The Byronic Hero characterization is the element that strikes me the most from "Adieu, Adieu, My Native Shore."  One part of the Byronic Hero is the forlorn condition present, one in which the character is rootless in their state in the world.  This is seen in the opening of the third stanza:

And now I'm in the world alone, /Upon the wide, wide sea:

But why should I for others groan,/ When none will sigh for me?

The condition of being at odds with the world is one of the critical elements of the Byronic Hero.  This can be seen in these lies, especially in the idea of the "alone" notion and the "none will sigh for me." Byron was conscious of emphasizing this element of Childe Harold's characterization.  It helps to construct one of Romanticism's most lasting characters.