In the present study, we used the recently developed TempestExtremes framework that provides an intuitive and versatile command line interface dedicated to atmospheric features tracking. Tracking algorithms typically consist of two steps. The first one aims at detecting candidate points at any given time. The second step links these candidates together to build the actual tracks. It is known as the stitching step. For the particular case of extratropical cyclones, it is well known that the number of detected tracks is sensitive to the tracking algorithm used e.g.,, the choice and calibration of which must thus be made with care. In the literature, two fields are commonly used to detect candidate points, namely the mean Sea Level Pressure (SLP) e.g., or the relative vorticity field at 850 hPa e.g.,. When using the SLP, the raw field does not need any spatial filtering since it is sensitive to synoptic-scale and planetary-scale circulation systems. In contrast, vorticity is more adapted to detect smaller-scales structures and needs a filtering step before being used, as it would otherwise be prone to multiple candidate point detections inside a unique cyclone, especially along fronts.

In this paper, we used SLP for the candidate detection step. This choice was driven by the fact we study large structures and we also wanted to avoid any kind of filtering. Candidate points are defined as locations where the SLP is a local minimum and where there exists a closed contour characterized by a SLP larger by at least 100 Pa than the minimum and located within a 6° great-circle distance

All the distances measured in degrees are great-circle distance.

The core temperature of a cyclone is hereafter defined as the difference between the temperature at the core of the cyclone and the temperature at its surroundings: 1DTL=TC-TB, where TC (K) is the temperature at the center of the cyclone and TB (K) is the mean temperature along a circle centered on the cyclone center at a radius of 5° (∼ 500 km). This radius value was chosen to be in agreement with the two main studies using CPS, i.e. and . Sensitivity tests were made with radii up to 10° and did not show any major differences in the results (Fig. S1a in the Supplement). DTL is hereafter estimated at 750 hPa. It is to be noted that other pressure levels could be taken and there is sensitivity associated to that choice (Fig. S1b). For example, in , the diagnostic is made in the middle troposphere. In this work, 750 hPa level is chosen in order to make a fair comparison with 's parameters and because we were interested in a lower-level diagnostic. When DTL is positive, the cyclone has a lower-level warm core and, conversely, when it is negative, the cyclone has a lower-level cold core. It is instructive to write DTL in terms of geopotential height Z using hydrostatic balance: 2DTL=gR∂ZB-ZC∂ln⁡p, where g and R are respectively the gravitational acceleration (m s⁻²) and the specific gas constant of the air (J K⁻¹ kg⁻¹). The vertical derivative can be estimated using finite differences of the geopotential height between 600 and 900 hPa. Analogously to the temperature, ZC and ZB stand respectively for the geopotential heights (m) at the cyclone center and averaged along a 5°-radius circle centered on the cyclone center.

defined the parameter -VTL as follows to estimate the core temperature: 3-VTL=∂ZMAX-ZMIN∂ln⁡p900hPa600hPa, where the minimum and maximum of Z (denoted as ZMIN and ZMAX respectively) are taken within a radius of 500 km. The vertical bar in Eq. () denotes a linear regression between 600 and 900 hPa. As we will see in the next section, the difference between the two parameters -VTL and DTL can be large for extratropical cyclones.

, inspired by 's work, built for their CPS another parameter called ROT, the thermal Rossby number, that is dimensionless and computed using the vertical derivative of the relative vorticity. They show that their parameter is closely proportional to TC-TB, so to DTL, but the diagnostic is made in the middle of the troposphere while the DTL diagnostic is applied at low levels (typically 750 hPa).

