# On ST6 Source Terms Model Assessment and Alternative

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Models

#### 2.2. Theoretical Background: Self-Similar Solutions

#### 2.3. Theoretical Background: Nonlinear Interactions

#### 2.4. ST6 Model Numerical Simulation

^{2}is in a good agreement with the target dependence KC1992 for the 5–40 km fetch span as well as the numerical results of [15]. The situation is different, however, for other curves corresponding to lower shore total wave energy levels $2.2\xb7{10}^{-5}$ m

^{2}and $1.2\xb7{10}^{-7}$ m

^{2}, which exhibit noticeable deviation from the KC1992 target dependence. Additionally, in the simplified model Equation (2), we were not able to reach the “steady sea state” observed in the numerical experiments by [15] for the 200–600 km fetch range. One more difference (idealization) with WW3 and other wave models is disactivating the linear input term, which is important at the initial stage of wave development. The effect of this is seen as a difference between PGZ-2 and other models for short dimensionless fetches. We plan to perform more detailed tuning of parts of our model to minimize the effect of numerical problems [37].

## 3. Results

#### 3.1. Alternative PGZ-2 Model: Wind Energy Input and Wave Dissipation Source Functions

#### 3.2. Alternative PGZ-2 Model: Tuning and Numerical Results

- Fourier space domain ${f}_{low}<f<{f}_{high}$, resolved by 72 logarithmically positioned discrete frequencies with ${f}_{low}=0.1$ Hz and ${f}_{high}=2.0$ Hz;
- Angular domain $-\pi <\theta \le \pi $ with 36 discrete angular directions, given the wind angle ${\theta}_{wind}=0$;
- Wind velocity ${U}_{10}=10$ m/s at a 10 m height above the ocean surface

