Constraints for the aperiodic O-mode streaming instability
CConstraints for the aperiodic O-mode streaming instability
M. Lazar , , R. Schlickeiser , , S. Poedts , A. Stockem and S. Vafin ∗ Center for Plasma Astrophysics, Celestijnenlaan 200B, 3001 Leuven, Belgium Institut f¨ur Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik,Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany Research Department Plasmas with Complex Interactions,Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany (Dated: September 24, 2018)In plasmas where the thermal energy density exceeds the magnetic energy density ( β (cid:107) > A = T ⊥ /T (cid:107) < (cid:107) and ⊥ denote directions relative to the uniform magnetic field). Whenstimulated by parallel plasma streams the instability conditions extend to low beta states, i.e., β (cid:107) <
1, and recent studies have proven the existence of a new regime, where the anisotropy thresholddecreases steeply with lowering β (cid:107) → β (cid:107) < β (cid:107) > PACS numbers: 52.25.-b — 52.25.Mq — 52.25.Xz — 52.35.FpKeywords: magnetized plasma – electromagnetic instabilities – counterstreams – temperature anisotropy –space plasmas
I. INTRODUCTION
There is an increased interest for understanding themechanisms that can destabilize the aperiodic or weaklypropagating modes in anisotropic plasmas. Of thesethe purely growing (aperiodic, i.e., (cid:60) ( ω ) = 0) ordi-nary (O) mode instability has recently received particu-lar attention owing to its potential applications in spaceplasmas . In a plasma at rest the O-mode instabil-ity (OMI) can develop only if the plasma beta is suffi-ciently high ( β ≡ πnk B T /B > B ), i.e., A = T ⊥ /T (cid:107) < .In the presence of streams, propagating along the or-dered magnetic field, the activity of this instability ex-tends to low beta β < (although theinstability is inhibited by the magnetic field by limit-ing the range of unstable wavenumbers ). Thus, fortwo symmetric counterstreams of electrons (subscript e )the conditions necessary for the O-mode instability are β e, (cid:107) > / (1 + 2 V e /u e, (cid:107) ) and A e < V e /u e, (cid:107) , where u e, (cid:107) = (2 k B T e, (cid:107) /m e ) / is the electron thermal velocityin parallel direction, β e, (cid:107) = 8 πnk B T e, (cid:107) /B , and V e is thestreaming velocity. (Symmetric counterstreams enable toanalyze the O-mode decoupled from the extraordinary(X) mode, which is less susceptible to the instability.Fig. 1 presents a schematic with the possible configu-rations of symmetric counterstreams.) These conditions ∗ Electronic address: [email protected] simply show that instability is also predicted for low val-ues of β e, (cid:107) < V e > u e, (cid:107) / √ A e > | Ω e | .A plasma system with counterstreaming ions can bemore susceptible to the O-mode instability than one inwhich only electrons are streaming . The streamingions enlarge the range of unstable wavenumbers but af-fect only slightly the maximum growth rates. However,for large growth rates (of the order of | Ω e | ), high electronbeta β e, (cid:107) ∼ T e, (cid:107) (cid:29) T e, ⊥ , are needed. On the other hand,only sufficiently high streaming velocities V e > u e, (cid:107) canpredict the occurrence of instability at large A e > . It is now straightforward to determinethe instability conditions for the whole range of plasmabeta, including the low-beta regime where the O-modeis driven unstable only by the relative motion of theplasma streams . For low β (cid:107) < β (cid:107) → , in order to confirm the marginalinstability condition in analytical forms, and implicitlythe existence of the new regime in the low-beta limit.Counterstreaming plasmas are also subject to the elec-trostatic two-stream instabilities (TSI), e.g., electron- a r X i v : . [ phy s i c s . p l a s m - ph ] D ec electron, electron-ion and ion-ion, of which, the insta-bilities driven by electrons (with a growth rate of theorder of electron plasma frequency ω pe = (cid:112) πne /m e )appear to be faster . Moreover, the electrostatic two-stream instability is in most scenarios faster than theO-mode instability (OMI), except for streaming veloc-ities very near or below the threshold for the onset ofthe two-stream instability . The instability of the O-mode remains to be established only for streaming ve-locities below the threshold of the two-stream instabilityeven for low β e, (cid:107) <
1. In this paper we propose to delim-itate these regimes on the basis of the results in Ref. 13,where the marginal condition has been derived systemati-cally for different types and characteristics of two-streaminstabilities. Thus, in Sec. II we revisit the instabilityconditions of the O-mode for three specific cases of sym-metric counterstreams. These are confronted in Sec. IIIwith the instability conditions of the electrostatic two-stream modes, providing the existence conditions for theO-mode streaming instability. The new criteria are dis-cussed along with our final conclusions in the last section.
