1. Introduction

The classic pattern of the meridional structureof global atmospheric circulation can be attributed tothe combined effect of the uneven distribution of so-lar radiation,earth's rotation, and mass continuity.Westerly winds prevail in the mid-high latitudes,s and -wiched by easterly trade winds in the low latitudes and easterlies in the polar latitudes. The uneven distribu-tion of l and and ocean, and the complex topographyof earth's surface result in complicated and zonallyasymmetric patterns of atmospheric general circula-tion,responsible for the distinct weather and climatein different parts of the world. Earth's atmosphere canbe regarded as an integrated and continuous fluid, and disturbances in one specific region will certainly affectthe atmospheric motion in other regions. For exam-ple,strong convective disturbance is almost perennialin the tropics. But to what extent do these tropicaldisturbances affect the midlatitudes? Similarly,howdoes frontal activity in the midlatitudes influence at-mospheric motion in the tropics, and even that in the other hemisphere? If links exist between the atmospheric motion in different regions,do they follow thesame general rules? These questions form the focusof this review article. In addition,underst and ing thelinks and physical mechanisms of atmospheric activityin different regions will undoubtedly improve the accuracy of weather and climate prediction(Ding,2009).

Since the invention of radiosonde in 1928,thequantity and availability of upper atmospheric observations has grown markedly. A specific wave patternwith a scale of several thous and s of kilometers wasfound in the mid-upper troposphere over the midlatitude regions. Rossby(1939)first pointed out thatthe formation of this type of wave is due to earth'srotation and the variation in the Coriolis effect withlatitude(β effect). This type of wave is thereby knownas the Rossby wave. Based on the differences on spatial and temporal scales,Rossby waves can be divided into synoptic- and planetary-scale Rossby waves(Tan,2008). Jaw(1946),Charney(1947), and Eady(1949)investigated synoptic-scale Rossby waves and argued that their genesis and development are dueto atmospheric baroclinity. Charney and Eliassen(1949),Bolin(1950), and Smagorinsky(1953)stud-ied planetary-scale Rossby waves and found that topo-graphic forcing and diabatic heating are the triggersfor stationary planetary waves. Yeh(1949)made asignificant contribution to the theory of Rossby wavepropagation in a pioneering study of Rossby wave en-ergy dispersion and the upstream-downstream effectsof one-dimensional Rossby wave propagation. Thetheory of energy dispersion has important implicationsfor how the atmospheric disturbance in a specific re-gion can affect the activity in other remote regions,via wave propagation and energy dispersion.

By the mid 1960s,studies of midlatitude atmo-spheric wave dynamics had already reached a ma-ture stage. However,studies of tropical atmosphericwave dynamics had only just begun,due partly to thelack of observations over the tropical oceans at thattime. Matsuno(1966)obtained an analytical solu-tion of tropical waves based on shallow-water equa-tions. Soon after,based on the observational anal-ysis,Yanai and Maruyama(1966),Maruyama and Yanai(1967), and Wallace and Kousky(1968)veri-fied the existence of mixed Rossby-gravity and Kelvinwaves. Since then,rapid advancements have beenmade in tropical wave dynamics. The majority of re-search carried out indicates that tropical atmosphericwaves play a crucial role in the development of con-vection,in troposphere-stratosphere interaction, and in the ENSO cycle(Lindzen and Holton, 1968; Holton and Lindzen, 1972; Hirota,1978; Jin, 1997a,b; Bald-win et al., 2001).

Alongside the above progress,research has alsofocused on the interaction between the tropics and ex-tratropics. Based on scale analysis,Charney(1963)explored the large-scale atmospheric motion in thetropics,arguing that perturbation energy in the trop-ical upper troposphere largely comes from the eddyactivity within subtropical westerly jets,while pertur-bations in the tropical lower troposphere mainly arisefrom the unstable basic flow itself or from convectiveheating in the intertropical convergence zone(ITCZ).In a study of large-scale circulation over the tropicalPacific,Yanai et al.(1968)found two separate fre-quency spectral peaks in the troposphere{one locatedclose to the sea surface, and the other at the top ofthe troposphere,consistent with Charney(1963). Mak(1969)also pointed out that the mechanism for thegeneration of disturbances in the tropical upper tro-posphere is a response of the stable barotropic tropicsto the unstable baroclinic temperate zone. Charney(1969)further indicated that meridional propagationof both synoptic- and planetary-scale waves can oc-cur only when the zonal phase speed of Rossby wavesis less than the speed of mean zonal flow. Whena synoptic-scale midlatitude perturbation propagatesequatorward and reaches a specific latitude where thezonal phase speed of perturbation is equal to the speedof mean zonal flow,it will stop propagating towardthe equator. This latitude is the so-called critical lat-itude/layer. Numerous studies have investigated theinteraction between waves and mean flow within thecritical layer. For example,Dickinson(1968,1970),Bennett and Young(1971), and Geisler and Dickinson(1975)explored the linear interaction between Rossbywaves and the critical layer; Béland (1976) and Warn and Warn(1978)studied the nonlinear interaction be-tween Rossby waves and the critical layer; Killworth and McIntyre(1985) and Brunet and Haynes(1996)suggested that Rossby waves can be absorbed,re-flected or even super-reflected at critical latitudes.

In other words,the critical latitude acts like a\barrier" that blocks the interaction between the trop-ics and extratropics. However,there is much observa-tional evidence of significant links between the atmo-spheric motion of the tropics and extratropics, and even between the Northern Hemisphere(NH) and theSouthern Hemisphere(SH). Radok and Grant(1957)argued that westerlies sometimes appear in the up-per troposphere over Australia during boreal winter,associated with synoptic activity north of the equa-tor. Based on observational data from tropical sta-tions,Tucker(1965)found large meridional variationin time-averaged zonal winds. Weak westerlies occurin the upper troposphere in the Western Hemisphere and are accompanied by strong cross-equatorial mo-mentum flux,while the Eastern Hemisphere is un-der the control of strong easterlies. Based on bal-loon observations in the SH upper troposphere,Web-ster and Curtin(1975)found a large-scale slow-movingtrough in the midlatitudes,which tilts in a southeast-northwest direction horizontally, and is quite distinctover the Pacific. This feature indicates that there is alarge amount of momentum flux transported from thetropics to the midlatitudes. Murakami and Unnina-yar(1977)used U.S. National Meteorological Centerobservational data to compare the time-mean distri-bution of zonal winds and turbulent kinetic energy, and found the location of perturbation kinetic energyin the equatorial area to be highly correlated with thezonal winds.

The need to better underst and observational re-sults led to the development of many theoretical mod-els that explore the physical mechanisms underpinningthe interaction of atmospheric motion in different lat-itudes. In this review,we focus on great circle theory,westerly duct theory,energy accumulation and waveemanation theory,equatorial wave lateral expansiontheory, and meridional basic flow theory,in explain-ing the atmospheric interaction between different lat-itudes,especially that between the mid and low lati-tudes, and between the NH and SH. In addition,wealso briefly discuss the advancements that have beenmade in the use of wave activity flux,a powerful toolfor diagnostic analysis of wave energy propagation and wave-mean flow interaction.

