1. Introduction
In a traditional framework based on the separation of eddies from a zonally uniform basic flow, the conventional Eliassen–Palm (E–P) flux has been proven to be a powerful tool for diagnosing propagation of Rossby waves and their interactions with a zonal-mean flow on the meridional plane (e.g., Andrews and McIntyre 1976;Edmon et al. 1980; McIntyre 1982; Andrews et al. 1987). The E–P flux is a flux of wave-activity pseudomomentum on the meridional plane. Its meridional and vertical components include zonally averaged eddy momentum and temperature fluxes, respectively, which ensure the phase independency of the flux. Since the flux requires no time averaging, it can by nature represent a “snapshot”1 of the wave propagation on the meridional plane. However, it cannot represent the propagation in the zonal direction.
If one is interested in the evolution of a locally forced wave packet that influences local weather and climate, its zonal propagation needs to be diagnosed explicitly. Even the zonal inhomogeneities in a basic flow may need to be taken into account in representing the wave packet propagation. In the troposphere, especially in the wintertime Northern Hemisphere (NH), the mean westerlies meander in the presence of thermally and topographically forced planetary waves. This meander substantially modulates amplitudes and propagation characteristics of synoptic-scale, migratory cyclones and anticyclones, including the localization of major storm tracks (e.g., Blackmon et al. 1977; Wallace et al. 1988). Furthermore, it has been indicated that cyclogenesis along storm tracks may be caused by the “downstream development” of baroclinic wave packets (e.g., Chang 1993). Even the propagation of quasi-stationary disturbances with longer wavelengths is complicated by zonal asymmetries in the background westerlies (e.g., Simmons et al. 1983; Hoskins and Ambrizzi 1993; Naoe et al. 1997). Recent studies suggested that local absorption of stationary Rossby wave packets is instrumental in the formation of blocking phenomena at certain geographical locations where the background westerlies are weaker than the zonal average (e.g., Nakamura et al. 1997).
For transient, migratory eddies, time averaging is appropriate for eliminating the half-wavelength oscillatory component in A and F, which admits the zonal component in F that represents the zonal propagation. The extended E–P flux as formulated by Hoskins et al. (1983) and Trenberth (1986) has been widely used, since it can represent the zonal propagation of a (small-amplitude) wave packet relative to the time-mean flow. Plumb (1986, hereafter P86) defined a flux of wave-activity pseudomomentum that can delineate three-dimensional propagation (relative to the earth) of transient eddies embedded on a zonally asymmetric basic flow. The flux includes products of velocity and temperature perturbations. The phase independency is ensured for the flux of P86 and extended E–P flux by time averaging, but therefore they are not applicable to a snapshot analysis.
For stationary eddies, of course, time averaging is not equivalent to phase averaging and therefore inappropriate. Hence, for an analysis of stationary eddies or a snapshot analysis of migratory eddies, a conservation relation meant to represent three-dimensional wave propagation with a wave-activity flux that is free from any oscillatory component should be derived without any averaging. Plumb (1985, hereafter P85) was the first to derive such a conservation law for small-amplitude stationary eddies on a zonally uniform basic flow.2 The wave-activity flux Fs based upon his conservation relation is phase independent but it includes no terms explicitly averaged. Therefore, it is suited for an analysis of stationary eddies. In fact, its usefulness has been demonstrated by him and others (e.g., P85; Karoly et al. 1989) in applications to large-scale stationary disturbances observed in the troposphere. Kuroda (1996) extended Plumb’s formula to an axially symmetric flow on a sphere. Its generalization to finite-amplitude eddies has been achieved by Brunet and Haynes (1996). Since the flux of P85 and its generalized forms mentioned above were defined for a zonally uniform basic flow, however, the usefulness and applicability to the real atmosphere are somewhat limited, especially for the NH wintertime troposphere. Furthermore, in each of their derivations a supplementary nondivergent flux (e.g., G in P85) was introduced rather heuristically, in order to render the flux independent of wave phase. Yet, the physical meaning of such a supplementary flux as G is not totally clear, nor a physical interpretation of each term that composes such a phase-independent wave-activity flux.
In this study, we attempt to generalize Fs of P85 and its conservation law so as to be applicable to small-amplitude quasigeostrophic (QG) disturbances, either stationary or migratory, that are superimposed on a zonally varying basic flow. We derive an approximate conservation relation of the wave-activity pseudomomentum and its phase-independent flux through an approach different from and more straightforward than that of P85. Our flux will be shown to be parallel to the local three-dimensional group velocity of Rossby waves, and hence to be suited for a snapshot diagnosis of the three-dimensional propagation of wave packets of migratory and stationary eddies on a zonally varying basic flow. We argue through our derivation that phase-independent fluxes of ours and P85 may be interpreted as a superposition of two dynamical aspects of the QG wave packet propagation, though P85 did not clarify physical meanings of individual components of Fs. Furthermore, a physical meaning of the supplementary flux G in P85 becomes clear through our particular approach, although such a flux does not appear explicitly in our derivation. It is also verified that our flux is indeed nearly phase-independent in applications to simulated and observed atmospheric data. Finally, we will derive formulas that present an instantaneous feedback from a propagating wave packet upon a basic flow on which it is embedded and the associated ageostrophic residual circulation as well. A summary of our results for stationary disturbances on a zonally varying basic flow has been published in Takaya and Nakamura (1997, hereafter TN97).
