var fruit = ['cherries', 'apples', 'bananas'];
fruit.sort(); // ['apples', 'bananas', 'cherries']

var scores = [1, 10, 2, 21]; 
scores.sort(); // [1, 10, 2, 21]
// Watch out that 10 comes before 2,
// because '10' comes before '2' in Unicode code point order.

var things = ['word', 'Word', '1 Word', '2 Words'];
things.sort(); // ['1 Word', '2 Words', 'Word', 'word']
// In Unicode, numbers come before upper case letters,
// which come before lower case letters.

// the array to be sorted
var list = ['Delta', 'alpha', 'CHARLIE', 'bravo'];

// temporary array holds objects with position and sort-value
var mapped = list.map(function(el, i) {
 return { index: i, value: el.toLowerCase() };
})

// sorting the mapped array containing the reduced values
mapped.sort(function(a, b) {
 return +(a.value > b.value) || +(a.value === b.value) - 1;
});

// container for the resulting order
var result = mapped.map(function(el){
 return list[el.index];
});

alert(result);