Effect of wave directional spread on the radiation stress: Comparing theory and observations

Transcription

1 1 Effect of wave directional spread on the radiation stress: Comparing theory and observations Falk Feddersen Integrative Oceanography Division, Scripps Institution of Oceanography, La Jolla, California, USA Short title: EFFECT OF DIRECTIONAL SPREAD ON THE RADIATION STRESS REVISED TO COASTAL ENG.

2 2 Abstract. Nearshore circulation and shoreline setup are forced through the wave radiation stress, which is a function of the frequency-directional wave-spectrum. The radiation stress components often are approximated by assuming a spectrum that is narrow-band in frequency and direction. These narrow-band approximations over-estimate the true radiation stress components in 8-m water depth on the Outer Banks N.C. where locally generated (broad in frequency and direction) sea conditions are prevalent. Using the empirical Pierson-Moskowitz spectrum, frequency-spread is shown not to be the cause of this overestimation. Analytic expressions derived by Battjes [1972] for the ratio of true to narrow-band approximated radiation stress, which depend only on the directional spread, are compared to field data. The observed ratio of true to narrow-band approximated radiation stress is a strong function of the observed bulk (energy-weighted) directional spread, and closely follows the analytic expressions. Thus the narrow-band radiation stresses approximation can be augmented with a directional-spread dependence. These results are expected to apply to other geographic areas where waves are often locally generated. KEYWORDS: NEARSHORE ENVIRONMENT, OCEAN WAVES, OCEAN CURRENTS, SURFZONE

3 3 1. Introduction Wave radiation stress gradients drive alongshore currents in the surfzone and lead to shoreline setup [Longuet-Higgins and Stewart, 1964; Bowen et al., 1968; Longuet-Higgins, 197]. Accurate offshore estimates of the radiation stress components S and S are required in alongshore current and setup models, respectively. The off-diagonal component S in a directionally-spread and frequency-spread (i.e., random) wave field is, according to linear theory [Battjes, 1972] (hereinafter ), S = E(f, θ) c g(f) sin(θ) cos(θ) dθdf, (1) c(f) where E(f, θ) is the frequency-directional (f-θ) energy spectrum, and c g (f) and c(f) are the frequency dependent group and phase velocities, respectively. For a monochromatic (single frequency) and unidirectional (single wave-angle) plane wave of height H, (1) reduces to S = 1 8 ρgh2 c g c sin θ cos θ, (2) where ρ is the water density, g is gravity, and c g and c are evaluated at the monochromatic wave frequency. For simplicity, alongshore current models that incorporate random waves [e.g., Church et al., 1993; Ruessink et al., 21] often assume that the wave spectrum is narrow banded in both frequency and direction, resulting in an approximation for S similar to (2) (hereinafter S ), S = 1 c g ( f) 8 ρgh2 rms c( f) sin θ cos θ, (3) where c g and c are evaluated at the peak or mean (energy-weighted) wave frequency f, and

4 4 θ is a peak or mean (energy-weighted) wave angle. The S because it depends on a few components of the wave field. formulation (3) is appealing The effect of directional spread on the radiation stress has been considered previously and mathematical expressions for corrections to the narrow-band approximations have been derived [Battjes, 1972]. Using parameterized E(f, θ), Battjes [1972] showed that the narrow-band approximation can result in significant error in estimating S. Indeed, in 8-m water depth at Duck N.C., the S was consistently 6% larger than [Appendix A, Ruessink et al., 21], which was suggested to result from directional spread of the wave field, but was not quantified. As discussed by Battjes [1972], errors in offshore S will result in modeled alongshore current errors or, when fitting a model to observations, in biased alongshore current model parameters such as the drag coefficient. The radiation stress component S drives shoreline setup, and in a frequency- and directionally-spread wave field, according to linear theory [Battjes, 1972], is (hereinafter ) S = E(f, θ) [ cg (f) c(f) (cos2 (θ) + 1) 1 ] dθdf. (4) 2 Similar to S, the narrow-band in frequency and direction approximation to (4) is (hereinafter S ) S = 1 8 ρgh2 rms [ cg ( f) c( f) (cos2 ( θ) + 1) 1 ]. (5) 2 This S or other simplifications (e.g., zero directional-spread, shallow-water, and θ = ) often are used in studies of the cross-shore momentum (i.e., setup) balances [e.g., Guza and Thornton, 1981; Nielsen, 1988; King et al., 199; Lentz and Raubenheimer, 1999; Lentz et al., 1999; Raubenheimer et al., 21]. As discussed by Battjes [1972], the narrow-band S may

