Coronal ion-cyclotron beam instabilities within the multi-fluid description
AAstronomy & Astrophysics manuscript no. Mecheri-beam c (cid:13)
ESO 2018October 30, 2018
Coronal ion-cyclotron beam instabilities within the multi-fluiddescription
R. Mecheri and E. Marsch Max-Planck Institut f¨ur Sonnensystemforschung, Max-Planck Strasse 2, 37191 Katlenburg-Lindau, Germanye-mail: [email protected]
Received xxxx; accepted xxxx
ABSTRACT
Context.
Spectroscopic observations and theoretical models suggest resonant wave-particle interactions, involving high-frequencyion-cyclotron waves, as the principal mechanism for heating and accelerating ions in the open coronal holes. However, themechanism responsible for the generation of the ion-cyclotron waves remains unclear. One possible scenario is that ion beamsoriginating from small-scale reconnection events can drive micro-instabilities that constitute a possible source for the excitationof ion-cyclotron waves.
Aims.
In order to study ion beam-driven electromagnetic instabilities, the multi-fluid model in the low- β coronal plasma isused. While neglecting the electron inertia this model allows one to take into account ion-cyclotron wave effects that are absentfrom the one-fluid MHD model. Realistic models of density and temperature as well as a 2-D analytical magnetic field modelare used to define the background plasma in the open-field funnel region of a polar coronal hole. Methods.
Considering the WKB approximation, a Fourier plane-wave linear mode analysis is employed in order to derive thedispersion relation. Ray-tracing theory is used to compute the ray path of the unstable wave as well as the evolution of thegrowth rate of the wave while propagating in the coronal funnel.
Results.
We demonstrate that, in typical coronal holes conditions and assuming realistic values of the beam velocity, thefree energy provided by the ion beam propagating parallel the ambient field can drive micro-instabilities through resonantion-cyclotron excitation.
Key words.
Sun: corona – waves – instabilities
1. Introduction
Observations made by the Ultraviolet CoronagraphSpectrometer (UVCS) and other instruments on the Solarand Heliospheric Observatory (SOHO) have significantlyincreased our knowledge of the kinetic properties ofcharged particles close to the Sun in the source regionof the fast solar wind. Spectroscopic determination of thewidths of ultraviolet emission lines in coronal holes indi-cate that heavy ions are very hot and have high temper-ature anisotropies, and that heavier ions have a highertemperature than the protons by at least their mass ra-tio, i.e. T i /T p > m i /m p (Kohl et al. 1997; Cranmer et al.1999). These observations strongly suggest resonant ion-cyclotron wave-particle interaction as a major mechanismfor the heating and acceleration of ions in the magneticallyopen corona. This notion led to a renewed interest in mod-els involving ion heating by high-frequency ion-cyclotronwaves (Isenberg et al. 2000; Hollweg 2000; Marsch & Tu Send offprint requests to : R. Mecheri a r X i v : . [ a s t r o - ph ] J un R. Mecheri & E. Marsch: Coronal ion-cyclotron beam instabilities through ion-cyclotron resonance and the Cerenkov effect.The possible origin of the ion beams observed in the solarwind from reconnection jets and explosive events in thecorona has been proposed by Feldman et al. (1996).Indeed, detailed spectroscopic studies of the so-calledhigh-velocity events and explosive events, using spec-tra obtained with the Coronal Diagnostic Spectrometer(CDS; Brekke et al. (1997)) and the Solar UltravioletMeasurement of Emitted Radiation instrument (SUMER;Innes et al. (1997)) both on SOHO, revealed a new charac-ter of the lower corona as a highly dynamic medium. Theysignify the omni-presence of transient explosive events anda wide variety of plasma jets with velocities ranging from afew tens of a kilometer per second up to several hundredsof kilometers per second. Since these plasma jets have beenobserved to evolve in a similar way as predicted by thetheory of magnetic reconnection (Innes et al. 1997), ex-plosive events and plasma jets have been associated withthe highly-dynamic small-scale reconnections which aresupposed to take place in the chromospheric network, ap-proximately at heights of 1000 − Fig. 1.
Magnetic field geometry of a funnel as obtainedfrom the 2-D potential-field model derived by Hackenberget al. (2000). The field lines emerge from the boundarybetween two adjacent supergranules ( x = 0) and expandrapidly to fill the corona. The photospheric level is at z =0. Coordinate axes, wave vector and beam drift velocityare shown in red color.cussed in Sect. 4, and finally we give our conclusions inSect. 5.
