2 Model Equations

Below we provide the typical basic equations used in the three classes of models reviewed here: MHD, multi-fluid, and hybrid.

2.1 Single fluid MHD

The single-fluid, normalized visco-resistive MHD equations with gravity are (see, e.g., Priest, 1982Jump To The Next Citation Point; Ofman, 2005)

∂ρ- ∂t + ∇ ⋅ (ρv) = 0, (1 ) ∂v β 1 J × B ---+ (v ⋅ ∇ )v = − --∇p − ---2-+ ------+ Fv, (2 ) ∂t 2ρ Frr ρ ∂B--= ∇ × (v × B ) + S −1∇2B, (3 ) ∂t ( ∂ ) p ---+ v ⋅ ∇ --γ = (γ − 1)(Sh + Sl), (4 ) ∂t ρ
where ρ is the fluid density, v is the fluid velocity, B is the magnetic field, J = ∇ × B is the current density, and p is the plasma pressure. The normalization of the magnetic field is by the typical magnetic field B0 at the base of the corona, distances (r) are normalization by the solar radius R ⊙, the density normalization is by the typical density ρ0 at the coronal base, the velocity normalization is by Alfvén speed VA = B0∕(4π ρ0)1∕2, and the time normalization is by the Alfvén time defined as τA = L∕VA, where L is the typical lengthscale of the problem (for convenience, L = R⊙ is used). The pressure is normalized by p0 = kBn0T0, where kB is the Boltzmann constant, n0 = ρ∕(μmp ) is the typical number density, where mp is the proton mass, and μ is the average mass number density of the coronal plasma, and T 0 is the typical temperature at the base of the corona. The Froude number is 2 Fr = VAR ⊙∕(GM ⊙ ), G is the gravitational constant, R ⊙ is the solar radius, M ⊙ is the solar mass, β = 2c2s∕(γV 2A) is the ratio of thermal to magnetic pressure, where cs = (γp ∕ρ)1∕2 is the sound speed. The Lundquist number S = τA∕ τr is the ratio of the typical Alfvén time to the resistive diffusion time (τr = 4πL2 ∕νc2, where ν is the resistivity, and c is the speed of light), and Fv is the viscous force term (see Braginskii, 1965Jump To The Next Citation Point).

The heating and loss terms are Sh, and Sl, respectively, where Sh is the coronal heating function (assumed, or obtained empirically), and Sl 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 γ = 1 in isothermal plasma, γ = 1.05 in a commonly used polytropic model of the solar wind, and γ = 5∕3 for solar wind models with explicit heating terms. Variable γ 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 p = nkBT, and by the solenoidality condition ∇ ⋅ B = 0. 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 (2004aJump To The Next Citation Point) to model the fast solar wind in coronal holes accelerated by nonlinear MHD waves. Neglecting electron inertia (me ≪ mp), relativistic effects (V ≪ c), assuming quasi-neutrality (ne = np + Zni), where Z is the charge number, the normalized three-fluid MHD equations can be written as

∂nk- ∂t + ∇ ⋅ (nkVk ) = 0, (5 ) [ ] nk ∂Vk--+ (Vk ⋅ ∇ )Vk = − Euk ∇pk − Eue-Zknk ∇pe − -nk--er ∂t Akne Frr2 +Ωknk (Vk − Ve ) × B + Fv + nkFk,coul, (6 ) ∂B--= ∇ × (Ve × B ) − 1∇ × ∇ × B, (7 ) ∂t S -1- Ve = ne (npVp + ZiniVi − b∇ × B), (8 ) ∂Tk- = − (γk − 1)Tk∇ ⋅ Vk − Vk ⋅ ∇Tk + Ckjl + (γk − 1)(Hk ∕nk + Sk ), (9 ) ∂t
where the index k = p, i (in Equation (9View Equation) k = e,p,i), Hk is the heat conduction term, Sk is the heating term of each fluid, C kjl is the energy coupling term between the various fluids (Li et al., 1997; Ofman, 2004aJump To The Next Citation Point), Fv = ∇ ⋅ Π is the viscous force term due to ions, where Π the viscous stress tensor and the Coulomb friction terms Fk,coul given by Braginskii (1965), γk = 5∕3 is the polytropic index of each species, Ak is the mass number, and Ωk = ZAkkemBp0cτA is the normalized gyrofrequency.

