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Barotropic Regeneration of Upper-Level Synoptic Disturbances in Different Configurations of the Zonal Weather Regime

Barotropic dynamics of upper-tropospheric midlatitude disturbances evolving in different configurations of the zonal weather regime (i.e., in different zonal-like large-scale flows) were studied using observational analyses and barotropic model experiments. The contraction stage of upper-level disturbances that follows their elongation stage leads to an increase of eddy kinetic energy that is called the barotropic regeneration process in this text. This barotropic mechanism is studied through notions of barotropic critical regions (BtCRs) and effective deformation that have been introduced in a previous paper. The effective deformation field is equal to the difference between the square of the large-scale deformation magnitude and the square of the large-scale vorticity. Regions where the effective deformation is positive correspond to regions where the large-scale flow tends to strongly stretch synoptic disturbances. A BtCR is an area separating two large-scale regions of positive effective deformation, one located upstream and on the south side of the jet and the other downstream and on the north side. Such a region presents a discontinuity in the orientation of the dilatation axes and is a potential area where the barotropic regeneration process may occur. Winter days presenting a zonal weather regime in the 40-yr ECMWF Re-Analysis dataset are decomposed, via a partitioning algorithm, into different configurations of the effective deformation field at 300 hPa. A six-cluster partition is obtained. Composite maps of the barotropic generation rate for each cluster exhibit a succession of negative and positive values on both sides of the BtCRs. It confirms statistically that the barotropic regeneration mechanism occurs preferentially about BtCRs. Numerical experiments using a forced barotropic model on the sphere are performed. Each experiment consists of adding a synoptic-scale perturbation to one of the zonal-like jet configurations found in observations, which is kept fixed with time. The combined effects of the effective deformation and nonlinearities are shown to be crucial to reproduce the barotropic regeneration process about BtCRs.

The contraction stage of upper-level disturbances that follows their elongation stage leads to an increase of eddy kinetic energy and is called barotropic regeneration process in the text. This barotropic mechanism is studied through the notions of barotropic critical regions (BtCR) and effective deformation that have been introduced in a previous paper. The effective deformation field is equal to the difference between the square of the large-scale deformation magnitude and the square of the large-scale vorticity. Regions where the effective deformation is positive correspond to regions where the large-scale flow tends to strongly stretch synoptic disturbances. A BtCR region is an area separating two large-scale regions of positive effective deformation, one located upstream and on the south side of the jet and the other downstream and on the north side. Such a region presents a discontinuity in the dilatation axes orientation and is a potential area where the barotropic regeneration process may occur.
Winter days presenting a zonal weather regime in the ECMWF 40-year reanalysis data set are decomposed, via a partitioning algorithm, into different configurations of the effective deformation field at 300 hPa. A 6-cluster partition is obtained.
Composite maps of the barotropic generation rate for each cluster exhibit a succession of negative and positive values on both sides of the BtCR regions. It confirms statistically that the barotropic regeneration mechanism occurs preferentially about BtCR regions.
Numerical experiments using a forced barotropic model on the sphere are performed. Each experiment consists in adding a synoptic-scale perturbation initially to one of the zonal-like jet configuration found in the observational part that is maintained fixed with time. The combined effects of the effective deformation and nonlinearities are shown to be crucial to reproduce the barotropic regeneration process about BtCR regions.

