Tidal anomaly correlations (TACs) in the Pacific for M2, K1 and δ-HAT, along with 95% confidence limits.
Global sea level rise understanding is critical for coastal zones, and estuaries are particularly vulnerable to water level changes. Sea level is increasing worldwide due to several climactic factors, and tidal range may also change in estuaries due to sea level rise and anthropogenic harbor improvements that may modify friction and resonance, increasing risks to population centers. Tidal range changes may further complicate the risks of sea level rise, increasing the frequency of nuisance flooding, and may affect tide-sensitive ecosystems. Higher total water levels threaten to increase flood zone areas in estuarine regions, which can impact the infrastructure, industry, and public health of coastal populations, as well as disrupting sensitive biological habitats. Therefore, it is of critical interest to analyze how tidal range changes under sea level changes. This chapter describes the tidal anomaly correlation (TAC) methodology which can quantify the tidal evolution related to sea level changes. A basin-wide survey of Pacific and Atlantic Ocean tide gauges is detailed, showing that tidal changes due to sea level rise is present at most locations surveyed. A focused regional study of Hong Kong is also described as an example of how tidal evolution can impact high population density coastal zones.
- ocean tides
- tidal variability
- sea level rise
- coastal flooding
- nuisance flooding
Ocean tides are a manifestation of the response to the gravitational forcing induced by astronomical bodies; namely, the Sun and Moon. The lunar forcing is approximately twice the magnitude of the solar forcing, since the closer distance of the Moon is more important than the larger mass of the Sun, as the universal law of gravitation is directly proportional to mass but inversely proportional to the square of the distance between heavenly bodies. However, there are also interactions between the Sun and Moon that modulate the distance of both bodies, which in turn influences the forcing felt at any point on Earth as a linear combination of tidal frequencies with forcing frequencies that range from twice-daily to decadal. Thus, though gravitation between two bodies is straightforward and definite, the true expression of tidal forcing experienced on Earth is an example of complex relation known as the “three-body problem” , which is only numerically calculable. However, this forcing is well-known, and essentially constant over short timescales.
Logic would dictate that due to this predictable “celestial clockwork”, the ocean tides on Earth should be equally predictable at all locations. However, this would only be true if Earth’s oceans had a constant depth and simple coastlines, as originally assumed by LaPlace in his tidal equations in the eighteenth century. This, of course, is not the case. Earth has a complex and highly variable ocean depth, with undersea ridges, trenches, plateaus, and valleys. Coastlines are also highly complex. Both factors can modulate the response of tidal forcing, with shallow coastal areas being the most sensitive. Thus, coastal tides are much larger and more variable than those seen in the deep ocean. The tides in coastal regions are also highly sensitive to changes in the shape and depth of shallow water regions. Some semi-enclosed coastal regions can amplify the resonant response of tidal forcing, such as in the Bay of Fundy in Canada, where tides can exceed tens of meters. Changes in local water depth can also influence the response of tides. Since recent decades have experienced the most rapid rise in mean sea levels (MSL) in millennia [2, 3], due to the steric rise of the ocean from ice melt and the thermal expansion of ocean water , both due to climate-change induced factors, future tidal range evolution is likely.
The changes in MSL are most pronounced in shallow coastal regions, especially in developed population centers, such as estuaries. Changes in MSL may lead to a change in local water depth in coastal regions, which have a first-order impact to coastal zones as rising background water levels. In turn, changes in water depth can modulate the response of tides as the resonant behavior changes. Small changes in water depth can lead to large changes in tidal range, which leads to a second-order impact to coastal zones. Estuaries are among the most vulnerable areas to these changes, since these regions are where large population centers are located, as well as sensitive ecosystems. Both natural and anthropogenic systems are highly dependent on tides. Biological habitats such as mangrove forests rely on constant tidal range, as do the complex food webs seen in estuarine regions. Consequently, changes in biology and ecology can have serious detrimental effects on human society, as much of the economy and industry of estuarine population may be dependent on stable ecosystems, e.g., fisheries, farming, and tourism. Changes in tidal range, tidal currents and tidal energy distribution can amplify these factors.
There are significant physical risks to estuarine cities and population centers that can be brought about by changing tides related to MSL rise. A large percentage of human settlements are in estuarine regions, as the abundance of fresh water and easy access to the open ocean allows civilization to easily thrive in these regions. Throughout history, estuarine cities have existed at the mercy of both the river and the sea. Extreme floods or extreme droughts can lead to extreme responses of the riverine aspects of estuaries, with implications for local farming and public health factors. On the oceanic side, storm events such as hurricanes and typhoons or tsunamis can be disastrous to estuarine cities, with extensive infrastructure damage and disruption to the local economy. However, both types of extreme events tend to be short-lived, and population centers in estuaries have developed knowing that even though such events can happen anytime, the average properties of the coastal zones, such as mean sea level and local tidal range, remain relatively constant. These assumptions have determined the planning and design of estuarine developments, such as harbors, roads, residence centers, and other infrastructure. Flooding due to inland storms and river surge might be occasionally extreme, but it could be predicted to only reach certain maximum flood levels. However, under scenarios of sea level rise, and the resultant changes in ocean tides, modern times are now producing changes in this “stable background”, and previous assumptions of the worst-case scenarios may no longer be valid, rendering past coastal defense efforts inadequate to resist future extreme events. Changing tides on top of sea-level rise also allow the possibility of nuisance flooding, also known as “sunny-day flooding”, in which flood levels can be exceeded at exceptional high tides without the influence of a storm of river surge event [5, 6]. Other impacts possible under rising sea-levels and tidal evolution besides local flooding include disruptions in shipping and other coastal-based logistic factors. Most importantly, the coupled changes in MSL and ocean tides may be occurring rapidly, and across multiple spatial and temporal scales, making it a complex problem to predict with certainty, as each coastal location may experience a much different response.
