water

Article A Cusp Catastrophe Model for Alluvial Channel Pattern and Stability

Yi Xiao 1,*, Shengfa Yang 1 and Mi Li 2

1 National Inland Waterway Regulation Engineering Research Center, Chongqing Jiaotong University, Chongqing 40074, China; [email protected] 2 Sichuan Energy Industry Investment Group Co.,Ltd., Chengdu 610000, China; [email protected] * Correspondence: [email protected]; Tel.: +86-023-6265-4621



Received: 17 February 2020; Accepted: 9 March 2020; Published: 11 March 2020


Abstract: The self-adjustment of an alluvial channel is a complicated process with various factors influencing the stability and transformation of channel patterns. A cusp catastrophe model for the alluvial channel regime is established by selecting suitable parameters to quantify the channel pattern and stability. The channel patterns can be identified by such a model in a direct way with a quantified index, which is a 2D projection of the cusp catastrophe surface, and the discriminant function is obtained from the model to distinguish the river state. Predictions based on this model are consistent with the field observations involving about 150 natural rivers of small or medium sizes. This new approach enables us to classify the channel pattern and determine a river stability state, and it paves the way toward a better understanding of the regime of natural rivers to assist decision-making in river management.

Keywords: fluvial process; channel stability; channel pattern; cusp catastrophe model

1. Introduction The self-adjustment of an alluvial channel is a complicated process with various factors influencing the stability and transformation of channel patterns. Although the formation of alluvial channels has been extensively studied, the problem of the alluvial channel stability still remains to be systematically resolved. Analysis of channel stability entails the determination of how a channel adjusts its river geometry in response to the water and sediment influxes imposed by its watershed [1]. The most important parameters for determining channel stability are its top width, average flow depth, flow velocity, and longitudinal slope. Therefore, efforts to understand the sustainability concept and present hydraulic geometry relations have been made. Several previous works have obtained a series of relationships for predicting a stable channel. In general, there are three approaches to determine the channel status: (1) empirical relationships and the regime method [2,3]; (2) the mechanical method of tensile force [4,5]; and (3) extremal or limiting methods [6–8]. The main difference between these theories is in the hydraulic mechanisms used to explain how flow balance can be achieved. In fact, the fundamentals of the regime theory lie in statistical methods with large numbers of observations. However, the basic assumption in all theories is constant and uniform flow intended to achieve equilibrium and semi-equilibrium balance. The identification of geomorphic and hydrologic thresholds that determine alluvial channel patterns (e.g., braided, meandering, and straight) has long been an important research focus in fluvial systems; it is now well known that alluvial channel patterns are influenced by at least seven primary controlling variables: channel slope, discharge, valley confinement, sediment supply, grain diameter, bank strength, and wood loading [9,10]. While the role of each of these variables is generally understood, this fundamental link between form and process has provided the theoretical basis upon

Water 2020, 12, 780; doi:10.3390/w12030780 www.mdpi.com/journal/water Water 2020, 12, 780 2 of 19 which numerous river classification schemes have been developed [11–14]. However, there are just a few attempts to incorporate these variables into a predictive model of channel pattern [10]; some researchers statistically analyzed large and complex geospatial data sets and developed predictive models of channel patterns [10,15,16]. The diversity of these methods and the selection criteria for one or another is influenced by different reasons, for example, objectives and scale of work (site, network, and watershed), time requirements and data, statistical and programmer expertise, or the complexity of the area of study and its analysis and interpretation. Catastrophe theory is a mathematical method for describing system behavior where a gradually changing force can produce a sudden dramatic effect. Rene Thom published the first paper on it in 1968, and then the first book in 1972 [17], and now it has found applications in a wide variety of disciplines, including physics, biology, psychology, economics, and geology [18–22]. With the development of nonlinear mathematics, some researchers have adopted catastrophe theory to describe the changes in river morphology. Thornes studied the characteristics of Spanish rivers based on the nonlinear mathematics theory [23]; Richards brought catastrophe theory into the mechanism of fluvial processes to discuss the variables in the transition [24]; Graf described the transformation among straight, meandering and braided river with the cusp catastrophe theory, and pointed out that rivers may result in different patterns although under the same sediment-discharge and boundary conditions [25]. The research results illustrate the effectiveness of the stability analysis of river patterns using catastrophe theory. Nevertheless, the catastrophe theory only explains the phenomenon for channel stability and their transition and lackeds quantitative results. This paper aims to propose an equilibrium equation of channel regime to investigate channel stability and then establish a function to distinguish the alluvial channel patterns, based on the cusp catastrophe model. The model enables the discrimination and the prediction of the equilibrium state for channel patterns based on the appropriate control and state variables. One hundred fifty natural rivers are used to test the reliability of the model, and then the model is applied in the river reach in the upstream reaches of the Yangtze River to investigate the temporal change of channel pattern and its statue. The results indicate that this approach can be used to judge river patterns and states when the control factors can be obtained in the field, to study the regime of natural rivers, and to give some suggestions to river management.

2. The Cusp Catastrophe Model for Channel Regime

2.1. The Cusp Catastrophe Model Behaviors of dynamical systems can usually be described by differential equations called entropy function. In applying science, it is customary to divide it into two groups: first, the space is of finite dimension (LPS), and second, it is of infinite dimension (DPS). In this paper, we adopt the first type, called lumped-parameter systems (LPS). The phase’s space of LPS is a finite-dimensional space, which is defined as an n-dimensional Euclidean space. The LPS can be usually written in the form:

dzi dS = = f (z , z zn, α , α αm), i = 1, 2 n (1) dt 1 2 ··· 1 2 ··· ··· where zi denotes the state variable, t is time, and αi is generally a system parameter. In this study, there are two assumptions for the dynamical system: one is autonomous to make sure the time variable does not occur on the right-hand sides (RHS) explicitly, and the other is continuous by using the continuously differentiable functions for their arguments. The steady-state (stationary) solution z of the system satisfies the nonlinear equation as follows:

f (z , z zn, α , α zm) = 0, i = 1, 2 n (2) 1 2 ··· 1 2 ··· ··· There are many types of approaches to investigating dynamic systems. The most general, qualitative, and topological techniques are used to classify topologically distinct types of behavior Water 2020, 12, 780 3 of 19 possible in the given model. Catastrophe theory belongs to this category. It was created by Rene Thom [17]. The theory is based on new theorems in multi-dimensional geometry, which classify the way that the discontinuities can occur in terms of a few archetypal forms. Of the seven elementary catastrophes in catastrophe theory, the cusp model is the most widely applied. It is often used to model the behavior of a system with two control parameters and a non-linear state parameter. The potential function E for the cusp catastrophe is [26]

1 1 E = z4 + xz2 yz (3) 4 2 − Its equilibrium points, as a function of the control parameters x and y are solutions to the equation as follows: dE = f (x, y, z) = z3 + xz y = 0 (4) dz − where x, y are defined as the control parameters; z is the state parameter of the system. Based on the similarity between the entropy function of dissipative structures and the potential function in the catastrophe theory, it can be described as the steady-state equation to express the state of a system with the cusp catastrophe model:

dE dS = (5) dz Cardan’s discriminant can be used to describe the state of the dynamic system, which is written as

∆ = 4x3 + 27y3 (6) where Equation (6) has one solution if > 0, and has three solutions if < 0. The set of values of x and 4 4 y for which = 0 demarcates the bifurcation set. These solutions are depicted as a two-dimensional 4 surface in a three-dimensional space, the floor of which is the two-dimensional (x, y) coordinates system called the "control plane".