Introduction
Synoptic disturbances in the upper troposphere take the form of coherent wave packets with particularly intense mobile troughs surrounding the globe in midlatitudes and play a crucial role in the dynamics of various weather systems. First, they are involved in the triggering of extratropical surface cyclogenesis via baroclinic interaction (Petterssen and Smebye 1971) and second, they act onto the large-scale atmospheric circulation through eddy momentum fluxes (Hoskins et al. 1983). In other words, they form with the surface cyclones the two main components of the so-called storm tracks and their life cycles are thus particularly important to document and analyze. Among the processes involved in the dynamics of upper-tropospheric disturbances are baroclinic interaction, downstream development and barotropic modulation (e.g., Nielsen-Gammon and Lefevre 1996; Chang et al. 2002). The aim of the present study is to focus on the latter process, i.e on the barotropic effects resulting from the horizontal variations of the background large-scale flow.
The climatology study of Sanders (1988) was among the first to notice that upper-level mobile troughs are usually generated east of the major mountain chains, i.e east of the Rocky mountains in North America and east of the Himalayas in Asia. Their births are particularly favoured in presence of a northwesterly flow over these regions (Schultz and Sanders 2002). By contrast, the deaths of upper-level mobile troughs occur preferentially over the oceans. Lackmann et al. (1997), Lackmann et al. (1999) and more recently Schultz and Sanders (2002) have shown that the genesis of upper-tropospheric troughs east of the mountains can be explained by barotropic processes. An initially elongated trough evolving in a northwesterly flow, say over North America, is rapidly compacted east of the Rockies leading to a deepening of the trough as well as an increase of eddy kinetic energy. This mechanism is confirmed by the cli-matological maps of barotropic generation rate made independently by Lee (2000) and Black and Dole (2000). The barotropic generation rate characterizes the amount of kinetic energy extracted by the eddies from the background flow; it is positive east of the mountains and strongly negative over the Pacific and Atlantic leading to an increase and a loss of eddy kinetic energy in the respective regions. Lee (2000) explains this result by the fact that meridionally elongated waves in the diffluent region of the Pacific jet for example are contracted when arriving in the confluent region at the entrance of the Atlantic jet. Furthermore, another area of positive barotropic generation rate exists over the North African subtropical jet; eddies evolving along the extratropical Atlantic jet get a northeast-southwest tilt in its exit region, propagate equatorward and are compacted on the cyclonic side of the subtropical jet. Following all these studies, the elongation-contraction process (called hereafter barotropic regeneration process) takes place preferentially east of the mountains in a confluent region or north of the subtropical jets. It is worth noting that this process is a transient nonmodal phenomenon that may occur without the presence of barotropic instability as shown theoretically in simple linear barotropic models by Farrell (1989) and Lee (1995).
The transition from elongation to contraction stages needs a rapid change of the environment in which disturbances are embedded and in particular of the dilatation axes orientation. This is the case from a diffluent to a confluent region or from the anticyclonic side of the eddydriven jet to the cyclonic side of the subtropical jet. Despite the lack of climatological evidences of such process over the ocean, case studies of Rivière and Joly (2006a) (hereafter denoted as RJ06a) show that it may also occur sometimes over the Atlantic when an upper disturbance crosses the extratropical jet from the south to the north. This crossing takes place in specific regions of the large-scale extratropical flow called barotropic critical regions (denoted hereafter as BtCR). Following theoretical results of Rivière et al. (2003) (hereafter denoted as RHK03), RJ06a defines a new quantity called effective deformation and based on the stretching, shearing and vorticity components of the large-scale deformation tensor. This quantity is shown to be more relevant than the classical deformation magnitude to analyze deformation effects. A perturbation is strongly stretched in a region of strongly positive effective deformation but not necessarily in a region of large deformation magnitude. BtCRs regions are defined from horizontal maps of the effective deformation field. They separate two large-scale regions of positive effective deformation and are characterized by a rapid change of the dilatation axes orientation.
Note finally that BtCRs regions are shown in RJ06a to play an important role in the regeneration of both upper-level synoptic disturbances and surface cyclones.
The notions of weather regimes have been introduced by Vautard (1990) to represent quasistationary circulations of the large-scale atmospheric flow. Four weather regimes are identified in the Atlantic; the Atlantic ridge, the zonal regime, the Greenland anticyclone and the blocking. The zonal regime correspond to a zonal-like jet along the latitude 45 • N. Two distinct periods corresponding both to a zonal weather regime are studied in RJ06a. Despite the apparent similarities between the two zonal-like jets, they present very different configurations of the effective deformation fields. For one jet (mid-February 1997), a BtCR region and its associated barotropic regeneration process is present at the jet exit whereas for the other (end of December 1999) it is at the jet entrance only. Two apparently similar zonal jets lead therefore to very different behavior of the synoptic eddies. The aim of the present study is first to validate statistically these results and to analyze systematically the different configurations of the zonal weather regime in terms of their effective deformation fields and their impact onto synoptic upper-tropospheric disturbances. To perform this statistical study, ERA40 reanalysis data of the European Centre for Medium-Range Weather Forecast is used and analyzed for all the winter periods from 1957 to 2001. The second purpose of the study is to be able to reproduce the same mechanism as that shown in reanalysis data in idealized barotropic simulations and to identify key factors that explain the whole barotropic regeneration process about a BtCR region.
The paper is decomposed as follows. Barotropic dynamics related to the notions of effective deformation field and barotropic critical regions are briefly recalled in section 2. Section 3 is dedicated to the observations of the different configurations of the zonal weather regime.
The dynamical cluster algorithm of Michelangeli et al. (1995) (hereafter denoted as MVL95) is introduced to separate zonal weather regimes into different configurations of the effective deformation field at 300 hPa. In the same section, a statistical analysis of the barotropic regeneration process about a BtCR region is described. In section 4, numerical experiments in a forced barotropic model on the sphere are performed. Each experiment consists in adding a synoptic-scale perturbation initially to one of the zonal-like jet configuration found in section 3 that is maintained fixed during the whole simulation. Section 4 completes the results of section 3 and show the crucial importance of the combined effects of the effective deformation and the nonlinear terms to reproduce the barotropic regeneration process about a BtCR region. A conclusion is provided in section 5.

