3.1. Flux and mass conservation

To simplify the analysis, solar rotation is neglected, and the magnetic field is taken to be radial, except in the corona, where open magnetic-field lines fan out to fill the space above closed magnetic loops at lower altitudes. Mathematically,

(3.2)\begin{equation} B(r) = \frac{\bar{B} \eta(r) R_{{\odot}}^2}{r^2}, \end{equation}

where $\eta (r)$ is the local super-radial expansion factor, which approaches 1 when $r/R_{\odot } \gg 1$, and $\bar {B}$ is the magnetic-field strength that would arise at the photosphere in the absence of super-radial expansion (i.e. if $\eta (r)$ were unity everywhere.) Given (3.2), the magnetic-field strength at the coronal base is

(3.3)\begin{equation} B_\textrm{b} \equiv B(r_\textrm{b}) = \bar{B} \eta_\textrm{b} \psi, \end{equation}

where here and in the following a ‘b’ subscript indicates that the subscripted quantity is evaluated at $r=r_\textrm {b}$, and

(3.4)\begin{equation} \psi \equiv \frac{R_{{\odot}}^2}{r_\textrm{b}^2} = 0.9901. \end{equation}

Because rotation is neglected, the steady-state solar-wind outflow velocity is aligned with the background magnetic field (Mestel Reference Mestel1961):

(3.5)\begin{equation} \boldsymbol{v} = U \frac{\boldsymbol{B}}{B}. \end{equation}

The density and velocity satisfy the steady-state continuity equation,

(3.6)\begin{equation} {\boldsymbol{\nabla}\,\,\boldsymbol{\cdot}}\,\,(\rho \boldsymbol{v}) = 0. \end{equation}

It follows from (3.5), (3.6), and ${\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, \boldsymbol {B}=0$ that $\boldsymbol {B} \cdot \boldsymbol {\nabla }( \rho U / B) = 0$, and, hence,

(3.7)\begin{equation} \frac{\rho U}{B} = \mbox{constant}. \end{equation}

Equation (3.7) is equivalent to

(3.8)\begin{equation} \dot{M} \equiv A(r) \rho(r) U(r) = \mbox{constant}, \end{equation}

where

(3.9)\begin{equation} A(r) = \frac{4 {\rm \pi}r^2}{\eta(r)} \end{equation}

is the cross-sectional area of the flow, which satisfies $A(r) = \mbox {constant}/ B(r)$, as required by flux conservation. Equations (3.8) and (3.9) follow the convention of expressing the mass outflow rate as the total solar mass-loss rate that would arise if the local mass flux at some $r$ characterised the entire outflow at that $r$, even though the actual solar wind is comprised of different wind streams with different properties. All of the results of this paper can be applied to an individual flux tube accounting for some fraction $\nu$ of the total flow area by multiplying the right-hand side of (3.9) by $\nu$.

It follows from (2.1) and (3.7) that $\rho ^{1/2} U/v_\textrm {A}$ is a constant. Because $U(r_\textrm {A}) = v_\textrm {A}(r_\textrm {A})$, this constant must be $\rho _\textrm {A}^{1/2}$, where

(3.10)\begin{equation} \rho_\textrm{A} \equiv \rho(r_\textrm{A}). \end{equation}

Thus,

(3.11)\begin{equation} v_\textrm{A} = y U, \end{equation}

where

(3.12)\begin{equation} y \equiv \left(\frac{\rho}{\rho_\textrm{A}}\right)^{1/2}. \end{equation}

With the aid of (3.3), (3.4), (3.9) and (3.11), (3.8) can be rewritten as

(3.13)\begin{equation} \dot{M} = A_\textrm{b} \rho_\textrm{b} U_\textrm{b} = \frac{4 {\rm \pi}R_{{\odot}}^2}{\psi \eta_\textrm{b}} \times \rho_\textrm{b} \times \frac{v_\textrm{Ab}}{y_\textrm{b}} = \frac{R_{{\odot}}^2 \bar{B} (4 {\rm \pi}\rho_\textrm{b})^{1/2}}{y_\textrm{b}} = \bar{B} R_{{\odot}}^2 (4 {\rm \pi}\rho_\textrm{A})^{1/2}. \end{equation}

Thus, $\dot {M}$ is determined uniquely by the value of $\rho _\textrm {A}$ and the single-hemisphere open magnetic flux, $2 {\rm \pi}R_{\odot }^2 \bar {B}$.

3.2. Reflection-driven AW turbulence

Dmitruk et al. (Reference Dmitruk, Matthaeus, Milano, Oughton, Zank and Mullan2002) (hereafter D02) developed an analytic model of reflection-driven AW turbulence in the solar corona valid in the limit of small $L_\perp$, where $L_\perp$ is the correlation length of the AW fluctuations measured in the plane perpendicular to the background magnetic field. Chandran & Hollweg (Reference Chandran and Hollweg2009) (hereafter CH09) generalised this model by accounting for the solar-wind outflow velocity. Section 3.2.1 summarises the main results of the CH09 model, and § 3.2.2 uses the CH09 model to show that heating by reflection-driven AW turbulence causes the sub-Alfvénic region of the solar wind (at $r<r_\textrm {A}$, where $U< v_\textrm {A}$) to become approximately isothermal. Section 3.2.3 presents a modified version of the CH09 model that is easier to work with analytically, which is used to incorporate AW turbulence into the solar-wind model developed in this section.

3.2.1. The Chandran & Hollweg (2009) model of reflection-driven AW turbulence

In classical mechanics, a simple harmonic oscillator with frequency $\omega$ and energy $E$ possesses an adiabatic invariant $E/\omega$. If the parameters of the oscillator vary on a time scale $t_0$ satisfying $t_0 \gg \omega ^{-1}$ (e.g. if the length of a pendulum is slowly varied), then $E/\omega$ is almost exactly conserved. As $(\omega t_0)^{-1} \rightarrow 0$, changes in $E/\omega$ vanish faster than any power of $(\omega t_0)^{-1}$ (Landau & Lifshitz Reference Landau and Lifshitz1960).

An AW is like a space-filling collection of harmonic oscillators, and the wave action is analogous to the harmonic oscillator's adiabatic invariant. The wave action per unit volume per unit $\omega$ is ${\mathcal {E}}_\omega / \omega ^\prime$, where $\omega$ is the AW frequency in an inertial frame centred on the Sun, $\omega ^\prime$ is the AW frequency in the local plasma frame, and ${\mathcal {E}}_\omega$ is the AW energy per unit volume per unit $\omega$. In the Wentzel-Kramers-Brillouin (WKB) limit, in which the wave period is much shorter than the time scale on which the plasma parameters vary appreciably and the wave length is much shorter than the length scales over which the background plasma varies appreciably, the wave action satisfies the conservation law

(3.14)\begin{equation} \frac{\partial }{\partial t} \left(\frac{{\mathcal{E}}_\omega}{\omega^\prime}\right) +{\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, \left(\frac{\boldsymbol{c} {\mathcal{E}}_\omega}{\omega^\prime}\right) = 0, \end{equation}

where $\boldsymbol {c}$ is the group velocity of the waves (Bretherton & Garrett Reference Bretherton and Garrett1968; Dewar Reference Dewar1970). For outward-propagating AWs in the solar wind, $\omega = k_r(U+v_\textrm {A})$, $\omega ^\prime = k_r v_\textrm {A}$, and $\boldsymbol {c} = (U+v_\textrm {A})\boldsymbol {\hat {r}}$, where $\boldsymbol {\hat {r}}$ is the radial unit vector and $k_r$ is the radial component of the wave vector. In a steady-state solar wind, $\omega$ depends on neither position nor time. Upon multiplying (3.14) by $\omega$, integrating over $\omega$, and assuming a steady state, one obtains

(3.15)\begin{equation} {\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, \left[ \frac{\boldsymbol{\hat{r}} (U+v_\textrm{A})^2 {\mathcal{E}}_\textrm{tot}}{v_\textrm{A}}\right] = 0, \end{equation}

where

(3.16)\begin{equation} {\mathcal{E}}_\textrm{tot} = \int {\mathcal{E}}_\omega \,\textrm{d} \omega = \tfrac{1}{4} \rho z_+^2 \end{equation}

is the total AW energy density,

(3.17)\begin{equation} z_\pm\ {\equiv}\ \langle |\boldsymbol{z}^\pm|^2 \rangle^{1/2}, \end{equation}

$\langle \dots \rangle$ indicates a time average,

(3.18)\begin{equation} \boldsymbol{z}^\pm\ {=}\ \delta \boldsymbol{v} \mp \frac{\delta \boldsymbol{B}}{\sqrt{4{\rm \pi} \rho}} \end{equation}

are the Elsasser variables, $\delta \boldsymbol {v}$ and $\delta \boldsymbol {B}$ are the fluctuating velocity and magnetic field, and $\boldsymbol {z}^+$ ($\boldsymbol {z}^-$) corresponds to AW fluctuations propagating away from (toward) the Sun.Footnote ¹ In (3.17) and the following, a $\pm$ sign is used as a subscript (as opposed to a superscript) when the subscripted quantity is an r.m.s. value. As ${\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, (\rho U \boldsymbol {\hat {r}}) = 0$, (3.15) can be rewritten as

(3.19)\begin{equation} \frac{\mathrm{d}}{\mathrm{d}r}g^2 = 0, \end{equation}

where

(3.20)\begin{equation} g^2 = \frac{(U+v_\textrm{A})^2 z_+^2}{Uv_\textrm{A}} = \frac{(1+y)^2 z_+^2}{y} \end{equation}

is the wave-action flux per unit mass flux per unit $\omega$ times $4 \omega$ integrated over $\omega$, or, for brevity, the ‘wave action flux per unit mass flux’.

When the finite radial wavelength of the AWs is taken into account, the radial gradient in $v_\textrm {A}$ causes partial non-WKB reflection of $\boldsymbol {z}^+$ fluctuations, leading to the production of $\boldsymbol {z}^-$ fluctuations. Counter-propagating AWs then interact, causing fluctuation energy to cascade to small scales and dissipate. In the CH09 model, this loss of fluctuation energy causes $g^2$ to decrease with radius according to the equationFootnote ²

(3.21)\begin{equation} (U+v_\textrm{A}) \frac{\mathrm{d}}{\mathrm{d}r}g^2 ={-} \frac{z_-}{L_\perp} g^2. \end{equation}

Equation (3.21) states that in a reference frame that follows an outward-propagating AW, the wave action flux per unit mass flux decays on the eddy turnover time scale $L_\perp / z_-$. The reason that only $z_-$ (and not $z_+$) appears in this eddy turnover time scale is that ${\boldsymbol z}^+$ fluctuations are not sheared or distorted by other $\boldsymbol {z}^+$ fluctuations, but they are sheared and distorted by counter-propagating $\boldsymbol {z}^-$ fluctuations (Iroshnikov Reference Iroshnikov1963; Kraichnan Reference Kraichnan1965). The radial decay of $g^2$ is accompanied by turbulent heating at the rate

(3.22)\begin{equation} Q = \frac{\rho z_+^2 z_-}{4 L_\perp}, \end{equation}

which is the energy density of the outward-propagating AWs $\rho z_+^2 / 4$ divided by their eddy turnover time scale $L_\perp / z_-$. Equation (3.22) drops a term $\rho z_-^2 z_+ / (4 L_\perp )$ that is normally included in the turbulent heating rate because of CH09's assumption that

(3.23)\begin{equation} z_-\ {\ll}\ z_+ , \end{equation}

an inequality that holds in the small-$L_\perp$ limit, as can be seen later in (3.25).

