Submesoscale tidal eddies in the wake of coral islands and reefs: satellite data and numerical modelling

Abstract

Interaction of tidal flow with a complex topography and bathymetry including headlands, islands, coral reefs and shoals create a rich submesoscale field of tidal jets, vortices, unsteady wakes, lee eddies and free shear layers, all of which impact marine ecology. A unique and detailed view of the submesoscale variability in a part of the Great Barrier Reef lagoon, Australia, that includes a number of small islands was obtained by using a “stereo” pair of 2-m-resolution visible-band images that were acquired just 54 s apart by the WorldView-3 satellite. Near-surface current and vorticity were extracted at a 50-m-resolution from those data using a cross-correlation technique and an optical-flow method, each yielding a similar result. The satellite-derived data are used to test the ability of the second-generation Louvain-la-Neuve ice-ocean model (SLIM), an unstructured-mesh, finite element model for geophysical and environmental flows, to reproduce the details of the currents in the region. The model succeeds in simulating the large-scale (> 1 km) current patterns, such as the main current and the width and magnitude of the jets developing in the gaps between the islands. Moreover, the order of magnitude of the vorticity and the occurrence of some vortices downstream of the islands are correctly reproduced. The smaller scales (< 500 m) are resolved by the model, although various discrepancies with the data are observed. The smallest scales (< 50 m) are unresolved by both the model- and image-derived velocity fields. This study shows that high-resolution models are able to a significant degree to simulate accurately the currents close to a rugged coast. Very-high-resolution satellite oceanography stereo images offer a new way to obtain snapshots of currents near a complex topography that has reefs, islands and shoals, and is a potential resource that could be more widely used to assess the predictive ability of coastal circulation models.

Appendices

Appendix A: Total vorticity flux in the free shear layer

In this section, it is shown that the total vorticity flux in the lee of a backward facing step computed using two different finite element methods, respectively a P1 continuous Galerkin (CG) method and a P1 discontinuous Galerkin (DG) method (such as SLIM), is insensitive to the numerical method and mesh resolution if computed correctly. This appendix is inspired from Deleersnijder (2016) and D.P. Marshall (2006, personal communication to E. Deleersnijder), who studied the sensitivity of the total vorticity flux to various finite difference numerical schemes, obtaining the same conclusions as for the finite element schemes detailed hereinafter.

Let assume a flow propagating alongside a wall (see Fig. 9), with a velocity u. This velocity is zero on the wall (y = 0). A boundary layer develops in the y-direction. The vorticity is ω = −∂ u/∂ y and the total vorticity flux is as follows:

$$ {\Phi} = {\int}_{0}^{\infty} u\; \omega \;dy = -\frac{u_{\infty}^{2}}{2}. $$

(8)

1.1 CG formulation

The 1D domain is discretized into linear elements. On element e, the two nodes are numbered e and e + 1 and the finite element solution is $\hat {u} = {\sum }_{i=0}^{1} U_{e+i}{\phi ^{e}_{i}}$, where ${\phi _{0}^{e}}$ and ${\phi _{1}^{e}}$ are the piecewise linear shape functions, equal to 0 everywhere except at node e and e + 1, respectively. $\hat {\phi _{0}}$ and $\hat {\phi _{1}}$ are the shape functions on the parent element. The vorticity is defined as $\hat {\omega } = -{\sum }_{i=0}^{1} U_{e+i}\partial {\phi ^{e}_{i}}/\partial y$. The vorticity flux $\hat {\Phi }$ is then as follows:

$$\begin{array}{@{}rcl@{}} \hat{\Phi} &=& -\sum\limits_{e=0}^{\infty} {\int}_{{\Omega}_{e}} \hat{u} \frac{\partial \hat{u}}{\partial y} dy \end{array} $$

(9)

$$\begin{array}{@{}rcl@{}} &=& -\sum\limits_{e=0}^{\infty} {\int}_{{\Omega}_{e}} \sum\limits_{i=0}^{1}\sum\limits_{j=0}^{1} U_{e+i} {\phi^{e}_{i}} \; U_{e+j} \frac{\partial {\phi^{e}_{j}}}{\partial y} dy \end{array} $$

(10)

$$\begin{array}{@{}rcl@{}} &=& -\sum\limits_{e,i,j} U_{e+i} U_{e+j} {\int}_{\hat{\Omega}_{e}} \hat{\phi}_{i} \frac{\partial \hat{\phi}_{j}}{\partial \xi} d\xi \end{array} $$

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$$\begin{array}{@{}rcl@{}} &=& -\sum\limits_{e=0}^{\infty} \left( \frac{U_{e+1}^{2}}{2} - \frac{{U_{e}^{2}}}{2} \right) \end{array} $$

(12)

$$\begin{array}{@{}rcl@{}} &=& -\frac{U_{\infty}^{2}}{2}. \end{array} $$

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1.2 DG formulation

Again, the domain is discretized into linear elements. On element e, the two nodes are numbered 2e and 2e + 1. In this time, the vorticity cannot be defined as for the CG formulation, since the nodes are discontinuous at the element interfaces. This discontinuity has to be accounted for properly. On element e, vorticity W _2e and W _2e+1 are computed resolving:

$$\begin{array}{@{}rcl@{}} \sum\limits_{j=0}^{1} W_{2e+j}\! {\int}_{\omega_{e}}\! {\phi^{e}_{j}} {\phi^{e}_{i}} dy\! &=&\! \sum\limits_{j=0}^{1} U_{2e+j}\! {\int}_{\omega_{e}}\! {\phi^{e}_{j}} \frac{\partial {\phi^{e}_{i}}}{\partial y} dy - \large[ u^{*} \phi_{i} \large],\\ i &=& 0,1. \end{array} $$

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The only way to obtain the correct vorticity flux is by defining the following u ^∗ at the interfaces as the mean of the two nodal values:

$$ u^{*} = \frac{U_{\text{top}}+U_{\text{bot}}}{2}, $$

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where U _top and U _bot are the nodal values above and below the interface, respectively. Now the vorticity flux can be computed easily:

$$\begin{array}{@{}rcl@{}} \hat{\Phi} &=&\sum\limits_{e,i,j} U_{2e+i}W_{2e+j}{\int}_{{\Omega}_{e}} {\phi^{e}_{i}}{\phi^{e}_{j}} dy \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} &=&\sum\limits_{e=0}^{\infty} \frac{1}{2} \left[ -\left( U_{2e+1}U_{2e+2} \right) + \left( U_{2e-1}U_{2e} \right) \right] \end{array} $$

(17)

$$\begin{array}{@{}rcl@{}} &= &-\frac{U_{\infty}^{2}}{2}. \end{array} $$

(18)

Provided that the velocity at the element interfaces is computed correctly and that the velocity outside the boundary layer is correct, the DG finite element method computes accurately the vorticity flux, even if the local velocity in the boundary layer is under-resolved or even wrong.

Appendix B: Extra figures

Fig. 10

Effect of a 2% wind shift on the velocity map of PIV data. While top map shows the original velocities computed with the PIV method (same as in Fig. 7), the bottom map removes a 2% contribution of the wind velocity to the PIV velocity

Fig. 11

Small eddies in the free shear layer downstream of Renou Island, located on the east side of gap A (Fig. 3). Inset shows 0.5-m-resolution panchromatic imagery within a zoomed area (red box). The pan data reveal clockwise swirls of algae, consistent with shear-induced eddies of about 10-m diameter