(1) It does not look like rainfall data. There is both a trend and a yearly cyclical component.
(2) The residual pattern is a result of using linear regression rather than a time series approach
(3) You could capture some of the cyclical pattern - and thus maybe reduce residuals to "white noise" (no pattern) - by introducing dummy variables. Either trigonometric or standard dummies. This approach may be challenging in this instance since (1) your time series is fairly short (2) there is not clear hypothesis (pre-specified concept of cycle length) to generating dummies (usually one uses month, season or quarter etc) (3) trig variables are a tad painful and also assume you have some concept of cycle duration.
(4) But more generally - leaving the time series aspect aside - you're assuming a linear relationship where one clearly does not exist. The residual pattern here is commonly seen when that assumption is violated. Normally a non-linear function would be sufficient (polynomial being the easiest), but that isn't going to be a great way to capture the cyclical nature of data (the data has a specific type of non-linearity).
(5) The model does appear to capturing the overall trend over time.
