Analysis of Vertically Propagating Convectively Coupled Equatorial Waves Using
Observations and a Non-Hydrostatic Boussinesq Model on the Equatorial Beta Plane
Paul E. Roundy
Matthew A. Janiga
University At Albany
State University of New York
Submitted to
The Quarterly Journal of the Royal Meteorological Society
January 2011
Corresponding Author Address
Department of Atmospheric and Environmental Sciences
DAES-ES351
Albany, NY 12222
roundy@atmos.albany.edu
0
Abstract
A non-hydrostatic model on the equatorial beta plane is solved for the most basic
solutions of vertically and zonally propagating internal plane waves. Solutions tilt in the
vertical and propagate upward, with buoyancy as a principal restoring force. Results indicate
that when frequency is of the same order of magnitude as a reduced buoyancy frequency, the
shallow water model equivalent depth depends on frequency. One consequence of this
dependence is that Kelvin waves become dispersive at high frequencies. In a complementary
observational analysis, linear regression and a space-time wavelet spectrum analysis of
observed convectively coupled mixed Rossby gravity (MRG) waves are applied to estimate
vertical wavelengths that are consistent with the strongest signals associated with observed
convectively coupled waves at specific zonal wave numbers and frequencies. Substitution of
these dispersion parameters into the model yields theoretical structures characterized by
vertical and horizontal wavenumbers and frequencies similar to those observed in the middle
troposphere. These model solutions demonstrate that the Coriolis terms associated with the
horizontal component of the earth’s angular momentum explain substantial meridional tilts
and phase shifts between quantities associated with observed convectively coupled waves,
especially proximate to the surface of the earth.
1
1. Introduction
Convectively coupled equatorial waves interact with and organize moist deep
convection in the tropics. They influence tropical cyclogenesis (e.g., Frank and Roundy
2006; Bessafi and Wheeler 2006) and interact with the Madden Julian Oscillation (MJO, e.g.,
Zhang 2005; Roundy 2008). Many authors have drawn analogs between these waves and
zonally propagating wave solutions of the shallow water model on the equatorial beta plane
(Matsuno 1966; Lindzen 1967; Kiladis et al. 2009 and references therein). These solutions
provide horizontal structures and dispersion characteristics that are remarkably similar to
many observed waves, given the crude approximations made.
In spite of the successes of these shallow water models in predicting the basic patterns
of horizontal structure and dispersion, they cannot directly represent many aspects of the
observed waves without substantial modification (e.g., Yang et al. 2007 a, b, c; Roundy
2008). For example, wave structures tend to tilt meridionally against the direction of
propagation with increasing latitude (e.g., Kiladis et al. 2009, their Figure 12). Many
observed waves would rely on buoyancy as a principal restoring force in an atmosphere
statically stable to moist deep convection on the spatial scales of the waves, (e.g., Kiladis et
al. 2009 and references therein). Lindzen (1974), Holton (2004), Kiladis et al. (2009), and
others have overcome this limitation by modifying the shallow water model to include
vertical variation of density. If a rigid lid is not applied at the top of a model troposphere and
the atmosphere is assumed to be stably stratified on the spatial scales of the waves, solutions
include wave structures that tilt in the vertical, with dispersion characteristics and horizontal
structures that depend on vertical wavenumber and the buoyancy frequency (Lindzen 2003).
2
The horizontal structures and dispersion properties of waves in these models can also be
obtained from a shallow water model by choosing a particular equivalent depth. This paper
investigates how the vertical structure of convectively coupled waves changes with the zonal
scales of the waves, through analysis of theory and observations. This work also provides a
convenient opportunity to analyze through a combination of simple theory and observations
the influence of the complete Coriolis force on a subset of observed convectively coupled
waves.
Recently, Kasahara (2003) postulated that the horizontal component of the earth’s
angular velocity would be relevant to the dynamics of non-hydrostatic, diabatically driven
circulations in the tropics, through the cosine Coriolis terms in the momentum equations (see
Gerkema et al. 2005 for a review of these terms). For simplicity, we label these terms the
“non-traditional Coriolis terms”. These terms couple zonal and vertical flow, and would be
balanced in the vertical by buoyancy in an atmosphere statically stable to deep convection on
the spatial scales of planetary to synoptic scale waves.
Recent works have demonstrated that the non-traditional Coriolis terms play
important roles in the dynamics of deep atmospheres such as that of Jupiter (e.g., Yano 1994,
1998, 2002; Yano et al., 2003, 2005). Due to a relatively weak density stratification of the
deep oceans, others have recently suggested they could be equally important in the equatorial
oceans. For example, Fruman et al. (2009) discuss their role in generation of equatorial jet
currents. Even in the Earth's atmosphere, which is neither deep nor weakly stratified, the
possibility that the non-traditional Coriolis terms play critical roles has not been excluded.
This possibility was recently investigated by Kasahara (2003a, b, 2007, 2009). Due to a
drastic reduction of effective stratification by diabatic heating associated with moist deep
3
convection and maximization of the cosine of latitude near the equator, the non-traditional
Coriolis component might be important in the tropical atmosphere.
Recently, Fruman (2009) investigated equatorial wave modes in a simple Boussinesq
model on the equatorial beta plane that included these terms without analyzing observations.
He found that these terms do not influence the dispersion properties of equatorial waves, but
that they do influence horizontal structures of waves when the ratio /N is large (where  is
the angular velocity of the earth, and N is the buoyancy frequency). This modification of
horizontal structure is expressed as a meridional tilt imposed on the solutions. Fruman’s
model is hydrostatic, suggesting that his result is useful only with respect to low frequency
motions. To date, no previous works have attempted to diagnose the magnitude of the
contributions of these terms to any type of observed convectively coupled waves. The present
paper analyzes the influence of these terms on observed vertically propagating convectively
coupled waves in observations, in the context of a simple model.
We first discuss a non-hydrostatic model on the equatorial beta plane similar to the
Boussinesq model of Kasahara (2003), and the simplest forms of its solutions. We apply this
model to generalize the analysis of Fruman (2009) to include high frequency waves, and we
briefly summarize the consequences of our solutions for such waves. We further generalize
the result of Fruman (2009) to include simple vertical structure. We then apply this simple
generalization to analyze a subset of observed synoptic to planetary scale convectively
coupled mixed Rossby gravity waves that tilt in the vertical and propagate upward through
the troposphere. Finally, we estimate the magnitude of the contribution of the non-traditional
Coriolis terms to such observed waves.
4
2.1 The Model
We begin with the Boussinesq model of Kasahara (2003). We assume a volume mean
density  that is constant in space and time across a layer in the lower to middle
troposphere. This assumption would be applicable to waves characterized by small vertical
 displacements in the flow near the height of the selected layer, even if their characteristic
vertical wavelengths are substantially larger than the thickness of the layer. Local density,  ,
is allowed to vary in a given layer only in terms multiplied by g (the constant acceleration
due to gravity). The perturbation velocity equations (u’, v’, w’) from a resting basic
 state are:


u'
1 p'
 yv'2w'
 0,
t
 x
(1)
v'
1 p'
 yu'
 0,
t
 y
(2)
w'
1 p'
 2u'
 s' ,
t
 z
(3)
the buoyancy equation is

s'
 N 2 w' 0 ,
t
(4)
and the continuity equation is

u' v' w'