Barotropic dynamics a. Barotropic generation rate and effective deformation
The following theoretical aspects are described in details in RHK03 within the quasigeostrophic framework. They have been then applied to the real atmosphere in RJ06a and are briefly recalled in this section.
The barotropic energy exchanges between the high-frequency component (denoted with primes) and the low-frequency component (denoted with subscript m) of the atmospheric flow are usually diagnosed by the so called barotropic generation rate that can be expressed following Mak and Cai (1989) as the scalar product E.D m where (2) u and v are the horizontal velocity components, ϕ is latitude, a is the earth radius, ∂/∂x ≡ (a cos ϕ) −1 ∂/∂λ and ∂/∂y ≡ a −1 ∂/∂ϕ are the horizontal derivatives and λ is longitude. When E.D m is positive (negative), the perturbation extracts energy from (loses energy to) the reference flow. The barotropic generation rate is maximum (minimum) when the perturbation velocity vector aligns with the contraction (dilatation) axis. This leads to E.D m = +K ′ e .σ m (−K ′ e .σ m ) where σ m ≡ |D m | is the deformation magnitude and K ′ e ≡ 1 2 (u ′2 + v ′2 ) is the perturbation kinetic energy.
However, the dilatation/contraction axes are not usually the equilibrium axes along which the perturbation orientation tends to align. In fact, two equilibrium axes exist whose exact formulation are provided in RHK03; one induces perturbation kinetic energy growth and is an unstable equilibrium while the other perturbation decay and is stable. These orientations lead to the following approximate expression of the barotropic generation rate in the case of low-frequency flows with weak curvatures: E.D m = ±K ′ e . √ ∆ m . ∆ m is called effective deformation and is defined as where ζ m ≡ ∂v m /∂x − ∂u m /∂y − u m tan ϕ/a is the rotation rate relative to the basis set (i, j) (i and j being the unit vectors directed eastward and northward). The sign of ∆ m is crucial as it determines regions where equilibrium axes exist and where they do not. Positive values of ∆ m correspond to regions where synoptic eddies can be strongly stretched due to the deformation action, and where barotropic processes can be very important. Note in particular that in the extreme case where ∆ m >> 0, the equilibrium axes are close to the dilatation/contraction axes.
By contrast in regions of negative ∆ m , the rotation of the large-scale flow is so strong that the perturbation cannot be elongated along specific orientations. In the latter environment, the perturbation main axis keeps rotating and in average barotropic processes are quite weak.

