Divergence of tropical cyclone hazard based on wind-weighted track distributions in the Coral Sea, over 50 years

Tropical cyclones (TCs) with genesis in the Coral Sea present significant hazards to coastal regions in their surroundings. In addition, the erratic nature of TC tracks is not well understood in this region. Therefore, this study grouped Coral Sea TC tracks over the last fifty years based on K-means clustering of the maximum wind-weighted centroids. This was done in order to extract valuable new cyclone power, track curvature and location related information from their historical track records and to predict their behaviour in the light of a changing climate. TC track variance and curvature (sinuosity) were assessed. Three well-defined clusters of TC tracks were identified, and the results showed differing predominant directions of TC movement by cluster. Track sinuosity was shown to increase from east to west. Only one cluster showed a statistically significant trend (decreasing) in TC frequency. The TC power dissipation index (PDI) was used to reveal that two of the clusters have diverging trends for PDI post-2004. Based on the location of cyclone maximum intensity, only one cluster showed a statistically significant trend (towards the equator). All these findings demonstrated a clear variance in between-cluster hazard and show that TC trends discovered for the southwest Pacific are not manifest or consistent across all clusters.


Introduction
Tropical cyclones (TCs) can be extremely hazardous to areas in their vicinity (Kreussler et al. 2021;Tauvale and Tsuboki 2019) and, in addition, have the capacity to cause devastation from a distance through the waves they generate (Taupo and Noy 2017). The consensus from recent global studies has predicted more intense TC activity due to anthropogenic climate change which may exacerbate inundation in association with sea level rise (Knutson et al. 2020;Walsh et al. 2019).

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TCs near the east coast of Australia have historically caused significant damage to natural landscapes and anthropogenic infrastructure (Anderson-Berry 2003; Asbridge et al. 2018). Of the top five most expensive natural disasters in Australia from 1966 to 2017, three were due to TCs (McAneney et al. 2019). Of these three TCs, two tracked in the eastern basin of Australia-more specifically, in the Coral Sea shown in Fig. 1 (Bureau of Meteorology 2021). TCs can therefore pose significant risk to the east coast of Australia. Recent severe flooding that impacted the most easterly sections of this coastline (occurring February and March 2022 and triggering the declaration of a national emergency) has highlighted the urgent requirement for understanding future extreme weather threats at a regional level.
The future change in the risk profile of these cyclones is uncertain, with an apparent decrease in frequency of occurrence, along with an increase in the proportion of severe cyclones for the years 1980 to 2018 in Australia, especially in the eastern basin (Chand et al. 2019). An increase in TC intensity and destructiveness due to climate change is therefore a potential future scenario to be considered, and research into future TC trends is becoming an urgent requirement for risk mitigation (Bloemendaal et al. 2022).
When exploring recent research on Southern Hemisphere (SH) TCs collectively, clear differences are evident between the hemispheres (Pillay and Fitchett 2021b). As an example, the trend of increasing power was not reported as significant for TCs in the SH by Klotzbach and Landsea (2015), which has been ascribed to a lack of consistency in the presatellite-era (1980) records (Klotzbach and Landsea 2015). In addition, the poleward shift of TC occurrence is more significant for the SH (Kossin et al. 2014) and marked differentiation in seasonality and lifespan results have been shown (Kossin 2018). The mean number of TCs per year from 1980 to 2016 in the SH is 13 with the greatest percentage occurring in the South Indian Ocean (45%) followed by the South Pacific (21%), East Australian (19%) and West Australian (15%) regions (Pillay and Fitchett 2021b). Examining model projections of TCs in response to anthropogenic-based warming of a hypothetical 2 °C, Sea, 1970Sea, -2020 showing the exposure of the east coast of Australia to these TCs the South Indian Ocean shows an average decrease in TC frequency (over 115 estimates) of approximately 18% and the Southwest Pacific approximately 20% (over 122 estimates). For both of these SH basins, the decrease is slightly larger than the global average decrease of about 15% (Knutson et al. 2020). Intensity change projections for SH TCs show a large average increase for the South Indian Ocean (5% over 17 estimates) relative to the Southwest Pacific Ocean (1% over 18 estimates). The projected global increase is around 5% (Knutson et al. 2020).

