BUG: JX1 regularisation on the axis

[aborderies]

Describe the issue:

Hi all,
I have in my simulations an asymmetric behaviour of the toroidal magnetic field component (BX3 in spherical coordinates) between the north and south poles of the spherical grid.
Looking at the axis boundary condition, I noticed this part that regularises the current JX1 on the axis:

idefix/src/fluid/boundary/axis.cpp

Lines 130 to 138 in 54482d5

	// Use the circulation around the pole of Bphi to determine Jr on the pole:
	// Delta phi r^2(1-cos theta) Jr = int r sin(theta) Bphi dphi
	idefix_for("fixJ",0,data->np_tot[KDIR],0,data->np_tot[IDIR],
	KOKKOS_LAMBDA(int k,int i) {
	real th = x2(jc);
	real fact = sign*sin(th)/(deltaPhi*x1(i)*(1-cos(th)));
	J(IDIR, k,js,i) = BAvg(i)*fact;
	});

I think the relation
$\Delta \varphi r^2(1-\cos \theta) J_r = \int r \sin(\theta) B_{\varphi} \mathrm{d}\varphi$
cannot be true, since $1-\cos \theta$ is not (anti)symmetric between $\theta = 0$ and $\theta = \pi$.
By recalculating the Amper's law around the axis, I get
$\frac{1}{2}\Delta \varphi r^2 \sin(\theta)^2 J_r = \int r \sin(\theta) B_{\varphi} \mathrm{d}\varphi$
or
$\frac{1}{4}\Delta \varphi r^2 (1-\cos(2\theta)) J_r = \int r \sin(\theta) B_{\varphi} \mathrm{d}\varphi$
which converge to the same result as the current relation near $\theta=0$, but not near $\theta=\pi$.

So line 136 should be replaced by
real fact = sign*4.*sin(th)/(deltaPhi*x1(i)*(1-cos(2*th)));
or
real fact = sign*2./(deltaPhi*x1(i)*sin(th));.

Is this correct or am I missing something?

Error message:

runtime information:

idefix V2.2.00

[glesur]

Hi @aborderies , thanks for spotting this. I believe you're right, the polar contour integration is messed up.

[glesur]

This was addressed and merged in v2.2.01

[neutrinoceros]

Did you mean 2.2.01 ?

[glesur]

yep, corrected.