## 1. Introduction

It has been known for some time that the momentum transport by tropical eddies plays a first-order role for the determination of the mean tropical winds in the terrestrial atmosphere (Lee 1999). Dima et al. (2005) performed a comprehensive analysis of the tropical momentum budget in NCEP reanalysis and showed that the stationary (climatological seasonal-mean) tropical eddies transport westerly momentum into the equator in the annual mean and from the winter to the summer hemisphere during the solstices. They proposed that the momentum transport could be associated with the atmospheric response to asymmetric tropical heating. This is supported by findings from aquaplanet simulations with zonally varying SST in the Northern Hemisphere subtropics, which also produce cross-equatorial eddy momentum transport in a solstitial setting (Shaw 2014).

The fact that zonally localized tropical heating may give rise to equatorward momentum transport is also well known from idealized studies (Suarez and Duffy 1992; Kraucunas and Hartmann 2005). Beyond the terrestrial atmosphere, this forcing is thought to be responsible for the superrotation of tidally locked planets due to their strong day–night heating contrast (Merlis and Schneider 2010; Showman and Polvani 2011). Although the eddy acceleration has been traditionally attributed to the meridional propagation of Rossby waves forced by the heating (Held 1999; Laraia and Schneider 2015), Showman and Polvani (2011, hereafter SP11) proposed an alternative mechanism involving the interaction between the Rossby and Kelvin components of the Matsuno–Gill response. This interaction has also been found to induce equatorward momentum transport and superrotation in the absence of external forcing (Zurita-Gotor and Held 2018).

A key difference between tropical and extratropical eddy-driven jets is the role played by the divergent flow. Extratropical eddy-driven jets are driven by vorticity mixing and vorticity fluxes and can be understood in terms of purely rotational dynamics (Vallis 2017). In contrast, in both SP11 and Zurita-Gotor and Held (2018) the zonal acceleration is due to vertical rather than meridional momentum advection (see also Showman and Polvani 2010). Zurita-Gotor (2019a) showed that the same is true in the deep Earth tropics, in which the eddy acceleration is dominated by the convergence of the *divergent* momentum flux *r* and *d* subscripts indicate rotational and divergent components).

Motivated by these observations, Zurita-Gotor (2019b, hereafter Z19b) studied the determination of the divergent momentum flux. The direction of this flux depends on the phase relation between the rotational and divergent anomalies rather than on the meridional tilt of the streamlines as is the case for the traditional rotational momentum flux

Figure 1 shows the observed structure of the *k* = 1 stationary wave using ERA-Interim data (Dee et al. 2011) during the years 1979–2016, integrated between 150 and 300 hPa. The left panels show with contours the seasonal-mean geopotential height and with shading the large-scale divergence, coarse grained to eliminate the fine ITCZ structure (meridional wavenumbers |*l*| > 3 have been filtered out). We can see that the divergence field tilts eastward with latitude moving away from its maximum near the equator in the summer hemisphere during all seasons. To our knowledge, this robust divergence pattern is not broadly recognized in the literature, except for its regional manifestations during austral summer: the South Pacific and South Atlantic convergence zones (e.g., Van Der Wiel et al. 2015). However, as Fig. 1 shows, a similar tilt is detectable in the Northern Hemisphere and/or during other seasons when the fine ITCZ structure is filtered out. The robustness of the divergence tilt suggests that it has a dynamical rather than geographical origin, which is consistent with idealized studies of the South Pacific convergence zone (Van Der Wiel et al. 2016).

The observed seasonal cycle of the eddy momentum flux is consistent with this structure. During DJF, when the wave source shifts into the Southern Hemisphere, the southward cross-equatorial momentum flux into that latitude (Fig. 1c) is consistent with the southwest-to-northeast divergence tilt found over the propagation region. Reversibly, when the wave source moves to the Northern Hemisphere during JJA, the northward cross-equatorial momentum flux (Fig. 1f) is consistent with the northwest-to-southeast tilt to its south. During both seasons, the momentum flux is dominated by its divergent component (red lines) as noted above.

The weak eddy momentum flux sensitivity on the mean flow found by Z19b is at odds with previous nonlinear shallow-water results by Kraucunas and Hartmann (2007, hereafter KH07), who found that the interhemispheric eddy momentum flux in their model was enhanced by the cross-equatorial Hadley cell. Z19b speculated that his different results might be due to the use of a weak temperature gradient approximation (WTG; Sobel et al. 2001), in which the eddy divergence is kept fixed as the meridional flow is changed. The impact of the Hadley cell on the eddy momentum flux might be larger if the meridional flow also affected the divergence, for instance modulating its tilt. A cursory inspection of the KH07 results lends some credence to this idea, as their simulations with large cross-equatorial momentum transport have a more pronounced divergence tilt (cf. their Figs. 2f and 5f to their Figs. 2c and 5c). The eddy momentum flux and the divergence tilt are both enhanced with a Hadley cell (their Fig. 5f).

