Super-rotation independent of horizontal diffusion reproduced in a Venus GCM

Abstract

Horizontal diffusion and resolution are important factors to generate and maintain the super-rotation in the general circulation model (GCM) because waves which transport angular momentum are sensitive to them. Here, we investigated how the super-rotation depends on the magnitude of horizontal hyper diffusion using a Venus atmospheric GCM with medium and high resolutions. In both the runs, we found a parameter range where the structure of fully developed super-rotation is almost independent of the magnitude of horizontal diffusion. Spectral analysis shows that unrealistically strong super-rotation is developed when medium-scale disturbances are dissipated by stronger horizontal diffusion. On the other hand, artificially weak super-rotation is also realized because spurious small-scale disturbances are accumulated when the horizontal diffusion is too weak.

Graphical Abstract

Introduction

In the Venus atmosphere, fast zonal wind has been observed (e.g., Sánchez-Lavega et al. 2017; Read and Lebonnois 2018), which is called super-rotation, because it is much faster than the rotation period of the solid part of the planet. So far, Venus general circulation models (VGCMs) have been used to study the super-rotation and several studies obtained fully developed super-rotation faster than 100 m/s (e.g., Yamamoto and Takahashi 2003; Lee et al. 2005; Takagi and Matsuda 2007; Lebonnois et al. 2010; 2016). Sugimoto et al. (2019; hereafter STM2019) also reproduced fully developed super-rotation using a relatively high resolution VGCM named AFES-Venus (Sugimoto et al. 2014a), which is based on AFES (an atmospheric GCM for the Earth Simulator) (Ohfuchi et al. 2004), with weak vertical viscosity.

Recently, Yamamoto and Takahashi (2022; hereafter YT2022) pointed out that the super-rotation developed in their VGCM crucially depends on the magnitude of horizontal “hyper” diffusion, where “hyper” means that it is not normal Laplacian diffusion term of ${\nabla }^{2}(=\Delta )$. They suggested that the fully developed super-rotation obtained in STM2019 could be realized because strong horizontal diffusion was used in AFES-Venus. In the numerical simulation, it is desirable to set the magnitude of horizontal diffusion to be as weak as possible within the range where the energy is not accumulated at the maximum wavenumber. In AFES-Venus, the second-order horizontal hyper diffusion (Laplacian squared, ${\Delta }^{2}={\nabla }^{4}$) is used and the damping times for the maximum wavenumber component are determined by the spectral analysis. For example, the damping times are 0.1 Earth days for T42L60 (Sugimoto et al. 2014a; Ando et al. 2016; STM2019), 0.03–0.05 Earth days for T63L120 (Sugimoto et al. 2014b; Takagi et al. 2018), 0.01 Earth day for T159L120 (Kashimura et al. 2019), and 0.0003 Earth days for T639L260 (Sugimoto et al. 2021) runs, where T and L denote the triangular truncation number for spherical harmonics and vertical levels, respectively. These values are comparable to those used in the Earth GCM (MIROC; Model for Interdisciplinary Research on Climate) (K-1 model developers 2004). YT2022, however, used very weak horizontal diffusion (the relaxation times are more than one Earth day for T42L50 runs) and then the super-rotation was not fully developed. Furthermore, they did not propose an appropriate damping rate.

In the present study, we examine how the super-rotation depends on the magnitude of horizontal diffusion and propose the appropriate damping rate clearly by the spectral analysis. The fully developed super-rotation is maintained for both the medium (T42L60) and high (T159L120) resolution runs in a same range of the magnitude of horizontal diffusion. Note that since horizontally smaller disturbances tend to have smaller vertical scales, it is usual to increase the vertical resolution according to the horizontal resolution; though the increasing rates are usually not equal due to computational cost. Although we did not conduct sensitivity experiments on the vertical resolution, we have confirmed that there is no significant change in the atmospheric structures of T42L60 (Sugimoto et al. 2014a; Ando et al. 2016; STM2019), T42L120 (Ando et al. 2020), T63L120 (Sugimoto et al. 2014b; Ando et al. 2018; Takagi et al. 2018), T159L120 (Kashimura et al. 2019), and T639L260 (Sugimoto et al. 2021) runs.

