of *K _{V}* = 0.15 m

^{2}s

^{−1}are applied to the model (

*K*is the diffusion coefficient). Horizontal flow is dissipated by the Rayleigh friction of 30 days near the uppermost layer in order to reduce the boundary effect. In addition, the eddy components of horizontal flow are dissipated by the Rayleigh friction with the same time constant as that of the Newtonian cooling.

_{V}The physical processes near the surface are incorporated by the following simple parameterization. The latitudinal contrast of the surface temperature is newly incorporated in the present model (Yamamoto and Takahashi 2004). The equator–pole contrast of the surface potential temperature *dθ*_{sfc} is set at 10 K, though it is somewhat larger than the lower-atmospheric contrast (≤5 K). The value of 10 K was given as the tunable parameter value in Joshi and Young (2002). The latitudinal dependence of *dθ*_{sfc} is given as cos^{1.4}*ϕ* (*ϕ* is the latitude). Since the vertical resolution near the surface is higher than that in YT03 and Yamamoto and Takahashi (2003b), the frictional-drag time constant is changed. The surface drag, which relaxes the temperature and horizontal flow toward the surface values, has a time constant of 3 days at the undermost layer with a thickness of ∼100 m. The time constant of the surface drag is estimated from the velocity of ∼0.1 m s^{−1} and the drag coefficient of ∼4 × 10^{−3} (Del Genio et al. 1993). In addition to the above parameters (the standard case shown in section 3), sensitivities to the surface temperature and frictional drag are examined in section 4, since the real dynamical processes in the boundary layer at 92 atm are not sufficiently understood.

## 3. Results

The time integration was started from a motionless state with horizontally uniform reference temperature and continued until the result came into equilibrium. Figure 2 shows time histories of longitudinally averaged zonal flows. The mean zonal flows of ∼120 m s^{−1} at 69 km and of ∼8 m s^{−1} at 21 km are maintained at 2.8° latitude after 45 000 days (Earth days).

**) and the Eliassen–Palm (EP) flux (*w

*F*

^{ϕ}_{EP},

*F*

^{z}_{EP}) were defined aswhere

*u*is the zonal flow,

*υ*is the meridional flow,

*w*is the vertical flow,

*θ*is the potential temperature,

*ρ*

_{0}is the mean atmospheric density,

*a*is the planetary radius,

*f*is the Coriolis parameter,

*ϕ*is the latitude, and

*z*is the height. Overbar and prime represent the zonal mean and eddy components, respectively. The residual mean meridional circulation is introduced in order to infer the circulation directly forced by the diabatic heating in the TEM equation system (e.g., Andrews et al. 1987). Instead of momentum flux, the EP flux is used since longitudinally averaged zonal flow is modified by not only eddy momentum flux but also eddy heat flux. The EP flux represents the angular momentum flux including the momentum transport induced by the horizontal eddy heat flux. It should be noted that this EP flux (

*F*

^{ϕ}_{EP},

*F*

^{z}_{EP}) has the opposite sign to the EP flux familiar to atmospheric dynamics on earth (Andrews et al. 1987). In this definition, the equatorward eddy angular momentum flux is expressed as the equatorward vector, and the upward eddy angular momentum flux is expressed as the upward vector. In short, positive

*F*

^{ϕ}_{EP}indicates the horizontal EP flux transporting angular momentum in the positive latitudinal direction, and positive

*F*

^{z}_{EP}indicates the vertical EP flux transporting angular momentum upward. The convergence of the EP flux represents the zonal-flow acceleration caused by both the eddy momentum flux and the eddy horizontal heat flux.

*θ*in the above equations is defined bywhere

*T*is the temperature,

*P*is the atmospheric pressure, and

*P*is the standard surface pressure. The relationship between

_{s}*h*and

*T*(where

*h*is the geopotential height) is determined by the hydrostatic equation,where

*ψ*=

*gh*and

*σ*is the sigma level.

To investigate the structures of waves with a wide range of phase velocities, *u*, *υ*, *w*, *T*, *θ*, and *h* are decomposed by fast Fourier transformation for time and longitude. From the Fourier coefficients, the amplitude and the EP flux of each mode are calculated.

