During the field experiment of the Coastal Sediment Transport Assessment using SAR imagery project of the Marine Science and Technology program of the European Commission an Air–Sea Interaction Drift Buoy (ASIB) system was equipped with special sensors and instruments to measure the position, the water depth, the surface current velocity and direction, the modulation characteristics of short-wave energies, and relevant air–sea interaction parameters due to undulations in the seabed. The ASIB system was operated from on board a research vessel and the data were measured while the buoy drifted in the tidal currents across sand waves of the study area. All buoy measurements were analyzed by computing frequency spectra of low and high frequency waves (scalar spectra between 0.1 and 50 Hz). The whole range of short water waves was recorded by these in situ measurements on board the buoy, which is responsible for the backscattering of commonly used air- and spaceborne imaging radars. A comprehensive dataset of wave energy density spectrum modulations above sand waves was produced. Normalized Radar Cross Section (NRCS) modulations of a selected P-band airborne Experimental-Synthetic Aperture Radar (E-SAR) image were compared with wave energy density spectrum variations at the appropriate short surface gravity Bragg-wave frequency measured along the drift path of the ASIB system. The NRCS and wave energy density modulation depths agreed within a factor of 2.
Using the obtained in situ measurements from the ASIB system the relaxation rate μ of short water waves due to current variations above submarine sand waves was calculated by applying a first-order weak hydrodynamic interaction theory. The relaxation rate μ dependence on several responsible hydrodynamic air–sea interaction parameters was calculated as a function of wavenumber k in the range of P-, L-, C-, and X-band radar Bragg waves for three different mean wind speed regimes of Uw = 0.8 m s−1, Uw = 3.8 m s−1, and Uw = 7.4 m s−1. Several published parameterizations of μ showed that this parameter increases with wavenumber and wind speed. Results show that μ increases also with wind speed Uw but decreases with wavenumber k. This can possibly imply that the wind growth relaxation rate μw is not equivalent with the relaxation rate μ of short waves due to current variations above submarine sand waves as a function of k. The analysis can also imply that the Bragg scattering mechanism seems to be insufficient to explain completely alone the NRCS modulation due to the seabed via surface current gradients especially at higher radar frequencies.
It is well known that radar images of the sea surface reveal information of bathymetry and other oceanographic and atmospherical phenomena. The visibility of submarine sand waves on radar imagery in coastal waters was discovered by De Loor in 1969 (De Loor 1981). First models for the radar imaging mechanism of the seabed were published by Alpers and Hennings (1984) and Shuchman et al. (1985). The imaging mechanism that they proposed, consisting of three steps, has been generally accepted: Water depth variations over sea bottom topography lead to modulations of the (tidal) flow. The resulting surface current gradients give rise to a hydrodynamic modulation of the wind-generated spectrum of water waves, which is known via the variation of the height of the short surface waves to affect the intensity of the backscattered radar signal. During recent years, the understanding of the radar imaging mechanism of sea bottom topography has been refined in several research projects and experiments as well as theoretical investigations (Holliday et al. 1986; Thompson 1988; Hennings 1990; Cooper et al. 1994; Hennings et al. 1994a; Van der Kooij et al. 1995; Donato et al. 1997; Romeiser and Alpers 1997; Vogelzang 1997, 2000; Vogelzang et al. 1997). Recently, a brief historical overview of radar imagery of sea bottom topography during the last 30 years was presented by Hennings (1998).
In one of the Marine Science and Technology (MAST) projects of the European Commission (EC) different imaging methods in the visible and radar parts of the electromagnetic spectrum have been studied to map the sea bottom topography with air- and spaceborne sensors. It turned out from the analysis of the data that mapping with imaging radars, like the Synthetic Aperture Radars (SARs) carried by the European Remote Sensing satellites ERS-1/-2 and the Canadian RADARSAT have the greatest potential in turbid coastal waters like the North Sea. Possibilities and limitations for practical use of radar in bathymetric applications was discussed by Greidanus (1997). That submarine bottom topography in coastal seas of ≤50 m water depth becomes visible on air- and spaceborne radar images suggests that it may be possible to employ radar remote sensing techniques for a cost-effective retrieval of depth charts from such imagery. A so-called Bathymetry Assessment System (BAS) developed by a research group in the Netherlands inverts the water depth–radar backscatter relation by using a data assimilation scheme.
Nevertheless, acquisitions of experimental data are still needed in order to gain more insight into the wave–current interaction mechanism, to calibrate and validate the existing radar imaging models, and thus enable the improvement of inverse bathymetry modeling in coastal waters. One of the main aims of this paper is to try to determine the relaxation rate of short water waves in the open ocean above submarine sand waves by using 1) in situ data of a recently developed Air–Sea Interaction Drift Buoy (ASIB) system of the Forschungsanstalt der Bundeswehr für Wasserschall und Geophysik (FWG), Kiel, Germany; and 2) calculating the lacking parameters by applying a first-order weak hydrodynamic interaction theory.
