Low-Frequency Equatorial Waves in Vertically Sheared Zonal Flow. Part I: Stable Waves

Bin Wang and Xiaosu Xie
Department of Meteorology, University of Hawaii,
Honolulu, HI 96822

J. Atmos. Sci., 53, 449-467

Abstract | Introduction | Summary


The mechanism by which a vertically sheared zonal flow affects large-scale, low-frequency equatorial waves is investigated with two-level equatorial -plane and spherical coordinates models.

Vertical shears couple baroclinic and barotropic components of equatorial wave motion, affecting significantly the Rossby wave and westward propagating Yanai wave but not the Kelvin wave. This difference results from the fact that the barotropic component is a modified Rossby mode and can be resonantly excited only by westward propagating internal waves. The barotropic components emanate poleward into the extratropics with a pronounced amplitude, while the baroclinic components remain equatorially trapped. A westerly vertical shear favors the trapping of Rossby and Yanai waves in the upper troposphere, whereas an easterly shear tends to confine them in the lower troposphere. As such, their westward propagation is slowed down by both westerly and easterly shears. When the strength of the vertical shear varies with latitude, both the vertical modes are locally enhanced in the latitudes of strong shear.

The theory suggests that the vertical shear plays an essential role in emanation of heating-induced internal equatorial Rossby waves into the extratropics with a transformed barotropic structure. It may also be partially responsible for trapping perturbation kinetic energy in the upper-troposphere westerly duct and the lower-troposphere monsoon trough.



a. Conclusion

Effects of a basic zonal flow with vertical and meridional shears on vertically standing, low frequency equatorial waves (the Kelvin wave, the Rossby wave with a gravest meridional structure and westward propagating Yanai wave) are investigated using simple two-level models on an equatorial $\beta$-plane and spherical coordinates. The simplicity of the $\beta$-plane model allows for a better understanding of the mechanism by which basic flows alter wave dynamics.

The vertical shear of a mean zonal flow was found to have profound influences on the equatorial Rossby wave and westward-propagating Yanai wave. The meridional shear, however, have only a moderate modification to low frequency equatorial waves. This is due to the fact that vertical shears couple the baroclinic and barotropic modes; their interaction causes marked changes in wave characteristics, whereas meridional shears do not.

In the presence of a moderate vertical shear, the baroclinic Rossby mode acts to force the barotropic mode. With a westerly (easterly) shear, positive (negative) barotropic vorticity is generated in the region of longitudinal stretch of baroclinic vorticity. The intensity of the barotropic mode increases with increasing vertical shear. In contrast, with the same shear, a baroclinic Kelvin mode can only stimulate a rather weak barotropic mode. This fundamental difference results from the rotational nature of the barotropic mode which, in the absence of forcing, is a Rossby wave maintained by an effective $\beta$-parameter ($\beta -\bar{u}_{yy}$). Because of the intrinsic nature of the barotropic mode, it can only be resonantly excited by a westward propagating baroclinic mode, but not an eastward propagating one.

In the presence of vertical shear, the baroclinic Rossby mode is slightly less equatorially trapped. However, the barotropic Rossby mode extends poleward with zonal wind and geopotential extrema occurring in the extratropics. This implies that tropical heating-induced internal Rossby mode may generate, with the add of the vertical shear, a conspicuous barotropic Rossby wave response in the extratropics.

The vertical shear can change vertical modal structure. This is because the coupling of the two vertical modes depends on the sign of the vertical shear. The barotropic and baroclinic Rossby modes are nearly in phase in a westerly shear (i.e., westerlies increasing with height) whereas precisely $180^\circ$ out of phase in an easterly shear. It follows that the vertical shear creates a vertically asymmetry in the structures of the Rossby wave and westward propagating Yanai waves: The waves in a westerly shear have a larger amplitude in the upper troposphere, whereas the waves in an easterly shear have a greater amplitude in the lower troposphere. This results in a raised (lowered) steering level for the waves in a westerly (easterly) shear. The mean flow steering is thus eastward in both westerly and easterly shears if the vertical mean flow vanishes. The propagation of the Rossby wave is thus slowed down by vertical shear regardless of its sign.

