2.1 Single fluid MHD

where is the fluid density, is the fluid velocity, is the magnetic field, is the
current density, and is the plasma pressure. The normalization of the magnetic field is by the typical
magnetic field at the base of the corona, distances () are normalization by the solar radius ,
the density normalization is by the typical density at the coronal base, the velocity normalization is by
Alfvén speed , and the time normalization is by the Alfvén time defined as
, where is the typical lengthscale of the problem (for convenience, is used). The
pressure is normalized by , where is the Boltzmann constant, is the
typical number density, where is the proton mass, and is the average mass number
density of the coronal plasma, and is the typical temperature at the base of the corona. The
Froude number is , is the gravitational constant, is the solar
radius, is the solar mass, is the ratio of thermal to magnetic pressure,
where is the sound speed. The Lundquist number is the ratio of
the typical Alfvén time to the resistive diffusion time (, where is the
resistivity, and is the speed of light), and is the viscous force term (see Braginskii,
1965).

The heating and loss terms are , and , respectively, where is the coronal heating function
(assumed, or obtained empirically), and represents the losses due to thermal conduction
and radiation (for example, Landi and Landini, 1999; Colgan et al., 2008, for optically thin
plasma). The polytropic index in isothermal plasma, in a commonly used
polytropic model of the solar wind, and for solar wind models with explicit heating
terms. Variable was used by Cohen et al. (2007) to match solar wind properties at 1 AU.
The above set of equations is supplemented by the equation of state , and by the
solenoidality condition . The visco-resistive single fluid MHD model was used recently to
reproduce the global emission properties of the solar corona (Lionello et al., 2009; Downs et al.,
2010).

2.2 Multi-fluid models

Here, we describe the multi-fluid equations and model utilized by Ofman (2004a) to model
the fast solar wind in coronal holes accelerated by nonlinear MHD waves. Neglecting electron
inertia (), relativistic effects (), assuming quasi-neutrality (),
where is the charge number, the normalized three-fluid MHD equations can be written
as

where the index (in Equation (9) ), is the heat conduction term, is the
heating term of each fluid, is the energy coupling term between the various fluids (Li et al., 1997;
Ofman, 2004a), is the viscous force term due to ions, where the viscous stress
tensor and the Coulomb friction terms given by Braginskii (1965), is the
polytropic index of each species, is the mass number, and is the normalized
gyrofrequency.

The three-fluid equations are normalized by , where is the solar radius; ;
; ; ; . The following parameters enter in the above
equations: the Lundquist number; the electron and proton Euler number;
the ion Euler number; the Froude number; is the atomic mass of species
; , is the normalization temperature, is the mass of the particles,
is the normalization magnetic field. The heat conduction term in Equation (9) is normalized by
.

In Ofman (2004a) model the fast solar wind is produced by a broad band spectrum of waves. The
linearly polarized Alfvén waves are driven at the base of the corona as follows:

where , for the spectrum, the discrete frequencies are given by
, and the range is defined by , where is the number
of modes, is the random phase that depends on the solar latitude . Typically, the frequencies are
in the mHz range, the driving amplitude is few percent of , with an order of 100 modes used to
model the desired spectrum (see Figure 4). As described by Ofman (2004a) dissipation of the waves occurs
through viscous and resistive terms in the momentum and inductance equations, respectively. The
dissipation coefficients used in that model are hyper-viscosity and hyper-resistivity, i.e., their values are
much larger than the classical resistivity and viscosity, accounting empirically for kinetic and turbulent
effects.

Figure 4: The typical form of the driving spectrum of Alfvén waves used in the 3-fluid model to
drive the solar wind.

In addition to the waves, an empirical heating function is introduced in this model to heat the ions:

where is the amplitude of the heat input in normalized units, and is the length scale of
the heating in . This is necessary, since the Alfvén wave spectrum constrained by available
observations can not account for the observed (e.g., Kohl et al., 1997) preferential acceleration and heating
of heavy ions. The asymptotic solar wind parameters (speed of various ion species, mass flux, temperature,
etc.), can be matched by fitting the parameters of the heating function, combined with the parameters of
the driving Alfvén wave spectrum.

2.3 Hybrid models

In the hybrid model the ions are represented as particles, neglecting collisions, while the electrons are
described as a finite temperature massless fluid to maintain quasineutrality of the plasma. This method
allows one to resolve the ion dynamics and to integrate the equations over many ion-cyclotron periods,
while neglecting the small temporal- and spatial- scales of the electron kinetic motions. In the hybrid model,
the three components of order million particle velocities are used to calculate the currents, and the fields in
the 1D, 2D, or, in some cases, 3D grid. Note that each numerical particle represents large number of real
particles, determined by the density normalization. The required number of particles per cell is determined
by the required limitation on the overall statistical noise, and could be increased by an order of
magnitude as needed. The following equations of motion are solved for each particle of species (k):

where is the particle mass, is the charge number, is the electron charge, and is the
speed of light. The electron momentum equation is solved by neglecting the electron inertia

where is used for closure, and quasi-neutrality implies , where is the
number density of electron, protons, and ions, respectively. The above equations are supplemented with
Maxwell’s equations

where the displacement current is neglected in non-relativistic plasma, and

The field solutions are obtained on the 1D, 2D, or 3D grid, and the proton and ion equations of motions are
solved as the particle motions respond to the fields at each time step. The method has been tested and used
successfully in many studies.

In Ofman and Viñas (2007) 2D study 128 × 128 grid with 100 particles/cell/species were used. The
particle and field equations were integrated in time using a rational Runge–Kutta (RRK) method
(Wambecq, 1978) whereas the spatial derivatives were calculated by pseudospectral FFT method. When
non-periodic boundary conditions are applied finite difference method is used for the field solver.
The hybrid model allows computing the self-consistent evolution of the velocity distribution of
the ions that includes the nonlinear effects of wave–particle interactions without additional
assumptions. Moreover, the hybrid model is well suited to describe the nonlinear saturated state of the
plasma.

Since the hybrid models usually describe a region of several hundred ion inertial length
() across, the method is limited to model local small scale structures in the corona.
For a typical solar plasma density of 104 cm–3 at and a simulation box with a side
of we get about 1000 km for the extent of the simulated region in each dimension.
At 1 AU the plasma density is much lower and the modeled region covers about 45 000 km
in each dimension. A way to overcome the computational limitation to small scales is to use
an ‘expanding box’ model (e.g., Grappin and Velli, 1996; Liewer et al., 2001; Hellinger et al.,
2005). This approach employs transformation of variables to the moving solar wind frame that
expands together with the size of the parcel of plasma as it propagates outward from the Sun. In
particular, it is assumed that a small packet of plasma of length and width ,
where expands in the lateral direction only as it moves away from the Sun at
constant speed . The initial distance is . Thus, the modeled regions position is
, and the width . Using these transformations the coordinates
are transformed as , , and , and the equations of
motions together with the field equations are transformed to the moving and expanding frame.
Although, the method requires several severe simplifying assumptions (i.e., lateral expansion
only, constant solar wind speed) and approximations (the original spherical coordinates and
the mean magnetic field transformed to new coordinates using second order expansion (see
Liewer et al., 2001) to remain tractable, it provides qualitatively good description of the solar
wind expansions, thus connecting the disparate scales of the plasma in the various parts of the
heliosphere.