Prediction of Coral Sea tropical cyclone power and latitude of maximum intensity using climate indices

Tropical cyclone (TC) wind power, often highly destructive, can be quantified using the power dissipation index (PDI) and in this study, the PDIs for Coral Sea TC tracks, as well as the latitude of maximum intensity (LMI) were investigated for correlation with climatological indices. Multiple linear regression with collinearity removed and an overall Pearson correlation of 0.7 or above was used for this. The results for all tracks showed that three indices dominated for PDI: Niño 4 Sea Surface Temperature (SST), the Dipole Mode Index (DMI) and the Madden Julian Oscillation (MJO). Coral Sea TC tracks clustered by maximum windspeed-weighted locations were then examined: For cluster 1 (located more south-east), the additional influence of the Southern Annular Mode (SAM) was apparent, whilst for cluster 2 (located more north-west), the same indices dominated as for the all-tracks model. For LMI, four indices were identified: the Indian Ocean East SST Anomaly (DMI E), the MJO, the Southern Oscillation Index (SOI) and the SAM. Only TCs clustered in the northeast of Australia had a valid model for LMI, with correlation 0.8, using three indices: DMI E, DMI and the SOI. Overall, raised Niño 4 SST combined with a negative DMI and low MJO amplitude were shown to predict large increases in TC power, whilst a combination of increasing DMI E temperature anomaly with a positive SOI moves LMI equator-wards. The models compiled in this study identified the most significant climatic indices and successfully predicted TC power dissipation and LMI.


