## 1. Introduction

Prior to the second world war, many atmospheric scientists believed the tropics to be very stable outside of the occasional occurrence of tropical cyclones [see prologue in Riehl (1954)]. As interest in the tropics grew and observations increased, scientists showed that the tropics contain a diversity of “tropical weather systems” that exhibit a wide variety of spatial and temporal scales. Near the equator, studies have documented the existence of convectively coupled equatorial waves (Matsuno 1966; Kiladis et al. 2009) and the Madden–Julian oscillation (Madden and Julian 1972). Away from the equator, we see tropical cyclones and a variety of tropical depression–like (TD-like) systems that include monsoon low pressure systems and easterly waves (Riehl 1954; Chang 1970; Mooley 1973).

Interestingly, such a diversity of systems exists within an atmosphere that exhibits little spatial and temporal variation in temperature (Charney 1963; Held and Hoskins 1985), leading to the development of the weak temperature gradient (WTG) approximation (Sobel and Bretherton 2000; Sobel et al. 2001). They also coexist with the mean circulations: the Hadley and Walker cells, and the monsoons. These large-scale systems are known to export energy from the source regions to net sink regions (Riehl and Malkus 1958; Held and Hou 1980; Hartmann 2015).

Analysis of the governing thermodynamics of these systems has indicated that tropical weather systems exist in a spectrum (Adames et al. 2019; Inoue et al. 2020; Adames and Maloney 2021; Mayta and Adames 2023). Systems that propagate quickly exhibit thermodynamics that are governed by temperature fluctuations, as in gravity waves (Raymond et al. 2007; Herman et al. 2016). These are the systems that are responsible for maintaining WTG balance over the tropics (Bretherton and Smolarkiewicz 1989; Wolding et al. 2016). Slower-evolving convectively coupled systems are often in WTG balance and their thermodynamics are governed by moisture, hence the moniker “moisture modes” (Yu and Neelin 1994; Raymond and Fuchs 2009; Adames and Maloney 2021).

Initially, only the MJO was hypothesized to be a moisture mode (Raymond and Fuchs 2009; Sugiyama 2009; Sobel and Maloney 2013). Numerous advances in our understanding of the MJO have resulted from its study under the moisture mode lens [see reviews by Zhang et al. (2020), Jiang et al. (2020), and Adames et al. (2021)]. While the idea of the MJO being a moisture mode remains a topic of debate (Powell 2017; Chen 2022a; Mayta and Adames Corraliza 2023a), other studies have indicated that moisture modes may be a broader feature of the tropics (Adames 2022; Inoue et al. 2020; Maithel and Back 2022; Mayta and Adames 2023). On the basis of scale analysis, Adames et al. (2019) and Adames (2022) showed that systems with propagation speeds near 5 m s^{−1} are likely to be moisture modes. Such systems include most “rotational” tropical systems such as convectively coupled equatorial Rossby waves and TD-like systems.

Over the Western Hemisphere, observations indicate that equatorial Rossby waves and oceanic TD-like waves are moisture modes (Mayta et al. 2022; Mayta and Adames 2023). The same results were identified over the eastern and western Pacific and over the Indian Ocean (Gonzalez and Jiang 2019; Nakamura and Takayabu 2022a,b; Chen 2022b; Mayta and Adames Corraliza 2023b). Only African easterly waves do not fully exhibit moisture mode properties (Wolding et al. 2020; Núñez Ocasio and Rios-Berrios 2023; Vargas Martes et al. 2023). Wavenumber–frequency analysis also supports this hypothesis (Inoue et al. 2020). Storm-permitting simulations on an aquaplanet also reveal a high variance of moisture mode–like behavior in the form of easterly wave–like disturbances (Rios-Berrios et al. 2023). More recently, Maithel and Back (2022) found that convective recharge–discharge cycles behave like moisture modes. All these results allude to the potential commonality of moisture modes.

In spite of their geographical and structural differences, all these moisture modes are driven by the same processes. Their moist static energy (MSE) anomalies are governed by moisture and they propagate westward via horizontal moisture advection (Yasunaga et al. 2019; Inoue et al. 2020; Mayta and Adames Corraliza 2023b). Many studies have shown that vertical MSE advection dissipates the systems while longwave radiative supports their growth (Andersen and Kuang 2012; Sobel et al. 2014; Mayta et al. 2022). A less documented feature of these systems is that the advection of background-mean MSE by the anomalous meridional winds also contributes to their growth (Mayta and Adames Corraliza 2023b). This type of growth was initially documented in the balanced moisture waves of Sobel et al. (2001) and was later coined by Adames and Ming (2018) as “moisture-vortex instability”. While this instability was initially posited to explain the growth of monsoon low pressure systems (Adames and Ming 2018), a companion study (Mayta and Adames Corraliza 2023b, hereinafter MA) suggests that this mechanism is present in TD-like waves and equatorial Rossby waves around the globe. Most of these systems also have the common feature that they are advected westward by the mean trade winds (Russell et al. 2020; Gonzalez and Jiang 2019; MA).

