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Vertical Wavenumber Spectra of Gravity Waves in the Venus Atmosphere Obtained from
Venus Express Radio Occultation Data: Evidence for Saturation
HIROKI ANDO AND TAKESHI IMAMURA
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Kanagawa, Japan
TOSHITAKA TSUDA
Research Institute for Sustainable Humanosphere, Kyoto University, Uji, Kyoto, Japan
SILVIA TELLMANN AND MARTIN PÄTZOLD
Abteilung Planetenforschung, Rheinisches Institut f€
ur Umweltforschung, Universit€
at zu K€
oln, Cologne, Germany
BERND HÄUSLER
Institut f€
ur Raumfahrttechnik und Weltraumnutzung, Universit€
at der Bundeswehr M€
unchen, Neubiberg, Germany
(Manuscript received 19 October 2014, in final form 19 February 2015)
ABSTRACT
By using the vertical temperature profiles obtained by the radio occultation measurements on the European
Space Agency (ESA)’s Venus Express, the vertical wavenumber spectra of small-scale temperature fluctuations that are thought to be manifestations of gravity waves are studied. Wavenumber spectra covering
wavelengths of 1.4–7.5 km were obtained for two altitude regions (65–80 and 75–90 km) and seven latitude
bands. The spectra show a power-law dependence on the high-wavenumber side with the logarithmic spectral
slope ranging from 23 to 24, which is similar to the features seen in Earth’s and Martian atmospheres. The
power-law portion of the spectrum tends to follow the semiempirical spectrum of saturated gravity waves,
suggesting that the gravity waves are dissipated by saturation as well as radiative damping. The spectral power
is larger at 75–90 km than at 65–80 km at low wavenumbers, suggesting amplitude growth with height of
unsaturated waves. It was also found that the wave amplitude is larger at higher latitudes and that the amplitude is maximized in the northern high latitudes. On the assumption that gravity waves are saturated in the
Venusian atmosphere, the turbulent diffusion coefficient was estimated. The diffusion coefficient in the
Venusian atmosphere is larger than those in Earth’s atmosphere because of the longer characteristic vertical
wavelength of the saturated waves.
1. Introduction
Internal gravity waves are vertically propagating,
small-scale waves for which buoyancy is the main restoring force. Gravity waves are known to have a variety
of sources (topography, convection, fronts, jet stream,
etc.) and are thought to be one of the most essential
Denotes Open Access content.
Corresponding author address: Hiroki Ando, Institute of Space
and Astronautical Science, Japan Aerospace Exploration Agency,
3-1-1 Yoshinodai Chuo-ku, Sagamihara, Kanagawa, Japan.
E-mail: [email protected]
DOI: 10.1175/JAS-D-14-0315.1
Ó 2015 American Meteorological Society
elements driving the atmospheric circulation in Earth’s
stratosphere and mesosphere (e.g., Fritts and Alexander
2003). Among various dissipation processes of gravity
waves, saturation is considered to be of particular importance in Earth’s atmosphere. The amplitude growth
with height of gravity waves leads to wave breaking via
convective and/or shear instability, and then the amplitude is limited by turbulent diffusion of wave energy;
this process is called saturation (e.g., Lindzen 1981;
Fritts and Alexander 2003). Smith et al. (1987) and
Tsuda et al. (1991) derived a semiempirical formula for
the vertical wavenumber spectrum of saturated gravity
waves. In terms of the temperature perturbation T 0 , it is
written as
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ANDO ET AL.
1
N4
FT 0 /T 5 2
,
0
4p 10g2 k3z
(1)
where kz is the vertical wavenumber with the unit of
cycles per meter, T0 is the background temperature, N is
the Brunt–Väisälä frequency, and g is the gravitational
acceleration. Another expression, which adopts a conventional unit for kz, is given in Ando et al. (2012).
Observations using radars, radiosondes, and GPS radio
occultation measurements showed that the spectra of
gravity waves in Earth’s atmosphere are roughly consistent with (1) (i.e., a logarithmic spectral slope of 23
for wavelengths shorter than 2–5 km) and show flattening at longer wavelengths (e.g., Fritts et al. 1988; Tsuda
et al. 1989, 1991; Tsuda and Hocke 2002). While retaining such spectral features indicative of saturation,
the amplitude varies over space and time. For example,
the spectral density is usually larger in the equatorial
region than in higher latitudes, and this equatorial enhancement is attributed to strong cumulus convection
(e.g., Tsuda and Hocke 2002; Sato et al. 2003).
