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1. PHYSICS AUXILIARY _ PUBLICATION SERVICE Document No PFLDA 29 1558 22 Journal Reference PHYSICS OF FLUIDS Title User s Manual for the Flora Equilibrium amp Stability Code Authors Cohen 5 Freis Current Physics Microform Reference 8605c 0001 A service of the American Institute of Physics UCID 20400 User s Manual for the FLORA Equilibrium Laboratory Work performed under the auspices of the U S Department of Energy by the Lawrence Livermore Laboratory under Contract W 7405 Eng 48 This is an informal report intended primarily for internal or limited external distribution The opinions and conclusions stated are those of the author and may or may not be those of the and Stability Code Robert Freis Bruce I Cohen April 1 1985 DISCLAIMER This document was prepared as an account of work sponsored by an agency of the United States Government Neither the United States Government nor the University of California nor any of their employees makes any warranty express or implied or assumes any Tegal liability or responsibility for the accuracy completeness or usefulness of any information apparatus product or process disclosed or represents that its use would not infringe privately owned rights Reference herein to any specific commercial products process or service hy trade name trademark manufacturer or otherwise does nct necessarily constitute2. p X RO Q 72 XIO 2 ud G XRO r XIO 1 linebending B r B X10 2 2 z z 87 pe XRO B r x10 curvature Pi y Brriztr Ple y P1 y XRO XIO fir m 1 X RO X10 x RO r B r 2 X10 r B r X10 X RO and XIO are the real and imaginary parts of the perturbation Y Y Y is the quasi elastic term Q B 4 p m is the azimuthal mode number and subscripted quantities are derivatives with respect to the subscript e g axRO The accurracy of this check is limited because the derivatives are calculated by finite differences If the relative energy is defined as kinetic f potential Pel cs eed eee f kinetic f potentzal where potential linebending curvature flr then typical results have Rel z a few percent The relative energy error can be significantly worse in extreme cases of very low beta for which the line bending terms involve the products of a relatively large quantity Q and very small quantities the z derivatives of the flute mode amplitudes 12 INPUT DATA Apendix B is a list of the input data All input is in format free Namelist mode and is echoed to the output one dimensional plot file Most input is Data loaded with default values located mainly in subroutines Input and Inputtm The input file name should be INFLMS or else the code
3. vacuum B fields electric potential Tandem mirror systems are simulated by assuming symmetry around the midplane z 0 and calculating half of the total system The half system cam consist of either two or three cells referred to as the center cell the choke cell and the end plug cell The cell boundaries are defined by solenoids which generate the vacuum B fields Within each cell several pressure components both perpendicular and parallel can be specified which will satisfy pressure balance equations and together with the vacuum B fields will generate the self consistent finite B fields Densities and potentials are defined by analytic functions Appendix A The potentials are not self consistent with the equilibria 3 VACUUM B FIELDS The vacuum B at any point in space is NCOIL BVAC z bs 2sc 2 BCENTER z 8 1 independent of because of the paraxial model Ncoil is 2 or 3 b is th on axis B field of solenoid located at z Appendix A BCENTER is a constant from z 0 to a specified transition region z ztrans beyond which it rapidly falls away see Appendix A It is designed to represent the center cell vacuum magnetic field For a 3 region case 3 Bmax chake cell end plug cell 2 zimin 22C z2min For 2 region case NCOIL 2 21min 22c Note 1 BMAX BVX2 BVX3 BVO are the resultant values due to all the sources
