SHEAR FLOW INSTABILITIES IN LOW-BETA SPACE PLASMAS TARAS SIVERSKY, YURIY VOITENKO∗ and MARCEL GOOSSENS Centre for Plasma Astrophysics, K.U. Leuven, Heverlee, Belgium (∗ Author for correspondence: E-mail:
[email protected]) (Received 25 August 2005; Accepted in final form 3 February 2006)
Abstract. We study instabilities driven by a sheared plasma flow in the low-frequency domain. Two unstable branches are found: the ion-sound mode and the kinetic Alfv´en mode. Both instabilities are aperiodic. The ion-sound instability does not depend on the plasma β (gas/magnetic pressure ratio) and has a maximum growth rate of about 0.1 of the velocity gradient d V0 /d x. On the other hand, the kinetic Alfv´en instability is stronger for larger β and dominates the ion-sound instability for β > 0.05. Possible applications for space plasmas are shortly discussed. Keywords: shear flows, plasma instabilities, kinetic Alfv´en wave, ion-sound wave
1. Introduction Shear plasma flows (i.e., the flows where the velocity gradient is normal to the flow velocity) are widely observed in space. The flow of the solar wind around the Earth magnetosphere is an obvious example. There is a strong velocity gradient between the steady magnetosphere and the solar wind. During a magnetic reconnection event in the solar corona strongly sheared flows arise between the inflowing and outflowing plasma. Winebarger et al. (2002) have shown that even without reconnection coronal loops can contain steady flows with velocities up to 40 km s−1 . Another example of a sheared flow is a slow magnetoacoustic wave (King et al., 2003) in the presence of a temperature gradient perpendicular to the background magnetic field. The wave phase velocity is different on different magnetic field lines and the phase mixing occurs. The cross-field velocity gradients arise between different wave phases and can cause a micro-instability. Sheared flows are usually considered as a source of the MHD Kelvin–Helmholtz instability (Mikhailovskii, 1974). However, the MHD KH instability requires strongly inhomogeneous flows. On the other hand, a velocity gradient can produce local micro-instabilities. D’Angelo (1965) used a local electrostatic model and found that the ion-sound wave (ISW) can be excited by a sheared flow. Ganguli et al. (2002) have studied the influence of a sheared flow on the current driven ion-acoustic and ion-cyclotron instabilities. They found a significant reduction of the threshold current in the presence of a velocity gradient. Earlier, Pu and Zhou (1986) have shown that the electro-magnetic kinetic Alfv´en wave (KAW) can be driven by shear plasma flows. KAW is the extension of the Space Science Reviews (2005) 121: 343–351 DOI: 10.1007/s11214-006-7182-6
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Alfv´en wave mode in the range of large perpendicular wave numbers (Hasegawa and Chen, 1976), where the perpendicular wave length approaches kinetic plasma length scales, such as the ion thermal gyroradius or electron inertial lengths, and kinetic effects become pronounced. Because of their ability to interact with plasma particles, KAWs and KAW solitons provide the means for plasma heating and particle acceleration in space and are under intense investigation (see, e.g., recent papers by Wygant et al., 2002, Wu and Chao, 2004, and Voitenko and Goossens, 2005, and references therein). In this paper, we further develop the kinetic electro-magnetic theory for a lowbeta plasma with a sheared flow. New analytical and numerical results are obtained for shear flow driven instabilities, and a comparative analysis of kinetic Alfv´en and ion-sound instabilities is given.
2. Dispersion Equation We consider a uniform, fully ionized hydrogen plasma with a background magnetic field B0 and a sheared flow V0 parallel to the magnetic field. Because we are interested in a situation where the plasma beta is much smaller than unity, we do not take into account the coupling of KAWs (ISWs) with compressional magnetic waves. The unperturbed distribution function Fs0 for species s (s = e for electrons and s = i for ions) is written as a function of motion constants: kinetic energy m s V2s /2, velocity along the ambient magnetic field Vz , and the x-coordinate of the particle guiding center X s = x + Vy /s (where s = es B0 /(m s c) is the cyclotron frequency). 2 Vx + Vy2 + (Vz − V0 (X s ))2 n0 0 Fs (V, r) = (1) exp (2π)3/2 VT3s 2VT2s √ where VT s = Ts /m s is the thermal velocity. We choose the reference frame so that (Figure 1): – B0 = B0 ez ; – V0 (x) = V0 (x)ez ; – V0 (0) = 0. In what follows we limit our investigation to a local analysis applicable for perturbations that have short wavelengths in the x direction, d ln V0 (x) . dx This allows us to regard V0 (x) as a linear function, in the vicinity of x = 0, kx
(2)
V0 (x) = αx.
(3)
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Figure 1. The reference frame selection.
