purposelessly

Often at that moment she would vaguely, purposelessly, smile over to him, and he would smile in absolute reciprocity.

Also, why is that banner persistently and purposelessly waved during the whole of the great Soldiers' Chorus?

She only floundered and flounced about in the water, struggling madly and purposelessly in the waves.

This went on so long, and apparently so purposelessly, that even the constant Si and Shorty were shaken a little by it.