The tropical tropopause layer (TTL), spanning from 150 to 70 hPa or 14 to 18.5 km, is the main entry region for air into the stratosphere from the troposphere e.g.. The properties of this region are influenced by the presence of a prominent annual cycle in temperatures which is clear in, for example, radiosonde measurements and GPS radio occultation measurements . Figure  shows the structure of the annual temperature cycle in a month-by-month climatology from the ERA-Interim reanalysis dataset , constructed using data from 1991 to 2010. Consistent with earlier studies, the annual cycle in temperatures is coherent over the layer from 130 to 40 hPa (Fig. a), with relatively cold temperatures in Northern Hemisphere (NH) winter, relatively warm temperatures in NH summer only and early autumn, and weak latitudinal gradients over the tropics (20∘ N–20∘ S; Fig. b–d). The maximum peak-to-peak amplitude of the annual cycle is 8 K near 70 hPa, decreasing to 6 K at 90 hPa and 3 K at 100hPa, below which the amplitude reduces very rapidly (Fig. a). Above about 30 hPa (not shown), temperature variations are dominated by the semi-annual oscillation.

The temperature variations at the tropical cold point near 100 to 90 hPa regulate the water vapour entering the stratosphere on annual and interannual timescales by modulating the freeze drying of upwelling air e.g.. In particular, the regular annual cycle in temperatures shown in Fig. c at 90 hPa and panel (d) at 100 hPa leads to a substantial annual cycle in water vapour. The additional effect of upward transport in the lower-stratospheric Brewer–Dobson circulation creates the well-known water vapour tape recorder signal e.g.. Note that the larger annual temperature cycle at 70 hPa (Fig. b) does not have a direct effect on water vapour because overall temperatures are higher than at the cold point below.

Despite the potential significance of the annual cycle in TTL temperatures, both in its role in determining stratospheric water vapour mixing ratios and also simply as a conspicuous and persistent aspect of temperature variation, the mechanisms responsible for the cycle are not yet completely clear. Furthermore, state-of-the-art climate models, e.g. within the CMIP5 dataset, still exhibit large inter-model differences in the amplitude of the annual cycle at 100 hPa with peak-to-peak amplitudes ranging from ∼1 to ∼5 K compared to ∼4 K for the 15∘ N–S average in ERA-Interim . This indicates that many current climate models do not capture correctly the processes that drive the TTL annual temperature cycle. Moreover, the lack of an understanding of the quantitative impact of different physical mechanisms on TTL temperatures precludes the development of a set of general principles for improving models .

To consider the annual cycle further, it is useful to begin by introducing the thermodynamic equation in the transformed Eulerian mean framework , neglecting eddy terms: ∂tT‾=Q‾rad-w‾∗S‾-v‾∗∂yT‾=Q‾rad+Q‾dyn; this predicts the rate of change of zonal mean temperature, T‾, with time, t, where (.)‾ represents a zonal mean. The dynamical heating, Q‾dyn, is defined by the second equality in Eq. (). v‾∗ and w‾∗ are the horizontal and vertical components of the mean residual velocity respectively. y is the meridional coordinate. S‾=∂zT‾+κT‾/H is a measure of the static stability, where z is the log-pressure height. z=-Hlog⁡(σ), where H is a scale height taken to be 7 km, and σ=p/p0, where p is pressure and p0=1000 hPa. κ=R/cp≃2/7, where R is the gas constant for dry air and cp is the specific heat at constant pressure. The radiative heating, Q‾rad, depends in general on the temperature and the distributions of various radiatively active components including clouds, aerosols, and trace gases.

Combined with the equations for zonal wind, continuity, and thermal wind balance, Eq. () determines the zonally symmetric response to imposed heatings and mechanical forcing Chap. 3. This is particularly important in considering the response to an imposed forcing on the right-hand side of Eq. () or indeed an imposed forcing term in one of the other equations.

