## 1. Introduction

The intertropical convergence zone (ITCZ) is a region of near-surface convergence of horizontal winds, typically located north of the equator in the Pacific ocean. It is characterized by ascending vertical velocities and, as a result, high climatological rainfall. It is also an important region for the generation of tropical cyclones (TCs), which can result from a process known as “ITCZ breakdown.” This breakdown can occur from a combination of barotropic instability of the ITCZ itself and in association with westward-propagating disturbances known as easterly waves. The breakdown can be viewed as a transition from linearly growing waves via barotropic instability to finite-amplitude vortices that can subsequently intensify into TCs. A classic example is described by Agee (1972), who suggested that Tropical Storm Anna formed as a result of ITCZ instability. Wang and Magnusdottir (2006) found that 65 TCs in the central and eastern Pacific were produced between 1999 and 2003 with an equal number of TCs formed due to easterly waves and barotropic instability. In another observational study, Berry and Reeder (2013) showed that, between 1979 and 2010, more than 50% of all TCs globally formed within 600 km of the ITCZ. There are, of course, important mechanisms other than barotropic instability in TC production (e.g., Montgomery et al. 2006; Emanuel 2003; Wing et al. 2016). Our focus here, however, is on barotropic instability.

The latitudinal location of ITCZ varies seasonally with the annual average being slightly north of the equator (Waliser and Gautier 1993; Berry and Reeder 2013; Marshall et al. 2014; Wodzicki and Rapp 2016). The impact of latitudinal location on TC frequency was discussed in Merlis et al. (2013) and Ballinger et al. (2015) in the context of aquaplanet global climate models at TC-permitting resolution. In these studies, sensible and latent heat fluxes associated with an anomalously warm zonal band of sea surface temperature (SST) led to ascending vertical motion and convergent surface winds analogous to those of Earth’s ITCZ. When the ITCZ was shifted poleward, the frequency of simulated TCs increased (see also Hsieh et al. 2020). In aquaplanet simulations with an imposed SST generating an ITCZ, Hsieh et al. (2020) proposed that increases in the large-scale vorticity leads to an increase in the probability of TC formation. However, the underlying mechanism that causes increases in large-scale vorticity as the ITCZ shifts poleward in these simulations is unclear. In this study, we present an analysis of how barotropic instability—the first phase of an important TC development process—depends on the latitudinal position of an imposed heat source representing the zonal-mean latent heat release in the ITCZ region.

To examine the influence of latitudinal position on the barotropic instability associated with ITCZ breakdown, we choose the simplest model that captures this phenomenon. In particular, we use a single-layer shallow-water model similar to that of Nieto Ferreira and Schubert (1997). They used a prescribed heat source to produce a zonal-wind profile similar to the low-level winds of the ITCZ. This profile was found to be barotropically unstable and simulations showed the development of vortices analogous to TCs. More recently, Asaadi et al. (2016) used a shallow-water model with prescribed heating to study effects of parameterized latent heat release associated with easterly waves and the ITCZ has also been studied in Matsuno–Gill-type models (e.g., Adam 2018). Results similar to those of Nieto Ferreira and Schubert (1997) have also been reproduced in more comprehensive models. For example, Yokota et al. (2012, 2015) used a nonhydrostatic mesoscale model with a prescribed warm belt of SST that produced an ITCZ-like wind profile. An energy budget analysis showed the initial breakdown of the ITCZ to be through low-level barotropic instability (similar to the observational results in Cao et al. 2012). This produced vortices that later intensified through latent heat release. Wang and Magnusdottir (2005) used a primitive equation model with prescribed heating that generated ITCZ-like winds at the beginning stages of a simulation. They studied the resulting ITCZ breakdown with and without a prescribed background flow and found that barotropic instability increased if the background flow reinforced the ITCZ wind shear produced by the heat source (e.g., Dritschel 1989). In short, the ITCZ may undergo breakdown via dry barotropic instability, and latent heat release can play a role in near-neutral conditions or in the subsequent finite-amplitude growth.

This study seeks to answer two questions:

How does the latitudinal position of the ITCZ affect the barotropic instability of the associated horizontal flow?

How does flow-dependent moisture and its associated latent heat release impact the ITCZ breakdown in our idealized framework?

To address the first question, we force our shallow-water model using a prescribed latitudinally dependent heat source whose center latitude we vary. This heat source is an idealized representation of the zonal-mean latent heat release associated with the ITCZ, similar to that used by Nieto Ferreira and Schubert (1997). We run a suite of spinup simulations with the heat source located at different latitudes in order to generate a set of balanced base-state wind profiles. These base-state profiles depend on latitude only. As such, zonal dependencies in the base state—which are important in the context of monsoon depressions (Diaz and Boos 2019)—are not represented. The barotropic instability of our base-state profiles is then assessed in a series of dry, unforced, initial value simulations, from which we assess growth rates and determine how these vary with the latitudinal position of the model ITCZ. We find that, for an equivalent amount of heating, a poleward shift of the ITCZ leads to an increased growth rate. A simple mechanism is presented to explain this result; essentially, heating at higher latitudes results in a stronger wind shear.

To address the second question, we include humidity as a tracer variable that precipitates and releases latent heat upon supersaturation (Bouchut et al. 2009). [This model is similar to Gill (1982).] Here, we use the barotropically unstable profiles from question 1 and assess how barotropic growth rates vary as a function of latent heat release strength. For most of our simulations, a spatially constant saturation humidity is assumed. However, we also examine the sensitivity of growth rates to different spatially varying prescriptions of saturation humidity, as it has been noted that gradients in saturation humidity can have a strong influence on tropical dynamics (e.g., Sobel et al. 2001; Sukhatme 2014; Suhas and Sukhatme 2020).

