The Zonal Wavelength of the Quasi-Stationary Rossby Waves. Trapped in the Westerly Jet. By Toru Terao

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1 Journal of the Meteorological Society of Japan, Vol. 77, No. 3, pp , The Zonal Wavelength of the Quasi-Stationary Rossby Waves Trapped in the Westerly Jet By Toru Terao Disaster Prevention Research Institute, Kyoto University, Uji, Japan (Manuscript received 3 October 1997, in revised form 26 February 1999) Abstract Properties of the quasi-stationary Rossby waves along the westerly jets are investigated with the spacetime spectral analysis of 200hPa meridional wind velocity for four regions in the Northern Hemisphere summer using ECMWF data for 1980 to The observed zonal wavenumber of eastward (westward) propagating disturbances increases (decreases) as the frequency increases. The eastward propagating disturbances are stronger than the westward propagating ones. The quasi-stationary Rossby waves are seen not only in 10-to 30-day time scales but also in 30-to 90-day time scales. These observed zonal wavenumber-frequency relationships are reproduced by a B-channel model with step-like basic states. Calculated zonal wavenumber of eastward (westward) propagating solution of this model increases (decreases) as the frequency increases. The eastward propagating solutions are more strongly trapped in the westerly jet than the westward propagating ones. For these four waveguides mentioned, the results of the spectral analysis agree with the properties of the solutions deduced from the B-channel model with basic states derived from the climatological basic flow near these waveguides. 1. Introduction It has been shown that quasi-stationary disturbances with a 10-to 30-day time scale are strongly trapped, mainly in the westerly jets in the extratropics. The comprehensive statistical data analyses using the global long-term datasets elucidated the basic properties of these disturbances (Blackmon et al. (1984), Hsu and Lin (1992) and Kiladis and Weickmann (1992) for the Northern Hemisphere winter, Berbery et al. (1992) for the Southern Hemisphere, and Ambrizzi et al. (1995) and Terao (1995) for the Northern Hemisphere summer). A zonally oriented wave train-like pattern with zonal wavelength at about 4,000 to 6,000km, which corresponds to the zonal wavenumber 5-7 in the midlatitudes, is commonly observed. An eastward group velocity at a speed comparable to the zonal velocity of the basic flow is fairly evident in most cases. The barotropic vertical structure is dominant in the midlatitudes with a maximum amplitude at about 200 to 300hPa. These findings were also confirmed by statistical studies using the regional datasets (Shapiro and Goldenberg, 1993; Terao, 1998). Corresponding author: Toru Terao, Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto , Japan. teraoadpac.dpri.kyotou.ac.jp (c) 1999, Meteorological Society of Japan Some evidence that shows the dependency of the wave properties on the direction of the phase propagation and frequency change has been obtained. In most cases, these waves exhibit eastward phase propagations (for examples, see Figs. ha and lib of Berbery et al. (1992) and Figs. 5, 6 of Hsu and Lin (1992)) or standing patterns (Fig. 7 of Hsu and Lin (1992)). The westward phase propagations are rarely seen. The numerical calculation by Yang and Hoskins (1996) showed that the wavenumber of an eastward phase propagating trapped wave was larger than that of a westward phase propagating one. This relationship becomes more significant for larger frequencies. For the investigation of such property changes of the trapped waves according to the frequency change to be possible, the space-time spectral analysis (Hayashi, 1971) along the waveguides should be effective. In Section 2, the properties of these disturbances are investigated with this method for regions along the westerly jets. It was applied by Terao (1998) to the trapped quasi-stationary disturbances, in which, however, the analysis was limited to the region along the subtropical westerly jet over the Eurasian continent. On the other hand, some diagnostic studies with the ray-path analysis on the Rossby wave in the barotropic atmosphere (Hoskins and Karoly, 1981) showed that the propagation paths of observed

