A number of recent publications have dealt with cyclone identification and tracking. Following on, this paper extends the typical cyclone life cycle back in time to embrace a new feature called a “diminutive frontal wave.” One aim is to improve predictability by extending tracks. This is particularly important for small, cyclonic windstorms, which can often be missed in postprocessed output from operational, ensemble, and climate runs.
The recognition of diminutive waves requires a new, front-relative, low-level vorticity partition. The parts are labeled “frontal vorticity” and “disturbance vorticity” and are computed, respectively, from front-parallel and cross-front low-level wind components. A diminutive frontal wave then lies wherever there is a local, along-front maximum in the disturbance vorticity. Computations require local coordinates; these are conveniently provided, at all grid points, by objective front diagnostics.
Analysis of cyclone-type transitions over the North Atlantic in operational numerical model data confirms the validity of adding the diminutive wave stage to the revised cyclone life cycle. Examples then suggest that nonmodal growth of diminutive waves can occur, albeit with a sometimes complex interplay between separate cyclonic features. In all cases, model resolution is necessarily higher than the 100–500 km typically used in previous work.
Cyclonic disturbances in the extratropics vary considerably in shape and size. Despite this, many have tried to devise conceptual models of a “typical” vigorous cyclone to highlight the salient features, and these models are still widely quoted. It was arguably Jinman, in 1861, who developed the first lifelike cyclone model (see Ludlam 1966). This was expanded on in the life cycle portrayals found in Bjerknes and Solberg (1922) and, later, Shapiro and Keyser (1990): in Fig. 1, stages 3 to 6 come from the Shapiro–Keyser model. Because adverse weather prevails in the later, more developed stages (4 to 6), these have generally received the most attention (e.g., George 1972; Bosart 1981; Reed and Albright 1986; Ulbrich et al. 2001; Browning 2004). Many have also addressed the “pre-cyclone stage,” wherein a front (e.g., stage 0), which is the main breeding ground for cyclones, forms through frontogenetic mechanisms. Lamb (1951a, b,c) presents a useful descriptive standpoint, and Hoskins (1982) provides dynamical theory. Between frontogenesis and a vigorous frontal cyclone lie the first stages of cyclonic development on a front. These have been considered much less, and are the main focus of this study.
The new term “diminutive frontal wave” is introduced here to refer to the very first (or sometimes last) signs of a cyclonic disturbance on a front (Fig. 1, stage 1). This term improves upon “potential frontal wave,” first used by Hewson (1998b), by avoiding a clash with temperature phraseology. It is also preferred to the term “incipient frontal wave” because transitions through at least the early cyclone stages are reversible, as will be seen.
Stage 2 has also been added to the conceptual model to denote instances when a frontal wave is not associated with a mean sea level pressure minimum. This stage will not always occur, depending in part on the surrounding geopotential environment, but its inclusion is viewed as pragmatically important, in part because of the tendency for cyclonic systems responsible for many western European windstorms to often traverse large distances in this phase in the central North Atlantic (e.g., see Figs. 25 and 26 in Shapiro et al. 1999). This is also an important consideration for automated cyclone tracking.
It is true that not all cyclones form on fronts; for example, polar lows and the upper-level forcing-dominated (type C) cyclones in Deveson et al. (2002) apparently provide counterexamples. Results in Ayrault and Joly (2000), based on 160-km-resolution data, suggest that about 28% of North Atlantic cyclonic systems form in a nonfrontal environment. However, because composites (which they present) always exhibit a weaker gradient than do, on average, their constituent members, it is expected that a case-by-case analysis would yield a markedly lower percentage. Also, when higher model resolution and greater data coverage permit closer scrutiny, it transpires that many polar lows do in fact exhibit frontal and even life cycle characteristics that, although of different scales, are structurally very similar to those of their larger, warmer-airmass counterparts (e.g., Rasmussen and Aakjær 1992, Hewson et al. 2000). The discussion presented in the current paper should thus find wide application.
The main aim of this particular paper is to introduce the concept of diminutive frontal waves and place it in a synoptic and dynamical setting. Section 2 provides an overview of some past work on frontal waves; section 3 defines the term diminutive frontal wave more fully. Section 4 then uses operational model data to assess real-world diminutive wave behavior and the validity of the revised conceptual model in Fig. 1. Specific examples are given in section 5, focusing on developmental mechanisms. A summary is offered in section 6.
