© The Authors 2023

1 Introduction

The vertical eddy diffusion coefficient (K_zz) is a parameter characterizing the efficiency of vertical atmospheric mixing, incorporating turbulence, convection, and wave-induced processes. These eddy motions could occur across a wide variety of spatial scales, making them challenging to trace directly, and thus poorly constrained. Nevertheless, K_zz is widely used in various planetary atmospheric studies, especially in 1D photochemical and cloud models that are unable to resolve 3D eddies.

Determining the planetary atmospheric K_zz typically relies on analysis of the vertical distributions of atmospheric species. Inert gases or species with low chemical activity, such as He and CO, are commonly selected as trace species. By tracking the vertical eddy velocities of these species, atmospheric K_zz has been estimated on several planets (von Zahn et al. 1980; Mahieux et al. 2021; Swenson et al. 2019; Yoshida et al. 2022). However, Venus, covered globally by a thick layer of sulfuric acid clouds, poses challenges due to significant radiation absorption and scattering, restricting the detection of typical tracers in this region. Consequently, our knowledge of Venusian cloud K_zz remains limited. A recent model study has proposed that K_zz governs the H₂SO₄ cycle in the middle and lower clouds, leading to the mass accumulation in these regions (Dai et al. 2022b). Observations (Ando et al. 2020a; Hinson & Jenkins 1995; Imamura et al. 2017) and models (Ando et al. 2020b, 2022) have indicated a region of low static stability in the Venusian clouds, suggesting the presence of a convection layer. Notably, Ando et al. (2015) report significant latitudinal variability in the static stability distribution. Given the role of the Venusian clouds in regulating the global radiative transfer and subsequent climate change (Esposito et al. 1983; Titov et al. 2018; Turbet et al. 2021), a comprehensive understanding of K_zz in the clouds is crucial.

Due to the scarcity of observational constraints on the cloud K_zz, a variety of K_zz profiles have been used in the literature (see Table 1). Based on the in situ wind speed observations of Vega balloons (Sagdeev et al. 1986), the K_zz at 54 km has been estimated at 10⁷ cm² s⁻¹ (Blamont et al. 1986). Pioneer Venus radio scintillation measurements estimated the K_zz to be 2 × 10³ cm² s⁻¹ at 45 km (Woo et al. 1982) and 4 × 10⁴ cm² s⁻¹ at 60 km (Woo & Ishimaru 1981).

Most microphysical models empirically assumed the cloud K_zz. The K_zz has been estimated to be 10⁴−10⁶ cm² s⁻¹ in 50–56 km (Imamura & Hashimoto 2001; Gao et al. 2014) and 10⁴−10⁷ cm² s⁻¹ in 50–57 km (McGouldrick & Toon 2007). This indicates an enhancement of the convection in the lower and middle clouds, corresponding to the observed strong wind shears near the cloud base (Kerzhanovich & Marov 1983) and cloud top (Imamura et al. 2020). The latest 3D large-eddy simulation indicates an enhancement of K_zz from 6 × 10⁴ to 1 × 10⁸ cm² s⁻¹, extending from 48 to 56 km (Lefèvre et al. 2022). Besides, empirical cloud K_zz profiles have commonly been employed in 1D photochemical models. Krasnopolsky (2012) assumed a constant K_zz of 1 × 10⁴ cm² s⁻¹ below 55 km. Zhang et al. (2012a) and Shao et al. (2020) estimate the K_zz in 58–80 km to be 4–5×10⁴ cm² s⁻¹ by linearly interpolating the observed values at 60 and 80 km. Bierson & Zhang (2020) examined a weak cloud K_zz in the 45–65 km region, assuming a constant value of 1 × 10³ cm² s⁻¹ to interpret the SO₂ depletion in the clouds. However, this weak cloud K_zz contradicts the low static stabilities observed by Venus Express (Ando et al. 2015) and Akatsuki (Imamura et al. 2017).

This study aims to address three main objectives: (a) proposing a novel approach to estimating the Venusian cloud K_zz by utilizing the H₂SO₄ vapor mixing ratio profiles from Venus Express measurements; (b) examining the variability of the cloud K_zz in both the equatorial and polar regions; and (c) evaluating the derived K_zz through the implementation of a 1D photochemical model. The details regarding the data and methodology employed in this study are presented in Sect. 2. The results obtained are presented in Sect. 3. Furthermore, the evaluation through the 1D photochemistry-diffusion simulation is discussed in Sect. 4. Finally, we conclude the study in Sect. 5.

2 Data and method

2.1 Data

The H₂SO₄ vapor profiles used in estimating the K_zz were obtained from observations conducted by Venus Express (Oschlisniok et al. 2021). The presence of globally distributed thick clouds and the unknown UV absorber on Venus result in significant opacity in the visible and ultraviolet wavelengths, obstructing the visibility of most cloud features (Limaye et al. 2018; Titov et al. 2018). However, H₂SO₄ vapor has been proposed to be a strong absorber of radio waves (Steffes & Eshleman 1982), allowing its transmission through the clouds. Oschlisniok et al. (2021) derived over 800 H₂SO₄ vapor abundance profiles in the cloud regions using dual-frequency (X/S-band) radio science open loop data from the radio occultation measurements (VeRa) onboard the Venus Express, collected between 2006 and 2014. The derived distributions of H₂SO₄ vapor abundance, temperature, and pressure are presented in Figs. 1a–c. In the polar regions, H₂SO₄ vapor exhibits a concentration within the altitude range of 40–44 km, while at the equator this peak shifts to 45–48 km. The temperature distribution exhibits an arch-shaped pattern, while the pressure shows minimal latitudinal variations. We note that the presented H₂SO₄ vapor measurements have removed the SO₂ absorption by using an estimated SO₂ distribution, according to Oschlisniok et al. (2021). Its uncertainty is discussed in Sect. 5.

