Downstream Development of Baroclinic Waves in the Midlatitude Jet Induced by Extratropical Transition: A Case Study

As a tropical cyclone (TC) moves northward, it often transforms into an extratropical cyclone. This is known as an extratropical transition (ET) process. During an ET event, a complex interaction between the TC and the midlatitude flow it encounters takes place, which may lead to the TC modifying the midlatitude flow (Jones et al., 2003). There are two mechanisms by which a TC can modify the midlatitude flow during an ET process. Through action at a distance, the TC outflow excites Rossby wave trains in the upper level, known as upper-level potential vorticity (PV) gradients, associated with the midlatitude jet (Bishop and Thorpe, 1994). The jet acts as a waveguide for the trapped Rossby waves (Schwierz et al., 2004), and the trapped Rossby waves propagate or disperse downstream of the jet (Simmons and Hoskins, 1979). (Riemer et al., 2008) and (Riemer and Jones, 2010) used idealized simulations to investigate the interaction of a TC with the midlatitude flow during an ET. They found that as the TC approached the jet, a ridge-trough couplet and a distinct jet streak formed in the upper level. A midlatitude cyclone developed rapidly downstream of the ET system and its further evolution was characterized by a downstream baroclinic development. The second mechanism involves the modification in the upper levels by low PV originating in the TC outflow. (Atallah and Bosart, 2003) found that the diabatic outflow of a TC could assist in the formation of a tropopause fold. Many studies have described how the diabatic reduction of PV in the upper troposphere enhanced downstream ridging (e.g. Davis et al., 1993; Stoelinga, 1996; Massacand et al., 2001). The enhanced ridge can be viewed as a steepened tropopause.

The complex interaction in an ET event not only changes the local atmospheric state but also affects a much larger geographical region that can extend to a near-hemispheric scale (Harr et al., 2008). In particular, the downstream impact can be obvious, and can result in high-impact weather such as explosive cyclogenesis far from the TC. Observations have shown that a hurricane in the North Atlantic Ocean can lead to the development of a windstorm or severe flooding event in Europe (Hoskins and Berrisford, 1988). As indicated above, an important mechanism of the downstream impact is the excitation and propagation of Rossby wave trains in the midlatitude jet. Through targeting studies on the ET of Hurricane Fabian (2003), (Chen and Pan, 2010) (hereafter referred to as CP2010) found that Rossby energy dispersed downstream in the midlatitude jet under the effects of the TC outflow. (Harr and Dea, 2009) chose four typhoon cases to represent the wide spectrum of variability in ET, and then examined the impact of each case on downstream development across the North Pacific. Their results showed that the variability in downstream response to each ET process was related to specific physical characteristics associated with the evolution of the ET process and the phasing between the TC and the midlatitude flow. (Riemer et al., 2008) and (Riemer and Jones, 2010) indicated that the TC undergoing an ET acted as a sustained forcing for the wave trains, and the structure of the ET system influenced the development most significantly one to two wavelengths downstream of the ET. Both the cyclonic circulation and the outflow of the TC are important contributors to the formation and amplification of the ridge-trough couplet. (Harr et al., 2008) and (Anwender et al., 2008) demonstrated that predictability was typically reduced across the North Pacific following an ET event over the western North Pacific. In addition, smaller-scale upstream developments can take place on the lower-level temperature gradient (Thorncroft and Hoskins, 1990; Wernli et al., 1999), which may be important in some cases because the maximum amplitude of TC circulation is located at lower levels. However, actual upstream impacts are weak over land due to surface friction.

Based on the diagnosis of local eddy kinetic energy, Orlanski and his colleagues proposed a conceptual picture of the dynamics of eddy activity along a storm track, which they called "downstream baroclinic development" (DBD) (Orlanski and Katzfey, 1991; Chang and Orlanski, 1993; Orlanski and Chang, 1993; Orlanski and Sheldon, 1993; Danielson et al., 2006a, b). That is, ageostrophic geopotential fluxes from a pre-existing, large-amplitude system are the major source for the growth of a downstream eddy. Baroclinic conversion becomes the primary energy source only after the development is well under way. In the mature stage, the eddy radiates ageostrophic geopotential fluxes downstream to cause its own decay, which triggers the growth of further downstream eddies. In some situations, the downstream eddies can develop into strong extratropical cyclones.

