1. Introduction
Rossby waves are a fundamental ingredient of atmospheric dynamics. For synoptic-scale Rossby waves, the strong gradient of potential vorticity (PV) associated with the midlatitude jet effectively constitutes the waveguide (Platzman 1949; Schwierz et al. 2004; Martius et al. 2010). A local perturbation of this waveguide may lead to the formation of distinct wave packets (e.g., Simmons and Hoskins 1979; Hakim 2003; Riemer et al. 2008). The significance of such Rossby wave packets (RWPs) for weather forecasting has long been recognized (Cressman 1948; Hovmöller 1949). The role of RWPs as precursors to high-impact weather events has more recently been given special attention (Shapiro and Thorpe 2004; Chang 2005; Martius et al. 2008; Wirth and Eichhorn 2014; Piaget et al. 2015). The investigation of RWPs along the midlatitude waveguide is the focus of this study.
It is generally expected that RWPs, as large-scale flow features obeying balanced dynamics, will exhibit a large degree of predictability that may then be inherited by smaller-scale weather features (Anthes et al. 1985; Grazzini 2007; Grazzini and Vitart 2015). Challenging this expectation, however, is the fact that it has been demonstrated that PV forecast errors exhibit maximum amplitude precisely along the midlatitude waveguide, that is, within RWPs (Dirren et al. 2003). Furthermore, forecast errors and uncertainties associated with specific weather systems may propagate and grow along the waveguide, severely affecting predictability in the downstream region (Anwender et al. 2008; Harr et al. 2008; Riemer and Jones 2010; Rodwell et al. 2013). Also, Grams et al. (2011) demonstrated the prominent role that moist processes, with low intrinsic predictability, may play in modifying the PV distribution in the vicinity of the waveguide. Indeed, the reliable prediction of RWPs constitutes a challenge for state-of-the-art numerical forecast systems (Glatt and Wirth 2014; Gray et al. 2014). The important question therefore arises of under which conditions RWPs exhibit high and low predictability, respectively. To address this question, we here develop a diagnostic framework to quantify the different processes governing the dynamics of real-atmospheric RWPs.
Our framework will build on the well-known PV perspective of Rossby waves and their baroclinic coupling (Rossby 1940; Eady 1949; Hoskins et al. 1985; Hoskins 1990) and, in particular, on Rossby’s early notion of the displacement of PV contours. An illustration of a high-amplitude Rossby wave pattern along the midlatitude jet is provided in Fig. 1 in terms of PV on an isentropic level intersecting the midlatitude tropopause. An RWP apparently extends from the North Pacific Ocean into Europe and is identified by the undulation of the strong PV gradient between the stratospheric, high-PV air and the tropospheric, low-PV air. Clearly associated with the wave pattern are high and low pressure systems in the lower troposphere, reflecting the well-known baroclinic nature of RWPs. Since both PV and potential temperature θ are conserved for adiabatic motion, displacement of PV contours is governed to first order by PV advection on isentropic surfaces when diabatic, frictional, and nonhydrostatic effects are small. The partition and the physical interpretation of the advective PV tendencies, complemented by the examination of diabatic PV tendencies, will be at the heart of our diagnostic.
Illustration of an apparent RWP extending from the North Pacific into eastern Europe in terms of PV on an isentropic (325 K) surface (color shaded; 0000 UTC 28 Oct 2008). The black contour depicts the dynamic tropopause defined as the 2-PVU surface. The RWP can be identified by the synoptic-scale undulation of the tropopause. Gray contours depict 850-hPa geopotential (every 5 gpdam) to indicate low-level pressure systems.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Illustration of an apparent RWP extending from the North Pacific into eastern Europe in terms of PV on an isentropic (325 K) surface (color shaded; 0000 UTC 28 Oct 2008). The black contour depicts the dynamic tropopause defined as the 2-PVU surface. The RWP can be identified by the synoptic-scale undulation of the tropopause. Gray contours depict 850-hPa geopotential (every 5 gpdam) to indicate low-level pressure systems.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Illustration of an apparent RWP extending from the North Pacific into eastern Europe in terms of PV on an isentropic (325 K) surface (color shaded; 0000 UTC 28 Oct 2008). The black contour depicts the dynamic tropopause defined as the 2-PVU surface. The RWP can be identified by the synoptic-scale undulation of the tropopause. Gray contours depict 850-hPa geopotential (every 5 gpdam) to indicate low-level pressure systems.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Our PV–θ framework will focus on the evolution of individual troughs and ridges, which may help to link our analysis to a synoptician’s view of RWPs. In addition to the previously studied processes of barotropic wave propagation and baroclinic interaction, reviewed in some detail below, our framework will explicitly diagnose the impact of 1) upper-tropospheric divergent flow, which can often be attributed to latent-heat release below, and 2) diabatic PV modification. Recent studies using a related PV–θ framework demonstrated that the divergent flow may have a significant impact on the wave evolution, in particular on ridge building (e.g., Riemer and Jones 2010; Archambault et al. 2013; Grams et al. 2013). The importance of diabatic PV modification near the midlatitude waveguide has also been emphasized by recent studies (e.g., Chagnon et al. 2013; Chagnon and Gray 2015). Our PV–θ framework is also motivated by the abovementioned fact that PV errors maximize along the midlatitude waveguide. Also on the basis of the PV–θ perspective, Davies and Didone (2013) have derived a tendency equation for PV forecast errors. In combination with this tendency equation, the herein-developed diagnostic can be applied to examine the dynamics of forecast error amplification (M. Baumgart et al. 2015, unpublished manuscript). A further motivation for us to consider a PV–θ framework arises from recent developments in wave activity formulations, which are based on the topology of PV contours on isentropic surfaces (Nakamura and Solomon 2011; Methven 2013). While the diagnostic presented in the current study will not be formally related to wave-activity conservation laws, we hope that such a link can be established in future work.
Eddy kinetic energy (EKE) and individual contributions to its tendency provide a well-established framework with which to analyze the evolution of RWPs (e.g., Orlanski and Sheldon 1995; Chang 2000; Chang et al. 2002). Within that framework, the paradigm of downstream baroclinic development has been established. Starting from a preexisting upper-level EKE center, usually identified as a jet streak in the vicinity of a trough, downstream development is described by the downstream energy dispersion due to ageostrophic geopotential fluxes. Subsequent to such initial growth, the new EKE center downstream is further amplified by baroclinic energy conversion. Then, downstream radiation of energy initiates the decay of the new EKE center. The cycle may then repeat itself in the region farther downstream. In addition, barotropic conversion may play a notable role, especially during the initiation of a wave packet and during wave breaking. Chang (2000) used the EKE framework to investigate RWPs in the Southern Hemisphere over one winter season. He found that, while the paradigm of downstream baroclinic development is, in general, valid, the dynamic processes may exhibit a large variability within an individual wave packet.
An alternative framework to examine the evolution of the upper-tropospheric Rossby wave pattern has been developed by Nielsen-Gammon and Lefevre (1996). These authors identified individual contributions to a geopotential-height-tendency equation by using piecewise inversion of quasigeostrophic PV. Nielsen-Gammon and Lefevre likened the advective tendencies associated with anomalies of low-level PV and boundary θ to baroclinic conversion and the tendencies associated with upper-level PV anomalies to the energy transfer by the divergence of the ageostrophic geopotential flux. They applied the framework to the life cycle of an upper-level mobile trough and found results that were consistent with those from the EKE framework.
Commonalities and important differences between our PV–θ framework and the EKE and height-tendency frameworks will be discussed in some detail below (section 3f). Before that, section 2 will briefly review the PV perspective of Rossby waves. The methods and data used to identify and quantify individual processes are presented in section 3. This section also introduces an integrated measure of the impact of the individual processes on RWP evolution. Section 4 provides an illustrative case study of an RWP in October 2008. Section 5 contains our summary and a discussion of our results in light of recent studies on forecast errors and predictability of RWPs.