Although vertical wavenumber spectra provide fundamental information about the sources and dissipation
processes of gravity waves, studies of the vertical wavenumber spectrum are limited in other planetary atmospheres. Recently, Ando et al. (2012) obtained vertical
wavenumber spectra of gravity waves in the Martian atmosphere using radio occultation data taken by Mars
Global Surveyor. They found that the spectral shape is
qualitatively similar to that of Earth’s gravity waves: the
spectra are roughly consistent with (1) for wavelengths
shorter than 3–8 km. This indicates that saturation plays
an important role, although radiative damping should also
be important in the Martian atmosphere, which is composed of CO2. Spectral densities tend to be maximized in
the equatorial region in any seasons, suggesting a contribution of convectively generated waves.
Wavenumber spectra have never been obtained for
Venus, which is another terrestrial planet with a thick
atmosphere. To reproduce the realistic atmospheric dynamics in Venusian general circulation models (GCMs),
the momentum transport by gravity waves and the eddy
diffusion associated with gravity wave breaking should be
included similarly to Earth’s GCMs. At the present stage,
however, the lack of information about the gravity wave
spectrum prevents realistic parameterizations of such
gravity wave effects (e.g., Lebonnois et al. 2010).
Many observations have been made for Venusian gravity
waves. Small-scale periodic structures attributable to gravity
waves have been observed in temperature profiles obtained
by entry probes (Seiff et al. 1980; Counselman et al. 1980)
and radio occultation measurements (Kliore and Patel 1980;
2319
Kolosov et al. 1980; Yakovlev et al. 1991; Hinson and
Jenkins 1995; Tellmann et al. 2012). The Vega balloon detected vertical wind oscillations over large topographic rises,
suggesting propagation of topographically generated gravity waves (Sagdeev et al. 1986; Young et al. 1987; 1994).
Recently, the European Space Agency (ESA)’s Venus
Express observed gravity wave–like features by a variety of
methods. Peralta et al. (2008) identified gravity waves at
cloud heights using images taken by the Visible and Infrared Thermal Imaging Spectrometer (VIRTIS) onboard
Venus Express. They found that the observed locations of
the waves are not correlated with topographic features or
the local time, implying that convection in the cloud layer
generates the waves. Garcia et al. (2009) detected gravity
waves using images of nonlocal thermodynamic equilibrium
(non-LTE) CO2 emission taken by VIRTIS and suggested
that the polar vortex might be the source of gravity waves
based on the geographical distribution of the waves and the
orientation of wave fronts. Markiewicz et al. (2007)
found, using cloud images taken by the Venus Monitoring
Camera (VMC), that wave trains, located perpendicularly
to the direction of cloud streaks, are ubiquitously present
at the low-latitude side of the bright polar band. Tellmann
et al. (2012) studied small-scale temperature perturbations detected by radio occultation measurements and
found that gravity wave activity increases with latitude.
They also showed that the greatest wave activity occurs in
the vicinity of Ishtar Terra, the highest topographical
feature on Venus, and that gravity wave activity in the low
latitudes is enhanced near noon.
The present study obtains vertical wavenumber spectra
of gravity waves in the Venusian atmosphere for the first
time and compares them with those in Earth’s and Martian atmospheres. The spectra are calculated from the
temperature profiles obtained by the Venus Radio Science experiment (VeRa) performed during the Venus
Express mission (Häusler et al. 2006; Pätzold et al. 2007;
Tellmann et al. 2009, 2012). Radio occultation has an
advantage of providing high-vertical-resolution and highprecision temperature profiles, which are essential to
studies of gravity waves. We also estimate the turbulent
diffusion coefficients in the Venusian atmosphere based
on the wave saturation theory. Section 2 introduces the
procedure for obtaining vertical wavenumber spectra,
and section 3 gives the result. Section 4 compares the
wave characteristics among the Venusian, Martian, and
Earth’s atmospheres. Section 5 estimates turbulent diffusion coefficients. Section 6 gives the summary.
2. Dataset and analysis procedure
Venus Express was launched in November 2005 and
inserted into a polar orbit with the period of 24 h. The
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FIG. 2. Normalized temperature perturbation profile obtained from
the difference between the two curves in Fig. 1.
