This study uses ERA-Interim (ERA-I) reanalysis data (Dee et al., 2011) provided by the European Centre for Medium-Range Weather Forecasts on a 1∘×1∘ spatial grid and 6-hourly temporal resolution for 1979–2018. The datasets used are meridional wind velocity, 2 m temperature (T2m), and PV. Daily-maximum 2 m temperature is derived from T2m data, and the anomalies in daily-maximum T2m data are calculated with respect to the day-of-year mean for the period 1979–2018. The datasets are freely available to download from https://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=pl/ (last access: 25 September 2019). PV fields are used as it is for calculating the blocking fields. However, for rest of the analysis, the PV fields are multiplied by minus one, which implies that negative (positive) PV anomalies represent anticyclones (cyclones) similar to the NH.

The R metric, developed by Röthlisberger et al. (2019), is used to quantify the recurrence of synoptic-scale Rossby wave packets. For the SH, we use the same R-metric data as in Ali et al. (2021). First, 6-hourly meridional winds at 250 hPa are averaged between 35 and 65∘ S as vma(λ,t). To the resulting longitude–time data vma(λ,t), a 14.25 d (day) running mean, is applied to isolate signals with timescales longer than the synoptic timescale. This results in a longitude–time field of temporally smoothed, meridionally averaged 250 hPa meridional wind vtf(λ,t). The envelope of the synoptic wavenumber contribution to the time-filtered vtf(λ,t) is extracted following Zimin et al. (2003) as follows: the vtf(λ,t) is transformed into the frequency domain for each t using a fast Fourier transform over longitude, yielding Fourier coefficients v^tf(k,t) for zonal wavenumber k at the time step t. Finally, an inverse Fourier transform is applied to calculate the envelope of the wave while only considering contributions from a selected band of synoptic wavenumbers k= 4–15. Thus, R(λ,t) for each longitude λ and time t is calculated as 1R(λ,t)=2∑k=4k=15v^tf(k,t)e2πiklλ/N, where k is the wavenumber, lλ denotes the longitudinal grid point index for longitude λ, and N=360 denotes the number of longitudinal grid points.

In most cases, large values of R reliably identify situations in which amplified waves (of distinct wave packets) recur in the same phase. However, the definition of R does not contain a criterion for recurrence of distinct wave packets. Thus, in a few cases, high values of R over a few days may result from stationary synoptic-scale troughs or ridges (see Röthlisberger et al., 2019, for a discussion on the R metric). Figure A1 shows the day-of-year climatology of the R metric in the Southern Hemisphere and compares it to that of the Northern Hemisphere. The code for calculating the R metric is freely available (see “Code and data availability”).

For the phase–amplitude information used in Sect. 3.3, it is extracted using the Fourier decomposition along the longitude of meridionally averaged (35 and 65∘ S) 250 hPa meridional wind vma(λ,t) as used for calculating the R metric above. After applying the fast Fourier transform, one obtains Fourier coefficients in the form of complex numbers v^(k,t). Plotting the complex number on a complex plane, i.e. real vs. imaginary (Img) axis, provides information on the phase and amplitude at a given time step t for a particular wavenumber k.

Atmospheric blocking data are computed following the methodology of Schwierz et al. (2004) as in Rohrer et al. (2020) and Lenggenhager and Martius (2019). The detection scheme identifies persistent anticyclonic PV anomalies vertically averaged (VAPV) between 500 and 150 hPa vertical levels. First, the VAPV anomaly is computed from the 30 d running mean climatology of the corresponding time step of the year for the years 1979–2018. An additional 2 d running mean filter is applied to smooth out high-frequency transients. Then the algorithm identifies areas with VAPV ≥1.3 PVU (potential vorticity unit) in the SH. The identified areas having a persistence criterion of 5 d and a minimum overlap of 0.7 between consecutive time steps are classified as blocks. Blocking fields identified with this algorithm are available at 6-hourly temporal resolution and 1∘×1∘ spatial resolution. We tested the blocking fields with a less stringent threshold of VAPV ≥1.0 PVU for the two case studies and did not find blocking directly over SEA. The code used to calculate blocks is available on GitHub (see “Code and data availability”).

