How to measure the time dependent correlation of two signals
Your signals look like they have about the same ordinal scale, and little lag. Hence, you can perform local correlations: take a segment of both signals centered at a time location, you can normalize them in amplitude  and dispersion (divise by mean and standard deviation, or median and median absolute deviation). Apply a smoothing window: if you do not do that, remember you are already applying a rectangular window. Then do a cross-correlation, and repeat over time. 

Such a process creates a 2D image from two 1D signals. It  could be called windowed cross-correlation (see also Windowed cross correlation in Matlab), or local cross-correlation. From the latter:


  We address the problem of capturing and tracking local correlations
  among time evolving time series. Our approach is based on comparing
  the local auto-covariance matrices (via their spectral decompositions)
  of each series and generalizes the notion of linear cross-correlation


If you are unsure about scales, or want of more generic setting, some perform cross-correlation in a multiscale wavelet domain. You can check the concept of wavelet coherence (Matlab), illustrated by Torrence et al. Interdecadal Changes in the ENSO–Monsoon System
, with Matlab software