GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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54 1.8 1.6 1.4 Ks 1.2 1.0 0.8 Example: For a deep water wavelength λ 0 = 156 m and T 0 = 10 s, then k 0 = 0.04 m –1 , and for T 0 = 15 s, k 0 = 0.018 m –1 . Table 4.10 shows the shoaling factor K s for a number of shallow depths. It also shows that the height of shoaling waves is reduced at first, but finally increases up to the point of breaking which also depends on the initial wave height in deep water (see also Figures 4.7(a) and (b)). 4.5.1.2 Variation of wave height due to refraction In the previous example, no refraction was taken into account, which implies propagation of waves perpendicular to parallel bottom depth contours. In natural conditions this will rarely occur. So the angle of incidence with respect to the bottom depth contours usually differs from 90°, which is equivalent to α, the angle between a wave crest and the local isobath, being different from 0°. This leads to variation of the width between wave rays. Using Snell’s law, H = H The refraction factor is: 0 cg c g cos α . cos α 0 0 Kr = cos α0 cos α GUIDE TO WAVE ANALYSIS AND FORECASTING 0.01 0.1 1 h/λo (4.13) with α 0 the angle between a wave crest and a local isobath in deep water. Figure 4.8, taken from CERC (1984), is based on Equation 4.13. For a given depth and wave period the shallow-water angle of incidence (solid lines) and the refraction factor (broken lines) can be read off easily for a given deep-water angle of incidence α 0. It is valid for straight parallel depth contours only. Problem: Given an angle α0 = 40° between the wave crests in deep water and the depth contours of the sloping bottom, find α and the refraction at h = 8 m for T = 10 s. h TABLE 4.10 Wavenumber k and shoaling factor Ks for two wave periods, at several depths, h, in metres, using Equations 4.11 and 4.12 T = 10 s, k 0 = 0.04 m –1 Figure 4.7(b) — A graph of shoaling factor K s versus h/λ 0 (derived from CERC, 1984) T = 15 s, k 0 = 0.018 m –1 (m) k (m –1 ) K s k (m –1 ) K s 100 0.040 1.00 0.018 0.94 50 0.041 0.95 0.021 0.92 25 0.046 0.91 0.028 0.98 15 0.055 0.94 0.035 1.06 10 0.065 1.00 0.043 1.15 5 0.090 1.12 0.060 1.33 2 0.142 1.36 0.095 1.64
α0, degrees Solution: 80° 70° 60° 50° 40° 30° 20° 10° 0 10-4 0.50 0.60 0.70 h/(gT 2 ) = 8/(9.8 x 100) = 0.0082, and from Figure 4.8 the refraction factor is 0.905, and α = 20°. 4.5.1.3 Dorrestein’s method Due to the fact that in reality the bottom depth contours rarely are straight, we generally find sequences of convergence and divergence (see also Section 1.2.6 and Section 7.3). Dorrestein (1960) devised a method for manually determining refraction where bottom contours are not straight. This method requires the construction of a few wave rays from a given point P in shallow water to deep water, including all wave directions that must be taken into account according to a given directional distribution in deep water. We assume that, in deep water, the angular distribution of wave energy is approximately a uniform distribution in the azimuthal range α1´ and α2´. These angles correspond, respectively, to the angles of incidence α1 and α2 at point P. Rays must at least be 5° Lines of equal Kr WAVE FORECASTING BY MANUAL METHODS 55 0.80 0.85 0.90 10° 0.95 2 5 2 5 2 5 15° 0.97 Lines of equal α (angle between wave crest and contour) 10-3 10-2 0.1 h/(gT 2 ) Figure 4.8 — Changes in wave direction and height due to refraction on slopes with straight parallel bottom contours. α 0 is the deep water angle of incidence, measured between the wave crest and the local isobath. The continuous curves are lines of equal incidence for various combinations of period and depth. To estimate the refraction as a wave moves into shallower water, starting from a given α 0 follow a horizontal line from right (deep water) to left. The broken lines are lines of equal refraction factor, K r (derived from CERC, 1977) 20° 0.99 constructed for waves at these outer limits of the distribution. It may be sufficient to assume straight isobaths and use Figure 4.8 to calculate these angles. Then, according to Dorrestein, the refraction factor is: (4.14) with c 0 and c the phase velocities in deep water and at point P, respectively. Problem: As in the example in Section 4.5.1.2, h = 8 m at point P. T = 10 s, so h/(gT2 ) = 0.0082. With the help of Figure 4.8, α = 20° for α0 = 40°. Find Kr by Dorrestein’s method. Solution: Snell’s law gives: c c 30° 5° K r 40° 50° 60° c0 α1 – α2 = ⋅ c α′ – α′ 1 2 ° = = 188 20° = sin sin . . sin α sin 0 α0 40 70°
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WORLD METEOROLOGICAL ORGANIZATION G
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© 1998, World Meteorological Organ
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IV Chapter 7 - WAVES IN SHALLOW WAT
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ACKNOWLEDGEMENTS The revision of th
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VIII has been included particularly
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X considerable attention. Annex III
