Geography Reference
In-Depth Information
eddy and mean flow compensation occurs in the Eulerian mean continuity equation
for a long-lived tracer so that tracer transport calculations are also inefficient in
this formulation.
10.2.2
The Transformed Eulerian Mean (TEM)
An alternative approach to analysis of the zonal-mean circulation, which provides a
clearer diagnosis of eddy forcing and also provides a more direct view of transport
processes in the meridional plane, is the
transformed Eulerian mean
formulation
introduced by Andrews and McIntyre (1976). This transformation takes account
of the fact that in (10.12) there tends to be a strong cancellation between the eddy
heat flux convergence and adiabatic cooling, while the diabatic heating term is a
small residual. Since in the mean an air parcel will rise to a higher equilibrium
altitude only if its potential temperature is increased by diabatic heating, it is the
r
esidual meridional circulation
associated with diabatic processes that is directly
related to the mean meridional mass flow.
The TEM equatio
ns
c
an
be obtained from (10.11) and (10.12) by defining the
residual circulation
v
∗
, w
∗
as follows:
RH
−
1
∂
ρ
0
v
T
N
2
∂z
v
∗
=
ρ
−
1
0
v
−
(10.16a)
RH
−
1
∂
v
T
N
2
∂y
w
∗
=
w
+
(10.16b)
The residual vertical velocity defined in this manner clearly represents that part of
the mean vertical velocity whose contribution to adiabatic temperature change is
not canceled by the eddy heat flux divergence.
Substituting from (10.16) into (10.11) and (10.12) to eliminate (v, w) yields the
TEM equations
∂u
∂t
f
0
v
∗
=+
ρ
−
1
0
−
∇·
F
+
X
≡
G
(10.17)
∂T
∂t
J
c
p
N
2
HR
−
1
w
∗
=
+
(10.18)
∂v
∗
∂y
∂
ρ
0
w
∗
∂z
ρ
−
1
0
+
=
0
(10.19)
where
F
k
F
z
, the
Eliassen-Palm flux
(EP flux), is a vector in the meridional
(y, z) plane, which for large-scale quasi-geostrophic eddies has the components
≡
j
F
y
+
ρ
0
f
0
Rv
T
N
2
H
ρ
0
u
v
,
F
y
=−
z
=
(10.20)
and G designates the total zonal force due to both large- and small-scale eddies.
The TEM formulation clearly shows that the eddy heat and momentum fluxes
do not act separately to drive changes in the zonal-mean circulation, but only in