Following D02, CH09 determined $z_-$ by balancing the rate at which $z_-$ is produced through reflections against the rate at which $z_-$ cascades to small scales through nonlinear interactions in the small-$L_\perp$ limit, obtaining

(3.24)\begin{equation} \frac{(U+v_\textrm{A})}{v_\textrm{A}}\left| \frac{\mathrm{d}v_\textrm{A}}{\mathrm{d}r}\right| z_+\ {=}\ \frac{z_+}{L_\perp} z_-, \end{equation}

or, equivalently,

(3.25)\begin{equation} \frac{z_-}{L_\perp} = \frac{(U+v_\textrm{A})}{v_\textrm{A}}\left| \frac{\mathrm{d}v_\textrm{A}}{\mathrm{d}r}\right|. \end{equation}

Upon substituting (3.25) into (3.21), assuming that $v_\textrm {A}(r)$ has a single maximum at $r_\textrm {m} > r_\textrm {b}$, and solving for $g(r)$, CH09 found that

(3.26)\begin{equation} g^2 = g_\textrm{b}^2 h(r) , \end{equation}

where

(3.27)\begin{equation} h(r) = \left\{ \begin{array}{@{}ll} v_\textrm{Ab}/v_\textrm{A}(r) & \mbox{ for } r_\textrm{b} < r < r_\textrm{m} \\ v_\textrm{Ab} v_\textrm{A}(r) / v_\textrm{Am}^2 & \mbox{ if } r > r_\textrm{m} \end{array}\right., \end{equation}

where $v_\textrm {Am} = v_\textrm {A}(r_\textrm {m})$ is the maximum value of $v_\textrm {A}$. Conceptually, (3.26) and (3.27) state that an appreciable fraction of the local AW action flux per unit mass flux dissipates within each Alfvén-speed scale height, which causes $z_+^2(r)$ to drop below the WKB value that would be predicted from (3.20) with constant $g^2$.

Upon substituting (3.26) into (3.22) and making use (3.20), CH09 obtained

(3.28)\begin{equation} Q = \rho (\delta v_\textrm{b})^2 \left(\frac{U+v_\textrm{A}}{v_\textrm{A}}\right) \left|\frac{\mathrm{d}v_\textrm{A}}{\mathrm{d}r}\right| \left(\frac{y}{y_\textrm{b}}\right)\left(\frac{1 + y_\textrm{b}}{1+y}\right)^2 h(r), \end{equation}

where

(3.29)\begin{equation} (\delta v)^2 \equiv \langle | \delta \boldsymbol{ v}|^2 \rangle = \frac{z_+^2}{4}. \end{equation}

The second equality in (3.29) follows from (3.23). As

(3.30)\begin{equation} y_\textrm{b} \gg 1, \end{equation}

equation (3.28) can be rewritten to a good approximation in the following simplified form with the aid of (3.11):

(3.31)\begin{equation} Q = \rho (\delta v_\textrm{b})^2 y_\textrm{b}\left(\frac{U }{U+v_\textrm{A} }\right) \left| \frac{\mathrm{d}v_\textrm{A}}{\mathrm{d}r}\right| h(r). \end{equation}

3.2.2. Approximate isothermality between the transition region and Alfvén critical point

In steady state, the plasma internal-energy equation takes the form

(3.32)\begin{equation} {\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\,\left[\boldsymbol{v} \left(\frac{p}{\gamma-1}\right)\right] ={-}p {\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, \boldsymbol{v} -{\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\,\boldsymbol{q} + Q - R, \end{equation}

where $p$ is the pressure, $p/(\gamma -1)$ is the internal-energy density, $-p {\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, \boldsymbol {v}$ is the rate at which $p\,\textrm {d} V$ work is done on the plasma per unit volume, $Q$ is the rate of turbulent heating per unit volume, $\boldsymbol {q}$ is the heat flux, which is written in the formFootnote ³

(3.33)\begin{equation} \boldsymbol{q} = q_r \frac{\boldsymbol{B}}{B}, \end{equation}

and $R$ is the rate of radiative cooling per unit volume. In the corona and solar wind, the density is sufficiently small that radiative cooling can be neglected, and, to a good approximation, (3.32) can be rewritten with the aid of (3.6) as

(3.34)\begin{equation} - c_\textrm{s}^2 U \frac{\textrm{d} \rho}{\textrm{d} r} + \frac{\rho U}{\gamma-1} \frac{\mathrm{d}}{\mathrm{d}r} c_\textrm{s}^2 ={-} \frac{1}{A} \frac{\mathrm{d}}{\mathrm{d}r}\left(A q_r\right) + Q . \end{equation}

A generic consequence of heating by reflection-driven AW turbulence is that, when other forms of heating (including conduction) are subdominant, the flow becomes quasi-isothermal at $r_\textrm {b} < r<r_\textrm {A}$, meaning that

(3.35)\begin{equation} \left| \frac{1}{c_\textrm{s}^2} \frac{\mathrm{d}c_\textrm{s}^2}{\mathrm{d}r}\right| \ll \left| \frac{1}{\rho} \frac{\mathrm{d}\rho}{\mathrm{d}r}\right|, \end{equation}

whereas (3.35) is not satisfied at $r>r_\textrm {A}$. This can be demonstrated by (1) neglecting conductive heating in (3.34), (2) assuming that (3.35) is satisfied so that the second term on the left-hand side of (3.34) can be neglected, and (3) solving (3.34) for $c_\textrm {s}^2$. If the resulting expression for $c_\textrm {s}(r)^2$ satisfies (3.35), then the neglect of the second term on the left-hand side of (3.34) is self-consistent, and this expression for $c_\textrm {s}(r)^2$ is a reasonable approximation for the full solution of (3.34). On the other hand, if the resulting value of $c_\textrm {s}^2(r)$ does not satisfy (3.35), then (3.34) does not possess a quasi-isothermal solution.

Carrying out this procedure and equating the first term on the left-hand side of (3.34) with the turbulent heating term on the right-hand side (which is given by (3.31)), one obtains

(3.36)\begin{equation} c_\textrm{s}^2 = y_\textrm{b} (\delta v_\textrm{b})^2 \left(\frac{ L_\rho}{L_{v_\textrm{A}}}\right) \left(\frac{v_\textrm{A}}{U+v_\textrm{A}}\right)h(r), \end{equation}

where

(3.37)\begin{equation} L_\rho = \rho\left|\frac{\mathrm{d}\rho}{\mathrm{d}r}\right|^{{-}1} \qquad L_{v_\textrm{A}} = v_\textrm{A}\left|\frac{\mathrm{d}v_\textrm{A}}{\mathrm{d}r}\right|^{{-}1} \end{equation}

are the density and Alfvén-speed scale heights. These scale heights are generally some constant of order unity times $r$, with $L_\rho /L_{v_{\textrm {A}}}$ increasing by a factor of a few between the low corona and $r_\textrm {A}$. On the other hand, $h(r)$ decreases by a factor of a few between the low corona and $r_\textrm {A}$, so that the product $(L_\rho/L_{v_{\rm A}})h(r)$ varies quite weakly with $r$.Footnote ⁴ The $v_\textrm {A}/(U+v_\textrm {A})$ term in (3.36) likewise exhibits little variation in the sub-Alfvénic region, ranging from $\simeq 1$ at $r=r_\textrm {b}$ to $0.5$ at $r=r_\textrm {A}$. On the other hand, $\rho$ varies by a factor of $\sim 10^5$ between the coronal base and $r_\textrm {A}$ (see, e.g. Cranmer et al. Reference Cranmer, van Ballegooijen and Edgar2007). Thus, the approximate solution for $c_\textrm {s}^2$ in (3.36) satisfies (3.35) at $r_\textrm {b} < r < r_\textrm {A}$, and AW heating indeed causes the sub-Alfvénic region to become quasi-isothermal. In contrast, at $r>r_\textrm {A}$, $U$ asymptotes toward a constant value, $h \propto v_\textrm {A} \propto \rho ^{1/2} U \propto \rho ^{1/2}$, $v_\textrm {A}/(U+v_\textrm {A}) \propto \rho ^{1/2}$, and the right-hand side of (3.36) becomes proportional to $\rho$, contradicting (3.35).

3.2.3. A modified version of the Chandran & Hollweg (2009) model

There are two difficulties with incorporating the CH09 model into an analytic solar-wind model that includes the region immediately above the transition region. First, (3.25) is consistent with (3.23) only if

(3.38)\begin{equation} \frac{v_\textrm{A} L_\perp}{z_+} \ll \frac{v_\textrm{A}}{|\textrm{d} v_\textrm{A}/ \textrm{d} r|}. \end{equation}

The left-hand side of (3.38) is the characteristic distance a $\boldsymbol {z}^-$ fluctuation at scale $L_\perp$ (measured perpendicular to the background magnetic field) propagates along the magnetic field before cascading and dissipating, and the right-hand side is the Alfvén-speed scale height. Just above the transition region, the magnetic-field strength has a scale height of order $10^{-2} R_{\odot }$ (Hackenberg, Marsch & Mann Reference Hackenberg, Marsch and Mann2000; Cranmer et al. Reference Cranmer, van Ballegooijen and Edgar2007), the $v_\textrm {A}$ scale height has a similar value, and (3.38) is not satisfied (see, e.g. equation (3.13-a) and table 3 of Chandran & Perez Reference Chandran and Perez2019). On the other hand, beyond the low corona, the $v_\textrm {A}$ scale height grows to values $\sim r$ that are $\gg L_\perp$. Thus, the CH09 model should be applied only beyond some minimum heliocentric distance, and some other approximation is needed to treat the region immediately above the transition region.

The second difficulty with incorporating the CH09 model into an analytic solar-wind model is that the function $h(r)$ in (3.27) depends on the number of local extrema in the Alfvén-speed profile and the locations of these extrema. In models that account for the rapid increase in $B(r)$ as $r$ drops below $\simeq 1.01 R_{\odot }$, $v_\textrm {A}$ typically has two local extrema within the corona: a local minimum just above the coronal base and a local maximum a few tenths of a solar radius above the coronal base. (See, e.g. figure 3 of Cranmer & van Ballegooijen (Reference Cranmer and van Ballegooijen2005) or figure 9 of Cranmer et al. (Reference Cranmer, van Ballegooijen and Edgar2007).) In the context of the analytic solar-wind model developed in the following, each local extremum introduces two new unknowns: the location of the extremum, and either the density or flow speed at the extremum.

