Power Law Wave-Number Spectra of Scum on the Surface of a Flowing Fluid

An initial cloud of particles floating on the surface of a flowing fluid will often tend to a fractal pattern. If the wave-number spectrum of the pattern has an observable power law dependence $k^{-\rho }$, then the exponent $\rho$ is predicted to be $\rho {=D}_2-1\mathrm{}$, where $D_2$ is the correlation dimension of the fractal attractor. Numerical, experimental, and theoretical results are shown to support this prediction, but it is also found that, when the observable range in $k$ is limited, the predicted power law scaling can be obscured by fluctuations in the $k$ spectrum. The expected behavior can, however, be restored by use of averaging.