Introduction
Projected TC threat is often presented as a function of frequency of occurrence (e.g., Bloemendaal et al. 2022;Knutson et al. 2020;Chand et al. 2022).For the Australian region (including the Coral Sea), for instance, there is a well-known relationship between TC frequency and the El-Niño Southern Oscillation (ENSO)-El Niño (La Niña) events result in a reduction (increase) of TC activity in the Coral Sea, with TCs occurring further eastward (westward) (Dowdy et al. 2012).However, there is limited information on cyclone hazard to be acquired through an understanding of TC frequency trends alone, since this metric is often largely driven by storms with relatively low destructive potential (Emanuel 2021).A therefore arguably more useful metric to present when evaluating future TC hazard is wind-sourced energy (or power).TCs can release large amounts on kinetic energy as they decay-the devastating consequence to the surrounds is more accurately correlated to the cyclone's kinetic energy than to maximum windspeed alone, since energy measures consider the total volume of space in which the wind blows (Kreussler et al. 2021).The power dissipated (intensity) and the variability of location of peak power are primary factors when considering the magnitude of the hazard posed by TC energy.Moreover, a changing climate and the resultant raising of SST (the mean warming trend is 0.09 °C per decade globally (Bulgin et al. 2020)), raises the urgency and importance of understanding the relationships of TC power and latitude of the lifetime maximum intensity (LMI) to changes in SST and commonly used climatological indicators such as ENSO.Interestingly, Klotzbach (2006) showed a positive correlation between SST and TC power for only two of six oceanic basins in a global study, 1986-2005, suggesting that other ocean and climate factors play a major role, including perhaps relative SST across basins.
Globally, there has been considerable variability in the reported measures of cyclone power on annual and interdecadal timescales (Bell et al. 2000) and global averages of power metrics have been found to be of limited use for revealing the impact of climate change on TC power characteristics (Wehner 2021).Miller et al. (2023) examined TCs with genesis in the Coral Sea and grouped their tracks over the last fifty years based on K-means clustering of the maximum windweighted centroids.TC track variance and curvature (sinuosity) were assessed with three well defined TC track clusters were produced.Two of the clusters demonstrated diverging trends for power dissipation post 2004.One of the three clusters showed a statistically significant trend (towards the equator) of location of maximum intensity.A clear variance in between-cluster hazard was demonstrated and in addition the study showed that TC trends discovered for the southwest Pacific are not manifest or consistent across all the clusters.
In the South Pacific there has been limited research on the link between ENSO and TC power/intensity (Chand and Walsh 2011).This is despite the critical link between sea surface temperature (SST) and TC intensity in the developing stages (Diamond and Renwick 2015b).In addition, with the arguable exception of Magee and Kiem (2020), that focused on TC frequency, the authors have not discovered any studies relating ENSO together with other concurrently active tele-connected climate indices to TC intensity/ power and indeed also to LMI in this region.
Another climate index besides ENSO with a major influence on TC behaviour in the Australian region, including the Coral Sea, is the Indian Ocean SST index-the Indian Ocean Dipole (IOD) and the associated dipole mode index (DMI) (Magee and Verdon-Kidd 2018).Magee and Verdon-Kidd (2018) showed that, for the Southwest Pacific (SWP), the two climate modes ENSO and DMI can enhance or attenuate the singular effects of each in isolation, when considering cyclogenesis.
The intensity of a cyclone is a function of the difference between the SST and the average tropospheric temperature (Emanuel 2005).Although tropospheric warming occurs on multidecadal timescales (Santer et al. 2017), the relationship between tropospheric changes and TC intensity suggests the influence of another climate index on cyclone power, the Madden Julian Oscillation (MJO).The MJO can be described as a disturbance of various features of the tropical troposphere, including pressure, zonal winds, tropospheric temperature and water-vapour mixing ratios moving easterly with a period of 40-50 days and a zonal wavenumber 1 (Madden and Julian 1972;Hall et al. 2001).A zonal wavenumber is the number of wavelengths fitting within a full circle around the globe at a given latitude (Glossary 2015).The MJO influences numerous aspects of the tropical flow, including temperature (Hall et al. 2001) and is known to modulate the genesis of TCs (Camargo et al. 2009).An index to quantify the MJO was developed by Wheeler and Hendon (2004) which included a division of the state of the MJO into eight separate phases in two-dimensional phase space.Diamond and Renwick (2015a, b) showed that in the SWP, TC frequency decreased during MJO phases 2 and 3, whilst both overall TC frequency and major TC frequency increased for phases 6 and 7.Besides this finding on major TC frequency, to date studies dedicated to the influence of DMI and MJO on cyclone power metrics in this region do not appear to be published, to the authors' knowledge.
Another climate driver related to the troposphere is the Southern Annular Mode (SAM).Diamond and Renwick (2015b) have found a relationship between the SAM and cyclogenesis which suggests a possible link between the SAM and TC intensity in the Coral Sea-the SAM was therefore included, along with DMI and MJO, as one of the climate indices whose influence on TC power in the Coral Sea was investigated.The SAM, sometimes referred to as the Antarctic Oscillation, exerts its influence mainly in the high to mid latitudes of the Southern Hemisphere (Thompson and Wallace 2000).The SAM is an index of sea level or geopotential height pressure and is either positive or negativepositive SAM demonstrates low air pressure anomalies in the Antarctic and high air pressure anomalies in the mid latitudes, with correspondingly strong westerly winds, shifted poleward.In negative SAM occurrences, the westerly winds are relatively weaker and shift towards the equator (Lee et al. 2019).The SAM can exert a significant effect upon midlatitude SST (Diamond and Renwick 2015b).Since SST is a primary influence on TC genesis and intensity (Dowdy et al. 2012;Emanuel 2005), the SAM is worthy of inclusion into this study, especially when considering the power indices of the more southerly tracking TCs.
The low frequency variability in the Pacific Ocean-the Interdecadal Pacific Oscillation (IPO) should also be considered.This oscillation in Pacific Ocean SST affects the relationship between climate variations in Australia and ENSO.When the IPO index is positive (negative) the relationship is tenuous (well defined) (Power et al. 1999).Due to the long period of this oscillation (13-35 years), compared to the time period of consistent TC track records, relationships between TC variability in the SWP and IPO are not well researched (Magee et al. 2017).However, additional studies have demonstrated that a positive (negative) IPO index results in more frequent El Niño (La Niña) events (Verdon and Franks 2006).In addition, (Flay and Nott 2007) noted that TC numbers in eastern Australia increased during the co-incidence of La Nina conditions and the negative phase of IPO.
Since the investigation of TC intensity and power is central to this study, a well-defined measure of TC power dissipation was needed.Currently there are two accepted power metrics as the cyclone tracks towards decay: the Accumulated Cyclone Energy (ACE) index and the Power Dissipation Index (PDI).They are alike in that each uses the cyclone peak surface windspeeds, integrated over the lifetime of the TC track, to derive a measure of dissipated energy.They differ in that ACE calculates the square of this windspeed, whilst PDI uses the cube of the windspeed (Wehner 2021).Both ACE and PDI measures are sensitive to intense storms, due to this aforementioned non-linear relationship to windspeed, with PDI the more sensitive to windspeed due to the cube non-linear relationship (Wehner 2021).It is worth mentioning here that the financial consequences of TC damage have been shown to be a function of the maximum windspeed cubed (Southern 1979).In addition, since the windspeed-cubed relationship corresponds to the wind frictional force over an area, it is assumed this relationship also applies to damage of eco-systems-therefore the PDI index was selected for this study.
Specific Coral Sea environmental factors influencing the formation and sustainment of TCs have been identified in previous research: Sufficient (above 26.5 °C) SST is a principle thermodynamic requirement for TC development (Dowdy et al. 2012).Globally, the MJO has also been shown to have a clear influence on TC behaviour by modulating atmospheric circulation (Balaguru et al. 2021).In the SWP and the Australian region (and indeed globally) it has been shown that the active phase of the MJO raises cyclo-genesis (Diamond and Renwick 2015a, b;Hall et al. 2001;Liebmann et al. 1994), although the influence of the MJO on TCs is not direct but rather though the increase in depressions during the convective phase of the MJO (Liebmann et al. 1994).Camargo et al. (2009) found that the principal influence of MJO on TC genesis was due to modulation of mid-level relative humidity (high humidity increases TC formation potential)-though this mid-level humidity is itself a function of tropospheric temperature.The SOI appears to be a minor factor in TC incidence in the Coral Sea- Basher and Zheng (1995) found the correlation between SOI (in isolation to Niño SST) and TC incidence in the Coral Sea to be weak, with SST alone being a far stronger indicator of cyclone incidence in this specific area of the SWP.This study investigates the effect of SOI on TC power, as a separate metric to TC incidence.
A critical consideration in the evaluation of future TC threat due to wind-power, in the light of anthropogenic climate change, is the latitude of maximum intensity (LMI).A change in LMI could potentially have devastating effects on communities and ecosystems located at latitudes other than the historical means of LMI.Recent studies at a global scale have shown a poleward migration of this point ( (Kossin et al. 2014)Sharmila and Walsh 2018) which is often attributed to expansion of the tropics due to global warming (e.g., Ramsay 2014; Tennille et al. 2017;Sharmila and Walsh 2018).However, a review for the Australian region (Chand et al. 2019) showed LMI trending in the opposite direction since the 1960s (by comparison of two 30-year climatological normals).In order to further understand the trend in LMI for this basin, this paper will analyse the correlation of LMI to the various climatological metrics mentioned above.
Only TC tracks with genesis in the Coral Sea are considered.The Coral Sea has the Australian coast as its western boundary; in the north it is bounded by Papua New Guinea and the Solomon Islands; the eastern boundaries are the Vanuatu archipelago, Pine Island, New Caledonia and the Elizabeth and Middleton Reefs.The southern boundary is the 30° south latitude (Ceccarelli et al. 2013).TC tracks are clustered over the area they traverse from genesis until their maximum windspeed falls below the TC threshold of 17.5 m/s (refer Fig. 2 and (Miller et al. 2023)).
The 9 climatological indicators will be compiled into a predictor model.The model will then be used to achieve the following aims for this study: 1) The modelling of TC power dissipation in the Coral Sea using a minimum number of climate indicators but having an overall correlation above 0.7 to the observed power dissipation index (PDI) values.2) The modelling of TC latitude of maximum intensity (LMI) in the Coral Sea using a minimum number of climate indicators but having an overall correlation above 0.7 to observed LMI values.

