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A numerical study on matching relationships of gravity waves in nonlinear interactions

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Abstract

Applying a fully nonlinear numerical scheme with second-order temporal and spatial precision, nonlinear interactions of gravity waves are simulated and the matching relationships of the wavelengths and frequencies of the interacting waves are discussed. In resonant interactions, the wavelengths of the excited wave are in good agreement with the values derived from sum or difference resonant conditions, and the frequencies of the three waves also satisfy the matching condition. Since the interacting waves obey the resonant conditions, resonant interactions have a reversible feature that for a resonant wave triad, any two waves are selected to be the initial perturbations, and the third wave can then be excited through sum or difference resonant interaction. The numerical results for nonresonant triads show that in nonresonant interactions, the wave vectors tend to approximately match in a single direction, generally in the horizontal direction. The frequency of the excited wave is close to the matching value, and the degree of mismatching of frequencies may depend on the combined effect of both the wavenumber and frequency mismatches that should benefit energy exchange to the greatest extent. The matching and mismatching relationships in nonresonant interactions differ from the results of weak interaction theory that the wave vectors are required to satisfy the resonant matching condition but the frequencies are permitted to mismatch and oscillate with amplitude of half the mismatching frequency. Nonresonant excitation has an irreversible characteristic, which is different from what is found for the resonant interaction. For specified initial primary and secondary waves, it is difficult to predict the values of the mismatching wavenumber and frequency for the excited wave owing to the complexity.

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References

  1. Lindzen R S. Turbulence and stress owing gravity wave and tidal breakdown. J Geophys Res, 1981, 86: 9707–9714

    Article  Google Scholar 

  2. Holton J R. The role of gravity wave induced drag and diffusion in the momentum budget of the mesosphere. J Atmos Sci, 1982, 39: 791–799

    Article  Google Scholar 

  3. Xu J, Smith A K, Collins R L, et al. Signature of an overturning gravity wave in the mesospheric sodium layer: Comparison of a nonlinear photochemical-dynamical model and lidar observations. J Geophys Res, 2006, 111, doi: 10.1029/2005JD006749

  4. Hu X, Zhang X, Zhang D. Effects of atmospheric gravity waves on the mesosphere and lower thermosphere circulations. Chin J Space Sci, 2005, 25: 111–117

    Google Scholar 

  5. Fovell R, Durran D R, Holton J R. Numerical simulations of convectively generated stratospheric gravity waves. J Atmos Sci, 1992, 49: 1427–1442

    Article  Google Scholar 

  6. Alexander M J, Holton J R, Durran D R. The gravity wave response above deep convection in a squall line simulation. J Atmos Sci, 1995, 52: 22299–22309

    Article  Google Scholar 

  7. Yue J, Vadas S L, She C Y, et al. Concentric gravity waves in the mesosphere generated by deep convective plumes in the lower atmosphere near Fort Collins, Colorado. J Geophys Res, 2009, 114, doi: 10.1029/2008JD011244

  8. Vadas S L, Fritts D C, Alexander M J. Mechanism for the generation of secondary waves in wave breaking regions. J Atmos Sci, 2003, 60: 194–214

    Article  Google Scholar 

  9. Vadas S L, Liu H L. Generation of large-scale gravity wave and neutral winds in the thermosphere from the dissipation of convectively generated gravity waves. J Geophys Res, 2009, 114, doi: 10.1029/2009JA014108

  10. Fritts D C, Alexander M J. Gravity wave dynamics and effects in the middle atmosphere. Rev Geophys, 2003, 41, doi: 10.1029/2001RG000106

  11. Vincent R A, Alexander M J. Gravity waves in the tropical and lower stratosphere: An observational study of seasonal and interannual variability. J Geophys Res, 2000, 105: 17971–17982

    Article  Google Scholar 

  12. Bian J, Chen H, Lu D. Statistics of gravity waves in the lower stratosphere over Beijing based on high vertical resolution radiosonde. Sci China SerD-Earth Sci, 2005, 48: 1548–1558

    Article  Google Scholar 

  13. Alexander M J, Teitelbaum H. Observation and analysis of a large amplitude mountain wave event over the Antarctic peninsula. J Geophys Res, 2007, 112, doi: 10.1029/2006JD008368

  14. Zhang S D, Yi F. Latitudinal and seasonal variations of inertial gravity wave activity in the lower atmosphere over central China. J Geophys Res, 2007, 112, doi: 10.1029/2006JD007487

  15. Hines C O. The saturation of gravity wave in the middle atmosphere. Part II: Development of Doppler-spread theory. J Atmos Sci, 1991, 48: 1360–1379

    Google Scholar 

  16. Weinstock J. Theoretical gravity wave spectra in the atmosphere: Strong and weak interactions. Radio Sci, 1985, 20: 1295–1300