4. execution line must be extended to account for a different name on the data file 13 CODE EXECUTION At execution time the user s private file list must contain the controllee XFLM8 and a data input file To execute XFLMS simply type its name if the input file is INFLMS Otherwise type XFLM8 INFLMS8 input file 8 where input file stands for the name of the input file After completion of the run there will be two new plot files in the users private file list unless one or both are suppressed by input data both files have names that begin with F3 Both the executable file XFLMS and the fortran source file FLRMS can be obtained from Filem storage by typing FILEM READ 326 HANDOUT XFLM8 FLRM8 14 SOME OBSERVATIONS BASED ON EXPERIENCE The equilibrium calculations are relatively inexpensive in time a few seconds for meshes of 2500 points so an obvious strategy is to optimize the equilibrium as much as possible by a number of short runs of one or two time steps before doing a full stability run of typically a few hundred time steps The equilibrium runs can be further economized by turning off the 3 D plots whenever possible NO3D 1 The diagnostic 1 D plot of FLUTES is very useful even for low m high 8 cases FLUTE3 is the line average of the square of the curvature driven MHD growth rate in the limit of high m and low 8 usually referred to as 42 Any design changes e g pressure profiles e ring positions etc whi
5. p T im 5 subscripts mean derivatives and all coeffi cients are presumed to depend 2 and 15 APPENDIX B The input data is entered via four namelists Within each namelist the order is arbitrary Default values are preset for most data see the code listing Namelist 1 Namelist 2 input aname bias ex0 exl fil fizx fji fjrx fpsi fu fv fz jfour kplotm kzs Imax ndiag nen input description nold turns off on 1 D plots no3d 1 0 turns off on 3 D plots description problem identification up to 5 fields of 8 letters each Time centering parameter bias 0 for fully centered bias 1 for fully forward bias initial perturbation coefficient set 1 for random initialization initial perturbation coefficient set 1 for cosine initialization minimum z boundary condition set 1 for 0 slope set 1 for 0 value maximum z boundary condition set 1 for 0 slope set 1 for 0 value minimum psi boundary condition set 1 for 0 slope set 1 for 0 value For fj1 0 mm 1 results in 0 slope mm gt 2 results in 0 value maximum psi boundary condition set 1 for 0 slope set 1 for 0 value grid stretching parameter see sec 9 default grid stretching parameter see sec 9 default grid stretching parameter see sec 9 default grid stretching parameter see sec 9 default w index at which XROis Fourier analyzed in 2 index o
6. 1 0510 DS max 9 MASS DENSITY p The general form for each componenet s is ps p es fs B 95 amass The coefficients f g are are based on the assumption that each density component is proportional to its related perpendicular pressure with the proportionality constant determined from input data The total mass density is p p t po 8 where is a constant calculated from input data po ncenter x cold X amass see Appendix B 10 GRID STRETCHING Nonuniformly spaced physical grids z are analytically mapped onto a uniform com putational grid u v to improve numerical accurracy in the finite differencing scheme The mappings have the form l zru z 2 u hmaz where _ lafz In fu vy Die In fv fz fu f v fv are input data within the range O to 1 These relations have the properties zmar umar Ymar vmaz The z value 2 fz x zmaz maps to the u value u fu x umaz and similarly for y and v For example if fz 5 and fu 7 then the inner 50 of the physical space will be represented by 7096 of the computational points with the concentration of points increasing at smaller values 1i ENERGY CHECK The energy constant 2 eq 3 ref 2 is calculated by a two dimensional numerical integration over the total z v space In FLORA variables d d ES kinetic linebending curvature flr 7 where kinetic
7. 2 2 BV _ ppasi ppas2 2 BMAX BMAX BMN1 _ ppas3 ppas2 5 a BV X2 BM N1 BVX2 BMN 5 BV X24 BMN14 BMN1 bg 5 4 BV X2 Zmn2 c a BM AX ppasl H 0 21 b4 zic Zmn1 bs H 2mn1 22 X24 ppas2 BM AX n prn BMAX BMN Ia 6 c5 ppas2 a5 BMNI 65 BMN1 H zmn1 22 BMN 4 BV X2 H z2c Zmn2 a 4a BMX 2 b pslosht pslosh psloshe Note as without ie ag a5 H 2mni Z2c Likewise for b and c A 3 HOT ELECTRON PRESSURE For long 0 pperpe B abf B bbf B cbf 12 abf LES 3 H 22 24 1 bbf 2abf cbf abf for long 1 and AC gt fring BV X2 priri bbf 2abf B d pring BV X2 BV X2 Be abf BV X2 2 B H z3 z4 for long 1 and BV AC lt abf 0 bbf 0 cbf pring H z3 24 A 4 SOLENOIDAL VACUUM B FIELD P AL Zsc 2 Sc X ae 212 217 zat f a je 2e Az 2 s 1 NCOIL FLORA determines K for each coil by simultaneous solution of this equation at z 21 22 23c for NCOIL z Zic Z2c for NCOIL 2 given the desired vacuum magnetic field amplitudes as input 5 CENTER C
