JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi:10.1029/2005jc002893, 2007 Cross-shore sediment transport on natural beaches and its relation to sandbar migration patterns: 1. Field observations and derivation of a transport parameterization I. J. Mariño-Tapia, 1,2 P. E. Russell, 1 T. J. O Hare, 1 M. A. Davidson, 1 and D. A. Huntley 1 Received 21 January 2005; revised 19 July 2006; accepted 1 September 2006; published 2 March 2007. [1] Cross-shore sediment transport processes are investigated with measurements of horizontal velocity and sediment suspension obtained with electromagnetic current meters and optical backscatter sensors on five different beaches across Europe. Data were gathered under a wide variety of hydrodynamic and morphological conditions spanning the swash, surf, and shoaling zones. Results show that the near bed velocity moments, normalized by the local energy level (hu t 2 i n ), have consistent shapes (shape functions) when plotted against normalized cross-shore depth (h/h b ). According to the energetics approach, near bed velocity moments are good descriptors of sediment transport processes, consequently, the shape functions describe and quantify the cross-shore distribution of the most relevant cross-shore sediment transport processes. The pattern consists of net onshore transport in the swash zone, offshore transport in the surf zone, and onshore transport in the shoaling zone thereby producing divergence of sediment in the inner surf/swash zone and convergence in the breaker zone. In consequence, the shape function supports the breakpoint hypothesis for sandbar generation. This behavior is a product of the balance between multiple opposing mechanisms including undertow, coupling between mean flows and short (long) wave stirring, short (long) wave skewness, and long wave coupling with short wave variance. The cross-shore structure of the measured and normalized cross-shore sediment fluxes is consistent with the velocity moment shape functions, showing the robustness of the approach. Apart from the shortcomings bound to the energetics approach, the present parameterization is also limited due to the difficulty of defining with confidence a breaker depth (h b ) in bar-trough systems. Citation: Mariño-Tapia, I. J., P. E. Russell, T. J. O Hare, M. A. Davidson, and D. A. Huntley (2007), Cross-shore sediment transport on natural beaches and its relation to sandbar migration patterns: 1. Field observations and derivation of a transport parameterization, J. Geophys. Res., 112,, doi:10.1029/2005jc002893. 1. Introduction [2] Long term (years to decades) morphological studies support the idea that shore parallel sandbars have a dominant cross-shore movement [e.g., Lippmann et al., 1993; Ruessink and Kroon, 1994; Wijnberg and Terwindt, 1995; Plant et al., 2001b; Kuriyama, 2002] with cycles of generation close to shore, net offshore migration and eventual degeneration (bar decay). It is accepted that this behavior is not an apparent longshore migration of obliquely oriented bars, but essentially a cross-shore redistribution of sediment. Consequently the processes of cross-shore sediment transport must play a 1 Centre for Coastal Dynamics and Engineering, School of Earth Ocean and Environmental Sciences, University of Plymouth, Drake Circus, Plymouth, UK. 2 Now at Coastal Processes and Physical Oceanography Laboratory, Centre for Scientific Research and Advanced Studies (CINVESTAV), Mérida, Yucatán, Mexico. Copyright 2007 by the American Geophysical Union. 0148-0227/07/2005JC002893 crucial role in the generation and evolution of such morphological features, and should be better understood if modeling of their medium (weeks to months) to long term (years to decades) behavior is to be improved. [3] Over the last three decades, field observations in the nearshore have shown that short waves (frequencies of 0.05 0.4 Hz), infragravity waves (frequencies of 0.004 0.05 Hz) and mean flows, such as the breaking induced undertow, are the main hydrodynamic processes involved in suspending and transporting sediment in the cross-shore direction. There is accumulated evidence in such studies of the existence of a pattern, which suggests that the magnitude and direction of the suspended sediment transport depends on the cross-shore location relative to the breaking point. For example, numerous studies have shown that under nonbreaking conditions (e.g., seaward of the surf zone) onshore sediment transport, produced by short wave skewness, tends to dominate [Guza and Thornton, 1985; Hanes and Huntley, 1986; Doering and Bowen, 1987; Roelvink and Stive, 1989; Beach and Sternberg, 1991; Osborne and Greenwood, 1992a, 1992b; Thornton 1of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 et al., 1996; Gallagher et al., 1998; Russell and Huntley, 1999]. Net onshore transport outside the surf zone is predominant even if a reverse (offshore) transport induced by bound long waves is present [Hanes and Huntley, 1986; Ruessink et al., 1998; Russell and Huntley, 1999]. [4] Inside the surf zone, wave breaking generates mean flows and energy transfers towards lower and higher frequencies that modify wave characteristics and drastically affect sediment transport. For example, near-bottom offshore directed mean flows, known as undertow currents, play a crucial role in transporting sediment in the offshore direction [Osborne and Greenwood, 1992a, 1992b; Russell, 1993; Ruessink et al., 1998; Russell and Huntley, 1999], and several field studies have associated the gradients in the undertow current with offshore sandbar migration, especially during high energy conditions [Thornton et al., 1996; Gallagher et al., 1998; Aagaard et al., 1998]. [5] Despite a sediment transport pattern consisting of onshore transport outside the surf zone and offshore transport inside, has been commonly found in previous field studies, its consistency has not been fully tested and its detailed behavior has not been properly characterized. This is largely because natural variability in surf zone processes is very complex, and this complexity challenges the regularity of such simple pattern. For instance, in the very shallow waters of the inner (saturated) surf