7
7.1
The General Circulation
The axisymmetric state
At the beginning of the class, we discussed the nonlinear, inviscid, axisymmetric theory of the meridional structure of the atmosphere. The important
axisymmetric equations can be written as
@m
1@
+ u rm =
;
@t
@z
@T
@T
+v
+ wS = Q ;
@t
@y
(1)
(2)
where m = a2 cos2 ' + ua cos ' is the speci…c absolute angular momentum,
is the frictional stress, S = (p=p0 ) @ =@z, and Q = J= cp , where J is the
diabatic heating rate. We discussed how, in the inviscid free atmosphere, (1)
implies that in steady state u rm = 0, and in turn this led us to conclude
that a nonzero meridional circulation can exist only in a tropical region where
m is uniform; in the extratropics, there can be no meridional circulation, as
depicted in Fig. 7.1. As we also discussed, in the tropics this produces a ‡at
Q>0
Q<0
v=w=0
-> Q=0 -> T=T
-> τs=0 -> us=0
EQ
POLE
temperature distribution, with diabatic heating in the upwelling region and
diabatic cooling in the subtropical downwelling, with a westerly subtropical
jet in the upper troposphere, at the outer wings of the Hadley cell. Outside
1
the Hadley cell, the absence of a meridional circulation implies radiative
equilibrium with Q = 0 in steady state, and T = Te , the radiative equilibrium
temperature. Further, since there is no angular momentum transport in
steady state, the vertical integral of (1) gives s = 0: there can be no surface
stress whence the zonal velocity us at the surface must vanish.
Especially in the extratropics, this picture is not quite right. For one
thing, temperatures are not in radiative (or radiative-convective) equilibrium; e.g., the poles are much warmer than they would then be. For another,
the meridional circulation is shown in Fig. 1; while it is indeed strongest
Figure 1: Annual and zonal mean meridional streamfunction [Oort]
in the tropics, there are noticeable reverse (thermally indirect) circulations
2
in middle latitudes— the “Ferrel” cells1 . Further, as Fig. 2 shows, the dis-
Figure 2: Annual and zonal mean zonal wind [Peixoto & Oort].
tribution of zonal winds is also inconsistent with the axisymmetric picture;
the well-known surface extratropical westerlies are at odds with the prediction of zero surface stress, and the upper tropospheric jets, while being in
more-or-less the right place, are much weaker than they would be if there
were no gradient of absolute angular momentum across the tropical upper
1
The Ferrel cells, unlike the Hadley cells, are not robust with respect to changes in how
one does the zonal averaging. E.g., averages along isentropes produce an extratropical
circulation in the same sense as, but weaker than, the Hadley circulation (the same is true
of the “residual” circulation.)
3
troposphere. Missing from our picture are the ubiquitous extratropical eddies.
7.2
The extratropical mean state in the presence of
eddies
Let’s consider the impact of extratropical eddies. To be consistent with our
treatment of eddies, we’ll assume QG theory is valid and (for the moment)
we’ll ignore spherical geometry. Separate everything into its zonal mean and
eddy component:
Z
1 L
a(y; z; t) =
a(x; y; z; t) dx ;
L 0
a0 (x; y; z; t) = a(x; y; z; t) a(y; z; t) :
The zonally averaged QG equations are then
@u
1@
@
fv =
u0 v 0
@t
@z @y
@
@T
+ Sw = Q
v0T 0
@t
@y
@v 1 @
+
( w) = 0
@y
@z
and
f
@u R @T
+
=0:
@z H @y
(3)
(4)
(Note that v; w are ageostrophic, because vg = @ =@x = 0, and so terms like
v@ u=@y and v@ T =@y are formally negligible.) Thus, the impact of the eddies
on the mean state is felt through only 2 terms23 : an e¤ective zonal force (per
unit mass) and an e¤ective diabatic heating, which are represented by the
convergence of the eddy ‡uxes of momentum and of heat, respectively. It is
tempting, then, to regard the eddy momentum ‡ux as impacting the mean
2
There could be less direct impacts, e.g., if rain is produced within the eddies, they
will have an impact on Q.
