# Modeling solar wind variations over an 11 year cycle with Alfvén Wave dissipation: A parameter study

Soumitra Hazra, Victor Réville, Barbara Perri, Antoine Strugarek, Allan

Sacha Brun, Eric Buchlin

► To cite this version:

Soumitra Hazra, Victor Réville, Barbara Perri, Antoine Strugarek, Allan Sacha Brun, et al. Modeling solar wind variations over an 11 year cycle with Alfvén Wave dissipation: A parameter study. The Astrophysical Journal, 2021, 910 (2), pp.90. 10.3847/1538-4357/abe12e. hal-03381403

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# Modeling Solar Wind Variations over an 11 Year Cycle with Alfvén Wave Dissipation: A Parameter Study

Soumitra Hazra<fnref n="1" />,<fnref n="2" />✉, Victor Réville<fnref n="3" />✉, Barbara Perri<fnref n="1" />,<fnref n="2" />✉, Antoine Strugarek<fnref n="1" />✉, Allan Sacha Brun<fnref n="1" />✉, and Eric Buchlin<fnref n="2" />✉

Received 2020 December 19; revised 2021 January 25; accepted 2021 January 26; published 2021 March 30

## Abstract

We study the behavior and properties of the solar wind using a 2.5D Alfvén wave (AW)-driven wind model. We first systematically compare the results of an AW-driven wind model with a polytropic approach. Polytropic magnetohydrodynamic wind models are thermally driven, while AWs act as additional acceleration and heating mechanisms in the AW-driven model. We confirm that an AW-driven model is required to reproduce the observed bimodality of slow and fast solar winds. We are also able to reproduce the observed anticorrelation between the terminal wind velocity and the coronal source temperature with the AW-driven wind model. We also show that the wind properties along an 11 yr cycle differ significantly from one model to the other. The AW-driven model again shows the best agreement with observational data. Indeed, solar surface magnetic field topology plays an important role in the AW-driven wind model, as it enters directly into the input energy sources via the Poynting flux. On the other hand, the polytropic wind model is driven by an assumed pressure gradient; thus, it is relatively less sensitive to the surface magnetic field topology. Finally, we note that the net torque spinning down the Sun exhibits the same trends in the two models, showing that the polytropic approach still correctly captures the essence of stellar winds.

Unified Astronomy Thesaurus concepts: Solar cycle (1487); Solar wind (1534); The Sun (1693); Sunspots (1653); Alfvén waves (23); Magnetohydrodynamics (1964)

## 1. Introduction

It is now assumed that the large-scale magnetic field of most solar-like stars is created by large-scale convective flows acting inside the stellar interior (Brun et al. 2015; Hazra & Nandy 2016). The solar large-scale magnetic field varies cyclically with an average periodicity of 11 yr; this is known as the solar cycle. When the Sun is at the maximum of the solar cycle, there are large number of solar storms (flares and coronal mass ejections) that affect the space and ground-based vulnerable infrastructures. The solar magnetic field is multi-polar and complex at the cycle maximum, while it is dipolar at the cycle minimum. This large-scale magnetic field is responsible for the structure of the solar or stellar corona. Parker (1958) argued that it is not possible to have a static equilibrium corona due to the high temperature of the solar corona and low pressure of the interstellar medium. A supersonic expansion driven by the pressure gradient (known as solar wind) is then the only possible steady state for the outer solar atmosphere. Later, standard magnetohydrodynamic (MHD) solar and stellar wind models were developed considering the effect of both rotation and the magnetic field, also known as the theory of magnetic rotators (Schatzman 1962; Parker 1963; Weber & Davis 1967; Mestel 1968). As the solar wind is accelerated outward from the solar surface, it carries information regarding the complex topology of the solar magnetic field (see for instance Wang & Sheeley 2003; Pinto et al. 2011; Réville et al. 2015a). The solar wind indeed exhibits significant variation in speed and density and has a spiral shape due to 28 days of solar rotation. There is also a large variation in the density and speed of the interplanetary medium during solar storms. The solar wind is important from the point of view of the Sun's evolution as it carries the angular

momentum (AM) away from the Sun (Weber & Davis 1967; Schatzman 1959, 1962). Because of the magnetized nature of the solar wind, it also exerts a magnetic torque that slows down the Sun (Skumanich 1972). In summary, it is necessary to have a clear understanding of the behavior and structure of the solar wind for perspective on space weather and stellar evolution.

Various in situ spacecraft have also indicated that the structure and speed of the solar wind are largely controlled by the solar magnetic activity. Ulysses' first orbit observations showed that near the solar minimum, the solar wind has a relatively simple bimodal structure in the outer heliosphere. Fast solar wind emanates from the high-latitude polar coronal holes, while slower, high-density solar wind is found at low latitudes (McComas et al. 1998, 2000). Ulysses' second orbit, which occurred over the maximum of cycle 23, found a complex magnetic structure of the solar wind at all latitudes (McComas et al. 2002a, 2002b). The complex solar wind structure at the solar maximum is primarily due to the mixture of different flows arising from different sources, such as streamers, coronal holes, active regions, etc., at various latitudes (McComas et al. 2002a, 2008; Neugebauer et al. 2002). However, the reason behind the behavior of the solar wind plasma in the solar corona is still unclear.

Different studies have been performed to explain Ulysses' observations by solving MHD equations in two and three dimensions with the observed photospheric vector magneto-gram as a boundary condition (Keppens & Goedbloed 1999; Usmanov et al. 2000; Usmanov & Goldstein 2003; Pinto et al. 2011; van der Holst et al. 2010; Tóth et al. 2012; Riley et al. 2015; Réville & Brun 2017). Many of these studies assumed that the solar wind is driven by the pressure gradient from an approximately $10^6$ K corona, which is modeled through a polytropic equation of state (Washimi & Shibata 1993;

<footnote n="1">¹Département d'Astrophysique/AIM, CEA/IRFU, CNRS/INSU, Université de Paris-Saclay, Université de Paris, CEA Paris-Saclay, F-91191 Gif-sur-Yvette, France; soumitra.hazra@cea.fr, soumitra.hazra@gmail.com, barbara.perri@cea.fr, antoine.strugarek@cea.fr, sacha.brun@cea.fr</footnote>

<footnote n="2">²Université Paris-Saclay, CNRS, Institut d'astrophysique spatiale, F-91405, Orsay, France; eric.buchlin@ias.u-psud.fr</footnote>

<footnote n="3">³IRAP, Université Toulouse III—Paul Sabatier, CNRS, CNES, Toulouse, France; victor.reville@irap.omp.eu</footnote>
Keppens & Goedbloed 1999; Matt et al. 2012; Réville et al. 2015a, 2015b). These polytropic MHD models are also successful in explaining some of the observational properties of the solar wind (Pinto et al. 2011; Matt et al. 2015; Réville et al. 2016; Réville & Brun 2017; Perri et al. 2018; Finley & Matt 2018; Finley et al. 2018). However, it has been shown that coronal density has to be sufficiently low to match the observed interplanetary density (Hundhausen 1972; Leer et al. 1982; Stewart & Bravo 1997). If one assumes the observed coronal density, a polytropic wind will lead to an interplanetary density too high compared to observations (see Hundhausen 1972). In summary, we need an additional source of energy and momentum to produce a relatively low-density fast wind while keeping the coronal density consistent with observations (Munro & Jackson 1977; Barnes et al. 1995). Riley et al. (2001) used an ad hoc mechanism to incorporate an additional source of acceleration in their solar wind model and found good agreement with Ulysses' observations.

Alfvén waves (AWs) have recently been proposed to be one of the best mechanisms for additional acceleration of the solar wind plasma (Belcher 1971; Jacques 1977). While the AW pressure acts as a mechanism for the solar wind acceleration, the dissipation of waves can be responsible for heating (Hollweg 1986). The Helios 1 and 2 missions have given us detailed observations regarding the behavior of the velocity and magnetic field fluctuations ($\delta v$ and $\delta B$, respectively) in the solar system. Peculiar radial evolution of the $\delta v$ and $\delta b$ at the time of maximum cross helicity can be explained by the properties of outward propagating AWs (Tu & Marsch 1993, 1995). Cross helicity basically describes the cross-correlation between the fluid velocity fluctuations ($\delta v$) and magnetic field fluctuations ($\delta b$) (Yokoi 2011). Observations from these missions indicate that fluctuations of the velocity and magnetic field have a clear radial evolution and they are highly correlated (Marsch & Tu 1990; Tu & Marsch 1993). Both their correlation and their radial evolution can be explained by the properties of outward propagating AWs (Belcher & Davis 1971). High-resolution observations have also found the signature of AWs in the solar atmosphere (De Pontieu et al. 2007; Tomczyk et al. 2007).

One can characterize the radial evolution of magnetic and velocity field fluctuations by different parameters, namely, the fluctuation energy, cross helicity, and the Alfvén ratio (Matthaeus & Goldstein 1982). The Alfvén ratio basically represents the ratio between the kinetic and magnetic fluctuation energy. Wentzel-Kramers-Brillouin (WKB) theory explains the decrease in fluctuation energy with distance as a consequence of the AW propagation in the solar wind (Hollweg 1974; Marsch & Tu 1990; Tu & Marsch 1993, 1995). It is thus necessary to include the basic essence of the AW propagation and dissipation mechanisms in the solar wind model. In the AW-driven solar wind model, footpoints of flux tubes are shaken randomly by the convection at the solar surface and generate wave-like fluctuations (usually AWs) that propagate up to the corona. It is also believed that these waves are partially reflected back toward the Sun and generate MHD turbulence that is responsible for the gradual wave energy dissipation into the corona (Velli et al. 1989; Cranmer 2012). Several one-, two-, and three-dimensional solar wind models with AW propagation and a dissipation mechanism have been developed over the years to explain different properties of the solar wind (Tu & Marsch 1993; Usmanov et al. 2000; Verdini et al. 2009, 2010; Sokolov et al. 2013; van der Holst et al. 2014; Lionello et al. 2014; Réville et al. 2018; Shoda et al. 2018, 2020).

In this paper, we aim to characterize and test the behavior and properties of the AW-driven solar wind model (Réville et al. 2020b) over an 11 yr solar cycle. So far this particular model has been tested for only one epoch and only on the Parker Solar Probe trajectory, showing an encouragingly excellent comparison with in situ observations. Please note that we denote the MHD solar wind model with AW propagation and dissipation mechanisms as the AW-driven solar wind model (AW model) in the rest of the paper. One can refer to van der Holst et al. (2010, 2014), Sokolov et al. (2013), Lionello et al. (2014), Cohen (2017), and Réville et al. (2020a, 2020b) for similar approaches. Here we model the AWs' propagation and dissipation mechanisms following the WKB theory (Alazraki & Couturier 1971; Belcher 1971; Whang 1973; Hollweg 1974), which consists of solving two extra equations—one for parallel AWs ($\delta v$ is in the direction of $\delta B$) and another for antiparallel AWs ($\delta v$ is in the opposite direction than $\delta B$). We furthermore use the Wilcox synoptic maps of photospheric magnetic field as an inner boundary of our model. In this study, we first compare the properties of the solar wind structures obtained from the polytropic and AW models to see how the AW model performs compared to the polytropic one. Next, we try to understand the impact of various coronal parameters on the AW model solution for a few specific time periods. Finally, we test the property of the AW model over cycle 23.

Section 2 describes the details of our solar wind models, both polytropic and AW. We present a detailed comparison between the results obtained from both models in Section 3. We also investigate there the relationship between solar wind terminal velocity and the coronal source temperature. In Section 4, we characterize how the mass loss, AM loss, and averaged Alfvén radius depend on the coronal parameters in the AW model. We also derive a simple scaling law for the mass loss and the net torque applied to the star (Réville et al. 2015b). In Section 5, we study the variation of mass and AM loss during cycle 23. Finally, we present a summary of this study and our conclusions in the last section.

## 2. Solar Wind Models

In this section, we provide the basic governing equations for the two classes of models used in this study, namely, the polytropic wind model and the AW-driven MHD solar wind model. We use the PLUTO MHD code (Mignone et al. 2007) to solve the equations described in the following.

### 2.1. Polytropic MHD Solar Wind Model

In the polytropic model, the MHD equations are written as usual in their conservative form:

$$ \frac{\partial}{\partial t} \rho + \nabla \cdot (\rho \mathbf{v}) = 0, \qquad (1) $$

$$ \frac{\partial}{\partial t} \mathbf{m} + \nabla \cdot \left( \mathbf{m}\mathbf{v} - \frac{1}{\mu_0} \mathbf{B}\mathbf{B} + \mathbf{I}\mathbf{p} \right) = -\rho \nabla \Phi, \qquad (2) $$

$$ \frac{\partial}{\partial t} (E + \rho\Phi) + \nabla \cdot \left( (E + p + \rho\Phi)\mathbf{v} - \frac{1}{\mu_0} \mathbf{B}(\mathbf{v} \cdot \mathbf{B}) \right) = 0, \qquad (3) $$
$$ \frac{\partial}{\partial t} \mathbf{B} + \nabla \cdot (\mathbf{v}\mathbf{B} - \mathbf{B}\mathbf{v}) = 0, \quad (4) $$

where **B** is the magnetic field, ρ is the mass density, **m** = ρ**v** is the momentum, E = ρe + ρ**v**²/2 + B²/2 is the total energy, **v** is the velocity field, p = p<sub>th</sub> + B²/2 is the total (thermal and magnetic) pressure, and **I** is the identity matrix.