8. 231 1973 W A Newcomb J Plasma Phys 26 529 1981 W A Newcomb Mirror Theory Monthly LLNL Sept 1981 D B Nelson C L Hedrick Nuclear Fusion 19 283 1979 D A D Ippolito J R Myra J M Ogden Plasma Phys 24 707 1982 10 APPENDIX A A l PRESSURE NORMALIZATION Input quantities are beta s which are converted to pressures as follows pcenter betcent 252 ions pcentee betcene T warm electrons 1 sum over species pltrap betrap ASX sum over species pslosh betslsh Au ions psloshe betslse A warm electrons ppas2 and ppas3 are inputed as fractions of betpasl A 2 EQUATIONS FOR a b PRESSURE FORMULAS Define a function 21 z2 as zd afa e ES A z 22 0 otherwise Then pperp B pcen B ppas2 B ppas3 B ptrap B pslosh B abp B bbp B chp 6 abp js bbp b Ja cbp t 1 pcenter pcentee 1 BV0 BMAXY BMAX4 0 zie 11 ptrap az m du ae gt G BMN BV Bv xii Ce pslosht ee hls ESI NR svxs fi ppas2 pp ppas2 Bins BMAX BM ba 1 BMN1 2 os 3BMNE 7 ng 2a BV XP th BVXP BMN BV X27 BM 2 b 2 BMAX b 2 BV X
9. ELL VACUUM FIELD b 3 if z 2trans 3 BCENTER ee 2 5 bceng exp otherwise A 6 SELF CONSISTENT B FIELD 13 Define U1 P2 v abp PA V abf U2 P2 bbp 1 bbf BV AC z U3 P2 p chp P1 cbf 5 then 1 U2 uzy vni nn son 3 For very low pressures B z BV AC z 2 U3 2 U2 BV AC z 2 U1 BV AC z f U1 where f U1 4U3 UX 8ULU3 BV AC z AU2 BV AC z 12 U1 U2 BV AC z 8 U1 BV AC z 1 a U2 2 03 U3 A 7 ELECTRIC POTENTIAL z v o1 z v where phipl phi argl Pilem PERCE exe er arg2 1 y 0 2 argi zxpot 1 H 0 21 2 1 20 2 arg2 wpot 20 22 14 8 FINITE LARMOR RADIUS TERMS XX X z v p z v QogxB wvp w 8 6 YYY z v 2 ugxB eExB w sf8 where UExB Op Pi z v OVB wei p z v B 2 v P i Wet p z 2 is the ion mass density P is the ion perpendicular pressure we is the ion larmor radius and c is the speed of light A 9 THE EQUATION OF MOTION FOR THE PERTURBED RADIAL DISPLACE MENT x The equation of motion for the perturbed radial displacement is Bs 1 m ET X V m Pi x B n r Bx r B r J 50 where
10. ag value of at which and are set constant value to which pg is set if p lt p2flag mirror ratio of the inner component of the sloshing profile sec 6 wall radius in cm slightly less within one grid cell or equal to wall radius rw exponent coefficient in end plug cell potential exponent coefficient in center cell potential power of polynomial in potential dependence outer axial position where hot electrons go to 0 description three element array of z location of each solenoid center three element array of radius of each solenoid magnetic field at the choke coil solenoid Gauss sec 3 magnetic field at the inboard end plug solenoid Gauss sec 3 18 Namelist 4 continued input bmx3 bceng dt dphi epsp kin ltrans ncoil phicen phiplg pfudge theta0 zmax zic z2c z3c description magnetic field at the outer end plug solenoid Gauss sec 3 magnetic field in center cell Gauss time step sec ignore minimum pressure normalized to 1 below which B is calculated by an expansion see sec A 4 number of points used in Simpson s quadrature for rzz default 23 transition length for central cell vacuum number of solenoid coils also regions maximum electric potential in center cell maximum electric potential in end plug cell parameter on sloshing shape default 0 ignore maximum z of the domain z location of choke solenoid z location of end plug inboard solenoid z locatio
11. cell minimizes at the minimum B field of the choke cell peaks at the entrance to the end plug cell and goes to zero at the minimum B field of the end plug cell In contrast to the trapped pressures the passing pressures in all cells are related and the input for the pressure in the choke and end plug cells are expressed as fractions of the center cell passing pressure Passing pressure 21 6 zimin z2min 5 8 RADIAL v PRESSURE PROFILES ions and warm electrons have the same p w Hot electrons have a separate pi For ions the form is 9 ps v dip p2wide p3a p3b p p3c p3d y This allows hollow profiles The constants p3a p3d are calculated such that has a maximum at 1 p2wide and p3 w goes to zero at Setting the input quantity dip to zero removes the hollowness 1 pat v 2 1 tanh Hot electrons have a p of the form _ J if 0 Y lt va pi 9 if Yme lt V lt Vnaz where rem E em el pe10 bel cel del De1 1 Vme Pme 2 ss 21 1 Woe tan fer R p2ewide pei and pez are matched to give continuous pressure and slope at Yme p2ewide is an input and Yme Poe 1 p2ewide pero an input sets the normalized value of pe at 0 and is designed to adjust the profile from disk shaped pe1o 1 to ring shaped peio 0 Pil vs v 5