zone, phase coupling between large short-waves and long-wave crests can drive sediment onshore [Abdelrahman and Thornton, 1987], but during storm conditions infragravity waves have been reported to drive large amounts of sediment offshore if they are negatively skewed [Russell, 1993; Butt and Russell, 1999]. Aagaard and Greenwood [1994] have also shown that under an infragravity standing wave, sediment transport can change direction and magnitudes across-shore, as the velocity structure can drive sediments towards the antinodes. [6] Given the complexity in cross-shore sediment transport processes, a robust sediment transport parameterization that captures the relative strengths and cross-shore structure of the most important processes driving onshore and offshore transport could be highly valuable for simplifying the midterm to long-term modeling of sandbar migration patterns, and ideally, of profile evolution. This has been recognized for some time and a number of field-based parameterizations are now available in the literature [Russell and Huntley, 1999; Plant et al., 2001a; Aagaard et al. 2002]. The parameterization developed in the present investigation is an improvement of that proposed earlier by Russell and Huntley [1999]. [7] Following the approach of Russell and Huntley [1999], this study confirms that the normalized cross-shore velocity moments derived from field data have a consistent cross-shore structure (the shape function ) that depends on the position relative to the breakpoint. The consistent structure in the new shape function and the details of its behavior are evident due to the addition of a substantial amount of field data (288 time series) that encompass swash, surf, and shoaling regions. As sediment transport is proportional to the velocity moments, the shape function operates as a parameterization of the most important crossshore sediment transport processes, within the limitations of the energetics approach, and includes the effects of crossshore mean flows, short waves, infragravity waves, and interactions between them. 2. Field Experiments [8] A unique data set acquired on five different beaches across Europe over a time span of eleven years provides a continuum of data points covering the swash, surf and shoaling zones. Four beaches are located on the coast of the UK (Llangennith, Perranporth, Teignmouth, and Spurn Head) and the remaining beach is located on the Dutch coast (Egmond aan Zee). Figure 1 shows the geographic location of the field sites. The data were gathered during the British Beach And Nearshore Dynamics experiment (B-BAND) experiment [Russell et al., 1991; Davidson et al., 1992], an experiment in the swash zone [Butt, 1999], and during the European project Coastal Study of Three Dimensional Sand Transport (COAST3D) [Soulsby, 2001]. The general characteristics of the field sites are presented in Table 1. Figure 2 shows the beach profile for each site, the position of the instrument rigs on the profile, and the maximum tidal elevation observed on the day of the measurements. [9] For details of previously reported data and associated field sites (i.e., Llangennith, Perranporth, and Spurn Head) the reader is referred to Russell and Huntley [1999], Butt and Russell [1999], and Tables 2 and 3. In the following paragraphs only the information related to newly added data will be covered. [10] Teignmouth, one of the COAST 3D sites, is located on the coast of South Devon, UK (Figure 1e). Data from the pilot (March 1999) and main (October November 1999) campaigns are included (see Tables 2 and 3). During the pilot campaign the instrument rigs were located close to the slope break on the steep part of the beach. Figure 2d shows a beach profile across the instruments and the maximum tidal elevation observed during the dates of the observations. Data from the Teignmouth main campaign were recorded on both, the flat low tide terrace (labeled Teignmouth main F and G on Tables 2 and 3, and herein after referred to as LTT), and on the steep beach. Figure 2e shows the beach profile across the instruments and the maximum tidal elevations observed during the dates of the observations. The second COAST3D site is Egmond aan Zee, located on the central coast of the Netherlands (Figure 1d). Its cross-shore morphology is characterized by the presence of two nearshore breaker bars. Figure 2f shows the beach profile across the instruments and the observed high water mark. 3. Measurements 3.1. Instrumentation [11] Suspended sediment concentration (c), horizontal current velocities (u and v) and surface elevations (h) were measured with optical backscatter sensors (OBS), electromagnetic current meters (EMCM), and pressure transducers (PT) respectively. The instruments were installed at the low tide mark, on self-logging mode (in Perranporth, Teignmouth and Egmond) or hard-wired to a shore-based station (e.g., Llangennith, Spurn Head). All the data sets were recorded on the intertidal region with a single rig of instru- 2of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Figure 1. Geographic location of the field sites. ments and as the water level rose (or fell) over the rigs, measurements were acquired from different regions of the nearshore relative to the shoreline. The instruments were colocated so that velocity matched the phase of the wave at which suspension events were recorded, but in most cases the OBSs were deployed as close to the bed as possible to capture significant sediment suspension events whereas the current meters were kept further away to minimize the risk of signal distortion from the boundary (see Table 3). The height of the sensors above the bed was adjusted every low tide to compensate for bed level changes. The data were logged continuously at 2, 3 or 4 Hz (see Table 3) and subsequently subdivided into time series of 17 min to ensure stationarity of conditions and avoid adverse effects Table 1. General Characteristics of the Field Sites Field Site Beach Type a d 50 /Sorting Wave Climate Tidal Range Spring/Neap Llangennith D 0.23/well sorted Atlantic swell and local wind 8.5/4 m Perranporth D 0.24/very well sorted Atlantic swell and local wind 7/5.5 m Egmond D/I 0.25 0.5/well sorted North Sea swell and local wind 2/1 m Spurn Head I 0.35/well sorted North Sea swell and local wind 7/3 m Teignmouth R/I 0.17 0.5/well sorted to poorly sorted Local wind, protected from swell 4/1.7 m a D, dissipative; I, intermediate; R, reflective. 