Geography Reference
In-Depth Information
4.4.1
Cartesian Coordinate Form
For motions of synoptic scale, the vorticity equation can be derived using the
approximate horizontal momentum equations (2.24) and (2.25). We differentiate
the zonal component equation with respect to y and the meridional component
equation with respect to x:
∂u
∂t
+
∂
∂y
u
∂u
v
∂u
w
∂u
1
ρ
∂p
∂x
∂x
+
∂y
+
∂z
−
=−
fv
(4.14)
∂v
∂t
+
∂
∂x
u
∂v
v
∂v
w
∂v
1
ρ
∂p
∂y
∂x
+
∂y
+
∂z
+
fu
=−
(4.15)
Subtracting (4.14) from (4.15) and recalling that ζ
=
∂v/∂x
−
∂u/∂y, we obtain
the vorticity equation
f)
∂u
∂ζ
∂t
+
u
∂ζ
v
∂ζ
w
∂ζ
∂v
∂y
∂x
+
∂y
+
∂z
+
(ζ
+
∂x
+
=
∂w
∂x
∂ρ
∂x
∂v
∂z
−
∂w
∂y
∂u
∂z
v
df
1
ρ
2
∂p
∂y
−
∂ρ
∂y
∂p
∂x
+
+
dy
=
(4.16)
Using the fact that the Coriolis parameter depends only on y so that Df/Dt
=
v(df/dy), (4.16) may be rewritten in the form
f)
∂u
D
Dt
(ζ
∂v
∂y
+
f)
=−
(ζ
+
∂x
+
∂w
∂x
∂ρ
∂x
∂v
∂z
−
∂w
∂y
∂u
∂z
1
ρ
2
∂p
∂y
−
∂ρ
∂y
∂p
∂x
−
+
(4.17)
Equation (4.17) states that the rate of change of the absolute vorticity following
the motion is given by the sum of the three terms on the right, called the divergence
term, the tilting or twisting term, and the solenoidal term, respectively.
The concentration or dilution of vorticity by the divergence field [the first term
on the right in (4.17)] is the fluid analog of the change in angular velocity resulting
from a change in the moment of inertia of a solid body when angular momentum
is conserved. If the horizontal flow is divergent, the area enclosed by a chain
of fluid parcels will increase with time and if circulation is to be conserved, the
average absolute vorticity of the enclosed fluid must decrease (i.e., the vorticity
will be diluted). If, however, the flow is convergent, the area enclosed by a chain
of fluid parcels will decrease with time and the vorticity will be concentrated.
This mechanism for changing vorticity following the motion is very important in
synoptic-scale disturbances.
The second term on the right in (4.17) represents vertical vorticity generated by
the tilting of horizontally oriented components of vorticity into the vertical by a
