I would not expect the core to 'melt', but it could be severely damaged. I would expect the pressure vessel to contain the damage core. The pressure vessel at TMI-2 contained the core, but it was damaged. It now sits shutdown, and the water in containment is just sitting there. Unit 2 is more or less mothballed.
Ideally, they will cool the reactor. At the moment, we can only wait and see what occurs and what actions the government will take with respect to future operation.
Update: Here is another article on Yahoo.
http://finance.yahoo.com/news/Japan-quake-causes-apf-1121920057.html
That's not necessarily significant, but it could indicate that some fuel is breached, or there is a lot of crud (activated corrosion products) that got released in the coolant. Crud would normally be cleaned up on filters. The problem with the article is that it does not indicate what activity, gas or coolant, or the radioisotope, so we don't know the basis of the statement or how accurate it is.
Meanwhile -
http://www.publicradio.org/columns/marketplace/business-news-briefs/2011/03/mid-day_update_youtube_web_bri.html
I'm constantly amazed at the people who simply drive into the flood water. They do not seem to know a route inland.