2. Key processes for real-atmospheric RWPs
a. Quasi-barotropic dynamics
The essence of Rossby waves is described by linear barotropic dynamics. The advection of upper-level PV by the wind field associated with the upper-level PV distribution itself will here be interpreted as the real-atmospheric analogy to barotropic dynamics (cf. Nielsen-Gammon and Lefevre 1996). We will hereinafter refer to the associated term as “quasi barotropic.” Recall that the upper-level PV distribution is used in this study to characterize real-atmospheric RWPs (cf. Fig. 1). Our specific definition of upper (and lower) levels will be given below.
The linear barotropic dynamics comprises two processes: advection of PV anomalies by a background flow and advection of background PV by the flow associated with the PV anomalies (see section 6a of Hoskins et al. 1985). The latter process describes intrinsic wave propagation and is illustrated in Fig. 2a. Amplification or decay of PV anomalies usually occurs when there is an asymmetry in the amplitude of existing anomalies. This asymmetry leads to the generation or amplification of PV anomalies at the leading edge of the wave packet and the decay of PV anomalies at the trailing edge. Amplification and decay of anomalies therefore relate to the intrinsic group propagation of the wave. It is well known that the group speed is faster than the phase speed for the midlatitude RWPs under consideration (Hoskins 1990). If upstream development at upper levels were observed, such development would be associated with baroclinic coupling (Simmons and Hoskins 1979).
Schematic illustration of three key processes for the evolution of RWPs: (a) linear wave propagation, (b) baroclinic interaction, and (c) modification by divergent outflow; (a) and (c) are horizontal sections, whereas (b) is a three-dimensional depiction. Black (green) contours depict the wave pattern at a time t0 (t1, with t1 > t0). The associated PV anomalies, assuming a zonal background state, in (a) and (b) are displayed by the shading in the respective color. The sign of the PV anomalies is given, and the associated cyclonic and anticyclonic circulations, assuming PV anomalies in the Northern Hemisphere, are indicated by the circular arrows. The green arrows depict the meridional component of these circulations, i.e., their projection on the zonal background PV gradient in (a) at the wave’s inflection points and in (b) within troughs and ridges. The green dotted line represents the PV tendencies due to advection of the background PV by these circulations.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Schematic illustration of three key processes for the evolution of RWPs: (a) linear wave propagation, (b) baroclinic interaction, and (c) modification by divergent outflow; (a) and (c) are horizontal sections, whereas (b) is a three-dimensional depiction. Black (green) contours depict the wave pattern at a time t0 (t1, with t1 > t0). The associated PV anomalies, assuming a zonal background state, in (a) and (b) are displayed by the shading in the respective color. The sign of the PV anomalies is given, and the associated cyclonic and anticyclonic circulations, assuming PV anomalies in the Northern Hemisphere, are indicated by the circular arrows. The green arrows depict the meridional component of these circulations, i.e., their projection on the zonal background PV gradient in (a) at the wave’s inflection points and in (b) within troughs and ridges. The green dotted line represents the PV tendencies due to advection of the background PV by these circulations.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Schematic illustration of three key processes for the evolution of RWPs: (a) linear wave propagation, (b) baroclinic interaction, and (c) modification by divergent outflow; (a) and (c) are horizontal sections, whereas (b) is a three-dimensional depiction. Black (green) contours depict the wave pattern at a time t0 (t1, with t1 > t0). The associated PV anomalies, assuming a zonal background state, in (a) and (b) are displayed by the shading in the respective color. The sign of the PV anomalies is given, and the associated cyclonic and anticyclonic circulations, assuming PV anomalies in the Northern Hemisphere, are indicated by the circular arrows. The green arrows depict the meridional component of these circulations, i.e., their projection on the zonal background PV gradient in (a) at the wave’s inflection points and in (b) within troughs and ridges. The green dotted line represents the PV tendencies due to advection of the background PV by these circulations.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
The consideration of PV anomalies requires the definition of a background state. A uniform background flow, as is often used in simple models, is not a suitable representation of real-atmospheric flows. Specification of a representative background will be discussed in more detail below. Horizontal shear in a nonuniform background flow implies the deformation of PV anomalies and may eventually lead to wave breaking. Although wave breaking is beyond the scope of this work, a caveat of the diagnostic developed below is that it will not capture the barotropic deformation of PV anomalies, for reasons that will be discussed in section 3c. The same is true for the nonlinear barotropic dynamics: that is, the advection of the PV anomalies by the flow that is associated with the PV anomalies themselves.
b. Baroclinic interaction
In the PV–θ framework, baroclinic interaction is represented by the modification of the upper-level PV anomalies by the wind field associated with low-level PV anomalies, and vice versa (e.g., Eady 1949; Heifetz et al. 2004). This viewpoint is not limited to small-amplitude, modal perturbations but can be applied as long as suitable low-level and upper-level PV anomalies can be identified (e.g., Davis and Emanuel 1991).
In general, the low-level anomalies influence both the amplitude and the phase of an upper-level wave. This influence depends on the phase relation between upper- and low-level anomalies (as detailed in section 6b of Hoskins et al. 1985). The focus in this study is on the impact on wave amplitude. An amplification of the upper-level wave by baroclinic processes can be expected when the respective low-level anomalies lead the upper-level pattern by less than one-half wavelength (exemplified in Fig. 2b).
c. Divergent flow
The above-noted theoretical frameworks of Rossby wave dynamics do not consider the impact of the irrotational flow (referred to as “divergent” hereinafter). Under nonlinear balance (Charney 1955), a good approximation for synoptic-scale flows, the divergent flow component is, in general, an order of magnitude smaller than the nondivergent component. PV advection by the divergent flow, however, can be considerable because the projection of the divergent flow onto the PV gradient may be large (e.g., Riemer et al. 2008). The secondary circulation associated with jet streaks is one well-known source of upper-level divergence in the midlatitudes (e.g., Shapiro 1981). Arguably, however, pronounced upper-level divergence is largely due to secondary circulations associated with latent-heat release in clouds. Some preliminary evidence in support of this statement can be found in Riemer and Jones (2010), Riemer et al. (2014), and Quinting and Jones (2015, manuscript submitted to Mon. Wea. Rev.). The following specific sources of pronounced upper-level divergence have been identified: recurving tropical cyclones (e.g., Archambault et al. 2013), organized convection (e.g., Stensrud 2013; Rodwell et al. 2013), and warm conveyor belts of midlatitude cyclones (e.g., Grams et al. 2011).
The significance of the divergent flow in modifying an upper-level trough–ridge pattern has been noted by Davis et al. (1993). Ridge amplification (e.g., Riemer and Jones 2010; Grams et al. 2013) and the impact on the structure and phase propagation of troughs (e.g., Pantillon et al. 2013; Riemer and Jones 2014) have more recently been investigated in some detail. The prototypical pattern of divergence associated with a warm-conveyor-belt outflow ahead of a trough is illustrated in Fig. 2c. As indicated, such a pattern can be expected to yield ridge amplification and a hindrance and deformation of the upstream trough.
d. Direct diabatic modification
Diabatic processes have a direct impact on the PV distribution by modifying θ gradients—a mechanism that is represented by diabatic terms in the PV-tendency equation. Different contributions to the diabatic generation of PV anomalies in an operational forecast model have been investigated in detail by Chagnon et al. (2013) and Chagnon and Gray (2015). For the purpose of this study, we focus on diabatic PV tendencies due to latent-heat release and longwave radiation. Latent-heat release in the midlatitudes is most prominently associated with the warm conveyor belt of cyclones (Pfahl et al. 2014). The role of diabatically generated, low-level PV anomalies in the intensification of cyclones has been given much attention (e.g., Davis et al. 1993), with a more recent emphasis also on the formation of upper-level, negative PV anomalies (e.g., Ahmadi-Givi et al. 2004). Longwave radiative PV tendencies are prominent near the tropopause, where the large vertical water vapor gradient implies large (climatological) gradients in radiative heating (e.g., Zierl and Wirth 1997). From a Lagrangian perspective, Chagnon et al. (2013) have shown that positive (negative) PV tendencies are found, in general, just above (below) the tropopause. Similarly, a dipole of prominent PV tendencies due to longwave radiation can be expected at the top of clouds. So far, however, the impact of these radiative tendencies on Rossby wave dynamics is not well understood.
