A Comparison Between Random and Deterministic Tree Networks for River Drainage Basins

César A. Marin Universidade Federal do Paraná

Curitiba, Brazil cesaugmarin@gmail.com

Marcelo R. Errera Universidade Federal do Paraná

Curitiba, Brazil merrera@gmail.com

Abstract—A fully determinisitc erosion model recently developed was used to explain the natural creation of river networks, which was assumed to be due to the optimal reduction of the flow resistance in a area to point flow. This work adds to this discussion, modifying the model to a fully random approach. The results showed that randomness does not create networks, but the global reduction of the flow resistance follows the same trend of the deterministic networks. This suggest that local factors also are important to the explanation of the very existance of natural networks.

Tree networks; drainage basins; erosion; Constructal Theory

I. INTRODUCTION Tree networks are forms omnipresent in nature on a

broad range of processes, e.g.: blood vessels configuration, nerves in leaves, dendritic solidification, roots distribution, bronchial trees, cracks in shrinking solids. The nature´s resilience in maintain and make use of tree-like networks can be interpreted as a consequence of some global principle. Therefore, the dynamic origin of these structures has been an important subject of study.

The aspects of river basins are widely known, mainly the morphology, that has been documented in various geophysics and hydrology treatises (e.g. [1]-[4]). One important feature documented is that river architectures exhibit similarities among a wide range of basins, computed on the form of scaling laws as the Horton’s Laws, Melton’s Law, and Hack’s Law as pointed out in [2]. The fractal approach recently developed brought other scaling properties, which are based on statistical factors [5]. Hack’s Law accounts for the elongation trend of basins, which is a regular criterion of evolution that has been interpreted as an evidence of randomness [4].

The first models of river network formation were also designed in a completely random framework ([1] and [6]). Afterward, it was shown that the assumption of equal likelihood of any tree-like configuration, inherent in these models, leads to the creation of networks that do not reproduce well the scaling characteristics verified in real river basins [7]. Using the concept of minimum energy expenditure, the random models were adapted to a concept now known as Optimal Channel Networks (OCN). The OCNs are created in an algorithm that starts with a random network, and make random changes channel by channel in the structure – a similar principle as Darwin’s Natural

Selection for organisms traits. The changes are accepted only if the new configuration reduces the value of a functional related to the energy expenditure ([7] and [8]). The configurations obtained with this method are able to reproduce well the scaling laws parameters found in nature.

But one important challenge remains unsolved after these achievements. The OCN creation uses a physical concept, but starts with the prior assumption that the networks exists – this assumption, though, does not comply with Darwin’s theory. In other words, it fails to explain why the nature “chooses” to use a network instead of another geometrical form (e.g. a lake or a pond, or a single wider channel). There are also deterministic models of fluvial landform evolution (e.g. [9], [10]) that consider the effects of mass diffusion, erosion, and tectonic uplift in the shape of the landforms, but the networks only appear because of the initial conditions or because of random erodibility [10].

The origin of networks in river basins was approached in [11], where concepts of Constructal Theory (CT) were used. This theory was first presented in [12] as the following: “For a finite-size system to persist in time (to live), it must evolve in such way that it provides easier access to the imposed (global) currents that flow through it”. It deals with the generation of flow configuration in nature, in a deterministic fashion [13], mainly with the global characteristics, that are not driven by local restraints [14]. References [15], [16] are recent reviews that show a wide array of configuration of natural phenomena that can be predicted by this theory, in both inanimate (e.g. duct cross sections, open channel cross sections, turbulent flow structure, global circulation and climate) and animate (e.g. allometric laws of body size, breathing, flying running and swimming) systems, along with applications to engineering (e.g. heat conduction, heat exchangers, fluid flow). Therefore, it can be seen that CT has been able to explain the very existence of the various types of flow configurations in nature, in a completely deterministic approach. Eventually one will be able to obtain, as a clear and plain result, very similar forms of river basins by CT.