2. Formulation
Our approach to formulate a phase-independent wave-activity flux for QG eddies without any averaging is based on a simple idea. If a perturbation streamfunction (ψ′) is proportional to the sine of the wave phase, wave enstrophy and wave energy are proportional to the sine squared and cosine squared, respectively. Then, an appropriate linear combination of these two quantities (one proportional to the wave enstrophy and the other to the wave energy) can be phase-independent even without averaging. In practice, a quantity A, enstrophy divided by the magnitude of the basic potential vorticity (PV) gradient, and another quantity
3. Examples
a. Stationary eddies in numerical simulations
As the first example of applications of our wave-activity flux W, we briefly refer to Enomoto and Matsuda (1999), who examined the behavior of stationary Rossby waves around critical latitudes through numerical integrations of the two-dimensional nondivergent barotropic vorticity equation on a sphere. In one of their experiments, Rossby waves are forced by a localized divergence centered at (40°N, 90°E) in the exit region of a midlatitude westerly jet (Fig. 1a). On the map of the response streamfunction (ψ′) 14 days after the activation of the forcing (Fig. 1b), a wave train emanating downstream from the forcing region appears to be split into two branches, each of which is approximately along a great circle as in a theoretical argument by Hoskins and Karoly (1981). The wave-activity flux W, estimated from ψ′ with CP = 0 prescribed in (C5), clearly illustrates the wave propagation along these two branches (Fig. 1c). Along the northern branch, W across the diffluent westerlies converges into the weak-westerly region along the southern flank of the jet entrance. The flux along the southern branch delineates that the wave activity indeed propagates through an equatorial westerly duct and then across the diffluent Southern Hemisphere (SH) westerlies until it finally converges into a weak-westerly region along the northern flank of the southern jet. Our flux diverges out of the forcing region in the NH and traces positive and negative ψ′ centers along the branch even beyond the equator, which clearly indicates that the ψ′ centers in the SH are indeed associated with the wave train forced in the midlatitude NH.7 Since zonal asymmetries in the background flow in that experiment are not very strong, Fs derived in P85 for a zonally uniform basic flow can depict the above-mentioned propagation over the extratropics in a very similar manner (not shown). However, Fs cannot be applied, in theory, to the tropical westerly duct where the zonal-mean flow is easterly, and hence it cannot illustrate the wave-packet propagation across the Tropics. An important aspect evident in Figs. 1c and 1d is that W and ∇ · W are almost free from the half-wavelength component of stationary Rossby waves. Another important aspect is that W is, in general, almost perpendicular to phase lines of the waves as represented by the ψ′ = 0 contours. Apparently, W tends to be parallel to the local group velocity of stationary Rossby waves, as shown theoretically in the previous section.
Another application of W to a model-simulated stationary response is found in Honda et al. (1999), who examined a stationary Rossby wave train forced thermally by anomalous surface heat fluxes in association with abnormal sea-ice cover within the Sea of Okhotsk. They compared W with Fs of P85 both estimated from the stationary response (Fig. 2). In the upper troposphere, W is strongly divergent right over a pair of the primary cooling and heating sources in the Sea of Okhotsk and to the east of it, respectively. Compared to ∇H · W, ∇H · Fs is shifted to the west and spreads over eastern Siberia, where no significant heating or cooling source is present. Hence, in this example, ∇ · W leads to a more reasonable estimation of the wave forcing region than ∇ · Fs of P85 does.
b. Observed stationary eddies associated with blocking
Blocking highs are associated with high-amplitude, quasi-stationary anticyclonic anomalies that give rise to prolonged abnormal weather situations. In most cases a blocking anticyclone decays by releasing accumulated wave activity toward downstream in the form of a stationary Rossby wave train. A number of studies indicated that blocking formation is due primarily to local feedbacks from migratory synoptic-scale eddies. Recent studies have demonstrated, however, that in some locations a converging wave-activity flux associated with an incoming stationary Rossby wave train is of primary importance in blocking formation (e.g., Nakamura et al. 1997). Hence, diagnosing ∇H · W associated with stationary Rossby waves on a meandering mean flow may be insightful for understanding the dynamics that underlies the blocking formation. One may argue that applying a wave-activity flux to a blocking phenomenon is inappropriate because a blocking high itself consists of high-amplitude anomalies. With their nonlinearities, the plane-wave assumption utilized in deriving W must break down in the vicinity of the blocking center. Still, a stationary Rossby wave train emanating from a blocking ridge or an incoming one from upstream to the ridge should exhibit nonlinearity to a much lesser degree, and hence they may be regarded as linear waves suited for applying W.
