NASA-CR-ZO_IZ3
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. D1, PAGES 1571-1588, JANUARY 20, 1996
A simulated spectrum of convectively generated
gravity waves: Propagation from the tropopause to
the mesopause and effects on the middle atmosphere
M. Joan Alexander
Department of Atmospheric Sciences, University of Washington, Seattle
Abstract. This work evaluates the interaction of a simulated spectrum of
convectively generated gravity waves with realistic middle atmosphere mean winds.
The wave spectrum is derived from the nonlinear convection model described by
Alezander et al. [1995] that simulated a two-dimensional midlatitude squall line.
This spectrum becomes input to a linear ray tracing model for evaluation of wave
propagation as a function of height through climatological background wind and
buoyancy frequency profiles. The energy defined by the spectrum as a function
of wavenumber and frequency is distributed spatially and temporally into wave
packets for the purpose of estimating wave amplitudes at the lower boundary of the
ray tracing model. A wavelet analysis provides an estimate of these wave packet
widths in space and time. Without this redistribution of energies into wave packets
the Fourier analysis alone inaccurately assumes the energy is evenly distributed
throughout the storm model domain. The growth with height of wave amplitudes
is derived from wave action flux conservation coupled to a convective instability
saturation condition. Mean flow accelerations and wave energy dissipation profiles
are derived from this analysis and compared to parameterized estimates of gravity
wave forcing, providing a measure of the importance of the storm source to global
gravity wave forcing. The results suggest that a single large convective storm
system like the simulated squall line could provide a significant fraction of the
zonal mean gravity wave forcing at some levels, particularly in the mesosphere.
The vertical distributions of mean flow acceleration and energy dissipation do not
much resemble the parameterized profiles in form because of the peculiarities of
the spectral properties of the waves from the storm source. The ray tracing model
developed herein provides a tool for examining the role of convectively generated
waves in middle atmosphere physics.
/'
1. Introduction
Gravity waves transport energy and momentum from
the troposphere to the middle atmosphere where, it is
widely recognized, they can have a profound effect on
the general circulation patterns, temperature structure,
and spatial distributions of mixing ratios of the atmo-
spheric gases. The importance of wave drag and diffu-
sion in the middle atmosphere was clearly demonstrated
in zonal mean model studies in the 1980s [e.g. Holton,
1982, 1983; Dunkerton, 1982; Garcia and Solomon,
1985] that utilized the gravity wave parameterization
developed by Lindzeu [1981]. Lindzen's parameteriza-
tion required assumptions about the phase speeds and
source distributions of gravity waves which are to date
still not well characterized. Holton's [1982] work, as well
Copyright 1996 by the American Geophysical Union.
Paper number 95JD02046.
0148-0227/96/95JD-02406505.00
as that of Matsuno [1982], established the importance of
high phase speed waves to explain the observed mean
zonal wind structure and thereby stressed the impor-
tance of wave sources other than flow over topography.
These wave-driven processes are also important in
three-dimensional global circulation models where pa-
rameterization of gravity wave effects is complicated by
the models' sensitivity to the geographical and tempo-
ral distributions of wave sources. Planetary scale waves
can be resolved explicitly in these models, however the
effects of smaller-scale waves (of the order of 100 km
and less) will likely be treated only via parameterization
for some time to come. Orographically excited waves
have been successfully parameterized in such models
and have been shown to affect circulation in the tropo-
sphere as well as the middle atmosphere [Palmer et al.,
1986; McFarlane, 1987; Bacmeister, 1995]. Other wave
sources have been more difficult to characterize. Specif-
ically, waves excited by convective activity are likely
very important in the tropics and southern hemisphere
where there are few orographic wave sources, and con-
1571
1572 ALEXANDER:PROPAGATIONOFCONVECTIVELYGENERATEDGRAVITYWAVES
vectionmaybea source of the high phase speed waves
known to be important in the mesosphere.
Recent modeling efforts reported by Fovell et al.
[1992], Holton and Durran [1993], and Alezander e_ al.
