An approximate analytic solution to the coupled problems of coronal heating and solar-wind acceleration
aa r X i v : . [ a s t r o - ph . S R ] J a n Under consideration for publication in J. Plasma Phys. An approximate analytic solution to thecoupled problems of coronal heating andsolar-wind acceleration
Benjamin D. G. Chandran
Department of Physics and Astronomy, University of New Hampshire, Durham, NewHampshire 03824, USA(Received xx; revised xx; accepted xx)
Between the base of the solar corona at r = r b and the Alfvén critical point at r = r A ,where r is heliocentric distance, the solar-wind density decreases by a factor & , butthe plasma temperature varies by a factor of only a few. In this paper, I show thatsuch quasi-isothermal evolution out to r = r A is a generic property of outflows pow-ered by reflection-driven Alfvén-wave (AW) turbulence, in which outward-propagatingAWs partially reflect, and counter-propagating AWs interact to produce a cascade offluctuation energy to small scales, which leads to turbulent heating. Approximating thesub-Alfvénic region as isothermal, I first present a simplified calculation showing thatin a solar or stellar wind powered by AW turbulence with minimal conductive losses, ˙ M ≃ P AW ( r b ) /v , U ∞ ≃ v esc , and T ≃ m p v / [8 k B ln( v esc /δv b )] , where ˙ M is the massoutflow rate, U ∞ is the asymptotic wind speed, T is the coronal temperature, v esc isthe escape velocity of the Sun, δv b is the fluctuating velocity at r b , P AW is the powercarried by outward-propagating AWs, k B is the Boltzmann constant, and m p is the protonmass. I then develop a more detailed model of the transition region, corona, and solarwind that accounts for the heat flux q b from the coronal base into the transition regionand momentum deposition by AWs. I solve analytically for q b by balancing conductiveheating against internal-energy losses from radiation, p d V work, and advection withinthe transition region. The density at r b is determined by balancing turbulent heating andradiative cooling at r b . I solve the equations of the model analytically in two differentparameter regimes. One of these solutions reproduces the results above for ˙ M , U ∞ ,and T to leading order. Analytic and numerical solutions to the model equations matcha number of observations.
1. Introduction
Pioneering works by Parker (1958, 1965), Hartle & Sturrock (1968), and Durney(1972) modeled the solar wind as a steady-state, spherical outflow powered by theoutward conduction of heat from the base of the corona. These models succeeded inproducing a supersonic wind, but were unable to explain the large outflow velocitiesmeasured in fast-solar-wind streams near Earth. They also had little predictive powerfor the mass outflow rate from the Sun, ˙ M , because they specified the temperatureof the coronal-base as a boundary condition, and ˙ M is highly sensitive to the coronaltemperature (Hansteen & Leer 1995).A possible solution to these problems was proposed almost as soon as the difficultiesbecame apparent, namely that the solar wind is powered by an Alfvén-wave (AW)energy flux (Parker 1965, p. 686; Hollweg 1973, 1978). This idea received strong supportfrom the discovery of large-amplitude AWs in the interplanetary medium that propagateaway from the Sun in the local plasma frame (Belcher & Davis 1971) as well as theremote observation of AW-like motions in the low solar atmosphere carrying an energyflux sufficient to power the solar wind (De Pontieu et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. ˙ M and the outflow speed far from the Sun ( U ∞ ) remain elusive. A numberof studies have obtained a single equation that constrains the two unknowns ˙ M and U ∞ .For example, Sandbaek et al. (1994) pointed out that if the energy flux far from the Sunis mostly in the form of bulk-flow kinetic energy, and if the energy flux at the coronalbase is dominated by the flux of gravitational potential energy, heat, and some additionalform of mechanical energy (from, e.g., AWs), then energy conservation implies that ˙ M = ˙ E m0 + ˙ E q ( v + U ∞ ) , (1.1)where ˙ E m0 is the mechanical-energy input into the solar wind at the corona base, ˙ E q is the energy input at the coronal base from thermal conduction (which is negative inmodels that include the lower solar atmosphere), v esc = (cid:18) GM ⊙ R ⊙ (cid:19) / = 617 . km s − (1.2)is the escape velocity of the Sun, G is the gravitational constant, M ⊙ is the solarmass, and R ⊙ is the solar radius. Schwadron & McComas (2003) derived a variantof (1.1) that explicitly relates ˙ E q to the altitude of the coronal-temperature maxi-mum. Hansteen & Leer (1995) and Hansteen et al. (1997) showed that ˙ E q ≪ ˙ E m0 , andHansteen & Velli (2012) made use of this finding to further refine (1.1), obtaining ˙ M = ˙ E m012 ( v + U ∞ ) . (1.3)Another important result was obtained by Leer & Holzer (1980), who found that heatinginside the sonic critical point enhances ˙ M but has little effect on U ∞ , whereas heatingbeyond the sonic critical point increases U ∞ but has little effect on ˙ M .The main goal of the present paper is to obtain approximate analytic solutions for ˙ M , U ∞ , the temperature of the corona, the heat flux from the coronal base into the lowersolar atmosphere, and the plasma density at the coronal base under the assumption thatAW turbulence is the primary energization mechanism of the solar wind. Section 2 takes afirst step towards this goal by presenting a simplified approximate calculation of ˙ M , U ∞ ,and the coronal temperature. Section 3 develops a more detailed solar-wind model thataccounts for physical processes neglected in Section 2. Approximate analytic solutionsto the equations of this model are presented in Section 3, and numerical solutions arepresented in Section 4. Section 5 discusses and summarises the main results of the paper.
2. Heuristic calculation of ˙ M , U ∞ , and the coronal temperature inAW-driven winds with minimal conductive losses Approximate expressions for ˙ M , U ∞ , and the coronal temperature can be quicklyobtained by modeling the solar wind as a spherically symmetric, steady-state outflowand assuming that: (1) AW turbulence is the dominant heating mechanism, (2) solarrotation can be neglected, so that the magnetic field B and flow velocity are aligned, (3) B ∝ r − ˆ r , where ˆ r is the radial unit vector (i.e., a split-monopole, with B r > in onehemisphere and B r < in the other), (4) momentum deposition by Alfvén waves canbe neglected between the coronal base and sonic critical point, and (5) p d V work is thedominant sink of internal energy in the sub-Alfvénic region of the solar wind, in whichthe solar-wind outflow velocity U is smaller than the Alfvén speed v A = B √ πρ , (2.1)where ρ is the mass density. Assumptions (3) through (5) are relaxed in the next section.In steady state, given assumption (2) above, mass and flux conservation imply (seeSection 3.1) that v Ab = y b U b , (2.2)where y b ≡ (cid:20) ρ b ρ ( r A ) (cid:21) / , (2.3) ρ b = ρ ( r b ) , r b is the radius of the coronal base, which in this section (but not the next)is simply set equal to R ⊙ , r A is the Alfvén critical point at which U = v A , v Ab = v A ( r b ) ,and U b = U ( r b ) . The mass outflow rate can thus be written in the form ˙ M = 4 πR ⊙ ρ b U b = 4 πR ⊙ ρ b v Ab y − . (2.4)As shown in Section 3.2, heating by reflection-driven AW turbulence causes the sub-Alfvénic part of the solar wind at r < r A to become quasi-isothermal, in the sense that c (cid:12)(cid:12)(cid:12)(cid:12) d c d r (cid:12)(cid:12)(cid:12)(cid:12) ≪ ρ (cid:12)(cid:12)(cid:12)(cid:12) d ρ d r (cid:12)(cid:12)(cid:12)(cid:12) , (2.5)where c ≡ pρ = 2 k B Tm p (2.6)is the square of the isothermal sound speed, k B is the Boltzmann constant, T is thetemperature, and m p is the proton mass. This argument is consistent with observationsand models of the solar wind, which suggest that the temperature varies by a factor ofonly a few between r b and r A , whereas ρ varies by a factor ∼ (see, e.g., Cranmer et al. r < r A can be written in the form U d U d r = − c ρ d ρ d r − v R ⊙ r . (2.7)Since ˙ M = 4 πr U ρ is independent of r , U − d U/ d r = − ρ − d ρ/ d r − /r . Upon substi-tuting this relation into (2.7) and rearranging terms, one obtains ( c − U ) ρ d ρ d r = 2 U r − v R ⊙ r . (2.8)In order for (2.8) to have a smooth transonic solution, the right-hand side of (2.8) mustvanish at the radius r c at which U = c s , so that d ρ/ d r remains finite. This leads to thetwo critical-point conditions U c = c s r c R ⊙ = v c , (2.9)where U c = U ( r c ) . Integrating (2.7), one obtains the Bernoulli integral U + c ln (cid:18) ρρ b (cid:19) − v R ⊙ r = constant = − v , (2.10)where the second equality in (2.10) results from evaluating the left-hand side of (2.10)at r = r b and dropping the U / term, which is ≪ v / . After evaluating the left-handside of (2.10) at r = r c and using (2.9) to rewrite U c and r c in terms of c , one obtains ln (cid:18) ρ c ρ b (cid:19) = − v c + 32 , (2.11)where ρ c = ρ ( r c ) . Upon setting ˙ M = 4 πr ρ c U c and using (2.9) and (2.11) to rewrite r c , ρ c , and U c in terms of c s , one obtains (Hansteen & Velli 2012) ˙ M = πR ⊙ v ρ b c exp (cid:18) − v c + 32 (cid:19) . (2.12)The exponential appearing on the right-hand side of (2.12) reflects the fact that, at r < r c ,the flow is subsonic and the density drops off approximately as in a static atmosphere(Hansteen & Velli 2012). Equating the right-hand sides of (2.4) and (2.12), one finds that ln y b = v c − ln c ≃ v c , (2.13)where c ≡ e / v / (16 c v Ab ) .When radiative cooling and thermal conduction are neglected, the internal-energyequation becomes − c U d ρ d r + ρUγ − c d r = Q, (2.14)where Q is the turbulent heating rate, and γ is the ratio of specific heats. In thequasi-isothermal approximation, the second term on the left-hand side of (2.14) canbe neglected. Multiplying (2.14) by πr and integrating over the quasi-isothermal sub-Alfvénic region, one obtains − c ˙ M Z r A r b ρ d ρ d r d r = 4 π Z r A r b r Q ( r ) d r ≃ P AW ( r b ) , (2.15)where the approximate equality assumes that the volume-integrated turbulentheating rate between r b and r A is comparable to P AW ( r b ) , the power carried byoutward-propagating AWs at the coronal base, consistent with direct numericalsimulations (Perez et al. submitted). Since ln( ρ b /ρ ( r A )) = 2 ln y b , (2.15) yields ˙ M ≃ P AW ( r b )2 c ln y b ≃ P AW ( r b ) v , (2.16)where the second relation in (2.16) follows from (2.13).Equations (2.15) and (2.16) can be understood as follows. Within the quasi-isothermalsub-Alfvénic region, each time the density of a parcel of plasma decreases by a factorof e , its thermal energy must be replaced via heating to offset internal-energy lossesfrom p d V work. The quantity c ln( ρ b /ρ ( r A )) = 2 c ln y b ≃ v is thus the heating costper unit mass for plasma to transit the sub-Alfvénic region. The reason that the product c ln( ρ b /ρ ( r A )) is approximately constant is that increasing c leads to an exponentialincrease in ˙ M and the solar-wind density and an exponential reduction of ρ b /ρ ( r A ) ,leaving c ln( ρ b /ρ ( r A )) approximately unchanged. Equations (2.15) and (2.16) statethat ˙ M is the net heating power within the sub-Alfvénic region (which is taken tobe ≃ P AW ( r b ) ) divided by the heating cost per unit mass.Assuming that the AW-energy flux and gravitational-potential-energy flux are thedominant mechanical-energy fluxes at the coronal base and that the kinetic-energy flux isthe dominant mechanical-energy flux at large r , and equating the mechanical luminositiesat r = r b and at large r , one obtains P AW ( r b ) −
12 ˙
M v = 12 ˙
M U ∞ . (2.17)Substituting (2.16) into (2.17) yields U ∞ ≃ v esc . (2.18)Reflection-driven AW turbulence thus changes the energy per unit mass from ≃ − v / at the coronal base to ≃ v / far from the Sun.In the absence of super-radial expansion, P AW ( r b ) = 4 πR ⊙ v Ab ρ b ( δv b ) , where δv b isthe root-mean-square (r.m.s.) amplitude of the fluctuating velocity at the coronal base.This relation, in conjunction with (2.4) and (2.16), implies that y b ≃ v / ( δv b ) , whichleads via (2.6) and (2.13) to the approximate value of the coronal temperature, T ≃ m p v k B ln( v esc /δv b ) . (2.19)A number of factors can cause ˙ M , U ∞ , and the coronal temperature to deviate fromthe estimates in (2.16) (2.18), and (2.19). For example, some of the AW power survivesout to r A , which reduces the total turbulent heating in the sub-Alfvénic region appearingon the right-hand side of (2.15), which in turn reduces ˙ M . The heat that is conductedfrom the corona into the transition region adds a negative sink term to the right-handsides of (2.14) and (2.15), which likewise acts to reduce ˙ M . On the other hand, the AWpressure force helps drive plasma away from the Sun at the critical point, which acts toincrease ˙ M . These effects, as well as super-radial expansion of the magnetic field, areincluded in the more detailed solar-wind model developed in the next section.