Figure 3 shows latitude–height distributions of mean zonal flow and absolute angular momentum. A strong zonal flow of more than 100 m s^{−1} is reproduced near the cloud top (65–70 km). The weak wind region is seen near 80 km. The maximum of absolute angular momentum is located near 20 km. Figure 4 shows latitude–height distributions of residual mean meridional and vertical flows. A single cell in the vertical direction is dominated in a latitude–height plane between 0 and 80 km, though the noisy structure appears in the lower atmosphere. The strong poleward flows are seen near 75 km. The vertical flow is formed continuously from the bottom to the 80-km altitude.

### a. Angular momentum transport process

The angular momentum of the simulated superrotation is supplied from the surface through the surface drag. Since the atmospheric density is very low and the mean zonal flow * u* is positive near the top boundary, the angular momentum (proportional to atmospheric density) is slightly lost (not supplied) through the top-boundary Rayleigh friction (section 2). Except for near the top boundary, the angular momentum lost by the Rayleigh friction is neglected. Although another Rayleigh friction is set for eddy horizontal flow (section 2), it never supplies zonal mean angular momentum from the planetary rotation frame. Thus, the Rayleigh frictions do not produce artificial superrotation.

Figure 5a shows a latitude–height section of the angular momentum flux by residual mean vertical flow. The meridional circulation transports angular momentum upward at low latitudes and downward at high latitudes. The global mean flux is upward. Accordingly, the meridional circulation pumps up angular momentum from the lower to the middle atmosphere.

Figure 5b shows a latitude–height section of the vertical EP flux. The vertical eddy angular momentum flux is predominantly downward at high latitudes. This indicates that retrograde (prograde) momentum is transported upward (downward), where “retrograde (prograde)” means the opposite (same) direction to the superrotation. In the height range between 5 and 25 km, the positive *F ^{Z}*

_{EP}is seen at high latitudes (|

*ϕ*| > 60°) but is cancelled by the negative one at mid- and low latitudes (|

*ϕ*| ≤ 60°) when the globally averaged flux is calculated. Since the global-mean flux of

*F*

^{Z}_{EP}is negative above 5 km, waves transport the cloud-top angular momentum (pumped up by the meridional circulation) toward the lower atmosphere. The vertical eddy diffusion of

*K*= 0.15 m

_{V}^{2}s

^{−1}, instead of

*F*

^{Z}_{EP}, predominantly transports angular momentum downward in the region below 5 km, where the global mean fluxes of

*F*

^{Z}_{EP}have slightly positive or negative values.

The large downward eddy angular momentum flux at high latitudes is caused by planetary-scale waves with zonal wavenumbers of 1 and 2. Thermally induced waves and mixed Rossby–gravity waves with periods of 10–100 days predominantly transport angular momentum vertically. Here, thermally induced waves are defined as planetary-scale waves rotating with solar heating in this article and strictly include both thermal tides directly forced by solar heating and the planetary-scale waves with periods of ∼1 Venusian day. In addition to these planetary-scale waves, many gravity waves with wavenumbers of 5 to 12 also transport angular momentum vertically, though the vertical flux of each of these gravity waves is smaller than one-tenth that of the largest mode. These waves decelerate the zonal flow above the cloud top, and their zonal-flow deceleration leads to the enhancement of zonal mean meridional flow above 70 km.

At low latitudes, the upward momentum flux by waves (such as the Kelvin wave and thermal tide) in Fig. 5b (∼5 × 10^{4} kg s^{−2}) is 20 times smaller than that by the meridional circulation in Fig. 5a (∼1 × 10^{6} kg s^{−2}). It is emphasized that the Kelvin wave and thermal tide do not pump up a large amount of angular momentum in the equatorial region in comparison with the meridional circulation, though these waves enhance the equatorial superrotation. Accordingly, the meridional circulation is a primary angular momentum transporter supplying the angular momentum to the middle atmosphere.