II. THE O-MODE INSTABILITY
We first reanalyze the O-mode instability invoking therecent results in Refs. 2 and 4. These results are hereapplied for three specific cases of symmetric counter-streams with each component modeled by a drifting bi-Maxwellian distribution function. A schematic of theseplasma systems is presented in Fig. 1: I. The streams areneutral with electrons and ions having the same stream-ing velocity V e = V i ; II. The electron streaming velocity ishigher than the ion streaming velocity, V e > V i ; and III.Ions are at rest, V i = 0, and only electrons are streaming.Symmetric counterstreams minimize the number of em-ployed parameters, and enable to analyze the O-mode de-coupled from the extraordinary (X-) mode, which is lesssusceptible to being unstable. Only counterstreams ofthe same species need to be symmetric to satisfy this con-dition, but the electron and ion properties (e.g., stream-ing velocity, parallel or perpendicular temperature) arenot necessary the same. We are dealing with a singlespecies of ions, namely, protons. A. Preliminary conditions: general case
We assume two-component streams of electron-protonplasmas, and start their stability analysis from the ana-lytical Eq. (49) in Ref. 4 µ y + (1 + µ ) (cid:20) − β (cid:107) − P e µ µ (cid:18) νµ (cid:19)(cid:21) y = β (cid:107) − (cid:16) µν (cid:17) P e , (1) e (cid:45) e (cid:45) i (cid:43) i (cid:43) e (cid:45) e (cid:45) e (cid:45) e (cid:45) V e = V i V i = 0 i (cid:43) i (cid:43) V e > V i IIIIII
FIG. 1: Schematic of three possible cases of symmetric coun-terstreams: I. Electrons and ions have the same streamingvelocity V e = V i ; II. The electron streaming velocity is higherthan the ion streaming velocity V e > V i ; III. Ions are station-ary V i = 0. with the same streaming parameters ν ≡ (cid:15) e (cid:15) p − (cid:15) p − (cid:15) e V e V p , P e ≡ πm e n V e B (cid:15) e − (cid:15) e , introduced in Ref. 4, and y / ≡ x being thelimit value of the squared normalized wavenumber x e ≡ k u ⊥ ,e / (2Ω e ) = k c A e β (cid:107) / (2 ω pe ) required by themarginal condition of instability ( (cid:61) ( ω ) ≡ γ = 0), Thisequation is obtained based on the improved approxima-tions (36) and (37) in the same reference, and here in thenext will be refined by neglecting 1 (cid:28) µ (or µ − (cid:28) µ = m p /m e = 1836 is the proton-electron massratio, and removing the restriction to very high values ofthe parameter ν . Because of the symmetry of the coun-terstreams of each species (with the same relative density (cid:15) e = (cid:15) e = 1 / (cid:15) p = (cid:15) p = 1 /
2, and the same streamingvelocity V e = V e = V e , V p = V p = V p ) the quantities ν and P e simplify as follows ν = V e V p , P e = ω pe Ω e V e c . (2)Also for simplicity, the electron and ion temperaturesare assumed equal ( T e, (cid:107) = T p, (cid:107) = T (cid:107) , T e, ⊥ = T p, ⊥ = T ⊥ ),implying β e, (cid:107) = β p, (cid:107) = β (cid:107) .Equation (1) then reads y + ay + b = 0 , (3)with a = 1 − β (cid:107) − P e (cid:18) νµ (cid:19) , (4) b = 1 µ (cid:104) − β (cid:107) − (cid:16) µν (cid:17) P e (cid:105) , (5)This equation admits a positive solution y > a or b is negative. Ina low β < P e are large enough. Thus, a < P e > − β (cid:107) νµ , (6)and b < P e > − β (cid:107) µν . (7)For a low β (cid:107) < − β (cid:107) νµ > − β (cid:107) µν , (8)leading to the necessary condition b <
0, also found inRef. 4, Eq. (50). Looking to the solutions of Eq. (3)we can easily observe that this condition b < y = − a + √ a − b > , (9)that yields, explicitly, x = 12 (cid:20) P e (cid:18) νµ (cid:19) + β (cid:107) − (cid:21) + 12 (cid:34)(cid:18) β (cid:107) − (cid:19) + P e (cid:18) νµ (cid:19) − P e (cid:18) − β (cid:107) (cid:19) + 2 P e νµ (cid:18) β (cid:107) (cid:19)(cid:21) . . (10)Otherwise, for a high β (cid:107) > b < P e , and the same solution (9) remains positive. B. Interlude: condition b < Here we analyze in detail the necessary condition b <