Many studies have focused on the role of wave-induced forces in driving the annual cycle in temperature through their effects on upwelling in the TTL and hence on Q‾dyn in Eq. (), although uncertainty remains about what types of waves are the most important and references therein. However, radiative contributions to the annual cycle have also been suggested, principally in connection with the strong annual cycle in TTL ozone mixing ratios . The quantitative effect of ozone on TTL temperatures was first investigated by , who used a one-dimensional radiative convective model representing a tropical average profile. They concluded that at 70 hPa, about 3 K of the observed 8 K peak-to-peak variation in temperature might be caused by the radiative effects of the annual cycle in ozone, reducing to about 1 K of the observed 3 K peak-to-peak variation at 100 hPa. used a seasonally evolving fixed dynamical heating (SEFDH; see Sect. ) calculation and found a slightly smaller contribution from ozone of 2 K at 70 hPa.

Both and asserted that annual variations in TTL water vapour mixing ratios only have a small role in determining the annual cycle in temperatures; quantitative details, however, were omitted. There has been recent significant interest in the radiative effect of variations in stratospheric water vapour, both in the effect on the radiative balance of the troposphere e.g. and also in the effect on the lower stratosphere. For example, used a set of radiative calculations to show that a uniform increase in stratospheric water vapour gives rise to a cooling that is largest in the lower stratosphere at all latitudes.

In this work, we investigate, first using the SEFDH approach, the individual and combined radiative effects of the annual cycles in ozone and water vapour on TTL temperatures, including at 70 hPa where the amplitude of the annual cycle is at a maximum and at 90 hPa near the cold point which is crucial for determining stratospheric water vapour mixing ratios. The radiative calculations required for this investigation also allow us to examine carefully how vertical structure in the background radiative environment combines with the variations in radiative and dynamical heating to determine the vertical structure of the annual cycle in temperatures. All the calculations presented here neglect any cloud effects and assume clear-sky conditions.

For a more complete assessment of the effect of seasonal variations in ozone and water vapour on the annual cycle in TTL temperatures, which goes beyond the simplifying assumptions of the SEFDH approach, it is necessary to take account of dynamical changes. The seasonal cycle in radiative heating induced by ozone and water vapour variations will in part be balanced by a change in the meridional circulation e.g.. This is shown in Sect.  below to modify strongly the latitudinal structure of the temperature response.

The structure of the paper is as follows. Section  describes the data and the radiative calculations. Section  describes the results of SEFDH calculations to quantify the effect of annual variations in ozone and water vapour on the annual cycle of temperature in the TTL. This section includes a detailed discussion of the radiative effects of water vapour variations omitted by previous authors. These calculations are complemented by a set of illustrative fixed dynamical heating (FDH) radiative calculations in Appendix . The SEFDH temperature changes are also sensitive to the background ozone mixing ratios and a set of further calculations is presented in Appendix . (The results presented in Sect.  are in broad agreement with those from similar work by , which we became aware of during the review process.) Section  discusses the vertical structure of the annual cycle in temperature, distinguishing the role of the background radiative environment from that of the radiative and dynamical heating in determining this structure. Details regarding the estimates of uncertainty associated with the calculations in Sects.  and are given in Appendix . Section  then goes beyond the SEFDH calculation reported in Sect.  to consider how the temperature response to variations in ozone and water vapour is modified by the zonally symmetric dynamical response to the radiative heating. The final section discusses the results and their implications and reviews the various simplifying assumptions that have been made.

We use temperature and dynamical fields from the ERA-Interim reanalysis dataset covering the period 1991 to 2010, using data at a horizontal resolution of 1∘, at 6-hourly analysis time intervals (00:00, 00:06, 00:12, and 00:18 UTC) and at 60 model levels. The mean residual vertical velocity in the transformed Eulerian mean framework, calculated using the same method as , and the dynamical heating used in Sect.  are both computed on the original grid from the ERA-Interim data and then smoothed by linearly interpolating the monthly averages to daily values. The temperatures are also linearly interpolated to the grids relevant for the calculations described below and in Sect. .