2.2. Determination of the Control Parameters and the State Parameter The morphology of natural river channels is determined by the interaction of fluid flow with the erodible materials in the channel boundary [27]. The interrelationships between the (1) oncoming bed material load and the sediment carrying capacity of the flow, and (2) the erodibility of the river banks and the erosive power of the water decide essentially the general direction of the self-adjustment. Besides a variety of available methods for predicting channel stability parameters, a point of high research interest is the effect of hydraulic parameters and factors, such as the discharge, median grain size, Shields parameter, vegetation type, average roughness, and sediment composition [28–32]. On the basis of the preliminary analysis of the multitude of parameters reported in the literature, the synthetic indexes of river-bed stability determined by the longitudinal river-bed stability in contrast to the transverse river-bed stability are adopted in this study, the longitudinal index of B.и.AcrpaxaниeB, and the transverse index of C. T. Aлтунинare expressed in the form [27]:

B BJ0.2 φ = = (7) b 0.5 B1 Q

(γs γ)D50 φ = − (8) h γhJ where φh is the longitudinal index, and φb is the transverse index; h, J are defined as the local water depth and channel slope; Q denotes the bank-full discharge; B is the channel width under the bank-full discharge; D50 is the median bed material grain size; γ,γs is the volumetric weight of the water and sediment, respectively. Water 2020, 12, 780 4 of 19

The transverse index can reflect the channel planform changes; we decide φ1 as the dominant control parameter and φ2 as the second control parameter that determines the channel scouring and deposition. The control parameters can be written as:

a φ1 = (φb) (9)

b φ2 = (φh) (10) where a, b are the weight coefficients, satisfied with a + b = 1. Considering the importance of the oncoming bed material and the erodibility in the river process, we assumed a = b = 1/2 when a + b = 1. The state parameter is often used to describe the behavior of a system, which is influenced by the control parameters [19]. The sinuosity l is defined as the ratio of channel length to valley length, and responses to the changes of the control parameters. So we chose it as the state parameter in this study.

2.3. The Cusp Catastrophe Model for Alluvial Channel Regime Based on the features of the topological relationship, the characteristics and the continuity of the equation can be maintained with the coordinate transformation. Translate the coordinate into the O0 (φ1, φ2, l), and obtain the following relationship for control parameters in the form:

x = m l + m φ + m φ φ c 1 2 1 3 2 − 1 y = n l + n φ + n φ φ c (11) 1 2 1 3 2 − 2 z = b l + b φ + b φ lc 1 2 1 3 2 − here x, y, z is the value of the coordinate in O0(φ1, φ2, l); bi, mi, ni are the direction cosines between the original and new coordinate system. Substitute Equations (9)–(11) to Equation (4), the equation of the cusp catastrophe model under the new coordinate system O0(φ1, φ2, l) can be expressed as:

f (l, φ , φ ) = (b l + b φ + b φ lc)3 + (m l + m φ + m φ φc ) 1 2 1 2 1 3 2 − 1 2 1 3 2 − 1 (12) (b l + b φ + b φ lc) (n l + n φ + n φ φ c) = 0 1 2 1 3 2 − − 1 2 1 3 2 − 2 As shown in Figure1, the rotation around the z-axis of the original coordinate to the coordinate system O0(φ1, φ2, l) contributes to b1 = 1; and bi, mi, ni satisfy the additional condition as:

2 + 2 + 2 = m1 n1 b1 1 2 + 2 + 2 = m2 n2 b2 1 (13) 2 + 2 + 2 = m3 n3 b3 1

The coordinate transformations satisfy the right-handed system:

m1 m2 m3

n1 n2 n3 = 1 (14)

b1 b2 b3

Substitute Equation (13) to Equation (14), and obtain:

m1 m2 m3 0 cos γ sin γ c γ = arctanl n1 n2 n3 = 0 sin γ cos γ (15) − b1 b2 b3 1 0 0

There is no external force when the control parameters (φ1, φ2, l) equal to zero, the channel should maintain its pattern as Equation (12) and can be expressed as:

lc3 lcφ c φc = 0 (16) − 1 − 2 Water 2020, 12, x FOR PEER REVIEW 5 of 18

φ φ = − c 3 + φ θ −φ c −φ θ − θ + c3 = f (l, 1, 2 ) (l l ) l( 1 cos 1 ) 2 (sec l cos ) l 0 (17)

θ = arctanl c (18) Water 2020, 12, 780 5 of 19 where φc1, lc is the critical index, assuming the critical value of state parameter denotes lc = 1, and the critical transverse river-bed stability φc1 = 1 when the rivers attain the equilibrium state. EquationSubstitute (17) denotes Equations the (14)–(16)equilibrium into Equation state of (12), the we channel have regime under the coordinate system φ φ O′ ( 1, 2, l). The surfacef (l ,ofφ ,equilibriumφ ) = (l lc)3 channel+ l(φ cos stateθ φ inc) aφ 3-dimensional(sec θ l cos θ) +spacelc3 = 0is shown in (17) Figure 2: 1 2 − 1 − 1 − 2 − the pointsWater located 2020, 12 on, x FOR the PEER upper REVIEW or lower sheet are minimal (steady equilibrium), a smooth5 of change 18 in θ = arctanlc (18) (φ1, φ2) maintains the equilibrium state. The middle sheet, often referred as the inaccessible region, c c φ φ = − c 3 + φ θ −φ c −φ θ − θ + c3 = c representswhere an unstableφ 1, l is the maximum criticalf (l, index,1, 2[14];) assuming(l ifl )thel the(rivers1 cos critical lie1 value ) in 2this(sec of state area,l cos parameter any) l slight denotes0 alterationl = 1, and by(17) the a control c critical transverse river-bed stability φ 1 = 1 when the rivers attain the equilibrium state. parameter could result in the behavior being trθansferred= arctanl c to either the upper or lower sheet.(18)

where φc1, lc is the critical index, assuming the critical value of state parameter denotes lc = 1, and the critical transverse river-bed stability φc1 = 1 when the rivers attain the equilibrium state. Equation (17) denotes the equilibrium state of the channel regime under the coordinate system O′ (φ1, φ2, l). The surface of equilibrium channel state in a 3-dimensional space is shown in Figure 2: the points located on the upper or lower sheet are minimal (steady equilibrium), a smooth change in (φ1, φ2) maintains the equilibrium state. The middle sheet, often referred as the inaccessible region, represents an unstable maximum [14]; if the rivers lie in this area, any slight alteration by a control parameter could result in the behavior being transferred to either the upper or lower sheet.

Figure 1. The cusp catastrophe model in the new coordinate system. Figure 1. The cusp catastrophe model in the new coordinate system. Equation (17) denotes the equilibrium state of the channel regime under the coordinate system O0 (φ1, φ2, l). The surface of equilibrium channel state in a 3-dimensional space is shown in Figure2: the points located on the upper or lower sheet are minimal (steady equilibrium), a smooth change in (φ1, φ2) maintains the equilibrium state. The middle sheet, often referred as the inaccessible region, represents an unstable maximum [14]; if the rivers lie in this area, any slight alteration by a control parameter could resultFigure in the 1. The behavior cusp catastrophe being transferred model in tothe either new coordinate the upper system. or lower sheet.

Figure 2. The equilibrium channel state surface (the dotted line is the intersection between φ2 = 4 and the equilibrium channel state surface).

Figure 2. The equilibrium channel state surface (the dotted line is the intersection between φ2 = 4 and Figure 2. The equilibrium channel state surface (the dotted line is the intersection between φ2 = 4 and the equilibrium channel state surface). φ φ To investigatethe equilibrium the applicability channel state surface).of the cusp model presented herein, the control parameters 1, 2 are obtained from a large dataset of natural rivers, including different types of channels as straight, To investigate the applicability of the cusp model presented herein, the control parameters φ1, φ2 To investigate the applicability of the cusp model presented herein, the control parameters φ1, φ2 meandering,are obtained and braided. from a largeThe datasetdetails of of natural the river rivers, data including are shown different in Table types of 1. channels as straight, are obtained from a large dataset of natural rivers, including different types of channels as straight, meandering, and braided. The details of the river data are shown in Table1. meandering, and braided. The details of the river data are shown in Table 1. Table 1. Observed data of natural rivers.

