Sea State Analysis. Topics. Module 7 Sea State Analysis 2/22/2016. CE A676 Coastal Engineering Orson P. Smith, PE, Ph.D.

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1 Sea State Analysis Module 7 Orson P. Smith, PE, Ph.D. Professor Emeritus Module 7 Sea State Analysis Topics Wave height distribution Wave energy spectra Wind wave generation Directional spectra Hindcasting Extreme sea states Corresponding to Chapter 6 in text (Sorensen) Module 7 Sea State Analysis Orson P. Smith, PE, Ph.D., Instructor

2 Wave height distribution Natural waves are rarely regular or sinusoidal in appearance Many interacting waves give sea surface an irregular appearance Analysis of irregular (t) : Sum of 5 sinusoidal waves Zero down crossing id of a wave Module 7 Sea State Analysis 3 Rayleigh Distribution Seas generally assumed to have wave height Rayleigh distribution Swell does not, bus often described by Rayleigh parameters For, wave heights relative to mean probability density function: px ( ) xe probability distribution function, probability that a particular wave height exceeds a specified height 4 P( x) e ", " x 4 x p P H mean = m x H mean = m x.456 Module 7 Sea State Analysis 4 Orson P. Smith, PE, Ph.D., Instructor

3 Wave period No such well accepted distribution of wave periods Average period is commonly observed and reported As with wave heights, visual observations miss shorter periods Sometimes called dominant wave period, which relates to energy.6.6 spectrum characteristics Rayleigh H distribution.6 How many waves in a sea state? x x Suppose the average wave period was 6 seconds and conditions were stationary (statistics remained constant) for hour 6 waves Stationarity is often assumed to exist from one to 3 hours in wave monitoring strategies i.e., a measurement every to 3 hours: order of 3 waves per sea state Encountering a wave height x H s doesn t seem far fetched! Module 7 Sea State Analysis 5 p.4. P Wave energy spectrum Single wave energy per unit surface area: Energy in physics generally proportional to amplitude squared For example: Wave energy flux (Power): A function of /T (frequency) Spectral analysis associates wave energy and frequency Either (Hz, cycles per sec) or (radians per sec) Sometimes period is used, usually as a secondary x axis Module 7 Sea State Analysis 6 Orson P. Smith, PE, Ph.D., Instructor 3

4 Energy in ocean waves Wind induced freesurface gravity waves: < T < 3 seconds > f >.3 Hz Note narrow peaks at diurnal and semidiurnal periods Engineering applications call for a spectrum of an individual sea state Module 7 Sea State Analysis 7 Fourier transformation a i Decompose a sea state into constituents Each with an amplitude (H/) and a frequency n (ft) time (seconds) Associate constituent energy with its corresponding frequency i a i a3 i a4 i a5 i Module 7 Sea State Analysis 8 Orson P. Smith, PE, Ph.D., Instructor 4

5 Fast Fourier Transformation FFT of 5 wave sum shows the original frequencies, FFT y value proportional to t 5 5 time series of sum of 5 waves FFT.8.6 t a i a i a3 i a4 i a5 i f (Hz) Module 7 Sea State Analysis 9 Sea State Spectra Wind energy wave energy correspondence Predict wave spectrum from wind conditions Distribution is continuous Steep on low frequency side Tapered on high frequency side Stronger winds Higher peak energy Lower peak frequency Longer peak period Module 7 Sea State Analysis Orson P. Smith, PE, Ph.D., Instructor 5

6 Empirical spectra 5 Phillis (958) E( f ) g f E(f) is energy (m or ft ) as a function of frequency, f Pierson Moskowitz (964) 4 5 E( f ) g f e JONSWAP (Joint North Sea Wave Project, 973) E( f ) g 4 f f f m 4 5 f m f ln e m 5 4 f = equilibrium range constant or Phillips constant f m = peak frequency = peak enhancement factor increases peak energy above Pierson Moskowitz = shape parameter = a for f < f m = b for f > f m e 4 g Uf Module 7 Sea State Analysis CEM wind wave prediction JONSWAP based: independent variables Wind speed: an average value, generally over an hour Not gust speeds for most coastal engineering purposes Adjusted to equivalent value at m above water Duration: measured resolution rarely less than an hour Values measured for shorter spans adjusted to hr equivalent Fetch: Distance over water across which the wind blows Straight line in dominant direction of wind Winds measured over land adjusted to over water equivalent Depth: Depths other than deep water constrain wave growth Coastal Engineering Manual, Part II, Chapter Module 7 Sea State Analysis Orson P. Smith, PE, Ph.D., Instructor 6

