Funny birds!

These birds fascinate me. They hang out along the shore with the sea gulls, and although they share some coloration with the gulls, they are clearly a different type of bird.

Standing, their legs seems to be running in place, kneading the sand below a thin layer of water. What's down there? Why the quick movements?

And then... they're off! Time to run across the sand. Funny...

I think they are some kind of plover, but I'm not sure. Do you know?