Modeling Coastal River, Wetland, and Shoreline Dynamics

Home , Aggradation, Base level, Deposition (geology), Erosion, Pocket beach, Progradation

From the River to the Sea: Modeling Coastal River, Wetland, and Shoreline Dynamics

Katherine Murray Ratliﬀ

Earth & Ocean Sciences Duke University

Date: Approved:

A. Brad Murray, Supervisor

Marco Marani

Peter Haﬀ

James Heﬀernan

Dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Earth & Ocean Sciences in the Graduate School of Duke University 2017 Abstract From the River to the Sea: Modeling Coastal River, Wetland, and Shoreline Dynamics

Katherine Murray Ratliﬀ

Earth & Ocean Sciences Duke University

Date: Approved:

A. Brad Murray, Supervisor

Marco Marani

Peter Haﬀ

James Heﬀernan

An abstract of a dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Earth & Ocean Sciences in the Graduate School of Duke University 2017 Copyright c 2017 by Katherine Murray Ratliﬀ All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract

Complex feedbacks dominate landscape dynamics over large spatial scales (10s – 100s km) and over the long-term (10s – 100s yrs). These interactions and feedbacks are particularly strong at land-water boundaries, such as coastlines, marshes, and rivers. Water, although necessary for life and agriculture, threatens humans and infrastructure during natural disasters (e.g., floods, hurricanes) and through sea-level rise. The goal of this dissertation is to better understand landscape morphodynamics in these settings, and in some cases, to investigate how humans have influenced these landscapes (e.g., through climate or land-use change). In this work, I use innovative numerical models to study the larger-scale emergent interactions and most critical variables of these systems, allowing me to clarify the most important feedbacks and explore large space and time scales. Chapter 1 focuses on understanding the shoreline dynamics of pocket (embayed) beaches, which are positioned between rocky headlands and adorn about half the world’s coastlines. Previous work suggested that seasonality or oscillations in climate indices control erosion and accretion along these shorelines; however, using the Coastline Evolution Model (CEM), I find that patterns of shoreline change can be found without systematic shifts in wave forcings. Using Principal Component Anal- ysis (PCA), I identify two main modes of sediment transport dynamics: a shoreline rotation mode, which had been previously studied, and a shoreline “breathing” mode, which is newly discovered. Using wavelet analysis of the PCA mode time series, I

iv find characteristic time scales of these modes, which emerge from internal system dynamics (rather than changes in the wave forcing; e.g., seasonality). To confirm the breathing mode’s existence, I retroactively identified this mode in observations of pocket beach shoreline change from different parts of the world. Characterization of these modes, as well as their timescales, better informs risk assessment and coastal management decisions along thinning shorelines, especially as climate change affects storminess and wave energy variations across the world. Chapter 2 moves slightly inland to examine how coastal marshes, which provide numerous ecosystem services and are an important carbon sink, respond to climate change and anthropogenic influences. Specifically, I focus on how increasing concentrations of atmospheric CO2 affect marsh resilience to increased rates of sea-level rise relative to inorganic sediment availability and elevated nitrogen levels. Using a meta-analysis of the available literature for marsh plant biomass response to elevated levels of CO2 and nitrogen, I incorporated these effects into a coupled model of marsh vegetation and morphodynamics. Although nitrogen’s effect on biomass and marsh accretion rates is less clear, elevated CO2 causes a fertilization effect, increasing plant biomass, which enhances marsh accretion rates (through increased rates of both inorganic and organic sedimentation). Findings from the model experiments suggest that the CO2 fertilization effect significantly increases marsh resilience to sea-level rise; however, reduced inorganic sediment supply (e.g., through land-use change or damming) still remains a serious threat to marsh survival). Almost half a billion people live on or near river deltas, which are flat, fertile landscapes that have long been ideal for human settlement, but are increasingly vulnerable to flooding. These landscapes are formed by the repeated stacking of sedimentary lobes, the location and size of which are formed by river channel avulsions, which occur when the river changes course relatively rapidly. Despite the importance of avulsions to delta morphodynamics, we do not fully understand their dynamics

v (specifically, avulsion location and timing). In order to investigate the relative influence of rivers and waves on delta morphology and avulsion processes, I develop the River Avulsion and Floodplain Evolution Model (RAFEM) and couple it to CEM to create a new morphodynamic river delta model. In Chapter 3, I use the new coupled fluvial-coastal model to examine the upstream location of avulsions over a range of sea-level rise rates and wave energies. In model experiments, the longitudinal river profile adjusts as the river progrades, causing a preferential avulsion location where the river aggradation relative to the floodplain topography is most rapid. This avulsion length scale is a function of the amount of in-channel sedimentation required to trigger an avulsion, where a larger amount of aggradation required necessitates a greater amount of pre-avulsion progradation. If an avulsion is triggered once aggradation reaches half bankfull channel depth, the preferential length scale is around a backwater length, which scales well with laboratory and field observations. In Chapter 4, I explore how a wide range sea-level rise rates and wave climates affect both delta morphology and avulsion dynamics with the coupled model. Sur- prisingly, I find that increasing sea-level rise rates do not always accelerate avulsions. In river-dominated deltas, avulsion time scales tend not to decrease, as upslope river mouth transgression counteracts base-level driven aggradation. I also find that both the sign and magnitude of the wave climate diffusivity affects both avulsion dynamics and large-scale delta morphology. My findings highlight not only important differences between river and wave-dominated deltas, but also prototypical deltas and those created in the lab. Because the wave climate, sea-level rise rate, and amount of in-channel aggradation required to trigger an avulsion all affect rates of autogenic variability operating within the delta, each of these forcings has important implication for avulsion dynamics and stratigraphic interpretation of paleo-deltaic deposits.

vi To my parents, who have provided their unconditional support throughout my academic endeavors

vii Contents

Abstract iv

List of Tablesx

List of Figures xi

Acknowledgements xiii

1 Modes and emergent timescales of embayed beach dynamics1

1.1 Abstract...... 1

1.2 Introduction...... 2

1.3 Background and Methods...... 5

1.4 Results and Discussion...... 8

1.4.1 Primary Modes of Variability...... 8

1.4.2 Characteristic Mode Time Scales...... 10

1.4.3 Implications...... 14

2 Spatial response of coastal marshes to increased atmospheric CO2 16 2.1 Abstract...... 16

2.2 Introduction...... 17

2.3 Vegetation Responses to Changing Environmental Conditions.... 18

2.4 Results and Discussion...... 23

3 Avulsion dynamics in a coupled river-ocean model 29

3.1 Abstract...... 29

viii 3.2 Introduction...... 30

3.3 Results and Discussion...... 33

4 Exploring wave climate and sea-level rise eﬀects on delta morphodynamics with a coupled river-ocean model 39

4.1 Abstract...... 39

4.2 Introduction...... 40

4.3 Coupling River-Ocean Processes...... 42

4.4 Results and Discussion...... 44

4.4.1 Wave climate diﬀusivity aﬀects delta morphology...... 44

4.4.2 Avulsion Dynamics...... 47

4.4.3 Autogenic Processes...... 53

4.4.4 Implications and Next Steps...... 56

A Coupled River-Ocean Model Description 58

A.1 River Avulsion and Floodplain Evolution Model...... 58

B Supporting Information and Figures 65

B.1 Aggradational Proﬁle Adjustment with Sea-Level Rise...... 65

Bibliography 74

Biography 86

ix List of Tables

2.1 Predicted marsh loss under RCP scenarios...... 25

x List of Figures

1.1 Prototypical embayed (pocket) beach...... 3

1.2 Plan-view Coastline Evolution Model (CEM) schematic...... 4

1.3 Spatial shape of PCA modes...... 7

1.4 Stability diagram of primary PCA mode...... 10

1.5 Example time series of PCA modes...... 11

1.6 Wavelet power spectra of mode time series...... 12

1.7 Characteristic time scales of PCA modes...... 13

2.1 Biomass response to nitrogen fertilization...... 21

2.2 Marsh model equilibrium proﬁle lengths in response to current and elevated atmospheric CO2 for a range of rates of RRSLR...... 24 2.3 Modeled marsh equilibrium proﬁle lengths for changing forcings... 27

3.1 Planview delta morphology with increasing wave height...... 34

3.2 Proﬁle view before and after avulsion...... 35

3.3 Avulsion length for SER=1 and SER=0.5...... 37

4.1 Planview delta morphology varying wave height and wave climate diffusivity...... 46

4.2 Time to avulsion, varying wave height and wave climate diﬀusivity.. 48

4.3 Path length diﬀerence before and after an avulsion...... 54

4.4 Autogenic variability in sediment ﬂux through time...... 55

A.1 RAFEM model schematic...... 59

xi A.2 Normalized SER diagram...... 61

B.1 River mouth trajectory as it adjusts to parallel background slope.. 67

B.2 Schematic illustrating deep shoreface and hypothetical adjustment.. 69

B.3 Planview delta morphology through time...... 70

B.4 Time to avulsion with imposed steeper shoreface...... 71

B.5 Comparing time to avulsion for SER=1 and 0.5...... 72

B.6 Sediment ﬂux for SER=0.5...... 73

xii Acknowledgements

First, I’d like to express my sincere thanks and appreciation for my advisor, Brad Murray. It is because of his support and guidance that I now more clearly see the beauty in complexity. I’d also like to thank Marco Marani, who motivated and supervised a chapter of this dissertation. Eric Hutton provided tremendous support for all things related to the technical aspects of the latter portion of my research, and I’m forever grateful for his programming help and assistance with distributed computing techniques. I’m thankful for my countless insightful conversations with Peter Haff, who has been a great mentor during my time at Duke, and for Jim Heffernan, who has provided helpful feedback on my research. I would also like to thank my extended lab group for invaluable feedback and encouragement over the past five years; specifically, Rebecca Lauzon, Evan Goldstein, Kenny Ells, Pat Limber, Laura Moore. I thank the faculty, staff, and students of the Earth and Ocean Sciences division, who have made the past five years as enjoyable as they have been educational. I also thank Anna Braswell for fruitful collaboration and camaraderie. Last but not least, I’d like to thank my family. My parents, John and Kathy, have unwaveringly supported my academic endeavors since the beginning. Words can’t express how thankful I am to have had my husband, Will, as part of my support system over the past five years. I’m additionally very grateful for my brother, John, who provided many patient hours of technical support. I also must acknowledge my

xiii kitties, who have been my faithful research and writing companions, and my horses, who have kept me sane and balanced throughout my studies. This work has been supported by a National Science Foundation Graduate Re- search Fellowship (Grant DGF1106401), NSF EAR 13-24114, and the Duke Univer- sity Graduate School Jo Rae Wright Fellowship for Outstanding Women in Science.

xiv 1

Modes and emergent timescales of embayed beach dynamics

Ratliﬀ, K. M., and A. B. Murray (2014), Modes and emergent time scales of embayed beach dynamics, Geophysical Research Letters, 41, doi:10.1002/2014GL061680. Re- produced with permission from the American Geophysical Union.

1.1 Abstract

In this study, we use a simple numerical model (the Coastline Evolution Model, CEM) to explore alongshore-transport driven shoreline dynamics within generalized embayed beaches (neglecting cross-shore effects). Using Principal Component Anal- ysis (PCA), we identify two primary orthogonal modes of shoreline behavior that describe shoreline variation about its unchanging mean position: the rotation mode, which has been previously identified and describes changes in the mean shoreline orientation, and a newly identified breathing mode, which represents changes in shoreline curvature. Wavelet analysis of the PCA mode time series reveals characteristic timescales of these modes (typically years to decades) that emerge within

1 even a statistically constant white noise wave climate (without changes in external forcing), suggesting that these timescales can arise from internal system dynamics. The timescales of both modes increase linearly with shoreface depth, suggesting that the embayed beach sediment transport dynamics exhibit a diﬀusive scaling.

1.2 Introduction

Embayed (or pocket) beaches are positioned between rocky headlands (Figure 1.1), and they exhibit morphologic changes over many length and time scales. Although about 51% of the world’s coastlines exhibit headland-bay morphology (Short and Masselink, 1999), the sediment transport dynamics within these beaches are not fully understood. Day-to-day wave forcing drives alongshore sediment transport, which leads to patterns of erosion and accretion. Storms and wave energy variations cause cross-shore sediment exchange, which can also affect shoreline position. Beach rotation occurs when enough sediment moves from one side of the embayment to the other to shift the mean orientation of the beach. A better understanding of the processes that drive beach rotation and the timescales of these dynamics will inform better management decisions, such as whether beach nourishment or the construction of hard structures will be effective along thinning shorelines. Previous studies of embayed beach morphology have focused on empirical or modeling data from specific locations (Harley et al., 2011; Jeanson et al., 2013; Turki et al., 2013b), rather than considering the processes that affect embayed beaches generally. The mean shoreline shape of embayed beaches typically exhibits a distinct curvature, which has been attributed to both wave diffraction and variations in the wave climate, where a higher directional variance of waves more strongly affects curvature (Daly et al., 2014). Many studies have focused on understanding the processes that cause variation about or changes in this mean position, and specifically, beach rotation has been a primary focus of research (da Fontoura Klein et al., 2002;

2 N

1 km

Figure 1.1: Typical embayed (pocket) beach in Carmel, CA. The headland amplitude is 0.5 km, beach length is 2 km, and beach width is 100 m. Image downloaded from Google Earth on July 16, 2014.