So-called polytropic models (Washimi & Shibata 1993; Keppens & Goedbloed 1999; Matt et al. 2012; Réville et al. 2015a) are built using the ideal equation of state ρε = p<sub>th</sub>/(γ - 1), where ε represents the internal energy per mass and p<sub>th</sub> is the thermal pressure. The adiabatic exponent γ is then chosen to be <5/3, which is the value expected for an adiabatic expansion of the solar wind. Decreasing γ artificially is equivalent to creating an upward free-streaming heat flux (q ∝ p<sub>th</sub>**v**), down to the (unreached) limit γ = 1, where the heat flux is infinitely fast, and the plasma isothermal. Polytropic winds are thus taking advantage of a value of γ close to 1 (in the range of [1.05, 1.2]), in combination with an already hot corona at the base of the domain, to create the pressure gradient necessary to the flow acceleration. Our solar wind model uses the value γ = 1.05 and is initialized with the solution of a one-dimensional, hydrodynamic polytropic wind.

We perform 2.5D axisymmetric (around the rotation axis of the Sun) polytropic wind simulations in the meridional plane R<sub>☉</sub> ≤ r ≤ 20R<sub>☉</sub> and 0 ≤ θ ≤ π with a resolution of 512 × 512 (i.e., N<sub>r</sub> = N<sub>θ</sub> = 512). Please note that we use a uniform grid in latitude and a stretched grid in radius. Our grid spacing in radius varies from 0.001 solar radii at the surface of the star to 0.01 solar radii at the outer boundary. We use the Harten–Lax–van Leer (HLL) Riemann solver (see Einfeldt 1988) to solve the MHD equations in spherical polar coordinates. We maintain ∇·**B** = 0 using an hyperbolic divergence cleaning method (Dedner et al. 2002). We describe the initial and boundary conditions for both the polytropic and AW models in Sections 2.3 and 2.4, respectively.

## 2.2. AW-driven MHD Solar Wind Model

AW-driven models aim to account for both the coronal heating and the additional source of energy and momentum necessary to create the observed velocity bimodality. The turbulent dissipation, a consequence of the observed cascade (see, e.g., Tu & Marsch 1993, 1995), is one of the main factors of the coronal heating in these models, while the AW pressure (Alazraki & Couturier 1971; Belcher 1971) is an additional acceleration term for the wind.

The model used in this work is fully described in Réville et al. (2020b), and in this section we focus on the differences with the polytropic model. First, two equations are solved in addition to the usual MHD equations. These equations read as

$$ \frac{\partial \mathcal{E}^{\pm}}{\partial t} + \nabla \cdot ([\mathbf{v} \pm \mathbf{v}_A] \mathcal{E}^{\pm}) = -\frac{\mathcal{E}^{\pm}}{2} \nabla \cdot \mathbf{v} - Q_w^{\pm}, \quad (5) $$

where **v**<sub>A</sub> is the Alfvén speed and

$$ \mathcal{E}^{\pm} = \rho \frac{|z^{\pm}|^2}{4} \quad (6) $$

is the energy density for parallel and antiparallel Alfvénic perturbations in their Elsässer form:

$$ z^{\pm} = \delta v \mp \text{sign}(\mathbf{B}_r) \frac{\delta \mathbf{b}}{\sqrt{\mu_0 \rho}}, \quad (7) $$

where δ**b** and δ**v** are the fluctuations in the magnetic field and velocity field (transverse velocity) respectively. These equations essentially follow the Wentzel–Kramers–Brillouin theory (WKB; Alazraki & Couturier 1971; Belcher 1971), with a dissipation term Q<sub>w</sub> defined as

$$ Q_w = Q_w^+ + Q_w^-, \quad (8) $$

where each term due to different populations of AWs is prescribed as

$$ Q_w^{\pm} = \frac{\rho |z^{\pm}|^2}{8 \lambda} (\mathcal{R}|z^{\pm}| + |z^{\mp}|), \quad (9) $$

which follows the Kolmogorov phenomenology. It was previously suggested that Kolmogorov-type dissipation can take place only if there are counter-propagating AWs (Cranmer & van Ballegooijen 2005; Chandran et al. 2009). In coronal holes, partial reflection due to inhomogeneities in the solar corona are responsible for these counter-propagating AWs (Verdini et al. 2009). Keeping these facts in mind, we set a constant reflection coefficient R = 0.1 for the sake of simplicity. This coefficient is close to that obtained in the analytical model of Chandran & Hollweg (2009), which leads to very similar heating rates. λ is the dissipation length scale and varies with distance to the Sun as the inverse of the square root magnetic field strength (or the width of a given flux tube). The value at the base of the corona is set to λ<sub>☉</sub> = 0.025R<sub>☉</sub>, which is the size of supergranules (see Réville et al. 2020a for a more detailed discussion).

These equations are coupled to the rest of the MHD system through two terms. First, we add in the total pressure the contribution of AWs:

$$ p = p_{\text{th}} + B^2/2 + \mathcal{E}/2, \quad (10) $$

where E = E<sup>+</sup> + E<sup>-</sup>. This terms acts, in particular, directly in the momentum equation to add an additional outward pressure gradient. Then, we modify the energy equation, which becomes

$$ \frac{\partial}{\partial t}(E + \mathcal{E} + \rho\Phi) + \nabla \cdot [(E + p + \rho\Phi)\mathbf{v} - \mathbf{B}(\mathbf{v} \cdot \mathbf{B}) + \mathbf{v}_g^{\pm}\mathcal{E}^{\pm} + \mathbf{v}_g^{-}\mathcal{E}^{-}] = \mathcal{Q}, \quad (11) $$

where **v**<sub>g</sub><sup>±</sup> = **v** ± **v**<sub>A</sub> denotes the group velocity of the AW packets. Here γ = 5/3, which calls for a new term Q, that does not exist in the polytropic model, to control the thermodynamics of the AW-driven model. It can be decomposed in four contributions (sources and sinks):

$$ \mathcal{Q} = Q_w + Q_h - Q_c - Q_r, \quad (12) $$

The sources are made of the AW heating term Q<sub>w</sub> described above, and an ad hoc term following Withbroe (1988):

$$ Q_h = F_h/H \left(\frac{R_{\odot}}{r}\right)^2 \exp\left(-\frac{r-R_{\odot}}{H}\right), \quad (13) $$
with $H \sim 1R_\odot$, the heating scale height, and $F_h$ is the energy flux from the photosphere (in erg cm$^{-2}$ s$^{-1}$). This additional term is used to render heating in the chromosphere and lower corona that could find a different origin than AWs (nanoflares for instance, see Parker 1988).

The sinks are made of the usual thin radiation cooling term

$$Q_r = n^2 \Lambda(T), \quad (14)$$

with $n$ the electron density, $T$ the electron temperature, and $\Lambda(T)$ defined as in Athay (1986), and of the thermal conduction flux $Q_c$. The latter combines a collisional and a collisionless prescription:

$$Q_c = \nabla \cdot (\alpha \mathbf{q}_s + (1 - \alpha) \mathbf{q}_p), \quad (15)$$

$\mathbf{q}_s = -\kappa_0 T^{5/2} \nabla T$ being the usual Spitzer-Härm collisional thermal conduction with $\kappa_0 = 9 \times 10^{-7}$ cgs, and $\mathbf{q}_p = 3/2 p_{th} \mathbf{v}_e$ the free-stream heat flux (Hollweg 1986). The coefficient $\alpha = 1/(1 + (r - R_\odot)^4/(r_{coll} - R_\odot)^4)$ creates a smooth transition between the two regimes at a characteristic height of $r_{coll} = 5R_\odot$.

As in the polytropic model, we performed all our axisymmetric AW-driven solar wind simulations in the meridional slab $R_\odot \le r \le 20R_\odot$ and $0 \le \theta \le \pi$ with a resolution of $256 \times 256$ (i.e., $N_r=N_\theta=256$). We use a uniform grid in the latitudinal direction and a stretched grid in the radial direction. Note that resolution is different in the AW and polytropic scenario. The equations are solved using an improved HLL Riemann solver (see Einfeldt 1988), combined with a parabolic reconstruction method and minmod slope limiter. We use an hyperbolic divergence cleaning method (Dedner et al. 2002) to maintain $\nabla \cdot \mathbf{B} = 0$.

## 2.3. Initial Conditions

We initialize our solar wind models with the solutions from a hydrodynamic polytropic wind and from a potential field source surface (PFSS) magnetic field extrapolations (Schatten et al. 1969) obtained from Wilcox solar magnetic maps.

The Wilcox Solar Observatory (WSO) has provided continuous line-of-sight magnetogram observations since 1976 May 27 (Scherrer et al. 1977). We decompose the Wilcox surface radial magnetic field at a given time $t_0$ in terms of spherical harmonic coefficients for each Carrington rotation (CR), as in DeRosa et al. (2012):

$$B_r(R_\odot, \theta, \phi, t_0) = \sum_{\ell,m} \alpha_{\ell,m}(R_\odot, t_0) Y_{\ell,m}(\theta, \phi), \quad (16)$$

where $Y_{\ell,m}$ are the normalized spherical harmonics. We use $\ell_{max} = 40$ for the decomposition. Figure 1 shows the properties of the solar surface magnetic energy ($\sum_{\ell,m} \alpha_{\ell,m}^2$) for the last 45 years. We define a parameter, referred to as $f_{axisym}$, to quantify the energy of the axisymmetric component over the total surface energy following the equation:

$$f_{\text{axisym}} = \frac{\sum_{\ell} \alpha_{\ell,0}^2}{\sum_{\ell,m} \alpha_{\ell,m}^2}. \quad (17)$$

Similarly, we also define a quantity

$$f_{\text{non-axisym}} = \frac{\sum_{\ell,m \neq 0} \alpha_{\ell,m}^2}{\sum_{\ell,m} \alpha_{\ell,m}^2} = 1 - f_{\text{axisym}} \quad (18)$$

to quantify the energy of the non-axisymmetric component over the total surface energy ($\frac{1}{2\mu_0} \sum_{\ell,m} \alpha_{\ell,m}^2$). Figure 1 shows the variation in time of the total surface magnetic energy. We observe a clear anticorrelation between the total surface magnetic energy and $f_{axisym}$ over the whole period. This simply means that the axisymmetric surface energy is the main contributor to the total surface energy (typically more than 80%) at the cycle minimum. On the other hand, a major contribution to the total energy (typically more than 80%) comes from the non-axisymmetric components of the solar magnetic fields at the solar maximum (as $f_{non-axisym} = 1 - f_{axisym}$). In summary, non-axisymmetric surface energy is the major part of the total surface energy at the cycle maximum, while axisymmetric surface energy is at the cycle minimum.

We have also defined two other quantities, namely, the parity factor ($P$) and the map complexity to characterize the properties of the synoptic magnetic maps:

$$P = \frac{\sum_{\ell+m=\text{even}} \alpha_{\ell,m}^2 - \sum_{\ell+m=\text{odd}} \alpha_{\ell,m}^2}{\sum_{\ell,m} \alpha_{\ell,m}^2}, \quad (19)$$

where $\sum_{\ell+m=\text{even}} \alpha_{\ell,m}^2$ corresponds to the symmetric surface energy with respect to the equator and $\sum_{\ell+m=\text{odd}} \alpha_{\ell,m}^2$ corresponds to the antisymmetric surface energy with respect to the equator. When the parity factor ($P$) is closer to 1, the magnetic field is more symmetric, and closer to -1, it is more antisymmetric. Parity actually quantifies the nonlinear coupling between symmetric and antisymmetric modes of solar magnetic fields across the hemisphere (see DeRosa et al. 2012; Hazra & Nandy 2019, for more details). Please note that if one considers axisymmetry and ignores higher order modes, then parity basically represents the nonlinear coupling between the dipolar and quadrupolar modes of solar magnetic fields across the hemisphere. The middle panel of Figure 1 shows the evolution of the parity function with time. We note that the parity factor is mostly negative throughout the time; that indicates the antisymmetric nature of the magnetic field across the equator (DeRosa et al. 2012). We define the characteristic harmonic degree as

$$l_c = \frac{\sum_{\ell,m} \ell \alpha_{\ell,m}^2}{\sum_{\ell,m} \alpha_{\ell,m}^2}. \quad (20)$$

A higher $l_c$ generally indicates the presence of more complex magnetic fields at the solar surface. The bottom panel of Figure 1 shows the time evolution of complexity. If we compare the time evolution of complexity with the time evolution of total surface energy, we find a good positive correlation between them.