A station-based temperature dataset is used to identify extreme and persistent heatwaves in SEA. Following the methods developed in Parker et al. (2014a) and refined in Quinting and Reeder (2017), heatwaves in SEA in December–February (DJF) are detected from temperatures observed at the Australian Bureau of Meteorology's (BoM) monitoring stations (Fig. 1). The BoM's Australian Climate Observations Reference Network – Surface Air Temperature (ACORN-SAT, available at http://www.bom.gov.au/climate/data/acorn-sat/#tabs=ACORN-SAT, last access: 1 May 2020) is a high-quality temperature dataset used to monitor long-term temperature trends from 112 weather stations across Australia. The dataset provides a daily-maximum temperature (TMAX) for each station. These TMAXs are extracted for stations in SEA as defined here for DJF from 1979 to 2018. The 90th-percentile TMAX (T90) is then calculated for each station for each month in DJF. A heatwave is defined as any period of at least 4 consecutive days for which the TMAXs at three or more stations equal or exceed the T90 for that station and month. From here on, the term heatwave refers to the heatwave in SEA. This criterion results in 57 heatwaves, on average 8 d long, with the most prolonged heatwave lasting 22 d starting in December 1990. Note that the heatwave identification scheme aims to identify the most intense and persistent heatwaves in SEA and thus serves a different purpose than the hot-spell identification scheme described in the next section. Following Parker et al. (2014a), a day that is part of the SEA heatwaves is termed a SEA heatwave day (SEA HD). For evaluating the co-occurrence of SEA HDs with RRWP conditions, high-RSEA days are defined as days exceeding the 90th percentile of the daily-mean R averaged over the longitudinal extent of SEA (between 130 and 153∘ E). The 90th-percentile threshold is a subjectively chosen threshold consistent with the threshold for TMAX. A sensitivity test with a threshold of 85th percentile did not change the conditional probability reported in Sect. 3.3 and Table C1.

Hot spells are identified for all SH grid points between 20 and 70∘ S for 1980–2016 using 2 m temperatures (T2M) from the ERA-I fields at 6-hourly temporal resolution and 1∘ spatial resolution. The hot-spell definition follows that of Röthlisberger et al. (2019), in which a hot spell is calculated for each grid point as consecutive values exceeding the 85th percentile from the linearly detrended T2M fields. Spells separated by less than a day are merged to form a single uninterrupted spell. Spell durations of less than 36 h are excluded from further analysis. Contrary to the SEA heatwave identification scheme, the hot-spell identification scheme aims to identify many warm periods that are not necessarily overly extreme, which can then be used for statistical analyses of the factors that determine the duration of these events. This statistical analysis (see next section) will be used to quantify the effect of RRWPs on the persistence of hot surface weather. To ensure a large sample size for robust statistical results, we identify hot spells for the period of November to April. Figure 2a shows the spatial distribution of the number of hot spells at each grid point between 20 and 70∘ S. Over land, many hot spells are seen over parts of SEA, South Africa, and South America, having 350 or more spells. The 95th percentile for hot-spell duration varies from 6 d to more than 2 weeks (Fig. 2b). Over SEA, the 95th-percentile duration varies from a week to roughly 2 weeks.

To quantify the effect of RRWPs on the persistence of hot surface weather, we extend an analysis from Röthlisberger et al. (2019) to the SH, including SEA, using the same statistical model setup, a Weibull regression model. This model allows us to model the distribution of the duration of hot spells at each grid point. An advantage of the model of Röthlisberger et al. (2019) is that we do not need to subjectively define the duration of a “significant” spell because the model quantifies the changes in all quantiles of the spell duration modelled. The null hypothesis tested here at each grid point is that RRWPs have no effect on the duration of hot spells at the respective grid point. The Weibull model is only briefly introduced here. Please refer to Röthlisberger et al. (2019) for further details and their Supplement for a detailed introduction to the Weibull model.

To fit the Weibull model to the observed spell duration distribution, a representative value of the R metric needs to be assigned to each hot spell. This is achieved in the following way: for each hot spell i at grid point g with a duration Dg,i, the raw R metric R(λ,t) is longitudinally averaged within a 60∘ longitudinal sector centred at the grid point g with longitude λg to yield Rlon(λ,t). Then, a median of Rlon(λ,t) is calculated for the lifetime of the hot spell to assign a representative value of R as R̃λg,i for each spell. The model is formulated as 2ln⁡Dg,i=α0,g+α1,gR̃λg,i+∑j=26αj,gmjtg,istart+σgϵg,i;i=1,…,ng.

