1. Introduction
The Brewer–Dobson circulation (BDC) is a global-scale overturning circulation in the stratosphere, characterized by air that ascends into and within the tropical stratosphere, spreading poleward and eventually downward in the extratropical winter hemisphere. This stratospheric circulation can significantly impact tropospheric climate, most notably through its modulation of the distribution of stratospheric ozone, which absorbs harmful ultraviolet radiation from the sun (Dobson 1956). The widely accepted mechanism that explains the existence of the BDC is the principle of “downward control” (Haynes and McIntyre 1987; Haynes et al. 1991). This principle states that for steady circulations, the upward mass flux across a specified vertical level is solely a function of the zonal momentum sources (wave drag) and sinks above that level; thus, processes in the middle and upper stratosphere can exert a “downward” influence on flow in the lower stratosphere and troposphere. The theoretical findings of Haynes et al. (1991) have been well supported by numerical modeling evidence and withstood the test of time (Butchart 2014, and references therein). Thus, in the “downward control” paradigm, wave dissipation drives the circulation.
The BDC is typically separated into two branches: a slow and deep equator-to-pole overturning branch, and a faster shallow branch in the lower stratosphere extending to about 50° latitude (Plumb 2002; Birner and Bönisch 2011). In this study, references to the BDC refer to the shallow branch circulation. The shallow branch is thought to be driven by subtropical wave dissipation in the lower stratosphere (Plumb and Eluszkiewicz 1999, hereafter PE99; Plumb 2002).
In our opinion, there are a few characteristics of the shallow branch circulation that remain unresolved. First, calculations of residual vertical velocities at 70 hPa indicate off-equator maxima in shallow branch upwelling in the summertime hemisphere (Randel et al. 2008; Seviour et al. 2012). Even though wave drag can force circulations nonlinearly and nonlocally, wave drag is at its annual maximum in the winter hemisphere, which is thus at odds with the observation of tropical upwelling maximizing in the summertime hemisphere (Holton et al. 1995; PE99). In fact, all of the experiments performed in PE99 showed that as long as wave drag maximizes in the winter hemisphere, upwelling maximizes in the winter hemisphere. Only when thermal forcing was included, did PE99 observe that upwelling maximizes in the summer hemisphere. Furthermore, at low latitudes, a weak flow-dependent force (such as momentum diffusivity or linear damping) can be of leading-order importance in determining the steady circulation; as PE99 showed, these weak forces, which can arise from thermal forcing, undermine the underlying hypothesis of downward control, namely, that the force can be specified independently of the applied heating. All of this together implies that thermal forcing may be important in determining tropical stratospheric upwelling.
In the tropical stratosphere, the observed upwelling strength is strongly correlated with temperature (Randel et al. 2006; Kerr-Munslow and Norton 2006), since a cold anomaly that slowly varies in time must be maintained by adiabatic cooling against the effect of radiative heating. Therefore, via downward-control arguments, wave dissipation has been historically linked with tropopause temperature. For instance, an annual cycle in subtropical wave dissipation of equatorward-propagating extratropical waves has been suggested as responsible for the annual cycle in tropical tropopause temperature (which is much larger in amplitude than that of the tropical troposphere) (Yulaeva et al. 1994; Holton et al. 1995; Randel et al. 2002; Taguchi 2009; Garny et al. 2011; Kim et al. 2016). Other studies have also attempted to understand how waves originating in the tropics can explain various aspects of the tropopause region, including the annual cycle in temperature (Boehm and Lee 2003; Norton 2006; Randel et al. 2008; Ryu and Lee 2010; Ortland and Alexander 2014; Jucker and Gerber 2017). In this view, the strength of zonally symmetric upwelling in the lower stratosphere is the primary control on zonally symmetric temperature near the tropopause.
In contrast, many observational studies have found that, on a variety of space and time scales, strong cold anomalies occur above regions of deep convection—in essence, local and regional tropopause cooling is associated with local and regional tropospheric (Johnson and Kriete 1982; Gettelman et al. 2002; Dima and Wallace 2007; Holloway and Neelin 2007; Kim and Son 2012; Grise and Thompson 2013; Virts and Wallace 2014; Kim et al. 2018). There also seems to be some spatial correlation between tropospheric warming and stratospheric cooling trends on global warming time scales (see Fig. 1 of Fu et al. 2006). In general, the cold anomalies in the lower stratosphere have been interpreted to be caused by convection itself, or forced from the “bottom up.” Since convection warms the troposphere, there is strong observational evidence of an anticorrelation between tropospheric temperature and lower-stratospheric temperature.
This oft-observed link between tropopause cooling and tropospheric warming has a number of theoretical explanations. First, there is the hypothesis that convective overshooting (of the level of neutral buoyancy) can cool the tropopause (Danielsen 1982; Sherwood 2000; Kuang and Bretherton 2004), emphasizing the role of convection in determining the mean temperature of the tropopause. Holloway and Neelin (2007) offer an alternative hypothesis, and propose that a convective cold top forms via hydrostatic adjustment above tropospheric convective heating. This theory requires that the associated pressure perturbation vanishes at some arbitrary level. Note that there is no dependence of the temperature anomaly on the horizontal scale in this theory. Separately, some authors have also argued that deep convection can excite a large-scale Kelvin wave response, which also has a vertically tilted signature of tropopause cooling (Kiladis et al. 2001; Randel et al. 2003). Finally, the anticorrelation in tropospheric temperature and lower-stratospheric temperature has also been explained through the vertical propagation of Rossby waves (Dima and Wallace 2007; Grise and Thompson 2013), though this theory is focused on subtropical regions, rather than on the deep tropics. Regardless, most of these studies focus on daily to monthly time scales, and do not consider how the observed lower-stratospheric cold anomalies might affect lower-stratospheric upwelling more broadly. This is not trivial—while changes to the tropopause temperature that project onto the zonal mean could theoretically induce changes in shallow branch upwelling, a corresponding, self-consistent change in the momentum budget must also occur to balance the changes in the meridional circulation (Ming et al. 2016a).