Experimental setup

AFES-Venus is a spectral GCM for the Venus atmosphere solving nonlinear primitive equations on a sphere (Sugimoto et al. 2014a). The truncation wavenumbers are 42 for medium resolution runs (T42L60) and 159 for high resolution runs (T159L120), then the numbers of horizontal grids are 128 × 64 and 480 × 240 for the T42L60 and T159L120 resolutions, respectively. The atmosphere is divided into 60 and 120 levels from the ground to ~ 120 km at regular spacings of 2 and 1 km in height for the T42L60 and T159L120 resolutions, respectively. The vertical eddy viscosity is included and its coefficient is fixed to 0.0015 m²/s to maintain the super-rotation (STM2019). The horizontal eddy diffusion is the same as STM2019, which is represented by the second-order horizontal hyper diffusion (Laplacian squared, ${\Delta }^{2}={\nabla }^{4}$). For the sensitivity experiments, its damping time for the maximum wavenumber component is swept through 10 values (0.01, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 30 Earth days) for T42L60 runs and 12 values (0.0001, 0.0002, 0.0005, 0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.5, 1 Earth day(s)) for T159L120 runs. Note that because the strength of horizontal diffusion is expressed by the damping time for the maximum wavenumber component, the horizontal diffusion coefficient is about 4⁴ = 256 times larger in T42L60 than that in T159L120 when the same value is set for the damping time (see Additional file 1: S1); also note that, in this paper, the word “damping time” is always used to indicate that for the maximum wavenumber component. Rayleigh friction is used as the surface friction at the lowest level and its relaxation time is set to 0.5 Earth days. A sponge layer is applied only to eddy components above 80 km and its coefficient gradually increases with height.

The solar heating is based on Tomasko et al. (1980). In contrast to the previous studies STM2019 and YT2022, the thermal tides are excited in the present model because its diurnal component is included. The infrared radiative process is simplified by the Newtonian cooling with relaxation time based on Crisp (1989). A horizontally uniform temperature field, which is based on VIRA (Venus international reference atmosphere, Seiff et al. 1985) and observed static stability (Tellmann et al. 2009) with a weakly stratified layer extending from 55 to 60 km, is used as a relaxation field of the Newtonian cooling. The initial condition is an idealized super-rotating flow in solid-body rotation. At the equator the zonal flow increases to 100 m/s linearly with height from the ground to 70 km. Above 70 km it is constant. The horizontal distribution of temperature field is in gradient wind balance with this zonal flow. From this initial state, nonlinear numerical simulations are conducted for 4 Earth years. The model output is sampled at every 1 Earth day (24 h). Note that AFES-Venus has realistically reproduced the super-rotation with planetary-scale waves (Sugimoto et al. 2014b; Takagi et al. 2022), polar vortex with cold collar (Ando et al. 2016, 2017), thermal tides (Takagi et al. 2018; Suzuki et al. 2022), thermal structure in equatorial region (Ando et al. 2018), and planetary-scale streak structure (Kashimura et al. 2019) by the similar settings of this study. The directions of planetary rotation and basic zonal flow are set to be eastward (positive) in the present study. Details of experimental settings are described in Sugimoto et al. (2014a, b).

