Water retention on random surfaces

Image:Wateronsurface.png

Image:30x30fiveLevelWaterRetention.png

One system in which the retention question has been studied is a surface of random heights. Here one can map the random surface to site percolation, and each cell is mapped to a site on the underlying graph or lattice that represents the system. Using percolation theory, one can explain many properties of this system. It is an example of the invasion percolation model in which fluid is introduced in the system from any random site.{{cite journal

| last = Chayes

| first = J. T.

|author2=L. Chayes |author3=C. M. Newman

| title = The stochastic geometry of invasion percolation

| journal = Communications in Mathematical Physics

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| issue = 3

| year = 1985

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| bibcode=1985CMaPh.101..383C

| doi = 10.1007/BF01216096| s2cid = 120780380

| url = http://projecteuclid.org/euclid.cmp/1104114182

}}

{{cite journal

| last = Damron

| first = Michael

|author2=Artëm Sapozhnikov |author3=Bálint Vágvölgyi

| title = Relations between invasion percolation and critical percolation in two dimensions

| journal = Annals of Probability

| volume = 37

| issue = 6

| year = 2009

| pages = 2297–2331

| doi = 10.1214/09-AOP462| arxiv = 0806.2425| s2cid = 15969464

}}

{{cite journal

| last = van den Berg

| first = Jacob

|author2=Antal Járai |author3=Bálint Vágvölgyi

| title = The size of a pond in 2D invasion percolation

| journal = Electronic Communications in Probability

| volume = 12

| year = 2007

| pages = 411–420

| doi = 10.1214/ECP.v12-1327| arxiv = 0708.4369

| bibcode = 2007arXiv0708.4369V

| s2cid = 115164537

}}

In hydrology, one is concerned with runoff and formation of catchments.{{cite journal|last1=Tetzlaff|first1=D.|author-link=Dörthe Tetzlaff|last2=McDonnell|first2=J. J.|last3=Uhlenbrook|first3=S.|last4=McGuire|first4=K. J.|last5=Bogaart|first5=P. W.|last6=Naef|first6=F.|last7=Baird|first7=A. J.|last8=Dunn|first8=S. M.|last9=Soulsby|first9=C.|year=2011|title=Conceptualizing catchment processes: simply too complex?|journal=Hydrological Processes|volume=22|issue=11|pages=1727–1730|bibcode=2008HyPr...22.1727T|doi=10.1002/hyp.7069|s2cid=129570624 }}

The boundary between different drainage basin (watersheds in North America) forms a drainage divide with a fractal dimension of about 1.22.{{cite journal

| last = Fehr

| first = E.

|author2=D. Kadau |author3=N. A. M. Araújo |author4=J. S. Andrade Jr |author5=H. J. Herrmann

| title = Scaling Relations for Watersheds

| journal = Physical Review E

| volume = 84

| issue = 3

| year = 2011

| pages = 036116

|bibcode=2011PhRvE..84c6116F

| doi = 10.1103/PhysRevE.84.036116| pmid = 22060465

|arxiv = 1106.6200 | s2cid = 14658854

}}

{{cite journal

| last = Schrenk

| first = K. J.

|author2=N. A. M. Araújo |author3=J. S. Andrade Jr |author4=H. J. Herrmann

| title = Fracturing Ranked Surfaces

| journal = Scientific Reports

| volume = 2

| year = 2012

| pages = 348

| doi = 10.1038/srep00348|arxiv = 1103.3256 |bibcode = 2012NatSR...2E.348S | pmid=22470841 | pmc=3317236}}

{{cite journal

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|author2=D. Kadau |author3=J. S. Andrade Jr |author4=H. J. Herrmann

| title = Impact of Perturbations on Watersheds

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| volume = 106

| issue = 4

| year = 2011

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| bibcode=2011PhRvL.106d8501F

| doi = 10.1103/PhysRevLett.106.048501|arxiv = 1101.5890 | pmid=21405368| s2cid = 17763945

