Nonlinear wave-particle interactions: Hybrid ...

Proceedings of ISSS7, 000–000, 2005

Nonlinear wave-particle interactions: Hybrid expanding box simulations Petr Hellinger1 , Roland Grappin2 , Andr´e Mangeney2 , and Pavel Tr´avn´ıcˇ ek1 , 1 Institute

of Atmospheric Physics, Prague, Czech Republic 2 Paris Observatory, Meudon, France

A new tool, Hybrid Expanding Box (HEB) model is presented and applied to study wave-particle interactions in collisionless plasmas in two cases. First, the HEB code is used to investigate the effect of a slow compression on protons in the magnetosheath. The simulation results verify the hypothesis of the magnetosheath marginal stability path with respect to the Alfv´en ion cyclotron instability. Second, the HEB code is used to study the role of (proton) fire hose instabilities in the expanding solar wind plasma. While the parallel fire hose instability exhibits the marginal stability evolution as a parallel to the magnetosheath case, the oblique fire hose gives rise to a different paradigm. The system oscillates between the stability and the marginal stability owing to special character of the oblique fire hose instability. The numerical HEB method, its properties and its simulation results are discussed.

1. Introduction

with the signatures of ion cyclotron waves and empty circles show the cases with the signatures of the mirror waves. Color scale denotes the growth rate of the ion cyclotron instability in the corresponding bi-Maxwellian plasma. As a consequence, the magnetosheath protons are characterized by an anticorrelation (Fuselier et al., 1994; Gary et al., 1994) between the proton temperature anisotropy Tp⊥ /Tpk and the proton parallel beta βpk These results suggest that the magnetosheath plasma follows a marginal stability path (Manheimer and Boris, 1977) in the parameter space (βpk , Tp⊥ /Tpk ).

Properties of collisionless plasmas are largely determined by wave-particle interactions. The collisionless plasma of the solar wind and the magnetosheath is never stationary; the large-scale inhomogeneous flow translates locally as a plasma expansion, compression, velocity shear, etc. In an ideal collisionless plasma the first and the second adiabatic invariants are conserved (Chew et al., 1956) and these conservations lead to the following (CGL) relation between the magnitude B of the magnetic field B and the density n and the parallel and perpendicular (with respect to B) particle temperatures Tk and T⊥ , respectively:

10.0

(1)

1

1.0

0.10

1.00

βpk

10.00

vd/vA βαk T⊥α/Tkα − 1

0.1 0.01

PSfrag replacements

However, nonideal effects as the heat flux (Hau, 1996) and/or an important wave activity (Belmont and Mazelle, 1992) break the invariants and lead to a behavior quite different from that predicted by CGL Equation (1). An important wave activity may be present in or injected to the system or it may be locally generated when the plasma becomes unstable with respect to an instability. The waves generated by the instability strongly interacts with particles and create a feed-back that suppresses the source of the instability. When the wave amplitudes are modest one expects a quasi-linear evolution: the waves keep the system near the marginal stability. The marginal stability condition determine the system evolution (Manheimer and Boris, 1977). A clear example of the marginal stability evolution is the day-side magnetosheath behind the quasi-perpendicular portion of the Earth’s bow shock dominated by locally generated waves (Lacombe and Belmont, 1995; Schwartz et al., 1996; Hubert et al., 1998): Transverse Alfv´en ion cyclotron waves are observed in low-beta regions whereas compressional mirror waves are observed in high-beta regions. These waves are generated by ion temperature anisotropies T⊥ > Tk , and are usually observed near the marginal stability of the corresponding instability. Fig. 1 shows an example of AMPTE/CCE in situ observations (Anderson and Fuselier, 1993; Gary et al., 1993): the observations are shown in the parameter space (βpk ,Tp⊥ /Tpk ) full circles show the cases

Tp⊥/Tpk − 1

T⊥ ∝ B and Tk ∝ n2 /B 2 ,

Fig. 1. AMPTE/CCE observations (Anderson and Fuselier, 1993; Gary et al., 1993) in the parameter space (βpk ,Tp⊥ /Tpk ): full circles denote the cases the ion cyclotron waves and empty circles denote the cases with the mirror waves. Color scale denotes the growth rate of the ion cyclotron instability in the corresponding bi-Maxwellian plasma.

