GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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OCEAN SURFACE WINDS 29 TABLE 2.6 The effects of stability on wind profiles Wind speed profiles are shown (in m/s) for a range of stabilities (characterized by the air-sea temperature difference). Profiles are given both for a fixed wind speed at 50 m and for a fixed surface friction velocity (0.36 m/s) Height Air-sea temperature difference (°C) (m) –10 –8 –6 –4 –3 –2 –1 0 1 2 3 4 6 8 10 Fixed wind speed at 50 m 50 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 45 10.0 10.0 10.0 10.0 10.0 10.0 9.9 9.9 9.8 9.8 9.7 9.7 9.7 9.7 9.7 40 9.9 9.9 9.9 9.9 9.9 9.9 9.9 9.8 9.6 9.5 9.4 9.4 9.4 9.4 9.4 35 9.9 9.9 9.9 9.9 9.8 9.8 9.8 9.7 9.4 9.2 9.1 9.1 9.1 9.1 9.1 30 9.8 9.8 9.8 9.8 9.8 9.8 9.7 9.6 9.2 8.9 8.8 8.7 8.7 8.7 8.7 28 9.8 9.8 9.8 9.8 9.7 9.7 9.7 9.5 9.1 8.8 8.6 8.6 8.5 8.5 8.6 26 9.8 9.8 9.7 9.7 9.7 9.7 9.6 9.4 9.0 8.6 8.5 8.4 8.3 8.4 8.4 24 9.7 9.7 9.7 9.7 9.7 9.6 9.6 9.4 8.9 8.5 8.3 8.2 8.2 8.2 8.2 22 9.7 9.7 9.7 9.6 9.6 9.6 9.5 9.3 8.7 8.4 8.1 8.0 8.0 8.0 8.0 20 9.7 9.6 9.6 9.6 9.6 9.5 9.5 9.2 8.6 8.2 8.0 7.8 7.8 7.8 7.8 18 9.6 9.6 9.6 9.5 9.5 9.5 9.4 9.1 8.5 8.0 7.8 7.6 7.5 7.5 7.6 16 9.6 9.5 9.5 9.5 9.4 9.4 9.3 9.0 8.4 7.9 7.6 7.4 7.3 7.3 7.3 14 9.5 9.5 9.5 9.4 9.4 9.3 9.2 8.9 8.2 7.7 7.4 7.2 7.0 7.0 7.0 12 9.4 9.4 9.4 9.3 9.3 9.2 9.1 8.8 8.1 7.5 7.1 6.9 6.7 6.7 6.7 10 9.3 9.3 9.3 9.2 9.2 9.1 9.0 8.6 7.9 7.3 6.9 6.6 6.4 6.4 6.4 9 9.3 9.2 9.2 9.1 9.1 9.0 8.9 8.5 7.8 7.2 6.7 6.5 6.2 6.2 6.2 8 9.2 9.2 9.1 9.1 9.0 8.9 8.8 8.4 7.7 7.0 6.6 6.3 6.1 6.0 6.0 7 9.1 9.1 9.1 9.0 8.9 8.8 8.7 8.3 7.6 6.9 6.4 6.2 5.9 5.8 5.7 6 9.0 9.0 9.0 8.9 8.8 8.7 8.6 8.2 7.4 6.7 6.3 6.0 5.7 5.5 5.5 5 8.9 8.9 8.8 8.7 8.7 8.6 8.5 8.0 7.3 6.6 6.1 5.8 5.4 5.3 5.2 4 8.8 8.7 8.7 8.6 8.5 8.4 8.3 7.8 7.1 6.4 5.9 5.5 5.2 5.0 4.9 3 8.6 8.5 8.5 8.4 8.3 8.2 8.0 7.6 6.9 6.2 5.6 5.3 4.9 4.7 4.5 2 8.3 8.2 8.1 8.0 8.0 7.9 7.7 7.2 6.6 5.9 5.3 5.0 4.5 4.3 4.1 1 7.7 7.6 7.5 7.4 7.4 7.3 7.1 6.7 6.1 5.4 4.9 4.5 4.0 3.7 3.6 Friction velocity fixed (0.36 m/s) 50 8.2 8.3 8.5 8.7 8.8 9.0 9.3 10.0 11.1 12.0 12.8 13.5 14.5 15.2 16.1 45 8.1 8.3 8.4 8.6 8.8 9.0 9.2 9.9 10.9 11.8 12.5 13.1 14.1 14.9 15.6 40 8.1 8.3 8.4 8.6 8.7 8.9 9.2 9.8 10.7 11.5 12.1 12.7 13.7 14.5 15.1 35 8.1 8.2 8.4 8.6 8.7 8.9 9.1 9.7 10.4 11.2 11.8 12.3 13.2 13.9 14.6 30 8.0 8.2 8.3 8.5 8.6 8.8 9.0 9.6 10.2 10.8 11.4 11.9 12.7 13.4 13.9 28 8.0 8.2 8.3 8.5 8.6 8.8 9.0 9.5 10.1 10.7 11.2 11.7 12.5 13.1 13.7 26 8.0 8.1 8.3 8.5 8.6 8.7 9.0 9.4 10.0 10.6 11.1 11.5 12.2 12.9 13.4 24 8.0 8.1 8.2 8.4 8.6 8.7 8.9 9.4 9.9 10.4 10.9 11.3 12.0 12.6 13.1 22 8.0 8.1 8.2 8.4 8.5 8.7 8.9 9.3 9.7 10.2 10.7 11.1 11.7 12.3 12.8 20 7.9 8.1 8.2 8.4 8.5 8.6 8.8 9.2 9.6 10.1 10.5 10.8 11.5 12.0 12.5 18 7.9 8.0 8.2 8.3 8.4 8.6 8.7 9.1 9.5 9.9 10.3 10.6 11.2 11.7 12.1 16 7.9 8.0 8.1 8.3 8.4 8.5 8.7 9.0 9.3 9.7 10.1 10.4 10.9 11.4 11.8 14 7.8 8.0 8.1 8.2 8.3 8.4 8.6 8.9 9.2 9.5 9.8 10.1 10.6 11.0 11.4 12 7.8 7.9 8.0 8.2 8.2 8.4 8.5 8.8 9.0 9.3 9.6 9.8 10.2 10.6 11.0 10 7.7 7.9 7.9 8.1 8.2 8.3 8.4 8.6 8.8 9.1 9.3 9.5 9.9 10.2 10.5 9 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.5 8.7 8.9 9.1 9.3 9.7 10.0 10.2 8 7.6 7.8 7.8 8.0 8.0 8.1 8.2 8.4 8.6 8.8 9.0 9.1 9.4 9.7 10.0 7 7.6 7.7 7.8 7.9 8.0 8.0 8.1 8.3 8.5 8.6 8.8 8.9 9.2 9.5 9.7 6 7.5 7.7 7.7 7.8 7.9 8.0 8.0 8.2 8.3 8.5 8.6 8.7 9.0 9.2 9.4 5 7.4 7.6 7.6 7.7 7.8 7.8 7.9 8.0 8.1 8.3 8.4 8.5 8.7 8.9 9.1 4 7.3 7.5 7.5 7.6 7.6 7.7 7.8 7.8 7.9 8.0 8.1 8.2 8.4 8.5 8.7 3 7.2 7.3 7.3 7.4 7.4 7.5 7.5 7.6 7.7 7.8 7.8 7.9 8.0 8.1 8.3 2 7.1 7.1 7.1 7.1 7.2 7.2 7.2 7.2 7.3 7.4 7.4 7.5 7.6 7.6 7.7 1 6.6 6.6 6.6 6.7 6.7 6.7 6.7 6.7 6.8 6.8 6.8 6.8 6.9 6.9 7.0