If one persists with the assumption that the same mechanism responsible for local- and regional-scale anticorrelations between tropospheric warming and tropopause cooling can manifest itself at the zonally symmetric scale (which is not a given), then it is perhaps unsurprising that there also exists a tight coupling between tropospheric warming and the BDC shallow branch mass flux, at least when using SST to characterize the tropical troposphere. In general circulation models (GCMs) and reanalyses, there are strong correlations between tropical-mean SST and the BDC shallow branch mass flux, across a wide variety of time scales (Lin et al. 2015; Orbe et al. 2020; Abalos et al. 2021). Fluctuations in tropical stratospheric upwelling have also been tied to El Niño–Southern Oscillation (ENSO), one of the dominant sources of interannual tropical SST variability (Randel et al. 2009). In fact, interannual variations in tropical mean SST explain 40%–50% of the interannual variability of the 70-hPa vertical mass flux (Lin et al. 2015; Abalos et al. 2021). In addition, nearly 70% of the CMIP6 model spread in the long-term trend of shallow branch mass flux is explained by the spread in tropical warming (Abalos et al. 2021).
The tight coupling between tropical SST and BDC shallow branch upwelling on interannual to climate change time scales has been explained through changes to the wave drag, in light of the downward-control paradigm: surface warming leads to upper-tropospheric warming and modification of the subtropical jets, which alters the upward propagation and dissipation of midlatitude waves in the subtropics (Garcia and Randel 2008; Calvo et al. 2010; Shepherd and McLandress 2011; Lin et al. 2015). While these theories can explain how SST and shallow branch mass flux are correlated, they were not constructed to also explain the oft-observed local-scale anticorrelation between SST and tropopause temperature.
In this study, we put forth an alternative explanation for the anticorrelation between tropospheric and lower-stratospheric temperature. To start, consider the simplified atmospheric state shown in Fig. 1, which has a troposphere in radiative convective equilibrium, with an overlying stratosphere at rest. Here, we assume that the tropopause acts as an infinitesimally small boundary between the troposphere and stratosphere, which neglects the existence of the tropical tropopause layer (TTL) (Fueglistaler et al. 2009), as further discussed in the conclusions. The TTL’s role in the broader climate should not be neglected, especially since the TTL temperature has been linked with the concentration of water vapor in the stratosphere (Jensen and Pfister 2004; Fueglistaler et al. 2005; Randel et al. 2006; Randel and Park 2019).
Schematic of a troposphere in radiative–convective equilibrium, with an overlying stratosphere that is at rest. The troposphere is forced with a steady warm SST anomaly in the ocean. The troposphere warms (indicated by color shading) following a moist adiabat, the surface pressure falls, and the geopotential rises at the tropopause. How does the stratosphere respond to an imposed tropopause geopotential anomaly?
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
This approximation notwithstanding, suppose we impose a steady patch of positive SST anomaly in the ocean. The increased surface enthalpy flux warms the troposphere, following a moist adiabat. The surface pressure falls, and the geopotential at the tropopause rises. Since there cannot be a pressure discontinuity across the tropopause, the pressure must also rise in the lower stratosphere. How far up does it extend, and what is the steady response in the stratosphere?
Section 2 tries to answer this conceptual question by introducing the concept of SST forcing of the tropopause and building a zonally asymmetric framework to understand the processes that control the upward extent of tropopause anomalies. It is shown that there is a quasi-steady, quasi-balanced response of the stratosphere to tropospheric thermal forcing. Section 3 extends the analysis to the zonally symmetric case, using a steady, coupled troposphere–stratosphere system to show how zonally symmetric SST anomalies (or zonally symmetric tropospheric heating) can influence tropical upwelling in the lower stratosphere. Section 4 uses reanalysis data to argue for the real-world presence of the processes described in the proposed theory. Section 5 concludes the study with a summary and discussion.
2. Stratospheric response to a tropopause anomaly
In this section, we introduce a simple conceptual model that will 1) illuminate how SST forcing can induce a tropopause geopotential anomaly and 2) understand what parameters modulate the upward extent of the tropopause anomaly into the stratosphere.
Since the tropopause is colder than the mean troposphere temperature,
The effect of radiative damping on stratospheric circulations has been thoroughly explored in a number of early theoretical studies (Garcia 1987; Haynes et al. 1991; Haynes and Ward 1993). In particular, the seminal work of Haynes et al. (1991) showed that in zonally symmetric, radiatively damped, time-dependent systems whereby a steady mechanical forcing is instantaneously applied, there is an adjustment to a barotropic state (in u) above the level of forcing. Our setup is similar to the model outlined in section 3 of Haynes et al. (1991), except here the steady forcing is restricted to the tropopause geopotential—the forcing is neither wave driven nor thermal in origin.
The geopotential associated with a boundary PV anomaly of q = −1 (ϕb) (red), a constant PV anomaly of q = −1 in the interior (ϕq) (blue), and the sum of the two (ϕ = ϕq + ϕb) (yellow). The corresponding total PV is shown in purple. Here we assume km = 2, and ztop = 1 + 2π.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
A simple physical picture is painted with this conceptual model that can provide a rather straightforward answer to the schematic shown in Fig. 1. If the troposphere is forced with a steady positive SST anomaly, a positive geopotential anomaly forms at the tropopause. A positive tropopause geopotential anomaly is initially accompanied with a cold anomaly in the stratosphere, which is associated with radiative heating and rising motion. If this process is allowed to proceed toward a steady state back to radiative equilibrium, the geopotential and PV must eventually become constant with height (i.e., barotropic), as implied by Eq. (18), and the temperature anomaly in the stratosphere disappears. In this way, the troposphere can force the stratosphere, at least on the steady time scales considered here. This also shows that the geopotential does not have to go to zero at the upper boundary. The only requirement is that the energy density goes to zero. Thus, the assumption of the geopotential going to zero at the upper boundary in Holloway and Neelin (2007) seems arbitrary.
