Effects of Low-Frequency Tropical Forcing on Intraseasonal Tropical ...

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JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 54

Effects of Low-Frequency Tropical Forcing on Intraseasonal Tropical–Extratropical Interactions LONG LI

AND

TERRENCE R. NATHAN

Atmospheric Science Program, Department of Land, Air, and Water Resources, University of California, Davis, Davis, California (Manuscript received 15 June 1995, in final form 12 July 1996) ABSTRACT A spherical nondivergent barotropic model, linearized about a 300-mb climatological January flow, is used to examine the extratropical response to low-frequency tropical forcing. A two-dimensional WKB analysis shows that the energy propagation depends on the sum of three vectors: the basic state wind vector, a vector that is parallel to the absolute vorticity contours, and the local wave vector. The latter two vectors are functions of the slowly varying background flow and forcing frequency v. As v decreases, the ray paths approach that of the local wave vector, so that the energy propagates in a direction perpendicular to the wave fronts. The extratropical jet streams have a stronger influence on the long period (.30 day) ray paths than on those of intermediate period (;10–30 day). Global and local energetics calculations show that the energy conversion from the zonally varying basic flow increases as v decreases. The local energetics show that for the long period disturbances, both the energy conversion and energy redistribution due to advection and pressure work are significant along the North African– Asian jet stream. The long period disturbances are less sensitive to the location of the tropical forcing than those of intermediate period. This provides a plausible explanation for the observations showing that the long period oscillations tend to be geographically fixed at the exits of the extratropical jet streams, whereas those of intermediate period are zonally mobile wave trains. The long (intermediate) timescale disturbances dominate in the Northern (Southern) Hemisphere, where the zonal variations in the basic flow are more (less) pronounced.

1. Introduction The major goal of this study is to examine the role of time-varying tropical forcing on the generation and maintenance of atmospheric low-frequency variability (LFV) during the Northern Hemisphere (NH) winter. Emphasis is placed on timescales that exceed deterministic predictability but which are less than seasonal; we will focus on intraseasonal timescales within the 10–60 day range. Phenomena associated with LFV on the intraseasonal timescale have been well documented for both the Tropics (e.g., Madden and Julian 1971) and extratropics (e.g., Ghil and Mo 1991a,b); they have equivalent barotropic structures and well-defined teleconnection patterns in the midlatitudes (Wallace and Gutzler 1981; Blackmon et al. 1984a,b). Moreover, LFV is not uniformly distributed in space, having the largest signature over the North Pacific, especially during winter (Wallace and Blackmon 1983). Blackmon et al. (1984a,b) have also demonstrated that the long-time scale (k30 day peri-

Corresponding author address: Dr. Terrence R. Nathan, Atmospheric Science Program, University of California, Davis, Hoagland Hall, Davis, CA 95616-8627. E-mail: [email protected]

q 1997 American Meteorological Society

ods) teleconnections are characterized by geographically fixed dipole patterns over the jet exit regions, whereas those of the intermediate timescales (10–30 day periods) are mobile, zonally oriented wave trains. Among the mechanisms proposed for LFV are (i) energy dispersion from localized forcing associated with sea surface temperature (SST) anomalies (Hoskins and Karoly 1981, hereafter HK; Branstator 1983, hereafter B83; Li and Nathan 1994, hereafter LN); (ii) linear instability of zonally varying flow (Simmons et al. 1983; Branstator 1985b, 1990; Howell and Nathan 1990; Anderson 1991); (iii) spatial baroclinic instability (Pierrehumbert 1986; Nathan 1993); and (iv) nonlinear topographic instability (Nathan 1988, 1989; Jin and Ghil 1990; Nathan and Barcilon 1994). Mechanism (i), which is particularly relevant to this study, is briefly reviewed below. The global atmospheric response to localized forcing by tropical SST anomalies is well established (e.g., Bjerknes 1969; Horel and Wallace 1981). As demonstrated in several theoretical studies (e.g., HK; Simmons 1982; B83; Lau and Lim 1984; Branstator 1985a), these anomalies generate localized, external energy sources that produce steady two-dimensional Rossby waves that disperse energy to remote regions of the globe. The major findings of these time mean forcing studies are

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(i) the remote response of (steady) tropical forcing has an equivalent barotropic structure in the upper-level extratropical troposphere, (ii) a large response in the extratropics is produced when the forcing is within the subtropical westerlies, (iii) Rossby wave propagation out of and through the Tropics is greatly enhanced (reduced) when it encounters subtropical westerlies (tropical easterlies), and (iv) the energy propagation is influenced by both zonal and meridional variations of the basic-state flow. Although the external forcing studies cited above have provided valuable insight into the role that external forcing plays in LFV and tropical–extratropical interactions, several important issues have yet to be fully resolved. For example, most external forcing studies (e.g., HK) have focused on localized steady forcing, yet observations show that SST anomalies have associated with them a strong fluctuating component in the upperlevel divergence field (e.g., Sardeshmukh and Hoskins 1985; Nakazawa 1986). These observations have indicated significant low-frequency variations of upper-tropospheric divergence in the Tropics, which have been confirmed by comparison with satellite data of outgoing longwave radiation. Pronounced signatures in the 15– 25 day and the 30–60 day bands have been identified. The influence of low-frequency periodic forcing on the extratropics has been examined theoretically by Karoly (1983) and LN for zonally averaged basic flows. Both studies have shown that the low-frequency waves generated by localized, periodic forcing can result in cross-equatorial propagation. Li and Nathan have demonstrated that for realistic damping, tropical low-frequency forcing with periods of 10–20 day has the largest influence on the extratropical circulation. This is due to the combination of energy dispersion from the forcing region and energy extraction from the equatorward flank of the midlatitude jet. However, it is unclear to what extent these results hold for the observed zonally varying flows. The zonally varying extratropical jet streams not only act as strong waveguides for the stationary waves during NH winter (B83; Hoskins and Ambrizzi 1993), but they also serve as local energy sources for the low-frequency barotropically unstable modes, termed SWB modes after Simmons et al. (1983). Using the same spherical, nondivergent barotropic model as B83 and Simmons et al., Ferranti et al. (1990) demonstrated that the forced responses having structures similar to the SWB modes can be easily excited by a periodic forcing in the 40– 50 day range. However, many questions remain regarding the role of such periodic forcing in tropical–extratropical interactions on the intraseasonal timescale. For example, is there an optimal frequency that maximizes the midlatitude response to low-frequency tropical forcing? Is the position of the forcing relative to the jet streams important for forced low-frequency disturbances? How does the waveguide effect of the NH winter jet stream depend on forcing frequency? Are there

