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  • View in gallery

    Clear-sky longwave cooling rate and spectral cooling rate as computed from the LBLRTM v12.1, a line-by-line radiative transfer code (Clough et al. 2005), with input of typical tropical sounding profiles (McClatchey et al. 1972): (left) the total longwave cooling rate (K day−1) and (right) the longwave spectral cooling rate (K day−1 10 cm−1). Positive value indicates (spectrally) radiative cooling and negative value indicates (spectrally) radiative warming.

  • View in gallery

    The diffusive weighting functions (ddiff/dz) of the water vapor far-IR band (0–600 cm−1) based on mean temperature and humidity profiles over the inner tropics for the 20C period (1971–2000, blue line) and the 21C period (2071–2100, red line) of the GFDL CM2.1 simulation. The calculation was done using the moderate resolution atmospheric transmission (MODTRAN) version 5.3 (Berk et al. 2005), a narrowband radiative code with easy configuration for computing weighting functions.

  • View in gallery

    (a) The 30-yr mean of clear-sky net radiative cooling rate (longwave + shortwave) over the inner tropics for the 20C (blue line) and the 21C (red line) from the GFDL CM2.1 simulation; only 200 to 925 hPa shown here. These cooling rates were computed using the GFDL radiation code and archived part of simulation output. (b) The vertical velocity ωest estimated from the modeled cooling rates and lapse rates for the two periods using Eq. (1).

  • View in gallery

    (a) The mean profile of vertical velocity for the large-scale ascending (asterisks) and descending (open diamonds) branches of the inner tropics in the CM2.1 simulation for the 20C period (blue lines) and the 21C period (red lines). Refer to the context for the way of averaging. (b) The mean fractional area for the large-scale ascending and descending branches. Color coding and notation are as in (a).

  • View in gallery

    Differences in the vertical velocity of the descending branch between the 21C and 20C periods as calculated using the CM2.1 model (blue ×) and estimated from Eq. (1) (open green circle).

  • View in gallery

    Estimated change of vertical velocity from Eq. (1). The green line is the same as the one in Fig. 5: the change when both the clear-sky net radiative cooling rate and the lapse rate change are taken into account. The blue line is due to the clear-sky net radiative cooling rate change alone. The red line is due to the lapse rate change alone. The black dash line is the summation of the blue and red lines, which agrees well with the green line for all of the levels.

  • View in gallery

    Change of vertical velocity at three different pressure levels from the year 2000 based on Eq. (1) when both the clear-sky net radiative cooling rate and lapse rate are considered (green) or when only one factor is considered (blue for the clear-sky radiative cooling rate change and red for the lapse rate change).

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A Radiative–Convective Equilibrium Perspective of Weakening of the Tropical Walker Circulation in Response to Global Warming

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  • 1 Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, Ann Arbor, Michigan
  • | 2 Department of Atmospheric Sciences, Texas A&M University, College Station, Texas
  • | 3 Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, Ann Arbor, Michigan
  • | 4 Department of Physics, New Mexico Institute of Mining and Technology, Socorro, New Mexico
  • | 5 NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey
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Abstract

Both observational analysis and GCM simulations indicate that the tropical Walker circulation is becoming weaker and may continue to weaken as a consequence of climate change. Here, the authors use a conceptual radiative–convective equilibrium (RCE) framework to interpret the weakening of the Walker circulation as simulated by the GFDL coupled GCM. Based on the modeled lapse rate and clear-sky cooling rate profiles, the RCE framework can directly compute the change of vertical velocity in the descending branch of the Walker circulation, which agrees with the counterpart simulated by the GFDL model. The results show that the vertical structure of clear-sky radiative cooling rate QR will change in response to the increased water vapor as the globe warms. The authors explain why the change of QR is positive in the uppermost part of the troposphere (<300 hPa) and is negative for the rest of the troposphere. As a result, both the change of clear-sky cooling rate and the change of tropospheric lapse rate contribute to the weakening of circulation. The vertical velocity changes due to the two factors are comparable to each other from the top of the planetary boundary layer to 600 hPa. From 600 to 300 hPa lapse rate changes are the dominant cause of the weakening circulation. Above 300 hPa, the change due to QR is opposite to the change due to lapse rate, which forces a slight increase in vertical velocity that is seen in the model simulation.

