## 1. Introduction

The theory of baroclinic instability, initially devised at the end of the forties (Charney 1947; Eady 1949), appears in many sources as the favored physical explanation of cyclogenesis. Many atmospheric dynamics articles contain such a statement, and textbooks, such as Gill (1982) or Pedlosky (1987), set it as one of the most accomplished achievements of theoretical meteorology. The initial linear core has been extended to nonlinear life cycles and frontogenesis on the sphere by the now classical work of Simmons and Hoskins (1978).

Nonetheless, in spite of the undoubtable importance of this original body of work, it has not closed the problem. For one thing, quite a few meteorologists have been and still remain skeptical about the ability of the theory to account for the actual process of midlatitude cyclogenesis. Criticisms were voiced very early when the theory became popular amongst dynamicists, mostly from scientists that also made real forecasts. Sutcliffe (1947) formulates an alternative that shares some ideas with the theory but does not rely on the linear assumption. Although fundamental to the theoretical framework of baroclinic instability, the fact that storms emerge from weak amplitude perturbations has been put in doubt explicitly by Palmén (1951) and Petterssen (1955). An indication of the fact that the theory is perceived quite differently in the theoretical and operational worlds is that it is seldom used in forecast rooms. The relationship between many elegant predictions of baroclinic instability and real cyclones is confused enough for Pierrehumbert and Swanson (1995, p. 420) to note in the introduction of their review that there is a “stream of inquiry [that] deals with the matter of what, if anything, such results [from linear instability studies] have to do with the observed midlatitude synoptic eddies, and here, the state of affairs is rather less satisfactory.”

Farrell (1985, 1989) considered closely some of the criticisms that were addressed to the original normal mode theory. The underestimation of the growth rates is an example. The fact that the structure of a real cyclone keeps changing as it evolves is another one. Worse, he casts serious doubts on the assertion that the atmosphere is actually unstable since even a weak surface friction dramatically reduces the normal mode growth rates. Farrell and Ioannou (1996a,b) and Farrell (1999) critically review the theory, but mostly they propose a generalization that can be considered now as the current standard model to explain cyclogenesis.

Rather than associate the genesis of cyclones to emerging normal modes (eigenmodes of the linear tangent model or propagator 𝗥, not always well defined), this revised theory links it to singular modes (eigenmodes of the operator 𝗥*****𝗥, where 𝗥***** is the adjoint of 𝗥, nearly always well defined). Relative to normal modes, singular modes bring several benefits, such as more rapid growth (at least temporarily), a time-varying structure, and a capacity to develop even in the presence of surface friction.

However, this renewed framework also meets difficulty as far as gaining universal acceptance is concerned: the structure and properties of singular vectors critically depend on the norm necessary to define them [see, e.g., Joly (1995) for a wide panel of norms and their influence]. So far, it is difficult to pretend that a particular norm is the intrinsic measure of strength or growth in the atmosphere. Moreover, these structures, depicted by their potential vorticity, for example, have many times been found to be disconcerting. Finally, this approach keeps the original linear framework and the idea that a system related to a mechanism of instability emerges spontaneously from a noisy basic state or large-scale flow, although this has shifted to a more statistical viewpoint (Farrell and Ioannou 1995).

Indeed, the cornerstone of the linear theory is to split a situation into two parts. One part is a smooth and slowly evolving large-scale flow, representative of a possible equilibrium of the general circulation. The incipient cyclone is in the other part. It is generally represented as a set of plane waves and it is supposed to keep a small enough amplitude so as not to affect the large-scale flow throughout the time it turns from nondescript noise or small amplitude initial condition to the recognizable most unstable structure.

In a textbook or on the blackboard, waves seem to come up naturally and elegantly. With this, ideas of wavelengths and growth rates appear to be familiar. Nonetheless, none of these operations, namely, splitting the flow into two parts interacting linearly and measuring the wavelengths of a storm or its growth rate, are well defined when it comes to considering actual cases. As a result, although the theory is often considered to be verified in terms of broad orders of magnitudes, it is difficult to find sources that actually attempt to verify it as a predictive tool, as any theory in physics should be tested. Randel and Stanford (1985) extracted some properties, such as the tilt with height from a few cases in the Southern Hemisphere, and compared them to the results of Simmons and Hoskins (1978). Davis and Emanuel (1991) have developed a diagnostic framework based on the conservation and invertibility of Ertel’s potential vorticity; they applied it to a strong extratropical cyclogenesis case and discussed the development in terms of existing theoretical models of baroclinic growth. Another group of studies attempt to simplify verification by jumping from a real case or composite of cases to an idealized situation allowing the computation of all kinds of modes, such as Rotunno and Bao (1996) or Hodyss and Grotjahn (2001). This jump severs the connection between theoretical calculations and the real case. Orlanski and Katzfey (1991) already note that “considerable quantitative differences remain between the flows that are observed and the idealized flows used in theoretical studies” and take an energy budget viewpoint. None of these previous studies have consistently computed the unstable modes of an actual case and compared them to that case.

The purpose of the present work is to test the statement by Farrell (1999, p. 113) that the “stability theory explains individual examples of cyclogenesis” by devising an approach that enables an actual case to be cast into the framework postulated by the theory, namely, splitting the flow into a basic state and the cyclone case, with the latter a perturbation to the former, in a reasonably objective way. This is performed at the initial appearance of the storm since it is generally accepted that “the baroclinic instability theory indicates how depressions form in the atmosphere and how their initial structure is determined” (Gill 1982, p. 578) or that “linear theory . . . only accurately describes the *initial* evolution and structure of the disturbance” (Pedlosky 1987, p. 590, his emphasis). How accurately? This is the result sought in the present approach. The theory provides predictions, such as scales and structures, even locations, amplifications, and 3D structures, all of which can be *measured* against the actual cyclone once it is separated from the rest of the flow. This is achieved either by tabulating these properties or by calculating a well-defined distance between the actual and theoretical structures.

