Bathymetric measurements show that a deep, subtidal channel in San Pablo Bay, California, has consistently narrowed during the past 150 years. This raises general questions on the seasonal and intertidal morphodynamic processes acting at the subtidal channel-shoal interface. The current work addresses these questions using a process-based morphodynamic model (Delft3D). Model results reveal considerable morphodynamic activity during a tidal cycle. Deposition on the channel margin is largest during flooding of the shoals. Erosion rates (mainly occuring during ebb) remain relatively small, so that net accretion occurs on much of the channel margin. A remarkable finding is that locally generated wind waves are responsible for shoal extension and depositional channel narrowing. High suspended sediment concentration (SSC) in the channel is a critical factor. Apart from sediment supply during high river flow, wind waves suspending sediment on the shoals cause high SSC levels in the channel at ebb. Sensitivity analysis shows that wind direction even determines the location of channel margin accretion. Fluvial sediment supply is another cause of high SSC in the channel. Density currents, 3-D circulation flows, sea level rise, or varied sediment characteristics only have a limited effect on the erosion and sedimentation patterns. A 30 year forecast shows that deeper shoals and decreasing fluvial sediment supply lower SSC levels in the channel, limit channel margin accretion, and even lead to net channel margin erosion in some areas. Channel shape thus remains subject to dynamic processes related to local variations in sediment supply, albeit to a more limited extent than in earlier decades.
Estuarine channel-shoal systems are of economic and ecologic interest. Channels provide natural, deep water access to ports, whereas mudflats often form a feeding ground for migratory birds and may evolve into vegetated salt marshes with unique ecosystems [Allen, 2000]. It is important to understand the morphodynamics in estuaries to develop tools that are able to predict the impact of anthropogenic forcing (e.g., channel dredging, variations in sediment supply by upstream mining or dam construction, or sea level rise) on future morphological developments.
The morphology of alluvial estuaries is characterized by channels and shoals [Perillo, 1995; Hibma et al., 2004a, 2004b]. Tidal channels convey a large part of the tidal flows. Because of the high prevailing velocities, the channel bed generally consists of sandy material. In contrast, shoals act mainly as water storage at higher water levels. Velocities on shoals typically are low, and as a result, shoals are often covered by fine, muddy sediment [Pethick, 1984]. Shoals may eventually evolve to intertidal flats or vegetated salt marshes depending on forcing conditions [Perillo and Iribarne, 2003; Friedrichs, 2011]. Estuarine morphology may seem to be in equilibrium on a short time scale but can evolve on a decadal time scale due to land reclamation works [Elias et al., 2006], natural evolution of the basin, or slowly changing forcing conditions such as sediment supply [Barnard et al., 2013] or sea level rise [Pethick, 1993; Van Goor et al., 2003].
Sequentially measured bathymetries provide an important tool for investigating decadal morphodynamic developments [Pye and Van der Wal, 2000a, 2000b; Jaffe et al., 2007]. The longest records are generally in man-made and maintained access channels of ports [see for example Lane, 2004] and do not document natural channel development. Measurements typically are taken at irregular intervals and focus on the deepest parts of the channels, further limiting their use in a thorough analysis of channel evolution.
Reynolds  was among the first to study pattern development in a tidal environment in flume experiments. Recent literature addresses the generation and evolution of channel-shoal patterns under highly schematized conditions in laboratory experiments [among others Tambroni et al., 2005; Stefanon et al., 2012] or by using mathematical approaches [among others Schuttelaars and De Swart, 1999; Seminara and Tubino, 2001; Marciano et al., 2005; Van der Wegen and Roelvink, 2008, 2012]. However, these studies are limited because they use a single grain size, typically sand, and do not account for more complex conditions typically observed in nature such as graded sediments ranging from mud to sand, salt-fresh water density currents, or wind waves.
Many schematized 1-D modeling studies have applied a predefined channel width or cross-sectional shape without allowing for morphodynamic adjustment. Partly, this is because detailed knowledge of the dominant hydrodynamic and sediment transport processes at the channel-shoal interface is lacking. Several studies have included channel-shoal interaction in different hydraulic environments, such as levee development along river channels during floods [Brierley et al., 1997; Adams et al., 2004; Rowland et al., 2009], deep water levee-type features along channels in alluvial ocean fans developed during high-turbidity flow events [Piper and Normark, 1983; Normark et al., 2002; Fildani et al., 2006; Straub and Mohrig, 2008], and levee development along tidal creek systems in salt marshes and mudflats [Wells et al., 1990; Perillo and Iribarne, 2003; Temmerman et al., 2005]. Most of these papers presented measured developments and provided hypothetical explanations of their origin. Fildani et al. , Straub and Mohrig , and Temmerman et al.  used process-based, morphodynamic modeling to explain morphological development.
Brand et al.  and Mariotti and Fagherazzi  investigated the exchange of sediment at the channel-shoal interface using a correlation analysis between measured velocities, water levels, wind waves, and suspended sediment concentrations. Ralston et al.  expanded on earlier studies by including the impact of measured salinity gradients on modeled bathymetric variation across the main channel and adjacent shoals during several spring-neap tidal cycles. Focusing on a much longer time scale, Ganju et al. , Ganju and Schoellhamer , Van der Wegen et al. , and Van der Wegen and Jaffe [2013a] used coupled hydrodynamic/sediment transport models that were able to reproduce observed decadal morphodynamic developments in a (partially) submerged, muddy environment. Van der Wegen and Jaffe [2013a] further indicated significant model skill both for a 30 year period of net deposition and net erosion. Uncertainty of model input parameters only had a limited impact on model skill and volume change.
