Surface Fluxes of Momentum, Heat, and Water Vapor
67
3.1.4 Temperature variance
The mean dissipation rate of the temperature variance (co) is related to the vertical temperature
flux « wO > ) through the temperature variance budget equation, (actually the budget for
! < 0 2 », which for steady, horizontally homogenous flow is written
o ae 1 a < w0 2 >
- < w > az - 2
az
= co
(3.13)
The first term is the average rate of production of temperature variance by interaction of the
vertical heat flux with the vertical gradient of mean potential temperature. The second term
represents the flux divergence of the variance. Non-dimensionalizing (3.13) by u.e~/kz yields
(3.14)
where Tr (= ~a<~~2» is the dimensional transport, iPH is the dimensionless production of
temperature variance from (3.5), and iP •• (= ~0~ kz) represents the dimensionless dissipation rate
.u.
of temperature variance. Hogstrom (1990) found measurements of the transport term to contain
large scatter, as with WC71, but found no systematic deviation from zero.
The classic Businger-Dyer empirical formula for the normalized production in the near neutral
and unstable region is (Businger, 1966; Dyer, 1967)
for
0< -::.. < 2
L
(3.15)
although the empirical constants are subject to some uncertainty and varying interpretation.
1
However, several recent studies have found that iPH scales convectively (i.e. ex: (-z/Lf a) for
-z/ L » 0 (e.g. Kader and Perepelkin, 1984). An important point is that under convective
scaling, iP.. is also proportional to (- z / L t ~, and hence co is independent of surface shear
stress. This simplifies greatly the calculation of H from co.
Following the approach described above for the three sublayer model we obtain the following
scaling form for the temperature variance dissipation
iP ••
iPH = B1 ~ 1
z
-"L < 0.04
(3.16a)
iP ••
B2 (-7)-~
z
0.12 < -"L < 1.2
(3.16b)
iP ••
( zr
L
B3 -"L •
z
-"L >2.
(3.16c)
Note that the production and dissipation rates scale with (-z/ L )-1/3 for all but the most neutral
region of the convective boundary layer, and not with (-z/ L )-1/2 as was suggested in earlier
research (e.g. WC71). This -1/3 scaling simplifies greatly the process by which heat fluxes are
computed from inertial subrange measurements of scalar dissipation rates. The constants in
(3.16) are determined below from the results of recent experiments (Kiely et aI., 1996).
67
3.1.4 Temperature variance
The mean dissipation rate of the temperature variance (co) is related to the vertical temperature
flux « wO > ) through the temperature variance budget equation, (actually the budget for
! < 0 2 », which for steady, horizontally homogenous flow is written
o ae 1 a < w0 2 >
- < w > az - 2
az
= co
(3.13)
The first term is the average rate of production of temperature variance by interaction of the
vertical heat flux with the vertical gradient of mean potential temperature. The second term
represents the flux divergence of the variance. Non-dimensionalizing (3.13) by u.e~/kz yields
(3.14)
where Tr (= ~a<~~2» is the dimensional transport, iPH is the dimensionless production of
temperature variance from (3.5), and iP •• (= ~0~ kz) represents the dimensionless dissipation rate
.u.
of temperature variance. Hogstrom (1990) found measurements of the transport term to contain
large scatter, as with WC71, but found no systematic deviation from zero.
The classic Businger-Dyer empirical formula for the normalized production in the near neutral
and unstable region is (Businger, 1966; Dyer, 1967)
for
0< -::.. < 2
L
(3.15)
although the empirical constants are subject to some uncertainty and varying interpretation.
1
However, several recent studies have found that iPH scales convectively (i.e. ex: (-z/Lf a) for
-z/ L » 0 (e.g. Kader and Perepelkin, 1984). An important point is that under convective
scaling, iP.. is also proportional to (- z / L t ~, and hence co is independent of surface shear
stress. This simplifies greatly the calculation of H from co.
Following the approach described above for the three sublayer model we obtain the following
scaling form for the temperature variance dissipation
iP ••
iPH = B1 ~ 1
z
-"L < 0.04
(3.16a)
iP ••
B2 (-7)-~
z
0.12 < -"L < 1.2
(3.16b)
iP ••
( zr
L
B3 -"L •
z
-"L >2.
(3.16c)
Note that the production and dissipation rates scale with (-z/ L )-1/3 for all but the most neutral
region of the convective boundary layer, and not with (-z/ L )-1/2 as was suggested in earlier
research (e.g. WC71). This -1/3 scaling simplifies greatly the process by which heat fluxes are
computed from inertial subrange measurements of scalar dissipation rates. The constants in
(3.16) are determined below from the results of recent experiments (Kiely et aI., 1996).