interhemispheric differences in the far-field response to localized, low-frequency tropical forcing? If so, what factors determine the differences? These questions are among those that will be addressed in this study. The paper is organized as follows. Section 2 describes the model, the basic state, and the forcing structure. Section 3 provides an analytical analysis of the group velocity and ray paths based on WKB theory. Section 4 presents a global and local energetics analysis. The possible application of the results to observed flows in the atmosphere is discussed in section 5. The last section summarizes the main conclusions of the study. 2. The model a. Governing equations To address the issues raised in the introduction, and to facilitate comparison with previous studies (e.g., HK; B83; LN) examining the extratropical response to tropical forcing, we consider a nondivergent barotropic flow on a sphere. The flow is governed by the barotropic vorticity equation, which is linearized about the steady two-dimensional basic-state streamfunction, c¯ (l, f). In the presence of an external vorticity source and dissipation, the nondimensional equation governing the perturbation streamfunction c9(l, f, t) can be written as (B83; LN) ]¹ 2c 9 ¯ · =(¹ 2c 9) 1 v9 · =(¹ 2c¯ 1 f ) 1 V ]t 1 a¹ 2c 9 1 K¹4(¹ 2c 9) 5 R9,

(2.1)

where length, time, and streamfunction have been scaled by ae, (2V)21, and 2Va2e , respectively, where ae is the earth’s radius and V is its rotation rate; f 5 sin f is the dimensionless Coriolis parameter; l is longitude and f is latitude; ¹2 is the horizontal Laplacian; v9 5 (u9, y9) ¯ 5 (U, V) are the perturbation and basic-state and V vector winds, respectively, where the overbar denotes a time average. The dissipation processes are modeled as Rayleigh damping and fourth-order diffusion. For the former we choose a 5 0.011, corresponding to an effective spindown time of about 7 days, whereas for the latter we set K 5 1027, yielding a diffusive timescale of about 1 day for the smallest wave scales resolved in our model. Our choices for a and K are identical to those used by B83 and LN. The spatial and temporal variation of an upper-tropospheric vorticity source, induced by anomalous tropical heating, for example, is modeled as R9(l, f, t,) 5 R(l, f)[1 1 gvcos vt],

(2.2)

where R is a steady vorticity source, v is the (real) forcing frequency, and gv is the ratio of the steady to periodic forcing strength, which, for simplicity, is chosen to be independent of frequency. The spatial structure for R is identical to that used by LN [their equation

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(2.7)]. In the subsequent analysis, attention will be focused on intraseasonal forcing periods in the 10–60 day range. b. Basic states Because time-averaged basic flows generally do not satisfy the steady, barotropic vorticity equation, a forcing is required to maintain the basic state against dissipation; that is, ¯ ·=(¹2c¯ 1 f ) 1 a¹2c¯ 1 K¹6c¯ 5 F¯(l, f), (2.3) V where the basic-state forcing F¯(l, f) can be thought of as representing a zonally asymmetric vorticity forcing that would arise, for example, from localized external heating in a baroclinic atmosphere. Although Andrews (1984) and Howell and Nathan (1990) have shown that the detailed nature of the forcing that maintains the basic state may have an impact on the linear stability characteristics of the disturbance field, we opt instead to follow B83 and Manney and Nathan (1990), among others, and simply specify the basic state, c¯ (l, f). The basic state used in this study is the 300-mb climatological January rotational flow. The data are from the European Centre for Medium-Range Weather Forecasts (ECMWF) analysis, on a 2.58 3 2.58 grid that is initialized twice daily at 0000 and 1200 UTC. Based on the model data spanning the years 1980–88, we constructed the time-mean wind field for each year, which was then interpolated to the Gaussian grid of our spectral model (see section 4a). The climatological basic-state wind field was obtained by time averaging individual January flows. Figure 1 displays the basic-state streamfunction, zonal wind, absolute vorticity, and absolute vorticity gradient for the January climatological flow. The basic-state fields shown in Fig. 1 have several features that are of particular importance. For example, the NH flow is characterized by strong zonal asymmetries. In contrast, the SH flow is approximately zonal, so the results obtained by LN for zonally averaged basic states should apply there. In the NH extratropics, there are two prominent jet regions: the stronger of the two is located off the east coast of Asia, having a maximum zonal wind speed of about 64 m s21; the other is located off the east coast of North America, having a maximum speed of about 32 m s21 (Fig. 1b). The former jet has a sharper meridional variation of absolute vorticity than the latter (Fig. 1c), whereas the meridional gradient of absolute vorticity changes sign at both flanks of the Asian jet exit region (Fig. 1d). The Tropics are dominated by easterlies with a maximum wind speed of about 28 m s21 over the western tropical Pacific, although there is a relatively small region of tropical westerlies over the central eastern Pacific (1508W–908W). The tropical easterlies meander around the equator, exhibiting the greatest penetration into the Southern Hemisphere (SH) over the Atlantic Ocean and the greatest penetration into the NH over the western Pacific.

FIG. 1. Climatological January 300-mb basic flow derived from time-averaged ECMWF data from 1980 to 1988. (a) Streamfunction; contour interval: 1.2 3 107 m2 s21. (b) The zonal component u of the basic wind field; contour interval: 8 m s21. (c) Absolute vorticity; contour interval: 7.29 3 1026 s21. (d) Meridional gradient of the absolute vorticity; contour interval: 1.14 3 10211 m21 s21.