2.2 Method of K_zz estimation

In this study, the equilibrium between the vertical transports and the other sources and sinks of H₂SO₄ vapor serves as the foundation for the calculation of K_zz. Both observations (Imamura et al. 2017; Oschlisniok et al. 2021) and models (Dai et al. 2022b,a; Krasnopolsky 2015) indicate a rapid decrease in the H₂SO₄ vapor mixing ratio above the cloud base. The vertical diffusion flux of H₂SO₄ vapor becomes negligible above 55 km (Gao et al. 2014; Imamura & Hashimoto 2001). Thus, we chose 55 km as the upper boundary of our domain and assumed that the diffusion flux of H₂SO₄ vapor was zero at this altitude. The lower boundary of our domain is at the cloud base altitudes ranging from 43 to 47 km, varying with latitudes. The definition of the cloud base is described later.

Ahead of handling, the H₂SO₄ data was vertically smoothed by a Savitzky-Golay Filter (Savitzky & Golay 1964; Schafer 2011, impulse response length L = 11, polynomial order ω = 1), for the purpose of decreasing the influences of statistical fluctuation and the convergence or divergence of horizontal H₂SO₄ vapor flow. In our domain, the H₂SO₄ vapor mixing ratio could be simultaneously influenced by diffusion, condensation, evaporation, and chemistry. The differential diffusion flux versus altitude is defined as $$\frac{\text{d}\Phi }{\text{d}z}=S_{\text{evap }}-S_{\text{cond }}+S_{\text{prod }}-S_{\text{loss }},$$(1)

where Φ = vn_s (v is the diffusion velocity and n_s the number density of H₂SO₄ vapor), z represents the altitude, and the terms at the right-hand side of the equation represent the rates of evaporation, condensation, chemical production, and loss, respectively. The net condensation or evaporation rate of H₂SO₄ vapor is written as (Seinfeld & Pandis 2016, p. 538–540) $$S_{\text{\hspace{0.17em}net}}=S_{\text{\hspace{0.17em}evap}}-S_{\text{\hspace{0.17em}cond}}=-{\displaystyle {\sum }_i\left[\frac{2\pi {\text{Θ}}_i^{\text{p}}D}{kT}{f}_i\left(P-P^{\text{svp}}{\text{e}}^{\frac{4\sigma M}{RT\rho {\text{Θ}}_i^{\text{p}}}}\right)n_i^{\text{p}}\right],}$$(2)

where D is the molecular diffusion coefficient, k the Boltzmann constant, T the atmospheric temperature, P the pressure of H₂SO₄ vapor, and P^svp the saturation vapor pressure. The factor of P^svp represents the Kelvin Effect, where σ=71.11 erg cm⁻² is the surface tension of sulfuric acid (Myhre et al. 1998), R is the gas constant, ρ = 1.8 g cm⁻³ is the mass density of the droplets. The distributions of particle diameter and number density are adopted from the cloud particle size spectrometer (LCPS) onboard the Pioneer Venus probes (Knollenberg & Hunten 1980), which cover several latitudes and are considered to be global mean distributions. We divided the observed diameters (ranging from 0.01 to 15 µm) into 32 bins (in log scale), counted as i. ${\text{Θ}}_i^{\text{p}}$ and $n_i^{\text{p}}$ represent the characteristic diameter and number density of bin i, respectively. D was calculated following Hamill et al. (1977): $$D=\frac{\lambda }{3}{\left(\frac{8kT{N}_a}{\pi M}\right)}^{\frac{1}{2}},$$(3)

where N_a is the Avogadro constant, M = 98 g mol⁻¹ is the molecular weight of H₂SO₄, and λ is the mean free path of H₂SO₄ molecule, written as (Hamill et al. 1977) $$\lambda ={\left[\pi n_{\text{atm}}{\left(\frac{d_{\text{s}}+d_{\text{atm}}}{2}\right)}^2{\left(\frac{M_{\text{atm}}}{M+M_{\text{atm}}}\right)}^{\frac{1}{2}}\right]}^{-1},$$(4)

where n_atm is the total atmospheric number density, d_s = 4.4 × 10⁻⁸ cm (Hanson & Eisele 2000) and d_atm = 3.3 × 10⁻⁸ cm (Mehio et al. 2014) represent the molecular diameters of H₂SO₄ and the atmospheric major component (CO₂), respectively, and M_atm = 44 g mol⁻¹ is the molecular weight of CO₂. The flux-matching factor, f_i, is adopted from Zhang et al. (2012b): $$f_i=\frac{1}{1+\frac{2\lambda }{{\text{Θ}}_i^{\rho }\alpha }},$$(5)

where α = 1 is the accommodation coefficient (Seinfeld & Pandis 2016, p. 500–502).