However, none of these studies explicitly identified the processes and conditions at an intertidal time scale governing morphodynamic developments at a submerged channel-shoal interface. Furthermore, a detailed analysis of the (potential) impact of wind waves on tidal channel-shoal morphodynamics was lacking.
1.1 Aim and Methodology
The aim of the current work is to investigate the processes and conditions responsible for morphodynamic evolution on the tidal channel-shoal interface with special emphasis on the impact of wind waves. By means of a process-based model, our study will thus explore mechanisms that control channel morphodynamics on a time scale range of minutes to decades. The study area is San Pablo Bay, California.
The reason we selected San Pablo Bay is that the pronounced narrowing of its main channel is documented by a large and unique bathymetric data set collected over the past 150 years at 30 year intervals [Jaffe et al., 2007]. In addition, San Pablo Bay has a relatively simple bathymetry with a single major channel whose position is fixed by rocky outcrops at its landward and seaward ends. Measured developments are thus not the consequence of channel migration or redistribution of the tidal prism over adjacent, emerging channels. This study will focus on an analysis of the erosional period with a prospect on future changes, since these are typical conditions currently experienced in San Pablo Bay.
We hypothesize that the processes responsible for the decadal-scale channel narrowing and deepening occur at time scales of a (semidiurnal) tidal cycle or the few hours of a wind event. To test this hypothesis, we would need a model that is able to address time scales that range from minutes to several years (which is about 6 orders of magnitude). We apply a numerical 3-D process-based model (Delft3D) that, by its typical minute-scale time step, allows for a detailed analysis of the intertidal conditions that may lead to channel narrowing and deepening while runs may cover annual to decadal time scales reflecting changes in channel-shoal geomorphology. In this paper, we first investigate the sensitivity of the model results to different forcing conditions and process descriptions. We then use the model as a virtual laboratory to investigate the intertidal conditions responsible for morphodynamic development at the channel-shoal interface.
1.2 Description of Study Area
San Pablo Bay, a subembayment in the northern San Francisco Estuary (Figure 1), is circular with an area of about 270 km2 and an average tidal range of about 1.5 m. It is rather shallow (average depth of 3.7 m, two thirds of the Bay is less than 2 m depth at mean lower low water) and muddy apart from the main sandy channel, which is from 11 to 24 m deep (average 12 m), that transects the Bay from east to west conveying tidal flows back and forth and the river flow seaward. To the southwest, the Bay connects to Central Bay, which in turn is connected to the Pacific Ocean via the Golden Gate. To the east, the Bay connects to Suisun Bay via the Carquinez Strait. Suisun Bay is connected to the inland delta of the Sacramento and San Joaquin Rivers, the area referred to as the Delta. River flow has a pronounced seasonal signal with peak discharges as great as 18,000 m3/s during winter and spring and much lower discharge (200–500 m3/s) during the rest of the year. Flow hydrographs also vary considerably from year to year. The Sacramento River accounts for 70–80% of the total flow. Kimmerer  and references therein provide a more detailed overview of the physical and biological characteristics of the Bay-Delta system.
Hayes et al.  reported a predominant wind direction from the west, although the main direction changes to the east during winter months. The wind has an important diurnal variation with maxima around noon and limited wind activity during the night [Schoellhamer et al., 2008]. For the neighboring Suisun Bay, Ganju et al.  found, apart from the diurnal signal, significant weekly and seasonally fluctuating wind amplitudes, with summer wind speeds being twice as high as those in winter. The local wind initiates wind waves that resuspend sediment on the shoals. Schoellhamer et al.  reported waves in San Pablo Bay that had a maximum height of 0.6 m with a period of 4 s from March to May 2006 when daily averaged wind speeds were 6 m/s and gusts were up to 10 m/s.
Sediment loads and morphodynamics of the Bay-Delta system have changed dramatically in the past 150 years [Gilbert, 1917; Porterfield, 1980; Cappiella et al., 1999; Foxgrover et al., 2004; Jaffe et al., 1998, 2007; Fregoso et al., 2008; Barnard et al., 2013]. Hydraulic mining during the Gold Rush in the midnineteenth century created more than a billion cubic meters of tailings in the Sacramento River watershed, much of which moved into the Delta and San Francisco Estuary. At the end of the nineteenth century, hydraulic mining was stopped, and reservoir dam construction took place in the watershed. As a result, sediment loads decreased considerably. Wright and Schoellhamer  report a decrease in measured sediment load of about 50% in the Sacramento River during the period 1957–2004.
A series of six bathymetric surveys conducted from the 1850s to 1980s at ~30 year intervals documents large morphologic changes in the San Francisco Estuary. In San Pablo Bay, deposition of more than a quarter billion cubic meters of hydraulic gold mining debris reduced the average depth by 85 cm in the mid- to late 1800 s. In the late 1900s, the intertidal flats attached to the Bay's edges narrowed, and both the submerged shoals and the main channel in the Bay deepened as more sediment was lost to the sea than entered from rivers. Locally generated wind waves are probably responsible for the mudflat erosion [Krone, 1979; Schoellhamer et al., 2008; Van der Wegen et al., 2011; Bever and MacWilliams, 2013]. Processes of sediment redistribution caused the main channel to become narrower as well, a trend observed over the last 150 years [Jaffe et al., 1998, 2007, Figure 2]. This trend is remarkable since it is persistent throughout periods of net deposition and erosion.
