Dynamical constraints on monsoon circulations

Transcription

1 Dynamical constraints on monsoon circulations R. Alan Plumb Dept. of Earth, Atmospheric and Planetary Sciences Massachusetts Institute of Technology August 17, 2005 Abstract Monsoon circulations are particular clear examples of divergent ows driven by spatial variations in thermodynamic boundary conditions. Simple application of the circulation theorem yields a powerful constraint on steady, inviscid, divergent ows, which in dry, symmetric, cases implies a simple criterion for the existence of such ows in terms of the external forcing. This paper reviews recent work addressing the applicability, and limitations, of this constraint to more realistic moist, non-symmetric cases. 1 Introduction While much has been written on monsoon circulations, a comprehensive theory of their dynamics is still lacking. It is clear that, given the distribution of precipitation, many of the observed characteristics of monsoon ows can be understood on the basis of the linear response to diabatic heating (e.g., Gill [1980], Hoskins and Rodwell [1995]). Nevertheless, such solutions fall short of a complete theory. For one thing, in general the distribution of precipitation is not determined a priori by the externals of the problem, but is in itself a part of the atmosphere s response, and this is especially true in the presence of the heterogeneous surface conditions typical of monsoon situations. Explaining the distribution of precipitation over land is a key part of the monsoon problem. A second issue not resolved by linear theory involves placing the monsoon problem into the broader context of the theory of tropical dynamics. The widely accepted paradigm for the tropical Hadley circulation, the theory of Held and Hou [1980], hinges on nonlinear angular momentum advection in the upper troposphere, assumed inviscid, creating uniform absolute angular momentum right across the Hadley cell. To some extent, monsoon circulations form a part of the seasonal Hadley circulation. This is especially true in northern summer, when the zonally averaged upward motion in the tropics is dominated by the contribution over south Asia, so that in this season the Hadley circulation if by that one means the zonally averaged meridional overturning is essentially the zonal average of the south Asian 1

2 2 SUSTAINED DIVERGENT FLOWS: THE CIRCULATION CONSTRAINT IN THE UPPER TROPOS monsoon. Placed in that context, it is di cult to reconcile linear theories of monsoon ows with nonlinear theories for the Hadley cell. In fact, applying the principles of the Held-Hou theory to zonally symmetric cases with applied forcing o the equator leads to the conclusion that a threshold forcing must be exceeded before radiative-convective equilibrium is replaced by a deep (non-viscous) meridional circulation [Plumb and Hou, 1992; Emanuel, 1995], although under moist dynamics, the relevant criterion involves the distribution of boundary layer moist entropy and is more diagnostic than predictive. In principle, extending the zonally symmetric constraint of uniform angular momentum to a more general requirement of zero potential vorticity (PV) in the absence of zonal symmetry is straightforward [Schneider, 1987]. However, a patch of zero PV localized o the equator may become unstable; during northern summer, this is manifested in the shedding of eddies from the Tibetan anticyclone [Hsu and Plumb, 2000; Popovic and Plumb, 2001]. A distinct, better known, instability of the low level easterly jet that in simple models is ubiquitous in monsoon circulations (and is in fact strongest prior to monsoon onset) further complicates the picture. Nevertheless, the low level moist entropy distribution seems to hold the key to the existence of strong monsoonal ows, and to the distribution of continental precipitation. In this paper, these issues will be discussed in greater detail. There will be no attempt to give a comprehensive account of monsoon dynamics. Rather, what follows will be a personal view of the circulation constraint on divergent tropical ows, the way in which this constraint is violated, and its implications for monsoon dynamics. 2 Sustained divergent ows: the circulation constraint in the upper troposphere Consider a divergent, monsoonal, ow such as that depicted schematically in Fig. 1. We C θ Figure 1: Schematic of a divergent circulation. The focus of discussion in the text is on the upper level potential vorticity budget within the isentropic layer enclosed by the contour C: focus (following the arguments of Schneider [1987]) on the upper level out ow, where the ow might reasonably be assumed to be inviscid and adiabatic. The inviscid equation of