30 a general sense, and can be applied to the atmosphere. These concepts have been developed to formulate diagnostic models that allow us to deduce surface turbulent flow fields from the free atmospheric flow. The marine boundary layer itself can be separated into two regimes: the constant flux (or constant stress) layer (from the surface to about 50 m) and the Ekman layer (from about 50 m up to the free atmosphere ≈ 1 km). In the surface layer it can be assumed that the frictional forces contributing to turbulence are constant with height, and the effects of the Coriolis and pressure gradient forces, as well as the horizontal gradient of turbulent fluxes, are negligible. The wind direction is consequently constant with height. Using the mixing length theory developed by Prandtl, it can be shown that the flow in the constant flux layer (or Prandtl layer) depends only on the surface roughness length. 2.4.1 Constant flux layer Under neutral conditions, Prandtl’s solution shows that the horizontal flow over the ocean surface follows the well known “log” (logarithmic) profile in the vertical direction, u = u * κ ln (2.8) where κ is the von Kármán constant, z 0 is the constant of integration, known as the roughness length, and u * is the friction velocity, which has magnitude: u * = ⎛ z ⎞ ⎜ ⎟ ⎝ ⎠ , (2.9) τ is the magnitude of the surface stress, and ρa the density of air. u * can be thought of as a proxy for the surface stress. It is common also to express stress (τ) through the bulk transfer relation: τ = ρa Cd u2 , (2.10) where Cd is the drag coefficient. In general, Cd and u are both functions of height. Determining Cd has been the objective of many field research programmes over the years. One of the problems of specifying the wind in the turbulent layer near the ocean is the formulation of z0 and its relationship to u *. Using dimensional argument, Charnock (1955) related the roughness length of the sea surface to the friction velocity of the wind as follows: z0 = (2.11) where α is the Charnock constant, with a value as determined by Wu (1980) of 0.0185 , and g is the acceleration by gravity. If the boundary layer is in a state of neutral stratification, the drag coefficient, which is a function of height, can be expressed as: αu * 2 , g τ ρ a z 0 , GUIDE TO WAVE ANALYSIS AND FORECASTING or C z d( )= κ 2 Cd ( z) = ⎛ ⎡ z ⎤⎞ / ⎜ln ⎢ ⎥ ⎝ ⎣ z ⎟ , 0 ⎦⎠ ⎛ u * ⎞ ⎜ ⎟ ⎝ uz⎠ ( ) (2.12) (2.13) However, the boundary layer over the ocean is not necessarily neutral. A stability dependence was originally derived by Monin-Obukhov (1954) from profile similarity theory. This is used to modify the simple log relation provided above: u = (2.14) u * κ ln ⎡ ⎛ z ⎞ ⎜ ⎟ – ψ ⎛ z ⎞ ⎤ ⎢ ⎝ z0 ⎠ ⎝ L ⎠ ⎥ ⎣⎢ ⎦⎥ ⎛ ⎡ ⎛ z ⎞ C z ⎛ z ⎞⎤⎞ d( )= ⎜κ / ⎢ln ⎜ ⎟ – ψ (2.15) ⎝ z ⎠ ⎝ L⎠ ⎥⎟ . ⎝ ⎣⎢ 0 ⎦⎥ ⎠ The function ψ has been derived for both stable and unstable conditions. L is the Monin-Obukhov mixing length. For neutral conditions ψ (z/L) = 0. Businger et al. (1971) proposed functional relationships between the non-dimensional wind shear and z/L which can be used to determine the stability function ψ (z/L) in Equation 2.14. 