Exact canonical drift Hamiltonian formalism with pressure anisotropy and finite perturbed fields

Transcription

1 PHYSICS OF PLASMAS 14, Exact canonical drift Hailtonian foralis with pressure anisotropy and finite perturbed fields G. A. Cooper a epartent of Physics, University of the South, Sewanee, Tennessee 37383, USA and Ecole Polytechnique Fédérale de Lausanne, Centre de Recherches en Physique des Plasas, Association Eurato-Suisse, CH1015 Lausanne, Switzerland M. Jucker, W. A. Cooper, and J. P. Graves Ecole Polytechnique Fédérale de Lausanne, Centre de Recherches en Physique des Plasas, Association Eurato-Suisse, CH1015 Lausanne, Switzerland M. Yu. Isaev Nuclear Fusion Institute, RRC Kurchatov Institute, Moscow, Russia Received 10 August 2007; accepted 27 August 2007; published online 11 October 2007 A Hailtonian forulation of the guiding center drift orbits is extended to pressure anisotropy and field perturbations in axisyetric systes. The Boozer agnetic coordinates are shown to retain canonical properties in anisotropic pressure plasas with finite electrostatic perturbations and electroagnetic perturbed fields that solely affect the parallel coponent of the agnetic vector potential. The equations of otion developed in the Boozer coordinate frae are satisfied by direct verification of the drift velocities. A nuerical application illustrates the significance of retaining all second order ters Aerican Institute of Physics. OI: / I. INTROUCTION When driven by the neutral bea injection or ion cyclotron resonance heating required to achieve high teperatures in agnetic confineent systes, energetic particles behave like independent test particles whose otion can be accurately tracked using the guiding center drift approxiation. In addition, these energetic particles can generate a significant level of pressure anisotropy in the background equilibriu state. The Lagrangian for particle drift otion constitutes an invaluable tool in deterining canonical variables. 1,2 However, the canonical angular variables derived in Refs. 1 and 2 do not satisfy toroidal periodicity conditions. White and Zakharov have developed a drift Hailtonian forulation in generalized, nonstraight equilibriu agnetic field line coordinates that satisfies periodicity. 3 Yet, in practice this forulation appears to be highly cubersoe as ost guiding center orbit codes are ipleented in Boozer coordinates. The principle issue that lies in the transforation to Boozer coordinates, 4 where periodicity is guaranteed, usually involves neglecting higher order ters. 2,5 However, Wang 6 has developed a technique for axisyetric equilibriu fields in which the transforation to the Boozer coordinate frae retains all the relevant higher order ters neglected in previous treatents. The orbits can then be exactly identified with the drift velocity equation satisfying Liouville s theore, 7 V d e B + c B K B 1+ c 0, 1 B 2 where 0 KB. In particular, the corrections see to be of iportance when the plasa is at high and the particle has relatively large energy. In this article, we extend the ethod developed by Wang to deonstrate that the Boozer agnetic coordinates also retain their canonical properties in equilibriu fields sustained with anisotropic pressure and containing both finite electrostatic and nearly incopressible electroagnetic field perturbations. The Lagrangian of the syste is anipulated in Sec. II in order to define a set of canonical variables applicable to guiding center otion. In Sec. III, these variables for the basis of a Hailtonian forulation, obtaining equations of otion that are then transfored to a canonical Boozer coordinate frae. A nuerical application of the guiding center drifts is presented in Sec. IV, with a conversion to the notation of the VENUS code 8 outlined in the Appendix. Section V contains the conclusions and discussion. II. LAGRANGIAN ETERMINATION OF CANONICAL VARIABLES We begin by establishing, respectively, the covariant and contravariant representations of the equilibriu agnetic field and deriving its vector potential fro the latter, a Electronic ail: coopega0@sewanee.edu B = g + I + g,, X/2007/1410/102506/7/$ , Aerican Institute of Physics

