The thermal tides in the lower atmosphere of Venus are examined. It is shown that both the diurnal and semidiurnal tides excited in the cloud layer propagate to the ground, and the existence of thermal tides below the cloud bottom can be attributed almost to the solar heating in the cloud layer. At altitudes of 0–10 km, the atmospheric superrotation is accelerated and decelerated by the momentum transport associated with the diurnal and semidiurnal tides, respectively. The effect of diurnal tide is much smaller than that of the semidiurnal tide, so that the mean zonal flow is accelerated in the direction opposite to the Venus rotation there. It is argued that this momentum transport by the semidiurnal tide is balanced with the surface friction in the atmospheric layer adjacent to the ground, and the net momentum is supplied from the solid Venus to maintain the atmospheric superrotation.
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 In the Venus atmosphere, thermal tides are strongly excited by the solar heating in the cloud layer, and propagate both upward and downward. Since angular momentum is transported by their propagation, the thermal tides exert a significant influence on the general circulation of the entire Venus atmosphere, especially a zonal retrograde superrotation extending from the ground to 90–100 km levels [Schubert, 1983].
Fels and Lindzen  discussed the interaction of the thermal tides and the mean flows theoretically. It is shown that the atmospheric superrotation can be generated in the heating layer by the momentum transport associated with the vertical propagation of the thermal tides. In their model, a pair of positive and negative zonal momentum is generated in the atmosphere, but net momentum as observed in the Venus atmosphere cannot be produced by this mechanism only. In the work of Plumb  on the generation mechanism of the atmospheric superrotation by the thermal tides, significance of momentum supply from the solid part by friction with the planetary surface is emphasized. It is theoretically shown that the strong mean flows with net momentum can be generated by the co-operation between the momentum transport associated with the thermal tides and the momentum pumping from a rigid boundary.
 By using a numerical model which covers altitudes from 30 to 110 km, Newman and Leovy  examined the dynamical effect of the thermal tides on the generation and maintenance of the Venus atmospheric superrotation. It is shown that the atmospheric superrotation near the cloud top may be maintained by horizontal transports of angular momentum due to the diurnal tide and by vertical transport due to the semidiurnal one. However, it seems that increase of the net angular momentum which leads to the generation of the superrotation near the cloud levels is produced by Rayleigh friction near the upper boundary in their result. The Rayleigh friction is balanced mainly with the deceleration of the mean zonal flow associated with the upward propagation of the semidiurnal tide. An appropriate boundary condition should be used to examine correctly the possibility of the maintenance of the superrotation. Moreover, since the lower boundary is located at 30 km in their model, the downward propagation of the thermal tides (i.e., downward momentum transport) cannot be represented correctly.
 In order to elucidate the dynamical effect of the diurnal and semidiurnal tides on the atmospheric superrotation of Venus, especially on the supply of net angular momentum from the solid planet to the atmosphere, the downward propagation of the thermal tides excited in the cloud layer is numerically examined in the present study.
 The numerical model used in the present study is based on that constructed by Takagi and Matsuda . The basic equations are primitive equations in spherical and log p coordinates. Conventional notation is used in the following. The thermal tides are described by the perturbation equations linearized about a basic field. In the present study, it is assumed that the mean zonal flow is in solid body rotation: (θ, ζ) = aΩ(ζ) cos θ, where a is the radius of Venus, θ latitude, and Ω(ζ) angular velocity of the solid body rotation at an altitude of ζ. The zonal wind speed increases monotonically with altitude, and reaches about 100 m s−1 near 60 km on the equator. Above 70 km, it decreases to 50 m s−1 at 100 km. It is also assumed that the pressure field (or temperature field) is balanced with the mean zonal flow by the relation of gradient wind. The horizontally averaged temperature and static stability are determined by the Venus international reference atmosphere (VIRA) data [Seiff et al., 1985].
 The distribution of solar heating is based on the observational and theoretical works of Tomasko et al.  and Crisp . Their results show that the solar heating has a local maximum near 65 km because of the absorption of solar flux in the cloud layer. It is also shown by Tomasko et al.  that about 12% of the solar energy absorbed by Venus reaches the ground. The solar heat absorbed by the ground is redistributed to the lower atmosphere by convection and radiation. However, it is assumed in the present study that the solar heat absorbed by the ground is distributed directly to the lowest layer, since the vertical structure of heat distribution is almost unknown.