Corresponding author address: Xianglei Huang, Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109-2143. E-mail: xianglei@umich.edu

Abstract

Both observational analysis and GCM simulations indicate that the tropical Walker circulation is becoming weaker and may continue to weaken as a consequence of climate change. Here, the authors use a conceptual radiative–convective equilibrium (RCE) framework to interpret the weakening of the Walker circulation as simulated by the GFDL coupled GCM. Based on the modeled lapse rate and clear-sky cooling rate profiles, the RCE framework can directly compute the change of vertical velocity in the descending branch of the Walker circulation, which agrees with the counterpart simulated by the GFDL model. The results show that the vertical structure of clear-sky radiative cooling rate QR will change in response to the increased water vapor as the globe warms. The authors explain why the change of QR is positive in the uppermost part of the troposphere (<300 hPa) and is negative for the rest of the troposphere. As a result, both the change of clear-sky cooling rate and the change of tropospheric lapse rate contribute to the weakening of circulation. The vertical velocity changes due to the two factors are comparable to each other from the top of the planetary boundary layer to 600 hPa. From 600 to 300 hPa lapse rate changes are the dominant cause of the weakening circulation. Above 300 hPa, the change due to QR is opposite to the change due to lapse rate, which forces a slight increase in vertical velocity that is seen in the model simulation.

Corresponding author address: Xianglei Huang, Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109-2143. E-mail: xianglei@umich.edu

1. Introduction

Since the 1990s there have been debates about how atmospheric circulation patterns could change in a warmer climate. While many modeling studies have predicted a weakening of both zonal and meridional circulations in the tropics (Betts and Ridgway 1989; Knutson and Manabe 1995; Mitas and Clement 2006), some reanalysis products show a strengthened Hadley circulation (Tanaka et al. 2004; Quan et al. 2004; Mitas and Clement 2005). In addition, a decadal trend in outgoing longwave radiation (OLR) that has been observed by Earth Radiation Budget Experiment (ERBE) satellites could be explained by postulating a strengthened tropical circulation in response to global warming (Chen et al. 2002; Wielicki et al. 2002). However, later it was shown that the decadal trend in the observed OLR was likely due to instrumentation issues instead of real physical signals (Wong et al. 2006). Since 2005 further studies about the circulation change in response to global warming have been carried out, partly owing to the availability of high-quality reanalysis datasets and the large amount of climate model output from the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4) archives. For example, using the west–east gradient of sea level pressure (SLP) along the equatorial Pacific region as a measure of the strength of the atmospheric circulation over the Pacific, Zhang and Song (2006) analyzed ship-based observations and reanalysis dating back to 1950 and found a trend of decreasing SLP, which suggested a weakening of the Walker circulation. They further discussed that the change of lapse rate (thermal stability) was the physical reason for such a change in circulation. Vecchi et al. (2006) showed a trend of decreasing SLP from an even longer period (~1860 to present) of observations over the tropical Pacific as well as from five coupled GCM simulations over the same period when anthropogenic forcings were included in the simulations. Similar trends were not seen when the simulations were done only with natural forcing. Vecchi and Soden (2007) further examined 22 IPCC AR4 GCMs in the A1B scenario and showed that all of the models consistently project a weakening of tropical circulation under the global warming scenario. An ensemble mean of the 22 GCMs predicted that the strength of the tropical circulation would weaken as the climate warms and that such a weakening would be dominated by the zonally asymmetric circulation (i.e., the Walker circulation) instead of meridional circulation. Merlis and Schneider (2011) used an idealized atmospheric GCM to examine the Walker circulations over a wide range of climates and provided a scaling estimate of the weakening of Walker circulation as the climate warms. They also found that the asymmetric component of the vertical velocity change may have a different relationship to tropical precipitation than the symmetric component.