Using a similar model with constant saturation humidity, Lambaerts et al. (2011) found that precipitation led to a decrease in the growth rate of barotropically unstable jet. Here, we find that whether an increased latent heat release strength leads to stronger or weaker growth rates depends on how saturation humidity is specified. If it is either taken to be constant or to be a function of temperature (i.e., proportional to the interface height field), a decrease results. Instead, taking saturation humidity to have a fixed meridional profile mimicking the base-state temperature field leads to faster growth. We present mechanisms that explain these results.

Section 2 presents the model, the energy budget of our system, and the numerical model and parameter values used. Section 3 describes the results of shifting the latitudinal position of ITCZ and the subsequent dry barotropic instability generated by those base states. Section 4 describes the flow-dependent moisture effects on the linear instability phase of the barotropic instability and interprets the results using the simulations’ energy budget. Section 5 presents our conclusions.

## 2. Methods

### a. Shallow-water model

*β*plane with heating, represented as a mass sink, in the thickness equation. The equations of motion are given by

**u**= (

*u*,

*υ*) is the horizontal velocity,

*g*is the gravitational acceleration, and

**f**= (

*f*

_{0}+

*βy*)

**k**is the Coriolis parameter. The thickness of the fluid column is

*h*=

*H*+

*η*, where

*H*is the mean-layer depth with

*η*being deviations from the mean. That is, ⟨

*h*⟩ =

*H*and ⟨

*η*⟩ = 0, where ⟨⋅⟩ denotes the domain average. The mean-layer depth

*H*is chosen to represent a typical equivalent depth of the tropical atmosphere (e.g., Kiladis et al. 2009).

^{1}

*f*

_{0}and

*β*according to

*R*

_{e}is Earth’s radius. Different base-state profiles are produced by varying the central latitude of heating

*ϕ*

_{c}.

The forcing term *F* in (2) represents heating. Positive *F* corresponds to a thinning of the layer. Our model can be understood as the lower layer of a two-layer model where the upper layer is quiescent and has a rigid lid. Heating is then represented as a mass transfer from the lower, cooler layer into the upper, warmer layer.

*ϕ*

_{c}in section 3a. For this, a set of spinup simulations taking the forcing term in (2) to be a prescribed mass sink are carried out. That is,

*F*=

*Q*, where

*Q*can be thought of as a zonally averaged representation of convection associated with the ITCZ (Nieto Ferreira and Schubert 1997). This provides a convenient way to spin up our base states. The heat source is defined as the product of a latitudinally dependent top-hat function

*r*(

*t*). Specifically,

*Q*are shown in the top panels of Fig. 1. The ramping function

*r*(

*t*) slowly increases in magnitude over several days and then decays to zero smoothly. This is done to limit the amount of gravity waves that are produced during this process. Following Nieto Ferreira and Schubert (1997),

*Q*

_{0}= 55.2 m day

^{−1}and the width of the heating is 504 km. This choice of heat source is similar in effect to Merlis et al. (2013) and Ballinger et al. (2015), where a typical half-width of the GCM-simulated zonal-mean precipitation is ≈500 km. As in Nieto Ferreira and Schubert (1997), we use

*F*to be proportional to precipitation

*P*. To do this, we follow Lambaerts et al. (2011) and include a tracer representing a layer-averaged specific humidity

*q*(g kg

^{−1}):

*q*

_{s}represents the saturation specific humidity,

*τ*represents the time scale of precipitation, and

*q*

_{s}takes several forms in our simulations. In particular, we consider the case of a spatially and temporally constant

*q*

_{s}(as in Lambaerts et al. 2011), a fixed meridional profile

*q*

_{s}=

*q*

_{s}(

*y*) (similar to Suhas et al. 2017), and

*q*

_{s}=

*q*

_{s}(

*η*) following the Clausius–Clapeyron relation (as in Vallis and Penn 2020). The latent heat release associated with precipitation implies a mass sink in (2),

*F*=

*αP*, where

*α*corresponds to the strength of latent heating. Precipitation thus couples the thickness and

*q*equations [(2) and (5), respectively] in these simulations. Our focus is on how variations in

*α*impact growth rates for the ITCZ profiles generated in section 3a. A similar study, but for forced-dissipative simulations in a two-layer quasigeostrophic

*β*plane was carried out by Lapeyre and Held (2004). Their main result was to show a transition from jet-dominated to vortex-dominated turbulence as the strength of latent heat release increased. In another two-layer shallow-water version of the problem, Bembenek et al. (2020) varied

*q*

_{s}and found a similar transition in part of parameter space.

In summary, in section 3a, we set *F* = *Q* in spinup simulations to generate ITCZ-like wind profiles at different latitudes. In section 3b, we set *F* = 0 and run initial value problem simulations of these ITCZ profiles to calculate the dry growth rate. Section 4 then takes *F* = *αP* to investigate how precipitation [determined by Eqs. (5) and (6)] impacts the growth. This is done by varying *α*, which relates precipitation to a mass sink in (2).