It is important to note that diabatic processes have a broader impact on RWPs than the direct impact represented by diabatic terms in the PV equation. The significance of latent-heat release for upper-level divergence has already been noted above. Furthermore, the diabatic formation of PV anomalies implies a modification of the balanced wind field and thus an impact of diabatic processes on subsequent advective tendencies. Disentangling these complex interactions between diabatic processes and advective tendencies is beyond the scope of this study.
3. Quantifying contributions to RWP evolution
As noted above (Fig. 1), RWPs in the PV–θ framework can be identified by the synoptic-scale undulation of the strong PV gradient on an isentrope intersecting the midlatitude tropopause. Dividing the PV distribution into a steady background state 
a. Data
Before introducing the diagnostic framework, we briefly describe the unique “Year of Tropical Convection” (YOTC) dataset (e.g., Waliser et al. 2012; Moncrieff et al. 2012) that is used in this study. The YOTC data cover the period from May 2008 to April 2010 and are based on the integrated forecast system of the European Centre for Medium-Range Weather Forecasts (ECMWF). The spatial resolution is 1°, and data are available on 17 pressure levels (1000, 950, 925, 900, 850, 800, 700, 600, 500, 400, 300, 250, 200, 150, 100, 70, and 50 hPa). The unique aspect of the YOTC data is that, in addition to standard variables, the model tendencies from the individual physical parameterization schemes are provided. These tendencies are computed from 36-h forecasts starting daily from the 1200 UTC analysis. The accumulated values of the physical tendencies are available every 3 h. The heating rates contained in the YOTC data are used to estimate the direct diabatic impact [i.e., to evaluate the diabatic term 
b. Quantification of physical processes
The basic idea of our diagnostic is to quantify individual contributions to ∂PV′/∂t in Eq. (1) and attribute these tendencies to the physical processes discussed in section 2. To this end, the advective tendencies are partitioned into contributions attributable to quasi-barotropic dynamics, baroclinic interaction, and the divergent flow, respectively. For the diabatic term, contributions from latent-heat release and longwave radiation are evaluated. The partitioning and the applied methods are described in detail below. An overview of the procedure is provided in Fig. 3.
Diagram summarizing our procedure to quantify the individual contributions to RWP evolution. The blue boxes contain the generic terms in the PV equation [Eq. (1)], the red-outlined boxes contain the employed techniques and data, and the green boxes contain the individual processes that are diagnosed in our framework.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Diagram summarizing our procedure to quantify the individual contributions to RWP evolution. The blue boxes contain the generic terms in the PV equation [Eq. (1)], the red-outlined boxes contain the employed techniques and data, and the green boxes contain the individual processes that are diagnosed in our framework.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Diagram summarizing our procedure to quantify the individual contributions to RWP evolution. The blue boxes contain the generic terms in the PV equation [Eq. (1)], the red-outlined boxes contain the employed techniques and data, and the green boxes contain the individual processes that are diagnosed in our framework.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
1) Selection of background state
The partitioning PV = 
We here define 
2) Partitioning of advective tendencies
The advective tendency is partitioned into individual contributions by partitioning the wind field. Using a Helmholtz partitioning, the wind is partitioned into its nondivergent and divergent components as in Riemer et al. (2008). We consider the divergent component only to represent the divergent flow vdiv. The nondivergent wind is further partitioned using piecewise PV inversion under nonlinear balance (Charney 1955), as described in Davis and Emanuel (1991) and Davis (1992). The inversion domain extends from 20° to 80°N, with vertical boundary conditions specified at 875 and 125 hPa. Zonally, the domain is approximately centered on the RWP and usually spans the hemisphere.
3) Evaluation of diabatic terms
We focus on the diabatic impact of latent-heat release and radiation in this study. To estimate the former, the heating rates in the YOTC data due to the convection parameterization and the cloud scheme are used. The cloud scheme usually dominates. To estimate the latter, the heating rate due to the longwave radiation scheme is used. Contributions from shortwave radiation are negligible near the tropopause. To approximate the tendency at a specific time, we use the difference quotient over the subsequent 6-h period. From the available data, this approximation is arguably the best representation of the diabatic processes between two specific times when we evaluate the advective tendencies.
While latent-heat release can usually be associated with specific weather systems, the same is not the case for radiative heating. Radiative heating exhibits a prominent climatological signal due to the pronounced water vapor gradient across the tropopause. Because of the conceptual difficulties involved in identifying the contributions of individual weather systems to radiative heating, no attempt is made in this study to separate out the climatological signal. It is thus not clear to what extent the radiative tendencies presented below modify the background state or to what extent they modify the RWP under consideration. These tendencies should thus be considered as only a crude estimate of the impact of radiation on the RWP and should not be overinterpreted.
c. Integrated metric for wave amplitude
The individual contributions to ∂PV′/∂t provide a detailed view of the PV evolution associated with RWPs. Examples of ∂PV′/∂t maps are given below in association with the case study in section 4 (Fig. 9). The goal of our diagnostic, however, is to provide a succinct representation of the governing processes over the life cycle of an RWP. For this purpose, examination of a number of individual PV-tendency maps is too laborious.
Instead, we propose to examine the tendencies of PV′ integrated spatially over individual troughs and ridges, that is, to examine the amplitude evolution of the PV anomalies. A similar procedure has successfully been applied in Riemer and Jones (2014). In linear theory, wave amplitude is related to wave energy and the propagation of wave energy is given by the group velocity. In this sense, the amplitude evolution of individual PV anomalies of a real-atmospheric, finite-amplitude wave packet can be associated with the concept of group propagation (cf. Fig. 2a). The spatial integration of PV tendencies highly compresses the dynamical information but no longer captures every aspect of the evolution. The tendencies of the integrated anomalies, however, allow for the desired compact description of the RWP dynamics over the timeframe of a week or longer (section 4; see also Piaget et al. 2015).
advection of background PV,
divergence within the anomaly, implying a change in the area of the anomaly,
direct diabatic PV modification, and
a boundary term that contains the net flux of PV′ across the boundary
It is of interest to consider the idealized case of nondivergent, adiabatic flow and PV anomalies with boundaries defined by a PV′ = 0 isoline. Under these conditions, on the right-hand side of Eq. (4) all terms but one vanish and the evolution of the integrated amplitude is governed solely by advection of background PV. In this sense, the dynamics of the RWP amplitude is similar to linear dynamics (cf. Fig. 2a and associated discussion).
We note that the deformation of PV anomalies, contained in the term v ⋅ ∇PV′, does not contribute to the evolution of the integrated amplitude and thus is not captured by our (integrated) diagnostic. This is not to say that this term is not important for the overall evolution. The consequences of the elimination of this deformation term will be discussed in more detail in section 3f.
To the extent that v ⋅ ∇PV′ is associated with a net PV flux across the boundary that defines the PV anomaly, this term is contained in the boundary term. The calculation of the boundary term is described in detail in section 3d below.