This chapter will explore the dynamics and details of changing ocean tides. A background will first be given about past research that has identified secular (long-term) non-astronomical changes in ocean tides as well as a summary of past studies of MSL rise. Next will be a description of the methodology of newer efforts that have analyzed the correlated changes in sea levels and ocean tides in the Pacific and Atlantic Oceans, based primarily on the work of Devlin et al. [7, 8, 9, 10, 11]. Following this will be a summary of significant results in the Pacific and Atlantic basins, as well as results from a focused study of the Hong Kong region, where some of the strongest magnitude changes have been observed. Next, there will be a discussion section about the implications of coupled MSL and tidal variability for estuaries and coastal zones including effects like nuisance flooding, and finally, conclusions.
2.1 Sea level changes
Mean sea level (MSL) is increasing nearly everywhere on Earth, with a globally-averaged rise of +1.7 ± 0.2 mm year−1 as estimated from coastal and island tide gauge measurements from 1900 to 2009 [2, 12, 13], and at a rate +3.4 ± 0.4 mm year−1 for 1993–2016 as estimated from satellite altimetry (
However, there is a wide spatial variability to these rates [3, 20], attributed to the combined effects of spatially variable wind and warming, and different vertical rates of land subsidence. In the Western Pacific, MSL rise often is larger than +10 mm year−1 in some locations, whereas Eastern Pacific rates are near zero or sometimes slightly negative because of tectonic and weather factors . The anomalously rapid sea level rise observed in the western Pacific tends to be underestimated in many models. This may be because of low variability in tropical zonal wind stress . However, the extreme rate in the Western tropical Pacific is unlikely to persist unabated , and a reversal of this Pacific asymmetry may be imminent soon.
2.2 Tidal changes
Ocean tides have classically been considered stationary because of their close relationship to celestial motion of the Moon and Sun . However, many studies have clearly demonstrated that tides are evolving at different rates in different regions of the world, and these changes are not related to astronomical forcing [25, 26, 27, 28]. Early studies discovered that long-term tidal changes are present at some stations such as at Brest, France, which has been steadily recording tidal levels for hundreds of years [29, 30]. It has also been shown that tidal changes can be a result of harbor modifications [31, 32, 33, 34, 35, 36] through mechanisms such as channel deepening and land reclamation. Alternatively, long-term tidal changes can be due to modulations in the internal tide [37, 38]. Regionally focused studies have discovered changes in the major diurnal and semidiurnal tides in the Eastern Pacific , in the Gulf of Maine, , in the North Atlantic [26, 41], in China [42, 43], in Japan , and at certain Pacific islands .
2.3 Coupled changes in tides and MSL
Mean sea levels may influence tidal evolution directly, or it may be correlated with tidal variability through secondary mechanisms in a multitude of ways; some may be acting locally, and others may be active on basin-wide (amphidromic) scales. One way is through changes in water depth (e.g., due to climate-change induced sea level rise), which may influence tides on a large geographic scale via a “coupled oscillator” mechanism between the shelf and the deep ocean [46, 47]. Water depth changes can also modify the propagation and dissipation of tides [48, 49] by directly altering wave speed in shallow areas, or by changing the effect of bottom friction. The warming of the upper ocean  may lead to internal changes to stratification properties and a modulation of thermocline depth. Both mechanisms can yield a steric sea level signal which may modify the surface manifestation of internal tides, thus producing a detectable change at tide gauges. Such changes have been observed at the Hawaiian Islands . On a shorter timescale, seasonal tidal variations can be due to rapid changes in water column stratification [50, 51], or by seasonal river flow characteristics [52, 53]. The shifting of the amphidromic points, e.g., as seen around Britain and Ireland , is possibly associated with changes in regional tidal properties . In harbors and estuaries, increased water depths can alter the tidal prism, local resonance, and frictional properties [34, 56].
2.4 Dynamical relations of MSL and tides
A tidal constituent amplitude can be expressed as a function of multiple variables:
Here, H is the water depth (which includes MSL, waves, storm surge, ocean stratification, river discharge, winds, etc.), r represents friction, and Ψω is the frequency-dependent tidal response to astronomical tidal forcing. The “…” indicates other variabilities not considered here, e.g., wind. For a constituent amplitude to experience change (i.e., ΔAmptidal ) it is necessary that one or more of these variables change, expressed by:
Subsequently, each of the variables that can change the tidal amplitudes depend on multiple factors:
The depth-averaged tidal response function is therefore a function of astronomical forcing, water depth and the local frictional properties. Additionally, water depth may depend on vertical land movement , global sea-level rise , and other location-dependent environmental factors such as the local water density (ρ), local river discharge, Qr , and local and far field wind forcing  effects. The effective frictional damping will be dependent on water depth, stratification, and mixing induced at the boundaries (bottom and surface). Finally, density ρ, as well as changes in buoyancy and stratification, are a function of water temperature, Tw , water salinity, Ts, river discharge, Qr , and mixing, mx :
The chain rule can be applied to Eq. (2), and considering the possible changes of all factors yields a general expression for the variability in tidal amplitudes:
It can be seen from this derivation that many of the variabilities can be correlated to each other. Figure 1 shows a simple cartoon displaying the possible mechanisms that can affect MSL and tides, based on the derivations and references given above. Hence, the existence of multiple mechanisms, many of which may be correlated with each other, can make it difficult to discern the causes of observed variability. Yet, understanding these correlations is still vital, with the best strategy being to consider each location’s dominating factors individually instead of relying on globally averaged solutions.