We use twice-daily gridded fields of geopotential height at the 250-hPa level, based on the operational analyses by the National Meteorological Center [now known as the National Centers for Environmental Prediction (NCEP)] for 1965–92. The dataset was obtained from the National Center for Atmospheric Research (NCAR) Data Library. As in Nakamura et al. (1997), these fields were composited relative to the peak times of the 15 strongest blocking events observed around a given location in the 27 winter seasons (mid-November–mid-March). Before compositing, a low-pass filter with a cutoff period of 8 days was applied to the data time series, in order to isolate quasi-stationary eddies from migratory, higher-frequency transients. The basic state for the quasi-stationary eddies was defined as the 27-winter mean (Fig. 3). Departures of the filtered data from this mean state represent circulation anomalies associated with a blocking high and accompanied stationary wave trains. We use W on the pressure coordinate whose expression is given in appendix C. For simplicity, wind fields were approximated by the local geostrophic winds.
In Fig. 4, we plot the horizontal component of W and its divergence (∇H · W) based on the composite blocking flow centered around (54°N, 100°E). It is obvious that W and ∇H · W both exhibit little oscillatory component with one-half wavelength. During the development (day −2) of the blocking ridge, quasi-stationary height anomalies are evident near (60°N, 40°E) upstream of the ridge. The associated W is dominantly eastward and nearly perpendicular to the height anomaly contours. The flux is converging into the amplifying blocking ridge and divergent upstream, which is suggestive of a particular importance of a converging wave-activity flux associated with an incoming stationary Rossby wave train in the blocking formation over Siberia, as Nakamura (1994) and Nakamura et al. (1997) demonstrated for a European blocking ridge.8 During the breakdown of the block, W diverges out of the blocking center and converges into newly developing cyclonic anomalies downstream.
In Fig. 4, we also compare W with Fs defined in P85 for a zonally uniform basic flow. Neither Fs nor our flux W apparently exhibits an oscillatory component on a scale of one-half wavelength, and Fs and W are distributed similarly at a first glance. A close inspection reveals, however, that W exhibits a stronger meridional component than Fs over central and eastern Siberia (Fig. 4), where V is significantly southward upstream of a climatological-mean trough (Fig. 3). The lack of contributions of V seems to cause an unrealistic undulation in the stream of Fs (i.e., corresponding Cg) around the blocking ridge. Farther to the east, Fs penetrates into the trough, which is again unrealistic, whereas W is suppressed within the trough and its main stream follows the jet axis detouring to the south of the trough. Hence, W follows the meandering basic flow better than Fs, thus better representing the advective nature of Cg of stationary Rossby waves. The tendency that W is systematically stronger than Fs over Siberia reflects the fact that the background westerlies are substantially stronger there than the zonal average.
Another comparison between W and Fs was presented in TN97, who applied these fluxes to the low-pass-filtered circulation anomalies composited for the 15 strongest blocking anticyclones around (56°N, 160°W). They showed that W and Fs, including their divergence patterns, were nearly phase-independent. It was shown again that W follows the meandering mean westerlies better than Fs.
c. Observed migratory eddies: A baroclinic wave packet
In this subsection, we present an example of our wave-activity flux applied to migratory eddies. The data we use are 12-hourly gridded fields of geopotential height and temperature based on the NCEP–NCAR reanalyses. A high-pass filter with a cutoff period of 8 days was applied to the data time series, in order to extract migratory, high-frequency transient disturbances. As in the previous subsection, W was evaluated on pressure surfaces. We focus on the period around 20 November 1983, when strong high-frequency disturbances were observed over the North Pacific.
Since the static stability of basic state is assumed to be uniform on a pressure surface in QG scaling, interpretations of W may be somewhat complicated at the 250-hPa level where a tropopause intersection often occurs. In the lower tropopause levels, on the other hand, a tropopause intersection is much less frequent. However, a region where M and hence W are well defined is much narrower horizontally, because these pressure levels are close to the steering level of the baroclinic disturbances. Thus, the 300-hPa level was chosen for plotting the horizontal distribution of W, in order both to suppress influence of the intersecting tropopause to some extent and to depict horizontal wave propagation reasonably well within substantial part of the analysis domain.
Figure 5 shows the 8-day low-pass-filtered 300-hPa height for that day, regarded as the basic state on which those high-frequency disturbances are embedded. A strong westerly jet over the midlatitude North Pacific, where (|U| − CP) exceeds a certain positive value, say 2.0 (m s−1), approximately defines a “wave guide” for the migratory eddies.