In the case of the solar wind the situation seems to be more complicated. There observational evidences that local ion (and electron) instabilities play an important role in constraining the plasma properties (e.g. in the case of proton and alpha particle beams Marsch and Livi, 1987; Reisenfeld et al., 2001). Concerning the proton temperature anisotropy in the solar wind, Wind/SWE in situ observations (Kasper et al., 2002, 2003) indicate that both the Alfv´en cyclotron and mirror instabilities constrain the proton

HELLINGER ET AL.: HYBRID EXPANDING BOX SIMULATIONS

Spherical expansion

Expanding box

R1

velocity U so that the distances R0 and R1 at times t0 and t1 , respectively, are related through the relation

t1

R1 = R0 + U (t1 − t0 ) t0

R0

(2)

(this model replaces the spatial dependence, R0 → R1 , by the temporal one, t0 → t1 ). The transverse dimensions of the expanding box x⊥1 and x⊥2 are proportional to the distance R from the Sun and increase linearly with time   U R1 = x⊥ (t0 ) 1 + (t1 − t0 ) . (3) x⊥ (t1 ) = x⊥ (t0 ) R0 R0

PSfrag replacements

The ratio τ = R0 /U defines a characteristic expansion time. The simple idea of the expanding box may be straightFig. 2. Schematic view of the approximation and the coordinate transfor- forwardly extended to a general expansion (or a compresmation used to derive the expanding box model (cf. Liewer et al., 2001, sion with τ < 0) using the co-moving coordinates ξ = Fig. 1): (bottom) Sketch of the Sun and its magnetic field lines (Parker (ξ1 , ξ2 , ξ3 ) so that the expanding box sizes (physical coorspiral). (left) Solar wind with a constant radial velocity expands in the dinates) x = (x1 , x2 , x3 ) are given as xi = ξi (1 + t/τi ) transverse directions as it propagates away from the Sun. (right) Expandfor i ∈ {1, 2, 3}. In fact the co-moving coordinates ξ are ing box model neglects the curvature and follows the solar wind. the coordinates at time t = 0. Now if we define the expansion matrix L as a diagonal matrix with Lii = 1 − t/τi temperature anisotropy in the solar wind (though the clear for i ∈ {1, 2, 3} we may express Vlasov equation in the comagnetosheath-like marginal stability path is not observed). moving coordinates ξ and ν = dξ/dt in the following form These observations also indicate that the (proton) parallel (Hellinger and Tr´avn´ıcˇ ek, 2005): fire hose instability Quest and Shapiro (1996); Gary et al. ∂f q ∂f ∂f ∂f +ν · + (E + ν × B)·L−2 · = 2ν ·V· (4) (1998) constrains the proton temperature anisotropy in the ∂t ∂ξ m ∂ν ∂ν case of Tp⊥ < Tpk . These observations leave an open question whether the oblique fire hose instability (Hellinger and where V = L−1 · dL/dt, and B = (detL)L−1 · B and Matsumoto, 2000, 2001) plays a significant role in the solar E = L · E are the modified magnetic and electric fields, wind. The oblique fire hose is typically more efficient in re- B and E, respectively. In these expression q and m denote moving the proton temperature anisotropy than the parallel the particle charge and mass, respectively, L−1 is the inverse one. matrix of L, and detL = L11 L22 L33 . In this presentation we describe a new numerical tool, a Since we are interested in the low-frequency, ion kinetic hybrid expanding box model and apply it to study the prop- physics we implemented the expanding box model to a hyerties of the plasma under an expansion/compression. First, brid code (Matthews, 1994), where electrons are considered we present the numerical tool (section 2), then we investi- as a massless, charge neutralizing fluid whereas the ion kigate the marginal stability path in the magnetosheath in the netics is fully resolved (Matthews, 1994). In this Hybrid Excase of a slow compression (section 3). In section 4 we study panding Box (HEB) code the ions follow Equation (4) in the the effect of the parallel and oblique fire hose instabilities on co-moving coordinates instead of the standard Vlasov equathe expanding solar wind properties. Finally in section 5 we tion. The electrons are an adiabatic fluid and define an Ohm’s discuss the numerical method and the simulation results. law and the magnetic evolution is given by Faraday equation transformed to the co-moving coordinates

