## 1. Introduction

Monsoon circulations consist of highly three-dimensional flow forced by heat sources localized in regions with irregular coastlines and topography. Nevertheless, it has become common to describe monsoons as zonally symmetric meridional overturning circulations with precipitating ascent in the summer hemisphere, cross-equatorial flow, and subsidence in the winter hemisphere (Webster and Fasullo 2003). Monsoon circulations do, indeed, project strongly onto the zonal-mean Hadley circulation during solstice seasons, with most cross-equatorial mass flux in the Hadley circulation composed of flow through monsoon regions (e.g., Peixoto and Oort 1992; Findlater 1969; Boos and Emanuel 2009). While it is not obvious that a highly three-dimensional circulation can be adequately described in terms of zonal-mean quantities, even when the zonal-averaging operator is applied over a limited range of longitudes, abstracting monsoons as zonal-mean circulations has facilitated much study of monsoon dynamics.

In the past 30 years, multiple theoretical advances made in understanding Earth’s Hadley circulation have been extended to solstice-like conditions representative of monsoons. Held and Hou (1980) developed an axisymmetric^{1} theory for the strength and meridional extent of the annual-mean Hadley circulation, and Lindzen and Hou (1988) extended this theory to cross-equatorial flow driven by off-equatorial thermal forcings. Central to this axisymmetric theory is the idea that meridional flow conserves absolute angular momentum about Earth’s rotational axis, a constraint that leads to a nonlinear dependence of circulation strength on parameters such as the strength and meridional structure of the thermal forcing.

*f*is the Coriolis parameter,

*υ*is the meridional wind, and

*ζ*is the vertical component of relative vorticity. If the geostrophically balanced zonal flow produced by the thermal forcing is not strong enough to reduce the absolute vorticity to zero, then (1) shows that

*υ*must be zero. In the presence of viscosity and friction, nonzero

*υ*occurs for such subcritical forcings but intensifies nonlinearly once the forcing exceeds the threshold where

*S*:where brackets denote a zonal mean and an overbar denotes a time mean. Nonzero

While the observed annual-mean Hadley circulation is thus not well described by the axisymmetric, AMC theory of Held and Hou (1980), the cross-equatorial Hadley circulation in boreal summer does lie closer to the AMC limit, particularly in the longitudes of the South Asian monsoon (Schneider and Bordoni 2008). However, even if the AMC theory could be applied to the mean summer state of the South Asian monsoon, the transition between the equinox and solstice states cannot be exactly that described by Plumb and Hou (1992), because baroclinic eddy momentum fluxes, rather than linear viscosity, are a central part of the circulation dynamics for subcritical off-equatorial thermal forcings. Walker and Schneider (2006) and Schneider and Bordoni (2008) used dry idealized GCMs with zonally symmetric thermal forcings to show that the transition between equinox and solstice states seems to be between a weak equinox regime in which eddy-momentum fluxes control the strength of Hadley flow {i.e., *S* and *S* and

However, all of the Hadley circulation theories discussed above assume zonally symmetric forcings, whether or not they allow for the transport of momentum by transient, zonally asymmetric eddies. Observed monsoon flows are produced by thermal forcings with large zonal asymmetries, as evidenced by the strong zonal asymmetry in upper-tropospheric temperature and absolute vorticity observed in the Asian summer monsoon (Fig. 1). Linear theory for the steady response to an off-equatorial, zonally confined heat source has long served as a useful tool for understanding the observed zonally asymmetric distribution of winds and rain in monsoons, as in the highly influential papers by Gill (1980) and Hoskins and Rodwell (1995). But Plumb (2007) noted that it is difficult to reconcile such linear theory for the response to a zonally asymmetric forcing with the nonlinear theories for zonal-mean Hadley flow described by either Held and Hou (1980) or Walker and Schneider (2006).