We hypothesize that the self-similarity of these balanced waves may be a fundamental feature of their interaction with the tropical mean state. This hypothesis is further strengthened by examining a snapshot of column-integrated water vapor for a typical day of tropical wave activity (Fig. 1). In it, we see that the humid regions of the tropics appear wavy, even when there is no obvious sign of extratropical forcing. This large-scale behavior hints at the possibility that tropical waves may actively “stir” column water vapor, mixing it throughout the tropics.

Observational support for this hypothesis already exists. Sherwood (1996) and Pierrehumbert (1998) showed that the subtropics can be moistened by horizontal transports of water vapor from the convecting regions. Conversely, many studies have shown that tongues of dry subtropical air can intrude into the rainy regions of the tropics, suppressing rainfall (Mapes and Zuidema 1996; Parsons et al. 2000; Jensen and Del Genio 2006; Kerns and Chen 2014). Inoue et al. (2021) found that horizontal moisture advection is the primary cause of precipitation variability in the tropics. Furthermore, analysis of the poleward energy transport in the tropics shows a nonnegligible poleward latent energy transport by transients at the submonthly time scale (Trenberth and Stepaniak 2003; Peters et al. 2008; Rios-Berrios et al. 2020), suggesting that this stirring is also important for the global energy balance. These eddy moisture transports have been found to behave as a moisture diffusion (Sobel and Neelin 2006; Peters et al. 2008), weakening the ITCZ and moistening the midlatitudes. MA showed that a statistically significant signal in poleward moisture transport exists in all the westward-propagating moisture modes they examined.

The similarity of moisture modes across the tropics becomes more intriguing when we compare them to unstable baroclinic waves. One of the main features of dry baroclinic instability is that they grow from meridional temperature advection. The phasing between meridional winds and the temperature anomalies causes the growing waves to exhibit a poleward heat flux, a major piece of midlatitude wave–mean flow interactions as crystalized in the Eliassen–Palm formulation of wave activity (Andrews and Mcintyre 1976; Edmon et al. 1980). The heat flux acts to weaken the horizontal temperature gradient, therefore weakening the jet stream via thermal wind balance. If not for these fluxes, the temperature difference between the tropics and the poles would be larger than observed.

This study aims to examine the possibility that an analogous wave–mean flow interaction exists in the tropics. In this analogy the moisture modes are akin to the baroclinic waves, interacting with the mean state via moisture advection rather than temperature advection. The Hadley cell is the analog of the midlatitude jet stream. As in baroclinic instability, we hypothesize that *moisture modes grow by extracting energy from the mean meridional moisture gradient (i.e., MVI), flattening it and therefore weakening the Hadley cell*.

To test this hypothesis, we will employ a simple two-layer model to examine how an idealized circulation could lead to the existence of moisture modes (section 2). Then, we will examine the properties of these waves, including their propagation and growth (section 3). Interactions between moisture modes and the Hadley cell are discussed in section 4. A concluding discussion about the main findings of this study is offered in section 5.

## 2. Two-layer model

### a. Model setup

We are particularly interested in capturing the moisture mode behavior of oceanic TD-like waves and how these may interact with an idealized Hadley cell. Studies using observations and reanalysis show that these waves exhibit little tilt in height in their vertical velocity field (Serra et al. 2008; Inoue et al. 2020; Feng et al. 2020a,b; Huaman et al. 2021). This lack of tilting implies that the two-layer model with a simplified profile of vertical velocity used by Adames (2021, hereafter A21) may adequately capture their behavior. This is the same model shown in Holton and Hakim (2012) except modified to include a prognostic moisture budget. This model allows us to include the effects of vertical wind shear but is also sufficiently simple to allow for a tractable analysis of its wave solutions. The main variables in this study are outlined in Table 1, and constants with their values are shown in Table 2.

The main variables and definitions used in this study.

Constants and their values.