3. WKB analysis To analytically examine the differences between the extratropical response to stationary versus low-frequency tropical forcing, we follow, for example, HK, Karoly (1983), and LN and examine the group velocity and ray path characteristics of locally generated disturbances using WKB and ray tracing techniques. The analysis is simplified by neglecting diffusion and introducing the Mercator coordinates (HK), x 5 l, and

15 JANUARY 1997

y 5 ln[(1 1 sin f)/cos f]

¯ M 1 k 3 =Mh¯ A5V zKr z 2

so that (2.1) becomes

1

2

] ¯ M · =M ¹M2 c 9 1 v9M · =Mh¯ 5 2a¹M2 c 9, 1V ]t

(3.1)

where ¯ M 5 (u¯M, y¯ M) 5 V

1cos f , cos f 2 U

V

(3.2)

is the basic-state velocity and h¯ is the basic-state absolute vorticity. The operators appearing in (3.1) are =M 5 (]x, ]y) and ¹2M 5 ]xx 1 ]yy. The WKB analysis hinges on the assumption that the basic state varies slowly compared to that of the disturbance field. Although this assumption may not be satisfied everywhere, particularly near the jet regions, the WKB analysis will, at the very least, provide qualitative information regarding the propagation of energy through the zonally varying background flow. The WKB results will be confirmed a posteriori by comparison with our numerical results in section 4. A WKB solution to (3.1) is chosen of the form (Bender and Orszag 1978)

c9 5 C exp[ie21F0(X, Y) 1 F1(X, Y) 1 ···]exp(2ivt),

(3.3)

where C is a constant and X 5 exx, Y 5 eyy, where O(ex) 5 O(ey) K 1 measure, respectively, the ratio of the zonal and meridional length scales of the perturbation to that of the basic flow. In (3.3) we retain the first two terms in the WKB series, that is, F0 and F1, which are generally complex and represent the phase and amplitude modulations of the wave packet, respectively. The (real) forcing frequency, v, may be positive or negative, corresponding to westward or eastward propagating disturbances. By substituting (3.3) into (3.1), we obtain at lowest order the complex dispersion relation

v 5 u¯Mk 1 y¯ M l 1

h¯ x l 2 h¯ y k 2 ia, (k 2 1l 2)

(3.4)

where the local wavevector is defined as K 5 (k, l) 5 (]F0/]X, ]F0/]Y) 5 =MF0. In the absence of damping (a 5 0), (3.4) reduces to the dispersion relation for twodimensional Rossby waves. In the presence of damping (a ± 0), k and/or l must be complex because the forcing frequency, v, is real; thus, K 5 Kr 1 Ki, where the subscripts r and i denote the real and imaginary parts of the local wavevector, respectively. Assuming that the damping and imaginary wavenumbers are small, corresponding to zKiz 5 O(a) K 1, (3.4) yields the inviscid dispersion relation

v 5 A · K r, where

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(3.5)

(3.6)

¯ M and is the sum of two vectors: the basic wind vector V a vector that is parallel to the absolute vorticity contour. In (3.6) k is a vertical unit vector. For the small zKiz and a limit, the group velocity concept holds (Gaster 1962); that is, cg 5 A 2

2Kr [(k 3 =Mh¯ ) · Kr , zK r z4

c g · K i 5 a,

(3.7) (3.8)

where cg 5 (ug, yg) is the two-dimensional group velocity. If the scalar version of (3.7) is used, it becomes identical in form to the expression obtained by Karoly (1983). Equation (3.8) states that the damping of the disturbances, measured by Ki, is inversely proportional to the group velocity. We note that the vector notation used for the dispersion relation (3.5) and the group velocity (3.7) are physically clearer than the scalar expressions. For example, (3.7) states that the group velocity cg is determined by A and Kr, whose projection on each other is determined by the forcing frequency [see Eq. (3.5)]. Here, A vanishes for stationary waves embedded in zonally averaged basic flows; thus, in this case cg is locally parallel to Kr (HK). For stationary waves embedded in zonally varying basic flows, however, A is either a zero or perpendicular to Kr [from (3.5)]. The former condition can only be satisfied if the two terms forming A offset each other; in general cg and Kr are not parallel in zonally varying flows. For nonstationary Rossby waves, cg and Kr are not parallel to each other because v ± 0 [see (3.5)]. To examine this point in greater detail, it is convenient to define a coordinate system based on the local wave vector. Let s and n represent unit vectors that are parallel and normal to Kr, respectively. Then cg 5 c0 s 1 c⊥ n, for which ¯ M z cos b 2 v , c0 5 2zV zKr z c⊥ 5 zAz sin u 5

v tan u, zKr z

(3.9)

¯M where b and u are the angles of Kr with respect to V and A, respectively. The dispersion relation (3.5) becomes v/zKrz 5 zAzcos u. Equation (3.9) shows that in the direction of Kr, the group velocity for the stationary Rossby waves (v 5 0) is twice the basic flow speed (Hoskins and Ambrizzi 1993), which is larger (smaller) than that of the eastward (westward) moving Rossby waves because v . (,) 0. However, the energy of the low-frequency waves can also propagate in the direction parallel to the wave fronts (normal to Kr) (c⊥ is generally nonzero). The exception is when u 5 0 (p), which oc-

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curs when A is in the same (opposite) direction of Kr. Equation (3.9) also states that for a fixed u, c⊥ is proportional to v and inversely proportional to zKrz, which means that the energy propagation in the direction parallel to the wave front is most significant for waves with higher frequency and longer wave scale. However, some care must be taken when interpreting these results because the wave scales and u depend on v through the dispersion relation; that is, zKrz 5 zKrz (X, Y; v, u). The path of energy propagation can be obtained by defining a ray as a trajectory that is locally tangent to cg; thus, a ray is governed by (Lighthill 1978) dy y 5 g. dx ug

(3.10)

From (3.5)–(3.7), we see that the orientation of a ray depends on the zonal and meridional variations of the ¯ M, =Mh¯ , and on the basic state, measured explicitly by V local wave vector, Kr, which depends implicitly on the basic state. For zonally uniform basic flows, only the meridional wavenumber lr varies along the ray. However, for zonally varying basic flows, both kr and lr vary along the ray. The variation of the local wavenumbers along the ray can be determined by kinematic wave theory (Whitham 1960):

