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We present a conceptual model for simulating the temporal adjustments in the banks of the Lower Yellow River (LYR). Basic conservation equations for mass, friction, and sediment transport capacity and the Exner equation were adopted to simulate the hydrodynamics underlying fluvial processes. The relationship between changing rates in bankfull width and depth, derived from quasiuniversal hydraulic geometries, was used as a closure for the hydrodynamic equations. On inputting the daily flow discharge and sediment load, the conceptual model successfully simulated the 30-year adjustments in the bankfull geometries of typical reaches of the LYR. The square of the correlating coefficient reached 0.74 for Huayuankou Station in the multiple-thread reach and exceeded 0.90 for Lijin Station in the meandering reach. This proposed model allows multiple dependent variables and the input of daily hydrological data for long-term simulations. This links the hydrodynamic and geomorphic processes in a fluvial river and has potential applicability to fluvial rivers undergoing significant adjustments.

The bankfull characteristics of alluvial rivers are basic research topics in fluvial processes [

Two kinds of approaches have been developed to quantify the variation in bankfull characteristics in terms of the timescales at which the channel adjustment is explored: geomorphic and hydrodynamic approaches. The geomorphic approach is usually based on geomorphic laws expressed by power law, hyperbolic, and exponential equations or relatively complex differential equations [

The hydrodynamic approach reproduces channel response processes by accounting for microscale river dynamics. That is, the Saint Venant equations (continuity and momentum equations) for river flow, the sediment transport equation, and the river bed or bank deformation equations are set up to describe the instantaneous relations between channel adjustment and the incoming flow and sediment conditions. One-, two-, and three-dimensional models for either steady flow [

In nature, the channel response is achieved by the cumulative effects of fluvial erosion/deposition processes, which relate to both geomorphic processes and hydrodynamic events. The geomorphic approach, particularly the rate law method, can capture the overall behaviour of the channel response to seasonal or annual flow and sediment data, but it cannot account for the effects of flood events. The channel response can be characterised by a set of variables, but the coordination of multiple dependent variables has not been explored. By contrast, the hydrodynamic approach is capable of representing the details of the channel response process at small timescales, such as bank erosion, bar migration, and channel shallowing/widening. When applied on large timescales, the computational limitations and requirements for accurate resistance parameters and boundary conditions may stop it from producing reliable simulation and predictions.

Here we propose a physics-based model linking hydrodynamic and geomorphic scales. It is intended to represent channel response processes subject to both hydrodynamic and geomorphic controls. The model adopts hydrodynamic equations and boundary equations with daily data as input. It enables the characterisation of flood events in large rivers (typically lasting for around 10 days) and the simulation of continuous response behaviour over a long-term period (years or decades). In a scientific sense, linking across hydrodynamic and geomorphic scales is an attempt to overcome the shortcomings of each scale by considering the flood details and obtaining the long-period channel response processes. It may be applied both to simulate fluvial processes over the past few decades and to forecast channel-forming processes with potential flow and sediment series in the future for large alluvial rivers like LYR.

The Yellow River is the second longest river in China and supports 12% of the Chinese population (Figure ^{2}. The Yellow River is famous for its excessive sediment load and deficient flow. The long-term average sediment load at the Sanmenxia Dam is 1.6 billion tons per year, which ranks first in the world.

Schematic of the Yellow River basin.

The LYR stretches from Huayuankou to Lijin, as shown in Figure ^{3} during the period from 1950 to 1999, of which 60% was deposited in the wandering reach.

Map of the Lower Yellow River.

Here, the Huayuankou and Lijin Stations were chosen to validate the effects of the model, because these two stations are representative of typical wandering and meandering reaches, respectively. They have different bankfull characteristics as a function of flow and sediment load.

For Lijin Station in the meandering reach, the bankfull characteristics are easy to determine by observing when the water fills the main channel without overtopping the banks of the floodplain (Figure

Typical cross-sectional profile at Lijin Station.

Typical cross-sectional profile at Huayuankou Station.

The deviation of the inner-annual distributions of flow and sediment discharge can cause different channel forms in the LYR. These two chosen reaches had similar annual mean values of flow and sediment discharge, but different annual distributions (Figure ^{3}/s at Huayuankou Station and 1036 m^{3}/s at Lijin Station, while the annual peak flow discharge (daily peak data) was 6014 and 4713 m^{3}/s, respectively. The annual mean sediment discharge from 1950 to 2002 was 32 m^{3}/s at Huayuankou Station and 26 m^{3}/s at Lijin Station, while the annual peak sediment discharge (daily peak data) was 653 and 330 m^{3}/s, respectively.

Flow and sediment discharge data at Huayuankou and Lijin Stations.

Using time series analysis, a decreasing trend was detected for the annual mean and daily peak flow and sediment discharge for the two stations over the past 50 years using the Mann-Kendall method [

Figure

Correlation matrix of the bankfull characteristics at Lijin Station.

