176 J. Og´ee et al. Agricultural and Forest Meteorology 106 2001 173–186
Fig. 3. Variation of mean canopy leaf area index during the ex- periment. Experimental data triangles and parameterisation used
to calculate transmitted net radiation solid line. The bars repre- sent the standard deviations calculated over a different number of
points at each date.
temperatures. We use the litter temperature at the air-litter interface for the ground and the air tempera-
ture at 14 m for canopy temperature. The transmitted net radiation is then computed with
R
n,t
= R
′ n,t
+ 1 − f
2
σ {T
4 a,14m
− T
4 a,l
} 2
where σ = 5.67 × 10
− 8
W m
− 2
K
− 4
, T
a,14m
is the air temperature at 14 m above ground, a surrogate for the
mean foliage temperature, T
a,l
is the air temperature at the air-litter interface, and f and R
′ n,t
are given by equations 9a and 11a of Berbigier and Bonnefond
1995, respectively. The mean canopy leaf area index was measured reg-
ularly by an optical method based on the interception of the solar beam Lang, 1987. We observed contin-
uous variations from about 2.8 in winter to almost 3.0 in summer Fig. 3. The standard deviations shown
in Fig. 3 are a consequence of the heterogeneity of the canopy rather than measurement uncertainties: at a
given time of the year, different days of measurement would provide almost the same means and standard
deviations.
3. Methodology
Several methods have been suggested to estimate the ground heat flux at the surface from tempera-
ture, humidity and bulk density data. A review can be found in Kimball and Jackson 1979. Most of
these methods require extra information such as the soil thermal conductivity at some depth, or a value of
soil heat flux as measured with a flux plate. The ad- vantage of the null-alignment method Kimball and
Jackson, 1975 is that no additional measurement is needed.
3.1. The null-alignment method Kimball and Jackson, 1975
We start with the continuity equation for heat Kondo and Saigusa, 1994:
C ∂T
∂t = −
∂Q
h
∂z −
LE
soil
3 In this equation t is time s, z the depth defined
positive downwards m, C the volumetric soil heat capacity J m
− 3
K
− 1
, L the latent heat of vaporisa- tion J kg
− 1
, T the local soil temperature K, Q
h
the downward soil heat flux W m
− 2
and E
soil
the evaporation source strength kg m
− 3
s
− 1
. Integration of Eq. 3 between z and a reference depth z
r
, defined later, gives
Q
h
z = Q
h
z
r
+ Z
z
r
z
C ∂T
∂t dz +
Z
z
r
z
LE
soil
dz 4
The null-alignment method is based on two assump- tions, whose validity is discussed further: 1 the soil
heat flux at the reference level is only transported by conduction and water vapour, which amounts to
neglecting the transport of heat by the mass flow of liquid water, and 2 no evaporation occurs in
the soil between the surface and the reference level. Assumption 1 leads to
Q
h
z
r
= −λz
r
∂T ∂z
z
r
5 where λz
r
is the effective soil thermal conductivity at the reference depth W m
− 1
K
− 1
. Assumption 2 allows us to neglect the last term on the right hand
side of Eq. 4. Hence, we have Q
h
z = −λz
r
∂T ∂z
z
r
+ Z
z
r
z
C ∂T
∂t dz
6
J. Og´ee et al. Agricultural and Forest Meteorology 106 2001 173–186 177
Soil heat capacity can be estimated from de Vries, 1963:
C = C
m
1 − ε + C
w
θ ≈
2 × 1 − ε + 4.18 × θ × 10
6
7 where C
m
≈ 2 and C
w
≈ 4.18 MJ m
− 3
K
− 1
are the heat capacities for minerals and water, respectively,
and θ is soil volumetric moisture. In Eqs. 7 and 1, the organic matter is included with the minerals
because it represents less than 5 in volume, even though its heat capacity and density are different
from those of the minerals C
o
≈ 2.5 MJ m
− 3
K
− 1
and ρ
o
≈ 1300 kg m
− 3
for organic matter. Hence, from temperature, humidity and bulk density profiles,
we are able to estimate all terms but the soil thermal conductivity at z
r
in Eq. 6. For this, the method developed by Kimball and Jackson 1975 makes use
of the fact that shortly after sunrise and sunset there is a well-defined level z
near the surface called ‘null-point’ by the authors, where the temperature
gradient is zero Fig. 4. Hence, for those times of the day when a null-point exists, Eq. 6 gives:
λz
r
= + R
z
r
z
C∂T ∂t dz
∂T ∂z|
z
r
8 In the original version of the method λz
r
was es- timated several times a day each time a null-point
Fig. 4. Temperature profiles at different times of the day on 4th of September evening and 23rd of September morning of year
1997. Time is indicated on each profile. Null-points are represented by closed circles. Temperature gradients below the null-point are
positive downward in the morning, and positive upward in the evening.
occurred above the reference depth and daily aver- ages were computed. These daily values of thermal
conductivity were then used to compute the soil heat flux at any time of the day, assuming that λz
r
did not change much throughout the day Kimball and
Jackson, 1975. In the present paper we adopt an- other scheme, that is better suited to long-term energy
budget studies. It operates in two steps.