2 Cooper et al. Phys. Plasas 14, B = + q = q 3 c,,p, 11 = =, where B is the equilibriu agnetic field,,, are the Boozer agnetic coordinates and +q. The basis for the difference in the representations of Eq. 2 between the anisotropic and isotropic pressure liits lies with the description of Apère s law. For anisotropic pressure plasas, Apère s law is H= 0 K, where HB while in the isotropic liit where =1, it is written as B= 0 j. The fields K and j correspond to the current density. Therefore, in the covariant Boozer representation, the agnetic field intensity H rather than its induction B is expressed in Eq. 2 under anisotropic pressure conditions. A valid interpretation is that the pressure anisotropy effectively odifies the pereability fro that of free space 0 to 0 /. We also find that the equilibriu vector potential of the agnetic field strength, A e, is. The drift approxiation of the oenta is defined as P = P B B + ea, where the parallel oentu P v, v is the particle velocity projection along the equilibriu field lines, while and e correspond to the particle ass and charge, respectively. The vector potential consists of equilibriu and perturbed coponents A=A e +A p, where A p is a finite perturbed function. We shall consider electroagnetic perturbations parallel to the equilibriu agnetic field only. Hence we define A p V,,,tB, 2,8 11 where =1 0 p p /B 2 is the pressure anisotropy paraeter with p and p representing the parallel and perpendicular pressures, respectively. 5 Thus the Lagrangian can be specified, noting that P /eb, L = P dx H =e c B + A e dx H, where c is defined as +V. By substituting for in the covariant representation of the agnetic field, we deterine B = h + g, g. We have defined, as,+q and h as I+gq, with the sybol denoting the derivative of a flux surface quantity with respect to. The Lagrangian becoes L e = cgd + c hd c g,d H e, 9 P = c g. 10 We know that the Lagrangian ust be of the for L = i p i dq i H in order to eet the conditions of a canonical coordinate syste. 1 Thus it is necessary to eliinate the d ter. This is facilitated by the introduction of the following: 6,,P P such that w,dw, d c = d +,d P,dP + d, L e = cgd c +,d + c h + P d P P,dP The equations of otion are invariant to the addition of a full tie differential ter to the Lagrangian. 1 Thus we introduce 6 S P L e w w,dw, ds = cgd c + c h + P P w w dwd By setting, Q,, 6 we deduce that the poloidal coponent of the oentu in the covariant representation is P = c h + c gq, +P Qw,dw. 17 We now reconsider and c as functions of the canonical variables P, P,, where c is an ignorable variable. Thus fro Eqs. 10 and 17, and setting gq+i +gi c gi c g 2,, we calculate that P = g, P = g q + QP, Q, + I g, = g c g Q P Q w dw, c = 1 cg, 21 P c = 1 cgq + QP, Q, + I P g 1 g, 22

3 Exact canonical drift Hailtonian foralis Phys. Plasas 14, c = 1 cg c g Q P Q d c dw. = e2 B 2 w 1 c g + g, 23 Recalling that =q c,, P, we also calculate the partial derivatives of the toroidal angle with respect to the canonical variables =,, P P =, + P,, P P = q, Q, + QP,, with the c differential of equal to the trivial negative unity. III. HAMILTONIAN FORMULATION OF RIFT ORBITS The Hailtonian associated with these canonical variables is actually H/e=e 2 2 B 2 /2+B/e+H e where the equations of otion are defined as dp = H e P,P, c, d = H e P P,, c, d c dp = H e, cp,p, = H e P P,, c. 27 The variation of H e with respect to the canonical variables is deterined using Eqs , dp = e2 B 2 1 c g + g, c g Q Q P w q + QP, Q,, dw 28 d = e2 B 2 1 c g + g,, 29 dp =, 30 q + QP, Q, e2 B 2 q + c I c g + I, + P,, 31 where the,, and ters are equal to 1 + 2, 1 + 2, and 1 + 2, respectively: 1 = e2 B 2 V, 2 = eb 2 + e B 1 = e2 B 2 2 = V, + eb2 eb 2 + e B +, 1 = e2 B 2 2 =. V, 2 +, Rather than following the oenta and the coordinate c which does not trivially satisfy periodicity, it is both ore convenient and physically intuitive to evaluate the particle otion in the Boozer coordinate frae and the parallel gyroradius. 1 By expanding the tie differential of P, P, and calculating the tie differential of =q we find that d = I g, d = e2 B 2 q + c I c g + g, I Furtherore we calculate the otion equation along by subtracting the tie differential of V,,,t fro that of c P, P,, d c = 1 cg q + I g, 40

