Revisiting the zonally asymmetric extratropical circulation of the Southern Hemisphere spring using complex empirical orthogonal functions

1 Abstract

The large-scale extratropical circulation in the Southern Hemisphere is much more zonally symmetric than that of the Northern Hemisphere, but its zonal departures, albeit highly relevant for regional impacts, have been less studied. In this study we analyse the joint variability of the zonally asymmetric springtime stratospheric and tropospheric circulation using Complex Empirical Orthogonal Functions (cEOF) to characterise planetary waves of varying amplitude and phase. The leading cEOF represents variability of a zonal wave 1 in the stratosphere that correlates slightly with the Symmetric Southern Annular Mode (S-SAM). The second cEOF (cEOF2) is an alternative representation of the Pacific-South American modes. One phase of this cEOF is also very highly correlated with the Asymmetric SAM (A-SAM) in the troposphere. Springs with an active ENSO tend to lock the cEOF2 to a specific phase, but have no consistent impact on its magnitude. Furthermore, we find indications that the location of Pacific Sea Surface Temperature anomalies affect the phase of the cEOF2. As a result, the methodology proposed in this study provides a deeper understanding of the zonally asymmetric springtime extratropical SH circulation.

2 Introduction

The large-scale extratropical circulation in the Southern Hemisphere (SH) is much more zonally symmetric than that of the Northern Hemisphere, but departures from the zonal mean are associated with regional impacts (e.g. Hoskins and Hodges 2005). They strongly modulate weather systems and regional climate through promoting longitudinally varying meridional transport of heat, humidity, and momentum (K. E. Trenberth 1980; M. N. Raphael 2007) and could even be related to the occurrence of high-impact climate extremes (Pezza, Rashid, and Simmonds 2012).

The zonally asymmetric circulation is typically described by the amplitude and phase of zonal waves obtained by Fourier decomposition of geopotential height or sea-level pressure at each latitude (e.g. van Loon and Jenne 1972; K. E. Trenberth 1980; Turner et al. 2017). This approach suggests that zonal waves 1 and 3 explain almost 99% of the total variance in the annual mean 500 hPa geopotential height zonal anomalies at 50ºS (van Loon and Jenne 1972). K. F. Trenberth and Mo (1985) concluded that wave 3 plays a role in the development of blocking events. In addition, previous works have identified wave-like patterns with dominant wavenumbers 3-4 at extratropical and subpolar latitudes with distinctive regional impacts. M. N. Raphael (2007) showed that variability in the planetary wave 3 projected onto its climatological location is associated with anomalies in the Antarctic sea-ice concentration.

Fourier decomposition relies on the assumption that the circulation can be meaningfully described in terms of zonal waves of constant amplitude along a latitude circle. However, this is not valid for meridionally propagating waves or zonal waves with localised amplitudes. Addressing this limitation, the Fourier technique can be generalized to integrate all planetary wave amplitude regardless of wave number by computing the wave envelope (Irving and Simmonds 2015). The wave envelope can represent planetary waves with different amplitude at different longitudes, but lacks information about phase and wave number. Using this method, Irving and Simmonds (2015) showed that planetary wave amplitude in general is associated with Antarctic sea-ice concentration and temperature, as well as to precipitation anomalies in regions of significant topography in SH mid-latitudes and Antarctica.

Another extensively-used approach to characterise the SH tropospheric circulation anomalies, is computing Empirical Orthogonal Functions (EOF, also known as Principal Component Analysis). Within the EOF framework, the Southern Annular Mode (SAM) appears as the leading mode of variability of the SH circulation (Fogt and Marshall 2020). Spatially, the SAM is characterised by a centre of geopotential anomalies over Antarctica surrounded by anomalies of the opposite sign at middle latitudes. Embedded in this zonally symmetric structure is a wave 3 pattern that is more prominent in the Pacific sector. The 2nd and 3rd EOFs, usually known as Pacific–South American Patterns (PSA) 1 and PSA2 patterns, respectively, describe meridionally propagating wave trains that originate in the eastern equatorial Pacific and Australian-Indian Ocean sector, and travel towards the South Atlantic following a great-circle arch along the Antarctic coast (Mo and Paegle 2001). These patterns influence precipitation anomalies in South America (Mo and Paegle 2001). Although these patterns are usually derived by applying EOF to temporal anomalies, M. Raphael (2003) also applied EOF methods specifically to zonal anomalies. Irving and Simmonds (2016) proposed a novel methodology for objectively identifying the PSA pattern using Fourier decomposition. More recently Goyal et al. (2022) created an index of amplitude and phase of zonal wave 3-like variability by combining the two leading EOFs of meridional wind anomalies.

