Wave mud interaction over the muddy Atchafalaya subaqueous clinoform, Louisiana, United States: Wave processes

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jc006644, 2011 Wave mud interaction over the muddy Atchafalaya subaqueous clinoform, Louisiana, United States: Wave processes A. Sheremet, 1 S. Jaramillo, 2 S. F. Su, 1 M. A. Allison, 3 and K. T. Holland 4 Received 7 September 2010; revised 12 January 2011; accepted 14 March 2011; published 15 June [1] Observations of wave and sediment processes collected at two locations on the Atchafalaya inner shelf show that wave dissipation in shallow, muddy environments is strongly coupled to bed sediment reworking by waves. During an energetic wave event (2 m significant wave height in 5 m water depth), acoustic backscatter records suggest that sediment in the surficial bed layer evolves from consolidated mud through liquefaction, fluid mud formation, and hindered settling to gelled, under consolidated mud. Net swell dissipation increases steadily during the storm from negligible prestorm values, consistent with bed softening, but shows no correlation with detectable fluid mud layers. Remarkably, the maximum dissipation rate occurs poststorm, when no fluid mud layers are present. In the waning stage of the storm, the contribution of different wave forcing processes to wave dissipation is analyzed using an inverse modeling approach based on a nonlinear three wave interaction model. Although wave mud interaction dominates dissipative processes, nonlinear three wave interactions control the shape of the frequency distribution of the dissipation rate. In the wake of the storm, the viscosity values predicted by the inverse modeling converge toward measured values characteristic for gelled mud in a trend that is consistent with a fluid mud entering dewatering and consolidation stages. Citation: Sheremet, A., S. Jaramillo, S.-F. Su, M. A. Allison, and K. T. Holland (2011), Wave mud interaction over the muddy Atchafalaya subaqueous clinoform, Louisiana, United States: Wave processes, J. Geophys. Res., 116,, doi: /2010jc Introduction [2] The dissipative effect of muddy seabeds on wave propagation is well known. Field observations [e.g., Wells and Coleman, 1981; Jiang and Mehta, 1996; Mathew et al., 1995; Elgar and Raubenheimer, 2008; Rogers and Holland, 2009], and laboratory experiments [e.g., Gade, 1957, 1958; Jiang and Mehta, 1995; De Wit, 1995; Hill and Foda, 1999; Chan and Liu, 2009; Holland et al., 2009] show that as much as 80% of wave energy can dissipate due to wave mud interaction over a distance of just a few wavelengths. [3] Theoretical efforts to understand wave mud interaction have resulted in a range of rheological models and dissipation mechanisms. Mud has been described as a viscous Newtonian fluid [Dalrymple and Liu, 1978; Ng, 2000; De Wit, 1995]; viscoelastic solid [Jiang and Mehta, 1995]; viscoplastic Bingham material [Mei and Liu, 1987; Chan and Liu, 2009]; or poroelastic material [Yamamoto et al., 1978; 1 Civil and Coastal Engineering, University of Florida, Gainesville, Florida, USA. 2 Ocean and Resources Engineering, School of Ocean and Earth Science and Technology, University of Hawai i atmānoa, Honolulu, Hawaii, USA. 3 Institute for Geophysics, University of Texas at Austin, Austin, Texas, USA. 4 Seafloor Sciences Branch, Naval Research Laboratory, Stennis Space Center, Mississippi, USA. Copyright 2011 by the American Geophysical Union /11/2010JC Yamamoto and Takahashi, 1985] (other mud mediated processes have also been hypothesized to contribute to wave damping, such as nonlinear interactions between surface and interfacial waves at the water mud interface [Jamali et al., 2003]). [4] In addition to the practical difficulty to obtain direct measurements of relevant physical quantities such as mud layer thickness, mud density, and viscosity, the applicability of most of these models for field data is also limited by their relatively simple description of rheological behavior of mud (i.e., linear, with characteristic parameters such shear modulus or shear viscosity assumed independent of strain rate amplitude). The complexity of nonlinear rheological models [Chou, 1989; Chou et al., 1993; Mei and Liu, 1987] has precluded their application to field observations. However, there is considerable observational evidence [Sheremet et al., 2005; Jaramillo et al., 2009; Robillard, 2009], that bed reworking by waves results in significant changes in the bed rheology. [5] Waves soften the muddy bed, mobilize and resuspend sediment, which then settles at the end of the storm, forming fluid mud layers that slowly dewater and consolidate. If given enough time, the bed eventually recovers to the prestorm solid state. Jain and Mehta [2009] proposed a classification of the range of applicability of different rheological models based on the characterization of sediments using the Peclet number and the fractional volume of solids. Although their goal was slightly different, their Figure 11 can be read as a map of possible states sediment can go 1of14

2 through during a storm. Conceptually, between the extremal states of solid (noncompliant) bed (low Peclet numbers, high solids volume fraction) and Newtonian viscous fluid (high Peclet number, low sediment content) there is a gradation of transitional states, including Bingham plastic, viscoelastic solid/fluid, viscous fluid. Dynamically, any of these states could be realized, provided that bed evolution results in the corresponding characteristics (Peclet number, sediment content). Only some of these states strictly qualify as fluids, but all have dissipative properties. Moreover, because shear stresses vary with the distance from the bed, it seems reasonable to assume that various rheological states can coexist and possibly form stacked layers, with the softer states on top of the stiffer ones. [6] Based on analysis of acoustic backscatter, Jaramillo et al. [2009] identified (or conjectured, in some cases) several distinct stages in the response of bed sediment to wave activity: consolidated mud state; initial sediment mobilization (possibly through liquefaction, sediment water mixing and erosion), resuspension, and hindered settling, leading to the formation of a fluid mud layer of gradually decreasing thickness. This sequence of processes is seen typically during energetic swell events, and might not be seen for short wave events of low energy swells. For