WidthTable 1. ObservedDischarge data of naturalD50 rivers. Slope Depth Sinuosity Data Source No × (m)Width (mDischarge3/s) (mm)D50 Slope( 1000) Depth (m) Sinuosity (-) Data Source No 1.Andrews [33] 18 5.21–83.8(m) 2.21–255(m3/s) 23–122(mm) (× 1.74–22.1000) 23 (m) 0.29–1.85 (-) 1.07–1.98 2.Church and 1.AndrewsRood [34] [33] 17 18 5–104 5.21–83.8 2.7–354 2.21–255 23–122 33–89 1.74–22. 0.97–13.723 0.29–1.85 0.65–3.06 1.07–1.98 1.0–1.65 2.Church and Rood [34] 17 5–104 2.7–354 33–89 0.97–13.7 0.65–3.06 1.0–1.65

Water 2020, 12, x FOR PEER REVIEW 6 of 18

3.Hey and Thorne [35] 18 6.5–76.5 3.9–424 13.9–83.8 2.108–11.4 0.68–3.21 1.33–2.5 Water4.Kellerhals 2020, 12, x FOR et al.PEER [36] REVIEW 24 18–442 7.93–2606 0.2–145 0.12–16.5 0.58–7.2 1.01–2.26 of 18 5.LambeekWater 2020, 12 [37], 780 4 111–450 1841–5320 12–73 0.22–1.6 3.8–10 6 of 19 1.1–1.7 3.Hey6.McCarthy and Thorne et al. [35] [38] 18 1 6.5–76.5 137 3.9–424 630 13.9–83.8 0.35 2.108–11.4 0.4 0.68–3.21 4.1 1.33–2.5 1.86 7.Monsalve4.Kellerhals and et al. Silva [36] [39] 24 3 18–442 140–160 7.93–2606 830–1200 0.2–145 0.4–6.2 0.12–16.5 0.24–0.8 0.58–7.2 3–5 1.01–2.2 1.2–2.5 Table 1. Observed data of natural rivers. 8.Morton5.Lambeek and Donaldson[37] 4 111–450 1841–5320 12–73 0.22–1.6 3.8–10 1.1–1.7 2 Width 57–122 Discharge 70–389 D50 0.3–0.45Slope 0.74–1.01Depth 7–7.6Sinuosity 1.37–2.24 6.McCarthy etData al. Source[38] 1 No 137 630 0.35 0.4 4.1 1.86 [40] (m) (m3/s) (mm) ( 1000) (m) (-) × 7.Monsalve9. Taylor andand SilvaWoodyer [39] 3 140–160 830–1200 0.4–6.2 0.24–0.8 3–5 1.2–2.5 1.Andrews [33]1 18 5.21–83.8 40 2.21–255 210 23–122 0.15 1.74–22.23 0.05 0.29–1.85 5.0 1.07–1.98 2.3 8.Morton and2.Church Donaldson and Rood [34 ] 17 5–104 2.7–354 33–89 0.97–13.7 0.65–3.06 1.0–1.65 3.Hey[41] and Thorne [35]2 18 57–122 6.5–76.5 70–389 3.9–424 13.9–83.8 0.3–0.45 2.108–11.4 0.74–1.01 0.68–3.21 7–7.6 1.33–2.5 1.37–2.24 10. Mosley[40]4.Kellerhals [42] et al. [36 ] 66 24 4.2–1753 18–442 7.93–2606 1.73–3112 0.2–145 1.0–189 0.12–16.5 0.62–28.3 0.58–7.2 0.33–2.96 1.01–2.2 1.0–1.77 5.Lambeek [37] 4 111–450 1841–5320 12–73 0.22–1.6 3.8–10 1.1–1.7 9. Taylor11. Williams and6.McCarthy Woodyer [43,44] et al. [ 38]4 1 13.7–57.9 137 12.2–365.3 630 0.35 2.7–42 0.4 1.52–6.9 4.1 0.7–3.38 1.86 1.32–1.93 7.Monsalve and Silva [39]1 3 140–160 40 830–1200 210 0.4–6.2 0.15 0.24–0.8 0.05 3–5 5.0 1.2–2.5 2.3 8.Morton[41] and Donaldson [40] 2 57–122 70–389 0.3–0.45 0.74–1.01 7–7.6 1.37–2.24 10. Mosley9. Taylor [42] and Woodyer [41]66 1 4.2–1753 40 1.73–3112 210 0.15 1.0–189 0.05 0.62–28.3 5.0 0.33–2.96 2.3 1.0–1.77 Figure 310. shows Mosley [42 the] distribution 66 4.2–1753 of these 1.73–3112 rivers in 1.0–189the equilibrium 0.62–28.3 surface: 0.33–2.96 most 1.0–1.77 points are on the 11. Williams11. Williams[43,44] [43 ,44]4 4 13.7–57.9 13.7–57.9 12.2–365.3 12.2–365.3 2.7–42 2.7–42 1.52–6.9 1.52–6.9 0.7–3.38 0.7–3.38 1.32–1.93 1.32–1.93 other side of the surface, the structures are far from the equilibrium. It indicates that the internal adjustments are undergoing to reach the steady state in the dynamic system (Figure 3). Figure 3Figure shows3 shows the distribution the distribution of ofthese these rivers rivers in in the the equilibriumequilibrium surface: surface: most most points points are on are on the other sidethe of other the side surface, of the surface, the structures the structures are arefar far fr fromom the equilibrium.equilibrium. It indicates It indicates that the that internal the internal adjustments are undergoing to reach the steady state in the dynamic system (Figure3). adjustments are undergoing to reach the steady state in the dynamic system (Figure 3).

Figure 3. The distribution of natural rivers in the equilibrium channel state surface.

Combined withFigure the 3. The dissipative distribution ofstructure natural rivers theory in the equilibriumand Equation channel state(17), surface. we can obtain the entry function, whichFigure implies 3. The distribution that the system of natural should rivers acin hievethe equilibrium a stable channelsurface stateat the surface. expense of energy Combined with the dissipative structure theory and Equation (17), we can obtain the entry flowing into the system. Figure 4 shows that the sinuosity l will increase until the system reaches the Combinedfunction, whichwith impliesthe dissipative that the system structure should achieve theory a stableand surfaceEquation at the (17), expense we of can energy obtain flowing the entry surface. It indicates that the meandering river is a relatively stable pattern in nature, and this function,into which the system. implies Figure that4 shows the system that the sinuosityshould acl willhieve increase a stable until thesurface system at reaches the expense the surface. of energy conclusionIt indicates is proven that the and meandering accepted riverby most is a relatively of the re stablesearchers. pattern If in the nature, entry and function this conclusion dS > 0, is the system flowing into the system. Figure 4 shows that the sinuosity l will increase until the system reaches the shouldproven be in a and chaotic accepted state by mostat the of thesecond researchers. area; the If the system entry function would dS reach> 0, the the system lower should sheets be as in adecreasing surface. It indicates that the meandering river is a relatively stable pattern in nature, and this the sinuosity,chaotic state it indicates at the second the area; transition the system to wouldunstea reachdy braid the lower river sheets patterns as decreasing with thethe sinuosity, expense it of energy conclusionindicates is proven the transition and accepted to unsteady by braidmost river of the patterns researchers. with the expenseIf the entry of energy function flowing dS out > of0, thethe system flowing out of the system (Figure 5). should besystem in a chaotic (Figure5). state at the second area; the system would reach the lower sheets as decreasing the sinuosity, it indicates the transition to unsteady braid river patterns with the expense of energy flowing out of the system (Figure 5).

Figure 4. The first zone for the river stability state. Figure 4. The first zone for the river stability state.

Figure 4. The first zone for the river stability state.

Water 2020, 12, x FOR PEER REVIEW 7 of 18 Water 2020, 12, 780 7 of 19 Water 2020, 12, x FOR PEER REVIEW 7 of 18

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Figure 5. The second zone for the river stability state. Figure 5. The second zone for the river stability state. Figure 5. The second zone for the river stability state. As shown in Figure6, we can see a few points in the middle sheet, which is often referred to as the inaccessible region, representing an unstable maximum. All points along the fold curve are semi-stable points of inflection. The sinuosity l may either increase or decrease to reach the upper or lower sheet.