1. Counterstreams of electrons
When ions are at rest, V i = 0, then ν → ∞ , and theinstability condition (7) becomes (also see Eq. (50) fromRef. 4) P e > − β (cid:107) . (11) Using the explicit form in Eq. (2), the instability condi-tion (11) requires V e > (1 − β (cid:107) ) c Ω e /ω p,e or β (cid:107) >
11 + V e u e, (cid:107) (12)which is less constrained than the condition derived inRef. 9 β (cid:107) >
21 + V e u e, (cid:107) >
11 + V e u e, (cid:107) (13)For cold beams we recover the same condition derived inRef. 14 V e > c Ω e /ω p,e .
2. Neutral counterstreams
When both the electrons and ions are streaming withthe same velocity V e = V p = V , implying ν = 1, the samecondition (7) becomes P e > − β (cid:107) µ/ν = 1 − β (cid:107) µ . (14)Writing P e in terms of Alfven speed V A = c Ω p /ω p,p P e ≡ ω p,e V e Ω e c = V e µV A (15)implies in condition (14) V > V A (1 − β (cid:107) ) / , (16)which is the same with condition (7) from Ref. 11. Forcold beams we find necessary V > V A , the same con-dition derived in Ref. 15. Since V A = c Ω p /ω p,p = c ( m e /m p ) . (Ω e /ω p,e ) < c Ω e /ω p,e , it follows that a sys-tem with counterstreams of protons (ions) is much moresusceptible to the instability than one with only counter-streams of electrons. C. The instability condition
With x derived in Eq. (10) the marginal instabilitycondition is readily found from Eq. (34) in Ref. [4] A < W ( x ) + W ( µx ) µ + 2 P e β (cid:107) (cid:20) W ( x ) + W ( µx ) ν (cid:21) − x β (cid:107) , (17)where W ( z ) = 1 − e − z I ( z ). We use this condition to de-rive the marginal stability against the O-mode instability.This condition is displayed with solid lines in Figs. 2 and3 for a number of relevant cases. The O-mode must beunstable in the gray shading below the solid lines. InFig. 3 counterstreams are chosen to be neutral, i.e., with V e = V p (case I in Fig. 1), while in Fig. 2 the electron parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) FIG. 2: Marginal instability for the OMI from Eq. (17) (solid line), and for the TSI from Eq. (26) (dashed line), when ν = V e /V i = 10. The OMI can develop only in the darkest shading where streaming velocity is below the threshold requiredfor the onset of TSI. streams are faster than ions ( V e > V p , i.e., case II inFig. 1) with ν = 10.For the third case, when ions are stationary and onlyelectrons are counterstreaming (case III in Fig. 1), theparameter ν → ∞ becomes very large, reducing the ex-pression in Eq. (10) x = P e + β (cid:107) − , (18)and the instability condition from Eq. (17) A < W ( x ) + W ( µx ) µ + 2 β (cid:107) [ P e W ( x ) − x ] . (19)Notice that by comparison to Ref. 2, where the instabilitycondition is derived by neglecting the ion effects, here the ion nonstreaming effects are still present in the right-side (second term) of inequality (19). This new formin Eq. (19) is used in the present paper to derive themarginal stability of the electron counterstreams, whichis displayed with solid lines in Fig. 4. III. INTERPLAY WITH THE ELECTROSTATICTWO-STREAM INSTABILITY