The theory is presented in section 2. In section 3 we describe the experiment and measurement configuration in the study area of the southern North Sea. The results of general oceanographic and meteorological observations, the current and wave energy density modulation measurements, as well as other measured hydrometeorological parameters, a comparison between experimental-SAR (E-SAR) Normalized Radar Cross-Section (NRCS) and wave energy density data of the ASIB system, and the calculation of the relaxation rate are presented in section 4. Finally, section 5 contains the discussion and conclusions.
The relaxation rate or wave growth rate μ is one of the most crucial parameters in weak hydrodynamic interaction theory. The reciprocal value μ−1 is called relaxation time τ. The short surface water waves responsible for the radar backscatter are located in that part of the wavenumber spectrum called the “saturation range.” This quasi-steady state that can be parameterized by a relaxation timescale of a balance between wind forcing, nonlinear interaction among different wave components, and dissipation by wave breaking is poorly understood. In general, the wave energy density falls with the fourth power of the wavenumber in this range. The relaxation time specifies the time in which the disturbed action spectrum of short water waves returns back to its quasi-steady state. In the relaxation approximation, it is assumed that the response of the wave system to straining is a simple exponential relaxation back to equilibrium. The relationship between variations in NRCS modulation and the sum of contributions from receding and advancing Bragg waves that result from modulations in the energy spectral density has its limitations and applicabilities. P-band Bragg waves can have rather high relaxation rates, and the modulations are proportional to the gradient in the surface current and, via the continuity equations, proportional to the sea bottom slope. The angular distribution of this wave equilibrium spectrum is peaked into the wind direction. At L- or C-band the wave energy density modulations can be dominated by waves moving against the wind direction that have small to zero relaxation rates. The modulations of these waves are dominated by advection and are proportional to the surface current itself. In this case the angular distribution of these Bragg waves is much more isotropic.
The discussion outlined by Zimmerman (1985) comparing the paper of Alpers and Hennings (1984) with that of Phillips (1984) is still topical, even now. One important conclusion was that careful observations of relevant physical parameters in tidal areas are recommended in both papers to settle the dispute over the relaxation timescale. Therefore, the C-STAR experiment in the southern North Sea was set up in April 1996 (see section 3) using a newly developed measurement configuration (Stolte 1994; Hennings et al. 1994b) to tackle these uncertenties.
To estimate at least the order of magnitude and the shape of the relaxation rate as a function of wavenumber due to variable currents over large submarine sand waves, the simple Alpers and Hennings (1984) model was used in combination with in situ measurements of the ASIB system. Of course, one may also question here whether such a simple theory may be inverted analytically for calculating relaxation rates. However, this procedure could be the starting point of a more comprehensive determination of the relaxation rate as a function of different oceanographic and meteorological parameters. Because of the longer wavelength that applies at P- and L-band radar systems, the Bragg model probably has greater applicability than at the shorter wave lengths associated with C- and X-band radar, where a composite scattering (or more sophisticated) model seems to be required.
According to Alpers and Hennings (1984) the action balance or radiation balance equation can be expressed as
is the action spectrum of short water waves, F(x, k, t) is the wave energy spectrum, ω′ is the intrinsic frequency of the wave in a reference system that is locally at rest, x = (x, y) is the space variable, k is the wavenumber, t is the time, δN is the time-dependent perturbation of the action spectrum, and μ is the relaxation rate parameter. The relaxation rate is determined by the combined effect of wind excitation, energy transfer to other waves due to conservative resonant wave–wave interaction, and energy loss, due to dissipative processes like wave breaking. An empirical expression for the wave growth–relaxation rate μw has been obtained by Hughes (1978) and is based on measurements of the growth of waves by wind. In the more exact results of Hughes (1978) and Valenzuela and Wright (1979), it has been established that the wave growth–relaxation rate μw is equal or less than the exponential relaxation at the rate μ due to the response of the wave field to perturbation by variable currents. Several parameterizations of the relaxation rate parameter are discussed by Caponi et al. (1988). Although there is, at present, considerable uncertainty about the exact functional form of μw, there is general agreement that it increases with wavenumber and wind speed. To prove this statement, in situ measurements were realized in the open ocean above submarine sand waves. To our knowledge, such a procedure using measurements in combination with an already known theory did not exist until now.
is the absolute value of the short-wave group velocity, U0 is the mean (tidal) current velocity, U is the local current velocity, g is the acceleration of gravity, ρ is the water density, τ is the surface tension, N0 is the equilibrium action density, and δN = N − N0. The first term of the left-hand side of (3) is the local time and is of the order of the period of the semidiurnal tide. The second term represents the advection of the current and the third term is the relaxation rate. Inserting (2) in (3) for N0 and δN and neglecting the first term against the second and third term in (3) yields
For the slope of the unperturbed energy spectral density the following expression was derived by Hennings et al. (1998):
where ap is the Phillips constant.