b. Barotropic critical regions
As the stable equilibrium axis leads to destruction of kinetic energy, the high-frequency kinetic energy necessarily decreases after a while if the environment of the high-frequency eddy does not change. Barotropic development has therefore very good chances to occur when there is a rapid change of the environment and this can happen around barotropic critical regions.
A barotropic critical region is defined as a specific area separating two regions of strongly positive effective deformation where the dilatation axes (or the stable equilibrium axes) are almost perpendicular upstream and downstream of it. The mechanism is the following about a BtCR region. Upstream of this area, the eddy is strongly stretched along the dilatation axes and lose energy. As it crosses the BtCR region, its orientation becomes perpendicular to the new dilatation axes and gains temporarily energy slightly donwstream of the BtCR region. After a while evolving in this new environment, it is stretched again and loses energy. The aim of next section is to prove that this mechanism is statistically relevant.

Observations of different configurations of zonal weather regimes a. Data
The Data used in this section are the geopotential and the horizontal wind components ex-  (2) and (3). Low-and high-frequency components of the flow correspond hereafter to periods greater and less than ten days respectively and have been obtained from a Lanczos temporal filter.

b. Methodology
The principles of the dynamical cluster algorithm used in this section are briefly recalled here but more details are provided in MVL95. In what follows, the algorithm is applied to two different atmospheric fields, the geopotential and the effective deformation fields.

1) DYNAMICAL CLUSTER ALGORITHM
Before using the partitioning algorithm, an EOF analysis is performed on a given field X in a given domain in order to reduce the number of degrees of freedom. The aim is to choose an appropriate number of EOFs that explain a sufficient part of the total variance of X. For example, if X is the geopotential field, 20 EOFs are enough to explain about 95 % of the variance. The partitioning algorithm is then applied to the components of the field X in this reduced EOFs subspace whose dimension is much smaller than the grid-point space.
Let us now define the partitioning algorithm itself. Given a number of clusters k, the principle of the algorithm is to partition the sample of fields considered into k clusters such as the sum of the variances inside each cluster is minimized. The iterative algorithm starts with k random fields chosen among the data and looks for a minimization of the sum of the variances. The minimum found is generally not the global one but is close to it. As the result of each partition depends on the initial random fields, this step is repeated few times (50 times as in MVL95) and the best partition is chosen among them.
The next step is to define a classifiability index to find the right number of clusters k as described in MVL95 and Plaut and Simonnet (2001). The classifiability index c(k) is a measure estimating the resemblance of the different partitions obtained with the different random initializations. When this quantity is close to 1, it means that the different partitions are close to each other. The right number of clusters is however not that leading to the largest classifiability index because c(k) usually decreases with k for any sample. The best way is to compare for each k the classifiability index of the sample we wish to partition with the classifiability indexes obtained from randomly built samples. We then choose the index c(k) that is significantly greater than the classifiability indexes of the random samples.

2) CLASSIFICATION INTO FOUR WEATHER REGIMES
The previously described algorithm has been applied to the low-frequency (periods greater than 10 days) geopotential field at 500 hPa for all the winter days of the period 1958-2001, i.e to 7837 days, and to the following Atlantic domain (28.5 • N-79.5 • N, 79.5 • W-28.5 • E). By applying the test based on the classifiability index, k=4 and k=8 were found to be the two best suitable numbers of clusters k. Note that k=4 was also the number found by MVL95 and Plaut and Simonnet (2001) using the same method but applied to other datasets. k=4 is the number chosen in the present study. It leads to the so-called four weather regimes described initially by Vautard (1990); the Atlantic ridge, the zonal regime, the Greenland anticyclone and the blocking that represent respectively 1856, 2364, 1709 and 1908 days (see Figs.1.a-d).

