Icarus 278 (2016) 38–51

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Wave analysis in the atmosphere of Venus below 100-km altitude,
simulated by the LMD Venus GCM
Sébastien Lebonnois a,∗, Norihiko Sugimoto b, Gabriella Gilli aabLaboratoire de Météorologie Dynamique (LMD/IPSL), Sorbonne Universités, UPMC Univ Paris 06, CNRS, 4 place Jussieu, Paris 75252, France
Department of Physics, Hiyoshi campus, Keio University, Hiyoshi 4-1-1, Kouhoku-ku, Yokohama, 223-8521, Japana r t i c l e

i n f oArticle history:
Received 21 January 2016
Revised 2 June 2016
Accepted 6 June 2016
Available online 14 June 2016
Keywords:
Venus, atmosphere
Atmospheres, dynamics
Experimental techniques
Image processing
Ultraviolet observations
Numerical modelinga b s t r a c t
A new simulation of Venus atmospheric circulation obtained with the LMD Venus GCM is described andthe simulated wave activity is analyzed. Agreement with observed features of the temperature structure,static stability and zonal wind ﬁeld is good, such as the presence of a cold polar collar, diurnal and semidiurnal tides. At the resolution used (96 longitudes × 96 latitudes), a fully developed superrotation is
obtained both when the simulation is initialized from rest and from an atmosphere already in superrotation, though winds are still weak below the clouds (roughly half the observed values). The atmosphericwaves play a crucial role in the angular momentum budget of the Venus’s atmospheric circulation. In
the upper cloud, the vertical angular momentum is transported by the diurnal and semi-diurnal tides.
Above the cloud base (approximately 1 bar), equatorward transport of angular momentum is done bypolar barotropic and mid- to high-latitude baroclinic waves present in the cloud region, with frequencies
between 5 and 20 cycles per Venus day (periods between 6 and 23 Earth days). In the middle cloud, justabove the convective layer, a Kelvin type wave (period around 7.3 Ed) is present at the equator, as well
as a low-latitude Rossby-gravity type wave (period around 16 Ed). Below the clouds, large-scale mid- tohigh-latitude gravity waves develop and play a signiﬁcant role in the angular momentum balance.
© 2016 Elsevier Inc. All rights reserved.

1. Introduction
The general circulation in Venus’s atmosphere is dominatedby the phenomenon called superrotation, with most of the atmosphere rotating in the same direction but about sixty times fasterthan the solid surface. The mechanism that controls this phenomenon combines transport of angular momentum by the mean
meridional circulation with compensation done by planetary-scale
waves. This idea was originally proposed by Gierasch (1975) andRossow and Williams (1979), and has been demonstrated in recent
years with the study of numerical simulations conducted byGeneral Circulation Models (GCMs) of the Venus’s atmosphere (e.g.
Yamamoto and Takahashi, 2003; Lee et al., 2007). These tools arevery useful to study the wave activity and its role in the angularmomentum budget. In addition to the balance between transport
by the mean meridional circulation and transport by the horizontal
planetary waves, the role of thermal tides that transport angularmomentum vertically in the low latitudes was conﬁrmed by recent
works (Lebonnois et al., 2010; Takagi and Matsuda, 2007). The∗

Corresponding author.
E-mail address: Sebastien.Lebonnois@lmd.jussieu.fr (S. Lebonnois).

http://dx.doi.org/10.1016/j.icarus.2016.06.004
0019-1035/© 2016 Elsevier Inc. All rights reserved.

most recent and realistic GCMs either use a full radiative transfer
module to compute temperature self-consistently (Ikeda, 2011;
Lebonnois et al., 2010; Lee and Richardson, 2011; Mendonca et al.,2015), or force the temperature structure with carefully preparedheating rate proﬁle and Newtonian cooling (Sugimoto et al., 2014a;
2014b).
A variety of waves in Venus’s atmosphere has been observedin the cloud region, in the middle cloud from infrared observations or at the cloud-top from reﬂected visible and ultraviolet sunlight (e.g. Belton et al., 1976; Rossow et al., 1980; Del Genio andRossow, 1990; Peralta et al., 2008; Piccialli et al., 2014). Theseobserved waves range from planetary-scale waves to small-scale
gravity waves. Variability observed in the wind reveals periods
from 4 to 5 Earth days to 1 Venus day, or even longer (Khatuntsev
et al., 2013; Kouyama et al., 2013; Rossow et al., 1990). Small-scale
gravity waves are observed in Venus-Express datasets, with VIRTIS images (Peralta et al., 2008), VMC images (Piccialli et al., 2014),
or VeRa radio-occultations (Tellmann et al., 2012). The theoreticalanalysis of waves present in Venus’s atmosphere is different fromthe Earth case, because of the cyclostrophic regime and the crucialrole played by the mean zonal wind ﬁeld. Most analytical works
studying Venus’s atmospheric waves use a realistic vertical windS. Lebonnois et al. / Icarus 278 (2016) 38–51

39

Table 1
Values of the terms in the total angular momentum budget, averaged over the last 2 Vd (units are
1018 kg m2 s−2 ).