Inserting (6) into (5), we obtain
Solving (7) for μ in terms of the components of the flow velocity and the short-wave group velocity perpendicular to the sand wave crest yields
where |U| is the absolute value of the local current velocity, θ is the plane polar coordinate angle (counting anticlockwise) between the x0 axis (E direction) and the direction of the local current velocity U, ϕC0 is the mean angle, and ϕC is the local angle, respectively, between the current direction and the component of the current direction perpendicular to the sand wave crest, and ϕw is the local angle between the short-wave group velocity and its component perpendicular to the sand wave crest. The definition of symbols and angles used in (8) are shown in Fig. 1. The components parallel to the sand wave crest were neglected as a first approximation. All quantities of the right-hand side of (8) are known, either by parameters that can be calculated or by in situ measurements.
To produce a water depth map, the Bathymetry Assessment System developed in the Netherlands (Calkoen et al. 2001) needs one or more SAR images and a limited number of conventional (echo sounding) depth measurements. Only water depth variations can be detected with SAR and echo soundings are needed because the absolute water depth or very smooth slopes of the seabed cannot be determined. The conventional depth measurements are also used to calibrate the imaging model and to generate a first guess water depth map to start the data assimilation inversion.
A simulated radar image is calculated from the first guess water depth map using the imaging model suite. The simulated radar image is compared with the real radar image by evaluating the penalty function. This is achieved by minimizing the penalty function that contains three terms: one giving the difference between simulated and recorded radar image, one giving the difference between available depth information and model depth, and one containing a smoothness criterion for the model water depth map. The smoothness criterion is needed in order to prevent speckle noise in the radar images to be interpreted as water depth variations. Now the first guess water depth map is adjusted by minimizing the penalty function. This leads to an iterative procedure. When it has converged, the resulting model water depth map is the best estimate for the real water depth map.
3. Experiment and measurement configuration
The field experiment of the Coastal Sediment Transport Assessment using SAR imagery (C-STAR) project of the MAST-III program of the EC took place in an area off the coast of the Netherlands (Hoek van Holland test site) in the southern North Sea on 9–26 April 1996. The following institutes have participated: Forschungsanstalt der Bundeswehr für Wasserschall und Geophysik, Kiel, Germany; GEOMAR Forschungszentrum für marine Geowissenschaften der Christian-Albrechts-Universität zu Kiel, Kiel, Germany; Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR), Oberpfaffenhofen, Germany; and the Department of Oceanography, University of Southampton, Southampton, United Kingdom. The experiment was coordinated at the home base of the Advisory and Research Group on Geo Observation Systems and Services, Vollenhove, the Netherlands. Additional measurement campaigns were carried out in the same study area by the University of Gent, Research Unit for Marine and Coastal Geomorphology, Gent, Belgium, during two periods on 3–10 May 1996 and 6–11 April 1997, respectively. In the following mainly those measurements are described that have been performed on board the R.V. Planet and the drift buoy of FWG. But our investigation concentrates also on a selected P-band Experimental-Synthetic Aperture Radar image obtained quasi simultaneously on board the DO 228-200 aircraft of the DLR.
The location of the C-STAR study area is shown in Fig. 2 indicated as a dark square marked by M. It is located 28 km northwesterly of Hoek van Holland, the Netherlands, in the Southern Bight of the North Sea in the vicinity of a large shipping route to Rotterdam. This area shows a minimum water depth of 18 m and is covered by asymmetric large sand waves with a height between 2 and 10 m. The bathymetry of the test site, which is a 5 km by 5 km square is presented in Fig. 3. The tracks 9, 10, 13, and 14 of the Air–Sea Interaction Drift Buoy system (see also section 4) are also indicated.
The R.V. Planet was equipped with the following main sensors that were used and operated during the C-STAR experiment: the ASIB system, an acoustic Doppler current profiler (ADCP), an ELAC 30-kHz echo sounder for navigation, a Differential Global Positioning System (DGPS), a SELESMAR X-band navigation radar, oceanographic and meteorological sensors, and handheld cameras.
The positions of the ASIB system were measured by the Global Positioning System (GPS data rate 3 s) installed on board the buoy. In addition, the buoy positions were also obtained from DGPS measurements (data rate 10 s) on board R.V. Planet in combination with radar bearings of the buoy (data rate 5 min) to check the accuracy of the GPS position of the ASIB system. The expected absolute surface current velocity accuracy based on the accuracy of the DGPS/ASIB measurements is ±15 cm s−1. There was no cable connection between the ASIB system and R.V. Planet. The ship drifted most of the time in the tidal current and followed the buoy at a distance of about 400 m, controlled by radar bearings. Disturbances by the wake of the ship due to brief manuevering of R.V. Planet were not observed.
Three different bathymetric measurements were performed during each run: 1) echogram data measured on board the R.V. Planet using a 30-kHz ELAC echo sounder for navigation, 2) digital depth data of lower spatial resolution measured by the ADCP on board the R.V. Planet, and 3) high spatial resolution digital depth data measured on board the drift buoy using a VDO LOGIC depth 200-kHz echo sounder. The water depths measured by the echo sounder on board the buoy and obtained by the ADCP on board R.V. Planet have been compared with the bathymetric data presented in Fig. 3 to check any inconsistency in positional information or depth accuracy.
