Non-deterministic effects in modelling the tidal currents in a high-energy coastal site

Numerical models are commonly employed for predicting tidal stream resource, since harmonic analysis is typically insufficient for representing tidal currents. This is particularly true in regions of high-energy flow and complex bathymetric and coastline features, resulting in highly nonlinear dynamics. Within this work, we demonstrate that nondeterministic effects pose a further barrier to resource predictability in such regions. We take the Inner Sound of the Pentland Firth as a case study region, situated between mainland Scotland and the island of Stroma, where nondeterministic effects are introduced by eddy shedding off islands and headlands in the region. We set up a numerical model of the region using the Thetis finite element coastal ocean model. We first demonstrate the presence of nondeterministic effects, by forcing the model with a single tidal constituent and observing the departure from purely periodic behaviour within the Inner Sound, and also by comparing an ensemble of fully-forced model runs generated with perturbed initial conditions. We then take steps towards model calibration with respect to the model bottom friction and viscosity parameters, comparing a variety of choices of model-observation error metrics for this purpose. The non-deterministic component not only places a limit on the predictability of the flow (and hence the tidal stream resource), but also on the calibration process. Focusing on calibration with respect to the bottom friction parameter, we quantify the identifiability limit which the non-deterministic behaviour places on the parameter. We explore the sensitivity of this identifiability limit with respect to the viscosity parameter and the choice of error metric. We find that calibration based on matching the modelled and observed mean kinetic power density is a good choice, and is relatively robust with respect to the non-deterministic behaviour. Based on these results, and on the sensitivity of the error metrics to each model parameter and the mesh resolution, we make recommendations for how to proceed with model calibration in the presence of non-deterministic flow. Since the non-deterministic effect is due to complex, high-energy flow leading to highly nonlinear dynamics, we expect these results to generalise to other regions of high tidal stream resource.


Introduction
Pentland Firth (Rahman and Venugopal, 2015), with 60% of the energy entering the Pentland Firth estimated to be lost through bottom friction (Easton et al., 2012). 40 In the context of the Pentland Firth, there exists a variety of sources of data which can be used for model calibration inference approaches (Sraj et al., 2013(Sraj et al., , 2014b and evolutionary algorithms (Huybrechts et al., 2021)), which to date 50 have not been employed within models of the Pentland Firth. However, the complexity of the dynamics within the 51 Pentland Firth poses an additional challenge to the implementation of such methods; the above calibration strategies 52 have not previously been applied in such a high-energy and complex domain. 53 As a step towards a more sophisticated model calibration, in this study we set up a numerical model of the Pentland 54 Firth within the finite element coastal ocean model Thetis, with the aim of quantifying the predictability of the tidal 55 stream resource within the Inner Sound. We find that eddies shed off islands and headlands in the region induce a 56 departure from purely harmonic behaviour, and furthermore that there is a non-deterministic component to the flow.

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This non-deterministic behaviour introduces an additional barrier to resource predictability and model calibration. 58 The aim of this paper is therefore to demonstrate the existence of this non-deterministic component, and investigate 59 its implications for resource predictability and model calibration. 60 The numerical model and data sources used within this work are described in section 2. In section 3 we present where η is the surface elevation, H = η + h the total water depth, h the bathymetry, u the depth-averaged velocity, 82 F C the Coriolis force, g the acceleration due to gravity, τ b the bottom stress due to friction, ρ the density of seawater, 83 and ν the eddy viscosity. The bottom stress is parameterised here using Manning's n formulation, given by where n is the (spatially varying) Manning coefficient. We introduce wetting and drying via the bathymetry modifi- Database (GSHHG) (Wessel and Smith, 1996). The primary mesh used within this study is shown in figure 1a, which  The governing equations (1) are solved using a P DG 1 -P DG

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The bathymetry used within this work combines data from several sources, which are indicated in figure 2  at the location indicated in figure 3. The ADCP was deployed for 33 days, and we make comparisons with data which 131 have been time-averaged to 10 min and depth-averaged. We apply the same time-averaging to the model outputs.

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Since we are motivated by tidal stream energy, within this work we make comparisons with the ADCP data only where u and v are the x-and y-components of the velocity, respectively. The use of only a single (scalar) velocity a proxy for this variability, we compute the standard deviation of the peak-to-peak current amplitudes within each 152 tidal cycle. We compute this standard deviation over a simulated period of 36 days (or 69 M2 tidal cycles), after an 153 initial spin-up period of five days. Figure 5 shows how this standard deviation depends on n and ν. We emphasise 154 that for a fully deterministic flow, this standard deviation should be equal to zero, but that the presence of a non-155 deterministic component means that the flow field is inherently unpredictable, and therefore varies between each tidal 156 period. The non-deterministic behaviour can be suppressed by increasing the viscosity ν, or the friction coefficient n.

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Within the parameter ranges considered here, the viscosity has a stronger influence than the friction coefficient on the between the vorticity fields from two ensemble members generated with n = 0.03 s m −1/3 and ν = 1 m 2 s −1 . We find 173 that the differences between the two simulations are substantial, and easily visible by eye. Figure 8 shows the results 174 from an equivalent experiment, with ν = 40 m 2 s −1 . In this higher-viscosity case, the differences between the two 175 ensemble members are too small to see by eye. This is consistent with the results of figures 5 and 6, where we find that 176 a viscosity ν = 40 m 2 s −1 strongly suppresses the non-deterministic variability. However, comparing figures 7 and 8, 177 we see that much of the finer-scale flow structure is removed by the use of the higher viscosity value.