2. Great circle theory

Based on one-point correlation analyses,Horel and Wallace(1981) and Wallace and Gutzler(1981)found a significant teleconnection pattern in the NHwinter at the 500-hPa geopotential height. Due to theimportance of atmospheric teleconnection for weather and climate prediction,scientists have since been at-tempting to underst and the nature of atmosphericteleconnections. The two-dimensional wave propaga-tion theory proposed by Hoskins et al.(1977) and Hoskins and Karoly(1981)is a representative work ofsuch studies.

Atmospheric energy dispersion emerged as a the-ory following the early work of Yeh(1949). How-ever,Yeh's formula is limited to one-dimensionalwave propagation,when the real atmosphere is ona two-dimensional sphere. To underst and planetarywave propagation on a spherical surface,Hoskins etal.(1977)investigated the propagation characteristicsof perturbation vorticity using the linear barotropicvorticity equation. Grose and Hoskins(1979)sub-sequently studied the propagation of linear Rossbywaves triggered by large-scale topography in the zonalmean basic flows. They found that a topography-forced Rossby wave tends to split into northward- and southward-propagating wave trains downstream of thetopography. The northward-propagating wave trainnorth of 40±N possesses a longer wavelength than thesouthward-propagating one when using the 300-hPaboreal winter zonal mean zonal flow as the basic state.Note,however,that Grose and Hoskins(1979)only in-vestigated barotropic wave propagation in their study.Considering the strong atmospheric baroclinicity inmidlatitude winter,Hoskins and Karoly(1981)useda five-layer baroclinic model to further investigate theatmospheric responses to thermal forcing and large-scale topography.

Sea surface temperature(SST)anomalies can of-ten lead to vertical motion(Bjerknes,1966). Gill(1980)studied the tropical atmospheric response todiabatic heating by using a simple theoretical model.It indicated that eastward-propagating Kelvin waves and westward-propagating Rossby waves are triggeredif there is a localized heat source symmetrical with re-spect to the equator. Hence,it is important to studythe remote atmospheric responses to, and the propa-gation of disturbance triggered by local heating.

Hoskins and Karoly(1981)imposed a negativevortex anomaly in the upper troposphere to repre-sent atmospheric thermal forcing. As shown in Fig. 1,they found that the perturbation vorticity initiallypropagated northeastward, and then the wave laterstarted to split. The relatively longer of the resultanttwo waves continued to propagate poleward and evencrossed the north pole to the other hemisphere. Mean-while,the shorter wave turned to propagate equator-ward. A large number of experiments(Hoskins and Karoly, 1981)indicated that the basic propagationpattern of the perturbation vorticity is insensitive tothe location and intensity of the thermal forcing,al-though the phase and amplitude of the wave may beaffected. That is to say,the wave train will only propa-gate northward,eastward,or equatorward,consistentwith results derived from barotropic models.

In addition to thermal forcing,the effect of large-scale topography cannot be ignored. Due to the fieffect,extra-long Rossby waves can generate cyclonesover mountain ridges. In contrast,for other relativelyshort-length waves,anticyclones are produced overmountains due to zonal vorticity advection. Hence,topographic forcing exerts different effects on waves ofdifferent wave lengths, and thus the phases of telecon-nections may be different(Smith,1979). Therefore,large-scale topography is another source for triggeringatmospheric disturbance.

Hoskins and Karoly(1981)proposed that vortic-ity and geopotential height disturbances would ap-pear at 300 hPa if large-scale topography was in-serted into the westerlies. The propagation of thewave train would split around 40°N,with the rela-tively long waves propagating poleward while shortwaves move equatorward,again consistent with resultsfrom barotropic models.

Numerical modeling indicates that large-scaleRossby waves propagating in a slowly varying mediumare quasi-barotropic. Using the non-divergent lin-ear barotropic vorticity equation,Hoskins and Karoly(1981)obtained the Rossby wave dispersion relationon a two-dimensional spherical surface as follows:

where ω is frequency,k and l are zonal and meridional wave numbers, and um = u/cosφ is the time-meanzonal-mean wind,u,weighted by the cosine of latitude.This is an expansion of the one-dimensional Rossbydispersion relationship proposed by Yeh(1949). Us-ing a wave ray tracing technique,Hoskins and Karoly(1981)obtained the following relationship to describeRossby wave propagation in a spherical atmosphere:where α is the so-called turning latitude,where themeridional propagation of the Rossby wave will re-verse its direction; λ₀ is the longitude where the initialperturbation is located; and ε² = ε[2(Ω+ ω)]^-1. Ac-cording to the ray theory for wave propagation,thisequation actually yields a great circle path(Longuet-Higgins,1964). Along this circular path,Rossby waveenergy propagates at a speed of C_g = 2εak(aω).Therefore,once the location of the initial perturbationis decided,the propagation paths for various wave-length Rossby waves can be calculated.

Figure 2 shows the relationship between theRossby wave propagation path,zonal wave number, and the location of the initial perturbation. Figure 2a indicates that the maximum wave amplitude oc-curs at higher latitudes with smaller zonal wave num-bers. Figures 2b-d show that the Rossby wave prop-agation path is almost the same when the initial per-turbation is located at 15°,25°,or 35°N. The Rossbywave splits at a specific latitude during its propaga-tion,with the longer of the two resultant waves prop-agating poleward, and the shorter wave moving equa-torward,forming great circle paths.

The great circle theory of Rossby wave propaga-tion in a spherical atmosphere has been widely applied.Numerous observational and numerical studies haveproven the existence of the great circle path. Manystudies have shown that the midlatitude zonal west-erly jet plays a waveguide role in Rossby wave propa-gation(Simmons and Hoskins, 1979; Hoskins and Am-brizzi, 1993; Ambrizzi et al., 1995). The wave trainpropagates along the jet axis, and strong disturbanceoccurs at the jet stream exit. The westerly jet waveguides are highly correlated with the East Asian "SilkRoad" teleconnection pattern and NH teleconnectionpattern(Enomotol et al., 2003; Ding and Wang, 2005;Liu et al., 2014). Zhou et al.(2012)found that aPacific-North America-like wave train appears in EastPacific in the NH winter,which propagates eastward and upward, and reaches the stratosphere over NorthAmerica. Zheng et al.(2013)argued that wintertimeprecipitation anomalies over the Indian-western Pa-cific Ocean significantly affect atmospheric circulationin Asia. Tan et al.(2015)further pointed out thatthe dipole pattern of heating in the tropical westernPacific can trigger a Rossby wave train,which prop-agates toward the midlatitude region of the northernPacific along the extratropical jet before turning to-ward East Asia and then Alaska Bay,where it finallyforms the teleconnection wave train through wave-mean flow interaction over the eastern Pacific. In addi-tion,blocking and large-scale ridges and troughs alsoaffect the great circle path of Rossby wave propaga-tion(Trenberth,1986; Magaña and Yanai, 1991; Suo et al., 2008). Rossby wave propagation is of great im-portance in modulating regional climate(Ding et al., 2014; Trenberth et al., 2014). As suggested by Chen et al.(2005),Wang et al.(2009), and Tan and Chen(2014),stationary wave activity in the NH affects thewinter monsoon in Asia, and subsequently,tempera-ture anomalies in East Asia.

