15. SIMPLE MHD EQUILIBRIA

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1 15. SIMPLE MHD EQUILIBRIA In this Section we will examine some simple examples of MHD equilibium configuations. These will all be in cylinical geomety. They fom the basis fo moe the complicate equilibium states in tooial geomety. Many MHD equilibium configuations incluing tokamaks, spheomaks, an RFPs) ae base on the pinch effect, which esults fom the attactive natue of paallel cuents exceptions ae mios an stellaatos). Consie two elements caying cuents J 1 an J 2 in the z-iection, as shown in the figue. The magnetic fiel B 1 pouce by J 1 encicles element 1 accoing to the ight han ule, an similaly fo B 2 an J 2. The Loentz foce J 1 B 2 acting in element 1 is iecte towa element 2. Similaly, the Loentz foce J 2 B 1 acting on element 2 is iecte towa element 1. If the cuent J is istibute continuously in space, the net effect of the Loentz foce will be to pull the flui togethe, compessing it an theeby inceasing the pessue. This pocess will cease when the incease in the pessue foce tening to expan the flui just balances the Loentz foce tening to compess the flui, o p = J B. If the cuent flows in a column, the column will ten to contact, o pinch, in a iection pepenicula to its axis until the equilibium conition is eache. This is calle the pinch effect, an is shown in the figue. 1

2 Equilibium configuations base on the pinch effect ae name afte the iection of the cuent, not the magnetic fiel. We will now examine seveal of these in cylinical geomety. In the Theta Pinch o -Pinch), the cuent flows only in the azimuthal, o, iection. This pouces a magnetic fiel in the z-iection, as shown in the figue. The magnetic fiel is a combination of a unifom, extenally geneate fiel an the fiel pouce by the cuent. We take B z to be in the positive z-iection, an J to be in the negative -iection. We assume that all quantities ae functions of only. The Loentz foce is then aially inwa, i.e., J B = J ê B z ê z = J B z ê. 15.1) We will geneally use thee equations to analyze an equilibium configuation. These ae: 1. B = 0 ; J = B ; an, 3. Foce balance, p = J B. These ae now consiee fo the case of the Theta-Pinch. 1. B = 0. In cylinical geomety, this is 1 B ) + 1 B + B z z = ) Since the configuation epens only on, an B = 0, we equie B z z which is satisfie automatically. = 0, 15.3) J = B. Une these conitions, this becomes J = B z. 15.4) 3. Foce balance, p = J B. This becomes 2

3 = J B z. 15.5) Substituting Equation 15.4) into Equation 15.5), we have o = B z 1 B z $ z 2 ' = 2 B z $ 2 ', $ = ) The secon tem in paentheses is the magnetic pessue. This can be integate to yiel z = B 2 0, 15.7) 2 2 so that B z = B 0 when p = 0, i.e., outsie the flui. The constant B 0 is thus the extenally geneate component of the axial magnetic fiel. Note that Equation 15.7) is a single equation containing two unknowns, B z an p. We ae fee to specify one an then etemine the othe. This will be a geneal popety of MHD equilibia. An example of a solution of Equation 15.7) is an p) = p 0 e 2 /a 2, 15.8) B z ) = B e 2 /a ) 2 1/2, 15.9) whee 0 = 2 p 0 / B 0 2 an = a is the aius of the oute bounay. These solutions ae sketche vey oughly) in the figue. We now consie the linea Z-pinch. The cuent now flows in the z-iection, an the magnetic fiel is in the -iection, as shown in the figue 3

4 We again assume that thee is only, an pocee as with the -pinch. 1. B = 0. 1 B ) + 1 B + B z z We have B = B z = 0, so we equie = ) B = 0, 15.11) which is automatically satisfie if B = B ). J = B. J z = 1 3. Foce balance, p = J B. Using Equation 15.12), o B ) ) = J zb ) = B B ), = B B B 2, $ 2 ' = B ) This looks like the esult fo the -pinch, Equation 15.16), with the aition of a tem on the ight han sie. This tem is calle the hoop stess, an aises fom the cuvatue of the magnetic fiel lines. In the -pinch, the fiel lines ae staight.) 4

5 Consie the cuve shown in the figue. Let s be the istance along the cuve, an efine t as a unit vecto tangent to the cuve. The cuvatue vecto is then efine as = t, 15.15) s the ate of change of the tangent vecto as we move along the cuve. The aius of cuvatue at a point s is efine as R c = ) If the cuve is a magnetic fiel line, the unit tangent vecto is ˆb = B / B, an / s = ˆb, so the cuvatue of a magnetic fiel line is = ˆb ˆb ) A staightfowa calculation yiels ) = ˆb $ˆb = ) ˆb ˆb Then anothe staightfowa calculation using Ampée s law an foce balance leas to ) = ˆb $ ˆb o = B $p B $ B, = B $ p + B2 ) 2 ' 2 * +, 5

6 p + B2 $ 2 ' = B ) If = 0, as in the -pinch, we obtain Equation 15.6). Fo the case of the z-pinch, we have = ˆb ˆb = B $ê$ B $ê$ B $ ' ) *, = ê $ ê $ = ê $ ' = 1 Then Equation 15.17) becomes ê $ +ê $ + B $ ) *, ê = ê ) p + B2 $ 2 ' = B2 ê, o, fo ou one-imensional configuation, $ 2 ' = B 2, 15.19) which agees with Equation 15.14). The hoop stess, o tension foce, balances the gaient of the total pessue. Again, this is one equations in two unknowns. One can be specifie abitaily. The case that contains both B ) an B z ) an, consequently, both J an J z ) is calle the geneal scew pinch, because the fiel lines wap aoun the cyline in a helical fashion, like the theas on a scew. Fo this configuation: 1. B = 0. 1 B + B z z = 0, 15.20) which is tivially satisfie if the fiels ae functions of only. an J = B. J = B z, 15.21) J z = 1 3. Foce balance, p = J B. B ) ) 6

7 o = J B z J z B, = 2 B z $ 2 ' 2 B $ 2 ' B 2, 2 + B z $ 2 ' = B ) We now have one equation in thee unknowns, so that two of functions can be specifie. We will follow the same poceue fo analyzing the moe complicate situation of tooial equilibium. Finally, we emak that in each of the examples consiee in this Section, the cyline is infinitely long in the z-iection, i.e., each example is puely two-imensional. Recall that the Viial Theoem poven in Section 14 assume that a suface of integation coul be taken completely outsie the flui. This is clealy impossible if the flui extens to infinity in some iection. We thus o not expect the Viial Theoem to apply to these simple examples. 7