Two-fluid numerical simulations of the origin of the fast solar wind

TL;DR

This study uses advanced 2.5D simulations with the JOANNA code to model plasma outflows from the solar surface, providing insights into the origin of the fast solar wind by matching observed mass and energy loss rates.

Contribution

The paper introduces realistic 2.5D numerical simulations of plasma outflows using the JOANNA code, linking solar granulation dynamics to the fast solar wind origin.

Findings

Simulated outflows match observed mass and energy loss rates.

Outflows originate from the upper chromosphere, transition region, and inner corona.

Results support the hypothesis that solar granulation drives the fast solar wind.

Abstract

With the use of our JOANNA code, which solves radiative equations for ion + electron and neutral fluids, we perform realistic 2.5D numerical simulations of plasma outflows associated with the solar granulation. These outflows exhibit physical quantities consistent to the order of magnitude with the observational findings for mass and energy losses in the upper chromosphere, transition region and inner corona, and they may originate the fast solar wind.

Equations

$$\frac{\partial {\varrho }_{\mathrm{n}}}{\partial t}+\nabla \cdot ({\varrho }_{\mathrm{n}}{\mathbf{V}}_{\mathrm{n}})=0,$$

$$\frac{\partial {\varrho }_{\mathrm{i}}}{\partial t}+\nabla \cdot ({\varrho }_{\mathrm{i}}{\mathbf{V}}_{\mathrm{i}})=0,$$

$$\begin{array}{r}\frac{\partial ({\varrho }_{\mathrm{n}}{\mathbf{V}}_{\mathrm{n}})}{\partial t}+\nabla \cdot ({\varrho }_{\mathrm{n}}{\mathbf{V}}_{\mathrm{n}}{\mathbf{V}}_{\mathrm{n}}+p_{\mathrm{n}}\mathbf{I})=\\ {\alpha }_c({\mathbf{V}}_{\mathrm{i}}-{\mathbf{V}}_{\mathrm{n}})+{\varrho }_{\mathrm{n}}\mathbf{g},\end{array}$$

$$\begin{array}{r}\frac{\partial ({\varrho }_{\mathrm{i}}{\mathbf{V}}_{\mathrm{i}})}{\partial t}+\nabla \cdot ({\varrho }_{\mathrm{i}}{\mathbf{V}}_{\mathrm{i}}{\mathbf{V}}_{\mathrm{i}}+p_{\mathrm{ie}}\mathbf{I})=\\ \frac{1}{\mu }(\nabla \times \mathbf{B})\times \mathbf{B}+{\alpha }_c({\mathbf{V}}_{\mathrm{n}}-{\mathbf{V}}_{\mathrm{i}})+{\varrho }_{\mathrm{i}}\mathbf{g},\end{array}$$

$$\frac{\partial \mathbf{B}}{\partial t}=\nabla \times ({\mathbf{V}}_{\mathrm{i}}\times \mathbf{B}),\nabla \cdot \mathbf{B}=0,$$

$$\begin{array}{r}\frac{\partial E_{\mathrm{n}}}{\partial t}+\nabla \cdot [(E_{\mathrm{n}}+p_{\mathrm{n}}){\mathbf{V}}_{\mathrm{n}}]={\alpha }_c{\mathbf{V}}_{\mathrm{n}}({\mathbf{V}}_{\mathrm{i}}-{\mathbf{V}}_{\mathrm{n}})\\ +Q_{\mathrm{n}}^{\mathrm{in}}+q_{\mathrm{n}}+{\varrho }_{\mathrm{n}}\mathbf{g}\cdot {\mathbf{V}}_{\mathrm{n}},\end{array}$$

$$\begin{array}{r}\frac{\partial E_{\mathrm{i}}}{\partial t}+\nabla \cdot \left[\left(E_{\mathrm{i}}+p_{\mathrm{ie}}+\frac{{\mathbf{B}}^2}{2\mu }\right){\mathbf{V}}_{\mathrm{i}}-\mathbf{B}(\mathbf{V}\cdot \mathbf{B})\right]=\\ {\alpha }_c{\mathbf{V}}_{\mathrm{i}}({\mathbf{V}}_{\mathrm{n}}-{\mathbf{V}}_{\mathrm{i}})+Q_{\mathrm{i}}^{\mathrm{in}}+Q_R^i+q_{\mathrm{i}}+{\varrho }_{\mathrm{i}}\mathbf{g}\cdot {\mathbf{V}}_{\mathrm{i}},\end{array}$$