The P^svp (in the unit of bar) depends on the atmospheric temperature and the droplet acidity, measured by $$\ln P^{\text{svp}}(\text{bar})=\ln P_{\text{p}}^{\text{svp}}(\text{bar})+\frac{\mu -{\mu }^0}{RT},$$(6)

where µ − µ⁰ is the difference in the chemical potential of H₂SO₄ between the solution and the pure condensate, adopted from the experimental data (Zeleznik 1991). The previously estimated cloud acidities from the observations have large uncertainties for either assuming a constant acidity throughout the clouds or that the observed spectrum was linearly interpolated among three acidity data points from experimental data (Arney et al. 2014; Palmer & Williams 1975), which showed a strong nonlinear dependence of the imaginary index of refraction on the cloud acidity (Barstow et al. 2012). Therefore, we adopted an acidity profile that reduces from 98% at the cloud base to 85% at 55 km, based on recent model results (Dai et al. 2022b; Krasnopolsky 2015; Stolzenbach et al. 2023). $P_{\text{p}}^{\text{svp}}$ is the SVP above the surface of pure H₂SO₄ condensate, adopted from the parameterization of 98% weight percentage H₂SO₄ (Kulmala & Laaksonen 1990): $$\begin{array}{l}\ln P^{\text{svp}}\text{ (bar) }=16.259-\frac{10156}{T_0}\\ +10156[-\frac{1}{T}+\frac{1}{T_0}+\frac{0.38}{T_{\text{c}}-T_0}\\ \times \left(1+\ln \frac{T_0}{T}-\frac{T_0}{T}\right)],\end{array}$$(7)

where T₀=360 K is a reference temperature and T_c=905 K is the critical temperature.

Three illustrative cases of the H₂SO₄ vapor mixing ratio and saturation vapor mixing ratio profiles in different latitudes are presented in Figs. 2a,d,g. The lowest intersections of the H₂SO₄ vapor mixing ratios and the saturation profiles occur at 47, 47, and 43.5 km, respectively, indicating the onset of condensation. These layers are defined as the cloud base in this study. Above the cloud base, the mixing ratios increase slightly to the peaks and then generally decrease as the altitude increases, with the presence of some fluctuations. The H₂SO₄ vapor mixing ratios are higher than the saturation lines at most altitudes except that an undersaturation region appears near 51 km in the polar case, attributed to the horizontal transport of H₂SO₄ vapor or a statistical fluctuation in the data. Additionally, Fig. 2b,e,h display the corresponding profiles of H₂SO₄ condensation or evaporation rates. These rates reach 10¹¹−10¹² cm⁻³ s⁻¹ in the middle and lower clouds. Here we note that such high rates might lead to latent heat exchange, contributing to local temperature variation. However, since the temperature data we used was adopted from direct observations, and K_zz was calculated by vertical transport rather than static stability, the latent heat is not what this study needs to focus on. Chemical models (Bierson & Zhang 2020; Krasnopolsky 2007, 2012) suggest that the net production or thermal-decomposition rates of H₂SO₄ vapor in this region are lower than 1 × 10⁵ cm⁻³ s⁻¹. Therefore, the chemical processes were neglected in our investigation of K_zz in the middle and lower clouds of Venus. That is, S_prod-S_loss ≈ 0 cm⁻³ s⁻¹. Then, the H₂SO₄ vapor flux at z altitude could be calculated by $$\text{Φ}(z)=-{{{\displaystyle \int }}^{\text{}}}_z^{55(\text{km})}{S}_{\text{net }}\text{d}z.$$(8)

Subsequently, the diffusion velocity of H₂SO₄ vapor was calculated by: $$v(z)=\frac{\text{Φ}(z)}{n_{\text{s}}(z)}.$$(9)

We note that this is the total diffusion velocity, that is, the combination of eddy diffusion and molecular diffusion. The total diffusion flux could be described as $$\text{Φ}=-D{n}_{\text{s}}\left(\frac{1}{H_{\text{s}}}+\frac{\text{d}T}{T\text{d}z}+\frac{\text{d}{n}_{\text{s}}}{n_{\text{s}}\text{d}z}\right)-K_{zz}{n}_{\text{atm}}\left[\frac{\text{d}}{\text{d}z}\left(\frac{n_{\text{s}}}{n_{\text{atm}}}\right)\right],$$(10)

where H_s = kT/m_sg is the scale height of H₂SO₄ vapor, m_s is the molecular weight of H₂SO₄, and g = 870 cm s⁻² is the Venusian gravity acceleration. The general expression for the molecular and eddy diffusion velocities of H₂SO₄ vapor are $$v_{\text{mol}}(z)=-D\left(\frac{1}{H_{\text{s}}}+\frac{\text{d}T}{T\text{d}z}+\frac{\text{d}{n}_{\text{s}}}{n_{\text{s}}\text{d}z}\right)\text{, }$$(11) $$v_{\text{eddy }}(z)=v(z)-v_{\text{mol }}(z)\text{. }$$(12)

The eddy diffusion coefficient could be derived by $$K_{zz}=-\frac{v_{\text{eddy }}{n}_{\text{s}}}{n_{\text{atm}}\left[\frac{\text{d}}{\text{d}z}\left(\frac{n_{\text{s}}}{n_{\text{atm}}}\right)\right]}.$$(13)

