A Formulation of a Phase-Independent Wave-Activity Flux for Stationary and Migratory Quasigeostrophic Eddies on a Zonally Varying Basic Flow

Koutarou Takaya Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan

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Hisashi Nakamura Department of Earth and Planetary Science, University of Tokyo, and Institute for Global Change Research, Frontier Research System for Global Change, Tokyo, Japan

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Abstract

A new formulation of an approximate conservation relation of wave-activity pseudomomentum is derived, which is applicable for either stationary or migratory quasigeostrophic (QG) eddies on a zonally varying basic flow. The authors utilize a combination of a quantity A that is proportional to wave enstrophy and another quantity E that is proportional to wave energy. Both A and E are approximately related to the wave-activity pseudomomentum. It is shown for QG eddies on a slowly varying, unforced nonzonal flow that a particular linear combination of A and E, namely, M ≡ (A + E)/2, is independent of the wave phase, even if unaveraged, in the limit of a small-amplitude plane wave. In the same limit, a flux of M is also free from an oscillatory component on a scale of one-half wavelength even without any averaging. It is shown that M is conserved under steady, unforced, and nondissipative conditions and the flux of M is parallel to the local three-dimensional group velocity in the WKB limit. The authors’ conservation relation based on a straightforward derivation is a generalization of that for stationary Rossby waves on a zonally uniform basic flow as derived by Plumb and others.

A dynamical interpretation is presented for each term of such a phase-independent flux of the authors or Plumb. Terms that consist of eddy heat and momentum fluxes are shown to represent systematic upstream transport of the mean-flow westerly momentum by a propagating wave packet, whereas other terms proportional to eddy streamfunction anomalies are shown to represent an ageostrophic flux of geopotential in the direction of the local group velocity. In such a flux, these two dynamical processes acting most strongly on the node lines and ridge/trough lines of the eddy streamfunction field, respectively, are appropriately combined to eliminate its phase dependency. The authors also derive generalized three-dimensional transformed Eulerian-mean equations with the residual circulation and eddy forcing both expressed in phase-independent forms.

The flux may not be particularly suited for evaluating the exact local budget of M, because of several assumptions imposed in the derivation. Nevertheless, these assumptions seem qualitatively valid in the assessment based on observed and simulated data. The wave-activity flux is a useful diagnostic tool for illustrating a“snapshot” of a propagating packet of stationary or migratory QG wave disturbances and thereby for inferring where the packet is emitted and absorbed, as verified in several applications to the data. It may also be useful for routine climate diagnoses in an operational center.

Corresponding author’s address: Koutarou Takaya, Dept. of Earth and Planetary Sciences, Graduate School of Science, University of Tokyo, Sci. Bldg. 1, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.

Abstract

A new formulation of an approximate conservation relation of wave-activity pseudomomentum is derived, which is applicable for either stationary or migratory quasigeostrophic (QG) eddies on a zonally varying basic flow. The authors utilize a combination of a quantity A that is proportional to wave enstrophy and another quantity E that is proportional to wave energy. Both A and E are approximately related to the wave-activity pseudomomentum. It is shown for QG eddies on a slowly varying, unforced nonzonal flow that a particular linear combination of A and E, namely, M ≡ (A + E)/2, is independent of the wave phase, even if unaveraged, in the limit of a small-amplitude plane wave. In the same limit, a flux of M is also free from an oscillatory component on a scale of one-half wavelength even without any averaging. It is shown that M is conserved under steady, unforced, and nondissipative conditions and the flux of M is parallel to the local three-dimensional group velocity in the WKB limit. The authors’ conservation relation based on a straightforward derivation is a generalization of that for stationary Rossby waves on a zonally uniform basic flow as derived by Plumb and others.

A dynamical interpretation is presented for each term of such a phase-independent flux of the authors or Plumb. Terms that consist of eddy heat and momentum fluxes are shown to represent systematic upstream transport of the mean-flow westerly momentum by a propagating wave packet, whereas other terms proportional to eddy streamfunction anomalies are shown to represent an ageostrophic flux of geopotential in the direction of the local group velocity. In such a flux, these two dynamical processes acting most strongly on the node lines and ridge/trough lines of the eddy streamfunction field, respectively, are appropriately combined to eliminate its phase dependency. The authors also derive generalized three-dimensional transformed Eulerian-mean equations with the residual circulation and eddy forcing both expressed in phase-independent forms.

The flux may not be particularly suited for evaluating the exact local budget of M, because of several assumptions imposed in the derivation. Nevertheless, these assumptions seem qualitatively valid in the assessment based on observed and simulated data. The wave-activity flux is a useful diagnostic tool for illustrating a“snapshot” of a propagating packet of stationary or migratory QG wave disturbances and thereby for inferring where the packet is emitted and absorbed, as verified in several applications to the data. It may also be useful for routine climate diagnoses in an operational center.

Corresponding author’s address: Koutarou Takaya, Dept. of Earth and Planetary Sciences, Graduate School of Science, University of Tokyo, Sci. Bldg. 1, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.

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  • Andrews, D. G., 1983: A conservation law for small-amplitude quasi-geostrophic disturbances on a zonally asymmetric basic flow. J. Atmos. Sci.,40, 85–90.

  • ——, 1984: On the existence of nonzonal flows satisfying sufficient conditions for instability. Geophys. Astrophys. Fluid Dyn.,28, 243–256.