3. Methodology of tidal variability analysis
3.1 Tidal analysis
Recent studies have developed a reliable methodology to analyze tidal variability related to MSL variability [7, 8, 9, 10, 11]. These methods have been applied to many tide gauge locations worldwide, with a twofold approach. The first technique involves analyzing individual tidal constituents, while the second involves the consideration of the combination of multiple tides. For any individual tide gauge, water levels are typically recorded hourly as a continuous time series. Harmonic analysis of this data yields individual time series of multiple tidal constituents, each corresponding to an individual component of astronomical motion of the Sun and Moon, and their co-interactions. The largest parts of the tidal energy concentrate in the once-a-day (diurnal) and twice-a day (semidiurnal) frequency bands, with several closely spaced tidal constituents being important in each band. In practice, however, only a small number of these contain most of the tidal energy. For the purposes of our discussions of past studies we will only need to mention a few. The major twice-daily (semidiurnal) tide due to the Moon is denoted M2, and the twice-daily tide due to the Sun is denoted S2. Two important lunisolar interaction tides that define the once-a-day (diurnal) tides are denoted K1 and O1. Most of the past analyses only consider these four components, however, some of the locations considered (in the Atlantic Ocean) also consider two more semidiurnal components, denoted N2 and K2, and two more diurnal components, P1 and Q1 ( Table 1 ).
|Station name||Country||Lat.(N)||Long. (E)||K1||(±)||M2||(±)||δ-HAT||(±)|
|French Frigate Sh.||USA||23.87||−166.28||−8.52||4.44||−33.37||9.36||−52.62||16.89|
|Cabo San Lucas||Mexico||22.88||−109.92||−8.83||3.06||39.78||12.45||57.75||64.52|
|Kodiak Island, Alaska||USA||57.73||−152.52||12.96||10.03||20.84||7.79||12.27||18.80|
|Dutch Harbor, Alaska||USA||53.88||−166.53||−44.37||14.73||−2.59||8.26||−61.00||17.63|
|Nawiliwilli Bay, Hawaii||USA||21.97||−159.35||−3.63||5.87||61.28||13.80||55.61||12.27|
|San Francisco, California||USA||37.8||−122.47||−70.20||10.99||−80.34||21.57||−146.56||35.20|
|La Jolla, California||USA||32.87||−117.25||16.03||12.40||28.83||12.57||59.50||34.60|
|Crescent City, California||USA||41.75||−124.18||−15.27||10.44||−9.50||11.40||−48.44||12.73|
|Neah Bay, Washington||USA||48.37||−124.62||4.50||8.21||−11.95||6.26||9.80||14.06|
|Port San Luis, California||USA||35.18||−120.77||−10.55||12.57||12.68||6.89||4.49||18.17|
|Los Angeles, California||USA||33.72||−118.27||−11.03||10.87||−7.38||7.33||−21.39||17.05|
|San Diego, California||USA||32.72||−117.17||−13.71||7.29||−0.87||7.02||−17.77||17.18|
|Santa Monica, California||USA||34.02||−118.5||−2.46||12.81||−38.31||6.77||−51.72||47.75|
|South Beach, Oregon||USA||44.63||−124.05||−18.06||9.08||5.08||8.58||−16.05||40.56|
|Papeete (Tahiti)||Fr. Poly.||−17.53||−149.57||−10.12||7.75||−136.07||33.43||−95.29||43.32|
|Juan Fern Island||Chile||−33.62||−78.83||−4.57||6.50||−16.53||4.92||−28.68||11.47|
|Lobos de Afuera||Peru||−6.93||−80.72||3.46||4.21||38.64||10.79||23.92||16.59|
|Nauru||Rep of Nauru||−0.53||166.92||−2.33||6.78||−20.22||9.46||−53.89||7.83|
|Majuro||Rep Marshall Is||7.12||171.37||0.77||7.57||−2.52||16.65||−26.84||20.39|
|Malakal||Rep of Belau||7.33||134.47||50.30||5.19||−27.23||7.01||4.34||24.61|
|Yap||Fd St Micronesia||9.52||138.13||24.44||6.77||−39.92||7.01||−57.78||25.15|
|Rabaul||Pap. New Guinea||−4.2||152.18||−30.61||3.82||103.43||9.22||30.36||7.86|
|Christmas Island||Rep of Kiribati||1.98||−157.47||−7.59||3.68||−50.00||8.77||−70.51||8.38|
|Saipan||N. Mari. Islands||15.23||145.75||7.19||12.54||34.13||17.96||−5.84||14.93|
|Kapingamarangi||Fd St Micronesia||1.1||154.78||−21.95||6.02||36.75||8.44||15.71||15.00|
|Manus Island||Pap. New Guinea||−2.02||147.27||−15.77||10.06||−26.10||13.99||0.36||51.64|
|Ko Taphao Noi||Thailand||7.83||98.43||−2.85||4.33||−62.23||14.66||−70.35||32.54|
|Lord Howe Island||Australia||−31.52||159.07||4.80||7.23||−7.22||12.74||10.24||27.38|
3.2 Tidal admittance calculations
Investigations of tidal trends are carried out using a tidal admittance method. An admittance is the unit-less ratio of an observed tidal constituent to the corresponding tidal constituent in the astronomical tide generating force (ATGF) expressed as a potential, V, divided by the acceleration due to gravity, g, to yield a quantity, Zpot (t) = V/g, with units of length that can be compared to tidal elevations Zobs (t) on a constituent by constituent basis. Because nodal and other low-frequency astronomical variability is present with similar strength in both the observed tidal record and in V, its effects are eliminated in the yearly analyzed admittance time series. Yearly tidal harmonic analyses are performed at monthly time steps on both the observed tidal records and the hourly Zpot (t) at the same location, using the r_t_tide MATLAB package , a robust analysis suite based on t_tide . The tidal potential is determined based on the methods of Cartwright and Tayler , and Cartwright and Edden . The result from a single harmonic analysis of Zobs (t) or Zpot (t) determines an amplitude, A, and phase, θ, at the central time of the analysis window for each tidal constituent, with error estimates. Analyzing the entire tide gauge record produces time-series of amplitude and phase. From amplitude A(t) and phase θ(t) time series, one can construct complex amplitudes Z(t):
The harmonic analysis that generates the As and Ps also provides an MSL time-series. For each resultant dataset (MSL, A and P), the mean and trend are removed from the time series, to allow direct comparison of their co-variability around the trend. Applying trend removal also reduces the effects of land motions (e.g., glacial isostatic adjustment (GIA), subsidence, and tectonic effects. All of these are assumed linear on the time scale of tidal records) that occur on longer time scales, whereas we are concerned with short-term variability.
