October 1984 T. Yamagata and Y. Hayashi 709 A Simple Diagnostic Model for the 30-50 Day Oscillation in the Tropics By Toshio Yamagata* and Yoshikazu Hayashi Geophysical Fluid Dynamics Program, Princeton University, Princeton, N, J. 08540 (Manuscript received 18 June 1984 in revised form 30 July 1984) Abstract A theoretical discussion is given of the structure of observed 30-50 day atmospheric oscillation in the tropics by the use of a simple linear diagnostic model based on primitive, shallow water, *-plane equations with the long wave approximation. It is assumed that this oscillation is forced by a localized heat source pulsating with a 40-day period. The response of the model shows that the zonal wind oscillation consists of a standing wave with a phase jump in the region of heating and traveling waves with slow zonal phase variations elsewhere. The pressure field is, however, not associated with a phase jump in agreement with observations. 1. Introduction The 30-50 day oscillation1) in the tropical troposphere was detected by Madden and Julian (1971, 1972) and Parker (1973). The existence has been further confirmed by several observational works (Yasunari, 1979, 1980; Julian and Madden, 1981; Maruyama, 1982; Krishnamurti and Subrahmanyam, 1982). This oscillation is most prominent in the zonal component, which is out of phase between the upper and lower troposphere. This baroclinic structure is confined to low latitudes (10*S- 10*N) with the maximum amplitude located along the equator. According to Madden and Julian (1972) and Parker (1973), the above aspect of this tropical oscillation may be interpreted as an eastward propagation of the Walker circulation (cf. Bjerkness, 1969). In other words the 30-50 day oscillation may be regarded as a miniature * Permanent affiliation : Research Institute for Applied Mechanics, Kyushu University, Kasuga 816, Japan. 1) This oscillation can also be found in the spectral analysis of Wallace and Chang (1969). of the interannual Southern Oscillation, since the time scales of these oscillations differ by more than an order of magnitude. Madden and Julian (1972) also suggested that the short term variations in the Walker circulation are related to the enhanced large-scale convection processes over the Indian ocean and the "maritime continent" of Borneo and Indonesia. Yasunari (1979) confirmed a 30-50 day oscillation in convective heating by analysing the cloud cover over India. Major convective heating also exists over South America and equatorial Africa (Krishnamurti et al., 1973). Orlanski and Polinsky (1977) and Yasunari (1979), however, pointed out that the variation in cloud cover over equatorial Africa has a period of a few days. Such an analysis over South America is not, however, available. Nevertheless, the above spectral difference in the cloudiness over India and Africa suggests the importance of monsoon activity in the generation of the 30-50 day oscillation. Using the cloud data set (May 1978-December 1980) available from the Japanese geostationary satellite located over the tropical western Pacific (0*N, 140E), Maruyama (1982a)
710 Journal of the Meteorological Society of Japan Vol. 62, No. 5 has recently showed by a spectral analysis (Fig. 1) that the 30-50 day oscillation is associated with a "node" (i. e. fast zonal phase variation and slow local phase speed) over the western equatorial Pacific (140*E-180*E). This node is detectable in Madden and Julian's (1972) spectral analysis (Fig. 2) of the 150mb zonal wind over 10 years, although the long term average may have smoothed a sharp local phase variation if the node drifts from Fig. 1 Observed spatial distribution (May 1978- December 1980, 150-200mb) of phase differences (solid) and coherences (dashed) between zonal wind field (period range 25-60 days) and the reference zonal wind indicated by the cross. Phase difference of *0.5 cycle corresponds to *180 degrees, while coherence of 100% corresponds 1.0 (after Maruyama, 1982). Fig. 2 Observed phase angle (period range 40-50 days) between given variables at given stations and the station pressure at Canton Island as given by a cross spectral analysis of data over 10 years (after Madden and Julian, 1972). year to year. This node is also detectable in the spectral analysis by Maruyama (1982b) of FGGE data (April-June, 1979). Parker (1973) tried to explain the eastward zonal propagation within the framework of the equatorial Kelvin wave theory (Matsuno, 1966). However, according to the dispersion relation of Kelvin waves, the slow phase speed measured in the neighborhood of the node gives a very short vertical and meridional scales which are inconsistent with those observed. Chang (1977) and Stevens and White (1979) pointed out that the observed scales can be explained, if the effects of Rayleigh friction and Newtonian cooling are taken into account. However, the present paper suggests that the observed scales can also be explained, if the effects of heating is taken into account in the region of the phase jump. Away from the region of heating, the observed fast speed is indeed consistent, even in the absence of dissipation, with the observed vertical and meridional scales. There is another aspect of the 30-50 day oscillation, Yasunari (1979, 80) noticed, by use of daily satellite cloud images, the meridional propagation (about 1* latitude per day) of cloud cover in addition to the eastward propagation. A synoptic perspective of the oscillation during MONEX was recently provided by Krishnamurti and Subrahmanyam (1982). They claimed that a train of troughs and ridges is formed near the equator, migrates away from the equator and dissipates near the Himalayas. Recently, Murakami (1984) also showed the meridional propagation of the 30-50 day mode by using the activity index of the deep convective clouds. These recent observational results suggest that the 30-50 day oscillations have an important influence on extratropics, as well as the interannual Southern Oscillation (Horel and Wallace,1981). On the basis of the above observational analyses, we shall propose a simple hypothesis that the 30-50 day oscillation is the response of the tropical troposphere to a localized heat source over the equatorial Asian monsoon region which pulsates with the observed periodicity. It is of interest to examine this possibility by the use of a simple model. It is,
October 1984 T. Yamagata and Y. Hayashi 711 however, beyond the scope of the present paper to explain why the heat source pulsates with such a periodicity. In Section 2, a diagnostic model is described, while the results are shown in Section 3. Summary and remarks are given in Section 4. 2. Description of a diagnostic model In order to present a simple solution representing the 30-50 day oscillation, we shall modify a simple model of Gill (1980) for tropical stationary waves by oscillating the imposed localized heating with a 40-day period. The crucial assumption of Gill's model is that the vertical structure can be described by a single vertical mode. Geisler and Stevens (1982) demonstrated that the altitude dependence of the response amplitude in the troposphere is very close to that of the tropospheric forcing for *<* where * is the frequency of the periodic forcing and * is the dissipation coefficient. Therefore, if the forcing is of a single dominant mode, the shallow water equations for this mode can describe the response fairly well within the layer of forcing. Above this layer, however, many vertical modes must be superposed as discussed by Hayashi (1976). The vertical distribution of the forcing used in several studies (Stevens et al., 1977 ; Geisler,1981; Geisler and Stevens, 1982) is a reasonable fit to the first baroclinic mode with the equivalent depth of 400m. This is mainly because the heating has a single maximum at about 500mb (e. g., Yanai et al., 1973; Hantel and Baader, 1978) in the tropics. Thus problem can be reduced to solving the shallow water equations with an imposed localized periodic forcing. It is of course necessary to include a barotropic mode to more adequately explain the extratropical responses. The basic equations with the equatorial beta-plane approximation and the "long wave approximation2" are 2) Recently, Lim and Chang (1983) showed that this approximation distorts the responses especially in the mean westerlies. However, this distortion is negligible in the present problem. where u, *, p and Q are the zonal velocity, the meridional velocity, the pressure, the mass source or sink (which corresponds to the heating rate). *and * are the Rayleigh friction and Newtonian cooling coefficients which are assumed to be equal for the moment. The above system is nondimensionalized by using as a length scale the equatorial Rossby radius (Co/2*)1/2 and the time scale (2*Co)-1/2, where Co(=(gH)1/2) is the long gravity wave speed with the equivalent depth of H. In the present