Method
A high-level method description is as follows (Fig. 1

Data preparation
The TC track records, 1970-2020, for TCs with genesis in the Coral Sea, stored in the Australian Bureau of Meteorology (BOM) Best Track Tropical Cyclone database were used as baseline source data.Several a-priori steps were then carried out to prepare the required TC power data (as detailed in Miller et al. 2023): 1) These records were filtered to control windspeed recording instrument bias (especially relevant to track records prior to 1978) and to define TC genesis and decay endpoints.
2) The tracks were clustered, by maximum windspeed weighted track centroids, using the K-means statistical method.Three spatially well-defined clusters were produced, as shown in Fig. 2. 3) Three year and five-year low pass filtered PDI values were produced.( , where V max is the maximum windspeed and is the lifetime of the track, following Emanuel (2005)).The method is detailed in Miller et al. (2023).Low pass filtered PDI values were produced for all tracks combined and by cluster.4) Three year and five-year low pass filtered values of the LMI were produced.The method is detailed in Miller et al. (2023).The LMI low pass filtered values were produced for all tracks combined and by cluster.

Model selection
MLR was selected to model both PDI values and LMI values for each track (of 120 tracks in total).MLR is capable of representing all of the potentially important predictors in one model, whilst potentially providing an accurate and quantified understanding of the effect of each predictor individually on the predictand.It has, in addition, been previously used effectively both for studies involving climate indices (e.g., He et al. 2014) and for studies of cyclone intensity (e.g., Kotal et al. 2008;Kaplan & DeMaria 2003).The MLR equations are shown in the appendix (Eqs. 1 and 2).

Climate indices
In order implement the aims of this study-to model TC PDI and LMI using a minimum number of climate indicators-it was firstly required to select specific indices as covariates (independent variables).
For each TC track, climate index covariates (for the selected indices, as discussed in the introduction and listed in Table 1) were chosen to represent both recordwise (monthly or yearly) and time averaged (3-year and 5-year low pass filtered) index values.Emanuel (2007) showed a degree of correlation (r 2 = 0.33) between TC PDI and SST, with fluctuations on timescales of three years or longer in the western North Pacific and in addition, preliminary modelling showed low overall correlation using 6 monthly lagged indices as variables: For these reasons, each index was assigned the 2 covariates: record-wise, to investigate instantaneous climate variable effects and 3-yearly/5-yearly time averaged to investigate time averaged effects on an annual scale.The selected covariates for the nine climate indices are detailed in Table 1.Note that for the ENSO SST indices, Niño 3 and Niño 4 were chosen to isolate the effects of the zonal differences of SWP SST related indices.They were then compiled into a data frame, together with the covariates for PDI and LMI.

Model initialisation and refinement
Initial MLR models were setup-Table 1 shows that the total number covariates to be used in both PDI and LMI MLRs is 20 (2 for each of 8 indices and then 4 for the MJO index, considering amplitude and phase).A primary object of this study is to model both PDI and LMI with a minimum number of covariates yet still achieve an overall robust correlation (Pearson correlation of 0.7 or above).In addition, a simple statistical examination showed a high degree of multicollinearity between the climate indices.This was an intuitive result since numerous studies showing teleconnection between these climate indices have been previously produced (e.g., Moon et al. 2010;Dong et al. 2018).For these reasons, a further two-step process, using Akaike Information Criterion (AIC) and detailed in the appendix, was followed to arrive at a final selection of primary indices, I 2, that drive both PDI and LMI.

