Geography Reference
In-Depth Information
where
V
j
v is the
horizontal
velocity vector. In order to express (3.1) in
isobaric coordinate form, we transform the pressure gradient force using (1.20)
and (1.21) to obtain
=
i
u
+
D
V
Dt
+
f
k
×
V
=−
∇
p
(3.2)
where
∇
p
is the horizontal gradient operator applied with pressure held constant.
Because
p
is the independent vertical coordinate, we must expand the total
derivative as
D
Dt
≡
∂
∂t
+
Dx
Dt
∂
∂x
+
Dy
Dt
∂
∂y
+
Dp
Dt
∂
∂p
(3.3)
∂
∂t
+
∂
∂x
+
∂
∂y
+
∂
∂p
=
u
v
ω
Dp/Dt (usually called the “omega” vertical motion) is the pressure
change following the motion, which plays the same role in the isobaric coordinate
system that w
Here ω
≡
Dz/Dt plays in height coordinates.
From (3.2) we see that the isobaric coordinate form of the geostrophic relation-
ship is
≡
f
V
g
=
k
×
∇
p
(3.4)
One advantage of isobaric coordinates is easily seen by comparing (2.23) and
(3.4). In the latter equation, density does not appear. Thus, a given geopotential gra-
dient implies the same geostrophic wind at any height, whereas a given horizontal
pressure gradient implies different values of the geostrophic wind depending on
the density. Furthermore, if
f
is regarded as a constant, the horizontal divergence
of the geostrophic wind at constant pressure is zero:
∇
p
·
V
g
=
0
3.1.2 The Continuity Equation
It is possible to transform the continuity equation (2.31) from height coordinates
to pressure coordinates. However, it is simpler to directly derive the isobaric form
by considering a Lagrangian control volume δV
=
δx δy δz and applying the
hydrostatic equation δp
=−
ρgδz (note that δp < 0) to express the volume element
as δV
δxδyδp/(ρg). The mass of this fluid element, which is conserved
following the motion, is then δM
=−
=
ρδV
=−
δxδyδp/g. Thus,
δxδyδp
g
1
δM
D
Dt
(δM)
g
δxδyδp
D
Dt
=
=
0