2. A brief review of frontal wave cyclones
a. Secondary cyclones
The development from front through frontal wave to cyclone is often classified as “secondary” cyclone development. On simple descriptive grounds, this is because the initial, trailing front connects to a parent “primary” cyclone. On theoretical grounds, it is because the competing physical mechanisms at work (see, e.g., Rivals et al. 1998) may differ from those responsible for larger primary cyclone development, such as baroclinic instability. Indeed, Parker (1998), in his comprehensive review of secondary cyclones, tends toward this view. Nevertheless, the distinction is at best somewhat blurred. One reason is that a so-called primary cyclone must, at the earliest point in its history, have been a vanishingly small cyclonic perturbation, most probably on a pre-existing front; as such it would then be secondary to any larger, older cyclones on its periphery. A second reason is that secondary cyclones, in certain parts of the world, frequently spiral in toward a primary cyclone, helping to reinforce its center. Indeed, this contributes to the dominance of the Icelandic Low in the North Atlantic and the Aleutian low in the North Pacific. In turn, this precipitates a contentious question: do secondary cyclones become primary cyclones? The analysis of Grotjahn et al. (1999), which shows substantial increases in scale of Pacific cyclones during their lifetime, would suggest that they can. The general view taken in this paper is that the cyclone spectrum is very broad, that it lacks clear discontinuities, and that similar identification methodologies can be usefully employed throughout. It is certainly not the intention to dismiss any categorization as a wasteful activity, but rather to develop an all-embracing framework of objectively identified cyclonic features that needs no previous distinctions other than frontal proximity and typing to proceed and that is based purely on the local low-level thermodynamic structure of the atmosphere. These methodologies have evolved by analogy with operational forecasting practice, in which cold and warm front waves (and of course cyclones) have long been plotted and recognized as portents of bad weather. Hewson (1998a, 1997a) describes the previous objective techniques. These techniques have now been further developed, in conjunction with this work (see Hewson 2001). For further discussion of the multifaceted extratropical cyclone spectrum, the reader is referred to Shapiro et al. (1999) and references.
b. Previous studies
A number of generic studies of cyclonic features have focused on or included the formative stages in a climatological context (e.g., Jones 1962). In the 1990s, interest increased with several papers investigating factors controlling initial development, from theoretical discussion in Bishop and Thorpe (1994a, b) to case application in Renfrew et al. (1997) and others. To highlight the cyclone identification methods used, key examples of these climatological and case-based investigations are listed in Table 1 in chronological order.
Evidently methods differ markedly. With time there has been a move toward greater use of automated model analyses. Also apparent is a slight shift of emphasis, away from using mean sea level pressure minima and toward using low-level vorticity maxima. It has long been recognized that vorticity maxima provide a better local measure of activity than pressure minima, which depend strongly on the background environment. And if Jones’ (1962) observation that 80% of (rain-producing) United Kingdom warm front waves had no closed isobar is typical, and one wants to capture such features, then this change of approach is fundamental. Note, however, that using the full vorticity for identifying cyclones in high-resolution data is not without its problems, in part because of frontal “contamination,” as highlighted by a number of authors (e.g., Blender et al. 1997; Sickmöller et al. 2000; Wernli and Schwierz 2006). More recently, Ayrault and Joly (2000) and Hoskins and Hodges (2002) have also filtered raw fields to remove longer wavelength and longer time scale variability prior to processing. This too would generally yield better coverage of smaller cyclonic features (especially for pressure, as they discuss), although arguably filtering has the disadvantage of breaking the direct link with synoptic charts. Section 3 will illustrate a new approach, using a vorticity partition, which retains the synoptic link yet avoids the frontal contamination.
c. Benefits of studying diminutive waves
The motivation for studying cyclonic systems when they are intense is clear-cut. The reasons for studying diminutive frontal waves are, at first inspection, less obvious. However, there are several. The first is predictability. Diminutive frontal waves can be the “seedlings” of major cyclonic storms, and these storms remain, even today, difficult to predict accurately (see, e.g., Wernli et al. 2002; McMurdie and Mass 2004). Such storms grow, by definition, in an environment conducive to rapid growth, and it follows that correct prediction will require, among other things, correct specification of initial conditions. Indeed, given Ayrault and Joly’s (2000) result that many frontal cyclones seem to form in the absence of an upper-level precursor, the low-level initial conditions around any diminutive waves could well be key. Thus, more routine identification of these focal points (for ensemble perturbation or even for observation targeting) may help in the quest for improved forecasts. The second reason is related and concerns cyclone tracking. Objective tracking combines a cyclone identification algorithm [such as that of Murray and Simmonds (1991)] with a tracking algorithm [such as that of Terry and Atlas (1996)]. This has attracted increasing attention in recent years and has been applied to reanalysis data to specify cyclone activity in the current climate (e.g., Hanson et al. 2004) and in climate runs to assess change (e.g., Geng and Sugi 2003; Lambert 2004). Of considerable practical importance would be changes in the “damaging cyclone climatology.” Because extreme cyclones tend to deepen very rapidly, investigation is far from trivial. Evidence for this can be seen in Fig. 10 in Hanson et al., where the first time the Great Storm of October 1987 (Burt and Mansfield 1988) was identified in either the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) or the National Centers for Environmental Prediction (NCEP) mean sea level pressure reanalysis was when it was at its peak.1 Other storms are almost certainly being missed or discarded on account of the short life cycle cut-off often applied. Extending the range of recognized cyclonic features can only help build better, longer tracks that much more closely resemble those identified daily within operational forecasting and that fully capture cyclone growth. The introduction of objective frontal waves (in Hewson 1997a) has already helped in this regard (causing a trebling of the lifetime in one example); adding diminutive waves, as in Fig. 1, will simply build on this trend. A third reason for introducing diminutive frontal waves concerns cyclone splitting. One key recent result is that splitting occurs regularly (Baehr et al. 1999; Ayrault and Joly 2000). As will be shown in section 5, the splitting process can involve development of a new diminutive wave and, soon afterward, decay of the old cyclone into another such wave. Diminutive waves thus provide a conduit for further study of this phenomenon (which in turn also has further implications for cyclone tracking). There is also considerable potential for the corollary, cyclone merger, to be investigated too.