3of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Figure 2. Beach profiles, instrument locations and maximum tidal level observed during the field experiments. on spectral estimates and other statistical properties caused by the tidal variations. [12] The pressure signals were corrected for atmospheric pressure by subtracting the pressure values measured when the instruments were dry at low tide. In situ zeroing of the EMCM signal was repeatedly made in buckets of still water during low tide, when the instruments were exposed. The OBSs were gain-calibrated in a recirculation tank using sand from the deployment locations, and any background offset levels were removed prior to analysis. The time series were visually inspected for data quality. Spiky velocity data with values above three standard deviations were corrected (adopting the mean value of the neighboring points), and time series with extreme noise levels or with evidence of burying, or wetting and drying were eliminated with the exception of data from the swash zone which was corrected for intermittent submersion using co-located pressure transducers buried on the sand to identify dry periods. Other data quality procedures include the time lag correction between the velocity and the suspended sediment concentration signal introduced by the use of analogue filters in the logging equipment, and time series with either low coherence or random phase spectrum between u and h were regarded as erroneous and were eliminated. 3.2. Measured Hydrodynamic Conditions [13] Figure 3 presents the morphodynamic stage of the studied beaches according to the classification system of Masselink and Short [1993]. From this figure it is clear that the data analyzed in the present study spans a wide spectrum of morphodynamic stages including reflective (Teignmouth), intermediate (Teignmouth and Spurn Head), dissipative barred (Egmond), and nonbarred beaches (Teignmouth LTT, Llangennith, Perranporth). A wide variety of hydrodynamic conditions is considered essential to test the hypothesis of this investigation. Hence, further proof of the wide coverage of hydrodynamic conditions is given below through the analysis of the wave transformation across-shore and the structure of the data in frequency space. Figure 4 left panels show the cross-shore transformation of wave height (H), and Figure 4 right panels show the cross-shore evolution of breaker index (g), for Egmond, as representative for dissipative conditions. The total (top), incident (middle) and infragravity (bottom) components of these two variables are shown. Dissipative beaches are usually characterized by a mild beach slope and broad saturated surf zones where wave height decreases monotonically with depth. The saturation law is evident in the 4of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Table 2. Breaking Wave Height (H b ), Peak Spectral Period (T p ), Beach Slope at the Instruments, and Span of the Data in the Nearshore for All Data Sets a Code of Data Set Dates H b,m T p, s Data Span tanb Llangennith 17/11/88 2.5* 14.28 inner surf 0.02 Spurn Head A 23/04/91 0.75** 10 surf/shoal 0.023 Spurn Head B 23/04/91 0.75** 10 surf/shoal Perranporth calm 25/04/98 1.1* 12.5 swash/surf 0.028 Perranporth storm 27/04/98 2.5* 12.5 swash Egmond main A 22/10/98 am 1.8** 6.25 inner surf 0.02 Egmond main B 22/10/98 pm 1.8** 6.66 inner surf Egmond main C 23/10/98 2.36** 7.14 inner surf Teignmouth pilot A 12/03/99 0.73** 5.55 surf 0.1 Teignmouth pilot B 12/03/99 0.73** 5.55 surf Teignmouth pilot C 19/03/99 0.16** 11.11 surf/shoal Teignmouth main A 29/10/99 0.35** 11.11 surf/shoal 0.08 Teignmouth main B 4/11/99 0.81** 5 surf/shoal Teignmouth main C 5/11/99 1.06** 5.88 surf/shoal Teignmouth main D 10/11/99 0.77** 7.14 surf/shoal Teignmouth main E 11/11/99 1.29** 7.40 surf/shoal Teignmouth main F 10/11/99 0.8** 5 surf/shoal 0.008 Teignmouth main G 11 13/11/99 1** 7.14 surf/shoal a H b estimated on the field (*), and from wave records (**). T p estimated from the spectrum of the offshore most time series. middle panels of Figure 4, where incident wave height is linearly dependent on water depth (R 2 1) and the breaker index of incident short waves is nearly constant (g s 0.45) throughout the surf zone. The data also shows that infragravity energy can substantially affect the cross-shore evolution of g especially close to the shore where values of g total, increase to 0.8. As expected, the cross-shore evolution of the spectral energy (not shown) also reflects this behavior with monotonic decay of the spectral peaks at incident wave frequencies and marked growth of infragravity frequencies near the shore. The peak of the infragravity energy is at surf beat frequencies (0.02 Hz) and is broad banded. The outliers observed on Figure 4 for g total and Hs total have their origin on the presence of substantial amounts of infragravity energy during the rising tide at Egmond. [14] Figure 5 presents the cross-shore transformation of wave height (left panels) and breaker index g (right panels) for an intermediate beach, Spurn Head. The morphological characteristics found for the reflective and intermediate beaches of this study, together with the milder energy conditions experienced, produce narrower surf zones where wave height is not linearly dependent on water depth. Wave height saturation is only evident in the inner surf zone. It is interesting to note that in these cases, filtering infragravity energy does not change the cross-shore behavior of H s or g substantially. The incident wave breaker index (g s ) never reaches a saturation (constant) value but increases from a shoaling value of 0.3 to a value close to shore of 1.5. The spectral evolution of Spurn Head s cross-shore velocity (not shown) has a fairly narrow and well defined incident spectral peak at 0.1 Hz (10 s) in deeper waters (3 m). From about 1.2 m depth, the incident spectral peak decreases monotonically showing a saturated inner surf zone. The presence of statistically significant peaks at the harmonic (0.2 Hz) and infragravity (0.02 Hz) frequencies is an important indication of strong nonlinear triad interactions in this region. The data from more reflective stages, shares Table 3. Heights of the Instruments Above the Bed (cm), Sampling Frequency (Hz), and Number of Time Series and Tides Used on Each Data Set Code of Data Set (Marker) Sampling Frequency, Hz Number of Time Series Instrument Heights (Centimeters Above Bed) PT OBS EMCM Llangennith () 3 12 8 4 12 Spurn Head A (*) 2 24 16 10 10 Spurn Head B (*) 2 24 16 25 17 Perranporth calm (?) 