In spite of the aforementioned studies, more theoretical explanations are still needed in order to fully understand the weakening of tropical Walker circulation as observed from SLP or as simulated by the GCMs. Using a mass balance equation between global-mean precipitation (P) and the product of convective mass flux (Mc) out of the planetary boundary layer (PBL) and the boundary layer mixing ratio (q), Held and Soden (2006) argued that Mc would become smaller in response to global warming since the increasing rates of P and q with respect to surface temperature are different. The argument rigorously holds only for the global average or to a lesser extent for the tropics in isolation. For particular regions such as regions featured with the tropical deep convective systems, a large portion of the moisture source for precipitation could be supplied by the horizontal convergence of water vapor from adjacent regions. Lin et al. (2006) used multiple satellite observations to study the effect of environmental conditions on the tropical deep convective systems and their analysis of moisture balance showed the dependence of such moisture convergence with SST. Moreover, this scaling argument does not explain how the vertical profiles of vertical mass flux (equivalently, the vertical velocity) above the PBL would change in response to the rising of global-mean surface temperature.

As far as the tropical mean Walker circulation is concerned, numerous studies have used a radiative–convective equilibrium (RCE) framework to understand its essential features (e.g., Schneider 1977; Sarachik 1978; Minschwaner and McElroy 1992; Satoh and Hayashi 1992; Sun and Lindzen 1993; Pierrehumbert 1995; Minschwaner and Dessler 2004), especially the gross features of its time-invariant component. Unlike zonally symmetric circulations (i.e., Hadley circulation) where the interactions between the tropics and extratropics may be an important consideration, the Walker circulation is confined within the tropics, so a two-box RCE model with ascending and descending branches is a reasonable first-order approximation for the free troposphere. Under such equilibrium, downward motion in the descending branch causes adiabatic warming of the air parcel, which is balanced by the clear-sky radiative cooling. Therefore, the downward mass flux in the descending branch can be derived based on the clear-sky radiative cooling rate and the lapse rate. Meanwhile, at each level the downward mass flux at the descending branch is balanced by the upward mass flux at the ascending branch. These constraints motivate us to explore the weakening of the Walker circulation within such a simplified conceptual framework. Indeed, in their radiative–convective model of the tropics, Fig. 1 in Minschwaner and Dessler (2004) already indicated that the mean overturning circulation of the tropics would weaken as the surface temperature increases, suggesting an explanation from this conceptual framework.

Though the lapse rate change is deemed as one of the reasons for the weakening of tropical circulation (Knutson and Manabe 1995; Zhang and Song 2006), the clear-sky radiative cooling rate will also change in response to global warming. The contribution of such radiative cooling rate change to the change of circulation has not been thoroughly discussed before. In this study, we explore the change of clear-sky radiative cooling rate in response to the global warming due to an increase in greenhouse gas abundance. We also explore to what extent the change of clear-sky radiative cooling contributes to the weakening of tropical Walker circulation under the RCE framework and compare its contribution to that from other factors, such as lapse rate changes. Section 2 presents a heuristic argument based on infrared radiative transfer, water vapor feedback, and the aforementioned balance relations in RCE to explain why and how the change of radiative cooling rate would affect the change of vertical velocity. In section 3, simulation results from a two-box RCE model are discussed. Section 4 discusses the estimated change of vertical velocity from the simple RCE framework, compares it with the Geophysical Fluid Dynamics Laboratory (GFDL) model simulation, and studies the relative importance of lapse rate change and clear-sky radiative cooling change for the weakening circulation. A discussion and conclusions are then presented in section 5.

2. Changes of clear-sky cooling rate and its relations with the change of vertical velocity

a. Vertical velocity in the RCE framework

In the framework of radiative–convective equilibrium (e.g., Minschwaner and Dessler 2004), the adiabatic warming of downward motion in the descending branch is balanced by the clear-sky radiative cooling. Since the hydrostatic approximation is applicable to the descending branch, the vertical velocity in the pressure coordinate (ω) can be written as
e1
where Cp is the heat capacity of dry air, T is the temperature, z is the altitude, Γd is the dry-adiabatic lapse rate, g is the gravity acceleration, and QR is the clear-sky radiative cooling rate. Note that positive ω means descending motion. If we use a subscript f to denote future climate and a subscript c to denote current climate, it is trivial to show that
e2
Note that the future temperature increase in the upper troposphere is calculated to be larger than that in the lower troposphere and at the surface; therefore, (dT/dz)f > (dT/dz)c, which is normally referred as the negative lapse rate feedback (Soden and Held 2006; Bony et al. 2006). As a result, the denominator in Eq. (2) is always larger than one. Thus, if QR remains unchanged, vertical velocity will decrease at all altitudes for which Eq. (1) is applicable (i.e., from the top of PBL to the averaged detrainment level of deep convection in the tropical troposphere). If the change of QR is negative at one altitude, then the decrease in vertical velocity would be even greater. If the change of QR is positive, then the vertical velocity could either decrease or increase depending on the relative change between the nominator and denominator of Eq. (2).