Targeted observations or adaptive strategies take additional observations in sensitive or targeted areas where large analysis errors exist and forecast errors grow significantly, to maximally improve the skill of the numerical forecast (Langland et al., 1999; Szunyogh et al., 2000; Morss and Emanuel, 2002; Mu and Jiang, 2008; Mu et al., 2009; Qin and Mu, 2011; Wang et al., 2011; Zhou and Mu, 2011). Singular vector (SV) is a method to identify targeted areas (Buizza and Montani, 1999; Buizza et al., 2007), and is currently used by the European Centre for Medium-Range Weather Forecasts (ECMWF). CP2010 studied the ET of Hurricane Fabian (2003) using the targeting method. They defined a signal as the forecast difference between the sensitivity experiment and the control experiment. The control experiment used all the observational data, but in the sensitivity experiment all the observational data were removed from the targeted area, which was determined by singular vectors. From the growth and propagation of signals, they found that the disturbances that originated from the TC reached the midlatitude jet through the interaction among the TC, jet, and an upstream trough. Thereafter, the disturbances propagated and developed in the midlatitude jet as Rossby wave trains. Although CP2010 revealed the processes and mechanisms of a small disturbance propagating from the TC to (or in) the midlatitudes, it is unclear how the downstream disturbances develop and how they intensify to strong cyclones under the effects of an ET system. Moreover, the dynamics and factors controlling the maintenance and intensification of the downstream disturbances need to be determined. The current study investigates how the ET event of Hurricane Fabian (2003) induced DBD and cyclogenesis in the midlatitude jet environment. First, we describe the observational analyses and targeting study to reveal some important characteristics of the DBD. Then, we perform a diagnosis of the local eddy kinetic energy to explore the dynamic mechanisms of these characteristics or developing methods.

The paper is organized as follows. Section 2 describes the data and methodology. Section 3 presents the characteristics of coupling development and boundary-trapping during the downstream development based on the targeting method and analyses of the actual synoptic situation. In section 4, we investigate the mechanisms of downstream development by diagnosing the local eddy kinetic energy. Section 5 examines the physical nature of boundary-trapping. And finally, section 6 provides a discussion and summarizes the main conclusions of the study.

This study continues the work of CP2010, and so uses the same datasets. For more detailed information on the experimental design, readers are referred to CP2010. The datasets are from the ECMWF, and comprise twice-daily operational analyses and forecast outputs on a 0.5^{°× 0.5°} grid. Forecast outputs are used for signal analyses, whereas operational analyses are used for diagnosing the eddy kinetic energy. The forecast base time is 1200 UTC 4 September 2003. Two forecast experiments are designed for this study. The control experiment starts from the operational analyses, which uses all the observational data. The sensitivity experiment, which is a partial data denial experiment, is different from the data injection experiment of targeted observations in which extra observations are added in the targeted area. In the sensitivity experiment, all the observational data are removed from the targeted area, which is determined by SVs and located near Hurricane Fabian's (2003) center. The SV method is an approach used to identify sensitive areas in targeted observation. The ECMWF data-assimilation and forecast experiments have been performed with the system used operationally between 21 November 2000 and 1 February 2006 (Mahfouf and Rabier, 2000; Kelly et al., 2007). The assimilation system is 4D variational with 12-h cycle window. It has analysis resolutions of outer loop of T _L511L60 and inner loops of T _L159/ T _L95L60. Forecast resolution is T _L511L60. Here, T _L is spectral triangular and L is the number of levels in the vertical. The signal is defined as the forecast difference (mainly for geopotential height) between the sensitivity experiment and the control experiment. In the sensitivity experiment, five types of observations are removed: 32 surface buoys (DRIBUs), 8 radiosondes temperatures at 500 hPa, 93 temperature observations from aircraft (250 hPa), 37 atmospheric motion winds (AMVs), and 58 satellite radiances from Advanced Microwave Sounding Unit AMSU-A channel 6. Vertical velocities at isobaric surfaces are from the National Centers for Environmental Prediction (NCEP) FNL (Final) reanalysis, interpolated to a 0.5^{°× 0.5°} grid. Local eddy kinetic energy analysis is an effective tool for studying DBD (Orlanski and Katzfey, 1991; Orlanski and Chang, 1993), and is employed in the current study to investigate the eddy kinetic energy budget of individual eddies.

The synoptic case in this study is Hurricane Fabian in 2003, which originated over western Africa on 25 August 2003. After moving westward and northwest for several days, Fabian (2003) turned toward the northeast, and transformed into an extratropical cyclone on 8 September. Corresponding to the ET event of Fabian (2003), a cyclone formed on 9 September and remained over Western Europe for two days.

