174 M.D. Novak et al. Agricultural and Forest Meteorology 102 2000 173–186
Novak et al., 2000a, b describe the wind and turbu- lence regimes and the transfer of heat and moisture
within and above a 10 t ha
− 1
mulch. This paper deals with the radiation distribution within the mulch for all
the application rates 2, 5, 10, and 15 t ha
− 1
that we studied in our field plots.
To simulate the radiation distribution, some re- searchers have treated the mulch as a single bulked
layer Ross et al., 1985; Chung and Horton, 1987; Hares and Novak, 1992. This approach, however,
does not provide details of the distribution within the residue layers, which is particularly critical for, say,
hay drying. Others have divided the mulch or hay into many layers Bristow et al., 1986; Tuzet et al., 1993;
Bussière and Cellier, 1994 and used standard radi- ation modelling techniques Kreith, 1973; Norman,
1979 to simulate the vertical profiles of radiation flux densities. Residue elements were usually assumed to
be randomly distributed although within-mulch ra- diation flux densities were not measured to test the
validity of this assumption. In some cases there is an indirect indication that the radiation simulation is in
error. For example, the model of Bussière and Cel- lier 1994 predicted small daytime evaporation rates
under a sugar-cane mulch but their measured values were quite large 150–200 W m
− 2
. One possible rea- son for this error is the underestimation of downward
solar radiation flux at the underlying soil surface. To improve the simulation of radiation distribution
in a mulch, features specific to mulches must be con- sidered Tanner and Shen, 1990, hereafter referred to
as TS90. An important attribute is whether the mulch elements are uniformly or randomly distributed, or
clumped, as found also in plant canopies Chen and Black, 1991. TS90 found that the residue elements
were nearly uniformly distributed in a flail-chopped corn mulch but Wagner-Riddle et al. 1996 found
a near random distribution in a rye mulch. The uniformity, randomness, or clumping of mulch ele-
ments probably depends strongly on the method of spreading and therefore can be altered. Dead residue
elements have much smaller transmissivities than live leaves, e.g., flail-chopped corn residue elements
have transmissivities of 0.005 and 0.02 in the visible and near-infrared bands, respectively, compared to
0.07 and 0.36, respectively, for senesced corn leaves TS90. There can be a large difference between the
temperatures of the upper and lower surfaces of a sunlit mulch element, in part because of the lack
of transpiration which is an important component in the temperature regulation of leaves Norman, 1979.
This difference can be accentuated by the low wind speeds within the mulch and the low thermal conduc-
tivity of dry residue elements.
The objective of this research was therefore to develop a relatively simple but physically-based radi-
ation model of a barley-straw mulch that incorporates these features. We present such a model in this paper
and test it against measurements of radiation fluxes made both above and below the mulches applied at
the various rates in our field plots. We then assess the sensitivity of the model to its input parameters, use
the model to determine the profiles of shortwave and longwave components within the mulch under field
conditions, and simulate the effects of uniformity, randomness, and clumping of mulch elements on the
shortwave radiation distribution.
2. Theory
2.1. Clumping index For a mulch or any canopy consisting of flat el-
ements with their flat sides in the horizontal plane, which we assume to be the case, the transmission of
radiation can be described as Ross, 1976
τ R
i
= 1 − 1R[1 − 1R]
i− 1
, i ≥
1, 1
where τ R
i
is the transmissivity through i vertically stacked ‘elemental’ layers within which no mutual
shading occurs among residue elements having a single-sided residue-area index R
i
= i1R
, 1R is the residue-area index of an elemental layer, and is
the clumping index, which describes the way in which the residue elements are arranged, as illustrated by
the following limiting and special cases Ross, 1976: 1. =0, which is when the highest degree of clump-
ing occurs. This implies that the residue elements in all layers below the first are arranged one be-
low the other. Such an arrangement provides maxi- mum mutual shading and, consequently, maximum
transmissivity. From Eq. 1 we obtain
τ R
i
= 1 − 1R,
i ≥ 1.
