This is something I keep forgetting so I think that I may as well add it here.
The directory where the alert.log found is determined by the background_dump_dest initialization parameter: So the ultimate way to find it is by issuing an SQL command like :
select value from v$parameter where name = 'background_dump_dest';
Needless to say that the alert.log's name is
alert and that it is a simple text file that can be opened using any text editor and that you can delete it or shrink it online.