In summary, we consider six different contributions to the tendency of the integrated amplitude [Eq. (4)]: the advection of background PV by the quasi-barotropic and baroclinic flow, respectively; the combined impact of the divergent flow; diabatic modification due to latent-heat release and longwave radiation, respectively; and a boundary term. To further discuss the boundary term, we consider next the definition and objective identification of PV anomalies employed in this study.
d. Automated identification of PV anomalies and the calculation of the boundary term
In an idealized wave pattern with a meridionally confined undulation of the PV gradient, individual troughs and ridges are characterized by positive and negative anomalies that are bounded by PV′ = 0 (Fig. 4a). Technically, individual troughs (ridges) can be identified as the positive (negative) anomaly in between two neighboring ridge (trough) axes, and axes can be identified by standard criteria [e.g., local extrema in the (smoothed) PV field (cf. Riemer and Jones 2014)]. We follow these idealized ideas to identify troughs and ridges of real-atmospheric RWPs. For these real cases, two important differences arise in comparison with an idealized wave pattern: 1) PV anomalies are often not completely bounded by a PV′ = 0 contour but may connect with neighboring, like-signed anomalies. This situation is schematically illustrated in Fig. 4. Ridges tend to connect in the subtropics, whereas troughs tend to connect in the polar regions. 2) PV anomalies are not only associated with the wave pattern, but the PV′ field contains additional PV′ features, which are often associated with previous wave-breaking events. Prominent examples are positive (negative) cutoff anomalies in the subtropics (polar region). Positive cutoff anomalies are illustrated in Fig. 5.
Schematic illustration of PV anomalies (PV′; red = positive, blue = negative, and darker shading indicates higher absolute values) associated with a wave pattern (a) in an idealized case with a meridionally confined PV gradient and (b) representative of the real atmosphere. Note that individual trough and ridge anomalies in the real atmosphere are usually not completely encompassed by PV′ = 0. Dashed lines indicate trough axes. The numbers are used in the main text to describe specific segments of the PV′ boundary.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Schematic illustration of PV anomalies (PV′; red = positive, blue = negative, and darker shading indicates higher absolute values) associated with a wave pattern (a) in an idealized case with a meridionally confined PV gradient and (b) representative of the real atmosphere. Note that individual trough and ridge anomalies in the real atmosphere are usually not completely encompassed by PV′ = 0. Dashed lines indicate trough axes. The numbers are used in the main text to describe specific segments of the PV′ boundary.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Schematic illustration of PV anomalies (PV′; red = positive, blue = negative, and darker shading indicates higher absolute values) associated with a wave pattern (a) in an idealized case with a meridionally confined PV gradient and (b) representative of the real atmosphere. Note that individual trough and ridge anomalies in the real atmosphere are usually not completely encompassed by PV′ = 0. Dashed lines indicate trough axes. The numbers are used in the main text to describe specific segments of the PV′ boundary.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Example of an automatically identified ridge region (stippled) between two trough axes (solid black) at 0000 UTC 28 Oct. Colors denote PV anomalies on the 325-K isentrope. The dashed line denotes the ridge axis.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Example of an automatically identified ridge region (stippled) between two trough axes (solid black) at 0000 UTC 28 Oct. Colors denote PV anomalies on the 325-K isentrope. The dashed line denotes the ridge axis.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Example of an automatically identified ridge region (stippled) between two trough axes (solid black) at 0000 UTC 28 Oct. Colors denote PV anomalies on the 325-K isentrope. The dashed line denotes the ridge axis.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Our algorithm to identify the PV anomalies associated with troughs and ridges works as follows: trough and ridge axes are identified as the lines at which the meridional wind anomaly vanishes (υ′ = 0) in data smoothed over 7° × 7°. For our purpose, these lines provide a reliable estimate of the trough (ridge) axes poleward (equatorward) of the 2 potential vorticity unit (PVU; 1 PVU = 10−6 K kg−1 m2 s−1) contour. Equatorward (poleward) of the intersection point with the 2-PVU contour, trough (ridge) axes are simply extended to the boundary of the domain by a straight line. The rationale for this simple extension is associated with the calculation of the boundary term and is given below. Troughs (ridges) are then identifies as a single connected positive (negative) PV′ region in between two ridge (trough) axes. Furthermore, this region is required to contain the intersection point of the trough (ridge) axis with the 2-PVU contour.
An automatically detected ridge anomaly is exemplified in Fig. 5. The method handles well the finite-amplitude nature and considerable deformation of anomalies (e.g., the troughs in Fig. 5) and continuously yields well-defined trough and ridge anomalies for the Rossby wave packet under consideration. We emphasize, however, that the method requires a basically wavelike underlying PV pattern. During wave breaking or trough filamentation, the method ceases to yield sensible results at some point in time. At that time, the examination of the affected PV anomaly is discontinued.
Figure 6 provides a comparison between the integrated amplitude tendency, that is, the sum of the terms on the rhs of Eq. (4), and the observed change of the PV anomaly over a 6-h period. Overall, the temporal evolution of the two metrics is reasonably similar. Differences between the two metrics can be expected as a result of 1) comparing an instantaneous tendency with a finite-time difference, 2) uncertainties in the calculation of vs, 3) the neglect of some subgrid-scale processes (e.g., PV tendencies due to turbulent mixing), and 4) nonlinearities in the piecewise PV inversion. At times, a 6-h phase shift can be observed between the metrics, which can partly be attributed to point 1 above. The largest differences are found when merging or breakup of PV anomalies occurs (not shown). These processes are not captured by the integrated amplitude tendency. At most other times, when PV anomalies are “well behaved,” the observed differences are small and the sum of the individual terms in Eq. (4), using the partition of the wind [Eq. (5)], provides a good representation of the actual evolution of individual troughs and ridges.
Comparison between the integrated amplitude tendency (solid) and the observed change of the PV anomaly over a 6-h period (dashed). Temporal evolution is shown from 0000 UTC 25 Oct to 1200 UTC 30 Oct 2008 for (a) R3 and (b) T4 from the case study in section 4.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Comparison between the integrated amplitude tendency (solid) and the observed change of the PV anomaly over a 6-h period (dashed). Temporal evolution is shown from 0000 UTC 25 Oct to 1200 UTC 30 Oct 2008 for (a) R3 and (b) T4 from the case study in section 4.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Comparison between the integrated amplitude tendency (solid) and the observed change of the PV anomaly over a 6-h period (dashed). Temporal evolution is shown from 0000 UTC 25 Oct to 1200 UTC 30 Oct 2008 for (a) R3 and (b) T4 from the case study in section 4.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
e. Choice of isentropic levels
Several waveguides for RWP propagation may exist on different isentropic levels (e.g., Martius et al. 2010). These waveguides coincide with the Arctic, midlatitude (or polar), and subtropical jets and their respective strong PV gradients. Our focus here is on RWPs along the midlatitude waveguide. Martius et al. suggest 320 K as a representative isentropic level for this waveguide. This value may exhibit some seasonal variability.
To minimize the sensitivity of our method to the precise choice of the isentropic level, we integrate the PV tendencies representing RWP evolution over a range of reasonable θ values. As illustration, the vertical distribution of PV anomalies (at 60°N) is shown in Fig. 7. PV anomalies with alternating signs representing the synoptic-scale RWP are most prominently found between 200 and 400 hPa. In terms of θ, the positive anomalies are approximately centered on the 320-K isentrope, whereas the negative anomalies tend to have maximum amplitude above this level, approximately on the 330-K isentrope. The strongest negative anomaly near 320 K extends well above this level. In general, however, the 320–330-K range is a reasonable choice, and thus, specifically, the PV tendencies for the following case study are calculated as a vertical average over the 320-, 323-, 325-, 327-, and 330-K isentropes.