Figure 6 shows latitude–height sections of the horizontal EP flux and its *σ*^{−1}-weighted flux. These EP fluxes are predominantly equatorward and thus deposit the angular momentum into the low-latitudinal regions, where the upward flow of the meridional circulation pumps up the angular momentum. The maximum of the *σ*^{−1}-weighted EP flux is located near the upper branch of the meridional circulation. Since the equatorward flux transports the angular momentum toward the equator, the equatorial superrotation is enhanced in the cloud-top region.

Thermally induced waves predominantly transport the angular momentum toward the equator in the middle atmosphere. A zonal wavenumber-1 thermally induced wave (corresponding to a diurnal tide), which has the tilting phase and divergent flow, produces the equatorward momentum transport in the longitude–latitude plane at the 77-km level (Fig. 2b of Yamamoto and Takahashi 2004). Equatorward momentum flux of diurnal tide was reported by Newman and Leovy (1992). Instead of barotropic waves (Rossow and Williams 1979), the thermal tide is a promising candidate for equatorward eddy momentum transport in the middle atmosphere.

The inertial instability condition (the change of the sign of the absolute vorticity in Fig. 7) is satisfied below 30 km. This instability is seen in the Venusian atmospheric model of Joshi and Young (2002). Since weak midlatitude jets are formed in the lower atmosphere, the inertial instability occurs in the equatorial side of the jet. As a result, the noisy structure of the mean meridional flow (Fig. 4a) is caused by the inertial instability in the equatorial regions below 30 km.

The planetary-scale gravity and Rossby waves transport angular momentum toward the equator in the lower atmosphere. At 29 km, low-latitude waves with equatorward momentum fluxes predominantly have phase velocities faster than the mean-flow velocity, while high-latitude waves with equatorward momentum fluxes predominantly have phase velocities slower than the mean-flow velocity (Fig. 3 in Yamamoto and Takahashi 2004). The planetary-scale waves have different structure between low and high latitudes. The slowly traveling high-latitude Rossby waves transport angular momentum toward the equator. For the planetary-scale waves with equatorial phase velocities faster than 20 m s^{−1}, the horizontal pattern of a Rossby wave (high-latitude vortical wave) and a Kelvin wave (equatorially trapped gravity wave) with the same frequency produces equatorward momentum flux. The momentum fluxes across the boundary of the two waves are equatorward (Fig. 4 of Yamamoto and Takahashi 2004). This pattern is similar to unstable modes of shear instability in a spherical shallow-water system, recently reported by Iga and Matsuda (2005). The energy conversions (see appendix) are calculated for a 10.7-day wave. The latitude–height distribution of the zonal-mean to eddy kinetic energy conversion term C1 in the appendix is shown in Fig. 8. Other terms, C2 to C6, are too small in comparison with C1 and thus can be neglected. Shear or barotropic instability occurs in the height range between 25 and 40 km, since the large positive C1 means the energy conversion from zonal-mean to eddy kinetic energy in the horizontally differential superrotation. In this region, it is difficult to see nondivergent vortical flow almost satisfying geostrophy in horizontally differential rotation (Yamamoto and Tanaka 2006). Accordingly, the simulated waves are generated by shear instability, rather than barotropic instability.

Geostrophic approximation is not satisfied at low and midlatitudes in the lower atmosphere, where the momentum fluxes are equatorward. Thus, the inertial instability (Joshi and Young 2002) and the shear instability (Iga and Matsuda 2005), in which the geostrophic and nondivergent approximations are not required, play a crucial role in the superrotation mechanism at low and midlatitudes in the lower atmosphere, rather than the barotropic or baroclinic instability based on the geostrophic or nondivergent assumption. The shear instability predominantly produces the equatorward momentum flux in our experiment, and the inertial instability is also generated. We suggest that shear and inertial instabilities coexist in the real lower atmosphere. Both the instabilities may contribute to the equatorial superrotation in the real Venus atmosphere, though small-scale eddies by inertial instability are not strictly simulated in our AGCM.

