A global analysis of the dry-dynamic forcing during cyclone growth and propagation

Mechanisms driving the intensification and propagation direction of extratropical cyclones are an active field of research. Dry-dynamic forcing factors have been established as fundamental drivers of the deepening and propagation of extratropical cyclones, but their climatological interplay, geographical distribution and relatedness to the observed cyclone deepening and propagation direction remains unknown. This study considers two key dry-dynamic forcing factors, the Eady Growth Rate (EGR) and the upper-level induced quasi-geostrophic lifting (QGω), and relates them to the surface deepening 5 rates and the propagation direction during the cyclones’ growth phase. To this aim, a feature-based cyclone tracking is used and the forcing environment is climatologically analyzed based on ERA-Interim data. The interplay is visualized by means of a forcing histogram, which allows one to identify different combinations of EGR and QGω and their combined influence on the cyclone deepening (12-hour sea-level pressure change) and propagation direction. The key results of the study are: (i) The geographical locations of four different forcing categories, corresponding to cyclone growth in environments characterized by 10 low QGω and low EGR (Q↓E↓), low QGω but high EGR (Q↓E↑), high QGω and low EGR (Q↑E↓) and high QGω and EGR (Q↑E↑), displays distinct hot spots with only mild overlaps. For instance, cyclone growth in a Q↑E↑ forcing environment is found in the entrance regions of the North Pacific and Atlantic storm tracks. Category Q↓E↑ is typical found over continental North America, along the southern tip of Greenland, over parts of East Asia and the western North Pacific. In contrast, category Q↑E↓ dominates the subtropics; (ii) the four categories are associated with different stages of the cyclones’ growth phase: 15 large EGR forcing occurs typically earlier, during the growth phase at genesis, while large QGω forcing attains its maximum amplitude later towards maturity; (iii) poleward cyclone propagation is strongest over the North Pacific and North Atlantic, and the poleward propagation tendency becomes more pronounced as the deepening rate gets larger; zonal, or even equatorward propagation, on the other hand, is characteristic for cyclones developing in the lee of mountain ranges, e.g., to the lee of the Rocky Mountains. The exact location of maximum QGω forcing relative to the surface cyclone center is found to be a good 20 indicator for the direction of propagation, while no information on the propagation direction can be inferred from the EGR. Ultimately, the strength of the poleward propagation and of the deepening are inherently connected and the two dry-dynamic forcing factors, :::::: which allow cyclone development in distinct environments to effectively be identified.

indirectly accounted for via their influence on the low-level baroclinicity. The influence of diabatic processes has already been analysed in previous climatological studies (Čampa and Wernli, 2012;Boettcher and Wernli, 2013;Büler and Pfahl, 2017).
The relevance of this research topic is highlighted by the fact that the environment and the different forcing factors, which 60 drive extratropical cyclone deepening and the direction of propagation, are expected to change as the climate warms (Shaw et al., 2016;Catto et al., 2019). Indeed, climate projections suggest a decrease in low-level baroclinicity due to an amplified warming at higher latitudes, a process known as the Arctic Amplification. On the other hand, the increased water storage capacity of the atmosphere in a warmer climate suggests a potential increase in the latent heat release and thus (positive) diabatic impact on cyclone development. Both changes seem to be engaged in a tug-of-war (Catto et al., 2019)  where V g denotes the geostrophic wind and Φ the geopotential. The forcing from upper levels only is obtained by setting the divergence ∇ · Q to zero for pressure values less than 600 hPa,.
The formal definition of EGR is (Lindzen and Farrell, 1980), where u is the horizontal wind speed, θ potential temperature and z height.
The forcing factors will be determined along all cyclone tracks, i.e. the geographical position of minimum SLP (see section 2.2 below). It is, however, not reasonable to only consider the values of QGω and EGR at the cyclone's center, which is defined by the location of the minimum SLP, because the cyclone growth and propagation is determined by its larger environment.
Hence, we calculated the mean value of EGR and QGω within a 1000 km radius around the cyclone center. For QGω, two 100 mean values were calculated: one by only considering negative values (corresponding to forcing of upward motion), and a second one considering only positive values (forcing of downward motion). In this way, we take into account that QGω often appears as a dipole, with the effect that the two dipole parts counterbalance each other if a simple mean is calculated.