2 4 0.0 10 15 Perranporth storm (?) 2 1 2 11 7 Egmond main A (6) 4 28 24 4.5 13 Egmond main B (6) 4 26 26.5 4 12 Egmond main C (6) 4 27 28 13.4 14.4 Teignmouth pilot A (5) 4 19 27 15.2 16 Teignmouth pilot B (5) 4 25 27 15 14.5 Teignmouth pilot C (5) 4 8 7.6 7.9 7 Teignmouth main A (D) 4 21 11 13 11.6 Teignmouth main B (D) 2 23 19.4 21 20 Teignmouth main C (D) 2 22 17.6 19.8 19.5 Teignmouth main D (D) 2 17 3.5 7.2 7.5 Teignmouth main E (D) 2 19 10.7 11.1 12.1 Teignmouth main F (D) 2 10 9 9.2 10.3 Teignmouth main G (D) 2 38 9 9.2 10.3 5of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Figure 3. Beach classification according to the conceptual model of Masselink and Short [1993] for the data sets of this study. most characteristics explained above, with the difference that long wave energy is narrow-banded and close to subharmonic frequencies, with no surf beat energy noticeable in the spectra. 4. Analysis Approach 4.1. Velocity Moments and Cross-Shore Sediment Transport Processes [15] One of the most robust sediment transport formulations used for surf zone conditions is the energetics approach [e.g., Bagnold, 1963; Bowen, 1980; Bailard, 1981]. This approach successfully replicates the offshore migration of sandbars [Thornton et al., 1996; Gallagher et al., 1998] and simple modifications of the Bailard formula to include the effects of pressure gradients on the bed caused by asymmetric waves [Drake and Calatoni, 2001], have shown better prediction of changes in seafloor topography including onshore bar migration [Hoefel and Elgar, 2003]. [16] According to the energetics approach, the time averaged sediment flux, expressed as immersed weight sediment transport per unit width, per unit time is given by [Bailard, 1981]: e b i t ¼ rc f tan f hju tj 2 U t i tan b tan f hju tj 3 i þ rcf e s W hju tj 3 U t i e s W tan bhju tj 5 i where r is water density, Cf is the bed drag coefficient, e b and e s are the bed and suspended load efficiency factors respectively, interpreted as the fraction of the energy dissipation rate that is spent in transporting the sediment, tan b is the local bed slope, f is the sediment angle of repose, W is the sediment fall velocity, U t is the instantaneous total velocity measured at the top of the boundary layer, where angle brackets represent time averaging. The first term of equation (1) is generally associated with bed load transport and the second term with suspended load transport. Both include the down slope transport contribution by gravity (modulus of the third and fifth moments). [17] The analysis made in this paper includes only the cross-shore component of the total velocity vector therefore the effects of the longshore component on the sediment ð1þ Figure 4. Cross-shore evolution of wave height (right) and breaker index g (left) for Egmond aan Zee (dissipative conditions). The total (top), incident wave (middle), and infragravity (bottom) components are shown. Open circles indicate data points, and the solid line represents the linear regression. 6of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Figure 5. Similar to Figure 4 for Spurn Head (intermediate conditions). transport processes are not included. When the complexity of nearshore hydrodynamics is reduced to the cross-shore dimension, it is usually assumed that the cross-shore dynamics dominate the velocity field, and the alongshore gradients in longshore sediment transport, if present, are not affecting the cross-shore dynamics either because they are negligible or because they are independent. Analysis of the data sets of this investigation show that the longshore oscillations (in terms of longshore velocity variance, s 2 v )are negligible for all the data sets (s 2 v < 0.1 m 2 s 2 ) and the magnitude of the alongshore currents inside the surf zone is generally small (<0.25 m/s) except for the Egmond data set where longshore currents were strong (1 m/s). The data sets where the longshore component is of importance (Egmond and Spurn Head), and where the cross-shore assumption is not strictly fulfilled, will be included to show that the pattern proposed by the shape function is also valid in such circumstances. 4.2. Cross-Shore Structure of the Velocity Moments [18] In order to make comparisons between data sets gathered under different energy conditions, the velocity moments were normalized so their values were insensitive to energy variations. Following Foote et al. [1994] and Russell and Huntley [1999] the velocity moments of equation (1) were normalized by a quantity proportional to the local cross-shore kinetic energy (hu 2 t i n ). In the present contribution, shape functions are presented for the four velocity moments in equation (1), this is the process-related momentshju t j 2 u t i andhju t j 3 u t i and the moments related to the gravity termshju t j 3 i and hju t j 5 i. [19] Bailard [1987] examined the spatial distribution of the short wave and current-related (mean) cross-shore transport velocity moments calculated from field data. He found that the velocity moments presented a systematic structure when plotted against normalized surf zone position and concluded that the surf zone width is a natural scaling factor for the velocity moments, as wave-induced radiation stress, the motor of the most important surf zone dynamics, becomes negligible outside the surf zone. The original shape function work [Russell and Huntley, 1999] and the present investigation, adopted this approach, where the location of a time series in the nearshore (i.e., swash, surf, or shoaling regions) was established by dividing the local depth by the estimated