2 688 Journal of the Meteorological Society of Japan Vol. 77, No. 3 or numerically calculated quasi-stationary disturbances trapped in the westerly jets agreed with theoretically predicted ray-paths (Hoskins and Ambrizzi, 1993; Yang and Hoskins, 1996; Terao, 1998). The diagnostic studies identified waveguides and the possible zonal waenumber ranges of the quasi-stationary Rossby waves trapped there. It should be noted here, however, that the WKB approximation on which the ray-path analysis is based is not applicable near the westerly jets, since the basic flow varies steeply in the meridional direction. Therefore, studies in which no meridional WKB assumption is employed should be carried out and compared with the results obtained from the ray-path analyses. One of the major purposes of the present study is to get new insights into the properties of quasistationary Rossby waves by a method without assuming the meridional WKB approximation. In Section 3, we will explicitly derive the meridional structures of the trapped waves using the tonally symmetric B-channel models for the jet-like basic flows. In Section 4, it is shown that the standing wave pattern frequently observed is reproduced by the composition of eastward and westward propagating solutions of the B-channel model. Concluding remarks are presented in Section 5. The basic equations and parameter definitions are given in Appendix A. The validity of the step-like basic state used in the present study is tested in Appendix B. The nine years of ECMWF/WMO initialized analysis ( ) and the five years of ECMWF/TOGA non-initialized analysis ( ) are used in the present study. 2. Space-time spectral analysis Space-time spectral densities (Hayashi, 1971) are calculated for the following four regions along the climatological westerly jets in both hemispheres in the boreal summer: Asian waveguide (WG-1): The subtropical westerly jet along 37.5oN over the Eurasian Continent and the western North Pacific from 0oE to 180oE. Atlantic waveguide (WG-2): The subtropical west erly jet along 45oN over the eastern North Pacific, the North American Continent and the Atlantic Ocean from 180oW to 0oW. Australian waveguide (WG-3): The subtropical westerly jet in the Southern Hemisphere along 25oS over the Indian Ocean, the Australian Continent and the South Pacific Ocean from 60oE to 120oW. Southern Hemispheric polar waveguide (WG-4): The polar jet along 52.5oS from 60oE to 120oW. Judging from the global distribution of the stationary total wavenumber KS defined by Eq. (3) (Fig. ib), these regions are waveguides for stationary Rossby waves. The KS distribution is calculated from the climatological ( ) zonal wind U at 300 hpa (Fig. la) for northern summer (June to September) average. See Appendix A.2 for the detailed method in which we find waveguides from a Ks distribution map. The regions of the waveguides are also indicated in Fig. la. They nearly correspond to the waveguides pointed out by Ambrizzi et al. (1995). The space-time spectral densities are calculated with a fast Fourier transform (FFT) method. The spectra of 14 summers, from 1980 to 1993, are averaged. The seasonal march components, which are assumed to be expressed by second-order trends, are removed from time series of each grid point. The 10% cosine tapers (see Subsection 6.4 of Percival and Walden, 1993) are used for the temporal and zonal directions to reduce the Gibbs phenomena. Spectra are calculated for all waveguides for 200 hpa meridional wind velocities, while only for WG-1 and WG-3 for 200 hpa streamfunctions (Fig. 2). 1 The spectral peaks for meridional wind velocities can be seen for periods up to 90 days. This implies that the quasi-stationary Rossby waves are systematically seen not only in 10- to 30-day time scales but also in up to 90-day time scales. In the streamfunction data, however, the spectral peaks are not so evident for the periods longer than 45 days. Most of the past statistical studies on these disturbances have mainly used geopotential height data (Blackmon et al., 1984) or streamfunction data (Hsu and Lin, 1992). This could be the reason why the quasi-stationary disturbances have been mostly observed within 10- to 30-day time scales. These peaks are seen at different zonal wavenumbers k; at k=5-7 for Fig. 2b, at k=4-6 for Figs. 2c and 2e, and at k=3-5 for Fig. 2f. Furthermore, a significant relationship is seen between the frequency and wavenumber of the spectral peaks. As the frequency increases, the wavenumber of peaks increases (decreases) for eastward (westward) prop- 1 The 200 hpa data are used for the spectral analysis, while in Fig. 1 the 300 hpa climatological flow was used. This is because the author presumed that the quasi-stationary Rossby waves have nearly barotropic vertical structures, as has been reported by many authors. We considered that the horizontal structure of the waves can be most prominently seen at the level where their amplitude is maximum; in this case it was 200 hpa. That is why the author used the 200 hpa data for the spectral analysis. On the other hand, Grose and Hoskins (1979) have shown that the motions of the real atmosphere are well modeled by a barotropic atmosphere at 300 hpa. This indicates that the 300 hpa climatological flow is more appropriate to diagnose the behavior of the nearly barotropic disturbances than the 200 hpa flow. Therefore, the author used the 300 hpa climatological flow in Fig. 1.

