5.5 Stationary-eddy thermodynamics
5.5.2 Surface vertical velocity
The essential difference between R45, R54, and R36 are the different responses of ω∗sfc to climate change, which we examine here in more detail. The spatial structure ofω∗sfc in R45 is shown in Fig.
5.12a. It can be split up into components due to the zonal-mean and stationary-eddy surface wind according to
ω∗sfc= ([usfc] +u∗sfc)· ∇p∗sfc+v∗sfc∂y[psfc]. (5.9)
The first two terms of this decomposition are shown in Fig. 5.12 b and c respectively. The third term on the right-hand side of Eq. 5.9 is negligible because|∇p∗sfc| |∂y[psfc]|1. The larger zonal-mean wind term can be further approximated by its zonal component [usfc]· ∇p∗sfc≈[usfc]∂xp∗sfc, shown in Fig. 5.12d. This term by itself gives a rough estimate of the magnitude of orographic forcing by ω∗sfcand can be calculated from the zonal-mean zonal surface wind climatology (Fig. 5.13a), which is mostly independent of the existence of the stationary wave (cf. Fig. 5.2).
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d c
a
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30 N 45 N 60 N
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Pa s -1
Figure 5.12: a) Map of ω∗sfc over topography (shading) in the reference climate of R45 (a). Also shown are the horizontal surface winds (usfc, arrows) and two contours (900 and 800 hPa) of psfc (black contours). (b), (c), and (d) show components ofω∗sfcbased on the decomposition in Eq. 5.9.
We would like to make a scaling for the zonally-averaged spatial variance ofω∗sfc, which is shown in Fig. 5.13d. Despite the importance of eddy winds in the phase structure of the response to orographic forcing (Chen and Trenberth, 1988), one might expect [usfc]∂xp∗sfc (Fig. 5.12d) to give a rough idea of changes in
ω∗2sfc
according to
δ ω∗2sfc
∼δh
[usfc]2(∂xp∗sfc)2i
. (5.10)
The rescaled spatial variance of [usfc]∂xp∗sfc is shown as a dashed line in Fig. 5.13d. The rescaling parameter is 0.4, so the eddy surface winds clearly play a role in reducing the amplitude of the response, but the goodness of fit for R45 and R54 suggests that the strength of eddy surface winds scale with the strength of the zonal-mean zonal surface wind. Based on this scaling, the two main
1A component due to the temporal correlations of the surface winds and the surface pressure gradients,u0sfc· ∇p0s, is also negligible.
280 290 300 310 320 3
4 5 6
Global−mean surface temperature (K)
Zonal−mean surface wind (m/s)
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Global mean temperature (K)
p0 − pm (hPa)
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1 2 3 4 5
Global−mean surface temperature (K)
Zonal−mean surface wind (m/s) 280 290 300 310 3200
1 2 3 4
Global−mean surface temperature (K)
Zonal−mean surface wind (m/s)
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Global mean temperature (K)
p0 − pm (hPa)
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Global mean temperature (K)
p0 − pm (hPa)
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0 0.5 1 1.5 x 10−4
Global−mean surface temperature (K) ω*sfc2 (Pa2 s−2)
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1 2 3 x 10−4
Global−mean surface temperature (K) ω*sfc2 (Pa2 s−2)
a
R45 R54 R36
b
c
2800 290 300 310 320
0.5 1 1.5 x 10−4
Global−mean surface temperature (K) ω*sfc2 (Pa2 s−2)
Global−mean surface temperature (K)
285 290 295 300 305 310 315
20 S 40 S 60 S
−6
−4
−2 0 2 4 6
d
a
m s -1
Figure 5.13: Influences on orographic forcing by ω∗sfc. a) Variation of zonal-mean zonal surface wind ([usfc]) with warming across the range of climates. Dashed lines indicate the latitude of the mountain center in each experiment. b) Average of [usfc] over the range of latitudes where the mountain reaches maximum height in each experiment. c) Difference in surface pressure from the top of the mountain to a reference location and a scaling based on Eq. 5.13. d) Zonal-mean ω∗2sfc averaged within 10◦ latitude of the mountain center and an estimate based on [usfc] (Eq. 5.10).
influences on orographic forcing are [usfc] at the latitude of the mountain and the zonal surface- pressure gradients at these latitudes. The change of the relevant surface zonal winds across the range of climates is shown for all 3 experiments in Fig. 5.13b, where [usfc] is averaged over the latitudes where the topography reaches maximum height (φ∈[42.5, 47.5] for R45). The rough sense of change in stationary-wave amplitude is set by the change of [usfc] over the mountain, with an increase then a decrease in R45 and a strong decrease in R54.