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Hasselmann, K. On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech.
**1962**, 12, 481–500. [Google Scholar] [CrossRef] - Hasselmann, K. On the non-linear energy transfer in a gravity wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irrevesibility. J. Fluid Mech.
**1963**, 15, 273–281. [Google Scholar] [CrossRef] - Komen, G.J.; Cavaleri, L.; Donelan, M.; Hasselmann, K.; Hasselmann, S.; Janssen, P.A.E. Dynamics and Modeling of Ocean Waves; Cambridge University Press: Cambridge, MA, USA, 1994. [Google Scholar]
- The WAVEWATCH III Development Group (WW3DG). User Manual and System Documentation of WAVEWATCH III Version 6.07; Tech. Note 333; NOAA/NWS/NCEP/MMAB: College Park, MD, USA, 2019; 465p+Appendices. [Google Scholar]
- Young, I.R.; Banner, M.L.; Donelan, M.A.; McCormick, C.; Babanin, A.V.; Melville, W.K.; Veron, F. An Integrated System for the Study of Wind-Wave Source Terms in Finite-Depth Water. J. Atmos. Ocean. Technol.
**2005**, 22, 814–831. [Google Scholar] [CrossRef] - Donelan, M.A.; Babanin, A.V.; Young, I.R.; Banner, M.L.; McCormick, C. Wave-Follower Field Measurements of the Wind-Input Spectral Function. Part I: Measurements and Calibrations. J. Atmos. Ocean. Technol.
**2005**, 22, 799–813. [Google Scholar] [CrossRef] - Pushkarev, A.; Zakharov, V. Limited fetch revisited: Comparison of wind input terms, in surface wave modeling. Ocean Model.
**2016**, 103, 18–37. [Google Scholar] [CrossRef][Green Version] - Zakharov, V.; Resio, D.; Pushkarev, A. Balanced source terms for wave generation within the Hasselmann equation. Nonlinear Process. Geophys.
**2017**, 24, 581–597. [Google Scholar] [CrossRef][Green Version] - Zakharov, V.E. Theoretical interpretation of fetch limited wind-drivensea observations. Nonlinear Process. Geophys.
**2005**, 12, 1011–1020. [Google Scholar] [CrossRef] - Badulin, S.I.; Pushkarev, A.N.; Resio, D.; Zakharov, V.E. Direct and inverse cascade of energy, momentum and wave action in wind-driven sea. In Proceedings of the 7th International Workshop on Wave Hindcasting and Forecasting, Banff, AB, Canada, 21–25 October 2002; pp. 92–103. [Google Scholar]
- Badulin, S.I.; Pushkarev, A.N.; Resio, D.; Zakharov, V.E. Self-similarity of wind-driven seas. Nonlinear Process. Geophys.
**2005**, 12, 891–946. [Google Scholar] [CrossRef][Green Version] - Badulin, S.I.; Babanin, A.V.; Zakharov, V.E.; Resio, D. Weakly turbulent laws of wind-wave growth. J. Fluid Mech.
**2007**, 591, 339–378. [Google Scholar] [CrossRef] - Zakharov, V.E.; Badulin, S.I.; Geogjaev, V.V.; Pushkarev, A.N. Weak-Turbulent Theory of Wind-Driven Sea. Earth Space Sci.
**2019**, 6, 540–556. [Google Scholar] [CrossRef] - Badulin, S.I.; Babanin, A.V.; Resio, D.; Zakharov, V. Numerical verification of weakly turbulent law of wind wave growth. In Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence; Springer: Berlin/Heidelberg, Germany, 2008; pp. 45–47. [Google Scholar]
- Liu, Q.; Rogers, W.E.; Babanin, A.V.; Young, I.R.; Romero, L.; Zieger, S.; Qiao, F.; Guan, C. Observation-Based Source Terms in the Third-Generation Wave Model WAVEWATCH III: Updates and Verification. J. Phys. Oceanogr.
**2019**, 49, 489–517. [Google Scholar] [CrossRef] - Gagnaire-Renou, E.; Benoit, M.; Badulin, S.I. On weakly turbulent scaling of wind sea in simulations of fetch-limited growth. J. Fluid Mech.
**2011**, 669, 178–213. [Google Scholar] [CrossRef] - Annenkov, S.Y.; Shrira, V.I. Effects of finite non-Gaussianity on evolution of a random wind wave field. Phys. Rev. E
**2022**, 106, L042102. [Google Scholar] [CrossRef] [PubMed] - Alves, J.H.G.M.; Banner, M.L. Performance of a saturation-based dissipation-rate source term in modelling the fetch-limited evolution of wind waves. J. Phys. Oceanogr.
**2003**, 33, 1274–1298. [Google Scholar] [CrossRef] - Tracy, B.; Resio, D. Theory and Calculation of the Nonlinear Energy Transfer between Sea Waves in Deep Water; WES Report 11; U.S. Army Engineer Waterways Experiment Station: Vicksburg, MS, USA, 1982. [Google Scholar]
- Webb, D.J. Non-linear transfers between sea waves. Deep-Sea Res.
**1978**, 25, 279–298. [Google Scholar] [CrossRef] - Pushkarev, A. Comparison of different models for wave generation of the Hasselmann equation. Procedia IUTAM
**2018**, 26, 132–144. [Google Scholar] [CrossRef] - Pushkarev, A.N.; Zakharov, V.E. Nonlinear amplification of ocean waves in straits. Theor. Math. Phys.
**2020**, 203, 535–546. [Google Scholar] [CrossRef] - Pushkarev, A. Laser-like wave amplification in straits. Ocean Dyn.
**2021**, 71, 195–215. [Google Scholar] [CrossRef] - Gagnaire-Renou, E. Amelioration de la Modelisation Spectrale des Etats de mer par un Calcul Quasi-Exact des Interactions Non-Lineaires Vague-Vague. Thèse pour L’obtention du Grade de Docteur, Université du Sud Toulon Var, Ecole Doctorale Sciences Fondamentales et Appliquées. 2009. Available online: https://theses.hal.science/tel-00595353/document (accessed on 24 February 2023).
- Hwang, P.; Wang, D.; Rogers, W.; Swift, R.; Yungel, J.; Krabill, A. Bimodal directional propagation of wind-generated ocean surface waves. arXiv
**2001**, arXiv:1906.07984. [Google Scholar] [CrossRef] - Hwang, P.A.; Wang, D.W.; Yungel, J.; Swift, R.N.; Krabill, W.B. Do wind-generated waves under steady forcing propagate primarily in the downwind direction? arXiv
**2019**, arXiv:1907.01532v1. [Google Scholar] - Hwang, P. Retondo080102Bimodal. 2002, Web Publication. Available online: https://www.researchgate.net/profile/Paul-Hwang-2/publication/275041443_Retondo080102Bimodal/links/55311a320cf20ea0a070df83/Retondo080102Bimodal.pdf (accessed on 24 February 2023).
- Hwang, P.A.; Wang, D.W.; Rogers, W.E.; Kaihatu, J.M. A Discussion on the Directional Distribution of Wind-Generated Ocean Waves. In Proceedings of the 6th International Workshop on Wave Hindcasting and Forecasting, Monterey, CA, USA, 6–10 November 2000; Resio, D.T., Ed.; Meteorological Service of Canada Environment Canada, Enviroment Canada: North York, ON, Canada, 2000; pp. 273–279. [Google Scholar]
- Simanesew, A.W.; Krogstad, H.E.; Trulsen, K.; Nieto Borge, J.C. Bimodality of Directional Distributions in Ocean Wave Spectra: A Comparison of Data-Adaptive Estimation Techniques. J. Atmos. Ocean. Technol.
**2018**, 35, 365–384. [Google Scholar] [CrossRef] - Cavaleri, L.; Abdalla, S.; Benetazzo, A.; Bertottia, L.; Bidlot, J.R.; Breivik, O.; Carniela, S.; Jensen, R.E.; Portilla-Yandune, J.; Rogers, W.E.; et al. Wave modelling in coastal and inner seas. Prog. Oceanogr.
**2018**, 167, 164–233. [Google Scholar] [CrossRef] - Long, C.; Resio, D. Wind wave spectral observations in Currituck Sound, North Carolina. JGR
**2007**, 112, C05001. [Google Scholar] [CrossRef][Green Version] - Zakharov, V.E.; Resio, D.; Pushkarev, A. New wind input term consistent with experimental, theoretical and numerical considerations. arXiv
**2012**, arXiv:1212.1069. [Google Scholar] - Hasselmann, S.; Hasselmann, K.; Allender, J.H.; Barnett, T.P. Computations and Parameterizations of the Nonlinear Energy Transfer in a Gravity-Wave Specturm. Part II: Parameterizations of the Nonlinear Energy Transfer for Application in Wave Models. J. Phys. Oceanogr.
**1985**, 15, 1378–1391. [Google Scholar] [CrossRef] - Zakharov, V.E.; Filonenko, N.N. The energy spectrum for stochastic oscillations of a fluid surface. Sov. Phys. Docl.
**1967**, 11, 881–884. [Google Scholar] - Phillips, O. Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech.
**1985**, 156, 505–531. [Google Scholar] [CrossRef] - Kahma, K.K.; Calkoen, C.J. Reconciling Discrepancies in the Observed Growth of Wind-generated Waves. J. Phys. Oceanogr.
**1992**, 22, 1389–1405. [Google Scholar] [CrossRef] - Badulin, S.; Zakharov, V. The Phillips spectrum and a model of wind-wave dissipation. Theor. Math. Phys.
**2020**, 202, 353–363. [Google Scholar] [CrossRef][Green Version] - Phillips, O.M. The Dynamics of the Upper Ocean; Cambrige University Press: Cambrige, MA, USA, 1966. [Google Scholar]
- Lenain, L.; Pizzo, N. The contribution of high-frequency wind-generated surface waves to the Stokes drift. J. Phys. Oceanogr.
**2020**, 50, 3455–3465. [Google Scholar] [CrossRef]

