Venus's major cloud feature as an equatorially trapped wave distorted by the wind

Abstract

The superrotation of the atmospheres of slowly rotating bodies is a long‐standing problem yet unsolved in atmospheric dynamics. On Venus, the most extreme case known of superrotation, this is accompanied and influenced by a recurrent planetary‐scale cloud structure, known as the Y feature. So far, no model has simultaneously reproduced its shape, temporal evolution, related wind field, nor the relation between its dynamics and the unknown UV‐absorbing aerosol that produces its dark morphology. In this paper we present an analytical model for a Kelvin‐like wave that offers an explanation of these peculiarities. Under Venus cyclostrophic conditions, this wave is equatorially and vertically trapped where zonal winds peak and extends 7 km in altitude, and its vertical wind perturbations are shown to produce upwelling of the UV absorber. The Y‐feature morphology and its 30 day evolution are reproduced as distortions of the wave structure by the Venus winds.

1 Introduction

Venus's large‐scale dark Y pattern (the Y‐tilted 90°, also described as a “Psi”) was discovered in the 1960s [Boyer and Camichel, 1961], and it has been repeatedly identified in ultraviolet images from many space missions [Belton et al., 1976; Rossow et al., 1980; Peralta et al., 2007a; Titov et al., 2012]. The Y pattern is a recurrent pattern whose lifetime seems to alternate between cycles of creation and destruction [Rossow et al., 1980]. It shows two arms symmetric relative to the equator, extends up to a latitude of 45° (Figure 1a), and it has a quasi‐uniform rotation with a period of about four terrestrial days, differing between 0.7 and 0.1 days from the period of the zonal wind at the equator [del Genio and Rossow, 1990]. A recent characterization of the wind field associated with the Y feature [Kouyama et al., 2012] showed a good correspondence with the cloud brightness distribution [Peralta et al., 2007a], with westward acceleration occurring in the dark regions (Figure 1b). Previous analytical studies of equatorial waves on Venus relied on the geostrophic formulation by means of a frame of coordinates fixed to the superrotating winds for which solid‐body rotation is assumed [Smith et al., 1992; Covey and Schubert, 1981; Imamura, 2006]. This is not realistic since the zonal wind profile is far from such a rigid rotation [Rossow et al., 1990; Peralta et al., 2007b; Smith et al., 1993]. Moreover, the properties of the equatorial waves in a cyclostrophic regime like the Venusian have never been studied analytically but with complex numerical models involving a difficult interpretation of the phenomenon, and in no case make a successful comparison with the time evolution of the Y feature reported with Pioneer Venus observations [Smith et al., 1992, 1993; Yamamoto and Tanaka, 1997, 1998; Imamura, 2006; Lee et al., 2010; Kouyama et al., 2015]. Most of the previous works interpret the Y feature as the combination of two modes: a Rossby mode at midlatitudes and a Kelvin mode at the equatorial region [del Genio and Rossow, 1990; Imamura, 2006; Kouyama et al., 2015], although classical Rossby and Kelvin waves are not possible in cyclostrophic regimes [Peralta et al., 2014b], and periodograms from zonal winds obtained with Venus Express observations do not show evidence of two simultaneous modes at equator and midlatitudes but a single equatorial mode extending to midlatitudes [see Kouyama et al., 2013, Figure 9] and with periods varying between 4 and 5 days [see Khatuntsev et al., 2013, Table 2]. For this reason, we investigate the Kelvin‐like wave that could arise as a solution in Venus's particular conditions.

Figure 1

Observations and model of Venus Y pattern. (a) Composite of Venus cloud tops during Galileo flyby [Peralta et al., 2007a]. (b) Zonal wind perturbations during Galileo flyby [Kouyama et al., 2012]. (c) Horizontal structure of our wave according to equations 4–6. Zonal wind disturbances (u′) are displayed with red arrows. Blue lines display perturbations on pressure (P′) and in‐phase component of vertical velocity (w′). Green lines show the 90°‐shifted component of w′. Lines are percentages of the maximum amplitude, and continuous/dotted lines denote positive/negative disturbances, respectively. (d) Zonal wind disturbances predicted by our wave model after 6 days of propagation within a realistic zonal mean flow.

