Thermally Forced Stationary Waves in a Quasigeostrophic System

Transcription

1 5 JUNE 00 CHEN 585 Thermally Forced Stationary Waves in a Quasigeostrophic System PING CHEN NOAA CIRES Climate Diagnostics Center, Boulder, Colorado (Manuscript received 6 April 999, in final form 3 November 000) ABSTRACT Analytical solutions of thermally forced stationary waves in a linear quasigeostrophic model are obtained. It is found that the zonal flow has a profound impact on the structure of the responses. The inviscid solutions on a resting basic state are the Sverdrup solutions that are confined to the heating region. The solutions on a westerly zonal flow are composed of a local and a vertically propagating part. The local response exists only in the heating region. The vertically propagating response exists in the far field as well as in the heating region. The thermally forced vertically propagating response can be conceptualized as a response to an equivalent topography, the height of which is proportional to the intensity and zonal scale of the heating, and inversely proportional to the strength of the zonal flow. Particular solutions forced by realistic summer heating fields reveal that, for weak westerlies, the height of the equivalent topography is much larger than that of the real topography, suggesting that heating is more important than topography in forcing the summer stationary waves in the subtropics. It is also found that Newtonian cooling has a significant effect on the structure of the thermally forced stationary waves.. Introduction The thermal forcing of atmospheric stationary waves is a classical problem in climate dynamics. Early works by Smagorinsky (953) and Manabe and Terpstra (974) suggested that diabatic heating is of comparable importance to topography in forcing the stationary waves in the troposphere although thermal effects become less important in the stratosphere. More recent works by Dickinson (980), Roads (98), and Held (983) indicated, however, that diabatic heating could play a more important role. Dickinson (980) and Held (983) noted that thermally forced stationary waves have a component that can be thought of as the forced response to an equivalent topography. Webster (97) investigated the atmospheric response to localized heating and found that latent heating is the dominant forcing in the Tropics. Gill (980) obtained analytical solutions of a simple model of the tropical atmosphere at rest. His solutions are quite similar to the numerical solutions of Webster (97), and provided a dynamical explanation for the origin of tropical stationary waves. The analytical solutions of Gill (980), Dickinson (980), and Held (983) provided significant insight into the nature of thermally forced stationary waves. The Gill (980) solutions are applicable to tropical regions of deep heating and weak zonal flow. The Dickinson Corresponding author address: Dr. Ping Chen, NOAA CIRES Climate Diagnostics Center, Mail Code: R/CDC, 35 Broadway, Boulder, CO pc@cdc.noaa.gov (980) and Held (983) solutions are quite accurate for stationary waves forced by shallow heat sources in the midlatitudes. The application of these analytical solutions are, however, limited in the subtropical westerlies where both horizontal and vertical advection are important. The purpose of the current paper is to present a class of new analytical solutions of thermally forced stationary waves on uniform zonal flows. These solutions are derived from the full linear quasigeostrophic system. The solutions clarify the effects of zonal flow, Newtonian cooling, and Ekman drag on the structure of stationary waves. The paper is organized as follows: section introduces the model; section 3 presents the general formula of the analytical solutions; sections 4 and 5 discuss some particular solutions in response to a class of specific forcing; and finally, section 6 includes a summary and discussion.. Model The model used in this paper is the quasigeostrophic system on a beta plane. The dynamics of the thermally forced linear stationary waves on a zonally symmetric basic state is governed by uqq x y[ 0 (0) z z] 0 (0Q) z (.a) u zx uz z Q, at z 0 (.b) 0, at y (W/), x (.c) where Q denotes diabatic heating, and coefficients 00 American Meteorological Society

