## 1. Introduction

The important role for wave forces (or “wave drag”) in the dynamics of the mean meridional circulation (e.g., the Brewer–Dobson circulation in the stratosphere) is now well understood. In particular, for a steady circulation that crosses angular momentum contours (which are to good approximation vertical outside of the tropics), there must be a corresponding wave force and the relation between the wave force and the circulation is expressed by the downward control principle (Haynes et al. 1991). In the context of a specified wave force this excludes the possibility of a change in the meridional circulation as a result of a zonally symmetric applied heating. However, in practice, in an atmosphere with an active wave field, such a heating may change the wave force and hence may change the mean meridional circulation. The change in the wave force due to an applied heating needs to be taken into account in seeking a causal explanation for the magnitude and structure of the change in mean meridional circulation. Correspondingly, the change in wave force may need to be taken into account as part of the response to a zonally symmetric mechanical forcing. A specific example has been discussed by Cohen et al. (2014) in considering intermodel differences in the driving of changes in the stratospheric Brewer–Dobson circulation. The imposed mechanical forcing in this example represents a change (or an intermodel difference) in the parameterized gravity wave momentum flux. The wave force is due to the large-scale resolved waves that may respond to the imposed change.

A specific problem, which served as initial motivation for the work reported in this paper, is to explain the double peaks in upwelling and heating in the lower stratosphere in reanalysis datasets such as ERA-Interim and in general circulation models. A preceding companion paper (Ming et al. 2016, hereafter M16) reports a diagnostic study of the angular momentum and radiative balance associated with the double peaks. The conclusion of M16 was that the ERA-Interim estimates of the upwelling and the resolved wave force give a self-consistent angular momentum balance. However, consistency by itself does not establish the cause of the structure in upwelling. Consider a case where external radiative influences are essentially homogeneous in latitude. In the Newtonian cooling approximation, this is equivalent to assuming that the radiative relaxation temperature and damping rates are independent of latitude. A double peak in upwelling could arise through the wave dynamics if that organizes the wave force to be suitably confined away from the equator. This would require the temperature field to have a double-peak structure but this is not observed. The temperatures are in fact almost constant across a broad equatorial region (see M16, their Fig. 1c). An alternative is that the radiative relaxation temperature includes a double-peak structure. Then, as is observed, there can be a corresponding double peak in upwelling with the actual temperature constant latitude. It is unlikely that the required correspondence between the latitudinal structure of the radiative relaxation temperature field and that of the wave force would arise by chance. A more plausible explanation is that the latitudinal structure of the radiative relaxation temperature field, which from purely dynamical considerations at least is externally imposed, determines the latitudinal structure of the wave force.

It was also shown in M16, using offline radiative calculations, that the double peak in the diabatic heating rates observed in ERA-Interim arises primarily from the latitudinal structure in ozone, with contributions from both the longwave and shortwave heating with smaller contributions from the latitudinal variation in the temperature structure below and above the level of the double peak. Specifically, the difference in the clear-sky longwave radiative heating rates between 20°N and the equator at 70 hPa can be attributed to the latitudinal gradient in ozone (~70%) and to the latitudinal difference in the temperature profile (~20%), with the remainder resulting from latitudinal differences in water vapor. The fact that the double peak arises from factors other than the local temperature is analogous to the case with Newtonian cooling where there is externally imposed latitudinal structure in the radiative relaxation temperature field.

The aim of this paper is to investigate further the hypothesis that the double-peak structure in upwelling observed in ERA-Interim and in other reanalyses and models, and the required structure in wave force, is caused by radiative effects that can be regarded as external (in the sense that they do not arise from the temperature structure in the 70-hPa layer). A heuristic dynamical discussion is presented in section 2 and identifies a potentially relevant dynamical regime in which a suitably narrow imposed heating (the “narrow heating” regime) is primarily balanced by upwelling rather than by the radiative relaxation associated with a change in temperature. Section 3 describes and discusses an idealized 3D model experiment that verifies this response and section 4 describes an extended set of experiments that, by varying the parameters defining the imposed heating field, clarifies the conditions under which it occurs.

