Geography Reference
In-Depth Information
Noting that the mean velocity fields satisfy the continuity equation (5.5), we can
rewrite (5.7), as
u
u
u
v
u
w
D
Du
Dt
=
u
Dt
+
¯
∂
∂x
∂
∂y
∂
∂z
+
+
(5.8)
where
D
Dt
=
∂
∂t
+
∂
∂x
+
∂
∂y
+
∂
∂z
u
v
w
is the rate of change following the mean motion.
The mean equations thus have the form
∂u
u
∂x
D
∂u
v
∂y
∂u
w
∂z
u
Dt
=−
¯
1
ρ
0
∂
p
∂x
+
¯
+
F
rx
f
v
¯
−
+
+
(5.9)
∂u
v
∂x
D
∂v
v
∂y
∂v
w
∂z
v
Dt
=−
¯
1
ρ
0
∂
p
∂y
−
¯
+
F
ry
f
u
¯
−
+
+
(5.10)
∂u
w
∂x
D
g
θ
w
Dt
=−
¯
1
ρ
0
∂
p
∂z
+
¯
∂v
w
∂y
∂w
w
∂z
+
F
rz
θ
0
−
+
+
(5.11)
∂u
θ
∂x
D θ
Dt
=−¯
∂v
θ
∂y
∂w
θ
∂z
w
dθ
0
dz
−
+
+
(5.12)
∂
u
∂x
+
¯
∂
v
∂y
+
¯
∂
w
∂z
=
¯
0
(5.13)
The various covariance
term
s in square brackets in (5.9)-(5.12) represent turbu-
lent fluxe
s. For
ex
amp
le, w
θ
is a vertical turbulent heat flux in kinematic form.
Similarly w
u
=
u
w
is a vertical turbulent flux of zonal momentum. For many
boundary layers the magnitudes of the turbulent flux divergence terms are of the
same order as the other terms in (5.9)-(5.12). In such cases, it is not possible to
neglect the turbulent flux terms even when only the mean flow is of direct interest.
Outside the boundary layer the turbulent fluxes are often sufficiently weak so that
the terms in square brackets in (5.9)-(5.12) can be neglected in the analysis of
large-scale flows. This assumption was implicitly made in Chapters 3 and 4.
The complete equations for the mean flow (5.9)-(5.13), unlike the equations for
the total flow (5.1)-(5.5), and the approximate equations of Chapter
s 3 and 4, ar
e
not a closed set, as in addition to the five unknown mean variables u, v, w, θ,p,
there are unknown turbulent fluxes. To solve these equations,
closure
assump-
tions must be made to approximate the unknown fluxes in terms of the five
known mean state variables. Away from regions with horizontal inhomogeneities