d. Model representation
As indicated above (Table 1, column 2), analysis of small-scale atmospheric features used to depend on manual interpretation of observations, and as a result was unreliable over the oceans, where data was sparse. Nowadays, numerical model analyses are used as “truth.” Although not perfect, this is unequivocally more reliable, thanks to increasingly sophisticated data assimilation, ever-improving model formulations that increase background field quality, and satellite data. This study utilizes the Met Office operational forecast model (Cullen 1993; Cullen et al. 1997). As regards its background field quality, Hewson (2002) tested for systematic errors and reported just a 20% underprediction of modest frontal waves developing by T + 48 h, suggesting that any intrinsic model bias contaminating background T + 6 h fields is small. As regards assimilated satellite data, the launch of, for example, the European Remote Sensing Satellite (ERS-1) in 1991 (see Bell 1994) and the QuikScat satellite in 1999 (see Pickett et al. 2003 and references) represented major steps forward in correctly specifying the lower tropospheric wind field (even allowing for ambiguity resolution issues). QuikScat provides comprehensive high-resolution surface wind observations over large ocean swathes twice daily—its true merits are well illustrated in Patoux (2003), where frontal wave evolution has been analyzed using this data almost in isolation. For these several reasons it is believed that recent formulations of the Met Office operational models provide data of sufficiently high quality, in general, for identifying diminutive frontal waves and tracking their behavior.
3. The identifying characteristics
a. Vorticity partitions
As will be seen, the usefulness of low-level vorticity can be increased by using vorticity partitions. A number of ways exist to decompose relative vorticity, or geostrophic relative vorticity, into constituent parts, many of which rely on a local coordinate system. For example, Saucier (1955, 303–372), Bell and Keyser (1993), and others refer to the shear–curvature partition, wherein local coordinates are defined by the velocity vector. Hewson (1997a), in developing an objective methodology for identifying frontal wave tips, utilizes instead the along-front vector to effectively define the vorticity of the cross-front (i.e., front-normal) wind and the vorticity of the front-parallel wind, with full vorticity being simply the sum of these terms. In this paper, use of this particular partition is extended to help define diminutive frontal waves. It will now be helpful to highlight the advantages, and ultimately the necessity, of this particular partitioning approach.
The advantages of Hewson’s (1997a) methodology are twofold: first, life cycles are extended further back in time; second, two key elements of synoptic practice are successfully incorporated. The first element is simply that the sign of the cross-front geostrophic wind is widely used for frontal typing.2 The second is that the along-front isobaric spacing provides a good estimate of frontal speed, and the cross-front geostrophic wind is (on an f plane) inversely proportional to this. A steady change, along a front, in frontal speed implies frontal rotation. An uneven change, if of sufficient magnitude, implies buckling. In general terms, cyclonic disturbances show up as frontal buckling caused by a local maximum in (cyclonic) frontal rotation. This is identified using the vorticity of the cross-front wind, not the full vorticity.
It is straightforward to use the objective front methodology to define a local, horizontal, front-relative Cartesian coordinate system (s, n) as represented in Fig. 2. After Hewson (1998a), the gradient of the thermal gradient magnitude field represents the cross-front (front-normal) axis s, with a unit axis ŝ obtained by normalizing:
Then n, the axis normal to s, is the along-front axis (n = k × s). This axis will be tangential to the front at the point in question (e.g., point X in Fig. 2). Evidently, n = n(x, y) and s = s(x, y). Although fronts are strictly not defined everywhere, the quantities in Eq. (1) clearly are (provided care is taken with vanishingly small gradients).
With the local coordinates defined, it is straightforward then to decompose the full horizontal vector wind field (V) into cross-front (Vs) and front-parallel (Vn) components:
Meanwhile, the full relative vorticity ζ can be decomposed into the vorticity of the cross-front wind, (namely, the disturbance vorticity ζdi) and the vorticity of the front-parallel wind (namely, the frontal vorticity ζfr):
Note that the word “disturbance” as used here is a convenient shorthand reference to “cyclonic disturbance.” The partition is illustrated schematically on the right of Fig. 2.
Finally, note for completeness that the local coordinate convention used above, in which axis n is along-front, while fully consistent with Hewson (1998a), does differ in tone from that used in the Q-vector partitioning studies of Keyser et al. (1988, 1992) and others. In these, the vector n is normal to an isentrope, which to a first approximation is cross-front.