Jaffe et al.  reported minor dredging activities on the northern shoal to maintain the channel toward Petaluma River before 1922. An area of dredging in the San Pablo Bay (SPB) main channel after 1922 is delineated by the narrowest section of the 9.14 m (30 ft) contour lines (Figure 2). Between 1955 and 1983, an average of about 0.2 × 106 m3/yr was dredged from the main channel near Pinole Point and released in SPB [Ogden Beeman and Associates and Ray Krone and Associates, 1992].
1.3 Model Setup
Our study applied the Delft3D software [Lesser et al., 2004; Deltares, 2013]. Van der Wegen et al.  and Van der Wegen and Jaffe [2013a] give details on the San Pablo Bay model setup. Here we present a brief summary of how we applied the model in this study. We distinguish (1) a period of net deposition (1856–1887), (2) a period of net erosion (1951–1983), and (3) a period with unknown “future” geomorphic change (1983–2013).
Figure 3 shows the curvilinear model grid with grid size varying from 150 to 800 m. For 3-D runs, we used 15 vertical sigma layers, which proportionally vary in height according to water level and bed level variations. Just west of Napa River, a breakwater was included by allowing no transport across the grid line following the northern channel bank for 2 km. The model is driven by boundary conditions derived from a larger model covering the full Bay-Delta area and includes salinity gradients. The hydrodynamic forcing at the seaward (tidal water level, salt water density 1025 kg/m3) and landward (tidally varying discharge, freshwater density 1000 kg/m3) boundaries was schematized by considering the major tidal constituents M2, M4, K1, and O1 only. Based on the principle of similar tide residual sediment transports and following Lesser  and Hoitink et al. , the K1 and O1 components were merged into a single artificial C1 constituent to create a regular diurnal forcing. The yearly varying river discharge is schematized, at the landward boundary, as 1 month of high river flow (HRF) at 5000 m3/s followed by 11 months of low river flow (LRF) at 350 m3/s. For the erosional and future periods, the HRF was set at 2100 m3/s to account for an expected lower river flow regime due to upstream reservoir construction [Enright and Culberson, 2010].
Sediment transport was modeled by a 3-D advection-diffusion equation with expressions for erosion and deposition as source terms [Deltares, 2013]. Fall velocities and formulations for the erosion and deposition rates depend on sediment size. The transport of cohesive mud is modeled by the Partheniades–Krone formulations [Krone, 1962, 1993; Ariathurai,1974; Winterwerp and Van Kesteren, 2004]. Because the majority of the modeled morphodynamic developments involve mud, we here explicitly present the mud transport formulations implemented in Delft3D
in which E is the erosional flux in kg/m2/s, M is the erosion parameter in kg/m2/s, D is the depositional flux in kg/m2/s, ws is the sediment fall velocity in m/s, cb is the near-bottom concentration in kg/m3, τcw is the maximum shear stress due to combined waves and currents in N/m2, τcr,e is the critical shear stress for erosion in N/m2, and
For the transport of noncohesive sediment (sand), we used the Van Rijn's  approach, which is based on the concept of an equilibrium concentration for stationary flow conditions. This equilibrium concentration was applied at the boundaries. We applied three sand fractions and five (three) mud fractions for the depositional (erosional and future) periods. The reason for the different number of mud fractions was that the two finest fractions in the depositional period runs did not contribute significantly to the morphodynamic developments and thus were omitted in the erosional period runs. In addition, we lowered the erosion coefficient (M) for the erosional period to account for mud consolidation in the preceding decades. Following the approach of Ganju et al. , the landward mud concentration was set at 300 mg/L for the depositional period and 30 mg/L for the erosional and future periods. At the seaward side of San Pablo Bay, the sediment concentration was set to zero, although the model applied a “Thatcher–Harleman” time lag of 120 min. This implies that sediment concentration entering the model domain during flood is determined during this time lag by the concentration leaving the model domain during ebb using concentration data from an earlier time. The initial distribution of the different sediment classes over the model domain was determined following the methodology decribed by Van der Wegen et al. , where the model is allowed to redistribute all size classes and results in sand classes in the channel and mud classes on the shoals and channel margins.
The wind schematization was based on the seasonal variations in wind field following Hayes et al.  and additional analysis of wind data from http://wwwcimis.water.ca.gov/cimis. Sensitivity analysis on wind conditions showed that model results are sensitive to variations in wind velocity and direction. We varied the wind schematization to optimize morphodynamic model results, which led to a slightly different wind schematization for HRF and LRF forcings as well as for the depositional and erosional periods. The former difference in schematization may be justified by changing the observed seasonal conditions (see section 1.2), but literature provides no indication of a changed historic wind climate. The latter different wind schematization thus results from using the wind field as calibration parameter.
Wind was uniformly distributed over the model domain with a diurnal sinusoidal signal varying from 0 at midnight to 7 (9) m/s at noon for the depositional (erosional and future) period. For the 1 month HRF forcing, wind direction was from the west (southwest) for the depositional (erosional and future) period. For the LRF forcing, the wind during the first and final half month came from the west (southwest), and for the remaining months, the wind came from the southeast for the depositional (erosional and future) period.