How long does it take to reach the barotropic state? Haynes et al. (1991) showed that in the zonally symmetric case, the adjustment toward a barotropic state above the level of forcing occurs with an upward propagation speed of
This long relaxation time makes it unlikely that the barotropic state is ever reached in the real stratosphere, since unsteady processes can disrupt the simple state assumed in this model. For instance, tropospheric thermal forcing does not remain steady on the order of years, as there is a seasonal cycle in heating. Furthermore, since the β effect is not included in this simple framework, we ignore the possibility of the excitation of large-scale waves (and their corresponding effects) as a part of the response to tropospheric thermal forcing.
Indeed, the vertical propagation of planetary waves into the stratosphere has been cited as one potential reason for the observed anticorrelation between tropospheric and lower-stratospheric temperature (Dima and Wallace 2007; Grise and Thompson 2013). Here, we offer an alternative perspective, by returning to the schematic shown in Fig. 1. In the case that there is constant Coriolis force everywhere, there would be no stationary Rossby wave associated with tropospheric heating. But, at least according to the proposed theory, a cold anomaly (that is not related to convective overshooting) would still form above the tropopause. Of course, in the real world, β allows for a steady wave response (Gill 1980) that could disrupt the simple atmospheric state we have proposed. In this case, the quasi-balanced response of the stratosphere could occur in tandem with the vertical propagation of planetary waves (which are excited as part of the tropospheric thermal forcing), though a thorough investigation of this is left to future work.
In light of this, the intermediate states between the fast stratospheric response (ϕb in Fig. 2) in which the anomaly decays exponentially with height, and the barotropic steady-state response in which the boundary anomaly is communicated throughout the depth of the stratosphere (ϕ in Fig. 2), could be important. For practical purposes, the geopotential anomaly is not as important as the associated radiative heating, which is potentially important for tracer transport into the stratosphere. Figure 3 shows the nondimensional diabatic heating profiles with height after 30 days of integration, for a stratosphere subject to an imposed tropopause geopotential anomaly that is associated with a unitary nondimensional anticyclonic PV, under varying magnitudes of stratospheric radiative relaxation rates. The diabatic heating profiles are normalized by the radiative relaxation rate. For comparison purposes, we show the temperature anomaly associated with the (time-independent) zero perturbation PV geopotential solution (i.e., an infinite radiative-relaxation time scale), even though there is no associated diabatic heating, by definition. Figure 3 shows that after 30 days, there is nontrivial lifting (in height) of the diabatic heating anomaly over time. The stronger the strength of radiative relaxation, the faster the diabatic heating anomaly is communicated into the stratosphere.
(left) The diabatic heating profile (Q/αr) with height in the stratosphere after 30 days of integration, subject to a steady tropopause boundary forcing with a horizontal scale of around 28 000 km, and 5 (blue), 20 (red), and 40 days (yellow). The vertical derivative of the geopotential for the zero-PV stratospheric response to a tropopause forcing (infinite radiative relaxation time scale) is shown in black. (center),(right) As in the left panel, but for a horizontal scale of around 9500 and 4500 km, respectively. We assume a latitude of 10°, a scale height of 8 km, and a tropopause height of 16 km to convert to dimensional height. Note the vertical scale varies in each subplot for detail.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
These calculations show that tropospheric heating imposes a positive tropopause geopotential anomaly, which elicits a quasi-balanced response in the stratosphere. The fast stratospheric response is simply an anomaly that decays in the vertical according to the Rossby penetration depth. On slower time scales, radiative relaxation induces an upward migration of the anomaly. The radiative relaxation rate, the horizontal scale of the anomaly, and the Coriolis parameter all determine the upward migration rate, as shown in Haynes et al. (1991). Thus, the ensuing, time-dependent temperature response in the stratosphere is also tied to these parameters. In the next section, we will elaborate on the ideas put forth in this conceptual model in a zonally symmetric framework, and analyze, in detail, the sensitivity of the stratospheric response to tropospheric forcing, with regards to these parameters.
3. Troposphere–stratosphere response to SST
In the previous section, we used a simple QGPV framework to understand how an SST anomaly can impose a tropopause geopotential anomaly and therefore elicit a quasi-balanced response in the stratosphere. However, we used the tropopause as a lower boundary condition for the stratosphere when in reality, the tropopause and stratosphere are coupled. In this section, we develop a simple, zonally symmetric, coupled troposphere–stratosphere model, and explore how radiation and wave drag can modulate the response of the stratosphere to SST forcing.
a. Model formulation
As formulated, the tropospheric system represents an atmosphere in which temperature anomalies in the vertical are restricted to follow the moist adiabat. The associated baroclinic mode, which is forced through surface enthalpy fluxes (
b. Stratospheric response to tropopause forcing
Figure 4 shows the stratospheric response to a tropopause geopotential anomaly, under varying values of ξ. Here, the numerical calculations confirm the mathematical analysis. Indeed, for ξ = 0.01 (i.e., when wave drag is very weak), radiation acts to create a nearly barotropic stratosphere, in which motion is confined to constant angular momentum surfaces. The vertical structure of the vertical velocity in this case is qualitatively similar to the thermally forced vertical mode calculated in PE99 (see their Fig. 11). When the time scale of wave drag is faster than radiation (ξ = 100), the vertical penetration of the tropopause geopotential anomaly is significantly muted. In fact, the vertical velocity anomalies only extend on the order of a few kilometers into the stratosphere. In this sense, the relaxational wave drag acts to both mute the vertical scale of the tropopause geopotential anomaly, and sustain a meridional overturning circulation.