Figure 2 shows hypothetical examples of resolved cross-front (Vs,1 and Vs,2) and front-parallel (Vn,3 and Vn,4) winds, from which a qualitative visual estimate of ζfr and ζdi at point X could be made utilizing Eqs. (3) and (4). Evidently both ζfr and ζdi are positive here, but ζfr has a much greater magnitude. This difference in magnitude is typical. In Fig. 2 the grid points line up perfectly with the front orientation; in practice, of course, this would rarely be the case. Thus, to facilitate simple calculation for all orientations, on a Cartesian grid Eqs. (3) and (4) can be rewritten as shown:
Computation can then proceed, as follows, at each grid point:
In the first step, the calculation of a mean axis—not a mean vector—is critical. This avoids discontinuous transitions in the axis field near fronts. On a standard Cartesian grid, simple five-point mean axes are computed [for details, see appendix 2 in Hewson (1998a)].
Higher-order derivatives are evaluated using the same approach described above. This method for computing local coordinate derivatives on a Cartesian grid is intrinsically the same as that found in Bell and Keyser’s (1993) appendix, which dealt with the shear-curvature vorticity partition. The difference lies in the local coordinate definition. There is no correspondence between the components in the two partitions, as examples in Fig. 3 illustrate.
The ambiguous situation shown on the right-hand side of Fig. 3 is also evidence that using the shear-curvature partition in a low-level frontal environment would be unhelpful. Just a tiny change to the isobaric pattern here could commute all the vorticity to either shear vorticity or curvature vorticity. This is because the velocity vector, which defines the local coordinates, is ill defined in the zero wind environment at the front, in addition to which the average of the adjacent velocities is also zero. The ζfr − ζdi partition exhibits no such ambiguities around fronts, because for the front to exist it must, by definition, be well defined, which in turn means that the front-relative local coordinates will also be well defined. So (from a mathematical perspective, at least) this new vorticity partition is much better suited to analysis in the low-level frontal environment. Another advantage of the new partition is that it is Galilean invariant whereas the shear–curvature partition is not (see Viúdez and Haney 1996 for discussion).
A standard assumption when calculating partitioned derivatives is that local coordinate orientation does not vary much within the computation’s spatial range (note how the n axis is tangential to the front in Fig. 2). Highly curved fronts are more problematic. Fortunately, this scenario is rare, and often when it does occur numerical representation will be suboptimal anyway because of model resolution.
b. An operational model example
Figure 4 depicts vorticity values calculated from high-resolution (∼60 km) gridded numerical model output, using the methodology of section 3a(3) above. Referring specifically to the cold front (dashed line), note how the full geostrophic vorticity (left-hand side; darkest red = 14–15 × 10−5 s−1) is considerably more than the geostrophic disturbance vorticity (right-hand side; lightest orange = 1–1.5 × 10−5 s−1), as one would expect for any incipient cyclonic features, and is in qualitative agreement also with the schematic vectors on the left-hand side of Fig. 2. The geostrophic frontal vorticity represents the difference [Eq. (2)] and is therefore large. It is the inclusion of the front parallel wind component into the vorticity shown on the left that leads to a strip-like pattern (mirroring the frontal orientation) and its exclusion on the right that leads to a much more cellular structure. In essence, this is because only the front-parallel component of the wind is defining the (pre-existing) front. Excluding the front-parallel wind from the vorticity calculations (right-hand side of Fig. 2) thus conveniently acts, of itself, like a local filter. It removes a signal that is relatively large and could thus contaminate and retains just that part of the local wind field that, to a good approximation, dictates subsequent frontal motion, and which is thereby of singular importance in determining the frontal buckling that characterizes frontal waves. Thus, the vorticity of the cross-front wind becomes a much better discriminant of wavelike features on fronts than the full vorticity, further supporting the earlier introduction of the term disturbance vorticity.
A preliminary (albeit incomplete) diminutive wave definition arises from the above: the tip of a diminutive frontal wave lies wherever there is a local maximum in the along-front direction in the disturbance vorticity. Synoptically, diminutive wave formation is probably best thought of as the time when isobaric spacing along a previously two-dimensional front (Fig. 1, stage 0) first becomes, locally, nonuniform (stage 1). The gray spot in stage 1, lying just on the low pressure side of the stronger along-front pressure gradient, denotes the tip of the diminutive wave. Although not shown in Fig. 1, it is also perfectly acceptable for an isobar to go straight through the tip of a diminutive wave.
Reference to the cold front example in Fig. 4 shows that locations of the along-front maxima in the vorticity of the cross-front geostrophic wind (right panel; clear-cut at A1 and A3, more marginal at A2 and A4) can differ from the locations of along-front maxima in the full geostrophic relative vorticity (left panel; A1 only). Indeed, there is no particular reason to expect, in general, that the corresponding maxima will be exactly collocated (although sometimes they will be).3 These differences provide further support for developing the aforementioned vorticity partition, as opposed to using full vorticity, to define and identify diminutive waves. Note how the maxima in the right panel also correspond visually (and implicitly) with regions where there is a step in geopotential height contour spacing (which would correspond closely with isobaric spacing). For example, along the front, south of A3 but before minimum B4, the height contours are slightly closer together, implying a stronger cross-front geostrophic wind. Just to the north of A2 and A3 they are slightly further apart, implying a lighter cross-front geostrophic wind. The spacing differences are small because we are, of course, implicitly looking for small-scale incipient features. With more height contours, it might have been possible to see local changes in spacing corresponding to A2 and A3, although at this level of detail numerical model resolution is a key limiting factor (grid spacing is marked on the left panel).
