1. Introduction

Among the many concerns related to recent and predicted climate change is the trend of rising sea levels. At global scales, studies such as Douglas (1997) show global sea level rising approximately 0.20 m in the past 100 yr, and predictions for sea level rise in the twenty-first-century range from 0.2 to 2.0 m (Parris et al. 2012). Adding to trends in the global-mean sea level, observations in some regions also show a pattern of increasing tidal amplitudes and increasing nontidal variations in sea surface height. The combined effects of sea level rise and potentially increasing tidal ranges will have far-reaching impacts on coastal inundation as many low-lying areas either become uninhabitable or require massive mitigation measures to fend off higher sea levels. Regional studies such as Grinsted et al. (2013) also point toward increases in inundation due to more frequent and more extreme weather events. As inundation is a consequence of peak sea level, not mean sea level, it is essential to consider both coastal ocean trends in mean sea level and how those trends will couple with local tidal dynamics to affect the peak sea surface height adjacent to areas in danger of flooding. At the same time, inundated areas add to the available tidal prism and the overall tidal energy dissipation, such that one must consider the whole system in order to accurately capture the coupling of tidal dynamics and inundation.

Nearshore regions are also influenced by management decisions, which in turn rely on predictions of flooding and sea level rise. Relevant management actions fall into two main categories. The first category, shoreline “hardening,” describes the construction of hydrodynamic barriers such as concrete sea walls or levees. These projects may be motivated by flood risks, “reclamation” of shallows into dry land, or creation of ponds for salt harvesting. In many areas shoreline hardening is widespread and significantly alters the dynamics of the basin, such as in San Francisco Bay, California, where upward of 85% of historic marshlands have been filled or fundamentally altered (Collins and Grossinger 2004). Shoreline hardening decreases the tidal prism and often leads to greater tidal amplification. The second, generally opposite, category of shoreline modifications could be labeled shoreline “softening,” but because it is often attempting to reverse the effects of earlier hardening projects, these actions may also be termed restorations. Typical restoration projects include breaching old levees or dredging new channels. Returning tidal action to these areas serves a number of purposes including reestablishing highly productive marsh ecosystems, improving water quality, and even mitigating flood risks. These projects often increase the area available to tidal action and introduce softer, natural shorelines and slough networks that are effective at dissipating tidal energy. Between the growing number of restoration projects and the potential for widespread sea level rise mitigation efforts, it will be important in the coming century to quantify the range of shoreline modifications and the effects those changes will have both on localized inundation and basinwide tidal dynamics.

a. Tidal amplification

Variations in tidal range within a basin come primarily from four physical processes: standing wave resonance from the reflection of the incident tidal wave, frictional effects, converging geometry (i.e., a landward decrease in the cross-sectional area), and inertial effects (van Rijn 2011). Resonance and converging shoreline geometry lead to an increase in tidal amplitude away from the open-ocean boundary of a basin, while diverging shorelines and friction lead to attenuation. Inertial effects are typically negligible and are ignored in most analyses that do not target shallow macrotidal systems.

Standing wave resonance is easily understood in terms of a simple prismatic channel, in which the tides can be described by the superposition of an incident wave and a reflected wave. Standing tidal waves occur when the incoming tidal wave is fully reflected, such as in a nonfrictional basin with a nondissipative landward boundary, and in this case the superposition of the two waves simplifies to

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where η is the space- and time-varying free surface perturbation, x is the distance from the close end of the basin, k is the wavenumber, t is time, ω is the angular frequency of the tidal forcing, and η₀ is the tidal range at the closed end of the estuary. Given the length L from the closed, reflective landward boundary of a basin to the open, ocean-forced mouth, the amplification is simply α = sec(kL). In systems where L is near a quarter wave node, the amplification approaches a resonant peak, such as the famous tides of the Bay of Fundy. The resonant period of a basin depends on the phase speed of the tidal wave; if the resonant period is altered toward the dominant period of the tidal forcing, one would expect that the net result would be an increase in tidal range. A fundamental parameter characterizing the degree to which a standing wave is present is the velocity phase lead ϕ, which we define here as the phase offset between peak flood velocity and peak high water for a specific tidal constituent. A progressive wave tide in which reflected energy is vanishingly small will have a phase lead approaching zero, while a standing wave system will see ϕ ≈ 90°. The velocity phase lead provides a useful local estimate of basinwide tidal energy dynamics. Aside from diagnosing standing wave or progressive wave dynamics, ϕ is also relevant for residual scalar transport and sediment dynamics, because Stokes transport is greatest for ϕ = 0 and negligible when ϕ = 90°.

The geometry of a basin can lead to amplification or attenuation through converging or diverging shorelines. van Rijn (2011) investigated the competing roles of convergence (both in depth and width), friction, and reflection. He found that in sufficiently long, deep, and converging estuaries, the amplifying effects dominate and tidal amplitude increases toward the head of the estuary. Shallow converging channels are dominated by friction, resulting in an attenuated tidal range landward of the mouth. In broad terms, he found that the reflected wave, if one exists, affects roughly the landward third of the basin, as the reflected wave is both dissipated by friction and attenuated by diverging shorelines as it travels seaward. In strongly converging channels, the phase lead of the peak flood velocity ahead of high water approaches 90°, independent of the presence of a closed landward boundary. Savenije et al. (2008) and van Rijn (2011) term this condition an apparent standing wave.