We will modify the model of A21 to account for the presence of an idealized Hadley cell, and examine motions under the WTG approximation, rather than the quasigeostrophic approximation. For simplicity, we will consider the evolution of motions on an *f* plane. Readers interested in how the system behaves on a beta plane are referred to section 1 of the online supplemental material. The layout of the model is shown schematically in Fig. 2. The idealized Hadley cell comprises equatorward and easterly winds in the lower troposphere and poleward and westerly winds in the upper troposphere. The equatorward side of the cell is ascending and humid, while the poleward side is descending and dry. Since we are interested in understanding the interactions between waves and the Hadley cell in isolation, we will bind our model with rigid walls at a distance *L _{H}* from the center of the domain.

*ω*) attains a maximum amplitude in layer 2 and becomes zero at the top and bottom boundaries. For simplicity, we assume that

*ω*increases linearly from

*p*

_{0}to

*p*

_{2}and decreases linearly thereafter. As a result,

*ω*

_{3}can be written in terms of

*ω*

_{2}as

*ω*has a simple structure, we can invoke mass continuity in each discrete layer in order to relate it to the horizontal divergence (

*δ*). Hence, we can write the divergence in layers 1 and 3 as

*p*is half the depth of the atmosphere in this model. Many of the variables used here will be column integrated, zonally averaged, and/or meridionally integrated. For any variable

*X*, these operations are defined as

*a*= 6378 km is the radius of Earth, and

*φ*= 10°N is our reference latitude. Deviations from this zonal average will be denoted with primes; that is,

*Q*

_{1}and

*Q*

_{2}, respectively, in terms of surface processes and column radiative heating. Following Yanai et al. (1973), we can column integrate

*Q*

_{1}and

*Q*

_{2}, yielding

*P*is the surface precipitation rate,

*E*is the surface evaporation, and ⟨

*Q*⟩ is the column radiative heating rate. The ⟨

_{r}*Q*⟩ is decomposed into a cloud-radiative heating component and a clear-sky radiative cooling

_{r}*R*

_{cs}:

*r*is the cloud-radiative feedback parameter (Peters and Bretherton 2005) or the greenhouse enhancement factor (Kim et al. 2015). Note that the effect of water vapor on radiative heating is implicitly included in

*r*.

*W*= ⟨

*q*⟩, where

*q*is the specific humidity). We will do a Taylor series expansion centered at

*P*and

*W*is nonlinear (Bretherton et al. 2004), we will use a constant value of

*τ*to obtain simpler results. We have found that a value of

_{c}*τ*of 4 days yields a reasonably realistic mean state in this model. This time scale is much larger than what empirical data suggest (Bretherton et al. 2004; Rushley et al. 2018; Adames et al. 2017). However, it is important to note that the

_{c}*τ*is usually defined for regions of high precipitation, not for the tropical average. While the large and fixed value of

_{c}*τ*is a limitation of this study, our main findings are not sensitive to the value of

_{c}*τ*that is chosen for the analysis.

_{c}### b. Domain means

^{−2}, assuming that

*r*= 0.1.

*L*:

_{H}*s*=

*C*+ Φ is the dry static energy,

_{p}T*T*is the temperature, and Φ is the geopotential. If

*W*=

*q*

_{3}Δ

*p*/

*g*is 50 mm at −

*L*,

_{H}^{1}and approximating

*q*

_{4}≃ 2

*q*

_{3}, we find that

*s*

_{0}] − [

*s*

_{4}] = 5 × 10

^{4}J kg

^{−1}.

### c. Zonal mean state

*y*:

In Fig. 3a we show the meridional profile of *L _{υ}W* from ERA5 (Hersbach et al. 2020) are also shown for a domain that is comparable to that of the two-layer model. A further description of the ERA5 data is found in the caption of Fig. 3. This domain extends from the ITCZ to near the poleward edge of the tropics. The chosen linear profile of

*L*reasonably represents the moisture decrease from the ITCZ near 7°N to the domain’s edge near 25°N.

_{υ}W*L*≃ 1000 km and

_{H}*L*. The parabolic profile of

_{H}*ϵ*is the dissipation coefficient. These momentum balances are similar to the WTG Hadley cell model of Polvani and Sobel (2002). In this study, we will use a value of

*ϵ*

_{1}= 1 × 10

^{−6}s

^{−1}for the upper troposphere and

*ϵ*

_{3}= 2 × 10

^{−6}s

^{−1}for the lower troposphere. These values of

*ϵ*

_{1}and

*ϵ*

_{3}correspond to dissipation time scales of 11.6 and 5.8 days, respectively, within the range of what previous studies have used [see Table 1 in Kim and Zhang (2021)]. In section 2 of the online supplemental material we discuss the sensitivity of the mean state to various values of

*ϵ*.