3 June 1999 T. Terao 689 Fig. 1. (a) Climatological ( ) summer (from June to September) mean of the 300 hpa tonal wind. Contour interval is 5ms-l. Rectangles with thick solid lines show four regions listed in the text (WG-1 to 4), being likely to correspond to waveguides for quasi-stationary Rossby waves. (b) Global distribution of KS calculated from (a) using Eq. (3) in Appendix A. Dashed and dotted contours indicate the wavenumber 3 and 6, respectively. Areas where Ks2<0 are hatched. agating disturbances, although for westward propagating disturbances in WG-4 it is not so obvious. In WG-1 (Fig. 2b), for example, spectral peaks in higher frequency ranges (about 10- to 20-day periods) are seen at about k = 7 for eastward propagating disturbances. For westward propagating ones, they are at about k=5. Particularly for shorter periods, the eastward propagating disturbances have higher peak intensities than the westward propagating ones. For example, in Fig. 2b, the eastward propagating disturbances are stronger than the westward propagating ones for periods shorter than 30 days. In Fig. 2c, the peak values of eastward propagating disturbances at 20- and 30-day periods are about 1,400 and 1,600, respectively, while corresponding values of westward propagating ones are about 1,200 and 1,000, respectively. As can be seen in Figs. 2b, 2c, 2e and 2f, the westward propagating disturbances are suppressed for the periods shorter than 30 days. 3. Disturbances in a B-channel model In the previous section, dependencies of the characteristics of the quasi-stationary disturbances in the westerly jet on the direction of the phase propagation and the frequency change were addressed by use of the space-time spectral analysis. We will now compare these results with a tonal dispersion relationship calculated from normal mode solutions of zonally symmetric basic flows in a /3-channel model. In former studies, properties of such disturbances have been discussed mainly in the context of the raypath analysis of the Rossby wave, which is based on the meridional WKB approximation. However, as is noted in Section 1, this approximation is not applicable in the region along the westerly jet where the basic flow changes steeply in the meridional direction. In this section, an alternative approach that does not require the meridional WKB approximation is examined.

4 690 Journal of the Meteorological Society of Japan Vol. 77, No. 3 Fig. 2. Space-time spectral densities for 200 hpa streamfunctions (a) along WG-1 and (d) along WG-3, and those for 200 hpa meridional wind velocities along (b) WG-1, (c) WG-2, (e) WG-3 and (f) WG-4. Contour intervals are 2.0X1014 m4s-2.deg.day and 2.0X102 m2s-2.deg.day for the streamfunction and the meridional wind velocity, respectively. Dot-dashed lines indicate c=+3ms-1 and c=+5ms Solutions under step-like zonally symmetric flows Zonally sinusoidal solutions in simplified step-like basic states in the B-channel model are calculated here. The validity of the simplification of the basic state is discussed in Appendix B. Details of the model and the method to derive solutions are given in Appendix A. In Table 1, the cases used in the present study are listed. Here, the meridional profile of U and KS named case-1 is examined. Its basic state (Fig. 3a) is derived from a sinusoidal basic state (see Fig. 11 and

5 June 1999 T. Terao 691 (a) basic state (b) solutions(case-1) Fig. 3. (a) Basic state of case-1. Meridional profiles of the zonal velocity U(y) and stationary total wavenumber K3 (y) are shown by solid and dashed lines, respectively. The vertical axis shows latitudinal distance from the center of the channel, while the horizontal axis shows the zonal wavenumber. Horizontal long dashed lines indicate the latitude at y=+2b=+1110km and y=+b=+550km. Thick horizontal lines at y=+7b=+3870km show locations of rigid boundaries. (b) Dependencies of c on k of solutions for case-1 are represented by small circles (o). Horizontal long dot-dashed lines indicate c=+3ms-1 and +5ms-1. Alphabet symbols S, E and W indicate the stationary, eastward propagating (c=+3ms-1) and westward propagating (c=-3ms1) solutions, respectively. The 2nd mode solution at k=3 is labeled by A in this figure. Solutions with relatively slow zonal phase velocities (lcl)<5ms-1) are linked by a thick line. Eq. (9)) through Eq. (7), as is mentioned in Appendix B. The number of grids (N) and the distance of boundaries from the center of the westerly jet (Y) are set to 71 and 35, respectively, since