example, in the case discussed by Jaramillo et al. [2009] and reexamined below, sediment mobilization coincides with swell significant height exceeding 1 m. As such, the process appears to be dependent on the characteristic frequency and amplitude of the wavefield. [7] Sheremet and Stone [2003] reported unexpected shortwave (frequency higher than 0.2 Hz) dissipation in muddy environments, and hypothesized that this is due to triad (three wave) nonlinear coupling between short and long waves, the latter interacting efficiently with the bed. The implication that nonlinearities are not suppressed by the strong mud induced dissipation is supported by subsequent numerical experiments [Kaihatu et al., 2007], but has yet to be confirmed by observational data. Net wave dissipation estimates represent coupled effects of multiple dissipation/ growth mechanisms that can include, in addition to mudinduced dissipation, depth limited breaking, wind growth, whitecapping, and nonlinear triad interactions. Evaluating the contribution of each of these mechanisms is difficult without the use of numerical models. I turn, due to the wide range of wave processes that become active in shallow water (such as depth limited breaking, refraction, shoaling, nonlinear wave interactions, wave current interaction and others) the shallow water environment is in itself a challenging environment for numerical wave models. [8] The initial motivation of this study was the desire to understand the role and magnitude of the various constituents of the net dissipation. This, however, cannot be achieved without a quantitative description of mud induced wave dissipation. In the absence of direct observations of bed rheology, we estimate the equivalent viscosity of the bed sediment using the inverse modeling approach proposed by Rogers and Holland [2009], modified to use a nonlinear wave model (Nonlinear Mild Slope Equation (NMSE)) [Agnon et al., 1993; Agnon and Sheremet, 1997, 2000], that accounts for shallow water, nonlinear three wave interactions. The sediment characterization derived from the inverse modeling approach provides an opportunity to corroborate some of the proposed evolutionary sequence of bottom mud rheology. 2. Field Experiment 2.1. Site and Instrumentation [9] Details of the experimental site and instrumentation arediscussedinacompanionstudybyjaramillo et al. [2009]. Here, we review elements essential for the scope of this paper. The experiment site (Figure 1) is located shoreward of the 5 m isobath on the nearly flat (maximum slope less than 1:1000) topset of the Atchafalaya mud subaqueous delta [Neill and Allison, 2005] in southwestern Louisiana. In winter and early spring (November April), the Atchafalaya shelf and the adjacent muddy chenier coast are swept periodically (about 3 10 day intervals) by cold atmospheric fronts accompanied in the prefrontal phase by energetic onshore swells (8 10 s period, 1 2 m wave height). Due to basin wide influences, these storms usually coincide with rising or high Mississippi Atchafalaya River water and sediment discharge. On the shelf, wave induced bottom turbulence resuspends significant quantities of sediment. As wave activity decreases at the end of the storm, and the direction of propagation of short wave fields rotates to align with postfrontal seaward winds, hindered settling of suspended sediment leads to the formation of episodic fluid mud layers with a duration of about 12 h, and thickness less than 30 cm [Allison et al., 2000; Draut et al., 2005; Kineke et al., 2006; Jaramillo et al., 2009]. The inner Atchafalaya shelf was chosen with the purpose to observe large scale wave propagation in shallow water (the 10 m isobath is in some places about 50 km offshore), mud rich environments. [10] Observations of wave and suspended sediment concentration were collected by two instrumented platforms ( T1 and T2 ), deployed in a cross shore array near the 5 m isobath on the Atchafalaya shelf (Figure 1) between 15 February and 1 April The mean water level was approximately 5 m at T1, and 4.3 m at T2, and the platforms were separated by an approximate distance d = 3800 m. The instruments recorded continuously in intervals of approximately 14 days, interrupted by 1 2 days necessary for data downloading, cleaning of instruments and their redeployment. [11] The analysis presented here is based on wave observations (pressure, velocities, and acoustic surface track) and acoustic backscatter intensity collected by velocity profilers. Each platform was equipped with downlooking, 1.5 MHz PC ADPs (pulse coherent acoustic Doppler profiler, Sontek YSI), sampling the velocity at 2 Hz in 17 vertical bins (3 cm height) in the first 50 cmab (cm above the bed). At T1, an uplooking 1 MHz AWAC current profiler (Nortek) with AST capability (acoustic surface track, vertically oriented transducer echo ranging to the surface) was used to sample the water surface displacement at 4 Hz. An additional pressure transducer (Paroscientific) recorded pressure at 2 Hz. Salinity and temperature observations collected by a Seabird MicroCAT mounted at 115 cmab (cm above bed) were used to estimate water density. More details about instrument location and sampling scheme are given by Jaramillo et al. [2009]. 2of14

3 Figure 1. (a) Qualitative map of surficial sediments on the Atchafalaya inner shelf [Neill and Allison, 2005]. The approximate locations of the tripods deployed in 13 February to 14 March 2006 are marked by circles. (b) Smoothed estimate of bathymetry profile (solid line) along a transect through the two experiment sites (circles), based on NRL survey data (May 2007, dark gray dots), and fathometer readings from the R/V Pelican (March 2008, light gray dots). Maximum slope is , reached at crossshore distance of 5 km, slightly offshore of the offshore platform T Data Analysis [12] Pressure observations were processed using standard spectral analysis procedures. Time series segments of 20 min length were detrended and de meaned, then divided into 128 s blocks with 50% overlap, and tapered using a Hanning window. The resulting spectra have approximately 18 DOF (degrees of freedom), with a frequency resolution of Hz. Wave spectra were corrected to account for the mean depth of the sensors using linear wave theory, with a high frequency cutoff defined by a depth attenuation of wave variance larger than 95%. [13] Significant wave height values