The goal of this study is to investigate the sensitivity of the eddy momentum flux in simple variants of the classical Gill (1980) model, in an attempt to reconcile the conflicting results of Z19b and KH07. The model is described in section 2, and sections 3–5 present the main results. Section 3 addresses the WTG limit, in which the divergence field is constrained by the prescribed heating. Section 4 investigates the impact of thermal damping on the solutions–a key difference is that the divergence field is now internally determined. Section 5 moves away from the homogeneous framework to replicate and analyze the results of KH07. Section 6 concludes with a summary.

## 2. Model formulation

The Gill (1980) model consists of the shallow-water momentum and continuity equations on an equatorial beta plane, linearized about a resting basic state and forced by prescribed heating. In its original formulation, Gill’s equations were derived for the first baroclinic mode using a vertical modal decomposition. However, equivalent equations can be derived under alternative frameworks. For instance, Neelin (1988) applied these equations to model the boundary layer flow, in which context the use of strong friction (required for realistic solutions) is more palatable.

*V*motivated by the presumed impact of the Hadley cell on meridional propagation and eddy momentum fluxes (Kraucunas and Hartmann 2007). The governing forced, steady equations are

*u*′ and

*υ*′ are the horizontal eddy velocities and

*D*′ = ∂

_{x}

*u*′ + ∂

_{y}

*υ*′ is the horizontal eddy divergence,

*ϕ*′ =

*gh*′ is geopotential (

*h*′ is the shallow-water depth), and

*H*is the mean layer depth, which we may regard as a measure of the stratification. This parameter is expressed in terms of the gravity wave speed

*a*and

*b*are the mechanical and thermal damping rates, respectively. For simplicity, we only include the meridional advection by the Hadley cell in the momentum equations (sensitivity experiments suggest a weak impact on the continuity equation).

*ξ*′ = ∂

_{x}

*υ*′ − ∂

_{y}

*u*′ is relative vorticity. In the inviscid limit

*a*= 0 and with no meridional flow, this equation reduces to simple Sverdrup balance:

*υ*′ = −

*yD*′. In this limit, the solution can be calculated analytically as shown by SP11 (their appendix C).

*υ*′ and

*D*′:

Here, we consider two cases. With no Newtonian cooling (*b* = 0), Eq. (1c) reduces to *D*′ = *Q*′, which we can substitute above to solve for *υ*′ given *Q*′. Bretherton and Sobel (2003) refer to this as the WTG limit. Because the divergence field is directly prescribed, this limit provides a useful device for testing the sensitivity of the eddy momentum fluxes on the divergence tilt suggested by our previous work. This is done in section 3.

With nonzero *b*, we combine the momentum equations to find a relation between *ϕ*′ and *υ*′, which we can use to eliminate *ϕ*′ in Eq. (1c) following Bretherton and Sobel (2003). Substituting in Eq. (3), we then get a closed expression for *υ*′. In this general case, the divergence is internally determined and may develop meridional tilts even when the heating does not tilt. Section 4 studies this limit aiming to understand the determination of the divergence tilt in our model.

To replicate the results of KH07, we also consider in section 5 an inhomogeneous formulation with nonzero zonal wind and latitude-dependent *U*(*y*), *V*(*y*), and *H*(*y*) diagnosed from their climatologies. The resulting system of linearized equations must be solved numerically.

*υ*′ to calculate all other variables and the eddy momentum flux

*k*= 1) zonal harmonic (Zurita-Gotor 2019a), we force this single wavenumber instead of using localized heating as most previous studies. The forcing is defined by the following expression:

*λ*= (

*c*/

*β*)

^{1/2}is the equatorial deformation radius. We use a forcing amplitude

*Q*

_{0}= 1.5 × 10

^{−6}s

^{−1}, which is on the order of the observed

*k*= 1 eddy divergence in the upper troposphere.

To produce cross-equatorial eddy momentum fluxes *Q*′ needs to be asymmetric about the equator. Traditionally, this has been achieved by shifting the heating off the equator or adding an antisymmetric component to *Q*′. An alternative way to break the symmetry is by tilting the heating meridionally using a nonzero

We solve the above equations numerically, using a spectral decomposition in *x* and finite differences in *y* in a rectangular domain with size *L*_{x} × *L*_{y}, where *L*_{x} = 2*πa* and *L*_{y} = 60*λ*. Using *β* = 2Ω/*a* (Ω and *a* are the terrestrial rotation rate and radius) and a typical gravity wave speed *c* = 50 m s^{−1}, the deformation radius is *λ* ≈ 1500 km. These are the same parameters used by Bretherton and Sobel (2003). With *V* = 0 the solutions of these equations are known to be meridionally trapped (Matsuno 1966) but with nonzero *V* meridional propagation is allowed (Schneider and Watterson 1984), which may give rise to resonance with rigid walls. A sponge damps the eddies near the walls to prevent backward reflection and resonance. Sensitivity experiments changing the meridional domain size and resolution (NY = 760) and the sponge damping rate have shown our results to be robust. When thermal or frictional damping are included, we take as reference, control values the same damping rates as Bretherton and Sobel (2003): *a* = *b* = 0.15*c*/*λ*.