Wind drives the simulating waves nearshore (SWAN) wave model (http://www.swan.tudelft.nl/) that adds wave-induced shear stress to the shear stress developed by the tidal flow. The SWAN settings included (among others) wind growth, whitecapping [Komen et al., 1994], depth-induced breaking [Battjes and Janssen, 1978], and refraction.
Morphodynamic development was enhanced following the approach of Roelvink  by introducing a morphological factor (MF) which multiplied bed level changes every hydrodynamic time step, which was 1 min in the current model. Roelvink  suggested that this is valid as long as bed level changes per time step remain small compared to the water depth. For example, model results in Figure 4 were obtained by subsequently running a 0.5 month period with HRF boundary conditions with a MF of 80 and a (little more than) 2 month period with LRF conditions with a MF of 160. The smaller morphological developments per time step during LRF (with MF = 1) allowed for a higher LRF MF. The modeled bed level changes add up to 32 years of morphodynamic development. The results we present do not reflect the modeled morphodynamic development of 30 successive years. Rather, the modeled morphodynamic developments are the summation of enhanced developments during one HRF condition followed by one LRF condition. Modeling 30 successive years would take approximately 150 days on a 3.0 Ghz, 3.25 Gb RAM, quad core PC, whereas typical computation time for a “schematized” run was 1 day.
Table 1 summarizes the applied model parameter values for the different periods. The different settings between the depositional and erosional periods may be partly explained by altered conditions (lower river discharge and mud consolidation leading to higher critical shear stress and erosion coefficient for the erosioanl period) and partly by the sensitivity analysis carried out for each of both periods to gain the best comparison to measured developments (roughness values, wind amplitude, and direction); see Van der Wegen and Jaffe [2013a] for further details.
Table 1. Overview of Model Parameter Settings
1951–1983 and 1983–2013
Sand fraction diameter (µm), D
Lateral bed slope factor, αbn
Inflow concentrations (mg/L), c
Critical shear stress (N/m2), τcr
Erosion coefficient (kg/m2/s), E
1 × 10−4
5 × 10−5
Sediment fall velocity (mm/s), w
Horizontal eddy diffusivity (m2/s)
(s/m1/3) Manning, n
(m1/2/s) Chézy, C
River discharge (m3/s), Qr
West and southeast
Southwest and southeast
Hydrodynamic run period (months)
Morphological factor (−)
2 Model Results
We applied two approaches to increase our understanding of channel-shoal morphological change. The first approach is to use the model to assess the impact of varying forcing conditions and process descriptions on the net erosion and deposition patterns. The second approach is to analyze the model results of the intertidal processes at the channel-shoal interface.
2.1 Erosion/Deposition Pattern Sensitivity to Varying Forcing Conditions
Figure 4 shows the main modeling results earlier described by Van der Wegen et al. [2010, 2011] and Van der Wegen and Jaffe [2013a]. Appendix A in the supporting information shows the distribution of the different sediment fractions during the model runs. Erosion and deposition patterns are reproduced with significant skill, although model results are less accurate and more sensitive to model input variations in the erosional period. The depositional (erosional) period reaches a Brier skill score of 0.35 (0.2) which is considered reasonable to good in area modeling studies (see Sutherland et al.  for more details on the Brier skill score). Channel narrowing is reproduced during all three periods. Observed channel deepening is reproduced by the model in the erosional period but not in the depositional period (compare Figure 2b and Figure 4). The pronounced difference between the depositional and erosional period is attributed to a significantly decreased supply of sediment defined by a much lower suspended sediment concentration (SSC) [Ganju et al., 2008] and a 20% lower river flow at the landward boundary for the erosional period [Enright and Culberson, 2010].
The model clearly reproduces channel margin accretion on the northern margin for all periods, but deposition on the southern margin is underestimated. Van der Wegen and Jaffe [2013a] attribute this underestimation to the schematized wind and wave climate, of which the uncertainty is relatively high given the limited (historical) wind and wave measurements. Still, we consider the model skill convincing and the modeled channel narrowing pronounced enough to analyze the underlying processes in more detail in the current study.
Figure 5 presents the modeled erosion and deposition patterns for the depositional (Figures 5a–5c) and erosional (Figures 5d–5f) periods isolating development during HRF forcing (Figures 5a and 5d) and LRF forcing (Figures 5b and 5e), and the combined effect of both the HRF and LRF forcing (Figures 5c and 5f). All significant deposition (in the channel, on the channel margins, and on the shoals) is mud. Sand does not enter the shoal domain. At the end of LRF forcing, most of the deepest parts of the channel are covered by sand again. An important observation is that the eastern channel margins accrete during HRF forcing, which is attributed to sediment supply from the Delta watershed. Deposition during HRF forcing is most pronounced for the depositional period (Figures 5a and 5d) because of the larger sediment supply. During LRF forcing (Figures 5b and 5e), some parts of the channel margins erode again, and large parts of the channel deepen. The most significant morphodynamic development in the erosional period occurs during LRF forcing when the northern shoals erode and the northern margins accrete. The margin accretion occurs further west than during HRF forcing (Figures 5d and 5e).
