# Modeling of Joint Parker Solar Probe – Metis/Solar Orbiter Observations

L. Adhikari1, G. P. Zank1, D. Telloni2, and L.-L. Zhao1

###### Abstract

We present a first theoretical modeling of joint Parker Solar Probe (PSP) -
Metis/Solar Orbiter (SolO) quadrature observations (Telloni et al., 2022c).
The combined observations describe the evolution of a slow solar wind plasma
parcel from the extended solar corona ($3.5-6.3$ R⊙) to the very inner
heliosphere (23.2 R⊙). The Metis/SolO instrument remotely measures the solar
wind speed finding a range from $96-201$ kms-1, and PSP measures the solar
wind plasma in situ, observing a radial speed of 219.34 kms-1. We find
theoretically and observationally that the solar wind speed accelerates
rapidly within 3.3 – 4 R⊙, and then increases more gradually with distance.
Similarly, we find that the theoretical solar wind density is consistent with
the remotely and in situ observed solar wind density. The normalized cross-
helicity and normalized residual energy observed by PSP are 0.96 and -0.07,
respectively, indicating that the slow solar wind is very Alfvénic. The
theoretical NI/slab results are very similar to PSP measurements, which is a
consequence of the highly magnetic field-aligned radial flow ensuring that PSP
can measure slab fluctuations and not 2D. Finally, we calculate the
theoretical 2D and slab turbulence pressure, finding that the theoretical slab
pressure is very similar to that observed by PSP.

magnetohydrodynamics (MHD) – plasmas – turbulence – Sun: corona – Sun:
heliosphere – solar wind

11affiliationtext: Center for Space Plasma and Aeronomic Research (CSPAR), and
Department of Space Science, University of Alabama in Huntsville, Huntsville,
AL 35899, USA22affiliationtext: National Institute for
Astrophysics—Astrophysical Observatory of Torino Via Osservatorio 20, I-10025
Pino Torinese, Italy

##  1 Introduction

One of the main purposes of Parker Solar Probe (PSP) and Solar Orbiter (SolO)
is to understand coronal heating, and the acceleration of the solar wind.
Turbulence is thought to be a central element in addressing these questions.
PSP and SolO allow us to study the radial evolution of turbulence in the inner
heliosphere (e.g., Telloni et al., 2021; Adhikari et al., 2022; Telloni et
al., 2022c). Adhikari et al. (2022) studied the evolution of 2D and slab
turbulence in similar types of slow solar wind observed by PSP (Kasper et al.,
2016; Bale et al., 2016) and SolO (Horbury et al., 2020; Owen et al., 2020),
finding that 2D turbulence is the dominant component. Telloni et al. (2021)
studied the evolution of turbulence in the same plasma parcel measured by PSP
and SolO, using a radial alignment between PSP and SolO from 0.1 au (PSP’s
position) to 1 au (SolO’s position). This alignment facilitates the study of
turbulence characteristics on the same plasma parcel with different locations.

When PSP was at 0.11 au on January 18, 2021 at 18:59 UT during encounter 7,
PSP entered the plane of sky (POS) measured by the SolO coronagraph Metis on
January 17, 2021 at 16:30 UT (Telloni et al., 2022c, interval # 1). Five days
later, when PSP was at 0.26 au on January 23, 2021 at 17:02 UT, PSP again
crossed the POS corresponding to the longitude observed by Metis (Telloni et
al., 2022c, interval # 2). Telloni et al. (2022c) presented the first
observational study that combined in situ PSP data and remote Metis/SolO
observations. Their unique configuration allowed Telloni et al to follow the
evolution of a plasma parcel through the extended solar corona ($3.5-6.3$ R⊙,
where R${}_{\odot}=6.95\times 10^{5}$ km) to the very inner heliosphere.
Telloni et al. (2022c) therefore provide plasma observations from the sub-
Alfvénic corona to the super-Alfvénic solar wind.

Using a Potential Field Source Surface (PFSS) extrapolation (e.g., Panasenco
et al., 2020), Telloni et al. (2022c) found that PSP sampled (interval # 1)
plasma coming from the equatorial extension of the southern polar coronal
hole, and (interval # 2) plasma coming from the low latitude of a northern
coronal hole. During interval # 1, PSP observed almost exclusively outwardly
propagating Alfvén waves ($\sigma_{c}\sim 1$, where $\sigma_{c}$ is the
normalized cross-helicity), whereas interval # 2 contains switchbacks (Bale et
al., 2019; Telloni et al., 2022c). In this letter, we model the solar wind
plasma (interval # 1) measured by the PSP – SolO quadrature from the sub- to
super-Alfvénic solar wind, and compare the predicted solar wind radial profile
with the combined PSP (in situ) – Metis (remote data set). Our model describes
the (coronal) heating of the slow solar wind near the equatorial region and
the acceleration of the solar wind. This differs from heating and acceleration
in open field region (Matthaeus et al., 1999; Dmitruk et al., 2001; Chandran &
Hollweg, 2009; Cranmer et al., 2013; Verdini et al., 2010; Zank et al., 2018a;
Cranmer & van Ballegooijen, 2010; Adhikari et al., 2020; Telloni et al.,
2022a), although the heating mechanism is thought to be the same for the fast
and slow solar wind flow (Zank et al., 2021).