12. ch reduce FLUTE3 move the system toward stability Note that even with regions of positive FLUTES it is possible that the system is stable due to flr and wall effects To avoid numerical instabilities or intolerable inaccuracies the time step dt must be con strained conservative first guess is to satisfy the conditions U flr dt 1 FLUTE3 dt lt 1 is the real frequency due to the fir gyroscopic terms in the Lagrangian ref 1 2 Appendix If the flr term XXX is turned off 3 6 0 one of the constraints on dt is relaxed If YYY is also turned off 8 8 0 the system of P D E s is decoupled and the iterations can be dispensed with LM AX 0 In the general coupled case LMAX 4 has usually been required to insure numerical convergence 15 ACKNOWLEDGMENTS We are pleased to acknowledge our debt to W A Newcomb for developing the basic theory upon which this work rests We are also grateful to L L LoDestro T B Kaiser L D Pearlstein and J J Stewart for many helpful discussions and suggestions This work was performed under the auspices of the U S Department of Energy by the Lawrence Livermore National Laboratory under qontract number W 7405 ENG 48 9 REFERENCES B I Cohen R P Freis W A Newcomb Finite Orbit Corrections to Ballooning Interchange Stability of Long Thin Axisymmetric Systems Mirror Theory Monthly Nov Dec 1982 LLNL and LLNL report in preparation W A Newcomb Ann Phys 81
13. ems with finite pressure and finite ldrmor radius effects FLORA solves an initial value problem for interchange rota tional and ballooning stability 1 INTRODUCTION This user guide is a brief description of the FLORA code and is designed to be used in conjunction with the code lisitng The theory and general equations which this program solves are described elsewhere in detail FLORA solves in a 2 D domain 2 0 for the linearized stability of a long thin axisym metric equilibrium It uses an initial value method in which an equilibrium is given an initial perturbation to its magnetic B field and the time behavior of the perturbation is followed The perturbation has been Fourier expanded in the azimuthal 0 direction and each mode m must be examined separately The values of m can be arbitrary with an upper limit around O 10 because of accuracy consideration as the modes become more highly localized The complex partial differential equation of motion for the perturbed radial displacement of the field lines Appendix A is solved as a coupled system of two real p d e s and the solution consists of two parts the real part called XRO in FLORA and the imaginary part called XJO The system is solved by bringing the coupling terms in each equation to the right hand side and using an iterative technique 2 FLORA EQUILIBRIUM OVERVIEW FLORA equilbrium are specified by the following spatial quantities pressure P density
14. f spatial location of time history plots If set 0 center of region automatically chosen flute mode initialization kzs 1 ex0 1 ex1 0 sets initial condition r B XRO 0 and r B XIO 0 iteration parameter azimuthal mode number number of time steps between diagnositc plots number of time steps between energy checks tl i 16 Namelist 2 continued input nfourmax nfourp nmax sf6 sf8 swgl swg2 swg3 swe4 Namelist 3 input bceng betcene betcent betpasl betrap betring betslse betslsh cold dip dpasl dpsihrel ditrap echarg fring long ncenter nsloshin description no of times the buffer is read to the history file for Fourier analysis Fourier analyze XROevery nfourp th time step total number of time steps for problem arbitrary scaling factor on the gyroscopic flr term XXX arbitrary scaling factor on the quasi elastic flr term YYY arbitrary scaling factor on the curvature drive term default 1 arbitrary scaling factor on some of the line bending terms default 1 arbitrary scaling factor on some of the line bending terms default 1 arbitrary scaling factor on some of the line bending terms default 1 description center cell magnetic field in Gauss peak center cell electron 81 peak centew cell ion 8 center cell passing peak choke cell 8 peak hot electron 8 peak warm sloshing electron 81 peak sloshing ion 8 a global density m
15. inimum as a fraction of ncenter the center cell density parameter in pressure profile sec 8 center cell passing density width relative to maz of transition to halo region peak density in choke cell ion charge Default 4 8e 10 used for elongated hot electrons See sec A 3 switch which sets hot electron z length as elongated long 1 or regular long 0 center cell density particles cm peak plug cell density particles cm 17 Namelist 3 continued input 10 2 ppas3 psi0rel psiOerel psihrrel psislp psi3rel p2wide p2ewide plfloor p2flag p2floor rpl rw rwl wpot xpot ypot 23 Namelist 4 input als bmxl bmx2 description coefficient of hot electron radial pressure profile sec 8 minimum passing pressure in the choke cell expressed as a fraction of 1 sec 7 maximum passing pressure at the inboard mirror of the end plug cell V value relative to maz at which ion radial pressure is half the maximum v value relative to at which hot electron radial pressure 5 half the maximum v value relative to maz of halo coefficient of pe sec 8 v value relative to Wmaz beyond which electric field 0 parameter inversly proportional to ramp width of the ion radial pressure profile sec 8 parameter inversly proportional to ramp width of po the hot electron radial pressure profile sec 8 value to which p is set if po lt p2fl
