5.4 Kinetic microphysics

The theoretical models discussed in the previous two subsections mainly involved a “one-fluid” or MHD approach to the coronal heating and solar wind acceleration. However, at large heights in coronal holes, the collisionless divergence of plasma parameters for protons, electrons, and heavy ions allows the multi-fluid kinetic processes to be distinguished in a more definitive way. The UVCS measurements of strong O+5 preferential heating, preferential acceleration, and temperature anisotropy have spurred a great deal of theoretical work in this direction (see reviews by Hollweg and Isenberg, 2002Jump To The Next Citation PointCranmer, 2002aMarsch, 20052006Kohl et al., 2006Jump To The Next Citation Point). Specifically, the observed ordering of T ≫ T > T i p e and the existence of anisotropies of the form T > T ⊥ ∥ in coronal holes led to a resurgence of interest in models of ion cyclotron resonance.

The ion cyclotron heating mechanism is a classical resonance between left-hand polarized Alfvén waves and the Larmor gyrations of positive ions around the background magnetic field. If the wave frequency and the natural ion gyrofrequency are equal, then in the rest frame of the ion the oscillating electric and magnetic fields of the wave are no longer felt by the ion to be oscillating. The ion in such a frame senses a constant DC electric field, and it can secularly gain or lose energy depending on the relative phase between the ion’s velocity vector and the electric field direction. In a wave field with random phases, an ion will undergo a random walk in energy. Thus, on average the ions can be considered to “diffuse” into faster (i.e., wider) Larmor orbits with larger perpendicular energy (see Rowlands et al., 1966Galinsky and Shevchenko, 2000Isenberg, 2001Jump To The Next Citation PointCranmer, 2001Jump To The Next Citation PointIsenberg and Vasquez, 2009).

In the actual solar corona, however, it is not likely that the situation is as straightforward as summarized above. Instead of a population of pre-existing, linear cyclotron waves that are dissipated, there may be a rich variety of nonlinear plasma mechanisms at play. The observed ion heating is likely to be just the final stage of a multi-step process of energy conversion between waves, turbulent motions, reconnection structures, and various kinds of distortions in the particle velocity distributions. Table 1 surveys the field of suggested possibilities, and the remainder of this section discusses these ideas in more detail.

Table 1: Tabular outline of suggested physical processes for preferentially heating and accelerating minor ions in coronal holes.
Reconnection events in the low corona generate:
  Ion cyclotron resonant Alfvén waves Tu and Marsch (1997Jump To The Next Citation Point); Cranmer (2000Jump To The Next Citation Point, 2001Jump To The Next Citation Point)
  Electron beams → ion cyclotron waves Markovskii and Hollweg (2002Jump To The Next Citation Point, 2004Jump To The Next Citation Point)
  Fast collisionless shocks Lee and Wu (2000Jump To The Next Citation Point)
MHD turbulence in the extended corona generates:
  Ion cyclotron waves (“parallel cascade?”), with:
    ≫ Alfvén and fast-mode nonlinear coupling Chandran (2005Jump To The Next Citation Point)
    ≫ Three-wave (ion-sound/parametric) coupling Yoon and Fang (2008Jump To The Next Citation Point)
    ≫ Fermi-like diffusion between inward/outward wave resonances Isenberg (2001)
  Kinetic Alfvén waves (“perpendicular cascade”), with:
    ≫ Shear instabilities → ion cyclotron waves Markovskii et al. (2006Jump To The Next Citation Point)
    ≫ Nonlinear wave-particle resonances Voitenko and Goossens (2003Jump To The Next Citation Point, 2004Jump To The Next Citation Point)
    ≫ Debye-scale electron holes Matthaeus et al. (2003Jump To The Next Citation Point); Cranmer and van Ballegooijen (2003Jump To The Next Citation Point)
  Oblique MHD waves (high k∥, high k⊥) Li and Habbal (2001Jump To The Next Citation Point)
  Current sheets → coherent Fermi acceleration Dmitruk et al. (2004Jump To The Next Citation Point)
  Transverse density gradients → drift currents Markovskii (2001Jump To The Next Citation Point); Zhang (2003Jump To The Next Citation Point)
Low-frequency Alfvén waves in the corona directly undergo:
  Polarization drift → lower-hybrid waves Singh and Khazanov (2004Jump To The Next Citation Point); Khazanov and Singh (2007Jump To The Next Citation Point)
  Nonresonant stochastic heating Lu and Li (2007Jump To The Next Citation Point); Wu and Yoon (2007Jump To The Next Citation Point)
  Stochastic heating at fractional cyc. resonance Chen et al. (2001Jump To The Next Citation Point); Guo et al. (2008Jump To The Next Citation Point)
Heavy ion velocity filtration Pierrard and Lamy (2003Jump To The Next Citation Point); Pierrard et al. (2004Jump To The Next Citation Point)