### b. Energy budget

One can define an moist static energy for our model as *m* = *h* − *αq* or *m* = *η* − *αq*. This is both materially conserved (see also Gill 1982; Bouchut et al. 2009) and conserved in an area-integrated sense. In words, conservation of *m* simply states that the mass removed from our active shallow-water layer is proportional to the moisture removed by precipitation. Note that *m* is independently conserved: it does not represent a reservoir with which potential and kinetic energy can be exchanged. Because of this, and because we will compare moist and dry simulations, it is more convenient to use the dry energy norm. Precipitation then appears as a forcing term for potential energy, analogous to that for any mass source or sink driving a shallow-water system.

*η*equations gives the energy budget for the zonal and eddy components:

*η*and

*P*. The conversion between zonal and eddy components of energy is given by

Note that this budget is similar to that of Lambaerts et al. (2011), but we make a further decomposition into zonal and eddy components. Additionally, in the model used in Lambaerts et al. (2011), the free surface *η* represents the bottom of the atmosphere, so their precipitation forcing terms should be multiplied by −1 relative to ours.

### c. Numerical model and parameter values

To solve the system in (1)–(5), we use a finite-difference model on an Arakawa C grid with leapfrog time stepping. A bi-Laplacian dissipation (with a coefficient of *A*_{h} ≈ 9 × 10^{11} m^{4} s^{−1}) in the momentum and humidity equations removes energy at the smallest scales. The domain is a periodic channel with walls at the north and south boundaries, where free-slip boundary conditions are applied.

The physical parameters of the problem are stated in Table 1. The domain size is 13 440 km in *x* (≈120° in longitude) and 3360 km in *y* (≈30° in latitude) with 512 × 128 grid points. This implies that Δ*x* = Δ*y* ≈ 26 km. The gravitational wave speed is

Parameter values used in dry and constant *q*_{s} simulations.

There are several parameters associated with moisture: *τ*, *q*_{s}, and *α*. We fix *τ* = 3600 s ≈ 25Δ*t* to be the fastest time scale in the problem. For the case of constant saturation humidity, we take *q*_{s} = 1 g kg^{−1}. The parameter *α* is related to the moist gravity wave speed by *q*_{s} (Bouchut et al. 2009). Because of this, *α* cannot be chosen to be arbitrarily large as this would lead to exponentially growing moist gravity waves. We thus require *α* < *α*_{max}, where *α*_{max} ≡ *H*/*q*_{s} = 222 m kg g^{−1}. Specifically, we consider values of *α* corresponding to 25%, 50%, 75%, and 90% of *α*_{max}. Note that the dynamics are controlled by the product *αq*_{s} rather than by *α* or *q*_{s} per se. For example, simulations using *q*_{s} = 10 g kg^{−1} and *α*_{max} = 22.2 m kg g^{−1} gave similar results to simulations using *q*_{s} = 1 g kg^{−1} and *α*_{max} = 222 m kg g^{−1} (not shown). This suggests that increasing saturation humidity, as in climate change simulations, is equivalent to increasing the strength of latent heat release in this model. Therefore, we vary *α* in section 4. We will also explore the sensitivity of the system to different functional forms of *q*_{s} In particular, we set *q*_{s} to be a fixed meridional profile and *q*_{s} to be an exponential function of *η*, following the Clausius–Clapeyron equation in section 4b.

## 3. Dependence of dry dynamics on central forcing latitude *ϕ*_{c}

### a. Base-state profiles

Base-state profiles are spun up by forcing the thickness Eq. (2) using *ϕ*_{c} = 6°, 8°, 10°, 12°, and 14°. Variations in *ϕ*_{c} imply a variation in *f*_{0} and *β*. These spinup simulations are run for 20 days. As expected, given that *r*(*t*) → 0 near the end of the simulations, this produces a set of base-state profiles for which

Profiles of zonal-mean vorticity *ϕ*_{c} are shown in the bottom panels of Fig. 1. Vorticity increases with *ϕ*_{c} and has profiles similar to those found in Nieto Ferreira and Schubert (1997) and qualitatively match the lower-level zonal winds in three-dimensional models (Wang and Magnusdottir 2005; Yokota et al. 2012, 2015). In particular, the maximum zonal-wind shear of our *ϕ*_{c} = 10° case is ≈10 m s^{−1} and this is similar to those studies. Profiles of *ϕ*_{c}. That is, a dip in *y* = 0 is all but absent at *ϕ*_{c} = 6° and becomes increasingly pronounced at higher latitudes. We next examine how these differences impact the barotropic instability properties of these base states.

### b. Instability

To determine the changes in instability, we run simulations initialized using the above meridional profiles for *t* = 15 days for three cases of *ϕ*_{c}. The instability is not apparent in the case where *ϕ*_{c} = 6° and is only beginning to emerge at this time in the case where *ϕ*_{c} = 10°. For *ϕ*_{c} = 14°, on the other hand, the flow is much more developed, with nonlinear effects such as vortex roll up already apparent. For all *ϕ*_{c} tested, the instability shows a wavenumber 8 pattern corresponding to a length scale of around 1700 km, which is close to the equatorial Rossby radius,

Vorticity normalized by 10^{−5} s^{−1} for *ϕ*_{c} = (top) 6°, (middle) 10°, and (bottom) 14°. All snapshots are taken at *t* = 15 days.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Vorticity normalized by 10^{−5} s^{−1} for *ϕ*_{c} = (top) 6°, (middle) 10°, and (bottom) 14°. All snapshots are taken at *t* = 15 days.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Vorticity normalized by 10^{−5} s^{−1} for *ϕ*_{c} = (top) 6°, (middle) 10°, and (bottom) 14°. All snapshots are taken at *t* = 15 days.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