During all runs across the study area ADCP measurements have been obtained. The ADCP was mounted on the hull bottom of the R.V. Planet at 3.9-m water depth. An arrangement of four downward-looking concave transducers was performed in such a way that two pairs of transducers pointed in the fore–aft and starboard–port directions, respectively, with an incidence angle of 30°. Only three transducers were required to compute the three-dimensional current velocity (e.g., north, east, and vertical). The redundant information of the fourth transducer was used by the system to compute the error of the current velocity and to evaluate whether the assumption of horizontal homogeneity was reasonable enough during the experiment. The operating frequency was 300 kHz. During one tidal cycle (1 run per tidal cycle), flow velocity and water depth measurements were recorded continuously over a depth range of 32 m, separated into 32 cells (bins) of 1 m. Data were collected and processed after each pulse transmission and averaged over a sampling interval of one minute. This sampling interval did not limit the usefulness of the measurements. At the end of the sampling interval, a profile of the current velocity relative to the ADCP was performed as a function of range and then stored on floppy diskettes. According to information provided by the manufacturer, RD Instruments, the accuracy of the relative value of the current velocity is ±5 cm s−1. The accuracy of the estimated absolute current velocity is ±10 cm s−1. The measurement configuration, the instruments, and data rates are presented in Fig. 4.
The ASIB system was developed to measure the modulation characteristics of short-wave energies and relevant air–sea interaction parameters following the local currents and the elevations of long waves (Stolte 1994). During the C-STAR experiment the ASIB system was equipped with special sensors and instruments to measure the position, the water depth, and variations of hydrometeorological parameters due to undulations in the seabed. The configuration of the ASIB system during the experiment is shown in Fig. 5 (Stolte and Stolte 1999). A photo of the ASIB system during a test acquired at 0835 UTC 10 April 1996 at position 53°42.39′N, 5°15.10′E is presented in Fig. 6. All buoy measurements were analyzed by computing frequency spectra of low and high frequency waves (scalar spectra up to 50 Hz), spectra of horizontal and vertical wind speed components, temperature fluctuations, and mean values of boundary layer parameters. From the spectra the following parameters of the sea states were calculated: significant wave height, mean wave period, variance of slopes, and spectral constant and frequency decay in different frequency intervals above 0.8 Hz. Corresponding to the analysis intervals of the spectra mean values of the following quantities were calculated: orientation of the buoy to the north direction, wind speed and direction, air temperature, variances of air temperature fluctuations, friction velocity of the wind speed calculated from measurements of turbulence, friction coefficient, momentum flux, heat flux, and degree of turbulence for horizontal and vertical wind speed components. All parameters without the first one were measured at 2-m height. In addition, water temperature profiles measured at water depths of 0.1 m, 0.8 m, 1.7 m, and 2.5 m have also been obtained. An electromagnetic current meter of less than 0.1-s time constant (Marsh McBirney, Inc., Gaithersburg, model 511) was mounted on a small wave follower measuring surface currents at about 5-cm water depth. The currents result from current differences at the surface and at 1-m water depth, which are dominated by the wind-induced surface drift. The accuracy of the absolute relative current velocity is ±5 cm s−1. There was no potential problem of disturbance of the waves by mechanical parts of the buoy. The absolute surface current velocity was calculated by using the GPS measurements on board the ASIB system.
The whole range of the short water waves responsible for the backscattering of commonly used imaging radars was recorded by these in situ measurements. A unique dataset of wave-energy spectrum density modulations above sand waves was produced and compared with NRCS modulations of quasi simultaneously obtained multifrequency airborne E-SAR images (see section 4c).
The modulation of the sea surface roughness above submarine sand waves in the C-STAR study area was clearly visible by eye on board R.V. Planet. Figures 7a,b show two handheld camera images acquired on board R.V. Planet at 0817 UTC and 1410 UTC 19 April 1996 during southwesterly (SW) and northeasterly (NE) tidal current directions, respectively. To our knowledge, it was the first time that photographs were taken during both tidal phases, which show impressively the different roughness modulation patterns above asymmetric large sand waves. These visual observations are evidence of De Loor's (1981) real aperture radar images obtained in 1977 during opposite tidal flow directions in the Noordwijk study area of the southern North Sea. A first firm theoretical foundation to radar signatures of asymmetric sand waves was then given by Alpers and Hennings (1984).
a. General oceanographic and meteorological observations
Oceanographic and meteorological data were measured on board the R.V. Planet on 10–25 April 1996 in the C-STAR study area of Hoek van Holland. Measurements were also done in the study areas Noordwijk (the Netherlands) and Baltrum (Germany), but these data are not considered here. The measurements and visual observations were performed every three hours at 0600, 0900, 1200, and 1500 UTC. During the buoy measurements these data were collected every hour. Figures 8a–k show time series of the wind speed and direction at a 19-m anemometer height, the air pressure at 16-m height, the air temperature Ta at 16-m height, and water temperature Tw at 2 m below the sea surface, the estimated significant wave height, the direction of the wind waves, the sea surface height ζ relative to mean sea level of the central point of the Hoek van Holland study area, the estimated swell height, the swell direction, the dewpoint and relative humidity at 16-m height, and the estimated visibility. The times of the aircraft SAR flights during the experiment are also indicated in Fig. 8 as well as the different measurement periods in the study areas of Hoek van Holland, Noordwijk, and Baltrum.