## 3. The WTG limit

*a*= 0 in a resting basic state (

*V*= 0). With untilted heating, both the rotational and divergent components of

*l*the direction of

*ω*

_{0}

*l*, where

*ω*

_{0}(

*k*,

*l*) is the free mode frequency, westward for the dominant grave modes. Appendix A generalizes this result with a full meridional wave spectrum and shows that in a resting basic state the domain-integrated eddy momentum flux by zonal wavenumber

*k*is given by

Consistent with this result, with the control westward-tilted heating (shading in Fig. 2a) the divergent momentum flux is northward (Fig. 2b, red line), as the forcing biases the *l* (cf. the solid and dotted black lines in Fig. 2c). The meridional

However, the *full* eddy momentum flux has a very different structure due to the impact of the rotational component *υ*′ = −*yD*′ implies *υ*′ = 0. Cross-equatorial propagation requires violation of Sverdrup balance, or an equatorial vorticity source.

Kinematically, we can understand the sign of

In the next subsections, we discuss two processes that may be important for violating Sverdrup balance and weakening the rotational momentum flux in observations.

### a. Violation of Sverdrup balance: The impact of friction

It is well recognized that linear tropical models need strong friction to produce a realistic circulation, which has been traditionally attributed to the missing impact of nonlinearity (e.g., Ting and Held 1990). SP11 show that friction plays a key role for eddy momentum transport in the equatorially symmetric Gill problem. With nonzero friction, all their simulations produce eddy momentum transport into the equator due to the interaction between the Kelvin and Rossby components of the Gill response, as the differential propagation of these two waves produces an eastward tilt of the isohypses moving toward the equator (cf. Fig. 3d). Friction is essential to this mechanism because in its absence, there can be no steady Kelvin wave response^{1} as the zonal momentum balance equation [Eq. (1a)] requires geopotential anomalies to vanish at the equator (Fig. 2a).

Figure 3a describes the sensitivity of the eddy momentum fluxes to friction with the control westward tilt. As friction is increased, a region of meridional eddy momentum convergence encompassing the equator appears. The eddy momentum fluxes reverse to the south of the equator and we now observe cross-equatorial propagation from the northern to the Southern Hemisphere. Looking at the partition between the rotational and divergent components, we can see that both

This spatial structure is consistent with the arguments of SP11 when applied to a tilted heating field. In particular, the differential eastward propagation by the Kelvin wave at the equator increases the northwest-to-southeast tilt imparted by the forcing to the north of the equator but opposes that tilt to the south of the equator. This is illustrated in Fig. 3 for the responses with weak (half the control value, Fig. 3d) and strong (twice the control value, Fig. 3e) heating tilts. With weak heating tilt, the geopotential tilt is dominated by the Kelvin–Rossby differential propagation and approaches the symmetric solutions of SP11. With strong heating tilt, the geopotential tilt still reverses to the south of the equator but the tilt is much reduced due to the two competing effects. Figure 3f shows the eddy momentum flux for both cases.

To summarize, friction affects the phase tilt of the response, enhancing (reducing) the tilt imparted by the forcing on the summer (winter) side of the equator. As a result, the rotational momentum fluxes weaken on the winter side and the cross-equatorial momentum flux is dominated by the divergent component as in observations.

### b. Role of meridional flow

It is not clear that the use of strong friction in the tropical upper troposphere is justified. Additionally, although the direction of the eddy momentum fluxes in Fig. 3 is consistent with observations, the magnitude of these fluxes is too weak. An alternative process that could violate Sverdrup balance in the inviscid limit is the meridional vorticity advection by the Hadley cell. The meridional flow breaks the symmetry of Rossby’s dispersion relation and favors propagation in the direction of the flow (Schneider and Watterson 1984), which has been argued to be relevant for the cross-equatorial propagation of Rossby waves (Li et al. 2015). Z19b note that by affecting Rossby’s natural frequency, the meridional flow may also shift the forced response toward meridional modes with *Vl* > 0.

The meridional Hadley flow *V* has a profound impact on the model’s inviscid solution. Figure 4a describes the sensitivity of the momentum flux using the control westward tilt and varying southward meridional flow in the range *V* = 0–2 m s^{−1}. The negative *V* sustains southward propagation (positive momentum flux) that increases with *V*. As the solution is no longer meridionally trapped, boundary sponges are used to prevent reflection of the waves at the southern wall and ensuing resonant behavior. The sensitivity of the momentum flux is almost entirely due to the weakening of the rotational component *V* (Fig. 4c) consistent with Z19b. With realistic *V*, the rotational momentum flux becomes small and the solution is dominated by the northward divergent flux as in observations.

