Contribution of Thermal Tides to Venus Upper Cloud-Layer Superrotation

2.1 Data Reduction

The 1,016 temperature profiles were derived from RO observations by Venus Express/VeRa (Tellmann et al., 2009) and Akatsuki/RS (Imamura et al., 2011, 2017, 2025; Murakami et al., 2017) between July 2006 and December 2022, with a 2-year gap from April 2014 to February 2016. Figure 1 shows the spatial, local time, and latitudinal distribution of these profiles between 50 and 90 km. Retrieval uncertainties were estimated using a linear error propagation method (Lipa & Leonard Tyler, 1979; Tellmann et al., 2012), primarily reflecting uncertainties in the assumed upper boundary conditions, which decrease with altitude. Given the different upper boundary altitudes used in VeRa (100 km) and RS (95 km), only data points with temperature uncertainties below 1.0% (Figure 1c) and pressure uncertainties below 10% were used. The lowest five altitude points of each profile were excluded due to telemetry issues and inaccuracies in determining number density. The horizontal drift of VeRa radio waves caused by the polar orbit of Venus Express at low latitudes is limited to a few degrees (Figure 1a), which maintains the validity of the hydrostatic assumption (Pätzold et al., 2007; Tellmann et al., 2009); thus, this effect is not considered.

To remove potential long-term trends as previously observed in UV albedo and derived winds (Horinouchi et al., 2024; Khatuntsev et al., 2022; Lee et al., 2019), which may obscure tidal signals, we applied a least-squares polynomial fit to the RO data. The data were first averaged at daily, monthly, seasonal, and yearly scales to mitigate the influence of uneven temporal sampling and time-weighted effects on the long-term fitting. A second-order polynomial fit was applied to normalize data within each 1 km × 20° latitude × 4-hr local time bin, provided that the bin contained more than five yearly data points. The processed data were gridded using the inverse distance weighted method at 1 km altitude, 5° latitude, and 1-hr local time resolution, with interpolation limited to nearest neighbors to reduce the influence of distant points.

2.2 Model Summary

The Venus PCM is built on the LMDZ latitude-longitude grid finite-difference dynamical core (Hourdin et al., 2006), incorporating a soil model (Hourdin et al., 1993), a topographic boundary layer scheme (Lebonnois et al., 2018), and a radiative transfer module that incorporates latitudinally varying cloud properties. Shortwave solar heating rates are taken from a look-up table derived from the radiative transfer calculations of Haus et al. (2015). Longwave infrared cooling is computed using the correlated k-distribution method and the net exchange rate formalism (Eymet et al., 2009), with cloud aerosol density profiles consistent with those used by Haus et al. (2015). For detailed descriptions, see Garate-Lopez and Lebonnois (2018) and Lebonnois et al. (2016). The Venus PCM simulations were initialized from the Venus Climate Database version 2.2 state (Martinez et al., 2023) and analyzed over the last 5 Venus days after a 25 Venus day (∼2919 Earth days) spin-up. The horizontal resolution is 1.25° in both latitude and longitude, with 50 hybrid vertical levels extending up to ∼95 km. Each Venus day is output by 240 time steps. The simulated background fields show good agreement with observed zonal wind, temperature, and static stability (Garate-Lopez & Lebonnois, 2018; Lai et al., 2024; see Figure S1 in Supporting Information S1).

2.3 Methodology for AM Diagnostics

3.1 Thermal Tidal Structure

Although the middle atmosphere of Venus has been previously observed using infrared spectrometers (Schofield & Taylor, 1983; Zasova et al., 2002), UV and IR cameras/imagers (Fukuya et al., 2021; Grassi et al., 2010; Horinouchi et al., 2018), and RO experiments (Ando et al., 2020), a comprehensive view of the global thermal tidal structure, especially in the southern hemisphere, is still lacking. Figure 2 shows the local time-altitude structure of temperature derived from RO data after data reduction, focusing on southern low (0°–20°), middle (30°–50°), and high (60°–80°) latitudes. These observations are compared with simulations from the Venus PCM.

