General dispersion properties of magnetized plasmas with drifting bi-Kappa distributions. DIS-K: DIspersion Solver for Kappa plasmas
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Under consideration for publication in J. Plasma Phys. General dispersion properties of magnetizedplasmas with drifting bi-Kappa distributions.DIS-K: DIspersion Solver for Kappa plasmas
R. A. L´opez † , S. M. Shaaban and M. Lazar Departamento de F´ısica, Universidad de Santiago de Chile, Usach, 9170124 Santiago, Chile Institute of Experimental and Applied Physics, University of Kiel, Leibnizstrasse 11, D-24118Kiel, Germany Theoretical Physics Research Group, Physics Department, Faculty of Science, MansouraUniversity, 35516, Mansoura, Egypt Centre for mathematical Plasma Astrophysics, KU Leuven, Celestijnenlaan 200B, B-3001Leuven, Belgium Institut f¨ur Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universit¨atBochum, D-44780 Bochum, Germany(Received xx; revised xx; accepted xx)
Space plasma populations are known to be out of (local) thermodynamic equilibrium,as observations show direct or indirect evidences of non-thermal velocity distributions ofplasma particles. Prominent are the anisotropies relative to the magnetic field, anisotropictemperatures, field-aligned beams or drifting populations, but also, the suprathermalpopulations enhancing the high-energy tails of the observed distributions. Drifting bi-Kappa distribution functions can model all these features and enable for a kineticfundamental description of the dispersion and stability of these collision-poor plasmas,where process are conditioned mainly by wave-particle interactions. In the present paperwe derive the full set of components of the dispersion tensor for magnetized plasmapopulations modeled by drifting bi-Kappa distributions. A new solver called DIS-K(DIspersion Solver for Kappa plasmas) is proposed to solve numerically the dispersionrelations of high complexity. The solver is tested comparing to the unstable wave solutionsobtained with other codes, operating in the limits of drifting Maxwellian and non-driftingKappa models. The present results provide valuable tools for realistic characterizationof wave fluctuations specific to the complex configurations measured in-situ in spaceplasmas.
1. Introduction
In-situ measurements of the particle velocity distributions in collision-poor plasmasfrom heliosphere report a variety of non-thermal features, providing evidence of theout of thermodynamic equilibrium nature of these plasma systems. Non-equilibriumdistributions are local sources of kinetic instabilities and enhanced wave fluctuations,confirmed by the observations (Wilson et al. et al. et al. † Email address for correspondence: [email protected]
R. A. L´opez et al. (Bale et al. et al. et al. et al. et al. desiteratum , and requires understanding ofthe basic properties of wave fluctuations. To do that, we need to derive the dispersionand stability relations, and implicitly the dielectric tensor of the plasma system, whichdepends directly on the shape of the underlying velocity distributions (Stix 1992).The observed velocity distributions of plasma particles combine kinetic anisotropies,relative to the magnetic field direction, e.g., anisotropic temperatures (ˇStver´ak et al. et al. et al. et al. et al. a , b ). In the solar wind frame, i.e., fixed to protons, the major (anti-sunward) drift is associated with the beaming velocity of the electron stahl, while therelative drift between core and halo is in general modest (Wilson et al. a ) allowing toincorporate these two components in the same bi-Kappa (ˇStver´ak et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. rifting bi-Kappa plasmas: dispersion and stability et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al.
2. Theoretical formalism
Away from energetic events, i.e., quiet times in heliosphere, many plasma systems ofinterest can be considered sufficiently homogeneous on large enough time and spatialscales. 2.1.