Researchers in China have contributed greatlyto our underst and ing of Rossby wave propagation,through both theoretical and numerical studies. Chao(1977)analyzed the development and dynamical fea-tures of spiral-shaped Rossby waves and their ef-fects in maintaining atmospheric circulation. Lu and Zeng(1981)explored the evolutionary characteristicsof long waves in the barotropic atmosphere,suggest-ing that perturbation development depends entirelyon the structure of the perturbation itself,particu-larly the direction of the perturbation axis and itsposition relative to the basic flow. Zeng(1983a,b)used Wenzel-Kramers-Brillouin theory to investigatethe evolutionary features of Rossby wave packets ina three-dimensional baroclinic atmosphere and sug-gested that the development of a wave packet is re-lated to the nature of the wave packet itself and its location relative to the basic flow. Through study-ing the vertical propagation of Rossby waves,Huang(1983)found that two wave guides exist in the winter-time NH. Liu and Tan(1988a,b)investigated the im-pact of topographic forcing and the fi effect on Rossbywave propagation,suggesting that east-west orientedtopography is conducive to the formation of troughs and shear lines,while north-south oriented topogra-phy mainly leads to shorter wavelengths for the lead-ing waves and longer wavelengths for the draggingwaves. Tan(1990,1993)also studied the stabilityof Rossby waves and the interactions between nonlin-ear Rossby waves, and suggested that the stability ofRossby waves is not only related to the spatial distri-bution of basic flow,but also to the structure of thewave itself and its location within the basic flow.

3. Westerly duct theory

When studying the atmospheric interaction be-tween different latitudes,large discrepancies exist be-tween the theoretical critical latitude and observations Thus,what are the possible mechanisms involvedin breaking this critical latitude,allowing the propa-gation of midlatitude disturbance to lower latitudes and even into the other hemisphere?

Charney(1969)suggested that planetary Rossbywaves can only propagate to low-latitude regions whentheir westward phase speed is greater than the max-imum speed of tropical easterly wind. This is knownas the weak westerly waveguide. However,this typeof wave only contains limited energy and thus cannotexert any great impact on the atmospheric activityat low latitudes. Synoptic-scale Rossby waves cannotcontinue to propagate equatorward once they reachthe critical latitude,since the zonal wind speed is low.Charney(1969)also indicated that the propagation ofwave disturbance from the midlatitudes to the trop-ics often occurs in the upper troposphere and lowerstratosphere,since weak easterly and westerly windsat this level are conducive to the meridional propaga-tion of waves.

Note,however,that Charney(1969)only consid-ered the zonally averaged basic state,which is simply afunction of latitude and height. In reality,time-meanzonal flow varies with longitude as well. On the ba-sis of Charney's argument,Webster(1973)studied theresponse of zonally homogeneous basic flow to a sta-ble forcing and found that the response in the equa-torial region is a slowly varying Kelvin wave,whilein the midlatitude region a Rossby wave is triggered.These waves of different type are separated by a criti-cal latitude. Later,Webster and Curtin(1975)furtherstudied the meridional propagation of midlatitude dis-turbances and indicated that they dominate subtrop-ical atmospheric perturbations; while in the tropics,changes are mainly under the influence of local effects.However,despite exp and ing the y ¡ z plane to a two-dimensional spherical atmosphere,the basic state con-sidered by Webster and Curtin(1975)was still a sim-ple pattern,with easterlies in the low latitudes and westerlies in the midlatitudes.

Murakami and Unninayar(1977)found that inthe NH winter strong zonal westerly winds exist overthe central and eastern tropical Pacific and Atlantic.Figure 3 shows the distribution of average perturba-tion kinetic energy(PKE) and zonal mean winds. Ascan be seen,the PKE distribution corresponds well tothe distribution of zonal winds,with maximum PKEoccurring within the zonal westerly wind belt in theupper troposphere. In contrast,minimum PKE occursin the region of the tropical easterly wind belt. Web-ster and Holton(1982)named the mean zonal westerlyin the upper troposphere above the tropical central and eastern Pacific and Atlantic,the "westerly duct"for wave energy propagation, and established the west-erly duct theory for the cross-equatorial propagationof midlatitude disturbance.

Webster and Holton(1982)further consideredfour different types of zonal flow basic state in theirshallow-water model,as shown in Fig. 4. The zonalflow basic state in Fig. 4 ranges from weak easterlywinds across all latitudes between 15°N and 15°S(Fig. 4a),to medium strength easterly winds with a patch ofweak westerly winds in the deep tropics(Fig. 4b),toa similar pattern as that in Fig. 4b but with both theeasterly and westerly winds strengthened(Fig. 4c), and finally to a similar pattern as that in Fig. 4c butwith a wider westerly wind duct in the deep trop-ics(Fig. 4d). The basic state shown in Fig. 4a is thesame as that described by Charney(1963,1969),whilethat in Figs. 4b and 4c resembles the real atmosphere,as shown in Fig. 3. In the equatorial area,zonallyaveraged winds are usually easterly. Hence,the trop-ical zonal mean zonal winds in Figs. 4a-c are indeedeasterly([u(0)] < 0). To compare with the situationwhen zonal mean zonal winds are westerly([u(0)] >0),the westerly wind duct has been increased in Fig. 4d.

Webster and Holton(1982)indicated that whenthe basic flow is in the form of weak easterly winds(Fig. 4a),the wave propagation is largely restrictedto the latitudes north of zonal zero wind in the NH,no matter where the initial midlatitude disturbanceoriginated. At the location of the initial disturbance,the equatorward propagation weakens rapidly after itenters the easterly wind zone. This is because onlywavenumber-1 Rossby waves can cross the weak east-erly zone,which contain small amounts of energy and attenuate quickly(Charney,1963). When the basicstate is in the form of weak westerly winds,as shownin Fig. 4b,the location of the initial disturbancerelative to the westerly duct and its zonal scale be-comes important. On the one h and ,perturbations canonly propagate through the westerly duct to reach theother hemisphere; on the other h and ,only those dis-turbances with a zonal wavelength less than the zonalextension of the westerly duct can propagate acrossthe westerly duct. The shorter the wavelength,thehigher the propagation e±ciency. The results fromexperiments with different basic states(Figs. 4c and 4d)further indicate that the cross-equatorial propaga-tion of midlatitude disturbance is closely related to theintensity and zonal extent of the westerly duct. In gen-eral,a stronger westerly duct(i.e.,greater wind speed) and larger zonal extension are more conducive to cross-equatorial propagation of disturbance,which resultsin stronger atmospheric interaction between the twohemispheres.

Webster and Holton(1982)explained the impactof the westerly duct intensity on the propagation ofthe disturbance wave train theoretically. The wavedispersion relation for linear and barotropic Rossbywaves can be expressed as(Holton,1979)

For a stationary wave,the phase speed is zero; thus,Eq.(3)can be written asAccording to the dispersion relation,group velocitycan be written asUsing Eq.(4)in Eqs.(5) and (6)yields

Equation(8)shows clearly that meridional energypropagation is only possible when the zonal mean basicstate is westerly, and disturbances in strong westerlywinds can propagate equatorward and reach the otherhemisphere rapidly. Westerly duct theory is essentiallyan extension of critical latitude theory,which revealsthe laws governing how midlatitude perturbations inone hemisphere affect the atmospheric behavior in theother hemisphere. The perturbation wave train origi-nated in midlatitudes usually propagates along a greatcircle path, and is then absorbed or reflected when itreaches the critical latitude. However,if the pertur-bation wave train reaches the area where the westerlyduct is located,it will continue to propagate equa-torward, and even reach the other hemisphere. Themeridional propagation speed of the perturbation wavetrain depends on the intensity and zonal extension ofthe westerly duct.