The derived K_zz distributions in the specific scenarios are presented in Figs. 2c,f,i. In the lower cloud regions, the values reach magnitudes of 1 × 10⁸ cm² s⁻¹. We note that occasional negative values occur at certain altitudes, especially near the top boundary. The H₂SO₄ data near the top boundary is occasionally negative due to its low abundance. This results in negative K_zz in this region. The effect of this issue could rapidly decrease with decreasing altitude since the low condensation or evaporation rate contributes less to its column integration. Besides, parameterizing the complicated 3D transport of atmosphere by 1D diffusion needs a great simplification, which might overlook some potential effects such as horizontal transport. The same negative K_zz has been indicated by Zhang & Showman (2018) to be the result of the non-diffusive effects regulating the atmosphere. 1D model studies (Dai et al. 2022b; Gao et al. 2014) proposed that H₂SO₄ vapor diffuses upward in the cloud layer to contain the formation of the thick clouds. However, the effects of statistical fluctuation and horizontal convergence or divergence, which are carried by H₂SO₄ vapor data and not completely removed by the smooth process, increase the local H₂SO₄ vapor abundance and possibly lead to a downward flux. Then, negative K_zz could also appear in these regions to correct the direction of the vapor flux. Since these issues are not able to be included in the 1D studies and could be diluted by horizontal averages, we removed these data points.

Fig. 1 VeRa observations (Oschlisniok et al. 2021). The panels show (a) the H2SO4 vapor volume mixing ratio, (b) temperature, and (c) pressure, respectively. We note that these observations are divided into 36 latitudinal bins with a resolution of 5 degrees. The lack of measurements is caused by the orbit geometry of Venus Express.

Fig. 2 Example cases illustrating the method of Kzz estimation. The three rows refer to (a–c) −0.02°, (d–f) −52.65°, and (g–i) 87.8° latitude, respectively. The left column shows the original H2SO4 vapor mixing ratio (VMR, solid black line), smoothed H2SO4 vapor mixing ratio (solid red line), and saturation vapor mixing ratio (SVMR, dashed black line) profiles. The middle column shows the net condensation (blue line) and evaporation (red line) rate profiles. The right column shows the derived Kzz distributions. The red triangles represent negative values, caused by the statistical fluctuation and horizontal convergence or divergence of H2SO4 vapor.

Fig. 3 Global mean results. (a) The derived global mean Kzz by adopting LCPS particle distributions. The blue line represents the median value along with the upper and lower quartiles. The derived Kzz profile below 46.5 km is distinguished in red since it is basically constituted of data from the polar regions and better considered as the “polar mean.” The bottom-left subplot displays the Kzz below the clouds, which is based on the given column production of H2SO4 vapor. The red line represents the below-cloud Kzz estimated in polar regions and the green line represents the other regions. The estimations based on Vega balloon observations (fuchsia, Blamont et al. 1986), Pioneer Venus radio scintillation measurements (black, Woo et al. 1982), and model results of McGouldrick & Toon (2007, purple) and Lefèvre et al. (2022, orange) are also presented. (b) The distribution of H2SO4 vapor upward diffusion flux. (c) The derived H2SO4 vertical molecular (left) and eddy (right) diffusion velocities as functions of altitude. The observations of atmospheric convection velocities from Vega balloons (fuchsia, Linkin et al. 1986), as well as model estimations of Imamura et al. (2014, green), Lefèvre et al. (2018, gray), and Gierasch et al. (1997, orange triangles for the Hadley velocity and black stars for the convective velocity) are also presented.

3 Results

3.1 Global average

The derived global mean K_zz profile is presented in Fig. 3a. The particle distribution used for this estimation was adopted from LCPS observations. According to the results, the cloud K_zz has a peak of 5 × 10⁸ cm² s⁻¹ near 48 km. Between 50 and 54 km, it remains nearly constant at 2.5 × 10⁸ cm² s⁻¹ before exhibiting a rapid decrease as the altitude increases above 54 km. This profile is larger than the results obtained from the latest 3D convection-resolving model (Lefèvre et al. 2022) by a factor of three in the middle cloud region, and is roughly an order of magnitude larger than the estimation based on VeRa measurements (Blamont et al. 1986; Sagdeev et al. 1986) at 54 km and the microphysical model (McGouldrick & Toon 2007) above 50 km. The K_zz quickly decreases above 54 km, being consistent with the observed increase in the static stability reported by Akatsuki (Imamura et al. 2017) and the model estimations by Lefèvre et al. (2022). This altitude typically represents the upper boundary of the middle clouds, where the number density of large particles decreases rapidly (Knollenberg & Hunten 1980), thereby weakening the condensation of H₂SO₄. Notably, a significant enhancement of K_zz is presented within the altitude range of 46.5–49.5 km, corresponding to the altitudes of the lower clouds (Knollenberg & Hunten 1980). The derived K_zz profile below 46.5 km in Fig. 3a is distinguished by another color, for the cloud base in the polar regions (>60 or <−60 degree) could descend to 43 km, while it is averaged at 46.5 km in middle and low latitudes. Since the bottom boundary of our domain is the cloud base, the results in 43–46.5 km are basically made up of data in the polar regions. Thus, it is better considered as a “polar mean.” Detailed results about the latitudinal variations are posted later in this section.

