Streamer saturation: a dynamical systems approach

Transcription

1 Physics Letters A 351 (2006) Streamer saturation: a dynamical systems approach F. Jenko Max-Planck-Institut für Plasmaphysik EURATOM Association Garching Germany Received 26 September 2005; received in revised form 5 November 2005; accepted 11 November 2005 Available online 18 November 2005 Communicated by F. Porcelli Abstract Small-scale turbulence in toroidal magnetoplasmas driven by electron or ion temperature gradient modes is known for its tendency to form strongly anisotropic structures namely radially elongated vortices called streamers or (mostly) poloidal E B shear flows called zonal flows. The interaction of such patterns is studied here in the framework of a simple dynamical system focusing in particular on the initial saturation phase of a streamer. It is shown that the saturation amplitude in this reduced description is determined by a balance between the streamer s linear growth rate and the zonal mode s nonlinear growth rate Elsevier B.V. All rights reserved. PACS: Ra; Mw; Kt Keywords: Plasma turbulence; ETG; ITG; Streamer; Zonal flow 1. Introduction One of the most challenging topics in the physics of turbulent magnetoplasmas is the formation of strongly anisotropic structures and their interaction with the turbulent environment. In toroidal systems the turbulence is typically driven by a combination of background temperature/density gradients and magnetic curvature effects favoring the formation of radially elongated vortices called streamers [1 4]. The presence of such structures may lead to a substantial enhancement of the resulting radial fluxes with respect to mixing length expectations of the type χ γ/k 2 [5]. Here χ is the heat diffusivity γ is a characteristic linear growth rate of the underlying microinstabilities and k is a respective perpendicular wavenumber. On the other hand one often finds purely radial modes of the electrostatic potential at fairly high amplitudes (see Ref. [6] and references therein). These zonal modes are associated with E B shear flows pointing (more or less) in the poloidal direction. Their origin can be traced back to the fact that the electron density does not respond to fluctuations of the electrostatic po- address: fsj@ipp.mpg.de (F. Jenko). tential which are constant on magnetic surfaces breaking the electron adiabaticity constraint. The dynamics of streamers and zonal flows is of central importance for microturbulence driven by electron temperature gradient (ETG) modes [7] and ion temperature gradient (ITG) modes [8]. Neglecting Debye shielding effects these modes are symmetric with respect to the interchange of the species labels in the adiabatic electrostatic limit. Their nonlinear dynamics displays profound differences however. While the ITG system tends to spin up strong zonal flows which may even lead to self-suppression of the turbulence [9] the ETG system may be dominated by high-amplitude streamers [2]. Although the physical reason for this type of pattern formation is far from being fully understood there is significant evidence that streamers may be viewed as remnants of linear modes [10]. Thus the situation seems to resemble that in Rayleigh Bénard convection and related systems [11]. In the light of the relative weakness of zonal flows in the ETG case it is clear that there must be a different kind of nonlinear saturation mechanism at work. Following earlier work by Cowley and co-workers [12] it was shown that streamer breakup can be caused by Kelvin Helmholtz-like secondary instabilities [23]. Semi-analytical results obtained in the frame /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.physleta