the main results do not change significantly for larger values of Y and N. Dependencies of the zonal phase speed c on k for all modes in case-i are indicated by small circles in Fig. 3b. To distinguish trapped solutions from other solutions, the values ice and ki (see Eq. (8)) are plotted in Fig. 3b by thick and thin dotted lines, respectively. Figure 3b shows that solutions satisfying the trapping condition (8) are found only on the 1st mode solution curve indicated by the thick solid line. The 1st mode solution curve shows that the zonal wavenumber of the trapped quasi-stationary Rossby wave increases not only on this k-c plane (thick line in Fig. 3b) but also on the k-w plane (thick line in Fig. 4) with increasing wavenumber. If the frequency w (or the zonal phase speed c) is given, the wavenumber k of the corresponding trapped quasi-stationary Rossby wave is determined uniquely. Thus, for the quasi-stationary Rossby wave, the wavenumber of eastward (westward) propagating solution increases (decreases) as the frequency increases. In Figs. 5 and 6, the amplitude functions and the horizontal structures of the solutions of case-1 are shown, respectively. The stationary, eastward propagating (c=+3ms-1) and westward propagating (c=-3ms-1) 1st mode solutions are noted S, E and w, respectively. The solutions E and W can be regarded as representatives of eastward and westward propagating 1st mode solutions, respectively. The 2nd mode solution at k=3(a) has a meridionally anti-symmetric structure, unlike the symmetric 1st mode solutions (Fig. 5b). The energy concentration ratio E (defined by Eq. (6)) of solutions A and S are 0.24 and 0.42, respectively. This indicates that the solution S is more strongly trapped in the westerly jet than the solution A. The patterns protrude well out of the westerly jet, although the 1st mode solutions are relatively confined into the waveguide. The horizontal wave patterns (Figs. 6a to 6c) are similar to some of quasi-stationary disturbances observed in some data analyses and model calculations of the previous work. Examples of such disturbances obtained from the data analyses are seen in Fig. 7 of Hsu and Lin (1992), while those obtained from the model calculations are in Fig. 12 of Ambrizzi et al. (1995). In Fig. 7, energy concentration ratios are plotted for the 1st mode solutions of case-1 with a thick solid line. Generally speaking, energy concentration ratios of the eastward propagating solutions are greater than those of westward propagating ones, indicating that the eastward propagating solutions are more strongly confined in the waveguide than are the westward propagating ones.

6 692 Journal of the Meteorological Society of Japan Vol. 77, No. 3 Fig. 4. Dependencies of w on the zonal wavenumber k for the 1st-mode solutions of case-1 (small open circles). Solutions with relatively slow zonal phase velocities (lcl<5ms-1) are linked by a thick line. The symbols S, E and W indicate the same solutions as those shown in Fig. 3. Dot-dashed lines indicate c=+3ms-1 and +5ms Comparison between model and observation The results obtained with the B-channel model are qualitatively consistent with the results shown in Section 2 with the space-time spectral analysis. In Fig. 8, the k-w relationships of the 1st mode solutions of cases-2 to 5 (Table 1) deduced from the basic flows near the waveguides WG-1 to WG-4 are shown by the lines with boxes (D). These basic states are calculated using Eq. (7) where the observed meridional profiles are used as U(y) and Ks(y). Central latitudes (Q), widths of waveguides (b), and basic states (Ue, Ui, Ke and Ki) are as given in Table 1. The 1st mode solutions are compared with the results of the spacetime spectral analysis for the corresponding waveguides (see the contour in the figure). As a whole, the peaks in the space-time spectral densities are seen near the 1st mode solution curves of the B- channel model. It indicates that the B-channel model explains the observed characteristics of the zonal wavenumber of the quasi-stationary Rossby wave, such as the dependency of the wavenumber on the phase speed, although some exceptions can be pointed out (eastward-propagating disturbances in WG-4 with periods shorter than 20 days (Fig. 8d), for example). 