The method is presented in section 2, after a summary of the case study and the numerical model. Once a stormless basic flow close to the actual situation has been obtained, the theory can be employed rigorously. The singular vectors are computed along this trajectory, representing a fully realistic 3D evolving basic state. One key feature of the present approach is that the cyclone of interest has been extracted from an actual consistent time evolution close to observations at any time of that evolution. So we do have a complete representation of the system as a perturbation. The singular structures can then be compared directly to the storm, and this is done in section 3, in the linear approximation. Conversely, the storm can be projected onto singular vectors. Using this ability, section 4 contains results regarding nonlinear integrations of the resulting combination of these singular vectors. Finally, some conclusions are drawn.

## 2. Methodology

### a. Case study

The powerful cyclogenesis leading to the storm that stroked western Europe on 26 December 1999 is the focus of the present study. This storm, named Lothar by the media but hereafter referred to as T1, grew explosively just before reaching France, causing huge damage and many deaths in France, Germany, and Switzerland (Baleste et al. 2001). It definitely belongs to the “bomb” class as defined by Sanders and Gyakum (1980), with a central pressure fall average larger than 24 hPa day^{−1}. However, in this case, the pressure drop occurred in a very short time off the French coast. Furthermore, it took place within a jet flow that was exceptional both by its extension over the Atlantic Ocean (from Florida to France) and by its maximum magnitude (larger than 100 m s^{−1}).

Several distinct phases have been identified in the life cycle of T1. The first sign of the initial formation of T1 is an ascending area east of the North American coast at 40°N over the Atlantic Ocean as indicated by the operational analysis from the Météo-France then 3D variational data assimilation (3DVAR) system (Fig. 1a). Between 24 and 25 December, T1 develops slightly but remains confined at low levels (Fig. 1b) as it moves rapidly eastward below the anticyclonic side of the jet (Fig. 1c). Between 25 and 26 December, T1 travels faster and its trajectory takes a slight northerly component. This propagation phase is associated with a small decrease of intensity (Fig. 1d). On 26 December, T1 dramatically intensifies when reaching the jet eastern exit area. This explosive phase occurs between 0000 and 0600 UTC (Fig. 1e). During the morning of 26 December, T1 rushes into France and Germany, with wind gusts stronger than 150 km h^{−1}. It later ends its run in Poland. Figure 2 summarizes this trajectory.

This study is mainly focused on the initial development phase of T1 between 0000 UTC 24 December and 0000 UTC 25 December, as this incipient stage is primarily the one for which the framework of baroclinic instability theory has been proposed as recalled above. Wernli et al. (2002) and Rivière and Joly (2006) rather deal with mechanisms involved in the explosive phase. The former emphasizes the role of latent heat release that accompanied the intensification phase. The latter investigates the role of the large-scale deformation field on the rapid development phase in the jet exit region. Arbogast et al. (2001) explore hypotheses relating to the origin of the exceptional large-scale jet stream.

### b. Numerical model and singular vector calculations

The numerical experiments are performed with the joint European Centre for Medium-Range Weather Forecasts (ECMWF) and Météo-France operational forecasting system Action de Recherche Petite Echelle Grand Echelle (ARPEGE)/Integrated Forecast System (IFS; Courtier et al. 1991). It is run at a T63 horizontal resolution with full physics with 31 vertical levels. The choice of this resolution is the result of a compromise between the numerical cost of the computation of at least 10 singular vectors (SVs) and a reasonable representation of the storm in the simulations. The resolution over the Atlantic Ocean is about 50 km in the operational analyses and about 300 km in the numerical experiments. This study primarily focuses on the synoptic features of the storm T1. This resolution properly captures the synoptic signal and it does represent the successive phases of the development of T1. Moreover, SVs used in the construction of the initial perturbations of the ECMWF ensemble prediction system at the time of the event, in targeting observations (Bergot et al. 1999) or in order to study their structure, are often computed at this resolution (Buizza and Montani 1999) or even at coarser ones (e.g., T42, T21) (Buizza and Palmer 1995; Hoskins et al. 2000; Barkmeijer et al. 2001).

Singular vectors represent the structures with the largest superexponential growth over an optimization time interval in the linear approximation. However, their definition is tied to the norm that measures this growth. The proper choice of this norm is an open question, particularly because the key properties of singular vectors are strongly norm dependent (Palmer et al. 1998; Errico 2000; Kuang 2004). The so-called total energy norm is usually used for singular vectors computation with IFS (Hoskins et al. 2000; Buizza and Palmer 1995), but this is mostly motivated by practical reasons. Indeed, a large variety of norms exists in the atmosphere–ocean literature. In a quasigeostrophic box ocean model, Rivière and Hua (2004) compared forecast error statistics estimates derived from the leading singular vectors for the total energy, enstrophy, and palenstrophy norms. They found that the best spatial distribution is obtained for the latter. With a wide range of scalar products, Joly (1995) computed the most unstable singular modes for steady 2D atmospheric fronts in a uniform potential vorticity semigeostrophic model. He showed that the quasigeostrophic enstrophy norm was the one ultimately leading to the largest surface pressure drop in the nonlinear regime. This result is further explained in terms of the scale imposed on the initial perturbation by this choice of norm.

*T*

*p*), where

*p*is the pressure. Let

**e**stand for the state vector of a perturbation to that reference state. Taking into account the change of vertical coordinate from

*p*to the hybrid

*η*used in ARPEGE/IFS, the scalar product derived from the energy ||

*e*||

^{2}

_{TE}of state

**e**iswhere

*u*and

_{i}*υ*are the zonal and meridional wind components perturbations,

_{i}*T*is the temperature anomaly, and

_{i}*p*

^{surf}

_{i}is the surface pressure perturbation. The subscripts 1 and 2 are relative to the state vectors

**e**

_{1}and

**e**

_{2}. The constants

*g*,

*R*,

_{d}*C*, and

_{pd}*p*

_{ref}are, respectively, the gravity constant, the perfect gas constant for dry air, its specific heat at constant pressure, and a reference surface pressure. The weight

*S*

^{−1}on the thermal component of the energy isso that

*S*

*N*

^{2}when

*T*

*η*) varies:

*S*

*T*

^{2}/

*g*

^{2})

*N*

^{2}. The calculations performed here use the standard atmosphere as the reference profile providing temperature and static stability. Note that routine calculations performed with the default energy norm in IFS and published so far employ a constant temperature

*T*

_{ref}taken from the semi-implicit time integration scheme of the model, a choice which is not without consequences on the structures of the modes. By default, the domain

Singular vectors are perturbations to a given nonlinear model trajectory such that the amplification ^{2} = ||**e**(*t* = Δ*t*)||^{2}_{TE}/||**e**(*t* = 0)||^{2}_{TE} is a maximum, with **e**(*t* = Δ*t*) = 𝗥**e**(*t* = 0) and 𝗥 is the tangent linear model or propagator along the reference trajectory. The optimization period Δ*t* is taken here to be 24 h. This 24-h period corresponds with the initial development phase of T1.