breaking depth (h b ). The approach of normalizing of the cross-shore position using the nondimensional surf zone reflects the existence of two distinct regions of the nearshore, a shoaling zone of unbroken waves and a surf zone where most waves have broken. The fundamental concept underlying the shape function is that these regions have very different sediment transport characteristics induced by the differences in hydrodynamic processes found in them. The limitations of this approach for complex morphology (bar-trough systems), where the definition of a breaker depth is not straightforward is discussed in section 7. [20] The breaking depth (h b ) was calculated with the expression H b = g s h b, where H b is the incident breaking wave height and g s is the short wave breaker index (extracted from the data). Values of wave height at breaking (H b ) were estimated in the field [see Russell and Huntley, 1999] or calculated from offshore wave records (for the newly added data) by using the expression: H b ¼ 0:39g 1=5 TH0 2 2=5 ð2þ where g is the acceleration due to gravity, T is the wave period, and H 0 is the offshore wave height. Expression (2) was suggested by Komar and Gaughan [1972] using laboratory and field data. Rattanapitikon and Shibayama [2000] verified the reliability of 24 existing formulas for computing breaking wave height against a wide range of 7of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Figure 6. Shape function of the normalized third velocity moment. The x-axis is mean water depth normalized by breaker depth so that 0 is the shoreline, 1 the breaking point and values of x > 1 are outside the surf zone. On the y axis, positive values represent onshore transport and negative values represent offshore transport. wave and beach steepness conditions and large amount of published laboratory data (574 cases). They found that equation (2) presented the smaller values of the root mean square relative error, consistently for all the cases, making it the most reliable formula. [21] For those data sets that covered the whole surf zone and parts of the shoaling region (e.g., Teignmouth and Spurn Head), the breaker depth (h b ) was identified directly at the point where the maximum wave height existed (H max H b ). This is valid assuming H 0 does not vary drastically for the three to four-hour period that the instruments were immersed. Values of significant wave height were estimated from the surface elevation time series as H s = 4s h, where s h is the surface elevation standard deviation. To further aid the definition of H max, parameterizations from parametric wave transformation models were used [Battjes and Janssen, 1978; Baldock et al., 1998]. 5. Results 5.1. Shape Functions [22] Following the methodology explained above, each velocity moment term in equation (1) was normalized and plotted against normalized depth to give four shape functions. Figures 6 and 7 show the shape functions for the process-related sediment transport terms, and Figures 8 and 9 show the shape functions associated with the gravity terms of the transport equation. [23] The behavior of the process-related shape functions will be analyzed first. Figures 6 and 7 show the shape functions for the third (bed load) and fourth (suspended load) velocity moments respectively, the origin on the x-axis is at the shoreline, 1 at the breaking point and values of x > 1 are outside the surf zone. On the y-axis, positive values indicate onshore transport and negative values offshore transport. Each marker represents the value of the normalized cross-shore velocity moment averaged over an entire 17-min time series. The consistent structure in the shape functions of Figures 6 and 7 demonstrate that the differences in hydrodynamic conditions found in the data sets (from saturated surf zones with large quantities of broad banded surf beat energy, to unsaturated surf zones with narrow banded subharmonic energy) were successfully normalized. [24] Because of the substantial amount of data added, the improved shape function (Figures 6 and 7) confirms what Figure 7. Similar to Figure 6 for the normalized fourth velocity moment. 8of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Figure 8. Russell and Huntley [1999] observed with much less resolution due to limitations on their data coverage. The new added data helps to resolve much better the behavior of the normalized cross-shore velocity moments across the surf and shoaling zones. Net onshore sediment transport is observed for small values of h/h b, which represent swash/ inner surf zone conditions. As h/h b increases, the normalized moments become increasingly negative, suggesting the existence of a divergence of sediment transport in the outer border of the inner surf zone. For most of the surf zone the normalized moments are negative, indicating offshore transport, with a maximum at the mid surf zone. Data points become less negative in the outer surf zone to converge near the breaking point with onshore sediment transport coming from outside the surf zone. The scatter in Figures 6 and 7 is attributed mainly to the difficulty of accurately establishing the breaker depth where in reality a breaker zone exists. On the steep beaches (Teignmouth, in triangles) the problem is exacerbated, as a small error on the estimation of h b can result in a large deviation on the normalized x-axis by the fact that the surf zone is narrow. Also, the use of h b as a normalization factor makes it difficult to clearly separate data in the region where wetting and drying occurs, hence data points that should be closer to shore appear further offshore (i.e., swash data - stars in Figures 6 and 7). [25] The potential use of the shape function for the modeling of bar migration patterns [Mariño-Tapia et al., 2007] motivated a reparameterization of the shape function. The normalized third velocity moment (Figure 6) was parameterized mathematically with the following equation: ¼ sin 2p h 3 = 2 h b u 2 t ut u 2 t " # 0:275 0:14 e 0:45 h 1:9 h h b and for the normalized fourth velocity moment (Figure 7) the equation is: D E ju t j 3 " u t 2 ¼ sin 2p h # 0:275 4 h 0:14 e 0:45 h hb h b h b u 2 t Similar to Figure 6 for the bed load