Stability analysis from fourth order evolution equation for counter-propagating gravity wave packets in the presence of wind flowing over water

Fourth order nonlinear evolution equations are derived for two counter-propagating surface gravity wave packets in deep water in the presence of wind flowing over water. The resulting equations are asymptotically exact and nonlocal. Stability analysis is made for a uniform standing surface gravity wave train for longitudinal perturbation on the basis of these equations. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness for some different values of dimensionless wind velocity. Significant deviations are noticed between the results obtained from third order and fourth order nonlinear evolution equations. This paper has an application in rough waves. References T. Brooke Benjamin and J. E. Feir. The disintegration of wave trains on deep water part 1. theory. Journal of Fluid Mechanics, 27:417–430, 1967. doi:10.1017/S002211206700045X Sudebi Bhattacharyya and K. P. Das. Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline. The ANZIAM Journal, 39:214–229, 1997. doi:10.1017/S033427000000881X K. P. Das. On evolution equations for a three-dimensional surface gravity wave packet in a two-layer fluid. Wave Motion, 8(2):191–204, 1986. doi:10.1016/0165-2125(86)90024-7 A. Davey and K. Stewartson. On three-dimensional packets of surface waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 338(1613):101–110, 1974. doi:10.1098/rspa.1974.0076 Suma Debsarma and K. P. Das. Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water. The ANZIAM Journal, 43:513–524, 2002. doi:10.1017/S1446181100012116 A. K. Dhar and K. P. Das. A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water. Physics of Fluids A: Fluid Dynamics, 2(5), 1990. A. K. Dhar and K. P. Das. Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water. Physics of Fluids A: Fluid Dynamics, 3(12), 1991. A. K. Dhar and K. P. Das. Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear. The ANZIAM Journal, 35:348–365, 1994. doi:10.1017/S0334270000009346 V. D. Djordjevic and L. G. Redekopp. On two-dimensional packets of capillary-gravity waves. Journal of Fluid Mechanics, 79:703–714, 1977. doi:10.1017/S0022112077000408 J. C. Dungey and W. H. Hui. Nonlinear energy transfer in a narrow gravity-wave spectrum. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368(1733):239–265, 1979. doi:10.1098/rspa.1979.0126 K. B. Dysthe. Note on a modification to the nonlinear schrodinger equation for application to deep water waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 369(1736):105–114, 1979. doi:10.1098/rspa.1979.0154 Tetsu Hara and Chiang C. Mei. Frequency downshift in narrow banded surface waves under the influence of wind. Journal of Fluid Mechanics, 230:429–477, 1991. doi:10.1017/S002211209100085X Tetsu Hara and Chiang C. Mei. Wind effects on the nonlinear evolution of slowly varying gravity-capillary waves. Journal of Fluid Mechanics, 267:221–250, 1994. doi:10.1017/S0022112094001163 S. J. Hogan. The fourth-order evolution equation for deep-water gravity-capillary waves. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 402(1823):359–372, 1985. doi:10.1098/rspa.1985.0122 Peter A. E. M. Janssen. On a fourth-order envelope equation for deep-water waves. Journal of Fluid Mechanics, 126:1–11, 1983. doi:10.1017/S0022112083000014 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water. I. superharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 360(1703):471–488, 1978. doi:10.1098/rspa.1978.0080 M. S. Longuet-Higgins. The instabilities of gravity waves of finite amplitude in deep water II. subharmonics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 360(1703):489–505, 1978. doi:10.1098/rspa.1978.0081 R. D. Pierce and E. Knobloch. On the modulational stability of traveling and standing water waves. Physics of Fluids, 6(3), 1994. Michael Stiassnie. Note on the modified nonlinear Schrodinger equation for deep water waves. Wave Motion, 6(4):431–433, 1984. doi:10.1016/0165-2125(84)90043-X M. A. Weissman. Nonlinear wave packets in the Kelvin–Helmholtz instability. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 290(1377):639–681, 1979. doi:10.1098/rsta.1979.0019 V. E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics, 9(2):190–194, 1968. doi:10.1007/BF0091318