Numerical Calculations of Steady State Microave Plasma Parameters by J C Sprott and A C England PLP 489 Plasma Studies University of isconsin These PLP Reports are informal and preliminary and as such may contain errors not yet eliminated They are for private circulation only and are not to be further transmitted ithout consent of the author or major professor
The ork described here as motivated by a desire to predict the plasma parameters in the ELMO Bumpy Torus presently under construction at Oak Ridge The basic method is a zero-dimensional computer calculation in hich the particle and poer balance equations are simultaneously solved n the course of the development of the computer code, it became apparent that it could be applied to any steady state cylindrical plasma such as a mirror or a large aspect ratio torus n fact, ith only minor modifications, the technique can be extended to time dependent plasmas as ell Shon belo are the three differential equations that are simultaneously solved dn err = n LON n -1- dt P T - T e = e 1 at nv- L el T e 3 LDFF LON dt T -T 1 _ e 1 at - L el T T 1 1 ----- LDFF LCX The first is a particle balance equation The electron and ion densities are assumed equal Particles are produced by ionization and lost by diffusion The ionization time is approximated by a complicated analytic function of electron energy The electrons and ions are assumed to be Maxellian The neutral pressure outside the plasma is specified explicitly, and the pressure inside the plasma is calculated using the mean free path for neutral atoms Both thermal and Franck-Condon neutrals are considered Various diffusion
- 2 - mechanisms have been considered including neo-classical and Bohm The electrons are heated by microaves The heating is assumed to be 1% efficient so that the heating rate is just the input microave poer divided by the number of particles Energy is lost from the electrons through Coulomb collisions ith the ions, through diffusion to the alls, through excitation and ionization of neutrals, and by Bremsstrahlung and synchrotron radiation The electrons are assumed to lose 3 ev of energy per ionization to the background neutrals The neutral gas is assumed to be hydrogen, and impurities hich may be liberated from the alls have been neglected The ions are heated by collisions ith electrons and cooled by diffusion to the alls and by charge exchange ith neutrals These three equations are solved by successive iteration The initial conditions are set arbitrarily, and the iteration proceeds until a steady state is reached The microave poer can then be sloly increased in time so as to maintain a steady state This is done by requiring that dt e dt be less than 1% of P l-l!nv The density and the electron and ion temperature can then be plotted as a function of microave poer Fig 1 shos a typical result for the ELMO mirror device ith classical diffusion into the loss cone The ambipolar electrostatic potential has been included, but the method by hich this as done ill not be discussed The density and electron temperature increase monatonically ith poer hile the ions remain cold n the real experiment, most of the stored energy is in a relativistic component of electrons, and these have been ignored in the computer calculation The numbers agree ithin about a factor of to ith the parameters of the cold plasma component actually observed in ELMO Fig 2 shos ho the electrons are losing energy as a function of microave poer Diffusion dominates the losses over most of the range except at
-3- lo poer here the temperature is lo and collisions ith neutrals are important Fig 3 shos the result for the ELMO Bumpy Torus assuming neo-classical diffusion e assume a stable equilibrium ith a density gradient scale length equal to the minor radius of the torus Ambipolar potentials have also been neglected e don't take the numbers too seriously, but there are some interesting features such as the discontinuity at 5 k that marks the transition into the collisionless regime Fig 4 shos here the energy is lost for this case The dependences are a bit complicated, but note that none of the loss mechanisms, except perhaps bremsstrahlung, are negligible over the hole range Fig 5 shos the result of adding Bohm diffusion to the neo-classical diffusion in the Bumpy Torus The density is considerably loer, and the electron temperature runs aay hen the microave poer reaches about 3 k Fig 6 shos here the poer is lost for this case onization and diffusion are the dominant loss mechanisms ith Bohm diffusion These cases are only examples of more than 1 cases that have been run for various combinations of parameters n particular, e have been interested in determining scaling las for the bumpy torus configuration e have been rather encouraged by the results, although e don't take the numbers too seriously because classical confinement seems to be the exception rather than the rule in toroidal devices e intend no to apply the calculation to a variety of existing confinement devices and to make the calculation as realistic as possible by refining the numerous approximations that have been made e are also in the process of developing a one-dimensional version of the code so that e can study the radial dependence of the plasma parameters
E'LMO 5 MULRT ON -- CL L -:' o z,, (, :' :, '::" :,: : 1 t1 : ', ", (Y) * * L U """- m )(( )(( o x "-' - - t--t en :z " 13 12 " - " " : : ' :, "', "! : ;' 11 1 12 13 14 MCRRVE PER(ATTS) Figure 1
ELMO SMULRTON D, 1-:-1 18 U a: t- U -J t- cc -J : 1-1 ' 1-2 1-3 1-1! 1-5, --- - -- -- -- _ -- - - Mirror Loss -- -- --- -- ' onization "", " ' '' -!" on Heating -- ---=- ",, --- Synchrotron Radiation "- ', : J, ' Radia tion,brems s trahlung 12 13 14 M CRAVE PER (ATTSL Figure 2
--- EBT S MULRT ON ' ::t: t- 15 Neoclassical Diffusion " cr: (Y') )!( 3(( L U "- en *, a -,x - t- +-t (f) 1L! 13 12 11 T e -==---_ ---- -=-- l -,- r-'- - ( ---- T, e T l 1 Figure 3
, -------- ---- onization, EBT SMULRTON r- -- ' (!) ' 1-2 1-1 C- -----, ' :,, : :, ';,: " ";', g' -1;"3 : ' on Heating ':': - --,---,J} V',,,,'J " " ', t,j (_,,"-! 'r"- " ) ", :, : : ''':,:4r -t : ' ' f,k- " ''< ''-'" ", '- ": o, : - ' X, ' ',:', : ' ' D ', ', ' ':',',,, ' : " a: J a: 1-11 1-5 - ync rot ron, S h ----- ),', ',,- o '{, 9 o' -vp -' ) --, Yv f'e, (Q Neoclassical Diffusion 1 1 MCRfjAVE PfjER Figure 4
EBT S MULRT ON, CL :E J- T 15 Bohm Diffusion ' " Density ",," : ;':'" -4 X, - J- t-- (J) 12 11 1 1 2 13 14 MCRRVE PER(RTTS) Figure 5
EBT SMULRTON ------- ---- -- ' onization - ' - -- ' Bobm Diffusion D 1--1 -' D o U '-- on Heating ----- Diffusion _, ' z : t; 1-3 ' -1 1--1 - : -1 cc Synchrotron - - -, -- " - - -- --, ' --- "-, " " ',,, Bremsstrahlung,, ", ', ", ' ' =-L 1 1 2 1 3 MCRRVE PER (RTTS) Figure 6