The first constructal approach to tree networks was developed analytically, based on several simplifying assumptions: e.g. right angles between stems and tributaries, fully developed laminar flow, a construction sequence, constant-thickness branches ([12] and [17]). Subsequently, most of the assumptions were relaxed, and the same problem was solved numerically [18]. The networks in those resulted

2010 Third Southern Conference on Computational Modeling

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2010 Third Southern Conference on Computational Modeling

978-0-7695-3976-8/09 $26.00 © 2009 IEEEDOI 10.1109/MCSUL.2009.11

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from the need for connecting an infinit amount of points (e.g. a volume, a area or a line) to a single point (a source or a sink), balancing two flow schemes, that experience different resistances, to create the least global resistance possible [15].

In the river basins case, the higher resistance flow is the flow in porous media, and the lower resistance flow is the open channel flow (the network). Recently, using an analytical treatment similar to [12] applied to river networks [13], it was shown that the principal scaling laws of river networks can be deterministically deduced, including the tendency of elongation reported through Hack’s Law [19]. In this case, the ordering scheme was still assumed.

On the other hand, in [11] the network appears based only on an erosion model, without any assumption about ordering scheme or initial conditions. That is, the basin was self-constructing to a network. It is shown that this process leads to a reduction of the global resistance. However, it is not possible to conclude that this results is a consequence of optimization, because the placing of lower flow resistance paths on the space previously occupied by higher flow resistance is a process that reduces the global resistance, no matter where it is done.

When applying an optimization procedure at every step of change, it was verified that the configurations created in the erosion model (real nature) was very close to the optimum (not necessary a natural ocurrence), that is, by using time, a deterministic figure. But it must be cleared what would happen if the configuration was purely random. One could think that the random creation of a network could also lead to configurations with global characteristics close to the erosion patterns, or even to the optimum, what is similar to the arguing of the random approach ([1], [6]-[8]).

Therefore, the purpose of this article is to change the erosion model of [11] to a completely random model, and compare the forms and the resistances created with this approach with the ones created in the deterministic models.

The results obtained show that the global performance of the random configurations follow the same trend of decrease of the erosion configurations, being slightly poorer than the last ones. The patterns created are similar to diffusion process, with the same order of magnitude of the global permormances. This opens the discussion about the need of considering also local gains when dealing with shape in nature, instead of only global gains.

This article also adds a counter argument to the called “irreducible complexity” theory, showing that networks can be created from less complex figures.

II. SIMULATION METHODS In this work, the erosion model of [11], that is fully

deterministic, was adapted to reproduce a fully random dynamic.

The model used in [11] is represented in Fig. 1. A surface area of A = HL and shape H/L fixed is initially coated with a homogeneous porous layer of permeability K, with a very small thickness W « (H,L).

This area receives a uniform mass flow rate of an incompressible Newtonian fluid (in the case of water, this is similar to rainfall) of value ṁ´´A, that is discharged through

a small opening of size D x W placed over the origin of the system. Thus, it is created a pressure field P(x,y) that drives the fluid to the outlet.

Figure 1. Two-dimensional model of an area-to-point flow in a porous medium with the soil simulated as square blocks of side D, that can be

dislodged to form an open channel flow, adapted from [11]

If the flow through the K medium is a Darcy flow, than the pressure field established is obtained by a Poisson equation:

02

2

2

2

WKm

yP

xP

where ν is the kinematic viscosity of the fluid. Spaces

with open channel flow can be created because the soil blocks can be dislodged. This happens when the force created by the pressure field acting on the block (∆PDW) is stronger than the cohesion force of the soil (τD2, where τ is the maximum shear stress supported by the block). On these terms, Eq. (1) is changed to a dimensionless form:

0~

~

~

~2

2

2

2

MyP

xP

m

L

H

HLA

D

W

1919

where:

Dyxyx ,~,~

WDPP/

~KDmM

In this case, if s is the direction of the resultant force

acting on the block, the block is not dislodged as long as:

1~~

sP

which gives the condition to the removal of the blocks.

In every step of the simulation, this condition was applied to the blocks that had one of the sides facing to a previous removed block, which are called available blocks. The first block to be removed is the one facing to the outlet, and the dynamics follows from then. The blocks removed are then a new domain of open channel, that it was assumed to be also a Darcy flow, but with less flow resistance, translated in a higher permeability Kp.