The thermal asymmetry of a cyclone is hereafter defined as: 4BK=<T750hPa>R-<T750hPa>L, where the subscript L indicates the sector left of the current motion and the subscript R the sector right of the current motion and <>R and <>L denote areal averages over a semi-circle of radius 5° (∼ 500 km) respectively to the right and left sides of the cyclone motion. BK vanishes for a perfectly symmetric thermal structure. In contrast, cyclone frontal structures with a warm sector to the right of the direction of motion result in positive values and, conversely, a warm sector to the left of the direction of motion leads to negative BK values. BK can also be negative, especially at the end of the life cycle of extratropical cyclones during the rolling up of the fronts around the cyclone center. Using both hydrostatic balance and finite differences, BK can also be written as: 5BK≃-gRΔln⁡p(<Z600hPa-Z900hPa>R-<Z600hPa-Z900hPa>L), where Δln⁡p=ln⁡p600hPa-ln⁡p900hPa.

defined the thermal asymmetry as: 6B=<Z600hPa-Z900hPa>R-<Z600hPa-Z900hPa>L The two parameters BK and B are proportional to each other but we prefer to use BK to express the thermal asymmetry in Kelvin, as is DTL.

As mentioned in the introduction, the baroclinic instability is a key driver of extratropical cyclones genesis and growth. Quantifying its importance during the life cycle of cyclones is commonly done using the so-called Baroclinic Conversion rate (hereafter referred to as BC, in m² s⁻³) which measures the conversion from the mean flow available potential energy to the eddy potential energy . To compute BC for any given cyclone, one needs to separate the flow between a background flow and a perturbation. Usually this is made using time filtering , the background flow corresponding to the low-frequency part and the perturbation to the high-frequency part. To keep the CPS parameters as simple as possible and to stay as close as possible to the tracking algorithm outputs, our strategy is to define the background flow with a spatial average made around the cyclone. More precisely the background flow is defined as follows: 7u‾=<u>v‾=<v>T‾(x)=<T>+<∇T>⋅x-xC, where <> denotes a surface average over a circle of radius 10° (∼ 1000 km) whose center is the cyclone center. u = (u,v) is the horizontal wind vector (m s⁻¹), x is the position vector (m) and xC is the position vector of the cyclone center (m). The background flow definition is set to satisfy the thermal wind balance. Indeed, because the geostrophic wind ug satisfies the thermal wind balance: 8∂ug∂ln⁡p=-Rfk∧∇T, where f is the Coriolis parameter (s⁻¹), and because the averaged wind over the circle is close to the averaged geostrophic wind <ug>≃<u>, one can write: 9∂u‾∂ln⁡p≃∂<ug>∂ln⁡p≃-<Rfk∧∇T>≃-RfCk∧<∇T>≃-RfCk∧∇T‾, fC being the value of f at the cyclone center. In other words, the background flow has a constant horizontal temperature gradient balanced by a constant vertical wind shear. Using that definition for the background flow, the perturbation of a variable X will be denoted with primes such that X = X‾+X′ for X = {u,v,T}. More details on the computation of the background flow in the stereographic projection of TempestExtremes are given in Appendix .

Using these definitions for the perturbations, we recover the usual form of BC in the eddy potential energy equation (see Appendix ): 10BC=-θ′Su′⋅∇θ‾, where θ is the potential temperature (K) and S = -1h∂θ∂p (K² s² m⁻²) the static stability in pressure coordinates, with h = Rppp0κ (m³ kg⁻¹ K⁻¹). It is a source term in the equation governing the Eddy Total Energy (ETE, m² s⁻²) defined in and as: 11ETE=EKE+EPE=12u′2+v′2+12θ′2S. EKE and EPE are the Eddy Kinetic and Potential Energies. It is thus relevant to normalize BC by ETE to obtain the baroclinic growth rate σ = BC/ETE, which reduces also to the product between the environmental baroclinicity (proportional to the Eady Growth Rate) and the efficiency of the cyclone to extract the available potential energy from its environment . We will use σ in addition to BC and ETE when studying the energy life cycle of midlatitudes cyclones in the following.

All baroclinic parameters are defined in a 10°-radius circle around the cyclone center (∼ 1000 km) and, unless otherwise stated, they are averaged between 900 and 300 hPa.

