## 1. Introduction

Tropical rainfall is typically organized by waves that move in the zonal direction close to the equator (Wallace and Chang 1972; Takayabu 1994; Wheeler and Kiladis 1999). While idealized studies of these zonally propagating waves are often based on shallow-water models (e.g., Matsuno 1966; Gill 1980; Silva Dias et al. 1983), many theories exist for how to include the coupling between wave dynamics and moist convective processes (e.g., Lindzen 1974; Emanuel et al. 1994; Neelin and Zeng 2000; Mapes 2000; Lindzen 2003; Raymond and Fuchs 2007). In this paper, we use a linear shallow-water model with a crude parameterization for large-scale precipitation anomalies to study the behavior of moist equatorial waves in terms of their propagation speed, amplitude, and horizontal structure.

From the observational standpoint, spectral analysis of satellite cloudiness data suggests a strong connection between equatorial waves and moist convection (Takayabu 1994; Wheeler and Kiladis 1999). In particular, the wavenumber–frequency power spectrum of tropical cloudiness data, after removing an estimate of the background power, reveals spectral peaks that tend to line up with the equatorially trapped shallow-water dispersion curves (Matsuno 1966). This result is illustrated by the diagram shown in Fig. 1, which is adopted from Kiladis et al. (2009) showing an updated version of the diagram introduced by Wheeler and Kiladis (1999). These propagating disturbances are usually called convectively coupled equatorial waves (CCEWs) because they possess spectral characteristics similar to Matsuno's modes, and composite analyses have demonstrated that their horizontal structures are also in reasonable agreement with that expected from theory (Wheeler et al. 2000). In addition, spectral peaks consistent with CCEWs are also seen in dynamical variables such as geopotential and zonal winds (Hendon and Wheeler 2008; Gehne and Kleeman 2012). Aside from CCEWs, enhanced power is also seen in association with the planetary-scale Madden–Julian oscillation (MJO). However, unlike CCEWs, spectral signals of the MJO do not match to any linear wave mode, implying that the dynamics of the MJO is more complex than that of CCEWs.

An important difference between the waves observed in cloudiness data and the theoretical dry equatorial waves is that the CCEWs spectral peaks imply equivalent heights in the range 12–50 m. This is much shallower than the peak projection response to deep convective heating (e.g., Fulton and Schubert 1985), which implies an equivalent height of about 200 m. The concept of equivalent height comes from a vertical modal decomposition of linearized atmospheric motion equations, and depends essentially on the vertical mode and the static stability of the atmosphere [see a review in the context of equatorial waves in Kiladis et al. (2009)]. Recent cloud-resolving model studies (Tulich et al. 2007; Tulich and Mapes 2008; Kuang 2010) and theoretical work (Mapes 2000; Majda and Shefter 2001; Khouider and Majda 2006; Raymond and Fuchs 2007; Kuang 2008) suggests that the observed shallow equivalent heights are a result of convection interacting with relatively short vertical wavelengths at low levels. In this view, shallower equivalent heights are dominant in the case of CCEWs because the wave structure is triggered by low-level moist convective processes (e.g., shallow convection) that typically lead deep convection. This theory is an alternative to an earlier idea in which convection can interact with deeper vertical wavelengths through compensation between latent heat release and adiabatic cooling within deep convective regions. In this theory, the gross moist stability (GMS) sets the wave speed (Neelin and Held 1987; Neelin et al. 1987; Neelin and Yu 1994; Emanuel et al. 1994; Frierson et al. 2004; Raymond et al. 2009). Specifically, the effect of convection on large-scale waves is to reduce the effective static stability of the atmosphere, which, in turn, reduces the phase speed of the waves, along with altering their scale and horizontal structure (Neelin and Zeng 2000; Frierson et al. 2004; Kiladis et al. 2009). Although determining the dominant mechanism that controls the propagation of CCEWs is beyond the scope of the present work, it is worthwhile noting that the two theories for the phase speed of CCEWs are not mutually exclusive. For instance, it is possible that CCEW phase speeds are set by the second baroclinic mode, and then modulated by first baroclinic GMS effects. Conversely, observed phase speeds may be primarily due to GMS associated with the first baroclinic mode, being then modulated by the higher modes.

Not surprisingly, CCEWs in general circulation models (GCMs) are very sensitive to convective and other physical parameterization schemes, which makes GCMs difficult to use in order to access the physical mechanisms underlying CCEW dynamics. For instance, a model that uses a convective parameterization that does not reproduce low-level convective heating adequately will tend to develop CCEWs with deeper vertical wavelengths and faster phase speeds in comparison to observations. To better understand these issues, studies such as Lin et al. (2008), Frierson et al. (2011), and Benedict et al. (2013) have investigated the impacts of the convective parameterization on simulated CCEWs, showing a tight relationship between equivalent height and GMS in both cases of mass flux and moisture convergence convective schemes (e.g., Frierson et al. 2011, their Fig. 9). In particular, Frierson et al. (2011) show that GMS from the first baroclinic mode seems to play a dominant role in explaining the phase speed of their modeled Kelvin waves (KWs). On the other hand, Straub et al. (2010) show that GCMs with well-simulated KWs tend to have realistic second-baroclinic-mode structures.

