# Best measure of similarity between multivariate Gaussian distributions?

I am working to describe differences in covariances between a "baseline" multivariate Gaussian random process $$\mathbf{x}_0$$ and other processes $$\mathbf{x}_i$$. All are discrete and approximated as multivariate Gaussian (they are generated by various climate models). An important characteristic of these processes is that their covariances have steep eigenvalue spectra, meaning that a handful of fixed covariance eigenvector patterns dominate the total process variance, while the remainder are effectively noise in the system (although where to draw that line is not always clear).

So far, I have used mean-removed processes $$\mathbf{x}_0-\left<\mathbf{x}_0\right>$$, $$\mathbf{x}_i-\left<\mathbf{x}_i\right>$$ to compute Kullback-Leibler and Jensen-Shannon divergences and the Mahalanobis distance. But it appears these metrics weight differences for large and small eigenvectors the same way, i.e. a difference in a small-amplitude noise process will contribute to metric size as much as a proportional difference in a large-amplitude, presumably physics-based process.

Is there another metric (Bhattacharyya, Helliman?) that prioritizes larger eigenvalues? More generally, what are the pros and cons for various metrics between distributions?