Models and Monte Carlo Simulations of the Mean Sinuosity of Major Meandering Rivers ()

Pirooz Mohazzabi^{*}, Qinghua Luo^{}

Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, USA.

**DOI: **10.4236/jamp.2022.107161
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Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, USA.

The purpose of this research is to investigate the sinuosity of major rivers in the United States and the world, and to compare them to that predicted by the existing theories. It is shown that the average sinuosity of meandering rivers deviates considerably from what has been reported previously as *π*. Calculations of the mean value of actual sinuosities of major rivers in the United States and in the World show that this average is very close to 2. Exact models as well as a Monte Carlo simulation for meandering rivers that is based on Gaussian probability distribution function are also presented, and the possibility of composite meandering is discussed.

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Mohazzabi, P. and Luo, Q. (2022) Models and Monte Carlo Simulations of the Mean Sinuosity of Major Meandering Rivers. *Journal of Applied Mathematics and Physics*, **10**, 2368-2380. doi: 10.4236/jamp.2022.107161.

1. Introduction

As a river or stream flows, as a result of various disturbances it normally does not flow in a straight path, but winds snakelike so that the curvilinear length (actual length along the curve) of the river *L* is longer than its Euclidean distance (straight end-to-end distance) *D*. This phenomenon is known as “meandering” [1].

The dynamics of meandering rivers have been a subject of interest to geologists and geographers alike. However, due to the complexity of the phenomenon and nonlinearity of the governing equations, understanding the process from theoretical and computational points of view has remained a challenge. Nevertheless, many computational models have been developed to simulate the dynamics of such rivers involving various assumptions and approximations [2] - [10].

Various aspects of meandering rivers have been subject of discussions for many years. For example, the fractal nature of these rivers has been suggested by Mandelbrot [11] and further studied by Snow [12], Montgomery [13], and Stolum [14]. Another characteristic of a river is its sinuosity *s*, defined as the ratio *L*/*D*,

$s=\frac{L}{D}$ (1)

where, clearly $s\ge 1$.

The actual profile of a river can have infinitely many shapes, and its sinuosity can have any value greater than or equal to unity. However, various models have been suggested to approximate the shape of a river. This includes circular and other types of smooth curves [15], as well as simulation models [16] [17] [18]. Based on the assumption of fractal geometry and computer simulations, it has been suggested that the mean sinuosity of rivers should have a value of *π* [18]. However, this result has not been verified by the actual sinuosities of rivers. In fact, a simple examination of the data reveals that, on the average, sinuosity of major rivers is substantially different from *π*.

In this article, we evaluate the actual sinuosities of major rivers in the United States and around the World, and calculate their average. We then suggest two Monte Carlo models, a parabolic and a zig-zag model, each involving a single-parameter Gaussian probability density function to calculate the average theoretical sinuosity of rivers. The parameter of the models is then adjusted to produce the observed mean sinuosity of the rivers in each case.

2. Observed Sinuosities of Rivers

Using the available data for the major US and World rivers, as well as the information obtained from the Google Maps, we calculate the sinuosity of each river, and then we find the average value of the sinuosities for each category.

2.1. Major US Rivers

Table 1 shows the curvilinear lengths, Euclidean distances, and the sinuosities of major rivers in the United States [19]. Figure 1 shows sinuosity as a function of river number as listed in Table 1. Since curvilinear lengths of the rivers decrease with river number, we see from the figure that there is no correlation between curvilinear length and sinuosity of the rivers. The mean and the standard deviation of these sinuosities are $\langle s\rangle =2.10\pm 0.49$. This mean value is shown by the horizontal line in Figure 1.

2.2. Major World Rivers

Table 2 shows the curvilinear lengths, Euclidean distances, and the sinuosities of major rivers in the World, including some of those in the United States listed in Table 1 [20]. The sinuosities of these rivers as a function of river number are shown in Figure 2. Since according to Table 2, curvilinear length of the rivers decrease with river number, Figure 2 shows that again there is no correlation

Table 1. Major rivers of the United States and their lengths *L*, straight end-to-end distances *D*, and sinuosities *L*/*D*.

Table 2. Major rivers of the world and their curvilinear lengths *L*, Euclidean distances *D*, and sinuosities *L*/*D*.

