22.615, MHD Theory of Fusion Systems Lecture 4: Toroidal Equilibrium and Radial Pressure Balance Basic Problem of Toroidal Equilibrium 1. Radial pressure balance 2. Toroidal force balance Radial Pressure Balance 1. The largest forces are usually associated with radial pressure balance 2. The magnetic field must confine the plasma radially so it is isolated from the vacuum chamber. 3. There are several ways to do this using toroidal and/or poloidal fields. This is not very difficult to accomplish. Toroidal Force Balance 1. Smaller forces are also present associated with unavoidable outward toroidal expansion forces 2. We now consider two opposing limits to demonstrate the basic problem of toroidal equilibrium. Even though the forces are smaller, toroidal force balance is the crucial issue. Purely Poloidal Field 1. 2. Outward Force #1 a. flux conservation: b. fields: B, ! B,, areas: A, A,, c. force: I, B,2 A, > B,, A,, I,, or B, A, B,, A,, d. this leads to net outward hoop force 3. Outward force #2 a. The pressure force is outward. It results from a larger area on the outside compared to the inside at fixed pressure. This is similar to a rubber tire tube force. 4. Restoring force #1 a. plasma shifts outward b. flux is compressed against the perfectly conducting shell c. Bp increases on the outside of the torus d. The increased magnetic pressure produces a force to balance the outward toroidal force. This is a conducting shell equilibrium. 5. Restoring force #2 a. A vertical field produces a J u B forces b. With the proper magnitude and sign, Bv can balance the outward force c. This is the vertical field force Fv = BvIpL 6. Conclusion: It is easy to provide toroidal equilibrium using purely poloidal magnetic fields 7. But, we shall see that such systems are very unstable to macroscopic MHD modes (in the purely poloidal case) 8. However, we shall also see that systems with large toroidal fields have much better stability properties. Purely Toroidal Field 1. There is no hoop force since ITOR = 0 2. The rubber tire tube force is present for the same reasons as before 3. There is an additional force due to Jp 4. Outward force #3 5. Note: BT v 1 R : ³ B dl BT P0 Icoil : 2SBT R P0 Icoil P0 Icoil 2SR 6. Now BI ! BII AI AII 2 BI2 AI ! BII AII Since B2 dominates, there is a net outward force 7. This is called the 1 R force 8. Can a conducting shell balance outward force in purely toroidal case? No! Magnetic flux is not trapped. Lines are free to slide around plasma 9. Can a vertical field balance outward force in purely toroidal case? No! There is no net inward force because of the basic field directions 10. Conclusion A purely toroidal field cannot hold a plasma in toroidal equilibrium. The toroidal force cannot be balanced. 11. The time scale for loss of equilibrium is in the P sec range. For a = 1 m, B = 5 T, n = 1021 m-3, E 1, R 1km o tequil 18P sec. 12. Single particle picture 13. Effect of transform o poloidal field Basic Problem of Toroidal Equilibrium 1. Poloidal fields: Good equilibrium o poor stability 2. Toroidal fields: Poor equilibrium o good stability Our goal is to optimize the advantages and minimize the disadvantages. This is the challenge of creating desirable fusion geometries. General Approach to MHD Equilibria 1. The MHD equilibrium problem separates into two parts for most configurations of interest: a. radial pressure balance: zero order in a R b. toroidal force balance: first order in a R 2. We examine radial pressure balance in a 1-D geometry: straight cylinder (no problems with toroidal force balance) 3. The 1-D radial pressure balance relation is valid for tokamaks, stellarators, RFP’s, pinches, and EBT’s 4. We introduce toroidal effects as an aspect ratio expansion to see what must be done to achieve toroidal force balance. This requires 2-D calculations. Radial Pressure Balance 1-D T pinch 1. T pinch o analog of the purely toroidal case. 2. Configuration 3. Nonzero components B= Bz (r)ez, p= p(r), J=J(r) eT 4. Solution of the MHD equilibrium equations a. B = 0 wBz wz 0 automatically b. P0 J uB P0 JT J×B = p c. wBz wr wp wr er Bz P0 wBz wrer J×B = JǇ Bz er w Bz2 2P wr er d. T pinch pressure balance relation w wr § B2 · ¨p z ¸ ¨ 2P ¸¹ © 0 5. Integrate 6. Example: We are free to give one function arbitrarily, say p(r) p(r ) p0 e r 2 a2 Then Bz2 B02 1 2P p0 B02 er 2 a2 Define E { 2P0 p0 B02 “ E on axis” Bz2 (r ) B02 1 E0 e r 2 a2 Note 0 E0 1 for real solutions 7. In a T pinch any value of E is possible (good for fusion) including very high E , E 1 8. Concepts in which radial pressure balance corresponds to a T pinch a. T pinch b. stellarator c. high E stellarator d. high E tokamak e. EBT f. central cell of a tandem mirror 1-D Z Pinch 1. Z pinch – analog of the purely poloidal case 2. Configuration 3. Nonzero components: B BT (r )eT , p p(r ), J Jz (r )e z 4. Solution of MHD equilibrium equations a. B 0 b. P0 J uB c. JuB p p JuB wBT wT 1r P0 Jz 1r 0 automatically satisfied w rBT wp er wr Jz BT er BT w rB P0 r wr T d. Z pinch pressure balance relation wp BT w rBT wr P r wr 5. Alternate form 0 wr Tension causes field lines to collapse. This produces an inward radial force 6. Typical profiles: for large r , BT 1 r o Jz 0 7. The plasma is confined by magnetic tension. The magnetic pressure and plasma pressure each produce positive (outward forces) near the outside of the plasma. 8. Example (Bennett Profiles) p0 p 1 r 2 a2 2 , BT Then BT J p P0 I r 2 2S r a2 I a2 S r 2 a2 2 P I 2 a2 8S2 r 2 a2 2 P0 I r 2 2 2S r a o 2ap0 12 I 2S 9. Define: E (r) = 2 P0 p (r ) ª2P0 p(r ) BT2 (r )º ¬ ¼ E (r) = 1 1 + r2 a2 10. Note: E (0) = 1 is always true for a pure Z pinch. Although high E is good, it is more desirable to have some flexibility in order to avoid instabilities. 11. Bennett Pinch Relation ³ 2Spr dr 2SP0 I 2 2 8S ³ P0 I 2 1 r dr a2 r 2 a2 2 4S f 2 ³0 dx 1 x 2 1 2S³ pr dr P0 I 2 8S This, we shall see, is more general than only for the Bennett profiles 12. Configurations with Z pinch radial pressure balance a. ohmically heated tokamak b. reversed field pinch

Lecture 4 Notes: Toroidal Equilibrium and Radial Pressure Balance

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