Another model utilized in [11] is the optimization model. In this model, at every step, it was examined all the available blocks, and it was calculated the new global resistance when each block was removed alone. So, the block to be removed is the one that leaded to the largest drop in the overall resistance.

The changes made in this work are also directly related to the condition of removal. Instead of using the physics of the erosion problem, or the optimization procedure, the free blocks now have equal likelihood of being removed. They are randomly dislodged based on series of uniform distribution created to every step of the dynamics. In a mathematical viewpoint, the removal criterion is based on the following relationship:

NP 1

where ω is the block to be removed, Ω is the set of all the

available blocks at a time, and NΩ is the size of this set. At this point, it must be clear that Ω is a discrete set of random variables.

Then, the only deterministic characteristic of the problem turns to be the determination of the free blocks. Doing that, the rest of the dynamics is fully random. It was assumed a symmetry of the figure in reference to the x axis solely for computational convenience. The problem was also approached in a completely random framework, in which the blocks to be dislodged are selected randomly, between all the blocks present (not only the free ones), also assuming an

equal likelihood of removal, but the results are just commented.

In [11], it is shown that local differences are created due to the sequence in that the value of M is increased (though the global resistance is almost insensitive to this). In the work done here, the configuration doesn’t depend of physical factors, thus it is not affected by the values of M.

The final objective of the analysis is to compare the configurations obtained here (random) to the ones obtained using the erosion model (“real” world) and the optimization model. Among a realm of simulations carried out by [11], it was chosen two for the comparison: erosion model considering ∆M = 0.001, and the optimization model. Both were done considering H = L = 51D, that gives a first critical value Mc as 0.00088932 to the erosion model (the value to the removal of the first block).

The flow resistance is measured as the ration between the peak pressure and the flow forcing (M), that is:

MP

MP peak

~~maxResistance Flow

and this ration is used to compare the former cases within an optimization model basis.

.

III. RESULTS AND DISCUSSION Prior to the random computations, the results of [11],

namely the ones of Fig. 2a, were completely reproduced, to make the validation of the random model. Then, a total of six simulations were done, using different sets of random variables. The results are presented in this section.

The configurations obtained are shown in Fig 2, along with some configurations obtained by [11]. In Fig 2c, it is shown a superposition composing of the six results of the random model, where the darkness is representative of the number of simulations in that the related block was dislodged. The black ones were removed in all the simulations, while the white space means no change in any of the simulations. Fig 2a and Fig 2b where taken from [11], to elucidate the difference between the two models results.

It is clear that the configuration generated by the random model is very different from a network. It is similar to the pattern verified in the molecular diffusion process, that is created also with the erosion model, when K/Kp ~ 1 [11]. That is, assuming that the blocks have the same probability of removal is the same as assuming that the flow resistance created by the network paths is very close to the flow resistance of the body being drained, and so no network needs to be created. Therefore, no network is generated if its presence is not a prior assumption, as it is done in [1] and [6].

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Figure 2. Fig. 2 Comparison between the three models: a) The evolution of the system of the Fig. 1, obtained in [11], considering K/Kp = 0.1 and M increasing in steps of size 10-3; b) the evolution of the optimization model [11], simulated up to n = 100, also with K/Kp = 0.1; c) the evolution of 6

simulations of the random free block choose model, where the intensity of the square filling is related to the number of times that block was removed

(black means six times and white means none). It can be seen that, different from the physical model, if the network isn’t previous assumed, then a random selecting of blocks spreads the Kp domain along a nearly half

circle, centred in the origin, and no network is created.

It is important to verify the existence of “fingers” in some steps of the random configurations. Most of the “fingers” appear at some steps and vanish soon. And then other ones appear. Thus, it can be said that the mere occurrence of “fingers” is not necessarily due to a physical law.

Other important fact is that all the configurations in Fig. 2 clearly show that networks, that are complex geometrical forms, can be derived from simpler ones, as the diffusion patterns. The “irreductible complexity” theory says that, in the case of river networks, they can only be consequence of other networks. The results obtained here are a clear proof that this is a inconsistent argument.