3 2 SUSTAINED DIVERGENT FLOWS: THE CIRCULATION CONSTRAINT IN THE UPPER TROPOS motion can be t u + a u = where a is absolute vorticity and B is the Bernouilli function. circulation around any xed closed contour C, then If = R u dl is the C t + a u dl = 0 : C Now if the contour lies within an isentropic surface and the ow at the level of interest is locally adiabatic, then I I a u dl = P u n dl ; C where = g p is the isentropic density and n the outward unit normal. If, furthermore, we choose the contour C to be one of constant potential vorticity P, we nally have, for steady ow (@ t = 0), I P u n dl = 0 : (1) In inviscid, adiabatic, steady ow, therefore, there can be no net divergent ow across any contour of constant PV on an isentropic surface, unless the value of PV on that contour is zero. On the face of it, this statement represents a major constraint on divergent tropical circulations in the upper troposphere, where one might regard the ow as nearly inviscid. Fig. 2 illustrates the point. Consider a localized region of upper level divergence, represented rb P P P 1 2 y (a) (b) x Figure 2: Schematic of the implications of (1). In each case, the shaded ellipse represents a region of upper level divergence; the contours represent PV on an upper tropospheric isentropic surface. In frame (a), there are closed PV contours enclosing the divergent region; in (b), the PV contours span the globe. by the shaded regions in the gures. In the rst case, frame (a), it is envisaged that the region of divergence lies within a closed P contour, in which case a steady conservative ow of this kind can exist, with nonzero net divergence within the contour, only if P = 0 on all closed contours enclosing the region of divergence, and thus on all such contours except those

4 2 SUSTAINED DIVERGENT FLOWS: THE CIRCULATION CONSTRAINT IN THE UPPER TROPOS Figure 3: July mean PV (upper) and Montgomery streamfunction (lower) at 370 K. [Popovic and Plumb, 2001; from NCEP reanalysis data.] that enclose all the compensating convergence as well as the divergence. If the PV contours span all longitudes, as in case (b), then if the region between the two contours P = P 1 and P = P 2 contains the region of divergence (and not all of the compensating convergence) there must be net divergent ow across at least one of those contours, in which case P must be zero on that contour. This constraint is well known in the theory of the zonally averaged Hadley circulation (Held and Hou, 1980; Lindzen and Hou, 1988). Under zonal symmetry, a = (a 2 cos ) m, where m = a 2 cos 2 ' + ua cos ' is the absolute angular momentum per unit mass, and so vanishing of upper tropospheric absolute vorticity over the region of the divergent Hadley circulation amounts to uniform absolute angular momentum there. In reality, absolute vorticity does not vanish across the tropical upper troposphere, although in much of the region relative vorticity is anticyclonic, making the absolute vorticity weaker than the planetary term alone. Thus, while the theory goes some way toward explaining the observed circulation, it is clear that angular momentum is not well conserved in reality: air is permitted to cross angular momentum surfaces through the agency of eddy momentum transport, whose e ects are neglected in the foregoing theory. Monsoon circulations comprise a major component of the divergent tropical ow, especially in northern summer when the circulation is dominated by the ow of the South Asian monsoon. Four-year means of upper tropospheric absolute vorticity and Montgomery potential M for July are shown in Fig. 3. The huge Tibetan anticyclone that so dominates M is

5 3 CRITERIA FOR SUSTAINED DIVERGENT CIRCULATIONS 5 also evident in the absolute vorticity distribution; the core of the anticyclone is a local minimum in a (which coincides with the elevated tropopause over the region). Nevertheless, summertime-mean a is signi cantly di erent from zero across south Asia, especially between the monsoon region and the equator. How, then, does the extensive, cross-equatorial divergent circulation satisfy the circulation constraint? Vorticity budget analysis in the region of the Tibetan anticyclone shows that, as in the zonally averaged picture, transient eddy uxes play a leading order role. These do not, however, appear to be eddies of midlatitude origin crashing into the topical upper troposphere; rather, both models and observations indicate that the anticyclone itself is the source of these eddies. 3 Criteria for sustained divergent circulations The circulation constraint states that, if the upper level PV remains cyclonic, there can be no sustained upper level divergence, assuming the ow there to be steady and inviscid. Plumb and Hou [1992] argued, on the basis of a dry, zonally symmetric, model with diabatic heating and cooling represented by Newtonian relaxation to an equilibrium temperature T e possessing a localized subtropical maximum, that this implies that a threshold of the magnitude of the T e maximum must be exceeded before a divergent circulation can be sustained. This follows from consideration of the radiative equilibrium state, when T = T e and the zonal wind u (assumed zero at the ground) is in gradient wind balance with the temperature eld. It is straightforward to show that absolute vorticity is cyclonic at the tropopause (z = D), and therefore that the radiative equilibrium solution is regular, provided gd T r sin cos cos 3 ^T e < f 2 a 2 ; (2) where f = 2 sin is the Coriolis parameter, ^Te is the vertically averaged radiative equilibrium temperature and T r a reference temperature. When (2) is satis ed, no upper level divergent ow is permitted and numerical simulations show only weak (viscously driven) meridional circulations. On the other hand, when the forcing is strong enough for (2) to be violated, the equilibrium solution becomes irregular, and numerical solutions show a strong meridional circulation spanning an extensive region of near-zero upper level PV. Emanuel [1995] extended the argument to a moist atmosphere, in which case the radiativeconvective equilibrium state is dictated by the distribution of boundary layer moist entropy, and (2) is replaced by sin cos 3 cos sin s b < f 2 a 2 (3) where T is the temperature di erence between the surface and tropopause, and s b the boundary layer moist entropy. The relevance of (3) has been demonstrated in zonally symmetric moist models [Emanuel, 1995; Zheng, 1998]. It has even been found to be relevant in three-dimensional models where, moreover, the location of the s b maximum appears to

