Geography Reference
In-Depth Information
5.1.1
The Boussinesq Approximation
The basic state (standard atmosphere) density varies across the lowest kilometer of
the atmosphere by only about 10%, and the fluctuating component of density devi-
ates from the basic state by only a few percentage points. These circumstances
might suggest that boundary layer dynamics could be modeled by setting den-
sity constant and using the theory of homogeneous incompressible fluids. This is,
however, generally not the case. Density fluctuations cannot be totally neglected
because they are essential for representing the buoyancy force (see the discussion
in Section 2.7.3).
Nevertheless, it is still possible to make some important simplifications in
dynamical equations for application in the boundary layer. The
Boussinesq approx-
imation
is a form of the dynamical equations that is valid for this situation. In this
approximation, density is replaced by a constant mean value, ρ
0
, everywhere except
in the buoyancy term in the vertical momentum equation. The horizontal momen-
tum equations (2.24) and (2.25) can then be expressed in Cartesian coordinates as
Du
Dt
=−
1
ρ
0
∂p
∂x
+
fv
+
F
rx
(5.1)
and
Dv
Dt
=−
1
ρ
0
∂p
∂y
−
+
fu
F
ry
(5.2)
while, with the aid of (2.28) and (2.51) the vertical momentum equation becomes
Dw
Dt
=−
1
ρ
0
∂p
∂z
+
g
θ
θ
0
+
F
rz
(5.3)
Here, as in Section 2.7.3, θ designates the departure of potential temperature from
its basic state value θ
0
(z).
1
Thus, the total potential temperature field is given by
θ
tot
=
θ(x, y, z, t)
+
θ
0
(z), and the adiabatic thermodynamic energy equation has
the form of (2.54):
Dθ
Dt
=−
w
dθ
0
dz
(5.4)
Finally, the continuity equation (2.34) under the Boussinesq approximation is
∂u
∂x
+
∂v
∂y
+
∂w
∂z
=
0
(5.5)
1
The reason that we have not used the notation θ
to designate the fluctuating part of the potential
temperature field will become apparent in Section 5.1.2.