 0.
x y z
(5)
Buoyancy is represented as

s
g 

;
the Brunt-Väisällä or buoyancy frequency is

(6)
5
N2  
g d
 dz .
(7)
 is the angular velocity of the Earth, and  is the meridional gradient of the Coriolis force

on the equator. The standard equatorial beta plane approximation for 2sin() (where  is
latitude) is y. We extend the beta plane approximation by including the non-traditional
Coriolis terms, approximated by 2cos()2; this approximation generates an error of less
than 14% at +/-30 latitude. p’ represents the part of the pressure not in hydrostatic balance
with the uniform background density . Our use of (3) and (4) allows us to analyze high
frequency waves that were excluded from the analysis of Fruman (2009) (See his Section 2)

because these equations represent
a relaxation of the hydrostatic approximation. We
generally assume that the vertical acceleration term is negligible, but we retain it to diagnose
a more general solution. We analyze that solution in comparison with that obtained after
applying the quasi hydrostatic assumption (e.g., White and Bromley 1995) to diagnose the
scale of the impacts of this term on the propagation of convectively coupled waves in varying
background environments.
For simplicity, we assume a background state at rest, although previous works have
shown that vertical and meridional shear of the zonal wind (e.g., Xie and Wang 1996) and
the vertically averaged zonal wind influence the structures of observed waves.
2.2 Solutions
We assume zonally and vertically propagating internal plane wave solutions of the
form u',v',w', p',s'  uˆ, vˆ , wˆ , pˆ , sˆexp ikx  lz  t ,
(8)
where the hatted variables are each functions of y only. k and l are zonal and vertical

wavenumbers, respectively;  represents angular frequency. Since the troposphere is not
6
capped to vertical propagation and since we assume that the troposphere is statically stable to
deep convection on the spatial scales of the waves, an infinite number of vertical modes
would exist (e.g., Lindzen 2003). We assume that wave planes propagate upward into the
layer (thus we assume purely radiative boundary conditions at both the top and bottom of the
selected layer). The observational analysis (below) demonstrates that patterns revealed by
this simple theory successfully represent previously unexplained patterns in the horizontal
structures of convectively coupled waves, suggesting that although crude assumptions are
made, the result improves the representation of convectively coupled waves over shallow
water models that do not account for the non-traditional Coriolis terms. Substitution of (8)
into equations (1)-(5) yields the following:
ˆ  ik pˆ  0
iuˆ  y vˆ  2w
dpˆ
0
dy
(10)
iwˆ  2uˆ  ilpˆ  sˆ
(11)
i vˆ  yuˆ 


(9)
ikuˆ 

dv
 ilwˆ  0
dy
(12)
isˆ  N 2 wˆ  0


(13)
To find a solution to (9)-(13) in terms of vˆ analogous to the shallow water model of
Matsuno (1966), we begin by solving (13) for sˆ , substitute the result into (11)
iwˆ  2uˆ  ilpˆ 
iN 2 wˆ


,

(14)
and solve (14) for uˆ


uˆ 


i
 2  N 2 wˆ  lpˆ .
2
(15)
7
ˆ , substitute the result into
Substitute (15) into (9), (10), and (12), solve the resulting (9) for w
(10) and (12), solve (12) for pˆ , and substitute the result into (10) to get an equation in vˆ

only:
 
k 2
2 2
2
2
2
2 
d 2 vˆ
4iyl
d vˆ l   k N     N   
2il
l 2 2 y 2
 2
 
vˆ  2
vˆ  0
vˆ  2
2
2
2
2
2
2
2
2
dy
N  4   dy 
N  4  
N  42   2
 N  4  



(16)

Next, apply the transformation
 i

vˆ  (y)exp y 2 ,
 2

(17)
where


2l
,
N  42   2
2
(18)
This transformation is applied because it eliminates the first order derivative and the

imaginary part of the zeroth order derivative; it also modifies the term in y2 and generates an
equation in  for which there exist familiar solutions. This transformation is similar to that
applied by Fruman (2009) and by Gerkema and Shrira (2005). The similar transformation of
Fruman (2009) does not depend on  as a consequence of his assumption of quasihydrostatic
balance (White and Bromley 1995). When considered in the context of (8), the complex
exponential portion of (17) results in a latitude-dependent, zonal phase shift  in solutions for
v’, where  is the object of the complex exponential in (17) with the –i omitted (consistent
with the phase shift diagnosed by Fruman 2009). After applying the transformation, (16)
reduces to
 2 2

k 2
2
2
2
2
2 2 2
2
2
d 2 l   k N     N    y  l N    
 

0
2 
dy 2 
N 2  42   2
N 2  42   2  




(19)
8
Equation (19) has the same form as the final equation for vˆ of Matsuno (1966), and is known
as a Weber equation (Polyanin and Zaitsev 2003). This equation has a broad set of solutions,
 discrete meridional modes familiar from
but one particular class of solutions includes the
equatorial beta plane shallow water theory (e.g., Matsuno 1966). This class of solutions
occurs only if
 N 2   2 1/ 2 
  k 2  k  l 2 2  2n  1,





l
 N 2   2 


(20)
where n is an integer. Equations (19) and (20) generalize the result of Fruman (2009) to

include simple vertical structure and a correction in terms of  that appears due to relaxation
of the hydrostatic approximation in this model.
Discrete solutions to (20) for  are the numerically calculated roots of the quadratic
in k after assuming a range of values for k and specific values of the other parameters. Figure

1 shows the associated dispersion diagram giving the solutions for n= –1 through n=1, all
assuming N =4.6x10-3s-1 and l-1=2x104m. Analogous solutions to those of shallow water
theory are labeled on Fig. 1, and include the Kelvin, equatorial Rossby (ER), mixed Rossby

gravity
(MRG), eastward inertio-gravity (EIG), and westward inertio-gravity (WIG) modes.
The dashed curve is associated with a negative root of the Kelvin wave, and it has previously
been deemed unphysical because its associated zonal wind anomalies do not taper to zero
with latitude, thus making it inconsistent with the equatorial beta plane assumption (Matsuno
1966).
Although the values of l and N applied to obtain Fig. 1 result in dispersion curves that
are roughly consistent with the shallow water equivalent depths of observed convectively
coupled equatorial waves (e.g., Liebmann and Hendon 1990; Wheeler and Kiladis 1999;
9
Roundy and Frank 2004; Yang et al. 2007a,b,c), substantially different values of N and l also
yield reasonable dispersion solutions. Comparison of this dispersion relation with the shallow
water model of Matsuno (1966) reveals that the shallow water model equivalent depth
he 
N2 2
l 2g
.
(21)
This result is equivalent to that of Fruman (2009) except for inclusion of 2. This inclusion

indicates that relaxation of the hydrostatic approximation yields dependence of he on
frequency. This dependence would become important when 2 is of the same order of
magnitude as N2.
If we let
 2 l 2 (N 2   2 
N  4  
2
2

