Fractal aircraft trajectories and nonclassical turbulent exponents.

The dimension (D) of aircraft trajectories is fundamental in interpreting airborne data. To estimate D, we studied data from 18 trajectories of stratospheric aircraft flights 1600 km long taken during a "Mach cruise" (near constant Mach number) autopilot flight mode of the ER-2 research aircraft. Mach cruise implies correlated temperature and wind fluctuations so that DeltaZ approximately Deltax (H(z) ) where Z is the (fluctuating) vertical and x the horizontal coordinate of the aircraft. Over the range approximately 3-300 km , we found H(z) approximately 0.58+/-0.02 close to the theoretical 5/9=0.56 and implying D=1+ H(z) =14/9 , i.e., the trajectories are fractal. For distances <3 km aircraft inertia smooths the trajectories, for distances >300 km , D=1 again because of a rise of 1 m/km due to fuel consumption. In the fractal regime, the horizontal velocity and temperature exponents are close to the nonclassical value 1/2 (rather than 1/3 ). We discuss implications for aircraft measurements as well as for the structure of the atmosphere.

Large particle number limit in rain.

The way we conceptualize rain is fundamental in many branches of science since it provides the basis not only for rain modeling notably in meteorology and hydrology, but also for interpreting rain data (from gauges and radars). In order to empirically address this question, we use stereophotographic data to measure the positions and volumes of raindrops from approximately 10 m(3) regions containing 5000-15,000 of these drops. By determining the drop statistics in spheres of increasing size, we conduct a basic continuum mechanics thought experiment. We show that-presumably due to turbulence-there is no microscale-macroscale separation. We find that the large particle number (N) limit in rain is not a homogeneous continuum, but rather it is nonclassical, strongly inhomogeneous, and approaching a multifractal discontinuum.

Direct evidence of multifractal atmospheric cascades from planetary scales down to 1 km.

We use 909 satellite images spanning the scale range 1-5000 km at both visible and infrared wavelengths to show that the variability at all observed scales and at all levels of intensity is very close to that predicted for a direct multiplicative scale invariant cascade starting at planetary scales. To within 1.6%/octave in scale, the observed type of (multi)scaling is very close to that theoretically predicted for universal multifractals, including multifractal phase transitions. Because of the strong vertical stratification, the scaling cannot be isotropic; these findings thus give strong support to the anisotropic "unified scaling" model of atmospheric dynamics.

Functional box-counting and multiple elliptical dimensions in rain.

Many physical systems that have interacting structures that span wide ranges in size involve substantial scale invariant (or scaling) subranges. In these regimes, the large and small scales are related by an operation that involves only the scale ratio. The system has no intrinsic characteristic size. In the atmosphere gravity causes differential stratification, so that the scale change involves new elliptical dimensions (d(el)). Fields that are extremely variable, such as rain, involve multiple scaling and dimensions that characterize the increasingly intense regions. Elliptical dimensional sampling and functional box-counting have been used to analyze radar rain data to obtain both the multiple dimensions of the rain field and the estimate d(el) = 2.22 +/- 0.07.