2 418 F. Jenko / Physics Letters A 351 (2006) work of a simple fluid system were later confirmed and refined by numerical experiments based on the nonlinear gyrokinetic euations [5]. Here it was assumed that the saturation amplitudes of the streamers are set by the condition that their linear growth rates are balanced by the nonlinear growth rates of the respective secondary instabilities. The main goal of the present work is to test this criterion by means of a dynamical systems approach. The remainder of this Letter is organized as follows. In Section 2 a system of three coupled ordinary differential euations describing the interaction of streamers and zonal flows is derived from Hasegawa Mima type fluid models of plasma turbulence driven by ETG and ITG modes. This simple dynamical system is then used in Section 3 to analyze the nonlinear saturation of streamers. Here we will put particular emphasis on the initial saturation phase trying to answer the uestion of how to best determine the streamer s saturation level. Finally some conclusions will be presented in Section Streamers and zonal flows in Hasegawa Mima type fluid models 2.1. Basic euations While studies of small-scale turbulence in a magnetized plasma should generally be based on gyrokinetics [13] certain basic features of the nonlinear dynamics can be considered in the framework of simple fluid models like the one proposed by Hasegawa and Mima [14]. Their well-known and widely used euation can be written as d ( φ 2 φ x ) = 0 (1) dt where d dt = t φ y x + φ x y. Here the time variable t the space variables x and y and the electrostatic potential φ = φ(xyt) are normalized appropriately [14]. E.(1) describes the advection of the generalized vorticity φ 2 φ x by the E B velocity v E = φ ẑ.assuming a finite and doubly periodic domain in r = (x y) space one may expand the potential in a Fourier series: φ(rt)= Φ k (t) exp(ik r). (3) k Since φ is a real uantity the complex-valued Fourier coefficients satisfy Φ k = Φ k. In Fourier space E. (1) reads ( 1 + k 2 ) Φ k + ik y Φ k = [ẑ (k1 k 2 ) ( 1 + k2 2 )] Φk1 Φ k2. k 1 +k 2 =k Here the dot denotes the derivative with respect to time and the right-hand side of this euation is the Fourier representation of (2) (4) (5) the uadratic vorticity nonlinearity describing three-wave coupling. The sum runs over all wavevectors k 1 and k 2 which obey the constraint k 1 + k 2 = k. A few years ago several authors noted that a more realistic description of plasma turbulence on ion-gyroradius scales calls for a slight generalization of the classic Hasegawa-Mima euation [215 17]. As it turns out E. (1) is to be replaced by d ( φ φ 2 φ x ) = 0 (6) dt where the angular brackets denote spatial averaging in the y direction. The Fourier space representation of this generalized Hasegawa Mima euation is obtained by replacing 1 + k 2 k 2 for k y = 0 and 1 + k2 2 k2 2 for k 2y = 0inE.(5). One can take the classic and generalized Hasegawa Mima euations as simple model systems describing respectively turbulence driven by electron temperature gradient (ETG) modes and ion temperature gradient (ITG) modes. Whereas the ITG system is known for its tendency to self-generate strong zonal flows which are characterized by k y = 0 [6] ETG turbulence is able to self-sustain radially elongated streamers with k x = 0(or more general k x k y ) [23]. The physics of these structures is certainly too complicated to be captured by two-dimensional fluid models like the ones introduced above but the latter can still be used to gain some insight into basic aspects of the nonlinear dynamics of zonal flows and streamers. In the following we shall further reduce the complexity of the model by restricting to only four modes. The basic idea here is to focus on the competition of a single streamer and a single zonal flow mode involving two sidebands. This procedure leads to a basic version of the secondary instability approach advanced by Cowley and co-workers [12] as one possible streamer saturation mechanism A simple dynamical system describing streamers and zonal flows (ETG case) To study the generation of zonal flows and saturation of streamers we now reduce the Hasegawa Mima euation to a simple nonlinear dynamical system. We assume that only four modes are excited a streamer Φ with (k x k y ) = (0) a zonal flow Φ 0 with (k x k y ) = (p 0) and two sidebands Φ + with (k x k y ) = (p ) and Φ with (k x k y ) = (p ). In this case E. (5) yields: Φ 0 = p3 ( Φ 1 + p 2 Φ Φ Φ ) + Φ + + iω + Φ + = p(2 p 2 ) p 2 Φ Φ 0 Φ + iω Φ = p(2 p 2 ) p 2 Φ Φ 0 Φ + iω Φ = 3 p ( Φ Φ Φ +Φ0 ) where the Ω s are given by Ω k = k y 1 + k 2 (7) (8) (9) (10) (11)