Fig. 1 TC tracks with genesis in the Coral
Numerous examples of Southern Hemisphere TC activity analysis (including the South Pacific), using historical cyclone track data, exist in recently published scientific literature (e.g. Magee et al. 2017Magee et al. , 2020Bell et al. 2020;Chand et al. 2019;Ramsay et al. 2012;Sharma et al. 2020, Sharmila andWalsh 2018;Zhao et al. 2018). Also, several recent studies using TC track data have been conducted in the Northern Hemisphere (e.g. Bell et al. 2020;Bhaskar Rao et al. 2019;Ge and Colle 2019;Kelly et al. 2018;Liu et al. 2019;Zhang et al. 2020). However, whilst historical records of TC tracks, as recorded by geospatial location, have the potential to yield new insight into TC behaviour, the body of research into TC track spatial geometry and curvature and the evolution of these characteristics post-genesis is far less prevalent (Sharma et al. 2021).
A set of geo-spatial TC track records yields information on numerous inter-related temporal and spatial track characteristics, including shape parameters such as overall degree of curvature and variance in the nominated directions of the total area transgressed by a particular TC and the TC average speed and duration. These characteristics can be used to index and group cyclones in order to assess and understand the movements of cyclones within a region and to predict future changes to TC behaviour at a sub-regional level, based on defined statistical groupings of chosen track parameters. Studies of this type conducted to date can be categorised into a few types: (A) Fitting polynomial regression curves to the TC tracks and then grouping by similarity of the polynomial regressions (e.g. Ramsay et al. 2012;Sharma et al. 2021); (B) K-means clustering of a selected n-dimensional vector compiled from chosen TC characteristics (e.g. Yan et al. 2018); (C) K-means clustering of TC mass moments (e.g. Nakamura et al. 2009); (D) Grouping using the Fuzzy C-means algorithm (e.g. Liu et al. 2019).
Analysis types (A) and (D) have proved effective in grouping TC tracks by the shape characteristics of the geographical trajectories though they do not (without potential method adjustment in future efforts) consider change in cyclone power through the cyclone lifetime and involve mathematically complex models of each TC track. A commonly stated limitation of the grouping method B) above, using K-means of selected TC characteristics, is that it does not directly consider track length or shape . Although Terry and Gienko (2011), for instance, have developed an index (of sinuosity) that characterises tracks based on their shape that can be used in K-means and therefore, to a degree, overcomes this limitation. The grouping method in (C) indirectly considers track length in the first mass moment (with maximum wind speed or minimum central pressure being a measure of the mass of the cyclone at a particular point along its track) and directly considers length, orientation and curvature if the second mass moment (incorporating variance in three directions) is applied. The grouping of TCs by considering the evolution of both their power (maximum wind speed or minimum central pressure) and track geometry is another powerful advantage of this method (C), since it can then be used to investigate the underlying drivers of regional climatology such as the EL Nino Southern Oscillation (ENSO), for instance. It is noted, however, that genesis point information is lost after applying the algorithms with this method. Grouping cyclones by genesis location and then investigating the TC track characteristics by group is another potential method.
As cyclone intensity and power are of primary concern when identifying cyclone trends over the last fifty years (the period most affected by anthropogenic climate change forcing), the cyclone power dissipation index (PDI) is defined and then trended by group. Trends of TC PDI using track data arranged in clusters illustrate that trending by cluster group facilitates the identification of climatological trends that differ or diverge when comparing across clusters-a new dimension to climatological trend analysis using track data is therefore created. The trend in location of cyclone maximum intensity (LMI) or lowest central pressure (LCP) during its evolution is a critical characteristic in order to understand TC behaviour and the relationship to climate change. Recent studies have shown a poleward shift in the LMI or LCP location for many regions across the globe (e.g. Kossin et al. 2014;Sharmila and Walsh 2018). However, another study has shown the opposite movement for the East Coast of Australia (Chand et al. 2019). Trending of LMI per cluster will give a new understanding of this risk critical characteristic at a regional level.
The aims of this study are (1) to investigate the frequency trends, directional and curvature (sinuosity) properties of TCs with genesis in the Coral Sea, within well-defined groups, and (2) to trend TC power and location of maximum intensity within the defined groups.

Study area
The selected study area is the Eastern Basin of Australia (Coral Sea), as shown in Fig. 1. Much of this coastline is north of the Tropic of Capricorn and so is mostly situated in the tropics and experiences the Australian Summer Monsoon, the strongest monsoon in the Southern Hemisphere (Gallego et al. 2017). This monsoon is associated with high winds and rainfall and storms, including TCs (Dare and Davidson 2004). A storm track was included in the study as a TC track if the storm reached TC category 1 maximum mean wind speed at any point in its lifetime.
There are several environmental conditions that need to co-exist to facilitate TC genesis and propagation: Sea surface temperature (SST) above 26.5 °C (Dowdy et al. 2012), moderately small wind shear between the lower and upper troposphere, low-level convergence and a conditionally unstable atmosphere (Dare and Davidson 2004). Large energy differences between the low-energy (modal) and extreme events leaves many areas along this coast vulnerable to damaging erosion from TCs (Mortlock et al. 2018). The region also has high biodiversity that is particularly vulnerable to damage by TCs (Anderson-Berry 2003) and the western boundary (Australia) of this region of the South Pacific has a high population density and is therefore a suitable area in the South Pacific to focus on.
This region is a subset of several previous studies on TC tracks in the South Pacific (e.g. Magee et al. 2017Magee et al. , 2020Bell et al. 2020;Chand et al. 2019;Sharma et al. 2020;Song et al. 2018;Tauvale and Tsuboki 2019;Zhao et al. 2018) and so will provide additional detailed information to this existing literature. A probable future scenario in this region is rising TC power dissipation and strength, with more frequent severe TCs (Parker et al. 2018) and therefore further insight into TC behaviour here will be invaluable.
A dominant influence on the climate here, and on the entire southwest Pacific (SWP), is ENSO, which has variability on an inter-annual scale (Magee et al. 2017). ENSO interacts with the Walker Circulation and has three phases: El Nino, La Nina and Neutral (Callaghan and Power 2010). On a decadal scale, the Interdecadal Pacific Oscillation (IPO) has a frequency of influence in the range of 15-25 years. The absence of reliable wind speed and central pressure records in TC databases before 1970 (prior to satellite technology) makes analysis of TC behaviour in relationship to this driver unfeasible, except for TC climatological characteristics that are purely geomorphic (Magee et al. 2017). Also, of significant influence on the climate in this region are the South-east Trade Winds and the South Pacific Convergence Zone (McGregor et al. 2012).