2. Background plasma configuration
For the background plasma density and temperature weuse the model parameters of Fontenla et al. (1993) for thechromosphere and Gabriel (1976) for the lower corona.The 2-D potential-field model derived by Hackenberg et al.(2000) is used here to define the background magnetic field(Fig. 1) in the funnel. Analytically, the two componentsof this model field are given by: B x ( x, z ) = ( B max − B ) L π ( L − d ) ln cosh πzL − cos ( πdL + πxL ) cosh πzL − cos ( πdL − πxL )(1) B z ( x, z )= B + ( B max − B ) (cid:20) − dL − d + L ( L − d ) π × (cid:32) arctan cosh πzL sin πd L + sin ( πd L + πxL ) sinh πzL cos πd L + arctan cosh πzL sin πd L + sin ( πd L − πxL ) sinh πzL cos πd L (cid:33)(cid:35) (2)The typical parameters relevant for this model are: L =30 Mm , d = 0 .
34 Mm , B = 11 . , and B max = 1 .
3. Linear perturbation analysis
To describe wave propagation in the funnel we use themulti-fluid equations and subject them to a linear pertur-bation analysis. The fluid equations associated with thepolytropic gas law for any particle species j are given by: ∂n j ∂t + ∇ · ( n j v j ) = 0 , (3) . Mecheri & E. Marsch: Coronal ion-cyclotron beam instabilities 3 m j n j ( ∂ v j ∂t + v j · ∇ v j ) + ∇ p j − q j n j ( E + v j × B ) ++ m j n j (cid:88) j (cid:48) ν jj (cid:48) ( v j − v j (cid:48) ) = , (4) p j n − γ j j = const , (5)where m j , n j , v j , p j and γ j (=5/3) are respectively themass, density, velocity, pressure and the adiabatic poly-tropic index of a species j . Subscript j stands for electron e , proton p or alpha particle α (He ). The quantity ν jj (cid:48) isthe collision frequency of a particle of species j with par-ticles of species j (cid:48) (only the electron-proton collisions aretaken into account). The electric field E and the magneticfield B are linked by Faraday’s law: ∇ × E = − ∂ B ∂t . (6) The linear perturbation analysis is performed by express-ing all the quantities in the fluid equations as a sum of anunperturbed stationary part (with subscript 0) and a per-turbed part (with subscript 1) that is much smaller thanthe stationary part: n j = n j ( z ) + n j , T j = T j ( z ) + T j , p j = p j ( z ) + p j , v j = v j + v j , B = B ( x, z ) + B , E = E + E , with: n j (cid:28) n j , T j (cid:28) T j , p j (cid:28) p j , | v j | (cid:28) | v j | , | B |(cid:28) | B | , | E | (cid:28) | E | and v j × B = . (7)We assume charge neutrality and a current-free state forthe unperturbed stationary plasma, i.e., (cid:80) j q j n j = 0and (cid:80) j q j n j v j = 0. The zero-order terms cancel outwhen Eq. (7) is inserted into the multi-fluid Eqs. (3)-(5).Neglecting the nonlinear products of the first-order terms,we get a system of coupled linear equations: i ( ω − k · v j ) n j n j − i k · v j = 0 , (8) i ( ω − k · v j ) v j + Ω j (cid:18) E | B | + v j × B | B | + v j × B | B | (cid:19) − iC sj p j p j k − (cid:88) j (cid:48) ν jj (cid:48) ( v j − v j (cid:48) ) = , (9) p j p j − γ j n j n j = 0 , (10) i k × E = iω B , (11)where all the perturbed quantities have been expressedin form of a plane wave. This Fourier analysis turns allderivatives into algebraic factors, i.e. ∂/∂t → − iω and ∇ → i k , where ω is the wave frequency and k the wavevector. In the above equations, C sj = (cid:112) γ j k B T j /m j is theacoustic speed and Ω j = eB /m j the cyclotron frequencyof species j , and k B is the Boltzman constant. To derive the dispersion relation, the above linearizedequations have to be combined in order to obtain a linearrelation between the current density J and the electricfield E : J = σ · E , (12)where σ is the conductivity tensor which is related to thedielectric tensor (cid:15) through the following relation: (cid:15) ( ω, k , r ) = I + iωε σ ( ω, k , r ) . (13)Finally, the local dispersion relation is obtained using thetheory of electrodynamics (see, e.g., Stix 1992): D ( ω, k , r ) = Det (cid:20) c ω k × ( k × E ) + (cid:15) ( ω, k , r ) · E (cid:21) = 0 , (14)where c is the speed of light in vacuum and r is the large-scale position vector. We choose the wave vector k to liein the x − z plane, with k = k ( sinθ, , cosθ ). In the framework of the WKB approximation, the raytracing problem consists in solving a system of ordinarydifferential equations of the