TRUBA User Manual - Laboratorio Nacional de Fusión
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1. Jw i e the normalized effective a 4 Wans Y lt Yi lt 7 Wo 2 radius line number from 1 to y Ir where y W 1 400 I 1 200 TRUBA User Manual 17 To calculate the net radial profile of power deposition per unit volume dWaps dV from the results of a multiple ray tracing simulation one has to sum up the 2nd column vectors for all the rays of a bunch correspondingly weighted then multiply the result by the total power of the beam and divide it by the plasma volume or more strictly by dv dy Damp_profileOT dat Damp_profileiT dat Damp_profile2T dat etc The same as above for the rays underwent tunneling around the q 1 layer O mode only The transmission efficiency is taken into account here except for a damping of the ray before the tunneling B_U_G_S Intended for output of any data helpful in analysis of results or debugging if F8 2 If F8 1 only the changes of the wave mode along the ray trajectory are registered successively for all the rays traced 18 TRUBA User Manual Internal structure of the code job assignment loading the magnetic configuration initiation of beam rays vacuum lt em gt Yes pre tracing GD setting the boundary values calculation of power deposition profile Yes finding the origin for the reflected ray Damp_profile dat REFLECTION Figure 2 Flow chart of TRUBA showing the2. 2 m1 O nn A 0 n i 0 nn 0 ni 1 gO K po Raw Ri Te 25 OMe y l IK Ry a RY 2A Ro gt ER 1 1 JA 1 0 1p To Rojo LR Rs 5Rin j where Ry a2 Ry 2s0 AoIRD p 1 xJu J 0 1 and a yenn 2 er Atat A dt R z a A K if exp it er Ja ay 26 is the Robinson s relativistic plasma dispersion function 8 This function may be expanded co 2 via usual Shkarofsky functions F z a i exp izt at dt v a p 1 at 1 t AS a _ 2k 2j 1 R 2 0 4 kK Xd 29 Far 2 a H l Tag In 27 Keeping only the leading term in 27 will reduce the dispersion tensor 25 to the standard A 1l weakly relativistic expression The so called moderately relativistic approximation 9 corresponds to retaining some higher order terms of this expansion Except for A 1 use of 27 is computationally expensive and suffers from subtraction errors In order to speed up the rate of the Hamiltonian evaluation the following quite accurate approximation is implemented in the code Rulz a A K exp A 7 A C A TR 2 a CAA 1 h 5 2z h a hF z 1 h a where C A A L 0 and h J2 1 2 In order to preserve the peculiar behavior of the fundamental harmonic K 1 contributions to the combinations 28 8 P A Robinson Relativistic plasma dispersion functions J Math Phys 27 1986 1206 9 D G Swanson Exact and moderately relativistic plasma dispersi
3. while shaping the number of step Whenever diagnoses of abnormalities RAYO DAT RAY1 DAT RAY2 DAT etc line in record output values and explanation block In vacuum pre tracing 0 During tracing in plasma In vacuum 0 In plasma the number of step In vacuum post tracing 1 current value of A 1 except the first record after exit 2 Also 2 in the N u u Type real case of preconditioned stop in plasma Type integer 2 Rx Ry Rz Current coordinates in cm of the reference ray Type real 3 Nx Ny Nz Components of the vector N on the reference ray Type real Current values of N N q U preceded with the titles if F6 1 In vacuum the 4 first three are zeroes Type real Current value of H H max 1 Nf preceded with the title if F6 1 In vacuum 3 0 Type real The files which are not necessarily output will henceforth be colored blue RAYOT DAT RAY1T DAT RAY2T DAT etc The same as above for the rays underwent tunneling through the q 1 layer O mode only The first record block is for the point just before tunneling found from minimum N lt 0 01 on the ray trajectory the second block is for the first point of the transmitted ray The 5th line in these two blocks contains N instead of H No more peculiarities The step numbering continues the original one 16 TRUBA User Manual POWERO DAT POWER1 DAT P