Modeling aside, while the observed power spectrum of observed CCEWs is spread over a rather narrow range of equivalent heights between 12 and 50 m for any mode considered, there is observational evidence that CCEW phase speeds are modulated by the background environment. For instance, Yang et al. (2007b) show that KWs propagating over the Eastern Hemisphere (EH) propagate more slowly than over the Western Hemisphere (WH), and both Roundy (2008) and Roundy (2012) show that KW phase speeds are modulated by the MJO, varying from about 11 to 17 m s^{−1}. Dias and Pauluis (2011) provide some observational evidence along with theoretical arguments that KW phase speeds should be related to the position and meridional extent of the intertropical convergence zone. Because there is also observational evidence that GMS varies in time and space (Yu et al. 1998; Back and Bretherton 2006), the present work is based on the hypothesis that CCEW phase speed anomalies are controlled by the large-scale background moisture. Specifically, we use an idealized model to study how small perturbations of background moisture affect the structure, phase speeds, and amplitude of equatorial waves. The model consists of a single shallow-water system with a variable coefficient representing a variable equivalent height. The model is based on a GMS argument, but we allow the effective static stability of the atmosphere to vary with longitude, mimicking large-scale zonal background moisture variations, such as a moister EH in comparison to the WH. The underlying assumption is that there is a first-order mechanism that reduces the equivalent height from the theoretical dry value (~200 m) to a uniform moist value (~25 m). We then investigate the secondary circulation that develops as a result of a small perturbation around this reduced equivalent height. Our analysis demonstrates that the secondary flow depends not only on the scale of the background perturbation, but is also strongly first-order-mode dependent. We show that this approach can provide insight into the coupling between moisture and wave dynamics in the context of our particular choice of parameterization that is also consistent with observations of CCEWs. Moreover, since GCMs are frequently based on GMS concepts, this framework can be used to interpret some of the behavior of these models. We return to these issues in the final section.

The paper is organized as follows. The modeling framework and the asymptotic solutions are presented in sections 2 and 3. Section 4 documents the horizontal structure of the asymptotic solutions in the case of planetary-scale variations of the equivalent height. In section 5, we investigate the relationship between the varying equivalent height and the coupled wave divergence. Section 6 discusses the similarities between our idealized coupled waves and observed CCEWs. The final section summarizes and discusses the main results.

## 2. Modeling framework

*u*,

*υ*) and temperature

*T*and the main parameters are the gravity wave speed

*H*

_{eq}, and a heat source

*Q*. The parameter

*β*= 2Ω/

*R*is fixed, where Ω and

*R*are Earth's angular velocity and radius. Unless stated otherwise, we assume

*c*≈ 16 m s

_{m}^{−1}, which is based on the dispersion diagram shown in Fig. 1, where

*H*

_{eq}= 25 m appears to be the best fit to the data. The model used here is based on a Galerkin truncation of the linearized Boussinesq equations into a finite set of vertical modes, where a single baroclinic mode is retained. In the context of the tropical atmosphere, Matsuno (1966) derived a complete set of orthonormal modes for (1) with

*Q*= 0, which are typically denoted equatorial shallow-water modes, or “Matsuno modes.” For reference, these solutions are detailed in appendix A. An important feature of these normal modes is that they are zonally propagating and their amplitude decays with distance from the equator. The trapping scale is proportional to

*H*

_{eq}, moist convection not only acts to slow down the waves, but also causes the wave structures to become more trapped about the equator. To further explore how geographical variations of the moisture background affect equatorial modes, we make the following approximation:where

*ε*is a small positive constant. By conservation of mass, (2) implies that diabatic heating is proportional to vertical motion, and modulated by some background profile

*h*(

*x*), here representing geographic variations of the background moisture. This assumption is based upon the effective static stability concept (Neelin and Held 1987), and within our modeling framework, it is analogous to considering a shallow-water model with a varying moist gravity wave speedwhere

*h*is prescribed. To simplify the notation, we nondimensionalize the system using the equivalent gravity wave speed

*c*and the equatorial value of

_{m}*β*= 2.28 × 10

^{−11}m

^{−1}s

^{−1}, which yields the equatorial length scale

*T*=

_{E}*L*/

_{E}*c*≈ 14 h. The nondimensionalization yieldswhere we defined

_{m}*g*(

*x*) = g

*h*(

*x*)/

*c*, and the parameter

_{m}*ε*is such that

*ε*

*g*(

*x*) ≪ 1.

## 3. Asymptotic solutions

*ε*= 0) problem, the system of equations in (4) can be expressed aswith the following definitionsAssuming that

*ε*is small, we look for asymptotic solutions of the formwhere (

**X**

_{0},

*ω*

_{0}) are an eigenvector and eigenvalue of the unperturbed system for a fixed

*m*=

*m*

_{0}and

*k*=

*k*

_{0}(details are given in appendix A). We then need to obtain the pair (

**X**

_{1},

*ω*

_{1}), which represents the secondary flow and frequency adjustment due to the varying

*H*

_{eq}. Taking the time derivative in (10) yieldsDenoting

**Λ**

_{0}=

*iω*

_{0}and

**Λ**

_{1}=

*iω*

_{1}, we have a perturbation eigenvalue problem (Kato 1995)The leading order term in (12) is solved exactly:and the

*O*(

*ε*) system isorNote that because

**X**

_{0}spans the kernel of

**Λ**

_{0},

**Λ**

_{1}has to remove the projection of

**X**

_{0}onto

**X**

_{0}so that the right-hand side of (15) does not project onto

**X**

_{0}. To determine

**Λ**

_{1}, notice that in the case where

**X**

_{0}is a free normal mode we havewherethus,where

*P*

_{0}is the projection onto

**X**

_{0}. Applying Fourier transform in

*x*,

**Λ**

_{1}is determined explicitly as detailed in appendix B. Notice that by Fourier transforming (15), its right-hand side can be further decomposed using normal modes from the unperturbed system, here denoted by

