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  • View in gallery

    Overview of ωbal (shaded, 10−3 Pa s−1, negative values denote ascent) in the idealized ET scenario at (a) 132, (b) 144, (c) 156, and (d) 168 h into the experiment. The black contours depict surface pressure for 995 hPa and lower, every 5 hPa. The dashed contours show the streamfunction in the region of the jet stream at 270 hPa, every 107 m2 s−1. The hatching indicates the jet streak by highlighting wind speeds > 40 m s−1 at 270 hPa.

  • View in gallery

    Comparison of ωbal (contours) and the full vertical motion field (shaded, 10−3 Pa s−1) in the numerical experiment at 156 h. The values for the contours are the same as the color shading with dashed (dotted) contours showing negative (positive) values. For reference, surface pressure is shown every 5 hPa below 995 hPa (solid line).

  • View in gallery

    As in Fig. 1, but for the individual contributions to vertical motion in the idealized ET scenario: (a),(b) curvature term, (c),(d) stretching term, and (e),(f) shear-advection term at (left) 144 and (right) 168 h.

  • View in gallery

    The vertically averaged vorticity spinup [from Eq. (15), shaded, 10−10 s−2] due to (a) the total balanced ascent, (b) the curvature term, and (c) the sum of the stretching and shear-advection terms at 156 h. The contours depict the Lagrangian pressure tendency (with a contour interval of 10−3 Pa s−1) and the thick contour depicts the 60% threshold of the maximum pressure fall (see text for details) with an absolute value of −6.25 × 10−3 Pa s−1.

  • View in gallery

    Relative contribution by the curvature term to the vertically and spatially averaged vorticity spinup (see text for details) during the time period of cyclone development.

  • View in gallery

    Illustration of the cyclone development downstream of the ET of (a) Hanna at 0000 UTC 9 Sep 2008, (b) Choi-wan at 0000 UTC 9 Sep 2009, and (c) Jangmi at 1200 UTC 1 Oct 2008. Near-surface development is illustrated by geopotential height at 950 hPa (solid contours, every 4 gpdm). The upper-level flow is illustrated by the streamfunction (dashed contours, every 107 m2 s−1) and wind speeds above a threshold value (hatched) at 250 hPa. The threshold value is 45 m s−1 for Hanna, 50 m s−1 for Choi-wan, and 60 m s−1 for Jangmi. In addition, for Jangmi, wind speeds > 70 m s−1 are hatched in light gray. Color shading depicts the balanced vertical motion (10−3 Pa s−1) at 500 hPa. The arrows point to the respective downstream cyclone and “ET” marks the ET system.

  • View in gallery

    As in Fig. 6a, but for vertical motion associated with (a) the stretching term, (b),(d) the curvature term, and (c) the sum of the stretching and shear-advection terms for the case of Hanna at (top) 1200 UTC 8 Sep and (bottom) 1200 UTC 9 Sep.

  • View in gallery

    As in Fig. 6b, but for vertical motion associated with (a) the curvature term and (b) the sum of the stretching and shear-advection terms for the case of Choi-wan at 0000 UTC 22 Sep. The asterisk marks the position of the surface center of the downstream cyclone 12 h later. The pattern shown is representative for the evolution of the downstream cyclone from 0000 UTC 21 Sep to 1200 UTC 23 Sep.

  • View in gallery

    As in Fig. 6c, but for vertical motion associated with (a) the curvature term and (b) the stretching term for the case of Jangmi at 0000 UTC 3 Oct. In (a), the streamfunction is shown at 500 hPa and the black curved line indicates the axis of a shortwave trough. The downstream cyclone is approximately in the center of the domain.

  • View in gallery

    As in Fig. 6c, but for vertical motion associated with (a) the stretching term at 1200 UTC 1 Oct, (b) the shear-advection term at 0000 UTC 3 Oct, and (c) again the stretching term at 1200 UTC 4 Oct for the case of Jangmi. The downstream cyclone is approximately in the center of the domain.

  • View in gallery

    Vertical motion associated with latent heat release (shaded) as seen by the YOTC data (see text for details) and balanced ascent with values <−0.12 hPa s−1 (hatched) at (a) 1200 UTC 2 Oct and (b) 0000 UTC 3 Oct. Strong ascent in the full vertical motion field as seen by ECMWF data is illustrated by the −0.44 hPa s−1 isoline (thick contour). The displacement between maxima in this full ascent and maxima in ascent associated with latent heat release is likely due to the fact that heating terms averaged over the subsequent 3-h period are employed.

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Cyclogenesis Downstream of Extratropical Transition Analyzed by Q-Vector Partitioning Based on Flow Geometry

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  • 1 Institut für Physik der Atmosphäre, Johannes Gutenberg-Universität Mainz, Mainz, Germany
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Abstract

During extratropical transition (ET), tropical cyclones exert a significant impact on the midlatitude circulation. Archetypical features of this impact are jet streak formation, amplification of the downstream trough, and modification of the associated downstream cyclogenesis. This study investigates the relative importance of the jet streak and the upper-level trough for cyclone development by quantifying the respective contributions to midtropospheric vertical motion using the Q-vector partitioning by J. C. Jusem and R. Atlas. Their framework is here extended from quasigeostrophic theory to alternative balance. The Q vector under alternative balance involves the nondivergent wind, instead of the geostrophic wind, and therefore represents more accurately the balanced dynamics associated with vertical motion, in particular downstream of ET where the flow often exhibits significant curvature associated with the amplified trough.

An idealized ET scenario and three real cases, the cyclones downstream of Hanna (2008), Choi-wan (2008), and Jangmi (2009), are analyzed. In all cases, the trough plays a prominent role in cyclone development. The jet streak plays a prominent, favorable role in the idealized ET scenario and downstream of Hanna. In contrast, the role of the jet streak downstream of Choi-wan is clearly of secondary importance. Interestingly, downstream of Jangmi the jet streak has a prominent but detrimental impact. It is concluded that amplified jet streaks associated with ET have the potential to be of significant importance for downstream cyclone development. The few cases considered in this study, however, point to a large case-to-case variability of the role of the jet streak.

Current affiliation: Laboratory for Climatology and Remote Sensing, Philips Universität Marburg, Marburg, Germany.

Corresponding author address: Michael Riemer, Institut für Physik der Atmosphäre, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany. E-mail: mriemer@uni-mainz.de

This article is included in the Predictability and Dynamics of Weather Systems in the Atlantic-European Sector (PANDOWAE) Special Collection.

Abstract

During extratropical transition (ET), tropical cyclones exert a significant impact on the midlatitude circulation. Archetypical features of this impact are jet streak formation, amplification of the downstream trough, and modification of the associated downstream cyclogenesis. This study investigates the relative importance of the jet streak and the upper-level trough for cyclone development by quantifying the respective contributions to midtropospheric vertical motion using the Q-vector partitioning by J. C. Jusem and R. Atlas. Their framework is here extended from quasigeostrophic theory to alternative balance. The Q vector under alternative balance involves the nondivergent wind, instead of the geostrophic wind, and therefore represents more accurately the balanced dynamics associated with vertical motion, in particular downstream of ET where the flow often exhibits significant curvature associated with the amplified trough.

An idealized ET scenario and three real cases, the cyclones downstream of Hanna (2008), Choi-wan (2008), and Jangmi (2009), are analyzed. In all cases, the trough plays a prominent role in cyclone development. The jet streak plays a prominent, favorable role in the idealized ET scenario and downstream of Hanna. In contrast, the role of the jet streak downstream of Choi-wan is clearly of secondary importance. Interestingly, downstream of Jangmi the jet streak has a prominent but detrimental impact. It is concluded that amplified jet streaks associated with ET have the potential to be of significant importance for downstream cyclone development. The few cases considered in this study, however, point to a large case-to-case variability of the role of the jet streak.

