 
Summary: Turbulent Rivers
Bjšorn Birnir
Center for Complex and Nonlinear Science
and
Department of Mathematics
University of California, Santa Barbara
Abstract
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 Hšolder continuous with exponent 3/4. The
scaling of the solutions' second structure (or width) function gives rise to
Hack's law [16]; stating that the length of the main river, in mature river
basins, scales with the area of the basin l Ah, h = 0.568 being Hack's
exponent.
1 Introduction
The flow of water in streams and rivers is a fascinating problem with many appli
cation that has intrigued scientists and laymen for many centuries, see Levi [21].
Surprisingly it is still not completely understood even in one or twodimensional
approximation of the full threedimensional flow. Erosion by water seems to de
