1 Introduction

The solar wind plasma plays a major role in the physical connection between the Sun and the Earth, and affects the plasma conditions in the heliosphere. The solar wind is an important component of space weather, and forms the background upon which solar disturbances, such as CMEs and energetic particles propagate towards the Earth. However, the exact physics of formation of the solar wind is poorly understood. Remote sensing and in-situ observations indicate that there are two types of solar wind: slow and fast. The fast solar wind, reaching the speed of ∼ 800 km s–1 at 1 AU, is steady and of low density (few particles per cm3 at 1 AU); the slow solar wind reaches half the above speed with an order of magnitude higher density (see Figure 1View Image). At solar minimum the solar wind speed is a clear function of latitude as measured by Ulysses’ SWOOPS instrument with the fast wind emerging from polar coronal holes, and the slow wind confined to the equatorial regions. Near the solar maximum the corona is dominated by streamers and by slow solar wind. The heavy ion composition of the solar wind is correlated with the fast and slow wind, indicating their coronal origin.
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Figure 1: The solar wind speed as a function of latitude (in km s–1) measured by Ulysses’ SWOOPS instrument near solar minimum (left panel) and near solar maximum (right panel). The direction of the magnetic field is marked (red-outward; blue-inward). The typical composite solar images near minimum (8/17/96) and maximum (12/07/00) are shown using data from SOHO/LASCO, EIT, and Mauna Loa K-coronameter images (McComas et al., 2003).

The twin Helios spacecraft investigated the heliosphere in the region 0.3 to 1 AU and found evidence for magnetic fluctuations spectrum, and investigated the radial dependence of plasma parameters and the velocity distributions in the solar wind. These measurements provide the basis of many theoretical studies of solar wind acceleration and heating (see the review by Marsch, 2006Jump To The Next Citation Point). Recent in-situ observations by ACE (Stone et al., 1998), Wind (e.g., Lepping et al., 1995), Ulysses, (e.g., Balogh et al., 1992), and other spacecrafts confirm that the solar wind contains magnetic fluctuations that have power law dependence with frequency (see Figure 2View Image adapted from Smith et al., 2006Jump To The Next Citation Point). The magnetic fluctuations exhibit high correlation with velocity fluctuations indicating their Alfvénic nature. Remote sensing observations show that the solar wind is heated and accelerated close to the Sun within 10 R ⊙ (SOHO/UVCS; Kohl et al., 1997Jump To The Next Citation Point). The observed kinetic and compositional properties provide clues on coronal origin (i.e., coronal holes, active regions) (e.g., Ko et al., 2008), and on the acceleration and heating mechanism. However, the interpretation of observations requires theoretical and computational modeling of the global and kinetic properties of the solar wind to understand the physics and the dynamics of the multi-ion solar wind plasma acceleration and heating, and improve the accuracy of space weather forecasting.

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Figure 2: Example of magnetic spectrum from the ACE database. Year/day of sample (decimal day of year in 1998) and Power Spectral Density (PSD) are shown. The top curve represents the summed power in the two components perpendicular to the mean magnetic field, and the lower curve represents the power in the parallel component. The fit functions with power exponent –1.56 ± 0.3 (upper) and –1.8 ± 0.03 (lower) are shown (see Smith et al., 2006, for further details).

In the region between 1 R ⊙ and 1 AU the corona and the heliosphere span many orders of magnitudes in relevant temporal-spatial scales, density, magnetic field strength, and in other physical parameters. As a result, the physical processes in the plasma and the modeling approaches are dramatically different in the collisional lower corona with densities on the order of 108–9 cm–3, magnetic field strength of 1 to 100 G, and the heliospheric plasma near 1 AU with densities of few particles per cm–3 and magnetic field strength of few nano-tesla (few tens of micro-Gauss). Correspondingly, the plasma frequency, collision frequency, and the proton gyrofrequency vary over many orders of magnitude between 1 R ⊙ and 1 AU. This disparity of scales and physical regimes leads to difficulty in the physical modeling of the heliospheric plasma with a single ‘all-inclusive’ model. For example, solving the Boltzmann equations in three spatial dimensions and six degrees of freedom is not practical with current or foreseeable future computational resources. A practical modeling approach that is applicable on large scales (usually, MHD) can not resolve the physics of the small scale (kinetic) phenomena. The global models provide little or no information on the kinetics of the heating and acceleration processes. The solution of this difficulty is a multi-level modeling approach, where global models are used for large scale structures, with small scales parameterized as external (to MHD) diffusion coefficients and heating/loss functions. Kinetic models are used to study the physics of the local heating and dissipation phenomena on small spatial-temporal scales. The output of the kinetic models can serve as a guide of the diffusion processes that need to be included in global models.