In Fig. 3d we see that ^{−1}. However, *ϵ* used and the application of rigid walls at the edges of the Hadley cell. Decreasing *ϵ* will increase the upper-tropospheric winds and accentuate a jet stream at the poleward edge of the Hadley cell but will also increase the meridional temperature gradient (see section 2 of the online supplemental material).

^{−1}. When compared to the mean moisture gradient we see that the temperature gradient is roughly 6 times weaker (Fig. 3a). In ERA5 data we also see that the

*W*gradient is much steeper than the temperature gradient, although the gradient is flatter on the equatorward edge and steeper at the poleward end of the domain.

### d. Simplified basic equations

We can substantially simplify the model of A21 if the following conditions are satisfied for the motions of interest.

#### 1) Thermodynamic variations are in WTG balance

*L*is the meridional scale of the system, and

_{y}*c*is the gravity wave phase speed. As we will show below, the meridional scale of the motion is determined by

*L*, and the largest value that

_{H}*L*can take is 2

_{y}*L*/

_{H}*π*∼ 6.4 × 10

^{5}m. For the motions we are interested in, we find that

*N*∼ 0.1.

_{w}#### 2) Moisture modes are the main synoptic-scale motion

*L*must govern the distribution of anomalous column moist enthalpy (

_{υ}W*L*′ +

_{υ}W*C*⟨

_{p}*T*′⟩). A small value of

*N*

_{mode}implies that |

*L*′| ≫ |

_{υ}W*C*⟨

_{p}*T*′⟩|. Since we are considering motions at high moisture values, it is convenient to define

*N*

_{mode}following Adames et al. (2019):

*N*

_{mode}∼ 0.02.

*ζ*

_{3}and

*W*. Hence, we will drop it and note that it is still used to determine the vertical structure of the resulting waves. Thus, we can combine Eqs. (21b)–(21d) to obtain the following equations for vorticity and moisture:

*L*/

_{υ}W*S*is the so-called Chikira parameter

*α*, while 1 −

*L*/

_{υ}W*S*is similar to the so-called gross moist stability (Neelin and Held 1987).

There are three processes on the right-hand side (rhs) of Eq. (22a): horizontal vorticity advection, vortex stretching, and frictional dissipation. There are four processes on the rhs of Eq. (22b): horizontal moisture advection and three column processes: vertical MSE advection by convection (including cloud-radiative feedbacks), vertical moisture advection by clear-sky radiative cooling, and surface evaporation.

## 3. Linear wave solutions

### a. Linearization and scaling

*ζ*

_{3}is much smaller than

*f*

_{0}over most of the domain. Given that the perturbations in vorticity are much smaller than the mean values, we can simplify (

*f*

_{0}+

*ζ*

_{3}) ≃

*f*

_{0}in Eq. (22a). These assumptions and approximations lead to the following pair of linearized equations:

*R*

_{cs}from the moisture budget. It is also worth noting that the last two terms on the rhs of Eq. (24b) are the anomalous column processes (

*C*′ = −Γ

*′/*

_{e}W*τ*+

_{c}*E*′).

Over the entirety of our idealized Hadley cell *L _{y}* ≥

*L*). This is a reasonable approximation since

_{x}*L*∼

_{y}*L*for the two conditions outlined above to be valid.

_{x}*f*

_{0}, the nondivergent wind will be much stronger than the irrotational wind [Eq. (37) in Sobel et al. 2001]. This approximation is obtained from Eq. (29a) from the following condition:

*τ*is the time scale of the wave relative to the mean flow (i.e., the Doppler-shifted time scale). If

_{d}*f*

_{0}

*τ*≫ 1 it follows that |

_{d}*ζ*′| ≫ |

*δ*′|. Thus, we can write the anomalous horizontal winds in terms of a streamfunction (

*ψ*′):

We are particularly interested in the special case when the meridional advection of mean moisture by *υ*′ governs *E*′. Additionally, *τ _{d}* must be on the order of 10

^{6}days and Γ

*must be much smaller than unity for*

_{e}*to be small, the environment must be humid and cloud-radiative feedbacks sufficiently strong to offset any large-scale drying driven by convection. These conditions are discussed in more detail in section 4 of the supplemental material. While we note that these conditions are not necessarily realistic, they will allow us to investigate instabilities driven by horizontal moisture advection in isolation.*

_{e}*L*and that the Γ

_{H}*term is dropped.*

_{e}### b. Necessary conditions for instability

*k*is the zonal wavenumber,

*ϖ*is the wave frequency, and

*ψ*′ → 0 at the boundaries. This is the same procedure done by Charney and Stern (1962). Doing so yields the following condition for the imaginary component of the resulting equation:

*ϖ*=

*ϖ*+

_{r}*iϖ*), defining

_{i}*ϖ*is the contribution to the propagation, while

_{r}*ϖ*describes the growth rate.