$$Q_{\mathrm{n}}^{\mathrm{in}}={\alpha }_c(\Delta \tilde{V}+\Delta \tilde{T}),$$

$$Q_{\mathrm{i}}^{\mathrm{in}}={\alpha }_c(\Delta \tilde{V}-\Delta \tilde{T})$$

$$\Delta \tilde{V}=\frac{1}{2}|{\mathbf{V}}_{\mathrm{i}}-{\mathbf{V}}_{\mathrm{n}}{|}^2,$$

$$\Delta \tilde{T}=\frac{3k_{\mathrm{B}}}{m_{\mathrm{H}}({\mu }_{\mathrm{i}}+{\mu }_{\mathrm{n}})}\left(T_{\mathrm{i}}-T_{\mathrm{n}}\right),$$

$$E_{\mathrm{n}}=\frac{p_{\mathrm{n}}}{\gamma -1}+\frac{{\varrho }_{\mathrm{n}}{\mathbf{V}}_{\mathrm{n}}^2}{2},$$

$$E_{\mathrm{i}}=\frac{p_{\mathrm{ie}}}{\gamma -1}+\frac{{\varrho }_{\mathrm{i}}{\mathbf{V}}_{\mathrm{i}}^2}{2}+\frac{{\mathbf{B}}^2}{2\mu }.$$

$$p_{\mathrm{n}}=\frac{k_{\mathrm{B}}}{m_{\mathrm{H}}{\mu }_{\mathrm{n}}}{\varrho }_{\mathrm{n}}{T}_{\mathrm{n}},p_{\mathrm{ie}}=\frac{2k_{\mathrm{B}}}{m_{\mathrm{H}}{\mu }_{\mathrm{i}}}{\varrho }_{\mathrm{i}}{T}_{\mathrm{i}}.$$

Full text

Sun: activity - Sun: corona - Sun: solar wind - magnetohydrodynamics (MHD) - methods: numerical

††journal: ApJ

1 Introduction

The solar wind is a stream of energized, charged particles, primarily electrons and protons from hydrogen, along with atomic nuclei like helium, alpha particles, flowing outward from the Sun (Parker, 1965; Bierman, 1951). There is a fast, widely uniform wind, emanating from polar coronal holes and at the distance of 1 AU from the Sun traveling at about 750 km s*-1*, and a slow, sporadic one, pouring from active equatorial regions and moving with about half smaller speed. The magnetic field lines stretch out radially in coronal holes, and do not loop directly back to the Sun, providing an open path for the fast plasma to escape the gravity grasp. As the corona expands, these winds must be replaced by plasma moving up from below to feed them.

In the modern era of high resolution space-born and ground-based observations, special attention has been paid for studying the origin of plasma outflows which form the solar wind. The early models of the wind assumed the inner corona as its origin (e.g. Tu, 1987). Recently, Tu (2005) detected such outflows in coronal funnels at altitudes between 5 Mm and 20 Mm above the photosphere, and found that they reach the speed of up to 10 km s*-1* at the height of 20 Mm. The emphasis was also given on searching for the outflows in the chromosphere/transition region (e.g. Marsch et al., 2008; McIntosh, 2012; Yang et al., 2013; Kayshap et al., 2015). It was found that these outflows can be generated by variety of jets (Wedemeyer-Böhm et al., 2012; Kayshap et al., 2013; Tian et al., 2014; Martínez-Sykora et al., 2017), and injection of often twisted magnetic field and its subsequent reconnection may also contribute to the bulk plasma outflows in coronal holes (e.g. Krieger et al., 1973; Zirker, 1977; Yang et al., 2013; Kayshap et al., 2013).

It was showed theoretically that magnetohydrodynamic waves may be responsible for providing the momentum to the upwardly moving plasma (e.g. Ofman, 2005; Suzuki & Inutsuka, 2005; Srivastava & Dwivedi, 2006; Marsch, 2006; Arber et al., 2016). Among others, Hollweg (1986), Kudoh & Shibata (1999), Matsumoto & Suzuki (2012) concluded that Alfvén waves possess a potential to drive plasma outflows. He et al. (2008) developed a model of plasma outflows in the coronal funnels, which includes Alfvén waves (Ofman et al., 1995). Along this line of investigation, Yang et al. (2016) found that Alfvén waves are able to form fast plasma outflows. See also Ofman et al. (1995) and Shestov et al. (2017) for a similar analysis. However, Alfvén waves are difficult for detection, particularly when high-frequency waves are concerned (Srivastava et al., 2017); in an inhomogeneous and structured medium, these waves can experience reflection, mode coupling, phase-mixing and resonant absorption (Ofman et al., 1995; Nakariakov et al., 1997; Zaqarashvili & Roberts, 2006; Goossens et al., 2012).