Martins and de Mahiques, 2006; Thomas et al., 2010). In the prior location-speciﬁc studies, rotation has typically been attributed to seasonality or oscillations in climate indices, which can shift the distribution of wave-approach angles (Ranasinghe et al., 2004; Thomas et al., 2011). In some locations, cross-shore processes account for the majority of embayed beach shoreline variability (Harley et al., 2011). We use the Coastline Evolution Model (CEM) (Ashton et al., 2001; Ashton and Murray, 2006a) to investigate morphologic changes that occur in a generalized embayed beach (Figure 1.2) so that our results are broadly applicable to embayed beaches in a variety of settings. We focus on the dynamics arising from alongshore sediment ﬂuxes, omitting changes in mean shoreline position that arise from cross-

3 incoming waves headland sediment φ" transport shadow shoreline zone

sand embayed beach

rock

Figure 1.2: Plan-view schematization of Coastline Evolution Model (CEM) algorithm. Incoming offshore waves result in sediment transport except in the wave- shadow zone, where no transport occurs. shore sediment fluxes. A statistically-constant wave climate (which drives alongshore sediment transport) and a particular headland-bay geometry (which dictates the amount of wave shadowing behind the headlands) produces a curved mean shoreline shape for each model experiment. This shape, in which each local shoreline orientation adjusts to the value that produces 0 net alongshore sediment flux, results from the 0-flux boundary conditions imposed by the headlands. The local 0-flux local orientations of the mean shoreline depend on the local wave climates, which are determined by cumulative wave shadowing as well as the global wave climate. Stochastic shifts in wave-approach angles cause the quasi-steady shoreline shape to fluctuate about its static mean position. We then use Principal Component Analysis (PCA) to analyze the variation of the shoreline over time about its mean position. Using wavelet analysis on the time series of shoreline-variation modes, we then determine the characteristic time scales of these shoreline changes (e.g. beach rotation).

4 1.3 Background and Methods

In the one-line CEM (Ashton et al., 2001; Ashton and Murray, 2006a), gradients in

wave-driven alongshore sediment transport, Qs (as calculated by the CERC equation, e.g. Ashton and Murray(2006a)), cause erosion and accretion down to the shoreface depth Dsf . Assuming conservation of nearshore sediment, and an approximately constant long-term shoreface proﬁle geometry (implicitly averaged over the return period of storms or seasonal cycles), the shoreline change is calculated by

dη 1 dQ “ ´ s (1.1) dt Dsf dx

where η is shoreline position, t is time, and x is the alongshore direction. Offshore wave height, angle, and period are transformed into breaking-wave height and angle (assuming wave transformation over shore-parallel bed contours and iteratively applying Snell’s law and conservation of energy until depth-limited breaking occurs; Ashton and Murray(2006a)), which are used to calculate alongshore sediment transport Qs. In this study, the rock elements incorporated in the model (Valvo et al., 2006; Barkwith et al., 2014), which form headlands and back the beach, are not erodible so as to maintain a constant sediment volume within the embayed beach. Although this method neglects headland and beach-backing cliff or bluff erosion (Limber and Murray, 2011), it allows us to focus purely on the sediment transport dynamics within a closed system. The model used a 0.1 day time step, but the beach width time series used in PCA and wavelet analysis were discretized to one day reso- lution. Each of the simulations ran for 500 years. In a ‘one-line’ model such as CEM, shoreline changes are driven solely by gradients in alongshore transport (Equation 1.1). Cross-shore sediment fluxes, which could change the mean shoreline position, are not addressed.

5 Offshore wave-approach angles (representing conditions at the seaward limit of approximately shore-parallel contours; Ells and Murray(2012); van den Berg et al. (2012)) are chosen randomly each model day, based on a probability distribution function representing the wave climate. If a headland blocks a beach cell from the current offshore wave direction, then that cell is within the wave-shadowing zone, where sediment transport does not occur (illustrated in Figure 1.2). This simplified approach neglects diffraction and refraction around the headland, and wind wave generation within the shadowed region, but it serves to approximate the effect of decreased wave energy within the shadowed zone. Patterns of embayed beach shoreline change arise from gradients in alongshore sediment transport related to wave-shadowing effects, as well as to gradients related to shoreline curvature. The wave climate is defined by its asymmetry (the fraction of waves approaching from the left, A) and the fraction of waves approaching from ‘high’ angles (fraction of approaching waves with angles between offshore wave crests and the shoreline

trend >„45 degrees, U), which produce antidiffusive, unstable shoreline-change pat- ternsn(Ashton et al., 2001). A value of 0.5 for A indicates equal influences from waves approaching from the left and right, whereas a value of 0.2 for A would represent an asymmetric wave climate, with 20% of waves approaching from the left and 80% from the right. A value of 0 for U indicates that all the waves come from ‘low’ angles, which produces the greatest shoreline-smoothing influence, represented by the greatest shoreline diffusivity. A value of 0.5 for U indicates equal influences from high- and low-angle waves, which produces a diffusivity of 0. (In results reported here, we do not consider U values above 0.5.) (In a one-line model such as CEM, the wave height only affects the rate of shoreline changes. To enhance the clarity of our experiments involving variations in wave-angle distributions, we hold the offshore wave height constant. However, the probabilities for waves to approach from different angles represents the proportion of the influence on alongshore sediment transport

6 rotation a) b) breathing

0.5 white noise wave climate 0.5 low angle wave climate A = 0.5, U = 0.5 A = 0.5, U = 0.2 0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

Loading 0 0

−0.1 −0.1

−0.2 −0.2

−0.3 −0.3

500 1000 1500 2000 500 1000 1500 2000 Alongshore position (m) Alongshore position (m) Figure 1.3: Spatial shape of the orthogonal PCA modes, which represent variance about the mean shoreline orientation, for a) white noise wave climate and b) low angle wave climate. In a), the rotation mode (mode 1) contains 40.0% of the variance, and the breathing mode characterizes 31.4%. For b), the rotation mode describes 64.3%, and the breathing mode represents 15.3%.

from those angles, which in an observed wave climate is a function of the frequency and heights of waves coming from those angles; Ashton and Murray(2006b).) We explored the effects of a white noise wave climate (A = 0.5, U = 0.5, Figure 1.3a), as well as more commonly observed diffusive, low-angle dominated wave climates (e.g. A = 0.5, U = 0.2, Figure 1.3b). Changes in either the wave climate or the headland-bay geometry (rock elements) affect the gradients in alongshore transport and therefore lead to differing patterns of shoreline change. Using Principal Component Analysis of the de-trended beach width time series from each alongshore position in the embayed beach (PCA, using the Statistics Tool- box in Matlab 2012b), we find the distinct (orthogonal) empirical modes, which are amplitude-free shapes that characterize the shoreline variation about its mean position (Jolliffe, 2005). These modes are ranked in terms of the amount of variance that they explain in the shoreline change data. In addition, we use PCA to find the

7 time series of the amplitude of these modes, which we further analyze using a con- tinuous wavelet transform (Yajnik(2013), using the Haar wavelet and the Wavelet Toolbox in Matlab 2012b). Squaring the wavelet transform coeﬃcients and averag- ing the variation over each wavelet scale (in days) creates a wavelet power spectrum (essentially the equivalent of a Fourier transform power spectrum, e.g. Lazarus et al. (2011)), where peaks represent characteristic time scale for each mode.

1.4 Results and Discussion

1.4.1 Primary Modes of Variability

The first two principal components, or modes, explain the majority (>70%) of the variation about the mean shoreline position (Figure 1.3). One of these modes, which we call the rotation mode, describes the shifting of sand from one end of the beach to the other, changing the mean shoreline orientation. This principal component has been previously recognized (Harley et al., 2011; Short and Trembanis, 2004). How- ever, the second significant mode, which we call the breathing mode, has not yet been identified. The breathing mode characterizes changes in shoreline curvature, as sand moves from the middle of the pocket beach to the edges and back. Across a series of model runs using a variety of wave climates, including both the “white noise” forcing (Figure 1.3a) and the more naturally occurring low-angle diffusive wave climate (Figure 1.3b), these two modes retain their respective qualitative characteristics and remain easily distinguishable as the primary modes of shoreline variation. We interpret the rotation mode as a response to stochastic variations in wave- climate asymmetry. With a symmetric long-term wave climate, during some time periods the effective asymmetry will be slightly >(<) 0.5, tending to produce a reorientation of the beach, so that the average offshore direction faces slightly to the left (right) looking offshore, as movement of sediment from the left to the right (v.v.) brings local orientations closer to those that would produce 0 net alongshore

8 flux for the temporary effective value of A. We interpret the breathing mode as a response to stochastic variations in U. The local curvatures of the average shoreline shape arise from the interactions between wave-shadowing effects and the diffusive influence of alongshore transport gradients. During periods where the effective value of U deviates above (below) the long-term average value, the smoothing influence weakens (strengthens), and curvatures tend to increase (decrease). We have retrospectively observed the breathing mode in observational data from two studies of embayed beaches at different locations. In Figure 6 of Short and Trembanis(2004), the spatial shape of the third PCA mode (second after cross- shore change has been removed) from observational time series of Narrabeen Beach in Australia is similar to that of the breathing mode from our modeling results. In Figures 6 and 9 from Loureiro et al.(2012), the spatial shape of the first PCA modes of shoreline variability in two Portuguese embayed beaches also resembles that of the breathing mode identified here. Whether the rotation mode or the breathing mode is responsible for more of the variance (i.e. is the first principal component) in our model results depends on the wave climate (Figure 1.4). Under symmetric, low-angle-dominated wave regimes, the rotation mode is typically the first principal component, but with increasing asymmetry or frequency of high angle waves, the breathing mode captures more of the variance and is the first mode. In Figure 1.3, the rotation mode is responsible for about three times more variation than the breathing mode (64.3% versus 15.3%), whereas in a white noise wave climate, the amount of variation described by the two modes is more similar (40.0% for rotation mode and 31.4% for breathing mode). The mean orientation of the shoreline varies less with asymmetric wave climates (e.g. A = 0.2), and the amplitude of the rotation mode decreases with greater asymmetry. As a result, the breathing mode may characterize a larger fraction of the variance, making it the first mode. In a wave climate with a greater fraction of high-angle

9 R = Rotation 0.5 C C B C R C = Combination B = Breathing

0.4 C C B R R

0.3 C B C R R Highness (U) 0.2 BBR R R

0.1 B C R R R

0.1 0.2 0.3 0.4 0.5 Asymmetry (A) Figure 1.4: Stability diagram showing which PCA mode (rotation, indicated by red “R”; breathing, indicated by blue “B”; or a mode exhibiting the combined features of both, indicated by green “C”) has the most variance as a function of the asymmetry (A, the fraction of waves approaching from the left) and highness (U, the fraction of waves approaching from high angles) of the wave climate. The headland-bay aspect ratio for these experiments is 1:4. Note that the rotation mode is typically the first principal component in symmetric, low-angle wave climates. The breathing mode (or a combination of the two modes) contains more variance in less diffusive, higher-angle wave climates. waves, sediment transport is less diffusive, and the relative dearth of the smoothing, low-angle waves leads to higher-amplitude changes in shoreline curvature. As a result, the breathing mode also contains more variance when U is high (Figure 1.5).

1.4.2 Characteristic Mode Time Scales

For both the breathing and rotation PCA modes, the first peak in the wavelet coefficient mean variance indicates a characteristic time scale of that mode (Figure 1.6). These time scales are affected by changing the wave climate and by varying the headland-bay geometry. For the rotation mode, the characteristic time scale gener-

400 low angle (A = 0.5 U = 0.2)

200

−200 Rotation Breathing −400

400 white noise (A = 0.5 U = 0.5)

200 mode amplitude (meters)

−200

−400

0 500 1000 1500 2000 2500 3000 3500 time (days) Figure 1.5: Time series of PCA mode amplitudes for low angle and white noise wave climates. In the white noise wave climate, a larger fraction of the waves are coming from high angles (U “ 0.5 vs. U “ 0.2 in the low angle experiment). Note that the amplitude of the breathing mode in the white noise case features greater variance versus the low angle experiment. The peaks in the rotation mode time series are fairly similar. This similarity arises because both model runs feature the same sequence of ‘left’ and ‘right’ approach angles; both runs use A “ 0.5 and the same initial seed, and the random draw determining whether the next wave approaches from ‘left’ or ‘right’ is executed separately from the draw determining whether the next wave comes from high or low angles (Ashton and Murray, 2006a). ally occurs on the order of thousands of days („1000 to „3000 days, few years to decades), depending on the headland-bay geometry, wave climate, and other model parameters (Figure 1.6). This time scale is shorter within a low-angle wave climate (A = 0.5, U = 0.2), but with a white-noise wave climate (A = 0.5, U = 0.5), the peak is more clearly deﬁned and has a higher mean variance. The characteristic timescale of the breathing mode is generally shorter than that of the rotation mode for equivalent wave climates (Figure 1.7), suggesting that changes in the shoreline curvature can occur on shorter time scales than changes in the mean shoreline orientation. After an initial peak, the wavelet mean variance

11 6 x 10 3.5

2.5

2 M1: Rotation (low angle) M2: Breathing (low angle) 1.5 M1: Rotation (white noise) M2: Breathing (white noise) 1

Wavelet coefficient mean variance 0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 Wavelet scale (days) x 10 Figure 1.6: Wavelet power spectra of rotation and breathing mode time series for low angle and white noise wave climates. The first peaks in the spectra represent characteristic time scales of the modes. remains relatively constant over larger wavelet time scales (Figure 1.6). As expected, we see a greater amount of variance in the breathing mode of the white-noise wave climate versus that of the low-angle wave climate, because the vanishing shoreline diffusivity allows for greater-amplitude fluctuations in shoreline curvature. The variance for the rotation mode is also higher for the white-noise wave climate, compared with low-angle climate. This occurs because the directional spread of wave-approach angles increases as the proportion of high-angle waves increase, producing greater deviations in shoreline orientations for a given stochastic variation in wave-approach asymmetry. We interpret the peaks in the spectra (Figure 1.6) to represent the amount of

12 2200 Rotation mode 2000 Rotation timescale trend

1800 Breathing mode Breathing timescale trend 1600 y = 56.3x + 731.9 1400 R−squared = 0.93

1200

1000

800 y = 12.6x + 745.3 R−squared = 0.97 600

Characteristic timescale (days) 400

200

0 2 4 6 8 10 12 14 16 18 20 Shoreface depth (m) Figure 1.7: Characteristic time scales for the rotation and breathing modes in a low angle wave climate (A = 0.5, U = 0.2) with changing shoreface depth. Time scale is directly proportional to shoreface depth, suggesting a diffusive scaling. time that it takes the beach to fully adjust from one equilibrium shape to another in response to a stochastic shift in wave climate. Although previous studies suggest that these decadal time scales for rotation result from shifts in climate indices (Thomas et al., 2010), here we find that these variable time scales arise within statistically constant wave climate, showing that these time scales can emerge from internal system dynamics, rather than systematic shifts in the forcing. We find that the characteristic timescales of both rotation and breathing increase (apparently linearly) with shoreface depth (Figure 1.7). As the shoreface depth shallows, shoreline changes tend to occur more rapidly (holding the gradient in sediment flux constant; Equation 1.1), and the shoreline is able to respond more quickly to stochastic shifts in wave climate. Because coastline diffusivity is inversely

13 proportional to shoreface depth (e.g. Ashton and Murray, 2006a) the apparent linear relationship between the characteristic timescale and shoreface depth is broadly consistent with a diffusive scaling. The diffusive scaling implies that increasing either spatial scales (i.e. the alongshore length of embayed beaches), or the decreasing wave heights, will tend to result in nonlinear increases in the characteristic time scales. (Changing shoreline sediment characteristics, represented by changes to the empirical coefficient in the CERC relationship for alongshore sediment transport, will also change the timescales.) Moreover, changes in the aspect ratio of the headland-bay system will likely affect characteristic timescales through changes in wave shadowing. Future work exploring these relationships further would be useful. The diffusive time scaling in the modeling results here agrees with the analytical findings of previous studies (e.g. Turki et al., 2013a).