b. Change of the clear-sky radiative cooling rate in the future climate

In terms of the change of clear-sky radiative cooling rate (QR) in response to the global warming, a radiative–convective equilibrium framework can be built upon the following principles:

  1. Clear-sky QR in the troposphere is almost entirely due to water vapor absorption and emission. In the tropical troposphere above ~800 hPa, it is mainly due to the absorption and emission of the water vapor pure rotational band (a.k.a. the far-IR band, <600 cm−1). Below 800 hPa it is mainly due to the water vapor continuum absorption and emission from the IR window region between 800 and 1100 cm−1. These can be clearly seen from the plot of spectral cooling rate in Fig. 1.
  2. Because of (1), clear-sky QR for a spectral band can be well approximated by the cooling-to-space approximation (CTS) (Rodgers and Walshaw 1966); that is,
    e3
    where ν is frequency, Bv(T) is the Planck function, and the overbar indicates average over a spectral band with bandwidth of Δν (e.g., water vapor band or part of it), and , in which Δν(z; μ) is the band-averaged transmissivity from the top of atmosphere to altitude z at the given zenith angle θ.
  3. For a band with large opacity such as the water vapor far-infrared band (i.e., diff approaches zero at the surface), the derivative of diff with respect to z (usually termed as the weighting function) is zero at both the top of atmosphere and at the surface. This is because
    e4
    where ρ is the mass density of absorber and kΔv is the band-averaged absorption coefficient. As concisely summarized in Stephens (1994), Eq. (4) indicates that, at the top of the atmosphere, the derivative is zero because density is zero there and, at the surface, the derivative is also zero because Δv is essentially zero for all zenith angles. Meanwhile, it is trivial to show that WΔv(z) is nonnegative within the atmosphere. Therefore, WΔv(z) must have a convex curvature with the maximum somewhere in the atmosphere and zero at both ends, in both current climate and in future climate. Furthermore, it can be shown that WΔv(z) attains its maximum when the optical depth τΔv ≈ 1 (Goody and Yung 1989; Stephens 1994). For the water vapor far-IR band, WΔv(z) peaks in the troposphere as shown in Fig. 2.
  4. As the surface temperature increases due to the increasing abundance of greenhouse gases such as CO2, the water vapor feedback will lead to an increase of water vapor in the entire troposphere. More water vapor means that the altitude for the peak of WΔv(z) (i.e., the altitude for τΔv ≈ 1) will become higher than that in the current climate. If we denote the altitude for the peak of WΔv(z) for the entire far-IR band as zc for current climate and zf for the global warming scenario, then zf > zc. However, at both the surface and the top of atmosphere, WΔv(z) is zero regardless of the climate regime. Therefore, the shape of WΔv(z) must change in following way:
    1. Above an altitude of z0 that is between zf and zc, WΔv(z) of future climate is always larger than that of current climate.
    2. Below z0, WΔv(z) of future climate is always smaller than that of current climate.
      • Such changes of the shape of WΔv(z) can indeed be seen from GCM simulations. Here the GFDL simulations are used and we primarily analyze output from two periods. One period is 1971–2000 in the IPCC AR4 historical run simulated by the GFDL Climate Model version 2.1 (CM2.1) (Delworth et al. 2006). The other period is 2071–2100 in the IPCC AR4 Special Report on Emission Scenarios (SRES) A1B run (Solomon et al. 2007) simulated by the same model. For brevity, hereafter the first period will be referred to as 20C and the second period as 21C. Since our focus is on the tropical Walker circulation, we limit the analysis to the inner tropics from 17°N to 3°S (a 20° zonal belt symmetric to the modeled ITCZ climatological position at ~7°N). Figure 2 shows the weighting function of the water vapor far-IR band as computed from the 30-yr averages of temperature and humidity profiles in the 20C as well as its counterpart computed from the 30-yr averages in the 21C. The two weighting function profiles do change as explained above.
  5. If everything else were unchanged, such changes in WΔv(z) would directly translate to following conclusions:
    1. Above z0, QR will be larger than the current value as global surface temperature increases and, given Eq. (1), a larger downward velocity would be expected above z0.
    2. Below z0 the change is just opposite: QR will become smaller than the current value as global surface temperature increases, and therefore a smaller downward velocity would be expected (i.e., a weakening of large-scale circulation).
Fig. 1.
Fig. 1.