3) CLASSIFICATION OF THE EFFECTIVE DEFORMATION FIELDS FOR THE ZONAL REGIME
Our aim is to focus on the zonal regime and to find different configurations in terms of the effective deformation field. We apply to the 2364 days of zonal regime the same cluster algorithm but to the effective deformation field at 300 hPa. 300 hPa is chosen as it corresponds to the upper troposphere where the low-frequency jet is stronger and its horizontal gradients too (500 hPa has been also tested and does not change the result of the partition). The number of EOFS needed to represent a sufficient part of the variance of the effective deformation field is much larger than in the case of the geopotential as the former is a much smaller-scale field than the latter. In our study, 100 EOFs are kept and represent 85 % of the total variance.
The classifiability index obtained after applying the partitioning algorithm is shown in Fig.2.
k=2 and k=6 are the two best choices for the number of clusters as for these specific values of k the classifiability index of our sample (denoted with a circle) is significantly larger than that computed from randomly built samples (shaded areas). We keep k=6 in order to have stronger correlations inside each cluster and also because k=2 leads systematically to very high classifiability index which is usually artificial (see MLV95).

c. Composite patterns
The present section is dedicated to the presentation of the 6-cluster partition of the zonal and Christmas 1999, are respectively included in clusters 1 and 6. As noticed in the previously mentioned paper, there are some jets that exhibit a BtCR region at the jet exit and others that do not. In the present 6-cluster partition, clusters 1, 2, 3 and 4 present a BtCR region at the jet exit east of 40 • W while clusters 5 and 6 do not.
The effective deformation field is compared to the classical deformation magnitude for the particular case of cluster 6 (see shaded contours of σ 2 m in Fig.4 and ∆ m in Fig.3f). Similarities exist as both fields reach their maximum values north of the jet between 40 • W and 0 • W but discrepancies are also present as strong values of σ 2 m are visible in regions of negative effective deformation south of the jet between 50 • W and 30 • W (it is a region of strong vorticity due to the anticyclonic curvature of the jet at these longitudes). Furthermore, the BtCR point called 6.A in Fig.3f is not a remarkable point of the field σ 2 m and cannot be detected by this classical diagnostic. the BtCR region seems therefore to be a crucial point to reproduce the barotropic regeneration process. This aspect will be analyzed in next section. The exception of this rule is the point 2.B which exhibits a crossing of ∆ m from the south to the north without presenting evidences of the regeneration process (values of E.D m are very weak downstream of 2.B). This may be due to the fact that 2.B is located far downstream in the jet-exit region and that there is no much kinetic energy crossing this region (see Fig.7b).
Composites of high-frequency eddy kinetic energy (EKE) for all clusters are presented in To conclude, statistical evidences of barotropic regeneration have been shown for some particular BtCR regions (those denoted with letter A) but not for all of them. The aim of next section is to reproduce in a barotropic model the underlying barotropic regeneration mechanism to better understand what is the essential ingredient about a BtCR region that makes possible the occurrence of this mechanism.

Simulations in a forced barotropic model a. Model
In the different numerical experiments, synoptic-scale perturbations are initially added to a basic state having a non trivial effective deformation field. The basic state streamfunction is the low-frequency streamfunction composite of one of the 6 clusters described in the previous section. As the basic state is generally not stationary in our simplified model and its shape changes rapidly within few days, additional terms are added in the vorticity equation to make the basic state stationary. The total streamfunction ψ is the sum of a perturbation streamfunction ψ ′ , and a basic state streamfunction ψ and the time evolution of ψ ′ is determined from the following nonlinear barotropic vorticity equation on the sphere where f denotes the Coriolis parameter. The model has a T81 resolution and ν =2 10 35 m 8 s −1 .
The barotropic generation rate is now denoted as E.D to make the difference with E.D m of the previous section where the decomposition eddy/mean flow was different and based on a time filtering method.