**Figure 1.**Limited fetch problem geometry. Constant wind of 10 m/s is blowing orthogonally to the fetch.

**Figure 6.**Spectral wave energy density $\epsilon $ as the function of the frequency $\omega $ and angle $\theta $ for ST6 case.

**Figure 7.**Decimal logarithm of the angle averaged wave energy spectral density $\frac{1}{2\pi}{\int}_{0}^{2\pi}\epsilon (\omega ,\theta )d\theta $ as the function of the decimal logarithm of frequency f for ST6 case. Dashed line—the KZ spectral fit $\phantom{\rule{-0.166667em}{0ex}}\sim \phantom{\rule{-0.166667em}{0ex}}{\omega}^{-4}$; dashed/dotted line—the Phillips spectral fit $\phantom{\rule{-0.166667em}{0ex}}\sim \phantom{\rule{-0.166667em}{0ex}}{\omega}^{-5}$.

**Figure 8.**Spectral wave energy density $\epsilon $ as the function of the frequency $\omega $ and angle $\theta $ for PGZ-2 case.

**Figure 9.**Decimal logarithm of the angle-averaged wave energy spectral density $\frac{1}{2\pi}{\int}_{0}^{2\pi}\epsilon (\omega ,\theta )d\theta $ as the function of the decimal logarithm of frequency f for PGZ-2 case. Dashed line—the KZ spectral fit $\sim \phantom{\rule{-0.166667em}{0ex}}{\omega}^{-4}$; dashed/dotted line—the Phillips spectral fit $\sim \phantom{\rule{-0.166667em}{0ex}}{\omega}^{-5}$.

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**MDPI and ACS Style**

Pushkarev, A.; Geogjaev, V.; Zakharov, V.
On ST6 Source Terms Model Assessment and Alternative. *Water* **2023**, *15*, 1521.
https://doi.org/10.3390/w15081521

**AMA Style**

Pushkarev A, Geogjaev V, Zakharov V.
On ST6 Source Terms Model Assessment and Alternative. *Water*. 2023; 15(8):1521.
https://doi.org/10.3390/w15081521

**Chicago/Turabian Style**

Pushkarev, Andrei, Vladimir Geogjaev, and Vladimir Zakharov.
2023. "On ST6 Source Terms Model Assessment and Alternative" *Water* 15, no. 8: 1521.
https://doi.org/10.3390/w15081521