It is in the regions just north of each of the A1–A4 maxima on Fig. 4, rather than in the regions just to the south, that any subsequent general change in the cross-front wind would be most likely to lead to flow reversal. For any diminutive wave, such a flow reversal would imply simultaneous conversion of a short segment of the cold front, just north of the wave tip, to a warm front (assuming frontal typing by geostrophic wind) and thereby conversion of that diminutive wave into a frontal wave. In turn, this suggests that the evolution sequence implied by frames 0 through 2 in Fig. 1 is not unreasonable, and also that for real cases it should be devoid of track jumps.
c. Published examples
Experienced forecasters have long recognized that a weakness or slack area in the pressure pattern along a front is a sign that frontal wave development may be imminent. Uncovering published examples of features that could be termed diminutive frontal waves is, however, more problematic. Unsurprisingly, this is because it is unusual to publish data for times prior to when a studied cyclonic disturbance was deemed to have formed. Coles (1962), however, provides one good example: he marks a wavelike feature on a warm front that has no attendant cold front, yet that satisfies the criteria for diminutive wave outlined here and that subsequently evolves into a full frontal wave (his Figs. 4b and c). Figures in Rivals et al. (1998), focusing on a cold front, also show pressure weaknesses developing prior to frontal wave formation; one wave in particular, not discussed but emerging out of their box B2, can be clearly traced back as a distinct pressure weakness for at least 18 h prior to the point at which it appeared to first develop its own warm front (their Fig. 4). Ayrault and Joly (2000) provide arguably the most comprehensive work yet on early cyclone stages by taking all instances of cyclone formation (defined by appearance of an 850-hPa relative vorticity maximum of scale ≥380 km) 6 h back in time and producing composites for types defined statistically. Notably, three of the four frontal cyclone types they illustrate “begin” at T − 6 h, with a strip-like relative vorticity structure (akin to the left panel on Fig. 4). Considered in conjunction with the current discussion, and notably the right panel on Fig. 4, this result seems to further support using different vorticity diagnostics to try to identify the earliest cyclone stages.
Consistent with their inclusion in Fig. 1, diminutive frontal waves are intrinsically synoptic-scale features. As such, they can be characterized by length scales on the order of 200–500 km (i.e., slightly smaller than typical frontal waves). This concurs with the size of features closely resembling diminutive waves that develop in the idealized 12-km primitive equation model runs of Dacre and Gray (2006) (∼300 km; their Fig. 3g) and also with enlargement during early development as illustrated in Grotjahn et al. (1999). Current-generation operational global models have a resolution of 60 km or less, which makes identification tractable (based on minimum feature separation of four grid lengths; e.g., see Fig. 4).
e. Three types of cyclonic feature
Because this study focuses on features that develop on fronts, the discussion will now exclude disturbances characterized by a maximum in the full low-level vorticity but no front or pressure minimum. This leaves three intrinsic types of cyclonic feature:
(i) Diminutive frontal waves—local along-front maxima in the disturbance vorticity;
(ii) frontal waves—meeting points of cold and warm objective fronts where the disturbance vorticity is positive; and
(iii) (barotropic) lows—mean sea level pressure minima.
It is important to note that these preliminary definitions are not mutually exclusive. Depending on the geometric configuration of isobars and fronts there may be overlap; that is, one point satisfying one criterion may lie very close to a second point that satisfies another (or, in extreme circumstances, one point in space might satisfy all three). Note how the gray spot in Fig. 1, representing the notional center of the cyclonic disturbance, slowly moves away from the front in the later stages. As this happens, features defined independently by (ii) and (iii) above would separate slightly. However, at the same time this is symptomatic of the eventual decay of thermal gradients at the low center and thus, through thermal wind arguments, the development of an axisymmetric vortex at upper levels, implying a low that is becoming increasingly barotropic, or, in the rarer case of a warm core system, equivalent barotropic. As the front loses its definition, any signature of frontal waves (and diminutive frontal waves) disappears, leaving just a “barotropic” low center. Because of the overlap problem, a consistent objective approach to naming systems, using (i), (ii), and (iii) above, requires the application of a hierarchy, whereby centers identified within specified small radii of one another defer to each other according to type. In order of precedence, the hierarchy recommended is: frontal wave → barotropic low → diminutive frontal wave.
The fact that a frontal wave tip takes precedence over a low pressure center reflects the desire to focus on substantive systems that grow on fronts, and also supports use of the term barotropic low. Diminutive waves are placed last because of the greater synoptic significance of the other features, as reflected by their widespread use over the years. Referring back to Fig. 1, with the above hierarchy now applied, the correspondence between stages is as follows:
Stage 1→ Diminutive wave
Stages 2, 3, 4, 5 → Frontal wave
Stage 6 → Barotropic low.
With objective techniques for each of these three cyclone types in place (Hewson 2001), there now exists a framework for objectively identifying the full life cycle of a typical extratropical cyclone. This becomes especially powerful when one begins to consider the degree to which real cyclones, in different parts of the world, actually evolve according to the extended Shapiro–Keyser model in Fig. 1. Transitions in and around the North Atlantic are assessed in section 4, with more detailed case studies presented in section 5.
