Low Frequency Equatorial Waves in Vertically Sheared Zonal Flow:
Part II: Unstable Waves

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

J. Atmos. Sci., 53, 3589-3605

Abstract | Introduction | Summary


The stability of equatorial Rossby waves in the presences of mean flow vertical shear and moisture convergence-induced heating is investigated with a primitive equation model on an equatorial beta plane.

A vertical shear alone can destabilize equatorial Rossby waves by feeding mean flow available potential energy to the waves. This energy transfer necessitates unstable waves' constant phase lines tilt both horizontally (eastward with latitude) and vertically (against the shear). The preferred most unstable wavelength increases with increasing vertical shear and with decreasing heating intensity, ranging typically from 3000 to 5000 km. The instability strongly depends on meridional variation of the vertical shear. A broader meridional extent of the shear allows a faster growth and a less-trapped meridional structure. When the shear is asymmetric relative to the equator, the unstable Rossby wave is constrained to the hemisphere where the shear is prominent. Without boundary layer friction the Rossby wave instability does not depend on the sign of the vertical shear, whereas in the presence of the boundary layer, the moist Rossby wave instability is remarkably enhanced (suppressed) by easterly (westerly) vertical shears. This results from the fact that an easterly shear confines the wave to the lower level, generating a stronger Ekman-pumping-induced heating and an enhanced meridional heat flux, both of which reinforce the instability.

The moist baroclinic instability is a mechanism by which westward propagating rotational waves (Rossby and Yanai waves) can be destabilized, whereas Kelvin waves cannot. This is because the transfer of mean potential energy to eddy requires significant magnitude of barotropic motion. The latter is a modified Rossby wave and can be resonantly excited only by the westward propagating rotational waves. The common features and differences of the equatorial Rossby wave instability and midlatitude baroclinic instability, as well as the implications of the results are discussed.


In Part I (Wang and Xie 1996a, hereafter WX96) of this paper, the mechanism by which a vertically sheared zonal flow affects the structure and propagation of equatorial waves was examined. It was shown that a moderate vertical shear has little impact on equatorial Kelvin waves but remarkable effects on equatorial Rossby and westward propagating Yanai waves. This is due to the fact that modification of wave properties depends on the magnitude of the barotropic motion component. The barotropic component of the equatorial wave motion is a modified Rossby mode and can be resonantly excited only by westward propagating internal waves. It is not clear whether this selective excitation of barotropic motion may also affect the instability of the equatorial waves. WX96 also found that in the tropics a westerly shear favors trapping of Rossby waves in the upper troposphere, while an easterly shear confines Rossby waves in the lower troposphere. The different effects of vertical shears on wave structure are expected to have a major impact on the strength of the boundary layer moisture convergence. The latter may in turn influence the Rossby wave through organized convective heating, changing the wave stability.

In this paper, we will further investigate the instability of the equatorial waves and address the issue of how a vertically sheared zonal flow and boundary layer moisture convergence-induced heating destabilize equatorial waves.

The present study is also motivated by the need to interpret the nature of the observed low-frequency westward-propagating disturbances which are reproducible in numerical models as well. Whereas the westward propagation is sometimes episodic and the intensity is relatively weak , it is a prevailing mode in off-equatorial monsoon domain (see Fig. 1 of Wang and Xu 1996). The westward propagating disturbances were found over the South Asia and the western North Pacific regions on 10-20 day and 30 day time scales (e.g., Murakami 1980; Cadet 1983; Nitta 1987, Wang and Rui 1990a). They exhibit a typical westward phase speed of a few \ms and a zonal scale (one quarter of wavelength) of the equatorial Rossby radius of deformation (1,000-1,500 km). The behavior of the synoptic (about 8-9 day) westward-propagating waves with a typical wavelength of 4,000 km and a speed of 5 \ms were documented in detail by Lau and Lau (1990). These disturbances are characterized by an eastward tilt with latitude of the constant phase lines and by low-level cyclonic (anticyclonic) vorticity accompanying by enhanced (suppressed) convection. In the troposphere, their troughs and ridges tilt eastward with height, against the direction of the vertical shear. These observed features call for explanations.

Whereas observations show a strong evidence of the linkage between the westward-moving low-frequency modes and the South Asian summer monsoon circulation, theoretical explanation of the phenomenon has been lacking. Based on their numerical experiments, Lau and Peng (1990) suggested that an equatorial intraseasonal mode interacting with the Asian monsoon flows can generate westward-moving synoptic disturbances in monsoon regions. They interpreted these disturbances as resulting from quasi-geostrophic baroclinic instability of the monsoon flows. Zhang and Geller (1994) investigated the impacts of vertical shears on CISK-waves. Their analysis for the Rossby waves is also based on a quasi-geostrophic $\beta$-plane model. Since the tropical atmospheric motions consist of a significant ageostrophic component, a primitive equation model is more relevant for study of equatorial Rossby waves which have large-amplitude pressure fluctuation at one Rossby radius of deformation ($10-15^\circ$ latitude) and strong zonal wind along the equator. Kuo (1978) studied combined barotropic and baroclinic instability in the tropics and concluded that the tropical disturbances develop primarily through barotropic instability whereas baroclinic instability has trivial impact on the growth rate. His model atmosphere is free of heating and in a regime in which the barotropic instability is dominant. The condensational heating, however, plays an essential role in the tropical wave dynamics and tropical circulation. Static stability can be substantially reduced by latent heating, and thus critical vertical wind shear required for baroclinic instability can be much smaller than its adiabatic counterpart.