FIG. 1. Example of the vertical temperature profile (thick solid
curve) and the cubic function fitted to 65–80-km altitudes of the
observed profile (thin solid curve). The measurement was conducted at 0414 LT 16 Aug 2006 (location: 45.48S, 145.78W).
radio occultation experiments are conducted using radio
waves with wavelengths of 3.6 (X band) and 13 cm
(S band) transmitted by the spacecraft and received on
Earth. From the frequency variation of the received
signal during each occultation, the angle of the ray
bending due to the atmosphere of the planet is obtained
as a function of the impact parameter, and then the
vertical density profile is obtained by Abel inversion.
The vertical profile of the atmospheric pressure is obtained from the density profile under the assumption of
hydrostatic equilibrium. Finally, the atmospheric temperature profile is retrieved from the density and pressure
profiles by using the ideal gas law. A broad latitudinal
coverage from the equatorial region to the polar region is
achieved, with a dense sampling in the high latitudes because of the polar orbit of the spacecraft. Details of the
measurements are given in Häusler et al. (2006), Pätzold
et al. (2007), and Tellmann et al. (2009, 2012).
In this study, we analyzed X-band open-loop data
obtained by 286 occultations during the period from July
2006 to June 2010. Unlike the closed-loop receiver, which
obtains the frequency and amplitude in real time by means
of phase-lock loop technology, the open-loop receiver
digitizes the received signal after down conversion to
;100 kHz, and the data are analyzed offline by a phaseunwrapping technique (Imamura et al. 2011). An advantage of using open-loop data is its high-frequency sampling:
thanks to the wider spectral coverage, one can examine the
influence of the Fresnel diameter as shown below.
The method of analyzing temperature profiles is
similar to Ando et al. (2012). A cubic function is fitted to
each profile (Fig. 1), and the fitted function is regarded
as T0 . The subtraction of T0 from the original temperature profile gives T 0 , which is considered as a manifestation of gravity waves (Fig. 2). The result given in the
next section is virtually unchanged by the change of the
order of fitting to 1 or 2, suggesting that the uncertainty
in the determination of the background does not have a
significant influence on the result. The minimum vertical
wavelength resolved in this study is ;1.4p
km
ffiffiffiffiffiffiffibecause the
diameter of the first Fresnel zone is lL ’ 0.7 km,
where L ’ 12 000 km is the maximum distance between
the transmitter and the point of the ray’s closest approach to Venus during the measurements and l ’
3.6 cm is the wavelength of the radio wave (e.g., Häusler
et al. 2006; Tellmann et al. 2012).
Temperature profiles in the altitude regions 65–80 and
75–90 km are analyzed separately with a 1024-point fast
Fourier transform (FFT) including a Welch window.
The gravest Fourier components are excluded from the
analysis because of the possible influence of the finite
Fourier length, and thus the maximum wavelength analyzed is ;7.5 km. Figure 3 shows typical examples of
the spectra obtained in the northern and southern highlatitude occultations corresponding to small and large L
cases, respectively. The vertical wavenumber is defined
by kz 5 1/lz in the rest of this paper, where lz is the
vertical wavelength. A near power-law dependence
which is consistent with (1) is seen at low wavenumbers
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ANDO ET AL.
TABLE 1. Number of data values and the mean Brunt–Väisälä
frequencies for seven latitude bands and two altitude regions. The
mean Brunt–Väisälä frequency is obtained by averaging the Brunt–
Väisälä frequencies over the altitude region and all the profiles in
each latitude bin.
Altitude
range
Latitude
range
Number of
data values
Brunt–Väisälä
frequency
(rad s21)
75–90 km
758–908N
458–758N
158–458N
158S–158N
158–458S
458–758S
758–908S
758–908N
458–758N
158–458N
158S–158N
158–458S
458–758S
758–908S
122
24
9
31
49
31
20
119
27
11
31
47
31
20
0.018
0.018
0.019
0.020
0.020
0.018
0.019
0.018
0.019
0.017
0.017
0.017
0.019
0.018
65–80 km
FIG. 3. Examples of the vertical wavenumber spectra in the
altitude range of 65–80 km. Averages for the latitude bands of
908–758S (red) and 758–908N (blue) are plotted. Semiempirical
saturation curve given by (1) (dotted) is also plotted.