300 Vd(from rest)
190 Vd(from superrotation)

dMr /dt

T (T + )F (F + )

D

S



∗

ξ

1.5

−28.3 (42.5)−7.6 (3.0)

1.6

−3.2

39.0

37.4

0.46

6.8

−28.1 (41.6)−7.0 (1.8)

1.9

−3.5

43.5

41.9

0.53

Mr = Relative part of the total atmospheric angular momentum, due to zonal wind u. T = Mountaintorque on the atmosphere due to topography (T + is its positive (source) component). F = Surface torqueon the atmosphere due to friction (F + is its positive (source) component). D = Residual torque due toconservation errors in the horizontal dissipation parameterization. S = Torque on the atmosphere due
to upper boundary conditions (sponge layer).  = Residual numerical rate of total angular momentum∗
variation due to conservation errors
in the dynamical core. = S + D +  , should theoretically be zero.
ξ = Ratio between | ∗ | and Max T + + F + , |T − + F − | (Lebonnois et al., 2012b).

proﬁle, but with a solid-body rotation approximation for the latitudinal wind proﬁle (e.g. Covey and Schubert, 1982; Schinder et al.,
1990; Smith et al., 1993). A detailed theoretical analysis of waves
in the context of a realistic zonal wind ﬁeld for Venus has been
recently undertaken by Peralta et al. (2014a); 2014b); 2015), exploring wave solutions and their classiﬁcation.
Kouyama et al. (2015) explore the possibility of retroactive
interactions between a low-latitude Kelvin type wave and midlatitude Rossby type waves at the cloud top: the variations of thezonal wind induced by each wave favor the vertical propagation of
the other one, that process yields long-term variability of the windover periods of several Venus days.
Using a high-resolution Venus GCM starting from superrotationand forced by observed heating rate proﬁle and Newtonian cooling, Sugimoto et al. (2014a, b) analyzed the wave activity producedin the cloud region. The large vertical zonal wind shear and latitudinal temperature gradient generate the basic state of baroclinic
instability in the cloud region. Baroclinic waves develop, and atcloud-top, Rossby type waves are produced by this baroclinic activity. However, further studies of Venus waves using various GCMs
are required to get a comprehensive understanding of Venus’s atmospheric dynamics.
In this work, we present the recent evolutions of the LMD
Venus GCM (Lebonnois et al., 2010), that simulates realistic temperature and zonal wind ﬁelds (Section 2), with a detailed analysisof the waves produced in this simulation (Section 3).
2. Simulations and validation against observations
2.1. The LMD Venus GCM
The model developed at LMD for the study of Venus’s atmosphere has been described in details in Lebonnois et al. (2010). It
is based on the LMDZ latitude–longitude grid ﬁnite-difference dynamical core (e.g. Hourdin et al., 2006), including a longitudinalpolar ﬁlter.
Most of the features of this GCM are similar to those presented in Lebonnois et al. (2010). Among the physical parameterizations, the main difference is the use of a boundary layer scheme
taken from Mellor and Yamada (1982) to compute the eddy diffusion coeﬃcient and the time evolution of the mixed variables.
The equations used are described in Appendix B of Hourdin et al.
(2002). This boundary layer scheme is based on a more physicalrepresentation of the unstable regions. It was successfully used
for other planetary applications of the LMD GCM (e.g. Lebonnois
et al., 2012a, for Titan). At surface, the drag coeﬃcient is similar tothe one used in the previous parameterization: Cd = (0.4/ ln(1 +z1 /z0 ))2 , where z1 is the altitude of the center of the ﬁrst layer ofthe model (roughly 10 m in our case) and z0 is the roughness coeﬃcient, taken equal to 1 cm. Otherwise, compared to Lebonnois
et al. (2010) we use the same hybrid vertical coordinates with topography (50 vertical levels), the soil model is unchanged, the temperature dependence of the speciﬁc heat is taken into account, theradiative transfer includes solar heating rate proﬁles as a functionof solar zenith angle taken from a look-up table based on Crisp(1986), and the infrared net-exchange rate (NER) matrix formulation discussed in Eymet et al. (2009). For the computation of the
IR NER matrix, opacity sources (gas, clouds) are taken horizontally uniform and properties are the same as in Lebonnois et al.
(2010).
The second main difference is the horizontal resolution. For the
study that is presented here, the resolution was increased to 96
longitudes by 96 latitudes (3.75° × 1.875°). A few years ago, a simulation was performed with this model with a horizontal resolution of 48 longitudes by 32 latitudes and run for up to 190 Venus
days (1 Venus day (Vd) = 117 Earth days (Ed)) starting from an atmosphere already in superrotation. Modeled temperature ﬁeld wascompared with VIRTIS/Venus-Express temperature retrievals to discuss thermal tides above the clouds (Migliorini et al., 2012). It must
be noted that when starting from rest, the superrotation was not
fully evolved in this version of the GCM, with weaker zonal windin the deep atmosphere and a peak around 80 m/s at the cloudtop, after more than 10 0 0 Vd. In the present work, the simulations
were started either from rest (for 300 Vd) or from a zonal windﬁeld already in superrotation (for 190 Vd). Both simulations converged toward very similar wind ﬁelds, meaning that the superrotation is fully evolved by the LMD VGCM from motionless state forthe ﬁrst time. The present work will focus on the analysis of thesimulation started from superrotation.
Increasing the resolution had a drawback: it has induced alarger residual term in the angular momentum budget as indicated in Table 1 (that can be compared to simulations presented in
Lebonnois et al., 2012b). As mentioned in Lebonnois et al. (2012b),
the lack of angular momentum conservation is a problem for simulations of Venus’s atmosphere, and Table 1 shows that in this conﬁguration, our GCM has a bias in the angular momentum budget.
This is currently investigated and needs to be improved in future
simulations. This numerical inaccuracy alters the balance of angular momentum transport, the non-zero term due to dynamics inthe angular momentum budget being compensated by a non-zerobalance of the surface momentum exchanges. However, the convergence between simulations started from rest and started fromsuperrotation gives conﬁdence in the achieved wind distributions,
except near the surface where the winds must be biased. This bias
should be taken into account when comparing the resulting simulated circulation to the observations near the surface.