The resistances generated by the configurations of Fig. 2 are plotted against the number of dislodged blocks, inf Fig. 3, considering the ratio K/Kp = 0.1. This shows the trend of flow resistance decrease to the erosion and random models, including the one that is not shown in Fig. 2, the fully random model. In this model, the gain with the removal of one block is almost the same along the dynamic steps.

On the other way, both the erosion model and the random choose model create configurations that have a lot of performance increase in the first steps, and the gain with any new block dislodged is more significant. In the case of the erosion model, the gain is the most significant, but there is only a slight difference between them.

Figure 3. Fig. 3 The monotonic decrease in the overall flow resistance in

both erosion and random models, showing that, up to N~50, the performance of the system is almost the same, and after that the dynamics follow the same trend. This observation, however, is not valid to the fully

random model.

a

b

c

Random Choose Model

Erosion Model

MPpeak~

Fully Random Model

1.0pK

K

2121

This can be better seen when the global flow resistances are compared with the optimization model results. In Fig 4, the ration between the flow resistances created by the erosion or random models and the ones created by the optimization model are plotted also against the number of dislodged blocks, as in Fig. 3. It can be seen that the fullly random model performances are very poor, and the resistance is constantly increasing in relation to the optimal performance values. When N is close to 100, the fully random resistance is almost the double of the optimum. This shows that the simple removal of low permeability blocks cannot explain the flow resistance reduction trend observed in the optimization model.

However, the dynamics of the flow resistance of the random choose model and the erosion model, apart for being close to each other, as noted before, are also close to the optimum values. Compared to the fully random model, these ones have an important difference: the blocks to be dislodged are selected only from the ones that are free to move, and their number is always increasing, while in the former one they are selected from all blocks available, and this number is always decreasing. Thus, the organization helps on the global performance, and this is obtained by reducing the degree of freedom.

Figure 4. Fig. 4 Dynamics of the ration between the flow resistance

obtained in the erosion and random models and the ones obtained with the optimization models, for K/Kp = 0.1. The dashed line represents the unity.

The fact that the random choose model follows dynamics close to the erosion and optimum models (that are deterministic), in terms of the global performance, suggests that this is not the only feature in the networks creation. Based on the results of Fig. 4, it cannnot be said that the network was created only because of the need of the global flow resistance reduction, as stated by CT. It must have at least one more factor that is important to the dynamics, which can be the need of local performance gain, as represented by the erosion model, that is, the changing of parts subjected to high local pressures.

It is important to see in Fig.3 that the dispersal of the monotic resistance reduction created by the random simulations are almost the same. This shows that the number of six random simulations done here is good enough for the purposes of this work.

IV. CONCLUSIONS The constructal theory, applied in a geomorphology

context, states that the configuration of river networks are resultant from a need to the maximum reduction of the global flow resistance. The results obtained here clearly indicates that the global performance is significant in the shaping of these forms, but does not explain all the phenomenon by itself.

The dynamics created by the fully random model are very poor, and this shows that the removal of low permeability blocks is not the underlying cause of the high global flow reduction verified in the erosion model. But the random choose model, which is not capable of creating networks, have a global flow reduction trend very close to the deterministic ones. This can be easily saw on Figs. 3 and 4, what indicates that must be another principle, acting on a local scale, that contributes to a specific and unique form as a network.

The degrees of freedom play a special role in this subject. The networks are only formed when, in any step, there is only one block that can be removed, that is the most “stressed” one (on the deterministic models). In the random choose case, this number gets higher at every step, while in the fully random model, although it reduces, the first number is very high (all the blocks in the basin) and, in both cases, the networks are not created.

These conclusions are important because they add to the Constructal Theory model, in the hope of creating necessarily insights about some points where the theory is still incomplete. It can be seen that the networks can be explained by CT, but not by CT alone.

At the end of this work, it is important to realize that the results obtained here, together with the ones of [11], prove that complex structures like the networks can be derived from simpler ones. This shows that the doctrine of the “irreductible complexity” is very inconsistent when putting in the river networks example.

ACKNOWLEDGMENT We wish to acknowledge Coordenação de

Aperfeiçamento de Pessoal de Ensino Superior (CAPES) for the educational and financial support; and Duke University for the license to the use of the FIDAP® software

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