Proceedings of the Institute of Acoustics 

 

Maritime applications of synthetic aperture radar; recap, advances, future directions and a shallow water case study

 

N. R. Stapleton, Dstl, UK

 

Overview

 

This paper is separated into two parts: (1) a brief recap of Synthetic-Aperture Radar (SAR) for ocean applications, some key advances followed by future possibilities for SAR for ocean observation and (2) surface effects of in-water turbulence resulting from flow over shallow variable seabed topography. This paper provides an overview of what will be presented at the Keynote Lecture.

 

1. Recap, Advances and Future Direction in Synthetic-Aperture Radar for Ocean Observation

 

1.1. Introduction

 

Space-based, satellite SAR offers several advantages for observing the Earth’s surface compared with other sensors. Radar can see the surface regardless of cloud, fog or smoke cover and as an active sensor, it can image both day and night. There are disadvantages however, as an active instrument requiring power, the opportunity to image may be restricted due to overall satellite power requirements. Therefore, spatial and temporal coverage is not as good as many wide-field passive sensors (in visible and infrared (IR), note: these are cloud- and light-limited). SAR coverage can be improved by using modern wide-swath, ScanSAR sensors and using multiple (constellations) of smaller, more affordable SARs and satellites. In the case of imaging the ocean that is a moving surface, the formation of the synthetic-aperture can be problematic in that wave motion leads to de-focussing (smearing) when imaging waves. Furthermore, the mapping of moving scatterers in the processed image does not, in general, lead to a direct mapping of the wave-field; an artefact known as velocity-bunching, where the imaging process is in general non-linear and wave imaging information can effectively be lost.

 

It should be noted that the topic of interest here concerns radar imaging of the sea surface. The ocean radar imaging community is smaller than the land imaging community and there are many advances in the land domain including multi-frequency and multi-polarisation methods to classify land cover and even to differentiate crop type that are not discussed here. Across-track interferometry SAR (INSAR) has also been used over land, notably on the Space Shuttle Radar Topography Mission, to map much of the Earth’s surface to produce Digital Elevation Maps (DEMs).

 

1.2. Satellite SAR Views the Ocean: A Recap

 

It is surprising, remarkable even, what can be seen when looking at the sea surface with SAR. The first satellite, Seasat, dedicated to the use of microwave sensors for observation of the Earth’s oceans, was launched on June 28, 1978, by the United States National Aeronautics and Space Administration (NASA); the project was managed by the Jet Propulsion Laboratory (JPL) of the California Institute of Technology. Fu and Holt (1982) ‘Seasat Views Oceans and Sea Ice’ provides a comprehensive review of what the SAR instrument observed during the short 98 day period that Seasat was operational.

 

1.2.1. Shallow-water Bathymetry

 

Figure 1-1 shows an image where the SAR appears to ‘see’ underwater to around 25 m depth, although this is physically impossible given the radar’s centre frequency of 1.28 GHz, with a corresponding wavelength of 23.5 cm such that the skin-depth in sea-water is around 1.5 cm. However, the reason the SAR can ‘see’ or image the underwater topography (seabed relief) is that the sea surface roughness (waves) are altered as tidal flow changes speed as it passes over the variable relief (conserving mass transport). The surface waves are strained such that they become steeper in regions of current convergence (decelerating flow) and smoother in divergent, accelerating flow regions. Alpers and Hennings (1984) is a seminal paper that describes a radar-imaging model for shallow water topography. Part two of this paper returns to the subject of imaging shallow underwater topography as, for certain flow-bed geometries, a more thorough understanding of the tide-bathymetry interaction is necessary.

 

 

Figure 1-1: Seasat SAR image (left) of Nantucket Shoals to the south of Nantucket Island south off Cape Cod, MA, USA. The SAR image reflects closely the bathymetric patters shown on the map (right). Copied Fu and Holt (1982).

 

1.2.2. Oceanic Internal Waves

 

For the same reason that SAR is able to image underwater shallow bathymetry, it is also able to image oceanic internal waves, i.e. waves that occur beneath the surface when there are changes in density (salinity and temperature) with depth. Internal waves (like surface waves) have associated with them a spatially variable orbital current field. This field can be appreciable at the surface especially for internal waves where there is a strong, shallow, seasonal thermocline/pycnocline (vertical density gradient) in the ocean. In the case of large amplitude, non-linear, oceanic internal waves, the spatial gradient (strain rate) at the surface can be sufficiently large to cause surface waves to become so energetic that they break in convergent current regions (the internal waves have a surface acoustic signature in these extreme cases). Such an example of large amplitude internal waves is discussed by Osborne and Burch, (1980). Figure 1-2 shows a Seasat SAR image off the Florida straits where large, rank-ordered, non linear, packets of internal wave solitons have a surface expression as bright and dark bands at the sea surface. Also visible in the image near the top is a ship-wake that is likely an internal wave wake. Other examples of ship-generated internal wave wakes will be shown in the Keynote Lecture presentation.

 

1.2.3. Oceanic Fronts and Eddies

 

Ocean fronts and eddies are also very often apparent in SAR imagery. This is for several reasons; the most common imaging mechanism arises as there is usually shear in the current field at the boundary between two water masses whether the boundary is a frontal zone or the periphery of an eddy. Mathematically, a shear will resolve into a divergent (smooth/dark in radar backscatter intensity) or convergent (bright intensity) current gradient depending on the orientation of the shear relative to the radar look direction, as illustrated in Figure 1-3. Fronts and eddies may also be visible due to sea surface temperature changes, as this affects the wind-drag coefficient over the sea surface and, as a result, atmospheric stability (due to the air-water temperature difference). Provided the wind speed is not too high, around 8 m/s or less, cold water appears relatively darker (smoother) and warm water appears brighter and also sometimes has greater variation in texture – this is polarisation dependent. At higher wind speeds, these subtle changes are increasingly are washed out.