2. Expanding Box Model

∂B ∂ =− × E. ∂t ∂ξ

In order to study the collisionless plasma reaction to the expansion (or compression) it is convenient to use the expanding box model which was used in magnetohydrodynamic context to study the wave evolution in the solar wind (Grappin et al., 1993). A schematic view of the approximation and the coordinate transformation used in deriving the expanding box model is shown in Fig. 2 (cf. Liewer et al., 2001, Fig. 1). Fig. 2 shows a sketch of the Sun and its magnetic field lines (Parker spiral). Fig. 2 (top left) displays the transverse expansion of the solar wind (with a constant radial velocity) as it propagates away from the Sun. Fig. 2 (top right) shows the approximation of the expanding box model: the model neglects the curvature and follows the solar wind. The expanding box model treats the expansion as a linearly driven evolution: One assumes a solar wind with a constant

(5)

The characteristic spatial and temporal units used in the HEB code are c/ωpp0 and 1/ωcp0 respectively, where c is the speed of light, ωpp0 = (n0 e2 /mp 0 )1/2 is the initial proton plasma frequency, and ωcp0 = eB0 /mp is the initial proton gyrofrequency (B0 is the initial magnitude of the ambient magnetic field B 0 , n0 is the initial density, e and mp are the proton electric charge and mass, respectively; finally, 0 is the dielectric permittivity of vacuum). The generalized HEB code has the three parameters τi , the characteristic expansion (or compression) times in the three (Cartesian) directions. The HEB code in one-dimensional (1-D) and twodimensional (2-D) implementations was successfully used in the solar wind and the magnetosheath contexts. 2

HELLINGER ET AL.: HYBRID EXPANDING BOX SIMULATIONS 10.0

0.10

1.00

βpk PSfrag replacements

0.1 0.01

1.0

0.1 0

10.00

2

4

βpk

6

8 PSfrag replacements

Tp⊥/Tpk

1.0

vd/vA βαk T⊥α/Tkα − 1

Tp⊥/Tpk − 1

10.0

Fig. 4. Evolution in the parameter space (βpk , Tp⊥ /Tpk ) for 1-D simulation (dashed curve) and for 2-D simulation (solid curve). Color scale denotes the growth rate of the parallel fire hose instability in the corresponding bi-Maxwellian plasma.

Fig. 3. Evolution during the plasma compression in the parameter space (βpk , Tp⊥ /Tpk ). Color scale denotes the growth rate of the ion cyclotron instability in the corresponding bi-Maxwellian plasma.