[ [

] ]

dgkr ]v ] ¯ k 3 =Mh¯ 52 52 V 1 · Kr , dt ]X ]X M Kr2

(3.11)

dglr ]v ] ¯ k 3 =Mh¯ 52 52 V 1 · Kr , dt ]Y ]Y M Kr2

(3.12)

where dg/dt 5 ]/]t 1 cg·=M. The ray path is obtained numerically by integrating (3.10)–(3.12) using a fourthorder Runge–Kutta scheme with a grid space of 1.28. The initial wavenumbers are given, and unless stated otherwise, the rays start from different locations along the equator. In all cases the rays are terminated prior to reaching a turning point. For the zonally uniform basic flow considered in Li and Nathan (1994), a turning point occurs where the local meridional wavenumber l approaches zero. At the turning point, the assumptions upon which the WKB analysis is based are violated. For the zonally varying flows considered here, our numerical calculations show that in the vicinity of the jet streams the (local) meridional wavenumber l becomes so small that the WKB analysis becomes questionable. Also, as discussed later, near the jet streams there is strong local energy conversion from the basic flow to the disturbance. For such regions, the ray tracing approach becomes physically less relevant. Therefore, carrying out the WKB analysis beyond the turning points would provide little additional insight into the effects of tropical forcing on the middle latitude circulation, insight that is obtained from the energetics analysis carried out in section 4. Before carrying out detailed numerical calculations, it is instructive to first consider the general role of the

VOLUME 54

extratropical jet streams on the local wavenumbers of the disturbance. For the climatological basic flow shown in Fig. 1, the zonal variation is generally much weaker than the meridional variation. For example, the North African–Asian jet stream has a maximum wind speed of about 64 m s21 at (308N, 1508E), which decreases by half over 208 meridionally and 2108 zonally. Based on (3.11) and (3.12), the changes of kr along a ray are much smaller than that of lr and thus can be regarded as a constant to a first approximation. However, at the jet exits, the zonal variations of the basic-state wind field and absolute vorticity gradient are significant (Figs. 1b and 1d), so that considerable change in kr downstream of those regions is anticipated. Such zonal-scale changes in the vicinity of jet exit regions have been noted by Williams et al. (1984) for easterly jets and by Finley and Nathan (1993) for westerly baroclinic jets. Figure 2 displays the rays of the stationary and 10–50 day waves of integer zonal wavenumber 3 generated at the points marked along the equator. At A (08, 708E), which is located in the tropical easterlies over the Indian Ocean, the low frequency waves can propagate to both hemispheres, whereas the stationary waves are trapped due to the presence of the tropical easterlies. Although all of the 10–50 day waves can propagate to the north, only the 10–20 day waves can propagate to the south; the other lower-frequency waves are also trapped by the tropical easterlies. In both hemispheres, the low-frequency waves propagate westward to about 58N (208S), then turn back to the east. The westward propagation increases with decreasing frequency, consistent with LN. North of 208N, the waves with periods .30 day propagate more zonally than the higher-frequency waves and extend farther downstream along the Asian jet stream. At B (08, 1208E), which is over the western Pacific, both the stationary and low-frequency waves are trapped in the Northern Hemisphere. The low-frequency waves (10–50 day) can propagate through the tropical easterlies to the Southern Hemisphere (SH), although the stationary waves remain trapped to the south. The rays initially extend southwest of the forcing, eventually bending to the east at 208S toward Australia. From Fig. 1b we find that the tropical easterlies extend north from the Indian Ocean to the western Pacific, reaching a maximum over the tropical western Pacific of 28 m s21. Such strong tropical easterlies prevent even the lowfrequency waves from propagating to the north. Energy propagation to the SH is easier over the western Pacific than over the Indian Ocean. The other two points shown in Fig. 2, C (08, 1208W) and D (08, 308W), are both within the tropical westerlies, that is, at the tropical eastern central Pacific and the western Atlantic, respectively. At C, the stationary and low-frequency waves propagate to both hemispheres due to the tropical westerlies over this region. This region has been referred to as the ‘‘westerly duct’’ by Webster and Holton (1982). Thus, it is more likely that waves generated in this region can propagate to the

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337

FIG. 2. The ray paths of wave trains with periods of 10 (long dashed), 20 (medium dashed), 30 (short dashed), 40 (dotted), 50 (dash–dotted), and ` day (solid) for zonal wavenumber k 5 3. The point sources are located at A (08, 708E), B (08, 1208E), C (08, 1208W), D (08, 308W), respectively.

extratropics. The rays to the Southern Hemisphere are very close and difficult to distinguish, but the rays to the Northern Hemisphere are widely separated, especially north of 158N. The 10–20 day waves propagate to the north, but the waves with periods .30 day tend to follow the North American jet stream toward the northeast. At D, all of the rays extend northward to 258N, while the 10–20 day waves go even farther to the north; the long period (.30 day) waves bend northeast toward North Africa. Only the low-frequency waves can propagate through the easterlies of the tropical Atlantic into the SH. To demonstrate how the extratropical jet streams can influence both the stationary and low-frequency waves, we show in Fig. 3 the rays of waves generated at points E (158N, 708E), F (158N, 1808E), and G (158N, 608W). At E, which is located at the upstream side of the subtropical Asian jet stream, the low-frequency and stationary waves propagate downstream along the jet. Downstream of the North Africa–Asian jet stream, at F, the rays are mostly directed to the north. Thus, the lowfrequency and stationary waves tend to follow the jet stream. Previous studies (e.g., Hoskins and Ambrizzi 1993) have shown that this jet stream is a strong waveguide for stationary waves. Therefore, the accumulation of wave perturbation energy is favored downstream of the jet, irrespective of the forcing location. For point G, which is at the upstream end of the North American jet, all of the rays are along the jet toward the northeast. However, the ray paths are more widely separated than those starting from E. This is due to the fact that the North Atlantic jet is much weaker than the Asian jet. The ray tracing analysis presented above only provides information on the path of energy propagation. It is often of interest to determine how the amplitudes of

the wave packets change along the rays. For zonally averaged basic flows, such information can be obtained by either considering the conservation of wave action along rays for inviscid flow (Bretherton and Garrett 1969) or by considering the O(e) analysis of the WKB solution for viscous flow (LN). Both approaches give the same results for inviscid, zonally uniform flow; that is, the modulation of the disturbance amplitude will depend on changes in its meridional scale, l21/2 , as shown r by HK. For viscous zonal flow, however, LN have pointed out that the disturbance amplitude also depends on the damping strength and the meridional group velocity vg [their (Eq. 3.14)]. For zonally varying viscous flows, an expression for the disturbance amplitude can be found by inserting (3.3) into (3.1) and solving the O(e) problem. However, due to the complexity of the zonally asymmetric basic flow used in this paper, deriving a simple mathematical expression describing the relationship among wave amplitude, wave scale, and wave frequency is precluded. Thus, we discuss in the next section the amplitude modulation of the forced wave solutions by using a local energetics analysis. 4. Global and local energetics a. Energetics equations A global energy budget analysis has been carried out by Branstator (1985a) to demonstrate the relative importance of external forcing versus energy conversion for steady forced solutions in a zonally varying basic flow. Li and Nathan carried out a local energetics analysis for low-frequency forced disturbances with zonally uniform basic states. Here we extend LN’s analysis by considering the effects of zonal variations of the basic