4. 13 refs Abstract The TRUBA pipeline in Russian code is a computational tool for studying thepropagation of Gaussian sha ped microwave beams in a prescribed equilibrium plasma This manual covers the basic material hended to use the implementation of TRUBA version 3 4 interfaced with the numerical library of the TJ Il stellarator The manual provides a concise theoretical background of the problem specifications for setting up the input files and interpreting the output of the code and some information useful in modifying TRUBA Manual de Usuario de TRUBA Tereshchencko M A Castejon F Cappa A 24 pp 2figs 13 refs Resumen El c digo TRUB tuber a en ruso es una herramienta computacional para estudiar la propagaci n de haces gaussianos de mircroondas en un equilibrio de plasma prescrito El presente manual muestra los materiales m nimos necesarios para usar la versi n 3 2 del programa TRUBA interconectada con la librer a que provee las caracter sticas del equilibrio del stellarator TJ II El manual aporta tambi n un repaso te rico conciso al problema las especificaciones de los ficheros de entrada y de salida del c digo y la informaci n necesaria para las posibles futuras modificaciones de TRUBA 1 Prokhorov Institute of General Physics Russian Academy of Sciences 119991 Moscow Russia 2 Laboratorio Nacional de Fusi n por Confinamiento Magn tico Asociaci n EURATOM CIEMAT para Fusi n 28040 Madri
5. 3 3 Z4 3 3 3 3 Input r 1 3 x y z Cartesian coordinates of a point in cm Output B i B in tesla modB B in tesla q qe U Ue t H Dq i dq dr in cm Dt i dyu dr in cm Du i du dr in cm DD4q i j d q dr dr in cm DDt i j d y dr dr in cm DDu i j d u dr dr in cm Z2 i j d B B dr in cm Z3 i j k d B B B dr in cmt ZA i j k l d B B x d B B dr dr in cm SUBROUTINE DIELECTR_NR q u t nl2 nt2 K 3 3 Input q qe U Ue t U N2 n nt2 n Output K 3 3 non relativistic dielectric tensor as it appears in 19 SUBROUTINE DIELECTR_WR q u t nl2 nt2 K 3 3 Input q qe U Ue t H7 ni2 n nt2 n Output K 3 3 weakly relativistic dielectric tensor as it appears in 25 with the use of 28 TRUBA User Manual 21 SUBROUTINE DIELECTR_WR1 q u t nl2 nt2 K 3 3 Input q qe U Ue t H7 I2 n nt2 n Output K 3 3 weakly relativistic dielectric tensor as it appears in 25 with the use of the expansion 27 retaining only those terms of lower orders that would be necessary and sufficient for frequencies up to the 2 EC harmonic in the Ae 1 limit SUBROUTINE ZERO_DTH m q u t nl2 nt2 nt2m Input mode m of the wave q qe U Ue t 1 N2 n nt2 n Output nt2m n the root of the dispersion equation corresponding to the given mode if m 1 or 2 and u gt 1 and qe lt 0 1 or u lt 1
6. and q lt 1 n 1 Ju otherwise closest to the m input SUBROUTINE DAMP O 6 g Input O 1 6 containing O 1 3 R in cm and 0 4 6 N Output g lim Im det A det A H det A damping coefficient so that 11 takes the form dP ds 229P SUBROUTINE QDAMP a b Oa 6 Ob 6 G Input a and b s coordinates of adjacent points of the ray trajectory Oa 1 6 containing Oa 1 3 R a in cm 0a 4 6 N a and Ob 1 6 containing R b N b Output G integral of the damping coefficient g see above from a to b taken along the spline interpolated hyper trajectory in 6D phase space SUBROUTINE ZERO_F F a b Fa Fb c Input F user supplied real function of which a zero will be found between a and b Fa F a and Fb F b must be opposite in sign Output c computed zero SUBROUTINE D1H O 6 01 6 Input O 1 6 containing O 1 3 R in cm and O 4 6 N Output O1 1 6 containing O1 1 3 OH OF in cm and O1 4 6 0 H 0n 22 TRUBA User Manual SUBROUTINE D2HX O 6 X 6 6 Input O 1 6 containing O 1 3 R in cm and 0 4 6 N Output X 1 6 1 3 0 01 1 6 0F and X 1 6 4 6 0 01 1 6 0m at given point O1 1 6 is the output of D1H SUBROUTINE FRHSO X Y 6 DY 6 Input X independent variable Y 6 solution at X Output DY 6 RHS of the system of ODEs dy dx f x Y Ys 1 6 assembled of 10 and 12 SUBROUTINE FRHS1 X Y 12 DY 12 Input X indepe
7. each step Type real Control flag F2 for the calculation mode F2 0 ray tracing F2 1 or 2 beam tracing F2 1 is valid only for vacuum launch Type integer 10 If F2 0 or 1 d P Pz where amp is the rotation angle in degrees which is defined in Fig 1 and associated text p and p2 are the principal radii of the beam attenuation ellipse in cm If F2 1 these values are used to initialize Log If F2 0 these values are used to initialize a parallel bunch of 4 characteristic surrounding rays to be traced right after the central ray if one of ps is zero corresponding two rays will be taken away if both ps are zeros only central ray will be traced If F2 2 Lyx Lxy Luz Lyy Lyz Lzz in cm Type real TRUBA User Manual 13 11 If F2 0 this line is ignored If F2 1 Om 1 P 1 p2 where Um is the rotation angle in degrees which ensures the diagonal form of Mog 1 p and 1 pz are the inverse principal focal parameters of the wave front in cm If F2 2 Max Myy Myz Myy Myz Mzz in cm Type real 12 Control flag F3 for the dielectric tensor model F3 0 non relativistic F3 1 weakly relativistic F3 11 lite weakly relativistic valid if Ae 1 up to the 2 EC harmonic Type integer 13 Control flag F4 for the choice of ODE solver F4 1 DO2EJF NAG F4 2 LSODE non stiff ODEPACK F4 3 LSODA ODEPACK Type integer 1
