Geography Reference
In-Depth Information
a positive zonal wind in the middle latitudes despite the momentum lost to the
surface.
It is convenient to analyze the momentum budget in terms of absolute angular
momentum. The absolute angular momentum per unit mass of atmosphere is
M
=
(a cos φ
+
u) a cos φ
where as before a is the radius of the earth. The crucial role of zonal eddy drag in
maintaining the observed latitudinal profile of the mean zonal wind can be seen
by evaluating the mean zonal velocity that would arise if a zonal ring of air at
rest on the equator were to be displaced poleward conserving M. In that case
u (φ)
a sin 2 φ cos φ , so that in an angular momentum-conserving Hadley
circulation, u
=
130 ms 1 at 30 latitude, which is far greater than is observed.
Clearly, the absolute angular momentum must decrease as air parcels are advected
poleward in the Hadley circulation. The absolute angular momentum of an indi-
vidual air parcel can be changed only by torques caused by the zonal pressure
gradient and eddy stresses. In isobaric coordinates, Newton's second law in its
angular momentum form is thus
a cos φ
g ∂τ E
∂p
DM
Dt =−
∂x +
(10.27)
where τ E is the zonal component of the vertical eddy stress, and it is assumed that
horizontal eddy stresses are negligible compared to the vertical eddy stress.
10.3.1
Sigma Coordinates
In neither the isobaric nor the log-pressure coordinate system does the lower bound-
ary exactly coincide with a coordinate surface. In analytical studies it is usual to
assume that the lower boundary can be approximated as a constant pressure surface
and to apply the approximate condition
≈−
ω (p s )
ρ 0 gw (z 0 )
as the lower boundary condition. Here we have assumed that the height of the
ground z 0 is coincident with the pressure surface p s (where p s is usually set equal
to 1000 hPa). These assumptions are of course not strictly valid even when the
ground is level. Pressure does change at the ground, but, more importantly, the
height of the ground generally varies so that even if the pressure tendency were
zero everywhere, the lower boundary condition should not be applied at a constant
p s . Rather, we should set p s =
p s (x, y). It is very inconvenient for mathematical
analysis, however, to have a boundary condition that must be applied at a surface
that is a function of the horizontal variables.
Search WWH ::




Custom Search