To explore governing intertidal conditions that cause the erosion and deposition patterns, we performed further sensitivity analysis for the erosional period; see Figures 6a–6i. Figures 6b–6e show that the erosion and deposition patterns are not fundamentally different for different forcing conditions. Neglecting diffusive transport by putting a very low value for the diffusion coefficient in the advection-diffusion equation for sediment transport does not lead to a significantly different pattern (not shown). There is little deposition on the channel margins for a bed composed entirely of sand (not shown). The sensitivity analysis suggests that a major factor influencing channel margin accretion is a high SSC in the channel. A bed composed entirely of mud leads to greater erosion of the channel bed and higher channel SSC that results in greater margin accretion (Figure 6f). When waves are excluded (Figure 6g), the northern shoals do not erode, and channel accretion does not take place. A possible explanation is that sediment suspended by wave action on the northern shoals is taken by ebb currents seaward into the channel, whereas subsequent flood conditions with a high SSC deposit the material on the channel margins.
2.2 Intertidal Processes on the Channel-Shoal Interface
This section explores the hydrodynamic conditions governing the observed channel narrowing in more detail. Figure 7a shows the bed level at different modeled and measured points in time along the cross section indicated by the eastern red line in Figure 5f. We selected six points of interest covering the channel margin (two points) and four adjacent locations in the channel and on the mud flat. Figure 7b shows bed level changes at these six locations. The elongated tidal period reflected in Figure 7b is the result of multiplying the diurnal tidal signal by the morphological factor. One can clearly distinguish short-term intertidal variations from a long-term net development. It takes more than a year (or more than 4.5 hydrodynamic days) before development takes place during HRF forcing because the high sediment load defined at the landward boundary needs time to arrive in San Pablo Bay. More significant developments take place during subsequent LRF forcing with net erosion of the shoals and part of the channel and significant accretion of the margins and (to a lesser extend) the deepest part of the channel. The abrupt change at the end of the LRF forcing is due to a change in wind direction.
To determine the cause for geomorphic change, we decomposed the flow velocity into along-channel and transverse-channel components (perpendicular to each other). These directions correspond to the grid orientation, which roughly follows the orientation of the channel. Flood is positive landward for along-channel velocities and northward for transverse-channel velocities.
Figure 8 shows intertidal hydrodynamic conditions at the locations indicated in Figure 7a. Velocities are depth averaged. Water level and maximum along-channel velocity are 0.1–0.2 m and 0.1–0.2 m/s higher during HRF than LRF forcing (Figures 8a and 8b). Along-channel velocities are higher in the channel than on the shoal and remain largely directed seaward during HRF forcing because of the river flow. The river flow not only has a direct impact by adding a net seaward flow but also an indirect impact by changing the tidal wave characteristics. Figures 8c and 8d show that transverse velocities are low in the channel and increase considerably toward the margins and shoal and become even greater than the along-channel flow. This suggests that, at this location, shoal flooding is caused by transverse flow.
Depth averaged SSC levels are highly variable during the tidal cycle and between HRF and LRF forcing (Figures 8e and 8f). Peak SSC levels occur just after the along-channel ebb flow under HRF forcing (Figure 8e, around 14 h) as the result of local mud resuspension and river supply. The highest SSC on the shoals occurs during flooding for HRF forcing (Figure 8e, between 17 and 20 h) and during ebb flow for LRF forcing (Figure 8f, between 10 and 15 h). The relatively low SSC in the channel during LRF forcing is because mud has been washed out (see Appendix A and Figure A2 in the supporting information).
For HRF forcing, the majority of shoal deposition (Figure 8g, red lines, 19–21 h) occurs during flood tide when both the along-channel and transverse velocities decelerate. The channel margin (black lines) and the channel (blue lines) reflect similar but more limited deposition during that period. The majority of the deposition there takes place during ebb tide as the along-channel flow decelerates and transverse velocities are low (Figure 8g, 2.5–5 h and 14–17 h). Channel erosion takes place around peak along-channel ebb flows (13–14 h) and peak along-channel flood flows (17–20 h). For LRF forcing, the majority of the channel deposition (Figure 8h, blue lines) occurs at relatively low-along-channel velocities around slack water (2–4 h) and (7–12 h). Channel erosion takes place during relatively high-along-channel flood flow (5–7 h and 17–19 h) and high-ebb flow (24–26 h). Shoal erosion (Figure 8h, red lines, 10–13 h) occurs during high transverse ebb velocities. Generally, the relationship between the hydrodynamic conditions and the bed level development is less evident than during HRF forcing.
Explaining bed level development by local hydrodynamic conditions only is difficult. Mud deposition rates are a function of the fall velocity and SSC and do not instantly react to decreasing flow velocities (settling lag). The finer the material, the larger the lag and its influence on geomorphic development. Mud erosion only occurs when a critical threshold of the shear stress is exceeded (scour lag). Additionally, intertidal developments make a detailed analysis even more complex. For example, locations may significantly erode and accrete during different phases of the tidal cycle, but small net (tide residual) bed level changes determine longer-term morphological development. By further categorizing hydrodynamic conditions, we may obtain a better understanding of the governing processes.
