Distributed Turing Machine?
[Is there] a formal representation of
  a distributed system on top of a
  turing machine? 


Regarding this, the discussion (see link posted by Jukka on comments) is the way to look. The way, I see it, how you would formally represent a distributed system largely depends on how you view them, and that depends on "your favourite system assumptions" (i.e., the assumptions on synchrony (i.e., relative timing of actions in the distributed system), on communication (message passing vs. shared memory), on faults (of processes and/or links, benign or Byzantine, etc). As the community does not agree on this point, there is also no agreement on the basic formalism.


  [Is] it
  possible to extend (make a variant)
  the concept of a turing machine to
  take advantage of distributed
  computing?


I guess it is entirely possible, but no-one (that I know of) has looked into it. 
What I know of are these:


Timed IO Automata also used in Lynch's Distributed Computing book 
Communicating Sequential Processes 
Temporal logic of actions
Pi-Calculus (also already mentioned by Alex)
And more (have been and will be mentioned here)...