(top) The zonally symmetric geopotential response to an imposed tropopause geopotential anomaly, as shown in Eq. (38), for varying values of ξ. (bottom) As in the top row, but for the zonally symmetric vertical velocity response. The red line is the zero vertical velocity isoline. Tropopause height is 16 km, and stratospheric-scale height is 8 km.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
As elaborated on earlier, there is much existing theoretical work that shows the response of the stratosphere to an external forcing is dependent on the strength of wave drag, the strength of radiative relaxation, and the aspect ratio of the tropopause anomaly (Garcia 1987; Haynes et al. 1991; Ming et al. 2016b). This work is mathematically similar to and agrees with the aforementioned studies. Unlike the others, this work emphasizes the role of tropopause forcing on the stratosphere, and introduces the idea that there is a quasi-balanced response in the stratosphere to tropopause forcing, via tropospheric heating.
c. Tropospheric forcing of stratospheric upwelling
To set the nondimensional parameters of the model, we use Earthlike parameters of N2 = 6 × 10−4 s−2, H = 16 km, Hs,t = Hs,s = 8 km, β = 2.3 × 10−11 s−1 m−1, Ly = 1200 km (such that y = 1 represents approximately 10° of latitude), Cd = 10−3, and |V| = 3 m s−1. Furthermore, we choose Tb = 303 K, a surface pressure of 1000 hPa, and a tropopause pressure of 100 hPa. The vertical temperature profile in the troposphere follows a pseudoadiabatic lapse rate (neglecting changes to heat capacity; see Eq. 4.7.5 of Emanuel 1994), such that
Since αrad and
For now, we restrict the analysis to “Earthlike” parameters, with
(left) The zonally symmetric response to an SST (
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
What is the sensitivity of the stratospheric circulation to
(a) Vertical profiles of nondimensional geopotential and (b) vertical velocity, at y = 1.5, for varying values of radiative relaxation, at a fixed Rayleigh damping (wave drag) of (25 days)−1. Dashed lines show the geopotential and vertical velocity associated with a pure baroclinic mode (normalized so that the peak vertical velocity is 0.02). (c),(d) As in (a) and (b), respectively, but for varying values of stratospheric Rayleigh damping, at a fixed radiative relaxation rate of (25 days)−1. Tropopause is defined at 16 km, and tropospheric Rayleigh damping is fixed at (25 days)−1.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
The stratospheric response to a steady tropopause geopotential anomaly also shows a strong dependence to
The vertical shape of the geopotential profiles above the tropopause also allows for an estimate of the magnitude of the tropopause temperature cold anomaly as a function of tropospheric heating. Figure 7, left, shows the temperature anomaly right above the tropopause, per degree of warming in the boundary layer, as a function of the radiative damping and Rayleigh damping time scales. In general, the longer the radiative damping time scales, the larger the temperature anomaly (as pointed out by Randel et al. 2002). In addition, there is also a strong dependence of the tropopause temperature anomaly on the Rayleigh damping time scale: the faster the damping, the larger the magnitude of the temperature anomaly. It is clear that both the magnitudes of the Rayleigh damping (wave drag) and radiative damping play significant roles in modulating the temperature anomaly above the tropopause.
(left) Temperature anomaly right above the tropopause, per degree of warming in the boundary layer, as a function of the radiative relaxation and Rayleigh damping (wave drag) time scales. Rayleigh damping time scale is fixed in the troposphere and varied in the stratosphere. Both the abscissa and ordinate axes are in log coordinates. (right) Temperature anomaly right above the tropopause, per degree of warming in the boundary layer, as a function of the meridional length scale, Ly (km), for fixed Rayleigh damping and radiative relaxation. Ordinate axis is logarithmic.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
Interestingly, for “Earthlike” estimates of the time scale of Rayleigh damping and radiative relaxation [O(10 days)−1], the temperature anomalies just above the tropopause are around 2–3 times the magnitude of the boundary layer anomalies, slightly larger than what is observed in convecting regions of the tropical atmosphere (see Fig. 5a in Holloway and Neelin 2007). This theory thus provides a scaling argument for the degree of tropopause cooling that is expected per degree of boundary layer warming. Note that the derivative of the geopotential is discontinuous across the tropopause in this model, since we assume an instantaneous transition between quasi-equilibrium thermodynamics in the troposphere, and dry, passive dynamics in the stratosphere.
These theoretical results provide a potential explanation for the observed correlation between tropical-averaged SST anomalies and tropical stratospheric upwelling (Lin et al. 2015), as well as the anticorrelation between SST and tropopause temperature (Holloway and Neelin 2007). First, an SST anomaly is communicated throughout the depth of the troposphere through moist convection. Indeed, observations have found strong positive correlations between the tropopause geopotential anomaly and the boundary layer temperature anomaly (Holloway and Neelin 2007). The tropopause geopotential anomaly is initially associated with cold temperature anomalies just above the tropopause. The strength of radiative relaxation then determines the time scale at which the geopotential anomaly rises in the stratosphere through diabatic heating. In the zonally symmetric case, the presence of wave drag, through conservation of angular momentum, disrupts this process and induces a meridional overturning circulation that mediates the vertical scale at which the geopotential anomaly can rise in the stratosphere.
Our work shows that, at least in the zonally symmetric case, the ratio between the strength of radiative relaxation and that of Rayleigh damping are significant factors in determining the response of the stratosphere to an SST anomaly. However, there are a number of other quantities unveiled through the nondimensionalization that are also important. Surface friction, for instance, factors into γ. In general, increasing the magnitude of F does little to change the behavior of the stratospheric response to tropospheric forcing when ξ is large, since F only enters in γ and ξγ is what matters for the tropopause boundary condition. The tropospheric and stratospheric stratification, as well as the shape and length scale of the SST (or tropopause) perturbation (Ly), also factor into the nondimensional parameters that control the vertical decay scale of tropopause geopotential anomalies. The horizontal scale of the SST anomaly can also be quite important, due to the dependence of S on
4. Tropopause forcing in reanalysis data
In this section, we evaluate the proposed theory using the ERA5 (Hersbach et al. 2019b,a). We use monthly fields of SST, geopotential, and temperature, over the years 1979–2022. The quasi-biennial oscillation (QBO) is regressed out of the geopotential and temperature fields, by using the 50-hPa zonal wind averaged over the tropics. In particular, we will analyze correlations between metrics of tropospheric warming and stratospheric cooling, on the global scale and the local scale.
