Interactions Between Mean Sea Level, Tide, Surge, Waves and Flooding: Mechanisms and Contributions to Sea Level Variations at the Coast

Coastal areas epitomize the notion of ‘at-risk’ territory in the context of climate change and sea level rise (SLR). Knowledge of the water level changes at the coast resulting from the mean sea level variability, tide, atmospheric surge and wave setup is critical for coastal flooding assessment. This study investigates how coastal water level can be altered by interactions between SLR, tides, storm surges, waves and flooding. The main mechanisms of interaction are identified, mainly by analyzing the shallow water equations. Based on a literature review, the orders of magnitude of these interactions are estimated in different environments. The investigated interactions exhibit a strong spatiotemporal variability. Depending on the type of environments (e.g., morphology, hydrometeorological context), they can reach several tens of centimeters (positive or negative). As a consequence, probabilistic projections of future coastal water levels and flooding should identify whether interaction processes are of leading order, and, where appropriate, projections should account for these interactions through modeling or statistical methods.

We took into account almost all the remarks of the reviewer. In the appendix are the responses to reviewer comments.
In addition, we updated the manuscript in sections 3.1.2 and 3.3.2, regarding the paper of Harker et al. (2019) on Australia. Indeed, in the previous version, we were referring to Harker et al. (2018) which corresponds to the same paper, but, at that time, was in open discussion, not yet revised, and was focusing only on the north of Australia. Here please find our responses to the reviewers comments. The reviewer comments (in between "***Begin" and "***End" character chains).