study H=400m and Co=62ms-1. As in Gill (1980), we prescribe the heating Q which is localized and symmetric about the equator as where K=*/(2L). Gill (1980) assumed a steady forcing (*=0) and a zonally infinite domain. In the present model, we assume a periodic forcing with frequency * and a zonally periodic domain (x1, x2). Our solution is different from that of Lau and Lim (1982) who solved a transient solution approaching the stationary solution. Following Gill (1980), we can readily write down the solution of our problem as and where the time factor exp(i*t) has been omitted for convenience. The above solution consists of the Kelvin (q0) and Rossby (q2) modes which are given by
712 Journal of the Meteorological Society of Japan Vol. 62, No. 5 and where **+i* and D= x2-x1. The above solution can be obtained by formally replacing * in Gill's stationary wave solution by **(=*+i*). Unlike Gill's solution, both eastward and westward propagating waves are allowed outside the region of heating. These waves propagate around the earth, although they are almost dissipated for large *. The real part of u(x, y, t) is rewritten as time section (Fig. 3b). In particular, a node is seen near x=0 in Fig. 3a, while the time phase varies rapidly between 120*-180*E in Fig. 3b. Fig. 4 illustrates the velocity field at t=0 in the lower troposphere for the case 2*/*= 40 days, *=0.1, L=4. The damping time corresponds to 2.2 days. L*4 (40*) is chosen, since Yasunari (1979) showed the high spatial correlation of the cloud cover extends between 60*E and 140*E. At t=0, when heating attains the maximum, the velocity field is quite similar to Gill's solution. As Gill pointed out, a localized heating symmetric about the equator produces easterly flow in the lower troposphere east of the heating due to the Kelvin mode (q0-component). West of the heating, westerly flow is generated in the lower troposphere due to the Rossby mode (q2-component). The zonal group velocities of the Kelvin and Rossby modes are 1 and -1/3, while their zonal decay rates are * and 3*, respectively. When *D is much larger than unity (as is the case in the present model), it turns out that there is little difference between the cyclic domain (present model) and the zonally infinite domain (Gill's model). where and The above A and * denote the time amplitude and phase of u(x, y, t), respectively. 3. Results Fig. 3a shows the equatorial longitude-time section for the u-component. This figure resembles Parker's (1973) observed longitude- Fig. 3a Equatorial longitude-time section of u- component for *=0.1 and L=4. Negative areas are shaded.
October 1984 T. Yamagata and Y. Hayashi 713 Fig. 3b Equatorial longitude-time section of observed u-component at 100mb for part of 1966 (after Parker, 1973). Fig. 4 The velocity vector field in the lower troposphere at the time of maximum heating. Lowest pressures are marked as L. Fig. 5 Ampitude (upper) of u-component with contour intervals of 0.3. Phase (lower) of the u-component with contour intervals of 0.09. Dark shade >0.48, light shade <0, *0.5 cycle corresponds to *180 degrees. The phase difference is given by the difference between two contour labels.
714 Journal of the Meteorological Society of Japan Vol. 62, No. 5 Fig. 5 and Fig. 6 show the spatial distribution of the amplitude and phase of the present solution. It is seen that the amplitude of the u-component (Fig. 5) attains its maximum over the equator, while the amplitude of the pressure (Fig. 6) attains its maxima off the equator except at longitudes east of the heating. The model's phase difference of u (Fig. 5) shows a slow zonal phase variation except for a sharp jump (almost out of phase) near the heating region both in zonal and meridional directions. From (9b) and (10b), the position of the phase jump over the equator is approximately given by x*l/3. This position depends on the shape and the zonal extent of the forcing. This phase jump may explain the observed rapid variation of phase. On the other hand, the pressure field (Fig. 6) is not associated with a phase jump. This feature is consistent with the fact that a phase jump is not detectable in the observed pressure Fig. 6 Amplitude (upper) and phase (lower) of pressure with contour intervals of 0.02. The phase difference is given by the difference between two contour labels. Fig. 7 Latitude-time section of pressure at x=0.5 with constant contour intervals for *=*=0.1 (a) and *=0.1, *=0 (b).