Illustration of the vertical structure of PV anomalies in a pressure–longitude cross section along ϕ = 60°N at 0000 UTC 28 Oct. Shading denotes PV anomalies, gray contours are isentropes, and the black contour is the 2-PVU contour representing the tropopause.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Illustration of the vertical structure of PV anomalies in a pressure–longitude cross section along ϕ = 60°N at 0000 UTC 28 Oct. Shading denotes PV anomalies, gray contours are isentropes, and the black contour is the 2-PVU contour representing the tropopause.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Illustration of the vertical structure of PV anomalies in a pressure–longitude cross section along ϕ = 60°N at 0000 UTC 28 Oct. Shading denotes PV anomalies, gray contours are isentropes, and the black contour is the 2-PVU contour representing the tropopause.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
f. Comparison with previous diagnostics
In this section, we discuss commonalities and important differences between the integrated tendencies of our PV–θ framework and the terms in the height-tendency and EKE frameworks noted in the introduction. Our framework is very similar to that of Nielsen-Gammon and Lefevre (1996) with respect to the partitioning and piecewise inversion of PV. Following these authors, we identify tendencies due to upper-level PV anomalies with (quasi barotropic) wave propagation and tendencies due to the low-level PV anomalies, including lower-boundary θ anomalies, with baroclinic interaction.
In addition, Nielsen-Gammon and Lefevre’s diagnostic captures the impact of barotropic deformation (
In comparison with the EKE framework (see references in the introduction), our quasi-barotropic term is closely related to the divergence of the ageostrophic geopotential flux. Both terms diagnose the effect of the wave’s group propagation/downstream development (Chang 2000). Our baroclinic term is closely related to baroclinic energy conversion. In a moist atmosphere, however, the baroclinic conversion term contains the impacts of diabatic heating (Gutowski et al. 1992), which are analyzed separately in our PV framework. On the other hand, our divergent term not only contains the secondary circulation associated with diabatic heating but also contains the secondary circulation associated with the (adiabatic) baroclinic coupling of the wave. A (small) part of the baroclinic interaction is therefore contained in our divergent term. As there is some evidence that the diabatic secondary circulations clearly dominate (see section 2c), it seems reasonable to associate the divergent term in our framework mostly with diabatic processes.
Divergent flow due to the secondary circulations associated with baroclinic conversion and, in particular, diabatic processes contributes to the ageostrophic geopotential flux term in the EKE framework. A drawback of the EKE framework is that the impact of both types of secondary circulation is interpreted as part of the wave’s group propagation. Notwithstanding other caveats, one strength of our PV–θ framework is that the impact of the divergent flow (and diabatic PV modification) is diagnosed separately.
Barotropic energy conversion, which is closely related to barotropic deformation, has no equivalent in our framework. As discussed above, this is a consequence of considering the integrated amplitude of PV anomalies. Barotropic conversion may play a notable role, especially during the initiation of wave packets and during wave breaking. Neglecting this process is certainly a caveat of our diagnostic. As the focus of our current study is on the propagation phase of RWPs, however, the elimination of barotropic deformation in our integrated PV metric does not appear to be a major drawback. The extension of our diagnostic in the future will attempt to incorporate a metric for the deformation of PV anomalies.
4. Illustrative case study of wave-pattern evolution
On 28 October 2008, a Rossby wave pattern can be observed that spans most of the Northern Hemisphere, extending from the central North Pacific Ocean into eastern Europe (Figs. 1, 8c). As part of this wave pattern, a high-amplitude ridge–trough couplet developed over the eastern North Atlantic Ocean and western Europe (illustrated on 31 October in Fig. 8e). Associated with this couplet, which was predicted poorly in ECMWF medium-range forecasts (P. Giannakaki 2014, personal communication), heavy precipitation occurred over England. One motivation to consider this specific Rossby wave pattern is the characterization of the dynamics underlying a synoptic-scale weather pattern with low (practical) predictability.
Evolution of an RWP in terms of PV anomalies (shaded, on the 325-K isentrope) from 0000 UTC 25 Oct to 0000 UTC 31 Oct 2008, every 36 h. Dark-gray contours depict the 2-, 3-, and 4-PVU contours (increasing from south to north) of the background PV (in this case a 30-day temporal average from 13 Oct to 11 Nov). Light-gray contours depict the geopotential at 850 hPa (every 5 gpdam). Labels highlight the PV anomalies that constitute the wave pattern under consideration.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Evolution of an RWP in terms of PV anomalies (shaded, on the 325-K isentrope) from 0000 UTC 25 Oct to 0000 UTC 31 Oct 2008, every 36 h. Dark-gray contours depict the 2-, 3-, and 4-PVU contours (increasing from south to north) of the background PV (in this case a 30-day temporal average from 13 Oct to 11 Nov). Light-gray contours depict the geopotential at 850 hPa (every 5 gpdam). Labels highlight the PV anomalies that constitute the wave pattern under consideration.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Evolution of an RWP in terms of PV anomalies (shaded, on the 325-K isentrope) from 0000 UTC 25 Oct to 0000 UTC 31 Oct 2008, every 36 h. Dark-gray contours depict the 2-, 3-, and 4-PVU contours (increasing from south to north) of the background PV (in this case a 30-day temporal average from 13 Oct to 11 Nov). Light-gray contours depict the geopotential at 850 hPa (every 5 gpdam). Labels highlight the PV anomalies that constitute the wave pattern under consideration.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
a. Overview of evolution
The starting point of our considerations is the distribution of PV anomalies at the time at which the wave pattern appears most prominently (0000 UTC 28 October; Fig. 8c). The wave pattern can be identified as the succession of PV anomalies with alternating signs in the vicinity of the background PV gradient, extending from the ridge near the date line (labeled as R1) to the ridge over eastern Europe (R7). The disorganized anomaly pattern upstream of R1 and downstream of R7 clearly contrasts with this wavelike pattern. Note that the background state exhibits a (zonal) wavenumber-1 pattern with an approximate (meridional) amplitude of 10°, with a wavenumber-5 pattern with varying amplitude superimposed. The magnitude of the background PV gradient exhibits some zonal variability also.
The evolution of the wave pattern is illustrated by a sequence of the PV anomalies (Fig. 8). In tracing the PV anomalies back in time, the large-amplitude ridge R1 on 0000 UTC 25 October can be identified as the apparent origin of a near-hemispheric RWP (Fig. 8a). Upstream of this ridge, we could not identify a coherent positive PV anomaly associated with a potential upstream trough. At this time, a downstream trough (T2) and a further nascent downstream ridge (R3) are evident. Farther downstream, the apparent remnants of a previous RWP can be found (approximately from −90° to 30°).
The flow in this downstream region has reorganized into a wavelike pattern 36 h later (Fig. 8b; 1200 UTC 26 October). Identifying the leading edge of an RWP as the trough near −30° (T6) at this time and, 36 h earlier, as the nascent ridge R3, an intrinsic group velocity of approximately 75 m s−1 is implied, which seems to be excessively fast. In fact, the analysis below will further suggest that the wave pattern visible at this time is not a single RWP but that cyclogenesis associated with trough T4 might excite a second RWP. Three days later, the trailing edge of the RWP (R1 and T2) has disintegrated (Fig. 8d). The trough T4 starts to break cyclonically while in the region farther downstream the wave pattern still amplifies. Near the end of the time period under consideration, the high-amplitude ridge–trough couplet over the eastern North Atlantic and western Europe (R5 and T6; Fig. 8e) is evident. The downstream ridge R7 marks the end of the RWP as a further coherent dispersion into the downstream region cannot be discerned. As an interesting aside, we note that, at this time, a large part of the Northern Hemisphere is void of a distinct Rossby wave pattern (from 100° to −60°).
b. Illustration of PV tendencies associated with individual processes
Before diagnosing the dynamics of the Rossby wave pattern, an illustration of the horizontal pattern of individual contributions to the tendency of the integrated amplitude [Eq. (4)] is provided. The magnitude of the PV tendencies varies on different isentropic levels. For illustrative purposes, the tendencies are shown on the respective θ level at which their signal is most prominent and at the time of the largest extent of the wave pattern (at 0000 UTC 28 October; cf. Fig. 8c).