The findings of sections 2–4 are also relevant to understanding the response to an imposed mechanical forcing, rather than an imposed heating, when the response can include an adjustment in the wave force and this problem is discussed in section 5. In this case there is a dynamical regime in which a suitably narrow imposed mechanical forcing (the “narrow force” regime) is balanced by a change in wave force rather than by the Coriolis torque associated with a change in meridional circulation. This relates to the recent discussion of the Brewer–Dobson circulation by Cohen et al. (2013, 2014). Finally, section 6 contains a discussion of some of the main findings and gives some conclusions.

## 2. Dynamical considerations

*f*is the Coriolis parameter;

*a*is the radius of Earth;

*ϕ*is latitude;

*z*is log-pressure height;

*H*is a scale height taken to be 7 km;

*R*is the gas constant for dry air;

*T*is the temperature;

*c*

_{p}is the specific heat at constant pressure. The terms on the right-hand side of (1d) represent radiative heating, with

An important aspect of the Eliassen–Palm flux term on the right-hand side of (1a), which we will refer to as the wave force, is that it depends on the mean flow *G* for the real atmosphere remains unknown.

We now proceed to analyze the response to a given imposed heating *G*, will be considered in section 5. Note that in the context of the response of a preexisting circulation to an imposed heating or mechanical forcing, the dependent variables must be interpreted as changes in the physical quantities. As a result of the imposed heating, all the dependent variables in (1) will change (including *G*, because of its dependence on *G*. For the moment we do not attempt to relate

From (1a), it follows that

*N*, the relative sizes of the vertical advection term and the Newtonian cooling term, in (1d), are given by the quantity

*K*is a typical value of the ratio

*G*to the velocity).

*f*must be replaced by

*z*level, implying that the assumed balance cannot be perfect for any arbitrary

For given *α*, and *K*, the heating is “narrower” at low latitudes than at high latitudes in the sense that *f* is smaller and the aspect ratio is larger. The quantities in the (2) and (3) and the distinction between the broad and narrow response have been identified and discussed in many previous papers including Dickinson (1971), Fels et al. (1980), Garcia (1987), Plumb and Eluszkiewicz (1999), and Haynes (2005). These discussions generally make specific assumptions about the form of *G*—essentially, that it can be represented by a Rayleigh friction so that *κ* then replaces *K* in (2) and (3). However, it is generally accepted that Rayleigh friction is a nonphysical and poor representation of the wave forces that operate in the stratosphere. The difference here is that we are taking *K* to be a rough quantitative description of a more general *G* that, as emphasized previously, is an unknown, nonlocal, and possibly very complicated function of *K* as the ratio *G* and there will be some *K* that captures that proportionality. In this analysis, it is not necessary to know precisely how *G* varies with *U*. Provided *K* is large enough, the heating or at least the latitudinally varying part of the heating will be balanced by an upwelling.

We have noted previously in M16, from their Fig. 1c and sections 3 and 4, that the double-peak structure in heating is not matched by a corresponding structure in temperature. In the light of the scaling above, this suggests that the double-peak structure is described by the narrow-heating regime. In the next section, we examine the response to a narrow imposed heating in idealized 3D model calculations.

## 3. Model calculations of response to a double-peak applied heating

### a. Model description

We now describe detailed model simulations of the response to an applied heating. Bearing in mind the arguments in the previous section, we expect the change in the wave force will be an important part of the overall response. We use a 3D model in which the wave field, and hence the wave force, are free to vary. In particular we choose the well-known idealized system first defined by Held and Suarez (1994), in which there is a simple thermal relaxation to a specified temperature field

*p*is the pressure and

*p*is set to 1000 hPa).