In this study, we have performed our wind calculations in 2.5D, assuming axisymmetry. This allows us to perform a large number of simulations using a reasonable amount of computational resources. However, as evident from the discussion above, we know that non-axisymmetric components play a significant role at the maximum of the solar cycle. Finley & Matt (2018) found that both axisymmetric and non-axisymmetric field components impact the radial decay of magnetic flux in a similar pattern. Garraffo et al. (2016) also showed that
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Top panel: comparison between the total surface magnetic energy $\sum_{\ell,m} \alpha_{\ell,m}^2$ (black line) and the axisymmetric component $f_{\text{axisym}}$ obtained from the Wilcox magnetic maps (red line). Middle panel: comparison between the total surface energy $\sum_{\ell} \alpha_{\ell,m}^2$ (black line) and the parity factor (P) obtained from the Wilcox magnetic maps (red line). A negative parity factor indicates that the surface magnetic field is mostly antisymmetric across the equator, while positive indicates the opposite. Bottom panel: comparison between the total surface energy $\sum_{\ell} \alpha_{\ell,m}^2$ (black line) and the map complexity ($l_c$) obtained from the Wilcox magnetic maps (red line). This indicates that complexity is highly correlated with the total surface energy.</figcaption></figure>

torque generated by both axisymmetric and non-axisymmetric components are comparable. Motivated by these studies, we chose to alter the field strength for each $l$ mode by including the strength of both the axisymmetric and non-axisymmetric components following the quadrature addition formula:

$$B_r^{\text{sim}}(R_\odot, \theta) = \sum_l \left( \sum_{m=-l}^{l} \alpha_{l,m}^2 \right)^{1/2} Y_{l,0}(\theta), \quad (21)$$

where $\alpha_{l,m}$ is the strength for each $l$ and $m$ mode (see Finley et al. 2018 for more details on this formulation).

We reconstructed the fields inside our PLUTO grid by combining these coefficients with spherical harmonics. Finally, we used the PFSS extrapolation (Schatten et al. 1969; Altschuler & Newkirk 1969; Schrijver & DeRosa 2003) method to extrapolate the magnetic fields inside our computational domain. We chose the source surface radius $R_{ss}$ as 100$R_\odot$ for the PFSS extrapolation method. We used this extrapolated magnetic field as an initial magnetic field for our model that is then left to evolve self-consistently in the solar corona (see Appendix B).

## 2.4. Boundary Conditions

**Boundary conditions for the polytropic setup**—In the polytropic setup, we set an outflow boundary condition at the top boundary for all variables except the radial magnetic field, while we fix an axisymmetric boundary condition at the

latitudinal boundaries ($\theta=0$ and $\pi$). We ensure the opening of magnetic field lines at the top boundary and the divergence-free condition of the magnetic field by enforcing the condition $\frac{\partial}{\partial z}(r^2 B_r) = 0$ at the top boundary. We set the conditions as prescribed by Zanni & Ferreira (2009) for all variables at the bottom radial boundary.

**Boundary conditions for the AW-driven setup**—We use the same boundary conditions as those of the polytropic setup at the top and latitudinal boundaries. We also set the outflow boundary condition at the top boundary and the axisymmetric boundary condition at the latitudinal boundaries for the wave components ($\mathcal{E}^{\pm}$). The inner boundary condition is, however, different since the domain starts at the top of the transition region. We fix the plasma density, pressure, and the radial component of the magnetic field $B_r$. We want to model the star as a perfect conductor rotating with an angular speed of $\Omega_\odot$. This implies that the electric field should be zero in the reference frame co-rotating with the star. We enforce the zero electric field condition at the bottom boundary by setting $v_p \parallel B_p$. The pressure (or temperature) is modified dynamically to remain close to a hydrostatic equilibrium. Finally, the wave amplitude of the outward component is set to a fixed value $\delta z^+ = 2 \delta v$, where $\delta v$ is the input transverse velocity. The inward component of the waves is set to 0.

The resulting Poynting flux through our lower boundary is

$$F_p = \langle \rho_\odot v_{A,\odot} \delta v_{\odot}^2 \rangle \approx \rho_\odot \langle v_{A,\odot} \rangle \delta v_{\odot}^2. \quad (22)$$
This acts as an energy input to the solar wind. The input Poynting flux is chosen to remain near $10^5$ erg cm$^{-2}$ s$^{-1}$. An additional heating term is also added, with a heating flux $F_h$ and a scale height of 1 solar radius, which acts essentially in the transition region and the low corona. We varied the values of $F_h$ and $\delta v_\odot$ for the parameter space study described in Section 4. The total input source flux in our model remains around $1.5 \times 10^5$ erg cm$^{-2}$ s$^{-1}$, which corresponds to the energy flux of the solar wind in the outer heliosphere (see Réville et al. 2018).

### 3. Comparison between the Polytropic Solar Wind Model and the AW-driven Solar Wind Model

In this section, we aim to compare the results obtained from a polytropic solar wind model and an AW model. We selected CR 1975 (near the maximum of solar cycle 23) and CR 1919 (near the minimum of solar cycle 23) for this purpose. We initialize both polytropic and AW simulations with the decomposed surface magnetic field data obtained from the synoptic maps. We discuss the significant impact of non-axisymmetric components on the wind solutions in detail in Appendix A. Motivated by the results described in Appendix A, we chose to initialize the model considering the contribution of both axisymmetric and non-axisymmetric components following Equation (21) (see Appendix B for the initial magnetic field topologies obtained from these two maps (Figure 16)). We now have four cases for the purpose of comparison: CR 1919 (the polytropic and AW cases) and CR 1975 (the polytropic and AW cases). We have denoted these four cases as 1919mP, 1919mAW, 1975mP, and 1975mAW in the rest of this paper.

As the value of the mass-loss rate is better known from the observations, we chose to compare the polytropic and AW-driven wind models when both models will give an almost similar mass-loss value. We define the mass-loss rate as

$$ \dot{M} = \int \rho v_r dS, \qquad (23) $$

where $\rho$ is the density and $v_r$ is the radial wind speed. We compute this integral over a spherical surface S beyond the largest closed magnetic loop. Regardless of the choice of integration surface, one will get the constant mass-loss value as long as the previous condition is satisfied. The mass-loss rates obtained from the polytropic solar wind model for CR 1919 and CR 1975 (the 1919mP and 1975mP cases) are $2.67 \times 10^{-14} M_\odot \text{yr}^{-1}$ and $2.75 \times 10^{-14} M_\odot \text{yr}^{-1}$, respectively. The mass-loss rate in the AW model is $2.38 \times 10^{-14} M_\odot \text{yr}^{-1}$ for CR 1919 and $2.68 \times 10^{-14} M_\odot \text{yr}^{-1}$ for CR 1975 (the 1919mAW and 1975mAW cases). Mass-loss rates are almost similar for all cases (see Table 4 in Appendix A, which summarizes the value of the mass loss in different scenarios). Despite this common global property, we nevertheless found significant differences between the results obtained from the polytropic and AW-driven solar wind models for both CR 1975 and 1919.

#### 3.1. On the Wind Speed Bimodality

Figures 2(a) and (b) show the poloidal velocity in the meridional plane obtained from the polytropic and AW-driven

solar wind models initialized with the CR 1919 WSO map (cycle minimum). Figures 2(c) and (d) show the same when the models are initialized with the WSO map of CR 1975 (cycle maximum). One will be able to understand the impact of solar wind dynamical pressure by comparing these steady-state solutions (Figure 2) with their corresponding initial magnetic field line topology (Appendix B). Initial magnetic field lines become open due to the dynamical pressure of the solar wind. When the solution achieves the steady state, we find closed magnetic field lines at latitudes where we have the zero radial magnetic field. These regions are known as dead zones (or streamers). By comparing the steady-state solutions at the maximum (CR 1975) and minimum (CR 1919) phases of the solar cycle, we find that the number of dead zones is controlled by the initial magnetic field topology. In summary, there is a strong influence of the initial magnetic field topology on the solar wind structure.

Figure 2 indicates that we can obtain higher poloidal velocity in the AW-driven wind model compared to the polytropic model. We are able to reproduce the observed bimodality in the AW-driven wind model scenario. However, we are not able to reproduce the same in the polytropic wind model scenario. Poloidal velocity distribution is much wider in the AW-driven scenario compared to polytropic one. Earlier studies have indicated the importance of large-scale energy deposition beyond the critical point in the supersonic regime to explain the observed velocity distribution (Leer & Holzer 1980; Leer et al. 1982; Stewart & Bravo 1997). Since magnetic field lines are mostly open at 0.1 au, we have also estimated the open flux from all of our models:

$$ \Phi_{\text{open}} = \int |B_r| dS, \qquad (24) $$

where $B_r$ is the radial magnetic field. Here, we also compute the integral over a spherical surface beyond the largest closed magnetic loop. The magnetic flux becomes constant beyond a certain distance as field lines become open, and this constant value is the open flux. The open flux obtained from the polytropic solar wind model is $1.37 \times 10^{22} M\text{s}$ for CR 1919 and $1.36 \times 10^{22} M\text{s}$ for CR 1975. On the other hand, open flux obtained from the AW-driven solar wind model is $2.81 \times 10^{22} M\text{s}$ for CR 1919 and $3.29 \times 10^{22} M\text{s}$ for CR 1975. The open flux is therefore higher for the AW model compared to the polytropic one, and we also find that the open flux is higher at the cycle maximum (CR 1975) than at the cycle minimum (CR 1919) in the AW model. This is not the case in the polytropic model, which shows almost no variation in open flux between the two CRs.

Next, we focus on the latitudinal distribution of the wind speed. We find slower wind around the dead zones (streamers) compared to the open magnetic field regime. Streamers are generally located at latitudes where magnetic forces are strong enough to counteract the thermal and dynamic pressure. Away from the streamer, we find the flux tubes that allow the wind to reach its maximum speed. On top of the dead zone, the magnetic field reaches its minimum value (zero in ideal MHD scenario). It is thus expected that the Alfvén surface, the surface at which the speed of the solar wind is equal to the local Alfvén speed ($v_A = \sqrt{\|B\|/(\mu_0\rho)}$), will touch the top of the dead zone. Slow wind around the dead zones extends the Alfvén surface slightly on both sides from this minimum of the
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Wind simulations considering the contribution of both axisymmetric and non-axisymmetric components by Equation (21). The top two panels show the poloidal velocity in the meridional plane for CR 1919 (cycle minimum) obtained from the (a) polytropic and (b) AW-driven solar wind models, respectively, while the bottom two panels show the poloidal velocity in the meridional plane for CR 1975 (cycle maximum) obtained from the (c) polytropic and (d) AW-driven models. In each plot, the white solid lines indicate the magnetic field lines in the meridional plane, and the black line corresponds to the Alfvén surface. The poloidal velocity (color scale) is in kilometers per second. This plot suggests that poloidal velocity is higher in AW-driven scenario compared to the polytropic, albeit the simulations achieve a similar mass loss.</figcaption></figure>

Alfvén surface. In these extended regions (both sides from the minimum of the Alfvén surface) thin current sheet structures are formed. At the solar minimum (CR 1919), we find streamers at the low latitudes (near the equator) and open field line structures at high latitudes (see Figures 2(a) and (b)). Because of this, we mostly find the slow wind at the low latitude (near the equator) and fast wind at the high latitude (mostly from the polar coronal hole regime) during the solar minimum. On the other hand, we find a complex coronal structure at the maximum of solar activity (CR 1975), with streamers distributed across different latitudes (see Figures 2(c) and (d)). The only difference in the coronal structure between the cycle maximum and minimum is that streamers are differently located in latitude. We find that the streamer size is bigger in the polytropic wind scenario compared to the Alfvén-driven one (see Figure 2). We also note a drop in the wind speed at the high latitude at the cycle maximum due to the presence of closed magnetic loop structures at high latitudes. This kind of scenario is also found in the Ulysses data and has been interpreted as a high-latitude conic current sheet in the solar wind (Khabarova et al. 2017).

Let us now compare the profile of the Alfvén surface obtained at both the activity maximum and minimum. As we already discussed, the Alfvén surface in all of our four cases touches the top of the dead zone, when there is a current sheet formation (see Figure 2). As the number of dead zones is higher at the cycle maximum (due to a more complex topology), we find the Alfvén surface more irregular at the activity maximum compared to the activity minimum (see Figure 2(d)). In the steady state, all closed magnetic field lines reside within the Alfvén radius. We find almost a similar geometrical average Alfvén radius for both the AW-driven and polytropic wind model setups (5R⊙ and 5.5R⊙, respectively). Interestingly, we find a large asymmetry in the Alfvén radius for the 1975mP case. The Alfvén radius is higher in the southern hemisphere compared to northern hemisphere (see Figure 2(c)). However, we do not find the same in the 1975mAW case. This is because there is an increase in both the poloidal velocity and Alfvén velocity in the AW-driven scenario.

Finally, we study the distribution of the poloidal velocity amplitude at 20 solar radii (0.1 au) obtained from our four selected models, as shown in Figure 3. We confirm here again that the AW model (right panel) reaches higher wind speeds
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Histograms of the poloidal velocity distribution at $20R_⊙$ (≈0.1 au) for CR 1975 (blue) and CR 1919 (orange), for the polytropic model (left panel) and for the AW-driven model (right panel). The AW-driven wind model shows the bimodal distribution, while the polytropic model does not.</figcaption></figure>

than the polytropic one (left panel). The polytropic model shows hints of a bimodal distribution spanning a narrow range of about 100 km s⁻¹ and reaching a maximum of 280 km s⁻¹. Conversely, the AW model strikingly exhibits a solar-like bimodal distribution, especially at the solar maximum, with a slow component near 350 km s⁻¹ and a fast component close to 800 km s⁻¹. Note that the solar wind speed usually lies between 400 and 900 km s⁻¹ at 1 au. Near the solar minimum (the CR 1919 case), the AW model achieves maximum wind speeds up to 650 km s⁻¹ at 0.1 au. In summary, we are able to better reproduce observed bimodality and distribution breadth of wind speeds with the AW model.