1.Reviewer #3
First, we thank the reviewer for his positive comments and recommendations. ***Begin I thought this was a useful overview of the various interactions, generally nonlinear, that impact sea levels at the coast. Except for one place, it was well organized, although there is some amount of repetition throughout the paper. I particularly liked Section 3.4 on wave effects. The review stimulated me to think of ways that certain problems can be better attacked, and perhaps this is what we should hope for in a good review.
I do have some criticisms and suggestions before I recommend final acceptance. These are arranged by importance, according to UPPER CASE LETTERS below.
MOST IMPORTANT: 1. I was surprised there is no discussion (except one brief mention on page 18) of the Bay of Bengal and the problem of storm surges on the coasts of Bangladesh and India. There has been a lot of work done on tide-surge interaction there and there are many papers that could be cited. In fact, one could argue this is the most critical area to study because of the terrible loss of life caused by storm surges in Bangladesh. I think some of these many papers should be discussed. ***End The correction has been done The following paragraph has been added in section 3.2.2 (lines 371-383): "The Bay of Bengal is another region where tide-surge interactions are known to be very relevant (Johns et al. 1985). Krien et al. (2017b) performed numerical experiments during cyclone Sidr (2007), focusing on the head of the bay. They investigated the interaction between the tide and the total storm surge (including wave setup). As the computed wave setup ranged from 0.2 to 0.3 m and varies little over a tidal cycle, the tide-surge interactions analyzed by these authors mostly corresponds to the effect of the tide on atmospheric surge and vice versa. They showed that tide-surge interactions in the range ±0.6 m develop in shallow areas of this large deltaic zone. In addition, such interactions occurred at a maximum of 1-2 hours after low tide due to the combination of a stronger wind contribution during periods of shallow depth and a faster propagating tide compared to a situation without surge. These findings corroborate those of Johns et al. (1985), Antony and Unnikrishnan (2013) and Hussain and Tajima (2017)." Figure 2 has been updated accordingly. 2. On page 17 (middle of page), an important point is made about the need for high resolution modeling and how it is not computationally feasible to obtain realistic simulations in some cases. I agree with this point, but it seems (perhaps) to invalidate some of the studies that the authors cite throughout the paper. To me, this is a point that could have been made up front. And then the authors might have mentioned cases where they think certain investigations are possibly inadequate because of computational/resolution limitations ---assuming they think this is actually so (and if not, then why make the point at the end of the paper?). ***End The correction has been done The main limitation of low resolution (let say few hundred meters) is related with the local wave setup which cannot be captured properly at this resolution. To the author knowledge, in many cases, the few hundred meter resolution is sufficient to capture tide, atmospheric surge and their interaction (see e.g. Idier et al., 2012, Muller et al., 2014, in which 2 km resolution models are used), while, depending on environments, tens meters are required for regional wave setup (Bertin et al, 2015) and few meters or local wave setup. A second limitation arise when flood occurs. The paragraph has been modified to make this point clearer, by adding the following sentence (lines 776-788): "Indeed, even if in many cases a resolution of few hundred meters is sufficient to capture tide, atmospheric surge and their interaction (see e.g. Idier et al. 2012;Muller et al. 2014, in which 2 km resolution models are used), it is not sufficient to capture regional and local wave setups, which, depending on the environment, require resolutions of tens and few meters, respectively, but also high quality bathymetry (see e.g. Bertin et al. 2015; Pedreros et al. 2018). The quality of the topography (especially on the coastal defenses) is also crucial to account for the flooding effect on coastal water level. As highlighted here and in section 4.4 (penultimate paragraph), there are downscaling challenges, but there are also upscaling issues (indeed, local processes as for instance dissipation by floods or on intertidal areas or tidal bedforms may also affect the regional dynamic). In addition, to account for the uncertainties in the future climate projections, many simulations should be done" ***Begin 3. Along these same lines, some of the modeling, such as the effect of sea level rise on tides, is difficult for readers to assess. It would be good if some assessments could be added, and in particular I suggest that the two or three models that have been run for the European Shelf could be intercompared more clearly to see if they are consistent or not. The first author has written on this topic, so it should not take great effort to make some comparisons. ***End The correction has been done The following paragraphs have been added in sections 3. Bight and the Irish Sea. The main discrepancy is the positive trend along the Danish coast given in (Pickering et al. 2012), probably due to the closed model boundary to the Baltic Sea in that study. As an additional comparison, we can also refer to the study of Palmer et al. (2018), which shows strikingly similar spatial patterns of increase and decrease to the one of Pickering et al. (2012), except for the region that spreads out from the Bristol Channel. For a more detailed comparison on the modeling studies of the SLR effect on tides over the European shelf, see Idier et al. (2017)." •"It should be noted that there is a strong consistency between these modeling studies. Idier et al. (2017) and Pelling and Green (2014) provide very similar M2 amplitude changes in terms of order of magnitude and patterns. The main discrepancy is in the Bristol Channel, which may be due to the differences in spatial resolution and quality of the topographic data used in each case." •"Finally, whilst the agreement in local to regional scale tidal changes from differing models (e.g. on the NW European Shelf) provides some confidence, it would be of value to conduct further investigations of macro-tidal regions (such as the Bristol Channel) with consideration of interim 'partial recession' or 'flood' scenarios depending on coastal management priorities and sensitivities to the resulting change." ***Begin 4. Much of Section 3.3 repeats parts of Section 3.1, which I initially found confusing. The authors wish to emphasize the difference between flooding and no-flooding in simulations. But it means, for example, the work of Holleman & Stacey gets mentioned in each section, but in different ways with different results, which is confusing, especially since in Section 3.1 a reader won't realize that Holleman & Stacey also examined flooding scenarios until their work is brought up again in Section 3.3. I'm not sure this split is justified. In fact, the authors may have been prompted to do it based simply on limitations of how some current models are implemented. However, it does follow several recent papers in the literature, so if that route is taken then maybe repetition cannot be avoided. I agree that any change would require considerable rewriting, so I won't press the issue. But I found it confusing. ***End The correction has been partially done We agree that the reviewer suggestion makes sense, as the flooding we are mainly referring to in section 3.3 is related with the sea-level rise. However, there are so many uncertainties on future coastal defenses strategies that we preferred to really distinguish section 3.1 and 3.3, regarding the aim of the paper which is to provide orders of magnitude of different interactions. Thus, we kept the present structure, but add the following sentence at the end of section 3.1.1 (lines 201-205): "Finally, as a counterbalancing effect, SLR will also result, in some locations, in new flooded areas, which will act as additional dissipative areas for tide and surge. In the present section, for sake of clarity, we do not consider the effect of these additional wet areas. The effect of flooding on tides and on still water level is discussed in sections 