Hello Maarit and others,

I just wanted to let you know that I redid our SOCA calculation 
from the 2015 PRE, but removing the pressure force with constant 
desnity (i.e., burgulence). I actually found exactly zero for 
both the kinetic and magnetic shear-current effects, for arbitrary 
viscosity or resistivity (although, being SOCA, it's only really 
valid when they are large) .There is still a nonzero rotational 
(Radler effect) contribution for both, but it looks quite different 
from the case with turbulence – happy to provide the expression if 
you're interested for some reason.

The difference compared to the result in your appendix is probably
because our calculation is perturbative in the shear. So it is a 
bit of a different limit, and simpler. We did notice some 
differences between our CE2 calculations (which are effectively 
SOCA but non-perturbative in the shear) and the analytic ones, 
so this is not altogether surprising. 

One other thing worth mentioning: without the pressure you 
indeed see a magnetic contribution to the standard (diagonal) 
beta effect (i.e., turbulent diffusion), whereas with the 
pressure this goes away to zero, as well known. This isn't 
surprising (it fits with previous results and our JPP cartoon) 
but it's nice to see in the formal calculation.

Best,
Jono