Current affiliation: Laboratory for Climatology and Remote Sensing, Philips Universität Marburg, Marburg, Germany.

Corresponding author address: Michael Riemer, Institut für Physik der Atmosphäre, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany. E-mail: mriemer@uni-mainz.de

This article is included in the Predictability and Dynamics of Weather Systems in the Atlantic-European Sector (PANDOWAE) Special Collection.

1. Cyclogenesis downstream of extratropical transition

Recurving tropical cyclones, in particular those that undergo extratropical transition (ET; Jones et al. 2003), exert a profound impact on the midlatitude circulation. Several recent studies have investigated this impact in some detail and documented a distinct modification of the upper-level jet stream (e.g., Riemer et al. 2008; Riemer and Jones 2010, 2014; Archambault et al. 2013; Grams et al. 2011, 2013a; Pantillon et al. 2013). One signature impact of ET on the midlatitude, upper-level flow is the formation of a jet streak and the amplification of the downstream trough. Both features are well known to support rapid cyclogenesis (Uccellini 1990). Rapid cyclogenesis downstream of ET is a ubiquitous feature in idealized ET scenarios (Riemer et al. 2008; Riemer and Jones 2010) and has been observed in the real atmosphere also (Hoskins and Berrisford 1988).

Severe weather and the associated potential for a high societal impact are one motivation to examine cyclogenesis downstream of ET in more detail. Further motivation stems from the important role of the downstream cyclone in the midlatitude dispersion of the impact of ET. This larger-scale impact exhibits high variability, may reach near-hemispheric scale, and is often associated with enhanced uncertainty in medium-range forecasts (Harr et al. 2008; Anwender et al. 2008). The dispersion of the downstream impact is governed by the modification of midlatitude, baroclinic Rossby wave trains (e.g., Harr and Dea 2009; Riemer and Jones 2010; Pantillon et al. 2013; Grams et al. 2013b; Archambault et al. 2013). In an idealized ET scenario, Riemer and Jones (2010) identified the modification of the downstream cyclogenesis as one important first step in the chain of events governing the downstream dispersion. Their result is consistent with the concept of downstream baroclinic development (Orlanski and Sheldon 1995).

In this study, we focus on one specific aspect of cyclogenesis downstream of ET, namely the relative importance of the jet streak as compared to that of the upper-level trough. Forecasters often discuss jet streaks and troughs as distinct features. In an idealized baroclinic wave, jet streak formation can be viewed as an intrinsic part of the wave’s amplification (Rotunno et al. 1994; Wandishin et al. 2000). During ET, however, there is additional “external” forcing of the jet streak, largely due to the outflow of the recurving tropical cyclone (e.g., Riemer and Jones 2010, section 5.2.2). It is therefore plausible to hypothesize that jet streaks downstream of ET play a particularly prominent role for cyclogenesis. The purpose of the current study is to test this hypothesis.

To quantify the contributions of the upper-level trough and the jet streak to downstream cyclogenesis, the vertical motion attributable to these features will be analyzed. Section 2 recalls the relationship between vertical motion and cyclogenesis, the use of an ω equation to diagnose vertical motion, and the concept of partitioning the dynamical forcing term, the Q vector, to attribute vertical motion to individual processes. As discussed further below, the Q-vector partitioning introduced by Jusem and Atlas (1998) provides a suitable framework to identify the individual contributions from upper-level troughs and jet streaks, respectively. Previously, this framework has been applied to analyze contributions to subsidence during the formation of a deep tropopause fold (Donnadille et al. 2001).

The Jusem and Atlas partitioning has been developed for the quasigeostrophic (QG) ω equation. Section 3 will detail the straight-forward extension of their partitioning to so-called alternative balance (AB; Davies-Jones, 1991), a balance approximation that is less restrictive than QG. The AB ω equation has been used previously to study cyclogenesis by Mallet et al. (1999). The extension of the Q-vector partitioning to AB has been outlined by Jusem and Atlas (1998) at the end of their summary. In the AB ω equation, vertical motion is diagnosed from the (full) nondivergent horizontal wind, as compared to the geostrophic wind in QG. The geostrophic wind systematically overestimates (underestimates) the wind speed for cyclonic (anticyclonic) curvature because geostrophic balance neglects centrifugal forces. This implies an overestimation of cyclonic curvature in the base of the trough and an overestimation (underestimation) of cyclonically (anticyclonically) curved jet streaks and their associated shear and stretching patterns. The nondivergent wind does not exhibit such systematic discrepancies. In particular downstream of ET, where the upper-level flow is often highly amplified and thus exhibits significant curvature, the AB ω equation can therefore be expected to give a more accurate description of the balanced vertical motion.

To illustrate the utility of the Jusem and Atlas partitioning in the ET context, we first consider a highly idealized scenario (section 4). In this scenario, a tropical cyclone interacts with an initially straight jet stream. As a consequence of this interaction, a pronounced jet streak and trough develop and rapid midlatitude cyclogenesis occurs downstream of ET. By design of the numerical experiment, the early development of the downstream cyclone is driven entirely by upper-level forcing. The relative importance of the trough and the jet streak for downstream cyclogenesis may thus be analyzed in isolation from other physical processes.

In section 5, cyclogenesis downstream of three ET systems in the real atmosphere [Hanna (2008), Choi-wan (2009), and Jangmi (2008)] is investigated. These ET events and their downstream impact have been studied recently by Grams et al. (2011), Grams (2011), and Grams et al. (2013b), respectively. These cases have been selected also because they occurred during the “Year of Tropical Convection” (YOTC; Waliser et al. 2012) period. For this period, diabatic tendencies from the physical parameterization schemes of the Integrated Forecast System (IFS) at the European Centre of Medium-Range Weather Forecasts (ECMWF) are available. The tendency terms are here used to provide a quantitative estimate of vertical motion attributable to latent heat release in clouds, in particular for the Jangmi case. Our focus on cloud diabatic processes is motivated by a number of previous studies that have shown the important role that these processes may play for cyclone development (e.g., Davis et al. 1993; Moore and Montgomery 2004).

The summary and our conclusions are presented in section 6.

2. Diagnosis of vertical motion for cyclogenesis

The importance of vertical motion for surface development has long been recognized (e.g., Sutcliffe 1947). Here, the following simplified framework [summarized by Hoskins (1990), section 5.3] is applied. Under geostrophic scaling, the vorticity equation on an f plane in pressure (p) coordinates reads
e1
where Dg/Dt denotes the material derivative following the geostrophic wind, ζg is the geostrophic vertical vorticity, f0 is the constant Coriolis parameter, and ω is the vertical motion. Equation (1) states that vortex-tube stretching due to upward motion increasing with height is the only process that may (materially) increase geostrophic vorticity on the f plane. Applying the usual condition ω = 0 at the lower boundary, upward motion at a midtropospheric level (600–500 hPa) indicates the spinup of low-level vorticity, and thus cyclogenesis. Note that low-level vorticity spinup not only governs the amplification of a cyclone but also its movement. According to Eq. (1), cyclones tend to move in the direction of the maximum midtropospheric ascent. This simplified framework is a reasonable approximation for the development of cyclones that are not dominated by diabatic processes.
The vertical motion that is associated with the balanced dynamics of a flow can be diagnosed by an appropriate ω equation. For quasigeostrophic theory, the Q-vector form of the ω equation (Hoskins et al. 1978) on an f plane1 in p coordinates can be written as (Sanders and Hoskins 1990)
e2
with the usual simplification of a horizontally uniform static stability , where α0 and θ0 are reference vertical profiles of specific volume and potential temperature, respectively, and p denotes the horizontal gradient on a constant-p surface. The Q vector is given by
e3
where vg denotes the geostrophic wind. This standard QG version of the ω equation has been applied to cyclogenesis downstream of ET by Riemer and Jones (2010) and Grams et al. (2013b). Hereafter, vertical motion associated with the divergence of Q (i.e., with the balanced dynamics of the flow) will be referred to as balanced vertical motion ωbal.