3 F. Jenko / Physics Letters A 351 (2006) for the respective wavenumbers: Ω + = Ω = p 2 Ω = (12) If the amplitude of the streamer is much larger than that of the three remaining modes E. (10) may be linearized i.e. its right-hand side may be neglected. One thus obtains the following differential euation for Φ 0 : [ p 4 ( 2 p 2 ] ) Φ0 (13) (1 + p 2 )( p 2 ) Φ 2 Ω 2 Φ 0 = 0. Here Ω = Ω Ω + is the freuency mismatch between the streamer and the sidebands. For Φ 0 (t) exp(γ 0 t) one thus gets 2 γ 0 = 2 p 4 ( 2 p 2 ) (14) (1 + p 2 )( p 2 ) Φ 2 Ω 2 for the zonal flow growth rate in accordance with a similar calculation presented in Ref. [18]. In this context it is important to note that E. (14) is closely related to the secondary instability analysis in Ref. [3]. This is not by accident. Given a streamer with k y = the mode structure of a secondary instability with k x = p has the nonzero Fourier components k y = N where N is an integer (positive zero or negative). Since in the ETG system the k y = 0 component does not play a special role (this is not true for the ITG case) the nonlinear growth rate of a streamer is close to that of a corresponding secondary mode. Therefore the terms zonal flow and secondary instability can be used more or less interchangeably whenever the ETG case is considered. Of course our present model cannot replace a complete secondary instability analysis since is misses effects like spatial localization of the secondary s mode structure in regions where the drive is largest. Nonetheless it is expected to be at least ualitatively right. Actually E. (14) turns out to yield a fairly accurate representation of Fig. 4 in Ref. [4] which takes the secondary s real mode structure into account. In the following we will neglect the differences in the real parts of Ω Ω + and Ω. This approximation which allows us to remove the oscillatory part in the dynamics of the respective Φ s will be justified later. On the other hand it turns out to be of some interest to introduce damping terms in the euations for the zonal flow and the sidebands via the replacements t t + ν 0 in E. (7) and t t + ν ± in Es. (8) and (9) [18]. Furthermore we continue to take into account the exponential increase of the streamer amplitude Φ through a finite linear growth rate γ. Thus we get Φ 0 = p3 ( Φ 1 + p 2 Φ Φ Φ ) + Φ + = ν ± Φ + + p(2 p 2 ) p 2 Φ Φ 0 Φ = ν ± Φ p(2 p 2 ) p 2 Φ Φ 0 Φ = γ Φ + 3 p ( Φ Φ Φ +Φ0 ). (15) (16) (17) (18) Taking the limit p 1 expressing the time variable t in terms of streamer growth times γ 1 and introducing the uantities Γ 0 = ν 0 /γ Γ ± = ν ± /γ and (19) (A 0 A + A ) = 2 p 2 (Φ 0 Φ + Φ ) A = p3 Φ (20) γ γ we arrive at Ȧ 0 = Γ 0 A 0 A A + A A + Ȧ + = Γ ± A + + αa A 0 Ȧ = Γ ± A αa A 0 Ȧ = A + A 0 A A +A 0 (21) (22) (23) (24) where α = 2 /p 2 1. Any solution of this set of four amplitude euations will be real if the initial conditions are chosen to be real. Therefore one may neglect the imaginary parts of Es. (21) (24). Replacing A + and A by A s = A + A we finally obtain Ȧ 0 = A A s Γ 0 A 0 Ȧ s = 2αA A 0 Γ ± A s Ȧ = A A s A 0 A t = A + + A (25) (26) (27) (28) together with the euation Ȧ t = Γ ± A t which has the trivial solution A t (t) exp( Γ ± t) and may thus be neglected. The dynamical system (26) (28) has been derived before by Holland and Diamond [18] for an ETG-type Hasegawa Mima model. In the next subsection we will show that one obtains (basically) the same result for the ITG case A simple dynamical system describing streamers and zonal flows (ITG case) The ITG-like Hasegawa Mima system is described by E. (6). It differs from its ETG counterpart by a different treatment of zonal flow modes with k y = 0. Here Es. (7) (10) are replaced by Φ 0 = p ( Φ Φ Φ Φ +) Φ + + iω + Φ + = p(1 + 2 p 2 ) p 2 Φ Φ 0 Φ + iω Φ = p(1 + 2 p 2 ) p 2 Φ Φ 0 Φ + iω Φ = p ( Φ 0 Φ Φ +Φ0 ) (29) (30) (31) (32) and the zonal flow growth rate in the presence of a largeamplitude streamer is given by 2 γ 0 = 2 p 2 (1 + 2 p 2 ) (33) ( p 2 Φ 2 Ω 2 )

