Coordinate transformation -- adding lateral variation

[saurabh shukla]

Hi all,

I am new to Dedalus and trying to modify the standard 2D Rayleigh–Benard convection example (ivp_2d_rayleigh_benard.py). The task I have is the following:

The way to do it is to map the domain to a constant thickness one, i.e. rewrite the equations with a new vertical coordinate

Z = z/d(x),

where z is the depth and d(x) is the thickness of the ice/fluid layer. These equations can then be implemented in Dedalus, starting from the Rayleigh–Bénard example.

In other words, the goal is to implement the lateral variations of the thickness in the convection example, by scaling the vertical coordinate as above.

Since I am using Dedalus for the first time, I am not sure how to proceed. My main questions are:

1. How should I implement this coordinate transformation in Dedalus v3?

   • Should I explicitly rewrite the PDEs with the chain rule in terms of Z, or

   • Is there a way to use Dedalus’s coordinate system / metric features to handle this?

2. What modifications would I need to make to the existing 2D Rayleigh–Bénard example code to incorporate this scaling?

Any guidance, explanation, or example code would be really helpful.

Thanks!