Geography Reference
In-Depth Information
The along-front velocity is in geostrophic balance with the cross-front pressure
gradient with error of order 1%, but geostrophy does not hold even approximately
for the cross-front velocity. Therefore, if the geostrophic wind components are
defined by
∂/∂x
and we separate the horizontal velocity field into geostrophic and ageostrophic
parts, to a good approximation u
fu
g
=−
∂/∂y,
fv
g
=
=
u
g
,butv
=
v
g
+
v
a
, where v
g
and v
a
are the
same order of magnitude.
The x component of the horizontal momentum equation (9.2), the thermody-
namic energy equation (9.4), and the continuity equation (9.6) for frontal scaling
can thus be expressed as
Du
g
Dt
−
fv
a
=
0
(9.7)
Db
Dt
+
wN
2
=
0
(9.8)
∂v
a
∂y
+
∂w
∂z
=
0
(9.9)
Here, (9.8) is obtained by using (9.5) to replace by b in (9.4), and N is the
buoyancy frequency defined in terms of potential temperature as
g
θ
00
∂θ
0
∂z
N
2
≡
Because the along-front velocity is in geostrophic balance, u
g
and b are related by
the thermal wind relationship:
f
∂u
g
∂b
∂y
∂z
=−
(9.10)
Note that (9.7) and (9.8) differ from their quasi-geostrophic analogues; although
zonal momentum is still approximated geostrophically, and advection parallel to
the front is geostrophic, the advection of momentum and temperature across the
front is due not only to the geostrophic wind, but to the ageostrophic (v
a
,w)
circulation:
v
a
D
Dt
=
D
g
Dt
+
∂
∂y
+
∂
∂z
w
where D
g
/Dt was defined in (6.8). Replacement of momentum by its geostrophic
value in (9.7) is referred to as the
geostrophic momentum
approximation, and the
resulting set of prediction equations are called the
semigeostrophic
equations.
1
1
Some authors reserve this name for a version of the equations written in a transformed set of
coordinates called geostrophic coordinates (e.g., Hoskins, 1975).