That the flow is more developed for larger *ϕ*_{c} indicates that the growth rate of the instability σ increases with increasing *ϕ*_{c}. This is confirmed in Fig. 3, which shows time series of eddy energy, (9), for each *ϕ*_{c}. Evident in all simulations is a period of exponential growth, and estimation of the growth rates (stated in parentheses in the legend of Fig. 3) show the expected increase with *ϕ*_{c}. The largest growth rate found is approximately 0.92 day^{−1}, which is similar to Yokota et al. (2015), who found a rate of 0.896 day^{−1}. Most of our simulations have a *σ* comparable to the 2-day *e*-folding time found in Nieto Ferreira and Schubert (1997).

Time series of eddy energy *E*_{e} as a function of latitude *ϕ*_{c}. Least squares estimate of the growth rate *σ* indicated in parentheses in the legend (with units of days^{−1}). To quantify *σ*, we calculate a least squares fit to the slope of these curves for the periods where ln(*E*_{e}) grows linearly, indicated by the vertical ticks in each time series.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Time series of eddy energy *E*_{e} as a function of latitude *ϕ*_{c}. Least squares estimate of the growth rate *σ* indicated in parentheses in the legend (with units of days^{−1}). To quantify *σ*, we calculate a least squares fit to the slope of these curves for the periods where ln(*E*_{e}) grows linearly, indicated by the vertical ticks in each time series.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Time series of eddy energy *E*_{e} as a function of latitude *ϕ*_{c}. Least squares estimate of the growth rate *σ* indicated in parentheses in the legend (with units of days^{−1}). To quantify *σ*, we calculate a least squares fit to the slope of these curves for the periods where ln(*E*_{e}) grows linearly, indicated by the vertical ticks in each time series.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

*f*+

*ζ*)/

*h*, evolves according to

*η*≪

*H*has been assumed. Ignoring the advection term in the material derivative, we integrate this expression in time, and since

*y*,0) =

*f*/

*H*from both sides and approximating

*f*≈

*f*

_{0}, we can estimate that the base-state

*Q*, the perturbation PV will be larger when the center latitude of the forcing profile is at higher latitudes (so that |

*f*

_{0}| is larger). For forcing profiles with horizontal length scales small compared to the deformation radius (as is the case for the latent heat release in a tropical convergence zone), the partition between relative and stretching vorticity heavily favors relative vorticity. For example, Fig. 1 shows that the magnitude of

The growth rate is determined by the shear. This, in turn, depends on the forcing strength, forcing duration, and on *ϕ*_{c}. Because *β* varies weakly with *ϕ*_{c} in low latitudes, the *ϕ*_{c} dependence can be thought of as a dependence on *f*_{0}. Thus, for example, a spinup at *ϕ*_{c} = 6° with *r*(*t*) [see Eq. (4)] redefined so that forcing is applied for nine instead of five days produces a vorticity profile similar to our *ϕ*_{c} = 14° (not shown). By contrast, the stretching vorticity profiles remain qualitatively different. Since the dynamics is dominated by relative vorticity, differences in the stretching profiles do not appreciably impact the growth rates. In other words, our result that a poleward shift in the ITCZ implies faster growth rates assumes the position, but not the strength of the heating to be varying with latitude. It is a combination of the two that determines the base-state shear, and by extension, the growth rate.

Vertical wind shear can break up the vertical coherence of developing vortices, potentially inhibiting growth. To explore the robustness of our barotropic results to this process, we ran similar simulations in a two-layer shallow-water model with a prescribed heat source for *ϕ*_{c} ∈ [6°, 14°] configured to produce comparable barotropic shear in our one-layer model. Results showed that the ITCZ breakdown process remained dominated by barotropic instability (not shown). Similar results were found by Wang and Magnusdottir (2005), who studied three-dimensional ITCZ breakdown in a primitive equation model with prescribed heating. Specifically, they found that ITCZ breakdown was dominated by barotropic instability.

This barotropic framework allows us to propose a possible interpretation for the increased frequency of TCs in Merlis et al. (2013) and Ballinger et al. (2015) as a result of an increase in PV forcing as the heating moves to higher latitudes. In those GCM simulations, the latent heating in the ITCZ is internally determined, unlike the prescribed heating here. However, the tropical-mean precipitation in GCMs is constrained by energy budgets (O’Gorman et al. 2012) that are insensitive to ITCZ latitude. In this context, our use of a prescribed heating is reasonable. We can similarly interpret some of the results in Chavas and Reed (2019), who examined aquaplanet simulations with uniform thermal forcing. They showed that increased rotation rates led to an increased frequency of TCs. This is consistent with the point we emphasize above whereby the diabatic PV source term is proportional to the rotation rate. In their case, the forcing was due to storms rather than to an imposed meridional profile. Nevertheless, a similar logic applies: for the same magnitude of latent heat release, faster rotation implies stronger PV forcing. Finally, in Earthlike aquaplanet simulations, Hsieh et al. (2020) suggest that the increase in TC formation results from an increased probability of TC seeds forming when *f*_{0} gets larger and this is consistent with the enhanced diabatic PV generation mechanism that we described.