It may result in two different stable states. These rivers may transfer to the meandering or unstable braided river in a slight alteration.Figure 5. The second zone for the river stability state.

Figure 6. The third zone for the river stability state. Figure 6. The third zone for the river stability state. As shown in Figure 6, we can see a few points in the middle sheet, which is often referred to as the inaccessibleAs shown inregion, Figure representing 6, we can see an a fewunstable points maximum. in the middle All sheet,points which along is theoften fold referred curve toare as semi-stable points of inflection. The sinuosity l may either increase or decrease to reach the upper or the inaccessible region, representing an unstable maximum. All points along the fold curve are lowersemi-stable sheet. pointsIt may ofresult inflection. inFigure two The di 6.fferent Thesinuosity third stable zone l may states. for either the These river increase stabilityrivers ormay state. decrease transfer to to reach the meanderingthe upper or orlower unstable sheet. braided It may riverresult in inFigure a twoslight di6. fferentalteration.The third stable zone states.for the riverThese stability rivers state.may transfer to the meandering φ or unstableFigureFigure 7braided shows thatriver 10 in points a slight are alteration. just on thethe equilibriumequilibrium channelchannel statestate surfacesurface (the(the valuesvalues ofof φ11, , φφ2,2 l, satisfyl satisfy Equation Equation (17)), (17)), either either on on the the upper upper sheet sheet or or lower lower sheet sheet (the (the global global minimum). minimum). However, However, AsFigure shown 7 shows in Figure that 6,10 we points can seeare ajust few on points the equilibrium in the middle channel sheet, state which surface is often (the referred values ofto φas1, therethere are nono pointspoints in in the the middle middle sheet sheet (unstable (unstabl maximum);e maximum); it implies it implies that the that adjustments the adjustments of alluvial of theφ2, l inaccessiblesatisfy Equation region, (17)), representing either on the an upper unstable sheet maximum. or lower sheet All (thepoints global along minimum). the fold curveHowever, are alluvialsemi-stablechannelsthere are channels tendno points points to thetend of minimuminflection.in tothe the middle minimum state,The sinuositysheet not anstate, (unstabl unstable l maynote aneithermaximum); maximum. unstable increase maximum. Theit orimplies conclusiondecrease thatThe to agrees theconclusionreach adjustments the qualitatively upper agrees orof qualitativelylowerwithalluvialthe sheet. channels minimum Itwith may thetend result entropy minimum to in the two of minimum the dientropyfferent channel ofstate,stable patternthe notchanstates. an theoriesnel Theseunstable pattern [rivers28]. theoriesmaximum. There may transfer is[28] no The. straightThere toconclusion the is meandering riverno straight onagrees the riverorequilibriumqualitatively unstable on the braided surface, equilibriumwith the river which minimum in surface, demonstrates a slight entropy whichalteration. that ofdemonstr the straightchanatesnel patternthatpattern the should theoriesstraight not [28] bepattern included. There should is as no one straightnot of thebe φ includedtypicalriverFigure on self-formed asthe one7 showsequilibrium of the channel that typical 10 surface, patternspoints self-formed are which [45 just]. channel ondemonstr the equilibriumpatternsates that[45]. channel the straight state surfacepattern (the should values not of be1, φ included2, l satisfy as Equation one of the (17)), typical either self-formed on the upper channel sheet patterns or lower [45]. sheet (the global minimum). However, there are no points in the middle sheet (unstable maximum); it implies that the adjustments of alluvial channels tend to the minimum state, not an unstable maximum. The conclusion agrees qualitatively with the minimum entropy of the channel pattern theories [28]. There is no straight river on the equilibrium surface, which demonstrates that the straight pattern should not be included as one of the typical self-formed channel patterns [45].

Figure 7. The distribution of rivers on the equilibrium channel state surface.

Figure 7. The distribution of rivers on the equilibrium channel state surface. The wandering river is wide, and the mainstream swings frequently; there are a large number Figure 7. The distribution of rivers on the equilibrium channel state surface. of sandbarsThe wandering in the river river channel, is wide, but and the the vegetation mainstream is scarce swings [46 frequently;]. Based on there the aforementioned are a large number cusp catastrophe model of alluvial channel regime, the data set of wandering rivers [46] are added to the of sandbarsThe wandering in the river river channel, is wide, but and the the vegetation mainstream is scarce swings [46]. frequently; Based on thethere aforementioned are a large number cusp catastrophechannel state model surface, of alluvial and some channel suggestions regime, about the da theta channel set of wandering stability can rivers be obtained [46] are fromadded Figure to the8. of sandbars in the river channel, but the vegetation is scarce [46]. Based on the aforementioned cusp catastrophe model of alluvial channel regime, the data set of wandering rivers [46] are added to the

Figure 7. The distribution of rivers on the equilibrium channel state surface.

The wandering river is wide, and the mainstream swings frequently; there are a large number of sandbars in the river channel, but the vegetation is scarce [46]. Based on the aforementioned cusp catastrophe model of alluvial channel regime, the data set of wandering rivers [46] are added to the

Water 2020, 12, x FOR PEER REVIEW 8 of 18

Water 2020, 12, 780 8 of 19 channel state surface, and some suggestions about the channel stability can be obtained from Figure 8.

Figure 8. Projection forward and backward about 0.50.5 unitsunits fromfrom thethe sectionsection φφ22 = 44 in the equilibrium

channel state surface with naturalnatural rivers in the space of l–φ1.1.

Choose the section φ = 4 forward and backward about 0.5 units from the equilibrium surface Choose the section φ22 = 4 forward and backward about 0.5 units from the equilibrium surface (Figure7), and the projection of the river points along the l-φ plane is shown in Figure8. The natural (Figure 7), and the projection of the river points along the l-φ1 plane is shown in Figure 8. The natural channels areare scatteredscattered on on either either side side of of the the intersection intersection line line which which defines defines the controlthe control plane plane into threeinto threeparts: parts: the meandering, the meandering, straight, straight, andbraided and braided rivers rivers belong belong to the to upper, the upper, middle, middle, and lower and lower sheet, sheet,respectively; respectively; they may they tend may to achieve tend theto intersectionachieve the lineintersection by adjusting line the by control adjusting parameters the controlφ1, φ2. Straight channels lied on the middle sheet represent an unstable maximum [39]; bifurcation may occur, parameters φ1, φ2. Straight channels lied on the middle sheet represent an unstable maximum [39]; bifurcationand result in may the meanderingoccur, and result or braided in the channel meanderi patternng or without braided external channel interference. pattern without external interference.Over the short time scale, a channel can adjust its non-fluid boundaries to impose conditions in orderOver to obtain the short and maintaintime scale, steady-state a channel can equilibrium. adjust its non-fluid Fluctuations boundaries occur during to impose the self-adjustment conditions in orderof alluvial to obtain channels and maintain [11]. From steady-state Figure8, weequilibr can assistium. Fluctuations in some suggestions occur during for riverthe self-adjustment management offrom alluvial the diagram. channels When [11]. theFrom meandering Figure 8, channelwe can withassist high in some sinuosity suggestions is far from for theriver intersection management line, fromthe stability the diagram. can no When longer the be maintainedmeandering andchannel a cut-o wiffthmay high occur sinuosity to reach is far anew from balance. the intersection In order line,to prevent the stability the unfavorable can no longer impact be ofmaintained river stability and changea cut-off in may practice, occur we to shouldreach a enhancenew balance. channel In orderstability to toprevent control the the unfavorable pattern. If the impact channels of areriver just stability around change the intersection in practice, line, we the should channel enhance pattern channelcan be maintained stability to if control the condition the pattern. of flow If discharge the chan andnels sediment are just isaround constant. the intersection line, the channelBased pattern on the can definition be maintained of the ifstate the conditio parametern of asflow sinuosity dischargel, the and region sediment in the is constant. lower sheet is inaccessibleBased on (Figure the definition8); the channels of the belong state parameter to braided patterns,as sinuosity the widthl, the region and the in width-depth the lower sheet ratio ofis inaccessiblethese channels (Figure are relatively 8); the channels large according belong to to br theaided field patterns, data (Table the 2width). Deviations and the fromwidth-depth equilibrium ratio ofconditions these channels will trigger are the relatively decrease large of the controlaccording parameter to the φfield2 to attaindata the(Table available 2). Deviations balance with from the equilibriumresponse between conditions inflowing will trigger and outflowing the decrease water of andthe control sediment parameter discharge, φ2 suchto attain as decreasing the available the balanceslope or with width-depth the response ratio. between inflowing and outflowing water and sediment discharge, such as decreasing the slope or width-depth ratio.