The core temperature diagnostic DTL, the thermal asymmetry BK and the baroclinic conversion rate BC form the new CPS adapted to the study of extratropical cyclones and is hereafter denoted as ETC-CPS.

Storm Ciarán was initiated as a diabatic Rossby wave in the western North Atlantic over the Gulf Stream and then moved eastward along the Atlantic stormtrack. It hit Western Europe during fall 2023 while reaching a deep low SLP of 954 hPa (Fig. a). At the time it reached its minimum SLP (Fig. b), the warm air mass of the center was isolated from the warm sector while the cold front is visible south of the cyclone but detached from the warm front (see the temperature gradient in black contours). The presence of the frontal fracture zone and the warm-air seclusion is typical of a Shapiro–Keyser cyclone type. These characteristics are consistent with the recent results of and .

Two CPS diagrams are presented for Ciarán: 's CPS based on B and -VTL (Fig. c) and the ETC-CPS based on BK and DTL (Fig. d). Both B and BK start from finite, positive values at the beginning of the storm life cycle, and progressively evolve towards lower values until they almost vanish. Such an evolution indicates that Ciarán starts being asymmetric and gradually evolves towards a symmetric structure. The relatively high values of BK (10 K) and B (100 m) at the beginning of its life cycle can be explained by the fact that Ciarán formed over the Gulf Stream in a region of high baroclinicity. Later in the life cycle, the decreases in BK and B are due to colder air moving southward relatively to the cyclone center and warmer air simultaneously approaching that center.

DTL is positive during Ciarán's entire life cycle, decreasing from an initial value of about 5 K to nearly 0 K (Fig. d). Ciarán can thus be classified as a warm-core cyclone, a result confirmed by a visual inspection of the temperature at 18:00 UTC on 1 November 2023 (Fig. 1b), which clearly shows that the core is warmer than its environment. Interestingly, -VTL parameter (Fig. c) exhibits a different evolution, transitioning from a negative value (-500 m), suggestive of a cold core, to a nearly vanishing value, suggestive of a near-neutral core. This is in striking contrast with the evolution of DTL during the storm life cycle and in disagreement with its structure shown in Fig. b.

Storm Domingos formed a few days after Ciarán in the same area. However, in contrast with Ciarán, it developed through a baroclinic interaction with a pre-existing upper-level trough (not shown). Domingos then followed a track similar to Ciarán, moving eastward from the western North Atlantic and hitting Europe with a deepening up to 960 hPa (Fig. a). The ends of Ciarán and Domingos life cycles were very different in terms of frontal structures. At the time it reached its minimum SLP, Domingos core structure did not show a coherent isolated warm air mass (Fig. b), in contrast with Ciarán (Fig. b). The cold front is well visible south of the cyclone and connected to the warm front (see the temperature gradient in black contours). This connection and the fact that the temperature gradient is weak around the cyclone center fits the last stage of the Norwegian cyclone type.

Despite these structural differences, the trajectories of Domingos in both CPS diagrams are very similar to that of Ciarán (compare Fig. c with Fig. c and Fig. d with Fig. d). Both B and BK start large and positive before decreasing to nearly vanishing values. -VTL is initially negative before evolving towards small values while DTL starts from positive values before also evolving towards small values. Near the time at which the central SLP reaches its minimum value, the value of DTL is smaller for Domingos (around 0–2.5 K) than for Ciarán (around 2.5–5 K). This is consistent with the presence of a coherent secluded air mass in Ciarán (Fig. b) and a more diluted warm air mass around the cyclone center in Domingos (Fig. b). To conclude, both cyclones have a warm core temperature during their whole life cycle.

Why are ETC-CPS and 's CPS diagrams so different for both Ciarán and Domingos? As shown above, the evolutions of B and BK in the two CPS diagrams are similar for the two storms but the ones of DTL and -VTL are dramatically different, particularly during the early stage of their life cycle, to the point that they have opposite signs in their respective CPS diagrams. There are a priori two main reasons that could explain why DTL and -VTL significantly differ: (i) the positions where Z = ZMIN and Z = ZMAX are not the same at every vertical level and the vertical derivatives of ZMAX and ZMIN cannot be associated with a temperature, (ii) if the positions where Z = ZMIN and Z = ZMAX at the different levels are co-located, ZMAX is not necessarily equal to ZB and/or ZMIN is not necessarily equal to ZC.