Cai et al. (2012) derived an analytic model of basin amplification applicable to basins with varying depth, convergence, friction, tidal forcing, and off-axis storage. The resulting expressions allow a classification of basins based on how the actual depth compares to the ideal depth (i.e., producing zero amplification) and the critical depth (i.e., producing maximum amplification). The model also includes prognostic equations for the phase lead and nondimensional amplification factor, as a function of basin geometry, tidal forcing, friction, and mean depth. The flexibility of the input parameters and wide range of behaviors that can be predicted make this model particularly relevant for sea level rise forecasts. In section 5a, we apply it to a portion of the study area and compare analytic and numerical predictions for the M₂ tide in order to understand the degree to which the analytic approach captures the necessary physics.

2. Physical domain

San Francisco Bay has one of the longest continuous tidal records on the Pacific Ocean at 160 yr (Talke and Jay 2013), showing sea level trends of 0.22 m rise per century (Flick et al. 2003). Another recent analysis (Bromirski et al. 2011) has found a reversal in the trend of sea level off the coast of California since 1997, though they attribute this to the Pacific decadal oscillation and note that in due time the trend of increasing sea level is likely to return. A comprehensive evaluation of the projected change in ocean forcing for San Francisco Bay is detailed in Knowles (2010). Shifts in regional climate (as well as future management decisions) will undoubtedly affect river inputs as well. For the present purposes, though, we take freshwater forcing, nontidal sea level, and ocean tidal range as constants.

San Francisco Bay is a bifurcated, mesotidal estuary. Tides at the mouth are mixed diurnal and semidiurnal, with a great diurnal range at the mouth of 1.8 m (NOAA 2013). The mouth of San Francisco Bay is the Golden Gate (star in Fig. 1), a 100-m-deep, constricted channel, connecting Central Bay to the Pacific Ocean. Along-channel distances throughout this paper are referenced to the Golden Gate with negative distances denoting the southern transect and positive denoting the northern transect. The southern branch of the bay, typically referred to as SSFB, has little freshwater inflow, is roughly funnel shaped, and is characterized by a single channel 12–20 m deep and broad shoals tapering from 5 m deep to intertidal. Tides in SSFB are close to a standing wave, with a velocity phase lead for the M₂ constituent in the channel of approximately ≈ 75° (i.e., peak flood velocity leads the peak sea surface elevation by 75°). The perimeter of SSFB, particularly to the south and southeast, is dominated by tidal sloughs and salt ponds. Ongoing restoration projects are returning many of the salt ponds to tidal action by levee breaches.

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Overview of model domain. Shaded depth contours are shown at −5- and −12-m NAVD88. The solid gray line outlines the present-day MHHW shoreline in SSFB and SPB. The solid black outlines the model domain including three false deltas. Dashed black line is the end-to-end thalweg, with the origin at the Golden Gate marked by a star and 10-km intervals marked by solid dots.

Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0214.1

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The northern branch of San Francisco Bay connects through two largely self-contained bays, consecutively San Pablo Bay (SPB) and Suisun Bay, before reaching the Sacramento–San Joaquin River Delta. A large fraction of the northern borders of both San Pablo Bay and Suisun Bay are tidal marshland joined to the respective bays via networks of tidal sloughs. A large number of wetland restoration projects, in various phases of planning or completion, are also targeted at the northern perimeter of San Pablo Bay.

3. Numerical model

To quantify the tidal processes and contributions to variation in high water across the domain, we employ the Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS) hydrodynamic model (Fringer et al. 2006; Wang et al. 2009) to simulate a range of sea level and shoreline configurations in San Francisco Bay. While estuarine circulation and other 3D processes undoubtedly will be altered by sea level rise, we have chosen to run the model in the less computationally expensive depth-averaged mode. Recent modeling studies have shown sea level rise–driven changes in the salinity intrusion length of 10%–20% of seasonal changes (Delta Science Council 2013; Chua 2011). Given that seasonal variation in amplification, presumably driven by seasonal river flows, is on the order of 0.05 m, we expect that changes in stratification will have minimal effects on inundation and the barotropic tidal response. The model solves the shallow water equations,

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where u is eastward velocity, υ is northward velocity, u is the vector-valued horizontal velocity, g denotes gravitational acceleration, and f = 2Ω sinΦ is the Coriolis parameter with angular velocity of the earth Ω and latitude Φ. Elevation of the free surface is given by η, measured as a departure from the North American Vertical Datum of 1988 (NAVD88) geopotential surface, and h gives the total height of the water column. These equations are discretized on a prismatic, finite-volume grid comprising unstructured triangles in the horizontal. The drag coefficient C_D is derived from a bottom roughness z₀ that in turn is calculated from the water column height based on relationships from previous modeling efforts in San Francisco Bay (Gross 1997). At each time step, the bottom roughness for each cell is found by linear interpolation over the values in Table 1. Following MacWilliams et al. (2008), roughness throughout the false delta regions is set to a minimal 10⁻⁵ m. At each time step n, an effective drag coefficient is calculated from the edge-local bed roughness as

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where is the height of the water column on edge j at time step n, and κ = 0.42 is the von Kármán constant.

Table 1.

Bottom roughness as a function of water column height [values taken from Gross (1997)].