*V*affects Rossby’s dispersion relation:

Figure 4d shows the westward Rossby frequency −*ω*_{0}(*k* = 1, *l*; *V*) for all values of *V* considered. For the longest waves *ω*_{0} is dominated by the *β* term and insensitive to *V*, but for |*l*| ≳ 0.5 × 10^{−6} m^{−1} modes with negative (positive) *l* have smaller (larger) |*ω*_{0}| and hence a stronger (weaker) response to forcing. The resonant/propagating wavenumber, defined by *ω*_{0}(*l*_{0}) = 0, is negative and shifts to smaller values of |*l*| as the southward flow strengthens.

Figure 4f shows the implications of this structure for the rotational response. The dotted black line shows the normalized divergence spectrum and the solid black line the spectrum of the Rossby wave source *l* due to the imparted southeast–northwest tilt. With a resting basic state (and no friction) the spectrum of the streamfunction response agrees with that of the Rossby wave source up to a normalization constant. However, the two spectra diverge as |*V*| increases due to its impact on *ω*_{0}. The most obvious effects are a reduction in the response at positive *l* and the emergence of a strong peak at the propagating wavenumber *l*_{0}. The eddy momentum flux cospectra in Fig. 4g show that the former is key to the reduction in the southward rotational momentum flux (thick blue line) when |*V*| increases (*V* = −1.5 m s^{−1} here) relative to the control resting state (thin blue line). The divergent momentum flux (red lines) is much less affected. The strong positive peak at the propagating wavenumber reflects the large contribution (sensitive to domain size) of this scale to the meridionally integrated momentum flux.

The above analysis implies that advection by the Hadley cell modulates the tilt of the rotational response, favoring modes that tilt westward in the direction of *V* (*Vl* > 0). When the large-scale divergence field is dominated by modes with *Vl* < 0, the tilt of the response and hence the rotational momentum fluxes weaken, or even reverse, with increasing |*V*|. The impact on the geopotential field with *V* = −1.5 m s^{−1} can be appreciated in Fig. 4e. It is apparent that the geopotential tilt is much reduced compared to the *V* = 0 solution in Fig. 2a. Note that although the equatorial geopotential anomalies no longer vanish with nonzero *V* (zonal geopotential gradients can now be balanced by meridional momentum advection), the equatorial Kelvin wave is still very weak.

## 4. Newtonian cooling results

The sensitivity of the rotational eddy momentum fluxes to meridional flow uncovered in the previous section may help reconciling the conflicting results of Z19b and KH07 on the Hadley cell impact on tropical eddy momentum fluxes. Another factor that could play a role is the modulation of the eddy divergence field by the Hadley cell. As noted in the introduction, the eddy divergence is qualitatetively different in the simulations of KH07 with and without a Hadley cell. In this section we investigate the relevance of the above arguments when the divergence field is internally determined. The key questions that we address are the following. How is the divergence field internally determined in the full model? Can the meridional flow impact the divergent momentum flux through the modulation of the divergence field? To answer these questions, we add thermal damping to the previous setup and let the divergence field be internally determined forcing with a nontilted *Q*′.

Interestingly, inclusion of Newtonian damping makes the divergence field tilt eastward from the equator in both hemispheres as observed when the heating *Q*′ does not tilt. This is shown in Fig. 5a for the solution with no friction and no meridional flow. Figure 5b describes in more detail the phase relation between *D*′ and *Q*′ for this solution and shows that *D*′ leads *Q*′ at all latitudes, with their phase difference increasing from 0 at the equator to *π*/2 at high latitudes (see also SP11). We can understand the phase lag between *D*′ and *Q*′ by noting that for a Rossby wave, we expect positive geopotential anomalies to the west of the divergence maximum and negative geopotential anomalies to the east. As a result, to the west of the *D*′ maximum, Netwonian cooling compensates part of the prescribed heating *Q*′ and *D*′ < *Q*′, while to the east, Netwonian heating adds up to *Q*′, leading to the observed eastward shift of *D*′. In this context, the eastward tilt of *D*′ with latitude reflects the increasing impact of Newtonian cooling moving poleward (SP11, their appendix C).

Although the divergence field tilts in the right direction, the eddy momentum fluxes are very different from observations (Fig. 5c) for reasons discussed in section 3. As before, there is strong compensation between the rotational and divergent contributions to the eddy momentum flux, with the former dominating. This produces poleward eddy momentum fluxes in both hemispheres, indicating equatorward propagation of the off-equatorial Rossby waves. However, there can be no acceleration at the equator in this limit, as noted in section 3. When the symmetry of the problem is broken by tilting or shifting the heating the momentum flux becomes asymmetric but there is still no cross-equatorial propagation consistent with Sverdrup balance (not shown).

