a. Shallow-water model

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.

We begin by producing ITCZ-like base states for different 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 $\tilde{Q}\left(y\right)$ and a ramping function r(t). Specifically, $Q\left(y,t\right)=\tilde{Q}\left(y\right)r\left(t\right)$, where

$\tilde{Q}\left(y\right)={\displaystyle \frac{Q_0}{2}}\left[\tanh \left({\displaystyle \frac{y+252\hspace{0.17em}\text{km}}{25\hspace{0.17em}\text{km}}}\right)-\tanh \left({\displaystyle \frac{y-252\hspace{0.17em}\text{km}}{25\hspace{0.17em}\text{km}}}\right)\right]$(3)

and

$r\left(t\right)={\displaystyle \frac{1}{2}}\left[\tanh \left({\displaystyle \frac{t-4\hspace{0.17em}\text{days}}{1\hspace{0.17em}\text{day}}}\right)-\tanh \left({\displaystyle \frac{t-9\hspace{0.17em}\text{days}}{1\hspace{0.17em}\text{day}}}\right)\right].$(4)

The latitudinally and time-varying components of 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), Q₀ = 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 $\tilde{Q}\left(y\right)$ to generate ITCZ-like horizontal winds: the mass sink is balanced by mass convergence of the layer’s horizontal winds, as is found in the near-surface flow of the ITCZ region. We note that theories of the tropical general circulation of the atmosphere typically do not take the latent heating as external to the flow (e.g., Held and Hou 1980; Neelin and Held 1987; Emanuel et al. 1994; Merlis 2015) because of its intimate link to the vertical velocity (Sobel and Bretherton 2000). Here, our aim is singly to spin up idealized ITCZs for a subsequent instability analysis done in section 3b and this is an expedient approach.

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(top left) Heating function $\tilde{Q}\left(y\right)$ and (top right) ramping function r(t). (bottom left) Zonal-mean vorticity $\overline{\zeta }$ and (bottom right) free surface $\overline{\eta }$ of the spun-up base state. Meridional position y is R_e(ϕ − ϕ_c), where R_e is Earth’s radius. Vertical dotted lines indicate position of the equator.

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

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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.

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 × 10¹¹ m⁴ 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 Δx = Δy ≈ 26 km. The gravitational wave speed is $c_d=\sqrt{gH}\approx 47\hspace{0.17em}\text{m}\hspace{0.17em}{\text{s}}^{-1}$. As a result, the time step is limited to $\text{Δ}t=0.25\text{Δ}x/c_d\approx 140\hspace{0.17em}\text{s}$.

Table 1.

Parameter values used in dry and constant q_s simulations.

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There are several parameters associated with moisture: τ, q_s, and α. We fix τ = 3600 s ≈ 25Δt 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 $c_m=\sqrt{g\left(H-\alpha q_s\right)}$ for constant 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.

a. Base-state profiles

Base-state profiles are spun up by forcing the thickness Eq. (2) using $F=Q=\tilde{Q}\left(y\right)r\left(t\right)$ [see (3) and (4)] for ϕ_c = 6°, 8°, 10°, 12°, and 14°. Variations in ϕ_c imply a variation in f₀ 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 $\overline{\upsilon }\left(y\right)\approx 0$. Our reference base states are then defined by $\overline{u}\left(y\right)$ and $\overline{\eta }\left(y\right)$ at the end of these simulations.

Profiles of zonal-mean vorticity $\overline{\zeta }=-{\overline{u}}_y$ and $\overline{\eta }$ for different choices of 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 $\overline{\eta }$ show a qualitative change with ϕ_c. That is, a dip in $\overline{\eta }$ centered on 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 $\overline{\zeta }$ and $\overline{\eta }$ in Fig. 1 as base states to which we add white noise. Figure 2 shows snapshot of the vorticity at 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, $\sqrt{c_d/\beta }\approx 1400\hspace{0.17em}\text{km}$.

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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

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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).

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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

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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 f₀. 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 f₀ gets larger and this is consistent with the enhanced diabatic PV generation mechanism that we described.

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 $\left(\mathrm{\lesssim }10\text{\%}\right)$. In other words, precipitation is seen to have a weak de-energizing effect on the instability. For fixed 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.

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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

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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, ${\mathcal{P}}_z$ and ${\mathcal{P}}_e$ [(12) and (13), respectively] as well as the zonal-to-eddy conversion term ${\mathcal{C}}_z$. During the initial instability, ${\mathcal{P}}_e<0$ and ${\mathcal{P}}_z>0$. That is, precipitation generates zonal potential energy and reduces eddy potential energy. After the linear phase of the instability, an opposite-signed effect is evident, although this is weaker by comparison. The amplitudes of these tendencies increase markedly with α. This contrasts the zonal-to-eddy conversion term ${\mathcal{C}}_z$, (14), which is essentially independent of 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 ${\mathcal{P}}_e$ is a net sink, however, implies a reduction in barotropic growth rates.

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Energy budget terms for simulations with flow-dependent precipitation (α > 0) and the corresponding dry case (α = 0). (top) Precipitation forcing on eddies ${\mathcal{P}}_e$, (13); (middle) precipitation forcing on the zonal component ${\mathcal{P}}_z$, (12); and (bottom) conversion from zonal to eddy kinetic energy ${\mathcal{C}}_z$, (14), for the ϕ_c = 10° case.

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

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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 $\overline{\eta }$ at 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 $\eta \approx \overline{\eta }$ which does not vary significantly. However, we will show that the formulation of q_s has a significant impact on how precipitation affects the growth rate.

We choose two formulations of saturation humidity,

$q_s=q_s\left(y\right)=q_0\exp \left[-\gamma {\displaystyle \frac{\overline{\eta }\left(y,t=0\right)}{H}}\right],$(17)

$q_s=q_s\left(\eta \right)=q_0\exp \left(-\gamma {\displaystyle \frac{\eta }{H}}\right),$(18)

where γ = L_υ/R_υT₀ ~ 20 is representative of Earth’s tropics (L_υ = 2.4 × 10⁶ J kg⁻¹, R_υ = 462 J kg⁻¹ K⁻¹, and T₀ = 300 K) (Vallis and Penn 2020) and q₀ = 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 $\overline{q_s},$ for the simulation with q_s = q_s(η) and γ = 20 at different times. The solid curves denotes $\overline{q_s}$ during the initial instability (t ≤ 16 days) and shows a profile that is remarkably fixed in time. At later times (shown in dotted and dashed lines), $\overline{q_s}$ has more pronounced variations as is to be expected when eddies have a stronger influence on the flow. Given that $\overline{q_s}$ is essentially fixed in time over the course of the instability, one would expect the impact of precipitation in these simulations to be similar that in simulations using 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.

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(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

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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

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To understand why this occurs, recall that the sign of ${\mathcal{P}}_e$ determines whether precipitation increases or decreases the growth rate. Recall also that whether ${\mathcal{P}}_e$ acts as energy source or sink depends on the sign of correlations between 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.

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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

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c. Mechanisms leading to precipitation

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).

For the constant q_s and q_s = q_s(η) cases, precipitation is associated with convergence. In the growth phase of the instability, ${\eta }_t^{\prime }$ (and thus 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.