6 4 VIOLATION OF THE CIRCULATION CONSTRAINT IN THE UPPER TROPOSPHERE6 determine the poleward limit of the divergent circulation, and thus the maximum inland penetration of the monsoon [Privé and Plumb, 2005a,b], consistent with the ndings of Chou and Neelin [2003]. Thus, the essential dynamical constraint of the upper level circulation budget manifested in (2) does seem to extend into the moist, three-dimensional case, though with some caveats. The rst caveat is that, in the moist case, the threshold criterion is less useful as a predictor than in the dry case, simply because s b is not always a simple function of the external boundary conditions. Advection by even a weak divergent circulation in the boundary layer (which is always present, even in subcritical cases) can have a substantial impact on the distribution of s b near the coast. A second caveat is that, as we have seen, time-averaged upper level PV does not vanish over extensive regions (though it may become weak). Three-dimensional dynamics i.e., the presence of eddies allows the assumptions leading to (1), and thence to (2) and (3), to be violated. These eddy processes, which appear to be inevitable in a monsoon circulation, will be discussed in the following sections. Their existence means that, in realistic situations, (3) can only be regarded as approximate. 4 Violation of the circulation constraint in the upper troposphere How the steady circulation constraint can be broken can be understood from simple arguments, and is revealed by simple models. Let us represent the divergent upper tropospheric as the out ow from a localized mass source in a single layer on a beta-plane, as depicted in Fig. 4. As zero PV air spreads out from the source region, it does so asymmetrically, Figure 4: Schematic of shallow-water ow away from a localized source (marked by the heavy dot) on a beta-plane. North is at the top. spreading preferentially westward as a beta-plume (Rhines, 1983). As the plume extends westward, it becomes unstable, as expected for an elliptical vortex; a wave forms along its edges, rolls up as an anticyclonic eddy, and is shed westward from the now-diminished source vortex. An example of this occurring in a shallow water model is shown in Fig. 5. In such

7 4 VIOLATION OF THE CIRCULATION CONSTRAINT IN THE UPPER TROPOSPHERE7 Figure 5: Periodic shedding of an anticyclone from a mass source on a shallow-water betaplane. Color shading is absolute vorticity (scale at right); contours are free surface height; arrows show velocity. One period of the cycle is shown; legend gives time in fractions of a period. [Hsu and Plumb, 2000.]

8 4 VIOLATION OF THE CIRCULATION CONSTRAINT IN THE UPPER TROPOSPHERE8 simple cases, the behavior is very periodic, and it su ces to show a single period. Low PV air emanating from the source is shed from the source region within the detached eddy. Thus, the source ow diverges away from the source region, into an environment of nonzero PV, violating the circulation constraint by virtue of the sustained time-dependence of the ow. Alternatively, from a time-averaged perspective, net divergent ow occurs through the agency of transient eddy PV uxes, the eddies produced, inevitably, as a consequence of the instability of the vorticity distribution created by the divergent ow itself. This kind of shedding behavior is indeed evident in the upper troposphere, at least in the vicinity of the Tibetan anticyclone in northern summer. Fig. 6 shows one example Figure 6: Illustrating eddy shedding at 370 K (just below the tropopause). Time sequence of PV eld (color shading) and the geopotential height at 200 hpa (contours) over the Asian summer monsoon at successive 18-h time intervals from 0000 UTC 11 Jul to 0600 UTC 13 Jul The color contour interval is 0.05 PVU. The black contour is 25 m. [Hsu and Plumb, 2000.] of this, an event that took place during July Over the 54-hr period shown, the main anticyclone elongated westward and split, shedding a large anticyclone as far as 300 E. Subsequently (not shown) the detached eddy becomes caught up in the westerly jet immediately to its north, sheared out rapidly eastward into an lament that appears, to some extent, to re-merge with the main anticyclone. This kind of event is not an isolated occurrence; typically, three or four such events can be identi ed over the course of a summer

