ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus GmbHGöttingen, Germany10.5194/angeo-33-1479-2015Gravity wave transmission diagramTomikawaY.tomikawa@nipr.ac.jpNational Institute of Polar Research, Tokyo,
JapanSOKENDAI (The Graduate University for Advanced Studies),
Tokyo, JapanY. Tomikawa (tomikawa@nipr.ac.jp)1December20153312147914845September201524November201524November2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/33/1479/2015/angeo-33-1479-2015.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/33/1479/2015/angeo-33-1479-2015.pdf
A new method of obtaining power spectral distribution of gravity waves as a
function of ground-based horizontal phase speed and propagation direction
from airglow observations has recently been proposed. To explain gravity
wave power spectrum anisotropy, a new gravity wave transmission diagram was
developed in this study. Gravity wave transmissivity depends on the
existence of critical and turning levels for waves that are determined by
background horizontal wind distributions. Gravity wave transmission diagrams
for different horizontal wavelengths in simple background horizontal winds
with constant vertical shear indicate that the effects of the turning level
reflection are significant and strongly dependent on the horizontal
wavelength.
Atmospheric composition and structure (airglow and aurora) – meteorology and atmospheric dynamics (waves and tides)Introduction
An airglow imager is a unique instrument that can detect the horizontal
structures of gravity waves around the mesopause region with high temporal
and horizontal resolutions (i.e. ∼ 1 min and ∼ 1 km,
respectively). Nearly monochromatic gravity waves have been extracted
from airglow data by visual inspection and examined in previous studies
(e.g. Taylor et al., 1993; Nakamura et al., 1999; Hecht et al., 2001).
However, this method is time consuming and relies upon the ability of data
analysts to detect the wave-like structures from the airglow data. Thus,
methods that are more objective and universally applicable for describing
the characteristics of gravity waves in airglow images are required.
Matsuda et al. (2014) proposed a new method (hereafter, Matsuda's method) to
obtain a power spectral distribution of gravity waves as a function of
ground-based horizontal phase speed and propagation direction from airglow
data. They applied a three-dimensional (i.e. zonal, meridional and temporal)
fast Fourier transform (3-D FFT) to the airglow emission intensity data at
Syowa Station, Antarctica (69∘ S, 40∘ E), after some
corrections and converted the obtained 3-D power spectrum into the polar plot
with ground-based horizontal phase speed as the radius and propagation
direction as the polar angle. The peaks in the obtained power spectral
distributions closely corresponded to the gravity waves obtained using a
conventional event analysis based on visual inspection. Matsuda's new method
enabled rapid parallel analysis of airglow data with different temporal and
spatial resolutions, which is highly beneficial for the comparison of
gravity wave statistics between different locations.
Gravity waves observed by airglow imagers generally have short horizontal
wavelengths (i.e. < 100 km) and propagation times shorter than a few
hours from the troposphere up to the mesopause region (e.g. Stockwell and
Lowe, 2001). Thus, it is often assumed that the propagation times and
horizontal propagation distances of gravity waves during their vertical
propagation from a source region up to the mesopause are small such that
temporal and horizontal changes of the background horizontal winds can be
ignored. Based on this assumption, using a vertical distribution for the
background horizontal winds, Taylor et al. (1993) introduced a critical
level blocking diagram that represents a restricted region of gravity wave
propagation from the source region up to the mesopause due to critical level
filtering. Gravity waves with ground-based horizontal phase speeds and
propagation directions within this restricted region would encounter the
critical level during their upward propagation and thus would be unable to
reach the mesopause region. Such blocking diagrams enable assessment of
whether the power spectrum anisotropy is due to critical level filtering.