3
In fact, it is possible to reorganize (3) such that the eddy term disappears from the
heat budget, in which case the eddy term in the momentum budget becomes just 1 r F,
the divergence of the EP ‡ux.
4
zonal wind, and the eddy heat ‡ux as impacting mean T , but that would be
a mistake, since thermal wind shear balance (4) ensures that the mean wind
and temperature impacts are linked (the linkage occurs via the meridional
circulation). So one needs to be careful deciding what does what.
First, let’s look at the eddy ‡uxes in a climatology. The eddy heat ‡uxes,
shown in Fig. 3(a) and (b) for the transient and stationary waves respectively,
are generally poleward except in the subtropics, where they are weak. The
eddy momentum ‡uxes, Fig. 4(b) and (c), are also generally poleward, except
poleward of 50 60o latitude, where they become equatorward.
Why do the ‡uxes look like this? The eddy heat ‡uxes are relatively
straightforward. Recall our energetic argument that for baroclinic eddies
v 0 T 0 must be directed generally poleward; we saw that we expect the same to
be true for upward-propagating stationary Rossby waves. The momentum
‡uxes are a little less obvious. We can actually understand them in simple
terms, by invoking some of the principles we discussed earlier in the class,
including our …nding that the EP ‡ux,
u0 v 0 ; f
F = (Fy ; Fz ) =
v0T 0
S
;
(5)
indicates the direction of wave activity propagation. As just noted, we expect
the EP ‡uxes to be generally upward (v 0 T 0 poleward) for both types of eddy,
as is con…rmed by observations (Fig. 5). The ‡uxes are generally upward
except in summer, when the stationary wave ‡ux is actually downward, but
the ‡uxes are actually weak then (note the arrow scale on the upper right of
the …gure) and so are of little consequence. Note, especially for the transients,
the ‡uxes “splay out”in the upper troposphere, producing a predominantly
equatorward component, but poleward in higher latitudes. Through (5), we
can see that this is consistent with the behavior of the momentum ‡ux.
To understand the gross aspects of the ‡uxes, consider Fig. 6. On a
plane with a spatially homogeneous baroclinic zone [Fig. 6(a)] the eddy heat
‡ux is poleward whence Fz > 0: there is a source of wave activity at the
lower boundary, where the surface temperature gradient allows the stability
criterion to be violated. But spatial homogeneity requires Fy = 0 whence
u0 v 0 = 0 everywhere in this case4 . Suppose now, however [Fig. 6(b)], that
the baroclinic zone is localized in latitude. Then the source of wave activity
is similarly localized; the EP ‡uxes point upward (v 0 T 0 poleward) out of the
4
Hence, e.g., the momentum ‡ux is zero in the classic Eady problem.
5
Figure 3: Stationary and eddy heat ‡uxes (annual and zonal means). [Peixoto
& Oort]
6
Figure 4: Eddy angular momentum ‡uxes for (b) stationary waves and (c)
transient eddies. [Peixoto & Oort]
7
8
Figure 5: Northern hemisphere EP ‡uxes for transient and stationary waves.
[Peixoto & Oort]
TROPOPAUSE
(a)
v'T' > 0
z
u' v' = 0
SURFACE
(b)
u' v' < 0
u' v' > 0
v'T' > 0
POLE
EQ
(c)
u' v' < 0
u' v' > 0
v'T' > 0
POLE
EQ
Figure 6: Fluxes associated with baroclinic eddies (schematic). Arrows show
EP ‡ux vector, ‡uxes of momenum and heat represented by light and dark
shading, repectively. (a) homogeneous case, (b) localized baroclinic zone on
a -plane, (c) localized baroclinic zone on a sphere.
