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**Multifractal Analysis of Radiation in Clouds: 5000km to 50cm**

Lovejoy, S., McGill University

Atmospheric Thermodynamics and Vertical Structures

Cloud Properties

Lovejoy, S., D. Schertzer, J. D. Stanway, 2001: "Direct Evidence of planetary scale atmospheric cascade dynamics," *Phys. Rev. Lett.* 86(22): 5200-5203.

##### Left: Power spectrum of the 5 different aircraft measured liquid water data sets from the FIRE experiment (averaged over 10 equally logarithmically spaced points on the k-axis and vertically offset). The absolute slopes with Β = 1.45 is indicated (straight line on top of graph) for reference. The number of sets used to compute the average from top to bottom: 4, 3, 1, 2, 5. A constant aircraft speed of 100m/s has been assumed. Right: Ensemble averaged power spectrum (averaged over 100 equally logarithmically spaced points per magnitude on the k axis). The ensemble average of the squared moduli of the 15 equal sized data sets yields a power spectrum with a spectral slope Β ≈1.45.

##### Top, Left: This is the average of the power spectra of 29 GMS visible images (0.65 µm) over the western pacific. The images were taken from 21 March to 10 April 1996, all at 0424 GMT. The images have a resolution of 5.0 km and are of dimension 1024 x 1024 pixels. The absolute slop of the graph is Β=1.32. Note the flattening in both this and the one to the right at the high wavenumber extreme (factor=2); this is due to the inadequate quantization (and hence dynamical range) of the data which effectively introduces noise and breaks the scaling. Top, Right: This is the average power spectrum of 97 NOAA 12 (AVHRR) images, channel 1, taken from January to September, 1996. The images were all taken between 1330 and 1430 GMT over the ARM site in Oklahoma. The images were taken using the visible channel 1 (0.58-0.68 µm) with a resolution of 1.1 km. All images are 256x256 pixels. The average Β=1.51 is used for comparison on other figures, is obtained from this graph. Bottom, Right: The average spectrum for each month’s NOAA 12 (AVHRR) data is shown here. The spectra area all displaced from each other and each month is indicated, along with the number of spectra used in the average next to the relevant plotted data. Each spectrum is shown from comparison next to a line of the average Β over all 97 images used, 1.51. Bottom, Left: The spectra of half of the NOAA 12 (AVHRR) images obtained in May 1996 are shown here displaced from each other and compared to the slope of the average spectrum for all 97 images Β= 1.51. From top to bottom, the dates of the images are 1, 3, 8, 11, 15, 21, and 30 May.

##### Top: Spectral analysis of 18 downward propagating radiation fields, including power law fits for a random collection Montreal clouds near local noon (during the period March – July, the highest resolution ones are presented here). The spectra are offset in the vertical for clarity: the resolution was determined by the heights of the cloud base. Wavenumber units are in m^{-1}. The resolutions were estimated via knowledge of the cloud base heights and are probably only valid to within ±50%. Bottom: Same as 4am but for the 18 lowest resolution clouds. The value of Β was fairly constant (Β=2.1 ±0.1); close to the SPOT value (1.9), but was larger than the AVHRR and GMS values. This is probably due to the fact the background (the blue sky) is totally homogenous and is hence smoother (higher Β) than expected for upwelling radiances which will be affected surface albedo variability. Full analysis of another 50 or so pictures is currently in progress, We conclude that the visible radiances are quite remarkably scaling over the entire range of 5000 km to <1 m; it likely continues down the dissipation scale (<1 cm). The weak evidence for scaling breaks at 60 m (Davis et al. 1997; presented on the basis of a single-factor of 2 on a single LANDSAT images) were not corroborated.