4 Cooper et al. Phys. Plasas 14, FIG. 1. Color A bird s eye view of the torus showing the orbits of a trapped particle coencing at the vertical idplane. The blue trajectory corresponds to the standard forulation of guiding center otion, while the red orbit is integrated using Eqs with perturbed fields neglected. Under these conditions, the inclusion of full second order coponents solely affects the toroidal drift. This particular perspective gives a clear presentation of the toroidal displaceent of the two orbits. q + I g dv = 1 cg 1 + I V 2 V 2+ V t, 1 + g V 2 V 2 +, V 2 V 2 41 I V d = gv g V V g, 1+ cg g, V IV q ci + c g 2 V t. 2 42

5 Exact canonical drift Hailtonian foralis Phys. Plasas 14, FIG. 2. Color A full toroidal view illustrates another perspective of the toroidal displaceent of the two orbits. The blue trajectory corresponds to the standard forulation of the guiding center otion, while the red orbit is integrated using Eqs with perturbed fields neglected. Under these conditions, the inclusion of full second order coponents solely affects the toroidal drift. An evaluation of the projections of the drift velocity, V d, with respect to,,, and recovers precisely the Hailtonian forulation of the Boozer coordinate equations of otion calculated and expressed in Eqs. 29, 38, 39, and 42. The drift velocity that satisfies Liouville s theore is given by Eq It can be deonstrated that =V d, =V d, =V d, and =t +V d, where t = t V. ds = 0Js + 0Is, 43 v v d = e2 B 2 s + c 0 Is v 0Is s B s, v v 44 IV. APPLICATION WITH THE VENUS COE In order to ipleent Eqs. 29, 38, 39, and 42 into the VENUS code, 8,5 they ust first be converted to the notation eployed in the progra. When the required transforations are ade see the Appendix, the equations of otion becoe d = e2 B 2 e2 B 2 s + c 0 Js v 0Js s + B s v v c B s, 45 v