3. Steady-state model of the transition region, corona, and solar wind
The lowest layer of the solar atmosphere is the chromosphere, which extends two tothree thousand km above the photosphere with a temperature T ranging from severalthousand K to ∼ K. Bounding the chromosphere from above is the transition region,a narrow layer ∼ km thick, in which the density ρ drops (and T increases) byapproximately two orders of magnitude. Above the transition region lies the corona,which extends out a few solar radii from the Sun and has a temperature of ∼ K.The corona contains both closed magnetic loops, which connect back to the Sun atboth ends, and open magnetic-field lines that connect the solar surface to the distantinterplanetary medium. Regions of the corona permeated by open magnetic-field lineshave lower densities than closed-field-line regions and are referred to as coronal holes. Inthe analysis to follow, r b denotes the radius of the coronal base just above the transitionregion in Sun-centred spherical coordinates ( r, θ, φ ) , which is taken to be r b = 1 . R ⊙ . (3.1)The steady-state model developed in this section describes the outflowing plasmawithin an open magnetic flux tube from the transition region to beyond the Alfvéncritical point r A , at which the plasma outflow velocity U equals v A . Figure 1 providesa schematic overview of the model, which determines five unknowns — the density atthe coronal base ρ b , the temperature of the quasi-isothermal sub-Alfvénic region, themass outflow rate ˙ M , the asymptotic outflow velocity U ∞ , and the flux of heat from thecoronal base into the transition region q b — through the following five steps:1. balancing turbulent heating at r = r b against radiative cooling at r = r b ;2. balancing the total turbulent heating between r b and r A against the twoprimary sinks of internal energy in this region: pdV work and the flux ofheat into the transition region;3. balancing, within the transition region, conductive heating against internal-energy losses from p d V work, advection, and radiation;4. equating the mass outflow rate at r = r b with the mass outflow rate at thewave-modified sonic critical point r = r c ; and5. equating the wave-modified Bernoulli integral at r = r b and r = r c .These five steps are detailed in Sections 3.4 through 3.7. Section 3.1 reviews someidentities that follow from the conservation of mass and magnetic flux. Sections 3.2and 3.3 review previous results on reflection-driven AW turbulence in the solar windand present the simplified model of reflection-driven AW turbulence that is used in thispaper. 3.1. Flux and mass conservation
To simplify the analysis, solar rotation is neglected, and the magnetic field is taken tobe radial, except in the corona, where open magnetic-field lines fan out to fill the spaceabove closed magnetic loops at lower altitudes. Mathematically, B ( r ) = Bη ( r ) R ⊙ r , (3.2)where η ( r ) is the local super-radial expansion factor, which approaches 1 when r/R ⊙ ≫ ,and B is the magnetic-field strength that would arise at the photosphere in the absence Quantity Meaning First use ˙ M mass outflow rate (1.1) U ∞ asymptotic wind speed (1.1) v esc escape velocity at photosphere (1.2) R ⊙ solar radius (1.2) v A Alfvén speed (2.1) ρ plasma density (2.1) U solar-wind outflow velocity (2.2) y b [ ρ ( r b ) /ρ ( r A )] / (2.2) r b radius of coronal base (2.3) r A radius of Alfvén critical point (2.3) c s isothermal sound speed (2.6) T temperature (2.6) m p proton mass (2.6) k B Boltzmann constant (2.6) r c radius of wave-modified sonic critical point (2.9) γ ratio of specific heats (5/3) (2.14) Q turbulent heating rate (2.14) P AW power of outward-propagating AWs (2.15) δv b r.m.s. amplitude of fluctuating velocity at r b (2.19) η ( r ) local super-radial expansion factor (3.2) B field strength at r = R ⊙ in absence of super-radial expansion (3.2) ψ ( R ⊙ /r b ) ≃ . (3.3) A ( r ) cross-sectional area of the outflow (3.8) y ( r ) [ ρ ( r ) /ρ ( r A )] / (3.12) z ± r.m.s. amplitude of z ± (3.16) z + , z − Elsasser variables (3.17) q heat flux (3.33) l b AW dissipation length scale at r b (3.39) σ dimensionless coefficient in turbulent heating rate (3.40) χ H fraction of P AW ( r b ) that dissipates between r b and r A (3.42) f chr AW transmission coefficient, P AW ( r b ) /P AW ⊙ ( R ⊙ ) (3.46) ρ ⊙ plasma density at the photosphere, ≃ m p cm − (3.46) δv ⊙ r.m.s. amplitude of fluctuating velocity at R ⊙ (3.46) δv ⊙ eff f / δv ⊙ (3.48) Λ ( T ) optically thin radiative loss function (3.50) c R numerical constant in approximate formula for Λ ( T ) (3.51) c s constant value of c s in the sub-Alfvénic region (3.55) B ref reference value for magnetic-field-strength, ≃ . G (3.57) B ∗ B/B ref (3.57) ˜ ρ ⊙ πρ ⊙ v /B ≃ . × (3.57) ξ the parameter combination [ ǫ ⊙ η b / ( B ∗ ˜ l b )] / (3.57) x dimensionless temperature ( c s /v esc ) (3.58) ǫ ⊙ ( δv ⊙ eff /v esc ) (3.60) ˜ l b l b /R ⊙ (3.62) ǫ ( δv b /v esc ) (3.64) w dimensionless heat flux (3.77) γ B ( r ) − ( r/ B ) d B/ d r (3.86) Table 1.
Glossary transition region total turbulent heating balances total cooling from conduction and expansion r A heat flux non-isothermal,super-Alfvénic windradiation Alfvén-wave fluxchromosphere quasi-isothermal wind from transition region to Alfvén critical point r A conduction balances cooling from radiation and expansion r b Figure 1.
Schematic overview of model. of super-radial expansion (i.e., if η ( r ) were everywhere unity.) Given (3.2), the magnetic-field strength at the coronal base is B b ≡ B ( r b ) = Bη b ψ, (3.3)where here and below a ‘b’ subscript indicates that the subscripted quantity is evaluatedat r = r b , and ψ ≡ R ⊙ r = 0 . . (3.4)Because rotation is neglected, the steady-state solar-wind outflow velocity is alignedwith the background magnetic field (Mestel 1961): v = U B B . (3.5)The density and velocity satisfy the steady-state continuity equation, ∇ · ( ρ v ) = 0 . (3.6)It follows from (3.5), (3.6), and ∇ · B = 0 that B · ∇ ( ρU/B ) = 0 , and hence ρUB = constant . (3.7)Equation (3.7) is equivalent to ˙ M ≡ A ( r ) ρ ( r ) U ( r ) = constant , (3.8)where A ( r ) = 4 πr η ( r ) (3.9)is the cross-sectional area of the flow, which satisfies A ( r ) = constant /B ( r ) , as requiredby flux conservation. Equations (3.8) and (3.9) follow the convention of expressing themass outflow rate as the total solar mass-loss rate that would arise if the local mass fluxat some r characterised the entire outflow at that r , even though the actual solar windis comprised of different wind streams with different properties. All of the results of thispaper can be applied to an individual flux tube accounting for some fraction ν of thetotal flow area by multiplying the right-hand side of (3.9) by ν .It follows from (2.1) and (3.7) that ρ / U/v A is a constant. Because U ( r A ) = v A ( r A ) ,this constant must be ρ / , where ρ A ≡ ρ ( r A ) . (3.10)Thus, v A = yU, (3.11)where y ≡ (cid:18) ρρ A (cid:19) / . (3.12)With the aid of (3.3), (3.4), (3.9) and (3.11), (3.8) can be rewritten as ˙ M = A b ρ b U b = 4 πR ⊙ ψη b × ρ b × v Ab y b = R ⊙ B (4 πρ b ) / y b = BR ⊙ (4 πρ A ) / . (3.13)Thus, ˙ M is determined uniquely by the value of ρ A and the single-hemisphere openmagnetic flux, πR ⊙ B .3.2. Reflection-driven Alfvén-wave turbulence
Dmitruk et al. (2002) (hereafter D02) developed an analytic model of reflection-drivenAW turbulence in the solar corona valid in the limit of small L ⊥ , where L ⊥ is thecorrelation length of the AW fluctuations measured in the plane perpendicular to thebackground magnetic field. Chandran & Hollweg (2009) (hereafter CH09) generalisedthis model by accounting for the solar-wind outflow velocity. Section 3.2.1 summarisesthe main results of the CH09 model, and Section 3.2.2 uses the CH09 model to showthat heating by reflection-driven AW turbulence causes the sub-Alfvénic region of thesolar wind (at r < r A , where U < v A ) to become approximately isothermal. Section 3.2.3presents 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 thissection.3.2.1. The Chandran & Hollweg (2009) model of reflection-driven AW turbulence
In classical mechanics, a simple harmonic oscillator with frequency ω and energy E possesses an adiabatic invariant E/ω . If the parameters of the oscillator vary on a timescale t satisfying t ≫ ω − (e.g., if the length of a pendulum is slowly varied), then E/ω is almost exactly conserved. As ( ωt ) − → , changes in E/ω vanish faster than anypower of ( ωt ) − (Landau & Lifshitz 1960).An AW is like a space-filling collection of harmonic oscillators, and the wave actionis analogous to the harmonic oscillator’s adiabatic invariant. The wave action per unitvolume per unit ω is E ω /ω ′ , where ω is the AW frequency in an inertial frame centredon the Sun, ω ′ is the AW frequency in the local plasma frame, and E ω is the AW energyper unit volume per unit ω . In the Wentzel-Kramers-Brillouin (WKB) limit, in which thewave period is much shorter than the time scale on which the plasma parameters varyappreciably and the wave length is much shorter than the length scales over which thebackground plasma varies appreciably, the wave action satisfies the conservation law ∂∂t (cid:18) E ω ω ′ (cid:19) + ∇ · (cid:18) c E ω ω ′ (cid:19) = 0 , (3.14)where c is the group velocity of the waves (Bretherton & Garrett 1968; Dewar 1970).For outward-propagating AWs in the solar wind, ω = k r ( U + v A ) , ω ′ = k r v A , and c = ( U + v A ) ˆ r , where ˆ r is the radial unit vector and k r is the radial component of the0wave vector. In a steady-state solar wind, ω depends on neither position nor time. Uponmultiplying (3.14) by ω , integrating over ω , and assuming a steady state, one obtains ∇ · (cid:20) ˆ r ( U + v A ) E tot v A (cid:21) = 0 , (3.15)where E tot = Z E ω d ω = 14 ρz (3.16)is the total AW energy density, z ± ≡ h| z ± | i / , (3.17) h . . . i indicates a time average, z ± = δ v ∓ δ B √ πρ (3.18)are the Elsasser variables, δ v and δ B are the fluctuating velocity and magnetic field, and z + ( z − ) corresponds to AW fluctuations propagating away from (toward) the Sun. † In(3.17) and below, a ± sign is used as a subscript (as opposed to a superscript) when thesubscripted quantity is an r.m.s. value. Since ∇ · ( ρU ˆ r ) = 0 , (3.15) can be rewritten as dd r g = 0 , (3.19)where g = ( U + v A ) z U v A = (1 + y ) z y (3.20)is the wave-action flux per unit mass flux per unit ω times ω integrated over ω , or, forbrevity, the ‘wave action flux per unit mass flux’.When the finite radial wavelength of the AWs is taken into account, the radial gradientin v A causes partial non-WKB reflection of z + fluctuations, leading to the productionof z − fluctuations. Counter-propagating AWs then interact, causing fluctuation energy tocascade to small scales and dissipate. In the CH09 model, this loss of fluctuation energycauses g to decrease with radius according to the equation ‡ ( U + v A ) dd r g = − z − L ⊥ g . (3.21)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 ⊥ /z − .The reason that only z − (and not z + ) appears in this eddy turnover time scale is that z + fluctuations are not sheared or distorted by other z + fluctuations, but they are shearedand distorted by counter-propagating z − fluctuations (Iroshnikov 1963; Kraichnan 1965).The radial decay of g is accompanied by turbulent heating at the rate Q = ρz z − L ⊥ , (3.22) † The use of ∓ on the right-hand side of (3.18) instead of ± implies that z + fluctuationspropagate parallel to the background magnetic field, and z − fluctuations propagate anti-parallelto the background magnetic field. The identification of z + with outward-propagating AWs thuscorresponds to the case B r > . If B r < , the same analysis goes through by replacing the ∓ on the right-hand side of (3.18) with ± . ‡ CH09 derived (3.21) starting from the MHD equations; the alternative derivation presentedhere is given for brevity. ρz / divided by their eddyturnover time scale L ⊥ /z − . Equation (3.22) drops a term ρz − z + / (4 L ⊥ ) that is normallyincluded in the turbulent heating rate because of CH09’s assumption that z − ≪ z + , (3.23)an inequality that holds in the small- L ⊥ limit, as can be seen in (3.25), below.Following D02, CH09 determined z − by balancing the rate at which z − is producedthrough reflections against the rate at which z − cascades to small scales through nonlinearinteractions in the small- L ⊥ limit, obtaining ( U + v A ) v A (cid:12)(cid:12)(cid:12)(cid:12) d v A d r (cid:12)(cid:12)(cid:12)(cid:12) z + = z + L ⊥ z − , (3.24)or, equivalently, z − L ⊥ = ( U + v A ) v A (cid:12)(cid:12)(cid:12)(cid:12) d v A d r (cid:12)(cid:12)(cid:12)(cid:12) . (3.25)Upon substituting (3.25) into (3.21), assuming that v A ( r ) has a single maximum at r m >r b , and solving for g ( r ) , CH09 found that g = g h ( r ) , (3.26)where h ( r ) = (cid:26) v Ab /v A ( r ) for r b < r < r m v Ab v A ( r ) /v if r > r m , (3.27)where v Am = v A ( r m ) is the maximum value of v A . Conceptually, (3.26) and (3.27) statethat an appreciable fraction of the local AW action flux per unit mass flux dissipateswithin each Alfvén-speed scale height, which causes z ( r ) to drop below the WKB valuethat would be predicted from (3.20) with constant g .Upon substituting (3.26) into (3.22) and making use (3.20), CH09 obtained Q = ρ ( δv b ) (cid:18) U + v A v A (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) d v A d r (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) yy b (cid:19) (cid:18) y b y (cid:19) h ( r ) , (3.28)where ( δv ) ≡ h| δ v | i = z . (3.29)The second equality in (3.29) follows from (3.23). Since y b ≫ , (3.30)(3.28) can be rewritten to a good approximation in the following simplified form withthe aid of (3.11): Q = ρ ( δv b ) y b (cid:18) UU + v A (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) d v A d r (cid:12)(cid:12)(cid:12)(cid:12) h ( r ) . (3.31)3.2.2. Approximate isothermality between the transition region and Alfvén critical point
