Take a look at the Woods Hole website - see what their staff have degrees in.
Here is a list of the physical oceonography group:
http://www.whoi.edu/main/po/people
there is also an applied ocean physics and engineering group that may also fit the bill. Their group list is
http://www.whoi.edu/main/aope/people
In both of those you can select to only see the scientific staff, research staff, etc. I glanced quickly and found lots of undergrad degrees in physics, but (at least for the more applied group) a fair number of civil/environmental/mechanical engineers as well.
Hopefully a physical oceanographer will read your post and give you more specific advice on choices of major / coursework.
In any case I'm guessing that if you are interested, you can take the mechanical engineering year-long sequence in fluids as a physics major (the primary question is whether it has a time conflict with required physics classes). Whether or not you want to consider one of the engineering disciplines instead of physics is a very complicated decision only you can make.
My best advice would be to discus your interests with your faculty advisor when you start in the fall.
EDIT. I thought I would add a related anecdote. My sophomore year I fell in love with electromagnetic theory, and thought that whatever my major I wanted to eventually do something related to electromagnetics. I had a hard time choosing my major - it was a toss-up between applied physics and electrical engineering (I was in an engineering college - physics was in arts and sciences and would have been hard to switch to). My father, an economist, asked, 'would there be a fundamental difference between the two when it comes to how prepared you would be for your future career goals?" my answer was NO. So he suggested EE since jobs would likely be easier to find. Anecdotal evidence tells me he was correct - when I graduated the applied physics majors I knew had a much harder time finding jobs than the EEs. I went to grad school then went into the workforce - while I use electromagnetics in my job, it is not my specialty, and my EE background is VERY useful to me now!
best regards,
jason