Some of the zonally asymmetric patterns of the SH circulation variability described previously appear to have experienced secular changes. For instance, M. Raphael (2003) suggests that the amplitude of the zonal wave 1 experienced a large increase and that the zonal wave 3 experienced changes in its annual cycle between 1958 and 1996. However, little is known yet about variability and trends of these patterns.

Patterns resulting from EOF analysis are more flexible than Fourier decomposition-derived modes in that they can capture oscillation patterns that cannot be characterised by purely sinusoidal waves with constant amplitude. Nonetheless, they are restricted to standing oscillation modes and cannot properly represent propagating or phase-varying features such as zonal waves. A single EOF can also represent a mixture of two or more physical modes.

A third methodology commonly used to describe circulation anomalies consists on identifying particular features of interest and creating indices using simple methods such as averages and differences. Examples of this methodology are the SAM Index of Gong and Wang (1999), the SH wave 3 activity index defined by M. N. Raphael (2004) and the SH zonally asymmetric circulation index from Hobbs and Raphael (2010). These derived methods are grounded on other methods such as Fourier decomposition or EOF to identify the centres of action for the described phenomena and can be useful to characterise features that are not readily apparent with these methods. These kinds of indices are generally easy to compute, but they usually do not capture non-stationary patterns.

An alternative methodology that has been proposed to study travelling and standing waves is complex Empirical Orthogonal Functions (cEOF; Horel (1984)). This method extends EOF analysis to capture oscillations with varying amplitude and phase and has been applied to the time domain. For instance, Krokhin and Luxemburg (2007) applied cEOF to station-based monthly precipitation anomalies and monthly temperature anomalies in the Eastern Siberia and the Far East region to characterise the main modes of variability and their relationship with teleconnection indices. Similarly, Gelbrecht, Boers, and Kurths (2018) applied cEOF to daily precipitation from reanalysis to study the propagating characteristics of the South American Monsoon. To our knowledge, cEOF analysis has not been applied in the spatial domain to capture the phase-varying nature of planetary waves in the atmosphere.

The general goal of this study is to improve the description and understanding of the zonally asymmetric extratropical SH circulation using cEOF, which can describe phase varying planetary waves with variable amplitude along a latitude circle. In addition, we try to expand the knowledge of the simultaneous behaviour of SH asymmetric circulation in the troposphere and the stratosphere.

We restrict this work to the September-October-November (SON) trimester. During this season the tropical teleconnections over South America are maximised (Cazes-Boezio, Robertson, and Mechoso 2003), and the SH zonal winds associated with the stratospheric polar vortex increase to peak in October and extend downward after that (Lim, Hendon, and Thompson 2018).

In Section 3 we describe the methods. In Section 4.1 we analyse the spatial patterns of each complex EOF. In Section 4.2 we study the spatial regressions with geopotential height, temperature, and ozone anomalies. In Section 4.3 and 4.4 we analyse the relationship between cEOF2, the PSA and SAM modes. In Section 4.5 we study tropical forcings that explain the variability of each cEOF. In Section 4.6 we show the relationship between these modes of variability and precipitation and surface temperature anomalies in South America and Oceania. In Section 5 we compare our results with previous studies and discuss the benefits of our methodology.

