Campbell Scientific Bowen Ratio Instrumentation - Manual (Page 5)

1-1
Section 1. System Overview
1.1 Review of Theory
By analogy with molecular diffusion, the flux-gradient approach to vertical
transport of an entity from or to a surface assumes steady diffusion of the
entity along its mean vertical concentration gradient.
When working within a few meters of the surface, the water vapor and heat
flux densities, E and H, may be expressed as:
E k
z
v
v =
∂ρ
∂
(1)
H C k
T
z
p H
= ρ
∂
∂
(2)
Here
ρ
v
is vapor density,
ρ is air density, C
p
is the specific heat of air, T is
temperature, z is vertical height, and k
v
and k
H
are the eddy diffusivities for
vapor and heat, respectively. Air density and the specific heat of air should
account for the presence of water vapor, however, use of standard dry air
values usually causes negligible error. The eddy diffusivities are functions of
height. The vapor and temperature gradients reflect temporal and spatial
averages.
Applying the Universal Gas Law to Eq. (1), and using the latent heat of
vaporization,
λ, the latent heat flux density, λ, can be written in terms of vapor
pressure (e).
L
k
P
e
z
e
v =
λρε ∂
∂
(3)
Here P is atmospheric pressure and
ε is the ratio of the molecular weight of
water to the molecular weight of dry air.
In practice, finite gradients are measured and an effective eddy diffusivity
assumed over the vertical gradient:
( )
( ) L
k
P
e e
z z
e
v =
−
− λρε
1 2
1 2
(4)
( )
( ) H C k
T T
z z
p H
=
−
−
ρ 1 2
1 2
. (5)
In general, k
v
and k
H
are not known but under specific conditions are assumed
equal. The ratio of H to L
e
is then used to partition the available energy at the
surface into sensible and latent heat flux. This technique was first proposed by
Bowen (1926). The Bowen ratio,
β, is obtained from Eq. (4) and Eq. (5).