Table 2. Observed data of natural braided rivers.

Width to Control Parameters Width Slope Depth River Reach × Depth Ratio (M) ( 1000) (M) φ1 φ2 (-) 1. Rakaia 1753 4.04 0.84 2086 3.69 3.50 2. Grey 124 3.41 1.86 66.7 1.34 4.26 3. Waiau-Uha 1156 4.9 0.78 1482 3.4 3.90 4. Hakataramea 390 5.9 0.67 573.5 2.8 3.50 5. Ohau 109 6.9 0.84 129.7 1.71 3.55

2.4. The Cusp Catastrophe Model for the Channel Pattern Classification

Water 2020, 12, 780 9 of 19

Table 2. Observed data of natural braided rivers.

Width Slope Depth Width to Control Parameters River Reach (M) ( 1000) (M) Depth Ratio (-) φ φ × 1 2

1. Rakaia 1753 4.04 0.84 2086 3.69 3.50

2 Grey 124 3.41 1.86 66.7 1.34 4.26

3 Waiau-Uha 1156 4.9 0.78 1482 3.4 3.90

4 Hakataramea 390 5.9 0.67 573.5 2.8 3.50

Water 2020, 12, 5x FOROhau PEER REVIEW 109 6.9 0.84 129.7 1.71 3.55 9 of 18

Numerous descriptive classification schemes have been proposed [47] as well as differences in 2.4. The Cusp Catastrophe Model for the Channel Pattern Classification stability and sedimentation controls in alluvial rivers. Based on the cusp model of alluvial channel regime, a newNumerous channel descriptive pattern classification classification schemes is pres haveented been according proposed [to47] the as well channel as differences stability in in this section. Thestability discriminant and sedimentation function controlss are obtained in alluvial from rivers. the Based model on the to cusp distinguish model of alluvialthe channel channel patterns. regime, a new channel pattern classification is presented according to the channel stability in this The validitysection. of the The proposed discriminant method functions is aretested obtained with from the theriver model data to from distinguish Table the 1. channel patterns. The cuspThe catastrophe validity of the model proposed of method alluvial is tested channel with re thegime river can data fromreflect Table the1. stability of the rivers; one can see the channelsThe cusp achieve catastrophe to the model prescribed of alluvial channelminimum regime of equilibrium can reflect the stabilitystate when of the L rivers; = 1; the one value of sinuositycan in seeexcess the channels of 1.5 is achieve often to used the prescribed to discri minimumminate the of equilibrium meandering state and when braided L = 1; the rivers value of[48]. The cross-sectionssinuosity of L in = excess 1 and of L 1.5 = is1.5 often are used used to discriminateto divide the the equilibrium meandering and surface, braided as rivers shown [48]. in The Figure 9, cross-sections of L = 1 and L = 1.5 are used to divide the equilibrium surface, as shown in Figure9, and and broughtbrought into into Equation Equation (17), (17), thethe discriminators discriminators are derivedare derived as: as: When L = 1: φ −φ = 0 When L = 1 : φ1 φ2 = 0 (19) (19) 1 − 2 φ −φ /3 = 0.35 WhenWhen L = 1.5:1.5 : φ1 φ2 /3 = 0.35 (20) (20) 1 − 2

Figure 9. Classification of alluvial channel patterns based on the cusp catastrophe model (Line A is Figure 9. Classification of alluvial channel patterns based on the cusp catastrophe model (Line A is the projection of the intersection line between L = 1.5 and Equation (14) in φ1–φ2 space; line B is the the projection of the intersection line between L = 1.5 and Equation (14) in φ1–φ2 space; line B is the projection of the intersection line between L = 1.0 and Equation (14) in φ1–φ2 space). projection of the intersection line between L = 1.0 and Equation (14) in φ1–φ2 space).

The 2D projection of the natural channel (Figure 6) along the control φ1–φ2 plane can be sketched in Figure 10. Discriminator (20) represents a reasonable lower bound for wandering channels (the data for wandering channels come from Zhang et al. [46]). Discriminator (19) forms a reasonable upper bound for the meandering channels; braided rivers by the definition used in this analysis, or should at least be regarded as transitional forms based on the dynamic features, and they are in the region between Discriminators (19) and (20) in this diagram. The control plane is divided into three zones: the upper zone, which can be regarded as the stable zone; the middle zone, defined as the transitional stable zone; and the unstable lower zone. It indicates that the classification diagram is in agreement with the known channel pattern theories from the stability feature. Substitute Equations (6) and (7) to the Discriminators (19) and (20), the discriminant function that predicts the channel pattern can be derived as follows: Meandering channel:

1/ 2 1/ 2  (γ -γ )D   BJ 0.2   s 50  > 3  −1.05 (21)  γhJ   Q 0.5      Braided channel:

1/ 2 1/ 2 1/ 2  BJ 0.2   (γ -γ )D   BJ 0.2    <  s 50  < 3  −1.05 (22)  Q0.5   γhJ   Q0.5        Wandering channel：

Water 2020, 12, 780 10 of 19 Water 2020, 12, x FOR PEER REVIEW 10 of 18

The 2D projection of the natural channel (Figure6) along the control φ1–φ2 plane can be sketched in Figure 10. Discriminator (20) represents a1 reasonable/ 2 lower bound1/ 2 for wandering channels (the data  BJ 0.2   (γ -γ )D  for wandering channels come from Zhang et al.<  [46s]). Discriminator50  (19) forms a reasonable upper (23)  Q0.5   γhJ  bound for the meandering channels; braided rivers by the definition used in this analysis, or should at least be regarded as transitional forms based on the dynamic features, and they are in the region As shownbetween in Figure Discriminators 10, the (19) andchannel (20) in thispatterns diagram. ar Thee difficult control plane to isdistinguish divided into threenearby zones: the origin, according tothe the upper catastrophe zone, which can theory: be regarded the as region the stable around zone; the middlethe origin zone, defined of the as thefold transitional line could have remarkably differentstable zone; outcomes and the unstable of the lower system zone. It beha indicatesvior. that For the although classification the diagram two ispaths in agreement may begin close with the known channel pattern theories from the stability feature. Substitute Equations (6) and (7) to together, theirthe paths Discriminators are divergent (19) and (20), [26], the discriminantwhich can functionbe used that to predicts explain the channelthat the pattern different can be planforms may result fromderived the as same follows: sediment-discharge and boundary conditions.

Equation (20)

Equation (19)

Figure 10. The diagram classifying alluvial channel patterns in the control space of φ1–φ2. Figure 10. The diagram classifying alluvial channel patterns in the control space of φ1–φ2. Meandering channel: !1/2 0.2 !1/2 (γs γ)D50 BJ 3. The Discriminant Function for the Alluvial− Channel> 3 Stability1.05 (21) γhJ Q0.5 − 3.1. EstablishmentBraided of the channel:Discriminant Function 0.2 !1/2 !1/2 0.2 !1/2 BJ (γs γ)D50 BJ The stability of a channel is related

0.2 φ ' = φ c −φ = − BJ 1 h h 1 (25) Q0.5 where φ1’ reflects the traversal stability of a river. γhJ φ ' = φ c −φ = θ − 2 b b c γ −γ (26) ( s )D50 where φ2’ reflects the longitudinal stability of a river.