## 4. Moisture effects on barotropic instability

### a. Sensitivity to latent heat release strength α

Here, we consider how flow-dependent moisture affects the barotropic instability of our model ITCZ. The thickness equation, (2), is forced by *F* = *αP* where *P*, defined by Eq. (6), is determined by the evolution of the humidity equation, (5). We first treat the case of constant *q*_{s} and later explore how the dynamics are impacted if *q*_{s} is a fixed meridional profile and if *q*_{s} has a temperature dependency. Latent heat release strength is varied in a manner similar to Lapeyre and Held (2004). In our system, latent heat release strength is controlled by *α*. As described in section 2c, *α* < *α*_{max} ≡ *H*/*q*_{s} is required in order to ensure that moist gravity wave speeds remain real. We consider simulations with *α*/*α*_{max} = 0.25, 0.5, 0.75, and 0.9. We will consider these values for each *ϕ*_{c} profile shown in Fig. 1. In other words, for each of the ITCZ base states from section 3a, we repeat the growth rate analysis of section 3b, this time allowing for precipitation effects.

Snapshots of vorticity are similar to those in the dry simulations (not shown), suggesting that precipitation has only a modest effect on the dynamics. To quantify this effect, we calculate growth rates, *σ*, and compare them with results from the previous section. Results are shown in Fig. 4. Overall, the reduction compared to dry simulations is modest *ϕ*_{c}, the growth rate shows a larger fractional decrease for larger *α* (i.e., for stronger precipitation forcing). This effect is more pronounced at larger values of *ϕ*_{c} but remains modest overall. The wavenumber of the most unstable mode remains the same as the dry case (a wavenumber-8 pattern, ≈1700 km) and is insensitive to *α*.

Percentage change in moist growth rate *σ* with respect to the dry (*α* = 0) simulation for each *ϕ*_{c}. See Fig. 3 for dry growth rates.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Percentage change in moist growth rate *σ* with respect to the dry (*α* = 0) simulation for each *ϕ*_{c}. See Fig. 3 for dry growth rates.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Percentage change in moist growth rate *σ* with respect to the dry (*α* = 0) simulation for each *ϕ*_{c}. See Fig. 3 for dry growth rates.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Although the effect is modest, that precipitation has a damping effect on barotropic instability merits some discussion. Results in the literature have been mixed. In the context of easterly waves, Thorncroft and Hoskins (1994) found a modest increase in the growth rate in a moist simulation using a three-dimensional primitive equation model. Kwon (1989), in contrast, found negligible differences in growth rates between a dry and moist case of easterly waves using a quasigeostrophic model. Differences in the base-state flow and parameterization of latent heat release are both likely important in comparing these layered or three-dimensional model results to those in our one-layer model. Finally, using a model similar to our own, Lambaerts et al. (2011) studied the barotropic instability of a Bickley jet in simulations taking *q*_{s} to be constant. Unlike us, they found that moisture led to an increase in the growth rate. One difference in our analysis relative to theirs is the definition of zonal and eddy components. Specifically, in their analysis, time-varying zonal averages were considered part of the eddy field. Repeating our analysis using the same base states and methodology as theirs, however, we continue to find that precipitation leads to a decrease in the growth rate. It is thus unclear why Lambaerts et al. (2011) find a growth rate increase in their simulations. In section 4c, we present a physical mechanism that explains how precipitation leads to an increase or a decrease in the growth rate, depending on how saturation humidity is parameterized.

We next analyze the energy budget associated with the zonal and eddy components [(10) and (11), respectively] of the flow. Figure 5 shows terms associated with precipitation, *α*. This contrasts the zonal-to-eddy conversion term *α*. As expected for barotropic instability, this conversion term is positive. Overall, then, precipitation adds zonal PE and removes eddy PE. The zonal PE added, however, is small compared to that present initially (not shown). Related to this, changes in both the base-state shear and the eddy conversion term do not change appreciably in response to this forcing. That

Energy budget terms for simulations with flow-dependent precipitation (*α* > 0) and the corresponding dry case (*α* = 0). (top) Precipitation forcing on eddies *ϕ*_{c} = 10° case.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Energy budget terms for simulations with flow-dependent precipitation (*α* > 0) and the corresponding dry case (*α* = 0). (top) Precipitation forcing on eddies *ϕ*_{c} = 10° case.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Energy budget terms for simulations with flow-dependent precipitation (*α* > 0) and the corresponding dry case (*α* = 0). (top) Precipitation forcing on eddies *ϕ*_{c} = 10° case.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

### b. Nonconstant saturation humidity

So far, our experiments have considered *q*_{s} to be spatially and temporally constant. More realistically, saturation humidity follows the Clausius–Clapeyron relation which suggests an exponential dependence on temperature (proportional to −*η* in our model). In the tropics, this implies that saturation humidity peaks near the equator and decays meridionally. We model this in two ways. First, following Suhas and Sukhatme (2020), we set saturation humidity to a fixed meridional profile, *q*_{s} = *q*_{s}(*y*). Specifically, we take *q*_{s} to depend on the base-state *t* = 0. Second, following Vallis and Penn (2020), we set saturation humidity to evolve with the flow, *q*_{s} = *q*_{s}(*η*). Intuitively, during the initial instability, one would expect similar results between *q*_{s} = *q*_{s}(*y*) and *q*_{s} = *q*_{s}(*η*) since *q*_{s} has a significant impact on how precipitation affects the growth rate.