Westerly duct theory suggests that the atmo-spheric activity in the SH should be considered whenpredicting the weather and climate of the NH. The ac-curate prediction of the intensity and location of thewesterly duct could help in predicting the meridionalpropagation of perturbation energy. The westerly ductdemonstrates distinct monthly,seasonal, and interan-nual variations. It is strongest in winter and springin the NH,weakening and diminishing in the summer and autumn. The interannual variation of the west-erly duct is mainly related to ENSO activity. Dur-ing El Niño years,the westerly duct often weakens and disappears in the upper troposphere of the cen-tral and eastern Pacific and is sometimes replaced bymean easterly winds. In contrast,westerly winds in-tensify significantly in the upper troposphere over theAtlantic during El Niño(Arkin,1982; Arkin and Web-ster, 1985; Tomas and Webster, 1994; Webster and Chang, 1998). During the NH summer,the low lati-tudes are often under the control of tropical easterlies.However,in some years,troughs over the central NorthPacific and North Atlantic may intrude into the lowlatitudes,leading to strong westerly winds. As a re-sult,midlatitude disturbances in the NH summer maypropagate across the equator via this route(Maga~nna and Yanai, 1991; Knippertz,2007).

4. Energy accumulation-wave emanation theoryIn their study of the meridional propagation of

midlatitude disturbances,Arkin and Webster(1985) and Webster and Yang(1989)found that the meanperturbation kinetic energy(PKE)in the midlatitudesis rather uniformly distributed in the vertical direc-tion,in contrast to that in the tropics where the PKEcenters and convective activity in upper and lower tro-posphere are out of phase. A large PKE center in thetropical upper troposphere corresponds to weak con-vective activity in the lower troposphere,while strongconvective activity in the lower levels corresponds to asmall PKE center and strong easterly wind in the up-per levels. Therefore,Arkin et al.(1985)argued thatlarge PKE values in the tropical upper troposphereare not caused by local convection. Rather,they are aconsequence of the meridional propagation of midlat-itude transient wave disturbance.

Webster and Chang(1988)further pointed outthat a large PKE in the tropical upper tropospherederives from the meridional propagation of midlati-tude disturbances,as well as nonlocal convective ac-tivity. They hypothesized that equatorial waves aretriggered by tropical convection. During their zonalpropagation,these equatorial waves transport PKE tozonal westerly wind belts in the upper troposphere,where PKE accumulates and results in a large PKEvalue.

Webster and Chang(1988)argued that the follow-ing three conditions must be satisfied for accumulating kinetic energy in average westerly wind belt:

(1)Physical processes that can produce transientkinetic energy in the tropics;

(2)Physical mechanisms that are able to trans-port transient kinetic energy from its origin to wherePKE accumulates;

(3)A physical mechanism responsible for the ac-cumulation of PKE in the equatorial westerly windzone.

Usually,points(1) and (2)are easily satisfied.Convective activity often occurs in the tropics,suchas that in the Asian-Australian monsoon region and within the ITCZ. Well-organized regional convectioncan trigger equatorial waves,which then transportPKE from their source region to other regions throughzonal propagation. In terms of point(3),Webster and Chang(1988)indicated that the zonally varying basicflow in the tropics could be responsible for PKE accu-mulation.

Generally,the distribution of basic flow in the realatmosphere varies zonally, and the meridional gradi-ent of the zonal flow(uy) and the zonal gradient ofthe meridional flow(vx)are often referred to as basicflow shear. Here,to distinguish from the concept ofconventional shear,we refer to the zonal gradient ofthe zonal flow(ux)as stretching deformation,whichdescribes the elongation of the zonal basic flow in theeast-west direction. Assuming that the zonal basicflow changes slowly with time,Webster and Chang(1988)argued that when equatorial waves propagatezonally,the zonal wavenumber change must satisfy thefollowing condition:

where ω_r is the modal frequency in a frame of referencewhere u= 0. Figure 5 shows the relationship betweengroup velocity and the zonal wavenumber of equatorialRossby waves. The group velocity is zero when n = 3(n is the latitudinal nodal number for the equatoriallytrapped waves) and k = 6,negative(westward propa-gating)when k = 4(A), and positive(eastward prop-agating)when k = 8(B). Equation(9)illustrates thatwhen equatorial waves reach areas where du=dx < 0,then dk=dt > 0,i.e.,the zonal wavenumber increases.For long waves with k = 4,increases in zonal wave-number will lead to decreases in westward groupvelocity. For short waves with k = 8,increasesin zonal wavenumber will also lead to decreasesin the eastward group velocity. Therefore,no matterwhat the initial wavenumber of the equatorial Rossbywave,its group velocity change will always depend onthe sign of the zonal basic flow stretching. Areas wheredu=dx < 0 behave like a natural buffer zone,whereequatorial wave energy propagation will slow down.

Based on the studies of Whitham(1965) and Bretherton and Garrett(1968),the wave-activity den-sity along the propagation path in slowly varying basicflow is conserved,i.e.,

where ζ =ε/ω_r is the wave-activity density,ε=ρgh²=2 is the wave energy density, and C_gd =u+ cis the local Doppler velocity during wave propagation.The parameters ζ and ε both reflect wave energy,butthe former represents the average wave energy undera specific frequency while the latter represents totalwave energy. Webster and Chang(1988)obtained thefollowing equation: and suggested that the negative stretching deforma-tion of the zonal mean flow,i.e.,du=dx < 0, and the zero Doppler group velocity Cgd for the zonal propa-gation of the equatorial wave at one specific location,within the area du/dx < 0,are two important con-ditions for energy accumulation. When an equatorialRossby wave reaches a location within du=dx < 0,thewave propagation begins to slow down. At the timewhen the Doppler group velocity Cgd reduces to zero ata certain location within the area du=dx < 0,the waveenergy density " will increase exponentially accordingto Eq.(11). That is to say,zonally propagating waveenergy will accumulate rapidly in this region. Webster and Chang(1988)named such a region as the energyaccumulation region. Apparently,the largest energyaccumulation rate occurs where(du=dx|_max)< 0. Fora simple zonal basic flow with a sinusoidal pattern,thelargest energy accumulation rate occurs at the pointu = 0,i.e.,to the east of the maximum westerly wind.