Results

Figure 1 shows zonal-mean zonal flow and temperature in a latitude–height cross section for the cases with (a) 0.01, (b) 0.1, (c) 1, and (d) 10 Earth days for the damping time of horizontal diffusion obtained for T42L60 run (see also Additional file 1: S2 (Figure S1) for the cases with 0.05, 0.2, 0.5, 2, 5, and 30 Earth days). They are temporally averaged over the last 1 Earth year (360 datasets of daily data). The fully developed super-rotation of ~ 130 m/s with mid-latitude jets appears at ~ 70 km for the cases with moderate damping time (0.1 and 1 Earth day(s)) of horizontal diffusion (Fig. 1b, c); these super-rotational flows are similar to those obtained in the previous studies (Sugimoto et al. 2014b; Takagi et al. 2018) and consistent with observations (e.g. Machado et al. 2012). For the case with stronger horizontal diffusion (0.01 Earth days), the faster super-rotation of ~ 180 m/s without mid-latitude jets appears (Fig. 1a). For the case with weaker horizontal diffusion (10 Earth days), the slower super-rotation of ~ 110 m/s without mid-latitude jets appears. These results are qualitatively consistent with that shown in YT2022. Because the super-rotation is very similar between the two cases with 10 times different magnitude of horizontal diffusion (Fig. 1b, c), it is suggested that the super-rotation is maintained by the resolved eddies and the dependence on unresolved eddies (i.e., horizontal diffusion) is negligible in this range.

Fig. 1

Zonal-mean zonal flow (colour, m/s) and temperature (contour, K) obtained for T42L60 run. Latitude–height cross sections for the cases with a 0.01, b 0.1, c 1, and d 10 Earth days for the damping time of horizontal diffusion are shown. They are temporally averaged over the last 1 Earth year (360 datasets of daily data)

In order to check the accumulation of spurious small-scale disturbances, the spectral analysis of kinetic energy of horizontal wind is performed for T42L60 run. Figure 2a shows how it depends on the damping time of horizontal diffusion at 70 km. It is clearly seen that the spectra in large-scales (with total wavenumber less than 5) are almost unchanged for the cases with moderate damping time (from 0.1 to 2 Earth days). For the cases with stronger horizontal diffusion (0.01 (orchid) and 0.05 (purple) Earth days), the spectra in small-scales decrease with total wavenumber quite rapidly and those in large-scales are smaller than those obtained for the moderate horizontal diffusion. For the cases with weaker horizontal diffusion (5 (lime green), 10 (khaki), and 30 (mocha) Earth days), the spectra in small-scales increase with total wavenumber, indicating the accumulation of spurious small-scales. The spectra in large-scales (with total wavenumber 2 and 3) are also smaller than those obtained for the moderate horizontal diffusion.

Fig. 2

Kinetic energy of horizontal wind per unit mass per unit wavenumber for T42L60 run. a Dependency on the damping time of horizontal diffusion at 70 km. b Dependency on the height for the case with 0.1 Earth days for the damping time of horizontal diffusion. Gradient of n^−5/3 and n⁻³ are shown by solid lines. The analysed data is the final day of the numerical simulation

Figure 2b shows how the kinetic energy of horizontal wind depends on height of for the case with the moderate damping time (0.1 Earth days) of horizontal diffusion. Though the spectral slope in small-scales tend to be shallower above 80 km because of wave propagation from 60 to 70 km altitudes (the upper cloud layer), the accumulation of spurious small-scales does not occur. For weaker horizontal diffusion (with the damping time longer than 0.2 Earth days), the accumulation of spurious small-scales occurs in the upper atmosphere above 80 km even though the obtained super-rotation is similar in each case at the cloud top level. This result suggests that influence of the accumulation of the kinetic energy in small-scales above 80 km might be cancelled out by the sponge layer. However, to be on the safer side, we used 0.1 Earth days for the damping time of horizontal diffusion for T42L60 run in Sugimoto et al. (2014a) and STM2019.