General dispersion relation
Without loss of generality we assume cartesian coordinates ( x, y, z ) with z -axis parallelto the magnetic field B , and with the wave vector k in the ( x − z ) plane, such that k = k ⊥ ˆx + k k ˆz (2.1)where k , ⊥ are defined with respect to the magnetic field direction. From Vlasov-Maxwellequations one can derive the general expression of the dispersion relation (Stix 1992) Λ · E = 0 (2.2)in terms of the electric field of the wave fluctuation E ( k , ω ) and the dispersion tensor Λ .For arbitrary (but still gyrotropic) velocity distribution functions F a ( v ⊥ , v k ) of plasmaspecies of sort a (e.g., a = e, p, i for electrons, protons and ions, respectively) the R. A. L´opez et al. components of the dispersion read as follows Λ ij ( k , ω ) = δ ij − c k ω (cid:18) δ ij − k i k j k (cid:19) + X a ω pa ω Z d v ∞ X n = −∞ V ni V n ∗ j ω − k k v k − nΩ a (cid:18) ω − k k v k v ⊥ ∂F a ∂v ⊥ + k k ∂F a ∂v k (cid:19) + ˆB i ˆB j X a ω pa ω Z d v v k (cid:18) ∂F a ∂v k − v k v ⊥ ∂F a ∂v ⊥ (cid:19) . (2.3)Here V n = v ⊥ nJ n ( b ) b ˆe − iv ⊥ J ′ n ( b ) ˆe + v k J n ( b ) ˆe , ˆB = B ˆe , b = k ⊥ v ⊥ Ω a , (2.4)and J n ( b ) is the Bessel function with J ′ n ( b ) its first derivative.2.2. Drifting bi-Kappa distribution
For magnetized plasmas in space environments realistic model distributions able toreproduce all the main departures from thermal equilibrium (i.e., anisotropies, but alsosuprathermal populations), are drifting bi-Kappa velocity distribution functions, in theform F a ( v ⊥ , v k ) = 1 π / α ⊥ a α k a Γ ( κ a + 1) κ / a Γ ( κ a − / " v k − U a ) κ a α k a + v ⊥ κ a α ⊥ a − κ a − , (2.5)where the parameters α k , ⊥ , known as the most probable speed (Vasyliunas 1968), α k a = (cid:18) k B T k a m a (cid:19) / , α ⊥ a = (cid:18) k B T ⊥ a m a (cid:19) / , (2.6)correspond to the thermal speeds of the Maxwellian limit that approximately describethe low-energy core out of the Kappa distribution (Lazar et al. T ( κ ) k a = Z d v v k F a ( v ⊥ , v k ) = m a k B κ a κ a − α k a , (2.7) T ( κ ) ⊥ a = Z d v v ⊥ F a ( v ⊥ , v k ) = m a k B κ a κ a − α ⊥ a , (2.8)requiring κ a > /
2. These kinetic temperatures are lower than the correspondingtemperatures of the Maxwellian core, through T k , ⊥ = lim κ →∞ T ( κ ) k , ⊥ < T ( κ ) k , ⊥ .
3. Dispersion tensor
Replacing the gyrotropic drifting bi-Kappa distribution, Eq. (2.5), in the generalexpression for the dispersion tensor, Eq.( 2.3), and integrating in cilindrical coordinates,we can define similar special functions as in Gaelzer & Ziebell (2016); Gaelzer et al. (2016), obtaining the following expressions for each element of the dispersion tensor, Λ ij = D iD D − iD D iD D − iD D ij (3.1) rifting bi-Kappa plasmas: dispersion and stability D = 1 − c k k ω + X a ω pa ω ∞ X n = −∞ n λ a (cid:20) ξ a Z (1 , n,κ ( λ a , ζ na ) + A a ∂∂ζ na Z (1 , n,k ( λ a , ζ na ) (cid:21) , (3.2) D = 1 − c k ω + X a ω pa ω ∞ X n = −∞ (cid:20) ξ a W (1 , n,κ ( λ a , ζ na ) + A a ∂∂ζ na W (1 , n,k ( λ a , ζ na ) (cid:21) , (3.3) D = X a ω pa ω ∞ X n = −∞ n (cid:20) ξ a ∂∂λ a Z (1 , n,κ ( λ a , ζ na ) + A a ∂ ∂λ a ∂ζ na Z (1 , n,k ( λ a , ζ na ) (cid:21) , (3.4) D = c k ⊥ k k ω + 2 X a q a | q a | ω pa ω s T k a T ⊥ a U a α k a ∞ X n = −∞ n √ λ a ξ a Z (1 , n,κ ( λ a , ζ na ) − X a q a | q a | ω pa ω s T k a T ⊥ a ∞ X n = −∞ n √ λ a (cid:20) ξ a − A a (cid:18) ζ na + U a α k a (cid:19)(cid:21) ∂∂ζ na Z (1 , n,κ ( λ a , ζ na ) , (3.5) D = X a ω pa ω | q a | q a s T k a T ⊥ a r λ a ∞ X n = −∞ (cid:18) ξ a − A a (cid:18) ζ na + U a α k a (cid:19)(cid:19) ∂ ∂λ a ∂ζ na Z (1 , n,κ ( λ a , ζ na ) − X a ω pa ω | q a | q a s T k a T ⊥ a r λ a U a α k a ∞ X n = −∞ ξ a ∂∂λ a Z (1 , n,κ ( λ a , ζ na ) , (3.6) D = 1 − c k ⊥ ω + 2 X a ω pa ω T k a T ⊥ a (cid:18) − κ a (cid:19) U a α k a + 2 X a ω pa ω T k a T ⊥ a U a α k a ∞ X n = −∞ ξ a Z (1 , n,κ ( λ a , ζ na ) − X a ω pa ω T k a T ⊥ a ∞ X n = −∞ " ( ξ a − A a ζ na ) (cid:18) ζ na + 2 U a α k a (cid:19) − A a U a α k a ∂∂ζ na Z (1 , n,κ ( λ a , ζ na ) . (3.7)Here we have used the following definitions (Gaelzer & Ziebell 2016; Gaelzer et al. et al. Z ( α,β ) n,κ ( λ, ξ ) = 2 Z ∞ dx xJ n ( x √ λ )(1 + x /κ ) κ + α + β − Z ( α,β ) κ ξ p x /κ ! , (3.8) Y ( α,β ) n,κ ( λ, ξ ) = 2 λ Z ∞ dx x J n − ( x √ λ ) J n +1 ( x √ λ )(1 + x /κ ) κ + α + β − Z ( α,β ) κ ξ p x /κ ! , (3.9) W ( α,β ) n,κ ( λ, ξ ) = n λ Z ( α,β ) n,κ ( λ, ξ ) − λY ( α,β ) n,κ ( λ, ξ ) , (3.10) Z ( α,β ) κ ( ξ ) = 1 π / κ β +1 / Γ ( κ + α + β − Γ ( κ + α − / Z ∞−∞ ds (1 + s /κ ) − ( κ + α + β − s − ξ , (3.11) λ a = k ⊥ α ⊥ a Ω a , (3.12) ξ a = ω − k k U a k k α k a , (3.13) ζ na = ω − k k U a − nΩ a k k α k a , (3.14) A a = 1 − T ⊥ a T k a . (3.15)The dispersion relation for nontrivial solutions is obtained from the determinant of the R. A. L´opez et al. dispersion tensor, det Λ = 0, which can be written explicitly as0 = D D D − D D − D D − D D − D D D . (3.16)3.1. Maxwellian limit
In the limit κ a → ∞ we recover the dispersion tensor for a drifting bi-Maxwellianplasma (Roennmark 1982; Vi˜nas et al. κ a →∞ Z ( α,β ) n,k ( λ a , ζ na ) = Λ n ( λ a ) Z ( ζ na ) (3.17)lim κ a →∞ Y ( α,β ) n,k ( λ a , ζ na ) = Λ ′ n ( λ a ) Z ( ζ na ) (3.18)lim κ a →∞ (cid:0) y /κ a (cid:1) − ( κ + α ) = e − y . (3.19)Then, the dispersion tensor reduces to D = 1 − c k k ω + X a ω pa ω ∞ X n = −∞ n λ a Λ n ( λ a ) A n , (3.20) D = 1 − c k ω + X a ω pa ω ∞ X n = −∞ (cid:18) n λ a Λ n ( λ a ) − λ a Λ ′ n ( λ a ) (cid:19) A n , (3.21) D = X a ω pa ω ∞ X n = −∞ nΛ n ( λ a ) A n , (3.22) D = c k ⊥ k k ω + 2 X a q a | q a | ω pa ω s T k a T ⊥ a ∞ X n = −∞ n √ λ a Λ n ( λ a ) B n , (3.23) D = − X a ω pa ω | q a | q a s T k a T ⊥ a r λ a ∞ X n = −∞ Λ ′ n ( λ a ) B n , (3.24) D = 1 − c k ⊥ ω + 2 X a ω pa ω T k a T ⊥ a U a α k a (cid:18) U a α k a + 2 ξ a (cid:19) + 2 X a ω pa ω T k a T ⊥ a ∞ X n = −∞ Λ n ( λ a ) C n , (3.25) A n = − A a + ( ξ a − A a ζ na ) Z ( ζ na ) , (3.26) B n = ξ a + (cid:18) ζ na + U a α k a (cid:19) A n (3.27) C n = ξ a ζ na + (cid:18) ζ na + U a α k a (cid:19) A n . (3.28)These expressions agree very well with those in Roennmark (1982); Vi˜nas et al. (2000);Swanson (2003) 3.2. Non-drifting bi-Kappa
In the nondrifting limit U a = 0, we recover the results for bi-Kappa plasmas, see alsoGaelzer et al. (2016) and Kim et al. (2017). D = 1 − c k k ω + X a ω pa ω ∞ X n = −∞ n λ a (cid:20) ξ a Z (1 , n,κ ( λ a , ζ na ) + A a ∂∂ζ na Z (1 , n,k ( λ a , ζ na ) (cid:21) , (3.29) D = 1 − c k ω + X a ω pa ω ∞ X n = −∞ (cid:20) ξ a W (1 , n,κ ( λ a , ζ na ) + A a ∂∂ζ na W (1 , n,k ( λ a , ζ na ) (cid:21) , (3.30) rifting bi-Kappa plasmas: dispersion and stability D = X a ω pa ω ∞ X n = −∞ n (cid:20) ξ a ∂∂λ a Z (1 , n,κ ( λ a , ζ na ) + A a ∂ ∂λ a ∂ζ na Z (1 , n,k ( λ a , ζ na ) (cid:21) , (3.31) D = c k ⊥ k k ω − X a q a | q a | ω pa ω s T k a T ⊥ a ∞ X n = −∞ n √ λ a ( ξ a − A a ζ na ) ∂∂ζ na Z (1 , n,κ ( λ a , ζ na ) , (3.32) D = X a ω pa ω | q a | q a s T k a T ⊥ a r λ a ∞ X n = −∞ ( ξ a − A a ζ na ) ∂ ∂λ a ∂ζ na Z (1 , n,κ ( λ a , ζ na ) , (3.33) D = 1 − c k ⊥ ω − X a ω pa ω T k a T ⊥ a ∞ X n = −∞ ζ na ( ξ a − A a ζ na ) ∂∂ζ na Z (1 , n,κ ( λ a , ζ na ) . (3.34)