As indicated in CP2010, signal propagation characteristics can reveal the "negative impact" of the sensitivity experiment (the denial experiment excluding all data from the targeted area). On the other hand, signal propagation can reveal the propagation of a positive impact of the targeted observations, and can explain why dropwindsonde data have apparent influences only on certain areas of the model domain. Since signals can be thought of as geopotential disturbances; their propagation can indicate how a disturbance propagates and develops downstream or how an ET process results in DBD. The propagation of signals (i.e., the forecast difference of geopotential height) can be seen in CP2010. As CP2010 concluded, the signal first appeared at the TC location, and then it propagated to the midlatitude jet through the interaction between the TC and the jet itself. Thereafter, the signals propagated downstream in the midlatitude jet and downward to the lower troposphere at the same time as Rossby wave trains during the ET event of Hurricane Fabian (2003). Szunyogh et al. (2000, 2002) also found that downstream development played an important role in spreading the effects of dropsonde data from the targeted area. Figure 1 shows the time series of the vertical cross section of the signals. To plot the section, we selected those signals whose absolute values were larger than 10 geopotential meters between 45^°N and 55^°N, and then performed meridional averaging. Figure 1 may indicate the coupling development process between downstream disturbed systems in the upper and lower troposphere induced by an ET process. From Fig. 1, we can see that the disturbances first developed and propagated downstream in the upper troposphere (CP2010), and then triggered the generation and development of disturbances in the lower troposphere. At the early stage, a lower-level disturbance was half a wavelength out of phase with a disturbance in the upper troposphere, which was directly above and induced the lower-level disturbance. So, the potential anomaly of the lower-level disturbance generally had an opposite sign to that in the upper troposphere. All disturbances propagated and moved downstream, but the upper-level disturbances moved faster than those at the lower levels due to the stronger basic flow in the midlatitude jet. Thus, the cyclonic (anticyclonic) upper-level disturbance gradually caught up with the cyclonic (anticyclonic) lower-level disturbance ahead of it. When the upper-level disturbance approached the lower-level one, they influenced each other or coupled with each other. Although the disturbances developed before they coupled with each other, the coupling process could phase-lock the two disturbances and promote their mutual intensification. In the end, the two disturbances merged to form a strong system that extended through the whole troposphere. In addition, the downstream development showed clear characteristics of being boundary-trapped; that is, at the early stage the upper-level and lower-level disturbances all developed and propagated near the boundaries (i.e., the tropopause or surface), and they had the largest amplitudes at the tropopause or the surface and decayed away from the boundaries.

Figure 1.  Time series of vertical cross section of signals. Here, signals are defined as the forecast difference of geopotential height between the sensitivity and the control experiment. Those signals with absolute values that are larger than 10 gpm between 45^°N and 55^°N were selected and meridional averaging was performed. The forecast initial time is 1200 UTC 4 September 2003, and VT is the forecast valid time with 24-h increments.

Figure 2.  Synoptic situation and calculation areas for the eddy kinetic energy analysis at 200 hPa. The rectangles shown are the calculation areas for the upstream trough (red), downstream ridge (blue), and downstream trough (dark yellow). The jet regions where wind speeds are >40 m s^-1 are outlined by green solid lines. Fabian (2003) is marked by the red symbol.

The targeting method revealed small disturbances that originated from a TC and then developed in the midlatitude jet and had the characteristics of coupling development and being boundary-trapped. Since the targeted area was located near the TC center, the signal may be an indicator of TC, and an indicator of the variation of the effects of the TC on the midlatitude flows. On the other hand, as the TC moved into the midlatitudes, the whole TC actually became an intense disturbing source to the midlatitude circulations. The TC might have acted on the midlatitute flows more intensely and induced much stronger DBD than those small disturbances. The signal propagated into the midlatitude jet through the outflow transporting low PV air into the midlatitudes. By the same approach, the TC induced the downstream systems in the jet. Moreover, the downstream systems had the same forcing source as the TC, and had the same developing environment of the midlatitude jet with the signals. Their developing processes thus should have been similar to those of the signals. After examining Fabian (2003), we found that downstream development was boundary-trapped and coupling development did exist in an actual synoptic situation. In the case of Fabian (2003), a strong ridge developed just downstream of Fabian (2003) at the upper levels after 0000 UTC 5 September, and reached its peak amplitude at around 1200 UTC 9 September. After 0000 UTC 8 September, a deep trough developed just downstream of the ridge, and moved over Western Europe from 0000 UTC 9 through 0000 UTC 11 September, with its peak amplitude at 0000 UTC 10 September 2003 (Fig. 2). The ridge and trough thus composed a ridge-trough couplet. While the downstream ridge developed in the upper troposphere, an eddy began to develop north of the subtropical high in the lower troposphere at 0000 UTC 6 September (Figs. 3 and 4). The eddy was located around (47^°N, 32^°W) and was beneath the upper-level ridge. The eddy moved eastward while developing, and moved faster than the upper-level ridge-trough couplet because the couplet was very large. The eddy, therefore, caught up with the upper-level trough, and was located exactly beneath the trough at 1200 UTC 8 September. Thereafter, the lower-level eddy coupled with the upper-level trough, and developed further into a cyclone that extended through the whole troposphere (Figs. 2-4). This cyclone stayed over Western Europe for about two days. The downstream ridge-trough couplet reached its peak amplitude at the tropopause and decayed away from it (Fig. 4). In the lower troposphere, the eddy developed earlier and was strongest at the surface (not shown); but it was weakest at 700 hPa (Fig. 4). So, the systems at the upper and lower levels all showed distinct boundary-trapped features. Moreover, prior studies also confirm that downstream baroclinic systems could develop as a TC interacts with the midlatitude flows. For example, (Riemer et al., 2008) used full-physics numerical experiments with idealized initial conditions to investigate the impact of ET on the downstream flow. They found that a ridge-trough couplet and a distinct jet streak formed at the upper levels just downstream of a TC as the TC approached the midlatitude jet. The preceding targeted studies and observational analyses all revealed the characteristics of coupling development and boundary-trapping of the DBD induced by an ET event. So, how do the downstream baroclinic waves develop? In the following, we explore the mechanisms of these characteristics, or why the downstream baroclinic waves develop. From the PV perspective, the downstream ridge is the result of the TC outflow transporting low a PV air mass into the midlatitude flow (Massacand et al., 2001; Riemer et al., 2008), but a different viewpoint may arise from the analyses of the local eddy kinetic energy budget.