To begin, we regress the anomalous tropical-averaged geopotential, at different vertical levels, onto the tropical-averaged SST anomaly. Anomalies are generated by subtracting the linear trend in each field, as well as the seasonal cycle. Figure 8, solid lines, shows the coefficients of the linear regressions of geopotential and temperature onto SST. We first observe an approximate moist-adiabatic structure in the tropical tropospheric geopotential, consistent with quasi equilibrium and the findings of previous studies (Holloway and Neelin 2007). We also see a large, significant correlation (r ≈ 0.75) between tropical-averaged SST and the corresponding 100-hPa geopotential. The magnitude of the geopotential anomaly maximizes at 100 hPa, which is interpreted as an approximate tropopause level, since below this level there is warming, and above this level there is cooling (this is not exact, since the cold-point tropopause could occur above this level). Note the similarity to the geopotential profile shown in Fig. 6, which also maximizes around the climatological tropopause. This is indicative of a tropopause geopotential anomaly that is induced by an SST anomaly. The coefficient magnitudes and correlations decay with increasing height in the stratosphere, but are still statistically significant and nonnegligible even at 20 hPa. Note, for a pure baroclinic mode anomaly, the surface geopotential would be anticorrelated with the upper-troposphere anomaly (and the SST). Thus, when the surface geopotential is positively correlated with the upper-tropospheric anomaly, there is a significant barotropic component to the geopotential profile. We indeed observe that the tropical-averaged surface geopotential is positively correlated with both SST and the upper-tropospheric geopotential, highlighting the role of the barotropic mode and the troposphere’s communication with the stratosphere.
(left) Linear coefficient of geopotential at varying levels, regressed onto regionally averaged SST anomaly. Above 500 hPa, significant correlations at the 1% level (two sided) are denoted by upside-down triangles. (center) As in the left, but for temperature. (right) Vertical dependence of the correlation coefficients for geopotential (blue) and temperature (red). The regions are the entire tropics (20°S–20°N;solid), the Indo-Pacific region (40°–120°E;dashed), the east Pacific region (180°–260°E; dot–dashed), and the Atlantic region (80°–0°E; dotted). Vertical level is scaled as the logarithm of pressure.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
The temperature structure of the tropical troposphere is also approximately moist adiabatic, as also shown in Holloway and Neelin (2007). Figure 8 also shows that the tropics-averaged 70-hPa temperature is modestly but significantly anticorrelated (r ≈ −0.34) with surface temperature. We also observe temperature anomalies at 70 hPa (lower stratosphere) to be approximately 2 times larger in magnitude than that of the surface, which is in agreement with the estimates shown in Fig. 7. This is not exactly equivalent with the quantity derived in the left portion of Fig. 7, as the regridded, pressure-interpolated output for ERA5 does not have many vertical levels near the tropopause, such that sharp reversals in the temperature response might be smoothed out. While data on the underlying model levels are available at a much higher vertical resolution, the ensuing analysis is very data intensive and left for future work.
The same relationships are also observed on regional scales (the Indo-Pacific, east Pacific, and the Atlantic), as shown in Fig. 8. The geopotential anomalies maximize at 100 hPa in the Indo-Pacific, at 125 hPa in the Atlantic, and at 150 hPa in the east Pacific. Thus, the level at which the geopotential anomaly maximizes is influenced by the mean SST of the region (the east Pacific has the coldest climatological SSTs, while the Indo-Pacific has the warmest). In addition, the cold anomaly associated with SST warming maximizes above the level of maximum geopotential. The regional-scale geopotential anomalies persist upward to around 50 hPa, though the correlations drop significantly in magnitude, and the statistical significance ceases around 50 hPa. This means that regional- and local-scale variations in the lower-stratospheric geopotential (50 and 70 hPa) are strongly influenced by the tropopause geopotential in the same region. In general, the temperature anticorrelations are strongest in the east Pacific region—this may because there are large SST perturbations in this region as a consequence of El Niño–Southern Oscillation variability, increasing the signal of the relationship.
Of course, this analysis is not definitive proof that there is a quasi-balanced response of the stratosphere to tropopause forcing. After all, if stratospheric temperature is modulated by tropical heating through changes to wave drag (Garcia and Randel 2008; Calvo et al. 2010; Lin et al. 2015), then one would also expect the geopotential to decay with height in the stratosphere, as is shown in Fig. 8. Perhaps what would serve as stronger evidence for the processes described in this study is if the spatial signature of tropospheric warming is retained in that of stratospheric cooling. If true, this implies that lower-stratospheric temperature is also influenced by “bottom-up” processes (Garfinkel et al. 2013; Fu 2013)—not just “top-down” processes.
In the tropics, the surface temperature need not always be connected to tropospheric warming, especially if the boundary layer moist static energy is lower than the saturation moist static energy of the free troposphere. This is possible since temperature gradients in the tropical atmosphere are weak, owing to the smallness of the Coriolis force, such that convecting regions more effectively modulate the free-tropospheric moist static energy (Sobel and Bretherton 2000). Thus, we use 500-hPa temperature as a proxy for local tropospheric warming.
Figure 9 shows a map of the DJF-averaged 500-hPa climatological temperature, a proxy for tropospheric heating, and the climatological temperature at 100 and 70 hPa in the lower stratosphere [these maps are well known and have been shown before, for instance, in Dima and Wallace (2007), Fueglistaler et al. (2009), and Grise and Thompson (2013), but with different interpretations]. Here, we observe the warmest 500-hPa temperatures are in regions typically associated with active convection (the west Pacific warm pool, equatorial South America, and equatorial Africa). Note that tropospheric heating is a by-product of convection. Furthermore, these same regions are where the coldest 100- and 70-hPa temperatures are also observed. Importantly, the coldest temperatures in the lower stratosphere occur right on or close to the equator, where the Coriolis force is small. At 70 hPa, the signature of the equatorial 100-hPa cold anomalies disappears. This may be a manifestation of the shallow vertical Rossby penetration depth of anomalies on the equator.