}}

The retention problem can be mapped to standard percolation.{{cite journal

| last = Knecht

| first = Craig

|author2=Walter Trump |author3=Daniel ben-Avraham |author4=Robert M. Ziff

| title = Retention capacity of random surfaces

| journal = Physical Review Letters

| volume = 108

| issue = 4

| year = 2012

| pages = 045703

| url = https://arxiv.org/trackback/1110.6166

| doi = 10.1103/PhysRevLett.108.045703| pmid = 22400865

|arxiv = 1110.6166 |bibcode = 2012PhRvL.108d5703K | s2cid = 12419148

}}

{{cite journal

| last = Baek

| first = Seung Ki

|author2=Beom Jun Kim

| title = Critical Condition of the Water-Retention Model

| journal = Physical Review E

| volume = 85

| issue = 3

| pages = 032103

| year = 2012

| doi = 10.1103/PhysRevE.85.032103| pmid = 22587136

|arxiv = 1111.0425|bibcode = 2012PhRvE..85c2103B | s2cid = 7438828

}}

{{cite journal

| last = Schrenk

| first = K. J. |author2=N. A. M Araújo |author3=R. M. Ziff |author4=H. J. Herrmann

| title = Retention capacity of correlated surfaces

| arxiv=1403.2082

| year = 2014 | doi=10.1103/PhysRevE.89.062141 | pmid = 25019758 | volume=89 | issue = 6 | pages = 062141 | journal=Physical Review E|bibcode=2014PhRvE..89f2141S| s2cid = 18058921 }}

For a system of five equally probable levels, for example, the amount of water stored R₅ is just the sum of the water stored in two-level systems R₂(p) with varying fractions of levels p in the lowest state:

: R₅ = R₂(1/5) + R₂(2/5) + R₂(3/5) + R₂(4/5)

Typical two-level systems 1,2 with p = 0.2, 0.4, 0.6, 0.8 are shown on the right (blue: wet, green: dry, yellow: spillways bordering wet sites). The net retention of a five-level system is the sum of all these. The top level traps no water because it is far above the percolation threshold for a square lattice, 0.592746.

The retention of a two-level system R₂(p) is the amount of water connected to ponds that do not touch the boundary of the system. When p is above the critical percolation threshold p _c, there will be a percolating cluster or pond that visits the entire system. The probability that a point belongs to the percolating or "infinite" cluster is written as P_∞ in percolation theory, and it is related to R₂(p) by R₂(p)/L² = p − P_∞ where L is the size of the square. Thus, the retention of a multilevel system can be related to a well-known quantity in percolation theory.

To measure the retention, one can use a flooding algorithm in which water is introduced from the boundaries and floods through the lowest spillway as the level is raised. The retention is just the difference in the water level that a site was flooded minus the height of the terrain below it.

Besides the systems of discrete levels described above, one can make the terrain variable a continuous variable say from 0 to 1. Likewise, one can make the surface height itself be a continuous function of the spatial variables. In all cases, the basic concept of the mapping to an appropriate percolation system remains.

A curious result is that a square system of n discrete levels can retain more water than a system of n+1 levels, for sufficiently large order L > L*. This behavior can be understood through percolation theory, which can also be used to estimate L* ≈ (p − p_c)^−ν where ν = 4/3, p = i*/n where i* is the largest value of i for which i/n < p_c, and p_c = 0.592746 is the site percolation threshold for a square lattice. Numerical simulations give the following values of L*, which are extrapolated to non-integer values. For example, R₂ < R₃ for L ≤ 51, but R₂ > R₃ for L ≥ 52:

As n gets larger, crossing become less and less frequent, and the value of L* where crossing occurs is no longer a monotonic function of n.

The retention when the surface is not entirely random but correlated with a Hurst exponent H is discussed in Schrenk et al.

• Drainage divide

{{reflist|2}}

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• https://commons.wikimedia.org/wiki/Category:Associative_magic_squares_of_order_4
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• [https://www.youtube.com/watch?v=ftcIcn8AmSY] Water retention histogram as a computing problem
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Category:Random matrices

Category:Critical phenomena

Category:Percolation theory