Location, Area, Curvature and Temporal Trends of Coral Sea Tropical Cyclone Tracks, Over 50 Years

Tropical Cyclones (TCs) with genesis in the Coral Sea, often near the east coast of Australia, present significant hazards to coastal regions in their surroundings. There has, therefore, been significant recent efforts to extract information from records of their historical tracks in order to help predict their future behaviour in the light of a changing climate. In this study, the Australian Bureau of Meteorology (BOM) database of TC tracks over the last fifty years were grouped based on K-means clustering of the maximum wind-weighted centroids. Track shape variance and track curvature (sinuosity) were assessed. Three well defined clusters of TC tracks were identified, and the results showed 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 concept of TC power dissipation index (PDI) was introduced, revealing that two of the clusters have diverging trends for PDI post 2004. The location of cyclone maximum intensity (LMI) was trended, and only one cluster showed a statistically significant trend (towards the equator) for LMI. All these findings demonstrated a clear variance in risk between the clusters and suggests that this method of cluster analysis is a useful and productive complementary tool when establishing future impacts of TCs the method identifies divergent TC characteristics and trends at a finer scale (cluster) level which then aids in assigning specific and differing TC risk mitigation strategies to different areas of the Australian east coast.


Introduction
Tropical cyclones (TCs) can be extremely hazardous to areas in their vicinity and, in addition have the capacity to cause devastation from a distance through the waves they generate. The consensus from recent global studies have predicted more intense TC activity due to anthropogenic climate change which may exacerbate inundation in association with the sea level rise (Knutson et al. 2020;Walsh et al. 2019).
TCs near the east coast of Australia have historically caused significant damage to natural and anthropogenic infrastructure. Of the top five most expensive natural disasters in Australia from 1966 to 2017, three were due to Tropical Cyclones (TC's) (McAneney et al. 2019). Of these three TCs, two tracked in the eastern basin of Australia (reference). TCs can therefore be said to pose significant risk to the east coast of Australia.
The future change in the risk profile of these cyclones is uncertain, though there has been an increase in the proportion of severe cyclones in the last 30 years 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 and research into future TC trends is becoming an urgent requirement for risk mitigation.
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. The studies of this type that have been done 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, Camargo, & Kim 2012;Sharma et al. 2021); B) K-means clustering of a selected n-dimensional vector compiled from chosen TC characteristics (e.g., Yan, Xu, Ma, & Ma 2018); C) K-means clustering of TC mass moments (e.g., Nakamura, Lall, Kushnir, & Camargo 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 & 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 windspeed 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 windspeed or minimum central pressure) and track geometry is another powerful advantage of this method (C), since it can then be used to investigate 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 methodology.
As cyclone intensity and power is of primary concern when identifying cyclone trends over the lasty fifty years (the years most affected by anthropogenic climate change forcing), cyclone power dissipation index (PDI) is defined and then trended by group.
Trends of TC power dissipation index (PDI) using track data arranged in clusters illustrates 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 to understand in relationship to climate change. Recent studies have shown a poleward shift in this location for many regions across the globe (e.g., Kossin, Emanuel, & Vecchi 2014;Sharmila & 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 to:1) To investigate the frequency trends, directional and curvature (sinuosity) properties of TCs with genesis in the Coral Sea, within welldefined groups. This includes relating the results to ENSO over the same time period (fifty years). 2) To trend TC power and location of maximum intensity within the defined groups.  (Gallego, García-Herrera, Peña-Ortiz, & Ribera 2017). This monsoon is associated with high winds and rainfall and storms, including TCs (Dare & Davidson 2004).

Study area
In addition to the tropical location, there are several other 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 & Davidson 2004). The TC season on the east coast of Australia is from late spring to mid-autumn (November to April) when sea surface temperatures are high enough for cyclogenesis. The ocean beaches of the region have a tide range of 2 to 4 m, with a predominantly low-energy hydrodynamic regime and experience infrequent and intermittent high-energy TC events. Large energy differences between the low-energy (modal) and extreme events leaves many beaches along this coast vulnerable to damaging erosion from TCs (Mortlock et al. 2018). The effects of this erosion endure and have impacts on coastal eco-systems, infrastructure and tourism. 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. 2017;Bell et al. 2020;Chand et al. 2019;Magee et al. 2020;Sharma et al. 2020;Song et al. 2018;Tauvale & 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, Bruyère, Mooney, & Lynch 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