[1995] have described properties of vertically propagat-
ing waves generated by deep convection in the form of
a two-dimensional squall line. In the work of Alezar_der
et al. [1995; hereafter referred to as AHD] the spec-
tral properties of the waves in the stratosphere in their
simulation were characterized and compared to observa-
tions. They noted a strong response at high frequencies
(corresponding to periods of 8 min to 1 hour) and at
long vertical wavelengths (6-10 km), which show sim-
ilarity to observations of stratospheric motions above
convective sources [Larsen e_ al., 1982; Sato, 1993], and
which may be characteristic of waves associated with
deep convection. Additional modeling studies and ob-
servations will eventually clarify the generality of these
spectral characteristics.
In this work, the interaction of this spectrum of con-
vectively generated waves with realistic middle atmo-
sphere winds is examined via a linear ray tracing tech-
nique which uses conservation of wave action flux to
predict wave amplitudes as a function of height. Wave
interactions with the mean flow are included as satu-
ration effects when amplitudes exceed convective insta-
bility limits and via the filtering effects of critical level
absorption and wave reflection. This method is chosen
as a means of estimating the importance of the contri-
bution of waves from such a convective source to the
estimated global gravity wave forcing. It is unrealis-
tic to extend the domain of the nonlinear simulation of
AHD to the altitudes and horizontal distances required
to make this estimate. Durran [1995] also highlights
the difficulty in using traditional diagnostic methods for
evaluating gravity wave dissipation effects in a domain
of limited size. The approach here is to take the two-
dimensional power spectrum derived from the nonlin-
ear model (AHD) as input to a linear wave propagation
calculation. The power in this spectrum is distributed
into packets of finite width in horizontal distance, z,
and time, $, and the packet widths are estimated from a
wavelet analysis of the nonlinear model results. The lin-
ear and nonlinear resultsare testedfor consistencybe-
low 32-km altitudewhere the models overlap.The ray
tracingmodel is in many ways simpler than the three-
dimensionalmodels describedby Eckerrnann [1992]and
Mar_ and Echermann [1995] that were designed to
study global propagation characteristics over the full
range of possible gravity wave frequencies. For the spec-
trum of convectively generated waves considered here, a
number of simplifying assumptions are possible for the
purpose of estimating the mean flow forcing.
The results of the linear wave propagation include
estimates of the mean flow acceleration and energy dis-
sipation rates associated with the input wave spectrum
and specified climatological mean wind profiles. Com-
parison to the spectral gravity wave parameterization
of Fri_s and La [1993] for the same mean wind profiles
provides an estimate of the importance of the single
storm source to globalgravitywave forcingand also
givesinsightinto how the peculiarspectralcharacter-
isticsassociatedwith the convectivelygenerated waves
affectthe profileof wave drag and dissipation.The
method developed here can be used to compare future
simulationsto the midlatitude case of AHD and also
provides an avenue for testing simpler parameteriza-
tionsof convectivelygenerated waves againstthismore
complete spectraldescription.
The followingsectionbrieflyreviews the convection
simulation and determination of the two-dimensional
power spectrum describedby AHD. The method ofcon-
vertingpower spectraldensityto wave amplitude isalso
derived,includinga wavelet analysisof the nonlinear
model resultsto estimate wave energy packet dimen-
sions in space and time. This then establisheslower
boundary conditionsforthe linearray tracinganalysis
described in section 3, which includes determinations
of wave amplitudes as a function of height as well as
effectson the mean state. Expressions for mean flow
accelerationand rateof wave energy dissipationare de-
rived.Section 4 isdevoted to checking the linearprop-
agation model againstthe fullnonlinearmodel results
between 13 and 32 km where the two models overlap,
testingmany of the simplifyingassumptions in the lin-
ear model. In section5 the wave spectrum interaction
with realisticmean wind and buoyancy frequency pro-
filesis examined and compared to the Frittsand Lu
[1993]parameterized resultsfor the same background
state.A concluding summary followsin section6.