To better understand the cause of the large difference between DTL and -VTL for Ciarán and Domingos, we looked at the positions of ZMAX and ZMIN at different levels between 600 and 900 hPa. For both storms and for most of the time during their life cycle, ZMIN is systematically located very close to the SLP minimum and ZMAX is generally found south of the cyclone center (see the positions where Z = ZMIN and Z = ZMAX in orange and purple symbols respectively in Figs. b and b). Therefore, reason (i) cannot explain the discrepancies between DTL and -VTL for Ciarán and Domingos. Since ZMIN is found very close to the cyclone center, we have ZMIN≃ZC but ZMAX is very different from ZB. ZMAX is situated in a relatively warm region of the 5°-radius circle, south of the storm center. This preferred location is due to the fact that geopotential values (as well as temperatures) tend to increase equatorward in the midlatitudes. For Ciarán, the location where Z = ZMAX even falls in the warm front of the cyclones, exacerbating the warm temperature at this location.

There is no guarantee that the picture revealed by the two storms is not an exception. We therefore performed a systematic comparison between DTL and -VTL by tracking all Northern Hemisphere midlatitude cyclones (Fig. S2). We found that the cases of Ciarán and Domingos are not peculiar. More precisely we obtained the following results: (i) BK and B are correlated, (ii) differences between DTL and -VTL can be very large and the two parameters can even have opposite signs, especially during the initiation stage of the cyclones, where DTL is generally positive and -VTL generally negative, and (iii) the higher BK (or B), the higher the difference between DTL and -VTL. A large B (or BK) corresponds to a strong thermal and geopotential asymmetry so that the point of maximum geopotential ZMAX is not representative of the cyclone environment and differences between -VTL and DTL are large in that situation.

We conclude that -VTL is not an appropriate measure of the core temperature with respect to that of the environment, as opposed to DTL. This is because -VTL is more representative of the difference between the core temperature and the warmest temperature in the vicinity of the cyclone, which is likely to be negative, whereas DTL is representative of the difference between the center and the average around a 5°-radius circle.

To validate the tracking algorithm calibration and our methodology, this subsection displays seasonal track density maps of midlatitude cyclones and compares them with the existing literature. In DJF (Fig. a), cyclones are mainly found over the oceans in the North Atlantic and North Pacific. These locations correspond to the well-known Northern Hemisphere “storm tracks” . The entrances of the two storm tracks are associated with strong lower-level westerlies as shown by the high values of zonal wind climatology at 300 hPa. It reflects high values of baroclinicity and the usual key role of baroclinic interaction in those regions. Secondary spots of cyclone activity are also visible and well known in the Mediterranean Sea and downstream of mountain ranges Rockies, Ural, Himalayas; see.

In JJA (Fig. b), cyclones are again mainly located over the North Atlantic and the North Pacific, highlighted as well by the high upper-tropospheric zonal wind values, but they are slightly less frequent and the track densities are more zonally oriented than in DJF. The continental cyclone activity becomes of equivalent importance as the oceanic one. The Mediterranean Sea spot disappears.

There are thus important seasonal variations of the localization of tracks with more oceanic than continental tracks in winter and a more homogeneous spatial distribution in summer. These findings are consistent with previous studies on extratropical cyclones , the main difference being that we only consider tracks initiated between 30 and 60° N which suppresses the representation of many Arctic cyclones in our study. The consistency with existing literature confirms that the tracking methodology is accurate.