In this paper, the first of the two aforementioned difficulties is handled by replacing the CH09 result for $Q(r)$ at $r=r_\textrm {b}$ with the expression

(3.39)\begin{equation} Q_\textrm{b} =\frac{\rho_\textrm{b} (\delta v_\textrm{b})^2 v_\textrm{Ab} }{l_\textrm{b}}, \end{equation}

where $l_\textrm {b}$ is a free parameter, which is set equal to $0.3 R_{\odot }$ in the numerical examples presented in § 4. The second difficulty is resolved by replacing (3.25) with the expression

(3.40)\begin{equation} \frac{z_-}{L_\perp} ={-} \sigma (U + v_\textrm{A}) \frac{\mathrm{d}}{\mathrm{d}r} \ln(1+y) \end{equation}

at $r_\textrm {b} < r < r_\textrm {A}$, where $\sigma$ is a free parameter, whose value is approximately 0.1–0.5 in the numerical examples in § 4. Whereas $z_- / L_\perp = (U+v_\textrm {A})/L_{v_\textrm {A}}$ in (3.25), (3.40) takes $z_- / L_\perp$ to be $\simeq (U+v_\textrm {A})/L_{\rho }$ times a free parameter.Footnote ⁵ Because $\rho$ (and, hence, $y$) is a monotonically decreasing function of $r$, the minus sign on the right-hand side of (3.40) ensures that $z_- > 0$. The right-hand side of (3.40) is taken to be proportional to $\ln (1+y)$ instead of $\ln y$ simply to make some of the expressions encountered later on easier to integrate analytically. Another motivation for using (3.40) instead of (3.25) is that the parameter-free CH09 result in (3.25) overestimates the rate at which AWs cascade and dissipate in the solar wind (Chandran & Hollweg Reference Chandran and Hollweg2009), and the introduction of a free parameter in (3.40) in principle makes it possible to correct for this.

Upon substituting (3.40) into (3.21), solving for $g^2(r)$, and rewriting $g^2$ in terms of $z_+^2$ using (3.20), one finds that

(3.41)\begin{equation} z_+^2 = z_{+\textrm{b}}^2 \frac{y}{y_\textrm{b}} \left(\frac{1+y_\textrm{b}}{1+y} \right)^{2-\sigma} . \end{equation}

Equations (3.22), (3.29), (3.40), and (3.41) can then be used to show that

(3.42)\begin{equation} H = \chi_\textrm{H} P_\textrm{AW}(r_\textrm{b}), \end{equation}

where

(3.43)\begin{equation} H = \int_{r_\textrm{b}}^{r_\textrm{A}} Q(r) A(r) \,\textrm{d} r \end{equation}

is the total turbulent heating power between $r_\textrm {b}$ and $r_\textrm {A}$,

(3.44)\begin{equation} P_\textrm{AW}(r) = \rho (\delta v)^2 (U + v_\textrm{A}) A \end{equation}

is the power (energy flux times area) of outward-propagating AWs at radius $r$, and

(3.45)\begin{equation} \chi_\textrm{H} = 1 - 2^{\sigma -1} \left(\frac{2-\sigma}{1-\sigma}\right) \frac{(1+y_\textrm{b})^{1-\sigma}}{y_\textrm{b}} + \frac{1}{y_\textrm{b}(1-\sigma)} \end{equation}

is the fraction of $P_\textrm {AW}(r_\textrm {b})$ that is dissipated between $r_\textrm {b}$ and $r_\textrm {A}$.

3.3. Relating the AW amplitudes at the coronal base and photosphere

In the absence of wave reflection and dissipation, $P_\textrm {AW}$ (defined in (3.44)) would have almost exactly the same value at $r_\textrm {b}$ and $R_{\odot }$.Footnote ⁶ However, the steep Alfvén-speed gradient in the transition region leads to strong AW reflection, and a vigorous energy cascade in the chromosphere leads to substantial AW dissipation (van Ballegooijen et al. Reference van Ballegooijen, Asgari-Targhi, Cranmer and DeLuca2011; Chandran & Perez Reference Chandran and Perez2019). This reduces $P_\textrm {AW}(r_\textrm {b}) / P_\textrm {AW }(R_{\odot })$ to some value $f_\textrm {chr} < 1$, where $f_\textrm {chr}$ is an effective AW transmission coefficient for the chromosphere and transition region. Equivalently, because $U\ll v_\textrm {A}$ at $r\leqslant r_\textrm {b}$,

(3.46)\begin{equation} P_\textrm{AW}(r_\textrm{b}) = \rho_\textrm{b} \delta v_\textrm{b}^2 v_\textrm{Ab} A_\textrm{b} = f_\textrm{chr} \rho_{{\odot}} \delta v_{{\odot}}^2 v_{\textrm{A}{\odot}} A_{{\odot}}, \end{equation}

where a $\odot$ subscript indicates that the subscripted quantity is evaluated at the photosphere. For the numerical calculations in § 4, I set

(3.47)\begin{equation} \rho_\odot\ {=}\ 10^{17} m_\textrm{p} \mbox{ cm}^{{-}3}. \end{equation}

As $B A = \mbox {constant}$ and $v_\textrm {A} = B /(4{\rm \pi} \rho )^{1/2}$, (3.46) can be rewritten as

(3.48)\begin{equation} \delta v_\textrm{b}^2 = \delta v_{{\odot}\textrm{eff}}^2 \left(\frac{\rho_{{\odot}}}{\rho_\textrm{b}}\right)^{1/2} , \end{equation}

where

(3.49)\begin{equation} \delta v_{{\odot}\textrm{eff}} = f_\textrm{chr}^{1/2} \delta v_{{\odot}}. \end{equation}

The difference between $\delta v_{\odot }$ and $\delta v_{\odot \textrm {eff}}$ is that $\delta v_{\odot }$ gives rise to the fluctuating velocity $\delta v_\textrm {b}$ at $r=r_\textrm {b}$ when reflection and dissipation are accounted for, whereas $\delta v_{\odot \textrm {eff}}$ would give rise to the same value of $\delta v_\textrm {b}$ via WKB propagation (i.e. without reflection or dissipation).

The value of $\delta v_{\odot \textrm {eff}}$ can be constrained in different ways. For example, observations fix $\delta v_{\odot }$ at a value of $\simeq 1 \mbox { km} \mbox { s}^{-1}$ (Richardson & Schwarzschild Reference Richardson and Schwarzschild1950), and numerical simulations suggest that $f_\textrm {chr}$ is $\simeq 0.04 \text {--} 0.08$ (van Ballegooijen et al. Reference van Ballegooijen, Asgari-Targhi, Cranmer and DeLuca2011; Chandran & Perez Reference Chandran and Perez2019). Alternatively, if the solar wind is assumed to be powered primarily by an AW energy flux, then $\delta v_{\odot \textrm {eff}}$ can be inferred directly from measurements of the mass flux and energy flux far from the Sun. This latter method is used to determine $\delta v_{\odot \textrm {eff}}$ in some of the numerical solutions presented in § 4 and is described in more detail in § 4.2. It is worth noting that in contrast to $\delta v_{\odot }$, $f_\textrm {chr}$, and $\delta v_{\odot \textrm {eff}}$, which are plausibly independent of the properties of the coronal plasma and coronal magnetic field, $\delta v_\textrm {b}$ depends upon $\rho _\textrm {b}$, which varies between different flux tubes with different super-radial expansion factors $\eta _\textrm {b}$, as shown later in (3.55).

3.4. Balancing turbulent heating and radiative cooling at the coronal base

Within the corona and transition region, the plasma is optically thin, and the rate of radiative cooling is given by

(3.50)\begin{equation} R = \left(\frac{\rho}{m_\textrm{p}}\right)^2 \varLambda(T), \end{equation}

where $m_\textrm {p}$ is the proton mass, and $\varLambda (T)$ is the optically thin radiative loss function. Figure 2 shows three different approximations to $\varLambda (T)$. The dashed lines correspond to equation (A1) of Rosner, Tucker & Vaiana (Reference Rosner, Tucker and Vaiana1978). The solid line is a plot of

(3.51)\begin{equation} \varLambda(T) = c_\textrm{R} T^{{-}1/2}, \end{equation}

where

(3.52)\begin{equation} c_\textrm{R} = 1.549 \times 10^{{-}19} \mbox{ erg} \mbox{ cm}^3 \mbox{ s}^{{-}1} \mbox{ K}^{1/2}. \end{equation}

Equation (3.51) is the temperature derivative of equation (A3) of Rosner et al. (Reference Rosner, Tucker and Vaiana1978). The dotted line in figure 2 is a piecewise-continuous linear approximation to the low-temperature range of $\varLambda (T)$ in figure 1 of Cranmer et al. (Reference Cranmer, van Ballegooijen and Edgar2007), which is included to illustrate that optically thin radiative cooling becomes extremely weak at $T\lesssim 10^4$ K.

Figure 2. Three approximations to the optically thin radiative loss function $\varLambda (T)$; see the text for further discussion.

Throughout most of the corona, $\rho$ is sufficiently small that radiative cooling is negligible. However, as $r$ decreases within the corona, $\rho$ increases by orders of magnitude, which causes $R/Q$ to increase, because $R\propto \rho ^2$. There is thus some radius $r_\textrm {b}$ at which

(3.53)\begin{equation} R(r_\textrm{b}) = Q(r_\textrm{b}), \end{equation}

which marks the transition between the corona, in which radiative cooling is negligible, and the low solar atmosphere, in which radiation is thermodynamically important. In this paper, the radius $r_\textrm {b}$ is identified as the base of the corona, as already noted in (3.1). As $r$ decreases below $r_\textrm {b}$, $\rho (r)$ increases above $\rho _\textrm {b}$, $R/Q$ increases to values $\gg 1$, and the temperature gradient length scale must decrease so that conductive heating can balance radiative cooling (as well as internal-energy losses from advection and $p\,\textrm {d} V$ work). This shortening of the temperature gradient length scale gives rise to the transition region, which is discussed further in § 3.6. To facilitate an analytic solution, I neglect turbulent heating at $r<r_\textrm {b}$ and radiative cooling at $r>r_\textrm {b}$.