Gravity wave reflection at a turning level has never been considered in
previous studies using airglow data. This is probably because the horizontal
wavelength of a gravity wave is required to determine its turning level in
addition to the background horizontal winds. However, the horizontal
wavelength can be obtained using the 3-D FFT in Matsuda's method, which makes
it possible to seek the turning level of the wave. A gravity wave
transmission diagram, which represents a restricted region of gravity wave
propagation due to critical level filtering and turning level reflection, is
presented in this article. The relative effects of critical level filtering
and turning level reflection on the vertical propagation of gravity waves
from their source region to the mesopause are also demonstrated. The effects
of additional critical level filtering, which would occur before the
intrinsic horizontal phase speed of the wave becomes zero, are also
discussed.
The theoretical backgrounds of the critical and turning levels are described
in Sect. 2. Section 3 shows how a restricted region of gravity wave
propagation due to critical level filtering and turning level reflection
could change in relation to the horizontal wavelength of a wave under simple
background horizontal wind conditions. The applicability of the gravity wave
transmission diagram is discussed in Sect. 4. Concluding remarks are given
in Sect. 5.
Theoretical backgroundCritical level filtering
Vertically propagating gravity waves are absorbed at a critical level at
which the background horizontal wind speed component parallel to a
horizontal wavenumber vector is equal to the ground-based horizontal phase
speed of the wave (Booker and Bretherton, 1967). In other words, the
intrinsic horizontal phase speed of the wave becomes zero at the critical
level. This intrinsic horizontal phase speed is given by
c^h=ch-ucosθ+vsinθ,
where ch and c^h are the ground-based and intrinsic
horizontal phase speeds, respectively, u and v are the background zonal
and meridional winds, respectively, and θ is the direction of the
horizontal wavenumber vector (and ground-based horizontal phase speed)
(i.e. degrees anticlockwise from eastward). Thus, c^h=0 at the
critical level. It should be noted that the intrinsic phase speed can be in
the opposite direction to the ground-based phase speed when the background
wind along the horizontal wavenumber vector is sufficiently strong. In
short, c^h can take either sign, while ch is positive
definite. The convention, where ch is positive definite, simplifies
the definition of source waves whose transmission is to be considered and
eases the interpretation of the transmission diagrams. However, it is noted
that this convention differs to that used in Fritts and Alexander (2003)
where ω^ (and thus c^h) is defined as positive
definite. Stationary gravity waves (i.e. mountain waves), for which the
direction of the ground-based phase speed cannot be determined, require an
alternative treatment (Dunkerton and Butchart, 1984; Whiteway and Duck,
1996).
When a vertical distribution of background horizontal winds is given, Eq. ()
can be used to determine the height of the critical level where the
intrinsic horizontal phase speed changes sign with height. Although Taylor
et al. (1993) defined the height of the critical level to be where the
intrinsic horizontal phase speed changes from positive to negative with
height, their definition excludes gravity waves for which the directions of
the ground-based and intrinsic horizontal phase speeds are opposite (e.g.
Thomas et al., 1999). The effects of such a difference in the definition of
the critical level are discussed in Sect. 3.
Additional critical level filtering
An intrinsic horizontal phase speed of zero, which is generally used as the
definition of the critical level, corresponds to a zero intrinsic wave
frequency. However, from the dispersion relation of inertia–gravity waves
given by
m2=N2-kh2c^h2c^h2-f2/kh2-14H2
(cf. Fritts and Alexander, 2003), where N and f are buoyancy and
inertial frequencies, respectively, kh and m are the horizontal and
vertical wavenumbers of the wave, respectively, and H is a scale height,
m2→∞ at c^h=±fkh. Note that
kh is positive definite. Thus, the lowest intrinsic frequency of a
vertically propagating gravity wave is the inertial frequency in a rotating
fluid. A critical level of a gravity wave in a rotating fluid exists at the
height at which c^h=±fkh and provides more
gravity wave filtering than does the critical level defined by
c^h=0.
Turning level reflection
In linear theory, all waves are absorbed and their vertical wavelengths
become infinitely small at the critical level defined in Sect. 2.2;
however, the waves are totally reflected at the height at which m2=0.