2 2
 K 2  0,
(22)
then the corresponding solutions for the meridional structure of v anomalies are of the

form y   H n

 Ky 2 
K y exp
,
 2 

(23)
where Hn is the n’th Hermite polynomial. Therefore, solutions for vˆ may be obtained from 

by applying (17).
 only the appearances of  that
Setting explicit appearances of  equal to 0 (i.e., retaining
are implicit in ) eliminates the contributions of the non-traditional Coriolis terms and
reduces the result to the Boussinesq model analog to the shallow water model of Matsuno
(1966).
We assume that a Boussinesq plane wave solution would approximate the structure of
some convectively coupled atmospheric waves propagating upward through a layer of the
lower to mid troposphere, from below. In the context of the MRG wave, we assume a wave
10
source in the upper mid troposphere (such as, perhaps, a tropical cyclone) from which wave
energy disperses downward and eastward across the mid to lower troposphere and upward
and eastward across the upper troposphere and the stratosphere. We assume pure radiative
boundary conditions at the bottom and top of the layer.
2.3 Kelvin Wave
Solutions for the Kelvin wave are found by setting v’ = 0 everywhere and for all time in
equations (1)-(5), and by solving a selection of four of the resulting equations for uˆ . Since
four equations can be selected from the original five, we found that two forms of the same
solution exist in terms of different combinations of parameters. 
The same gamma
transformation applied above for v' also applies to the Kelvin wave solution for uˆ , and one
form of the solutions for the corresponding  is identical to (23), except that uo (the

 relationship for the
maximum value of the equatorial zonal wind) replaces Hn. The dispersion

Kelvin wave can be found by substituting
roots for
 –1 for n into (20). Of the three resulting
, the positive root of



N
1 
 2  2 k 2
2 
l
1 k 
 l 2 
2
(25)
is the non-hydrostatic analog to the shallow water Kelvin mode. The same result can also be

obtained by comparing the combination of parameters in the two solutions for the Kelvin
wave noted above. Equation (25) implies that the Kelvin wave is dispersive at wavenumbers
much higher than those plotted in Fig. 1. This dispersion is a consequence of relaxation of the
hydrostatic approximation, and it is relevant only at high zonal wavenumber. Wheeler and
Kiladis (1999) and many others have demonstrated that observed convectively coupled
11
Kelvin waves exhibit some dispersion, but this result demonstrates that relaxing the
hydrostatic approximation cannot explain this observed behavior.
2.4 Reduced Moist Stability
Values for N2 required to match the dispersion curves to those of observed convectively
coupled waves are much smaller than observed values of N2. Previous works have suggested
that the actual value of N2 felt by a wave is reduced by the influence of the release of latent
heat in deep convection coupled to the wave. These works suggest that an effective N’2 can
be obtained from the actual N2 as
N 2  1  N 2
(26)
(e.g., Neelin and Held 1987; Emanuel 1987; Yano and Emanuel 1991; Emanuel et al. 1994;

Kiladis et al. 2009). Here,  is considered a property of the environment in which vertical
velocity is positively correlated with the release of latent heat, reducing the effective static
stability under vertical motion imposed by a wave. Although some waves might evolve
differently, the focus of this work is on waves that propagate in an atmosphere statically
stable to moist deep convection on the spatial scales of the waves themselves, with unstable
conditions possible on local scales. Such waves would induce wave-scale envelopes of
enhanced and suppressed convection without wave-scale vertical overturning circulations.
Our use of  under these assumptions is consistent with its application by Emanuel et al.
(1994). Although this parameterization is a simplification of the observed system, it has some
observational support and allows for simple analytical solutions to the model that might offer
insight into the behaviors of some observed waves. In order to obtain real-valued dispersion
parameters from equation (20), it is necessary to assume
 2
0    1    1 .
N 

(27)
12
We thus focus our analysis on a particular subset of atmospheric waves for which  is
everywhere less than one. Consistent with this view, previous works have suggested that  is
close to, but less than 1 for most convectively coupled waves (e.g., Gill, 1982; Kiladis et al.,
2009). Observed values of N2 are substantially higher in the stratosphere than in the
troposphere. Parameterized latent heating effectively reduces N2 in the troposphere through
, which further increases this difference. Hereafter, we focus only on waves of the lower to
middle troposphere that tilt in the vertical against the direction of propagation. This
distinction of the class of waves that are the focus of this work is important in light of the
suggestion Yano (2007) that  might vary substantially in space and time and occasionally
even take on values greater than 1 in association with some disturbances. This condition
would lead to imaginary values of N’ that are incompatible with this model because
buoyancy would no longer serve as a restoring force.
2.5 Meeting the Surface Boundary Condition
It is useful to consider a simple thought experiment how such waves might interact with a
deeper layer including the surface of the Earth. It is necessary to consider vertical variation of
the effective N2 to obtain the full vertical structure. If we assume that its value is constant in
one layer, but allow it to vary between layers, it is possible to analyze vertical structure. For a
given  and k (which we can assume to be fixed in a given uniform horizontal background
environment for a given wave), l varies with N2. Thus if we know the effective N2 for a given
layer, given  and k, we can find the corresponding value of l and match the phase to the
layers above and below. N2 would be effectively zero at the surface of the Earth, because a
surface that does not move in the vertical has an infinite buoyancy period. The effective N2
experienced by a large-scale wave might decrease more gradually toward the surface.
13
Decreases in effective N2 toward the surface imply that a wave would produce progressively
smaller vertical displacements with approach toward the surface, if we assume conservation
of energy. Thus vertical variation in N2 is sufficient to effectively meet the boundary
condition at the surface for a plane wave solution. To further clarify the influence of
propagation toward the surface on wave structure, Equation (20) can be restated


 2 k 
l
l 2 2



.
k   2n  1
N 2   2 1/ 2  N 2   2

 


(24)
For low frequency waves, if we hold  and k constant (thus making the left hand side

constant) and allow N2 to approach 2, the vertical wavenumber l must also approach zero to