Method
To achieve the aims of this paper the methods were divided into five parts: first we present the TC track dataset used, then derive the parameters to be clustered through K-means, then present the clustering methodology, following which TC track sinuosity is introduced as a track characteristic to be investigated. Finally, the methodology for producing TC trends by cluster is described. However, the Australian BOM database, with complete cyclone maximum windspeed 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) 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.
Fifty years of TC track data from 1970 onwards was used in this study which is a sufficient time range to investigate decadal trends and at the same time has a sufficient percentage of consistent maximum windspeed and minimum pressure records which are needed for this methodology. Complete cyclone tracks with genesis in the Coral Sea were selected for inclusion into the study.

TC track weighted centroids and variances
The power dissipated by a TC is a function of its surface windspeed over the recorded timespan (Emanuel 2005). The maximum sustained windspeed (with tenminute 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 windspeed 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 Vmax is the maximum sustained windspeed and  is the lifetime of the track (Emanuel 2005), it is appropriate to use Vmax (not √ ) 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 reduce the scale of expected PDI values and to align with BOM TC track record capture, which has in the order of 2 -10 captured records per day.
The latitude and longitude co-ordinates of the weighted centroids of each TC track were calculated, using standard weighted centroid equations (2) and (3): Where xi is the latitude, and yi 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 section 2.2).
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.

Cluster Analysis
Since the K-means clustering of TC mass moments (e.g., Nakamura, Lall, Kushnir, & Camargo, 2009) accommodates track geometry and is also expandable to include additional meta track shape indexes (such as sinuosity), it has been selected to use for this study.
TC tracks in the Eastern Basin of Australia (Coral Sea) were grouped by power (maximum windspeed) 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 interpretedcurvature (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 (1st 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, Ying-qing, Qi-shu, Jingan, & Zhen-bin 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 windspeed 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 applicationsthe 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 & Wong 1979): For n TC track data sets with p variables per track x(i,j) for i = 1,2,...,n; j = 1,2,...,p; Kmeans assigns each track to one of K groups or clusters to minimize the withincluster sum of squares.
where ( , ) 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 withincluster sum of squares by moving tracks from one cluster to the next (Hartigan & Wong 1979).
A suitable range from which to select the optimal number (n) of clusters was estimated using the Elbow Method (Kaufman and Rousseeuw 1990). From this range, the final selection of n was done using the Silhouette Method (Kaufman and Rousseeuw 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 as used by Terry & Gienko (2011) has been used to compare track curvature across the clusters:

=
Eq. 8 The distances are at a resolution of the BOM TC track readings (readings are recorded at six hourly intervals) 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.
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.
Cyclone intensity totalled over its lifespan is a measure of overall cyclone destructiveness and risk. Therefore, the PDI, as defined in equation 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 (1st 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 is selected for the study.

Temporal TC trends by cluster results
A timeseries of count per year for TCs that have genesis in the Coral Sea is shown in Fig. 4 below. There is a statistically significant linear downward trend in TC frequency over the last fifty years for the complete set (not clustered) as well as for Cluster 2. A further analysis of the sinuosity results in Table 3 indicated a temporal increase in sinuosity for cluster 1 (Fig. 5), 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.  year rolling mean (data courtesy of Australian BOM) 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 timeseries has been re-represented with a five-year rolling mean per cluster in  TCs is shown in Fig. 8. There is a demonstrated linear trend (at 99% confidence) towards the equator.  Fig. 9). There is no overall linear trend over the entire timespan 1970 -2020 for this cluster. Cluster 2 (Fig. 9) does have an overall linear trend (at 99% confidence) towards the equator for 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. 0.09896). Cluster 3 is not trend able due to low record count