3.3 and 4.2, respectively". In addition, at the beginning of section 3.1.2, the following sentence is added (lines 212-214): "In addition, the orders of magnitude provided below are extracted from model results obtained assuming a fixed shoreline (i.e. impermeable walls along the present day shoreline)." ****Begin MINOR ITEMS: 5. Throughout the paper the authors use the term 'magnitude orders'. In English this is more commonly stated as 'orders of magnitude'. So I found 'magnitude order' somewhat jarring. However, please do not simply perform a global change to 'orders of magnitude' because, depending on the surrounding text, the result can be unsatisfactory (e.g., too many prepositions piled atop one another). You'll have to make a judgement call, case by case. ***End The correction has been done ***Begin 6. As suggested by the previous point, some of the writing is a little awkward, which makes me wonder whether the coauthors who are native English speakers spent enough time on the paper. ***End The correction has been done One of the two native English speakers carefully checked and improved the English writing. ***Begin I wouldn't mind pointing out many of these wording problems, but since the authors did not use line numbers in their manuscript, this is an annoying task for a reviewer. So the authors can count lines, as I had to. ***End We are sorry for the omission of line numbers, they are included in the revised version, for the next review. ***Begin -last sentence of Abstract, trough > through ***End The correction has been done ***Begin -p 2, line 7, the comma after 'storm tides' should be a semicolon ***End The correction has been done ***Begin -p 3, middle, 'sea level lied' > 'sea level lay'. 'Lie' is irregular in English (unless it means to tell a falsehood), with past tense being 'lay'. ***End The correction has been done ***Begin -2 lines later, of > by ***End The correction has been done ***Begin -p 6, line 18, " Figure 2a herein" > this should be either ' Figure 2a therein' OR 'their Figure 2a'. This same problem occurs several other times in the paper (search "herein" to find them). ***End The correction has been done ***Begin -p 7, line 8, At the South > In the southern hemisphere ***End The correction has been done ***Begin -p 9, line 8 from bottom, "The results analysis" --needs reworded. ***End The correction has been done ***Begin -p 10, line 4, anthropic is wrong word. (In English it refers to a philosophical concept.) ***End The correction has been done (replacing "anthropic" by "man-made") ***Begin -p 11, line 3 from bottom, celerity. From the French. The term is nowadays archaic in English and rarely used. (Except it IS very commonly used by French authors when writing in English, in fact so often that English readers now readily understand it! So use it if you like.) ***End The correction has been done (replacing "celerity" by "velocity") ***Begin -p 14, line 4, (e.g. for instance) --use one or the other, but not both. ***End The correction has been done ***Begin 7. p 5, 12 lines up from bottom, contributes to an increase in tidal range --but this depends on whether nor not flooding is allowed. I realize that topic is deferred to Section 3.3 later, but it would be useful to acknowledge the point here. ***End The correction has been done We add the following sentence at the end of the third paragraph of section 3.1.1(lines 201-205): "Finally, as a counterbalancing effect, SLR will also result, in some locations, in new flooded areas, which will act as additional dissipative areas for tide and surge. In the present section, for sake of clarity, we do not consider the effect of these additional wet areas. The effect of flooding on tides and on still water level is discussed in sections 3.3 and 4.2, respectively". ***Begin 8. p 6, line 15, this change in MHW is apparently IN ADDITION to the 2 meter rise? Similarly, on p 10, middle, the change of -0.69 in Rotterdam presumably does not include the sea level rise? ***End The correction has been done The reviewer is right. The MHW changes and the -0.69 does not include the sea-level rise. To avoid misunderstanding, the following sentence has been added at the beginning of section 3.1.2 (lines 209-212), and reminded at the beginning of section 3.3.2: "The values given in the next paragraphs correspond to changes in the tidal component alone (i.e. relative to the mean sea level) and not to the absolute tide level (which includes the mean sea level and thus SLR itself)." ***Begin 9. p 11, I found the top dozen lines or so very difficult to follow. Could this be made clearer? ***End The correction has been done We agree with the reviewer that these lines were unclear. We completely rewrite these lines as follows, reformulating the first sentence (concerning the Holleman & Stacey 2014 study) and focusing more on the effect of flooding on tide alone, still relying on the work of Wang et al. (2017) (see lines 463-480) "In San Francisco Bay, Holleman and Stacey (2014) and Wang et al. (2017) also investigated the effect of coastal defense scenarios. We discussed in section 3.1 that a SLR of 1 m induces an increase of 6 and 5 cm in the southern and northern bay in the case of hardened shoreline scenario based on the study of Holleman and Stacey (2014). These authors made the same experiment with present coastal defenses and topography, finding a decrease of the high tide level (relative to the mean sea level) ranging from a few centimeters to 13 cm, i.e. an opposite change compared to results obtained with the hardened shoreline scenarios. Wang et al. (2017) also investigated the effect of coastal defenses on tides, considering two scenarios: existing topography and full-bay containment that follows the existing land boundary with an impermeable wall. Comparing the model results obtained for both scenarios, they found that the semidiurnal mode exhibits local changes at the shoreline up to 2 mm, changes in the diurnal mode extend into the bay (reaching values of about 1 mm), and overtide changes exhibit a significant spatial variability (with changes exceeding locally 2 mm). But the most important impact of the full-bay containment appears to be in the longterm process, with the changes in the long-term tidal mode being almost uniform in space, and exceeding 5 mm. The correction has been done It was changes in 100-year surge levels. The sentence has been extended as follows to include a bit of physical explanations (lines 730-735): "They show that a 1 m SLR leads to changes in the 100-year surge levels ranging between -0.3 and +0.5 m in areas of coral reef and shelf. Most of the domain (especially between the eastern shoreline and coral reefs) is subject to a decrease (larger water depths induce a decrease in wave setup and wind induced storm surge), while increase is observed in very local low-lying regions where the inundation extent is strongly enhanced by the sea level rise." ***Begin 10. p. 17, lines3-5, I did not understand this, especially: ±1 decade of SLR ***End The correction has been done The text in question has been edited as follows (lines 746-757): "The obvious implication of the nonlinear relationship between SLR and tidal amplitude is an enhancement or reduction of future water level at high tide. From a planning perspective, however, it is also important to consider how the nonlinear response of tidal amplitude to SLR affects the timing of impacts. Rates of SLR will be on the order of 10 cm per decade under the plausible scenario that GMSL rises by 1 m during the 21st century. The nonlinear response of tidal amplitude to 1 m of SLR is also on the order of ±10 cm in some locations (Pickering et al. 2017). Thus, the effect of tidal amplification on the MHW datum can be roughly equivalent to the effect of ±1 decade of SLR. Depending on the sign and magnitude of the tidal amplification at a given location, optimal planning horizons may need to be adjusted earlier or later to account for the nonlinear response of tidal amplitude and its effect on the frequency of high-tide flooding."     The present review focuses on water level resulting from mean sea level, tide 51 and surge (atmospheric surge and wave setup) and investigates how this water level 52 can be altered by interaction processes occurring between SLR, tides, storm surges, 53 waves and flooding. Indeed, many mechanisms can affect the still water level (e.g.   Thus, the present review is mainly based on modeling studies.