maintain equality. If l approaches zero, it implies that the wave planes approach vertical. The
solutions confirm that reduced effective N2 is associated with smaller vertical velocities and
wave planes that approach the vertical. Thus, a MRG wave packet with a source in the mid
troposphere would have a downward and eastward group velocity, except near the surface of
the Earth, where refraction associated with decreasing N2 yields a group velocity that is
eastward only and the surface boundary condition is met. A more careful analysis of a WKB
approximation (e.g., Kundu 1990) to equations 1-6 would provide greater insight about the
details of this thought experiment, but it is beyond the scope of this paper.
3. Diagnosing Model Parameters Consistent with Observed Waves
We analyze observational data in the context of (7), (8), (17), and (20) to estimate the
values of  near particular layers of the lower to middle troposphere associated with a subset
of observed convectively coupled waves. We now consider the above Boussinesq solutions
to be linearizations of the observed convectively coupled waves about specific layers of the
troposphere. The approximate parameters and the model solutions allow us to estimate the
14
zonal phase shifts attributable to the non-traditional Coriolis terms, which can be compared
with similar phase shifts in observed waves. We apply wave number-frequency wavelet
spectrum analysis (Kikuchi and Wang, 2010) along with simple linear regression (Wheeler at
al., 2000) to diagnose the structures of MRG waves at specific wave numbers and
frequencies for direct comparison with the theoretical patterns. The corresponding spatial
structures and vertical tilts are measured directly from regressed waves (Wheeler et al.,
2000).
3.1 Data
Daily-interpolated outgoing longwave radiation (OLR) data on a 2.5-degree grid are
applied as proxy for moist deep convection (Liebmann and Smith, 1995). These OLR data
have been updated every few months since 1995 following the original algorithm. Daily
mean zonal and meridional wind, temperature, specific humidity, and geopotential height
data are obtained for standard pressure levels from the NCEP/NCAR reanalysis (e.g., Kalnay
et al., 1996). The mean and first four harmonics of the seasonal cycle are subtracted from the
OLR and wind data to generate anomalies. All data are analyzed for the period June 1, 1974
through December 2009, except that OLR data are missing during 17 March through 31
December 1978.
Prior to wavelet analysis, to reduce data storage and computation requirements, we first
average OLR anomalies between 15S and 15N, filtered by symmetry across the equator.
Symmetric OLR signals are obtained by adding OLR signals at the same latitudes across the
equator, and antisymmetric signals were obtained by subtracting the signals, as by Wheeler
and Kiladis (1999).
15
3.2 Space-time Wavelet Decomposition
Wheeler and Kiladis (1999, hereafter WK99) and many others have demonstrated that
signals obtained from the regions of the wave number frequency domain near equatorial beta
plane shallow water model dispersion curves of 25m equivalent depth are associated with
spatial patterns similar in some respects to those obtained from the shallow water theory. To
set the context, Figure 2 shows dispersion lines of 8 and 90m equivalent depths from the
shallow water model superimposed on an OLR spectrum (calculated by Fourier transforms in
a similar manner to WK99, normalized by dividing by the same spectrum smoothed by 40
applications of a 1-2-1 filter). All subsequent observational analyses in this study are reported
for MRG waves at the 25m equivalent depth, nearly between the 8 and 90m curves in Fig. 2,
near the maximum power in the MRG band. The above theoretical analysis provides the
corresponding space-time structures of Boussinesq wave solutions for specified zonal
wavenumbers and frequencies. Most previous observational analyses of wave structures, in
contrast, conglomerate structures of a broad range of wavenumbers and frequencies
associated with the waves (e.g., Wheeler et al. 2000). Interpretation of such results is
complicated because the vertical and meridional structures of the observed waves might vary
with wavenumber and frequency. Thus, clear comparison with the theoretical patterns
requires extraction of observed wave signals associated with specific wavenumbers and
frequencies. We apply zonal wave number-frequency wavelet analysis to extract such signals
from OLR data. A detailed description of space-time wavelet analysis is beyond the scope of
this paper, but Kikuchi and Wang (2010) and Wong (2009) offer overviews of the technique.
16
The space-time wavelet transform is the wavelet transform in longitude of the wavelet
transform in time of the antisymmetric 15S to 15N mean OLR anomalies. We apply the
Morlet wavelet
 (s) 
1
exp isexp s2 /B
1/ 2
B
(28)
where s represents x or t for the spatial or temporal transforms, respectively, and  represents

angular frequency  or wavenumber k. B, the bandwidth parameter, was assigned a value of
20/k for the spatial transform and 50/ for the temporal transform. Conclusions are not
sensitive to these arbitrarily assigned values of B, but much larger values reduce the
amplitude contrast of signals and enhance a ringing effect, and substantially smaller values
do not sufficiently resolve large-scale or low frequency waves. The transform is obtained by
taking the time-centered dot product of the wavelet and all daily consecutive overlapping
time series segments at each grid point around the globe, then applying a similar transform to
the result in longitude by the same approach. The final result is calculated at every day and
longitude grid point in the OLR data from 1975 to 2009, inclusive. The product of the result
and its complex conjugate, averaged in time and longitude, yields a power spectrum similar
to that of Wheeler and Kiladis (1999) except near wavenumber zero, where amplitude is lost
(not shown, but see Wong, 2009). We limit our analysis to wave numbers 3 and higher to
avoid the apparently degraded signals at the largest spatial scales.
Wavelet coefficients were calculated for zonal wavenumbers 3-20 and for periods
consistent with the original frequency resolution of Wheeler and Kiladis (1999), which
include all harmonics of 96 days. For convenience, we select frequencies at a given wave
number along the dispersion curves near the peaks in the normalized spectra of Wheeler and
17
Kiladis (1999), including 12, 25, and 90m equivalent depths, but for brevity, we report only
results for the MRG wave at h=25m.
3.3 Linear Regression Models
Simple linear regression is applied to diagnose coherent structures that tend to be
associated with observed wave signals (e.g., Hendon and Salby, 1994; Wheeler et al., 2000).
Using the space-time wavelet transform at a selected wavenumber and frequency, we extract
a base index time series for each grid point over a range of longitudes. Either the real or
imaginary parts of the transform work for the base index. We chose the imaginary part for
convenience of the associated zonal phasing, but the conclusions are the same regardless of
this choice. We select the geographical points from the region that shows the greatest OLR
variance in the wavenumber-frequency band of the MRG wave (Wheeler et al., 2000;
Roundy and Frank, 2004), from 160E to 170W. The time series from each of those points
serve as predictors in regression models at each grid point across a broad geographical
domain to diagnose the associated structures. One grid of regression models is calculated for
each base point time series. To illustrate, we model the variable Y at the grid point S as
Ys  Px As ,
(29)
where Px is a matrix whose first column is a list of ones and second column is the base index

at the longitude grid point x. As is a vector of regression coefficients at the grid point S. After
solving for As at each grid point by matrix inversion, (29) is then applied as a scalar equation
to diagnose wave structure by substituting a single value for the second column of Px that is
representative of a crest of a wave located at the base longitude (we set its value at +1