First, velocity or any scalar can be partitioned into two parts, zonal mean and eddy:

\begin{eqnarray} \label{eq1} \bm{V}=\bm{V}_m+v ,\\[-1mm] \label{eq2} Q=Q_m+q . \end{eqnarray}

Here, V is velocity, Q is a scalar, and the subscript m indicates zonal mean; the lower case indicates it is an eddy variable, i.e., v is eddy velocity. The zonal mean is calculated between 100^°W and 40^°E at every time step.

The zonal-mean kinetic energy and eddy kinetic energy are defined as

\begin{equation} \label{eq3} K_{\rm m}=\dfrac{1}{2}(U_m^2+V_m^2),\quad K_{\rm e}=\dfrac{1}{2}(u^2+v^2) , \end{equation}

where (U_m,V_m) and (u,v) are the horizontal components of zonal-mean velocities and eddy velocities, respectively.

Figure 3.  The same as Fig. 2, except for 1000 hPa and the lower-level downstream trough.

Figure 4.  The upper-level downstream ridge-trough couplet (200, 300, 500 hPa) and the lower-level downstream trough (1000, 850, 700 hPa) shown at different levels and demonstrating distinct boundary-trapping.

Following (Orlanski and Katzfey, 1991) and (Chang and Orlanski, 1993), the eddy kinetic energy equation in the pressure coordinates is defined as

\begin{eqnarray} \label{eq4} \dfrac{\partial K_{\rm e}}{\partial t}&=&-\bm{V}_3\cdot\nabla K_{\rm e}-(\nabla\cdot\bm{v}_3\varphi)-\omega\alpha-\bm{v}\cdot(\bm{v}_3\cdot\nabla \bm{V}_m)+\nonumber\\[-0.5mm] &&\bm{v}\cdot\overline{(\bm{v}_3\cdot\nabla \bm{v})}+{\rm diss} ,\\[-3.5mm]\nonumber \end{eqnarray}

where the vector quantities with subscript 3 represent 3D vectors, and those without the subscript refer to the horizontal components. ∇ is a 3D gradient operator, and the overbar indicates zonal mean; ω is the vertical velocity on an isobaric surface; α is specific volume; and φ is geopotential. The term on the left is the time tendency of K _e. The first term on the right is the advection of K _e. The second term on the right is the geopotential flux divergence term, which is entirely due to ageostrophic geopotential fluxes (Orlanski and Katzfey, 1991). The third term on the right represents the baroclinic conversion. The fourth and fifth terms on the right represent barotropic conversions; and the final term represents friction dissipation and diffusion.

Figure 5.  Evolution of volume-mean eddy kinetic energy budget terms (m² s^-3) for individual eddies. The curves shown are contributions from the ageostrophic geopotential flux term (solid), barotropic conversion (long dash), advection (dotted), and baroclinic conversion (dot dot dash). The volume integrated mean of all terms for the TC were calculated throughout the whole troposphere.

The rectangles shown in Figs. 2 and 3 delineate the volumes in which the volume mean of every eddy kinetic energy budget term except friction dissipation is calculated for those developing individual eddies. In general, the volumes are located at the centers of the eddies, and have side lengths of 15^° latitude and 15^° longitude. But for the upper-level downstream ridge, the ridge line is set as the western boundary for the calculation less the volume overlapping with that of the TC. The side length is adjusted to 8^{°× 8°} for Hurricane Fabian (2003) and to 10^{°× 10°} for the lower-level eddy because of their smaller extents, and the zonal length is adjusted to 30^° for the upstream trough after 1200 UTC 7 September because of its larger zonal extent. This study focuses on thin layers because the eddies developed and propagated near the boundaries at the early stage and were distinctly boundary-trapped. Therefore, in the vertical, a volume is confined in a thin layer that is bounded by two levels, i.e., 200 hPa and 300 hPa for the upper-level eddies, and 1000 hPa and 925 hPa for the lower-level eddies. Volume-mean energy budget terms are computed following the volumes.