At high southern latitudes, the structure is less coherent but exhibits predominantly diurnal tidal features in the upper cloud layer (60–70 km), transitioning to a mixture of diurnal and semidiurnal components above. These features align with Venera-15 IR results, showing diurnal tidal amplitude peaks of ∼6 K in cloud for latitudes 55–75°N, as well as both semidiurnal and diurnal tidal amplitude peaks of ∼3 K in these latitudes (Zasova et al., 2002). A similar pattern is reproduced in the Venus PCM simulations. The irregularities in the RO data at high latitudes may be attributed to the zonal drift of the sampled locations, particularly in the southern hemisphere (Figure S2 in Supporting Information S1), and the degraded longitudinal resolution resulted from the reduced zonal circumference at high latitudes (Figure 1a). Although the native resolution is sufficient to resolve gravity waves with wavelengths of ∼100 km (Tellmann et al., 2012), these limitations affect the interpretability of the data. Therefore, high-latitude results are not analyzed further. The tidal structure below 60 km diminishes rapidly due to the lower static stability of the atmosphere, as shown in Figure S1 in Supporting Information S1. Therefore, we do not compare the observed structures at these altitudes in detail with the model, as they may result from sparse sampling. To date, no reliable observations have confirmed the tidal characteristics below 60 km, although some modeling studies have investigated tidal transport in the deep atmosphere (Newman & Leovy, 1992; Takagi & Matsuda, 2006).

At low latitudes, wavenumber-2 structures in local time (semidiurnal tides) dominate above 60 km, with eastward phase tilts indicating upward energy propagation. This pattern aligns with previous observations from LIR (Akiba et al., 2021), OIR (Schofield & Taylor, 1983), and RO (Ando et al., 2018) at 0°–30° latitudes. Above 75 km, the peak-to-peak temperature variations in the 12–24 hr local time sector are approximately twice those in the 0–12 hr sector, suggesting the superposition of strong wavenumber-1 (diurnal) components onto the semidiurnal structure. This pattern is consistent with Venus PCM simulations and earlier OIR observations at low northern latitudes (Schofield & Taylor, 1983), although the mixed peak in the PCM is ∼3 K larger and ∼1 K smaller in the OIR.

Figures 3 and 4 show the amplitude and phase of the diurnal and semidiurnal components from both RO data and Venus PCM simulations. In the RO data, the phase of the diurnal tide exhibits three main stages of upward propagation, from 60 to 64 km, from 64 to 77 km, and from 77 to 90 km. The Venus PCM shows limited propagation between 65 and 70 km, coinciding with the region of strong net radiative heating near the upper cloud layer (65–70 km; Haus et al., 2016), and consistent with recent LIR observations (Akiba et al., 2021). The diurnal tide primarily propagates above 70 km, originating from the upper cloud region. The phase differences between RO observations and Venus PCM simulations at altitudes of around 70 km should be interpreted with caution, as the simulations exhibit a pronounced phase shift that is not evident in the RO data. This discrepancy may be related to the relatively smooth static stability profile observed by RO between 65 and 75 km. However, Pioneer Venus RO measurements (Paul & Huber, 2020) suggest that a layered structure in static stability may exist in this region (Figure S1 in Supporting Information S1). Such layering is also physically expected, as relatively unstable regions may occur near the cloud base and above the cloud top, leading to discontinuities in static stability. The observed diurnal amplitude between 60 and 75 km exhibits a single peak of ∼2 K near 68 km, whereas the PCM simulation yields two peaks of ∼1 K near 62 and 68 km. A similar double-peak pattern has been suggested by LIR observations, although with smaller amplitudes (∼0.5 K). Above 85 km, the simulated diurnal amplitude in the PCM can be nearly twice as large as in the RO data, likely due to limitations in representing processes near the model's upper boundary and its coupling with the upper atmosphere. For semidiurnal tides at low latitudes (Figures 4a, 4c, and 4e), upward propagation with eastward phase tilt and increasing amplitude is evident above 70 km, with a maximum around 87 km. There is no propagation between 65 and 70 km, consistent with LIR observations. Another GCM, AFES-Venus, also shows a similar semidiurnal tidal structure at the equator, with limited propagation within the upper cloud layer and clear upward propagation above it (Suzuki et al., 2022). However, their sensitivity tests on the static stability profiles within the Newtonian cooling scheme indicate that the assumed stability significantly influences the peak amplitude, altitude, and tilt rate of the tide. Both diurnal and semidiurnal components have vertical wavelengths of ∼20–30 km, consistent with previous observational and modeling studies (Akiba et al., 2021; Schofield & Taylor, 1983; Suzuki et al., 2022).