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Fig. 2. Temporal variations of the temperature ﬁeld at pressure p = 2×103 Pa (near
70 km altitude, cloud top), and longitude 0, for the last 2 Vd of the simulation.

Fig. 1. Temperature structure obtained in the simulation started from superrotation,
after 190 Vd: (a) vertical proﬁle of globally averaged temperature; (b) latitudinal
proﬁles of temperature (zonal and temporal average) at roughly 50, 60 and 70 km
altitude. The dashed lines are from the simulation started from rest, after 300 Vd.

Fig. 3. Temperature ﬁelds at pressure p = 3×103 Pa (∼67 km altitude) in the northern polar region. In each panel, the temperature ﬁeld is averaged over 1/100 Vd(1.17 Ed), and the panels are separated by this time interval. The black contours
show the sub-solar area and the terminator. The latitude and longitude divisions
are 30°.

2.2. Temperature and winds ﬁelds
The temperature structure obtained both in the simulation
started from superrotation (after 190 Vd) and in the simulation
started from rest (after 300 Vd) are displayed in Fig. 1. Though
the vertical proﬁle is close to the VIRA reference atmosphere (Seiffet al., 1985), the modeled surface temperature is colder than observed, and the temperature in the cloud slightly higher (e.g.
Tellmann et al., 2009; Migliorini et al., 2012; Grassi et al., 2014).
The temporal variation of the temperature at a ﬁxed longitude,
at cloud top, is shown in Fig. 2. It is dominated by the thermaltides, but high-frequency variations are also clearly present. As
in previous simulations (Lebonnois et al., 2010; Migliorini et al.,2012), the semi-diurnal tide is dominant at low latitudes, while
the diurnal tide is dominant at high latitudes. The quasi-bidiurnal
oscillation that was present in L10 is not present in these simulations. This mode of oscillation, which is not understood yet andwhich is clearly not present in the observations, appeared in previous simulations only when starting from rest, and in simulations
when the zonal wind speed in the cloud region remained weaker
than observed.
The horizontal structure of the temperature at 3 × 103 Pa(roughly 67 km, just below the cloud-top) is shown in Fig. 3,
with four panels separated by 1/100 Vd (1.17 Ed). Though thezonally and temporally averaged latitudinal proﬁles show a small

cold-collar signature at cloud-top, an equator-to-pole contrast
around 30 K is obtained near 60 km altitude (Fig. 1b). The structure of the cold polar collar is much more visible in the maps
shown in Fig. 3 (without zonal and temporal averaging). This structure is very similar to the cold polar collar structure discussed inAndo et al. (2016), which is obtained with the GCM named AFES
(Sugimoto et al., 2014a, b). A similar feature was also obtained
previously in the works of Lee et al. (2005) and Yamamoto and
Takahashi (2012). The model of Lee et al. (2005) obtained quite
a different circulation compared to the present results and to the
AFES model, so comparison is diﬃcult. Yamamoto and Takahashi(2012) showed a partially formed cold collar with a warm core,
but did not investigate the mechanism producing this feature. In adetailed investigation, Ando et al. (2016) demonstrated the role ofthe thermal tide in this temperature distribution.
The shape of the polar temperature distribution (cold collar andwarm pole) obtained in our results is affected both by the thermal tide and by high-frequency wave activity, though the variations illustrated in Fig. 3 are weak. The contrasts inside the polar regions are also weaker than in the AFES simulations of Andoet al. (2016). Note that AFES is a spectral model used at higher resolution. These two signiﬁcant differences, spectral model and highresolution should improve the description of the polar region. It
must be noted that the latitudinal variation of the cloud structure

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41

Fig. 4. Vertical proﬁle of globally averaged static stability obtained in the simulation started from superrotation, after 190 Vd, compared to proﬁle retrieved fromthe Vega 2 entry probe dataset (adapted from Zasova et al., 2007), shown in gray.
The dashed line is from the simulation started from rest, after 300 Vd.

Fig. 5. Distribution of the mean zonal wind ﬁeld (white contours show the mean
meridional stream function, in units of 109 kg/s) obtained in the simulation started
from superrotation, after 190 Vd.

is not taken into account in any of these works. This could play arole in the shape and strength of the cold collar, since the radiative impact of the clouds in the polar region is signiﬁcant, but theresults discussed here show that this interaction is of second orderfor the presence of the cold collar.
The static stability proﬁle is shown in Fig. 4. It may be compared to the stability proﬁles deduced from the Pioneer Venus
data (Schubert, 1983), from the Vega 2 entry probe (also reproduced in Fig. 4, Zasova et al., 2007), and from the VeRa/Venus
Express data (Tellmann et al., 2009). The dominant features are
the convective region located in the lower and middle clouds (theregion with very small stability around 50 km altitude), the stable layer below the clouds, and the low-stability deep atmosphere(below 30 km altitude). Compared to the simulations done in
Lebonnois et al. (2010), the change of boundary layer scheme has
clearly improved the agreement with observations close to the surface. The thickness of the modeled convective region is smaller
than in the observations. As mentioned in Lebonnois et al. (2015),
the cloud model used to computed the radiative transfer plays
a role in this feature. Simulations based on a new cloud model(Haus et al., 2014) in which the thickness of the convective layer