4. Reproducing a standing wave pattern The quasi-stationary disturbances observed along the westerly jet often exhibit standing wave patterns. As a representative, the wave observed over the Eurasian Continent in 1983 investigated by Terao (1998) is shown in Fig. 9. This figure shows the longitude-time cross section of the 25-to 60-day band-passed meridional wind velocity at 200 hpa. The Lanczos filter (Duchon, 1979) is used. A clear standing wave pattern is seen from late June to early September. However, this pattern is not perfectly standing but somewhat oblique. That is, the wavepackets are moving rapidly eastward as is indicated by thick solid arrows. Such oblique standing wave pattern can be reproduced by adding the eastward and westward propagating 1st mode solutions at a certain period. In Fig. 10, the spatial and temporal evolution of the standing wave is shown for the meridional wind velocity of case-1 at 16-day period. Here, the amplitudes of the eastward and westward propagating solutions to be added are adjusted so that they have the same kinetic energy integrated over the whole area in the channel. On day-0, the wavepacket is at the center of the figure. It moves eastward and disappears on day-4. On day-5, another wavepacket comes from the west and it reaches the center of the figure on day-8. At that time, the sign of the meridional wind velocity pattern is opposite to that on day-0. The similarity between the reproduced and observed (Fig. 9) patterns is obvious. The velocity of the eastward packet propagation can be explained by the k-w relation of the 1st mode solution as follows. This relationship results in a small but finite difference of the wavenumber between the eastward and westward propagating solutions, which is noted by 6k here, at certain w. They, ok and w, would determine the eastward group velocity of the wave as cg=2w/6k. Since the frequency w of the 1st mode solution is a monotonically increasing function of the tonal wavenumber k

7 June 1999 T. Terao 693 Fig. 5. Meridional structures of the solutions for case-i. (a) Amplitude functions of the streamfunction (W(y)) and (b) the meridional wind velocity (V(y)). The stationary (S), the eastward propagating (c=+3ms-l, E) and the westward propagating (c=-3ms-1, W) 1st mode solutions are plotted with thick solid, thick dashed and thick dotted lines, respectively. The 2nd mode solution at k=3 (A) is shown by a thin solid line. The magnitudes of these amplitude functions are adjusted so that they have the same kinetic energies integrated over the whole channel. (a) S(1st mode) (b) W(1st mode) (c) E(1st mode) (d) A(2nd mode) Fig. 6. Horizontal structures of the meridional velocity of the solutions for case-1. Solutions 5, W, E and A are plotted in (a), (b), (c) and (d), respectively. Contour interval is 1ms-1. Zero contours are not shown. (see Fig. 4), bk is, and therefore cg also is, positive. 5. Conclusion Properties of quasi-stationary disturbances along the westerly jets are investigated with the spacetime spectral analysis for the 200 hpa meridional wind velocity. Spectral densities are calculated for four regions along the climatological westerly jets in the Northern Hemisphere summer. The zonal wavenumber and intensity of disturbances have systematic dependencies on the frequency change and the direction of phase propagation as follows. First, the tonal wavenumber of eastward (westward) phase propagating disturbances increases (decreases) as the frequency increases. Second, the eastward propagating disturbances tend to be stronger than the westward propagating ones. These results are in good agreement with the results obtained by many

8 694 Journal of the Meteorological Society of Japan Vol. 77, No. 3 Fig. 7. Dependencies of E on the zonal wavenumber k for the 1st mode solutions of case-1 (small open circles). Solutions with relatively slow zonal phase velocities (lcl<5ms-1) are linked by a thick line. The symbols S, E and W indicate the solutions same as those shown in Fig. 3. Fig. 8. Panels showing the k-w relationships of the trapped quasi-stationary Rossby wave solutions calculated for the basic states generated from the observed basic states near four waveguides WG-1 to WG-4 defined in Section 2. Superimposed on them, the space-time spectral densities for 200 hpa meridional wind velocities for corresponding waveguides are plotted (same as Figs. 2b, 2c, 2e and 2f). The solid lines with boxes (O) show the calculated solution curves. The thick parts of these lines show the solutions whose zonal phase speeds are less than 5ms-1. The sizes of these boxes are proportional to the corresponding value of e (see Eq. (6) in Appendix A). Dot-dashed lines indicate c=+3ms-1 and +5ms-1. Contour interval is 2.0X102m2s-2 deg day.