Usually, ωbal exhibits a smooth, coherent vertical structure with maximum absolute values in the midtroposphere. Toward the tropopause and the surface, ωbal approximately vanishes. Midtropospheric ωbal is usually dominated by dynamical processes in the upper troposphere. We have verified that these general characteristics hold in our idealized experiment analyzed in section 4 and in the real cases discussed in section 5. Therefore, focusing on midtropospheric ωbal to examine cyclone development and relating midtropospheric ωbal to upper-tropospheric features such as troughs and jet streaks is justified.

Q-vector partitioning is a commonly used tool that aims at isolating the contributions of different dynamical processes to the total vertical motion field (e.g., Hoskins et al. 1978; Keyser et al. 1992). Writing , the vertical motion ωi associated with an individual term Qi can be quantified due to the linearity of the QG ω equation. The partitioning of Q by Jusem and Atlas is based on the geometry of the flow and will be discussed in detail below.

3. Jusem and Atlas’s Q-vector partitioning for alternative balance

a. The ω equation under alternative balance

Alternative balance has been proposed by Davies-Jones (1991) as a less restrictive balance approximation than QG. In contrast to QG, however, AB is not derived by an asymptotic expansion of the primitive equations but is based on heuristic approximations. The key approximation is that the material derivative of the thermal wind imbalance can be omitted, instead of omitting the thermal wind imbalance itself, as is done in QG. To derive an ω equation for AB, two further heuristic approximations are made. First, the vertical wind shear is assumed to be in thermal wind balance. Second, the horizontal wind gradients are approximated by the horizontal gradients of the nondivergent component of the full wind field. These approximations neglect, inter alia, the interaction between the primary and the secondary circulation and nonlinear interactions of the secondary circulation. As shown by Xu (1990) and Xu (1994), these interaction terms may lead to a slantwise intensification of the secondary circulation.

The ω equation in pseudoheight (z) coordinates is in QG-like form [cf. Eq. (2)] and given by
e4
[Davies-Jones 1991, his Eq. (7.4)], with Brunt–Väisälä frequency N2fb/∂z, where b ≡ (g/0)θ serves as the thermal variable, and H denotes the horizontal gradient on a constant-z surface. Note that Eq. (4) allows for a nonuniform horizontal distribution of N2 and that w is the vertical motion in z coordinates. With the AB approximations given above
e5
where vψ denotes the nondivergent horizontal wind (Davies-Jones 1991, end of p. 507). As is the QG version, the AB ω equation is linear and thus partitioning of is a useful tool for identifying individual contributions to vertical motion.2 Using the identities given at the end of appendix B in Davies-Jones (1991), Eq. (4) can be converted into pressure coordinates3
e6
where cV and cp are the specific heat at constant volume and pressure, respectively. On the right-hand side, has been substituted by
e7
noting that
e8
with the gas constant R and κ = R/cp. In this study, Eq. (6) is solved numerically for ω using a standard successive over-relaxation scheme (Press et al. 1992, p. 860). The standard condition of ω = 0 is applied at the boundaries. Inaccuracies associated with this simplified condition decay with distance from the boundary. Our analysis of ω in the midtroposphere near the center of the inversion domain should thus be affected to minimal extent. Importantly, the focus will be on the relative importance of individual contributions to vertical motion. Considering the ratio of individual solutions of the partitioned ω equation further reduces inaccuracies introduced by the boundary conditions.

b. Jusem and Atlas’s partitioning for QAB

The Q-vector partitioning introduced by Jusem and Atlas (1998) is based on a natural coordinate system in which, locally, one axis is oriented along isolines of the geopotential (i.e., along streamlines of the geostrophic wind). This axis is called the s axis with associated unit vector t. The vertical unit vector is denoted by k. The axis normal to t is called the n axis with associated unit vector n. The direction of n is such that the triplet (t, n, and k) is right handed. It is straight forward to apply the same natural coordinate system to the streamlines of vψ, instead of the streamlines of the geostrophic wind.4 We here detail the partitioning of QAB for the sake of completeness. Our presentation follows very closely that of Jusem and Atlas (their section 2).

With the above coordinate transformation
e9
where Ks and Kn denote curvature terms [cf. Eq. (2.8) in Jusem and Atlas 1998]. The curvature of the streamlines is denoted by Ks. For counterclockwise motion Ks > 0, and Ks < 0 for clockwise motion. Thus, the curvature is cyclonic if the sign of Ks is the same as that of f0, and anticyclonic if the sign is the opposite to that of f0. The curvature of the lines orthogonal to the streamlines is denoted by Kn. This curvature term denotes diffluence (Kn > 0) and confluence (Kn < 0).
The only further prerequisite in Jusem and Atlas’s derivation is the nondivergence of the geostrophic wind. This prerequisite, of course, is met by the (full) nondivergent horizontal wind also. In natural coordinates, the continuity equation then reads
e10
that is, the change of the wind speed in the along-flow direction is compensated by confluence/diffluence. Using Eq. (10) to substitute for ∂|vψ|/∂s in the first term of Eq. (9), QAB can be partitioned into formally the same four terms as given in Jusem and Atlas’s Eqs. (2.10)–(2.13), except that here we write |vψ| instead of sg. The first term in Eq. (9),
e11
describes the change of the along-flow thermal gradient by confluence/diffluence and has been named alongstream stretching by Jusem and Atlas. The second term,
e12
describes the change of the cross-stream thermal gradient by the curvature of the flow. The third term,
e13
describes the change of the along-flow thermal gradient by the horizontal shear of the flow. Following Keyser and Pecnick (1985) this term has been named shear advection by Jusem and Atlas. The last term,
e14
describes the change of the cross-stream thermal gradient by confluence/diffluence and has been named cross-stream stretching by Jusem and Atlas.

The two stretching terms, Qalst and Qcrst, are directly coupled by the nondivergence of the horizontal flow. Because of this immanent interdependence, it is reasonable to combine both terms into a single stretching term: QstQalst + Qcrst. The results of Jusem and Atlas (1998) and Donnadille et al. (2001), as well as our own results, indicate that Qalst is, in general, small compared to Qcrst. Hence, Qst is dominated by the cross-stream stretching term, Qcrst.

c. Relationship of the partitioned Q terms to upper-level troughs and jet streaks

The relationships between the curvature, stretching, and shear advection term with upper-level troughs and jet streaks, respectively, have been discussed in the introduction of Jusem and Atlas (1998). We here reemphasize these relationships because they are in the focus of the current study.

It has long been recognized that the midlatitude upper-level wave pattern is associated with an alternating pattern of ascent and subsidence (Bjerknes and Holmboe 1944). Vertical motion is also an intrinsic feature of the standard QG theory for developing baroclinic waves. The curvature term in the upper troposphere is directly related to the upper-level wave pattern (i.e., troughs and ridges). It will be clear from the results below that vertical motion associated with the curvature term (ωcurv) exhibits the well-known pattern of ascent ahead of a trough and subsidence in the back of a trough.

Jet streaks exhibit pronounced confluence and diffluence in their entrance and exit region, respectively (e.g., Fig. 10.6a in Shapiro and Keyser 1990), and thus project prominently on the stretching term. For straight jet streaks, the four-quadrant model—with ascent in the equatorward entrance and poleward exit region and subsidence in the poleward entrance and equatorward exit region—constitutes a standard conceptual model for the distribution of vertical motion associated with jet streaks (e.g., Fig. 6.6 in Uccellini 1990). In our analyses below, the vertical motion pattern associated with the stretching term (ωst) frequently exhibits similarities with the classic four-quadrant model.

