Bers, A.

Fuchs, V.

Ko, K.

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THE COUPLING APPROXIMATIONOF LOCAL DISPERSION RELATIONS*V. Fuchs**, K. Ko and A. BersPFC/RR-81 -10 February 1981* To appear in Proceedings of the 4th Topical Conference on RF Heatingof Plasma, University of Texas at Austin, February 1981.** IREQ, Varennes, Quebec, JOL 2P0.THE COUPLING APPROXIMATION OF LOCAL DISPERSION RELATIONS*V. Fuchs, (I EQ, Varennes, Quebec, ,OL2POK. Ko an(l A. Bers, NUT Plasma Fusion Center, Cambridge, MA 02139The propagation of waves of a given frequency in one-dimensional njon-uniformmedia (coordinate x) with general local dispersion functions, e.g. D(k, x) (such asthose from the Vlasov eqLarion), and not expressible as a polynomial in wavevec-tor k has not previously been solved due to the difficulty of treating mode cou-pling regions where k values come together rapidly, and the eikonal approxima-tion must fail. Providing the modes couple only two at a time, a general methodis given for generating appropriate and unambiguous second-order differentialequations for each coupling. The differential equation requiring solUtion is essen-tially the Schrodinger equation with potential -2(D(k)/O 2D/Ok2)k=;(x), wherekc(x) satisfies3D(k, x)/3k = 0, x real.Lower-hybrid heating of plasmas relies on the mode-conversion of incident electromagnetic power to ai ormal plasma mode which is more readily absorbed via non-collisional damping processes. The usual mode-coupling theory' based on the fourth- or sixth-order differential equations corresponding to the warm-plasmaapproximation of the Vlasov dispersion relation is inapplicable in tile presence of kinetic damping. Specifically,tMe appearance, due to damping, of terms which are not powers of the wavenumber k breaks down the pos-sibility of direct representation of the dispersion relation by a differential equation. Nevertheless, the conceptcan be recovered, and we present here a method of analysis resulting in a system of coupled-mode differentialequations, representing any dispersion relation which is an entire analytic function of k. We are dealing herewith "local" dispersion relations, valid for waves in a non--uniform medium (along x) satisfying the eikonalapproximation.Consider a plasma which is weakly nonuniform in one spatial dimension x, supporting waves excited ata'fixed frequency u). The local dispersion relation D(k, z; w) = 0 representing these waves as they propagateunder the influence of nonuniformities, defines a many valued mapping of the complex spatial variable z =x + iy onto the complex wavenumber plane k = k,. + iki. We have defined k = k,, and we have suppressedthe explicit dependence on parameters such as the other two components of the wavevector, etc.; y and z arenot to be confused with spatial dimensions. We will assume that the inverse mapping k -+ z is single-valued.It poses no principal difficulty to relax this assumption, and it is useful to do so on occasion, since the case ofmany valued z is frequently encountered in practice. An arbitrary contour C in the z-plane thus maps onto nbranches, n > 1, k = f(z), and with such a mapping there are associated saddle and branch points, k, andzD, respectively. The branch points z13 are those for which 1 diverges, and they are, by definition, the onlyones which map onto a single point k, in the k-plane, the latter being the saddle points of the inverse mappingz = f~1(k).The general procedure of mapping z --* k through the dispersion relation involves two steps. The first isto determine the saddle points k, and branch points zB. The second is to specify the branch lines. The saddleand branch points are inherent to the mapping. Each saddle point defines a distinct coupling event between thebranches- we will assume throughout that the branches couple pair-wise, which occurs when*Work supported by National Science Foundation Grant No. 77-00340-A01.Idf -t d2f- o.dk dk 2The crucial step in the analysis of the dispersion relation regards the construction of branch cuts. On theone hand, the branch cuts define the process of mode coupling and the process of analytic continuation of abranch. but on the other, the dispersion rel ition alone does not provide any ready-made in formation on how to:pcCify the branch lines. This is, of course, 1 source of major concern since mode coupling occurs, by definition,O'vcry time the wa\e-propagation contour C (the real axis x = Re z) crosses a branch cut. The intuitive reactionin this situation would be to trace the bratich cuts in some arbitrary manner away from the real axis, to avoidcrossings other than through the branch points themselves. But what if all branch points lie away from thereal axis? Does this imply that now no niode coupling takes place? The answer to these questions lies in theI roper definition of mode-coupling and of a mode itself. The relevant question to ask is which are the physicalL onditions for coupling and find their anal tical counterparts. Branches couple, by definition, whenever the real;,-axis crosses a branch cut (Fig. 1). The asual definition of a mode, in terms of a branch of the dispersionrelation, is thus not complete without associating with it a properly cut Riemann sheet. The Riemann sheet thenmaps onto a simply connected region of th- k-plane, whose boundaries (given by the maps of the branch cuts)limit the extent of wavenumbers accessilble to the branch under deformations of the x-axis. The required branchcuts CQ and the mode-boundaries k, arc given by the mappings2D(ke, C) =- 0, OD(k,, z) = 0, z E C, (2)where C is