As a result, there are two main types of solar wind models: (1) observationally driven global MHD models that provide the overall global structure of the solar wind as shaped by the interaction with the solar magnetic field, rotation, gravity, and heating; (2) local models that deal with the physical processes of heating and acceleration of the solar wind magnetized plasma with the initial state and boundary conditions not necessarily tied to a particular observation. The output of these models provides the solar wind speed as a function of position, density, temperature, and other parameters. Usually, the acceleration of the solar wind is modeled by introducing simplifying assumption into the energy equation, such as taking the plasma to be isothermal at coronal temperature, or assuming polytropic index below the adiabatic γ = 5∕3, that can also vary with heliocentric distance (Cohen et al., 2007Jump To The Next Citation Point). The first isothermal and polytropic solar wind expansion models were developed by Parker (1958, 1963) in one spatial dimension. Present global models provide 3D structure of the solar wind and try to match empirically the observations at 1 AU by adjusting the model parameters (e.g., Mikić et al., 1999; Linker et al., 1999; Usmanov et al., 2000Jump To The Next Citation Point; Roussev et al., 2003; Usmanov and Goldstein, 2003; Cohen et al., 2007Jump To The Next Citation Point). Recently, 2D MHD models that use observations to constrain the solar wind heating function and momentum input were developed (Sittler Jr and Guhathakurta, 1999; Vásquez et al., 2003; Guhathakurta et al., 2006; Sittler Jr and Ofman, 2006Jump To The Next Citation Point). The empirical heating function was recently included in 2.5 solar wind MHD models (Sittler Jr and Ofman, 2006; Airapetian et al., 2011).

On the other end of the scale are the kinetic models which provide the description of the small scale interaction in the solar wind plasma between the waves, the ions, and the background magnetic field. These models usually do not provide the information on the global structure of the solar wind in the heliosphere. However, the kinetic models are well suited for the investigation of the kinetic processes and instabilities involved in heating of solar wind-like plasma. Due to the complexity of the kinetic models, and the necessity to resolve the fine scale on the proton, or even electron Larmor radius scale, it is computationally difficult to include global scale structures.

The range in between the above two modeling approaches is occupied by MHD models that include explicit heating and acceleration of the solar wind by MHD waves. Following Osterbrock (1961) study suggesting MHD waves for the heating of the solar chromosphere and corona, the acceleration of the solar wind by Alfvén waves was studied in the past (Barnes, 1969; Alazraki and Couturier, 1971; Belcher, 1971; Belcher and Davis Jr, 1971; Heinemann and Olbert, 1980). Recently, one-dimensional (Cranmer and van Ballegooijen, 2005Jump To The Next Citation Point; Suzuki and Inutsuka, 2005Jump To The Next Citation Point, 2006; Cranmer et al., 2007) and three-dimensional MHD models (e.g., Usmanov et al., 2000Jump To The Next Citation Point; Evans et al., 2009) were developed (see the review by Ofman, 2005Jump To The Next Citation Point). However, due to the requirements of time step and resolution, the waves are not included explicitly in global models. They are modeled by an additional wave-pressure term, and wave-energy equations. Only few models consider the resolved waves explicitly in 2.5D MHD models (Ofman and Davila, 1997Jump To The Next Citation Point, 1998Jump To The Next Citation Point). The next level of plasma approximation of fully resolved wave driven wind is via multi-fluid models, that describe each particle species as separate fluid (Ofman and Davila, 2001Jump To The Next Citation Point; Ofman, 2004aJump To The Next Citation Point). The fluids interact through momentum and energy exchanges, and through electromagnetic interactions resulting from quasi-neutrality condition, and through their contribution to the net current. These models can be tested directly by comparing to observations that contain heavy ion emission (e.g., Ofman, 2004b; Abbo et al., 2010).

Observations with SOHO Ultraviolet Coronagraph Spectrometer (UVCS) show that heavy ions such as O5+, and Mg9+ undergo preferential perpendicular heating, causing large temperature anisotropy (T⊥ ∕T∥ > 10), are hotter and flow faster in coronal holes than protons (Kohl et al., 1997Jump To The Next Citation Point, 1998; Li et al., 1998; Cranmer et al., 1999Jump To The Next Citation Point). Enhanced perpendicular heating of ions compared to protons has also been observed in streamers (Strachan et al., 2002; Uzzo et al., 2007). The magnitude of this effect is significant, but smaller in streamers than in coronal holes. Ulysses and Helios in-situ measurements in the heliosphere have shown that minor ions flow faster than protons by the local Alfvén speed, and are preferentially heated as well, and proton distributions often appear double-peaked with an average relative drift parallel to the background magnetic field (e.g., Marsch et al., 1982bJump To The Next Citation Point; Feldman et al., 1996Jump To The Next Citation Point; Neugebauer et al., 1996).