standard deviation). These regression models are applied to create ‘composite’
anomalies of
OLR, u and v winds, and density across the global tropics at standard pressure levels. Results
18
are calculated for the region 90 to the east and west of each base longitude, and then the set
of results from all base points are averaged to reduce noise and geographic effects.
Corresponding regressed geopotential heights are also calculated, and results are used in
plotting the other regressed data to simplify comparison with the Boussinesq solutions, which
are expressed in terms of height instead of pressure. The statistical significance of the
difference from zero of the result at each point on the map is assessed based on the
correlation coefficient (e.g., Wilks 2005), and we analyze and discuss only those regressed
signals that are deemed to be significantly different from zero at the 90% level. Since the
regression is accomplished in the time domain, some signal from wavenumbers other than
the target wavenumber can appear in results. However, we found such contributions to be
small in comparison with signals at the target wavenumber.
3.4 Comparison of Observations with the Boussinesq Wave Solutions
We estimated the best fit of the Boussinesq wave solutions to observed waves by
measuring the characteristics of regressed waves and substituting the resulting parameters
into the model. We then applied the model to predict other characteristics of the regressed
waves, and we compare those characteristics with direct measurement of the regressed
waves. The zonal wave number k and the frequency  are specified in the regressed waves. l
is estimated by measuring the vertical slopes in the longitudinal direction of anomalies of the
lower to mid troposphere in the equatorial vertical cross-sections of meridional wind in a
regressed MRG wave, then by multiplying the result by the corresponding value of k.
Meridional wind was chosen because of its strong signal at the equator relative to other
variables. The reanalysis data do not allow us to directly measure the vertical slopes of the
waves close to the surface. We applied time lag regression analysis of sounding data from
19
Tarawa island (1.35N,172.92E) to examine whether MRG wave planes approach vertical
near the surface of the earth. N2 is calculated from temperature and specific humidity data
from the NCEP/NCAR reanalysis, and is assumed to be everywhere equal to its value
averaged over the troposphere from 850 to 200 hPa and from 10N to 10S (0.0323 s-1). n is
set equal to 0 for MRG waves. Given these values of k, l, , n, and N2, we find the solution of
equation (20) for  (given 26) that yields a wave structure consistent with the selected value
of n.
After tuning the model to these observed parameters, we check whether the upward
motion implied in the solutions is consistent with the locations of the most negative OLR
anomalies in the corresponding regressed waves. We further check that model vertical
velocities are consistent with regressed vertical velocities in the NCEP/NCAR reanalysis.
Although we have low confidence in the specific magnitudes of the reanalysis vertical winds,
agreement between those winds and our results might increase confidence. Lastly, we
estimate the magnitude of the phase shift (y) at +/– 20 latitude associated with the
inclusion of the non-traditional Coriolis terms from the model, given measured values of l.
To verify these predictions, we measured zonal phase shifts from the regression maps by first
finding the longitude of the equatorial center of a gyre. Then, at 20N within the same gyre,
the corresponding phase of the wave is the point of minimum absolute value in the
meridional wind. The phase shift is calculated from the zonal distance between the point on
the equator and the point at 20N.
4. Results: Observed and Boussinesq MRG Waves
The horizontal structures of regressed MRG waves including OLR anomalies
(shaded) and 1000 hPa winds are plotted in Fig. 3 a-d for k = 3 through 6, respectively.
20
Positive and negative OLR anomalies form patterns that are antisymmetric about the equator.
Convective anomalies associated with MRG waves originate within the eastern sides of the
cyclonic portions of the circulations and extend poleward and eastward across part of the
low-level anticyclonic portions of the wave (consistent with Liebmann and Hendon, 1990
and Kiladis et al., 2009). Circulation patterns are also highly distorted toward the east with
latitude. Heavy black lines are drawn on Fig. 3 to highlight this distortion. The north-south
lines locate the equatorial centers of select gyres. Heavy slanted lines approximate the trough
or ridge axes, defined as the longitude within the gyre at a given latitude at which the
meridional wind is closest to zero. These results suggest that at 1000 hPa, zonal phase shifts
of roughly 25% of the wavelength are observed.
Figure 4 shows regressed horizontal and NCEP/NCAR reanalysis vertical winds
(shading) at 850 hPa. Comparison of the horizontal wind patterns with the corresponding
patterns in Fig. 3 suggests that meridional tilts in the horizontal wind structures are much less
pronounced near 850 hPa than at 1000 hPa. Some more substantial meridional tilts are
apparent on wavenumber 5 and 6 waves or on the western and southern portions of the
domains. Measureable eastward shifts in the horizontal winds near the center of the domain
are at most a few percent of the wavelength in the Northern Hemisphere, larger in the
Southern. We also analyzed wave structure near 500 hPa (not shown), and results suggest
even less distortion of the horizontal winds, but the vertical wind anomalies show more
meridional tilt in comparison with the horizontal wind anomalies. Some of the distortion in
the vertical wind anomalies observed at 850 hPa and above might be forced by convection
originating from the more distorted horizontal wind patterns observed closer to the surface.
21
Figure 5 shows the vertical structure of the regressed MRG wave at zonal
wavenumber 4 along 7.5N, including v’ and ’ in panel (a) and v’ and w’ in panel b.
Regressed MRG waves at other wave numbers are similar, but with wave planes
characterized by different slopes in the vertical (not shown). Negative (positive) ’ anomalies
occur with southward (northward) flow. Anomalous upward motion occurs roughly in
quadrature with anomalous ’ and v’, with maximum upward motion immediately to the
west of the region of anomalously dense air.
Figure 6 a-d shows the composite longitude-height cross-sections for equatorial
meridional wind, with northward flow shaded, for wave numbers 3-6, respectively, for
comparison with Figs (3-4). Anomalies in the eastern portion of the domain tilt eastward with
height through the troposphere, become more vertical near the tropopause, and tilt westward
with height through the stratosphere. Anomalies across the western portion of the domain
show distinctly opposite signs between the lower and upper troposphere instead of tilted
structures. These vertical dipole patterns are consistent with the first baroclinic vertical mode,
or with the structure of merdional flow associated with tropical cyclones, from which MRG
waves might disperse eastward. Such patterns might also suggest that the atmosphere
becomes unstable to moist deep convection on the spatial scales of those waves, implying
failure of (20).
Lines drawn through the tropospheric anomalies in Fig. 6 are used to estimate vertical
tilt, which is used to find l as discussed in Section 3.4. These lines are not intended to provide
specific values, but instead outline a reasonable range of values representative of the
observed waves near particular altitudes. Changes in slope near the surface are not well
resolved in Fig. 6, but a time lag regression analysis of Tarawa meridional wind sounding
22
data suggests that the slope does approach the vertical proximate to the surface (not shown).
When the slope is nearly vertical, small changes in the slope yield large changes in the
estimated value of l. Thus direct measurement can yield no more specific conclusion than
that the vertical wavelength exceeds 105m. For such slopes, we noted first that the slope was
nearly vertical (consistent with the theory for wave planes near the surface), then we
estimated the more specific value of l from equations (17-18) by first measuring the zonal
phase shifts in the regressed horizontal wind anomalies at 1000 hPa. Table I shows parameter
values estimated from regressed waves and the corresponding predictions of  and  based
on the theory (with the exception of values at 1000 hPa, for which vertical wavelength is
made more precise by applying the theoretical result after measuring ).
Values from Table I are substituted into the Boussinesq solutions for the MRG wave
to get the results plotted in Figs. 7 and 8 for 1000 and 850 hPa, respectively. Only the
wavenumber 4 MRG wave is plotted because the results for other wavenumbers are similar.
Shading in Figs. 7-8 represents upward motion. These solutions compare favorably with the
similar OLR and w anomalies in Fig. 3-4 b. The Boussinesq solutions show similar phase
relationships between u, v, w, and  anomalies to those shown in Fig. 5. Some of the
similarity between the tropospheric portions of the cross sections of the meridional wind
anomalies in regressed MRG waves east of the dateline and the Boussinesq solutions for
meridional wind is by design, since the parameter l was measured from the regressed waves
and k is pre-defined. However, these parameters do not individually predetermine the
meridional width of the solution in Fig. 6-7, nor do they individually predetermine the zonal
phase relationships between solutions for u, v, and w. Correspondence between vertical wind
anomalies in the solutions and regressed OLR anomalies of opposite sign suggest that the
23
Boussinesq waves are good analogs to the regressed observed waves, especially near the
center of the composite domain where results are most reliable. The vertically tilted solutions
of the Boussinesq waves show similar phase relationships at a given height between the
winds and buoyancy to the patterns shown from observations and reanalysis data in Fig. 5.
Accounting for vertical variation in alpha, the theory is consistent with the observation that
upward motion begins at lower levels in cyclonic flow anomalies in MRG waves, and this
upward motion tilts eastward, reaching a maximum at mid levels over the regions of
strongest poleward low level flow. Where wave planes are nearly vertical proximate to the
surface, the theory predicts that vertical motion vanishes, consistent with the surface
boundary condition. Ascent in the upper levels occurs over low-level anticyclonic flow
because of vertical tilt through the mid levels.
These results suggest that the vertical structures of observed waves are similar to the
Boussinesq MRG wave, taking into account the varying influence of convection
parameterized in  and proximity to the surface (here also parameterized in ). The most
negative OLR anomalies occur at roughly the same point in the horizontal wave structure as
low to mid level upward motion in the Boussinesq wave. The tilted vertical motion in the
Boussinesq solution is consistent with a progression from shallow convection beginning
within the low-level cyclonic flow, to deep convection located in the vicinity of low-level
poleward flow, to stratiform and cirrus in the vicinity of low level anticyclonic flow. These
patterns are consistent with previous observations of the convectively coupled MRG wave
(e.g., Kiladis et al. 2009).
24
5. Analysis of the Non-Traditional Coriolis Terms
A first glance at equation (20) reveals that this dispersion relationship does not
depend on the non-traditional Coriolis terms (consistent with Fruman 2009). Although (20)
does not include these terms, the structural solutions in equation (24) do. A tracer of
dependence on these terms is the explicit appearance of  (i.e., ignoring appearances of  in
).  appears explicitly in both  and K. Its appearance in K demonstrates that these terms
modify the meridional widths of the wave structures. The explicit dependence in  results in

