On another concept of Hasselmann equation source terms

Transcription

1 On another concept of Hasselmann equation source terms An exploration of tuning - free models Vladimir Zakharov, Donal Resio, Andrei Pushkarev Waves and Solitons LLC University of Arizona University of North Florida Novosibirsk State University Lebedev Physical Institute RAS

2 Klaus Hasselmann (1962) ε + ω k ε =S +S +S nl in diss t k r 1/ 3 P 0+0=Snl +0+0 ε=c K 4 ω Snl derived from free surface Euler equations Sin multiple versions, differences up to 500% Sdiss multiple LF and HF versions Detailed discussion in Pushkarev, Zakharov 2016

3 Motivation : Build Sin properties requiring consistent of with mathematical Hasselmann equation and minimal tuning of the model

4 Field Experiments Theory Numerics 1/ 3 P S nl =0 ε=c K 4 ω 4 ε ω ε χ p, ω χ 0.74< p<1 0.2<q<0.3 q Zakharov, Filonenko 1968 ε χ p, ω χ q p=1, q=0.3 4 ε ω ε χ p, ω χ q p 1, q 0.3 Pushkarev, Resio, Zakharov 2003 Badulin, Babanin, Zakharov 2005 Badulin, Pushkarev, Resio, Zakharov 2008 Zakharov, Resio, Pushkarev Resio, Zakharov

5 Experiment Black Sea (Babanin & Soloviev 1998b) Walsh et al. (1989) US coast Kahma & Calkoen (1992) unstable Kahma & Calkoen (1992) stable Kahma & Pettersson (1994) JONSWAP by Davidan (1980) JONSWAP by Phillips (1977) Kahma & Calkoen (1992) composite Kahma (1981, 1986) rapid growth Kahma (1986) average growth Donelan et al. (1992) St Claire Ross (1978), Atlantic, stable Liu & Ross (1980), Michigan, unstable JONSWAP (Hasselmann et al. 1973) Mitsuyasu et al. (1971) ZRP numerics p q Exponents of wind-wave growth in fetch-limited experiments. Adapted from Badulin, Babanin, Zakharov, Resio 2007

6 Theoretical approach

7 ωk ε =S nl +S in k r Fetch limited case: ε =Snl +Sin Duration limited case: t Sin ε ω s+1 Existence of self-similar solutions is no guarantee of their realization! Example wave collapse in NLS

8 Duration Limited Case ε=t p+q ε=ε0 t p q F (ω t ) ω =ω0 t Fetch Limited Case ε=χ q p+q ε=ε0 χ q G (ω χ ) p q ω =ω0 χ 9 q 2 p=1 10 q 2 p=1 s=s( p, q) s=s( p, q)

9 Experimental approach

10 Resio-Long regression line 1/3 P 5/2 ε=c K 4 F (k )=β k ω 1 2 1/3 1/2 β= α4 [(uλ c p ) u0 ] g 2

11 Duration Limited Case p=10/7 q=3 /7 s=4 /3 Fetch Limited Case p=1 q=3 /10 s=4 /3

12 ZRP wind input term: 4/3 ρair ω Sin (ω, θ)= A ρ ω ( ω ) f (θ) ε(ω, θ) 0 water 2 cos (θ), for π/ 4<θ<π /4 f (θ)= 0, otherwise g ω0 = U 10 { }

13 Numerical approach

14 The model still misses 2 features: - the coefficient in front of ZRP - dissipation function S diss S in

15 f 5 implicit dissipation for f 1.1 Hz Low-pass filter

16 WAM3 dissipation

17 Duration limited case Wind speed 10 m/sec

18 Dimensionless energy versus dimensionless solid line. Self-similar solution - dashed line.

19 Total energy index as the function of dimensionless time - solid line. Self-similar index p = 10/7 - dashed line.

20 Dimensionless frequency versus dimensionless - solid line, self-similar solution - dashed line.

21 Mean frequency index versus dimensionless time - solid line. Self-similar index q = 3/7 - dashed line

22 "Magic number" 9q 2p versus dimensionless time solid line. Self-similar target - dashed line.

23 Decimal logarithm of the angle averaged spectrum versus decimal 4 logarithm of the frequency - solid line. Spectrum f - dashed line, 5 spectrum f - dash-dotted line.

24 Compensated spectrum versus frequency f.

25 Angle averaged wind input function (dotted line) and angle averaged spectrum (solid line) versus frequency f.

26 Experimental (dotted line), theoretical (dashed line) and numerical (diamonds) 1000β versus specific velocity for wind speed 10 m/sec.

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28 Limited fetch case Wind speed 5 and 10 m/sec

29 Total energy versus fetch: wind speed 10 m/sec - solid line, 5 m/sec - dash-dotted line. Self-similar solution - dashed line

30 Local energy index versus fetch.

31 Mean frequency versus the fetch for wind speed 10 m/sec (solid line) and 5 m/sec (dashed line). Self-similar dependence - dash-dotted line.

32 Local mean frequency exponent q = dln<ω> dlnx as the function of dimensionless fetch xg/u2 for fetch limited case. Wind speed 10 m/sec - solid line, wind speed 5 m/sec - dashed line. Horizontal dashed line - target value of the self-similar exponent -q = -0.3.

33 Magic number" 10q 2p versus the fetch. Wind speed 10 m/sec - solid line, wind speed 5 m/sec - dash-dotted line. Self-similar target 1 dashed line.

34 Decimal logarithm of the angle averaged spectrum versus decimal 4 logarithm of the frequency - solid line. Spectrum f - dashed line, 5 spectrum f - dash-dotted line.

35 Angle averaged wind input function (dotted line) and angle averaged spectrum (solid line) versus frequency f.

36 Compensated spectrum versus frequency f.

37 Experimental (dotted line), theoretical (dashed line) and numerical (diamonds) 1000β versus specific velocity for wind speed 10 m/sec.

38 Energy versus fetch, adapted from Young 1999

39 Frequency versus fetch, adapted from Young 1999

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41

42 CONCLUSIONS:

43 1. New set of source terms: XNL self-similarity experiments implicit dissipation 2. Conceptual difference with WAM3: HF vs spectral peak dissipation no wind input and dissipation overlapping 3. Physically based model: reproduction of HE theoretical properties reproduction of field experiments 4. No tuning for wind speed change

44 What we call long-wave, or spectral peak dissipation?

45 Is our simulation trustworthy?

46 Komen, S. Hasselmann, K. Hasselmann JPO (1984)