Clear-sky longwave cooling rate and spectral cooling rate as computed from the LBLRTM v12.1, a line-by-line radiative transfer code (Clough et al. 2005), with input of typical tropical sounding profiles (McClatchey et al. 1972): (left) the total longwave cooling rate (K day−1) and (right) the longwave spectral cooling rate (K day−1 10 cm−1). Positive value indicates (spectrally) radiative cooling and negative value indicates (spectrally) radiative warming.

Citation: Journal of Climate 26, 5; 10.1175/JCLI-D-12-00288.1

Fig. 2.
Fig. 2.

The diffusive weighting functions (ddiff/dz) of the water vapor far-IR band (0–600 cm−1) based on mean temperature and humidity profiles over the inner tropics for the 20C period (1971–2000, blue line) and the 21C period (2071–2100, red line) of the GFDL CM2.1 simulation. The calculation was done using the moderate resolution atmospheric transmission (MODTRAN) version 5.3 (Berk et al. 2005), a narrowband radiative code with easy configuration for computing weighting functions.

Citation: Journal of Climate 26, 5; 10.1175/JCLI-D-12-00288.1

Figure 2 shows that the modeled z0 is at ~300 hPa (i.e., ~10 km) for the water vapor far-infrared band, which is then confirmed to hold even for the clear-sky net radiative cooling rate (longwave + shortwave) directly computed by the GFDL model, as shown in Fig. 3a. This suggests that the downward vertical velocity would be reduced over a majority of the troposphere in the descending branch of the Walker circulation. In an equilibrium state the upward vertical velocity in the ascending branch must decrease accordingly at each level. Therefore, a weakening of the Walker circulation below z0 would be expected under a global warming scenario. Note that from the surface to 700 hPa the water vapor continuum absorption at the window region dominates the radiative cooing (as shown in Fig. 3a). The window region is not included in Fig. 2 and in the above heuristic argument because its transmissivity from the top of atmosphere to the surface is close to one instead of zero. However, based on the GCM result shown in Fig. 3a, the net cooling rate in this portion of the troposphere will decrease as the surface warms in simulations of future climate, so it behaves in the same way as the rest troposphere below z0.

Fig. 3.
Fig. 3.

(a) The 30-yr mean of clear-sky net radiative cooling rate (longwave + shortwave) over the inner tropics for the 20C (blue line) and the 21C (red line) from the GFDL CM2.1 simulation; only 200 to 925 hPa shown here. These cooling rates were computed using the GFDL radiation code and archived part of simulation output. (b) The vertical velocity ωest estimated from the modeled cooling rates and lapse rates for the two periods using Eq. (1).

Citation: Journal of Climate 26, 5; 10.1175/JCLI-D-12-00288.1

In summary, a general increase in water vapor will lead to the changes in the vertical structure of clear-sky radiative cooling rates in the troposphere. To compensate for such a change under radiative–convective equilibrium, the downward vertical velocity must be reduced below a certain altitude z0 and increased above z0. However, in reality the assumption in (5) is not rigorously held because the temperature change will also contribute to the change of QR. Moreover, as illustrated in section 2a, the lapse rate change also can contribute to the weakening of circulation. Therefore, it is meaningful to explore the following two questions in the next two sections:

  1. Is the vertical velocity change calculated from Eq. (1) consistent with the change simulated by the 3D coupled GCM?
  2. If the answer to (i) is yes, then what is the relative contribution of lapse rate and QR changes to a weakening circulation based on Eq. (1)?