This height is called the turning level (Fritts and Alexander, 2003). The
intrinsic phase speed at the turning level is given by
c^h2=N2+f24H2kh2kh2+14H2,
and, therefore, the vertically propagating gravity waves must have an
intrinsic phase speed slower than that in Eq. (). The intrinsic wave
frequency at the turning level is equal to the buoyancy frequency under the
condition H→∞ (i.e. the Boussinesq approximation).
Results
Matsuda et al. (2014) extracted gravity waves with horizontal wavelengths
(λh≡2πkh) of 5–100 km in their method.
Thus, transmission diagrams for gravity waves with λh= 100, 50 and 10 km are considered in this study.
The buoyancy frequency squared, which is used in the subsequent analysis, is
taken to be
N2=2×10-4s-250km≤z≤90km(mesosphere)4×10-4s-210km≤z<50km(stratosphere)1×10-4s-20km≤z<10km(troposphere).
In this treatment, it is assumed that the airglow emission layer exists at
an altitude of 90 km. The inertial frequency is taken to be the value at
Syowa Station (69∘ S; i.e. the inertial period is about 12.8 h),
but would be lower at lower latitudes.
When background horizontal winds and a horizontal wavelength of the wave are
given, gravity wave transmissivity is obtained as follows: first, the
intrinsic horizontal phase speed (c^h) and vertical wavenumber
squared (m2) at each height are computed for specified ground-based
horizontal phase speed and propagation direction using Eqs. () and (),
respectively. If c^h or c^h∓fkh
change sign with height between starting height and 90 km, the wave is
considered to encounter the critical level of c^h=0 or
c^h=±fkh, respectively. On the other hand, if
m2 becomes zero or negative between starting height and 90 km, the wave
is considered to encounter the turning level. This calculation is performed
for every phase speed and propagation direction, such that the transmission
diagram is constructed.
Figure 1a–e show transmission diagrams for gravity waves with λh= 100 km in background horizontal winds with constant vertical
shear (shown in Fig. 1f). The transmissivities for propagation to 90 km of
the gravity waves emitted at 0 km (i.e. ground), 5 km (i.e. middle
troposphere), 10 km (i.e. tropopause), 30 km (i.e. middle stratosphere) and
60 km (i.e. lower mesosphere) were computed. The black shaded regions
represent the restricted vertical propagation of gravity waves due to the
critical level filtering (outlined in Sect. 2.1). The restricted regions
exhibit circular structures around the background horizontal winds that vary
with height, similar to critical level blocking diagram results (Taylor et
al., 1993; Medeiros et al., 2003; Dowdy et al., 2007). However, a region of
allowed vertical propagation appears inside the region of restricted
vertical propagation, especially in Fig. 1d and e. This region of allowed
vertical propagation corresponds to those gravity waves for which the
directions of the intrinsic and ground-based phase speeds are opposite, as
mentioned in Sect. 2.1. For example, a gravity wave with an eastward
ground-based phase speed of 50 m s-1 emitted at 60 km can propagate to
the mesopause region, because its intrinsic phase speed is always negative
between 60 and 90 km and it does not encounter the critical level (see
Fig. 1e). Such gravity waves can exist in regions of strong background
horizontal winds, such as the polar-night jet, and they have not been
considered in previous studies using critical level blocking diagrams.
Gravity wave transmission diagrams for gravity waves
propagating to the mesopause region with λh= 100 km starting
from (a) 0 km, (b) 5 km, (c) 10 km, (d) 30 km and
(e) 60 km, and (f) vertical distributions of background (solid) zonal and (dashed) meridional
winds. Black, red and blue shading represents restricted regions of gravity
wave propagation due to critical level filtering, additional critical level
filtering and turning level reflection, respectively.
The red shaded regions in Fig. 1a–e represent the restricted vertical
propagation of gravity waves due to additional critical level filtering
(outlined in Sect. 2.2). As fkh≈ 2 m s-1 for λh= 100 km at Syowa Station, the region of
critical level filtering is extended outward by about 2 m s-1 compared
to the c^h=0 case. Since this effect becomes smaller for
shorter horizontal wavelengths, its impact on gravity wave propagation is
not expected to be significant for λh≤100 km.