5 5 over- or under-estimate the true S (tr), resulting in over- or under-estimated setup. Although, it is noted that Raubenheimer et al. [21] found small differences in setup predictions using or S. In principle a directional wave buoy or current-meter provides the information required to estimate (tr) and S [e.g., Longuet-Higgins et al., 1963; Higgins et al., 1981; Elgar et al., 1994], although this capability is not always utilized. Here, the true and narrow-band S and S formulations are examined. In 8-m water depth, the narrow-band S and S are consistently larger (by 6% and 16%, respectively) than the true and. The breakdown of the narrow-band approximation is examined by separately studying the effects of frequency and directional spread. Using an empirical frequency spectrum, the effect of frequency spread is shown to be small and unlikely to explain the error in the narrow-band approximation. Assuming zero frequency-spread and symmetric directional spectra, analytic expressions [Battjes, 1972] for the S (tr) /S and S (tr) /S ratios are functions of the directional spread. The observed ratios are strong functions of the observed directional spread and the analytic expressions reproduce this dependence, demonstrating that the true can be accurately parameterized by augmenting the S directional-spread dependence. and S and representations with a 2. Observations Table 1. Wave observations were taken near Duck N.C. on the northern part of the Outer Banks in the mid-atlantic Bight with a pressure sensor array in 8-m water depth [Long, 1996] maintained by the U.S. Army Corps of Engineers Field Research Facility during the Duck94

6 6 (August October, 1994) and the SandyDuck (August November, 1997) field experiments. Hourly (tr) and S were calculated using a moment-estimation technique [Elgar et al., 1994], and hourly rms wave height H rms, bulk (energy-weighted) mean angle θ, and frequency f (Appendix A) were used to the calculate narrow-band approximations S (3) and S (5). Additionally, bulk (energy-weighted) directional spread σ θ (A2) [Kuik et al., 1988] was calculated (Appendix A). A total of 222 and 2635 hourly estimates were collected for Duck94 and SandyDuck, respectively. A wide range of wave-conditions occurred in 8-m water depth (Table 1), with the average condition being a locally generated sea typical of the mid-atlantic Bight. Figure 1. As noted by Ruessink et al. [21] for Duck94, during both experiments the estimates are correlated (r =.98), but the narrow-band S true and S is consistently larger than the (Fig 1a & b). The best-fit slopes (.63) are nearly identical for the two experiments, and indicate a systematic 6% over-estimation of S by using S. This over-estimation with the narrow-band approximation applies also to S (Figure 2). The Duck94 and SandyDuck true and narrow-band S are correlated (r =.99), but the narrow-band S is consistently larger than with similar best-fit slopes (Duck94:.86 and SandyDuck:.89). Use of the narrow-band S would result in over-predicted modeled shoreline setup. Figure 2. Thornton and Guza [1986] recognized the potential error in using S. They drove an alongshore current model with a S conserving wave angle ˆθ [Higgins et al., 1981] defined so that, in 9-m water depth at a beach near Santa Barbara CA, 1 c g ( f) 8 ρgh2 rms c( f) sin ˆθ cos ˆθ = S (tr),

7 7 where the peak frequency of the spectrum was used for f and was calculated with an array of pressure sensors. Small differences ( ) were found between ˆθ and the mean angle at the peak frequency (analogous to θ). Perhaps this agreement led to a more general adoption of the narrow-band S approximation (3). However, Thornton and Guza [1986] observed very long period ocean swell, and thus the narrow-band (for both f and θ) approximations were likely satisfied. In contrast, the Duck wave field is often locally generated, with a broad frequency-directional spectrum, resulting in typical 1 2 differences between θ and ˆθ (not shown). Analogous differences occur with a S conserving angle. 3. Radiation Stress Parameterizations The breakdown of the narrow-band approximation is examined by considering a separable frequency-directional spectrum, i.e., E(f, θ) = F (f)d(θ θ), (6) where D(θ θ) is normalized so that D(θ θ)dθ = 1 and F (f)df = E where E is the total wave energy (i.e., E = ρghrms 2 /8). Although real oceanic E(f, θ) are rarely separable, this provides a springboard to examine separately the effects of frequency-spread and directional-spread on S and S. With (6), the expressions for (tr) and S become, = = F (f) c g(f) c(f) df F (f) c g(f) c(f) df [ 1 + D(θ θ) sin(θ) cos(θ) dθ ] D(θ θ) cos 2 (θ)dθ 1 2 E. (7a) (7b)