Cyclone climatology and time normalization
To obtain the cyclone climatology, we used the identification and tracking algorithm by Sprenger et al. (2017), which is a 105 slightly modified version of the algorithm introduced by Wernli and Schwierz (2006). First, the algorithm scans the grid points for SLP minima defined as having a lower value than the eight neighbouring SLP values on the grid. Secondly, the cyclone extent is determined by the outermost closed SLP contour encompassing the identified SLP minima (assuming a 0.5 hPa interval). To exclude spurious, small-scale SLP minima, the enclosing contour has to exceed a minimum length of 100 km, otherwise the SLP mimimum is discarded. In some cases the outermost SLP contour contains more than one SLP minimum.

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If the distance between two SLP minima within the same outermost enclosing SLP contour is less than 1000 km, they are attributed to the same cyclone cluster, creating a multi-center cyclone. In this case only the lowest SLP minimum is kept and the others are disregarded. The central SLP value and its geographical coordinates are stored, and subsequently used to determine cyclone tracks with cyclogenesis and lysis at the first and final time step of the track, respectively. As in Sprenger et al. (2017), the tracks have to exceed a minimum lifetime of 24 h from genesis to lysis.

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During their life cycle, cyclones can undergo a process called 'cyclone splitting', which occurs when a cyclone (or a multicenter system) breaks up and forms two (or more) cyclones that then extend from the same origin as two separate cyclone tracks. As a consequence, the newly formed cyclone typically experiences only decay characterised by an increasing SLP over the course of its life cycle. To eliminate this subcategory of cyclones from the data, we disregard all cyclones that exhibit their minimum SLP value either at genesis or at genesis + 6 hours. In this way, we restrict the analysis to cyclones with an archetypal 120 pressure evolution, i.e., starting with higher pressure at genesis than attained during maturity.
Because the lifetime of cyclones can extend from 24 h (by the requested minimum duration; see above) to several days and therefore it is difficult to compare different cyclone life cycles, we applied the method of Schemm et al. (2018) to normalise the cyclone lifetimes. More specifically, three main time stamps are determined: t genesis (time of genesis), t max (time of maximum intensity, i.e., minimum SLP), and t lysis (time of lysis). We then calculate ∆t 1 , which is the difference between t genesis and 125 t max and results in a negative value: Next, the difference between t lysis and t max (indicated as ∆t 2 ) is determined, which results in a positive value: For a certain time t between t genesis and t max we can calculate a normalised time t norm : The same is done for time t between t max and t lysis : Hence, in this normalized time frame, t norm = −1 corresponds to cyclogenesis, t norm = 0 to the time instance of minimum SLP, and t norm = +1 to cyclolysis. In the remainder of the study, t always refers to this normalized time. All of the analysis 135 in section 3 and 4 will be restricted to the phase with normalized times between -1 and 0, i.e., the focus is on the cyclone's life cycle between genesis and the time instance of deepest SLP. To be attributed to one of these three target regions, a cyclone at the time of genesis must be located inside the respective latitude-longitude 140 box.