Jet streaks exhibit also very pronounced horizontal shear on their flanks and thus project prominently on the shear-advection term also. The shear-advection term yields ascent (subsidence) for warm- (cold-) air advection by the jet streak, given that the cross-stream thermal gradient is sufficiently small [cf. Eq. (13) and Fig. 1c in Jusem and Atlas 1998]. Temperature advection by horizontal (geostrophic) shear has been shown to play an important role for the distribution of vertical motion during upper-level frontogenesis (e.g., Shapiro 1981; subsequently dubbed the “Shapiro effect”). In fact, a clear relation between the vertical motion associated with the shear-advection term (ωshdv) and a jet streak is often found in our analyses. The interested reader may find an interpretation of the Shapiro effect in terms of shear-vorticity advection in Martin (2014).

4. Downstream cyclogenesis in a highly idealized ET scenario

In this section, the utility of the Jusem and Atlas partitioning to investigate downstream cyclogenesis is illustrated in the least complex, idealized ET scenario: the interaction of a tropical cyclone with an initially straight jet (Riemer et al. 2008). This high degree of idealization excludes the sensitivity of the downstream development to the phasing between the ET system and the midlatitude wave pattern. First, aspects of the experimental setup are described. Then, a brief overview of the synoptic evolution and a comparison of the balanced vertical motion and the (full) model vertical motion is given. The more detailed analysis based on Q-partitioning is presented in sections 4d and 4e.

a. Experimental setup

The idealized numerical experiment was performed with the fifth-generation Pennsylvania State University (PSU)–National Center for Atmospheric Research (NCAR) Mesoscale Model, version 3 (MM5V3; Grell et al. 1994) on a 60-km grid with 288 × 141 grid points. A vortex-following, higher-resolution (20 km) two-way nest was employed to better resolve the tropical cyclone and the resulting ET system. The model was run in a channel configuration with periodic boundary conditions in the zonal direction on a Cartesian grid. The Coriolis parameter was specified to vary with the geometry of Earth. The meridional location of the initial jet maximum corresponds to 42°N. The jet maximum is 40 m s−1 at 175 hPa. The initial jet profile is similar to that in the seminal work by Simmons and Hoskins (1980) but the meridional scale is contracted by a factor of 0.6. The wind profile is in thermal wind balance resulting in a tropospheric-deep baroclinic zone. The lower boundary is specified as an ocean surface with time-invariant sea surface temperature, which is specified as 28°C south of the low-level baroclinic zone and varies in the meridional direction in accordance with the near-surface atmospheric temperature.

In a separate model run, an incipient tropical cyclone vortex is spun up in a quiescent environment on an f plane (corresponding to 21°N) until a mature storm stage is reached. The mature model tropical cyclone is then inserted into the domain of the straight jet at initial time and 1620 km south of the jet axis. The model setup and the initial conditions are described more comprehensively in section 2 of Riemer et al. (2008).

For the current study, the streamfunction Ψ, potential temperature θ, the specific volume α, and static stability are interpolated to 18 equally spaced pressure levels from 920 to 70 hPa. For the calculation of the specific volume, virtual potential temperature is used. Output from the numerical experiment is available every 12 h.

b. Overview of cyclone development and balanced vertical motion

In the first 132 h of the experiment, the tropical cyclone gradually moves poleward toward the initially straight jet. During this time, the tropical cyclone’s outflow starts to impinge on the jet stream and a ridge–trough couplet and a jet streak begin to develop downstream (Fig. 1a). At 132 h, the tropical cyclone commences ET and the interaction with the jet stream increases. Subsequently, during ET, the ridge–trough couplet and the jet streak amplify (144–156 h; Figs. 1b and 1c). Downstream cyclogenesis then occurs in association with the downstream trough and in the poleward exit region of the jet streak (168 h; Fig. 1d). Subsequently, the cyclone intensifies further and the development extends into the farther-downstream region as a baroclinic Rossby wave train [see Riemer et al. (2008) for more details].

Fig. 1.
Fig. 1.

Overview of ωbal (shaded, 10−3 Pa s−1, negative values denote ascent) in the idealized ET scenario at (a) 132, (b) 144, (c) 156, and (d) 168 h into the experiment. The black contours depict surface pressure for 995 hPa and lower, every 5 hPa. The dashed contours show the streamfunction in the region of the jet stream at 270 hPa, every 107 m2 s−1. The hatching indicates the jet streak by highlighting wind speeds > 40 m s−1 at 270 hPa.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

Strong subsidence and ascent of midtropospheric ωbal is found in the vicinity of the ET system. In the downstream region, an alternating pattern of subsidence and ascent is found, as expected for a developing baroclinic wave. The focus of this study is on the region of ascent (ωbal < 0) in the downstream region. This ascent is associated with a decrease in surface pressure and subsequent cyclogenesis, as is evident from the evolution of the surface isobars. Concurrently with the amplification of the jet streak and the ridge–trough couplet, the ascent in the downstream region amplifies gradually. The location of the ascent indicates that both the jet streak and the upper-level trough may provide important contributions. Before the individual contributions obtained by Q-partitioning are analyzed, a brief comparison between ωbal and the full model vertical motion is provided.

c. Local amplification within the envelope of balanced vertical motion

Figure 2 depicts the balanced and the full model vertical motion (ωmodel) at a representative time (156 h). The smooth, synoptic-scale pattern of ωbal provides an envelope in which ωmodel exhibits finer-scale structure. In the subsidence regions, the values of ωbal and ωmodel are very similar. In the ascent regions, ωmodel is amplified locally. In the region of the developing downstream cyclone, ωmodel exceeds −0.2 Pa s−1 whereas ωbal hardly exceeds −0.075 Pa s−1. To the south of the developing downstream systems, in the region where ωbal is very weak, ωmodel exhibits the characteristics of unorganized (model) convection.

Fig. 2.
Fig. 2.

Comparison of ωbal (contours) and the full vertical motion field (shaded, 10−3 Pa s−1) in the numerical experiment at 156 h. The values for the contours are the same as the color shading with dashed (dotted) contours showing negative (positive) values. For reference, surface pressure is shown every 5 hPa below 995 hPa (solid line).

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

The local amplification of ascent can mostly be attributed to the release of latent heat [cf. section 5d(4)]. Overall, however, Fig. 2 demonstrates that ωbal represents well the synoptic-scale vertical motion in the region of downstream development.5 Amplification of ascent occurs only within the ωbal envelope in this region. We have verified that the same is true for the three real cases examined in section 5. An important exception to this notion occurs during a period of the development downstream of Jangmi and will be examined in more detail in section 5d(4).

d. Jusem and Atlas partitioning

The individual contributions to ωbal are illustrated in Fig. 3 during the early part of the development (at 144 h) and during the intensification of the downstream cyclone (at 168 h). In general, the curvature term yields the strongest contribution to vertical motion downstream of ET. This term clearly shows the signature of the developing upper-level wave pattern with pronounced subsidence within the downstream ridge, ascent ahead of the downstream trough, and subsidence associated with ridge building in the farther downstream region (Figs. 3a,b). The vertical motion associated with stretching exhibits characteristics of the four-quadrant model for straight jet streaks. In the exit region of the jet streak, ascent is found underneath the poleward side and subsidence underneath the equatorward side (Figs. 3c,d). In the entrance region, subsidence is found underneath the poleward side. There is ascent underneath the equatorward entrance region but this region is apparently dominated by the ET system. The ascent associated with shear advection is clearly related to the jet streak (Figs. 3e,f). Beyond the vicinity of the ET system, maximum ascent is approximately aligned with the jet streak. This ascent is associated with warm-air advection in the ridge [cf. Eq. (13); see also Fig. 2c in Jusem and Atlas] and counteracts the pronounced subsidence associated with the curvature term. As a consequence, the total subsidence in the downstream ridge is relatively weak (Fig. 1).

Fig. 3.
Fig. 3.