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FIG. 3. As in Fig. 2 but for point sources located at E (708E, 158N), F (1808E, 158N), G (1208W, 158N), respectively.

state. In this case, the equation for the local energetics can be written as (see appendix) ]E9 ¯ · =E9 2 = · v9p9 1 CKX 1 CKY 5 2V ]t 1 D9 1 G9,

(4.1)

where the perturbation pressure field p9(l, f, t) is obtained from the diagnostic equation (A.5). In (4.1) E9 5 (u92 1 y92)/2 is the perturbation kinetic energy (KE) per unit mass. Equation (4.1) states that the local time change of KE is due to (i) advection of KE; (ii) work done by the perturbation pressure field; (iii) barotropic energy conversion between the basic state and perturbation, which is represented by

[

CKY 5 2u9y9 cos f

1

2

] U 1 ]V 1 ]f cos f cos f ]l

]

(4.2a)

and CKX 5 2(u92 2 y92)

[

]

1 ]U 2 V tan f , cos f ]l

(4.2b)

which represent, respectively, energy conversion due to shearing and stretching deformation; (iv) dissipation of KE; and (v) generation of KE by spatially and temporally varying external forcing, G9(l, f, t). The local energetics Eq. (4.1) thus provides important information regarding how much energy generated locally (e.g., Tropics) propagates to the far field (e.g., extratropics) via energy advection and pressure work, and how much energy is converted locally from the basic state to the disturbance. To ease comparison between the steady and periodically forced energetics solutions, we consider a timemean version of (4.1) and define, as in LN, an ‘‘ensemble average.’’ For a periodically forced solution of the

form c(l, f) 5 cc(l, f)cos(vt) 1 cs(l,f)sin(vt), we can write Ê9& 5 (EC 1 ES)/2, where the angle brackets denote ensemble averaging, and EC 5 (u9C2 1 yC92)/2 and ES 5 (u9S2 1 y9S 2)/2 are the two-phase quadrature components of the periodically forced solutions. Thus, the local as well as global Ê9& tendency vanishes and the ensemble averaged terms on the rhs of Eq. (4.1) balance. We note that the contribution of the periodically forced solutions to the time-averaged perturbation kinetic energy is readily obtained as Ê9& p 5 Ê9& 2 E9, s where E9, E9s , and E9p represent the perturbation kinetic energy of the total, steady, and periodically forced disturbances, respectively. Before proceeding with detailed energetics calculations, we note that several studies have carried out a wave activity flux vector analysis (Plumb 1986) to examine two- and three-dimensional wave energy propagation. In this paper, however, we use a local energetics analysis rather than the wave activity flux vector analysis for the following reasons: The definition of the flux vector is valid under the condition that the basic-state absolute vorticity gradient not vanish (Plumb 1986). Because our realistic background flows are such that the basic-state absolute vorticity vanishes in the vicinity of the jet streams, regions emphasized in our analysis, application of the wave activity flux vector is inappropriate. Moreover, in contrast to our local energetics analysis, the wave activity formalism cannot distinguish between how much energy is due to conversion versus how much is due to propagation, a distinction that is physically important for our basic state. b. Numerical procedures Variables in (2.1) are expanded in spherical harmonics:

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LI AND NATHAN

O OcPe M

(*)9 5

zmz1N

m n

m iml n

1 c.c.,

339

(4.3)

m50 n5zmz

where Pmn is the associated Legendre function and c.c. denotes the complex conjugate of the preceding term. The numerical solutions are obtained by solving (2.1) using the spectral transform method (Haltiner and Williams 1979) and a Gaussian elimination algorithm (see appendix B of LN for details). The grid space used in the calculations consists of 128 longitudes and 150 Gaussian latitudes (or 2.818 3 1.28). Due to the sharp meridional gradient of the basic state, high meridional resolution is required; thus M 5 15, N 5 60 is used for all of the calculations. Further increases in resolution produced no discernible changes in the results. In this section, we calculate, using finite differences, the energy budget described by (4.1) and, as in the previous section, focus on the far-field response to forcing in the following five regions: 1) the tropical Indian Ocean (708E–908E), 2) the western Pacific (1208E), 3) the central Pacific (the date line, 1808), 4) the eastern Pacific (1208W–908W), and 5) the tropical Atlantic Ocean (308W). c. Results 1) TROPICAL INDIAN OCEAN The tropical Indian Ocean is located within the tropical easterlies, at the upstream side of the extratropical Asian jet stream. Figure 4 shows the KE for the steady forced response and the forced low-frequency responses having periods of 40, 20, and 10 day, respectively. For the steady solution (Fig. 4a), most of the KE generated by the forcing propagates along the Asian jet stream to the central North Pacific, continuing to the Gulf of Alaska and the Great Plains of the United States. The steady response in the SH is very weak. This is due to the fact that the tropical easterlies in this region have a much wider extent to the south (208S) than to the north (78N), so that the energy propagation to the SH is almost completely prevented. The very long period (.30 day) responses are similar to the steady solution (cf. Figs. 4a,b). As the forcing frequency increases, the forced response in the NH becomes less significant, whereas more energy propagates to the SH, especially for the 10-day disturbances (Fig. 4d). As Fig. 4 clearly shows, the lower-frequency forced disturbances exhibit strong downstream propagation along the Asian jetstream in the NH, whereas the higher-frequency forced disturbances, especially the 10-day waves, are more pronounced in the SH. These energetics results are in qualitative agreement with the ray tracing results shown in Fig. 2 (case A). However, when making such a comparison, we remind the reader that the ray tracing analysis in section 3 is obtained under the assumption that the background state is slowly varying and that the ray path is for an individual wavenumber; in contrast, the energetics analysis

FIG. 4. Shown are the (a) steady kinetic energy, E9, s and the ensemble averaged kinetic energy Ê9& p of the (b) 40, (c) 20, and (d) 10 day periodically forced solutions. The forcing is centered at (708E, 08); contour interval: 40 m2 s22.

in this section is for an ensemble of waves having the same frequency, where both energy propagation and energy conversion are taken into account. Despite these differences in analysis approaches, we find very good agreement between the results shown in Figs. 2 and 4. 2) WESTERN PACIFIC The western Pacific is also located within the tropical easterlies, a little upstream of the Asian jet stream. This

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FIG. 5. As in Fig. 4 but for the forcing centered at (1208E, 08).

