Turbulent Rivers

The existence of solutions describing the turbulent flow in rivers is proven. The existence of an associated invariant measure describing the statistical properties of this one dimensional turbulence is established. The turbulent solutions are not smooth but H\"older continuous with exponent $3/4$. The scaling of the solutions' second structure (or width) function gives rise to Hack's law \cite{H57}; stating that the length of the main river, in mature river basins, scales with the area of the basin $l \sim A^{h}$, $h = 0.568$ being Hack's exponent