 Newtonian cooling rates for the temperature deviations are taken from the work of Crisp . As shown by Takagi and Matsuda , the structures of thermal tides are strongly affected by Newtonian cooling. It is very important to take the scale-dependence of radiative damping into account. In the present study, the diurnal (semidiurnal) tides are calculated by using the vertical distributions of Newtonian cooling rates for temperature anomaly with vertical scales of 7 (30) km. Vertical eddy viscosity and vertical eddy thermal diffusion are also introduced into the perturbation equations. Prandtle number is assumed to be 1. The coefficient is set to a constant value, 2.5 m2 s−1, except in the lowest layer. By calculating radiative-convective equilibrium of the Venus atmosphere, Matsuda and Matsuno  showed that a convective layer appears in altitudes of 0–20 km. Since the dissipating effect of convection on the thermal tide is not clearly understood for the Venus atmosphere, this effect is simply assumed to be represented by the vertical eddy viscosity and diffusion. Several values of the vertical eddy viscosity, that is, 2.5–250 m2 s−1, are used in the lowest layer.
3.1. Downward Propagation
 In order to examine the property of downward propagation of the thermal tides excited in the cloud layer, the thermal tides are calculated for two profiles of the solar heating: profile (a) is obtained from the works of Tomasko et al.  and Crisp , and profile (b) is obtained by neglecting heating in a layer of 0–40 km altitudes (below the cloud bottom). The solar flux absorbed at the ground is neglected in these cases.
Figure 1 shows the vertical profiles of T′ × (temperature deviations multiplied by the square root of the basic state density) and phase of T′ associated with the diurnal and semidiurnal tides calculated for the two profiles of solar heating (a) and (b). The coefficients of the vertical eddy viscosity and thermal diffusion in the lowest layer, EV, are set to be 2.5 m2 s−1. T′ × can be regarded as energy density associated with the thermal tides. If the thermal tide is not damped, amplitude of T′ × is constant. Therefore, Figure 1 indicates that both the diurnal and semidiurnal tides are strongly damped in the upper layer above the cloud top, and are less so in the lower layer below the cloud bottom. It should be noted here that the amplitude of T′ × is not completely damped in the lower layer below 40 km for both the diurnal and semidiurnal tides. The distribution of T′ × obtained for the heating profile (a) (the dashed lines) is almost overlapped with that for (b) there. This means that the diurnal and semidiurnal tides excited in the cloud layer propagate in the lower layer, and predominates there over disturbance induced directly by the heating at 0–40 km altitudes. It is argued from this result that the angular momentum is transported from the cloud layer to the lower atmosphere by the thermal tides, and the transported momentum accelerates the lower atmosphere in the direction opposite to the Venus rotation.
 The thermal tides are calculated also for other values of EV. Though the amplitude of the thermal tides near the ground becomes smaller for the larger values of EV, the results remain qualitatively similar. It is confirmed that the thermal tides excited in the cloud layer propagate downward and reach the lowest atmosphere near the ground.
3.2. Effect of the Ground Heating
 While most of the solar energy absorbed by Venus is concentrated in the cloud layer, about 12% of it is absorbed by the ground. The globally averaged net solar flux is about 17 W m−2 at the ground [Tomasko et al., 1980]. The solar energy absorbed by the ground is redistributed to the lower Venus atmosphere, and may excite the thermal tides. Here, it is assumed that the heat is redistributed equally to the lowest atmospheric layer with thicknesses of 5, 10 or 20 km. The vertical distributions of T′ × associated with the thermal tides calculated for these heating profiles are shown in Figure 2. EV is set to be 2.5 m2 s−1. In the case of diurnal tide (Figure 2a), the obtained distributions are very different in a layer of 0–15 km altitudes. It is also confirmed that the phase structures are different there (not shown). It appears, however, that above 15 km the diurnal tide is not affected by the ground heating. It may be suggested that the diurnal tide response is almost in the form of trapped wave. In the case of the semidiurnal tide (Figure 2b), the difference in the amplitude of T′ × among the heating profiles is much smaller than that in the case of diurnal tide. The waves excited by the ground heating propagate upward, and reach the cloud layer. The difference in the vertical propagation property between the diurnal and semidiurnal tides may be attributed to difference of their vertical group velocities. It is inferred from the dispersion relation of internal gravity wave and the vertical wave numbers shown in Figure 1 that the vertical group velocity of the semidiurnal tide is about 10 times larger than that of the diurnal one in a layer of 0–40 km altitudes.
3.3. Mean Flow Acceleration
 As shown in Figure 1, the thermal tides excited in the cloud layer propagate upward and downward to reach the ground. The semidiurnal tide induces mean flow acceleration in the upper cloud layer (60–70 km), and deceleration above the cloud top [Newman and Leovy, 1992]. It may be expected here that the mean flow deceleration (i.e., acceleration in the direction opposite to the retrograde atmospheric rotation) is induced in the lower atmosphere by the downward propagation of the thermal tides.