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Wetting and drying in the model is handled by deactivating cells for which the water column height falls below a threshold height. We choose a threshold of 5 mm, as it is sufficiently small to avoid significant artificial storage, but large enough that the model is stable at a reasonable time step of 10 s.

The ocean boundary of the model domain is approximately 50 km beyond the Golden Gate, coinciding with a long-term tidal gauge at Point Reyes (see Fig. 1). The focus of the present study is SSFB and San Pablo Bay. Within these basins, the model domain extends up to the 3.5-m NAVD88 contour to accommodate tidal amplification and sea level rise. Upstream of San Pablo Bay and seaward of the Golden Gate, the model extends up to the present-day MHHW shoreline. Beyond Suisun Bay, the Sacramento–San Joaquin Delta is represented by a pair of hypsometry-matched false deltas, as is the slough and marsh network north of Suisun Bay. For each of the three false deltas, hypsometry (the relationship between planform area and free surface elevation) is extracted from a 10-m digital elevation model (DEM), excluding areas already accounted for in the original grid. A length for each false delta is determined by the along-channel length of the primary channel in each region. The width is determined by this length and the maximum area found in the hypsometry calculations. A regular triangular grid is constructed to match these dimensions, with a resolution of approximately 400 m. The hypsometry is then binned by depths such that each successive depth bin corresponds to an increase in the planform area equivalent to the area of one cell. These ordered depths are assigned to cells starting at the seaward end of the false delta, proceeding first along a strip of cells to the landward end, and then proceeding laterally to the next strip of cells until all cells have been assigned a depth. This simple approach ensures that a subtidal channel extends the length of the false delta (consistent with known river features), with a bed that slopes up in the landward direction and has a degree of lateral bathymetry variation. By matching hypsometry and length, the false deltas approximate the tidal response of the more complex physical channel network with a substantial reduction in the number of grid cells compared to a fully resolved delta.

For the purpose of this study, the portion of the domain at elevations between 0 and 3.5 m NAVD88 is the most relevant, as this is the intertidal range across the imposed sea level rise scenarios of 0–1.0 m. The grid resolution in the area between these contours is set to a uniform nominal length scale of 50 m (resolving the broad intertidal areas at finer resolution than this runs into practical computational limits). Additionally, five dynamically important channels outside this region are also given increased resolution: the Golden Gate (resolved at a scale of 125 m), Carquinez Strait and Suisun main channel (100 m), Suisun Cutoff (100 m), and New York Slough (200 m). In all other areas, the grid resolution is allowed to increase at a rate of 10%, up to a maximum grid scale of 3 km at the open-ocean boundary. The resulting grid has 937 759 cells.

Bathymetry data are derived from a range of sources covering subtidal, intertidal, and supertidal areas up to the 3.5-m NAVD88 contour. The base elevation data source is a 10-m seamless topography–bathymetry product designed for inundation studies (Carignan et al. 2011). Bathymetry at 10-m resolution for the Sacramento–San Joaquin Delta is taken from Foxgrover et al. (2013). Given the impact of small levee and slough features on inundation and hydrodynamic connection, special care is taken to assemble up-to-date and high-resolution intertidal topography. This includes the preprocessed 2-m bathymetry from Foxgrover et al. (2011), as well as gridded bare earth lidar datasets from Foxgrover and Jaffe (2005), NCALM (2003), and NOAA (2012). Missing data in the lidar datasets in small regions are filled via interpolation from nearby lidar data, or, in cases where the lidar was missing data over a span greater than 10 m, data are filled in from Carignan et al. (2011).

Though the intertidal regions are resolved at 50 m, the length scale of many channels and levee features, essential for the inundation characteristics of the marshes, is 5–25 m. As in previous studies, such as Bates et al. (2003), we have found that simple averaging of the DEM along each edge was insufficient to resolve either channels or levees robustly. Given the importance of narrow channels and levees in quantifying inundation, and the difficulty in applying the method of Bates et al. (2003) to an orthogonal finite-volume grid, we instead have developed a method that calculates the overtopping elevation for each edge. The algorithm and comparisons between the simple averaging of bathymetry and this connectivity-preserving method are described in Holleman (2013).

Calibration of the model (see the appendix) has been performed with observed tides and winds s_Obs. Periodic tides are used for all subsequent analysis in order to avoid the need for spring–neap duration runs of each scenario and to allow the analysis to focus on the individual effects of a single tidal constituent. The ocean-free surface is forced with an M₂ period (12.42 h) sinusoidal signal with a peak-to-peak amplitude of 1.64 m. The amplitude was chosen to match the great diurnal range observed at the Golden Gate. The imposed M₂ amplitude, larger than the observed M₂ constituent, allows for a range of inundation similar to the combined tides. This avoids the complications of nonlinear interactions between constituents, but retains and resolves the interplay between tidal dynamics and inundation regions, which is not affected by these interactions, and allows the development of higher-frequency harmonics.