As in Fig. 1, but for the individual contributions to vertical motion in the idealized ET scenario: (a),(b) curvature term, (c),(d) stretching term, and (e),(f) shear-advection term at (left) 144 and (right) 168 h.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

The partial compensation between ωcurv and ωshdv found here is not a systematic feature of the Jusem and Atlas partitioning in baroclinic development, in contrast to the large compensation between the warm-air and vorticity-advection terms in the traditional form of the QG ω equation. In a baroclinic wave, in contrast to our ET scenario, there is cold-air advection between the ridge and the trough (e.g., Fig. 8.10 in Holton 2004) and thus ascent associated with the shear-advection term can be expected.

Comparison of Fig. 3 with Fig. 1 shows that the general pattern of balanced vertical motion downstream of ET is dominated by the curvature term. In particular, the largest values of ascent downstream of ET are found in ωcurv. The remaining contributions by ωst and ωshdv tend to be of larger relative importance on the upstream side of the developing downstream cyclone than in the center of the cyclone. To provide a quantitative estimate of the relative importance of the trough and the jet streak to the cyclone development, we now consider the lower-tropospheric vorticity spinup attributable to these features.

e. Estimate of vorticity spinup

A close linkage between the vorticity and pressure distributions exists for all balanced flows. According to geostrophic scaling [Eq. (1)], the spinup term is the leading-order term in the vorticity equation. We therefore write
e15
where D/Dt is the material derivative following the total flow, η is the vertical component of absolute vorticity, and contains the tilting term and effects of friction. The term is neglected in our analysis below and ω is approximated by ωbal = ωcurv + (ωst + ωshdv). Furthermore, the spinup term is averaged over the lower troposphere (p > 520 hPa).

In the early stage of the development (from 132 to 156 h), the vorticity spinup due to ωbal exhibits excellent agreement with the (Lagrangian) pressure tendency of the developing surface low (illustrated at 156 h in Fig. 4a). The surface pressure tendency is estimated from the 12-h difference between two consecutive output data. Consistent with our use of Eq. (15), the average wind speed in the lower troposphere (p > 520 hPa), estimated from the initial jet profile (Fig. 1 in Riemer et al. (2008)) as ≈8 m s−1, is used to track the developing surface low. After the onset of diabatic processes along the developing warm front around 168 h, the excellent agreement slightly deteriorates (not shown).6

Fig. 4.
Fig. 4.

The vertically averaged vorticity spinup [from Eq. (15), shaded, 10−10 s−2] due to (a) the total balanced ascent, (b) the curvature term, and (c) the sum of the stretching and shear-advection terms at 156 h. The contours depict the Lagrangian pressure tendency (with a contour interval of 10−3 Pa s−1) and the thick contour depicts the 60% threshold of the maximum pressure fall (see text for details) with an absolute value of −6.25 × 10−3 Pa s−1.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

Further general characteristics of the vorticity spinup due to ωcurv and (ωst + ωshdv) are exemplified in Figs. 4b and 4c. The spinup due to ωcurv has a close relationship with the maximum in the surface pressure fall. The spinup due to (ωst + ωshdv), on the other hand, is most pronounced upstream of the maximum pressure fall. Note that upstream of the developing cyclone, ascent due to (ωst + ωshdv) is counteracted by subsidence in ωcurv (Fig. 3).

As a concise estimate of the contribution to cyclone development, we consider the vorticity spinup integrated over the region where the surface pressure fall exceeds 60% of the maximum pressure fall. The threshold of 60% identifies a region that is clearly associated with the cyclone development at all analysis times (132–180 h, exemplified in Fig. 4). The relative contribution of ωcurv to the integrated vorticity spinup is depicted in Fig. 5. Early during the development (132–156 h), the relative contribution by ωcurv is roughly 50%. Subsequently, the relative contribution increases to 70% at 168 h and 85% at 180 h.

Fig. 5.
Fig. 5.

Relative contribution by the curvature term to the vertically and spatially averaged vorticity spinup (see text for details) during the time period of cyclone development.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

The values shown in Fig. 5 exhibit some sensitivity to the choice of the relative threshold. Relative thresholds greater than 60% tend to increase the relative contribution by ωcurv because the integration domain is more confined to the center of the pressure fall (cf. Fig. 4b). Relative thresholds smaller than 60% extend the integration domain farther upstream of the developing cyclone and tend to increase the relative contribution by (ωst + ωshdv) (cf. Fig. 4c). We have verified that the result shown in Fig. 5 holds for reasonable choices of relative thresholds.

Attributing (ωst + ωshdv) to the jet streak and ωcurv to the upper-level trough, we conclude that both features are of similar importance for cyclone development in the early stage of the evolution (132–156 h). In the subsequent intensification stage, starting at 168 h, the contribution of the jet streak to vorticity spinup is below 30% and the development is apparently dominated by the upper-level trough.

5. Application to real-atmospheric ET events

In the following, we examine the relative contribution of jet streaks and upper-level troughs to cyclone development downstream of three ET events in the real atmosphere: Hanna (2008), Choi-wan (2009), and Jangmi (2009). The three cases exemplify the wide range of possible scenarios that may occur downstream of ET in the real atmosphere (see below). The downstream impact of these ET systems have recently been examined by Grams et al. (2011), Grams (2011), and Grams et al. (2013b), respectively, by performing numerical experiments in which the respective tropical cyclone was artificially removed from the initial conditions before ET and model forecasts were rerun (referred to as the “no-TC scenario” below). Our brief summary of the downstream impact of the three ET systems is based on these differences in Grams et al.’s TC and no-TC scenarios.

As in the idealized scenario, it will be found in the real cases that the curvature term can be associated with the upper-level wave pattern and the stretching and shear-advection terms with jet streaks. A vorticity-spinup analysis (as in section 4e) will not be attempted because it is unclear to us how to reliably estimate the Lagrangian pressure tendencies for the real atmospheric downstream cyclones. Before presenting our analysis based on the Jusem and Atlas partitioning, a brief description of the data used in this section is provided.

a. YOTC data and diabatic forcing term

For the analysis of the real-atmospheric ET events, the Year of Tropical Convection data from ECMWF are used. Besides providing standard variables, the YOTC data contain also the model tendencies due to the physical parameterizations employed in the ECMWF forecast system. The tendencies are provided as 3-hourly accumulated tendencies in an (Eulerian) grid box. For the current study, we use the diabatic temperature tendencies due to latent heat release from the microphysics scheme and the schemes for deep and shallow convection. In addition to these tendency terms, the same variables as described in section 4a are calculated. Data are available on a horizontal grid of 1° × 1° and 17 unevenly distributed pressure levels (1000, 950, 925, 900, 850, 800, 700, 600, 500, 400, 300, 250, 200, 150, 100, 70, and 50 hPa).

Following the derivation of the QG ω equation (e.g., Holton 2004, section 6.4.1), it is straightforward to include a diabatic forcing term in the AB ω equation as well (cf. Mallet et al. 1999). Denoting, for the sake of brevity, the linear operator on the left-hand side of Eq. (6) by , we can write
e16
where , with the material temperature tendency This material tendency is approximated by the 3-hourly accumulated YOTC tendencies. This approximation usually implies a downstream displacement of the extrema in ω derived from Eq. (16) as compared to the instantaneous vertical motion field as represented in the YOTC data. This displacement is reasonably small and does not affect the interpretation of our results [section 5d(4)].