Figure 5a shows that all the terms were small in the early stage, and the barotropic conversion term was even negative, indicating a conversion from eddy kinetic energy to mean kinetic energy. After 0000 UTC 5 September 2003, the ageostrophic geopotential flux divergence term increased quickly to reach a peak at 0000 UTC 9 September. This occurred in conjunction with the development of the downstream ridge (Fig. 2b). The barotropic conversion term became positive only after 0000 UTC 6 September, and increased quickly afterwards. At this time, the barotropic conversion and ageostrophic flux became the main sources of eddy kinetic energy of the downstream ridge. The advection term was always small before 1200 UTC 8 September, but had a large negative value around 0000 UTC 9 September, indicating a strong advection of eddy kinetic energy from the ridge to the downstream trough. The baroclinic conversion was very small all the time.

According to the above results, the ageostrophic geopotential flux term dominated the ridge growth just downstream of the TC. Thus, the question naturally arises as to where the ageostrophic geopotential fluxes came from. The upstream systems that influenced the ridge directly were the upstream trough (Fig. 2) and Fabian (2003). Did the ageostrophic geopotential fluxes responsible for the downstream ridge growth originate from the upstream trough or Fabian (2003)?

In Fig. 6, the ageostrophic geopotential flux term for Fabian (2003) was calculated in a layer in the same way as that for the downstream ridge or the upstream trough and from 0000 UTC 4 through 1200 UTC 9 September 2003. It was negative all the time, which indicates that Fabian (2003) radiated energy downstream via ageostrophic flux divergence. Another important point to note is that the energy gained by the downstream ridge via ageostrophic flux convergence was roughly equal to the energy lost by Hurricane Fabian (2003) via ageostrophic flux divergence. In other words, the absolute value of the ageostrophic flux term for the downstream ridge was roughly equal to that for Fabian (2003). However, its development dropped behind that for Fabian (2003) for a little time, although they had opposite signs. In Fig. 6, two minima of the flux term for Fabian (2003) appeared at 1200 UTC 6 September and 1200 UTC 8 September indicating the maximum downstream radiation of eddy kinetic energy from Fabian (2003). Corresponding to the two minima in the fluxes from Fabian (2003), two maxima of the flux term for the downstream ridge appeared at 0000 UTC 7 September and 0000 UTC 9 September, and they indicated the maximum import of eddy kinetic energy from upstream. A local minimum of the flux term for the downstream ridge appeared at 1200 UTC 7 September. It corresponded to the maximum of the flux term for Fabian (2003) at 0000 UTC 7 September, and occurred 12 hours later than the maximum. For the upstream trough, the ageostrophic flux term was positive and very small most of the time, indicating no fluxes radiated downstream from the upstream trough. In Fig. 7, the strong ageostrophic geopotential flux vectors at 200 hPa at 0000 UTC 06 September clearly originated from Hurricane Fabian (2003). The directions and covering region of the vectors corresponded well to those of the outflow [high θ region to the north and northeast of Fabian (2003) in Fig. 7], and most of the covering region coincided with the downstream ridge. According to the above analysis, it is clear that the ageostrophic geopotential fluxes that were responsible for the downstream ridge growth did originate from Fabian (2003). As a weather system generated in the tropical area with strong rotation, the flows inside a TC are highly ageostrophic. When a TC moves northward and interacts with an upper-level trough, its outflow can be strengthened by the strong southwesterly flow ahead of the trough or it can be destroyed by shear. The outflow transported the ageostrophic geopotential fluxes from Fabian (2003) into the midlatitude jet, and the fluxes led to the downstream ridge growth. As Fabian (2003) moved northward, it radiated the ageostrophic flux downstream, and the radiation led to the decay of Fabian (2003) itself. It should be noted that while the ageostrophic flux term for the ridge increased, the barotropic conversion term also increased. The ageostrophic fluxes from Fabian (2003) might have triggered barotropic conversion growth in the downstream ridge, and barotropic conversion also became the main energy source for the ridge development.

Figure 6.  Comparison of ageostrophic flux divergence evolution among the downstream ridge (solid line), TC (dashed line), and upstream trough (dotted line). The flux divergence for the TC was calculated in a layer the same as that for the ridge or the trough.

Figure 7.  Ageostrophic geopotential flux vectors in the area (30^°-60^°N, 80^°-20^°W) at 200 hPa at 0000 UTC 06 September 2003. The solid line is the 200 hPa geopotential height. Shaded areas are potential temperature (θ) (^°C) on a dynamic tropopause.