*η*provides a constraint on the time-mean divergent flow, as described by Hsu and Plumb (2000) and reviewed by Plumb (2007):Here, primes represent deviations from the time mean and

*A*is the area enclosed by a contour of constant

*η*. The integral on the left-hand side represents the net mass flux across the chosen

*η*contour, while the right-hand side represents the transient eddy flux of vorticity across the contour. If the flow is steady and there is no transient eddy flux, a thermal forcing can produce a divergent circulation only if

*η*produced by zonally asymmetric heatings on rotating spheres are known to become dynamically unstable, spreading westward from the source in an anticyclonic

*β*plume that periodically breaks down into low-vorticity eddies that propagate away from the source region (Rhines 1983; Hsu and Plumb 2000). These transients provide a vorticity flux into the region of the heating that allows the time-mean

*η*to remain nonzero even while there is divergent flow. Regions of low–potential vorticity air have been observed to break off from the Asian summer monsoon anticyclone and propagate westward (Popovic and Plumb 2001). Sobel and Plumb (1999) showed that the transient vorticity fluxes vanish in (3) if the integral is taken over a contour that is allowed to deform in time, and any time-mean net mass flux across the contour must be balanced by viscous effects. Viscous effects become more important as transient eddies stretch the vorticity field into fine filaments with sharp horizontal gradients (Plumb 2007).

An open issue is how the strength of the divergent response to an off-equatorial, zonally confined thermal forcing varies with the strength of that forcing. In other words, as the strength of a zonally asymmetric heating is increased, does the shedding of low-vorticity air from the heating region alter the rate at which the divergent circulation strengthens? Or can scalings developed for zonally symmetric forcings (e.g., Walker and Schneider 2006) be applied even though the time-mean response will contain large zonal asymmetries? Our hypothesis is that the divergent mass flux produced by a zonally asymmetric heating will not exhibit the abrupt, nonlinear transitions seen in the response to a zonally symmetric heating, because the eddy flux of vorticity into the heating region will prevent the circulation from achieving the nonlinear, zero-vorticity state. Even if the absolute vorticity becomes small enough to produce nonlinear flow, we hypothesize that eddy activity will reduce the rapidity with which the transition to the AMC state occurs.

The influence of thermal forcing asymmetries on cross-equatorial monsoon circulations was previously examined by Privé and Plumb (2007b), but they focused on other aspects of the moist response and did not examine how the meridional circulation scaled with the strength of a zonally asymmetric forcing. Shaw (2014) analyzed the role played by planetary-scale waves in the onset of the boreal summer general circulation and did examine how the circulation changed as a wavenumber-2 thermal perturbation was increased in amplitude relative to a zonally symmetric basic state. Once the wavenumber-2 forcing in her model exceeded a threshold, the circulation transitioned to a state dominated by stationary waves and anticyclonic absolute vorticity in the equatorial upper troposphere in some longitudes. This change in sign of the absolute vorticity occurred as part of a southward migration of negative relative vorticity from the Northern Hemisphere as the stationary wave amplitude was increased (T. A. Shaw 2015, personal communication) and is distinct from the regime transitions marked by absolute vorticity reaching zero in the region of the off-equatorial heating (e.g., Plumb and Hou 1992; Hsu and Plumb 2000). Nevertheless, the results of Shaw (2014), as well as those of Kelly and Mapes (2011, 2013), illustrate the important role played by stationary eddy momentum transports in the boreal summer general circulation. This study is complementary to that work in that we also force an idealized model with a zonally asymmetric thermal forcing, but here we focus on the scaling of the total vertical mass flux, integrated over all longitudes, instead of just the zonal mean.

One particular goal of this paper is to examine how both the zonally symmetric and asymmetric parts of the divergent response vary with the strength of the forcing. When Privé and Plumb (2007b) examined the response to heating from a zonally asymmetric continent, they noted that much of the ascent that occurred over the off-equatorial continent was accompanied by subsidence to the west-northwest, which is qualitatively consistent with the linear response to a zonally confined heating (e.g., Gill 1980). We know of no study that clearly shows how the total vertical mass flux (zonally symmetric and asymmetric components) produced by an off-equatorial heating varies with the amplitude of that heating. This question is presumably highly relevant to the seasonal evolution of precipitation in monsoons, because that precipitation will increase with the total ascent, not just its zonally symmetric component.

