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Geostrophic Wave Circulations
Yong Zhu
Chapter I Introduction
The time-averaged circulations in the Earth's atmosphere are characterized by the large-scale perturbations, called the large-scale atmospheric waves, at middle and high latitudes. These waves were recognized generally as the Rossby waves, illustrated firstly by Rossby (1939) in terms of the variation of horizontal Coriolis force on the b-plane in a barotropic, non-divergent and statically stable atmosphere. The wave mechanics revealed by Rossby relied on the vorticity conservation which guided the studies on theoretical meteorology until the present time.
The studies of Rossby waves have been extended to different circumstances in dealing with different problems. For example, Charney and Drazin (1961) presented the vertical propagation of forced Rossby waves in the compressible atmosphere. Before them, the wind divergence had already been taken into account for studying the baroclinic instability of Rossby waves (Phillips, 1954; Rosenthal, 1964). The propagation of Rossby wave on a sphere was studied by Hoskins and Karoly (1981) and Karoly and Hoskins (1982) using the ray-tracing theory. Although the inclusion of wind divergence could affect greatly the wave propagation mechanism, the waves in the compressible atmospheres were still called Rossby waves because they were elaborated on a b-plane.
The most fundamental property of large-scale atmospheric circulations is that they are constrained to be in geostrophic balance and hydrostatic equilibrium (Holton, 1979). A theoretical model applied for the study of large-scale circulations in the extratropical regions must possess these properties. However, the geostrophic balance of Rossby waves has never been examined adequately. The observed basic circulation patterns, such as the planetary stationary waves and low-frequency variabilities in the troposphere and stratosphere are typical examples of these circulations in geostrophic and hydrostatic equilibrium. Neither the classical nor modern Rossby wave theories can give successfully the interpretations for the developments and geographic distributions of these circulation patterns.
The heat and momentum transport required for heat and momentum balances in the atmosphere are completed mostly by eddy fluxes at middle and high latitudes, as shown by the statistical surveys of Oort and Rasmusson (1971) and Newell et al. (1972). Usually, these eddy fluxes are evaluated with observed or simulated perturbation fields. Although the large-scale perturbations in the extratropical atmosphere are represented by Rossby waves and the fluxes may be evaluated analytically from the wave solutions, the results obtained cannot explain the major features of the perturbations, the eddy transport processes and the forced mean meridional circulations.
The present study is started by examining the propagation mechanism and geostrophic property of classical Rossby wave in the next chapter. It will be found that the vorticity advection, which was considered (Rossby, 1940) as the basic mechanism of Rossby wave propagation in the non-divergent atmosphere, depends itself on wave propagation. Also, the classical Rossby wave on a planetary scale exhibits a pronounced departure from the geostrophic balance on a b-plane especially at lower-middle latitudes.
Therefore, an alternative description of the large-scale wave circulations is proposed in Chapter 3. The large-scale perturbations will be represented by the geostrophic waves solved from the geostrophic perturbation equations introduced. To ensure the geostrophic balance of the wave solutions, we use scale analysis to derive the particular perturbation equations. The derivation is supported physically by the small-oscillation approximation introduced. The new wave pattern accommodates a three-dimensional structure and possesses horizontal divergence, but manifests a better geostrophic property compared with the non-divergent Rossby waves. The dynamic and thermal properties as well as propagation mechanism will be discussed in Chapter 4.
Although the large-scale waves may be illustrated approximately by the wave solutions of linearized perturbation equations, the atmospheric circulations are governed by the nonlinear primitive equations. When the geostrophic perturbation solutions are applied for the primitive equations, we obtain the remainder equations containing nonlinear terms of perturbations, which are generally smaller in the order of magnitude than the linear terms in the perturbation equations. However, they may have important influences on the climatological circulation patterns, such as the planetary stationary waves and their low-frequency variabilities, in the extratropical troposphere and stratosphere. While, the linear terms in the perturbation equations make no contribution to the long-term balances, because their integrations over a wave period or wavelength vanish.
Thus, the time-averaged circulations cannot be illustrated completely by simplified linear equations, and the heat and momentum balances in the time-averaged circulations should be calculated using the primitive equations. For the geostrophic wave circulations discussed, we shall use the remainder equations to calculate in Chapter 5 the long-term heat and momentum balances dominated by nonlinear processes. The results show that the perturbation fields can produce, by themselves, the zonally symmetric and asymmetric acceleration and heating, which act as internal forcings to the wave circulations and their variations.
The zonally asymmetric topography may produce external asymmetric forcings to atmospheric circulations. The time averaged perturbations forced by the asymmetric forcings must be asymmetric too, so that they may produce stationary internal forcings to balance the asymmetric external forcings. The typical example of the time-averaged asymmetric circulation pattern is the stationary waves in the atmosphere. According to the phase relations between the forced stationary waves and internal or external forcings, we may explain the observed geographic distributions of stationary waves at different latitudes in both hemispheres. The seasonal changes of stationary waves may then be interpreted in terms of the variations in the external thermal forcing. The stationary waves forced orographically and thermally are called the orographic and thermal stationary waves and investigated in Chapters 6 and 7 respectively.