The optimization problem is cast into an eigenvalue problem for ^{2}; the solutions being the eigenvectors of the symmetric operator 𝗥^{E*}𝗥, where 𝗥^{E*} is the adjoint of 𝗥 for the scalar product defined by (1). At least 10 leading singular vectors are computed over the 24-h optimization period that begins at 0000 UTC 24 December 1999. This is performed iteratively using a Lanczos algorithm as in Buizza et al. (1993).

### c. From a reference trajectory to a trajectory without storm

The model shows a good ability to represent the trajectory of T1. Indeed, the comparison between the 48-h forecast (Fig. 3) starting from the analysis at 0000 UTC 24 December 1999 and the sequence of analyses (Fig. 1) suggests that the main characteristics and timing of events are captured. This forecast will be referred to as the reference trajectory (Table 1). The next key requirement for this study, as mentioned in the introduction, is to have a trajectory without the storm T1 while retaining all the main features of its large-scale environment, such as the jet stream intensity, location, and extent.

To obtain a stormless trajectory, the idea is to remove precursor anomalies linked to the storm in the reduced phase space of potential vorticity and of boundary potential temperature. Potential vorticity inversion is then employed to convert these anomalies into balanced increments of the primitive equation model variables. These increments are in turn removed from the initial conditions and the model can then be integrated in time and the solution checked for the disappearance of the storm. The two most critical steps of that sequence of actions are (i) a reasonably objective way of catching the relevant anomalies and (ii) the potential vorticity inversion algorithm. Step (i) is addressed through time filtering of the sequence of relevant analyzed fields. A Gaussian-type 10-day temporal filtering is applied to the sequence of analyzed 1.5-PVU [where 1 potential vorticity unit (PVU) = 10^{−6} K m^{2} kg^{−1} s^{−1}] and 850-hPa geopotential heights. (The 1.5-PVU potential vorticity surface defines the dynamical tropopause.) The filtering is centered on 0000 UTC 24 December 1999, at a very early stage of the life cycle. The anomalies are recovered from the difference between the low-pass filtered fields and the analyzed ones. By construction, then, the low-frequency fields that form the most critical part of the environment of the storm are little or not changed. From the global fields of anomalies, only the relevant ones must be inverted and removed. Relevance is simply defined by the impact of removing a subset of anomalies from the analysis, which must be small at the initial time as recalled above and which should give a stormless forecast. The time-filtering technique enables, in the area of interest corresponding to the broad confluence zone of the North Atlantic jet stream, to bring out anomalies that can barely be seen in the total fields. After inversion, this area is roughly bounded by latitudes 30° and 50°N (line A–B in Fig. 2a) to the west and 50°W to the east. Their amplitudes are about 50 gpm at 850 hPa (for a 40-gpm standard contour interval in operations) and 100 gpm at 300 hPa (for a 120-gpm standard contour interval). There is an element of trial and error when selecting anomalies but on the objective grounds set by the time filtering.

Step (ii) employs an evolved version of the potential vorticity inversion for ARPEGE/IFS of Chaigne and Arbogast (2000) outlined in Arbogast et al. (2001). The anomalies selected in the 850-hPa geopotential field are used directly as the Dirichlet lower boundary condition. The potential temperature increments at 850 hPa are applied uniformly in the boundary layer below, while the wind increments are vertically interpolated between their 850-hPa values and zero at the surface. The anomalies at 1.5 PVU are given a vertical scale according to a simple regression built from previous inversion cases from a larger set from which the cases of Chaigne and Arbogast are taken. As a result, purely tropospheric potential vorticity anomalies are barely affected, while selected anomalies astride the tropopause are essentially removed. The balance condition in this inversion is close to the quasigeostrophic one.

Figure 4 shows a composite vertical cross section of the changes of potential vorticity and potential temperature between the operational analysis and the modified initial state. Potential vorticity indicates removal of an upper-level anomaly. The potential temperature anomaly inverted from the previous fields bears the corresponding expected signature. This cross section also enables to see the low-level precursor of storm T1, associated with the low-level positive anomaly on the southern (right) side. Although its amplitude at this stage is small compared to the upper-level anomaly, it is the most critical component of the structure highlighted by this methodology.

A forecast is then run from this modified initial state. This trajectory without the storm in the initial state is referred to as the modified or NOSTORM trajectory (Table 1). The comparison between the reference trajectory (left column in Fig. 3) and the NOSTORM trajectory (right column in Fig. 3) shows that most of the signal of T1 has been successfully removed. Arrows indicates the locations of T1. The initial jet stream is weaker in its western part, a less desirable side effect, but most of the large-scale environment is preserved elsewhere. At 0000 UTC 25 December, at the end of its first development phase, T1 is clearly identified in the reference trajectory with a low-level relative vorticity maximum associated with an ascending area (Fig. 3c), whereas only some weak ascending zone remains in the modified trajectory (Fig. 3d). During the explosive phase, the lack of any storm over France in the modified trajectory (Fig. 3f) contrasts with the intensification of T1 in the reference trajectory (Fig. 3e).

Being able to generate this realistic but stormless trajectory provides two benefits. One is to allow the proper computation of unstable perturbations favored in this highly baroclinic environment, as in the theoretical framework. The other benefit is that, at each time step along the duration of the life cycle of T1, the difference between the two trajectories, reference and NOSTORM, yields a representation of the storm itself as an anomaly or perturbation in a natural way. Although this difference is not strictly limited to T1, so that a number of spatially integrated properties are slightly affected, T1 is by far the main signal. This alone provides a relatively original picture of a real midlatitude cyclone.