Hamiltonian form (Weinberg1962). The ray-tracing equations, which represent theequations of motion for the wave frequency ω , the wavevector k , and the space coordinate r , have been formu-lated by Bernstein & Friedland (1984). In the simple caseof a hermitian dielectric tensor, they are given by:d ω d t = − ∂D ( ω, k , r ) /∂t∂D ( ω, k , r ) /∂ω = 0 , (15)d k d t = ∂D ( ω, k , r ) /∂ r ∂D ( ω, k , r ) /∂ω , (16)d r d t = − ∂D ( ω, k , r ) /∂ k ∂D ( ω, k , r ) /∂ω . (17)A generalization of these equations to the case of an anti-hermitian dielectric tensor was also proposed in the paperof Bernstein & Friedland (1984). In this case, in additionto the ray path, the growth rate of the instability can becomputed as well. Note that Eq. (15) can be set to zero,because the dispersion relation does not explicitly dependon the time t (the background plasma is stationary). Theabove set of differential equations represents an initial-value problem which can be solved by using the initialconditions obtained from the local solutions of the disper-sion relation (Eq. 14). R. Mecheri & E. Marsch: Coronal ion-cyclotron beam instabilities x = 7 . z = 2 . ϕ ≈ ◦ Fig. 2.
Two-fluid (e-p plasma on the left panels) and three-fluid (e-p-He plasma on the right panels) dispersionsurfaces (top panels) and single curves (for θ = 30 ◦ at the bottom) for the case of a zero beam speed, i.e. v α = 0.The alpha-particle (He ) density is n α = 0 . n p and the plasma beta values are β e = 0 . β p = 12 β α . Here ω and k are normalized, respectively, to the proton cyclotron frequency, Ω p , and the inertial length, Ω p /V Ap , where V Ap =B / √ µ n p m p is the proton Alfv´en speed ( µ is the magnetic permeability in the vacuum). Here T e = T p = T α = 1 . × K , n p = 3 . × m − , B = 15 .
54 Gauss.
4. Numerical Results
We assume that the ion-beam particles are generated atthe funnel location at x = 7.5 Mm and z = 2.2 Mm, pre-sumably by small scales reconnection events. This loca-tion is characterized by a magnetic field inclination an-gle of ϕ = 82 ◦ with respect to the normal on the solarsurface. According to the observations, the calculationswill be performed for a beam velocity equal to 320 km/s.We consider the case of an alpha-particle (He ) beamplasma configuration, propagating parallel to the ambientfield. For comparison purpose, the dispersion diagrams inthe case of a plasma without the beam are also presented.We first present the results obtained from the local solu-tions of the dispersion relation (14) and then the resultsobtained from the non-local wave analysis using the ray-tracing equations. For v α = 0 and θ = 30 ◦ , the dispersion diagram in thecase of the two-fluid (e-p) model (left panels of Fig. 2)shows the presence of three stable modes, each one ofthem is represented by an oppositely propagating ( ω > ω <
0) pair of waves. These modes represent the ex- tensions of the usual Slow (red line), Alfv´en (green line)and Fast (blue line) MHD modes into the high-frequencydomain around ω = Ω p (= eB /m p ), where the wavesare dispersive (for details see Mecheri & Marsch 2006). Inthe case of the three fluid (e-p-He ) model (right panelsof Fig. 2) five stable modes are present (each one repre-sented by an oppositely propagating pair of waves). Thesemodes are: two slow modes (yellow and red lines), two ion-cyclotron modes (gray and green lines) and one fast mode(blue line). The waves are subject to mode conversion orcoupling, a phenomenon associated with the appearanceof a cut-off frequency concerning the fast mode (for detailssee Mecheri & Marsch 2006). The alpha particle beam configuration consists of threespecies: electrons (with density n e ), protons (with den-sity n p ) and a tenuous beam of alpha particles, He indicated by α , with a velocity v α parallel to the ambi-ent magnetic field B and a density n α = 0 . n p . Theprotons are considered to be at rest and the electrons arein motion with a velocity v e . Alphas and electrons sat-isfy thus the zero-current condition, v e = 2( n α /n e ) v α .The alpha-particle-beam relative density is taken fromin-situ observations made by the Helios spacecraft, with n α /n e ≈ n α /n p = 0 . − . . Mecheri & E. Marsch: Coronal ion-cyclotron beam instabilities 5 x=7.5 Mm, z=2.2 Mm, ϕ ≈ ◦ Fig. 3.