2 586 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 of Newtonian cooling and Ekman drag, and (W/) and (W/) the south and north ends of the beta-plane channel. Other symbols are defined as in Holton (99). The system is closed by the radiation condition at infinity. 3. Analytical solutions When u is constant, (.) can be solved analytically. The solutions of (.) with u 0 and 0 will be called the Sverdrup solutions, because in this case (.a) reduces to the simple Sverdrup balance. The velocity field of a Sverdrup solution is determined solely by the heating field: the vertical velocity is the heating rate divided by a constant and the meridional velocity is proportional to the vertical gradient of the heating. Thus, ascending motions occur in regions of heating and descending motions occur in regions of cooling, whereas northward (southward) motion occurs when the heating increases (decreases) with height. The Sverdrup solutions are decoupled in the vertical and do not satisfy the lower boundary condition if heating does not vanish at the surface. If either u or is nonzero, the solutions of (.) may be sought in the form of [ ] (z) rz ikx Ree e cos(ly), (3.) Q [ (z) ] where k and l are the zonal and meridional wavenumbers, i, r (H), and satisfies where zz m f (3.a) b, at z 0, (3.b) z m [(iku )] {(ik)[ u(k l )]} r (3.3a) rz rz f (iku ) [e (e ) z] (3.3b) r (iku ) [(k l )] (3.3c) b (iku ) ( z0). (3.3d) The solution of (3.) can be written as where b imz (z) F(z) A e, (3.4) im F(z) F (z) F (z) (3.5a) im A F 0(0) F (0) and (3.5b) im [ ] [ im ] imz imz im z imz imz F (z) e e f(z) dz (3.6a) F (z) e e f(z) dz. (3.6b) z The parameter m plays a critical role in determining the properties of the solution (e.g., Pedlosky 987). If it is a pure imaginary number, the solution is vertically trapped; otherwise, the solution possesses a vertically propagating component. Since A and b are constants, (3.4) suggests that a thermally forced stationary wave is generally composed of a local and a vertically propagating part. The local response is nonzero only in regions of heating. The amplitude of the vertically propagating response is determined by the heating at the surface via b and the vertical integral of the heating field via A. If u 0, the solution in (3.4) can be formally written as where imz (z) F(z) C(ih E )e, (3.7) B iku C and (3.8a) im iku he (Bku ) [ z0 ( im)(iku )A], (3.8b) and B N / f 0. The constant h E can be called the height of an equivalent topography induced by the heating as its role is equivalent to that of the height of the real topography. 3 The height of the equivalent topography is determined by the zonal flow as well as the heating field. For a given heating field, the forced response on a weaker zonal flow will be stronger than that on a stronger zonal flow since the magnitude of h E is inversely proportional u (see Fig. 6 for an example). For a given zonal flow, the forced response will be stronger if the heating is stronger and its zonal scale is larger because h E is proportional to the intensity of the heating and inversely proportional to the zonal wavenumber of the heating. It is noted from (3.8b) that the heating at the surface and the interior of the atmosphere both have a contribution to the equivalent topography. Thus, h E is generally nonzero even if there is no heating at the surface. It can be inferred from (3.7) that the vertically prop- A Rayleigh friction was used in an earlier version of the model. Calculations have shown that a Rayleigh friction with a timescale of 5 days has only a minor effect on the solutions. If Newtonian cooling and Ekman drag are ignored, (.a,b) reduces to (6.3.6, 7) in Pedlosky (987). 3 The concept of equivalent topography was first introduced in Dickinson (980) and Held (983). The h E defined in (3.8b) of this paper is, however, more general than that defined in Dickinson (980) and Held (983) because either the horizontal or the vertical advection term in the thermodynamics equation was ignored in their derivation.