_{r}From an examination of Eq. (32), we see that if *ϖ*_{rd} is nonzero then *ϖ _{i}* must also be nonzero for the integral to vanish. Either

### c. Dispersion relation

*l*is the meridional wavenumber. Note that

*L*, it follows that the wave solutions have meridional structures that are restricted to oscillate an integer amount of times over twice the width of the Hadley cell (4

_{H}*L*). Thus,

_{H}*l*is defined as

*K*

^{2}=

*k*

^{2}+

*l*

^{2}is the horizontal wavenumber, and

*β*to facilitate its interpretation. Note that this definition differs from that of A21 as it is multiplied by

*L*(1 +

_{υ}W*r*)/

*S*.

The dispersion relation in Eq. (35a) has two wave solutions, with one of them being unstable, as shown in Fig. 4. The square root term is equal parts real and imaginary since *ω*_{rd} = *ω _{i}*. A close examination of Eq. (35a) shows that regardless of the sign of

*β*, one of the wave solutions will always be unstable.

_{q}Two opposing processes contribute to the propagation of the wave. The first is advection by the mean lower-tropospheric winds, which act to propagate the system westward. The second is vortex stretching arising from meridional moisture advection. Lower tropospheric southerlies advect moist, precipitating air to the east of the wave center (bottom panel of Fig. 5). The vortex stretching associated with the advected air acts to propagate the wave eastward. From Fig. 4, we see that the latter process is dominant at small *k*. As *k* increases, the phase speed asymptotically approaches the ^{−1}. One can interpret the growth of synoptic-scale systems as propagation against the mean winds, akin to midlatitude Rossby waves, albeit in the opposite direction of motion.

Vortex stretching arising from meridional moisture advection is the only process that contributes to growth. Growth increases from wavenumber 1 and peaks near zonal wavenumber 10, decreasing afterward. It is worth pointing out that the instability is identical to the one that is obtained from Sobel et al. (2001) and Adames and Ming (2018) when the dispersion relation is not simplified. The main distinction is that the moistening is in the direction of propagation relative to the mean flow. In the case we are considering, the waves are traveling eastward relative to the mean easterly winds. Thus, these systems grow from MVI.

In Fig. 5 we see that the growing mode exhibits a horizontal structure that is mostly confined to the lower troposphere. Little signature is seen in the upper troposphere. The positive *W* and hence the *P* anomalies are shifted to the east of the center of low pressure (*P* anomalies are shifted east of, but not in quadrature with *P* anomalies are contributing to both the propagation and growth of the system, as Eq. (35a) indicates.

When examining the column-integrated moisture budget of the growing waves, which is equivalent to their MSE budget under WTG balance, we see that the fastest-growing zonal scale propagates westward nearly entirely from advection by the mean zonal winds (top panel in Fig. 6). They grow from meridional advection of mean moisture (bottom panel in Fig. 6), as was found for many TD-like and equatorial Rossby waves in MA.

## 4. Moisture mode–Hadley cell interactions mediated by moisture transports

### a. Moisture mode activity

*W*′ is largely determined by the meridional advection of

*W*′ and

*W*′ and zonally average it, yielding a budget for the zonally averaged column moisture variance:

*W*′ are coupled. Furthermore, when comparing with QG theory, we can see that Eq. (36) resembles, and behaves similarly to the QG enstrophy equation, while Eq. (37) resembles the mean-state PV budget (Vallis 2017).

*L*to

_{H}*y*. Noting that there are no eddy moisture fluxes at the edges of the Hadley cell, the resulting equation is written as

The relationship in Eq. (40) implies that moisture modes in this model grow by extracting ALE via MVI and hence flattening the mean moisture gradient. Thus, the *moisture modes in this model grow at the expense of the Hadley cell*.

*κ*is a moisture diffusivity. Using this approximation we can re-express Eq. (39) as

_{W}*κ*suggests that it is roughly 3 × 10

_{w}^{−5}m

^{2}s

^{−1}, a value similar to that used by Neelin and Zeng (2000). Using this value we find that

*τ*is roughly on the order of 20 days. Thus, while the consumption of ALE is slow compared to the time scale of the moisture modes, it is fast enough that it can have a significant impact at the intraseasonal time scale.

