5
10
15
20
5
10
15
20
5
10
15
20
Cloud coverage : 100%
4
2
2
1
5
10
15
20
10
15
20
5
10
15
20
5
10
15
20
Figure 4: Contour of the biase on radiative fluxes δF (a) Approx=PPH, (b)
Approx=IPA for cloud coverage C = 100%, as a function of mean optical depth
and geometrical thickness. Two cloud base altitudes are represented: 1 km and
4 km. Cloud coverage : 100%
b)
10
15
20
5
3
10
0
15 20
2
2
1 1
1
5
10
15
20
0. 5
5
10
15
Fractional coverage cloud
β
=1.
78
Overcast Cloud
Figure 3 : 2D tdMAP clouds examples, with some pertinent
statistic properties (Optical depth distribution and energy in
frequency space. ρτ is an inhomogeneity parameter (Szczap et
al., 2000), β is the spectral slope.
ρτ =
σ (τ )
τ = 0.4
β
=1.
47
16
2
1 .5
2
1
1
1
10
15
1
20
5
10
15
20
δFup (W/m2)
4
Zb = 1 km
1
2
3
4
5
Local geometrical thickness (h)
6
7
τ
α
τ m ax
where τ is the optical depth of pixel , τmax the
maximum of the cloud field optical depth, α is an exponent and
H0 is a constant value computed in such a way that mean
geometrical thickness of bumpy cloud equals geometrical
thickness of the corresponding flat-top cloud. Distribution 1 and 2
corresponds to α= 0.5 and = 0.7 respectively.
Zb = 4 km
0
3
4
2
0.4
1
0
5
1
0
10
15
20
4
3
1
0.2
0.2
1
3
2
2
2
0.2
600
0.6
3
Geometrical thickness
Number
τ = 1.6
20
3
3
800
Figure 5: Geometrical thickness histogram of bumpy cloud (C =
0.58 %). Mean geometrical depth = 2 km.
σ (τ )
10
3
Figure 5: Same as figure 10 but for cloud with fractional cloud coverage C =
0.25%. (a) Approx=PPH, (b) Approx=FCPPH and, (c) Approx=IPA.
400
ρτ =
5
4
2
δFdown (W/m2)
: Distribution 1 (σ = 2.0)
: Distribution 2 (σ = 2.3)
1000
h = Ho
20
Mean optical depth
4
0
15
2.5
5
15
1
2
20
20
2
1
10
15
2
1.5
5
10
3
1
3
1200
Figure 2: a) Illustration of the tdMAP model for tree values of Hurst
parameter H (0.8, 0.5, 0.2 respectively). Bernoulli parameter is set to p = 1.
b) Same as Figure 2.a), but space dimension D= 2.
5
4
0.5
1
4
1. 5
1
20
15 20
2
2
4
3
Bumpy Top Cloud Case
Figure 1: a) Illustration of the tdMAP model for tree values of Bernoulli
parameter p (1, 0.9, 0.8 respectively). Hurst parameter is set to H = 0.5 and
space dimension D = 1. For p=1, it is indistinguishable from Fractional
Brownian Motion (Benassi et al. ,1997). b) Same as Figure 1.a, but space
dimension D=2. The basic shape of the morphlet (a Gaussian) can be seen
clearly at the largest scales. ξ are a family of independent and identically
distributed normal random variables.
10
4
5
10
15
4
3
0.4
Clouds characteristics :
Mean visible optical depth : 2, 5, 10, 20.
Fractional cloud coverage : 100%, 58 %, 25%
Cloudy mean geometrical depth : 0.5, 1 ,2 ,4 km
Cloud base altitude : 1 km and 4 km.
5
Cloud coverage : 25%
0.4
Radiative transfer characteristics :
Infrared band : [10.2 – 12.5] µm, k-distribution method
Pixel size and number: 0.5 km × 0.5 km and 128 × 128
Ground temperature and emissivity : 288.1°K and 1
Methodology : 1) Simulation of 3D infrared radiative transfer
in US standard atmosphere with SHDOM (Spherical Harmonic
Discrete Ordinate Method) in realistic inhomogeneous clouds
generated with tdMAP (Tree Driven Mass Accumulation
Process). 2) Comparison to the “Approximate radiative
transfer” model used in Global Climate Model.
