The Evolution of the Surface of the Earth - UCSBcnls.math.ucsb.edu/files/cnls/research-documents/Courant_erosion_… · Ted Welsh Mathematics, Duke Andrea Bertozzi Mathematics, UCLA

River BasinsErosion

The Evolution of the Surface of the Earth

Björn Birnir

Center for Complex and Nonlinear ScienceDepartment of Mathematics

UCSB

Courant Oct. 2011

Birnir The Evolution of the Surface of the Earth

River BasinsErosion

Collaborators

Terrence R. SmithGeography and Computer Science, UCSB

Jorge Hernandez

Probability and Statistics, UCSB

Russel SchwabMathematics, UCSB

Kristen MeekerGeography and Computer Science, UCSB

Ted WelshMathematics, Duke

Andrea BertozziMathematics, UCLA


River BasinsErosion

The Different Stages

Channelization: The initial process wherein many smallrivulets merge to form larger ones. As the rivulets join toform streams, they carve out the originally smoothlandsurface and form a basic network of streams andrivers.

Adolescence: The second process where young, smoothsurfaces evolve into an initially concave and eventuallyconvex shaped surface. This metamorphosis is effected byshocks first in the water flow and then in the sediment flow.

Maturation: The final process in which the rough, maturelandsurfaces develop into a distinctly convex shapedsurface which afterwards retains its overall appearance butjust decreases in elevation.


River BasinsErosion

Landsurface AgeMount Sugarloaf in Massachusetts (left) vs. Rolling Hills of California


River BasinsErosion

Properties of Rivers

Outline

1 River BasinsProperties of Rivers

2 ErosionLandsurface CharacteristicsThe Evolution of Rivers


River BasinsErosion


The Cache la Poudre River Basin near Ft. Collins, CO


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Hack’s Law

The length ` of the main river in a river basin scales withthe area A of the river basin as

` ∼ A0.58


River BasinsErosion


The Amazon River Basin


River BasinsErosion


Strahler’s Order of Streams

Streams without tributaries have order 1.

When two streams of order n join, they form a stream oforder n+1

When two streams of different orders join, the resultingstream inherits the higher order of the two.


River BasinsErosion


Strahler’s Order


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Horton’s Bifurcation Ratios

Horton discovered that if N(n)denotes the number ofstreams of order n and L(n) is their mean length then thelength and bifurcation ratios

Rl = N(n)/N(n + 1)Rb = L(n + 1)/L(n)

are constant over the entire river basin.


River BasinsErosion


Probability of Exceedance

The probability that the area of a subbasin exceeds acertain value a, scales with a

P(area of subbasin > a) ∼ a−0.42


River BasinsErosion

Landsurface CharacteristicsThe Evolution of Rivers

Outline




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Erosion


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River Valleys with Uplift


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The Variogram

How do we measure the roughness of the surface?

V (x, t) = < |u(y + x, t)− u(y, t)|2 >1/2

∼ |t |βf (t/|x|z)

for t small, and for t large

V (x, t) ∼ |x|χf(t/|x|z)


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River BasinsErosion



River BasinsErosion


The Surface Seeks a Fractal Equilibrium

There is an equivalence between time and distance inspace t ∼ |x|z given by the dynamic exponent z.

The system (width function) roughens initially as a powerof t given by the temporal roughness coefficient β.

Eventually the system gets into a statistically stationarystate where is does not roughen any more, but spatialfluctuations scale with a power of the lag variable given bythe spatial roughness coefficient χ = zβ.


River BasinsErosion


Outline




River BasinsErosion


The Merging of the Rivulets


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The Evolution of a River


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Water Depth


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The Channelization Process


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Maturation Scaling in the Stationary State


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Scaling Exponents

For the first process the values of the exponents of thewidth functions are

β = 14 , χ = 1

2 , z = 2

For the second process the exponents areβ = 0.5, χ = 0.75, z = 1.5


River BasinsErosion


Hack’s Law

A ∼ `D

The avalanche dimension D = 1 + χ, ` being the length ofthe main river, then the width of the basin in the direction`χ, perpendicular to the main river, is close to theseparable solution χ = 0.75 hence

` ∼ A1

1+χ

≈ A0.58


River BasinsErosion


The Surface Determines the River Network

Everything is determined by the Hack’s exponent h = 47

The exponent for the probability of exceedance ist = h − 1 = −3

7 ∼ −0.43

h = LogRLLogRB

Hack’s exponent is determined by the roughnesscoefficient of the surface.


River BasinsErosion


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 isalways turbulent.

Most of the sediment is carried by turbulent flow.

But turbulence is still not understood.


River BasinsErosion


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 coefficient1/(1 + 3/4) = 4/7

Thus the roughness coefficient of the surface seems to becaused by sediment transported by turbulent flow in rivers.

This also explains all the scalings of the river network.


Documents

The Evolution of the Surface of the Earth - UCSBcnls.math.ucsb.edu/files/cnls/research-documents/Courant_erosion_… · Ted Welsh Mathematics, Duke Andrea Bertozzi Mathematics, UCLA