2. Determination of Gravity Wave
Amplitudes and Propagation
Characteristics
2.1. The Convection Simulation
The convectivelygenerated gravitywave spectrum in
thisanalysisis derived from the resultsof a numerical
midlatitudesqualllinesimulationpreviouslydescribed
by AHD [Alezander et al.,1995].Some of the features
of the model willbe brieflydescribed here. For more
detailsthe reader isalso referredto previous applica-
tionsof the squalllinemodel describedby Fovelle$ al.
[1992] and Holton and Durran [1993].
The convection simulation from which the gravity
wave spectrum is derived is a two-dimensional, nonlin-
ear, compressible, nonhydrostatic squall line model. It
resolves a deep stratosphere layer, from the tropopause
at .._12 km up to 32 km altitude. The full model domain
is 840 km in the horizontal and 32 km in the vertical and
includes wave permeable boundaries at the sides and
top. The model reference frame translates eastward at
16 m s -1 to track the motion of the storm, keeping it in
the center of the domain. Winds in the stratosphere in
this model are constant in height at 16 m s -1, the same
as the reference frame translation speed, and so the
stratospheric waves produced by the storm are viewed
in their Uintrinsic" frame of reference (i.e., the observed
wave frequencies are the intrinsic frequencies). A rich
ALEXANDER: PROPAGATION OF CONVECTIVELY GENERATED GRAVITY WAVES 1573
30
25
N 20
15
7
6
_5
i
_: 4
. (m/s) 6 (°K)
, , i , , i , , , i
200 600 800
2OO 400
X (KM)
, , , L ,
400
X (KM)
,,,(x,t) (m/s)
6O0 8OO
Figure 1. Contours of vertical velocity in the stratospheric portion of the nonlinear convection
model. Contours are plotted at 0.5 m s -1 intervals. Dotted contours represent negative values.
(a) w(_, z) at _ = 5 hours. Horizontal lines represent surfaces of constant potential temperature
at 25 K intervals. The storm center, which is the main source region for these waves, lies in the
troposphere below the figure and is located at $ __ 420 kin. (b) w(_, $) at z = 30 km. The slopes
of surfaces of constant phase indicate the intrinsic phase speeds of the wave motions.
spectrum of gravity waves is generated in the strato-
sphere, as can be seen in Figure I. Figure la shows a
single time frame of vertical velocity contours and po-
tential temperature surfaces above 13 km. In Figure
lb, contours of w(z,_) at z = 30 km are shown. The
slope of the phase surfaces in Figure lb is an indication
of the separation east and west of storm center (at _ .._
420 kin) of eastward and westward propagating waves,
respectively, and points to the storm center region as
the location of the primary source of the wave energy.
A strong preference for forcing of westward propagat-
ing waves is observed in this and other similar simu-
lations due to a westward tilt with height of the main
tropospheric updraft and the westward propagation of
convection cells in the troposphere. These features have
been described in earlier work with this model [Fovell et
al., 1992; AHD], and have been observed in squall lines
in nature [e.g. Houze, 1993]. A spectral analysis of the
stratospheric waves was described by AHD and revealed
some distinctive spectral signatures which may be char-
acteristic of waves generated by deep convection. AHD
describe wave forcing mechanisms and their signatures
in the spectral response and also review similarities to
observations of wave motions above convective sources.
2.2. Fourier Spectral Analysis
The two-dimensional power spectrum in horizontal
wavenumber and intrinsic frequency P(/_, w) computed
from the simulation results of AHD will be used as lower
boundary input to the linear propagation analysis to
follow. The spectrum shown in Figure 2 separates
power into eastward and westward phase with positive
and negative frequencies, respectively. The power in
this spectrum defines wave energies at an altitude of
13 km. The superimposed white contour surrounds re-
gions of the spectrum with power greater than or equal
to 5 x 104(m/s)2(cycle/m)-1(cycle/s) -I. Regions out-
side this contour with power lessthan this threshold will
be excluded from the following analysis. The contour
includes over 93% of the total energy in the spectrum,
and spectral points outside this region are very likely
heavily contaminated by spectral bias from regions of
high power. Each pixel inside the white contour will
be treated as a monchomatic wave packet. The wave
amplitude can be derived from (I) the power spectral
density at that point, together with (2) the definition
of the wave packet width in time and space, and (3) an
estimate of power aliasing to other regions of the spcc-
1574 ALEXANDER: PROPAGATION OF CONVECTIVELY GENERATED GRAVITY WAVES
?