3.2. A two-step version of the null-alignment method At a given depth the thermal conductivity is usually
modelled as a function of temperature and volumetric moisture de Vries, 1963. Such a parameterisation re-
quires large ranges of soil moisture and temperature, that are normally encountered during a long-term ex-
periment. In a first step, we use the whole data set and Eq. 8 every time a null-point occurs to estimate and
model λz
r
as a function of soil volumetric moisture. Its temperature dependence is neglected. During the
investigated year soil moisture ranges from dry values 5–6 to saturation, providing us with the full range
of variation of the thermal conductivity. In this first step, only the temperature profiles fulfilling the basic
assumption of the null-alignment method are selected.
In a second step, the whole data set is used again to calculate the heat flux throughout the year at z
r
, then at the surface, using the modelled conductivity.
3.3. The reference depth z
r
In the original version of the method the reference depth was chosen ‘shallow enough so that temperature
changes can be measured accurately and deep enough so that thermal conductivity changes little during the
day’ Kimball and Jackson, 1975, and the authors rec- ommended z
r
= 20 cm. Here, the recommendations
should be similar because we also use daily estimates of soil thermal conductivity from daily measurements
of soil moisture. However, the main difference for the choice in z
r
is that our soil is covered by vegeta- tion, so that at a given depth the daily amplitudes of
soil temperature and thermal conductivity are smaller than for a bare soil. For temperature, this amplitude
may not be large enough for a correct evaluation of the heat storage term; for the thermal conductivity these
daily variations may be small in the upper soil. Con- sequently we compared three reference levels: z
r
=
178 J. Og´ee et al. Agricultural and Forest Meteorology 106 2001 173–186
10 cm case A, z
r
= 15 cm case B and z
r
= 20 cm
case C. The results are presented in Section 4.2. 3.4. Selection of temperature profiles to estimate soil
thermal conductivity As mentioned above, a selection was operated on
the temperature profiles in order to satisfy the basic assumptions of the null-alignment method and get ac-
curate estimates of the soil thermal conductivity. According to the first assumption the heat trans-
ported by the mass flow of liquid water must be negligible at the reference depth z
r
. This hypothesis is not very restrictive and is often made for soil heat
flux measurements de Vries and Philip, 1986; Cellier et al., 1996 as well as in soil models Kondo and
Saigusa, 1994. As pointed out by de Vries and Philip 1986, ‘sensible heat transport produced by infiltra-
tion of large quantities of water is potentially more important but this will occur through identifiable and
anomalous events’, mainly during heavy rains. Hence, the temperature profiles recorded during heavy rains
and within 2 h after them have been systematically removed from the set of profiles used to parameterise
λ
z
r
. The second assumption of the null-alignment
method states that no evaporation occurs between the null-point and the reference depth. This assump-
tion is more stringent than the previous one and can in fact lead to unrealistic results if no precaution is
taken de Vries and Philip, 1986. In fact, if evapo- ration is not neglected, we obtain from Eqs. 4, 5
and 8:
λz
r
= + R
z
r
z
C∂T ∂t + LE
soil
dz ∂T ∂z|
z
r
= λ
na
z
r
+ R
z
r
z
LE
soil
dz ∂T ∂z|
z
r
9 and
Q
h
0 = Q
h,na
0 + Z
z
LE
soil
dz 10
where λ
na
is the thermal conductivity derived from the null-alignment method and Q
h,na
the correspond- ing soil heat flux previously denoted as λ and Q
h
, respectively. The temperature gradient at z
r
is posi- tive in the late morning, when evaporation is likely to
be positive, and negative in the late evening, when no evaporation but rather condensation occurs Fig. 4.
Therefore, in all cases, neglecting evaporation should lead to an underestimation of λ:
λ
na
z
r
≤ λz
r
11 The temperature gradient at 10 cm varies between
− 20 and +20 K m
− 1
. Hence, an evaporation of 2 W m
− 2
, a reasonable value for a soil under a litter and a canopy, would lead to an underestimation of λ
of 0.1 W m
− 1
K
− 1
if the null-point z were so close
to the surface that all evaporation would occur below this level. This represents between 5 and 10 of the
value of λ for a sand with a medium range humidity. At other times of the day, when the temperature gra-
dients are lower, the error could be more important. Fortunately, soil evaporation usually occurs only in
the top centimetres. Hence, if the null-points are deep enough and if the gradients at the reference level are
large enough, the conductivity estimates should be accurate. For this reason, we only selected temper-
ature profiles with null-points deeper than 2 cm and temperature gradients larger than 4 K m
− 1
. This is in agreement with de Vries and Philip’s comments
on the null-alignment method de Vries and Philip, 1986.
Regarding now the computation of soil heat flux, it is highly desirable to keep as many values as possible,
even during heavy rainfall or when soil evaporation is not a priori negligible. Our results show that the heat
flux at z
r
only represents 20–30 of the surface flux, so that an error of 10 on the flux at z
r
would lead to a 2–3 error on the flux at the soil surface, which is
negligible in this type of energy budget studies. On top of this, soil evaporation is always very low in our con-
ditions. At field capacity, when drainage is negligible e.g. at the end of March 1998, the rate of decrease in
soil moisture is of the order of −0.25 kg m
− 3
d
− 1
not shown. Assuming a uniform distribution of soil evap-
oration with time and depth down to 20 cm, this leads to a maximum evaporation sink of about 1.5 W m
− 2
, a negligible value compared with the instantaneous
soil heat flux values. Previous chamber measurements of soil evaporation confirmed this result Loustau and
Cochard, 1991. It was therefore decided to keep all soil heat flux values to perform short — as well as
long-term energy budgets.
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