### b. Maintenance of superrotation

In our simulation, upward global-mean angular momentum flux of meridional circulation pumps up angular momentum and balances downward global-mean angular momentum flux of waves in the middle and lower atmospheres. The formation of large equatorial flow (i.e., the existence of large equatorial angular momentum), which results from eddy momentum fluxes, is needed in order to maintain large upward angular momentum flux of meridional circulation. If the zonal flow *U* is not developed at the equator in Fig. 9a, the vertical angular momentum flux proportional to *UW*cos*ϕ* is predominantly negative. On the other hand, large positive *UW*cos*ϕ* (i.e., large upward angular momentum flux) is realized under the condition that the equatorial zonal flow is fully developed (Fig. 9b). In this situation, the angular momentum is efficiently transported upward.

As discussed above, the formation of the equatorial flow with large velocity (Fig. 9b) is important in the upward angular momentum transport required to maintain the superrotation. According to Hide’s theorem (Hide 1969), nonaxisymmetric eddy stress is indispensable to the formation of such an equatorial zonal flow. Figure 10 shows the vertical profiles of acceleration/deceleration at the equator. The deceleration due to the vertical EP flux is somewhat larger than the acceleration due to the vertical advection above the maximum of the equatorial zonal flow (>69 km). The excess deceleration (the difference between the vertical EP flux and advection) is cancelled by the acceleration due to the horizontal EP flux (∼1 m s^{−1} day^{−1}), which has the 20% contribution near 75 km in Fig. 10. The horizontal EP flux is dominantly produced by diurnal tide (Yamamoto and Takahashi 2004) and modifies the balance between the vertical EP flux and the vertical advection.

The advection decelerates the equatorial flow below the zonal wind maximum (<69 km). Both the horizontal and vertical EP fluxes accelerate the equatorial zonal flow. Figure 11a shows the horizontal EP flux spectrum at 65 km. The thermally induced waves produce the equatorward EP flux at the cloud top. This indicates that the thermal tides contribute to the zonal-flow acceleration due to the horizontal EP flux near 65 km in Fig. 10. Figure 11b shows the vertical EP flux spectrum at 2.8°. The thermally induced waves have negative vertical angular momentum flux above 60 km, while they have positive angular momentum flux in the region from 45 to 60 km. The thermal tides contribute to the cloud-top equatorial acceleration due to the vertical EP flux in Fig. 10, since the vertical EP flux of the thermally induced waves is largely changed near 60 km in Fig. 11b. The sign of the vertical angular momentum flux of the thermally induced waves is changed with height in the lower atmosphere. This is seen in the numerical experiment of Covey et al. (1986) on the conditions of weak Rayleigh friction and Newtonian cooling.

The eddy angular momentum transport due to waves with phase velocities from 25 to 100 m s^{−1} is upward (red in Fig. 11b) below the critical levels. These predominant signals mostly correspond to vertically propagating Kelvin waves (i.e., equatorially trapped gravity waves). In addition to thermal tides, Kelvin waves transport angular momentum upward in the equatorial region from 20 to 60 km.

*F*

_{CRV}is defined asThe vertical EP flux accelerates the zonal flow near 60° latitude at 29 km and decelerates it at 77 km. This results from the downward eddy angular momentum flux at midlatitudes. The horizontal EP flux accelerates the equatorial flow and decelerates the midlatitude flow at 29 and 77 km. The poleward momentum flux of the meridional circulation is much larger than the equatorward EP flux at the poleward branch of the meridional circulation (∼77 km). However, since the deceleration due to the mean meridional flow is mostly cancelled by the acceleration due to the mean vertical flow, the net acceleration/deceleration of the meridional circulation (the meridional and vertical flows) is not extremely large. The small acceleration of the horizontal EP flux can easily modify the balance among the residual mean meridional circulation, the curvature force, and the vertical EP flux in the acceleration/deceleration profile. This is similar to the modification of zonal mean flow due to the horizontal eddy diffusion (required in the Gierasch mechanism), since the EP flux transports angular momentum toward the equator.