Counterstreaming plasmas are also subject to the elec-trostatic streaming instabilities, i.e., electron-electron( e − e ), electron-proton ( e − p ), and proton-proton ( p − p )instabilities, with a maximum growth in the streamingdirection. Of these, the most efficient (or faster) are the parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν (cid:61) P e (cid:61) A e (cid:61) FIG. 3: Marginal instability for the OMI from Eq. (17) (solid line), and for the TSI from Eq. (26) (dashed line) for neutralcounterstreams, i.e., V e = V i = V ( ν = 1). The OMI can develop only in the darkest shading where streaming velocity is belowthe threshold required for the onset of TSI. instabilities driven by electrons, i.e., e − e or e − p two-stream instabilities . Notice that no acoustic modecan be excited if the plasma populations and compo-nents are isothermal ( T e ∼ T p ). The two-stream instabil-ity is also faster than the O-mode instability , exceptfor streaming velocity very near or below the thresholdfor the onset of the two-stream instability. The electro-static instability is only inhibited by the thermal spreadof plasma particles in the streaming direction (i.e., theparallel temperature), and it is therefore expected to be-come even more competitive against the O-mode instabil-ity in the low β < . For that reason, the existence of the O-mode insta-bility can be well established only below the marginalcondition for the two-stream instability. Here we pro-pose to delimitate these regimes based on the results inRef. [13], where the marginal condition has been derivedsystematically for different two-stream instabilities. Thesymmetry conditions imposed in our present study forthe counterstreaming plasmas (enables decoupling of theO-mode from the X-mode) along with T e (cid:39) T p (ubiq-uitous in space plasmas) lead to a reduced number oftwo-stream instabilities specified in Table I. Relevant forus here are only the plasma counterstreams a − b of thesame species a = b = e, p or different species a = e , b = p , parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν(cid:174)(cid:165) ; P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν(cid:174)(cid:165) ; P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν(cid:174)(cid:165) ; P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν(cid:174)(cid:165) ; P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν(cid:174)(cid:165) ; P e (cid:61) A e (cid:61) parallel beta Β (cid:254) A n i s o t r opy A e (cid:61) T e , (cid:166) (cid:144) T e , (cid:254) Ν(cid:174)(cid:165) ; P e (cid:61) A e (cid:61) FIG. 4: Marginal instability for the OMI, from Eq. (19) (solid line), and for the TSI from Eq. (23) (dashed line), driven onlyby the counterstreams of electrons (stationary ions, V i = 0). The OMI can develop only in the darkest shading where thestreaming velocity is below the threshold required for the onset of TSI. but satisfying Λ ≡ (cid:18) (cid:15) a T b (cid:15) b T a (cid:19) / = 1 , (20)(since (cid:15) a = (cid:15) b = 1 /
2, and T a = T b , see previous section).In this case the marginal condition of stability for twogeneric counterstreams a − b is | (cid:126)V b − (cid:126)V a | u b, (cid:107) (cid:54) . (cid:18) ω p,b ω p,a (cid:19) , (21)We analyze this condition for each type of two-streaminstability found relevant in Table I, and then comparethem to derive the lowest threshold condition. A. Electron-electron streaming instability
In this case a = b = e , the relative velocity reads | (cid:126)V b − (cid:126)V a | = 2 V e and the marginal condition (21) becomes V e (cid:54) . u e, (cid:107) . (22)Now, to adapt within a A ∝ β − (cid:107) -dependence (tempera-ture anisotropy vs. inverse plasma beta) we use definition(2), and express condition (22) in terms of the (electron)streaming parameter P e and the parallel plasma beta P e (cid:54) . β (cid:107) . (23) B. Proton-proton streaming instability