3. Vertical velocity from the RCE framework and comparison with coupled GCM simulations

We use the monthly-mean temperature and clear-sky cooling rate profiles from the GFDL simulations as input to Eq. (1) to compute vertical velocities that can be compared with those directly simulated by the model. As mentioned in section 2, the 20C and 21C are used to denote the two 30-yr periods, 1971–2000 and 2071–2100, from the CM2.1 historical run and SA1B run, respectively, and confine our analysis to the tropical belt from 3°S to 17°N. Using the 30-yr average of each period, two mean profiles are constructed, one for the 20C and one for the 21C.

Figure 3a shows the clear-sky net radiative cooling rates (both shortwave and longwave) for the 20C and the 21C. As discussed in section 2b, the change of the clear-sky cooling rate resembles the change of the weighting function (Fig. 2). Above 300 hPa the 21C cooling rate is larger than that of 20C, consistent with the reasoning presented in section 2. Figure 3b shows the vertical velocity computed from Eq. (1) (hereafter ωest) using clear-sky net radiative cooling rates in Fig. 3a and the simulated temperature profiles as input. Above 250 hPa the ωest in the 21C is larger than that in the 20C (note that the lapse rate change is included in the calculation). Thus, for the RCE framework, the increase of downward velocity in the uppermost layer of the troposphere is still true even when the lapse rate change is taken into account. Below 250 hPa ωest is reduced from the 20C to the 21C cases.

It is not straightforward to obtain the exact counterpart of ωest in the GCM. A typical grid box of a GCM is ~200 km by 200 km in the tropics. For any given GCM gridbox at any given time step, regardless its grid-averaged vertical velocity being upward or downward, it could contain both upward and downward motions within the grid box, which are parameterized and not resolved by the model. Monthly averages further complicate the issue since only monthly-mean fields are available to us for the analysis. We therefore estimate the monthly mean large-scale downward vertical velocity (ωdn) by merely averaging over all the grid cells with a monthly-mean downward motion and estimate the monthly-mean large-scale upward vertical velocity (ωup) in the similar way.

Compared to a two-box RCE representation of the tropics, ωup estimated in this way would be smaller than the actual ωup in the RCE model because the true area (~4%–6% in the tropics) covered by the ascending branch should be much smaller than the area of such monthly mean large-scale upward motion, as shown in Fig. 4b. Meanwhile, ωdn estimated in this way can be comparable to the ωdn in the RCE framework as long as the characteristic scale of spatial variation of ωdn is comparable or larger than the horizontal resolution of the GCM (i.e., 2.5° × 2° in the GFDL CM3). Therefore, it is more reliable to study the ωdn estimated here instead of the ωup. Figure 4a shows profiles of ωdn and ωup for the 20C and 21C periods. From 925 to 300 hPa both ωdn and ωup become weakened in the 21C simulation. The corresponding fractional areas of ascending and descending branches are shown in Fig. 4b. Above ~600 hPa the area of descending branch is slightly larger than that of the ascending branch in the middle and upper troposphere. The differences become smaller in the 21C case. Below 700 hPa the area of ascending branch is noticeably larger than that of descending branch and there is little change from the 20C to the 21C.

Fig. 4.
Fig. 4.

(a) The mean profile of vertical velocity for the large-scale ascending (asterisks) and descending (open diamonds) branches of the inner tropics in the CM2.1 simulation for the 20C period (blue lines) and the 21C period (red lines). Refer to the context for the way of averaging. (b) The mean fractional area for the large-scale ascending and descending branches. Color coding and notation are as in (a).