Before computing W based on (C5) with
In Fig. 6, we plot W to show snapshots of a “baroclinic wave packet” at the 300-hPa level. Each of the wave components (i.e., highs and lows) move eastward with CP ≈ 10 (m s−1), while their envelope appears to propagate eastward much faster accompanied by the rapid decay of a cyclonic anomaly center over the western Pacific. These features may be indicative of downstream development of the disturbances (e.g., Chang and Orlanski 1993) although it may be more or less underestimated here in the high-pass-filtered fields. Over the Gulf of Alaska the stream of W splits into two branches, following the split mean flow. While part of the wave activity is propagating into higher latitudes, most of the activity propagates southeastward along the main branch of the westerly jet. A region of strong upper-level divergence (∇H · W) almost coincides with that of the strongest upward W at the 600-hPa level9 (Fig. 6). This“wave source” region gradually weakens, as it quickly moves eastward following the wave packet toward the region where the basic-state baroclinicity is weaker. Another wave source region seems to be around (50°N, 150°W), just upstream of the split of the jet. In Fig. 6, W and ∇H · W appear to be somewhat noisier than in the applications to stationary eddies (as in Fig. 4). Nevertheless, the suppression of half-wavelength noise and the dominance of the wave-packet signal on scales of a wavelength or larger are both apparent in Fig. 6, which manifest the greatest advantage of our flux W.
It is clear in the zonal and meridional cross sections (Figs. 7a and 7b) that the wave-activity flux are dominantly upward in the mid- to upper troposphere over the primary wave source region between the date line and 160°W, indicating conversion of the available potential energy from the mean flow to the disturbances. From this “source region” the wave activity propagates dominantly eastward along the upper-tropospheric jet. Part of the wave activity appears to be propagating slightly upward and downward above and beneath of the 350-hPa level, respectively, around 155°W as if there were another wave source region, which is consistent with the distribution of ∇H · W shown in Fig. 6.
d. Validity assessment of the approximations
First, we evaluated the two sides of (39) at each grid point, based on the barotropic simulation by Enomoto and Matsuda (1999, see Fig. 1). In the evaluation, only the horizontal component was used and PV was replaced by the absolute vorticity. The patterns of
The same assessment as above was performed based on the composited height anomalies for the Siberian blocking. At each grid point, ∇H · W,
The pattern of our flux W and its three-dimensional divergence (∇ · W) at the 300-hPa level associated with migratory synoptic-scale eddies are shown in Fig. 10a. Again, we use the data at 20 November 1983. We compare ∇ · W with
4. Physical interpretations of phase-independent wave-activity fluxes
In section 2, we formulate a phase-independent wave-activity flux W. Though shown to be parallel to the local three-dimensional group velocity Cg of a Rossby wave packet, a rather complicated expression of the flux may make it difficult to intuitively relate it to the wave packet propagation mechanisms. In this section, we present an interpretation in which each term of the individual components of our flux or Fs of P85 is related to an explicit physical process involved in the wave packet propagation. We also formulate a set of generalized transformed Eulerian-mean (TEM) equations that represent instantaneous feedbacks from a propagating Rossby wave packet upon a basic flow on which the packet is embedded.
We next consider the first term of each component of Fs (or Ws). We begin with an interpretation of υ′2, relating it to wave-packet propagation in the zonal direction (in Fig. 11c). A meridional flux of the meridional momentum (υ′2) at point E on a node line of the ψ′ field acts to induce the southward and northward geostrophic motions to the south (point B) and north (point D), respectively. Due to the nondivergent nature of the geostrophic flow, compensating westerly and easterly accelerations must occur to the west (point A) and east (point C). These accelerations at points A and C manifest the westward transport of the second-order westerly momentum, acting most strongly on node lines of the ψ′ field, in the direction opposite to the eastward group velocity of the wave packet. An interpretation of u′υ′ in the meridional component of Fs (or Ws) is more straightforward. It represents the meridional transport of the second-order westerly momentum in the direction opposite to the meridional group velocity (Fig. 11c). The vertical component of Fs (or Ws) includes the meridional temperature flux υ′θ′. For an upward propagating wave packet, the flux is poleward that acts to reduce the vertical westerly shear in a basic state (Fig. 11b), resulting in the downward transport of the second-order mean westerly momentum. Therefore, the first term of Fs including eddy momentum and heat fluxes represents the systematic “backward” transport of the mean westerly momentum that occurs mainly around the node lines of the ψ′ field associated with a propagating Rossby wave packet.
The above arguments can readily be extended to W (or Ws) applicable to wave-packet propagation through a zonally asymmetric basic flow. Thus, a snapshot of a Rossby wave-packet propagation can be represented as a superposition of the aforementioned two “complementary” dynamical processes involved in the propagation. It also becomes apparent that they exhibit different dependencies of the wave phase. Specifically, in the almost plane-wave limit of ψ′ = ψ0 sin(kx + ly + mz − ωt), the ageostrophic flux of geopotential on ridge/trough lines of the ψ′ field and the systematic backward transport of the mean “westerly” momentum on the node lines are proportional to sine squared and cosine squared of the wave phase, respectively. Hence, an appropriate combination of those two processes as in Fs and W leads to the elimination of their phase dependencies.