The [usfc] is relevant to the extent that the surface winds near the mountain are dominated by the zonal-mean zonal component. This is not the case for R36 and the warmest climates of R54,
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b c
e
d f
a α = 1.0 α = 1.0 α = 1.0
α = 3.0 α = 3.0 α = 3.0
R45 R54 R36
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30 N 45 N 60 N
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Pa s -1
Figure 5.14: Maps of ω∗sfc over topography (shading) in a,b) R45, c,d) R54, and e,f) R36, in the reference climate (a,c,e) and in the 3x optical depth simulations (b,d,f). Also shown are the horizontal surface winds (usfc, arrows) and two contours (900 and 800 hPa) ofpsfc(black contours).
where [usfc] at the mountain is small. This can be seen from Fig. 5.14, which shows usfc and the resultingω∗sfcforα= 1 andα= 3 for all 3 experiments. Here it can be seen that the main difference between the surface winds and the zonal-mean values is due to the tendency of the wind to be deflected around the mountain following contours of surface pressure. The strength of the wind over the mountain is similar to the zonal-mean zonal wind in R45 and R54. In the R36 experiment, it is not valid to use the zonal-mean zonal surface wind as a scaling for
ω2sfc
; the meridional surface wind on the equatorial side of the mountain plays a significant role (Fig. 5.14c,f).
The surface pressure variance part of the scaling is given by the difference in surface pressure between the top of the mountain (pm) and a reference point at sea level (p0),
h
(∂xp∗sfc)2i
= (pm−p0)2/π2r2ecos2φ. (5.11)
The orographic pressure difference,pm−p0, decreases across the range of climates in all 3 experiments (Fig. 5.13d). This change is dominated by an increase in pm which gives a tendency towards weaker orographic stationary waves in warmer climates. Since∇p∗sfcalso shows up in the nonlinear orographic forcing by stationary-eddy surface winds, the influence of pm on the total orographic forcing should remain, no matter the structure of the surface winds.
Using hydrostatic balance, ∂zlnp = g/rdT, we may obtain an expression for the pressure at
heightzm (the height of the mountain) over a point where the surface pressure isp0by integrating vertically from the surface
lnp(zm)≈lnp0− Z zm
0
g
rdTdz. (5.12)
Assuming that horizontal pressure differences atzm(normally of order 10 hPa) are much less than p0−pm ∼ 250 hPa, then p(zm) ≈ pm. Substituting an approximate temperature profileT(z) = T0−Γz, we obtain an expression for the top of mountain pressure in terms of a lapse rate Γ and a reference surface temperatureT0,
pm≈p0
1−Γzm
T0
g/rdΓ
. (5.13)
Evaluating T0 as an average surface temperature at the latitude of the mountain in the eastern hemisphere (away from the mountain) and Γ as an average lapse rate over the bottom 250 hPa of the troposphere in the same region, we obtain the approximate p0−pm shown as a dashed line in Fig. 5.13c. By fixing T0 and Γ independently, we can see that the increase in top-of-mountain pressure is a direct influence of the increasing surface temperature via the equation of state.
The orographic vertical surface winds, which set the initial orographic perturbation to the atmo- sphere, are sensitive to changes in the zonal-mean zonal surface winds, the top-of-mountain surface pressure, and the stationary-eddy modification of the surface winds. When topography is fully in the surface westerlies, the orographic stationary Rossby wave amplitude reflects the strength of zonal-mean zonal surface winds. Additionally, there is a tendency towards weaker forcing in warmer climates as the top-of-mountain surface pressure increases, a consequence of the equation of state.
It is worth noting that the R54 experiment was designed to get the maximum zonal surface-wind response to climate change at the latitude of the mountain, with the hypothesis that as the zonal winds go to zero at this latitude, the stationary-wave response would also go to zero. The experiment verified this hypothesis. In this sense, this theory leads to semi-quantitative predictions of sEKE in terms of zonal surface-wind changes.