Method
To achieve the aims of this paper, the methods were divided into five parts: (1) we present the TC track dataset used; (2) derive the parameters to be clustered through K-means; (3) present the clustering methodology; (4) TC track sinuosity is introduced as a track characteristic to be investigated and (5) the TCs were trended by cluster. The sections below describe each step, in detail.

Available data
Several cyclone track databases are available for the SWP-some examples for region include the Southwest Pacific Enhanced Archive of Tropical Cyclones (SPEArTC), the International Best Track Archive for Climate Stewardship (IBTrACS) (Knapp et al. 2010(Knapp et al. , 2018, the Joint Typhoon Warning Center (JTWC) best-track database and the Australian Bureau of Meteorology (BOM) tropical cyclone track dataset. The SPEArTC (Diamond et al. 2012) database can be considered to be the most complete TC track database for the SWP (Magee et al. 2016). However, the Australian BOM database, with complete cyclone maximum wind speed data available from 1970, was selected as the preferred data source for the following reasons: (1) Only a subset of the SPEArTC database spatial data, the area adjacent to Australia, is being considered in this study and for the Australian basins, the BOM database is a source of TC data for the SPEArTC database (Magee et al. 2016); (2) Holland (1981) has discredited cyclone count statistics using SWP data before 1960 and use of cyclone intensity data before 1970, with a few exceptions-thus caution is advised in using pre-satellite (1970) era data in studies involving cyclone power (Magee et al. 2016) and so the advantage of the extra-temporal scope of the SPEArTC is lost for the purposes of this exercise.
An important consideration for the study dataset was the selected timespan and the inverse relationship of this on data quality. The publication of the Dvorak technique in 1972 has enabled significant improvements in cyclone intensity estimates since then (Chand et al. 2019). Although routine infra-red polar orbiting satellites have been in use in Australia for cyclone measurements since 1972, routine geostationary meteorological satellites have only covered this region since 1978 (Harper et al. 2008). A reduction of intensity measurement instrumental bias of around 10% was achieved in the 1970s (Harper et al. 2008). For Australia, use of intensity data from 1978 onwards is recommended by Harper et al. (2008). However, this study will group (cluster) tracks from 1972 onwards (the start year of routine infra-red polar satellites and the Dvorak technique), since: • The intensity (wind speed) is used as a track specific weighting factor for track clustering and therefore across-track wind speed biases are not a concern.
1 3 • The particular intensity estimation technique deployed in the 1970's is assumed to be consistent within each track, as the region under consideration falls under only one BOM Tropical Cyclone Warning Centre (Brisbane). • Approximately fifty years of TC track data from 1972 onwards is a sufficient time range to investigate decadal trends for TC characteristics unrelated to intensity, e.g. TC frequency.
In summary, complete BOM cyclone tracks with genesis in the Coral Sea, from 1972 onwards, were selected for inclusion into the study for clustering. For two of the resulting analyses-overall and within-cluster cyclone power and curvature (sinuosity), tracks from 1978 onwards were considered as within-cluster subsets, to reduce instrument bias from the results of these two analyses.

TC track weighted centroids and variances
The power dissipated by a TC is a function of its surface wind speed over the recorded timespan (Emanuel 2005). The maximum sustained wind speed (with ten-minute means) was therefore selected as a weighting factor for calculating track centroids and variance, as it is analogous to the inertial mass of the cyclone at its position in space, as defined by its geographical co-ordinates. Weighted centroids calculated using this maximum wind speed weighting factor could then be described as first mass moments of the TC track (Nakamura et al. 2009).
A previous study by Jin-hua et al. (2016) applied the square root of the maximum sustained wind as the weighting factor for their cluster analysis, to decrease the weighting factor assigned to each position along the track.
In order to trend TC power, the power dissipation index (PDI) was chosen as an indicator over the entire life of the cyclone.
Since PDI is defined as where V max is the maximum sustained wind speed and τ is the lifetime of the track (Emanuel 2005); it is appropriate to use V max (not √ V max ) as the weighting factor in this study (following Emanuel 2005). This aligns the weighting factor at each recorded track position with the measure of power to be trended, post-grouping.
PDI was integrated using days as the timescale in order to reduce the scale of expected PDI values and to align with BOM TC track record capture, which has on the order of 2-10 captured records per day. Track PDI calculations were constrained at genesis and decay-both time t = 0 and t = τ were defined as the closest track record to when the ten-minute mean wind speed first exceeds/last reduces to 17.5 m/s (the minimum tropical cyclone 10-min mean maximum wind speed as per Dare and Davidson 2004). In addition, to control data quality as previously discussed, only tracks from 1978 onwards were considered for PDI analysis.
The latitude and longitude co-ordinates of the weighted centroids of each TC track were calculated, using standard weighted centroid Eqs. (2) and (3): where x i is the latitude, and y i is the longitude of the ith track position. N is the total number of track position records and w(i) is the weight at the ith track position (as defined in the first paragraph of Sect. 3.2). An inclusion of track co-ordinate directional variance introduces shape characteristics over the track lifetime into the cluster properties in addition to the already included location of "mass" centre. The first directional variance is in the x (latitudinal or zonal) direction, the second is in the y (longitudinal or meridional) direction and the third is a variance in the xy (diagonal) direction. The directional variance equations used are detailed in the appendix.