One potential obstacle to the idea of ion cyclotron heating is that the required gyroresonant wave frequencies in the corona are of order 102 to 104 Hz, whereas the dominant frequencies of Alfvén waves believed to be emitted by the Sun are thought to be much lower (i.e., less than 0.01 Hz, corresponding to periods of minutes to hours). Axford and McKenzie (1992Jump To The Next Citation Point) suggested that the right kinds of high-frequency waves may be generated in small-scale reconnection events in the chaotic “furnace” of the supergranular network. These waves could propagate upwards in height – and downwards in magnetic field strength – until they reached a location where they became cyclotron resonant with the local ions, and thus would damp rapidly to provide the ion heating (see also Schwartz et al., 1981Tu and Marsch, 1997).

The above scenario of “basal generation” of ion cyclotron waves has been called into question for several reasons. Cranmer (20002001Jump To The Next Citation Point) argued that the passive sweeping of a pre-existing fluctuation spectrum would involve ions with low gyrofrequencies (i.e., small ratios of charge to mass; qi∕mi ≈ 0.1 to 0.2 in proton units) encountering waves of a given frequency at lower heights than the ions that have been observed to exhibit preferential heating, like O+5 (qi∕mi = 0.31) and Mg+9 (qi∕mi = 0.37). Thus, the resonances of many minor ion species may be strong enough to damp out a base-generated spectrum of waves before they can become resonant with the observed species. Furthermore, Hollweg (2000) found that a base-generated spectrum of ion cyclotron waves would exhibit a very different appearance in interplanetary radio scintillations than the observed radio data. There remains some uncertainty about these criticisms of a basal spectrum of ion cyclotron waves, and definitive conclusions cannot yet be made (see discussions in Tu and Marsch, 2001Hollweg and Isenberg, 2002Jump To The Next Citation Point).

There are several other interesting consequences of the Axford and McKenzie (1992) idea of rapid reconnection events at the coronal base. It is possible, for example, that such microflaring activity would give rise to intermittent bursts of parallel electron beams that propagate up into the extended corona. Sufficiently strong beams may be unstable to the growth of wave power at the ion cyclotron frequencies (Markovskii and Hollweg, 20022004Voitenko and Goossens, 2002), and these waves would then go on to heat the ions. Also, Lee and Wu (2000) suggested that small-scale reconnection events could produce fast collisionless shocks in the extended corona. For shocks sufficiently thin and strong (i.e., with a bulk velocity jump of at least ∼ 0.3 times the Alfvén speed), ions that cross from one side of the shock to the other remain “nondeflected” by the rapid change in direction of the magnetic field. Thus, they can convert some of their parallel motion into perpendicular gyration. Mancuso et al. (2002) suggested this mechanism may be applied to understanding UVCS measurements of ion heating in shocks associated with CMEs. However, it is unclear to what extent the open magnetic regions in coronal holes are filled with sufficiently strong shocks to enable this process to occur (see also Hollweg and Isenberg, 2002Jump To The Next Citation Point).