The estimated K_zz profiles below the cloud base (distinguished by polar and other latitudes) are displayed in the subplot of Fig. 3a. The H₂SO₄ vapor is continuously produced by a chemical process near the cloud top and thermally decomposed in the lower atmospheres (Krasnopolsky 2007, 2012), leading to a net downward flux of total H₂SO₄. This flux is suggested by a photochemical model (excluding cloud formation) as about 1 × 10¹² cm⁻² s⁻¹ in our domain (Yung & Demore 1982). In the cloud regions, this net flux is contributed to by both the downward transport of liquid H₂SO₄ and the upward diffusion of the vapor. A recent microphysical model study (Gao et al. 2014) derived both the downward and upward fluxes on the order of 1 × 10¹⁴ cm⁻² s⁻¹ in the clouds. They are dramatically larger than the net flux, indicating the strong internal H₂SO₄ cycle. This difference could even increase by adopting larger cloud K_zz values in their model. In the region shortly below the cloud base, the liquid H₂SO₄ is not able to contribute to this flux. Then we could expect the total flux to be completely made up of the vapor flux, given that both the chemical production and loss of vapor are weak in this region. To calculate these profiles, we adopted a constant downward H₂SO₄ vapor flux of 1 × 10¹² cm⁻² s⁻¹ in a short distance below the cloud base. In this particular region, the H₂SO₄ vapor abundance is primarily controlled by the vertical transport and its column production, as its thermal decomposition becomes significant below 38 km (Jenkins et al. 1994). The results show that the K_zz is around 1 × 10³ cm² s⁻¹ below the cloud base, increasing slightly with increasing altitude, agreeing with the estimate based on Pioneer Venus radio scintillation measurements at 45 km (Woo et al. 1982). We note that this K_zz profile is displayed for the purpose of indicating that there could be a sharp decrease of K_zz below the cloud base and supporting the comparison of different latitudes later. Since the transition region near the cloud base is not considered in this study, this profile is not connected to the profile in the clouds.

To investigate the vertical motions in the cloud region more comprehensively, we have illustrated the upward diffusion flux of H₂SO₄ vapor in Fig. 3b, along with its eddy and molecular diffusion velocities shown in Fig. 3c. In 46.5–48.5 km (the lower cloud region), H₂SO₄ vapor exhibits an upward flux on the order of 10¹⁷ cm⁻² s⁻¹. Here, we provide a simple estimate to elucidate this outcome. The diameter of mode-3 particles, which are concentrated in this region, is approximately ten times larger than that of mode-1 and mode-2 particles that dominate in the upper clouds. We note that the Stokes velocity of sedimentation is proportional to the square of the particle diameter. Additionally, the mass loading of the lower clouds is roughly 100 times greater than that of the upper clouds (Knollenberg & Hunten 1980). These factors suggest that the sedimentation flux of liquid H₂SO₄ in the lower clouds is significantly enhanced by four orders of magnitude compared to the upper clouds. Given that the liquid H₂SO₄ has a sedimentation flux of 1 × 10¹² cm⁻² s⁻¹ at the transition between the middle and upper clouds (Imamura & Hashimoto 2001; Yung & Demore 1982), an estimated sedimentation flux of liquid H₂SO₄ in the lower clouds would be approximately 1×10¹⁶ cm⁻² s⁻¹. Generally, for mass conservation, the downward liquid H₂SO₄ flux resulting from both sedimentation and diffusion of the particles is equal to the upward vapor diffusion flux. The estimated sedimentation flux is an order of magnitude smaller than our results. This discrepancy can be attributed to the effect of the diffusion of different-sized particles, which it is hard to physically estimate.

The eddy diffusion of H₂SO₄ vapor yields a velocity on the order of 10²−10³ cm s⁻¹. It is consistent with the model-estimated Hadley velocity in 52–54 km (Gierasch et al. 1997), but it is two to eight times faster than the convection velocities measured by Vega balloons (Linkin et al. 1986) and convection models (Imamura et al. 2014; Lefèvre et al. 2018). Here, we note that the eddy diffusion in this study characterizes varying processes that drive the atmospheric mixing, incorporating convection, turbulence, and waves. Such a discrepancy might indicate that other processes besides convection also contribute to the eddy mixing of H₂SO₄ vapor. Eddy diffusion plays a crucial role in the formation of these cloud layers. The H₂SO₄ vapor generated by evaporation below the cloud base is rapidly transported upward by eddy diffusion, and subsequently condenses back into the droplets within this region (Dai et al. 2022b). Therefore, strong eddy diffusion contributes to a high condensation efficiency in the clouds. This internal cycle is essential for the observed accumulation of mass loading (Knollenberg & Hunten 1980) in the lower and middle clouds (Imamura & Hashimoto 2001; McGouldrick 2017).

The molecular diffusion velocity of H₂SO₄ vapor ranges from 1 × 10⁻⁶ to 1 × 10⁻⁵ cm s⁻¹. Its magnitude is significantly weaker compared to eddy diffusion in the cloud region, as expected, since our domain is far below the homopause (120–140 km).

3.2 Latitudinal variation

To investigate the latitudinal variation in the Venusian cloud eddy diffusion, we used particle distributions from the latest microphysical model (McGouldrick & Barth 2023), specifically tailored for different latitudes. Although LCPS observations are more reliable than model results, they cannot distinguish latitudinal variations in particle distributions that could influence the condensation rate. Using LCPS data might overlook some critical features. We categorized the latitudes into three bins, covering both hemispheres: 0°–30°, 30°–60°, and 60°–90°. The corresponding K_zz, flux, and velocity profiles generated for each bin are illustrated in Figs. 4a,b,c, respectively, while the altitude-latitude distribution is depicted in Fig. 4d.