2 An Equatorial Wave for Cyclostrophic Regimes

Our analytical model starts from the primitive equations for a cyclostrophic regime [Peralta et al., 2014a, 2014b] (see Appendix A), neglecting equatorward of midlatitudes the meridional shear of the zonal wind, and incorporating the vertical shear of the zonal wind [Schubert, 1983; Gierasch et al., 1997; Peralta et al., 2007b]. While on the Earth, what traps atmospheric waves along the equator is the meridional variation of the Coriolis parameter f ≈ β · y (with f = 2Ω · sinφ, β = df/dy, with Ω being Earth's angular rotation velocity and y the meridional coordinate) [Sánchez‐Lavega, 2011], we find that on Venus this role is played by the centrifugal force through a centrifugal frequency [Peralta et al., 2014a, 2014b] Ψ = (u₀· tanφ)/a (where φ is the latitude, u₀ is Venus background zonal wind, and a is the planetary radius of Venus). Equivalently, Ψ is found to vary linearly between the equator and midlatitudes with (Ψ≈β_*·y with β_* = ∂Ψ/∂y) [Peralta et al., 2014b]. Under these conditions, the wave dispersion relation and the e‐folding decays for our Kelvin‐like wave amplitude are (see Appendix A for demonstration)

(1)

(2)

(3)

where and are the horizontal and vertical e‐folding decays, is the horizontal phase velocity, k and m are the horizontal and vertical wave numbers, N is the Brunt‐Väisällä frequency, du₀/dz is the vertical shear of the zonal wind, and Γ² > 0 (with units m⁻²) is a constant that appears as a consequence of the vertical wind shear, du₀/dz ≠ 0.

The e‐folding decays are expressed relative to the equator and the altitude z₀∼67km where the zonal wind peaks (Figure 2b). According to our reference atmosphere (see Figure 2 and Peralta et al. [2014a]), for z < z₀ we have that N² ≈ 3.6 · 10⁻⁴ s⁻², du₀/dz ≈ −6.5 · 10⁻³ s⁻¹ and β_* ≈ −2.5 · 10⁻¹² s⁻¹ m⁻¹; while for z > z₀ we obtain du₀/dz ≈ 3.5·10⁻³s⁻¹, β_* ≈ −2.6·10⁻¹² s⁻¹ m⁻¹, and nearly the same value for N². Equation 2 predicts that at the altitude level z = 65 km the absolute intrinsic phase velocity must fulfill the condition and indicates that the meridional width is quite sensitive to the phase velocity. And using the horizontal e‐folding 2882 ± 427 km obtained during Galileo flyby of Venus [Kouyama et al., 2012] (Figures 1a and 1b and Table 1), we obtain , which compares well with the phase velocity 23 ± 6 m s⁻¹ of the Y feature during the Galileo flyby [Kouyama et al., 2012].

Figure 2

Meridional profile for the centrifugal frequency Ψ at three height levels of (a) the Venus atmosphere and (b) vertical profiles for the background zonal wind and (c) Brunt‐Väisällä frequency at several latitudes. The grey area corresponds to the reference altitude z₀ where du₀/dz = 0 and the zonal wind peak. Observe that the centrifugal frequency behaves as linear between the equator and midlatitudes. The zonal wind data were taken from our Venus reference atmosphere combining and interpolating horizontal profiles from cloud tracking with vertical profiles obtained with in situ measurements by Pioneer Venus probes [Peralta et al., 2014a].

Table 1. Venus Y‐Feature Wave Properties

Wave Description	Measurementsa^a Measurements derived from cloud albedo [del Genio and Rossow, 1990] and winds [Kouyama et al., 2012].	Model
Period (ground based)
(τ = 2π/ω)	3.94–4.00 days	3.7 days
Zonal wave number
(s = k·a cosφ)	1	1
Vertical wavelength
(λz = 2π/m)	—	7.6 km
Zonal velocity (65 km)	95 ± 2m s^−1	100 m s^−1
Intrinsic phase velocity
(relative to zonal flow)	16 ± 5 m s^−1	23 m s^−1
	9 ± 1 m s^−1	10 m s^−1
	—	32.9 Pa
	—	1.6·10^−3 m s^−1
Horizontal e‐folding	2882 ± 427 km	2880 km
Vertical scale	—	7 km

• ^a Measurements derived from cloud albedo [del Genio and Rossow, 1990] and winds [Kouyama et al., 2012].

If we take [Kouyama et al., 2012], equation 2 gives Γ² ≈ 1.99 · 10⁻¹⁴. Then, equation 3 gives and above and below the altitude z₀, respectively, implying for the wave a net vertical envelope of ∼7 km. This vertical extent can be compared with the vertical wavelength given by equation 1 for the Y feature having zonal wave number 1 [Schubert, 1983; Gierasch et al., 1997; Peralta et al., 2007a] (m ≫ k). At 65km height we have λ_z ≈ 7.6km, which is compatible with our predicted vertical length. This value also constrains the nature of the UV absorber since its weighting function should be narrower than λ_z/2 for the Y feature to be visible [del Genio and Rossow, 1990].

