Wave models in the single fluid MHD are limited to frequencies much smaller than the proton gyrofrequency and correspondingly to wavelengths that are much larger than the proton gyroradius. The acceleration of the solar wind plasma by the waves due to momentum transfer (wave reflection) and gradient of the wave pressure is modeled. The dissipation of the waves by Ohmic and viscous dissipation terms is included. The multi-fluid models allow including waves with frequencies in the MHD and in the range of proton and ion gyrosresonant frequency. The multi-ion cyclotron resonant dispersion of waves is reproduced by this model even in the linear regime. It has been shown that multi-fluid dispersion relation is equivalent to Vlasov’s dispersion for cold plasma (for example, see Ofman et al., 2005). In addition, the multi-fluid models can include separate heating and dissipation processes for electrons, protons, and each ion species with different dissipation coefficients. The fluids are coupled through collisional energy exchange terms and Coulomb friction, and through electromagnetic interactions. The multi-fluid models provide the next level of plasma approximation between the MHD and the kinetic descriptions.
The hybrid simulations extend the modeled physics of the solar wind plasma to even smaller scale in time and space to the kinetic regime, and the wave frequencies in the ion- and proton-gyroresonant scale are resolved. In addition, ion velocity space instabilities and ion kinetic processes are modeled fully. The hybrid models are limited to waves with frequencies below the electron gyroresonant frequency, since the electrons are treated as fluid in these models. Other limitations of these models are outlined in Winske and Omidi (1993). The 1D hybrid models are limited to parallel propagating waves in one spatial direction, or oblique waves with fixed angle of propagation. The more general 2D hybrid models include the description of waves with arbitrary propagation direction and can be used to model inhomogeneous plasma in two spatial dimensions.
It is well known that thermally driven Parker’s solar wind model with typical coronal temperature of 1 – 2 MK can produce the slow solar wind asymptotic speed of about 400 km s–1, but can not explain the fast solar wind that is observed to reach 800 km s–1 within and is associated with coronal holes with typical temperatures < 1 MK (Aschwanden, 2004). The common approach is to include an additional source of momentum in the MHD equations, such as Alfvén waves as an empirical WKB momentum addition term (e.g., Usmanov et al., 2000). This approach was recently extended to include the effects of turbulence dissipation in a global wave driven solar wind model, and implemented in the Space Weather Modeling Framework (SWMF) (Tóth et al., 2005) coronal 3D MHD code (Evans et al., 2012; Sokolov et al., 2013; Oran et al., 2013). Two-temperature Alfvén wave driven fast solar wind models were also developed in SWMF (van der Holst et al., 2010).Update
Lau and Siregar (1996) studied the acceleration of the solar wind by resolved nonlinear Alfvén waves in 1.5D MHD model. Ofman and Davila (1997) were the first to use the single fluid 2.5D MHD model to study resolved Alfvén wave driven fast solar wind in a coronal hole. In their model the Alfvén waves were launched at the solar boundary of a coronal hole, and were resolved throughout the coronal hole to . The acceleration of the solar wind occurs through momentum transfer from the waves to the solar wind plasma. The heating of the solar wind plasma was not included explicitly in this model and an isothermal approximation was used (). However, wave dissipation does occur through resistive dissipation with finite value of . In Figure 6, a snapshot of the spatial dependence of the solutions in terms of , , , and is shown at . The velocities and are in units of = 1527 km s–1, and the density is in units of 108 cm–3. The monochromatic Alfvén waves launched in this model are evident in and in . The nonlinear longitudinal waves produced by the gradient of the compressions associated with the Alfvén wave, , are evident in and . The large amplitude, long wavelength compressional velocity and density fluctuation propagate in-phase (Ofman and Davila, 1998). Ofman and Davila (1998) found that low-frequency (0.35 mHz) Alfvén waves with amplitude of 46 km s–1 can produce the fast solar wind in coronal holes.
Grappin et al. (2002) were the fist to study resolved Alfvén waves driven wind that include both, closed and open field regions using 2.5D MHD model. They found that onset of Alfvén wave flux in one hemisphere generates a stable global circulation pattern in the closed loops region that can lead to global north-south asymmetry of the solar corona.