and a deviation from the power law is observed at high
wavenumbers. The change of the slope occurs around
2.0 km21 (wavelength ’ 0.5 km) for the northern highlatitude case and 1.0 km21 (wavelength ’ 1.0 km) for the
southern high-latitude case. This is attributed to blurring
by the finite vertical resolution, because L ’ 5000 km for
the northern high latitudes and L ’ 12 000 km for the
southern high latitude mean their respective Fresnel diameters are ;0.4 and ;0.7 km. The change of the spectral
slope at this wavenumber suggests that the spectrum for
wavelengths longer than twice the Fresnel diameter is free
from the influence of the vertical resolution; herein the
analysis is limited to wavelengths longer than 1.4 km. Obtained spectra are classified into seven latitude bands (908–
758S, 758–458S, 458–158S, 158S–158N, 158–458N, 458–758N,
and 758–908N) and then are averaged separately. The
number of profiles used in the analysis and the background
Brunt–Väisälä frequencies are summarized in Table 1.
It should be noted that atmospheric waves other
than gravity waves can also contribute to the spectra.
The wavenumber-1 Kelvin wave seen at the cloud top
is considered to have a vertical wavelength of ;7 km
(Del Genio and Rossow 1990), although the wave is
considered to be significantly damped above clouds
(e.g., Imamura 2006). Thermal tides have a vertical
wavelength of ;30 km according to the infrared radiometer measurement in the Pioneer Venus mission (Schofield
and Taylor 1983). The wavenumber-1 Rossby wave observed at the cloud top will not influence the spectra since
the wave is considered to have a vertical wavelength
of ;30 km (e.g., Del Genio and Rossow 1990).
3. Result
a. General features of the wavenumber spectra
Figure 4 shows the obtained vertical wavenumber
spectra divided by N 4. The semiempirical spectrum of
saturated gravity waves given by (1) is also plotted in
Fig. 4. The accuracy of each spectrum is evaluated based
on the scatter of the spectral densities before averaging;
the half-width of the 95% confidence interval is given by
2 times the standard deviation divided by the square root
of the number of data and is typically 8% for the number
of data of 100 and 12%–25% for the number of 10. Based
on this estimate, the spectral features described below are
considered statistically significant. The contribution of the
measurement noise is evaluated on the assumption that
the temperature measurement error is random with a
magnitude of ;0.1 K (200 K)21 (Häusler et al. 2006) as
5
(DT/T0 )2 1
; 1 km s4 ,
Dkz
N4
(2)
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FIG. 4. Vertical wavenumber spectra of the normalized temperature fluctuation divided by
N 4 within the altitude range of 65–80 (red) and 75–90 km (blue) for seven latitude bins.
Semiempirical saturation curve given by (1) (dotted) is also plotted for comparison.
where Dkz is the observation bandwidth that is taken to
be equal to the maximum wavenumber of ;0.7 km21.
This noise level is considered to be low enough.
We see a general tendency that the spectral density
decreases with the wavenumber similarly to those in
Earth’s and Martian atmospheres. The logarithmic
spectral slope is close to 23 at high wavenumbers in all
spectra and becomes flatter at low wavenumbers. The
inflection wavenumber is kz ’ 0.2 km21 (wavelength ’
5 km) in the altitude range of 65–80 km, while it is not
clearly seen in 75–90 km. The observed spectral densities are closer to the saturation curve in the high
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ANDO ET AL.
latitudes than in the low latitudes. More specifically, for
the altitude region 65–80 km, the spectra observed in
758–908N, 458–758N, and 908–758S almost coincide with
the saturation curve at kz 5 0.3–0.7 , 0.12–0.7 , and 0.2–
0.7 km21, respectively. In other latitudinal bins the
spectral densities do not reach the saturation value. For
75–90 km, on the other hand, the spectra observed in
758–908N, 458–758N, 458–158S, 758–458S, and 908–758S
show almost perfect agreement with the saturation
curve at kz 5 0.12–0.7, 0.12–0.7, 0.12–0.3, 0.12–0.5, and
0.12–0.4 km21, respectively. This implies that saturation
contributes to wave dissipation in these wavenumber
ranges.
At low wavenumbers of kz , 0:3 km21 (wavelength .