Wave frequency (black) and growth rate (red)in a alpha particle beam plasma ( n α /n p = 0 . v α = 320 km/s ≈ . V Ap ) versus wave vector for twopropagation angles θ . Note, that for θ = 15 ◦ (i.e., quasi-perpendicular propagation, top panel), the left-hand reso-nant ion-cyclotron mode ( ω L ) is excited, and that for θ =65 ◦ (i.e., quasi-parallel propagation, bottom panel) theright-hand resonant fast mode ( ω R ) is excited, wherebyboth satisfy the condition (19) for the anomalous Dopplereffect. Here T e = T p = T α .choice is justified since it has been argued by Feldmanet al. (1996) that the alpha beams might originate fromreconnection events at the base of the expanding solarcorona. In this case (i.e. v b (cid:54) =0) Fig. 3 shows that the dis-persion curves are strongly modified by the appearance ofregions of instability associated with two cyclotron beammodes, which in the cold-plasma case are characterized bythe dispersion relations (Cap 1978): ω b ≈ (cid:26) kv α cos ( ϕ − θ ) + Ω α cyclotron beam 1 kv α cos ( ϕ − θ ) − Ω α cyclotron beam 2 (18)Therefore, the value of electron-proton collision fre-quencies in this region of the solar atmosphere (i.e. ν ep / Ω p ≈ ν pe / Ω p ≈ θ = 15 ◦ (top panel), the instability results fromthe intersection between the ion-cyclotron mode ( ω L ) andthe cyclotron beam mode 2. This region of instability isindicated by (L) in reference to the left-hand polarizationcharacterizing the ion-cyclotron mode. Indeed, these two distinct and initially stable modes merge, within a certainrange (corresponding to the red curve) of the wave num-ber k , into one single unstable mode which satisfies theresonance condition: ω ≈ kv α cos ( ϕ − θ ) − Ω α . (19)This condition corresponds to a left-hand resonant cy-clotron excitation of the ion-cyclotron mode through theanomalous Doppler effect (see, Gary 1993). As shownin the left panel of Fig. 4, this instability extends fromsmaller k , at propagation angles around θ ≈ ◦ withsmall growth rate, i.e. γ ≈ . p , to higher k andto smaller angles of propagation with a larger growthrate, i.e. γ ≈ . p . Since the location at x=7.5 Mmand z=2.2 Mm is characterized by a B -inclination angle ϕ = 82 ◦ , we can therefore say that as k increases this in-stability tends to appear at increasingly oblique propaga-tion angles with respect to the ambient magnetic field, andits growth rate tends to maximize for perpendicular prop-agation. On the middle panel of Fig. 4 the left-hand reso-nant instability is shown as a function of θ and n α , for thecase of v α = 320 km/s ≈ . V Ap and a normalized wavenumber kV Ap / Ω p = 0 .
5. It is clearly seen that the growthrate of the instability increases with increasing n α andgradually is also covering a wider range of θ , but it staysbelow approximately θ ≈ ◦ . The maximum growth rateis γ ≈ . p for n α = 0 . n p and θ ≈ ◦ . This instabilityis also presented on the right panel of Fig. 4 as a functionof θ and v α and for n α /n p = 0 . kV Ap / Ω p = 0 . v α below which it does not occur. Thisthreshold depends on θ and increases from v α ≈ . Ap at θ ≈ ◦ to v α ≈ Ap for quasi-perpendicular propa-gation θ ≈ ◦ (knowing that the inclination angle of B is ϕ ≈ ◦ ).On the other hand, for v α (cid:54) = 0 and an angle of prop-agation θ = 65 ◦ (bottom of Fig. 3), the results show thedisappearance of the left-hand resonant instability involv-ing the ion-cyclotron mode and the appearance of anotherkind of instability involving the right-handed polarizedfast mode, from which the name right-hand resonant in-stability is derived. This instability is indicated by (R)and results from the intersection of the fast mode ( ω R )with the cyclotron beam mode 2 satisfying the resonancecondition (19). As shown on the left panel of Fig. 5 thisinstability is mainly centered around a normalized wavenumber kV Ap / Ω p ≈ .