The forcings used in this study are all confined to off-equatorial regions and vary only in amplitude, retaining a constant spatial structure similar to that used in Plumb and Hou (1992). This contrasts with the approach of Walker and Schneider (2006) and Schneider and Bordoni (2008), who examined the transition between equinox and solstice states by moving the peak of the equilibrium temperature distribution from the equator into the subtropics. While we eventually wish to examine the effect of forcing asymmetries in such a configuration, as it is likely more relevant to real monsoon circulations with cross-equatorial SST gradients, addition of zonal asymmetries to the Plumb–Hou-type forcing seems more straightforward as a first step. All forcings are applied in a dry, idealized GCM by relaxing the temperature toward a prescribed equilibrium distribution. The effects of moisture, time variations in the forcing, and other complications will be explored in future work.

The next section of this paper provides details on our numerical model and further information on the forcing. Results are then presented, and we close with a summary and discussion of caveats and implications.

## 2. Model setup

We used the finite-volume dynamical core of the Community Atmosphere Model, version 5.0 (CAM5.0), developed by the National Center for Atmospheric Research (NCAR; Neale et al. 2012). We used a resolution of 1.9° latitude by 2.5° longitude, with 30 vertical levels from the surface pressure of about 1000 hPa to the model top near 0 hPa. A few integrations were conducted at a finer resolution, as described in the results section below. There is no moisture or topography.

The model circulation is forced by a scheme very similar to that described by Held and Suarez (1994), which employs linear relaxation of temperatures toward a prescribed equilibrium distribution over a time scale of 40 days throughout most of the troposphere. We modified the equilibrium temperature distribution as described below, but all other aspects of the forcing are as described in Held and Suarez (1994). For example, boundary layer dissipation is represented by linear relaxation of winds toward zero in the lowest 30% of the atmosphere’s mass. We used a 30-min time step with the model state output as 6-hourly averages. All output was transformed to pressure coordinates for analysis.

We conducted two sets of integrations: one forced by zonally symmetric equilibrium temperature profiles and the other by zonally asymmetric equilibrium temperatures. All integrations used time-invariant equilibrium temperature distributions and were run for 10 years with the last 9 years used for analysis.

### a. Zonally symmetric forcing

### b. Zonally asymmetric forcing

*λ*is longitude (radians), the zonal boundaries of the

## 3. Results

We first present an analysis of the zonal-mean circulation resulting from both the zonally symmetric and zonally asymmetric forcings. Then we examine how the total vertical mass flux, including its zonally asymmetric component, scales with the strength of the forcing. Finally, we analyze the upper-tropospheric vorticity budget in order to explore the relevance of transitions to AMC states.

### a. Spatial structure of the circulation

For both weak and strong zonally symmetric forcings, the zonal-mean circulation consists of three cells: a thermally direct tropical cell that for strong forcings crosses the equator, a subtropical cell that is also thermally direct, and a thermally indirect cell centered near 40°N (Figs. 3a,b). The last of these corresponds to Earth’s Ferrel cell, while the first two are qualitatively similar to the observed boreal summer Hadley circulation and so are referred to herein as the winter and summer cells, respectively. For the weakest forcings (e.g., *A*_{e} = 3 K), the winter cell is confined almost entirely within the boundary layer and within the region where meridional gradients of