3 5 JUNE 00 CHEN 587 but their amplitudes increase exponentially with height (the wave energy still remains finite). The solution for m r 0 and m i r is a free-mode resonant response because the imposed heating is exactly balanced by the Newtonian cooling. Marshall and So (990) referred to a similar solution in their threelevel quasigeostrophic model as the thermally equilibrated solution. This solution is vertically trapped and its amplitude is constant above the heating region. The phase lines of this solution are vertical but include a 80 jump somewhere in the heating region. From the dashed curves in Fig. it is seen that Newtonian cooling has a significant effect on stationary waves. The effect is especially strong when the zonal flow is weak. It is particularly interesting to note that m r in the dashed curve is always positive and finite, even if u 0, which implies that a thermally damped wave always propagates upward and has a finite vertical wavelength. It is clear from Fig. that the effect of thermal damping on the vertically propagating part of the solution depends on u: it is very strong when u is around zero and much weaker when u 0 m s. When u 0, thermal damping is critical for the exis- tence of vertically propagating waves, because if 0 the solutions of (3.) would be the Sverdrup solutions that are confined to the heating region and vertically decoupled. 4. Particular solutions forced by a zonally harmonic heating FIG.. Line graphs of (a) real and (b) imaginary part of m/r as functions of zonal wind. The value of is 0 and (5 days) for the solid and dashed curves, respectively. agating response forced by heating has the same vertical structure as that forced by topography but lags the latter by a quarter wavelength. Thus, the far-field effect of diabatic heating and topography on stationary waves can be readily assessed by simply comparing the magnitude of h E with that of the real topography. Significant insights about the effects of zonal flow and Newtonian cooling on the structure of stationary waves can be obtained by studying the dependences of m on u and. Figure shows line graphs of m r and m i as functions of zonal wind. The solid curves represent the famous Charney and Drazin (96) criterion for the vertical propagation of Rossby waves: if u 0 then m r 0 and m i r, and thus the solutions would be trapped and their amplitudes decrease exponentially with height; if 0 u u c where uc is about 38 m s then m r 0 and m i 0, and therefore the solutions could propagate in the vertical and their amplitudes increase exponentially with height; if u u c then m r 0 and 0 m i r, and therefore the solutions would be trapped In this section, we discuss the solutions forced by a heating specified by Q Qˆ cos(kx) cos(ly), (4.a) where k (/L) and l (/W) where L and W are the length and width of the beta plane, and Q0 sin(z), z z ˆQ(z) (4.b) 0, z z, where /z with z 4 km, and the constant Q 0 is defined as [ ] Q00 R (r) Q0, (4.c) rz c Hf ( e ) p 0 where Q 00 0 W m is the column-integrated heating rate. This heating field, with a maximum heating rate of about 3 per day at 7 km, is confined to the altitude range of 0 4 km. Using the general formulas in section 3, it can be shown that the vertical structure of the stationary waves forced by this heating is given by where imz P(z) F P(z) Ae P, (4.)

4 588 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 f0 V(T T ) U(T3 T 4 ), z z 0, z z F (z) im U V P (4.3a) [ ] Vz Uz f U( e ) im V( e 0 ) P A, im V im U (4.3b) where f 0 (iku ) Q 0, U r im, V r im, and im(zz) rz T (z) e e (4.4a) ru rz V T (z) cos(z) sin(z) e (4.4b) im(zz) rz T (z) e e 3 (4.4c) rv rz U T (z) cos(z) sin(z) e. (4.4d) 4 The height of equivalent topography for u given by 0 is im iku hep A P, (4.5) B ku where B is defined earlier. Figure shows the amplitude and phase of three inviscid solutions of (3.) forced by this heating. It is seen that the Sverdrup solution (solid line) is vertically confined to the region of nonzero heating between 0 and 4 km. The streamfunction is a local maximum at the surface and about.5 km, and it is zero at z (4.5 km). The phase of the streamfunction has no vertical tilt but exhibits a 80 jump at z : below (above) z, the ridge in streamfunction is a quarter wavelength east (west) of that in heating. The solutions on nonzero zonal flows (dashed and dot dashed lines) extend vertically beyond the region of nonzero heating. When u 0 (dashed line), the re- sponse is vertically trapped. Within the heating region, the response is weaker than that of the Sverdrup solution, although its phase is nearly the same as that of the Sverdrup solution. When u 0 (dot dashed line), the response is much stronger, and above the heating region, the amplitude increases exponentially with height. The phase progresses gradually westward with height, an indication of upward energy propagation. The surface ridge is located about 00 downstream of that of the Sverdrup solution. Further calculations reveal that, as the zonal flow increases, the amplitude of the response decreases and the phase moves progressively eastward (not shown). It is clear from Fig. that zonal flow significantly impacts the thermally forced stationary wave solutions. FIG.. Line graphs of (a) amplitude and (b) phase for the inviscid solutions of (.) for u 0ms (solid), 3ms (dashed), and 3 ms (dotted), respectively. The amplitudes are normalized by 0 8 m s. The phase is the longitude of the ridge. Figure 3 shows the amplitude and phase of two solutions of (3.) forced by the same heating on a resting basic state. As described earlier, the Sverdrup solution (solid line) is vertically confined to the region of nonzero heating. The structure of the solution with Newtonian cooling (dashed line) is significantly different from that of the Sverdrup solution. The response is no longer confined to the heating region and the phase variation is smooth. The surface trough/ridge is over 40 east of that of the Sverdrup solution. Thus, Newtonian cooling has a significant effect on the thermally forced stationary wave solutions. The height of the equivalent topography is a sensitive function of zonal flow as well. Figure 4 plots h E as a function of u for stationary waves forced by the heating in (4.). It is seen that as u decreases, h E increases nearly exponentially. For example, h E is about km for u 0ms and about 40 km for u ms. It is clear that, for u 0ms, the height of the equivalent topography is much larger than that of the real topog-