FIG. 6. As in Fig. 4 but for the forcing centered at (1808, 08).

area is the so-called ‘‘warm pool’’ region, where warm SSTs produce significant convection and upper-tropospheric divergence during normal years. Figure 5 is the same as Fig. 4, except the forcing is centered at (1208E, 08) over Indonesia. The steady response remains significant downstream of the Asian jet in the NH, that is, near the date line at 358N. However, the influence of the steady forcing over the western Pacific cannot reach as far downstream as that over the Indian Ocean (see Fig. 4a). The low frequency responses (Figs. 5b–d) are similar to those in Fig. 4, although the 20-day forced solution becomes stronger in the SH. The 10-day forced response remains

dominant in the SH, particularly over Australia. By comparing Fig. 5 with Fig. 4, we find that regardless of the position of the tropical forcing, the long period (.30 day) responses always have large amplitudes downstream of the jet streams in the NH. However, the intermediate period (;10–20 day) disturbances shift eastward with the forcing. This result implies that the jet streams in the NH play a more important role for the long period disturbances, while the intermediate timescale disturbances are more sensitive to the position of the tropical forcing. Comparison of Figs. 4 and 5 shows that nonstationary waves have a greater tendency to propagate into the SH

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FIG. 7. As in Fig. 4 but for the forcing centered at (1208W, 08).

FIG. 8. As in Fig. 4 but for the forcing centered at (308W, 08).

when they are generated over the western Pacific than when they are generated over the tropical Indian Ocean, which is consistent with the ray tracing solutions shown in Fig. 2 (cases A and B). The difference between the ray tracing solutions shown in Fig. 2 (case B) and the energy solutions shown in Fig. 5 in the NH is due to the fact that a point source is used for the ray tracing calculations, whereas a finite-sized source is used for the energy calculations.

stream. The steady response (Fig. 6a) propagates energy northeast of the forcing, reaching the west coast of North America. Although the forcing amplitude is the same as in the previous cases, the steady response shown in Fig. 6a is much weaker than those generated at the upstream side of the Asian jet (Figs. 4a and 5a). This is also true of the low-frequency forced responses (Fig. 6b). A local energetics analysis reveals that for the cases shown in sections 1 and 2 there is significant energy conversion from the basic flow to the perturbation along the Asian jet stream, which acts like an amplifier for the disturbances propagating through it. As shown by Eq. (4.1), the energy conversion depends on the

3) CENTRAL PACIFIC The central Pacific is located within the tropical easterlies, but on the downstream side of the Asian jet

basic-

342


TABLE 1. Listed below is the ratio of ensemble-averaged energy conversion to dissipation (in percent) for the global kinetic energy budget for both the steady and periodically forced solutions. Period (days) ` 40 20 10

Forcing locations (08, 708E) (08, 1208E) (08, 1808) (08, 1208W) (08, 308W) 56 54 51 40

60 58 60 46

56 48 55 50

49 45 38 32

40 36 33 29

state wind shear CKY and horizontal deformation CKX, which are much weaker on the downstream side than on the upstream side of the jet, the result being that the perturbations generated downstream of the jet stream are much weaker. However, the 10-day response still strongly influences the extratropics of both hemispheres. Its influence in the NH is more longitudinally extended, reaching as far as the southwest coast of the United States; its influence in the SH is more meridionally extended, reaching northeastern Australia. This is further evidence that the intermediate timescale responses are less influenced by the extratropical basic flows. 4) EASTERN

CENTRAL

PACIFIC

Tropical westerlies exist over the eastern central Pacific (1408W–1008W). Figure 7 displays the corresponding KE generated by a forcing centered at (1208W, 08). The steady response (Fig. 7a) propagates both to the northeast and to the southeast of the tropical forcing. The northeast branch follows the North Atlantic jet stream to the northeast coast of the United States, crosses the North Atlantic Ocean toward Europe, ultimately reaching central Russia. The southeast branch reaches 308S where it bends toward central Brazil. The former branch is much stronger and reaches farther downstream than the latter. This is due to the fact that the North Atlantic jet stream is much stronger than the basic flow in the SH. The low-frequency forced responses are similar to, but weaker than, that of the steady response and can propagate to both hemispheres. Thus, when the forcing is located within tropical westerlies, the steady responses dominate over the low-frequency responses. However, the contributions of the low-frequency forced disturbances to the total KE are considerable. Finally, we note that comparison of Fig. 7 and Fig. 3 (case C) reveals consistency between the energetics and WKB results; that is, both stationary and nonstationary waves generated in the tropical westerlies over the eastern Pacific propagate into both hemispheres. 5) ATLANTIC OCEAN The tropical easterlies over the Atlantic Ocean are located between 58S and 208S. Figure 8 shows the corresponding KE for a forcing centered at (08, 308W). Both

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TABLE 2. Listed below is the ratio of the ensemble-averaged energy conversion to dissipation (in percent) for the local kinetic energy budget in the Northern Hemisphere for both the steady and periodically forced solutions. Period (days)