 

 

Figure 1-2: NASA Seasat SAR image of Florida Straits off Cape Canaveral covering an area roughly 32 km x 38 km. The groups of stripes in the image correspond to packets of internal waves and at the top of the image, a ship and its wake are visible.

 

 

Figure 1-3: Schematic showing where bright (convergent) and dark (divergent) areas appear at the periphery (shear boundary) for eddies in the northern hemisphere given a radar looking from either left or right across the page. The same imaging principle would apply for a curvilinear frontal boundary. Copied from Stapleton et al. (1996).

 

A third effect is due to changes in biology that exist in different water masses (or just at the boundary due to local upwelling of nutrients) delineated by the front or eddy boundary. Phytoplankton during its life cycle produces fatty acids and oils and these rise to the surface as they are buoyant – they are also transported by micro-bubble interfaces to the surface from the bulk. These natural films are very efficient at damping surface waves and produce local dark, smooth areas on the sea surface. These are discussed in the following sub section.

 

Figure 1-4 shows two images. Figure 1-4 (a) shows a NASA Seasat SAR image. Note: the bright and dark narrow curvilinear features (filaments) often separating brighter and darker regions of radar intensity. The effects are due to water masses of different temperature moving relative to one another. Figure 1-4 (b) is a European Space Agency (ESA) ERS-1 SAR image showing eddies traced by dark filaments due to natural films at the sea surface that damp waves/smooth the surface. They appear dark as less radar energy is backscattered.

 

Figure 1-4: (a) Seasat image of an oceanic front. The vertical striping is due to optical processing of the raw data. Taken near the region of the North Pacific subtropical front from Fu & Holt (1982). (b) ERS-1 SAR image of chains of small-scale cyclonic eddies (size 3 km to 5 km in the Sea of Japan, 22 October 1992, © ESA, from Ivanov and Ginzburg (2002).

 

There is an issue that can confuse image interpretation when imaging ocean fronts, this is due to collocated /near-coincident atmosphere fronts, e.g. a change in wind velocity. The ocean and atmosphere are a coupled system, where the ocean provides a bottom boundary condition for the atmosphere. It is therefore not surprising that atmospheric effects occur where there are changes in water temperature. If temperature changes are large across a frontal boundary, the ocean may be warmer than the overlaying air; in this case the near surface air is unstable and near-surface active convection arises. The resulting convection cells lead to small, km to 10 km scale, variations in wind speed and gives rise to quasi-organised patterns (image texture) in radar intensity. Due to this issue, and to aid SAR image interpretation, it is instructive to examine near temporally coincident images from sensors other than SAR.

 

In order to validate and improve the interpretation of SAR imagery for supporting oceanographic forecasting, SAR imagery are often combined with other satellite sensor data. This includes radar altimeter (these measure sea surface height from which large-scale currents can be inferred), IR, ocean colour, passive microwave and radar scatterometers data. Figure 1-5 shows a near simultaneous ocean colour and SAR image. A cyclonic eddy is apparent near the top in both images, traced by chlorophyll in the colour image and by natural films in the SAR image. An ocean front is also evident in both images although this is difficult to see in the SAR image reproduced in the figure. Satellite data are used routinely to correct/nudge both global ocean and meteorological models; a process known as data assimilation.

 

 

Figure 1-5: SeaWiFS chlorophyll image (left) and ERS-2 SAR image (right) acquired on March 8th, 1999 in the Gulf of Oman, at 8:14 and 6:49 UTC, respectively. A spiral eddy is evidently traced by phytoplankton concentration in the SeaWiFS image and by surface films on the SAR image. Reproduced from Stapleton et al. (2000).

 

1.2.4. Sea Surface Slicks

 

Sea surface slicks fall into two categories; (i) minerogenic and (ii) natural or biogenic slicks. Minerogenic slicks are composed of petro-chemical fractions, they are familiar as pollution due to oil spills (although natural seepages do occur) from accidental and deliberate discharges from vessels. Figure 1-6 shows an example of minerogenic oil pollution; the dark patches demark regions of the surface affected by pollution. Minerogenic films are usually thick (cm to mm) and can become emulsified, whereas natural films are monomolecular, or close to, nm thick. Figure 1-4 (b) provides an example of natural films revealing currents due to eddies; both types of slick damp surface waves. However, natural films are more efficient at damping through the Marangoni effect (see below) and are more widespread than minerogenic slicks.

 

Natural, biogenic surface films (fatty acids and oils) that are surface-active (surfactants) have different surface tension and elasticity compared to sea or pure water. Because these films are surface-active they often spread to just monomolecular thickness on the surface. Surfactants can alter interfacial flow and energy dissipation significantly through the Marangoni effect because of gradients generated in surface tension. The Maragoni effect is important for cm- to mm-wavelength short gravity, gravity-capillary and capillary waves as discussed by Batchelor et al., (2003) and Hühnerfuss et al., (1987). These short waves are also key in determining radar backscatter. Slicks are not visible at very low wind speeds (~< 2 m/s) as there has to be waves/roughness present on the sea surface to damp, to reveal the presence of a slick. Low wind areas are sometimes referred to as ‘wind slicks’ in the community; this term applies whether a surface film is present or not.