With the aid of (3.39), (3.46), and (3.50), (3.53) can be written in the form

(3.54)\begin{equation} \frac{\rho_\textrm{b}^2}{m_\textrm{p}^2} \varLambda(T_\textrm{b}) = \frac{\rho_\textrm{b} (\delta v_\textrm{b})^2 v_\textrm{Ab} }{l_\textrm{b}} = \frac{\rho_{{\odot}} (\delta v_{{\odot}\textrm{eff}})^2 v_{\textrm{A} \odot} B_\textrm{b} }{l_\textrm{b} B_{{\odot}}}. \end{equation}

Solving for $\rho _\textrm {b}$, one finds via (3.51) that

(3.55)\begin{equation} \rho_\textrm{b} = \left(\frac{\delta v_{{\odot}\textrm{eff}}^2 \psi \bar{B} \eta_\textrm{b} m_\textrm{p}^{5/2} \overline{c_\textrm{s}}}{c_\textrm{R} l_\textrm{b}}\right)^{1/2} \left(\frac{\rho_{{\odot}}}{8 {\rm \pi}k_\textrm{B}}\right)^{1/4}, \end{equation}

where

(3.56)\begin{equation} \overline{ c_\textrm{s}} = \mbox{constant} \end{equation}

is the sound speed (2.6) in the sub-Alfvénic region ($r_\textrm {b} < r < r_\textrm {A}$), which is approximated as a constant for the reasons discussed in § 3.2.2. Equation (3.55) can be rewritten as

(3.57)\begin{equation} \rho_\textrm{b} = \frac{B_\textrm{ref}^2}{4 {\rm \pi}v_\textrm{esc}^2} \tilde{\rho}_{{\odot}}^{1/4} B_\ast^2 \xi^2 \psi^{1/2} x^{1/4}, \end{equation}

where

(3.58)\begin{equation} x = \frac{\overline{ c_\textrm{s}}^2}{v_\textrm{esc}^2} \end{equation}

is the dimensionless temperature of the sub-Alfvénic region,

(3.59)\begin{equation} B_\textrm{ref} = \left(\frac{4 {\rm \pi}m_\textrm{p}^{5/2} v_\textrm{esc}^6}{c_\textrm{R} R_{{\odot}} \sqrt{2 k_\textrm{B }}}\right)^{1/2} = 118.8 \mbox{ G} \end{equation}

is the magnetic-field strength for which $v_\textrm {A} = v_\textrm {esc}$ when the radiative cooling time $\rho \overline { c_\textrm {s}}^2 / R$, free-fall time $R_{\odot }/v_\textrm {esc}$, and sound-crossing time $R_{\odot }/\overline {c_\textrm {s}}$ are all equal,

(3.60)\begin{gather} \epsilon_{{\odot}} = \frac{(\delta v_{{\odot}\textrm{eff}})^2}{v_\textrm{esc}^2}, \end{gather}

(3.61)\begin{gather} \tilde{\rho}_{{\odot}} = \frac{4 {\rm \pi}v_\textrm{esc}^2 \rho_{{\odot}} }{B_\textrm{ref}^2}, \end{gather}

(for reference, $\tilde {\rho }_\odot = 5.7 \times 10^5$ given (3.47)), and

(3.62)\begin{equation} \xi = \left(\frac{\epsilon_{{\odot}} \eta_\textrm{b}}{B_\ast^3 \tilde{l}_\textrm{b}}\right)^{1/4} \qquad B_\ast\ {=}\ \frac{\bar{B}}{B_\textrm{ref}} \qquad \tilde{l}_\textrm{b} = \frac{l_\textrm{b}}{R_{{\odot}}}. \end{equation}

Equation (3.57) can used to express $v_\textrm {Ab}$ in the form

(3.63)\begin{equation} v_\textrm{Ab} = v_\textrm{esc} \psi^{3/4} \xi^{{-}1} \eta_\textrm{b} (x\tilde{\rho}_\odot)^{{-}1/8}, \end{equation}

and (3.48), (3.57), and (3.60) can be used to write

(3.64)\begin{equation} \epsilon \equiv \frac{(\delta v_\textrm{b})^2}{v_\textrm{esc}^2} = \epsilon_{{\odot}} \tilde{\rho}_\odot^{3/8} B_\ast^{{-}1} \xi^{{-}1} \psi^{{-}1/4} x^{{-}1/8}. \end{equation}

3.5. Internal-energy equilibrium within the sub-Alfvénic region

In the quasi-isothermal approximation (3.35), the second term on the left-hand side of (3.34) is negligible, and the volume integral of (3.34) between $r=r_\textrm {b}$ and $r=r_\textrm {A}$ yields

(3.65)\begin{equation} -\!\dot{M} \overline{ c_\textrm{s}}^2 \int_{r_\textrm{b}}^{r_\textrm{A}} \frac{1}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}r} \textrm{d} r ={-} \left(A q_{r}\right) |^{r_\textrm{A}}_{r_\textrm{b}} + \chi_\textrm{H} \dot{M} \delta v_\textrm{b}^2 (1 + y_\textrm{b}), \end{equation}

where (3.42)–(3.44) have been used to rewrite the volume integral of the turbulent heating rate, (3.8) has been used to pull a constant factor of $\dot {M} = \rho A U$ out of the integral on the left-hand side, and (3.11) has been used to write $U_\textrm {b} + v_\textrm {Ab} = U_\textrm {b} (1 + y_\textrm {b})$.

The heat flux at $r=r_\textrm {A}$ is a small fraction of the total energy flux in AW-driven solar-wind models and is thus neglected in (3.65). Upon evaluating the integral on the left-hand side of (3.65), noting that $q_{r\textrm {b}} = - |q_{r \textrm {b}}| \equiv - q_\textrm {b}$, making use of (3.12), and rearranging terms, one obtains

(3.66)\begin{equation} \chi_\textrm{H} \dot{M}\delta v_\textrm{b}^2 (1 + y_\textrm{b}) = 2 \dot{M}\overline{ c_\textrm{s}}^2 \ln y_\textrm{b} + A_\textrm{b} q_\textrm{b}. \end{equation}

The left-hand side of (3.66) is the total turbulent heating rate within the sub-Alfvénic region, which represents the source of internal energy between $r_\textrm {b}$ and $r_\textrm {A}$. The two terms on the right-hand side of (3.66) are the dominant sinks of internal energy in the sub-Alfvénic region: $p\,\textrm {d} V$ work and thermal conduction into the transition region. Dividing (3.66) by $\dot {M} v_\textrm {esc}^2$ leads to

(3.67)\begin{equation} \epsilon \chi_\textrm{H} (1 + y_\textrm{b}) = 2 x \ln y_\textrm{b} + \frac{ q_\textrm{b}}{\rho_\textrm{b} U_\textrm{b} v_\textrm{esc}^2}. \end{equation}

3.6. The flux of heat from the corona into the transition region

The temperature structure within the transition region can be determined using a method similar to the methods of Rosner et al. (Reference Rosner, Tucker and Vaiana1978) and Schwadron & McComas (Reference Schwadron and McComas2003). The Knudsen number $N_\textrm {K}$ (the electron Coulomb mean free path $\lambda _\textrm {mfp}$ divided by the temperature gradient scale length $l_T$) is approximately $10^{-3}$ in the low corona, approximately $10^{-3}$ at the upper end of the transition region, and approximately $10^{-6}$ at the lower end of the transition region.Footnote ⁷ Because $N_\textrm {K} \ll 1$, the radial component of the heat flux in the transition region is well approximated by the Spitzer & Härm (Reference Spitzer and Härm1953) formula,

(3.68)\begin{equation} q_r ={-}\alpha T^{5/2} \frac{\mathrm{d}T}{\mathrm{d}r}, \end{equation}

where

(3.69)\begin{equation} \alpha =\frac{1.84 \times 10^{{-}5} \mbox{ erg} \mbox{ cm}^{{-}1} \mbox{ s}^{{-}1} \mbox{ K}^{{-}7/2}}{\ln \varLambda_\textrm{Coul}}, \end{equation}

and $\ln \varLambda _\textrm {Coul}$ is the Coulomb logarithm. In the numerical examples of § 4,

(3.70)\begin{equation} \ln \varLambda_\textrm{Coul} = 18.1, \end{equation}

the value for electron-electron collisions in a proton-electron plasma with $\rho /m_\textrm {p} = 10^{9} \mbox { cm}^{-3}$ and $T = 10^6 \mbox { K}$ (Huba Reference Huba2013). The magnetic field near $r=r_\textrm {b}$ is taken to be approximately radial, and the width of the transition region ($\sim 10^2 \mbox { km}$) is so narrow that $p$, $B$, and $A$ are treated as constants within the transition region, with

(3.71)\begin{equation} {\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, \boldsymbol{q} = \frac{\mathrm{d}q_r}{\mathrm{d}r}. \end{equation}

As mentioned previously, turbulent heating is neglected at $r< r_\textrm {b}$. Within the transition region, (3.32) thus becomes

(3.72)\begin{equation} \left(\frac{\gamma}{\gamma-1}\right) p {\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, \boldsymbol{v} ={-} \frac{\mathrm{d}q_\textrm{r}}{\mathrm{d}r} - \frac{c_\textrm{R} p^2}{4 k_\textrm{B}^2 T^{5/2}}, \end{equation}

where $\rho$ has been expressed in terms of $p$ and $T$ using (2.6). The velocity divergence in (3.72) can be expressed in terms of the heat flux via

(3.73)\begin{equation} {\boldsymbol{\nabla}\,\boldsymbol{\cdot}}\, \boldsymbol{v} ={-} \frac{\boldsymbol{v}}{\rho} \boldsymbol{\cdot} \boldsymbol{\nabla} \rho = \frac{\boldsymbol{v}}{T} \boldsymbol{\cdot} \boldsymbol{\nabla} T = \frac{U}{T} \left(-\frac{q_r}{\alpha T^{5/2}}\right). \end{equation}

where the first equality follows from (3.6). With the aid of (3.73), and using (3.68) to write

(3.74)\begin{equation} \frac{\mathrm{d}q_r}{\mathrm{d}r} ={-} \frac{\mathrm{d}q_r}{\mathrm{d}T} \frac{q_r}{\alpha T^{5/2}} , \end{equation}

one can rewrite (3.72) in the form

(3.75)\begin{equation} -\!a_2 q_r = q_r \frac{\mathrm{d}q_r}{\mathrm{d}T} - a_1, \end{equation}

where

(3.76)\begin{equation} a_1 = \frac{\alpha p^2 c_\textrm{R}}{4 k_\textrm{B}^2} \qquad a_2 = \left(\frac{\gamma}{\gamma-1}\right) \frac{p U}{T} \end{equation}

are both constants. Upon defining

(3.77)\begin{equation} w ={-} \frac{a_2}{a_1} q_\textrm{r}, \end{equation}

which is positive, one can rewrite (3.75) as

(3.78)\begin{equation} \frac{a_2^2}{a_1} \textrm{d} T = \frac{w }{1 + w} \textrm{d} w, \end{equation}

which can be integrated from the chromospheric values of $T$ and $w$, denoted $T_\textrm {chr}$ and $w_\textrm {chr}$, to the values of $T$ and $w$ at $r=r_\textrm {b}$, denoted $T_\textrm {b}$ and $w_\textrm {b}$. As $T_\textrm {chr} \ll T_\textrm {b}$ and the chromospheric heat flux is negligible ($q$ scaling as $T^{7/2}$ divided by the temperature-gradient scale length), the chromospheric terms are dropped, and the integral becomes

(3.79)\begin{equation} w_\textrm{b} - \ln(1+w_\textrm{b}) = a_3, \end{equation}

where

(3.80)\begin{gather} a_3 = \frac{a_2^2 T_\textrm{b}}{a_1} = \frac{2}{I_1^2} \left(\frac{\gamma}{\gamma-1}\right)^2 \left(\frac{U_\textrm{b}}{\overline{ c_\textrm{s}}}\right)^2, \end{gather}

(3.81)\begin{gather} I_1 = \left(\frac{\alpha c_\textrm{R} m_\textrm{p}}{4 k_\textrm{B}^3}\right)^{1/2} = 0.1581, \end{gather}

and the numerical value on the right-hand side of (3.81) is calculated using (3.70). Equation (3.79) can be solved in terms of the lower branch of the Lambert $W$ function, $W_{-1}$. This solution, in conjunction with (3.77), yields

(3.82)\begin{equation} q_\textrm{b} ={-}q_{r{\rm b}} ={-} \frac{a_1}{a_2}[1 + W_{{-}1}({-}e^{-(1+a_3)})], \end{equation}

which is positive since $\displaystyle W_{-1}(- e^{-(1+a_3)})<-1$.