Cluster analysis
Since the K-means clustering of TC mass moments (e.g. Nakamura et al. 2009) accommodates track geometry and is also expandable to include additional meta track shape indexes (such as sinuosity), this method was selected to use for this study.
TC tracks in the Eastern Basin of Australia (Coral Sea) were grouped by power (maximum wind speed) weighted centroids. The centroid for each track is a two-dimensional vector of latitude and longitude. The meta characteristics of each group or cluster of TCs were then determined and interpreted-curvature (sinuosity) and area are examples of such meta characteristics. Unlike previous mass moment studies (e.g. Nakamura et al. 2009), K-means clustering was done firstly on power weighted centroids alone (first moment) and then track variance and curvature (sinuosity) was examined for each cluster. This method produced improved definition and spatial separation between the clusters.
The application of the K-means method to the clustering of cyclones has been used several times previously in the northern hemisphere, since it allows grouping by cyclone track spatial characteristics (Elsner 2003;Jin-hua et al. 2016). Some of these studies have clustered TC tracks based on chosen points in a track evolution, for instance positions of final and maximum intensities (Elsner 2003). This, however, does not use track length as a grouping parameter and so grouping by weighted first moments is preferred.
The grouping method used was K-means clustering of cyclone track centroids, weighted by the maximum wind speed at each measured co-ordinate along the track. It is analogous to the first mass moment of the cyclone, with maximum wind speed being a measure of the mass of the cyclone at a particular point along its track.
The separate application of the first moment (track weighted centroids) as the clustering variable, without the second moment (track weighted variances) as a clustering variable, is an approach that was not previously undertaken (Nakamura et al. 2009) and is based on the rationale that the two moments have distinct and different applications-the first is a measure of (weighted) centroid position in space and the second of track area, track axis orientation and shape.
When using the K-means clustering approach, initial partitions are established and then cluster centres (not to be confused with TC track centroids, of which latitude and longitude form two of the variable set subject to clustering) are progressively recalculated and updated after each change to the cluster (MacQueen 1966). The algorithm selected for K-means clustering in this study is the one developed by Hartigan and Wong (Hartigan and Wong 1979): For n TC track datasets with p variables per track x(i, j) for i = 1,2,…,n; j = 1,2, …, p; K-means assigns each track to one of K groups or clusters to minimise the within-cluster sum of squares.
where x(k, j) is the mean variable j of all elements in group K.
A starting K x p matrix with starting cluster centroids for the K clusters is calculated. The tracks are then assigned to the cluster with the closest cluster centroid. The iterative calculation then attempts to find the best partition with the least within-cluster sum of squares by moving tracks from one cluster to the next (Hartigan and Wong 1979).
A suitable range from which to select the optimal number (n) of clusters was estimated using the Elbow Method (Kaufman et al. 1990). From this range, the final selection of n was performed using the Silhouette Method (Kaufman et al. 1990).
The study will produce a set of clusters and associated results for each. In addition to trending of track characteristics such as frequency, power and location of maximum intensity, the clustering will enable analysis on a group (cluster) level of track geometric properties such as sinuosity.

TC track sinuosity
Although the three variances in TC track positions describe the area covered, orientation and shape of the area, these do not give a direct measure of overall curvature of the tracks. Therefore, the definition of sinuosity, S, as used by Terry and Gienko (2011) has been used to compare track curvature across the clusters: The distances are at the resolution of the BOM TC track readings and are the summed segment lengths between the points as recorded in the readings.
The displacements are the shortest distances between the first and last readings of a track. Sinuosity for unconstrained tracks was calculated and then, as with PDI, sinuosity calculations were constrained at genesis and decay-first and last records were defined as the closest track record to when the ten-minute mean wind speed first exceeds / last reduces to 17.5 m/s. Also, as with PDI, only tracks from 1978 onwards were considered for this second calculation.
The sinuosity categories for the entire southwest Pacific (SWP) that were developed by Sharma et al. (2021) are used here to describe the degree of track curvature. These are based on applying a normal distribution to the sinuosity index value, SI.
The SI range is then divided into quartiles (Table 1).

TC trends by cluster
TC frequency trends over the last fifty years, grouped by cluster, were investigated on a yearly and decadal scale. The p test was used to determine statistical significance of any detected trends in TC frequency over the timescales, using a Poisson regression model. Sinuosity trends were also investigated, using linear regression. Cyclone intensity totalled over its lifespan is a measure of overall cyclone destructiveness and risk. Therefore, the PDI, as defined in Eq. 1, has been selected as a climatological characteristic to be analysed. The cross-cluster trends can then be compared to other non-clustered trend analyses that have been produced in previous studies on Australian TC characteristics over the last 50 years. A cyclone has the potential to cause the most destruction at peak intensity. The position of peak intensity corresponds to the location at lowest central pressure. The temporal trend in the lowest central pressure (LCP) is examined by cluster group.
This analysis of TC tracks grouped by centres and then co-ordinate variances will give insight into changes in the frequency of cyclones, curvature of cyclones and temporal trends in power and location of maximum intensity of the TCs. These observations will be on a group level in the Eastern Basin of Australia and will contribute to the understanding of the historical behaviour of cyclones in this region.