In contrast to the ideas of base-generation of ion cyclotron waves, there have been several proposed mechanisms for “gradual generation” of these waves over a range of distances in the corona and solar wind. A natural way to produce such an extended source of fluctuations is MHD turbulent cascade, which continually transports power at large scales to small scales via the stochastic shredding of transient eddies. A strong turbulent cascade is certainly present in interplanetary space (see reviews by Tu and Marsch, 1995Goldstein et al., 1997). It is well known, though, that in both numerical simulations and analytic descriptions of Alfvén-wave turbulence (with a strong background “guide field” like in the corona) the cascade from large to small length scales (i.e., from small to large wavenumbers) occurs most efficiently for modes that do not increase in frequency. In other words, the cascade acts most rapidly to increase the perpendicular wavenumber k⊥ while leaving the parallel wavenumber k∥ largely unchanged (e.g., Strauss, 1976Shebalin et al., 1983Goldreich and Sridhar, 1995Cho et al., 2002Oughton et al., 2004). This type of cascade is expected to generate so-called kinetic Alfvén waves (KAWs) with k⊥ ≫ k∥, but not ion cyclotron waves.

Under typical “low plasma beta” conditions in the corona and fast solar wind, the linear dissipation of KAWs would lead to the preferential parallel heating of electrons (Leamon et al., 1999Cranmer and van Ballegooijen, 2003Jump To The Next Citation PointGary and Borovsky, 2008). This is essentially the opposite of what has been observed with UVCS. However, there have been several suggestions for more complex (nonlinear or multi-step) processes that may be responsible for ions to receive perpendicular heating from KAW-type fluctuations.

  1. Markovskii et al. (2006) discussed how a perpendicular turbulent cascade produces increasingly strong shear motions transverse to the magnetic field, and that this shear may eventually be unstable to the generation of cyclotron resonant waves that can in turn heat protons and ions (see also Mikhailenko et al., 2008). This effect may also produce a steepening in the power spectrum of the magnetic field fluctuations that agrees with the observed “dissipation range” (Smith et al., 2006).
  2. Voitenko and Goossens (20032004) suggested that high-k⊥ KAWs with sufficiently large amplitudes could begin to exhibit nonlinear resonance effects (“demagnetized wave phases”) leading to rapid ion perpendicular heating. There are definite thresholds in the KAW amplitude that must be exceeded for these mechanisms to be initiated, and it is unclear whether the actual coronal turbulent spectrum has enough power in the relevant regions of wavenumber space (see also Wu and Yang, 20062007).
  3. The damping of low-frequency KAWs may give rise to substantial parallel electron acceleration. If the resulting electron velocity distributions were sufficiently beamed, they could become unstable to the generation of parallel Langmuir waves. In turn, the evolved Langmuir wave trains may exhibit a periodic electric potential-well structure in which some of the beam electrons can become trapped. Adjacent potential wells can then merge with one another to form isolated “electron phase space holes” of saturated potential. Ergun et al. (1999), Matthaeus et al. (2003), and Cranmer and van Ballegooijen (2003Jump To The Next Citation Point) described how these tiny (Debye-scale) electrostatic structures can heat ions perpendicularly via Coulomb-like quasi-collisions.
  4. Obliquely propagating MHD waves with large perpendicular and parallel wavenumbers – including KAWs and fast-mode waves – can interact resonantly with positive ions via channels that are not available when either k∥ or k⊥ are small. Li and Habbal (2001) found that oblique fast-mode waves with large wavenumbers may be even more efficient than Alfvén waves at heating ions under coronal conditions. Hollweg and Markovskii (2002) discussed how the higher-order cyclotron resonances become available to obliquely propagating waves with large wavenumbers, and how these can lead to stochastic velocity-space diffusion for ions.
  5. On the smallest spatial scales, the plasma in numerical simulations of MHD turbulence is seen to develop into a collection of narrow current sheets undergoing oblique magnetic reconnection (i.e., with the strong “guide field” remaining relatively unchanged). Dmitruk et al. (2004) performed test-particle simulations in a turbulent plasma and found that protons can become perpendicularly accelerated around the guide field because of coherent forcing from the perturbed fields associated with the current sheets (see also Parashar et al., 2009). It remains to be seen whether this process could lead to more than mass-proportional energization for minor ions.
  6. If the plasma contains sufficiently small-scale density gradients transverse to the magnetic field (∇ ⊥ρ), then drift currents can be excited that are unstable to the generation of high-frequency waves (Markovskii, 2001Zhang, 2003Vranjes and Poedts, 2008Mecheri and Marsch, 2008). These instabilities depend on both the amplitudes and scale lengths of ∇ ⊥ρ. To measure the latter, it is important to take into account both remote-sensing measurements of coronal density inhomogeneities (e.g., Woo, 2006Pasachoff et al., 2007) and constraints from radio scintillation power spectra at larger distances (Bastian, 2001Spangler, 2002Jump To The Next Citation PointHarmon and Coles, 2005Jump To The Next Citation Point).