Fig. 6. Zonal and temporal averaged proﬁles of the zonal wind: (a) vertical proﬁles
at three different latitudes, compared to observed proﬁles from Venera and Pioneer
Venus probes (gray, adapted from Schubert, 1983); (b) latitudinal proﬁles at roughly
50, 60 and 70 km altitude, compared to averaged cloud-tracking zonal wind proﬁles
obtained with VIRTIS-M images at UV (blue diamonds), visible (green triangles) andnear-IR (magenta circles) wavelengths. They correspond to altitudes 66–72 km for
UV spectral range, and a few kilometers below that level for visible/near-IR wavelengths (adapted from Hueso et al., 2015). The dashed lines are from the simulation
started from rest, after 300 Vd. (For interpretation of the references to color in thisﬁgure legend, the reader is referred to the web version of this article.)

is improved compared to observations will be studied in futurework.
The zonally and temporally averaged zonal wind ﬁeld obtainedin the simulation is presented in Fig. 5, together with the mean
meridional circulation. As in Lebonnois et al. (2010), the meridional circulation is divided in three regions along the vertical: adominant pair of equator-to-pole Hadley-like cells in the lower andmiddle cloud (roughly 105 –104 Pa), a similar pair above 104 Pa(up to the model top), and another below the clouds, though inthe deep atmosphere the cell structure is affected by perturbations
from topography. The comparison between the modeled zonal
wind ﬁeld and observations has improved compared to Lebonnois
et al. (2010) as shown in Fig. 6, where the vertical and latitudinal
proﬁles of the zonal wind are compared to in-situ vertical proﬁles(adapted from Schubert, 1983) and cloud-tracking latitudinal proﬁles (taken from Hueso et al., 2015). The improved treatment of

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Fig. 7. Frequency analysis of the temperature, zonal and meridional wind time series as a function of pressure, at the equator and 70°N. The solid black line indicates thefrequency corresponding to the zonal and temporal average of the zonal wind speed.

the boundary layer yields more eﬃcient pumping of angular momentum in the deep atmosphere, and thus the zonal wind below the clouds can reach more than 20 m/s. Although it is approximately half of the observed values (Fig. 6), this is a signiﬁcant step forward compared to Lebonnois et al. (2010) where thedeepest atmosphere was moving with only very weak zonal winds.
However, the increased horizontal resolution plays also a role.
The simulations previously mentioned, done with the improved
boundary layer but at lower horizontal resolution, led to weakerzonal wind (especially in the deep atmosphere) when started fromrest. One possible explanation is that the increase in the model
grid resolution have improved the representation of the planetaryscale waves at the smallest resolved scales, with therefore an effect on the global angular momentum balance (see in particular
Section 3.5 below). This will be investigated in order to conﬁrm
the present results.

In the upper region of the clouds, the modeled zonal wind isclose to the observations. However, several discrepancies must be
mentioned: (1) above the clouds, the zonal wind does not decreaseas fast with altitude as the thermal wind computations suggest(e.g. Piccialli et al., 2012) and values around 100 m/s extends up
to 90 km altitude; (2) the equatorial jet is too strong compared
to mid-latitudes – it must be noted that this region is extremely
sensitive to radiative forcings, and to details in the model (suchas cloud structure), which affect the balance in angular momentum and the zonal wind distribution; (3) the vertical gradient between the lower and upper cloud is higher than observed; (4) the
latitudinal gradient in the polar region is too steep compared tothe cloud-tracking zonal wind proﬁles. This is related to the temperature ﬁeld in the polar vortex (position and amplitude of thecold collar). Several investigations are on-going in order to improve on these discrepancies, mainly regarding the potential role of

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43

Fig. 8. Frequency analysis of the temperature, zonal and meridional wind time series as a function of latitude, at pressure levels 2 × 104 Pa and 1 × 106 Pa. The solid blackline indicates the frequency corresponding to the zonal and temporal average of the zonal wind speed.

small-scale processes (gravity waves, convective mixing) and improvements in the radiative transfer. Despite these discrepancies,
the overall structure of the zonal wind ﬁeld is close to the observed structure, and we focus now on the analysis of the waves
produced in the GCM simulation.
3. Wave analysis
3.1. Analysis technique
To study the waves developing in the modeled atmosphericcirculation, we use Fast Fourier Transform (FFT) to analyse frequency spectrum of temperature, zonal and meridional wind ﬁelds.

Modeled atmospheric ﬁelds are output with a frequency of 100
points per Venus day. For each grid point in the GCM (given latitude, longitude and pressure level), the FFT is applied to this timeseries over the last four Venus days of the simulation. The amplitude of the FFT signal at a given frequency is averaged over thelongitudes before study. To minimize the impact of the 4-Vd time
window, the time series is multiplied by a triangular window (0 att = 0 and t = 4 Vd, 2 at t = 2 Vd) (Harris, 1978). Resolution of the
FFT spectra is 0.25/Vd, over the range 0.25–50/Vd.
The amplitude of the FFT of temperature, zonal and meridional
wind ﬁelds as a function of pressure and frequency are displayedin Fig. 7, for the equator and 70°N. Several groups of waves appear
in these plots:

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Fig. 9. Filter functions applied to the FFT spectra to separate the different waves
present in the temperature and wind ﬁelds.