5 5 JUNE 00 CHEN 589 FIG. 4. Height of the equivalent topography as a function of zonal wind for stationary waves forced by a zonal wavenumber heating. The {, } are {0, 0}, {0, (5 days) }, and {0, (5 days) } for the solid, dashed, and dot dashed curves, respectively. This heating field is confined zonally to 50 30E and vertically to 0 4 km (see Fig. 8a). The maximum heating rate of about.5 per day occurs at (90E, 7 km). This heating field is intended to simulate the latent heating associated with the summer Asian monsoon and the sensible heating over the Asian highland. FIG. 3. As Fig. but for two solutions on u 0ms with 0 (solid) and (5 days) (dashed), respectively. raphy (e.g., at 35N the amplitude of the first zonal harmonic of the topography is about km). This result strongly suggests that heating is more important than topography in forcing the summer stationary waves in the subtropics where the zonal flow is weak. 5. Particular solutions forced by a zonally localized heating In this section, we discuss the solutions forced by a localized heating field. The heating is specified analytically by Q Q(z)X(x) ˆ cos(ly), (5.a) where Qˆ and l are the same as in (4.), and x 90 cos 50 x 30 X(x) 80 (5.b) 0 otherwise. a. Effects of zonal-mean flow Figure 5 shows the streamfunction and vertical velocity fields for three inviscid solutions of (3.) forced by the heating in (5.). The solution with u 0ms is the Sverdrup solution and the response is totally confined to the vertical range where heating/cooling exists (Figs. 5a,b). This solution exhibits a streamfunction response having a baroclinic structure that is antisymmetric in longitude about the center of the heating (Fig. 5a). The lower-level circulation is cyclonic to the west of the heat source and anticyclonic to the east, with the surface cyclone centered around 50E and the surface anticyclone centered around 30E. The vertical velocity of this solution is proportional to the imposed heating (Fig. 5b). The solution with u 3ms (Figs. 5c,d) is sim- ilar, albeit weaker, to the Sverdrup solution and is representative of all the inviscid solutions with easterly zonal flows. Unlike the Sverdrup solution, however, this solution is not confined to the altitude of nonzero heating, although it is vertically trapped. As the easterly flow increases, the phase of the solution does not change, though the amplitude decreases further. The solution with u 5ms (Figs. 5e,f) is fun- damentally different from those on a basic state at rest or easterly. In particular, the solution is much stronger, exhibits a distinctive phase tilt in the vertical, and is asymmetric in longitude. At the surface, for example, the cyclone is much stronger than the downstream anticyclone and its center is confined to the longitude of heating. In contrast, the surface anticyclone spans nearly 0 of longitude.

6 590 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 FIG. 5. Longitude height cross sections of streamfunction (left column) and vertical velocity (right column) for the inviscid solutions of (.) on zonal flows of u 0ms (a),(b), 3 ms (c),(d), and 5ms (e),(f), respectively. The contour interval is m s for streamfunction and ms for vertical velocity. The region of heating is shaded. The vertical velocity field (Fig. 5f) now differs greatly from the imposed heating. In the troposphere, there is a strong downward motion on the western flank of the heating whereas the strong upward motion is displaced to the eastern flank of the heating. The vertical velocity is almost zero at the central longitude of the heating and in particular is near zero in the center of the cyclone in the lower troposphere. The distribution of this vertical velocity field is qualitatively similar to the results in Rodwell and Hoskins (996), who investigated the atmospheric response to localized heating in a primitive equation model. The vertical structure of solutions with westerly basic states differs strikingly from that with easterly or resting basic states. First, note that the amplitude of the solution increases with height. Between the surface and 4 km where heating exists, the solution can be understood as being composed of a local and a vertically propagating part (see the discussions in previous sections). The gradual phase change in streamfunction in this region is due to the propagating response. Above 4 km, the solution is composed solely of the propagating part, and therefore, its amplitude increases exponentially with height, as predicted by (3.4). The solution Figs. 5e and 5f is representative of the inviscid solutions on weak westerly zonal flows. As the zonal wind increases, the amplitude of the response decreases and the surface cyclone and anticyclone move progressively eastward. As shown in Fig. 6, the response with u 5ms is much weaker than that on u