a zonal phase shift in the wave structures that depends on latitude, l, N2, and  that is

expressed as a westward tilt with distance from the equator (consistent with Fruman 2009).
For example, the poleward ends of circulations in MRG waves occur to the east of the zonal
center of the disturbances on the equator. Dependence of  on  is a result of relaxation of
the hydrostatic approximation and thus was excluded from the result of Fruman (2009). 
varies more than a few percent with  in moist waves near the surface of the earth only for
periods shorter than about 4 days.

Explicit dependence of K on  disappears when (1  )N 2   2  42 . By this
interpretation, the non-traditional Coriolis terms might be irrelevant to dry waves unless the
 except proximate to the surface of the
wave frequency is close to the buoyancy frequency,
earth. Further, reduced moist stability or close proximity to the surface are required to
generate values of N2 sufficiently small for these Coriolis terms to be relevant to lower
frequency convectively coupled waves. Further, as N2 approaches  2 , K approaches zero,
implying that the equatorial trapping radius goes to zero, and the solutions to equation (20)

become undefined. Nevertheless, the solutions indicate a broad range of parameter values
25
over which the impacts of the non-traditional Coriolis terms matter, and the analysis of
observations demonstrates the anticipated impacts.
Observed vertical tilts of regressed waves characterized by particular wavenumbers
and frequencies suggest values of parameters representative of observed waves that evolve in
a manner consistent with our original assumption of static stability on the spatial scales of the
waves. These parameters are applied in Table I to diagnose the magnitudes of  and the
zonal phase shift  associated with observed waves. Values of  are of order 10-13m-2. Zonal
phase shifts associated with such values of gamma would be zero on the equator and would
range from roughly 0.1 to 30% of the zonal wavelengths at +/–20 of latitude (See Table I).
Waves propagating in less statically stable large scale environments (i.e.,  approaching 1)
would exhibit greater sensitivity to the non-traditional Coriolis terms, but wave structures
proximate to the surface of the earth show even stronger contributions from these terms.
These observed large zonal phase shifts near 1000 hPa are consistent with the near vertical
wave planes near the surface suggested in the sounding data from Tarawa and from the
theoretical contraints discussed in association with equation (24). These results demonstrate
that the non-traditional Coriolis terms strongly influence the horizontal structure of
convectively coupled equatorial waves near the earth’s surface, and since flow near the
surface influences the organization of deep convection, these terms would contribute
substantially to the overall structure of convectively coupled waves. These results do not
reveal the spread of values of  and  across the full population of waves, suggesting that
higher or lower values may be observed in association with individual wave events. Note that
equation 20 fails as 2 approaches N2. However, the difference between these two quantities
never becomes closer than 99.7% of N2 for the values of the parameters given in Table I. Yet
26
this difference is associated with a pronounced eastward distortion of the wave structure near
20 latitude equal to about 26% of the wavelength.
The zonal phase shift of distortion implied by  is not the only phase shift imposed by
the non-traditional Coriolis terms on the solutions. Although the phase shift imposed by 
applies to all fields, these terms also impose additional phase shifts between different
quantities. To illustrate, the solution for wˆ for the MRG wave is
wˆ 
i l  2ik dvˆ iklyvˆ


dy  
(30)
where

   2 l 2  k 2 2  k 2 N 2 .
(31)
In the absence of the non-traditional Coriolis terms, the quantity 2ik disappears. In

