Chapter 9 Summary and outlook

Summary and outlook

This thesis aimed to address two problems of plasma astrophysics: how are cosmic plasmas isotropized (A 7→ 1), and why does the equipartition of the magnetic field energy density and parallel kinetic energy density (β = 1) hold? As cosmic plasmas are often collisionless, binary collisions cannot con- tribute to their relaxation, and the idea is that instead, temperature anisotropy driven instabilities play a crucial role. A plasma deviating too much from A = β = 1 will drive an instability, and the arising fluctuations will scatter the plasma to a stable configuration (Schlickeiser et al. (2011)

). In this work, we investigated various instabilities with special emphasis on the threshold conditions.

In order to compare our theoretical results to reality, we examined the solar wind plasma, which is the only space plasma allowing for direct mea- surements. To do so, it is useful to visualize the data distributions in β-A parameter space. The data used in this thesis is based on 17 years of WIND SWE/MFI measurements (Kasper et al. (2006)

), which provide moments of the thermal core populations of the protons obtained by bi-Maxwell fits (Chapter 4). It is known for a while that firehose and mirror type instabilities constrain the solar wind for β

> 1, although the fit is not particularly well.

The main part of the thesis consists of refining the understanding of the kinetic firehose instability. The proton firehose instability (PFHI), driven by anisotropic protons (A

< 1), and the electron firehose instability (EFHI), which is excited by electrons with A

< 1, are commonly studied in the literature. In most works studying a firehose driven by one particle species, the other species is assumed isotropic. The more general case of A

6= ¹ and A

6= 1 was briefly discussed for the PFHI by Kennel and Scarf (1968)

and remained neglected since then. We changed this situation by continuing

with a rigorous analysis of both PFHI for anisotropic electrons, and EFHI for

anisotropic protons. In the general case, the growth rate w

as a function of the

9 . Summary and outlook

wavenumber κ shows a two peak structure. The first peak at lower normalized wavenumbers κ ≈ 1 has a right-handed polarization and we identified it as the PFHI peak, while the second peak at larger wavenumbers κ . 10 is left- hand polarized and is the maximum of the EFHI.

In Chapter 5 we derived analytical solutions describing both firehose in- stabilities. To the best of the author’s knowledge, no analytical approxima- tions which correctly describes the higher wavenumber behavior, i.e. that can reproduce a peak or cutoff, could be found in the literature prior to this anal- ysis, except for the EFHI solution by Pilipp and Völk (1971)

. Our analysis was based on an empirical examination of the resonance terms | ^ξ

| ^{and real} frequency to growth rate ratios w

/ | ^w

| of numerical solutions of the full par- allel kinetic dispersion relation, which has been derived in Chapter 3. A first result was that the exponential term ∼ ^exp ( − ^ξ

) , arising from the analytic continuation in approximations of the plasma dispersion function Z, cannot be neglected in general as it provides the cutoff at higher wavenumbers and is crucial for forming the growth rate peak.

This analysis allowed us to identify three limiting cases for which the dispersion relation could be approximated to derive analytical solutions. The PFHI peak needs two separate approximations, one for the large wavelength limit and small growth rates, where both particle species are non-resonant,

| ^ξ

| > 1, and the weak amplification limit ( | ^w

|  | ^w

| ) can be invoked, and another approximation for larger growth rates with an existing EFHI, where the protons are resonant | ^ξ

| ≈ 1 and the growth rate is of the same order as the real frequency. The first approximation allowed us to derive a concise and accurate formula for the growth rate cutoff, κ

= √

ApCe

.

The EFHI can be described in one model where | ^ξ

| < ^{1 and} | ^ξ

| > ^1, which has been done already by Pilipp and Völk (1971)

. However, we extended their model by using the weak amplification expansion which con- siderably improved the peak approximation for large growth rates, and by including additional Z ( ξ

) expansion term which enables a peak description for small growth rates, as then the electrons approach the resonance condition

| ^ξ

| & 1. Unfortunately, all analytical solutions are too complex to derive the maximum growth rate, so that we could not derive analytical expressions for the threshold conditions. However, we found a criterion to decide whether the PFHI or EFHI will have the larger growth rate which is valid if both instabilities exist, and depends only on electron parameters. The condition A

< 1 − ^2/β

indicates a dominating EFHI and looks like an MHD firehose threshold

. The measured solar wind electron data seems to indicate that the PFHI always dominates.