Model finalisation
As an initial output, the Index covariant with the highest individual Pearson correlation coefficient was identified (for PDI and LMI, unclustered and for clusters).
The MLR models for PDI and LMI as independent variables, before clustering and for clusters 1 and 2 (refer Fig. 2) were then examined and a set I 2 was produced for each.The limited number of tracks in cluster 3 prevented analysis for this cluster (refer Miller et. al 2023).The observational data for the set I 2 was also normalised to enable comparison between the climate indices for relative influence on the dependant variable (PDI or LMI) and the final models were configured and run using original and normalised I 2 data.
These MLR models using the I 2 indices were then checked for validity and reliability using the following criteria as described by Townend (2002) and as detailed in the appendix.

Model validation and output
Preparatory model checks using the complete I 2 data set indicated a tendency for bias in the latter years (around the period 2006-2009, and after 2017).(The reasons for this are examined in the discussion section).Leave-one-out-crossvalidation (LOOCV) was therefore chosen to validate the models as it is the cross-validation method that best minimises bias (Maleki 2020) and in addition, the size of the training set is maximised when using this validation technique (Cheng et al. 2017).
The quality of the final set I 2 to model three-year means of PDI (and I 2 for LMI) was evaluated using the following: • Model Pearson correlation coefficient r • Model Coefficient of determination (R 2 ) and its corresponding p -value for the F statistic.

Results
All the final models met the criteria for MLR validity and reliability (the criteria are detailed in the methods) although, due to detected autocorrelation, the Newey-West adjusted standard errors of the covariant coefficients are reported for each model.These are the Heteroskedasticity and Autocorrelation Corrected (HAC) standard errors-they show an improved estimation of the standard errors for each independent variable.
The 5-year low pass filtered values of the independent variables showed better correlation results than 3-year low pass filtered values.However, the model prediction value is raised by lowering the time averaging windows of its output, the 3-year means for the dependent (output) variables were considered a better result than 5-year means and were selected for the dependant variables.The target overall Pearson correlation (0.7) was achieved using 3-year low pass filtered values of the dependant variables (PDI and LMI) and 5-year low pass filtered values of the independent variables.

PDI for all tracks
The PDI model for all TC tracks (the base model) demonstrated that of the nine indices available, three (Niño 4, MJO and DMI) significantly predicted PDI (see Table 2 and  Appendix Table 6).This was with multicollinearity reduced to an acceptable level (VIF less than 5 as described in the methods).For both Niño 4 and MJO, two covariates-the monthly value concurrent with the cyclone and the 5-year low pass filtered value, were required to produce the level of correlation the study was aiming to achieve.All covariates had statistically significant intercepts (coefficients).
The 5-year low pass filtered value of Niño 4 was the only predictor with a positive intercept and examination of the normalised coefficients (Table 2) shows that it is a strong predictor of PDI for the base model (Niño 4 values are a measure of temperature not temperature anomaly and are an order of magnitude higher than the other covariates).
Table 3 shows the cross-validation results for the PDI test dataset, with a small reduction in correlation statistics.

TC cluster 1 PDI
The PDI model for cluster 1 TC tracks demonstrated that of the nine indices available, four (Niño 4, MJO, DMI, SAM) significantly predicted PDI (see Table 2 and Appendix  Table 6), with multicollinearity reduced to an acceptable level.All four indices were statistically significant as covariates in the model.The emerging influence of the Marshall SAM on TC power for this more southerly located cluster is apparent.This is also consistent with the shape of the cluster 1 variance ellipse (refer Fig. 2), showing that these cluster 1 TCs have a higher probability of tracking into the midlatitudes, where the SAM has an influence on atmospheric pressure (Diamond and Renwick 2015b), compared to the cluster 2 TCs, which track more along latitudes.In addition, it can be seen from Table 2 that the phase of the MJO plays a role in the prediction of TC power for this cluster.
The Niño 4, 5-year low pass filtered, and the SAM were the two predictors with a positive intercept (coefficient).Once again, the normalised coefficients in Table 2 show that Niño 4 is a strong predictor of PDI.

TC cluster 2 PDI
As with the base model, the PDI model for cluster 2 TC tracks demonstrated that of the nine indices available, three (Niño 4, MJO, DMI) significantly predicted PDI (see Table 2 and Appendix Table 6), with multicollinearity reduced to an acceptable level.The same level of correlation (r = 0.72) was achieved as the base model.
As with the base model, for TC cluster 2, Niño 4 required two covariates: the monthly value concurrent with the cyclone occurrence and the 5-year low pass filtered.
As with the baseline PDI model, the Niño 4, 5-year low pass filtered was the only predictor with a positive intercept (coefficient) and the normalised coefficients (Table 2) show that it is a strong predictor of PDI. a few individual TCs of unusually high power during this period and PDI modelling results in relationship to theses TCs is further explored in the discussion section.

