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A probabilistic approach to an upper-body region simulation in a bedload-dominated delta: implications for neck area morphology and plant colonization

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Abstract

The uncertain effects of water–sediment input on the delta morphology should be carefully considered given the current high occurrence frequency of extreme climate events. To determine the formation and evolution of the upper-body region in a bedload-dominated shallow-water delta, jet theory was fundamentally utilized and resolved into 2-D plane flow fields based on the similarity solution method, and a newly parameterized bedload transport rate equation, including both the partial bedload within relatively small suspended loads and the flow regime effects, was combined with grain-size transformation between transporting and bed material to establish a bed deformation equation in this study. In a lab-scale case, the coupled deterministic model of the upper-sectional delta topography was extended based on two independent stochastic events, considering the impacts of the transverse position deviation from the longitudinal centerline in the delta and the deviation of the (least-square optimal) flow regime influencing parameter from its classic default value during diffusion and the drift term of the established stochastic differential equations. The stochastic solutions were highly consistent with limited measurements (though without reasonable parameter estimation), especially the median solution in the two-sided parts of sections and the 75th percentile solution in the central parts of the sections. Furthermore, the prediction calculations corresponding to different roughness conditions resulting from a possible improvement in the ecological environment revealed that the neck area of a mature upper delta contains both the riskiest and safest parts when faced with a storm, and that plant colonization may generate a local positive effect along with a greater accumulation of real data and a more certain determination of flow-sediment interactions in the delta, pointing a way to disaster risk reduction such as the 2021 heavy rainfall disaster in Henan Province (China) to some extent.

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Abbreviations

b :

Characteristic half thickness of jet section

C :

Integration constant

C m :

Maximum sediment concentration of the sediment-laden flow for maintaining the overall fluid characteristics

\(D_{{{\text{T}}50}}\) :

Median size of transporting sediment

d T l :

Sediment size of the lth sediment group in transport

d l :

Bed material size of the lth sediment group

e :

2.718281828

f x :

Unit mass force in x direction

f y :

Unit mass force in y direction

G 1, G 2::

Parameters in diffusion terms of SDEs of \(z_{s}^{ * }\),\(k_{3}^{ * }\) respectively

g :

Gravitational acceleration

h :

Water depth

h a :

Water column height representing the atmospheric pressure

h 0 :

Height of deposit

k 1, k 2, k 3::

Parameters in bedload transport rate formula

\(k_{3[t]}^{ * }\) :

Randomizing variable of k3 at any time t

L :

Average distance of deltaic progradation

N :

Total number of sediment groups

n :

Roughness coefficient

P bl :

Sediment composition coefficient of the lth sediment group

Q :

Water discharge

Q s :

Sediment discharge

\(q_{bl}\) :

Total bedload transport rate

\(q_{blx}\) :

Bedload transport rate of the lth sediment group along the x direction

\(q_{bly}\) :

Bedload transport rate of the lth sediment group along the y direction

R :

Independent standard normal random variable

r :

Polar radius

\(\overline{S}\) :

Average sediment concentration

\(S^{\prime}\) :

Pulsating sediment concentration

S v :

Qs/Q

U :

Velocity magnitude

U cl :

Incipient velocity of bedload of the lth sediment group

u m :

Longitudinal axial velocity

u x :

Velocity in x direction

\(u^{\prime}_{x}\) :

Pulsating velocity in x direction

u y :

Velocity in y direction

\(u^{\prime}_{y}\) :

Pulsating velocity in y direction

u 0 :

Incident velocity

W :

Uncorrelated multiplicative standard Gaussian white noise

z s :

Bed surface elevation

\(z_{s[t]}^{ * }\) :

Randomizing variable of delta surface elevation at any time t

z 0 :

Bottom height of bed sediment

Δρ :

Density difference between sediment and clear water

δ :

Water molecular thickness

ε :

0.154

ε x :

Turbulent diffusion coefficient in x direction

ε y :

Turbulent diffusion coefficient in y direction

\(\eta {\kern 1pt}\) :

\(- \frac{{y^{2} }}{{b^{2} }}\)

θ :

Bed slope angle

\(\mu_{{k_{1} }}\), \(\mu_{{k_{2} }}\), \(\mu_{{k_{3} }}\) :

Mean of estimation of k1, k2, k3

\(\zeta_{l}\) :

Coefficient of turbulent effect on the sediment of the lth group in transport

ρ m :

Sediment-laden flow density

ρ s :

Sediment density

ρ 0 :

Mixed density of bed sediment and saturated pore water

ρ :

Water density

σ g :

Geometrical standard deviation of bed material

τ sx :

Wind stress in x direction

τ sy :

Wind stress in y direction

\(\upsilon\) :

Dynamic viscosity coefficient

φ :

e η

ψ :

Stream function

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Acknowledgements

We would like to thank Springer Nature Author Services for their professional manuscript services, especially the English polishing.

Funding

This research is supported by the National Natural Science Foundation of China (Grant No. 51879182), and the Science and Technology Planning Program of Tianjin, China (Grant No. 21JCQNJC00480).

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Authors and Affiliations

  1. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin, 300350, China

    Xiaolong Song, Haijue Xu & Yuchuan Bai

  2. Institute for Sedimentation on River and Coastal Engineering, Tianjin University, Tianjin, 300350, China

    Xiaolong Song, Haijue Xu & Yuchuan Bai

  3. Tianjin Research Institute for Water Transport Engineering, Ministry of Transport, Tianjin, 300456, China

    Weiyan Xin

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Contributions

Weiyan Xin: Flow Model Methodology; Xiaolong Song: Bed Deformation Model Methodology, Data Curation and Analysis, Innovative Deltaic Perspective on Heavy Rainfall Disaster Affected Areas in Henan Province of China in 2021, Writing; Haijue Xu: Resources, Methodology; Yuchuan Bai: Conceptualization, Supervision, Project Administration.

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Correspondence to Xiaolong Song.

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Xin, W., Song, X., Xu, H. et al. A probabilistic approach to an upper-body region simulation in a bedload-dominated delta: implications for neck area morphology and plant colonization. Stoch Environ Res Risk Assess 37, 4141–4160 (2023). https://doi.org/10.1007/s00477-023-02498-x

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