The temperature anisotropy of protons deduced from remote sensing and in-situ observations of fast solar wind streams provides indirect evidence for the presence of the ion-cyclotron waves in coronal plasma, since the anisotropy can be produced by the resonant absorption of the ion-cyclotron waves. Purely adiabatic expansion of the solar plasma is expected to result in an opposite effect: T⊥ < T∥ due to the conservations of magnetic moment of the expanding ions in the decreasing radial magnetic field (e.g., Marsch, 2006Jump To The Next Citation Point). However, T⊥∕T ∥ > 1 is observed in the heliosphere (e.g., Marsch et al., 1982bJump To The Next Citation Point; Gazis and Lazarus, 1982). In the past, several theories of ion-cyclotron resonance have been developed and applied to the heating of the solar corona and the solar wind (e.g., Axford and McKenzie, 1992; Marsch, 1992; Tu and Marsch, 1997; Li et al., 1999; Hollweg, 2000; Hu et al., 2000; Cranmer, 2000Jump To The Next Citation Point; Hollweg and Isenberg, 2002). However, there are theoretical difficulties with the application of the ion-cyclotron mechanism for coronal heating, and its role is not yet fully understood (e.g., Cranmer, 2000; Isenberg, 2004). Most such theories may be classified as either fluid-like or quasi-linear kinetic models. The limitation of the fluid or quasi-linear kinetic models is the assumption of fixed-shape ion velocity distribution and quasi-linear limits (i.e., small magnetic fluctuation amplitude allowing simplified description of wave–particle interactions). In the hybrid models the electrons are treated as a fluid, and the ions are treated fully kinetically as particles. Hybrid simulations (see Section 2.3 below) allow relaxing many approximations used in the fluid, multi-fluid, and in linear or quasi-linear kinetic theory. The model is nonlinear, and can describe both the brief initial linear evolution of the plasma, as well as the nonlinear saturated state. Recently, new kinetic models of heating and acceleration of solar coronal plasma in inhomogeneous magnetic field by Alfvén waves were developed (Galinsky and Shevchenko, 2013a,b). The generation of the solar wind by parallel (Isenberg, 2012; Mecheri, 2013) and oblique (Chandran et al., 2010; Isenberg and Vasquez, 2011) ion-cyclotron waves were also studied. The possible role of kinetic Alfvén waves (KAW) (e.g., Voitenko and Goossens, 2006; Dwivedi et al., 2012) and Alfvén waves turbulence (Chandran, 2010; Chandran et al., 2011; Li et al., 2011; Cranmer and van Ballegooijen, 2012) on the acceleration and heating of the solar wind was recently considered.UpdateJump To The Next Update Information

Recently, Cranmer and van Ballegooijen (2005Jump To The Next Citation Point) solved the linearized wave equation for Alfvén waves in the heliosphere. The dependence on heliocentric distance of the frequency-integrated Alfvénic velocity amplitude obtained from the solution of the linearized Alfvén wave equation driven by a spectrum of transverse photospheric fluctuations with an amplitude of 3 km s–1 at the photosphere is compared to Alfvén wave amplitude inferred from spectroscopic observations (SOHO/SUMER and UVCS), IPS measurements, and in-situ data in Figure 3View Image. The study shows that there is generally good qualitative agreement between observations and theoretical prediction of the Alfvén wave evolution in the heliosphere, even for a linearized model.

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Figure 3: Height dependence of the frequency-integrated velocity amplitude obtained from the solution of the linearized Alfvén wave equation driven by a spectrum of transverse photospheric fluctuations. The solution was obtained in the thin flux tube approximation by Cranmer and van Ballegooijen (2005Jump To The Next Citation Point). Solid lines give the undamped value of ⟨δV ⟩ (the dashed line is for different model parameters). The red line is √ ---- ⟨δV ⟩B = ⟨δB ⟩∕ 4π ρ for the 3 km s –1 driving amplitude case. The symbols correspond to observational constraints of the Alfvén waves amplitudes from various observations discussed by Cranmer and van Ballegooijen (2005) (reproduced by permission of the AAS).