TC cluster 3 PDI
No correlation above 0.7 was found which was expected due to the limiting size of the dataset (no. of TCs (n) = 17).Therefore, it was not possible to undertake this analysis for this cluster.

LMI
The LMI model for all TC tracks (the base model) demonstrated that of the nine indices available, four (SOI, DMI E, SAM and MJO) significantly predicted LMI (see Table 4 and Appendix Table 7).This was with multicollinearity reduced to an acceptable level (VIF less than 5 as described in the methods).Only one covariate for each of the four indices, the 5-year low pass filtered, was required to produce the level of correlation the study was aiming to achieve.All covariates had statistically significant intercepts (coefficients) at the 95% confidence level.
The coefficients of DMI E and SOI were positive in sign (increasing values move LMI equatorward), whilst the coefficients of SAM and MJO (phase) were negative (increasing values move LMI poleward).Examination of the normalised coefficients in Table 4 shows that DMI E, SOI and SAM are approximately of equal strength as predictors of LMI, with MJO (phase) exerting about half as much influence as each of the previous three, individually.
The number of indices required to significantly predict LMI (above) the target Pearson correlation of 0.7 for Cluster 1 TCs was reduced by one, in comparison to the base LMI model.DMI, DMI E and SOI significantly predicted LMI (see Table 4 and Appendix Table 7), at a high Pearson correlation of 0.82 (with multicollinearity reduced to an acceptable level).For DMI and SOI, two covariates were  2023)) and MLR predicted (shown in blue).The other clusters did not show any overall correlation above 0.7 for their MLR models required to produce this level of correlation-the monthly value concurrent with the cyclone and the 5-year low pass filtered.DMI E only required a 5-year low pass filtered value.All 5 covariates had statistically significant intercepts (coefficients) at the 95% confidence level.
Interestingly, all coefficients except DMI 5-year low pass filtered were positive.Examination of the normalised coefficients (Table 4) shows that DMI E is the dominating predictor of LMI for this cluster, with the SOI also exerting a strong influence (since the SOI coefficients are both of the same sign-positive).
Figure 5 shows the comparison between the data calculated LMI trend and the validated MLR model of LMI trend, for all tracks in the Coral Sea. Figure 6 shows this for cluster 1 tracks only.
For cluster 2, no model with Pearson correlation close to 0.7 could be formulated with statistically significant covariate coefficients.
For cluster 3, no correlation above was found which was expected due to the limiting size of the dataset (no. of TCs (n) = 17).
Table 5 shows the cross-validation results for the LMI test dataset, with a small reduction in correlation statistics.