_{D}### b. Mass streamfunction perspective

*p*

_{2}= 500 hPa. By combining Eq. (42) with Eq. (14) we obtain the following:

### c. Steady-state interactions under the presence of dissipation

*C*′ will also be included. As a result, the modified equation for

*latent energy follows a direct energy cascade, as in kinetic energy*.

Furthermore, if we assume that the horizontal scale of the eddies remains fixed, we find that

### d. Predator–prey cycles in moisture mode activity and Hadley cell strength

*κ*is not a constant, but is instead proportional to

_{W}*γ*is a constant with a rough value of 4 × 10

^{−19}kg (m s)

^{−1}, which we obtain from Table 2 and the scales discussed in section 3.

*being positive. Thus, it is reasonable to assume that*

_{e}^{13}J m

^{−1}. With the estimated value of

*γ*we find that the oscillation has a time scale of ∼35 days. Given that this value was derived from an estimation with a high degree of uncertainty, it is possible that this value could vary by a factor of 2. Nonetheless, the oscillation would still occur at the intraseasonal time scale.

## 5. Discussion and conclusions

### a. Summary

In recent decades our understanding of tropical atmospheric motions has grown expeditiously (Emanuel 2018; Jiang et al. 2020; Adames and Maloney 2021). Part of this growth has been due to the increased awareness of the importance of water vapor in these motions (Held et al. 1993; Brown and Zhang 1997; Raymond 2000; Sobel et al. 2001; Wolding et al. 2020; Maithel and Back 2022, among others). As our understanding continues improving, we have recognized that “moisture modes” may exist in the tropics (Yu and Neelin 1994; Fuchs and Raymond 2002). Initially used to describe the MJO (Raymond and Fuchs 2009; Sobel and Maloney 2013), recent studies have posited that moisture modes may be a more common feature of the tropics (Adames and Maloney 2021; Inoue et al. 2021; Maithel and Back 2022; Adames 2022). There is increasing evidence that convectively coupled equatorial Rossby waves and tropical depression–like waves exhibit properties of moisture modes (Inoue et al. 2020; Gonzalez and Jiang 2019; Nakamura and Takayabu 2022a; Mayta et al. 2022).

On the basis of these findings, we investigated the possibility that moisture modes interact with the Hadley cell. To do this, we employed the two-layer model of Adames (2021) and adapted it to accommodate an idealized Hadley cell. The strength of the Hadley cell is assumed to be proportional to the mean meridional moisture gradient. As in previous works (Charney 1963; Sobel et al. 2001; Adames 2022), the overturning circulation is in WTG balance so long as the square of its meridional half-width (

If the mean state is simplified further by assuming the mean lower-tropospheric winds are constant, we obtain a system of equations that are nearly identical to those of Sobel et al. (2001), but with a restricted meridional scale. Wave solutions to this idealized mean state yield a pair of wave solutions, one of which is unstable. The unstable solution propagates westward from a combination of zonal advection by the mean flow and meridional advection of the mean moisture by the anomalous meridional winds. It is destabilized by the latter process. The most unstable wave grows near zonal wavenumber 10 and propagates westward at roughly the same speed as the trade winds.

An interesting result of Eq. (35) is that one of the wave solutions is always unstable, so long as the square root is nonzero. For example, a growing solution still exists if the moisture gradient is poleward, as we observe in the South Asian monsoon (Clark et al. 2020). This result means that a meridional moisture gradient in the presence of rotation (nonzero *f*) will always be unstable so long as convection is sensitive to water vapor fluctuations.

We then return to the original Hadley cell model to examine if transients that grow from meridional moisture advection interact with the Hadley cell. Our results show that moisture mode activity increases when there are poleward moisture fluxes. The pair of equations that describe the interaction between moisture modes and the Hadley cell [Eqs. (38) and (39)] are analogous to the Eliasen–Palm flux formulation (Andrews and Mcintyre 1976; Edmon et al. 1980), with column water vapor taking the place of potential vorticity.

Examination of this pair of equations reveals that the eddy moisture fluxes flatten the mean meridional moisture gradient, therefore weakening the Hadley cell, as summarized in Fig. 8. Conversely, a stronger ITCZ is associated with drier subtropics, consistent with previous work (Hohenegger and Jakob 2020; Popp et al. 2020). Furthermore, we found that the amplification of moisture modes at the expense of the Hadley cell is transient, as the available latent energy (ALE) for moisture mode activity decreases with a weakening Hadley cell. Such a transient behavior leads to a cycle, where moisture modes deplete the ALE and hence weaken the Hadley cell, followed by a period of weak moisture mode activity where the Hadley cell and the ALE rebuild. This oscillation occurs at the subseasonal time scale, and it is not clear if it may be associated with the MJO. It is, however, reminiscent of the ITCZ breakdown discussed by Ferreira and Schubert (1997). While we emphasize different mechanisms from theirs, it is possible that they are nonetheless related.