The meridional and vertical scales of the winter cell increase greatly as the amplitude of the zonally symmetric forcing is increased so that, for strong forcings, it extends through the full depth of the troposphere and encompasses the entirety of the tropical belt in both hemispheres (e.g., Figure 3b). The circulation is strong enough to homogenize the absolute angular momentum field and create a region of low absolute vorticity in much of the tropical upper troposphere, indicating that these strongly forced circulations are in a nonlinear, AMC regime. The cross-equatorial flow seen for strong forcings exhibits an equatorial jump similar in appearance to that discussed by Pauluis (2004), so it is presumably caused by boundary layer friction and the absence of strong cross-equatorial gradients in the

Now we examine the zonal-mean circulation that results from the zonally asymmetric forcings. For weak forcings, we expect the zonal-mean response to depend only on the zonal-mean *A*_{e} = 12 and 48 K closely resembles the zonal-mean response to symmetric forcings with *A*_{e} = 3 and 12 K, respectively (Fig. 3; recall that *A*_{e} for the zonally asymmetric forcing must be 4 times as large as for the symmetric forcing to achieve the same zonal-mean

It is not surprising that the zonally asymmetric forcings produce stronger circulations for the same zonal-mean

The above zonal-mean view masks large differences between the circulations produced by the zonally symmetric and asymmetric forcings. For weak forcings, both symmetric and asymmetric forcings produce zonal-mean ascent near the

The response to the strong asymmetric forcing also exhibits many of these features, although changes in spatial structure suggest the presence of at least some nonlinearity (Fig. 7d). Compared to the response to the weak asymmetric forcing, the peak ascent is located farther toward the equator, near 20°N at a latitude of zonal-mean ascent rather than zonal-mean subsidence. The peak subsidence is shifted poleward together with the westerly branch of the upper-tropospheric anticyclone. Thus, we expect less cancellation of ascent and subsidence when the zonal mean is taken than we saw in the case of the weak forcing. We also expect the response to the strong forcing to lie closer to the zonally asymmetric AMC limit in which nonlinear meridional advection of zonal momentum becomes important; this is consistent with the fact that we no longer find the linear Sverdrup vorticity balance with ascent and subsidence on the eastern and western sides of the upper-tropospheric anticyclone.

### b. Mass flux scaling

To study how circulation strength changes with forcing amplitude, we examine several measures of the mass flux in the time-mean overturning circulation. The first index,

We examine one other index to estimate the strength of the summer Hadley cell: the absolute value of the minimum (i.e., the most negative) zonal-mean mass streamfunction value between 20° and 50°N, denoted by

We also applied (7) to the zonally asymmetric component of the vertical velocity field (i.e.,

^{2}Assuming a weak temperature gradient approximation (Sobel and Bretherton 2000), in which

*Q*is balanced entirely by adiabatic cooling in the ascending branch of the circulation, allows the vertical mass flux to be expressed as a linear function of the forcing amplitude

*Q*. We therefore present two quantities: a calculation of

In summary, the zonal-mean response to both symmetric and asymmetric forcings becomes more sensitive to the forcing once

### c. Upper-level vorticity distribution

Several questions naturally arise regarding the response to our forcings. First, is the abrupt increase in sensitivity of the zonal-mean mass flux to the forcing that occurs near *η*.

We find the minimum value of *η* within a three-dimensional volume spanning the equilibrium temperature maximum (10°–40°N, 40°–160°E, and 175–250 hPa) and plot this as a function of the forcing amplitude *η*, since a zonally asymmetric forcing can achieve a zero vorticity state locally even though the zonal-mean circulation is far from this limit. Since the local *η* distribution, we refrain from normalizing

For the symmetric forcing, upper-tropospheric *η* in the heating region decreases sharply as the strength of the forcing is increased (Fig. 12a). While it does not reach zero, it does approach that limit for the strongest forcings, which produce a local minimum of *η* near 15°N (Fig. 12b). When *η* nears its lowest values, the zonal-mean mass flux in the cross-equatorial Hadley cell increases more rapidly with the strength of the forcing, as evidenced by the fact that the change in slope of the red line in Fig. 8 occurs when the minimum *η* drops to about *η*, which is found near 15°N, corresponds to a local Rossby number *η* in our three-dimensional volume is larger than the minimum value of *f* in that volume. This occurs because the circulation response is local to the forcing and exports *η* from the center of the heating region to its periphery (e.g., red line in Fig. 12b).