These relations (45), (46), (47), (48), and (49) are in correspondence to (4.3), (4.4), (4.5), and (4.6) in P86, which are expressed in a time-averaged form and therefore inappropriate for stationary eddies. Our relations, in contrast, are expressed in a phase-independent form without any averaging required. They can therefore be used for evaluating instantaneous feedback forcing on the mean flow induced by a wave packet propagating through it, regardless of whether the packet consists of migratory or stationary eddies.
Our expression of a three-dimensional residual ageostrophic circulation is somewhat different from the corresponding expressions derived in P86 and Trenberth (1986), with respect particularly to the vertical component of R and its counterparts. The differences can be attributed partly to a distinction between our instantaneous phase-independent form and their time-averaged form. It can be attributed also to another distinction that eddy feedbacks in our formulation are evaluated on a coordinate system moving with wave phase speed, whereas the feedbacks in their formulations are on a coordinate system moving with a basic flow. Note that the vertical component R contributes to the horizontal residual motion in the TEM framework. Thus, a specific expression of the residual circulation does depend upon a coordinate system on which the eddy feedbacks are evaluated and upon a particular definition of a wave-activity flux as well, although these expressions of the residual circulation reduce to a single form when zonally averaged.
5. Conclusions
We have derived an approximate conservation relation of the wave-activity pseudomomentum for QG eddies on a zonally varying basic flow through averaging neither in time nor in space. We have shown that a linear combination of quantities A and
Furthermore, we have presented a physical interpretation of individual terms of our flux W and Fs in P85. They illuminate the following two dynamical aspects of Rossby wave-packet propagation: one is the systematic transport of basic-state westerly momentum in the form of eddy momentum and heat fluxes that are strongest along node lines of an eddy streamfunction field, and the other is an associated ageostrophic flux of geopotential strongest along trough/ridge lines, which is related to the rate of working by the pressure force in the direction of the local group velocity of a Rossby wave packet. Although only the former aspect appears explicitly in the conventional and extended E–P fluxes and the flux of P86, they can still represent wave-packet propagations because they are expressed in phase-averaged forms by zonal or time averaging. In the wave-activity fluxes of P85 and ours, those two aspects of the wave-packet propagation are combined in an explicit manner so as to eliminate their phase dependencies. We have shown that the flux Fs of P85 can be derived, as our flux W, through manipulations of both the eddy enstrophy and energy equations, whereas the conventional and extended E–P fluxes were derived only on the basis of the enstrophy conservation form.
Generally, pseudoenergy obeys an exact (local) conservation law even for finite-amplitude disturbances on a zonally varying basic flow (McIntyre and Shepherd 1987; Haynes 1988). Conservation of pseudoenergy reflects the time invariance of a basic flow, whereas that of pseudomomentum reflects the translational invariance of a basic state in the zonal direction (e.g., Held 1987). Unlike pseudoenergy, an exact conservation of pseudomomentum for finite-amplitude perturbations is realized only for a zonally uniform basic flow. Hence, a conservation relation of pseudomomentum is necessarily an approximate one when extended to a zonally inhomogeneous basic state, as we needed to assume on several occasions during our derivation that the basic state is unforced and slowly varying in space, and that perturbations are small in amplitude as well. However, as P85 pointed out, pseudoenergy, if averaged over the wave phase, vanishes for any stationary waves even in the generalized Lagrangian-mean theory (Andrews and McIntyre 1978). For example, it is easily checked that [
Applying M and W to the simulated and observed atmospheric data, we have verified their phase-independency and ability to depict the wave-packet propagation of both stationary and migratory QG eddies. In particular, W is the first diagnostic tool capable of illustrating an instantaneous status of the three-dimensional propagation of a packet of migratory waves in a phase-independent manner. The price we ought to pay for it is that we need a priori knowledge of the local phase speed of the eddies, which should be either prescribed theoretically or estimated statistically in advance. It should be kept in mind that this estimation is apt to induce certain errors in the evaluation of W. For stationary eddies, we have shown that W can depict wave-packet propagation on a zonally varying basic flow in a more realistic manner than such a wave-activity flux as Fs defined for a zonally uniform basic flow, in a sense that W better represents the advective nature of Cg. We have also shown that ∇H · W leads to a more realistic estimation of a wave source region than ∇H · Fs. Of course, it may be enough to use Fs for diagnosing the planetary waves, although W applied to a zonally uniform basic flow is equivalent to Fs. We claim that W is a useful diagnostic tool for disturbances embedded on a basic flow, which is asymmetric in the zonal direction in the presence of the planetary waves.