7 Wind speed adjustments. Adjust wind speed to equivalent speed at m U = wind speed adjusted to m U z = wind speed measured at height z (m). Adjust wind speed to hr average per graph See CEM II for more details Module 7 Sea State Analysis 3 Wind speed adjustments (continued) 3. Adjust for air sea temperature difference (C) If variable (usual case), assume zero See CEM II for more details Module 7 Sea State Analysis 4 Orson P. Smith, PE, Ph.D., Instructor 7

8 Wind speed adjustments (continued) 4. Adjust for effects of land drag Module 7 Sea State Analysis 5 Fetch limited sea state Is the sea state fetch limited?., , = time for waves over fetch X to become fetch limited If wind duration is at least,, the sea state is fetch limited If less, it s duration limited X = straight line fetch (distance over which the wind blows) H mo = zero moment wave height H sig u * = friction velocity = C D = drag coefficient U = adjusted m wind speed T p = period of peak wave energy Module 7 Sea State Analysis 6 Orson P. Smith, PE, Ph.D., Instructor 8

9 Duration limited sea state t < t x,u Find equivalent fetch, X t, for t (duration limited) 5.3 Use X n fetch limited equations to estimate H mo and T p Module 7 Sea State Analysis 7 Fully developed sea state No fetch or duration limitations Deep water Module 7 Sea State Analysis 8 Orson P. Smith, PE, Ph.D., Instructor 9

10 Whas H mo? A moment of a wave energy spectrum E(f): n = for the zero moment In general, the zero moment wave height 4 Quick estimate from water level data: Since m variance of a water level time series 4 = standard deviation of water level time series, relative to mean Module 7 Sea State Analysis 9 Directional spectra Wave energy spreads laterally outward from source Dominant direction prevails S(f,) = directional spectral density (m /Hz/deg) = S(f)G(f, ) = direction of wave propagation (CCW is positive, right hand rule) G(f,) = directional spreading function (non dimensional), relative magnitude of S(f) at each e.g., a ½ plane cosine distribution:.8 G( f, ) G( ) cos for G( ) for spreading function, G direction (degrees) Module 7 Sea State Analysis Orson P. Smith, PE, Ph.D., Instructor

11 Measured directional spectra The energy density spectrum in any direction is associated with the color code Dominant direction is from NW Module 7 Sea State Analysis Hindcasting wave conditions Reconstruct surface winds from isobaric charts Model generation and propagation of waves Useful for compilation of wave climate statistics Useful for forensic investigations of disastrous storms Alaska wave hindcast database at stations: western Alaska Review WIS documentation for procedure background Wind and wave statistics, plus individual event parameters Nome wave rose wind rose Module 7 Sea State Analysis Orson P. Smith, PE, Ph.D., Instructor

12 Extreme sea state statistics Use representative H s ; only extremes, not mild conditions Need multi year record; the longer, the better Cumulative probability function, F(x) Probability that x x (a value of interest) Extremal Type I (Gumbel): Fx () ( x ) e e Log normal: Weibull: Fx ( ) e x Fx ( ). e C x Module 7 Sea State Analysis 3 Extremal analysis. Segregate extremes from record; e.g., H s >.5 m = extremes per year; choose H s for = 3. n values in ascending order; k is place in order (, n) k 3. Estimate F by: F k n 4. Transform Extremal Type I by: F x ln( ln( ˆ ( ))) x This form is the equation of a line: y ax b (slope a, y intercept b) H s values = x; y ln( ln( Fˆ ( x))) Module 7 Sea State Analysis 4 Orson P. Smith, PE, Ph.D., Instructor

13 Extremal Analysis (continued) 5. Compute function parameters from data: Fx () ( x ) e e 6. Return period (years): T return ( Fx ( )) 7. Non encounter probability: NE( x) e T return probability for design life L, return period T return, largest condition x For L = T return, NE(x) =.37; i.e., 63% chance x will be exceeded L Module 7 Sea State Analysis 5 Extreme wave estimate, Kenai, AK Analysis by UAA grad student Heike Merkel sponsored by PND, Inc., and City of Kenai Module 7 Sea State Analysis 6 Orson P. Smith, PE, Ph.D., Instructor 3

14 Conclusion Module 7 Sea State Analysis 7 Orson P. Smith, PE, Ph.D., Instructor 4