1.4.3 Implications

The findings of this study are of interest to coastal developers, as the amplitudes of shoreline rotation and breathing excursions are critical for understanding the patterns of erosion and accretion along the beach. Both the amplitude of the excursions about the mean shoreline orientation (how much sediment moves) and the timing of the variation (how often sediment moves) are important elements for understanding sediment transport dynamics in embayed beaches. We are able to differentiate between the amplitude and frequency of the shoreline variations in a modeled pocket beach by using both PCA (which describes the variation amplitude) and wavelet analysis (which identifies the time scales of the nonstationary PCA mode time series, see Figure S2). Recognition of the time scales of these variations is relevant for both risk assessment and management of developed shorelines. A better understanding of the processes that drive alongshore sediment transport and the resulting patterns of erosion and accretion along embayed beaches will improve management decisions,

14 especially as global anthropogenic change affects storm behaviors (e.g. Bender et al., 2010; Knutson et al., 2010) and can result in shifting wave climates (defined here as angular distributions of wave influences on alongshore sediment transport), affecting both mean shoreline shape and variability.

15 2

Spatial response of coastal marshes to increased atmospheric CO2

Ratliﬀ, K.M., Braswell, A, and M. Marani (2015) Spatial response of coastal marshes to increased atmospheric CO2. Proceedings of the National Academy of Sciences, U.S.A., 112(51), 15580-15584. doi:10.1073/pnas.1516286112. Copyright (2015) Na- tional Academy of Sciences. Reproduced with permission.

2.1 Abstract

The elevation and extent of coastal marshes are dictated by the interplay between the rate of relative sea-level rise (RRSLR), surface accretion by inorganic sediment deposition, and organic soil production by plants. These accretion processes respond to changes in local and global forcings, such as sediment delivery to the coast, nutrient concentrations, and atmospheric CO2, but their relative importance for marsh resilience to increasing RRSLR remains unclear. In particular, marshes up-take atmospheric CO2 at high rates, thereby playing a major role in the global carbon cycle, but the morphologic expression of increasing atmospheric CO2 concentration, an im-

16 minent aspect of climate change, has not yet been isolated and quantified. Using the available observational literature and a spatially explicit ecomorphodynamic model, we explore marsh responses to increased atmospheric CO2, relative to changes in inorganic sediment availability and elevated nitrogen levels. We find that marsh vegetation response to foreseen elevated atmospheric CO2 is similar in magnitude to the response induced by a varying inorganic sediment concentration, and that it increases the threshold RRSLR initiating marsh submergence by up to 60% in the range of forcings explored. Furthermore, we find that marsh responses are inherently spatially dependent, and cannot be adequately captured through 0-dimensional representations of marsh dynamics. Our results imply that coastal marshes, and the major carbon sink they represent, are significantly more resilient to foreseen climatic changes than previously thought.

2.2 Introduction

Coastal marsh extent and morphology are directly controlled by rate of relative sea- level rise (RRSLR) and the soil accretion rate, the latter associated with inorganic sediment deposition and organic soil production by plants. Previous studies observed that CO2 fertilization increases marsh plant biomass productivity through increased water use efficiency and photosynthesis(McKee et al., 2012), and hypothesized that, as a consequence, marsh resilience should increase via increased organic accretion (Cherry et al., 2009; Langley et al., 2009). However, this hypothesis has not yet been tested, and the observed increased plant productivity in response to the CO2 fertilization effect has not been translated into its actual geomorphic effects. In fact, direct CO2 effects on vegetation and marsh accretion (as opposed to its indirect effects, e.g., via the increase in temperature) have not yet been incorporated into marsh models, and their importance relative to other leading forcings of marsh dynamics (e.g., inorganic deposition, RRSLR, nutrient levels) remains unknown. Here

17 we use existing data and a 1D ecomorphodynamic model to assess the direct im-

pacts of elevated CO2 on marsh morphology, relative to ongoing [e.g., RRSLR, and suspended sediment concentration (SSC)] and emerging [nutrient levels (Darby and Turner, 2008; Morris et al., 2013; Turner et al., 2009)] environmental change.

2.3 Vegetation Responses to Changing Environmental Conditions

We use published observations of aboveground and belowground biomass productivity in coastal marshes under ambient and elevated CO2 conditions to derive a range of possible marsh plant biomass responses to changing atmospheric CO2 (Langley et al., 2013; Curtis et al., 1990, 1989; Drake, 1992; Erickson et al., 2007). Because the photo- synthetic pathway of C3 plants is CO2-limited for a wider range of atmospheric CO2 concentrations than in C4 plants, CO2 fertilization effects are expected to be more ev- ident in C3 species; elevated CO2 concentrations may also give C3 species a competitive advantage over C4 species, and hence possibly favor an expansion of the former at the expense of the latter (McKee et al., 2012; Henderson et al., 1995). However, enclosure field experiments specifically focusing on marsh vegetation (Langley et al., 2013; Curtis et al., 1990, 1989; Drake, 1992; Erickson et al., 2007) find both C3 and

C4 marsh species to be responsive to increases in atmospheric CO2. A meta-analysis of this literature (Curtis et al., 1990, 1989; Drake, 1992; Erickson et al., 2007) [See Table S1, Ratliﬀ et al.(2015)], when the responses of C3-only, C4-only, and mixed communities are collectively considered, gives a median 39% increase in aboveground

biomass and a median 33% increase in belowground biomass as CO2 is elevated by 400 ppm with respect to current conditions. For comparison, the very limited available data documenting C4-only belowground productivity (including rhizomes and roots, both contributing to organic soil production) report increases between 13% and 47% (Curtis et al., 1990). Because of the positive belowground response of marsh C4 vegetation, the expected competitive advantage of C3s, and the largely positive

18 response of the latter to increased CO2, we consider all of the available data to pa-

rameterize biomass response to elevated CO2 (i.e., not separating C3s and C4s). Our parameterization expresses the biomass in the changed scenario as a linear function of CO2 change (∆CO2), according to the median responses observed in the compiled dataset for both the aboveground [Bap∆CO2q “ p1 ` p0.39{400ppmq ¨ ∆CO2q ¨ Bap0q,

Bap0q being the current aboveground biomass productivity] and the belowground

compartment [Bbp∆CO2q “ p1 ` p0.33{400ppmq ¨ ∆CO2q ¨ Bbp0q]. To explore the

range of potential vegetation responses to CO2 changes, we also consider the ﬁrst quartile (13% increase in aboveground biomass and a 20% increase in belowground biomass) and the third quartile (101% increase in aboveground biomass and a 94% increase in belowground biomass) in the compiled dataset. This range in vegetation responses provides an estimate of the uncertainty in our evaluation of marsh

responses to elevated CO2. Salinity, ﬂooding, water stress, and eutrophication can all potentially inﬂuence

the eﬀects of CO2 fertilization on marsh plants (Langley et al., 2009, 2013; Lenssen et al., 1993; Owensby et al., 1999; Rozema et al., 1991; Ward et al., 1999). Overall,

the available evidence is coherent in pointing to a significant CO2 fertilization effect on organic accretion and plant production even in the presence of other simultane- ous environmental changes (Cherry et al., 2009; Langley et al., 2009; Langley and Megonigal, 2010; Rasse et al., 2005). However, we are here interested in isolating the effects of different forcings. Hence, to avoid the introduction of confounding factors, we do not use in our parameterization data that include the interactions between

the above environmental conditions and CO2 fertilization [e.g., Rasse et al.(2005); Cherry et al.(2009)].

To quantify the geomorphic response of marshes to elevated CO2 in the wider con- text of the main environmental changes expected over the next century, we compare it to the responses induced by changes in RRSLR, SSC, and nitrogen availability. Al-

19 though the effects of the first two forcings are relatively well constrained, the effects of nitrogen are the subject of some controversy. In fact, an analysis of the extensive literature on elevated nitrogen levels in coastal marshes reveals large uncertainties in plant responses [See Table S2, Ratliff et al.(2015)]. Fertilization experiments usually find that increased nitrogen promotes aboveground biomass productivity (Darby and Turner, 2008; Langley et al., 2009), but a wide scatter characterizes the observations, even within the same study (Figure 2.1). Nitrogen’s effect on belowground biomass productivity is even less clear, such that even the sign of the change in productivity remains undetermined (Morris et al., 2013; Turner et al., 2009) and may be dependent on the levels of nitrogen addition (Darby and Turner, 2008). Such scatter can be attributed to the fact that existing studies used different methods (e.g., the forms of nitrogen adopted for fertilization, such as ammonium sulfate, ammonium nitrate, etc.), and were conducted in different regions and under different environmental conditions. Species composition, baseline ambient nitrogen loading (Wigand et al., 2003; Graham and Mendelssohn, 2010), salinity, flooding frequency, and the concentration of other nutrients also vary widely across the existing literature, and can significantly affect the outcome of nitrogen addition experiments (Morris et al., 2013; Boyer et al., 2001; Crain, 2007; Deegan et al., 2007; Gough and Grace, 1999; Loomis and Craft, 2010; McFarlin et al., 2008; Pennings et al., 2002; Ryan and Boyer, 2012; Vivanco et al., 2015; Wigand et al., 2004). Hence, to isolate the potential impacts of increased nitrogen concentrations, we use the minimum (55.2gN ¨m´2 ÿ´1) and maximum (892.8gN ¨ m´2 ¨ y´1) nitrogen addition levels and the corresponding biomass responses from one study that explored a wide range of nitrogen addition levels at a single site (Darby and Turner, 2008). The minimum addition level corresponds to a 15% increase in both the aboveground and belowground biomass, whereas the maximum addition experiment caused a 134% increase in aboveground biomass and a 24% decrease in belowground biomass. Using these two cases to parameterize biomass re-

20 Figure 2.1: Change in aboveground and belowground biomass in response to nitrogen fertilization. Data shown in ﬁgure are from sources listed in Table S2, Ratliﬀ et al.(2015). sponse to nitrogen addition, we explore two end-member response scenarios with our experiments. The corresponding changes in aboveground and belowground productivity are representative of the range of responses observed in the dataset compiled (Figure 2.1), yet avoid confounding factors from cross-study methodological hetero- geneity, and preserve the observed belowground to aboveground biomass (root to shoot) ratios. We use a spatially explicit 1D model of marsh ecomorphodynamic processes to evaluate the CO2, nitrogen, and suspended sediment concentration impacts on marsh morphology (Marani et al., 2013). The model describes the evolution of a marsh transect perpendicular to a tidal channel, which delivers inorganic sediment, by focusing on accretion-submersion processes rather than on marsh lateral erosion or expansion (Donnelly and Bertness, 2001; Mariotti and Fagherazzi, 2010). Changes in the

21 marsh surface elevation, z, at a position x along the marsh transect are dictated by

the sum of the rate of inorganic soil deposition, Qspxq, the rate of sediment trapping

by marsh plants, Qtpxq, the rate of organic soil production by vegetation, Qopxq, and the RRSLR,

Bz px, tq “ Q px, tq ` Q pB pxq, tq ` Q pB pxq, tq ´ R (2.1) Bt s t a o b where Bapxq and Bbpxq are aboveground and belowground biomass (linear functions of ∆CO2), and R is the RRSLR [See Table S3, Ratliﬀ et al.(2015)]. Surface erosion is set to zero in our experiments, as it is negligible in vegetated marshes (Marani et al., 2007). Settling of suspended sediment, which is transported from the channel across the marsh over a tidal cycle with period T through advection-dispersion, determines inorganic soil deposition. This sedimentation rate is calculated as

Qspz, Bapxqq “ ws{pT ¨ ρbq Cpz, Bapxq, tqdt (2.2) żT ´4 where ws is the settling velocity (here 2 ˚ 10 m{s), ρb is the bulk density (1,325

3 kg{m ), and C is the instantaneous SSC. The amount of belowground biomass Bbpxq, modulated by a factor γ (2.5˚10´3m3{kg) that accounts for compaction and decomposition processes, governs the rate of organic soil production by Qopxq “ γBbpxqfipzq.

The fitness function fipzq (0 ď fipzq ď 1) describes how biomass productivity and the competitive abilities of species i vary as a function of elevation z (Marani et al., 2013; Morris, 2006). We note that, although decomposition is accounted for in a simplified manner through a refractory fraction coefficient (embedded in γ), the value of this parameter is defined on the basis of observations (Marani et al., 2007). Deposi- tion by sediment trapping is calculated as a power-law function of aboveground plant biomass Bapxq (D’Alpaos et al., 2007; Mudd et al., 2010). We assume here that three marsh species (representative of low-, middle-, and high-marsh vegetation) compete

22 for space, each adapted to a particular elevation range relative to mean sea level. Marani et al.(2013) have shown that the biogeomorphic evolution of a marsh under these assumptions leads to geomorphic and ecological patterns along the marsh transect that are consistent with observed ones (Marani et al., 2006; Silvestri et al., 2005).