Figure 9 gives a possible explanation for the generation of transverse velocities. Similar to measurements [Kimmerer, 2004], the modeled tidal wave in San Pablo Bay is a mixed progressive/standing wave with water levels lagging behind velocities about 2 h (compare blue and cyan lines in Figure 9 between 11 and 14 h). Flood flow in the channel is already decelerating when the maximum water level is reached. Another important observation is that shoal water levels lag behind the water levels in the channel by about 15 min because of a water level difference that is about 15 cm over 8 km during the highest transverse velocities (gray area in Figure 9 around 8–9 h). Water level gradients and transverse flow are nearly in phase and change sign/direction just before high and low water. The water level difference between the channel and the shoal is the result of relatively high friction on the shoals. This analysis leads to the following conceptual model for flow patterns (subsequent periods are given by roman letters indicated in Figure 9 and Figure 10). As the water level in the channel starts to rise after low water, a transverse water level gradient develops that induces transverse flooding of the shoals even before flood occurs in the main channel (period I). When flood flow enters San Pablo Bay from the southwest, “overshooting” of the channel banks occurs because of the curvature of the main channel, and transverse velocities are enhanced (period II). Starting at about 12 h, the channel water level becomes lower than the shoal water level, and the transverse channel-shoal water level gradient shifts inducing draining of the shoals. The shoals continue draining from 12 to 15 h, and the channel flow shifts from flood to ebb at 15 h (period III). Ebb occurs in the channel while the shoals drain (period IV). Just after maximum channel ebb velocities, the channel water level exceeds the shoal water level again leading to transverse flooding of the shoal during ebb channel flow (like in period I).
We can now systematically relate the governing hydrodynamic conditions and processes of deposition and erosion at different locations. After combining the model results of both HRF and LRF forcings, we calculated 10 min interval erosion or deposition levels according to the hydrodynamic conditions (periods I–IV) and added the contributions for different locations. We define “deposition" (“erosion”) as an increase (decrease) in bed level during a 10 min interval. “Accretion” or “net erosion” refers to the net result. Figure 11 shows the results along the western (Figures 11a, 11c, and 11e) and eastern (Figures 11b, 11d, and 11f) cross sections. As already observed (Figure 7b), the intertidal deposition (Figures 11a and 11b) and erosion (Figures 11c and 11d) levels are large compared to the net (Figures 11e and 11f) morphodynamic development. Net bed level changes may be the result of a high deposition but also due to low erosion rates. For example, the low erosion level at the upper channel margin (Figure 11d) probably causes the net margin accretion at the higher shoal in the eastern cross section (Figure 11f). Another important observation is that all conditions contribute significantly to deposition and erosion, except for the erosion in the western cross section (Figure 11c) where out-of-phase conditions (periods I and III) are absent.
However, conditions of shoal flooding (adding periods I and II, solid lines in Figure 11) and channel flood (adding periods II and III, gray lines in Figure 11) are responsible for the majority of deposition on the higher margin and shoals (Figures 11a and 11b), whereas these flooding conditions contribute to a minor part of the total erosion (Figures 11c and 11d). The small difference between the gray and black lines in (Figures 11a and 11b) suggests that channel flooding and shoal flooding are in phase for the western cross section. The pronounced difference between the dashed and solid lines in (Figure 11 b) indicates that deposition in the channel and lower margin also occurs during a combination of channel flood and ebbing of the shoals in the eastern cross section.
In summary, the intertidal deposition and erosion levels are large compared to the net morphodynamic development. All hydrodynamic conditions (i.e., shoal and channel ebbing or flooding) contribute to significant erosion and deposition. Still, conditions of shoal flooding and channel flood are responsible for the majority of deposition on the higher margin and shoals.
3.1 Conditions for Channel Margin Accretion
In the previous sections, we analyzed processes responsible for the channel margin accretion observed in San Pablo Bay. Our model reveals a clear seasonal signal (HRF versus LRF conditions), but we could not define clear governing intertidal hydrodynamic conditions. Unlike river levee formation taking place during flooding of the river forelands, both ebbing and flooding in the channel or on the shoal can cause deposition. Still, model results reveal that the most significant deposition on the channel margins occurs during conditions of along-channel flood and flooding of the shoals. Furthermore, net (tide residual) accretion may be the result of locally limited erosion and not only high deposition rates.
Our analysis shows that high SSCs are the main parameter responsible for channel margin accretion. High SSC levels may result from river supply, but also from wave-induced resuspension on shoals attached to the channel. The eastern portion of the San Pablo Bay channel margins primarily accretes during HRF forcing when sediment entering San Pablo Bay from Carquinez Strait increases suspended sediment concentrations in the eastern part of the channel significantly (tidally averaged value of 80 mg/L with peaks up to 180 mg/L). This imported suspended sediment mainly settles in the eastern channel region and barely reach the western part of the bay where the SSC remains low (on average 20 mg/L). In contrast, the western part of the northern channel margin and shoals primarily accrete during LRF forcing while wind waves are present. Wind waves resuspend mud on most of the northwestern shoals. Ebb flows transport the mud into the channel where it causes high SSC levels (tidally averaged value of 85 mg/L with peaks up to 190 mg/L) and partly deposits on the bed. Subsequent flood flow resuspends the mud from the channel bed and transports high SSC water toward the western shoals even reaching the eastern part of the bay where it covers part of the imported mud.
3.2 Importance of Tidal Variations and Timing of (Extreme) Wind Events
Investigating sediment dynamics in San Pablo Bay with a similar type of model, as was applied in the current study, Bever and MacWilliams  confirmed measurements by Schoellhamer et al.  that wind waves are the primary source of sediment resuspension on the shoals. Bever and MacWilliams  further point to the effect of spring-neap tidal variations and timing of wind events on tide residual sediment transport mechanisms. During a 1.5 month period, spring tidal conditions lead to larger sediment fluxes from shoals to the channel, whereas neap tidal conditions occasionally even lead to a small transport from channel to shoals. Bever and MacWilliams  showed that wind events have a temporal, relatively high impact on local SSC, but a more limited impact on residual transports. A possible explanation is that the fine sediment remains in suspension over a tidal cycle [Bever and MacWilliams, 2013], so that, after ebb currents transported sediment toward the channel, flood flows transport a significant portion of sediment back to the shoal.
