Geoscience Reference
In-Depth Information
This duality in the solution has large practical implications in the modelling of
turbulent luxes under stable conditions (van de Wiel et al.,
2007
; Basu et al.,
2008
).
As long as the demand for energy, imposed by surface cooling
Q*
-
G
, is smaller (in
magnitude) than the possible supply of energy by turbulent transport (set by a ixed
wind shear) two equilibria are possible: one with a small vertical temperature differ-
ence (to the left of the heat lux minimum in
Figure 3.21
) and one with a large vertical
temperature difference. However, if the energy demand exceeds the possible turbu-
lent heat lux, no equilibrium can be reached. Cooling of the surface will proceed (as
Q* - G - H
is not equal to zero) and runaway cooling will occur (van de Wiel et al.,
2012b
). As stability increases (the conditions move towards the right in
Figure 3.21
),
turbulence will be suppressed eventually.
Summarizing: if for a given shear the energy demand is larger than the maximum
magnitude of downward lux that can be delivered by turbulence, there is no equilib-
rium possible and runaway cooling will occur. This cooling will continue until radia-
tive equilibrium is reached (
Q* = G
) or may halt when the shear increases such that
turbulence can sustain a larger (in magnitude) heat lux.
Question 3.28:
Make a graph similar to
Figure 3.21
, but now for the momentum trans-
port (i.e.,
u
2
) as a function of the vertical temperature difference, at a given, ixed ver-
tical wind speed difference. Explain the shape of this graph (as compared to that of
Figure 3.21
).
3.6.5 The Schmidt Paradox
In some applications one is interested in daily averaged luxes, rather than in instan-
taneous or hourly averaged luxes. It would then be tempting to apply the similarity
relationships developed in this chapter using 24-hour averaged input data (tempera-
tures, wind speed, etc.) to determine the 24-hour averaged luxes. For this approach
to be valid, it would be required at least that the sign of the 24-hour averaged lux
would be consistent with the 24-hour averaged vertical gradient (e.g., if the mean lux
is positive, the mean temperature should decrease with height).
In
Table 3.4
we investigate this requirement using the data from the same day as
those used in
Figure 3.3
. We see that whereas the mean temperature proile is stably
stratiied, the mean heat lux is positive: the mean lux goes
against
the mean gra-
dient. This feature has long been recognized and is known as the Schmidt paradox
(Schmidt,
1921
; see also Lettau,
1979
).
To illustrate this further, the 24-hour average turbulent diffusivity has been cal-
culated from the observed sensible heat lux and the observed vertical temperature
difference in two different ways:
•
The mean diffusivity is calculated from the mean lux and the mean vertical temperature
difference. This yields a negative diffusivity, which is inconsistent with the concept that
a lux should low down the gradient.
Search WWH ::
Custom Search