Usually, results obtained with lower resolutions are validated by those with higher resolutions. Figure 3 shows zonal-mean zonal flow and temperature in a latitude–height cross section for the cases with (a) 0.0001, (b) 0.001, (c) 0.01, (d) 0.1, and (e) 1 Earth day(s) for the damping time of horizontal diffusion obtained for T159L120 high resolution run. They are again temporally averaged over the last 1 Earth year (360 datasets of daily data). The fully developed super-rotation of ~ 130 m/s with mid-latitudes jets appears at ~ 70 km for the cases with a wide range of damping time (from 0.001 to 0.1 Earth days) of horizontal diffusion (Fig. 3b–d), which is very similar to those obtained for T42L60 run with the moderate damping time (0.1 and 1 Earth day(s)) of horizontal diffusion (Fig. 1b, c). For the case with stronger horizontal diffusion (0.0001 Earth days), the faster super-rotation of ~ 160 m/s without mid-latitude jets appears (Fig. 3a). For the case with weaker horizontal diffusion (1 Earth day), the slower super-rotation of ~ 100 m/s without mid-latitude jets appears. These results are also consistent with that obtained for T42L60 run. Again, because the super-rotation is very similar among the cases in a very wide parametric range of the horizontal diffusion strength with different resolutions of T159L120 (Fig. 3b–d) and T42L60 (Fig. 1b, c), it is strongly suggested that the super-rotation is maintained by the resolved eddies and the dependence on horizontal diffusion is negligible in this range.

Fig. 3

Zonal-mean zonal flow (colour, m/s) and temperature (contour, K) obtained for T159L120 run. Latitude–height cross sections for the cases with a 0.0001, b 0.001, c 0.01, d 0.1, and e 1 Earth day(s) for the damping time of horizontal diffusion are shown. They are temporally averaged over the last 1 Earth year (360 datasets of daily data)

The spectral analysis of kinetic energy is also performed for T159L120 run. Figure 4a shows its dependency on the damping time of horizontal diffusion at 70 km. It is clearly seen that for the cases with a wide range of moderate damping time (from 0.0005 to 0.1 Earth days) of horizontal diffusion, the spectra in large- and medium-scales (with total wavenumber less than 20) are almost unchanged among the cases and similar to those obtained in T42L60 run with a damping time of 0.1 Earth days (black) of horizontal diffusion. For the cases with stronger horizontal diffusion (0.0001 (purple) and 0.0002 (blue) Earth days), the spectra rapidly decrease with total wavenumber more than 10. For the cases with weaker horizontal diffusion (0.5 (orange) and 1 (pink) Earth day(s)), the spectra in small-scales increase with total wavenumber, indicating the accumulation of spurious small-scales. The spectra in large-scales (with total wavenumber less than 10) are also smaller than those with the moderate horizontal diffusion. Figure 4b shows the dependency of kinetic energy on height for the case with the moderate damping time (0.01 Earth days) of horizontal diffusion. Though the spectral slope in small-scales tend to be shallower above 80 km because of the wave propagation from the upper cloud layer (from 60 to 70 km), the accumulation of spurious small-scales does not occur, as in T42L60 run. For weaker horizontal diffusion (larger than 0.02 Earth days), even though the super-rotation is similar in each case at the cloud top level, the accumulation of spurious small-scales occurs in the upper atmosphere above 80 km. With the same reason mentioned above for previous studies with the T42L60 resolution, we used 0.01 Earth days for the damping time of horizontal diffusion for T159L120 resolution in Kashimura et al. (2019).

Fig. 4

Kinetic energy of horizontal wind per unit mass per unit wavenumber for T159L120 run. a Dependency on the damping time of horizontal diffusion at 70 km. b Dependency on the height for the case with 0.01 Earth days for the damping time of horizontal diffusion. Gradient of n^−5/3 and n⁻³ are shown by solid lines. They are temporally averaged over the last 1 Earth year (360 datasets of daily data)

Finally, zonal wind deviation (color, m/s) from its zonal mean at 70 km is shown in a longitude–latitude cross section for the cases with (a) 0.01, (b) 0.1, (c) 1, and (d) 10 Earth days of the damping time of horizontal diffusion obtained for T42L60 run (Fig. 5). In all the cases, the diurnal (semi-diurnal) tide with wavenumber 1 (2) in the mid- to high-latitudes (low-latitude) are predominant. For the case with moderate horizontal diffusion (0.1 Earth days), there are also medium-scale disturbances (Fig. 5b). However, for the case with stronger horizontal diffusion (0.01 Earth days), only the large-scale structure can be seen; this indicates that disturbances of medium-scales are dissipated or decreased by the horizonal diffusion (Fig. 5a). For the cases with weaker horizontal diffusion (1 and 10 Earth days; Fig. 5c, d), small-scale disturbances appear randomly, reflecting the accumulation of the kinetic energy in the smallest scales shown by magenta and purple lines in Fig. 2a.