Two fluctuating electromagnetic potentials are used to describe the wave electric (E) and magnetic (B) fields: E = −∇ϕ −
1 ∂A , c ∂t
B = ∇ × A.
(4) (5)
Here ϕ is the scalar potential and A the vector potential. To obtain the dispersion relation of the kinetic Alfv´en waves we assume as Voitenko (1998) that A = {0, 0, A z } and use the z-component of Ampere’s law 4π js = −∇ 2 A z , (6) c s and the quasi-neutrality condition ρs = 0,
(7)
s
where js and ρs are currents and charge densities, respectively. Using the linearized Vlasov equation we derive the dispersion relation V2 1 D ρ D j + A2 μi2 = D ρ D j , Vk Ti where k⊥ VT s ω B02 , Vk = , V A2 = , s kz 4π n 0 m i 1 ky α = sn 1 + σs0 Z sn − (1 + σsn Z sn ) , k z s s Ts n
μs = D ρ
(8)
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1 ky α ns σs0 Z sn − = sn (1 + σsn Z sn ) , ω k z s s Ts n 1 ky α ns , 1− = sn (1 + σsn Z sn ) 1 − k z s ω s Ts n 1 ky α ns ns = sn (1 + σsn Z sn ) 1 − , 1− ω k z s ω s Ts n
and
Z sn ≡ Z σsn
1 =√ π
+∞ exp(−z 2 ) dz , z − σsn
−∞
ω − ns σsn = √ , 2VTs k z sn ≡ n μ2s = In μ2s exp − μ2s , In is the modified Bessel function of index n. Note that the shear parameter α is everywhere multiplied by k y . Hence, k x does not affect the instability. This allows us to choose k x so small that k⊥ ≈ k y . In this paper, we consider low frequency perturbations, |ω| i . In that domain we can simplify the dispersion relation (8) to 1 1 2 2 2 2 ky α , ( 0e − 0i ) + Vk = V A μ −VT i k z i τ (1 − 0e ) + 1 − 0i τ De + Di (9) where τ = Ti /Te and
ky α Ds = s0 (1 + σs0 Z s0 ) 1 − k z s
.
(10)
In the absence of shear (α = 0) Equation (9) describes two branches: KAW and ISW. In the case of a low-beta plasma these branches are decoupled and we investigate them separately in the next two sections.
3. Kinetic Alfv´en Instability If the phase velocity satisfies the condition VT e |Vk | VT i ,
(11)
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and m i /m e μi2 1, we can transform Equation (9) into a shear-modified KAW dispersion relation 2 2 2 2 ky α Vk = V A μ − VT i (1 + τ −1 ). (12) k z i It is obvious that this mode is purely unstable if k⊥ α V2 > 2A μ2 . k z i VT i
(13)
The maximum of the instability growth rate in the wave number space (k y , k z ) is 1
γmax = α β. 2 Here β = (1 + τ −1 )
(14)
VT2i V A2
(15)
is the plasma beta. The maximal value (14) for the increment is reached on the line αi k y kz = . (16) 2V A2 To find the point of maximal increment more accurately, Equation (9) has to be solved. We have done this numerically for β = 0.5, and the result is shown on Figure 2. It is clear from Figure 2 that the left part of the condition (11) is not satisfied. Thus we can roughly assess the value of the wave vector for which the instability increment reaches its maximum by finding the point on the line (16) where |σe0 | = 1. The instability growth rate attains its maximum (14) for the wave vector components
k (max) y k z(max)
i = VA
2 mi , 1 + τ me α 1 + τ me = . 2V A 2 mi
(17)
(18)
4. Ion-Sound Instability In this section, we discuss the ion-sound branch of the dispersion relation (9). This mode has the phase velocity |Vk | VT i V A .
(19)
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Figure 2. The contour plot of the KAW growth rate γ (normalized by 102 i ) in the k z – k y plane. The plasma parameters are: τ = 1, α/i = 0.15, β = 0.5.
We can transform the dispersion relation (9) into ky α 1 + τ − i0 + i0 1 − (1 + σi0 Z i0 ) = 0. k z i Then we use the following Pade approximation: √ 3i π(4 − π) Z (σi0 ) ≈ √ 3(4 − π) − 2i πσi0
(20)
(21)
and i0 ≈
1 . 1 + μi2
(22)
The analysis of Equation (20) confirms the conclusion by D’Angelo (1965) that the ion-sound mode is purely growing. The increment of the ion-sound instability is found by putting ω = iγ in (20): 2 3(4 − π)(1 + ) γ = k z VT i , (23) π (10 − 3π) − 2 where
kyα 1 = 1− . k z i τ + (1 + τ )μi2
(24)
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Figure 3. The contour plot of the ISW growth rate γ (normalized by 102 i ) in the k z – k y plane. The plasma parameters are: τ = 1, α/i = 0.15, β = 0.02.