 

1.2.5. Recap Summary

 

As discussed above, many effects can alter sea surface roughness. The wind is key in creating waves and hence roughness on the surface, so to first-order, radar is sensitive to wind speed. This effect is exploited by non-imaging radar; namely, satellite scatterometers, to provide routine measurements of wind speed over the ocean. In the case of imaging radar, like SAR, detailed changes and patterns are often observed in sea surface roughness and these can reveal underlying oceanography and shallow bathymetry as shown above. Some other effects not discussed above include the appearance of ship-wakes in SAR imagery (although touched upon in Figure 1-2). Wakes take on several forms and a whole paper could be devoted to this one area. Some examples of ship wakes will be presented and discussed in the Keynote Lecture. Other effects not discussed include rain, as rain impacts the sea surface it can either damp (through an induced turbulence interaction) or roughen the surface by creating spikes, crowns and droplets. Hence, rain can appear bright or dark, depending on rain-rate and also the carrier frequency of the observing radar. As just alluded to, turbulence can also be observed. Turbulence may roughen or damp the surface. Part 2 of this paper discusses the effect of turbulence at the surface in relation to imaging shallow water bathymetry.

 

 

Figure 1-6: From NOAA-NESDIS, SAR Marine User’s Manual (https://www.sarusersmanual.com/). Oil polluted sea area in a busy shipping lane off the coast of Malaysia (near Kuantan). Wind is from the East – causing a feathering of the oil trails. ERS-2 C VV SAR image was acquired 4 April 1998 at 0325 UTC over the South China Sea. Imaged area is 100 km x 100 km. © ESA 1997.

 

1.3. Advances

 

Following on from the recap, it is evident that currents at or near the sea surface are associated with underlying oceanographic phenomena and shallow seabed interactions. Changes in the current field strain and advect surface waves and these can lead to changes in sea surface roughness and hence produce changes in radar backscatter intensity that allow us to observe the aforementioned phenomena of interest.

 

There have been several useful advances in SAR over the past two decades or so, either in terms of improved hardware, processing methods or combinations of the two. An example of a hardware improvement is the use of what is known ScanSAR to achieve larger swath widths and hence greater coverage. By periodically sweeping the antenna beam through different sub swaths in range, a ScanSAR can achieve a wide swath (~700 km) at the expense of sacrificing azimuth resolution. A key advance in terms of processing methods is an application termed ‘Doppler centroid analysis’. In this approach, differences between observed mean Doppler frequencies (Doppler centroids) and nominal values corresponding to the relative speed between satellite and rotating Earth in the radar look direction, are interpreted to be the effect of surface currents and mean contributions of wave motions. Within the last decade or so, others have demonstrated the soundness of this approach for a variety of test sites and SAR data from different satellites; mostly Envisat, e.g. Hansen et al., (2011) and TerraSAR-X, e.g. Romeiser et al., 2013. A drawback is that the extra Doppler information is obtained at low spatial resolution, usually in the order of kilometres, because Doppler centroids need to be estimated from raw data corresponding to clusters of hundreds to thousands of full-resolution pixel samples. An example of the use of combined hardware and processing methods is presented in the next sub-section.

 

1.3.1. SAR Along-Track Interferometry

 

What if there was a way to observe surface currents directly at high resolution using SAR? Well it turns out there is a way and it uses two synthetic apertures displaced in the azimuth or flight direction to collect two images, the second later than the first. The method is known as Along Track-Interferometric SAR (ATI-SAR). Relative radial (line-of-sight) velocity between the scatterers in an imaged scene is already exploited by SAR to synthesise and achieve fine resolution in azimuth. However, if two, time-displaced images are collected, an interferogram can be used to estimate the phase difference between each image and this can be related to movement of surface that occurs in the time interval between successive images in time. In practice, it is the time-dependent change in the mean Doppler for each image that matters, however, in many cases, it is just the radial component of the velocity of scatterers in the scene that affects the mean Doppler. As such, often there is a straightforward means to observe surface currents using ATI-SAR. Goldstein and Zebker (1987) explained the basic concept of ATI-SAR and presented example results from a first experiment with an airborne system; the NASA JPL DC-8 AirSAR.

 

ATI-SAR is a versatile tool that is able to provide new insight and information about the ocean surface and underlying phenomena that are associated with currents that both advect and modulate sea surface waves. The full potential of ATI-SAR has yet to be realised. It has been demonstrated using airborne SAR, notably with the NASA JPL DC-8 AirSAR, Goldstein and Zebker (1987) and through limited space borne SAR experiments. The first space borne demonstration occurred in February 2000 when the Space Shuttle Endeavour was deployed with ATI capability. For the Shuttle Radar Topography Mission (SRTM), Endeavour was equipped with antennas in the cargo bay and optimised for topography measurements overland. An offset in the flight direction necessitated for mechanical reasons meant that the SRTM had, nonetheless, a nominal, if sub-optimal, ATI capability. Figure 1-7 shows an image of the Wadden Sea, The Netherlands together with a corresponding reference current field from the numerical circulation model, KUSTWAD. Due to suboptimal system parameters (short ATI baseline), the interferogram had to be averaged over many pixels to reduce phase noise. The effective spatial resolution of the resulting SRTM-derived line-of-sight current field was found to be on the order of 1 km, with a remaining root-mean-square (rms) uncertainty for the currents of about 10 cm/s. Apart from the relatively low resolution, the agreement between the SRTM and model-derived current fields was found to be good and consistent with theoretical expectations.