Bankfull discharge | Bankfull area | Bankfull width | Bankfull depth | |
---|---|---|---|---|

Bankfull discharge | 1 | 0.80 | 0.45 | 0.74 |

Bankfull area | 1 | 0.44 | 0.97 | |

Bankfull width | 1 | 0.23 | ||

Bankfull depth | 1 |

Observed bankfull characteristics at Lijin Station: (a) bankfull discharge and area; (b) bankfull width and depth.

The same analysis for Huayuankou in the wandering reach is shown in Figure

Correlation matrix of the bankfull characteristics at Huayuankou Station.

Bankfull discharge | Bankfull area | Bankfull width | Bankfull depth | |
---|---|---|---|---|

Bankfull discharge | 1 | 0.41 | 0.35 | 0.23 |

Bankfull area | 1 | 0.52 | 0.79 | |

Bankfull width | 1 | −0.08 | ||

Bankfull depth | 1 |

Observed bankfull characteristics at Huayuankou Station: (a) bankfull discharge and area; (b) bankfull width and depth.

In previous geomorphic models, channel response to disturbances can be described using nonlinear decay functions, for which the rate law has been most widely used to describe relaxation paths and recovery times. The rate law takes the form of a negative exponential equation:

As mentioned above, the rate law is used to illustrate the channel forming macroprocess (10^{1}-10^{2} years) without any physical mechanism. It smoothes the flood details and cannot account for the effects of the deviation of flood events. This paper focuses on the dynamic processes of the channel geometry of alluvial rivers. Bankfull parameters such as the bankfull discharge

The cross-sections of the LYR are mostly wide and shallow with 400–4000 m river top widths and 1–5 m river depths. As a result, the wetted perimeter ^{2}) and ^{3}/s),

Equation (^{3}), ^{2}/s), and

These four governing equations involve two independent variables, ^{0}-10^{1} years,

For an alluvial river like the LYR,

We rewrite (^{3}/s). With the assumption of a rectangular channel, we have

For a nonstationary state,

A plot of the channel response processes is shown in Figure

Channel response processes for the change of water and sediment flow.

Equation (

The computational model in a reach of the LYR is shown in Figure _{0} and ^{3}) of the two stations; ^{3}/s),

Computational process of the bankfull area computational model.

From Jinan Station to the Lijin Station, the model uses the daily water and sediment discharge data from the 1964 to 2000: the bed slope

Grain size of

Year | Huayuankou/mm | Lijin/mm |
---|---|---|

1964 | 0.009 | 0.014 |

1965 | 0.027 | 0.014 |

1966 | 0.015 | 0.015 |

1967 | 0.02 | 0.018 |

1968 | 0.027 | 0.021 |

1969 | 0.016 | 0.009 |

1970 | 0.018 | 0.012 |

1971 | 0.02 | 0.017 |

1972 | 0.02 | 0.019 |

1973 | 0.018 | 0.018 |

1974 | 0.019 | 0.021 |

1975 | 0.021 | 0.026 |

1976 | 0.015 | 0.023 |

1977 | 0.019 | 0.014 |

1978 | 0.018 | 0.016 |

1979 | 0.016 | 0.016 |

1980 | 0.018 | 0.017 |

1981 | 0.018 | 0.027 |

1982 | 0.018 | 0.023 |

1983 | 0.022 | 0.029 |

1984 | 0.022 | 0.021 |

1985 | 0.023 | 0.021 |

1986 | 0.018 | 0.012 |

1987 | 0.015 | 0.008 |

1988 | 0.02 | 0.015 |

1989 | 0.024 | 0.018 |

1990 | 0.018 | 0.015 |

1991 | 0.017 | 0.016 |

1992 | 0.028 | 0.016 |

1993 | 0.022 | 0.021 |

1994 | 0.029 | 0.018 |

1995 | 0.024 | 0.022 |

1996 | 0.022 | 0.02 |

1997 | 0.027 | 0.016 |

1998 | 0.021 | 0.021 |

1999 | 0.02 | 0.012 |

2000 | 0.005 | 0.029 |

Bankfull computational results at Lijin Station for 38 years: bankfull (a) discharge; (b) area; (c) width; and (d) depth.

Figure ^{3} per year or 10% less than the mean value in the past 50 years. In 1997, the annual sediment load at Lijin Station was 0.016 billion tons per year or 98% less than the mean value in the past 50 years. The annual flow volume was 1.86 billion m^{3} per year, with 227 days in the zero-flow state. In these years, deposition along the river occurred more heavily at the outlet. This was the main reason why the observed bankfull width was reduced and the calculated value was not. The second type was years with overflood events, such as what occurred in 1967 and 1988. In 1967, the maximum daily discharge was 8510 m^{3}/s and the observed bankfull discharge (after the flood season) was 7500 m^{3}/s. In 1988, the maximum daily discharge was 5220 m^{3}/s and the observed bankfull discharge (after the flood season) was 5000 m^{3}/s. However, overbank flood events were less important than the larger sediment load and smaller flow volume from the error analysis at Lijin Station. One explanation for the smaller impact might be the fact that the overbank flood volume was not much larger than the bankfull flood at the river outlet.