*γ*=

*L*

_{υ}/

*R*

_{υ}

*T*

_{0}~ 20 is representative of Earth’s tropics (

*L*

_{υ}= 2.4 × 10

^{6}J kg

^{−1},

*R*

_{υ}= 462 J kg

^{−1}K

^{−1}, and

*T*

_{0}= 300 K) (Vallis and Penn 2020) and

*q*

_{0}= 1 g kg

^{−1}. We tested the sensitivity of the growth to variation in

*γ*with

*ϕ*

_{c}= 10° and

*α*= 100 m kg g

^{−1}. Profiles of

*q*

_{s}(

*y*) are shown in the left panel of Fig. 6 for different values of

*γ*. Saturation humidity peaks near the center of the domain and decays away from

*y*= 0, similar to Suhas and Sukhatme (2020). As in the constant

*q*

_{s}case, simulations are initialized with humidity near saturation throughout the domain so as not to bias where barotropic instability triggers precipitation. Larger

*γ*increases humidity and leads to a stronger gradient in

*q*

_{s}. The right panel of Fig. 6 shows the zonally averaged

*q*

_{s}, denoted

*q*

_{s}=

*q*

_{s}(

*η*) and

*γ*= 20 at different times. The solid curves denotes

*t*≤ 16 days) and shows a profile that is remarkably fixed in time. At later times (shown in dotted and dashed lines),

*q*

_{s}=

*q*

_{s}(

*y*). As seen from Fig. 7, however, this is not the case. Specifically, assuming

*q*

_{s}=

*q*

_{s}(

*y*) leads to an increase in

*σ*, whereas assuming

*q*

_{s}=

*q*

_{s}(

*η*) leads to a decrease.

(left) Profiles of saturation humidity, *q*_{s} = *q*_{s}(*y*), as a function of *γ*. (right) Zonally averaged *q*_{s} for simulations with *q*_{s} = *q*_{s}(*η*) and *γ* = 20 at *t* = 0, 10, 15, 20, and 30 days. The solid lines indicate *q*_{s} during linear instability (for *t* ≤ 16 days) and the dotted lines indicates *q*_{s} during nonlinear phase of simulation.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

(left) Profiles of saturation humidity, *q*_{s} = *q*_{s}(*y*), as a function of *γ*. (right) Zonally averaged *q*_{s} for simulations with *q*_{s} = *q*_{s}(*η*) and *γ* = 20 at *t* = 0, 10, 15, 20, and 30 days. The solid lines indicate *q*_{s} during linear instability (for *t* ≤ 16 days) and the dotted lines indicates *q*_{s} during nonlinear phase of simulation.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

(left) Profiles of saturation humidity, *q*_{s} = *q*_{s}(*y*), as a function of *γ*. (right) Zonally averaged *q*_{s} for simulations with *q*_{s} = *q*_{s}(*η*) and *γ* = 20 at *t* = 0, 10, 15, 20, and 30 days. The solid lines indicate *q*_{s} during linear instability (for *t* ≤ 16 days) and the dotted lines indicates *q*_{s} during nonlinear phase of simulation.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Percentage change in growth rate *σ* as a function of *γ* for simulations with *q*_{s} = *q*_{s}(*y*) (black) and *q*_{s} = *q*_{s}(*η*) (gray). The nondimensional parameter *γ* controls the magnitude of the derivative of *q*_{s}, see Eqs. (17) and (18). The simulation with *γ* = 0 corresponds to the constant *q*_{s} case with *α* = 100 m kg g^{−1} and *ϕ*_{c} = 10°.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Percentage change in growth rate *σ* as a function of *γ* for simulations with *q*_{s} = *q*_{s}(*y*) (black) and *q*_{s} = *q*_{s}(*η*) (gray). The nondimensional parameter *γ* controls the magnitude of the derivative of *q*_{s}, see Eqs. (17) and (18). The simulation with *γ* = 0 corresponds to the constant *q*_{s} case with *α* = 100 m kg g^{−1} and *ϕ*_{c} = 10°.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Percentage change in growth rate *σ* as a function of *γ* for simulations with *q*_{s} = *q*_{s}(*y*) (black) and *q*_{s} = *q*_{s}(*η*) (gray). The nondimensional parameter *γ* controls the magnitude of the derivative of *q*_{s}, see Eqs. (17) and (18). The simulation with *γ* = 0 corresponds to the constant *q*_{s} case with *α* = 100 m kg g^{−1} and *ϕ*_{c} = 10°.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

To understand why this occurs, recall that the sign of *P* and *η*′. For constant *q*_{s}, precipitation occurs where *η*′ > 0 (see top panel of Fig. 8) and this leads to a reduction in eddy energy (see top panel of Fig. 5). When *q*_{s} = *q*_{s}(*y*), precipitation shifts to the northeastern and southwestern flanks of negative *η*′ (see middle panel of Fig. 8), thus implying an energy source (and an increased growth rate). Finally, when *q*_{s} = *q*_{s}(*η*), precipitation reverts to a pattern similar to that seen for constant *q*_{s} (see bottom panel of Fig. 8), so that once again, a decrease the growth rate results. In the next section, we propose mechanisms by which the location of precipitation is determined for each of these cases.

