Bers, A.

Ram, A.K.

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PFC/JA-91-8COMMENTS ON ABSOLUTE ANDCONVECTIVE INSTABILITIESAbhay K. Ram and Abraham BersMarch 1991Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139 USAThis work was supported by NASA Grant No. NAGW-2048 and NSFGrant No. ECS-88-2475. Reproduction, translation, publication, useand disposal, in whole or part, by or for the United States Governmentis permitted.Submitted to Geophysical Research Letters.iCOMMENTS ON ABSOLUTE ANDCONVECTIVE INSTABILITIESAbhay K. Ram and Abraham BersTABLE OF CONTENTSABSTRACT .......... ................................. 1INTRODUCTION .............................. 1SPACE-TIME EVOLUTION OF INSTABILITIES ...................... 2DISCUSSION OF THE EXAMPLES USED IN OSCARSSON [1986] ........... 3ACKNOWLEDGEMENTS ......... ........................... 5REFERENCES . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . 6iiABSTRACTWe point out the misconceptions in the arguments put forth by T. E. Oscarsson and K.G. R5nnmark (Geophys. Res. Lett. 13, 1384, 1986) that question the validity and usefulnessof the well-known theory of absolute and convective instabilities. The solid basis of thewell-known theory is clarified.iiiCOMMENTS ON ABSOLUTE AND CONVECTIVE INSTABILITIESA. K. Ram and A. BersPlasma Faion Center and Researh Laboratory of ElectrmnicsMassachuaetts Inatitute of TechnologyCambridge, MA. 02139ABSTRACTWe point out the misconceptions in the arguments put forth by T. E. Oscarsson and K. G.R~nnmark (Geophys. Res. Lett. 13, 1384, 1986) that question the validity and usefulnessof the well-known theory of absolute and convective instabilities. The solid basis of thewell-known theory is clarified.INTRODUCTIONAbsolute or convective evolutions of instabilities can produce very different signaturesof observed radiation from unstable plasmas. This is particularly useful in space plasmaswhen correlating experimentally observed emissions with theoretical models describingthe source regions [Ram 1991]. A letter [Oscarsson 1986] has questioned the basis andusefulness of the theory of absolute and convective instabilities by using some very simpleminded examples. However, the arguments put forth by these authors are based upon somesingular examples that are misleading. Furthermore, the notion of a time-asymptotic limitis treated in a trivialized manner by these authors.The theory of absolute and convective instabilities [Briggs 1964, Bers 1983] describesthe linear evolution of instabilities from an initially localized source in an infinitely homo-geneous medium. In this paper we clarify the conditions for the validity of the theory ofabsolute and convective instabilities, and, furthermore, show that the examples discussedin the aforementioned letter are not generic for continuous media and do not invalidatethe theory of absolute and convective instabilities.In what follows, we shall consider the case of evolution of instabilities in time and in1one spatial dimension. The generalization to higher spatial dimensions is more complicatedbut can be carried out in a straightforward manner.SPACE-TIME EVOLUTION OF INSTABILITIESThe theory of absolute and convective instabilities is based on a Green's functionanalysis of equations describing the space-time dynamics of a small-amplitude perturbationin a spatially homogeneous and a time-invariant medium. The equation for the Green'sfunction, G(z, t), is given by:Z G(z, t) = 6(z) 6(t) (1)where Z is, in general, a linear integro-partial-differential operator with constant coeffi-cients. Using complex Fourier-Laplace transforms, the solution to Eq. (1) can be writtenas:G(z, t) = IL doF S D(,w) exp(ikz - iwt) (2)where D(k, w) is the dispersion function for the system - directly related to the transformsof L, and L and F are the appropriate Laplace and Fourier contours, respectively, chosento satisfy causality [Bers 1983].The time-asymptotic evolution of G(z, t) can belong to one of only two possible cate-gories [Landau 1953, Bers 1983 and references therein]:(a) an absolute instability, where the response grows in time and encompasses more andmore of the space as a function of time - the response always including the spatiallocation of the initial perturbation; thus, every spatial point eventually becomingunstable, i.e. having temporally growing fields;(b) a convective instability, where the initial response grows in time but propagates awayfrom its point of origin; thus, any spatial point eventually becoming stable, i.e. havingtemporally decaying (or oscillatory) fields.As shown by Bers and Briggs [Briggs 1964, Bers 1983], for an unstable medium,fthis distinction in the time-asymptotic behavior of Eq. (2) is obtained by determining thet An unstable medium is one for which D(k,.,w) = 0 (k, being the real-k) gives at leastone branch, w(k,), that has a positive imaginary part, w,(k,.) > 0, for some k,..2pinch-point saddles in k (at k = k.), and the associated branch points in w (at w = w,)given by:9D (k., wo)D(k, wO) = 0 , k = 0 (3)The time asymptotic Green's function is dominated by the pinch point associated with thelargest value of w0a (w 1 being the imaginary part of w,). If that w0, is positive then theinstability evolves as an absolute instability. If wj is negative for all the pinch points thenthe instability will evolve as a convective instability.DISCUSSION OF THE EXAMPLES USED IN OSCARSSON [1986]Before we consider the examples in Oscarsson and Rnnmark [1986], two remarksare in order. First, the notion of time-asymptotics should not be considered, trivially, assimply t --+ oo; in this limit any linear instability will have long violated