_{s}The first term on the right-hand side of (4) is used in the Held–Suarez configuration with a Newtonian cooling term proportional to the difference between the actual temperature and the temperature specified by the thermal relaxation state. The value of the Newtonian cooling coefficient, outside of a shallow boundary layer, is 0.025 day^{−1}. The term

### b. Response to imposed heating

Figures 2a–e show the imposed diabatic heating perturbation *t* test (see appendix). The response in the upwelling has a spatial structure that broadly resembles the imposed heating. The upwelling term

To demonstrate that there is a consistent angular momentum balance in the response, we calculate the upwelling using the downward control integral (see M16 for details). Figure 2f shows that upwelling from the model is in agreement with that inferred from the quasigeostrophic and full downward control equations at 78 hPa. The solid line shows the upwelling calculated from the wind and temperature response in the model. The downward control integral is calculated using the time-averaged changes in *J* term (Scott 2002), which represents the time-averaged angular momentum advection terms due to the time varying part of the flow (see also M16, their section 2). Unlike the ERA-Interim data, the model runs are sufficiently long to allow the inclusion of this term. While this term does not make a significant difference for this particular heating, it becomes more important the closer to the equator the heating is located.

The response to applied heating found here is in broad agreement with the combination of double peak in upwelling and flat temperature structure observed in ERA-Interim, previously described in section 3 of M16.

The impression from Figs. 2a and 2b is the balance comes primarily from the change of vertical velocity; however, this has to be consistent with the constraint that *z* level.

Figures 3a and 3b show respectively the quantities

The balance in the thermodynamic equation in (6) can be assessed fully only by considering the height–latitude variation of the various quantities shown in Figs. 3a–c. However, it is convenient to find a simple quantitative measure of the extent to which the applied heating is balanced by a response in upwelling. Here, we will use the ratio

### c. Zonally symmetric simulations with Rayleigh friction

To illustrate the importance of the sensitivity of the wave field *K*, we provide some results for the case where the wave force *G* is represented by Rayleigh friction. This was the basis for many of the earlier studies of the driving of the mean meridional circulation (e.g., Dickinson 1971; Fels et al. 1980; Garcia 1987) and incorporates the *u* dependence of *G*, but in a highly simplified manner that is not believed to be realistic. Analytical progress using Hough functions (e.g., Garcia 1987) is possible under certain simplifying assumptions, which essentially require weak departures from a latitudinally independent state. Here it is most convenient simply to calculate the response using a zonally symmetric version (2D) of the full idealized 3D model described previously. This allows incorporation of latitude-dependent temperature structure, for example.

The zonally symmetric model is relaxed toward the Held–Suarez configuration. Rayleigh friction with a constant friction coefficient *κ* is added throughout the whole domain. All dynamical fields are constrained to be zonally symmetric. This means, for example, that there is no baroclinic instability. The first 1000 days of the model run are discarded as spinup and the model state after this is analyzed.

A set of experiments were performed in the zonally symmetric model with values of

For large values of *κ* (1/*κ* = 1 and 10 days), the system adjusts such that the dominant balance in the thermodynamic equation is between the upwelling and the heating (cf. Fig. 2a and 4c,f). Given the vertical wind response, the continuity equation implies the required change in *κ* regime, while the vertical velocity (and hence the latitudinal velocity) change very little as *κ* varies, at least in the region of the applied heating, there is substantial change in *κ* and a horizontal structure that is broad for large *κ* and narrows as *κ* decreases.

We can think of *κ* as setting the width of some region over which the circulation generated by the heating can spread. As *κ* is decreased, this region becomes smaller and the temperature and zonal wind changes become increasingly confined to the region of heating. This can also be seen in the location of the regions of downwelling in the right column of Fig. 4. More of the heating is balanced by a temperature change and less of it by the circulation change.

For a weaker Rayleigh drag (1/*κ* = 30 days), there are clear quantitative differences between the applied heating Fig. 2a and the vertical upwelling (Fig. 4i). (Note, for example, that the magnitude of the equatorial downwelling relative to the subtropical upwelling is much larger than for smaller values of 1/*κ*.) It follows that the Newtonian cooling term is an important part of the balance in the thermodynamic equation. For reference, the value of the dynamical aspect ratio (2), setting *κ*: 1, 10, and 30 days is respectively 20, 2, and 0.7, confirming that the scaling arguments given previously are consistent with the calculated response with the “narrow regime” no longer applying when *κ* is sufficiently small.