The validity of the approximations we used has also been assessed. In our applications, the residual term
Acknowledgments
Discussion with Dr. H. Sakuma of IGCR, the Frontier Research System for Global Change, was very helpful in clarifying our argument on the relationship between pseudomomentum and pseudoenergy. We thank Drs. Y. Matsuda, Y. Wakata, M. Takahashi, and Y.-Y. Hayashi of the University of Tokyo, Drs. Y. Kuroda, S. Maeda, and S. Takano of the Japan Meteorological Agency, and Dr. Yamamoto of Wakayama University for their constructive comments and suggestions. Dr. P. Kushner of the Princeton University and anonymous reviewers gave us critical comments, which also led to substantial improvement of our paper. Also, we thank Dr. T. Enomoto and Dr. M. Honda for providing us numerical data simulated in their latest works. This study is supported in part by the Grant-in-Aid for Scientific Research on Priority Areas (08241104) of the Japanese Ministry of Education, Science, Sports and Culture.
REFERENCES
Andrews, D. G., 1983: A conservation law for small-amplitude quasi-geostrophic disturbances on a zonally asymmetric basic flow. J. Atmos. Sci.,40, 85–90.
——, 1984: On the existence of nonzonal flows satisfying sufficient conditions for instability. Geophys. Astrophys. Fluid Dyn.,28, 243–256.
——, and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen–Palm relation and the mean zonal acceleration. J. Atmos. Sci.,33, 2031–2048.
——, and ——, 1978: On wave action and its relatives. J. Fluid. Mech.,89, 647–664.
——, J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. Academic Press, 489 pp.
Blackmon, M. L., J. M. Wallace, N.-C. Lau, and S. L. Mullen, 1977:An observational study of the Northern Hemisphere wintertime circulation. J. Atmos. Sci.,34, 1040–1053.
——, Y.-H. Lee, J. M. Wallace, and H.-H. Hsu, 1984: Time variation of 500-mb height fluctuations with long, intermediate and short time scales. J. Atmos. Sci.,41, 981–991.
Bretherton, F. P., and C. J. R. Garret, 1968: Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. London,A302, 529–554.
Brunet, G., and P. H. Haynes, 1996: Low-latitude reflection of Rossby wave trains. J. Atmos. Sci.,53, 482–496.
Chang, E.-K. M., 1993: Downstream development of baroclinic waves as inferred from regression analysis. J. Atmos. Sci.,50, 2038–2053.
——, and I. Orlanski, 1993: On the dynamics of a storm track. J. Atmos. Sci.,50, 999–1015.
Edmon, H. J., B. J. Hoskins, and M. E. McIntyre, 1980: Eliassen–Palm cross sections for the troposphere. J. Atmos. Sci.,37, 2600–2616; Corrigendum, 38, 1115.
Enomoto, T., and Y. Matsuda, 1999: Rossby wavepacket propagation in a zonally-varying basic flow. Tellus,51A, 588–602.
Haynes, P. H., 1988: Forced, dissipative generalizations of finite-amplitude wave-activity conservation relations for zonal and nonzonal basic flows. J. Atmos. Sci.,45, 2352–2363.
Held, I. M., 1987: New conservation laws for linear quasi-geostrophic waves in shear. J. Atmos. Sci.,44, 2349–2351.
Holton, J. R., 1992: An Introduction to Dynamic Meteorology. 3d ed. Academic Press, 511 pp.
Honda, M., K. Yamazaki, H. Nakamura, and K. Takeuchi, 1999: Dynamic and thermodynamic characteristics of atmospheric response to anomalous sea-ice extent in the Sea of Okhotsk. J. Climate,12, 3347–3358.
Hoskins, B. J., and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci.,38, 1179–l196.
——, and T. Ambrizzi, 1993: Rossby wave propagation on a realistic longitudinally varying flow. J. Atmos. Sci.,50, 1661–1671.
——, I. N. James, and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci.,40, 1595–1612.
Karoly, D. J., R. A. Plumb, and M. Ting, 1989: Examples of the horizontal propagation of quasi-stationary waves. J. Atmos. Sci.,46, 2802–2811.
Kuroda, Y., 1996: Quasi-geostrophic 3-dimensional E–P flux of stationary waves on a sphere. J. Meteor. Soc. Japan.,74, 563–569.
McIntyre, M. E., 1982: How well do we understand the dynamics of stratospheric warmings? J. Meteor. Soc. Japan,60, 37–65.
——, and T. G. Shepherd, 1987: An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnold’s stability theorems. J. Fluid Mech.,181, 527–565.
Nakamura, H., 1994: Rotational evolution of potential vorticity associated with a strong blocking flow configuration over Europe. Geophys. Res. Lett.,21, 2003–2006.
——, M. Nakamura, and J. L. Anderson, 1997: The role of high- and low-frequency dynamics in the blocking formation. Mon. Wea. Rev.,125, 2074–2093.