In conclusion, returning to the case of diminutive waves, application of the above hierarchy, together with scale considerations, now enables the definition of a diminutive frontal wave to be extended, into its final form:
The tip of a diminutive wave exists wherever, on a low-level front, the disturbance vorticity reaches, in the along-front direction, a local maximum—that is, provided there is neither a frontal wave nor a barotropic low in the vicinity. Vorticity calculations should accord with the typical diminutive wave length scale of 200–500 km.
The various objective techniques developed in conjunction with this study (Hewson 2001) have been so designed to ensure that only points which satisfy all facets of the above definition are recognized as diminutive frontal waves.
4. Evolution over the North Atlantic
The top of Fig. 5 shows a snapshot from one 72-h model forecast (call this time T72) of all cyclonic features (spots) identified objectively using the methodology outlined in previous sections and covered in detail in Hewson (2001). Isobars and objective fronts for the same time are included. The main cyclonic synoptic features are successfully identified and typed over most of the domain. Around high topography a few problems remain, with fronts being more fragmented and some, though by no means all, cyclonic features being suspect. The primary reason for this is probably that traditional concepts of cyclones and fronts, which the methodology here tries to replicate, tend to break down near steep slopes. Similar problems have been noted in other cyclone identification and tracking studies.
The lower part of Fig. 5 shows the tracks, within the model forecast run window of T72 ± 72 h, of all the substantive (large solid circle) cyclonic features from the top panel. Open circles were produced automatically; arrows were added manually, using animation as a tracking aid. Isolated solid circles thus denote T72 features for which no tracks could be created; most are tied to high topography. The many tracks that do exist are generally smooth and coherent, implying that the locating methodology could be successfully coupled to a tracking algorithm to capture the full cyclone life cycle, including the diminutive wave stage. Compared to Fig. 1, many cyclones move through the relevant stages in the prescribed order. For example, the leftmost track over Canada and the track crossing northern Scotland both show two “correct” transitions: diminutive wave to frontal wave cyclone followed by frontal wave cyclone to barotropic low (= green to orange and orange to black). Several other features show one of these. The barotropic low “graveyard” near Iceland is also well represented by two decelerating lows. There are also a few counterexamples. For instance, diminutive waves can represent a decaying cyclonic feature (e.g., over Iberia) and they can also develop and decay without reaching the frontal wave cyclone stage (e.g., the two features south of Newfoundland). The barotropic low east of Florida is Hurricane Jeanne. Although not apparent here, many examples have been seen of tropical systems such as this “colliding” with an objective front and thereby changing from a barotropic low to a frontal wave around the time of extratropical transition. Evidently, then, there is a sense in which all the transitions depicted on Fig. 1 are reversible, although the majority of real cases, as subsequent discussion shows, are from left to right.
To gain further insight into cyclone longevity, cyclone-type transitions, and the general applicability of Fig. 1, subjective tracking similar to that illustrated in Fig. 5 above has been applied to 12 cases between 2000 and 2005. One randomly selected 6-day forecast was analyzed for each of the 12 calendar months. A manual check confirmed that the range of atmospheric regimes encountered was not atypical; dates are given in the Table 2 caption. A total of 341 separate cyclonic entities and 1674 transitions were recognized. This sample is believed to be large enough to be reasonably representative of the population over the domain considered for the substantive features. Note that features in the thermally “weak” category were not themselves tracked from T72, although they were included in the analysis if they formed part of the life cycle of a substantive T72 feature. This explains the smaller number of recorded weak feature transitions (184); accordingly, result robustness for such features is proportionately lower.
One could question the use of model forecasts rather than analyses. However, these are relatively short-range forecasts from a reputable operational forecast model, so although the potential for contamination by biases in wave handling (see section 2d) needs to be borne in mind, overall this should be small. More importantly, this approach avoids any “jumpiness” in the handling of minor features that can be associated, in analysis sequences, with differences between observations and background model fields.
The results are illustrated in Fig. 6, which shows the lifetime of the cyclonic disturbances, and Table 2, which shows sample percentage probability of different cyclone type transitions in any given 12-h time step.
For Fig. 6, identification on one time frame corresponds to a lifetime of 12 h. A notable feature of Fig. 6 is that there is no modal lifetime, just a gradual decay (with what appears to be some sampling noise superimposed). The mean is 66 h, permitting on average four to five feature transitions with the 12-hourly data. Interestingly, if the 2-day minimum cutoff used by Hoskins and Hodges (2002) is applied, the recalculated mean lifetime here exactly matches their quoted value for this region (108 h).
Table 2 is best explained with an example: refer to the DW–W row, representing a substantive diminutive warm front wave. The probabilities of such a feature, 12 h later, being in the same category or evolving into a frontal wave are, respectively, 44% and 19%. Other useful calculations can be performed: for example, the probability P that a feature of a given type A will evolve into a feature of another type B at some future point without any intermediate transitions, is given by the geometric series
In the warm front diminutive wave example above, the probability of transition at some future point to a frontal wave is thus 0.19/(1 − 0.44) = 34%. Probabilities can then be combined to evaluate the likelihood of the Fig. 1 sequence being replicated, insofar as it is represented by two successive transitions (assuming no dependence of probabilities on earlier history). Note also that 12 h appears to be a small enough interval given that direct transitions of diminutive wave to barotropic low are limited. Some key results arising from Table 2 and related calculations include the following:
One-half of substantive cyclonic features start out as diminutive waves.