4 420 F. Jenko / Physics Letters A 351 (2006) Fig. 1. Simple model for streamer dynamics: time evolution of A 0 and A for Γ 0 = Γ ± = 2α = 1and(A 0 A s A ) = ( ) at t = 0 indicating that the fix point of this system at (A 0 A s A ) = (1 1 1) is unstable. instead of E. (14) in accordance with Ref. [19]. Neglecting once again the differences in the real parts of Ω Ω + and Ω and introducing drive/damping terms for all four modes we arrive at Φ 0 = p ( Φ Φ Φ Φ +) Φ + = ν ± Φ + + p(1 + 2 p 2 ) p 2 Φ Φ 0 Φ = ν ± Φ p(1 + 2 p 2 ) p 2 Φ Φ 0 Φ = γ Φ + p ( Φ 0 Φ Φ +Φ0 ). (34) (35) (36) (37) If one then takes again the limit p 1 and normalizes the time variable t and drive/damping coefficients with respect to γ or its inverse one obtains Es. (21) (24) for the uantities (A 0 A + A A ) = p γ (Φ 0 Φ + Φ Φ ) (38) with the parameter α set to unity. Conseuently the dynamical system (26) (28) also applies to the ITG case. The key difference is the meaning of the normalized amplitudes (A 0 A + A A ). We will return to this point later. In the remainder of this Letter we will study the physics of streamer saturation in the framework of the above three-amplitude model. 3. Dynamical systems analysis of streamer saturation 3.1. Fix point stability The goal of this section is to investigate the nonlinear saturation of streamers via coupling to zonal modes. Our study will be based on the simple dynamical system (26) (28) which has a nontrivial fixed point at A 0 =  0 Γ ± /2α A s =  s Γ0 and A =   0  s [18]. Although one might expect that this fix point determines the long-time behavior of the system we will show that this is not the case. Some insights into the dynamics described by Es. (26) (28) can be obtained by solving them numerically for a number of parameter sets (Γ 0 Γ ± α) and initial conditions (A 0 A s A ) at t = 0. Choosing e.g. Γ 0 = Γ ± = 1 and a = 1/2 it follows that  0 =  s =  = 1. Therefore if all three amplitudes coincide with these numbers for t = 0 they will stay at these values as time advances. However if A (t = 0) = 0.99 is chosen instead (keeping all other parameters fixed) one obtains a totally different scenario. The time histories of A and A 0 are shown in Fig. 1. (Note that for a = 1/2 and Γ 0 = Γ ± the dynamics of A 0 and A s are identical if their initial values are the same.) Between t 6 and t 9 A first rises from 1to 1.5 before it suddenly drops and becomes negative. A 0 on the other hand starts to rise a little later (at t 7) and at t 11 it drops to zero. A change of a single parameter by just one per cent was sufficient to create this large a difference. This result suggests that the fixed point is not stable. The following analysis confirms this suspicion. To examine the stability of the fixed point solution of Es. (26) (28) we linearize the right-hand sides about the fixed point. Introducing A (A 0 A s A ) T  ( 0  s  ) T and à A  we obtain t à = Mà (39) where ( ) Γ0 Γ± Γ 0 /2α Γ0 2αΓ± M = Γ 0 Γ ± 2αΓ± (40) Γ 0. Γ ± /2α 1 This matrix has the property det M = 4Γ ± Γ 0 i.e. its determinant is nonzero as long as both Γ ± and Γ 0 are nonzero. Its eigenvalues are given by the solutions of the cubic euation λ 3 + (Γ 0 + Γ ± 1)λ 2 + 4Γ 0 Γ ± = 0 (41) which turns out to be independent of the parameter α. The fix point is stable if and only if the real part of all three solutions of E. (41) is negative (or zero). For Γ ± = Γ 