The effect of bed roughness uncertainty on tidal stream power estimates for the Pentland Firth

Uncertainty affects estimates of the power potential of tidal currents, resulting in large ranges in values reported for sites such as the Pentland Firth, UK. Kreitmair et al. (2019, R. Soc. open sci. 6, 180941. (doi:10.1098/rsos.191127)) have examined the effect of uncertainty in bottom friction on tidal power estimates by considering idealized theoretical models. The present paper considers the role of bottom friction uncertainty in a realistic numerical model of the Pentland Firth spanned by different fence configurations. We find that uncertainty in removable power estimates resulting from bed roughness uncertainty depends on the case considered, with relative uncertainty between 2% (for a fully spanned channel with small values of mean roughness and input uncertainty) and 44% (for an asymmetrically confined channel with high values of bed roughness and input uncertainty). Relative uncertainty in power estimates is generally smaller than (input) relative uncertainty in bottom friction by a factor of between 0.2 and 0.7, except for low turbine deployments and very high mean values of friction. This paper makes a start at quantifying uncertainty in tidal stream power estimates, and motivates further work for proper characterization of the resource, accounting for uncertainty inherent in resource modelling.


Comments to the Author(s)
This is a well written paper about tidal power in the Pentland Firth following a previous paper using idealised models. This paper uses more realistic depth-averaged shallow water tidal model for the region. I do have some comments. The paper refers to power potential but it is not clear from the start that is about 'removable power' rather than 'extractable power' clarified later. Initially I thought it was about power extracted by turbines. I realise this cannot be simply undertaken but some estimate could be given, e.g. 50% based on some idealised cases. If it was 10% for example the exercise would be rather pointless. In the Conclusions powers of GW level are discussed but this is for removable power.
Going through the paper, in the Introduction it is stated that 'there is uncertainty associated with the correct value of bed roughness coefficient that should be applied.' It should be clear that this is associated with depth-averaged models used here. 3-D models with hydrostatic pressure are more accurate and indeed show that 2-D depth-averaged models with strong curvature, recirculating flows in the extreme, can be rather misleading (Stansby et 2016). Soon after values of Cd for a rough surface are referred to but I think these are for steady flow. In oscillatory they can be different. This should be clarified.
In section 2a it is said that 'little change to the natural currents at the boundary occurs'. Little should be quantified in some way, e.g. 1% or 10% in change in tidal level or volume flux. 'little' means different things to different people! I would like to see fig 1c enlarged so the fences can be more easily seen. Why not include in fig  1b? In eq 2.4 is ubar with or without the fence present?
The actual analysis of uncertainty is quite comprehensive and dense.
With these relatively minor points addressed I think the paper will be suitable for publication. Stansby, P. Chini,N. and Lloyd,P. 2016 Oscillatory flows around a headland by 3D modelling with hydrostatic pressure and implicit bed shear stress comparing with experiment and depthaveraged modelling, Coastal Engineering 116, 1-14

Review form: Reviewer 3
Is the manuscript scientifically sound in its present form? Yes

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes Do you have any ethical concerns with this paper? No

Recommendation?
Accept with minor revision (please list in comments) Comments to the Author(s) Tidal stream power estimates often rely on the results of hydrodynamic numerical models that are calibrated using bed friction coefficients. This way of doing is subjected to high uncertainties. Interestingly, this paper evaluates the effect of these uncertainties on the resource assessment by considering different scenarios of turbine deployments. The paper, written following the IMRAD structure, is well written and contains high quality figures. The methodology is clearly described and the conclusions are well supported by the results. I therefore strongly recommend its publication.
My only (minor) comment concerns the omission of modelling studies with non-uniform roughness coefficients (in the introduction, you suggest that, in existing studies, Cd is always uniform). As using a uniform Cd is not always applicable, especially in tidal sites where the seabed is highly heterogeneous, I wonder how is the seabed in the Pentland Firth. Showing a sedimentary map could help justifying the assumption Cd = constant.