Dry-dynamic forcing categories
In this study, we frequently employ a 2D histogram, which has EGR on the x-axis and QGω on the y-axis. The histogram consists of 49 bins defined by a range of EGR and QGω values such that each bin is characterized by a specified EGR and QGω forcing. Each six-hour time interval during a cyclone growth phase is classified according to its EGR and QGω values 145 such that each bin is populated by a multitude of cyclone time segments from various cyclone tracks. The color shading in each of the bins represents the mean value over all time steps of a specific cyclone characteristic, for example, the deepening rate. Figure 1 shows such a histogram where the mean represents the average growth rate defined by the 12-hour change in mean sea-level pressure (∆SLP). If, for instance, a 12-hour ∆SLP value of 6 hPa is associated with an EGR of 1.2 day −1 and a QGω value of -0.01 Pa s −1 , it contributes to the lower right corner of the histogram. The 12-hour ∆SLP values in every bin will vary 150 between different cyclones and each bin is therefore populated by a distribution of 12-hour ∆SLP values. This is illustrated is of the order of several thousands, except for the most extreme corner bins of the 2D histogram. The detailed numbers can be seen in Fig. S1 in the Supplement.
The lower-left corner of the histogram represents low QGω and low EGR forcing. Conversely, high QGω and high EGR forcing is located in the upper-right corner. The lower-right (upper-left) corner represents cyclone growth in environments characterized by low (high) QGω and high (low) EGR forcing. The four corners of the histogram are used to define four 160 forcing categories, which we study in more detail in section 3. More specifically, the box in the lower left corner, for example, is referred to as Q↓E↓ (cyclone growths in a low QGω Q↓ and low EGR E↓ environment) and the other boxes are labelled accordingly.
3 Dry-dynamic forcing during cyclone growth In this section, we study the regional variability of the forcing mechanisms during the growth phase of cyclones. We start with 165 the geographical distribution of the four categories Q↓E↓, Q↓E↑, Q↑E↓ and Q↑E↑ that were introduced in section 2.3, then proceed in section 3.2 with a detailed analysis of the forcing histogram and finally, in section 3.3, we discuss cyclone-centered composites of QGω and EGR.

Geographical distribution of dry-dynamic forcing
In this section, consideration is given to the geographical distribution of the four forcing categories. Density plots are created 170 by considering all time steps during the cyclones' growth phase and by computing their inclusiveness to one of the four forcing categories (Q↓E↓, Q↓E↑, Q↑E↓ or Q↑E↑) and the corresponding latitude-longitude location. The outcome is a remarkably distinct geographical distribution of the occurrence of the four forcing categories (Fig. 2). The plots show a probability density distribution, which integrates to 1. Each forcing category has unique hot spots, which are discussed in the following.
We start our discussion of the forcing-based categories ( Fig. 1) with category Q↑E↓ (Fig. 2a), which is mostly confined 175 within a latitudinal band between 25 • and 40 • N. The major hot spot is located over the North Atlantic off the west coast of Northern Africa and spanning over the Mediterranean. A second but less dense region is discernible in the Pacific off the U.S.
west coast, reminiscent of kona lows (Simpson, 1952;Morrison and Businger, 2001;Moore et al., 2008). Over the Atlantic, this category comprises the deepening of subtropical cyclones, which form under strong QGω forcing, provided by equatorward pushing intrusions of high-PV air (Caruso and Businger, 2006) and a weak baroclinic zone.

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The next category Q↑E↑ (Fig. 2b) has two distinct hot spots: one northeastward orientated band reaching from North America to Norway with the maximum frequency northeast of Nova Scotia in the North Atlantic, and the other hot spot off the coast of Japan. Both are located slightly poleward of the identified hot spot in Q↓E↑ (Fig. 2d). We hypothesize that early during the life cycle time steps are categorized into Q↓E↑ (Fig. 2d), while afterward during the main deepening period both forcings contribute to the deepening and the corresponding time steps are categorized into Q↑E↑ (Fig. 2b). We will come back to this 185 hypothesis in the next section.
For category Q↓E↑ (Fig. 2d), the regions where the forcing occurs most frequently are over North America and partly over the western North Atlantic, and further the southern tip of Greenland and parts of central Asia. Another prominent hot spot is located over the Pacific ocean off the coast of Japan. Over North America, the maximum is located near 50 • N and therefore north of the cyclogenesis region downstream of the southern Rocky Mountains in the U.S. (see Fig. 5c in Hoskins and Hodges,190 2002). It is connected to cyclone deepening in the lee of the Canadian Rocky Mountains and the formation of 'Alberta clipper' cyclones (Chung and Reinelt, 1973;Thomas and Martin, 2007). The southern tip of Greenland is a well-known cyclogenesis hot spot (Hoskins and Hodges, 2002;Wernli and Schwierz, 2006) and high baroclinicity in this region is connected to the steep QGω maximum. The EGR maximum, on the other hand, is found to the southwest of the cyclone center (Fig. 5b) where we expect the trailing surface cold front. Given the strong forcing and the archetypal flow situation that is well known for many 255 developing cyclones, it is no surprise that this category is characterized by the largest deepening rates (as seen in Fig. 3a).
Category Q↓E↓ displays a minor upper-level PV structure (Fig. 4c), resembling a PV cutoff centered above the surface cyclone's center attaining only a small amplitude of 1.5 pvu. The barotropic structure and small amplitude of the upper-level PV points to small deepening rates, in particular together with a weak QGω forcing (Fig. 5c) and a uniform low EGR environment (Fig. 6c). Indeed, category Q↓E↓ exhibits the weakest deepening rates (Fig. 3a).