2.4 Results and Discussion

An elevated atmospheric CO2 concentration increases aboveground and belowground biomass productivity, which, in turn, increases marsh accretion rates and resilience to RRSLR. Increased belowground productivity enhances soil organic production, whereas increased aboveground productivity boosts suspended sediment trapping by standing biomass. For a marsh adjacent to a tidal creek with SSC = 20 mg/L and a tidal amplitude of 0.5 m, the threshold beyond which partial marsh drowning is initiated increases from an RRSLR of 6.5 mm/y in current conditions to 7.5 mm/y for both the first quartile and median response to elevated atmospheric CO2 (Fig- ure 2.2A). The third quartile of plant responses, which nearly doubles vegetation biomass, extends the threshold for partial drowning to 10.5 mm/y. The values of these drowning thresholds will vary in marshes across the globe, depending on local environmental factors (e.g., SSC, tidal amplitude), but the significant increase in marsh resilience resulting from CO fertilization observed here has broad global implications. Even before the onset of marsh submergence, the marsh transects from the current and elevated CO2 (median response) scenarios under the same RRSLR exhibit different morphologies (e.g., see RRSLR = 6.5 mm/y, Figure 2.2B); the average elevation of the marsh is lower for the current CO2 concentration than in the elevated

CO2 case, and the plant species distribution is shifted toward low-lying and more ﬂooding-tolerant species. The two transect morphologies are most diﬀerent beyond

Ambient CO 0.4 RRSLR = 6.5 mm/yr 1 2 Elevated CO 2 0.2 (median) b 0.8 Elevated CO 0 2 5 10 15 20 25 30 35 40 (first quartile) Elevated CO 0.4 RRSLR = 7.5 mm/yr 0.6 2 (third quartile) 0.2 c 0.4 0 5 10 15 20 25 30 35 40 Elevation, Z (m)

Fractional marsh length 0.4 RRSLR = Ambient CO 0.2 8.5 mm/yr 2 Elevated CO 0.2 2 a d (median) 0 0 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 35 40 Rate of relative sea−level rise (mm/yr) Space (m) Figure 2.2: Marsh model equilibrium profile lengths in response to current and elevated atmospheric CO2 for a range of rates of RRSLR. (A) Fractional marsh length (equilibrium profile length divided by initial length, where a value of 1 represents no partial marsh drowning and 0 is completely submerged) for a range of RRSLR. Marsh partial drowning is initiated at RRSLR = 7.5 mm/y in the current CO2 scenario, versus RRSLR = 8.5 mm/y in the median and first quartile elevated CO2 cases, and 11.5 mm/y in the third quartile response case, where the CO2 fertilization effect increases marsh resilience to higher RRSLR. Marsh equilibrium profiles [where space (m) represents distance from a tidal creek that delivers sediment from the left side of the figures] for both current and median biomass response to elevated CO2 are shown for (B) RRSLR = 6.5 mm/y, before the drowning threshold has been crossed in either case; (C) RRSLR = 7.5 mm/y, where the current CO2 scenario shows drowning initiation, but not the elevated CO2 profile; and (D) RRSLR = 8.5 mm/y, where both marsh profiles are partially drowned.

the threshold RRSLR for drowning in the current CO2 scenario but before partial drowning of the elevated CO2 case (Figure 2.2C). For RRSLR beyond the elevated

CO2 drowning threshold (i.e., both marshes are becoming shorter due to the progres- sive submergence of their interior zones), the marsh morphologies under the current and elevated CO2 concentrations converge toward similar profiles (Figure 2.2D). In both cases, the lowered marsh elevation increases the duration of submergence, thus increasing the amount of inorganic sediment deposition (which only occurs during flooding periods). Hence, at high RRSLR, the profile is mostly maintained by inorganic sedimentation, which is almost independent of the atmospheric CO2 concentra-

24 Table 2.1: Predicted marsh loss under IPCC Fifth Assessment Report (IPCC, 2013) RCP mean scenarios

RCP2.6 RCP4.5 RCP6 RCP8.5 Atmospheric CO2 (ppm) 421 538 667 930 Rate of sea-level rise, mm/yr 4.4 6.2 7.5 12 Marsh loss no CO2 effect, % 0 0 40 66 Marsh loss first quartile CO2 effect, % 0 0 0 65 Marsh loss median CO2 effect, % 0 0 0 63 Marsh loss third quartile CO2 effect, % 0 0 0 0 tion (except indirectly, through the effect of enhanced sediment trapping by increased standing plant biomass). This explains the observed weak dependence of marsh resilience on atmospheric CO2 concentration at high RRSLR, and the similarity of the submerging marsh profiles for both CO2 scenarios. We also tested the possible effects of the end-of-century concurrent atmospheric

CO2 concentration and RRSLR values projected in the Representative Concentration Pathways (RCP) adopted in the latest Intergovernmental Panel on Climate Change (IPCC) report (IPCC, 2013). We note that we are here assuming the local subsidence rate and mean sea-level changes to be negligible, and take RRSLR to be equal to global eustatic SLR. Under the environmental conditions examined here, our analyses show that, without the CO2 fertilization eﬀect, about 40% of the marsh will be lost under the mean RCP6 in 2100 (end-of- century RRSLR = 7.5 mm/y), whereas the marsh loss becomes 66% under the mean RCP8.5 (end-of-century RRSLR = 12 mm/y). No marsh loss occurs under the remaining RCPs (Table 2.1). First quartile, median, and third quartile plant biomass responses to increased atmospheric

CO2 show no marsh loss under the RCP2.6, RCP4.5, and RCP6 scenarios. The median and ﬁrst quartile biomass responses in the RCP8.5 scenario result in 65% and

63% marsh loss, respectively. The third quartile CO2 fertilization eﬀect completely negates marsh loss in all RCP scenarios.

25 The absolute values of marsh loss in the RCP scenarios will vary depending on other forcings that differ across tidal environments. For example, in marshes with high sediment supply and inorganic sedimentation rates, the effects of CO2 fertilization on marsh elevation and extent may be reduced (Mudd et al., 2009). How- ever, our results show that the CO2 fertilization effect plays a central role in determining marsh resilience globally and that it can significantly offset marsh drowning induced by an accelerated RRSLR. Previous models of marsh dynamics under environmental change have neglected this effect (or heuristically assumed it is implicitly represented in temperature responses), and, although additional observational work is urgently needed to further improve our understanding and modeling representations, this mechanism is established as an important marsh response. Suspended sediments play a fundamental role in marsh survival (Kirwan et al., 2011), and marsh responses to changing SSC serve to evaluate the relative importance of and the interplay with the CO2 fertilization impacts. Doubling the available SSC (C0) from 20 mg/L to 40 mg/L increases inorganic sedimentation rates and therefore the threshold RRSLR for partial drowning in all cases. The increase is about 1 mm/y (Figure 2.3), which is of the same order as the increase in the threshold RRSLR caused by a doubling of CO2 concentration (for the median fertilization response). In the “sediment-poor” case (C0 = 20 mg/L), just beyond the threshold RRSLR for drowning, marsh loss is between 39% and 61% (Figure 2.3A). Marsh loss occurs more gradually in the higher-SSC scenarios, where marsh extent is only reduced by 1 – 8% just beyond the initiation of partial drowning (Figure 2.3B). Damming, the construction of levees, and other anthropogenic manipulations of upstream catchments have largely decreased sediment delivery to coastal environments (Syvitski et al., 2005). Our results confirm that this trend, likely to continue at the global scale, will significantly diminish marsh resilience to increasing RRSLR. In model experiments with nitrogen addition, elevated levels of nitrogen showed com-

1 1 C = 20 mg/l C = 40 mg/l 0 0

0.8 0.8

0.6 0.6

0.4 0.4

Current CO 2

Fractional marsh length Fractional marsh length Elevated CO (median) 0.2 0.2 2 Minimum N addition a b Maximum N addition 0 0 4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20 Rate of relative sea−level rise (mm/yr) Rate of relative sea−level rise (mm/yr) Figure 2.3: Modeled marsh equilibrium profile lengths for changing forcings: varying CO2, nitrogen addition, sediment delivery, and the rate of RRSLR. Fractional marsh lengths (equilibrium profile length divided by initial length, where a value of 1 represents no partial marsh drowning and 0 is completely submerged) as a function of RRSLR for two different CO2 scenarios (current and median biomass response to el- ´2 ´1 evated CO2), two different nitrogen addition scenarios (minimum = 55.2 g Nm y and maximum = 892.8 g Nm´2y´1), and two different sediment delivery scenarios: (A) C0 = 20 mg/L and (B) C0 = 40 mg/L. Increased sediment supply increases the threshold for partial marsh profile drowning in all forcing scenarios. peting effects on marsh resilience (Figure 2.3). A minimum nitrogen addition (55.2 g Nm´2y´1) slightly increased the drowning initiation threshold RRSLR by 1 mm/y in both SSC cases, and the maximum addition scenario (892.8 g Nm´2y´1) slightly decreased the drowning threshold (also by 1 mm/y for both SSC scenarios). Our results suggest that the impacts of increased nitrogen concentrations on marsh stability could be significant, but that they remain uncertain, due to the wide range of observed plant responses to nitrogen fertilization. Our results emphasize that marsh responses to environmental change are spatially variable. Many empirical studies and field experiments (Morris et al., 2013; Langley

et al., 2013) have focused on measuring the eﬀects of elevated CO2 and nitrogen addition in single plots or chambers. Similarly, numerous modeling studies have used a point model framework to explore marsh dynamics (Marani et al., 2007;

27 Kirwan and Mudd, 2012; Morris et al., 2002; D’Alpaos et al., 2011; Fagherazzi et al., 2012). These approaches made it possible to identify marsh response mechanisms and feedbacks but do not capture actual marsh responses as they unfold in time and space (Marani et al., 2013; Da Lio et al., 2013; D’Alpaos and Marani, 2016). In particular, point models cannot reproduce the observed marsh drowning dynamics, moving from the inner part of a marsh toward the creeks that feed it (DeLaune et al., 1994; Erwin et al., 2004), or the persistence of the marsh portion adjacent to the tidal creek (maintained by inorganic sedimentation) even at very high RRSLR,

after a majority of the marsh has drowned. The CO2 fertilization eﬀect increases the lateral extent of marshes (as well as their average elevation), resulting in marshes that are more resilient to increasing RRSLR. In turn, more-expansive marshes provide a greater area for sequestering more carbon (Kirwan and Mudd, 2012; Mcleod et al., 2011) than those that are retreating and drowning. Consequently, the fertilization eﬀect may also contribute to a stabilizing feedback within the climate system, where increasing biomass production and organic deposition consequently sequester greater

amounts of CO2. The marsh resilience evidenced by our results suggests that this feedback will be robust with respect to foreseeable values of the RRSLR.

Although higher levels of atmospheric CO2 are inevitable (IPCC, 2013), sediment delivery to the coast continues to decline globally (Syvitski et al., 2005). Elevated

CO2 may possibly partially balance the negative eﬀects of land use change and damming, but reduced sediment supply is likely to remain the most serious threat to marsh survival.

28 3

Avulsion dynamics in a coupled river-ocean model

3.1 Abstract

River deltas grow through repeated stacking of sedimentary lobes, the location and size of which are determined by river channel avulsions, relatively rapid changes in river course. Despite the importance of avulsions in delta morphodynamics, debate exists about what processes are necessary and sufficient to cause them. We use a new model of river and floodplain evolution coupled to a one-line coastline model to explore avulsion dynamics under a range of wave energies and sea-level rise rates. As a delta progrades and the longitudinal river profile adjusts in our model experiments, a preferential avulsion location arises, where river aggradation relative to the surrounding floodplain topography is most rapid. The avulsion length scale varies with the critical superelevation ratio required to trigger an avulsion. With a higher critical ratio, a greater amount of pre-avulsion progradation occurs, leading to a longer characteristic avulsion length. For a superelevation ratio of 0.5, avulsion length is on average slightly less than one backwater length, scaling well with laboratory and field observations. However, in contrast to recent laboratory experiments, this length

29 scale in our model arises for geometric rather than hydrodynamic reasons.

3.2 Introduction

More than half a billion people live on or near deltas, as their low-lying, fertile landscapes have long been ideal locations for human settlement (Syvitski et al., 2009). Deltaic landscapes and their inhabitants have become increasingly vulnerable to submergence and natural disasters as sea-level rise, accelerated subsidence, and decreased upstream sediment supply all lower deltaic elevations relative to sea level (Tessler et al., 2015; Blum and Roberts, 2009). Rivers typically deliver sediment only to one part of a delta at a time, successively building distinct lobes. River avulsion dynamics control both the location and size of delta lobes. Recent advances (Chatanantavet et al., 2012; Ganti et al., 2014, 2016a) have furthered our understanding of avulsion dynamics, yet this understanding is incomplete. To humans inhabiting and managing delta environments, avulsions pose hazards to infrastructure, and better understanding of natural avulsion dynamics could facilitate improved sustainable management practices (Paola et al., 2011). In the lowermost portion of the river (near the mouth), ﬂow is aﬀected by the presence of the receiving lake or ocean basin. The length of this “backwater zone”

is approximated by LB « D{S, where D is a characteristic flow depth and S is the riverbed slope (Paola, 2000). Avulsion nodes (where the river leaves the previous channel) on deltas in the field and lab tend to occur at a distance from the river mouth that scales with a characteristic backwater length LB (Jerolmack, 2009; Chatanantavet et al., 2012; Ganti et al., 2014, 2016b). Researchers have proposed differing theories as to why avulsion length, LA, which controls delta size (Ganti et al., 2016b), scales with the backwater length. Recent work (Chatanantavet et al., 2012; Ganti et al., 2014, 2016b) credits the interplay between temporally varying river discharges and the offshore river plume as the primary control of avulsion lo-