Webster and Chang(1988) and Chang and Web-ster(1990,1995)subsequently conducted numerousnumerical experiments and confirmed that the resultswere consistent with those of theoretical analysis. Nomatter where the initial perturbation originates inthe tropics,the PKE transported via equatorial wavepropagation always accumulates where du/dx < 0.In addition,they found that areas of PKE accumula-tion are also areas of wave emanation from the tropicsto the mid and high latitudes. Thus,tropical con-vective PKE is first transported to locations withindu/dx < 0 via Rossby wave propagation and accu-mulation. Then,perturbation wave trains emanatefrom the energy accumulation region to high lati-tudes,realizing atmospheric interaction between thetropics and extratropics. There appear to be twophases of tropical-extratropical teleconnection: first,through the zonal teleconnection along the equator, and then the meridional teleconnection through merid-ional wave emanation. This explains why there tendto be geographically fixed modal patterns in the ex-tratropical responses to tropical disturbance,i.e.,theextratropical responses are relatively insensitive to thelocation of the tropical forcing(Branstator,1983; Sim-mons et al., 1983; Geisler et al., 1985; O'Lenic et al., 1985; Ting and Yu, 1998; Zhou et al., 2012).

Using the ray-tracing method,Chang and Web-ster(1990)found that the energy accumulation pro-cess can be divided into forward and backward ac-cumulation. In general,forward accumulation hap-pens when a long Rossby wave propagates westward and reaches the energy accumulation area. However,short Rossby waves tend to be sensitive to their ini-tial location; when located to the east of the maxi-mum westerly wind speed or critical latitude,back-ward accumulation happens, and vice versa. MixedRossby-gravity waves often experience backward accu-mulation,whereas non-dissipative Kelvin waves oftenpossess high phase speed and thus energy accumula-tion does not happen easily.

Figure 6 is a schematic diagram showing the en-ergy accumulation-wave train emanation process. InLa Niña and El Niña years,the location and intensityof the westerly wind belt in the equatorial upper tro-posphere change significantly. During La Niña years(Fig. 6a),the central-eastern Pacific is an area of largeenergy accumulation and wave train emanation. InEl Niña years(Fig. 6b),however,large energy accu-mulation and wave train emanation occurs in the upper levels over the tropical Atlantic. In addition,theupper atmosphere over Indonesia is another region ofdistinct energy accumulation and wave emanation.

Many advancements have been made in re-cent years in the study of energy accumulation-waveemanation theory. It has been found that the extra-tropical atmospheric response to equatorial diabaticheating anomalies is closely related to the basic statestructure(Ting and Sardeshmukh, 1993; Ting,1996;Zheng et al., 2013; Tan et al., 2015). In addition,Kuoet al.(2001)found that,under a linear assumption,energy mainly accumulates via zonal advection. If theeffects of diabatic heating are not considered,the ac-cumulated energy will gradually dissipate by friction.However,under a nonlinear assumption,energy accu-mulation is maintained through the contraction of thescale of perturbation waves. For example,in the NHsummer,westerly wind dominates west of the north-western Pacific,while easterly wind dominates overthe east,which makes it easy to satisfy the conditiondu/dx < 0 in the northwestern Pacific. When tropi-cal perturbations pass by the area of du/dx < 0,theymay accumulate energy via scale contraction. Kuo etal.(2001)suggested that the energy accumulation inthe northwestern Pacific actually provides the initialperturbation vorticity for the formation of tropical cy-clones,which is consistent with the conclusions of Hol-l and (1995) and Sobel and Bretherton(1999),in thatwave energy accumulation provides favorable condi-tions for the genesis of tropical cyclones over the west-ern Pacific. Using reanalysis data,Tam and Li(2006)investigated the formation of synoptic-scale perturba-tion in the western Pacific and found that the accu-mulation of perturbation energy due to zonal scale-contraction is more important than that due to zonalflow advection. Based on energy accumulation the-ory,Done et al.(2011)investigated the relationshipbetween easterly waves in the tropical Atlantic and hurricane formation. Chen et al.(personal communi-cation,2014)discussed the impact of monsoon troughsover the South China Sea on the genesis of tropical cy-clones. In addition,many researchers have tried to ex-plain the mechanisms involved in the formation,varia-tion, and distribution of the South Pacific convergencezone(SPCZ),using energy accumulation-wave emana-tion theory. These studies have revealed that areasnear the SPCZ meet the condition of du/dx < 0, and are thus conducive to energy accumulation. The ori-entation of the SPCZ also suggests the existence ofenergy transport from the tropics to midlatitudes inthe SH(Widlansky et al., 2011).

5. Equatorial wave lateral expansion theory

A key characteristic of energy accumulation-waveemanation theory is that the midlatitude atmosphericresponse to tropical forcing always occurs on bothsides of the energy accumulation region. However,the transitional process from energy accumulation towave emanation is not entirely clear. For example,the initial meridional group velocity is zero when theequatorial waves are triggered by the tropical forcing;however,when waves emanate from the energy accu-mulation region,their meridional group velocity is ap-parently non-zero. One possible mechanism for theformation of the new wave trains may be nonlinearwave-wave interaction. However,Chang and Webster(1990,1995)indicated that linear and nonlinear pro-cesses are rather similar. In other words,wave emana-tion can be realized under a linear assumption,with-out the nonlinear wave-wave interaction in the energyaccumulation area. What,therefore,is the mecha-nism behind the transport of PKE from the tropics tothe midlatitudes,which often results in geographicallyfixed modal responses?

In their study of the teleconnection between trop-ical perturbation and global climate,Lau and Lim(1984)proposed that equatorial Rossby waves can af-fect atmospheric activity in the midlatitudes only ifthey are located in the equatorial westerly wind belt.Wilson and Mak(1984)investigated the impact ofmidlatitude atmospheric activity on tropical weather and climate. They found that equatorial Rossby wavescan experience "trapping" when they propagate in azonally sheared basic flow. That is to say,in additionto the zonal stretching deformation,zonal shear in thebasic flow can also affect the propagation of Rossbywaves. Zhang and Webster(1989)considered the effect of shear in the basic flow, and obtained the generalexpression for a wave equation:

where Γ²(y)is the square of refractive index,which iswidely applied in the study of planetary wave propa-gation(Bennett and Young, 1971; Wilson and Mak, 1984). Generally,when Γ²(y)> 0; V(y)yields a wavesolution; when Γ²(y)< 0; V(y)attenuates. For anequatorially trapped wave,it needs to satisfy the fol-lowing condition:where y_t is the turning latitude for the equatoriallytrapped wave. When y < y_t; V(y)yields a wave solu-tion; when y > y_t; V(y)attenuates and disappears. Alarger turning latitude indicates a wider range of lati-tudes under the influence of the trapped waves, and isthus more conducive to interaction between the trop-ics and midlatitudes.

Zhang and Webster(1989)suggested that theshear effects of zonal flow would increase the turn-ing latitude to some degree. They explained that thisis possibly because the shear effects of zonal flow in-creases the north-south geopotential gradient,thus in-tensifying the β effect and resulting in a decrease inthe "trapping" of equatorial waves. There is no signif-icant difference between the turning latitude in east-erly and westerly sheared flows for eastward propagat-ing Kelvin waves,gravity waves,or westward prop-agating mixed Rossby-gravity waves. However,theturning latitude for westward-propagating equatorialRossby waves is much larger in sheared westerliesthan in sheared easterlies,while the turning latitudefor westward-propagating gravity waves is larger insheared easterlies than in sheared westerlies. Hence,on the timescale of tropical convection,the transientgravity waves triggered by latent heat release in thezonally sheared easterly wind belt are more able toaffect the mid-high latitudes. In contrast,on thetimescale of Rossby waves(i.e.,synoptic and low-frequency timescales),interaction between the low- and midlatitudes is mainly realized through westward-propagating Rossby waves in zonally sheared westerlywind.