4. Fire hose instabilities in the solar wind Following Hellinger et al. (2003a) we present the application of the 1-D and 2-D HEB simulations for study of fire hose instabilities in the expanding solar wind. The 1-D simulation had ∆x = 0.5, 512 grid points with 1024 particles per cell whereas the 2-D simulation had ∆x = ∆y = 1, 256 × 256 grid points with 256 particles per cell. The initial parameters are βpk = 1, Tp⊥ /Tpk = 0.5 and the ambient magnetic field is B 0 = (B0 , 0, 0). A slow expansion is set in the second and third directions (τ2 = τ3 = τ = 2000/ωcp0) which leads to a decrease of the density and of the magnitude of the magnetic field n, B ∝ 1/(1 + t/τ )2 . The simulation results are shown in Fig. 4 in the parameter space (βpk , Tp⊥ /Tpk ). Fig. 4 shows the evolution in the (dashed) 1-D and (solid) 2-D simulations; Color scale denotes the growth rate of the parallel fire hose instability in the corresponding bi-Maxwellian plasma. Fig. 4 shows that initially the plasma evolves adiabatically (following the CGL Equation (1)) and the ratio Tp⊥ /Tpk decreases whereas βpk decreases. In the 1-D simulation (Fig. 4, dashed curve) the parallel fire hose waves appear after the system crossed the instability threshold. The Tp⊥ /Tpk decrease is stopped and after a transition the system evolves around the marginal stability as a counterpart to the Alfv´en ion cyclotron marginal stability path in the magnetosheath. In the 2-D simulation (Fig. 4, solid curve) the system behavior is qualitatively different. The parallel fire hose instability appears as well (the threshold of the parallel fire hose is typically lower than that of the oblique fire hose, cf. Hellinger and Matsumoto, 2000) but soon after its appearance the oblique fire hose instability sets in. The essentially non-quasilinear property of the oblique fire hose (Hellinger and Matsumoto, 2001) leads to a decay of the wave activity and consequently to an important increase of Tp⊥ /Tpk . The evolution repeats, the system behave nearly adiabatically and after the crossing of the threshold the parallel and oblique fire hoses reappear. The system (instead of following the marginal stability path) oscillates between the stable and marginally stable regions. The oscillations are however

3. Magnetosheath marginal stability path Following Hellinger et al. (2003b) and Hellinger and Tr´avn´ıcˇ ek (2005) we present the application of the 2-D HEB code with a negative characteristic time inducing the plasma compression for these parameters: the resolution ∆x = 0.25, ∆y = 1, 512 × 256 grid points with 256 particles per cell, the particle time step ∆t = 0.02/ωp0 , the field substep ∆tB = ∆t/10. The initial ambient magnetic field is B 0 = (B0 , 0, 0) and the continuous compression in the second direction (τ2 = −τ = −4000/ωcp0) leads to an increase of the density and of the magnitude of the magnetic field, n, B ∝ 1/(1 − t/τ ). The properties of the simulated plasma being compressed is shown in Fig. 3. This figure shows the evolution of the simulated system in the parameter space (βpk , Tp⊥ /Tpk ); Color scale denotes the growth rate of the ion cyclotron instability in the corresponding bi-Maxwellian plasma (see Fig. 1). The evolution may be split into three phases. During the first phase, the plasma is stable with respect to the proton cyclotron and mirror instabilities, and it evolves doubleadiabatically in agreement with Equation (1). This adiabatic evolution translates into an upward motion in the space (βpk , Tp⊥ /Tpk ) of Fig. 3. This evolution leads to the development of an important proton temperature anisotropy. The second phase starts when the anisotropy becomes stronger than the threshold for the proton cyclotron instability. During this phase the generated waves heat the plasma and the evolution departs from the double adiabaticity. After a short transition the system follows a constant value of the maximum linear growth rate γ ∼ 0.045ωcp of the ion cyclotron instability. This phase is a clear example of the marginal stability path (with respect to the ion cyclotron instability) and is in agreement with observations, see Fig. 1. The third phase starts around βpk ∼ 1 when the mirror instability become important. The marginal stability path along the constant (ion cyclotron instability) growth rate is then modified by the presence of the mirror waves. 3

HELLINGER ET AL.: HYBRID EXPANDING BOX SIMULATIONS

damped and in the high beta region βpk & 3.5 the 1-D and 2-D simulations give similar results.

the magnetosheath: Hybrid simulations, Geophys. Res. Lett., 23, 2887– 2890, 1996.