Ozone and water vapour mixing ratios are obtained from the Stratospheric Water and OzOne Satellite Homogenized dataset SWOOSH;. This record is formed from a combination of measurements from various limb and solar occultation satellite instruments from 1984 to 2015, namely SAGE-II/III, UARS HALOE , UARS MLS, and Aura MLS instruments. The measurements are homogenized by applying corrections that are calculated from data taken during time periods of instrument overlap. Using the same data, SWOOSH also provides combined monthly climatologies of ozone and water vapour which we make use of in this work. SWOOSH is chosen for this study because it provides a homogenized record useful for climate studies and has been used previously to study both stratospheric water vapour and stratospheric ozone variability. The pressure at the lowest altitude level in SWOOSH is 316 hPa. The results presented in this paper are not sensitive to mixing ratios of water vapour and ozone below 316 hPa (within plausible limits), and for convenience the vertical profiles below 316 hPa were simply defined by linear interpolation between the SWOOSH values at 316 hPa and the surface values taken from ERA-Interim.

The radiative calculations were performed using a modified version of the radiation scheme, which includes updates to the longwave absorption properties of water vapour . All calculations were performed on zonal mean data at 5∘ intervals in latitude and at 100 pressure levels (which are the same as those listed in Appendix  for the FDH calculations). Shortwave heating rates are calculated as diurnal averages and the surface albedo is taken from ERA-Interim data. Carbon dioxide is assumed to be well mixed and the volume mixing ratio is set to 360 ppmv. All calculations in this study assume clear-sky conditions (i.e. neglecting radiative effects of clouds).

To study the radiative contributions of seasonal variations in ozone and water vapour to the annual cycle in TTL temperatures, we make use of the seasonally evolving fixed dynamical heating calculation . This method calculates the time-varying temperature change due to a specified radiative perturbation (e.g. a change in a trace gas) and takes into account the specified time dependence of temperature and trace gas concentration profiles in a background state to which the perturbation is applied.

Given time-varying background profiles (at a specified latitude) of temperatures, T‾0, and mixing ratios of trace gases, χ‾O30 and χ‾H2O0 (where (⋅)0 denotes the background state), the dynamical heating, Q‾dyn0, is first calculated by assuming the balance in Eq. (), i.e.

∂tT‾0=Q‾rad(T‾0,χ‾O30,χ‾H2O0)+Q‾dyn0. A perturbation is applied to the trace gas mixing ratios (Δχ‾O3,Δχ‾H2O), and the new time evolving equilibrium temperature state, T‾0+ΔT‾, is obtained from ∂t(T‾0+ΔT‾)=Q‾rad(T‾0+ΔT‾,χ‾O30+Δχ‾O3,χ‾H2O0+Δχ‾H2O)+Q‾dyn0. Equation () is integrated forward in time with a daily time step until the perturbed temperature field, T‾0+ΔT‾, is also annually repeating. Five years is found to be sufficient for accurate convergence (see Appendix  for the criteria for convergence). The radiative transfer calculation couples vertical levels but not latitudes so that each calculation is local in latitude. Following a similar method to , Eq. () is applied to update the temperature only above a certain level taken here to be 130 hPa, on the basis that there are distinct processes determining temperature variations in the troposphere below. The choice of this level is further justified in Sect. . In setting up the calculation we verified that the results could be reproduced. The SEFDH technique reduces to the more standard and widely used FDH technique if the imposed background temperature and species mixing ratios are constant in time and Eq. () is integrated to a steady state (Appendix ).

The background state is taken to be the annual average ERA-Interim temperature (which implies ∂tT‾0=0) and the annual mean SWOOSH constituent mixing ratios. The latter are then perturbed to their annually varying climatologies. One could alternatively use the annually varying temperature climatology as the base state e.g., but this was found to have a negligible impact on the simulated temperature response. The use of a time-independent background state was also easier to implement in the dynamical calculations reported in Sect. .