9
surface baroclinic zone, but spread out laterally away from the source. Thus,
Fy points away from the baroclinic zone (and thus away from the jet), whence
the eddy momentum ‡ux u0 v 0 = Fy , on both sides, is directed into the jet.
From the cautionary note above, we cannot immediately conclude from this
that the eddies strengthen the jet; we’ll see below that it does mean.
On a sphere, we saw that for simple geometric reasons there is a bias
toward equatorward propagation. This means [Fig. 6(c)] that , compared to
case (b), the region of equatorward EP ‡uxes and poleward eddy momentum ‡ux is enhanced, while that of poleward EP ‡ux and equatorward eddy
momentum ‡ux is weakened. The scenario depicted in Fig. 6(c) is indeed
similar to what is observed, both for EP ‡uxes, and for heat and momentum
‡uxes.
What does the eddy transport do? In steady state (such as we expect to
encounter near the solstices) the momentum budget is straightforward:
fv =
1@
@z
@
u0 v 0 :
@y
(6)
If we integrate this in the vertical, given that mass balance demands no net
northward mass ‡ux, i.e.
Z 1
v dz = 0 ;
0
and the impossibility of any frictional stress at in…nity, we …nd that the
surface stress must be balanced by the eddy momentum transport:
Z 1
@
u0 v 0 dz :
(7)
s =
@y 0
This column balance, illustrated in Fig. 7.2(a), simply states that the only
contributions to the column are from eddy momentum ‡uxes and surface
stress. The mean circulation makes no contribution because the absence of a
net mass ‡ux means no net advection of planetary angular momentum, and
because the QG assumption renders advection of relative angular momentum
negligible here5 . Hence, in the vicinity of the generation of eddy activity—
the midlatitude baroclinic zone— where the momentum ‡uxes are convergent
in the upper troposphere, the most direct impact is a positive surface stress,
and hence surface westerlies.
5
Unlike in the tropical Hadley cell.
10
(a)
u' v'
W
τs
(b)
(u'v')y balanced by fv
WARMING COOLING
EQ
POLE
fv balanced by surface stress
(c)
COOLING
EQ
WARMING
v'T'
11
POLE
If we go beyond the vertical integral, if we make the reasonable assumption that frictional stresses are negligible outside the boundary layer, then in
the free atmosphere (6) tells us that
fv =
@
u0 v 0
@y
(8)
so the region of momentum ‡ux convergence in the upper troposphere must
be balanced by mean equatorward ‡ow, as depicted in Fig. 7.2(b). Applying
continuity, this must become vertical motion at the edges of the region; which
way does it go. We could integrate the continuity equation upward or
downward to get w, but going downward involves the boundary layer, where
(8) is not valid. If instead we integrate from in…nity and apply the constraint
that w ! 0 at in…nity, we have
Z 1
@
w(z) =
v dz
@y z
So the vertical motion must be as shown, upward in low latitudes and downward at higher latitudes. The return ‡ow then occurs in the boundary layer,
where the Coriolis term is balanced by surface stress. Thus, we see that
the thermally indirect Ferrel cell must be driven by eddy momentum ‡uxes:
heat ‡uxes have not entered the argument.
There is a thermal impact of the momentum ‡uxes, however: adiabatic
warming/cooling associated with the Ferrel cell will produce warming in low
latitudes and cooling in high latitudes! But this is where the eddy heat
‡uxes come into play, as depicted in Fig. 7.2(c). The poleward transport of
heat represented by the eddy ‡ux v 0 T 0 dominates over the Ferrel cell e¤ect
(as it must, in a gross sense) and the net e¤ect of the eddies is to reduce
the pole-to-equator temperature gradient overall, and correspondingly to reduce the baroclinic shear of the mean ‡ow. How the strength of the upper
tropospheric jet is impacted depends on the competing e¤ects of barotropic
acceleration/baroclinic deceleration, which depends on many factors. [These
issues were addressed by Robinson, J. Atmos. Sci., 51, 2553 (1994).]
12