30 cation; repeat flooding events and Froude-subcritical flows creates a peak in net channel sedimentation within the backwater zone, driving avulsions at this location of maximum deposition as a result of this “hydrodynamic backwater effect”. Prior work (Hoyal and Sheets, 2009) attributes preferential avulsion locations to the “morphodynamic backwater effect”, where a hydraulic backwater effect drives a negative feedback: decelerating channelized flow near the river mouth causes sediment deposition, which results in further decelerating flow and an upstream-migrating wave of deposition. Here, using a simple model of coupled river-ocean processes, we find a preferential avulsion location and delta size that scales with the backwater length without explicitly including hydrodynamic backwater effects or varying flows. For a range of modeled wave influences and sea-level rise rates (SLRR˚, see Appendix A for nondimensionalization), the preferential avulsion length arises from river bed aggradation and the surrounding floodplain geometry. The newly-developed river delta model couples the River Avulsion and Floodplain Evolution Model and the Coastline Evolution Model (Ashton et al., 2001; Ashton and Murray, 2006a), using the framework and tools of the Community Surface Dy- namics Modeling System (Peckham et al., 2013). It has not been designed with any specific delta in mind, but rather its processes and components are generic and broadly applicable to a range of river and wave-dominated delta types. The model is scale-invariant, although it has been designed to simulate large space and time scale processes (e.g., multiple avulsions and delta lobe-building events). In the River Avulsion and Floodplain Evolution Model (RAFEM), cell widths are larger than channel widths, such that the channel belt is contained within a single cell, and within-channel processes are not resolved. Natural levees form as a result of overbank deposition adjacent to the river channel (Pizzuto, 1987; Walling and He, 1998; Aalto et al., 2003), and levee elevation is maintained at one bankfull channel depth above the river bed elevation (Jerolmack and Paola, 2007; Hoyal and

31 Sheets, 2009). Herein, ‘floodplain elevation’ refers to the distal floodplain (adjacent cells), not the levee topography. The river course is determined using a steepest- descent algorithm (Jerolmack and Paola, 2007) that compares the elevations of the three downstream and two cross-stream cells (i.e., no upstream flow is permitted). Erosion and deposition along the river channel are modeled as a linear diffusive process (Paola, 2000), and the bed elevation at the mouth of the river is a held at a constant channel depth below sea level, such that river progradation or base-level rise will cause a diffusive wave of aggradation to migrate upstream. An avulsion is triggered when a river cell meets or exceeds the critical superelevation ratio (SER, the elevation difference between top of the levees and the surrounding floodplain, normalized by bankfull channel depth (Mohrig et al., 2000)), and a new steepest-descent path to sea-level is determined. If the new path is shorter than the prior river course, the avulsion occurs, and the river course follows the new shorter path (Slingerland and Smith, 2004; Ganti et al., 2016a). However, if the new path is longer than the previously-existing one, the avulsion does not occur, representing the result that an avulsion would not be successful along a shallower channel gradient with a decreased sediment transport capacity (Slingerland and Smith, 2004; Hoyal and Sheets, 2009). Instead, a crevasse splay is deposited (Shen et al., 2015; van Toorenenburg et al., 2016) around the failed avulsion branch, adjacent to the existing channel. See Appendix A for a more complete description of RAFEM. The bedload sediment flux from the RAFEM river mouth is retained in the shoreline. This sand is distributed along the shoreline using the Coastline Evolution Model (CEM) (Ashton et al., 2001; Ashton and Murray, 2006a), a one-line model in which gradients in wave-driven alongshore sediment transport, Qs, cause erosion and accretion of the shoreline and nearshore seabed. Erosion and accretion extend offshore to the shoreface depth, Dsf (below which wave-driven sediment transport becomes negligible). Assuming an approximately constant long-term shoreface profile geometry

32 and conservation of nearshore sand leads to:

dη 1 dQ “ ´ s (3.1) dt Dsf dx

where η is shoreline position, t is time, and x is the alongshore direction. In CEM, offshore wave-approach angles change every model day, and coastline- shape evolution depends on the mix of influences from different angles (the ‘wave climate’ (Ashton and Murray, 2006a)). In experiments reported here, the net effect of the wave climate is to diffusively smooth plan view coastline shape. The wave climate features a symmetric mix of influences from the left and right (looking in the globally offshore direction). If part of the delta lobe blocks a shoreline cell from the current offshore wave direction, then no sediment transport occurs in this “wave-shadowed zone”, which approximates the effect of decreased wave energy within and adjacent to this region (from wave refraction and diffraction). Wave height, representing an effective average value (Ashton and Murray, 2006a), remains constant during each model run.

3.3 Results and Discussion

Studies of avulsions in the field (Mohrig et al., 2000) indicate that avulsions generally occur at a superelevation ratio of 0.5-1 (Hajek and Wolinsky, 2012), but this value is significantly less for some systems (e.g. Mississippi, ă0.1) (Törnqvistand Bridge, 2002). The critical SER in prototypical channels can be affected by a number of factors, including levee grain size, cohesiveness, and vegetation. In lab experiments (Ganti et al., 2016a), avulsions occurred when the channel aggraded approximately 0.3D. In our model experiments, we use a critical superelevation ratio of either 0.5 or 1, to explore how avulsion dynamics depend on this parameter. In our experiments, we hold river sediment input fixed across all runs. We vary

33 WH*=0.1 WH*=0.3 WH*=0.5 ocean

land increasing wave influence Figure 3.1: Planview morphology of deltas in a diffusive wave climate with increasing wave influence from left to right. Critical superelevation ratio to trigger an avulsion is one. Higher wave heights (WH*, nondimensionalized by channel depth) are more effective at spreading the fluvial sediment alongshore, leading to smoother shorelines. Blue line represents the river cells. wave height (WH˚, nondimensionalized by channel depth D) between runs. Increas- ing wave height (with a diffusive angular distribution of wave influences) tends to smooth coastline shape more rapidly. With small wave heights, the river delivers sand to the shoreline more rapidly than waves can spread it along the coast, leading to more rugose shorelines (‘river dominated’ (Nienhuis et al., 2015); Figure 4.1). With increasing wave influence, alongshore sediment transport redistributes the river sand more rapidly, forming more cuspate to nearly flat shorelines (‘wave dominated’; Figure 4.1). In the river-dominated case, progradation is not inhibited, and avulsions occur relatively quickly resulting from in-channel aggradation associated with progradation (and base-level rise). Increasing wave height decreases progradation rates, and avulsions do not occur as rapidly as in the river-dominated case because of this inhibited progradation (and more limited upstream aggradation) (Swenson, 2005). New land created by the growth of a delta lobe maintains a small elevation relative to sea level, representing marshes that aggrade at the rate of sea-level rise, even at high sea-level rise rates (Kirwan et al., 2016; Ratliff et al., 2015). As the river bed aggrades (driven by either progradation or sea-level rise), the river will tend to become superelevated most quickly at the break in slope between the terrestrial

34 adjacent ﬂoodplain (leV) adjacent ﬂoodplain (right) river bed 5 sea level

0 pre-avulsion

supereleva5on ra5o 5 approaching 1

0 channel depths

5 post-avulsion (shorter course) 0

0 20 40 60 80 100 120 river cells Figure 3.2: Profile view of river bed and surrounding floodplain geometry. As the river progrades, aggradation causes the channel to become super-elevated relative to the floodplain at the break in slope between the background landscape and horizontal delta lobe. An avulsion is triggered when SER ě 1 at this location (river bed elevation equal to or higher than floodplain elevation), and the river switches course down a new shorter and steeper path.

landscape and the horizontal delta lobe (Figure 3.2). Avulsions consistently occur at this location because of the surrounding geometry of the floodplain, and because they are rarely limited by the shortest path criterion (since the new course is typically shorter for the experiments shown here). Natural and laboratory delta floodplains are unlikely to exhibit a distinct break in slope. Floodplain deposition (including crevasse splay deposition) will tend to diffuse the landscape. In addition, nonuniform subsidence will influence where floodplain elevations tend to be lowest relative to an aggrading river profile. However, the

35 tendency for the depositional river lobes to have vanishingly small slopes is likely robust on natural deltas. In addition, the transition zone in natural landscapes between the approximately horizontal lobes and the higher-slope terrestrial landscape will feature concave-upward curvature analogous to the break in slope in our model experiments. Our model results suggest that avulsions will occur preferentially where this concave curvature is greatest. Most fundamentally, superelevation tends to be greatest where the floodplain concavities are sharpest because the diffusion of the longitudinal profile will tend to be more rapid than the diffusion of the floodplain topography. In the coupled model, the critical superelevation ratio is a first-order control of

LA (Figure 3.3). If the river bed elevation must aggrade to that of the surrounding distal ﬂoodplain elevation (SER=1, Figure 3.3A), then the avulsion length is between 1.5 and two backwater lengths over several orders of magnitude in SLRR˚. When

˚ ˚ SLRR is very small, LA is closer to two, and for high SLRR , LA is closer to 1.5. If SER=0.5 (Figure 3.3B), this value is close to one backwater length. When

˚ SLRR is low, LA scales approximately linearly with SER. This scaling arises because a higher SER requires more aggradation, and therefore more progradation. More progradation leads to a greater length between the preferred avulsion location (spatially ﬁxed when SLRR˚ is low) and the river mouth. In lab experiments (Ganti et al., 2016a), avulsions occurred when the channel ﬁlled approximately 0.3D, and

LA was approximately 0.5 LBW . In our model, lowering the critical SER to 0.3 would decrease LA to close to 0.5 LBW . Because LA controls the extent to which the delta progrades into the basin(Ganti et al., 2016b), our results suggest that delta size also scales with SER. Over a range of wave energy and several orders of magnitude in SLRR˚, a preferential location for avulsions in the coupled river-ocean model arises (Figure 3.3), without including the varying discharge, and associated variations in backwater dy-

36 Figure 3.3: Avulsion length LA for critical SER=1 (left) and 0.5 (right) for a range of wave influence and SLRR˚. Envelopes represent range of values from five sets of numerical experiments, and lines represent average values. LA is slightly less than one backwater length LB for SER=0.5, and between 1.5 and 2 LB for SER=1. namics thought to be essential in explaining avulsion locations (Chatanantavet et al., 2012; Ganti et al., 2014, 2016a). These model experiments suggest an alternative explanation involving the relationship between floodplain geometry and the longitudinal pattern of channel bed aggradation.

The fact that the relationship between LA and SER in laboratory experiments (Chatanantavet et al., 2012; Ganti et al., 2014, 2016a) is consistent with the relationship in our model raises the question of which potential explanation for avulsion locations is relevant in the laboratory experiments. Given that channels are not always well defined and sheet flow outside of the channels is prevalent in the laboratory experiments (Ganti et al., 2016b,a), the ‘floodplain’ deposition likely keeps pace with the aggrading river profile more effectively than it does in our numerical model. Thus, the hydrodynamic explanation seems more likely to be relevant in the laboratory case than the geometrical explanation for the avulsion dynamics

37 in our model. Determining which explanation is relevant for avulsions in natural delta settings awaits future investigation – and the answer might depend on how well channelized and conﬁned the river and its deposition tend to be.

38 4

Exploring wave climate and sea-level rise eﬀects on delta morphodynamics with a coupled river-ocean model

4.1 Abstract

Deltas, which are often densely populated, have become increasingly vulnerable to flooding due to increasing rates of sea-level rise and decreasing sediment supply (Blum and Roberts, 2009; Tessler et al., 2015). These landscapes grow through the successive stacking of delta lobes, as abrupt changes in the river’s course episodically alter the river mouth’s location. The dynamics of these “avulsions” determine the location and size of delta lobes. We use a newly developed coupled model of river and coastal processes to explore how changing wave climates and a range of sea-level rise rates affect river avulsion processes and large-scale delta morphology. Although the relative influences of the fluvial sediment flux and wave influence have long been studied (Galloway, 1975) and more recently quantified (Nienhuis et al., 2015), we find that the sign and magnitude of wave-climate diffusivity also plays an important role in determining not only delta morphology (Nienhuis et al., 2015), but also avulsion

39 dynamics. We also find, surprisingly, that increasing sea-level rise rates do not always accelerate avulsions. In river-dominated deltas, upslope river mouth transgression counteracts aggradation driven by base-level rise – a finding that highlights important differences between wave and river-dominated deltas, as well as between prototypical deltas and experimental deltas created in the lab. The wave climate, sea-level rise rate, and critical superelevation required to trigger an avulsion all play important roles in determining the rates of autogenic variability in delta systems, which has implications for both avulsion dynamics and stratigraphic interpretation of delta stratigraphy.

4.2 Introduction

River deltas and their low-lying, fertile landscapes have long been attractive locations for human settlement. More than half a billion people live on or near deltas, often in large cities (Syvitski et al., 2009; Seto, 2011). Deltas have been a central focus of agriculture and resource extraction (e.g. oil, gas, groundwater), and their channels are useful for transportation and trade. Although society has become increasingly reliant on these landforms, the processes and feedbacks that determine delta morphology are not fully understood, especially over large space and time scales (i.e., space and time scales for multiple river avulsions and the evolution of multiple delta lobes). Deltas and their inhabitants have become increasingly vulnerable to submergence and natural disasters (e.g. flooding, storm surges) as sea level continues to rise and/or sediment delivery rates decrease (Ericson et al., 2006; Syvitski et al., 2009; Tessler et al., 2015). Low subaerial slopes, along with compaction and subsidence processes, lead to widespread flooding on deltas. These floods deliver sediment to the floodplain, where deposition tends to counteract subsidence, allowing the delta to maintain elevation relative to sea level. Through the construction of dams (which

40 trap sediment upstream), artificial levees, and other engineering structures which inhibit sediment delivery to the floodplain (Meade and Moody, 2010; Syvitski and Kettner, 2011), humans have effectively lowered subaerial delta landform elevations relative to sea level, making large areas even more susceptible to flooding or even drowning (Blum and Roberts, 2009; Ericson et al., 2006; Restrepo, 2012; Syvitski et al., 2009). Untangling the relative importance of various natural influences (e.g. waves, rivers, tides, grain size, vegetation, subsidence) has long been the focus delta morphology research (Galloway, 1975; Orton and Reading, 1993; Edmonds and Slinger- land, 2010; Syvitski and Saito, 2007), yet our understanding of the processes driving delta formation and evolution is incomplete. For example, recent work has focused on better understanding the conditions that give rise to avulsions, which occur relatively rapidly and could be catastrophic for human infrastructure (Sinha, 2009). In-channel aggradation causes the river to become super-elevated relative to the surrounding floodplain, prompting flow to relocate to a new or different channel (Mohrig et al., 2000; Slingerland and Smith, 2004). The way river flow interacts with the water in the lake or sea, combined with temporally varying river discharge, can cause a preferential location for in-channel sedimentation. In laboratory deltas, this locus of deposition can drive avulsions at a distance from the river mouth that scales with the backwater length (LB “ D{S, where D is channel depth and S is slope) (Chatanan- tavet et al., 2012; Ganti et al., 2014, 2016a). Aggradation associated with base-level rise (e.g., sea-level rise in the receiving ocean basin) can also increase avulsion frequency (Törnqvist, 1994; Martin et al., 2009). Where waves disperse river sediment alongshore, channel progradation rates and the associated channel aggradation rates decrease, slowing the frequency of avulsions (Swenson, 2005). A better understanding of how changing storminess [affecting offshore wave energy, e.g., Komar and Allan (2008)] and increasing rates of relative sea-level rise (Blum and Roberts, 2009) will

41 affect delta morphology and avulsion processes is increasingly critical for sustainable and effective flood-risk management in these often densely-populated landscapes. Recent numerical modeling work focusing on models in which wave-driven alongshore sediment transport redistributes fluvial sediment has revealed basic insights about wave-influenced delta morphodynamics (Ashton et al., 2013; Nienhuis et al., 2013). These studies have used a fixed river mouth location and static sea-level. Here, we use a complementary new model that couples river processes (including longitudinal profile evolution, floodplain deposition, and dynamic avulsions) and coastline processes (including wave-driven redistribution of sediment and shoreline response to sea-level rise). We use this model to explore how wave climate and sea-level rise affect delta morphology, avulsion dynamics, and autogenic variability [associated with sediment storage and release, e.g., Jerolmack and Paola(2010); Kim et al.(2006)].