For constant du₀/dz and N² (see Figures 2b and 2c), the wave's amplitude on the pressure, and zonal and vertical velocity is given by (see Appendix A for demonstration)

(4)

(5)

(6)

where and ρ₀ is the density of the basic state. Equation 4 indicates that the altitude z₀ represents an upper limit for this equatorial wave propagation: du₀/dz > 0 for z > z₀ means that the wave amplitude exponentially decreases with altitude, while du₀/dz < 0 for z < z₀ implies that the wave amplitude decreases exponentially but with depth. Accordingly, the wave stays vertically trapped in the layer of maximum background wind speed, which is also where more than a half of the solar energy is absorbed [Gierasch et al., 1997]. On the other hand, from 5 wave disturbances u′ and P′ are phase shifted 180° implying that the phase of westward acceleration (u′ < 0) corresponds to increasing pressure (P′ > 0) and decreasing geopotential height. Our estimates for the wave amplitude (Table 1) can be checked to have values less than 1% of the nominal value in the case of the pressure and vertical velocity, while in the case of the zonal wind perturbations they reach up to 10%, as predicted by Seiff et al. [1985], barely affecting the cyclostrophic balance.

3 Differences With the Geostrophic Kelvin Wave

Even though the equatorial wave hereby deduced keeps similarities with the terrestrial Kelvin waves (it propagates along the west‐east direction and is equatorially trapped, its wave amplitude is maximum at the equator and decreases away from it), it also presents distinct properties that arise from the absence of the Coriolis factor on Venus and clearly set its different nature. Equation 1 indicates that the wave propagates in the same sense as the mean zonal flow, thus implying that it moves to the west while Kelvin waves on the Earth propagate eastward. The polarization relation 4 contains information about both the meridional and vertical modulations of the wave amplitude. In the presence of vertical shear (Γ ≠ 0) with a local maximum (Figure 2b), our wave is also vertically trapped about a certain altitude. Moreover, when the horizontal e‐folding decay for the Kelvin waves [Sánchez‐Lavega, 2011] is confronted with equation 2, we notice that they have different expressions. This is reasonable as the nature of the agent responsible of trapping the waves about the equator is completely different on both planets, i.e., the change of sign for the Coriolis factor f on the Earth, while in the Venus case that role is played by the centrifugal frequency, Ψ. Moreover, the dependence between the phase velocity and meridional width clearly differs from that for the Kelvin waves since on Venus we have different values of β_* and a wider range of validity for the β‐plane approximation (see Figure 2a).

4 The Dark Appearance of the Y Feature

It is still unclear whether the cause for the dark contrasts observed in ultraviolet is due to differences in cloud top altitude, temperature contrasts, or compositional variations, with the later being probably related with small aerosol concentrations of reflectors [Yamamoto and Tanaka, 1998] or enhanced concentrations of an UV absorber [Rossow et al., 1980; del Genio and Rossow, 1990; Titov et al., 2008]. Based on recent evidence from Venus Express observations [Titov et al., 2008], here we assume that contrasts are caused by upwelling of an absorber from a depth which is yet to be determined accurately [Molaverdikhani et al., 2012] and that convective mixing cannot be the only agent for this upwelling since a permanent solar‐locked dark feature attached to the subsolar point would be observed then [Rossow et al., 1980]. Figures 1c and 1d show that our model's P′ and u′ match the observed cloud albedo field and the horizontal wind field (Figures 1a and 1b). This evident correlation supports previous works' interpretation of dark features as the result of upwelling of the ultraviolet absorber by vertical wind perturbations over a half cycle of the wave, while bright features are the result of downwelling of absorber‐depleted air over the other half cycle [Belton et al., 1976; del Genio and Rossow, 1990; Kouyama et al., 2012]. Nevertheless, maximum pressure and vertical velocity amplitudes (equations 4 and 6) are and (Table 1), so vertical displacements are ∼280m over half a cycle of the equatorial wave (∼2 days), which is a short distance (<< vertical scale height H) to justify the upwelling causing the high concentrations of ultraviolet absorber. Alternatively, we propose that this upwelling can be provided by the 90°‐shifted component of w′. This component is inversely proportional to the vertical shear of the wind (equation 6), thus allowing to shoot up close to the altitude where the zonal wind peaks and du₀/dz→0. The lag between the dark region and the phase for maximum upwelling seen in the observations can be explained as a result of the strength of atmospheric molecular diffusion [del Genio and Rossow, 1990], not included in this model.

