Geography Reference
In-Depth Information
This condition is measured by a quantity called the
flux Richardson number
.Itis
defined as
Rf
≡−
BP L/MP
If the boundary layer is statically unstable, then Rf < 0 and turbulence is sus-
tained by convection. For stable conditions, Rf will be greater than zero. Observa-
tions suggest that only when Rf is less than about 0.25 (i.e., mechanical production
exceeds buoyancy damping by a factor of 4) is the mechanical production intense
enough to sustain turbulence in a stable layer. Since
MP
depends on the shear, it
always becomes large close enough to the surface. However, as the static stability
increases, the depth of the layer in which there is net production of turbulence
shrinks. Thus, when there is a strong temperature inversion, such as produced by
nocturnal radiative cooling, the boundary layer depth may be only a few decame-
ters, and vertical mixing is strongly suppressed.
5.3
PLANETARY BOUNDARY LAYER MOMENTUM EQUATIONS
For the special case of horizontally homogeneous turbulence above the viscous
sublayer, molecular viscosity and horizontal turbulent momentum flux divergence
terms can be neglected. The mean flow horizontal momentum equations (5.9) and
(5.10) then become
D
∂u
w
∂z
u
Dt
=−
¯
1
ρ
0
∂
p
∂x
+
¯
f
v
¯
−
(5.16)
D
∂v
w
∂z
v
Dt
=−
¯
1
ρ
0
∂
p
∂y
−
¯
f
u
¯
−
(5.17)
In general, (5.16) and (5.17) can only be solved for u and v if the vertical distribution
of the turbulent momentum flux is known. Because this depends on the structure
of the turbulence, no general solution is possible. Rather, a number of approximate
semiempirical methods are used.
For midlatitude synoptic-scale motions, Section 2.4 showed that to a first approx-
imation the inertial acceleration terms [the terms on the left in (5.16) and (5.17)]
could be neglected compared to the Coriolis force and pressure gradient force
terms. Outside the boundary layer, the resulting approximation was then simply
geostrophic balance. In the boundary layer the inertial terms are still small com-
pared to the Coriolis force and pressure gradient force terms, but the turbulent
flux terms must be included. Thus, to a first approximation, planetary boundary
layer equations express a three-way balance among the Coriolis force, the pressure
gradient force, and the turbulent momentum flux divergence:
f
¯
v
g
−
∂u
w
∂z
v
−¯
=
0
(5.18)