4.3 Coupling River-Ocean Processes

Using the framework and tools of the Community Surface Dynamics Modeling Sys- tem (Peckham et al., 2013), we couple the newly-developed River Avulsion and Floodplain Evolution Model (RAFEM) to the Coastline Evolution Model (CEM) (Ashton et al., 2001; Ashton and Murray, 2006a) to explore the processes that shape delta morphology and aﬀect avulsion behavior. The scale-invariant model has been designed to simulate the time scales of multiple avulsions and delta lobe-building events, and to be broadly applicable to a range of river and wave-dominated delta types. In RAFEM, the initial river course is determined using a steepest-descent algorithm that compares the elevations of the two cross-stream and three downstream cells (Jerolmack and Paola, 2007). Cell widths are larger than river channel widths, such that within-channel processes are not resolved, and the channel belt is contained within a single cell. Within the channel cells, natural levees form as the result

42 of overbank deposition adjacent to the river channel (Pizzuto, 1987; Walling and He, 1998; Aalto et al., 2003) and maintain the elevation of bankfull channel depth above the river bed (Jerolmack and Paola, 2007; Hoyal and Sheets, 2009). Combin- ing a slope-dependent sediment flux and a one-dimensional mass balance, erosion and deposition along the river profile are modeled using a linear diffusion equation (Paola, 2000). Because the river mouth elevation is fixed relative to sea level, either progradation or base-level rise will drive channel aggradation. Avulsions in RAFEM occur when two criteria are met. First, a river cell must meet or exceed a critical superelevation ratio (SER), the elevation difference between top of the levees and the surrounding floodplain, normalized by bankfull channel depth (Heller and Paola, 1996; Mohrig et al., 2000). Second, a new steepest-descent path from the avulsion location to sea-level is determined, and if that path is shorter than the existing course, the avulsion occurs (Slingerland and Smith, 2004; Ganti et al., 2016a). If the new steepest path is not shorter than the existing path, then a crevasse splay is deposited around the failed avulsion branch, adjacent to the existing channel (Shen et al., 2015; van Toorenenburg et al., 2016). This deposition approximates the effect of a decreased sediment transport capacity along the new branch (associated with the shallower channel gradient) (Slingerland and Smith, 2004; Hoyal and Sheets, 2009). See Appendix A for a more complete description of RAFEM. The sand-sized sediment flux from RAFEM is reworked along the coastline by CEM (Ashton et al., 2001; Ashton and Murray, 2006a), where gradients in along-

shore sediment transport, Qs, drive erosion and accretion along the shoreline and nearshore bed. Erosion and accretion extend oﬀshore to the shoreface depth, Dsf (below which wave-driven sediment transport becomes negligible). Assuming conservation of nearshore sand and approximately constant long-term shoreface geometry, sediment transport is calculated by:

43 dη 1 dQ “ ´ s (4.1) dt Dsf dx

where η is shoreline position, t is time, and x is the alongshore direction. In CEM, the offshore wave approach angle changes daily, and this mix of influences from different angles (the ‘wave climate’) (Ashton and Murray, 2006a) determines the evolution of coastline shape. Here we use simple wave climates defined by asymmetry (the relative influences of waves approaching from the left and the right, looking offshore) and the fractional influence of offshore waves approaching from high angles (U, where U ă 0.5 has a positive diffusivity and smooths the coastline, and U ą 0.5 has a negative diffusivity, causing coastline perturbations to grow). Both wave height (and to a small degree, wave period) and the spread of wave angles affect wave climate diffusivity, but here we separate these two effects as wave height and the distribution offshore wave-approach angles (Ashton and Murray, 2006b). In the experiments reported here, we use an equal mix of wave influences from the left and right (a “symmetric” wave climate) and a range of U values spanning from 0.1 to 0.7 (from strongly diffusive to anti-diffusive). If the growing delta lobe blocks part of the shoreline from the current offshore wave direction, then that section of shoreline is within the “wave-shadowed zone”, where no sediment transport occurs in order to approximate the effect of decreased wave energy (from wave diffraction and refraction) in and adjacent to in this region. Wave height, representing an effective average value (Ashton and Murray, 2006a), remains constant during each model run.

4.4 Results and Discussion

4.4.1 Wave climate diﬀusivity aﬀects delta morphology

Varying wave heights and wave climate diffusivity (holding background fluvial characteristics constant) leads to different planform delta morphologies (Figure 4.1).

44 Nienhuis et al.(2015) introduced the nondimensional fluvial dominance ratio, R, to express the balance between the fluvial and wave influence:

Q R “ r (4.2) Qs,max

where Qr is the fluvial sand flux and Qs,max is the combined maximum possible alongshore transport to the left and the right of the river mouth. Nienhuis et al.(2015) show that for R values greater than one, fluvial input dominates delta morphology, leading to protruding, crenulated shorelines (no deltaic shoreline angle would be able to transport the provided fluvial sediment). For R less than one, deltas are wave- dominated, with cuspate shorelines were deltaic shoreline angles are generally less than the angle that maximizes alongshore transport.

With small wave heights (left panels in Figure 4.1, R ą 1, where WH˚ is nondimensionalized by channel depth), sediment is delivered to the coastline more quickly than waves can spread it alongshore, leading to initially rugose shorelines (Figure B.3; consistent with Nienhuis et al.(2015)). River progradation is not inhibited by waves, and progradation-driven avulsions cause the river mouth (and locus of fluvial sediment flux to the shoreline) to jump to various locations along the shoreline. The delta platform progressively fills out a smoother, elliptical shape (Figure B.3). Changing the mix of offshore wave approach angles (i.e. changing the wave-climate diffusivity) has little effect on morphology, as the relatively small wave influence is overwhelmed by the incoming fluvial sediment flux.

With larger wave heights (right panels in Figure 4.1, R ă 1), wave-driven alongshore sediment transport redistributes the ﬂuvial sediment as fast as it is delivered, dispersing sand more broadly across the domain Swenson(2005). When diﬀusive

(“low-angle”) waves dominate the wave climate (U “ 0.1), increasing WH˚ decreases the aspect ratio of the cuspate shoreline, which becomes nearly ﬂat as waves

45 WH*=0.1 WH*=0.3 WH*=0.5 U=0.1 U=0.1 U=0.1 R=11.5 R=0.9 R=0.3 diﬀusive

WH*=0.1 WH*=0.3 WH*=0.5 U=0.3 U=0.3 U=0.3 R=16.1 R=1.2 R=0.4

WH*=0.1 WH*=0.3 WH*=0.5 U=0.5 U=0.5 U=0.5 R=19.4 R=1.5 R=0.4

WH*=0.1 WH*=0.3 WH*=0.5 Increasing frac5on of high-angle waves (U) U=0.7 U=0.7 U=0.7 R=17.7 R=1.4 R=0.4 an5-diffusive Increasing wave height (WH*) Figure 4.1: Planview delta morphology with no sea-level rise, critical superelevation ratio (SER) to trigger an avulsion equal to one, varying wave height (WH˚, nondimensionalized by channel depth, increasing from left to right) and fraction of anti-diffusive offshore waves approaching from high angles (U, increasing from top to bottom, A “ 0.5 for all scenarios). The fluvial dominance ratio, R, is greater than one when fluvial sediment delivery dominates and less than one for wave-dominated deltas. Each frame is 20 backwater lengths (LB “ D{S, where D is channel depth and S is slope) wide and 12 backwater lengths high.

spread the sand laterally more effectively. With flatter shorelines, river progradation becomes increasingly inhibited, and associated in-channel aggradation slows. In this case, abandoned delta lobes diffuse away after an avulsion causes the river mouth to relocate elsewhere (Nienhuis et al., 2013). When anti-diffusive high-angle waves

dominate the wave climate (U “ 0.7), alongshore transport aids in driving river progradation, generating large, smooth delta lobes. Relict delta lobes persist long after an avulsion relocates the river mouth, creating cape-like features as a result

46 of the anti-diﬀusive wave climate (Ashton and Murray, 2006a; Ashton et al., 2001) (e.g., bottom right panel of Figure 4.1). In addition to considering R, the sign of wave climate diﬀusivity is important for understanding delta morphology – and, as we discuss next, avulsion processes.

4.4.2 Avulsion Dynamics

The directional spread of oﬀshore wave energy also inﬂuences avulsion dynamics.

If the offshore waves are predominantly approaching from low angles (U ă 0.5, Figure 4.2A,B) and wave energy is sufficiently high, alongshore sediment transport diffuses coastline shape, inhibiting progradation. In this case, progradation-related aggradation is also inhibited, and avulsions tend to take longer to occur. If wave heights are too small to effectively redistribute sediment along the shoreline (i.e.,

fluvially-dominated, R ą 1), avulsions occur relatively rapidly, since progradation is not inhibited with smaller waves. On the other hand, if a larger fraction of wave influences come from high-angle offshore waves (U ą 0.5), even when wave heights are large enough to significantly influence the shoreline (R ă 1), avulsion time scales approach those of the river- dominated deltas (Figure 4.2D). For U ą 0.5, waves can act anti-diffusively to drive transport toward perturbations (e.g., the prograding river mouth). Even for similar, wave-dominated values of R ă 1, avulsion time scales can be quite different (e.g., rightmost column of Figure 4.1) depending on the sign and magnitude of wave climate diffusivity. The length of time before the river avulses can also be affected by the sea-level rise rate, SLRR˚ (see Appendix A for nondimensionalization, Figure 4.2. When waves have a significant influence (R ă 1) and the angular mix of wave influences is diffusive (U ă 0.5), increasing SLRR˚ accelerates in-channel aggradation and causes avulsions to occur more rapidly (Figure 4.2A,B). This result is consistent with laboratory delta

47 Figure 4.2: Time to avulsion (in channel-filling timescales, TCF ) with varying wave height WH˚ and wave climate diffusivity (represented by the fraction of waves approaching from anti-diffusive high-angles, U). Increasing either U or nondimensional sea-level rise rate, SLRR˚, decreases the time to avulsion for wave-dominated deltas (R ă 1), and the time to avulsion is relatively unaffected by either U or SLRR˚ for the fluvially-dominated cases. Envelopes represent 5 sets of model experiments, and the solid line represents the mean. experiments (Kim et al., 2006; Martin et al., 2009), and makes intuitive sense, based on consideration of how base level tends to affect channel aggradation. Since the river mouth elevation is fixed to a channel depth below sea level in our model, rising sea level raises the downstream end of the river longitudinal profile, tending to drive in- channel aggradation as the shape of profile diffuses (Paola, 2000). Increasing SLRR˚ thus tends to increase the rate of channel aggradation, causing the critical SER to be achieved more rapidly. However, unexpectedly, we find that increasing SLRR˚ does not affect the timescale

48 for avulsions for river-dominated cases (R ą 1), and it has very little effect in wave- dominated cases (R ă 1) with an anti-diffusive angular mix of wave influences. This results raises the question of why increasing SLRR˚ has the expected effect on avulsion timescale for wave-dominated, diffusive wave climates (and laboratory deltas), but not for river dominated or anti-diffusive cases. We suggest that the answer in- volves the lateral movement (transgression) of the shoreline and river mouth related to sea-level rise (Bruun, 1962; Wolinsky and Murray, 2009). This transgression has the opposite effect of progradation and tends to inhibit channel aggradation upstream of the river mouth. If, under some conditions, the aggradation-inhibiting effect of transgression exactly counteracts the aggradation-promoting effect of rising base level, SLRR˚ will have no effect on avulsion timescale. Analyzing what determines the ratio of an increment of base-level rise and the amount of resultant transgression leads to a possible explanation for why the aggradation- promoting and aggradation-inhibiting effects will tend to cancel out in some circum- stances. First, when cross-shore sediment fluxes maintain a constant cross-shore profile shape (averaged over storm or seasonal timescales), measured relative to sea level, conservation of mass dictates that the ratio of base-level rise and the resultant transgression distance tends to equal the average slope of the cross-shore profile (Bruun, 1962; Wolinsky and Murray, 2009). This relationship arises because raising the profile requires sediment that that is made available by a corresponding horizontal shift in the profile. For a depositional setting, the horizontal shift can be a decrease in progradation, relative to what would occur in the absence of sea-level rise. (For simplicity, we still refer to a sea-level-rise related decrease in progradation as ‘transgression’.) If the average slope of the profile is steep – e.g., if the vertical height of the profile is large – making available enough sediment to raise the profile induces a relatively small change in horizontal position of the profile. This average slope tends to adjust to match the slope of the landscape over which the system is

