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River networks as emergent characteristics of open
dissipative systems
Kyungrock Paik and Praveen Kumar
4th IAHR Symposium on River, Coastal and Estuarine Morphodynamics
Department of Civil and Environmental Engineering
University of Illinois at Urbana-Champaign
AcknowledgementsNational Science Foundation grant no. EAR 02-08009
Dissertation Completion Fellowshipsfrom the University of Illinois at Urbana-Champaign
River networks exhibit apparent self-similarity, or more broadly fractal
Rogue River in Oregon
Self-similar topological organization[e.g., Peckham, 1995]
What causes this regularity?
Is this a result of an optimization process?
Optimal channel network (OCN) [Rodriguez-Iturbe et al., 1992b; Rinaldo et al., 1992]
Minimum energy expenditure
lN
iii LQEMinimize
1
5.0
Is this a result of an optimization process?
Figures from
Stevens[1974]
Is this a result of an optimization process?
Figures from
Stevens[1974]
“This ‘law-like’ feature seems to emerge as the inevitable result of a dynamic process that minimizes the dissipation of energy”- Mark Buchanan
[Nature 419, 787 (24 October 2002) ]
However, the mechanism that enables the systems to find these extremal states remains elusive
Hypothesis
Evolutionary dynamics driven by a flow gradient and subject to proximity constraint, that is, the matter and energy can traverse only through a continuum, in the presence of inherent randomness of media properties, give rise to a tree topological organization.
Grid size: 500m × 500m Domain: 40401 cells (201
× 201) Effective cells: 31397
(=7849.25 km2)
< Puerto Rico t =2 years =0.5 s=2650 kg/m3
Dynamic equilibrium condition
To test the proposed hypothesis, a deterministic numerical model is built
To test the proposed hypothesis, a deterministic numerical model is built
Regard river networks as streamlines.Streamlines: orthogonal to the geographic contourD8 method [O'Callaghan and Mark, 1984] for flow path decision• Governing eq:
• Schoklitsch [1934]
• ie=0.1mm/hr (=876mm/yr) de AiQ
01
xQ
tz
W ss
QSd
Qs2/37000
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
2 4 6 8 10 12 14 16 18
Simulation result Time: 0 years
101
102
103
104
1055
5.5
6
6.5
7
7.5
8x 10
6
Time (years)
E ( )
101
102
103
104
1050.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
Time (years)
Tec
toni
c up
lift r
ate
(mm
/yr)
Simulation result Time: 10 years
101
102
103
104
1055
5.5
6
6.5
7
7.5
8x 10
6
Time (years)
E ( )
101
102
103
104
1050.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
Time (years)
Tec
toni
c up
lift r
ate
(mm
/yr)
Simulation result Time: 102 years
101
102
103
104
1055
5.5
6
6.5
7
7.5
8x 10
6
Time (years)
E ( )
101
102
103
104
1050.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
Time (years)
Tec
toni
c up
lift r
ate
(mm
/yr)
Simulation result Time: 103 years
101
102
103
104
1055
5.5
6
6.5
7
7.5
8x 10
6
Time (years)
E ( )
101
102
103
104
1050.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
Time (years)
Tec
toni
c up
lift r
ate
(mm
/yr)
Simulation result Time: 104 years
101
102
103
104
1055
5.5
6
6.5
7
7.5
8x 10
6
Time (years)
E ( )
101
102
103
104
1050.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
Time (years)
Tec
toni
c up
lift r
ate
(mm
/yr)
Simulation result Time: 105 years
294 Watersheds
101
102
103
104
1055
5.5
6
6.5
7
7.5
8x 10
6
Time (years)
E ( )
101
102
103
104
1055
5.5
6
6.5
7
7.5
8x 10
6
Time (years)
E ( )
No tectonic upliftDynamic equilibrium
101
102
103
104
1050.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
Time (years)
Tec
toni
c up
lift r
ate
(mm
/yr)
Power law relationships can be measures of self-similarity
Figures from [Maritan et al., 1996] and [Rigon et al., 1996]
Power law relationships can be measures of self-similarity
P(Ad ) =0.43±0.03[Rodriguez-Iturbe et al., 1992]
x h
h=0.6±0.1[Hack, 1957]
P(L l) l- =0.68±0.24[Rigon et al., 1996; Crave and Davy, 1997]
Figures from [Maritan et al., 1996] and [Rigon et al., 1996]
Simulated networks exhibit these power law distributions
20 40 60 80 100 120 140 160 180 2000
20
40
60
80
100
120
140
160
180
200
100
101
102
100
101
102
Ad (km2)
x (k
m)
61.0dAx
100
101
102
10-2
10-1
100
(km2)
P(A
d
)
46.0 dAP
100
10110
-3
10-2
10-1
100
l (km)
P (
L
l )
P(L l) l-
Key results are robust regardless of the shape of islands
The insights gained here may be extended to explain the formation of other networks
Images from [Merrill, 1978], http://www.lightningsafety.noaa.gov/photos.htm, [Huber et al., 2000], and [Jun and Hübler, 2005]
20 40 60 80 100 120 140 160 180 2000
20
40
60
80
100
120
140
160
180
200
Evolutionary dynamics driven
by gradient
under spatial proximity constraint
In summary, evolutionary dynamics driven by a flow gradient and subject to proximity constraint, that is, the matter and energy can traverse only through a continuum, in the presence of inherent randomness of media properties, give rise to a tree topological organization
The minimization of energy expenditure is not the cause but a consequent signature
Findings may serve as a motif for the formation of other networks.
Questions?