The annual cycle in tropical lower-stratospheric temperature is largest over a shallow layer from 100 to 50 hPa, with a maximum amplitude at 70 hPa (Fig. c). This vertical structure has been attributed by to the presence of long radiative timescales in this region. In this section, we reconsider the question of whether the location of the maximum variation in tropical temperatures over the annual cycle is due to the structure of the major radiative and dynamical forcings and/or to the structure of the background radiative environment.

Figure a shows the variation in tropical averaged w‾∗, in height and time, and reveals a systematic decrease in amplitude with increasing height from 150 to 50 hPa. Figure b shows the full dynamical heating term, w‾∗S‾, and Fig. c shows the same quantity with the annual mean, 〈w‾∗S‾〉, removed. The annual cycle in dynamical heating is larger above 100 hPa compared to below and rather uniform in amplitude over a deep layer that extends from 90 hPa up to about 40 hPa (Fig. c). This behaviour is strongly influenced by the annual cycle in upwelling (Fig. d), and in the region 120 to 90 hPa, it is further modified by the annual cycle in static stability (Fig. e), which causes a reduction in the annual cycle in dynamical heating around 100 hPa. The peak-to-peak amplitude of the annual cycle in dynamical heating in the lower stratosphere is around 0.15 to 0.2 K day-1. This amplitude does not decrease below the tropopause as rapidly as does the observed peak-to-peak amplitude of temperature (Fig. a).

To probe the effects of these heating structures on the annual temperature cycle, we now consider a set of SEFDH-like calculations forced by a set of specified dynamical heating structures. This is achieved by imposing an additional dynamical heating, ΔQ‾dyn, on the right-hand side of Eq. (). To determine whether the localization of temperature variation is a result of the structure of the radiative environment, we first consider an idealized dynamical heating perturbation with no vertical structure: ΔQ‾dyn=-0.1cos⁡(2πt/365) K day-1. For this and the next few calculations, we remove the constraint on temperatures below 130 hPa, permitting them to evolve freely in response to the radiative perturbations. Any vertical dependence in the response to this heating will therefore be solely determined by vertical structure in the temperature-dependent part of the radiative heating.

The resulting temperature change for this case is shown in Fig. a. The amplitude of the response is largest in a layer centred on 100 hPa. The phase lag with respect to the imposed heating is also largest in this layer and equal to about 60 days. This is consistent with a Newtonian cooling model in which the radiative relaxation timescale was a maximum of about 60 days at 100 hPa, roughly at the cold point, consistent with theoretical expectations . The implied radiative timescales peak over a broader height range than those found by and in particular do not show such a strong reduction below 100 hPa. inferred damping timescales from the cross-correlation between the annual components of analysed T‾ and w‾∗, which implicitly includes non-local effects such as those of non-radiative processes operating in the upper troposphere. This is also true of the supporting radiative calculations they performed on the basis of observed temperature anomalies. As demonstrated below, the tropospheric processes have a substantial effect on the relaxation of temperature anomalies even in the lower stratosphere, in part because of the strong dependence of radiative timescales on the vertical scale of the imposed temperature perturbation .

The response to the annual cycle in w‾∗S‾ from ERA-Interim (Fig. c) is now considered. For convenience, we set the dynamical heating below 450 hPa to have the same value as at 450 hPa. This does not affect the main conclusions of this calculation. The corresponding temperature response is shown in Fig. b. The vertical structure in the dynamical heating significantly modifies the vertical structure in the temperature response. In particular, the fact that the dynamical heating is larger at 70 hPa than at 100 hPa leads to a larger temperature response at 70 hPa than at 100 hPa, in contrast to the response to the uniform dynamical heating shown in Fig. a. Therefore, the vertical structure in amplitude of the annual temperature cycle driven by dynamical heating is determined by both the background radiative environment and by the vertical structure of the dynamical heating itself.