If the instability is driven only by protons, i.e., a = b = p , the marginal condition (21) provides a lower thresholdfor the streaming velocity of protons (also discussed byStringer ) V p (cid:54) . u p, (cid:107) . (24)since u p, (cid:107) = u e, (cid:107) /µ / < u e, (cid:107) ( µ = m p /m e = 1836).The first two cases in Fig. 1 include conditions whenthe proton streaming velocity is high enough to drivethe two-stream p − p instability. However, developing ofthis instability (with a growth rate of the order of pro-ton plasma frequency) is not realistic since the O-modeinstability (with a growth rate of the order of electrongyrofrequency) is expected to be much faster. C. Electron-proton streaming instability
In this case a Buneman-like instability isinduced , and the lowest threshold is foundfrom the same condition (21) if we consider the case a = p and b = e V e (cid:54) . u e, (cid:107) µ / ± ν / . (25)Furthermore, we can distinguish between two growingmodes distinctively assigned to ” ± ” in the denominator.Thus, the instability can be driven by electrons and pro-tons streaming either in the same direction, when therelative drifting speed is only | (cid:126)V e − (cid:126)V p | = V e − V p , or inopposite directions, when the relative drifting speed ishigher | (cid:126)V e − (cid:126)V p | = V e + V p and determines a lower thresh-old, asigned to ”+” in the denominator in Eq. (25). Interms of the streaming parameter P e and the parallelplasma beta this condition reads P e (cid:54) . β (cid:107) (cid:32) µ / ν / (cid:33) . (26)Now, comparing conditions (23) and (26), we find thatthe lowest threshold is defined by (23) only if ν > µ =1836, condition well satisfied when ions are almost sta-tionary, e.g., the limit case III. Indeed, in the limit caseIII, the ions are stationary ν → ∞ , and the two-stream e − e instability presents the lowest marginal conditiongiven by (23). In the other limit of neutral counter-streams, when the electrons and ions move with the samestreaming velocity, ν = 1 and the lowest marginal con-dition is that against the electron-proton instability re-duced to P e (cid:54) . β (cid:107) (cid:18) µ / (cid:19) (cid:39) . β (cid:107) < . β (cid:107) . (27) TABLE I: Types of electrostatic two-stream instabilities rel-evant for our cases I, II, and III, and the lowest marginalcondition of stability. ν Instabilities Marginal conditionI. ν = 1 e − e , e − p , p − p Eq. (27)II. ν < µ e − e , e − p , p − p Eq. (26) ν > µ e − e , e − p , p − p Eq. (23)III. ν → ∞ e − e , e − p Eq. (23)
The lowest marginal conditions of stability against theelectrostatic two-stream instabilities are summarized inTable I, indicating three different cases, which are notnecessarily related to our first classification, i.e., cases I,II and III.Thus, a new distinction can be made function of the pa-rameter ν as it takes values less or higher than the proton-electron mass ratio µ . If ν < µ is satisfied the lowestmarginal condition against any electrostatic instability isgiven by (26). For an arbitrary value ν = 10 (case alsostudied in Ref.[4]) this condition becomes P e (cid:54) . β (cid:107) ,and it is displayed with dashed lines in Fig. 2. For thelimit case when ν = 1, the same marginal condition sim-plifies to (27), which is displayed with dashed lines inFig. 3. In the opposite case, when ν > µ is satisfied,including the limit case ν → ∞ of stationary ions, thelowest marginal condition against any electrostatic insta-bility is given by (23). This condition is displayed withdashed lines in Fig. 4.The regimes where only the O-mode instability canoperate are always found in the right-hand side of thesedashed lines, i.e., the lighter gray shading. The super-position with the conditions for O-mode instability indi-cates for the existence of this instability only the dark-est shading regions. In the high beta ( β (cid:107) >
1) plas-mas the existence of the O-mode instability is not sig-nificantly affected, unless the streams are very energetic( P e > β (cid:107) <
1) regimes.