    Article  Google Scholar 

  17. Yeh K C, Liu C H. Evolution of atmospheric spectrum by processes of wave-wave interactions. Radio Sci, 1985, 20: 1279–1294

    Article  Google Scholar 

  18. Smith S A, Fritts D C, VanZandt T E. Evidence for a saturated spectrum of atmospheric gravity waves. J Atmos Sci, 1987, 44: 1404–1410

    Article  Google Scholar 

  19. Yeh K C, Liu C H. The instability of atmospheric gravity waves through wave-wave interactions. J Geophys Res, 1981, 86: 9722–9728

    Article  Google Scholar 

  20. Dong B, Yeh K C. Resonant and nonresonant wave-wave interactions in an isothermal atmosphere. J Geophys Res, 1988, 93: 3729–3744

    Article  Google Scholar 

  21. Klostermeyer J. Two- and three-dimensional parametric instabilities in finite amplitude internal gravity waves. Geophys Astrophys Fluid Dyn, 1991, 64: 1–25

    Article  Google Scholar 

  22. Fritts D C, Sun S J, Wang D Y. Wave-wave interactions in compressible atmosphere: 1. A general formulation including rotation and wind shear. J Geophys Res, 1992, 97: 9975–9988

    Article  Google Scholar 

  23. Yi F, Xiao Z. Evolution of gravity waves through Resonant and nonresonant interactions in a dissipative atmosphere. J Atmos Sol-Terr Phys, 1997, 59: 305–317

    Article  Google Scholar 

  24. Xiong J G, Yi F, Li J. The nonlinear interaction between Rossby wave and inertial gravity wave in the middle atmosphere, (1) The equations of nonlinear interaction. Chin J Geophys, 1995, 38: 150–157

    Google Scholar 

  25. Xiong J G, Yi F. Nonlinear interaction among planetary wave and inertial gravity wave in the middle and upper atmosphere. Chin J Space Sci, 2000, 20: 121–128

    Google Scholar 

  26. Müller P, Holloway G, Henyey F, et al. Nonlinear interactions among internal gravity waves. Rev Geophys, 1986, 24: 493–536

    Article  Google Scholar 

  27. Dunkerton T J. Effect of nonlinear instability on gravity-wave momentum transport. J Atmos Sci, 1987, 44: 3188–3209

    Article  Google Scholar 

  28. Vanneste J. The instability of internal gravity waves to localized disturbances. Ann Geophys, 1995, 13: 196–210

    Article  Google Scholar 

  29. Yi F. Resonant interactions between propagating gravity wave packets. J Atmos Sol-Terr Phys, 1999, 61: 675–691

    Article  Google Scholar 

  30. Zhang S D, Yi F. A numerical study on the propagation and evolution of resonant interacting gravity waves. J Geophys Res, 2004, 109, doi: 10.1029/2004JD004822

  31. Huang K M, Zhang S D, Yi F. A numerical study on nonresonant interactions of gravity waves in a compressible atmosphere. J Geophys Res, 2007, 112, doi: 10.1029/2006JD007373

  32. Huang K M, Zhang S D, Yi F. Gravity wave excitation through resonant interaction in a compressible atmosphere. Geophys Res Lett, 2009, 36, doi: 10.1029/2008GL035575

  33. Huang K M, Zhang S D, Yi F. Propagation and reflection of gravity waves in a meridionally sheared wind field. J Geophys Res, 2008, 113, doi: 10.1029/2007JD008877

  34. Hu Y Q, Wu S T. A full-implicit-continuous-Eulerian (FICE) multidimensional transient magnetohydrodynamic (MHD) flows. J Comput Phys, 1984, 55: 33–64

    Article  Google Scholar 

  35. Tsuda T, Kato S, Yokoi T, Inoue T, et al. Gravity waves in the mesosphere observed with the middle and upper atmosphere radar. Radio Sci, 1990, 25: 1005–1018

    Article  Google Scholar 

  36. Huang K M, Zhang S D, Yi F. Reflection and transmission of atmospheric gravity waves in a stably sheared horizontal wind field. J Geophys Res, 2010, 5, doi: 10.1029/2009JD012687

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Authors and Affiliations

  1. School of Electronic Information, Wuhan University, Wuhan, 430072, China

    KaiMing Huang, ShaoDong Zhang & Fan Yi

  2. Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan, 430072, China

    KaiMing Huang, ShaoDong Zhang & Fan Yi

  3. State Observatory for Atmospheric Remote Sensing, Wuhan, 430072, China

    KaiMing Huang, ShaoDong Zhang & Fan Yi

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  1. KaiMing Huang
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  2. ShaoDong Zhang
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  3. Fan Yi
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Correspondence to KaiMing Huang.

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Huang, K., Zhang, S. & Yi, F. A numerical study on matching relationships of gravity waves in nonlinear interactions. Sci. China Earth Sci. 56, 1079–1090 (2013). https://doi.org/10.1007/s11430-012-4522-0

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