Despite the above mentioned achievements, the origin of the solar wind still remains one of the central issues of heliophysics. We investigate here the role of granulation in generation of chromospheric ejecta and associated plasma outflows in coronal holes. We are motivated by the fact that a base of the corona is filled with dynamic jets propelled from below the transition region upwards at speeds of about 25 km s*-1* into higher layers, and carry a significant amount of momentum (e.g. Sterling, 2000; Zaqarashvili & Erdélyi, 2009).

This paper is organized as follows. A physical model is presented in Sect. 2 and the corresponding numerical results are shown in Sect. 3. Our paper is concluded by discussion and summary of the numerical results in Sect. 4.

2 Two-fluid model of a partially-ionized coronal hole

We consider a solar coronal hole that is magnetically structured and gravitationally stratified, and its dynamics is described by 2-fluid equations for ions + electrons treated as one fluid and neutrals regarded as second fluid. These equations can be written as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the heat production terms are

[TABLE]

[TABLE]

with

[TABLE]

With the use of our JOANNA code, which solvesradiativeequations for ion + electron and neutralfluids, we perform realistic 2.5D numerical simulations of plasma outflows associated with the solargranulation. These outflows exhibit physical quantities consistentto the order of magnitude with theobservational findings for mass and energy losses in the upper chromosphere, transition region andinner corona, and they may originate the fast solar wind.

[TABLE]

and the energy densities are given by

[TABLE]

[TABLE]

Here subscripts i, n and e correspond to ions, neutrals and electrons, respectively. The symbols $\varrho_{\rm i,n}$ denote mass densities, ${\bf V}_{\rm i,n}$ velocities, $p_{\rm ie,n}$ ion+electron and neutral gas pressures, ${\bf B}$ is magnetic field and $T_{\rm i,n}$ are temperatures specified by ideal gas laws,

[TABLE]

A gravity vector is ${\bf g}=[0,-g,0]$ with its magnitude $g=274.78$ m s*-2*, $\alpha_{\rm c}$ is the coefficient of collisions between ion and neutral particles (e.g. Oliver et al., 2016; Ballester et al., 2018, and references cited therein), $Q^{\rm i}_{R}$ radiative losses term which is implemented here in the framework of Abbett & Fisher (2012) in the low atmospheric regions and of thin radiation (Moore & Fung, 1972) in the top atmospheric layers, $q_{\rm i,n}$ are thermal conduction terms (Spitzer, 1962), $\mu_{\rm i}=0.29$ and $\mu_{\rm n}=1.21$ are the mean masses of respectively ions and neutrals, which are taken from the OPAL solar abundance model (e.g. Vögler et al., 2004), $m_{\rm H}$ is the hydrogen mass, $k_{\rm B}$ is the Boltzmann constant, $\gamma=1.4$ is the specific heats ratio, and $\mu$ is magnetic permeability of the medium. The other symbols have their standard meaning.

We consider the case of $z-$invariant system and start our simulations at $t=0$ s with the hydrostatic equilibrium being supplemented by transversal and vertical magnetic field given as $\mathbf{B}=[B_{x},B_{y},B_{z}]=[0,B_{0},B_{0}]$, where $B_{0}=5/\sqrt{2}$ Gs. The transversal component, $B_{z}$ results in Alfvén waves being linearly coupled to magnetoacoustic waves. The presence of Alfvén waves is essential in the model, as in the nonlinear regime they are capable of driving vertical flow (e.g. Hollweg, 1986; Murawski, 1992; Shestov et al., 2017).

In the framework of the implemented magnetic field model we set at $t=0$ s identical hydrostatic temperature for ions and neutrals, $T_{\rm i}(y)=T_{\rm n}(y)=T(y)$ (Oliver et al., 2016; Martínez-Gómez et al., 2016, 2017; Soler et al., 2017; Srivastava et al., 2018). This temperature is determined by the semi-empirical model of Avrett & Loeser (2008) that is extrapolated into the corona.

The hydrostatic equilibrium is restructured in time by the solar granulation. This granulation appears naturally in the convection zone which is convectively unstable. First signs of granulation are seen already after about $5$ min from the start of the simulations with a fully developed state occurring after about 3000 s of the solar time.