At middle latitudes above 70 km, the temperature structures exhibit a combination of diurnal and semidiurnal tides, similar to those at low latitudes, and are broadly consistent with Venus PCM simulations, although the peak amplitudes remain ∼2–3 K higher (Figure 2). Previous infrared observations have reported peak-to-peak variations of up to ∼8 K near 35° at 85 km (0.1 mbar), which is comparable to the PCM results (Zasova et al., 2002). Between 60 and 70 km, the semidiurnal structures resemble those observed by LIR at 40–50° latitudes, with peak amplitudes around 4 K near 65–70 km. Both the diurnal and semidiurnal components show good agreement between the RO data and PCM simulations in terms of amplitude and phase, with limited propagation within the cloud layer and upward propagation above it (Figures 3 and 4). These features are also consistent with earlier observations, which reported diurnal and semidiurnal tidal amplitudes of ∼1–3 K within the cloud layer (Akiba et al., 2021; Peralta et al., 2012; Schofield & Taylor, 1983; Zasova et al., 2002). At high latitudes, diurnal tides dominate, with temperature peaks around 65–75 km (∼6 K), 80 km (∼4 K), and above 85 km (∼8 K) (Figure S2 in Supporting Information S1). This is consistent with AFES-Venus simulations (Suzuki et al., 2022), although the peak altitudes and amplitudes differ by 1–2 km and 2–3 K, respectively, likely due to differences in the infrared cooling scheme and dynamical core.

The tidal structures observed in the RO data and simulated by the Venus PCM are generally consistent with previous measurements at the cloud top (Akiba et al., 2021; Kouyama et al., 2019) and in the northern midlatitudes (Schofield & Taylor, 1983; Zasova et al., 2002), exhibiting the expected equatorial symmetry of thermal tides on Venus. The hemispheric similarities and differences in thermal tides are crucial for understanding background atmospheric features, such as the polar vortex (Piccioni et al., 2007) and the slightly faster cloud-top zonal wind in the southern hemisphere (Horinouchi et al., 2018). The diurnal tide phase differences in the middle cloud layer (50–60 km) remain uncertain, though large amplitudes should not be expected in the RO data due to the low static stability at these altitudes. For the semidiurnal tide, the simulated phase in the Venus PCM tends to lie between those observed in the RO and LIR data sets within the cloud layer (50–70 km). The main discrepancy lies in the slightly larger amplitudes simulated above 85 km in the PCM, which may result from both observational limitations and model uncertainties. In the Venus PCM, the radiative transfer scheme is based on cloud models constrained by Venus Express atmospheric data. The net radiative heating, derived from Venus Express/VIRTIS observations, captures realistic tidal source regions and vertical propagation patterns (Haus et al., 2016), indicating that the radiative processes are reasonably well represented. However, heating rates above 85 km are more uncertain in the cloud model (Haus et al., 2015). Temporal variability of thermal tides, particularly for the diurnal component (Lai & Li, 2023), may also affect the RO data's tidal fitting.