7 5 JUNE 00 CHEN 59 FIG. 6. Longitude height cross sections of streamfunction (a) and vertical velocity (b) for the inviscid solution of (.) on u 5ms. The contour interval is 0 6 m s for streamfunction and 0 3 ms for vertical velocity. 5ms (note the smaller contour intervals in Fig. 6). The surface cyclone (anticyclone) is now on the eastern (western) flank of the heating, and the vertical motion is downward in the heating region. The results in Figs. 5 and 6 demonstrate clearly that the zonal flow has a profound impact on the structure of the thermally forced stationary waves. These results also clarify the origin for differences between the Webster (97) and Gill (980) solutions forced by a localized heating in the subtropics (Fig. of Webster and Fig. 3 of Gill). The center of the surface cyclone in the Webster solution is almost coincident with the heating maximum. That in the Gill solution, on the other hand, is displaced to the west of the heating maximum. This difference is often explained as due to the different damping rates used in the models (e.g., White 98). Our results (Fig. 5) show that the central location of the surface cyclone is quite sensitive to the strength of the background wind and it moves eastward as the zonal-mean wind increases. It can thus be understood that, while damping may have an effect, the difference in the location of the surface cyclone between the Webster and Gill solutions is due mostly to the difference in the background zonal wind, which is zero in the Gill model and westerly in the Webster model. b. Effects of nonconservative processes Figure 7 shows longitude height cross sections of streamfunction and vertical velocity for three solutions of (3.) with a 5-day Ekman drag and a 5-day Newtonian cooling. Compared to the solutions in Fig. 5, the current solutions with u 0ms (Figs. 7a and 7b) and 3 ms (Figs. 7c and 7d) are no longer antisymmetric in longitude: the streamfunction on the west flank of the heating is much stronger and much less expansive than that on the east flank. The phase changes in the vertical are more gradual than those for the corresponding inviscid solutions (Fig. 5). In addition, the solution with u 0ms is no longer vertically confined to the heating region. While its amplitude is weaker, the solution with u 5ms (Figs. 7e and 7f) is quali- tatively similar to the corresponding inviscid solution (Figs. 5e and 5f). It should be noted that Ekman drag only has a very minor effect on the solutions. The solutions without the Ekman drag (not shown) are essentially the same as those shown here. c. Heat budget It is useful to examine how the prescribed diabatic heating is balanced by other physical processes in the system. For linear stationary waves, the thermodynamic equation can be written as Qdh Q dc Q adv Qw 0, (5.) where Q dhand is the prescribed diabatic heating, and Q dc, Q adv, Q w are the induced heating or cooling due to thermal damping, horizontal temperature advection, and vertical motion, respectively. This equation states that diabatic heating is balanced by Newtonian cooling, horizontal temperature advection, and adiabatic cooling of ascent. In the special case of u 0ms and 0, (5.) reduces to Qdh Qw 0, which states that, for Sverdrup solutions, the specified diabatic heating is totally balanced by the adiabatic cooling associated with the vertical motion. Figure 8 shows the four terms in (5.) for the solution of (3.) with a 5-day Ekman drag and a 5-day Newtonian cooling on a zonal flow of u 5ms (see Figs. 7e and 7f for the streamfunction and vertical velocity fields of this solution). The prescribed diabatic heating/cooling is confined to the region between the surface and 4 km with heating between 50 and 30E and weak uniform cooling in other longitudes (Fig. 8a). The thermal damping generates some weak cooling in much of the shaded region and some heating above 4 km (Fig. 8b). The horizontal advection term produces a pattern of alternating adiabatic cooling and heating (Fig. 8c). The adiabatic cooling/heating due to horizontal advection is balanced mainly by adiabatic heating/ cooling associated with descent/ascent (Fig. 8d). It can be concluded from the heat budget that, while the imposed diabatic heating is vital for the existence of the

