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Magnetic ring evolution

Consider the magnetic field to be similar to an atmosphere where we balance magnetic pressure gradients with magnetic tension forces? Than we have

\frac{d B^2}{2 d r} = -\frac{B^2}{r}

which gives us

B^2=B_0^2\frac{r^2_{o}}{r^2}

We can then integrate this to get the net tension force

F_T=h\displaystyle \int_{r_i}^{r_o}{\frac{B(r)^2}{r} 2\pi r dr} = \pi h B_0^2 r_{o}^2 \left ( \frac{1}{r_i} - \frac{1}{r_o} \right )

and dividing by the inner area gives

P_T=B_0^2 \frac{r_o \Delta r}{r_i^2} = P_{ram}

which we can combine with flux freezing

2 \pi B_0 r_o \Delta r = B_0 \pi \left ( r_o^2 - R^2 \right )

and

\Delta r = r_o-r_i

Which can be reduced to

2 r_o \Delta r = \left ( r_o^2 - R^2 \right )

B_0^2 \frac{r_o \Delta r}{\left ( r_o-\Delta r \right )^2} = P_{ram}

And then can be combined to

\frac{2 P_{ram}}{B_0^2}\left ( r_o-\Delta r \right )^2 = \left ( r_o^2 - R^2 \right )

\frac{\Delta r}{r_o} = \frac{1}{2} \left ( 1 - \frac{R}{r_o}^2 \right )

\frac{\beta_{ram}}{4} \left ( R^2 + r_o^2 \right ) = \left ( r_o^2 - R^2 \right )

which eventually gives (if I haven't made any mistakes…)

r_0 = R \sqrt{\frac{4+\beta_{ram}}{4-\beta_{ram}}}

although clearly I have since \beta_{ram} > 4

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