Evidence of coastal trapped wave scattering using high-frequency radar data in the Mid-Atlantic Bight

Coastal trapped waves (CTWs) become scattered when they encounter irregular coastlines and bathymetry during propagation. Analytical and modeling studies have provided some information about the different types of shelf geometries that can induce scattering, but much of the CTW scattering process generally remains a large knowledge gap. Furthermore, CTW scattering has never before been directly identified with observations. High-frequency radar surface velocity data covering the Mid-Atlantic Bight (MAB) continental shelf provides unprecedented observations of CTWs within a region with a highly 5 complex coastline and bathymetry. A combination of velocity vector maps from real vector empirical orthogonal function (REOF) analysis and phase maps from complex empirical orthogonal function (C-EOF) analysis allow the identification of CTW scattering by assuming each EOF mode corresponds to a CTW mode. Abrupt jumps in phase in association with magnitude amplification/reduction or directional rotation of velocity vectors are indications of scattering. Using these guidelines, Georges Bank, Hudson Shelf Valley, Delaware Bay mouth, Chesapeake Bay mouth, and the North Carolina shelf are identified as high 10 scattering regions within the MAB. Furthermore, stratification is confirmed to increase scattering into progressively higher order modes through a cascading process by comparing winter and summer cases, which supports previous theoretical and numerical model predictions. The simple methodology used here can be applied to observations of CTWs on other coastlines around the world to identify additional scattering regions and help close the knowledge gap.


Introduction
Coastal trapped waves (CTWs) were first observed around the coast of Australia in the 1960's (Hamon, 1962(Hamon, , 1966. Many observational and theoretical studies on the topic were conducted throughout the 1970's and 1980's (notably Buchwald and Adams 1968;Gill and Schumann 1974;Gill and Clarke 1974;Clarke 1977;Brink 1982;Freeland et al. 1986; see summary in Brink 1991), but advancements in furthering our understanding of this coastal phenomena have largely stagnated since 20 then. During this time, several analytical studies examined the role of more complex, but highly idealized shelf geometries on CTW scattering (Chao et al., 1979;Wang, 1980;Huthnance et al., 1986). A more robust modeling study including the single velocity component. Second, we eliminate data in water depth greater than 500 m as we are interested in the circulation on the shelf and data quality tends to decrease with distance offshore. Lastly, we remove data from a thin coastal region along the New Jersey, Delaware, and Maryland coastline that we have determined to have large directional uncertainty (Brunner and Lwiza, 2020).

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We linearly interpolate over gaps less than six hours in length to get as complete a data set as possible without introducing significant interpolation errors. The range of periods of CTWs within the study region is identified using continuous wavelet power spectra analysis as in Camayo and Campos (2006). The data are then bandpass filtered between 3 and 12 days to isolate the identified CTW signal. Other signals may exist within this range of periods, but their contribution is small compared to the observed CTW velocities which are O(10 cm s -1 ).

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Following Kaihatu et al. (1998) and Barnett (1983), we perform real vector empirical orthogonal function (R-EOF) analysis and complex empirical orthogonal function (C-EOF) analysis, respectively, on the CTW velocity data. Similar to standard EOF analysis, these methods are used to determine the variability of the CTW velocity by decomposing the time series into independent spatial modes or eigenvectors (EOFs) and their corresponding time series (principal components, PCs). The first mode explains the highest percentage of the variance of the original data being analyzed, with each successive mode explaining 80 less of the variance (North, 1984). We assume that each of the EOF modes correspond to a CTW mode.
Unlike standard EOF analysis, R-EOF analysis accepts the coupled vector velocity components as [u v] and is a better method for decomposing propagating signals. R-EOF results in velocity vector maps and their corresponding PCs. This method is able to resolve correct directionality within the velocity vector maps as it maintains the vector nature of the velocity field but provides no propagation information as it cannot determine phase. The vector maps, therefore, provide CTW flow direction 85 and magnitude information for each mode. Similarly, C-EOF analysis is good at decomposing propagating signals but only decomposes each velocity component individually. Due to the natural propagation direction of CTWs and the orientation of the coastline, we focus on the meridional velocity component (v). Obtaining the Hilbert transform of the data before calculating the covariance matrix results in phase maps that provide propagation information but are unable to provide vector information.
for use.
We, therefore, use a combination of C-EOF and R-EOF results -vector velocity maps and vector PCs from R-EOF and phase spatial modes and phase PCs from C-EOF. The methods provide different, but complimentary information about the CTW modes. The phase maps provide propagation information while the vector maps provide CTW flow direction and magnitude information for each mode. We further assume that each of the modes from the two different methods are approximately the 95 same, i.e. mode 1 from C-EOF corresponds to mode 1 from R-EOF, etc. The percentage of the variance that each mode explains as well as the spatial correlation help validate this assumption.

Observations of CTWs
After using wavelet analysis to identify the range of CTW periods and bandpass filtering to isolate the identified signal, we 100 first plot both the zonal and meridional CTW surface velocity components at select points along the coastline to confirm the presence of CTWs and observe their propagation ( Figure 2). The filtered velocity signals are consistent with previously observed CTWs  although the signal is carried in both velocity components rather than just the meridional component due to the coastline orientation. Identified surface velocities typically fall between -35 to 20 cm s -1 , with a negative sign indicating typical, southward propagation. Velocity magnitudes are reduced in summer compared to winter, also consistent 105 with previous observations. The phase speed is too fast to reasonably estimate from hourly observations; at such coarse time resolution, the signal appears to be largely in phase or slightly shifted right from one location to the next. It is, however, apparent that the signal transforms or becomes modified as it travels along the coast, likely due to changing bathymetry and scattering. Sometimes a large signal becomes damped as it propagates (e.g., mid-March or end of July, Figure 2) and other times it becomes enlarged 110 (e.g., mid-January).
We begin to investigate the role of scattering at these locations by decomposing the winter CTW velocity signal into its modes using the dominant velocity component. Standard EOF analysis (North, 1984) is conducted on the observational data whose grid points fall along the indicated cross-sections (Figure 2 left). For the observations, the cross-sections end where the lines intersect with the 500 m isobath due to the applied cutoff criteria. These modes are compared to theoretical EOF 115 modes obtained for each of the cross-sections using K. Brink's dynamic mode analysis software (available at: whoi.edu ;Brink 2006). The dynamic mode analysis software solves for CTW free wave solutions to the Boussinesq, hydrostratic, linearized momentum equations, where x is the cross-shelf direction, y is the alongshelf direction with positive northward, and z is depth positive upward; u(x, z, t), v(y, z, t), and w(x, y, z, t) are the velocities; v 0 (x, z) is the mean flow; ρ(x, y, z, t) = ρ 0 + ρ 1 (x, z) + ρ 2 (x, y, z, t) is the density as a combination of the mean density, ρ 0 , background density, ρ 1 , and wave perturbation density, ρ 2 ; P (x, y, z, t) is 130 the perturbation pressure; and τ tx (x, z) and τ ty (y, z) are the turbulent stresses. Additional details can be found on the software website or in Brink (2006) and . This formulation and others rely on an assumption of a straight coastline with shelf similar bathymetry so they are incapable of capturing CTW scattering. Therefore, differences between observed and theoretical modes are indications of possible scattering as the model is unable to capture this behavior.
Generally, the shape of each of the observed modes agrees with the modeled modes but overall agreement is very poor 135 ( Figure 3). Agreement is better for MA, NJ, and DE, which show reasonable mode 1 agreement while secondary peaks in mode 2 and 3 are shifted toward the shelfbreak. All of these locations are on relatively straight coastlines and also have the greatest percentage of the variance explained by mode 1 (Table 1), indicating these may not be high scattering regions.
Secondary peaks in mode 2 and 3 are shifted toward the coast for NC and mode 1 peaks offshore rather than at the coast, as it does for HC and DE as well. Percentage of the variance explained indicates that a great deal of energy has been shifted from 140 mode 1 to mode 2, likely due to scattering. Finally, observed HC modes are very complex and show very little agreement with the modeled modes. Scattering is expected in this highly complex region Lentz, 2017, 2018).