Friction has a profound impact on the solution by allowing a steady Kelvin wave response (Fig. 5d). Near the equator, the Kelvin response is associated with positive (negative) geopotential anomalies to the east (west) of the *D*′ maximum. Using similar arguments to those above, it follows that *D*′ must lag *Q*′ over this region (Fig. 5e) in contrast with what we found for the Rossby response in the inviscid case. On the other hand, away from the equator we still observe a Rossby response with positive geopotential anomalies to the west of the divergence maximum, which implies that *D*′ again leads *Q*′ over this region. The westward (eastward) shift of *D*′ relative to *Q*′ near (away from) the equator is then associated with an eastward tilt of the divergence field moving poleward, most evident^{2} in Fig. 5e. The phase lag between *ϕ*′ and *D*′ now exceeds *π*/2 at high latitudes, where the weak negative

As discussed by SP11, the interaction between the Kelvin and Rossby waves produces equatorial eddy momentum flux convergence in this problem. Figure 5f shows that this convergence is due to

Larger momentum fluxes are obtained when Sverdrup balance is violated through the addition of meridional flow. Figures 5j–l show the inviscid solution with *V* = −1 m s^{−1}. With no friction, the Kelvin response is very weak so that positive *ϕ*′ anomalies are found westward of the *D*′ maximum except very close to the equator. Consistent with this, the divergence field leads the heating *Q*′, with the eastward *D*′ shift increasing with latitude away from the equator in both hemispheres as in Fig. 5a. However, the divergence field is also affected by the meridional flow, becoming asymmetric about the equator. The divergence phase tilt is enhanced to the south of the equator and reduced to the north (cf. the solid and dotted red lines in Fig. 5k), and a secondary *D*′ maximum appears at high southern latitudes collocated with regions of negative height (Fig. 5j). Since this divergence maximum extends beyond the forcing region, it must be forced by the circulation. As discussed in appendix B, the negative

Associated with these changes in the divergence field, there is now northward cross-equatorial divergent momentum flux. This stands in contrast with the results of Z19b, who found that *V* and vanished at the equator in his model *when the divergence field was kept fixed as V was changed* (see his Fig. 3f). On the other hand, the rotational eddy momentum flux *V* = 0 case, the geopotential phase lines now tilt in the opposite direction to the divergence instead of sharing the same tilt as in Fig. 5b.

Figure 6a describes the sensitivity of the two eddy momentum flux components when *V* is changed from −3 to 3 m s^{−1} with the control diabatic damping. Meridionally integrated, both *V*. For small |*V*| the *V*| increases. The cross-equatorial momentum fluxes exhibit a similar dependence, but the divergent momentum flux catches up with *V*| (≈1 m s^{−1}, not shown).

*V*, Eq. (6) is no longer exact and must be replaced by (see appendix A):

*α*=

*f*(

*l*/

*l*

_{0}) is a correcting factor that enhances the contribution of scales near

*l*

_{0}to the integrated momentum flux. The sensitivity of

*V*may arise from changes in this correcting factor with finite

*l*

_{0}and/or from changes in the

*l*with negative (positive)

*V*, enhancing the eastward tilt of the divergence in the direction of

*V*(

*Vl*< 0).

Figure 6c shows that Eq. (8) (red line) reproduces very well the sensitivity of the simulated *α* = 1, equivalent to neglecting *V* and using Eq. (6). The prediction is quite good, suggesting that changes in the *V* play the dominant role for the simulated dependence. In contrast, the prediction computed using a *V*-dependent *α* with constant

*K*

_{d}-weighted mean meridional wavenumber:

*V*in Fig. 6b (thick red line). Inspection of the divergent response suggest two factors that may explain the

*V*found in our simulations. First, inclusion of

*V*makes the divergence tilt asymmetric about the equator as noted above, so that the divergence phase changes more rapidly in the downstream direction of

*V*(Fig. 6d, contours). Additionally,

*K*

_{d}itself becomes equatorially asymmetric and also reaches its maximum in the downstream direction of

*V*, over the region with rapid phase changes (Fig. 6d, shading).

## 5. Comparison with KH07

We next investigate the relevance of our findings for the eddy momentum flux sensitivity described by KH07 in nonlinear shallow-water simulations. The first question is whether our simple linear model can replicate their results. With this aim, we now use an inhomogeneous model with latitude-dependent *U*(*y*), *V*(*y*), and *H*(*y*), diagnosed from the climatologies in Fig. 1 of KH07. We take into account the *U* structure to compute the background absolute vorticity and its gradient, and include the vorticity damping by the zonal-mean divergence ∂_{y}*V*. We use the same localized heating and the same frictional and diabatic time scales (*a*^{−1} = *b*^{−1} = 10 days) as KH07. The only, important difference between the two models is our use of a linear beta-plane formulation.