Contributors:

S. Lovejoy, Department of Physics, McGill University,
Canada

D.Schertzer, Laboratoire de Modelisation en Mecanique, France

J. D. Stanway, Department of Physics, McGill University, Canada

D. Sachs, Department of Physics, McGill University, Canada

Existing General Circulation Models (GCMs) rely heavily on strong cloud homogeneity assumptions; the solar radiation is assumed to interact with highly unrealistic plane parallel (horizontally homogenous) clouds. In contrast, the ARM analyses discussed here show that real clouds exhibit fractal structures and multifractal statistics over wide ranges of scale. Since spectral analysis is a very sensitive indicator of scale invariance, the remarkably good scaling - even on individual cloud pictures - contradicts the standard model of the atmosphere which is based on a hypothetical "meso-scale gap" ("dimensional transition") separating two-dimensional isotropic turbulence at large scales and three-dimensional isotropic turbulence at small scales. However, since the dynamical velocity field is strongly nonlinearly coupled with the cloud field, such scaling was, in fact, predicted by the "unified scaling model" of the atmosphere (based on scaling stratification; Schertzer and Lovejoy 1985; Lovejoy and Schertzer 1985, 1986) and gives it strong support.

Since scaling cloud radiances imply strong (power law) resolution dependences in radiation measurements, any estimate of the earth's radiation budget requires a good knowledge of both the limits and types of scaling. Hence, such knowledge is fundamental to ARM. It is, therefore, important not only to better characterize the scaling in the range greater than 1 km (i.e., the range accessible to meteorological satellites), but also to discover the inner scale of the scaling. Below the (apparently nonexistent) mesoscale gap, the only other theoretically predicted fundamental length scale in clouds is the dissipation scale (roughly 1-10 mm).

Preliminary multifractal analysis results are presented in Figure 3 below for
the GMS visible data. Recall that, in general, scaling fields will be multifractal,
characterized by an infinite hierarchy of scaling exponents (each intensity/singularity
level will have a different exponent). This infinite number of parameters
would render scaling quite unmanageable if it were not for the existence
of stable, attractive (hence "universal") multifractal processes
(e.g., Schertzer and Lovejoy 1997). We confirmed that the radiance fields
are indeed very close to what is expected for universal multifractals. In
the latter, the infinite hierarchy of exponents (e.g., dimensions/codimensions)
is described by only three universal exponents (H, C_{1}, alpha):
the nonconservation exponent (H) of the mean field; the mean singularity
(C_{1}) and (Levy) index of multifractality (alpha). Recall that
alpha=0 corresponds to the monofractal minimum while alpha=2 to the "lognormal"
maximum. Figure 3 shows the relatively small variation in these parameters
for the various GMS scenes (at at roughly constant sun and viewing angles).
The variation seems directly related to the amount of cloud cover as shown
by the steady variation of the parameters as functions of the average scene
brightness.

Although it may seem surprising, there appears to be no systematic study
of the scaling of cloud radiances at scales below roughly the resolution
of the AVHRR data. The partial exceptions are Cahalan and Snider (1989) who
analysed about 1% of a single LANDSAT picture (down to 30 m), Lovejoy et
al. (1993) who analyzed three LANDSAT MSS pictures (160 m resolution), Barker and
Davies (1992) and Davis et al. (1997) who each analyzed a single LANDSAT^{TM }image (30 m), and the SPOT analysis (20 m) presented
above. Aside from the nearly prohibitive cost of the necessary large quantities
of LANDSAT^{TM }and SPOT data, these sensors
were not designed for clouds and are prone to saturation even in the presence
of only moderately thick clouds. We, therefore, decided to take our own photographs,
digitize them, and analyse the results. Aside from the low cost, a further advantage
is that resolutions of less than 1 m are readily achieved. The results (including
the technical details) will soon be published; suffice it to say that the
fundamental limitation is the low (8-bit) dynamical range of available commercial
scanners. A simple model for this quantization effect (checked using multifractal
simulations) is that it roughly mimicks the introduction of a (near) white
noise of unit amplitude, breaking the scaling at a scale corresponding to
the point where the spectral variance is reduced by (2^{8})^{2}
from the low frequency maximum. Hence for example, if beta=2 (close to that
found here), then the maximum range of scales available before the quantization
spoils the variance estimates (and hence the scaling) is =2^{8}
(the variance at wavenumber 2^{8} will be (2^{8})^{2}
times lower, i.e., it will already be at the quantization level).

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