Cartesian coordinates version of SphericalEllProduct operator

[Peter Rau]

Hi all,

I am working on an MHD problem in 3D that is periodic in the x and y directions. I'm thus using RealFourier bases in the x and y directions and the ChebyshevT basis in the z direction. I wish to implement the potential boundary condition (Eq. (25) of https://arxiv.org/abs/1804.09283) at one of the boundaries in the z-direction (say z=Lz). I believe the correct Cartesian form of the boundary condition is

dV_{kx,ky}/dz + sqrt(kx^2+ky^2)*V_{kx,ky} = 0

where V is a vector field, with the left-hand side evaluated at the boundary z=Lz. Here kx and ky are the wave numbers in the x and y directions, so the boundary condition is applied to the individual coefficients of the spectral basis. In spherical coordinates, the equivalent boundary condition depends on the spherical harmonic degree ell and is imposed in Dedalus v3 using the SphericalEllProduct operator. What is the best way to impose this boundary condition in Cartesian coordinates in Dedalus v3?

Thanks,

Peter