 

 

Figure 1-7: (a) Line-of-sight current field in the Dutch Wadden Sea derived from a SRTM image acquired on February 15, 2000, 12:34 UTC. (b) Reference current field from a numerical circulation model. Area size = 70 km × 70 km. Radar look direction was toward northwest. Flight direction was toward northeast. From Romeiser, 2013). http://dx.doi.org/10.5670/oceanog.2013.37

 

Romeiser and Runge (2007) assessed the theoretical performance of ATI using a longer interferometer baseline (distance between antennas) and predicted an antenna separation of 40 m to 100 m (time lag 3 ms to 7 ms) for TerraSAR-X SAR would be optimal. TerraSAR-X has an experimental divided antenna mode, but the baseline is very short at ~ 1 m. Performance of the divided mode was tested and current and spatial resolutions similar to SRTM were achieved, however, after launch of a second satellite, TanDEM-X, the opportunity arose collect data with a near optimal baseline with the two satellites orbiting in close formation. The orbits were refined in February and March 2012 to allow high-quality, near optimal baseline separation to be achieved. Data were collected over several selected ocean test sites including the Pentland Firth where strong tidal currents flow between the Scottish mainland and the Orkney Islands. Figure 1-8 shows the full potential of what could be achieved using space borne ATI. Note: the TerraSAR-X/TanDEM-X dataset shows clear signatures of orbital wave motions for swell wave with wavelengths on the order of 200 m using the German TerraSAR-X and TanDEM-X satellites. At these fine spatial scales, the SRTM and TerraSAR-X divided-antenna mode would be dominated by phase noise. Detailed analysis indicated that the effective resolution of the TanDEM-X derived Doppler velocities in Figure 1-8 is around 33 m, for a residual rms velocity uncertainty of 10 cm/s. At this level of fidelity, ATI data are suited for direct measurement of wave motion and high-resolution measurements of surface current variations over internal waves or in narrow rivers.

 

 

Figure 1-8: A 10 km × 10 km sub-image of the Pentland Firth region showing signatures of surface waves in (a) the intensity image, and (b) the full-resolution Doppler velocity image derived from the interferogram. http://dx.doi.org/10.5670/oceanog.2013.37

 

1.4. Future Direction

 

A particularly interesting future development for space borne SAR is the opportunity to extend ATI-SAR such that it can be used routinely and extract full current velocity vectors, not just the velocity projected into the radar line-of-sight. Torpokov et al., (2005) demonstrated an experimental airborne ATI system with two pairs of antennas in different look-directions relative to platform flight direction that allows full two-dimensional vector current measurements to be made. A similar system based on squint-ATI is currently (2023) being demonstrated and assessed using an aircraft test-bed as a candidate for an ESA Earth Explorer 11 Mission. The SEASTAR concept uses a squinted ATI-SAR at Ku-band with two squinted beams pointed ±45° fore and aft of broadside at VV polarisation and a single broadside beam at VV and HH polarisation with a physical baseline of 15 m and a swath width of 170 km, Gommenginger et al, (2019).

 

SEASTAR is important as it observes sub-mesoscale currents, i.e. scales ranging from 1 km to 10 km. It will provide high-accuracy total surface current vector (TSCV) data, of 10 cm/s rms error or less, at 1 km resolution; synoptic current field maps for wider dynamical context; TSCV collocated with high-resolution wind vectors and directional wave-spectra. These quantities are of particular importance to validate fine-scale numerical coastal models. They are also relevant where coastal erosion, sea-level rises and pollution (oil and plastics) are a concern for many densely populated regions around the world. Sub-mesoscale dynamics are critical to understanding ocean-atmosphere exchanges that are important for upper ocean mixing and vertical transport; these in turn influence marine ecosystems and long-term climate change. Figure 1-9 shows the proposed satellite antenna configuration (top) and the present airborne test-bed; antenna (left) and aircraft (right).

 

 

Figure 1-9: SEASTAR satellite antennas concept (top), airborne test-bed antennas (left), aircraft with belly radome (right). Top – Gommenginger et al., (2018), bottom – McCann (2023).

 

The SEASTAR concept is undergoing presently, 2023, a validation/assessment campaign for which an airborne test-bed, Ocean Surface Current Airborne Radar (OSCAR), is being used. Figure 1-10 shows a comparison of the current field obtained from OSCAR off Brest, west of Ushent Island compared with a coastal X-band radar and a high-resolution numerical coastal model, MARS2D. This area has strong sub-mesoscale hydrography. It can be seen there is good agreement between OSCAR, the coastal radar and the numerical model.

 

 

Figure 1-10: Cross-comparison: OSCAR (left); X-band coastal radar (centre); model surface current fields (right). From McCann (2023).

 

1.5. Conclusions

 

There has been continual progress since the days of SeaSat 1978, where the SAR revealed a surprising wealth of information regarding open-ocean and coastal processes. A key limitation that has hampered SAR, and continues to some extent, is the limited spatio-temporal coverage it achieves. It is however a high-resolution, active sensor and this is to be expected. Coverage has already improved markedly the past decade or so and with the advent of satellite constellations and cheaper, smaller SARs and satellites coverage will likely improve still further. An area to watch is the development of high-altitude pseudo-satellites (HAPS) in the stratosphere, where ultra-persistent observation using SAR could be realised. Furthermore, the extension of ATI-SAR to measure sub-mesoscale sea surface current vectors from space with the proposed SEASTAR mission is an exciting prospect for ocean modellers, earth remote sensing scientists and oceanographers alike.