3 Numerical simulations of 2-fluid plasma outflows

To solve 2-fluid equations numerically, we use JOANNA code (Wójcik et al., 2018). We set in our numerical experiments the Courant-Friedrichs-Lewy number equal to $0.9$ and choose a second-order accuracy in space and a four stage, third-order strong stability preserving Runge-Kutta method (Durran, 2010) for integration in time, supplemented by adopting the Harten-Lax-van Leer Discontinuities (HLLD) approximate Riemann solver (Miyoshi & Kusano, 2005) and Global Lagrange Multiplier (GLM) method of Dedner et al. (2002). The simulation box extends from the convection zone (2.56 Mm below the bottom of the photosphere) to the corona (up to 30 Mm above the photosphere) in $y-$direction and horizontally from $x=-2.56$ Mm to $x=2.56$ Mm. This box is divided into several patches. The bottom region, specified by $-2.56$ Mm $\leq y\leq 7.68$ Mm, is covered by the $1024\times 1024$ identical cells, leading to the spatial resolution of $10$ km. Within the layer of $7.68$ Mm $\leq y\leq 30$ Mm, we implement several patches of progressively larger cells along $y-$direction. At the left- and right- sides of the simulation box we set periodic boundary conditions, while the top and bottom ghost cells are filled by plasma quantities equal to their equilibrium values. The layer for the optical depth greater than 10 is additionally heated by implementing extra source term in the energy equation of ions that balances the energy losses there.

Figure 1 (colormap) shows typical spatial profiles of $\log\varrho_{\rm i}$ (top) and vertical component of ion velocity (bottom). The granulation excites a wide range of waves deep in the photosphere. Some of these waves steepen into shocks while propagating upwards. This steepening results from wave amplitude growth with height, and chromospheric jets are excited (top). The plasma above the apices of these jets moves upwards reaching its maximum speed of around $100$ km s*-1* (bottom). The outflowing plasma essentially follows open magnetic field lines (black lines) of magnetic funnels that are formed by the granulation. The footpoints of these funnels are rooted deep in the photosphere between granules, while higher up, above the transition region, the magnetic field lines remain essentially vertical. The plasma outflows form strand-like structures along magnetic field lines in the corona, with subsiding plasma taking place at lower altitudes.

We have run the code with extra non-adiabatic terms such as thermal conduction and magnetic diffusion included along with radiation. However, due to the computational effort we have not yet obtained satisfactory results. We have also run the code with the spatial resolution of 20 km x 20 km and the results are showed in Fig. 2. It follows from them that the results are close to these shown in Fig. 1 (bottom), which confirms that the chosen spatial resolution of 10 km x 10 km is sufficient to resolve plasma outflows.

Figure 3 displays time-distance plot of vertical component of ion velocity that is averaged over the whole horizontal distance, $<V_{{\rm i}\,y}>$. The plasma jets emerge from the chromospheric background and move into the corona. Some of the injected plasma subsides rapidly after reaching its maximum phase (e.g. Kuźma et al., 2017; Srivastava et al., 2018). This entire process is driven by ongoing granulation in the photosphere. Analyzing the $<V_{{\rm i}\,y}(y,t)>$ we find that the solar corona experiences about $1-3$ minute periods oscillations and $<V_{{\rm i}\,y}>$ reaches a magnitude of $10-20$ km s*-1* at $y=8$ Mm and it grows with height. The physical properties of these outflows are aikin to the flow characteristics reported by Tu (2005).

Figure 4 (top) illustrates time-distance plot of the horizontally averaged total vertical ion mass flux, $F_{m}(y,t)=<\varrho_{\rm i}V_{{\rm i}\,y}>$, which attains its maximum in the photosphere and lower chromosphere and falls off with height due to rapidly decreasing ion mass density. However, even above the transition region the estimated magnitude of this mass flux lays within the range of $10^{-6}$ and $10^{-5}$ g cm*-2* s*-1* and it matches the prediction for solar mass losses in the low corona (Withbroe & Noyes, 1977).

Vertical component of ion energy flux transported through the medium can be calculated as $F_{\rm E}(x,y,t)=\varrho_{\rm i}\mathbf{V}_{\rm i}^{2}V_{\rm iy}/2$. The time-distance plot of horizontally averaged vertical energy flux, $<F_{\rm E}(y,t)>$, is shown in Fig. 4 (bottom). We see that plasma escaping into the corona above the transition region carries a significant amount of energy (orange and yellow patches). By comparison with time-distance plots of $<V_{{\rm i}\,y}>$ (Fig. 3) we infer that the energy flux associated with the upflowing plasma is higher than for the descending plasma. Note that the obtained values lay within the range of theoretical findings for energy losses in the upper chromosphere, transition region and low corona (Withbroe & Noyes, 1977).