pffiffiffiffiffiffiffiffiffiffiffiffiffiffi were calculated based on the first spectral moment, 4 SðfÞdf, where S(f) is the spectral density of wave variance and f the frequency. Bispectral estimates B(f 1, f 2 ) were calculated for 1 h segments of free surface time series derived from the AWAC surface track, with the same frequency resolution, and approximately 85 DOF. The real part of the normalized bispectrum [Herbers et al., 1994] Bf ð 1 ; f 2 Þ bf ð 1 ; f 2 Þ ¼ ; ð1þ 1=2 ½Sf ð 1 Þðf 2 ÞSf ð 1 þ f 2 ÞŠ was used to detect nonlinear phase coupling (effects of nonlinearities) between different frequency bands. The imaginary part of the bispectrum is theoretically zero. [14] The standard definition of the modal energy flux F(f, x) dissipation rate is the relative change of the energy flux per unit distance x dfðf ; xþ dx ¼ 2Fðf; xþ; ð2þ where is the attenuation rate of the amplitude of mode with frequency f, and is often introduced as an imaginary modal wave number. In general, is a complicated function of the position vector x, the wave frequency and amplitude, and bed sediment characteristics (e.g., viscosity, density, thickness, yield stress etc). The modal energy flux is F(f, x) = S(f, x)c(f, x)df, where C is the modal group velocity, and df is the frequency bandwidth. [15] A simple measure of swell dissipation can be derived by applying a finite difference discretization of equation (2), and a mean value theorem for integrals P Swell Swell Dissipation ¼ 1 F ð f; x 1Þ P Swell F ð f; x 2Þ 2 SwellDx; ð3þ where Swell can be regarded as the characteristic (band averaged) swell dissipation, and x n is the position of platform Tn, with n = 1, 2. In equation (3), the summation is done over the swell band, 0.05 < f 0.2 Hz. The frequency distribution of the dissipation rate was estimated as ðf Þ ¼ 1 2Dx ln F ð f; x 2Þ Fðf; x 1 Þ : ð4þ [16] The distance D x = dcosa is the effective propagation distance between T1 and T2, where d is the distance between tripods, and a is the propagation angle with respect to the T1 T2 axis of the peak of the directional spectrum at frequency f. [17] The estimates defined in equations (3) and (4) are measures of net wave dissipation/growth, as they do not distinguish between different processes such as wind growth, 3of14

4 Figure 2. Wind and wave observations at T1 during the 10 March storm, versus time: (a) wind direction and intensity (thick line, with color coded directions; wind speed axis is the right axis), significant heights of sea (frequency > 0.2 Hz, red) and swell (frequency 0.2 Hz, blue), (b) evolution of normalized (total variance = 1 m 2 ) frequency spectra, and (c) propagation direction of the peak of the directional spectral density for each frequency band (derived from acoustic Doppler current profiler measurements). Wind and wave propagation directions are represented as flow directions (e.g., N means propagating northward). Surface wind is numerically simulated using COAMPS [Hodur, 1997], Naval Research Laboratory, Stennis Space Center, Mississippi. whitecapping, breaking, mud induced damping, and nonlinear interactions. [18] For the event studied here, the propagation angle held steady at approximately a = 40, yielding D x = 0.77d = 2,926 km. The use of the effective distance is justified by the small bottom slope and the negligible swell refraction, which resulted in a change of propagation direction of less than 2 between the two platforms (based on numerical simulations, Appendix A). To reduce roughness due to use of low DOF spectra, (f) estimates were further smoothed by band averaging (in 21 frequency bands with Hz frequency width), and time averaging using a running mean with a 3 h window width. [19] Because the analysis below is based on data collected at only two measurement sites, with no information available regarding the spatial variability of various processes (sediment transport advection, convergence, or divergence effects, distribution of fluid mud layers, etc), the conclusions of this study should be regarded as preliminary. However, the relatively short (about 1 day) period analyzed here and the flat bathymetry of the area, suggest that fluctuations in the large scale circulation or lateral sediment influx might not play an important role in wave dynamics. The spatial structure of the bottom sediment state is probably more important, and might present specific variability issues (e.g., fingering of fluid mudflows). Based on previous survey data and our observations at the two sites [Kineke et al., 2006; Jaramillo et al., 2009], we assume a spatially nearly uniform distribution of fluid muds, with a thickness slightly decreasing onshore. 3. Observations [20] The major event of the 2006 experiment was the frontal storm that passed over the site on 10 March, with sustained 10 m/s southerly winds out of the South and seas (waves with frequency f > 0.2 Hz) between 0.5 and 1 m significant height (Figures 2a and 2b). These conditions lasted for more than 4 days and resulted in swells that peaked 1.5 m significant height reached in 5 m water depth. Throughout the storm, seas propagation direction was approximately northward (Figure 2c). Swell intensity was correlated with (slightly lagging) the two peaks in wind intensity. Swells (f 0.2 Hz) arrived at the T1 site early 9 March, peaked the evening of 9 March (1.5 m significant height) and again briefly at midnight 10 March (1.0 m significant height). Mean swell propagation direction was N NNW, Figure 2c). [21] The response of bed sediment to wave activity has been discussed in detail by Jaramillo et al. [2009]. In summary, Figures 3a and 3b (see labels) suggests that, for strong enough events, the bed sediment responds to swell activity in several distinct stages: starting from a consolidated mud state, the bed sediment is rapidly mobilized (in this particular case when swell approaches 1 m height) and resuspended, forming a lutocline at the peak of the storm 4of14