## 3. Singular vectors versus the storm as a perturbation

At this point, an actual nonlinear solution of the model equations—unlike some average state, as sometimes used in previous studies—is available. It features the exceptionally intense baroclinic zone over the North Atlantic but does not include storm T1. This situation plainly sets us in the framework of linear baroclinic instability yet with a real, complex flow, probably for one of the first times. Following this framework, a number of total energy singular vectors are computed over the period 0000 UTC 24 December–0000 UTC 25 December. To test the theory as much as possible, only a weak geographical constraint has first been chosen for this calculation: the wide domain A in Fig. 2b is used, covering about half of the Northern Hemisphere. This section presents a number of basic properties of the most unstable singular vectors, with a focus on the most unstable one. These properties are computed and shown under the tangent linear assumption, that is, the results here are strictly linear. Nonetheless, as explained above, the present work offers the unique opportunity to compare quantitatively and yet simply the predictions of linear theory with the real storm T1 represented as a perturbation and subsequently called perturbation T1.

### a. Amplifications and phase speeds

The amplification and the mean growth rate during the 24-h period 0000 UTC 24 December–0000 UTC 25 December are first examined. Note that at the initial time, the horizontally averaged total energy norm of each singular vector is 1 J m^{−2} and that of perturbation T1 is about 3 × 10^{5} J m^{−2}. The left-hand part of Table 2 shows the amplification

Figure 5b shows the evolution of the phase speeds of T1 and SV1. Perturbation T1 moves faster than SV1, but both the magnitude (between 20 and 25 m s^{−1}) and its evolution during the period considered are remarkably similar. The initial phase speed increases in both cases, probably as a consequence of T1 and SV1 moving zonally closer to the rapid core of the jet. The phase speed is underestimated with a relative error of about 12%. This is definitely a behavior that could be improved by including moist processes in the linear model, since in that case the phase speed would be increased (see, e.g., Joly and Thorpe 1989).

### b. Structure of the perturbations

Vertical cross sections of SV1 and T1 along their trajectory are given by Figs. 6 –8. The behavior of the most unstable singular vector SV1 is similar to the one discussed by Badger and Hoskins (2001) and termed “unshielding” by the authors. The potential vorticity sequence of sections best reveals this characteristic evolution. The large westward tilt with height rapidly changes, as the anomalies grow, to an upright and even eastward vertical tilt. Other fields accompany this change, also unfolding but with important contributions from the boundaries. The main initial signature of perturbation T1 is not in the potential vorticity but in the low-level potential temperature. Derived fields, such as the wind or vorticity (Fig. 7), emphasize the importance of the maximum east of 70°W compared to the various other anomalies. The other anomalies do not develop and mostly mark the impact of surface fluxes on the boundary layer potential vorticity. The other fields clearly bring out perturbation T1 as a horizontal dipole at low levels, with larger amplitudes there than corresponding dipoles and varying phase shifts at upper levels. Figure 7 reveals that a significant lower boundary component also exists in the singular vector vorticity.

Consider now the vertical structure evolution of T1 (Figs. 6 and 8). The main apparent modification is the strong positive potential vorticity anomaly that forms during the first 12 h, reaching up to 500 hPa but with maximum amplitude between 900 and 800 hPa. In other words, there is no significant midtropospheric anomaly at the earliest stage: this is true for both the anomaly resulting from the difference field shown here as well as for the raw high-pass filtered field. It is not an artifact of the two-level selection part of the otherwise 3D methodology. In the next 12 h, the overall shape moves eastward, but its vertical extent and slight eastward tilt with height remain the same. The low-level amplitude amplifies actively throughout the 24 h. This feature is linked to diabatic processes. The very rapid buildup of that slanted potential vorticity column, followed by its mere advection, is another sign of an initial short-lived pulse-like growth. The growth is helped by the incipient storm moving into the diabatically active ascent zone related to the confluence. These cross sections show the initial formation of the “potential vorticity tower” that plays a central role in the depiction of the explosive phase of cyclogenesis at a later stage by Wernli et al. (2002). It must be noted that the growth of this feature is not continuous in time, and therefore there is something more in this process than the idea that “diabatic processes build the tower up and up, the tower reaches the tropopause, it induces back rapid development.” The building up is stepwise rather than continuous: something allows the steps to take place or not.

Nonetheless, compared to the continuously evolving singular vector, the changes in perturbation T1 appear simpler. Looking at the other fields, apart from the eastward motion, the main signal is the steady amplification of the features already present at initial time, with little changes in their tilts with height. Upper- and low-level anomalies, although connected at the onset, do not exhibit significant tilts with height when considered separately. Several of these features match the strongest points noted by Grotjahn (1996) in a composite of 27 western Pacific systems. The amplitudes of the low-level features grow more than the upper-level ones. The impression of a familiar westward tilt with height between the upper- and low-level meridional wind features must be balanced by the uneven development of the low- and upper-level components in favor of the former, not predicted by elementary theory. Unlike perturbation T1, very strong downstream developments accompany the evolution of the singular vector.

A summary of the most important features of the vertical structures is given by Fig. 9. It shows the vertical distribution of the horizontally averaged total energy of SV1, T1, and the average of the first 10 vectors. The total energy norm has been computed over a reduced domain (10° × 10°) around the center of the structures. The energy has been normalized by their respective total norm computed at initial time, reducing the amplification differences. Looking at the vertical shape of the distribution, the low-level trapped structure of T1, with some boundary layer influence, comes up very clearly, as well as the lack of change. As pointed out by Badger and Hoskins (2001), singular vectors have an energy density profile that initially peaks around 800 hPa but later evolves into a profile with two maxima, one around 450 hPa and the other at the surface. The minimum in between recalls uniform potential vorticity systems. The surface maximum is exaggerated by the lack of a proper boundary layer in the linear model.

Given that both types of perturbations favor the lower troposphere, the horizontal location and organization are best seen by looking at some low-level fields. Figure 10 therefore shows the 24-h evolution of potential temperature at 850 hPa. The environment within which the perturbations develop is summarized by the isotachs.