49 transgressing (Wolinsky and Murray, 2009). Considering our river-dominated model experiments, the relevant cross-shore profile includes both the shoreface and the portion of the river profile experiencing significant aggradation. (The upstream and offshore limits of this ‘active profile’ are both gradational.) The average slope of this composite profile adjusts through changes in how far seaward the delta extends, relative to the background landscape slope (Figure B.1). If this adjustment is achieved rapidly (relative to the timescale for avulsions), then further increments of sea-level rise, by themselves, will not tend to change the seaward extent of the delta relative to the landscape slope. In other words, the sea-level-rise related changes to the river geometry will not tend to increase the SER by themselves; the aggradation-promoting effect of base-level rise and the aggradation-inhibiting effect of the consequent transgression cancel each other out. Net progradation will still produce avulsions, but in this scenario increasing SLRR˚ will not affect their timescale. If this cancelation is occurring in our river-dominated experiments, what is different in the diffusively wave dominated experiments? In our river-dominated experiments, the shoreface makes up a relatively small portion of the relevant cross-shore profile. If the shoreface extended to a much greater depth, that would tend to increase the average slope of the cross-shore profile, requiring more adjustment of the profile geometry before the average slope equals the landscape slope (where aggradation- promoting and inhibiting effects cancel; Figure B.1). In diffusively wave dominated experiments, the river mouth is connected to the shoreface not just directly offshore of the river mouth, but also the shoreface along the stretch of coastline over which river sediment is redistributed. This connection, in terms of the conservation of mass, acts exactly like a very deep shoreface would; it decreases the amount of transgression that results from the sediment demand of aggrading the river profile in unison with sea-level rise. In other words, the greatly extended area of seabed connected to the

50 river profile greatly increases the ratio of sea-level rise and transgression distance. A higher value of this ratio favors the aggradation-promoting effect of base level rise, over the aggradation-inhibiting effect of transgression. Presumably, if avulsions did not occur, the relevant river-and-shoreface profile would ultimately adjust to equal the slope of the landscape (Wolinsky and Murray, 2009). However, in the diffusively wave-dominated cases, avulsions occur before such an adjustment is completed (Figure B.2). In other words, the aggradation-promoting effect of base-level rise dominates over the transgression effect, and increasing SLRR˚ decreases avulsion timescales. Changing the slope assumed for the nearshore-seabed portion of the cross-shore profile, the shoreface, will tend to change the average slope of whole profile. In model experiments, imposing a steeper shoreface slope (10 times and 100 steeper than that of the background landscape slope) does slightly decrease the time to avulsion at high SLRR˚, in most cases (Figure B.4). This effect is relatively limited, though, as the shoreface represents a small fraction of the profile, which consists mostly of the aggradational portion of the longitudinal river profile. This finding – that increasing SLRR˚ does not necessarily decrease avulsion timescales in model experiments – raises questions about important differences between laboratory and prototypical deltas. Using a morphodynamic depth of closure defined by a depth-dependent diffusivity timescale and wave characteristics (Ortiz and Ashton, 2016), we calculate the shoreface depth appropriate for these model experiments to be four times the channel depth. In laboratory experiments, channel depths are typically much smaller than ‘shoreface’ depths (the depth of the avalanche-slope foreset beds between the shorelines and the basin floor), resulting in relatively very deep surfaces over which progradation is spread. The deep surface means that slight changes in progradation represent large volumes of sediment (analogous to our wave-diffused experiments), so that sea-level-rise related aggrada-

51 tion causes relatively little transgression (decrease in progradation). The fact that laboratory deltas typically have very deep ‘shorefaces’ (relative to the river scales) may explain why laboratory experiments have not yet captured the effect we see in our model, where transgression offsets base-level-rise driven aggradation in the fluvially-dominated cases. In large coastal deltas where the wave influence is relatively small, equilibrium shoreface depths are on the order of tens of meters (Friedrichs and Wright, 2004) and are often not much deeper (if at all) than the corresponding delta channel depth. For example, channel depth at the of the Mississippi River at the Head of Passes, where the river birfurcates in three directions into the Gulf of Mexico, is about 20 meters (Nittrouer et al., 2011), and the nearby shoreface depth is slightly shallower than that (Jaffe et al., 1997). Natural deltas in which offshore deposition of coarse, river bed sediment is restricted to relatively shallow shorefaces will, in this respect, more closely resemble our numerical model than they do laboratory models. Natural deltas in which coarse sediment is directly deposited across deeper foreset beds will more closely resemble laboratory deltas, in terms of the relationship between SLRR˚ and avulsion timescales. The timescale of avulsions also depends on the SER required to trigger an avulsion (Figure B.5), because increasing the critical value of the SER increases the amount of channel aggradation needed. However, the relationship is nonlinear. In- creasing the critical SER from 0.5 to 1 (the range of values estimated from field studies) doubles the amount of aggradation required, but the avulsion timescale in-

creases by a factor of „ 2.5 for R ą 1 and nearly 5 for R ă 1. This nonlinear response arises because as the river progrades, the longitudinal slope decreases, reducing the sediment ﬂux to the shoreline and slowing progradation rates.

52 4.4.3 Autogenic Processes

After an avulsion, the new river course must be shorter (and therefore steeper) than its prior path. This diﬀerence between path lengths before and after an avulsion is a function of wave characteristics and SLRR˚. For ﬂuvially-dominated deltas (small

WH˚ and R ąą 1), this difference in path length is about one backwater length (Figure 4.3), even as SLRR˚ is varied several orders of magnitude. Conversely, increasing SLRR˚ does increase this path length difference for wave-dominated deltas; with very little or no SLRR˚, this difference in river length is on average less than half a backwater length for R ă 1. At higher SLRR˚, however, this change in path length becomes more similar to the fluvially-dominated cases, approaching one backwater length. The path-length difference is a proxy for shoreline shape. The new path from the avulsion node (where the river bed aggrades most rapidly relative to the surrounding floodplain) to the closest location along a rugose shoreline in the river-dominated case is typically shorter than the possible new paths along a wave-dominated cuspate or flat shoreline (Figure 4.3). At high SLRR˚, the shoreline transgresses rapidly into the landscape, tending to bring the shoreline closer to the avulsion node. This transgression effect plays a larger role in wave-dominated cases, because in river- dominated cases, some portion of the rugose shoreline is typically already close to the avulsion node. These differences in path length before and after an avulsion determine the amount of variability in the sediment flux discharged from the river mouth to the shoreline and seabed (Figure 4.4). Peaks in the sediment flux time series represent avulsions, as shortening the river course steepens the channel gradient, causing an abrupt increase in sediment delivery rates. The greater the path difference before and after an avulsion (Figure 4.3), the larger the peak in flux. With in-

53 Figure 4.3: Difference between river course lengths before and after an avulsion for A) critical SER “ 1 and B) critical SER “ 0.5 for a diffusive wave climate (U “ 0.3 in both panels.) Envelopes represent 5 sets of model experiments, and the solid line represents the mean. creasing wave height (in a diffusive wave climate), avulsions occur less frequently and with a smaller path difference, resulting in a sediment flux time series with smal/Users/kmratliff/Box Sync/Home Folder ktm19/Dissertation/ch4/pnas-latex- template/PNAS-template-main.texler and less frequent peaks. In our experiments, the allogenic sediment delivery (externally-forced) is con-

54 Figure 4.4: Sediment flux over time for U “ 0.3 (diffusive wave climate), critical SER “ 1 for three different sea-level rise rates SLRR˚ and wave heights. Peaks in sediment flux correspond to avulsions and sediment fluxes are normalized to the background rate. stant, as are the wave climate characteristics and SLRR˚. Therefore, the variability in sediment flux at the river mouth is all autogenic (from dynamics within the system). For diffusively wave-dominated deltas with little or no SLRR˚, avulsions and changes in slope are infrequent, and the river-mouth sediment flux deviates very little from the mean allogenic sediment discharge (Figure 4.4A,B, WH˚ “ 0.3 or 0.5). In contrast, with increasing fluvial dominance, avulsions and changes in slope occur more frequently, and produce large cyclic peaks in sediment discharge (over three times the allogenic mean; Figure 4.4, WH˚ “ 0.1). Moreover, greater SLRR˚ increases the frequency and magnitude of sediment flux deviations about the mean

55 flux in wave-diffused deltas (Figure 4.4C), resulting from the greater path difference at high SLRR˚ (Figure 4.3). Changing SER also affects autogenic variability; path-length differences are smaller with lower SER (Figure 4.3), but avulsions occur more frequently (Figure B.5), so sediment fluxes will have smaller but more frequently-occurring peaks (Figure B.6) relative to same conditions with a higher critical SER. In summary, changing SER, wave height, or SLRR˚ all affect autogenic variability in the fluvial sediment delivery. Because autogenic processes may overwhelm the allogenic signals recorded in stratigraphy (Kim et al., 2006; Jerolmack and Paola, 2010), understanding the controls on autogenic variability will improve interpretations of paleo-deltaic processes in sedimentary deposits.

4.4.4 Implications and Next Steps

The coupled model omits some of the processes that influence delta morphology and avulsion dynamics. For example, increased rates of floodplain deposition or subaerial landscape diffusion will affect how quickly the riverbed super-elevates relative to the surrounding landscape, therefore affecting avulsion rates. Tidal influence, which affects channel size and stability, as well as variable subsidence rates, can also affect delta morphodynamics. The drowning or erosion of abandoned delta lobes would also affect avulsion dynamics (in river-dominated deltas). In the results presented here, we assume a sufficient inorganic sediment supply for marshes to keep pace with sea-level rise, but the drowning of old lobes post-avulsion is common in natural deltas. If drowning of abandoned lobes were included in the model, we would expect the shoreline to maintain a greater rugosity – and consequently larger path differences –over the timescale of many avulsions (as the river visits a greater and greater proportion of the shoreline; Figure B.3). Peaks in the sediment flux would persist in magnitude

56 (rather than gradually decreasing after repeated avulsions; Figure 4.4), leading to greater autogenic variability. Despite the model simplifications, the basic insights gained from our experiments regarding the effects of sea-level rise and wave climates on long-term, large-scale delta morphodynamics, avulsions, and autogenic signals are likely to be relevant to a broad range of actual river and wave-dominated deltas over large space and time scales. Our findings also highlight the need for further work to understand the most critical similarities and differences in avulsion dynamics between our model, experimental deltas, and deltas in natural settings. Further couplings with additional model components will broaden the scope of questions that can be addressed with the model presented here.

57 Appendix A

Coupled River-Ocean Model Description

A.1 River Avulsion and Floodplain Evolution Model

The River Avulsion and Floodplain Evolution Model (RAFEM) is discretized into fixed square cells, which track either floodplain, river bed, or seafloor elevations. Cell widths are larger than river channel widths (Figure A.1), such that the mean- der or braid-belt is contained within a single river cell, and channel processes are not resolved (Jerolmack and Paola, 2007). Natural levees form as a result of overbank deposition adjacent to the river channel and within the corresponding river cell (Pizzuto, 1987; Walling and He, 1998; Aalto et al., 2003), and the levee elevation is maintained at one bankfull channel depth above the river bed elevation (Hoyal and Sheets, 2009) (Figure A.1B). The river course is determined by a steepest-descent algorithm (Jerolmack and Paola, 2007) which compares the elevations of the three downstream and two cross-stream cells (i.e., no upstream flow is permitted) and it- erates until the river course reaches sea level. In the experiments described here, we initialize a downstream-sloping landscape with a small amount of random variability. Assuming uniform flow, constant boundary shear stress, and a bed shear stress

58 .Fragvnsto ieaeae uniiscaatrsi fagvnchannel, by: given calculated a be of can characteristic coefficient quantities diffusion time-averaged the of set given a For 2007). and channel, river the where process: diffusive and linear erosion a as scales, modeled time are sediments and bed space river large the of over deposition product depth-slope a by balanced cross-section, scale). cell River to B) (not levees width. adjacent channel the and than channel wider containing significantly are cells river A.1 Figure η RB B)# A)# stervrbdelevation, bed river the is oe ceai.A lniwwt ie el eoe nge,where grey, in denoted cells river with Planview A) schematic. Model : cell,cross%sec5on, ν

JrlakadPaola, and Jerolmack 2000; (Paola, coefficient diffusion a is down%stream, floodplain,cells, B cross%stream, η B RB t River,channel, t levees, stime, is “ 59 ν B river,cells, 2 B η 2 RB x x stedwsra oiinalong position downstream the is plan%view, floodplain, (A.1) ? ´8xqyA cf ν ” (A.2) C0ps ´ 1q

where xqy is the long-term average water discharge, A is a river-type dependent constant (A “ 1 for meandering rivers or A ă 1 for braided rivers), cf is a drag coefficient, C0 is the concentration of sediment along the river bed, and s is the sediment specific gravity (Hutton and Syvitski, 2008). The bed elevation at the mouth of the river is a constant channel depth below sea level, such that river progradation or base-level rise will cause a diffusive wave of aggradation to migrate upstream. As the river bed aggrades in absence of equivalent rates of distal floodplain sedimentation (that beyond coarse deposition that forms channel-flanking levees), the river channel will become super-elevated relative to its adjacent floodplain cells (Heller and Paola, 1996; Jerolmack, 2009; Hajek and Wolinsky, 2012). At each time step, the normalized superelevation (Mohrig et al., 2000) is calculated for each river cell:

η ` D ´ η SER “ RB FP (A.3) D

where D is measured as bankfull channel depth and ηFP is the adjacent ﬂoodplain cell elevation. A superelevation ratio of 1 indicates that the channel bed elevation is equal to that of the adjacent ﬂoodplain cells (Figure A.2). Once the critical superelevation ratio is met or exceeded within a particular river cell, a new steepest-descent path to sea-level is determined. If the new path is shorter than the prior course, an avulsion occurs, and the river course switches to the new shorter path (Slingerland and Smith, 2004; Ganti et al., 2016a). If the new path is longer than the previously-existing one, an avulsion does not occur. The longer path to sea level will have a shallower channel bed gradient, and therefore an avulsion is unlikely to be successful (entirely switch