We compared the horizontal structure of tidal temperature and winds at the cloud top between LIR observations and PCM simulations (Figure 5) and found similar patterns, although LIR amplitudes are slightly smaller, likely due to the smoothing effects of its contribution functions (Akiba et al., 2021) and underestimation of infrared radiance (Nishiyama et al., 2025). The brightness temperature data are sourced from the Level 3d data set of the Akatsuki Science Data Archive (Taguchi et al., 2025), calibrated for limb darkening (Lai & Li, 2023). The horizontal winds are derived using the method described in previous studies (Fukuya et al., 2021; Lai & Li, 2023), based on the UVI cloud-tracking technique (Horinouchi et al., 2017, 2018). Results poleward of 65°S/N should be interpreted with caution, as they are mostly located at the image edges due to Akatsuki's equatorial orbit. The cloud-top structure of thermal tides has also been investigated using AORI and AFES-Venus GCMs. While the AFES-Venus model exhibits differences in the cloud-top temperature field, integrating brightness temperature from 45 to 85 km using LIR's weighting function and spectral response yields results that are roughly consistent with LIR observations, showing a 1–2 LT phase difference (Suzuki et al., 2022). Thermal tides in the AORI GCM have been decomposed into diurnal and semidiurnal components (see Figure 1 in Yamamoto et al. (2024)). However, for semidiurnal tides, the AORI model exhibits midlatitude peaks centered at lower altitudes (around 50–60 km; Yamamoto et al., 2024). Although such midlatitude peaks are not clearly seen in LIR data, similar structures were observed by the Pioneer Venus OIR instrument, which detected midlatitude peaks centered around 65 km at ∼70°N with amplitudes of 1.5–2.5 K (Pechmann & Ingersoll, 1984). For diurnal tides, the OIR also showed peaks of 2–5 K near 60–65°N at 65 km, suggesting that the relatively higher amplitudes in PCM simulations compared to LIR may be plausible.

Overall, the Venus PCM remains sufficiently robust for investigating the momentum transport of thermal tides in the 50–80 km cloud region. This conclusion is further supported by previous studies that validated the PCM's thermal tide simulations against Venus Express/VIRTIS observations (Scarica et al., 2019). Although the basic radiative transfer scheme is similar to that used in Scarica et al. (2019), their lower model resolution likely limited the excitation of gravity waves by thermal tides, which may also influence the superrotation (Sugimoto et al., 2021).

3.2 Thermal Tidal AM Budget

Akatsuki's observations (Fukuya et al., 2021; Horinouchi et al., 2020) and Venus PCM simulations (Lai et al., 2024) highlight the key role of thermal tides in sustaining equatorial cloud-top superrotation. However, the contributions of individual tidal components throughout the cloud layer remain unclear. Figure 6 shows the Eliassen-Palm (EP) flux and its divergence, with the divergence scaled by the reciprocal of the product of atmospheric density and cosine latitude for the total thermal tides (tidal 24-, 12-, 8-, and 6-hr components combined) as well as their diurnal and semidiurnal components (see Section 2.3). The EP flux represents the transport of AM and wave activity, while its divergence quantifies the contribution of these waves to the zonal wind.

The EP flux reveals two main source layers for thermal tides: one near the cloud base (∼50 km) and another near the cloud top (∼67 km). These tides propagate away from their source regions, transporting AM against the superrotation and thus accelerating the source layers. Downward propagation is largely inhibited below 60 km at low latitudes due to low static stability. Tidal forcing near the cloud base is weaker, primarily due to lower solar heating (<0.5 K day⁻¹ compared to ∼1–2 K day⁻¹ at the cloud top; Haus et al., 2015) and higher atmospheric density. Diurnal tides dominate the momentum transport relative to semidiurnal components, with similar source regions and impacts on the background wind as the total tidal forcing. The diurnal tide accelerated the upper cloud (65–70 km) with 0.3–0.5 m s⁻¹ day⁻¹, largely offsetting the deceleration induced by meridional circulation (∼0.3–0.6 m s⁻¹ day⁻¹; Horinouchi et al., 2020; Lai et al., 2024) at low latitudes. The semidiurnal tide is primarily excited at ∼62 km and propagates upward and poleward, contributing ∼0.1 m s⁻¹ day⁻¹ of acceleration near 65 km in the equatorial and subtropical regions.