The first important result revealed here is that, although only a weak constraint has been imposed on the geographical location of the unstable mode, the most amplifying one is remarkably collocated with perturbation T1: taking the central pole as the one representing the location of the low, both perturbations are similarly located at the final time and the singular vector is slightly ahead of the actual perturbation at the initial time, consistent with its slower phase speed. Both structures take the form of a tripole wavepacket at both times. As it has been noted by previous studies (e.g., Lee and Held 1993), wavepackets are better models of real cyclones than the classical wave model. Although these structures have definite horizontal scales, none corresponds to a wavelength in the sense of classical theories. Now, considering the distance between the two extreme poles as the characteristic closest to a wavelength, it appears that it is about 3000 km at the initial time for perturbation T1, *twice* that of SV1. Indeed, the latter is significantly confined zonally. At final time, defining a wavelength for perturbation T1 appears even more arbitrary, while the typical upscale energy transfer of the singular vector SV1 (Hoskins et al. 2000) now gives it a wavelength of about 3000 km. Despite the apparent agreement on such raw figures and locations, there are many differences on structures. For one, these figures pertain to the zonal extent: their horizontal shapes change significantly. The core of perturbation T1 (the positive anomaly alone) at the initial time and the local zonal axis are at a 30°–40° angle with one another. At the final time, perturbation T1 is clearly stretched zonally and seems to align along the southern part of the jet stream core. On the other hand, the singular vector is stretched meridionally at the initial time, which means that it is close to being at right angles with the jet axis. At the final time, its structure has become much more isotropic. These shapes and their changes are closely related to the large-scale deformation field and the way the perturbations interact barotropically with it (see, e.g., Rivière et al. 2003). Kinetic energy is clearly gained in this way by the singular vectors, since their large amplification is explained in part by the fact that they use all available energy sources (see Joly 1995), and the significant reshaping is a sign of this. Conversely, when a structure tends to evolve and keeps close to the local dilatation axis, the sense of the energy transfer is difficult to guess, especially as this property changes at a jet entrance. At final time, the orientation taken implies barotropic loss (Rivière and Joly 2006) at least at 850 hPa.

### c. Energy diagnostics

The discussions of both vertical and horizontal structures invite an analysis of the growth mechanisms. Indeed, keeping with the classical framework of linear instability theories, the natural next step is to consider energy conversions as a first indication of these mechanisms. Following a “local energy budget” approach [Orlanski and Sheldon (1995) expanding on Lorenz (1955)], the various terms involved in the energy budgets of SV1 and perturbation T1 have been computed. The definition of the energy conversions can be found in, say, the above source or in Rivière and Joly (2006). The computation domain is constant for SV1 and perturbation T1 and its size is chosen large enough to encompass their trajectories during the 24-h optimization period. The resulting evolution of these energy conversions appears in Fig. 11 and in Table 3.

The primary message conveyed by the full column budget is that the energy cycles of both perturbation T1 and SV1 are baroclinic in nature, namely, throughout the optimization period, both the baroclinic tapping from the thermal gradient of the environment Ca and the internal conversions Ci dominate the growth. Except at the initial time, the baroclinic tapping is always larger than the internal conversion in the case of the unstable vector. This is as expected from more idealized studies and results from the constraint that such a perturbation has to develop entirely on its own. For perturbation T1, the period with a larger internal conversion lasts longer than the initial time. This can be explained either by the uncomputed contribution of the net budget of diabatic processes or by some part of the ageostrophic circulation at the jet entrance benefiting the very early growth of the cyclone. This advantage to the internal conversion remains rather modest, though, and does not last. A similarly modest positive contribution, but one not expected from classical studies, is provided by a small but definite initial impulse from the geopotential fluxes Cf that Orlanski and Sheldon link to downstream development. However, immediately after the initial time for perturbation T1, and from the onset for SV1, the geopotential fluxes become the main energy sink of the systems considered. Given that this budget is performed with open lateral boundaries and that the baroclinic mechanism is extremely active, this behavior is to be expected: the growing cyclone exports some of its baroclinic gain downstream. Coming back to the initial time, perturbation T1 appears to lose energy from both barotropic conversions horizontal Ck and vertical Cv, whereas, as recalled previously, the unstable mode makes use of everything local to grow. The initial importance of the vertical barotropic conversion Cv is to be noticed: this term is negligible in classical studies, at least as long as the instability occurs at a small Rossby number. In the present situation, the jet stream is both exceptionally intense and yet remains confined, so that this environment is characterized by exceptional shears both horizontally and vertically. This may explain why this term becomes so visible in the budget. Unlike the suggestion of its horizontal shape, perturbation T1 appears to gain energy from the barotropic source. To understand this would require a more detailed analysis of where these conversions take place with respect to the core of the system in the three dimensions.

From the current budget and assuming 0000 UTC 24 December is a proper choice of its initial time, the cause of the appearance of perturbation T1 does not seem to be the same one as the mechanism identified by Arbogast (2004) in another warm side low-level cyclone. Indeed, this should be indicated by a positive horizontal barotropic source. Instead, the positive initial conversions suggest that either a baroclinic interaction in a Sutcliffe–Petterssen type of scenario or an effect of downstream development or both may explain the initial growth. Note that, at the end of the optimization period, the various sources and sinks of the perturbations are ordered similarly (with the exception of the vertical barotropic term Cv). Furthermore, the budgets at that time are quite close to one another. In perturbation T1, the baroclinic tapping Ca is about 1.3Ci, the barotropic one Ck is 0.3Ci, and the loss by geopotential fluxes Cf is −0.6Ci. In SV1, Ca is 1.8Ci, Ck is 0.2Ci, and Cf is −0.7Ci: from this point of view, the structural differences appear to lead to systems that will continue to grow in similar ways.

