Outflows

Particles can launch outflows following the description in ​Federrath et al 2014

The basic problem is to apply the following outflow function, but in a discrete way that is numerically symmetric and exact.

Let's assume that the density and velocity have the following form

\rho_{\pm} = \alpha_{\pm} \rho(x)

p_{\pm} =\alpha_{\pm} \left ( a_\pm v_r(x) + dv_{\perp,\pm} + b_\pm R_\pm v_\Omega \right)

\alpha, a, and b are scalar adjustment factors, R is a rotation matrix to align the spin, and dp is a vector normal to the spin axis used to control the total momentum.

We begin by transforming the coordinates of the cell centers into the coordinate system of the star using a rotation matrix.

We then calculate the mass correction factors \alpha_\pm trivially by summing up the predicted mass injection into each hemisphere and then calculating the normalization factors.

We then estimate the velocity of each cell using

a_\pm v_r(x) + dv_{\perp,\pm} + b_\pm R_\pm v_\Omega

with the scalars set to 1, the rotation matrix set to the identity, and dp=0.

We then calculate the spin and momentum in the top and bottom hemispheres. We then adjust b_\pm to adjust the magnitude of the spin, and we calculate the rotation matrix R necessary to rotate the spin to the jet axis which we apply directly to the v_\Omega. We also calculate the total momentum in the top and bottom hemispheres and update the magnitude of the z component using a_\pm, and zero out the transverse components using the shift dv_{\perp,\pm}.

This process is iterated a few times and then a final velocity scaling and shift is applied to make the momentum injection correct to machine precision

v_\pm = a_{\pm,f} \left [ \left (a_\pm v_r(x) + dv_{\perp,\pm} + b_\pm R_\pm v_\Omega \right) + dv_{\perp,\pm,f} \right ]