 

2. Surface Effects of In-Water Turbulence Resulting from Flow over Shallow Variable Seabed Topography, Neil Stapleton

 

2.1. Introduction

 

Modelling flow over shallow bathymetry to better understand in-water effects was motivated to try and explain why radar signatures of shallow-water bathymetry appear regardless of the direction of tidal flow relative to long quasi-one dimensional sand-wave, seabed forms. Shallow-water signatures also do not appear to depend strongly on radar viewing (radial) direction relative to tidal flow velocity or seabed forms or wave crest directions. The same is the case for oceanic internal wave crest directions relative to radar radial direction; both these observations are difficult to explain in terms of existing, established imaging models. To reiterate, the reason hydrodynamic modelling was undertaken is because traditional, well-established models, a key example being Alpers and Hennings (1984), hereafter abbreviated as AH 1984, are unable to explain all SAR observations. Figure 2.2-1 shows such a case, where fine-scale (approximately 200 m) bright and dark stripes (modulations) in the image, oriented North-West to South-East, can be explained by the model of AH 1984, whereas, the larger scale changes in radar backscatter, oriented North East to South-West, in the image cannot.

 

To address this model shortfall, rather than attempt to refine the radar modelling aspect of models like AH 1984 to include multi-scale scattering effects from long and short Bragg-scale waves (as has been attempted by others), detailed modelling of the hydrodynamics of flow over one dimensional sand-waves with wall roughness was undertaken instead. Addressing the hydrodynamic aspect of the problem is key as this is the foundation for subsequent radar modelling; the starting point for the modelling needs to capture the essential physics else subsequent modelling steps will not be built on a solid foundation. The model of AH 1984 performs well for its intended application where tidal flow is perpendicular, or nearly so, to the crests of sand-wave bed forms and for a limited range of radar wavelengths (only L-band, 24 cm radar wavelength; Seasat SAR data is analysed in the paper by AH 1984). However, the AH 1984 model cannot account for situations where the tidal flow is parallel or nearly parallel to the wave crests and this is the case for long-wavelength sand-waves in the southern North Sea as shown in Figure 2.2-1. It should be noted, however, that this tidal/bathymetry geometry also occurs elsewhere in the world.

 

2.2. Large-Eddy Simulation Modelling (Hydrodynamics)

 

As per the rationale stated above, hydrodynamic modelling of the tidal/bathymetric geometry of interest was undertaken using a Large-Eddy Simulation (LES) model owned by the School of Engineering Sciences, University of Southampton. The LES code was run on the UK’s super computing capability, namely the Cray T3E computer based at the University of Manchester. The modelling was performed in 2001 funded by the Ministry of Defence (MOD). LES modelling was relatively new in 2000 and opened up the possibility to model ‘large-scale’ geophysical flows with Reynolds numbers (Re) of order 10⁸, 𝑅𝑒= 𝑉𝐿/𝜈, where 𝑉 is the speed of the flow, here ~1 m/s, 𝐿 is the length scale of the flow, ~ 100 m (sand-wave wavelength order of magnitude) and 𝜈 is the kinematic viscosity of water ~10^-6 m²s.

 

 

Figure 2.2-1: Image of the southern North Sea off the Netherlands coast. Note: flow parallel to large bed forms annotated upper left and perpendicular to finer-scale bed forms throughout the image. Image processed by TNO-FEL, NL. © ESA 1994

 

2.2.1. Geometry

 

The bathymetric features that are of interest normally occur in water up to 40 m in depth and range in length from tens of metres to several kilometres. For efficiency of the computational model, while maintaining the ability to model features of large-scale turbulence, the sand-wave modelled was restricted to one 160 m long and 10 m high, with a total water depth of 30 m. This is smaller than those presented in experimental data and often seen SAR images, but is expected to produce similar large-scale turbulent structures. The chosen computational domain was 160 m by 30 m by 320 m in the lateral (y), vertical (z) and stream-wise (x) directions respectively as illustrated in Figure 2.2.2-1.

 

The domain was initially represented using 160 cells in the y direction, 48 in the z direction and 320 in the x direction and then divided into 180 blocks for parallel domain decomposition. Because of the Cartesian mesh format of the multi-block LES code, the sloping sides of the sand-wave are approximated by a series of steps (following standard practice). The number of cells was doubled in all directions for increased accuracy after the flow had converged to a statistically steady state.

 

2.2.2. Boundary Conditions

 

It was assumed that the turbulence statistics are horizontally homogeneous so the computational domain has been made periodic in both the x (stream-wise) and y (lateral) directions. The model therefore simulates adjacent rows of identical sand-waves of infinite length. The free surface was simulated by a free slip boundary. The seabed is characterised by a uniform roughness height of 0.003 m equivalent to the drag effect of a sandy bottom, Heathershaw, (1988).

 

 

Figure 2.2.2-1: Sand-wave geometry used in the LES simulations. Top: 3-D view. Bottom: end view indicating a broad, developed secondary-flow cell geometry and sand-wave height and depth.

 

The tidal flows of interest (as found in the Southern Bight of the North Sea) generally have velocities between 0.5 m/s and 1.5 m/s. A bulk flow velocity of the required magnitude in the free stream (x) direction was implicitly specified by imposing a pressure gradient in the x direction. This pressure gradient was calculated to balance the bed shear stress generated by a bulk flow of approximately 1 m/s. In reality, the tidal flows are time-dependent but the flow adjustment time is considered small enough compared with the time of a tidal cycle for the steady results to be representative.