8. ie ar A Ape ep n N s 7 with A s SA N 6n exp iF n R d3 n Superscripts H and A denote Hermitian and anti Hermitian parts of the tensor respectively Evidently A ReA and Ap eme eg ilmA We posit that the relation ReA 0 is conserved along the reference ray hypercurve in the 6D phase space Within the ansatz of 1 the 3D trajectory r R s is uniquely determined by the direction of P So it is easily seen that the tangent to this trajectory i e the dR ds is directed along the OH n nis where H r n fReA and f f r n is an arbitrary non vanishing real function As a result one can define H Rea det A 8 j 0 12 However when more than one eigenvalue tends to zero it is necessary to somehow isolate the root corresponding to the required mode The aforesaid power balance equation in the ray coordinates takes the form dP dR dR w P 9 ds sen ds RR ai 9 Therefore choosing the s norm such that dR OH ie on a we obtain 2 P fim A en 11 where as before f H ReA The invariant H 0 holds all along the reference ray and the essential condition dH ds 0 together with 10 brings to AN _ oH ds OT 12 1 M D Tokman E Westerhof M A Gavrilova Wave power flux and ray tracing in regions of resonant absorption Plasma Phys Control Fusion 42 2000 91 TRUBA User Manual 3 Clearly the partial derivatives in 10 and 12 are to b
9. of practical use are given by py Ma Vacuum solution for the beam shape In any homogeneous medium 2 De 0 q Mao Lua gy y Mar Hr Mio 1 5 42 If the medium is also isotropic as is the case with vacuum then H H n so that dR H oH H oH do Ole A AN L 43 ds con On Ang Pon Pan Y aa with I being the unit dyadic Let us transform the coordinate system for example using the same generating matrix 38 as employed above but with 4 0 and e N N N N N 8 S l T N N N n s J1 N3 44 S S 5 o Ns Now the non vanishing beam coefficients are governed by the matrix Riccati equation dQ PA Q jk 45 TRUBA User Manual 11 where Latin indices range from 1 to 2 Q T M iL T and the subscript 3 has been omitted from W The analytic solution of 45 is 1 Q w Q w Ty Qsw 1 Qw Q w Qw dw 46 where Q Q w Q w Qi w and 6W W W Then at the vacuum plasma interface one should restore the original coordinate representation M iL TQT that will be passed to the numerical solver 12 TRUBA User Manual Input files for the code TRUBA INI line input parameters and explanation Rx Ry Rz Cartesian coordinates in cm for the origin of the central ray the coordinate system coincides with that of TJ II Library Type real Control flag F1 for interpreting of the next line see below Ty
10. 4 Relative tolerance parameter for ODE solvers Type real 15 Control flag F5 for the choice of the ray tracing termination condition F5 1 P s P 0 lt 10 F5 2 N gt 10 and u 1 lt 10 F5 3 N u u gt 50 F5 4 any of the last two criteria fulfilled By default tracing goes on till the ray comes upon the wall of the vacuum vessel Type integer 16 Control flag F6 for the choice of output style of the RAYx DAT files F6 1 all the values in the lines 4 and 5 are preceded with its titles see below F6 0 the same without the titles Type integer 17 Control flag F7 for the choice of output mode for the POWERx DAT files F7 1 two more columns are output if m 1 or 2 see below F7 0 optional output is suppressed Type integer 18 Control flag F8 for the choice of output mode for the B_U_G_S file F8 2 detailed auxiliary debugging information F8 1 only checkpoint data F8 lt 0 no output If F8 1 all the output to the screen is suppressed except for the error messages Type integer TJ2_B INI line input parameters and explanation Name of the file containing the coefficients of a given configuration To be passed to the initialization routine of the TJ II Library Type character Name of the file containing the namelist with the currents flowing trough the different coils of the device To be passed to the initialization