3.3 Tidal anomaly correlations (TAC)
Tidal range changes are quantified using tidal anomaly correlations (TACs), the relationships of detrended short-term tidal variability to detrended short-term MSL fluctuations. These are used to determine the sensitivity of individual constituents to a sea-level perturbation, and the result is expressed as a millimeter change in constituent amplitude per meter change in sea-level. The M2, S2, K1, and O1 tidal constituents are first considered separately, and later in combination as a proxy for the change in the highest astronomical tide (δ-HAT). We assume that the interannual variability captured by TACs can be extrapolated to the longer time scales, subject to the qualification that the changes remain “small-amplitude”, meaning a 0.5–1 m change in MSL and a change in tidal amplitude of a few tens of cm. Thus, we report TACs in units of mm m−1. The detrended time series of A and P can each be compared to detrended MSL, but herein, only the absolute magnitudes of the A for major constituents will be considered, because of their direct role in changing high water levels. The slope of the regression between A and MSL is the definition of the TA, deemed significant if the signal to noise ratio (found from comparing the magnitude of the TAC to the 95% confidence interval (CI) of the robust fitting error) is greater than 2.0.
3.4 Example of a TAC
The M2 TAC results at Honiara in the Solomon Islands exhibits one of the clearest signals in our data inventory. Figure 2 shows the M2 TAC at Honiara on the island of Guadalcanal (Solomon Islands, 9.4167°S and 159.950°E). The M2 tide amplitude is relatively small at this location (∼50 mm), but the anomaly correlation is large, +58.9 ± 3.7 mm m−1 (118% of the local M2 amplitude), and very coherent. Since the trend is reasonably linear over such a large range (>100% in terms of tidal amplitude and ∼0.45 m MSL), our analysis approach is demonstrated to be valid, even in cases where the small amplitude assumption is stretched.
3.5 Change in the highest astronomical tide (δ-HAT)
We also consider a combined tidal variability besides the individual TACs. The M2, S2, K1, and O1 variabilities are summed to produce a combined tidal variability that is compared to MSL (δ-HAT). The δ-HAT is a proxy for the change in the highest astronomical tide, which is estimated by combining the complex time series of the yearly analyzed M2, S2, K1, and O1 tides, approximately 75% of the full tidal height. “Complex” means, in this context, that each constituent is considered as a complex number (accounting for both amplitude and phase), the complex vectors are added, and the total amplitude is resolved from the complex sum. The detrended time series of δ-HAT is then compared to the detrended MSL variability. The magnitude of the slope of the regression is the definition of the δ-HAT, and, like the TACs, we report δ-HATs in units of mm m−1. Theoretically, the four constituents will not be exactly in phase more than once during every 18.6-year nodal cycle, though the constituents may be approximately aligned more often; therefore, this summation provides a suitable proxy for the envelope of possible tidal amplitudes. A detailed description of the step-by-step method, with additional figures showing the intermediate steps in the process, are provided in the supplementary materials of Devlin et al. . The δ-HAT analyses performed for the Atlantic tide gauge stations  employed an eight-tide combination, which adds the N2, K2, P1 and Q1 tidal constituents to the δ-HAT sum ( Table 2 ).
|Station||Country||Lat (N)||Long(E)||M2 TAC||IQR||K1 TAC||IQR||δ-HAT||IQR|
|Port Isabel, Texas||USA||26.06||−97.22||−8.6||1.9||1.6||10.0||−31.1||27.6|
|Corpus Christi, Texas||USA||27.58||−97.22||3.1||3.4||4.9||8.8||14.5||24.0|
|Galveston Pl. Pier, Texas||USA||29.29||−94.79||−0.1||6.6||7.9||16.9||−5.7||48.3|
|Galveston Pier 21, Texas||USA||29.31||−94.79||−57.6||8.6||9.0||13.7||−116.4||49.5|
|Sabine Pass N, Texas||USA||29.73||−93.87||−25.1||8.9||−1.0||12.9||−49.9||37.2|
|Grand Isle, Louisiana||USA||29.26||−89.96||0.1||3.8||25.8||6.4||39.4||9.3|
|Dauphin Island, Alabama||USA||30.25||−88.08||−1.6||18.1||33.3||21.8||47.8||50.8|
|Panama City Beach, Florida||USA||30.21||−85.88||12.2||2.8||0.8||22.7||109.5||59.7|
|Saint Petersburg, Florida||USA||27.76||−82.63||102.4||23.3||36.3||21.5||296.5||70.6|
|Key West, Florida||USA||24.55||−81.81||−4.9||9.9||−1.1||14.3||−6.9||39.1|
|Virginia Key, Florida||USA||25.73||−80.16||21.4||5.8||−42.1||17.4||−48.3||29.4|
|Port Canaveral, Florida||USA||28.42||−80.59||−15.0||23.0||−10.7||9.9||−56.8||41.8|
|Fernandina Beach, Florida||USA||30.67||−81.47||−54.7||5.9||4.6||10.6||−91.6||9.2|
|Fort Pulaski, Georgia||USA||32.03||−80.90||−87.9||14.0||−10.4||8.0||−132.5||61.0|
|Charleston, S. Carolina||USA||32.78||−79.93||−107.6||5.6||−11.7||6.0||−157.7||30.3|
|Springmaid, S. Carolina||USA||33.66||−78.92||−42.0||20.3||27.5||3.5||4.5||10.7|
|Wilmington, N. Carolina||USA||34.23||−77.59||−211.4||60.5||−38.0||12.1||−413.3||141.2|
