Thank you for your diligent work in revising the text. The introduction now helps to describe the general problem, and the table of variable names is very useful. I appreciate also your response regarding the scope of the paper and the analytical solution, and wish you luck in your follow-up to the in-review work for Geology. I appreciate your clarification regarding viscous dissipation. The detailed boundary conditions and equations describing your set-up are extraordinarily helpful; when I first read your paper, I went through several sheets of my own paper trying to rederive the connections among your equations, being unfamiliar with your methods. These I think will help your work better reach the broad audience of GMD. Towards this, I have read the "Mathematical model" section closely, and it is really helpful and represents considerable work on your part, which I appreciate. Within this section, the update "Figure 2" is likewise a strong asset to a reader such as myself.

While I suggest "publish as is", I have two small comments:

Lines 73-74: Suggest for correctness noting that lava transfers substantial heat; other flows do also exchange heat with their surroundings

Software link: I would suggest that you link "https://zenodo.org/record/3863306" instead, as this will be the version of record associated with this paper. Therefore, if you make your anticipated modifications and expansions to the model, readers will be able to (without cross-referencing version numbers) arrive at the proper release of your code.


Finally, on a personal note, reading your paper has opened my eyes to a new way to simulate complex flows, and your connections to the Navier-Stokes and shallow-water equations and derivations helped me to understand this connection. If I can assume that I am not alone in this more basic fluid-mechanics background, I think that your paper might help to educate some of us on the DG approach (and make us wish that we had taken more numerical methods courses back when we could!).