Figure 7 shows results for some select KH07 cases: (i) a resting basic state with equatorially symmetric heating (cf. to their Figs. 2a,c), (ii) a solstitial basic state with equatorially symmetric heating (their Figs. 7a,c), (iii) a solstitial basic state with off-equatorial heating (their Figs. 7d,f), and (iv) a solstitial setting with no Hadley cell (their Fig. 9b). The left panels show the divergence and geopotential responses, using the same contour intervals and shading scheme as KH07 to facilitate the comparison. The center column of Fig. 7 shows the phase tilt for *k* = 1 (although KH07 use zonally localized heating, this zonal wave provides the dominant

Away from the equator, the agreement between the two models is remarkably good, both in structure and magnitude (except for the solstitial no-Hadley simulation). The main deficiency is found at the equator, where the Kelvin wave is nearly absent. This is consistent with the argument in section 3a relating the amplitude of the Kelvin wave response to friction, for KH07 use very weak friction. The stronger Kelvin wave in their simulations suggests that nonlinear advection is important for balancing the zonal pressure gradient at the equator in those simulations, consistent with the notion that nonlinearity provides the main source of friction in the tropics. This deficiency, however, does not seem to be important for the eddy momentum fluxes, which our model reproduces reasonably well. The failure of our model to capture the nearly symmetric KH07 response in the solstitial no-Hadley case suggests that friction/nonlinearity might be more important in the absence of a Hadley cell, consistent with the Sverdrup constraint.

KH07 did not separate the momentum fluxes into their rotational and divergent components. In our model, *U* to zero and using a constant *V* and *H* characteristic of the solstitial state) the rotational momentum flux is much enhanced, while the divergent momentum flux has a similar magnitude (Figs. 8a–c). This simulation also exhibits a much more pronounced geopotential phase tilt than found in the inhomogeneous model.

Consistent with the dominance of the divergent component, the eddy momentum flux enhancement with a Hadley cell is almost entirely due to *k* = 1 component of the solution, while Fig. 8e shows the coarse-grained divergence, filtered meridionally to retain only wavelengths longer than 4000 km (|*l*| ≤ 1.6 × 10^{−6} m^{−1}). As in observations (see Fig. 1 and Z19b), the eastward divergence tilt moving away from the equator only becomes apparent with this filtering. Note that the bulk of the eddy momentum flux is due to these long scales, as shown by the meridional cospectra.

To investigate the impact of *V* in more detail, Fig. 8f shows the sensitivity of *V*(*y*) of KH07 by a constant factor *γ* keeping everything else unchanged, with *γ* ranging from 0 to 2. The eddy momentum flux is more than doubled over this range, with the divergent component explaining the bulk of the change. We also show in this figure the prediction by Eq. (8) for a homogeneous model constructed taking the average *V* and vorticity gradient from 10°N to 30°S (dashed red line). This prediction can reproduce qualitatively the *V*. To assess the extent to which changes in the *V* contribution to *α* in Eq. (8) [friction is still included, essentially we take *r* = 0 in Eq. (A11)]. This prediction can capture the sensitivity of the momentum flux. The *K*_{d}-weighted meridional wavenumber *V* in similar fashion (Fig. 8g).

## 6. Conclusions

This paper has investigated the dynamics of cross-equatorial eddy momentum transport in variants of the Gill model. We summarize our main findings below:

- We extended the analysis of Z19b on the relation between the sign of the divergent eddy momentum flux and the divergence phase tilt for a single wave to the case of a localized divergence field with a full meridional spectrum. We showed that in that case, the direction of the momentum flux depends on the sign of the mean meridional wavenumber weighted by the divergent eddy kinetic energy spectrum.
- In the absence of a vorticity source, Sverdrup balance represents a barrier to cross-equatorial propagation and the rotational and divergent momentum fluxes compensate. Two processes that may be important for violating Sverdrup balance and allowing cross-equatorial propagation are friction and meridional vorticity advection by the Hadley cell. Both processes are able to reduce or reverse the geopotential tilt imparted by a tilting divergent field, limiting the compensation by the rotational momentum fluxes.
- Away from the WTG limit, the heating is no longer locally balanced by the divergence. Newtonian cooling makes the divergence lag (lead) the heating for a Kelvin (Rossby wave). For the latter, the divergence phase lead increases with latitude as Newtonian cooling becomes more efficient balancing the heating poleward. This makes the divergence field tilt eastward with latitude as observed, though the relevance of this mechanism for the observed tilt is unclear because the heating is also affected by the circulation in a moist atmosphere.
- The divergence phase tilt is also affected by the mean meridional flow away from the WTG limit. With a Hadley cell, the eastward divergence tilt is enhanced in the downstream direction of the mean flow and the divergent eddy momentum flux increases in the opposite direction. Changes in the divergent momentum flux are well explained by the shift in the divergent eddy kinetic energy spectrum toward negative
*Vl.* - Our linear model reproduces reasonably well the nonlinear shallow-water results of Kraucunas and Hartmann (2007) when linearized about their climatology. The main deficiency of the linear model is the disappearance of the Kelvin wave with weak friction, as either friction or nonlinearity is needed to balance the zonal pressure gradient by the Kelvin wave. Despite this deficiency, the model can capture the eddy momentum fluxes of the nonlinear model. The sensitivity of the divergent momentum fluxes to the meridional flow in this model is consistent with the homogeneous theory, but the rotational momentum fluxes are much weaker.
- Our findings help reconcile the conflicting results of KH07 and Z19b on the sensitivity of cross-equatorial momentum transport to the Hadley cell. Consistent with KH07, the eddy momentum flux increases with Hadley cell strength in our model. This sensitivity was not observed by Z19b because the Hadley cell impacts the momentum transport through a modulation of the eddy divergence field, kept fixed in Z19b by virtue of the WTG approximation.