**X**

_{1}is bounded, the denominator in (18) suggests that singularities may exist. To study these singularities we first investigate the parameter space (

*m*,

*k*) where the denominator in (18) is close to zero. This parameter space is relatively simple because when

**X**

_{1}has no projection onto

*m*=

*m*

_{0}− 2,

*m*

_{0}, and

*m*

_{0}+ 2; therefore,

**X**

_{1}is made up of a superposition of only these meridional modes. It can also be shown that in the case where

*g*(

*x*) = sin(

*lx*) (

*l*is the background scale),

*k*=

*k*

_{0}−

*l*and

*k*=

*k*

_{0}+

*l*. In addition, we consider only the cases where

*k*>

*l*because we are interested in how large-scale moisture affects equatorial waves. Given these constraints, Fig. 2 summarizes the potential singularities by displaying the minimum distance between

**Λ**

_{0}among all cases where

*k*=

*k*

_{0}−

*l*,

*k*

_{0}+

*l*, and

*m*=

*m*

_{0}− 2,

*m*

_{0},

*m*

_{0}+ 2):Each panel in Fig. 2 corresponds to a different initial mode

**X**

_{0}as defined in Table 1; thus, for each

**X**

_{0},

*d*

_{0,1}is a function of only the secondary flow wavenumber

*k*and the moisture background scale

*l*. Figure 2 shows that while the qualitative behavior of

*d*

_{0,1}as a function of wavenumber is similar when comparing

*l*= 1, 2, and 3 for each mode, the magnitude of

*d*

_{0,1}is sensitive to the initial mode

**X**

_{0}. For example,

*d*

_{0,1}in the case of KWs is larger at low wavenumbers because within this region the KW dispersion curve is well separated from those of the other modes (e.g., MRGs, ERs, etc.). In contrast, at high wavenumbers the dispersion curves for ERs (

*m*

_{0}= 1, 2, 3, …) are relatively close to one another; hence,

*d*

_{0,1}is expected to be small. Interestingly, using the magnitude of

*d*

_{0,1}as a metric for the suitability of the asymptotic solution, the wavenumbers corresponding to the observed spectral peaks (see spectral peaks in Fig. 1) coincide with where the

*d*

_{0,1}is larger. For example, KW, ER, MRG, and EIG0 spectral peaks are located at relatively low wavenumbers and

*d*

_{0,1}decreases with increasing

*k*

_{0}in these cases, whereas the spectral peak associated with WIG1s is at a larger wavenumber (

*k*

_{0}≈ 12) and

*d*

_{0,1}increases with increasing

*k*

_{0}.

Summary of Matsuno's shallow-water normal modes: Kelvin (KW), mixed Rossby–gravity (MRG), eastward inertio-gravity (EIG), westward inertio-gravity (WIG), and equatorial Rossby (ER) waves.

*d*

_{0,1}, the amplitude of the secondary flow also depends on the magnitude of the numerator in (18). To illustrate this effect, Fig. 3 shows the total secondary wave energy defined aswhere the angle brackets represent integration over the entire domain. The first-order wave energy is set to

*E*

_{0}(

*k*,

*l*) = 1 in all cases. Similarly to Fig. 2, each panel in Fig. 3 corresponds to the initial eigenmodes defined in Table 1. Figure 3 shows that the magnitude of the secondary circulation is not very sensitive to the initial wavenumber in the case of KWs, while it decreases with increasing wavenumbers for MRGs, EIG0s, EIG1s, and WIG1s and it increases with increasing wavenumbers for ER1s. The ER1 amplitude in Fig. 3 goes off the scale at planetary wavenumbers

*k*≈ 12–14, which, along with the small values of

*d*

_{0,1}, suggest that the solution has singularities at those wavenumbers. However, the precise wavenumber location of these singularities shifts depending on the choice of

*H*

_{eq}.

Interestingly, the amplitude of *E*_{1} strongly depends on the initial mode. For instance, within the range of observed scales, KWs and MRGs develop a secondary flow of approximately the same order of magnitude as *E*_{0}. Meanwhile, EIG0s, EIG1s, WIG1s, and ER1s develop a secondary flow 5–10 times larger than *E*_{0}, which suggests that our asymptotic expansion is more suitable to the former modes. Nevertheless, Figs. 2 and 3 show that the asymptotic solutions are bounded within the range of observed scales of CCEWs and, while the figures are shown only in the case *H*_{eq} = 25 m, we found that this overall behavior is robust within the range *H*_{eq} = 12–50 m. Because the asymptotic solutions are well behaved at the scales of interest, we now turn to the analysis of the structure of the secondary flow when *g*(*x*) = sin(*lx*), focusing on first-order modes consistent with the observed zonal scales highlighted in Fig. 1.

