The responses to the referees were thorough and thoughtful, and the changes to the paper reflect these responses. I am grateful in particular to the review from referee #1, whose understanding of the issues at hand regarding reverse-mode operator overloading is far far beyond my own. As the issues raised seem to be acknowledged and addressed, I believe the amendments are appropriate and sufficient.

I raise one small point which I would like the authors to consider or comment on before publication, and also make an additional minor suggestion. The authors are free to ignore as I will still allow publication.

-- In response to referee 2 the authors have referenced Goldberg et al 2016, but i think the reason for its mention by referee 2 was not the fact that it is source-to-source but that it implements the fixed-point algorithm of christianson (1994), which does make use source-transformed code, but solves a slightly different fixed-point problem for the adjoint of the velocity solver, which is distinct from the reverse approach used here. I am not sure whether this would even be possible with OL, but outwith that I just want to flag that the source-to-source approach was not the innovation to which the referee referred.

-- A very small point: I have recently learned that the canonical reference for the in-text equations at lines 236-7 is

Mike B. Giles. Advances in Automatic Differentiation, chapter Collected Matrix Derivative Results for Forward
and Reverse Mode Algorithmic Differentiation, pages 35–44. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008