The fact that ω was very small at the boundaries resulted in a baroclinic conversion (the conversion between eddy potential energy and eddy kinetic energy) always being small near the tropopause and the surface. Actually, the baroclinic conversion was not small at mid-levels, and it was responsible for the eddy growth together with the ageostrophic flux convergence (not shown). We also calculated the energy budget for Hurricane Fabian (2003), but the vertical average is defined through the whole troposphere. We found that baroclinic conversion played a major role in maintaining the TC during its ET process together with barotropic conversion, while ageostrophic fluxes and advection dispersed the TC's energy (Fig. 5d). Strong barotropic or inertial instabilities originally existed in the jet because of its strong basic flow and shear, and these instabilities would be enhanced because the jet was strengthened by the outflow (discussed later). Therefore, the barotropic conversion was much larger than the baroclinic conversion at the upper levels in the midlatitude jet (especially in the outflow region). We performed an energy budget calculation on the vertical integration in thin layers near the boundaries because the ET system first disturbed the midlatitude flows at the levels below the tropopause and the downstream development at the early stage mainly occurred at the boundaries. Moreover, the downstream development occurred in a weak baroclinic and jet environment. Therefore, it was barotropic conversion, not baroclinic conversion, that played an important role at the upper levels, since eddy development might be very different in some specific layers and under some specific synoptic conditions. But did baroclinic conversion really contribute little to the eddy kinetic energy growth at the upper levels? What was its role during the downstream development?

As indicated in CP2010, the outflow of a TC strengthened by the upstream trough transports air mass with a negative PV anomaly into the midlatitude jet below the tropopause, which leads to a raised tropopause that constructs a strongly sloping tropopause together with tropopause depression to the north and west of it. Therefore, the outflow of the TC contributes to the steep PV gradient in the region of the sloping tropopause. The baroclinicity in this region is enhanced, and the jet becomes stronger. The strongest jet is always located just inside the region, and moves along with the TC. To investigate the changes of baroclinicity and average velocity in the outflow region, we selected an area (45^°-55^°N, 65^°-45^°W), which is located in the jet region, for calculation. The outflow moved into the area at about 0000 UTC 5 September and moved out at about 1200 UTC 8 September according to CP2010. The maximum Eady growth rate was used to measure the baroclinicity of the atmosphere (Chang and Orlanski, 1993; Orlanski and Sheldon, 1993), which is defined in isobaric coordinates as

\begin{equation} \label{eq5} \sigma_{\rm BI}=\dfrac{0.31f\left|\dfrac{\partial U_h}{\partial P}\right|\sqrt{\dfrac{P}{R\overline{T}}}}{\sqrt{\left|\dfrac{\partial\ln\bar{\theta}}{\partial P}\right|}} , \end{equation}

where U_h denotes the magnitude of the horizontal velocity vector, and is also used to calculate the average velocity in the outflow region. Variables f,P,T,θ are geostrophic parameter, pressure, temperature and potential temperature respectively. Strong baroclinicity in a region indicates there may be more potential energy, strong baroclinic instability, and strong conversion of potential energy to kinetic energy. The Eady growth rate was calculated at 200 hPa. In Fig. 8, the Eady growth rate and velocity are normalized by their time-mean values; that is, the values in Fig. 8 are only multiples of the time-mean values of the Eady growth rate or velocity. The Eady growth rate increased as the outflow moved into the selected area, and decreased as the outflow moved out of the area. The maximum value was four times larger than the initial value. The mean velocity at 200 hPa also increased along with the Eady growth rate, and the maximum value was 1.4 times larger than the initial value. The velocity increase at 300 hPa was small and the velocity at 400 hPa did not increase at all. These results indicate that the baroclinicity in the outflow region was enhanced greatly by the outflow. At the same time, the midlatitude jet close to the tropopause was also strengthened, and a strong jet streak formed at the location of the downstream ridge. The above results may indicate that it is difficult for eddy potential energy to be converted to eddy kinetic energy directly in the upper-level jet. However, potential energy can be converted to mean kinetic energy of the jet first, which in turn is converted to eddy kinetic energy through barotropic conversion.

Figure 8.  Evolution of baroclinicity and average velocities in the jet region (45^°-55^°N, 65^°-45^°W): maximum Eady growth rate (solid line); and average velocity at 200 hPa (dashed line), 300 hPa (dotted line), and 400 hPa (dot-dashed line). The outflow moved into the area at about 0000 UTC 5 September 2003 and out at about 1200 UTC 8 September 2003.

As a component of the ridge-trough couplet, the downstream trough developed much later, compared to the ridge. It began to develop only after the ridge was well developed. At that time, the strong jet streak had already formed and coincided with the ridge (Fig. 2d). After 0000 UTC 10 September 2003, the trough began to decay. It is clear from Fig. 5b that the term dominating the growth was the advection term. The ageostrophic flux term and the barotropic conversion term were both small during the whole growth process. Barotropic conversion was negative most of the time, and its absolute value was large after 0000 UTC 10 September, indicating that it played a primary role during the trough's decay. The advection term was negative after 0000 UTC 10 September. Therefore, it was also responsible for the trough's decay, but played a secondary role. The ageostrophic flux term was negative most of the time during the growth stage, and was positive during the decay stage. However, its absolute value was never large; thus, it was not important during either stage. Similar to the above, the baroclinic conversion term was very small all the time.