Discussion
Niño 4 was found to be the best of the ENSO indices for describing PDI in the Coral Sea and was the primary covariate, both for all tracks combined and for the clusters (although cluster 2 did include SOI as an additional ENSO covariate in the MLR).A similar result was obtained by Magee and Kiem (2020) for their Poisson regression model of TC frequency on the east coast of Australia (referencing their cross-validated Niño 4 ENSO index model 4).Interestingly, their cross-validated Niño 3 ENSO index model showed a higher correlation for frequency, which demonstrates the improved performance of the closer Niño (Niño 4) index to the Coral Sea for the TC power regression models in this study.The strong positive correlation of Niño 4 to PDI is an understandable result when considering that TC power is a function of the difference between localised SST and average tropospheric temperature (Emanuel 2005).Pillay and Fitchett (2021) found that raised SST was a primary factor for increased TC power in the Southern Hemisphere.Potential TC power dissipation as measured by PDI is possibly raised during El Niño warm pool events (as described by Kug et al. 2009)-these El Niño warm pool events are associated with positive Niño 4 anomalies.The Niño 4 results indicate that the relationship between the SST's experienced by the TCs as they track in the Coral Sea and the Niño 4 climate indices is best defined by the 5-year low pass filtered temperature measurement moderated by the monthly average value at the time of the TC.Each of these two Niño 4 coefficients appears to be required in order to model the two components of this Coral Sea/ Niño 4 relationship: (1) The SST at the time and location of TC genesis.For this region, the years with increased cyclone activity have been shown to be preceded by high SST in the north Australia region (close to the Niño 4 region) (Basher & Zheng 1995;Nicholls 1984).( 2) The SST changes as the TCs track to decay.Evolving tropical cyclone intensity is often nonlinear in nature, including complex and opposing thermodynamic processes with concurrent SSTs ahead of the storm and SSTs near the storm being key factors (Hendricks 2012).Figure 7 shows the trend in Niño 3 and Niño 4 SST-the effect of Niño 4 in driving cluster 2 PDI upwards is apparent when comparing this to Fig. 4.
Unlike the other relatively more static climate indexes, the MJO can be described as intraseasonal, with a period of 40-50 days and so the identification of the monthly MJO amplitude as an influencing index on 3-year low pass filtered PDI is an unexpected result (refer to the introduction on the influence of the MJO on TC formation).There has also, however, been shown to be clear interannual variability in MJO signal, with periods of strong influence interspersed by times of weak MJO activity (Diamond and Renwick 2015a, b).This interannual variability is reflected in the results through the presence of the 5-year low pass filtered MJO oscillation.The MJO amplitude is a measure of the degree of disturbance of the troposphere, which affects tropospheric temperature but research has shown that it also lowers SST, not only through atmospheric cooling, but through ocean currents and subsurface mixing after moving through a region.(Balaguru et al. 2021;  showed a significant reduction in the maximum 10 m wind profile (and therefore PDI) in the wake of a MJO event as opposed to the modelled wind profile without the MJO event.(Balaguru et al. 2021).This explains the emergence of MJO amplitude as a second controlling covariate in the PDI MLR model.TC PDI decreases with strong MJO amplitude (the MLR model shows a negative correlation), suggesting the cooling effect on SST by MJO events, as described.
The third climate index that appears as a covariate in the PDI MLR (unclustered and by cluster) is DMI-the correlation is negative, therefore a negative Indian Ocean Dipole or DMI (warm SST in the east Indian Ocean) will increase the PDI index.The explanation for this seems to be that the probability of TC genesis is increased due to warmer SST (above 26.5 °C) in the Coral Sea-a higher TC frequency results in a raised PDI index for a particular year.Liu & Chan (2012) found that a negative DMI, in combination with La Niña, increased the probability of TCs in the east Australian basin, due to enhanced negative cyclonic relative vorticity from stronger and more easterly extended westerly winds.Miller et al (2023) showed that cyclones further westward in cluster 2 show PDI trends closely matched to SST in the Northern Tropics-i.e.SSTs closer to the east Indian Ocean.This reinforces the finding by Emanuel (2005) that localised SST affect TC power more than entire ocean scale equatorial SST trends.Figure 8 shows the closely correlated relationship between SST in the Coral Sea and SST in the Northern Tropics.The Northern Tropics are located close to the DMI E area, which explains the influence of the DMI index in the model.An examination of the Niño 4 SST in Fig. 7 and the DMI SST anomaly in Fig. 9, for the years 1996-1998, shows spikes for both, with the spike in DMI SST being particularly dramatic.This historical SST feature is mirrored in the PDI trend for all tracks and the within-cluster tracks, between 1996 and 1998 (refer Figs. 3 and 4).The strong correlation between TC PDI and both Niño 4 SST and DMI SST anomaly is evident in this time period (refer to Figs. 7  and 9).
For the more southerly located cluster 1, the SAM plays a role-positive SAM index values are associated with high air pressure anomalies in the mid latitudes (Ho et al. 2012).Diamond and Renwick (2015b) showed a relationship between the SAM and tropical cyclogenesis in the South Pacific-positive phases of SAM yielded an increase in TC frequency.The PDI MLR model shows a positive correlation with the SAM, so it seems the same holds true for TC intensity, as measured by PDI, for the more southerly located tracks in the Coral Sea.This is intuitive also because yearly PDI is dependent on frequency of cyclogenesis per year.
When comparing the all-track PDI model to the two within-cluster models of interest is the repeat of the same indices and covariates for cluster 2 as were in the all-track model.The magnitude (and sign) of the cluster 2 covariates (refer to Tables 2 and 3) were very similar to the all-track model with two Niño 4 coefficients required to reflect the two (suggested) components of the Coral Sea/ Niño 4 SST relationship.
Location (or latitude) of maximum intensity (or lowest central pressure) studies to date have shown considerable inter-basin variability across the globe (Song et al. 2018) and although a poleward migration of this characteristic is statistically significant and clear on a global and hemispherical level, this trend on an oceanic basin or regional level is typically statistically insignificant (Kossin et al. 2014).In the eastern basin of Australia for instance, a movement in the opposite direction (towards the equator) has been detected when comparing the two 30-year climatological normals, 1960-1989and 1980-2009(Chand et al. 2019)).The question of how these high level (global/hemispherical) poleward statistically significant trends can occur without statistically significant trends at a basin level has been investigated and one proposed theory is that a global poleward trend of LMI can be identified simply from the frequency changes in each basin (Moon et al. 2015).The results of this study can be seen to both support this theory and expand on it: Our results show that a sub-basin grouping of TCs (cluster 1), at this higher resolution, formed by statistical analysis with corresponding statistically significant trends in both LMI (equator-wards) and frequency (decreasing) (Miller et al. 2023), can be modelled by climate indices-variables that are separate to TC frequency.In addition, it is understood that El Niño events can cause a decrease of TC occurrence in the Coral Sea (Dowdy et al. 2012).This implies that the trend towards increasing El Niño events in this region (Dowdy et al. 2012) is matched by the trend of LMI towards the equator for cluster 1.This aligns with the understanding that for the Coral Sea, TCs migrate NE during El Niño (Ramsay et al. 2014).To investigate the underlying drivers of this movement of LMI equatorward during conventional El Niño, it is useful to consider a few environmental parameters known to effect TC behaviour: SST, vertical wind shear, lower tropospheric relative vorticity and midtropospheric relative humidity and the South Pacific convergence zone.Of these, it appears that for the Coral Sea, the relative humidity differences between La Niña and El Niño climate states is the principal driver of this LMI latitude trend.Dowdy et al. (2012) illustrates the stark contrast in relative humidity between El Niño and La Niña years, with high relative humidity only available at low latitudes during El Niño years.This supports the finding by Ramsay et al. (2014) that of several environmental parameters, only mid-tropospheric relative humidity consistently affects TC counts (and as described previously, TC counts are further correlated with LMI in this region).
When considering characteristics besides frequency, it can be seen from the LMI model for all tracks in the Coral Sea that LMI is driven in the direction of the equator by SST as measured in the east Indian Ocean (DMI E) (refer to the DMI E coefficient in Table 4).Although research on climate index related mechanisms and causes of LMI movement on a sub-basin level is sparse, Nicholls (1984) showed the correlations of TC activity (frequency) with the north Australian region (120-160°E) SST depended on the point in time of the TC season.The SST index region closest to the north Australian region is DMI E (90°E to 110°E)-it appears from these results that the correlation of TC activity with SST extends east towards the DMI E region.One TC characteristic analogous to LMI is location of cyclo-genesis.Magee and Verdon-Kidd (2018) showed that warmer (cooler) east Indian Ocean SST results in a statistically significant north/east (south/west) displacement of TC genesis in the SWP.It should be noted though that IOD/DMI SST change is related to, and dependant on, ENSO SST change (it leads or lags ENSO), with only around 32% of IOD events occurring independently of ENSO events (Stuecker et al. 2017).The regression modelling can be said to select the most statistically useful of the SST indices, but the indices are, to one degree or another, correlated to each other.
The results also show (refer to Table 4) that the phase of SOI, SAM and MJO all support or counter this movement, depending on sign.In particular, the SOI plays a dominating role.Table 4 shows a positive coefficient for SOI which implies northerly (southerly) movement of LMI for the positive (negative) phase of SOI.This should be compared to the findings of Dowdy et al. (2012), who found a westward contraction of very low cyclone central pressure in the eastern Australian region during La Niña events with the region of maximum intensity located around 20°S, 160°.The results for LMI of this study, however, are not related to ENSO events, but rather events pertaining to DMI(E)SO-DMI, as mentioned previously, could be considered leading or lagging Niño region SST.In addition, as described in the introduction, the correlation between SOI (in isolation to Niño SST) and TC incidence in the Coral Sea is weak, with SST alone being a far stronger indicator of cyclone incidence in this specific area of the SWP.It appears that a smoothed SOI index (in conjunction with DMI E) in this region is a more powerful predictor of TC LMI than it is a predictor of TC incidence.
A comparison between the cluster 1 LMI results and the baseline (all tracks) results suggest that the SAM and MJO oscillations have historically prevented an overall trend for LMI for all tracks in the Coral Sea (since the SAM and MJO indices are opposite in sign to the DMI and SOI indices).However, for those tracks in cluster 1, the SAM and MJO covariates are not needed in the MLR to achieve the target level of correlation and are statistically insignificant if included.This cluster of cyclones does show a statistically significant trend in LMI, towards the equator.As previously mentioned, this is in stark contrast to the often-cited global poleward LMI trend and cluster 1 has a corresponding statistically significant decrease in TC frequency.
For both PDI and LMI modelling, a comparison between data-calculated (observed) values and MLR predicted values shows large overall variance for the years 2005-2010 (refer Figs. 3 and 5), with the model tending to under-estimate values for this timeframe.This can be explained by the formation of three cyclones of anomalous/ outlier strength during this time period: Cyclone Larry ( 2006) was the most severe cyclone to impact the wet tropical coast since 1918 (Turton 2008); cyclone Hamish (2009) was the only cyclone to become category 5 in the Coral Sea during the timeframe of this study and cyclone Ului (2010) reached category 5 before moving into the Coral Sea.