In conclusion, our analysis of the Gill model supports the arguments of Z19b that propagation from a tropical wave source is typically associated with an eastward tilt of the divergence in the direction of propagation. Z19b discussed this relation in the context of a single wave, and we have shown here that this relation also holds in an integral sense with localized heating. As a note of caution, this relation may not apply locally and may also not be obvious when the divergence field has fine meridional scale as shown in section 5. Additionally, the divergence field only tells part of the story when the rotational momentum fluxes are important.

As noted in the introduction, there is some evidence that the observed, robust eastward tilt of the divergence moving away from the equator is not forced by the geography. Indeed, idealized aquaplanet models appear to develop similar precipitation tilts when forced with untilted SST anomalies (see, e.g., Fig. 4 of Ting and Held 1990). Understanding what determines this tilt and its relation to propagation is an important open question. We have shown in this paper that Newtonian cooling alone can produce an eastward divergence tilt from the equator as observed, and that this tilt is enhanced in the downstream direction of a Hadley cell. However, it is likely that in the atmosphere the coupling between heating, moisture, and circulation plays the dominant role.

## Acknowledgments

This work was motivated by comments and discussions with Ming Cai and Olivier Pauluis. I am grateful to Isaac Held for making me aware of the existence of the SPCZ as well as for sharing his insights on the tropical circulation. I am also grateful to the reviewers for comments that improved the content and clarity of this manuscript. A reviewer had a major impact on this work by stressing the role of the divergence changes for the momentum flux sensitivity, an aspect poorly emphasized in a first draft of the manuscript. She enlightened us on the sensitivity of the divergence tilt in the KH07 simulations and suggested the detailed comparison to their work in section 5. We acknowledge financial support by Grant CGL2015-72259-EXP by the State Research Agency of Spain and NSF funding for a summer visit to Princeton under Grant AGS-1733818.

## APPENDIX A

### Relation between the Divergent Eddy Kinetic Energy Spectrum and the $\overline{{u}_{r}^{\prime}{\upsilon}_{d}^{\prime}}$ Direction

Z19b discusses the relation between the *l* for a single wave. He shows that the sign of the momentum flux forced by the divergent advection of planetary vorticity, *ω*_{0}*l*, where *ω*_{0} is the free Rossby frequency. For waves with large meridional scales, the same is true for

With localized heating, waves with different *l* contribute to the momentum transport. Depending on the sign of *ω*_{0}*l*, these contributions may or may not have the same sign as *l*. In this appendix, we derive an expression for the meridionally integrated momentum flux

*ω*

_{0}=

*Uk*+

*Vl*−

*βk*(

*k*

^{2}+

*l*

^{2})

^{−1}. We may then express

*k*can be written as

Figure A1a shows that this prediction agrees very well with the simulated cospectrum for a sample simulation with *V* = −3 m s^{−1} and the control diabatic damping. Figure A1b describes the cospectrum sensitivity to *V* predicted by this expression, also in excellent agreement with the numerical results (not shown). With nonzero *V* it is no longer true that the sign of the cospectrum is determined by *l* alone. As discussed by Z19b, with *Vl* > 0 and |*l*| shorter than the resonant/propagating waveumber *l*_{0}, *ω*_{0} changes sign and becomes eastward, and the divergent momentum flux has opposite sign to *l*. However, the contribution of this spectral region to

Note that although Eq. (A5) has a singularity at the resonant frequency *ω*_{0} = 0, the prediction is well behaved because

*W*(

*l*) = −∂/∂

*l*[

*βl*

^{2}/

*ω*

_{0}(

*k*

^{2}+

*l*

^{2})] is a weighting function.

*ω*

_{0}≈ −

*βk*(

*k*

^{2}+

*l*

^{2})

^{−1}, the weighting function is particularly simple and we can write

*K*

_{d}-weighted meridional wavenumber. Figure 6c shows that this can provide a reasonable approximation even when

*V*≠ 0.