0 Γ the exact solutions of E. (41) are λ 1 = 2Γ λ 2/3 = 1 2( 1 ± 1 8Γ ) (42) satisfying the relations λ 2 λ 3 = 2Γ and λ 2 + λ 3 = 1 and the corresponding eigenvectors are given by ( ) ( ) 1 2αλ3 λ3 v 1 = 2α v 2 = 0 2 Γ ( ) 2αλ2 λ2 v 3 = (43) 2. Γ The Γ dependence of the real part of these eigenvalues is shown in Fig. 2. ForΓ>0 the real parts of λ 2 and λ 3 are always positive. For Γ 1/8 we find Re(λ 2 ) = Re(λ 3 ) = 1/2 and for 0 <Γ <1/8 we have Re(λ 2/3 ) = λ 2/3 > 0. For Γ = 0 we find Re(λ 2 ) = 1 and Re(λ 3 ) = 0. This means that for any choice of Γ the fix point is unstable. If the initial conditions are chosen such that à 1 and the v 2 component of à does not vanish the envelope of à grows with time as exp[re(λ 2 )t]. For Γ ± Γ 0 the stability analysis becomes more involved. One option is to inspect Cardan s solution of the cubic euation (39). Alternatively one can study the solutions numerically. Given the symmetry between Γ ± and Γ 0 with respect to interchange it is sufficient to vary Γ 0 holding Γ ± constant. One

5 F. Jenko / Physics Letters A 351 (2006) Fig. 2. Simple model for streamer dynamics linearized about the nontrivial fix point: real part of all three eigenvalues as a function of the damping parameter Γ.ForΓ>1/8 the two positive solutions coincide. Fig. 4. Reduced dynamical system: flow vectors ( t A 0 t A ) in (A 0 A ) space for Γ = 0.01 and Γ = 1. The fix points are located at (0 0) and at ( ΓΓ). Fig. 3. Simple model for streamer dynamics linearized about the nontrivial fix point: real part of all three eigenvalues as a function of the damping parameter Γ 0 for Γ ± = 1. The two positive solutions coincide. The respective asymptotic limits are shown as dashed lines. example (for Γ ± = 1) is shown in Fig. 3 along with the analytical solutions of E. (41) for Γ 0 : λ 1 (Γ 0 + Γ ± 1) λ 2/3 2Γ ± ± 2i Γ ±. (44) Γ 0 Again one finds that Re(λ 2 ) = Re(λ 3 )>0. We may thus conclude that for any choice of (Γ 0 Γ ± α) the fix point  is unstable Long-time behavior In order to study the time evolution of the system for à 1 we cannot use the above linearization procedure anymore. Thus we have to return to Es. (26) (28). We simplify this problem by choosing Γ ± = Γ 0 = Γ. Employing Es. (26) and (27) one can show that ( t + 2Γ) ( A 2 s 2αA2 0) = 0. (45) Therefore A 2 s 2αA2 0 for t 1/Γ i.e. the variable A s can be removed from the system. Scaling out the parameter α via 2αA0 A 0 and 2αA A we are then left with the following two euations: Ȧ 0 = A A 0 ΓA 0 Ȧ = A A 2 0. (46) (47) Fig. 5. Reduced dynamical system: time evolution of A 0 and A for Γ = 0.2 and (A 0 A ) = (10 5 1) at t = 0. The cusp in the upper curve signals a change of the sign of A. The resulting flow vectors ( t A 0 t A ) in (A 0 A ) space for Γ = areshowninFig. 4. Initializing the system such that 0 <A 0 1 and 0 <A 1 the trajectory of the system will first move more or less parallel to the A axis in the positive direction. This corresponds to an exponential growth of A. Meanwhile A 0 does not change much. Once A gets sufficiently large A 0 starts to grow rapidly. Correspondingly the trajectory moves to the right in the (A 0 A ) plane. In the follow-up phase the system evolves such that A 0 0 and A. The time evolution of A 0 and A is illustrated in Fig. 5. Here we took Γ = 0.2 and (A 0 A ) = (10 5 1) at t = 0. Before and after the turning point A grows like exp[t] while A 0 exhibits superexponential growth/damping. The cusp in the upper curve signals a change of the sign of A. Numerical tests show that these findings also apply to the three-amplitude model described by Es. (26) (28). Therefore we may conclude that the above dynamical system (both in its reduced form and in its three-amplitude version) is not capable of describing the long-time behavior of a streamer. No (uasi-)stationary saturated state is reached. In the following we therefore concentrate on the initial saturation of the streamer.