Citation: Journal of Climate 26, 5; 10.1175/JCLI-D-12-00288.1

Figure 5 summarizes differences in vertical velocity over the downward branch from Eq. (1)ωest, differences of two curves in Fig. 3b) and from the GCM simulation (Δωdn, differences of two curves in Fig. 4a). Both sets show reduced downward vertical velocity below 300 hPa. In the free troposphere between the boundary layer (above 900 hPa) and below the mean detrainment level of deep convection (~200–250 hPa), the amplitudes of Δωest and Δωdn are comparable to each other, though Δωest tends to be larger than Δωdn at all levels except 700 hPa. Above 300 hPa the Δωest is positive (increases in downward vertical velocity) owing to the change of radiative cooling profiles mentioned in section 2. Meanwhile, the Δωdn is only slightly negative at 250 hPa and slightly positive at 200 hPa. The Δωest is computed with the RCE being the only physical mechanism controlling the vertical velocity, thus having limited applicability at and above the detrainment level of the deep convection, which is about 200–250 hPa in the CM2.1 (Chuang et al. 2010). Furthermore, in the full GCM, other mechanisms such as wave propagation and tropical stratosphere–troposphere exchange can also affect the vertical velocity at these tropospheric levels above the mean detrainment levels. Therefore, it is not a surprise to see discrepancies between Δωest and Δωdn exist at these levels. Nevertheless, the change of sign in Δωdn from 250 to 200 hPa is consistent with the discussion in section 2 about the change of QR in the uppermost part of the troposphere. Below 900 hPa the RCE is not applicable either since the boundary layer processes dominate these layers. Therefore, generally speaking, the GCM simulation and analytic results based on the simple RCE framework are consistent in the magnitude and the sign of vertical velocity changes over the free troposphere where the RCE concept is applicable.

Fig. 5.
Fig. 5.

Differences in the vertical velocity of the descending branch between the 21C and 20C periods as calculated using the CM2.1 model (blue ×) and estimated from Eq. (1) (open green circle).

Citation: Journal of Climate 26, 5; 10.1175/JCLI-D-12-00288.1

4. Sensitivity tests

Section 3 shows that, to a large extent, Eq. (1) is capable of capturing the change of vertical velocity profile in the free troposphere from the 20C to the 21C periods as simulated by the GFDL GCM. Then we can further use it to explore the sensitivity of vertical velocity to changes in QR and to changes in lapse rate separately. We take ωest computed from the QR and lapse rate of the 20C period as the control value. We then can adjust QR or the lapse rate separately to the 21C values and compare ωest to the control ωest. Such sensitivity results are summarized in Fig. 6. Above 300 hPa the QR change dominates over the lapse rate change. This is largely because the lapse rate at such high altitude is always close to the dry-adiabatic lapse rate and further possible changes are very limited. From 300 to 600 hPa, the contribution of the lapse rate change dominates over that of the cooling rate. This is also the vertical layer where the lapse rate changes most. Below 600 hPa (including 600 hPa) the contributions from the QR change and the lapse rate change are comparable to each other. As mentioned in sections 1 and 2, if only the lapse rate were changed, there would be no increase of downward vertical velocity in the uppermost part of the troposphere. This effect is shown in Fig. 6 as well. Moreover, the impacts on Δωest due to the QR change and the lapse rate change are additive so that their sum is nearly the same as the Δωest due to changes of the two factors simultaneously.

Fig. 6.
Fig. 6.

Estimated change of vertical velocity from Eq. (1). The green line is the same as the one in Fig. 5: the change when both the clear-sky net radiative cooling rate and the lapse rate change are taken into account. The blue line is due to the clear-sky net radiative cooling rate change alone. The red line is due to the lapse rate change alone. The black dash line is the summation of the blue and red lines, which agrees well with the green line for all of the levels.

Citation: Journal of Climate 26, 5; 10.1175/JCLI-D-12-00288.1

We also extend such sensitivity test to the 300-yr simulation (2000–2300) done by the CM2.1 under the SA1B scenario. The 5-yr mean profiles of QR and lapse rate are used for each calculation. For sensitivity test, the values of QR (or lapse rate) of the first 5-yr segment are used. Figure 7 shows the time series of Δωest for 250 hPa, 400 hPa, and 700 hPa. It can be seen the relative importance of QR change and lapse rate change indicated in Fig. 6 is also applicable to longer time series and a time-dependent Δωest.

Fig. 7.
Fig. 7.

Change of vertical velocity at three different pressure levels from the year 2000 based on Eq. (1) when both the clear-sky net radiative cooling rate and lapse rate are considered (green) or when only one factor is considered (blue for the clear-sky radiative cooling rate change and red for the lapse rate change).

