**1. Introduction**

The conflicting definition of the braided pattern raises the issues concerning (a) the difference between midchannel bars and islands, (b) the precise nature of the interaction between flow stage and bars or islands, and (c) the differences between the mechanisms of channel divergence that lead to river patterns termed as "braided" and those defined as "anastomosing." Consideration is given to the factors involved in determining the shear stress distribution at the flow boundary layer. The experimental results are presented in two parts. Experimental observations of meander evolution are described qualitatively. The most important parameter is the shear stress distribution, because of the inhomogeneous distribution of boundary layer meander features. At the wavy boundary layer, the shear stress distribution, measured with WTG-50 hot – film –anemometer is given graphically and theoretically.

Bankfull discharge is generally considered to be the dominant steady flow that would generate the same regime channel shape and dimensions as the natural sequences of flows would. This is because investigation on the magnitude and frequency of sediment transport has determined that for stable rivers the flow that transports most materials in the longer term has the same frequency of occurrence as bankfull flow. For stable gravel-bed rivers, this is considered to be the 1.5-year flood [1].

The objective of regime theory is to predict the size, shape, and slope of a stable alluvial channel under given conditions. The distribution parameter at the boundary layer is a meander feature curvature with the scope of inhomogeneous shear

stress. A channel is characterized by its width, depth, and slope. The regime theory relates these characteristics to the water and sediment discharge transported by the channel empirically. Empirical measurements are taken on channels and attempts are made to fit empirical equations to the observed data. The channel characteristics are related primarily to the discharge, but allowance is also made for variations in other variables, such as sediment size.

For practical purposes, rivers are preserved to be in equilibrium (in regime) or in quasi-equilibrium with these characteristics, which have not changed over a long period of time. Canals usually maintain constant discharge, and regime relations may, therefore, be established using field data. However, field measurements for rivers are not usually suitable for establishing laws for rivers in regime.

If you use a ruler of k = 1000 m, you will need K rulers to run the entire river meander curvature. If you use a ruler of l = 500 m, you need L rulers and successively. What is the physical meaning of fractal dimension?

The number of rulers necessary to measure a meander curvature line M is proportional to the length of ruler m with an exponent D, where D is a constant that defines the dependence between the number of rules and the length of ruler and is known as a fractal dimension for measuring a river meander curvature.

It is intended to calculate fractal dimension slightly undulating line. It is found one code from net on boxcounting method (by [2]) and used for slightly undulating surface that is not given correct answer. Having x and z value of corresponding line. Is it possible to calculate from these values by any software/code [2]?

A characteristic feature of fractals is their fine structure. An object is known to have fine structure if it has irregularities at arbitrarily small scale. 'Fractal dimension' attempts to quantify the fine structure by measuring the rate at which the increased detail becomes apparent as we examine a fractal ever more closely. Fractal dimension indicates the complexity of the fractal and of the amount of space it occupies when viewed at high resolution. All definitions of dimension depend on measuring fractals in some way at increasingly fine scales.

A fractal, strictly speaking, has no "physical meaning." It is like asking about some curve we see on some Cartesian 2D coordinate frame-"what is its physical meaning? Or the curve, the function which we may have available to help us understand it and the frame of reference are all constructs of what we can now say… for purposes of brevity…is our intuition and our urge to express ourselves in ways that somehow help us deal with or cope with actions we have to take either now or in the future.

Thus, the lines we see on the graph paper have no physical meaning, per se. But that does not mean they have no "use." In fact, "use" is perhaps the best notion of "meaning." Their use if those who may be able to co further with those constructs and incorporate them into models they might work with in regard to various inquiries in science. Unfortunately, there has been little inquiry into just how and in which way and why fractals may be of use. We only tend to "look at the computer screens" and think that we "are seeing" something beyond some interesting calculations in complex number space.

In the end, complex numbers and their spaces are of far more use than real numbers and Euclidean-style geometries. Hopefully, we will be able to hone our intuitions to make use of them and of fractals in a wide range of pursuits, and, among them would be those understandings of ourselves and matters of human engagement that cannot begin to be approached with real number spaces and Euclidean assumptions about "reality."

"Reality" itself is an entirely flawed concept, which is rooted in our intuitions and imaginations being locked into a limited "real number/Euclidean/Cartesian" model for thinking and expressing ourselves. When we then speak of "reality," we

**85**

lime.

**Figure 1.**

*River Plume in Sediment-Laden Rivers*

to see what real numbers cannot [3].

ated with irregular geometry [2].

**3. River meander curvature fractals**

*Examples of naturally occurring fractal patterns in nature [5].*

**2. Method**

other engagements with the so-called physical world.

*DOI: http://dx.doi.org/10.5772/intechopen.90089*

expressly bring up the intrinsic nonsense and paradox of Cartesian coordinates and the real numbers. Fractals are the first message or signal to us that we can, in the long run, learn more about the universe and about ourselves via the creative "use; of complex numbers and indeed of complex number spaces and those number spaces further down the road of honing of intuitions such as quaternions and octonions… as well. Their beauty is a great lure and clue that there is much more than meets the eyes in our numbers…and that complex numbers can enable us and our mind's eye

That is then the first step to using them and using fractal awareness within our

The body is where most of the objects found in nature possess irregular shapes that cannot be quantified with the help of standard Euclidian geometry. In many cases, these objects have a peculiar character of self-similarity where a part of the object looks like the whole [4]. Such objects are known as fractals and the associated degree of complexity of shape, structure, and texture is quantified in terms of fractal dimension (**Figure 1**). Natural fractals do exhibit self-similarity and scale invariance; however, this is present to a limited extent [5]. For example, a part of a cauliflower may look like the whole, but if further division is made, the resulting part may not resemble much the original cauliflower after several steps. The concept of fractal was first introduced by Mandelbrot in the year 1980; he showed that the concept of fractal can be used to quantify the complexity of shape associ-

Fractal dimension of the curve is found from the slope of the best fitting straight

line to the data (fractal dimension = 1 – m), where m is the slope of the straight

Richardson's plot technique using rulers or segments of different sizes [6]. It is seen from **Figure 2** that for a given line with irregular shape, the number of segments or rulers of a given size increases as the size of the ruler is decreased. This results in different measures of the length of the curved line and the complexity of

*River Plume in Sediment-Laden Rivers DOI: http://dx.doi.org/10.5772/intechopen.90089*

expressly bring up the intrinsic nonsense and paradox of Cartesian coordinates and the real numbers. Fractals are the first message or signal to us that we can, in the long run, learn more about the universe and about ourselves via the creative "use; of complex numbers and indeed of complex number spaces and those number spaces further down the road of honing of intuitions such as quaternions and octonions… as well. Their beauty is a great lure and clue that there is much more than meets the eyes in our numbers…and that complex numbers can enable us and our mind's eye to see what real numbers cannot [3].

That is then the first step to using them and using fractal awareness within our other engagements with the so-called physical world.