To further illustrate this, Fig. c shows the temperature response to the ERA-Interim dynamical heating assuming a constant radiative relaxation timescale of 40 days. The response is a good approximation to that in Fig. b around 70 hPa, suggesting that the radiative timescale appropriate for the dynamical heating perturbation is around 40 days, somewhat shorter than that inferred from Fig. a and consistent with the smaller vertical length scale of the imposed perturbation. However, there remains below 100 hPa a peak-to-peak amplitude in the temperature that is significantly larger than is observed. If we assume that this calculation of the dynamical heating provides a reasonable estimate of the magnitude of the dominant terms in the thermodynamic budget of the upper troposphere, this suggests that upper-tropospheric processes provide a stronger constraint on temperature perturbations than do clear-sky radiative processes. Further calculations (results not shown) suggest (subject to the preceding assumption) that the effective timescale of this constraint is approximately 10 days.

To illustrate the implications of the observed tropospheric constraint on temperatures, we reintroduce the clamp on the temperatures below 130 hPa in the SEFDH calculation as a simple representation of these processes. Figure e shows the resulting temperature response to the same dynamical heating perturbation imposed in Fig. b. The tropospheric constraint causes the maximum amplitude of the response to shift upwards from around 80 hPa to about 70 hPa and reduces the magnitude of the peak response compared to Fig. b. It is clear, therefore, that this upper-tropospheric constraint has a significant radiative effect on the region above.

We now add the radiative heating perturbations from the annual ozone and water vapour cycles to the ERA-Interim dynamical heating to produce Fig. f. The net effect of the annual ozone and water vapour cycles, as shown in Sect. , is to increase the amplitude of the temperature response. This produces an annual cycle with a structure that is in better agreement than that in Fig. e with the ERA-Interim annual cycle, Fig. d, with a more pronounced peak at 70 hPa.

In summary, the calculations reported in this section suggest that the vertical structure of the peak-to-peak amplitude in the annual cycle of temperatures arises from a combination of several effects. In the absence of the implied upper-tropospheric constraint, we find that clear-sky radiative processes produce long radiative timescales over a deep layer centred around 100 hPa and would, in the absence of other effects, imply a similarly deep structure in the amplitude of the annual cycle. The vertical structure in the dynamical heating and radiative heating from constituent changes, both of which exhibit a peak in the region around 80 to 70 hPa, combined with the tropospheric constraint, lead to a shallower vertical structure with a stronger response at 70 than at 100 hPa.

We will now consider the temperature response to annual cycles in ozone and water vapour, relaxing the SEFDH assumption to include zonally symmetric dynamical adjustment. This approach assumes no change in the zonally averaged wave force, which might well be a significant part of the full dynamical response in a three-dimensional atmosphere, even if the imposed annual cycles in ozone and water vapour are zonally symmetric. We discuss the implications of this simplifying assumption in Sect.  below. The zonally symmetric dynamical response problem has been considered in many previous papers e.g.. The expectation from this previous work is that the response to the heating implied by an imposed change in constituents will occur in part through dynamical heating, modifying the vertical and latitudinal structure of the temperature response. One important difference in our approach from these previous studies is that, rather than approximating the temperature-dependent part of the radiative heating by Newtonian cooling, we continue to use the modified Morcrette–Zhong and Haigh radiation code.

We have analysed radiative aspects of the prominent annual cycle in temperature in the TTL and tropical lower stratosphere, which has a maximum peak-to-peak amplitude at 70 hPa of ∼8 K. Building on previous work , we have applied the seasonally evolving fixed dynamical heating (SEFDH) method to calculate the temperature response to the annual cycle variations in zonal mean ozone and water vapour, derived here from the SWOOSH satellite dataset . We extend the previous work by presenting explicit results for the effects of water vapour variations and by paying particular attention to the vertical structure of the temperature response and the role of variations in the trace gas mixing ratios in different vertical layers. In our first approach, we have used an SEFDH calculation in which the temperature response to annual variations in a trace gas is calculated independently at each latitude, assuming that the dynamical heating at each height is unchanged from its value in a control state in which the trace gas mixing ratios are constant (and equal to their annual mean values).