For example, for a comparable environment in the southern embayment of San Francisco Bay (South Bay), Schoellhamer  measured increased SSC in the channel during ebb and attributed this to wind wave suspension on the shoal, which is a process identified in the current study. However, Brand et al.  observed high SSC for the same area during flood and relatively high-wave conditions, resulting in landward transport of sediments. Apparently, timing and duration of high-wind wave events relative to the semidiurnal and spring-neap tidal cycles can have a significant impact on the transport direction and morphodynamic development.
Our modeling approach does not explicitly describe a spring-neap tidal cycle or (extreme) wind events. However, the effect of such conditions is probably implicitly included in the tuning of model parameters during model calibration. For example, our applied wind amplitude may be high compared to prevailing average conditions, but this high value includes the impact of occasional extreme wind velocities.
Still, it may be questioned whether or not the impact and timing of extreme wind events play a governing role in the morphodynamic development of SPB. These events may have a temporal, high impact, but average and longer duration conditions could be leading eventually. The fact that our model approach is able to reproduce realistic developments by prescribing average conditions over long time scales is only an indication of limited importance of extreme events. Exploring the impact of (timing of) extreme events and spring-neap tidal cycles in more detail would be an intriguing topic of future research.
3.3 The Impact of Wind and Wave Climate Schematization
The wind schematization highly influences the development of wind waves which have considerable impact on the shoal dynamics and margin accretion. The impact of wind-induced water level setup or flow patterns is insignificant. For a cross section similar to the eastern cross section in Figure 5f, Jaffe et al.  (see also Figure 3) show that the shoal profile vertically accreted about 1 m from 1856 to 1898 after which it became smoother as it slowly eroded again (on average less than 1 m). The post-1898 profile progression seems to confirm general theory on mudflat/shoal morphodynamics [Bearman et al., 2010; Friedrichs, 2011] that assumes a dynamic equilibrium under prevailing wind, wave, and tidal conditions.
The wind climate (magnitude and direction) was an important calibration parameter for fitting morphodynamic model results to measurements, but it also includes a high degree of uncertainty. Both the the spatial distribution and the number of wind observations over the past 150 years are limited. Although the model probably does not describe the wind climate in an optimal way, model results have enough skill to show the relative importance and impact of wind conditions on the morphodynamic developments.
In our model, the wind schematization and the associated wave climate have a significant impact on SSC levels in the channel through resuspension of sediment on the shoals and on the location of morphodynamic development. For example, the reason for limited channel narrowing in the western part of the channel during HRF forcing is the schematized wind direction from the southwest, whereas the wind direction during LRF forcing varies between southwest and southeast. Also, water levels are typically 0.1 m higher during HRF forcing because of the river flow. Further analysis indicates that this leads to slightly higher waves on most of the shoals, but that shear stresses decrease due to the larger water depth. Figure 12 shows that different wind directions (continuously from the east (Figure 12a), southwest (Figure 12b), or northwest (Figure 12c)) lead to different erosion/deposition patterns for the erosional period. A striking observation is that major channel margin accretion takes place near (within the distance of a tidal excursion) the area where the largest erosion on the shoals (and highest SSC) occurs due to wave action. A probable explanation is that suspended sediments on the shoals are taken toward the channel by ebb currents (see also Bever and MacWilliams ), where they are advected by along-channel flows and deposit during low-flow (flood or ebb) conditions. The deposition location thus depends on the area where sediments are suspended, where the shoal ebb currents discharge into the channel and the local tidal excursion.
3.4 Channel Dynamics or Shoal Extension?
It is difficult to assign the processes responsible for the change in channel cross section because of a continuous feedback. The shoal extension could be a dominant process that narrows the channel and enhances channel velocities so that the channel further deepens. On the other hand, the channel deepening could be a process independent of accretion of the channel margins, i.e., purely a process of the channel dynamics itself. The deeper parts of a channel will attract more flow because of the reduced friction. Flow velocities will be lower in shallower parts of the channel (i.e., the channel margins) favoring deposition (or less erosion) at these locations.
However, we have some arguments that it is rather the extension of the shoals than the dynamics of the channel that result in narrowing of the channel. First, channel margin accretion is governed by mud, which is the same material as the tidal shoals. Second, when waves are excluded (Figure 6g), there is limited margin accretion and limited channel deepening. Major deepening of the channel is apparently not caused by tidal movement alone. Third, channel width remains still large compared to observations in other estuaries. Prandle et al.  observed estuarine lateral margin values in the range of 1:27–90. Observations [Jaffe et al., 2007] (see also Figure 3) also indicated that the channel narrowed about 2.5 km at the −4 mean sea level (msl) depth contour and deepened by about 4 m in the deepest location apart from a 1 m accretion during the period 1856–1887. As a result, the channel margin has steepened from 6 m over 7 km (1:1167) in 1856 to 10 m over 1 km (1:100) in 1983. This suggests that the current channel dimensions are still too large and that further steepening of the cross-sectional profile is possible.