DJF-averaged climatological temperature at (top) 70, (middle) 100, and (bottom) 500 hPa. Note the strong anticorrelation in troposphere and lower-stratospheric temperature.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
To further emphasize spatial variability, we compute monthly anomalies by subtracting the climatological monthly zonal mean from the climatological monthly mean, and then average these across December–February (DJF). Figure 10 shows maps of the DJF-averaged temperature anomalies at 500, 100, and 70 hPa. Note the difference in the color scale at 100 hPa. It is evident that 500-hPa temperature is an excellent predictor of both the 100- and 70-hPa temperature anomaly, though the strongest patterns are observed in the subtropical regions and associated with Rossby wave–like features. Still, spatial variability in the tropospheric temperature anomaly is remarkably retained in the spatial variability of the stratospheric temperature. Furthermore, the lower-stratospheric temperature anomalies can be rather large (upward to around 4° in magnitude at 100 and 70 hPa), though the total area encompassed by these large anomalies is small. There is also some qualitative evidence from the maps in Fig. 10 that suggests that the magnitude of the lower-stratospheric temperature anomalies is dependent on the horizontal scale of the tropospheric anomaly. For instance, from 60°W to 30°E in the Northern Hemisphere, there is a large-scale tropospheric cold anomaly of peak magnitude around 2°. The associated temperature anomaly at 100 hPa is around 4°. There is also a large-scale tropospheric warm anomaly of peak magnitude around 3° in the Asian region (90°E–180°), with 100-hPa temperature anomalies of around −6°. In contrast, smaller-scale tropospheric anomalies (10°–30°S, 150°–90°W and 45°–15°W, 10°–25°S) with comparatively weaker peak temperature anomalies are associated with 100-hPa temperature anomalies that are of similar magnitude to the 100-hPa temperature anomalies of the stronger, large-scale anomalies. This is in agreement with the proposed theory. In addition, at 70 hPa, the most prominent temperature anomalies are those associated with the large-scale tropospheric anomalies (i.e., over the northeast African and Asian regions). This is also in agreement with the theory, in that the vertical depth of the tropopause anomalies increases with the horizontal scale of the tropospheric anomaly. Of course, the analysis here is mostly qualitative, and more substantial analysis is required to further quantify the scale dependence of the lower-stratospheric temperature anomalies, which will be pursued in future work.
DJF-averaged temperature anomaly at (top) 70, (middle) 100, and (bottom) 500 hPa. Note the strong anticorrelation in troposphere and lower-stratospheric temperature. Anomalies are calculated by subtracting the climatological monthly zonal mean, and averaging across the entire year. The color scale at 100 hPa is different than that at 70 and 500 hPa.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
The remarkable correlation between tropospheric heating and stratospheric cooling can be further quantified by regressions of 500-hPa temperature against lower-stratospheric temperature, among all grid points shown in Fig. 10. Figure 11, top row, shows 2D density histograms between the 500-hPa climatological temperature and the 100-, 70-, and 50-hPa climatological temperature, as well as the linear regressions. We have subsetted the latitudinal region in this analysis to 15°S–15°N, in order to focus on the tropical regions. Per degree of warming at 500 hPa, the cooling response is around 2.0° at 100 hPa (r = −0.84), 0.72° at 70 hPa (r = −0.64), and 0.21° at 50 hPa (r = −0.43). The correlations are all significant, and generally decrease in strength as one moves up further in the stratosphere. The linear regressions of 500-hPa anomalous temperature against lower-stratospheric anomalous temperature tell a similar story, as shown in Fig. 11, bottom row. Per degree of anomalous 500-hPa temperature, there is a cooling response of around 2.1° at 100 hPa (r = −0.86), 1.03° at 70 hPa (r = −0.90), and 0.14° at 50 hPa (r = −0.61). Note that while this paper focuses on the tropics, the proposed mechanism need not only apply to the tropics (though Rossby wave excitation can be important outside of the tropics). In fact, the correlations are even stronger if one extends the region of analysis to 30°S–30°N.
(top) Gridpoint 2D histograms between the 500-hPa climatological temperature and the (left) 100-, (center) 70-, and (right) 50-hPa climatological temperature, during DJF and from 15°S to 15°N. (bottom) As in the top row, but for anomalous temperatures at each pressure level. Color scale is logarithmic, and indicates the bin count. Linear regressions are plotted as the dashed blue lines, with correlation coefficients shown on the lower left of each panel.
Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0081.1
While the monthly anomalies shown in Fig. 10 are averaged across DJF, there is significant seasonal variability in the pattern of 500-hPa tropospheric temperature (not shown). The analysis can be repeated by separating into various seasons, and we find that the local-scale anticorrelation are generally strongest during boreal winter, and weakest during boreal summer (not shown). Still, the results and interpretation remained unchanged: 500-hPa temperature is strongly anticorrelated with lower-stratospheric temperature. It is important to note that these correlations do not suggest that there are correlations on significantly smaller horizontal scales; as suggested by Fig. 10, the correlations merely reflect the large-scale structure of the temperature anomalies.
Therefore, the observational data suggest that there might be a quasi-balanced response of the stratosphere to tropospheric thermal forcing in the real world. However, there is reason to remain cautious. As detailed in section 2, separating the effect of the vertical propagation of planetary waves from that of the quasi-balanced response of the stratosphere is nearly impossible in observational data. While we restricted our analysis to 15°S–15°N, further insight into the relative contribution of each proposed mechanism to the anticorrelation between tropospheric and lower-stratospheric temperature is left for future work.
5. Summary and discussion
In this work, we present theoretical evidence for how tropopause geopotential anomalies, generated through tropospheric thermal forcing, can modulate upwelling in the stratosphere. Using a conceptual model based on the linearized QGPV equations, we show that tropospheric thermal forcing can induce a tropopause geopotential anomaly, which subsequently elicits a quasi-balanced response in the stratosphere. The tropopause anomalies initially have vertically shallow structures scaled by the Rossby penetration depth (i.e., the fast adjustment of the stratosphere). Afterward, radiative relaxation in the stratosphere acts to increase the vertical penetration of these anomalies. In the steady-state limit, where radiative equilibrium is again satisfied, the stratospheric PV becomes barotropic, though it takes on the order of years to be achieved. The solutions are akin to those of Haynes et al. (1991), who found that the stratosphere becomes barotropic above the level of forcing (in this case, the tropopause). This theory provides another potential explanation for why cold stratospheric anomalies form above areas with local tropospheric warming. Despite the focus on the tropics in this study, this proposed mechanism need not be confined to the tropics. However, the excitation of planetary waves as a response to tropospheric heating, which was ignored for simplicity in this study, ought to be taken into account. This will be the subject of future research.