The typical cloud-base altitude in polar regions is expected to be 43 km, notably lower than that projected in middle and low latitudes. This is consistent with the measurements obtained by VIRTIS on board Venus Express (Barstow et al. 2012). At the cloud regions in middle and low latitudes, the K_zz is generally comparable, except for a layer situated near 48 km, where the K_zz in middle latitudes exceeds that in low latitudes by a factor of 50%. In 46.5–50 km, the cloud K_zz in polar regions is lower than that in low latitudes, with a typical value of ~8 × 10⁷ cm² s⁻¹. This seems to contradict some model studies (e.g., Ando et al. 2020b) in which equatorial static stability is higher than that in polar regions. It is reasonable, since K_zz characterizes varying processes that contribute to the atmospheric mixing, including convection. The correlation between K_zz and static stability depends on its relative contribution to the atmospheric motion. For instance, at low latitudes, the global circulation might have some contribution and increase the cloud K_zz in this region. Besides, the derived K_zz in polar regions exhibits a notable enhancement at deeper altitudes, with a peak of 3×10⁸ cm² s⁻¹ at 43 km, corresponding to the altitude of the lower clouds in the polar regions (Barstow et al. 2012). Although our domain only extends to the cloud base, we can anticipate that the K_zz below this level is several orders lower than the cloud K_zz, as previously mentioned. The bottom of the high-K_zz region significantly shifts from 46.5 km at the equator to 43 km in polar regions (as depicted in Fig. 4d). This altitudinal variation is consistent with the static stability distributions obtained from VeRa measurements (Ando et al. 2015, Ando et al. 2020a), where the low static stability region extends down to 42 km at high latitudes.

The upward diffusion fluxes of H₂SO₄ vapor above 46.5 km exhibit a significant decrease as the latitude increases. As we have mentioned above, this might be related to the effect of global circulation. The flux basically decreases with increasing altitude at all latitudes. It peaks at 47.5 km in low latitudes, with a value of 1.5 × 10¹⁷ cm⁻² s⁻¹. Below 46.5 km, the fluxes in middle and low latitudes could be expected to rapidly decrease as K_zz decreases, while the flux in polar regions reaches a peak at 44 km, with a value of 1 × 10¹⁷ cm⁻² s⁻¹. The agreement with observed low polar static stability in 43–46.4 km (Ando et al. 2015, Ando et al. 2020a) indicates that convection might regulate the H₂SO₄ vapor mixing in this region. The latitudinal variation in the eddy diffusion velocity is not significant. It increases as the altitude increases on the order of several m s⁻¹ and reaches a peak at 53 km on the order of 10 m s⁻¹. It inverses above 53 km as the decreasing particle size reduces the condensation rate.

Fig. 4 Latitudinal variation. (a) Kzz, (b) H2SO4 vapor upward diffusion flux, and (c) H2SO4 vapor eddy diffusion velocity profiles in different latitude bins were derived by adopting particle distributions from the microphysical model results (McGouldrick & Barth 2023). The red, green, and blue lines represent the latitudinal bins of 0°–30°, 30°–60°, and 60°–90°, respectively. Panel d presents the altitude-latitude distribution of derived Kzz. We note that the missing middle-latitude results near 47 km are caused by a lack of data.

4 Discussion

A 1D photochemistry-diffusion model (Rimmer et al. 2021) has been used to evaluate the measured K_zz in the Venusian clouds. Here we only summarize the key points of the model. More details can be found in Rimmer et al. (2021). This model includes over 3000 reactions involving 480 species. It solves the continuity equations containing eddy mixing and chemistry at each altitude. Furthermore, it performs the cloud formation and tracks the SO₂ dissolution based on Henry’s Law and the presence of hydroxide salts. By incorporating our derived global mean cloud K_zz (46.5–55 km) into their nominal profile, we made comparisons of several species. The results are shown in Fig. 5. We note that the discontinuous patchwork of K_zz profiles in Fig. 5a has a 4-order of magnitude enhancement in a 10-km region that does not take the transition regions near the intersections of two K_zz profiles into account. A similar pattern of K_zz distribution is indicated by the results of a 3D convection-resolving model (Lefèvre et al. 2022).

By adopting a high-K_zz profile in the Venusian clouds, the simulated vapor profiles of SO₂, H₂O, SO, and CO are in good agreement with observations, with the exception of a slightly lower CO mixing ratio above 65 km. These indicate that the presence of a high-K_zz layer in the clouds may not significantly affect the overall profiles of these species. Comparing these results to the nominal case presented in Rimmer et al. (2021, hereafter R21), we gain an insight into the role played by cloud K_zz. While previous studies such as Bierson & Zhang (2020) emphasize the significant influence of cloud K_zz on the vertical transport of SO₂, our × 10⁴ enhancement of cloud K_zz in R21 does not induce a suppression of the SO₂ depletion at 57 km (Fig. 5b). This can be attributed to the fact that the depletion is primarily regulated by the dissolution of SO₂ into the droplets, a process accounted for in R21 but not in the study by Bierson & Zhang (2020), which does not include cloud formation. A similar behavior has occurred to H₂O depletion (Fig. 5c), indicating its connection to cloud formation.