5 Figure 3. (a) Significant height of seas and swells, (b) intensity of PC ADP acoustic backscatter observed at T1, (c) net swell dissipation rate between T1 and T2 (equation (3)), and (d) normalized (peak value = 1) frequency distribution of the swell dissipation rate ((f), equation (2)), versus time. In Figure 3b, black lines mark the position of peak backscatter intensity (top line), and the position where the PC ADP records zero velocity (hydrodynamic bottom, bottom line). As discussed by Jaramillo et al. [2009], a distance between these markers larger than 3.2 cm (height of PC ADP measurement bin) indicates the presence of a mobile mud layer. In Figure 3c, dots are hourly estimates of swell dissipation, solid line is a running average, and dashed line with circles is the mean water level at T1 (axis on right). Before 8 March 0900 UT, swell variance is too low to yield reliable dissipation rates. In Figure 3d, circles mark the position of the spectral peak. In Figures 3c and 3d, positive values represent dissipation and negative values represent growth. Labels are discussed in the text. The black rectangle marks the period analyzed using numerical models. (fluid mud event 1 of Jaramillo et al. [2009]). Subsequent hindered settling results in the formation of a well defined fluid mud layer of gradually decreasing thickness (fluid mud event 2 of Jaramillo et al. [2009]). [22] Here, fluid muds are identified in PC ADP observations as the mobile layer between the topmost maximum backscatter surface and the zero mean flow surface. This is a narrow, practical definition of fluid muds, based on the capability of our instruments to detect density gradients and motion. The fluid muds identified this way will be referred to in the sequel as detectable fluid muds, to distinguish them from the physical fluid mud concept. Detectability reflects instrument limitations. For example, very dilute suspensions with weak backscatter, layers with thickness of the order of the instrument measurement bin (3.2 cm), or very dense layers (opaque to sound) might not be detectable. Such dense, thin (undetectable) fluid muds could conceivably exist in the field during our field experiment, their presence expressed in measurable wave dissipation. [23] Swell dissipation (equation (3)) increases steadily during the storm (Figure 3c), qualitatively consistent with the assumption that dissipation is mainly driven by bed softening (reworking) by waves [e.g., Sheremet and Stone, 2003; Elgar and Raubenheimer, 2008; Rogers and Holland, 2009]. Hourly estimates are correlated to the tidal phase (higher dissipation at low tides) likely due to increased wave bottom interaction and increased wave breaking. The detided evolution of swell dissipation shows no significant correlation with the two fluid mud events [Dalrymple and Liu, 1978; Ng, 2000], possibly due to the detectable mud layers being too dilute, and/or having a discontinuous spatial distribution (unknown beyond monitoring sites). Remarkably, swell dissipation continues to increase in the wake of the storm, even after any detectable fluid muds have disappeared (peaks at 50% incoming energy flux lost from T1 to T2). [24] At this stage, wave mud interaction is the dominant wave dissipation mechanism, with whitecapping and breaking likely weak due to low swell energy. Because the acoustic instruments do not detect a fluid mud layer, but wave dissipation is significant, and taking into account that bottom sediment is in a poststorm consolidation stage, we speculate that the mud is in a space filling (gel) state at incipient stages of dewatering. This state, routinely observed 5of14

6 Figure 4. Wave dissipation rate predicted by the Newtonian model [Ng, 2000] versus wave frequency and mud kinematic viscosity, for water depth h = 5 m and a fluid mud thickness of d m = 10 cm and mud density r m = 1050 kg/m 3 (values characteristic of Atchafalaya conditions). The distribution is unimodal (peaks marked by black lines) in both frequency and viscosity. Dashed lines mark the frequencies (constant viscosity) for which the dissipation decreases to 50% of the peak. For a fixed frequency, there are two values of viscosity that result in the same dissipation rate, one in Region 1 (low viscosity domain) and the other in Region 2 (high viscosity domain, grayed). in the laboratory and in field cores taken after storms [see, e.g., Elgar and Raubenheimer, 2008; Robillard, 2009], is probably best characterized as either a viscoelastic fluid or solid, depending on its characteristic Peclet number and fractional volume of solids (see Appendix C [also Jain and Mehta, 2009]). However, rather than speculate on the properties of this state, we will refer it using the nonspecific terms of soft, or under consolidated bed (Figure 3). The underlying assumption is that the bed participates in the motion of the overlying water column, but is too dense to be probed by acoustic backscatter. The strong wave dissipation associated with this state suggests that this stage in the evolution of bed rheology is highly relevant for wave dynamics. This stage is not mentioned by Jaramillo et al. [2009], because it is not detectable from acoustic backscatter data, and that paper did not examine wave dissipation effects. [25] Despite the apparent domination of mud induced wave dissipation, the complex structure of the frequency distribution of the dissipation rate (Figure 3d) cannot be explained by bottom interaction alone. The peak of the distribution is strongly correlated with the spectral peak, and its width shows rapid and significant changes. The width of the dissipation band ( > 0) is on average 0.2 Hz, however, between noon and midnight 10 March, it decreases from 0.3 Hz to 0.15 Hz. In general, the dissipation band appears to narrow when swell activity peaks (e.g., evening of 9 and 10 March). This should be contrasted with the comparatively wider distribution predicted by the Newtonian fluid mud model [e.g., Ng, 2000], shown in Figure 4 (see Appendix B), for typical Atchafalaya conditions (5 m water depth, 10 cm mud layer thickness, and fluid mud density 1,050 kg/m 3 ). In contrast with the observations, the modeled width of, say, the frequency band (n) > 0.5 max f (f, n)) is approximately 0.3 Hz and shows almost no variability in the lower viscosity range (Region 1, Figure 4). Also, in the same domain, the position of the dissipation peak only changes by approximately 0.05 Hz, much less than the shift observed in the field (Figure 3d). Following the discussion of the potential complexity of the reworked bed state, the Newtonian viscous fluid model is probably a good approximation for bottom processes before the initial sediment mobilization and during the fluid mud events (but not for the end of the settling period and the final soft mud stage of the storm, Figure 3b). [26] The variability of the observed swell dissipation rate (Figure 3d) suggests that other