b. Experiments
Our experiments consist in initializing a perturbation upstream of the Atlantic domain and indicating that the barotropic regeneration process is a transient phenomenon as expected from the theory. The maximum of perturbation kinetic energy is located more or less along the jet axis during each simulation (see heavy solid lines in Fig.9) even if a slight displacement from the south to the north about each BtCR region is visible. It suggests that the barotropic regeneration mechanism can occur without a frank crossing of the synoptic perturbations from the south to the north.
The time evolution of the perturbation embedded in the cluster-4 environment is presented in the left column of Fig.10. In the right column, it is the same simulation but the perturbation amplitude has been initially divided by 10. The aim is to distinguish between linear and nonlinear effects: simulations on the left and on the right are respectively called the nonlinear and linear runs. At day 0.5, there is no real difference between the two simulations (see Figs.10a and b). 12 hours later, at day 1.0, the vorticity maximum in panel c is over Newfoundland while in panel d it is located south of Newfoundland. Furthermore, E.D is essentially negative at that time for both simulations with stronger negative values for the linear run (Fig.10d). At day 1.5, the differences become more pronounced; the perturbation maximum is around the jet axis for the nonlinear run (Fig.10e) while it is clearly located on the south side of the jet for the linear run (Fig.10f). Both have still negative values of E.D due to the stretching of the perturbations along the dilatation axes. Note that although the perturbation vorticity maximum is located around the jet axis in the nonlinear run, the perturbation (in particular the anticyclone) is elongated on the anticyclonic side of the jet in regions where ∆ m is strongly positive and gets a NE-SW tilt. From day 1.5 to day 2.0, the vorticity maximum in the nonlinear run is rapidly pushed poleward (compare Fig.10e and g) during its crossing of the BtCR region and at day 2.0, E.D is positive with isolines separating the cyclone from the anticyclone perpendicular to the new dilatation axes orientation. By contrast, at the same time, the perturbation in the linear run is farther southward, it did not cross the BtCR region at all and still loses energy (Fig.10h). The same results (not shown here) have been found for all the other clusters. To conclude, nonlinear effects are crucial to reproduce the barotropic regeneration mechanism about a BtCR region.
The time average of E.D for the nonlinear and linear runs are respectively shown in Figs.11ab. There are strong negative values southwest of the BtCR region for both runs. But no prominent signal appears farther eastward for the linear run (Fig.11b) while a positive maximum of E.D is well visible northeast of the BtCR region for the nonlinear run (Fig.11a). It confirms the comments made in the previous paragraph. The time average of E.D of another simulation is displayed in Fig.11c. It is the same simulation as in panel a but the sign of the initial perturbation has been reversed. By changing sign, the perturbation reaches its maximum amplitude in the center of an anticyclone and not of a cyclone as before. In Fig.11c, the behavior is very different from Fig.11a with a tendency for the perturbation to go farther southward during the whole simulation and there is no crossing of the BtCR region. It suggests again the crucial role played by the nonlinearities. The barotropic regeneration mechanism can only occur in presence of a cyclone with significant amplitude.