*U*and focus on long zonal waves, we expect

*l*≫

*k*for any

*l*range for which

*V*is not negligible. Thus, we can approximate

*α*(

*l*/

*l*

_{0}) is a correcting factor defined by

*r*=

*Vl*

^{3}/(

*βk*) = (

*l*/

*l*

_{0})

^{3}, with

*l*

_{0}= (

*βk*/

*V*)

^{1/3}the resonant wavenumber. This expression is shown with thick black line in Fig. A1c. Since the correcting factor is generally positive, it is still true that the sign of the momentum flux is determined by the weighted meridional tilt. But with nonzero

*V*, the contribution by modes with large |

*l*| is reduced, while contributions near the resonant frequency are greatly enhanced.

In fact, the integral using this weighting no longer converges due to the increased order of the singularity and the fact that unlike *a* and make this damping suitably small. More generally, we would also like to generalize our results for the dissipative case.

*a*, the Rossby frequency

*ω*

_{0}is imaginary. In that case, the cospectrum

*l*space) of the

*a*is small this dependence can be neglected and the above derivation is easily generalized to produce the following

*α*:

*f*=

*al*

^{2}/(

*βk*) =

*a*/(

*Vl*

_{0})

*r*

^{2/3}. Figure A1c shows the structure of

*α*for a few choices of the dimensionless damping parameter

*a*/(

*Vl*

_{0}). The red line in Fig. 6c, computed using a frictional time scale

*a*

^{−1}= 20 days, produces nearly perfect agreement with the inviscid model

The main limitation of the diagnostics presented above is the assumption of a homogeneous basic state with constant *U*, *V*, and *β*. However, the diagnostics still work reasonably well in the more realistic, nonhomogeneous situations studied by KH07. This is illustrated in Fig. A1d for KH07’s solstitial setting with shifted heating. The red line shows the model cospectrum *C*_{k}(*l*) and the two blue dashed lines show predictions based on Eq. (A5) using two different homogeneous basic states (the maximum southward *V* and an equatorial *β*, or the mean *V* and absolute vorticity gradient averaged within 10° of the equator). The two predictions are similar and reproduce reasonably well the model results [the largest differences, near the resonant frequency, are in fact due to neglecting the imaginary component of *ω*_{0} in the derivation of Eq. (A5)]. Despite some errors, Fig. 8f shows that Eq. (A9) can capture qualitatively the sensitivity of the divergent momentum flux to the mean flow in the inhomogeneous model.

## APPENDIX B

### Energy Conversions and the *ϕ*′–*D*′ Phase Lag

*ϕ*

In this appendix, we discuss the closure of the eddy energy balance and its relationship with the *ϕ*′–*D*′ phase lag. As discussed in section 4, this lag has implications for the determination of the divergence phase tilt.

*Q*′ can be dissipated diabatically or converted into eddy kinetic energy and then dissipated by friction (possibly nonlocally). The well-known role of

*ϕ*′ −

*D*′ phase relation) using the energy cycle.

For instance, consider the inviscid limit *a* = 0. In this limit, all the eddy dissipation must be diabatic. When *V* = 0 too, the zonal momentum equation [Eq. (1a)] implies that *D*′ and *ϕ*′ are in quadrature (Fig. 5b). All the EAPE generated by the heating is dissipated locally by Newtonian cooling (Fig. B1a) and the phase shift between *Q*′ and *D*′ increases with Newtonian dissipation as discussed in section 4.

With nonzero *a*, we expect part of the EAPE generatedy by the heating to be converted into EKE and dissipated by friction, hence a positive *D*′ and *ϕ*′ to be less than *π*/2 whether *D*′ leads *ϕ*′ (as for a Rossby wave) or lags it (as for a Kelvin wave). This is the case over most of the domain in Figs. 5e and 5h. However, in both cases rapid changes in the phase of *D*′ are found on the sides and the *D*′ − *ϕ*′ phase lead appears to asymptote *π* at large distance. Figures B1d–f show the EKE, EAPE and total eddy energy balance for the solution in Fig. 5d. Although

Finally, we can also get nonzero *a* = 0 in the presence of a Hadley cell. In that case, *D*′–*ϕ*′ phase lag between *π*/2 and *π*) is now observed over regions with significant eddy amplitude. This is manifested in Fig. 5j in the form of a second divergence maximum at high negative latitudes collocated with regions of negative height.

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^{1}

Strictly speaking, when the response is decomposed in terms of the free modes a weak Kelvin wave component is required to balance the zonal pressure gradient by the Rossby wave at the equator. However, this response is weak and by construction cannot produce geopotential tilts.

^{2}

With the large control value of friction, the Kelvin wave is so prominent that *ϕ*′ is found to lie eastward of *D*′ (so that *D*′ lags *Q*′) over a wide tropical region and the transition to westward *ϕ*′ occurs over latitudes where the heating and divergence are already very weak. The subtle *D*′ tilt in Fig. 5d becomes much more obvious with weaker friction and/or with broader *Q*′ (not shown).