It is shown that both the barotropic and baroclinic Rossby modes respond sensitively to the meridional variation of the vertical shear. The responses are enhanced in the latitudes where the vertical shear is strengthened, suggesting the importance of the regional vertical shear in modification of the in-situ wave characteristics.

The vorticity and meridional vorticity gradient associated with a meridional shear can alter wave refraction index and thus the degree of equatorial trapping for equatorial waves. Longer waves are less affected than shorter waves. Due to the wavelength-dependent modification of the meridional structure and non-uniform zonal advection by meridionally sheared flows, Kelvin waves become weakly dispersive and the Rossby waves also experience noticeable changes in their dispersivity.

b. Discussion

One of the fundamental impacts of the vertical shear is the excitation of prominent barotropic Rossby wave motion through an interaction with the gravest baroclinic Rossby mode. This is relevant to explanation of the {\bf emanation} of equatorial waves toward midlatitude. The energy emanation was suggested resulting from the reduction of equatorial trapping scale of the Rossby waves in equatorial westerlies (Zhang and Webster 1989) . We have shown that the direction of the mean flow does not affect meridional structure of the internal waves. The meridional shear enhances the trapping of the equatorial Rossby waves and can not generate a barotropic Rossby wave component. Based on our analysis, we infer that the emanation of equatorial waves is associated with an excitation of barotropic Rossby mode which is caused by a combined effect of vertical shear and the zonal stretch of the vorticity associated with baroclinic mode. The baroclinic Rossby mode is presumably stimulated directly by the equatorial internal heating. The meridional variation of divergence and longitudinal variation of vorticity associated with the baroclinic mode acting on the vertical shear of the mean flow represent an excitation mechanism for the barotropic (external) motion. This view of emanation explains how an internal equatorial heating can generate salient extratropical barotropic response. Although tropical heating may initiate both eastward propagating Kelvin wave and westward propagating Rossby waves on a synoptic scale or longer, the intrinsic rotational nature of the barotropic motion endorses a resonant mode selection, i.e., only westward propagating baroclinic Rossby mode can activate a large amplitude barotropic motion that extends into the extratropics..

Another fundamental impact of the vertical shear on the equatorial Rossby wave is that a westerly (easterly) vertical shear favors trapping wave kinetic energy to the upper (lower) troposphere. This may be pertinent to interpretation of the {\bf in-phase} relationship between the transient kinetic energy and the equatorial mean zonal flow in the upper troposphere as observed by Murakami and Unninayer (1977) and Arkin and Webster (1985). Over the regions of upper-level westerlies (easterlies), the mean zonal flow is characterized by westerly (easterly) shear because of the dominance of the upper tropospheric zonal winds. The vertical structure of the Rossby wave can be so modified by the vertical shear that the wave kinetic energy tends to be confined in the upper tropospheric westerlies. On the other hand, in a region of upper-level easterlies (easterly shear) the upper-level perturbation kinetic energy may be reduced due to the concentration of the perturbation in the lower troposphere. It is plausible, therefore, that the in-phase relationship between upper tropospheric perturbation kinetic energy and time mean zonal flow may be a result of the transformation in vertical modal structure between the regions of easterly and westerly shear along the equator. In order to verify the present theory, it would be interesting to compute transient kinetic energy in the lower troposphere and check whether they have maxima in easterly shear region and minima in westerly shear region. It would also be interesting to explore the transition of the vertical structure of equatorial waves from regions of easterly to westerly shear.

The investigation in this paper is confined to neutral waves in a moderate vertical shear. It should be pointed out that strong vertical shears can result in dynamically unstable Rossby waves even in a adiabatic, stably stratified atmosphere. This type of dynamic instability and the effects of sheared flow on diabatic equatorial waves will be reported in part II.

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