2) ANALYTICALLY PRESCRIBED PERTURBATIONS
In the previous section, perturbations are defined from linear regressions using reanalysis data. As the perturbation spatial structure obtained from the regression map depends on the grid point considered and as we want to test the sensitivity to the location independently of the perturbation shape, it is more suitable to analytically define perturbations. It allows to initialize the same perturbation at different locations. Another purpose is simply to confirm the results with a different definition for the perturbations. The perturbation streamfunction is now defined as and is centered about the grid point (λ 0 , ϕ 0 ). L determines the length scale of the perturbation and is hereafter equal to 700 km.
Results of simulations with different amplitudes and locations for the perturbations and different basic states are displayed in Fig.12. In the left and right columns, the basic states are clusters 4 and 6 respectively. Top panels correspond to a perturbation located at the same place  Fig.12c and no crossing of the jet axis of the perturbation kinetic energy maximum by contrast with Fig.12a. However, by keeping the same location (λ 0 =90 • W, ϕ 0 =37 • N) and by slightly increasing the initial amplitude (ψ 0 =20 10 6 m 2 s −1 ) (see Fig.12e), we recover positive values northeast of the BtCR region similarly to Fig.12a and a clear crossing of the BtCR region by the perturbation. Similar results are visible when the basic state is cluster 6 (see right panels). To conclude, the more southward the initial position of the perturbation, the stronger its amplitude should be in order to make possible the barotropic regeneration process about the BtCR region. In other words, when the perturbation is initially located far south of the jet axis, a large amplitude is essential to make possible the crossing of the jet and of the BtCR region from the south to the north.
The aim is now to compare the time evolution of two distinct simulations differing only from their initial basic states. Left and right panels of Fig.13 correspond respectively to the same simulations as in Fig.12a and b, i.e for which the basic states are clusters 4 and 6. They are particularly interesting to analyze together as cluster 4 presents a BtCR region at the jet exit around 25 • W whereas cluster 6 at the jet entrance around 55 • W. At day 1.0, the two simulations exhibit distinct behavior regarding the values of the barotropic generation rate; for cluster 4 (Fig.13c), E.D is essentially negative while for cluster 6 (Fig.13d), it is strongly positive north of the cyclone and negative south of it. At that time, in the latter case, there is already barotropic regeneration and crossing of the BtCR region by the perturbation which is located in the western Atlantic around Newfoundland. At day 1.5, in the cluster-4 case (Fig.13e), the perturbation formed by a cyclone upstream and an anticyclone downstream is still entirely located upstream of the BtCR region, it is strongly stretched on the anticyclonic side of the jet and E.D is largely negative. At the same time, in the cluster-6 case (Fig.13f), the perturbation is essentially composed of an anticyclone and two cyclones. The more upstream cyclone corresponds to the main cyclone initialized in the beginning of the simulation, while the more downstream one is formed during the simulation, probably via downstream development (see Orlanski and Sheldon 1995).
The former cyclone moves to the north of the jet and by day 2.0 ( Fig.13h) is entirely outside the jet axis while the latter propagates along the jet axis and is stretched in the diffluent region of the jet. This elongation explains the strong negative values of E.D around longitude 0 • W.
The negative-positive-negative pattern of the time average of E.D in Fig.12b is not due to an elongation-contraction-elongation of the same cyclone. The second elongation takes place for the secondary cyclone that appears and grows during the simulation. Concerning the cluster-4 case, a barotropic regeneration stage finally happens northeast of the BtCR region at day 2.0.
Indeed, positive values of E.D are clearly visible at that time (Fig.13g) whereas there were still negative values 12 hours before (Fig.13e). To conclude, the same barotropic regeneration process that appears at day 1.0 in the cluster-6 case occurs one day later in the cluster-4 case and is due to the farther downstream location of the BtCR region in the latter case relative to the former.
The time lag between the two barotropic regeneration stages of the two different simulations is also revealed by plotting spatially averaged energy conversion rates with time in Fig.14 To conclude this section, the same perturbation embedded in two distinct configurations of the zonal regime can lead to very different behavior for the perturbation. The original point of our approach is that we are able for a given jet to determine apriori the area where barotropic perturbation growth may occur.