4. Numerical Results
In this section we present the new results obtained with our new solver DIS-K(DIspersion Solver for Kappa plasmas). In order to optimize the performance of thedispersion solver, we can further reduce the expressions of the elements of the dispersiontensor in Eqs. (3.2)–(3.7) to minimize the number of integrals that need to be performed.Then, the elements of the dispersion tensor can be written as follows, D = 1 − c k k ω + X a ω pa ω ∞ X n = −∞ n λ a I ( λ a , ζ na ) , (4.1) D = 1 − c k ω + X a ω pa ω ∞ X n = −∞ I ( λ a , ζ na ) , (4.2) D = X a ω pa ω ∞ X n = −∞ nI ( λ a , ζ na ) , (4.3) D = c k ⊥ k k ω + X a q a | q a | ω pa ω s T k a T ⊥ a ∞ X n = −∞ n √ λ a I ( λ a , ζ na ) , (4.4) D = X a | q a | q a ω pa ω s T k a T ⊥ a r λ a ∞ X n = −∞ I ( λ a , ζ na ) , (4.5) D = 1 − c k ⊥ ω + 2 X a ω pa ω T k a T ⊥ a (cid:18) − κ a (cid:19) U a α k a + 2 X a ω pa ω T k a T ⊥ a ∞ X n = −∞ I ( λ a , ζ na ) . (4.6)Now, each element of the dispersion tensor depends on a single integral of the form I ( λ a , ζ na ) = Z ∞ dx xJ n ( x √ λ a )(1 + x /κ a ) κ a +3 / H na ( x ) , (4.7) I ( λ a , ζ na ) = Z ∞ dx x (cid:16) n λ a J n (cid:0) x √ λ a (cid:1) − x J n − (cid:0) x √ λ a (cid:1) J n +1 (cid:0) x √ λ a (cid:1)(cid:17) (1 + x /κ a ) κ +3 / H na ( x ) , (4.8) I ( λ a , ζ na ) = r λ a Z ∞ dx x J n (cid:0) x √ λ a (cid:1) (cid:2) J n − (cid:0) x √ λ a (cid:1) − J n +1 (cid:0) x √ λ a (cid:1)(cid:3) (1 + x /κ ) κ +3 / H na ( x ) , (4.9) I ( λ a , ζ na ) = 4 Z ∞ dx xJ n ( x √ λ a )(1 + x /κ a ) κ a +3 / K na ( x ) , (4.10) R. A. L´opez et al.
Figure 1.
Dispersion relation of the aperodic electron firehose instability at θ = 50 ◦ . Black linesare the solutions obtained with the present code, DIS-K, while red dashed lines correspond toDSHARK. Left panel shows the real part of the normalized frequency, ω r /Ω e vs. the normalizedwave number ck/ω pe , and right panel shows the normalized growth rate, γ/Ω e vs. ck/ω pe . I ( λ a , ζ na ) = − √ λ a Z ∞ dx x J n (cid:0) x √ λ a (cid:1) (cid:2) J n − (cid:0) x √ λ a (cid:1) − J n +1 (cid:0) x √ λ a (cid:1)(cid:3) (1 + x /κ a ) κ a +3 / K na ( x ) , (4.11) I ( λ a , ζ na ) = 2 Z ∞ dx xJ n ( x √ λ a )(1 + x /κ a ) κ a +3 / Q na ( x ) , (4.12)with the following functions H na ( x ) = − (cid:18) − κ a (cid:19) A a + ( ξ a − A a ζ na ) p x /κ a Z (1 , κ ζ na p x /κ a ! , (4.13) K na ( x ) = (cid:18) − κ a (cid:19) ξ a + (cid:18) ζ na + U a α k a (cid:19) H na ( x ) , (4.14) Q na ( x ) = ξ a (cid:18) − κ a (cid:19) (cid:18) ζ na + 2 U a α k a (cid:19) + (cid:18) ζ na + U a α k a (cid:19) H na ( x ) . (4.15)Note that in this way, we only need to implement the modified plasma dispersionfunction Z (1 , κ .Here we show some illustrative examples of solutions obtained with the new solver DIS-K, and compare them with the same solutions obtained with DSHARK and NHDS. Sinceneither of both codes can handle drifting bi-Kappa distributions, we have to compareusing some limit cases. As we know, DHSARK (Astfalk et al. et al. (2019); L´opez et al. (2019), as a first test case. Here we consider an anisotropic electrondistribution with T ⊥ e /T k e = 0 .