Full shelf scattering results
Having confirmed the presence of CTWs and identified some possible regions of scattering, we utilize the full shelf R-EOF and C-EOF results to properly identify regions of CTW scattering. Amplification or reduction in vector magnitude, rotation 145 of vectors away from an alongshelf direction, and large, sudden jumps in phase are all indications of scattering (Wang, 1980).
We focus first on the winter results ( Figure 4), which we assume have a well-mixed water column. Vector and phase PCs are largely in phase and have a similar pattern for each mode despite obvious magnitude differences, allowing us to interpret the selected R-EOF and C-EOF results together. The first three modes of R-EOF explain 45.5%, 17.9%, and 7.9% of the variance and requires a cumulative 17 modes to reach the 90% variance threshold. PCs are similar to the zonal and meridional wind components (Figure 4g), indicating that these are wind-forced features that we are observing. The first three modes of C-EOF explain 51.7%, 14.5%, and 8.7% of the variance.
Mode 1 flow is unidirectional in the alongshelf direction over the vast majority of the shelf (Figure 4a). Flow north of Georges Bank (GB) in the Gulf of Maine (GOM) is also unidirectional and alongshelf, but pointing in the opposite direction.
This area is also approximately 180 o out of phase with the rest of the shelf, which is entirely in phase ( Figure 4d). Vector 155 magnitude in the GOM is reduced compared to the rest of the shelf, but is amplified after crossing into GB. These factors indicate that this is a scattering region for mode 1 CTWs. Vector magnitude is also reduced within the Hudson shelf valley (HSV) and along the North Carolina (NC) coast. Although there is very little change in phase associated with this reduction in amplitude, this is also indicative of some mode 1 scattering and absorption.
Mode 2 flow is more complicated. Flow is no longer unidirectional but cross-shelf or perpendicular to bathymetric contours Amplitude is also reduced along the NC coast. GOM/GB, HSV, and NC are common regions that show scattering in all three modes, suggesting that these are high scattering regions. Rotation and slight funneling near Delaware Bay in mode 2 suggests that estuaries, which are an interruption to the waveguide, may also induce scattering.

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It has been proposed that stratification increases scattering into a higher proportion of higher order modes (Wilkin and Chapman, 1990), so we compare the well-mixed winter case to a stratified summer case ( Figure 5). The first three modes of R-EOF explain 41.6%, 12%, and 8% of the variance and requires 30 modes to reach the 90% variance while the first three modes of C-EOF explain 45%, 15.6%, and 9% of the variance. Percentage of the variance explained by the first three modes has decreased and the number of modes required to reach the 90% variance threshold has increased. This indicates that all modes, 175 not just mode 1, are scattering into progressively higher modes in a cascade (Chao et al., 1979) with increased stratification.
Mode 1 is very similar to the winter case with flow unidirectional in the alongshelf direction. Flow north of GB is again pointing in the opposite direction (Figure 5a), although this area has decreased. Absence of data in this region does not allow us to determine if there is a phase jump (Figure 5d). Amplitude is not reduced over the HSV, suggesting stratification prevents the CTW from feeling the effect of the canyon as much. Reduced amplitude near Long Island Sound and Chesapeake Bay also 180 suggests that increased freshwater inflow from the estuaries may also induce scattering.
Mode 2 and 3 vector maps also look quite similar to the winter case. In mode 2 (Figure 5b), flow is again cross-shelf in GB, HSV, and Delaware Bay although there is more apparent funneling toward the Delaware and Chesapeake Bay estuary mouths.
The reflection near NC is reduced in mode 2 but greatly amplified in mode 3 (Figure 5c), suggesting that this interaction may have been scattered into a higher order mode by stratification. Notably, the influence of HSV is absent where it was so prevalent for the winter case. This is also true of the phase maps (Figure 5e and f), which are more complex than for the winter case although phase jumps of nearly 180 o are still present in these common areas.
We also evaluate the seasonal CTW contribution to the total kinetic energy over the continental shelf. Following Takikawa et al. (2003), the ratio of CTW to total kinetic energy is calculated as where u c and v c are the seasonal CTW velocities, u and v are the annual mean currents, and the overbar indicates a temporal average. The average summer contribution of CTWs to the total kinetic energy is 23.3%, while the winter contribution is 30.3%. This supports the seasonality of the CTW velocity magnitude that we observed in . The average CTW winter contribution is also approximately the same as the tidal contribution (Brunner and Lwiza, 2020), indicating that CTWs are an equally important component of the overall shelf circulation and together they contribute ∼60% of the total 195 kinetic energy. Spatial patterns indicate that the winter contribution is higher along the SNE, on either side of HSV, and along the NC coast ( Figure 6).