The numerical experiments cover three variations in sea level rise and four shoreline configurations, with a naming convention outlined in Table 2. The range of ocean boundary conditions comprises (i) present-day mean sea level, (ii) an increase of 0.6 m, and (iii) an increase of 1.0 m. These values were chosen to roughly bracket the middle of the range of predictions for conditions in 2100 (Parris et al. 2012). Multiple shoreline configurations are used to simulate the effects of mitigation efforts such as the construction of levees at present-day MHHW shorelines and how shoreline hardening in one portion of the domain affects tidal range in other portions of the domain. The first shoreline scenario, “soft” (s), does not include any explicit shoreline protection, only present-day topography and bathymetry. The completely hardened scenarios h_NSx limit flows in both San Pablo Bay and SSFB to present-day MHHW shorelines. Two additional scenarios represent shoreline hardening limited to either San Pablo Bay (h_N) or SSFB (h_S). In all periodic cases, the model is allowed to spin up for 4 days before the data are extracted for a single M₂ period.

Table 2.

Naming convention for numerical experiments: Obs denotes observed tides and M₂ denotes 12.42-h periodic tides with amplitude matched to spring range of observed tides. (The N denotes hardening in the northern reach, S denotes hardening in the southern reach, and NS indicates hardening in both the northern and southern reach.)

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4. Energy flux and tidal phase analysis

Energy flux and tidal phasing provide fundamental information about the spatially variable dynamics in each basin. While the one-dimensional tidal propagation problem is well described by the velocity phase lead ϕ, in two dimensions the direction of wave propagation and the sense of land- or seaward can become ambiguous. For a more robust description, we switch to a comparison of depth-averaged tidal energy flux and an effective tidal phase lag derived from the energy flux. The east- and northward energy fluxes are defined as the pressure work done by the flow on an imaginary surface normal to the respective coordinate direction, averaged over the mean depth of the water column (Kundu and Cohen 2004):

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where u_i is the velocity component in the ith coordinate direction (assumed constant in the vertical), p = −ρgz is the hydrostatic pressure due to fluid density ρ, h is the time-varying height of the water column, and angle brackets denote a tidal average. For a tidal constituent with frequency ω_c, vertical amplitude η_c, easting and northing velocity amplitudes U_c and V_c, respectively, and velocity phase shifts ρ_x,c and ρ_y,c relative to the free surface phase, the harmonic velocity and depth are given by

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where H is the mean depth. Substituting (7)–(9) into (6), the energy flux for constituent c is then

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Consistent with the energy flux estimates, a measure of the tidal phase can then be determined as the inverse cosine of the ratio of the energy flux to the maximum possible flux for a fully progressive wave with the same H, U_c, V_c, and η_c:

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In the case of unidirectional tidal flow aligned with the x coordinate, (12) simplifies to ϕ_c = ϕ_x,c. However, compared to considering the tides to be unidirectional along the principal axis, (12) is robust to amphidromes and rotary tides, where the principal axis is poorly defined.

Energy in higher harmonics is relatively small in the majority of the domain, and our initial analysis is focused on the M₂ constituent. The numerical experiments do predict a significant M₄ overtide, which is later considered in section 6. From each of the periodic scenarios in Table 2, the M₂ phase and amplitude of the sea surface height, eastward velocity, and northward velocity were extracted by a least squares approach over exactly one tidal period. Changes in M₂ energy flux between pairs of scenarios elucidate how the tides change in response to shoreline hardening and sea level rise and how this response differs between the two bays.

a. South San Francisco Bay

Figure 2a shows the M₂ energy flux and tidal phase for SSFB. The dominantly standing wave tidal dynamics are clear, with the majority of the embayment showing > 80°. The channelized portion of the bay to the southwest is slightly more progressive, but still close to a standing wave, especially in comparison to the channel in San Pablo Bay (discussed below). Though the tidal phase is generally close to standing throughout the bay, the energy flux magnitudes are still substantial due to the large tidal range.

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SSFB: (a) and tidal energy flux for scenario h_NS0. Incremental change in and tidal energy flux are shown for (b) h_NS100 − h_NS0 and (c) s₁₀₀ − h_NS100.

Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0214.1

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Interestingly, in much of the bay the easternmost portion of the shoals shows slightly “overstanding” tides with a seaward energy flux. At the most eastward margins of the bay, the seaward-directed progressive component is sufficient to reduce below 80°. This seaward energy flux has been observed in SSFB (Lacy et al. 2014) and is reminiscent of the seaward residual transport in Li and O’Donnell (2005). The intratidal analytic model of Li and Valle-Levinson (1999), when evaluated with a bathymetry profile extracted from SSFB [detailed in Holleman (2013)], closely mimics the overstanding tides observed in the present model data. Qualitative agreement in ϕ between that analytic model and the present 2D model is a good indicator that a channel–shoal bathymetry profile and a reflective basin are sufficient to drive the overstanding wave in the shoals.

Having discussed the present-day M₂ dynamics, we now move to how these dynamics are altered with sea level rise and inundation. To approximately separate the effects of deepening from inundation, we consider first the changes due to sea level rise with hardened shorelines throughout the domain. The change in M₂ phase and energy flux between the h_NS0 and h_NS100 scenarios is shown in Fig. 2b. The choice of the 1.0-m scenarios is motivated by the characteristics of inundation in south San Francisco Bay, where a majority of the inundation occurs above a sea level rise of 0.6 m. The changes in phase in the bulk of the bay are minimal. The landward end of the bay has a convergent geometry and is nearly closed, such that, absent any dissipation from inundation, tidal energy has nowhere to go, and the landward energy flux is constrained to be near zero. The southern half of the bay shows only small and scattered changes in the energy flux, while the northern half shows a distinct seaward shift in the energy flux. This shift is consistent with a deeper SSFB, which is less frictional and more reflective. Portions of the eastern shoals become more progressive, departing slightly from the bulk of the bay, but notably the change in energy flux is actually seaward, showing that the overstanding tidal phasing of the shoals is accentuated by the deeper sea level and greater tidal range in the far south end.

