!( r) =!( r)e i(m" + kz)!!!!. (30.1)

Transcription

1 3 EXAMPLES OF THE APPLICATION OF THE ENERGY PRINCIPLE TO CYLINDRICAL EQUILIBRIA We now use the Enegy Pinciple to analyze the stability popeties of the cylinical! -pinch, the Z-pinch, an the Geneal Scew Pinch Since these equilibia epen only on the aial cooinate, ae peioic in the! - an z -cooinates, an the fiels ae inepenent of the peioic cooinates, the isplacement can be witten as in tems of the Fouie ecomposition! ) =! )e im" + kz)!!!! 31) Since the equilibia have no aial component of the magnetic fiel, B! " = im B + ikb z 3) In Equation 31), m is an intege in the ange!" m " It is calle the poloial moe numbe If the system is infinitely long, k is a continuous vaiable Howeve, it is quantize if the cyline has finite length Fo example, if a tous with cicula coss section a an majo aius R has aspect atio R / a ) is cut an staightene into a cyline of length L =! R, then peioicity equies k = n / R, whee n is an intege in the ange!" n " It is calle the axial, o tooial, moe numbe Typical isplacements of the plasma column fo iffeent values of m an k ae shown in the figue 1

2 The moe with m =,!k! is colofully calle the sausage moe Since m =, the isplacement is azimuthally symmetic ie, inepenent of! ) The moe with m = 1,!k = is just a shift of the column with espect to the axis The moe with m = 1,!k! is calle a kink moe It istots the column helically We now consie the thee cylinical equilibia The! -Pinch The equilibium fo the! -pinch is ) + B z p = B!!!, 33) whee B is the magnetic fiel stength outsie the flui It is pouce extenally Since B! =, Equation 3) becomes B! " = ikb z, so that this opeato can be invete as long as k! In that case the minimizing conition fo is! " =, o 1! ) + im! + ik! " z =!!! 34) We can theefoe use Equation 34) to eliminate!! =! z fom accoing to! z == i k! ) " + im! '!!!, 35) ) whee )! enotes iffeentiation with espect to This is vali as long as k! In this case we have Q! = ikb z "! = ikb z " ê + " ê )!!!, 36) an! " = 1 ) + im!!!, 37)! = ˆb " ˆb =!!!, 38) J! =!!! 39)

3 In cylinical geomety the potential enegy pe unit length is a L = " W )!!! 31) Fo the case k!, using Equations 35) 39), W ) = B z ) + 1! k! +! ) " + m!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+ im *! ) "! im! ) "! * '!!! 311) Then afte a consieable amount of algeba, Equation 31) can be e-witten as L = " a B z k im -/ k ) + k ' k ) + k * 1/ 4 ) +,!!!, 31) / 3/ whee k = m / + k We note that! " appeas only in the fist tem in the integan Setting this tem to zeo, we fin that he minimizing tial function must have im! " = m + k so that Equation 31) becomes! )!!!, 313) L = " a k B z m + k ) + m + k ), ) *!!! 314) ' + We can aw two conclusions fom Equation 314): > fo k!, so that the! -pinch is stable fo all finite k " fo k! ", so that the stability becomes maginal fo vey long wavelengths The MHD stability of the! -pinch is explaine as follows: J! = J = J! ê!, B = Bê z ), so that thee ae no cuent iven moes! = no fiel line cuvatue), so! " p pessue iven moes )! " ) = an thee ae no Theefoe, thee is no MHD souce of fee enegy to ive instability Howeve, eal! -pinch have finite length an theefoe have fiel line cuvatue, so this can ive instability in the laboatoy This is the eason we have not consiee the case k = hee Also, thee ae also seveal non-mhd instability ives that have enee this concept poblematical fo magnetic confinement fusion Hence,! -pinches ae not a mainsteam concept The Z-Pinch 3