9 June 1999 T. Terao to 60-day v (1983, 200hPa, 40-45oN) Fig. 9. Longitude-time cross section of the 25- to 60-day band-passed 200 hpa meridional wind velocity at 40-45oN in 1983 summer. Only the Eastern Hemispheric data are shown. Contour interval is 4ms-1. Negative areas are hatched. Arrows indicate the propagation of wavepackets. other statistical studies (Hsu and Lin, 1992; Berbery et al., 1992; Ambrizzi et al., 1995) and model calculations (Yang and Hoskins, 1996). Additionally, the space-time spectral analysis for the 200 hpa meridional wind velocity shows that the quasi-stationary disturbances are seen not only in 10- to 30-day time scales as are already reported by many authors, but also in 30- to 90-day time scales. Furthermore, these observed dispersion relationships of the quasi-stationary disturbances trapped in the westerly jets can be reproduced by the B- channel models with step-like basic states. This basic state represents essential characteristics of basic states around the westerly jets; both the tonal wind velocity U and stationary total wavenumber Ks defined in Eq. (3) are larger inside the jet than outside. The solutions of this model with the smallest zonal phase velocities correspond to the quasi-stationary disturbances. Their properties agree with those observed in the space-time spectral analysis. The zonal wavenumber of eastward (westward) propagating solutions increases (decreases) as the frequency increases. The eastward propagating solutions are more strongly trapped in the westerly jet than the westward propagating ones. The solutions of the step-like basic states deduced from the observed climatological flows are obtained for four regions along the westerly jets and compared with the results of the space-time spectral analyses. They show that the observed tonal wavenumberfrequency relationships coincide with those calculated with the simple models not only qualitatively but also quantitatively. A standing wave pattern with an eastward rapid group velocity frequently observed is easily reproduced by adding the eastward and westward propagating solutions at a specified frequency. Acknowledgments The author would like to thank Professor Hisafumi Muramatsu at Meijo University for constant encouragement and fruitful discussions on this research. He is also grateful to the anonymous reviewers for their valuable suggestions. The GFD- DENNOU Library was used for drawing the figures and the numerical calculations. The data were provided by the European Centre for Medium-Range Weather Forecasts (ECMWF). In order to solve the eigenvalues and eigenvector of matrix, the LAPACK (Linear Algebra Package) subroutines were used. Appendix Deriving A solutions A.1 The basic equations The barotropic perturbation vorticity equation linearized about the tonally symmetric basic flow U(y) in the B-plane is written in the following form, where W' is the perturbation streamfunction and B* (=B-d2U/dy2) is the gradient of the absolute vorticity of the basic flow. Assuming that the perturbation with the zonal wavenumber k(>0) and the zonal phase speed c is in the form W'=W(y)eik(x-ct), the vorticity equation (Eq. (1)) reduces to the following second order ordinary differential equation; where Ks is the stationary total wavenumber defined by This is the same as Ks defined in Hoskins and Ambrizzi (1993). The frequency of the disturbance w can be written as w=ck. If c=u, another parameter k, which is equivalent to K defined in Yang and Hoskins (1996), can be defined by

10 696 Journal of the Meteorological Society of Japan Vol. 77, No. 3 (a) day-0 (e) day-4 (b) day-1 (f) day-5 (c) day-2 (g) day-6 (d) day-3 (h) day-7 (i) day-8 Fig. 10. Time evolution of the standing wave pattern shown by the meridional wind velocity. Contour interval is 1ms-1. See the text for more details. A.2 The ray-path analysis The ray-path analysis to diagnose the behavior of the quasi-stationary Rossby wave is based on Hoskins and Ambrizzi (1993) and Yang and Hoskins (1996). The coefficient c defined by Eq. (4) is used to diagnose the behavior of the quasi-stationary Rossby wave on the basic flow U. Now, the propagation route of its energy is diagnosed by the spatial distribution of k as follows.. When the quasi-stationary Rossby wavepacket with zonal wavenumber k propagating in the region where k>k reaches the turning latitude where k=k, it turns to the ic-increasing direction. On the other hand, as the wavepacket approaches the critical latitude where U=c, it refracts toward the critical latitude and its meridional wavelength and meridional group velocity of the wavepacket tend to zero. The linear wave theory is invalid near such a latitude.. When the wave is trapped in the region with turning latitudes both northward and southward, it cannot escape from this region, i.e., it is guided. Such a region is called waveguide. A.3 Solving the model Solving the disturbances in the B-channel model begins with Eq. (2). First, this differential equation is discretized about latitude y, where y is the