2
M.D. Novak et al. Agricultural and Forest Meteorology 102 2000 173–186 175
It follows that the penetration of radiation is deter- mined by the first layer and lower layers have no
effect. 2. =1, which is when residue elements are randomly
distributed. Eq. 1 becomes τ R
i
= 1 − 1R
i
, i ≥
1. 3
3. =11R for R
i
1, which is when the lowest de- gree of clumping or the highest degree of unifor-
mity exists. This is the situation where no mutual shading exists between elements in different layers,
which yields
τ R
i
= 1 − R
i
, 0 R
i
≤ 1,
0, R
i
1. 4
Because each residue layer supports the weight of the overlying layers this case is physically impos-
sible but represents an upper limit for .
2.2. Mulch radiation model The mulch is divided into N elemental layers, la-
belled 1. . . N, layer 0 denotes the atmosphere, and layer N+1 denotes the soil Fig. 1. The mulch ele-
ments are assumed to be opaque to both shortwave and longwave radiation and their longwave emissiv-
ities are assumed to be 1 because the reflectivity for thermal fluxes is negligible Norman, 1979; Bristow
Fig. 1. Schematic showing both downward subscript d and up- ward subscript u flux densities of shortwave, S, and longwave,
L, radiation in the soil–mulch–atmosphere system.
et al., 1986. The longwave emissivity of the soil is similarly assumed to be 1. The R
i
function is assumed to be the same for both shortwave and
longwave radiation, as found for flail-chopped corn residue by TS90 and Shen and Tanner 1990. Short-
wave radiation is not separated into beam and diffuse components. According to TS90 both components
are transmitted similarly through a mulch. Forward scattering of shortwave radiation is assumed to be
negligible so that all internal reflections within the mulch occur from the flat sides of the mulch ele-
ments. Penumbra effects are also neglected, which is reasonable for such a short canopy Ross, 1976.
From Eq. 1, the ‘primary’ downward transmitted direct solar radiation above the ith layer, S
′ d,i
, is given by
S
′ d,i
= S
τ
i− 1
= S
1 − 1R[1 − R
i− 1
1R ]
i− 2
, for 2 ≤ i ≤ N +1,
5 where the term primary refers to the fact that no sec-
ondary reflections downward are included, S is the
solar irradiance above the mulch, and τ =
1 which corresponds to i=1. These downward fluxes are re-
flected upward by all the mulch layers and the un- derlying soil surface yielding a primary upward flux
above the ith layer, S
′ u,i
, given by S
′ u,i
=
N
X
j =i
α
m
S
′ d,j
− S
′ d,j +1
τ
j −i
+ S
′ d,N +1
α
s
τ
N + 1−i
, for 1 ≤ i ≤ N,
6 withS
′ u,N +1
= S
′ d,N +1
α
s
, where α
m
is the shortwave reflectivity of the mulch elements and α
s
is the short- wave reflectivity of the soil. Note that the τ
i
R
i
func- tion is used for upward and downward transfer, which
is appropriate for mulch elements which appear the same whether looked at from above or below.
Similarly, the ‘secondary’ downward shortwave ra- diation above the ith layer, S
′′ d,i
, is given by S
′′ d,i
=
i− 1
X
j = 1
α
m
[S
′ u,j +1
− S
′ u,j
− α
m
S
′ d,j +1
− S
′ d,j
]τ
i− 1
, for 2 ≤ i ≤ N + 1,
7 with S
′′ d,1
= 0 and the secondary upward shortwave
radiation above the ith layer, S
′′ u,i
, is given by
176 M.D. Novak et al. Agricultural and Forest Meteorology 102 2000 173–186
S
′′ u,i
=
N
X
j =i
α
m
[S
′′ d,j
− S
′′ d,j +1
− α
m
S
′ u,j
− S
′ u,j +1
]τ
j −i
+ S
′′ d,N +1
α
s
τ
N + 1−i
, for 1 ≤ i ≤ N ,
8 with S
′′ u,N +1
= S
′′ d,N +1
α
s
. The second terms in the square brackets in Eqs. 7 and 8 ensure that fluxes
determined in a previous step are not counted twice. ‘Tertiary’ reflections were also determined similarly
but were found to be less than 1 W m
− 2
and so are neglected. The net shortwave radiation flux density
above the ith layer, S
n,i
, is therefore S
n,i
= S
′ d,i
− S
′ u,i
+ S
′′ d,i
− S
′′ u,i
, for 1 ≤ i ≤ N + 1.
