Equilibria of a cylindrical plasma

Transcription

1 // Miscellaneous Execises Cylinical equilibia Equilibia of a cylinical plasma Consie a infinitely long cyline of plasma with a stong axial magnetic fiel (a geat fusion evice) Plasma pessue will cause the axial fiel insie to be smalle than the fiel outsie as a esult of the iamagnetic ift cuent If this plasma, in aition, caies an axial cuent J, the J x foce will "squeee" the plasma an the fiel insie can be lage than the fiel outsie (paamagnetism) In this execise we will examine equilibia of cylinical plasmas The fist example is a plasma with pessue, but no axial cuent The secon example is a plasma with axial cuent but no pessue The thi example contains both axial cuent an pessue Cylinical plasma column I Plasma cyline with pessue an no axial cuent In equilibium, the J x foce must cancel the ga P plasma foce The vecto equation fo equilibium is: J P plasma We will assume that the plasma is cylinically symmetic (/= an / = ) The applie magnetic fiel is an the gaient in P plasma is in the aial iection The aial component of the vecto equation above is J J P plasma We assume also that thee is no axial cuent J The applie fiel is assume to be Tesla Tesla The subscipt eo fo vaiables is use hee to inicate a bounay conition, except fo Define: μ 4π 7 The extenal magnetic pessue is then: P mag P μ mag 3979 N/m Assume a plasma aius of m: a metes Let the plasma pessue be 3 of this an vay paabolically with aius: P plas ( ) 3P mag N/m a If thee is no axial cuent J, then the equilibium equation gives a simple esult fo J : J P plas

2 // Miscellaneous Execises Cylinical equilibia Ampee's law is: J an the component is: J Afte substituting fo the iamagnetic cuent, Ampee's law becomes: P plas The poblem is thus euce to a single iffeential equation with a solution that can be foun analytically We will, howeve, fin the solution by integating the iffeential equation, in oe to see how the poblem can be solve computationally The Runge-Kutta metho equies that we efine the eivative of as a function D(, ): D μ P plas () The applie magnetic fiel appeas at the maximum aius = a an is the bounay conition The fiel () is then foun by integating fom = a to eo This is one using the Runge-Kutta outine in Mathca an the answe matix will be M ans M ans kfixe a D We integate fom a to in steps Recall that the fist columns of M ans will have the values of an the secon column will have the values of M ans M ans The plot shows the iamagnetic euction in

3 // Miscellaneous Execises Cylinical equilibia 3 Fo plotting, we will nee the plasma pessue as a function of aius: The new vaiable P plasma is efine as a vecto with the same numbe of elements as the aius We similaly efine a vecto with the magnetic pessue: P magnetic P plasma P magnetic P plasma P magnetic P plasma P plasma P plas () P magnetic μ Ty it: Compae the pessues above to the pessue of stana atmosphee II Plasma cyline with axial cuent an no pessue gaient In the absence of a pessue gaient, the equilibium equation becomes: J ( ) This equation is satisfie if J is paallel to The equation can be ewitten: ( ) whee is a coefficient of popotionality that can be a function of aius J is foun fom: ( ) J The component of the equation is: ( ) These two equations can be combine: ( ) ( ) The component is: ( ) If () is simply a constant, then a solution fo is the eo oe essel function J ()an a solution fo is the fist oe essel function J ()

4 // Miscellaneous Execises Cylinical equilibia 4 Define a new vaiable theta, witten as one wo, which is efine as the pouct of an The equilibium equations ae: ( theta ( ) ) ( theta) ( ) ounay conitions will be specifie on the axis athe than at the plasma suface The integation will begin at = athe than = a The stating values will be: theta The paamete will be a constant: λ 3 The vaiable is a vecto with the two components of : theta The equilibium equations in the fom use by the Runge-Kutta outine ae: D( ) λ λ In oe to avoi ivision by eo, we will begin the integation at = athe than eo: The integation goes fom = to = with steps M kfixe theta D The columns of the answe matix will be put into vaiable with names that we ecognie: M M theta M M theta theta A vectoie equation