Here, α0,g is the intercept, α1,g is the regression coefficient for R̃λg, and αj,g,j=2,…,6 are regression coefficients for dummy variables mjtg,istart that take the value 1 if spell i starts in month mj and are zero otherwise. The coefficients αj,g, therefore, account for possible seasonality in the spell duration distribution at grid point g (e.g. longer hot spells in May compared to e.g. September), while σg is a scale parameter and the ϵg,i are error terms. Exponentiated regression coefficients, e.g. exp⁡(α1), are usually referred to as the acceleration factor (AF). The exp⁡(α1) is of particular interest here, as it quantifies the factor of change in all quantiles of the distribution of spell duration distribution at grid point g per unit increase in R̃ (Hosmer et al., 2008; Zhang, 2016; Röthlisberger et al., 2019). An AF >1 implies an increase in all spell duration quantiles with increasing R̃ (i.e. during RRWPs) and conversely for an AF <1.

Furthermore, fitting Eq. (2) to spell durations at all grid points results in a spatial field of the AF. The statistical significance of the AF values is evaluated in a two-step approach. First, a p value for the above null hypothesis is computed exactly as in Zhang (2016). Then, the false-discovery-rate (FDR) test of Benjamini and Hochberg (1995) is applied to the resulting field of p values. The FDR test controls for type I errors, i.e. falsely rejecting the null hypothesis that can occur substantially in analyses like this one where multiple tests are being performed independently from each other at each grid point (e.g. Wilks, 2016). Here we follow the recommendation of Wilks (2016) and allow for a maximum false discovery rate αFDR of 0.1.

The Weibull analysis reveals that RRWPs are significantly correlated with the duration of hot spells in several regions within the SH and including over SEA (Fig. 3). Recall that an AF larger than 1 means that an increase in R is related to an increase in hot-spell duration and conversely for an AF smaller than 1. Thus, several parts of central and southern Australia, including the states of SA, VIC, NSW, and TAS, experience longer hot spells during periods when RRWPs occur. Northern Australia, however, does not show such a correlation with RRWPs, which agrees with previous studies showing different dynamical pathways for northern and southern Australian heatwaves (Risbey et al., 2018; Quinting and Reeder, 2017; Parker et al., 2020). Other statistically significant areas over land include parts of South America: southern Brazil, Bolivia, and parts of Argentina and Chile. For the Northern Hemisphere summer half-year, the significant AFs, larger than 1, form a wavenumber 7 pattern (Röthlisberger et al., 2019). In contrast, no clear wave pattern emerges for the SH in the significant AFs in Fig. 3. The difference in AF patterns between the two hemispheres is consistent with different climatological stationary wave patterns. The spatial pattern in Fig. 3 highlights areas where the transient waves building up the RRWPs have a predominant phasing in summer. In summary, the regression analysis shows that RRWPs are significantly associated with the duration of hot spells in several SH regions over land, including SEA. However, the Weibull analysis does not provide any information about the processes and hence potential causal link between RRWPs and the most intense SEA heatwaves. Accordingly, we next focus on SEA heatwaves and elucidate the role of RRWPs and blocks for two selected cases studies of SEA heatwaves and investigate further co-occurrence of SEA heatwaves and days with high R.

First, we note the co-occurrence of high-RSEA days and SEA heatwave days (SEA HDs) as defined in Sect. 2.3. Out of 352 d with high RSEA, 67 co-occur with SEA HDs, and 285 do not co-occur (Table C1). Thus, the conditional probability of a SEA HD given high RSEA is 0.19 (67/352=0.19), which is higher than the climatology (457/3520=0.13). The conditional probability further increases to 0.34 on filtering out the high-RSEA days containing a cyclonic PV anomaly over SEA (Table C1). Many high-RSEA days do not co-occur with SEA HDs, which indicates that R is not a sufficient condition for SEA heatwaves on its own. We, therefore, further explore why some high-RSEA days co-occur with SEA HDs, while others do not.

High-RSEA days co-occurring with SEA HDs feature a large anticyclonic PV anomaly over SEA (Fig. 8a) on the 350 K isentropic surface. The 2 PVU isoline on the 350 K isentropic surface, indicating the dynamic tropopause, is also located over SEA, thereby indicating a suitable choice of the isentropic surface. Areas upstream and downstream of the anticyclonic PV anomaly over SEA feature cyclonic PV anomalies that are also located equatorward of the highest blocking frequencies (black contours in Fig. 8a). These may correspond to the cyclonic PV anomalies surrounding omega-type blocking or the cyclonic PV anomalies of the dipole blocks. Since blocking is a binary dataset, the blocking frequency in Fig. 8 indicates the percentage of days on which a grid point features a block. Thus, for high-RSEA days co-occurring with SEA HDs, blocks are more frequent over the Indian and the South Pacific oceans close to the Antarctic coast (Fig. 8a) compared to high-RSEA days without co-occurring SEA HDs (Fig. 8b, c) and are less frequent over the 60∘ S latitude, the latitudinal band featuring a high blocking frequency in the DJF climatology (Fig. 8c, f).