gravity term. hb ð3þ ð4þ where u t is the instantaneous cross-shore velocity, angle brackets denote time averaging, h is the local depth and h b is the depth at the breaking point. [26] The equations were fitted to the data using a Gauss- Newton nonlinear technique forcing the function to start and end at zero to allow for sediment conservation on the profile. The correlation coefficients of these two equations are 0.54 and 0.44 for the third and fourth moments respectively. Equations (3) and (4) capture the behavior suggested by the data. [27] Guza and Thornton [1985] have shown that the most important terms in the cross-shore transport equation are those included in the third (ju t j 2 u t ) and fourth (ju t j 3 u t ) velocity moments. However, it is considered important to investigate the behavior of the ju t j 3 and ju t j 5 terms (gravity contribution) in the Bailard equation (1) with the same field data. Figures 8 and 9 present the results. Guza and Thornton [1985] determined that the theoretical value for Gaussian waves (i.e., in deep water) for the normalized ju t j 3 is 1.6 and for the normalized ju t j 5 is 6.38. There is evidence in Figures 8 and 9 that the gravity-related moments oscillate around these values, but their spatial structure is not consistent and difficult to discern (e.g., fifth moment). As no physical justification for the definition of a particular shape was found, shape functions were not defined for these two terms. It must be emphasized that both statistical uncertainty and the sensitivity of the calculations to bad data points increase with increasing order of the calculated moment. This explains the amplified scatter on the fifth moment. 5.2. Processes Contributing to the Structure of the Shape Function [28] Russell and Huntley [1999] presented a detailed analysis to examine the relative contributions of incident waves, long period motions, mean flows, and interactions between the three to the total structure of the process-related shape functions using the normalized third velocity moment only. The third moment was chosen because its cross-shore structure is clearer (see Figure 6), statistically more robust than the fourth moment, and its expansion into individual terms is easily coupled with well-known sediment transport mechanisms. A similar analysis was carried out in this Figure 9. Similar to Figure 6 for the suspended load gravity term. 9of15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 study, arriving at comparable conclusions. Nevertheless, the improved data resolution brings more detail on the crossshore structure of each term which is presented in Figure 10. The definition of the expanded terms for the total third velocity moment is slightly different to that used by Russell and Huntley [1999] and is presented in Table 4. [29] Offshore transport inside the surf zone is dominated by the combined effect of short and long wave stirring and transport by the mean current (terms 4 and 5). Figure 10d shows that term 4 is important not only in terms of magnitude but also in cross-shore structure, showing that it is this term which gives the total shape function most of its cross-shore behavior. Term 5 (long wave stirring and mean flows) is not usually recognized as a crucial component of the total third velocity moment, but in these data sets, as in Russell and Huntley [1999], term 5 (Figures 10e) accounts for an important proportion of the offshore sediment transport inside the surf zone. [30] Outside the surf zone short wave skewness (term 2) produces most of the onshore transport. As expected, Figure 10b shows maximum values of velocity skewness before the breaking point and gradual decrease as the surf zone is approached. Nonetheless, the scatter in Figure 10b shows that normalized depth might not be ideal for the parameterization of short wave skewness. [31] Term 4 also has an important contribution to the onshore sediment transport outside the surf zone. The correlation between wave stirring and mean flows (term 4) outside the surf zone is generally considered insignificant for sediment transport because of its magnitude, but Figure 10d suggests that this weak mean flow in combination with short wave stirring might become quite important outside the surf zone for driving sediment onshore. This onshore directed mean flow could be generated by boundary layer streaming and/or by second-order Stokes drift. Sediment transport by mean onshore flows (streaming) was initially suggested by Trowbridge and Young [1989] and measurements outside the surf zone have reported such a process [Osborne and Greenwood, 1992b; Aagaard and Greenwood, 1994]. Recent investigations by Henderson et al. [2004] using an eddy-diffusive boundary layer model show that asymmetry dependent transport is not sufficient to explain shoreward bar migration, and nonlinear advection of sediment generated mainly by boundary layer streaming and Stokes drift is required to fully predict shoreward bar migration. 5.3. Observed Cross-Shore Sediment Fluxes [32] If the shape function is a reasonable parameterization of the mechanisms that transport sand in the cross-shore direction, then the patterns observed in the shape functions of Figures 6 and 7 should be at least qualitatively similar to the observed sediment fluxes. In order to compare the crossshore structure of the measured sediment flux to the shape function of Figures 6 and 7, it will be necessary to normalize the sediment fluxes in a similar way. [33] Normalization of the sediment fluxes requires a mathematical expression proportional to the magnitude of the sediment flux, which keeps the information about direction of transport and relative importance intact. In other words, a stirring term is needed in analogy to thehu 2 t i n term used for normalizing the velocity moments. Plant et al. [2001a] suggested an expression that fits our needs. According to Plant et al. [2001a] the local time averaged cross-shore sediment transport rate could be defined as Q ¼ s u c H rms p h ffiffi þ a1r uc ð5þ 2 where R cu is the cross-correlation between cross-shore velocity u and sediment concentration c, s u is the crossshore velocity standard deviation and s c is the sediment load