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Steady-state nonlinear internal gravity-wave critical layers satisfying an upper radiation condition

  • Kevin G. Lamb (a1) and Raymond T. Pierrehumbert (a2)


We consider the behaviour of an internal gravity wave encountering a critical level in a stratified fluid, assuming the critical-level flow to be dominated by nonlinear effects. The background flow is a shear layer, and the stratification is sufficiently strong to support wave propagation everywhere. Incident and reflected waves are permitted below the critical level, and a radiation condition is imposed far above it. For this geometry we construct, by a combination of asymptotic and numerical means, steady, nonlinear solutions, and discuss the associated transmission coefficients, reflection coefficients, phase shifts, and resonance positions when the system is forced from below.

The inviscid solutions we exhibit have continuous density and velocity everywhere, and so do not require the introduction of internal viscous boundary layers. Further, the streamlines bounding the recirculating cat's-eye regions have corners, just as in the unstratified case. For weak stratification, the transmitted wave is nearly as strong as the incident wave, and there is accompanying strong over-reflection. As the stratification increases, the critical level becomes a nearly perfect reflector. The amount of transmission depends on wave amplitude, and the sensitivity increases with increasing stratification.

There are regions of parameter space for which steady solutions could not be found. The critical-layer structure appears to break down by unbounded thickening when the stratification becomes too strong, suggesting that in these cases some neglected physical process must intervene to limit growth of the recirculating region.



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Bacmeister, J. T. & Pierrehumbert, R. T. 1988 On high drag states of nonlinear stratified flow over obstacles. J. Atmos. Sci. 45, 6380.
Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181204.
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513539.
Bretherton, F. P. 1966 The propagation of groups of internal gravity waves in a shear flow. Q. J. R. Met. Soc. 92, 466480.
Brown, S. N. & Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Part II. Geophys. Astrophys. Fluid Dyn. 10, 124.
Brown, S. N. & Stewartson, K. 1980 On the nonlinear reflexion of a gravity wave at a critical level. Part 1. J. Fluid Mech. 100, 577595.
Brown, S. N. & Stewartson, K. 1982 On the nonlinear reflection of a gravity wave at a critical level. Part 2. J. Fluid Mech. 115, 217230.
Fritts, D. C. 1984 Gravity wave saturation in the middle atmosphere. A review of theory and observations. Rev. Geophys. Space Phys. 22, 275308.
Graham, E. W. 1982 On the steady-state relations between disturbances above and below a critical level. J. Fluid Mech. 115, 395410.
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51, 139160.
Kelly, R. E. & Maslowe, S. A. 1970 The nonlinear critical layer in a slightly stratified shear flow. Stud. Appl. Maths 49, 301326.
Killworth, P. & McIntyre, M. E. 1985 Do Rossby wave critical levels absorb, reflect or overreflect? J. Fluid Mech. 161, 449492.
Lamb, K. G. 1989 Nonlinear internal gravity wave critical layers. Ph.D. thesis, Princeton University.
Maslowe, S. A. 1972 The generation of clear air turbulence by nonlinear waves. Stud. Appl. Maths 51, 116.
Maslowe, S. A. 1973 Finite-amplitude Kelvin-Helmholtz billows. Boundary-Layer Met. 5, 4352.
Maslowe, S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 405432.
Moore, D. W. & Saffman, P. G. 182 Finite-amplitude waves in inviscid shear flows. Proc. R. Soc. Lond. A 382, 389410.
Palmer, T. N., Shutts, G. J. & Swinbank, R. 1986 Alleviation of a systematic westerly bias in general circulation and numerical weather prediction models through an orographic gravity wave drag parameterization. Q. J. R. Met. Soc. 112, 10011039.
Peltier, W. R. & Clark, T. L. 1983 Nonlinear mountain waves in two and three spatial dimensions. Q. J. R. Met. Soc. 109, 527548.
Stewartson, K. 1981 Marginally stable inviscid flows with critical layers. IMA J. Appl. Maths 27, 133175.
Winters, K. & D'Asaro, E. A. 1989 Two-dimensional instability of finite-amplitude internal gravity wave packets near a critical level. J. Geophys. Res. 94, 1270912719.
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Steady-state nonlinear internal gravity-wave critical layers satisfying an upper radiation condition

  • Kevin G. Lamb (a1) and Raymond T. Pierrehumbert (a2)


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