## 4. The primary and secondary flow

Briefly summarizing the model parameters, the choice of the first-order wave **X**_{0} fixes the parameters (*m*_{0}, *k*_{0}) corresponding to the first-order wave meridional mode and zonal wavenumber (see appendix A). The first-order wave frequency *ω*_{0} is determined from the dispersion relation (A2). The background is defined by the parameter *l* where a moister (drier) region corresponds to longitudes where *g*(*x*) = sin(*lx*) > 0 (<0). Given the set of parameters {*m*_{0}, *k*_{0}, *l*}, the secondary solution **X**_{1} is determined explicitly (details in the previous section and appendix B). The full solution depends on the arbitrarily small constant *ε*, which we chose such that *ε**E*_{1} = 0.1*E*_{0}, and the coupled flow refers to **X** = **X**_{0} + *ε***X**_{1}.

Here we focus on the two idealized backgrounds that are illustrated in Fig. 4. The first configuration (Fig. 4a) resembles the zonal extent of deep convection patterns over the Indian Ocean and western Pacific in contrast to the relatively drier WH. Because this configuration misses peaks in deep convection over the WH such as the Amazon basin we also consider the case where *l* = 2 (Fig. 4b). We detail the secondary flow response in the cases *l* = 1 and 2 for the six primary modes shown in Table 2. Note that these particular choices are a compromise between satisfying the scale separation *k* > *l* and the location of the observed spectral peaks.

Summary of the modes that are illustrated in section 4.

### a. Horizontal structure

#### 1) KW: *m*_{0} = −1, *k*_{0} = 4, *l* = 1

The phasing between *u*_{1} and *u*_{0} for the KW (Fig. 5a) indicates that the coupled zonal wind is enhanced within moister regions and attenuated within the drier regions, with *u*_{1} maximized where the background zonal gradients are zero. In this case *u*_{1} is about an order of magnitude smaller than *u*_{0}. Figure 5c shows that the KW develops a weak secondary meridional circulation *υ*_{1}, which is about one order of magnitude weaker than *u*_{1}. Thus, moisture variations in this context alter the “pure Kelvin” character of the mode, as was also seen by Dias and Pauluis (2009) through interaction with a moisture interface. This aspect will be discussed further below. As with *υ*_{1}, the maximum amplitude of *T*_{1} (Fig. 5e) is not in antiphase with *u*_{1} and lies near the zero crossings of *g*(*x*), and its maximum amplitude is also an order of magnitude weaker than *u*_{1}. It turns out that the zonal structure of the secondary flow is made up of a superposition of pure modes of *m* = 1 (e.g., ER1, EIG1, and WIG1), so the interaction of the Kelvin mode with the background moisture field can be viewed as exciting those modes.

#### 2) EIG0: *m*_{0} = 0, *k*_{0} = 4, *l* = 1

As opposed to KWs, the right panels of Fig. 5 show that the EIG0 secondary flow is in near quadrature with the primary flow. The secondary flow has a strong *k* = *k*_{0} + *l*_{0} = 5 and *m* = 0 component, but its amplitude is modulated by *k* = *k*_{0} − *l*_{0} = 3. All fields are of comparable amplitude and their maxima are near the zero crossings of *g*(*x*)—that is, at the transition zones in the moisture field. Near the maximum background moisture anomalies (moister regions), the secondary flow *u*_{1} acts to weaken *u*_{0} (Fig. 5b) while *T*_{1} strengthens *T*_{0} amplitudes (Fig. 5f). Meanwhile the converse effect is observed near minima of *g*(*x*) (drier regions). In addition, the meridional structure of *υ*_{1} and *υ*_{0} are very similar (Fig. 5d). In fact the *m* = 0 mode seems to dominate the structure of the secondary flow.

#### 3) MRG: *m*_{0} = 0, *k*_{0} = −4, *l* = 1

The left panels in Fig. 6 show that for MRG waves *u*_{1} is roughly in phase with *u*_{0} (Fig. 6a) and *υ*_{1} is roughly in antiphase with *υ*_{0} (Fig. 6c) within the moister region; thus, the secondary flow acts to reinforce the zonal velocity while weakening the meridional circulation. The converse effect is seen within the drier region. In addition, *T*_{1} is nearly in phase (antiphase) with *T*_{0} (Fig. 6e) within the moister (drier) region, so that the coupled temperature amplitude is enhanced (attenuated) within moister (drier) regions. All fields are of comparable amplitude and, unlike EIG0s, their maxima are near the extreme values of *g*(*x*). Similarly to EIG0s, the *m* = 0 mode seems to dominate the structure of the secondary flow.

#### 4) ER1: *m*_{0} = 1, *k*_{0} = −4, *l* = 1

The right panels in Fig. 6 show evidence of projection onto higher meridional modes (*m* = *m*_{0} + 2 = 3) in the ER1 secondary flow. Owing to the more complex superimposition of modes, the phasing between the secondary flow and first-order flow is not as clear as in the previous cases. For instance, near the equator the ER1 secondary flow is in near quadrature with the primary flow. However, farther from the equator, *υ*_{1} (Fig. 6d) and *T*_{1} (Fig. 6f) are roughly in phase (antiphase) with *υ*_{0} and *T*_{0} where *g*(*x*) > 0 [*g*(*x*) < 0]; meanwhile, *u*_{1} (Fig. 6b) is roughly in antiphase (phase) with *u*_{0} where *g*(*x*) > 0 [*g*(*x*) < 0].

