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On the Wind Generation of Water Waves

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Abstract

In this work, we consider the mathematical theory of wind generated water waves. This entails determining the stability properties of the family of laminar flow solutions to the two-phase interface Euler equation. We present a rigorous derivation of the linearized evolution equations about an arbitrary steady solution, and, using this, we give a complete proof of the instability criterion of Miles [16]. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air–sea interface). We are thus able to give a unified equation connecting the Kelvin–Helmholtz and quasi-laminar models of wave generation.

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References

  1. Amick C.J., Turner R.E.L.: A global theory of internal solitary waves in two-fluid systems. Trans. Am. Math. Soc. 298, 431–484 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold V.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16, 319–361 (1966)

    Article  MATH  Google Scholar 

  3. Bates P., Lu K., Zeng C.: Approximately invariant manifolds and global dynamics of spike states. Invent. Math. 174, 355–433 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Beale J.T., Hou T.Y., Lowengrub J.S.: Growth rates for the linearized motion of fluid interfaces away from equilibrium. Commun. Pure Appl. Math. 46, 1269–1301 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenier Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Math. 52, 411–452 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caponi E.A., Caponi M.Z., Saffman P.G., Yuen H.C.: A simple model for the effect of water shear on the generation of waves by wind. Proc. R. Soc. Lond. Ser. A 438, 95–101 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Cheng C.-H.A., Coutand D., Shkoller S.: On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity. Commun. Pure Appl. Math. 61, 1715–1752 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chow S.-N., Lin X.-B., Lu K.: Smooth invariant foliations in infinite-dimensional spaces. J. Differ. Equ. 94, 266–291 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Chow S.-N., Lu K.: Invariant manifolds for flows in Banach spaces. J. Differ. Equ. 74, 285–317 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2nd edn., 2004. With a foreword by John Miles

  11. Ebin D.G., Marsden J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. (2(92), 102–163 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hristov T., Miller S., Friehe C.: Dynamical coupling of wind and ocean waves through wave-induced air flow.. Nature 422, 55–58 (2003)

    Article  ADS  Google Scholar 

  13. Hur V.M., Lin Z.: Unstable surface waves in running water. Commun. Math. Phys. 282, 733–796 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. James G.: Internal travelling waves in the limit of a discontinuously stratified fluid. Arch. Ration. Mech. Anal. 160, 41–90 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Janssen P.: The Interaction of Ocean Waves and Wind. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  16. Miles J.W.: On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204 (1957)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Miles J.W.: On the generation of surface waves by shear flows. II. J. Fluid Mech. 6, 568–582 (1959)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Miles J.W.: On the generation of surface waves by shear flows. III. Kelvin-Helmholtz instability. J. Fluid Mech. 6, 583–598 (1959) (1 plate)

    Article  MATH  Google Scholar 

  19. Morland L., Saffman P.: Effect of wind profile on the instability of wind blowing over water. J. Fluid Mech. 252, 383–398 (1993)

    Article  ADS  MATH  Google Scholar 

  20. Plant, W.J.: A relationship between wind stress and wave slope. J. Geophys. Res. Oceans (1978–2012) 87, 1961–1967 (1982)

  21. Shatah J., Zeng C.: A priori estimates for fluid interface problems. Commun. Pure Appl. Math. 61, 848–876 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shatah J., Zeng C.: Local well-posedness for fluid interface problems. Arch. Ration. Mech. Anal. 199, 653–705 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shnirelman, A.I.: The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid. Math. Sb. (N.S.) 128(170), 82–109, 144 (1985)

  24. Temam R.: Navier–Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam, 1977. Studies in Mathematics and its Applications, Vol. 2

  25. Thomson W.L.K.: Hydrokinetic solutions and observations. Philosophical Magazine Series 4, 362–377 (1871)

    Google Scholar 

  26. Walsh S., Bühler O., Shatah J.: Steady water waves in the presence of wind. to appear, SIAM J. Math. Anal (2013)

  27. Wasow W.: Asymptotic expansions for ordinary differential equations, Dover Publications, Inc., New York 1987

  28. Wheless G., Csanady G.: Instability waves on the air–sea interface. J. Fluid Mech. 248, 363–381 (1993)

    Article  ADS  MATH  Google Scholar 

  29. Young W.R., Wolfe C.L.: Generation of surface waves by shear-flow instability. J. Fluid Mech. 739, 276–307 (2014)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Oliver Bühler & Jalal Shatah

  2. Department of Mathematics, University of Missouri, Math. Sciences Building, Columbia, MO, 65211, USA

    Samuel Walsh

  3. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA, 30332, USA

    Chongchun Zeng

Authors
  1. Oliver Bühler
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  2. Jalal Shatah
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  3. Samuel Walsh
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  4. Chongchun Zeng
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Corresponding author

Correspondence to Samuel Walsh.

Additional information

Communicated by V. Šverák

O. Bühler: Work supported in part by NSF-DMS 1312159.

J. Shatah: Work supported in part by NSF-DMS 1363013.

C. Zeng: Work supported in part by NSF-DMS 1362507.

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Bühler, O., Shatah, J., Walsh, S. et al. On the Wind Generation of Water Waves. Arch Rational Mech Anal 222, 827–878 (2016). https://doi.org/10.1007/s00205-016-1012-0

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  • DOI: https://doi.org/10.1007/s00205-016-1012-0

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