We find substantial contributions to the peak-to-peak amplitude of the tropical average (20∘ N–20∘ S) annual cycle in temperatures from ozone (3.5±0.4 K at 70 hPa, 3.3±0.5 K at 90 hPa, and 2.6±0.2 K at 100 hPa) and from water vapour (0.9±0.1 K at 70 hPa, 1.1±0.1 K at 90 hPa, and 1.0±0.03 K at 100 hPa). Whilst the ozone contribution maximizes around 70 hPa and is roughly in phase with the observed annual temperature cycle, the water vapour contribution maximizes around 90 hPa and is of the opposite phase. Despite the cancellation, the net effect of variations in ozone and water vapour together is substantial and amounts to about 35 % of the observed annual cycle at both 70 and 90 hPa and about 45 % at 100 hPa (Fig. ). Our results are broadly consistent with the recent independent work of .

Further SEFDH calculations showed that in the region where the ozone has the largest temperature change, 70 hPa, the ozone-induced temperature variation is caused primarily (80 %) by local ozone variations (Fig. ). In contrast, the water-vapour-induced temperature variation is largest at 90 hPa and is caused by both local and non-local water vapour variations. Overall, 60 % of the water-vapour-induced temperature variation at this level comes from water vapour variation in the region 100 to 80 hPa and 40 % from the region 130 to 100 hPa (Fig. ). This upward non-local radiative effect is seen throughout the lower stratosphere and has important implications for cold point temperatures. For example, if the amplitude of the annual cycle in water vapour below the cold point was to increase, then the radiative effect would reduce the amplitude in the annual cycle in cold point temperatures and hence reduce the amplitude of the annual cycle in water vapour at and above the cold point.

All of the calculations make use of a clear-sky assumption. A rough SEFDH calculation taking into account an estimate of the annual mean climatological high cloud cover shows that the peak-to-peak annual cycle temperature change due to ozone at 70 hPa decreases by 5–10 % at all latitudes between 20∘ N and 20∘ S. The effect on the annual water vapour cycle at the same level is negligible. The clouds lead primarily to a reduction in the amount of upwelling longwave radiation reaching 70 hPa of about 0.05 K day-1, which in turn decreases the ozone temperature response. A full assessment of the cloud effect is beyond the scope of this work and further work is needed to establish its precise contribution.

We also examined the factors controlling the vertical structure of the amplitude of the annual cycle in temperatures. The observed maximum centred on 70 hPa and, largely restricted to the 50 to 100 hPa layer, arises from a combination of several factors. The vertical structure cannot be explained by clear-sky radiative damping timescales alone, which maximize over a deep region, centred near the cold point at 100 hPa. However, both the dynamical and radiative forcings maximize above the cold point, and in combination with an inferred upper-tropospheric constraint active below 130 hPa, these lead to the observed maximum at 70 hPa. We have not attempted to provide an explanation for the inferred upper-tropospheric constraint and highlight this as an area for further study.

Finally, we investigated the effect on the temperature response of relaxing the SEFDH assumption, thereby going beyond the work of and . We do this by incorporating the radiative code used for the SEFDH calculations within a 2-D (height–latitude) dynamical model. Consistent with dynamical expectations, part of the heating associated with annual cycle variations in both ozone and water vapour drives an annual cycle in the upwelling and that may play a non-negligible role in determining the latitudinal structure of the observed annual cycle in upwelling. This has the effect of reducing latitudinal gradients in the SEFDH-predicted temperature response, particularly across the tropics. However, this modification of the response leaves the tropical (20∘ N–20∘ S) average temperature response essentially unchanged. Therefore, the conclusion that the net effect of ozone and water vapour contributes about 35 % of the annual cycle peak-to-peak amplitude at 70 and 90 hPa from the SEFDH calculations is robust to including the dynamical adjustment. The detailed latitudinal structure predicted by the SEFDH calculation, however, is not robust to this adjustment.