The present analysis will address some interesting questions regarding the equatorial wave instability. For instance, why does the Northern Hemisphere summer monsoon circulation favor the development of the westward propagating disturbances rather than the eastward-propagating equatorial Kelvin waves? In particular, what roles does the monsoon easterly vertical shear play in the development of the low-frequency equatorial waves? If the westward-propagating disturbances are interpreted as equatorial Rossby waves, how are the equatorial Rossby waves destabilized by a vertical wind shear? What is the fundamental difference between the tropical Rossby wave instability and the midlatitude baroclinic instability? The vertical easterly shears in the boreal summer monsoon region exhibit strong asymmetry with respect to the equator. How does the latitudinal variation of the vertical shear affect the equatorial Rossby wave instability? These questions are the targets of the present study.


The equatorial Rossby waves with gravest meridional structure are shown to be unstable when the vertical shear of the mean zonal flow exceeds certain critical value. Although the growth rate does not depend on the sign of the vertical shear, the vertical structure of the unstable waves does: Easterly (westerly) shears confine the unstable waves in the lower (upper) troposphere. For given strength of vertical shear and heating intensity there exists a most unstable wave. The preferred unstable wavelength increases with increasing vertical shear and decreases with increasing heating intensity, ranging typically from 3000 to 5000 km.

There are a number of distinct characteristics in the structure of the unstable equatorial Rossby waves compared with their stable counterparts: (a) The constant phase lines exhibit an eastward tilt with latitude; (b) There is no meridional emanation toward the extratropics; both the baroclinic and barotropic modes remain trapped at the equator, displaying a single peak in each hemisphere in the geopotential and meridional winds; while the temperature field is less trapped than its counterpart of a stable wave; and (c) The constant phase lines tilt with height against the direction of the vertical shear.

Such a wave structure favors the perturbations to extract available potential energy from the basic flow, because the conversion rate is proportional to the Coriolis parameter (latitude), vertical shear, and the covariance between temperature (thickness) and barotropic meridional wind. The horizontal tilt of the ridges and troughs of the unstable wave ensures that the temperature field leads (lags) the barotropic meridional wind by more (less) than a quarter wavelength in an easterly (westerly) shear. This warrants a conversion of mean flow energy to the waves. In contrast, for a stable Rossby wave the temperature is precisely in quadrature with barotropic meridional wind so that no energy is transferred from mean flow to eddy.

The present study suggests that vertical wind shears have profound influences on the instability of equatorial rotational waves. The instability mechanism selects only those westward-propagating rotational waves, because transfer of the mean potential energy requires significant barotropic motion, which is excited by zonal stretch of vorticity associated with baroclinic motion acting on the vertical shear of the basic flow (WX96). The barotropic mode is rotational in nature, therefore, resonant barotropic motion can only be excited by the westward-propagating rotational waves but not the eastward-propagating divergent Kelvin waves. In this regard, baroclinic instability is a mechanism though which a tropical westward-propagating rotational wave can be destabilized, whereas eastward-propagating waves can not.

The meridional variations of vertical shear have significant influences on the moist Rossby waves. When the vertical shear is suppressed south of the equator, the unstable Rossby waves are strongly asymmetric with respect to the equator and large perturbations are confined to the northern hemisphere where the vertical shear is prominent. If a vertical shear is confined to the tropics, the instability is much weaker than that in a meridionally invariant vertical shear. Structure of unstable waves in a tropical vertical shear differs dramatically from that in a constant shear, especially the barotropic component. To have the mean potential energy transfer to eddies, i.e., $U_T \overline{yv_+ \phi_-}>0$, both the baroclinic and barotropic components should have large amplitude sufficiently far away from the equator. Strong meridional shear of the mean zonal wind causes tight trapping of the rotational waves near the equator and does not favor baroclinic instability in the tropics.

In the presence of an Ekman boundary layer, the tropical Rossby wave instability is more favorable in an easterly shear than in a westerly shear due to a combined effect of boundary layer Ekman pumping and vertical shear. The asymmetric tropical moist Rossby wave instability with respect to the sign of vertical shear is attributed to the differences in the wave structure and surface friction-induced moisture convergence. Firstly, in an easterly (westerly) shear, the phase shift between the barotropic meridional wind and the thickness favors (unfavors) the conversion of potential energy from the mean flow to eddies. Secondly, the frictional moisture convergence in an easterly shear has larger contribution to the generation of eddy available potential energy than in a westerly shear, because easterly (westerly) shears trap perturbations in the lower (upper) troposphere, inducing stronger (weaker) Ekman pumping.

The destabilization of moist Rossby waves by regional easterly vertical shears has important implication. During the northern summer season, large easterly shears exist over the Indian monsoon and western North Pacific regions. It provides a favorable large-scale environment for westward-propagating moist Rossby wave disturbances to develop only north of the equator. The tropical Rossby waves have two energy sources: vertical shear of the mean zonal flow via baroclinic instability and the surface friction-induced moisture convergence. Both processes contribute to the development of tropical Rossby waves. The disturbances take the form of asymmetric tropical Rossby waves modified by convective heating. The most unstable Rossby waves predicted by the model compare favorably to the observed vorticity waves (Lau and Lau 1990) in their horizontal and vertical structures, preferred wavelength and propagation speed, and their seasonality.

It should be pointed out that although the present study is based on an equatorial $\beta$-plane model, its qualitative validity is confirmed by an analogous study using a spherical coordinate model. Furthermore, the present study is the first step to perceive how the realistic basic state regulates the intraseasonal disturbances. The impacts of more complicated three-dimensional monsoon mean flows on the low-frequency disturbances during the boreal summer will be reported in a separate paper (Wang and Xie, 1996b).

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