## 3.2. On the Wind Speed-source Temperature Relationships

One important observational feature is the anticorrelation between the final solar wind speed and the corresponding coronal source temperature (Geiss et al. 1995). Many studies have tried to explain this observed anticorrelation. Schwadron & McComas (2003) explained the observed anticorrelation as a consequence of strong temperature dependence of the thermal conduction term. In summary, slow wind loses more energy from the thermal conduction due to higher coronal temperature (see Schwadron et al. 2011, 2014 for the details regarding this). However, other studies with the AW models have indicated that a higher electron temperature at the source of the slow solar wind may cause flux tubes with different expansion factors to have different Parker critical points and different radial distributions of heating rate (Leer & Holzer 1980; Cranmer 2005, 2010, 2010; Cranmer et al. 2017). Here, we want to study the relationship between the terminal wind velocity and the coronal source temperature from our wind models.

We have already shown that both of our wind models can generate faster and slower solar wind streams but differ by the difference in the amplitude between the fast and slow wind streams (see Figure 3). We now want to figure out the relationship between the final wind velocity and the coronal

source temperature from the polytropic wind model. As the solar wind accelerates along the open field lines, we consider the temperature at the beginning point of an open field line (near the coronal base) as the source temperature ($T_s$), and the wind velocity at the endpoint of the open field line (near 20 solar radii) as the final wind velocity ($V_∞$). Please note that the endpoint of the field line is not the physical endpoint of the field line but the computational one. We find a positive correlation (Spearman correlation coefficient 0.86) between the final wind velocity and the coronal source temperature from the polytropic wind model (see Figure 4). Sometimes we also noted a weaker positive correlation for some of the magnetic maps. The polytropic (thermally driven) wind model always shows a positive correlation between the final wind velocity and the temperature, even if we consider the temperature away from our lower coronal boundary. Previous studies based on the polytropic assumption have also suggested a positive correlation between the final wind velocity and the coronal source temperature (e.g., Leer et al. 1982). This kind of relationship is also expected in the Parker wind scenario. However, observations show the opposite. In summary, we are not able to explain the observed anticorrelation between the final wind velocity and the source temperature from the polytropic solar wind model.

Next, we aim to characterize this same relationship using our AW model. The left panel of Figure 5 shows an anticorrelation (correlation coefficient -0.89) between the final solar wind speed ($V_∞$) and the coronal source temperature ($T_s$). The final solar wind speed decreases from 810–300 km s⁻¹ with the increase of surface temperature. Our AW model is therefore able to reproduce the observed anticorrelation between the final wind velocity and the coronal source temperature. Probably different radial distributions of heating rate or the strong temperature dependence of thermal conduction is responsible for this result in our model (Schwadron & McComas 2003; Cranmer & van Ballegooijen 2005). However, if we consider the temperature away from the bottom boundary, after some
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Relationship between the final wind velocity ($V_\infty$) and the surface temperature ($T_s$) obtained from the polytropic solar wind model. The Spearman correlation coefficient is 0.86.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** The left panel shows the relationship between the final wind velocity $V_\infty$ and the surface temperature $T_s$ obtained from the AW model. The Spearman correlation coefficient is -0.89. The right panel shows how the Spearman correlation coefficient between the final wind velocity $V_\infty$ and temperature $T_s$ changes if we consider the temperature at different positions of the solar corona. The black-dashed line (line of no correlation) indicates the transition from negative correlation to positive correlation. The colors correspond to different magnetic maps. We note that the transition takes place at a larger distance from the solar surface at the cycle minimum than at the cycle maximum. Please note that we use the same initial parameters, namely, coronal base density $\rho_\odot = 3$ in units of $10^8$ cm$^{-3}$, input transverse velocity 30 km s$^{-1}$, and heating flux $5 \times 10^4$ erg cm$^{-2}$ s$^{-1}$, as shown in the legend of the left panel.</figcaption></figure>

distance we get a positive correlation between the final wind velocity and the temperature (see right panel of Figure 5).

To illustrate this point, we chose two magnetic maps near the maximum of solar cycle 23 (CR 1975 and 1994) and two magnetic maps near the minimum of solar cycle 23 (CR 1916 and 2071). The right panel of Figure 5 indicates that at the maximum of the solar cycle, the shifting from negative to

positive correlation takes place very close to the solar surface (only at 1.05 solar radii). However, this is not the case at the minimum of the solar cycle. The shift from the negative correlation to positive correlation takes place at a comparatively larger distance (1.3 solar radii for CR 1916 and 1.8 solar radii for CR 2071). The shifting distance lies between 1.04 and 2 solar radii in our AW-driven setup. We did not identify
<table><caption>**Table 1**

Values of the Mass-loss Rate, AM-loss Rate, and Average Alfvén Radius for Different Combinations of Coronal Parameters</caption><thead><tr><th colspan="7">Cycle Maximum CR 1975</th></tr><tr><th>Base Density ρ<sub>☉</sub><br>(10<sup>8</sup> cm<sup>-3</sup>)</th><th>Transverse<br>Velocity δv<sub>☉</sub><br>(km s<sup>-1</sup>)</th><th>Heating Flux F<sub>h</sub><br>(10<sup>4</sup> erg cm<sup>-2</sup> s<sup>-1</sup>)</th><th>Poynting Flux F<sub>p</sub><br>(10<sup>4</sup> erg cm<sup>-2</sup> s<sup>-1</sup>)</th><th>γ<sub>open</sub>/(4π)</th><th>Mass Loss<br>(10<sup>-14</sup><br>M<sub>☉</sub> yr<sup>-1</sup>)</th><th>AM Loss<br>(10<sup>29</sup> erg)</th><th>Alfvén<br>Radius<br>(R<sub>☉</sub>)</th></tr></thead><tbody><tr><td>3</td><td>30</td><td>0.05</td><td>3.73</td><td>303.71</td><td>1.18</td><td>3.31</td><td>5.956</td></tr><tr><td>3</td><td>30</td><td>0.1</td><td>3.72</td><td>302.71</td><td>1.20</td><td>3.34</td><td>5.948</td></tr><tr><td>1</td><td>30</td><td>2.0</td><td>2.04</td><td>338.11</td><td>0.85</td><td>2.29</td><td>5.877</td></tr><tr><td>3</td><td>30</td><td>0.54</td><td>3.69</td><td>286.2</td><td>1.31</td><td>3.47</td><td>5.801</td></tr><tr><td>0.5</td><td>30</td><td>2.0</td><td>1.48</td><td>423.46</td><td>0.55</td><td>1.71</td><td>6.278</td></tr><tr><td>3</td><td>30</td><td>2.0</td><td>3.61</td><td>240.73</td><td>1.69</td><td>3.84</td><td>5.392</td></tr><tr><td>0.3</td><td>30</td><td>2.0</td><td>1.16</td><td>507.85</td><td>0.40</td><td>1.37</td><td>6.635</td></tr><tr><td>3</td><td>10</td><td>2.0</td><td>0.44</td><td>395.02</td><td>0.48</td><td>1.99</td><td>7.343</td></tr><tr><td>0.3</td><td>10</td><td>2.0</td><td>0.13</td><td>737.53</td><td>0.19</td><td>1.06</td><td>8.408</td></tr><tr><td>3</td><td>30</td><td>10.0</td><td>3.38</td><td>136.29</td><td>3.80</td><td>5.24</td><td>4.186</td></tr><tr><td>3</td><td>30</td><td>5.0</td><td>3.51</td><td>187.18</td><td>2.4</td><td>4.49</td><td>4.831</td></tr><tr><td>1</td><td>30</td><td>5.0</td><td>1.99</td><td>251.7</td><td>1.42</td><td>3.04</td><td>5.207</td></tr><tr><td>1</td><td>30</td><td>10.0</td><td>1.97</td><td>176.73</td><td>2.44</td><td>3.79</td><td>4.439</td></tr><tr><td>1</td><td>30</td><td>0.1</td><td>2.09</td><td>532.64</td><td>0.49</td><td>1.84</td><td>6.947</td></tr><tr><td>1</td><td>40</td><td>2.0</td><td>3.55</td><td>255.25</td><td>1.36</td><td>2.88</td><td>5.183</td></tr><tr><td>2</td><td>40</td><td>2.0</td><td>4.96</td><td>195.32</td><td>2.35</td><td>4.37</td><td>5.168</td></tr><tr><td>0.5</td><td>40</td><td>2.0</td><td>2.58</td><td>343.72</td><td>0.78</td><td>1.87</td><td>5.898</td></tr><tr><td>0.3</td><td>40</td><td>2.0</td><td>2.04</td><td>433.76</td><td>0.52</td><td>1.43</td><td>6.296</td></tr><tr><td>2</td><td>10</td><td>2.0</td><td>0.35</td><td>508.39</td><td>0.95</td><td>2.24</td><td>7.948</td></tr><tr><td>1</td><td>10</td><td>2.0</td><td>0.24</td><td>634.9</td><td>0.35</td><td>1.97</td><td>8.452</td></tr><tr><td>0.5</td><td>10</td><td>2.0</td><td>0.17</td><td>778.04</td><td>0.21</td><td>1.23</td><td>8.727</td></tr><tr><td>1</td><td>30</td><td>2.0</td><td>2.03</td><td>338.11</td><td>0.85</td><td>2.29</td><td>5.877</td></tr><tr><td>3</td><td>25</td><td>2.0</td><td>2.59</td><td>284.15</td><td>1.23</td><td>3.28</td><td>5.825</td></tr><tr><td>3</td><td>20</td><td>2.0</td><td>1.69</td><td>337.83</td><td>0.89</td><td>2.81</td><td>6.353</td></tr><tr><td>3</td><td>15</td><td>2.0</td><td>0.97</td><td>313.78</td><td>0.61</td><td>1.85</td><td>6.253</td></tr><tr><td>1</td><td>30</td><td>0.54</td><td>2.07</td><td>451.01</td><td>0.60</td><td>2.03</td><td>6.583</td></tr><tr><td>1</td><td>10</td><td>2.0</td><td>0.24</td><td>634.9</td><td>0.36</td><td>1.97</td><td>8.763</td></tr><tr><td>1</td><td>15</td><td>2.0</td><td>0.53</td><td>563.01</td><td>0.42</td><td>2.01</td><td>7.769</td></tr><tr><td>1</td><td>25</td><td>2.0</td><td>1.44</td><td>398.27</td><td>0.67</td><td>2.16</td><td>6.396</td></tr><tr><td>2</td><td>30</td><td>2.0</td><td>2.85</td><td>284.83</td><td>1.35</td><td>3.44</td><td>5.686</td></tr></tbody></table><table><thead><tr><th colspan="7">Cycle minimum CR 1919</th></tr></thead><tbody><tr><td>1</td><td>30</td><td>0.54</td><td>1.72</td><td>490.83</td><td>0.65</td><td>2.34</td><td>6.775</td></tr><tr><td>3</td><td>30</td><td>0.54</td><td>2.81</td><td>283.50</td><td>1.15</td><td>2.76</td><td>5.535</td></tr><tr><td>2</td><td>30</td><td>2.0</td><td>2.29</td><td>257.81</td><td>1.33</td><td>2.91</td><td>5.265</td></tr><tr><td>2</td><td>40</td><td>2.0</td><td>4.17</td><td>169.33</td><td>2.03</td><td>2.91</td><td>4.304</td></tr><tr><td>0.5</td><td>40</td><td>2.0</td><td>2.02</td><td>339.96</td><td>1.02</td><td>2.46</td><td>5.557</td></tr><tr><td>0.3</td><td>40</td><td>2.0</td><td>1.58</td><td>444.66</td><td>0.76</td><td>2.27</td><td>6.149</td></tr><tr><td>2</td><td>10</td><td>2.0</td><td>0.27</td><td>593.08</td><td>0.56</td><td>2.94</td><td>8.156</td></tr><tr><td>1</td><td>10</td><td>2.0</td><td>0.19</td><td>674.16</td><td>0.52</td><td>3.02</td><td>8.637</td></tr><tr><td>1</td><td>20</td><td>2.0</td><td>0.73</td><td>508.52</td><td>0.69</td><td>2.80</td><td>7.204</td></tr><tr><td>1</td><td>15</td><td>2.0</td><td>0.42</td><td>603.19</td><td>0.57</td><td>2.82</td><td>7.940</td></tr><tr><td>3</td><td>15</td><td>2.0</td><td>0.73</td><td>466.13</td><td>0.72</td><td>2.87</td><td>7.130</td></tr><tr><td>3</td><td>20</td><td>2.0</td><td>1.30</td><td>365.71</td><td>0.94</td><td>2.96</td><td>7.519</td></tr><tr><td>3</td><td>25</td><td>2.0</td><td>1.98</td><td>287.46</td><td>1.21</td><td>2.96</td><td>5.601</td></tr><tr><td>0.5</td><td>10</td><td>2.0</td><td>0.13</td><td>774.22</td><td>0.44</td><td>2.76</td><td>8.979</td></tr><tr><td>0.3</td><td>10</td><td>2.0</td><td>0.11</td><td>799.41</td><td>0.31</td><td>1.96</td><td>8.921</td></tr><tr><td>1</td><td>25</td><td>2.0</td><td>1.12</td><td>411.25</td><td>0.85</td><td>2.79</td><td>6.485</td></tr><tr><td>3</td><td>10</td><td>2.0</td><td>0.32</td><td>580.91</td><td>0.56</td><td>2.82</td><td>7.988</td></tr><tr><td>1</td><td>30</td><td>2.0</td><td>1.61</td><td>336.10</td><td>1.03</td><td>2.77</td><td>5.838</td></tr><tr><td>0.5</td><td>30</td><td>2.0</td><td>1.14</td><td>463.17</td><td>0.74</td><td>2.51</td><td>6.563</td></tr><tr><td>0.3</td><td>30</td><td>2.0</td><td>0.91</td><td>564.57</td><td>0.59</td><td>2.31</td><td>7.069</td></tr><tr><td>3</td><td>30</td><td>2.0</td><td>2.79</td><td>221.17</td><td>1.57</td><td>2.99</td><td>4.953</td></tr><tr><td>1</td><td>10</td><td>2.0</td><td>0.19</td><td>674.16</td><td>0.52</td><td>3.02</td><td>8.637</td></tr><tr><td>3</td><td>30</td><td>0.1</td><td>2.83</td><td>305.44</td><td>1.02</td><td>2.48E</td><td>5.625</td></tr><tr><td>3</td><td>30</td><td>0.05</td><td>2.83</td><td>306.32</td><td>0.98</td><td>2.44</td><td>5.627</td></tr><tr><td>3</td><td>30</td><td>10.0</td><td>2.74</td><td>97.37</td><td>3.75</td><td>3.32</td><td>3.357</td></tr><tr><td>3</td><td>30</td><td>5.0</td><td>2.79</td><td>149.80</td><td>2.38</td><td>3.17</td><td>4.118</td></tr><tr><td>1</td><td>30</td><td>5.0</td><td>1.58</td><td>199.50</td><td>1.80</td><td>3.0</td><td>4.610</td></tr><tr><td>1</td><td>30</td><td>10.0</td><td>1.55</td><td>121.65</td><td>3.01</td><td>3.10</td><td>3.629</td></tr><tr><td>1</td><td>30</td><td>0.05</td><td>1.73</td><td>591.87</td><td>0.51</td><td>2.06</td><td>7.189</td></tr></tbody></table>
<table><caption>**Table 1**
(Continued)</caption><thead><tr><th colspan="7">Cycle Maximum CR 1975</th></tr><tr><th>Base Density ρ<sub>☉</sub> (10<sup>8</sup> cm<sup>-3</sup>)</th><th>Transverse Velocity δv<sub>☉</sub> (km s<sup>-1</sup>)</th><th>Heating Flux F<sub>h</sub> (10<sup>4</sup> erg cm<sup>-2</sup> s<sup>-1</sup>)</th><th>Poynting Flux F<sub>p</sub> (10<sup>4</sup> erg cm<sup>-2</sup> s<sup>-1</sup>)</th><th>γ<sub>open</sub>/(4π)</th><th>Mass Loss (10<sup>-14</sup> M<sub>☉</sub> yr<sup>-1</sup>)</th><th>AM Loss (10<sup>29</sup> erg)</th><th>Alfvén Radius (R<sub>☉</sub>)</th></tr></thead><tbody><tr><td>1</td><td>30</td><td>0.1</td><td>1.73</td><td>554.95</td><td>0.54</td><td>2.12</td><td>7.055</td></tr><tr><td>3</td><td>40</td><td>2.0</td><td>5.01</td><td>138.57</td><td>2.45</td><td>3.04</td><td>3.978</td></tr><tr><td>1</td><td>40</td><td>2.0</td><td>2.83</td><td>283.60</td><td>1.46</td><td>2.73</td><td>4.873</td></tr></tbody></table>

