You can certainly calculate the correlation between two time series. That's a short answer.
When, as true here and as true often, there is a marked trend in both cases, the correlation is likely to be extremely high. In general, it's not especially helpful. It's not as if there was serious doubt that there would be an apparent association; that's easily imagined from looking at the graphs of time series and thinking about the corresponding scatter plot. The P-value from conventional calculations is certainly not applicable, as independence of observations clearly does not hold.
The correlation throws absolutely no light on questions of process or causation. It's just a descriptive measure of strength of linear association.
What appear to be the same or similar data as in the question appear at How to interpolate a variable with frequency of 5 years to annual data? As an exercise I calculated the correlation between the variables parea
and urea
there as 0.9957; and between their logarithms as 0.9911.
In fact, many of the classic examples of high but spurious correlations arise from situations where two time series both show marked trends, but for quite different reasons, including apocryphally the price of rum and the number of Methodist ministers. Here there seems likely to be substantive association, but that's not the main question.
Best Answer
Your understanding of Spearman's rank-correlation measure seems wrong. There is no monotonicity assumption in applying it. In fact it was designed exactly for the purpose of measuring how monotonic the relationship is: if it is 1 (resp. -1), then a higher x value means a higher (resp. lower) y value.
The closer the value is at 0, the more you can say "the relationship (if any) is non-monotonic".
Now if you want to distinguish a weak non-monotonic relationship from a strong non-monotonic relationship, we need to get a little bit creative: One option is to compare the square of Spearman's rank correlation with the R-squared from a rank-regression with non-linear parts such as squares or splines: $$ \text{rank}(y_i) = \alpha + \beta_1 \text{rank}(x_i) + \beta_2 \text{rank}(x_i)^2 + \varepsilon_i $$ A low squared Spearman's rank correlation but high R-squared from such regression indicates a strong non-monotonic relationship.
A small example for illustration:
So here, we would speak of a strong non-linear relationship. A quick look at the scatter plot: