3D numerical computation of the tidally induced Lagrangian residual current in an idealized bay

Abstract

A numerical model that solves 3D first-order Lagrangian residual velocity (u_L) equations is established by modifying the HAMSOM model. With this model, u_L is studied in a wide, idealized bay. The results show that the vertical eddy viscosity term of Stokes’ drift (π₁) in the tidal body force determines the overall flow state of u_L, and the contribution of the advection term (π₂) is responsible for the small correction. In addition, two types of Coriolis effects introduced into the residual current system not only enhance the lateral flow and break the symmetry of the flow regime in the bay but also slightly correct the flow state driven by the entire tidal body force. It is also found by numerical sensitivity experiments that the increase in the aspect ratio δ, implying a decrease in the topographic gradient, can simplify the residual flow state. The increase in tidal amplitude at the open boundary significantly enhances the intensity of u_L and causes the residual flow regime to be more complicated in the bay. This can be ascribed to the disproportionate increase in the tidal body force. The proportion of the vertical eddy viscosity term of Stokes’ drift in the tidal body force also varies with the vertical eddy viscosity coefficient, which leads to different residual current states. Compared with the influence of incoming tidal strength on the residual current, the effect of the bottom friction coefficient on the residual current is relatively mild. An increase in the quadratic bottom friction coefficient induces an unbalanced decrease in the tidal body force. Therefore, u_L decreases, but the flow regime is more complex. The influence of the nonlinear effect of the bottom friction decreases from the bay head towards the bay mouth. The residual current only changes in magnitude near the bay mouth but changes in pattern near the bay head for different bottom friction coefficients. By keeping the bottom friction coefficient in the zeroth-order tidal equations constant, the sensitivity experiment shows that u_L is insensitive to the change in bottom friction coefficient in the governing equations of u_L.

Appendix. The difference schemes of each term in Eqs. 23 and 24

The index sequences k, i, and j are grid numbers along the x-, y-, and z-directions, respectively. The grid numbers k, i, and j in three directions increase from west to east, north to south, and top to bottom, respectively.

The difference scheme of each term in Eq. 23 is listed as follows.

$$ -\underset{z}{\overset{z+\varDelta h}{\int }}\frac{\partial }{\partial z}\left(\upsilon \frac{\partial \left({\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {u}_0\right)}{\partial z}\right) dz=\upsilon \frac{\partial \left({\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {u}_0\right)}{\partial z}{\left|{}_z-\upsilon \frac{\partial \left({\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {u}_0\right)}{\partial z}\right|}_{z+\varDelta h} $$

The difference scheme of Stokes’ drift in the x-direction at grid point (j, i, k) is

$$ {\displaystyle \begin{array}{l}{\left({\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {u}_0\right)}_{j,i,k}={0.5}^{\ast }{\xi}_{0j,i,k}\left[{u}_{0,j,i,k+1}-{u}_{0,j,i,k-1}\right]/\varDelta x\\ {}\kern1.7em +{0.125}^{\ast }{\left[{\eta}_{0,j,i,k}+{\eta}_{0,j,i,k+1}+{\eta}_{0,j,i-1,k+1}+{\eta}_{0,j,i-1,k}\right]}^{\ast}\left[{u}_{0,j,i-1,k}-{u}_{0,j,i+1,k}\right]/\varDelta y\\ {}\kern1.7em +{0.125}^{\ast}\left[{\varsigma}_{0,j,i,k}+{\varsigma}_{0,j,i,k+1}+{\varsigma}_{0,j+1,i,k}+{\varsigma}_{0,j+1,i,k+1}\right]\ast \left[{u}_{0,j-1,i,k}-{u}_{0,j+1,i,k}\right]/\varDelta h\end{array}} $$

Then, the difference scheme of the first term in Eq. 23 at any point in time is

$$ -\underset{z}{\overset{z+\varDelta h}{\int }}\frac{\partial }{\partial z}\left(\upsilon \frac{\partial \left({\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {u}_0\right)}{\partial z}\right) dz={\upsilon}^{\ast}\left({2}^{\ast }{\left({\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {u}_0\right)}_{j,i,k}-{\left({\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {u}_0\right)}_{j+1,i,k}-{\left({\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {u}_0\right)}_{j-1,i,k}\right)/\varDelta h $$

