Instruction manual

A-1
APPENDIX A. USING A KRYPTON HYGROMETER TO MAKE
WATER VAPOR MEASUREMENTS
A.1 WATER VAPOR FLUXES
The krypton lamp used in the hygrometer emits
a major line at 123.58 nm (line 1) and a minor
line at 116.49 nm (line 2). Both of these
wavelengths are absorbed by water vapor and
oxygen. The equation below describes the
hygrometer signal in terms of absorption of both
lines by water vapor and oxygen.
()
V V xk xk
ho wv oo
=−−
111
exp
ρρ
()
+−−
Vxkxk
owvoo222
exp
ρρ
(1)
where V
h
is the signal voltage from the
hygrometer, V
o1
and V
o2
are the signals with no
absorption of lines 1 and 2 respectively, x is the
path length of the hygrometer, k
w1
and k
w2
are
the absorption coefficients for water vapor on
lines 1 and 2, k
o1
and k
o2
are the absorption
coefficients for oxygen, and
ρ
v
and
ρ
o
are the
densities of water vapor and oxygen.
If V
o1
>> V
o2
and k
w1
∼
k
w2
, Eq. (1) can be
rewritten by approximating the individual
absorption of the two lines with a single
effective coefficient for either water vapor or
oxygen.
()()
[
V V xk xk
ho wv oo
=− −
11
exp exp
ρρ
()()
]
+−
VV xk
oo oo21 2
exp
ρ
(2)
Note that the quantity
()
VV
oo21
0
→
, thus the
above takes on the form below.
()()
V V xk xk
ho wv oo
=− −
exp exp
ρρ
(3)
Taking the natural log of Eq. (3) and solving for
ρ
v
yields Eq. (4).
ρρ
v
h
w
o
w
o
w
o
InV
xk
InV
xk
k
k
=
−
−
−
+
−
(4)
Applying the rules of Reynolds averaging, the
covariance between the vertical wind speed and
water vapor can be written as Eq. (5).
′′
=−
www
vv v
ρρρ
(5)
Substituting Eq. (4) into (5) yields the equation
below. Note that lnV
o
is a constant.
()
′′
=
−
−
w
w InV w InV
xk
v
h
h
w
ρ
()
+
−
−
k
k
ww
o
w
oo
ρρ
(6)
The first term in Eq. (6) is the water vapor flux
and second is the oxygen correction. The
density of oxygen is not directly measured. It
can, however, be written in terms of measured
variables using the ideal gas law. The density
of oxygen is given by Eq. (7) below.
ρ
o
oo
PC M
RT
=
(7)
where P is atmospheric pressure, T is air
temperature, C
o
is the concentration of oxygen,
M
o
is the molecular weight of oxygen, and R is
the universal gas constant. Substituting Eq. (7)
into Eq. (6) gives the equation below.
()
′′
=
−
−
w
w InV w InV
xk
v
h
h
w
ρ
+
−
−






−−
k
k
CMP
R
wT w T
o
w
oo
11
(8)
Using a relationship analogous to Eq. (5), the
numerator in the first term and the term within
the brackets of Eq. (8) can be rewritten. Note
that the atmospheric pressure over a typical flux
averaging period is constant, thus pressure can
be treated as a constant. Finally, the latent heat
flux can be written as follows.