These results for the zonally symmetric dynamics with Rayleigh friction may be compared with the full three-dimensional dynamical response shown in Figs. 2b–d. The three-dimensional response is very similar to that with large Rayleigh friction in that there is a balance between applied heating and upwelling, but the temperature and zonal wind responses are quite different. In this dynamical regime, the change in zonal wind (and, hence, through thermal wind balance, the change in temperature) is determined by the requirement that there is balance in the momentum equation and this change therefore depends on the details of the function *G*. The same point has been noted above with respect to different values of the Rayleigh friction coefficient. In other words, provided that *G* is sufficiently sensitive to *G*, but the changes in zonal wind and temperature are not.

Note in particular that for the Rayleigh friction case the change in force has, by definition, the same shape in the latitude–height plane as the change in velocity since they are proportional to each other. In contrast, this correspondence does not hold for the three-dimensional case where the change in the acceleration due to the waves (Fig. 2e) differs from the zonal wind change (Fig. 2d).

## 4. Model response to different types of applied heating

Having established a dynamical regime in which the applied heating is balanced by upwelling and the wave force adjusts to balance the corresponding Coriolis torque, we now investigate the implications of variations in the latitude, width, and strength of the heating. The next subsections report details of a set of experiments in the 3D model in which these quantities are varied. The parameters for this set of experiments are listed in Table 1. Whereas the standard case (**marked by an asterisk**) was motivated specifically by the radiative calculations in M16, the wider set of experiments are intended to explore the range of dynamical behavior rather than to model specific aspects of the real atmosphere.

Parameter values used in (4) for three sets of experiments that test the response of the idealized 3D model when the latitude (group A), strength (group B), or width (group C) of the applied heating is changed. The standard case (boldface), discussed in section 3, is repeated for completeness.

### a. Varying the latitude of the maximum heating (group A)

Figure 5 shows the temperature, upwelling, and divergence of the Eliassen–Palm flux response to a heating perturbation located at 10°, 20°, 40°, and 60°N/S (group A in Table 1). It is worth noting that although the experiment is set up such that the response should have hemispheric symmetry in the statistically steady state, there are some asymmetries that are especially prominent in

As the applied heating is moved away from the equator, the dominant structure of the upwelling response continues to match that of the applied heating, even when the latter is located in the extratropics. Further quantitative detail is given by Fig. 6, which shows the ratio *f* dependence implied by (2), it is consistent with (2) provided that *K* is sufficiently large; that is, the wave force is sufficiently sensitive to the zonal velocity. Indeed, the same behavior is observed in the Rayleigh drag case for large enough values of *κ* as may be seen from the additional gray curves in Fig. 6. These were obtained through a calculation involving the use of Hough function expansions (e.g., Garcia 1987). The numerical calculation requires a large number of Hough functions to converge close to the equator and adequate convergence was obtained using the first 1000 eigenfunctions. This method is a simplification of the Rayleigh drag calculation performed using the model (described in section 3c) since the buoyancy frequency is set to a constant value typical of the stratosphere in the Held–Suarez case (*κ* is increased. The black dotted–dashed curve from the full model further shows that the contribution from changes in static stability that give rise to the term

### b. Varying the strength of the heating (group B)

With the peak heating anomalies centered at 15°N/S, the strength of the heating is varied from 0.025 to 1 K day^{−1} (group B in Table 1). Typical heating perturbations that are observed in reanalysis diabatic heating rates are about 0.3 K day^{−1} (see M16, their Fig. 4). The response to the heating in all the cases is similar in structure with a near-linear relationship between the maximum heating ^{−1} (Fig. 7a).

The ratio of the imposed heating to the upwelling term (Fig. 7b) is close to 1, which is consistent with the fact that the dynamical aspect ratio (2) does not include a dependence on magnitude of the heating. For small ^{−1}), the behavior starts to become nonlinear, because the term

The conclusion from this set of experiments is that for heating amplitudes less than 0.5 K day^{−1}, the response is essentially linear and the upwelling consistently provides the dominant balance to the applied heating (for this particular heating structure). Larger amplitudes of applied heating give rise to nonlinear effects but are not likely to be relevant to the real tropical lower stratosphere.