processes. such as wind input, whitecapping, depth limited breaking, cannot be neglected. The growth rates ( < 0) at high frequencies are probably due to wind input. The growth in the infragravity (f < 0.05 Hz) band, the correlation with spectral peak, and the variability of the width of the dissipation band can in fact be explained as effects of nonlinear triad interactions (see below). [27] Bispectral estimates (a measure of wave nonlinearity [e.g., Herbers et al., 1994]) during the waning stage of the storm (Figure 5) do indicate various intensities of nonlinear coupling. On 10 March 1500 UT, the bispectrum (Figure 5b) shows a positive peak at (f Peak, f Peak ), and a weaker negative one at (f Peak, 0.05 Hz), with f Peak the frequency of the spectral peak. The positive bispectral peak indicates coupling between the spectral peak and its second harmonic (0.1 and 0.2 Hz frequency bands), the negative bispectral peak indicates coupling between the spectral peak and lower frequencies. Nonlinear interaction between the spectral peak and its second harmonic strengthens as the storm intensifies on 10 March 2100 UT (Figure 5d) and decreases to negligible values on 11 March 1100 UT (Figure 5f). In contrast, the coupling between the peak and the lower frequencies remains about the same in the first two examples and strengthens in the last one. 4. Inverse Modeling of Bottom Mud State [28] The goal of this study is to separate the constituents of net wave dissipation, while deriving a quantitative description of bed sediment state (through the inverse modeling approach used). This method also provides an opportunity to investigate the proposed evolutionary sequence of bottom mud rheology. [29] Untangling the effects of the evolving bed state from other wave growth/dissipation mechanisms might not be trivial, in particular due to the possibility of wave nonlinearities (see section 3) modifying the balance of growth/dissipation terms. For example, mud induced dissipation is typically limited to the swell band (e.g., Figure 4 for Newtonian muds), while wind input/whitecap dissipation are active at the highfrequency end of the spectrum [Van der Westhuysen et al., 2007]. Despite their relatively well separated spectral domains, mud induced dissipation can partially replace whitecapping 6of14

7 Figure 5. (a, c, e) Observed spectral density of wave variance versus frequency (vertical axis scaling is not the same in Figures 5a, 5c, and 5e). (b, d, f) Corresponding real part of the normalized wave bispectrum (equation (1)). The imaginary part of the bispectrum is statistically negligible. The estimates are for 10 March 1500 UT (Figures 5a and 5b), 10 March 2100 UT (Figures 5c and 5d), and 11 March 1100 UT (Figures 5e and 5f); events marked by 1, 2, and 3 in Figure 6. effects, due to triad interactions transferring energy from high frequencies to the swell band. [30] The effects of triad interactions are accounted for in this study by the use the stochastic Nonlinear Mild Slope Equation [Agnon and Sheremet, 1997, 2000]. Based on preliminary SWAN (Simulating Waves Nearshore) runs (Appendix A) showing weak swell refraction, numerical simulations were performed using the unidirectional version of the NMSE model, which allowed for a significant increase of computing speed. While the results show a good agreement with observations, neglecting the directional spread is expected to lead to an overestimation of energy transfer toward low frequencies. To simplify the description of bed rheology (below, and Appendix C), the numerical analysis presented here focuses on the period from 10 March 1200 UT to 11 March 1800 UT (rectangle in Figure 3), when the bed state can be identified with some certainty as fluid mud. The model was initialized using spectral observations at T1, and simulated spectra at T2 were compared with observations. Tide elevations and surface currents were derived from observations (1 h averages, assumed spatially constant over the integration domain). Surface winds were produced by the COAMPS (Coupled Ocean/Atmosphere Mesoscale Prediction System) model [Hodur, 1997], run at the Naval Research Laboratory, Stennis Space Center. [31] The NMSE model was supplemented with numerical modules for wind input, whitecapping and mud induced dissipation (Appendix B). SWAN overall balance of wave energy sources (Appendix A) indicates that depth limited breaking was not active during the period studied here. The Newtonian fluid description proposed by Ng [2000] was used to describe mud induced wave dissipation. While two layer Newtonian models [Dalrymple and Liu, 1978; Ng, 2000] are numerically efficient mud representations that capture essential features of mud induced wave dissipation, it is unlikely that field reality (see discussions in sections 1 and 3) is accurately described by a vertically uniform, Newtonian viscous fluid mud over an absolutely rigid bed. We speculate that effects such as increasing nearbed sediment density, as well as high density erosional layers generated by fluctuating bed stresses (e.g., due to tidal oscillations) could produce multilayer mobile muds, with non Newtonian, soft mud components in the vicinity of the bed. In such cases, the use of two fluid Newtonian models for forecasting wave dissipation processes should be interpreted as a vertically averaged, equivalent description (e.g., yielding an equivalent viscosity). The mud characteristics obtained this way should represent well the mud state for nearly Newtonian layers. However, if the mud state is far from Newtonian, a correction for high densities becomes necessary. The model proposed by Ng [2000] was modified by introducing an effective kinematic viscosity n eff (r m ) that depends only on the density of the fluid mud r m, yielding for the dissipation rate a dependency of the form (Appendices B C) M ¼ M f ; h; d m ; eff ð m Þ ; ð5þ where h is the local depth and d m is the mud layer thickness (subscript m refers to mud quantities). From equation (5), if the mud layer thickness can be estimated from observations, the dissipation rate is only a function of a single mud parameter, the effective viscosity. [32] In general, the dissipation rate is strongly dependent on pd m ffiffiffiffiffiffiffiffiffiffi and n. However, in the thick mud layer regime (d m > 1.5 2=! ) the dissipation is only controlled by the thickness of the Stokes boundary layer and is not sensitive to d m. Because the focus of this inversion exercise is the last half of the March event, when the mud thickness does not appear to change significantly, and due to significant uncertainties related to the spatial structure of the fluid mud event, d m was assumed constant in time. This implies that density changes in the fluid mud density/viscosity that occur in the model should be attributed to the added sediment mass due to the influx of sediment from the upper water column, consistent with the character of the fluid mud event (e.g., section 3). Based on observations fluid mud layer observations [Jaramillo et al., 2009], the mobile mud thickness for 7of14