3) INTERPRETATION OF NONLINEARITIES EFFECTS
The nonlinear term involved in the perturbation vorticity equation of eq.(4) is shown in Fig.15a for the cluster-4 simulation after 36 hours. It has a positive maximum north of the perturbation vorticity maximum and makes the cyclone move northward. The linear term due to the advection of the basic-state absolute vorticity by the perturbation winds (Fig.15b) is essentially negative between the cyclone and the anticyclone which is logical because of the presence of positive absolute vorticity gradients around the jet core. The sum of the two terms shown in In presence of background zonal flows, we have checked (not shown here) that an initial cyclone located south of the jet is able to cross the jet if it has enough amplitude. Being initially on the anticyclonic side of the jet, it is first stretched along the SW-NE direction and moves northward because of the background vorticity gradient and nonlinearities as discussed previously.
The barotropic regeneration process and the crossing of the jet from the south to the north by the cyclone can thus be reproduced in our model in presence of zonal flows, i.e when there is no BtCR area and the effective deformation is zero. However, the initial stretching of the cyclone is less strong and the regeneration process is less pronounced than in the more realistic case of nonzonal flows that are characterized by nontrivial effective deformation fields. Furthermore, in the latter case we have shown that this process is likely to occur about BtCR areas. Further studies will investigate more precisely the role of the nonlinearities and its combination with the effective deformation field.

Conclusion
Barotropic Indeed, anticyclonic vortices tend to go southwestward while cyclonic vortices northwestward.
The more southward the initial position of the main cyclone, the stronger its amplitude should be to cross the jet about the BtCR region. In some cases, because of its initial position too much in the south and despite having a reasonable amplitude, the cyclone cannot be regenerated and stays in the south.
In the different experiments, the main initial vortex of the perturbation is not itself regener-  (2000). By contrast, the barotropic regeneration mechanism of the present study is intrinsically a nonlinear phenomenon. It is interesting to note however the common point of all the barotropic regeneration processes; they all need a discontinuity of the dilatation axes orientation. It is the case from the diffluence to a confluence (e.g., east of the Rockies), from the anticyclonic side of the extratropical jet to the cyclonic side of the subtropical jet (e.g., over North Africa) as described in Lee (2000) but also from the anticyclonic to the cyclonic sides of the extratropical jet as described in the present study.
In further studies, the role of BtCR areas will be investigated in the case of stratified baroclinic models. We expect that the qualitative picture will not be modified at least in the upper levels as nonlinearities will increase in presence of baroclinicity.

Automatic detection of barotropic critical regions
The principle of the algorithm is to consider all the points along the jet core close to a separation area between two large-scale regions of positive effective deformation fields having perpendicular dilatation axes. The main steps of the algorithm are as follows; (i) at each longitude λ 0 , we determine the latitude ϕ 0 where the background low-frequency wind speed reaches its maximum, then (ii) a square of center (λ 0 , ϕ 0 ) and length 2.dα is considered and the number of times the effective deformation field changes its sign along the square is counted, (iii) if this number is four, we determine the averaged orientations of the dilatation axes in the two segments of the square where the effective deformation is positive and if the angle formed by these two orientations is between 60 and 120 degrees, i.e close to the orthogonality, the point (λ 0 , ϕ 0 ) is retained (see Fig.16 for an illustration of such a point). (iv) if (iii) is satisfied for all the squares with dα varying from 6 • to 12 • , the point (λ 0 , ϕ 0 ) is considered as being about a BtCR region. The last step (iv) consists in averaging the longitudes and latitudes of all these detected points (λ 0 , ϕ 0 ) that are close to each other to determine the averaged position of the BtCR point. If there is no detected point between two detected ones being 12 • longitude apart, they are considered as belonging to two distinct BtCR points. The choice of 12 • is deduced heuristically from the effective deformation fields and corresponds approximately to half the distance in degrees between two closest BtCR points. The BtCR points detected by the algorithm that has been applied in the Atlantic domain (90 • W-0 • W,35 • N-90 • N) for 6 distinct effective deformation fields are numbered in Fig.3a-f. We do not consider latitudes less than 35 • N to avoid detection along the subtropical African jet that is not the focus of the present study.