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D. G. Andrews and M. F. McIntyre


A simple multiple-scale expansion procedure is given for calculating corrections to the structure of equatorial planetary waves in the presence of weak shear and dissipation. For upward-propapting Rossby-gravity (Yanai-Maruyama) and Kelvin (Wallace-Kousky) waves, explicit results are obtained for the case of Newtonian cooling and Rayleigh friction, correct to the first two orders in the ratio μ of wave to mean-flow height scales. The results are used in a direct calculation of the horizontal Reynolds stress uv′¯ and demonstrate the strong dependence of u&primev′¯ on the ratio of friction to cooling coefficients.

In certain parameter regimes of interest in the tropical stratosphere, a slight north-south asymmetry in the y profile of ū can cause changes in the wave structure such that the mean zonal acceleration ∂ū/∂t tends to have the same asymmetry. That is, there may be a tendency for asymmetries in ū(y) to amplify in the presence of dissipating waves.

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T. Dunkerton, C-P. F. Hsu, and M. E. McIntyre


Some new diagnostics are presented for a wavenumber-2 sudden warming, simulated by a version of Holton's semi-spectral, primitive-equation model. First, Eliassen-Palm cross sections exhibiting the Eliassen-Palm (EP) planetary-wave flux together with contours of the corresponding flux divergence, are presented for selected days of the simulation. Second, a description of zonal-mean-flow evolution in the model, simpler than the conventional Eulerian-mean description and qualitatively like Lagrangian- mean descriptions in some respects, is constructed from the transformed Eulerian-mean equations presented by Andrews and McIntyre (1976). In this description the mean warming is brought about by a thermally direct “residual meridional circulation” arising as an essentially adiabatic response to a wave-induced torque about the earth's axis. The torque itself is equal to the divergence of the EP wave flux and approximately proportional to the northward flux of quasi-geostrophic potential vorticity. Third, some true Lagrangian means and related diagnostics are presented and discussed.

The EP cross sections strikingly display the effect of the mid-stratospheric zero-wind line which invades middle latitudes from the tropics during the first stage of substantial evolution of the mean state. This zero-wind line develops into a partial reflector of planetary waves, splitting the EP wave flux into two branches and deflecting one of them equatorward and the other to high polar altitudes. The consequent focusing of waves into a smaller horizontal area in the polar cap and into altitudes with lower densities helps bring about the reversal of the polar westerlies in the second stage of mean evolution. Focusing of planetary waves into the high-altitude polar cap should be similarly important for real warmings, but there is no evidence that subtropical zero-wind lines play any important role. Possible mechanisms leading to focusing and hence to warnings in the real atmosphere are discussed.

To picture the model warming in Lagrangian terms, we first compare the shape of an isentropic surface near the level of maximum warming with the computed behavior of sets of air parcels. The isentropic surface is approximately a material surface over the short times concerned. As the warming develops the surface dips down over the pole and rises at the equator (and in the real atmosphere this leads to widespread cooling in the summer stratosphere as has often been observed). Thermally direct motion similarly appears in the residual, generalized Lagrangian-mean, and modified Lagrangian-mean meridional circulations near the level of maximum warming, as might have been expected from the theoretical results of Matsuno and Nakamura (1979). The divergence effect, or non-solenoidality of Lagrangian-mean motion, neglected in their study, is strong here because of the large north-south dispersion of air parcels accompanying the highly transient wave activity. Implications for modeling tracer transport are noted.

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