8 Figure 6. Evolution of normalized (peak value = 1, darker values are larger) frequency distribution of the swell dissipation rate ((f), equation (2)) between 10 March 1200 UT and 11 March 1800 UT (marked with a rectangle in Figure 3): (a) observations and (b) numerical simulations based on the stochastic NMSE, using the best fit fluid mud viscosity and density. Positive values correspond to wave dissipation and negative values correspond to wave growth. Figure 7. Wave dissipation rates for 10 March 1500 UT (event 1 in Figure 6). (a) Spectral density of wave variance (circles, observations; line, model simulations) and (b) frequency distribution of dissipation rates; (circles, net dissipation, observations; line, net dissipation, model simulations; dashed line, mudinduced dissipation rate [Ng, 2000] using best fit fluid mud viscosity and density; crosses, net linear dissipation rate, including contributions of wind input, whitecapping, and mud induced dissipation). Positive values correspond to wave dissipation and negative values correspond to wave growth. 8of14

9 Figure 8. Wave dissipation rates for 10 March 2100 UT (event 2 in Figure 6). (a) Spectral density of wave variance and (b) frequency distribution of dissipation rates (see explanations in Figure 7). the March event was assumed to vary linearly between cm at T1 and about 5 cm at T2. [33] The inverse modeling approach was used to determine the effective viscosity of the bed that results in a best fit between the observed and simulated spectra at T2. Based on the speed of the FORTRAN implementation of the NMSE model (about 10 s on an average desktop computer), the problem was formulated as a constrained nonlinear least squares minimization. The algorithm (coded using the MATLAB fmincon function) seeks the value of the kinematic viscosity that minimizes the deviation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X N u 2 F m f j ; x 2 F fj ; x 2 ¼ t ð6þ N F 2 f j ; x 2 j¼1 of the energy flux spectrum F m predicted at T2 from the observed spectrum F, with N the number of spectral modes used. A best fit kinematic viscosity value was obtained for each spectral estimate, within a typical 30 iterations, and with typical spectral error of 3 5%. [34] The similarity between the observed and the numerically simulated, best fit frequency distribution of the dissipation rate (Figure 6) suggests that the numerical simulations can provide some insight into the structure of the dissipative processes. Figures 7 9 show the energy flux spectra and dissipation rates for events 1 to 3 in Figure 6. Mud induced dissipation is the main dissipative process; wind growth and whitecapping modify the highfrequency end of the spectrum, bending the dissipation curve toward growth; finally, nonlinearities modulate it by further transferring energy from the peak of the spectrum toward the lower frequency (infragravity) band and the second harmonic of the spectral peak. The numerical results are in agreement with the estimates of bispectra for the same events (Figure 5). Nonlinear effects intensify from 10 March 1500 UT to 10 March 2100 UT, but decrease significantly on 11 March 1100 UT, when the evolution is nearly linear. [35] Because nonlinearities conserve the energy of the system, their effects are only represented (aliased) as dissipation. The numerical results show that important features of the time frequency shape of wave dissipation (Figure 3) are due to nonlinear interactions; the peak of the dissipation distribution follows the spectral peak because of transfer of energy toward both the infragravity band, and high frequencies; narrower distributions of dissipation rate occur when nonlinear interaction is stronger, e.g., for larger waves. Growth rates in the infragravity band are exclusively due to transfer of energy from the spectral peak, an essentially nonlinear process [Sheremet et al., 2002]. [36] The evolution of the best fit kinematic viscosity (Figure 10) is qualitatively consistent with the available information about the rheological properties of Atchafalaya mud [e.g., Robillard, 2009]. For every pair of spectral estimates at T1 and T2, the inverse model returns two solutions for mud viscosity, corresponding to the high and lowviscosity domains (Figure 4). Based on the slow settling trend observed, which imply an increase in density (and viscosity), and on the measured characteristic values of Atchafalaya mud at gel (space filling) state, (r Gel = 1,100 kg/m 3,andn Gel = Figure 9. Wave dissipation rates for 10 March 2100 UT (event 3 in Figure 6). (a) Spectral density of wave variance and (b) frequency distribution of dissipation rates (see explanations in Figure 7). 9of14

10 Figure 10. Best fit viscosity versus time, for the period 10 March 1200 UT to 11 March 1800 UT (rectangle in Figure 3). Values corresponding to the low viscosity domain (Region 1, Figure 4) are marked by circles; those corresponding to the high viscosity domain (Region 2) are marked by crosses. The high viscosity values are unrealistic for this event. The value of the kinematic viscosity corresponding to the gelling point for Atchafalaya mud (n Gel = m 2 /s [Robillard, 2009]) is marked by a dashed line m 2 /s), only the low viscosity values are assumed to be physical. The steady increase is consistent with the hindered settling, followed by dewatering and consolidation. Remarkably, the estimated values approach those characteristic of gelled mud. [37] Due to uncertainties related to the spatial distribution of fluid muds (section 2.2), we cannot make a definitive statement about the state of the bed sediment, however, the results are consistent with the hypothesis that the fluid mud layer evolves at the end of the event toward a soft, underconsolidated state. 