We refer to vertical motion associated with the diabatic term in Eq. (16) as diabatic vertical motion, ωdiab. It is noted that latent heat release may have a more complex impact on vertical motion than implied in Eq. (16). Latent heat release at previous times may alter the stratification [σ in Eq. (6)] and the thermal field (α) and may project on the balanced flow (ψ). Therefore, both the operator and Q may be modified by latent heat release. The goal of our analysis, however, is to assess whether latent heat release merely modifies balanced vertical motion, as exemplified in Fig. 2, or whether a “diabatic mode” of vertical motion exists that is largely uncoupled from ωbal. Examples of such a diabatic mode are organized convective systems or a diabatic Rossby vortex. In this context, the distinction between ωbal and ωdiab implied in Eq. (16) is a reasonable concept.

b. Modest, short-lived cyclone downstream of Hanna (2008)

1) Brief overview of downstream impact

The modification of the upper-level, midlatitude flow downstream of the ET of Hanna exhibits archetypical features identified in idealized numerical experiments (Riemer et al. 2008; Riemer and Jones 2010): an amplified ridge–trough couplet downstream of ET and a strong jet streak in between the ridge and the trough (Grams et al. 2011). The downstream cyclone developed near the poleward exit region of this jet streak and in association with the amplified upstream trough (illustrated in Fig. 6a). In this sense, the evolution is similar to the idealized experiment discussed above, although the flow configuration downstream of Hanna is obviously more complex.

Fig. 6.
Fig. 6.

Illustration of the cyclone development downstream of the ET of (a) Hanna at 0000 UTC 9 Sep 2008, (b) Choi-wan at 0000 UTC 9 Sep 2009, and (c) Jangmi at 1200 UTC 1 Oct 2008. Near-surface development is illustrated by geopotential height at 950 hPa (solid contours, every 4 gpdm). The upper-level flow is illustrated by the streamfunction (dashed contours, every 107 m2 s−1) and wind speeds above a threshold value (hatched) at 250 hPa. The threshold value is 45 m s−1 for Hanna, 50 m s−1 for Choi-wan, and 60 m s−1 for Jangmi. In addition, for Jangmi, wind speeds > 70 m s−1 are hatched in light gray. Color shading depicts the balanced vertical motion (10−3 Pa s−1) at 500 hPa. The arrows point to the respective downstream cyclone and “ET” marks the ET system.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

The modification of the upper-level flow led to the more pronounced development of a modest, short-lived surface cyclone (5–10 hPa deeper as compared to the no-TC scenario; Grams et al. 2011). The impact on the location of the cyclone was small. The downstream cyclone affected the British Isles with widespread precipitation of 10–20 mm and a maximum of 50 mm in Wales. Subsequently, the downstream trough developed into a potential vorticity streamer that triggered heavy precipitation in the Mediterranean Sea.

2) Continued modification by amplified jet streak

Cyclogenesis downstream of Hanna (from 1200 UTC 8 September to 1200 UTC 9 September; see Figs. 7a,b, 6a, and 7c,d, respectively) is associated with the merger of two local minima in the 950-hPa geopotential height field. At 1200 UTC 8 September (Figs. 7a,b), one of these minima is located near the poleward exit of the jet streak and the other minimum is located just ahead of the downstream trough. In the exit region of the jet streak, vertical motion associated with stretching exhibits a distinct dipole pattern. The ascent in ωst in the poleward exit region constitutes the strongest individual contribution to ascent in the downstream region at this time and is collocated with the local minimum in geopotential. The second local minimum in geopotential is located near a maximum in ascent associated with curvature. In this region, there is ascent in ωst also, with an average value of approximately one-third of ωcurv. Vertical motion associated with shear advection is significantly weaker than ωst and ωcurv and slightly counteracts ascent in ωst in the development region (not shown).

Fig. 7.
Fig. 7.

As in Fig. 6a, but for vertical motion associated with (a) the stretching term, (b),(d) the curvature term, and (c) the sum of the stretching and shear-advection terms for the case of Hanna at (top) 1200 UTC 8 Sep and (bottom) 1200 UTC 9 Sep.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

The described pattern of vertical motion is similar at 0000 UTC 9 September (not shown) but starts changing at 1200 UTC 9 September. At this time, both the stretching and the shear advection term contribute to ascent over the cyclone center. The combined ascent (Fig. 7c), spatially averaged over the development region, is approximately equal to ascent in ωcurv in this region (Fig. 7d). The maximum ascent in ωcurv is located southeast of the cyclone at this time and starts to decouple. Subsequently, this distinct maximum moves farther to the east while the cyclone itself does not develop further but merges with Hanna’s ET system (not shown).

As in the idealized scenario, ωcurv and ωst can clearly be associated with the amplified trough and the amplified jet streak, respectively, downstream of Hanna. Ascent in the development region attributable to the jet streak is estimated to be 25%–50% of the total ascent. Therefore, it can be concluded that the amplified jet streak has a notable impact throughout the short life cycle of the downstream cyclone.

c. Amplified development downstream of Choi-wan (2009)

1) Brief overview of downstream impact

The interaction of Choi-wan with the midlatitude flow led to a considerable amplification of the ridge–trough couplet downstream of ET (Grams 2011). In this case, however, the amplitude of the jet streak downstream of ET was not notably modified. As in the case of Hanna, the downstream cyclone developed near the poleward exit region of the jet streak and in association with the upper-level trough (illustrated at the time of a first closed contour in the 950-hPa geopotential at 0000 UTC 22 September in Fig. 6b). In this sense, the development is comparable to the idealized experiment discussed above. In contrast to the idealized experiment, the developing cyclone subsequently moved into the entrance region of a second jet streak located further downstream (labeled J2 in Fig. 6b). Throughout the development, however, the cyclone remained ahead of the amplified upper-level trough (not shown).

The downstream cyclone played an important role in the coherent amplification of a Rossby wave train downstream of Choi-wan. Associated with the amplified downstream trough, the downstream cyclone developed significantly deeper (approximately 25 hPa) and moved along a more northerly track [as compared to the no-TC scenario of Grams (2011)]. Both modifications promoted tremendous ridge building farther downstream that resulted in an early-season cold-air outbreak in the western United States.

2) Dominance of the amplified upper-level trough

The development of the downstream cyclone is accompanied by a pronounced maximum of ascent associated with the curvature term during the whole time period considered (from 0000 UTC 21 September to 1200 UTC 23 September). Generally, this maximum is ahead of the cyclone and appears to govern the movement of the cyclone. This general pattern of ωcurv is illustrated in Fig. 8a (at 0000 UTC 22 September, the asterisk depicts the cyclone center 12 h later). The maximum in ascent exceeds −0.52 Pa s−1 at all times except at 1200 UTC 22 September when maximum ascent is approximately −0.45 Pa s−1.

Fig. 8.
Fig. 8.

As in Fig. 6b, but for vertical motion associated with (a) the curvature term and (b) the sum of the stretching and shear-advection terms for the case of Choi-wan at 0000 UTC 22 Sep. The asterisk marks the position of the surface center of the downstream cyclone 12 h later. The pattern shown is representative for the evolution of the downstream cyclone from 0000 UTC 21 Sep to 1200 UTC 23 Sep.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

Clearly, the sum of the stretching term and the shear advection term makes a minor contribution to vertical motion in the area of cyclone development (exemplified in Fig. 8b), as compared to ωcurv. Our estimate of the spatially averaged, relative contribution to ascent by ωst + ωshdv in the region of the downstream cyclone is less than 20% throughout the development.

We conclude that both the higher intensity of the downstream cyclone and its northerly track are prominently associated with the amplified upper-level trough downstream of the ET of Choi-wan. The jet streak has only a minor impact on the evolution of the downstream cyclone in this case.

d. Fast-moving cyclone downstream of Jangmi (2009)

1) Brief overview of downstream evolution

In contrast to Hanna and Choi-wan, the downstream impact of Jangmi was weak. During ET, Jangmi remained near the southern tip of the upstream trough and gradually decayed. Ahead of this trough, instead, a new downstream cyclone developed. The interaction of Jangmi’s outflow with the midlatitude jet formed a prominent jet streak immediately downstream of Jangmi. The location of downstream cyclogenesis coincided with the equatorward entrance region of this jet streak (Grams et al. 2013b, section 3). In these regards, the development is distinct from the idealized scenario and from the Hanna and Choi-wan scenarios. Furthermore, the distance between the ET system and the downstream cyclone in the case of Jangmi was significantly smaller than in the previous two cases (illustrated at 1200 UTC 1 October in Fig. 6c): The downstream cyclone was located 15°–20° east of Jangmi as compared to 35°–40° east of Hanna and Choi-wan, respectively. The cyclone underwent a brief (12–24 h) period of intensification and then moved quickly across the Pacific as a frontal wave without considerable intensity change (Grams et al. 2013b). On 4 October, the cyclone began to weaken gradually.