|Duck Pier, N. Carolina||USA||35.18||−75.75||−12.6||3.5||−18.9||6.3||−13.4||39.5|
|Sewells Point, Virginia||USA||36.95||−76.33||−21.7||1.6||−30.2||6.2||−50.6||29.2|
|Chesapeake BBT, Virginia||USA||36.97||−76.11||−10.1||5.7||−33.5||8.2||−39.0||18.0|
|Cambridge II, Maryland||USA||38.57||−76.07||31.1||9.7||−2.9||26.7||70.6||55.7|
|Cape May, New Jersey||USA||38.97||−74.96||19.9||29.1||1.6||13.9||104.3||15.1|
|Reedy Point, Maryland||USA||39.56||−75.57||97.6||48.4||−3.9||17.1||142.8||62.2|
|Atlantic City, New Jersey||USA||39.36||−74.42||−20.5||6.0||−6.0||18.5||−59.7||21.3|
|Sandy Hook, New Jersey||USA||40.47||−74.01||23.4||13.7||−36.0||7.5||54.1||35.1|
|New York City, New York||USA||40.70||−74.01||50.3||26.0||−0.8||18.7||57.3||36.2|
|Montauk, New York||USA||41.05||−71.96||4.9||6.1||−14.6||15.4||13.8||16.4|
|New London, Connecticut||USA||41.36||−72.09||15.0||3.2||−23.3||12.1||14.5||22.5|
|Newport, Rhode Island||USA||41.51||−71.33||−0.1||9.5||−13.0||22.9||33.2||31.7|
|Providence, Rhode Island||USA||41.81||−71.40||−4.7||8.1||−4.6||13.4||2.5||21.4|
|Woods Hole, Massachusetts||USA||41.52||−70.67||38.6||7.6||−7.3||8.8||69.2||34.4|
|Bar Harbor, Maine||USA||44.39||−68.21||62.4||16.0||0.9||12.4||243.0||63.2|
|Port Ellen (Islay)||UK||55.63||−6.19||−9.5||14.3||−14.8||7.7||−100.4||13.2|
|Hoek van Holland||Netherlands||51.98||4.12||27.3||22.7||−62.1||21.1||−4.7||51.5|
|St. Helier (Jersey)||UK||49.18||−2.12||−145.7||70.8||8.6||18.9||−475.2||163.6|
|Les Sables d’Olonne||France||46.50||−1.79||−17.2||9.7||−0.5||7.3||−44.1||66.3|
|Saint Jean de Luz||France||43.40||−1.68||−9.5||17.3||−1.2||8.0||−52.1||25.0|
3.6 Error analysis and autocorrelation handling
Our approach uses one-year harmonic analyses at a one-month step to yield smooth time-series. However, this approach must be taken with caution, as there may be autocorrelation in the regression due to data overlap. Thus, calculations of regressions and associated statistics (i.e., the p-values) are based on a sub-sampled dataset of one determination per year. The definition of the “year window” used for harmonic analysis may influence the value of the TAC or δ-HAT, i.e. calendar year (Jan–Dec) vs. water year (Oct–Sep). Thus, we use an ensemble of TACs and δ-HATs using 12 distinct year definitions (i.e., Jan–Dec, Feb–Jan, …). We take the average of this set as the magnitude of the TAC or δ-HAT. For an estimate of the confidence interval of the trend, the interquartile range (middle 50% of the set range) is used. We consider correlations to be significant if they have a p-level of <0.05, the trend is greater than the interquartile range by at least a factor of 1.5, and the magnitude is greater than ±10 mm m−1 for individual TACs, and greater than ±50 mm m−1 for the δ-HATs. Some determinations had unexpected errors or grossly insignificant statistics (especially for the shorter records) that made them unreliable. These were not included in averaging process, though at all locations, a minimum of eight of 12 determinations was required to deem a result significant. For more detailed descriptions of the TAC and δ-HAT determinations, please refer to Devlin et al. , and the supplementary material of Devlin et al. [8, 9].
4. Selected results
4.1 Data inventory and sources
We now present some of the best results from past studies. Pacific Ocean locations were analyzed in Devlin et al.  for individual TACs, whereas Devlin et al.  analyzed the combined tidal variability of the δ-HATs, 152 total locations were analyzed in both Pacific studies. Locations are shown in Figures 3 and 4 (Eastern Pacific and Western Pacific, respectively) with major water basins labeled. The Atlantic Ocean was analyzed in Devlin et al. , which calculated both TACs and δ-HATs, considering a total of 170 locations. Atlantic locations are displayed in Figure 5 and major basins are labeled. Most of the hourly tide gauge records in the Pacific and Atlantic are obtained from the University of Hawaii’s Sea Level Center (UHSLC), with additional data from the following agencies: The Japanese Oceanographic Data Center (JODC); Canada’s Fisheries and Ocean office (FOC); Australia’s National Tidal Center (AuNTC), and the remainder from the Global Extreme Sea Level Analysis dataset, 2nd edition (GESLA ;
In all Pacific plots, the magnitude of the TACs or δ-HATs are indicated by the color intensity of the dots according to the scale shown in the legend; positive TACs are in shades of red, negative TACs are in shades of blue. For a gauge with an insignificant TAC (signal-to-noise ratio less than 2.0), the dots are white. In the Atlantic and Hong Kong results, red markers indicate positive TACs and blue markers indicate negative TACs, with magnitudes proportional to marker size, as shown in the legend, and insignificant results are shown as black dots. These values provide a measure of the tidal response, normalized to a 1 m MSL rise. In the TAC plots, the green and yellow background fields show the mean value of tidal amplitudes over the satellite altimetry record (1993–2014), using a tidal solution from the TPXO7.2 tide model [62, 63].