The differences in structure seen from the vertical distribution of potential vorticity and total perturbation energy can be once more highlighted and summarized by considering a simple parameter, the ratio of kinetic to thermal energy. Given definition (1), it is a measure of the ratio of the wind component of the perturbation to the temperature component. Figure 12 shows the evolution of this ratio between 0000 UTC 24 December and 0000 UTC 25 December, the optimization period. In perturbation T1, the kinetic energy dominates the thermal energy throughout. Furthermore, the ratio is around 1.4 on average, while the initial 6-h period leads to a peak close to 1.7, implying that kinetic energy both dominates and increases more than available energy. In the unstable mode SV1, the ratio is very similar to that of the actual storm at initial time. Afterward, two things turn to be notably different. During most of the first 18 h, the weighted *temperature* perturbation is more important in the singular vector than in the actual storm. The other difference is that this ratio undergoes important changes during the optimization period, and these changes are the opposite of what occurs to the actual T1. From previous work in our group (such as Joly 1995 or Bergot et al. 1999), such a characteristic is to a large extent controlled by the norm employed to compute the singular vector. The larger importance of the thermal component is increased when the default “total energy norm” of ARPEGE/IFS is used (it assumes an isothermal reference state), and it is shown here that this is not a property of real storms.

The theory of baroclinic instability in the sense of Farrell (1989, 1999 has been applied to a realistic NOSTORM trajectory representing all the main features of the situation that led to storm T1, except that storm. The unstable vectors can be compared to T1, itself represented as a perturbation. The theoretical and real systems are located in the same area, move at roughly the same phase speed, and grow mostly baroclinically. The growth rates, the scales, and the shapes, as well as the changes in structure, on the other hand, just do not compare with one another.

## 4. Nonlinear behavior of singular vectors triggered by T1

This new step addresses two important issues of the relation between baroclinic instability theory and real-world cyclogenesis. In the continuation of the previous section, one issue is the extent to which linear theory is applicable. The other issue is of a more conceptual nature. It is the idea that cyclogenesis, while possibly imperfectly explained by the most unstable singular vector, still results from the triggering of some unstable modes at initial time. This idea is examined by extending the simulations to 48 h, to 0000 UTC 26 December, when the real storm reached the eastern jet exit and was on the verge of entering its explosive phase. All these simulations use the ARPEGE/IFS model at T63 resolution with full physics and the NOSTORM initial state perturbed by some 24-h total energy singular vector-derived anomaly, depending on the question addressed. Table 4 lists these experiments.

### a. Measure of nonlinearity

The theory of baroclinic instability is based on the linear assumption. Despite their strong and rapid development, singular vectors are supposed to evolve within the “linear regime” at least during the optimization time. This should happen with a range of initial amplitudes that reaches the values either of an actual perturbation when singular vectors are used as an explanation of cyclogenesis, as tested here, or of the magnitude of analysis errors when used to perturb initial conditions for predictability studies. Along the same line, the duration of the linear regime for a given initial amplitude is of interest and is documented in the present section.

**e**

*μ*

**e**, where

**e**is the normalized anomaly structure and

*μ*is its initial amplitude, expands to

**+**e

*μ*

**e**) =

**) +**e

*μ*𝗥

**e**+

**,**e

*μ*

^{2}

**e**

^{2}), where 𝗥 is the resolvant or propagator. Letthe ratio of the second- to first-order term, where ||·||

_{TE}is the total energy norm derived from definition (1). As long as ε is significantly small, the linear assumption is verified in the sense of that norm.

Figure 13a represents the evolution of ε as a function of the amplitude of the most unstable singular vector SV1 and as a function of the simulation range. The perturbation amplitude is measured by its temperature anomaly at 850 hPa. Note that at the initial time, 0000 UTC 24 December, the amplitude of perturbation T1 is 2.5 K. For 18- and 24-h simulations, ε is always greater than 0.4. The nonlinear terms cannot be considered as negligible for a wide set of amplitudes.

For an initial amplitude similar to the standard deviation of analysis error statistics, namely, 2 K, the linear limit is reached after only 12 h of evolution. Beyond this amplitude or range, the nonlinear development of the singular vector is weakened compared to the linear prediction, and the structural changes that should take place can be modified whenever the nonlinear regime is entered during the optimization time. Figure 13b provides direct evidence that nonlinear effects do reduce the growth of the singular vectors from a very early stage. The large-scale pattern at the end of December 1999 with, in particular, the exceptional jet stream intensity may partly explain those results. Another factor is the choice of norm, through the scales it selects. The total energy norm is identified by Joly (1995) as the worst choice from the point of view of linearity. In fact, as noted by Gilmour et al. (2001), it seems that for singular vector perturbations of a realistic magnitude the duration of the linear regime is often less than a day.

### b. Nonlinear life cycle of a triggered unstable initial condition

Section 3 reveals many critical discrepancies between perturbation T1 and the most unstable singular vector. It could be, however, that these differences are not significant: perturbation T1 as revealed by the methodology given in section 2 has a structure that hides the part of it that is actually important to the subsequent growth. (A possible analogy is medicine and the small fraction of *active ingredients* in most medicines.) It is possible that, by capturing and retaining only that part of perturbation T1 that is unstable, its life cycle is still correctly anticipated. This is sometimes termed as triggered or forced instability: the detailed structure of a given perturbation is not important per se, it is important in that it triggers some unstable modes.

**e**represent perturbation T1, and

**s**

*be the*

_{i}*i*th normalized total energy singular vector computed along the NOSTORM trajectory. The complete set of singular vectors forms a basis of the system phase space, but since they are orthogonal for total energy, the projection of

**e**on any truncated set of

*N*singular vectors at initial time

**u**

*(*

_{N}*t*= 0) is simple:Equation (3) defines the unstable part of perturbation T1(

**u**

*) and gives to each of the singular vectors included in the unstable subspace a well-defined amplitude with respect to the known proper initial condition through a basic use of the scalar product. Note that ||*

_{N}**s**

_{n}(

*t*= 0)||

^{2}

_{TE}= 1 J m

^{−2}but is dimensionally useful.

The left-hand part of Table 5 shows *α _{n}* computed for the 10 most unstable singular vectors at 0000 UTC 24 December 1999 using domain A. This is where singular vectors are actually measured against the real storm. Not surprisingly, the values of

*α*show that T1 is very poorly projecting onto the 10 singular vectors. The horizontally averaged total energy norm of the part of perturbation T1 that projects onto the 10 most unstable singular vectors is about 1800 J m

_{n}^{−2}. This only represents about 0.6% of the horizontally averaged total energy norm of perturbation T1. Also notice the nonmonotonic behavior of the suite of

*α*, with larger coefficients for orders as high as nine or five depending on the domain. The theoretical amplification of

_{n}**u**

*is 5.98, which is twice the observed amplification of T1 (2.80). Table 5 actually quantifies the fact that perturbation T1 does*

_{N}*not*look like a singular vector, with a resulting predicted growth that is also quite different. These few numbers fully confirm the findings of section 3.