3 km) the spectral densities are higher at the higher level
with the amplitude roughly doubled per two scale
heights (;10 km), suggesting amplitude growth with
height due to the density decrease with height. On the
other hand, at high wavenumbers of kz . 0:3 km21
(wavelength , 3 km), the spectral densities are almost
invariant in this height region, suggesting that the waves
are subject to saturation and other dissipative processes.
region. This feature is also observed in Fig. 4: for 65–
80 km, the spectral density at kz 5 0.25–5.0 km21
(wavelengths , 4 km) is maximized in 458–758N. Such a
feature is not seen for the 75–90-km altitude region.
b. Latitudinal tendency
where R is the radius of the planet and z is the altitude of
the tangential point along the ray path. In the Venus
Express radio occultation experiments, substituting d ’
700 m, R 5 6052 km, and z 5 70 km into (3) yields h ’
200 km.
Figure 6 shows the ratio of the apparent amplitude
to the true one for a 200-km-width running average
applied to sinusoidal waves with horizontal wavelengths
of 200–800 km. The ratio is ;80% for a relatively long
wavelength of 600 km and decreases to ;40% for a
wavelength of 300 km. Though the horizontal wavelengths of the observed gravity waves are unclear, the
result suggests that the smoothing effect might explain
the apparent shortage of the amplitude relative to the
saturation level.
For both 65–80- and 75–90-km altitude regions, the
spectral densities in the high latitudes are larger than
those in the low latitudes. This tendency is qualitatively
consistent with Tellmann et al. (2012), who showed that
the potential energy of gravity waves with wavelengths
shorter than 4 km increases with latitude in the altitude
range of 65–80 km. They attributed this poleward increase of the wave amplitude to the latitudinal difference in the strength of wave generation by convection in
the cloud layer, based on the fact that the depth of the
neutrally stable layer in the middle and lower cloud regions tends to be greater at higher latitudes (Tellmann
et al. 2009). If deeper convection accompanies stronger
vertical winds, wave generation by the penetration
of updraft into the overlying stable layer should be
stronger at higher latitudes. Imamura et al. (2014) argued that stronger convection occurs at higher latitudes
owing to a unique nature of the cloud-level convection:
heating of the cloud base by upwelling infrared radiation
drives convection, while solar heating of the upper
cloud region suppresses convection. Imamura (1997)
and Tellmann et al. (2012) also suggested that the midlatitude (or polar) jet provides a strong vertical shear in
the cloud and that this shear generates gravity waves via
shear instability in the low-stability layer.
Tellmann et al. (2012) also showed that the potential
energy of gravity waves with wavelengths shorter than
4 km in the altitude range of 65–80 km is largest around
608N probably because of the large topography in this
c. Underestimation of the wave amplitude due to
horizontal smoothing
Despite the spectral features indicative of saturation
in the Venusian atmosphere as shown above, neutral
stability layers are rarely observed in temperature profiles. As shown in Fig. 5, the observed amplitudes of the
static stability are typically 50%–80% of the amplitude
at which the waves would saturate and marginally cause
neutral stability layers in the wave field. Here we examine the possibility that the horizontal smoothing explains the apparent insufficient amplitude.
The length of horizontal averaging in radio occultationp
h ffiffiffiffiffiffi
is related
to the diameter of the first Fresnel zone,
ffi
d 5 lL, as (Kursinski et al. 1997)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h 5 2 2d(R 1 z) ,
(3)
d. Contribution of radiative damping
Hinson and Jenkins (1995) and Tellmann et al. (2012)
suggested that radiative damping is the principal process
to dissipate gravity waves having vertical wavelengths
shorter than 4 km, which have been observed by radio
occultation. The apparent saturated spectra obtained in
this study seem to contradict those arguments. However,
considering the wavelength dependence of the radiative
damping rate, longer-vertical-wavelength waves might
not be strongly dissipated.
The amplitude growth with height can be written
locally as
T 0 } exp(bz) ,
(4)
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FIG. 5. Typical vertical profiles of the atmospheric stability in the Venusian atmosphere obtained by Venus Express radio occultation.
where the growth rate b is given by (e.g., Imamura and
Ogawa 1995)
b5
1
1 2p lx 1
2
,
2H 2 N lz t
(5)
where H is the scale height, lx is the horizontal wavelength, and t is the radiative relaxation time. Positive b
means that the amplitude grows with altitude overcoming radiative damping, while negative b means that
the amplitude decreases with height. Here we estimate b
as a function of the altitude and the horizontal wavenumber for lz 5 5 and 2.5 km. The background atmosphere is assumed to have H 5 5 km and N 5 0.02 rad s21.