The diurnal and semi-diurnal tides are the strongest waves visible above roughly 105 Pa (see Section 3.3 and Fig. 11 below).
• In the cloud region, between 105 Pa and 103 Pa, a large groupof waves is visible at high latitudes, with frequencies between
5 and 20/Vd.
• At 2 × 104 Pa, in the equatorial region, a wave is visible around
16/Vd, mostly on the zonal wind. Its harmonics are visible too.
• In the deep atmosphere, below the clouds, waves are mostly
visible at high latitudes, with frequencies up to 30/Vd, and apeak near 8/Vd. Additional waves are also visible in the equatorial region, at very low frequencies.
•

Latitudinal dependence of the spectra are displayed in Fig. 8 forthe cloud region (pressure of 2 × 104 Pa) and for the deep atmosphere (pressure of 1 × 106 Pa). At 2 × 104 Pa, the diurnaland semi-diurnal tides are visible, as well as the high- to midlatitude group of waves, with maxima between 10 and 16/Vd. This
broad activity from mid- to high-latitudes, without a typical frequency, is certainly resulting from instability (barotropic or baroclinic) in this region (see Section 3.4). Near 16/Vd, the equatorial
wave mentioned previously is also visible in the zonal wind (with
its harmonics) and temperature. In the meridional wind, this wave
is seen at low latitudes but not on the Equator. These are characteristics of a Kelvin type wave (as conﬁrmed by its horizontal

Fig. 11. Amplitude of the FFT spectrum of the temperature at the frequency (a)1/Vd, i.e. diurnal tide and (b) 2/Vd, i.e. semi-diurnal tide. The white contours show
the mean zonal wind ﬁeld (in m/s).

structure, see Section 3.4 below). Another low-latitude wave isseen at a frequency around 8/Vd, with the opposite characteristics: visible only in the meridional wind at the equator, and in thezonal wind at low latitudes. This could indicate a Rossby-gravity
type wave. At 1 × 106 Pa, the distribution of the waves in the midto high-latitudes is displayed.

Fig. 10. Meridional (a) and vertical (b) transport of angular momentum, separating MMC contribution from transients and stationaries. The net horizontal transport iscompensated by exchanges with the solid surface. However, the net vertical transport should be zero and the residual is indicative of the numerical inaccuracies discussed
at the end of Section 2.1.

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45

Fig. 12. Amplitude of the FFT spectrum of (a, b) the zonal and (c, d) meridional winds, at the frequency 7.25/Vd (left) and 16/Vd (right). The white contours show (a, b) themean zonal wind ﬁeld (in m/s), (c, d) dq/dφ (in m/s).

It is also possible to apply the FFT bi-dimensionally (longitudetime) for a given latitude and pressure level. In this case, the direction of the wave propagation and the longitudinal wavenumber are obtained. Note that the mean zonal ﬂow is westward, as
Venus’s surface is in retrograde rotation compared to the Earth’s
rotation. Apart from the thermal tides that are eastward (alongwith the Sun), all the previously mentioned waves are westward,
with wavenumber 1 for the group in the clouds and wavenumber 2
or 3 for the group in the deep atmosphere. In the cloud, the Kelvintype wave is moving almost at the same speed as the mean zonal
ﬂow, while its harmonics are faster. The other group is slower, thus
propagating eastward relatively to the mean ﬂow. In the deep atmosphere, waves both faster and slower than the zonal wind arepresent.
To separate the different waves, we apply ﬁlters on the FFTspectra, before applying a reverse FFT. At a given frequency f0 , theshape of the lowpass ﬁlter is: 1 for f < f0 − δ f, 0 for f > f0 + δ f
and 0.5 + sin(π /2 × ( f0 − f )/δ f )/2 in between, where δ f is 1/Vd.
A similar shape is applied for a highpass ﬁlter, and a bandpass ﬁlter is obtained between two frequencies f1 and f2 by combining ahighpass ﬁlter at f1 and a lowpass ﬁlter at f2 . Filtering is done atthese cutting frequencies: 4, 10 and 22/Vd, as illustrated in Fig. 9.

3.2. Angular momentum transport
The speciﬁc angular momentum M is computed as M = u ×
a cos φ , where a is Venus’s radius, φ is the latitude, and u is thezonal wind. The total meridional transport [vM] of the angularmomentum can be decomposed as:[vM] = [v][M] + [v∗ ][M∗ ] + [v M ],(1)

where [M] is the zonal mean, M the temporal average, M∗ = M −[M], and M = M − M. [v][M] is the contribution from the mean
meridional circulation (MMC), [v∗ ][M∗ ] is the contribution fromstationary waves, and [v M ] is the contribution from transientswaves. The same equation applies for vertical transport of M, using
the vertical wind w (in Pa/s) instead of the meridional wind v. The
stationary waves component appears to be negligible.
The transient term can be computed either with the complete
perturbation ﬁeld M , or with the perturbations obtained after
wave ﬁltering. This allows to separate the contribution of eachgroup of waves in the horizontal or vertical transport.
The total horizontal transport is obtained with a vertical integration of each term, weighted by the mass of each cell. The total
vertical transport is done with a horizontal integration, weighted
only by cos φ . Fig. 10 shows these total horizontal and vertical transports of angular momentum. In the horizontal, the poleward MMC transport is compensated by the equatorward transport by the waves. Note that this plot is similar if the vertical integration is done only down to the cloud bottom, whichmeans that this balance is obtained both in the cloud region andin the deep atmosphere. In the vertical, the upward MMC transport is compensated by downward transport by the waves. In thecloud region and above, this transport is mostly done by the thermal tides. This is consistent with our previous analysis done in
Lebonnois et al. (2010). Therefore, in our simulations, superrotation is obtained and maintained through the Gierasch–Rossow–