6 Cooper et al. Phys. Plasas 14, FIG. 3. Color A graph of the difference between the toroidal drifts with respect to c the toroidal drift using the conventional forulation of the orbit equation illustrates the deviation that arises when certain second order ters are ignored. The difference is calculated as c subtracted by n, where n is the toroidal drift otion calculated using Eq. 45 with perturbed fields neglected. V 0Js d = + IsV v s2 s + 0 Is V s + c 0 Is v v 2 0 Js V s + s + c 0 Js 2 V t + V V 2 v 2 v B s + c 2 v B s. 46 The ters containing B s, in soe for or another, were factored out because they are essentially those that have not previously been included in the VENUS code. These ters are all of second order and while previous treatents contained soe ters of this order, this forulation is significant as it contains all ters of O. In order to deonstrate the significance of retaining all second order ters, the equations of otion were stripped of all perturbations before being ipleented into the code. In the absence of perturbed fields, the full second order drift contributions to the guiding center orbits only alters the equation of otion for the toroidal drift. Our nuerical application with the VENUS code was considered for a 10 MeV trapped proton orbit at a high of 19%. In addition, the equilibriu agnetic field was obtained fro the VMEC code 12 with a central value of 5.6 T. We estiate that a 3.5 MeV particle in a 6.6 T field would recover the sae difference in orbits as that investigated. The toroidal displaceent is best observed fro the perspective of Fig. 1. The orbit described by the red curve has been integrated using Eqs while the blue curve represents the standard forulation of these equations of otion in which certain higher order ters are neglected. The axiu deviation between the two orbits occurs about halfway between the vertical idplane and the turning points of the orbits. A full toroidal view is presented in Fig. 2. Figure 3 is a graph of the difference of the two toroidal drifts with respect to the conventional toroidal drift in which certain second order ters are ignored. The iportance of including all second order ters is deonstrated as the percent difference plotted here reaches up to 7%. V. ISCUSSION The ethod developed by Wang 6 is extended to deonstrate that Boozer coordinates are also canonical for guiding center drift orbit otion in the presence of both arbitrary electrostatic perturbed fields and electroagnetic perturbations with vector potential parallel to the equilibriu agnetic field. The forulation is further extended to conditions where energetic particles drive an anisotropic pressure plasa. The equations of otion we have derived in our Hailtonian foralis are recovered exactly fro direct verification of the corresponding projections of the guiding center drift velocity. In addition, a nuerical application in the VENUS code was presented to illustrate the significance of retaining all second order ters. The introduction of S into our Lagrangian eliinates the d ter and establishes c and as our canonical coordinate variables. The Hailtonian derivation of these coordinates and their respective oenta develops a set of equations of otion that, while canonical, do not satisfy other criteria such as periodicity. Iportantly, when the required conversion to the Boozer reference frae is ade the equations of otion retain their canonical nature, facilitating the ipleentation of nuerical schees. It is to be noted that our forulation of the perturbed vector potential rests on the assuption ade extensively in existing codes of a relatively low ratio of kinetic to ag-

7 Exact canonical drift Hailtonian foralis Phys. Plasas 14, netic pressures,. 2,8 11 Furtherore, the forulation is calculated in axisyetry and thus is only applicable to tokaak and reverse field pinch conditions. ACKNOWLEGMENTS This research was partially sponsored by the Fonds Nationale Suisse de la Recherche Scientifique and Eurato. We thank r. S. P. Hirshan for the use of the VMEC code. APPENIX: CONVERSION TO VENUS NOTATION The following transforations were ade to Eqs. 29, 38, 39, and 42: g 0 I, I 0 J, g, B, gq + I gb 2. The VENUS code does not include as a coordinate variable. Instead the code is written in ters of s, where =s. The ter s is usually proportional to the plasa volue enclosed and varies fro 0 at the agnetic axis to unity at the plasavacuu interface. The following transforations were ade with representing any flux surface quantity, s, Also, s s. B B s s. Note that the ter becoes s where s s. Furtherore, our Jacobian in the VENUS notation becoes g = 0 J Iq/B 2. However the Jacobian of the VENUS code is in fact gv = 0 sjs sis/b 2, where sqs. Thus, v s. 1 R. G. Littlejohn, Phys. Fluids 28, R. B. White, Phys. Fluids B 2, R. B. White and L. E. Zakharov, Phys. Plasas 10, A. H. Boozer, Phys. Fluids 25, W. A. Cooper, J. P. Graves, M. Jucker, and M. Yu. Isaev, Phys. Plasas 13, S. Wang, Phys. Plasas 13, R. B. White, A. H. Boozer, and R. Hay, Phys. Fluids 25, O. Fischer, W. A. Cooper, M. Yu. Isaev, and L. Villard, Nucl. Fusion 42, R. B. White and M. S. Chance, Phys. Fluids 27, R. B. White and A. H. Boozer, Phys. Plasas 2, S.. Pinches, L. C. Appel, J. Candy, S. E. Sharapov, H. L. Berk,. Borba, B. N. Breizan, T. C. Hender, K. I. Hopcraft, G. T. A. Huysans, and W. Kerner, Coput. Phys. Coun. 111, S. P. Hirshan and J. C. Whitson, Phys. Fluids 26,