How does rain turn into floods? A chaos theory perspective.

Rain, runoff, strange attractors and power laws.

What paths does the rain take in becoming runoff? What paths does the rain take to turn into floods? What parameters may inhibit those floods from happening?

If we run a hose spouting out water at the top of a cement driveway, then the water arriving at the bottom of the driveway is reasonably correlated in rate with the water coming out of the hose. If we turn the hose on for ten minutes, and then off for ten minutes at regular intervals, we will also find the water arriving at the bottom of driveway coming in roughly ten minute intervals.

If we run a hose spouting water at the top of a highly vegetated hill, how the water arrives at the bottom is not so correlated in rate with the water coming out of the hose. Water may slow down in a patch of vegetation, it may collect for a little while in an indentation in the soil, and then break free of that indentation, and then rush off down the hill. If we turn the hose on and off every ten minutes at the top of the hill, we will probably find that the runoff at the bottom does not have this oscillating rhythm. There might be a continuous trickle of water coming out, or there might be aperiodic flows of water.

Bellie Sivakuma, a professor of water resources, and his colleagues study how the rain generates runoff patterns, and whether they exhibit chaotic behavior [ref 1]. In one study they looked at the Gota River Basin in Sweden.

Here is data they collected. The top graph is that of the monthly rainfall in the Gota River Basin over a period of 1600 months, and the bottom graph is that of the monthly runoff over a period of 1600 months [2]. You can see that the rainfall oscillates with a higher frequency. The runoff does not oscillate as much. The landscape dampens the rainfall oscillations.

Sivamuka and his colleagues study various dynamical behaviors of the rainfall and runoff. The autocorrelation function they calculate for the runoff is how much the runoff at one point in time correlates with that at a later point in time. So for instance the amount of runoff in a watershed basin today probably will correlate with the amount of runoff tomorrow. It will correlate a little less with the amount of runoff in two days. Sivamuka and his colleagues found that the runoff stayed correlated for about 20 months, the correlation dropping slowly over time. The rainfall autocorrelation on the other hand dropped off much faster in about 3 months. [note: these autocorrelation functions depend on the size of the chunks of time being correlated with each other]

In general researchers studying different places around the world have found that the rain is correlated over much shorter period of time than the runoff. Which means that the rain varies quickly, while the runoff varies more slowly. The landscape smears the rain from a short interval to flow through the landscape over a wider stretch of time.

Sivamuka was curious about whether the dynamics of the water in going from from rain to runoff was chaotic.

Chaos theory was a theory that had a revolutionary impact on the scientific world in the 1980’s. Before that it was generally thought that a small variation in input would not make a big variation in a system. Edward Lorenz and other scientists found that a flap of a butterfly wings in one country could ostensibly cause a tornado in another country. There was sensitive dependence on initial conditions in the differential equations that described the dynamical behavior in these nonlinear systems. Such nonlinear systems appeared in many fields of science - ecology, neurology, cell behavior, chemistry, astronomy, condensed matter physics, meteorology, hydrology etc…

Chaos theory stated that a simple system with very few dimensions of freedom could generate behavior that appeared random . So for instance if you hung a pendulum from the bottom of another pendulum, the subsequent behavior of it will get very chaotic, the double pendulum will careen back and forth, sometimes oscillating quietly, and then all of a sudden it will start swinging both its limbs violently.

[The first diagram shows the double pendulum along with the path the end of the double pendulum takes. The second diagram has a flourescent light shining from the end of the double pendulum. ]

The behavior of a faucet turned on ever so slightly can exhibit chaos. Water droplets might drip out slowly one drop at a time with a somewhat regular beat, then suddenly switch to dripping out quickly for a while, and then begin dripping slowly again aperiodically.

If the path of water from rain to runoff was chaotic, it would mean that the slightest shift in amount of rain could lead to quite different amounts of runoff at another time.

In the case of a hose at the top of a vegetated slope scenario, if the system if is not chaotic then the more info we get about the situation then the better predictions we can make of what the runoff will be at a later time. If the system is chaotic though, we may no longer be able to make such predictions even if we have a lot of information.

If the vegetated slope scenario is chaotic, then if we allow the hose at the top of the hill to put out a milligram more of water over a millisecond, it could change the runoff at a later time quite drastically. This might be because a drop or two might cause the pressure on some soil, that was holding back water, to become great enough that the soil falls to the side, causing a small rush of water downhill, that might then influence a larger rush of water. That small increment of water from the hose might lead to a mini-landslide that leads to larger mini-landslides.

By analysing the time series patterns scientists can begin to discern whether dynamical behavior is chaotic or not. Sivamuka and his colleagues put the time series through a bunch of calculations and determined that the runoff in the Gota River Basin, and also for other geographic areas, was chaotic. (There was some controversy around this).

By analysing the autocorrelation function of a dynamical system one can determine how many variables matter in determining the behavior of the dynamical system. This was done by whats called the Grassberger-Procacia algorithm, named after the two physicists who had discovered the algorithm. In the Gota river basin, it seems about 5 variables would be enough to determine the behavior of the runoff. Example of variables might be the velocity of the water in the x, y, and z directions, or the amount of soil moisture. It wasn’t an infinite amount of variables that needed to be be understood to understand the runoff behavior. If one was chooseing possible variables that might matter to runoff behavior one might consider wind speed, evapotranspiration rate, branches on the ground, geomorphological shape, mycelia amount, microbial state, tree root type etc…. We don’t need to know all those variables, just a subset of them, in order to determine the general dynamical nature of the runoff. Which variables are important, though is not determined by the Grassberger-Procacia algorithm.