17-Oct-2019
Dear Dr Kreitmair On behalf of the Editors, I am pleased to inform you that your Manuscript RSOS-191127 entitled "The effect of bed roughness uncertainty on tidal stream power estimates for the Pentland Firth" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referee suggestions. Please find the referees' comments at the end of this email.
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Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. Comments to the Author(s) Manuscript ID RSOS-191127 entitled "The effect of bed roughness uncertainty on tidal stream power estimates for the Pentland Firth" for Royal Society Open Science.
The manuscript describes a study examining the impact of varying bed roughness in a real-world tidal energy setting. It uses 2D shallow water model solutions to examine the impact on energy extraction through variation in bottom friction.
It is a useful result if very incremental, so in that sense -fits the purpose of the journal.
It is clearly written.
A number of points really should be followed up upon to consolidate the worth of the manuscript.
The results appear to suggest minimal impact of varying Cd -but the same value of Cd can have a difference effect in different model configurations (scale etc). How does this sensitivity compare?
I really struggle with 2D perspectives on anything other than the very broad scale. The energy extraction will be highly heterogeneous in the vertical. How important is this likely to be?
By the same token, does the Cd variation fall within the impact of moving to 3D simulation?
Further, what's the meaning of 3% variation in power production in the context of the model. Is it thought that the present model is less than 3% from "truth"?
Is there some evidence that the present result is extendable to other systems and why?
How does the conclusion compare with the varying predictions mentioned in the abstract?
Reviewer: 2 Comments to the Author(s) This is a well written paper about tidal power in the Pentland Firth following a previous paper using idealised models. This paper uses more realistic depth-averaged shallow water tidal model for the region. I do have some comments. The paper refers to power potential but it is not clear from the start that is about 'removable power' rather than 'extractable power' clarified later. Initially I thought it was about power extracted by turbines. I realise this cannot be simply undertaken but some estimate could be given, e.g. 50% based on some idealised cases. If it was 10% for example the exercise would be rather pointless. In the Conclusions powers of GW level are discussed but this is for removable power.
Going through the paper, in the Introduction it is stated that 'there is uncertainty associated with the correct value of bed roughness coefficient that should be applied.' It should be clear that this is associated with depth-averaged models used here. 3-D models with hydrostatic pressure are more accurate and indeed show that 2-D depth-averaged models with strong curvature, recirculating flows in the extreme, can be rather misleading (Stansby et 2016). Soon after values of Cd for a rough surface are referred to but I think these are for steady flow. In oscillatory they can be different. This should be clarified.
In section 2a it is said that 'little change to the natural currents at the boundary occurs'. Little should be quantified in some way, e.g. 1% or 10% in change in tidal level or volume flux. In eq 2.4 is ubar with or without the fence present?
The actual analysis of uncertainty is quite comprehensive and dense.
With these relatively minor points addressed I think the paper will be suitable for publication. Stansby, P. Chini,N. and Lloyd,P. 2016 Oscillatory flows around a headland by 3D modelling with hydrostatic pressure and implicit bed shear stress comparing with experiment and depthaveraged modelling, Coastal Engineering 116, 1-14 Reviewer: 3 Comments to the Author(s) Tidal stream power estimates often rely on the results of hydrodynamic numerical models that are calibrated using bed friction coefficients. This way of doing is subjected to high uncertainties. Interestingly, this paper evaluates the effect of these uncertainties on the resource assessment by considering different scenarios of turbine deployments. The paper, written following the IMRAD structure, is well written and contains high quality figures. The methodology is clearly described and the conclusions are well supported by the results. I therefore strongly recommend its publication.
My only (minor) comment concerns the omission of modelling studies with non-uniform roughness coefficients (in the introduction, you suggest that, in existing studies, Cd is always uniform). As using a uniform Cd is not always applicable, especially in tidal sites where the seabed is highly heterogeneous, I wonder how is the seabed in the Pentland Firth. Showing a sedimentary map could help justifying the assumption Cd = constant.