11 June 1999 T. Terao 697 meridional distance from the assumed central latitude of the B-channel. If the meridional profiles of U(y) and Ks(y) and the boundary conditions are provided, the possible structures of the disturbances and corresponding tonal phase velocities can be obtained for arbitrarily given k's. If the area between two boundaries is discretized into N+2 grids at regular intervals including the boundaries, N pairs of the amplitude function Wj(y) and the zonal phase speed cj(j=1,..., N and c1<c2<...<cn) are obtained. Here, the j-th solution is noted as j-th mode solution. The boundary conditions are set at two latitudes y=+y, where W=0. This method is the same as those of Haltiner and Song (1962) and Yanai and Nitta (1968). It should be noted that Eq. (2) has singularities at critical latitudes where U=c. Since only the solutions corresponding to the quasi-stationary Rossby waves (lcl=0ms-1) are examined in the present study, only those with slow zonal phase velocities need to be considered. All solutions discussed in the present study can be assumed not to suffer the influence of the critical latitudes, since the minimums of tonal flow of all the basic states used are sufficiently large (see Table 1). As a measure that shows how strongly the solution is trapped in the westerly jet, the kinetic energy concentration ratio (e) is introduced as follows; where b is the half-width of the westerly jet. The perturbation zonal and meridional wind velocities u' and v' are calculated from the perturbation streamfunction W' of each solution. Therefore, e is evaluated for each mode. In the present study, only results with boundaries put at +35o(Y=35o) are shown, since it was confirmed that the results relevant to the conclusions are not significantly changed for larger Y. A.4 The step-like basic states For simplicity, the step-like basic states, U'(y) and Ks'(y), described following are mainly used in the present study. Two averaging operators, (.)) and (.)), are defined as the averaging over the domains lyl<b(the internal domain) and b<lyl<2b (the external domain), respectively. From an original basic state, U(y) and Ks (y), which satisfies Eq. (3), the step-like basic state is generated by the following equations; The basic flow with U0=18.0ms-1 and Ud= 12.0ms-1 (Fig. 11) is considered here in detail. In this case, the minimum zonal wind velocity becomes As with the original basic state, any possible basic states such as the sinusoidal basic states or the basic states derived from the observational dataset can be used. Hereafter, U' and Ks' are noted simply by U and Ks, if there is no confusion. For step-like basic states, there is another distinct measure other than E to distinguish the solutions trapped in the westerly jet. This measure is related to k inside and outside the westerly jet noted by ki and ice, respectively. Equation (5) indicates that the meridional structure of the solution at a certain latitude becomes locally sinusoidal (exponential) if k2-k2<0(>0). In order for the solution whose zonal phase speed is c to be trapped in the westerly jet, its tonal wavenumber k must satisfy since the wave must be sinusoidal inside the westerly jet and must decay exponentially outside. Consequently, utilizing values of ki and,ke, we can determine whether the solution with zonal wavenumber k and tonal phase speed c is trapped in the westerly jet or not. Appendix B Validation of the step-like basic state It should be noted that the definition of the steplike basic state given in appendix A.4 is somewhat tricky. Some supplementary explanations must follow. Equation (3) indicates that the profile of Ks depends upon the meridional profile of U. However, U' and Ks' determined by Eq. (7) no longer satisfy Eq. (3), because Ks'2=(B-d2U/dy2)/U= (B-d2U/dy2)/U=B/U'=B/U', where over bar denotes the averaging over any range of y. Besides, the profiles of Ks' and U' are determined only by the original Ks and U profiles near the westerly jet (lyl<2b). The original profiles far from the westerly jet (lyl>2b) are ignored. Therefore, in the following, the solutions corresponding to the quasistationary Rossby wave obtained by the step-like basic states are compared with results obtained from sinusoidal basic states. The sinusoidal basic flow is defined as

