# Stationarity and Autocorrelation of Time Series

I am currently approaching time series analysis and I am trying to find a stationary time series. On this side, I found the Southern Oscillation Index (SOI) data and plotted it in R.

In testing for stationarity, I have used both the graphical analysis and the Augmented Dickey-Fuller test (see images below):

On my side, graphically they look stationary, and it is confirmed by the Augmented Dickey-Fuller (the unit root test). However, I saw something strange in the autocorrelation function:

I see there is autocorrelation (significant) until the 10th lag. I was wondering if this result may affect stationarity. I was thinking that the ADF tells if there is a trend in the series, but what about the autocorrelation must be the same throughout the series and the variance as well? How can I test that?

Thank you so much!