 

2.2.3. Numerical Solution

 

The LES equations were approximated to second-order in space and time using finite differences on a staggered grid and then solved by marching forward in time, Thomas and Williams, (1999). The pressure field was calculated from a solution of Poisson’s equation using a multi-grid technique to aid efficiency. Continuity errors were limited to 3.5x10^-6. To maintain stability and accuracy, the Courant-Friedrichs-Lewy (CFL) number was set to approximately 0.15 using a time step of 0.075 s.

 

2.2.4. LES Results

 

The principal features of the calculated flow are the resulting secondary flows generated in the y,z (lateral) plane due to the action of the turbulent normal stress differentials and the marked anisotropy of the turbulence. The secondary flows in the y,z plane are shown in Figure 2.3-1 below, where the stream function ψ (y,z), in the y,z plane is potted. Several, definable secondary circulations are evident. Plots of flow velocity vectors (not shown) indicate the presence of upwelling at the sand-wave crest. Dashed lines indicate clockwise flow and solid lines anti clockwise flow.

 

2.3. Expressions of Variations in the Flow Field at the Surface

 

The primary mechanism by which the bed bathymetry can be expressed at the surface in this flow configuration is via lateral variability of the statistical characteristics of the flow. Figure 2.3-2 shows the variation of the stream-wise averaged mean velocity and root-mean-square (rms) fluctuating velocity, with lateral position.

 

Figure 2.3-1: Secondary circulation stream function ψ(y,z), plotted contours (in m²/s).

 

 

Figure 2.3-2: Variation of the surface velocity statistics with lateral position, y: (a) Velocity relative to the global mean; —, U component (stream-wise); – –, V component (lateral); (b) rms fluctuating velocity: —, U_rms; – –, V_rms. Velocity measured in m/s and lateral position (y) measured in metres. The sand-wave’s crest at y=160 m corresponds with a band of low mean velocity fluid above and to the right (y >160 m) of the crest and over the steep-sided slope of the sand-wave. Note: a 200 m section of a doubled domain of 320 m in the lateral direction is depicted to show more than one wave to reveal the flow variation clearly.

 

There is a significant variation in velocity above the steep, right-hand side of the sand-wave, associated with upwelling due to a long-lived secondary circulation cell. These cells are of an intermittent nature but are very long lived relative to the time scale of the turbulence. The effect is to create a narrow band (about 25 m wide) of slower moving fluid with a diverging lateral velocity corresponding to slightly more than ± one standard deviation in the mean velocity (standard deviation in velocity not relative to mean velocity). The appearance of these narrow, slower moving bands of variable current can be expected to interact with surface waves straining them. Hence, a modified surface wave-field could provide a means to relay a surface expression of the underlying large-scale turbulent structure. Although the lateral variation of the rms fluctuating velocities is less pronounced than for the variation in mean flow, there is a significant increase in velocity fluctuations in the upwelling bands. This is caused by the upward, advective, transport of turbulent kinetic energy from deeper within the flow. The average rms intensities are 0.051 m/s and 0.036 m/s in the stream-wise and lateral directions, respectively.

 

2.4. Wave-Turbulence Interaction

 

We anticipate that the standard radar-bathymetry imaging models do not include the appropriate physics to predict the observed intensity variations¹ observed in radar images over sand-waves in the case of the geometry considered, i.e. where flow is parallel to the crests (Figure 2.2.2-1). This is because the surface waves are not strained directly by the flow as it changes speed due to changes in water depth (the flow direction is closely aligned with the sand-wave crest and hence there is no change in depth in the direction of the flow). This geometry arises in the Southern Bight of the North Sea and yet sand-waves are still evident in satellite radar imagery of this area. Note: the geometry considered is also where the radar is looking in a direction aligned closely with the sand-wave crests so that the radar is imaging surface waves also propagating nearly parallel to the sand-wave crests. Hence, the surface waves also travel close to the direction of the mean tidal flow (and, as such, the surface wave crests are perpendicular to the sand-wave crests) it is these specific waves that are important for radar backscatter; a geometry similar to that shown in Figure 2.2-1.

 

To investigate the importance of turbulence as a possible mechanism for the imaging of sand waves, the interaction of the turbulent velocity field with surface waves was simulated. From the simulation, a rough estimate of scattering from the surface at microwave frequencies can be made.

 

2.4.1. Numerical Approach

 

Small-amplitude, linear surface gravity waves of length λ propagating on a non-uniform medium that is moving and deforming with a prescribed velocity U(x,t)=(U,V), where x=(x,y) in the horizontal plane were considered. The surface velocity is assumed to vary only slowly relative to the wave-field and the characteristic length-scale, L_u, of order 10 m, of the surface movements is large compared with wavelength of the surface waves (L_u >> λ). The wavelengths of the surface waves are restricted to be short relative to the water depth such that they do not interact directly with the bottom bathymetry, but only indirectly, through the spatial variation of the surface velocity field. The co-ordinates were preserved so that x is stream-wise (flow is along the sand-wave crest), y is lateral (transverse) to the flow, and z decreases downward with 𝑧 = 0 defining the mean position of the free surface. The effects of surface tension are neglected, but viscosity is included, although its influence on the simulation is negligible.