Using the newly introduced ETC-CPS, we present in this section a statistical analysis of all detected cyclones in winter and summer. Figure  shows the 2D density distribution functions (black contours) of midlatitude cyclones in a BK vs. DTL diagram, at the time they reach their maximum intensity (i.e minimum value of the SLP) during DJF and JJA. Most cyclones are asymmetric and have warm cores during both seasons (see the large densities on the upper-right quadrants in black contours). In addition, the lower the minimum SLP, the higher DTL (see shading). This is also true for both seasons but the amplitude spread is larger in winter. Hence, the most intense cyclones have very pronounced warm cores whatever the season.

Warm-core DJF cyclones (Fig. a) primarily belong to the two oceanic storm tracks in the North Atlantic and North Pacific (20 cyclones per year), which are characterized by high values of 300 hPa zonal wind at their entrance. Secondary track density maxima (with about 10 cyclones per year) are also visible over continental regions, such as central North America, Russia and China, as well as over the Mediterranean Sea. Winter cold-core cyclones are mainly continental (Fig. c). The highest track density regions are situated over the Great Lakes of North America and downstream of Ural mountains (2.4 cyclones per year). Three additional patches (1.2 cyclones per year) are visible in the Mediterranean Sea, in the western North Pacific near China and in the eastern North Pacific.

During JJA, warm-core and cold-core cyclones follow similar track density patterns to first order, contrarily to DJF (Fig. b and d). For both types of cyclones, the highest frequencies of occurrence are more zonally oriented and more evenly distributed between oceanic and continental regions in summer than in winter. The continental spots over northeastern America, Russia and China are relatively more active than in DJF and reach frequencies similar to the oceanic storm tracks. Regions with high-density cold-core cyclones are roughly the same as those of warm-core cyclones albeit with a slight northward shift for the former compared to the latter, in particular in the oceanic regions. The Mediterranean patch disappears for both types of cyclones.

Figure a shows the distribution of the minimum SLP reached by the cyclones for the four subsets. In DJF, warm-core cyclones are much more intense than cold-core cyclones: their median minimum SLP is lower by 14 hPa and their distribution is skewed toward small SLP values. In JJA, warm-core cyclones are less intense and cold-core cyclones are more intense than in DJF. Both types of cyclones display the same intensity distribution in summer, centered on 998 hPa. Cyclones lifetimes are longer in JJA than in DJF (Fig. b). The median duration is 0.5 to 1.25 d higher and there is also a greater proportion of long tracks: tracks lasting more than a week represent 3 % of DJF tracks but 13 % of JJA tracks. Differences in lifetime between warm-core and cold-core cyclones are smaller than differences between seasons. Warm-core tracks are slightly longer than cold-core tracks in DJF but the ordering is reversed in JJA.

To highlight the cyclones structural differences, Fig.  shows a composite analysis at the time they reach their minimum SLP for the four subsets and for various variables. The first obvious result is that differences between warm-core and cold-core cyclones are larger than differences between seasons. The radial composites of temperature anomalies at 750 hPa (first row) show that warm-core cyclones exhibit a warm anomaly at the cyclone center and a cold anomaly to the southwest in DJF (Fig. a) and to the northwest in JJA (Fig. c). Such a feature is related to the westward tilt of the geopotential height anomalies Z′ with height: Z′ indeed decreases with height to the west of the surface cyclone but increases with height near the center and to the east of the cyclone (see Fig. i, j). On the contrary, for cold-core cyclones, temperature anomalies are negative over the whole cyclone area. The coldest anomalies are found near the cyclones centers and their southwest flanks (Fig. b, d). This is in agreement with the funnel-like shape of the geopotential anomalies of cold-core cyclones shown in Fig. k, l: Z′ indeed decreases with height over the whole cyclone area. Consistent with the highly negative Z′ values at upper levels in cold-core cyclones, the potential vorticity is high at upper levels and greater than in warm-core cyclones (compare Fig. i with Fig. k and Fig. j with Fig. l). A stronger lower-level potential vorticity is also visible in warm-core cyclones, both in DJF and JJA, suggesting a stronger diabatic heating in warm-core cyclones. Interestingly, although the cold-core and warm-core separation is made at 750 hPa, it holds over the largest part of the troposphere. Cold-core cyclones exhibit cold temperature anomalies between 900 and 350 hPa, i.e. over almost the whole tropospheric column. Warm anomalies are found but the are confined to the boundary layer or in the stratosphere (see e.g., Fig. k). Warm-core cyclones are characterized by warm anomalies over the whole column with the largest anomalies near the surface (Fig. i, j).