The quasi-barotropic advective tendencies (Fig. 9a) are mostly aligned along the band of the strong gradient of the background PV. The tendencies clearly exhibit a wavelike pattern with maximum absolute magnitude along the flanks of the troughs and ridges. Such a pattern can be expected, recalling that here only the advection of background PV contributes to the advective tendencies. The tendency pattern implies a westward shift of the wave pattern and thus reflects the (linear) intrinsic phase propagation of the Rossby wave. For mere propagation, the positive and negative tendencies cancel when integrated over an individual trough or ridge. Near the leading (and trailing) edge of the RWP, this is clearly not the case. At the leading edge, for example, between −30° and 30°, positive tendencies prevail within the trough and negative tendencies prevail within the developing ridge (T6 and R7, respectively; cf. Fig. 8). These tendencies amplify the respective integrated amplitudes. The amplification of PV anomalies at the leading edge and the decay of PV anomalies at the trailing edge are a signature of the group propagation of the RWP.
Illustration of the individual contributions to the PV tendency (10−5 PVU s−1) (shaded, at 0000 UTC 28 Oct, on the θ level at which the respective signal is most prominent): advection of PV background by (a) the quasi-barotropic flow (330 K), (b) the baroclinic flow (320 K), and (c) the divergent flow (330 K); (d) the divergence term [PV′(∇ ⋅ v); 330 K]; the direct diabatic modification due to (e) latent-heat release and (f) longwave radiation (both on the 325-K isentrope). The black contour denotes the 2-PVU isoline. The dark-gray contours in (a)–(c) denote the background PV (2, 3, and 4 PVU, increasing from south to north). The light-gray contours in (c)–(f) denote 850-hPa geopotential (every 5 gpdam). The arrows in (a)–(d) denote the respective flow, with values smaller than 3 m s−1 omitted for visual clarity. Note the different scales of the color bars.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Illustration of the individual contributions to the PV tendency (10−5 PVU s−1) (shaded, at 0000 UTC 28 Oct, on the θ level at which the respective signal is most prominent): advection of PV background by (a) the quasi-barotropic flow (330 K), (b) the baroclinic flow (320 K), and (c) the divergent flow (330 K); (d) the divergence term [PV′(∇ ⋅ v); 330 K]; the direct diabatic modification due to (e) latent-heat release and (f) longwave radiation (both on the 325-K isentrope). The black contour denotes the 2-PVU isoline. The dark-gray contours in (a)–(c) denote the background PV (2, 3, and 4 PVU, increasing from south to north). The light-gray contours in (c)–(f) denote 850-hPa geopotential (every 5 gpdam). The arrows in (a)–(d) denote the respective flow, with values smaller than 3 m s−1 omitted for visual clarity. Note the different scales of the color bars.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Illustration of the individual contributions to the PV tendency (10−5 PVU s−1) (shaded, at 0000 UTC 28 Oct, on the θ level at which the respective signal is most prominent): advection of PV background by (a) the quasi-barotropic flow (330 K), (b) the baroclinic flow (320 K), and (c) the divergent flow (330 K); (d) the divergence term [PV′(∇ ⋅ v); 330 K]; the direct diabatic modification due to (e) latent-heat release and (f) longwave radiation (both on the 325-K isentrope). The black contour denotes the 2-PVU isoline. The dark-gray contours in (a)–(c) denote the background PV (2, 3, and 4 PVU, increasing from south to north). The light-gray contours in (c)–(f) denote 850-hPa geopotential (every 5 gpdam). The arrows in (a)–(d) denote the respective flow, with values smaller than 3 m s−1 omitted for visual clarity. Note the different scales of the color bars.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
The baroclinic advective tendencies exhibit also a wavelike pattern aligned along the band of the strong gradient of background PV (Fig. 9b). An alternating pattern of cyclonic and anticyclonic circulations is indicated in the associated wind field. This wind field is dominated by the distribution of θ anomalies on the lower boundary (not shown), suggesting that the low-level θ “wave” leads the upper-level Rossby wave, a phase shift that is favorable for baroclinic growth (cf. section 2). This baroclinic amplification is reflected by the negative tendencies prevailing within the ridges and the positive tendencies prevailing within the troughs between 180° and −30°. Note that the baroclinic tendencies are an order of magnitude smaller than the quasi-barotropic tendencies. Because of the prominent phase shift, however, baroclinic tendencies may contribute significantly to the integrated amplitude tendency of individual troughs and ridges.
The tendency associated with the divergent flow comprises two contributions: advection of the background PV and divergence within the PV anomaly [cf. Eq. (4)]. In contrast to the quasi-barotropic and baroclinic tendencies, the divergent tendencies do not exhibit a wavelike pattern (not shown) and are here illustrated within one approximate wavelength only (Figs. 9c,d). Upper-level divergent flow is most prominently found within ridges and is, in general, associated with an upstream surface cyclone. The tendency due to advection of background PV is, in general, relatively small and in this example is positive within the ridges [i.e., reducing ridge amplitude (Fig. 9c)]. The tendencies due to divergence within PV anomalies exhibit larger absolute values (Fig. 9d). Within the example ridge, the negative tendencies clearly dominate, implying ridge amplification. In general, the integrated tendencies due to the divergent flow tend to be dominated by the divergence term.
The diabatic PV tendencies due to latent-heat release and radiative heating are illustrated in Figs. 9e and 9f, respectively, for the same region as the divergent tendencies. The PV tendencies associated with latent-heat release are rather localized. One distinct minimum is found just to the east of the surface cyclone, and an elongated region of positive tendencies is found to the south (Fig. 9e). The minimum can be associated with cloud processes along the cyclone’s warm front, and the positive tendencies are associated with a warm conveyor belt. The depicted isentropic level (325 K) is mostly below the maximum of diabatic heating in the region of the warm conveyor belt, and thus the diabatic PV tendencies are positive (not shown). The radiative PV tendencies (Fig. 9f) are dominated by positive PV tendencies that are mostly located along and within the 2-PVU contour, that is, along the tropopause and inside the stratosphere. These tendencies mostly contribute to the evolution of troughs. Within the ridge in this example, negative PV tendencies prevail. The distinct maximum is partly collocated with the minimum as a result of latent-heat release, indicating that this maximum is associated with deep clouds. In general, however, as discussed briefly in section 3b, it is very difficult to separate the role of individual weather systems from the large climatological signal of the radiative tendencies. The current study thus provides only a very general indication of the role of longwave radiative heating in the dynamics of RWPs.
c. Characterizing the dynamics of the Rossby wave pattern
The dynamics of the Rossby wave pattern is characterized by discussing time series of the individual contributions to the integrated amplitude tendency of individual troughs and ridges. The associated PV anomalies are tracked in time as long as our automated algorithm is able to identify respective anomalies as coherent features (cf. section 3d). In the current case, individual anomalies can be tracked over 5–6 days.
Time series of the integrated tendencies for the individual troughs and ridges are depicted in Fig. 10. The tendencies are multiplied with the sign of the respective anomaly such that positive tendencies always denote amplification. A first important notion is that the advective tendencies are, in general, an order of magnitude larger than the diabatic tendencies from latent-heat release. The radiative tendencies can exhibit large values because of the spatially extended climatological signal of positive PV tendencies along the tropopause. The radiative tendencies therefore tend to amplify troughs and decrease ridges. The diabatic tendencies will not be examined in much more detail below and are shown mostly for the sake of completeness.