Note that the advection term was negative most of the time after 0000 UTC 6 September 2003 and had a very large absolute value around 0000 UTC 9 September (Fig. 5a), indicating strong transportation of eddy kinetic energy out of the ridge during this period. The negative peak of the advection term in Fig. 5a was ahead of the positive one in Fig. 5b by about half a day. Therefore, the advection of energy responsible for the trough growth can be traced to the eddy kinetic energy that was advected out of the ridge. In other words, the eddy kinetic energy that was transported out of the ridge by advection led to the development of the downstream trough. Furthermore, it did not lead to the ridge's decay because it was compensated for by the convergence of the ageostrophic geopotential flux. Therefore, the downstream trough had a life cycle of growth (due to advection) and decay (due to barotropic conversion), and the advection term also played a role during the trough's decay. In the DBD theory, the ageostrophic flux term is important for both eddy growth and decay, but in the case of Fabian (2003), it was small during both stages of the downstream trough. The mechanism of a strong system radiating ageostrophic fluxes downstream to dominate the downstream eddy growth should be investigated further. Since a system usually radiates ageostrophic fluxes outward as it decays, the energy source for the downstream trough growth was not the ageostrophic fluxes from the ridge because the ridge did not decay during this period.

We already know that the downstream ridge and the strong jet streak formed almost at the same time due to the outflow of Fabian (2003). It is clear from Fig. 2d and Figs. 4a and b that the strong flow in the jet streak might have intersected the isotachs (especially that of 60 m s^-1) by a large angle in the ridge under this synoptic situation; thus, a strong downstream advection of eddy kinetic energy formed around the exit region of the jet streak. The ridge triggered the development of the downstream trough through the advection of eddy kinetic energy. From the above analyses, we found that the relative importance of the eddy kinetic energy budget terms can be very different under different synoptic situations. Every term can dominate eddy growth or decay.

In Fig. 5c, the only term that dominated the eddy growth was the ageostrophic geopotential flux term. Contrary to the situation in the upper troposphere, the barotropic conversion did not contribute to the growth at all. Barotropic conversion was small most of the time because the weak flow at the lower levels resulted in weak barotropic instability, and it had a large negative value prior to 1200 UTC 7 September 2003, indicating a conversion from eddy kinetic energy to mean kinetic energy. The baroclinic conversion term was still small and the advection term was also small. To identify the source of the eddy energy, we performed horizontal and vertical decompositions for the ageostrophic geopotential flux. In Fig. 9, the horizontal component was positive prior to 0000 UTC 7 September. Thereafter, it changed to negative and decreased continuously with time (its absolute value increased). Contrary to the horizontal component, the vertical component was positive all the time and increased continuously with time.

Figure 9.  Decomposition for lower-level flux divergence: total (solid line); horizontal component (dashed line); vertical component (dotted line).

Figure 10.  (a) Vertical cross section of vertical ageostrophic geopotential flux (m² s^-2) at 47^°N at 0000 UTC 09 September 2003. Panel (b) is the same as (a), except for its divergence.

To further clarify the effect of vertical processes, we show the vertical sections along 47^°N of vertical ageostrophic geopotential flux (Fig. 10a) and its divergence (Fig. 10b) at 0000 UTC 9 September 2003. The two sections all went through the center of the developed cyclone, which remained over Western Europe at 0000 UTC 9 September (Fig. 3). In Fig. 10b, strong flux divergences are apparent at the upper levels and flux convergences at the lower levels at around the 0^° meridian where the cyclone was located. Both the divergences at the upper levels and the convergences at the lower levels had the largest amplitudes at the boundaries (i.e., the tropopause or the surface), and decayed away from the boundaries. Figure 10a indicates that strong vertical ageostrophic fluxes also existed in the area around the 0^° meridian and extended throughout the troposphere, and their directions were downward (the vertical velocity used here is that at the isobaric surface). Combining the analyses based on Fig. 10 with those based on Figs 5c and 9, we can deduce that both the genesis and the development of the lower-level eddy were the result of the upper-level systems (i.e., the downstream ridge and trough), which transported vertical ageostrophic geopotential fluxes downward. Vertical fluxes were also the crucial dynamical factor for the eddy's strong coupling with the trough. The horizontal component of the ageostrophic flux dispersed the eddy energy, and became the main energy sink for the lower-level eddy most of the time during the eddy growth.