Cross sections also suggest marked differences between the two categories of cyclones in terms of Eddy Kinetic Energy (EKE). Although the geopotential height anomaly minima of the warm-core and cold-core cyclones near the surface are similar in JJA (Fig. j, l), the horizontal gradients of geopotential height are much larger for cold-core cyclones across the troposphere, indicating higher values of the vertically averaged EKE for the latter cyclones. In DJF, the geopotential height minima are lower and the geopotential height gradients are higher for warm-core cyclones than cold-core cyclones in the lower troposphere but the opposite happens in the upper troposphere (Fig. i, k).

Radial composites of the vertical velocity ω (second row in Fig. ) provide additional information on the differences between cold-core and warm-core cyclones. For all cyclones, an ascending zone is visible northeastward of the center and a descending zone southwestward, respectively corresponding to the warm conveyor belt (WCB) and the dry intrusion (DI) air masses. When comparing warm-core and cold-core cyclones for a given season, the ascents associated with WCB are stronger in warm-core cyclones (negative ω lower) whereas the descents associated with DI are more intense in cold-core cyclones (positive ω higher). Such a result is not surprising as cold-core cyclones are dominated by upper-level geopotential height anomalies (Fig. k, l) from which DI descents start while warm-core cyclones are dominated by lower-level geopotential height anomalies (Fig. i, j) from which the WCB ascending motions start.

To summarize, warm-core cyclones are characterized by a clear vertical westward tilt of the geopotential height anomalies typical of unstable baroclinic waves while cold-core cyclones have a funnel-like vertical structure in geopotential height anomalies. The latter structure suggests that baroclinic conversion might not be strong in the latter cyclones. This hypothesis is investigated in the following subsection.

In Fig. , we show the seasonal 2D density distribution function in a DTL vs. BC parameter space, including all tracks points for each cyclone. An alternative representation in a BK vs. BC parameter space is also shown in Fig. S4. In both seasons, almost 80 % of them have a positive BC (black contours). The lifestage (shading) indicates that most cyclones tend to begin their life cycle with large positive values of DTL and BC and end their life cycle with values of DTL and BC close to zero. There is a minority of cyclones starting with near-zero and negative BC. These negative values are found in the upper troposphere where the geopotential height anomaly isolines are tilted eastward (not shown). The contours indicate that the main pathway is however from high and positive values of BC to lower ones and corresponds to a typical baroclinic life cycle, during which the baroclinic conversion peaks before the cyclone maximum intensity and weakens afterwards. This is consistent with the thermal asymmetry being positive during early stages of the life cycle and close to zero at the end (Fig. S4) as baroclinic systems show a well marked westward tilt with height of geopotential height anomaly isolines which necessarily leads to a warm temperature at low levels see e.g., Fig 6.9. The CPS diagrams shown here thus reveal that a large majority of midlatitude cyclones are driven by baroclinic processes but give no information about the distinction between cold-core and warm-core cyclones.

Figure  shows the time evolution of various mean (i.e. averaged over the different subsets) quantities along the tracks. The analysis of these composites is focused on the deepening of the cyclones, i.e. the part of the lagged composite before they reach their minimum SLP represented as lag 0 in the figure.

The lagged composites of SLP shown in Fig. a are consistent with Fig. a: warm-core cyclones are more intense than cold-core cyclones in DJF while they have the same intensity in JJA. It is noticeable that 48 h before the SLP reaches its minimum value, DJF warm-core cyclones already stand out by their SLP value 10 hPa lower than that of the other subsets. This can be explained by geographical and seasonal differences in climatological SLP. Warm-core winter cyclones deepen in particularly low-pressure regions, like the well-marked wintertime Icelandic and Aleutian lows (see tracks in Fig. a) while the other subsets develop in regions with higher climatological pressure. During the deepening phase, DJF warm-core cyclones additionally have a higher deepening rate than others as they exhibit the steepest SLP curve slope.