11. Informes T cnicos Ciemat 1134 Febrero 2008 TRUBA User Manual M A Tereshchenko F Castej n A Cappa Asociaci n EURATOM CIEMAT para Fusi n 108 Laboratorio Nacional de Fusi n por Confinamiento Magn tico Toda correspondencia en relaci n con este trabajo debe dirigirse al Servicio de In formaci n y Documentaci n Centro de Investigaciones Energ ticas Medioambientales y Tecnol gicas Ciudad Universitaria 28040 MADRID ESPANA Las solicitudes de ejemplares deben dirigirse a este mismo Servicio Los descriptores se han seleccionado del Thesauro del DOE para describir las ma terias que contiene este informe con vistas a su recuperaci n La catalogaci n se ha hecho utilizando el documento DOE TIC 4602 Rev 1 Descriptive Cataloguing On Line y la cla sificaci n de acuerdo con el documento DOE TIC 4584 R7 Subject Categories and Scope publicados por el Office of Scientific and Technical Information del Departamento de Energia de los Estados Unidos Se autoriza la reproducci n de los resumenes analiticos que aparecen en esta pu blicaci n Cat logo general de publicaciones oficiales http www 060 es Deposito Legal M 14226 1995 ISSN 1135 9420 NIPO 654 08 010 6 Editorial CIEMAT CLASIFICACI N DOE Y DESCRIPTORES S70 PLASMA HEATING BERNSTEIN MODE ELECTRON PLASMA WAVES TOKAMAK DEVICES STELLARATORS T CODES TRUBA User Manual Tereshchencko M A Castejon F Cappa A 24 pp 2 figs
12. OWER2 DAT etc 1st column 2nd column 3rd column last column w P P with P1 P in The number of step Im N ex 22 Im N aw P j c Zz a the first line These columns are output only if F7 1 and m 1 or 2 POWEROT DAT POWER1T DAT POWER2T DAT etc The same as above for the rays underwent tunneling through the q 1 layer O mode only The only difference the 3rd column and the last column are multiplied by the transmission coefficient damping before the tunneling is out of account here BEAMO DAT line in record output values and explanation block 1 The number of step or 0 1 2 synchronized with the RAYO DAT Type integer 2 Coordinates of the e basis vector of the beam attenuation ellipse Type real 3 Coordinates of the e basis vector of the beam attenuation ellipse Type real 4 P1 P2 Principal radii of the beam attenuation ellipse in cm Type real 5 Myx May Mxz Myy Myz Mzz in cm Type real REFLECTION line in record output values and explanation block 1 Rx Ry Rz Cartesian coordinates in cm of the emergent ray spot on the vessel 2 Ns Ny Nz Components of the vector N for the ray reflected from the vessel Such information on all rays traced is successively collected in this file Damp_profileO dat Damp_profilei dat Damp_profile2 dat etc 1st column 2nd column
13. aining the cubic spline coefficients y x y C 1 x x C 2 x x i 3 x x x i lt x lt x i 1 i 1 n 1 24 TRUBA User Manual Calls to external libraries TJ2LIB 11 or its substitute init_tj2 lib b field car flux car grad flux _ car tj2 dependencies NAG 12 DO2EJF dependencies ODEPACK 13 LSODE LSODA dependencies 11 V Tribaldos B Ph van Milligen A L pez Fraguas TJ II Library Manual Informes Tecnicos CIEMAT 963 2001 12 NAG Fortran Library Manual http www nag com numeric fl manual html FLlibrarymanual asp 13 A C Hindmarsh ODEPACK A Systematized Collection of ODE Solvers in Scientific Computing R S Stepleman et al eds North Holland Amsterdam 1983 55 64
14. be Cartesian In the above notation r 7 R 5 where R s is the space curve of the central reference ray of the beam with s being a parameter of this ray The treatment 1 is also known as the paraxial WKB expansion of the wave field phase Matrices M and L in 1 possess several properties worthy of being noted Evidently they are symmetric Me EM gis LL 2 Then the vector Gi Nq 8 Mag s iLag 8 Org which is the gradient of the complex eikonal is necessarily a function of the position r Hence d _ dR and thus we arrive at the additional constraints dk dN dR _B RER A eb ds ds Lop ds 0 4 Power transfer along the ray trajectory The slowly varying amplitude E of the wave field 1 for the mode m under consideration can be decomposed into a continuous superposition of plane waves by E r f IN n e N n exp ion r d 6n 5 where A is the spectral density e is the unit eigenvector corresponding to the m eigenvalue A of the dispersion tensor A up n Lag NN y Ko n is the normalized wave vector I is the unit dyadic and K is the dielectric tensor The dispersion relation for 2 TRUBA User Manual the given mode is A 0 As shown in 1 for the case of narrow spectrum the total wave power flux is A 2 E AE e mal m 6 ard ap a B n N s and the sink term in the stationary power balance equation V P w is given by _ 9 2 A my m