As explicitly illustrated by Fig. d (for ozone), the differences between the temperature responses to ozone and water vapour calculated through the SEFDH approach and those calculated using the 2-D dynamical model demonstrate that low-latitude temperature features with small latitudinal scales predicted by SEFDH calculations are unlikely to be reproducible when the SEFDH assumption is relaxed because these features will be smoothed out by the dynamical response. (When considering annual variations in the TTL, “small latitudinal scales” means less than about 20∘ of latitude.) This applies to Figs. b and b–d in this paper, to previous SEFDH calculations of the temperature response to annual variations in ozone their Fig. 5b, and to similar calculations of the effect of recent interannual variations in ozone and water vapour their Fig. 6.

Within the 2-D zonally symmetric dynamical formalism presented here, we do not take account of changes in wave-induced forces. This effect has been discussed by several authors over the last 30 years or so, including and , usually making the assumption that the wave force can be represented by Rayleigh friction (so that the local wave force is proportional to and opposite of the local zonal velocity). However, it is generally accepted that Rayleigh friction is a poor representation of the wave forces that operate in the upper troposphere and stratosphere. analyse the effect of the change in wave force in the response to imposed steady localized zonally symmetric heating in a simple 3-D model (where the waves are resolved and no Rayleigh friction assumption is necessary) and argue that the effect is to broaden the temperature response, particularly at low latitudes. Latitudinal structure in the imposed heating tends to be balanced by the dynamical heating associated with the meridional velocity response, and the change in wave force provides the necessary angular momentum balance. A similar effect is seen in the zonally symmetric problem with Rayleigh friction e.g. their Fig. 6. There is an analogous effect in the time-dependent zonally symmetric response problem, without any change in wave force or Rayleigh friction, considered in Sect. , with the angular momentum balance including the zonal acceleration. Therefore, the effect of including the change in wave force in the dynamical problem is, broadly speaking, expected to be similar; in addition to that already seen in the time-dependent zonally symmetric problem, at low latitudes, the dynamical adjustment will smooth the temperature response to latitudinally varying heating. If the change in wave force is weak then the additional effect will be small. If the change in wave force is strong then the result will be that the smoothing is over a larger range of latitudes. The fact that the observed annual cycle in temperature is coherent over the latitude range 20∘ N–20∘ S (Fig. b–d), but no more than that, suggests that the wave force effect cannot be too strong. Therefore, we expect that our conclusions from the zonally symmetric dynamical problem studied here would not be changed too much if the change in wave force was included. Furthermore, we expect that similar dynamical principles will allow extension to the fully three-dimensional case, implying that an SEFDH calculation will have limited ability to predict geographical (i.e. latitudinal and longitudinal) variations in temperature resulting from geographical latitudinal variations in radiatively active gases or in other relevant quantities such as clouds or aerosol.

Current comprehensive global (chemistry–)climate models show a large spread in the amplitude of the TTL annual cycle in temperature e.g., but the quantitative causes of these differences are not well understood. The results of this study show that an erroneous representation of the climatology of ozone and water vapour, as is commonplace amongst such models e.g., is likely to be a major contributor to poor model performance for capturing the TTL annual temperature cycle. Similar conclusions are likely to apply to interannual variations, e.g. in the 2010–2013 period investigated by using SEFDH calculations. Progress in improving the representation of the TTL in comprehensive global models therefore requires consideration of the coupling through transport and radiative effects between dynamics, ozone, and water vapour in the TTL. Specific aspects highlighted by our results include a strong sensitivity of ozone radiative effects to mean ozone mixing ratios in the 90 to 70 hPa region, for which models with interactive chemistry simulate a range of values and for which a range of observation-based gridded datasets exist for climate models that do not include chemistry . Furthermore, because of the importance shown here of non-local radiative effects for water vapour in the TTL, modelled cold point temperatures are also likely to be sensitive to the representation of water vapour mixing ratios in the upper tropical troposphere.