4.2 Pacific results
4.2.1 Pacific M2 TACs
The M2 TACs do not reveal any coherent basin-wide patterns of variability, however, there are localized features of interest. In the Eastern Pacific ( Figure 7 ), river-influenced gauges, such as San Francisco, California, Astoria, Oregon (labeled as “SF” and “AST” on the map) exhibit strong negative TACs. Strongly positive M2 TACs are observed at many Hawaiian and Alaskan gauges. Other locations show only weak or isolated correlations. At one gauge of note, Puerto Montt in far southern Chile (labeled as “PM” in Figure 7 ), there is an exceptionally large negative M2 TAC, greater than −500 mm m−1. There are a higher number M2 TACs in the Western Pacific than in the Eastern Pacific, but again the relevant variability is local ( Figure 8 ). M2 TACs are negative for the majority of Japan and Taiwan, some exceeding −100 mm m−1. However, some large positive M2 TACs are at isolated locations, such as at Okada (labeled as “OKA” in Figure 8 ) and in Tokyo harbor (“TOK”). At Hong Kong (“HK”), one of the largest positive M2 TACs is found (discussed in further detail below). Most significant positive TACs are south of the equator, and most negative TACs are north of the equator. The correlations at nearly all gauges in Malaysia (−40 to −150 mm m−1) and in the Philippines (−80 to −145 mm m−1) are strongly negative. Finally, Honiara (“HON”) in the Solomon Islands and Rabaul, Papua New Guinea (“RAB”) have small mean M2 amplitudes (∼50 mm) but display large relative correlations.
4.2.2 Pacific K1 TACs
K1 tidal anomaly correlations also reveal some regions of regionally coherent behavior but no basin-scale patterns. In the Eastern Pacific ( Figure 9 ), the river-influenced gauges of San Francisco, California (labeled “SF” in Figure 9 ), and Astoria, Oregon (“AST”) show TACs that are strongly negative, as was true for M2. However, slightly negative or insignificant K1 TACs are observed along the rest of the US west coast ( Figure 9 ). Fewer significant K1 TACs are observed in Alaska and Hawaii than was seen for M2. In South America, Puerto Montt, Chile (“PM”) shows a very strong negative K1 TAC, like M2. A larger number of significant K1 TACs are found in the Western Pacific ( Figure 10 ). As with M2, most coastal Japan gauges exhibit negative TACs, and there is a very large positive TAC in Hong Kong (labeled “HK” in Figure 10 ). In the Southwest Pacific, large positive TACs occur at both island and shelf stations, while significant negative TACs are mainly observed at island gauges, and in Malaysia and Thailand. Almost all significant negative TACs are north of ∼10 degrees South.
4.2.3 Pacific δ-HATs
In the Eastern Pacific ( Figure 11 ), significant δ-HATs are isolated. San Francisco, California (labeled “SF” in Figure 11 ) and Astoria, Oregon (“AST”) exhibit δ-HAT values of −146- and −257-mm m−1, respectively. Honolulu (“HONO”) and Hilo, Hawaii have positive values of +139- and +147-mm m−1, respectively. Along the coast of South America, there is an anomalously large negative anomaly correlation at far-southern Puerto Montt, Chile (“PM”), with a δ-HAT value of −963 mm m−1. Elsewhere in the Eastern Pacific of note, Johnston Island (“JOHN”) and Papeete on the island of Tahiti (“TAH”), exhibit large negative δ-HATs of −117 mm m−1 and −95 mm m−1, respectively.
The δ-HAT correlations are more significant in the Western Pacific ( Figure 12 ). Eleven gauges in Japan show negative δ-HATs, seven of which are greater than −100 mm m−1, with a maximum negative value of −351 mm m−1 occurring at Maisaka (labeled “MAI” in Figure 12 ). Only two significant positive δ-HATs are observed in Japan, at Okada (“OKA”) with a value of +159. At Western Pacific islands, results are mixed, with moderate positive δ-HATs and moderate negative δ-HATs both observed. Within the South China Sea, the distribution of δ-HATs is complex. An anomalously large positive δ-HATs is observed at Hong Kong (“HK”; +665 mm m−1) and at Bintulu, Malaysia (“BIN”; +615 mm m−1). Both sides of the Malay peninsula exhibit strongly negative δ-HATs. The Malacca Strait on the west side of the peninsula has δ-HATs of approx. −70 to −220 mm m−1, and the Gulf of Thailand on the eastern side shows δ-HATs of approx. −130 to −290 mm m−1; a common feature is that both sides show gradual increases in magnitude from the northern reaches to the southern tip of the Malay Peninsula.
4.3 Atlantic results
4.3.1 Atlantic M2 TACs
Figure 13a–c show the M2 results. In North America ( Figure 13a ), the M2 TACs show a dipole-like pattern. Both sides of the Florida Peninsula show TACs that are consistently positive. However, they are moderately negative in the Western Gulf of Mexico. Farther north along the US Atlantic coast, a strong concentration of negative TACs are seen from the Florida panhandle to Virginia in the Sargasso Sea, with the strongest correlation in Wilmington, North Carolina (−211 mm m−1; labeled “WIL” in Figure 13 ). Positive TACs are seen in the Chesapeake and Delaware Bays, New York City, New York (“NYC”), and Bar Harbor, Maine (“BH”). Finally, TACs in the Gulf of Maine and into the Bay of Fundy are strongly negative, reaching a magnitude of −259 mm m−1 at St. John, Canada (“STJ”). In Europe, the TAC patterns are somewhat more consistent ( Figure 13b , c ). Negative TACs are seen in most of the English Channel and the Irish Sea, and in the eastern parts of the North Sea. Conversely, positive TACs are found at inland semi-enclosed locations in England such as the Severn estuary; being largest at Avonmouth (“AVON”), farther north at Liverpool (“LIV”), and along the southern coast of the North Sea.
4.3.2 Atlantic K1 TACs
Diurnal components are generally smaller than semidiurnal constituents in the Atlantic, and TACs are also generally lower magnitude (usually <±100 mm m−1) and less often significant. However, there are still some regions of interest. The K1 tide ( Figure 14a ) has TACs that are consistently negative along the North American coast from Florida through maritime Canada. However, the TACs in the Gulf of Mexico are mainly positive, as well as in the Caribbean. K1 has some isolated positive TACs in Europe in the western English Channel, while the eastern Channel has a concentration of negative TACs ( Figure 14b ). There is a negative TAC at Bakar, Croatia (labeled “BAK” in Figure 14 ) at the end of the Adriatic Sea, where a large diurnal amplification occurs, and a small positive TAC at the end of the Gulf of Finland, where a diurnal amplification of the otherwise small Baltic tides also occurs ( Figure 14c ).