Centroid (first moment) clustering
For TC track centroid clustering, the elbow method suggests a suitable choice of n is in the range of three to five (refer "Appendix"). Within this range, n = 3 has a high silhouette width (refer "Appendix") and was selected for this study. Figure 2 shows that the three clusters have clear separation without overlap, whilst the variance ellipses (a graphical depiction of the data in Table 2) highlight the different and divergent shape characteristics of each cluster. Clusters 2 and 3 lie along the same latitude approximately and cluster 1 has its centroid located about 3 degrees further south. The long axis of each ellipse gives an indication of the predominant direction of cyclone travel: Cluster 2 cyclones often track along easterlies, Cluster 1 cyclones move (on average) in a South-East direction, and for Cluster 3, a typical direction is West-South-West. TCs in both clusters 2 and 3 move more laterally than those in cluster 1, with the North to South variance in cluster 1 almost twice that of the other clusters. 1 3

TC frequency
A time series of count per year for TCs that have genesis in the Coral Sea is shown in Fig. 3 below. There is a statistically significant downward trend (with fitted Poisson distribution) in TC frequency over the last fifty years for the complete set (not clustered) as well as for Cluster 1. Analysis of TC frequency temporal trends by cluster shows that only cluster 1 TCs show a linear downward trend-cluster 1 alone is thus responsible for the overall downward trend in cyclone frequency for the East Coast of Australia. Cyclones in clusters 2 and 3 do not show any evidence of decreasing frequency in the last fifty years.   Table 3 shows the sinuosity results per cluster for unconstrained tracks-nearly 80% of cyclones with genesis in the Coral Sea are curving or sinuous. Cluster 1 has the highest proportion of straight or quasi-straight-moving cyclones (28.5%). Clusters 2 and 3 have extremely high proportions of curving or sinuous tracks, 85% and above. Note that the Sinuosity value range does not follow a normal distribution, since it was characterised using the index developed by Sharma et al. (2021) for the entire SWP. A comparison of these results with the genesis and decay intensity (maximum wind speed) constrained results in Table 4 shows an approximate 9% decrease (increase) in curving or sinuous (straight or quasi-straight) for all the constrained tracks and a similar decrease (increase) in curving or sinuous (straight or quasi-straight) TC tracks in the clusters, compared to unconstrained track clusters. It can be inferred from this that fully formed TCs move in a less convoluted fashion than entire tropical storm tracks including TC pre-genesis and post-decay records. A further analysis of the sinuosity results in Table 3 indicated a small, statistically significant temporal increase in sinuosity for all tracks (Fig. 5). A temporal increase in sinuosity for cluster 2 (Fig. 4) was evident, although the statistical level of significance was below 90% (p value: 0.1523). The other clusters did not show any visible trends or statistically significant trends in sinuosity. The intensity constrained tracks did not show any trends in sinuosity at the 95% or 90% significance level.

TC track PDI
Comparing the wind speeds along the tracks as shown in Fig. 2 with the summary results of the PDI boxplots for the clusters as shown in Fig. 5 confirms that cluster 3, the cluster with the most westerly position, has the highest mean of power dissipated per track. The two most extreme PDI values for cluster 3 correspond to TC tracks that have moved so far west as to be situated over the Timor Sea and then even the Indian Ocean. As can be seen in Fig. 2, these have achieved maximum wind speeds of over 40 m/s for significant periods of time. When considering PDI, Cluster 3 also has a large percentage of outliers (24%), compared with the other two clusters which have less than 8% of their TCs as outliers. Outlier PDI values would be due to extreme track lengths or extreme maximum winds or both. In contrast to their differing directional (variance ellipse orientation), shape (variance ellipse) and curvature (sinuosity) properties, cluster Sinuosity trends for unconstrained tracks-firstly for all TCs with genesis in Coral Sea, with linear increase at 90% confidence (p value: 0.0991) and then for cluster 2 with fitted second-order polynomial regression line (no statistical significance above 90% for cluster 1, p value: 0.1523). Clusters 1 and 3 did not yield visible trends in sinuosity 2 with most of its centroids over the Coral Sea and cluster 1, with all centroids over the Coral Sea, have very similar mean PDI values and 25th, 75th percentiles. They therefore cannot be easily differentiated in terms of their power distributions (without considering temporal trends). Six monthly means of cyclone power dissipation index (PDI) together with the Southern Oscillation Index (SOI) are shown in Fig. 6 for all TCs with genesis in the Coral Sea. For both PDI and SOI, there is an apparent 6-8 year frequency between maxima and minima.
An increase in peak PDI over the 50-year timespan can be seen from Fig. 6, although there is no statistically significant linear trend in PDI over this timespan. The PDI time series has been re-represented with a five-year rolling mean per cluster in Fig. 7 in order to analyse low-frequency variations in the time series.
The clusters show strongly opposite and divergent trends from 2004 onwards-the cluster 2 TCs are increasing in power whilst cluster 1 TC power is decreasing. No statistically significant trend is observed for either cluster over the 50-year timespan.