If $U_\textrm {b}/ \overline { c_\textrm {s}}$ is sufficiently small, then $a_3 \ll 1$. In this low-Mach-number limit, (3.82) becomes, to leading order in $a_3$,

(3.83)\begin{equation} q_\textrm{b} = I_1 \rho_\textrm{b} \overline{ c_\textrm{s}}^3. \end{equation}

Equation (3.83) is equivalent to equation (3.15) of Rosner et al. (Reference Rosner, Tucker and Vaiana1978) when thermal conduction is the only source of heating in the transition region, i.e. when $f_\textrm {H}$ is set equal to zero in their equation (3.15). Equation (3.82) follows Schwadron & McComas (Reference Schwadron and McComas2003) in generalising the work of Rosner et al. (Reference Rosner, Tucker and Vaiana1978) to include internal-energy losses from $p\,\textrm {d} V$ work and advection. However, the analysis leading to (3.82) differs from that of Schwadron & McComas (Reference Schwadron and McComas2003) in that (3.82) completely neglects turbulent heating at $r<r_\textrm {b}$ and does not assume that $\int q(T) \,\textrm {d} T \propto T$.Footnote ⁸

3.7. Constraints associated with the wave-modified sonic critical point

In the presence of mostly outward-propagating AWs, a radial background magnetic field, and a radial outflow, the momentum equation in the approximately isothermal sub-Alfvénic region takes the form

(3.84)\begin{equation} \rho U \frac{\mathrm{d}U}{\mathrm{d}r} ={-} \overline{ c_\textrm{s}}^2\frac{\mathrm{d}\rho}{\mathrm{d}r} - \frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{\rho z_+^2}{8}\right) - \frac{G M_{{\odot}} \rho}{r^2}, \end{equation}

where $\rho z_+^2 / 8$ is the AW pressure, which is one-half the AW energy density (Dewar Reference Dewar1970). Equation (3.7) implies that

(3.85)\begin{equation} \frac{1}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}r} + \frac{1}{U} \frac{\mathrm{d}U}{\mathrm{d}r} - \frac{1}{B}\frac{\mathrm{d}B}{\mathrm{d}r} = 0. \end{equation}

Equations (3.41) and (3.85) can be used to rewrite (3.84) in the form

(3.86)\begin{equation} \frac{1}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}r} \left\{{-}U^2 + \overline{ c_\textrm{s}}^2 + \frac{(\delta v_\textrm{b})^2 (1+y_\textrm{b})^{2-\sigma}[y^2(1+\sigma) + 3y]}{4 y_\textrm{b}(1+y)^{3-\sigma}}\right\} = \frac{2}{r}\left(\gamma_B U^2 - \frac{v_\textrm{esc}^2 R_{{\odot}}}{4 r}\right), \end{equation}

where

(3.87)\begin{equation} \gamma_B \equiv{-} \frac{r}{2B} \frac{\mathrm{d}B}{\mathrm{d}r} = 1 - \frac{r}{2\eta} \frac{\mathrm{d}\eta}{\mathrm{d}r}, \end{equation}

and $\eta$ is defined in (3.2). In order for (3.86) to possess a transonic-wind solution for $U$, the quantity in braces on the left-hand side must be positive near the Sun and negative far from the Sun; i.e. it must pass through zero at some radius $r_\textrm {c}$ (the wave-modified sonic critical point). In order for $\textrm {d} \rho / \textrm {d} r$ to remain finite at $r=r_\textrm {c}$, the right-hand side of (3.86) must also vanish at $r_\textrm {c}$. Together, these conditions yield the constraints

(3.88)\begin{equation} r_\textrm{c} = \frac{v_\textrm{esc}^2 R_{{\odot}}}{4 \gamma_{B{\rm c}} U_\textrm{c}^2} \end{equation}

and

(3.89)\begin{equation} U_\textrm{c}^2 = \overline{ c_\textrm{s}}^2 + \frac{(\delta v_\textrm{b})^2 (1+y_\textrm{b})^{2-\sigma}[y_\textrm{c}^2(1+\sigma) + 3y_\textrm{c}]}{4 y_\textrm{b}(1+y_\textrm{c})^{3-\sigma}} , \end{equation}

where, here and in the following, a ‘c’ subscript indicates that the subscripted quantity is evaluated at $r=r_\textrm {c}$. An implicit assumption underlying (3.88) and (3.89) is that

(3.90)\begin{equation} r_\textrm{c} < r_\textrm{A}, \end{equation}

so that the quasi-isothermal approximation applies at $r_\textrm {c}$.

Mass and flux conservation (i.e. (3.7)) imply that

(3.91)\begin{equation} \frac{\rho_\textrm{c} U_\textrm{c}}{B_\textrm{c}} = \frac{\rho_\textrm{b} U_\textrm{b}}{B_\textrm{b}} = \frac{\rho_\textrm{b} v_\textrm{Ab}}{B_\textrm{b} y_\textrm{b}}, \end{equation}

where the second equality in (3.91) follows from (3.11). Equation (3.2) implies that

(3.92)\begin{equation} \frac{B_\textrm{b}}{B_\textrm{c}} = \frac{r_\textrm{c}^2}{R_{{\odot}}^2} \psi \frac{\eta_\textrm{b}}{\eta_\textrm{c}} = \frac{v_\textrm{esc}^4}{16 \gamma_{B{\rm c}}^2 U_\textrm{c}^4} \psi \frac{\eta_\textrm{b}}{\eta_\textrm{c}}, \end{equation}

where the second equality in (3.92) follows from (3.88). Upon substituting (3.92) into (3.91) and evaluating $v_\textrm {Ab}$ using (3.63), one obtains

(3.93)\begin{equation} U_\textrm{c}^2 = v_\textrm{esc}^2 \left[\frac{y_\textrm{c}^2 \psi^{1/4} \xi (x \tilde{\rho}_{{\odot}})^{1/8}}{16 \gamma_{B{\rm c}}^2 \eta_\textrm{c} y_\textrm{b}}\right]^{2/3}. \end{equation}

Substituting (3.93) into (3.89) yields

(3.94)\begin{equation} \left[\frac{y_\textrm{c}^2 \psi^{1/4} \xi (x \tilde{\rho}_{{\odot}})^{1/8}}{16 \gamma_{B{\rm c}}^2 \eta_\textrm{c} y_\textrm{b}}\right]^{2/3} - x - \frac{\epsilon (1+y_\textrm{b})^{2-\sigma}[y_\textrm{c}^2(1+\sigma) + 3 y_\textrm{c}]}{4 y_\textrm{b}(1+y_\textrm{c})^{3-\sigma}} = 0. \end{equation}

The integral over $r$ of $\rho ^{-1}$ times the momentum equation (3.84) yields the Bernoulli integral,

(3.95)\begin{align} \frac{U^2}{2} + \overline{ c_\textrm{s}}^2 \ln\left(\frac{\rho}{\rho_\textrm{b}}\right) - \frac{v_\textrm{esc}^2 R_{{\odot}}}{2 r} - \frac{(\delta v_\textrm{b})^2 (1+y_\textrm{b})^{2-\sigma}}{2y_\textrm{b}} \left[ \left(\frac{1+\sigma}{1-\sigma}\right) (1+y)^{\sigma - 1} + (1+y)^{\sigma -2}\right] = \varGamma, \end{align}

where $\varGamma$ is independent of $r$. Evaluating (3.95) at $r=r_\textrm {b}$ leads to the equality

(3.96)\begin{equation} \varGamma = \frac{U_\textrm{b}^2}{2} - \frac{v_\textrm{esc}^2 \psi^{1/2}}{2} - \frac{(\delta v_\textrm{b})^2}{2}\left[ \frac{1+\sigma}{1-\sigma} + \frac{2}{y_\textrm{b}(1-\sigma)}\right]. \end{equation}

Evaluating (3.95) at $r=r_\textrm {c}$, rewriting $r_\textrm {c}$ and $U_\textrm {c}$ using (3.88) and (3.89), rewriting $\varGamma$ using (3.96), and multiplying the resulting equation by $2/v_\textrm {esc}^2$ yields

(3.97)\begin{align} &\psi^{1/2} + x\left[4 \ln\left( \frac{y_\textrm{c}}{y_\textrm{b}}\right) + 1 - 4 \gamma_{B{\rm c}}\right] + \frac{\epsilon (1 + y_\textrm{b})^{2-\sigma} \varPhi}{4 (1-\sigma)y_\textrm{b} (1+y_\textrm{c})^{3-\sigma}}\nonumber\\ &\quad - \frac{\eta_\textrm{b}^2 \psi^{3/2}}{y_\textrm{b}^2 \xi^2 (x \tilde{\rho}_{{\odot}})^{1/4}} + \epsilon \left[\frac{1+\sigma}{1-\sigma} + \frac{2}{y_\textrm{b}(1-\sigma)}\right] = 0 , \end{align}

where

(3.98)\begin{equation} \varPhi \equiv y_\textrm{c}^2[(1-4\gamma_{B\textrm{c}})(1-\sigma^2) - 4(1+\sigma)] + y_\textrm{c}[3(1-4\gamma_{B\textrm{c}})(1-\sigma) - 12 - 4\sigma] - 8. \end{equation}

3.8. Mathematical structure of the model and approximate analytic solutions

The various quantities appearing in the model equations can be divided into five groups: (1) quantities that are determined observationally ($R_{\odot }$, $M_{\odot }$, $v_\textrm {esc}$, $\rho _{\odot }$, $\tilde {\rho }_{\odot }$, $\psi$, $\delta v_{\odot \textrm {eff}}$, $\bar {B}$); (2) free parameters ($\sigma$, $l_\textrm {b}$); (3) the super-radial expansion factor $\eta (r)$, which takes on different values in different magnetic flux tubes and in different models for the solar magnetic field; (4) the three principal unknowns, $y_\textrm {b}$, $y_\textrm {c}$, and $x$; and (5) additional unknowns that can be determined once $y_\textrm {b}$, $y_\textrm {c}$, and $x$ are found ($\dot {M}$, $\chi _\textrm {H}$, $U(r)$, $\rho _\textrm {b}$, $q_\textrm {b}$, $v_\textrm {Ab}$, $r_\textrm {c}$, $r_\textrm {A}$). The three principal unknowns $y_\textrm {b}$, $y_\textrm {c}$, and $x$ are determined by solving the three simultaneous equations (3.67), (3.94), and (3.97), where it must be remembered that $\epsilon$ is itself a function of $x$ via (3.64). The additional unknowns $\dot {M}$, $\chi _\textrm {H}$, $\rho _\textrm {b}$, $v_\textrm {Ab}$, $q_\textrm {b}$, and $r_\textrm {c}$ then follow immediately from (3.13), (3.45), (3.57), (3.63), (3.82), and (3.88), respectively. For example,

(3.99)\begin{equation} \dot{M} = \frac{R_\odot^2 \bar{B}^2}{v_\textrm{esc}} y_\textrm{b}^{{-}1} (x \tilde{\rho}_\odot)^{1/8} \xi \psi^{1/4} . \end{equation}

The procedures for determining $r_\textrm {A}$ and $U(r)$ involve a few more steps, which are described in Appendix A.