In steady state, the plasma internal-energy equation takes the form ∇ · (cid:20) v (cid:18) pγ − (cid:19)(cid:21) = − p ∇ · v − ∇ · q + Q − R, (3.32)where p is the pressure, p/ ( γ − is the internal-energy density, − p ∇ · v is the rate atwhich p d V work is done on the plasma per unit volume, Q is the rate of turbulent heating2per unit volume, q is the heat flux, which is written in the form † q = q r B B , (3.33)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 goodapproximation, (3.32) can be rewritten with the aid of (3.6) as − c U d ρ d r + ρUγ − r c = − A dd r ( Aq r ) + Q. (3.34)A generic consequence of heating by reflection-driven AW turbulence is that, whenother forms of heating (including conduction) are subdominant, the flow becomes quasi-isothermal at r b < r < r A , meaning that (cid:12)(cid:12)(cid:12)(cid:12) c d c d r (cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:12)(cid:12)(cid:12)(cid:12) ρ d ρ d r (cid:12)(cid:12)(cid:12)(cid:12) , (3.35)whereas (3.35) is not satisfied at r > r A . This can be demonstrated by (1) neglectingconductive heating in (3.34), (2) assuming that (3.35) is satisfied so that the second termon the left-hand side of (3.34) can be neglected, and (3) solving (3.34) for c . If theresulting expression for c s ( r ) satisfies (3.35), then the neglect of the second term on theleft-hand side of (3.34) is self-consistent, and this expression for c s ( r ) is a reasonableapproximation for the full solution of (3.34). On the other hand, if the resulting valueof c ( 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)), oneobtains c = y b ( δv b ) (cid:18) L ρ L v A (cid:19) (cid:18) v A U + v A (cid:19) h ( r ) , (3.36)where L ρ = ρ (cid:12)(cid:12)(cid:12)(cid:12) d ρ d r (cid:12)(cid:12)(cid:12)(cid:12) − L v A = v A (cid:12)(cid:12)(cid:12)(cid:12) d v A d r (cid:12)(cid:12)(cid:12)(cid:12) − (3.37)are the density and Alfvén-speed scale heights. These scale heights are generally someconstant of order unity times r , with L ρ /L v A increasing by a factor of a few between thelow corona and r A . On the other hand, h ( r ) decreases by a factor of a few between thelow corona and r A , so that the product ( L ρ /L v A ) h ( r ) varies quite weakly with r . ‡ The v A / ( U + v A ) term in (3.36) likewise exhibits little variation in the sub-Alfvénic region,ranging from ≃ at r = r b to . at r = r A . On the other hand, ρ varies by a factorof ∼ between the coronal base and r A (see, e.g., Cranmer et al. c in (3.36) satisfies (3.35) at r b < r < r A , and AW heatingindeed causes the sub-Alfvénic region to become quasi-isothermal. In contrast, at r > r A , † As in Section 3.2.1, B is taken to point radially outwards. If B in fact points toward theSun, the same analysis goes through if one introduces a minus sign in front of the right-handside of (3.33). ‡ An exception to this statement arises in the vicinity of the Alfvén speed maximum r m ,where the right-hand side of (3.36) vanishes. This vanishing is an artifact of the CH09 model,which determines the amplitude of inward-propagating AWs through a purely local balancingof wave reflections against cascade and dissipation. In numerical simulations of reflection-drivenAW turbulence, in which inward-propagating AWs travel some distance before cascading anddissipating, the heating profile varies smoothly with r without any strong reduction at r = r m (Perez & Chandran 2013). U asymptotes toward a constant value, h ∝ v A ∝ ρ / U ∝ ρ / , v A / ( U + v A ) ∝ ρ / ,and the right-hand side of (3.36) becomes proportional to ρ , 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 v A L ⊥ z + ≪ v A | d v A / d r | . (3.38)The left-hand side of (3.38) is the characteristic distance a z − fluctuation at scale L ⊥ (measured perpendicular to the background magnetic field) propagates along the mag-netic field before cascading and dissipating, and the right-hand side is the Alfvén-speedscale height. Just above the transition region, the magnetic-field strength has a scaleheight of order − R ⊙ (Hackenberg et al. et al. v A scale heighthas a similar value, and (3.38) is not satisfied (see, e.g., equation (3.13-a) and table 3of Chandran & Perez 2019). On the other hand, beyond the low corona, the v A scaleheight grows to values ∼ r that are ≫ L ⊥ . Thus, the CH09 model should be applied onlybeyond some minimum heliocentric distance, and some other approximation is needed totreat the region immediately above the transition region.The second difficulty with incorporating the CH09 model into an analytic solar-windmodel is that the function h ( r ) in (3.27) depends on the number of local extrema inthe Alfvén-speed profile and the locations of these extrema. In models that accountfor the rapid increase in B ( r ) as r drops below ≃ . R ⊙ , v A typically has two localextrema within the corona: a local minimum just above the coronal base and a localmaximum a few tenths of a solar radius above the coronal base — see, e.g., figure 3 ofCranmer & van Ballegooijen (2005) or figure 9 of Cranmer et al. (2007). In the contextof the analytic solar-wind model developed below, each local extremum introduces twonew unknowns: the location of the extremum, and either the density or flow speed at theextremum.In this paper, the first of the two difficulties mentioned above is handled by replacingthe CH09 result for Q ( r ) at r = r b with the expression Q b = ρ b ( δv b ) v Ab l b , (3.39)where l b is a free parameter, which is set equal to . R ⊙ in the numerical examplespresented in Section 4. The second difficulty is resolved by replacing (3.25) with theexpression z − L ⊥ = − σ ( U + v A ) dd r ln(1 + y ) (3.40)at r b < r < r A , where σ is a free parameter, whose value is ∼ . − . in the numericalexamples in Section 4. Whereas z − /L ⊥ = ( U + v A ) /L v A in (3.25), (3.40) takes z − /L ⊥ tobe ≃ ( U + v A ) /L ρ times a free parameter. † Because ρ (and hence y ) is a monotonicallydecreasing function of r , the minus sign on the right-hand side of (3.40) ensures that z − > . 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 integrateanalytically. Another motivation for using (3.40) instead of (3.25) is that the parameter- † More precisely, ( d / d r ) ln(1 + y ) = (1 + y ) − d y/ d r , which is ≃ − / (2 L ρ ) when y ≫ and ≃ − / (4 L ρ ) near r = r A where y = 1 . g ( r ) , and rewriting g in terms of z using (3.20), one finds that z = z yy b (cid:18) y b y (cid:19) − σ . (3.41)Equations (3.22), (3.29), (3.40), and (3.41) can then be used to show that H = χ H P AW ( r b ) , (3.42)where H = Z r A r b Q ( r ) A ( r ) d r (3.43)is the total turbulent heating power between r b and r A , P AW ( r ) = ρ ( δv ) ( U + v A ) A (3.44)is the power (energy flux times area) of outward-propagating AWs at radius r , and χ H = 1 − σ − (cid:18) − σ − σ (cid:19) (1 + y b ) − σ y b + 1 y b (1 − σ ) (3.45)is the fraction of P AW ( r b ) that is dissipated between r b and r A .3.3. Relating the AW amplitudes at the coronal base and photosphere
In the absence of wave reflection and dissipation, P AW (defined in (3.44)) wouldhave almost exactly the same value at r b and R ⊙ . † However, the steep Alfvén-speedgradient in the transition region leads to strong AW reflection, and a vigorous energycascade in the chromosphere leads to substantial AW dissipation (van Ballegooijen et al. P AW ( r b ) /P AW ( R ⊙ ) to some value f chr < ,where f chr is an effective AW transmission coefficient for the chromosphere and transitionregion. Equivalently, since U ≪ v A at r r b , P AW ( r b ) = ρ b δv v Ab A b = f chr ρ ⊙ δv ⊙ v A ⊙ A ⊙ , (3.46)where a ⊙ subscript indicates that the subscripted quantity is evaluated at the photo-sphere. For the numerical calculations in Section 4, I set ρ ⊙ = 10 m p cm − . (3.47)Since BA = constant and v A = B/ (4 πρ ) / , (3.46) can be rewritten as δv = δv ⊙ eff (cid:18) ρ ⊙ ρ b (cid:19) / , (3.48)where δv ⊙ eff = f / δv ⊙ . (3.49)The difference between δv ⊙ and δv ⊙ eff is that δv ⊙ gives rise to the fluctuating velocity δv b at r = r b when reflection and dissipation are accounted for, whereas δv ⊙ eff would give riseto the same value of δv b via WKB propagation (i.e., without reflection or dissipation). † When U is nonzero, it is g in (3.20) that is invariant in the absence of dissipation andreflection, not P AW . However, between r = R ⊙ and r = r b , U ≪ v A , and P AW is to an excellentapproximation proportional to g . T (K) −31 −29 −27 −25 −23 −21 Λ ( e r g c m s − ) Figure 2.
Three approximations to the optically thin radiative loss function Λ ( T ) ; see text forfurther discussion. The value of δv ⊙ eff can be constrained in different ways. For example, observationsfix δv ⊙ at a value of ≃ km s − (Richardson & Schwarzschild 1950), and numerical simu-lations suggest that f / is ≃ . − . (van Ballegooijen et al. δv ⊙ eff can be inferred directly from measurements of the mass flux andenergy flux far from the Sun. This latter method is used to determine δv ⊙ eff in some of thenumerical solutions presented in Section 4 and is described in more detail in Section 4.2. Itis worth noting that in contrast to δv ⊙ , f chr , and δv ⊙ eff , which are plausibly independentof the properties of the coronal plasma and coronal magnetic field, δv b depends upon ρ b ,which varies between different flux tubes with different super-radial expansion factors η b ,as shown in (3.55) below.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 ofradiative cooling is given by R = (cid:18) ρm p (cid:19) Λ ( T ) , (3.50)where m p is the proton mass, and Λ ( T ) is the optically thin radiative loss function.Figure 2 shows three different approximations to Λ ( T ) . The dashed lines correspond toequation A1 of Rosner et al. (1978). The solid line is a plot of Λ ( T ) = c R T − / , (3.51)where c R = 1 . × − erg cm s − K / . (3.52)Equation (3.51) is the temperature derivative of equation A3 of Rosner et al. (1978).The dotted line in figure 2 is a piecewise-continuous linear approximation to the low-temperature range of Λ ( T ) in figure 1 of Cranmer et al. (2007), which is included toillustrate that optically thin radiative cooling becomes extremely weak at T . K.Throughout most of the corona, ρ is sufficiently small that radiative cooling is neg-ligible. However, as r decreases within the corona, ρ increases by orders of magnitude,which causes R/Q to increase, because R ∝ ρ . There is thus some radius r b at which R ( r b ) = Q ( r b ) , (3.53)which marks the transition between the corona, in which radiative cooling is negligible,6and the low solar atmosphere, in which radiation is thermodynamically important. Inthis paper, the radius r b is identified as the base of the corona, as already noted in (3.1).As r decreases below r b , ρ ( r ) increases above ρ b , R/Q increases to values ≫ , and thetemperature gradient length scale must decrease so that conductive heating can balanceradiative cooling (as well as internal-energy losses from advection and p d V work). Thisshortening of the temperature gradient length scale gives rise to the transition region,which is discussed further in Section 3.6. To facilitate an analytic solution, I neglectturbulent heating at r < r b and radiative cooling at r > r b .With the aid of (3.39), (3.46), and (3.50), (3.53) can be written in the form ρ m Λ ( T b ) = ρ b ( δv b ) v Ab l b = ρ ⊙ ( δv ⊙ eff ) v A ⊙ B b l b B ⊙ . (3.54)Solving for ρ b , one finds via (3.51) that ρ b = δv ⊙ eff ψBη b m / c s c R l b ! / (cid:18) ρ ⊙ πk B (cid:19) / , (3.55)where c s = constant (3.56)is the sound speed (2.6) in the sub-Alfvénic region ( r b < r < r A ), which is approximatedas a constant for the reasons discussed in Section 3.2.2. Equation (3.55) can be rewrittenas ρ b = B πv ˜ ρ / ⊙ B ∗ ξ ψ / x / , (3.57)where x = c s2 v (3.58)is the dimensionless temperature of the sub-Alfvénic region, B ref = πm / v c R R ⊙ √ k B ! / = 118 . G (3.59)is the magnetic-field strength for which v A = v esc when the radiative cooling time ρc s2 /R ,free-fall time R ⊙ /v esc , and sound-crossing time R ⊙ /c s are all equal, ǫ ⊙ = ( δv ⊙ eff ) v , (3.60) ˜ ρ ⊙ = 4 πv ρ ⊙ B , (3.61)(for reference, ˜ ρ ⊙ = 5 . × given (3.47)), and ξ = (cid:18) ǫ ⊙ η b B ∗ ˜ l b (cid:19) / B ∗ = BB ref ˜ l b = l b R ⊙ . (3.62)Equation (3.57) can used to express v Ab in the form v Ab = v esc ψ / ξ − η b ( x ˜ ρ ⊙ ) − / , (3.63)7and (3.48), (3.57), and (3.60) can be used to write ǫ ≡ ( δv b ) v = ǫ ⊙ ˜ ρ / ⊙ B − ∗ ξ − ψ − / x − / . (3.64)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 b and r = r A yields − ˙ M c s2 Z r A r b ρ d ρ d r d r = − ( Aq r ) (cid:12)(cid:12)(cid:12)(cid:12) r A r b + χ H ˙ M δv (1 + y b ) , (3.65)where (3.42) through (3.44) have been used to rewrite the volume integral of the turbulentheating rate, (3.8) has been used to pull a constant factor of ˙ M = ρAU out of the integralon the left-hand side, and (3.11) has been used to write U b + v Ab = U b (1 + y b ) .The heat flux at r = r 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-handside of (3.65), noting that q r b = −| q r b | ≡ − q b , making use of (3.12), and rearrangingterms, one obtains χ H ˙ M δv (1 + y b ) = 2 ˙ M c s2 ln y b + A b q b . (3.66)The left-hand side of (3.66) is the total turbulent heating rate within the sub-Alfvénicregion, which represents the source of internal energy between r b and r A . The twoterms on the right-hand side of (3.66) are the dominant sinks of internal energy inthe sub-Alfvénic region: p d V work and thermal conduction into the transition region.Dividing (3.66) by ˙ M v leads to ǫχ H (1 + y b ) = 2 x ln y b + q b ρ b U b v . (3.67)3.6. The flux of heat from the corona into the transition region