Discussion
This clustering method uses maximum windspeed as a weighting factor to calculate track centroids, therefore it can only be deployed on datasets with reliable and consistent records for this TC characteristic (or a another suitable analogous one like central pressure)this limits its applicability to track database records post 1970 (satellite-era records only). It is a limitation to 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 characteristicsthe long lateral axis of cluster 3, indicates long straightmoving 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).
The orientation of their variance ellipses shows that the TCs in clusters 1 and 3 track in a predominantly westerly direction and the cluster centres both have a closer location to land compared to cluster 2. 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 1 and 3.
This risk to the east coast of Australia is amplified by the fact that clusters 1 and 3 have a high proportion of curving or sinuous tracks. This is due an increased probability of tracks making landfall or situating 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.
In addition, when considering trends: Cluster 1 shows an increase in sinuosity, a strong increase in PDI from 2004 and a southwards movement of LMI from 2004.
When considering these findings and those in the previous paragraphs together -a clear separation of risk between clusters 1 and 3 and cluster 2 is present. It should be noted that the marginal overall increase in track sinuosity for the entire Coral Sea indicates a slight increase in cyclone risk for this entire region.
As shown in the results (Table 3)  It is interesting to compare the findings of Terry & Gienko (2011), who identified a possible intermittent increase of sinuous TCs in SWP, with the findings in this study, which show that the cluster with the highest proportion of straight or quasi-straight cyclones (cluster 2) demonstrates a decreasing frequency trend which dominates the overall decreasing trend, whilst the other more curving or sinuous clusters demonstrate no frequency trend at all. 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.
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). Decomposing this trend by cluster is insightful as it indicates that the more south-east located tracks of cluster 2 are the primary drivers of this reducing PDI post 2004. In contrast, PDI is increasing post 2004 for cluster 1. 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 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).
Although a clear understanding of the underlying drivers in TC trends in this area has not yet been secured (Chand et al. 2019), a few papers have included the influence of the Indian Ocean Dipole (IOD) to the influence of ENSO events in this region (Ham, Choi, & Kug 2016). As an extension of this idea, it appears that those cyclones further westward (cluster 1) 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). cluster 1 (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)   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 & Davidson 2004), due to trough interactions. A future overall equatorward 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. It is noted that the IPO moved into a negative phase post 2004 approximatelythis IPO phase and a simultaneous positive SOI has co-incided with peak TC activity in the southern region of the Coral Sea (Levin 2011). Both this and the links between the weakening of the Walker Circulation (the shift to a El-Nino dominant climate) as described by Callaghan & 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 windspeed weighted track centroids was shown 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 will greatly add to the ability to prepare for and manage the numerous threats TCs pose on a finer resolution along the east coast of Australia and in an area-focused manner in the Coral Sea in general. It provides a method to uncover TC climatological behaviours and trends on a sub-regional level, using geo-locational grouping.
Each of the three clusters produced showed a predominant direction of TC movementthe directions being westerly, south-easterly and west-south-westerly for clusters 1, 2 and 3 respectively. Clusters 1 and 3 showed a high degree of lateral (westerly) movement. The clustering showed that TC tracks in this region increase in sinuosity moving east to west, which is a phenomenon that has been previously reported in the SWP. TC risk increases with sinuosity (as described previously) therefore the more westerly situated clusters (1 and 3) can be regarded as carrying the most risk. Magnifying this risk is the fact that clusters 1 and 3 track in a westerly direction, have their ellipse centres much closer to land and that cluster 3, the cluster of most westerly position, has the highest median of power dissipated per track. The regional changes in TC wind power and maximum intensity have been further classified by regional groups (clusters) in this studyan understanding of these changes on a localised level is vital to protect human life and infrastructure as well as precious and exposed natural systems. 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 2,1 and 3 in order of risk.
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 2, and when considering the climatological characteristics described in the paragraphs above, this cluster can be considered to carry the least risk of the three.
Future risk due to apparent decline in frequency of TCs near the east coast of Australia may need to be re-assessed, considering this finding. 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 re-enforced by the results that both the number of TC tracks characterised as sinuous and PDI has been increasing for cluster 1 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 1, shown a dramatic reversal around 2004.
This will also lead to increased cyclone risk for more southern locations along the east coast of Australia.
This work 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 in a specific sub-region (the Coral Sea) and by identifying the group of historical TCs within the Coral Sea that this applies to (cluster 1), in a manner that is repeatable as the historical dataset of TC tracks post 1970 grows. It also repeats the finding of previous studies that the impact of ENSO on the selected TC climatological trends, within the fifty-year time bracket, is not clearly apparent.
There remains a substantial opportunity to determine if there is any demonstratable relationship between climate variability cycles such as ENSO, IPO and IOD to the trends in these characteristics as shown in this paper. In addition, although this subject has received increasing attention recently, exactly how TC genesis has responded, and future response to the forcing of global warming remains without consensus (Sharmila & Walsh 2018). The hypothesis that the statistically significant trends in frequency and TC LMI over the half-century timeframe are due to anthropogenic climate change and/ or the weakening of the Walker Circulation needs to be further investigated and tested.