Geography Reference
In-Depth Information
4.6.1
Equations of Motion in Isentropic Coordinates
If the atmosphere is stably stratified so that potential temperature θ is a mono-
tonically increasing function of height, θ may be used as an independent vertical
coordinate. The vertical “velocity” in this coordinate system is just
θ
Dθ/Dt .
Thus, adiabatic motions are two dimensional when viewed in an isentropic coordi-
nate frame. An infinitesimal control volume in isentropic coordinates with cross-
sectional area δA and vertical extent δθ has a mass
≡
δA
−
δθ
δp
g
δA
g
∂p
∂θ
=
=
−
=
=
δM
ρδAδz
σ δAδθ
(4.29)
Here the “density” in (x, y, θ) space (i.e., as shown in Fig. 4.7 the quantity that
when multiplied by the “volume” element δAδθ yields the mass element δM)is
defined as
g
−
1
∂p/∂θ
σ
≡−
(4.30)
The horizontal momentum equation in isentropic coordinates may be obtained by
transforming the isobaric form (4.19) to yield
∂t
+
∇
θ
V
∂
V
·
V
=−
θ
∂
V
+
+
(ζ
θ
+
f )
k
×
V
∂θ
+
F
r
(4.31)
2
where
∇
θ
is the gradient on an isentropic surface, ζ
θ
≡
k
·∇
θ
×
V
is the isen-
tropic relative vorticity originally introduced in (4.11), and
is the
Montgomery streamfunction (see Problem 10 in Chapter 2). We have included a
frictional term
F
r
on the right-hand side, along with the diabatic vertical advection
term. The continuity equation can be derived with the aid of (4.29) in a manner
analogous to that used for the isobaric system in Section 3.1.2. The result is
≡
c
p
T
+
∂θ
σ θ
∂σ
∂t
+
∇
θ
·
∂
=−
(σ
V
)
(4.32)
The and σ fields are linked through the pressure field by the hydrostatic equation,
which in the isentropic system takes the form
c
p
p
p
s
R/c
p
∂
∂θ
=
T
θ
(p)
≡
=
c
p
(4.33)
where is called the
Exner function
. Equations (4.30)-(4.33) form a closed set
for prediction of
V
, σ , , and p, provided that
θ and
F
r
are known.
