Topics in Fusion and Plasma Studies 2 (459.667)

Midterm Examination

6 November, 2009

1. (1) What are the assumptions made to derive the ideal MHD equations?

(2) Discuss the validity of ideal MHD model in plasmas of fusion interest.

(3) Write down the ideal MHD equations.

(4) Derive the ideal MHD equilibirium equations.

(5) Write down the basic equation describing axisymmetric toroidal equilibrium.

2. The Grad-Shafranov equation can be expressed as

In the finite aspect ratio limit with following assumptions, R→∞, r/R→0, cylindrically symmetric,
(1) Derive the basic radial pressure-balance relation (zeroth-order solution) of the Grad-Shafranov equation and find A and B shown below.

(Using  )

(2) What kinds of forces are acting and how are they balanced in this toroidal system?

3. Describe pros and cons of the θ-pinch and the Z-pinch briefly and explain how to overcome the drawbacks.

4. The Shafranov shift is expressed as follows,

,

(1) Explain the physical meaning of terms and by this explain the meaning of the Shafranov shift.

(2) Calculate how much vertical field is needed to keep the plasma centered.

"Look at the birds of the air; they do not sow or reap or store away in barns, and yet your heavenly Father feeds them. Are you not much more valuable than they?" (Mathew 6:26)