8 Frequency Spread Frequency-spread results in variable c g /c which in intermediate water depths could potentially affect the radiation stress. For example, in 8-m water depth, c g /c spans over the frequency range.7.2 Hz. To isolate frequency spread effects, consider a narrow-banded directional spectrum E(f, θ) = F (f)δ(θ θ) where δ is a Dirac delta function. The expressions for (tr) (7a) and S (7b) become = F (f) c g(f) c(f) df sin( θ) cos( θ), S (tr) = F (f) c g(f) c(f) df [ 1 + cos 2 ( θ) ] 1 2 E. The amount the frequency-spread contributes to error in S and S is given by the ratios S = F (f) cg(f) E c g( f) c( f) c(f) df, (8a) S = F (f) cg(f) df [ 1 + cos 2 ( θ) ] 1 E c(f) 2 E [ cg( f) c( f) (1 + cos2 ( θ)) 1 2 ]. (8b) Although the possible permutations of oceanic F (f) are limitless, for analysis purposes a Pierson and Moskowitz [1964, hereinafter PM] spectrum is considered F PM (f) = αg2 f 5 (2π) 4 exp [ 5 4 ( ) ] 4 f where f p is the peak frequency, g represents gravity, and α sets the total wave energy. Note that there is a difference between the peak frequency f p and the energy-weighted mean frequency f ( f = /4 Γ(3/4)f p 1.3f p ). Although the PM spectrum is intended for deep-water waves, it is not much different from empirical finite-depth spectral forms [e.g., Bouws et al., 1985], and is useful because it has one free parameter f p. The S (tr) /S and S (tr) /S ratios f p

9 9 (8a and 8b) with the PM spectrum are efficiently numerically integrated [Abramowitz and Stegun, 1965] and are functions soley of f Directional Spread In examining the effects of directional spread, the analysis of Battjes [1972] is followed. With frequency spectrum F (f) = E δ(f f) and symmetric D(θ θ), The expression for (7a) becomes (with θ = θ θ) S (tr) = E c g ( f) c( f) sin( θ) cos( θ) D(θ ) cos(2θ )dθ. }{{} R The theta-integral R is the ratio S (tr)/s which is 1, and using circular moment definitions reduces to [Battjes, 1972] S = 1 2σ θ 2, (9) where σθ is the directional-spread1 [Kuik et al., 1988]. The expression for S (tr) (7b) becomes = E The theta integral S in (1) reduces to c g ( f) (1 + c( f) D(θ θ) cos 2 (θ)dθ) 1 2 (1) }{{} S D(θ θ) cos 2 (θ)dθ = cos 2 ( θ) D(θ ) cos 2 (θ )dθ + sin 2 ( θ) D(θ ) sin 2 (θ )dθ = cos 2 ( θ)(1 σ θ 2 ) + sin 2 ( θ)σ θ 2. (11) 1 Kuik et al. [1988] defines two directional spreads: σ θ 2 = sin2 (θ )D(θ )dθ and σ 2 θ = 4 sin2 (θ /2)D(θ )dθ.