Fig. 5

Zonal wind disturbances (color, m/s) from its zonal mean obtained for T42L60 run. Longitude–latitude cross sections at 70 km for the cases with a 0.01, b 0.1, c 1, and d 10 Earth days of the damping time of horizontal diffusion are shown. The analysed data is the final day of the numerical simulation

Summary and discussion

In the present study, we investigated how the super-rotation depends on the magnitude of horizontal diffusion using AFES-Venus (atmospheric general circulation model for the Earth Simulator for Venus) with medium (T42L60) and high (T159L120) resolutions. For a range of the magnitude of horizontal diffusion (from 0.1 to 1 Earth days for T4260 and from 0.0005 to 0.1 Earth days for T159L120 runs, respectively), fully developed super-rotation was obtained for several cases in both the resolutions, which is almost independent of the magnitude of horizontal diffusion and similar among these cases. For the cases with stronger horizontal diffusion, unrealistically strong super-rotation was developed and medium-scale disturbances are vanished. It is suggested that equatorward momentum transport caused by strong horizontal hyper diffusion can produce the strong super-rotation artificially, namely, Gierasch-Rossow-Williams mechanism (Gierasch 1975; Rossow and Williams 1979) works too much in these cases. For the cases with weaker horizontal diffusion, weak super-rotation was artificially realized and the accumulation of spurious small-scale disturbances occurred. It seems that spurious small-scale disturbances and inverse cascades from small-scales work to weaken the super-rotation and this effect is larger than that of resolved disturbances which contribute to the maintenance of super-rotation. Therefore, the results strongly suggested that the damping rate of 0.1 Earth days for the horizontal diffusion in T42L60 is reasonable and disturbances is resolved to maintain the super-rotation for that value.

Remember that we have expressed the strength of horizontal diffusion (represented by the second-order Laplacian) by the damping time for the maximum wavenumber component of the resolution setting. The value of horizontal diffusion coefficient that corresponds to the damping time of 0.1 Earth days for the horizontal wavenumber 42 (in Venus) is ~ 4.8 × 10¹⁶ m⁴/s (see Additional file 1: S4). With this coefficient value, the damping time for the wavenumber 159 is ~ 0.0005 Earth days. The reproduced super-rotation in T42L60 run with the damping time of 0.1 Earth days could be verified by T159L120 run with the damping time of 0.0005 Earth days (Figure S2a) and it is also confirmed that the two spectra match very well in the spectral analysis (Fig. 4a; black and blue lines). Similarly, the magnitude of horizontal diffusion with the damping time of 10 Earth days for the case of T42L60 is comparable to that with ~ 0.05 Earth days for the case of T159L120. However, since the reproduced super-rotation in T42L60 run is totally different from that reproduced in the cases of T159L120 run, T42L60 run with the damping time of 10 Earth days cannot be reliable.

Similarity of the obtained super-rotation with different resolutions (T42L60 and T159L120) and different strength of horizontal diffusion suggests that disturbances smaller than ~ 900 km is less important at least for maintaining the super-rotation. Although the slope of the energy spectra is out of focus in the present study, there seems to be a region of energy cascade with –5/3 slope for total wavenumber less than ~ 20 (~ 50) in T42L60 (T159L120) runs, which is consistent with that for the Earth (Takahashi et al. 2006). It is also well known that the direct energy cascade from large-scales of motion (quasi-2D flow) to small-scales with –3 slope (Nastrom and Gage 1985) in larger scales, though it is difficult to find such kind of slope in Figs. 2 and 4. In conclusion, we should carefully check the magnitude of horizontal diffusion by the spectral analysis and higher resolution runs before conducting the long-term Venus GCM simulations to reproduce the super-rotation.