standard deviation, a1 is a constant of O(1), H rms is the root mean squared wave height, and h is the local water depth. The term outside the brackets in (5) scales the potential magnitude of the transport, and might be thought of as a sediment stirring term. This term is the mathematical expression that will be used for normalization of the observed total sediment fluxes. The nondimensional terms inside the brackets control, primarily, the direction of the transport, describing the balance between several competing transport mechanisms. The terms in brackets would be analogous to a sediment flux shape function. [34] Figure 11 shows the normalized measured sediment fluxes plotted against normalized depth. As with the other shape functions (Figures 6 and 7), the origin on the x-axis represents the shoreline, 1 is the location of the breaking point, and values greater than one are located outside the surf zone. On the y-axis, positive values indicate net onshore-directed sediment fluxes, and negative values are offshore-directed fluxes. Every marker represents an average of a 17-min time series. Despite the limitations on the determination of sediment fluxes and scatter in Figure 11, the measured normalized sediment fluxes show a very similar spatial structure to that observed in the normalized velocity moments of Figures 6 and 7 with only a few points violating the general trend. [35] Nevertheless important differences exist between the velocity moments shape functions and Figure 11. For example, the sediment flux shape function shows a more defined convergence around the breakpoint (less scatter), and the predominant onshore sediment transport in the innermost surf and swash zones is observed in more data points. This onshore transport in the swash/inner surf is stronger and clearer than the one observed in the velocity moments shape functions. Recent investigations in the swash zone [Masselink and Russell, 2006] have found that the ability of the velocity moments to predict sediment transport in the swash zone is limited. In this region of the nearshore, there are several processes not included in the velocity moments approximation such as the effects of pressure gradients and breaking induced turbulence which could dominate the suspension and transport of sediments. Because this and other known limitations regarding the relationship between velocity moments and sediment concentration, a sediment transport shape function (rather than a velocity moments shape function) seems to be the following logical step into this approach. 6. Implications of the Shape Function for Sandbar Migration and Profile Development [36] The cross-shore structure of the normalized velocity moments (Figures 6 and 7) and sediment fluxes (Figure 11) 10 of 15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Figure 10. (a e) Nonzero components of the shape function on Table 4. The format of the figures is analogous to that of Figures 6 9. Table 4. Components of the Third Cross-Shore Velocity Moment and Associated Sediment Transport Mechanisms Term Sediment Transport Process 1 u 3 mean velocity cubed (e.g., undertow current inside the surf zone) 2 hũ 3 s i short-wave velocity skewness 3 hũ 3 l i long-wave velocity skewness 4 3hũ 2 s iu stirring by short waves and transport by mean flow 5 3hũ 2 l i u stirring by long waves and transport by mean flow 6 3hũ 2 s ũ l i correlation of short wave variance and long wave velocity 7 3hũ 2 l ũ s i correlation of long wave variance and short wave velocity 0 8 6h ũ s ũ l i u three way correlation 0 9 3hũ s iu 2 time average of oscillatory component 0 10 3hũ l i u 2 time average of oscillatory component 0 11 of 15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 Figure 11. Table 3. Normalized sediment flux plotted against normalized depth with markers according to reflect three distinct regions of the nearshore that are clearly noticeable to any observer at the coast. These are a shoaling zone of unbroken waves (with onshore transport), a surf zone where most waves are broken (and transport is predominantly offshore), and the wetting and drying region of the swash zone (where transport is again onshoredirected). This has several implications for profile development, the most important of which is perhaps the existence of a convergence of sediment around the breakpoint and the associated concept of sandbar generation and migration. [37] Plant et al. [1999, 2001b] used bathymetric surveys to study the long term (O(decades)) behavior of a shoreparallel sandbar system at Duck, North Carolina. These studies show that bars tend to migrate consistently towards a position that coincides with the onset of wave breaking, confirming the importance of this dramatic change in hydrodynamic conditions caused by the breaking waves. The shape function proposed in this study supports the hypothesis of bars being formed at the breakpoint by convergence in sediment transport, and it also provides an integrated mechanism (which includes undertow, short and infragravity waves) by which the sandbar evolution is linked to the small scale hydrodynamic forcing. For example, in an initial featureless beach, sediment will be transported offshore inside the surf zone and onshore outside the surf zone to be accumulated around the breaking point and form a sandbar (Figure 12a). If energy levels increase, the surf zone will be broaden as the larger waves break in Figure 12. Schematic representation of shape function effect on profile morphology. (a) Regions of the nearshore (shoaling, surf, and swash zones) and related regions of the shape function showing generation of a bar at the convergence of sediment transport (breaking point). (b) Offshore transport dominance during storms. (c) Onshore sediment transport will be acting over the bar during low wave conditions. 12 of 15
