*3.1. Kelp-Type Module Validation*

The model predicted velocity and turbulent stress profiles were compared with laboratory data digitized from Plew [15]. Figure 3a shows the model predicted vertical velocity profiles for Validation Run A which has the highest cylinder density among all the runs. In general, the velocity profiles with cylinders showed a significant change from that of the baseline condition with no cylinders present. Flows were reduced within the canopy while increased beneath the canopy. As expected, the change to flow structure is more significant for flows in-line with the cylinder arrays than between arrays. Figure 3b shows the corresponding model-data comparison of the normalized (by mean free-stream) velocity profiles. The model predicted mean velocity profile agrees with the laboratory measurements reasonably well. A drag coefficient (*Cd*) of 1.75 was found to provide good model-data comparison in this validation run.

The model-data comparisons for all four validation runs are presented in Figure 4, and the corresponding error statistics are listed in Table 2. The model results have been normalized/non-dimensionalized so that they can be directly compared with laboratory data presented in Plew [15]. While the velocity profiles can be directly extracted from the model output, the turbulent stress was calculated using (9):

$$
\mu' w' = -K\_m \frac{\partial u}{\partial z} \tag{9}
$$

where *uƍwƍ* = turbulent stress (m2 /s2 ), and *Km* = vertical eddy viscosity (m2 /s). **Figure 3.** (**a**) FVCOM predicted vertical velocity profiles for Validation Run A. (**b**) Normalized velocity profiles for validation Run A compared to the experimental measurements.

**Table 2.** Error statistics for kelp-type structure module validation runs (*R*<sup>2</sup> denotes the coefficient of determination and RE stands for relative error defined by ܴܧ ൌ ͳͲͲΨ ൈ <sup>σ</sup> ൫ఎ ିఎ ൯ ಿ <sup>మ</sup> సభ σ ቂ൫ఎ ିఎതതതത൯ మ ା൫ఎ ିఎതതതത൯ మ ቃ ಿ సభ , where *Ș<sup>m</sup>* and *Ș°* stand for model predictions and laboratory observations, respectively, ߟതതത is the mean of observations).


There is an overall good agreement between model predictions and direct measurements for all the validation runs (Figure 4). The good model-data comparison is further confirmed by error statistics which show a high coefficient of determination (*R*<sup>2</sup> ) and low relative error (RE) between predicted and measured velocity and turbulent stress values (Table 2). In general, the model captures the vertical structure for both velocity and turbulent stress. For instance, the velocities were significantly attenuated for flows through the cylinders and the maximum turbulence was generated near the middle depth of the water column at the interface between the bottom of the cylinder canopy and the flow immediately below. In addition, as evident from model predictions and laboratory data, higher cylinder density (Parameter a in Table 1) tends to exert a stronger impact on flows. The maximum differences between the model and data occurred in the middle

depths of the water column. A better parameterization may be needed at canopy-water interfaces to account for this difference. For example, additional skin friction contributed by the bottom of the cylinder may be considered. This discrepancy could also be contributed by the inadequate characterization of the canopy's effects on turbulence in the current module, as suggested by other studies [16,17].

**Figure 4.** (**a**–**d**) Model-data comparisons of normalized mean vertical velocity profiles for Validation Runs A, B, C, and D, respectively. (**e**–**h**) Model-data comparisons of normalized mean turbulent stress profiles for Validation Runs A, B, C, and D, respectively.

The model results also indicated that the drag coefficient (*Cd*) is a function of canopy density and increases with higher density. For example, the calibrated *Cd* values for Runs A, B, C and D are 1.75, 1.5, 1.25, and 1.0, respectively. This is also consistent with the findings in other similar studies [9,15,18]. Wu and Wang [18] and Struve *et al.* [9] reported that Cd values greater than 3.0 were needed to reach a good agreement between model and data. As discussed by Wu and Wang [18], this is presumably due to the inherent dependency of *Cd* on the Reynolds number (Re). At higher canopy densities, reduced flow velocity and Reynolds number in the model domain lead to larger drag coefficients according to the relationship between *Cd* and Re for a single cylinder [19]. The validation tests also suggest this. Although the actual *Cd* values varied with canopy density, a spatially uniform *Cd* represented the major conditions reasonably well.

#### *3.2. Module Applications—Kelp Forest and Braid Adsorbent Farm Simulation*

The validated kelp-type structure module was used to simulate the effects of a kelp forest and uranium braid adsorbent farm on coastal flows. Numerous studies have documented the effects of kelp forest on coastal currents. The current measurements conducted by Jackson [6] off the coast of California in the Point Loma kelp forest were selected as the reference for kelp forest simulation in this study. The oceanographic conditions including 7-km-long and 1-km-wide stretch of real kelp forest were setup in the FVCOM model. The kelp fronds were 0.2 m in diameter and occupied the full 15 m of water depth. The model grid used was a simplified rectangular channel which carried a mean longshore ambient current of 2 cm/s reproducing the conditions observed during the field survey. The kelp density (Kd) used for this numerical experiment was 0.08 fronds/m2 , which is the average of the observed kelp density range of 0.02 to 0.14 fronds/m2 .

The model results for the Point Loma kelp forest shown in Figure 5 suggest that the drag provided by kelp forest results in a reduction of ambient currents from 45% to 55% for typical drag coefficient values ranging from 0.2 to 0.5. This is consistent with field observations that indicated a significant reduction in longshore currents with dense kelp forest [6,20,21]. Moreover, this prediction serves as an additional qualitative model validation study.

**Figure 5.** (**a**) Map of the Point Loma kelp forest offshore of the California coast. The dotted line represents a general outline of the kelp canopy (map adapted from Tegner *et al.* [22] with permission from Elsevier). (**b**) Predicted potential reductions in ambient currents by the kelp forest and braid adsorbent farm with typical canopy/mooring densities.

The kelp forest in the above model configuration was substituted with braid adsorbent moorings to estimate if a braid adsorbent farm for seawater uranium extraction would have a similar effect. The physical dimensions of the moorings were set identical to the kelp fronds (0.2 m diameter and 15 m long occupying the full water column). The proposed braid adsorbent farm design calls for moorings on 8 m × 70 m centers. This corresponds to an adsorbent mooring density (*Kd*) of 0.00178 moorings/m2 , much less dense than the kelp forest (0.08 fronds/m2 ). The model results suggest that with typical drag coefficient values of 0.2 to 0.5, the reduction in ambient currents by braid adsorbent farms could range from 4% to 10%.

## **4. Conclusions**

A kelp-type structure module was incorporated into the coastal ocean model FVCOM based on the commonly used momentum sink approach in which the resistance force exerted by kelp-type structures is parameterized as additional form drag in the momentum equations. The module was reasonably validated using observations from both laboratory flume experiments and field surveys conducted in the kelp forest near Pt. Loma off the Californian coast. Model results suggest that the reduction in ambient currents could range from 4% to 10% for a farm of uranium adsorbent material having configurations for mooring density described previously by Japanese scientists [5] and employing a typical drag coefficient. This study demonstrates that a momentum sink approach based on structure module is capable of characterizing the general hydrodynamic impact of kelptype structures on coastal flows. Improvements in the current module could be made by including other processes such as the effect of canopy on turbulence [16,17,23] and calibrated drag coefficients by braid adsorbents.

#### **Acknowledgments**

This work was funded by the Office of Nuclear Energy, U.S. Department of Energy. The authors also thank David Plew for providing additional information on the flume experiment configurations.

#### **Conflicts of Interest**

The authors declare no conflict of interest.

#### **References**

