methodology_pwi.tex

\subsubsection{Index of planetary wave activity}

To define and calculate an appropriate metric of planetary wave activity, $E(t,\lambda,\phi)$ was calculated from the 500 hPa meridional wind. The latitudinal dimension of $E(t,\lambda,\phi)$ was eliminated by calculating the meridional maximum over the range 40-70$^{\circ}$S, and then the zonal median was taken to eliminate the longitudinal dimension and arrive at a single `Planetary Wave Index' (PWI) value for each data time. Since wavenumbers 1-9 were retained during the process (i.e. essentially all wavenumbers), the PWI represents an integrated measure of the `waviness' of the hemispheric circulation.

The meridional wind was used in calculating the PWI because it fundamentally reflects the presence of waves in the zonal flow. If the flow is purely zonal there are no waves and $v = 0$, while the magnitude of $v$ reflects the activity of the waves. The meridional wind is also directly involved with meridional transports of heat and moisture, which impacts directly on surface temperature, precipitation and sea ice. In fact, many studies have shown that $v$ (either filtered or unfiltered) contains much more dynamic information about synoptic processes than alternatives like the geopotential height or streamfunction \citep[e.g.][]{Berbery1996,Hoskins2005,Petoukhov2013}, neither of which has a \textit{direct} involvement with meridional exchanges. 

While better suited to the purposes of our study, the selection of $v$ has important implications for the Fourier analyses we present. From the geostrophic relation we know that $v \propto dZ / dx$, which means that $Z$ tends to be dominated by longer wavelengths (or smaller wavenumbers) than $v$. In particular, since $Z$ is a sinusoidal function of $x$ in Fourier space, it follows that $v_k \propto k Z_k$ for any given wavenumber $k$, meaning more of the variance in $v$ is explained by the synoptic (and shorter) waves than it is for $Z$. This is an important distinction that is discussed further in the Results section. Besides the selection of the meridional wind, a number of other factors were taken into consideration in devising this methodology:
\begin{itemize}
\item The results show little sensitivity to the choice of atmospheric level because planetary waves are equivalent barotropic. 500 hPa was selected as it represents a mid-to-upper tropospheric level that is below the tropopause in all seasons and at all latitudes of interest.
\item The wave envelope is slightly smoother if wavenumbers greater than 9 are left out of the Hilbert transform, but otherwise the result is not appreciably different from when all wavenumbers are retained.
\item The meridional maximum (over 40-70$^{\circ}$S) was taken to allow for slight north/south variations in the mean latitude of planetary wave activity and also for the fact that the waveform is not perfectly zonally oriented. 
\item The zonal median (as opposed to the mean or integral) was taken to guard against large values in one part of the hemisphere overly influencing the end result.
\end{itemize}

In order to be consistent with much of the existing literature, the majority of the analysis focuses on the monthly timescale. Monthly mean data were obtained by applying a 30-day running mean to the daily (i.e. diurnally averaged) ERA-Interim data, so as to maximize the monthly information available from the dataset. As noted by previous authors \citep[e.g.][]{Kidson1988}, potentially useful information may be lost if only twelve (i.e. calendar month) samples are taken every year. Dates are labeled as the 16th day of the 30-day period (e.g. the labeled date 1979-01-16 spans the period 1979-01-01 to 1979-01-30).