We structure the letter as follows. Section 2 discusses a solar wind model
that incorporates nearly incompressible magnetohydrodynamic (NI MHD)
turbulence. Section 3 discusses the data analysis. Section 4 compares the
theoretical and observed results. Finally, Section 5 is the conclusions.

##  2 A turbulent solar wind model

In the letter, we use a superradial expansion turbulent driven solar model
(Telloni et al., 2022b) to study the coronal plasma in the slow solar wind
that the hot plasma emerges from the closed loop into the open field region by
interchange reconnection, after which it expands superradially. We consider a
steady flow in a one-dimensional, superradially expanding open flux tube of
cross-sectional area $A(r)(=r^{2}f(r)$, $f(r)$ is a superradial expansion
factor), and $A(r)$ is inversely proportional to the magnetic field strength
$B(r)$,

$B_{r}A(r)=B_{r}r^{2}f(r)=const.$ (1)

The super radial expansion term $f(r)$ is given by (Kopp & Holzer, 1976),

$f(r)=\frac{f_{m}\exp(\frac{r-r_{a}}{\sigma})+1-(f_{m}-1)\exp(\frac{R_{\odot}-r_{a}}{\sigma})}{\exp(\frac{r-r_{a}}{\sigma})+1},$
(2)

where $f_{m}=2$, $r_{a}=2$ R⊙, and $\sigma=0.8$ R⊙. The steady flow in the
superradially expanding tube can be described by the continuity, inviscid
momentum, and pressure equations,

$\frac{dn_{s}}{dr}=-\frac{2n_{s}}{r}-\frac{n_{s}}{U}\frac{dU}{dr}-\frac{n_{s}}{\sigma
f(r)}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1};$
(3) $\rho U\frac{dU}{dr}=-\frac{dP}{dr}-\frac{GM_{\odot}}{r^{2}}\rho;$ (4)
$\frac{dP}{dr}=-\frac{\gamma P}{U}\frac{dU}{dr}-\frac{2\gamma
P}{r}-\frac{\gamma P}{\sigma
f(r)}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1}+(\gamma-1)s_{1}\frac{S_{t}}{U},$
(5)

where $n_{s}(\rho)$ is the solar wind (mass) density, $U$ the solar wind
speed, $P$ the thermal pressure, $G$ the gravitational constant, $M_{\odot}$
the solar mass, $S_{t}$ the turbulent heating term, and $\gamma(=5/3)$ the
polytropic index. The parameter $s_{1}$ denotes the fraction of turbulence
energy used to heat the coronal plasma (protons). We use $s_{1}=0.6$, meaning
that the 60% of the turbulent energy heats the coronal/solar wind plasma
(Cranmer et al., 2009; Breech et al., 2009; Engelbrecht & Strauss, 2018;
Chhiber et al., 2019; Adhikari et al., 2021b; Telloni et al., 2022b).
Combining the above equations yields

$\frac{C_{s}^{2}}{U^{2}}\bigg{(}M_{s}^{2}-1\bigg{)}\frac{dU}{dr}=\frac{2\gamma
P}{\rho
Ur}-(\gamma-1)s_{1}\alpha\frac{S_{t}}{m_{p}n_{s}U^{2}}-\frac{GM_{\odot}}{Ur^{2}}+\frac{\gamma
P}{\sigma f(r)\rho
U}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1},$
(6)

where $M_{s}=U/C_{s}$ is the sonic Mach number, and $C_{s}^{2}=\gamma P/\rho$
is the square of the sound speed. Equation (6) possesses a critical point,
where $M_{s}=1$ and the right hand side (rhs) is zero simultaneously. We use
L’H$\hat{\text{o}}$pital’s rule to solve Equation (6) in the vicinity of the
critical point. This solar wind model (Equations (3) – (5)) includes only the
thermal force, and not the ponderomotive force (see, Holzer & Axford, 1970;
Leer et al., 1982; Withbroe, 1988; Fisk et al., 1999; Cranmer et al., 2013;
Verdini et al., 2010; Cranmer & van Ballegooijen, 2010) or wave pressure
(McKenzie et al., 1995). The turbulent heating term $S_{t}$ can be derived
from a von Kármán phenomenology, and is given by (Verdini et al., 2010;
Adhikari et al., 2015; Zank et al., 2018b)