North Carolina shelf scattering results
Next, we repeat the above procedure for the high scattering region of the NC shelf identified above and in our previous study . By de-coupling it from the remainder of the shelf, we can determine if there are separate or additional 200 scattering processes occurring there and also observe smaller scale scattering features than we can for the entire study region.
In winter, the first three modes of R-EOF explain 44.4%, 15.5%, and 8.1% of the variance and requires a cumulative 13 modes to reach the 90% variance threshold and the first three modes of C-EOF explain 47%, 17.5%, and 11.1% of the variance.
In summer, the first three modes of R-EOF explain 48.4%, 16.2%, and 7% of the variance and requires a cumulative 11 modes to reach the 90% variance threshold and the first three modes of C-EOF explain 48.4%, 16.8%, and 10.1% of the variance.

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Unlike the entire study region results, we do not see a decrease in the mode 1 variance explained from winter to summer and the number of modes required to reach the 90% threshold does not increase.
Vector and phase maps are consistent with the entire study region results but show higher resolution features (Figures 7, 8).
Amplification of the vector magnitude near the southern extent of the domain in all three modes of both seasons is consistent with shelf narrowing. Mode 1 flow is unidirectional with reduction of the amplitude near the Chesapeake Bay mouth and 210 amplification near Cape Hatteras (Figures 7a, 8a) and has a consistent phase (Figures 7d, 8d). Divergence of the flow at the Chesapeake Bay mouth and cross-shelf flow near Cape Hatteras is stronger in summer than winter. These areas are associated with 180 o phase jumps. Lastly, the mode 3 vector maps show the reflected or backscattered CTW (Figures 7c, 8c) which is much stronger in summer and likewise associated with phase changes (Figure 8f).
We also examine the energy density, (u 2 + v 2 )/2H, within this region using the summer R-EOF vector map results for u us to assume that the velocities are consistent vertically. We have previously shown that CTWs are barotropic in this region even when the water column in stratified , so this is a valid assumption.
We do not know the maximum value of the incident wave to normalize the energy density per Wang (1980). Therefore, we are forced to use some other value. The calculated energy density distribution is strongly right skewed and its maximum value 220 is an outlier. Hence we normalize by the value corresponding to the third standard deviation from the average, µ + 3σ (i.e., representing a probability of 0.003 of exceeding that value). Since we are unable to normalize by the incident energy density, the normalized values do not have the physical meaning of indicating whether scattering is increasing or decreasing energy density transmission relative to the incident wave. Our analysis is, therefore, more qualitative but still highlights regions of energy modification.