That the latent energy fluxes are poleward also suggests that they may be important for global energy transport. Eddy energy fluxes in the tropics are smaller than those in the midlatitudes, but they are not negligible [Fig. 1 in Trenberth and Stepaniak (2003) and Fig. 5 in Rios-Berrios et al. (2020)]. Stoll et al. (2023) found that these fluxes correspond to synoptic-scale eddies (2000–8000 km across). In the tropics, eddy fluxes of energy are dominated by the latent energy component, with DSE transports being negligible, as would be expected from moisture modes.

### b. MVI and the growth of TD-like waves

In spite being one of the most well documented tropical transients, the key processes that drive the growth of TD-like waves remains a topic of extensive research (Lau and Lau 1992; Tam and Li 2006; Rydbeck and Maloney 2014; Alaka and Maloney 2014; Feng et al. 2016; Russell et al. 2020; Núñez Ocasio et al. 2020, among others). TD-like waves are observed in many areas of the world including, but not limited to, the northern Atlantic Ocean, the northeast and northwest Pacific, and over tropical Africa (Lau and Lau 1990; Kiladis et al. 2009; MA). Previous studies have suggested that these waves grow from barotropic, baroclinic, or inertial instability, or a combination thereof that is amplified by deep convection (Lau and Lau 1992; Thorncroft and Hoskins 1994; Rydbeck and Maloney 2014; Torres et al. 2021). While our results do not reject the possibility that these processes are important, they hint at another potential explanation: TD-like waves are the result of the unstable horizontal moisture gradients that are present in the ITCZ and the monsoons. They may be fundamental features of the tropics as they may act to stabilize the Hadley cell, preventing the moisture gradient along the flanks of the ITCZ from steepening. This is an attractive possibility since strong horizontal moisture gradients are common in the regions where TD-like waves are active (see Fig. 2 in MA). That the linear waves described here resemble observed TD-like waves supports this possibility, although more work is needed to further test it.

### c. Potential relevance to tropical cyclones

The structure of the growing waves discussed here also bears resemblance to tropical cyclones (TCs). Both systems have circulations that are strongest in the lower troposphere, and water vapor plays an important role in the thermodynamic processes of both. In particular, it is well known that free tropospheric moisture is important for TC genesis (Tang et al. 2016; Yoshida et al. 2017; Raymond and Kilroy 2019; Tang et al. 2020). Thus, it is possible that processes akin to MVI may be at play in TC formation. Such a possibility was hinted at by Narenpitak et al. (2020) and Núñez Ocasio and Rios-Berrios (2023). Narenpitak et al. (2020) found that flux of moisture toward the vortex center differentiated vortices that developed into TCs versus those that did not. While they did not mention it explicitly, developing waves exhibited properties of MVI. A qualitatively similar result was found by Núñez Ocasio and Rios-Berrios (2023) for an African easterly wave that developed into Hurricane Helene. If TC development has elements of MVI in it, it is possible that the mechanisms described here also apply to TCs. This transport could occur in conjunction with the oceanic heat transport hypothesized by Emanuel (2001). This possibility should be examined in future work.

### d. Potential relevance to convective self-aggregation

An interesting interpretation of the results of this study comes from examining them from the perspective of convective self-aggregation (Bretherton et al. 2005; Wing and Emanuel 2014; Windmiller and Craig 2019). We can interpret the tendency of the Hadley cell to amplify through column processes as a tendency for the ITCZ to aggregate through radiative-convective instability (Emanuel et al. 2014). In this case, the MSE/moisture variance in the ITCZ will increase, consistent with an increase in ALE. Under this lens, moisture mode activity would act to disaggregate the ITCZ by diffusing the moisture distribution. Maithel (2023) found that aggregation/disaggregation cycles exist in reanalysis and are driven by horizontal MSE advection. Examining the results of this study from the point of view of aggregation could be a fruitful direction for future research.