o
5,0
2.0
Power Spectrum of w(x,t)
! I I I I I I
Figure 2. Two-dimensional power spectrum of the ver-
tical velocity field as a function of horizontal wavenum-
ber and intrinsic frequency. The spectrum is derived
from the two-dimensional Fourier transform of the field
w(z,L) like that shown in Figure lb. Spectra at each
altitude in the model have been averaged by using an
exp [-(z- 13 km)/H] weighting factor such that the
spectral power represents amplitudes at 13 km near the
tropopavse.
trum. Parameter (2) will be estimated from a wavelet
analysis of the vertical velocity field described in the
next section.
The power spectrum in Figure 2 resolves wavenum-
bets and frequencies,
n N=
k -- AzN., n : O,1, ..., -_-
in N_
w - AtN_ n = O, 1,..., y
with Az = 1.5 kin, N_ = 256, AI : 2 min and Nt =
128. So pixel dimensions are Ak = 2.6 x 10 -s cycle
m -1 and Aw = 6.5 x 10 -s cycle s-1. Wave energy with
wnvenumbers and frequencies smaller than the mini-
mum nonzero values (k < Ak and w < Aw) is also
omitted from the linear propagation analysis but repre-
sents less than 0.2% of the total energy in the spectrum.
The power spectral density in Figure 2 can be re-
lated to the vertical velocity amplitude (in m s -1) if it
is assumed that a given pixel represents a wave mode
with the frequency and wavenumber associated with
that pixel. The two-dimensional power spectral density
Pk_ is given by
2AzAf,
Pk< - IWk< l (1)
N®N,
where Wk_, is the discrete Fourier transform of w(z,_)
at wavenumber k and frequency w. The power at (/c,w)
describes the mean square amplitude of the wave with
those characteristics averaged over the (z,_) domain,
Pk,, llAzN_atNt,= ½1A <
or
Ak_ : [2Pk_A/cAw]X/l. (2)
This amplitude would be exactly the wave amplitude
if the assumptions (inherent in Fourier analysis) of sta-
tionarity throughout the (z, _) domain and periodicity
on the z and L intervals were both satisfied. On the con-
trary, both of these assumptions are violated, as can be
seen in Figure 1. There is significant spatial and tempo-
ral variation in the spectral properties. The waves can
be thought of as concentrated in wave packets in both
space and time; i.e., the areal extent of a given mode
over the (z,t) domain is limited in both dimensions.
These effects make the amplitude predicted from (2)
smaller than the true amplitude at any point in (z,_).
The power Pk_ in equation (2) must be adjusted by two
multiplicative factors:
1. The first factor is equal to the areal extent of
the whole domain divided by the energy-weighted areal
extent of the wave packet. This factor arises because the
spectrum gives a measure of the mean square amplitude
in the domain, not the amplitude in the wave packet.
2. The wave packet envelope can be thought of as
a taper function which creates bias in the spectrum,
reducing the value of the power at the peaks in the
spectrum and spreading that power over a broad range
of wavelengths. For a rectangular wave packet the bias
would be described by the Fejer kernel [Percival and
Walden, 1993, Section 6.4]. The power at the central
peak of the kernel is proportional to the length of the
window, so this second factor is also proportional to
the ratio of the domain area to the envelope area, just
as in factor 1. The proportionality constant will vary
between 1 and 2 depending on the shape of the wave
packet envelope. A value of 1 corresponding to a rect-
angular shape is assumed for simplicity.