Figure 1. Sinuosities of major rivers in the United States as a function of river number (circles) according to Table 1. The horizontal line represents the mean value of the sinuosities.

Figure 2. Sinuosities of major World rivers as a function of river number (circles) according to Table 2. The horizontal line represents the mean value of the sinuosities.

between the curvilinear length and the sinuosity of these rivers. The mean and the standard deviation of these sinuosities are $\langle s\rangle =2.17\pm 0.65$. This mean value is shown by the horizontal line in Figure 2.

3. Stochastic Models and Monte Carlo Simulations

3.1. Zig-Zag Paths

The zig-zag path of a meandering river is simulated by a Monte Carlo method [21], using random numbers drawn from a Gaussian (or normal) probability density function [22] [23],

$f\left(y\right)=\frac{1}{\sigma \sqrt{2\pi}}\mathrm{exp}\left[-\frac{{\left(y-\mu \right)}^{2}}{2{\sigma}^{2}}\right]$ (2)

with $\mu =0$, and adjustable standard deviation $\sigma $.

We choose a river with straight end-to-end distance of 2000 km, and a unit length of 1 km. We draw random numbers according to the Gaussian probability density function, which can be done by either the Box-Muller method or the acceptance-rejection method [24]. These random numbers are taken to be the heights *h* in Figure 3. Then the length of each section of the zig-zag, and hence the total length of the river *L* is calculated. Finally, if *D* is the straight-line distance between the two ends of the river, the sinuosity of the river is calculated from

$s=\frac{L}{D}$ (3)

We repeat this Monte Carlo experiment 1000 times and calculate the average value of the sinuosity
$\langle s\rangle $, and adjust the value of *σ *to obtain the observed value of the sinuosity. The results are shown in Table 3 for the United States and World rivers along with the corresponding adjusted value of *σ *in each case.

Figure 3. A zig-zag path model for meandering rivers.

Table 3. The observed and simulated (MC) values of sinuosities of the United States and World rivers. The simulated values are based on the zig-zag model with the Gaussian standard deviations *σ *shown.

3.2. Parabolic Paths

Consider a parabolic curve shown in Figure 4, whose equation is

$y=ax\left(x-1\right)$ (4)

where *a* is a constant. But in terms of the height of the parabola*h*, the equation of this parabolic curve can be written as

$y=-4hx\left(x-1\right)$ (5)

The length of the parabolic curve described above is given by [25]

$l={\displaystyle {\int}_{0}^{1}}\sqrt{1+{\left(\frac{\text{d}y}{\text{d}x}\right)}^{2}}\text{\hspace{0.05em}}\text{d}x={\displaystyle {\int}_{0}^{1}}\sqrt{1+16{h}^{2}{\left(2x-1\right)}^{2}}\text{\hspace{0.05em}}\text{d}x$ (6)

Evaluation of this integral gives

$l=\frac{1}{2}\sqrt{1+16{h}^{2}}-\frac{1}{8h}\mathrm{ln}\left(-4h+\sqrt{1+16{h}^{2}}\right)$ (7)

We choose a river of length 2000 km, and take the unit of length to be 1 km. We assume that the river meanders on a parabolic curve of height *h* in each unit of distance along the straight line from the beginning to the end of the river, as shown in Figure 4. The height of the parabolic curve in each step *h* is randomly chosen from the Gaussian distribution function,

$f\left(h\right)=\frac{1}{\sigma \sqrt{2\pi}}\mathrm{exp}\left(-\frac{{h}^{2}}{2{\sigma}^{2}}\right)$ (8)

The sinuosity of the river is then calculated by adding the length of the parabolic curves in each step and dividing it by the straight end-to-end distance of the river. This Monte Carlo experiment is then repeated 1000 times and the average sinuosity of the river is calculated which, of course, a function of the parameter *σ *of the Gaussian probability density function (8). We then adjust the value of *σ *to obtain the observed value of the sinuosity. The results are shown in Table 3 for the United States and the World rivers along with the corresponding adjusted value of *σ *in each case.

Figure 4. A parabolic curve with equation $y=-4hx\left(x-1\right)$.