1

Note that no MHD limit of the EFHI exists, only for the PFHI.

result was a confirmation of the analytical prediction that the PFHI only depends on one combined electron parameter of the form C

= 1 − ( ¹ − A

) β

/2. An increase of C

decreases the growth, while a decrease of C

increases growth. Consequently, decreasing C

moves the threshold to lower β

and increasing increasing to higher beta values. This change is sub- stantial, we found that the bad fit of the data for isotropic electrons can be markedly improved for C

> 1. The effect of the electrons is based on a change of the phase velocity of the mode, which moves the position of the proton cyclotron resonance closer or further away from the bulk of the dis- tribution which gives or removes energy from the mode. We found that the situation for the EFHI is more complicated, as the proton effects depend on both parameters β

and A

. In the case of A

< 1, the EFHI growth can be enhanced by the arising PFHI, which counterbalances the damping due to changes of the proton resonance term. However, the effects on the EFHI threshold are negligible and do not considerably improve the A

= 1 thresh- old.

In order to examine oblique modes, a double-polytropic MHD model was introduced in Chapter 7. Using this approach, the MHD equations are closed with a condition which is an interpolation of double-adiabatic and double- isothermal closure, controlled by two exponents γ

and γ

. The motiva- tion was to account for both adiabatic and isothermal processes in the solar wind. We found three possibly unstable modes, the parallel firehose, the oblique mirror and a third oblique mode which is constrained by the same threshold as the parallel firehose. Unfortunately, this oblique firehose solu- tion only exists for exponents beyond the isothermal-adiabatic mixture and is not properly physically explained. However, when examining the thresh- old conditions, we found that the MHD firehose threshold also depends on a parameter C

and, while showing the same qualitative dependence as in the parallel kinetic analysis, can exactly reproduce the kinetic oblique firehose threshold found numerically by Hellinger et al. (2006)

.

The compressible mirror mode additionally depends on the polytropic

exponents. We showed that the mirror condition fits the data quite well for

an isothermal closure, but can be improved by mixing in a small adiabatic

component, γ

= γ

= 1.06. We also found analytical expressions for the

maximum growth rate and cutoff angle. Then, we demonstrated that the

mirror instability increases with the anisotropy while the MHD firehose ap-

proaches a constant, which may explain why the mirror poses a threshold on

the solar wind distribution for A

> 1 while the data is rather constrained by

the kinetic firehose at A

< 1.

9 . Summary and outlook

10

10

10

10

10

Parallel plasma beta β

10

10

T emp erature Anisotrop y A

= T

/T

50 97 189 367

714 1390

2703 5256

10221

Double-polytropic mirror LH Alfv´en/cyclotron

Kinetic firehose Ce= 2 Kinetic firehose Ce= 1.5 MHD firehose Ce= 0.67

Figure 9.1: Comparison of the thresholds found in this work together with WIND data (resolution 60x60, cuto at 50, see Chapter 4).

In the last Chapter 8, the low beta regime β

< 1 was investigated. It was not clear what the constraints in this regime are until Schlickeiser and Škoda (2010)

proposed weakly unstable Alfvén/cyclotron solutions which may, in principle, constrain the solar wind proton distribution. Here, we refined their results by including collisional damping and using a better approximation.

Figure 9.1 compares the thresholds obtained in Chapters 6-8 in one plot. While the mirror can be adjusted to fit the data perfectly, the LH Alfvén/cyclotron is only a qualitatively right constrain. However, we made some very limiting assumptions when deriving this threshold, a more rigor- ous approach might improve the fit. The kinetic firehose solutions provide steeper thresholds compared to the MHD firehose. This could indicate a sce- nario where both instability types are important on different regimes: the MHD firehose fits better for β

 1 close to A

≈ 1, while the kinetic fire- hose better describes the isocontours closer to β

≈ 2. Although the kinetic firehose is much tighter than its MHD counterpart for equal C

, the MHD instability could be more effective in isotropizing the solar wind plasma, as it is a non-propagating aperiodic mode (Bale et al. (2009)

).

The effect of electrons improved the fit of the parallel PFHI and moved it

back to the same position in parameter space where the oblique firehose op-

erates. Consequently, their competition and influence has to be investigated,

especially including the non-linear evolution. As the full oblique kinetic dis-

persion relation is inaccessible analytically, this has to be done by means

changed growth rate and threshold due to anisotropies of another species has another implication. The current consensus is that the proton cyclotron instability (A

> 1) is independent of the electron parameters since the elec- trons are non-resonant, and that the electron whistler (A

> 1) is independent of proton temperatures since the protons are non-resonant (Gary (1993)

).

However, the PFHI is non-resonant with the electrons as well and neverthe- less markedly changed. It has to be checked whether especially anisotropic electrons can influence the proton cyclotron instability by changing the phase velocity so that it may compete with the mirror instability.

Obviously, the current model has to be refined by including more electron populations in order to account for the supra-thermal components. This may improve the fit by providing more anisotropic electrons, as C

= 2 might be too high using only the core population. Also the influence of the anisotropic He

population (Bourouaine et al. (2013)

) has to be investigated.

To conclude, we demonstrated that electron anisotropies can have a strong

influence on proton driven instabilities and cannot be neglected in a realistic

scenario. The resulting improved fit to the data can help in understanding

the isotropization of the solar wind and, thus, space plasmas in general.