Full Text - J

December
1985
I. Hirota
A Statistical
Study
in the
and T. Niki
of Inertia-Gravity
Middle
1055
Waves
Atmosphere
By Isamu Hirota and Torn Niki
GeophysicalInstitute, Kyoto University, Kyoto 606, Japan
(Manuscript received7 June 1985, in revised from 24 October 1985)
Abstract
An analysis was made of the structure and behaviour of small-scale motions in the
stratosphere and lower mesosphere with the aid of meteorologicalrocket observationsover
the period of six years from 1977to 1982, covering the wide range of latitudes.
By applying a filter to observed wind data with respect to height, wind fluctuations
with characteristic vertical scales close to 10km are separated from large-scalecomponents
such as the mean field, planetary waves and tides.
From the hodograph analysis it is found that at northern hemisphere stations most of
horizontal wind vectors show the clockwise rotation with increasing height while they
rotate anti-clockwise in the southern hemisphere. This strongly suggests that the wind
fluctuations are due mainly to upward propagating inertia-gravity waves.
On the basis of a simplifiedtheory of inertia-gravity waves, the wave-frequencydistribution is estimated statistically from the degree of elliptic polarizationof holographs, and
it is shown that the most predominant values of f*/* (f; the Coriolisparameter, *; the
intrinsic wave frequency) fall into a range of 0.20.4. Namely, the typical time scale of
these waves is of the order of several hours in middle and high latitudes and of a day in
low latitudes.
Further discussionsare made of the vertical profile of the wave energy density, and
it is suggested, from the uniform decay of the wave amplitudewith height that the wide
spectra of horizontal phase velocities should be taken into account.
1.
Introduction
In these several years, it has been widely
recognized
that vertically
propagating,
internal gravity
waves
in the middle atmosphere play an important
role in determining
the large-scale
wind field through
their
momentum
transport
and
deposition.
After
pioneering
work
of Lindzen
(1981) of the
effect of the gravity
wave
breaking
on the
mean zonal flow, there have been many
attempts
to simulate
the gravity
wave-mean
flow interaction
process in the middle atmosphere circulation
by using various
parameterization
methods in numerical
models (e.g.,
Matsuno,
1982; Holton,
1982, 1983; Holton
and Zhu, 1984; Miyahara,
1984).
On the other hand, there
have also been
numerous
observational
studies on small-scale
disturbances
in the real middle atmosphere
with the aid of various
techniques
such as
balloons,
rockets
and high-power
UHF and
VHF radars
(see the review of Fritts
et al.
(1984) for instance).
Although
these observations
have revealed
some characteristic
features
of gravity
waves in the stratosphere
and mesosphere,
the
evidence
is still fragmentary
mainly
because
the observations
have been made for only
limited
stations
and limited
periods.
Consequently,
in most
numerical
models
they
assume rather
arbitrarily
the nature
of internal gravity
waves so as to have the gross
feature
of mean fields in a qualitative
man-
1056
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of the
Meteorological
ner.
For the purpose of describing the gravity
wave-mean
flow interaction
process in a
global model, however,
it is necessary to
know the seasonal and geographical
dependency of gravity wave activities in a climatological sense. In this regard, Hirota (1984:
we will abbreviate this to H84 here) made a
preliminary attempt to show the climatology
of gravity waves in the middle atmosphere
on the basis of meteorological rocket observations, by paying attention to the wave
amplitude averaged in vertical as functions of
latitude and season.
In the present study we will expand our
statistics to the wave structure and frequency
in order to give further evidence for the
nature of gravity waves in the middle atmosphere.
2.
Theoretical
background
Among physical parameters
prescribing
the nature of gravity
wave motions, the
most informative one that can be obtained
from individual vertical soundings at a single
station is the vertical distribution of amplitude and phase of horizontal wind fluctuations.
In this connection, before discussing the
result of statistical analysis of rocket observations, let us briefly recall here the simplified
theory of gravity waves (e.g. Gossard and
Hooke, 1975), in order to have a basic idea
for the interpretation
of observed
wave
structure.