precisely why the shift from negative to positive correlation occurs at different distances depending on the structure of the initial magnetic map. It could originate from a decrease in the complexity of the map or in the total surface energy of the magnetic map. As the two are correlated (see the bottom panel of Figure 1), additional simulations would be required to disentangle their relative effects, which is beyond the scope of the present paper. In summary, the AW model can well reproduce the fact that fast wind streams originate from cool coronal holes, and slow wind streams from a near streamers-like structure.

## 4. Dependence of the Solar Wind Properties on the Coronal Parameters in the AW-driven Solar Wind Model

We now turn to the study of the influence of the coronal parameters of the AW model on the global properties of the modeled wind. We will characterize the global solution by three numbers: the mass-loss rate, the AM-loss rate, and the Alfvén radius of the wind solution. We have already defined the mass-loss rate in Equation (23). Here, we will define the other two quantities, namely, the AM-loss rate and the average Alfvén radius.

In this study, we define the rate of the AM loss following the equation:

$$j = \int_S \rho \Lambda v_r dS, \qquad (25)$$

where $\Lambda = R_0 \left( v_\phi - \frac{B_\phi}{\rho_0 \rho} \frac{B_p \cdot v_p}{v_p \|v_p\|^2} \right)$. Here, $B_p$ and $v_p$ represent the poloidal magnetic field and poloidal velocity, respectively. $R_0$ is the spherical radius where we want to calculate the AM-loss rate. This integral is computed over a spherical surface S (with radius $R_0$), which contains all closed magnetic field structures. The value of the AM loss does not depend on the value of $R_0$ as long as it contains all closed magnetic structures.

We can also describe the AM-loss rate in terms of Alfvén radius. Weber & Davis (1967) first quantified the AM carried by the plasma using their one-dimensional model and found the specific (per unit mass) AM ($I_w$) to be

$$I_w = \Omega_* R_A^2, \qquad (26)$$

where $\Omega_*$ is the stellar rotation rate and $R_A$ is the Alfvén radius. One can then express the stellar AM-loss rate in a steady state as a product of the specific AM ($I_w$) carried by the solar wind and the mass-loss rate:

$$j = \dot{M} \Omega_* R_A^2. \qquad (27)$$

We then define the average Alfvén radius following Equation (27):

$$\langle R_A \rangle = \sqrt{\frac{j}{\dot{M} \Omega_*}}. \qquad (28)$$

We computed the AM-loss rate $j$ and the mass-loss rate $\dot{M}$ from our simulations when the solution reached a steady state. We then calculated the average Alfvén radius following Equation (28).

### 4.1. General Solar Wind Properties with Respect to Different Sources of Heating

The total energy input to solar wind in the AW-driven solar wind model is basically controlled by the amount of input Poynting flux and the coronal heating flux $F_h$ at the base of the solar corona. The total input Poynting flux $F_p = \rho_\phi \langle v_{A,\phi} \rangle \delta v_\phi^2$ at the coronal base in the AW model depends on the input density at the base of the corona $\rho_\phi$, on the input transverse velocity ($\delta v_\phi$) and on the initial magnetic field structure. We have assessed the relative influence of these parameters in a systematic exploration. All the models we explored are listed in Table 1, along which we list the obtained rate of mass loss, the rate of AM loss, and the Alfvén radius. We performed this systematic exploration for the minimum and maximum phases of solar cycle 23 using the magnetic maps for CR 1975 and CR 1919.

Figure 6 shows the variation of the mass loss with the different coronal parameters for an initial magnetic map (CR 1975) taken at the maximum of solar activity. The top left panel of Figure 6 shows an exponential increase in the mass loss when we increase the input transverse velocity $\delta v_\phi$ from 2–40 km s$^{-1}$, while keeping the other two parameters, namely, the input coronal heating flux $F_h$ and the base coronal density $\rho_\phi$ fixed. The mass-loss rate also increases exponentially with the input coronal heating flux $F_h$ as seen in the top right panel. Conversely, we find in the bottom left panel a power law increase with a slope of ~0.5, in the rate of mass loss with the change in coronal base density $\rho_\phi$. The sublinear dependency of the mass loss on the base density can be understood by the increase of radiative losses with the density (see Equation (14)). In the bottom right panel, we find that the mass-loss dependency to the total Poynting flux is exponential. This exponential behavior of $\dot{M}$ with $F_h$ and $F_p$ is the result of the complex dynamics of the low corona inner boundary where the temperature evolves dynamically with the balance of heating terms. This is in agreement with earlier results that suggested that deposition of more energy at the coronal base leads to higher mass loss (Leer & Holzer 1980). We also find a similar kind of mass-loss variation with different coronal parameters if
<figure><img src="image_8.png" /><figcaption>Figure 6. Variation of the mass loss with different coronal parameters: input transverse velocity $\delta v_\odot$, input coronal heating flux $F_h$, input base density of the corona $\rho_\odot$, and input Poynting flux $F_p$. We initialize these AW-driven wind simulations with the WSO magnetic map of CR 1975. Lines with different colors (blue, black, and red) correspond to different combination of fixed coronal parameters. We mention these combinations in the legend of the figures.</figcaption></figure>

we initialize our solar wind solution with the magnetic map CR 1919 taken at the solar minimum. Please note that we have not considered the contribution of the non-axisymmetric components in this calculation, but we expect them to play a negligible role in the parameterization of the mass loss here.

## 4.2. Mass-loss Rate Scaling Law in the AW model

We have seen that the deposition of energies at the base of the solar corona in the AW model comes from the input coronal heating flux ($F_h$) and other initial parameters, namely, density ($\rho_\odot$), initial field strength ($B_*$), and the input transverse velocity ($\delta v$). This realization allows us to find a single scaling relationship between mass loss $\dot{M}$ and the input energy parameters. We are able to fit all our simulations listed in Tables 1 and 2 into one single law as shown in Figure 7.

$$ \ln \dot{M} = a + b F_h + c \ln \rho + d \ln B_* + e \delta v. \quad (29) $$

The numerical values of the parameters of this scaling law are: $a = -2.23 \pm 0.13$, $b = 1.53e^{-5} \pm 8.29e^{-7}$, $c = 0.46 \pm 0.024$, $d = 0.19 \pm 0.04$, and $e = 0.0403 \pm 0.001$. If one uses the value of the heating flux ($F_h$) in erg cm$^{-2}$ s$^{-1}$, density ($\rho$) in 10$^8$ cm$^{-3}$, transverse velocity ($\delta v$) in km s$^{-1}$ and $B_*$ in Gauss, then one will get the value of mass loss in units of $10^{-14}$ M$_\odot$ yr$^{-1}$ from the scaling law.

Our best-fit formulation indicates that mass loss is positively correlated with the magnetic field strength $B_*$. However, Réville et al. (2015a) used the polytropic solar wind model and found that mass loss is weakly dependent (negative correlation) on the magnetic field strength but increases strongly with the

rotation rate. Our model predicts exactly the opposite about the magnetic field dependency that one can expect from the polytropic wind model. In the case of their fixed temperature polytropic solar wind model, the mass-loss rate reduces with the strength of the magnetic field as more close magnetic loops are formed in their model, which confines the plasma. On the other hand, the energy source term responsible for wind driving changes with the magnetic field strength in the AW model. That is why mass loss is positively correlated with the magnetic field strength in our model. Cohen & Drake (2014) also found a similar trend with their AW model. Finally, note that we perform all our wind simulations with a fixed rotation rate. Because of this, our scaling law does not have any dependency on the rotation rate.

## 4.3. Alfvén Radius Scaling Law in the AW Model

In this paragraph, we derive a scaling law for the Alfvén radius, depending on the value of the open flux instead of the surface magnetic field strength. We recall that we use here an average Alfvén radius defined in Equation (28). As the mass loss and the AM loss occur through the open field lines, it is expected that Alfvén radius and the open flux are closely related.