#### 5) EIG:* m*_{0} = 1, *k*_{0} = 8, *l* = 2

The left panels in Fig. 7 show that, similarly to EIG0s, all EIG1 fields are of comparable amplitude and their maxima are near the zero crossings of *g*(*x*). Also similar to EIG0s, the meridional structure has a clear *k* = *k*_{0} + *l*_{0} = 10 and *m* = 1 component resembling the first-order flow. Close to the maximum (minimum) *g*(*x*) and nearby the equator, the secondary flow acts to strengthen (weaken) *u*_{0} (Fig. 7a) while weakening (strengthening) *T*_{0} (Fig. 7e). At the same time, *υ*_{1} (Fig. 7c) is roughly in quadrature with *υ*_{0}. The *m* = 1 mode seems to dominate the structure of the secondary flow.

#### 6) WIG:* m*_{0} = 1, *k*_{0} = −8, *l* = 2

Unlike EIG1s, the right panels in Fig. 7 indicate that the amplitude of the WIG1 secondary flow is larger nearby the extreme values of *g*(*x*). In this case, near the maximum *g*(*x*), the secondary flow acts to strengthen *u*_{0} (Fig. 7b) while weakening (strengthening) *υ*_{0} (Fig. 7d) and *T*_{0} (Fig. 7f) because, as opposed to EIG1s, *u*_{1} is roughly in phase with *u*_{0}, and *υ*_{1} and *T*_{1} are in antiphase with *υ*_{0} and *T*_{0}. Like EIG1s, the *m* = 1 mode seems to dominate the structure of the secondary flow.

### b. Phase speed variability

The variable coefficient in (4) implies that the speed of propagation of its wavelike solutions should follow the prescribed modulation shown in (3), and because of our choice of asymptotic expansion (10), the coupled flow exhibits this behavior by construction. For example, Fig. 8 displays a time–longitude section of the KW divergence field for *l* = 1. The left panel shows that the secondary low-level divergence is minimized when the background moisture amplitude is close to zero [i.e., where *g*(*x*) = 0]. The coupled low-level divergence field is shown on the right panel and confirms that the total flow speeds up within drying regions [*g*′(*x*) < 0] and, conversely, it slows down within moistening regions [*g*′(*x*) > 0]. This modulation of the coupled flow speed is observed for all modes (not shown); however, a drawback of our asymptotic approach is that the magnitude of this modulation depends on the arbitrary choice of *ε*. That said, by setting *ε**E*_{1} = 0.1*E*_{0}, it is implied that for modes with larger *E*_{1}, *ε* will be smaller and the coupled flow speed has to vary less. By this argument, our results imply that at the observed scales, the modeled KW and MRG phase speeds vary more as they propagate from moister to drier regions than the other modes analyzed in the previous section.

## 5. Modulations of the coupled divergence field

*g*(

*x*) > 0, then low-level convergence can be interpreted as regions more favorable to moist convection. It is therefore instructive to investigate whether the secondary circulation reinforces or weakens the coupled divergence, which essentially depends on the phasing between the first- and second-order divergence fields. Interestingly, based on governing equation (4), we can anticipate some mode variability in the modulation of the divergence field. To simplify the argument, here we approximate the moisture background by a step function:that is, a moister region to the left of a drier region as it is illustrated in the schematic in Fig. 9a. Focusing on a primary flow convergence zone, the difference in temperature tendencies in each side of the moister–drier interface imply that the moister portion of the domain cools off slightly more slowly than the drier portion, which then induces a secondary zonal circulation from drier to moister (represented by the black solid arrow in Fig. 9b). Note that the secondary circulation is zonally convergent (divergent) and meridionally divergent (convergent) near the moister (drier) side of the interface (Fig. 9b). As result, whether the coupled divergence is amplified, or not, depends on whether the secondary divergence is dominated by its meridional or its zonal component. For instance, in the case of KWs,

*υ*

_{1}is much weaker than

*u*

_{1}; therefore, KW coupled divergence has to be enhanced (attenuated) to the left (right) of the interface. The MRG secondary flow is very similar to its primary structure, which is dominated by its meridional flow. Thus, in the case of MRGs, the coupled divergence is attenuated (enhanced) to the left (right) of the interface. In both cases, the argument is based upon the following first-order approximation:While this approximation has its merit because it provides a mechanism for the coupled divergence modulation, it holds only near the interface and is less applicable when the secondary flow structure is more dissimilar from the primary flow, as for example in the case of higher meridional modes. These issues are further illustrated next.

Figures 10–12 show the divergence modulation in the case of the six examples shown in section 4. Overall, comparison among the top panels indicates that, in agreement with our prediction, the phasing between the first- and second-order divergence varies strongly mode by mode. Specifically, the impact of **X**_{1} onto the coupled divergence can be seen in the middle and bottom panels of Figs. 10–12, showing that not only the amplitude, but also the horizontal structure of the coupled divergence within moister and drier regions differ. The middle panels show the coupled divergence and the bottom panels are similar, except that we use a 3-times-larger *ε* to better visualize the changes. Note that for KWs (left panels in Fig. 10), and as predicted, the secondary divergence is dominated by the ∂_{x}*u*_{1} component, which is similar to ∂_{x}*u*_{0} in terms of meridional structure; hence, apart from the amplitude modulation, the coupled divergence is fairly similar to the primary divergence. EIG0s and MRGs exhibit similar behavior. Because of the stronger *m*_{0} + 2 component in the case of ER1s, EIG1s, and WIG1s, changes in the coupled divergence are more clear, and in particular the coupled divergence develops stronger north–south asymmetries.

