Runs

Mach	Beta	Setup	Max Amplification
6.1	.185	2D By	15+
6.1	.185	2D Bz	2.5
6.1	.185	3D	7.4 +

Magnetic Tension

We can break down the different forces (magnetic pressure gradient, the magnetic tension, the convective acceleration and the thermal pressure gradient) and plot their magnitudes for the 2D cases.

Parker Spiral

The parker spiral defined in ​1958 paper gives the following velocity field in the corotating frame (for regions outside of b) - which he says is the result of a spherical wind - but transformed into the rotating frame.

v_r = v_p(b)

v_\theta = 0

v_\phi = \omega (r-b) \sin \theta

This assumes that the velocity is primarily radial out to b and that the radial acceleration is negligible beyond b.

However, in our HD209458b simulation we used

v_r = v_p(r)

v_\phi = \omega(r - R^2/r) \sin \theta

which takes into account the angular momentum of the wind as it leaves the surface of the planet. Then one just needs to use

v \times B = 0

and

\nabla \cdot B = 0

to solve for the steady state magnetic field. We can write B = \alpha v to satisfy the first constraint, and then the divergence of B in spherical coords to get

\alpha = \frac{R^2 B_r(R) }{r^2 v_p(r)}

which has nearly the same 1/r^2 dependence as in the Parker Spiral - although it takes into account the actual parker velocity.