The effects of inundation on M₂ dynamics are shown in Fig. 2c, comparing scenarios h_NS100 and s₁₀₀. SSFB is ringed by numerous tidal sloughs, connecting pond and slough networks to the main body of the bay. The greatest changes are at the mouths of sloughs, which function as gateways to the increased tidal prism when inundation is permitted. The sloughs are typically small, but when considered in the aggregate they are a significant sink of tidal energy in the M₂ band. The change in energy flux is everywhere landward, consistent with a basin transitioning toward a progressive wave. The eastern shoals actually show an increase in , toward a standing wave. In the scenarios with hardened shorelines, these shoals had a seaward energy flux, such that as the basin as a whole becomes more progressive, those areas that were originally overstanding (i.e., a seaward tidal energy flux) shift toward a standing wave. The flux changes of Fig. 2c are quite uniform across the width of the bay, especially compared to the baseline energy flux of Fig. 2a, which has significant lateral variation. This stems from the fact that in SSFB the majority of the inundation occurs in the southern portion of the bay. The primary effect of inundation for most of the bay is directly related to the amplitude of the reflected wave, rather than local or lateral dynamics.

b. San Pablo Bay

Figure 3a shows throughout the interior of San Pablo Bay, under present-day conditions (scenario h_NS0). San Pablo Bay has somewhat progressive tides along the main channel in the south and a partially standing wave across the shoals. Separated from the main channel by the broad shoals, the Petaluma River (to the northwest) and Sonoma Creek (to the north) connect to the northern shore. Along with the Napa River connecting to Carquinez Strait, these features are local sinks of tidal energy.

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SPB: (a) and tidal energy flux for the h_NS0 scenario. Incremental change in and tidal energy flux are shown for (b) h_NS100 − h_NS0 and (c) s₁₀₀ − h_NS100.

Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0214.1

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As with south San Francisco Bay, we first compare scenarios h_NS0 and h_NS100 in order to isolate the effects of a deeper basin interior, without significant change in inundation or tidal prism. Figure 3b shows the change in energy flux and phase between these two scenarios. The landward energy flux shifts from the channel to a proportionally greater flux in the shoals. The present-day mean depth of the off-channel areas of San Pablo Bay is quite shallow, making it highly frictional and a high impedance path for tidal propagation. A 1 m increase in mean sea level has a proportionally greater effect on the role of friction in the shoals, allowing a greater fraction of the tidal energy to propagate via the shoals. Another feature of the change in energy flux is the significant increase of energy leaving San Pablo Bay by way of Carquinez Strait to the east. This is likely due to a combination of less energy being lost in San Pablo Bay and greater dissipation in the false deltas beyond Carquinez Strait. In terms of the tidal phasing, the trend is clear that most of the bay shifts toward a progressive wave.

Figure 3c shows the incremental change in energy flux and phase between scenarios h_NS100 and s₁₀₀. The bulk effect in San Pablo Bay is an increase in tidal energy entering the bay from the south and a decrease in energy leaving the bay in the east (note that Fig. 3c shows depth-averaged fluxes; in the south, the incoming flux is in twice the depth as the outgoing flux). Based on these changes in energy fluxes, it is clear that the inundation of the soft shorelines leads to greater dissipation, and the bay has become a greater sink of tidal energy. The hot spots of energy flux and progressive phase at the mouths of all three rivers show that the bulk of the newly inundating areas is not directly connected to the main body of the bay but are instead connected via river and slough features.

5. Tidal amplification and damping

So far we have examined only the M₂ wave, but to more concisely address the potential for inundation, we now turn to a direct measure of the peak sea surface height. The high water elevation includes the combined effects of the M₂ wave and its harmonics, as well as constant offsets due to, for example, Stokes transport increasing mean free surface setup. To separate changes in dynamics within the basin from changes in ocean forcing, we use the relative high water,

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the height by which the high water level at a point in the domain exceeds or falls shy of a reference high water level at the coastal ocean boundary. Lateral variations in Δη are typically small, allowing comparisons between scenarios simply along the central thalweg of the bay, as shown in Fig. 4. The highly reflective and converging SSFB is evidenced by the tidal amplification on the left side of the plot. At its most extreme, high water at the far south end of the bay exceeds coastal ocean high water by nearly 0.6 m in the s₀ scenario. Interestingly, the highest rate of amplification in the southern reach occurs in the first 20 km, which is still essentially part of Central Bay. The amplification rate slumps for the diverging reach between 20 and 40 km, with a slight uptick south of 40 km as the shorelines converge.

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Transect of relative high water for soft shoreline scenarios s₀–s₁₀₀. Negative longitudinal distance indicates SSFB, positive is SPB and beyond.

Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0214.1

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The more transmissive and dissipative northern reach of the bay sees mild amplification for the first 35 km, up to the transition from San Pablo Bay to Carquinez Strait. The seaward half of that stretch has particularly complex geometry and bathymetry, leading to greater variability over short length scales, up to the transition into San Pablo Bay proper, at 25 km from the Golden Gate. The greatest amplification occurs in the middle of San Pablo Bay. In a sense, San Pablo Bay can be considered a “leaky” reflective basin, particularly at present-day sea level. The landward outlet for tidal energy, Carquinez Strait, is relatively small; the within the strait itself dips below 30° (Fig. 3a), but this depression extends only partially into San Pablo Bay, approximately to the point of maximum amplification at 25 km from the Golden Gate. The northern shoals are far enough from the main channel to see additional, minor amplification relative to the main channel (not shown). This effect can be traced to the wide aspect ratio of San Pablo Bay and how the northern shoals behave somewhat like an off-axis standing wave basin.

The progression of tidal amplification from scenario s₀ to s₁₀₀ in Fig. 4 demonstrates the combined, attenuating effect of inundation and sea level rise. Beyond Central Bay (from −20 to 15 km), the attenuation due to inundation more than offsets the amplification expected from a deeper basin. The locations at which the scenarios begin to diverge roughly correspond to where the inundatable regions occur, notably south of −40 km and north of 25 km. Figure 5 shows the incremental extent of inundation for each soft shoreline scenario. The most marked change in inundation in SSFB occurs when sea level rise approaches 1.0 m, compared to a relatively small change in the inundated area between the s₀ and s₆₀ scenarios. Consistent with the inundation distribution in the south, the greatest change in amplification is between the s₆₀ and s₁₀₀ scenarios. San Pablo Bay has a more even distribution of inundated area, both in terms of where these areas are located and at what rise in sea level they become inundated. There, the incremental difference in attenuation between s₀ and s₆₀ is similar to the difference between s₆₀ and s₁₀₀.

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Extent of inundation across scenarios, progressing from least to most inundation.

Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0214.1

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The comparisons between soft and hard shorelines approximately separate the effects of deepening from the effects of inundation, but also allow a comparison of local versus remote effects by selectively hardening only a subset of the shoreline. This demonstrates the dynamic interactions of the basins and at the same time informs practical management decisions regarding the degree to which mitigation efforts must be coordinated throughout a basin. The local and remote effects of shoreline hardening are quantified in Fig. 6, where the change in relative high water is shown for the four shoreline configurations. The baseline amplification of scenario s₀ has been subtracted out, as the changes are small relative to the baseline tidal amplification (i.e., the solid line of Fig. 4). When all shorelines are allowed to inundate, the model shows that a small portion of Central Bay is essentially unchanged, but everywhere else the tides are attenuated. With a maximum change in Δη of approximately −0.13 m, the attenuation is notable, though small compared to baseline tidal amplification. Hardening only the shorelines of SSFB adds 0.06 m to the 0.60-m baseline amplification of the far southern reach, and also affects the high water level in Central Bay. Hardening and sea level rise both shift SSFB toward a more reflective, standing wave environment, and it is apparent in Fig. 6 that the reflected tidal wave couples back into Central Bay, which in turn alters the seaward boundary condition for the northern reach of the bay.

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Longitudinal transect showing change in Δη for s₁₀₀, h_N100, h_S100, and h_NS100, relative to the baseline Δη of scenario s₀. Negative longitudinal distance indicates SSFB, positive is SPB and beyond.

Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0214.1

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Modifications to San Pablo Bay have similar local effects as in SSFB. With soft shorelines, the broad inundatable regions of San Pablo Bay and its adjacent marshlands become a greater sink of tidal energy and tidal amplitudes decrease. Hardening these shorelines leads to a minor increase in tidal amplitudes. In contrast to SSFB, though, hardening the shorelines of San Pablo Bay has a negligible effect on SSFB, as the progressive wave dynamics of the northern reach reflect little energy back to Central Bay.

Analytic approach for converging basin

Analyzing the whole of San Francisco Bay through the analytic lens of converging estuary hydraulics such as Savenije et al. (2008), van Rijn (2011), and Cai et al. (2012) is frustrated by the various branching, diverging, and reconverging features. Nonetheless, analytic approaches aid in identifying the dominant factors controlling the tidal response and can quickly predict the general response of a system without detailed observation or involved numerical approaches. Although the complex geometry of much of San Francisco Bay makes a large-scale application of analytic theory difficult, the central portion of SSFB has a smoothly convergent geometry. In this section, we apply the methods of Cai et al. (2012, hereafter CST) to this reach, between 30 and 55 km south of the Golden Gate (roughly the widest point of SSFB to the point at which the bay transitions to a broad slough), with a goal of understanding the predictive skill of the analytic model and its capacity to include inundation effects.