_{1}, m/m), moist/drying (R

_{2}, m/d), dry/drying (R

_{3}, d/d), and dry/moistening (R

_{4}, d/m), which are defined in Table 3. Specifically,where

**∇**⋅

**V**=

**∇**·

**V**

_{0}+

*ε*

**∇**⋅

**V**

_{1}and the parameter

*ε*varies mode by mode because

*ε*= 0.1

*E*

_{0}/

*E*

_{1}. Positive (negative)

*r*represents amplification (attenuation) of the coupled divergence in comparison to the free divergence.

_{i}Definition of four subregions depending on the background moisture anomalies.

The amplitudes shown in Figs. 13 and 14 highlight the difference in behavior among modes, while keeping the ratio between *E*_{0} and *ε**E*_{1} constant. In the case of KWs (Fig. 13a) the divergence enhancement occurs within moister backgrounds (i.e., regions R_{1} and R_{2}), whereas the divergence is attenuated within drier backgrounds (i.e., regions R_{3} and R_{4}). The pattern for EIG0 is nearly identical to that of the KW, but much weaker (Fig. 13c). MRGs show the opposite effect (Fig. 13b), with attenuation (amplification) within moister (drier) backgrounds. ER1 divergence is amplified everywhere (Fig. 13d), but more strongly within the moister (R_{1} and R_{2}) regions. Also, note that ER1s and MRGs develop the largest enhancements of the divergence. Importantly, the diagrams displayed in Tables 4 and 5 show *r _{i}* for a range of parameters

*k*

_{0},

*l*and the basic state

*H*

_{eq}, demonstrating that the modulation described above is robust except in the case of ER1s where divergence modulations are fairly sensitive to all parameters (see Tables 4 and 5).

Percentages corresponding to the ratio defined in (23) for *l* = 1 and *H*_{eq} = 12, 25, and 50 m and *k*_{0} = 1−8 for KW, EIG0, MRG, and ER1. Negative values are in boldface.

As in Table 4, except that *l* = 2.

Figure 14 shows that for EIG1 and WIG1 *k*_{0} = 8 and *l* = 2 the secondary circulation has a much weaker effect in enhancing the coupled divergence in comparison to the modes shown in Fig. 13, especially in the case of EIG1s (note the change in the scale), even in the case with *l* = 1 (see Table 6). Similarly to KWs and EIG0s, the secondary EIG1 flow enhances (attenuates) divergence within the moister (drier) regions. While differences are small, the enhancement–attenuation is larger in the drying phase of the background (i.e., regions R_{2} and R_{3}). In contrast, and similarly to MRGs, the secondary WIG1 flow attenuates (enhances) divergence within the moister (drier) regions. As opposed to EIG1s, and noting that differences are small, the enhancement attenuation is larger in the moistening phase of the background (i.e., regions R_{1} and R_{4}).

As in Table 5, but for EIG1 and WIG1.

Interestingly, Figs. 13 and 14 also demonstrate that the divergence modulation is much more sensitive to the sign of the moisture anomaly than to the sign of its gradient because the budgets within R_{1}–R_{2} and R_{3}–R_{4} are nearly the same.

## 6. Relationship to observed CCEWs

As mentioned in the introduction, studies such as Roundy (2008), Yang et al. (2007b), Dias and Pauluis (2011), and Roundy (2012), using different approaches, have found that KWs tend to propagate more slowly in the EH in comparison to the WH. This is consistent with our modeling framework because the EH tends to be moister than the WH, but this is not the only similarity to observed CCEWs phase speeds. Yang et al. (2007b) estimated CCEW phase speeds based on both their convective and associated circulation signals (zonal or meridional winds). While their estimated phase speeds (see their Table 1) at upper levels are consistent with the convective phase speed and tend to be slightly slower than phase speeds at lower levels, all modes that were analyzed (KWs, MRGs, and ER1s) propagate more slowly in the EH. Moreover, the phase speed difference between WH and EH is smaller in the case of ER1, which is consistent with our model prediction because the larger amplitude of the secondary circulation in the case of ER1s suggests the need for a smaller expansion parameter *ε*, which, in turn, implies a weaker modulation of the phase speed.

Our model also predicts coupled wave amplitude zonal asymmetries (Figs. 13 and 14). In a companion paper, Yang et al. (2007a) compare composites of the horizontal structure associated with KWs, MRGs, and ER1s within the EH and WH. Their analysis (see their Figs. 6–8) shows that, in the case of KWs, both upper- and lower-level circulations are enhanced in the EH in comparison to the WH. Conversely, in the case of MRGs, the circulation is enhanced within the WH. Once again, because the WH is generally drier than the EH, this modulation is consistent with our model results shown in Figs. 5 and 6. In the case of ER1s, Yang et al. (2007a) show that the upper-level circulation is enhanced in the WH, but at lower levels it is enhanced within the EH; thus, the model prediction is in agreement with their ER composites only at the lower level. However, when comparing their vertical structure composites, the differences between EH and WH in the case of ER1s are much more dramatic. In particular, ER1s have a more barotropic structure in the WH (see also Kiladis and Wheeler 1995 and Kiladis et al. 2009), which indicates a more complex interaction with moisture and/or the basic state flow in that case (Kasahara and da Silva Dias 1986). While these vertical structure asymmetries merit further investigation, they cannot be addressed in context of our single layer model. It is also important to note that a drawback of the Yang et al. (2007a) analysis is that their composites were computed during a single Northern Hemisphere summer, so that their results may not be robust to a longer period analysis.

