A simple method, useful both for exploration and hypothesis testing, takes the Durbin Watson statistic and the semivariogram as its point of departure: denoting the sequence of residuals by $e_t$ (with $t$ the "index" of the plot), compute
$$\gamma(t) = \frac{1}{2}(e_{t+1}-e_t)^2.$$
The data (as presented in the question) result in this plot:

(The residual values have been uniformly rescaled compared to what is shown on the vertical axis in the original plot. This does not affect the subsequent analysis.) Typically there will be scatter, as shown here. To visualize this better, smooth it:

This plot is on a square root scale (let's call this the "dispersion") to dampen the extreme oscillations. The wiggly blue lines are a modest smooth (medians of 3 followed by a lag-13 moving average). The solid red line is an aggressive smooth of that. (Here, it's a Gaussian convolution; in general, though, I would recommend a Lowess smooth of $\sqrt{\gamma(t)}$.) Together, these lines trace out detailed fluctuations in dispersion and intermediate-range trends, respectively. The yellow and green thresholds, shown for reference, delimit (a) the lowest smoothed dispersion in the first 225 indexes and (b) the mean smoothed dispersion from index 226 onwards. I chose the changepoint of 225 by inspection.
Clearly almost all the smoothed dispersion values above 6 occurred during the first 225 indexes. From index=225 to index=350 or so, the smoothed dispersions decrease and then stay constant around 2.7.
The change in dispersion is now visually obvious. For most purposes--where the change is so clear and strong--that's all one needs. But this approach lends itself to formal testing, too. To identify a point where the dispersion changed, and to estimate the uncertainty of that point, apply a sequential online changepoint procedure to the sequence $(\gamma(t))$. An even more straightforward check, when you suspect there has been just one major change in volatility, is to use regression to check the dispersion for a trend. If one exists, you can reject the null hypothesis of homogeneous volatility.