The temperature structure within the transition region can be determined using amethod similar to the methods of Rosner et al. (1978) and Schwadron & McComas(2003). The Knudsen number N K (the electron Coulomb mean free path λ mfp dividedby the temperature gradient scale length l T ) is ∼ − in the low corona, ∼ − at theupper end of the transition region, and ∼ − at the lower end of the transition region. † Because N K ≪ , the radial component of the heat flux in the transition region is wellapproximated by the Spitzer & Härm (1953) formula, q r = − αT / d T d r , (3.68)where α = 1 . × − erg cm − s − K − / ln Λ Coul , (3.69)and ln Λ coul is the Coulomb logarithm. In the numerical example of Section 4, ln Λ Coul = 18 . , (3.70) † These estimates follow from setting ρ ∼ m p cm − , T ∼ K, and l T ∼ R ⊙ in the lowcorona, ρ ∼ m p cm − , T ∼ × K, and l T ∼ . R ⊙ at the upper end of the transitionregion, and ρ ∼ m p cm − , T ∼ K, and l T ∼ km at the lower end of the transitionregion. ρ/m p =10 cm − and T = 10 K (Book 1983). The magnetic field near r = r b is taken tobe approximately radial, and the width of the transition region ( ∼ km) is so narrowthat p , B , and A are treated as constants within the transition region, with ∇ · q = d q r d r . (3.71)As mentioned above, turbulent heating is neglected at r < r b . Within the transitionregion, (3.32) thus becomes (cid:18) γγ − (cid:19) p ∇ · v = − d q r d r − c R p k T / , (3.72)where ρ has been expressed in terms of p and T using (2.6). The velocity divergencein (3.72) can be expressed in terms of the heat flux via ∇ · v = − v ρ · ∇ ρ = v T · ∇ T = UT (cid:16) − q r αT / (cid:17) . (3.73)where the first equality follows from (3.6). With the aid of (3.73), and using (3.68) towrite d q r d r = − d q r d T q r αT / , (3.74)one can rewrite (3.72) in the form − a q r = q r d q r d T − a , (3.75)where a = αp c R k a = (cid:18) γγ − (cid:19) pUT (3.76)are both constants. Upon defining w = − a a q r , (3.77)which is positive, one can rewrite (3.75) as a a d T = w w d w, (3.78)which can be integrated from the chromospheric values of T and w , denoted T chr and w chr , to the values of T and w at r = r b , denoted T b and w b . Since T chr ≪ T b andthe chromospheric heat flux is negligible ( q scaling as T / divided by the temperature-gradient scale length), the chromospheric terms are dropped, and the integral becomes w b − ln(1 + w b ) = a , (3.79)where a = a T b a = 2 I (cid:18) γγ − (cid:19) (cid:18) U b c s (cid:19) , (3.80) I = (cid:18) αc R m p k (cid:19) / = 0 . , (3.81)and the numerical value on the right-hand side of (3.81) is calculated using (3.70). Equa-tion (3.79) can be solved in terms of the lower branch of the Lambert W function, W − .9This solution, in conjunction with (3.77), yields q b = − q r b = − a a h W − (cid:16) − e − (1+ a ) (cid:17)i , (3.82)which is positive since W − (cid:16) − e − (1+ a ) (cid:17) < − .If U b /c s is sufficiently small, then a ≪ . In this low-Mach-number limit, (3.82)becomes, to leading order in a , q b = I ρ b c s3 . (3.83)Equation (3.83) is equivalent to equation (3.15) of Rosner et al. (1978) when thermalconduction is the only source of heating in the transition region, i.e., when f H is set equalto 0 in their equation (3.15). Equation (3.82) follows Schwadron & McComas (2003)in generalising the work of Rosner et al. (1978) to include internal-energy losses from p d V work and advection. However, the analysis leading to (3.82) differs from that ofSchwadron & McComas (2003) in that (3.82) completely neglects turbulent heating at r < r b and does not assume that R q ( T ) d T ∝ T . † Constraints associated with the wave-modified sonic critical point
In the presence of mostly outward-propagating AWs, a radial background magneticfield, and a radial outflow, the momentum equation in the approximately isothermalsub-Alfvénic region takes the form ρU d U d r = − c s2 d ρ d r − dd r (cid:18) ρz (cid:19) − GM ⊙ ρr , (3.84)where ρz / is the AW pressure, which is one-half the AW energy density (Dewar 1970).Equation (3.7) implies that ρ d ρ d r + 1 U d U d r − B d B d r = 0 . (3.85)Equations (3.41) and (3.85) can be used to rewrite (3.84) in the form ρ d ρ d r (cid:26) − U + c s2 + ( δv b ) (1 + y b ) − σ [ y (1 + σ ) + 3 y ]4 y b (1 + y ) − σ (cid:27) = 2 r (cid:18) γ B U − v R ⊙ r (cid:19) , (3.86)where γ B ≡ − r B d B d r = 1 − r η d η d r , (3.87)and η 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 negativefar from the Sun; i.e., it must pass through zero at some radius r c (the wave-modifiedsonic critical point). In order for d ρ/ d r to remain finite at r = r c , the right-hand sideof (3.86) must also vanish at r c . Together, these conditions yield the constraints r c = v R ⊙ γ B c U (3.88) † Schwadron & McComas (2003) integrate the internal-energy equation from the upperchromosphere all the way out to the coronal temperature maximum, where the heat flux vanishes.The value of q b in (3.82) corresponds to r = r b , at which turbulent heating and radiative coolingbalance, which, in the low-Mach-number limit, corresponds to the maximum of the heat flux.This heat-flux maximum is lower down in the solar atmosphere than the temperature maximum. U = c s2 + ( δv b ) (1 + y b ) − σ [ y (1 + σ ) + 3 y c ]4 y b (1 + y c ) − σ , (3.89)where, here and below, a ‘c’ subscript indicates that the subscripted quantity is evaluatedat r = r c . An implicit assumption underlying (3.88) and (3.89) is that r c < r A , (3.90)so that the quasi-isothermal approximation applies at r c .Mass and flux conservation (i.e., (3.7)) imply that ρ c U c B c = ρ b U b B b = ρ b v Ab B b y b , (3.91)where the second equality in (3.91) follows from (3.11). Equation (3.2) implies that B b B c = r R ⊙ ψ η b η c = v γ B c U ψ η b η c , (3.92)where the second equality in (3.92) follows from (3.88). Upon substituting (3.92)into (3.91) and evaluating v Ab using (3.63), one obtains U = v (cid:20) y ψ / ξ ( x ˜ ρ ⊙ ) / γ B c η c y b (cid:21) / . (3.93)Substituting (3.93) into (3.89) yields (cid:20) y ψ / ξ ( x ˜ ρ ⊙ ) / γ B c η c y b (cid:21) / − x − ǫ (1 + y b ) − σ [ y (1 + σ ) + 3 y c ]4 y b (1 + y c ) − σ = 0 . (3.94)The integral over r of ρ − times the momentum equation (3.84) yields the Bernoulliintegral, U c s2 ln (cid:18) ρρ b (cid:19) − v R ⊙ r − ( δv b ) (1 + y b ) − σ y b (cid:20)(cid:18) σ − σ (cid:19) (1 + y ) σ − + (1 + y ) σ − (cid:21) = Γ, (3.95)where Γ is independent of r . Evaluating (3.95) at r = r b leads to the equality Γ = U − v ψ / − ( δv b ) (cid:20) σ − σ + 2 y b (1 − σ ) (cid:21) . (3.96)Evaluating (3.95) at r = r c , rewriting r c and U c using (3.88) and (3.89), rewriting Γ using (3.96), and multiplying the resulting equation by /v yields ψ / + x (cid:20) (cid:18) y c y b (cid:19) + 1 − γ B c (cid:21) + ǫ (1 + y b ) − σ Φ − σ ) y b (1 + y c ) − σ − η ψ / y ξ ( x ˜ ρ ⊙ ) / + ǫ (cid:20) σ − σ + 2 y b (1 − σ ) (cid:21) = 0 , (3.97)where Φ ≡ y [(1 − γ B c )(1 − σ ) − σ )] + y c [3(1 − γ B c )(1 − σ ) − − σ ] − . (3.98)3.8. Mathematical structure of the model and approximate analytic solutions
The various quantities appearing in the model equations can be divided into fivegroups: (1) quantities that are determined observationally ( R ⊙ , M ⊙ , v esc , ρ ⊙ , ˜ ρ ⊙ , ψ ,1 δv ⊙ eff , B ); (2) free parameters ( σ , l b ); (3) the super-radial expansion factor η ( r ) , whichtakes on different values in different magnetic flux tubes and in different models for thesolar magnetic field; (4) the three principal unknowns: y b , y c , and x , and (5) additionalunknowns that can be determined once y b , y c , and x are found ( ˙ M , χ H , U ( r ) , ρ b , q b , v Ab , r c , r A ). The three principal unknowns y b , y c , and x are determined by solving the threesimultaneous equations (3.67), (3.94), and (3.97), where it must be remembered that ǫ isitself a function of x via (3.64). The additional unknowns ˙ M , χ H , ρ b , v Ab , q b , and r c thenfollow immediately from (3.13), (3.45), (3.57), (3.63), (3.82), and (3.88), respectively. Forexample, ˙ M = R ⊙ B v esc y − ( x ˜ ρ ⊙ ) / ξψ / . (3.99)The procedures for determining r A and U ( r ) involve a few more steps, which are describedin Appendix A.Two approximate analytic solutions to (3.67), (3.94) and (3.97), valid in two differentparameter regimes, are derived in Appendix B. Both solutions rely on the approximations y b ≫ ψ = 1 ǫ ≪ η c = γ B c = 1 . (3.100)The last equality in (3.100) amounts to taking the magnetic-field lines to be purely radialat r > r 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-Alfvénicregion is the flux of heat from the corona into the transition region, rather than p d V work. As discussed further below, this regime arises only for values of δv ⊙ eff much smallerthan the solar value. This limit is thus not directly relevant to solar-wind observations. Toleading order in ǫ ⊙ , the mass outflow rate ˙ M (cond) and asymptotic flow velocity U (cond) ∞ in this parameter regime are given by ˙ M (cond) = R ⊙ B v esc I / I h ǫ − σ ⊙ ( η b B ∗ ) − σ ˜ l σ b ˜ ρ − σ ⊙ i / (7 − σ ) , (3.101)where I is a numerical constant given in (B 12), and U (cond) ∞ = v esc " − σ ζ (1 + σ ) I σ/ (cid:18) − σ − σ (cid:19) (7 − σ ) / ǫ − (7 − σ ) / ⊙ B (7 − σ ) / ∗ η σ/ ˜ ρ ( − σ ) / ⊙ ˜ l − σ/ / . (3.102)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 d V work resulting from expansion is the dominant sink of internal energyin the sub-Alfvénic region. To leading order, the mass outflow rate in this parameterregime is given by ˙ M (exp)0 = ǫ ⊙ BR ⊙ p πρ ⊙ = P AW ( r b ) v , (3.103)where P AW ( r b ) is the AW power (3.44) evaluated at the coronal base. The asymptoticwind speed in the expansion-dominated regime is to leading order given by U (exp) ∞ , = v esc , (3.104)2and the coronal temperature in the expansion-dominated regime is to leading order T = m p v k B ln( ǫ − ⊙ ˜ ρ − / ⊙ ˜ l − η b B ∗ ) . (3.105)Equations (3.103) and (3.104) reproduce the approximate scalings of the simplifiedcalculation presented in Section 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 themodel developed in this section, δv b has a weak dependence on the coronal temperaturevia (3.48) and (3.57) that is not accounted for in Section 2. Appendix B presents higher-order corrections to (3.103), (3.104), and (3.105) that account for conductive losses, wavemomentum deposition inside the wave-modified sonic critical point, and the fact that onlypart of P AW ( r b ) is dissipated within the sub-Alfvénic region. Second and fourth-orderapproximations to ˙ M and U ∞ are shown in figure 5 below.Analytic estimates for the ranges of ǫ ⊙ values corresponding to the conduction-dominated and expansion-dominated limits are given in Appendix B. There are threeconstraints on ǫ ⊙ in the conduction-dominated limit, the most stringent of which isthat the wave-energy term dominate over the internal-energy term in the Bernoulliintegral at the wave-modified sonic critical point. The resulting range of ǫ ⊙ values ismuch smaller than the solar value, as illustrated in figure 4. The expansion-dominatedlimit corresponds to a finite range of ǫ ⊙ values that is sufficiently large that p d V workdominates over conduction as the primary internal-energy sink within the sub-Alfvénicregion, and sufficiently small that the sound speed makes the dominant contribution tothe outflow velocity at the wave-modified sonic point in (3.89). This range of ǫ ⊙ valuesis relevant to the solar case, as shown in the next section.
4. Numerical examples
This section presents several numerical solutions and approximate analytic solutions tothe equations of the model developed in Section 3. The numerical solutions are obtainedby solving (3.67), (3.94), and (3.97) for y b , y c , and x using Newton’s method. † Once y b , y c , and x are determined, ˙ M , χ H , ρ b , v Ab , q b , r c , r A , and U ∞ are computed from (3.13),(3.45), (3.57), (3.63), (3.82), (3.88), (A 3), and (A 13), respectively. The approximateanalytic solutions are derived in Appendix B.4.1. Magnetic-field model
The equations in Section 3 are compatible with any model for the radial profile ofthe magnetic-field strength, or, equivalently, any choice of η ( r ) . On the other hand, theapproximate analytic solutions derived in Appendix B assume that η c = γ B c = 1 . (4.1)To maintain consistency between the numerical and analytic solutions, and to avoid intro-ducing additional complexity and free parameters, I take (4.1) to hold when computingthe numerical solutions presented in this section. In essence, (4.1) amounts to assumingthat all of the super-radial expansion of the magnetic field occurs inside the wave-modifiedsonic critical point. In measurements from the FIELDS experiment on the Parker Solar † In practice, rather than evaluating q b in (3.67) using the Lambert W function in (3.82),I evaluate q b using (3.77), treat w b as a fourth unknown, and include (3.79) as a fourthsimultaneous equation to be solved numerically. The resulting value of q b is identical to thevalue from (3.82). et al. | B r | ≃ . nT (1 au /r ) (Sam Badman, private communication), where au stands for astronomical unit. In orderto match this B r profile, I set B ∗ = 0 . ←→ B r ( ) = 2 . nT (4.2)when computing the results shown in figures 3 through 6.4.2. Values of the Alfvén-wave power at the coronal base and δv ⊙ eff Equations (3.46) and (3.49) and the flux-conservation relation A b B b = A ⊙ B ⊙ implythat δv ⊙ eff = (cid:18) πρ ⊙ (cid:19) / (cid:18) P AWb B b A b (cid:19) / . (4.3)When the solar wind is powered primarily by Alfvén waves, P AWb is approximately equalto the value of ˙ E m0 in (1.3), i.e., P AWb ≃
12 ˙ M (cid:0) v + U ∞ (cid:1) . (4.4)Upon evaluating the right-hand side of (4.4) using the data in table 1 of Schwadron & McComas(2008) for Ulysses ’ third northern polar pass (‘3NPP’), one obtains P AWb = 3 . × erg s − , (4.5)which corresponds to P AWb / (4 πR ⊙ ) = 0 . × erg cm − s − . I use the 3NPP databecause this is the part of Ulysses ’s first three orbits during which the average scaledradial magnetic field |h B r i| · ( r/ au ) was most consistent with (4.2) (in the 3NPP data, |h B r i|· ( r/ au ) = 2 . ± . nT), and because ˙ M is strongly correlated with the averagescaled radial magnetic field (Schwadron & McComas 2008). Equations (3.47), (4.2), (4.3),and (4.5) and the relation B b A b = | B r (1 au ) | π (1 au ) imply that δv ⊙ eff = 0 . km s − . (4.6)This value of δv ⊙ eff is used in figures 3, 5, and 7.4.3. Free parameters
As discussed in Section 3.8, there are two free parameters in the model: l b (the AWdissipation length scale at r b ) and σ (the dimensionless coefficient in the turbulent heatingrate). The solutions to (3.67), (3.94), and (3.97) are not very sensitive to the value of l b .Throughout the rest of this section, l b is thus simply fixed at a value that seems physicallyreasonable: l b = 0 . R ⊙ . (4.7)On the other hand, the solutions depend sensitively on the value of σ . Larger valuesof σ cause a larger fraction of the Alfvén-wave power at the coronal base P AWb to bedissipated in the quasi-isothermal sub-Alfvénic region, which increases ˙ M (see, e.g., (2.15)and (2.16)). For fixed P AWb , increasing ˙ M decreases U ∞ . In some of the examplesto follow, σ is varied to optimise agreement between the model and observations, asdescribed further below. The super-radial expansion factor at the coronal base, η b , alsovaries in the model, but this quantity is in principle observable for individual flux tubes.The value of η b is thus treated as an input into the model rather than an adjustableparameter.4 r (R ⊙ ⊙ U ( k m s − ⊙ UPSP E1 r (R ⊙ ⊙ ρ / m p ( c m − ⊙ PSP E1 r (R ⊙ ⊙ δ v ( k m s − ⊙ PSP E1Hinode/SOT r (R ⊙ ⊙ T ( K ⊙ PSP E1
Figure 3.