4.3.3 Atlantic δ-HATs
The calculation of the δ-HATs in the Atlantic use an eight-tide combination instead of the four-tide combination used in the Pacific, as detailed in Devlin et al. . Results in North America ( Figure 15a ) are generally positive in the eastern Gulf of Mexico, Puerto Rico, and a large magnitude δ-HAT is seen in the Bahamas (+323 mm m−1; labeled “BAH” in Figure 15 ). New York City, New York (“NYC”), Boston, Massachusetts (“BOS”), Bar Harbor, Maine (“BH”), and parts of the Delaware Bay also are strongly positive. Strong negative δ-HATs are found from Florida to Wilmington, North Carolina (−413 mm m−1; “WIL”), and at St. John at the head of the Bay of Fundy (−378 mm m−1; “STJ”). In Europe ( Figure 15b , c ), strong positive δ-HATs are found in the southern North Sea at Cuxhaven, Germany (+219 mm m−1; “CUX”) and in the Netherlands. Three locations within semi-enclosed regions of the Irish Sea show the strongest positive δ-HATs; at Avonmouth located at the head of the Severn Estuary (+256 mm m−1; “AVON”), and at Heysham (“HEY”) and Millport (“MIL”; +385 and +341 mm m−1, respectively). However, there are mainly negative δ-HATs seen in the rest of the UK, including most of the English Channel.
4.4 Hong Kong results
4.4.1 TAC and δ-HAT results
We now move from basin-wide surveys to a more tightly focused regional analysis of the Hong Kong waters, where 12 closely spaced tide gauges are available. For gauge names and locations, the reader is directed to refer to Figure 6 . The strongest positive M2 TACs are seen at Quarry Bay (+218 mm m−1), and at Tai Po Kau (+267 mm m−1), with a smaller positive TAC seen at Shek Pik ( Figure 16 ). In the waters west of Victoria Harbor, all gauges except Kwai Chung exhibit moderate negative TACs. The diurnal TACs in Hong Kong generally exhibit a larger-magnitude and more spatially coherent response than semidiurnal TACs. Like M2, the strongest K1 values in Hong Kong ( Figure 17 ) are seen at Quarry Bay (+220 mm m−1) and Tai Po Kau (+190 mm m−1).
The TACs are widely observed in Hong Kong, but the δ-HATs are only of significance at a few locations ( Figure 18 ). Five stations exhibit significant δ-HAT values, with Quarry Bay and Tai Po Kau having very large positive magnitudes (+665 mm m−1 and +612 mm m−1, respectively), and Shek Pik having a lesser magnitude of +138 mm m−1. Conversely, Ma Wan and Chi Ma Wan exhibit moderate negative δ-HAT values (approx. −100 mm m−1). The remainder of gauges (which are mainly open-water locations) have statistically insignificant results for the combined tidal amplitudes, even where some large individual TACs were observed. This shows that the combined tidal amplitude effect as expressed by the δ-HATs is most important in semi-enclosed harbors.
5. Summary and discussions
5.1 Summary of results
We have presented the salient features of several past studies, covering nearly all the global ocean, in efforts to understand the nature of tidal variability associated with mean sea level variability. Here, we will present a quick summary of the results, and then discuss some similarities of all results.
In the Pacific Ocean, out of 152 tide gauges considered, M2 TACs are significant at 89 gauge, and K1 TACs are significant at 76 gauges. For the combined tidal variability of these four tides, 54 stations (∼35%) are significantly correlated to variations in sea-level, with δ-HATs >±50 mm m−1 (i.e., >±5% of the sea level perturbation). In the Atlantic Ocean, 104 gauges have significant TACs, and for K1, 62 locations exhibit TACs that are significant. For the combination of the eight largest tides, there is a near even mix in the Atlantic of positive (40) and negative (47) δ-HATs. Finally, in Hong Kong, mixed results were seen for individual TACs, but δ-HATs were only important at a few semi-enclosed harbor locations, namely Quarry Bay (Victoria Harbor), and Tai Po Kau, where the largest positive feedbacks worldwide were seen.
5.2 TAC and δ-HAT distributions and patterns
There are a few commonalities seen in both the Pacific and Atlantic basins. First, the yearly averaged response of the tides due to correlated MSL changes (TACs) show an overall mixed pattern of positive and negative responses. There is no apparent ocean-wide pattern that might suggest a single cause to the variability, but some regionally coherent patterns of variability are apparent. While many gauges show an increase in tidal amplitudes with increasing MSL (positive TAC), many exhibit a decrease (negative TAC), which suggests a variety of mechanisms may be at play. Second, individual TACs are more widespread, being significant at a larger number of tide gauges, but δ-HATs are only significant at a smaller number of locations. This is because some individual TACs can be partially canceled out by other individual TACs, yielding an overall tidal variability that is less intense. However, some locations do see a reinforced trend when considering all major tidal constituents, yielding strong δ-HATs, and both positive and negative combinations are observed. Third, the largest δ-HATs tend to be located in coastal locations and not at open-ocean island locations. Many estuarine regions see the largest δ-HATs and most of the strongest individual TACs, as can be seen from the results in Hong Kong, which is located in the Pearl River Delta, and had the largest magnitude results. Other locations of note that are in estuarine environments include gauges in or near the Severn estuary in England, the Loire estuary in France, Astoria, Oregon, in the Columbia River Delta, and San Francisco Bay, California, which is fed by the Sacramento River.