5. Discussion and Conclusion We present a new tool, the hybrid expanding box model and its applications to investigation of wave-particle interactions. Two cases are presented: First in the case of magnetosheath compression. The HEB simulations directly verify the hypothesis of the marginal stability path with respect to the Alfv´en ion cyclotron instability. The simulation results are in agreement with the observations, linear and quasilinear expectation and support the bounded anisotropy model for a closure of the anisotropic magnetohydrodynamic equations (e.g. Gary et al., 1996). The case of fire hose instabilities in the solar wind is more complex. The parallel fire hose determine the marginal stability path in the 1-D HEB simulation and constitutes a parallel to the Alfv´en ion cyclotron instability in the magnetosheath context. The 2-D HEB simulation show another paradigm which is qualitatively different from the marginal stability path: the simulated system reaches the marginal stability region and (owing to special character of the oblique fire hose) jumps to the stable region. The system then oscillates between the stable and marginally stable regions. Whether this oscillatory behavior may explain some of the observations (Kasper et al., 2002, 2003) is to bee seen. The hybrid expanding box model represents an interesting and useful tool to investigate the role wave-particle interaction in determining the ion thermodynics in non-stationary plasmas for example in the case of the interaction with turbulent spectra (Liewer et al., 2001) or in the case of ion instabilities (Hellinger et al., 2003a,b; Hellinger and Tra´ vn´ıcˇ ek, 2005). Acknowledgments. Authors thank C. Lacombe and S. P. Gary for useful discussions and acknowledge grants GA AV IAA3042403 and ESA PRODEX contract No. 14529/00/NL/SFe(IC)

Gary, S. P., H. Li, S. O’Rourke, and D. Winske, Proton resonant firehose instability: Temperature anisotropy and fluctuating field constraints, J. Geophys. Res., 103, 14,567–14,574, 1998. Grappin, R., M. Velli, and A. Mangeney, Nonlinear-wave evolution in the expanding solar wind, Phys. Rev. Lett., 70, 2190–2193, 1993. Hau, L. N., Nonideal MHD effects in the magnetosheath, J. Geophys. Res., 101, 2655–2660, 1996. Hellinger, P., and H. Matsumoto, New kinetic instability: Oblique Alfv´en fire hose, J. Geophys. Res., 105, 10,519–10,526, 2000. Hellinger, P., and H. Matsumoto, Nonlinear competition between the whistler and Alfv´en fire hoses, J. Geophys. Res., 106, 13,215–13,218, 2001. Hellinger, P., and P. Tr´avn´ıˇcek, Magnetosheath compression: Role of characteristic compression time, alpha particle abundances and alpha/proton relative velocity, J. Geophys. Res., 110, in press, doi: 10.1029/2004JA010687, 2005. Hellinger, P., P. Tr´avn´ıˇcek, A. Mangeney, and R. Grappin, Hybrid simulations of the expanding solar wind: Temperatures and drift velocities, Geophys. Res. Lett., 30, 1211, doi:10.1029/2002GL016409, 2003a. Hellinger, P., P. Tr´avn´ıˇcek, A. Mangeney, and R. Grappin, Hybrid simulations of the magnetosheath compression: Marginal stability path, Geophys. Res. Lett., 30, 1959, doi:10.1029/2003GL017855, 2003b. Hubert, D., C. Lacombe, C. C. Harvey, M. Moncuquet, C. T. Russell, and M. F. Thomsen, Nature, properties, and origin of low-frequency waves from an oblique shock to the inner magnetosheath, J. Geophys. Res., 103, 26,783–26,798, 1998. Kasper, J. C., A. J. Lazarus, and S. P. Gary, Wind/SWE observations of firehose constraint on solar wind proton temperature anisotropy, Geophys. Res. Lett., 29, 1839, doi:10.1029/2002GL015128, 2002. Kasper, J. C., A. J. Lazarus, S. P. Gary, and A. Szabo, Solar wind temperature anisotropies, in AIP Conf. Proc. 679: Solar Wind Ten, edited by M. Velli, R. Bruno, and F. Malara, pp. 538–541, AIP, New York, 2003. Lacombe, C., and G. Belmont, Waves in the Earth’s magnetosheath: Observations and interpretations, Adv. Space Res., 15, 329–340, 1995. Liewer, P. C., M. Velli, and B. E. Goldstein, Alfv´en wave propagation and ion cyclotron interaction in the expanding solar wind: One-dimensional hybrid simulations, J. Geophys. Res., 106, 29,261–29,281, 2001.

P. Hellinger (e-mail: [email protected])

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