' = − l lc l (27)

Water 2020, 12, 780 11 of 19

3. The Discriminant Function for the Alluvial Channel Stability

3.1. Establishment of the Discriminant Function The stability of a channel is related to the bed material movement [49]. When the river achieves a c stable state, the bed material begins the incipient motion. The critical longitudinal index φb can be expressed as: U 2 ρ c = c = φb ∗ θc (24) (γs γ)D − 50 where θc is the critical shield’s number for the incipient motion. The river width is equal to that under the bank-full discharge when the river is in an equilibrium c state. The critical traversal index (7) can be expressed as φh = 1. The relative traversal and longitudinal indexes φ10, φ20 are applied as the control factors, and the relative sinuosity l’ is the state parameter in the cusp catastrophe model to describe the river stability. The control factors and state parameter can be written as:

0.2 c BJ φ0 = φ φ = 1 (25) 1 h − h − Q0.5 where φ10 reflects the traversal stability of a river.

γhJ = c = φ20 φb φb θc (26) − − (γs γ)D − 50 where φ20 reflects the longitudinal stability of a river.

l0 = lc l (27) − where l0 reflects the variation in the sinuosity between the stable and the local statue, lc, l is the sinuosity of a river in the stable and local state, respectively. Bring Equations (25)–(27) to the equilibrium Equation (5), and the stability equation for the channel can be obtained as: 0.2 3 BJ γhJ 4l0 + 2l0(1 ) + (θc ) = 0 (28) − Q0.5 − γs γ − Cardan’s discriminant function (6) can be written as:

3 BJ0.2 γhJ 2 ∆ = 8(1 ) + 27(θc ) (29) − Q0.5 − (γ γ)D S − where ∆ can reflect the river state; if ∆ > 0, the river is in a stable stage; if ∆ < 0, the river is in a unstable condition; and if ∆ = 0, the river is in a critical stable state, a small disrupt may lead to a change in the channel geometry.

3.2. Verification of the Discriminant Function The data set of the natural rivers are from Table1 and Zhang et al. [ 46]. According to the developed classification of the channel pattern, the control factors φ10, φ20 for the meandering, straight, and braided channel were calculated and brought into discriminate function (29). The results can be seen in Figures 11–13. Figure 11a shows that most of the meandering channel are in the stable zone. That is, the river can maintain the current state for a long time. There are still a few meandering rivers nearby the critical line in Figure 11b; some of them are in the unstable zone. In practice, we can adjust the control factors to make all of the channels tend to the equilibrium state, for example, an increase of the flow discharge can lead to a decrease in the distance from the relative longitudinal index to the stable zone. Water 2020, 12, x FOR PEER REVIEW 11 of 18

where l’ reflects the variation in the sinuosity between the stable and the local statue, lc , l is the sinuosity of a river in the stable and local state, respectively. Bring Equations (25)–(27) to the equilibrium Equation (5), and the stability equation for the channel can be obtained as: BJ 0.2 γhJ 4l '3 + 2l ' (1− ) + (θ − ) = 0 0.5 c γ − γ (28) Q s

Cardan’s discriminant function (6) can be written as: BJ 0.2 γhJ Δ = 8(1 − )3 + 27(θ − ) 2 0.5 c γ − γ (29) Q ( S )D

where Δ can reflect the river state; if Δ > 0 , the river is in a stable stage; if Δ < 0 , the river is in a unstable condition; and if Δ = 0 , the river is in a critical stable state, a small disrupt may lead to a change in the channel geometry.

3.2. Verification of the Discriminant Function The data set of the natural rivers are from Table 1 and Zhang et al. [46]. According to the developed classification of the channel pattern, the control factors φ1’, φ2’ for the meandering, straight, and braided channel were calculated and brought into discriminate function (29). The results can be seen in Figures 11–13. Figure 11a shows that most of the meandering channel are in the stable zone. That is, the river can maintain the current state for a long time. There are still a few meandering rivers nearby the critical line in Figure 11b; some of them are in the unstable zone. In practice, we can adjust the control factors to make all of the channels tend to the equilibrium state, for example, an increase of the flow discharge can lead to a decrease in the distance from the relative longitudinal index to the stable zone. Figure 12 shows that almost the straight channels are nearby the critical stable lines; it implies that the river may enter into an unstable zone with a tiny change. We need to focus on the changes of the control factors to ensure river stability. It can be seen that the braided channels are almost in the Water 2020unstable, 12, 780 zone (Figure 13). However, there are still a few meandering channels in the unstable zone,12 of 19 and some of the straight channels in the stable area (Figures 11 and 12).

(a) (b) Water 2020, 12, x FOR PEER REVIEW 12 of 18 Water 2020, 12, x FOR PEER REVIEW 12 of 18 FigureFigure 11. Distribution 11. Distribution of naturalof natural meandering meandering riversrivers based on on the the cusp cusp catastrophe catastrophe model. model. (a) Field (a) Field data indata the in plane the plane of cusp of cusp catastrophe catastrophe model; model; ( b(b)) The The datadata nearby the the critical critical line. line.

(a) (b)

FigureFigure 12. Distribution 12. Distribution of natural of natural straight straight rivers rivers based based on on the the cusp cusp catastrophe catastrophe model. (a (a) )Field Field data data in the planein the of plane cusp of catastrophe cusp catastrophe model; model; (b) The (b) The field field data data nearby nearby the the critical critical lines. lines.

(a) (b) Figure 13. Distribution of natural braided rivers based on the cusp catastrophe model. (a) Field data FigureFigure 13. Distribution 13. Distribution of natural of natural braided braided rivers rivers based based on on the the cusp cusp catastrophe catastrophe model. (a (a) )Field Field data data in in the plane of cusp catastrophe model; (b) The field data nearby the critical lines. the planein the of plane cusp of catastrophe cusp catastrophe model; model; (b) The (b) The field field data data nearby nearby the the critical critical lines. lines.

We choose point A Selwyn River located in the critical line (Figure 11b) and point B Buller River in the stable zone (Figure 12b) to investigate the state of the channel. The related parameters for the stability of the two rivers can be seen in Table3. According to the research on the fluvial process in the Selwyn River, the river is in a change between the braided and meandering pattern because of the influence of the inlet flow-sediment discharge and bank vegetation [50,,51]. Buller River is the largest river in New Zealand; although the channel pattern is straight, the clay bank materials and vegetable cover contribute to the stability of the bank, and the river keeps the stable state. These results are in agreement with the field data, and illustrate that the channel pattern can not determine the river stability.

Table 3. Field data of Hamps channel. Table 3. Field data of Hamps channel.

River section Φ1 ‘ Φ2 ‘ Δ River section Φ1 ‘ Φ2 ‘ Δ － Selwyn River －0. 75 0.24 1.8

Water 2020, 12, 780 13 of 19

Figure 12 shows that almost the straight channels are nearby the critical stable lines; it implies that the river may enter into an unstable zone with a tiny change. We need to focus on the changes of the control factors to ensure river stability. It can be seen that the braided channels are almost in the unstable zone (Figure 13). However, there are still a few meandering channels in the unstable zone, and some of the straight channels in the stable area (Figures 11 and 12). We choose point A Selwyn River located in the critical line (Figure 11b) and point B Buller River in the stable zone (Figure 12b) to investigate the state of the channel. The related parameters for the stability of the two rivers can be seen in Table3. According to the research on the fluvial process in the Selwyn River, the river is in a change between the braided and meandering pattern because of the influence of the inlet flow-sediment discharge and bank vegetation [50,51]. Buller River is the largest river in New Zealand; although the channel pattern is straight, the clay bank materials and vegetable cover contribute to the stability of the bank, and the river keeps the stable state. These results are in agreement with the field data, and illustrate that the channel pattern can not determine the river stability.

Table 3. Field data of Hamps channel.

River Section Φ1 0 Φ2 0 ∆ Selwyn River 0.75 0.24 1.8 − − Buller River 0.48 0.64 10.33

It should, however, be noted that the state of the river is not illustrated in Table1. In order to test the discriminate function, we adopted the field data from 20 naturally stable channels in the UK [52]. The calculated values of the discriminate function can be seen in Table4. The values are all larger than 0, and the results imply that the rivers are in a stable state, which agrees well with the measured data.