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We find that there are three primary areas where energy is increased (Figure 9). These areas are the Chesapeake Bay mouth, the coast, and the southern extent of the domain by Cape Hatteras. These areas correspond to scattering identified using the vector and phase maps (Figures 7 and 8), thus we can attribute this increased energy to scattering. The shelf narrowing to a minimum shelf width around Cape Hatteras and the Gulf Stream, which opposes typical CTW propagation direction, clearly have a large effect on scattering and energy transmission as its influence is seen to some extent in all three modes. The

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Chesapeake Bay mouth and estuarine inflow appears to have the greatest effect on mode 2, while the narrowing shelf width influence of the coast has the greatest effect on both modes 1 and 2.

Discussion
Guided by scattering principles determined through theoretical (Wang, 1980) and modeling work (Wilkin and Chapman, 1990), we are able to unequivocally identify CTW mode scattering within observations for the first time. We previously theorized that 235 scattering explains the poor agreement between observed and theoretical CTW modes off the coast of NC  but did not have the necessary framework to demonstrate that scattering was occurring. We can now show that a combination of R-EOF and C-EOF analyses can be effectively utilized to identify CTW scattering within both full shelf and regional scales.

Reduction/amplification of vector magnitude and/or rotation of vectors into the cross-shelf direction on the R-EOF vector maps
and large, sudden jumps in phase on the C-EOF phase maps are indications of scattering.

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Using these guidelines, we identify the GOM/GB transition zone, HSV, Delaware Bay mouth, Chesapeake Bay mouth, and NC shelf as high scattering regions. Bathymetry in the GOM/GB transition zone is very complex. The shelf is extremely thin along the Cape Cod coast before broadening into shallow GB and the SNE shelf. Generally, broadening of a shelf induces scattering into higher order modes (Wilkin and Chapman, 1990) and reduction of the velocity amplitude (Wang, 1980) although the large bathymetric step from GOM to GB which increases velocity amplitude must also be considered. Accordingly, we 245 observe divergence of the flow at the GOM/GB transition zone as the incoming wave is reflected, amplification of the magnitude in shallower GB, and increased cross-shelf flow due to scattering into mode 2.
Canyons, and HSV in particular, have garnered previous attention (Wang, 1980;Lentz, 2017, 2018). Only part of the incoming wave is transmitted as a portion of the wave is reflected as short wave. This short wave reflection is largely the tip of the canyon (Wang, 1980). However, orientation of the canyon with respect to propagation and wind forcing direction results in asymmetrical flow with onshore flow developing either weakly over the canyon or stronger upstream of the canyon (Zhang and Lentz, 2017). While we do see onshore flow associated with HSV, the overall pattern differs from this expected behavior. This is likely because the location of HSV is also associated with a nearly 90 o turn in the coastline which will induce its own scattering, while these studies are for canyons on an idealized, straight shelf.