6 422 F. Jenko / Physics Letters A 351 (2006) Fig. 6. Reduced dynamical system: saturation amplitude of A for Γ = 1and different initial conditions. Fig. 7. Reduced dynamical system: saturation amplitude of A as a function of Γ for (A 0 A ) = (10 5 1) at t = Initial saturation To get an idea of the sensitivity of the saturation amplitude with respect to the initial conditions the former was determined for a number of different initial values of A 0 (A ) holding the initial values of A (A 0 ) constant. The result is shown in Fig. 6. One observes that the saturation amplitude tends to increase as the initial amplitudes are reduced. The overall change is moderate however. For (A 0 A ) = (10 5 1) at t = 0 we obtain a saturation amplitude of about 16.2 and all other points lie within 50% or so of this value. A more interesting aspect to study is the dependence of the saturation amplitude on the damping parameter Γ. A corresponding scan is shown in Fig. 7. One notes that Γ only has an effect if it is of the order of unity or even larger. This result can be understood in terms of flow diagrams in (A 0 A ) space like the ones shown in Fig. 4. As we have seen before the system has two fix points a trivial one at (A 0 A ) = (0 0) and a nontrivial one at (A 0 A ) = ( ΓΓ).ForΓ 1 those two points are very close to each other and the phase space flows are rather insensitive to the actual choice of Γ. Only as the nontrivial fix point approaches the turning point of the trajectory i.e. for Γ 1 does Γ have a significant affect on the system s dynamics. The flows are modified such that for A 0 (t = 0) Γ the saturation amplitude of A get larger. Thus the role of the nontrivial fix point is not to determine the long-time evolution of a streamer but to affect its initial saturation. This effect is only pronounced however if Γ is chosen to be unrealistically large. Since Γ is given by the ratio of the damping rate of the zonal flow (and sidebands) and the linear growth rate of the streamer situations with Γ 1 are not likely to occur in practice. In the more realistic case of Γ 1 the parameter Γ drops out of the problem altogether. This is because Ȧ 0 = (A Γ)A 0 A A 0 near saturation for sufficiently small values of Γ. An alternative view on the physics of streamer saturation as described by the present model can be obtained in the following way. Defining an energy-like uantity E via E A2 2 + A2 0 2 E + E 0 Es. (46) and (47) yield the relationships Ė = 2E A 2 0 A Ė 0 = A 2 0 A 2ΓE 0 Ė = 2E 2ΓE 0. (48) (49) (50) (51) This means energy is put into the system through the linear drive of the streamer (note that the linear growth rate is normalized to unity) and it is taken out through the damping of the zonal flow (secondary instability). The term A 2 0 A acts as a conservative energy transfer mechanism between E and E 0. Although in principle such a system could develop a (uasi-) stationary state we saw above that it does not. Moreover streamer saturation is not determined by a balance of the nonlinear drive and the linear/nonlinear damping of the zonal flow as suggested by Holland and Diamond [18]. This mechanism would lead to A sat Γ whereas we find that A sat is more or less independent of Γ for Γ 1. Instead A sat is set by the condition that the linear growth rate of the streamer balances the nonlinear energy transfer rate from the streamer to the zonal mode (secondary mode). This statement can be uantified by introducing the nonlinear streamer growth rate γ NL via γ NL Ė = 1 A (E 0 /E ). 2E Obviously saturation is characterized by γ NL = 0or A sat = Esat /Esat 0. (52) (53) For the present model we find A sat 10 up to a factor of two or so for a wide range of initial conditions and (reasonable) values of Γ. Using Es. (20) and (38) this result can be expressed as Φ sat Φ sat γ p 3 γ p (ETG case) (ITG case). (54) (55) A comparison with Es. (14) and (33) shows that in both cases the condition Φ sat : γ γ 0 (56)