By assuming a plane wave in the frictionless and motionless atmosphere under the
influence of the earth's rotation,
the horizonal wind perturbations
of linearized equations are given by
where
The
where
notations
dispersion
the
wave
are
as
relation
frequency
usual.
is given
by
w is assumed
of Japan
Vol.
63, No. &
small compared
with the Brunt frequency
N.
Thus
the vertical
group
velocity
Cgz is
obtained
from (2-3) as
Since we have assumed
that *>0,
wind
perturbations (2-1) and (2-2) show that for a
given time at a station the wind vector rotates
clockwise with increasing
height for m<0
and *>0
(i.e., in the northern hemisphere)
whereas it rotates anti-clockwise for m>0.
The direction of vector rotation is in a reverse
sense in the southern hemisphere
where
<0.
*
It can be also said from (2-4) that the sign
of the vertical group velocity Cgz is opposite
to that of m. Therefore, in the case of *<
0, the clockwise rotation of wind vectors,
i.e., m<0, means the upward propagation of
wave energy, which in turn indicates that
the wave has its energy source in the lower
atmosphere.
In the following sections we
will show that this is the case in a statistical
sense.
Another interesting point indicated by the
simplified theory of the inertia-gravity
wave
is that the wave frequency *
can be estimated directly from the observation of wave
amplitudes without having a time-series data
set : From the equation (2-2) we have
the r. h. s. of which is obtained statistically
from the elliptic polarization
of horizontal
wind vectors over a large number of observation days. The result will be shown later in
section 6.
Note that, although the equations (2-1)*
(2-5) are highly simplified for the special
case of north-south propagation,
the discussions given in this section are still valid for
other cases if we transform
the coordinate
system in a x-y plane.
3.
to be
Society
Data
The data used in the present study are
the same as those used in our previous statistics (H84), except that the period of
December
1985
I. Hirota
analysis is expanded to the 6 years from
1977 to 1982. The stations and total number
of observation days are shown in Table 1.
Used are rocket observations of zonal and
meridional wind components with the vertical
resolution of 1km.
From the original data,
the contribution
of large-scale components
i.e., the mean background
flow, planetary
waves and tides, are removed by applying a
high-pass filter with respect to height, so that
we obtain the wind fluctuations with vertical
scales less than about 15km. The height
range of the filtered wind data is between
30 and 60 km.
In the following, unless otherwise stated,
we deal with the horizontal winds (u, *)
obtained by this filtering.
4.
Hodograph
analysis
First of all, in order to have a general
picture of the vertical structure
of wind
fluctuations, holographs
are made for each
observation day at some stations.
Fig. 1 shows some typical examples of the
holograph,
where the abscissa denotes the
zonal component u and the ordinate is the
meridional component *, and the circle and
cross-mark indicate the wind vector at 30km
and 60km height, respectively.
Moreover, in these holographs three-point
running mean was operated to smooth the
pattern by removing small-scale noises.
Inspection of a large number of holographs reveals that;
(1) In general, the wind vector shows the
characteristics
of elliptic porlarization, the
axis being oriented rather randomly,
(2) At northern hemisphere stations, most of
the wind vectors rotate clockwise with
increasing height, regardless
of latitude
and season,
(3) Roughly speaking,
the vectors
rotate
about three times from 30 to 60km, indicating that the dominant vertical scale
is about 10km, and
(4) The day-to-day variability is large, not
only in magnitude but also in horizontal
structure, suggesting that the characteristic timescale of this phenomenon is of
the order of a day or less.
and T. Niki
1057
The fact that the direction of the wave
train
is randomly
distributed
(i,e.,
the
isotropy) has been pointed out by our previous study from the variability
of a ratio
between vertically averaged amplitudes of u
and * (see Figs. 4 and 5 of H84). The dayto-day variability has also been shown by
H84 in terms of the variance
around the
monthly mean.
