The vorticity budget of the Walker circulation, which was already addressed for the lower troposphere in Section 3.3.2, is a good place to start to understand controls on Walker circulation strength.
This is motivated in part by the results of Levine and Boos (2016), where the lower-tropospheric vorticity budget was used to relate stationary-eddy ascending motion forced by subtropical heating to lower-tropospheric zonal temperature gradients. Zonal temperature gradients in the tropics are understood to decrease rapidly with warming based on energetic constraints on evaporation (Merlis and Schneider, 2011). If the Walker circulation strength could be tied to zonal temperature gradients, then the decrease in Walker circulation strength would be a direct result of the energetic constraints on evaporation. This would also fit with the observed correlation between zonal SST gradients and the strength of Walker circulations (Bjerknes, 1969).
The conclusion of Section 3.3.2 was that stationary-eddy vertical motion is well described by a combined Sverdrup-Ekman balance in the lower troposphere, except within 2◦ latitude of the equator. The budget of Section 3.3.2 was focused on understandingP∗−E∗ and thus focused on a moisture-weighted vorticity budget analysis from the surface to 850 hPa. If our goal is to understand the strength of the Walker circulation, it is beneficial to focus instead on the vorticity budget controls onω∗in the free troposphere. We examine a budget forω∗based on the zonally anomalous vorticity equation integrated from the surface or from the top of the domain,
ω∗(p) =−g f
D
βv∗+T∗+N∗Eps
p −g
f∇ ×τ∗ (6.4a)
ω∗(p) =g f
D
βv∗+T∗+N∗Ep
0. (6.4b)
We focus on applying theω∗ (vorticity) budget (Eq. 6.4) at a representative 600 hPa level.
With two equations for ω∗600 and multiple experiments, it is best to first look at average contri- butions of the different vorticity-budget terms toω∗600, before looking at the spatial distribution of
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Global−mean surface temperature (K) ω*600 (Pa s−1)
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Global−mean surface temperature (K) ω*600 (Pa s−1)
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−1 0 1 2 3 x 10−3
Global−mean surface temperature (K) ω*600 (Pa s−1)
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Global−mean surface temperature (K) ω*600 (Pa s−1)
a b
c d
Lower troposphere Upper troposphere
Q8
Q5
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Global−mean surface temperature (K) ω*600 (Pa s−1)
stationary−eddy vertical velocity Sverdrup term
Ekman drag term Transient−eddy term Time−mean nonlinear term Time−mean vorticity fluxes
Figure 6.1: Vorticity budget controls on|ω∗600i=− hω∗600|(black line) across the range of climates in a,b) Q8 and c,d) Q5. Vorticity budgets are calculated integrated (left) below 600 hPa or (right) above 600 hPa.
these terms. To this end we look at vorticity budget terms averaged with the operator,
|·i ≡ Z
(·)H(ω∗600)dA, (6.5)
where the integration area is global. For terms that average to zero (stationary-eddy components), this differs only by a sign from
h·| ≡ Z
(·)H(−ω∗600)dA. (6.6)
The contributions toω∗600based on the heavyside-averaged upper- and lower- tropospheric vorticity equations are shown in Fig. 6.1 for experiments Q8 and Q5.
The experiments show similar vorticity balances in the upper troposphere, with time-mean non- linear terms (N∗, grey line), and particularly horizontal vorticity fluxes by stationary eddies (dashed grey line), governing the changes inω∗600for all climates except the coldest 2. However, in the lower troposphere, the vorticity budgets are drastically different. In Q8, the declining strength of ω∗600 with warming comes primarily from the combined Ekman-Sverdrup balance (red and blue lines),
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latitudinal width (degrees) ω*600 (Pa s−1)
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−2
−1 0 1 2 3 x 10−3
latitudinal width (degrees) ω*600 (Pa s−1)
a b
Lower troposphere Upper troposphere
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Global−mean surface temperature (K) ω*600 (Pa s−1)
stationary−eddy vertical velocity Sverdrup term
Ekman drag term Transient−eddy term Time−mean nonlinear term Time−mean vorticity fluxes
Figure 6.2: Vorticity budget controls on|ω∗600i=− hω∗600|(black line) across the range of latitudinal widths of forcing. Vorticity budgets are calculated integrated a) below 600 hPa or b) above 600 hPa.
with some contribution from transient-eddy vorticity fluxes. On the other hand, in Q5, most of the decrease inω∗600comes from the decrease in transient-eddy vorticity fluxes (pink line). Despite these differences, there is not much difference between the actual vertical velocities in these experiments.