The second term in Eq. 23 can be adapted for the finite difference as follows

$$ -\underset{z}{\overset{z+\varDelta h}{\int }}{\mathbf{u}}_{\mathbf{0}}\cdot \nabla {u}_0 dz=-\underset{z}{\overset{z+\varDelta h}{\int }}\left(\frac{\partial {u}_0{u}_0}{\partial x}+\frac{\partial {u}_0{v}_0}{\partial y}+\frac{\partial {u}_0{w}_0}{\partial z}\right) dz $$

$$ {\left.={u}_0{w}_0\right|}_{z+\varDelta h}^z-\varDelta h\left({u}_0\frac{\partial {u}_0}{\partial x}+{u}_0\frac{\partial {u}_0}{\partial x}+{u}_0\frac{\partial {v}_0}{\partial y}+{v}_0\frac{\partial {u}_0}{\partial y}\right) $$

$$ =-\varDelta h{u}_0\frac{\partial {u}_0}{\partial x}-\varDelta h{v}_0\frac{\partial {u}_0}{\partial y}+\varDelta h{u}_0\frac{\partial {w}_0}{\partial z}+{u}_0{w}_0{\left|{}_z-{u}_0{w}_0\right|}_{z+\varDelta h} $$

The difference scheme of each term at grid point (j, i, k) is

$$ {\left(-h{u}_0\frac{\partial {u}_0}{\partial x}\right)}_{j,i,k}={0.5}^{\ast }{\varDelta h}^{\ast}\kern0.1em {u_{0,j,i,k}}^{\ast}\left[{u}_{0j,i,k-1}-{u}_{0,j,i,k+1}\right]/\varDelta y $$

$$ {\left(-h{v}_0\frac{\partial {u}_0}{\partial y}\right)}_{j,i,k}={0.125}^{\ast }{\varDelta h}^{\ast}\kern0.1em {\left[{v}_{0,j,i,k}+{v}_{0,j,i,k+1}+{v}_{0,j,i,k+1}+{v}_{0,j,i,k+1}\right]}^{\ast}\left[{u}_{0,j,i-1,k}-{u}_{0,j,i+1,k}\right]/\varDelta y $$

$$ {\left(h{u}_0\frac{\partial {w}_0}{\partial z}\right)}_{j,i,k}={0.5}^{\ast}\kern0.1em {u_{0,j,i,k}}^{\ast}\left[{w}_{0,j,i,k}+{w}_{0,j,i,k+1}-{w}_{0,j+1,i,k}-{w}_{0,j+1,i,k+1}\right] $$

$$ {\left.{\left({u}_0{w}_0\right)}_{j,i,k}={u}_0{w}_0\right|}_z={0.125}^{\ast}\kern0.1em {\left[{u}_{0j,i,k}+{u}_{0,j-1,i,k}\right]}^{\ast}\left[{w}_{0j,i,k}+{w}_{0,j,i,k+1}\right] $$

$$ {\left.{\left({u}_0{w}_0\right)}_{j-1,i,k}={u}_0{w}_0\right|}_{z+\varDelta h}={0.125}^{\ast}\kern0.1em {\left[{u}_{0j+1,i,k}+{u}_{0j,i,k}\right]}^{\ast}\left[{w}_{0j+1,i,k}+{w}_{0j+1,i,k+1}\right] $$

Then, the difference scheme of the second term in Eq. 23 at any point in time is

$$ -\underset{z}{\overset{z+\varDelta h}{\int }}{\mathbf{u}}_{\mathbf{0}}\cdot \nabla {u}_0 dz={\left(-\varDelta h{u}_0\frac{\partial {u}_0}{\partial x}\right)}_{j,i,k}+{\left(-\varDelta h{v}_0\frac{\partial {u}_0}{\partial y}\right)}_{j,i,k}+{\left(\varDelta h{u}_0\frac{\partial {w}_0}{\partial z}\right)}_{j,i,k}+{\left({u}_0{w}_0\right)}_{j,i,k}-{\left({u}_0{w}_0\right)}_{j-1,i,k} $$

The third term in Eq. 23 can be adapted for the finite difference as follows

$$ -\underset{z}{\overset{z+\varDelta h}{\int }}{f}^{\ast }{\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {v}_0\kern0.2em dz=-f\underset{z}{\overset{z+\varDelta h}{\int }}\left(\frac{\partial {\xi}_0{v}_0}{\partial x}+\frac{\partial {\eta}_0{v}_0}{\partial y}+\frac{\partial {\varsigma}_0{v}_0}{\partial z}\right) dz $$

$$ {\left.=f{v}_0{\varsigma}_0\right|}_{z+\varDelta h}^z- f\varDelta h\left({\xi}_0\frac{\partial {v}_0}{\partial x}+{\eta}_0\frac{\partial {v}_0}{\partial y}+{v}_0\frac{\partial {\xi}_0}{\partial x}+{v}_0\frac{\partial {\eta}_0}{\partial y}\right) $$

$$ =- f\varDelta h{\xi}_0\frac{\partial {v}_0}{\partial x}- f\varDelta h{\eta}_0\frac{\partial {v}_0}{\partial y}+ f\varDelta h{v}_0\frac{\partial {\varsigma}_0}{\partial z}+f{v}_0{\varsigma}_0{\left|{}_z-f{v}_0{\varsigma}_0\right|}_{z+\varDelta h} $$