In Figure 7 a cut through the center of the coronal hole is shown. The and solar wind velocity components are shown for two Alfvén wave driving frequencies, and the green curve shows Parker’s isothermal solar wind solution. It is evident that the low frequency waves (f = 0.35 mHz) lead to significant acceleration of the fast solar wind above Parker’s solution, and produce the fast solar wind far from the Sun. The higher frequency waves provide acceleration close to the Sun below .
The 2.5D model discussed above includes only the coronal part of the solar wind, with the driving Alfvén waves applied at the lower coronal boundary. Recently, Suzuki and Inutsuka (2005) modeled the acceleration of the fast solar wind by Alfvén waves from the photosphere to 0.3 AU using a 1.5D model (Figure 8). This approach allowed connecting directly photospheric motions of observationally constrained magnitude to solar wind speed at 0.3 AU. Although the model does not include the effects of cross-field gradients, the model demonstrates that sufficient Alfvén wave energy flux reaches the corona to accelerate the solar wind. The 1.5D model results compare favorably to IPS observation (Grall et al., 1996; Canals et al., 2002) and to SOHO observations (see, Suzuki and Inutsuka, 2005, for the details).
In Figures 9 and 10 we show the results of the 3-fluid model of the Alfvén wave driven fast solar wind in a coronal hole obtained by Ofman (2004a). In this model a broad band spectrum of Alfvénic fluctuations was applied at the lower coronal hole boundary. The fast solar wind was produced by acceleration and heating with the spectrum of Alfvén waves, that were fully resolved in the model. In Figure 9 the Alfvén waves are evident in and the accelerating solar wind in velocity components for He++ ions (left panels) and protons (right panels) at are shown. Note that compressive fluctuations are also seen in due to the local variation of wave pressure gradient. The velocity is in units of 1527 km s–1. The distance is in units of , and the latitude is in radians.
The typical form of the magnetic fluctuations obtained with the 3-fluid model at is shown in Figure 10. It is interesting to note that the spectrum launched at the base of the coronal hole results in spectrum at larger distances. The steepening of the magnetic fluctuations spectrum is expected due to turbulence and dissipation that affects shorter wavelengths and correspondingly higher frequencies more than the long wavelength (low frequencies) fluctuations. The power law dependence is close to Kolmogorov’s turbulent power spectrum of . At frequencies higher than the spectrum steepens due to increased dissipation.
In Figure 11 the -averaged outflow speeds of protons, He++, and O5+ ion fluid are shown for four sets of model parameters. The parameters are (a) , , , (b) , , , and (c) , , , where is the heating rate per particle for an empirical heating term used in Ofman (2004a), and is the amplitude of the Alfvén wave spectrum. The corresponding temperatures and densities are shown in Figure 12. In Figure 11a the solutions of the 3-fluid model with empirical heating term [Equation (12)] in addition to Alfvén wave spectrum are shown. The heating term parameters were chosen to match the observed fast solar wind speed, and the faster outflow of He++ ions compared to protons, observed at 0.3 AU and beyond with Helios and Ulysses spacecraft (Marsch et al., 1982a,b; Feldman et al., 1996; Neugebauer et al., 2001). In Figure 11b the O5+ ions were included as the third fluid, and the heating function parameters for O5+ ions were adjusted to get faster than proton outflow. In Figure 11c the same heating per particle was deposited in protons and He++ ions. Evidently, in this case the He++ ions outflow speed is slower than the proton outflow speed, contrary to observations. In Figure 11d the solar wind protons are accelerated and heated solely by the Alfvén wave spectrum (i.e., ). This was achieved by increasing the input wave amplitude, compared to the values used in Figures 11a–c. Note that the temperature structure of protons and O5+ ions as seen in Figure 12b is in qualitative agreement with SOHO/UVCS observations (Kohl et al., 1997; Cranmer et al., 1999; Antonucci et al., 2000) (at present there are no observations of He++ temperature in this region). The model shows that in all cases electron heating can be achieved by thermal coupling between electrons and protons alone (through the thermal coupling term in Equation (9)).