5, and has maximum growth rate γ ≈ . p at a large angle of propagation θ ≈ ◦ . Theright-hand resonant instability vanishes for highly obliquepropagation with respect to the ambient field (knowingthat the inclination of B in this location is ϕ ≈ ◦ ), andthe wave amplitude grows strongly for decreasing obliquityof the propagation angle. On the middle panel of Fig. 5,the right-hand resonant instability is shown as a functionof θ and n α , for the case of v α = 320 km/s ≈ . V Ap and kV Ap / Ω p = 0 .
5. It is clearly seen that the growthrate of this instability increases with increasing n α . This R. Mecheri & E. Marsch: Coronal ion-cyclotron beam instabilities x = 7 . z = 2 . ϕ ≈ ◦ Fig. 4.
Growth rate of the left-hand resonant ion-cyclotron instability , in the case of an alpha-particle beamplasma.
Left : as a function of the angle of propagation θ and the normalized wave number kV Ap / Ω p and for an alphabeam density n α = 0 . n p and velocity v α = 320 km/s = 1 . Ap . Middle : as a function of θ and n α and for v α = 320 km/s ≈ . Ap and kV Ap / Ω p = 0 . Right : as a function of θ and v α /v Ap and for n α = 0 . n p and kV Ap / Ω p = 0 .
5. Here T e = T p = T α .behavior is more pronounced for higher θ , which corre-sponds to decreasingly oblique propagation. The maxi-mum growth rate, γ ≈ . p , is obtained for n α =0.2n p and θ ≈ ◦ . We can also notice that the instability fadesaway for small propagation angles, with θ (cid:46) ◦ , whichcorresponds to quasi-perpendicular propagation (with re-spect to the field). In the same figure (on the right panel)we also present the dependence of this instability upon θ and v α for n α /n p = 0 . kV Ap / Ω p = 0 .
5. It canbe seen, similarly to the left-hand instability, that this in-stability has a threshold in the beam velocity v α , belowwhich it does not occur. This threshold depends on θ andincreases from v α ≈ . Ap at large angle of propaga-tion, θ ≈ ◦ (quasi-parallel propagation since ϕ = 82 ◦ ),to v α ≈ . Ap at θ ≈ ◦ (quasi-perpendicular propa-gation). In this section we intend to go beyond the local treatmentof the waves and perform a non-local wave study usingthe ray-tracing equations. The ray-tracing equations aresolved employing the initial conditions obtained from thelocal solutions of the dispersion relation (14) at the loca-tion with x = 7 . z = 2 . ) beam plasma config-uration with a constant concentration, n α = 0 . n p , and aconstant beam velocity of v α = 320 km/s.The ray paths of the unstable waves as well as the vari-ation of their growth rates as a function of height z , when the wave is launched at the initial location ( x = 7 . z = 2 . ϕ ≈ ◦ ) of the magnetic field with re-spect to the normal on the solar surface, are illustrated inFig. 6. The results are presented for a different initial an-gle of propagation, θ , with which an initial wave number k (that is normalized to Ω p /V Ap ) is associated and chosenas to correspond to the maximum growth rate, γ max .Our results show that the ray path of the left-handunstable wave (Fig. 6, on the left) is strongly affected bythe closed-field geometry characterizing this funnel region.Indeed, this unstable wave starting from its initial posi-tion propagates upward in the coronal funnel to a certainheight, where it turns down again and starts propagatingdownward to return back to the initial height. The as-sociated instability growth rate decreases along that raypath. Since the direction of the group velocity is alwaysparallel to the ray path and indicates where the energy istransported, we can say that the energy associated withthe left-hand resonant instability does not reach high al-titudes in the funnel. The smaller θ is the higher up thisunstable wave propagates in the funnel.The right-hand unstable wave (Fig. 6, on the right) isalso found to be well guided and, depending on θ , canfollow both closed and open coronal field lines, which mayexist side by side in this region of the funnel. Indeed, for θ = 50 ◦ and θ = 55 ◦ the unstable wave propagates alongthe open magnetic field lines and reaches high altitudes inthe funnel up to 15 Mm, while for θ = 60 ◦ , 65 ◦ and70 ◦ , similarly to the left-hand instability, the right-hand . Mecheri & E. Marsch: Coronal ion-cyclotron beam instabilities 7 Fig. 5.