Table 4. Results with field data of natural stability channels in the UK.

River Section Φ1’ Φ2’ ∆ Teviot at Hawick 0.081652675 0.421672464 2.077705394 Exe at Stoodleigh 0.146498713 0.479981148 3.157327172 − Usk at Llandetty 0.306517002 0.531238705 4.799550439 North Tyne at Tarset 0.171287961 0.486915852 3.351635451 Tweed at Peebles 0.025386507 0.565425868 4.885953245 − Yscir at Pontaryscir 0.185434467 0.480383884 3.268241467 Sprint at Sprint Mill 0.151169956 0.353716859 1.377717415 − Yarrow Water at Philiphaugh 0.211474561 0.359206172 1.609168992 − Alwin at Clennel 1.279978326 0.254689138 13.55281793 − Coquet at Bygate 1.953615324 0.3461899 31.65313202 − Snaizeholme Beck 0.633915741 0.425806949 5.299293804 − Rede at Rede’s Bridge 0.924715868 0.427212477 8.946006101 − Glaslyn at Beddlegert u/s 0.037797355 0.37255358 1.407572393 − Glaslyn at Beddlegert d/s 0.893945136 0.37186106 7.78147533 − Tarsetburn at Greenhaugh 0.150349187 0.478402527 3.137109449 Fowey at Restormel 0.076519308 0.535205284 4.186122941 − Hodder at Hodderplace 0.045766839 0.480741662 3.016603416 − Esk at Cropple How d/s 0.109785644 0.486580798 3.206912053 − Erme at Ermington 0.200651184 0.269685306 0.85167211 Otter at Dotton 0.022693634 0.42746865 2.113119973 − Dyfyrdwy at New Inn 0.123405107 0.484106991 3.185118538 Alwen at Druid 0.189059856 0.427541334 2.396024985 − Lugg at Byton u/s 0.36688634 0.41404418 2.993322602 − Lugg at Byton d/s 0.048668144 0.418682887 2.000564222 − Manifold at Hulme End 0.803954047 0.373945434 6.582586586 − Glendaramackin at Threlkeld 0.302536456 0.376950845 2.178391725 Tweed at Lyneford 0.068990566 0.530561351 4.070542484 − Water 2020, 12, 780 14 of 19

According to the verification with the measured data of 120 natural rivers, the discriminate function can quantitatively distinguish river stability, and give some assistance in river management and regulation.

4. Application to the Upstream of the Yangtze River, China The model is used to investigate the channel pattern and state of the typical river reach in the upstream of the Yangtze River. The study area is approximately 140 km long, stretching from Fuling to Chongqing (Figure 14a). As we know, the Three Gorges Dam (TGD) was brought into operation in 2003; the river reach in Chongqing-Fuling (CF) is in the fluctuating backwater area (Figure 14b). This study is based on an extensive dataset of the mean annual water discharge Q, river width B, and river slope J from 1980 to 2018, collected from the control hydraulic stations along the river reach [53]. The Watergrain 2020 size, 12 D50, x FOR of thePEER river REVIEW bed material is obtained from the core documents [54]. 14 of 18

(a)

(b)

FigureFigure 14. 14. TheThe information information of of the the study study area area in in the the TGD. TGD. ( (aa)) The The location location of of the the study study area area in in the the backwaterbackwater area of the ThreeThree GorgesGorges DamDam (TGD);(TGD); ( b(b)) The The topography topography of of the the river river reach reach from from Fuling Fuling to toChongqing Chongqing in in 2013. 2013.

4.1.4.1. The The Temporal Temporal Chang Changeses of the Channel Pattern TheThe control factors φ1,1, φφ22 proposed proposed in in Section Section 2 are calculatedcalculated andand plottedplotted inin thethe classificationclassification diagram.diagram. Considering Considering the the TGD TGD construction, construction, we we di dividedvided the the years years into into the the series series of of 1980–1993, 1980–1993, 1994–2002, and and 2003–2018, 2003–2018, which which are are the the before, before, un undergoing,dergoing, and and operation operation stages stages of of the the TGD. TGD. FigureFigure 1515 shows the temporal change of the cha channelnnel pattern for the past three decades. The The three three groupsgroups locate locate in in the the meandering zone, zone, indicating indicating that that the the plane plane form form of of the the river river reach reach maintains its its channelchannel pattern forfor decades,decades, which which is is consistent consistent with with the the actual actual condition condition [54 ].[54]. The The result result indicates indicates that thatriver river engineering engineering can can not alternot alter the channelthe channel pattern patte inrn the in the decadal decadal timescale. timescale. As As shown shown in Figurein Figure 15, 15,the the position position of the of pointsthe points after after 2002 2002 is far is away far away from from the ones the beforeones before the TGD the operation.TGD operation. The result The result indicates that the water surface slope J of the study area decreases with the operation of the TGD, and leads to a decline in the water power in the backwater zone [55]. φ 2 20

18 Meandering 16

14 x de n12 Braided i al n10 di tu i 8 g Wandering on l he 6 T 4 Eq.(19) Eq.(20) 2 1980-1993 1994-2002 2003-2018 0 0 5 10 15 20 The transverse index .

Water 2020, 12, x FOR PEER REVIEW 14 of 18

(a)

(b)

Figure 14. The information of the study area in the TGD. (a) The location of the study area in the backwater area of the Three Gorges Dam (TGD); (b) The topography of the river reach from Fuling to Chongqing in 2013.

4.1. The Temporal Changes of the Channel Pattern The control factors φ1, φ2 proposed in Section 2 are calculated and plotted in the classification diagram. Considering the TGD construction, we divided the years into the series of 1980–1993, 1994–2002, and 2003–2018, which are the before, undergoing, and operation stages of the TGD. Figure 15 shows the temporal change of the channel pattern for the past three decades. The three groups locate in the meandering zone, indicating that the plane form of the river reach maintains its Waterchannel2020 ,pattern12, 780 for decades, which is consistent with the actual condition [54]. The result indicates15 of 19 that river engineering can not alter the channel pattern in the decadal timescale. As shown in Figure 15, the position of the points after 2002 is far away from the ones before the TGD operation. The indicates that the water surface slope J of the study area decreases with the operation of the TGD, and result indicates that the water surface slope J of the study area decreases with the operation of the leads to a decline in the water power in the backwater zone [55]. TGD, and leads to a decline in the water power in the backwater zone [55]. φ 2 20

18 Meandering 16

14 x de n12 Braided i al n10 di tu i 8 g Wandering on l he 6 T 4 Eq.(19) Eq.(20) 2 1980-1993 1994-2002 2003-2018 0 0 5 10 15 20 The transverse index . Figure 15. The position of the points in the classifying diagram.

