Kardar-Parisi-Zhang Scaling of the Height of the Convective Boundary Layer and Fractal Structure of Cumulus Cloud Fields

We present the cumulative frequency-area distribution of tropical cumulus clouds as observed from satellite and space shuttle images from scales of 0.1 to 1000 km. The distribution is a power-law function of area with exponent $-0.8\mathrm{}$. We show that this result and the fractal dimension of cloud perimeters implies that the top of the convective boundary layer (CBL) is a self-affine interface with roughness exponent or Hausdorff measure $H\approx 0.4$, the same value as that of the Kardar-Parisi-Zhang equation in $2+1$ dimensions. In addition, we identify dynamic scaling in a time series of the local altitude of the top of the CBL as measured with FM/CW radar backscatter intensity. A possible growth model is discussed.