*T*) Cloud Archive User Service (CLAUS) data (Hodges et al. 2000) onto Matsuno's orthonormal modes using 6° for the trapping scale [refer to Gehne and Kleeman (2012) for details on the technique and sensitivity to the trapping scale]. This meridional decomposition acts as an effective filter to remove

_{b}*T*signals that are not related to these waves. As expected, space–time spectral analysis for each parabolic cylinder function (PCF) shows that KWs project mostly onto PCF

_{b}_{1}and MRGs onto PCF

_{2}[see Fig. 4 in Gehne and Kleeman (2012)], which is consistent with the theoretical structure of their divergence fields. Next, focusing on KWs and MRGs, we retain only the spectral coefficients associated within the regions defined in Fig. 1 for KWs and MRG, separately. Mathematically,wherewhere the sum is only over those wavenumbers and frequencies in the “Kelvin” space–time region (Wheeler and Kiladis 1999) and

*a*(

_{j}*x*,

*t*) is the projection coefficient associated with PCF

_{j}and an analogous definition for

_{j}. For comparison, we also consider projection coefficients onto a given PCF

_{j}that includes contributions from all scales, regardless of mode:Figure 15 shows

*T*variance field for each season. The

_{b}*T*variance is used as proxy for the mean moist convective activity for each mode during these periods, and these fields are similar to those obtained in WK99 using outgoing longwave radiation. Focusing on DJF, regions of low mean

_{b}*T*(enhanced moist convection) are relatively close to the equator from about 60°E to 180°. The maximum in KW variance (Fig. 15a) roughly matches the location of minimum

_{b}*T*(Fig. 15e). In particular, peaks in

_{b}*T*than

_{b}*T*located at 120°E, and it is much attenuated west of that. Significantly, the variance of

_{b}*T*. The MRG and KW variance zonal asymmetries are in agreement with the model predictions shown above because KWs are amplified as they propagate through the warm pool, whereas MRGs decay from east to west as it propagates through the same moister region. While similar behavior is observed during JJA, there are some differences. For instance, contrasting KW variance (Fig. 15a) and PCF

_{b}_{1}variance (Fig. 16b), it appears that KWs variance is better collocated with minimum

*T*close to the equator. However, Straub and Kiladis (2003) has shown evidence of substantial KW activity along the eastern Pacific off-equatorial ITCZ, which is not seen in Fig. 15a because it would project weakly onto PCF

_{b}_{1}. With respect to MRGs, once more in agreement with our model, as the minimum

*T*shifts to the west in comparison to DJF, enhanced MRG variance extends much farther in this direction in comparison to DJF. However, in this case, JJA

_{b}Similarly to the modeled waves, but in the context of KWs and the MJO, Roundy (2012) has shown that the observed structure of equatorial waves is sensitive to *H*_{eq}. For instance, the KW secondary temperature has peaks north and south of the equator, which also show up in the geopotential height composites from Roundy (2012) at low *H*_{eq}. The modeled horizontal structure shown in Figs. 5–7 also show meridional tilts in the secondary circulation. While these north–south tilts are often seen in observations, comparisons are complicated by the fact that in our model the orientation of the tilt strongly depends on the gradient of the moisture background, which is difficult to assess in observations. The meridional extension of the modeled waves, however, shares some similarities with observed CCEWs. For instance, Fig. 5 in Kiladis et al. (2009) shows the annual-mean variance of CCEW brightness temperature, which is, once again, interpreted as proxy for wave activity. Note that all modes, except for the ER, are fairly trapped to the equator. More specifically, ER variance peaks close to 20°N—much farther from the equator than all the other modes, which peak between 5° and 10°N (which is in closer agreement with the meridional trapping scale associated with *H*_{eq} = 25 m). The model results are in agreement with observations because the secondary circulation associated with ER1 is the only one that has a strong higher-meridional component, which enhances the divergence farther from the equator. Note also that the observed WIG activity (Fig. 5f in Kiladis et al. 2009) peaks east and west of the warm pool as well as over Africa, which is consistent with the model prediction that WIG1 divergence is enhanced over drier regions. In contrast, it is important to note that there is observational evidence that, similarly to KWs, WIG wave amplitude is dependent on the large-scale background moisture, specifically within the MJO (e.g., Yasunaga 2011).

With respect to the observed cloudiness power spectrum, our results are in agreement with the relative amplitude of the peaks above the background seen in the dispersion diagrams of tropical convection shown in Fig. 1. For instance, assuming, once more, a positive relationship between divergence amplitude and precipitation associated with the various CCEWs, the MRG spectral peak is much weaker than the KW and ER peak, which is consistent with the larger enhancement of the coupled divergence over moister regions in the cases of ERs and KWs, as opposed to MRGs. On the other hand, the model does not explain why the spectral peaks associated with WIGs are stronger than EIGs. Not only are the modeled WIG1 and EIG1 much less sensitive to changes in the background moisture (Fig. 14), but also, WIG1s are weakly attenuated whereas EIG1s are weakly enhanced within moister regions. The observed bias toward westward-propagating modes has been investigated by Tulich and Kiladis (2012), where it is shown that, overall, there is enhanced westward-propagating wave activity in tropical cloudiness because of the effects of low-level background easterly vertical shear prevalent throughout much of the tropics.