In spectroscopic observations of emission lines the observed finite line width is the result of broadening by Doppler shift due to the motion of the emitting ions in the line of sight. The motions are usually attributed to two components: (1) thermal or kinetic motions due to the finite width of the ion velocity distribution; (2) non-thermal motions, arising from any unresolved macroscopic motions of the plasma in the line of sight. The effect of observationally unresolved Alfvén waves on the apparent emission line widths was modeled with the 3-fluid model by Ofman and Davila (2001) and Ofman (2004a). These models allow separating the contribution of waves to the observed line profiles. In Figure 13 the Doppler-broadened emission line, resulting from combined thermal and non-thermal motions calculated with the 3-fluid model is shown. The solid line shows the simulated emission line profile of protons at at a temperature of 3.5 MK, and the dashes show the emission line broadened by unresolved Alfvénic fluctuation. In Figure 14 the effective temperature calculated from the kinetic temperature and the Alfvénic wave motion contribution is shown for the wave-driven fast solar wind as a function of the heliocentric distance. The effective proton temperature is affected significantly by the non-thermal component, while the relative effect on He++ is smaller close to the Sun. By comparing the results of the 3-fluid model to observations, it is possible to evaluate better the thermal and non-thermal motions for protons and heavier ions in the observational data.
Recently, Ofman et al. (2002) used 1D hybrid model of initially homogeneous, collisionless plasmas to study the heating of solar wind plasma by a spectrum of ion-cyclotron waves. Motivated by observations the model was driven by circularly polarized Alfvénic fluctuations of the form and for a limited bandwidth. They found that the ion heating depends on the resonant power in the frequency range of the input spectrum. Preferential heating of minor ions, such as O5+, over protons was demonstrated in this model. In Figure 15 the evolution of the temperature anisotropy for protons and O5+ ions is shown. It is evident that after the perpendicular heating of the ions saturates at anisotropy level of 7, and the proton are not heated significantly. The level of saturated anisotropy is determined by the temperature dependent nonlinear balance between ion-cyclotron unstable ion velocity distribution that releases electromagnetic ion-cyclotron waves and the resonant absorption of magnetic fluctuations together with parallel heating of the ions. After inspecting the perpendicular and parallel temperatures of O5+ at the end of the run it is evident that the heating was predominantly in the perpendicular direction (Ofman et al., 2002).
In Figure 16 the velocity distribution of the protons and O5+ ions is shown at the end of the run. It is evident that the proton velocity distribution is isotropic and the O5+ ions are hotter in the perpendicular direction than in the parallel direction. The O5+ velocity distribution is close to bi-Maxwellian with small non-Maxwellian features in the parallel velocity distribution, likely produced by the small parallel heating due to nonlinear compressive modes driven by the Alfvénic fluctuations spectrum.
The relaxation of O5+ ion temperature anisotropy due to ion-cyclotron instability for the parameter range relevant to fast solar wind in coronal holes was recently studied using 1D hybrid model (Ofman et al., 2001) (see Figure 23). The study was motivated by SOHO/UVCS observations indicating large temperature anisotropy of O5+ ions (Kohl et al., 1997; Cranmer et al., 1999). It was found that the scaling of the relaxed with the final (full circles) and the scaling of the relaxation time, , with the initial (circles) agree well with the theoretical scaling law (Gary, 1993). The “x”’s mark the values at . The enhanced O5+ abundance relative to protons of 6 × 10–4 in this model was implemented in oder to shorten the computation times. Similar result was found in 2D hybrid model by Gary et al. (2003) (see Section 3.4 and Figure 22 below).
Recently, Ofman et al. (2005) investigated the effects of high-frequency (of order ion gyrofrequency) Alfvén and ion-cyclotron waves on ion emission lines by studying the dispersion of these waves in a multi-ion coronal plasma. The dispersion relation of parallel propagating Alfvén cyclotron waves in the multi-ions coronal plasma was determined using 1D hybrid model (see Figure 17) and compared with multi-fluid and Vlasov dispersion relation. It was found that the three methods are in good qualitative agreement in the weakly damped regime (). The ratio of the ion to proton fluid velocities perpendicular to the direction of the magnetic field was calculated for each wave modes for typical coronal parameters (see Figure 18). It was found that the O6+ perpendicular fluid velocity exhibits strong (factor of 20 – 100) enhancement and He++ perpendicular velocity is enhanced by a factor of 3.5 – 5 compared with proton perpendicular fluid velocity, in qualitative agreement with SOHO/UVCS observations of large perpendicular velocity of heavy ions in coronal holes (e.g., Kohl et al., 1997; Cranmer et al., 1999). The study demonstrated how the results of hybrid models can be used to better understand the observations of coronal ion emission.
