Some more thoughts on magnetic ring evolution

I will start with a definition of ram pressure for a cylindrical flow

where r_0 is the initial radius of the cylinder, \rho_0 is the post-shock density in the cylinder and c is the post-shock sound speed.

To get the field that is being stretched by the flow we use flux freezing

where S refers to the area the flux threads.

This gives us

If we now use the tension "pressure" going as

then we have ram pressure = magnetic tension leading to

or the balance radius r_b

Comments

1. Adam Frank -- 9 years ago

Ahhhh yes flux freezing. That was what i was missing.

So we have

B_o S_o = B_1 S_1

Where S is the area of the annulus through which the field threads which in this case is a ring of thicjness \delta r. The field here is B_x running in the direction of the cylinders axis.

So S = \pi r \delta r

Right?

2. Jonathan -- 9 years ago

yea… S_1=\pi r \delta r and S_0 would be the area originally threaded by the ambient field now swept up by the expanding shell . S_0 = \pi (r2 - r02).