Interestingly, empirical estimates of cloud K_zz in 1D chemical models that exclude cloud formation (e.g., Bierson & Zhang 2020) usually yield lower values compared to those obtained from cloud models (e.g., Gao et al. 2014; McGouldrick 2017). The vertical distributions of certain species such as SO₂ in the former studies exhibit a higher sensitivity to the cloud K_zz. It appears that the Venusian clouds act as a large reservoir for related vapors, buffering their abundances in response to variations in vertical transport. In order to satisfy mass conservation, the cloud mass loading must vary with the cloud K_zz. This hypothesis is supported by our simulation, in which the mixing ratio of liquid H₂SO₄ significantly increases with the enhancement of cloud K_zz (Fig. 5d). The derived several-ppm liquid H₂SO₄ in the clouds agrees with Dai et al. (2022b). To further validate this hypothesis, it is crucial to conduct additional investigations into the material exchange between the clouds and the atmosphere.

In contrast to SO₂ and H₂O, the distributions of both SO and CO in the cloud region appear to be more uniform in the case of enhanced cloud K_zz compared to R21 (Figs. 5e–f). It indicates that the cloud K_zz can still have an impact on these species independently of cloud formation in this region.

Fig. 5 Evaluation of measured Kzz by adopting a 1D photochemistry-diffusion model (Rimmer et al. 2021). (a) The Kzz profiles in this study (red) and in Rimmer et al. (2021, blue). (b–f) Simulated vapor mixing ratio profiles of SO2, H2O, H2SO4 (liquid), SO, and CO, respectively (credit: Sean Jordan). The observations are shown as gray lines and scatters.

5 Conclusions

We estimated the K_zz between the cloud base and 55 km using H₂SO₄ vapor mixing ratios derived from the observations of VeRa on board Venus Express. Our approach relies on the premise that H₂SO₄ vapor is primarily regulated by diffusion and condensation processes in this region. The obtained global mean cloud K_zz exhibits distinct characteristics: it peaks at 5 × 10⁸ cm² s⁻¹ near 48 km, remains relatively constant at 2.5 × 10⁸ cm² s⁻¹ between 50 and 54 km, and rapidly decreases above that altitude. The measured K_zz is larger than model results (Lefèvre et al. 2022) by a factor of three and larger than evaluations of Vega (Blamont et al. 1986) and a microphysical model (McGouldrick & Toon 2007) by an order of magnitude. Besides, the vertical eddy motion exhibits a mean velocity on the order of 10²−10³ cm s⁻¹, which is two to eight times larger than the measurements of Vega balloons (Linkin et al. 1986) and convection models (Imamura et al. 2014; Lefèvre et al. 2018), suggesting that large-scale motions contribute significantly to the effective vertical eddy motions.

Our results suggest that cloud K_zz exhibits significant latitudinal variation. Although equatorial K_zz is three times as large as polar K_zz at 48 km, the high-K_zz layer in the polar regions can extend down to 43 km, with a peak K_zz of 1 × 10⁸ cm² s⁻¹ at 43.5 km. This altitudinal variation is consistent with the static stability distribution obtained from VeRa measurements (Ando et al. 2015, 2020a). However, our polar K_zz in the region above 46.5 km is lower than the equatorial K_zz, which seems to contradict the static stability distributions in some model studies (e.g., Ando et al. 2020b). This could be explained by other processes besides convection contributing to the H₂SO₄ vapor mixing in this region, such as global circulation. The upward diffusion flux of H₂SO₄ vapor above 46.5 km significantly decreases as the latitude increases. The flux in middle and low latitudes peaks at 48 km with a value of 1.5 × 10¹⁷ cm⁻² s⁻¹ while polar flux reaches a peak at 44 km with a value of 1.5 × 10¹⁷ cm⁻² s⁻¹. The latitudinal variation of eddy diffusion velocity is not significant.

To evaluate our results, we integrated the derived global mean K_zz in 46.5–55 km into the latest 1D photochemistry-diffusion model developed by Rimmer et al. (2021). The simulated distributions of SO₂, H₂O, SO, and CO are in good agreement with observations, except for a slight underestimation of the CO mixing ratio above 65 km. Our simulations indicate that the enhancement of cloud K_zz does not suppress the depletions of SO₂ and H₂O, but does significantly mix SO and CO in the cloud regions. Additionally, the enhancement of cloud K_zz largely increases the cloud mass loading. This indicates that the clouds could act as reservoirs for related vapors, buffering their abundances by adjusting the cloud mass loading in response to variations in vertical transport. Further investigations on the material exchange between the Venusian clouds and atmosphere are essential to verify this hypothesis.

Acknowledgments

We thank Janusz Oschlisniok (University of Cologne) for providing H₂SO₄ vapor data from VeRa observations, Kevin McGouldrick (University of Colorado Boulder) for providing particle distribution data from their public model results, Sean Jordan (University of Cambridge) for providing the output of their photochemistry-diffusion model. We thank the anonymous reviewer for his thoughtful suggestions that greatly improve the quality of our manuscript. This work is supported by the National Natural Science Foundation of China (Grant 42305135, 42275060). Longkang Dai is supported by Natural Science Foundation of Hunan Province (Grant 2023JJ40664). The data in this study are publically available at https://doi.org/10.6084/m9.figshare.24160752.V1.