20
3
1
5
15
1
0.6
Scientific Objectives : In the short run, to
quantify inhomogeneity cloud effects on their
shortwave and longwave radiative properties.
In the medium term, to build fast and accurate
radiative transfer algorithms based on neural
networks for inhomogeneous clouds.
10
2
1
200
Scientific Context : Modification of the atmospheric radiative equilibrium, due to the increase of greenhouse effect, could
have dramatic climatic impacts. Most of experiments deal with consequences of CO2 doubling. Direct radiative effect is an
increase of 4W/m2 on low atmosphere infrared energy budget. This radiative perturbation is difficult to interpret in term of
climatic parameters, because of numerous atmospheric retroactions increasing or decreasing these perturbation effects.
Nevertheless, most of Global Climate Model display a surface temperature increasing of 1.5 to 4.5°K as a function of cloud
representations.
Clouds exhibit fluctuations of microphysical characteristics at different averaging scales. How this spatial inhomogeneity of
clouds properties affects radiative transfer is one of the major issue in order to reduce the uncertainties of actual estimations
of climatic sensitivity.
5
3
2
3
1
Mean optical depth
b)
20
4
c) 4
2
1
5
0.1
0.05
3
19
1
2
1
3
0
15
1
0.2 5
0.2
0.25
0.1
3
0.15 0.2
2
3
0
a)
10
3
2
4
0.3
4
18
2
0
4
5
4
b)
0
20
Geometrical thickness
0.2
0
0.8
1
15
1
13
10
2
1
16
2
1
4
3
2
0.5
5
3
27
δFup (W/m2)
4
12
4
1
2
δFdown (W/m2)
6
Geometrical thickness
3
1
2
1
4
0
H = 0.2
3
0.6
H = 0.5
H = 0.8
2
1
3
a)
Abstract : We analyze the effects of flat and bumpy top, fractional and internally inhomogeneous cloud layers on large
area-averaged thermal radiative fluxes. Inhomogeneous clouds are generated by a new stochastic model: the tree-driven
Mass Accumulation Process (tdMAP). tdMAP model is related to wavelet decomposition. But this analogy is too limited to
be really usefull. Indeed, in wavelet analysis, we need to use specific wavelet (Haar, Lemarié-Meyer, etc.) in an orthogonal
basis. Instead of this, tdMAP uses arbitrary function F (“morphlet”) shifted and scaled in the physical space as dictated by
the tree-structure. This model is able to provide stratocumulus and cumulus cloud fields with properties close to those
observed in real clouds. A sensitivity study of cloud parameters is done by analyzing differences between 3D fluxes
simulated by the Spherical Harmonic Discrete Ordinate Method and three standard models likely to be used in General
Circulation Models: Plane-Parallel Homogeneous cloud model (PPH), PPH with Fractional Cloud coverage model
(FCPPH) and Independent Pixel Approximation model (IPA). We show that thermal fluxes are strong functions of
fractional cloud coverage, mean optical depth, mean geometrical thickness, and cloud base altitude. Fluctuations of incloud horizontal variability in optical depth and cloud-top bumps have negligible effects in the whole. We also showed that
PPH, FCPPH and IPA models are not suitable to compute thermal fluxes of flat top fractional inhomogeneous cloud layer,
except for completely overcast cloud. This implies that horizontal transport of photon at thermal wavelengths is important
when cloudy cells are separated by optically thin regions.
b)
p = 0.8
4
3
0. 4
p = 0.9
4
0.1
p =1
0.05
2
0.15 0.1
0.005
1
3
4
0.6
4
3
a)
0.4
0.2
4
a)
&
0.1
5
%
ξ j,k
δFup(W/m2)
δFdown(W/m2)
δFup(W/m2)
δFdown(W/m2)
Zb = 4 km
δFup (W/m2)
14
#
$
δFdown (W/m2)
Zb = 10 km
21
Zb = 1 km
10
8
H
F
Zb = 1 km
Flat Top Cloud Case
.
24
!
