Nonlinear source at the connected boundaries
In flux tube simulations with twist-shift boundary conditions we have duplicate points representing the same location. Specifically, at a connected boundary we have the +pi and -pi connected points from two different kx (assuming nperiod = 1).
Our linked boundary condition ensures gnew is unique at these duplicated points and this should mostly ensure that the fields are unique here as well.
Despite the distribution function and fields being unique, the nonlinear source term has two different values at this duplicate point. This is for two reasons:
- Firstly, because the
kxvalue is different for the two points, thei kx chiandi kx gpart of the NL source give different values. - For fixed sigma only one of +/- pi has the zero incoming boundary condition enforced, whilst the other point has no b.c. applied. All
kxpoints with the same theta value contribute to the NL source at that theta point, hence even for doubly connectedkx, the difference in b.c. between the +/- pi points results in a difference in the source term.
It is possible to “fix” point one by replacing kx with ky*shat*(theta-theta0) for the non-zonal modes. Even with this change there’s a difference in the nonlinear source term due to 2. One may verify that all the inputs (g, chi and the i ky/x weighted versions of these) to the nonlinear source are unique at the duplicate points yet the resulting source is still multi-valued here.
A slightly related point is that the nonlinear source term is unlikely to be periodic for the zonal modes (as this is built from the non-periodic non-zonal modes) and so we can end up driving the periodic zonal modes with a non-periodic source (e.g. assuming the NL term dominates). Whilst our algorithm ensures the zonal modes are periodic, this combined with the non-periodic source/drive can lead to sharp features near the periodic boundary.
It’s not immediately obvious to me that this is incorrect, but it feels like this could cause problems. In particular, for extended modes where the amplitude of potentials isn’t low at the connected boundaries and for situations in which potentials and particularly gnew can retain significant amplitude at the outgoing boundary one might expect this to have a significant impact. For well localised modes this may be less noticeable.