Citation: Journal of Climate 26, 5; 10.1175/JCLI-D-12-00288.1

5. Conclusions and discussion

Using the highly simplified two-box RCE framework, a heuristic argument for how the change of water vapor with global warming can affect the tropical clear-sky cooling rate and, hence, alter the large-scale tropical circulation is presented. For the large-scale subsidence branch in a RCE model, the downward vertical velocity is related to the clear-sky radiative cooling rate and to the lapse rate. The clear-sky cooling rate can be largely related to the diffusive weighting function of the water vapor bands (primarily the far-IR band) based on the cooling-to-space approximation. The shape of the weighting function will change in such a way to reduce the clear-sky cooling rate throughout the most part of the troposphere but to increase it in the uppermost part of the troposphere. This in turn would produce a decrease in vertical velocity in most of the troposphere and an increase in the uppermost portion of the troposphere. Using the GCM-simulated QR and lapse rate as input, the changes of vertical velocity computed from this RCE framework generally agree with the changes directly computed by the fully coupled GCM for the free troposphere up to 300 hPa. Above this level, physical mechanisms other than RCE also modulate the vertical velocity, so the agreement becomes worse. The slight increase of vertical velocity in the uppermost part of the troposphere is also seen in the GCM result, which can be explained partly by the consequence of a change of the clear-sky net cooling rate but not by the change of the lapse rate. Sensitivity tests show that, at different parts of the troposphere over the inner tropics, the relative importance of lapse rate changes and clear-sky cooling rate changes on the circulation changes are different. The contribution from the lapse rate change dominates in the middle troposphere (~600–300 hPa), whereas the radiative cooling rate change dominates above 300 hPa. In the lower troposphere from 900 to ~600 hPa, the two effects are comparable. Such relative partitioning between two factors does not change with the time in the SA1B simulation.

Knutson and Manabe (1995) used the energy balance to explain why there is no enhanced Walker circulation when convection/condensation heat is enhanced over the warm pool in a coupled GCM. They found that three forms of energies are always in balance: the convection/condensation heating, radiative cooling, and dynamical cooling or heating (dependent on ascending or descending motion as well as the static stability). Over the convective region, most of the enhanced convection/condensation heating is counterbalanced by the enhanced radiative cooling, which results from the enhancement of both temperature and water vapor. Our study here, on the other hand, focuses on the changes and balance in the descending branch. We note here that the strength of the Walker circulation can be measured in different ways: for example, the Eulerian mass overturning or the sea level pressure gradient. The changes of pressure vertical velocity discussed here are closely related to the change of downward mass flux in the nonconvective region (Minschwaner and Dessler 2004). Therefore, the result obtained here should be largely applicable to the change of downward mass flux. One note of caution is that the RCE framework used here cannot compute the area fractions of upward and downward regions. Depending on the measure used for the strength of the Walker circulation, the changes of such area fractions of upward and downward regions do affect a particular measure of the strength.

This study points out the role of clear-sky radiative cooling rate in the weakening of tropical circulation in response to the global warming. As for the simulated or observed change of circulation over the entire tropics instead of over only the inner tropics, conclusions from the RCE framework should be used and extrapolated with caution. Even for an idealized zonal belt without diabatic flow across its boundary, it can be shown that the mean vertical velocity is related not only to the clear-sky cooling rate and lapse rate change but also to the eddy mass flux along the vertical direction (), where symbols are defined as they normally are in the atmospheric dynamics convention. In reality, the diabatic flow across the boundary (e.g., the exchange between the tropics and extratropics) should also be considered for examining changes in circulation. If the entire tropics (30°S–30°N) were chosen instead of only the inner tropics for the analysis, then the difference between Δωest from Eq. (1) and Δωdn from the CM2.1 simulation would be much larger than the results for the inner tropics. This highlights the contributions of other physical mechanisms that should be taken into account for a more thorough understanding of the tropical circulation changes as a whole.

Acknowledgments

We thank the GFDL modeling team for carrying out the simulations. We are thankful to two anonymous reviewers for their thoughtful comments to improve the clarity of the paper. The lead author is thankful to the GFDL for generously providing computing resources for analyzing the GFDL simulation output. This research is supported by NSF Grant ATM0755310 and the NASA Terra/Aqua program under Grant NNX11AH55G awarded to the University of Michigan, and A.D. is supported by NSF Grant AGS-1012665 to Texas A&M University.

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