5, with β e = 4 . ω pe /Ω e = 100, θ = 50 ◦ , κ e = 4, andprotons are modeled using an isotropic Maxwellian distribution with β p = 4 .
0. Fig. 1shows the numerical dispersion relation obtained solving the set of equations presentedin this section. Left panel of Fig. 1 shows the real part of the normalized frequency, ω r /Ω e vs. the normalized wave number ck/ω pe . As expected from the aperiodic natureof this instability, the real part of the frequency is zero for the entire range of unstablemodes. Right panel of this figure shows the growth rate, γ/Ω e vs. ck/ω pe , whose rangeof unstable modes coincides with the results presented in Shaaban et al. (2019). We haveoverplotted the solutions obtained using DSHARK, as red dashed lines. The solutions of rifting bi-Kappa plasmas: dispersion and stability Figure 2.
Dispersion relation of the oblique whistler heat-flux instability at θ = 60 ◦ . Black linesare the solutions obtained with the present code, DIS-K, while blue dashed lines correspond toNHDS. Left panel shows ω r /Ω i vs. ck/ω pi , and right panel γ/Ω i vs. ck/ω pi . both codes have a perfect match for this case, validating our new solver DIS-K in thenon-drifting bi-Kappa limit, U a = 0, showed in Sec. 3.2.Following, we test the drifting case. Unfortunately, DSHARK is not programmed tosolve the dispersion relation for drifting bi-Kappa distributions. On the other hand,NHDS (Verscharen & Chandran 2018) solver is only foreseen to operate with drifting bi-Maxwellian distributions, as in our Sec. 3.1. Therefore, we have to settle for this limitingcase. For this comparison we will focus on the oblique whistler heat-flux instability,as described in L´opez et al. (2020). We consider a plasma composed by two electronpopulations, a dense central core (subscript c) and a tenuous suprathermal beam (sub-script b), with a relative drift along the background magnetic field, with the followingproperties: n c /n = 0 . n b /n = 0 .
05 are the core and beam normalized number density,respectively, T k b /T k c = 4 . T ⊥ j /T k j = 1 . ω pe /Ω e = 100, β c = 8 πn T c /B = 2 . U b /c = 0 .
035 (or U b /v A = 150). Fig. 2 shows the dispersion relation obtained for θ = 60 ◦ .This time we overplot the solution obtained using NHDS solver as a blue dashed line. Wecan clearly observe the agreement between both codes, in the real and imaginary part ofthe frequency, for damped and unstable modes.Finally, assuming the same plasma conditions, we explore the same instability underthe influence of suprathermal electrons, showing the results obtained with DIS-K for drift-ing bi-Kappa electrons. Fig. 3 shows comparatively three different cases. For comparison,left panel shows the solution for drifting Maxwellian electrons, as in Fig. 2, but this timefor the entire range of angles of propagation. Middle panel displays a new case, whencore electrons are modeled by a Kappa distribution with κ c = 2, while the beam remainsMaxwellian ( κ b = ∞ ). Right panel shows the case when both, core and beam populationsare modeled by Kappa distribution with the same Kappa index, κ c = κ b = 2. We observethat suprathermal electrons suppress the oblique whistler heat-flux instability. When thecore is Kappa distributed, the range of unstable angles and wavenumbers is reduced, aswell as the level of the unstable modes. The suppressing effect is even more prominentwhen both populations are Kappa distributed, reaching lower growth rates and shiftingto lower angles. This is a natural effect given by the enhanced suprathermal tails whichreduce the effective excess of free energy in parallel (drifting) direction (with increasingthe tails the core and beam electrons tend to merge and combine with each other, reducingthe relative drift between them). Ideed, in the last case, we also obtain unstable modesin parallel and quasi-parallel (low angles) directions, specific to lower drifts.0 R. A. L´opez et al.
Figure 3.
Dispersion relation of the oblique whistler heat-flux instability for all angles ofpropagation, and for various core-beam configurations described in the text.
5. Conclusions
In the present paper we have presented the full set of components of the dispersiontensor for magnetized plasma populations modeled by drifting bi-Kappa distributions.This extended dispersion tensor has been implemented in a new dispersion solver, namedDIS-K, and capable to resolve the full spectra of stable and unstable modes of thesecomplex plasma distributions. In order to validate our results and our code, we carriedout illustrative cases enabling comparison with limiting conditions, e.g., nondrifting bi-Kappa and drifting bi-Maxwellian plasmas, resolved by the existing solvers. The unstableelectromagnetic solutions of nondrifting bi-Kappa electrons are resolved by DSHARK,and for the aperiodic (oblique) firehose we found a perfect agreement for the wave-numberdispersion of both the wave frequency and growth rate. The same remarkable agreementhas been obtained with the stable and unstable whistler like modes triggered by driftingbi-Maxwellian electrons populations and described by another solver, NHDS. Further,we have explored the influence of suprathermal electrons on the oblique whistler heat-flux instability, showing that the instability is inhibited, i.e., growth rates are diminished(and the range of unstable wave-numbers is reduced), when core and beam electrons arereproduced by drifting bi-Kappa distributions.In the present paper we provide valuable new theoretical and numerical tools, whichextend and improve the existing capacity of analysis of the wave fluctuations, stableor unstable modes, specific to the complex plasma configurations unveiled by the in-situ observations. Thus, our results can be considered in the context of space plasmas,e.g., solar wind and planetary environments, where all plasma species (electrons andions) exhibit multiple drifting components, namely core, halo and strahl, populated bysuprathermals and with intrinsic anisotropies (i.e., anisotropic temperatures).