In contrast, on high-RSEA days not co-occurring with SEA HDs (Fig. 8b), there is no clear anticyclonic PV anomaly over SEA. Weak zonally elongated PV anomalies are present over the ocean basins, which are co-located with the blocking frequency fields south of South Africa and in the Indian and South Pacific oceans (black contours in Fig. 8b). The difference in the spatial distribution of PV anomalies on the high-RSEA days not co-occurring with SEA HDs and the high-RSEA days co-occurring with SEA HDs suggests that only the RRWPs whose phase is conducive to forming ridges over SEA are important for SEA heatwaves. Furthermore, Fig. D1 shows the PV composite for all SEA HDs.

In addition to the ridge over SEA, circum-hemispheric zonal wavenumber 4 and 5 (WN4, WN5) patterns are present in the composite mean PV fields for high-RSEA days co-occurring on SEA HDs (Fig. 8a), where five distinct highs (negative PV anomalies) and four lows are visible in the 30 to 60∘ latitudinal band. We hypothesize that Rossby waves in a particular phase help to establish the anticyclonic PV anomalies over SEA (in Fig. 8a). Hence, we present Fourier decomposition of the composite mean PV field for high-RSEA days co-occurring with SEA HDs (Fig. 8d, e). To check for a preferred phasing during high-R and SEA HDs, we also present the phase–amplitude distribution of the meridionally averaged meridional wind vma later in Fig. 9.

The WN4 and WN5 components (Fig. 8d, e) of the composite mean PV field for high-RSEA days co-occurring with SEA HDs bring out the role of the phase of the waves in contributing to the anticyclonic anomalies over SEA in Fig. 8a more clearly. Over SEA, both WN4 and WN5 components have the same phase and contribute to the anticyclonic PV anomalies. The north-east and south-west orientation in the WN4 pattern over Australia may be associated with the anticyclonic wave breaking over Australia. In the South Pacific and Indian oceans, we observe cyclonic PV upstream and downstream of the blocking frequency contours (Fig. 8d, e).

Figure 8d and e suggest that high-RSEA days that co-occur with SEA HDs have a particular phase conducive to forming anticyclonic PV anomalies over SEA. Therefore, as stated earlier, we next present the phase–amplitude distribution for WN4 and WN5 components to test this hypothesis in Fig. 9 for the latitudinally averaged (35–65∘ S) daily meridional wind velocity at 250 hPa as vma. We compare the density distribution for days with high RSEA and SEA HDs with high RSEA and non-SEA HDs. Using a Fourier decomposition (see Sect. 2), we extract the WN4 and WN5 components and plot the phase and amplitude density distribution on a complex plane using a Gaussian kernel density estimate (KDE). To determine the amount of smoothening, we used the default bandwidth estimation method, the Scott method. Since we are interested in the qualitative spread of the WN components (e.g. unimodal, bimodal) rather than quantitative estimation of the probability density function, the choice of our smoothening parameters for the KDE is sufficient.

On high RSEA and SEA HDs, the density distribution of the WN4 component in the complex plane is unimodal (Fig. 9a), which points to a preferred phase of the waves. From Fig. 8d, we know that it predominantly forms an anticyclonic PV anomaly over SEA. The density distribution of WN4 for high RSEA and non-SEA HDs has a broader bimodal spread. The distance from the origin of the complex plane represents the amplitude; hence, the peak of the WN4 distribution of high-RSEA days has a higher amplitude (Fig. 9a, b) compared to the DJF climatology whose peak is almost centred at the origin (Fig. 9c). For WN5, the peak of the distribution for days with high RSEA and SEA HDs have a different phase and higher amplitude compared to high RSEA and non-SEA HDs (Fig. 9c, d) and the DJF climatology (Fig. 9e).

The phase distribution for WN4 and WN5 is shown here because they emerge as the dominant patterns in the composite mean (Fig. 8a), whereas the density distributions for other wavenumbers (k=3, 6, 7) do not exhibit a clear difference between high RSEA and SEA HDs compared to high RSEA and non-SEA HDs and DJF climatology (not shown). Overall, our results agree with the understanding of SEA heatwaves featuring upper-level anticyclonic PV anomalies over SEA (Marshall et al., 2014; Parker et al., 2014b; Quinting and Reeder, 2017), and we show how RRWPs in a particular phase (WN4 and WN5) are conducive to forming anticyclones over SEA.