Two approximate analytic solutions to (3.67), (3.94) and (3.97), valid in two different parameter regimes, are derived in Appendix B. Both solutions rely on the approximations

(3.100)\begin{equation} y_\textrm{b} \gg 1 \qquad \psi = 1 \qquad \epsilon \ll 1 \qquad \eta_\textrm{c} = \gamma_{B \textrm{c}} = 1. \end{equation}

The last equality in (3.100) amounts to taking the magnetic-field lines to be purely radial at $r\geqslant r_\textrm {c}$. The first of the two approximate analytic solutions is valid in the conduction-dominated regime, in which the dominant sink of internal energy in the sub-Alfvénic region is the flux of heat from the corona into the transition region, rather than $p\,\textrm {d} V$ work. As discussed further in the following, this regime arises only for values of $\delta v_{\odot \textrm {eff}}$ much smaller than the solar value. This limit is thus not directly relevant to solar-wind observations. To leading order in $\epsilon _\odot$, the mass outflow rate $\dot {M}^\textrm {(cond)}$ and asymptotic flow velocity $U_{\infty }^\textrm {(cond)}$ in this parameter regime are given by

(3.101)\begin{equation} \dot{M}^\textrm{(cond)}=\frac{R_{{\odot}}^2 B_\textrm{ref}^2}{v_\textrm{esc}} \frac{1}{I_1^{1/14} I_2} [ \epsilon_\odot^{14-4\sigma} (\eta_\textrm{b} B_\ast)^{{-}4\sigma} \tilde{l}_\textrm{b}^{3\sigma}\tilde{\rho}_\odot^{7-2\sigma}]^{1/(7-7\sigma)}, \end{equation}

where $I_2$ is a numerical constant given in (B 12), and

(3.102)\begin{align} U_\infty^\textrm{(cond)} &= v_\textrm{esc}\Bigg[ \frac{2^{9-4\sigma} \zeta} {(1+\sigma)^2 I_1^{\sigma/14}} \left( \frac{1-\sigma}{7-3\sigma}\right)^{(7-3\sigma)/2}\nonumber\\ &\quad\vphantom{ \frac{2^{9-4\sigma} \zeta} {(1+\sigma)^2 I_1^{\sigma/14}}}\epsilon_\odot^{-(7-2\sigma)/7} B_\ast^{(7-5\sigma)/7} \eta_\textrm{b}^{2\sigma/7} \tilde{\rho}_\odot^{({-}7 + 2 \sigma)/14} \tilde{l}_\textrm{b}^{{-}3\sigma/14}\Bigg]^{1/2}. \end{align}

The coronal temperature in the conduction-dominated limit follows directly from (2.6) and (B 8).

The second approximate analytic solution is valid in the expansion-dominated regime, in which the $p\,\textrm {d} V$ work resulting from expansion is the dominant sink of internal energy in the sub-Alfvénic region. To leading order, the mass outflow rate in this parameter regime is given by

(3.103)\begin{equation} \dot{M}^\textrm{(exp)}_0 = \epsilon_\odot \bar{B} R_\odot^2 \sqrt{4{\rm \pi} \rho_\odot} = \frac{P_\textrm{AW}(r_\textrm{b})}{v_\textrm{esc}^2}, \end{equation}

where $P_\textrm {AW}(r_\textrm {b})$ is the AW power (3.44) evaluated at the coronal base. The asymptotic wind speed in the expansion-dominated regime is to leading order given by

(3.104)\begin{equation} U_{\infty, 0}^\textrm{(exp)} = v_\textrm{esc}, \end{equation}

and the coronal temperature in the expansion-dominated regime is to leading order

(3.105)\begin{equation} T = \frac{m_\textrm{p} v_\textrm{esc}^2}{k_\textrm{B} \ln(\epsilon_\odot^{{-}3} \tilde{\rho}_{{\odot}}^{{-}3/2} \tilde{l}_\textrm{b}^{{-}1} \eta_\textrm{b} B_\ast)}. \end{equation}

Equations (3.103) and (3.104) reproduce the approximate scalings of the simplified calculation presented in § 2. Equation (3.105) matches the right-hand side of (2.19) to within five percent for Sun-like parameters, the difference arising because in the model developed in this section, $\delta v_\textrm {b}$ has a weak dependence on the coronal temperature via (3.48) and (3.57) that is not accounted for in § 2. Appendix B presents higher-order corrections to (3.103), (3.104), and (3.105) that account for conductive losses, wave momentum deposition inside the wave-modified sonic critical point, and the fact that only part of $P_\textrm {AW}(r_\textrm {b})$ is dissipated within the sub-Alfvénic region. Second- and fourth-order approximations to $\dot {M}$ and $U_\infty$ are shown in figure 5 below.

Analytic estimates for the ranges of $\epsilon _{\odot }$ values corresponding to the conduction-dominated and expansion-dominated limits are given in Appendix B. There are three constraints on $\epsilon _\odot$ in the conduction-dominated limit, the most stringent of which is that the wave-energy term dominates over the internal-energy term in the Bernoulli integral at the wave-modified sonic critical point. The resulting range of $\epsilon _\odot$ values is much smaller than the solar value, as illustrated in figure 4. The expansion-dominated limit corresponds to a finite range of $\epsilon _\odot$ values that is sufficiently large that $p\,\textrm {d} V$ work dominates over conduction as the primary internal-energy sink within the sub-Alfvénic region, and sufficiently small that the sound speed makes the dominant contribution to the outflow velocity at the wave-modified sonic point in (3.89). This range of $\epsilon _\odot$ values is relevant to the solar case, as shown in the next section.

4.1. Magnetic-field model

The equations in § 3 are compatible with any model for the radial profile of the magnetic-field strength, or, equivalently, any choice of $\eta (r)$. On the other hand, the approximate analytic solutions derived in Appendix B assume that

(4.1)\begin{equation} \eta_\textrm{c} = \gamma_{B\textrm{c}} = 1. \end{equation}

To maintain consistency between the numerical and analytic solutions, and to avoid introducing additional complexity and free parameters, I take (4.1) to hold when computing the numerical solutions presented in this section. In essence, (4.1) amounts to assuming that all of the super-radial expansion of the magnetic field occurs inside the wave-modified sonic critical point. In measurements from the FIELDS experiment on the Parker Solar Probe (PSP) (Bale et al. Reference Bale, Goetz, Harvey, Turin, Bonnell, Dudok de Wit, Ergun, MacDowall, Pulupa and Andre2016) during PSP's first few orbits, $|B_r| \simeq 2.2 \mbox { nT} (1 \mbox { a.u.}/r)^2$ (Sam Badman, private communication), where a.u. is the abbreviation for astronomical unit. In order to match this $B_r$ profile, I set

(4.2)\begin{equation} B_\ast\ {=}\ 0.00856 \hspace{0.3cm} \longleftrightarrow \hspace{0.3cm} B_r(\mbox{1 a.u.}) = 2.2 \mbox{ nT} \end{equation}

when computing the results shown in figures 3 through 6.

Figure 3. The outflow velocity $U$, density $\rho$, r.m.s. amplitude of the velocity fluctuation $\delta v$, and temperature $T$ as functions of heliocentric distance $r$ in a numerical solution to the model equations with $\eta _\textrm {b} = 100$, $\sigma = 0.5$, and the parameter values in (4.1), (4.2), (4.6), and (4.7). The dotted lines in the top-left panel are described in the text. In the top panels and the lower-right panel, the circles are measurements from a 1 h interval containing PSP's first perihelion on 6 November 2018, from figure 1 of Kasper et al. (Reference Kasper, Bale, Belcher, Berthomier, Case, Chandran, Curtis, Gallagher, Gary and Golub2019). The error bars around the PSP data points in these three panels indicate the approximate range of values with a relative occurrence rate of at least 50 % of the peak occurrence rate within that 1 h interval. In the top-right panel, the error bars lie within the data point. In the lower-left panel, the PSP data point is from Chen et al. (Reference Chen, Bale, Bonnell, Borovikov, Bowen, Burgess, Case, Chandran, de Wit and Goetz2020), and the error bars around that data point show the approximate range of measured values near $r=35.7 R_{\odot }$ in figure 7 of Chen et al. (Reference Chen, Bale, Bonnell, Borovikov, Bowen, Burgess, Case, Chandran, de Wit and Goetz2020). The triangle in the lower-left panel is the value obtained by De Pontieu et al. (Reference De Pontieu, McIntosh, Carlsson, Hansteen, Tarbell, Schrijver, Title, Shine, Tsuneta and Katsukawa2007) from an analysis of the motion of filamentary structures in the low solar atmosphere based on observations from the Solar Optical Telescope on the Hinode satellite.

Figure 4. Mass outflow rate as a function of the effective fluctuating velocity at the photosphere $\delta v_{\odot \textrm {eff}} = f_\textrm {chr}^{1/2} \delta v_\odot$ (see (3.49)) for the parameter values in (4.1), (4.2), (4.7), $\eta _\textrm {b} = 30$, and $\sigma =0.18$, where $f_\textrm {chr}$ is the chromospheric/transition-region AW transmission coefficient in (3.46). The corresponding r.m.s. photospheric velocity $\delta v_\odot$ is shown at the top of the plot for the case in which $f_\textrm {chr} = 0.05.$ The solid line plots (3.99) using the numerical solution to (3.67), (3.94), and (3.97). The dotted and short-dashed lines plot the approximate analytic results from (3.101) and (3.103), respectively. The vertical dash-dot line corresponds to $\epsilon _{\odot \textrm {cond}}$ in (B 14). The left and right edges of the shaded region correspond to $\epsilon _{\odot \textrm {exp}, \textrm {min}}$ and $\epsilon _{\odot \textrm {exp}, \textrm {max}}$ in (B 35) and (B 36), respectively. To the right of the vertical long-dashed line, $r_\textrm {c} > r_\textrm {A}$, which violates (3.90) and the assumptions of the model.