Wind speeds at 1.0 m s−1 ≤ Uw ≤ 12.6 m s−1 with variable directions were measured during the experiment (Fig. 8a) and air pressures pa in the range 1006.5 to 1027.0 hPa were observed (Fig. 8b). A decrease of the air pressure from 1027.0 hPa on 15 April 1996 to 1006.5 hPa on 17 April 1996 is associated with changing wind direction from NE to S-SE, respectively. The air temperature Ta shown in Fig. 8c varies between −0.9° and 15.2°C and the water temperature Tw shows a variation from 2.7° to ≤10.0°C. In general, a change from unstable conditions (ΔT < −2°C) on 12–13 April 1996 to neutral (−2°C < ΔT < +2°C) on 14 April 1996 and then to stable (ΔT > +2°C) conditions was observed during the experiment. Figure 8d shows the significant wave height (SWH) varying between 0.03 and 2.00 m associated with dominant wind wave directions between 20° and 260°. The influence of the wind field is mainly reflected in the time series of the SWH. The dominant signal in the sea surface height shown in Fig. 8f is the period of the semidiurnal M2 tide. The varying extrema of the amplitudes have values −1.070 m ≤ ζ ≤ 1.004 m. Low swell heights less than 0.4 m with varying directions were observed (Figs. 8g–h). The dewpoint presented in Fig. 8i varies between −9.6° and 6.0°C. Relative humidities between 38.2% and 99.8% (Fig. 8j) and visibilities between 0.2 and 20 km (Fig. 8k) were observed.
b. ADCP and ASIB measurements
An example of the ADCP measurements on board R.V. Planet obtained during southwesterly tidal current flow direction from run 9 at 0616–08.31 UTC 16 April 1996 is shown in Figs. 9a,b. The depth-averaged current velocity (Fig. 9a) and water depth (Fig. 9b) are plotted as functions of time. Only those depth cells (bins) were considered for the calculation that covered the depth range between 8.5 m and 85% of the total water depth. A mean current velocity of Umean = 70.2 cm s−1 was calculated from the measurements. It can readily be noticed from Fig. 9 that all sand waves cause significant changes between 5 and 18 cm s−1 in the depth-averaged current velocity. A significant anticorrelation was found between the current velocity and the water depth record. The quantitative evaluation of the agreement between the current velocity and the water depth is given in terms of the linear correlation coefficient r = −0.62. Maximum current velocities were observed at the crests of sand waves and minimum current velocities were calculated from the measurements at the troughs of sand waves. Also changes of current direction were observed in the ADCP data. For example, an anticlockwise change of current direction up to 16° was apparent at 0628–0639 UTC in the depth-averaged ADCP data after crossing the sand wave crest.
The time series of the absolute surface current velocity calculated from the ASIB measurements during run 9 at 0622–0830 UTC 16 April 1996 is shown in Fig. 10a to complete the ADCP measurements. In addition, the water depth measurements from the echo sounder on board the ASIB system are also presented in Fig. 10b. A mean surface current velocity of Umean = 94.9 cm s−1 was obtained from the ASIB measurements, which was 24.7 cm s−1 higher than the depth-averaged current velocity measured by the ADCP on board the R.V. Planet. Significant variations of the current velocity between 5 and 63 cm s−1 can be observed. A significant anticorrelation between the current velocity and the water depth as shown in Figs. 9a,b cannot be established, which is also expressed by r = −0.13. During the first half of this run a downcurrent shift of the maximum current velocity in relation to the sand waves can be observed. The ADCP measurements (depth-averaged current velocity) correlate considerably better with water depth than the surface current derived from the ASIB measurements. Even the shape of the current patterns seems to correlate well. The water depth profiles shown in Figs. 9b and 10b are quite different because the ship track was displaced 250 m northwest parallel from the buoy track. However, either there are some major changes in velocity at unexpected locations or it is possible that the associated measured changes at these locations exceed the upper limiting values (in the order of 5–15 cm s−1) of the measurement devices. The very large changes of velocity at the surface at 6.6 and 6.8 h (UTC) should severely affect the wave energy density spectrum. But large changes of the wave energy density modulation were not observed at these times (see Fig. 14). Although the reason for these very large velocity changes remains unknown, they conceivably could result from small-scale oceanographic fronts or patches of turbulence known to disturb the surface current field (Nimmo Smith et al. 1999). The perpendicular and parallel components relative to the sand wave crests of the absolute surface current velocity and the depth of the seabed are presented in Figs. 11a–c. The corresponding results of the linear correlation coefficients are r = 0.16 and r = −0.10, respectively. The variation of the direction of the surface current velocity relative to the changes of the seabed is shown in Fig. 12. A direction change between 5° and 33° was calculated from the components of the current velocity presented in Figs. 11a,b. The perpendicular component of the current velocity is significantly correlated with variation of the direction of the current velocity. Often, it is found that, if the perpendicular component of the current velocity accelerates, then the direction can change anticlockwise significantly (up to 33°) after passing the crests of sand waves. Several sidescan sonar records showed that on the sides of the sand waves megaripples are present (Terwindt 1971). The ripple height generally increases in the direction of the sand wave crest. The transit sonar record from the sea area west of Hoek van Holland published by Terwindt (1971) showed that the crestlines of the ripples in the troughs and on both slopes of the sand waves form an angle αs with the crestline of the sand waves. The angle αs may reach values up to 45° and seems to be a function of the sand wave slope. However, the existence of an angle between the crestlines of megaripples and sand waves indicates that the direction of tidal currents changes also across sand waves. A possible explanation of the current direction variation is that during times of large sand transport the direction of the flow over the sand wave forms an angle with the sand wave crest, generating a bending of flow lines. The resulting current shear then produces a torque that also changes the direction of the angular momentum of short waves along their rays.