Warm front diminutive waves are 50% more common than cold front diminutive waves.
One quarter of substantive diminutive waves decay to nothing in 12 h.
Cold and warm front diminutive waves are equally likely to evolve into a frontal wave 12 h later.
There is a one in three chance that any cold or warm front diminutive wave will ultimately evolve into a frontal wave.
A diminutive wave to frontal wave transition is 2 to 3 times more likely than the opposite transition, supporting Fig. 1.
One in 10 substantive diminutive waves follow the transition to frontal wave then barotropic low shown by Fig. 1. There is little difference between warm and cold front cases.
A barotropic low to frontal wave transition (in 12 h) is more likely than the converse. At first sight, this contravenes Fig. 1; however, part of the reason is that there are many frontal wave stages. Reacquisition of fronts may be attributable to cyclone passage over temperature discontinuities on the underlying boundary (e.g., at an ice edge). This warrants further investigation. Extratropical transition is another possible explanation.
The chances of a frontal wave decaying completely in 12 h are 1 in 9. As such, this is the most resilient cyclonic feature.
The transitions most commonly observed are, in descending order, frontal wave to barotropic low, barotropic low to frontal wave, and warm front diminutive wave to frontal wave.
In broad terms, therefore, this analysis supports the revised cyclone life cycle shown in Fig. 1. Geographical variations also warrant investigation, although this would require more data. Finally, note that Table 2 data have recently assisted with the development of an automated tracking scheme; see Watkin and Hewson (2006).
5. Case studies and growth mechanisms
A number of studies have illustrated the growth of frontal waves on a front in the absence of upper-level support. The reader is referred in particular to Dacre and Gray (2006) for examples of idealized primitive equation model simulations that show this growth. Given that diminutive waves represent some midpoint between a front and frontal wave, it is therefore a given that diminutive waves can also develop as intrinsic instabilities on a front in the absence of upper-level forcing. Indeed, this can be clearly seen in some of the aforementioned simulations (not shown).
Less clear is the extent to which upper-level forcing may be playing a role in the real-world development of diminutive waves. The next two examples use 3-hourly operational model data to begin to address this question.
Figure 7 illustrates the evolution of two diminutive warm front waves (arrowed) in a model forecast. Both move east, the northern one developing markedly (and producing significant rain over southern England). The northernmost feature’s parent cyclone (marked L in the left frame) decays through the period but at the same time acquires a thermal gradient, becoming a frontal wave in the middle frame and a diminutive cold front wave near 13°W on the final frame. The brown lines in the central panel relate to upper levels. The patterned line is a 400-hPa height contour segment. The remaining brown contours depict wULQG700—that is, the portion of the quasi-geostrophically forced vertical motion at 700 hPa that can be attributed solely to the thermodynamic structure of the atmosphere in an upper-level layer: in this case, 650 to 50 hPa (after Clough et al. 1996; see also Deveson et al. 2002). Accordingly, wULQG700 gives a clear and quantitative measure of the upper-level forcing, the zero line here being a notional upper trough axis, with forced ascent (due to upper levels) ahead and forced descent behind. Such diagnostics are presented in recognition of the difficulties of inferring forcing from basic fields (see Hewson 1997b). Until T + 0 the trough axis was west of the low center marked L, permitting the trough to contribute to the low’s development. Thereafter, the upper trough overtook the surface low and rotated anticyclonically, such that by T + 21 the low was decaying in a descent region, while the wave continued to develop on the warm front where upper-level forcing was maximized. Thus, the European wave development appears to have been largely driven from upper levels. The full evolution could also be described as the parent low splitting into two features—the new one developing, the old one decaying.
Figure 8 illustrates diminutive waves on a long-lived cold front evolving into frontal waves, based this time on analyses. Note how the tracks are broadly either front-parallel (A, C, D, E) or front-normal (B). It is a moot point as to whether feature B actually turns to the left and evolves into C and/or D and/or E or whether it spawns these features while continuing on to the southeast (indeed, the development of C and D could be described as feature splitting). Either way, this appears to be an example of hitherto unrecognized evolutionary behavior for cyclonic features on fronts. The right-hand side of Fig. 8 also shows oscillations in type, which are consistent with life cycle reversibility.
The arrowheads in Fig. 8 denote decay except for feature E, which became a substantial low over Scandinavia. Often the last cyclonic feature on a front will develop most, due in part to reductions in environmental flow and inhibiting deformation caused by continued weakening of the original parent low (Renfrew et al. 1997; Chaboureau and Thorpe 1999). The parent low here is northwest of Scotland. However, the interplay between features is complex, with D apparently having decayed to a diminutive wave before disappearing, in response to the continued development of E. Note also the variations in system speeds: D and E clearly moved more slowly than C. This reflects a general reduction in strength of both the frontal thermal gradient and the overlying upper-level flow—probably due, in turn, to cross-front sea surface flux gradients.