that case, the coefficients in (28) are pure imaginary. In the context of (8), these imaginary
coefficients imply that w’ and v’ are in quadrature. However, ifthe non-traditional terms are
retained and l becomes small, then a significant real component is added to the coefficients.
This real component implies a phase shift in w’ out of quatrature with v’. Similar phase shifts
occur between other quantities in the model. These phase shifts occur in both the theory and
in observations and reanalysis datasets on approach to the surface of the earth. For example,
Figure 1a shows anomalous v wind adjusting smoothly toward the vertical on approach to the
surface. However, the phase relationship between density and v changes dramatically on
approach to the surface. Although other mechanisms might also contribute to this
observation, the adjustment is consistent with the contributions of the non-traditional Coriolis
terms for small values of l.
27
6. Summary and Discussion
A linear non-hydrostatic model on the equatorial beta plane was solved for the simplest
form of its solutions for zonally and vertically propagating internal plane waves for layers of
the lower to mid troposphere. We extended the similar analysis of Fruman (2009) by relaxing
the hydrostatic approximation and by including a simple vertical structure. The dispersion
characteristics of the solutions are identical to shallow water model solutions of Matsuno
(1966) (given equation 21), and the horizontal structures are also identical except for
modifications caused by inclusion of the non-traditional Coriolis terms associated with the
horizontal component of the earth’s angular momentum (Kasahara 2003). Relaxation of
hydrostatic balance in the model reveals that the model quantity identical to equivalent depth
in the shallow water model must depend on frequency, but we demonstrated that this
dependence contributes less than 1% to the equivalent depths of observed synoptic to
planetary scale equatorial waves. Another consequence of relaxation of the hydrostatic
assumption is that Kelvin waves become dispersive at very high zonal wavenumber.
Solutions tilt in the vertical against the direction of zonal propagation. Solutions for
vertical structure cannot be compared directly with the shallow water model of Matsuno
(1966), but are consistent with solutions of similar models that have been modified to allow
density variations in the vertical (e.g., Lindzen 1974; Holton 2004; Kiladis et al. 2009). We
demonstrate that the boundary condition for zero vertical flow through the surface of the
earth can be met for internal plane waves simply by allowing the buoyancy frequency to vary
in the vertical, with a frequency of zero at the surface.
Mapes et al. (2006) considered why patterns in cloudiness associated with large-scale
waves evolve in a similar way to mesoscale convective systems, with shallow convection
28
developing first, followed by deep convection and then by upper level stratus. They
suggested that the waves are associated with patterns of anomalous density (controlled by
temperature and humidity) that tilt in the vertical in a manner that modifies the life cycles of
mesoscale convective systems embedded within. According to their view, such convective
systems would spend a greater amount of time in the stages of their life cycles that are
consistent with the local large-scale environment set by the wave (i.e., the deep convective
phases of mesoscale systems would last longer when the systems are embedded within the
portion of the wave characterized by the strongest upward motion through the mid
troposphere). Although the Boussinesq solutions do not shed light on the specific temporal
lifecycles of convective systems embedded within, the association of anomalies of nonhydrostatic pressure perturbations, buoyancy, and vertical motion on the spatial scales of the
waves would favor a general progression of convection across the wave structure from
shallow to deep convective to upper level stratus in spite of the scales over which the
convective elements are organized within the wave (i.e., both meso and microscale elements
would respond to the environment modulated by patterns of vertical motion tied to the largescale wave).
The Boussinesq theory supports this perspective because upward motion associated
with observed waves in which the non-traditional Coriolis terms are not important follows
roughly 90 of phase behind positive buoyancy anomalies that tilt in the vertical against the
direction of wave propagation. Anomalous downward flow begins to accelerate upward in
response to the advance of regions of positive buoyancy, and the first upward motion at a
given longitude occurs in the low levels, with subsidence remaining above. Such broad-scale
low-level upward motion capped by subsidence would favor development of a population of
29
convective systems dominated by shallow convection. As the wave progresses, the buoyancy
anomalies extend upward through the mid-levels, after which time model upward motion
would favor a population of convective systems characterized by enhanced moist deep
convection. Later, negative density anomalies occur in the upper troposphere with positive
density anomalies beneath, consistent with the observed adjustment toward stratiform rain.
The Boussinesq theory thus supports the perspective that the large-scale wave conditions the
environment for development of cloudiness in a progression from shallow to deep to
stratiform. Convection modifies the background stability felt by the waves through the α term
such that the waves and convection depend on each other for their overall behaviors.
Solutions of the equatorial beta plane shallow water model are frequently applied to
help students interpret the structures and propagation characteristics of convectively coupled
equatorial waves. In light of the Boussinesq model solutions and previous solutions of other
models that also depend on buoyancy as a restoring force, it is important to distinguish
observed waves of the tropical atmosphere from their single layer shallow water model
counterparts. For example, shallow water models and Boussinesq models induce vertical
motion in different ways. To illustrate, consider the MRG wave. It is interesting that both
shallow water and Boussinesq models induce upward motion over low-level poleward flow
in the MRG wave. At a given height, upward motion in observed and Boussinesq MRG
waves with occurs with cyclonic flow at any height. Since the region of anomalously low
density tilts eastward with height, the maximum upward motion at mid levels occurs roughly
over the low-level poleward flow. Upward motion in the shallow water MRG wave occurs
together with deep confluence of the wind. Variation of density with time in the Boussinesq
model implies that horizontal convergence and buoyancy are coupled, in contrast with
30
complete control of vertical motion by convergence in a shallow water model. It is thus
apparent that the shallow water model leads to the right result for the MRG wave for the
wrong reason, since observations suggest that buoyancy is a leading restoring force for the
waves. Our results thus support the previously suggested notion that density variations that
are continuous in the horizontal are required to correctly describe the vertical velocity fields
in observed convectively coupled waves that propagate upward through the lower to middle
troposphere (e.g., Holton 2004; Kiladis et al. 2009).
The Boussinesq solutions relate vertical tilts to patterns in the horizontal structures of the
waves, through inclusion of the non-traditional Coriolis terms. We tested these associations
through analysis of observed MRG waves. Regression models based on a space-time wavelet
analysis of OLR data for MRG waves characterized by specific zonal wavenumbers and
frequencies () show measurable vertical tilts above the boundary layer, with near vertical tilt
near the surface of the earth. The vertical wavenumber l is estimated from the zonal
wavenumber and the measured slope. Some of the effects of the release of latent heat in deep
convection are parameterized by  in equation (26), which is estimated from the dispersion
relationship (24), given these values of l and , and assuming the mean value of the
buoyancy frequency (N2) observed through the troposphere of the 10S to 10N band. We
estimated values of  associated with MRG waves that are characterized by different zonal
scales to be between roughly 0.9 and 1. Waves characterized by higher zonal wavenumbers
appear to be associated with higher values of  and thus greater sensitivity to the nontraditional Coriolis terms (Table I). The model suggests that the non-traditional Coriolis
terms cause eastward phase shifts with latitude in MRG wave structure. Waves that tilt more
toward the vertical show larger eastward phase shifts. The regression analysis demonstrates
31
that similar phase shifts are in fact apparent in observed MRG waves. The largest phase
eastward phase shifts in the horizontal flow are evident close to the surface of the earth,
where the theory suggests that wave planes must tilt toward the vertical. The solutions also
predict phase shifts between different quantities associated with the wave, such as between
meridional wind and buoyancy and meridional wind and vertical velocity.
For  less than about 0.95, the impacts of these Coriolis terms would be less than few
percent of the wavelength—such that their impacts would likely be statistically
indistinguishable from noise. As  approaches 1, the impacts of these terms become more
substantial. Since these terms apparently are largest close to the surface, they influence the
horizontal and vertical patterns of convection that couple to the wave (as demonstrated by
meridionally tilted regressed OLR anomalies in Fig. 3). The large zonal phase shifts with
latitude suggest that these Coriolis terms contribute substantially to convectively coupled
equatorial wave structure, and that models that include these terms should be able to better
simulate the waves.
Some caveats must be considered along with the above conclusions. Only internal
waves were analyzed. No attempt was made to analyze the effects of friction in the boundary
layer or the influence of changes in effective stability near and above the tropopause. These
model solutions do not explain wave behavior when structures similar to those of the first
baroclinic vertical mode become more prominent than structures that tilt in the vertical.
Density variations were considered without distinguishing between the separate contributions
of temperature and humidity. Further, the vertical wavenumber values were estimated from
regression analyses and were simply substituted into the Boussinesq solutions. Our results
therefore raise questions. For example, other than proximity to the surface, what processes
32
determine the vertical tilts of observed waves (including limits set by the vertical extent and
structure of moist deep convection)? Further, a broad range of estimates for the values of 
and  suggest the possibility that waves of different spatial scales might be characterized by
different efficiencies with which moist convection reduces the cooling rate associated with
moist ascent. The results do not offer any explanation for how such scale-dependent
differences might occur. In addition, although wave theories have gained broad support in the
community, distinctions between observed and theoretical waves have led some to consider
that some observed disturbances might evolve as balanced phenomena rather than waves
(e.g., Yano and Bonazzola 2009, Yano et al. 2009, Delayen and Yano 2009). Buoyancy
would not act as a restoring force in such disturbances. The Boussinesq model would only be
applicable to observed disturbances for which buoyancy is a principal restoring force. Our
regression results suggest that a subset of observed waves evolve in a manner consistent with
buoyancy as a leading restoring force.
Acknowledgments:
Discussions with George Kiladis, Stefan Tulich, John Molinari,
Matthew Wheeler, Kerry Emanuel, Andy Majda, M. Bessafi, J.-I. Yano, Boualem Khouider,
Juliana Dias, and an anonymous reviewer offered useful insight.
33
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39
List of Tables:
Table I: Parameter values for MRG waves characterized by specific zonal wavenumbers and
vertical wavelengths. The periods and zonal wavenumbers are consistent with a
shallow water model equivalent depth of 25m. The first two vertical wavelengths
listed from top to bottom for each zonal wavenumber correspond with the slopes
measured from the lines plotted on the regressed waves in Fig. 4. The third vertical
wavelength is estimated from equations (17-18) assuming the zonal phase shifts
evident in Fig. 3 for 1000 hPa flow. The right column shows the fractional change in
the equivalent depth parameter between the solution to the hydrostatic equations and
the solution to the non hydrostatic equations.
40
Table I, MRG Wave Statistics
Zonal
Period Vertical
Wavenumber (days) Wavelength
(x104m)