10 1 The second term in (11) is often negligible relative to the first term. For example, with even large θ = 3 and σ θ = 3 values (Table 1), the second term is only 13% of the first. Neglecting the second term, the S (tr) /S ratio becomes S = c g [(1 c σ 2 θ ) cos 2 ( θ) + 1] 1 2 c g c [cos2 ( θ) + 1] 1 2, and assuming small angles (cos( θ) 1) and shallow water waves (c g /c 1), the ratio reduces to (analogous to the deep water result in Battjes [1972]) S = σ θ 2. (12) 4. Results Figure 3. Although oceanic E(f, θ) are not in general separable and θ symmetric, the combined Duck94 and SandyDuck observed S (tr)/s and S (tr)/s are examined to determine whether frequency or directional spread can explain the systematic difference between the true and narrow-band radiation stress representations (Figure 1 and 2). Both S (tr) /S and S (tr)/s are binned as functions of the observed (Appendix A) f and bulk (energy-weighted) σ θ (A2) from which means and standard deviations are calculated. Binned observations are restricted ( S.6 m3 /s 2 and θ 5, and S /ρ.8 m3 /s 2 ) so that ratios are not biased by dividing by exceedingly small numbers. This corresponds to excluding cases with very small waves that are not of typical interest. Changing these restrictions does not change the results. The integral expressions for S (tr)/s and S (tr)/s based soley on frequency-spread (8a and 8b) using the PM spectrum are calculated with 8-m water depth and θ =. Varying the depth between 5 13 m and θ by 2 does not change the ratios

11 11 significantly. The analytic S (tr)/s and S (tr)/s based soley on directional spread (9 and 12) are calculated as a function of the bulk σ θ. The effects of frequency spread are examined first (Figure 3). The observed S (tr) /S means range between.55.8, with standard deviations about.15 (Figure 3a). The observed S (tr)/s means (between.85.95) do not strongly vary with f (Figure 3b). The PM based numerically-integrated S (tr)/s and S (tr)/s are near 1 (dash-dot curves in Figure 3), and, in contrast to the observed ratios, increases (slightly) with larger f. At f >.2 Hz, the PM based numerically integrated maximum ratios are S (tr) /S = 1.8 and /S = Thus, for a PM spectrum, frequency spread is unable to explain the observed S (tr)/s and S (tr)/s ratios, and although the PM spectrum is by no means universal - particularly in intermediate depths, this strongly suggests that frequency spread is not the cause of the observed S (tr) /S and S (tr) /S. Figure 4. The effect of directional-spread are examined next with the bulk (energy-weighted) σ θ (A2) used in the place of the single wave frequency σθ (tr). The S /S and S (tr)/s ratios both are strong functions σ θ (Figure 4). The S (tr) /S binned means range between closely following the analytical expression of Battjes [1972] (9). Most binned standard deviations are.1 (Figure 4a) indicating little scatter. At the mean σ θ = 23 (Table 1) the analytic S (tr)/s =.66, close to the best-fit value of.63 (light dashed line in Figure 4a). The S (tr)/s binned means range between also closely following the the analytic expression (12), and have standard deviations.6. The observed best-fit slopes from Figure 2 (.86 and.89, light dashed lines in Figure 4b) intersect the analytic curve within the observed σ θ mean ± one standard deviation (23 ± 4 ). This demonstrates that the observed

12 12 difference between the true and narrow-band radiation stresses is due to directional spread of the incident wave field, and shows that the true given the bulk wave parameters H rms, f, θ, and σ θ. (tr) and S can be accurately parameterized The directional spread decreases due to refraction as waves propagate toward shore. For zero frequency spread and small directional spread, the cross-shore (x) transformation of σ θ due to Snell s law is [Kinsman, 1965] σ θ (x) σ θ (x ) = c(x) cos[ θ(x )] c(x ) cos[ θ(x)] where x is an offshore location. Given the observed mean f =.15 Hz and θ = 16 (Table 1), the directional-spread in 3-m water depth is still 2/3 the 8-m depth σ θ. Furthermore with wave-breaking, σ θ can increase up to 5% and reach values up to 3 [Herbers et al., 1999]. Thus, the results in 8-m depth, appear to be relevant in shallower water. 5. Summary (S The narrow-band (in frequency and direction) approximation to the radiation stress and S ) often is used in modeling the alongshore currents and setup. However, in 8-m water depth at Duck, N.C. the narrow-band approximated radiation stress components are systematically larger than the true radiation stress components ( (tr) and S ) based on moments of the frequency directional spectrum. Clearly, the true radiation stress, which can be estimated from a current meter or directional wave buoy, should be used whenever possible. Use of the narrow band radiation stress approximations will result in biased alongshore current and shoreline setup, or result in biased alongshore current model parameters when fitting