It
will be revealed that the zonally asymmetric generation of momentum and heat
in the stationary waves depends highly on wave amplitudes. Usually, the
stationary wave amplitudes increase with height in the troposphere and the
external forcings weaken upward, so the forced
stationary waves may produce a residual amount of heat or momentum which may not
be compensated by external forcings at the high
levels. This process is referred to particularly as the over-generation of
asymmetric internal forcings. In addition, there is the
feedback of heat from orographic waves and momentum from thermal waves produced
by internal forcings. The heat and momentum balances may be broken down at high
levers, if the stationary circulation pattern is
unchanged in a long time period.
We shall argue in Chapter 8 that maintenance of the physical balances can be retained by changing the stationary wave phases and amplitudes in order to remove the residual generations of heat and momentum periodically. These adjustments in the planetary circulation patterns are responsible for the low-frequency variabilities, such as the blocking events in the troposphere. According to the time-averaged momentum and heat balances in the stationary waves with over-generation and self-feedback, we may explain in Chapter 8 the phase distributions of blocking waves and their seasonal changes in both hemispheres. Moreover, the interactions between orographic and thermal stationary waves give an interpretation for the zonally asymmetric distribution of synoptic disturbances in the troposphere, which are smaller in scale than the low-frequency anomalies.
The nonlinear transport processes resulting from the wave interactions in the atmosphere may also affect the time-averaged zonally symmetric circulations. According to the long-term balances of heat and momentum, we shall calculate, in Chapter 9, the gross structure of mean circulation fields, such as the zonal mean temperature, the mean zonal and meridional circulations and the mean eddy fluxes of heat and momentum in the geostrophic wave circulations. These mean fields obtained are comparable with those in the real atmosphere. We shall discuss also the different dynamic mechanisms of the direct and indirect mean meridional circulations in the troposphere.
The large-scale perturbations in the atmosphere frequently become unstable. The wave instabilities will be classified in this study into two categories. One is the stochastic instability of transient waves. The long-term heat or momentum balances may not be considered when this instability mechanism is discussed for individual episodes of disturbance development. The other category of instability concerns the long-term physical balances associated with nonlinear processes in the primitive equations, and is responsible for the low-frequency variabilities in the atmosphere.
The nonlinear planetary wave instability in the troposphere may cause development of blocking systems and is discussed in Chapter 10. The heat balance evaluated with the primitive equations may reveal the main features of nonlinear planetary wave instability. For example, it occurs only in the breaking layers where the zonal mean temperature decreases poleward and the gradient exceeds a certain limit. The development of planetary waves in the troposphere is characterized by the onset, development, maintenance and decay of blocks.
In the stratosphere, a breaking layer and nonlinear wave instability may occur only at high latitudes in the winter hemisphere. Their features may be applied to explain the winter time stratospheric variabilities such as the warm pools and sudden warmings. It will be found in Chapter 11 that the coherent heating produced by interference of waves in a stable breaking layer possesses the specific propagation velocity and thermal structure similar to those of warm pools observed in the stratosphere. While, the zonal mean acceleration and heating produced by unstable planetary waves in the breaking layers manifest the basic characteristics of stratospheric sudden warmings as shown in Chapter 12.
The stochastic instability in geostrophic wave circulations, which is responsible mostly for the variabilities with a life cycle less than a week will be investigated in Chapter 13. As the large-scale disturbances exhibit a quasi-geostrophic property, we shall adopt the same geostrophic perturbation equations to deal with the incipient instability. The further development will be studied with the general perturbation equations without using the small-oscillation approximation. In particular, the cyclogenesis in the extratropical troposphere will be distinguished from the general wave amplification by assuming that the zonal and meridional wavelengths are similar in the cyclonic disturbances.
The traditional theory of large-scale wave propagation and dispersion played a significant role in the studies of meteorological dynamics. This theory was based on the traditional expression of multidimensional group velocity. It will be argued in Appendix A that the traditional expressions of group velocity and the ray-tracing theory deduced may only be applied for isotropic waves, such as the spherical sound waves, of which the propagation speed is independent of the direction. While, the large-scale perturbations in the atmosphere are generally anisotropic. Thus, we discuss in Appendix A the group velocity and ray-tracing theory for anisotropic waves.
This study introduces, for the first time, the basic tactics and methods for solving the primitive equations analytically, by resolving meteorological fields into time and zonal means and their departures. The major difference from the previous studies is that the time-averaged circulation patterns and their variabilities are investigated in terms of the heat and momentum balances calculated using the primitive equations instead of linearized perturbation equations. This technology is successful for the study of large-scale circulations and their low-frequency variations related to the large-scale
perturbations in the extratropical atmosphere. While, further challenges remain in the studies on the circulations at low latitudes where the horizontal Coriolis force is relatively weak and the humidity is relatively high. The new theory of air engines developed currently by Zhu (2002) may be very helpful for studying the moist convective systems.