### c. Varying the width of the heating (group C)

The experiments in group C (Table 1) address the change in response as the width of the double peaks is increased. As shown by Fig. 8, for cases with *ϕ*_{max} = 15°N/S and *ϕ*_{max} = 60°N/S, as the width of the perturbation is increased, the upwelling term no longer provides the dominant balance to the applied heating and other terms make comparable contributions. For *ϕ*_{max} = 15°N/S, the change in the static stability term *ϕ*_{max} = 60°N/S both

This is illustrated by the response to a broad heating, for the case

### d. Idealized orography

The standard Held–Suarez configuration is a convenient vehicle for a first exploration of the response of a system with synoptic- and planetary-scale eddies to applied heating in the tropical lower stratosphere. However, it is not defensible as an accurate quantitative model of the troposphere–stratosphere system and it is important to establish whether the results reported so far are robust to changes in this idealized configuration. As a first step, a crude representation of orography is added to the Northern Hemisphere in the standard model run. A wave-1 perturbation is added to the surface geopotential height as a sine wave in longitude and a half sine wave in latitude between 25° and 45°N with an amplitude of 500 m. Figure 10 shows the same quantities as Figs. 2b–e but for the case with orography. Comparing Figs. 10a and 5b, we find that the upwelling response is qualitatively similar in the case with and without orography, again confirming that in this regime of “narrow heating,” the response is not sensitive to the details of the wave field. The ratio

Since both hemispheres are in the appropriate dynamical regime where the wave force is sufficiently sensitive to the zonal velocity that this applied heating is narrow according to (2), the dominant balance in the thermodynamic equation is that the upwelling balances the applied heating. Therefore, the upwelling response, in the region of the applied heating, is relatively insensitive to the details of the wave force and, as predicted by the scaling arguments given above, is similar between the two hemispheres. On the other hand, the response in the zonal wind, temperature, and divergence of the Eliassen–Palm flux are more sensitive to the details of the wave force, and these therefore differ between the two hemispheres. In particular, the response in

## 5. Response to an imposed force

Having observed a dynamical regime where there is significant compensation between an imposed narrow heating by the upwelling response, we explore the response to an imposed mechanical forcing. This is motivated by recent discussion (Cohen et al. 2013, 2014) of the dynamics of model-predicted increases in the strength of the Brewer–Dobson circulation due to increases in the concentrations of long-lived greenhouse gases (e.g., Butchart 2014). It has been previously noted (e.g., Cohen et al. 2013) that while the predicted rate of increase in the strength of the circulation is broadly consistent across many models at about 2% decade^{−1}, there are significant disagreements among the models regarding the quantitative contributions of changes in wave forces from different wave types (Butchart et al. 2011). In some models the change is primarily from parameterized gravity waves, while in others it is primarily from synoptic- and planetary-scale Rossby waves that are resolved by the model dynamics. Cohen et al. (2013) have characterized this as a “compensation” by which the Brewer–Dobson circulation response to changes to the parameterized gravity waves, for example, is compensated by the Brewer–Dobson circulation response to consequential changes in the resolved waves. Cohen et al. (2013, 2014) have discussed possible mechanisms for this compensation, including a role for barotropic instability; however, the relevance of the latter mechanism has been questioned by Sigmond and Shepherd (2014), who studied compensation in a general circulation model, and by Watson and Gray (2015), who studied it in a stratosphere–mesosphere model.

Returning to the arguments of section 2, consider the wave force *G* appearing in (1a) to be a function not only of the zonal-mean state but also of some set of external parameters *G* will be due in part to the change in the zonal mean state—that is, the zonal-mean zonal wind and temperature field. We could express this formally by writing the change in *G* as a part that involves partial derivatives with respect to **h** and a part involving partial derivatives (or functional derivatives) with respect to flow variables. A convenient simplification would be to approximate the change

Just as previously we have asked whether an applied heating perturbation is balanced by upwelling or by a change in Newtonian cooling (or more generally longwave radiative heating) due to a change in temperature, here we ask whether the change in force