Thus if the wave packets were rectangular in shape,
and covered an area n=nt within the entire domain (area
N_Nt), then the wave amplitude within that packet be-
comes
= , , (3)
with one factor of (N, Nt/n_nt) from each of factors
1 and 2. This equation will be used to estimate am-
plitudes at the lower boundary for the purpose of pre-
dicting breaking levels for each mode in the ray tracing
analysis. Parseval's theorem, however, demands that
the total power in the spectrum equal the power in the
original signal integrated over the (z, _) domain,
ALEXANDER:PROPAGATIONOFCONVECTIVELYGENERATEDGRAVITYWAVES 1575
7
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4
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o (D G©©o
o o
o 0 @ @Oo
o o © 0
o o©@@00
o 0000
o00
I I I
200 400 600
x (kin)
"'r'" r,,,r....
i
8OO
Figure 3. An example of the wavelet basis functions
centered in the analysis domain (z, _). Peaks and valleys
are plotted with solid and dotted contours, respectively.
This is the 96-km wavelength and 32-min period case.
The panels along the top and side show cross sections
in z and t through the center.
P(k, @= A=A )12. (4)
Therefore the wave packet amplitude adjustments in
(3) must be removed prior to evaluating any net ef-
fects of the waves on the mean flow to conserve en-
ergy. Wave packet dimensions will be estimated via the
wavelet analysis described below.
2.3. Wavelet Analysis
Wavelet analysis is a spectral analysis technique that
retains information about the spatial/temporal location
of variations in the spectral power. Without this infor-
mation the amplitudes derived solely from the Fourier
analysis cannot be used to predict realistic breaking lev-
els. For an energy-conserving orthogonal wavelet set,
the spatial/temporal information is gained at the ex-
pense of spectral resolution: Given a 128-point time
array, the number of resolved frequencies in the wavelet
spectrum is only log2(128) -- 7, compared to the 64
nonzero frequencies obtained from a Fourier analysis.
Therefore wavelet analysis cannot replace the Fourier
analysis for our purposes but can provide a rough esti-
mate of the spatial/temporal extent of a given wavenum-
ber/frequency signal and thus an estimate of an effective
wave packet width.
The 512×128 array of w(z,_) with 1.5-km and 2-
min resolution, as shown in Figure lb, is examined
via wavelet analysis. A two-dimensional, orthogonal
wavelet transform is applied and an energy spectrum
computed as the square of the resulting wavelet coef-
ficients. Figure 3 shows an example of the wavelet
basis functions employed here. The wavelet transform
utilizes the set of Daubechies wavelet filters with 20
coefficients summarized in the Numerical Recipes sub-
routines "pwtset" and "pwt" [Press e_ aL, 1993, sec.
13.10]. The set of basis functions employed in the anal-
ysis consists of translations and dilations/contractions
of the function shown in Figure 3. The wavelengths
and periods resolved in the analysis are summari_.ed in
Table 1, as well as the fractional energy contained in
each of these modes. For each point in Table 1, a corre-
sponding array of energy as a function of (z,_) can be
produced. The resolutions in z and t of each energy ar-
ray are equal to the wavelength and period of the mode,
respectively. Four examples are shown in Figure 4 for
the four modes containing the largest fractional energy.
These modes are also highlighted in boldface in Table
1. Together these four modes comprise 64% of the total
energy in the spectrum. Note that the left and right
halves of the figure display properties of the westward
and eastward propagating waves, respectively. The con-
tours describe how energy for the resolved mode with
wavelength and period (A=,T) is distributed in (z,$).
When these figures are compared to Figure lb, it can
be seen that the wavelet analysis describes how energy
in a given spectral mode is localized in (z, $).
Table 1. Percent Energy in the Wavelet Spectrum as a Function of Hori2ontal
Wavelength and Period
As, km
T, min 768 384 192 98 48 24 12 6 3
256 0.ii <O.Ol 0.01 0.03 0.04 0.06 0.02 <O.Ol <0.01
128 0.04 <0.01 0.02 0.05 0.06 0.08 0.03 <0.01 <0.01
64 0.04 0.05 0.20 1.50 1.73 0.44 0.07 0.01 <0.01
32 0.02 0.04 0.20 1.46 9.46 6.01 1.33 0.08 0.04
16 <0.01 <0.01 0.02 0.17 4.41 15.83 10.98 0.33 <0.01
8 <0.01 <0.01 <0.01 0.02 0.24 5.52 29.16 5.47 0.05
4 <0.01 <0.01 <0.01 0.02 0.07 0.20 1.25 2.77 0.12