Grams et al. (2013b) analyze the evolution of the downstream cyclone in terms of upper- and lower-level contributions using the QG ω equation. They conclude that the downstream cyclone formed in the favorable location of upper-level QG ascent ahead of the trough and argue that the cyclone decoupled from the upper-level trough when the cyclone started to move quickly to the east. Furthermore, they emphasize that the cyclone moved in conjunction with the equatorward entrance region of the jet streak and exhibited characteristics of a diabatic Rossby wave. Effectively, however, the downstream cyclone lacked support from upper-level QG forcing and thus did not intensify further.

The Jusem and Atlas partitioning allows for a more detailed examination of the role of the trough and the jet streak. Our analysis below will provide an extended and alternative interpretation of the role of these features in the development of the downstream cyclone.

2) Shortwave trough versus equatorward entrance of jet streak

Consistent with the analysis of Grams et al. (2013b), we find that the early intensification of the downstream cyclone is associated with balanced ascent ahead and poleward of the cyclone (exemplified in Fig. 6c at 1200 UTC 1 October). This ascent is strong until 0000 UTC 2 October (ωbal < −0.44 hPa s−1) and then decreases gradually until 1200 UTC 3 October. Thereafter, consistent with the weakening of the cyclone, balanced subsidence occurs over a broad region of the cyclone.

Consistent also with the analysis of Grams et al. (2013b), we find that the cyclone decouples from the region of ascent ahead of the upstream trough. The pronounced ascent associated with curvature in this region (around 160° in Fig. 9a, illustrated at 0000 UTC 3 October) is clearly separated from the downstream cyclone by a region of subsidence. Our analysis, however, reveals a second region of ascent associated with curvature. This region is linked to a shortwave trough in the northern part of the jet stream (indicated by the trough axis in Fig. 9a). This trough is not evident at 250 hPa (cf. Fig. 9b) and divergence of Qcurv between the shortwave trough and the crest of the ridge downstream is found only at 300–600 hPa (not shown). This second region of ascent decouples from the larger-scale upstream trough at 1200 UTC 2 October and accompanies the cyclone until the end of our analysis at 1200 UTC 4 October.

Fig. 9.
Fig. 9.

As in Fig. 6c, but for vertical motion associated with (a) the curvature term and (b) the stretching term for the case of Jangmi at 0000 UTC 3 Oct. In (a), the streamfunction is shown at 500 hPa and the black curved line indicates the axis of a shortwave trough. The downstream cyclone is approximately in the center of the domain.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

In contrast, Grams et al. argue that, after having decoupled from the (large scale) upstream trough, the cyclone is coupled to the equatorward entrance region of the jet streak. Vertical motion associated with stretching indeed indicates ascent in the equatorward entrance region of the jet streak. This ascent, however, is located in the back of the cyclone (Fig. 9b), a configuration that is found throughout the development. Weak ascent associated with stretching is found also ahead of the cyclone, just poleward of the jet axis from 1200 UTC 1 October to 1200 UTC 3 October (exemplified at 0000 UTC 3 October in Fig. 9b). Ascent in this region, however, is clearly dominated by curvature, with ωcurv ≥ 3 ωst at all times considered. We therefore conclude that the cyclone at this time is not coupled to the jet entrance region but is rather associated with the shortwave trough in the northern part of the jet stream.

3) Decoupling, fast movement, and demise in the equatorward exit of the jet streak

Arguably, ascent associated with the equatorward entrance of the jet streak plays a secondary role for cyclone development. Subsidence associated with the jet streak, on the other hand, may play a more significant but detrimental role for the evolution of the cyclone. A pronounced maximum of subsidence associated with stretching is found in the back of the developing cyclone at 1200 UTC 1 October and 0000 UTC 2 October (exemplified in Fig. 10a). This region of subsidence is clearly associated with strong confluence in the poleward entrance of the jet streak. At 0000 UTC 2 October, additional subsidence associated with shear advection develops in the back of the cyclone (Fig. 10b). This subsidence is associated with cold-air advection between the upstream trough and the crest of the ridge and intensifies in the next 12 h (not shown). Subsidence with ωshdv > 0.14 Pa s−1 persists from then on until the end of our analysis.

Fig. 10.
Fig. 10.

As in Fig. 6c, but for vertical motion associated with (a) the stretching term at 1200 UTC 1 Oct, (b) the shear-advection term at 0000 UTC 3 Oct, and (c) again the stretching term at 1200 UTC 4 Oct for the case of Jangmi. The downstream cyclone is approximately in the center of the domain.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

Therefore, from 1200 UTC 1 October to 1200 UTC 2 October, considerable subsidence attributable to the jet streak (ωst + ωshdv > 0.14 Pa s−1) exists in the back and close to the center of the cyclone. This subsidence discourages cyclonic development on the western edge of the incipient cyclone, therefore promoting the decoupling of the cyclone from the upstream trough. From 1200 UTC 2 October onward, subsidence associated with shear advection in the back of the cyclone dominates the ascent associated with stretching in the equatorward entrance of the jet streak (cf. Figs. 9b and 10b). This continued subsidence in the back of the cyclone may promote its fast movement.

Beginning at 1200 UTC 3 October, subsidence associated with stretching occurs ahead and over the center of the cyclone (illustrated at 1200 UTC 4 October in Fig. 10c). Figure 10c indicates that the cyclone progresses at the end of its life cycle into the equatorward exit region of the extensive jet streak. The broad region of subsidence in the equatorward exit quadrant of the jet streak compensates for the continuous weak ascent associated with curvature (see above). Subsidence associated with stretching thus governs the demise of the cyclone.

Apparently, the decoupling of the cyclone from the upstream trough and the fast movement and final demise of the cyclone are all associated with subsidence attributable to the jet streak. The jet streak therefore has a significant but detrimental impact on the overall evolution of the downstream cyclone.

4) Diabatic contribution to development

In this subsection, we exploit the heating terms available from the YOTC data to examine the diabatic vertical motion, ωdiab. We use ωdiab to briefly comment on the relative importance of diabatic processes (here: latent heat release as seen by the ECMWF model) for the evolution of the downstream cyclone.

From 1200 UTC 1 October to 1200 UTC 2 October, we find that ωdiab near the cyclone is generally located within the envelope of local maxima of balanced ascent. At these times, latent heat release localizes and amplifies ascent considerably but is apparently still strongly tied to the balanced dynamics. This situation is exemplified in Fig. 11a at 1200 UTC 2 October. At this time, in particular, a band of strong ωdiab ascent extends from the cyclone center eastward. This banded region of strong ascent7 agrees well with the full upward motion as seen by the ECMWF data and obviously lies within the region of considerable balanced ascent.

Fig. 11.
Fig. 11.

Vertical motion associated with latent heat release (shaded) as seen by the YOTC data (see text for details) and balanced ascent with values <−0.12 hPa s−1 (hatched) at (a) 1200 UTC 2 Oct and (b) 0000 UTC 3 Oct. Strong ascent in the full vertical motion field as seen by ECMWF data is illustrated by the −0.44 hPa s−1 isoline (thick contour). The displacement between maxima in this full ascent and maxima in ascent associated with latent heat release is likely due to the fact that heating terms averaged over the subsequent 3-h period are employed.

Citation: Journal of the Atmospheric Sciences 71, 11; 10.1175/JAS-D-14-0023.1

A qualitative change occurs within the next 12 h. The band of ωdiab ascent persists but now it is clearly decoupled from the maximum of balanced ascent (Fig. 11b). Without notable connection to the balanced dynamics, the band further persists another 12 h with similar intensity. At 0000 UTC 4 October, when balanced subsidence establishes in the region of the band, the band weakens and exhibits less vertical extent.