5 // Miscellaneous Execises Cylinical equilibia A foce-fee equilibium The equilibium that we have foun is calle a foce-fee equilibium because thee ae no foces Ou Runge-Kutta solution fo constant agees with the essel function analytic solutions The plotte pofile applies to the evese-fiel pinch fusion evice which has an axial fiel that is evese at the ege of the plasma theta J λ J λ 4 8 The solution above is not ealistic because the plasma is colest an most esistive at the wall Thus the cuent ensity shoul go to eo at the wall If we constuct a function () that goes to eo at the wall, then the cuent ensity at the wall is automatically eo Let's fin a moe ealistic equilibium with J = at the wall using the new efinitions: λ() λ a D( ) λ() λ() In oe to avoi ivision by eo, we will begin the integation at = athe than eo: The integation goes fom = to = with steps M ans kfixe theta D The columns of the answe matix will be put into vaiable with names that we ecognie: M ans M ans θ M ans θ theta An we will efine the cuents: Jtheta ( λ() ) μ J ( λ( ) theta) μ

6 // Miscellaneous Execises Cylinical equilibia M ans Plot of the components of the cuent J Jtheta J Plot of the equilibium magnetic pofile using a paabolic () pofile theta J λ J λ 4 8 The essel function solutions (soli lines) ae plotte fo compaison These apply if is a constant Ty it: Note that the axial magnetic fiel is no longe evese Incease the value so that the evese fiel pinch configuation ( evese at the wall) is ceate Fining the cuent I fom the cuent ensity J : We have foun the cuent ensity J at a set of aial locations The axial cuent can be foun by integating J ove the aea, but fo this we nee a continuous function not a tabulate function The tabulate function can be mae continuous using spline intepolation: Ceate the vecto of coefficients vs: vs cspline( J) Define the function: J( R) intep( vs JR ) Do the integation: a π JR ( ) RR 78 amps Ou plasma caies about million amps

7 // Miscellaneous Execises Cylinical equilibia 7 III Plasma cyline with cuent an a pessue gaient A voltage applie at the ens of a long plasma column will ive a cuent The conuctivity of the plasma is geatest paallel to the fiel lines thus we can assume ou applie cuent is paallel to The iamagnetic cuent that is equie fo equilibium is pepenicula to We will a the applie an iamagnetic cuent ensities to obtain the total cuent ensity, an then integate Ampee's law to fin the magnetic configuation A Diamagnetic cuents J Fom the equilibium equation, we obtain: J P The pessue gaient is only in the aial iection, so we obtain: J P plas J P plas Applie cuents We must make an assumption egaing the istibution of the applie cuent We will assume that the amplitue of the cuent ensity vaies paabolically In oe not to istub the foce balance, we will assume that the applie cuent flows paallel to the magnetic fiel The applie cuent ensity is then: Define a cental cuent ensity: J amps/m J applie () J a J applie J ( ) Japplie J ( ) Wite the components of Ampee's law: P plas [ J Japplie ] J ( ) applie P plas [ J Japplie ] J ( ) applie Again us the vaiable theta to epesent multiplie by Again efine a vecto with theta an as the eoth an fist compontents: theta

8 // Miscellaneous Execises Cylinical equilibia 8 C Integation of Ampee's law We will fin the equilibium magnetic fiel pofiles by integating Ampee's law using the iamagnetic cuents J that satisfy the equilibium elation an the applie cuents that ceate no aitional foce D3( ) μ μ P plas () P plas () J applie () J applie () Note that / in these expessions is The above efinition is the same as: P plas J ( ) applie P plas J ( ) applie Fin the fiels by integating Ampee's law fom to in steps: M3 ans kfixe theta D3 M3 ans Use familia vaiable names: M3 ans M3 ans θ M3 ans abs theta θ theta

9 // Miscellaneous Execises Cylinical equilibia 9 Plot of the equilibium pofile with a cuent of MA 8 theta This pofile with (a) << is nea to the pofile fo a tokamak plasma if it wee convete to a staight cyline Note that thee is an inflection point in the pofile fo this choice of P an J applie Ty it: Expeiment with iffeent values of cuent ensity an plasma pessue D Test the equilibium fo accuacy If thee is an equilbium, the J X an Ga P foces ae equal an opposite We can check that by plotting them iniviually an by plotting thei sum The equilibium equation is: P plasma J J J J applie applie The gaient in pessue that we will nee is: GaP( R) R P plas ( R) k ows() Inex fo the numbe of aial locations The J X foce at each ofthe k aial locations is: JX k theta k J abs applie k k k abs k GaP k k k J abs applie k k theta k abs k GaP k theta k

10 // Miscellaneous Execises Cylinical equilibia Plot showing that the J X foce an Ga P foce ae equal an opposite: 3 JX k GaP k JX k GaP k k The nea-eo sum at ight shows that the metho fo fining the equilibium is accuate JX k GaP k Refeence: Paul M ellan, Funamentals of Plasma Physics (Cambige Univesity Pess, Cambige, ), p 48

11 // Miscellaneous Execises Cylinical equilibia