Despite the fact that theory predicts a predominantly perpendicular cascade, there is some evidence that the turbulent fluctuations in the solar wind have some energy that extends up to large k∥ values in a power-law tail (see, e.g., Bieber et al., 1996Dasso et al., 2005MacBride et al., 2008). Whether or not this means that true “parallel cascade” occurs in the corona and solar wind is still not known. However, some progress has been made using a phenomenological approach to modeling the cascade as a combination of advection and diffusion in wavenumber space. In the model of Cranmer and van Ballegooijen (2003), the relative strengths of perpendicular advection and diffusion determine the slope of the power-law spectrum in k ∥, and thus they specify the amount of wave energy that is available at the ion cyclotron frequencies (see also Cranmer et al., 1999aLandi and Cranmer, 2009Jiang et al., 2009).

There have also been proposals for additional mechanisms that could allow a parallel cascade to occur in the corona and solar wind. Nonlinear couplings between the dominant Alfvén waves and other modes such as fast magnetosonic waves (Chandran, 2005Luo and Melrose, 2006) and ion-acoustic waves (Yoon and Fang, 2008) have the potential to enhance the wave power at high frequencies. It has also been known for some time that nonlinear coupling between Alfvén and fast-mode waves may help explain why the measured in situ magnetic field magnitude |B| remains roughly constant while its direction varies strongly (Barnes and Hollweg, 1974Vasquez and Hollweg, 19961998).

If MHD waves have sufficiently large amplitudes, they may undergo nonlinear wave steepening, which leads to density variations as well as oscillations in the parallel components of the velocity and magnetic field. These may generate progressively smaller scales along the magnetic field (e.g., Medvedev, 2000Suzuki et al., 2007). There is also a “bootstrap” kind of effect for ion cyclotron wave generation that was discussed by Isenberg et al. (2001). If some outward-propagating cyclotron waves exist, the resonant diffusion may act to produce proton velocity distributions that are unstable to the generation of inward-propagating cyclotron waves. In response, the proton distributions would become further deformed and thus could become unstable to the growth of both inward and outward waves. It is not yet known if this process could reach the point of becoming self-sustaining, but if so, it may also serve as an extended generation mechanism for high-k∥ waves.

In addition to the above ideas that involve large wavenumbers and kinetic effects, there have been other suggested physical processes that do not require high-k resonances to be initially present in order to heat the ions.