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This loss of finer-scale structure can also be visualised by comparing the power spectral densities (PSD) for the 179 modelled velocity at the ADCP location, for ν = 1 and ν = 40 m 2 s −1 . This is shown in figure 9 for the x-component 180 of the velocity, where we also plot the PSD of the ADCP data. The use of higher viscosity suppresses the PSD at 181 all frequencies, but has a stronger effect on the noise between the peaks than on the peaks themselves. Furthermore, 182 above a frequency of approximately five times the M2 harmonic frequency, the use of higher viscosity reveals peaks 183 which could not be distinguished above the background noise for ν = 1, but which are present in the observation data.     the modelled and observed amplitude and phase of these constituents is given by where A C andÂ C are the modelled and observed amplitudes of the constituent C, respectively, and φ C andφ C are  (iv) Perturbations due to weather events e.g. storm surge, which we assume are small.

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The variance of the non-harmonic component is, by definition, relatively more sensitive to the non-deterministic com- variance. This is in contrast to the explained variance of the full flow, which shows less sensitivity to ν. This is consis-247 tent with the results of section 3, where we found that the influence of the viscosity is stronger on the non-deterministic 248 component than on the underlying deterministic flow.

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Finally, we can also consider the ratio of the modelled to observed mean kinetic power density, over the ADCP 250 deployment period. This is shown in figure 14. The optimal value of this metric is 1, which can be achieved within  to the optimal parameter resolution achievable. That is, for sufficiently small perturbations to n and ν, the influence 280 of the perturbation will be indistinguishable above the 'noise' introduced by the non-deterministic effect. As a result, 281 the error metrics will not vary smoothly with the input parameters, preventing the precise identification of optimal 282 parameters and potentially limiting the choice of possible calibration methods. In this section, we investigate this 283 lower limit on achievable parameter resolution, focusing on the friction parameter, n. is no longer linearly proportional to the input perturbation, for the reasons described above. Specifically, we find that 296 for a model configured with ν = 1 m 2 s −1 , it is not possible to distinguish between Manning coefficients which differ 297 by much less than ∆n = 10 −3 s m −1/3 using this error metric. For ν = 40 m 2 s −1 , this threshold decreases to less than ∆n = 10 −7 s m −1/3 .

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However, as well as depending on ν, these parameter identifiability thresholds also depend on the selected error 300 metric. Figure 16 compares the thresholds for a variety of error metric choices, in combination with different viscosity 301 values. The mean KPD is the least affected by the non-deterministic behaviour. This is likely to be because the 302 time-averaging allows periods of under-and over-estimation to cancel out over time. The error metric based on the 303 velocity harmonic analysis (given by equation (5)) also performs well at low viscosity, because the harmonic analysis 304 acts to filter out the high-frequency variability from the signal. The R 2 and RMSE measures based on the KPD 305 perform the worst. However, the choice of viscosity has a far greater effect than the choice of error metric.

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The thresholds also depend on the time range over which the error metric is computed. Given a stationary   the use of coarse model resolution, which will save computational cost within a preliminary calibration phase.

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The results of this study also have implications for the selection of a suitable model calibration method. Many  Finally, we note that the findings of this study are unlikely to be unique to the Inner Sound of the Pentland Firth 403 as studied within this work. The presence of a non-deterministic flow component is likely to affect any high-energy 404 site with complex bathymetry and/or coastline geometry, and the results of this study generalise to any such site.

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With these recommendations in hand, in future work we will perform a more extensive model calibration, using 406 further sources of observation data. The use of additional data will facilitate the estimation of a spatially varying 407 friction parameter, resulting in a robustly calibrated model suitable for tidal stream resource assessment. for tidal stream resource assessment, and also introduces additional uncertainty into efforts to calibrate the model.

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Furthermore, we expect that this issue is not unique to the Inner Sound of the Pentland Firth as studied here, and 417 will be present in any other complex, high-energy site.

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In order to calibrate a numerical model with respect to its bottom friction parameter, it is necessary that small 419 perturbations in the parameter induce changes in model-observation error metrics which can be resolved above the non-420 deterministic effect. In practice therefore, the non-deterministic component places a limit on the optimal parameter 421 resolution which can be achieved by model calibration. We find that this limit depends on both the choice of error within this study, we have found that the optimal parameters do not depend strongly on the mesh resolution, and 429 therefore that this calibration at high viscosity can be performed at coarse resolution, thus saving computational cost.

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Subsequent mesh refinement and reduction in viscosity can then be used to improve model performance, and attain 431 good agreement with observations via other metrics, such as the flow variability.

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This study has used observation data from a single ADCP deployed in the Inner Sound of the Pentland Firth, and 433 considered only spatially uniform model parameters. In future work, we will utilise data from a variety of sources for 434 the purpose of calibrating the model with respect to a spatially varying bottom friction coefficient.