We follow the topology-independent formulation developed by Réville et al. (2015a, 2016) to derive the scaling law for the average Alfvén radius:

$$ \frac{\langle R_A \rangle}{R_*} = K_1 \left[ \frac{\Upsilon_{\text{open}}}{(1 + f^2 / K_2^2)^{1/2}} \right]^m, \quad (30) $$
<table><caption>**Table 2**

Value of the Mass-loss Rate, AM-loss Rate, and Average Alfvén Radius Obtained from the AW-driven Wind Simulations Initialized with the Different CR Maps Over Cycle 23</caption><caption>AW-driven Wind Model<br>(both axisymmetric and non-axisymmetric components)</caption><thead><tr><th>CR</th><th>Fractional Year</th><th>Poynting Flux (F<sub>p</sub>)<br>(10<sup>4</sup> erg cm<sup>-2</sup> s<sup>-1</sup>)</th><th>Υ<sub>open</sub>/(4π)</th><th>Mass Loss<br>(10<sup>-14</sup> M<sub>☉</sub> yr<sup>-1</sup>)</th><th>AM Loss<br>(10<sup>29</sup> erg)</th><th>Alfvén Radius<br>(R<sub>☉</sub>)</th></tr></thead><tbody><tr><td>1916</td><td>1996.86</td><td>3.04</td><td>120.85</td><td>2.19</td><td>2.60</td><td>3.833</td></tr><tr><td>1927</td><td>1997.68</td><td>6.56</td><td>98.37</td><td>2.71</td><td>3.10</td><td>3.805</td></tr><tr><td>1941</td><td>1998.73</td><td>6.57</td><td>51.17</td><td>2.78</td><td>2.42</td><td>3.324</td></tr><tr><td>1951</td><td>1999.48</td><td>9.52</td><td>43.01</td><td>3.14</td><td>2.8</td><td>3.363</td></tr><tr><td>1953</td><td>1999.63</td><td>11.52</td><td>174.88</td><td>2.75</td><td>4.82</td><td>4.712</td></tr><tr><td>1961</td><td>2000.23</td><td>9.31</td><td>77.51</td><td>2.97</td><td>3.59</td><td>3.919</td></tr><tr><td>1965</td><td>2000.52</td><td>8.59</td><td>79.76</td><td>2.88</td><td>3.66</td><td>4.017</td></tr><tr><td>1970</td><td>2000.89</td><td>10.38</td><td>78.47</td><td>3.10</td><td>3.69</td><td>3.886</td></tr><tr><td>1975</td><td>2001.27</td><td>9.01</td><td>178.73</td><td>2.90</td><td>4.91</td><td>4.638</td></tr><tr><td>1980</td><td>2001.64</td><td>11.62</td><td>136.91</td><td>3.27</td><td>4.79</td><td>4.317</td></tr><tr><td>1985</td><td>2002.02</td><td>12.87</td><td>112.78</td><td>3.36</td><td>4.69</td><td>4.210</td></tr><tr><td>1989</td><td>2002.32</td><td>8.57</td><td>81.05</td><td>3.09</td><td>3.41</td><td>3.748</td></tr><tr><td>1994</td><td>2002.69</td><td>10.45</td><td>99.45</td><td>3.02</td><td>4.31</td><td>4.265</td></tr><tr><td>2005</td><td>2003.51</td><td>6.87</td><td>131.96</td><td>2.80</td><td>3.80</td><td>4.159</td></tr><tr><td>2019</td><td>2004.55</td><td>7.93</td><td>55.46</td><td>2.79</td><td>2.34</td><td>3.260</td></tr><tr><td>2032</td><td>2005.52</td><td>5.17</td><td>66.34</td><td>2.56</td><td>2.41</td><td>3.456</td></tr><tr><td>2043</td><td>2006.35</td><td>3.27</td><td>98.06</td><td>2.23</td><td>2.48</td><td>3.767</td></tr><tr><td>2058</td><td>2007.47</td><td>3.14</td><td>100.63</td><td>2.16</td><td>2.52</td><td>3.848</td></tr><tr><td>2071</td><td>2008.47</td><td>1.71</td><td>56.69</td><td>2.05</td><td>1.39</td><td>2.948</td></tr></tbody></table>

<table><caption>AW-driven Wind Model<br>(only axisymmetric component)</caption><thead><tr><th>CR</th><th>Fractional Year</th><th>Poynting Flux (F<sub>p</sub>)<br>(10<sup>4</sup> erg cm<sup>-2</sup> s<sup>-1</sup>)</th><th>Υ<sub>open</sub>/(4π)</th><th>Mass Loss<br>(10<sup>-14</sup> M<sub>☉</sub> yr<sup>-1</sup>)</th><th>AM Loss<br>(10<sup>29</sup> erg)</th><th>Alfvén Radius<br>(R<sub>☉</sub>)</th></tr></thead><tbody><tr><td>1916</td><td>1996.86</td><td>2.33</td><td>113.54</td><td>2.13</td><td>2.37</td><td>3.760</td></tr><tr><td>1927</td><td>1997.68</td><td>3.21</td><td>127.01</td><td>2.25</td><td>2.72</td><td>3.921</td></tr><tr><td>1941</td><td>1998.73</td><td>3.18</td><td>72.06</td><td>2.18</td><td>2.14</td><td>3.533</td></tr><tr><td>1951</td><td>1999.48</td><td>3.15</td><td>60.88</td><td>2.25</td><td>2.14</td><td>3.478</td></tr><tr><td>1953</td><td>1999.63</td><td>3.98</td><td>72.39</td><td>2.44</td><td>2.63</td><td>3.709</td></tr><tr><td>1961</td><td>2000.23</td><td>2.93</td><td>79.23</td><td>2.27</td><td>2.52</td><td>3.756</td></tr><tr><td>1965</td><td>2000.52</td><td>2.96</td><td>80.09</td><td>2.23</td><td>2.42</td><td>3.713</td></tr><tr><td>1970</td><td>2000.89</td><td>2.67</td><td>71.34</td><td>2.17</td><td>2.14</td><td>3.545</td></tr><tr><td>1975</td><td>2001.27</td><td>3.51</td><td>187.86</td><td>2.45</td><td>4.49</td><td>4.834</td></tr><tr><td>1980</td><td>2001.64</td><td>3.26</td><td>37.22</td><td>2.37</td><td>3.21</td><td>4.146</td></tr><tr><td>1985</td><td>2002.02</td><td>3.72</td><td>111.87</td><td>2.41</td><td>3.25</td><td>4.145</td></tr><tr><td>1989</td><td>2002.32</td><td>2.34</td><td>81.15</td><td>2.11</td><td>2.11</td><td>3.573</td></tr><tr><td>1994</td><td>2002.69</td><td>3.64</td><td>90.73</td><td>2.37</td><td>2.91</td><td>3.961</td></tr><tr><td>2005</td><td>2003.51</td><td>2.62</td><td>91.38</td><td>2.14</td><td>2.39</td><td>3.772</td></tr><tr><td>2019</td><td>2004.55</td><td>2.17</td><td>60.15</td><td>1.99</td><td>1.54</td><td>3.141</td></tr><tr><td>2032</td><td>2005.52</td><td>1.77</td><td>59.69</td><td>2.01</td><td>1.54</td><td>3.116</td></tr><tr><td>2043</td><td>2006.35</td><td>1.96</td><td>58.24</td><td>2.06</td><td>1.36</td><td>2.901</td></tr><tr><td>2058</td><td>2007.47</td><td>1.94</td><td>59.05</td><td>1.95</td><td>1.36</td><td>2.985</td></tr><tr><td>2071</td><td>2008.47</td><td>1.55</td><td>48.98</td><td>2.01</td><td>1.26</td><td>2.829</td></tr></tbody></table>

where $Υ_{open}$ is a magnetization parameter, which takes the value of open flux into account:

$$ \Upsilon_{\text{open}} = \frac{\Phi_{\text{open}}^2}{R_*^2 \dot{M} v_{\text{esc}}} \qquad (31) $$

where $R_*$ is the solar (stellar) radius, $\dot{M}$ is the mass loss, and $v_{esc} = \sqrt{(2G_{\odot}M_{\odot})/R_{\odot}}$ is the escape velocity. $Υ_{open}$ carries the information regarding the open field region and dead zones. It actually quantifies the value of the open flux in our simulation compared to the surface magnetic flux. In Equation (30), $K_1$, $K_2$ are the constant and $f$ is the fraction of stellar breakup rate (see also Matt & Pudritz (2008); Matt et al. (2012) for this kind of

formulation). The fraction of the breakup rate is defined as the ratio between the stellar rotation rate at the equator and the Keplerian speed that is represented by the formula:

$$ f = \frac{\Omega_{\odot} R_{\odot}^{3/2}}{\sqrt{GM_{\odot}}} \qquad (32) $$

In this study, all our simulations are performed with the constant rotation rate $\Omega_{\odot}$, which means the fraction of the breakup rate $f$ is constant for this study. In our all simulations, we use the value of $f$ as 0.004 (solar value). As there is no variation in the value of breakup rate $f$ in our simulations, it is
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Single law fit for the mass loss obtained from the AW-driven solar wind model, following Equation (29).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Single law fit for the average Alfvén radius obtained from the AW-driven solar wind model (blue dots) according to Equation (30). The best-fitted parameters for the AW-driven model are: $K_1 = 0.20 \pm 0.004$ and $m = 0.41 \pm 0.002$. The scaling laws of Réville et al. (2016) and Pantolmos & Matt (2017) are shown in orange and green, respectively. We see that despite different fitted parameters, the results of the AW model are compatible with the law of Réville et al. (2016) and Pantolmos & Matt (2017), obtained with polytropic simulations.</figcaption></figure>

not necessary to fit the value of $K_2$ in Equation (30). Moreover, Réville et al. (2015a, 2016) found a value of $K_2 = 0.06$ and $\sqrt{1+f^2/K_2^2} \sim 1.0022$. In the following, we thus assume the denominator on the right-hand side of Equation (30) is unity.

We are now able to fit Equation (30) with all our simulations listed in Tables 1 and 2. The fit is shown in Figure 8 in blue. The value of the fitted parameters are: $K_1 = 0.20 \pm 0.004$ and

$m = 0.41 \pm 0.002$. As in Réville et al. (2015a, 2016), we find that the open flux law is able to fit well all simulations, independently of the (complex) topology of the surface magnetic field. Réville et al. (2016) found the fitted parameter values as: $K_1 = 0.55 \pm 0.05$, $K_2 = 0.06 \pm 0.01$ and $m = 0.31 \pm 0.02$. Despite the fact that the two sets of coefficients are different, we can see in Figure 8 that the law
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Snapshots of the coronal magnetic field evolution with the AW model at different times over cycle 23. Times *t* = 1997.69, 1998.73, 1999.48, 2000.89, 2001.64, 2002.32, 2003.51, 2004.55, 2005.52, and 2008.47 are given in fractional years, and start from the first minimum of cycle 23. The color in the background shows the quantity *u*·*B*/(c_s||*B*||), which is basically the solar wind velocity projected onto the magnetic field in units of the Mach number. Outgoing poloidal magnetic field lines are represented by white solid lines and ingoing poloidal magnetic field lines by white-dashed lines. Here, we only represent the first 4 solar radii.</figcaption></figure>

of Réville et al. (2016) remains consistent with our AW simulations. In the current study, the exponent *m* is driven to higher values (and *K*₁ to lower values) because of the higher range of the average Alfvén radius distributions. Pantolmos & Matt (2017) have shown that higher coronal base temperature could lead to a higher *m* exponent, and we plot the open flux law they obtained for a coronal base temperature of 2.4 MK (the base temperature in Réville et al. (2016) is set to 1.5 MK). Typical maximum temperatures in the AW model are close to 2.0 MK and this is fully consistent with our simulation data lying in between the two polytropic open flux laws (orange and green lines) for the AM. This study hence shows that the results obtained with polytropic models and AWs models are close, as far as the AM loss and its scaling with the open flux are concerned.

**5. Evolution of the Solar Corona and its Properties During Cycle 23**

In this section, we perform a quasi-static study of the evolution of the solar corona along cycle 23. We use both the polytropic and AW models, and we have selected 19 WSO synoptic maps at different epochs of solar cycle 23 for this purpose. We have already discussed the impact of non-axisymmetric components over the total surface energy (see Section 2.3) and we will consider them here using Equation (21). We show our selected initial magnetic field topologies in Appendix B. We fix the other initial coronal parameters, namely, density, transverse velocity, and the coronal heating flux for the AW model. We take the value of density as 3 × 10⁸ cm⁻³, transverse velocity as 30 km s⁻¹, and

coronal heating flux $F_h$ as 5 × 10⁴ erg cm⁻² s⁻¹. On the other hand, we fix the density of the solar corona to 3 × 10⁸ cm⁻³ for the polytropic solar wind model. As the initial assumptions and parameters are different for both models, we cannot directly compare the result in terms of absolute amplitude. However, we can compare the trends of different physical quantities, namely, the mass loss, AM loss, and average Alfvén radius along cycle 23. Please note that we compared the polytropic and AW models in the previous section when both models satisfied the same mass-loss criteria.

Figure 9 shows the time evolution of the coronal magnetic field at different epochs of solar cycle 23, zoomed close to the Sun. Figures 9(a–j) represent different epochs of solar cycle 23. The color background of Figure 9 represents the quantity *u*·*B*/(c_s||*B*||), which is basically the solar wind velocity projected on the magnetic field in units of the Mach number. One will be able to easily identify the change in magnetic field polarity using this quantity. The regions of the magnetic polarity transition correspond to current sheets. Figure 9 clearly shows the changes in the magnetic field topology over solar cycle 23. At the beginning of solar cycle 23 (Figure 9(a)), the magnetic field is mostly dipolar. Magnetic field lines are going inward in the northern coronal hole and going outward in the southern coronal hole. As the solar cycle progresses, the surface magnetic field structure becomes more complex, sculpting the corona. We find the occurrence of pseudo streamers at various latitudes as the cycle progresses (see Figures 9(b–i)) Coronal holes with magnetic field lines going inward also start to appear in the southern hemisphere as the cycle progresses (Figures 9(c, d)). Finally, near the end of solar
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Time–latitude diagram of the radial solar wind velocity over cycle 23 in km s⁻¹ at *r* = 20*R<sub>☉</sub>*. The top panel is from the AW model, while the bottom panel is from the polytropic model. Please note the large difference in velocity in the breadth of the amplitude in the AW case vs. the polytropic case (350 km s⁻¹ vs. 100 km s⁻¹).</figcaption></figure>

cycle 23, the magnetic field returns again to almost dipolar but
with a completely opposite configuration (see Figure 9(j)).
Magnetic field lines are now fully going inward in the southern
coronal hole and going outward in the northern coronal hole.