In a certain sense,an increase in the turning lati-tude of equatorial waves represents the lateral expan-sion of equatorial waves. Figure 7 is a schematic di-agram showing the propagation of Rossby waves insheared zonal flow. The thick dotted line indicatesthe range of Rossby wave expansion at various longi-tudes. For an initial perturbation in the midlatitudes,when the perturbation is located at X,the perturba-tion cannot directly affect the tropics,since y > y_t. However,if the perturbation is located at Y,wherey < y_t,it will be able to affect the tropics through thezonal propagation of equatorial waves. In other words,the expansion of equatorial waves not only enablestropical forcing to influence the midlatitudes,but alsocauses midlatitude perturbations to affect the trop-ics. Zhang and Wesbter(1989)pointed out that theimpact of tropical forcing on the mid-high latitudesis not necessarily realized through the emanation ofwave trains from the equatorial westerly wind belt.Instead,the interaction between the tropics and extra-tropics is more like a wave "expansion" phenomenon.Equatorial Rossby waves propagate zonally and reachthe equatorial westerly wind belt area. Meanwhile,changes in the zonal basic flow result in a rapid in-crease in their turning latitude. When the turninglatitude becomes su±ciently large,the tropical influ-ence will reach the temperate zone.

It is worth noting that the importance of equato-rial wave lateral expansion theory needs more atten-tion. In fact,equatorial wave lateral expansion the-ory is consistent with energy accumulation-wave em-anation theory. They both explain the atmosphericinteraction between tropical perturbations and mid-latitude activity from different perspectives. Since en-ergy accumulation theory has a solid mathematicalbasis to support its framework,people tend to ap-ply this theory more than others when explain low- and mid-latitude interaction. However,as mentionedpreviously,under energy accumulation theory,equa-torial waves need to transfer from having zonal groupspeed only to having meridional group speed only;however,the process of transition is not clear. Equa-torial wave lateral expansion theory overcomes thisproblem in energy accumulation-wave emanation the-ory, and can explain the two-way interaction betweenthe tropics and temperate zone well,without intro-ducing new wave trains. Additional theoretical studies and verification of numerical experiments are requiredfor equatorial wave lateral expansion theory to furtherprogress.

6. Meridional basic flow theory

A common feature among the previously intro-duced energy propagation theories is that the merid-ional propagation of Rossby waves can only be realizedin the zonal mean westerly wind belt. For example,Rossby waves can propagate equatorward and pole-ward according to great circle theory,but its propa-gation will be constrained once it reaches the criticallatitude. Similarly,under westerly duct theory,thewesterly wind belt over the tropical eastern Pacific and Atlantic behaves like a \bridge" for the interaction be-tween the tropics and temperate zone. According tothese theories,large areas in the low latitudes are un-der the control of easterly winds in the NH summer, and hence the atmospheric activity in the NH and SHwould be independent of one another.

However,many studies have revealed that distinctlinks exist between the two hemispheres,even in theNH summer. For example,Lu(1987)found that sum-mertime SST anomalies in the tropical western Pacificaffect the atmospheric circulation of the NH throughPacific-Japan waves. He(1989)investigated the in-fluence of SH 40-day quasi-oscillation on the summermonsoon in the NH, and suggested that the meridionalpropagation of this perturbation is a possible mech-anism for interaction between the two hemispheres.In recent years,links between anomalous atmosphericcirculation in the SH and weather and climate changein East Asia have attracted great attention. Studieshave shown that changes in the circulation of the SHare closely related to the East Asian summer mon-soon, and precipitation anomalies in South and NorthChina,as well as dust storm occurrence in northernChina(Shi et al., 2001; Zeng and Li, 2002; Nan and Li, 2005; Fan and Wang, 2006; Song and Li, 2009).However,questions remain unanswered as to how theatmospheric interactions between the SH and NH arerealized. Fan and Wang(2006)argued that the inter-action between the two hemispheres is possibly relatedto the meridional teleconnection of barotropic waves inthe high latitudes of the SH and coastal regions of EastAsia. In addition,many studies have revealed thatHadley circulation plays a critical role in the merid-ional propagation of stationary waves(Watterson and Schneider, 1987; Ji,1990; Krishnamurti et al., 1997;Esler et al., 2000; Ji et al., 2014).

Li and Li(2012) and Li et al.(2015)studied the interaction between the two hemispheres over theAsian-Australian monsoon region,with a focus on theeffects of meridional basic flow on the quasi-stationarywave propagation in easterly basic flow. Under the as-sumption of uneven horizontal flow,they obtained thefrequency spectrum expression

where ω= kc is angular frequency,q =is the basic flow of absolute vorticity, and uM and vMare the zonal and meridional basic flows,respectively.The group velocity for wave propagation in unevenbasic flow can be expressed aswhere γ=qy=qx,which reflects the wave structurein the zonal and meridional directions. Equation(15)shows clearly that the rate of wave energy dispersiondepends mainly on zonal and meridional basic flows,phase speed,absolute vorticity gradient, and the wavefeature itself. For a stationary wave with prescribedparameters(ω= 0),energy dispersion largely dependson the zonal and meridional basic flows and absolutevorticity gradient. No matter what the value of γ,the zonal basic flow has the maximum impact on thezonal propagation of wave energy,while the meridionalbasic state has the largest impact on the meridionalpropagation of wave energy. This indicates that understrong zonal wind and weak meridional wind condi-tions,zonal energy propagation is determined by zonalbasic flow. Under weak zonal wind and strong merid-ional wind conditions,meridional energy propagationis determined by meridional wind. Hence,meridionalbasic flow may play a critical role in meridional energydispersion(Li and Li, 2012; Li et al., 2015).

Figure 8 shows the wave ray-tracing results. Fig-ures 8a and 8b present the prescribed distribution ofthe zonal and meridional basic flows,respectively. Fig-ure 8c shows that no matter where the initial pertur-bation is located,perturbation waves propagate equa-torward and poleward when only the zonal basic flowis considered. The poleward-propagating wave turnsequatorward after it reaches the turning latitude, and is eventually trapped at the critical latitude. This isconsistent with results from previous studies in thatstationary waves cannot propagate in easterly windbelts in a zonally symmetric zonal flow basic state.Figures 8d and 8e show the situation when the merid-ional basic flow is considered. Large changes occur inthe perturbation propagation under this condition. InFig. 8d,the initial perturbation is located in the NH.The results indicate that in strong northerly winds,the perturbation can cross the weak tropical easterlywind belt and propagate to the mid-high latitudes ofthe other hemisphere. On the other h and ,when theinitial perturbation is located in the SH(Fig. 8e),the equatorward propagation of perturbation is stillconstrained. Li and Li(2012) and Li et al.(2015)conducted numerous experiments to confirm that thepropagation of such a perturbation type is one direc-tional, and the propagation direction depends on thedirection of the meridional basic flow. For example,ifthe background basic flow is in the form of northerlywind,then the meridional group speed of NH sta-tionary waves intensifies,which is conducive to cross-equatorial wave propagation. In contrast,the merid-ional group velocity of the stationary waves in the SHdecreases. Thus,northward propagation of SH per-turbations is constrained. The opposite is true whenthe background basic flow is in the form of southerlywind. Under this situation,perturbations in the SHcan propagate to the NH,while the equatorward prop-agation of perturbations in the NH is limited.

