1. Radial transport in bounded cylinders and physics of “universal” profiles Francis F. Chen UCLA With Davide Curreli, Univ. of Illinois, Champaign-Urbana Gaseous Electronics Conference, Austin TX, Monday, October 22, 2012

2. Why is n(r) peaked on axis?

3. Take a very simple model It has ends!

4. Normal sheath

5. Let the ions diffuse inwards The sheath has to adjust to neutralize tube 2 It is AS IF electrons had followed the ions! But ACTUALLY, they have not moved across B.

6. Electrons fall into a Maxwellian But they still keep their temperatures T e (r)

7. A little simple algebra Ion equation of motion: Ion equation of continuity: Use the Boltzmann relation: Simplify the collision terms:

8. 3 equations for 3 unknowns: n, v,  Eliminate n and  to get an equation for v(r): Non-dimensionalize:

9. An O.D.E. for plasma profiles We had: Rescale r: Finally: K contains the plasma information:

10. A universal profile! Solutions for three values of k Rescale  so that  a  1 in each case This profile is independent of pressure, size, and magnetic field. It depends on KT e, but is always peaked at the center.

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