12 698 Journal of the Meteorological Society of Japa n Vol. 77, No. 3 Fig. 11. Meridional profile of the sinusoidal basic state defined by Eq. (9). Profiles of U(y) and Ks(y) are plotted with thick solid and dashed lines, respectively. The step-like basic state of case-1 is plotted with thin solid and dashed lines, respectively, which are the same as those in Fig. 3a. They are deduced from the sinusoidal basic state through Eq. (7). U0-2/3Ud=10.0ms-1 and the tonal wind velocity at the center of the jet becomes U0+1/3Ud=22.0ms-1. For this basic state, the corresponding profile of K S is obtained by Eq. (3). From this sinusoidal basic state, the step-like basic state used in case-1 (Fig. 3a) is derived through Eq. (7). In Figs. 12a and 12b are shown the k-w and k- E relationships of the 1st mode solutions for sinusoidal and step-like basic states. The filled and open circles show the solutions of sinusoidal and steplike basic states, respectively. It is shown that the quasi-stationary trapped solutions of these two basic states agree with each other. The qualitative relationship that the eastward propagating solutions are more strongly trapped than the westward propagating solutions is also found for the sinusoidal basic state. Thus, the solutions corresponding to the quasistationary Rossby wave obtained from the step-like basic state are comparable to those from the sinusoidal basic flow. Although figures are not shown, the errors between the wavenumbers (k) of the stationary (c=0) solutions obtained from the sinusoidal basic states and corresponding step-like basic Fig. 12. Dependencies of (a) w and (b) E on k for 1st mode solutions of the sinusoidal and step-like basic states. The filled and open circles show the solutions of the sinusoidal and step-like basic states, respectively. Dot-dashed lines indicate c=+3ms-1 and +5ms-1. states are less than 5% for various realistic Uo and Ud values. These results show that the simplification of the basic flow by Eq. (7) does not spoil the important properties of quasi-stationary solutions obtained from more realistic basic flows. References Ambrizzi, T., B.J. Hoskins and H.-H. Hsu, 1995: Rossby wave propagation and teleconnection patterns in the austral winter. J. Atmos. Sci., 52, Berbery, E.H., J. Nogues-Paegle and J.D. Horel, 1992: Wavelike southern hemisphere extratropical teleconnections. J. Atmos. Sci., 49, Blackmon, M.L., Y.-H. Lee and J.M. Wallace, 1984: Horizontal structure of 500mb height fluctuations with long, intermediate and short time scales. J. Atmos. Sci., 41, Duchon, C.E., 1979: Lanczos filtering in one and two dimensions. J. Appl. Meteor., 18, Grose, W.L. and B.J. Hoskins, 1979: On the influence of orography on large-scale atmospheric flow. J. Atmos. Sci., 36, Haltiner, G.J. and R.T. Song, 1962: Dynamic instability in barotropic flow. Tellus, 14,

13 June 1999 T. Terao 699 Hayashi, Y., 1971: A generalized method of resolving disturbances into progressive and retrogressive waves by space Fourier and time cross-spectral analyses. J. Meteor. Soc. Japan, 49, Hoskins, B.J. and D. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci., 50, Hoskins, B.J. and T. Ambrizzi, 1993: Rossby wave propagation on a realistic longitudinally varying flow. J. Atmos. Sci., 50, Hsu, H.-H. and S.-H. Lin, 1992: Global teleconnections in the 250-mb streamfunction field during the northern hemisphere winter. Mon. Wea. Rev., 120, Kiladis, G.N. and KM. Weickmann, 1992: Circulation anomalies associated with tropical convection during northern winter. Mon. Wea. Rev., 120, Percival, D.B. and AT. Walden, 1993: Spectral analysis for physical applications. Cambridge Univ. Press, New York, 583pp. Shapiro, L.J. and S.B. Goldenberg, 1993: Intraseasonal oscillations over the Atlantic. J. Climate, 6, Terao, T., 1995: Extratropical day variations during the northern hemisphere summer. Annuals of the Disaster Prevention Research Institute Kyoto University, 38B-2, (in Japanese). Terao, T., 1998: Barotropic disturbances on intraseasonal time scales observed in the midlatitude over the Eurasian Continent during the northern summer. J. Meteor. Soc. Japan, 76, Yanai, M. and T. Nitta, 1968: Finite difference approximations for the barotropic instability problem. J. Meteor. Soc. Japan, 46, Yang, G.-Y. and B.J. Hoskins, 1996: Propagation of Rossby waves of nonzero frequency. J. Atmos. Sci., 53,