"
# $
Biases between 3D radiative fluxes computed by SHDOM and those computed by “approximate” methods, where Approx = PPH, FCPPH or IPA,(Plan
Parallel Homogeneous, PPH with Fractional cloud Coverage and Independent Pixel Approximation respectively). Direction = up or down. Note that where up
indicates the bias estimated far above the cloud layer (altitude =20 km) for flux going into space, and down indicates the bias estimated below the cloud layer
(altitude = 0 km). Positive bias in red, negative bias in blue.
{
{
(2) Los Alamos National Laboratory, Space and Remote Sensing Sciences Group, Los Alamos, NM, USA.
X
x
Tp
"
{
{
(1) Laboratoire de Météorologie Physique, OPGC, .Université Blaise Pascal, Clermont-Ferrand, France.
3 D − Approx
δ FDirection
:
0.0
5
Albert Benassi,(1)Frédéric Szczap,(1)Anthony Davis,(2) Céline Cornet, (1) Pascal Bleuyard and (1) Bernard Guillemet
(1)
!
The statistical theory of turbulence proposed by A. Kolmogorov was
the departure of an enormous amount of work. Supporting directly or
inspired by this progress in turbulence theory, two kinds of models
were proposed: first, the multiplicative cascade model (Kolmogorov,
1962; Novikov and Stewart, 1964) and then Fractional Brownian
Motions or FBMs (Mandelbrot and Van Ness, 1968). Our model,
called “tree-driven Mass Accumulation Process” (tdMAP) pays tribute
to those both classes of mathematical models. Following Benassi
(1995), Benassi et al. (1997) and Benassi and Deguy (1999), tdMAP
model is a generalization of an adaptated wavelet decomposition of
above FBM; tdMAP model can thus be written as :
0
0 .0
0.0 .04 6
2
Large averaging Thermal Radiative Fluxes through Inhomogeneous
Cloud Fields: A Sensitivity Study using the tdMAP cloud generator
0.4
0.2
0.2
2
20
3
2
2
2
1
1
1
1
0
5
0
10
15
0
20
5
10
15
20
Mean optical depth
Figure 6: Bias between fluxes of bumpy and flat clouds computed
with 3D SHDOM in both cases as a function mean optical depth
and mean geometrical thickness for two cloud base altitude (Zb = 1,
4 km). Cloud coverage (C = 58%) and distribution 2 (α = 0.7) are
considered.
References:
References
Benassi, A, 1995: Local self similar Gaussian process in Lecture notes in statistics, Wavelets and Statistics, Anestis Antoniadis Georges Oppenheim (editors),
Springer-Verlarg,103, 43-54.
Benassi, A., S. Jaffard, and D. Roux, 1997: Gaussian processes and pseudo-differential elliptic operator, Revista Mathematica Iberoamericana, 13, 19-89.
Benassi, A., and S. Deguy, 1999: Multi-scale fractional Brownian motion: Definition and identification. Technical report, 83, LLIAC (Laboratoire de Logique,
Algorithmique et Informatique de Clermont 1), Aubière, France.
Evans, K. F., 1998: The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer, J. Atmos. Sci., 55, 429-446.
Kolmogorov, A.N., 1962: A refinement of previous hypothesis concerning the local structure of turbulence in viscous incompressible fluid at high Reynolds
number, J. Fluid Mech., 13, 82-85.
Novikov, E. A., and R. Stewart, 1964: Intermittency of turbulence and spectrum of fluctuations in energy dissipation. Izv. Akad. Naut. SSSR, Ser. Geofiz., 3, 408412.
Szczap, F., H. Isaka, M. Saute, B. Guillemet, and Y. Gour, 2000 : Inhomogeneity effects of 1D and 2D bounded cascade model clouds on their effective radiative
properties, Phys. Chem. Earth, 25, 83-89.
Acknowledgements :This work was supported by the Programme Atmosphère et Océan à Moyenne Echelle (PATOM) and a French National Institute for Sciences
of the Universe (INSU) grant. A. Davis was supported financially by LANL’s PR&TL and LDRD programs. This investigation was supported by the PATOM
(Programme Atmosphère et Océan à Moyenne Echelle)