Fire bans & regulations aside, I seriously doubt bears are attracted to ashes.  Ashes are all that's left when I burn food.  Sometimes it's really neat looking.  A whole fish skelleton burned through, but intact.  One touch with the poker and poof, it's dust.

I'm also kinda nutty about peoples' fear of bears.  Black bears at least.  Grizzlies & Polars are a different matter & I've never had to worry about them.  Black bears are wussies and you're more likely to be hit by lightning & win the lottery on the same day than be worried by a black bear.  There are more reasonable fears to have about being out there, including squirrels chewing holes in your food bag.

20 years tripping.  1 wild bear sighting.