*K*is a typical value of the ratio

*G*to the velocity. This is simply (2) with the length scales

When the dynamical aspect ratio (9) is large,

*ϕ*

_{max}= 30°S. All other corresponding parameters describing the height and width of the forcing are set to those used in the heating case in (4). The amplitude

^{−1}s

^{−2}. The imposed force for the

Our findings are consistent with the results reported by Cohen et al. (2013), who, in seeking to explain the compensation in the driving of the meridional circulation between the resolved waves and the parameterized waves, carried out numerical experiments where a given force (corresponding to our

## 6. Discussion

In this paper, we have analyzed the response of the circulation to imposed zonally symmetric heating and mechanical forcing when the wave force can change as part of the response. A specific motivation for considering the response to the heating was to understand the double peak in upwelling in the tropical lower stratosphere. A specific motivation for considering the response to mechanical forcing was the relevance to recent discussions of dynamical compensation in the trends in the Brewer–Dobson circulation. In M16, we looked at diagnostic studies of the angular momentum balance and radiative heating and argued that the double peak in upwelling near 70 hPa and 20°N/S is likely to be caused by latitudinal structure in the radiative heating rather than being a response to latitudinal structure in the wave force alone in the absence of any externally imposed structure in the radiative heating. This hypothesis implicitly requires a mechanism by which a long-term change in the meridional circulation can be caused by a change in radiative heating. For such a change to be maintained, there has to be a self-consistent angular momentum balance and hence also a change in the wave force. In this paper we have investigated this hypothesis further in a simple three-dimensional dynamical model, set up in the Held and Suarez (1994) configuration, which we argue captures the essential wave dynamics relevant to the subtropical lower stratosphere. A radiative heating perturbation was imposed by adding two localized regions of heating to the Held–Suarez configuration. For a latitudinally confined diabatic heating perturbation, the dominant balance in the thermodynamic equation in the region of the heating perturbation is between the heating and the upwelling terms. The temperature change makes a relatively small contribution (through the Newtonian cooling term) to the thermodynamic equation in this region and the latitudinal scale of the overall temperature change is much broader than the scale of the heating perturbation, with weak temperature gradients across the tropics and subtropics. Angular momentum balance is maintained by a change in the Eliassen–Palm flux, so that the change in wave force balances the Coriolis force associated with the change in meridional circulation.

We set out scaling arguments to provide some dynamical insight into this circulation response. These arguments assume that the typical magnitude of the change in wave force is *K* times the typical magnitude of the change in zonal velocity, with *K* having the dimensions of inverse time. According to these arguments, an applied heating would be primarily balanced by an upwelling provided that the dynamical aspect ratio *K*, *N*, *f*, and *α*. In particular, large values of the parameter *K* and/or small values of the Coriolis parameter *f* (i.e., low latitudes) make it more likely that the condition is satisfied. The scaling arguments are similar to those applied by previous authors (e.g., Fels et al. 1980; Garcia 1987) in considering the zonally symmetric response to heating when the wave force is represented by Rayleigh friction (*K* is then simply the Rayleigh friction coefficient) but potentially have wider applicability.

We presented explicit zonally symmetric calculations with Rayleigh friction to explore in a crude way the dependence of the response on *K*. These calculations capture the balance between applied heating and upwelling seen in the 3D simulations when the heating is deep and narrow (cf. Fig. 2b and the third column in Fig. 4). However, they do not capture, for example, the response in zonal velocity (cf. Fig. 2b and the second column in Fig. 4). This response is determined by the details of the dependence of the wave force on the zonal velocity in the 3D simulations. This dependence is poorly represented by Rayleigh friction.

We have demonstrated by varying the width

Perhaps surprisingly, for an applied heating with a width of *K* was at each latitude sufficiently large to ensure large values of (2). Whether such “narrow” higher-latitude cases are an appropriate model of any specifically realized process in the real atmosphere is less clear, although one might consider their relevance to the diabatic effects of trends in extratropical ozone, particularly associated with the ozone hole.