c. E-SAR NRCS and ASIB wave energy density data
The E-SAR system of DLR was operated in narrow swath mode with off-nadir angle range between 25° and 55° at an altitude of 3000 m resulting in a scene size of about 4-km width (ground range) with an effective spatial resolution of 6 m. From a collection of 21 scenes a P-band (0.45-GHz, HH polarization) SAR image was selected that was acquired on 16 April 1996 during southwestward ebb tidal current. The SAR image was processed by TNO, Physics and Electronics Laboratory, The Hague, the Netherlands (details can be obtained from H. Greidanus). For comparison with in situ measurements, georeferencing was essential: this was accomplished using the aircraft track direction, information about the starting- and ending-times of the flight track, and the position of the vessel Albatros, as determined both by the DGPS and its position within the image (the vessel Albatros was chartered by the University of Southampton). Taking into account the Doppler shift due to ship speed, the E-SAR image shown in Fig. 13 was transformed into a georeferenced Mercator grid. The maximum Doppler shift (in azimuth direction) of the radar signatures on the ocean surface due to current velocities in range direction is about 25 m.
The wave energy density spectra provided by R.V. Planet are calculated over 32-s intervals from resistance wire data sampled at 128 Hz. Noise in the spectra was reduced by line fitting the short wave log-decay in the spectral range 0.8–10 Hz. From the fitted spectra the wave energy density was extracted at fixed frequencies with a time series resolution of 32 s corresponding to about 30-m spatial resolution (at a tidal current of 1 m s−1). The GPS position of the buoy was corrected using the precision DGPS navigation of R.V. Planet and the radar bearings between the ship and the nearby buoy at 5 minute intervals. With this correction the echo sounder depth coincided well with the bathymetric map of Fig. 3.
The ebb tide experiment took place at 0620–0830 UTC 16 April 1996 (drift distance 6.26 km). Track 9 of the drift buoy shown in Fig. 3 indicates a tidal current toward the southwest, almost perpendicular to the sand wave crests. The dark small streaks of reduced backscatter intensity in the E-SAR image (see Fig. 13) clearly correspond to the sand wave pattern of Fig. 3. The initial geocoding of the image (with the ship position as reference) had to be corrected (for reasons not obvious to us): in order to make the dark streaks coincide with the reduced wave energy density modulation measured by the ASIB system, the image was shifted 220 m toward northeast. The airborne E-SAR image was acquired during a 2.5 minute interval, beginning at about 0830 UTC, when the buoy was at the end of the drift track.
In situ measurements of the wave energy density spectrum modulation, the water depth, the surface current and direction, and the wind speed and direction derived on board the drift buoy as well as the P-band NRCS variation of the E-SAR image are shown in Fig. 14. The wave energy density is normalized with the mean value over the complete track, which is assumed to represent the equilibrium wave energy density for the present wind condition (mean wind speed of 4.3 m s−1). The wave energy density was chosen at 2 Hz, as it corresponds to P-band Bragg waves of 0.44 m wavelength, which mainly contribute to the radar backscattering. The NRCS profile of the SAR image is averaged in the sand wave direction over 50 m on both sides of the buoy track.