16. n of end plug outer solenoid 19 Technical Information Department Lawrence Livermore National Laboratory University of California Livermore California 94550
17. or imply its endorsement recommendation or favoring by the United States Government or the University of California The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California and shall not be used for advertising or product endorsement purposes a Printed in the United States of America Available from National Technical Information Service U S Department of Commerce 5285 Port Royal Road Springfield VAS 22161 Price Printed Copy Microfiche 4 50 Domestic Domestic Page Range Price Page Range Price 001 025 7 00 326 350 26 50 026 050 8 50 351 375 28 00 051 075 10 00 376 400 29 50 076 100 11 50 401 426 31 00 101 125 13 00 427 450 32 50 126 150 14 50 451 475 34 00 151 175 16 00 476 500 35 50 176 200 417 50 501 525 37 00 201 225 19 00 526 550 38 50 226 250 20 50 551 575 40 00 251 275 22 00 576 600 41 50 276 300 23 50 601 up 301 325 25 00 Add 1 50 for each additional 25 page increment or porti n thereof from 601 pages up User s Manual For The FLORA Equilibrium And Stability Code Robert P Freis and Bruce I Cohen Lawrence Livermore National Laboratory Livermore Ca 94550 USA Abstract This document provides a user s guide to the content and use of the two dimensional axisymmetric equilibrium and stability code FLORA FLORA addresses the low frequency stability of long thin axisymmetric tandem mirror syst
18. present 2 Each solenoid is specified by 4 input parameters B field strength Gauss axial length cm radius cm z location of center cm 21 22 23 The magnetic field strength input is actually the desired total vacuum magnetic field at the center of each solenoid on axis excluding the center cell field 4 PRESSURE The general form of the perpendicular pressure is pilz v B c p V Pie where Pie is due to hot electrons and the sum is over all other species This is solved together with the perpendicular pressure balance equation B z V 2 pi z p BVAC z Then the parallel pressure is obtained from the parallel pressure balance equation d pj B dB py b B bs B cs d or The coefficients a b c and d are calculated from the conditions for zero pressure and zero slope at appropriate axial positions In addition in the center cell there can be a z independent pressure component ppas1 with constant with respect to B In the outer cells the passing component has B dependence The passing and trapped groups will be described separately following a brief description of the hot electron pressure 5 HOT ELECTRON PRESSURE The perpendicular hot electron pressure Ple be Ce p is separated from the other species in order to properly treat it as a stiff component in the manner of
19. the rigid Elmo Bumpy Torus model It is included in the total perpendicular pressure only when calculating the magnetic B fields It is not included in other pressure dependent equilibrium quantities for example Q B p It is therefore not dynamically included as a source of instability In addiiion the hot electron density is assumed to be negligible compared to the warm electron and ion densities in order to satisfy charge neutrality Note that this does not imply a constraint on the hot electron pressure 6 TRAPPED PRESSURE Each cell can contain a trapped species whose p peaks at the B field minimum and goes to zero with zero slope at the cell limits In the case of unequal magnetic mirror peaks the smaller magnetic field peak determines the maximum magnetic field beyond which the pressure is zero In the plug cell a sloshing profile is constructed from the difference of two trapped profiles A hot electron pressure can exist in the cell adjoining the central cell i e the choke cell in the three region case the end plug cell in the two region case A code option permits the axial profile of the hot electrons to be elongated with a region of constant pressure 3 region case zic 22C 23 10 2 region zic 226 Hot electron pressure elongated regular ___ 7T PASSING PRESSURE For the three region case only a passing pressure can exist which is constant in the center
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