From the radiative calculations in section 4 of M16 and the dynamical calculations in section 2, we deduce that the ozone distribution (and its radiative implications) is an important part of the cause of the double peak in upwelling. In reality, of course, dynamics, radiation, and chemistry are fully coupled and the ozone distribution is determined by transport processes. This is not captured by our dynamical calculations, in which the structure of the applied heating is simply imposed, but for reasons explained in section 5 of M16, these calculations nonetheless seem to give significant insight to the double-peak structure of the low-latitude upwelling.

Note the additional important point that a fixed dynamical heating calculation, which is often used to infer temperature changes that result from changes in constituents such as ozone, would not be relevant here—it is precisely the dynamical heating that is the main response to the structure in the ozone field.

We have noted with respect to the response to an applied mechanical forcing, added to the wave force, that the dynamical discussion leading to (2) can be extended to consider this response. In this case when the applied force is narrow in the sense that the aspect ratio in (9) is large, the applied force is primarily balanced by an adjustment to the flow-dependent wave force rather than by the Coriolis torque. The response in the meridional circulation is therefore small. If it is accepted that differences in parameterized waves correspond to narrow applied forces, which is suggested by the results of Cohen et al. (2013, their Figs. 4b and 5d) and Sigmond and Shepherd (2014, their Fig. 2), then this potentially offers an overall dynamical principle that explains the compensation (between changes in the Brewer–Dobson circulation driven by resolved waves and changes driven by parameterized waves) observed in climate model simulations. Our analysis suggests that the compensation is, to leading order, independent of the details of the background wave force, similar to the case with an imposed heating, and does not rely on a specific mechanism for the dependence of the wave force on the mean state. Of course, if the aspect ratio, in (9) is to be of quantitative use, then the sensitivity *K* must be estimated. As we have emphasized previously, this is by no means straightforward, because it essentially requires a parameterization of wave force for an arbitrary mean flow.

## Acknowledgments

The authors thank Amanda Maycock for help with the radiation code and for helpful discussions. AM and PHi acknowledge funding support from the European Research Council through the ACCI project (Grant 267760) lead by John Pyle. PHi also acknowledges support from an NSERC postdoctoral fellowship. The authors are grateful to Stephan Fueglistaler and Tom Flannaghan for conversations that stimulated some of this work. We received detailed and helpful comments from three anonymous reviewers that improved this manuscript.

## APPENDIX

### Statistical Methods

*X*and

*Y*and tested the null hypothesis

*N*and calculating

*N*tends to

*n*.

An example of this calculation is shown in Fig. A1 for the temperature change when the heating is applied at latitude 15° as described in section 3. Let *X* be a time series for the unperturbed Held–Suarez run (with mean *Y* be a time series with a heating perturbation (with mean

## REFERENCES

Butchart, N., 2014: The Brewer-Dobson circulation.

,*Rev. Geophys.***52**, 157–184, doi:10.1002/2013RG000448.Butchart, N., and Coauthors, 2011: Multimodel climate and variability of the stratosphere.

,*J. Geophys. Res.***116**, D05102, doi:10.1029/2010JD014995.Cohen, N. Y., E. P. Gerber, and O. Bühler, 2013: Compensation between resolved and unresolved wave driving in the stratosphere: Implications for downward control.

,*J. Atmos. Sci.***70**, 3780–3798, doi:10.1175/JAS-D-12-0346.1.Cohen, N. Y., E. P. Gerber, and O. Bühler, 2014: What drives the Brewer–Dobson circulation?

,*J. Atmos. Sci.***71**, 3837–3855, doi:10.1175/JAS-D-14-0021.1.Dickinson, R. E., 1971: Analytic model for zonal winds in the tropics.

,*Mon. Wea. Rev.***99**, 501–510, doi:10.1175/1520-0493(1971)099<0501:AMFZWI>2.3.CO;2.Fels, S. B., J. D. Mahlman, M. D. Schwarzkopf, and R. W. Sinclair, 1980: Stratospheric sensitivity to perturbations in ozone and carbon dioxide: Radiative and dynamical response.

