Dust acoustic instability driven by solar and stellar winds

Dust acoustic instability driven by solar and stellar winds

J. Vranjes

Belgian Institute for Space Aeronomy, Ringlaan 3, 1180 Brussels, Belgium

Abstract. A quantitative analysis is presented of the dust acoustic wave instability driven by the solar or stelar wind. This is a current-less kinetic instability which develops in permeating plasmas, i.e.., when one quasi-neutral electron-ion plasma propagates through another quasi-neutral plasma which contains dust, electrons and ions. The cometary dusty plasma in the solar wind appears to be practically always unstable.

Keywords: Dust, acoustic, solar wind PACS: 52.27.Lw, 52.35.Fp, 96.50.Dj, 96.50.Ci

INTRODUCTION

In the presence of a macroscopic electron velocity relative to static singly-charged ions, a kinetic current-driven instability of the ion acoustic mode may develop. The instability may require rather high values for the electron current. A lower instability threshold may be obtained in the case of two interpenetrating (permeating) plasmas, which implies a current-less instability [1]. This will be demonstrated in the example of the solar wind propagating through the cometary dusty plasma.

DERIVATIONS

Using the linearized Vlasov-Boltzmann kinetic equation for the perturbed distribution function, the perturbed number density may be calculated fromn_j1=^R f_j1d³~v. For the general species jthis yields

nj1

n_j0=−qjφ₁

κT_j [1−Z(α_j)]. (1)

Here, for non-streaming species we haveα_j=ω/(kvTj), and Z(α_j) = αj

(2π)^1/2 Z

dξexp(−ξ²/2)/(α_j−ξ). (2)

For the streaming species (that flow with the common speedv₀) the derivation is similar and Eq. (1) is obtained, but instead ofα_jandξ now we haveβ_j= (ω−kv₀)/(kvTj)andζ = (v_z−v₀)/vTj. The quasi-neutrality in the perturbed statenwi1+nci1=nwe1+nce1+Zdnd1will directly yield the dispersion equation. The indicescandwstand for the cometary and wind plasma, respectively.

In Eq. (1) for dust the following expansion will be used Z(α_d)'1+ 1

α_d²⁺ 3

α_d^{4+. . .−i}

³π 2

´_1/2

α_dexp(−α_d²/2). (3)

This is valid if |α_d| À 1 and |Re(α)_d| À |Im(α_d)|. For the two electron populations we shall use Z(α_ce)'

−i(π/2)^1/2α_ce,Z(β_we)' −i(π/2)^1/2β_we, that is valid for

|αce| ≡ |ω|

kvTce ¿1, |βwe| ≡|ω−kv0| kvTwe ¿1.

The same will be used for the cometary ions, |αci| ≡ |ω|/(kvTci)¿1. As for the wind ions, the parameterβwi= (ω−kv₀)/(kvTwi)contains two terms, where for the first one we expect thatω/(kvTwi)¿1, while for the second one

Dusty/Complex Plasmas: Basic and Interdisciplinary Research AIP Conf. Proc. 1397, 407-408 (2011); doi: 10.1063/1.3659866

© 2011 American Institute of Physics 978-0-7354-0967-5/$30.00

407

0.2 0.4 0.6 0.8 1.0 500

1000 1500 2000

zd

k [1/m]

0.000 0.01444 0.02888 0.04331 0.05775 0.07219 0.08663 0.1011 0.1155

FIGURE 1. The growth rate of the DA mode driven by the slow solar wind.

in this case we assumev₀/vTwiÀ1. Hence, the expansion similar to (3) should be used. The dispersion equation in general form reads

z²_cinci0

Tci

· 1+i

³π 2

´_1/2 ω kvTci

¸ +nwe0

Twe

· 1+i

³π 2

´_1/2ω−kv0

kvTwe

¸ +nce0

Tce

· 1+i

³π 2

´_1/2 ω kvTce

¸

−z²_winwi0

T_wi

½ k²v²_Twi (ω−kv₀)²−i

³π 2

´_1/2ω−kv₀ kvTwi exp

·(ω−kv₀)² 2k²v²_Twi

¸¾

−z²_dnd0

T_d

½k²v²_Td ω^{2 −i}

³π 2

´_1/2 ω kvTdexp

· ω² 2k²v²_Td

¸¾

=0. (4) The real part of (4) yields the frequency of the dust acoustic mode (where the contribution of the wind ions is negligible)

ω_r²'Z_d²nd0

n_ce0 κTce

m_d

1 1+n_we0

nce0

T_ce

T_we+z²_cin_ci0T_ce n_ce0T_ci

. (5)

The growth rateγ' −Im∆(k,ω_r)/[∂(Re∆)/∂ ω]_ω'ωr is γ=³π

8

´_1/2 n_we0 Z_d²n_d0

m_dm^1/2_e (κT_we)^3/2

ω_r³ k²

( v₀−ω_r

k

"

1+z²_cin_ci0 nwe0

µT_we Tci

¶_3/2µ m_ci me

¶_1/2#)

. (6)

Hence, the instability sets in if

v0>ωr

k (1+a), a=z²_cinci0

n_we0 µTwe

T_ci

¶_3/2µ mci

m_e

¶_1/2

. (7)

In application to the solar wind interaction with cometary dusty plasma, the cometary plasma parameters are used from Ref. [2]. These include the following: T_ce=1.16·10⁵ K,T_ci =2.32·10⁴ K, T_d=1.16·10² K,n_ci0=10⁷ m⁻³,nd0=10 m⁻³,Zd=800,md=1.13·10⁻²⁰kg. As for the electrons and ions from the solar wind, we use the following parameters: T_ew=T_iw =1.5·10⁵ K,n_wi0=n_we0=5·10⁶ m⁻³. The solar wind speed is adopted to be v_we0=v_wi0=v₀=5·10⁵m/s. Singly charged ions are assumed in both systems, and the grains are negatively charged so thatnce0+Zdnd0=nci0. For these parameters, the threshold velocityvthfor the instability is in fact very low. Taking k=1 m⁻¹, andZ_d=800, from (7) we havev_th=5.3 km/s only. Therefore, the wind-driven dust acoustic oscillations are always growing. In Fig. 1 the growth rate is given in terms of the wave number and the dust charge number, showing a mode that is practically always growing. In such a multi-component system, the Debye length turns out to be of the meter size. So the largest wave frequency that should be attributed to the DA mode is expected to be of the order of 10 Hz.

REFERENCES

1. J. Vranjes, S. Poedts, and Z. Ehsan,Phys. Plasmas16, 074501 (2009).

2. N. D’Angelo,Planet. Space Sci. 46, 1671 (1998).

408

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