One of the most interesting
aspects of
wind fluctuations revealed by the holograph
analysis (Fig. 1) is the rotation of wind
vectors with increasing height, which is considered to be the manifestation of the Coriolis
effect.
To see the effect on the wave structure
in more detail, a statistical analysis is made
of the direction of the vector rotation for all
observation days throughout the 6 years. In
practice, the direction of the rotation (clockwise, or anticlockwise)
is defined for each
day by accumulating the phase angle change
from the bottom level to the top level with
interval of 1km, over the height
region
between 30 and 60km.
The result is presented in Table 1, with
the total number of days and their percentage
for the two directions.
As is expected from
Fig. 1, the clockwise rotation is indeed dominant at northern hemisphere
stations : In
middle and high latitudes, about 80% of the
days show the clockwise rotation, and 60-70
% in low latitudes.
On the other hand, at
the southern hemisphere station, though only
one (Ascension Island), the anti-clockwise
rotation has the majority (64%).
The statistical result given in Table 1
leads to the conclusion that the wind fluctuations observed in the middle atmosphere are
due mainly to inertia-gravity
waves, the
structure of which is affected by the earth's
rotation.
Moreover, as was discussed
earlier in
terms of the vertical group velocity, the
clockwise (anti-clockwise)
rotation for *>0
(*<0) implies that most of these waves are
upward propagating ones with their energy
sources in the lower atmosphere.
It is of interest to note here that recent
study of Vincent
(1984) based on partial
1058
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Fig.
of the
Meteorological
Society
of Japan
Vol. 63, No. 6
1. Some examples
of holographs
for three stations.
zonal
and meridional
wind
component
respectively.
Abscissa
and ordinate
indicate
Units
are m/sec.
Circle
and
cross-mark
height.
denote
the
wind
vector
at 30km
and
60km
December
1985
I. Hirota
POINT
MUGU *34N,
and T. Niki
119W)
Fig. 1(b)
1059
1060
Journal
KWAJALEIN
of the
Meteorological
〔9N,
168E〕
Fig. 1(c)
Society
of Japan
Vol.
63, No. 6
December
1985
I. Hirota
Table
1.
Statistical
C and AC
denote
results
the
of the
clockwise
direction
Vertical
profile
of
wind
and anti-clockwise
reflection measurements of upper mesospheric
winds at Adelaide (35*S) and Townsville
(19*S) shows that at least 65% of the vertical energy flux is upgoing for inertia-gravity
waves with typical vertical wavelength
of
12km. Thus our result reconfirms
Vin*
cent's
observation
and expands it to the
wider region over the globe in a climatological sense.
As regards the vertical wavelength, there
have been some observational studies of inertia-gravity
waves in the lower stratosphere
below 25km (Cadet and Teitelbaum, 1979;
Barat, 1983; Maekawa et al., 1984). Their
results indicate, however, that the dominant
vertical scale is close to or less than about
3km and wind fluctuations
are confined to
rather thin layers.
In view of the longer vertical wavelength
(*10km) in a deep layer up to the mesosphere,
the mechanism
for generation
and
propagation of the waves detected by our
observation is probably different from that
of those waves in the lower stratosphere,
even though the both are upgoing waves.
5.
and T. Niki
wind
1061
vector
rotation
rotation
the
present
study
with
the
vertical
tude
of
inertia-gravity
tosphere
amplitude
wave
height.
fluctuations.
In
of
with
respectively.
some
and
lower
profile
dynamical
generation,
we
distribution
are
waves
processes
propagation
the
in
mesosphere,
is indicative
concerned
of
ampli-
the
stra-
because
of
the
such
and
the
nature
as
the
attenua-
tion.
of amplitude
In H84 the intensity of gravity waves was
roughly measured in a statistical manner by
the vertical average of the magnitude of
Fig.
2. A schematic
profile
of wave
detail.
illustration
amplitude.
of the vertical
See the text in
1062
Journal
of the
Meteorological
Let us consider an idealized inertia-gravity
wave. Since the orientation of the elliptic
axis of wind vector rotation is arbitrary
in
a u-* plane (cf. Fig. 1), it is reasonable to
define the wave amplitude by the use of the
vector length r=(u2+*2)1/2 as a function of
height.
