From Shallow Water

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Rutgers University radars cover these coastal regions at multiple frequencies from 4. Their echoes contain information on both currents and waves from deep water up into the shallow coastal zone, providing an excellent archive for such studies. This paper describes the analysis of both simulated and measured radar echo to demonstrate the effect of shallow water on radar observations and their interpretation.

Radar sea-echo spectra consist of dominant first-order peaks surrounded with lower-energy second-order structure. Analysis methods presently in use assume that the waves do not interact with the ocean floor, see [ 1 , 2 , 3 ] for phased-array-antenna beam-forming systems; and [ 4 ] for systems with compact crossed-loop direction-finding antennas, such as the SeaSonde.

The assumption of deep water is often invalid close to the coast and for broad continental shelves, and is particularly inadequate to describe the second-order sea-echo used to give information on ocean waves. To interpret this echo correctly, we show that the effects of shallow water must be taken into consideration.

In Section 2, we give the basic equations describing radar echo from shallow water, expanding on the previous description given in [ 5 ]. In Section 3, simulations are used to illustrate the effects of shallow water on waveheight, Doppler shifts and spectral amplitudes in radar sea-echo spectra, to investigate limits on the existing theory and to define depth limits at which shallow-water effects must be included in the analysis.

The effects of shallow water on the radar spectrum are illustrated using measured spectra. In Section 4, methods are applied to the interpretation of measured radar echo from a Rutgers University radar to produce wave directional spectral estimates, which are compared with wave observations from a bottom-mounted Acoustic Doppler Current Profiler ADCP moored in the second radar range cell.

It follows from the solution of the equations of motion and continuity that long ocean waves are more affected by shallow water. We define the depth at which waves interact with the ocean floor by the approximate relation:.

The deep-water analysis must be modified to allow for shallow-water effects in the coupling coefficients, the dispersion equation refractive effects on wave direction, and the directional ocean wave spectrum itself. We only consider water of sufficient depth that effects of wave energy dissipation such as breaking and bottom friction may be ignored; thus we operate in the linear wave transformation regime.

In this document, a subscript or superscript s indicates a shallow-water variable; its absence indicates a deep-water variable. The analogous relations for second-order backscatter are:. The electromagnetic coupling coefficient has the same form as for deep water [ 5 ] but with shallow-water wavevectors:. The hydrodynamic coupling coefficient, derived by Barrick and Lipa [ 6 ] through solution of the equations of motion and continuity, is a function of water depth:. The deep- and shallow-water spatial wavenumbers are related as follows:. It can be shown from these equations that at constant wavenumber, the coupling coefficient increases as the water depth decreases, resulting in an increasing ratio of second- to first-order energy as the depth decreases.

In the following analysis, we assume that the deep-water directional wave spectrum is spatially homogeneous and that any inhomogeneity in shallow water arises from wave refraction. When energy dissipation can be neglected, it follows from linear wave theory that since the total energy of the wavefield, is conserved, the shallow-water wave spectrum expressed in the appropriate variables is equal to the deep-water spectrum [ 7 ]:. Figure 1 illustrates refraction at a contour between regions of differing depth. Schematic geometry of the radar beam and an ocean wave train at a depth contour, denoted by the dashed line.

Wave angles are measured counter-clockwise from the radar beam to the direction the wave is moving. Substituting 10 and 11 into 12 gives the following relations which are useful for deriving the shallow- from the deep-water wave spectrum and vice versa:.

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Common numerical multiplicative constants in 15 and 16 have been omitted. It can be shown from 4 that the wavenumbers of the scattering waves are related as follows:. In terms of these variables 16 becomes. To calculate the integral in 18 , it is first reduced to a single-dimensioned integral using the delta function constraint. The remaining integral is computed numerically.