Growth rate of the right-hand resonant ion-cyclotron instability , in the case of an alpha-particle beamplasma.
Left : as a function of the angle of propagation θ and the normalized wave number kV Ap / Ω p and for an alphabeam density n α = 0 . n p and velocity v α = 320 km/s = 1 . Ap . Middle : as a function of θ and n α and for v α = 320 km/s ≈ . Ap and kV Ap / Ω p = 0 . Right : as a function of θ and v b /v Ap and for n α = 0 . n p and kV Ap / Ω p = 0 .
5. Here T e = T p = T α .resonant unstable waves are affected by the closed-fieldgeometry and reflected back towards lower altitudes inthe funnel. Thus, the energy associated with this insta-bility is also transported along the magnetic field lines,but eventually to much greater altitude, i.e. z = 15 Mm,in the funnel as compared to the left-hand instabilities.For θ = 50 ◦ and θ = 55 ◦ the growth rate of the right-hand instability first increases until respectively altitudesof z ≈ ≈ ≈ ≈ θ = 60 ◦ , 65 ◦ and 70 ◦ , the growth is first slightly increases during theupward propagation phase, and then it rapidly decreasesto a zero value while the wave is propagating downward.
5. Conclusion
We have studied beam-driven electromagnetic instabili-ties near the ion-cyclotron frequency in a coronal fun-nel using the multi-fluid model. We have considered thecase of an alpha-particle beam propagating in the fun-nel parallel to the ambient magnetic field lines. In agree-ment with kinetic dispersion theory, the local solutionsof the dispersion relation revealed the presence of twokinds of instabilities: the left-hand and right-hand reso-nant instabilities. The left-hand and right-hand instabili-ties arise from the resonant excitation of, respectively, theleft-hand-polarized ion-cyclotron mode and right-hand-polarized fast mode through the anomalous Doppler ef-fect, see Eq. (19). For the studied coronal region, our re-sults indicate that the left-hand resonant instability devel- ops for strongly oblique wave propagation with respect tothe ambient magnetic field, with a maximum growth rateat a quasi-perpendicular propagation, and disappears forweakly oblique propagation. Oppositely, the right-hand in-stability develops for a weakly-oblique propagation to theambient magnetic field, with a maximum growth rate atquasi-parallel propagation, and ceases for highly obliqueor quasi-perpendicular propagation.The nonlocal ray-tracing analysis revealed that bothinstabilities are sensitively affected by the magnetic fieldgeometry and found to propagate closely along the fieldlines. The left-hand resonant instability is rapidly re-flected, thus obeying the constraints imposed by a closed-field configuration. The associated growth rate slightly de-creases and eventually cancels along the ray path. On theother hand, the right-handed resonant instabilities are alsovery well guided along the magnetic field lines of the fun-nel. This instability, for small initial angles of propaga-tion, appears to follow the open field lines and can propa-gate higher up in the funnel, yet with a rapidly decreasinggrowth rate.Consequently, fast ion beams in the magnetically opencorona can provide enough energy for driving micro-instabilities through resonant wave-particle interactions.These instabilities may constitute in turn an important en-ergy source for high-frequency ion-cyclotron waves whichhave been invoked to play a relevant role in the heating ofcoronal ions through cyclotron damping.
R. Mecheri & E. Marsch: Coronal ion-cyclotron beam instabilities x = 7 . z = 2 . ϕ ≈ ◦ Fig. 6.
The ray trajectory (top) and growth rate (bottom) of the left-hand (left) and the right-hand (right) resonantion-cyclotron instabilities in the coronal funnel. These unstable modes are launched at the initial location ( x = 7 . z = 2 . B -inclination angle ϕ ≈ ◦ ), for different initial angle of propagation θ and wave number k chosen at the maximum instability growth rate. The dashed lines represent the magnetic fieldlines in the funnel. References
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