Although boundary-trapping existed at both upper and lower levels, the mechanisms might have been quite different. In general, the jet strength is directly proportional to the geopotential height gradient. The strengthened jet streak and flux transportation due to the outflow were also located near the tropopause in the case of Fabian (2003). Corresponding to these dynamic characteristics, the barotropic conversion and ageostrophic flux convergence were also enhanced with height in the jet. Since the ageostrophic flux and barotropic conversion dominated the eddy growth in the upper level, the ridge-trough couplet was strengthened with height and reached its peak amplitude at the tropopause. In the lower troposphere, vertical ageostrophic geopotential fluxes converged strongly due to the obstruction of the surface when the downward fluxes encountered the surface. The closer to the surface, the more strongly the fluxes converged. The boundary-trapping of the lower-level eddies was just the response to the boundary-trapping of the vertical ageostrophic flux divergence. However, we can propose a question regarding the other aspect of boundary-trapping; that is, is it possible that well-developed systems in the lower troposphere transport vertical ageostrophic fluxes upward and lead to the development of upper-level systems?

This study investigated how the ET process of Hurricane Fabian (2003) induced DBD in the midlatitude jet environment. Eddy kinetic energy diagnosis following (Orlanski and Katzfey, 1991) and (Chang and Orlanski, 1993) and a targeting method were employed. The signals that were the forecast differences between the sensitivity experiment and the control experiment can be considered as geopotential disturbances. Regardless of the signal propagation or the actual synoptic situation of Fabian (2003), they all revealed that the downstream development had distinct characteristics of coupling development and being boundary-trapped. Under the effects of the ET system, disturbances first developed and propagated downstream in the upper troposphere and then triggered disturbance development in the lower troposphere. At the early stage, the disturbances all developed and propagated near the boundaries (i.e., the tropopause or surface), and decayed away from them. Thereafter, a lower-level eddy caught up with an upper-level eddy ahead of it (or vice versa), and they coupled to form a cyclone that extended through the whole troposphere.

The ageostrophic geopotential fluxes that were transported from Hurricane Fabian (2003) to the midlatitude by the outflow became the main source of eddy kinetic energy for the downstream ridge development. The TC undergoing ET also led to the baroclinicity enhancement in the outflow region. The midlatitude jet close to the tropopause was thus strengthened, and a strong jet streak formed at the location of the downstream ridge. Barotropic conversion also became the primary energy source after the development was well under way because of the strong barotropic or inertial instabilities in the jet streak. The strong downstream advection of eddy kinetic energy around the exit region of the jet streak was the energy sink of the downstream ridge, but it dominated the growth of a further downstream trough. Baroclinic conversion was negligible near the boundaries while being large enough to be responsible for eddy growth together with ageostrophic flux convergence at the mid-levels.In the upper-level jet, it was difficult for eddy potential energy to convert to eddy kinetic energy directly. However, potential energy could be converted to the mean kinetic energy of the jet, which thereafter was converted to the eddy kinetic energy through barotropic conversion. As the downstream ridge developed, it transported vertical ageostrophic geopotential fluxes downward. The vertical fluxes converged at the lower levels due to the obstruction of the surface. The lower-level eddy development was just the response to the vertical flux convergence. The mechanisms of boundary-trapping at the upper and lower levels might be quite different. Since the ageostrophic flux and barotropic conversion were enhanced with height due to vertical distribution of the jet strength, the ridge-trough couplet thus decayed from the tropopause. In the lower troposphere, the closer to the surface, the more strongly the vertical ageostrophic fluxes converged. Therefore, the boundary-trapping of the lower-level eddies was just the response to the vertical ageostrophic flux convergence.

Although the results of eddy kinetic energy analyses are at odds with classic DBD theory, they do not mean the results go against DBD theory. Some factors allow for the discrepancies between them to exist reasonably. The downstream development in this study was induced by the ET system and occurred in a weak baroclinic environment, but not in the upstream waves and strong baroclinic environment in the DBD. Since eddies developed near the boundaries at the early stage, a volume used to integrate the eddy kinetic energy was bounded vertically in a thin layer close to each of the boundaries, but not in the whole troposphere as in the DBD theory. Due to small ω near the boundaries, the baroclinic conversion there was very small; that is, the eddy growth in a specific layer may be markedly different from that in the whole troposphere. Other unique circulation backgrounds existed in the case of Fabian (2003), which could lead to other terms other than the baroclinic conversion term dominating the eddy growth. Due to strong barotropic instabilities in the upper-level jet streak strengthened by the TC, barotropic conversion became the main energy source. In addition, the strong downstream advection of eddy kinetic energy in the exit region of the jet streak was responsible for the ridge's decay and the further growth of the downstream trough.

It is important to note that this is just one case study. To confirm the mechanism of the downstream development, more cases should be examined. Moreover, how the downstream development works in a strong baroclinic environment should also be investigated. Based on the results, further questions arise. For instance, is it possible that vertical ageostrophic fluxes could be transported upward by well-developed systems in the lower troposphere and then converge near the tropopause to result in upper-level system development? And in what situation do systems radiate ageostrophic fluxes outward? The role of upstream impact during lower-level system development should be further investigated. The roles of other eddy energy budget terms in different synoptic situations also need to be investigated.