The vertically-averaged Eddy Total Energy (ETE) is another metric of the intensity of the synoptic systems. Its mean evolution is shown in Fig. b. All subsets have an ETE peak around the time of minimum SLP. The absolute values are 1.5–2 times higher in DJF than JJA. In DJF, warm-core and cold-core cyclones have similar absolute values in the beginning but the slope is steeper for warm-core cyclones during the last 24 h before reaching the minimum SLP. It is consistent with the steeper slope of SLP for warm-core cyclones compared to cold-core cyclones in DJF (Fig. a) and confirms that warm-core cyclones undergo a more explosive growth than cold-core cyclones in winter. In JJA, ETE of cold-core cyclones is larger than ETE of warm-core cyclones, but the slopes are similar for the two subsets, as are the SLP slopes.

As the baroclinic conversion rate BC plays a key role in the rate of change of ETE (see Appendix ), it is represented in Fig. c. All subsets show a maximum of BC around 12 h before the SLP min, which is in co-occurrence with the maximum ETE slope. In DJF, the BC of warm-core cyclones is twice as large as the value of cold-core ones which is around 6 × 10⁻⁴ m² s⁻³. This is consistent with a higher ETE rate of change and by extension a higher deepening of the DJF warm-core cyclones subset. In JJA, the cold-core peak of BC occurs slightly before the warm-core one but the two peaks reach approximately the same value (5 × 10⁻⁴ m² s⁻³). BC values explain thus well ETE slopes values at first order. In JJA, cold-core and warm-core cyclones peaks of BC being similar is quite surprising regarding their very different vertical structure (Fig. ). The 3D structure of BC shown in Fig.  provides an explanation for this apparent contradiction. Warm-core cyclones exhibit two positive peaks of BC to the west and east of the cyclone center (Fig. a, b), consistent with the well-marked westward tilt with height of the geopotential height anomaly contours. The composite of cold-core cyclones shows only one strong positive peak to the west and a negative peak to the east but much smaller in amplitude than the positive one, which is in agreement with the funnel-like structure of the geopotential height anomalies (Fig. c, d). The vertical tilt being more pronounced on the westward side, it explains the difference in amplitude between the two opposite-sign anomalies of BC in the cold-core cyclone composite. It also explains why the averaged BC is almost as large in cold-core cyclones as in warm-core cyclones in summer.

Finally, the baroclinic growth rate σ, which scales the baroclinic conversion with the eddy total energy, is shown in Fig. d. It is also the product of the background baroclinicity and a dimensionless efficiency term that measures the efficiency of the eddies to extract energy from the background flow . In DJF, warm-core cyclones have a higher σ than cold-core cyclones, which is not surprising because the baroclinicity on the western side of the oceanic basins where warm-core cyclones evolve is much larger than the baroclinicity in more continental regions where cold-core cyclones evolve (Fig. a, c). It is consistent with the DJF differences observed in BC between warm-core and cold-core cyclones in Fig. c. In JJA, σ is also higher in warm-core cyclones, which can be explained both by the localization of their tracks – slightly closer to high baroclinicity areas (Fig. b, d), and by their higher efficiency due to their more pronounced westward tilted vertical structure (Fig. ). The similar BC values between the two JJA subsets (Fig. c) are explained by higher ETE values in cold-core cyclones than in warm-core ones (Fig. b) despite their smaller σ.

To conclude, both cold-core and warm-core cyclones clearly show a pronounced baroclinic growth phase and a peak of baroclinic conversion rate before reaching their maximum intensity, but cold-core cyclones evolve in smaller baroclinicity and have a less clear westward-tilted vertical structure.