The diurnal tide with negative *F ^{Z}*

_{EP}propagates vertically above the cloud top and produces the equatorward angular momentum flux in the upper shear zone of the mean zonal flow. The basic flow, which largely changes with latitude and height, influences the structure of thermal tide on Venus. In this situation, since it is difficult that the thermal tide is decomposed into the vertical and horizontal structures over the whole vertical range, the horizontal structure of the vertically propagating tides changes with height in the region where the basic flow has large vertical and horizontal shears. The eddy horizontal flow and geopotential height spread toward the equator with height in the upper shear zone of the basic flow between 65 and 80 km, and the equatorward angular momentum fluxes are seen in this region. It appears that the thermal tide with downward angular momentum flux is refracted horizontally.

Planetary-scale Rossby and gravity waves transport angular momentum toward the equator in the lower atmosphere. In particular, the pair of planetary-scale Kelvin and Rossby waves with the same frequency (Iga and Matsuda 2005) produces the equatorward momentum flux. Although barotropic instability assuming nondivergent eddies has been expected to contribute to the Gierasch–Rossow–Williams scenario (Gierasch 1975; Rossow and Williams 1979), shear instability generating divergent eddies plays a crucial role in the lower atmosphere of our Venus-like AGCM.

### c. Planetary-scale waves in the cloud layer

Diurnal and semidiurnal thermal tides are predominant near the equatorial cloud base. In addition, planetary-scale Kelvin waves with periods of 5.3, 5.8, and 6.1 days are also seen at low latitudes, together with high-latitude Rossby waves with the same periods. The eddy geopotential amplitudes with periods of 5.3, 5.8, and 6.1 days are one-third of the most predominant signal (semidiurnal tide) at 45 km. Figure 13 shows a phase-velocity–latitude cross section of the horizontal EP flux of 45 km. Together with thermal tides, the planetary-scale waves with periods of 5–6 days have equatorward angular momentum fluxes near 40° latitude. Figure 14 shows latitude–height distributions of the horizontal structure and the energy conversion C1 for a 6.1-day wave. The Kelvin wave accompanies the high-latitude vortical eddies such as the Rossby wave (Fig. 14a). The latitude–height distribution of the energy conversion C1 defined in the appendix is shown in Fig. 14b. Other terms C2 to C6 are neglected in comparison with C1. These waves are generated by the shear instability near the cloud base, since the zonal-mean to eddy energy conversion occurs in the horizontally differential superrotation. Accordingly, the fast planetary-scale Kelvin waves corresponding to planetary-scale NIR oscillations with the 5–6-day periods are generated by shear instability near the cloud base. They propagate vertically and are absorbed by zonal mean flow in the cloud layer. Although small EP fluxes of fast gravity waves with phase velocities of ∼100 m s^{−1} exist near the cloud base, the 4-day wave does not appear at the cloud top (65–70 km) in this experiment, since fast mean zonal flow (>110 m s^{−1}) absorbs the 4-day wave below the cloud top.

Thermally induced waves are predominant near the cloud-top regions where the solar heating maximum is located. Figure 15 shows horizontal structures of a thermally induced wave with zonal wavenumber 1 near the cloud top. The geopotential height has the horizontal structure of the Hough mode of −1 (Lindzen 1966). The poleward and equatorward winds of ∼30 m s^{−1} are seen at the 150° and 330° longitude, respectively (Fig. 15a). As shown in Figs. 15a,b, the strong poleward (equatorward) wind blows from the upwelling area in low latitude (high latitude) to the downwelling area in high latitude (low latitude). It follows that the meridional circulations associated with the diurnal tide are formed in latitude–height cross sections at the 150° and 330° longitude. Since the vertical flows are opposite across the 50° latitude in the areas of strong meridional flow, the phase difference between low- and high-latitude vertical flows is ∼180° in the longitudinal direction in Fig. 15b. The temperature structure of the diurnal tide is shown in Fig. 15c. The temperature component has peaks of ∼3.0 and ∼1.2 K at the 40° and 70° latitudes at 65 km, and the phase difference between these two peaks is ∼180° in the longitudinal direction (Fig. 15c). The latitudinal temperature structure is similar to the result of the *Pioneer Venus* OIR channel 5 (Elson 1983), in which there are two peaks of ∼4.5 K at ∼60° latitude and ∼3.0 K at ∼85° latitude and their phase difference is ∼180°.