70
The paper is organized as follows. First, the main mechanisms leading to

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At the global scale, depending on the location, spring tidal ranges vary from a few 105 tens of centimeters to several meters, and can locally exceed 10 meters as in the 106 Bay of Fundy (Pugh 1987). The atmospheric storm surges can also be regarded 107 as long waves. They are generated by changes in atmospheric pressure and wind 108 stress acting on the sea surface. Similar to tides, storm surges propagate to the 109 coast as shallow-water waves, and are subject to sea-bed friction, Earth's rotation 110 (Coriolis force), and local enhancements due to resonance producing large surges.

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In a first approximation, tides and surge can be modeled by the shallow-water 120 equations, which can be written as follows, omitting the horizontal viscosity term 121 (A∇ 2 u) for the sake of clarity: with u the depth-integrated current velocity, ξ the free surface, D the total 124 water depth (equal to the sum of undisturbed water depth H and the free surface where C D s and C D b are the free surface and bottom drag coefficients, respec-132 tively, ρ a is the air density and U 10 is the wind velocity at z = 10 m. Finally, F 133 includes other forces such as the wave-induced forces leading to wave setup, and Π 134 includes tide-related forcing terms (e.g. self attracting load, tidal potential forces).

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In the case of pure tides, the second, third and fifth terms of the right hand side 2). Finally, as a counterbalancing effect, SLR will also result, in some locations, 201 in new flooded areas, which will act as additional dissipative areas for tide and 202 surge. In the present section, for sake of clarity, we do not consider the effect of The SLR effect on tides has been investigated at different scales (global, regional,

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The effect of SLR on tides in the San Francisco Bay has also been investigated.        that wave breaking is more intense on ebb shoals at low tide so that the associated 571 setup in the lagoon/estuary is higher at low tide.  hand, it is more likely that the beach profile will translate onshore due to SLR 604 (Bruun 1962), so that wave setup would be globally unchanged. Due to large un-605 certainties concerning the response of sandy coastlines to SLR, we focus the review 606 on studies investigating the effect of SLR on reef environment, assuming that the 607 reef will weakly change for metric SLR.

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Wave setup ranges between a few to tens of centimeters and up to about 1 m. 640 We cannot, however, exclude that larger wave setup could occur. Tides can mod-   intertidal areas or tidal bedforms may also affect the regional dynamic). In ad-   In the present review, we focus on SLR, tide, atmospheric surge, wave and 851 wave setup interactions. However, in estuaries and inlets, when not negligible, the 852 interaction between water level and river discharge should also be considered. Wa-853 ter depth changes are dependent on river discharge (Q r ) in estuarine locations.

854
An increase in Q r will increase mean sea level locally, but the increased friction projections of still water level or for coastal flood hazard assessment by allowing 875 areas sensitive to SLR, tide and atmospheric surge interactions to be identified.

876
As mentioned above, one of the main issues when focusing on nearshore water 877 level is the temporal evolution of the sea-bed topography/morphology. Indeed, 878 such changes, especially for sandy beaches exposed to waves, can have a significant  Union, Grant 690462). Contributions from Xavier Bertin were funded through the Regional Fig. 1 (a) Components of storm tide, terminology and sketch of interactions. (b) main interactions between mean sea level, waves, atmospheric storm surges, tide, wave setup and flooding. In bold and black: the focus of the present paper.    It should be noted that these orders of magnitude are provided 'at the coast' (i.e. at the waterline). This does not exclude larger effect in the nearshore (for instance for tide modulation of wave setup in the surf zone).