15. d Spain Contents Gaussian microwave beam in the complex eikonal form ssas 1 Power transfer along the ray trajectory eeceee escent ee eee eee nennen nenn 1 Shape of the Gaussian beam uuneseeenenennnennen nun ee eee eee ann ann nun nennen 3 Non relativistic ray Hamiltonian zserseennennnn nenn nenn ann ann ann n nennen nennen 4 Weakly relativistic ray Hamiltonian ersssrnenennn sone nn en nnnnn nun nn nennen nenn 5 Absorption Of the ray nenuaennnnnnnnnnnnnnennnnnnnnnn nn nun nenn nenn nn nn nenne nnnn nenn 8 Reduction of L to the diagonal form nidad 9 Reduction of M to the diagonal form in vacuum nssr 10 Vacuum solution for the beam shape nennnnenenenenennnnn ann nun nenn nennen 10 Input files for the code zssusssnnnnnnn nenn nun esse teen nun nn nn une nun une ai 12 Output Of the code zerz2ennnnnnnnnnnnnnonennnnennnnnnnn nennen rr rr 15 Internal structure of the code cece cece eee eee eee nen nnnnn nennen nennen 18 Synopsis of subroutines unenunenennnennenennnennnnnnnnnnn nenne nun une nnn nenn 20 TRUBA User Manual 1 Gaussian microwave beam in the complex eikonal form In what follows the wave field in a weakly inhomogeneous and stationary medium is sought in the form of a monochromatic Gaussian shaped beam E E s exp fi Nasr 5 Mpls iL y s Or 2 iat 1 Hereinafter the Einstein sum convention for repeated indices is implied and the coordinate system r is assumed to
16. e evaluated on the reference ray Equations 10 and 12 are usually referred to as ray tracing equations and the H r n function is known as the ray Hamiltonian Shape of the Gaussian beam Making use of the complex coupling Q M iL constraints 4 combined with 10 and 12 give rise to the second invariant of the reference ray A A 0 DH 0 D R 13 a a Or Qag Ong We note here that the total derivative along the reference ray can be written as R a 14 ds s ds The condition d D H ds 0 then leads to the sought beam shape matrix equation dO ad a D DH 15 which can be splitted to give dM CH PH PH PH Pe My t Mg May M gg Loy L 1 ds 0r rg Or 0n a tar n ETA EE E cts dL PH EH al PH DH a Ms L Lo ds n On Ong Er 9rg n n n PY 17 Equations 10 12 16 and 17 constitute the system of the so called beam tracing equations 2 The aforementioned constraints appearing now as i MA y i 18 Ong Ory Ong should be used when setting the boundary values at S 0 and later as a check of consistency The system of equations 16 and 17 can be solved separately from 10 and 12 provided that the trajectory of the reference ray is pre computed So it is useful in practice to decompose the beam tracing procedure into the reference ray tracing with the subsequent beam shaping 2 E Poli G V Pereverzev A G Pee
17. e to be traced using the linearized Hamiltonian m 1 2 H max Cu 1 adet A a n2 2 Prin ni ren 34 where Nn is the corresponding root computed numerically of the det A 0 equation while the other parameters i e qe Ue Me and Ny are kept fixed at their values 10 V Krivenski A Orefice Weakly relativistic dielectric tensor and dispersion functions of a Maxwellian plasma J Plasma Phys 30 1983 125 8 TRUBA User Manual Absorption of the ray In calculating the damping coefficient we make use of the relation Ima TT Rea jem Im det A det A 2 35 Real 0 where Nn is the numerically computed root recall Eq 34 so that equation 11 has the following versatile form dP gw H A RR Pas a eta an 2 Nim TRUBA User Manual 9 Reduction of L to the diagonal form The objective is to find the transformation such that 2 2 w w w L267 01 5 ee 37 c Pi P with p being the principal radii of the cross 1 sectional attenuation Let e TR then if ds ds ez FVz we can introduce a local Cartesian coordinate system W with basis vectors e and e lying in the cross section of the beam as shown in Fig 1 Both systems are related via r T Wg with the generating matrix Figure 1 s sin s s cosd _s cosd 8 8 8ind Attenuation ellipse in the F P cross section of the beam and 5 the associated ort