Figure 16 shows a horizontal structure of a thermally induced wave with a zonal wavenumber 2 (which corresponds to semidiurnal tide). The eddy components of geopotential height and zonal wind are almost in phase at low latitudes. The temperature amplitude of ∼1.8 K at the equator is somewhat smaller than that obtained from the OIR channel 5 (∼3 K; Elson 1983), and the midlatitude temperature peak (60° latitude) is much weak in comparison with the observed amplitude (∼2.5 K).

## 4. Sensitivity study

The mean zonal flow of more than 100 m s^{−1} is maintained at the cloud top in the case in which the Newtonian cooling rate is changed to that of YT03 in Fig. 1b (Yamamoto and Takahashi 2003c). The large Newtonian cooling (i.e., YT03) above the cloud top does not significantly influence the formation and maintenance of Venus’ superrotation.

The angular momentum is exchanged between the surface and atmosphere through the surface drag with time constant of *τ*_{sfc}, and the equator–pole surface potential temperature difference *dθ*_{sfc} is given as a surface thermal forcing of the atmospheric circulation. Since *dθ*_{sfc} of 10 K is larger than the observation in the lower atmosphere (<5 K), the sensitivity of the superrotation to *dθ*_{sfc} should be examined in this section. In addition, since *τ*_{sfc} is still unknown in the boundary layer of the dense CO_{2} atmosphere, the sensitivity to the surface drag should be also examined. Thus, four sensitivity experiments listed in Table 1 were conducted in this study. Figure 17a shows the results for the case that *dθ*_{sfc} is changed from 10 to 5 K (case I-a), and Fig. 17b shows the results for the case that *τ*_{sfc} is changed from 3 to 10 days (case II-a). The equatorial zonal mean flows reach the equilibrium state at 50 000 days, though the large temporal variations from 50 to 110 m s^{−1} with periods of ∼10 000 days are seen near the cloud top.

Figure 18a shows the results for the case in which *dθ*_{sfc} is changed from 10 to 0 K (Case I-b), and Fig. 18b shows the results for the case that *τ*_{sfc} is changed from 3 to 30 days (Case II-b). The equatorial zonal mean flows reach the equilibrium state at 30 000 days at 21 km. The temporal variations of the cloud-top zonal mean flow range from 20 to 50 m s^{−1}.

The superrotation becomes weaker with decreasing *dθ*_{sfc} or increasing *τ*_{sfc}. For case I-b, the weak superrotation of ∼30 m s^{−1} is seen in the equatorial upper cloud layer (∼60 km), as shown in Fig. 19. In comparison with the standard case (section 3), the meridional circulation is weakened in Figs. 20a,b. Since the meridional circulation is weak, the upward momentum flux is small. Thus, the superrotation cannot be fully developed in the middle atmosphere. Large downward momentum fluxes of planetary-scale waves are confined in the polar regions (Fig. 20c). Slowly propagating vortical eddies like Rossby and mixed Rossby gravity waves are predominant in these fluxes, though inertio-gravity waves are also seen. The eddy horizontal momentum flux is poleward at mid- and high latitudes above 40 km (Fig. 20d). Planetary-scale waves with periods longer than 30 days predominantly contribute to the poleward momentum transport. The predominant waves with large poleward momentum fluxes have the structure of a mixed Rossby gravity wave, which has downward EP flux in the lower atmosphere. In this case, since the eddy horizontal momentum flux is poleward at mid- and high latitudes near 60-km altitude, the equatorial mean zonal flow cannot be accelerated by eddies near the cloud top. Although the eddy horizontal momentum flux is equatorward below 40 km, the superrotation mechanism in section 3 does not efficiently work because of weak meridional circulation and poleward eddy momentum flux in case I-b. The same process is also seen in case II-b.