The formulations of CST include the following parameters in predicting the behavior of a basin: the length scale of the basin convergence a, the mean depth , the inverse of the Manning–Strickler friction coefficient K = n⁻¹, and the relative width of off-channel storage r_s. The ocean boundary condition is described by the tidal amplitude η₀ and angular frequency ω. These parameters are combined as

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such that c₀ is the effective celerity including off-channel storage, ζ is a nondimensional tidal amplitude, γ is an estuary shape factor relating convergence and tidal wavelength, and λ is the ratio of the frictionless celerity to the frictional celerity. The tidal response of the system is described in terms of the nondimensional numbers:

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where δ is the amplification factor, χ is the friction number, υ is the velocity scale, μ is the velocity number, and additionally ϵ = π/2 − ϕ describes the velocity phase lead. CST then derives the system of equations:

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the solution to which is a prediction of the tidal response. Application of CST to real bathymetry and shorelines, even with idealized tidal forcing, leaves considerable room for interpretation. While CST focuses on channels with a constant depth and converging width, the basis for the exponential convergence is in terms of the cross-sectional area, such as in Savenije (1992). In the present case, we have found that the cross-sectional area demonstrates a significantly smoother trend and bases the length scale a on this quantity rather than the width. In all cases, K was taken to be 36.9, based on the range of and corresponding roughness values in Table 1. We define the in-channel region as a contiguous set of cells on either side of the thalweg with a mean depth of at least 1.0 m and a tidal amplitude of at least 0.2 m. The across-channel distance between the bounds of this region then define the channel width (as a function of the along-channel distance), and similarly is defined as the cross-sectional area between these bounds. The storage ratio r_s is calculated as the ratio of the planform area of all tidally active cells (tidal amplitude greater than 0.2 m) to the planform area of all in-channel cells. Both the shallow intertidal margins of the main basin and all slough or pond features off the main channel are counted as off-axis storage. The open boundary tidal amplitude η₀ is taken from the model at a single cell along the thalweg. The mean depth at each cross section is the equivalent depth of a rectangular channel with area and width . While CST assumes a constant depth, SSFB has a trend of decreasing depth toward the head of the estuary, and is taken as the mean over all cross sections.

The input parameters for each scenario and a comparison of phase and amplification between the numerical and analytic models are shown in Table 3. The range of η₀ is fairly small, showing that, according to the model, dynamics seaward of this section account for about 0.05 m of change in tidal amplitude. The greatest differences in the inputs parameters are the variation of with sea level and the variation of r_s with inundated area. All scenarios fall within the amplified classification of CST, meaning that the net amplification is positive, and incremental increases in the depth would result in additional amplification. While the velocity phase lead is predicted well by CST, the amplification factor is uniformly underpredicted (e.g., Fig. 7). The results from the analytic model reinforce the idea of competing effects of deepening and inundation, where an increase in due to the sea level rise is partially offset by an increase in r_s due to inundation. In terms of the amplification, these competing effects are nearly in balance in the numerical model, while the analytic approach is more sensitive to deepening than inundation, suggesting that inundation is not fully captured by considering storage alone.

Table 3.

Application of method of Cai to numerical scenarios.

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Comparison of amplification factor δ in scenario h_NS100, as predicted by the model and by the analytic approach of Cai et al. (2012). The longitudinal transect is limited to the convergent portion of SSFB.

Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0214.1

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6. Overtides

The total change in high water between h_NS100 and s₁₀₀, at −60 km from the Golden Gate, is approximately 0.16 m (Fig. 6), but the M₂ amplitude explains only about 0.07 m of this difference. While the ocean boundary is forced only with an M₂ tide, local generation of overtides leads to nonnegligible M₄ amplitudes within the domain, shown in Figs. 8a–c. Previous analysis of the nonlinearities in the shallow water equations (Parker 1991) has shown that M₄ overtides are predominantly generated by the depth dependence of the friction term, depth dependence in continuity, and the nonlinear advection term. The depth-dependent generation mechanisms are likely significant throughout much of San Francisco Bay, given the O(1 m) tides and O(2 m) depths prevalent in shoals throughout the domain. Though the mean M₄ amplitude is small (up to about 0.1 m), the differences across scenarios of the M₄ amplitude is of the same order as the differences in M₂ amplitudes. In addition to varying amplitudes, the distribution of M₄ generation and the resulting phase relationships between the M₂ and M₄ vary greatly between scenarios.

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M₄ amplitude for scenarios (a) h_NS0, (b) h_NS100, and (c) s₁₀₀. M₄ generation in 10⁶ W, integrated by region, for scenarios (d) h_NS0, (e) h_NS100, and (f) s₁₀₀. Arrows in (d)–(f) show the direction of M₄ energy flux between regions.

Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0214.1

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The panels of Figs. 8d–f show that in all cases Central Bay is a significant source of M₄, but areas in the north and south may be sources or sinks of M₄ depending on the scenario. In all cases, M₄ appears to be generated in shallow, off-axis portions of the domain, and in most cases propagates seaward. Variation in M₄ generation appears to be driven by three factors: change in mean depth (i.e., h_NS0 vs h_NS100 or s₁₀₀), change in M₂ amplitude from which M₄ can be extracted, and local dissipation of the M₄. The M₄ dynamics are further complicated by the shorter wavelength that allows for standing wave nodes to exist within the basins, such as in h_NS0 at the widest point of SSFB and in the middle of San Pablo Bay.

Taking into account the amplitude as well as the phase relative to the M₂ phase, we estimate that in h_NS100 the M₄ adds roughly 0.04 m to high water in most of SSFB, compared to s₁₀₀ in which M₄ actually decreases high water by up to 0.04 m. Of the original h_NS100 − s₁₀₀ difference of 0.16 m (at −60 km), the combined M₂/M₄ wave then accounts for roughly 0.15 m.

7. Discussion

Within a particular estuary or bay, the dominant factors controlling the tidal and inundation response to sea level rise include geometric factors like aspect ratio, the baseline phasing of the tidal wave, and the spatial distribution of inundated areas.