There are some isolated locations do not fit this generalization, such as Hawaii, where the large significant positive values of δ-HATs at Honolulu and Hilo are mainly due to the M2 TAC, but this is likely related to the changing phase of the internal tide . On the western side of the South China Sea, gauges in Malaysia exhibit large negative δ-HATs related to the seasonal variability in tides due to stratification, seasonal monsoon winds and water depth . In other shallow, semi-enclosed regions, such as the North Sea, increasing sea-level has amplified tides on the German/Dutch coast over the past 50 years due to reduced frictional effects .
The regional case of Hong Kong is particularly interesting. Only a few locations showed strong δ-HATs, and these are in sensitive harbor locations. Hong Kong has had a long history of land reclamation to accommodate an ever-growing infrastructure and population, including the building of a new airport island (Chep Lap Kok), new land connections, channel deepening to accommodate container terminals, and many bridges, tunnels, and “new cities”, built on reclaimed land. All of these may have changed the resonance and/or frictional properties of the region. Tai Po Kau has also had some land reclamation projects that have changed the coastal morphology and may have modulated the tidal response. Other locations in Hong Kong did not show such extreme variations, so these variations appear to only be amplified in harbor areas.
5.3 Importance of combined tidal variability and nuisance flooding
The individual TACs reveal valuable information about the complex frequency-dependent response of the ocean. However, the metric that is most important for coastal planning is how all tidal components can combine and interact under changing MSL to increase local flood frequency and intensity. The δ-HATs provide an effective parameter to measure the full effect of changing tides, as they incorporate multiple tidal variabilities simultaneously. It is therefore of the greatest interest for the future of coastal flooding to find where all tides can change in the same direction, and the occurrence of the largest δ-HATs are likely dominated by local effects, such as a combination of natural and human-induced water level changes in sensitive harbor areas. If regional or amphidromic scales were dominant, then more coherent regional changes would be observed in the δ-HATs. If the changes in a local environment are favorable, all major tides can be enhanced and reinforced, and this may be via changes in tidal velocity and phase that better “tune” the response to yield higher water levels.
The impact of large δ-HATs on coastal and estuarine locations as sea level rises may be best demonstrated via the concept of nuisance flooding. Nuisance flooding refers to minor flooding events that happen at high tide without a strong storm surge, also called “sunny day flooding” . Such events may also be induced by minor storm tides, Rossby waves  or pluvial flooding . Nuisance floods are usually non-destructive individually, yet frequent occurrences can cause cumulative financial and societal impacts to coastal regions. Roads may flood more, disrupting logistics and supply chains . Sewers and drainage systems may overflow , increasing public health risks . Flood probabilities and cumulative hazards are likely to further increase under future sea level rise scenarios, with an increased effect seen during El Nino events [5, 6, 72]. Most previous examinations of nuisance flooding only consider a changing MSL and a static tidal range, but some studies have demonstrated the importance of tidal changes leading to increased inundation, such as at Boston . In some locations, secular changes to tidal range far outpace sea-level rise (e.g., Wilmington, North Carolina; ), and help drive flood risk. Moreover, since storm surge is a long wave, factors affecting tides can also alter storm surge [34, 74].
Figure 19 shows a simple representation of nuisance flooding with four cases presented. In the past when MSL was stable, it would take a storm surge (dark blue) to surpass local flooding levels (situation [a]), but under higher MSL conditions of the present day with unchanged tides, inundation can occur at high tide, especially on higher spring tides (situation [b]). If tides are not stationary with MSL rise, two additional situations are possible. If MSL increase leads to a slightly dampened high tide, then nuisance flooding will still happen, but will not be as extreme (situation [c]). However, if there is an additional increase of tidal range as MSL rises, then flooding will be more extreme, both with storm surge and without (situation [d]).
If tidal evolution related to MSL variability is present, then flood risk cannot accurately be assessed via superposition of present-day tides and surge onto a higher baseline sea-level, as such predictions would be insufficient at locations with a high tidal sensitivity to water levels. Long-term trends of tides and MSL can give a picture of the “slow and steady change” that will be most relevant for the future of coastal health, such as the unrecoverable loss of low-lying population zones such as estuaries under higher baseline MSL. On the other hand, short-term variabilities can indicate where “quick and sudden change” is important, which may increase the intensity of major storms as well as increasing the frequency of lower-impact yet more frequent high-water events (such as nuisance flooding) that can yield a cumulative degradation of coastal areas. Therefore, both tides and MSL should be considered to fully quantify future water level changes in coastal areas, and a regional-to-local approach is prescribed.
This chapter has strived to summarize the salient features of a suite of past studies [7, 8, 9, 10, 11] that have analyzed the issue of changing tidal evolution that is correlated with MSL variability, both of which are likely related to climate-change related factors. It has been demonstrated that these tidal changes are likely to have the greatest effect on coastal locations, especially estuarine regions, which are often highly developed, densely populated, and environmentally sensitive. Overall, in both the Pacific and Atlantic, over 90% of all locations surveyed show a significant TAC in at least one tidal constituent, and around one-third of all locations have a significant δ-HAT.
In general, coastal inundation is associated with peak water level, not mean sea level, and depends on the combined effects of tides, storm surge, sea-level variability, inland precipitation, river flow, and other factors which may lead to increases in extreme water level exceedance probabilities. MSL rise can affect the tidal dynamics directly, or the reasons for the observed changes can be related to secondary mechanisms, including, but not limited to: river flow, changing bed friction due to harbor development and other causes, barotropic friction effects, heat content, buoyancy, stratification, mixing and eddy viscosity, ocean currents, waves, storm surge, and indeed, any source of water or energy input.
Here, the effects of tidal evolution, and the impacts of these changes on nuisance flooding have been described. Identifying connections and correlations between tidal range and MSL is critical for making reliable predictions of coastal water levels and inundation risk. When combined with storm surge, larger tides and higher MSL may amplify flood risk, coastal inundation, damage to infrastructure, and population displacement. Even without the consideration of storm surge, changes in tides and sea-levels may lead to more occurrences of nuisance flooding.