9 Longwave radiation above each layer has components
originating at the soil surface, residue layers, and the atmosphere. The fraction of the view from a plane
above the ith layer occupied by the part of the jth over- lying layer that is illuminated by radiation emanating
uniformly upward from above the ith layer is given by
f
j,i
= τ
i−j − 1
− τ
i−j,
for 1 ≤ j i, 10
with the fraction of the atmosphere seen from this plane given by f
0,i
= τ
i− 1
, or 1− P
i− 1
j = 1
f
j,i
. Similarly, the fraction of the view from a plane above the ith
layer occupied by the part of the jth underlying layer that is illuminated by radiation emanating uniformly
downward from above the ith layer is given by
f
j,i
= τ
j −i
− τ
j −i+ 1
, for i ≤ j N + 1,
11 with the fraction of the soil seen from this plane given
by f
N + 1,i
= τ
N + 1−i
, or 1 − P
N j =i
f
j,i
. These are assumed to be the effective ‘view factors’ for longwave
transfer from the atmosphere, jth mulch layer, and soil to the plane above the ith layer.
The downward longwave radiation flux density above the ith layer, L
d,i
, is then given by L
d,i
= f
0,i
ε
a
σ T
4 a
+
i− 1
X
j = 1
f
j,i
σ T
4 d,j
, for 2 ≤ i ≤ N + 1,
12 with L
d,1
= ε
a
σ T
4 a
, where ε
a
is the atmospheric emissivity, σ =5.67×10
− 8
W m
− 2
K
− 4
is the Stefan– Boltzmann constant, T
a
is the absolute temperature of the air at ‘screen’ height, and T
d,j
is the abso- lute temperature of the lower surfaces of the jth layer
mulch elements. The first term on the right hand side of Eq. 12 is due to radiation originating from the at-
mosphere and the second term due to radiation origi- nating from overlying mulch layers. Similarly, the up-
ward longwave radiation flux density above the ith layer, L
u,i
, is given by L
u,i
= f
N + 1,i
ε
s
σ T
4 s
+
N
X
j =i
f
j,i
σ T
4 u,j
, for 1 ≤ i ≤ N,
13 with L
u,N +1
= σ T
4 s
, where T
s
is the absolute temper- ature of the soil surface and T
u,j
is the absolute tem- perature of the upper surfaces of the jth layer mulch
elements. The first term on the right hand side of Eq. 13 is due to radiation originating from the soil and the
second term due to radiation from underlying mulch layers. The net longwave radiation flux density above
the ith layer is given by
L
n,i
= L
d,i
− L
u,i
, for 1 ≤ i ≤ N + 1,
14 and the net radiation flux density above the ith layer
is given by R
n,i
= S
n,i
+ L
n,i
for 1 ≤ i ≤ N + 1. 15
For clear-sky conditions, the value of ε
a
is determined from T
a
K and water vapour pressure, e
a
mb, at screen height using Novak and Black, 1985
ε
a
= ε
ac
= a
+ 5.95 × 10
− 5
e
a
exp 1500
T
a
, 16
where ε
ac
is the clear-sky emissivity and a is a pa-
rameter that accounts for dust concentrations in the atmosphere Idso, 1980. For cloudy conditions, ε
a
is calculated from Bristow et al., 1986
ε
a
= 1 − 0.84Cε
ac
+ 0.84C,
17 where C is the mean daily fraction of cloud cover
which ranges from 0 to 1. C is estimated from χ the ratio of daily solar radiation flux density to total
hemispherical solar radiation flux density incident on a horizontal surface at the outer edge of the earth’s
atmosphere via
M.D. Novak et al. Agricultural and Forest Meteorology 102 2000 173–186 177
C =
1, χ
0.35, 2.4 − 4χ ,
0.35 ≤ χ ≤ 0.6, 0,
χ 0.6.
18 Because the model neglects higher reflections beyond
the secondary, all variables are calculated explicitly from these simple formulas, in contrast with more
standard models which are not approximate in this way but which require matrix inversion for their so-
lution Kreith, 1973. The input variables S
, T
a
, e
a
, T
u,i
, T
d,i
, and T
s
, are all given as functions of time, t, during the period of interest. The model calculations
were programmed in FORTRAN.
3. Experiments