As indicated from the buoy measurements the reduced wave energy density regions correspond to the steep slope at the upstream (northeast) side relative to the sand wave crests as well as to the crest region itself, where strong current divergence due to decreasing water depth is expected. The correlation of wave energy density with bottom topography is obvious in most parts along the track but the data also show some noise, probably caused by local wind effects. The reduced backscatter regions in the NRCS profiles of the SAR image are even more pronounced: the NRCS bands are smaller (30–60 m) and the modulation depth (typically 3–4 dB minimum to maximum) is larger than for wave energy density (typically 2 dB). Despite some noise in both profiles and the lower spatial resolution of the wave energy density, both profiles compare well with respect to the position of minima and maxima and the overall shape. The surface current in most cases changes direction when crossing the wave crests (see also section 4b), whereas the current acceleration in the steep slope regions of the sand waves is often masked by the general current variability.
d. Relaxation rate calculations
Exemplary time series of the calculated relaxation rate μ with associated hydrometeorological parameters measured on board the ASIB system from run 9 at 0622–0830 UTC 16 April 1996 are shown in Figs. 15a–f. The relaxation rate μ calculated according to Eqs. (4) and (8) is presented in Fig. 15a. The calculations are carried out for |U0| = 0.95 m s−1, |cg| = 0.43 m s−1, k = 2k0 sinα, where k0 = 2π/λ0 is the wavenumber of the P-band SAR and λ0 = 0.67 m is the corresponding electromagnetic wavelength, α = 45° is the radar incidence angle, g = 9.81 m s−2, τ/ρ = 73 × 10−6 m3 s−2, ϕC0 = 21°, and ϕW = 15.5°. The propagation direction of cg was assumed to be aligned in the wind direction of 15.5° (wind direction was from 195.5°T). This is a simplification because measurements performed by electrical resistance wires on board the ASIB system indicated that variations of the short and moderate wave directions were also associated with changes of the tidal current direction at the sea surface. A mean orientation of 123° for the crests of sand waves was used for all calculations. All other parameters of Eq. (8) were calculated from the measurements of the ASIB system. Figures 15b–f show the modulation of the wave energy spectrum density δF/F0 at the frequency of 2 Hz (corresponding to P-band radar Bragg waves), the negative gradient of the surface current velocity, the surface current velocity, the wind speed and direction at 2-m height, and the water depth. All parameters were time averaged over 32 seconds.
The relaxation rate μ as a function of wavenumber k in the range of commonly used imaging radars (P-, L-, C-, and X-band radar) for three different wind speed regimes of Uw = 0.8 m s−1, Uw = 3.8 m s−1, and Uw = 7.4 m s−1 is shown in Fig. 16. At first, all relaxation rates were time averaged over 32 seconds and calculated perpendicular to the sand waves using Eq. (8) for each run in the C-STAR study area as shown, for example, in Fig. 15a. A total of 11 runs were analyzed. For the light wind speed case of Uw = 0.8 m s−1 three runs (8, 10, and 14) were averaged. The mean wind speed for these runs ranged between Uw = 0.7 m s−1 and Uw = 0.9 m s−1. A total of five runs (5, 6, 7, 9, and 13) were used for the mean value of the moderate wind speed of Uw = 3.8 m s−1. During these runs the wind speeds varied between Uw = 2.3 m s−1 and Uw = 4.7 m s−1. Finally, for the higher wind speed case of Uw = 7.4 m s−1 three runs (2, 3, and 4) have been selected for averaging. The mean wind speed ranged between Uw = 7.3 and 8.1 m s−1. As shown for example in Fig. 15a, the calculated relaxation rate μ has negative as well as positive values for all analyzed runs. But negative or zero values of μ are not admissible if it is assumed that an equilibrium wave spectrum exists for all wave numbers and directions (see, e.g., the discussion in Romeiser and Alpers 1997). Negative calculated values of μ indicates that the wave spectrum is apart from an equilibrium wave spectrum. The reason why negative or zero values of μ were calculated can be caused by numerical noise in the measurements. This noise is amplified by calculating numerical derivatives, a well-known phenomenon described in any text on numerical mathematics. The result of Eq. (8) is very sensitive to the derivative of ∂U/∂x and ∂(δF/F0)/∂x, because both parameters are obtained from in situ measurements on board the ASIB system. The current velocity U and the modulation of the wave energy spectrum density δF/F0 were smoothed by using a three point running average procedure. It is obvious that the existing noise that was not caused by sand waves has not been filtered out completely. Therefore, only positive values of μ were considered for further analysis. However, we believe that this approach for calculating μ under open ocean condition is the best way to do it at the moment. Also the theory used here has its limitations.
The obtained results are in general agreement with the parameterizations of μ discussed by Caponi et al. (1988) that μ increases with wind speed. But there is a disagreement between our and their results with respect to the behavior of μ with increasing wavenumber k. Our results show a general decrease of μ with wavenumber. A possible reason for this is suggested by the handheld camera image, shown in Fig. 17, which was acquired on board R.V. Planet during run 13 at 0912 UTC 19 April 1996 at a time when the buoy was located within a region of enhanced water surface roughness. The tidal current direction was southwesterward during the acquisition time of Fig. 17, the wind speed had a value of Uw = 4.8 m s−1, and the wind direction was from 204°. Figure 17 shows enhanced breaking waves and sharp crested waves often just before breaking. Mainly, the longer waves of the short-wave spectrum were involved in the breaking process responsible for the backscattering at P- and L-band radar frequencies.