Instantaneous eddy free surface *η*′ for (top) constant *q*_{s}, (middle) *q*_{s} = *q*_{s}(*y*), and (bottom) *q*_{s} = *q*_{s}(*η*), with black contours indicating precipitation and dashed (solid) white contours indicate convergence (divergence). All simulations have *α* = 100 m kg g^{−1} and *ϕ*_{c} = 10° and fields are shown on day 15. The cases with *q*_{s} = *q*_{s}(*y*) and *q*_{s}(*η*) have *γ* = 20.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Instantaneous eddy free surface *η*′ for (top) constant *q*_{s}, (middle) *q*_{s} = *q*_{s}(*y*), and (bottom) *q*_{s} = *q*_{s}(*η*), with black contours indicating precipitation and dashed (solid) white contours indicate convergence (divergence). All simulations have *α* = 100 m kg g^{−1} and *ϕ*_{c} = 10° and fields are shown on day 15. The cases with *q*_{s} = *q*_{s}(*y*) and *q*_{s}(*η*) have *γ* = 20.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

Instantaneous eddy free surface *η*′ for (top) constant *q*_{s}, (middle) *q*_{s} = *q*_{s}(*y*), and (bottom) *q*_{s} = *q*_{s}(*η*), with black contours indicating precipitation and dashed (solid) white contours indicate convergence (divergence). All simulations have *α* = 100 m kg g^{−1} and *ϕ*_{c} = 10° and fields are shown on day 15. The cases with *q*_{s} = *q*_{s}(*y*) and *q*_{s}(*η*) have *γ* = 20.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0346.1

### c. Mechanisms leading to precipitation

*η*′. To this end, we consider the linearized eddy moisture equation for each of the three

*q*

_{s}formulations considered. Since simulations are initialized close to saturation, we substitute

*q*=

*q*

_{s}+

*q*′ into the humidity Eq. (5) for each form of

*q*

_{s}and neglect eddy correlation terms. This gives

*q*

_{s}=

*q*

_{0}exp(−

*γη*/

*H*):

For constant *q*_{s}, we establish a negative correlation between horizontal divergence and precipitation, i.e., precipitation in our model must be preceded by a convergent horizontal flow (at least during the initial instability). Since eddy humidity evolves according to (19), it follows that a shallow-water fluid column can only become supersaturated following a period in which it is in a convergent part of the flow. That is, (19) implies that −∇ ⋅ **u**′ > 0 is required for *q*′ to increase. It then follows that precipitation can only occur in regions of convergence. A similar argument follows for *q*_{s} = *q*_{s}(*η*). Even though Eqs. (21) and (22) are more complex, convergence remains the only mechanism that increases *q*′ in a parcel. As a result, precipitation is driven by convergent modes in simulations where *q*_{s} is constant or *q*_{s} = *q*_{s}(*η*). This mechanism is apparent in the top and bottom panels of Fig. 8 where *η*′ > 0 coincides with precipitation (black contours) and convergence (white dashed contours).

For *q*_{s} = *q*_{s}(*y*), on the other hand, there is an additional mechanism by which humidity increases. In particular, the meridional advection of humidity, −*υ*′*dq*_{s}/*dy*, in Eq. (20) increases humidity and leads to precipitation. This mechanism is apparent in the center panel of Fig. 8 which shows that precipitation shifts to the flanks of *η*′ < 0. That the location of precipitation is shifted indicates that rotational modes instead of convergent are driving precipitation as has been suggested for the Madden–Julian oscillation (Pritchard and Bretherton 2014) and moist equatorial Rossby waves (Wheeler et al. 2000).

*η*′. Since convergence drives precipitation for constant

*q*

_{s}and

*q*

_{s}=

*q*

_{s}(

*η*) this mechanism allows us to conclude that

*q*

_{s}are used. In addition to being a prerequisite for precipitation, convergence also implies a thickening of fluid columns. Convergence in the eddy velocity field then implies an increase in eddy thickness

*η*′. For example, in the linear growth phase of barotropic instability

*η*′ evolves according to

*σ*) implies that

*η*′. But since

*η*′. Put together, these arguments imply that regions of precipitation occur where

*η*′ > 0, implying

For the constant *q*_{s} and *q*_{s} = *q*_{s}(*η*) cases, precipitation is associated with convergence. In the growth phase of the instability, *η*′ itself) are also associated with convergence. As such, the mass sink due to precipitation occurs in regions of elevated *η*′, and a sink of potential energy results. Note that this reasoning also applies to the simulations of Lambaerts et al. (2011), despite having their free surface at the bottom of the fluid column. More generally, convergence implies a thickening fluid column and precipitation, the confluence of which leads to a destruction of potential energy and hence a reduction in the growth rate. For the *q*_{s}(*y*) case, advection of the base state *q*_{s}(*y*) profile provides an additional term in the moisture equation, cf. (20). Since *q*_{s} decreases from the center of the ITCZ, meridional flow away from the center of the jet serves to increase *η*′. As such, precipitation tends to occur in regions of meridional flow away from the jet center. That is, precipitation straddles regions of positive and negative *η*′, but with a slight bias toward negative regions, so that a net sink results.

## 5. Conclusions

In this study, we aimed to answer two questions:

How does the latitudinal position of the ITCZ affect the barotropic instability of the associated horizontal flow?

How does flow-dependent moisture and the associated eddy latent heat release impact the ITCZ breakdown in our idealized framework?

To answer these questions, we used a shallow-water model with a prescribed heat source similar to that of Nieto Ferreira and Schubert (1997) and a parameterization for latent heat release that makes use of a humidity tracer as in Bouchut et al. (2009) and Lambaerts et al. (2011).

