6.4 Gross moist stability and the slowdown of the Walker circulation
6.4.1 Estimating gross moist stability
Moist static energy decreases with height in the boundary layer then increases with height in the free troposphere and stratosphere, as can be seen in Fig. 6.7a for the full range of climates. The value of MSE increases monotonically with warming at all levels, shown explicitly for the surface, LCL, and tropopause in Fig. 6.7b. Because the MSE profile is non-monotonic, the GMS depends sensitively on the Ω1(p) that is used. It is only a useful quantity if Ω1(p) accurately represents the vertical velocity profiles in these simulations. For comparison, we compute stationary-eddy vertical velocity profiles by averaging over the convecting region (O∗<−10 W m−2) at each level. We will denote such averages with a ˆ(·). The vertical velocity profile−ˆω(p)∗ is shown in Fig. 6.8a.
Global−mean surface temperature (K)
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1000 3e5
4e5 5e5
2803 290 300 310 320
3.5 4 4.5 5 5.5 x 105
Global−mean surface temperature (K)
MSE (J kg−1) LCL
surface tropopause
a b
Figure 6.7: a) Vertical profile of zonal-mean moist static energy as a function of mean climate. A black line shows the height of the tropical tropopause. Both quantities are averaged for φ∈[−10 10]. b) Value of zonal-mean moist static energy at select vertical levels.
We compute the wind mode Ω1(p) following Levine and Boos (2016), who present a generalization of the model found in Neelin and Zeng (2000). The generalization simply relaxes some of the approximations on the relative importance of temperature, moisture, and geopotential for MSE, such that the Ω1(p) modes are relevant over the full range of climates in these idealized simulations.
The wind mode is related to the temperature modeA1(p) through the use of thermal wind balance and the continuity equation. The temperature mode relates the amplitude of deviations from the tropical-mean temperature profile to deviations of the sub-cloud MSE. This assumes a state of convective equilibrium, where saturated MSE is constant above the LCL (Emanuel et al., 1994;
Emanuel, 2007). We assume a fixed LCL height with warming, which is consistent with small changes in relative humidity (Schneider et al., 2010). The derivation of Ω1(p) can be found in Appendix 6A (Section 6.7). The result of this quasi-equilibrium theory (QET) is that Ω1(p) can be computed from the tropical-mean temperature profileTr(p). The resulting Ω1(p) are shown in Fig 6.8c for the full range of climates.
We can more easily compare the modeled vertical velocity profiles in the convecting region−ˆω(p)∗ with Ω1(p) if we normalize such that the maximum values are the same for each climate. The normalized vertical velocity profile,
Ω1e(p)≡ −ω(p)ˆ ∗ max(Ω1(p))
max(−ˆω(p)∗), (6.16)
is shown in Fig 6.8b. The resulting empirical vertical wind modes Ω1e(p) are quite similar to Ω1(p)
−0.01 0 0.01 0.02 0.03 0.04 0
200 400 600 800 1000
−ω(p) (Pa s−1)
0 0.1 0.2 0.3 0.4 0.5
0 200 400 600 800 1000
Ω(p)
0 0.1 0.2 0.3 0.4 0.5
0 200 400 600 800 1000
Ω(p)
a
b
c
Figure 6.8: a) Stationary-eddy vertical-velocity profiles averaged over the ascent region, diagnosed as the region where the imposed heating is above 10 W m−2. b) Non-dimensional stationary-eddy vertical-vertical velocity profiles that have been normalized such that the maximum value is the same as the QET Ω1(p) profile. We call this Ω1e(p). c) Non-dimensional QET vertical-velocity profile Ω1(p). In all figures, profiles are shown for 14 mean climate states color coded from coldest (blue) to warmest (red).
across the range of climates. Differences are largest near the surface, near the tropopause and in the warmest climates.
We compute the GMS from both Ω1e(p) and Ω1(p) using Eq. 6.15,
Me=−
Ω1e(p)∂ph , M =−
Ω1(p)∂ph .
(6.17)
Both show a pronounced increase with warming through 310 K global-mean surface temperature (Fig. 6.9a). The QET estimate (M, dashed line) shows the same magnitude of increase with climate
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−10 0 10 20 30
Global−mean surface temperature (K) Energy input (W m−2)
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0.01 0.02 0.03 0.04
Global−mean surface temperature (K)
−ω*max (Pa s−1)
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Global−mean surface temperature (K) Gross moist stability (J kg−1)
[Me] [M]
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0.01 0.02 0.03 0.04
Global−mean surface temperature (K)
−ω*max (Pa s−1)
a b
c 2800 290 300 310 320
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Global−mean surface temperature (K)
Energy input (W m−2) <ω∂ph>*
ocean heating longwave radiation advection term transient−eddy term
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0.01 0.02 0.03 0.04
Global−mean surface temperature (K)
−ω*max (Pa s−1)
−ω*(p)max
<ω∂p h>*Δh−1 Δh−1
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2000 4000 6000 8000 10000 12000
Global−mean surface temperature (K)
−ω*max (Pa s−1)
−ω*(p)max
<ω∂p h>*[Me]−1
<ω∂p h>*[M]−1 [M]−1
d
Figure 6.9: Controls on the strength of the Walker circulation. a) Zonal-mean GMS [M] com- puted using Ω1e(p) (solid line) and Ω1(p) (dashed-line). b) Terms of the zonally anomalous moist static energy equation averaged over the region where the imposed heating is above 10 W m−2. The vertical-motion term d
ω∂ph∗
is primarily balanced by the zonally anomalous ocean heat-flux convergence (black dashed line). c,d) Scalings for Walker circulation strength. The maximum of the stationary-eddy vertical velocity profile, averaged over the region where the imposed heating is above 10 W m−2and estimates c) based on the zonal-mean GMS [M] or d) tropopause–LCL moist static energy difference ∆h.
change as the more appropriate empiricalMe. Overall, M provides a good approximation toMe. Levine and Boos (2016) use the same method for computing Ω1(p), but proceed in a different way, for an application to subtropical circulations. Instead of computing the GMS, they combine the linear vorticity balance in the subtropical boundary layer with the baroclinic wind modes to form a theory for stationary-eddy vertical velocities. This theory is successful at describing the strength of subtropical stationary-eddy circulations. This does not work in the tropics, because of the compensation and nonlinearity in the vorticity budget, as discussed in Section 6.3. We proceed to constrain the stationary-eddy vertical velocities instead through the GMS and the dominant balance in the MSE budget (Eq. 6.11). There is nothing to suggest that the GMS constraint wouldn’t also work in the subtropics.