9 5 LOWER TROPOSPHERIC EDDIES AND THEIR IMPACT 9 [Popovic and Plumb, 2001]. This behavior, then, makes it apparent that, in reality, the circulation constraint can be violated in the upper troposphere, and the divergent ow can thus extend across nite regions of nonzero PV, and that it is the dynamics of the divergent ow itself that allows this to happen (rather than, say, incidental transport from eddies of extraneous origin). Does this upper tropospheric behavior in uence the overall characteristics of the monsoon ow? The direct signal of anomalies in PV and M extends down only as far as 340K (about 400hPa) [Popovic and Plumb, 2001], so there is no obvious feedback onto the lower tropospheric ow. Nevertheless, Randel and Park [2005] have identi ed a coherent relationship between anomalies in upper level PV anomalies and in OLR, suggesting a relationship with deep convection; however, since the OLR signal appears to lead the PV signal, it is not clear whether this implies an impact of upper tropospheric behavior onto the convection. As an aside (since our main focus here is on the monsoon dynamics), the upper tropospheric activity we have noted here may be relevant to issues of transport in the vicinity of the tropical tropopause. Note from Fig. 5 that in concert with the shedding of the anticyclone, cyclonic air is entrained equatorward immediately to the east of the source region and rolls up cyclonically to the west. If this occurs in reality (and there are hints of it in Fig. 6), this is tantamount to entrainment of stratospheric air into the tropical upper troposphere. Such transport would be complementary to the poleward transport of tropical tropopause air into the stratosphere noted by Dethof et al. [1999]. Randel and Park [2005] have recently discussed the observed characteristics of tracer transport into and in the vicinity of the Tibetan anticyclone. 5 Lower tropospheric eddies and their impact Eddy formation in the lower troposphere, quite distinct from what may happen near the tropopause, appears to be ubiquitous in summertime situations where land lies poleward of an equatorial ocean. In such situations, which are not restricted to those with an active monsoon, the easterly jet associated with the landward temperature gradient becomes unstable, generating easterly waves (e.g., Rennick [1976], Thorncroft and Hoskins [1994]). In fact, in a seasonally varying model with a simple continent, Xie and Saiki [1999] found monsoon onset to occur just after the appearance of such waves, and ascribed the onset to the in uence of these eddies. Xie and Saiki s observation gives grounds for doubting the relevance of the steady circulation constraint in the presence of eddies. Consider the schematic Fig. 7. Suppose that s b has a localized maximum at latitude 0 ; the distribution of a in the middle or upper troposphere is shown schematically for four di erent magnitudes of this maximum (including zero, the straight line). a rst reaches zero in case 3, but is evident that its gradient changes sign, thus raising the possibility of instability, long before this stage is reached. If the waves generated by the instability are responsible for monsoon onset, then, this may occur at lower forcings than (3) would suggest. The role of these waves, amongst other things, was investigated by Privé and Plumb

