Final Spring 2007

Transcription

1 .615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o an approximae calculaion of he Troyon limi. The analysis is grealy simplified by several reasonably good approximaions. Firs, he equilibrium is calculaed using he high-bea okamak ordering. Second he equilibrium profiles are a oroidal version of Shafranov s sep-funcion model consising of a plasma core wih an approximaely consan curren densiy surrounded by a force free plasma region carrying no curren. Surrounding he force free plasma is a vacuum ou o infiniy. All he relevan equilibrium soluions have been derived in class and are conained in he MHD exbook. Mos of he equilibrium work involves renormalizing he physical parameers appearing in he formulae aking ino accoun he presence of he force free plasma. If you do he problem righ, here is no much compuing required. However, if you do no hink abou he problem in a mahemaically convenien way here can be a large amoun of ieraion and compuer usage. The procedure described below requires no ieraion alhough as you will see i involves somewha uninuiive mahemaical parameers. In any even feel free o use whaever procedure you like jus make sure you ge he correc final answer. The sabiliy analysis is carried ou using a δw analysis. This can be a dauning ask aking monhs o years of compuaional effor o ge he exac answer, even using he simplifying high-bea okamak ordering. For he final exam he analysis will be grealy simplified by using a rial funcion for he displacemen wih only a few parameers. I will give you wha I hink is a reasonably good rial funcion bu you should feel free o do beer if you wan o ry. 1

2 Par 1 The Equilibrium Problem The sep-funcion, force-free model of ineres is skeched on he las page. The final goal of he equilibrium calculaion is o rewrie he soluions obained in class in erms of more appropriae physical parameers as described below. There are wo specific pieces of informaion ha you need o calculae. (1) A boundary curve in pressure-curren space showing where equilibria are possible. Each equilibrium soluion is subjec o he consrain ha he safey facor on axis q is held consan. () An evaluaion of he safey facor a he edge of he plasma core q a and a he edge of he force free plasma q a. The saring poin for he calculaion is he high-bea okamak equilibrium for a circular cross secion plasma. The flux funcion in he plasma core is given by Eq. 6.11, ψ core 1 = r 1 + ν r 3 r cos θ a B 3 q a a a All quaniies are idenical o hose given in he exbook and he class noes excep ha a, q,νεβ,, are now wrien wih a prime o indicae ha hey correspond o values on he boundary of he core and no he rue ouer plasma boundary which occurs a he edge of he force-free plasma region. The corresponding parameers a he edge of he force free region have no primes. Mos of he work in his par of he exam is o deermine he relaionships beween he primed and unprimed parameers. The equilibrium flux funcion given in he exbook for he vacuum region acually is valid for boh he force-free and vacuum regions in he exam problem. The reason is ha by assumpion zero curren flows in he force free region so ha he magneic flux is he same as for a vacuum. This flux is given by Eq ,

3 ψ ouer 1 r ν r = ln a + cos θ a B q a a r To begin he analysis, choose one of he vacuum flux surfaces inside he separarix and define his as he edge of he force free plasma. The lef radius of his surface is denoed by a 1 while he righ radius is denoed by a. In general a 1 a. The acual radius of he plasma is denoed by a and his is a basic geomeric quaniy of physical ineres. By definiion 1 a = (a 1 + a ) As you progress all quaniies should be expressed in erms of a raher han a. To specify an equilibrium soluion you will need o give numerical values for hree parameers. For mahemaical and compuaional convenience hese parameers are denoed by ζ 1, α,q. The definiions are as follows. The quaniy q on he magneic axis. We shall assume ha q = 1.5 for all equilibria under is he safey facor consideraion in order o avoid he inernal m = 1,n = 1 sawooh mode. The quaniy alpha is defined by α = a / a. I approximaely measures he raio of he core radius o he oal plasma radius i is inversely relaed o he peaking facor. Clearly < α < 1. The las quaniy of ineres is ζ 1 = a 1 / a. I is relaed o he shif of he plasma and hus indirecly o he plasma bea. Physically we expec ha ζ 1 > 1 since he plasma core is shifed ouwards wih respec o he plasma edge. Now carry ou he following calculaions: 1. Assuming ha values are given for ζ 1 and α, derive an expression for he value of ν. Ulimaely here are wo equilibrium limis o evaluae. The firs one is qualiaively similar o he one derived in class. I corresponds o he siuaion 3

4 where he separarix moves ono he ouer edge of he force-free plasma. This condiion yields anoher relaion beween ζ α 1,, and ν. Derive his relaion. 3. Using he resuls of pars (1) and () plo curves of ν vs ζ 1 and α vs ζ 1. This is one se of curves ha define he allowable equilibrium space. You should also find ha soluions exis only for ζ 1 < ζ cri. Find ζ cri. (Hin: ζ cri range 1 < ζ cri < ) lies in he You will evaluae he second equilibrium limi shorly. A his poin, however, i is more convenien o derive relaions beween he physical parameers a he edge of he plasma core and a he edge of he force-free plasma. To do his you will need o evaluae he oal cross secional area bounded by he ouer edge of he force free plasma. Do his as follows. 4. Assume allowable values of ζ 1 and α are specified. Wrie a subrouine ha evaluaes he normalized area  = A / πa. Noe ha  = Â(ζ1, α). A small amoun of algebra and compuaion is required o carry ou his ask. You should now be in a posiion o calculae he physical parameers of ineres. Again, assume ha values of ζ 1 and α are specified along wih he value q = 1.5. This can be done convenienly in he following order. 5. Derive expressions for a. ν (already done in par 1) b. q = πa B μri β μ R 1 c. ε = B a πa prdrdθ d. q = AB μri e. β = μ R 1 prdrdθ ε B a A β q f. ν = ε 4

5 You should be able o obain numerical values for each of hese quaniies once ζ 1 and α are specified. 6. Using hese relaions plo a curve of β / ε vs 1/ q for he special equilibrium bea limi consrain given in par (3). The nex par of he problem involves he second equilibrium limi menioned above. This is a limi on he smalles allowable value of q (i.e. larges allowable value of 1/q ). If we wan o hold q fixed we canno decrease q indefiniely because in general q < q. Since q A / I he smalles q a a fixed curren occurs for he smalles A. Clearly he smalles A occurs when he ouer surface of he force-free plasma moves inward unil i coincides wih he surface of he plasma core. 7. Assume he ouer surface of he force-free plasma coincides wih he surface of he plasma core. Using his consrain you should be able o deermine a relaionship beween β / and 1/q. ε Deermine his relaion and superimpose i on he plo of β / ε vs 1/q obained in par (6). If you did everyhing righ you should now have a closed region in β allowable. ε - 1/q space where equilibria are / The las par of he equilibrium problem involves he calculaion of safey facor q on cerain surfaces. This informaion will be needed o carry ou he sabiliy analysis. 8. Derive (bu do no evaluae) a general inegral expression for he safey facor on any flux surface, eiher in he core or he force free plasma, using he high bea okamak ordering. Now, explicily evaluae q = q ζ a a ( 1, α), he safey facor a he surface of he plasma core. This is easy we did i in class. Finally, wrie a subrouine ha evaluaes q a = q a (ζ1, α), he safey facor a he ouer edge of he force-free plasma. A small amoun of compuaion is required for his ask. NOW YOU ARE DONE WITH THE EQUILIBRIUM CALCULATION 5