 

The computational domain is a square L×L region of the (x,y) plane, with L=320 m, and periodic boundary conditions. The LES computed surface velocity field U(x,t) was stored in a database on a 320×320 grid, i.e. at 1 m grid spacing, and sampled every 0.75 s in time for a duration of ~700 s. The data were Fourier-interpolated onto finer computational grids and linearly interpolated between samples in time. Two, (a) standard N = 1280 and (b) fine N = 2048, results in a minimum grid size of 0.25 m and 0.16 m, respectively. The velocity was adjusted so that computations were performed in a reference frame moving with the mean velocity of the surface. This required not only removing the mean velocity component but also applying a linearly increasing x translation to the data so that flow structures within the sampled velocity field, for example turbulent eddies, are advected through the moving co-ordinate system at the correct velocity. This translation is built into the Fourier interpolation.

 

2.4.2. Wave Simulations

 

As the surface waves are too short to interact directly with the bed bathymetry, the effect of unsteady, bottom-generated, turbulent currents provides a mechanism by which the seabed bathymetry can be expressed at the surface. If the flow were across the crests, rather than, as here, along the crests, the surface expression would be much more direct, arising due to straining of the flow as it accelerates over the crests in order to conserve mass transport. However, for the geometry considered (Figure 2.2.2-1), the effect of secondary circulation causes intermittent regions of slower moving fluid to appear at the surface in the region close to the sand-wave crests that results in a semi-permanent, narrow band of slower moving fluid (Figure 2.3-2 (a)). Upwelling close to the sand-wave crests also contributes to increased levels of turbulent fluctuations (Figure 2.3-2 (b)), thus providing a direct mechanism to transport turbulent eddies from near the bottom, where they are produced, up to the surface, resulting in boil-like structures at the surface. The scale and lifetimes of energetic turbulent structures are such that they are expected to interact strongly with short, wind-generated gravity waves.

 

Specifically, the effect of the turbulence on short, gravity wind-waves that are readily generated at low-to-moderate wind speeds was considered. The surface of the model domain was initialised with a monochromatic, homogeneous, unit-amplitude, wave field with λ=1 m, 2 m, 4 m and 8 m (8 m is approaching the eddy scale, L_u, so L_u >>λ is not well satisfied. These wave fields were then allowed to be scattered and otherwise modified by the turbulent velocity field over short t=50 s and long t=500 s intervals. The longer time interval was chosen to be about the same as the turbulence turnover (decorrelation) time of around 10 minutes.

 

The surface velocity data represent an average velocity over the top 0.625 m depth within the upper most grid cell. Note: depth averaging was not performed to improve the simulation for the longer wavelengths (4 m and 8 m).

 

2.4.3. Wave-Turbulence Interaction Results

 

The initial, homogeneous wave field at time t=0 s is modified as it propagates over the moving surface, with significant increases in local amplitude occurring in distinct regions but with different spatial patterns occurring for different wavelengths. Figure 2.4.3-1 shows the wave elevation (amplitude) contours corresponding to a +40% change relative to the initial amplitude, after the long, 500 s wave simulation run. The contour value is chosen to mark those regions with wave elevations exceeding this value and to leave the remaining regions clear. This gives a visual representation of the spatial pattern of those regions where the wave amplitude is significantly altered from its initial value due to interaction with the turbulent velocity field and shows very clearly the different behaviour at different wavelengths.

 

 

Figure 2.4.3-1: Spatial pattern of wave elevation contours in horizontal plane η(x,y) at time t=500 s: (a) λ =1 m, (b) λ =2 m, (c) λ =4 m, (d) λ =8 m. Contour levels are shown at +40% modulation relative to the initial wave amplitude. Axes are: abscissa x/Λ sand-wave wavelength (stream wise) against ordinate y/Λ (lateral); mean flow and wave propagation is from left to right.

 

The 1 m waves become organised after 50 s (for brevity, not shown) into long, thin bands approximately aligned with the flow. Over the longer time interval of 500 s (Figure 2.4.3-1), the waves are noticeably scattered from the stream-wise, x-direction, so that random interference between the different components creates localised peaks in the wave field. However, even after this longer scattering time, a statistical structure is evident, with a band of higher and, more probably, large amplitude waves being localised within the band of slower moving fluid located over the steep side of the sand-waves. This behaviour is consistent with the band of slower moving fluid acting as a wave-guide and confining the waves.

 

Figure 2.4.3-2 shows the variation of rms wave amplitude, averaged along the stream-wise (x) direction, with lateral position y across the sand-wave for the 500 s run. The rms amplitudes are plotted normalised relative to the initial rms amplitude at t=0 s. Figure 2.4.3-2 also shows the relative position of the sand-wave bathymetry. In the short 50 s run (not shown) there is a significant increase in rms amplitude within the slower fluid band at both short and long wavelengths, but it is less pronounced for the 4 m waves. In the longer 500 s run, local fluctuations in wave amplitude are less pronounced. This is because the waves have had time to scatter off axis. The contrast is strongest for the short (1 m and 2 m) and long (8 m) waves, and is consistent with the spatial distribution of waves with altered amplitude (Figure 2.4.3-1).

 

 

Figure 2.4.3-2: Lateral variation in rms wave amplitude relative to initial amplitude at time t=500 s: (a) λ=1 m, (b) λ=2 m, (c) λ=4 m, (d) λ=8 m. Data are normalised relative to the initial amplitude at time t=0 s and averaged in the x-direction (stream-wise). Note: the sand-wave bathymetry is indicated schematically at the bottom of each plot.

