The problem is that jQuery doesn't trigger the native click event for <a> elements so that navigation doesn't happen (the normal behavior of an <a>), so you need to do that manually. For almost all other scenarios, the native DOM event is triggered (at least attempted to - it's in a try/catch).
To trigger it manually, try:
var a = $("<a>")
.attr("href", "http://i.stack.imgur.com/L8rHf.png")
.attr("download", "img.png")
.appendTo("body");
a[0].click();
a.remove();
DEMO: http://jsfiddle.net/HTggQ/
Relevant line in current jQuery source: https://github.com/jquery/jquery/blob/1.11.1/src/event.js#L332
if ( (!special._default || special._default.apply( eventPath.pop(), data ) === false) &&
jQuery.acceptData( elem ) ) {