The turbulence in the solar wind magnetized plasma has been studied for decades in the past (see the review by Velli, 2003), and recently (Verdini et al., 2009; Chandran and Hollweg, 2009; Chandran et al., 2009; Verdini et al., 2010; Markovskii et al., 2010Jump To The Next Citation Point) as the possible state that leads to cascade of energy from the observed large scale fluctuations and waves to small scale structures, down to dissipation scales that can heat the solar wind plasma. Observations of Alfvénic fluctuations in the solar wind by Helios and Ulysses spacecraft show that the turbulent energy carried by these fluctuations is distributed in frequency according to a power law, at high frequencies going as f −5∕3, a Kolmogorov spectrum, while at lower frequencies the spectrum flattens to − 1 f (where f is the fluctuation frequency) (Goldstein et al., 1995). Alfvénic turbulence is predominant in fast wind streams while in slow solar wind the turbulence is of more complex nature with low Alfvénicity (see the reviews by Tu and Marsch, 1995; Bruno and Carbone, 2013). Recent observations by ACE spacecraft of the solar wind protons at 1 AU indicate that the turbulent cascade rate agrees better with Kraichnan (− 3∕2 f), rather than with Kolmogorov (−5∕3 f) rate (Vasquez et al., 2007). Similar results were seen by the Wind spacecraft (Podesta et al., 2006). Part of the fluctuating power at low frequencies can be attributed to propagating structures in the solar wind. However, there is strong evidence that the fluctuations are Alfvénic at frequencies of milli-Hertz and higher.

Recent observations by Hinode satellite show that Alfvénic fluctuations are the likely energy source that drives the solar wind (e.g., De Pontieu et al., 2007; Ofman and Wang, 2008; Hahn et al., 2012; Hahn and Savin, 2013). A review of observational evidence for propagating MHD waves in coronal holes that may accelerate the solar wind is found in Banerjee et al. (2011).UpdateJump To The Next Update Information Recently launched NASA’s Solar Dynamics Observatory (SDO) provides unparalleled opportunity to look for the solar coronal wave spectrum over the entire disk of the Sun at high temporal and spatial resolution. The analysis of the data from the Atmospheric Imaging Assembly (AIA) onboard SDO will likely provide the constraint on the input wave spectrum that drives the solar wind. Although the observed spectrum is limited to the MHD frequency range, since the temporal resolution does not allow resolving high frequencies down to the gyroresonant scale (∼ kHz), the form of the spectrum is likely to provide clues on the relevant turbulent cascade processes.

The following level of solar wind plasma approximation is via hybrid models (see Section 2.3), that describe protons and other ions kinetically as particles, and electrons as neutralizing background fluid (Winske and Omidi, 1993Jump To The Next Citation Point). Hybrid simulations can represent more completely (than fluid model) and self-consistently the wave–particle interactions in the multi-ion solar wind magnetized plasma. The models can be used to describe the kinetic processes involved in heating by a spectrum of waves, and can be used to study the nonlinear and resonant interactions of the turbulent spectrum with the ions. Such numerical methods have the potential to model the wave–particle interactions, and the corresponding velocity distribution and magnetic fluctuations in the nonlinear saturated state that can be compared to in-situ observations. Recently, one dimensional hybrid simulations of multi-ion solar wind plasma were used to study the heating by wave spectrum, beams, and the stability of solar wind multi-ion plasma (Liewer et al., 2001Jump To The Next Citation Point; Ofman et al., 2001Jump To The Next Citation Point, 2002Jump To The Next Citation Point; Xie et al., 2004Jump To The Next Citation Point; Lu and Wang, 2005; Li and Habbal, 2005; Hellinger et al., 2005Jump To The Next Citation Point; Ofman et al., 2005Jump To The Next Citation Point). Due to the local nature of the hybrid models, they require special treatment taking into account the global properties of the solar wind, such as expansion of the solar wind into the heliosphere (Liewer et al., 2001Jump To The Next Citation Point; Hellinger et al., 2005Jump To The Next Citation Point; Ofman et al., 2011Jump To The Next Citation Point). We will review some of the recent results of hybrid simulation models of solar wind plasma heating.

Two-dimensional hybrid models of homogeneous multi-ion plasma heating were also studied recently (Gary et al., 2001Jump To The Next Citation Point, 2003Jump To The Next Citation Point; Kaghashvili et al., 2003; Hellinger and Trávníček, 2006; Gary et al., 2006). However, only few studies considered the effect of time dependent wave fluctuations on solar wind plasma heating with 2D hybrid models (Ofman and Viñas, 2007Jump To The Next Citation Point; Ofman, 2010Jump To The Next Citation Point; Markovskii et al., 2010). It was found that the input wave spectrum can heat the ions by resonant interaction, as well as through non-resonant parametric decay instability of Alfvén waves (e.g., Araneda et al., 2007Jump To The Next Citation Point). It was also found that the presence of small scale inhomogeneity in the background plasma can enhance the heating by the high frequency Alfvén wave spectrum (Ofman, 2010Jump To The Next Citation Point).

This review is focused on selected wave-acceleration models (both, MHD and kinetic) of the solar wind. Other reviews of solar wind models were recently published (Hansteen and Velli, 2012; Cranmer, 2012).UpdateJump To The Next Update Information

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