The radiative relaxation time (Fig. 7) is estimated from
the model result for lz 5 7 km given by Crisp (1989) by
assuming that the damping time is proportional to the
vertical wavelength. The calculated b shown in Fig. 8 is
mostly positive for lz 5 5 km in the 60–90-km altitude
range, while it is negative for lz 5 2.5 km. This implies
that the amplitude of a gravity wave can increase with
altitude and reach the saturation amplitude depending
on the vertical wavelength.
than the density around the cloud top of the Venusian
atmosphere. It is noteworthy that the three planetary
atmospheres exhibit similar spectral features that a
roughly power-law dependence with an index of
around 23 is seen on the high-wavenumber side and that
this portion of the spectrum is close to the semiempirical
saturation model. This suggests that common fluid
dynamical processes, including saturation, create the
gravity wave spectrum in these atmospheres. The inflection wavenumbers in the Venusian and Martian atmospheres seem to be lower than that in Earth’s
stratosphere of 0.2–0.4 km21 (Tsuda and Hocke 2002).
4. Comparison with Earth and Mars
We compare the spectra with those in the Martian
atmosphere (Ando et al. 2012) as well as those in Earth’s
atmosphere. The Martian atmosphere is mostly composed of CO2 like the Venusian atmosphere, and its
density near the surface is one order of magnitude lower
FIG. 6. Ratio of the apparent amplitude of the wave to the true
one for a 200-km-width running average applied to sinusoidal
waves with horizontal wavelengths of 200–800 km.
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ANDO ET AL.
FIG. 7. Radiative relaxation time in the Venusian atmosphere for
vertical wavelengths of 5 (solid) and 2.5 km (dashed) obtained by
extrapolating the model prediction by Crisp (1989).
This implies that the wave amplitude before being suppressed by saturation is larger in the Venusian and
Martian atmospheres than in Earth’s middle atmosphere and that the influence of saturation extends to
lower wavenumbers in the former than in the latter.
In the short-wavelength range, where the logarithmic
spectral slope is near 23, the height variation of the
spectral density is small in both the Venusian and
2325
Martian atmospheres. At longer wavelengths the spectral density tends to increase with altitude in the Venusian atmosphere, while it is almost constant with
altitude in the Martian atmosphere. This difference
might be explained in terms of radiative damping, which
is stronger in the Martian atmosphere than in the Venusian atmosphere because radiative damping is more
efficient in optically thinner atmospheres provided that
the atmospheric composition is the same (Imamura and
Ogawa 1995).
In Earth’s and Martian atmospheres the spectral
density is usually maximized in the equatorial region
where strong insolation drives intense convection (e.g.,
Tsuda and Hocke 2002; Creasey et al. 2006). In the
Venusian atmosphere, on the other hand, the spectral
density is maximized in the high latitudes. If the primary
source of Venusian gravity waves is cloud-level convection, then the unusual latitudinal dependence is attributed to the long radiative relaxation time of the
Venusian atmosphere, which allows convection to be
driven by infrared flux from the thermally homogenized
lower atmosphere (Imamura et al. 2014).
5. Turbulent diffusion coefficients
As shown in Fig. 4, the spectra in the low-altitude
range in the Venusian high latitudes show a tendency of
FIG. 8. Amplitude growth rate (km21) as a function of the altitude and the horizontal wavelength for the vertical
wavelengths of (left) 2.5 and (right) 5 km for gravity waves dissipating through radiative damping in the Venusian
atmosphere.
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TABLE 2. Parameters for the estimation of the turbulent diffusion coefficient and the estimated coefficients.
Venus (65–80 km)
Earth (20–30 km)
Earth (30–40 km)
300–450
4.3–6.1
0.018
0.86–1.2
4.7
2.7–31
300–450
2.5–5.0
0.022
20–40
6.4
0.56–14
300–450
2.5–5.0
0.022
6.5–20
6.9
0.42–13
Horizontal wavelength (km)
Vertical wavelength (km)
Brunt–Väisälä frequency (rad s21)
Radiative relaxation time (day)
Scale height (km)
Turbulent diffusion coefficient (m2 s21)
inflection around 0.2 km21 in both hemispheres and are
close to the semiempirical spectrum of saturated gravity
waves at higher wavenumbers. Here we tentatively estimate turbulent diffusion coefficients in the low-altitude
range on the assumption that saturation occurs in the
Venusian atmosphere. Although the mean spectrum
does not reach the semiempirical saturation curve in the
low latitudes, the scatter of the spectral density before
averaging is large and not a few spectra exceed the
semiempirical saturation curve (not shown here).