TC track LMI
Critical to cyclone risk is the point at which they reach peak intensity. This point corresponds to the point of lowest central pressure and the trend for all Coral Sea TCs is shown in Fig. 8. There is a demonstrated linear trend (at 99% confidence) towards the equator. Although cluster 2 cyclones have demonstrated a corresponding shift of LCP latitude towards the equator for the period 1970-2008, a dramatic reversal shows from the period 2008-2020 (refer Fig. 9). There is no overall linear trend over the entire timespan Latitude of lowest central pressure reached by any TC per year, for cluster 1 and then cluster 2 cyclones. Blue lines are fitted by smoothed second-order polynomial regressions. Cluster 1 dotted line shows a linear trend at 90% confidence (p value: 0.09896). Cluster 3 is not trendable due to low record count 1970-2020 for this cluster. Cluster 1 (Fig. 9) does have an overall linear trend (at 99% confidence) towards the equator for its latitude of LCP, which suggests that this cluster is responsible for the overall liner trend of LCP towards the equator for the entire set of TCs.

Discussion
The clustering method used in this paper uses maximum wind speed as a weighting factor to calculate track centroids; therefore, it can only be deployed on datasets with consistent records for this TC characteristic (or an another suitable analogous one like central pressure)-this limits its applicability to track database records post-1970. It is a limitation of this method which will become less significant with passing time.
The variance ellipses of the clusters (Fig. 2) give an indication of their general movement characteristics-the long lateral axis of cluster 3 indicates long straight-moving cyclones. This cluster has a higher median PDI and significantly higher outlier PDI values, compared to the other two clusters which is intuitive, considering the longer mean track length indicated by the length of the ellipse long axis. The long tracks that often occur seem to be because these westerly moving cyclones gain energy as they once again situate over water (The Gulf of Carpentaria this time). An alternative explanation for the long tracks evident in cluster 3 is overland re-intensification-TCs are typically sustained by heat flux from the ocean (Kleinschmidt 1951) and decay upon landfall (Kaplan and DeMaria 1995); however, occasionally those moving over Northern Australia have been shown to reintensify whilst retaining their warm-core structure (Emanuel et al. 2008). Emanuel et al. (2008) suggest that this is a unique phenomenon to Northern Australia and may occur over sand whereby thermal conductivity is increased due to the increase in moisture content from rain associated with the storms. TC Abigail, 2001, is an example of such a storm (Emanuel et al. 2008) and falls within cluster 3 of this study.
The orientation of their variance ellipses shows that the TCs in clusters 2 and 3 track in a predominantly westerly direction and the cluster centres both have a closer location to land compared to cluster 1. The westward tracking of these TCs is consistent with the findings of Sharma et al. (2021) and Dare and Davidson (2004), both of which reported westward tracking for TCs with genesis west of 160 °E. It can be inferred from this, and the orientation of their variance ellipses, that the risk of landfall and associated destruction is much higher for tracks in clusters 2 and 3.
This risk to the east coast of Australia is amplified by the fact that clusters 2 and 3 have a high proportion of curving or sinuous tracks. This is due to an increased probability of TCs making landfall or being situated nearshore, the tendency to loop back or traverse large areas (spatial risk driver) and the increase in track life (temporal risk driver) for curvy or sinuous tracks results in an increase in TC risk probability. A similar finding was shown by Sharma et al. (2021) who assert that the risk of landfall in Northern Australia is raised by complex TC tracks in the Coral Sea.
In addition, when considering trends, cluster 2 shows an increase in sinuosity (for unconstrained tracks), a strong increase in PDI from 2004 and a southwards movement of LMI from 2004. This shift in LMI has also been demonstrated for the South Pacific overall (Kossin et al. 2014;Moon et al. 2015;Sharmila and Walsh 2018;Pillay and Fitchett 2021b). In addition, the South Pacific has the largest poleward migration of LMI across the Southern Hemisphere basins (Daloz and Camargo 2018;Kossin et al. 2014). When considering these findings and those in the previous paragraphs together-a clear separation 1 3 of risk between clusters 2 and 3 and cluster 1 is present. It should also be noted that the marginal overall increase in track sinuosity for the entire Coral Sea (as shown in Fig. 4 and reported by Sharma et al. (2021)) indicates a slight increase in cyclone risk for this entire region.
As shown in the results (Table 3) and as reported by Terry and Gienko (2011), the Coral Sea has a much higher proportion of curving or sinuous TC tracks, so the distribution is highly positively skewed, i.e. the increasing number of tracks with a curving or sinuous property moving east to west and the corresponding decrease in straight or quasi-straight tracks (as shown by percentage in Table 3), is consistent with findings by Terry and Gienko (2011) on SWP TC sinuosity. Curving or sinuous tracks have more complex trajectoriesthey can recurve, trace back on themselves or loop (Sharma et al. 2021).
It is interesting to compare the findings of Terry and Gienko (2011), who identified a possible intermittent increase in sinuous TCs in the SWP, with the findings in this study, which show that the cluster with the highest proportion of straight or quasi-straight cyclones (cluster 1) demonstrates a decreasing frequency trend which dominates the overall decreasing trend of all clusters combined, whilst the other more curving or sinuous clusters (clusters 2 and 3) demonstrate no frequency trend at all. As previously discussed, the orientation of cluster 2 and 3 variance ellipses shows they are more likely to include landfalling TCs. At a cluster level, for the east coast of Australia at the latitudes of clusters 2 and 3, the results of this analysis are inconsistent with the findings of Callaghan and Power (2010) and Wang and Toumi (2021) of less landfalling cyclones in the Coral Sea. Murakami et al. (2020) did show variance in the TC frequency trend, depending on location in the Coral Sea for the period from 1980 to 2018, which is consistent with the findings of this study. In addition, a low increase in sinuosity at the 90% confidence level (p value: 0.0991) was found for the entire set of Coral Sea TC tracks (Fig. 5). Terry and Gienko (2011) found a similar low-grade increase in sinuosity for the entire SWP region. However, for max wind velocity constrained tracks (> 17.5 m/s), no statistically significant trend in sinuosity was detected (and therefore no reduction in the fully formed TC risk due to convolution for this region).
PDI is a primary metric for evaluating TC threat (Emanuel 2005). On the east coast of Australia, the overall decreasing trend in PDI as shown in Fig. 6 corresponds to the trend in a closely related climatological characteristic, accumulated cyclone energy, as presented by Chand et al. (2019). Conversely, Kossin et al. (2007) and Kuleshov et al. (2010) showed no conclusive trend for this characteristic in the SWP based on best-track records. In contrast to the decreasing trend found by Chand et al. (2019), Knutson et al. (2020) project, based on 18 climate models, an increase in TC intensity in the SWP for 2 °C of anthropogenic warming.