Based on the YOTC data, we conclude that a diabatic mode of ascent that is independent from the balanced ascent exists for about 24 h (beginning around 0000 UTC 3 October and decaying around 0000 UTC 4 October). This brief analysis supports Grams et al.’s hypothesis that the downstream cyclone exhibited characteristics of a diabatic Rossby vortex during this time.

6. Summary and conclusions

The formation of a jet streak and an amplified trough are archetypical features of the modification of the midlatitude flow during ET. Both features are well known to promote cyclogenesis. Downstream cyclogenesis may lead to high-impact weather and has been shown to constitute an important step in the further dispersion of the downstream impact. During ET, the jet streak is forced by the tropical cyclone. It is thus plausible to hypothesize that the jet streak here plays a more prominent role in cyclogenesis than in baroclinic development without such specific external forcing. This study investigates the relative importance of the jet streak and the upper-level trough for downstream cyclogenesis using quantitative estimates of their respective contributions to midtropospheric vertical motion.

These estimates are derived by Q-vector partitioning based on the geometry of the flow (Jusem and Atlas 1998). We here detail the straight-forward extension of this Q-vector partitioning from quasigeostrophic theory (QG) to alternative balance, as outlined by Jusem and Atlas in their summary. The Q vector under alternative balance involves the nondivergent horizontal wind, instead of the geostrophic wind as in QG, and can thus be expected to more accurately describe the balanced dynamics associated with vertical motion, in particular in the highly curved flow associated with amplified trough–ridge patterns. Jusem and Atlas’s partitioning yields terms that can be related to upper-level troughs (the curvature term) and to jet streaks (the stretching term and the shear-advection term). The current study corroborates these relationships. This Q-vector partitioning is thus particularly useful to investigate cyclogenesis downstream of ET.

For an idealized ET scenario, the vorticity spinup associated with midtropospheric balanced ascent is determined as a proxy for cyclone development. In the early stage of the cyclone development, over a period of 36 h, balanced ascent attributable to the upper-level trough and the jet streak make approximately equal contributions to the vorticity spinup in the development region. In the later stage of the development, the ascent associated with the trough clearly dominates vorticity spinup, in particular near the center of the cyclone.

Investigation of three real atmospheric cases demonstrates a wide range of possible scenarios for downstream cyclone development. In the case of Hanna (2008), analysis of the midtropospheric balanced ascent indicates that the amplified jet streak and the trough downstream of Hanna make similar contributions to the development of the short-lived downstream cyclone. In contrast, in the case of Choi-wan the development and further evolution of the downstream cyclone is clearly dominated by the amplified upper-level trough. In the case of Jangmi, balanced ascent is dominated by troughs also. Subsidence attributable to the jet streak is found to play a prominent but detrimental role in the evolution of the downstream cyclone. In the early phase of the evolution, the jet streak apparently promotes the decoupling of the cyclone from the favorable region of balanced ascent associated with the upstream trough. Later in the development, subsidence in the equatorward exit region the jet streak governs the demise of the cyclone.

Previously, Grams et al. (2013b) have investigated the evolution of the cyclone downstream of Jangmi based on the partitioning of the QG Q vector into upper- and lower-level contributions. Our analysis confirms their result that the downstream cyclone decouples from the upstream trough after the initial development. In addition, and possibly surprisingly, our analysis reveals a distinct second local maximum of ascent associated with a shortwave trough that accompanies the downstream cyclone throughout its development. In contrast to Grams et al.’s results, our analysis does not support the notion that the cyclone is coupled to the favorable equatorward entrance region of the jet streak after decoupling from the upstream trough. Instead, as noted above, we find that the jet streak has a detrimental impact on the cyclone development.

Diabatic tendency terms from the YOTC data are used to evaluate the diabatic contribution to vertical motion downstream of Jangmi. During the early part of the development, diabatic tendencies amplify and localize ascent within the envelope of balanced ascent. Subsequently, however, for a period of about 24 h, diabatic ascent clearly decouples from the region of notable balanced ascent. This decoupling indicates a diabatic “mode” of the cyclone that is independent from the (synoptic scale) balanced dynamics. Thus, our brief analysis supports a hypothesis of Grams et al. (2013b) that the downstream cyclone exhibits characteristics of a diabatic Rossby vortex during this time.

Jusem and Atlas’s Q-vector partitioning under alternative balance, complemented by diabatic tendencies (e.g., from the YOTC data), provides a useful quantitative framework to examine different contributions to synoptic- to meso-α-scale forcing for cyclone development. Downstream of ET, the upper-level trough plays an important role in all of our cases. The hypothesis that the jet streak plays a particularly prominent role in all cases has to be rejected. Amplified jet streaks associated with ET certainly have the potential to be of significant importance for downstream cyclogenesis. The few cases considered in this study, however, point to a large case-to-case variability of the role of the jet streak.

Acknowledgments

The basis for this work was established in 2005 during a visit of the first author to the University at Albany, State University of New York, which was possible due to support from the Rupert Ford Award, granted by the Royal Meteorological Society. The first author thanks the Department of Atmospheric and Environmental Sciences in Albany for their hospitality and for inspiring scientific discussions and is deeply grateful to Dan Keyser for hosting the visit. Dan Keyser made crucial contributions to the initial design and formulation of the research project documented in this paper. Comments by Qin Xu and two anonymous reviewers were very helpful to improve the initial manuscript. Furthermore, the first and second authors thank Franziska Gierth for downloading the YOTC data.

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1

The f-plane approximation is a good approximation for the diagnosis of synoptic-scale vertical motion (Hoskins et al. 1978, at the end of their section 4).

2

The AB ω equation shares the further advantage with the QG version that only data from a single time step and a single level is needed to calculate Q. For more accurate balance systems (e.g., nonlinear balance) calculation of Q in the respective ω equation involves vertical and/ or temporal derivatives.

3

We note that this version of the AB ω equation is not the same as the version in Mallet et al. (1999). Their version is not based on the pseudoheight version of the AB ω equation given in Davies-Jones (1991). Instead, Mallet et al. applied the AB approximations at appropriate steps during the derivation of an ω equation from the primitive equations in pressure coordinates (J.-P. Cammas 2011, personal communication). We verified the formal validity of their derivation and cannot offer a mathematical explanation for the differences in the two versions of the ω equation. We note, however, that the individual approximations were taken in different order as compared to Davies-Jones (1991). It may therefore be speculated that the differences in the versions of the AB ω equation are due to the fact that AB is not derived as the distinguished limit of an asymptotic expansion (as QG is), for which the order in which approximations are made does not matter, but by using heuristic approximations. We have solved numerically both versions of the AB ω equation and compared the results for the idealized experiment presented in section 4. The qualitative patterns of the respective ω fields are identical. The version in Eq. (4) produces absolute values of ω that are 5%–10% larger than those of the version by Mallet et al.

4

The partitioning based on streamlines is not Galilean invariant. Jusem and Atlas (1998, p. 2167) assert that the lack of Galilean invariance does not affect the diagnostic value of the partitioning.

5

We note, as an aside, that the strong ascent and sharp transition to subsidence poleward of the ET system is not captured by the pattern of balanced vertical motion. This discrepancy may hint to the dominance of moist processes and/ or the occurrence of symmetric instability along the developing warm front at this time.

6

Diabatic generation of low-level potential vorticity anomalies (cf. Fig. 2 in Riemer et al. 2008) then contributes to the pressure tendency.

7

It is of interest to note that the banded ascent is slantwise. At 700 hPa, a very similar band of ascent is found 2°–3° south of the band at 500 hPa. Recall from section 3c(1) that the AB approximations imply an underestimation of slantwise ascent.

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