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Williams mechanism (mean meridional circulation and horizontal
waves), with a signiﬁcant additional contribution of the thermaltides in the global angular momentum balance. It must be notedthat the residual poleward transport is a combination of the latitudinal distribution of the numerical inaccuracies discussed at theend of Section 2.1, and of latitudinal redistribution of angular momentum at the surface. In the vertical, the net transport shouldbe zero and the residual is also an indication of these numericalinaccuracies.
3.3. Thermal tides
Latitude vs pressure maps of the FFT spectrum of the temperature, for the frequencies 1/Vd and 2/Vd are plotted in Fig. 11.
They show that the tides are mostly visible above 104 Pa. The
semi-diurnal tide is dominating in the low- to mid-latitudes between 104 Pa and 102 Pa, while the diurnal tide dominates above
102 Pa, and at high latitudes. This is consistent with the analysis of
our previous simulations at lower resolution that was discussed in
Migliorini et al. (2012), in comparison with VIRTIS/Venus-Express
dataset analysis. The amplitude of the diurnal and semi-diurnal
tides (5–10 K in the pressure range 104 –102 Pa) is consistent withthe observed values (Grassi et al., 2014; Migliorini et al., 2012).
3.4. Waves in the cloud layer and above
To illustrate the distribution of the waves present in the cloud
region and above, Fig. 12 shows the latitude vs pressure maps of
the FFT spectra of the zonal and meridional winds for two frequencies, 7.25/Vd and 16/Vd. One group is located on the poles, where
the latitudinal gradient of the zonal wind speed is large, so thewaves would be caused by barotropic instability. Another group is
at mid-latitudes (40–70°), in a region dominated by a large vertical gradient of the zonal wind. The level at which the phase speedof these disturbances is equal to the background speed (steeringlevel) is deeper for the lower frequency waves, which explainsthat this mid- to high-latitude wave group is located deeper (5 ×
104 –1 × 105 Pa) than the higher frequency wave group (around2 × 104 Pa). On the equatorial region, the Kelvin type and mixed
Rossby-gravity type waves are also visible.
As seen in Fig. 1, the latitudinal gradient between the equatorand the poles at 1 × 104 Pa (roughly 60 km altitude) is around
30 K. This latitudinal gradient of temperature and the strong vertical zonal wind shear in the 105 –104 Pa region may be the sourceof baroclinic instabilities. A necessary condition for baroclinic instability has been used by Sugimoto et al. (2014a), following Younget al. (1984), though the validity of this criterion for Venus may betaken with caution. The following latitudinal gradient must change
sign for instabilities to occur:



∂
∂q
∂
1
= 2au cos φ −(cos φ u˜ )∂φ∂φ cos φ ∂φ
2
4a2 2u sin φ ∂p ∂ u˜−p∂ z N2 ∂ z(2)where u includes the rotation of the atmosphere at a reference level of 3 × 104 Pa, and u˜ is the mean zonal ﬁeld relative tothis reference level. N2 is the Brunt–Väisälä frequency, computedfrom the temperature lapse rate:

g
N =
T
2




g
∂T
+
.
∂ z cp(3)

Such a criterion appears to be fulﬁlled in the mid to high latitudes, between 105 and 104 Pa, as seen in Fig. 12c and d. Note alsothat criteria are also satisﬁed in the polar region because of theFig. 13. Temperature (colors, in K) and meridional wind (contours, in m/s) perturbations ﬁltered in the frequency band [10–22]/Vd, at (a) 45°N (baroclinic structures)and (b) 80°N (barotropic structures). Note that the zonal wind is moving westward.
These perturbations are shown at a ﬁxed point in time (without averaging). (Forinterpretation of the references to color in this ﬁgure legend, the reader is referredto the web version of this article.)second term in Eq. (2), suggesting that barotropic instability would
happen.
Fig. 13 a shows the vertical and longitudinal structure of the
waves at 45°N, for temperature and meridional wind ﬁelds ﬁltered in the [10–22]/Vd frequency range. Both present a tilt fromup-east to down-west. They are phase-shifted by approximately aquarter period. This structure is similar to the one obtained in the
GCM simulations of Sugimoto et al. (2014b) (their Fig. 2; note thatin their simulations, the zonal wind is ﬂowing from west to east
while it is ﬂowing from east to west in our plots) and it is characteristic of a baroclinic mode. At 80°N (Fig. 13b), the structure isdifferent and the wave activity is barotropic, rather than baroclinic.
These mid-latitude baroclinic activity may be related to the Rossby
waves obtained at mid-latitude in the analysis of cloud-tracking
observations by Del Genio and Rossow (1990) (Pioneer Venus UV
images) and by Kouyama et al. (2013) (VMC/Venus Express images). However, due to the model-dependence of the wave andmean-ﬂow interactions, further investigations are needed to assess
the robustness of possible correlations with observed waves.
The latitudinal transport of heat (C p T ) v in the cloud regionis plotted in Fig. 14a for ﬁltered ﬁelds in the frequency range[4–10]/Vd, and Fig. 14b for ﬁltered ﬁelds in the frequency range

S. Lebonnois et al. / Icarus 278 (2016) 38–51

47

Fig. 14. Horizontal transport of heat (upper row) and angular momentum (lower row) computed using zonal and meridional ﬁelds ﬁltered in the frequency bands [04–10]/Vd(left) and [10–22]/Vd (right). Units are 1 × 1016 J m s−1 for heat transport and 1 × 1021 kg m3 s−2 for angular momentum transport.