Acknowlegements
R.A.L acknowledges the support of ANID Chile through FONDECyT grant No. 11201048.We also acknowledge the support of the projects SCHL 201/35-1 (DFG-German ResearchFoundation), C14/ 19/089 (C1 project Internal Funds KU Leuven), G.0A23.16N(FWO-Vlaanderen), and C 90347 (ESA Prodex). S.M. Shaaban acknowledges theAlexander-von-Humboldt Research Fellowship, Germany. rifting bi-Kappa plasmas: dispersion and stability Appendix A. Summary of necessary functions Z (1 , n,κ ( λ a , ζ na ) = 2 Z ∞ dx xJ n ( x √ λ a )(1 + x /κ ) κ +2 Z (1 , κ ζ na p x /κ ! , (A 1) ∂∂ζ na Z (1 , n,k ( λ a , ζ na ) = 2 Z ∞ dx xJ n ( x √ λ a )(1 + x /κ ) κ +3 / Z ′ (1 , κ ζ na p x /κ ! , (A 2) Y (1 , n,κ ( λ a , ζ na ) = 2 λ a Z ∞ dx x J n − ( x √ λ a ) J n +1 ( x √ λ a )(1 + x /κ ) κ +2 Z (1 , κ ζ na p x /κ ! , (A 3) W (1 , n,κ ( λ a , ζ na ) = n λ a Z (1 , n,κ ( λ a , ζ na ) − λ a Y (1 , n,κ ( λ a , ζ na ) , (A 4) ∂∂ζ na Y (1 , n,k ( λ a , ζ na ) = 2 λ a Z ∞ dx x J n − ( x √ λ a ) J n +1 ( x √ λ a )(1 + x /κ a ) κ a +3 / Z ′ (1 , κ ζ na p x /κ ! , (A 5) ∂∂ζ na W (1 , n,k ( λ a , ζ na ) = n λ a ∂∂ζ na Z (1 , n,κ ( λ a , ζ na ) − λ a ∂∂ζ na Y (1 , n,κ ( λ a , ζ na ) , (A 6) ∂∂λ a Z (1 , n,κ ( λ, ζ na ) = 2 √ λ a Z ∞ dx x J n ( x √ λ a )[ J n − ( x √ λ a ) − J n +1 ( x √ λ a )](1 + x /κ ) κ +2 × Z (1 , κ ζ na p x /κ ! , (A 7) ∂ ∂λ∂ζ na Z (1 , n,k ( λ a , ζ na ) = 2 √ λ a Z ∞ dx x J n ( x √ λ a )[ J n − ( x √ λ a ) − J n +1 ( x √ λ a )](1 + x /κ ) κ +3 / × Z ′ (1 , κ ζ na p x /κ ! . (A 8) Appendix B. Some Useful Functions
The modified Z ( α,β ) κ function can be calculated in terms of the hypergeometric functionas (Gaelzer & Ziebell 2016) Z ( α,β ) κ ( ζ na ) = iκ β +1 Γ ( κ + α + β − Γ ( κ + α + β − / Γ ( κ ) Γ ( κ + α − / × F (cid:20) , κ + α + β − λ ; (cid:18) i √ κ − ζ na i √ κ (cid:19)(cid:21) , (B 1)then, using ddz F [ a, b ; c ; z ] = abc F [ a + 1 , b + 1; c + 1; z ] , (B 2)we have ∂∂ζ na Z ( α,β ) κ ( ζ na ) = − κ β +1 Γ ( κ + α + β ) Γ ( κ + α + β − / Γ ( κ + α + β + 1) Γ ( κ + α − / × F (cid:20) , κ + α + β −
1) + 1; κ + α + β + 1; (cid:18) i √ κ − ζ na i √ κ (cid:19)(cid:21) , (B 3)2 R. A. L´opez et al.