Decomposing the decreasing trend shown here by cluster is insightful as it indicates that the more south-east located tracks of cluster 1 are the primary drivers of this reducing PDI post-2004. In contrast, PDI is increasing post-2004 for cluster 2. SST is a dominant thermodynamic factor in the development of TC power, since a temperature of at least 26.5 °C is required for cyclone genesis (Emanuel 2003) and temperatures above this threshold also support cyclone (re)intensification, although they can travel long distances at lower temperatures before final decay (Dowdy et al. 2012). It is noted, however, that Pillay and Fitchett (2021a) conclude that the minimum SST at which Southern Hemisphere tropical cyclones form is around 24 °C, which is significantly below the usually stated threshold of 26.5 °C, whilst the average SST at genesis in the Southern Hemisphere is 28.1 °C, with a standard deviation of 0.7 °C. Variance in TC intensity by location has major implications for the surrounding environment-as an example, the climate driven changes in TC power have been shown to affect dune systems on Fraser Island (Levin 2011) and these would, using track direction and co-ordinate variance as criteria (for example), be affected by cluster 1, 2 and 3, in order of risk.
The yearly variation in TC frequency in the SWP and also the Coral Sea specifically is associated with the El Niño-Southern Oscillation (ENSO) (Ramsay et al. 2014;Vincent et al. 2011). Even though, on a larger timescale, a clear understanding of the underlying drivers in TC trends in this area has not yet been secured (Chand et al. 2019), a few previous studies have included the modulating influence of the Indian Ocean Dipole (IOD) to the influence of ENSO events in this region (Ham et al. 2016). As an extension of this idea, it appears that those cyclones further westward (cluster 2) show PDI trends related to SST in the Northern Tropics, closer to the Indian Ocean (for instance SST in the Gulf of Carpentaria). In the tropics, localised SST trends could affect TC power more than entire Pacific Ocean scale SST trends since TC intensity is a function of the difference between SST and the average tropospheric temperature (Emanuel 2005). Figure 10 shows a corresponding steep in increase in SST for the northern tropics (including the Gulf of Carpentaria) for the period 2004-2014. This pattern is not evident in the Coral Sea for the same time period. The PDI trend, 2004-2014, for cluster 2 (as shown in Fig. 7) mirrors this. In contrast, it has been commonly acknowledged that the entire Pacific has moved to a more El Nino like state in the last few decades (Dowdy et al. 2012) and, for cluster 1, the trend in PDI post-2004 aligns with this shift. This is suggestive, on a decadal scale, of divergent climatological influences, depending on transport locations, affecting TC intensity through the Coral Sea.
To date, limited long-term TC wind speed and/or power data has been available (from around 1960 only); the lack of available PDI analysis for the long distance westerly moving tracks in cluster 3, due to minimal wind speed data, is an example of this.  . Line graphs show five-year running averages. Courtesy Australian BOM 1 3 However as increasing data becomes available, a clearer understanding of the underlying mechanisms of TC intensity changes should emerge.
For the Coral Sea, Dowdy et al. (2012) demonstrated a clear boundary for TC intensification at around 20° S-once TCs track further south they weaken and decay. Using this boundary meridian, for the eastern basin of Australia, Levin (2011) showed an increase in high intensity cyclones above 20° S (comparing cyclones in years 1957-1981 to those in years 1982-2006), with a decrease in high intensity cyclones below 20° S for the same time period comparison which correlates strongly with the PDI results by cluster in this study.
The linear trend of LMI towards the equator corresponds to recent research by Chand et al. (2019) for this region and indeed for Australia as a whole. This is not, however, a typically reported direction of movement-numerous global and South Pacific studies (e.g. Kossin et al. 2014;Moon et al. 2015;Sharmila and Walsh 2018) report a poleward shift in LMI. A few suggested causes for this poleward shift include vertical wind shear structural and intensity change from climate change induced tropical zone expansion (Kossin et al. 2014) and changes in the Walker Hadley circulation (Sharmila and Walsh 2018). However, for the Coral Sea, once TCs track further south than the previously described intensification boundary at 20° S, they weaken and decay. In addition, Dowdy et al. (2012) found some evidence of this boundary shifting equator-wards during El Nino years. As with PDI, the clusters show divergent trends for LMI post-2004.
It appears that cluster 1 shows the equator-ward shift proposed by Dowdy et al. (2012) for the El Nino like state the entire Pacific is moving towards whilst cluster 2 LMI trend post-2004 is poleward, perhaps being more influenced by steep SST rises as demonstrated in Fig. 10 for the Northern Tropics. The reason this Australian region demonstrates the equator-ward LMI trend is yet unknown and further research is required. Figure 8 hints at a possible inflection/ turning point around 2010 with a possible reversal in trend-as with PDI, the patterns identified might become more apparent as the historical dataset increases with time.
The LMI trends have other implications-for instance for a track, the degree of sinuosity or (re-)curvature often increases soon after the LMI is reached (Dare and Davidson 2004), due to trough interactions. A future overall equator-ward movement of LMI would then be expected to move the risk of TCs due to sinuosity in a northerly direction.
Along with ENSO and the IOD, the Interdecadal Pacific Oscillation (IPO), an interdecadal oscillation in Pacific Ocean SST, as described by Power et al. (1999) might also play a role in the trends observed in this paper. Noting that the IPO moved into a negative phase post-2004 approximately-this IPO phase and a simultaneous positive SOI have coincided with peak TC activity (frequency) in the southern region of the Coral Sea (Levin 2011). The increase in TC frequency during the co-incidence of La Nina events (positive SOI) and negative phases of IPO has also been found in the east Australian basin as a whole (Salinger 2005;Flay and Nott 2007). In contrast to this, PDI is shown in this study to have a negative correlation with SOI (refer Fig. 6) with a Pearson correlation co-efficient of approximately − 0.4. It is hypothesised here that Nino 4 variability (the Nino SST anomaly closest to the Coral Sea) is the explanation for this, since cyclone power is strongly controlled by the localised SST in the tropics (Emanuel 2005). Further study is planned to confirm this hypothesis. Both this and the links between the weakening of the Walker Circulation (the shift to an El Nino dominant climate) as described by Callaghan and Power (2010) and anthropogenic climate change are areas in need of further research.

