22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 4: Toroidal Equilibrium and Radial Pressure Balance
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1 .615, MHD Theoy of Fusion Systems Pof. Feidbeg Lectue 4: Tooidal Equilibium and Radial Pessue Balance Basic Poblem of Tooidal Equilibium 1. Radial pessue balance. Tooidal foce balance Radial Pessue Balance 1. The lagest foces ae usually associated with adial pessue balance. The magnetic field must confine the plasma adially so it is isolated fom the vacuum chambe. 3. Thee ae seveal ways to do this using tooidal and/o poloidal fields. This is not vey difficult to accomplish. Tooidal Foce Balance 1. Smalle foces ae also pesent associated with unavoidable outwad tooidal expansion foces Pof. Feidbeg Page 1 of 13
2 . We now conside two opposing limits to demonstate the basic poblem of tooidal equilibium. Even though the foces ae smalle, tooidal foce balance is the cucial issue. Puely Poloidal Field 1.. Outwad Foce #1 a. flux consevation: o BA b. fields: B B B A aeas: A A c. foce: B A >B A Pof. Feidbeg Page of 13
3 d. this leads to net outwad hoop foce 3. Outwad foce # a. The pessue foce is outwad. It esults fom a lage aea on the outside compaed to the inside at fixed pessue. This is simila to a ubbe tie tube foce. 4. Restoing foce #1 a. plasma shifts outwad b. flux is compessed against the pefectly conducting shell c. B p inceases on the outside of the tous d. The inceased magnetic pessue poduces a foce to balance the outwad tooidal foce. This is a conducting shell equilibium. Pof. Feidbeg Page 3 of 13
4 5. Restoing foce # a. A vetical field poduces a J B foces b. With the pope magnitude and sign, B v can balance the outwad foce c. This is the vetical field foce F v = B v I p L 6. Conclusion: It is easy to povide tooidal equilibium using puely poloidal magnetic fields 7. But, we shall see that such systems ae vey unstable to macoscopic MHD modes (in the puely poloidal case) 8. Howeve, we shall also see that systems with lage tooidal fields have much bette stability popeties. Pof. Feidbeg Page 4 of 13
5 Puely Tooidal Field 1. Thee is no hoop foce since I TOR =. The ubbe tie tube foce is pesent fo the same easons as befoe 3. Thee is an additional foce due to J p 4. Outwad foce #3 5. Note: BT 1 R: B dl Icoil : BTR Icoil B T I coil R 6. Now B I B II A I A II B I A I B II A II Since B dominates, thee is a net outwad foce 7. This is called the 1 R foce Pof. Feidbeg Page 5 of 13
6 8. Can a conducting shell balance outwad foce in puely tooidal case? No! Magne tic flux is not tapped. Lines ae fee to slide aound plasma 9. Can a vetical field balance outwad foce in puely tooidal case? No! Thee is no net inwad foce because of the basic field diections 1. Conclusion A puely tooidal field cannot hold a plasma in tooidal equilibium. The tooidal foce cannot be balanced. 11. The time scale fo loss of equilibium is in the sec ange. Fo a = 1 m, B = 5 T, n = 1 1 m -3, 1, R 1km t equil 18 sec. Pof. Feidbeg Page 6 of 13
7 1. Single paticle pictue 13. Effect of tansfom poloidal field Basic Poblem of Tooidal Equilibium 1. Poloidal fields: Good equilibium poo stability. Tooidal fields: Poo equilibium good stability Pof. Feidbeg Page 7 of 13
8 Ou goal is to optimize the advantages and minimize the disadvantages. This is the challenge of ceating desiable fusion geometies. Geneal Appoach to MHD Equilibia 1. The MHD equilibium poblem sepaates into two pats fo most configuations of inteest: a. adial pessue balance: zeo ode in ar b. tooidal foce balance: fist ode in ar. We examine adial pessue balance in a 1-D geomety: staight cylinde (no poblems with tooidal foce balance) 3. The 1-D adial pessue balance elation is valid fo tokamaks, stellaatos, RFP s, pinches, and EBT s 4. We intoduce tooidal effects as an aspect atio expansion to see what must be done to achieve tooidal foce balance. This equies -D calculations. Radial Pessue Balance 1-D pinch 1. pinch analog of the puely tooidal case.. Configuation 3. Nonzeo components B= B z ()e z, p= p(), J=J() e 4. Solution of the MHD equilibium equations a. B= B z automatically z Pof. Feidbeg Page 8 of 13
9 b. J B J Bz c. J B = p p e J B = J Be B B e B e z z z z d. pinch pessue balance elation B p z 5. Integate 6. Example: We ae fee to give one function abitaily, say p() p () pe a Then 1 B B z p B e a Define p B on axis B () z B 1 e a Pof. Feidbeg Page 9 of 13
10 Note 1 fo eal solutions 7. In a pinch any value of is possible (good fo fusion) including vey high, 1 8. Concepts in which adial pessue balance coesponds to a pinch a. pinch b. stellaato c. high stellaato d. high tokamak e. EBT f. cental cell of a tandem mio 1-D Z Pinch 1. Z pinch analog of the puely poloidal case. Configuation Pof. Feidbeg Page 1 of 13
11 3. Nonzeo components: B B() e, p p(),j Jz()e 4. Solution of MHD equilibium equations a. B 1 B automatically satisfied b. J B Jz 1 B c. JB p z p p e B JB JzBe B d. Z pinch pessue balance elation p B B 5. Altenate fom Pof. Feidbeg Page 11 of 13
12 Tension causes field lines to collapse. This poduces an inwad adial foce 6. Typical pofiles: fo lage, B 1 Jz 7. The plasma is confined by magnetic tension. The magnetic pessue and plasma pessue each poduce positive (outwad foces) nea the outside of the plasma. 8. Example (Bennett Pofiles) p I 1 I p, B ap 1 a a Then B I a J I a a p 8 I a a Pof. Feidbeg Page 1 of 13
13 9. Define: () = p () p() B () 1 () = 1+ a 1. Note: () = 1 is always tue fo a pue Z pinch. Although high is good, it is moe desiable to have some flexibility in ode to avoid instabilities. 11. Bennett Pinch Relation I da I 1 dx p d 8 4 a 1 x 1 p d I 8 This, we shall see, is moe geneal than only fo the Bennett pofiles 1. Configuations with Z pinch adial pessue balance a. ohmically heated tokamak b. evesed field pinch Pof. Feidbeg Page 13 of 13
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