26-Nov-2019
Dear Dr Kreitmair, It is a pleasure to accept your manuscript entitled "The effect of bed roughness uncertainty on tidal stream power estimates for the Pentland Firth" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
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Thank you for your fine contribution. On behalf of the Editors of Royal Society Open Science, we look forward to your continued contributions to the Journal. My only very minor query, is regarding the location of Figure 7. Whilst I believe that it is first referenced in the "A Power Surfaces" section, it doesn't look right to see it placed mid-way through the References. Please consider amending this within your finalised paper during the proofing process. Thank you very much for your kind email concerning our manuscript titled "The effect of bed roughness uncertainty on tidal stream power estimates for the Pentland Firth" (RSOS-191127) which was submitted to Royal Society Open Science for possible publication. We have modified the paper taking full consideration of all the comments and suggestions from the Reviewers. Corresponding revisions are indicated by red font in the new version of our manuscript. We also provide a detailed list of the revisions, along with itemized responses to each comment and suggestion, which we believe have led to a significant improvement in the quality of the manuscript. Two copies of the pdf are submitted: one where changes to the manuscript are indicated in red, and the other the final version. We confirm that format, style and referencing style of the finalized manuscript should fit the requirements of the Journal.
We would like to thank you again for your detailed suggestions. Sincere thanks are also due to the anonymous reviewers for their very helpful comments.
With best regards.

Reviewer #1 Comments
The manuscript describes a study examining the impact of varying bed roughness in a real-world tidal energy setting. It uses 2D shallow water model solutions to examine the impact on energy extraction through variation in bottom friction. It is a useful result if very incremental, so in that sensefits the purpose of the journal. It is clearly written.
Response: We thank the reviewer for his/her insightful comments regarding the manuscript. We have attempted to address each of the comments, as per the detailed responses below.
A number of points really should be followed up upon to consolidate the worth of the manuscript. The results appear to suggest minimal impact of varying Cdbut the same value of Cd can have a difference effect in different model configurations (scale etc). How does this sensitivity compare?
Response: In this paper, we have solely considered parameter uncertainty and what effect it has within a particular model configuration (i.e. model scale, mesh refinement, etc.). Changes to the model would affect both the value of the calibrated bed roughness coefficient and the magnitude of the associated parameter uncertainty. For example, a finer mesh could resolve physical features which are treated by the roughness coefficient in a coarser mesh. There exists an interaction between the model accuracy and the parameter uncertainty. This subtle point has potentially interesting consequences. We have addressed this, as well as the issues raised in the next comment, in a few sentences in the conclusions dealing with limitations.
Neglecting the interplay between model and parameter uncertainty, the impact of changing the model configuration on the value of bed roughness coefficient used is likely to be comparable to the standard deviations in Cd used in the paper. I really struggle with 2D perspectives on anything other than the very broad scale. The energy extraction will be highly heterogeneous in the vertical. How important is this likely to be? By the same token, does the Cd variation fall within the impact of moving to 3D simulation?
Response: Three-dimensionality is important of course, particularly at the location of the turbines or other obstacles where the wake and recirculation zones can cause significant increase in the value of the bed roughness coefficient (perhaps by an order of magnitude over the background level, see e.g. Stansby, Chini, and Lloyd, 2016). The consequent vertical mixing due to secondary flows cannot be captured by depth-averaged models (Stansby, 2006). However, at the scale of the present study, vertical heterogeneity has an almost negligible effect, and so 2D models are sufficiently capable of simulating bulk flow through the head-driven channels of the Pentland Firth and providing a sensible estimate of the associated energy removal (Adcock, Draper, and Nishino, 2015). As the scale of the modelled domain decreases, the uncertainty in results stems increasingly from model discrepancy (i.e. the consequence of using an over-simplified model that cannot account for increasingly important three-dimensional features) rather than from parameter uncertainty (uncertainty associated with the correct value of a calibration parameter). We have added text to this effect in the Introduction and Conclusions of the revised manuscript. Further, what's the meaning of 3% variation in power production in the context of the model. Is it thought that the present model is less than 3% from "truth"? Response: No, the variation referred to in the paper is the change in expected power as we move from considering a deterministic bed roughness coefficient to one with a probability distribution (i.e. comparing the mean of the power values for a distribution of Cd values to the power value calculated at the mean Cd) and is a measure as to how much uncertainty affects the model. This is stated in the first paragraph of section 3a.
Is there some evidence that the present result is extendable to other systems and why?
Response: Yes, there is analogous work in hydrological engineering science using a method called the derived distribution approach (Ang and Tang, 1975) that uses an analytic method to derive the probability distribution of a dependent variable from uncertainty in the independent variables. Example applications include flood frequency analysis by Eagleson (1972), Hebson and Wood (1982), Díaz-Granados et al. (1984), urban stormwater runoff by Chan and Bras (1979), subgrid hydrological processes in