In what ways are the four natural elements used in The Tempest?

The four elements of earth, fire, water and air are used in this play with reference to the powers that Prospero commands through his control of spirits such as Ariel. From the very first meeting between Prospero and Ariel in the play in Act I scene 2, Ariel makes clear that his magic allows him to control each of the elements, and that Prospero's control of Ariel therefore gives him that same control:

I come

To answer thy best pleasure. Be't to fly,

To swim, to dive into the fire, to ride

On the curled clouds, to thy strong biddding task

Ariel and all his quality.

Note the only element that Ariel doesn't directly refer to is earth, but surely it is implied in the extent of his powers. Air, fire and water are clearly mentioned, and Ariel shows that he is able to manipulate those elements at his command. The four elements in this play therefore are used by Prospero to work his enchantments as his control of the various elves and spirits on the island gives him that ability. It is the manipulation of these four elements that allows Prospero to stage manage the action of this play and bring about the reconciliation that he desires amongst the different characters.