18. honormal T s5sind s s cos s cosd s s sind 38 vector basis red S S 2 i s cosd s sind S where 5 are the laboratory coordinates of z S 41 853 and QU is the rotation angle ensuring the diagonal form of L T LT This angle or rather its sine and cosine should be found from the equation L Lapal go 0 which gives tan20 2y 3 515 Li gt Lao s u 83 Ly si ala Ly 8 83 85 L 52s 8 Ly 28 8 1 s Lia Bila 28357 sL 8 03 X 7 39 and hence sind Dr Ji 1 J4x2 1 cosd 1 1 Jar 1 40 1 2 Having determined T we easily obtain e and pP 2 1 1 2 On restoring the laboratory representation for given values of Y and Pi the inverse transformation is straightforward T is an orthogonal matrix L TLT 10 TRUBA User Manual Reduction of M to the diagonal form in vacuum In plasma M is generally a nondegenerate matrix This makes the diagonalization of M useless On the contrary in vacuum the first of the constraints 4 is reduced to Map Ng Q so that the canonical transformation must give 1 102 2 A a 41 with p being the principal focal parameters of the wave front paraboloid they are equal to the principal radii of the wave front curvature on the beam axis In 41 use is made of N 1 The procedure of finding the diagonal form M T MT is the same as for L with e N The inverse focal parameters which are
19. ly equivalent to the limiting central trajectory of the transmitted part of the wave packet The wave vector of the launching ray is equated with its value at the reflection point The fraction of the transmitted power is calculated using the one dimensional O mode tunneling theory 5 6 n exp m 2 6 Nun 2 1 ry Tope Y 72 24 where A is the density gradient length and Nat Ju 1 Ju with all the parameters taken at the reflection point Weakly relativistic ray Hamiltonian The weakly relativistic approximation is referred to as the u gt gt 1 condition which is valid in most ECRH experiments We start from the expression 2 from Ref 7 that uses the 4 A V Timofeev Electromagnetic oscillations near the critical surface in a plasma Methodological note Plasma Phys Rep 27 2001 922 5 E Mjglhus Coupling to Z mode near critical angle J Plasma Phys 31 1984 7 6 A A Zharov Theory of the conversion of normal waves in a nonuniform magnetized plasma Sov J Plasma Phys 10 1984 642 7 I P Shkarofsky Dielectric tensor in Vlasov plasmas near cyclotron harmonics Phys Fluids 9 1966 561 6 TRUBA User Manual only extra assumption of A He thus still allowing for the short wavelength Bernstein waves which typically possess A lt 20 As before we disregard the motion of ions Following the technique of 7 but not restricting to A lt 1 we obtain the following
20. n 1 n 142q 1 u q qete 1 nf 42 1 4 21 whenever it would be used is replaced by the following single root Hamiltonian gy n max D Dinin n a gt F D 2a D yaz 4a 09 22 3 M Brambilla Kinetic theory of plasma waves homogeneous plasmas Oxford Univ Press Oxford 1998 TRUBA User Manual 5 where a 1 u Q dy 2q 1 q 1 17 u q 2 1 u g 23 a 1 9 1 ni Gyr nu are the coefficients of nt n and n in the polynomial 21 and the customized small parameter D gt 0 prevents the Hamiltonian from degradation when the modes are nearly degenerate at the vacuum plasma interface Both O and X modes at densities under the O mode cutoff are treated using the cold plasma Hamiltonian 22 The general hot plasma Hamiltonian derived from 19 is switched on for q gt 50a 1 Ju 1 ni so that to provide O X mode conversion and for n gt 3 thus allowing for electron Bernstein waves The TRUBA code proceeds with the O X conversion in the following ad hoc manner If the reflection point is revealed along the ray trajectory of the incident O mode and the width of the further evanescent layer is small enough the launching point for the transmitted ray is to be determined on moving from that point along the density gradient until the dispersion relation is fulfilled again It was shown in 4 that the ray trajectory continued from this point is asymptotical