Then, we chose the dataset of daily river conditions to investigate the change trends of the channel pattern in one year (2018), and the river bed material condition adopted three values for the dry, medium, and flood discharge stages [54]. Figure 16 shows that the scatter points are still in Waterthe meandering2020, 12, x FOR zone,PEER REVIEW although the river condition varies over the days. The aforementioned15 results of 18 imply that the change of the channel pattern depends mainly on the geological, climate, and tectonic processes [56]. Figure 15. The position of the points in the classifying diagram. φ 2 20

18 Meandering

16

14 ex nd12 Braided i al in10 ud it g 8 Wandering on l he 6 T 4

2 Eq.(19) Eq.(20) 2018

0 0 5 10φ 15 20 The transverse index 1 Figure 16. Variation in the points in one year. Figure 16. Variation in the points in one year. 4.2. The Temporal Changes of the River Status Then, we chose the dataset of daily river conditions to investigate the change trends of the channelBased pattern on the in one aforementioned year (2018), and dataset the ofriver the be riverd material conditions condition in the studyadopted area, three we values calculated for the the dry,relative medium, control and factors floodφ 1discharge0, φ20. Figure stages 17 shows [54]. Figure the status 16 shows of the riverthat the reach scatter over thepoints past are three still decades. in the meanderingAs seen, the zone, river although reach in thethe upstream river condition of the Yangtzevaries over River the kept days. its The stable aforementioned state in the past results three implydecades, that and the the change stable trendof the will channel be maintained pattern depends in the future. mainly The on points the beforegeological, 2002 areclimate, in astable and tectonicarea, far processes away from [56]. the threshold lines. When the TGD was in the construction stage between 1994 to 2003, the distances between the points and the critical lines decrease, implying that river engineering 4.2.can The influence Temporal the Changes stability of of the the River river Status state. After the establishment of the TGD, the points returned to the original position of the time-series from 1980 to 1994, indicating that the adjustment of the river Based on the aforementioned dataset of the river conditions in the study area, we calculated dynamic process tends to a stable condition, which agrees well with the in situ measurements [55]. the relative control factors φ1’, φ2’. Figure 17 shows the status of the river reach over the past three decades. As seen, the river reach in the upstream of the Yangtze River kept its stable state in the past three decades, and the stable trend will be maintained in the future. The points before 2002 are in a stable area, far away from the threshold lines. When the TGD was in the construction stage between 1994 to 2003, the distances between the points and the critical lines decrease, implying that river engineering can influence the stability of the river state. After the establishment of the TGD, the points returned to the original position of the time-series from 1980 to 1994, indicating that the adjustment of the river dynamic process tends to a stable condition, which agrees well with the in situ measurements [55]. 1 Critical line 1980-1993 0.8 1994-2002 2013-2018

0.6

0.4 ex d n i se0.2 er Stable zone sv an 0 tr e v ti la-0.2 Stable zone Stable zone re e h T-0.4

-0.6

-0.8 Unstable zone -1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 φ ’ The relative longitudinal index 2

Figure 17. Location of the points in the distinguish diagram.

Water 2020, 12, x FOR PEER REVIEW 15 of 18

Figure 15. The position of the points in the classifying diagram. φ 2 20

18 Meandering

16

14 ex nd12 Braided i al in10 ud it g 8 Wandering on l he 6 T 4

2 Eq.(19) Eq.(20) 2018

0 0 5 10φ 15 20 The transverse index 1 .

Figure 16. Variation in the points in one year.

Then, we chose the dataset of daily river conditions to investigate the change trends of the channel pattern in one year (2018), and the river bed material condition adopted three values for the dry, medium, and flood discharge stages [54]. Figure 16 shows that the scatter points are still in the meandering zone, although the river condition varies over the days. The aforementioned results imply that the change of the channel pattern depends mainly on the geological, climate, and tectonic processes [56].

4.2. The Temporal Changes of the River Status Based on the aforementioned dataset of the river conditions in the study area, we calculated the relative control factors φ1’, φ2’. Figure 17 shows the status of the river reach over the past three decades. As seen, the river reach in the upstream of the Yangtze River kept its stable state in the past three decades, and the stable trend will be maintained in the future. The points before 2002 are in a stable area, far away from the threshold lines. When the TGD was in the construction stage between 1994 to 2003, the distances between the points and the critical lines decrease, implying that river engineering can influence the stability of the river state. After the establishment of the TGD, the points returned to the original position of the time-series from 1980 to 1994, indicating that the adjustmentWater 2020, 12 of, 780 the river dynamic process tends to a stable condition, which agrees well with the16 of in 19 situ measurements [55]. 1 Critical line 1980-1993 0.8 1994-2002 2013-2018

0.6

0.4 ex d n i se0.2 er Stable zone sv an 0 tr e v ti la-0.2 Stable zone Stable zone re e h T-0.4

-0.6

-0.8 Unstable zone -1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 φ ’ The relative longitudinal index 2 Figure 17. Figure 17. LocationLocation of of the the points points in in the the distinguish distinguish diagram. diagram. We calculated the values of the discriminant function ∆ for river stability and obtained the diagram

to investigate the changing trend of the river stability with time (Figure 18). The result demonstrates that the change process of stability was influenced by the TGD project, and agrees well with the Water 2020 , 12, x FOR PEER REVIEW 16 of 18 aforementioned conclusion from Figure 17.

5.5 The value of the discriminant function

ue l 5 va n io Operation of the ct TGD in 2003 un t f an4.5 in im cr is D 4

3.5

3 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 Ye ars Figure 18. Change trends of the value of ∆ with time. Figure 18. Change trends of the value of Δ with time. 5. Conclusions We calculatedThis paper the presents values a cusp of the catastrophe discriminant model for function alluvial channel Δ for regime river with stability suitable and parameters obtained the diagram toto determine investigate the riverthe channelchanging and thetrend channel of the stability. river Equations stability of with equilibrium time (Figure state and 18). channel The result demonstratespatterns that are establishedthe change based process on the modelof stabilit of cuspy catastrophe was influenced surface by by selecting the TGD suitable project, parameters. and agrees The channel patterns can be identified by such a model in a direct way with a quantified index, which well with the aforementioned conclusion from Figure 17. is a cusp catastrophe surface in a translated 2D dimensional coordinate, and the discriminant functions are obtained from the model to distinguish channel stability. Predictions based on this model are 5. Conclusionsconsistent with field observations involving 150 natural rivers of small or medium sizes. Then, the proposed model was applied to the typical river reach in the upper reaches of the Yangtze River, and This paper presents a cusp catastrophe model for alluvial channel regime with suitable the results were compatible with the field measurements. The results indicate that this method may parametersbe appliedto determine to study the regimeriver ofchannel natural and rivers the and channel to assist decision-makingstability. Equations in river of management. equilibrium state and channelHowever, patterns some are assumptions established were based proposed on duringthe model the model of cusp establishment; catastrophe further surface studies by are selecting suitable parameters. The channel patterns can be identified by such a model in a direct way with a quantified index, which is a cusp catastrophe surface in a translated 2D dimensional coordinate, and the discriminant functions are obtained from the model to distinguish channel stability. Predictions based on this model are consistent with field observations involving 150 natural rivers of small or medium sizes. Then, the proposed model was applied to the typical river reach in the upper reaches of the Yangtze River, and the results were compatible with the field measurements. The results indicate that this method may be applied to study the regime of natural rivers and to assist decision-making in river management. However, some assumptions were proposed during the model establishment; further studies are needed to improve understanding of the patterning process and modify the control parameters in the cusp catastrophe model.

Author Contributions: The cusp catastrophe model for channel regime, Y.X.; analysis and manuscript preparation, Y.X.; manuscript review and editing, F.S.Y. and M.L.

Funding: This research is supported by the advanced and applied projects of Chongqing Science and Technology (cstc2018jscxmsybX0316).

Acknowledgments: We greatly appreciate an anonymous reviewer’s constructive comments, which helped to improve the quality of our manuscript.

Conflicts of Interest: The authors declare no conflict of interest.

References

1. Hossein, A.; Mohammad, A.; Singh,V.P. Hydraulic Geometry Relations for Stable Channel Design. J. Hydrol. Eng. 2010, 15, 859–864. 2. Van der Berg, J.H. Prediction of alluvial channel pattern of perennial rivers. Geomorphology 1995, 12, 259–279. 3. Millar, R. Theoretical regime equations for mobile gravel-bed rivers with stable banks. Geomorphology 2005, 64, 207–220.

Water 2020, 12, 780 17 of 19 needed to improve understanding of the patterning process and modify the control parameters in the cusp catastrophe model.

Author Contributions: The cusp catastrophe model for channel regime, Y.X.; analysis and manuscript preparation, Y.X.; manuscript review and editing, S.Y. and M.L. All authors have read and agreed to the published version of the manuscript. Funding: This research is supported by the advanced and applied projects of Chongqing Science and Technology (cstc2018jscxmsybX0316). Acknowledgments: We greatly appreciate an anonymous reviewer’s constructive comments, which helped to improve the quality of our manuscript. Conflicts of Interest: The authors declare no conflict of interest.

References

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