This is even more apparent when examining the vorticity budget as the aspect ratio is varied with the optical depth at a reference value (α= 1.0), as shown in Fig. 6.2. The lower-tropospheric transient eddies become less important for the vertical velocity as the latitudinal width of the heating increases. The Sverdrup balance vertical velocity,−gf|hβv∗iiincreases such that the vertical velocity at 600 hPa remains unchanged throughout the range of aspect ratios. In the upper troposphere, there is a decrease in the extent to which the Sverdrup balance vertical velocity is out of phase with the full vertical velocity. There is a corresponding change in the fraction of the vertical velocity that is balanced by stationary-eddy vorticity fluxes.
The fact that the vertical velocity is unchanged despite large changes in the underlying vorticity balance suggests that there must be independent constraints on the vertical velocity and that the vorticity budget is balanced in whatever way is easiest based on the background absolute vorticity.
This is our motivation for investigating energetic constraints on Walker circulation vertical motion in the next section. Despite this, we would like to examine some aspects of the linear and transient-eddy regimes of the lower-tropospheric vorticity balance.
The spatial distribution of vorticity terms is shown for the linear regime (represented by Q8) in Fig. 6.3. The spatial distribution of Sverdrup upwelling (−gfhβv∗i) is qualitatively similar to distribution of zonal temperature gradients, with poleward motion east of the heating and west
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30 S EQ 30 N
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30 S EQ 30 N
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30 S EQ 30 N
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30 S EQ 30 N
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30 S EQ 30 N
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30 S EQ 30 N
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Pa s -1
Figure 6.3: Lower tropospheric vorticity budget controls onω∗600 in the reference climate (α= 1.0) of Q8. Terms are labelled as defined in the text.
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30 S EQ 30 N
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30 S EQ 30 N
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30 S EQ 30 N
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30 S EQ 30 N
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30 S EQ 30 N
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Pa s -1
Figure 6.4: Lower tropospheric vorticity budget controls onω∗600 in the reference climate (α= 1.0) of Q5. Terms are labelled as defined in the text.
of the cooling and equatorward motion west of the heating and east of the cooling. However, we find that thermal wind balance is not accurate enough this close to the equator to quantitatively link Sverdrup vertical motion and zonal temperature gradients. Additionally, the Sverdrup vertical velocity is mostly phase shifted from the actual vertical velocity where stationary-eddy overturning is strongest. The positive contribution to stationary-eddy overturning mostly comes on the edges of the convecting region, where the Sverdrup term is in phase. In the region of heating, transient-eddies are important for the westward propagation of the zone of convection. This could be accomplished by vorticity tendencies associated with equatorial Rossby waves and Kelvin waves.
The regime where the lower-tropospheric vorticity budget is dominated by transient eddies (rep- resented by experiment Q5) is shown in 6.4. In this case, as in the linear regime, vertical motion is mostly confined to the region directly over the ocean heating/cooling. Here the Sverdrup term (−fghβv∗i) actually plays a role in reducing the vertical motion associated with transient-eddy vortic- ity tendencies. This results as ageostrophic meridional motion transports mass into the convergence
longitude
time (days)
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−6
−4
−2 0 2 4 6 8
longitude
time (days)
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−0.1
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a b
Pa s -1 m s -1
Figure 6.5: Hovm¨oller diagrams for one year of simulation in the transient-eddy dominated regime (Q5). a)ω600averaged within 5◦ of the equator and b)usfcaveraged within 2◦ of the equator.
zone on the equator. Vertical motion can thus only come from nonlinear vorticity fluxes. This is predominantly accomplished by transient eddies (−gfD
T∗Eps
p
).
To investigate the source of this transient-eddy vorticity tendency, we present in Fig. 6.5 Hovm¨oller diagrams for one-year of simulation in the reference climate of the Q5 experiment, where the transient-eddy tendency is strongest. The left panel shows the bursty nature of convection over the heat source (90◦ longitude) seen in the vertical velocity at 600 hPa. Many of these bursts are fed by westward propagating easterly anomalies in the zonal surface wind (right pannel). This is characteristic of equatorial Rossby waves.