Time series (every 6 h; the days of October 2008 are given on the abscissa) of the individual contributions to the tendency of the integrated PV anomalies (ordinate; 108 PVU m2 s−1), representing the individual troughs and ridges of the example RWP. The sign of the tendencies is defined such that positive (negative) values always indicate amplification (weakening) of the anomaly, regardless of whether it is a trough or ridge. Tendencies are calculated as a vertical average over the 320-, 323-, 325-, 327-, and 330-K isentropic levels. The color coding is given in (a). The label for the respective PV anomaly is given in each individual panel (cf. Fig. 8). Tendencies are given for the time period in which the PV anomaly can be identified automatically as a coherent feature. At 1800 UTC 30 Oct and 0000 UTC 31 Oct, the piecewise PV inversion did not converge. Respective values for quasi-barotropic propagation and baroclinic interaction are omitted. The shading next to the quasi-barotropic and baroclinic tendencies indicates the uncertainty of the result due to the nonlinearity of piecewise PV inversion. Specifically, the shading indicates the difference in the advective tendency due to the difference between the nondivergent wind and the sum of the quasi-barotropic and the baroclinic wind. This uncertainty does not compromise the physical interpretation of our results.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Time series (every 6 h; the days of October 2008 are given on the abscissa) of the individual contributions to the tendency of the integrated PV anomalies (ordinate; 108 PVU m2 s−1), representing the individual troughs and ridges of the example RWP. The sign of the tendencies is defined such that positive (negative) values always indicate amplification (weakening) of the anomaly, regardless of whether it is a trough or ridge. Tendencies are calculated as a vertical average over the 320-, 323-, 325-, 327-, and 330-K isentropic levels. The color coding is given in (a). The label for the respective PV anomaly is given in each individual panel (cf. Fig. 8). Tendencies are given for the time period in which the PV anomaly can be identified automatically as a coherent feature. At 1800 UTC 30 Oct and 0000 UTC 31 Oct, the piecewise PV inversion did not converge. Respective values for quasi-barotropic propagation and baroclinic interaction are omitted. The shading next to the quasi-barotropic and baroclinic tendencies indicates the uncertainty of the result due to the nonlinearity of piecewise PV inversion. Specifically, the shading indicates the difference in the advective tendency due to the difference between the nondivergent wind and the sum of the quasi-barotropic and the baroclinic wind. This uncertainty does not compromise the physical interpretation of our results.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
Time series (every 6 h; the days of October 2008 are given on the abscissa) of the individual contributions to the tendency of the integrated PV anomalies (ordinate; 108 PVU m2 s−1), representing the individual troughs and ridges of the example RWP. The sign of the tendencies is defined such that positive (negative) values always indicate amplification (weakening) of the anomaly, regardless of whether it is a trough or ridge. Tendencies are calculated as a vertical average over the 320-, 323-, 325-, 327-, and 330-K isentropic levels. The color coding is given in (a). The label for the respective PV anomaly is given in each individual panel (cf. Fig. 8). Tendencies are given for the time period in which the PV anomaly can be identified automatically as a coherent feature. At 1800 UTC 30 Oct and 0000 UTC 31 Oct, the piecewise PV inversion did not converge. Respective values for quasi-barotropic propagation and baroclinic interaction are omitted. The shading next to the quasi-barotropic and baroclinic tendencies indicates the uncertainty of the result due to the nonlinearity of piecewise PV inversion. Specifically, the shading indicates the difference in the advective tendency due to the difference between the nondivergent wind and the sum of the quasi-barotropic and the baroclinic wind. This uncertainty does not compromise the physical interpretation of our results.
Citation: Journal of the Atmospheric Sciences 73, 3; 10.1175/JAS-D-15-0162.1
The boundary term is usually small but may exhibit values comparable to the advective tendencies at individual times. The large values may occur for two different reasons: 1) Wave breaking or a large deformation of the anomaly occurs up- or downstream of the considered anomaly, which may lead to rapid changes of the axis of the associated anomaly and thus to large values of vs (see section 3d). 2) The boundary of an anomaly cuts through a region where the absolute value of PV′ is large. If this occurs, it usually occurs with a negative anomaly in the polar region. We did not find an indication that the processes at the boundary make significant contribution to the evolution of the RWP in the region of the large background PV gradient.
For the building of the ridge R1 (Fig. 10a), baroclinic growth and the divergent flow make the most prominent contributions. In contrast, the quasi-barotropic tendency consistently shows large, negative values, reflecting our earlier notion that this ridge is not initiated by downstream development from an upstream precursor. Consistent ridge building by baroclinic growth, amplified by divergent outflow, is rather indicative that this ridge constitutes the origin from which an RWP disperses downstream. Ridge building is further supported by a small but consistently positive contribution from latent-heat release. The large value of the boundary term at 1800 UTC 25 October is due to a large change in the location of the identified trough axis upstream of R1.
The trough T2 evidently amplifies as a result of downstream development: the quasi-barotropic tendency makes the only positive (advective) contribution (from 25 to 28 October). The baroclinic tendency is small and mostly negative throughout the evolution, clearly showing that baroclinic growth does not play a role with this trough but rather hinders the trough amplification. The divergent tendency is negative throughout in the development, canceling the amplification due to downstream dispersion of the wave packet to a large extent. The divergent flow is associated with a developing cyclone ahead of the trough, and inspection of the advection pattern reveals that the divergent flow hinders the eastward propagation of the trough (not shown), leading to thinning and decay of the trough (cf. Figs. 9b–d). In this sense, the trough evolution here is similar to the cases discussed in Pantillon et al. (2013) and Riemer and Jones (2014).
The evolution of the ridge R3 follows the concept of downstream baroclinic development. From 25 to 27 October, the quasi-barotropic tendency is initially positive. The amplifying ridge then starts coupling with the low-level θ anomalies (not shown), and baroclinic growth plays an important role beginning on 27 October. From 28 October onward, the quasi-barotropic tendency is negative, indicating that more wave energy is now dispersed downstream than is gained from upstream. The divergent flow supports ridge amplification, in particular on 25 and 26 October, with a less consistent impact afterward. Overall, the divergent amplification is less pronounced than for ridge R1.
Similar to ridge R1, the trough T4 is not amplified by downstream development, as evidenced by the negative quasi-barotropic tendencies throughout the evolution. The negative values imply that the PV anomaly associated with the downstream ridge (R5), which is a remnant of a preexisting RWP, is consistently of larger amplitude than the PV anomaly associated with the upstream ridge (R3). Baroclinic growth, in contrast, is large on 25–27 October and, together with minor contributions from the divergent tendencies, dominates over the quasi-barotropic tendencies early during the development (0000 UTC 25 October–0000 UTC 26 October). In this sense, it can be inferred that the baroclinic development associated with this trough triggers the dispersion of a second RWP. Within the EKE framework, a similar interpretation of an apparently continuous wave packet consisting of two single wave packets has been given by Chang (2000).
Ridge R5 exhibits the same qualitative evolution as R3—that is, downstream baroclinic development, but the overall amplification of this ridge is about twice as strong. Both baroclinic growth and the divergent flow make prominent contributions to the development of this high-amplitude ridge, in particular from 28 October onward. The identification of the anomaly becomes questionable from 1200 UTC 30 October onward, and thus the physical interpretation of the tendencies becomes questionable also.
Downstream of this ridge, quasi-barotropic downstream development dominates the evolution of wave amplitude. The trough T6 starts growing as a result of this process on 26 October, supported by a minor amplification due to baroclinic growth until 1200 UTC 29 October and an indication of a small negative baroclinic impact in the following 2 days. The variable, mostly small-amplitude divergent tendencies tend to hinder trough amplification.
The ridge R7 starts amplifying 36 h after the upstream trough. Baroclinic interaction consistently opposes ridge building but is clearly dominated by the large quasi-barotropic tendencies. Although the direct diabatic modification due to latent-heat release consistently exhibits positive values, this ridge is the only ridge in the considered wave pattern that does not exhibit a prominent amplification due to the divergent flow. Arguably, this is because an upstream surface cyclone, with an associated warm conveyor belt, does not develop until late in the development (cf. Fig. 8e). The boundary value peaks strongly on 1200 UTC 30 October, again because of a large change in the location of the identified trough axis upstream of R7.