Conclusions
Grouping or clustering TCs originating in the Coral Sea by maximum wind speed weighted track centroids was shown here to be an effective and productive classifying method, indirectly incorporating track total length (by modelling the track as an open curve) into the grouping algorithm. Clear geographical separation was visible between the clusters. This augmentation of existing TC track forecasting methods contributes towards TC threat management in this area and provides a method to uncover TC climatological behaviours and trends on a sub-regional level, using geo-locational grouping.
Each of the three clusters identified showed a predominant direction of TC movementthe directions being south-easterly, westerly, and west-south-westerly for clusters 1, 2 and 3, respectively, with Clusters 2 and 3 showing a high degree of lateral movement. TC tracks in this region increase in sinuosity moving east to west, which is a phenomenon that has been previously reported in the SWP. Clusters 2 and 3 track in a westerly direction, have their ellipse centres much closer to land and cluster 3, the cluster of most westerly position, has the highest median of power dissipated per track. Although there is a statistically significant downward trend overall for frequency of TCs in the Coral Sea, this trend was only evident in one of the three clusters: Cluster 1, when considering the paragraph above, can therefore be considered to carry the least risk of the three. It appears that future change in cyclone risk is a function of location, with change increasing in a northerly direction, on this coast. This standpoint is reinforced by the results that show that both the number of TC tracks characterised as sinuous and PDI have been increasing for cluster 2 from 2004 onwards. In addition, it appears that the general trend of movement towards the equator of LMI for the region (which is counter to the general poleward movement found generally in global studies) has, in the case of cluster 2, shown a dramatic reversal around 2004.
This work has shown that since around the start of the new millennium, TCs that track in the area defined by cluster 2 are showing a clear increase in hazard, when compared to the other clusters (when considering power, sinuosity, location of LCP and frequency). This work also broadens insight into the conclusion from previous studies that track sinuosity in the SWP has been increasing in the last half-century by identifying this trend as marginal for unconstrained tracks and not statistically significant for maximum wind speed constrained tracks in a specific sub-region (the Coral Sea). It confirms that the impact of ENSO on the selected TC climatological trends, within the fifty-year time frame, is not clearly apparent-therefore there remains a substantial opportunity to determine if there are any further demonstratable relationships between climate variability cycles such as ENSO, IPO and IOD and the trends in the TC characteristics as shown in this paper.

Cluster number optimisation
A suitable range from which to select the optimal number (n) of clusters was estimated using the Elbow Method (Kaufman et al. 1990): where k = a nominated range of test values for n; C k = the kth clustering variables dataset; W = the total within cluster C k sum of squares (or total within cluster C k variation).
A bend (elbow) in the plot of the total within-cluster sum of squares versus number of clustersis used to determine the suitable range for n.
From this range, the final selection of n was done using the Silhouette Method (Kaufman et al. 1990): where a i = the average distance from the ith data point to the other points in the same cluster; b i = the average distance from the ith data point to the other points in the other clusters. The highest mean S i value obtained for a range of cluster numbers k shows the best corresponding choice of clusters number n.
The elbow method (refer Fig. 11 below) suggests a suitable choice of n is in the range of three to five.
Within this range, n = 3 has a high silhouette width (refer Fig. 12 below) and was selected for the study. The choice of n = 2 as suggested by the absolute highest silhouette width in Fig. 12 was rejected because the total within-clusters sum of squares value is significantly higher, compared with n = 3 (and outside of the "elbow" range).