5. Discussion [38] Observations of wave dissipation over the muddy Atchafalaya subaqueous delta support the hypothesis that an effective coupling exists between surface waves and cohesive bed sediment. The dominant wave dissipation mechanism is wave bottom interaction; the process is triggered by the reworking of bed sediment by waves. Based on acoustic backscatter records, several stages can be distinguished in the evolution of the bed: consolidated mud at the beginning of the storm (density kg/m 3 in surficial sediments from cores at the sites), followed by sediment mobilization possibly through a combination of liquefaction, expansion due to water absorbtion, erosion and rapid resuspension as the storm intensifies, followed by hindered settling (fluid mud formation), consolidation and dewatering. [39] The detided net dissipation rate increases steadily during a storm, with no detectable correlation to fluid mud layers, possibly due the discontinuous spatial distribution, or low density of the fluid mud layers. Wave dissipation reaches a maximum of intensity of about 50% loss of incoming energy flux (over the distance between the two observation stations) in the wake of the storm, when no bed change is detected in acoustic backscatter data. The evolutionary trend of suspended sediment concentration suggests that the maximum dissipation efficiency corresponds to a soft, under consolidated state of bed sediment (gelling and slow dewatering). [40] The inverse modeling approach of Rogers and Holland [2009] was used to investigate this hypothesis. The numerical results suggest that, while mud induced dissipation dominates wave propagation processes in the swell band, nonlinear three wave interactions determine the shape of the frequency distribution of the dissipation rate. Nonlinear energy transfer from the spectral peak toward infragravity waves and toward the high frequency band register in the net dissipation estimates as dissipation at the spectral peak and growth in the infragravity and highfrequency band. The values of the effective kinematic viscosity returned by the inverse model are consistent with measurements and with the observed evolutionary trend of suspended sediment concentration. [41] The analysis supports only in general terms (i.e., triad interactions are active) the mechanism proposed by Sheremet and Stone [2003], that explains short wave dissipation as nonlinear energy transfer toward low frequency waves. However, during the period analyzed here, sustained wind speeds were over 10 m/s and picked up at the end (evening of 11 March 2006). If this mechanism was active during the event discussed here, the high frequency damping could have been compensated by wind induced growth. [42] One of the more intriguing questions related to wave mud interaction in shallow environments such as the Atchafalaya inner shelf is whether characteristics of bottom sediment processes (e.g., bed density, presence of fluid mud layers) can be inferred from surface wave observations. While bed information is complex and hard to obtain, wave observations are readily available, e.g., through remote sensing means. Despite the limitations of the inverse modeling approach used here (unidirectional wave modeling, simplified representation of fluid mud layers, minimal information about bed sediment characteristics), the results are encouraging, and point to a number of important conclusions. Wave bottom interaction is the dominant process, however, three wave interactions play an integral role in the interpretation of the frequency distribution of net wave dissipation. Neglecting the effect of nonlinearities leads to aliasing nonlinear energy transfer into dissipation effects, and distorts the representation of mud induced dissipation. To fully validate the scenario proposed here, further efforts are required for in situ observations of bed sediment evolution under waves. Appendix A: SWAN Simulations [43] Details of the SWAN based numerical simulations are given by Jaramillo [2008]. SWAN was run in nonstationary mode on a rectangular bathymetric grid (E W side: 40 km long, 400 m resolution; N S side: 20 km long, 460 m resolution) with T1 at the center of the southern boundary. The AWAC estimate of the directional wave spectrum at T1, sampled in 90 directions by 99 frequencies, was used as uniform southern boundary condition. The model was run using bathymetric data extracted for the model grid from the National Geophysical Data Center database. Tide elevations and surface currents were derived from observations (assumed spatially constant over the integration domain). Surface winds were produced by the COAMPS model [Hodur, 1997], run at the Naval Research Laboratory, Stennis Space Center. The simulations accounted for wind 10 of 14

11 Figure A1. Numerical simulations (SWAN) of the transformation of the directional spectrum shape between (a c) T1 and (d f) T2, for 10 March 1500 UT (Figures 11a and 11d), 10 March 2100 UT (Figures 11b and 11e), and 11 March 1100 UT (Figures 11c and 11f; events 1 3 in Figure 6). Spectra are normalized for maximum peak density 1 (darker values are larger); contour level scale is logarithmic. growth, whitecapping, depth limited breaking and nonlinear four wave interactions, run with default parameter values. Mud induced bottom dissipation was not available in the version used here, and the triad interaction module was not activated. [44] Preliminary SWAN [e.g., Booij et al., 1999; Zijlema and Van der Westhuijsen, 2005] runs were used to assess refraction effects and the overall balance of growth/ dissipation effects. The main feature of the spectral transformation from T1 to T2 (Figure A1) is the development of a directionally spread, local short wavefield (in agreement with observations). The narrow swell (f < 0.2 Hz) peak and negligible refraction between T1 and T2 (within the angular resolution of the simulations) suggest that swell propagation is largely unidirectional. Without mud induced swell dissipation, other growth/dissipation processes are approximately in balance, resulting in a slight growth (negative dissipation) between 10 March and 14 March (Figure A2b). After 9 March the only active processes are mainly in the high frequency tail of the spectrum (Figure A2c), where wave growth is balanced by whitecapping and quadruplets (four wave interactions). Appendix B: Stochastic Nonlinear Wave Propagation Model [45] The stochastic (phase averaged) equation is derived