5. Summary and discussion
A quantitative diagnostic has been developed to investigate the evolution of real-atmospheric, finite-amplitude Rossby wave packets propagating along the midlatitude jet. The diagnostic is based on the PV–θ perspective of the atmosphere. RWPs are identified as PV anomalies on isentropic levels intersecting the midlatitude tropopause. Identifying the wave in terms of anomalies requires the careful definition of a background state, for which a unique definition is lacking. We required the background state to be stationary and representative for a specific RWP. We found that a 30-day temporal mean, centered on the time period of interest, fulfilled these requirements to a good approximation.
In our framework, the evolution of the RWP is characterized by the evolution of the PV anomalies associated with individual troughs and ridges. Individual contributions to the tendency of these anomalies are identified by partitioning the terms in the PV-tendency equation. The advective tendencies are partitioned into contributions due to the divergent wind and due to the wind associated with low-level and upper-level PV anomalies, respectively. Advective tendencies associated with the upper-level PV anomalies are interpreted as the advection of the wave pattern by itself, that is, intrinsic wave propagation. This contribution is here referred to as quasi-barotropic propagation. Advective tendencies associated with the low-level PV anomalies are interpreted as baroclinic coupling and are here referred to as baroclinic interaction. The interpretation of these terms follows Nielsen-Gammon and Lefevre (1996). These terms are similar also to the ageostrophic geopotential flux and baroclinic conversion terms, respectively, in the eddy kinetic energy framework (Nielsen-Gammon and Lefevre 1996; Chang 2000).
Differences and commonalities between our PV–θ framework and the EKE and Nielsen-Gammon and Lefevre’s geopotential-height-tendency frameworks have been discussed in some detail (section 3f). In comparison with these frameworks, the physically novel aspect of our diagnostic is that the important impact of the divergent wind and the direct diabatic PV modification are diagnosed explicitly. As further discussed below, a caveat of our diagnostic is that the deformation of PV anomalies is not diagnosed.
The diabatic PV modification is evaluated using the heating rates due to latent-heat release and due to longwave radiation available from the unique YOTC data. Several recent studies have investigated the impact of diabatic processes on Rossby waves in some detail (e.g., Grams et al. 2011; Chagnon et al. 2013). Our diagnostic framework lacks a similar level of detail but has the important advantage of making easily accessible the relative importance of individual processes to the overall evolution.
Our goal is to diagnose the dynamics over the life cycle of an RWP, that is, on a time scale of 5–10 days. For this purpose, the examination of a series of individual PV-tendency maps is too laborious. Instead, we consider the tendency of the integrated amplitude of individual troughs and ridges. The associated spatial integration highly compresses the dynamical information but no longer captures every aspect of the evolution. In linear theory, wave amplitude is related to wave energy and the propagation of wave energy is given by the group velocity. In this sense, the amplitude evolution of individual PV anomalies of a real-atmospheric, finite-amplitude wave packet can be associated with the concept of group propagation.
An important caveat that arises from our spatial integration of PV tendencies is the elimination of information about the deformation of the PV anomaly. This deformation is related to barotropic energy conversion in the EKE framework and is diagnosed explicitly also in the framework of Nielsen-Gammon and Lefevre (1996). Barotropic energy conversion may contribute notably to the evolution of RWPs, in particular during the initiation and decay of individual disturbances (e.g., Chang 2000). However, as the focus of our current study is on the group propagation of RWPs, the elimination of barotropic deformation in our integrated PV metric does not appear to be a major drawback. The extension of our diagnostic in the future will attempt to incorporate a metric for the deformation of the PV anomalies.
The diagnostic is illustrated by the case study of a Rossby wave pattern spanning a large part of the Northern Hemisphere in October 2008. Not surprisingly, quasi-barotropic propagation makes a major contribution to the evolution of all considered ridges and troughs. Baroclinic interaction and, important, the divergent flow make, in general, contributions to the amplitude evolution that are comparable in magnitude to wave propagation. Consistent with the results of Chang (2000), we find that the relative role of the different processes exhibits considerable variability among individual ridges and troughs. The divergent flow diagnosed in the current study has, in general, a more prominent impact on the ridges than on the troughs, mostly supporting ridge building.
The spatially integrated diabatic PV tendencies are an order of magnitude smaller than the advective tendencies. Therefore, the direct role of these tendencies for the evolution of the wave pattern is clearly subordinate. This is not to say that diabatic processes per se are not of importance. First, we emphasize that the current study considers instantaneous PV tendencies. This is an approach that is different from other current studies, in which the integrated impact of diabatic processes on the PV distribution is investigated from a Lagrangian perspective (e.g., Chagnon et al. 2013; Chagnon and Gray 2015). Diabatically generated PV anomalies usually exhibit a nondivergent circulation that may contribute to the advective tendencies. In our diagnostic, after the diabatic generation of PV anomalies, the further impact of these anomalies is contained in the quasi-barotropic and baroclinic terms. Second, and very important, prominent upper-tropospheric divergent outflow is most often associated with latent heat below. We argue that the most prominent impact of latent-heat release on the Rossby wave pattern is communicated by such divergent outflow. In this sense, we consider the tendencies due to the divergent flow to be a prominent indirect diabatic impact (cf. section 2c and section 3f).
One motivation for the presented case study was that the high-amplitude ridge–trough couplet over Europe, which was associated with a severe-weather event, was poorly predicted in medium-range ECMWF forecasts. In this regard, our diagnostic revealed two relevant results: First, the ridge–trough couplet is not part of a long-lasting RWP propagating from the central North Pacific into Europe. Such a faulty conclusion could be drawn from a cursory inspection of PV maps (e.g., Figure 1) alone. Instead, a second RWP is triggered by cyclogenesis over eastern North America. Grazzini and Vitart (2015) very recently found, for the European region that “higher than average medium-range forecast skill scores are often associated with the presence of long-lasting RWPs…On the contrary, bad medium-range forecast skill scores tend to be associated with shorter RWPs coming from the central USA or the west Atlantic.” Our result thus appears to be consistent with this notion. It is not clear, however, if the RWP identification method used by Grazzini and Vitart identified one long-lasting RWP or two separate RWPs in our specific case.
Second, ridge building over the Atlantic is diagnosed to be strongly amplified by baroclinic growth and divergent outflow. In fact, the tendencies due to the divergent flow exhibit the largest values for this ridge, with maximum (instantaneous) tendencies that rival the maximum (instantaneous) quasi-barotropic tendencies of all considered ridges and troughs. Clearly, the divergent flow plays a leading-order role for the evolution of this ridge, and downstream dispersion from the ridge is the most important process for the amplification of the downstream trough. Because of the strong link of the divergent flow with latent-heat release below, it is plausible that the formation of the ridge–trough couplet was a challenge to predict in the medium range. A similar result was found by Piaget et al. (2015), who also traced back medium-range forecast uncertainty over Europe to the development of a cyclone and an associated warm conveyor belt in the western Atlantic.
Third, we hypothesize a possible reason for the low bias of Rossby wave amplitude in global forecast models (Gray et al. 2014; Glatt and Wirth 2014). Gray et al. emphasize the low bias in the amplitude of ridges. Consideration of the few ridges in our case study emphasizes the important role of the divergent flow for ridge amplitude. We thus hypothesize that, notwithstanding other uncertainties in the representation of moist processes, the convective parameterization schemes in global models need to produce a realistic magnitude of upper-tropospheric divergence to remove the observed forecast bias.
Acknowledgments
This study has been performed as part of the first author’s doctoral thesis, which is funded by the German Research Foundation (DFG) as part of the research unit PANDOWAE (FOR896). The thoughtful comments of two anonymous reviewers helped to improve the presentation of our results considerably. The authors are grateful to Volkmar Wirth for comments on an earlier version of this paper; to Sue Gray, John Methven, and Mark Rodwell for fruitful discussions; and to Christian Grams for assistance with adapting the PV inversion code to ECMWF data.
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