NUMERICAL AND THEORETICAL STUDIES

Transcription

1 SELF-SIMILARITY OF WIND-WAVE SPECTRA. NUMERICAL AND THEORETICAL STUDIES Sergei I. Baduli (1), Adrei N. Pushkarev (,4), Doald Resio (3), Vladimir E. Zakharov (4,5) (1) P.P.Shirshov Istitute of Oceaology Moscow, Russia, () Waves ad Solitos, LLC, USA, (3) Waterways Exerimetal Statio, USA, (4) Ladau Istitute for Theoretical Physics, Moscow, Russia, (5) Uiversity of Arizoa, USA Suorted by U.S. Army Cors of Egieers, RDT&E rogram, DACA 4-00-C0044, ONR N , INTAS 01-34, Russia Foudatio for Basic Research N ,

2 This is the oly slide where terms Rogue Waves, Freak Waves etc. aear

3 To study extreme, abormal waves oe has to kow what `a ormal state is Self-similarity is imlied i exerimetal arameterizatios of wid-wave sectra; Self-similarity is a iheret feature of `ormal wid-wave field

4 JONSWAP sectrum = E() κ β α α = g U h 0 ) ( ex ex σ γ 5 g α U h C «iteral» arameter to describe sectral form «exteral» - wave age - effect of exteral forcig No exlicit deedece o duratio (t) or fetch (x) E tot ~t (x ); mea ~t -q (x -q )

5 The Hasselma equatio ) ( ) ( ) ( k k k k k k k d d d T S l = δ δ π = dt d k S diss l S iut + S, iut S - emirical arameterizatios S diss +

6 What arameterizatio of wave iut is true? Youg waves U/C = Old waves U/C =0.9

7 Our key oit Noliearity domiates! S l >> S iut, S diss There is o characteristic scale (dee water waves), i.e. homogeeity of the collisio itegral gives S l ~ 3 k 19/4

8 Primitive comariso of terms i KE S l vs S diss +S i ; t=1.5 hours

9 Look for aroximate solutios i a self-similar form (quite similarly for fetch-limited case) = at α U β ( bkt β, t); After substitutio U0 t β l ξ = bkt [ U ] β(ξ β ξ ξu β αu β = S ) β ; 19β a = b 19/4 ; α = 4 S + + i at α if α > 1 the effect of S i ad S diss vaishes S diss 1 at t

10 Determie arameters α, β from the higherorder aroximatio N ~ t U( kt ) dk ~ r = α α β β r t r is a exoet of wave actio growth ad is the oly arameter of the family of S-S solutios r = 0 swell (o exteral iut) r = 1 costat wave actio iut r = 4/3 costat wave eergy iut

11 Algebra <-> Physics We slit wid-wave balace ito two arts d k dt = S l A form of the selfsimilar solutio d k dt = S i + S diss Itegral balace for the wid-drive waves Does this model work?

12 Kolmogorov s cascades (Zakharov, PhD thesis 1966) E (1) (, θ ) = C g P 4 4/3 1/3 Direct cascade (Zakharov & Filoeko 1967) E () (, θ ) = C q g 4/3 Q 11/3 1/3 Iverse cascade (Zakharov & Zaslavskii 1983)

13 Does umerical solutios satisfy self-similar scalig? Swell /11 = t U ( t 0 1/11, Θ)

14 Academic umig N tot ~t r (just to obtai self-similarity i the `ure state) Wid-wave icremets deed o time!!! For sea waves r 1 I our rus 1/3 < r < 4/3

15 Comare form fuctios U(ξ) for differet growth exoets r (`academic ) ( = at α U r ( b t β ), θ ) Swell (r=0) Numerical solutios kee self-similar scalig a~b 19/4 Form fuctios U(ξ) does ot deed o r?

16 Scalig of frequecy sectra (directio-averaged) for differet wave growth rates r ( k =at α U r (bkt b )) Swell JONSWAP Frequecy sectra kee self-similar scalig a~b 19/4 E()/E does ot deed o r?

17 Waves uder wave umig: No room for iertial iterval o room for self-similarity?

18 Form fuctios U(ξ) for waves uder wid umig ξ= t β, =t α U(ξ,Θ) Θ=0 ο Θ=30 ο Θ=60 ο Θ=90 ο Θ=150 ο Θ=180 ο

19 Dow-wid (left) ad frequecy (right) sectra for `real wave umig Swell Swell JONSWAP

20 Exerimetal sectra have self-similar forms!!! JONSWAP sectrum Self-similar solutios ca be exressed i terms of mea (eak?) frequecy ad mea (eak?) eergy quite similar to the exerimetal forms 5 ), ( g E Θ,Θ 0 R U κ α C U 10 π α 0 = ), ( ) ( ), ( 5 Θ = Θ r b U gu a g E κ α

21 Exoets of wid-wave growth How to comare correctly umerical solutios ad exerimetal arameterizatios? There is a self-similar `core ad o-self-similar backgroud E tot ~t E eak ~ t or mea ~ t q E eak ~ t

22 Exoets of eergy growth ad frequecy dowshift q for mea (left) ad eak (right) values. Toba Toba S-S solutios S-S solutios `Academic series Aisotroic geeratio Isotroic geeratio `Realistic iuts Wave iut Swell

23 Numerical frequecy sectra for youg (left, U/C =1.3) ad old waves (right, U/C=0.87). JONSWAP sectra (dashed) use stadard set of arameters. Characteristics of eak growth are give for the comariso. Wave iut Doela & Pierso-jr. (1987)

24 Summary There is a strog tedecy of wid-wave sectra to self-similar behaviour i a wide rage of wid-wave coditios; The forms of the sectra are close to uiversal that is cosistet with exerimetal arameterizatios of wid-wave sectra Self-similar `core of wid-wave sectra coexists with o-self-similar backgroud. Peak arameters are more adequate to sectra descritio