Surface Wave and Roller Dissipation Observed with Shore-based Doppler Marine Radar

,

Within the present paper, we propose a new approach to apply the roller concept 114 to Doppler radar data recorded by a shore-based, coherent-on-receive X-band marine radar. 115 Preliminary results of this work were presented by Streßer and Horstmann (2019). Un-116 like camera based methods, which estimate dissipation based on geometrical roller prop-117 erties, the proposed method is based on roller kinematics. More specifically the increase 118 from slow to fast surface speeds at the toe (the front edge) of the surface roller is related 119 to roller energy and dissipation. It can be obtained from the Doppler velocity measured 120 by the radar. The method is used to efficiently obtain mean dissipation rates with high 121 spatial resolution (7.5 m) along a cross-shore transect spanning the entire surf zone (> 122 1 km) of a double-barred, sandy beach. 123 The paper is structured as follows: The field measurements are described in sec-124 tion 2. In section 3, the scaling to obtain dissipation is first derived empirically through 125 a comparison to the in-situ observations and then theoretically from physical principles. 126 The evolution of radar-derived roller dissipation during a 3-day storm event is shown in 127 section 4. In section 5, the cross-shore transformation of the wave height is presented 128 and the performance of the method is studied by comparing it to in-situ measurements 129 and simulations. Section 6 contains an investigation of the wave energy flux budget with 130 an attribution of the observed dissipation to the morphological features. The transfer-131 ability of the results to other sites as well as the expected uncertainty are discussed in 132 section 7 and, finally, a conclusive summary is given in section 8.  At the study site, the coastline is oriented at a small inclination of 2°with respect 144 to North. The local coordinate reference system used in this paper has the origin at the  The sea state is mostly locally generated and grows rapidly from 0.5 m to ≈ 2 m 175 significant wave height on the second half of Sep 27. Simultaneously, the peak wave pe-  The radar used in this study is a coherent-on-receive marine radar developed at where λ el is the electromagnetic wave length of the radar signal and α is the projection where U Bragg is the Bragg waves' phase speed, U curr is the mean current U drif t is the 219 a drift velocity due to wind shear and Stokes drift, U break is the contribution of break-220 ing waves and U graz is an additional Doppler shift apparent at grazing incidence. The spatial difference in scatterer velocity can be used to infer dissipation. This is described 235 in detail in the following section 3.

236
To estimate the Doppler velocity, Doppler spectra were computed from the com-  The wave energy dissipation due to wave breaking is is related to the vertical ve-275 locity shear. It is determined by the velocity difference between water particle velocity 276 at the surface (within the roller) and the underlying water mass (e.g. Svendsen, 1984).

277
Our hypothesis is thus that the large positive spatial Doppler velocity difference dU D 278 observed at the toe of surface rollers can be used as a proxy for this vertical velocity dif-279 ference and is linked to energy dissipation. The following empirical scaling was tested 280 for the radar-derived wave energy dissipation: where B emp and the exponent p are calibration constants and the overbar indicates time 282 avaraging over the 10 min long radar record. The empirical calibration constants need 283 to be determined from in-situ observations of the dissipation rate that are deduced from 284 the pressure wave gauges as where x P GA and x P GB are the cross-shore location of the pressure gauges and F P GA and 286 F P GB are wave energy flux at each pressure gauge. The wave energy flux is computed 287 as F P G = ρ w g η 2 P G c g , where c g = √ gd is the group speed of the waves (in shallow wa-288 ter equal to the phase velocity), η P G is the surface elevation obtained from the pressure 289 gauges, ρ w is water density, g is gravitational acceleration, and d is the local water depth.

290
To determine the values empirical constants B emp and p, a cost function is computed 291 representing the root mean square difference between the radar estimate and the obser-

292
-8-manuscript submitted to JGR: Oceans vations: pirically derived dissipation rate D emp (eq. 3) was evaluated two radar range cells (15 298 m) further offshore than the location of the pressure gauges. This is needed because the 299 jump from slow to fast scatterers appears at the toe of the surface roller, but the point 300 where the wave energy is dissipated (the wave crest) is located slightly further offshore.

301
This is explained in more detail in sections 3.2 and 5. The relationship in eq. 3 is purely empirical and was found from comparisons to 304 the in-situ observations. To gain further insight into the geophysical processes, a deriva-305 tion based on physical principles is presented in the following. For long-crested waves, 306 the total (bulk) kinetic energy stored in the surface roller per unit span is given by where A r is the cross-sectional roller area, u r and w r are the bulk horizontal and ver-308 tical motions of the roller and the overbar indicates time averaging. The bulk density 309 of the roller, including both water and air, can be expressed as where β ρ represents the reduction of the water density ρ w according to the void fraction 311 inside the roller. Phase-averaging the total roller energy yields the roller energy per unit where L is the wave length. With the assumption that the vertical component of the roller 314 motion is small (w r ≪ u r ), the roller moves approximately with the same speed as the 315 breaking wave, and thus The roller area can be expressed as where H is the wave height and κ is a proportionality constant that varies between 0.06 318 and 0.07 (Okayasu et al., 1986;Svendsen, 2005). Combining eq. 9, 10, 6 and 8 yields for 319 the roller energy The wave height can not be measured directly by the radar. To substitute H, the de- The calibration coefficient α ad determines to what extent the amplitude dispersion is con-325 sidered. For α ad = 0, eq. 12 corresponds to the shallow water phase velocity accord-326 ing to linear wave theory, whereas for α ad = 0.5 it corresponds to solitary wave the-327 ory. The water depth at the breakpoint can be roughly estimated as eq. 12 and 13 yields the approximate expression relating the wave height of a breaking shallow water wave to its phase speed. Combin-331 ing equations 11, 16 and 14 finally yields a scaling for the roller energy as a function of 332 the phase speed c p of the breaker The Doppler velocity for the radar cell just before the front edge of the roller is small.

