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Liouville equation-based stochastic model for shoreline evolution

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Abstract

Long-term shoreline evolution due to longshore sediment transport is one of the key processes that need to be addressed in coastal engineering design and management. To adequately represent the inherent stochastic nature of the evolution processes, a probability density evolution model based on a Liouville-type equation is proposed for predicting the shoreline changes. In this model, the standard one-line beach evolution model that is widely used in coastal engineering design is reformulated in terms of the probability density function of shoreline responses. A computational algorithm involving a total variation diminishing scheme is employed to solve the resulting equation. To check the accuracy and robustness of the model, the predictions of the model are evaluated by comparing them with those from Monte Carlo simulations for two idealised shoreline configurations involving a single long jetty perpendicular to a straight shoreline and a rectangular beach nourishment case. The pertinent features of the predicted probabilistic shoreline responses are identified and discussed. The influence of the density distributions of the input parameters on the computed results is investigated.

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Abbreviations

a 1 :

Constant in the expression for q l0 (\( \frac{{{\text{m}}^{ 2} \;{\text{s}}^{ 2} }}{\text{kg}} \))

B :

Operator

C gb :

Wave group celerity (m/s)

C v :

Coefficient of variance, 0 ≤ C v ≤ 1, defined as the ratio of the standard deviation to the mean

d:

Difference operator

d b :

Breaker depth (m)

D b :

Standard deviation of H b, confirmed to a normal distribution (m)

d c :

Closure depth (m)

E b :

Wave energy evaluated at the breaking point (kN)

g :

Acceleration of gravity (m/s2)

h b :

Discrete wave height at breaking (m)

H b :

Wave height at breaking (m)

H s :

Significant wave height (m)

i :

Cell number alongshore

i b :

Node number in PDF of H b

j :

Node number in cross-shore

k :

Time step of one-line model

k 1 :

Sediment transport coefficient, or, dimensionless empirical coefficient

K b :

Mean of H b

l :

Distance alongshore (m)

m :

Number of time steps in TVD scheme

m y :

Number of the discrete distribution in cross-shore direction

N b :

Number of uniformly meshing discretisation for each time increment

N p :

Total number of the selected nodes, = 2m y  × N b

n r :

Dimension of excitation vector

n s :

Void ratio of sand, ≈0.4

n t :

Number of time steps

p :

Probability density function or PDF

\( \hat{p} \) :

Mixing ratio at the cell edges

q l :

Volumetric long-shore sediment transport rate (m3/s)

q l0 :

Amplitude of volumetric long-shore sediment transport rate (m3/s)

r c :

Lattice ratio

t :

Time (s)

t 0 :

Initial time (s)

t d :

Time token for running the deterministic model (s)

t k :

Time in kth step of one-line shoreline model (s)

t m :

Time in mth step (s) of TVD scheme

t LE :

Time consumed by solving LE-based model (s)

TVm :

Total variation at time step m

v :

\( \dot{Y} \)

v m :

Mean of v

x :

Distance alongshore (m)

y :

Shoreline position in cross-shore direction (m)

Y :

Stochastic process or random variable of shoreline position or random state (which is a component of state vector)

Y 0 :

Initial random state or random variable of shoreline position Y

Y F :

Extension distance seaward from the rectangular beach

y m :

Expected value or mean value of y

Y m0 :

Initial mean value of Y

y max :

Maximum discrete shoreline position

y min :

Minimum discrete shoreline position

y s :

Standard deviation of y

Y s0 :

Initial standard deviation of Y

\( \dot{Y} \) :

Time derivate of Y, = −H 5/2b ζ(y, t), or the ‘velocity’ of the response for a prescribed h b, called advection velocity coefficient

Z :

Random parameter with known PDFs

Z 0 :

Initial Z

α 0 :

Incident angle of breaking waves relative to x (°)

α b :

Angle of breaking waves to the local shoreline (°)

γ :

Ratio of wave height to water depth at breaking, \( = {\raise0.7ex\hbox{${H_{\text{b}} }$} \!\mathord{\left/ {\vphantom {{H_{\text{b}} } {d_{\text{b}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${d_{\text{b}} }$}} \)

γ LH :

Log-normal distribution location parameter of H b

γ LY :

Log-normal distribution location parameter of Y 0

γ WH :

Weibull distribution location parameter of H b

γ WY :

Weibull distribution location parameter of Y 0

Δh b :

Size increment in h b (m)

Δt k :

Time increment of one-line shoreline model (s)

Δt m :

Time increment of TVD (s)

Δx :

Size increment in along-shore direction (m)

Δy :

Size increment in cross-shore direction (m)

ζ :

\( = \frac{1}{8}\rho_{\text{w}} g^{3/2} \gamma^{ - 1/2} a_{1} \frac{1}{{d_{\text{c}} }}\frac{{\partial ({ \sin }(2\alpha_{\text{b}} ))}}{\partial x} \)

η :

Operator of shoreline position evolution, = −H 2.5b ζ(y, t)

η :

Operator vector of determination of a dynamical state, may be determined by an appropriate deterministic shoreline evolution model

θ :

\( = \frac{{\left( {p_{j,m} - p_{j - 1,m} } \right)}}{{p_{j + 1,m} - p_{j,m} }} \)

λ :

Ratio, \( = \frac{{\varDelta t_{m} }}{{\varDelta t_{k} }} > 1 \)

μ LH :

Log-normal distribution scale parameter of H b

μ LY :

Log-normal distribution scale parameter of Y 0

μ WH :

Weibull distribution scale parameter of H b

μ WY :

Weibull distribution scale parameter of Y 0

ρ s :

Mass density of the sediment grains (kg/m3)

ρ w :

Mass density of water (kg/m3)

σ LH :

Log-normal distribution shape parameter of H b

σ LY :

Log-normal distribution shape parameter of Y 0

σ WH :

Weibull distribution shape parameter of H b

σ WY :

Weibull distribution shape parameter of Y 0

φ :

Flux limiter

Ω z :

Distribution domain

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Acknowledgments

The support of the UK Engineering and Physical Science Research Council as part of a research project under Grant No. GR\L53953 is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Applied Science, University of Science and Technology, Beijing, 30 Xueyuan Road, Haidan District, Beijing, 100083, People’s Republic of China

    Xing Zheng Wu

  2. Division of Civil Engineering, School of Engineering, Physics and Mathematics, University of Dundee, Dundee, DD1 4HN, UK

    Ping Dong

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  1. Xing Zheng Wu
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Correspondence to Xing Zheng Wu.

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Wu, X.Z., Dong, P. Liouville equation-based stochastic model for shoreline evolution. Stoch Environ Res Risk Assess 29, 1867–1880 (2015). https://doi.org/10.1007/s00477-015-1029-1

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