The aspect ratio determines the relative importance of longitudinal versus lateral variation. In the longer, high aspect ratio SSFB, changes due to sea level rise were relatively consistent across lateral transects, and lateral dynamics appeared secondary. In contrast, San Pablo Bay, with a round, low aspect ratio footprint, showed significant shifts of tidal propagation from the channel to shoals.

Tidal phasing is important both in terms of local tidal amplification and how much the tides in one part of a basin feedback to other parts of the system. A standing wave system such as SSFB appears more sensitive to sea level rise, in both the case of deepening only and deepening with inundation. Additionally, standing wave systems tend to have greater tidal range such that even small changes in phasing or dissipation lead to large changes in energy flux and net amplification.

The quantity and relative location of inundatable areas also affects the tidal response. Greater expanses of inundatable areas relative to the subtidal area lead to greater attenuation of the incident tidal wave. The location of inundatable areas, along with the tidal phasing within a bay, affects the spatial extent of the attenuation due to inundation. In a purely progressive wave system, these effects are limited areas landward of the inundation/attenuation. In a reflective, standing wave system, though, inundation even at the head of the estuary can attenuate the tidal range throughout the bay and even in adjacent tidal basins.

In addition to the incoming tidal wave constituents, overtides generated within a basin may add to or subtract from the high water elevation and appear to be very sensitive to shoreline conditions and incident tidal wave amplitudes. Depth-dependent M₄ generation mechanisms are of particular interest in sea level rise scenarios as the change in mean sea level can drastically change overtides in shallow basins. Large tidal ranges and shallow depths at the head of an estuary can generate seaward-propagating overtides. With the complexity of a seaward-propagating M₄ combined with a landward-propagating M₂, along with the potential for M₄ resonance, modulation of overtides by sea level rise is a nonobvious but important aspect of predicting peak sea level within tidal basins.

The net physical response to coastal sea level rise clearly depends on a broad set of factors. We have considered only the M₂ forcing, but diurnal tides and interactions between diurnal and semidiurnal tides are likely significant. The long wavelength of diurnal tides leads to phasing closer to a standing wave, though net amplification is typically smaller at longer wavelengths (e.g., the analysis of section 5a when applied to the s₀ scenario yields 24% less amplification when the tidal period is doubled). Similarly, we expect that the dissipative effects of inundation are also less important for diurnal tides.

Perhaps the largest uncertainty in predicting what will happen in a particular estuary is the unknown evolution of morphology, whether by natural or managed actions. Understanding how basins respond to sea level rise when morphology is kept static is the first step toward understanding what natural changes are likely to occur and what management decisions may be deemed necessary.

8. Conclusions

Utilizing numerical experiments with a variable coastal sea level rise and varying shoreline configurations, we approximately separated the effects of deepening from inundation. Comparisons of phase and energy flux of the M₂ tidal wave show that deepening decreases the influence of friction, while inundation adds considerable dissipation in the perimeter areas.

Deepening allows additional tidal amplification [consistent with an amplified estuary in the parlance of Cai et al. (2012)], which was observed in all hard shoreline cases with sea level rise. The long, convergent southern arm becomes more reflective when deepened, while the shorter, transmissive northern arm shows the landward energy flux shifting from the channel to the shoals.

In both branches of the bay, inundation introduced large energy sinks at the bay margins. Most inundation occurred off perimeter sloughs and rivers, causing the most drastic changes in tidal phasing and tidal prism at points where these features join the larger bays. Energy sinks in newly inundated regions caused a progressive shift in tidal phasing, a decrease in tidal amplification, and an increase in the landward tidal energy flux.

In the case of SSFB, local changes in the shoreline alter both the tidal range within the basin and also the magnitude of the reflected wave. The reflected wave subsequently affects tidal range in other parts of the domain. In contrast, local changes in the shoreline of San Pablo Bay have limited effects on tidal range downstream of San Pablo Bay, because a smaller fraction of tidal energy is reflected.

A one-dimensional analytic model has been applied to a reach of SSFB, with moderate success. Predictions of amplification from this approach qualitatively agree with model output, though the hydrodynamic model shows a much greater effect of inundation than is captured in the analytic model. Other parts of the domain are likely too irregular and two-dimensional to be reasonably treated with a one-dimensional analytic model.

While the M₂ amplitude is much larger than the amplitudes of overtides, the variation in M₄ amplitude across numerical experiments is comparable to the variation in M₂ amplitude. Together with variation in the phase of the M₄ wave relative to the M₂, we conclude that overtides are an important component of the variation in high water. Depth-dependent nonlinearities in the shallow water equations are the most likely M₄ sources, consistent with extensive shoals in which the depth is of the same order as the tidal range.

Overall, the coupling between sea level rise, tidal amplification, and inundation is important and must be taken into account for accurate assessment of future restoration and mitigation questions. In many estuaries and bays, rising sea level in the coastal ocean will lead to newly inundated areas. To a degree this inundation can mitigate sea level rise by decreasing tidal amplification within the basins. Reinforcing and hardening impacted shorelines can increase flood risks in adjacent areas, and in highly reflective basins the effects can be far reaching. Restoration of tidal marshland and construction of new low-lying tidal areas offer significant protection from rising tides by dissipating incident tidal energy, and these benefits may extend well beyond the areas directly sheltered by marshland.