## 7. Discussion and summary

In this paper a conceptual single layer model for the tropical atmosphere is used to study the sensitivity of equatorial waves to large-scale moist convection. The model consists of *β*-plane shallow-water system with a reduced *H*_{eq} in comparison to a dry atmosphere. The mean reduced *H*_{eq} is chosen so that the dispersion curves of the wave solutions fit the observed spectral peaks of tropical convection, and we then study how zonal changes around this mean *H*_{eq} affect zonally propagating waves. Owing to the crude physical assumptions underlying this model, we focus on small-amplitude *H*_{eq} anomalies at planetary scales. In this case, we find approximate solutions where the first-order flow solves the system in the case of a constant *H*_{eq}, and a secondary circulation develops to adjust the flow to the varying *H*_{eq}. The suitability of this asymptotic approximation is discussed in section 3, where it is shown that the secondary circulation has no singularities within the observed CCEWs scales. More specifically, we analyze cases of *H*_{eq} variations at planetary wavenumbers *l* = 1 and 2 and mean *H*_{eq} between 12 and 50 m and zonal wavenumbers *k*_{0} between 2 and 10. We find that the amplitude of the secondary circulation, while finite, strongly depends on the first-order mode. In particular, first-order KWs and MRGs develop a secondary flow of about the same order of magnitude as the primary flow. Meanwhile, EIG0s, EIG1s, WIG1s, and ER1s develop a secondary flow 5–10 times stronger, depending on the first-order wavenumber *k*_{0}.

The secondary flow structure is documented in section 4 with choices of parameters that are roughly consistent with the observed CCEWs scales that are highlighted in Fig. 1. By design, the coupled flow propagates faster when the *H*_{eq} is maximized in comparison to when it is minimized. The amplitude of the change in propagation speed varies mode by mode because the expansion parameter *ε* has to be adjusted to the amplitude of the secondary flow. That implies that the speed of propagation of KWs and MRGs are the most sensitive to changes in background moisture, which is consistent with observations as discussed in section 6. The phase between the primary and secondary flow also varies mode by mode, impacting the amplitude and the horizontal structure of the coupled flow. This issue is addressed in section 5 in the context of the low-level divergence fields because of its implicit association with moist convection. Interestingly, over moister regions, we find that the coupled KW and ER1 divergence are the most intensified in comparison to their free counterparts, which is consistent with the largest peaks seen in the observed power spectrum of tropical cloudiness (with exception of the MJO peak). In contrast to KWs, MRGs are actually attenuated over moister regions. In section 5, we argue that this opposing behavior is due to the fact that zonal divergence is the dominant component of the KW divergence, as opposed to MRGs where the meridional divergence is dominant. Physically, this can be interpreted as MRGs being more efficient than KWs in advecting moisture to higher latitudes. While this issue merits further theoretical investigation, in section 6, we argue that the opposing behavior between the modulation of MRGs and KWs divergence is consistent with observations.

By increasing a convective trigger, and particularly in the case of moisture convergence schemes (Kuo parameterization), the GCM study by Frierson et al. (2011) suggests that not only *H*_{eq} decreases, but the spectral power along KW dispersion curves also decreases (see their Fig. 1). Our model is consistent with this result in the sense that the amplitude of the coupled KW divergence decreases with decreasing *H*_{eq} (see Tables 4 and 5). However, this relationship is not as clear for the other modes, except for MRGs. This mode sensitivity in the divergence amplitudes due to changes in *H*_{eq} raises the important question of whether the relationship between GMS and *H*_{eq} is applicable to all modes of CCEWs.

While the detailed behavior of CCEWs as observed in nature is likely better characterized by models with more accurate representations of moist convection (Khouider and Majda 2008; Tulich and Mapes 2008; Kuang 2010), our idealized model reveals some interesting disparities on how modes of equatorially trapped waves respond to a zonally varying *H*_{eq}. Because these dissimilarities are consistent with observations in a gross sense, it would be worthwhile to further explore this modal dependence on the coupling between waves and moist convection in the context of more complex models.

## Acknowledgments

We thank the anonymous reviewers for their thoughtful suggestions and comments. J. Dias acknowledges the support by NRC Research Associate fellowship and P. L. Silva Dias acknowledges the support by CNPQ (INCT-Climate Change).

## APPENDIX A

### Free Shallow-Water Modes

*D*are the parabolic cylinder functions and

_{m}*E*is a normalization constant such thatThe dispersion relation isThe solutions

_{m}*r*refers to each root of (A2) seen as a function of

*ω*. The parameter

*m*= 0 corresponds to the mixed Rossby–gravity wave, and

*m*≥ 1 to Rossby and gravity waves. KWs are obtained by setting

*m*= −1 in the dispersion relation and assuming

*υ*= 0, in which case

*T*= −

*u*.

## APPENDIX B

### The Eigenvalue Perturbation Problem

*x*to (15) yieldsNoting thatandwhere the angle brackets denote the inner product,Suppose that

**X**

_{j}are orthornormal and solve

**X**

_{j}are the normal modes found by Matsuno 1966). ThenBecause (B5) implies that the left-hand side of (B1) has no projection onto

*υ*

_{1}:whereThe right-hand side of (B8) can be further decomposed in terms of parabolic cylinder functions:which yieldswhereNoting that our choice of Λ

_{1}implies that

*k*=

*k*

_{0}and

*m*=

*m*

_{0}, thus

*ω*

_{0}, there can be pairs (

*m*,

*k*) that zero the denominator in (B12) as is illustrated in Fig. 2, which shows the magnitude of the denominator of (B12). This issue is discussed further in section 3.

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