Caponi et al. (1988) published several available models for the relaxation rate μ based mainly on different sets of wave growth data in the laboratory and in the open ocean. Figure 18 shows the models of Inoue (1966) (I), Miles (1959) (M), Snyder and Cox (1966) (S&C), Snyder et al. (1981) (Sea), Mitsuyasu and Honda (1982) (M&H), Hughes (1978) (H), and Plant (1982) (P) for a wind speed of Uw = 5 m s−1. The relaxation rate data presented in Fig. 16 are also included in Fig. 18 for comparison and are marked by dots for the three different wind speed regimes.
5. Discussion and conclusions
Although surface expressions of the seabed were discovered in radar imagery of the ocean 30 years ago there was still a need for experimental quantification to explain the coherent imaging mechanisms. This was realized within the C-STAR project where basic data of relevant hydrodynamic processes have been collected in the southern North Sea utilizing a SAR and a special buoy system, which drifted across large sand waves. SAR images and in situ wave energy density measurements were acquired quasi simultaneously. This offered, for the first time, the possibility to compare P-band normalized radar cross section (NRCS) modulations and corresponding spectral wave energy density variations to test if the first order Bragg scattering mechanism is applicable: both profiles compare fairly well, taking into account the lower spatial resolution of the wave energy density measurements. Although the narrow streaks of reduced sea surface roughness are undersampled, the profile of wave energy density shows that the position of these streaks coincides with the steep slope of the sand waves close to the crest. An independent georeference was not realized for the SAR image. But the superposition of the NRCS modulation with the wave energy density data showed that the SAR image can be positioned relative to the topography with some confidence. The conclusion is that SAR imagery and the in situ measurements provided a qualitatively consistent view of sea suface manifestations of sea bottom topography. Quantitatively, the P-band NRCS and wave energy density modulation depths agreed within a factor of 2, which is acceptable but has to be further improved. For future applications a precise georeference of the SAR image and in situ measurements of high spatial resolution would be desirable. On the other hand, other investigations showed that under weak currents and strong stratification it is currently not possible to accurately map sand wave fields using imaging radars. Therefore, it is also necessary to investigate the influence of the seabed topography on the surface wave field spectrum under stratified flow conditions.
The wind growth–relaxation rate μw was assumed to be equivalent to or less than the exponential relaxation at the rate μ due to the response of short water waves to perturbations by variable currents. In situ measurements were realized in the open ocean above very large submarine sand waves to quantify this statement. For the calculation of the relaxation rate μ an already known simple first-order theory was applied. There is general agreement obtained from the results that μ increases with wind speed. But there is disagreement concerning the behavior of μ with increasing wavenumber k in the shorter wavelength portions of the energy density spectrum. Our results indicate that μ decreases with increasing k for the various wind speed regimes analyzed. Therefore, one can also conclude that the measured wave energy density modulation in the C- and X-band Bragg waves is larger than that predicted by existing models using available parameterizations of the relaxation rate. This implies that the wind growth relaxation rate μw is not equivalent to the relaxation rate μ of short waves due to current variations above submarine sand waves as a function of k. A plausible reason for this may be that the rates for dissipation of the longer waves (associated with P and L band), over the sand wave regions of the study area, are enhanced through wave breaking to such a degree that these portions of the spectrum do not have sufficient time to equilibrate. This effect would not be as consequential in the shorter wave portions of the spectrum, which are responsible for X-band and C-band radar backscatter because these waves are less strongly influenced by wave breaking. As a consequence, depending on the strength of the wind, different scenarios may become possible. For example, the timescale associated with dissipation of these shorter waves may become sufficiently long that they may equilibrate, in contrast to the situation (involving nonequilibrium conditions) in the longer wavelength portions of the spectrum. However, even if this is the case, problems still arise, because at the higher frequencies (associated with C and X band), the Bragg scattering mechanism seems to be insufficient to explain completely alone the NRCS modulation due to current variations at the higher radar frequencies. A second mechanism in particular when entering the capillary wave range may play a part and can be responsible already for about 25% of the backscattered radar power.
S. Stolte Sr., S. Stolte Jr., J. Förster, W. Waeber, H.-P. Westphal, and the captain and crew of the Planet are gratefully acknowledged for their excellent cooperation and assistance during the project. We thank J. Vogelzang for providing the water level data at the center of the C-STAR test site. C. Vernemmen and G. Moerkerke are gratefully acknowledged for providing us with the bathymetric data of the study area. We thank P. De Loor and J. Vogelzang for very fruitful discussions and M. Metzner for technical support. This work has been supported by the European Community as a part of the Marine Science and Technology Programme under Contract MAS3-CT95-0035.
Corresponding author address: Ingo Hennings, GEOMAR, Forschungszentrum für Marine Geowissenschaften der Christian-Albrechts-Universität zu Kiel, Wischhofstraße 1-3, D-24148 Kiel, Germany. Email: email@example.com