4 Fo the Z-pinch, B = B! )ê! an J = J z )ê z, an the equilibium conition is p + B! B! ) =!!! 315) Theefoe J! = an!! =! " Fo m!, the opeato B!" = imb / is well behave an can be invete In the case the minimizing conition is! " =, so that! " = i m! ) + ik! z '!!!, 316) ) which is vali as long as m! If m =, then B! " = eveywhee an we must consie! " in the minimization When m!, we have Q! = imb " ê + " ê " )!!!, 317)! " + " = ), ' * + + ik z!!!, 318) an! = " ê!!!, 319) J! =!!! 3) Again, afte much algeba, we fin L = " a, p + m B ) + m B ) 3 - m + k ' * + 1!!! 31) / The last tem is minimize fo k! ", so the stability of the Z-pinch is etemine by the sign of a L = " p + m B )!!! 3) Suppose the integan of Equation 3) is negative in some inteval < < 1 Then we can choose the tial function such that! " insie this inteval, an! = outsie it Since this inteval is abitay, we conclue that a necessay an sufficient conition fo the stability of the Z-pinch when m! is at all points in the flui p + m B! >!!! 33) 4

5 We can use the equilibium conition, Equation 315), to eliminate the pessue gaient fom the Equation 33) The esult is B! B! ) < m B!!!! 34) This can be e-witten in eithe of two foms The fist is B! " B! ' < 1 m 4)!!! 35) Fo, B! ~, so B! / ~ constant In this limit, the stability conition is theefoe m > 4, so that the inteio of the Z-pinch is stable fo stable fo m > an maginal fo m = Fo! ", B! ~ 1 / an B! / ) / ~ 1 / 3 ", so the same conclusion hols in this limit The secon fom of the stability conition is 1 B! B! ) < m " 1!!! 36) Fo! ", B! ) / ~ "1 /, an the stability conition is m > 1 Theefoe, all m > 1 ae stable, an m = 1 is maginal, in this limit Fo!, B! ~ 3 an the left han sie of Equation 36) ~ >, so that the coe is always unstable to the m = 1 moe Note that, since J! =, this m = 1 moe is not a cuent iven moe Rathe, stability is etemine by a competition between fiel line bening stabilizing) an unfavoable cuvatue estabilizing) The latte wins out in the coe of the Z-pinch fo the moe with m = 1 We now consie the case m = Hee B! " = eveywhee, an so! " Afte a fomiable calculation, is foun to be with L = " a p B / ) - ' + p p + B *!!!, 37) / + i + B! z =! ' * - "p + B / ) + "p -,!, ) * / 38) fo the minimizing petubation Again, stability equies that the integan of Equation 37) be positive fo all, which leas to the stability conition fo m = moes:! p! p =! ln p ln < 4" + "!!!, 39) 5

6 whee the poloial beta is! " = p / B " The Z-pinch can suppot a pessue gaient as long as it is not too lage We emak that, since the conition 39) epens on the aiabatic inex!, it equies that the flui satisfy the aiabatic law This is selom the case fo eal plasmas In that case, all bets ae off as fa as m = stability is concene The Geneal Scew Pinch You may have obseve that the stability calculations become inceasingly fomiable as the equilibium becomes moe complex In this ega, the case of the Geneal Scew Pinch inheits some of the wost elements of the calculations fo the! - pinch an the Z-pinch It will theefoe be teate hee with even moe infomality Fo the Geneal Scew Pinch, B = B! )ê! + B z )ê z, an the equilibium conition is p + B "! + B z ' + B! =!!! 33) The minimizing petubation has B!"! ' B ) = *"! +!!! 331) In this case we can wite B! " = if), whee F)! k " B = mb / + kb z Theefoe, B! " is well behave eveywhee that F! In that case the minimizing petubation has!! = ib F "!!!! 33) The oots of the equation F = ae calle singula sufaces again associating the aial cooinate with flux sufaces), fo at these points B!" is singula an cannot be invete Fom ou iscussion in Section 9, we can still choose a well behave!! that still minimizes, so that we can aw eliable conclusions fom the Enegy Pinciple Physically, the sufaces ae associate with k! =, so that the fiel line bening tem is minimize We may expect these sufaces to play an impotant ole in etemining stability whee an Une these cicumstances, an afte some algeba, one fins L = " a f + g )!!!, 333) f = F!!!, 334) k 6