8 59 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 FIG. 7. As Fig. 5 but for solutions with (5 days) and (5 days). thermally forced stationary waves, the dynamics of the forced waves on a westerly zonal-mean flow is the primary factor that determines the distribution of vertical motion. 6. Conclusions and discussion In this paper, analytical solutions of thermally forced stationary waves with uniform zonal flows are obtained. These solutions are derived from the full linear quasigeostrophic equations. In this sense, this study extends the work by Dickinson (980), Roads (98), and Held (983), who obtained approximate solutions of the system. This work also extends the work of Gill (980), who obtained analytical solutions of thermally forced responses in a simple model of a resting tropical atmosphere. It is demonstrated that the zonal flow has a profound effect on the structure of the thermally forced stationary waves. The responses on a resting basic state are the Sverdrup solutions that are confined to the heating region. The responses for westerly zonal flows are composed of a local and a vertically propagating part. The local response is vertically confined to the heating region. The propagating response exists in the whole vertical domain, and can be regarded as the forced response by an equivalent topography. The height of the equivalent topography is proportional to the intensity and zonal scale of the heating and inversely proportional to the strength of the zonal-mean zonal wind. The heating at the surface and that of the interior of the atmosphere both have a contribution to the equivalent topography. Thus, the height of the equivalent topography is generally not zero even when the heating is zero at the

9 5 JUNE 00 CHEN 593 FIG. 8. Longitude height cross sections of (a) Q dh, (b) Q dc, (c) Q adv, and (d) Qw for the solution of (3.) with (5 days) and (5 days) on a zonal flow of u 5ms. The contour interval is 0.5 per day. surface. For an inviscid atmosphere, the height of the equivalent topography is a real number, and therefore, the thermally forced propagating response has the same vertical structure as that of the topographically forced response but lags the latter in longitude by a quarter wavelength. Because the vertically propagating component is the only solution above the heating region, the relative importance of topography and diabatic heating in forcing the stationary waves in the far field can be readily assessed by simply comparing the magnitude of the height of the real topography and that of the equivalent topography. When the zonal flow is easterly or at rest, the eddy streamfunction in response to a zonally localized heating is trapped in the vertical and exhibits a baroclinic structure. It is antisymmetric in longitude about the heating field, with a surface cyclone to the west and anticyclone to the east. When the zonal flow is moderately westerly, the response is a wave train propagating eastward and upward, and the solution becomes asymmetric in longitude. At the surface, the cyclone is much stronger and less zonally expansive than the anticyclone. As the zonal westerly increases, the wave train moves progressively eastward. Our results help understand the origin of the summer stationary waves, which are dominated by the continental cyclones and oceanic anticyclones in the lower troposphere, and continental ridges and oceanic troughs in the upper troposphere (e.g., White 98; Chen et al. 00). The results in this paper strongly suggest that the low-level anticyclones over the North Pacific and North Atlantic can be explained as forced by the latent and sensible heat sources over Asia. A more detailed study on the origin of the summer oceanic anticyclones is found in Chen et al. (000). Acknowledgments. I am grateful to Dr. Randall Dole for his interest in and encouragement and support of my research at CDC. I thank Dr. Congming Li for discussions on some of the mathematics, and Drs. Martin Hoerling, Walter Robinson, Klaus Weickmann, and Jeffrey Whitaker for their comments on the manuscript. I also thank Drs. Isaac M. Held and Theodore G. Shepherd for their constructive criticism and suggestions that led to much improvement on this paper. This research was supported by the NOAA Climate and Global Change Program. REFERENCES Chen, P., M. P. Hoerling, and R. M. Dole, 00: The origin of the subtropical anticyclones. J. Atmos. Sci., in press. Dickinson, R. E., 980: Orographic Effects in Planetary Flows. GARP Publication Series, No. 3, WMO, 450 pp. Gill, A. E., 980: Some simple solutions for heat-induced tropical circulation. Quart. J. Roy. Meteor. Soc., 06, Held, I. M., 983: Stationary and quasi-stationary eddies in the extratropical troposphere: Theory. Large-Scale Dynamical Pro-

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