3 Data and Methods

3.1 Data

We used monthly geopotential height, air temperature, ozone mixing ratio, and total column ozone (TCO) at 2.5º longitude by 2.5º latitude of horizontal resolution and 37 vertical isobaric levels from the European Centre for Medium-Range Weather Forecasts Reanalysis version 5 [ERA; Hersbach et al. (2019)] for the period 1979 – 2019. Most of our analysis is restricted to the post-satellite era to avoid confounding factors arising from the incorporation of satellite observations, but we also used the preliminary back extension of ERA5 from 1950 to 1978 (Bell et al. 2020) to describe long-term trends. We derived streamfunction at 200 hPa from ERA5 vorticity using the FORTRAN subroutine FISHPACK (Adams, Swartztrauber, and Sweet 1999) and we computed horizontal wave activity fluxes following Plumb (1985). Sea Surface Temperature (SST) monthly fields are from Extended Reconstructed Sea Surface Temperature (ERSST) v5 (Huang et al. 2017) and precipitation monthly data from the CPC Merged Analysis of Precipitation (CMAP, P. Xie and Arkin 1997), with a 2º and 2.5º horizontal resolution, respectively. The rainfall gridded dataset is based on information from different sources such as rain gauge observations, satellite inferred estimations and the NCEP-NCAR reanalysis, and it is available from 1979 to the present.

The Oceanic Niño Index (ONI, Bamston, Chelliah, and Goldenberg 1997) comes from NOAA’s Climate Prediction Center and the Dipole Mode Index (DMI, Saji and Yamagata 2003) from Global Climate Observing System Working Group on Surface Pressure.

3.2 Methods

The study is restricted to the spring season, defined as the September-October-November (SON) trimester. We compute seasonal means for the different variables, averaging monthly values weighted by the number of days in each month. We use the 200 hPa level to represent the upper troposphere and 50 hPa to represent the lower stratosphere.

We computed the amplitude and phase of the TCO wave 1 by averaging (area-weighted) the data of each SON between 75°S and 45°S, and then extracting the wave-1 component of the Fourier spectrum. We chose this latitude band because it is wide enough to capture most of the relevant anomalies of SH mid-latitudes.

We computed the level-dependent SAM index as the leading EOF of year-round monthly geopotential height anomalies south of 20ºS at each level for the whole period (Baldwin and Thompson 2009). We further split the SAM into its zonally symmetric and zonally asymmetric components (S-SAM and A-SAM indices respectively) following Campitelli, Díaz, and Vera (2022a). The method consists in first computing the leading EOF of monthly geopotential height anomalies at each level and then computing the zonal mean and the zonal anomalies from its spatial pattern. We then project each level’s monthly geopotential height fields onto the corresponding EOF field, the zonally symmetric field and the zonally asymmetric fields to obtain time series corresponding to the full SAM, the symmetric SAM and the asymmetric SAM, respectively.

Seasonal indices of the PSA patterns (PSA1 and PSA2) were calculated, in agreement with Mo and Paegle (2001), as the third and fourth EOFs of seasonal mean anomalies for 500-hPa geopotential heights at SH.

Linear trends were computed by Ordinary Least Squares (OLS) and the 95% confidence interval was computed assuming a t-distribution with the appropriate residual degrees of freedom (D. Wilks 2011).

3.3 Complex Empirical Orthogonal Functions (cEOF)

Figure 3.1: Spatial patterns of the four leading EOFs of SON geopotential height zonal anomalies at 50 hPa south of 20º S for the 1979 – 2019 period (arbitrary units).

In the standard EOF analysis, zonal waves may appear as pairs of (possibly degenerate) EOFs representing similar patterns but shifted in phase (Horel 1984). Figure 3.1 shows the four leading EOFs of SON geopotential height zonal anomalies at 50 hPa south of 20º S. It is clear that the first two EOFs represent a single phase-varying zonal wave 1 pattern and the last two represent a similarly phase-varying pattern with higher wavenumber and three centres of action shifted by 1/4 wavelength (90º in frequency space).