Appendix A Uncertainty due to the retrieved H₂SO₄ profiles

The uncertainties produced by zonal data statistics in our cloud K_zz estimation have been exhibited in the figures. However, additional errors might be carried by H₂SO₄ vapor data. According to Oschlisniok et al. (2021), the absorption of SO₂ has been removed from the raw data by the estimated SO₂ abundance, and the SO₂ absorption is small compared to the H₂SO₄ absorption. Further testing from the authors shows that a 60-ppm increase or decrease in SO₂ could lead to about a 6% increase or decrease in the H₂SO₄ vapor in the lower clouds. Here we adopted a column variation of 6% H₂SO₄ vapor in the same cases in Figure 2. The derived H₂SO₄ vapor diffusion flux, the condensation or evaporation rate, and K_zz profiles are presented in Figure A.1. It shows that the effect on the H₂SO₄ vapor diffusion flux increases as the altitude decreases. 6% H₂SO₄ vapor column uncertainty results in a lower-cloud flux deviation of 1×10¹⁷ cm⁻² s⁻¹ in low latitudes and 4×10¹⁶ cm⁻² s⁻¹ in high latitudes. The effect on K_zz also increases with both decreasing altitude and latitude. Its uncertainty due to the retrieved H₂SO₄ profiles reaches 22.88% at 47.8 km in low latitudes and 8.92% at 44 km in high latitudes.

Fig. A.1 Uncertainty tests on corresponding cases in Figure 2. The left, middle, and right columns show H2SO4 vapor diffusion flux, the condensation or evaporation rate, and Kzz profiles, respectively. The green lines represent the original data. The red lines represent a 6% increase in the H2SO4 vapor column. The blue lines represent a 6% decrease in the H2SO4 vapor column.

References

All Tables

All Figures

	Fig. 1 VeRa observations (Oschlisniok et al. 2021). The panels show (a) the H2SO4 vapor volume mixing ratio, (b) temperature, and (c) pressure, respectively. We note that these observations are divided into 36 latitudinal bins with a resolution of 5 degrees. The lack of measurements is caused by the orbit geometry of Venus Express.
In the text

Fig. 2 Example cases illustrating the method of Kzz estimation. The three rows refer to (a–c) −0.02°, (d–f) −52.65°, and (g–i) 87.8° latitude, respectively. The left column shows the original H2SO4 vapor mixing ratio (VMR, solid black line), smoothed H2SO4 vapor mixing ratio (solid red line), and saturation vapor mixing ratio (SVMR, dashed black line) profiles. The middle column shows the net condensation (blue line) and evaporation (red line) rate profiles. The right column shows the derived Kzz distributions. The red triangles represent negative values, caused by the statistical fluctuation and horizontal convergence or divergence of H2SO4 vapor.

In the text

Fig. 3 Global mean results. (a) The derived global mean Kzz by adopting LCPS particle distributions. The blue line represents the median value along with the upper and lower quartiles. The derived Kzz profile below 46.5 km is distinguished in red since it is basically constituted of data from the polar regions and better considered as the “polar mean.” The bottom-left subplot displays the Kzz below the clouds, which is based on the given column production of H2SO4 vapor. The red line represents the below-cloud Kzz estimated in polar regions and the green line represents the other regions. The estimations based on Vega balloon observations (fuchsia, Blamont et al. 1986), Pioneer Venus radio scintillation measurements (black, Woo et al. 1982), and model results of McGouldrick & Toon (2007, purple) and Lefèvre et al. (2022, orange) are also presented. (b) The distribution of H2SO4 vapor upward diffusion flux. (c) The derived H2SO4 vertical molecular (left) and eddy (right) diffusion velocities as functions of altitude. The observations of atmospheric convection velocities from Vega balloons (fuchsia, Linkin et al. 1986), as well as model estimations of Imamura et al. (2014, green), Lefèvre et al. (2018, gray), and Gierasch et al. (1997, orange triangles for the Hadley velocity and black stars for the convective velocity) are also presented.

In the text

Fig. 4 Latitudinal variation. (a) Kzz, (b) H2SO4 vapor upward diffusion flux, and (c) H2SO4 vapor eddy diffusion velocity profiles in different latitude bins were derived by adopting particle distributions from the microphysical model results (McGouldrick & Barth 2023). The red, green, and blue lines represent the latitudinal bins of 0°–30°, 30°–60°, and 60°–90°, respectively. Panel d presents the altitude-latitude distribution of derived Kzz. We note that the missing middle-latitude results near 47 km are caused by a lack of data.

In the text

	Fig. 5 Evaluation of measured Kzz by adopting a 1D photochemistry-diffusion model (Rimmer et al. 2021). (a) The Kzz profiles in this study (red) and in Rimmer et al. (2021, blue). (b–f) Simulated vapor mixing ratio profiles of SO2, H2O, H2SO4 (liquid), SO, and CO, respectively (credit: Sean Jordan). The observations are shown as gray lines and scatters.
In the text

	Fig. A.1 Uncertainty tests on corresponding cases in Figure 2. The left, middle, and right columns show H2SO4 vapor diffusion flux, the condensation or evaporation rate, and Kzz profiles, respectively. The green lines represent the original data. The red lines represent a 6% increase in the H2SO4 vapor column. The blue lines represent a 6% decrease in the H2SO4 vapor column.
In the text