21. ndent variable Y 12 solution at X Output DY 12 RHS of the system of ODEs dy dx f X Y1 Y2 Y12 I 1 2 12 assembled of 16 and 17 SUBROUTINE FRWRDT e 3 X 6 T 3 3 Input e 3 any vector aligned with the direction of degeneracy X 6 vector composed from the 11 12 13 22 23 and 33 elements of symmetric degenerate matrix of the quadratic form 37 to be reduced Output 7 3 3 generating matrix 38 of the transform required SUBROUTINE INVRST e 3 th T 3 3 Input e 3 any vector th angle in degrees Output 7 3 3 matrix transposed with respect to 38 can serve as a generating matrix of the transform restoring the laboratory coordinate representation SUBROUTINE YNR x n Y n Input x real argument n number of components in the output Output vector containing Y j e Jj 1 x where I is the modified Bessel function of the first kind J 1 n SUBROUTINE INSIDE x y z in Input x y z coordinates of a point in m for concordance with the tj2 routine Output in logical true if this point is inside the vessel false otherwise TRUBA User Manual 23 SUBROUTINE SEGMENT phi R1 Z1 R2 Z2 e1 3 e2 3 Input phi toroidal angle in radians of the section R1 Z1 and R2 Z2 two points in m in this section which are presumably on different sides of the vessel surface Output e1 3 e2 3 coordinates in m of the pair of adjacent ve
22. on functions Plasma Phys Control Fusion 44 2002 1329 TRUBA User Manual 7 K Ko 2iK and Kig iK the derivatives of the Robinson s function in 25 should be computed as K _ au Rec i AR IDR ral KD R aC KIH 29 at least for the fundamental harmonic The Shkarofsky function has a valuable asymptotic representation in terms of the Z function and its high order derivatives in a domain of large real a F z a h z Mtv h Zy u h Zy uh ZO 30 SUN ZW vh ZO evs Zy 00 with h 4a 2v y h 2 v and v a v j If z is real 30 is also valid in regions z gt gt 1 v and 2 a gt gt 1 A further useful property is that F 2 a of half integer index V can be expressed in terms of the classical plasma dispersion function 10 F 2 0 a Z a i z 0 Z a i 2 a z a l 31 Faja 2 0 SEE Z a i z a Z a i z a and for K 2 0 Rosp 20 1 6 2 Frp 2 0 Foran 0 32 Practically however only the K lt 3 functions can be safely treated using this method due to numerical instability of the recursion 32 Moreover in the case of a lt 1 this formula should be completely rejected Instead of 32 the following relations can be used here YE r Jv 20 0 Fo 2 0 5 1 2F 2 0 33 j 0 Similar to the splitting technique 22 of the non relativistic Hamiltonian O and X modes in low density plasma ar
23. pe integer Vector N If F1 0 Ny Ny N absolute components manually adjusted or taken from somewhere else If F1 1 N h N h N h relative components the value of h will be computed by the code If F1 2 N h N 2h N where N is the parallel component of N N and N gt are the perpendicular components directed along Vw and BxVy correspondingly the value of h will be computed by the code If F1 3 and q 1 lt 0 001 critical layer h hz hs and N will be constructed so as to N 0 001 h h h N sgn h Ju 1 Ju f Microwave frequency in GHz Type real Mode m of the wave to be launched m 1 O mode m 2 X mode m 0 hot plasma mode Type integer Ni N2 n3 Coefficients of density profile nm n 1 2 x10 cm Type real Or alternatively the key word tabular case sensitive preceding characters are ignored and then the name of the file containing tabulated density profile leading and trailing blanks are ignored t t tz Coefficients of electron temperature profile T t 1 y 2 keV Type real Or alternatively the key word tabular case sensitive preceding characters are ignored and then the name of the file containing tabulated electron temperature profile leading and trailing blanks are ignored As Increment of the ray parameter for output is also the interval of s for the ODE solver to integrate over during
24. principal steps involved in a run of the code TRUBA User Manual 19 Contents of the beam data array O 0 18 array component value in beam or ray tracing mode 0 0 P s P 0 o 1 R s 0 2 Ry s 0 3 R s 0 4 Nx s 0 5 N s 0 6 N s 0 7 M x S Rx 0 0 8 Lxx S Ry 0 0 9 Mx S Rz 0 o 10 Ly S Nx 0 0 11 Mx S N 0 0 12 L s N 0 0 13 M s amp 0 Vx 0 14 Lyy s e 0 Vy 0 15 Myz S amp 0 Vz 0 16 Lyz Ss 0 Vx 0 17 Mz S 0 Vy 0 18 Lzz S 0 Vz Here e and e are the principal orts of the beam attenuation ellipse 20 TRUBA User Manual Synopsis of subroutines SUBROUTINE Local_1 r 3 B 3 modB q u t fl Input r 1 3 x y z Cartesian coordinates of a point in cm Output B 1 3 B By Bz components of the magnetic field B in tesla modB B in tesla g Qe U Ue t pe fI normalized magnetic flux SUBROUTINE Local_2 r 3 B 3 modB q u t Dq 3 Dt 3 DB 3 3 Input r 1 3 x y z Cartesian coordinates of a point in cm Output B 3 magnetic field B in tesla modB B in tesla q qe U Ue t H Dq 3 gradient of qe in cmt Dt 3 gradient of u in cm DB 3 3 gradient of B in cm DB i j dB ar SUBROUTINE Local_3 r 3 B 3 modB q u t Dq 3 Dt 3 Du 3 DDq 3 3 DDt 3 3 DDu 3 3 Z2 3 3 Z3 3