The outflow velocity U , density ρ , r.m.s. amplitude of the velocity fluctuation δv ,and temperature T as functions of heliocentric distance r in a numerical solution to the modelequations with η b = 100 , σ = 0 . , and the parameter values in (4.1), (4.2), (4.6), and (4.7). Thedotted lines in the top-left panel are described in the text. In the top panels and the lower-rightpanel, the circles are measurements from a one-hour interval containing PSP’s first perihelionon November 6, 2018, from figure 1 of Kasper et al. (2019). The error bars around the PSP datapoints in these three panels indicate the approximate range of values with a relative occurrencerate of at least 50% of the peak occurrence rate within that one-hour interval. In the top-rightpanel, the error bars lie within the data point. In the lower-left panel, the PSP data pointis from Chen et al. (2020), and the error bars around that data point show the approximaterange of measured values near r = 35 . R ⊙ in figure 7 of Chen et al. (2020). The triangle inthe lower-left panel is the value obtained by De Pontieu et al. (2007) from an analysis of themotion of filamentary structures in the low solar atmosphere based on observations from theSolar Optical Telescope on the Hinode satellite. Fiducial solution matching measurements from Parker Solar Probe’s first perihelion
Figure 3 illustrates the r dependence of the outflow velocity, density, fluctuatingvelocity, and temperature in a numerical solution to the model equations that is designedto agree with measurements from PSP’s first perihelion encounter (‘E1’) on November 6,2018. The procedure for computing U ( r ) and T ( r ) is described in Appendix A. In orderto solve for U at some radius r , the value of η must be specified at that r . For figure 3,I set η ( r ) = 1 + ( η b −
1) exp( − ( r − r b ) / (0 . R ⊙ ) ) , which is effectively unity at r = r c ,consistent with (4.1). This choice of η ( r ) is not realistic for the low corona (see, e.g.,Cranmer et al. δv at r = r b (i.e., δv b ) shown in the lower-leftpanel of Figure 3 depends on η b through (3.48), (3.57), and (3.62), δv b does not on theway in which η ( r ) decreases from η b to 1.The solution in Figure 3 is based upon the somewhat arbitrary assumption that η b =100 in the magnetic flux tube encountered by PSP at the time of its first perihelion. The5 −3 −2 −1 δv ⊙ ⊙km s −1 ) −3 −2 −1 δv ⊙eff ⊙km s −1 ) −27 −23 −19 −15 ̇ M ⊙ M ⊙ y r − ) numeriċl̇M ⊙exp)0 ̇M ⊙cond) Figure 4.
Mass outflow rate as a function of the effective fluctuating velocity at the photosphere δv ⊙ eff = f / δv ⊙ (see (3.49)) for the parameter values in (4.1), (4.2), (4.7), η b = 30 ,and σ = 0 . , where f chr is the chromospheric/transition-region AW transmission coefficientin (3.46). The corresponding r.m.s. photospheric velocity δv ⊙ is shown at the top of the plotfor the case in which f chr = 0 . . 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 analyticresults from (3.101) and (3.103), respectively. The vertical dash-dot line corresponds to ǫ ⊙ cond in (B 14). The left and right edges of the shaded region correspond, respectively, to ǫ ⊙ exp , min and ǫ ⊙ exp , max in (B 35) and (B 36). To the right of the vertical long-dashed line, r c > r A , whichviolates (3.90) and the assumptions of the model. value of σ 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 η b . For example, the parameter combinations ( η b , σ ) = (300 , . and ( η b , σ ) = (30 , . lead to similar agreement with the data. Modeling of the solar magnetic field during PSPE1 suggests that the solar-wind stream encountered by PSP at the time of PSP’s firstperihelion originated in a small equatorial coronal hole (Bale et al. σ suggests that the model is reasonably successful at capturing the physical processes thatcontrol the heating and acceleration of coronal-hole outflows.The dotted lines in the top-left panel of Figure 3 are solutions of the Bernoulliequation (3.95) for different values of the Bernoulli constant Γ . One of the dotted linesintersects the solid line, and that intersection occurs at the wave-modified sonic criticalpoint, r = r c . That dotted line is an accretion-like solution of (3.95) with the same valueof Γ as in the model solution but with d U/ d r < at r = r c .4.5. Illustration of the conduction-dominated and expansion-dominated regimes
Figure 4 plots ˙ M as a function of δv ⊙ eff when η b = 30 and σ = 0 . . This figureincludes the numerical solution to (3.67), (3.94), and (3.97) as well as the approximateanalytic results ˙ M (cond) and ˙ M (exp)0 from (3.101) and (3.103). The vertical dash-dot linein this figure corresponds to ǫ ⊙ cond in (B 14). The conduction-dominated approximationthat gives rise to ˙ M (cond) in (3.101) assumes that ǫ ⊙ ≪ ǫ ⊙ cond . † The shaded regioncorresponds to ǫ ⊙ exp , min < ǫ ⊙ < ǫ ⊙ exp , max , where ǫ ⊙ exp , min and ǫ ⊙ exp , max are defined † It should be noted, however, that the solution for U (cond) ∞ in (3.102) exceeds the speed oflight when δv ⊙ eff . × − km s − , and a relativistic treatment would be needed to model theoutflow correctly in this limit. ǫ ⊙ exp , min ≪ ǫ ⊙ ≪ ǫ ⊙ exp , max . The vertical long-dashed line corresponds to thecondition r c = r A . To the right of this line, ˙ M and the solar-wind density become so largethat r A < r c , violating (3.90) and the assumptions underlying the model. In conjunctionwith (4.6), Figure 4 illustrates that the expansion-dominated regime is relevant to thesolar wind and that the conduction-dominated regime corresponds to values of δv ⊙ eff much smaller than the solar value.4.6. Anti-correlation between the coronal super-radial expansion factor and U ∞ Wang & Sheeley (1990) showed that the outflow velocity in a solar-wind stream at r = 1 au is anti-correlated with the super-radial expansion factor in the coronal mag-netic flux tube from which the solar-wind stream originated. To compare their re-sults with the model developed in Section 3, I follow Cranmer et al. (2007) by defin-ing the Wang & Sheeley (1990) coronal super-radial expansion factor f max , WS to be η (1 . R ⊙ ) /η (2 . R ⊙ ) . I then compute η (1 . R ⊙ ) and η (2 . R ⊙ ) using a magnetic-fieldmodel similar to that used by Cranmer et al. (2007). In particular, I employ the globalsolar-magnetic-field model of Banaszkiewicz et al. (1998) with the parameter valueslisted below their equation (2) ( K = 1 , M = 1 . , a (B98)1 = 1 . , and Q = 1 . ).I supplement this global magnetic field with the low-solar-atmosphere magnetic-fieldmodel of Hackenberg et al. (2000) using the same parameter values listed in their figure 1( L = 30 Mm, d = 0 . Mm, B = 11 . G, and B max = 1 . kG). I then integrate out alonga magnetic-field line at the edge of the polar corona hole in the Banaszkiewicz et al. (1998) model, compute η ( r ) , and obtain f max , WS = η b . . (4.8)I take (4.8) to hold even as η b is varied in Figures 5, 6, and 7. This is equivalent toassuming that if local magnetic structures on the Sun cause η b to increase or decreaseby some factor relative to the Hackenberg/Banaszkiewicz model just described, then η (1 . R ⊙ ) increases or decreases by the same factor, but η (2 . R ⊙ ) does not change.If σ is fixed while η b is varied, the model of Section 3 does not agree with resultsof Wang & Sheeley (1990) shown in the top-right panel of Figure 5. On the other hand,the model becomes consistent with those results if σ is taken to have a power-lawdependence on η b of the form σ = 5 . × − η . . (4.9)Equation (4.9) is used to determine σ in figures 5 through 7. The implication of (4.9)that σ is an increasing function of η b is plausible because increasing η b increases v Ab , ascan be seen from (3.63). This in turn increases the number of Alfvén-speed scale heightsbetween the coronal base and the Alfvén critical point ( R r A r b (1 /v A ) | d v A / d r | d r ), whichincreases the fraction of P AWb that is dissipated at r < r A (Chandran & Hollweg 2009).Further work, however, is needed to clarify how the structure of the coronal magneticfield influences the rate of turbulent dissipation along different magnetic flux tubes withdifferent values of η b and to test the extent to which (3.41) and (4.9) are consistent withmore 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 max , WS ranges from ≃ to ≃ , ˙ M varies from − M ⊙ yr − to × − M ⊙ yr − , values that are similar to the solar-mass-loss rates inferred from Ulysses and PSP measurements (McComas et al. et al. ˙ M (exp)4 from (B 32) reproduces the numerical solution to the model equations7 f max, WS ̇ M ̇ − M ⊙ y r − ⊙ numerical̇M ̇exp⊙4 ̇M ̇exp⊙2 f max, WS U ∞ ( k m s − ) numericalU (exp)∞,4 U (exp)∞,2 WS empirical f max, WS q b ( e r g c m − s − ) q b I ρ b c s3 ρ b (δv b ) v Ab f max, WS v A b , c s , U b ( k m s − ) v Ab c s U b f max, WS ρ b / m p ( c m − ) f max, WS r A , r c ( R ⊙ ⊙ r A r c Figure 5.
The dependence of various flow properties on the Wang-Sheeley super-radialexpansion factor f max , WS (4.8) for the parameter values in (4.1), (4.2), and (4.6) through (4.9).All quantities are evaluated using numerical solutions to the model equations, with the exceptionof ˙ M (exp)2 , ˙ M (exp)4 , U (exp) ∞ , , and U (exp) ∞ , , which are defined in (B 32) and (B 33), and the datapoints labeled ‘WS empirical’, which are taken from table 2 of Wang & Sheeley (1990). Thehorizontal error bars on these data points convey the half widths of the f max , WS data bins inthat table. The vertical error bars correspond to one-half of the km s − increment betweenthe discretised U ∞ values that define four of the five data bins. The quantity I ρ b c s3 in the leftpanel of the second row is the low-Mach-number approximation to q b given in (3.83). reasonably well. The second-order analytic approximation ˙ M (exp)2 is also reasonablyaccurate at at f max , WS & , but deviates markedly from the numerical solution at f max , WS . . A similar comment applies to the top-right panel of figure 5, which alsoshows that the model agrees fairly well with observational constraints on U ∞ ( f max , WS ) when (4.9) holds. The left panel of the second row of figure 5 shows that the low-Mach-number approximation to q b given in (3.83) is quite accurate at small f max , WS ,where U b /c s ≪ (see the right panel in the second row of this figure). However,8as f max , WS increases, U b increases, because the outflow is concentrated into a smallercross-sectional area at the coronal base. This increase in U b leads to larger losses ofinternal energy within the transition region from p d V work and advection, which in turncauses q b to increase above the low-Mach-number scaling so that conductive heatingwithin the transition region can balance the additional non-radiative cooling. This samepanel also shows that q b is significantly smaller than the AW energy flux at the coronalbase ( ≃ ρ b ( δv b ) v Ab = ρ / ⊙ ( δv ⊙ eff ) B b / √ π ), which increases with f max , WS becauseincreasing f max , WS increases η b and hence B b . The lower-left panel of figure 5 showsthat ρ b increases by a factor of ≃ as f max , WS increases from 1 to 100, consistent withthe ρ b ∝ η / x / scaling in (3.57) and (3.62), given that x = c s2 /v is approximatelyconstant over this range of f max . The lower-right panel shows that r c is typically severalsolar radii, whereas r A ranges from R ⊙ to R ⊙ for this set of parameters.4.7. Dependence of the flow properties on the Alfvén-wave power, super-radialexpansion factor, and strength of the interplanetary magnetic field
The contour plots in figure 6 show how several quantities vary as functions of f max , WS and δv ⊙ eff in numerical solutions to the model equations based on the parameter valuesin (4.1), (4.2), (4.7), and (4.9). The top panels of figure 6 show that ˙ M is an increasingfunction of both f max , WS and δv ⊙ eff , whereas U ∞ is a decreasing function of both f max , WS and δv ⊙ eff . The second row shows that r A is a strongly decreasing function of δv ⊙ eff andonly weakly dependent on f max , WS for values of δv ⊙ eff comparable to the Sun-like valuein (4.6). The right panel of this row shows that r c is an increasing function of f max , WS and,for the most part, a decreasing function of δv ⊙ eff . The lower-left panel of figure 6 showsthat c s varies only weakly with f max , WS and δv ⊙ eff . As shown in the lower-right panel, χ H varies by a factor of ≃ over the parameter range shown. If δv ⊙ eff is set equal to the valuein (4.6) and f max is restricted to the interval (2 , so that U ∞ in the upper-right paneltakes on fast-wind-like values of − km s − , then χ H ≃ . − . , consistent withdirect numerical simulations of reflection-driven AW turbulence (Perez et al. submitted).Figure 7 displays contour plots of the same quantities as in figure 6, but this time asfunctions of f max , WS and B ∗ , or, equivalently, B r ( ) , using the parameters in (4.1),(4.6), (4.7), and (4.9). The top panels show that ˙ M increases approximately linearlywith B r ( ) at fixed f max , WS , and that U ∞ varies only weakly with B r ( ) , con-sistent with measurements from the Ulysses spacecraft (Schwadron & McComas 2008;Riley et al. r A depends more stronglyon B r ( ) than on f max , WS , whereas the reverse is true for r c . As in figure 6, c s variesvery weakly across the entire parameter range observed. The lower-right panel showsthat, when (4.9) holds, χ H depends more strongly on f max , WS than on B r ( ) .
