2.2 Tidal changes

Ocean tides have classically been considered stationary because of their close relationship to celestial motion of the Moon and Sun [24]. 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 [25–28]. 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 [31–36] 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 [39], in the Gulf of Maine, [40], in the North Atlantic [26, 41], in China [42, 43], in Japan [44], and at certain Pacific islands [45].

### 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 [4] 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 [37]. 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 [54], is possibly associated with changes in regional tidal properties [55]. 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:

$$Amp\_{tidal} = f(H, r, \Psi\_{av}, \dots) \tag{1}$$

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:

$$
\DeltaAmp\_{tidal} = f(\Delta H, \Delta r, \Delta \Psi\_{av}, \dots) \tag{2}
$$

Subsequently, each of the variables that can change the tidal amplitudes depend on multiple factors:

$$
\Psi\_{oo} = f(H, r, \dots) \to \Delta \Psi\_{oo} = f(\Delta H, \Delta r, \dots) \tag{3}
$$

$$H = f(\rho, Q\_r, \dots) \to \Delta H = f(\Delta \rho, \Delta Q\_r, \dots) \tag{4}$$

$$r = f(H, \rho, \dots) \to \Delta r = f(\Delta H, \Delta \rho, \dots) \tag{5}$$

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 [57], global sea-level rise [2], and other location-dependent environmental factors such as the local water density (ρ), local river discharge, Qr [52], and local and far field wind forcing [21] 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:

$$\rho = f(T\_w, \mathbb{S}\_w, Q\_r, m\_x, \dots) \to \Delta \rho = f(\Delta T\_w, \Delta \mathbb{S}\_w, \Delta Q\_r, \Delta m\_x, \dots) \tag{6}$$

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:

$$
\DeltaAmp\_{tidal} = f(\Delta H, \Delta Q\_r, \Delta \rho, \Delta m\_{\text{x}}, \Delta r, \Delta \Psi\_{av}, \dots) \tag{7}
$$

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

#### Figure 1.

Schematic cartoon showing some of the mechanisms that can affect MSL and tides. See the text above for complete description of cartoon components.

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.