Naoe, H., Y. Matsuda, and H. Nakamura, 1997: Rossby wave propagation in idealized and realistic zonally varying flows. J. Meteor. Soc. Japan,75, 687–700.
Plumb, R. A., 1985: On the three-dimensional propagation of stationary waves. J. Atmos. Sci.,42, 217–229.
——, 1986: Three-dimensional propagation of transient quasi-geostrophic eddies and its relationship with the eddy forcing of the time-mean flow. J. Atmos. Sci.,43, 1657–1678.
Simmons, A. J., J. M. Wallace, and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci.,40, 1363–1392.
Takaya, K., and H. Nakamura, 1997: A formulation of a wave-activity flux for stationary Rossby waves on a zonally varying basic flow. Geophys. Res. Lett.,24, 2985–2988.
Trenberth, K. E., 1986: An assessment of the impact of transient eddies on the zonal flow during a blocking episode using localized Eliassen–Palm flux diagnostics. J. Atmos. Sci.,43, 2070–2087.
Uryu, M., 1974: Mean zonal flows induced by a vertically propagating Rossby wave packet. J. Meteor. Soc. Japan.,52, 481–490.
Vanneste, J., and T. Shepherd, 1998: On the group-velocity property for wave-activity conservation laws. J. Atmos. Sci.,55, 1063–1068.
Wallace, J. M., G. H. Lim, and M. L. Blackmon, 1988: Relationship between cyclone tracks, anticyclone tracks, and baroclinic waveguides. J. Atmos. Sci.,45, 439–462.
APPENDIX A
Derivation of Wave-Activity Conservation Law through “Local Coordinate Rotation”
APPENDIX B
Group Velocity of Rossby Waves on a Zonally Varying Basic Flow
APPENDIX C
A Conservation Law of M and Its Flux W on the Pressure Coordinates
(a) Zonal wind velocity of the basic state for a barotropic model experiment by Enomoto and Matsuda (1999). Contoured for every 10 m s−1 and the easterlies are shaded. (b) Day-14 streamfunction response ψ′ (every 106 m2 s−1; dashed for negative values) to divergence forcing centered at (40°N, 90°E). (c) Corresponding wave-activity flux W (arrow), superimposed on contours for ψ′ = ±2 × 106 m2 s−1. Scaling for the arrows is given near the lower-right corner (unit: m2 s−2). (d) Wave-activity flux W (arrow) superimposed on contours for divergence of W. Contoured for every ±2.0 × 10−6 m s−2; zero contours are omitted and dashed lines are for negative values (i.e., convergence). Each of the above evaluations was made by setting CP = 0.
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
Wave-activity fluxes associated with a stationary Rossby wave train forced thermally by anomalous surface heat fluxes in association with anomalous sea-ice cover within the Sea of Okhotsk, in a simulation by Honda et al. (1999). Horizontal components of Fs of P85 and W defined in (C5) were evaluated on pressure surfaces and then averaged between the 200- and 500-hPa levels: (a) Fs (arrows) and ∇H · Fs (contoured), (b) W (arrows) and ∇H · W (contoured). Scaling for the arrows is given between the panels. Light solid and dashed lines indicate the flux divergence and convergence, respectively. Contour interval: 1.0 × 10−5 m s−2; zero lines are omitted. In the shaded regions surrounded by heavy solid and dashed lines, anomalies of surface turbulent heat fluxes into the atmosphere are significantly positive and negative, respectively, exceeding 200 W m−2 in magnitude.
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
Climatological-mean 250-hPa height (m) in winter (mid-Nov–mid-Mar) for 1965–92. Contour interval is 100. Heavy lines indicate 10 000 and 10 500. The closed circle corresponds to the composited blocking center at (54°N, 100°E).
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
Composite time evolution associated with the 15 strongest blocking events at the 250-hPa level around (54°N, 100°E) for −2, 0, and +2 days relative to the peak blocking time (from top to bottom; negative and positive values signify the amplification and decay stages, respectively). Presented are (left column) the horizontal component of our flux W with arrows based on low-pass-filtered 250-hPa height anomalies Z′ as contoured, (middle column) W with arrows superimposed on its horizontal divergence with shading, and (right column) horizontal component of Fs of P85 with arrows and its divergence with shading. Here Z′ is normalized by sin(45°N)/sin(lat). Contour intervals: every 100 m for Z′ (dashed for negative values); ±0.25, ±0.75, . . . (10−4 m s−2) for flux divergence; zero contours are omitted in all panels. Heavy and light shading signify the flux convergence and divergence, respectively. Scaling for arrows is given near the lower-right panel (unit: m2 s−2).
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
Basic state for migratory disturbances, as defined by 8-day low-pass-filtered 300-hPa height field at 0000 UTC on 20 Nov 1983. Contour interval is 100 m. Heavy lines denote 8500, 9000, and 9500 m. Regions where |U| − CP < 2.0 m s−1 are shaded.
