24" Cinema display blanks when hooked up via Thunderbolt drive

I just bought a Thunderbolt drive (Western Digital 6TB) for my Mac Mini, which works fine, but the only place left to plug in my 24" Cinema display is the spare Thunderbolt connector on the drive. When hooked up through the drive the screen keeps blanking for 1-2 seconds when drive is active several times a minute. No problem without drive. How can I use both the drive and my display without all this blanking?

FYI, when I access the computer remotely there are, of course, no instances of blanking.

All software is up to date (Mountain Lion) including the drive.