IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. PS-13, NO. 3, JUNE 1985 Bounds on Instability Growth Rates for Cold Plasmas Streaming Along an Infinite Magnetic Guide Field ABRAHAM KADISH, SENIOR MEMBER, IEEE, DON S. LEMONS, AND GREG KRIEGSMANN Abstract-We derive two bounds on growth rates for streaming insta- bilities of cold plasmas with inhomogeneous density and velocity pro- files transverse to a unidirectional infinite magnetic field: 1) a uniform bound which is independent of wavenumber, and 2) a wavenumber- dependent bound which is less than the uniform bound for long wave- lengths. Here streaming instabilities include both multistream instabili- ties in which two or more streams with different velocities overlap in configuration space, and the slipping stream instability in which a stream has a transverse velocity gradient. The bounds obtained are functions of only global steady-state parameters and are useful in bounding growth rates in experimental devices. I. INTRODUCTION AND SUMMARY C HARGE-NEUTRALIZED systems of charged particles \_flowing relative to one another along strong magnetic guide fields occur in a variety of natural and artificial settings- magnetotail and stellar wind flows and galactic jets, as well as in accelerators and pulsed power devices and their applications. The relative flow is a source of free energy for plasma instabili- ties. Among these instabilities are multistream instabilities in which two or more streams with different velocities overlap in configuration space, and the slipping stream instability in which a single stream has a velocity gradient transverse to the beam direction [1]. In beam applications these instabilities are often deleterious, although they are sometimes exploited to heat plasmas [2]. In any case, it is necessary to determine pa- rameter regimes in which these instabilities are important by calculating stability boundaries and linear growth rates. This has been done for several specific configurations [1], [3] - [6] . In this paper we present bounds on the growth rate for the class of instabilities driven by relative streaming of cold rela- tivistic beams along an infinite unidirectional magnetic guide field. The effect of the infinite magnetic field is to suppress motion across the field. The steady-state beams we consider are uniform in the direction along the magnetic field, but are allowed to be of arbitrary finite cross section and have arbi- trary inhomogeneous density and velocity profiles transverse to the guide-field direction. We derive two bounds for the growth rate. One of these is dependent on the eigenfunction wavenumber in the direction of the guide field. The other is independent of this wave- Manuscript received October 22, 1984; revised January 24, 1985. This work was supported by the U.S. Department of Energy under contract W-7405-ENG-36. A. Kadish and D. S. Lemons are with the Los Alamos National Lab- oratory, Los Alamos, NM 87545. G. Kriegsmann is with Northwestern University, Evanston, IL 60201. number. Both bounds are independent of the local steady- state structure transverse to the guide field. Both are "best possible" bounds in the sense that each is exact in a special case. They are achieved in uniform beams [7]. The wave- number-dependent bound is also achieved in locally concen- trated beams. An earlier calculation by Rome and Briggs [3] obtained spec- tral bounds for the slipping stream instability. Their bound on the growth rate is proportional to the maximum velocity gradient in the profile and only corresponds to the actual growth rate of an unstable mode for special profiles. In con- trast, the bounds we derive are robust in that they do not de- pend explicitly on gradients but only on the parameters of the steady-state flow (i.e., maximum and minimum values of ve- locity and density). Local derivatives do not appear. Since we show that these bounds are realized in special cases, we con- clude that in the absence of specific profile information, these are the best bounds obtainable. Our bounds and those obtained by Rome and Briggs are not directly comparable. For instance, one can imagine a large de- vice in which transverse gradients are small while differences between maximum and minimum values are large. Similarly, one can imagine a small device in which those differences are small but transverse gradients are large. Both bounds are best in different special cases. In general, neither is an actual growth rate. However, in calculating spectral bounds from experi- mental or even theoretical profiles, our formulation has advan- tages over the Rome and Briggs formulation. First, parameters can be measured with greater accuracy than their derivatives. Second, even given very good measurements, physical velocity and density profiles, although always continuous, are often not very smooth. Thus while the Rome-Briggs bound may be useful for analyzing model profiles, its sensitivity to local structure limits its utility in providing a priori bounds for any given device. II. EQUATIONS AND NOTATION We consider a charge-neutral relativistic plasma flowing along a unidirectional guide field which is taken to be in the z direc- tion. The guide field is assumed to be arbitrarily strong. If there is no applied electric field, the analysis of linear stability of the plasma and electromagnetic fields may be reduced, when z is an ignorable coordinate of the steady state, to study- ing the equation -C2AjE = (W2 - 00C) 1 - X E 0093-3813/85/0600-0167$01.00 © 1985 IEEE 1 67