  1. Particles in large-amplitude Alfvén waves exhibit both E × B drift motions (i.e., their standard velocity amplitude) and a polarization drift velocity Vpol that is smaller than the former by the ratio ω ∕Ωi, where ω is the wave frequency and Ωi is the ion cyclotron frequency. A sufficiently large Vpol can lead to cross-field currents unstable to the generation of high-frequency waves, and to eventual equipartition between Vpol and the ion thermal speed (Singh and Khazanov, 2004Singh et al., 2007Khazanov and Singh, 2007). It is not yet known whether the effective V pol for the coronal fluctuation spectrum is large enough to provide a significant fraction of the ion thermal speeds.
  2. Recently there have been suggested some completely nonresonant mechanisms that depend on the stochasticity of MHD turbulence to produce an effective increase in random ion motions (Lu and Li, 2007Wu and Yoon, 2007Bourouaine et al., 2008). Questions still remain, though, concerning the spatial scales over which one should refer to particle motions as “heating” versus “wave sloshing.” This energization mechanism may be just a more chaotic form of the standard velocity amplitude that an ion feels when in the presence of a spectrum of Alfvén waves (see, e.g., Wang and Wu, 2009). In this case, the maximum amount of heating from this process would provide mass-proportional heating for minor ions and protons (i.e., Ti∕Tp = mi ∕mp), and it is clear that the UVCS measurements for O+5 show heating in excess of this amount (see Section 4.3).
  3. Both numerical and analytic studies of Alfvén waves show that, at sufficiently large amplitudes, there can be gyroresonance-like ion energization for sets of frequencies at specific fractions of the local ion cyclotron resonance frequency (e.g., Chen et al., 2001Guo et al., 2008). Like several other processes listed above, this effect becomes active only above certain thresholds of wave amplitude. Also, Markovskii et al. (2009) showed that mildly nonlinear Alfvén waves – with frequencies slightly below the local proton gyrofrequency and power in both the upward and downward directions along the field – can also undergo additional modes of dissipation and proton heating that are not anticipated in linear ion cyclotron resonance theories.

Finally, there has been some development of the so-called velocity filtration theory, which requires neither direct heating nor wave damping in order to energize coronal ions. Spacecraft measurements of plasma velocity distributions, both in the solar wind and in planetary magnetospheres and magnetosheaths, have revealed that “suprathermal” power-law tails are quite common. These observations led to the suggestion by Scudder (1992aJump To The Next Citation Point1994) of an an alternative to theories that demand explicit energy deposition in the low corona (see also Parker, 1958bLevine, 1974). A velocity distribution having a suprathermal tail will become increasingly dominated by its high-energy particles at larger distances from the solar gravity well. Thus an effective “heating” occurs as a result of particle-by-particle conservation of energy. The major unresolved issue is whether suprathermal tails of the required strength can be produced and maintained in the upper chromosphere and transition region – where Coulomb collisions are traditionally believed to be strong enough to rapidly drive velocity distributions toward Maxwellians. Whether the solar atmosphere actually plays host to strong nonthermal tails is still under debate, with some evidence existing in favor of their presence (e.g., Esser and Edgar, 2000Ralchenko et al., 2007) and other evidence against them (Anderson et al., 1996Ko et al., 1996Feldman et al., 2007).

The original (Scudder, 1992a,b) ideas about suprathermal velocity filtration were applied only to the primary (proton and electron) coronal plasma. More recently, Pierrard and Lamy (2003) and Pierrard et al. (2004) have shown that this mechanism can produce extremely high temperatures for heavy ions in the corona – providing they had suprathermal tails in the chromosphere. The primary quantities presented in these papers, however, were integrated isotropic temperatures T. No information was given about the predicted sense of the temperature anisotropy for the minor ions. For a collisionless exospheric model, there is a suspicion that a combination of several effects (e.g., the initial velocity filtration and subsequent magnetic moment conservation) would result in velocity distributions with T∥ ≫ T ⊥, which is not what is observed.

Since it is obvious that not all of the proposed mechanisms described above (and shown in Table 1) can be the dominant cause of the collisionless ion energization in coronal holes, there is a great need to “cut through the jungle” and assess the validity of each of these processes. For many of these suggested ideas, further theoretical development is required so that specific observational predictions can be made. However, there are also several types of measurement that have not been widely recognized or utilized as constraints on theoretical models. A prime example is the use of radio sounding (i.e., interplanetary scintillations and Faraday rotation) to measure the fine structure of the corona and solar wind in density, velocity, and magnetic field strength (see, however, Hollweg and Isenberg, 2002Spangler, 2002Harmon and Coles, 2005Chandran et al., 2009b). Another example is the use of high-resolution UV spectral line profiles to probe departures from Maxwellian or bi-Maxwellian ion velocity distributions (e.g., Cranmer, 2001).

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