MARIÑO-TAPIA ET AL.: CROSS-SHORE SEDIMENT TRANSPORT, 1 deeper water and offshore transport will dominate over the sandbar. The corresponding gradients in the offshore directed transport will move the bar in the offshore direction (Figure 12b). Conversely, under low energy conditions the sandbar will be experiencing onshore transport, and onshore bar migration would be likely to occur (see Figure 12c). In an accompanying paper [Mariño-Tapia et al., 2007], the capability of the shape function parameterization to replicate observed bar crest migration patterns is investigated. When the shape function is incorporated in a time-varying model, bar generation and the subsequent migration patterns are quantitatively modeled. Nevertheless, in its present form, the shape function model is not capable of replicating the whole profile morphology but only bar crest migration patterns. This is partly due to the use of h/h b as a parameterization, which is a difficult variable to evaluate, especially in well developed bar-trough systems (see discussion on section 7). [38] The existence of a shape function in natural surf zones also implies that, whenever available, sediment is constantly carried from the inner shelf towards the beach face, especially during storms when the onshore phase of the shape function (h/h b > 1) can extend a couple of kilometers offshore. Haines et al. [1999] examined a twelve year-long data set of profile measurements taken at approximately two week intervals at Duck, North Carolina (1981 1993) which include a considerable distance in the offshore direction. Their observations show that the profile responds to episodic inputs of sediment coming from an offshore source. The long term morphological evolution is not a simple redistribution of sediment across the profile, but there is a net increase in profile volume, the beach face steepens, and ultimately the shoreline accretes. 7. Limitations of the Shape Function [39] The determination of a breaking point is a limitation in natural surf zones, especially in morphologies that include multiple bars and troughs. At troughs, values of h/h b can exceed one, despite troughs can be located onshore of the breakpoint. This seems to imply that the hydrodynamics are the same at the trough as further seaward where values of h/h b are the same. It must be stressed that the cross-shore distribution of surf zone energy scales the shape function to give an estimation of the total sediment transport. Hence, the sediment transport direction at troughs where h/h b > 1 will indeed be similar to that occurring further seaward (i.e., onshore transport predominates at the trough), but the magnitudes of the transport (and hence the processes at work) will not be the same because the energy level is different. Measurements carried out on troughs of barred beaches show that waves can reform and processes such as wave skewness regain importance, producing net onshore transport [Thornton et al., 1996]. On the other hand, state of the art numerical models predict onshore sediment transport at the troughs even though waves have already broken at the bar crest [van Rijn et al., 2003]. For the shape function to properly accommodate the hydrodynamic behavior at troughs, the nature of the sediment transport convergences and divergences would have to be further investigated in the context of variables more robust than the breaker depth, such as intensity of wave breaking, or intensity of wave dissipation. [40] The shape function is not expected to hold true when the hydrodynamics are dominated by alongshore rhythmic patterns where the undertow current structure is replaced by cell circulation systems (3-D rip circulation). Nevertheless, situations with a degree of alongshore variability can also fit the pattern suggested by the shape function. There is important evidence of this in the Egmond data sets, which include the presence of strong longshore currents and energetic shear waves [Miles et al., 2002]. For the same period (Egmond main experiment) Ruessink et al. [2000] found that 85% of the morphology variance was associated with alongshore migrating bars and only 10% was associated with the alongshore uniform cross-shore bar migration. Yet, in spite of the observed alongshore variability, the Egmond data fits rather well the pattern suggested by the shape function structure. Results that broadly agree with the shape function were also found at Duck, North Carolina [Thornton et al., 1996] during times of mild three dimensional bathymetry and occasionally strong alongshore currents (1.5 m/s). [41] Reverse (offshore) transport at short wave frequencies can be observed over rippled beds due to sediment laden vortices in the wake of ripples ejected to the water column. However, several studies have shown that the presence of bed forms alone is not always an indicator of phase lags and reversed sand transport. For example, Osborne and Greenwood [1992b] observed onshore sand movement when waves shoaled over steep, three dimensional vortex ripples outside the surf zone, and recent investigations have shown that ripples tend to migrate in the direction of the velocity skewness [Crawford and Hay, 2001; Doucette et al., 2002]. [42] Although the aim of this paper is not the generation of improved parameterizations for individual processes, it is necessary to note that short wave skewness presents an important scatter when plotted against normalized depth (Figure 10b). Doering et al. [2000] have shown that a better prediction of wave skewness requires the use of a fairly complex expression that contains three independent variables including the Ursell number, the surf similarity parameter and normalized wavelength. 8. Conclusions [43] Several aspects of cross-shore sediment transport processes relevant to bar generation and profile development have been addressed in this study by using field data and the energetics approach. Despite the fact that this has been a common topic of study in nearshore research for more than 20 years and a vast body of knowledge about the effects of individual processes is already available, an integrated view on the relative importance, directional attributes, and cross-shore structure of the net cross-shore sediment transport in the nearshore, including most of the relevant processes, has only been partially quantified to date. This study addresses this uncertainty by proposing a field-based parameterization (shape function) in which the cross-shore structure of the balance between multiple opposing mechanisms of cross-shore sediment transport is 13 of 15