required to pass through the branch points and, therefore, might occasionally deviate from the realr-axis (Fig. 2). The only coupling events which materialize under these conditions are those defined by thebranch (and saddle) points of the dispersion relation. We emphasize that it is actually through the mapping= 0 that we make contact with the physical requirements for mode-conversion. The argument goes asfollows. Consider a wave A(x) exp [-iwt + ikr(x)x propagating along the x-direction, and characterized by aslowly varying amplitude A (incorporating the effect ofki(x)) and wavenumnber kr(x). The wave has associatedwith it an averaged energy flux proportional to vAA*, where A* is the conjugate to A, and v. is die local groupvelocitydw (3)evaluated implicitly from the dispersion relation D(k, z; w) 0. h'lie energy flux thus defined depends onposition not only through the amplitude A, but also via vg. Under these conditions, energy ceases to flow inthe form of a particular mode where either A -+ 0 or v. -+ 0. h'le former case is indicative of dissipativeprocesses and A vanishes gradually. We are concerned with tie latter case as that is the one which is indicativeof mode-conversion, at least as far as the coupling of energy between different wave-types is concerned. Thisis because when vg -+ 0 without A vanishing simultaneously, the global conservation of energy flux requiresthat tie energy be either transformed into a different wave form, or the amplitude diverges and we have aresonance. The branch points and branch cuts are instrumental in obtaining a representation of the dispersionrelation in tie form of a system of sccond-order differential equations. To proceed let us define the function3(k) = D(k, z) of the independent variable k, with z as a parameter. To obtain an approximation for the tworoots k of the dispersion relation which couple at k = k,, we expand !(k) around the corresponding mode-boundary ke,(z) to second-order in k: 9(k) = 9(k.) + 1 (k - kc)29"(k).2This approximation for U(k) is excellent near the coupling points k, and zB, by virtue of the definition of kc(z),providing that 9"(k,) = 0. Moreover, it is a good approximation away from k, and zt, as long as G"(ke) doesnot vanish in the region of interest. The latter condition is particularly important when the branch point is notreal, since then the approximation must be extended all the way from zjj to the rea' axis, which is where thewaves actually propagate. It follows that the particular path in the z-plane must avoid passing through a branchpoint of the mapping OU = 0, since G"(k,,) vanishes there and kj(z)joins onto a dilferent branch. Except forthis restriction, the path can be quite arbitrary, and the transition to the real axis can be therefore performedby substituting - for z and making sure that kc(x) belongs to the correct branch as specified by the appropriatesaddle point k.(zn) = ks. The dispersion rc lation '(k) = 0 on the real axis Re z is thvn approximated byk' - 2kc(x)k + k (x) - Q(x) = 0, (4)whereQ(r) = -2 6(k)/9"(k)k (x). (5)is the scattering potential. We now note thai Eq. (4) contains the ambiguous combination kek which permits theinterpretations (k -+ i-I) ike 3L) , i d(cF)/dx, or a linear combination thereof, where (D is the wave-potential.The criterion for deciding the existence of a unique combination of these terms is that the resulting equa-tion must have regular turning points wher Q = 0, as this is where, by definition, th, inapping represented bythe dispersion relation has branch points. Such a combination does indeed exist, and te differential equation is4," + ikcV' + i(kcf)' + (Q - ke)( - 0, (6)where the prime now denotes differentiation with respect to x. To prove that Eq. (6) has the desired turningpoints it suffices to write (D = p exp (-i f kc(x) dx) = (pF, giving finallyP" + Q(X)p = 0. (7)The system, composed of the elementary local dispersion relations (4), together with their correspondingdifferential equations (7) can be appropriately termed as the "coupling approximation" of the dispersion rela-tion. An application of the method pertaining to the mode-conversion of lower-hybrid waves in the presence ofion-cyclotron damping is given in a companion paper of these Proceedings.References[1] T. H. Stix, Phys. Rev. Leti. 15, 878 (1965); W. Wasow, Ann. Math. 52, 350 (1950); A. L. Rabenstein,Arch. Ration. Mech. Anal. 1 418 (1958); V. S. Chan, S. C. Chiu, and G. E. Guest, Phys. Fluids 23, 1250(1980).[2] V. Fuchs, K. Ko, and A. Bers, Phys. Fluids (to be published).3yBRANCH CUT CbC'C-xf (C) f (C)ks= UBf (Cb)k rFigure 1. For a contour C not crossing the branch cut. Q, the branches f(C) lie, unconnected, on either side ofthe map. f(C,). Of the cut. A branch cut itnclf maps onto the k-plane as extending from the saddle point k, intotwo directions, and represenms the boundary between the maps of the two Riemann sheets it connects. In theother case, of C' crossing th- branch cut, the branches connect through the boundary f(Cb) signifying a mode-coupling event.yCbz BBBf (Cb)A -< sxBA IBkrA.~---BFipmre 2. The assumption of pairwise cotIpling reduceshe problem of an n-valued mapping into a series ofIO wvalued maps. If the branch point lies away from' thereai x-axs. the branches appear uncoupled in both thek, versus kr, as well as in the inore farniliaki, k, versusr, representations.4Ikr kiksr

The coupling approximation of local dispertion relations

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The coupling approximation of local dispertion relations

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