Figure 5. The dependence of various flow properties on the Wang-Sheeley super-radial expansion factor $f_{\textrm {max}, \textrm {WS}}$ (4.8) for the parameter values in (4.1), (4.2), and (4.6)–(4.9). All quantities are evaluated using numerical solutions to the model equations, with the exception of $\dot {M}_2^\textrm {(exp)}$, $\dot {M}_4^\textrm {(exp)}$, $U_{\infty , 2}^\textrm {(exp)}$, and $U_{\infty , 4}^\textrm {(exp)}$, which are defined in (B 32) and (B 33), and the data points labeled ‘WS empirical’, which are taken from table 2 of Wang & Sheeley (Reference Wang and Sheeley1990). The horizontal error bars on these data points convey the half widths of the $f_{\textrm {max}, \textrm {WS}}$ data bins in that table. The vertical error bars correspond to one-half of the $100 \mbox { km} \mbox { s}^{-1}$ increment between the discretised $U_{\infty }$ values that define four of the five data bins. The quantity $I_1 \rho _\textrm {b} \overline {c_\textrm {s}}^3$ in the left panel of the second row is the low-Mach-number approximation to $q_\textrm {b}$ given in (3.83).

Figure 6. The dependence of various flow properties on the Wang-Sheeley super-radial expansion factor $f_{\textrm {max},\textrm {WS}}$ (4.8) and the effective fluctuating velocity at the photosphere $\delta v_{\odot \textrm {eff}}$ defined in (3.49).

4.2. Values of the AW power at the coronal base and $\delta v_{\odot {\rm eff}}$

Equations (3.46) and (3.49) and the flux-conservation relation $A_\textrm {b} B_\textrm {b} = A_{\odot } B_{\odot }$ imply that

(4.3)\begin{equation} \delta v_{{\odot} \textrm{eff}} = \left(\frac{4{\rm \pi}}{\rho_{{\odot}}}\right)^{1/4} \left( \frac{P_\textrm{AWb}}{B_\textrm{b} A_\textrm{b}} \right)^{1/2}. \end{equation}

When the solar wind is powered primarily by AWs, $P_\textrm {AWb}$ is approximately equal to the value of $\dot {E}_\textrm {m0}$ in (1.3), i.e.,

(4.4)\begin{equation} P_\textrm{AWb} \simeq \tfrac{1}{2} \dot{M} ( v_\textrm{esc}^2 + U_{\infty}^2). \end{equation}

Upon evaluating the right-hand side of (4.4) using the data in table 1 of Schwadron & McComas (Reference Schwadron and McComas2008) for Ulysses’ third northern polar pass (3NPP), one obtains

(4.5)\begin{equation} P_\textrm{AWb} = 3.6 \times 10^{27} \mbox{ erg} \mbox{ s}^{{-}1}, \end{equation}

which corresponds to $P_\textrm {AWb} / (4 {\rm \pi}R_{\odot }^2) = 0.59 \times 10^5 \mbox { erg} \mbox { cm}^{-2} \mbox { s}^{-1}$. I use the 3NPP data because this is the part of Ulysses's first three orbits during which the average scaled radial magnetic field $|\langle B_r \rangle | \cdot (r/ 1 \mbox { a.u.})^2$ was most consistent with (4.2) (in the 3NPP data, $|\langle B_r \rangle | \cdot (r/ 1 \mbox { a.u.})^2 = 2.1 \pm 0.08 \mbox { nT}$), and because $\dot {M}$ is strongly correlated with the average scaled radial magnetic field (Schwadron & McComas Reference Schwadron and McComas2008). Equations (3.47), (4.2), (4.3), and (4.5) and the relation $B_\textrm {b} A_\textrm {b} = |B_r(1 \mbox { a.u.})| 4 {\rm \pi}(1 \mbox { a.u.})^2$ imply that

(4.6)\begin{equation} \delta v_{{\odot} \textrm{eff}} = 0.22 \mbox{ km} \mbox{ s}^{{-}1}. \end{equation}

This value of $\delta v_{\odot \textrm {eff}}$ is used in figures 3, 5, and 7.

Figure 7. The dependence of various flow properties on the Wang-Sheeley super-radial expansion factor $f_{\textrm {max},\textrm {WS}}$ (4.8) and the strength of the radial magnetic field at $r= 1 \mbox { a.u.}$

4.3. Free parameters

As discussed in § 3.8, there are two free parameters in the model: $l_\textrm {b}$ (the AW dissipation length scale at $r_\textrm {b}$) and $\sigma$ (the dimensionless coefficient in the turbulent heating rate). The solutions to (3.67), (3.94), and (3.97) are not very sensitive to the value of $l_\textrm {b}$. Throughout the rest of this section, $l_\textrm {b}$ is thus simply fixed at a value that seems physically reasonable:

(4.7)\begin{equation} l_\textrm{b} = 0.3 R_{{\odot}}. \end{equation}

On the other hand, the solutions depend sensitively on the value of $\sigma$. Larger values of $\sigma$ cause a larger fraction of the AW power at the coronal base $P_\textrm {AWb}$ to be dissipated in the quasi-isothermal sub-Alfvénic region, which increases $\dot {M}$ (see, e.g. (2.15) and (2.16)). For fixed $P_\textrm {AWb}$, increasing $\dot {M}$ decreases $U_{\infty }$. In some of the examples to follow, $\sigma$ is varied to optimise agreement between the model and observations, as described further in §§ 4.4 and 4.6. The super-radial expansion factor at the coronal base, $\eta _\textrm {b}$, also varies in the model, but this quantity is in principle observable for individual flux tubes. The value of $\eta _\textrm {b}$ is thus treated as an input into the model rather than an adjustable parameter.

4.4. Fiducial solution matching measurements from Parker Solar Probe's first perihelion

Figure 3 illustrates the $r$ dependence of the outflow velocity, density, fluctuating velocity, and temperature in a numerical solution to the model equations that is designed to agree with measurements from PSP's first perihelion encounter (E1) on 6 November 2018. The procedure for computing $U(r)$ and $T(r)$ is described in Appendix A. In order to solve for $U$ at some radius $r$, the value of $\eta$ must be specified at that $r$. For figure 3, I set $\eta (r) = 1 + (\eta _\textrm {b} - 1) \exp (-(r-r_\textrm {b})^2/(0.02 R_{\odot } )^2)$, which is effectively unity at $r= r_\textrm {c}$, consistent with (4.1). This choice of $\eta (r)$ is not realistic for the low corona (see, e.g. Cranmer et al. Reference Cranmer, van Ballegooijen and Edgar2007), but the flow profile in the low corona is not a focus of this work. It should be noted that while the value of $\delta v$ at $r=r_\textrm {b}$ (i.e. $\delta v_\textrm {b}$) shown in the lower-left panel of figure 3 depends on $\eta _\textrm {b}$ through (3.48), (3.57), and (3.62), $\delta v_\textrm {b}$ does not on the way in which $\eta (r)$ decreases from $\eta _\textrm {b }$ to 1.

The solution in figure 3 is based upon the somewhat arbitrary assumption that $\eta _\textrm {b} = 100$ in the magnetic flux tube encountered by PSP at the time of its first perihelion. The value of $\sigma$ is set equal to 0.5 to optimise the agreement between the model and the data. Fits of comparable (in some cases superior) quality can be obtained for different values of $\eta _\textrm {b}$. For example, the parameter combinations $(\eta _\textrm {b}, \sigma ) = (300, 0.47)$ and $(\eta _\textrm {b}, \sigma ) = (30,0.55)$ lead to similar agreement with the data. Modelling of the solar magnetic field during PSP E1 suggests that the solar-wind stream encountered by PSP at the time of PSP's first perihelion originated in a small equatorial coronal hole (Bale et al. Reference Bale, Badman, Bonnell, Bowen, Burgess, Case, Cattell, Chandran, Chaston and Chen2019). The fact that all four quantities plotted in figure 3 can be matched by varying the single parameter $\sigma$ suggests that the model is reasonably successful at capturing the physical processes that control the heating and acceleration of coronal-hole outflows.

The dotted lines in the top-left panel of figure 3 are solutions of the Bernoulli equation (3.95) for different values of the Bernoulli constant $\varGamma$. One of the dotted lines intersects the solid line, and that intersection occurs at the wave-modified sonic critical point, $r=r_\textrm {c}$. That dotted line is an accretion-like solution of (3.95) with the same value of $\varGamma$ as in the model solution but with $\textrm {d} U / \textrm {d} r < 0$ at $r=r_\textrm {c}$.

4.5. Illustration of the conduction-dominated and expansion-dominated regimes

Figure 4 plots $\dot {M}$ as a function of $\delta v_{\odot \textrm {eff}}$ when $\eta _\textrm {b} = 30$ and $\sigma = 0.18$. This figure includes the numerical solution to (3.67), (3.94), and (3.97) as well as the approximate analytic results $\dot {M}^\textrm {(cond)}$ and $\dot {M}^\textrm {(exp)}_0$ from (3.101) and (3.103). The vertical dash-dot line in this figure corresponds to $\epsilon _{\odot \textrm {cond}}$ in (B 14). The conduction-dominated approximation that gives rise to $\dot {M}^\textrm {(cond)}$ in (3.101) assumes that $\epsilon _\odot \ll \epsilon _{\odot \textrm {cond}}$.Footnote ¹⁰ The shaded region corresponds to $\epsilon _{\odot \textrm {exp}, \textrm {min}}< \epsilon _\odot < \epsilon _{\odot \textrm {exp}, \textrm {max}}$, where $\epsilon _{\odot \textrm {exp}, \textrm {min}}$ and $\epsilon _{\odot \textrm {exp}, \textrm {max}}$ are defined in (B 35) and (B 36). The expansion-dominated approximation in § B.2 assumes that $\epsilon _{\odot \textrm {exp}, \textrm {min}}\ll \epsilon _\odot \ll \epsilon _{\odot \textrm {exp}, \textrm {max}}$. The vertical long-dashed line corresponds to the condition $r_\textrm {c} = r_\textrm {A}$. To the right of this line, $\dot {M}$ and the solar-wind density become so large that $r_\textrm {A} < r_\textrm {c}$, violating (3.90) and the assumptions underlying the model. In conjunction with (4.6), figure 4 illustrates that the expansion-dominated regime is relevant to the solar wind and that the conduction-dominated regime corresponds to values of $\delta v_{\odot \textrm {eff}}$ much smaller than the solar value.