The three-fluid equations are normalized by r → r∕R ⊙, where R ⊙ is the solar radius; t → t∕τA; V → V∕VA; B → B ∕B0; nk → nk∕ne0; Tk → Tk∕Tk0. The following parameters enter in the above equations: S the Lundquist number; Eue,p = (kBTe,p,0∕mp )∕V 2A the electron and proton Euler number; Eui = (kBTi,0∕mi )∕V 2 A the ion Euler number; Fr the Froude number; Ak is the atomic mass of species k; b = cB0 ∕(4πene0R ⊙VA ), Tk,0 is the normalization temperature, mk is the mass of the particles, B0 is the normalization magnetic field. The heat conduction term in Equation (9View Equation) is normalized by Hk → Hk(kBVAR ⊙ ∕T2k.05 ).

In Ofman (2004aJump To The Next Citation Point) 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:

B ϕ(t,𝜃,r = 1) = − Vd∕VA,rF (t,𝜃) (10 ) N∑ F (t,𝜃) = aisin(ωit + Γ i(𝜃)) (11 ) i=1
where ai = ip∕2, p = − 1 for the f− 1 spectrum, the discrete frequencies are given by ω = ω + (i − 1)Δ ω i 1, and the range is defined by Δ ω = (ω − ω )∕(N − 1) N 1, where N is the number of modes, Γ i(𝜃) is the random phase that depends on the solar latitude 𝜃. Typically, the frequencies are in the mHz range, the driving amplitude Vd is few percent of VA, with an order of 100 modes used to model the desired spectrum (see Figure 4View Image). As described by Ofman (2004aJump To The Next Citation Point) 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.
View Image

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:

S = S0,k(r − 1)e−r∕λk , (12 ) k nk
where S0,k ≡ s0,knk is the amplitude of the heat input in normalized units, and λk is the length scale of the heating in R⊙. This is necessary, since the Alfvén wave spectrum constrained by available observations can not account for the observed (e.g., Kohl et al., 1997Jump To The Next Citation Point) 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):

dxk ---- = vk, (13 ) dt ( ) m dvk- = Ze E + vk-×-B- , (14 ) k dt c
where mk is the particle mass, Z is the charge number, e is the electron charge, and c is the speed of light. The electron momentum equation is solved by neglecting the electron inertia
( ) ∂-- ve-×-B- ∂tnemeve = 0 = − ene E + c − ∇pe , (15 )
where pe = kBneTe is used for closure, and quasi-neutrality implies ne = np + Zni, where nk is the number density of electron, protons, and ions, respectively. The above equations are supplemented with Maxwell’s equations
4π- ∇ × B = c J , (16 )
where the displacement current is neglected in non-relativistic plasma, and
1 ∂B ∇ × E = − -----. (17 ) c ∂t
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 (2007Jump To The Next Citation Point) 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 (li = c∕ ωpi) 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 10 R⊙ and a simulation box with a side of 440 li 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, 1996Jump To The Next Citation Point; Liewer et al., 2001Jump To The Next Citation Point; 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 δr ≪ R0 and width a (t), where a(t)∕R ≪ 1 0 expands in the lateral direction only as it moves away from the Sun at constant speed U0. The initial distance R0 is O (R ⊙). Thus, the modeled regions position is R (t) = R⊙ + U0t, and the width a(t) = R(t)∕R0. Using these transformations the coordinates are transformed as x ′ = x − R (t), y′ = y∕a(t), and z′ = z∕a(t), 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., 2001Jump To The Next Citation Point) 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.

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