Previous semi-spectral models with idealized tidal structures suggested that vertical momentum flux convergence from the semidiurnal tide plays a key role in driving equatorial superrotation (Baker & Leovy, 1987; Newman & Leovy, 1992). Recently, vertical EP flux order estimations based on Akatsuki observations and classical tidal theory emphasize the role of the semidiurnal tide at the cloud top (∼70 km; Fukuya et al., 2021; Horinouchi et al., 2020). However, the applicability of classical tidal theory to a slowly rotating, optically thick atmosphere such as Venus's remains uncertain, particularly for diurnal tides, for which no robust theoretical framework currently exists (Auclair-Desrotour et al., 2017). Moreover, classical tidal analyses on Venus, such as Hough mode selection, often compute the Coriolis force based on the atmospheric superrotation period, whereas tidal structures are interpreted in the context of Venus's actual slow planetary rotation (Fukuya et al., 2021; Horinouchi et al., 2020; Kouyama et al., 2019). This inconsistency, rooted in observational limitations and the reliance on a highly variable and spatially inhomogeneous atmosphere to approximate planetary rotation, complicates the accurate characterization of tidal dynamics. To identify the primary tidal drivers of equatorial superrotation, it is essential to assess the contributions of AM and heat fluxes from each tidal component (see Section 2.3 for methods).

In the Venus PCM, the total tidal acceleration rate in the low-to-mid latitudes is primarily driven by AM flux relative to heat flux (Lai et al., 2024), for both diurnal and semidiurnal tides (Figure S5 in Supporting Information S1). Figure 7 shows the meridional and vertical AM fluxes associated with these tidal components and their resulting zonal acceleration. Above the cloud top (∼69–74 km), diurnal meridional AM flux enhances superrotation at low latitudes (∼0.1–0.2 m s⁻¹ day⁻¹), while vertical AM flux sustains superrotation in the upper cloud layer (∼65–70 km) at a higher rate (∼0.3–0.4 m s⁻¹ day⁻¹). For the semidiurnal tide, vertical AM flux contributes less than meridional flux at low latitudes, with peak zonal accelerations of ∼0.1 m s⁻¹ day⁻¹ occurring in the subtropics. Akatsuki UVI observations show that during the dayside (local time 07–16 hr), meridional AM fluxes accelerate the cloud-top zonal wind at low latitudes by ∼0.3–0.8 m s⁻¹ day⁻¹. This relatively strong acceleration reflects the active phase of diurnal tides on the dayside, where negative momentum is transported poleward (Figure S3 in Supporting Information S1, Horinouchi et al., 2018, 2020). The order estimated vertical EP flux of semidiurnal tides is thought to be the main driver with ∼0.075–0.15 m s⁻¹ day⁻¹ in LIR at ∼60 km (Fukuya et al., 2021) and ∼1 m s⁻¹ day⁻¹ or smaller in UVI at ∼70 km (Horinouchi et al., 2020).

Estimates based on classical tidal theory, specifically Hough mode solutions to Laplace's tidal equation (Longuet-Higgins, 1968) and the relationship between vertical EP flux and wave activity energy (Andrews et al., 1987; Fels & Lindzen, 1974), were applied in the Venus PCM to evaluate the latitudinal structure of wave forcing. These theoretical results were then compared with direct diagnostics from the TEM framework (Figure S6 in Supporting Information S1). The estimated semidiurnal tidal vertical EP flux at the cloud-top equator (∼0.03 kg s⁻²) is comparable to Akatsuki's observations (∼0.035 kg s⁻²; Horinouchi et al., 2020). However, the associated acceleration derived from UVI data is nearly an order of magnitude larger than that in the PCM. Several factors may account for this discrepancy. First, although Horinouchi et al. (2020) reported that semidiurnal tidal vertical EP fluxes could contribute up to ∼1 m s⁻¹ day⁻¹ to the zonal wind, their direct estimate for the semidiurnal tide was only ∼0.3 m s⁻¹ day⁻¹. Second, the estimation assumes a vertical EP flux divergence over a 10 km depth, while the actual vertical scale of divergence for the semidiurnal tide may be 2–3 times larger, as it freely propagates from ∼65 to 90 km (Figure 4). Consequently, although the total vertical EP flux is similar between PCM simulations and observations, the divergence calculated in the PCM is significantly smaller, leading to a weaker momentum acceleration.