Fig. 2 shows a schematic illustration
of
r(z), and the wave amplitude is given by
Society
of Japan
Vol. 63, No. &
the outer envelope of r(z) as is shown by
A0(z) in Fig. 2. Then the averaging is made
of the amplitude for each station and for
each season over the 6 years.
Fig. 3 illustrates some typical examples of
the wave amplitude profile, together with
the estimate of standard deviations around
the ensemble average.
For comparison, alsa
shown is a dashed line proportional to exp
Fig. 3. Averaged wave amplitude at four stations for each season.
Units are m/sec..
Horizontal bars denote the standard deviation and the dashed line represents the constantt
energy density * exp (z/2H) .
December
1985
I. Hirota
(z/2H), i, e., the wave amplitude for the case
of energy
conservation,
where the scale
height H=RT/g
is estimated with the use
of the observed temperature.
From these figures and those at other
stations given in Table 1, it can be seen in
general that the observed amplitude significantly decays with height compared with
the line of exp (z/2H).
The extent of the
decay seems to be larger in winter.
In addition, as was presented by H84, the amplitude shows an annual variation in middle
and high latitudes with the maximum in
Fig.
4.
Histograms
of the
and
T. Niki
1063
winter.
The
fact
energy
in
that
gestive
of
process
in the
the
the
their
the
decay
with
height
other
words,
it is
level
at
which
in
breaking
vertical
ratio
.*/*
we
a
at four
is
dissipation
difficult
the
stations.
rather
a
uni-
year.
In
to find a particular
start
show
the
sense,
specific
sug-
However,
is
waves
cannot
a
a
throughout
distribution
conserve
of
density
statistical
at
not
atmosphere.
energy
form
least
do
propagation
occurrence
middle
of
Therefore,
waves
vertical
for
level
of the
to
the
related
mean
decay.
evidence,
flow
att
wave
to
the
which
,
1064
Journal
of the Meteorological
varies with season.
Alternatively,
the uniform decay of the
statistically
averaged
amplitude is perhaps
caused by the wide distribution of horizontal
phase velocities, which in turn brings about
the wave breaking and dissipation at various
height for a given basic flow in each season.
6.
Wave frequency
Equation (2-5) indicates that for an idealized ineertia-gravity
wave the ratio */*
can
be estimated if we know the magnitude of
horizontal
winds.
The value of |*| / |*|
means the ratio ,between large and small
radii of the ellipse in holograph,
and the
ratio is obtained from the outer and inner
envelopes shown as A0(z) and Ai( z) in Fig. 2.
A statistical analysis of the ratio is made
in such a way that we estimate Ai(z)/A0(z)
for every 1 km level between z1 and z2 (see
Fig. 2) where the interpolation of envelopes
is made and summarize them for all observation days into a histogram at each station.
Fig. 4 shows the results at four stations
for each season. From these histograms we
can see that the most predominant values of
/* fall into a range of 0.20*4.
The peak
value seems to have a tendency to increase
with increasing latitude.
However, compared
with the increase of f with latitude, i.e.,
*sin * (*: latitude) this tendency is weak.
Therefore it can be said that the wave frequency *
itself increases with latitude on
the average : Roughly speaking, therefore,
the typical timescale is of the order of several hours in middle and high latitudes and
of a day in low latitudes.
This is consistent
with the day-to-day variability as revealed
by the holograph analysis in Section 4.
It is also noteworthy
that the horizontal
wavenumber
k can be crudely estimated
from the dispersion relation (2-3) : By assuming typical values of */*0.3,
N*2.2*
10-2 sec-1 and m*2*10-4m-1,
we have a
value of k*10-5m-1 for middle latitudes (f
10-4 sec-1), which corresponds to the horizontal wavelength of several hundred kilometers.
Another important
problem is the relationship between the wave frequency and
the seasonal variation of the mean flow.
Society
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Vol.