[10–22]/Vd. In the region where they develop (30 to 70° of latitude, pressures around 2 × 104 Pa for frequency range [10–22]/Vd
and deeper, around 5 × 104 to 1 × 105 Pa for frequency range[4–10]/Vd), the baroclinic waves transport heat poleward. Below,
equatorward transport is visible, so that these waves tend to transport heat as in a direct meridional cell, a behavior that was alreadymentioned in Sugimoto et al. (2014b).
The latitudinal transport of angular momentum by these waves
is shown in Fig. 14c and d. Equatorward transport is dominant in
the frequency range [10–22]/Vd around 3 × 104 Pa, but is also
noted near 1 × 105 Pa for frequency range [4–10]/Vd. In the region around 60° and 5 × 104 Pa, poleward transport of angularmomentum by the waves is visible, which can be related to the
local jet in this region (as seen in Fig. 5). In the equatorial region
around 2 × 104 Pa, the Kelvin like wave transfers both heat andmomentum equatorward.
In the equatorial region, at 2 × 104 Pa, the horizontal structureof the two low-latitude waves previously mentioned, with frequencies around 7.25 and 16/Vd, are shown in Fig. 15. Both waves are
westward, with a wavenumber 1. The typical structure of a Kelvintype wave is visible in Fig. 15b (frequency ∼16/Vd). This Kelvintype wave is propagating faster than the zonal wind, as seen in
Fig. 8c and e, except at the equator where it accelerates the ﬂow.
The structure of the other wave (Fig. 15a) has similar characteristics as a Rossby-gravity type wave of order n = 0 as in Fig. 6a of
Matsuno (1966). It is propagating slower than the zonal wind. The
pressure at which these two waves develop (2 × 104 Pa) corresponds to a layer where the stability becomes large, just above theconvective region in the cloud.
Both type of waves are suggested in the analysis of cloud-top
observations by Del Genio and Rossow (1990) from Pioneer Venus

Fig. 15. Temperature (colors, in K) and horizontal wind (vectors, in m/s according to rule) perturbations ﬁltered in the frequency bands (a) [04–10]/Vd and (b)[10–22]/Vd, at 2 × 104 Pa. These perturbations are shown at a ﬁxed point in time(without averaging). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

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S. Lebonnois et al. / Icarus 278 (2016) 38–51

Fig. 16. Meridional (top) and vertical (bottom) transport of angular momentum below the clouds computed using zonal and meridional (a, b) or vertical (c, d) wind ﬁelds
ﬁltered in the frequency bands [04–10]/Vd (left) and [10–22]/Vd (right). Units are 1 × 1021 kg m3 s−2 for panels a and b and 1 × 106 Pa m2 s−2 for panels c and d.

UV images and by Kouyama et al. (2013) from VMC/Venus Express
images. In Del Genio and Rossow (1990), the Kelvin wave is present
when the zonal ﬂow is faster. Kouyama et al. (2013) have a different conclusion with their analysis: the Kelvin wave appears when
the zonal wind is slower, while Rossby waves prevail when thezonal wind is faster. However, it must be noted that in our work,
these waves appear in the middle cloud and do not propagate upward to the cloud-top region. The vertical shape of the equatorial
zonal wind ﬁeld and its time variations are crucial to the development and propagation of these waves, and they are therefore very
model-dependent. In the simulation started from rest, the Kelvintype wave was not present after 200 Vd, but has developed after 300 Vd when the equatorial jet is stronger. This suggests thatwave and mean-ﬂow interaction may be signiﬁcant in the equatorial region, a conclusion consistent with the works of Del Genioand Rossow (1990) and Kouyama et al. (2013). Time-variation andinteraction of this wave with the mean ﬂow will be the focus of afuture more detailed study.

3.5. Waves below the clouds
In the deep atmosphere, the dominant waves are seen at midto high-latitudes, with frequencies up to 30/Vd, as seen in Fig. 8.
The dominant frequency in the spectra is around 7–8/Vd. Fig. 16
shows the contributions of these waves in the angular momentum transport. They transport momentum downward and equatorward in the mid- to high-latitude regions of the deep atmosphere.
This is the dominant contribution for transients, that balances thehorizontal and vertical transport of angular momentum by themean meridional circulation below the clouds.
Fig. 17 illustrates the meridional distribution of these waves.
Meridional wind perturbations are associated to temperature perturbations: when the meridional wind converges, the temperature
increases, and reversely, negative temperature perturbations areassociated to region where the meridional wind diverges. This ischaracteristic of gravity waves (e.g. Holton, 2004). Wave trains arevisible in Fig. 18, with wavenumbers 2–3. The wind ﬁeld presents
strong convergence and divergence, but rather low vorticity, whichis again in favor of gravity waves. These waves are therefore identiﬁed as gravity waves. The source region is the stable zone below the clouds (around 2–3 × 105 Pa), where these waves may be
excited through the perturbations in temperature induced in thislayer by the cloud-region baroclinic waves. Note that these waves
are completely different from the small-scale gravity waves thatare observed at and above the cloud-top region in the images andradio-occultation data from Venus-Express, which are supposed tobe generated in the convective region located in the middle cloudand propagate upwards. Small-scale gravity waves generated bythis convection and propagating downwards may also be present
in the deep atmosphere, but they cannot be resolved by the GCM
and need to be parameterized, which is not done in this work.
Fig. 19 shows the temporal evolution of these waves bothvertically and horizontally. Propagation is downward and equatorward, inducing momentum transport where the mean zonal
wind is smaller than in their source region. These large-scale
downward-propagating gravity waves have never been suggested
before. They were not present in the LMD Venus GCM simulations

