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different versions of the cylindrical source terms

I believe I have discovered the difference between the source terms I was deriving and those presented in Skinner & Ostriker. It is emphasized that the source terms presented in that paper are used only for the reconstruction step. Those source terms are derived from simplified versions of the MHD equations which make use of div B = 0 (see example at bottom of this post).

If you leave the MHD equations in their full flux-conserving form, then you get the source terms that I previously posted. Below, I have summarized all the possible source terms from these two methods. "full" corresponds to using the complete flux version of the MHD equations, and "divergence-free" corresponds to what you get when you first expand the MHD equations and take out div B.

Conservative Variables

From this point on, these source terms are written to conveniently compare with cylindrical.f90 in astrobear. The "actual" source terms are these values multiplied by a factor of -1/r. Also, the magnetic terms usually come with a factor of 1/4pi, but all these values are in computational units.

variable	full source term	divergence-free source term
\rho	\rho v_r	\rho v_r
\rho v_r	\rho(v_{r}^{2} - v_{\phi}^{2}) - B_{r}^{2} + B_{\phi}^{2}	\rho(v_{r}^{2} - v_{\phi}^{2}) + B_{\phi}^{2}
\rho v_\phi	2 \rho v_{r} v_{\phi} - 2 B_{r} B_{\phi}	2 \rho v_{r} v_{\phi} - B_{r} B_{\phi}
\rho v_z	\rho v_{r} v_{z} - B_{r} B_{z}	\rho v_{r} v_{z}
E	v_{r} (E + P + \frac{B^{2}}{2}) - B_{r} (\vec{v} \cdot \vec{B})	v_{r} (E + P + \frac{B^{2}}{2})
B_r	0	v_{r} B_{r}
B_\phi	0	v_{\phi} B_{r}
B_z	v_{r} B_{z} - v_{z} B_{r}	v_{r} B_{z}

Primitive Variables

Momentum —→ velocity which is not as simple as just dividing the momentum source terms by the density. Those have to take into account a product rule that introduces another source term. The density and magnetic field source terms remain the same. Energy —→ thermal pressure which also has to be derived on its own.

variable	full source term	divergence-free source term
\rho	\rho v_r	\rho v_r
v_r	- v_{\phi}^{2} - \frac{B_{r}^{2}}{\rho} + \frac{B_{\phi}^{2}}{\rho}	- v_{\phi}^{2} + \frac{B_{\phi}^{2}}{\rho}
v_\phi	v_{r} v_{\phi} - \frac{2 B_{r} B_{\phi}}{\rho}	v_{r} v_{\phi} - \frac{B_{r} B_{\phi}}{\rho}
v_z	- \frac{B_{r} B_{z}}{\rho}	0
P	\gamma P v_{r} + (\gamma - 1) B_{r} (\vec{v} \cdot \vec{B})	\gamma P v_{r}
B_r	0	v_{r} B_{r}
B_\phi	0	v_{\phi} B_{r}
B_z	v_{r} B_{z} - v_{z} B_{r}	v_{r} B_{z}

Now the only part that is inconsistent with Skinner & Ostriker is the divergence-free version of the Br source term.

MHD Contributions in the Momentum/Velocity Equation

In the momentum equation, there is a term that looks like: \vec{\nabla} \cdot ( \vec{B} \otimes \vec{B}) If you brute force this equation and take the gradient of this tensor, you get the magnetic contributions for the momentum/velocity source terms as shown in the "full" columns above. However, you can also use a tensor identity to rewrite this expression as follows: \vec{B}(\vec{\nabla} \cdot \vec{B}) + (\vec{B} \cdot \vec{\nabla})\vec{B} Then, you could get rid of the div B term since div B should = 0. However, there are source terms associated with div B, so your overall source terms have now changed. Using this method results in the "divergence-free" columns above.

Similarly, the MHD contributions for energy/pressure and B also change if you use other calculus identities to expand the magnetic terms and get rid of any div B terms.

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different versions of the cylindrical source terms

Conservative Variables

Primitive Variables

MHD Contributions in the Momentum/Velocity Equation

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