63, No. 6
The wave frequency w that we are here
concerned with is the intrinsic frequency
and, in the presence of the background flow
U, is related to the Doppler-shifted frequency
observed at a station as *=*-kU. *
Note
that the Doppler-shifted
frequency *
and
the horizontal
wavenumber
k themselves
cannot be obtained directly from our rocket
observations which are discrete in space and
time.
If the gravity waves in the middle atmosphere are generated by the effect of surface
topography,
i. e., *=0
and k=const.,
then
the intrinsic frequency co must be proportional to the mean velocity U. However, in
spite of the seasonal variation of the mean
circulation in the middle atmosphere, the frequency distribution (Fig. 4) shows no significant change from season to season. This
fact implies that
topographic
stationary
waves with constant k are not necessarily
dominant,
and rather the wide spectra of
horizontal phase velocities should be taken
into account, in harmony with the conclusion
in the previous section.
7.
Concluding
remarks
Throughout the present analysis of rocket
wind data in the stratosphere
and lower
mesosphere, we have obtained following conclusions ;
(1)
A considerable
portion of the wind
fluctuations with vertical scales close to
10km have a notable character
of upward propagating inertia-gravity
waves.
The horizontal direction of the waves is
distributed rather randomly.
(2) The
that
averaged
amplitude profile shows
the wave energy density decays
gradually with height,
suggesting
the
occurrence of wave breaking and dissipation at various height levels.
(3) The most predominant values of */* are
in a range of 0.2*0.4, regardless of latitude and season. This is indicative of
the wide spectra of horizontal phase velocities. There is no supporting evidence
for the dominance of stationary
gravity
waves.
Recent progress in remote sensing techni-
December
1985
I. Hirota
and
ques has made it possible to investigate
the
nature of gravity waves in the upper part
of the middle atmosphere.
Neverthless,
a
very little is known about small-scale motions in the upper stratosphere
and lower
mesosphere (the region between 30km and
60km),
because
of the lack of adequate
observation techniques.
In this regard, the
meteorological rocket network is still important
for the study of the dynamics in the central
part of the middle atmosphere.
Furthermore,
as was mentioned earlier,
the validity of making assumptions of gravity wave-mean flow interaction processes in
numerical modelling has not yet been fully
supported by observations.
Therefore
more
detailed observational studies are needed in
the near future to improve our understanding
of the middle atmosphere dynamics.
Acknowledgements
The present authors wish to thank Dr.
M. A. Geller and two anonymous reviewers
for their useful comments on the original
manuscript.
This research was supported by
Funds for the Middle Atmosphere Program
(MAP) from the Ministry of Education.
References
Barat, J., 1983: The fine structure of the stratospheric flow revealed by differential sounding.
J. Geophys. Res., 88, 5219-5228.
Cadet, D. and H. Teitlebaum, 1979: Observational
evidence of internal inertia-gravity waves in the
tropical stratosphere. J. Atmos. Sci., 36, 892907.
Fritts, D. C., M. A. Geller, B. B. Balsley, M. L.
T. Niki
1065
Chanin, I. Hirota, J. R. Holton, S. Kato, R. S.
Lindzen, M. R. Schoeberl, R. A. Vincent and R. F.
Woodman, 1984: Research status and recommendations from the Alaska workshop on gravity waves and turbulence
in the middle atmosphere. Bull. Am. Meteorol. Soc., 65, 149-159.
Gossard, E. E. and W. R. Hooke, 1975: Waves in
the atmosphere, Elsevier, New York.
Hirota, I., 1984: Climatology of gravity waves in
the middle atmosphere.
J. Atmos. Terr. Phys.,
46, 767-773.
Holton, J. R., 1982: The role of gravity waveinduced drag and diffusion in the momentum
budget of the mesosphere.
J. Atmos. Sci., 39,
791-799.
-, 1983: The influence of gravity wave
breaking on the middle atmosphere.
J. Atmos.
Sci., 40, 2497-2507.
and Xun
-, Zhu, 1982: A further study of
gravity wave induced drag and diffusion in the
mesosphere.