 

2.5. Summary for LES Modelling and Wave-Turbulence Interaction

 

The LES model resolved the near-surface turbulence field down to 1 m. This is too large to allow the short Bragg surface waves (typically a few cm in wavelength) to be directly incorporated. However, it defines the large scale turbulence structure found at the surface over the sand-wave field. The lifetime of eddies with 10 m to 20 m scale is around 10 minutes and hence short gravity waves will effectively experience a steady flow and their amplitudes will be enhanced or reduced depending on propagation direction. The effect of the turbulence on short gravity waves, readily generated at low-to-moderate wind speeds, was considered as these are important for radar imaging, Churyumov and Kravtsov, (2000) and Barber, (2000), as they both tilt and strain the primary Bragg scattering waves.

 

The LES simulation of flow over periodic sand-wave forms results in a surface current field with non-homogeneous structure in the turbulence statistics of the current. Turbulent pressure fluctuations below the surface can produce surface waves; however the amplitude of these is of order O (u g _rms /²), i.e. about 0.3 mm for the present flow and such effects can be ignored. The wave model permits interactions of the wave field with the surface current through refraction at current shear boundaries and through the straining of the surface. However, it is important to note that the surface velocity is not constant with time and the average strain and shear rates are the result of larger unsteady fluctuations.

 

The interaction between the waves and the turbulence depends on the ratio u_rms / c_g, where urms denotes a characteristic value of the fluctuating, turbulent surface velocity field and c_g denotes the group velocity of a surface wave. Although this ratio is only about 0.1 for the 1 m waves, and less for the longer waves, the surface velocity probability distribution indicates that the instantaneous values can be much larger (up to about ~0.3). Therefore, the shorter waves should exhibit significant localised shortening and steepening and the effect will be stronger in the surface upwelling region where the turbulence intensity is slightly greater.

 

The initial, homogeneous, wave field is modified as it propagates through the velocity field associated with turbulent motions. Significant changes in local wave amplitude occur in distinct regions but with different spatial patterns for different wavelengths (Figure 2.4.3-1). Over the longer time interval of 500 s, the waves are noticeably scattered from the stream-wise direction, with a band of higher and more probable large amplitudes being localised within the band of slower moving fluid located over the steep side of the sand-waves.

 

2.6. 2006 Experimental Observations in the Southern North Sea

 

The modelling work presented above was completed by 2002. More recently, the author has become aware of experimental work, where in-water (ADCP and CTD²) and radar measurements were made of tidal flow over sand-waves in the southern North Sea off the German coast by Hennings and Herbers (2006) (HH 2006). The work by HH 2006 is remarkable as it supports the results of the modelling. The flow geometry is oblique to the steep side of sand-waves, Figure 2.6-1, with flow along the crest being similar in magnitude to flow across the crest. HH 2006 note the flow veers as it flows over the bed forms. Boils or kolks form where there is vertical, upward transport just over the steep side of the crest (Figure 2.6-2), in the same location as our simulations, despite flow differences. Figure 2.6-3 is a handheld photograph showing turbulent eddies at the surface. They appear as reflective, slick-like patterns (indicated with arrows), where local surface roughness within the boils is reduced. Surface waves appear to be enhanced at the boundaries of the boils with localised wave breaking occurring, evident as white patches in the photograph at the periphery of the boils/slicks.

 

 

Figure 2.6-1: Schematic sketch of the characteristic fluid flow in the water column near Lister Tief, showing bed form and tidal current geometry, as well as coordinate systems and definition of symbols. Copied from HH 2006. © AGU 2006

 

 

Figure 2.6-2: Water depth–dependent vertical component of uvert the current velocity as a function of the horizontal space component perpendicular to the sand-wave crest Xperp. The ADCP data from the near– water surface to the seabed have been obtained on board R/V Ludwig Prandtl during ebb tidal phase at 0500–0515 UT 10 August 2002. Blue shows the downward and yellow shows the upward oriented vertical component uvert. Copied from HH 2006. © AGU 2006

 

 

Figure 2.6-3: Handheld camera picture of turbulence patterns (marked by arrows) in the sea area of the Lister Tief acquired from on board the R/V Ludwig Prandtl at 1519 UT 15 August 2002 during flood tidal current phase. Copied from HH 2006. © AGU 2006

 

2.7. Conclusions and Future Direction

 

The simulations revealed that the surface wave field is significantly altered above the sand-waves for all the wavelengths considered. The wave amplitude is altered most over the crest and the steep side of the sand-wave. This provides a mechanism for imaging the sand-waves via wave turbulence interaction. The findings are (and remain) novel in that work performed can explain the visibility of shallow, submerged bathymetry in microwave image data, which are not explicable using standard end-to-end imaging models that utilise potential flow theory, AH 1994. Of note, the results by HH 2006 support the LES results with their measurements made through the water column and observations of strong turbulent eddies occurring at the surface over the sand-wave crests. It would be of value and interest to simulate general, oblique tidal flow geometries to establish whether secondary flows develop similar to that for the along-crest flow case presented in this paper. Understanding how surface waves are altered due to changes in seabed relief is important and all the processes involved remain unclear, Griedanus et al., (1997). A better understanding would enable more reliable remote, synoptic, above-water monitoring of changes in bathymetry that can impact coastal flood defences and man-made seabed structures such as tidal turbines, Bell et al. (2015) to ensure their integrity.

 

Part 1 – References

 

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¹Standard radar imaging models grossly under-predict observed modulations in radar intensity over submerged shallow sand-waves where the tidal flow oriented is parallel, or nearly parallel, to the sand wave crests.

²ADCP: acoustic Doppler current profiler, CTD: conductivity, temperature and depth (sensor).