The decrease of *dθ*_{sfc} and the increase of *τ*_{sfc} result in the weakness of the meridional circulation. In the cases of small *dθ*_{sfc} and large *τ*_{sfc}, the bottom thermal forcing due to *dθ*_{sfc} and *τ*_{sfc} is weakened. As a result, the meridional circulation is weakened and thus the upward angular momentum transport becomes inefficient. The equatorial mean zonal flow cannot be accelerated by eddies with the poleward momentum fluxes near 60-km altitude. Accordingly, it is difficult to maintain the superrotation of more than 100 m s^{−1} in these cases.

## 5. Conclusions

A fully developed superrotation of more than 100 m s^{−1} is reproduced in our AGCM experiment. In addition to the superrotation, some waves observed in the Venusian cloud layer are also reproduced in the present study. The zonal wavenumber-1 Kelvin waves with periods of 5–6 days correspond to the planetary-scale 5.5-day NIR markings. The Kelvin wave accompanying the high-latitude Rossby wave with the same frequency is generated by shear instability near the cloud base. The horizontal temperature structure of thermally induced wave with zonal wavenumber 1 (which corresponds to diurnal tide) is similar to that observed in the OIR observation at the cloud top (65 km). The simulated wave has double peaks in the latitudinal distribution of the eddy temperature amplitude, and the phase difference of these two peaks is ∼180°.

The angular momentum is transported upward by the meridional circulation and downward by the vertically propagating waves. The momentum transport due to waves maintains the equatorial cloud-top superrotation and raises the efficiency of pumping up the lower-atmospheric angular momentum. In other words, the fully developed superrotation is maintained by the meridional circulation and the eddies. The horizontal and vertical angular momentum fluxes of thermally induced waves produce the cloud-top equatorial superrotation of more than 100 m s^{−1} near the height region where the solar heating is strongest, and planetary-scale waves with phase velocities of 0–50 m s^{−1} transport angular momentum toward the equator in the lower atmosphere (<40 km) far from the solar heating maximum. The slowly traveling high-latitude Rossby waves with the equatorial phase velocities smaller than 20 m s^{−1} contribute to equatorward angular momentum transport at 29 km. In coexistence of an equatorial Kelvin wave and a high-latitude Rossby wave with the same frequency, the planetary-scale pattern of the two different waves produces the equatorward flux at 29 km. These waves deposit enough angular momentum into the upward branch of the meridional circulation and thus contribute to the efficient pumping-up of angular momentum in the lower atmosphere. Although barotropic instability assuming nondivergence has been expected to play a crucial role in the Gierasch–Rossow–Williams scenario (Gierasch 1975; Rossow and Williams 1979), shear instability including horizontal divergence (Iga and Matsuda 2005) contributes to the equatorward momentum transport in the lower atmosphere where it is difficult for the geostrophic assumption to be satisfied.

The surface conditions of the equator–pole temperature difference and the frictional drag influence the intensity of the superrotation. In the cases of small *dθ*_{sfc} and large *τ*_{sfc}, the upward angular momentum transport caused by the meridional circulation is inefficient, and the poleward eddy momentum is predominant near 60-km altitude. The pumping-up mechanism of angular momentum (by the meridional circulation with the help of the equatorward momentum flux) does not efficiently work, and the superrotation of more than 100 m s^{−1} cannot be formed in these experiments.

Although the superrotation is produced in the simplified AGCMs, a real Venusian superrotation mechanism is still unknown at the present stage. In addition to the further observations, the further improvements of AGCMs are needed in order to elucidate the real superrotation mechanism. Together with the improvement of the radiation code, more realistic surface processes should be incorporated into Venus AGCM.

This study is supported by the cooperative research project of the Center for Climate System Research, University of Tokyo, and by the JSPS Grant-in-Aid for Young Scientists (B) (No. 14740278 and 17740313). Numerical experiment was conducted on the HITACHI SR8000 at information technology center of the University of Tokyo. The GFD-DENNOU library is used for drawing figures.

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# APPENDIX

*T*is the reference temperature and

_{S}*H*is the local scale height.

_{S}Experimental conditions.