following the procedure outlined by Agnon and Sheremet [1997], modified to account for dissipative processes. Because dissipation rates are typically much smaller than wave numbers, their effect is included in the equation for the spectral evolution but is neglected in the bispectrum equation. The resulting approximation of the unidirectional spectrum equation is df j dx ¼ 2 jf j þ 16 X 1 jq W <e J j;j q;q qb jþ1 2 c W j;j q;q F j q F q þ W j q; q;j F q F j þ W q;q j;j F j q F j ; ðb1þ where x is the spatial coordinate, j is the net dissipation rate of mode j, d is the Kronecker symbol, and F j is the average energy flux of Fourier mode j. The notation s is used for the integer part of the real number s; () j,p,q indicates that the quantity is evaluated for the frequency triplet f j, f p and f q ; <e{z} is the real part of the complex number z. The terms of the sum corresponding to modes jþ1 2 q < j are sometimes called sum interaction terms, and are typically associated with the generation of peak harmonics; terms with q > j are called difference interaction, associated with energy transfers toward lower frequencies. The interaction coefficient W j,pq is a function of the frequencies and wave numbers of the interacting modes [e.g, Agnon and Sheremet, 1997]. The function Z x R JðÞ¼ x e i s Dkdu ds ðb2þ 11 of 14

12 Figure A2. (a) Observed evolution of the significant height of swell and seas (see also Figure 3a) and SWAN simulations of (b) swell dissipation (equation (3)) and (c) frequency integrated active source terms at T2 (quadruplets are integrated in absolute value) versus time. In Figure 12b, dots are hourly estimates of swell dissipation and line is a running average. In Figures 12b and 12c, positive values of the dissipation rate correspond to wave growth and negative values correspond to wave dissipation. parameterizes the contribution of bispectral evolution, where the nonlinear wave number is D j;p;q k ¼ k j k p k q ; and the wave numbers k ±j satisfy the dispersion relation 2 j ¼ k j tanh k j h; ðb3þ with s ±j =± p! ffiffi j g, w j =2p f j the modal radian frequency, g the gravitational acceleration, and h the local depth. [46] Dissipation or growth processes are introduced through the net modal dissipation rate j ¼ W j þ M j þ; where j W represents the combined effect of wind input and whitecapping (implemented following Van der Westhuysen et al. [2007]), and j M represents wave dissipation induced by Newtonian fluid mud layers [Ng, 2000; Kaihatu et al., 2007]. The fluid mud model predicts the wave dissipation rate M ¼ M f j ; h; d m ; m ; m ðb4þ as a function of the wave frequency, the water depth, and three parameters that characterize the fluid mud layer: layer thickness d m, mud density r m, and kinematic viscosity n m. Appendix C: Mud Density and Kinematic Viscosity [47] For practical applications, the viscosity of the fluid mud should be allowed to change as the density of the mud increases due to consolidation. As the density of the mud increases, however, rheological properties change, and the sediment goes through nonlinear transitional regimes (e.g., from viscoelastic fluid to viscoelastic solid [Jain and Mehta, 2009]) characterized by a dependency on the frequency and amplitude of the strain rate _, e.g., * ð_ Þ ¼ ð_ Þ i ð_ Þ; ðc1þ 12 of 14

13 where h is the complex viscosity of the material, h the dynamic viscosity, and h the elastic component [Barnes et al., 2005], and the asterisk denotes complex conjugation. In the limit of low densities or high strain rates, the mud behaves as a viscous fluid (V), with the elastic component negligible h h ; at high densities or low strain rates, it behaves as a linear viscoelastic (VE) material, with the elastic component dominant h h. In both these states, h is independent of the strain rate amplitude [Christensen, 1971]. [48] In the transitional regime between these two extreme states, the viscous and elastic component are comparable in magnitude and depend on the strain rate. Robillard [2009] proposed an interpolation model similar to the Carreau model [e.g., Christensen, 1971; Barnes et al., 2005] for the magnitude of the viscous component (effective viscosity) in the transitional regime m eff ¼ ð_ Þ ¼ V _ þ VE _ 0 ; ðc2þ _ þ _ 0 where _ is the strain rate, the subscripts denote the values of the quantities in the low/high strain rate regimes, and _ 0 is defined by h ( _ 0 )=h ( _ 0 ) (equations (4) (10) of Robillard [2009]). In this model, the operating range is close to the crossover point g 0, that essentially separates the solid from fluid behavior. To account for viscosity dependency on strain rate and density in the transitional range, the kinematic viscosity was replaced in equation (B4) with the effective viscosity (equation (C2)) M ¼ M f j ; h; d m ; eff ð m ; _ Þ : ðc3þ [49] Assuming that the variability of the strain rate is weak in the waning stage of the storm, _ = _ c, where _ c is a characteristic value, the effective viscosity is a function of the density only, and the dissipation rate becomes M ¼ M f j ; h; d m ; eff ð m Þ : ðc4þ [50] The values for parameters h LV, h LEV, sind LEV, and _ 0 measured for several volume fractions [Robillard, 2009, Table 4 2] (poststorm conditions, 0.68 rad/s angular frequency), can be substituted in equation (C2) to estimate (e.g., through an interpolation procedure) the effective viscosity of the mud as a function of density. The density of the mud is related to its dry sediment density r s through the volume fraction of solids vs ¼ m w s w : [51] From equation (C4), if the mud layer thickness can be estimated from observations, the dissipation rate is only a function of a single mud parameter, the effective viscosity, and can be obtained through an inverse modeling approach. The characteristic strain rate was set to _ c = O(1 s 1 ), and the primary sediment particle density to r s = 2580 kg/m 3. [52] Acknowledgments. This work is supported by the Office of Naval Research awards N and N Data was collected with the support of Coastal Studies Institute, Louisiana State University. We are grateful to A. Mehta for suggestions and advice on mud modeling and for providing data to characterize the Atchafalaya mud, to Erick Rogers for discussions on the SWAN based inverse modeling approach, and to the anonymous reviewers, whose advice helped improve this manuscript. References Agnon, Y., and A. 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