To describe the phase-varying nature of these two wave patterns, one way is to combine each pair of EOFs into indices of amplitude and phase. So, for instance, the amplitude of the wave 1-like EOF could be measured as \(\sqrt{\mathrm{PC1}^2 + \mathrm{PC2}^2}\) and its phase as \(\tan^{-1} \left ( \frac{\mathrm{PC2}}{\mathrm{PC1}} \right )\) (where \(\mathrm{PC1}\) and \(\mathrm{PC2}\) are the time series associated with each EOF). However, this rests on visual inspection of the spatial patterns and only works properly if both phases appear clearly in different EOFs, which is not guaranteed by construction. In particular, this does not work with the wave 1 pattern depicted as the leading EOF in 200 hPa geopotential height zonal anomalies (not shown).

On the other hand, a better alternative for describing phase-varying waves is to use Complex Empirical Orthogonal Functions (cEOF) analysis (Horel 1984). Each cEOF is a set of complex-valued spatial patterns and time series. The real and imaginary components of the complex spatial pattern can be thought of as representing two spatial patterns that are shifted by 1/4 wavelength by construction, similar to EOF1 and EOF2 in Figure 3.1. In this paper we use the term 0º cEOF and 90º cEOF to refer to each part of the whole cEOF. The actual field reconstructed by each cEOF is then the linear combination of the two spatial fields weighted by its respective time series. This is analogous to how any sine wave can be constructed by the sum of a sine wave and cosine wave with different amplitude but constant phase. This means that cEOFs naturally represent phase-varying wave-like patterns that change location as well as amplitude.

For instance, when the phase of the wave matches the 0º phase, then the 0º phase time series is positive, and the 90º phase time series is zero. Similarly, when the phase of the wave matches the 90º phase, the 90º phase time series is positive, and the 0º phase time series is zero. The intermediate phases have non-zero values in both time series.

In traditional EOFs, the resulting modes are not unique, and instead are defined up to sign, which corresponds to a rotation in the complex plane of either 0 or \(\pi\). In the same way, cEOFs are defined up to a rotation in the complex plane of any value between 0 and \(2\pi\) (Horel 1984).

cEOFs are computed in the same way as traditional EOFs except that the data is first augmented by computing its analytic signal. This is a complex number whose real part is the original series and whose imaginary part is the original data shifted by 90º at each spectral frequency – i.e. its Hilbert transform. The Hilbert transform is usually understood in terms of time-varying signal. However, in this work we apply the Hilbert transform at each latitude circle, level, and at each SON mean (i.e. the signal only depends on longitude). Since each latitude circle is a periodic domain, this procedure does not suffer from edge effects.

We first applied cEOF analysis to geopotential height zonal anomalies south of 20ºS at 50 and at 200 hPa. Figure 4.1 a.1 shows the spatial patterns of the two leading cEOF. The 0º phase is plotted with shaded contours and the 90º phase, with black contours. The two phases of the leading cEOF are very similar to the two leading EOFs shown in Figure 3.1 and represent a zonal wave 1 pattern; the 0º phase is roughly the EOF1 and the 90º phase is roughly the EOF2).

Table 3.1: Coefficient of determination (\(r^2\)) between the time series of the absolute magnitude of complex EOFs computed separately at 200 hPa and 50 hPa (p-values lower than 0.01 in bold).

	50 hPa
200 hPa	cEOF1	cEOF2	cEOF3
cEOF1	0.29	0.01	0.03
cEOF2	0.00	0.59	0.02
cEOF3	0.00	0.00	0.01

Table 3.1 shows the coefficient of determination between time series of the amplitude of each cEOF across levels. There is a high degree of correlation between the magnitude of the respective cEOF1 and cEOF2 at each level. The spatial patterns of the 50 hPa and 200 hPa cEOFs are also similar (not shown).

Both the spatial pattern similarity and the high temporal correlation of cEOFs computed at 50 hPa and 200 hPa suggest that these are, to a large extent, modes of joint variability. This motivates the decision of performing cEOF jointly between levels. Therefore cEOFs were computed using data from both levels at the same time. In that sense each cEOF has a spatial component that depends on longitude, latitude and level, and a temporal component that depends only on time.

Because we are computing the cEOFs of zonal anomalies and not temporal anomalies, the cEOFs need to account for the time-mean zonal anomalies. These will tend to be represented by the leading cEOF, which therefore will have a non-zero temporal mean.