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Due to wave refraction, the shallow water angle and wavenumber have discontinuities when the deep-water wave moves parallel to the depth contour, i. Frequency contours are hence also discontinuous due to this effect at deep-water wave angles defined by Normalized components p , q are defined so that p is along the radar beam and q perpendicular:. Examples of frequency contours for water of depth 10m continuous lines compared with the corresponding contours for deep water dashed lines. The discontinuities in the frequency contours are more pronounced when the contour is drawn in shallow-water wavenumber space, as it follows from 10 , 11 that there are discontinuities in the shallow-water wave angle due to wave refraction.

It can be seen from Figure 2 that the deep-water ocean wave numbers corresponding to a given radar spectral frequency change with depth: they become either greater or smaller than the deep-water values, depending on the wave direction. This results in the frequency of second-order peaks in the radar spectrum changing with water depth.

The effects of shallow-water on measured radar spectra are illustrated in Figure 3 , which shows measured spectra from a 5 MHz radar in five radar range cells, with distances ranging from 18km to 60km. As the water depth decreases, the second-order energy increases relative to the first-order and the frequency displacement between the first- and second-order peaks decreases. In the outer ranges, the second-order structure is almost the same from range cell to range cell, as the water is effectively infinitely deep. Spectra from a 5MHz SeaSonde monopole antenna.

To gain insight into the effects of shallow water, simulated radar echo spectra were calculated for a narrow-beam radar, using the model directional wave spectrum defined in [ 8 ] which consists of the sum of two terms: a continuous high-frequency wind wave spectrum and a swell component that is an impulse function in both wavenumber and direction. The swell component is defined by.

For this model, four sharp spikes occur in the radar spectrum. For these values, it can be shown numerically that Doppler frequencies are always greater than the positive Bragg frequency. The radar beam is taken to be pointing perpendicular to parallel depth contours i. For our model it follows from 13 that the relationship between the shallow- and deep-water rms waveheights is given by:.

This relationship is of course independent of radar frequency and has many angle symmetries. Figure 4 shows the ratio plotted as a function of depth for different wave directions. The ratio of shallow- to deepwater waveheight plotted vs. It can be seen from Figure 4 that the waveheight initially decreases with decreasing depth as the wave enters shallow water but increases at depths below about 20m, which agrees with [ 7 ]. It follows from 3 that for a given radar frequency, the Bragg frequency decreases with depth, causing the Bragg peaks to move slightly closer together.

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Figure 5 shows the Bragg frequency plotted as a function of depth. Bragg frequency plotted as a function of depth. Radar transmit frequency: Red 5Mhz, Blue 25Mhz. It can be seen from Figure 5 that the change in the Bragg frequency with depth is small.

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Figure 6 shows the displacement of the second-order peak from the Bragg frequency plotted as a function of depth for an 11s wave moving at different angles with respect to the radar beam. The frequency shift of the second-order peak from the Bragg frequency for an 11s wave. It can be seen from Figure 6 that as the water depth decreases, the second-order peak shifts toward the Bragg frequency for waves moving toward the radar, and further away for waves moving away from the radar.

This is consistent with the two branches of the contour plot as shown in Figure 2. This effect is more marked for lower radar frequencies and can be seen in the measured spectra shown in Figure 3 in which the second-order peak moves closer to the first-order as the range from the radar and water depth decrease, with waves moving toward the radar. It is shown in [ 8 ] that for the impulse-function model defined by 24 , the ratio R of the second-order to first-order energy is given by:.

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Radar frequency: Red: 5Mhz, Blue: 25Mhz. The absolute value of the coupling coefficient vs.

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Radar transmit frequency: 5Mhz. Wave period: Red 15s, Blue 12s. Green 9s. Since the coupling coefficient increases as the depth decreases, it follows from 26 that the second-order energy will increase with respect to the to first-order. This effect can be seen in the measured radar spectra shown in Figure 3. Figure 9 shows the theoretical ratio of the second- to the first-order energy obtained from 26 using our model for an 11s wave.