 (20N, as
Fraction of
Wavelength)
h

nonhydrostatic  hhydrostatic




hnonhydrostatic


3
3.8 (850 hPa)
0.9928
0.029
-3.2 x 10-5
1.5 (500 hPa)
0.9540
0.011 
-5.1 x 10-6
60 (1000 hPa)
1-2.8991x10-5
0.27
-0.008
4.5 (850 hPa)
0.9958
0.041
-4.9 x 10-5
1.2 (500 hPa)
0.9405
0.011
-3.5 x 10-6
50 (1000 hPa)
1-3.4491x10-5
0.2885
-0.006
2.9 (850 hPa)
0.9894
0.026
-1.7 x 10-5
1.1 (500 hPa)
0.9262
0.0097
-2.5 x 10-6
40 (1000 hPa)
1-5.597x10-5
0.26
-.0033
5.0 (850 hPa)
0.9964
0.043
-4.5 x 10-5
0.92(500 hPa)
0.8923
0.008
-1.5 x 10-6
30 (1000 hPa)
1-5.714x10-5
0.26
-0.0028
4
5
6
4.67
4.98
5.30
5.63
Table I: Statistics of the h=25m equivalent depth MRG wave. Zonal wavenumber and period
are specified for the regressed waves. Vertical wavelengths less than 105m were estimated by
measuring the slopes of the regressed waves. Wavelengths larger than 105m were estimated
from the theory by measuring meridional phase shifts at 1000 hPa. All other values of  and
 were estimated by substituting measured vertical wavelengths into equations (20) and (1718).
41
List of Figures
Figure 1: Dispersion curves in the zonal wavenumber-frequency domain for solutions to
equation (24), as expressed in equation (24), characterized by the values of N and
vertical wavelength given in the figure title. The dashed solution has been deemed
unphysical, consistent with Matsuno (1966). Zonal wavenumber is the circumference
of the earth divided by the zonal wavelength.
Figure 2 Wavenumber-frequency spectrum of OLR anomalies from 15N to 15S. Results
are normalized by dividing by an estimated red background. Dark shades represent
power above the background. Dispersion curves from equatorial beta plane shallow
water theory are included for reference, for equivalent depths of 8 and 90m.
Figure 3 Regressed OLR (shading) and vectors for the horizontal wind (1000 hPa), based on
wavelet analysis of OLR anomalies at specific wavenumbers within the MRG band
along the shallow water dispersion curve at the equivalent depth of 25m. Negative
OLR anomalies consistent with enhanced moist deep convection are shaded. The
contour interval is 1 Wm-2. Panels a-d show results for zonal wavenumbers 3-6,
respectively. The longest wind vector is roughly 0.5 ms-1. Anomalies are small
relative to regressed MRG waves reported by Kiladis et al. (2000) because their
results included signal from a broad range of neighboring wave numbers and
frequencies.
Figure 4, Regressed NCEP/NCAR reanalysis w wind (shading) and vectors for the horizontal
wind, based on wavelet analysis of OLR anomalies at specific wavenumbers within
the MRG band along the shallow water equivalent depth of 25m. For comparison
with Fig. 3, wind anomalies are plotted at 850 hPa instead of 1000 hPa. Upward
42
motion is shaded. The contour interval is 0.2 cm s-1, with contours beginning at 0.6
cm s-1.
Figure 5 Vertical structure of the regressed MRG zonal wavenumber 4 MRG wave at 7.5N.
Panel a shows v wind (shaded) with density anomalies contoured (positive in red,
with the contour interval on the top right). Panel b shows v wind (shaded), with
NCEP/NCAR reanalysis vertical wind in contours (anomalous upward motion in red,
with the contour interval listed on the top right). k is expressed in terms of global
wavenumber.
Figure 6 Regressed equatorial v wind anomalies plotted with respect to height corresponding
to the MRG-wave regression results shown in Figs. 3 - 4. Lines drawn on some
anomalies were used to estimate vertical slope, and resulted in the estimates for
vertical wavelength written on the diagrams (in m). Positive values are shaded, the
zero contour is omitted, and the maximum amplitude is close to 0.5 ms-1. The
horizontal axis represents degrees of longitude with positive values east of the base
longitude.
Figure 7 u and v winds (vectors) and w at 96m for MRG wave solutions near zonal
wavenumber 4. l is expressed in terms of vertical wavelength.
Figure 8: Same as Figure 7, except for at 850 hPa, assuming the vertical wavelength in the
figure title.
43
Figure 1: Dispersion curves in the zonal wavenumber-frequency domain for solutions to
equation (20), characterized by the values of N and vertical wavelength given in the figure
title. Specific wave modes represented by different curves are labeled. The dashed curve has
been previously deemed unphysical because the associated zonal wind signal does not vanish
with latitude (Matsuno 1966).
44
Figure 2 Wavenumber-frequency spectrum of OLR anomalies from 15N to 15S, calculated
by Fourier transform. Results are normalized by dividing by an estimated red
background. Dark shades represent power above the background. Dispersion curves
from equatorial beta plane shallow water theory are included for reference, for
equivalent depths of 8 and 90m.
45
Figure 3 Regressed OLR (shading) and vectors for the horizontal wind (1000 hPa), based on
wavelet analysis of OLR anomalies at specific wavenumbers within the MRG band
along the shallow water dispersion curve at the equivalent depth of 25m. Negative
OLR anomalies consistent with enhanced moist deep convection are shaded. The
contour interval is 1 Wm-2. Panels a-d show results for zonal wavenumbers 3-6,
respectively. The longest wind vector is roughly 0.5 ms-1. Anomalies are small
relative to regressed MRG waves reported by Kiladis et al. (2000) because their
results included signal from a broad range of neighboring wave numbers and
frequencies.
46
Figure 4, Regressed NCEP/NCAR reanalysis w wind (shading) and vectors for the horizontal
wind, based on wavelet analysis of OLR anomalies at specific wavenumbers within the MRG
band along the shallow water equivalent depth of 25m. For comparison with Fig. 3, wind
anomalies are plotted at 850 hPa instead of 1000 hPa. Upward motion is shaded. The contour
interval is 0.2 cm s-1, with contours beginning at 0.6 cm s-1.
47
Figure 5 Vertical structure of the regressed MRG zonal wavenumber 4 MRG wave at 7.5N.
Panel a shows v wind (shaded) with density anomalies contoured (positive in red, with the
contour interval on the top right). Panel b shows v wind (shaded), with NCEP/NCAR
reanalysis vertical wind in contours (anomalous upward motion in red, with the contour
interval listed on the top right).
48
Figure 6 Regressed equatorial v wind anomalies plotted with respect to height corresponding
to the MRG-wave regression results shown in Figs. 3 - 4. Lines drawn on some
anomalies were used to estimate vertical slope, and resulted in the estimates for
vertical wavelength written on the diagrams (in m). Positive values are shaded, the
zero contour is omitted, and the maximum amplitude is close to 0.5 ms-1. The
horizontal axis represents degrees of longitude with positive values east of the base
longitude.
49
Figure 7 u and v winds (vectors) and w at 96m for MRG wave solutions near zonal
wavenumber 4.
50
Figure 8: Same as Figure 7, except for at 850 hPa, assuming the vertical wavelength in the
figure title.