Figure 3 shows the meridional-height distributions of the mean flow acceleration and deceleration rates produced by the diurnal and semidiurnal tides at 0–20 km altitudes. These rates are evaluated for the thermal tides calculated for the case with EV = 2.5 m2 s−1 and without the ground heating. It is found that the mean zonal flow is not accelerated nor decelerated in the lowest levels by the diurnal tide. However, it is decelerated by the semidiurnal tide at 0–10 km altitudes in low latitudes. The local maximum of the deceleration rates about 2.4 × 10−4 m s−1 day−1 is located at 5 km altitude on the equator. It might be speculated that this deceleration rate is negligibly small. However, if the acceleration rate is rescaled by the ratio of the basic atmospheric density, this rate corresponds to about 1.4 × 10−1 m s−1 day−1 at the cloud top (70 km), which is comparable to the mean zonal acceleration rate by the semidiurnal tide there. It is argued from this result that the acceleration of zonal flow in the direction opposite to the Venus rotation by the momentum transport due to the semidiurnal tide may be balanced with the deceleration by the surface friction, namely, the net angular momentum may be pumped up from the ground to the atmosphere. Since the tidal waves act perpetually, there is a possibility that the net angular momentum required for creating the superrotation existing in the Venus atmosphere is supplied from the ground by the pumping mechanism due to the downward propagation of the semidiurnal tide. At 10–40 km altitudes, the mean zonal flow is decelerated in low latitudes and accelerated in mid latitudes by both the diurnal and semidiurnal tides. This tendency is contrasted with that given by Newman and Leovy , who argued that equatorward momentum transport occurs in the cloud levels.
 The deceleration of the mean zonal flow in the lowest layer by the semidiurnal tide decreases for larger values of EV. The local maximum of the deceleration rates at 5 km is reduced to 0.7 × 10−4 m s−1 day−1 for EV = 25.0 m2 s−1 and 0.9 × 10−5 m s−1 day−1 for EV = 250.0 m2 s−1. It is confirmed, however, that the qualitatively similar results are obtained for the semidiurnal tide in the wide range of EV.
 Effects of the ground heating on the acceleration and deceleration of the mean zonal flow in the lower atmosphere are also examined. It is shown in Figure 4 that the ground heating has the accelerating and decelerating effects for the diurnal and semidiurnal tides at 0–7 km altitudes, respectively. However, even for the diurnal tide calculated with the ground heating distributed over the layer with 5 km thickness, its acceleration rate is much smaller than those due to the semidiurnal ones with or without the ground heating. It is confirmed that the effect of the semidiurnal tide is predominant over that of the diurnal one in the lower atmosphere and this effect is not so affected by the ground heating.
4. Concluding Remarks
 The diurnal and semidiurnal tides in the Venus atmosphere are calculated under the reasonable conditions, in which the vertical eddy viscosity and the ground heating are taken into account. It is shown that both the diurnal and semidiurnal tides excited in the cloud layer propagate downward to the ground. The amplitude and spatial structures of the disturbance in a layer of 0–40 km altitudes below the cloud bottom are determined almost by the thermal tides propagating downward from the cloud layer. Namely, the effect of solar heating both in the atmosphere below 40 km and at the ground on the thermal tides is very small even in the lower atmosphere except for the diurnal tide in 0–5 km.
 It is also demonstrated that the mean zonal flow is induced by the momentum transport associated with the vertical propagation of the thermal tides. At altitudes of 0–10 km, the mean zonal flow is strongly decelerated in the direction opposite to the Venus rotation by the semidiurnal tide. As pointed out by Plumb , the mean zonal flow induced by the semidiurnal tide is damped by the surface friction, and the net angular momentum is extracted from the solid Venus to the atmosphere. The deceleration effect measured by the change rate of angular momentum is estimated to be comparable to the acceleration effect of the mean zonal flow near the cloud top by the semidiurnal tide. In contrast to the effect of the semidiurnal tide, the mean zonal flow in 0–10 km is accelerated by the diurnal tide. However, the acceleration rates are much smaller than the deceleration rates due to the semidiurnal tide. This result indicates a possibility that the Venus atmospheric superrotation may be maintained by the momentum transport associated with the vertical propagation of the semidiurnal tide.
 In the present study, nonlinear interaction between the thermal tides and the mean zonal flow is not examined. Time-dependent problem should be studied in a future work.
 All the figures in this paper were produced by GNUPLOT and GFD-DENNOU Library.