Meridional basic flow theory overcomes the prob-lem in the current theory of planetary wave propaga-tion,which considers zonally symmetric basic flow and cannot explain the interaction between easterly and westerly winds. Meridional basic flow theory proposesthat meridional basic flow enables stationary waves tocross the easterly wind belt and propagate betweenthe two hemispheres. Compared to traditional wavepropagation theory,meridional basic flow theory considers a more realistic background flow. The newconcept that stationary waves can overcome the con-straints of critical latitudes and propagate to the otherhemisphere is helpful for a better underst and ing ofteleconnection patterns. Using reanalysis data,Li and Li(2012) and Li et al.(2015)analyzed the global tele-connection patterns in the upper troposphere stream-function field during the NH summer. They foundteleconnection patterns between northern and south-ern Africa and the southern Indian Ocean and Antarc-tic,as well as between Hawaii and the southern Pacific and Atlantic. Based on horizontal wave ray-tracing ina zonally asymmetric flow,they proposed that the en-ergy dispersion during barotropic Rossby wave prop-agation is a possible reason for the formation of tele-connection in these wave trains.

Presently,there are two theoretical explanationsfor cross-equatorial teleconnection patterns. Sardesh-nukh and Hoskins(1988)provided an explanation inwhich it was suggested that heating could result ina Rossby wave source in extratropical westerly windbelts, and then further trigger the Rossby waves.The other explanation is based on stationary Rossbywave propagation theory. When stationary Rossbywaves cross the tropical easterly wind belt and prop-agate from one hemisphere to the other,they canlead to teleconnection between the two hemispheres.This theory has been widely applied(Watterson and Schneider, 1987; Ji,1990; Krishnamurti et al., 1997;Fan and Wang, 2006). Meridional basic flow theoryprovides a theoretical basis for cross-equatorial wavepropagation.

7. Diagnosis of energy propagation and wave flow interaction

In atmospheric wave dynamics,wave-mean flowinteraction is an important topic. In early work,Rossby(1939)discussed the relationship betweenzonal basic flow and stationary waves in the midlati-tudes. He found that the Aleutian low shifted west-ward when the basic flow was weak, and vice versa.In addition,Charney and Stern(1962)investigatedthe link between the vertical propagation of quasi-stationary waves and upper-level jet streams. Theyfound that a meridional temperature gradient of zeroat the surface is a prerequisite for maintaining a sta-ble upper-level jet stream, and identified the rapid out-break of the polar night jet in the NH winter as a goodexample of such a baroclinic instability type. Theysuggested that this is possibly related to the downwardpropagation of unstable perturbations at the heightof 30 km,accompanied by conversion between kinetic and potential energy. These are classic studies of wave-mean flow interaction.

Before the 1960s,most diagnostic analyses ofwave-mean flow interaction were based on eddy heat and momentum flux diagnosis. Charney and Drazin(1961)proposed the concept of wave refraction in theirstudy of the vertical propagation of planetary waves.Waves can propagate over areas where the refractiveindex is real,while they will be reflected in areaswhere the refractive index is imaginary. Therefore,the path and coverage area of wave propagation canbe diagnosed based on the spatial distribution of thewave refraction index. In addition,another impor-tant diagnostic quantity,the Eliassen-Palm(EP)flux(Eliassen and Palm, 1961),was proposed. The EPflux is a two-dimensional wave activity flux that in-cludes both the heat and momentum fluxes,expressedas F = {F(y); F(p)},where

The overbars and primes indicate the zonal average and zonal deviation of the variable; u; v, and w arethe zonal,meridional, and vertical winds,respectively; and θ is the potential temperature. There are severaladvantages to using EP flux. For example,it can di-agnose heat and momentum fluxes simultaneously. Inthe y-z plane,the EP flux divergence indicates thenorthward transport of quasi-geostrophic perturbationpotential vorticity flux, and vice versa. However,ina sheared flow,wave energy is not conserved. Usingthe potential vorticity equation,Andrews and McIn-tyre(1976,1978)introduced the concept of \residualmeridional circulation" and the wave activity conser-vation equation:where A is the wave activity; F = isthe wave activity flux related to the momentum and heat fluxes, and is also called generalized EP flux; and D is a non-conservative term,such as surface frictionor diabatic heating. Edmon et al.(1980)first showedthe vertical cross-section of EP flux vectors in a so-called EP flux diagram. The EP flux diagram providesa straightforward way to diagnose the EP flux conver-gence and divergence,as well as wave propagations, and has become widely used.

EP flux is one of the most important tools forthe study of small-amplitude wave-mean flow interac-tion. In the area of EP flux convergence,waves develop and basic flow decelerates, and vice versa. Therefore,EP flux has been widely used in studies of verticalwave propagation,wave-mean flow interaction,quasi-biennial oscillation of stratospheric winds, and sud-den stratospheric warming(Lindzen and Holton, 1968;Dickinson,1969; Holton and Lindzen, 1972; Dunker-ton and Baldwin, 1991; Limpasuvan et al., 2004). Theoriginal EP flux formulation was developed for the caseof wave perturbation superimposed on the zonally av-eraged mean flow,which greatly simplifies the math-ematical formula. However,we are more concernedwith Rossby wave propagation in three-dimensionalspace,which cannot be described by the traditionalEP flux formulation.

Hoskins et al.(1983)studied the interaction be-tween transient waves and basic flows using a timeaveraged method. They defined a horizontal vectoras . An angle forms betweenE_H and the direction of the relative group velocityof the transient waves(C_g-u). E_H can be approx-imated as the easterly momentum flux and its con-vergence will slow down the basic westerly flow. Itis applicable only in slowly varying barotropic fluids, and requires the zonal scale of the transient vortexto be larger than its meridional scale(e.g.,a horizon-tally elongated storm track). Plumb(1986)defineda three-dimensional vector in the study of the quasi-geographic transient vortex in a slowing varying basicflow. The vector is expressed as:

where M_R is the wave activity flux,whose direction| and the direction of the relative wave group velocity|forms an angle. Similarly,▽.M_R can also indicatesources and sinks of transient wave activity. Com-pared to E_H,there is one extra baroclinic term in thex-direction for M_R. Plumb(1986)explained that un-der certain conditions,the baroclinic term would be-come large enough to change the sign of M_R in thex-direction. M_R can be applied in a quasi-geostrophicbaroclinic atmosphere, and in a slowly varying basicmean flow. In the same year,Trenberth(1986)defineda transient wave activity flux in his study of blockingin the SH,which is expressed asCompared to E_H and M_R,E_T is parallel to the direc-tion of the group velocity of the transient waves. E_Thas the same diagnostic capability of E_H and M_Rbut with a simpler form. It can be used in a non-geostrophic atmosphere, and thus has been widely ap-plied in the study of transient Rossby waves.