Forcing locations (08, 708E) (08, 1208E) (08, 1808) (08, 1208W) (08, 308W)

the steady and low-frequency responses are largely confined to the NH, except for the 10-day response, which has significant southward propagation. The steady response has two prominent branches of downstream propagation: one propagates across the North Atlantic toward Europe and central Russia; the other propagates toward North Africa and South Asia, eventually reaching the west coast of North America. The steady response generated over the tropical Atlantic Ocean propagates along both jet streams in the NH. The low-frequency responses, however, mostly follow the North African–Asian jet stream and cannot reach as far downstream as the steady responses. To further demonstrate the importance of the steady tropical forcing on the extratropical circulation, we have calculated the global and local energy budgets for the forced responses. Table 1 lists the ratio of energy conversion to energy dissipation in the global energy budget. As stated in LN, the redistribution terms, that is, energy advection and pressure work, make no contribution to the global energy budget; thus, the forcing and energy conversion are balanced by dissipation. For the steady disturbances, the energy conversion provides about as much energy as the forcing term, varying from 40% to 60%. This result is consistent with Branstator (1985a), although the energy conversion term has a slightly higher percentage here. We attribute the small difference to the fact that the basic states used in the two studies are different. For the low-frequency disturbances, the contribution from the energy conversion decreases as the forcing frequency increases, which is in agreement with the results obtained by LN for zonal-mean basic flows. For example, for the 10-day disturbances, the energy conversion provides only 30%–50% of total kinetic energy. Thus, the extratropical basic flow plays a more important role in providing KE for the long timescale disturbances than for those of intermediate timescale. Table 2 tabulates the domain-averaged energy budget over the NH. The domain-averaged advection and pressure work terms are very small; thus, the forcing, energy conversion, and dissipation terms approximately balance. The contribution from the energy conversion is larger for steady disturbances than for low-frequency disturbances, ranging from 56% to 72%. The energy conversion from the NH basic flow provides more en-

` 40 20 10

68 67 62 40

72 68 66 40

71 52 53 46

69 62 52 36

56 51 47 37

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5. Discussion TABLE 3. Listed below is the ratio of the ensemble-averaged perturbation kinetic energy of the Northern Hemisphere to that of the globe (in percent) for both the steady and periodically forced solutions. Period (days)

Forcing locations (08, 708E) (08, 1208E) (08, 1808) (08, 1208W) (08, 308W)

ergy for the steady responses than the tropical forcing. However, for the low-frequency responses, the energy conversion becomes less important as the forcing frequency increases. For instance, the energy conversion contributes only 35%–46% to the total budget for the 10-day forced disturbances, which is about the same as that obtained for the global energy budget (Table 1). Thus, for these disturbances, the tropical forcing and extratropical flow are approximately equal in providing energy for the disturbances. Table 3 lists the percentage of perturbation KE of the NH to that of the whole globe. It shows that for the long timescale (periods .30 day) disturbances about 60%–80% of the total KE is in the NH, whereas for the intermediate timescale (period ;10–20 day) disturbances about 60%–70% is in the SH. Calculations also show that the extratropical flow plays an important role for the long timescale disturbances in the NH, whereas the tropical forcing is more important for the intermediate timescale disturbances in the SH. Although the energy redistribution terms make little or no contribution to the hemispheric or global energetics, these terms may still be significant in the local energy budget. For example, Fig. 9 displays the local energetics for the steady disturbances generated by the tropical forcing over the Atlantic Ocean (see Fig. 8a). Both the redistribution terms (i.e., energy advection and pressure work) and the energy conversion term are significant along the jet streams in the NH. The energy conversion term produces (feedbacks) perturbation kinetic energy locally from (to) the basic flows. However, the forced disturbances remain steady locally due to transport of kinetic energy through the redistribution terms, which generally offset each other along the jet streams. The net effect of these terms is to maintain steady forced disturbances with large amplitude along the jet streams. A typical example for low-frequency forced disturbances is shown in Fig. 10 for a 10-day disturbance generated by forcing over the central Pacific (see Fig. 6d). The energy conversion and energy redistribution terms are significant north and south of the tropical forcing region and much less so downstream of the forcing. Thus the 10-day forced disturbances can propagate to both hemispheres (Fig. 6d), but less downstream of the tropical forcing region.

Previous research examining the stability of zonally varying flows has demonstrated that the climatological winter upper-tropospheric flow is barotropically unstable (Simmons et al. 1983; Branstator 1985b). The energy conversion terms, CKX and CKY, in Eq. (4.1) are crucial for the generation of the low-frequency unstable SWB modes, which have large amplitudes downstream of the two major jet streams in the NH. In this study, however, we have only considered steady or periodically forced solutions. Therefore, the time-(ensemble) averaged energetics requires that the terms on the rhs of Eq. (4.1) balance. We have demonstrated that for the long timescale disturbances, the energy conversion term is significant downstream of the two jet streams in the NH, and thus, perturbations with relatively large amplitudes can be sustained at those locations. However, for the intermediate timescale disturbances, such downstream energy conversion becomes less important. For these modes, the responses are more sensitive to the position of the forcing than are those of the low-frequency modes. Blackmon et al. (1984a,b) investigated statistically the horizontal teleconnections of the 500-mb height fluctuations based on one-point correlation maps by dividing LFV into two different bands: long period (k30 days) and intermediate period (;10–30 days). They have shown that the correlation patterns for the former band are geographically fixed, with meridionally oriented dipole structures at the jet exit regions, whereas for the latter band the patterns tend to have a more similar shape relative to the base grid point, regardless of where the base grid point is located. This might be regarded as another indication that the extratropical intermediate fluctuations are mainly generated from the migrating low-frequency forcing, rather than from the extratropical basic flows. However, one difference between the study of Blackmon et al. (1984a,b) and the present research is that the former indicates that the intermediate time period teleconnection patterns originate within the jet entrance regions and propagate toward the Tropics, whereas the latter emphasizes the tropical origin of the perturbations. One possibility for the difference between the observations and model results may be due to the fact that Blackmon et al. (1984a,b) have focused on the NH rather than the whole globe, and the patterns with base grid points in the Tropics were not addressed. Another possible reason for the differences between the models is that the tropical influence on the 10–20 day timescale is more pronounced in the SH than in the NH. Such conjectures need to be tested further. 6. Conclusions In this study we have examined the role of localized, low-frequency tropical forcing on the generation and maintenance of atmospheric LFV in the extratropics.