### e. Applicability to the Walker cell

Extending these results to zonal moisture gradients can yield some interesting insights into the nature of Rossby waves and the MJO. Since the MJO is strongest over the Indian Ocean, a region characterized by an eastward and equatorward moisture gradient, it is possible that the MJO is diffusing latent energy from the Maritime Continent. Hence, the MJO moistens the Indian Ocean at the expense of moisture over the Maritime Continent. Some western Pacific equatorial Rossby waves may work analogously. Thus, both these systems would gain energy while weakening the Walker cell.

### f. Caveats

We acknowledge that the results of this study were obtained using an idealized two-layer model with several assumptions and approximations. Convection is parameterized as a linear function of column water vapor only. Focusing on water vapor makes our analysis more amenable for analytical interpretation. In observations, precipitation increases exponentially with increasing moisture, rather than linearly (Bretherton et al. 2004; Rushley et al. 2018). Additionally, large-scale convection also depends on fluctuations in temperature and vertical wind shear (Anber et al. 2016; Ahmed and Neelin 2018; Ahmed et al. 2020). The large-scale control on deep convection is a topic of ongoing research (Schiro et al. 2018; Wolding et al. 2022), but we acknowledge that the simple representation of convection is a caveat of this study.

We also acknowledge that our mean state is highly idealized compared to observations. In observations, zonal and meridional asymmetries in the distribution of wind and moisture due to the presence of landmasses can play an important role in the growth of tropical waves (Rydbeck et al. 2017; Feng et al. 2020a; Torres et al. 2021). While making these assumptions facilitates the analytical tractability of the model, they are another limitation of this study.

Another caveat of this study is that by assuming the motions exist within a Hadley cell bounded by rigid walls we are neglecting the role the extratropics may have in determining the mean state in which the wave propagates. This assumption causes the winds to become zero at the edges of the Hadley cell if friction is not ignored, a feature that is inconsistent with observations. Furthermore, the two-layer model only accounts for the first baroclinic vertical motions. This was done since the oceanic TD-like waves that our study examines exhibit upright vertical velocities that are approximately first baroclinic (Serra et al. 2008; Inoue et al. 2020; Huaman et al. 2021). However, interactions between the first and second baroclinic mode can be important in TD-like waves (Núñez Ocasio et al. 2020; Russell et al. 2020; Torres et al. 2021). More work is needed to understand whether the results of this study are still valid in a more realistic setting.

### g. Conclusions

We conclude this study by returning to the first paragraph of section 1. Despite the acknowledgment that the tropics exhibit a diversity of motion systems, there is a lack of work done trying to understand how these systems interact with the mean circulation. Most studies assume the Hadley cell to be a stable circulation and the most important source of energy transport out of the tropics. This study challenges this notion. Results from previous works (Inoue et al. 2021; Maithel and Back 2022; MA) suggest that horizontal moisture advection may be important in understanding rainfall variability and the growth of large-scale waves. Results from this work agree with these studies and reveal that horizontal moisture advection may play a central role in interactions between the Hadley cell and the waves. These interactions are analogous to those of baroclinic eddies and the jet stream. While baroclinic instability weakens the jet stream by flattening the temperature gradient, moisture modes weaken the Hadley cell by flattening the moisture gradient. These interactions are mediated by a poleward eddy moisture flux, which may play an important role in the global energy balance. If these results are confirmed by future studies, they would imply that the diversity of weather systems seen in the tropics interact with the climate system in richer and more dynamic ways than previously thought.

This result is obtained by adding the average value of *q* between layers 2 and 3, and layers 3 and 4, respectively: *W* = (*q*_{3} + *q*_{2})(*p*_{3} − *p*_{2})/(2*g*) + (*q*_{4} + *q*_{3})(*p*_{4} − *p*_{3})/(2*g*) = *q*_{3}Δ*p*/*g*, recalling that *q*_{2} = 0 and *q*_{4} = 2*q*_{3}.

## Acknowledgments.

ÁFAC was supported by NSF CAREER Grant 2236433, and by the University of Wisconsin startup package. VM was supported by NOAA Grant NA22OAR4310611. ÁFAC would like to thank his late father, Ángel David Adames Tomassini, for listening to and motivating him to complete this project in spite of strenuous circumstances. The contents of the manuscript were significantly improved after conversations with Rosimar Ríos Berríos and Rich Rotunno. We thank Haochang Luo for reviewing an early version of this manuscript. Lastly, we thank Adam Sobel and two anonymous reviewers for feedback and comments that helped improve the contents of the manuscript.

## Data availability statement.

ERA5 data are available at https://www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era5/). Interpolated *T _{b}* data are provided by NOAA/ESRL (https://catalogue.ceda.ac.uk/uuid/c2112bdd5f0ad698e70be6ab54c9a2ac).

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