Ratio of second - to first-order energy for an 11s wave. Significant waveheight: 2. Radar transmit frequency: a 5 Mhz, b 25 Mhz. It can be seen from Figure 9 that the ratio of the second- to the first-order energy exceeds unity i. This subsection demonstrates an important point. Since we have shown that the waveheight itself actually decreases slightly upon moving into shallow water, while the second-order echo increases significantly due to the rapid growth of the coupling coefficient, wrongly using deep-water inversion theory to estimate waveheight will overestimate this important quantity.

We note that all previous treatments and demonstrations of wave extraction have been based on deep-water theory, even when in fact many of the radar observations have been made in shallow water. When the magnitude of the second-order energy approaches that of the first-order, it is apparent that the perturbation expansions on which 15 and 16 are based are failing to converge and they therefore cannot provide an adequate description of the radar echo. This effect is similar to the well known radar spectral saturation occurring when the waveheight exceeds a limit defined by the radar transmit frequency.

Above this waveheight limit, the radar spectrum loses its definitive shape and the perturbation expansions fail to converge. The deep-water saturation limit on the significant waveheight W Sat defined to be four times the rms waveheight is given approximately by the relation:. For shallow-water, the saturation of the radar spectrum is exacerbated by the increase of the coupling coefficient and the radar spectrum saturates for waveheights less than that defined by We here define the shallow-water saturation limit W Sat s for the model to be that waveheight for which the second-order energy equals the first-order, and the ratio R is given by:.

In practice the theory may fail before this limit is reached. W Sat and W Sat s are plotted vs. At depths of 30m the saturation limits are approximately equal. At depths less that 30m, the shallow-water limit drops off sharply, particularly for the lower transmit frequency.

Thus the radar spectrum can be expected to saturate at lower values of waveheight in shallow water. Significant waveheight saturation limits for an second wave coming straight down the radar beam. For waveheights above the saturation limit, the waveheight predicted by the theory will be too high. However the theory cannot be applied at all when the second-order spectrum merges with the first, as then separation is not possible.

Shallow-water wave theory

We estimate depths for which shallow-water effects become significant as follows: For first-order echo, the depth limit is defined by equality in 1. For second-order echo, we define the depth limit D S at which shallow-water effects become significant as the value at which the coupling coefficient defined by 8 exceeds 1. Figure 11 plots the depths D S vs radar transmit frequency for an 11s wave.

Depths at which shallow-water effects become significant vs. Red: second-order echo. Blue: First-order echo. However they are based on a wave model 24 , which is quite restrictive: waves of a single wavelength are assumed to come down the radar beam. Also Figure 11 applies only to an 11s wave. Performing similar studies for more general wave spectral models is beyond the scope of this paper.

Therefore shallow-water effects will be more marked at a given waveheight for a broad nondirectional spectrum that includes longer wavelengths e. These differences would probably not be large however, due to the sharp cutoff of wave-spectral models for long wavelengths. The opposite effects would be expected for spectra that include wave directions not directly down the radar beam e. To summarize these effects: W Sat s will be less and D S will be greater than the values shown in Figs.

The time period from December 29 to 30, , was chosen because simultaneous coverage provided by the SeaSonde and a bottom-mounted ADCP allowed a direct comparison to be made between results from the two sensors. The ADCP was located in the second radar range cell in water of depth 8m. The bathymetry in the area and the locations of the two sensors are shown in Figure In our analysis, depth contours near the radar are assumed to be parallel to shore and the depth profile is obtained from Figure Figure 13 shows measured spectra from the Breezy Point SeaSonde at three ranges: the second-order energy can be seen to increase relative to the first-order as the water depth decreases.

Lipa and Barrick [ 5 ] describe the extension of the narrow-beam theory described in Section 2 to apply to a broad antenna system such as the SeaSonde, assuming ideal antenna patterns. Categories: Underwater diving physiology Underwater diving physics Physical oceanography Commons Glossary Indexes: dive sites divers diving Outline. Categories : Water waves Oceanography stubs. Hidden categories: Articles with short description All stub articles. Namespaces Article Talk. Views Read Edit View history. Languages Simple English Edit links. By using this site, you agree to the Terms of Use and Privacy Policy.

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1. Introduction