Next, we study the variation of wind speed during solar
cycle 23. As wind speed is mostly radial at 20 solar radii, we
specifically focus on the radial wind speed variations. The top
panel of Figure 10 shows the radial wind speed evolution at 20
solar radii during solar cycle 23. At the beginning phase of the
solar cycle, fast wind originates from the high latitudes, while
slow wind originates at the lower latitudes. As the solar cycle
progresses, the slow wind starts to appear at higher latitudes,
and a fast wind structure is also observed at lower latitudes. At
the end phase of the solar cycle, again fast wind appears at high
latitudes and slow wind at low latitudes. The time–latitude
diagram of the radial wind speed (top panel of Figure 10) also
indicates that wind speed is often asymmetric. This is due to the
presence of strong hemispheric asymmetries in the surface
magnetic field of individual synoptic maps during solar cycle
23. The southern hemisphere is more active compared to the
northern hemisphere during solar cycle 23. We also note that
sometimes there is a significant drop in the wind velocity at the
poles in the two hemispheres. Khabarova et al. (2017)
interpreted this kind of drop (also observed in the Ulysses
data) as the presence of high-latitude current sheets. The
bottom panel of Figure 10 shows the same obtained from the
polytropic wind model. Although we find that the behavior of
radial wind speed variation in the polytropic scenario is almost
similar to the AW one, the breadth of the distribution of the
wind speed is in a much narrower range compared to the
AW one.

Finally, we focus on the time evolution of different integrated
quantities, namely, mass loss (Equation (23)), AM loss
(Equation (25)), and the average Alfvén radius (Equation (28))
over solar cycle 23. Their evolution is displayed in Figure 11 for
the polytropic (red circles) and AW (blue circles) models. We

have also overplotted the variation of monthly sunspot numbers
for a better understanding of our results.

A large number of studies have been performed to estimate the
solar mass-loss and AM-loss rates along the solar cycle (Usmanov
et al. 2000; Riley et al. 2001; Pinto et al. 2011; Réville &
Brun 2017; Perri et al. 2018). These earlier studies have reported a
wide range of values for solar mass-loss and AM-loss rates. This is
probably due to the fact that all these studies have different input
physics, such as the inclusion of specific coronal heating functions
or the polytropic assumption. However, we note that in all these
models it is possible to reproduce the observed solar mass loss by
adjusting the input parameters. In this study, we have estimated the
mass-loss and AM-loss rates from both the AW and polytropic
solar wind models. Tables 2 and 3 list the values of different
physical quantities obtained from both models along cycle 23.
Our calculated mass loss from the AW model varies
from 2.1 × 10⁻¹⁴ M☉ yr⁻¹ at the solar minimum to 3.5 ×
10⁻¹⁴ M☉ yr⁻¹ at the solar maximum (see the top panel of
Figure 11; blue curve). The average mass loss over solar cycle 23
using this AW-driven wind simulation is 2.8 × 10⁻¹⁴ M☉ yr⁻¹.
Our calculated mass loss is the minimum at the minimum of the
solar cycle and the maximum at the maximum of the solar cycle.
McComas et al. (2008) estimated the mass loss from the Ulysses
spacecraft data and found a maximum mass-loss value of
3.1 × 10⁻¹⁴ M☉ yr⁻¹ during 1992–1993 and a minimum value of
2.3 × 10⁻¹⁴ M☉ yr⁻¹ during 1991. Our minimum value of the
mass loss is approximately the same as the observed mass-loss
value, while the maximum value is higher than the observed mass-
loss value. We found an almost 75% variation in the mass loss
between the solar cycle minimum and maximum, which is quite
larger than that those found in earlier studies. Finally, we note that
the positive correlation between the mass loss and the solar cycle
in the AW model stems from the fact that the Poynting flux
directly depends on the surface magnetic energy of the
magnetic maps.
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Time evolution of the mass loss, AM loss, and the average Alfvén radius during cycle 23 obtained from the polytropic and AW-driven wind simulations. The blue lines correspond to the polytropic wind simulations, while the red lines correspond to the AW-driven wind simulation. The black lines represent the monthly sunspot number during cycle 23. Here, we have included the contribution of non-axisymmetric components in both the axisymmetric polytropic and AW-driven wind simulations with the observed magnetic maps following the quadrature addition method (Equation (21)).</figcaption></figure>

On the other hand, we find a completely different behavior in the polytropic scenario. The top panel of Figure 11 indicates that the mass loss weakly varies along the cycle in the polytropic scenario (red curve), with a slight decrease for the first 3 yr and a slight increase during the descending phase of the cycle. The mass loss varies from $1.65 \times 10^{-14}$ to $2.1 \times 10^{-14} M_{\odot} \text{yr}^{-1}$ (the variation is at most 30%) in the polytropic scenario. In recent times, different studies have included the effect of magnetic variability in their wind simulations. Pinto et al. (2011) and Perri et al. (2018) used a kinematic solar dynamo model to drive their axisymmetric polytropic solar wind simulations. Both of these studies found a smaller variation compared to that of our AW model between the mass-loss rates obtained from the solar maximum and minimum, respectively. We also find a lower mass-loss variation in the polytropic scenario compared to the AW one. In our study the polytropic model shows an anticorrelation between mass loss and solar cycle because a larger magnetic energy generally means that more close loops can be formed in the polytropic approximation, and thus the mass loss can be expected to be smaller at the cycle maximum. Pinto et al. (2011)

nevertheless found that a mass loss is correlated with the solar cycle, due to the surface energy-magnetic cycle phase correlations in the Babcock-Leighton dynamo models they used. Likewise, Perri et al. (2018) found yet another behavior with a phase quadrature correlation that comes from the phase quadrature between the cycle phase and surface magnetic field energy embedded in their $\alpha-\Omega$ dynamo model. When using observed magnetograms, we thus show here that the polytropic model is not good enough and tends to predict a probably wrong anticorrelation between the cycle phase and mass loss.

The middle panel of Figure 11 shows the time evolution of the AM loss during solar cycle 23 from both the AW and polytropic solar wind models. In the AW model, AM loss is correlated with the solar cycle (see the middle panel of Figure 11; blue curve). The AM loss varies from $10^{29}$ erg at the solar minimum to $5.2 \times 10^{29}$ erg at the solar maximum. The variations in the AM loss are also higher compared to earlier studies by Pinto et al. (2011) and Perri et al. (2018) (see the middle panel of Figure 11). The average AM loss obtained from our AW model during solar cycle 23 is around $4 \times 10^{29}$ erg. On the other hand, we found an anticorrelation
<table><caption>**Table 3**
Value of the Mass Loss Rate, AM-loss Rate, and Average Alfvén Radius Obtained from the Polytropic Solar Wind Simulations Initialized with the Different CR Maps Over Cycle 23</caption><thead><tr><th colspan="5">Polytropic Wind Model</th></tr><tr><th colspan="5">(both axisymmetric and non-axisymmetric components)</th></tr><tr><th>CR</th><th>Fractional Year</th><th>Mass Loss<br>(10<sup>-14</sup> M<sub>☉</sub> yr<sup>-1</sup>)</th><th>AM Loss<br>(10<sup>29</sup> erg)</th><th>Alfvén Radius<br>(R<sub>☉</sub>)</th></tr></thead><tbody><tr><td>1916</td><td>1996.86</td><td>2.03</td><td>2.32</td><td>3.818</td></tr><tr><td>1927</td><td>1997.68</td><td>1.98</td><td>2.75</td><td>3.822</td></tr><tr><td>1941</td><td>1998.73</td><td>2.01</td><td>1.71</td><td>3.283</td></tr><tr><td>1951</td><td>1999.48</td><td>1.89</td><td>0.93</td><td>2.506</td></tr><tr><td>1953</td><td>1999.63</td><td>1.90</td><td>1.71</td><td>3.384</td></tr><tr><td>1961</td><td>2000.23</td><td>1.65</td><td>1.61</td><td>3.512</td></tr><tr><td>1965</td><td>2000.52</td><td>1.91</td><td>1.38</td><td>3.031</td></tr><tr><td>1970</td><td>2000.89</td><td>1.70</td><td>1.68</td><td>3.542</td></tr><tr><td>1975</td><td>2001.27</td><td>2.00</td><td>2.42</td><td>3.941</td></tr><tr><td>1980</td><td>2001.64</td><td>1.98</td><td>2.39</td><td>3.926</td></tr><tr><td>1985</td><td>2002.02</td><td>1.97</td><td>1.90</td><td>3.515</td></tr><tr><td>1989</td><td>2002.32</td><td>1.81</td><td>1.92</td><td>3.671</td></tr><tr><td>1994</td><td>2002.69</td><td>1.96</td><td>1.74</td><td>3.359</td></tr><tr><td>2005</td><td>2003.51</td><td>2.02</td><td>1.96</td><td>3.540</td></tr><tr><td>2019</td><td>2004.55</td><td>1.89</td><td>1.77</td><td>3.469</td></tr><tr><td>2043</td><td>2006.35</td><td>2.02</td><td>1.84</td><td>3.418</td></tr><tr><td>2058</td><td>2007.47</td><td>2.01</td><td>1.74</td><td>3.303</td></tr><tr><td>2071</td><td>2008.47</td><td>2.02</td><td>1.73</td><td>3.293</td></tr></tbody></table>

of the AM loss with the solar cycle in the polytropic wind model scenario (see the middle panel of Figure 11; red curve). This means that the AM-loss rate is the minimum when the star is more multipolar. Earlier studies based on polytropic wind model also found the same (see Pinto et al. 2011; Perri et al. 2018). We also note a larger AM-loss variation in the AW model compared to the polytropic one.

The last panel of Figure 11 shows the time evolution of the Alfvén radius during solar cycle 23 from both models. In the AW model, we find a maximum average Alfvén radius of around 4.8 solar radii at the cycle maximum, while the minimum average Alfvén radius is around 3 solar radii at the cycle minimum. We note that our average Alfvén radius is smaller compared to what one can expect from Helios mission. Pizzo et al. (1983) and Marsch & Richter (1984) estimated the average Alfvén radius from the AM obtained from the Helios mission. They found the value of the average Alfvén radius to be 12–13.6 solar radii at the cycle minimum and 13.6–16.6 solar radii at the cycle maximum. However, we find different time evolution behavior of the Alfvén radius in the polytropic scenario. The Alfvén radius decreases in the first few years of the solar cycle (see the bottom panel of Figure 11; red curve) Pinto et al. (2011) and Perri et al. (2018) found that the Alfvén radius is anticorrelated with the solar cycle, which means that the Alfvén radius is the minimum at the cycle maximum and the minimum at the cycle maximum. However, Réville & Brun (2017) used a 3D wind model based on polytropic assumption and found that the Alfvén radius increased from $5R_{☉}$ at the cycle minimum to $7R_{☉}$ at the cycle maximum. We note that the average Alfvén radius is closely related to the ratio of $J/M$. As a matter of fact, both $J$ and $M$ can vary differently along the solar cycle, depending on the details of the wind model. It can thus be expected that the average Alfvén radius may show positive and negative correlations with the magnetic cycle, depending on the modeling details.

Finally, we note that our AW model predicts significantly different trends compared to wind models based on the polytropic assumption. In principle, this difference could originate either from the physical model itself (polytropic and AW assumptions), or from our choice of quadratically summing the non-axisymmetric contributions in our 2.5D approach. To test this, we also perform 2.5D simulations considering only the axisymmetric components of the surface magnetic field (see Appendix A for a discussion regarding the wind simulations where we only consider the axisymmetric components of the surface magnetic field). Specifically, we first note that the AW model predicts almost no variation of mass loss along the magnetic cycle when considering axisymmetric modes only (see Figure 15). This behavior then becomes more similar to the one found with the polytropic model (Figure 11), which shows very similar trends whether or not non-axisymmetric modes are taken into account. Interestingly, in this case the AM loss also becomes almost flat along the solar cycle, and thus the Alfvén radius trend is actually very similar with and without non-axisymmetric modes in the AW model. We therefore conclude that the difference in trends between the AW and polytropic models comes from a fundamental difference in how both models depend on the magnetic structure at the surface. Indeed, the surface magnetic field topology contributes directly to the input energy source in the AW model through the Poynting flux, which is not the case for the polytropic model where the wind is mostly driven by a fixed pressure gradient in our study. That is why the AW-driven wind model is much more sensitive to the complex surface magnetic field topology compared to the polytropic one.

## 6. Summary and Conclusions

We performed this study to understand the behavior and properties of the solar wind obtained from an AW-driven solar wind model. We compared the AW-driven wind model results with a polytropic approach. We found significant differences in the behavior and properties of the solar wind obtained from these two models. We found that the AW-driven model is able to reproduce both the observed speed bimodality and the observed breadth of the speed distribution of the solar wind, while the polytropic model is only able to produce a weakly pronounced bimodality. Our study reconfirms the necessity of an additional acceleration mechanism (like AWs’ dissipation) on top of the thermal one in a wind model to explain the observed speed bimodality and breadth of distribution, as predicted by earlier studies. AWs act here as additional acceleration and heating mechanisms in our model, which helps in reproducing the observed bimodality.