334
For the next radar cell, which is dominated by the roller, the Doppler velocity is close 335 to the phase speed of the breaking wave; thus, the spatial increase in Doppler velocity 336 dU D can be used to approximate the breaking phase speed as The calibration parameter β D was introduced to correct for the fact that the positive where the over-bar indicates time averaging over the full radar record (10 min for the 346 present study). Accordingly, the flux of roller energy is given by The dissipation of roller energy is related to the roller energy through where β s is a calibration coefficient related to the slope of the breaking wave front (Deigaard 349 & Fredsøe, 1989; Nairn et al., 1990). Therefore, the scaling for the roller dissipation de-350 rived from the radar is All calibration parameters that affect B r are listed in tab. Note that the empirically found scaling factor that provided the best match with are presented in terms of the significant wave height H s , which is expected to be more 395 conceivable than the wave energy flux for most readers. The distribution of H s along the 396 cross-shore transect is computed for both, the radar and the model, using a coupled wave 397 and roller energy flux balance as explained in the following.

398
For the simplified case of a stationary, unidirectional, normally incident, random 399 wave field, and in the absence of cross-shore currents, the cross-shore wave momentum 400 flux balance can be written as where F w = E w c g is wave energy flux and E w = 1 16 ρ w gH 2 s is the organized wave en- where F r = E r c p is the roller energy flux and D τ is the dissipation of roller energy (eq. 413 19) and D w is the wave energy dissipation from eq. 21. The wave dissipation couples eq. 414 21 with eq. 22. Once wave energy is dissipated, it is transferred to roller energy and is 415 finally dissipated by the shear stress between the roller and the underlying water body.

416
-12-manuscript submitted to JGR: Oceans This generates turbulence and drives wave-induced currents. The roller energy grows or 417 decays according to the difference of D w and D τ .

418
The roller energy E r , the flux of roller energy F r , and the dissipation of roller en-419 ergy D τ can be directly estimated from the radar measurements using eq. 17, 18 and 420 20. The dissipation of organized wave energy D w , at the location x ri+1 of the range range 421 cell ri + 1 is estimated numerically according to eq. 22 as where ∆r = x ri+1 − x ri is the distance between two adjacent radar range cells (here 423 7.5 m). The wave energy flux at x r,i+1 then follows from eq. 21 as 424 F w,ri+1 = F w,ri + D w,ri ∆r , and the wave energy and significant wave height along the full radar transect are given and 427 H s,ri = 16 E w,ri ρ w g .
The first order numerical integration scheme used in eq. 23 is sensitive to large gradi-428 ents. Since the radar observations naturally involve some high frequency noise, the radar- compensates abrupt changes in wave dissipation leading to a smoothing and and onshore 460 shift of the forcing of wave-induced currents (e.g. Goda, 2006).

461
To better quantify the overall performance of the proposed method, error statis-462 tics are computed for the locations where in-situ data is available. Figure 5   ing the significant wave height is comparable. The results show that the proposed radar 479 method can be applied with similar accuracy as numerical wave models, but without the 480 need to know the bathymetry or incident wave height. This is a major benefit in par-481 ticular for long-term observations. tems, e.g. arrays of wave buoys, bottom mounted ADCPs or pressure wave gauges.

507
Furthermore, for many research questions the surface stress is the quantity of in-508 terest because it is directly driving wave-induced currents and turbulence production by 509 breakers (e.g. Svendsen, 2005). The primary quantity that is observed by the radar is 510 the roller energy and dissipation, which is closely related to the Reynolds stress acting 511 at the water surface. Estimating roller quantities, as done here, is therefore a more di-512 rect measure of the drivers of nearshore circulation compared to wave height measure-513 ments.

514
A small disadvantage is the fact that the wave height must be known for at least 515 one location along the transect. This is not a problem, if the beach (where there is no 516 wave energy) is located inside the area covered by the radar. However, it could be prob- In the present work, eq. 14 was used to substitute the wave height. This scaling is only and thus match the results of Duncan (1981). This suggests that the proposed scaling 537 is also valid for breaking deep water waves. However, further research is required to con-538 firm this assumption. ing energy transfer when the jet hits the surface. This is expected to be of higher rel-544 evance for a plunging breaker type, since for spilling breakers, the roller is formed faster.

545
The roller concept was utilized in the context of the present study to provide a physi-546 cal basis for the proposed scalings to obtain roller energy (eq. 17) and dissipation (eq. are not able to provide reliable measurements in the surf zone, mostly due to the influ-626 ence of wave breaking and and increased spatial inhomogeneity. Strong rain and the ab-627 sence of surface roughness due to low winds are expected to negatively influence the method.

628
However, shore-based radar is relatively easy to install and maintain and is able to mea- where R = H b /H rms , H b = γd and B is a calibration factor representing the breaking strength and is set to one. f rep is the representative frequency of the wave field (often the peak frequency is considered). JB07 includes a slight modification of the empir- Radar raw data is available from the authors on request. 673 ter waves. Coastal Engineering, 13 (4), 357-378. Retrieved from https://