The authors presented an interesting study on a reduced-order model (ROM) for the simulation of wind-induced currents in the Gorgan Bay in Iran. The technique they used is based on data obtained by primitive-equation simulations using the three-dimensional flow equations, which are subsequently reduced by proper orthogonal decomposition. The method is valuable as it helps manage large sets of hydrodynamic simulation data. It is clear that the quality of the reduced data depends on the quality of the original data; in this case, the results of the simulation of the wind-induced currents. Given the semi-empirical nature of calibration and the difficulties in the validation of simulations through the comparison with field measurements, it is worthwhile to interpret the salient features of these simulations in terms of known theories. Herein, we offer a comment on a feature present in the simulations of the wind-induced currents.

The data from the primitive-equations simulations that have been exploited by the authors have been presented in greater detail by Ranjbar & Hadjizadeh Zaker (2018). In both papers, the strong, nearshore currents are noted, but in the latter, the salient features of the wind-induced circulation in the Gorgan Bay are more specifically interpreted within the framework of Csanady's (1973) theory, which provides an explanation of the development of strong currents in waters shallower than the mean depth in the direction of the wind and weak leeward currents in the deeper-than-the-mean-depth waters. The application of Csanady's theory, originally developed for an elongated lake, is appropriate for an almost enclosed elongated bay with one opening, since the essential requirement of that theory is that the flowrate, passing from a cross-section perpendicular to the wind's direction, be zero, while the wind set-up is steady (ignoring the occasional seiches). The same essential theory holds even for circulation in a flow-through gulf (i.e. with two openings), but with the appropriate modifications, as has been shown by Horsch & Fourniotis (2018). In this same paper, the effect of the Coriolis force on the wind-induced flow in an elongated lake is examined. Using an asymptotic method, which is strictly valid during the initial acceleration phase, it was shown that Csanady's solution can be considered to be the first order of a series, which if expanded to second order, shows that under the influence of the Coriolis force the streamwise velocity profile, given a typical cross-flow bathymetry, causes the cross-flow shape of the free surface to be S-shaped, as depicted in Figure 1:

Figure 1

The cross-flow shape of the free surface for parabolic-shaped cross-flow bathymetry (from Horsch & Fourniotis 2018).

Figure 1

The cross-flow shape of the free surface for parabolic-shaped cross-flow bathymetry (from Horsch & Fourniotis 2018).

This specific shape, in dimensionless variables, is in fact the exact solution for a parabolic-shaped cross-flow bathymetry, but the exact solution for any bathymetry can be found in Horsch & Fourniotis (2018). Even though, as noted, the solution provided in Figure 1 is strictly valid during a time interval within the initial acceleration phase, Csanady (1973) provided an argument explaining why the structure of the flow remains unaltered even when the steady state is approached, and this is also verified by numerical simulations. In order to get the free-surface field, we should add the second-order effects (i.e. the shape shown in Figure 1) to the first order, linear set-up predicted by Csanady's theory. Omitting the details of the analysis, which can be found in the discussers' 2018 paper, the qualitative features of the free-surface contours corresponding to the profile of Figure 1 (again for an idealized parabolic cross-flow bathymetry) are depicted in Figure 2:

Figure 2

The S-shaped free surface contours which form under the influence of the Coriolis force.

Figure 2

The S-shaped free surface contours which form under the influence of the Coriolis force.

Note that, even though the computational domain in Figure 2 looks like a flow-through gulf, for the simulations shown here, the entrance and the exit have been numerically closed so that the domain functions as a lake. The S-shaped contours in Figure 2 present the same S-like shape as those of Figure 6 in Ranjbar & Hadjizadeh Zaker (2018), and it is therefore reasonable to interpret the latter as resulting from the effect of the Coriolis force as well. It would have been interesting to further examine if the shape of the cross-flow flux is similar to that predicted by the asymptotic theory depicted in Figure 3 (in dimensionless variables).

Figure 3

The cross-flow shape of the cross-flow flux for parabolic-shaped cross-flow bathymetry (from Horsch & Fourniotis 2018).

Figure 3

The cross-flow shape of the cross-flow flux for parabolic-shaped cross-flow bathymetry (from Horsch & Fourniotis 2018).

It should be added, however, that this cross-flow is very weak and as a result its shape can be significantly distorted in a realistic bathymetry, as has been the case with the Gulf of Patras (Horsch & Fourniotis 2018).

We conclude that the primitive-equation-solutions used by the authors present the expected salient features of wind-induced flow in an elongated, semi-enclosed bay, and this provides additional positive indications of their good quality. Given this fact, and since these features are expected to be persistent during extended periods of time, we suggest that studying whether these same features are present in ROM results, such as those presented in figure 8 by the authors, could supplement any criterion used to appraise the adequacy of these ROM results.

REFERENCES

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Hadjizadeh Zaker
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