The prescribed heat source *Q* represents a zonal-mean idealization of the convection associated with the ITCZ and was used to generate the ITCZ-like winds presented in section 3a. Shifting the center latitude of the heating poleward (while keeping the temporal and spatial structure of *Q* fixed) led to increased shear at higher latitudes. It may be, of course, that in a more realistic setting, the heat source itself would also vary with its latitudinal position. Our choice of fixed *Q* was motivated by Merlis et al. (2013) and Ballinger et al. (2015), whose GCM simulations showed similar shifts in ITCZ position without substantial changes to the overall heating. We also performed simulations (not shown) using a stronger heat source at lower latitudes and found that the effect of increasing *Q* is similar to that of shifting the center latitude poleward. A caveat is that a stronger *Q* at lower latitude could be made to produce a shear profile similar to that for a weaker *Q* at higher latitude with the *q*_{s}.

That heating at higher latitudes should result in enhanced shear was explained by a simple integration of the shallow-water PV equation. This showed that an equivalent heat source produced a stronger PV response at higher latitudes and a stronger PV gradient implied a stronger shear. We also ran simulations in a two-layer shallow-water model to assess the relative importance of barotropic and baroclinic instability. For the range of latitudes considered, *ϕ*_{c} ∈ [6°, 14°], barotropic instability remained dominant. This is consistent with Wang and Magnusdottir (2005), who showed that their prescribed heating led to barotropic instability in a three-dimensional primitive equation model.

Then, for each of our base-state profiles, we ran a series of simulations to assess the effect of precipitation on barotropic growth rate *σ*. This was done by varying a parameter related to the strength of latent heat release. First, we considered *q*_{s} to be constant, and found that precipitation led to a decrease in *σ*. This results from a correlation between precipitation and column thickness, both of which increase in response to horizontal convergence. Because of this, precipitation is systematically found in regions of larger *η*′ and an energy sink results. Other prescriptions of *q*_{s} were also considered. For example, taking *q*_{s} = *q*_{s}(*y*) (similar to Suhas and Sukhatme 2020), we found precipitation to lead to an increase in the growth rate. In these simulations, meridional advection shifts precipitation to the flank of warm eddies (where *η*′ < 0), leading to an increase in available potential energy. Finally, if saturation humidity depends on *η* (or temperature, following the Clausius–Clapeyron relation), precipitation leads to a reduction in the growth rate. As in the constant *q*_{s} case, precipitation is driven by convergence which is collocated with cold eddies. In summary, if *q*_{s} is constant or *q*_{s} = *q*_{s}(*η*), precipitation is driven by convergent modes and leads to a growth rate decrease. If *q*_{s} = *q*_{s}(*y*), precipitation can be driven by rotational modes and can lead to a growth rate increase.

To summarize, we found that dry barotropic instability increases for ITCZs generated by a fixed amount of heating when the heat source was placed further from the equator. This was due to the increase in potential vorticity generation with latitude that generated stronger horizontal wind shear. Latent heating during the growth of the instability modestly reduces growth rate if *q*_{s} is constant or a function of *η*. On the other hand, if *q*_{s} is a fixed function of *y* (related to the initial temperature profile), then precipitation led to an increase in the growth rate. This suggests that the choice of saturation humidity prescription can have a significant impact on how precipitation affects the flow. Our results used a simple one-layer shallow-water model and focused on initial value problems and the linear growth phase of instability. Bridging the gap between this sort of simple analysis and one allowing both for more complex dynamics and that considers the more climate-relevant case of statistical equilibria with the ITCZ migrating, undergoing breakdowns, and subsequent reformation present interesting avenues for future research.

## Acknowledgments

This research was supported by the Natural Sciences and Engineering Research Council of Canada. We appreciate three perceptive reviewers, who prompted the systematic investigation of the form of the saturation specific humidity.

## APPENDIX A

### Energy Budget Derivation

*h*=

*H*+

*η*, the total potential energy can be written as ⟨

*gh*

^{2}/2⟩ = ⟨

*gH*

^{2}/2⟩ + ⟨

*gη*

^{2}/2⟩, where terms of the form ⟨

*gηH*⟩ = 0 since ⟨

*η*⟩ = 0. Since

*H*is constant in time, the dynamically interesting component of the total potential energy is the

*available*potential energy, ⟨

*gη*

^{2}/2⟩.

*h*=

*H*+

*η*with

*η*≪

*H*. A time series of the true total energy and approximate total energy in the system shows a less than 1% variation with respect to the initial energy (not shown). This suggests that the approximation does not lead to substantial errors.

*η*equation. In particular, under the true total energy definition, precipitation appears as

*h*dependence in the kinetic energy. Under the approximation

*h*≈

*H*, the first term vanishes, so that the approximated total energy budget reads

*η*equation into its zonal and eddy components as

*gη*′ and average the resulting equations over the domain:

Finally, combining the

## APPENDIX B

### Eddy Moisture Budget Derivation

*η*, i.e., Eq. (21). We linearize the humidity equation, Eq. (5), about the state

*q*=

*q*

_{s}(

*η*) +

*q*′(

*x*,

*y*,

*t*). Using the chain rule, we have

*η*equation and approximating

*h*≈

*H*, we replace the term in the square brackets and rearrange to find Eq. (21):

*q*

_{s}=

*q*

_{0}exp(−

*γη*/

*H*), and Eq. (22) results:

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^{1}

Note that in Lambaerts et al. (2011), *η* is at the bottom of the atmosphere, while our model setup has *η* at the top of the lower layer of the atmosphere (e.g., Lindzen and Nigam 1987).