For asymmetric forcings, Fig. 12a shows that the minimum value of *η* decreases much more slowly with *η*. The zonal-mean value of *η* drops below *A*_{e} = 6 K and an asymmetric forcing with *A*_{e} = 24 K. In contrast, even the weakest asymmetric forcing produces a maximum *η* occurs on the equatorial edge of our control volume, where zonal asymmetries in *η* are small, while the maximum *η* near the

What process prevents the subtropical circulation from achieving a low-vorticity state in response to strong asymmetric forcings? Hsu and Plumb (2000) showed that strong asymmetric forcings can produce a dynamically unstable upper-level anticyclone that periodically sheds low-vorticity air. This sort of transient behavior is readily seen in our model in response to strong asymmetric forcings: a Hovmöller diagram of *η* at 200 hPa shows low-vorticity air generated around the heating center (near 95°E) then propagating westward and being replaced by high-vorticity air in a cycle that recurs about every 10 days (Fig. 13a). The amplitude of these temporal variations is large, with *η* at 25°N in the heating region varying in time between values that typically only occur at 40°N and at the equator in the time-mean state (Fig. 13b). However, examination of the vorticity budget in the next section will show that the associated transient eddy fluxes of vorticity make a relatively small contribution to the vorticity budget in this region.

### d. Upper-level vorticity budget

*η*substantially below

*f*, it still has a locally cyclonic value in regions of upper-level divergence that necessitates an eddy flux of vorticity. Here, we examine the time-mean vorticity budget to better understand the nature of these eddy fluxes. We start with the flux form of the inviscid vorticity equation (e.g., Haynes and McIntyre 1987) but separate the term involving the advective flux of vorticity into terms representing horizontal advection and stretching by the time-mean flow and by transient eddies:The first two terms on the right-hand side represent horizontal advection of vorticity and vortex stretching by the time-mean flow, respectively. The third term represents convergence of the horizontal advective flux of vorticity by transient eddies, which is computed explicitly from 6-h-averaged data. The last term, which is the sum of the tilting and vertical advection terms for the total flow (mean plus transient eddies), is omitted from the plots presented below because it is comparatively small in the upper troposphere for all forcings examined here. We also calculate a budget residual, neglecting the Eulerian time tendency in our 9-yr averages, as the sum of all the terms on the right-hand side of (12).

For strong symmetric forcings, the upper-level vorticity budget behaves as expected for a Hadley circulation approaching the AMC limit. The symmetric thermal forcing produces ascent near 25°N, which in turn produces a negative vorticity tendency through the stretching term (i.e., vortex squashing; Fig. 14). In the AMC limit, this tendency would be zero because

*β*is the planetary vorticity gradient. Thus, transient eddies are a visually prominent feature of the upper-tropospheric response to an asymmetric forcing (e.g., Fig. 13), but they do not prevent the upper-level vorticity from reaching the nonlinear, AMC limit. Instead, vortex squashing produced by the thermally driven ascent is balanced by horizontal advection by the time-mean upper-level anticyclone. This makes the linear scaling of the total vertical mass flux seen for the asymmetric forcings unsurprising, because the overturning circulation is more like that described by the linear model of Gill (1980) than the nonlinear limit of Plumb and Hou (1992). Transient eddies play little role in maintaining high values of absolute vorticity in the heating region. The circulation constraint (3) may be satisfied because there is no local minimum of

*η*in the region of strong divergence, so one cannot draw a closed contour of

*η*locally around the divergent region (e.g., Fig. 13b). Contours of

*η*span all longitudes, and the asymmetric vertical mass flux captured by our metrics may occur in zonal overturnings confined between sets of

*η*contours.