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49

Fig. 18. Temperature (colors, in K) and horizontal wind (vectors, in m/s according to rule) perturbations ﬁltered in the frequency bands (a) [04–10]/Vd and (b)[10–22]/Vd, at 1 × 106 Pa. These perturbations are shown at a ﬁxed point in time(without averaging). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 17. Temperature (colors, in K) and meridional wind (contours, in m/s) perturbations ﬁltered in the frequency bands (a) [04–10]/Vd and (b) [10–22]/Vd, at 0°
longitude. These perturbations are shown at a ﬁxed point in time (without averaging). (For interpretation of the references to color in this ﬁgure legend, the reader
is referred to the web version of this article.)done at lower resolution. The increase in horizontal resolution may
have improved the representation of wave activity at the lowest resolved scale, and favored the development of these gravity waves. Since their role in the angular momentum budget issigniﬁcant in the deep atmosphere in the present simulations,
their presence needs to be conﬁrmed in future works, as wellas their role in the superrotation of the atmosphere below theclouds.
4. Conclusion
The LMD Venus GCM was used in this work to produce an updated simulation of Venus atmospheric circulation. At the resolution used here (96 longitudes × 96 latitudes), convergence is obtained when started from rest or from an atmosphere already insuperrotation, though numerical inaccuracies in the dynamical core
have increased with resolution and affect the angular momentum
budget. The temperature structure features a vertical proﬁle close
to observations, yet slightly colder in the deep atmosphere, a vertical proﬁle of static stability in agreement with observations, dominating thermal tides above the cloud-base, and a cold polar collar
structure in the upper cloud. To improve the ﬁt of temperatures
to observations, future work will include improvement in the radiative transfer properties (based on our recent work Lebonnois
et al., 2015), and taking into account the latitudinal variations of

the cloud. The modeled zonal wind distribution presents a fully
developed superrotation, though winds are still weak below theclouds (roughly half the observed values).
We have analyzed the waves present in this simulation, andtheir role in the balance of angular momentum budget in the atmosphere of Venus. The role of diurnal and semi-diurnal tides invertical angular momentum transport is conﬁrmed in the upper
cloud. Polar barotropic and mid- to high-latitude baroclinic waves
are present in the cloud region, with frequencies between 5 and20 cycles per Venus day (periods between 6 and 23 Ed), that redistribute angular momentum signiﬁcantly. In the middle cloud,just above the convective layer, a Kelvin type wave (period around
7.3 Ed) is present at the equator, as well as a low-latitude Rossbygravity type wave (period around 16 Ed), but the characteristicsof these wave activities and their interaction with the backgroundzonal wind may be very sensitive to the modeled circulation andcomparison with cloud-top observations should be done with caution. Below the clouds, wave activity that transport angular momentum both downward and equatorward is dominated by largescale mid- to high-latitude gravity waves. The presence and roleof these waves was never mentioned in previous works. However,their sensitivity to details of the model needs to be assessed more
completely, so that their role in the superrotation mechanism on
Venus may be robustly established.
To assess the robustness of the wave activity in the different
atmospheric region of Venus’s atmosphere and their role in themechanisms of superrotation, further investigations should always
explore how sensitive these different waves are to the backgroundcirculation and temperature structure, and to the model and conﬁguration used. In particular, the numerical bias in the angularmomentum budget seen in the present simulations might affectthe wave activity, and this analysis will need to be conﬁrmed byusing a more conservative dynamical core. However, the waves analyzed in the present work present many similarities with the ones

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Fig. 19. Temporal variations of temperature (colors, in K) and meridional wind (contours, in m/s) perturbations ﬁltered in the frequency bands [04–10]/Vd (left column) and[10–22]/Vd (right column), at 0° longitude, 60°N latitude (top) and 1 × 106 Pa (bottom). (For interpretation of the references to color in this ﬁgure legend, the reader isreferred to the web version of this article.)developing in the AFES Venus GCM (Ando, Sugimoto, Takagi, et al.,2016; Sugimoto et al., 2014a; 2014b), though forcing conditions arevery different between these two GCMs. This brings conﬁdence inthe robustness of these features and of their role in the mechanismof superrotation. Temporal variability over long timescales (several
Earth years, tens of Venus days) needs to be investigated too, sincehints of such variability are present in the analysis of observational
datasets.
Acknowledgments
This work was supported by the Centre National d’Etudes Spatiales (CNES). GCM simulations were done at CINES, France, under
the Project no. 11167.
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