Recently, Araneda et al. (2007, 2008) used Vlasov theory and one-dimensional hybrid simulations to study the effects of compressible fluctuations driven by parametric instabilities of Alfvén-cyclotron waves. They found that field-aligned proton beams are generated during the saturation phase of the wave–particle interaction, with a drift speed somewhat above the Alfvén speed. This finding agrees with typically observed velocity distributions of protons in the solar wind that contain a thermal anisotropic core and a beam component (see the review by Marsch, 2006). The expanding box model (Grappin and Velli, 1996; Liewer et al., 2001) was recently applied in 1.5D hybrid models of H+-He++ solar wind plasma heated by a spectrum of turbulent Alfvénic fluctuations and in solar wind plasma with super-Alfvénic ion relative drift (Ofman et al., 2011; Maneva et al., 2013). In particular, Maneva et al. (2013) studied the turbulent heating and acceleration of He++ ions by an initial self-consistent spectra of Alfvén-cyclotron waves in the expanding solar wind plasma using 1.5D hybrid simulations. They found that the He++ ions are preferentially heated by the broad-band initial spectrum, resulting in much more than mass-proportional temperature increase (see Figure 19). Maneva et al. (2013) also found that the differential acceleration of protons and He++ ions depend on the amplitude and spectral index of the magnetic fluctuation, while solar wind expansion suppresses the differential streaming. They also find that the expansion leads in general to perpendicular cooling for protons and alphas. However, the cooling effect of the expansion is small and the waves provide sufficient heating, maintaining significant temperature anisotropy, in agreement with observations. Inspection of the proton and alpha velocity distribution in the - plane shows the formation of non-Maxwellian features due to the effects of the broad band spectrum, such as perpendicular broadening (i.e., temperature anisotropy), as well as formation of a population of accelerated particles by the waves (see Figure 20.)Update
The 2D hybrid codes solve similar set of equations as in the 1D hybrid codes but in two spatial dimensions. This allows an additional degree of freedom for particle motions, and the wave propagation is not limited to parallel propagating waves, allowing oblique propagation. In addition, the parallel magnetic field component does not have to be constant in order to conserve . As a result, a broader range of possible wave modes, wave–particle, and wave–wave interactions are included in the 2D model compared to 1D model. Obviously, the 2D models are computationally intensive, and may require parallel processing for similar resolution in 2D and similar numbers of particles per cell as in the 1D models that can be run on a desktop workstation.
The 2D hybrid models have been used extensively in the past to model successfully the electromagnetic interactions in magnetized plasmas (McKean et al., 1994; Gary et al., 1997; Daughton et al., 1999; Gary et al., 2000, 2001, 2003; Ofman et al., 2001, 2002; Xie et al., 2004; Ofman and Viñas, 2007). Comparisons between one-and two-dimensional hybrid simulations often show qualitative agreement in the ion response (Winske and Quest, 1986; Ofman and Viñas, 2007). In addition to allowing oblique waves, the 2D code allows including spatial inhomogeneity of plasma density perpendicular to the magnetic field, as well as divergent magnetic field geometry. These features are needed to describe solar wind acceleration and heating more consistently with corona conditions.
Recently, Ofman and Viñas (2007) studied the heating and the acceleration of protons, and heavy ions by a spectrum of waves in the solar wind, as well as the nonlinear influence of heavy ions on the wave structure using the 2D hybrid model. They considered for the first time the heating and acceleration of protons and heavy ions by a driven input spectrum of Alfvén/cyclotron waves, and by heavy ion beam in the multi-species coronal plasma in two spatial dimensions. They found that in the homogeneous plasma the ion beams heat the ions faster than the driven wave spectrum constrained by solar wind parameters, and produce temperature anisotropy with in qualitative agreement with observation. The beam-heating model requires that the beam speed is larger than the local Alfvén speed. Since any reconnection process produces Alfvénic beams as an exhaust (e.g., Priest, 1982; Aschwanden, 2004), the beams could readily become super-Alfvénic as the plasma moves to regions of lower local Alfvén speed. Since the threshold of beam stability is the Alfvén speed, it is possible that remnants of this process that takes place close to the Sun in the acceleration region of the solar wind are seen in proton data beyond 0.3 AU (Marsch, 2006). Below, the results obtained recently by Ofman and Viñas (2007) are reviewed.
