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Implementing Dipole fields in Planetary Studies

Planetary magnetic fields are dipole fields of the form

\frac{r_0^3}{r^5} \left ( 3\mathbf{r}\left ( \mathbf{B_0} \cdot \mathbf{r} \right) - \mathbf{B_0} r^2 \right )

B_{\theta}=-B_0 \cos{\theta}\frac{r_0^3}{r^3}

and

B_r=2 B_0 \sin{\theta}\frac{r_0^3}{r^3}

which (if we assume that the dipole is oriented along z) gives a field of

2 B_0 \hat{z}

at the pole and a field of

-B_0 \hat{z}

at the equator.

This also corresponds to a vector potential

\mathbf{A}=r_0^3 \frac{\mathbf{B_0} \times \mathbf{r}}{r^3}

Now if we assume that

\mathbf{B_0} = B_0\hat{z}

Then our vector potential is

\mathbf{A}=r_0^3 \frac{B_0 \sin \theta}{r^2} \hat{\phi}

Also, the dipole field used in Matsakos was a constant field inside of the planetary core (r<r_p/2)

They quote a constant field of 16 B_0 \hat{z} at the center would require a potential of the form

\mathbf{A}=\frac{16 B_0 r \sin{\theta}}{2} \hat{\phi}

which agrees with the potential outside at r=r_0/2

So we don't need any smoothing… just a piecewise definition for the potential that switches at r=r_0/2

Now, if we want to truncate the field lines so they do not pass out the inflowing boundaries, we need to have the derivative of the potential go to zero. This is tricky since the curl of the potential needs to go to zero in both the theta and r directions. However, the following potential has the desired properties

A_\phi^*=r_0^3\frac{B_0 \sin{\theta}}{r^2} \left (1 -\frac{r}{r_s} \right ) ^2

For r > r_s the potential is just 0.

This gives

B_{\theta}^*=\frac{1}{r}\frac{\partial}{\partial r}\left ( r A_\phi \right ) = -B_0 \sin{\theta}\frac{r_0^3}{r^3} \left ( 1 - \frac{r^2}{r_s^2} \right )

and

B_r^*=\frac{1}{r \sin{\theta}}\frac{\partial}{\partial \theta}\left ( \sin{\theta} A_\phi \right ) = 2B_0 \cos{\theta}\frac{r_0^3}{r^3} \left ( 1 - \frac{r}{r_s} \right )^2

for r < r_s and \mathbf{B}=\mathbf{0} for r>r_s

	Smooth	UnSmoothed
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Posted: 9 years ago (Updated: 9 years ago)
Author: Jonathan
Categories: (none)

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Smooth5.png (103.4 KB) - added by Jonathan 9 years ago.
Smooth4.png (131.5 KB) - added by Jonathan 9 years ago.
UnSmooth5.png (112.6 KB) - added by Jonathan 9 years ago.
UnSmooth4.png (155.7 KB) - added by Jonathan 9 years ago.

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Implementing Dipole fields in Planetary Studies

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