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The effect of narrowing shelves similar to the NC shelf has also been previously examined (Wang, 1980;Wilkin and Chapman, 1990). Decreasing phase speed with decreasing shelf width amplifies the wave and the percentage of the incoming energy that gets reflected is proportional to the geometry of the shelf (Wang, 1980). Furthermore, the effect of scattering within this region can also be found upstream of the scattering region, particularly in the cross-shelf component (Wilkin and Chapman, 1990). We observe amplification, reflection, and cross-shelf scattering on the NC shelf, as expected. We also find an increase 260 in the energy density associated with these indicators of scattering. The proximity of the Chesapeake Bay mouth complicates scattering just upstream of the NC scattering region so it is not clear whether the cross-shelf flow is due to the upstream influence of the narrowing shelf or due to the estuary. The effect of estuaries on CTWs is not well understood, although we observe scattering at both Chesapeake Bay and Delaware Bay. Flow scattered into the cross-shelf direction is not directly perpendicular to bathymetry but tends to funnel towards the estuaries. This is also a behavior we have previously observed, but were unable 265 to attribute directly to scattering .
Most previous studies ignore the role of stratification due to the complexity of the problem (Chao et al., 1979;Wang, 1980).
The use of computational models allows the effect of stratification to be investigated rather simply. Wilkin and Chapman (1990) found that scattering by irregular topography is increased by stratification, while Zhang and Lentz (2018) found that asymmetrical flow generated by CTWs over a shallow shelf valley such as HSV is insensitive to stratification. Theoretical 270 CTW mode results from the Brink (2006) analytical model for the NC shelf also suggest that seasonal stratification is not an important factor in that high scattering region either .
Our results are able to rectify these contrasting views and fall more in line with Wilkin and Chapman (1990). Stratification does have an effect on the modes and has a greater effect on higher order modes, although mode 1 is also impacted. The number of modes required to reach the 90% variance explained threshold doubles from winter to summer for the entire shelf 275 results. This is likely due to a coupling of stratification increasing scattering to higher order modes (Wilkin and Chapman, 1990) and a net cascading process induced by the complex bathymetry (Chao et al., 1979). Interruption of the waveguide and increased freshwater input from estuaries also have a larger impact in summer, indicated by changes in phase as well as more prominent rotation and funneling toward the estuary mouths. Interestingly, the scattering role of HSV is diminished in summer and nearly disappears from mode 1. This directly contradicts what Zhang and Lentz (2018) found and suggests the need for a 280 more concentrated study in that area.
Our simple methodology can be easily applied to CTW observations from other complex coastlines around the world. Most observational studies have been conducted on relatively straight coastlines with smooth bathymetry, but several other studies have also found deviations between theory and observations. Battisti and Hickey (1984) and Chapman (1987) attributed these differences along the California and Oregon coastlines to error in the model wind forcing. However, we find it more likely that these differences arise due to scattering caused by complexities in the coastline near the Channel Islands and Point Conception, the San Francisco Bay mouth, Point Reyes, and Salish Sea. A modeling study of the shelf narrowing at Lofoten, Norway did not include the effects of scattering (Drivdal et al., 2016), but observations should show the effects of a narrowing and widening shelf. Observations of CTW sea surface height from the coast of Australia did not show large changes in phase or amplitude associated with topographic features such as shelf narrowing at Portland or complexity at Bass Straight (Woodham 290 et al., 2013), suggesting that CTW velocity may be more sensitive to scattering than sea surface height.

Conclusions
SeaSonde HFR surface velocity data allow us to take a detailed look at CTWs propagating on the MAB continental shelf. Poor agreement between theoretical and observed CTW modes (Figure 3; ) as well as the complex coastline and bathymetry in the MAB suggest that CTW scattering is prominent throughout this region. We use R-EOF and C-EOF to 295 decompose the CTW velocity into its modes. Results from these two analysis methods allow us to identify regions of CTW scattering within the MAB. This is the first time that scattering has been identified using observations rather than analytical or computational models. Changes in vector magnitude and rotation of vectors into the cross-shelf direction on the R-EOF vector maps and phase jumps on C-EOF phase maps are indications of scattering. We identify the GOM/GB transition zone, HSV, Delaware Bay, Chesapeake Bay, and NC shelf as high scattering regions using these guidelines. CTW scattering is, therefore, 300 common throughout much of the MAB continental shelf.
Mode 1 is largely unidirectional and typically accounts for approximately 50% of the variance. This percentage decreases in summer when stratification increases scattering into a greater percentage of higher order modes. Furthermore, a cascade of scattering into progressively higher modes requires collectively many more modes to describe 90% of the total variance.
Stratification also greatly increases the scattering influence of estuaries, which has not been previously examined. This is one 305 aspect of scattering that is not particularly well understood and requires additional attention. The scattering influence of HSV also does not follow what we expect based on previous studies (Wang, 1980;Lentz, 2017, 2018), perhaps due to its location within a large coastline turn, and should be studied further. However, the amplification, reflection, and cross-shelf scattering of the narrowing shelf off the coast of NC does align with expectations and explains poor agreement with theory in our previous study . Our methodology can be applied to other coastlines globally to identify additional 310 high scattering regions and gain a better understanding of the CTW scattering process. Possible future work involves the set-up of a hydrodynamic model such as ROMS or HYCOM to compare to observations and quantify the amount of scattering.