7 g = k ) + k p k! Some geneal emaks can be mae " 1 k ' F + k kb 4 z " mb ) k ' F!!! 335) 1 Fist, f! fo all, so the tem f "! is stabilizing Howeve, it vanishes at the singula sufaces whee k! B =, so that we may expect instability to be associate with these aii The sign of the tem g! is etemine by the sign of g The minimizing isplacement theefoe shoul have! > whee g <, an! = whee g > It tuns out that the analysis of pessue iven moes aising fom the p! tem equies a etaile analysis of the behavio of the solutions of the Eule equation the ieal MHD wave equation) nea its egula singula points This will be biefly outline in Section 31 Fo cuent iven moes p! = ), the sign of g is etemine pimaily but not completely) by the sign of F Fo the case of a staight tous when k is quantize, we can wite whee F = B! q m + nq)!!!, 336) ) = B z ) ) RB! 337) is calle the safety facto, fo easons that will become clea shotly Consie an unwappe flux suface, ie, a cyline of aius unwappe an laye out flat The magnetic fiel lines lie completely within such sufaces, as sketche in the figue The wapping angle is! = tan "1 B / B z We efine the pitch of the fiel lines in the suface as P) = B z / B! This is the istance that one woul tavel axially in the z- 7

8 iection) by following a fiel line though one cicuit fom! = to! = " The pitch is a function of aius, meaning that the wapping angle vaies vaies fom suface to suface The safety facto is theefoe the nomalize pitch, q = P / R The quantity q! / q " lnq / ln is calle the magnetic shea Fom Equation 336) we see that when F =, q =!m / n, ie, q is a ational numbe This is why these sufaces ae also calle ational sufaces The fiel lines close upon themselves afte m tuns in the poloial! ) iection an n tuns in the tooial axial, o z) iection, within the suface The safety facto at the oute bounay = a is qa) = ab z a) / RB! a) Since B! a) ~ I / a, whee I is the total tooial axial) cuent, qa) ~ a B z a) / I ~ total tooial flux)/total tooial cuent) The moe cuent fo a given flux, the smalle qa) The configuation is unstable if g <, which can only occu if F <, o q) <!m / n In the tokamak liteatue it is customay to wite n! "n, so that q < m / n Since it is only the elative oientation of k! = m / an k z = k = n / R that ente the theoy, it is also customay to consie only m! Hee we employ the stana mathematical notation fo the Fouie ecomposition an live with the minus sign) Theefoe, esticting the iscussion to m >, the configuation is stable if n >, an may be unstable if n < an q < m / n The m = 1 moe may be unstable if q) < 1 / n We theefoe must equie q) > 1 eveywhee to assue stability In paticula, at = a we equie B z a) / B! a) > R / a This is the Kuskal-Shafanov stability conition It means that the atio of the tooial axial) flux to the tooial axial) cuent cannot be too small It implies the necessity of a stong tooial magnetic fiel fo stability This fiel, which oes not contibute to confinement, must be supplie by extenal means The Kuskal- Shafanov conition foms the basis fo the esign of the tokamak We have seen that the minimizing isplacement must have! > when g <, an!! = when g > Fo the case when g is a monotonically inceasing function of, an to the extent that the sign of g is etemine by the sign of F, the situation is sketche in the figue 8

9 This is the geneal case fo a tokamak The top hat shape of the isplacement is typical of the m = 1 moe in a tokamak Note that the isplacement vanishes outsie the ational suface, so that it oes not feel the wall The most unstable m = 1 isplacement in a tokamak will consist appoximately) of a igi shift of the flux sufaces insie the ational suface, an no isplacement outsie the ational suface We emphasize that this iscussion, an the figue, ae heuistic an appoximate because of the stabilizing tem in g popotional to F ; the oot of g = oes not exactly coespon to the oot of F = Howeve, the pictue seves as a easonable guie to what happens when a moe etaile analysis is attempte If instea g is a monotonically eceasing function of, then the situation is like that shown in the figue The minimizing isplacement now must vanish insie the ational suface, an be nonzeo outsie of t Howeve, it must also satisfy the bounay conition! a) = Theefoe "!, an the fist tem in Equation 334) can contibute to stabilization The etails epen on the elative location of the wall an the ational suface This type of stabilization is calle wall stabilization A monotonically eceasing pofile is chaacteistic of a Revese-fiel Pinch o RFP) We will escibe this concept in moe etail when we iscuss plasma elaxation in a late Section This conclues ou iscussion of the Enegy Pinciple It has povie a means of eucing some vey geneal popeties of MHD stability, an we neve ha to solve a iffeential equation! This will not be the case fom now on As mentione, the popeties of the Eule equation nea a egula singula point must be investigate in oe to etemine the stability of pessue iven moes An, the Enegy Pinciple is no longe vali in the pesence of esistivity, so of necessity we must aess iffeential equations when we stuy esistive instabilities That is what follows Of couse, all of the calculations of this Section can be an have been) caie out in tooial geomety, but the pimay concepts emain unchange 9