Answers on the comments of Reviewer 2 of the paper

Reviewer 2: First, it suffers a lack of illustration. Indeed, a single figure appears, and intends to show the full geometry of the problem. For instance, from the figure, I cannot understand what is this “average current” (average in time, in ‘a’ coordinate? In ‘b’ coordinate?) Neither can I see a weak vorticity. Thus, the definition of vorticity is confusing. Another way to say the same thing is that the Euler to Lagrange coordinate transform is not clear. Is a background vertical flow included? Or do we only consider the vertical flow induced by the waves?