The Evolution of the Surface of the Earth

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1 Björn Center for Complex and Nonlinear Science Department of Mathematics UCSB Courant Oct. 2011

2 Collaborators River Basins Terrence R. Smith Geography and Computer Science, UCSB Jorge Hernandez Probability and Statistics, UCSB Russel Schwab Mathematics, UCSB Kristen Meeker Geography and Computer Science, UCSB Ted Welsh Mathematics, Duke Andrea Bertozzi Mathematics, UCLA

3 The Different Stages Channelization: The initial process wherein many small rivulets merge to form larger ones. As the rivulets join to form streams, they carve out the originally smooth landsurface and form a basic network of streams and rivers. Adolescence: The second process where young, smooth surfaces evolve into an initially concave and eventually convex shaped surface. This metamorphosis is effected by shocks first in the water flow and then in the sediment flow. Maturation: The final process in which the rough, mature landsurfaces develop into a distinctly convex shaped surface which afterwards retains its overall appearance but just decreases in elevation.

4 Landsurface Age Mount Sugarloaf in Massachusetts (left) vs. Rolling Hills of California

5 Outline River Basins Properties of Rivers 1 River Basins Properties of Rivers 2

6 Properties of Rivers The Cache la Poudre River Basin near Ft. Collins, CO

7 Hack s Law River Basins Properties of Rivers The length l of the main river in a river basin scales with the area A of the river basin as l A 0.58

8 The Amazon River Basin Properties of Rivers

9 Strahler s Order of Streams Properties of Rivers Streams without tributaries have order 1. When two streams of order n join, they form a stream of order n+1 When two streams of different orders join, the resulting stream inherits the higher order of the two.

10 Strahler s Order River Basins Properties of Rivers

11 Horton s Bifurcation Ratios Properties of Rivers Horton discovered that if N(n)denotes the number of streams of order n and L(n) is their mean length then the length and bifurcation ratios R l = N(n)/N(n + 1) R b = L(n + 1)/L(n) are constant over the entire river basin.

12 Probability of Exceedance Properties of Rivers The probability that the area of a subbasin exceeds a certain value a, scales with a P(area of subbasin > a) a 0.42

13 Outline River Basins 1 River Basins Properties of Rivers 2

14 River Basins

15 River Valleys with Uplift

16 The Variogram River Basins How do we measure the roughness of the surface? V (x, t) = < u(y + x, t) u(y, t) 2 > 1/2 t β f (t/ x z ) for t small, and for t large V (x, t) x χ f(t/ x z )

19 The Surface Seeks a Fractal Equilibrium There is an equivalence between time and distance in space t x z given by the dynamic exponent z. The system (width function) roughens initially as a power of t given by the temporal roughness coefficient β. Eventually the system gets into a statistically stationary state where is does not roughen any more, but spatial fluctuations scale with a power of the lag variable given by the spatial roughness coefficient χ = zβ.

20 Outline River Basins 1 River Basins Properties of Rivers 2

21 The Merging of the Rivulets

22 The Evolution of a River

23 Water Depth River Basins

25 The Channelization Process

26 Maturation Scaling in the Stationary State

27 Scaling Exponents River Basins For the first process the values of the exponents of the width functions are β = 1 4, χ = 1 2, z = 2 For the second process the exponents are β = 0.5, χ = 0.75, z = 1.5

28 Hack s Law River Basins A l D The avalanche dimension D = 1 + χ, l being the length of the main river, then the width of the basin in the direction l χ, perpendicular to the main river, is close to the separable solution χ = 0.75 hence l A 1 1+χ A 0.58

29 The Surface Determines the River Network Everything is determined by the Hack s exponent h = 4 7 The exponent for the probability of exceedance is t = h 1 = h = LogR L LogR B Hack s exponent is determined by the roughness coefficient of the surface.

30 What Determines the Roughness? Flow in rivers is unstable most of the time. This means that the flow is frequently turbulent. The flow during storms, floods and heavy rainfalls is always turbulent. Most of the sediment is carried by turbulent flow. But turbulence is still not understood.

31 Recent Results on Turbulent Flow in Rivers (1 dimension) There exist solutions that are not smooth. They scale with roughness component 3/4. This gives the Hack s coefficient 1/(1 + 3/4) = 4/7 Thus the roughness coefficient of the surface seems to be caused by sediment transported by turbulent flow in rivers. This also explains all the scalings of the river network.

FRACTAL RIVER BASINS

FRACTAL RIVER BASINS CHANCE AND SELF-ORGANIZATION Ignacio Rodriguez-Iturbe Texas A & M University Andrea Rinaldo University of Padua, Italy CAMBRIDGE UNIVERSITY PRESS Contents Foreword Preface page xiii