The poor projection of this actual storm on initial energy singular vectors has been noted in a highly idealized context by Snyder and Hakim (2005). These authors, however, found that their ideal incipient storm was far better captured at the end of the optimization period Δ*t* by their energy singular vectors. The better projection makes sense since the singular vectors at Δ*t* are unfolded. Indeed, the vectors **s*** _{n}*(

*t*= Δ

*t*)/

*, where*

_{n}*is the amplification of*

_{n}**s**

*, still form an orthonormal basis on which perturbation T1 at*

_{n}*t*= Δ

*t*can also be projected. Table 5 thus shows the projection coefficients

*γ*

_{n}= 〈

**e**(

*t*= Δ

*t*),

**s**

_{n}(

*t*= Δ

*t*)/

_{n}〉

_{TE}/||

**s**

_{n}(

*t*= Δ

*t*)/

_{n}||

^{2}

_{TE}of perturbation T1 at Δ

*t*. Unlike what Snyder and Hakim found, the projection at Δ

*t*of a real storm is as poor as at the initial time. Since a linear prediction of

*γ*can be derived (

_{n}*γ*=

_{n}*), an independent measure of nonlinearity can be obtained. It indicates an average 51% relative error over the first 10 singular vectors between the expected linear and actual nonlinear behavior of perturbation T1. These numbers quantify the poor comparison between both the structures and the 24-h evolution of perturbation T1 and the first 10 energy singular vectors of the same basic flow.*

_{n}α_{n}Now, a new initial perturbation is added to the NOSTORM initial condition: the projection of perturbation T1 on the first singular vectors. Contrary to the classical framework, this perturbation has a well-defined initial amplitude, which implies conversely that it is no longer independent of perturbation T1, unlike in the previous section: a huge piece of information from the real case is now imposed. In the experiment shown, *N* = 10 and **u**_{10} is denoted low S in the following (see also Table 4). Is that limited part of perturbation T1 enough to recover the first part of its life cycle?

Figure 14 gives the time evolution of the 850-hPa relative vorticity maximum for low S, the unstable part of T1, and for T1 itself. For comparison, results for a simulation starting from the unperturbed NOSTORM initial state and for a simulation D, which only retains the part of perturbation T1 that projects on SV1, are shown. It appears that the development of cyclone S is much weaker than that of T1. For example, at the end of the 48-h simulation, the amplitude of S is half that of T1. The phases of the life cycle are also less easy to identify with the S initial condition. The first growth phase of T1 should be split into two steps in the case of S, since the initial growth, surprisingly, is very weak.

Textbook presentations, such as Gill (1982), Pedlosky (1987), or Holton (2004), concentrate on the most unstable solution of the baroclinic instability problem. This means to limit the projection (3) to *N* = 1. This simplification appears to be relevant in the sense that the result for SV1 projected alone (D) is essentially the same as with 10 vectors, except that the last day is even more poorly handled. It is not relevant in that the result (the weak lows S and D) is far from representing the actual cyclone.

A further look at the horizontal organization of cyclones D and S is shown in Fig. 15. Few differences between the two “unstable” systems are observed but a comparison with T1 (see Figs. 3b,d,f) indicates that S and D are located farther south and have a weaker development in comparison to the real storm. They are also away from the jet exit, preventing the mechanism identified by Rivière and Joly (2006) to lead to the explosive phase. In spite of a theoretical amplification twice as large as the observed one, it is relevant to note that low S actually develops significantly less than perturbation T1.

In short, it appears that the nonlinear evolution of the parts of perturbation T1 that project onto the 10 most unstable singular vectors or the first one only reproduce neither the location nor the intensity of the actual storm. It cannot be said that the cyclogenesis results from an unstable subspace being triggered by perturbation T1, even when this subspace is enlarged to account for the possible limited discrimination offered by or limited relevance of the “total energy” norm.

Various sensitivity experiments, listed in Table 4, have been set up to study the effect of having a slightly different background flow or of crudely changing the structure of the most unstable singular vector or the effect of focusing and moving around a limited target area. First, the way the actual flow is split into a perturbation and a background flow is not unique in spite of the effort to do this as objectively as possible. The conclusions only hold so far as they do not depend on details of the background flow, especially in the complex jet entrance area. Figure 16 shows the most unstable singular vector for a different background flow, to be compared with Fig. 10. While the background jet has been much more affected by the separation than in the NOSTORM trajectory (the westward extension of the core is reduced), the singular vector structure and its other properties are essentially identical.

Consider now the sensitivity to features of the singular vectors. The large initial tilt with height is the most essential component for the growth of the vector (not shown). The second critical aspect is the midtropospheric part of the structure, which does not fit the troposphere climatology, as already noted by Horànyi and Joly (1996). Using the restricted area C in Fig. 2 forces the computed solution to be exactly where T1 was at 0000 UTC 25 December: the theory does not predict anymore where a storm may be, rather it is told where it is. Tables 2 and 5 summarize the stability properties. The targeted vectors are less unstable, and perturbation T1 does not project better on them. Figures 15e and 15f illustrate the nonlinear behavior of that projection. It evolves very much like the projection on the wide domain vectors. Finally, the sensitivity to the targeted domain location has been explored. Nonlinear amplitudes are all smaller than with the restricted domain C, which is the closest to the reference (not shown). Figure 17 shows that trajectories end very differently from their initial distribution. The final area, covering the Bay of Biscay again, is smaller than the initial one. This may suggest the existence of some geographical basin of attraction for lows forming in a broad area about the jet entrance on 24 December: they end up in a smaller, but still wide, zone.