The blue shaded regions in Fig. 1a–e represent the restricted vertical
propagation of gravity waves due to turning level reflection (outlined in
Sect. 2.3). Turning level reflection occurs where the magnitude of the
intrinsic phase speed of the wave is larger than the threshold value given
in Eq. (). Most gravity waves with ground-based horizontal phase speeds of
< 50 m s-1 would be able to propagate vertically, except
through the restricted region due to critical level filtering.
Same as Fig. 1 except for λh= 50 km.
Figure 2a–e present transmission diagrams for gravity waves with λh= 50 km.
As critical level filtering only depends on the horizontal
wavelength of the wave through fkh and this is small for this
case, the black and red shaded regions are nearly the same as those in Fig. 1.
However, the effects of the turning level reflection are distinctly
larger than those in Fig. 1. Most gravity waves with westward ground-based
phase speeds cannot propagate vertically, even if they are emitted in the
stratosphere and mesosphere (Fig. 2d–e). Therefore, turning level
reflection could play a significant role in the gravity wave power spectrum
anisotropy around the mesopause obtained by airglow observations. The
restricted region is larger for gravity waves emitted at 0 and 5 km than
for those emitted at 10, 30 and 60 km. This difference occurs because
gravity waves propagating in regions of smaller buoyancy frequency such as
in the troposphere have m2 values closer to zero (the turning level
threshold) and are thus more prone to restriction in vertical propagation.
This feature indicates that buoyancy frequency also has significant effect
on gravity wave transmissivity.
Transmission diagrams for gravity waves with λh= 10 km are
shown in Fig. 3. Almost all gravity waves are restricted to propagate
vertically, even if they are emitted in the stratosphere and mesosphere.
Therefore, it is difficult for gravity waves with short horizontal
wavelengths, such as 10 km, to propagate vertically in horizontal background
winds that vary considerably with height (cf. Marks and Eckermann, 1995).
Discussion
The ground-based horizontal group speed for gravity waves with medium
frequency (i.e. N>>ω^>>f,
where ω^ is the intrinsic wave frequency) becomes the same as
the ground-based horizontal phase speed (cf. Fritts and Alexander, 2003). If
the period for vertical propagation of a wave from its source region to the
mesopause is about 2 h (Stockwell and Lowe, 2001), then the horizontal
propagation distance is at most a few hundred kilometres for gravity waves
with ch≤50 m s-1. Thus, the assumption that
the horizontal variation of the background horizontal winds is negligible
within the horizontal propagation distance of the wave is valid for
background winds modified by large-scale waves, such as planetary waves and
tides. However, the effects of large-scale gravity waves with horizontal
wavelengths of several hundred kilometres or longer need to be considered
carefully.
Gravity wave transmission diagrams in simple background horizontal winds
with constant vertical shear showed the following:
Gravity waves with λh= 100 km and ch≤50 m s-1 can propagate vertically, except for those
filtered at the critical level.
Gravity waves with λh= 50 km can propagate vertically if the
direction of their ground-based phase speed is roughly normal to the
background horizontal winds.
Gravity waves with λh= 10 km cannot propagate vertically.
These features are true even in the real atmosphere above Syowa Station. As
the vertical variations of the zonal wind are dominant over those of the
meridional wind in winter, because of the polar-night jet, the gravity wave
transmission diagram becomes similar to Figs. 1–3. Stratospheric
westward winds become dominant, and the meridional wind is generally weak in summer.
Gravity waves with westward and eastward ground-based phase speeds are
restricted to propagate vertically by critical level filtering and turning
level reflection, respectively, in summer.
Same as Fig. 1 except for λh= 10 km.