The brown lines on the left of Fig. 8 show a broad upper-level trough, with embedded shortwave features, that is advancing toward the southeast in tandem with the cold front. The diminutive wave B was situated, at this and other times, in a region of anomalously large wULQG700. The associated upper troughs clearly pre-existed feature B, upstream near Greenland, suggesting that they were the main drivers for development. The remaining four surface features (A, C, D, E) also all showed a strong connection with forcing maxima during their lifetimes, due either to upper levels (e.g., A; see left panel) or the middle troposphere (e.g., C). As B spawned each new feature to the northeast, the total forcing wFULLQG700 elongated toward the northeast and then broke into two. Partly because of this, each feature showed a clear association, at least in model fields, with precipitation maxima (not shown), providing further practical motives for feature identification.
The final example in Fig. 9 shows a warm front diminutive wave evolving into a major cyclonic windstorm over Europe—and leading ultimately to forest destruction over Slovakia of apocalyptic proportions—just southwest of the low center on the right inset. This appears to be around the frontal fracture stage (4) in Fig. 1. Compelling arguments for the need to improve predictions of such events can be found in Browning (2004). It is encouraging, therefore, to see the objective techniques pick out the full life cycle of such a storm, extending it backward using the diminutive wave concept, in accordance with goals stated at the outset. The 300-hPa wind arrows suggest that a jet streak and “jet crossing” played a significant role in major deepening (the feature enters a left exit region around 0° longitude). This is consistent with Baehr et al.’s (1999) findings for other cases. It also tallies with quasigeostrophic diagnostics in the lower left panel, which show forced ascent increasing over the feature. At the same time, a strong dipole in wFULLQG700 develops across it, consistent with the appearance of large near-surface pressure gradients and damaging winds. At genesis, upper forcing, although positive, is much less (= 0.3 cm s−1), suggesting that barotropic rollup on the front may be involved.
The three examples clearly indicate that diminutive waves can undergo nonmodal development and growth due to forcing associated with pre-existing structures, such as troughs and jets, in the middle and upper troposphere. However, it is difficult to isolate the effect of the inherent instability of the frontal strip and this too may be playing a role, as idealized simulations testify and as the third case suggests. Wave development may also be damped by flow around pre-existing cyclones. Evidently the evolutionary behavior of diminutive frontal waves is rich in detail and of fundamental importance if associated adverse weather is to be correctly predicted.
This paper describes how a typical cyclone life cycle, such as that shown in the Shapiro–Keyser (Shapiro and Keyser 1990) conceptual model, can be extended back in time to embrace the earliest signs imaginable of a cyclonic development on a front. Key points include the following:
The term “diminutive frontal wave” is introduced to refer to the incipient cyclonic feature.
Diminutive waves can be recognized synoptically on a front by a weakness in the pressure pattern.
Once a front segment has changed type, signifying geostrophic wind reversal, the diminutive wave becomes a frontal wave. This is stage 2 in the revised conceptual model.
A new front-relative vorticity partition has been developed to facilitate objective identification, comprising “disturbance vorticity” and “frontal vorticity.”
Mathematically, the tip of a diminutive wave marks a local maximum in the along-front direction in the disturbance vorticity (i.e., the vorticity of the cross-front wind), computed in accordance with a typical diminutive wavelength scale of 200–500 km.
Assessment of cyclonic feature transitions suggests that real cyclones do follow the revised conceptual model evolution, or portions thereof, most of the time.
All the conceptual model transitions can, in certain circumstances, proceed in reverse; hence, the diminutive wave stage can also be followed by decay.
The average lifetime of a North Atlantic cyclonic feature is about 66 h; the 12-h survival rate is about 80%.
Case studies have shown rich variety in the behavior of diminutive waves on fronts, with both front-normal and front-parallel wave tracks apparent.
Examples indicate that upper-level forcing can exert a controlling influence on wave development and behavior, although the intrinsic instability of frontal vorticity strips to cyclonic wrap-up probably also plays a role.
Many thanks to Helen Dacre for providing plots from her idealized model simulations, and to the reviewers for their very helpful comments.
Corresponding author address: Tim Hewson, ECMWF, Shinfield Park, Reading RG2 9AX, United Kingdom. Email: email@example.com
For the objective feature-based representation of this storm track the reader is referred to the figure in the Bulletin of the American Meteorological Society, Vol. 89, No. 9, p. 1273, in which genesis occurs some 102 h earlier.
Manual typing of whether a front is cold or warm in Europe generally depends on cross-front geostrophic flow. In the United States practice differs, with typing depending on frontal movement. Frequently the two methods produce the same result, but on occasions when there is a large cross-front ageostrophic flow, such as in cold, stable air in winter, there can be a difference. Such scenarios are probably more common in the United States, which may explain the difference in approach.
Note that there also exist thermodynamic configurations in which, in the along-front direction, a maximum in the full geostrophic relative vorticity can coincide with a minimum in the vorticity of the cross-front geostrophic wind. The configuration in mind is based on a straight front aligned along the axis of an anticyclonic col.