Conclusions
The power dissipated by TCs (as measured by the PDI index), as they track in the Coral Sea, was effectively modelled using multiple linear regression of a selected set of statistically significant climatological indices.This was done by using the AIC statistical method and then removing collinearity, with an overall Pearson correlation above 0.7.For all TCs tracking in this region, three indices (in order of correlation) emerged as the principal correlates of 3-year low pass filtered PDI: Niño 4 (using both a monthly and a 5-year low pass filtered covariate), 5-year low pass filtered DMI and MJO (using both a monthly and a 5-year low pass filtered covariate of amplitude).By examining the MLR correlation coefficients it was shown that PDI in the Coral Sea is significantly raised during the co-incidence of warm pool El Niño, a negative DMI index and weak MJO events.
The analysis by cluster (using maximum wind-weighted track location clusters in the Coral Sea) indicated that, for cluster 2, the same 3 indices (Niño 4, DMI and MJO) are the best correlates.For the more southerly cluster 1, in addition to the indices mentioned above, the inclusion of the monthly SAM index raised the correlation.As mentioned in the introduction, a positive SAM index correlates with high air pressure anomalies in the mid latitudes, which in turn create strong pressure gradients when near the low pressures of TCs, raising TC PDI.
The latitude of TC LMI (or LCP) as they track in the Coral Sea, was modelled using MLR with DMI temperature indices, the SOI, the SAM and the MJO phase oscillations.A suggested combined index, DMI(E)SO (equivalent to ENSO) could be said to be a strong correlate of TC LMI in the Coral Sea, moderated by SAM and MJO in the more northerly areas.The more southerly located TC track cluster 1 did not, however need the inclusion of the SAM and MJO indices to produce a statistically significant trend in both LMI (equator-wards) and frequency of occurrence (reducing).For the southerly located cluster 1 cyclones, a combination of DMI and DMI(E)SO is a strong correlate of TC LMI in the Coral Sea.
This study has revealed the relationships between the climatological indices and TC power in the Coral Sea in a manner that reduces overlap of the effect of the indices to a minimum, despite the well-established teleconnections between them.It demonstrates that although power as measured by PDI is significantly correlated with equatorial SP SST, it is the most local SST index (Niño 4) that dominates in the Coral Sea, and it is reinforced by both raised east Indian Ocean SSTs and weak MJO indices.In contrast to this, the latitude of maximum intensity is most strongly affected by DMI/E and the SOI-locating closer to the equator during negative DMI events.It remains to be determined how climate change will affect the balance between the effects of both Niño 4 and DMI on TC power (location of maximum intensity) in this region.