7 F. Jenko / Physics Letters A 351 (2006) is satisfied. In other words Φ sat is determined via a balance between the linear growth rate of the streamer and the nonlinear growth rate of the zonal mode (which is proportional to Φ sat ). This is a central result of the present analysis. Assuming that Φ sat is an indicator of the respective uantity in the saturated state the turbulent transport in a strong turbulence regime should be proportional to the expressions in Es. (54) and (55). This is the approach used successfully in Ref. [5]. Finally we would like to compare the above estimates for Φ sat with the nominal threshold values for zonal flow growth as set by Es. (14) and (33). In the long-wavelength limit ( p 1) the latter are given by Φ lim 1/p (ETG case) (57) Φ lim p (ITG case). (58) Assuming γ these two relations can also be written in the form Φ lim p 2 Φ sat Φ sat. (59) This finding constitutes an a posteriori justification for neglecting threshold effects in the zonal flow growth rates as discussed e.g. in Refs. [19] and [18]. 4. Conclusions The nonlinear saturation of streamers in plasma microturbulence has been investigated by means of a reduced lowdimensional description allowing for a dynamical systems analysis. It has been shown that the nontrivial fix point of the resulting system of three coupled ordinary differential euations is unconditionally unstable. Furthermore unless one initializes the system in special ways the streamer amplitude grows exponentially towards in the long-time limit. These results show that the present model is clearly incomplete. Nevertheless it can be used to address the physics of the streamer s initial saturation phase. In this context it was demonstrated that the saturation amplitude is more or less independent of the initial conditions and model parameters provided they are chosen realistically. In particular the damping rates of the zonal mode and sidebands have little impact as long as they do not clearly exceed the linear growth rate of the streamer. Moreover threshold effects stemming from a freuency mismatch between the streamer and the sidebands tend to play only a minor role. These results imply that the saturation level of the streamer in this reduced description is determined by a balance between the streamer s linear growth rate and the zonal mode s nonlinear growth rate. Acknowledgements The author gratefully acknowledges stimulating discussions with R. Friedrich. References [1] J.F. Drake P.N. Guzdar A.B. Hassam Phys. Rev. Lett. 61 (1988) [2] F. Jenko W. Dorland M. Kotschenreuther B.N. Rogers Phys. Plasmas 7 (2000) [3] W. Dorland F. Jenko M. Kotschenreuther B.N. Rogers Phys. Rev. Lett. 85 (2000) [4] P. Beyer S. Benkadda X. Garbet P.H. Diamond Phys. Rev. Lett. 85 (2000) [5] F. Jenko W. Dorland Phys. Rev. Lett. 89 (2002) [6] P.H. Diamond S.-I. Itoh K. Itoh T.S. Hahm Plasma Phys. Controlled Fusion 47 (2005) R35. [7] Y.C. Lee J.Q. Dong P.N. Guzdar C.S. Liu Phys. Fluids 30 (1987) [8] W. Horton Rev. Mod. Phys. 71 (1999) 735. [9] A.M. Dimits et al. Phys. Plasmas 7 (2000) 969. [10] F. Jenko J. Plasma Fusion Res. Ser. 6 (2004) 11. [11] T. Hartlep A. Tilgner F.H. Busse Phys. Rev. Lett. 91 (2003) [12] S.C. Cowley R.M. Kulsrud R. Sudan Phys. Fluids B 3 (1991) [13] E.A. Frieman L. Chen Phys. Fluids 25 (1982) 502. [14] A. Hasegawa K. Mima Phys. Rev. Lett. 39 (1977) 205. [15] W.D. Dorland Ph.D. thesis Princeton University [16] A.I. Smolyakov P.H. Diamond M. Malkov Phys. Rev. Lett. 84 (2000) 491. [17] J.A. Krommes C.-B. Kim Phys. Rev. E 62 (2000) [18] C. Holland P.H. Diamond Phys. Lett. A 344 (2005) 369. [19] G. Manfredi C.M. Roach R.O. Dendy Plasma Phys. Controlled Fusion 43 (2001) 825.