10 5 LOWER TROPOSPHERIC EDDIES AND THEIR IMPACT 10 zeta_a y Figure 7: Schematic showing the middle or upper level absolute vorticity distribution with latitude for a sequence of cases with a localized s b maximum of increasing magnitude (corresponding to decreasing minima of a at y = = 0 = 1). The sign of the absolute vorticity gradient changes sign at lower magnitude than the sign of absolute vorticity itself. [2005a,b] in two- and three-dimensional models of the response of a moist atmosphere to the presence of continents with simple geometries. In the case shown in Fig. 8, a single continent was placed everywhere north of 16 0 N, with prescribed SST (with an equatorial maximum) south of the coast. The total surface heat ux on the continent was prescribed, and the model run (for several hundred days) to an equilibrated state. For su ciently large continental heat ux, a monsoonal state was produced. For weaker (subcritical) forcing, the continent was found to be very arid and hot almost everywhere (except very close to the coast) with relatively shallow dry convection; the corresponding mid-level easterly jet along the coast became unstable and generated eastward propagating disturbances. The case of Fig. 8 is a particularly simple example of this, with a single eddy propagating westward along the coast. The steadily translating cyclonic eddy has a rainfall maximum on its eastern ank, where moisture is advected onto the land from the ocean, and a very dry region to its west, where the low level ow is divergent (as a consequence of Rossby-wave-induced subsidence the Rodwell-Hoskins [1996] e ect). While, in such cases, the eddies are obviously in uential in organizing the rainfall pattern, do they in fact facilitate monsoon development? With this con guration of a zonally uniform continent, it is relatively easy to address this question directly, by comparing the results of the 3D model with those of a zonally symmetric, 2D, model, in which of course the eddies and their e ects are suppressed. Making such a comparison, we [Privé and Plumb, 2005b] found the impact of eddies to be surprisingly modest and, in fact, to be such as to inhibit monsoon development in the sense that creation of a monsoon required slightly larger surface heat uxes over the land in the presence of eddies rather than to encourage it. The reason for

11 5 LOWER TROPOSPHERIC EDDIES AND THEIR IMPACT 11 Figure 8: A particularly simple example of a westward propagating eddy proceed by instability of a coastal easterly jet. A continent, with prescribed surface heat ux, occupies the region everywhere north of 16 0 N; to the south is ocean with prescribed SST. This case is slightly subcritical in the sense that the continental heat ux is just too weak to produce a monsoon. The lower frame shows rainfall at the coastal grid point vs. longitude and time; the upper frame shows rainfall and 1000 hpa ow, composited with respect to the moving disturbance. [After Privé and Plumb, 2005b.]

12 6 CONCLUSIONS 12 this is, when framed in terms of boundary layer moist entropy, straightforward: air advected from the ocean onto the land by the eddies, while moist, has lower entropy than that in the continental boundary layer. Thus, the e ect of the eddies is to weaken the land surface maximum in s b, and thereby to suppress monsoonal circulations. While the generation of westward propagating disturbances weakens once a monsoonal circulation appears (when the low level temperature gradient decreases) the eddies weaken, though they do not entirely disappear. Fig. 9 shows results from a case with a northern hemisphere continent occupying the region north of 16 0 N, and extending from in longitude. Like the previous case, the surface heat ux is speci ed (as a function of latitude only) at the land surface; SSTs are speci ed, with a maximum at 8 0 N. Signi cant time-averaged precipitation on the continent is con ned to the extreme southeast corner, where it almost merges with the midlatitude rainband associated with the oceanic storm track. What little precipitation there is along the central and western parts of the south coast is associated with weak westward propagating disturbances, as shown in the bottom frame of the gure. (In fact, as also found by Cook and Gnanadesikan [1991], continental aridity appears di cult to avoid in models with such simpli ed geography.) The aridity of all but the eastern continent is not primarily a Rossby wave e ect (it extends too far for that) but a consequence of the low-level advection of low s b air eastward from the cool midlatitude ocean. Like Chou et al. [2001] and Chou and Neelin [2003], we nd that the extent of the monsoonal rainfall in longitude as well as in latitude is very much restricted by the requirement of high s b. In fact, if advection of low entropy air onto the continent is suppressed by the addition into the model of impermeable walls (extending up to 700 hpa) at the eastern and western coasts, intense rain occurs over the land all along the south coast. 6 Conclusions While the circulation constraint yields, for dry dynamics under zonal symmetry, a simple predictor for the existence of divergent circulations in terms of the external forcing (the radiative equilibrium temperature distribution), it becomes progressively less direct when moisture and three-dimensionality are added. While the constraint still exists, and can be framed in terms of boundary layer moist entropy for steady ow, it is no longer a simple statement of external conditions, and is at best approximate, because of the presence of eddies which arise from instability of either the monsoon ow itself (in the upper troposphere) or the low-level easterly jet that is a consequence of the very same land-sea contrast that produces the monsoon. The upper level instability ensures that the zero PV limit is never reached (on a time-average; instantaneously, analyses suggest the frequent occurrence of zero or nearzero PV). Whether the quasi-periodic shedding of eddies in the upper troposphere has any signi cant manifestations in the big picture of the monsoon, or is merely an upper-level curiosity, is not clear, though suggestions of a relationship between uctuations of upper level PV and outgoing longwave radiation, reported by Randel and Park [2005], are intriguing. The low-level disturbances seen in models with simple continents [Privé and Plumb, 2005b], which appear to be representative of easterly waves in pre-monsoon conditions, and

13 6 CONCLUSIONS 13 Figure 9: Illustrating a case with a continent lying north of 16 0 N and between 0 0 and in longitude, with prescribed SST peaking at 8 0 N. Top: Time-averaged 1000 hpa winds and rainfall; bottom: rainfall at 16 0 N vs. longitude and time, showing westward propagating disturbances.