It is important to note that this dynamical regime may be sensitive to the scale of the forcing. Hsu and Plumb (2000) found that transient eddy vorticity fluxes were central to the divergent response created by a forcing that was horizontally narrow compared to the deformation radius and the scale of the resulting anticyclone. Since our asymmetric forcings have the same zonal scale as the resulting anticyclone, this may make stationary eddy fluxes more important here.

## 4. Summary and discussion

Using a dry, idealized model of Earth’s atmosphere, we investigated the response of the tropical and subtropical tropospheric circulation to off-equatorial thermal forcings localized in both latitude and longitude. This effort was motivated by the fact that previous studies of the nonlinear intensification of meridional flow that occurs when circulations reach AMC states have used either axisymmetric models (e.g., Plumb and Hou 1992; Boos and Emanuel 2008a) or eddy-resolving models with zonally symmetric boundary conditions and forcings (e.g., Schneider and Bordoni 2008). Our work extends a few studies that examined the response to zonally asymmetric forcings (Privé and Plumb 2007b; Hsu and Plumb 2000; Shaw 2014) by examining how the vertical mass flux contained in both zonal and meridional overturnings scales with the strength of the forcing.

Zonally symmetric and asymmetric forcings with the same zonal-mean strength (i.e., the same zonal-mean *f*, indicating a regime transition of the zonal-mean circulation to an approximate AMC state. For the symmetric forcings, this behavior is consistent with that discussed in previous studies (e.g., Walker and Schneider 2006; Schneider and Bordoni 2008). For the asymmetric forcings, one might expect the zonal-mean value of the forcing to set the zonal-mean response for weak forcings, but it is remarkable that this is also true for strong asymmetric forcings. This might seem at odds with the results of Shaw (2014), who found that a subtropical wavenumber-2 SST anomaly that averaged to zero in the zonal mean substantially increased the strength and meridional extent of a cross-equatorial Hadley cell in an aquaplanet model. However, the cross-equatorial response to the wavenumber-2 forcing in Shaw (2014) was associated with a strong near-equatorial eddy momentum flux divergence created by stationary waves that propagated from the subtropical forcing across the equator. Our asymmetric forcings create large stationary waves in the subtropics, but these do not propagate into the deep tropics (e.g., Fig. 6b). We speculate that the amplitude of near-equatorial wave activity is greatly enhanced by interaction with moist convection, which is absent in our dry models. Indeed, Boos and Shaw (2013) found that the equatorial wave response to an applied subtropical torque was highly sensitive to the treatment of moist convection in idealized aquaplanet models. More generally, the strength of the zonal-mean, cross-equatorial meridional flow may be highly sensitive to cross-equatorial eddy momentum transports that are only indirectly controlled by subtropical thermal forcings and are sensitive to the details of moist convective coupling with the large-scale flow.

Our most novel result is that the total upward mass flux contained in both zonal and meridional overturnings increases linearly with the strength of the subtropical forcing. This total mass flux is as much as several times larger than the mass flux in the cross-equatorial Hadley cell and exhibits no nonlinear or threshold-like response to the forcing, except for a weakening of its rate of increase at the strongest forcings. These results hold whether one includes all upward motions in the measure of the mass flux or whether one includes only the thermally direct ascent. The linear scaling of this total mass flux is accompanied by an upper-tropospheric vorticity balance in which horizontal advection by the time-mean flow compensates for most of the vortex squashing caused by the thermally forced ascent. This means that the dynamics of the total upward mass flux produced by the asymmetric forcing is better described by the Sverdrup balance of the linear model of Gill (1980) than by the nonlinear, AMC regime of Plumb and Hou (1992) or Walker and Schneider (2006). The asymmetric forcing is less effective than the symmetric forcing in reducing the upper-level vorticity toward the nonlinear limit, and it is vorticity advection by time-mean flow rather than transports by transient eddies that provides the largest positive vorticity tendency in the heating region. Although transient eddies are prominent in the upper-tropospheric vorticity field in our models as well as in observations (e.g., Popovic and Plumb 2001), they do not play a dominant role in the vorticity budget in our model. This presents a markedly different view of the dynamical response to a strong subtropical forcing than that presented by Plumb (2007) and motivates a closer examination of observed monsoon circulations in future work. Brief examination of the observations shown in Fig. 1 shows that the northern branch of the Asian ascent maximum during boreal summer has relatively high vorticities (e.g., near