Also, a usefull expression is obtained from Gaelzer & Ziebell (2016), Z ′ ( α,β ) κ ( ζ na ) = − (cid:20) Γ ( κ + α + β − / κ β +1 Γ ( κ + α − /
2) + ζ na Z ( α,β +1) κ ( ζ na ) (cid:21) (B 4)We can obtain a simpler expression if κ is assumed integer, as in Summers & Thorne(1991). Then we have Z ( α,β ) κ ( ξ ) = i − λ ) κ β +1 / Γ ( λ − Γ ( λ − / Γ ( κ + α − / Γ [2( λ − (cid:18) κ + ξ κ (cid:19) − λ × ( − (cid:18) i √ κ + ξ i √ κ (cid:19) λ − Γ ( λ − λ − X ℓ =0 Γ [ ℓ + λ − Γ ( ℓ + 1) (cid:18) i √ κ − ξ i √ κ (cid:19) ℓ ) . (B 5)Let’s take a look to some particular values Z (1 , κ ( ξ ) = 2 iπ / κ / κ ! Γ ( κ − / (cid:18) κ + ξ κ (cid:19) − κ − ( − κ ! (cid:18) i √ κ + ξ i √ κ (cid:19) κ +1 κ X ℓ =0 ( ℓ + κ )! ℓ ! (cid:18) i √ κ − ξ i √ κ (cid:19) ℓ ) . (B 6)And we are also interested in Z (1 , κ ( ξ ) = 2 iπ / κ / ( κ + 1)! Γ ( κ − / (cid:18) κ + ξ κ (cid:19) − κ − ( − κ + 1)! (cid:18) i √ κ + ξ i √ κ (cid:19) κ +2 κ +1 X ℓ =0 ( ℓ + κ + 1)! ℓ ! (cid:18) i √ κ − ξ i √ κ (cid:19) ℓ ) . (B 7)For the derivative, we have Z ′ (1 , κ ( ζ na ) = − iζ na κ / Γ ( κ + 2) Γ ( κ + 3 / Γ ( κ − / Γ (2 κ + 3) (cid:18) κ + ( ζ na ) κ (cid:19) − κ − ( − √ κiζ na (cid:18) i √ κ + ζ na i √ κ (cid:19) κ +2 × Γ ( κ + 2) κ +1 X ℓ =0 Γ ( ℓ + κ + 1) Γ ( ℓ + 1) ( ℓ − κ − (cid:18) i √ κ − ζ na i √ κ (cid:19) ℓ ) . (B 8) REFERENCESAstfalk, P., G¨orler, T. & Jenko, F.
J. Geophys. Res. Sp. Phys. (9), 7107–7120.
Bale, S. D., Kasper, J. C., Howes, G. G., Quataert, E., Salem, C. & Sundkvist, D.
Phys. Rev. Lett. , 211101.
Basu, B.
Phys.Plasmas (5), 052106. Cattaert, T., Hellberg, M. A. & Mace, R. L.
Phys. Plasmas (8), 082111. Collier, M. R., Hamilton, D., Gloeckler, G., Bochsler, P. & Sheldon, R.
Geophys. Res. Lett. (10), 1191–1194. Gaelzer, R. & Ziebell, L. F.
Phys. Plasmas (2), 022110. rifting bi-Kappa plasmas: dispersion and stability Gaelzer, R., Ziebell, L. F. & Meneses, A. R.
Phys. Plasmas (6), 062108. Gary, S. P., Jian, L. K., Broiles, T. W., Stevens, M. L., Podesta, J. J. & Kasper,J. C.
J. Geophys. Res. Sp. Phys. , 30–41.
Hammond, C. M., Feldman, W. C., McComas, D. J., Phillips, J. L. & Forsyth,R. J.
Astron. Astrophys. , 350–354.
Hellberg, M. A. & Mace, R. L.
Phys. Plasmas (5), 1495. Jeong, S.-Y., Verscharen, D., Wicks, R. T. & Fazakerley, A. N.
Astrophys. J. (2), 128.
Kim, S., Schlickeiser, R., Yoon, P. H., L´opez, R. A. & Lazar, M.
Plasma Phys.Contr. F. (12), 125003. Klein, K. G. & Howes, G. G.
Phys. Plasmas (3), 032903. Lazar, M., Fichtner, H. & Yoon, P. H. κ -distributions. Astron. Astrophys. , A39.
Lazar, M., L´opez, R. A., Shaaban, S. M., Poedts, S. & Fichtner, H.
Astrophys.Space Sci. (10), 171.
Lazar, M., Pierrard, V., Shaaban, S., Fichtner, H. & Poedts, S.
Astron. Astrophys. , A44.
Lazar, M. & Poedts, S.
Astron. Astrophys. (1), 311–315.
Lazar, M., Poedts, S. & Fichtner, H.
Astron. Astrophys. , A124.
Lazar, M., Scherer, K., Fichtner, H. & Pierrard, V. κ -distributions. Astron. Astrophys. , A20.
Liu, Y., Liu, S. Q., Dai, B. & Xue, T. L.
Phys. Plasmas (3), 032125. L´opez, R. A., Lazar, M., Shaaban, S. M., Poedts, S. & Moya, P. S.
Astrophys. J.Lett. (2), L25.