Chaotic systems are not completely random, they are more in a state between order and randomness. One behavior it can exhibit is called a strange attractor, which limits the possible states that a system can be in. Take our previously mentioned example of the faucet that is leaking drops of water. We can plot the time between one drop coming out of the faucet and its successor coming out, versus the time between the drop’s successor coming out and its successors successor coming out. See the second diagram below. Here the strange attractor is plotted for water dripping out. The x axis is T(n), which is the time between the drop and its successor. The y axis is T(n+1), which is the time between the successor and the successor’s successor. If this faucet was displaying completely random behavior we would have dots everywhere on the graph. Instead dots only appear in a certain strange attractor shape.

We can ask whether runoff has a strange attractor. Zengyun Hu, from the lab of Desert and Oasis Ecology in China, and his colleagues, studied the river runoff from the Manas river basin in Central Asia dryland to see if it this kind of chaotic behavior.

They averaged the river runoff over a ten day period, and plotted that versus runoff over another ten day period. If the behavior of the runoff was purely random the graph should be more evenly filled in. Instead we find that there is a hole in the middle, where certain values don’t exist. In this graph below we can see that if there was 500 meters cube per second of water flowing through on average during a ten day period, in one of the following ten day periods it is very unlikely that it will have a flow of 500 meters cube per second. However if during one period it had a flow of about 1500 meters cube per second, it might then very well have a flow of 500 meters cube per second in a successive period. This is not random behavior. This picture suggests the runoff is behaving like a strange attractor.

How does the river runoff size vary?

Generally speaking one way sizes can vary is the way people’s heights vary. There is an average height of humans, with variation around that. This type of data behavior is called a Poisson distribution, or a bell curve. Scores on test at school also often have this bell curve.

Earthquakes, on the other hand, do not have this distribution, there is no average size of an earthquake. There are earthquakes 10 times as big, and 100 times as big, and 1000 times as big. Its different than height distribution as we do not see people 10 times as tall. Earthquakes have what is called a power scaling law. Earthquakes are fractal. Fractal means self-similar over all size scales.

There is less data on floods than earthquakes, but some analysis suggests that rivers also exhibit similar scaling behavior that earthquakes have. So there are river flows that are 10 times as big, 100 times as big, and 1000 times as big, which would suggest that there is no average amount of river flow, river flow instead is fractal in size scale.

In the diagram below [3] we can see that rivers in California, New Mexico, Washington and Nebraska obey this power scaling law. The maximum river flood discharge (labelled Q) is plotted versus how many years it is likely before such a size event happens. If we make both the axis a log scale, then a straight line of data points in this graph indicates the river runoff obeys a power scaling law. The slopes have different values for these different rivers, which means that there are different geographic parameters influencing the likelihood of different floods. (my conjecture: the slope magnitude is influenced by the ability of the soil to absorb water)

We can ask if rain itself obeys these scaling laws? There is not as much clear data about rain as with earthquakes. And these scaling laws depends on what property of the rain we are measuring. If we are measuring the size of a rain event, then it seems that it obeys one power scaling law up to an extent, and then it may changes its behavior after that. The diagram below is of data from different Chinese cities. [4] The length of a rain event is plotted versus number of rain events with that rain length. We can see after that after a rain duration of ten days the slope changes, suggesting that its no longer obeying a scaling law for rain events longer than ten days.

So the question arises how does the rain turn into runoff/floods when there are fractal multiscale behaviors happening? Does large sudden rains get spread out into runoff that happens over months? Is there always a large discharge of runoff after every large rain? (There is some evidence that is not always the case). How is the slope of the scaling law, called the power law exponent, affected by vegetation, soil, geomorphology, paved roads, and urban concrete?

Does the scaling law of floods/runoff have to do with chaotic dynamics? The rules governing fluid behavior, called the Navier Stokes equations, give rise to chaotic dynamics. The turbulent fluid of the Navier-Stokes equations can evolve whorls of all sizes. Does this turbulence multiscale behavior relate to runoff’s multiscale behavior?

[turbulence in a fluid]

Does the power law scaling of floods have to do with self-organized criticality, a theory that has been used to explain the power law behavior of earthquakes, wildfires, and neurons firing?

We will explore this more in upcoming essays.

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Reference

[1] Sivakumar, Bellie, Ronny Berndtsson, Jonas Olsson, and Kenji Jinno. "Evidence of chaos in the rainfall-runoff process." Hydrological Sciences Journal 46, no. 1 (2001): 131-145. https://www.tandfonline.com/doi/abs/10.1080/02626660109492805

[2] Sivakumar, B., R. Berndtsson, J. Olsson, K. Jinno, and A. Kawamura. "Dynamics of monthly rainfall-runoff process at the Gota basin: A search for chaos." Hydrology and Earth System Sciences 4, no. 3 (2000): 407-417 https://hess.copernicus.org/articles/4/407/2000/

[3] Hu, Zengyun, Chi Zhang, Geping Luo, Zhidong Teng, and Chaojun Jia. "Characterizing cross-scale chaotic behaviors of the runoff time series in an inland river of Central Asia." Quaternary International 311 (2013): 132-139 https://www.sciencedirect.com/science/article/abs/pii/S1040618213004308

Wang, Zhiliang, and Chunyan Huang. "Self-organized criticality of rainfall in central China." Advances in Meteorology 2012 (2012) https://www.hindawi.com/journals/amete/2012/203682/