Ofman and Viñas (2007) compared the evolution of O5+ ion anisotropy relaxation by ion cyclotron instability, by modeling the coronal plasma with both 1D and 2D hybrid codes. In Figure 21 the results of a 1D and 2D hybrid models runs are shown. The initial temperature anisotropy was set to 50 in both cases with parallel temperature of 1.4 × 106 K. It is evident that the temperature anisotropy has relaxed to similar values in 1D and 2D runs, close to the marginally stable value of 10 obtained for the parameters used with the Vlasov stability analysis (Gary, 1993). The agreement that was found between 1D hybrid and 2D hybrid evolution is consistent with the Vlasov dispersion relation that shows maximal growth of the ion cyclotron instability for parallel propagating modes.
In Figure 22 the results of the parametric study of He++ anisotropy relaxation obtained with 2D hybrid code by Gary et al. (2003) is shown. In that study the parameters were with initial , , and isotropic protons. The initial anisotropy of He++ was chosen to maximize the linear growth rate of ion-cyclotron instability. The crosses show the anisotropy at , while the squares show the anisotropy at magnetic energy saturation, and the dots represent later times. The dashed line shows the best fit of the anisotropies at , that produces the scaling law . Note the qualitative agreement between the 2D hybrid study for He++ anisotropy relaxation, the 1D hybrid study for the O5+ anisotropy relaxation shown in Figure 23, and the scaling law obtained analytically (Gary, 1993). Gary et al. (2003) have shown that the model results are consistent with Ulysses in-situ observations of solar wind protons and He++ ions.
Ofman and Viñas (2007) found that the perpendicular heating occurs for the beam-driven instability, which quickly saturates nonlinearly, and due to the driven spectrum of waves. In the driven wave spectrum case the amplitude of the magnetic field fluctuations was , and the frequency range of the driver was below the proton gyroresonance. It was found (see Figure 24) that the O5+ anisotropy grows quickly (within ) to , and than saturates nonlinearly remaining in the range 4 – 5 throughout rest of the evolution. The frequency range of the wave spectrum included the O5+ ion resonant frequency at rest. The anisotropy of the protons remains close to unity throughout the run. No significant net drift was found between the protons and O5+ ions in the wave driven case. Similar results were obtained for He++ ions.
The non-Maxwellian features of the ion velocity distribution are evident in the perpendicular to the magnetic field phase space plane. When the initial distribution is drifting Maxwellian with the drift velocity , the perpendicular velocity distribution of the O5+ is shell-like, with decreased phase-space density in the central part of the distribution, compared to the perimeter. When the initial drift velocity was increased to , the shell like structure of the phase-space density of O5+ ions becomes even more apparent (Figure 25). It is interesting to note, that the He++ perpendicular velocity distribution for the drifting case is nearly bi-Maxwellian, and does not exhibit the shell structure.
Recently, Ofman (2010) expanded this study and considered in a parametric study the effect of inhomogeneous background density on the heating by high frequency circularly polarized Alfvén waves with, and without drift between the protons and heavier ions. Ofman (2010) found that the inhomogeneity, and the drift lead to increased heating of the solar wind ions, compared to the homogeneous case, and the spectrum of magnetic fluctuations steepens beyond Kolmogorov’s slope of –5/3. In Figure 26 the magnetic energy fluctuations spectrum obtained in the 2D hybrid simulation with inhomogeneous background density is shown. The dashed-dotted line shows the best fit power law to the spectrum in the regions where the slope did not change considerably. Ofman (2010) found that in the low density region the slope was m = –1.66 for wave driven case, and m = –1.81 for the beam driven case. However, in the high density region the slopes were m = –2.53 for the wave driven case, and m = –2.80 for the beam driven case, indicating enhanced dissipation due to the refraction of Alfvén waves, and the generation of small scale magnetosonic fluctuations that dissipate more effectively than Alfvénic fluctuations. Ofman et al. (2011) explored additional forms of background inhomogeneity on the magnetosonic drift instability and solar wind plasma heating by a spectrum of Alfvénic fluctuations. The expansion of the distant solar wind plasma at 0.3 AU and beyond and the generation of the associated kinetic instabilities and waves was considered in 2D hybrid models by Hellinger and Trávníček (2011, 2013).Update