Lindzen (1981) proposed a parameterization of
the turbulent diffusivity induced by a steady, monochromatic, saturated gravity wave on the assumption that the eddy diffusion of heat and
momentum prevents the amplitude growth with
height and keeps the amplitude at the saturation
level. Holton and Zhu (1984) extended this formula
to gravity waves damped by the combination of saturation and radiative damping. The diffusion coefficient is given by
l2
D5 z3
16p
!
N l2z
›u lz 2p
2 6p
2
,
H lx
›z lx
t
(6)
where u is the zonal velocity and z is the altitude. The
values of N and H are obtained from radio occultation
data, and t is obtained from the estimate by Crisp
(1989) assuming that radiative relaxation time varies
with vertical wavelength linearly. The dominant wavelength lz is taken to be the inverse of the inflection
wavenumber. The possible range of lx is given by two
constraints: the lower limit is determined by considering
the effect of the horizontal smoothing in radio occultation measurements as described in section 3c, which is
;300 km, and the upper limit is the wavelength above
which radiative damping inhibits amplitude growth with
height; that is, b 5 0 (see Fig. 8). The adopted parameter values and the estimated diffusion coefficients are
summarized in Table 2. The vertical shear of the background wind is ignored in this study because the background wind profile in the Venusian mesosphere is not
well constrained. The possible effect of the vertical shear
is mentioned later.
The turbulent diffusion coefficient in the Venusian
atmosphere is D 5 2.7–31 m2 s21 for the altitude range
of 65–80 km. Combined with the value of 4 m2 s21
around 60 km estimated from the scintillation of
the received signal intensity by Woo and Ishimaru
(1981), the diffusion coefficient seems to increase
with altitude. Zhang et al. (2012) assumed in their
one-dimensional photochemical model an altitudedependent diffusion coefficient increasing monotonically from 4 m2 s21 at 60-km altitude to 100 m2 s21 at
100 km; such height dependence is roughly consistent
with our estimate.
The turbulent diffusion coefficient in Earth’s atmosphere has been estimated by a variety of methods.
Fukao et al. (1994) obtained diffusion coefficients
of 0.5–5 m2 s21 in the stratosphere and 2–10 m2 s21 in
the mesosphere by analyzing the echo power spectra
observed by an atmosphere radar. We also estimated
the diffusion coefficient in the same way as the Venusian atmosphere based on the wavenumber spectra
given by Tsuda and Hocke (2002). Radiative relaxation time in Earth’s atmosphere is estimated by
referring to Fels (1982). Substituting the inflection
wavelengths of 2.5–5.0 km into (6), we obtain D 5
0.42–14 m2 s21 for the stratosphere, which is comparable to the previous estimates (Fukao et al. 1994). The
turbulent diffusion coefficients in Earth’s atmosphere
are smaller than that in the Venusian atmosphere because of the higher inflection wavenumber (shorter
wavelength).
Recent observational results (e.g., Piccialli et al. 2012)
show that the vertical shear of the zonal wind is nearly
absent around the cloud-top level and that the negative
shear becomes larger with altitude reaching approximately 22 m s21 km21 in the middle latitudes. Then the
turbulent diffusion coefficient can increase by up to 3
times when we consider the contribution of the vertical
shear in (6). This is also true for Earth’s case.
Given the estimates above, the characteristic time of
vertical diffusion in the Venusian atmosphere is estimated to be H2/D 5 8.2–95 days at altitudes of 65–
80 km. The time scale of the meridional circulation of
;90 days around the cloud top evaluated by a diagnosis
JUNE 2015
ANDO ET AL.
of the temperature distribution (Imamura 1997) is
within the range of the diffusion time. Thus eddy diffusion seems to make nonnegligible contributions on
transport processes in the Venusian atmosphere. Also in
Earth’s stratosphere the turbulent diffusion and the
meridional circulation equally contribute: the time scale
of the former is estimated to be H2/D 5 0.26–2.6 years
when Fukao et al.’s (1994) diffusion coefficient is
adopted, while that for the latter is ;4 years (e.g., Fritts
and Alexander 2003).