We then formulate a zonally symmetric troposphere–stratosphere linear β-plane model, which couples a convecting troposphere to a dry and passive stratosphere. We show that zonally symmetric tropospheric thermal forcing (via SST anomalies) can directly force upwelling in the lower stratosphere, provided the wave response is modeled purely as a response to the forced circulation. The stratospheric response to tropospheric forcing is controlled by two nondimensional parameters: 1) ξ, a dynamical aspect ratio (Garcia 1987; PE99; Haynes 2005; Ming et al. 2016b), and 2) γ, a ratio between the stratospheric drag and tropospheric drag. The main role of the tropospheric drag is to excite the tropospheric barotropic mode, which couples the troposphere with the stratosphere. In the limit that the radiative relaxation is much stronger than wave drag, the stratospheric response to a tropopause forcing asymptotically becomes barotropic, while in the opposite limit, the vertical length scale of the tropopause forcing becomes extremely small. We find that the stratospheric response to zonally symmetric tropospheric forcing is largely dependent on the radiative relaxation rate, the Rayleigh damping time scale of wave drag, and the horizontal scale. Our analyses show that the tropopause temperature anomaly is also modulated by all of these quantities.
We also use reanalysis data to show that tropical and regionally averaged lower-stratospheric temperatures are modestly and negatively correlated with SSTs in the same areas. In general, the temperature anomalies per degree of warming in the boundary layer are approximately equivalent to the corresponding theoretical predictions, at least when using “Earthlike” estimates of the time scale of wave drag and radiative relaxation. Furthermore, we show that the spatial variability in lower-stratospheric temperature anomalies is strongly correlated with the spatial variability in 500-hPa tropospheric temperatures. Significant correlations are seen upward to 50 hPa, which suggests that there is a quasi-balanced response of the stratospheric to tropospheric forcing. This provides a scale-dependent theory for the oft-observed anticorrelation between tropospheric warming and stratospheric cooling (Johnson and Kriete 1982; Gettelman et al. 2002; Holloway and Neelin 2007; Kim and Son 2012; Virts and Wallace 2014; Kim et al. 2018).
The widely accepted theory of tropical stratospheric upwelling is that it is mechanically driven by subtropical wave drag (Haynes and McIntyre 1987; PE99). There is ample evidence from numerical modeling suggesting that wave dissipation is a dominant mechanism that modulates mean and interannual upwelling in both the lower stratosphere and TTL (Boehm and Lee 2003; Norton 2006; Calvo et al. 2010; Ryu and Lee 2010; Gerber 2012; Ortland and Alexander 2014; Kim et al. 2016; Jucker and Gerber 2017, among many others). Of course, it is theoretically impossible to have flow across angular momentum contours without some momentum source. We emphasize that in no way does this work attempt to disprove the role subtropical wave drag has in modulating tropical stratospheric upwelling. In this model, even though wave drag acts as a Rayleigh damping, as in the linear system described in PE99, it is an important modulator of the upwelling response.
As shown in this study, the vertical penetration of the geopotential anomaly (and the rate at which the stratospheric circulation crosses angular momentum surfaces) is strongly a function of the wave drag. If the wave drag is a function of the zonal mean state, which could vary in time in part due to wave forcing (Cohen et al. 2013; Ming et al. 2016b), then the vertical penetration of the tropopause anomaly (and thus, its subsequent effect on upwelling) would also vary in time. In this view, stratospheric wave drag is, as countless studies have shown, a significant modulator of tropical upwelling. However, wave drag alone may not suffice to explain certain features of the behavior of the lower stratosphere, the foremost of which is the inverse correlation between SST and lower-stratospheric temperature anomalies, in both the zonal and meridional directions.
Our work, like PE99, investigates how tropospheric thermal forcing can modulate stratospheric upwelling. In addition to mechanical and thermal forcing, this suggests a third way in which the stratosphere can be forced—through the tropopause via tropospheric thermal forcing. In fact, the theoretical analysis shown in PE99 finds that in the tropics, “the existence of a thermally driven circulation and the breakdown of downward control go together” (if one accepts that what they define as viscosity is representative of large-scale drag). However, their calculation of the linear response to tropospheric thermal forcing exhibited large and unrealistic vertical penetration of the tropospheric circulation into the stratosphere. This work shows that this is likely a result of their assumptions of the strength of radiative relaxation [αrad = (10 days−1)] and viscosity [
In general, it is difficult to infer causality from diagnostic relations. For example, in the transformed Eulerian mean equations, it is not clear how much of the wave drag is an external forcing, as opposed to a response to a circulation that has a different forcing. Of course, variations in wave drag that are independent of those of the circulation support the idea that waves can force the circulation. This aspect of the stratosphere has been well studied. But what if wave drag acted purely as a response to the circulation? (Note that these ideas are at opposite ends of the spectrum with regards to the extent waves drive the circulation.) Then, at least in our framework, the causality becomes very clear—SST forces the stratosphere by imposing a tropopause geopotential anomaly. Of course, one could take the wave drag term (−Dsus) and use it to diagnose the associated upwelling response. However, that does not imply that waves are the forcing mechanism of the circulation.
There are a few pieces of observational evidence that could be interpreted to be in favor of the proposed theory. As stated earlier, the spatial variability of lower-stratospheric temperature is strongly correlated with that of the troposphere, when considering both the climatological and anomalous temperatures. In contrast, wave drag, in its classical arguments, can only explain departures of temperature from the zonal mean (Andrews et al. 1987). This is by no means a small feat, since the annual cycle in tropical-averaged temperature near the tropopause is around 8 K, around a factor of 2 larger than the peak temperature anomalies shown in Fig. 10 (Chae and Sherwood 2007).