Page 13 and 14: 2 η Crest Zero level a H = 2a a Tr
Page 15 and 16: 4 Figure 1.5 — Paths of the water
Page 17 and 18: 6 When waves propagate into shallow
Page 19 and 20: 8 1.3.2 Wave groups and group veloc
Page 21 and 22: 10 wavelengths in a given sea state
Page 23 and 24: 12 for instance, a frequency of 0.1
Page 25 and 26: 14 in which E(f) is the variance de
Page 27 and 28: 16 2.1.1 Wind and pressure analyses
Page 29 and 30: 18 GUIDE TO WAVE ANALYSIS AND FOREC
Page 31 and 32: 20 Figure 2.2(a) (right) — Usual
Page 33 and 34: 22 As a quick approximation of ocea
Page 35 and 36: 24 GUIDE TO WAVE ANALYSIS AND FOREC
Page 37 and 38: 26 Gr G Gr ∇p C Cnf ∇p C Cnf
Page 39 and 40: 28 POINT C — The effect of warm a
Page 41 and 42: 30 a general sense, and can be appl
Page 43 and 44: 32 Free atmosphere Ekman layer Cons
Page 45 and 46: 3.1 Introduction This chapter gives
Page 47 and 48: ange of directions. Also, waves at
Page 49 and 50: small enough that swell can survive
Page 51 and 52: Figure 3.7 — Structure of spectra
Page 53 and 54: 4.1 Introduction CHAPTER 4 WAVE FOR
Page 55 and 56: necessary to forecast waves for a p
Page 57 and 58: TABLE 4.4 Additional wave informati
Page 59 and 60: TABLE 4.6 Ranges of swell periods a
Page 61 and 62: this situation, the angular spreadi
Page 63: the energy flux is c gH 2 . This is
Page 67 and 68: 5.1 Introduction National Meteorolo
Page 69 and 70: only once, since it is usual to sto
Page 71 and 72: Frequency 0.050 0.067 0.083 0.100 0
Page 73 and 74: The models may differ in several re
Page 75 and 76: The CH class may include many semi-
Page 77 and 78: 6.1 Introductory remarks Since the
Page 79 and 80: in some applications, the zero up/d
Page 81 and 82: OPERATIONAL WAVE MODELS 71 Figure 6
Page 83 and 84: give large differences in the compa
Page 85 and 86: Wind (m/s) Waves (m) Buoy/GSOWM Wav
Page 87 and 88: TABLE 6.2 Numerical wave models ope
Page 89 and 90: TABLE 6.2 (cont.) Country Name of m
Page 91 and 92: 7.1 Introduction The evolution of w
Page 93 and 94: water boundary into shallow water.
Page 95 and 96: eflected waves, although the latter
Page 97 and 98: Energy (cm 2 /Hz) Energy (cm 2 /Hz)
Page 99 and 100: 8.1 Introduction Wave data are ofte
Page 101 and 102: matter and timing the intervals bet
Page 103 and 104: a Directional Waverider (Barstow et
Page 105 and 106: 80°N 60°N 40°N 20°N 0° -20°S
Page 107 and 108: a combination of sea-surface temper
Page 109 and 110: 8.7.1 Digital analysis of wave reco
Page 111 and 112: 9.1 Introduction Chapter 8 describe
Page 113 and 114: Presentation of data and wave clima
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the monsoon season of June to Septe
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Probability of non-exceedance case
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individual wave height derived usin
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emote, data-sparse areas. However,
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TABLE 9.2 (cont.) Country Source of
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For a storm hindcast, select the mo
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TABLE 9.3 (cont.) Country Model use
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ANNEX I ABBREVIATIONS AND KEY TO SY
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Abbreviation/ Definition Symbol MOS
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CODE FORM: D . . . . D or IIiii* SE
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sensors, spectral peak wave period
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cs1cs1 The ratio of the spectral de
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M jM j n mn m n smn sm n bn bn b P
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1744 Im Indicator for method of cal
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The following distributions are bri
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4. Weibull distribution Probability
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8. Fisher-Tippett Type I (FT-I) (or
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ANNEX IV THE PNJ (PIERSON-NEUMANN-J
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ANNEX IV 141 Duration graph. Distor
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REFERENCES Abbott, M. B., H. H. Pet
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REFERENCES 145 Carter, D. J. T., P.
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REFERENCES 147 Gumbel, E. J., 1958:
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REFERENCES 149 Maat, N., C. Kraan a
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Slutz, R. J., S. J. Lubker, J. D. H
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SELECTED BIBLIOGRAPHY CERC, 1984: C
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156 Energy . . . . . . . . . . . .
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158 steepness . . . . . . . . . . .
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GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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