Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay, China

Abstract

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.

Appendix A. Definition of indexes

Ln in (Eq. 7) measures the intra-tidal variation of the Lagrangian residual velocity. It can also be used to represent the degree of nonlinearity of the M ₂ tidal system.

$$ Ln=\sqrt{{\frac{{\sum\limits_{p=1}^N {\left( {\varDelta {u_{lp}}^2+\varDelta {v_{lp}}^2} \right)} }}{N-1 }}} $$

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where p is the initial tidal phase, N (=12) is the total number of tidal phases, ∆u _l and ∆v _l represent the deviation between the Lagrangian residual velocity at different tidal phases and the time-averaged Lagrangian residual velocity, respectively.

$$ \varDelta {u_l}={u_l}-{{\bar{u}}_{la }} $$

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$$ \varDelta {v_l}={v_l}-{{\bar{v}}_{la }} $$

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where u _l and v _l are, respectively, the east and north components of the Lagrangian residual velocity (i.e. ${{\mathop{u}\limits^{ \rightharpoonup } }_{L}}$) at a given initial tidal phase; ${{\mathop{u}\limits^{ \rightharpoonup } }_{{La}}} = \left( {\sum\limits_{{p = 1}}^{N} {{{{\mathop{u}\limits^{ \rightharpoonup } }}_{{Lp}}}} } \right)/N,p = 1,2 \cdots N$ is the time average of ${{\mathop{u}\limits^{ \rightharpoonup } }_{L}}$; and ${{\bar{u}}_{la }}$ and ${{\bar{v}}_{la }}$ are, respectively, the east and the north components of ${{\mathop{u}\limits^{ \rightharpoonup } }_{{La}}}$.

The indexes ha, va, hv and vv quantitatively measure the effect of horizontal advection, vertical advection, horizontal viscosity and vertical viscosity, respectively. In fact, in the indexes va and vv, the effect of bottom friction has been taken into account due to the vertical gradient of the current induced by bottom friction.

$$ ha=\frac{1}{{n{T_{M_2 }}}}\int_{t_0}^{{t_0 +n{T_{M_2 }}}} {\frac{{\sqrt{{{{{\left( {u\frac{{\partial u}}{{\partial x}}+v\frac{{\partial u}}{{\partial y}}} \right)}}^2}+{{{\left( {u\frac{{\partial v}}{{\partial x}}+v\frac{{\partial v}}{{\partial y}}} \right)}}^2}}}}}{{\sqrt{{{{{\left( {\frac{{\partial u}}{{\partial t}}} \right)}}^2}+{{{\left( {\frac{{\partial v}}{{\partial t}}} \right)}}^2}}}}}\mathrm{d}t} $$

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$$ va=\frac{1}{{n{T_{M_2 }}}}\int_{t_0}^{{t_0 +n{T_{M_2 }}}} {\frac{{\sqrt{{{{{\left( {\frac{\omega }{D}\frac{{\partial u}}{{\partial \sigma }}} \right)}}^2}+{{{\left( {\frac{\omega }{D}\frac{{\partial v}}{{\partial \sigma }}} \right)}}^2}}}}}{{\sqrt{{{{{\left( {\frac{{\partial u}}{{\partial t}}} \right)}}^2}+{{{\left( {\frac{{\partial v}}{{\partial t}}} \right)}}^2}}}}}\mathrm{d}t} $$

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$$ hv=\frac{1}{{n{T_{M_2 }}}}\int_{t_0}^{{t_0 +n{T_{M_2 }}}} {\frac{{\sqrt{{{{{\left( {\frac{\partial }{{\partial x}}\left( {2{A_M}\frac{{\partial u}}{{\partial x}}} \right)+\frac{\partial }{{\partial y}}\left( {A_M \left( {\frac{{\partial u}}{{\partial y}}+\frac{{\partial v}}{{\partial x}}} \right)} \right)} \right)}}^2}+{{{\left( {\frac{\partial }{{\partial y}}\left( {2{A_M}\frac{{\partial u}}{{\partial x}}} \right)+\frac{\partial }{{\partial x}}\left( {A_M \left( {\frac{{\partial u}}{{\partial y}}+\frac{{\partial v}}{{\partial x}}} \right)} \right)} \right)}}^2}}}}}{{\sqrt{{{{{\left( {\frac{{\partial u}}{{\partial t}}} \right)}}^2}+{{{\left( {\frac{{\partial v}}{{\partial t}}} \right)}}^2}}}}}} \mathrm{d}t $$

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$$ vv=\frac{1}{{n{T_{M_2 }}}}\int_{t_0}^{{t_0 +n{T_{M_2 }}}} {\frac{{\sqrt{{{{{\left( {\frac{1}{D}\frac{\partial }{{\partial \sigma }}\left( {\frac{K_M }{D}\frac{{\partial u}}{{\partial \sigma }}} \right)} \right)}}^2}+{{{\left( {\frac{1}{D}\frac{\partial }{{\partial \sigma }}\left( {\frac{K_M }{D}\frac{{\partial v}}{{\partial \sigma }}} \right)} \right)}}^2}}}}}{{\sqrt{{{{{\left( {\frac{{\partial u}}{{\partial t}}} \right)}}^2}+{{{\left( {\frac{{\partial v}}{{\partial t}}} \right)}}^2}}}}}} \mathrm{d}t $$

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where ${T_{M_2 }}$ is the M ₂ tidal period; n is the number of tidal periods (in this study, n = 1); u and v are the horizontal tidal current components of the velocities; ω is the vertical velocity in the sigma coordinate system; D is the transient total water depth; A _M is the horizontal diffusivity; K _M is the vertical diffusivity; and t ₀ is the initial time at which the water parcels are released.

To separate the advection and the viscosity regimes, the above indexes are divided by their summation values and multiplied by 100 % to obtain percentage values.

$$ ha\%=\frac{ha }{ha+va+hv+vv}\times 100\% $$

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$$ va\%=\frac{va }{ha+va+hv+vv}\times 100\% $$

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$$ hv\%=\frac{hv }{ha+va+hv+vv}\times 100\% $$

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$$ vv\%=\frac{vv }{ha+va+hv+vv}\times 100\% $$

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