The diurnal tide tends to contribute less when both diurnal and semidiurnal components are estimated under the assumption of gravity wave modes (Horinouchi et al., 2020; Kouyama et al., 2019), primarily due to the smaller horizontal wavenumber of the diurnal tide. However, this assumption may not be appropriate. The diurnal tide appears Rossby-like in its horizontal structure (Figure 5), but it is also associated with strong vertical winds near the equator (Figures S3 and S4 in Supporting Information S1; Kouyama et al., 2019). Such vertical motions indicate significant divergence, likely arising from radiative forcing. The polarization relationship between temperature and vertical wind (She et al., 2016) further supports the interpretation of the semidiurnal tide as a gravity wave mode, while providing no such support for the diurnal tide. The vertical EP flux contribution from the diurnal tide is estimated based on its relationship with wave energy and group velocity (Text S3 in Supporting Information S1; Andrews et al., 1987), assuming Rossby-gravity and inertia-gravity modes, including the Rossby-mode-based estimates adopted in UVI observations (Horinouchi et al., 2020), to assess their consistency with the direct calculations from the TEM framework. It should be noted, however, that the UVI-based estimates of the diurnal vertical EP flux only account for the heat flux component. Horinouchi et al. (2020) estimated the AM transport at mid-to-high latitudes, assuming a Rossby wave structure, but also noted that gravity waves may dominate in low latitudes, where such estimates are not available due to data limitations. Among the tested assumptions, the Rossby-gravity mode yields the best agreement with direct EP flux divergence derived from the TEM equation in the equatorial region between 65 and 70 km (Figure S6 in Supporting Information S1), although notable discrepancies persist in other regions, likely due to differences in the assumed background wind structure, such as the use of solid-body rotation. This does not imply that the diurnal tide itself is a Rossby-gravity mode, but rather reflects its strong equatorial contribution. Given observational constraints, previous estimation-based studies may have underestimated or incompletely characterized the contribution of diurnal tidal vertical transport. The acceleration patterns of the diurnal tide simulated in the Venus PCM show good agreement with the distribution of net radiative heating derived from Venus Express VIRTIS observations (Haus et al., 2015, 2016), suggesting that the AM budget in the low-latitude cloud-top region (65–75 km) may be primarily contributed by the diurnal tide.

3.3 Rotational and Divergent Flow of Thermal Tides

Compared to Figures 7 and 8 shows that the meridional AM flux of the diurnal tide primarily arises from a combination of rotational and divergent components, while the vertical AM flux is mainly associated with the divergent component. This suggests that, at low latitudes and altitudes between 65 and 70 km, the divergent flow dominates the vertical EP flux-driven acceleration of the diurnal tide. In contrast, the horizontal EP flux appears to result from a rotational Rossby mode mixed divergent component and primarily contributes to momentum transfer in the 0–5 km layer above the cloud top in the low-latitude region. These findings support the interpretation that the diurnal tide on Venus may correspond to a Rossby mode mixed with radiatively forced divergent flow. For the semidiurnal tide, both horizontal and vertical AM fluxes are well represented by the divergent component of the horizontal wind, indicating that this tide is predominantly governed by the gravity wave mode. This result is consistent with previous studies and the discussion in Section 3.2. The findings are also in good agreement with the AORI Venus GCM results (Yamamoto et al., 2024). Despite differences in the radiative transfer schemes, the consistency across different GCMs suggests a robust wave mode characteristic of the diurnal tide. In particular, the vertical momentum flux associated with the divergent flow is significant in the AORI GCM, reaching approximately 0.8 m s⁻¹ day⁻¹, compared to over 0.5 m s⁻¹ day⁻¹ in the Venus PCM (Figure 8b).

The horizontal AM transport associated with thermal tides and the zonal-mean flow at the upper cloud level (65–70 km) is examined using both Venus PCM simulations and Akatsuki/LIR observations (Figure S7 in Supporting Information S1). The results are consistent with the mode decomposition of the diurnal and semidiurnal tides presented in Figure 8. In particular, the prominent divergent component combined with the Rossby mode of the diurnal tide appears to play a key role in maintaining the superrotation in the upper cloud layer. Although discrepancies between RO observations and Venus PCM simulations are evident, especially in the vertical structure, these differences require further investigation involving improved observations and more realistic cloud models, which are beyond the scope of this study. Nevertheless, these differences do not affect the conclusion regarding the significant role of the diurnal tide.