As mentioned before, the choice of phases is arbitrary and equally valid. But to make the interpretation easier, we chose the phase of each cEOF so that either the 0º cEOF or the 90º cEOF is aligned with meaningful variables in our analysis. This procedure does not create spurious correlations, it only takes an existing relationship and aligns it with a specific phase.

Preliminary analysis showed that the first cEOF was closely related to the zonal wave 1 of TCO and the second cEOF was closely related to ENSO. Therefore, we chose the phase of cEOF1 so that the time series corresponding to the 0º cEOF1 has the maximum correlation with the zonal wave 1 of TCO between 75°S and 45°S. Similarly, we chose the phase of cEOF2 so that the coefficient of determination between the ONI and the 0º cEOF2 is minimised, which also nearly maximises the correlation with the 90º cEOF2.

In Section 4.6 we show regressions of precipitation and temperature associated with intermediate phases. For those plots, we rotated the cEOFs by 1/4 wavelength by multiplying the complex time series by \(\cos(\pi/4) + i\sin(\pi/4)\) and computing the regression on those rotated timeseries.

While we compute these complex principal components using data from 1979 to 2019, we extended the complex time series back to the 1950 – 1978 period by projecting monthly geopotential height zonal anomalies standardised by level south of 20ºS onto the corresponding spatial patterns.

We performed linear regressions to quantify the association between the cEOFs and other variables (e.g. geopotential height, temperature, precipitation, and others). For each cEOF, we computed regression maps by fitting a multiple linear model involving both the 0º and the 90º phases. To obtain the linear coefficients of a variable \(X\) with the 0º and 90º phase of each cEOF we fit the equation

\[ X(\lambda, \phi, t) = \alpha(\lambda, \phi) \operatorname{cEOF_{0^\circ}} + \beta(\lambda, \phi) \operatorname{cEOF_{90^\circ}} + X_0(\lambda, \phi) + \epsilon(\lambda, \phi, t) \]

where \(\lambda\) and \(\phi\) are the longitude and latitude, \(t\) is the time, \(\alpha\) and \(\beta\) are the linear regression coefficients for 0º and 90º phases respectively, \(X_0\) and \(\epsilon\) are the constant and error terms respectively.

We evaluated statistical significance using a two-sided t-test and, in the case of regression maps, p-values were adjusted by controlling for the False Discovery Rate (Benjamini and Hochberg 1995; D. S. Wilks 2016) to avoid misleading results from the high number of regressions (Walker 1914; Katz and Brown 1991).

3.4 Computation procedures

We performed all analysis in this paper using the R programming language (R Core Team 2020), using data.table (Dowle and Srinivasan 2020) and metR (Campitelli 2020) packages. All graphics are made using ggplot2 (Wickham 2009). We downloaded data from reanalysis using the ecmwfr package (Hufkens 2020) and indices of ENSO and Indian Ocean Dipole (IOD) with the rsoi package (Albers and Campitelli 2020). The paper was rendered using knitr and rmarkdown (Y. Xie 2015; Allaire et al. 2020).

4 Results

4.1 cEOF spatial patterns

Figure 4.1: Spatial patterns for the two leading cEOFs of SON geopotential height zonal anomalies at 50 hPa and 200 hPa for the 1979 – 2019 period. The shading (contours) corresponds to 0º (90º) phase. Arbitrary units. The proportion of variance explained for each mode with respect to the zonal mean is indicated in parenthesis.

Figure 4.2: Time series of the two leading cEOFs of SON geopotential height zonal anomalies at 50 hPa and 200 hPa. cEOF1 (row a) and cEOF2 (row b) separated in their 0º (column 1) and 90º (column 2) phase. Dark straight line is the linear trend. Black horizontal and vertical line mark the mean value and range of each time series, respectively.