4.6. Anti-correlation between the coronal super-radial expansion factor and $U_{\infty }$

Wang & Sheeley (Reference Wang and Sheeley1990) showed that the outflow velocity in a solar-wind stream at $r=1 {\rm a.u.}$ is anti-correlated with the super-radial expansion factor in the coronal magnetic flux tube from which the solar-wind stream originated. To compare their results with the model developed in § 3, I follow Cranmer et al. (Reference Cranmer, van Ballegooijen and Edgar2007) by defining the Wang & Sheeley (Reference Wang and Sheeley1990) coronal super-radial expansion factor $f_{\textrm {max}, \textrm {WS}}$ to be $\eta (1.04 R_{\odot })/ \eta (2.5 R_{\odot })$. I then compute $\eta (1.04 R_{\odot })$ and $\eta (2.5 R_{\odot })$ using a magnetic-field model similar to that used by Cranmer et al. (Reference Cranmer, van Ballegooijen and Edgar2007). In particular, I employ the global solar-magnetic-field model of Banaszkiewicz, Axford & McKenzie (Reference Banaszkiewicz, Axford and McKenzie1998) with the parameter values listed below their equation (2) ($K=1$, $M=1.789$, $a_1^\textrm {(B98)} = 1.538$, and $Q=1.5$). I supplement this global magnetic field with the low-solar-atmosphere magnetic-field model of Hackenberg et al. (Reference Hackenberg, Marsch and Mann2000) using the same parameter values listed in their figure 1 ($L = 30 \mbox { Mm}$, $d = 0.34 \mbox { Mm}$, $B_0 = 11.8 \mbox { G}$, and $B_\textrm {max} = 1.5 \mbox { kG}$). I then integrate out along a magnetic-field line at the edge of the polar corona hole in the Banaszkiewicz et al. (Reference Banaszkiewicz, Axford and McKenzie1998) model, compute $\eta (r)$, and obtain

(4.8)\begin{equation} f_{\textrm{max}, \textrm{WS}} = \frac{\eta_\textrm{b}}{5.75}. \end{equation}

I take (4.8) to hold even as $\eta _\textrm {b}$ is varied in figures 5, 6, and 7. This is equivalent to assuming that if local magnetic structures on the Sun cause $\eta _\textrm {b}$ to increase or decrease by some factor relative to the Hackenberg/Banaszkiewicz model just described, then $\eta (1.04 R_{\odot })$ increases or decreases by the same factor, but $\eta (2.5 R_{\odot })$ does not change.

If $\sigma$ is fixed while $\eta _\textrm {b}$ is varied, the model of § 3 does not agree with results of Wang & Sheeley (Reference Wang and Sheeley1990) shown in the top-right panel of figure 5. On the other hand, the model becomes consistent with those results if $\sigma$ is taken to have a power-law dependence on $\eta _\textrm {b}$ of the form

(4.9)\begin{equation} \sigma = 5.7\times 10^{{-}2} \, \eta_\textrm{b}^{0.34}. \end{equation}

Equation (4.9) is used to determine $\sigma$ in figures 5 through 7. The implication of (4.9) that $\sigma$ is an increasing function of $\eta _\textrm {b}$ is plausible because increasing $\eta _\textrm {b}$ increases $v_\textrm {Ab}$, as can be seen from (3.63). This, in turn, increases the number of Alfvén-speed scale heights between the coronal base and the Alfvén critical point ($\int _{r_\textrm {b}}^{r_\textrm {A}} (1/v_\textrm {A}) | \,\textrm {d} v_\textrm {A}/ \textrm {d} r|\, \textrm {d} r$), which increases the fraction of $P_\textrm {AWb}$ that is dissipated at $r<r_\textrm {A}$ (Chandran & Hollweg Reference Chandran and Hollweg2009). Further work, however, is needed to clarify how the structure of the coronal magnetic field influences the rate of turbulent dissipation along different magnetic flux tubes with different values of $\eta _\textrm {b}$ and to test the extent to which (3.41) and (4.9) are consistent with more rigorous treatments of AW turbulence in the sub-Alfvénic region of the solar wind.

The top-left panel of figure 5 shows that as $f_{\textrm {max}, \textrm {WS}}$ ranges from $\simeq 3$ to $\simeq 30$, $\dot {M}$ varies from $10^{-14} M_{\odot } \mbox { yr}^{-1}$ to $2\times 10^{-14} M_{\odot } \mbox { yr}^{-1}$, values that are similar to the solar-mass-loss rates inferred from Ulysses and PSP measurements (McComas et al. Reference McComas, Barraclough, Funsten, Gosling, Santiago-Muñoz, Skoug, Goldstein, Neugebauer, Riley and Balogh2000; Schwadron & McComas Reference Schwadron and McComas2008; Kasper et al. Reference Kasper, Bale, Belcher, Berthomier, Case, Chandran, Curtis, Gallagher, Gary and Golub2019). The fourth-order analytic approximation $\dot {M}_4^\textrm {(exp)}$ from (B 32) reproduces the numerical solution to the model equations reasonably well. The second-order analytic approximation $\dot {M}_2^\textrm {(exp)}$ is also reasonably accurate at $f_{\textrm {max}, \textrm {WS}} >rsim 3$, but deviates markedly from the numerical solution at $f_{\textrm {max}, \textrm {WS}} \lesssim 2$. A similar comment applies to the top-right panel of figure 5, which also shows that the model agrees fairly well with observational constraints on $U_{\infty }(f_{\textrm {max}, \textrm {WS}})$ when (4.9) holds. The left panel of the second row of figure 5 shows that the low-Mach-number approximation to $q_\textrm {b}$ given in (3.83) is quite accurate at small $f_{\textrm {max}, \textrm {WS}}$, where $U_\textrm {b}/ \overline {c_\textrm {s}} \ll 1$ (see the right panel in the second row of this figure). However, as $f_{\textrm {max}, \textrm {WS}}$ increases, $U_\textrm {b}$ increases, because the outflow is concentrated into a smaller cross-sectional area at the coronal base. This increase in $U_\textrm {b}$ leads to larger losses of internal energy within the transition region from $p\,\textrm {d} V$ work and advection, which, in turn, causes $q_\textrm {b}$ to increase above the low-Mach-number scaling so that conductive heating within the transition region can balance the additional non-radiative cooling. This same panel also shows that $q_\textrm {b}$ is significantly smaller than the AW energy flux at the coronal base ($\simeq \rho _\textrm {b} (\delta v_\textrm {b})^2 v_\textrm {Ab} = \rho _{\odot }^{1/2} (\delta v_{\odot \textrm {eff}})^2 B_\textrm {b} /\sqrt {4{\rm \pi} }$), which increases with $f_{\textrm {max}, \textrm {WS}}$ because increasing $f_{\textrm {max}, \textrm {WS}}$ increases $\eta _\textrm {b}$ and hence $B_\textrm {b}$. The lower-left panel of figure 5 shows that $\rho _\textrm {b}$ increases by a factor of $\simeq 10$ as $f_{\textrm {max}, \textrm {WS}}$ increases from 1 to 100, consistent with the $\rho _\textrm {b} \propto \eta _\textrm {b}^{1/2} x^{1/4}$ scaling in (3.57) and (3.62), given that $x = \overline {c_\textrm {s}}^2/v_\textrm {esc}^2$ is approximately constant over this range of $f_\textrm {max}$. The lower-right panel shows that $r_\textrm {c}$ is typically several solar radii, whereas $r_\textrm {A}$ ranges from $12 R_{\odot }$ to $15 R_{\odot }$ for this set of parameters.

4.7. Dependence of the flow properties on the AW power, super-radial expansion factor, and strength of the interplanetary magnetic field

The contour plots in figure 6 show how several quantities vary as functions of $f_{\textrm {max}, \textrm {WS}}$ and $\delta v_{\odot \textrm {eff}}$ in numerical solutions to the model equations based on the parameter values in (4.1), (4.2), (4.7), and (4.9). The top panels of figure 6 show that $\dot {M}$ is an increasing function of both $f_{\textrm {max}, \textrm {WS}}$ and $\delta v_{\odot \textrm {eff}}$, whereas $U_\infty$ is a decreasing function of both $f_{\textrm {max}, \textrm {WS}}$ and $\delta v_{\odot \textrm {eff}}$. The second row shows that $r_\textrm {A}$ is a strongly decreasing function of $\delta v_{\odot \textrm {eff}}$ and only weakly dependent on $f_{\textrm {max}, \textrm {WS}}$ for values of $\delta v_{\odot \textrm {eff}}$ comparable to the Sun-like value in (4.6). The right panel of this row shows that $r_\textrm {c}$ is an increasing function of $f_{\textrm {max}, \textrm {WS}}$ and, for the most part, a decreasing function of $\delta v_{\odot \textrm {eff}}$. The lower-left panel of figure 6 shows that $\overline {c_\textrm {s}}$ varies only weakly with $f_{\textrm {max}, \textrm {WS}}$ and $\delta v_{\odot \textrm {eff}}$. As shown in the lower-right panel, $\chi _\textrm {H}$ varies by a factor of $\simeq 2$ over the parameter range shown. If $\delta v_{\odot \textrm {eff}}$ is set equal to the value in (4.6) and $f_\textrm {max}$ is restricted to the interval $(2, 8)$ so that $U_{\infty }$ in the upper-right panel takes on fast-wind-like values of 600–800 $\mbox { km} \mbox { s}^{-1}$, then $\chi _\textrm {H} \simeq 0.5 \text {--}0.7$, consistent with direct numerical simulations of reflection-driven AW turbulence (Perez et al. Reference Perez, Chandran, Klein and Martinovic2021).

Figure 7 displays contour plots of the same quantities as in figure 6, but this time as functions of $f_{\textrm {max}, \textrm {WS}}$ and $B_\ast$, or, equivalently, $B_r(\mbox {1 a.u.})$, using the parameters in (4.1), (4.6), (4.7), and (4.9). The top panels show that $\dot {M}$ increases approximately linearly with $B_r(\mbox {1 a.u.})$ at fixed $f_{\textrm {max}, \textrm {WS}}$, and that $U_\infty$ varies only weakly with $B_r(\mbox {1 a.u.})$, consistent with measurements from the Ulysses spacecraft (Schwadron & McComas Reference Schwadron and McComas2008; Riley et al. Reference Riley, Mikic, Lionello, Linker, Schwadron and McComas2010). The left panel of the middle row shows that $r_\textrm {A}$ depends more strongly on $B_r(\mbox {1 a.u.})$ than on $f_{\textrm {max}, \textrm {WS}}$, whereas the reverse is true for $r_\textrm {c}$. As in figure 6, $\overline {c_\textrm {s}}$ varies very weakly across the entire parameter range observed. The lower-right panel shows that, when (4.9) holds, $\chi _\textrm {H}$ depends more strongly on $f_{\textrm {max}, \textrm {WS}}$ than on $B_r(\mbox {1 a.u.})$.