The difference scheme of each term at grid point (j, i, k) is

$$ {\left(- f\varDelta h{\xi}_0\frac{\partial {v}_0}{\partial x}\right)}_{j,i,k}={0.5}^{\ast }{f}^{\ast }{h_j}^{\ast }{\xi_{0,j,i,k}}^{\ast}\kern0.1em \left[{v}_{0j,i,k}+{v}_{0j,i-1,k}-{v}_{0j,i,k+1}-{v}_{0j,i-1,k+1}\right]/\varDelta x\kern0.1em $$

$$ {\displaystyle \begin{array}{l}{\left(- f\varDelta h{\eta}_0\frac{\partial {v}_0}{\partial y}\right)}_{j,i,k}={0.125}^{\ast }{f}^{\ast }{h_j}^{\ast}\kern0.1em \left[{v}_{0,j,i,k}+{v}_{0,j,i,k+1}-{v}_{0,j,i-1,k+1}-{v}_{0,j,i-1,k}\right]\\ {}{\kern0.1em }^{\ast}\left[{\eta}_{0,j,i,k}+{\eta}_{0,j,i,k+1}+{\eta}_{0,j,i-1,k+1}+{\eta}_{0,j,i-1,k}\right]/\varDelta y\end{array}} $$

$$ {\displaystyle \begin{array}{l}{\left( f\varDelta h{v}_0\frac{\partial {\varsigma}_0}{\partial z}\right)}_{j,i,k}={0.125}^{\ast }{f}^{\ast}\kern0.1em \left[{v}_{0,j,i,k}+{v}_{0,j,i,k+1}+{v}_{0,j,i-1,k}+{v}_{0,j,i-1,k+1}\right]\\ {}{\kern0.1em }^{\ast}\left[{\varsigma}_{0,j,i,k}+{\varsigma}_{0,j,i,k+1}-{\varsigma}_{0,j+1,i,k}-{\varsigma}_{0,j+1,i,k+1}\right]/\varDelta z\end{array}} $$

$$ {\displaystyle \begin{array}{l}{\left(f{v}_0{\varsigma}_0\right)}_{j,i,k}={0.0625}^{\ast }{f}^{\ast}\kern0.1em \Big[{v}_{0,j,i,k}+{v}_{0,j,i,k+1}+{v}_{0,j,i-1,k}+{v}_{0,j,i-1,k+1}+{v}_{0,j-1,i,k}+{v}_{0,j-1,i,k+1}\\ {}+{v}_{0,j-1,i-1,k}+{v}_{0,j-1,i-1,k+1}\Big]\ast \left[{\varsigma}_{0,j,i,k}+{\varsigma}_{0,j,i,k+1}\right]\end{array}} $$

$$ {\displaystyle \begin{array}{c}{\left(f{v}_0{\varsigma}_0\right)}_{j-1,i,k}={0.0625}^{\ast }{f}^{\ast}\kern0.1em \Big[{v}_{0,j,i,k}+{v}_{0,j,i,k+1}+{v}_{0,j,i-1,k}+{v}_{0,j,i-1,k+1}+{v}_{0,j+1,i,k}+{v}_{0,j+1,i,k+1}\\ {}\kern0.5em +{v}_{0,j+1,i-1,k}+{v}_{0,j+1,i-1,k+1}\Big]\left[{\varsigma}_{0,j+1,i,k}+{\varsigma}_{0,j+1,i,k+1}\right]\end{array}} $$

Then, the difference scheme of the third term in Eq. 23 at any point in time is

$$ -\underset{z}{\overset{z+\varDelta h}{\int }}{f}^{\ast }{\boldsymbol{\upxi}}_{\mathbf{0}}\cdot \nabla {v}_0\kern0.2em dz $$

$$ ={\left(- f\varDelta h{\xi}_0\frac{\partial {v}_0}{\partial x}\right)}_{j,i,k}+{\left(- f\varDelta h{\eta}_0\frac{\partial {v}_0}{\partial y}\right)}_{j,i,k}+{\left( f\varDelta h{v}_0\frac{\partial {\varsigma}_0}{\partial z}\right)}_{j,i,k}+{\left(f\;{v}_0{\varsigma}_0\right)}_{j,i,k}-{\left(f\;{v}_0{\varsigma}_0\right)}_{j-1,i,k} $$

Finally, the three terms in Eq. 23 are averaged in a tidal period. Then, the difference scheme of the x-direction tidal body force is obtained.

Similarly, the difference scheme of each term in Eq. 24 can also be obtained, which are omitted here for brevity.