5. Discussion and conclusion
The main goal of this paper is to obtain an approximate analytic solution to thecoupled problems of coronal heating and solar-wind acceleration under the assumptionthat the solar wind is powered primarily by an AW energy flux. Section 2 presents afirst step toward this goal, namely, a simplified calculation of ˙ M , U ∞ , and the coronaltemperature in a spherically symmetric, steady-state solar wind in the absence of solarrotation. This calculation is based upon: (1) the assumption that p d V work rather thanthermal conduction is the primary sink of internal energy in the sub-Alfvénic region as awhole; (2) the result of Section 3.2.2 that a solar or stellar wind heated primarily by AWturbulence becomes approximately isothermal between the coronal base r b and Alfvén9 f max, WS −1 −2 −1 −1 −1 δ v ⊙ e ff ⊙ k m s − ) . e - . e - . e - . e - . e - . e - ̇M ̇M ⊙ yr −1 ) f max, WS −1 −2 −1 −1 −1 δ v ⊙ e ff ⊙ k m s − ) U ∞ ⊙km s −1 ) f max, WS −1 −2 −1 −1 −1 δ v ⊙ e ff ⊙ k m s − ) r A ⊙R ⊙ ) f max, WS −1 −2 −1 −1 −1 δ v ⊙ e ff ⊙ k m s − ) r c ⊙R ⊙ ) f max, WS −1 −2 −1 −1 −1 δ v ⊙ e ff ⊙ k m s − ) c s ⊙km s −1 ) f max, WS −1 −2 −1 −1 −1 δ v ⊙ e ff ⊙ k m s − ) . . . . . χ H Figure 6.
The dependence of various flow properties on the Wang-Sheeley super-radialexpansion factor f max , WS (4.8) and the effective fluctuating velocity at the photosphere δv ⊙ eff defined in (3.49). critical point r A ; and (3) the finding in recent direct numerical simulations that mostof the AW power at the coronal base P AW ( r b ) is dissipated between r = r b and r = r A (Perez et al. submitted). The calculation of Section 2 shows that ˙ M ≃ P AW ( r b ) v U ∞ ≃ v esc T ≃ m p v k B ln( v esc /δv b ) . (5.1)Deviations from (5.1) can be caused by a number of factors, including conductive lossesinto the transition region, wave momentum deposition inside the wave-modified sonic-critical point, and the fact that part of the AW power reaches the super-Alfvénic region,where it enhances U ∞ without contributing to the heating that helps drive the outflowof mass past the wave-modified sonic critical point. These factors are accounted for inthe more detailed solar-wind model developed in Section 3. It is worth noting that inthis model:0 f max, WS B r ( a u )( n T ) . e - . e - . e - . e - . e - . e - ̇M (M ⊙ ⊙̇ −1 ) f max, WS10 B r ( a u )( n T ) U ∞ (km s −1 ) f max, WS B r ( a u )( n T ) r A (R ⊙ ) f max, WS B r ( a u )( n T ) . . . . . . . r c (R ⊙ ) f max, WS B r ( a u )( n T ) c s (km s −1 ) f max, WS B r ( a u )( n T ) . . . . . χ H Figure 7.
The dependence of various flow properties on the Wang-Sheeley super-radialexpansion factor f max , WS (4.8) and the strength of the radial magnetic field at r = 1 au. • the plasma density at the coronal base ρ ( r b ) = ρ b is determined by equatingthe turbulent heating rate Q ( r b ) and radiative cooling rate R ( r b ) ; and• an analytic solution for the heat flux from the corona into the transitionregion q b is obtained by balancing, within the transition region, conductiveheating against internal-energy losses from p d V work, advection, and radiativecooling.The expression (3.57) for ρ b that results from setting Q ( r b ) = R ( r b ) contains a factorof x / , where x = c s2 /v is the dimensionless temperature. Thus, x must be foundbefore the exact value of ρ b can be determined. However, because ˙ M is so sensitive tothe coronal temperature, x / can to a good approximation be treated as a constant (see,e.g., figures 6 and 7), and (3.57) with x / = 0 . can be used to obtain the approximatevalue of ρ b without solving the full model equations.The equations of the solar-wind model developed in Section 3 are solved analytically1in two different parameter regimes. One of these is the conduction-dominated limit,in which heat conduction into the transition region is the dominant mechanism fordraining internal energy from the sub-Alfvénic region, wave pressure makes the dom-inant contribution to the critical-point velocity U c in (3.89), and the wave-energy termdominates over the plasma-internal-energy term in the Bernoulli equation (3.95). Thislimit corresponds to photospheric velocities much smaller than those of the Sun. Thesecond parameter regime is the expansion-dominated limit, in which p d V work is thedominant sink of internal energy in the sub-Alfvénic region, the sound speed makes thedominant contribution to the critical-point velocity U c in (3.89), and the plasma-internal-energy term in the Bernoulli integral (3.95) dominates over the wave-energy term. Asillustrated in Figure 4, the expansion-dominated regime is relevant to the solar wind.The leading-order solution in the expansion-dominated regime reproduces (5.1), with asmall difference in the coronal temperature arising from the fact that δv b has a weakdependence on the coronal temperature in the model of Section 3. Numerical solutionsto the model equations approach the approximate analytic solutions in the appropriateparameter regimes, match a range of solar-wind observations, and illustrate how theproperties of the solar wind depend upon the r.m.s. photospheric velocity, super-radialexpansion factor, and interplanetary-magnetic-field strength.5.1. Top-down causality for determining ˙ M , q b , and the pressure, temperature range,and altitude of the transition region within the solar atmosphere In this paper, as in Parker’s original model (Parker 1958, 1965), the average rate atwhich mass flows out through the lower solar atmosphere is determined in large part bythe outflow condition at the wave-modified sonic critical point r c , several R ⊙ out from theSun, at which the plasma transitions from being gravitationally bound to gravitationallyunbound. Since the gravitational force weakens with increasing r , there is no physicalmechanism that can prevent plasma at r c from flowing outward at approximately thewave-enhanced effective sound speed, c s , eff ≡ [( p + p wave ) /ρ ] / = [ c + 0 . δv ) ] / ,evaluated at r = r c , which is comparable to the square root of the right-hand sideof (3.89). This is why ˙ M ∼ A ( r c ) ρ c c s , eff ( r c ) .On the other hand, localised motions near the transition region at speeds ≪ v esc aregravitationally bound, and the mass flux that they carry is not determinative of ˙ M . Forexample, if at some time, such motions led to an overall mass outflow rate at r b exceedingthe rate ∼ A ( r c ) ρ c c s , eff ( r c ) at which mass flows past the critical point r c , then plasmawould build up in the corona. This would in turn weaken the pressure gradient relativeto the gravitational force per unit volume in the vicinity of the transition region, therebyreducing the amount of plasma flowing up from the chromosphere.Nevertheless, the transition region and chromosphere do affect ˙ M in two ways. First, q b reduces the net heating power within the quasi-isothermal sub-Alfvénic region, P net . Asdiscussed following (2.16), the heating cost per unit mass for plasma to transit the quasi-isothermal sub-Alfvénic region is c s2 ln( ρ b /ρ ( r A )) ≃ v , and ˙ M ≃ P net /v . By reduc-ing P net , the conduction of heat from the corona into the transition region reduces ˙ M . Thesecond way that the lower solar atmosphere affects ˙ M is via the chromospheric/transition-region transmission coefficient, f chr = P AW ( r b ) /P AW ( R ⊙ ) , whose value again influencesthe value of P net . To summarise this paragraph and the preceding paragraph, the chro-mosphere and transition region influence ˙ M thermodynamically, but not dynamically.The regulation of the mass flux at r b by the critical-point condition at r c is anexample of ‘top-down’ causality, in which physical processes at larger r control the plasmaproperties at smaller r . In the model of this paper, top-down causality also characterisesthe determination of q b , the pressure within the transition region, the altitude of the2transition region in the solar atmosphere, and the temperature jump across the transitionregion. As mentioned above, ρ b is determined by the condition that Q ( r b ) = R ( r b ) ,without reference to conditions in the chromosphere or the value of q b . The sound speedat r = r b , which is ≃ c s , is approximately determined by balancing turbulent heatingagainst internal-energy losses from p d V work within the corona and sub-Alfvénic solarwind. The outflow velocity at r = r b , U b , follows from the value of ˙ M , which is controlledby the critical-point condition, as described above. The values of ρ b , c s , and U b jointlydetermine q b via (3.82), which embodies the requirement that conductive heating offsetinternal-energy losses (from radiation, p d V work, and advection) within the transitionregion. The values of ρ b and c s are sufficient to determine the approximate transition-region pressure, p tr ≃ ρ b c s2 . The pressure within the upper chromosphere, p chr ( r ) , is anapproximately exponentially decreasing function of altitude. The altitude of the transitionregion is determined by setting p chr ( r ) = p tr . The shape of the radiative loss functionplotted in figure 2 constrains the temperature at the bottom of the transition region tobe ∼ K, so that radiative cooling within the comparatively dense upper chromospherecan be balanced by local heating mechanisms in the absence of strong conductive heating.Given this constraint, the factor by which the temperature changes across the transitionregion is determined by the value of T ( r b ) , which is controlled by the balance betweenheating and p d V work in the corona, as discussed above.Although the condition R ( r b ) = Q ( r b ) determines the density ρ b at the upperboundary of the transition region within the corona (subject to the caveats at theend of the paragraph following (5.1)), setting R ( r ) = Q ( r ) within the chromospheredoes not determine the density at the lower edge of the transition region, because smallchanges in T within the upper chromosphere lead to dramatic changes in the radiativeloss function Λ ( T ) , as shown in Figure 2. The density at the bottom of the transitionregion is instead largely determined by the values of ρ b and c s , the near constancy of thepressure across the transition region, and the above-mentioned constraint (arising fromthe shape of the radiative loss function) that T ∼ K in the upper chromosphere.5.2.