3.5 Future Morphological Change
An intriguing question (without a definite answer in this study) is how long the channel narrowing will continue. Closure of the channel seems unlikely, since the tidal prism and river flow need to be conveyed through the channel. The 1983–2013 forecast reflects a continuation of the developments during the erosional period 1951–1983 (Figure 4e), although the channel margin accretion is less than in the erosional period.
Model results suggest that the observed channel narrowing is a supply-driven process. Lower SSC levels due to projected lower fluvial sediment supply [Wright and Schoellhamer, 2004] would limit channel narrowing in the eastern part. Deepening of the mudflats/shoals (or sea level rise, see Figure 6c) also decreases SSC levels. For model runs using the 1983 bathymetry, waves are 5–10% higher than for runs using the 1951 bathymetry where mudflats/shoals are about 20% shallower. However, the wave and tide-induced bed shear stresses are 30–40% lower on large portions of the shoals. As a result sediment resuspension on the shoals and SSC levels in the channel become lower, causing lower deposition rates on the channel margin. If the shoals become deeper, shoal deepening and channel narrowing will probably continue at a decreased rate. In addition, Van der Wegen and Jaffe [2013b] suggest that the channel narrowing enhances channel velocities and increases seaward transport of sediment relative to the landward sediment supply. This implies that more sediment bypasses San Pablo Bay resulting in less sediment available to deposit on the margins and shoals.
We simulated limited SSC future conditions for model runs of 30 years, excluding waves and fluvial sediment supply and starting from the 1983 bathymetry (not shown). In the absence of wave-induced high SSC levels, the tidal currents start to erode the northwestern channel margin. At that location, the tidal currents are highest and enter San Pablo Bay and its shoals at an angle. Remarkably, the northeastern channel margins still slightly accrete, probably due to high SSC levels as the result of the eroding western channel margins. In the absence of waves, levee type of features develop with a bed level slightly exceeding the bed level on the shoals.
High SSC levels favor accretion; when SSC becomes too low, the channel margins start to erode because of subtle differences between intertidal deposition and erosion rates. One may argue that a certain SSC level exists that maintains equilibrium in San Pablo Bay. However, the Bay has shown net erosion over the last decades, implying continuously deeper shoals and lower SSC levels. It seems not unlikely that deposition volumes will decrease enough to shift an accretion trend into a net erosion at several locations along the channel margin reversing the trend of channel narrowing.
The current work addresses the process of subtidal channel narrowing and deepening observed over the past 150 years in San Pablo Bay, California. We applied a process-based approach to analyze in detail the conditions and processes that govern the accretion of channel margins observed from 1951 to 1983. Earlier work shows that the Delft3D model was able to reproduce long-term erosion and deposition patterns with significant skill. The current work differs from other research on channel-shoal morphodynamics in the sense that it considers tidally submerged conditions and decadal time scale developments.
The analysis of process-based model results presented in the previous sections leads to the following conceptual model for morphodynamic developments on the channel-shoal interface: Intertidal morphodynamic developments (deposition and erosion) are considerable, but the comparatively small net tide residual change in bed elevations is similar to observations. Categorizing deposition and erosion events in relation to hydrodynamic conditions shows that deposition occurs during all phases of the tidal cycle (ebbing and flooding in the channel and on the shoal), although flooding conditions are responsible for the majority of the deposition. Net channel margin accretion may also be the result of limited erosion rather than large amounts of deposition.
An important finding of the current research is that wind wave-induced sediment resuspension on the shoals is a process leading to significant channel margin accretion. Even more, the magnitude and location of channel margin accretion are highly sensitive to wind direction. For the western part of San Pablo Bay, the channel mainly accretes during high SSC because of wave action on the shoals. For the eastern part of the San Pablo Bay, high SSC in the main channel and resulting channel margin accretion occur by fluvial supply during HRF conditions. Grid resolution, mean sea level, density currents, 3-D circulation flows, or sediment characteristic definition only have a secondary, more limited effect on the patterns and quantities of erosion and deposition.
Although it is hard to clearly separate both processes, the channel narrowing seems to be an extension of shoals rather than the result of (tidal) dynamics in the channel itself. Margin accretion consists of mud rather than sand from the channel bed; the channel margin does not accrete in the absence of waves and fluvial sediment supply, and the channel dimensions remain wide compared to observations of channel dimensions in other estuaries.
Modeling forecasts with the anticipated lower fluvial sediment supply and a decreased impact of wave resuspension due to deeper shoals indicate that channel narrowing may continue for decades at a decreasing rate. Further lowering of prevailing SSC levels will eventually lead to margin erosion, probably starting along the western parts of the main channel. Together with the ongoing channel deepening, this suggests that a possible equilibrium channel shape is not reached. The dimensions of the channel remain subject to dynamic processes related to local variations in sediment supply.
The research is part of the U.S. Geological Survey CASCaDE climate change project (CASCaDE contribution 33). The authors acknowledge the U.S. Geological Survey Priority Ecosystem Studies and CALFED for making this research financially possible. Continuous discussions with Ad van der Spek (Deltares) have been an inspiration for the current work. Lissa MacVean (USGS) and Dano Roelvink (UNESCO-IHE, TU Delft, and Deltares) provided a thorough and stimulating review of the manuscript before submission to this journal. We highly appreciated review comments by Alexander Densmore (Editor), Giovanni Coco and Wonsuck Kim (Associate Editors), and four anonymous reviewers.