5 Shape and Evolution of the Y Feature

Previous attempts to explain the shape of the Y feature required the resonant interaction between two planetary waves [Belton et al., 1976; del Genio and Rossow, 1990; Yamamoto and Tanaka, 1997] or a single Kelvin wave with transient convective patterns that explain its morphology only partially [Smith et al., 1992, 1993]. The first hypothesis is supported by the presence of two periods of 4 and 5 days during Pioneer Venus observations, although these modes were shown to rule over different latitude ranges which overlap in a narrow region [see del Genio and Rossow, 1990, Figure 1]. All theories failed to explain the evolution of the Y along one or several of its life cycles as documented by Rossow et al. [1980]. Alternatively, we explain the temporal behavior of the Y feature as the result of the distortion that the equatorial wave (equations 4–6) undergoes when propagating within a realistic zonal flow for Venus [Peralta et al., 2014a]. Since this flow is far from solid‐body rotation [Peralta et al., 2007b], the wave's period is shorter at higher latitudes. As a consequence, the coherence of the wave suffers a progressive deformation as it circles the planet (see Movie S1 in the supporting information), consistently with the phase tilt previously reported by Covey and Schubert [1982] and Imamura [2006]. Figure 3 exhibits the temporal evolution of the wave predicted with our model compared with that of the observed Y feature [Rossow et al., 1980]. The simulation, carried out with time steps of 0.1 terrestrial days and spanning 30 days, covers the 20 day characteristic time based on the energy loss per period for a wave excited by a radiative‐dynamic cloud feedback [Smith et al., 1992] and not included in the model. For the background zonal wind between the equator and midlatitudes, we used the mean value −86m s⁻¹ during the data set of Pioneer Venus images [see Rossow et al., 1980, Figure 16a]. An intrinsic phase velocity of 14m s⁻¹ was found to provide an optimum fit to Pioneer Venus observations, also in accordance with the estimation 16 ± 5m s⁻¹ from the cloud brightness [del Genio and Rossow, 1990], and implied a horizontal e‐folding decay of about 2348km. Except for the slightly different tilt at midlatitudes apparent in Pioneer Venus images, Figure 3 displays quite a good agreement between the cloud morphology and the brightness contrast expected as the equatorial wave becomes distorted. The so‐called spiral dark streaks that can be seen at higher latitudes at mature states of the Y feature [see Belton et al., 1976, Figure 6] appear in our simulations at day 22 and are consistent with the increasing lag with latitude of the wave phase (see Figure 4).

Figure 3

Evolution of Venus Y pattern and the wave model. The Y feature during the Pioneer Venus mission [Rossow et al., 1980] is compared with the evolution of the wave's horizontal structure shown in Figure 1c when propagating within Venus' winds. To fit the geometry of observations, spacecraft projections of the wave model were made with line of sight intersecting the equator. The wave phase with high concentrations of the absorber due to the upwelling is displayed with a single tone of black, while the phase associated to the downwelling and where wave's amplitude is less than 1% of maximum are displayed in white to enhance contrast and improve comparison.

Figure 4

Comparison between (left) the dark circular streaks that appear in the UV images taken during the Mariner 10 flyby of Venus (also, see Belton et al. [1976, Figure 6]) and (right) the lag of the wave phase apparent in our simulations after 22 terrestrial days.

6 Conclusions

Despite the many works facing the physical interpretation of the Venus Y feature, none has inferred an analytical model based in the prevailing cyclostrophic conditions. In this paper, we provide a self‐consistent model for the equatorial mode of the Y‐feature morphology, its 30 day evolution and the appearance of subpolar dark streaks [Belton et al., 1976; Rossow et al., 1980], which are a result of the phase distortion of an equatorially trapped wave as it propagates within the Venus winds. Our wave model is a solution of primitive equations from a scale analysis with Venus Express data [Peralta et al., 2014a] which properly describe the Venus cyclostrophic regime. Along with other waves' solutions previously obtained [Peralta et al., 2014b], our wave model could potentially be extended to other slowly rotating bodies like Titan [Flasar et al., 2010] and the increasing number of extrasolar planets suspected to have, globally or locally, cyclostrophic conditions. Future improvements for this model must include Venus's meridional flow (which may improve the tilt of the dark streaks at midlatitudes), midlatitude interactions with a 5 day mode at midlatitudes in terms of centrifugal waves [Peralta et al., 2014b], as well as exploring the sources of excitation for these equatorial waves.

Acknowledgments

J. Peralta acknowledges the Spanish MICINN for funding support through the CONSOLIDER program “ASTROMOL” CSD2009‐00038 and also funding through project AYA2011‐23552. A.S.‐L. was supported by the Spanish MICIIN project AYA2012‐36666 with FEDER support, Grupos Gobierno Vasco IT765‐13 and UPV/EHU UFI11/55. P.M. acknowledges the support from Observatoire de Paris‐LESIA and the Portuguese Foundation for Science and Technology (FCT, PhD grant reference: SFRH/BD/66473/2009). D.L. and P.M. acknowledge FCT funding through project grants POCI/CTE‐AST/110702/2009, PEst‐OE/FIS/UI2751/2014, and EC project EuroVenus. Anonymous reviewers are thanked for their fruitful corrections which allowed to greatly improve this paper.

The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.