To describe the variability of the circulation zonal anomalies, the spatial and temporal parts of the first two leading cEOFs of zonal anomalies of geopotential height at 50 hPa and 200 hPa, computed jointly at both levels, are shown in Figures 4.1 and 4.2. The first mode (cEOF1) explains 82% of the variance of the zonally anomalous fields, while the second mode (cEOF2) explains a smaller fraction (7%). In the spatial patterns (Fig. 4.1), the 0º and the 90º phases are in quadrature by construction, so that each cEOF describe a single wave-like pattern whose amplitude and position (i.e. phase) is controlled by the magnitude and phase of the temporal cEOF. The wave patterns described by these cEOFs match the patterns seen in the standard EOFs of Figure 3.1.

The cEOF1 (Fig. 4.1 column 1) is a hemispheric wave 1 pattern with maximum amplitude at high latitudes. At 50 hPa the 0º cEOF1 has the maximum of the wave 1 at 150ºE and at 200 hPa, the maximum is located at around 175ºE indicating a westward shift with height. The cEOF2 (Fig. 3.1 column 2) shows also a zonal wave-like structure with maximum amplitude at high latitudes, but with shorter spatial scales. In particular, the dominant structure at both levels is a wave 3 but with larger amplitude in the pacific sector. There is no apparent phase shift with height but the amplitude of the pattern is greatly reduced in the stratosphere, which is consistent with the the fact that the cEOF2 computed separately for 200 hPa explains a bit more variance than the cEOF2 computed separately for 50 hPa (11% vs. 3%, respectively). This suggest that this barotropic mode represents mainly tropospheric variability.

There is no significant simultaneous correlation between cEOFs time series. Both cEOFs show year-to-year variability but show no evidence of decadal variability (Fig. 4.2). The 0º cEOF has a non-zero temporal mean which, as discussed in Section 3.3, is due to the fact that the temporal mean of zonal anomalies need to be captured by the cEOFs. The other indices have almost zero temporal mean, which indicates that only cEOF1 includes variability that significantly projects onto the mean zonal anomalies. This is consistent with the fact that the mean zonal anomalies of geopotential height are very similar to the cEOF1 (\(r^2\) = 98%) and not similar to the cEOF2 (\(r^2\) = 0%).

A significant positive trend in the 0º phase of cEOF1 is evident (Fig. 4.2a.1, p-value = 0.0037) while there is no significant trend in any of the phases of cEOF2. The positive trend in the 0º cEOF1 translates into a positive trend in cEOF1 magnitude, but not systematic change in phase (not shown). This long-term change indicates an increase in the magnitude of the high latitude zonal wave 1.

4.2 cEOFs Regression maps

4.2.1 Geopotential

In the previous section, cEOF analysis was applied to zonal anomalies derived by removing the zonally mean values in order to isolate the main characteristics of the main zonal waves characterizing the circulation in the SH. In this section we compute regression fields using the full fields of the variables in order to describe the influence of the cEOFs on the temporal anomalies.

Figure 4.3: Regression of SON geopotential height anomalies (\(m^2s^{-1}\)) with the (column 1) 0º and (column 2) 90º phases of the first cEOF for the 1979 – 2019 period at (row a) 50 hPa and (row b) 200 hPa. These coefficients come from multiple linear regression involving the 0º and 90º phases. Areas marked with dots have p-values smaller than 0.01 adjusted for False Detection Rate.

Figure 4.3 shows regression maps of SON geopotential height anomalies upon cEOF1. At 50 hPa (Figure 4.3 row a), the 0º cEOF1 is associated with a positive centre located over the Ross Sea. The correlation between the 0º cEOF1 and the zonal mean zonal wind at 60ºS and 10hPa is -0.59 (CI: -0.76 – -0.35), indicating a moderate relationship with the SON stratospheric jet. The 90º cEOF1 is associated with a distinctive wave 1 pattern with maximum over the coast of East Antarctica. At 200 hPa (Figure 4.3 row b) the 0º cEOF1 shows a single centre of positive anomalies spanning West Antarctica surrounded by opposite anomalies in lower latitudes, with its centre shifted slightly eastward compared with the upper-level anomalies. The 90º cEOF1 shows a much more zonally symmetrical pattern resembling the negative SAM phase (e.g. Fogt and Marshall 2020). Therefore, the magnitude and phase of the cEOF1 are associated with the magnitude and phase of a zonal wave mainly in the stratosphere.