` 40 20 10

80 71 57 38

67 54 38 34

64 49 36 40

65 59 50 42

84 79 73 45

344


FIG. 9. Local energetics of the steady-state solution generated by a forcing centered at (308W, 08): (a) advection, (b) pressure work, and (c) energy conversion. The contour interval is 1 3 1027.

This study directly extends Li and Nathan’s (1994) linear work on zonally uniform basic states by considering a longitudinally varying climatological January flow of the upper troposphere. In particular, we have examined the energy propagation of both steady and low-frequency forced disturbances using WKB theory and energetics analyses. The two-dimensional WKB approach has allowed an analytical analysis of the path of energy propagation using ray tracing. The energetics analysis has considered both the global and local energy budget. Following are the major findings of this research: ● The ray path is determined by the combination of three vectors: the basic-state wind, the absolute vorticity gradient, and the local wave vector. The relative importance of each vector depends on the wave scale, background flow structure, and forcing frequency v. As v decreases, the ray path and thus energy propagation tends to align itself with the local wave vector.

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FIG. 10. As in Fig. 9 but for the 10-day periodically forced solution where the forcing is centered at (1808, 08).

● The extratropical jet streams in the NH have a stronger influence on the rays of the long period (.30 day) forced disturbances than on those of intermediate period (;10–20 day). ● The North Africa–Asian jet stream is a stronger waveguide for the long period waves than the North Atlantic jet stream. ● The long period disturbances have a strong influence on the NH circulation and are less sensitive to the location of the tropical forcing than those of intermediate period, which have a more significant influence on the SH circulation and are more dependent upon the forcing location. In this research we have considered a climatological mean January basic flow. In reality, the basic-state flow may have significant interannual variations, thus yielding substantially different middle-latitude responses to low-frequency tropical forcing. For example, during an El Ninõ year, the large-scale patterns are generally quite

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different from those of normal years (Philander 1990), and it remains unclear how the results obtained here would be modified. Because our energetics calculations have been carried out for steady-state conditions, a logical and important extension of the present work would be to examine the time-dependent problem in order to calculate the time required for a tropical signal to set up in the extratropics. In addition, although our energetics analysis provides important physical information about the flow field, including, for example, the location of internal energy sources, we note that the traditional energetics formulation that we have adopted may be somewhat arbitrary in the way in which the various energy terms are partitioned (Plumb 1983). Thus, it may be of interest to carry out similar calculations using the transformed Eulerian formulation of Plumb, though we note that his formulation provides the clearest interpretation under nonacceleration conditions, conditions which are not met in the present study. The role of low-frequency tropical forcing should also be carried out in more realistic baroclinic models in order to test the robustness of the barotropic model results obtained in this study. It is likely that the propagation of low-frequency disturbances from the Tropics into middle latitudes may draw on the available potential energy of the basic state as well as its kinetic energy. Moreover, the effects of nonlinear interactions among the free unstable modes, the steady forced, and the periodically forced modes also need to be considered. These problems are currently under study. Acknowledgments. The authors thank Dr. Grant Branstator, Professor Terry Williams, Eugene Cordero, Bob Echols, Ann Evans, Dan Gottas, and Mary Parlange for their many constructive comments on the manuscript. The authors also thank Steve Mauget and Professor Bryan Weare for their assistance in providing the ECMWF data that was used in the construction of our basic states. This work was supported by the National Science Foundation (Grant ATM-90-96152), the National Aeronautics and Space Administration (Grants NAG8-871, NAG8-1054, NAG8-1143), the Air Force Office of Scientific Research (Grant AFOSR 90-0009), the University of California Institute for Collaborative Research (INCOR), and the Atmospheric Science Division of Lawrence Livermore National Laboratory.

APPENDIX

Derivation of the Local Energetics Equation We start with the linearized nondimensional momentum equations on the sphere:

]u9 U ]u9 ]u9 u9 ]U 1 1V 1 ]t cos f ]l ]f cos f ]l 1 y9

]U 2 y9sin f 2 (Uy9 1 u9V)tan f ]f

52

1 ]p9 1 R l 1 Fl cos f ]l

(A.1)

]y9 U ]y9 ]y9 u9 ]V 1 1V 1 ]t cos f ]l ]f cos f ]l 1 y9

]V 2 u9sin f 1 2u9U tan f ]f

52

]p9 1 R f 1 Ff , ]f

(A.2)

where R, 5 R, li 1 R, f j and F 5 Fli 1 Ff j are the damping and forcing terms, respectively. The nondivergent vorticity equation is obtained by combining (A.1) and (A.2), where R9 5

1 ]Ff 1 ]Fl cos f 2 [ 2¹ 2x. cos f ]l cos f ]f

(A.3)

Thus, for a given R9, the momentum forcing components in terms of x are Fl 5

]x , ]f

Ff 5 2

1 ]x . cos f ]f

(A.4)

The damping term can be determined in a similar way. A diagnostic equation for the pressure p9 can be derived by adding (]/cos f]l)(A.1) and (]/cos f]f)[(A.2) cos f]: ¹ 2p9 5

[

1 ](V¹ 2c 9 1 v9¹ 2c¯ ) cos f ]l 2

]

] cos f(U¹ 2c 9 1 u9¹ 2c¯ ) ]f

1 (sin f¹ 2c 9 2 u9cos f) 2 ¹ 2(Uu9 1 Vy9).

(A.5)

By adding (A.1) 3 u9 1 (A.2) 3 y9, we obtain the local kinetic energy equation (4.1). REFERENCES Anderson, J. L., 1991: The robustness of barotropic unstable modes to a zonally varying atmosphere. J. Atmos. Sci., 48, 2393–2410. Andrews, D. G., 1984: On the stability of forced zonal flows. Quart. J. Roy. Meteor. Soc., 110, 657–662. Bender, C. M., and S. A. Orszag, 1978: Advanced Mathematical Methods for Scientists and Engineers. McGraw Hill, 593 pp. Bjerknes, J., 1969: Atmospheric teleconnection patterns from the equatorial Pacific. Mon. Wea. Rev., 97, 162–172. Blackmon, M. L., Y.-H. Lee, and J. M. Wallace, 1984a: Horizontal structure of 500 mb height fluctuations with long, intermediate and short time scales. J. Armos. Sci., 41, 961–979. , , , and H.-H. Hsu, 1984b: Time variation of 500 mb

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