13 13 models to observations. This difference between the narrow band and true radiation stress is not the result of frequency-spread, but is due to non-zero directional spread [Battjes, 1972]. Assuming a narrow-frequency, directionally-symmetric frequency-directional spectrum leads to analytic expressions for the S (tr) /S and S (tr) /S ratios that accurately describe the observed ratios with the bulk σ θ. While these results are presented for only a single geographic location and one water depth, observations total seven months, and it is expected that these results are valid for other geographic locations where the waves are often locally generated. Appendix A: Definition of Energy Weighted Wave Statistics The mean frequency f is defined as f = ss ss fe(f, θ)dθ df E(f, θ)dθ df where the frequency integral is over the sea-swell (ss) frequency band (.44.3 Hz). Directional moments defined by Kuik et al. [1988] at a single frequency, are generalized for the entire sea-swell frequency band. With the lowest four bulk Fourier directional-moments of E(f, θ) [Longuet-Higgins et al., 1963; Herbers et al., 1999], a 1 = b 1 = a 2 = b 2 = ss ss ss ss ss ss ss ss cos(θ)e(f, θ)dθ df E(f, θ)dθ df, sin(θ)e(f, θ)dθ df E(f, θ)dθ df, cos(2θ)e(f, θ)dθ df E(f, θ)dθ df, sin(2θ)e(f, θ)dθ df E(f, θ)dθ df.

14 14 the energy-weighted mean wave angle θ and directional spread σ θ are defined as [Kuik et al., 1988], θ = arctan(b 1 /a 1 ), ( σ θ) 2 = 1 a 2 cos(2 θ) b 2 sin(2 θ) ss = sin2 (θ θ)e(f, θ)dθ df 2 E(f, θ)dθ df. (A2) ss Except for using θ = arctan(b 2 /a 2 ), the σ θ definition is that used by Herbers et al. [1999]. Acknowledgments. This research was supported by NOPP and WHOI. The Field Research Facility, Coastal Engineering Research Center, Duck, N.C. provided and the 8-m depth pressure array data, and Tom Herbers processed the wave observations. Bob Guza, John Trowbridge, and Jurgen Battjes provided valuable feedback.

15 15 References Abramowitz, M., and I. Stegun, Handbook of Mathematical Functions, Dover, New York, Battjes, J.A., Radiation stresses in short-crested waves, J. Mar. Res., 3, 56 64, Bouws, E., H. Gunther, W. Rosenthal, and C. L. Vincent, Similarity of the wind wave spectrum in finite depth water, I. Spectral Form, J. Geophys. Res., 9, , Bowen, A. J., D. L. Inman, and V. P. Simmons, Wave set-down and set-up, J. Geophys. Res., 73, , Church, J. C., and E. B. Thornton, Effects of breaking wave induced turbulence within a longshore current model, Coastal Eng., 2, 1 28, Elgar, S., T. H. C. Herbers, and R. T. Guza, Reflection of ocean surface gravity waves from a natural beach, J. Phys. Oceanogr., 24, , Guza, R. T., and E. B. Thornton, Wave set-up on a natural beach, J. Geophys. Res., 86, , Herbers, T. H. C., S. Elgar, and R. T. Guza, Directional spreading of waves in the nearshore, J. Geophys. Res., 14, , Higgins, A. L., R. J. Seymour, and S. S. Pawka, A compact representation of ocean wave directionality, Appl. Ocean Res., 3, , King, B. A., M. W. L. Blackley, A. P. Carr, and P. J. Hardcastle, Observations of wave-induced set-up on a natural beach, J. Geophys. Res., 95, 22,289 22,297, 199. Kinsman, B., Wind Waves: Their Generation and Propagation on the Ocean Surface, 676pp., Prentice-Hall, Englewood-Cliffs, N.J., 1965.