The two-step model refinement process:
(1) The Akaike Information Criterion (AIC) statistical method, stepping both forward and backwards when searching, was deployed on both models separately to nominate a first set of indices I 1 for both.AIC is used for comparison across models in terms of data fit.An AIC value is calculated, for each model, from the number of independent variables used and the log-likelihood estimate.Using this criterion, the best-fit model is that which explains the greatest amount of variation using the fewest independent variables.The AIC value should be a small as possible and the process relies on the AIC value to converge to a solution for I 1 (Venables and Ripley 2002).(I 1 is a subset of all 20 indices in the data frame).(2) A final selection of indices I 2 was compiled by calculating, for each index X i in I 1, a variance inflation factor VIF: where R 2 is the coefficient of determination for a linear regression with X i as the dependant variable and all the other indices in I 1 as independent variables (right hand (1) PDI(c) = 0 + 1 X 1 (m) + ... + n X n (m) + (2) LMI(c) = 0 + 1 X 1 (y, m) + ... + n X n (y, m) + (3) VIF i = 1∕ 1− R 2 i side of the equation).All indices in I 1 with a VIF greater than 5 were eliminated, as this reduces multicollinearity to an acceptable level (Menard 2002).
The remaining indices formed the final set I 2.

MLR model validity and reliability criteria
• The relationship between each independent variable and

Fig. 1
Fig. 1 High level method flow diagram (matching colours indicate the same procedure was employed, using different data)

Fig. 2
Fig. 2 TC tracks with centroids clustered by first moment (track centroids).TC tracks with maximum windspeed indicated by graduated colour scale and positions of individual track centroids with associated track cluster groups indicated by centroid point colour.Clus- Fig. 3 TC PD for all tracks in the Coral Sea-Showing both the plot of values calculated from TC track data (maximum windspeed, refer Miller et al. (2023)) and MLR predicted (shown in blue)

Fig. 4
Fig.4TC PD for tracks in the Coral Sea by cluster(refer  Miller et al (2023))-Showing both the plot of values calculated from TC track data (maximum windspeed, referMiller et al. (2023)) and MLR predicted (shown in blue).The other clusters did not show any overall correlation above 0.7 for their MLR models

Fig. 6
Fig. 6 TC latitude of maximum intensity (LMI)/lowest central pressure (LCP) for cluster 1 tracks in the Coral Sea (refer Miller et al. (2023))-Showing both the plot of values calculated from TC track data (minimum pressure and latitude, referMiller et al. (2023) and  MLR predicted (shown in blue)

Table 1
Indices and the covariates used for each in order to compile the multiple linear regression models (MRLs)

Table 2
PDI MLR Covariate coefficient (β) statistics for normalised data of TC tracks in the Coral Sea, 1978−2020.Presented for all tracks and by cluster

Table 4
LMI MLR Covariate coefficient (β)statistics for normalised data of TC tracks in the Coral Sea, 1978-2020.Presented for all tracks and by cluster Fig. 5 TC latitude of maximum intensity (LMI)/ lowest central pressure (LCP) for all tracks in the Coral Sea-Showing both the plot of values calculated from TC track data (minimum pressure and latitude, refer Miller et al. (2023) and MLR predicted (shown in blue)

Table 5
LMI test dataset statistics for leave-out-one-cross validation (shown per model) Moum et al. 2016).As an example of this phenomenon, a numerical experiment on TC Olga(West Australia, 2000)

Table 6
Results: PDI MLR statistics for TC tracks in the Coral Sea, 1978-2020.Presented for all tracks and by cluster r is Pearson correlation, RMSE is root mean square error The independent variables are not correlated with each other.• No autocorrelation between the observations of each variable.The models were tested for independence of observations using the Durbin-Watson statistic.Detected autocorrelation was then accounted for by reporting of the Newey-West (Newey and West 1987) adjusted standard errors of the covariant coefficients.

Table 7
Results: LMI(LCP) MLR statistics for TC tracks in the CoralSea, 1978Sea,   -2020Presented for all tracks and by cluster.Correlations are Pearson correlations, RMSE is root mean square error