J. Atmos. Sci., 41, 2653-2662.
Lindzen, R. S., 1981: Turbulence and stress owing
to gravity wave and tidal breakdown.
J. Geophys. Res., 86C, 9707-9714.
Maekawa, Y., S. Fukao, T. Sato, S. Kato and R. F.
Woodman, 1984: Internal inertia-gravity
waves
in the tropical lower stratosphere
observed by
the Arecibo radar.
J. Atmos. Sci., 41, 23592367.
Matsuno, T., 1982: A quasi-one-dimensional
model
of the middle atmospheric circulation interacting
with gravity waves. J. Met. Soc. Japan, 60,
215-226.
Miyahara, S., 1984: A numerical simulation of the
mean zonal circulation
of the middle atmosphere including effects of solar diurnal tidal
waves and internal gravity waves; Solstice conditions.
Dynamics of the Middle Atmosphere,
edited by J. R. Holton and T. Matsuno, 271-287,
Terra Pub. Tokyo.
Vincent, R. A., 1984: Gravity wave motions in
the mesosphere.
J. Atmos. Terr. Phys., 46,
119-128.
中層 大 気 中 の慣 性 重 力 波 に 関 す る統 計 的 研 究
廣
田
勇 ・二
木
徹
京都大学理学部地球物理学教室
成 層 圏 及 び 下 部 中間 圏 に おけ る小規 模 擾 乱 の 構 造 と振 舞 い を,1977-1982の6年
ケ ッ トデ ー タを 用 いて 統 計 的 に 解 析 した 。
間 に わ た る広 範 囲 の 気 象 ロ
まず,水 平 風 速 デ ー タに 対 し高 度 方 向 の ハイ パ ス フ ィル タ ー をか け て,平 均 風,プ ラネ タ リー波 及 び 潮 汐 波
の 成 分 を 除 去 し,鉛 直 ス ケ ール が10km程
度 の擾 乱成 分 を 分 離 して取 り出 す。 この よ うに して 得 られ た 風 の
変 動 成 分 に関 し,個 々 の観 測 日に つ い て ホ ド グ ラ フを 作 っ て み る と,一 般 に北 半 球 で は 高 さ と と もに 時 計 回
り,南 半 球 で は 反 時 計 回 りの楕 円 を 描 く こ とが わ か る。 これ は上 向 き に伝 播 す る慣 性 内 部 重 力 波 の 特 徴 を 良 く
1066
Journal
of the Meteorological
Society
of Japan
Vol. 63, No. 6
現 わ して い る 。
簡 単化 され た 慣 性 重 力 波 の 理 論 式 に 基 づ いて,ホ
ドグ ラ フに見 られ る楕 円偏 波 の程 度 か ら,コ リオ リ因 子 ノ
と波 の 振 動 数 ω との 比*/ω を 統 計 的 に 見 積 る と,緯 度 と季 節 に殆 ど よ らず*/ω=0.2∼0.4で
あ る こ とが わ か
る 。 これ は 観 測 され る慣 性 重 力波 の周 期 が 中 高緯 度 で 数 時 間,低 緯 度 で は1日 の オ ー ダ ー で あ る こ とを 意 味 す
る。
一
方
,振 幅 の鉛 直 分 布 に つ い て 見 る と,波 の エ ネル ギ ー密 度 は 季 節 に よ らず 高 さ と と も に ほ ぼ一 様 に減 少 し
て お り,特 定 高 度 で の 砕 波 や 上 層 で の急 激 な 減 衰 を 示 す 証 拠 は 見 当た ら な い。 平 均 風 系 の季 節変 化 を考 慮す る
と,こ の事 実 は,慣 性 重 力 波 の 水 平 位 相 速 度 が,た
と は限 らず,む
とえ ば 地 形 効 果 に よ っ て作 られ る も の の よ うに ゼ ロで あ る
しろ幅 広 く分 布 して い る と考 え るべ き で あ る こ とを示 唆 して い る。