Figure 5 shows vertical variation of the temporarily and horizontally averaged mass, $<F_{\rm m}>$, (top) and energy, $<F_{\rm E}(y)>$, (bottom) fluxes. Note that $<F_{\rm E}(y)>$ grows abruptly within the region of $0<y<0.5$ Mm, where the dense photospheric plasma experiences a push from the below operating granulation. Higher up, that is for $y>0.5$ Mm, $<F_{\rm m}(y)>$ declines with height attaining a value of $10^{-6}$ g cm*-2* s*-1* at $y=10$ Mm. On the other hand, $<F_{\rm E}(y)>$ remains close to $10^{7}$ erg cm*-2* s*-1* in the entire chromosphere and transition region, attaining its local maxima at $y=0.25$ Mm and $y=3.5$ Mm. Higher up, it slowly falls off with height, reaching its value of $5\cdot 10^{5}$ erg cm*-2* s*-1* at $y=10$ Mm.

4 Discussion and summary

Within the framework of 2-fluid equations for ion-neutral plasma, we performed numerical simulations of the origin of the solar wind which results from plasma outflows. These outflows are associated with jets excited by the solar granulation which develops in the medium with initially straight magnetic field overlaying a hydrostatic equilibrium. This configuration well mimics the expanding open magnetic field region in a polar coronal hole. Our simulations show that this configuration is later on restructured by granulation which operates in the photosphere. The whole scenario is associated with the energy and mass leakage into higher atmospheric layers in the form of plasma outflows. Our results successfully match the expected values of mass and energy losses in the upper chromosphere, transition region and low corona (Withbroe & Noyes, 1977).

It is noteworthy here that Tu (2005) obtained a correlation of the Doppler-velocity and radiance maps of spectral lines emitted by various ions (Ne VIII, C IV, Si II) with the force-free magnetic field that was extrapolated from the photospheric magnetogram (SOHO/MDI) in a polar coronal hole. Tu found that Ne VIII ions mostly radiate around the height of 20 Mm above the photosphere, where they reveal the outflow speed of about 10 km s*-1*, while C IV ions with no average flow speed form essentially around the altitude of 5 Mm. Hence, Tu inferred that the plasma outflows start in the coronal funnels at altitudes in between 5-20 Mm. Yang et al. (2013) proposed that magnetic reconnection, which took place in the open and closed magnetic field region, triggers the plasma outflows observed by Tu (2005). The results of our simulations performed with a novel 2-fluid model of a partially-ionized solar atmospheric plasma confirm these observational findings.

Ten years before the plasma outflows were announced by Tu (2005), there was essentially no report on finely-structured jets. The exceptions were spicules/macrospicules diversely filling the chromosphere and contributing to the mass cycle of the corona (e.g. Tian et al., 2014; Wedemeyer-Böhm et al., 2012). In the limit of current observational resolution, it is established that the overlaying plasma outflows in the corona must be originated due to the contribution from various plasma ejecta. Therefore, without emphasizing on a particular type of a jet, we simulated the solar granulation which resulted in jets and studied their contribution to formation of the solar wind.

In summary, we investigated formation of plasma outflows between $5$ to $10$ Mm above the photosphere in the open magnetic field region in a coronal hole as observed by Tu (2005). The outflows in such regions consist of continuous streaming of plasma particles from the lower solar atmosphere outward. We point out its linkage to the granulation and associated with them the ubiquitous chromospheric jets which lead to mass and energy leakage into the inner corona. Our model is based on gravitationally stratified and partially-ionized bottom layers of the solar atmosphere with adequate temperature and magnetic field conditions to mimic the ion-neutral plasma outflows. Our studies determine that multiple jets excited by operating in the photosphere granulation are able to stimulate continuous plasma outflows in the solar atmosphere which may result in the fast solar wind at higher altitudes in the solar corona.

The authors express their thanks to Drs. Ramon Oliver, Roberto Soler and David Martinéz-Gómez for their comment on the hydrostatic model of the solar atmosphere. We would like to thank the reviewer for their time and valuable remarks. The JOANNA code has been developed by Darek Wójcik. This work was done within the framework of the projects from the Polish National Foundation (NCN) Grant nos. 2017/25/B/ST9/00506 and 2017/27/N/ST9/01798. Numerical simulations were performed on the LUNAR cluster at Institute of Mathematics of University of M. Curie-Skłodowska, Lublin, Poland.