Although a polytropic wind model cannot explain the observed bimodality, both the polytropic and AW-driven wind models show similar kinds of wind speed distribution (slow wind near streamers and the fast wind away from streamers). Away from streamers, open magnetic field lines allow the wind to achieve a comparatively faster speed. Some earlier studies have also suggested that the open magnetic field in the cooler regime (coronal hole) of the corona reconnects with the closed magnetic loops near the base, imparting extra energy into the overlying corona. This extra energy can act as an additional acceleration mechanism and increases the wind speed in the cooler (coronal hole) regime of the solar corona (Fisk 2003; Gloeckler et al. 2003; Schwadron et al. 2014).
One interesting observational feature is the anticorrelation between the final solar wind velocity and the coronal source temperature (Geiss et al. 1995). However, thermally driven, polytropic winds show a positive positive correlation between the final wind velocity and the coronal source temperature, which is against observations (see Figure 4). We have investigated this issue for our AW-driven wind model. We find a clear anticorrelation between the final wind velocity and the coronal source temperature. This result is probably due to the consequence of the strong temperature dependence of the thermal conduction term or due to the different radial distributions of the heating rate as suggested before (Schwadron & McComas 2003; Cranmer & van Ballegooijen 2005; Cranmer 2010; van der Holst et al. 2010). However, we get a positive correlation between the final wind velocity and the temperature from the AW model if we consider the temperature away from the bottom coronal boundary. Shifting from a negative to a positive correlation in the AW model occurs at different heights of the solar corona, depending on the assumed initial coronal parameters and the surface magnetic field.

We find that the global properties of the modeled wind, like the mass-loss rate, the AM-loss rate, and the average Alfvén radius, are sensitive to the different coronal parameters. Specifically, the surface magnetic field topology plays an important role in the variation of the global properties through the Poynting flux. We also proposed a single scaling law to compute the mass loss (Equation (29)) and the average Alfvén radius (Equation (30)) in AW-driven wind models. We find that the mass-loss scaling law is very different from the one derived with the polytropic approach, but the average Alfvén radius scaling law (Réville et al. 2015b, 2016) seems actually relatively independent of the wind model. Finally, we note that that mass loss is positively correlated with the initial surface magnetic field strength in the AW-driven wind model.

The evolution of different magnetic modes of the surface magnetic field over an 11 yr solar cycle induces a different type of complexity in the solar corona at different epochs of the solar cycle. The surface magnetic field structure at the minimum of the solar activity is mostly dipolar. We also find that axisymmetric components of the surface magnetic fields are the major contributor to the total surface energy at the activity minimum. As the solar wind carries the topological information of the surface magnetic field, one can expect less complexity in the coronal structure at the activity minimum. We also find a very organized behavior of the solar wind at the cycle minimum—slow wind at the equator and fast wind at the high latitudes. However, the surface magnetic field is much more complex and multipolar at the cycle maximum. On the contrary to the activity minimum, non-axisymmetric components are the major contributor to the total surface energy at the cycle maximum and we find much a more complex wind distribution at the cycle maximum. This is consistent with Ulysses observations that revealed similar structures in the solar wind (McComas et al. 2008).

As the solar corona evolves over the 11 yr solar cycle, different global wind properties are also expected to evolve. We find that the mass-loss rate and the AM-loss rate are strongly correlated with the solar cycle, while this is not the case with the polytropic wind model. Previous studies based on the polytropic wind model found a weak anticorrelation of the AM-loss rate with the solar cycle (Pinto et al. 2011; Perri et al. 2018). We also find a large difference between the physical

quantities calculated at the cycle minimum and maximum from the AW-driven wind model when we considered the non-axisymmetric components of the surface magnetic field. However, this difference is much less when we consider only the axisymmetric components of the surface magnetic field. These differences between the polytropic and the AW-driven wind models are basically due to the difference in the fundamental assumption of these models. In the polytropic wind model, there is no direct contribution of the magnetic field on the input energy source. Input energies mostly come from internal energy through the pressure gradient. However, in the AW-driven wind model, the magnetic field enters directly into the input energy sources through the Poynting flux. That is why different physical quantities are much more sensitive to the assumed surface magnetic fields in the AW model.

Finally, we believe that the AW-driven wind model is a step toward understanding the global properties of the solar corona in a better way. Please note that we performed this study in 2.5D, assuming axisymmetry, yet accounting for the contribution of non-axisymmetric energy. It nevertheless lacks the information of the full three-dimensional topology of the solar surface magnetic field. Consideration of the full three-dimensional topology would bring additional information and help us better understand the global properties of the solar wind. We leave this aspect for our future studies.

We thank Rui Pinto and Susanna Parenti for useful discussions and suggestions. We thank the Université Paris-Saclay (IRS SPACEOBS grant), ERC Synergy grant 810218 (Whole Sun project), INSU/PNST, and CNES Solar Orbiter for supporting this research. We acknowledge access to supercomputers through GENCI (grants 40410133, 60410133, and 80810133).

# Appendix A

## Influence of Non-axisymmetric Surface Magnetic Field Components on the Wind Solutions

We followed two approaches to model the solar wind along the solar cycle with the AW and polytropic models. In one scenario, we performed an axisymmetric AW-driven wind simulation with decomposed surface magnetic field data obtained from CR 1975 where we had not included the contribution of non-axisymmetric components. In the other scenario, we performed the same set of simulations but considered the contribution of non-axisymmetric components following Equation (21).

First, let us compare the radial velocity variations at 0.1 au obtained with the AW models with the observed Ulysses radial velocity variation. Figure 12 shows the comparison, which suggests that we can explain the observed Ulysses radial velocity variation in a better way when we consider the contribution of both axisymmetric and non-axisymmetric components of the surface magnetic fields to initialize the wind simulations (lower panel).

We aim next to characterize the impact of non-axisymmetric surface magnetic field components on the wind solutions obtained from the both AW-driven wind model and polytropic solar wind model. Figures 13(a), (b) show the poloidal velocity in the meridional plane obtained from the polytropic and AW-driven solar wind models when initialized with the only decomposed axisymmetric components of the surface magnetic field for the map of CR 1975. Figures 13(c), (d) represent the
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Histograms of the radial wind velocity obtained from the Ulysses data (blue color) and our AW-driven wind simulations (orange color). For the figure in the top panel, we performed the axisymmetric wind simulations considering only the axisymmetric components of the initial magnetic map. While for the figure in the bottom panel, we performed the simulations considering the contribution of non-axisymmetric components following Equation (21).</figcaption></figure>

<table><caption>**Table 4**
Mass Loss for the Solar Cycle 23 Maximum (CR 1975) and Minimum (CR 1919)</caption><thead><tr><th rowspan="2">Mass Loss (M<sub>☉</sub> yr<sup>-1</sup>)</th><th colspan="2">CR 1975</th><th colspan="2">CR 1919</th></tr><tr><th>Only Axisymmetric Components</th><th>Considering Contribution of Non-axisymmetric Components</th><th>Only Axisymmetric Components</th><th>Considering Contribution of Non-axisymmetric Components</th></tr></thead><tbody><tr><td>Polytropic wind simulation</td><td>2.95 × 10<sup>-14</sup></td><td>2.75 × 10<sup>-14</sup></td><td>2.67 × 10<sup>-14</sup></td><td>2.67 × 10<sup>-14</sup></td></tr><tr><td>AW-driven wind simulation</td><td>2.14 × 10<sup>-14</sup></td><td>2.68 × 10<sup>-14</sup></td><td>2.38 × 10<sup>-14</sup></td><td>2.37 × 10<sup>-14</sup></td></tr></tbody></table>

same when the models are initialized with the decomposed axisymmetric field components for the map of CR 1919. The figure is the same as Figure 2, albeit now we consider only the axisymmetric modes of the synoptic maps. We obtain a mass loss of around $2.14 \times 10^{-14}$ and $2.38 \times 10^{-14}$ M$_{☉}$ yr$^{-1}$ from the AW-driven solar wind model when initialized with the CR 1975 map and CR 1919 map, respectively. On the other hand, we get a mass loss of around $2.95 \times 10^{-14}$ and $2.67 \times 10^{-14}$ M$_{☉}$ yr$^{-1}$ from the polytropic solar wind model when initialized with the CR 1975 map and the CR 1919 map, respectively (see Table 4 for a detailed summary of mass loss in all scenarios).

Interestingly, we note that the value of mass loss increases at the solar maximum (the CR 1975 case) and remains the same at the cycle minimum (the CR 1919 case) in the polytropic scenario when we only considered the

contribution of axisymmetric components. However, we find a decrease in the mass loss at the solar maximum (the CR 1975 case) and almost the same mass loss at the cycle minimum (the CR 1919 case) in the AW-driven model scenario. We have already explained that non-axisymmetric components is not very significant at the minimum of the solar cycle; that is why we get the same mass loss at the cycle minimum in both cases. However, the trend of the mass-loss change at the solar maximum is the opposite in the polytropic and AW-driven model scenarios when we compare the models with and without the contribution of non-axisymmetric components (see Table 4). This is the case even though we find the higher poloidal velocity for both the polytropic and AW-driven wind model scenarios when we consider the contribution of non-axisymmetric components (see Figure 14).
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Same as in Figure 2. However, here we perform the wind simulations considering only the axisymmetric components.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 14.** Histograms of the radial wind velocity obtained from both the polytropic and AW-driven wind simulations. Left panel: figure is obtained from the polytropic wind simulations, while right panel figure is from AW-driven wind simulations. In both categories, we consider two scenarios. In one scenario, we initialize the wind model considering only axisymmetric components of the decomposed surface magnetic field obtained from the magnetic map of CR 1975. In another scenario, we perform the wind simulation considering the contribution of non-axisymmetric components following Equation (21). In both categories (polytropic and AW), we find a higher poloidal velocity when we consider the contribution of non-axisymmetric components.</figcaption></figure>

We finally characterize the impact of non-axisymmetric components on the evolution of physical quantities obtained from our AW-driven wind simulations along the solar cycle. Figure 15 shows the time evolution of mass loss, AM loss, and the average Alfvén radius obtained from the AW-driven wind

models. In one scenario [Alfvén(O)], we performed the simulations where we did not consider the contribution of the non-axisymmetric component. In the other scenario [Alfvén (M)], we perform the wind simulations considering the contributions of non-axisymmetric components following the
<figure><img src="image_17.png" /><figcaption>**Figure 15.** Time evolution of the mass loss, AM loss, and the average Alfvén radius during cycle 23 obtained from the two AW-driven wind simulation scenarios. The red line [Alfvén (O)] corresponds to the scenario where we initialize the wind model considering only axisymmetric components of the decomposed surface magnetic fields. The blue line [Alfvén (M)] corresponds to the scenario where we perform the wind simulations considering the contribution of non-axisymmetric component following the quadrature addition method (Equation (21)). We get a significantly higher mass loss and AM loss at the maximum of the solar cycle when we consider the contribution of non-axisymmetric component.</figcaption></figure>

quadrature addition method (Equation (21)). We find that mass loss is correlated with the solar cycle in both cases. However, we found a much smaller variation in the mass loss (only 25%) between the cycle maximum and minimum in the Alfvén (O) scenario (red curve) compared to the Alfvén (M) scenario (blue curve). There are almost 75% variations in the mass loss between the cycle maximum and minimum in the Alfvén (M) scenario (blue curve). We note that mass loss is almost the same at the cycle minimum in both scenarios, while it increases more at the cycle maximum when we consider the contribution of non-axisymmetric components. This is because non-axisymmetric components play a significant role in the cycle maximum compared to the cycle minimum. Similarly, we also note that variation of the AM loss between the cycle maximum and minimum is much less in the model where we have not considered the contribution of non-axisymmetric components (Alfvén (O) scenario). The middle panel of Figure 15 also indicates a slightly decreasing trend of the AM loss in the Alfvén (O) scenario at the beginning of the solar cycle. The last panel of Figure 15 indicates that the Alfvén radius is the maximum at the cycle maximum and the minimum at the cycle minimum in both scenarios. On the one hand, the inclusion of

non-axisymmetric components has a significant impact on the time evolution of the mass loss and AM loss. On the other hand, the average Alfvén radius is only mildly affected. We have previously shown that one will underestimate the total surface energy significantly at the solar maximum if they do not consider the contribution of non-axisymmetric components. Thus, it is necessary to include the contribution of non-axisymmetric components.

## Appendix B: Initial Magnetic Field Topologies for Our Selected Magnetic Maps

In this appendix, we show the initial magnetic field topologies from our selected CR maps during solar cycle 23. Figure 16 shows how the initial magnetic field topology (computed with potential field extrapolation with a source surface) varies during cycle 23. At the maximum of the solar cycle (Figures 16 (i, j, k)), we find a complex magnetic field topology. On the other hand, the magnetic field structure is mostly dipolar at the minimum of the solar cycle. Please note that we considered the contribution of both axisymmetric and
<figure><img src="image_18.png" /><figcaption>**Figure 16.** Magnetic field lines of our initial topologies obtained from our selected CR maps during solar cycle 23. The color in the background shows the radial magnetic field ($B_r$) in G. Outgoing poloidal magnetic field lines are represented by a white solid line and ingoing poloidal magnetic field lines by a white-dashed line. We note that the initial magnetic field topologies are dipolar at the minimum of the solar cycle, while we note a complex magnetic topology at the maximum of the solar cycle. Here, we only represent the first 4 solar radius.</figcaption></figure>
non-axisymmetric components following Equation (21) to get
these initial magnetic field topologies.

ORCID iDs

Soumitra Hazra https://orcid.org/0000-0002-7069-1711

Victor Réville https://orcid.org/0000-0002-2916-3837

Barbara Perri https://orcid.org/0000-0002-2137-2896

Antoine Strugarek https://orcid.org/0000-0002-9630-6463

Allan Sacha Brun https://orcid.org/0000-0002-1729-8267

Eric Buchlin https://orcid.org/0000-0003-4290-1897

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