## 5. Conclusions

The idea that atmospheric cyclogenesis results from baroclinic instability is probably one of the most widely accepted consensuses amongst dynamicists and meteorologists. “The quarter wavelength . . . predicted by [the theory] is . . . *O* (1000 km) and is in such excellent agreement with the observed scale of synoptic atmospheric disturbances that the mechanism of baroclinic instability becomes at once the most plausible explanation,” writes an enthusiastic Pedlosky (1987, p. 527). This is also clear from recent review works, such as the book edited by Shapiro and Grønås (1999) or the more recent *Encyclopedia of Atmospheric Sciences*: “The leading theory of cyclogenesis, baroclinic instability, posits that the westerly jet streams . . . are unstable to infinitesimal disturbances” (Hakim 2002). However, finding references dedicated to a deeper verification than the one above is difficult.

The present study proposes a practical scheme to split an actual atmospheric flow into a stormless nonlinear trajectory, suitable for instability computations, and an anomaly representing the storm of interest. Within this framework, which was previously difficult to put together, the Farrell (1999) formulation of the theory of baroclinic instability can then be fully applied and its results checked.

The linear theory of baroclinic instability in its present form cannot be said to fully explain or to account for the cyclogenesis case considered in the present study. From the point of view of weather prediction, the stormless trajectory *is* turned into a stormy one, although underestimating the intensity, by adding the theoretically computed unstable modes to it. From a predictability perspective, it seems also that computing a sequence of unstable modes over a large zone enables one to narrow the area exposed to a storm risk downstream. Other results favorable to the theory are its ability to pick the initial location of the real case and a quantitatively correct estimate of the energy budget but only at the end of the optimization time. Otherwise, the theory has numerous failures. The predictions provided by classical studies (amplification, scales, spatial shapes) are severely off target, as quantitatively measured in this work by projection and other means. The structures are not comparable, the scales only broadly agree, and the compatibility with the linear assumption is questionable. The structures and their evolution differ strongly at both the initial and optimization times.

A large number of baroclinic instability studies [including the previously cited work of Eady (1949), Charney (1947), or Simmons and Hoskins (1976), and many more] target normal modes as the theoretical solution representative of cyclogenesis. Recent work starting from untilted, independent anomalies as observed precursors of cyclogenesis seems to bring normal modes back to the front stage as better able to account for the subsequent evolution of these anomalies (e.g., Hodyss and Grotjahn 2001; Hakim 2000; Rotunno and Bao 1996). What about normal modes of the present realistic background flow and their connection to the real cyclone T1? With this background flow being highly realistic, it has large zonal variations—both ends of the Atlantic jet core are essential (see Rivière and Joly 2006)—and it has significant time variation. Casting such critical features into the normal mode approach framework has always raised serious difficulties. Indeed, the abovementioned recent work simply jumps to highly simplified flows, for which normal modes are defined but are very far from real cases. This, unfortunately, considerably limits the scope of these studies. Among other things, they cannot explain location, whereas the framework employed here allows this. Joly and Thorpe (1991) or Bishop (1993) and a few others study the stability of time-dependent flows with a normal mode-like vision in mind. But this is cumbersome and related problems are often discarded by twisted and weak arguments, see, for example, Frederiksen (1982, p. 2315) or the discussion of what should be a basic flow in Pedlosky (1987, p. 491) “consistent with the external [solar, lunar] forcing . . . predictable with the appropriate astronomical period.” These problems are formally addressed by Farrell and Ioannou (1996b). Flatly, they recall that the temporal eigenmodes of a nonautonomous dynamical system—such as any realistic flow—“are not defined” (Farrell and Ioannou 1996b, p. 2042). The relevant concept is the first Lyapunov vector, which is different at each point along the phase space trajectory. It relates to the long-term asymptotic behavior of the system and is determinative “only at timescales long compared to that of feasible deterministic forecast” (Farrell and Ioannou 1996b, p. 2052). Therefore, the singular vector or generalized version of baroclinic instability as studied here is the only well-posed form for a realistic flow and fit for quantitative verification as done here through projection. It is able to provide local solutions sensitive to all features of the background flow, including horizontal deformation, a key feature of cyclogenesis ignored in many recent idealized studies. These solutions take the form of wavepackets, which are recognized as the most suitable description in the review of Pierrehumbert and Swanson (1995). Yet, in spite of all these favorable possibilities, it does not really describe an incipient cyclone.

The limits of these results are as follows. A key element of the theory is the norm used to extract the relevant perturbations: here, a form of energy has been employed as in most kin previous studies and it is clearly not optimal. Joly (1995, p. 3100) suggested that the uniform potential vorticity assumption should be seen as a constraint on the structure related to its likeliness, and Snyder and Hakim (2005) actually build a norm close to such a constraint: this improves the projection of their idealized precursor feature on the resulting singular modes. Diabatic processes have been included in all the nonlinear simulations. However, they have not been included in the calculation of the unstable modes. This has been done, on the same case but at a much later time when the cyclone is well into its nonlinear life cycle, by Hoskins and Coutinho (2005). Because of this, only a very limited comparison can be made. It does not seem that this would dramatically change the present results: small shifts of the location with respect to humidity may take place, but the early behavior of the perturbations seems quite similar and the linearity becomes even more critical. The role of humidity could be further enhanced by including it in the norm.

Perhaps the most important step accomplished here is the demonstration that it has become possible to use the physical and numerical technology now available to produce well controlled but highly realistic situations, opening the way to the quantitative verification of any theory that could be produced for atmospheric systems properly captured by current data assimilation and numerical models. These are very exciting opportunities for further research into the obviously still open problem of midlatitude cyclogenesis.

## Acknowledgments

Most of the raw results of this study have been obtained in 2003 as part of the first author’s equivalent MsC, with support from the Ecole Nationale de la Météorologie. We also thank our reviewers for very careful reading of the original typescript, which led to many improvements.

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Characteristics of model trajectories.

Amplification and mean growth rate for T1 and for the first 10 singular vectors computed over the period 0000 UTC 24 Dec–0000 UTC 25 Dec using a wide domain A or restricted domain C (see Fig. 2 for domain locations).

Energy conversions at 0000 and 0600 UTC 24 Dec 1999 for SV1 and T1. Values have been normalized by the initial value of internal conversions.

Nonlinear integration experiments with SV perturbing the NOSTORM initial condition.