Geography Reference
In-Depth Information
Mean wind hodograph for Jacksonville, Florida (
=
Fig. 5.5
30˚ N), April 4, 1968 (solid line) com-
pared with the Ekman spiral (dashed line) and the modified Ekman spiral (dash-dot line)
computed with De
=
1200 m. Heights are shown in meters. (Adapted from Brown, 1970.
Reproduced with permission of the American Meteorological Society.)
and the geostrophic wind from that characteristic of the Ekman spiral. A typical
observed wind hodograph is shown in Fig. 5.5. Although the detailed structure is
rather different from the Ekman spiral, the vertically integrated horizontal mass
transport in the boundary layer is still directed toward lower pressure. As shown
in the next section, this fact is of primary importance for synoptic and larger scale
motions.
5.4
SECONDARY CIRCULATIONS AND SPIN DOWN
Both the mixed-layer solution (5.22) and the Ekman spiral solution (5.31) indicate
that in the planetary boundary layer the horizontal wind has a component directed
toward lower pressure. As suggested by Fig. 5.6, this implies mass convergence in
a cyclonic circulation and mass divergence in an anticyclonic circulation, which
by mass continuity requires vertical motion out of and into the boundary layer,
respectively. In order to estimate the magnitude of this induced vertical motion,
we note that if v
g
=
0 the cross isobaric mass transport per unit area at any level
in the boundary layer is given by ρ
0
v. The net mass transport for a column of unit
width extending vertically through the entire layer is simply the vertical integral
of ρ
0
v. For the mixed layer, this integral is simply ρ
0
vh(kg m
−
1
s
−
1
), where h is
the layer depth. For the Ekman spiral, it is given by
De
De
M
=
ρ
0
vdz
=
ρ
0
u
g
exp(
−
πz/De)sin(π z/De)dz
(5.35)
0
0
where De
π/γ is the Ekman layer depth defined in Section 5.3.4.
Integrating the mean continuity equation (5.13) through the depth of the bound-
ary layer gives
=
De
∂u
∂x
+
dz
∂v
∂y
w(De)
=−
(5.36)
0
