Geoscience Reference
In-Depth Information
side of Equation (20.2) by (
u
*
)
2
, rearranging, then multiplying the resulting
equation by
r
to give:
kz du
u
(
−
)
∂
*
tr
=
u
2
=
r
(20.21)
*
f
∂
z
M
Notice that the sign in this equation is different because momentum flux is defined
positive when downward toward the surface, but the fluxes of sensible and latent
heat are defined to be positive upward, away from the surface.
Resistance analogues and aerodynamic resistance
Diffusion equations are usually used to describe surface-atmosphere exchanges in
equations and numerical models. In the case of vertical flow by turbulent diffusion
of momentum and energy in the surface layer, the representation in terms of K
Theory is given using the equations:
u
∂
(20.22)
tr
=
K
.
∂
z
M
Hc K
q
∂
(20.23)
v
=−
ρ
.
∂
p
H
ρ
γ
c
∂
e
(20.24)
p
l
E
=−
K
.
∂
z
V
with the eddy diffusivities
K
M
, K
H
and
K
V
in Equations (20.22), (20.23) and (20.24)
specified by comparison with Equations (20.21), (20.17) and (20.20), respectively.
As discussed in more detail in the next chapter, very near the ground or very near
vegetation (and, in the case of latent heat, also inside leaves), flux transfer is largely
controlled by the molecular diffusion process. Molecular flow is described by
diffusion equations similar to those given above describing turbulent diffusion,
but the eddy diffusivities are replaced by molecular diffusivities
D
M
, D
H
and
D
V
,
which are properties of the air through which diffusion occurs.
When writing equations for building numerical models of surface exchange, it
is very common to write the turbulent and molecular diffusion equations that
describe flow in integrated form. Consider, for example, sensible heat flow by
turbulent diffusion in the vertical direction described by Equation (20.23) which
can be rearranged into the form:
(20.25)
H
1
cK
q
∂
=
V
−
r
∂
ap H