14 6 CONCLUSIONS 14 perhaps of monsoon depressions in an active monsoon, are very e ective at organizing rainfall, and indeed in producing rainfall in otherwise arid regions. Nevertheless, they appear to play a rather modest role in monsoon onset, delaying it slightly by advecting low entropy air from the ocean onto the land. Acknowledgments. I thank Nikki Privé for many discussions and for providing some of the gures, and Bill Randel for sharing his results of upper tropospheric transport in advance of publication. This work was supported by the National Science Foundation. References Chou, C., and J. D. Neelin: Mechanisms limiting the northward extent of the northern summer convection zones. J. Clim., 16, (2003). Chou, C., J. D. Neelin, and H. Hsu: Ocean-atmosphere-land feedbacks in an idealized monsoon. Quart. J. R. Meteor. Soc., 127, (2001). Cook, K. H., and A. Gnanadesikan: E ects of Saturated and Dry Land Surfaces on the Tropical Circulation and Precipitation in a General Circulation Model. J. Clim., 4, (1991). Dethof, A., A. O Neill, J. M. Slingo, and H. G. J. Schmidt: A mechanism for moistening the lower stratosphere involving the Asian summer monsoon. Quart. J. R. Meteor. Soc., 125, (1999). Emanuel, K. A.: On thermally direct circulations in moist atmospheres. J. Atmos. Sci., 52, (1995). Gill, A. E.: Some simple solutions for heat-induced tropical circulation. Quart. J. R. Meteor. Soc., 106, (1980). Held, I. M., and A. Y. Hou: nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, (1980). Hoskins, B. J., and M. J. Rodwell: A Model of the Asian Summer Monsoon. Part I: The Global Scale. J. Atmos. Sci., 52, (1995). Hsu, C. J., and R.A. Plumb: Non-axisymmetric thermally driven circulations and upper tropospheric monsoon dynamics. J. Atmos. Sci., 57, (2000). Plumb, R. A., and A. Y. Hou: The response of a zonally-symmetric atmosphere to subtropical thermal forcing. J. Atmos. Sci., 49, (1992). Popovic, J. M., and R. A. Plumb: Eddy shedding from the upper tropospheric Asian monsoon anticyclone. J. Atmos. Sci., 58, (2001). Privé, N. C., and R. A. Plumb: Monsoon dynamics with interactive forcing. Part I: Axisymmetric studies. Submitted to J. Atmos. Sci., (2005a) Privé, N. C., and R. A. Plumb: Monsoon dynamics with interactive forcing. Part II: Impact of eddies and asymmetric geometries. Submitted to J. Atmos. Sci., (2005b). Randel, W. J., and M. Park: Deep convective in uence on the Asian summer monsoon anticyclone and associated tracer variability observed with UARS. Submitted to J. Geophys. Res. (2005).

15 6 CONCLUSIONS 15 Rhines, P.: Lectures on geophysical uid dynamics. Lect. Appl. Math., 20, 1 58 (1983). Rennick, M. A.: The Generation of African Waves. J. Atmos. Sci., 33, (1976). Rodwell, M. R., and B. J. Hoskins: Monsoons and the dynamics of deserts. Quart. J. R. Meteor. Soc., 122, (1996). Schneider, E. K.: A simpli ed model of the modi ed Hadley circulation. J. Atmos. Sci., 44, (1987). Thorncroft, C. D., and B. J. Hoskins: An idealized study of African easterly waves. I: A linear view. Quart. J. R. Meteor. Soc., 120, (1994). Xie, S.-P., and N. Saiki: Abrupt onset and slow seasonal evolution of summer monsoon in an idealized GCM simulation. J. Meteor. Soc. Japan, 77, (1999). Zheng, X.: The response of a moist zonally symmetric atmosphere to subtropical surface temperature perturbation. Quart. J. R. Meteor. Soc., 124, (1998).