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A completely different upper-level PV structure is discernible for category Q↓E↑ (Fig. 4d). The cyclone center is located on the flank of a southwest to northeast oriented band of enhanced PV gradients. The cyclone is likely located near the exit of a jet streak that forms upstream around the trough. This is in agreement with the existence of upper-level QGω forcing ( Fig. 5d) that is larger compared with Q↓E↓ ( Fig. 5c) but lower compared with Q↑↓ (Fig. 5b). In this meteorological scenario we expect enhanced EGR forcing, remembering that an upper-level jet by thermal wind balance must be associated with a 265 significant horizontal temperature gradient beneath its core and hence also with a corresponding EGR signal by definition (see section 2.1). Indeed, this is what we find in Fig. 5d. The normalized times associated with this category (lower-right in Fig. 3b) indicate that the cyclone development is rather in an early stage, as one would expect from the upper-level PV structure that displays only a weakly developed trough and ridge.
4 Dry-dynamic forcing, deepening rates and propagation direction

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While the previous section solely focused on the deepening rates of extratropical cyclones, consideration is now given to the connection between the dry-dynamic forcing, the deepening rates and the direction of propagation of the cyclone during its growth phase. The propagation direction at a time instance along a track is determined by taking the cyclone's six-hour displacement vector and determining the angle between this vector and a zonal vector, i.e., an angle 0 corresponds to eastward propagation and 90 • to northward propagation.  Figure 6 shows four different windroses for varying deepening rate regimes by means of the 12-hourly SLP changes. For example, Fig. 6a consists of propagation angles corresponding to time steps with deepening rates of 10 hPa 12 h −1 or more.

Propagation angle and deepening rates
The rings indicate the frequency of a specific angle range (i.e., the windrose petal). For instance, Fig. 6a includes several petals, the longest of which points in northeastern direction or 45 • . The corresponding petal reaches the outer ring of the plot 280 indicating that 34.2% of all cyclones that deepen at a rate of 10 hPa 12 h −1 or less propagate into the northeastern direction.
Therefore, the number and size of the windrose petals indicate the relationship between the cyclone deepening and the direction of propagation. The results for weaker deepening (-10 hPa < ∆SLP < -6.5 hPa) (Fig. 6b) indicate a dominant northeast-oriented propagation direction although the east-northeast petal has increased. For even weaker deepening (-6.5 hPa < ∆SLP < -3 hPa), most of the values are still within the northeastern direction, however, the largest petal is found in the east-northeast section ( Figure 6c). Moreover, the petal in eastern direction is now significantly larger than in Figs. 6a and 6b. Finally, this shift toward zonal (eastward) propagation angles is found for the weakest deepening rates (-3 hPa < ∆SLP < 0 hPa), where the two petals in east and east-northeastern direction are the most prominent ones, each representing approximately 20% of the angle values.
Overall, we can summarize that while during their growth phase cyclones go through different magnitudes of deepening rates, during the times a cyclone experiences increased deepening rates it tends to propagate more poleward.   angles. Downstream of these maxima, in particular for the North Atlantic storm track, the poleward tendencies steadily decrease and over Europe the tendencies attain rather zonal values. Besides the storm track regions, additional distinct regions exhibit 300 positive mean propagation angles: for instance, over California to the west of the Rocky Mountains, to the east of Greenland, over the Black Sea, to the east of Lake Baikal, over North East Siberia and the nearby the Arctic Sea. It remains to be studied in a refined analysis to which degree these tendencies are determined by orographic effects or other forcings.
Of course, as seen in the histograms of propagation angles for the outlined four regions in Fig. 7, each region is characterized by a rather wide spread of possible propagation angles. For instance, in the North Pacific the mean propagation angle peaks 305 near 45 • , but smaller and higher values often occur. Even higher northward propagation angles, in the mean, are found in the box over the western and central North Atlantic. This is in agreement with the fact that the North Atlantic storm track is more tilted towards the northeast with increasing longitude compared to the North Pacific storm track (Hoskins and Hodges, 2002;Wernli and Schwierz, 2006).