In addition,great advancements have beenachieved in the diagnostic study of stationary Rossbywave propagation. Plumb(1985)first explored theconservation relationship that needs to be satisfiedduring the propagation of small-amplitude stationarywaves in a zonally uniform basic flow. He providedthe three-dimensional Rossby wave activity flux F_s: F_sis parallel to the group velocity direction of the sta-tionary waves and represents the direction of energypropagation. By calculating the divergence of waveactivity flux,we can determine the sources and sinksof the stationary waves(Plumb,1986; Karoly et al., 1989). Since Plumb(1985)only considered the zon-ally uniform basic flow and small amplitude perturba-tion,many researchers exp and ed his study to obtainmore realistic stationary wave activity flux in real at-mosphere(Kuroda,1996). For example,Takaya and Nakamura(1997,2001)developed an exp and ed ver-sion of the Plumb equation(Plumb,1985)for the wavybasic flow that better represents the situation in thereal atmosphere.

In recent years,substantial progress has beenmade in the diagnostic study of inertia-gravity waves and Kelvin waves. For example,Miyahara(2006)es-tablished a three-dimensional wave activity flux equa-tion appropriate for inertia-gravity waves in time-averaged Boussinesq flow. Based on the theory ofsmall-amplitude waves in a slowly varying mean flow and the primitive equations,Kinoshita et al.(2010) and Kinoshita and Sato(2013a,b)derived a waveactivity flux equation that is similar to the three-dimensional Stokes drift. They suggested that thisthree-dimensional wave activity flux equation is ap-propriate for both inertia-gravity waves and Rossbywaves. In addition,Kinoshita and Sato(2014)ob-tained a suite of equations in the equatorial β plane,which can be easily applied for diagnostic analysis ofthe interaction between Kelvin waves and the basicflow. This is important for studying equatorial quasi-biennial oscillation and semi-annual oscillation.

It is worth noting that researchers in China havecontributed greatly to the study of wave-mean flowinteraction. For example,Huang(1983)exp and edthe EP flux formulation to spherical coordinates,sug-gesting that quasi-stationary planetary waves oftensplit into two branches during their propagation ina three-dimensional spherical atmosphere. This re-sult confirmed the conservation law of spherical plan-etary wave actions. Wu and Chen(1989)further ad-vanced EP flux diagnostics by applying it to moist at-mosphere,which greatly exp and ed the possibilities forEP flux application. In studying upper-troposphericwave-mean flow interaction,Gao et al.(1990) and Ran et al.(2005)exp and ed the EP flux formula to abaroclinic atmosphere and proposed a generalized EP flux formulation that can better explain the signif-icant acceleration that sometimes occurs in the upper-tropospheric jet stream.

8. Summary and discussion

This paper reviews and summarizes theoreticalstudies of energy propagation and atmospheric inter-action between different latitudes. Diagnostic studiesof energy propagation and wave-mean flow interactionare also discussed. The following key components and findings are covered within the paper:

(1)Great circle theory explains the stationaryRossby wave propagation in a spherical atmosphere.Rossby wave energy dispersion theory can successfullyexplain the links between the atmospheric activity atdifferent latitudes.

(2)Westerly wind duct theory indicates that themean zonal westerly wind belt in the upper tropo-sphere over the equatorial central-eastern Pacific and the Atlantic is a "corridor" for atmospheric interac-tion between the two hemispheres. The perturbationwaves triggered in the midlatitudes can cross the equa-tor and propagate to the other hemisphere through the"westerly duct". The e±ciency of the cross-equatorialpropagation depends on the zonal scale of the pertur-bation waves relative to the "westerly duct" and theintensity of the zonal westerly wind.

(3)Energy accumulation-wave emanation theorysuggests that negative stretching deformation of thebasic zonal flow can cause accumulation of perturba-tion energy. Tropical perturbations first reach cer-tain regions through the zonal transport of equato-rial waves,wherein wave energy accumulates and thenaffects midlatitude activity by means of wave emana-tion.

(4)Equatorial wave lateral expansion theory dis-cusses the possible mechanisms behind the interactionof the tropics and temperate zone from the perspec-tive of turning latitudes of equatorial waves. Sincethe turning latitude is large for westward-propagatingRossby waves in the sheared westerly wind belt,equa-torial waves will experience "expansion" when theyreach this latitude. As a result,tropical perturbationscan affect midlatitude regions and there is a two-wayinteraction between the tropics and midlatitudes.

(5)Meridional basic flow theory considers the im-pact of meridional mean flow on the meridional prop-agation of stationary waves. Certain meridional meanflows play a critical role in the propagation of station-ary waves from one hemisphere to the other. Thistheory is important for underst and ing the connectionbetween the activity in the two hemispheres,especiallythe energy exchange between the two hemispheres inthe Australian-Asian monsoon region.

(6)Diagnostic methods for wave-mean flow inter-action have been advancing over the years,from thesimple calculation of heat and momentum fluxes inearly work,to the development of EP flux and three-dimensional wave action flux in subsequent studies. Inrecent years,important progress has been made in thestudy of wave conservation and wave activity fluxesfor inertia-gravity waves and equatorial Kelvin waves.

From the 1940s to the present day,continuousefforts have been made to study the energy propaga-tion and atmospheric interaction between different lat-itudes. The continued development and improvementof energy propagation theory has helped answer manyscientific questions and deepened our underst and ing ofatmospheric teleconnections.

In the context of global warming,large-scale at-mospheric circulation has been changing. For exam-ple,Held and Soden(2006),Vecchi and Soden(2007), and Tokinaga et al.(2012)all found that the intensityof the Walker and Hadley circulations has decreasedunder global warming. In contrast,Meng et al.(2012),Wang et al.(2013), and England et al.(2014)arguedthat the Walker and Hadley circulations have intensi-fied. These conflicting results indicate that changesin large-scale circulation against the global warmingbackground might be complicated, and these changeswill inevitably lead to climate anomalies in many re-gions throughout the world. Lu et al.(2007)foundthat the Hardly circulation has shifted northward,re-sulting in a northward shift of dry regions in the extra-tropics. Williams and Funk(2011)also suggested thatthe westward shift of Hardly circulation is one reasonfor the drought anomalies in East Africa. As the two most important zonal and meridional circulation pat-terns in the atmosphere,changes in the Hadley and Walker circulations directly affect basic mean flows inthe zonal and meridional directions, and thus influ-ence the cross-equatorial propagation of perturbations and the interactions between the mid and low lati-tudes. In addition,while energy propagation theoriesqualitatively describe the interactions between the at-mospheric activity at different latitudes,we are moreconcerned about quantitatively analyzing these inter-actions, and determining to what extent the activityin one region affects that in other regions. Finally,an-other important question is: how does one apply thesetheories in practical numerical weather prediction and in improving the accuracy of weather forecasts? Theabove theoretical analyses have shown that pertur-bation propagations are sensitive to changes in basicflow. Hence,one important issue in modeling studiesis how to precisely simulate changes in basic flow,es-pecially the distribution of basic flow in the tropicalupper troposphere. From the perspective of pertur-bation propagation,it is necessary to improve theprediction of basic flow for the purpose of improvingweather and climate prediction. This is a challengingissue facing the numerical modeling community in thecoming years.

Acknowledgments: We appreciate the sugges-tions and comments from three anonymous reviewers,which are helpful for improving the overall quality ofthe article.