In principle, the gravity wave transmission diagram can be applied to
gravity waves with long horizontal wavelengths such as λh= 1000 km and
low intrinsic frequency (i.e. ω^≈f). While these gravity waves with low
intrinsic frequency hardly encounter the turning level, additional critical
level filtering significantly restricts their vertical propagation because
of fkh≈20 m s-1 (not shown).
However, an assumption of small horizontal propagation distance does not
hold for these low-frequency gravity waves, because the ratio of vertical to
horizontal group speed for these gravity waves is much smaller than those
for high- and medium-frequency gravity waves (cf. Eq. 39 of Fritts and
Alexander, 2003). Thus, the gravity wave transmission diagram should not be
applied to the gravity waves with long horizontal wavelengths and low
intrinsic frequency.
A ray tracing theory has also been used to describe the horizontal and
vertical propagation of gravity waves given varying background horizontal
winds (Lighthill, 1978). The effects of planetary waves and tides on gravity
wave propagation have been studied by Senf and Achatz (2011) and Kalisch et
al. (2014) using ray tracing models. As the ray tracing of gravity waves
provides accurate and realistic trajectories of the gravity wave packets, it
should be explored if possible. However, ray tracing calculations require 3-D
horizontal wind and temperature distributions throughout the calculation
period. Generally, such data are not available near the mesopause, except in
general circulation models. Thus, the gravity wave transmission diagram used
in this study remains useful for the interpretation of gravity waves
observed by airglow imagers, even if it assumes that the gravity waves only
propagate vertically.
Conclusions
Gravity waves with relatively short horizontal wavelengths (i.e. < 100 km)
near the mesopause have been studied using airglow imagers.
Recently, a new statistical method of obtaining the gravity wave power
spectrum as a function of ground-based horizontal phase speed and
propagation direction using airglow imager data was proposed (Matsuda et
al., 2014). The gravity waves observed around the mesopause generally
originate at lower altitudes, and these waves are affected by various
processes such as critical level filtering, turning level reflection and
turbulent and radiative dissipation during their upward propagation. A
critical level blocking diagram, which considers the effects of critical
level filtering, has been used to explain the gravity wave statistics
observed by airglow imagers (Taylor et al., 1993; Stockwell and Lowe, 2001).
However, the effects of turning level reflection have never been considered
in such accounts. This report proposed a gravity wave transmission diagram
that considers the effects of both critical level filtering and turning
level reflection.
Transmission diagrams for gravity waves propagating to the mesopause region
with different horizontal wavelengths and starting heights in simple
background horizontal winds were presented in this study. In previous
studies, critical level filtering has been taken into account for gravity
waves with the same sign for their ground-based and intrinsic horizontal
phase speeds. However, this study demonstrated that gravity waves with
opposite signs for their ground-based and intrinsic horizontal phase speeds
could reach the mesopause region when the gravity waves were emitted in the
stratosphere and mesosphere. Although the effects of additional critical
level filtering at c^h=±fkh were also
evaluated, they were small for gravity waves with relatively short
horizontal wavelengths observed by airglow imagers.
The effects of turning level reflection on gravity wave transmissivity were
observed to depend considerably on the horizontal wavelength of the gravity
waves. While most gravity waves with λh= 100 km and ch≤50 m s-1 could propagate vertically, except for
those filtered at the critical level, almost all gravity waves with λh= 10 km could not reach the mesopause region because of critical
level filtering and turning level reflection. In the case of gravity waves
with λh= 50 km, while gravity waves with eastward and
westward ground-based phase speeds were not allowed to propagate vertically,
because of critical level filtering and turning level reflection,
respectively, gravity waves with southward and northward ground-based phase
speeds could reach the mesopause region. These results indicate that, in
addition to critical level filtering, turning level reflection is an
important mechanism in determining gravity wave transmissivity.
Acknowledgements
The author thanks T. Nakamura, M. Tsutsumi, M. K. Ejiri, T. Nishiyama
and T. S. Matsuda for their helpful comments. Figures were drawn using
Dennou Club Library (DCL). The author would like to thank Enago
(www.enago.jp) for the English language review.