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
Time sequence of (left column) 8-day high-pass-filtered 300-hPa height (Z′), superimposed on the horizontal component of W with arrows, (middle column) horizontal component of W with arrows and its divergence/convergence with contours, and (right column) horizontal component of W with arrows superimposed on the vertical component of 600 hPa with contours. The sequence begins at 1200 UTC on 19 Nov 1983 with 12-h intervals (from top to bottom). Here Z′ is contoured for every 100 m from 50 m and dashed for negative values, and ∇H · W is contoured for every 8.0 × 10−4 m s−2. Light shading with solid lines and heavy shading with dashed lines denote the divergence and convergence, respectively. The vertical component of W is contoured for every 5.0 Pa m s−2. Solid and dashed lines indicate where the flux is upward and downward, respectively. Scaling of W is given near the lower panel (unit: m2 s−2).
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
(a) Zonal-vertical section along 50°N where the zonal and vertical components of W are plotted with arrows. (b) Meridional-vertical section along 170°W where the meridional and vertical components of W are plotted with arrows. In (a) and (b) the basic-state westerlies are superimposed with contours (every 5 m s−1). Scaling of W is given just below the individual panels. (units: for horizontal component m2 s−2; for vertical component 10−2 Pa m s−2). Note that M can be defined only in regions where |U| > CP.
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
(a) Divergence of W, (b)
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
(a) Horizontal component of W (arrows with scaling given near the lower-right panel; unit: m2 s−2) and its divergence (contoured and shaded), (b)
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
Same as Fig. 9 but based on 8-day high-pass-filtered data at 0000 UTC on 20 Nov 1983 (see Fig. 6). (a) Three-dimensional divergence is plotted. Contoured for every 8.0 (10−4 m s−2) and dashed for negative values. Zero contours are omitted. Heavy and light shading signifies negative (or convergence) and positive (or divergence) values, respectively.
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
Schematic diagrams for explanations of an instantaneous Rossby wave-packet propagation mechanism. In each diagram, thin solid lines denote node lines in the perturbation streamfunction (ψ′) field at which perturbation QG flows are strongest, and thin dashed lines denote ridge (H)/trough (L) lines. Arrows with U and Cg signify the mean westerly flow and group velocity of the wave packet, respectively. (a) Horizontal ageostrophic flows (ua) associated with a wave-packet propagation, which aligned coherently on the ridge/trough lines and parallel to the horizontal group velocity. Thin arrows indicate the direction of the perturbation geostrophic flows. (b) Zonal-vertical section of an upward propagating Rossby wave packet. Short arrows with wa indicate vertical (ageostrophic) motions aligned on the ridge/trough lines in association with wave packet. Open arrows signify the second-order acceleration of the mean westerly flow as a result of systematic poleward heat transport strongest on the node lines. (c) Second-order mean flow accelerations (indicated with open arrows) induced by eddy momentum transport (υ′2 on left and u′υ′ on right). See text for more details.
Citation: Journal of the Atmospheric Sciences 58, 6; 10.1175/1520-0469(2001)058<0608:AFOAPI>2.0.CO;2
Throughout this paper, we use the term “snapshot” to signify a particular phase of waves. It is used for referring not only to the status of waves at a particular moment but also to a composite or linear regression map that represents spatial distribution of typical local anomalies at a particular wave phase based on a number of realizations at different dates and/or times.
In his paper a factor of ½ is missing in the formula of Fs, which is apparently a typo.
The definition of a quantity
Starting with (10), one can even derive an approximate conservation relation (31), simply following P85 based only on the equation of A. As shown below, however, our derivation is more straightforward without any need of incorporating such an unknown nondivergent flux as G.
Additionally, a nonconservative term DT = (D0∇ · r′ − r′ · ∇D0)/2 is in the same form as each of the components of W. It may be possible to interpret DT as a “phase-independent forcing” of a phase-independent pseudomomentum M.
Since M and W are independent of wave phase, the space and time derivatives that explicitly appear in (31), in effect, represent slow variations on the spatial and temporal scales of a wave packet, respectively, which are much larger than the wavelengths and periods of components that compose the wave packet.
There are some streamfunction anomalies in part of the Tropics where the basic flow is easterly and hence stationary Rossby waves cannot exist in the linear theory. Those anomalies are not associated with the wave-packet propagation from the forcing region. They appear to be associated with patches of relative vorticity that are cut off from the westerly duct due to wave breaking and then advected by the background easterlies. See Enomoto and Matsuda (1999) for details.
We also applied W to the low-pass-filtered height anomalies composited for the 15 strongest blocking anticyclones observed over Europe (54°N, 10°E). We confirmed the findings of Nakamura (1994) and Nakamura et al. (1997) based on the flux of P86 artificially smoothed in space.