L´opez, R. A., Lazar, M., Shaaban, S. M., Poedts, S., Yoon, P. H., Vi˜nas, A. F. &Moya, P. S.
Astrophys. J. Lett. (2), L20.
Mace, R. L. & Hellberg, M. A.
Phys. Plasmas (6), 2098. Maksimovic, M., Zouganelis, I., Chaufray, J.-Y., Issautier, K., Scime, E., Littleton,J., Marsch, E., McComas, D., Salem, C., Lin, R. & others
J.Geophys. Res. Sp. Phys. (A9).
Marsch, E.
Living Rev. Sol. Phys. (1), 1–100. Mason, G. M. & Gloeckler, G.
Space Sci. Rev. (1-4), 241–251.
Micera, A., Zhukov, A. N., L´opez, R. A., Innocenti, M. E., Lazar, M., Boella, E.& Lapenta, G.
Astrophys. J. Lett. (1), L23.
Pierrard, V. & Lazar, M.
Sol. Phys. (1), 153–174. R. A. L´opez et al.
Pilipp, W. G., Miggenrieder, H., Montgomery, M. D., M¨uhlh¨auser, K. H.,Rosenbauer, H. & Schwenn, R.
J. Geophys. Res. (A2), 1075. Roennmark, K.
Shaaban, S., Lazar, M., Yoon, P. & Poedts, S.
Astron. Astrophys. , A76.
Shaaban, S. M. & Lazar, M.
Mon. Not. R. Astron Soc. (3),3529–3539.
Shaaban, S. M., Lazar, M., L´opez, R. A., Fichtner, H. & Poedts, S.
Mon. Not. R. Astron.Soc. (4), 5642–5648.
Shaaban, S. M., Lazar, M., L´opez, R. A. & Poedts, S.
Astrophys. J. (1),20.
Shaaban, S. M., Lazar, M. & Poedts, S.
Mon. Not. R. Astron Soc. (1), 310–319.
Stix, T. H.
Waves in Plasmas . AIP-Press. ˇStver´ak, ˇS., Maksimovi´c, M., Tr´avn´ıˇcek, P. M., Marsch, E., Fazakerley, A. N. &Scime, E. E.
J. Geophys. Res. Sp. Phys. (A5), A05104.
Summers, D. & Thorne, R. M.
Phys. FluidsB (8), 1835. Summers, D., Xue, S. & Thorne, R. M.
Phys. Plasmas (6), 2012–2025. Swanson, D.
Plasma Waves, 2nd Edition . Series in Plasma Physics . Institute of PhysicsPublishing.
Tong, Y., Vasko, I. Y., Artemyev, A. V., Bale, S. D. & Mozer, F. S.
Astrophys. J. (1), 41. ˇStver´ak, ˇS., Tr´avn´ıˇcek, P., Maksimovic, M., Marsch, E., Fazakerley, A. N. & Scime,E. E.
J. Geophys.Res. Sp. Phys. , A03103.
Vasyliunas, V. M.
J. Geophys. Res. (9), 2839–2884. Verscharen, D. & Chandran, B. D. G.
Research Notes of the AAS (2), 13. Verscharen, D., Chandran, B. D. G., Jeong, S.-Y., Salem, C. S., Pulupa, M. P. & Bale,S. D.
Astrophys. J. (2), 136.
Vi˜nas, A. F., Gaelzer, R., Moya, P. S., Mace, R. & Araneda, J. A.
Kappa Distributions , chap. 7,pp. 329–361. Elsevier.
Vi˜nas, A. F., Wong, H. K. & Klimas, A. J.
Astrophys. J. (1),509–523.
Wilson, L. B., Chen, L.-J., Wang, S., Schwartz, S. J., Turner, D. L., Stevens, M. L.,Kasper, J. C., Osmane, A., Caprioli, D., Bale, S. D., Pulupa, M. P., Salem,C. S. & Goodrich, K. A. a Electron energy partition across interplanetary shocks.i. methodology and data product.
Astrophys. J. Suppl. S. (1), 8.
Wilson, L. B., Chen, L.-J., Wang, S., Schwartz, S. J., Turner, D. L., Stevens, M. L.,Kasper, J. C., Osmane, A., Caprioli, D., Bale, S. D., Pulupa, M. P., Salem,C. S. & Goodrich, K. A. b Electron energy partition across interplanetary shocks.II. statistics.
Astrophys. J. Suppl. S. (2), 24. rifting bi-Kappa plasmas: dispersion and stability Wilson, L. B., Koval, A., Szabo, A., Breneman, A., Cattell, C. A., Goetz, K.,Kellogg, P. J., Kersten, K., Kasper, J. C. & Maruca, B. A.
J.Geophys. Res. Sp. Phys. , 5–16.
Woodham, L. D., Wicks, R. T., Verscharen, D., Owen, C. J., Maruca, B. A. &Alterman, B. L.
Astrophys. J.884