The topical editor Christoph Jacobi thanks two anonymous referees for help in evaluating this paper.
References
Booker, J. R. and Bretherton, F. P.: The critical layer for internal
gravity waves in a shear flow, J. Fluid Mech., 27, 513–539, 1967.Dowdy, A. J., Vincent, R. A., Tsutsumi, M., Igarashi, K., Murayama, Y.,
Singer, W., and Murphy, D. J.: Polar mesosphere and lower thermosphere
dynamics: 1. Mean wind and gravity wave climatologies, J. Geophys. Res.,
112, D17104, 10.1029/2006JD008126, 2007.
Dunkerton, T. J. and Butchart, N.: Propagation and selective transmission
of internal gravity waves in a sudden warming, J. Atmos. Sci., 41,
1443–1460, 1984.Fritts, D. C. and Alexander, M. J.: Gravity wave dynamics and effects in
the middle atmosphere, Rev. Geophys., 41, 1003, 10.1029/2001RG000106,
2003.Hecht, J. H., Walterscheid, R. L., Hickey, M. P., and Franke, S. J.:
Climatology and modeling of quasi-monochromatic atmospheric gravity waves
observed over Urbana Illinois, J. Geophys. Res., 106, 5181–5195,
10.1029/2000JD900722, 2001.Kalisch, S., Preusse, P., Ern, M., Eckermann, S. D., and Riese, M.:
Differences in gravity wave drag between realistic oblique and assumed
vertical propagation, J. Geophys. Res., 119, 10081–10099,
10.1002/2014JD021779, 2014.
Lighthill, M. J.: Waves in Fluids, Cambridge Univ. Press, Cambridge, UK,
1978.
Marks, C. J. and Eckermann, S. D.: A three-dimensional nonhydrostatic
ray-tracing model for gravity waves: Formulation and preliminary results for
the middle atmosphere, J. Atmos. Sci., 52, 1959–1984, 1995.Matsuda, T. S., Nakamura, T., Ejiri, M. K., Tsutsumi, M., and Shiokawa, K.:
New statistical analysis of the horizontal phase velocity distribution of
gravity waves observed by airglow imaging, J. Geophys. Res., 119,
9707–9718, 10.1002/2014JD021543, 2014.Medeiros, A. F., Taylor, M. J., Takahashi, H., Batista, P. P., and Gobbi,
D.: An investigation of gravity wave activity in the low-latitude upper
mesosphere: Propagation direction and wind filtering, J. Geophys. Res., 108,
4411, 10.1029/2002JD002593, 2003.
Nakamura, T., Higashikawa, A., Tsuda, T., and Matsushita, Y.: Seasonal
variations of gravity wave structures in OH airglow with a CCD imager at
Shigaraki, Earth Planets Space, 51, 897–906, 1999.Senf, F. and Achatz, U.: On the impact of middle-atmosphere thermal tides
on the propagation and dissipation of gravity waves, J. Geophys. Res., 116,
D24110, 10.1029/2011JD015794, 2011.Stockwell, R. G. and Lowe, R. P.: Airglow imaging of gravity waves 2.
Critical layer filtering, J. Geophys. Res., 106, 17205–17220,
2001. Taylor, M. J., Ryan, E. H., Tuan, T. F., and Edwards, R.: Evidence of
preferential directions for gravity wave propagation due to wind filtering
in the middle atmosphere, J. Geophys. Res., 98, 6047–6057,
10.1029/92JA02604, 1993.Thomas, L., Worthington, R. M., and McDonald, A. J.: Inertia-gravity waves
in the troposphere and lower stratosphere associated with a jet stream exit
region, Ann. Geophys., 17, 115–121, 1999, https://angeo.copernicus.org/articles/17/115/1999/.
Whiteway, J. A. and Duck, T. J.: Evidence for critical level filtering of
atmospheric gravity waves, Geophys. Res. Lett., 23, 145–148, 1996.