25. routine of the TJ II Library Type character Artificial scale factor for the magnetic field returned by the TJ II Library Type real 14 TRUBA User Manual File indicated in the 6 line of TRUBA INI if any line input parameters and explanation i j_n The number of tabulated density values type integer dne di y o and dne d y 1 in 10 cm type real 2 Values of y in the order of increasing Corresponding values of density ne in j_n 1 from 0 to 1 Type real 10 cm Type real File indicated in the 7 line of TRUBA INI if any line input parameters and explanation i J_t The number of tabulated electron temperature values type integer dT dy 1 0 and dTe d y in keV type real 2 Values of w in the order of increasing Corresponding values of electron j_t 1 from Otto 1 Type real temperature Te in keV Type real Files indicated in the first two lines of TJ2_B INI For any details see the init_tj2_lib routine documentation TRUBA User Manual 15 Output of the code Screen the asterisk x unit If F8 1 at the origin of the central ray q q U u flux y if F1 1 Nis multiplied by h if F1 2 Nt2 is set to N Starting ray tracing while tracing the number of step m N at the end y q if F2 1 2 Starting beam shaping
26. rtexes of the polygon representing the vessel section such that these vertexes enclose the point of intersection with the line joining given points If no intersection two zero vectors SUBROUTINE CROSS x1 y1 x2 y2 x3 y3 x4 y4 x y K Input the line segment between the points x1 y1 and x2 y2 and the line segment between another two points x3 y3 and x4 y4 on a plane Output x y coordinates of the intersection point if any k integer classifies the positional relationship of the given line segments 2 overlap 1 cross at a point O mutually disjoint and parallel 1 mutually disjoint and non parallel 2 mutually disjoint though aligned SUBROUTINE CLspline n x x1 x2 f1 n f2 n df1 n df2 n f n Input n number of dimensions x coordinate of a point between x1 and x2 f1 n and f2 n values of a function at x1 and x2 df1 n and df2 n values of a derivative at x1 and x2 Output f n the value of the spline interpolant f Fc F Fc Fo at the given point where Fe is the cubic interpolant FL is the linear interpolant and Fo max f1 f2 e max df1 df2 x1 x2 SUBROUTINE CSPL n x n y n Dy 2 C 3 n 1 Input n number of tabulated values x n array containing the data point abscissas must be increasing y m array containing the data point ordinates Dy 2 array containing the values of dy dx at the outmost points Output C 3 n 1 array cont
27. ters Paraxial Gaussian wave beam propagation in an anisotropic inhomogeneous plasma Phys Plasmas 6 1999 5 4 TRUBA User Manual Non relativistic ray Hamiltonian We cite here the well known see e g 3 expression for the Maxwellian hot plasma dispersion tensor in a local coordinate system such that B 0 0 B and n 1 00 ni l 0 nn A 0 n 1 0 i Ho K p Au o Fee iK Yo Zeo ao de 19 2 FI le gt SRY 5 Lig Yo 2 Ya Zor u t E Y Zu o K 00 de K A A pg Y i E Y Z de KO KO ut 9 KO KO KT Gee Ko where o runs over the plasma species sgn w qo W W A N Ugke Us fw Ug M01 Ty Leo NH m 1 SK Uy Yo IA exp A Zo Z Les I is the Kth order modified Bessel function of the first kind Z if exp igt t7 4 dt is the nonrelativistic plasma dispersion function which can be represented for real arguments as follows Zaye exp 2 iv 2f exp t dt 20 The primed variables in 19 denote the derivatives of the corresponding functions with respect to their actual arguments Since the use of TRUBA is focused upon the EC frequency range at present the ion contribution has been neglected in this version and the sum over K is truncated to lt lt 10 In order to avoid the u 1 singularity in the determinant of 19 persisted in the u oo limit the ray Hamiltonian is defined as H u 1 det A The cold limit expression H
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