However, the quasi-balanced response of the stratosphere to tropopause forcing could serve as a potential explanation for a few outstanding issues. For instance, it can explain why there is peak tropical upwelling on the summer-side equator (Rosenlof 1995). It could also help to explain the observed connection between boundary layer temperature anomalies and lower-stratospheric temperature anomalies, as well as the high correlations between tropical SST and the upwelling strength of the shallow BDC branch, which is observed on all time scales (Lin et al. 2015; Abalos et al. 2021). Numerical modeling suggests that strengthening of the subtropical jets changes the upward propagation of waves (Garcia and Randel 2008; Calvo et al. 2010; Shepherd and McLandress 2011), ultimately strengthening the wave-driven stratospheric upwelling, although the exact specifics seem to vary from model to model (Calvo et al. 2010; Simpson et al. 2011). In the zonally symmetric coupled troposphere–stratosphere theory analyzed in this work, an equatorial SST anomaly is not only associated with strengthening of the subtropical jets (which no doubt could change the subtropical distribution of wave drag in the real world), but also a strengthening of the tropopause geopotential. As such, the theory proposed in this work does not have to be mutually exclusive with those based on wave drag.
Besides the inclusion of a relaxational wave drag (shown to be a poor assumption), our work stays silent on how the momentum budget must change in order to balance changes in the meridional circulation (Ming et al. 2016b). However, there would undoubtedly be a large-scale wave response to steady tropospheric heating (Gill 1980). Thus, disentangling the effects of heating from the ensuing wave response is quite complicated, as the two occur in concert. While other studies have analyzed the wave response to tropospheric heating (Ortland and Alexander 2014; Jucker and Gerber 2017) (as well as its subsequent effects on the stratospheric circulation), we have instead focused on the steady response to tropospheric heating. In general, however, when tropical tropospheric heating is used to generate a wave response, it is difficult to separate the tropopause forcing mechanism described in this study from wave driving. For instance, Jucker and Gerber (2017) used idealized GCM simulations to show that the inclusion of a tropical warm pool significantly changed the annual-mean temperature of the tropical tropopause (and more importantly, more so than midlatitude land–sea contrast and orographic forcing). However, the imposition of a warm pool will both intensify the tropopause anticyclone over the region, and trigger a large-scale wave response. According to the analysis shown in this study, the increased tropopause geopotential will act to cool the tropopause and induce more upwelling (as would increased wave drag from the large-scale wave response). Separately, Ortland and Alexander (2014) forced equatorial waves by prescribing time-varying latent heating anomalies in a primitive equation model, and found that stationary waves and weakly westward-propagating waves are most responsible for driving residual-mean upwelling in the TTL. Again, tropospheric heating will induce a tropopause geopotential anomaly, such that the steady tropospheric forcing is not separated from the wave response. Regardless, both of the modeling results in Ortland and Alexander (2014) and Jucker and Gerber (2017) show that at least in numerical models, the seasonal cycle in upwelling in the tropical tropopause layer cannot be explained by tropospheric thermal forcing.
It is only fair for these conclusions to be discussed alongside the assumptions posited in this model. In this model, we assume that there is an instantaneous transition between tropospheric, quasi-equilibrium dynamics, and passive, dry stratospheric dynamics. In reality, the presence of the TTL could dampen the upward influence of tropospheric forcing. The assumption of a moist adiabatic lapse rate all the way to the tropopause is one that is has mixed observational evidence, which suggests that the free-tropospheric temperature anomalies, per degree of warming in the boundary layer, approximately follow a moist adiabat up to around 200 hPa, after which temperature anomalies transition to being out of phase with lower-tropospheric temperature anomalies (see Fig. 8 and Holloway and Neelin 2007) (though some of this may be owing to time averaging with a vertically moving tropopause). While the proposed theory can predict the magnitude of the tropopause temperature anomalies with respect to boundary layer warming, it does not include a transition layer. The presence of a transition layer could, in theory, dampen the vertical penetration of thermal forcing in the troposphere. This will be the subject of future research.
Finally, we also assume a fixed tropopause height that interfaces the two regimes, as in PE99. This makes the analysis mathematically tractable. Indeed, one would expect tropospheric temperature to affect tropopause height (Held 1982; Lin et al. 2017). The relaxation of both of these assumptions will be the subject of future research, but requires a theory for how moist convection interacts with the transition layer. More research is necessary to understand the role of convection in modulating the behavior of the transition layer.
The analysis carried out in section 4 uses the ERA5 dataset, which is not truly observational data. This could be mitigated by the use of GPS radio occultation (RO) measurements, provided by the COSMIC mission (Anthes et al. 2008). The high vertical resolution of GPS RO measurements could be leveraged in future work, as done in Grise and Thompson (2013). Furthermore, while we focused on large-scale tropospheric anomalies in this work, there are also numerous mesoscale convective systems, usually with anticyclones at their tops, that might also be able to contribute to tracer transport into the stratosphere. Higher-resolution observational data, such as that provided by GPS RO measurements, could also be useful to evaluate this possibility.
Acknowledgments.
The author thanks Adam Sobel and Peter Hitchcock for comments and suggestions on earlier versions of this work. The authors also thank two anonymous reviewers and Peter Haynes for their helpful suggestions, which greatly improved the manuscript. In particular, the authors are grateful for Peter Haynes’s suggestions on the formulation of the coupled boundary condition. J. Lin gratefully acknowledges the support of the National Science Foundation through the NSF-AGS Postdoctoral Fellowship, under Award AGS-PRF-2201441.
Data availability statement.
The monthly-mean ERA5 data for sea surface temperature are available at https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysis-era5-single-levels-monthly-means via DOI: 10.24381/cds.f17050d7 (Hersbach et al. 2019b). The monthly averaged ERA5 data for temperature and geopotential are available at https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysis-era5-pressure-levels-monthly-means via DOI: 10.24381/cds.6860a573 (Hersbach et al. 2019a). All code to generate the data from the theoretical models are available at https://github.com/linjonathan/steady_coupled_trop_strat.
APPENDIX
Details on Solutions
a. Solutions to conceptual model in section 2
b. Numerical solver for coupled troposphere–stratosphere
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