Propagation and Breakdown of Internal Inertio-Gravity Waves
near Critical Levels in the Middle Atmosphere

and Hiroshi Tanaka
Water Research Institute, Nagoya University, Chikusa, Nagoya 464, Japan

Journal of the Meteorological Society of Japan , vol.62, no.1, pp.1-17.

A part of M.Sc Thesis by M. D. Yamanaka submitted to Graduate School of Science, Nagoya University in February 1982, accepted in March 1982.
First manuscript received: December 28, 1982;
Final manuscript accepted: December 6, 1983
Published: February, 1984

The relation between IIGW and turbulence layers is calculated to compare with the turbulence layers observed in the stratosphere and to have information on IIGW's propagating upwards to the mesosphere and thermosphere. In general, thickness of the turbulence layers associated with IIGW's is thinner than that associated with non-inertial gravity waves for common mesoscale wavelength domain.

1. Introduction

2. Equations and Solutions

2.1. Governing Equations

2.2. Exact Solutions

2.3. Asymptotic Solutions

3. Vertical Propagation of IIGW in a Shear Flow

3.1. Local Dispersion Relation

3.2. Propagation of IIGW far from the Critical Levels

3.3. Propagation in the Vicinity of Jones' Critical Levels

4. Wave Behaviors near Jones' Critical Levels

4.1. Valve Effect

4.2. Turning Levels

4.3. Wave Absorption Associated with Jones' Critical Levels

5. Formation of Turbulence Layers

5.1. Local Convective Instability

5.2. Turbulence Layers Generated by IIGW Breakdown

6. Conclusion

Appendix A: Notations

Appendix B: Treatment for f = 0

Appendix C: Treatment for l = 0

Appendix D: Liouville-Green Method Developed by Olver (LGO Method)

Appendix E: Examples of LGO Method

Appendix F: Group Velocity near Jones' Critical Level

• Subsection 2.1 (p.3 left), Eq.(9): Denominator of the second term must be read as $\omega^{2} (\omega^{2} - f^{2})$
• Subsection 2.2 (p.3 right), Eq.(19):
• Caption of Fig.1 (p.5 left): `The' must be printed as `the'.
• Subsection 2.3 (p.5 right): The third case printed as (a) must be read as (c).
• Caption of Fig.2 (p.8): `$u_{z}$' must be printed as `$\overline{u}_{z}$'.
• Caption of Fig.4 (p.10): `with' must be inserted between `the basic flow' and `the Jones' critical level'.

• Fig. 1. Locations of turning levels for the basic Richardson number $J (= N^{2} / \overline{u}_{z}^{2})$ parameterizing $\nu (= l / k)$.
• Fig. 2. Local dispersion relation ($\omega$-$m$ diagram) for $\overline{u}_{z} < 0$ obtained by the LGO method.
• Fig. 3. Vertical profile of $\Re [ \tilde{w} ]$ of IIGW for $\overline{u}_{z} < 0$ calculated from the asymptotic solution based on the LGO method.
• Fig. 4. Schematic illustrations of the rays relative to the basic flow with the Jones' critical level.

• Barat, 1975a: Compt. Rend. Acad. Sci. Paris, B280, 691.
• Barat, 1975b: Compt. Rend. Acad. Sci. Paris, B281, 53.
• Booker and Bretherton, 1967: J. Fluid Mech., 27, 513.
• Bretherton, 1966: Quart. J. Roy. Meteor. Soc., 92, 466.
• Cadet, 1977: J. Atmos. Sci., 34, 137.
• Cadet and Teitelbaum, 1979: J. Atmos. Sci., 36, 892.
• Eady, 1949: Tellus, 1, 33.
• Geller et al., 1975: J. Atmos. Sci., 32, 2125.
• Grimshaw, 1975: J. Fluid Mech., 70, 287.
• Grimshaw, 1980: Geophys. Astrophys. Fluid Dyn., 14, 303.
• Hearth et al., 1974: Structure and Dynamics of the Upper Atmosphere, Verniani (ed.), Elsevie, p.131.
• Holton, 1982: J. Atmos. Sci., 39, 791.
• Jones, 1967: J. Fluid Mech., 30, 439.
• Kitchen and McIntyre, 1980: J. Meteor. Soc. Japan, 58, 118.
• Klostermeyer, 1980: J. Geophys. Res., 85, 2829.
• Lindzen, 1970: J. Atmos. Sci., 27, 394.
• Lindzen, 1981: J. Geophys. Res., 86, 9707.
• Matsuno, 1982: J. Meteor. Soc. Japan, 60, 215.
• Miyahara, 1976: J. Meteor. Soc. Japan, 54, 91.
• Miyahara, 1981: J. Meteor. Soc. Japan, 59, 902.
• Olbers, 1990: J. Phys. Oceanogr., 11, 1224.
• Olver, 1974: Asymptotics and Special Functions, Academic Press, 572pp.
• Sawyer, 1961: Quart. J. Roy. Meteor. Soc., 87, 24.
• Tanaka, 1975: J. Meteor. Soc. Japan, 53, 425.
• Tanaka, 1982: J. Meteor. Soc. Japan, 60, 1034.
• Tanaka, 1983a: J. Meteor. Soc. Japan, 61, 763.
• Tanaka, 1983b: J. Meteor. Soc. Japan, 61, 805.
• Tokioka, 1970: J. Meteor. Soc. Japan, 48, 503.
• Woodman et al., 1981: MAP Handbook, 2, 86.
• Yamanaka and Tanaka, 1984: in Dynamics of the Middle Atmosphere, Holton and Matsuno (eds.), D. Reidel Pub. Co., p.117.

• Dunkerton, 1984: J. Atmos. Sci., 41, 3396.
• Dunkerton, 1985: J. Atmos. Sci., 42, 1151.
• Matsuda, 198x: J. Meteor. Soc. Japan, xx, x.
• Nakamura, 199x: J. Atmos. Sci
• Palmer et al., 1986: Quart. J. Roy. Meteor. Soc., 112, 1001.
• Teitelbaum
• Ushimaru and Tanaka, 1990: J. Meteor. Soc. Japan, 68, 1.
• Wuertel et al., 1996: J. Atmos. Sci
• Yamanaka, 1985: J. Meteor. Soc. Japan, 63, 715.
• Yamanaka and Tanaka, 1984: in Dynamics of the Middle Atmosphere, Holton and Matsuno (eds.), D. Reidel Pub. Co., p.117.
• Yamanaka et al., 1989: Pure Appl. Geophys., 130, 481.
• Yamanaka et al., 1994: in Solar Terrestrial Energy Program: The Initial Results from STEP Facilities and Theory Campaigns, Baker et al.(eds.), Pergamon Press, London, p.511. (original: 1993b)
• Zhu and Holton, 1987: J. Atmos. Sci., 44, 620.