Available Energy: Net Radiation and Soil Heat Flux - Transport in the Atmosphere-Vegetation-Soil Continuum - page 67

The method underlying Eq. ( 2.51 ) was originally developed for the prediction of
the growth of ice on open water (Stefan, 1889 ). The long-term development of the
thickness of ice sheets indeed shows a development similar to Eq. ( 2.51 ): a decrease
of the growth rate with time due to the increased dificulty to remove the heat released
on freezing towards the atmosphere when the ice gets thicker. However, to remove the
need for empirical constants (like the a used in Eq. ( 2.51 )) and to allow for day-to-day
variations of meteorological conditions, the correct coupling between the atmosphere
and the ice surface is essential and hence more elaborate models are needed (see, e.g.,
DeBruin and Wessels, 1988 ; Ashton, 2011 ).

Question 2.27: Given the following observations of daily mean temperatures at the
Haarweg meteorological station (in ºC, starting on December 15, 2007):

Day 15 16 17 18 19 20 21 22 23

T 1.5 -0.8 -0.4 -1.2 -2.7 -3.8 -4.3 -3.0 1.8

Day 24 25 26 27 28 29 30 31

T 1.3 2.9 3.8 4.6 6.8 5.8 5.3 3.6

a) At which date did the maximum frost penetration occur?
b) What was the frost index at that date?
c) Estimate the depth of the frost penetration (assume a typical value of the empirical
constant a in Eq. ( 2.51 )).
d) On which date had frost disappeared again from the soil?

2.4 Summary
The energy that is available for transport of heat and water vapour into the atmosphere
is equal to the net supply of radiative energy, diminished with the transport of heat
into the soil (when storage terms, etc. are omitted).
Radiation exchange at Earth's surface can be decomposed based on the origin (the
Sun or the atmosphere or surface) and the direction of the radiation (upward or down-
ward), leading to four composing terms. Downwelling shortwave radiation is highly
affected by the geometry of the solar beam relative to the surface. This depends on
the geographical position, date and time, leading to seasonal and diurnal variation of
the radiation available at the top of the atmosphere. Subsequently, the composition
of the atmosphere and the presence of clouds modify this radiation on its way to the
surface: it leads to variations in the amount of radiation, the directional dependence
(direct versus diffuse) and spectral composition of the radiation. A large part of the
downwelling shortwave radiation is absorbed by the surface, but depending on the
type of surface (and to a lesser extent the solar angle) a certain fraction is relected.
Downwelling longwave radiation is emitted both by gases and by liquid (or solid)
water in the atmosphere. Emission by gases in the atmosphere closely follows the
spectral dependence of a black body, except for the wavelength region between 8 and