October 1984 T. Yamagata and Y. Hayashi 715 field (Fig. 2 ; see also Fig. 4 of Madden and Julian, 1972 and Fig. 7 of Muruyama, 1982b). It should be noted that the u and p of the Kelvin mode response to the east of the heating are nearly in phase, while those of the Rossby mode response west of the heating are nearly 180* out of phase near the equator. It can be proved from (6) and (8) that cu*p near the equator (c=local phase speed). This relation is consistent with the fact that these modes are associated with a small y* in (1) near the equator. Thus, u and p are in phase for the eastward moving (c=1) Kelvin mode, while they are out of phase for the westward moving (c=-1/3) Rossby mode. The present model (see Fig. 6) does not explain the observed meridional phase tilt. However, this tilt can occur even within the framework of this simple model. As shown in Yamagata and Philander (1984), the difference between the coefficient of Newtonian cooling and the coefficient of Rayleigh dampling * * causes a phase shift in the meridional structure of equatorial waves. For instance, the phase line of the Kelvin mode is given by where k is the tonal wavenumber. When <*, the phase lines slant backward on * both sides of the equator relative to the direction of zonal phase propagation.3) Fig. 7 shows the latitude-time section for p-component evaluated along the meridian which passes near the center of the heating region (i, e. x=0.5). This figure was produced by integrating linear shallow water equations on a sphere numerically. For *=* (Fig. 7), no meridional phase propagation of the p-field is observed. For (Fig. 7b), however, the phase propagation * is quite clear. If we adopt *=0.1 (*(2.2 days)-1) and *=0 as in Fig. 9b, the meridional phase speed at 20*N (or 20*S) is less than 2*/day, which is consistent with Yasunari (1979) and Krishnamurti and Subrahmanyam (1982). Thus, with the exception of the equatorial region, the meridional propagation of 3) It should be noted here that Geisler (1981) adopted *= (15 days)-1 and *= (5 days)-1 in his model of the Walker circulation. the cloudiness over the Asian monsoon area may be, to some extent, explained within the framework of the present simple model. It should be noted here, however, that the meridional propagation may be more adequately explained if one also takes barotropic responses into account. 4. Summary and remarks We have presented the results of a diagnostic model for the tropical 30-50 day oscillation to examine the response of the equatorial troposphere to a localized heat source pulsating with a 40-day period. The model shows that the zonal wind oscillation consists of a standing wave with a phase jump in the region of heating, while its time phase propagates like traveling waves elsewhere. This standing wave oscillation, however, is not a pure one such as is formed between two longitudinal boundaries as discussed by Gent (1981), since the former is a quasi-equilibrium response inside the oscillatory heat source. The phase jump in the zonal wind located on the eastern, half of the region of forcing is consistent with the observations. The model's pressure field is, however, not associated with a phase jump in agreement with observations. The zonal wind and pressure are in phase to the east of the heating, while they are out of phase west of the heating. A major defect of the present model is that the eastward and westward traveling waves are of comparable magnitude, whereas the observed wind oscillation is associated with a more eastward than westward moving component, particularly in the upper troposphere (Maruyama, personal communication, 1983). This defect may be alleviated by allowing the time phase of the heating to propagate while its amplitude is localized. The effect of local upper easterlies in a three-dimensional model may explain the suppression of the localized westward moving component in the upper troposphere. It has also been demonstrated that the observed meridional phase tilt can be, to some extent, explained by adopting a stronger Rayleigh friction coefficient than the Newtonian cooling coefficient. If nonlinear inter-