Cyclone-centered composites for poleward and eastward propagation
In the previous section, we showed that there is a general relationship between the deepening and the propagation of cyclones.
Cyclones tend to propagate poleward when the deepening is strongest. In this section, we link this result back to the drydynamic forcing QGω and EGR by means of cyclone-centered composites for cyclones that propagate predominantly poleward compared to eastward propagating cyclones. 325 Figure 9 shows the QGω forcing (color) for poleward propagating cyclones (Fig. 9a) and for cyclones propagating more eastward (Fig. 9b). The categories are defined according to a range of propagation angles: angles between 35 and 65 • for poleward propagation, and between -5 • and 25 • for eastward propagation. The black dot represents the SLP minimum (cyclone centre) and the black arrow in the center indicates the mean propagation direction of the cyclone within the following 6 h. To make the comparison easier, a white point indicates in both fields the position of the maximum QGω forcing for poleward 330 propagation and a white cross indicates the maximum in case of an eastward propagation.
In both cases, the cyclone center is located close to and just equatorward of the maximum QGω forcing. The maximum forcing is more pronounced for poleward propagating cyclones with values exceeding -0.24 Pas −1 (Fig. 9a). The QGω maximum for eastward propagating cyclones reaches -0.16 Pas −1 (Fig. 9b). The forcing in case of poleward propagation by QGω is purely poleward and the direction of propagation is northeastward (black arrow in Fig. 9b). In contrast, for eastward propa-335 gating cyclones the maximum of QGω forcing (white cross) is not only weaker but also located to the northeast of the cyclone center resulting in a more zonal direction of propagation. In summary, it seems that the weaker amplitude and eastward shifted QGω center leads to a more zonal cyclone propagation, whereas a QGω maximum to the north is able to deflect the cyclone path poleward.
The EGR environment for poleward and zonal cyclone propagation is shown in Fig. 9c and d. In case of poleward propagating 340 cyclones (Fig. 9c), the displacement vector is orientated essentially normal to the EGR field, pointing towards lower EGR values. This indicates that cyclones propagate away from high EGR values, which are found in this case in the southwestern sector of the cyclone where we expect the cold front. On the other hand, in the case of eastward propagating cyclones (Fig. 9d), the displacement vector is also locally normal to the EGR isolines, but the large-scale EGR environment has a stronger zonal orientation compared to the poleward propagating case. It is, however, difficult to judge what the exact contribution by the 345 EGR environment is to the cyclone's propagation.

Conclusions
The deepening and propagation of extratropical cyclones occurs within a remarkable wide range of environments. In this study, we analyse the environment during the cyclone growth period in terms of the dry-dynamic forcing, its variability and relationship with the cyclone propagation direction. To this aim, extratropical surface cyclones are identified and tracked 350 during the Northern Hemisphere cold season (October to March) based on six-hourly ERA-Interim data . Each time step along every cyclone track is characterized in terms of its 12-hour deepening rate (∆SLP), the upper-level QG forcing for ascent (QGω), the lower-tropospheric Eady Growth Rate (EGR) and the propagation direction. Since cyclone deepening helpful discussions on cyclone evolution, and to Timo Schmid whose excellent Bachelor thesis gave the inspiration for the 2D forcing diagrams.