2.4.2 The use of the drag coefficient In the above discussion, we developed the concept of the drag coefficient, as defined in Equation 2.10. There have been many studies to determine Cd under varying wind speeds and stabilities and a summary can be found in Roll (1965). More recently, Wu (1980, 1982) has shown through empirical studies that the drag coefficient at a given height depends linearly on the wind speed, and that the following formulation holds for a wide range of winds under near neutrally stable conditions: 103 C10 = (0.8 + 0.65 U10) (2.16) where: C10 = Drag coefficient at the 10 m level; and U10 = Wind speed (m/s) at the 10 m level. This simple linear empirical relationship, however, needs to be modified when temperature stratification is present. Stratification affects turbulent momentum transport, thereby causing the wind profile to deviate from the logarithmic form. Schwab (1978) determined Cd over water for a wide range of wind speeds and atmospheric stabilities. Figure 2.9 shows the results of his calculation. A crucial issue to be addressed at this juncture concerns the effect of changing stability and wind stress on the prediction of wave growth. It can be inferred from Figure 2.9 that, for a given 10 m level wind speed, unstable conditions result in higher drag coefficients (or surface stress) and hence larger wave growth than stable conditions. Liu et al. (1979) have developed a set of equations which compute the surface variables u * , z0 and the boundarylayer stability length (L), so that the wind profile of 2 . 2 2
Page 1 and 2: WORLD METEOROLOGICAL ORGANIZATION G
Page 3 and 4: © 1998, World Meteorological Organ
Page 5 and 6: IV Chapter 7 - WAVES IN SHALLOW WAT
Page 7 and 8: ACKNOWLEDGEMENTS The revision of th
Page 9 and 10: VIII has been included particularly
Page 11 and 12: 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: 28 POINT C — The effect of warm a
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 and 64: the energy flux is c gH 2 . This is
Page 65 and 66: α0, degrees Solution: 80° 70° 60
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
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7.1 Introduction The evolution of w
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water boundary into shallow water.
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eflected waves, although the latter
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Energy (cm 2 /Hz) Energy (cm 2 /Hz)
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8.1 Introduction Wave data are ofte
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matter and timing the intervals bet
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a Directional Waverider (Barstow et
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80°N 60°N 40°N 20°N 0° -20°S
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a combination of sea-surface temper
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8.7.1 Digital analysis of wave reco
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9.1 Introduction Chapter 8 describe
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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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