Several important caveats exist. First, in the spirit of Plumb and Hou (1992), our model examined the circulation response to thermal forcings confined to the subtropics. Circulations that are forced by cross-equatorial SST gradients, which often exist in monsoon regions, may be governed by different dynamics. Second, the summer cell in our simulations is too strong compared to the summer cell in Earth’s observed solstitial Hadley circulation [e.g., compare Figs. 3b and 3d with the observed mass streamfunctions presented in Dima and Wallace (2003)]. Thus, an overly large fraction of zonal-mean ascent in our model occurs in the summer cell, which might make the total upward mass flux behave more linearly in our models than in the real atmosphere. However, the summer cell in our model strengthens faster than linearly with the strength of the forcing, indicating that its zonal-mean behavior cannot account for the linear scaling of the total upward mass flux. The sublinear scaling needed to offset the nonlinear intensification of the cross-equatorial Hadley cell is seen only in the zonally asymmetric part of the circulation. Strong zonal overturnings do exist in the northern subtropics during boreal summer, and meridional advection of momentum and vorticity by the upper-tropospheric Asian anticyclone is large even when compared to transports by the zonal-mean flow and transient eddies (e.g., Kelly and Mapes 2013; Shaw 2014). Thus, the overly large strength of the summer cell in our model would not seem to make our results irrelevant to Earth’s observed summer circulation.

Our model also lacks moisture and was forced by relaxing temperatures to a prescribed equilibrium distribution. This model design was chosen for consistency with previous influential studies of the Hadley circulation (e.g., Held and Hou 1980; Plumb and Hou 1992), but of course the thermal forcing on Earth arises from surface fluxes of sensible and latent heat that are mixed vertically through the troposphere by dry and moist convection. The large-scale circulation influences this convection as well as the surface enthalpy fluxes, providing a rich set of interactions and potential feedbacks that were omitted from our model. Our prescribed forcings were also constant in time, and it is possible that the time-mean states produced by steady forcings will not be individually realized at each step of a seasonally varying forcing. And as mentioned in the previous section, transient eddy fluxes of vorticity may play a more dominant role in the response to forcings that are zonally narrower than those considered here. Nevertheless, our results raise important questions about how relevant studies of the zonal-mean response to a zonally symmetric forcing are to monsoon circulations, which have large zonal asymmetries in both the forcing and the time-mean responses. The importance of these zonal overturnings for monsoon dynamics merits further examination in both observations and models.

## Acknowledgments

Both authors gratefully acknowledge support from National Science Foundation Grant AGS-1253222 and Office of Naval Research Grant N00014-11-1-0617. This work was supported in part by the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center. Brian Dobbins provided assistance with model configuration.

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^{1}

Consistent with the literature discussed herein, we use “axisymmetric” to refer to circulations that have no variations about the planetary rotation axis. While “axisymmetric” can be a synonym for “zonally symmetric,” we will use the latter term to describe a thermal forcing that is invariant with longitude in order to be clear that we are not assuming the resulting circulation is axisymmetric.

^{2}

When expressed in this manner, the peak amplitude of *Q* ranges from 0 to 2 K day^{−1} for our asymmetric forcings. Since this forcing is applied over a region having a horizontal scale of thousands of kilometers, and since the Held–Suarez model has a stratification that is typically weaker than that of a moist adiabatic atmosphere, the largest forcing is quite strong.