the, ab-initio,assumption of small-amplitude fields. A time-asymptotic state is established as soon asthe contribution from the pinch point with the largest w; dominates over the contributionof the pinch point with the next largest w0 i; this gives a time-asymptotic time scale whichis quite finite. The importance of nonlinear effects over such time scales of evolution must,of course, be assessed separately.Second, the evolution of an arbitrary initial perturbation, 40 (z) at t = 0, is obtainedby convolving the Green's function with 0 (z) [Morse 1953]. Clearly, if 0 (z) is spatiallylocalized, 0 (z, t) will evolve in a manner determined by the Green's function. Thus, ifthe Green's function analysis indicates the medium is absolutely (convectively) unstable,0 (z, t) will evolve as an absolute (convective) instability. However, if &0 (z) extends all theway to ioo then ?(z, t) may not, in general, evolve in a manner indicated by the Green'sfunction.We now consider the examples treated in Oscarsson and R~nnmark [1986] and showthat they are not characteristic of space-time evolution of instabilities; they describe aninstability that evolves essentially in time only and, thus, are singular examples. Thedispersion function of Oscarsson and R6nnmark [1986] is:D(k, w) = w - kv. - i- (4)3where vo and -y are constants. The corresponding equation describing the evolution of theelectric field, E(z, t), is of the form:OE(z, t) 1 OE(z, t) Y t = 0 (5)+z - E(z, t)= ()This is a hyperbolic partial differential equation with its characteristics given by z - v(t =constant. It is easy to realize that Eq. (5) is a singular equation which is not generic ofspace-time evolution. If we replace z and t by ( = z - vot and r = i, respectively, then Eq.(5) becomes:E,) -E((,T) = 0 (6)Obviously, this equation describes the evolution of E in time and has nothing to do withspace-time. So the concept of an absolute or convective evolution of an instability, whichdefines the space-time response of a medium, is not defined for such an equation. Thesolution to Eq. (6) is:E( , r) = e7(7- 7-o) Eo(e, 0 ) 6(r -ro) (7)where E is the initial prescribed value of the field at r = r, and e is the Heaviside functionneeded to satisfy causality. Thus, the solution to Eq. (5) is:E(z, t) = e7(t - to) Eo(z - vot, to) 0(t - to) (8)This equation implies that the field at a point z at time t has grown uniformly in time (atthe rate given by y) from the value of the field at position z - vet at time to. There is onlya trivial spatial evolution of the field along the characteristics.The Green's function for Eq. (5) is:G(Z, t) = v*e t 6(z - vt)e(t), Vo 3 0eGt -(z)0(t), V. = 0For vo 0 0, this shows that the instability convects away from its initial point of origin.However, in a reference frame moving with velocity vo, the instability just grows in timewithout any spatial evolution. The case when v. = 0 does not represent any space-time4evolution; thus, it is irrelevant to speak of absolute or convective instability. This, again,points out the singular nature of the dispersion function in Eq. (4).It is important to realize that, for the dispersion function of Eq. 4 with v0 $ 0, thespatial width of any instability does not change with time. For plasmas described bynon-singular dispersion relations, this is not the case for either absolute or convective in-stabilities. Any arbitrary perturbation, in general, will spread spatially beyond its originalspatial width. By choosing initial perturbations that extend all the way to too, Oscars-son and R~nnmark [1986] have attempted to create the impression that the perturbationsevolved spatially (instead of just temporally). We highligbt that aspect by considering indetail one of the examples they used.By using the Green's function given above, the time evolution of an initial perturba,tion:E(z, 0) = exp(-z 2 /d 2 ) exp(-ikoz) (10)is given by [Morse 1953]:E(z, t) = exp {-(z - vot) 2/d 2} exp(7t) exp -ik.(z - vt)} (11)All points in space, in the coordinate system moving with velocity v0 , grow at the samerate. This can be further emphasized by choosing E(z, 0) to be initially localized:E(z, 0) = exp(-z 2/d2 ) exp(-ikz) {6(z - 1) + 6(z + l)} (12)where I can be chosen to be as large as one wants. Then it is easy to show that at anytime t > 0 the field will be confined to a spatial width of 21 and will not extend beyondthat width.ACKNOWLEDGEMENTSThis work was supported by NASA Grant No. NAGW-2048 and NSF Grant No.ECS-88-2475.5REFERENCESBers, A., Handbook of Plasma Physics, Vol. 1, M. N. Rosenbluth, and R. Z. Sagdeev,eds. (North-Holland, Amsterdam 1983), Chap. 3.2.Briggs, R. J., Electron-Stream Intemction tuith Plasmas, (MIT Press, Cambridge, MA1964).Landau, L D. and I. M. Lifshitz, Electrodynamic. of Continuous Media (in Russian,GITTL, Moscow 1953); see also their Fluid Mechanics (Pergamon Press, London1959).Morse, P. M. and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, NewYork 1953).Oscarsson T. E. and K. G. Rbnnmark, Comments on the theory of absolute andconvective instabilities, Geophys. Res. Lett. 13, 1384, 1986.Ram, A. K. and A. Bers, Absolute versus convective analysis of instabilities in spaceplasmas, to appear in Physics of Space Plasmas (1990), SPI Conference Proceed-ings and Reprint Series, Number 10, T. Chang, G. B. Crew, and J. R. Jasperse,eds. (Scientific Publishers, Cambridge, MA, 1991); also, M.I.T. Plasma FusionCenter Report PFC/JA-91-4, Cambridge, MA, 1991.6

Comments on absolute and convective instabilities

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Comments on absolute and convective instabilities

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