IV. DISCUSSION AND CONCLUSIONS
We have studied the marginal conditions for the O-mode instability by contrast to those of the electrostaticinstabilities. In the process of relaxation of the counter-streaming plasmas the electrostatic two-stream instabil-ities (with a growth rate of the order of electron plasmafrequency) are usually much faster than the O-mode in-stability (with a growth rate of the order of electron gy-rofrequency). The existence of the O-mode can there-fore be established only for streaming velocities belowthe threshold of the two-stream instabilities.Our present analysis is based on accurate analyticalexpressions of the marginal conditions of instability pro-vided in Ref. 4 for the O-mode instability, and in Ref.13 for the two-stream instabilities. The refined analysisin Secs. II and III aims to assess the robustness of theseanalytical expressions by the agreements found with par-ticular cases studied before, e.g., stationary ions, neutralstreams, e − e or e − p counterstreams. Despite the limi-tations imposed by the symmetry of the counterstreams,this seems to be the most convenient way to study theO-mode (decoupled from other plasma modes) and makeits properties more transparent. Presently there is anincreased interest for the O-mode instability, especiallyfor understanding its activity in the low-beta plasmas( β <
1) and for temperature anisotropies A = T ⊥ /T (cid:107) even larger than unity. As a mechanism of limitation ofthe kinetic anisotropies, this instability could provide aplausible explanation for the low-beta boundaries of sta-ble plasma configurations observed in the solar wind andterrestrial magnetosphere.Our investigation on the competition with the two-stream instabilities reveals that the parameter range ofthe O-mode instability is significantly restrained, espe-cially in the low-beta plasmas. Thus the low-beta regimeswhere only the O-mode instability can operate are sig-nificantly restrained by a minimum cutoff, given by β (cid:107) > β (cid:107) ,c ≡ P e . (cid:32) ν / µ / (cid:33) (28)if ν < µ = 1836 is satisfied, see Figs. 2 and 3, or by β (cid:107) > β (cid:107) ,c ≡ P e .
85 (29)in all the other cases, see Fig. 4. The activity of this in-stability in the high-beta plasmas can be also constrainedby the two-stream instability, if P e is large enough, seebottom panels in Figs. 2 and 3.Moreover, the existence of the O-mode instability be-comes limited only to sufficiently small A = T ⊥ /T (cid:107) , lessthan a maximum value A m given by Eqs. (17) and (10)at the plasma beta cutoff ( β (cid:107) ,c ) derived above, i.e., A m = W ( x c )+ W ( µx c ) µ + 2 P e β (cid:107) ,c (cid:20) W ( x c ) + W ( µx c ) ν (cid:21) − x c β (cid:107) ,c (30)with x c = 12 (cid:20) P e (cid:18) νµ (cid:19) + β (cid:107) ,c − (cid:21) + 12 (cid:34)(cid:18) β (cid:107) ,c − (cid:19) + P e (cid:18) νµ (cid:19) − P e (cid:18) − β (cid:107) ,c (cid:19) + 2 P e νµ (cid:18) β (cid:107) ,c (cid:19)(cid:21) . . (31)Estimations can be made if we, for instance, take a fewexamples of a less particular case when ν < µ , likethe ones displayed in Fig. 2 ( ν = 10). In this case β (cid:107) ,c (cid:39) . P e and x c = 0 . . P e − . P e + 1) . + TABLE II: Limit (maximum) values of the temperatureanisotropy ( A m ) given by Eq. (30) for the cases displayedin Fig. 2. P e β (cid:107) ,c x c ( × − ) 0.17 0.26 4.72 99.0 298 A m . P e − . (cid:39) P e −
1, and the limit values A m calculatedwith Eq. (30) are listed in Table II for different values ofthe parameter P e . Notice that values larger than unity A m > P e >
1, and only for sufficientlylarge values of the plasma beta parameter β (cid:107) > β (cid:107) < β (cid:107) > β (cid:107) (cid:62) β (cid:107) ,c . This restriction is particularly important forthe low-beta plasmas, since the existence of this insta-bility was claimed in the previous studies for any smallvalue (without limit) of the parallel plasma beta β (cid:107) → ν < µ and a streaming energydensity less than the magnetic energy density ( P e < A = T ⊥ /T (cid:107) >
1, thestimulation of this instability by the energetic streamsseems to be impossible (or unrealistic) in the low-betaplasmas. These situations appear to be resolved by amore realistic dissipation of the streaming free energy bythe electrostatic instabilities.
Acknowledgments
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