This problem is highly non-linear, and has several non-local couplings (in the y direction, note that u is a function of both x and y), which does not naturally fall under ufl.
What you first need to do is to find a way to linearize the problem (a newton iteration or piccard iteration, i would probably use Piccard on the non-local interaction).
Then secondly, you would have to build up an operation for the lon-local interaction. I would probably make my own quadrature scheme in y direction, and use Implementation — FEniCSx tutorial to make the contributions in y direction.