4. Exact Models

A meandering river can also be modeled by a deterministic curve that repeats itself. Consider a function $y=f\left(x\right)$ defined on the interval $\left[\mathrm{0,}d\right]$ as shown in Figure 5(a). The length of this curve is given by

$l={\displaystyle {\int}_{0}^{d}}\sqrt{1+{\left(\frac{\text{d}y}{\text{d}x}\right)}^{2}}\text{\hspace{0.05em}}\text{d}x$ (9)

If this curve (called the *basis*) repeats itself, alternating on two sides of a straight line, it generates the profile of a meandering river, as shown in Figure 5(b). If the river consists of *n* basis, its sinuosity is given by

$s=\frac{nl}{nd}=\frac{l}{d}=\frac{1}{d}{\displaystyle {\int}_{0}^{d}}\sqrt{1+{\left(\frac{\text{d}y}{\text{d}x}\right)}^{2}}\text{\hspace{0.05em}}\text{d}x$ (10)

Analytical evaluation of the integral in Equation (10) is not always possible. Nevertheless, it can always be evaluated numerically. However, in simple cases, the sinuosity can be obtained in closed form [15]. For example, consider a basis consisting of a circular arc as shown in Figure 6(a). The length of this arc is
$l=r\theta $, where *r* is the radius of the circular arc, and its Euclidean end-to-end distance is obtained from the cosine law,

$d=r\sqrt{2\left(1-\mathrm{cos}\theta \right)}$ (11)

Therefore, the sinuosity of a river obtained from this basis, shown in Figure 6(b), is given by

$s=\frac{l}{d}=\frac{\theta}{\sqrt{2\left(1-\mathrm{cos}\theta \right)}}$ (12)

which is independent of the radius of the circular arc. If the sinuosity of a river is

Figure 5. An exact model for meandering rivers. The profile of the river (b) is generated by alternating the basis curve (a) on two sides of a straight line.

Figure 6. Circular model for meandering rivers.

known, this equation can be solved numerically for $\theta $. For the major rivers of the United States and of the World with a mean sinuosity of about 2.0, we find $\theta =3.79\text{\hspace{0.17em}}\text{rad}={217}^{\circ}$.

5. Discussion and Conclusions

The analysis of the data for major rivers in the United States and in the World shows that the mean sinuosity of rivers is not *π* as suggested previously [18]. Instead, the data for both classes of rivers show that the mean sinuosity is closer to 2. This is further evidenced by the fact that *π *does not fall within one standard deviation from the mean sinuosities of the major United States and World rivers.

Monte Carlo simulations using random numbers from a Gaussian probability distribution, with fairly small standard deviations, generate the observed mean sinuosity of the rivers with either a zig-zag model or a parabolic model as examples. Exact curves can also be used to model meandering rivers, as we have shown by simple circular curves.

In the calculation of sinuosities, there were a couple of rivers with sinuosities greater than 4. We ignored those rivers in our calculations due to the fact that large sinuosities are caused by higher composite meandering effects. To see this, consider a river shown in Figure 7, where a small-scale meandering is superimposed on a large-scale one. The total sinuosity is given by

$s=\frac{L}{D}=\frac{L}{d}\frac{d}{D}$ (13)

where *d* is the length of the curve shown by the dotted line in the Figure 7. But
$L/d={s}_{1}$ where
${s}_{1}$ is the sinuosity of the small-scale meandering, and
$d/D={s}_{2}$ where
${s}_{2}$ is the sinuosity of the large-scale meandering. Therefore, we have

$s={s}_{1}{s}_{2}$ (14)

Examples of high composite meandering rivers are Kama River [26] and Kizilirmak River [27], with composite sinuosity of 6.78 and 4.66, respectively. It is, of course, possible for a river to have even higher composite sinuosity.

Figure 7. Composite meandering.

In conclusion, the observed sinuosities of major rivers in the United States and in the World have a mean value of about 2 (or more accurately 2.1 which is very close to 2*π*/3), and not *π* that has been suggested based on the assumption of fractal geometry and idealizing the river geometry as a perfectly symmetrical sequence of bends.

These meandering rivers can be modeled by either stochastic simulations or by exact curves. In each case, a parameter in the governing equation needs to be adjusted to reproduce the observed mean sinuosity.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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