Limitations and future work
The model of Section 3 has a number of shortcomings. First, the sub-Alfvénic region isnot truly isothermal, and hence the quasi-isothermal approximation in (2.5) leads to someerror. Second, for the solutions shown in figure 5, the temperature m p c s2 / (2 k B ) of the sub-Alfvénic region increases from ≃ × K when U ∞ ≃ km s − to ≃ . × K when U ∞ ≃ km s − . In contrast, in measurements from the Ulysses spacecraft, the coronalfreeze-in temperature increases from ≃ × K when U ∞ ≃ km s − to ≃ . × K when U ∞ ≃ km s − (McComas et al. σ that appears in the turbulent heating rate has a large impact on the solutionto the model equations, but is an adjustable free parameter. Further work is neededto provide a physical basis for determining σ and how it varies from one flux tube toanother.In a future study, the model developed in Section 3 could be used in conjunction withstudies that map the magnetic-field line traversed by the Parker Solar Probe (PSP) backto a source region on the Sun to rapidly predict flow properties at PSP’s location based onthe observed super-radial expansion factor within the source region (see, e.g., Bale et al. et al. Appendix A. Solving for U ( r ) and r A Once y b , y c , and x are determined by solving (3.67), (3.94), and (3.97), the valueof U ( r ) between r = r b and r = r A (which is as yet unknown) can be found by solvingthe Bernoulli integral (3.95) with Γ given by (3.96). An equation for r A can be obtainedby evaluating the Bernoulli integral (3.95) at r = r A , setting U ( r A ) = v A ( r A ) and y = 1 ,and rewriting Γ using (3.96), which leads to v A ( r A ) − c s2 ln y b + v (cid:18) ψ / − R ⊙ r A (cid:19) − ( δv b ) − σ ) (cid:20) σ − (3 + σ )(1 + y b ) − σ − y b − − σ (cid:21) − v y . (A 1)Upon setting v A ( r A ) = v Ab y b (cid:20) B ( r A ) B b (cid:21) = v Ab y b η ( r A ) R ⊙ η b r ψ (A 2)in (A 1) and rewriting v Ab in (A 1) using (3.63), one obtains the following equation for r A : (cid:20) y η ( r A ) ( x ˜ ρ ⊙ ) / ξ ψ / (cid:21) (cid:18) R ⊙ r A (cid:19) − R ⊙ r A − x ln y b + ψ / − ψ / η ( x ˜ ρ ⊙ ) / ξ y + ǫ − σ (cid:20) − σ − (3 + σ )(1 + y b ) − σ y b + 1 + σ (cid:21) . (A 3)At r > r A , the outflow velocity U ( r ) cannot be determined from the Bernoulli integral,because the quasi-isothermal approximation does not apply. In addition, (3.40) yields apoor approximation for z − at r > r A , because y approaches a constant when r ≫ r A . Abetter approximation for z − in the super-Alfvénic region can be obtained from (3.25) andthe simplifying approximation that | ( d / d r ) ln( v A ) | = − ( d / d r ) ln y , which holds when B ∝ r − and ρ ∝ r − (the latter scalings being fairly accurate for R ⊙ . r . R ⊙ , a regionin which the field lines are approximately radial and the flow speed is approximatelyconstant). In this case, (3.25) becomes z − L ⊥ = − ( U + v A ) dd r ln y. (A 4)Integrating (3.21) from r = r A out to larger r then gives z = [ z + ( r A )] y / (1 + y ) ,where the value of z + ( r A ) is obtained by setting y = 1 in (3.41). The approximate valueof c ( r ) can then be found in terms of ρ by solving the internal-energy equation (3.32)with Q given by (3.22) and neglecting radiative cooling and thermal conduction. This4leads to c ( r ) = (cid:20) ρ ( r ) ρ A (cid:21) γ − c s2 + K Z y y − γ d y y ! (A 5)at r > r A , where K = ( γ − z y b (cid:18) y b (cid:19) − σ . (A 6)Total-energy conservation implies that the mechanical luminosity L mech ( r ) = ˙ M (cid:20) U c ( r )2 − v R ⊙ r + z (cid:18)
32 + y (cid:19) + q r ρU (cid:21) (A 7)is independent of r , where the ratio of specific heats γ has been set equal to / . Thevalue of U ( r ) at r > r A can be obtained by setting L mech ( r ) = L mech ( r A ) , (A 8)and rewriting ρ ( r ) in terms of U ( r ) using (3.7) — i.e., ρ ( r ) U ( r ) /B ( r ) = ρ b U b /B b . Since U ( r A ) = v A ( r A ) and y ( r A ) = 1 , (A 7) implies that L mech ( r A ) = ˙ M (cid:20) v A ( r A ) c s2 − v R ⊙ r A + 5( δv b ) (1 + y b ) − σ − σ y b (cid:21) . (A 9)Determining U ( r ) at r > r A via (A 7) through (A 9) requires evaluating r A and v A ( r A ) .An alternative method that avoids this requirement results from noting that L mech ( r b ) = ˙ M ( v y + 5 c − v ψ / δv b ) (cid:20)
12 + (1 + y b )(1 − χ H ) (cid:21) + 2 c s2 ln y b ) , (A 10)where (3.66) has been used to eliminate q b . Replacing χ H in (A 10) with the expressionon the right-hand side of (3.45) leads to the consistency check that L mech ( r A ) = L mech ( r b ) . (A 11)Combining (A 8) and (A 11), one can find U ( r ) at r > r A by setting L mech ( r ) = L mech ( r b ) . (A 12)Equation (A 12) leads to a simple expression for the asymptotic wind velocity U ∞ , i.e., U ( r ) as r → ∞ . As r → ∞ , the kinetic-energy flux dominates the total energy flux, and L mech ( r ) → ˙ M U ∞ / . Setting ˙ M U ∞ / L mech ( r b ) yields U ∞ = (cid:20) v y + 5 c s2 − v ψ / + 2( δv b ) (cid:18)
32 + y b (cid:19) − q b y b ρ b v Ab (cid:21) / . (A 13) Appendix B. Approximate analytic solutions
As discussed in Section 3.8, the core of the solar-wind model developed in Section 3is a set of three simultaneous equations — (3.67), (3.94), and (3.97) — for the threeunknowns y b , y c , and x . In this section, two different approximate analytic solutions tothese equations are obtained in two different parameter regimes. Both solutions rely onthe simplifying approximations in (3.100), which are repeated here: y b ≫ ψ = 1 ǫ ≪ η c = γ B c = 1 . (B 1)5In particular, y b is taken to be sufficiently large that: (1) the U = v /y term in (3.96)can be dropped, which amounts to dropping the second-to-last term on the left-hand sideof (3.97); and (2) O ( y − ) terms can be dropped in (3.45), so that χ H = 1 − ζy − σ b , (B 2)where ζ = 2 σ − (cid:18) − σ − σ (cid:19) . (B 3)The y − σ b term in (B 2) is retained, despite discarding terms ∼ O ( y − ) , on the workingassumption that σ ∼ . − . , as is the case in the numerical examples in Section 4. Asmentioned in Section 3.8, the last equality in (B 1) amounts to taking all of the super-radial expansion of the field lines to occur inside the wave-modified sonic critical pointand the field lines to be completely radial at r > r c . With these approximations, (3.67),(3.94), and (3.97) become, respectively, ǫ ⊙ ˜ ρ / ⊙ χ H B ∗ ξx / − x ln y b y b − q b ρ b v Ab v = 0 , (B 4) y x / ξ ˜ ρ / ⊙ y b ! / − x − ǫ ⊙ ˜ ρ / ⊙ y − σ b [ y (1 + σ ) + 3 y c ]4 B ∗ ξx / (1 + y c ) − σ = 0 , (B 5)and − x (cid:20) (cid:18) y b y c (cid:19) + 3 (cid:21) − ǫ ⊙ ˜ ρ / ⊙ y − σ b [ y (1 + σ )(7 − σ ) + y c (21 − σ ) + 8]4(1 − σ ) B ∗ ξx / (1 + y c ) − σ = 0 . (B 6)B.1. Conduction-dominated limit
When ǫ ⊙ is sufficiently small, an approximate solution to (B 4), (B 5), and (B 6) can beobtained through the method of dominant balance (Bender & Orszag 1978), in which twoof the three terms in each equation are taken to be dominant, and the third term is takento be much smaller in magnitude. Neglecting the smaller term in each equation yieldsthe leading-order solution, with the smaller term producing higher-order corrections. Inthe present case, it is the second term on the left-hand side of each equation that can beneglected to leading order. In (B 4), this corresponds to balancing turbulent heating of thesub-Alfvénic region against the heat that is conducted from the corona into the transitionregion. In other words, conduction into the transition region rather than p d V work isthe dominant sink of internal energy in the sub-Alfvénic region as a whole. Neglectingthe second term in (B 5) amounts to assuming that the fluctuating velocity makes thedominant contribution to U c in (3.89), which is equivalent to taking δv ( r c ) ≫ c s . Thelatter equality appears paradoxical, since the small- ǫ ⊙ limit corresponds to small values ofthe fluctuating velocity at the coronal base. However, as ǫ ⊙ → , the coronal temperaturedrops, the density scale height in the corona decreases, and the wave amplitude at thecritical point grows as the AWs attempt to conserve wave action, which would leadto δv ∝ ρ − / when U ≪ v A in the absence of reflection and dissipation (see (3.19)and (3.20)). Neglecting the second term in (B 6) amounts to taking the wave pressure forceto have a larger cumulative effect than the thermal pressure force on the acceleration ofthe outflowing plasma between r b and r c , which can again be understood as a consequenceof the wave amplitudes growing rapidly with increasing r when the density scale heightin the corona is small.As already noted in (3.101) and illustrated in Figure 4, ˙ M is exceedingly small in the6conduction-dominated limit, and as a consequence U b ≪ c s . The heat flux at the coronalbase is thus approximately given by (3.83), and (B 4) can be rewritten as ǫ ⊙ ˜ ρ / ⊙ χ H B ∗ ξx / − x ln y b y b − I x / ξ ˜ ρ / ⊙ η b = 0 . (B 7)Balancing the first and third terms in (B 7) and dropping the y − σ b term in (B 2) yieldsthe leading-order solution for the dimensionless temperature in the sub-Alfvénic region: x = I − / ˜ ρ / ⊙ ( ǫ ⊙ η b B ∗ ˜ l b ) / . (B 8)Anticipating the solution for y c , it is useful to predict at the outset (as will shortly beconfirmed by (B 10) and (B 11)) that y c ≫ . (B 9)Balancing the first and third terms in (B 6) then gives y b y c = " − σ ) B / ∗ η / I / (1 + σ )(7 − σ ) ǫ / ⊙ ˜ l / ˜ ρ / ⊙ / (1 − σ ) . (B 10)Balancing the first and third terms in (B 5) and making use of (B 9) and (B 10), onearrives at the leading-order solution for y b : y b = I h ǫ − (12 − σ ) / ⊙ B (9 − σ ) / ∗ η σ ) / ˜ l − σ ) / ˜ ρ − (6 − σ ) / ⊙ i / (1 − σ ) , (B 11)where I = 16 I − (1+ σ ) / (14 − σ )1 (cid:18)
41 + σ (cid:19) / (1 − σ ) (cid:18) − σ − σ (cid:19) (7 − σ ) / (2 − σ ) . (B 12)Equations (3.13), (3.57), (B 8), and (B 11) then yield the leading-order mass outflow ratein the conduction-dominated regime, ˙ M (cond) , given in (3.101).The asymptotic wind velocity U ∞ is obtained by setting ˙ M U ∞ / equal to the me-chanical luminosity at the coronal base L mech ( r b ) as described in Appendix A, where L mech is defined in (A 7). When this procedure is carried out using (B 8) and (B 11), thetwo leading-order terms in L mech ( r b ) cancel. To obtain the leading-order non-vanishingterm in U ∞ , one must account for the next largest term in (B 7), which results fromthe χ H correction — i.e., the second term on the right-hand side of (B 2). When thisterm is retained, one obtains the leading-order asymptotic wind speed in the conduction-dominated regime, U (cond) ∞ , given in (3.102).The range of ǫ ⊙ values for which (3.101), (3.102), (B 8), (B 10), and (B 11) areapproximately valid can be determined by requiring that the neglected terms in (B 5),(B 6), and (B 7) be small compared to the other terms when y b , y c , and x are givenby (B 8), (B 10), and (B 11). The most stringent condition on ǫ ⊙ arises from carrying outthis procedure for (B 6), which results in the requirement that I ǫ / ⊙ (cid:20) I + 3 − − σ ) ln ǫ ⊙ (cid:21) ≪ , (B 13)where I = I − / ˜ ρ / ⊙ ( η b B ∗ ˜ l b ) / , and I equals the right-hand side of (B 10) withoutthe ǫ ⊙ term; i.e., I = ǫ / (7 − σ ) ⊙ y b /y c , with y b /y c given by the right-hand side of (B 10).Upon neglecting the quantity I + 3 on the left-hand side of (B 13), which is smallerin magnitude than the remaining term when ǫ ⊙ is in the conduction-dominated regime7but other parameters take on Sun-like values, one finds that the conduction-dominatedregime corresponds to ǫ ⊙ ≪ ǫ ⊙ cond ≡ exp (cid:18) W − (cid:18) σ − I (cid:19)(cid:19) , (B 14)where W − is the lower branch of the Lambert W function.B.2. Expansion-dominated limit
In the expansion-dominated limit, the first two terms on the left-hand sides of (B 4),(B 5), and (B 6) are treated as dominant. For this case, it is useful to define the variables u = A y b x / p = y b y c , (B 15)where A = ǫ / ⊙ ˜ ρ / ⊙ ˜ l / ( η b B ∗ ) / . (B 16)Rewriting (B 4), (B 5), and (B 6) in terms of u , p , and x rather than y b , y c , and x , oneobtains x ln A u = a, (B 17) ux / ξ ˜ ρ / ⊙ A p ! / − x = b, (B 18)and − x (4 ln p + 3) = c. (B 19)The quantities a , b , and c , which are treated as small, are functions of u , p , and x .In order to see how (B 17), (B 18), and (B 19) follow from (B 4), (B 5), and (B 6), it ishelpful to leave terms containing y b and y c in the expressions for a , b , and c , with theunderstanding that y b = x / u/A and y c = x / u/ ( A p ) : a ≡ xu ln( ux / ) + ζy − σ b + x / q b A ρ b v Ab v , (B 20) b ≡ ǫ ⊙ ˜ ρ / ⊙ y − σ b [ y (1 + σ ) + 3 y c ]4 B ∗ ξx / (1 + y c ) − σ , (B 21)and c ≡ ǫ ⊙ ˜ ρ / ⊙ y − σ b [ y (1 + σ )(7 − σ ) + y c (21 − σ ) + 8]4(1 − σ ) B ∗ ξx / (1 + y c ) − σ . (B 22)From (B 17), it follows that x = (1 − a ) u − A . (B 23)Equation (B 19) implies that p = exp (cid:18) − c x − (cid:19) . (B 24)After substituting (B 23) and (B 24) into (B 18), one finds that u − S ( u, a, b, c ) , (B 25)8where S ( u, a, b, c ) = cu − a − (1 − a )ln A (cid:20)
54 ln u + ln(1 − a )4 − ln( − A )4 − ln A + 32 −
32 ln (cid:18) b − (1 − a ) u A (cid:19)(cid:21) , (B 26)and A = 16 ξ ˜ ρ / ⊙ , (B 27)which is ≃ for typical solar parameters. The quantities a , b , and c are themselvesfunctions of u by virtue of (B 15) and (B 20) through (B 24).In the expansion-dominated regime u ∼ O (1) | S ( u, a, b, c ) | ≪ . (B 28)The latter inequality is achieved when − / ln A , | a | , b , and c are ≪ . When (B 28) issatisfied, (B 23), (B 24), and (B 25) can be solved perturbatively through the recursionrelations u n − (cid:26) if n = 0 S ( u n − , a n − , b n − , c n − ) if n > (B 29) x n = (1 − a n − ) u n − A (B 30) p n = exp (cid:18) − c n − x n − (cid:19) , (B 31)where n = 0 , , , . . . , and a − = c − = 0 . The values of a n , b n , and c n are obtained by re-placing ( a, b, c, u, y b , y c , x ) with ( a n , b n , c n , u n , y b ,n , y c ,n , x n ) in (B 20), (B 21), and (B 22).For reference below, the values of ρ b , v Ab , and q b that result from this substitution(via (3.57), (3.63), and (3.82)) are denoted ρ b ,n , v Ab ,n , and q b ,n . The values of y b ,n and y c ,n are obtained by replacing ( y b , y c , x, u, p ) with ( y b ,n , y c ,n , x n , u n , p n ) in (B 15).The n th -order approximation for ˙ M in the expansion-dominated regime, denoted ˙ M (exp) n , can be found by setting ˙ M = ˙ M (exp) n , x = x n , y b = y b ,n and ψ = 1 in (3.99): ˙ M (exp) n = R ⊙ B v esc y − ,n ( x n ˜ ρ ⊙ ) / ξ. (B 32)For n > , I define the n th -order approximation for U ∞ , denoted U (exp) ∞ ,n , to be the valueof U ∞ in (A 13) when ( x, y b , ψ, q b , ρ b , v Ab ) = ( x n , y b ,n , , q b ,n , ρ b ,n , v Ab ,n ) : U (exp) ∞ ,n = v esc " v ,n y ,n v + 5 x n − y b ,n ǫ ⊙ ˜ ρ / ⊙ B ∗ ξx / n − q b ,n y b ,n ρ b ,n v Ab ,n v / , (B 33)where I have invoked (B 1) to approximate / y b ,n as y b ,n . The leading-order ap-proximation for U ∞ in the expansion-dominated regime, denoted U (exp) ∞ , , is obtainedfrom (B 33) with n = 0 after dropping terms that were neglected in the calculation of u — in particular, the first, second, and last terms inside the brackets on the right-handside of (B 33). This yields U (exp) ∞ , = v esc . (B 34)As in the conduction-dominated limit, the range of ǫ ⊙ values for which (B 29), (B 32),and (B 34) are approximately valid can be determined by imposing the constraint that the9neglected terms in (B 4), (B 5), and (B 6) be small compared to the terms that are keptwhen y b , y c , and x are given by (B 15), (B 16), (B 23), (B 24), and u = 1 . Carrying outthis procedure for (B 4) and making the simplifying approximations that a is dominatedby the last term on the right-hand side of (B 20), that q b is given by (3.83), and that ln A ≃ ln (cid:16) ǫ / ⊙ (cid:17) , one obtains the requirement that ǫ ⊙ ≫ ǫ ⊙ exp , min ≡ exp W − − I /
21 ˜ ρ / ⊙ (˜ l b η b B ∗ ) / !! , (B 35)where W − is, as above, the lower branch of the Lambert W function. Equation (B 35)corresponds to the requirement that the conductive losses from the sub-Alfvénic regioninto the transition region be negligible compared to p d V work in this region. Carryingout the above procedure for (B 5) and again making the simplifying approximation that ln A ≃ ln (cid:16) ǫ / ⊙ (cid:17) , one obtains ǫ ⊙ ≪ ǫ ⊙ exp , max ≡ exp σ ) W − − / (1 + σ ) / e − σ ) / " η b B ∗ ˜ l b ˜ ρ / ⊙ (1+ σ ) / . (B 36)Equation (B 36) corresponds to the requirement that the sound speed make the dominantcontribution to the outflow velocity U c at the critical point in (3.89). As illustratedby the shaded gray rectangle in figure (4), for Sun-like parameters (and in particular,for η b = 30 ), ǫ ⊙ exp , max is approximately four orders of magnitude larger than ǫ ⊙ exp , min .There is thus a finite range of ǫ ⊙ values that satisfy both (B 35) and (B 36). However, itshould be noted that the approximations ln A ≃ ln( ǫ / ⊙ ) and q b = I ρ b c s3 cause (B 35)to underestimate the lower bound on ǫ ⊙ in the expansion-dominated regime. Also, forSun-like parameters, the assumption r c < r A that underlies the model of Section 3 breaksdown at values of ǫ ⊙ smaller than ǫ ⊙ exp , max , as illustrated in Figure 4. REFERENCESBale, S. D., Badman, S. T., Bonnell, J. W., Bowen, T. A., Burgess, D., Case, A. W.,Cattell, C. A., Chandran, B. D. G., Chaston, C. C., Chen, C. H. K., Drake,J. F., de Wit, T. D., Eastwood, J. P., Ergun, R. E., Farrell, W. M., Fong,C., Goetz, K., Goldstein, M., Goodrich, K. A., Harvey, P. R., Horbury, T. S.,Howes, G. G., Kasper, J. C., Kellogg, P. J., Klimchuk, J. A., Korreck, K. E.,Krasnoselskikh, V. V., Krucker, S., Laker, R., Larson, D. E., MacDowall,R. J., Maksimovic, M., Malaspina, D. M., Martinez-Oliveros, J., McComas,D. J., Meyer-Vernet, N., Moncuquet, M., Mozer, F. S., Phan, T. D., Pulupa,M., Raouafi, N. E., Salem, C., Stansby, D., Stevens, M., Szabo, A., Velli, M.,Woolley, T. & Wygant, J. R.
