# A Cusp Catastrophe Model for Alluvial Channel Pattern and Stability

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Cusp Catastrophe Model for Channel Regime

#### 2.1. The Cusp Catastrophe Model

_{i}denotes the state variable, t is time, and α

_{i}is generally a system parameter.

#### 2.2. Determination of the Control Parameters and the State Parameter

_{h}is the longitudinal index, and ϕ

_{b}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; D

_{50}is the median bed material grain size; γ,γ

_{s}is the volumetric weight of the water and sediment, respectively.

_{1}as the dominant control parameter and ϕ

_{2}as the second control parameter that determines the channel scouring and deposition. The control parameters can be written as:

#### 2.3. The Cusp Catastrophe Model for Alluvial Channel Regime

_{1}, ϕ

_{2}, l), and obtain the following relationship for control parameters in the form:

_{1}, ϕ

_{2}, l); b

_{i}, m

_{i}, n

_{i}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 O′(ϕ

_{1}, ϕ

_{2}, l) can be expressed as:

_{1}, ϕ

_{2}, l) contributes to b

_{1}= 1; and b

_{i}, m

_{i}, n

_{i}satisfy the additional condition as:

_{1}, ϕ

_{2}, l) equal to zero, the channel should maintain its pattern as Equation (12) and can be expressed as:

^{c}

_{1}, l

^{c}is the critical index, assuming the critical value of state parameter denotes l

^{c}= 1, and the critical transverse river-bed stability ϕ

^{c}

_{1}= 1 when the rivers attain the equilibrium state.

_{1}, ϕ

_{2}, 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 (ϕ

_{1}, ϕ

_{2}) 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.

_{1}, ϕ

_{2}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 Table 1.

_{1}, ϕ

_{2}, l satisfy Equation (17)), either on the upper sheet or lower 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].

_{2}= 4 forward and backward about 0.5 units from the equilibrium surface (Figure 7), and the projection of the river points along the l-ϕ

_{1}plane is shown in Figure 8. The natural channels are scattered on either side of the intersection line which defines the control plane into three parts: the meandering, straight, and braided rivers belong to the upper, middle, and lower sheet, respectively; they may tend to achieve the intersection line by adjusting the control parameters ϕ

_{1}, ϕ

_{2}. Straight channels lied on the middle sheet represent an unstable maximum [39]; bifurcation may occur, and result in the meandering or braided channel pattern without external interference.

_{2}to attain the available balance with the response between inflowing and outflowing water and sediment discharge, such as decreasing the slope or width-depth ratio.

#### 2.4. The Cusp Catastrophe Model for the Channel Pattern Classification

_{1}–ϕ

_{2}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:

## 3. The Discriminant Function for the Alluvial Channel Stability

#### 3.1. Establishment of the Discriminant Function

_{b}

^{c}can be expressed as:

_{c}is the critical shield’s number for the incipient motion.

_{h}

^{c}= 1.

_{1}′, ϕ

_{2}′ 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:

_{1}

^{′}reflects the traversal stability of a river.

_{2}

^{′}reflects the longitudinal stability of a river.

_{c}, l is the sinuosity of a river in the stable and local state, respectively.

#### 3.2. Verification of the Discriminant Function

## 4. Application to the Upstream of the Yangtze River, China

#### 4.1. The Temporal Changes of the Channel Pattern

#### 4.2. The Temporal Changes of the River Status

_{1}′, ϕ

_{2}′. 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].

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The equilibrium channel state surface (the dotted line is the intersection between ϕ

_{2}= 4 and the equilibrium channel state surface).

**Figure 8.**Projection forward and backward about 0.5 units from the section ϕ

_{2}= 4 in the equilibrium channel state surface with natural rivers in the space of l–ϕ

_{1.}

**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 ϕ

_{1}–ϕ

_{2}space; line B is the projection of the intersection line between L = 1.0 and Equation (14) in ϕ

_{1}–ϕ

_{2}space).

**Figure 10.**The diagram classifying alluvial channel patterns in the control space of ϕ

_{1}–ϕ

_{2}

_{.}

**Figure 11.**Distribution of natural meandering rivers based on the cusp catastrophe model. (

**a**) Field data in the plane of cusp catastrophe model; (

**b**) The data nearby the critical line.

**Figure 12.**Distribution of natural straight rivers based on the cusp catastrophe model. (

**a**) Field data in the plane of cusp catastrophe model; (

**b**) The field data nearby the critical lines.

**Figure 13.**Distribution of natural braided rivers based on the cusp catastrophe model. (

**a**) Field data in the plane of cusp catastrophe model; (

**b**) The field data nearby the critical lines.

**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.

Data Source | No | Width (m) | Discharge (m ^{3}/s) | D50 (mm) | Slope $(\times $1000) | Depth (m) | Sinuosity (-) |
---|---|---|---|---|---|---|---|

1.Andrews [33] | 18 | 5.21–83.8 | 2.21–255 | 23–122 | 1.74–22.23 | 0.29–1.85 | 1.07–1.98 |

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 |

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 |

4.Kellerhals et al. [36] | 24 | 18–442 | 7.93–2606 | 0.2–145 | 0.12–16.5 | 0.58–7.2 | 1.01–2.2 |

5.Lambeek [37] | 4 | 111–450 | 1841–5320 | 12–73 | 0.22–1.6 | 3.8–10 | 1.1–1.7 |

6.McCarthy et al. [38] | 1 | 137 | 630 | 0.35 | 0.4 | 4.1 | 1.86 |

7.Monsalve and Silva [39] | 3 | 140–160 | 830–1200 | 0.4–6.2 | 0.24–0.8 | 3–5 | 1.2–2.5 |

8.Morton and Donaldson [40] | 2 | 57–122 | 70–389 | 0.3–0.45 | 0.74–1.01 | 7–7.6 | 1.37–2.24 |

9. Taylor and Woodyer [41] | 1 | 40 | 210 | 0.15 | 0.05 | 5.0 | 2.3 |

10. Mosley [42] | 66 | 4.2–1753 | 1.73–3112 | 1.0–189 | 0.62–28.3 | 0.33–2.96 | 1.0–1.77 |

11. Williams [43,44] | 4 | 13.7–57.9 | 12.2–365.3 | 2.7–42 | 1.52–6.9 | 0.7–3.38 | 1.32–1.93 |

River Reach | Width (M) | Slope $(\times $1000) | Depth (M) | Width to Depth Ratio (-) | Control Parameters | |
---|---|---|---|---|---|---|

ϕ_{1} | ϕ_{2} | |||||

- 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 |

River Section | Φ_{1} ′ | Φ_{2} ′ | Δ |
---|---|---|---|

Selwyn River | −0.75 | 0.24 | −1.8 |

Buller River | 0.48 | 0.64 | 10.33 |

River Section | Φ_{1} ’ | Φ_{2} ’ | Δ |
---|---|---|---|

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 |

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Xiao, Y.; Yang, S.; Li, M.
A Cusp Catastrophe Model for Alluvial Channel Pattern and Stability. *Water* **2020**, *12*, 780.
https://doi.org/10.3390/w12030780

**AMA Style**

Xiao Y, Yang S, Li M.
A Cusp Catastrophe Model for Alluvial Channel Pattern and Stability. *Water*. 2020; 12(3):780.
https://doi.org/10.3390/w12030780

**Chicago/Turabian Style**

Xiao, Yi, Shengfa Yang, and Mi Li.
2020. "A Cusp Catastrophe Model for Alluvial Channel Pattern and Stability" *Water* 12, no. 3: 780.
https://doi.org/10.3390/w12030780