The contour plot of the instability increment as a function of the wave vector components is shown on Figure 3. The mode is unstable if ky α > (1 + τ ) 1 + μi2 . (25) k z i The maximal value of the instability increment as a function of k y and k z is given by Equation (23) with = (max) , where (max) is a negative root of 4τ + 2 2τ 2(max) + (4τ − 1) (max) − − 3 = 0. (26) 10 − 3π The wave vector components, which give the maximal increment, are τ (max) − 1 i (max) ky = , (27) VT i (max) (1 + τ ) −1/2 α k z(max) = (max) (1 + τ ) τ (max) − 1 . (28) 2VT i For an isothermal plasma, τ = 1, we obtain: γmax ≈ 0.11α, ≈ 0.8 k (max) y
i , VT i
(29) (30)
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k z(max) ≈ 0.09
α . VT i
(31)
5. Conclusions and Discussion In this paper, we have developed the kinetic electro-magnetic theory for a low-beta plasma with a sheared flow. In the low-frequency domain two unstable modes are found: the kinetic Alfv´en and ion-sound modes. Both modes are purely growing in the selected reference frame. The maximal growth rates are given by Equations (14) and (29), from which it follows that the transition between the kinetic Alfv´en and ion-sound instabilities occurs when the plasma β reaches a certain “transition” value. In an isothermal plasma βt ≈ 0.05. If β < βt the ISW instability dominates, and the KAW instability dominates in the opposite case. Equations (17), (18) and (30), (31) indicate that the anisotropy of the exited fluctuations is
λ⊥ λz
= KAW max
me α 2m i i
(32)
for the KAW instability, and
λ⊥ λz
= 0.11 ISW max
α i
(33)
for the ISW instability. This anisotropy could be useful in the interpretation of data obtained by interplanetary scintillation observations (Harmon and Coles, 2005). The shear flow instabilities that we study can have a profound influence on the shear flows in space plasmas. They reduce the velocity shears and slow down flows (effect of anomalous viscosity). On the other hand, perturbations are generated, which can be observed and used for plasma diagnostics. In particular, small-scale anisotropic perturbations of plasma density are observed in the solar corona at 2–5 solar radii (see Harmon and Coles, 2005, and references therein), where the acceleration of the solar wind is initiated. As the acceleration mechanism is not uniform, the plasma flow in this region should have a jet-mixing character with considerable cross-field velocity gradients. In turn, the velocity gradients should drive the KAW instability (where β > 0.05) and/or ISW instability (where β < 0.05). The anisotropic fluctuations of the plasma density, produced by these instabilities, can contribute to observed radio scintillations (Armstrong et al., 1990; Harmon and Coles, 2005). Since the anisotropies of the perturbations generated by the KAW and ISW instabilities are different, the observed variation of anisotropy with height can be used for the identification of regions in the solar corona where β > 0.05 and β < 0.05.
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Acknowledgements This work is supported by the FWO grant G.0178.03 and the OT/02/57 grant.
References Armstrong, J. W., Coles, W. A., Rickett, B. J., and Kojima, M.: 1990, Astrophys. J. 358, 685. Ganguli, G., Slinker, S., Gavrishchaka, V., and Scales, W.: 2002, Phys. Plasmas 9(5), 2321. D’Angelo, N.: 1965, Phys. Fluids 8, 1748. Harmon, J. K. and Coles, W. A.: 2005, J. Geophys. Res. 110, A03101. Hasegawa, A. and Chen, L.: 1976, Phys. Fluids 19, 1924. King, D. B., Nakariakov, V. M., Deluca, E. E., Golub, L., and McClemets, K. G.: 2003, Astron. Astrophys. 404, L1. Mikhailovskii, A. B.: 1974, Theory of Plasma Instabilities, Consultants bureau, New York, Vol. II, p. 141. Pu, Z. and Zhou, Y.: 1986, Scientia Sinica, Series A 29, 301. Voitenko, Y. M.: 1998, Sol. Phys. 182(2), 411. Voitenko, Yu. M., and Goossens, M.: 2005, J. Geophys. Res. 110, A10S01, DOI: 10.1029/ 2004JA010874. Winebarger, A. R., Warren, H., van Ballegooijen, A., DeLuca, E. E., and Golub, L.: 2002, Astrophys. J. 567(1), L89. Wu, D. J. and Chao, J. K.: 2004, J. Geophys. Res. 109, A06211, DOI: 10.1029/2003JA010126. Wygant, J. R., et al.: 2002, J. Geophys. Res. 107, A1201, DOI: 10.1029/2001JA900113.