16 16 Kuik, A. J., G. P. Van Vledder, and L. H. Holthuijsen, A method for the routine analysis of pitch-and-roll buoy wave data, J. Phys. Oceanogr., 18, , Lentz, S. J., and B. Raubenheimer, Field observations of wave setup, J. Geophys. Res., 14, 25,867 25,875, Lentz, S. J., R. T. Guza, S. Elgar, F. Feddersen, and T. H. C. Herbers, Momentum balances on the North Carolina inner shelf, J. Geophys. Res., 14, 18,25 18,226, Long, C.E., Index and bulk parameters for frequency-direction spectra measured at CERC Field Research Facility, June 1994 to August 1995, Misc. Pap. CERC-96-6, U.S. Army Eng. Waterw. Exp. Stn., Vicksburg, Miss., Longuet-Higgins, M. S., Longshore currents generated by obliquely incident sea waves 1, J. Geophys. Res., 75, , 197. Longuet-Higgins, M. S., D. E. Cartwright, and N. D. Smith, Observations of the directional spectrum of sea waves using the motion of a floating buoy, Ocean Wave Spectra, Prentice-Hall, , Longuet-Higgins, M. S., and R. W. Stewart, Radiation stress in water waves, a physical discussion with application, Deep Sea Res., 11, , Nielsen, P., Wave setup: A field study, J. Geophys. Res., 93, 15,642 15,652, Pierson, W. J. and L. Moskowitz, A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii, J. Geophys. Res., 69, , Raubenheimer, B., R. T. Guza, and S. Elgar, Field observations of wave-driven setdown and setup, J. Geophys. Res., 16, , 21.

17 17 Ruessink, B. G., J. R. Miles, F. Feddersen, R. T. Guza, S. Elgar, Modeling the alongshore current on barred beaches, J. Geophys. Res., 16, 22,451 22,463, 21. Thornton, E. B., and Guza, R. T., Surf zone longshore currents and random waves: field data and models, J. Phys. Oceanogr., 16, , Falk Feddersen, Integrative Oceanography Division, Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA ( Received compiled on May 7, 24 This manuscript was prepared with AGU s L A TEX macros v5, with the extension package AGU ++ by P. W. Daly, version 1.6b from 1999/8/19.

18 18 Figure Captions 1 (a) Duck94.5 PSfrag replacements /ρ (m 3 /s 2 ) /ρ (m 3 /s 2 ) S (tr) /ρ (m 3 /s 2 ) (b) SandyDuck S /ρ (m 3 /s 2 ) Figure 1. (a) Duck94 and (b) SandyDuck true /ρ vs. narrow-band S /ρ. The dashed line is the one-to-one relationship. The correlation for both is r =.98 and the best-fit slopes (solid lines) are both.63. Positive S corresponds to waves from the north.

19 19 PSfrag replacements /ρ (m 3 /s 2 ) S (tr) /ρ (m 3 /s 2 ) (a) Duck94 (b) SandyDuck /ρ (m 3 /s 2 ) S /ρ (m 3 /s 2 ) Figure 2. (a) Duck94 and (b) SandyDuck true /ρ vs. narrow-band S /ρ. The dashed line is the one-to-one relationship. The correlation for both is r =.99 and the best-fit slopes (solid lines) are.86 and.89, respectively.

20 (a).9 S (tr) /S (b) PSfrag replacements S (tr) /S f (Hz) Figure 3. Combined Duck94 and SandyDuck means (diamonds) and standard deviations (vertical bars) of (a) S (tr) /S and (b) S (tr) /S vs. mean frequency f. The dash-dot curves represent the effects of frequency-spread (8a and 8b) using the Pierson and Moskowitz [1964] spectrum. The horizontal dashed lines in panels a and b represent the best fit slopes in Figures 1 and 2, respectively.

21 (a) S (tr) /S (b).95 PSfrag replacements S (tr) /S σ θ (deg) Figure 4. Combined Duck94 and SandyDuck means (diamonds) and standard deviations (vertical bars) of (a) S (tr) /S and (b) S (tr) /S versus σ θ (tr). The dash-dot curves represent the analytic S /S and S (tr) /S dependence of σ θ (9 and 12). The horizontal dashed lines in panels a and b represent the best fit slopes in Figures 1 and 2, respectively.

22 22 Tables H rms (m) f (Hz) θ (deg) σ θ (deg) mean standard deviation maximum minimum Table 1. Combined Duck94 and SandyDuck hourly wave statistics in 8-m water depth.