Nature (7786), 237–242.
Bale, S. D., Goetz, K., Harvey, P. R., Turin, P., Bonnell, J. W., Dudok deWit, T., Ergun, R. E., MacDowall, R. J., Pulupa, M., Andre, M., Bolton,M., Bougeret, J.-L., Bowen, T. A., Burgess, D., Cattell, C. A., Chandran,B. D. G., Chaston, C. C., Chen, C. H. K., Choi, M. K., Connerney, J. E.,Cranmer, S., Diaz-Aguado, M., Donakowski, W., Drake, J. F., Farrell, W. M.,Fergeau, P., Fermin, J., Fischer, J., Fox, N., Glaser, D., Goldstein, M.,Gordon, D., Hanson, E., Harris, S. E., Hayes, L. M., Hinze, J. J., Hollweg,J. V., Horbury, T. S., Howard, R. A., Hoxie, V., Jannet, G., Karlsson, M.,Kasper, J. C., Kellogg, P. J., Kien, M., Klimchuk, J. A., Krasnoselskikh,V. V., Krucker, S., Lynch, J. J., Maksimovic, M., Malaspina, D. M., Marker,S., Martin, P., Martinez-Oliveros, J., McCauley, J., McComas, D. J., McDonald, T., Meyer-Vernet, N., Moncuquet, M., Monson, S. J., Mozer,F. S., Murphy, S. D., Odom, J., Oliverson, R., Olson, J., Parker, E. N.,Pankow, D., Phan, T., Quataert, E., Quinn, T., Ruplin, S. W., Salem, C., Seitz,D., Sheppard, D. A., Siy, A., Stevens, K., Summers, D., Szabo, A., Timofeeva,M., Vaivads, A., Velli, M., Yehle, A., Werthimer, D. & Wygant, J. R.
Sp. Sci. Rev. , 49–82.
Banaszkiewicz, M., Axford, W. I. & McKenzie, J. F.
Astron. Astrophys. , 940–944.
Belcher, J. W. & Davis, Jr., L.
J. Geophys. Res. , 3534–3563. Bender, C. M. & Orszag, S. A.
Advanced Mathematical Methods for Scientists andEngineers . Book, D. L.
Tech. Rep. . Bretherton, F. P. & Garrett, C. J. R.
Royal Society of London Proceedings Series A , 529–554.
Bruno, R. & Carbone, V.
Living Reviewsin Solar Physics , 4–+. Chandran, B. D. G., Dennis, T. J., Quataert, E. & Bale, S. D.
Astrophys. J. , 197, arXiv: 1110.3029.
Chandran, B. D. G. & Hollweg, J. V.
Astrophys. J. , 1659–1667, arXiv: 0911.1068.
Chandran, B. D. G. & Perez, J. C.
Journal of Plasma Physics (4),905850409, arXiv: 1908.00880. Chen, C. H. K., Bale, S. D., Bonnell, J. W., Borovikov, D., Bowen, T. A., Burgess,D., Case, A. W., Chandran, B. D. G., de Wit, T. D., Goetz, K., Harvey, P. R.,Kasper, J. C., Klein, K. G., Korreck, K. E., Larson, D., Livi, R., MacDowall,R. J., Malaspina, D. M., Mallet, A., McManus, M. D., Moncuquet, M.,Pulupa, M., Stevens, M. L. & Whittlesey, P.
Astrophys. J. Suppl. (2), 53, arXiv:1912.02348.
Coleman, P. J.
Astrophys.J. , 371.
Cranmer, S. R. & van Ballegooijen, A. A.
Astrophys.J. Suppl. , 265–293.
Cranmer, S. R., van Ballegooijen, A. A. & Edgar, R. J.
Astrophys. J. Suppl. , 520–551, arXiv: arXiv:astro-ph/0703333.
De Pontieu, B., McIntosh, S. W., Carlsson, M., Hansteen, V. H., Tarbell, T. D.,Schrijver, C. J., Title, A. M., Shine, R. A., Tsuneta, S., Katsukawa, Y.,Ichimoto, K., Suematsu, Y., Shimizu, T. & Nagata, S.
Science , 1574–7.
Dewar, R. L.
Physics of Fluids , 2710–2720. Dmitruk, P., Matthaeus, W. H., Milano, L. J., Oughton, S., Zank, G. P. & Mullan,D. J.
Astrophys. J. , 571–577.
Durney, B. R.
J.Geophys. Res. , 4042–4051. Gressl, C., Veronig, A. M., Temmer, M., Odstrčil, D., Linker, J. A., Mikić, Z. &Riley, P.
Sol. Phys. (5), 1783–1801, arXiv: 1312.1220. Hackenberg, P., Marsch, E. & Mann, G.
Astron. Astrophys. , 1139–1147.
Hansteen, V. H. & Leer, E.
J. Geophys. Res. , 21577–21594.
Hansteen, V. H., Leer, E. & Holzer, T. E.
Astrophys. J. (1), 498–509.
Hansteen, V. H. & Velli, M.
Sp.Sci. Rev. (1-4), 89–121.
Hartle, R. E. & Sturrock, P. A.
Astrophys. J. , 1155.
Heinemann, M. & Olbert, S.
J. Geophys.Res. , 1311–1327. Hollweg, J. V.
Astrophys. J. , 547–566.
Hollweg, J. V.
Reviews of Geophysics andSpace Physics , 689–720. Horbury, T. S., Forman, M. & Oughton, S.
Physical Review Letters (17), 175005–+, arXiv:0807.3713.
Iroshnikov, P. S.
Astron.Zh. , 742–+. Kasper, J. C., Bale, S. D., Belcher, J. W., Berthomier, M., Case, A. W., Chandran,B. D. G., Curtis, D. W., Gallagher, D., Gary, S. P., Golub, L., Halekas, J. S.,Ho, G. C., Horbury, T. S., Hu, Q., Huang, J., Klein, K. G., Korreck, K. E.,Larson, D. E., Livi, R., Maruca, B., Lavraud, B., Louarn, P., Maksimovic, M.,Martinovic, M., McGinnis, D., Pogorelov, N. V., Richardson, J. D., Skoug,R. M., Steinberg, J. T., Stevens, M. L., Szabo, A., Velli, M., Whittlesey,P. L., Wright, K. H., Zank, G. P., MacDowall, R. J., McComas, D. J., McNutt,R. L., Pulupa, M., Raouafi, N. E. & Schwadron, N. A.
Nature (7786), 228–231.
Kraichnan, R. H.
Physics of Fluids , 1385. Landau, L. D. & Lifshitz, E. M.
Mechanics . Oxford: Pergamon Press.
Leer, E. & Holzer, T. E.
J. Geophys. Res. ,4681–4688. Matthaeus, W. H. & Goldstein, M. L.
J. Geophys. Res. , 6011–6028. Matthaeus, W. H., Zank, G. P., Oughton, S., Mullan, D. J. & Dmitruk, P.
Astrophys. J. Lett. , L93–L96.
McComas, D. J., Barraclough, B. L., Funsten, H. O., Gosling, J. T., Santiago-Muñoz, E., Skoug, R. M., Goldstein, B. E., Neugebauer, M., Riley, P. &Balogh, A.
J. Geophys.Res. , 10419–10434.
McComas, D. J., Elliott, H. A., Gosling, J. T., Reisenfeld, D. B., Skoug, R. M.,Goldstein, B. E., Neugebauer, M. & Balogh, A.
Geophysical Research Letters , 1290. Mestel, L.
Mon. Not. R. Astron.Soc. , 473.
Parker, E. N.
Astrophys. J. , 664–676.
Parker, E. N.
Space Science Reviews , 666. Perez, J. C. & Chandran, B. D. G.
Astrophys. J. ,124, arXiv: 1308.4046. Perez, J. C., Chandran, B. D. G., Klein, K. G. & Martinovic, M. submitted , Heat orwork? How Alfvén waves energize the solar wind.
Journal of Plasma Physics . Richardson, R. S. & Schwarzschild, M.
Astrophys. J. , 351.
Riley, P., Lionello, R., Linker, J. A., Mikic, Z., Luhmann, J. & Wijaya, J.
Sol. Phys. (1-2), 361–377.
Riley, P., Mikic, Z., Lionello, R., Linker, J. A., Schwadron, N. A. & McComas,D. J.
Journal of Geophysical Research (Space Physics) , A06104.
Rosner, R., Tucker, W. H. & Vaiana, G. S.
Astrophys. J. , 643–645.
Sandbaek, O., Leer, E. & Hansteen, V. H.
Astrophys. J. , 390.
Schwadron, N. A. & McComas, D. J.
Astrophys. J. ,1395–1403.
Schwadron, N. A. & McComas, D. J.
Astrophys. J. Lett. , L33–L36.
Shoda, M., Suzuki, T. K., Asgari-Targhi, M. & Yokoyama, T.
The Astrophysical Journal (1), L2, arXiv: 1905.11685.
Smith, C. W., Matthaeus, W. H., Zank, G. P., Ness, N. F., Oughton, S. &Richardson, J. D.
J. Geophys. Res. , 8253–8272.
Spitzer, L. & Härm, R.
PhysicalReview , 977–981. Tu, C. & Marsch, E.
Space Science Reviews , 1–210. Usmanov, A. V., Goldstein, M. L. & Matthaeus, W. H.
Astrophys. J. , 43. van Ballegooijen, A. A. & Asgari-Targhi, M.
Astrophys. J. , 106, arXiv: 1602.06883. van Ballegooijen, A. A. & Asgari-Targhi, M.
Astrophys. J. , 10, arXiv: 1612.02501. van Ballegooijen, A. A., Asgari-Targhi, M., Cranmer, S. R. & DeLuca, E. E.
Astrophys.J. , 3–+, arXiv: 1105.0402. van der Holst, B., Sokolov, I. V., Meng, X., Jin, M., Manchester, IV, W. B.,Tóth, G. & Gombosi, T. I.
Astrophys. J. , 81, arXiv: 1311.4093.
Velli, M.
Astron. Astrophys. , 304–314.
Velli, M., Grappin, R. & Mangeney, A.
Physical Review Letters ,1807–1810. Verdini, A. & Velli, M.
Astrophys. J. , 669–676, arXiv: arXiv:astro-ph/0702205.
Verdini, A., Velli, M., Matthaeus, W. H., Oughton, S. & Dmitruk, P.
Astrophys. J. Lett. , L116–L120, arXiv: 0911.5221.
Wang, Y.-M. & Sheeley, Jr., N. R.
Astrophys. J. , 726–732.
Wicks, R. T., Horbury, T. S., Chen, C. H. K. & Schekochihin, A. A.
Mon. Not. R. Astron. Soc.407