60 A) SER < 1 ﬂoodplain River channel

B) D ηFP η bankfull SER > 1 RB depth Floodplain river bed eleva5on eleva5on

Figure A.2: Normalized superelevation ratio (SER) for two cases: A) SER ă 1, where the river bed elevation ηRB is lower than that of the distal floodplain ηFP and B) SER ą 1, where the river bed elevation is higher than distal floodplain. In the experiments described here, SER ě 1 would trigger an avulsion. course). Flow would decelerate as a result of the lower gradient, reducing sediment transport capacity in the new branch (Slingerland and Smith, 2004; Ganti et al., 2016a). Studies of avulsions in the field (Mohrig et al., 2000) and the lab (Chatanantavet and Lamb, 2014; Ganti et al., 2016a) and indicate that avulsions generally occur at a superelevation ratio of 0.5-1 (Hajek and Wolinsky, 2012), but this value is significantly less for some systems (e.g. Mississippi, ă0.1) (Törnqvistand Bridge, 2002). Other studies (Mackey and Bridge, 1995) have cited an increased ratio of cross-valley to down-channel slope in driving avulsions. Because we are not explicitly resolving levee topography in RAFEM, and greater river channel superelevation will also inherently cause a steeper transverse levee slope, avulsions are triggered using a critical normalized superelevation criterion in the model. Here, we explore model morphodynamics with two superelevation ratios that trigger avulsions, 0.5 and 1. Many field and modeling studies have demonstrated an exponential decay of floodplain deposition to some distance from the channel (Aalto et al., 2003; Pizzuto,

61 1987; Walling, 1999). This deposition of coarser material forms the natural channel levees, which exist within a river cell. A uniform “blanket” of fine material can also be deposited beyond the zone of coarser sediment deposition (i.e. levee deposition) (Karssenberg and Bridge, 2008; Törnqvistand Bridge, 2002), but no studies have constrained the extent of this blanket deposition. Remote sensing data (Syvitski et al., 2012) reveals highly varied floodplain deposit shapes over large spatial scales (e.g., Holocene Yellow River floodplain width up to 700 km), but there is very little data to guide large time and spatial scale floodplain deposition (Törnqvistand Bridge, 2002). In RAFEM, when the river is sufficiently super-elevated to trigger an avulsion, but the new steepest path to sea level has a lower gradient than the current river course, a crevasse splay is deposited (Shen et al., 2015; van Toorenenburg et al., 2016). Splay deposition occurs in the first ’failed’ river channel cell and its immediately adjacent cells at a spatially-averaged rate of in-channel aggradation just upstream of the splay. Although limiting floodplain deposition to crevassing in the model is a highly simplified way of representing variable and complex floodplain depositional structures, this approximation serves to deposit floodplain sediment in locations where overbank flow would be most likely over large space and time scales. In some systems, up to 90% of floodplain deposits consist of crevasse splays (Burns et al., 2017). On floodplains, remnant channel topography can either act as attractors, where overbank flow tends to reoccupy former river channels, or repellors, where remnant levee elevations serve as topographic highs that steer flow away from former channels (Mackey and Bridge, 1995; Jerolmack and Paola, 2007). Other suggest that remnant channel topography diffuses away quickly enough to either attract or repel flow (Ganti et al., 2016a). Here, we do not seek to address the role that abandoned channels play in driving channel path selection, and so abandoned river channels are filled to an

62 elevation that is an average of the adjacent floodplain cells. This diffusive smoothing of channel relicts is an appropriate approximation for the scales of interest in our experiments. Low-lying land on the delta lobe behind the shoreline also maintains a small elevation above sea level, representing organic accretion processes maintained by marshes (Kirwan et al., 2016). Sand-sized sediment that is delivered from the RAFEM river channel is retained in the shoreline. This fraction is calculated by a slope-dependent sediment flux:

dη q “ ´νC RB (A.4) s 0 dx where qs is the mean sediment flux per unit width of the river (Paola, 2000). This sand is reworked along the shoreline using the Coastline Evolution Model (CEM) (Ashton et al., 2001; Ashton and Murray, 2006a), a one-line model where gradients in wave-driven alongshore sediment transport cause erosion and accretion down to the shoreface depth (assuming an approximately constant long-term shoreface profile geometry and conservation of nearshore sand). The wave climate in CEM is based on a probability density function defined by A, the fraction of alongshore sediment transport arising from waves approaching from the left, and U, the fraction of alongshore sediment transport arising from waves approaching from “high” angles, (angle between offshore-approaching wave crests and the shoreline trend is greater than 45˝). For U ă 0.5, the wave climate is diffusive, and waves work to flatten shoreline features, whereas with U ą 0.5, the antidiffusive wave climate tends to grow perturbations along the coast. If a promontory blocks a shoreline cell from the current offshore wave direction, then that location lies within the wave-shadowing zone, where no sediment transport occurs to approximate the effect of decreased wave energy within this region(Ashton et al., 2001; Ashton and Murray, 2006a). Although RAFEM is written in Python and CEM is written in C, the two models

63 are coupled using the CSDMS Basic Model Interface (BMI), a framework developed to simplify the coupling of models (Peckham et al., 2013). Coupled RAFEM-CEM parameter sensitivity studies were performed using the Dakota toolkit (Adams et al.,

2014). We non-dimensionalize time with a channel-ﬁlling time scale, TCF , which is calculated as the amount of time it would take for the river channel within the

backwater length (LB) to aggrade a channel depth, if all the sediment were retained within the river:

D2 T “ (A.5) CF νS2

where S is slope, and LB “ D{S(Paola, 2000). Sea-level rise is imposed on the landscape assuming a quasi-equilibrium generalized Bruun Rule (Bruun, 1962; Wolinsky and Murray, 2009). We calculate sea-level rise rate SLRR˚ [-] by dividing dimensional sea-level rise by the number of TCF per year multiplied by a channel depth D.

64 Appendix B

Supporting Information and Figures

B.1 Aggradational Proﬁle Adjustment with Sea-Level Rise

In Figure B.1, we illustrate sea-level-rise related aggradation and transgression effects associated with the adjustment of the aggradational portion of the river profile and the shoreface (Wolinsky and Murray, 2009). With river sediment input, an additional component of progradation will also be superimposed, acting to adjust the river mouth location. However, for simplicity, here we only illustrate the components of the vertical and horizontal changes in the location of the profile arising from sea-level rise. In other words, in a delta setting, the landward transgression of the river mouth shown here represents a decrease in progradation – a decrease that is caused by sea-level-rise related aggradation. In Figure B.1A, the average slope of the composite accretional profile is steeper than the landscape slope. The ratio between an increment of sea-level rise and the resultant transgression (i.e. decrease in progradation) tends to equal the average slope of the aggradational profile (Bruun, 1962; Wolinsky and Murray, 2009). Thus, the river mouth moves along a trajectory with a slope equaling the average slope of the profile (purple arrow), which is steeper

65 than the landscape slope (orange arrow). In this case, sea-level rise drives a net elongation of the delta. Following the migrating delta in a Lagrangian frame of reference, this elongation corresponds to net aggradation in the river profile as sea-level rises. This in-channel aggradation will tend to raise the profile elevation relative to the background landscape slope as sea level continues to rise. In Figure B.1B, the average slope of the composite profile has adjusted to be closer to that of the landscape slope (but not parallel) after some further increment of sea-level rise. The river mouth trajectory is nearing that of the landscape slope as a result of the in- channel aggradation, which continues to elongate the delta and raise the composite profile relative to the background landscape slope. In Figure B.1C, after a sufficient amount of sea-level rise, the composite accretional profile slope has adjusted such that its average slope equals the background landscape slope. Here, further increments of sea-level rise will not change the position of the river mouth and accretional profile relative to the background landscape slope, thus increasing SLRR˚ does not promote avulsions. (River progradation unrelated to sea-level rise, not shown here, will still produce avulsions – but the rate of those avulsions will be independent of sea-level rise.) After the adjustment in the average profile slope is complete, the aggradation-promoting effect of sea-level rise and the aggradation-inhibiting effect of the consequent transgression cancel each other out. If the timescale for this adjustment in profile slope is shorter than the avulsion timescale, avulsions will not occur more rapidly with increasing SLRR˚. This case is most relevant to river-dominated deltas, where a relatively shallow shoreface allows for more rapid adjustment. In diffusively wave-dominated deltas (and in laboratory deltas), the effective shoreface is much deeper relative to the river channel depth (Figure B.2) and to the shoreface of the fluvially-dominated cases (Figure B.1). For a very deeper shoreface, the average slope of the composite aggradational profile (red line, Figure B.2) is significantly steeper than the background landscape slope (orange arrow). River mouth

66 A)

C) Figure B.1: River mouth trajectory as it adjusts to landscape scope as sea level rises. Brown line represents the background landscape slope, red line is a composite of the accretional portion of the river profile and shoreface, blue line is sea level, green block is marsh behind the shoreline. The purple dashed line represents the average slope of the accretional profile, which is the path of the river mouth trajectory, and the orange dashed line parallels the landscape slope. A) Composite profile steeper than background landscape slope; B) Composite profile adjusting to background landscape slope; C) profile trajectory has adjusted to parallel that of the background landscape slope. Not drawn to scale. Linear shoreface represented here is an approximation of convex aggradational shoreface profiles near prograding river mouths (Friedrichs and Wright, 2004).

67 trajectory will follow this steeper slope, where sea-level rise drives in-channel aggradation (relative to the surrounding landscape, in a Lagrangian frame of reference moving with the river mouth). Consequently, increasing SLRR˚ causes more rapid avulsions. Figure B.2B Illustrates the hypothetical situation where composite proﬁle with a very steep shoreface has adjusted to parallel that of the background landscape slope. In the limit that avulsions could not occur (or would only occur at very high

SER, e.g., SER ąą 1) this adjustment would eventually happen, given suﬃcient sea-level rise; however, because the in-channel aggradation required for this situation to occur is much greater than a channel depth, avulsion timescales are faster than the proﬁle adjustment timescale. In this case, increasing SLRR˚ will tend to drive faster avulsions, because base-level rise will drive in-channel aggradation that is not completely canceled out by the associated upslope transgression (Figure B.1C).

68 A)

SER=1

B) Figure B.2: In diffusively wave-dominated deltas, the effective shoreface is very deep relative to the river channel depth and consists of a greater proportion of the composite accretional profile relative to the fluvially-dominated case.. Brown line represents the background landscape slope, red line is a composite of the accretional portion of the river profile and shoreface, blue line is sea level, green block is marsh behind the shoreline. The purple dashed line represents the average slope of the accretional profile, which is the path of the river mouth trajectory, and the orange dashed line parallels the landscape slope. A) Example showing a very steep shoreface, where the slope of the composite profile is significantly steeper than the background landscape slope. B) Hypothetical situation where composite profile with a very steep shoreface has adjusted to parallel that of the background landscape slope. This adjustment is unlikely in natural deltas, as the amount of in-channel aggradation required to achieve this adjustment would be many channel depths. (For example, the amount of aggradation required to trigger an avulsion with the critical SER “ 1 is illustrated by the small black line.) Not drawn to scale.

69 hr eiettasotsrassdmn ln h hrln n niisprogra- inhibits and shoreline the along- right, along avulsions). the sediment (and progradation on spreads dation on delta wave-dominated transport limitation the wave sediment In little shore with avulsions). frequent left, relatively the (and on morphology delta dominated ( B.3 Figure U

=3), 5me t = 4587 T t = 3058 T t = 1529 T R=16.1 SER WH*=0.1 lniwdlamrhlg hog iefradﬀsv aeclimate wave diﬀusive a for time through morphology delta Planview : CF CF CF 1 n ag of range a and =1,

increasing wave influence R=1.2 H W WH*=0.3 ˚ ae r o ml oiflec river- influence to small too are Waves . 70 R=0.4 WH*=0.5 Figure B.4: Time to avulsion with an imposed shoreface steepness that is 10x or 100x steeper than that of the terrestrial landscape. At high SLRR˚, increasing shoreface steepness does slightly decrease the time to avulsion.

71 Figure B.5: Time to avulsion in a diﬀusive wave climate (U “ 0.3) for A) SER “ 1 and B) SER “ 0.5. Avulsions occur more rapidly in B) as half the amount of in- channel aggradation is required to trigger an avulsion. Envelopes represent 5 sets of model experiments, where the solid line represents the mean.

72 3.0 WH =0.1, R=16.1 SLRR ∗ = 0 ∗ WH =0.3, R=1.2 2.0 ∗ WH ∗=0.5, R=0.4

1.0

0.0 3.0 5 SLRR ∗ = 10− 2.0

1.0

0.0 3.0 normalized sediment flux 3 SLRR ∗ = 10− 2.0

1.0

0.0 0 200 400 600 800 1000 1200 1400

time (channel-filling timescale, TCF) Figure B.6: Sediment flux over time in a diffusive wave climate (U “ 0.3) with critical SER “ 0.5. Avulsions occur more frequently with smaller differences in path length relative to the SER “ 1 case (see main text), so the frequency of peaks here is greater, but the magnitude is smaller.

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85 Biography

Katherine Murray Ratliff was born in Nashville, Tennessee, on December 6th, 1988. She graduated from the University School of Nashville in 2007 and earned a B.A. with Highest Honors in Earth and Environmental Sciences from Vanderbilt University in December 2011. Katherine has published three first-author publications: Ratliff, K.M., Braswell, A.E., and M. Marani (2015), “Spatial response of coastal marshes to increased atmospheric CO2” in Proceedings of the National Academy of Sciences, U.S.A.; Ratliff, K.M. and A.B. Murray (2014), “Modes and emergent time scales of embayed beach dynamics” in Geophysical Research Letters; and Murray, K.T., Miller, M. F., and S. S. Bowser (2013), “Depositional processes beneath coastal multi-year sea ice,” in Sedimentology. Katherine was awarded the Orrin Pilkey Fellowship from Duke University in 2012, the National Science Foundation Graduate Research Fellowship (which funded the last three years of her Ph.D.) in 2014, and the the Jo Rae Wright Fellowship for Outstanding Women in Science from the Duke University Graduate School in 2016. Katherine was awarded the Best Poster Award at the Community Surface Dynamics Modeling System Annual Meeting (Boulder, CO) in 2015, the Best Student Poster Award at the 9th Symposium on River, Coastal, and Estuarine Morphodynamics (Iq- uitos, Peru) in 2015, and the Dean’s Award for Outstanding PhD Student Manuscript from the Nicholas School of the Environment in 2016.