Geography Reference
In-Depth Information
After differentiating, using the chain rule, and changing the order of the differ-
ential operators we obtain
1
δx
δ
Dx
δy
δ
Dy
δp
δ
Dp
1
1
1
+
+
=
0
Dt
Dt
Dt
or
δu
δx
+
δv
δy
+
δω
δp
=
0
0 and observing that δx and δy are evaluated at
constant pressure, we obtain the continuity equation in the isobaric system:
∂u
∂x
+
Taking the limit δx, δy, δp
→
p
+
∂v
∂y
∂ω
∂p
=
0
(3.5)
This form of the continuity equation contains no reference to the density field
and does not involve time derivatives. The simplicity of (3.5) is one of the chief
advantages of the isobaric coordinate system.
3.1.3
The Thermodynamic Energy Equation
The first law of thermodynamics (2.42) can be expressed in the isobaric system by
letting Dp/Dt
=
ω and expanding DT /Dt by using (3.3):
c
p
∂T
u
∂T
v
∂T
ω
∂T
∂p
∂t
+
∂x
+
∂y
+
−
αω
=
J
This may be rewritten as
∂T
∂t
+
u
∂T
v
∂T
∂y
J
c
p
∂x
+
−
S
p
ω
=
(3.6)
where, with the aid of the equation of state and Poisson's equation (2.44), we have
RT
c
p
p
−
∂T
∂p
=−
T
θ
∂θ
∂p
S
p
≡
(3.7)
which is the static stability parameter for the isobaric system. Using (2.49) and the
hydrostatic equation, (3.7) may be rewritten as
S
p
=
(
d
−
) /ρg
Thus, S
p
is positive provided that the lapse rate is less than dry adiabatic.
However, because density decreases approximately exponentially with height, S
p
increases rapidly with height. This strong height dependence of the stability mea-
sure S
p
is a minor disadvantage of isobaric coordinates.
1
From now on
g
will be regarded as a constant.