  • ——, and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen–Palm relation and the mean zonal acceleration. J. Atmos. Sci.,33, 2031–2048.

  • ——, and ——, 1978: On wave action and its relatives. J. Fluid. Mech.,89, 647–664.

  • ——, J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. Academic Press, 489 pp.

  • Blackmon, M. L., J. M. Wallace, N.-C. Lau, and S. L. Mullen, 1977:An observational study of the Northern Hemisphere wintertime circulation. J. Atmos. Sci.,34, 1040–1053.

  • ——, Y.-H. Lee, J. M. Wallace, and H.-H. Hsu, 1984: Time variation of 500-mb height fluctuations with long, intermediate and short time scales. J. Atmos. Sci.,41, 981–991.

  • Bretherton, F. P., and C. J. R. Garret, 1968: Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. London,A302, 529–554.

  • Brunet, G., and P. H. Haynes, 1996: Low-latitude reflection of Rossby wave trains. J. Atmos. Sci.,53, 482–496.

  • Chang, E.-K. M., 1993: Downstream development of baroclinic waves as inferred from regression analysis. J. Atmos. Sci.,50, 2038–2053.

  • ——, and I. Orlanski, 1993: On the dynamics of a storm track. J. Atmos. Sci.,50, 999–1015.

  • Edmon, H. J., B. J. Hoskins, and M. E. McIntyre, 1980: Eliassen–Palm cross sections for the troposphere. J. Atmos. Sci.,37, 2600–2616; Corrigendum, 38, 1115.

  • Enomoto, T., and Y. Matsuda, 1999: Rossby wavepacket propagation in a zonally-varying basic flow. Tellus,51A, 588–602.

  • Haynes, P. H., 1988: Forced, dissipative generalizations of finite-amplitude wave-activity conservation relations for zonal and nonzonal basic flows. J. Atmos. Sci.,45, 2352–2363.

  • Held, I. M., 1987: New conservation laws for linear quasi-geostrophic waves in shear. J. Atmos. Sci.,44, 2349–2351.

  • Holton, J. R., 1992: An Introduction to Dynamic Meteorology. 3d ed. Academic Press, 511 pp.

  • Honda, M., K. Yamazaki, H. Nakamura, and K. Takeuchi, 1999: Dynamic and thermodynamic characteristics of atmospheric response to anomalous sea-ice extent in the Sea of Okhotsk. J. Climate,12, 3347–3358.

  • Hoskins, B. J., and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci.,38, 1179–l196.

  • ——, and T. Ambrizzi, 1993: Rossby wave propagation on a realistic longitudinally varying flow. J. Atmos. Sci.,50, 1661–1671.

  • ——, I. N. James, and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci.,40, 1595–1612.

  • Karoly, D. J., R. A. Plumb, and M. Ting, 1989: Examples of the horizontal propagation of quasi-stationary waves. J. Atmos. Sci.,46, 2802–2811.

  • Kuroda, Y., 1996: Quasi-geostrophic 3-dimensional E–P flux of stationary waves on a sphere. J. Meteor. Soc. Japan.,74, 563–569.

  • McIntyre, M. E., 1982: How well do we understand the dynamics of stratospheric warmings? J. Meteor. Soc. Japan,60, 37–65.

  • ——, and T. G. Shepherd, 1987: An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnold’s stability theorems. J. Fluid Mech.,181, 527–565.

  • Nakamura, H., 1994: Rotational evolution of potential vorticity associated with a strong blocking flow configuration over Europe. Geophys. Res. Lett.,21, 2003–2006.

  • ——, M. Nakamura, and J. L. Anderson, 1997: The role of high- and low-frequency dynamics in the blocking formation. Mon. Wea. Rev.,125, 2074–2093.

  • Naoe, H., Y. Matsuda, and H. Nakamura, 1997: Rossby wave propagation in idealized and realistic zonally varying flows. J. Meteor. Soc. Japan,75, 687–700.

  • Plumb, R. A., 1985: On the three-dimensional propagation of stationary waves. J. Atmos. Sci.,42, 217–229.

  • ——, 1986: Three-dimensional propagation of transient quasi-geostrophic eddies and its relationship with the eddy forcing of the time-mean flow. J. Atmos. Sci.,43, 1657–1678.

  • Simmons, A. J., J. M. Wallace, and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci.,40, 1363–1392.

  • Takaya, K., and H. Nakamura, 1997: A formulation of a wave-activity flux for stationary Rossby waves on a zonally varying basic flow. Geophys. Res. Lett.,24, 2985–2988.

  • Trenberth, K. E., 1986: An assessment of the impact of transient eddies on the zonal flow during a blocking episode using localized Eliassen–Palm flux diagnostics. J. Atmos. Sci.,43, 2070–2087.

  • Uryu, M., 1974: Mean zonal flows induced by a vertically propagating Rossby wave packet. J. Meteor. Soc. Japan.,52, 481–490.

  • Vanneste, J., and T. Shepherd, 1998: On the group-velocity property for wave-activity conservation laws. J. Atmos. Sci.,55, 1063–1068.

  • Wallace, J. M., G. H. Lim, and M. L. Blackmon, 1988: Relationship between cyclone tracks, anticyclone tracks, and baroclinic waveguides. J. Atmos. Sci.,45, 439–462.

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