J. Og´ee et al. Agricultural and Forest Meteorology 106 2001 173–186 179
3.5. Numerical implementation All temperature profiles are smoothed using cubic
splines Press et al., 1992. They are used to interpolate the temperature at any depth, determine null-points
and calculate temperature gradients at the reference depth Fig. 4. Linear interpolation is used for humi-
dity profiles because it was found that applying cubic splines to such profiles with few measurement depths
can lead to unrealistic shapes.
Integrals in Eq. 8 first step and Eq. 6 sec- ond step are estimated using the trapezoidal rule.
To avoid errors in the computation of the integral in Eq. 8, we also consider temperature profiles for
which null-points are at 2 cm or more above the refer- ence level z
r
. Hence, only 701 case A: z
r
= 10 cm,
1187 case B: z
r
= 15 cm and 1479 case C: z
r
= 20 cm of the 17,400 or so temperature profiles fulfil
the conditions 2 cm ≤ z ≤ z
r
− 2 cm and ∂T ∂z|
z
r
≥ 4 K m
− 1
. Among them, only 458 case A, 773 case B and 1085 case C profiles had been recorded
during days when soil humidity was simultaneously measured. These profiles covered 237 case A, 250
case B and 219 case C days, so that we had about 2 case A, 3 case B and 5 case C profiles per day on
average. Case C then had better statistics but covered fewer days than the other two cases. Case B covered a
larger number of days with fewer profiles per day. The next section presents a comparison of the three cases.
4. Experimental results
4.1. Thermal conductivity versus soil volumetric moisture at three reference depths
The experimental λ–θ plots for the whole year and the three reference depths are presented in Fig. 5.
Only mean daily values are plotted. The differences observed between the three sets of values reflect the
spatial variations in bulk density that can be seen in Fig. 2.
These values are also compared with the model of de Vries 1963. To use it we assumed a quartz frac-
tion of 95, as suggested by Arrouays personal com- munication, and a mean temperature of 285 K. The
matric potential versus soil moisture curve required by the model was taken from Clapp and Hornberger
1978 for our type of soil. The form factor g
a
was parameterised as in the original model linear increase
with moisture. Tests were also performed by adding a slight concavity in the g
a
–θ relationship as suggested by de Vries 1963. Because this made negligible
improvement, no adjustment was made between the experimental data and the theoretical model. The vari-
ations in bulk density allow the model to reproduce well the observed differences in thermal conductivity.
The good agreement shows in particular that the selection made on the temperature profiles is correct.
According to de Vries 1963, the model gives a typi- cal uncertainty of 5–10, depending on the water
status. Hence, the results are within the expected ex- perimental error. In what follows the soil heat flux is
therefore computed using this model for λ, with the parameters described above.
4.2. A comparison of soil heat fluxes using the three reference depths
Before choosing a reference level, it is desirable to compare the soil heat fluxes computed with various
reference levels. A comparison is shown in Fig. 6. The computed values turn out to be similar over the
whole range of soil moisture so that for the rest of the study only soil heat fluxes computed with z
r
= 10 cm are used, since this depth requires the smallest
number of sensors. This result shows that an accurate computation of soil heat flux can be performed from
temperature probes located in a shallow soil layer.
4.3. Half-hourly values 4.3.1. Annual variations of the daily cycle
All 17,400 or so 30 min temperature profiles are kept to compute the soil heat flux; for days when the
soil moisture profiles were not recorded, linear inter- polation is performed between the preceding and fol-
lowing values. Fig. 7a shows that the soil heat flux has a daily amplitude from about 25 W m
− 2
in winter to more than 60 W m
− 2
in early summer, which rep- resents between 20 and 50 of the amplitude of the
transmitted net radiation as can be seen in Fig. 7b. In this figure, the amplitude of G has been normalised by
that of R
n,t
to suppress the seasonal trend: these vari- ations in amplitude then appear well correlated with
the variations in soil moisture.
180 J. Og´ee et al. Agricultural and Forest Meteorology 106 2001 173–186
Fig. 5. Soil thermal conductivity, λz
r
, vs. soil volumetric moisture, θ z
r
, at three different depths, z
r
, 10, 15 and 20 cm: experimental daily means closed circles and modelled curves solid lines using the model of de Vries 1963.
4.3.2. Energy budget The energy budget closure is a good test to check
whether or not flux measurements are correct. The soil is the most important sink or source of energy, as
compared with the other components, trunk, litter and air McCaughey and Saxton, 1988. In March 1998
the turbulent fluxes were measured just above the un- derstorey in addition to all the measurements already
J. Og´ee et al. Agricultural and Forest Meteorology 106 2001 173–186 181
Fig. 6. A comparison of soil heat fluxes computed with z
r
= 10
and z
r
= 20 cm for the full year of study N = 17,378 points. The
solid line is the linear regression line R
2
= 0.99, slope = 1.01,
intercept = 0.57.
performed above the canopy. It was therefore possi- ble to evaluate the energy budget of each vegetation
layer. The results are presented in a companion paper Lamaud et al., 2000 which shows that the storage
terms play an important role in the regulation of the energy cycle, especially in the understorey, and that
the soil heat flux values obtained in the present work strongly improve the closure of the understorey en-
ergy budget. Indeed, soil heat flux represents between 30 and 50 of transmitted net radiation at midday and
up to 80 at night-time.
4.4. Daily values At a time scale of the order of one day or more, the
soil heat flux is usually considered negligible on the basis that the heat stored during the day is released
during the night. Indeed the daily sums of soil heat flux range between –1 and 1.5 MJ m
− 2
during the pe- riod of study. Annual variations of these daily sums
do not seem correlated with changes in soil moisture not shown here, as is the case for daily amplitudes.
However, they are linearly related to the daily sums of transmitted net radiation Fig. 8. The negative value
of the zero-intercept cannot be entirely attributed to an overestimation of the sky longwave radiation by
Fig. 7. a Maximum and minimum values of soil heat flux through- out the year of study: daily values thin line and 30-days moving
averages thick line. b Ratio of the daily amplitudes of soil heat flux, 1G, and transmitted net radiation, 1R
n,t
, vs. near surface soil moisture content.
the Q-7 net-radiometer as discussed in Berbigier et al. 2000 and probably has a physical origin.
4.5. Monthly values Monthly values of soil heat flux are presented in
Fig. 9a. It can be seen that for long-term energy bud- get one to several months, neglecting soil heat flux
would generate an imbalance between turbulent fluxes and net radiation. From September 97 to March 98, the
cumulative value of transmitted net radiation reaches about 140 MJ m
− 2
Fig. 9b. During the same period, the cumulative value of soil heat flux reaches a min-
imum of −70 MJ m
− 2
, i.e. 50 of the energy budget of the understorey. Neglecting this term in long-term
environmental studies is therefore likely to produce incorrect results.
182 J. Og´ee et al. Agricultural and Forest Meteorology 106 2001 173–186
Fig. 8. Daily sums of soil heat flux vs. daily sums of transmitted net radiation. The solid line is the linear regression line R
2
= 0.59,
slope = 0.27, intercept = −0.47.
5. Modelling soil heat flux