716 Journal of the Meteorological Society of Japan Vol. 62, No. 5 actions are included in the model, it may not be necessary to assume strong Rayleigh friction, as suggested by Sardeshmukh and Held (1984) for tropical stationary waves from the analysis of the nonlinear terms of the timemean vorticity balance of a GFDL general circulation model. It is of importance to study the 30-50 day oscillation by the use of a general circulation model, since it gives valuable information on thermal forcing and will clarify whether or not atmosphere-ocean coupling is crucial for this periodicity, as suggested by an atmosphere-ocean model of Webster and Chou (1980) for the low frequency variability of the monsoon (see also Philander et al., 1984). In order to explain the periodicity theoretically, the convective heating must be somehow parameterized in terms of the large scale field as in the wave-cisk (Hayashi, 1970; Lindzen, 1974), since there is no predominant response at 30-50 day periods of tropical waves to random thermal forcing (Hayashi, 1976). It is also of importance to clarify the relations between the 30-50 day oscillation and the break and active monsoons (see Cadet, 1983). It is not clear whether the latter is merely a local and seasonal intensification of the former which, according to Maruyama (1982), exists all year around. Acknowledgements We are grateful to Drs. T. Maruyama, K. Miyakoda, M. Salby, S.G.H. Philander, K. Bryan and I. Held for their interest and helpful suggestions, to the GFDL Drafting Group for figure preparation and to Ms. J. Callan and Miss K. Sunabe for typing. The work was completed while T. Yamagata was a Visiting Scientist in the Geophysical Fluid Dynamics Program at Princeton University, which is supported by NOAA/Princeton University Grant 04-7-022-44017. References Bjerkness, J.A., 1969: Atmospheric teleconnections from the equatorial Pacific. Mon. Wea. Rev., 97, 163-172. Cadet, D., 1983: The monsoon over the Indian Ocean during summer 1975. Part II: Break and active monsoons. Mon. Wea. Rev., 111, 95-108. Chang, C.-P., 1977: Viscous internal gravity waves and low frequency oscillations in the tropics. J. Atmos. Sci., 34, 901-910. Geisler, J.E., 1981: A linear model of the Walker circulation. J. Atmos. Sci., 38, 1390-1400. and D.E. Stevens, 1982: On the vertical structure of damped steady circulation in the tropics. Quart. J. Roy. Meteor. Soc., 108, 87-93. Gent, P.R., 1981: Forced standing equatorial ocean wave modes. J. Mar. Res., 39, 695-709. Gill, A.E., 1980: Some simple solutions for heatinduced tropical circulation. Quart. J. Roy. Meteor. Soc., 106, 447-462. antel, M. and H.R. Baader, H 1978: Diabatic heating climatology of the zonal atmosphere. J. Atmos. Sci., 35, 1180-1189. Hayashi, Y., 1970: A theory of large-scale equatorial waves generated by condensation heat and accelerating the zonal wind. J. Met. Soc. Japan, 48, 140-160., 1976: Non-singular resonance of equatorial waves under the radiation condition. J. Atmos. Sci., 33, 183-201. Horel, J.D. and J.M. Wallace, 1981: Planetaryscale atmospheric phenomena associated with the Southern Oscillation. Mon. Wea. Rev., 109, 813-829. Julian, P. R, and R.A. Madden, 1981: Comments on a paper by T. Yasunari, a quasi-stationary appearance of 30 to 40-day period in the cloudiness fluctuations during the summer monsoon over India. J. Met. Soc. Japan, 59, 435-437. Krishnamurti, T. N. and D. Subrahmanyan, 1982: The 30-50 day mode at 850mb during MONEX. J. Atmos. Sci., 39, 2088-2095. M. Kanamitsu, W.J. Koss and J.D. Lee, 1973: Tropical east-west circulations during the northern winter. J. Atmos. Sci., 30, 780-787. Lau, K.-M. and H. Lin, 1982: Thermally driven motions in an equatorial *-plane: Hadley and Walker Circulations during the winter monsoon. Mon. Wea. Rev., 110, 336-353. Lim, H. and C.-P. Chang, 1983: Dynamics of teleconnections and Walker circulations forced by equatorial heating. J. Atmos. Sci., 40, 1897-1915. Lindzen, R.L., 1974: Wave-CISK in the tropics. J. Atmos. Sci., 31, 156-179. Madden, R.A. and P.R. Julian, 1971: Detection of a 40-50 day oscillation in the zonal wind in the tropical Pacific. J. Atmos. Sci., 28, 702-708. and, 1972: Description of globalscale circulation cells in the tropics with a 40-50 day period. J. Atmos. Sci., 29, 1109-1123. Maruyama, T., 1982a: Upper tropospheric zonal wind oscillation with a 30-50 day period over the equatorial western Pacific observed in cloud movement vectors. J. Met. Soc. Japan, 60, 172-182., 1982b: Fluctuations with a period of 30-50 days over the equatorial Indian ocean.
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