,*J. Atmos. Sci.***37**, 2265–2297, doi:10.1175/1520-0469(1980)037<2265:SSTPIO>2.0.CO;2.Forster, P. M., M. Blackburn, R. Glover, and K. P. Shine, 2000: An examination of climate sensitivity for idealised climate change experiments in an intermediate general circulation model.

,*Climate Dyn.***16**, 833–849, doi:10.1007/s003820000083.Fueglistaler, S., P. H. Haynes, and P. M. Forster, 2011: The annual cycle in lower stratospheric temperatures revisited.

,*Atmos. Chem. Phys.***11**, 3701–3711, doi:10.5194/acp-11-3701-2011.Garcia, R. R., 1987: On the mean meridional circulation of the middle atmosphere.

,*J. Atmos. Sci.***44**, 3599–3609, doi:10.1175/1520-0469(1987)044<3599:OTMMCO>2.0.CO;2.Haynes, P., 2005: Stratospheric dynamics.

,*Annu. Rev. Fluid Mech.***37**, 263–293, doi:10.1146/annurev.fluid.37.061903.175710.Haynes, P., M. E. McIntyre, T. G. Shepherd, C. J. Marks, and K. P. Shine, 1991: On the “downward control” of extratropical diabatic circulations by eddy-induced mean zonal forces.

,*J. Atmos. Sci.***48**, 651–680, doi:10.1175/1520-0469(1991)048<0651:OTCOED>2.0.CO;2.Held, I. M., and M. J. Suarez, 1994: A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models.

,*Bull. Amer. Meteor. Soc.***75**, 1825–1830, doi:10.1175/1520-0477(1994)075<1825:APFTIO>2.0.CO;2.Hoskins, B. J., and A. J. Simmons, 1975: A multi-layer spectral model and the semi-implicit method.

,*Quart. J. Roy. Meteor. Soc.***101**, 637–655, doi:10.1002/qj.49710142918.Ming, A., P. Hitchcock, and P. Haynes, 2016: The double peak in upwelling and heating in the tropical lower stratosphere.

,*J. Atmos. Sci.***73**, 1889–1901, doi:10.1175/JAS-D-15-0293.1.Plumb, R. A., 1982: Zonally symmetric Hough modes and meridional circulations in the middle atmosphere.

,*J. Atmos. Sci.***39**, 983–991, doi:10.1175/1520-0469(1982)039<0983:ZSHMAM>2.0.CO;2.Plumb, R. A., and J. Eluszkiewicz, 1999: The Brewer–Dobson circulation: Dynamics of the tropical upwelling.

,*J. Atmos. Sci.***56**, 868–890, doi:10.1175/1520-0469(1999)056<0868:TBDCDO>2.0.CO;2.Randel, W. J., R. Garcia, and F. Wu, 2008: Dynamical balances and tropical stratospheric upwelling.

,*J. Atmos. Sci.***65**, 3584–3595, doi:10.1175/2008JAS2756.1.Scott, R. K., 2002: Wave-driven mean tropical upwelling in the lower stratosphere.

,*J. Atmos. Sci.***59**, 2745–2759, doi:10.1175/1520-0469(2002)059<2745:WDMTUI>2.0.CO;2.Shepherd, T. G., and C. McLandress, 2011: A robust mechanism for strengthening of the Brewer–Dobson circulation in response to climate change: Critical-layer control of subtropical wave breaking.

,*J. Atmos. Sci.***68**, 784–797, doi:10.1175/2010JAS3608.1.Sigmond, M., and T. G. Shepherd, 2014: Compensation between resolved wave driving and parameterized orographic gravity wave driving of the Brewer–Dobson circulation and its response to climate change.

,*J. Climate***27**, 5601–5610, doi:10.1175/JCLI-D-13-00644.1.von Storch, H., and F. W. Zwiers, 2001:

*Statistical Analysis in Climate Research*. Cambridge University Press, 484 pp.Watson, P. A. G., and L. J. Gray, 2015: The stratospheric wintertime response to applied extratropical torques and its relationship with the annular mode.

,*Climate Dyn.***44**, 2513–2537, doi:10.1007/s00382-014-2359-2.