## 1. Introduction

[2] In 1922, Lewis Fry Richardson published the now celebrated book “Weather forecasting by numerical process” [*Richardson*, 1922] in which he daringly proposed that the weather could be forecast by brute force numerical integration of coupled nonlinear partial differential equations (PDE's). But the father of numerical weather predication was Janus-faced: his book contains a famous phrase in which he proposed that the complex nonlinear atmospheric dynamics cascaded scale after scale from planetary down to small viscous scales. Shortly afterwards [*Richardson*, 1926], he suggested that atmospheric particle trajectories might be Wierstrasse-like functions (fractals) with simple (but nonclassical) scale by scale regularity. Richardson apparently believed that messy complexity ought to give way to scale by scale simplicity: he is often considered the grandfather of modern cascade models.

[3] Today, numerical forecasting is a daily reality; but what about the dream of scale by scale simplicity embodied in cascades? For a long time after Richardson, cascades were inspirational and were regularly invoked in turbulence theories. However, it was not until the development of explicit multiplicative cascade models (starting in the 1960's and 70's [e.g., *Novikov and Stewart*, 1964; *Yaglom*, 1966; *Mandelbrot*, 1974]) that empirically verifiable cascade predictions could go much beyond the determination of (non intermittent) spectral exponents and of the up scale or down scale cascade direction [see, e.g., *Boer and Shepherd*, 1983; *Chen and Wiin-Nielsen*, 1978; *Strauss and Ditlevsen*, 1999].

[4] By the 1980's it was realized that multiplicative cascade models were the generic multifractal process. Subsequent developments have shown their great generality which have spawned applications throughout physics and the geosciences. In particular, while today there is a general consensus that at least over some scale range the atmosphere is multifractal, there have not yet been planetary scale investigations of the precise predictions of these explicit cascade models (equation 1 below). One of the reasons is that the dynamically most important fields must be measured in situ and this introduces numerous difficulties of interpretation (e.g., both (sparse) networks and aircraft trajectories can themselves be fractal [*Lovejoy et al.*, 1986, 2004; S. Lovejoy et al., Reinterpreting aircraft measurements in anisotropic scaling turbulence, submitted to *Atmospheric Physics and Chemistry*, 2008] and sonde outages can be multifractal (S. Lovejoy et al., The vertical cascade structure of the atmosphere and multifractal drop sonde outages, *Journal of Geophysical Research*, in press, 2008)). Consequently it is advantageous to use remotely sensed radiances: the largest relevant study [*Lovejoy et al.*, 2001] used nearly one thousand 256 × 256 pixel “scenes” of satellite visible and Infra red radiances over the range 2.2 to 280 km. While the fields accurately displayed cascade statistics, the largest scales - including the key outer scale of the variability - was only indirectly estimated by extrapolation well beyond the measured range. Up until now, these shortcomings have made it possible to dismiss the idea that scaling might hold up to near planetary scales or over wide ranges and to continue to pursue approaches incompatible with scaling.

[5] Although the study [*Lovejoy et al.*, 2001] had a hundred times the data content of the largest in situ turbulence experiment - it was small by today's standards. In this paper, we use about one thousand orbits of visible, infra red (IR), passive and active microwave data (11 bands in all) from the Tropical Rainfall Monitoring Mission (TRMM) satellite to directly extend these analyses to 20,000 km. Because of this wide range and the fact that each orbit comprises about the same amount of data as the entire previous study, this paper provides the first near “empirical proof” of wide range, planetary scale cascade scaling.