The daily water and sediment discharge data for Xiaolangdi and Huayuankou Stations from 1976 to 1997 was used in the model, together with the daily water discharge data of Heishiguan and Wuzhi Stations. The simulation results are shown in Figure

Bankfull computational results at Huayuankou station for 22 years: bankfull (a) discharge; (b) area; (c) width; and (d) depth.

The calculated errors at Huayuankou station were more obvious than those at Lijin station. Besides the first type, which included 1987 and 1992, the second type, which included ^{3}/s, respectively, and the bankfull discharge (after the flood season) was 5320, 6000, and 6800 m^{3}/s. Three years of overbank flood events caused obvious river width expansion, which the methods in this paper cannot reflect.

From Figures

Two tributaries in the Xiaolangdi-Huayuankou reach and the Yiluo and Qin Rivers affected the flow and sediment discharge in the reach. Given the lack of daily sediment concentration data, a simplified study is required.

According to the statistical data from 1960 to 1996, the annual average sediment discharge of the Yiluo River (Heishiguan station) is

Differences exist in the scour and silting mechanisms between overbank and normal floods. Scouring occurs when the water level rises, while silting occurs when the water level falls. Floodplain silting with channel scouring is common during overbank flood periods in the LYR.

For overbank floods with a low sediment load, for example, the flood that occurred from July 31 to August 8, 1982, the peak discharge at Huayuankou station was 15,300 m^{3}/s and the average sediment concentration was 67 kg/m^{3}. This caused 0.217 billion tons of sediment to be deposited in the floodplain and 0.15 billion tons to be scoured from the main channel in the Huayuankou-Aishan reach.

For overbank floods with high sediment load, for example, the flood that occurred in 1977, the peak discharge at Huayuankou station was 10,800 m^{3}/s and the average sediment concentration was 437 kg/m^{3}. This caused the amount of sediment deposited in the main channel to be three times that deposited in the floodplain in the Huayuankou-Aishan reach.

Consequently, the methods used in this paper are not suitable for overbank situations. The key issue is that the distribution of resistance in the floodplain and main channel is unknown. A simplified statistical method is used to deal with overbank flood events. For flows with a high sediment concentration, sediment is deposited all along the river, and a distributing coefficient ^{3}) and

For flows with a high sediment concentration, sediment is deposited all along the river ^{3}.

Here, the chosen river reach is assumed to be uniform so that the depositing or eroding sediment volume can be distributed along the reach evenly. This assumption is much more acceptable in the Jinan-Lijin reach than in the Xiaolangdi-Huayuankou reach, because there is a valley channel beyond Xiaolangdi Station that affects the sediment distribution along the Xiaolangdi-Huayuankou reach. This might be an important impact factor for the accuracy of the model applied to Huayuankou station.

Based on the equations for stable alluvial rivers, namely, the Exner equation, flow continuity, the momentum, and sediment concentration conservation equations and assuming that the riverbed slope is constant on middle timescales (10^{0}-10^{1} years), we found stationary geomorphic relations for time-varying fluvial processes. We obtained a time-dependent relation for the channel geometry given nonstationary alluvial processes.

The final solutions from the equations for stable alluvial rivers simply reflect the adjusted direction of channel geometry. The adjusted magnitude of the hydraulic geometry each day is determined by the daily input of depositing or eroding sediment volumes. The direction and magnitude give the final channel shape.

Using daily flow and sediment data to determine the adjustment direction and magnitude, the variation in the bankfull characteristics for Lijin and Huayuankou Stations in the LYR were calculated. The final results agreed closely

Compared with the rate-law and hydrodynamic methods, the proposed method enables multiple dependent variables and the input of detailed hydrological data for long-term simulations. However, because of those assumptions for simplification, the model we provided also has some limitations. Firstly, it cannot be applied to a dramatic time-varying fluvial procedure like an overbank flow or a hyperconcentration flood, only fitting for a quasisteady fluvial process. Secondly, the time-step of one day is still large for a physics-based model which smoothes the deviation of inner-daily flood process. Thirdly, the key equation in this model is derived by the experiential relations between channel geometry and hydraulic relations, which is not verified in theory. These limitations are the main sources of the model error and will be our next work in the future.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This study was supported financially by the China Ministry of Science and Technology through the Key Project in the National Science & Technology Pillar Program in the Twelfth Five-year Plan Period of China (Grant no. 2012BAB02B02). It was supported by the Open Program Fund of MWR Key Laboratory of Yellow River Sedimentation of the Yellow River Conservancy Commission.