Example 2 Let $A=\left\{a_{1}, a_{2}, a_{3}\right\}, B=\{-1,0,1\}$.
(1) How many different mappings are there from $A$ to $B$?
(2) Determine the number of mappings $f: A \rightarrow B$ that satisfy $f\left(a_{1}\right)>f\left(a_{2}\right) \geqslant f\left(a_{3}\right)$.