It should be noted that the Lindzen’s (1981) parameterization of turbulent diffusion is based on a
highly simplified situation. Achatz (2007) and Fritts
et al. (2009a,b) studied the temporal development of
turbulence generated by a breaking of gravity wave
and the attenuation of the wave by using nonlinear
models. They suggested that wave breaking occurs
even when the condition of convective or dynamical
instability is not satisfied and that, once a wave begins
to break, turbulence rapidly develops and reduces the
amplitude of the primary wave to ;1/ 3 of the original
value. This implies that a wave does not propagate
over a long vertical range, keeping its amplitude near
saturation level; rather, a wave packet rapidly decays
once its amplitude reaches a threshold that is lower
than the saturation level, and strong turbulence occurs
during a short time interval after the wave breaking.
Based on these results, the wave field observed in this
study is considered as a superposition of waves that
are about to break, waves undergoing breaking, and
waves significantly attenuated by turbulence. When a
certain amplitude level is considered for waves that
are about to break, the resultant turbulence in real
atmospheres might be stronger than the Lindzen’s
(1981) formula.
6. Summary
By using the temperature profiles obtained by the
radio occultation measurement on the Venus Express
mission, we obtained vertical wavenumber spectra of
small-scale temperature perturbations that are thought
to be associated with gravity waves. The obtained
spectra generally show remarkable similarities to those
in Earth’s atmosphere. The spectral density decreases
with the wavenumber with a characteristic slope of
around 23 at wavelengths shorter than about 5 km, and
the spectrum slightly flattens near the long-wavelength
end in the high latitudes. The power-law portion of the
spectrum is close to the semiempirical spectrum of saturated gravity waves. The remarkable similarity of the
gravity wave spectrum among Venusian, Martian,
and Earth’s atmospheres suggests that common
2327
dissipation processes, including saturation, are at work
in these atmospheres.
Temperatures in the two altitude ranges of 65–80 and
75–90 km are analyzed separately, and vertical wavenumber spectra covering wavelengths of 1.4–7.5 km are
obtained for each altitude region. The power is larger at
75–90 km than at 65–80 km for wavelengths larger than
3 km, suggesting amplitude growth with height. In the
Martian atmosphere, contrary to the case of Venus, the
power does not vary noticeably with altitude (Ando
et al. 2012). This might be due to weaker radiative
damping in the Venusian atmosphere than in the Martian atmosphere.
The gravity wave amplitude is enhanced in the high
latitudes in the Venusian atmosphere, while it is maximized in the equatorial region in Earth’s and the Martian atmospheres. This is attributed to the difference of
the latitudinal distribution of the convective activity
generating gravity waves. Recent studies suggest that
the cloud-level convection in the Venusian atmosphere
is more intense at higher latitudes (Tellmann et al. 2012;
Imamura et al. 2014). On the other hand, it is thought
that strong insolation drives more intense convection in
the lower latitudes in the Martian atmosphere as suggested in Creasey et al. (2006). Convection in equatorial
latitudes is thought to be an important source of gravity
waves also in Earth’s atmosphere (e.g., Tsuda and
Hocke 2002), although surface topography and stratospheric jet streams are also major sources (Wu and
Waters 1996).
The vertical eddy diffusion coefficient in the Venusian atmosphere was estimated based on the wave
saturation theory for the first time. The result shows
that the diffusion coefficient in the Venusian atmosphere is generally larger than those in Earth’s atmosphere. The characteristic time of vertical diffusion is
comparable to that of advective exchange in the
Venusian atmosphere, implying that vertical diffusion
plays a role as important as meridional circulation in
the transport of energy, momentum, and various atmospheric constituents.
We should note that the process determining the
wavenumber spectrum is still under debate. For example, Weinstock (1985), Hines (1991), and Dewan (1997)
suggested that the frequently observed power-law
spectrum might be created by interaction among
waves. Scale-dependent radiative damping can also
contribute to the power-law dependence (Zhu 1994).
Comparing the statistical characteristics of gravity
waves on various planets whose atmospheric environments are different from each other might be helpful
in constraining the mechanism to create the spectral
form.
2328
JOURNAL OF THE ATMOSPHERIC SCIENCES
Acknowledgments. Prof. Sato provided a lot of valuable comments on this study. Prof. Nakamura supported
the study of H. Ando during his graduate course. We
also appreciate the suggestions made by Prof. Iwagami,
Dr. Yoshikawa, Dr. Iga, and Prof. Takahashi.
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