$\begin{split}S_{t}&=\alpha
m_{p}n_{s}\bigg{[}2\frac{\langle{{z^{\infty}}^{+}}^{2}\rangle^{2}\langle{{z^{\infty}}^{-}}^{2}\rangle^{1/2}}{L_{\infty}^{+}}+2\frac{\langle{{z^{\infty}}^{-}}^{2}\rangle^{2}\langle{{z^{\infty}}^{+}}^{2}\rangle^{1/2}}{L_{\infty}^{-}}+2\frac{\langle{{z^{*}}^{+}}^{2}\rangle\langle{{z^{\infty}}^{+}}^{2}\rangle\langle{{z^{\infty}}^{-}}^{2}\rangle^{1/2}}{L_{\infty}^{+}}\bigg{]},\end{split}$
(7)

where $\langle{{z^{\infty}}^{\pm}}^{2}\rangle$ are the 2D outward and inward
Elsässer energies, $L^{\pm}_{\infty}$ the corresponding energy weighted
correlation lengths, and $\langle{{z^{*}}^{+}}^{2}\rangle$ is the NI/slab
energy in forward propagating modes. The parameter $\alpha(=0.01)$ is a von
Kármán-Taylor constant, and $m_{p}$ the proton mass.

The 1D steady-state transport equations for the majority 2D turbulence,
including the superradial expansion factor are given by (Zank et al., 2017;
Adhikari et al., 2020; Telloni et al., 2022b)

$\begin{split}U\frac{d\langle{z^{\infty}}^{\pm
2}\rangle}{dr}&=-\bigg{(}\frac{\langle{z^{\infty}}^{\pm
2}\rangle}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{D}^{\infty}\bigg{)}\frac{dU}{dr}-\frac{2U}{r}\bigg{(}\frac{\langle{z^{\infty}}^{\pm
2}\rangle}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{D}^{\infty}\bigg{)}-\frac{U}{\sigma
f(r)}\bigg{(}\frac{\langle{z^{\infty}}^{\pm 2}\rangle}{2}\\\
&+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{D}^{\infty}\bigg{)}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1}-2\alpha\frac{\langle{z^{\infty}}^{\pm
2}\rangle^{2}\langle{z^{\infty}}^{\mp
2}\rangle^{1/2}}{L^{\pm}_{\infty}}+S^{\langle{z^{\infty}}^{\pm
2}\rangle};\end{split}$ (8)
$\begin{split}U\frac{dE_{D}^{\infty}}{dr}&=-\bigg{(}\frac{E_{D}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{T}^{\infty}\bigg{)}\frac{dU}{dr}-\frac{2U}{r}\bigg{(}\frac{E_{D}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{T}^{\infty}\bigg{)}-\frac{U}{\sigma
f(r)}\bigg{(}\frac{E_{D}^{\infty}}{2}\\\
&+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{T}^{\infty}\bigg{)}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1}-\alpha
E_{D}^{\infty}\bigg{(}\frac{\langle{z^{\infty}}^{+2}\rangle^{1/2}\langle{z^{\infty}}^{-2}\rangle}{L_{\infty}^{-}}+\frac{\langle{z^{\infty}}^{-2}\rangle^{1/2}\langle{z^{\infty}}^{+2}\rangle}{L_{\infty}^{+}}\bigg{)}\\\
&+S^{E_{D}^{\infty}};\end{split}$ (9)
$\begin{split}U\frac{dL_{\infty}^{\pm}}{dr}&=-\bigg{(}\frac{L_{\infty}^{\pm}}{2}+\bigg{(}a-\frac{1}{4}\bigg{)}L_{D}^{\infty}\bigg{)}\frac{dU}{dr}-\frac{2U}{r}\bigg{(}\frac{L_{\infty}^{\pm}}{2}+\bigg{(}a-\frac{1}{4}\bigg{)}L_{D}^{\infty}\bigg{)}-\frac{U}{\sigma
f(r)}\bigg{(}\frac{L_{\infty}^{\pm}}{2}+\bigg{(}a-\frac{1}{4}\bigg{)}L_{D}^{\infty}\bigg{)}\\\
&\times\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1};\end{split}$
(10)
$\begin{split}U\frac{dL_{D}^{\infty}}{dr}&=-\bigg{(}\frac{L_{D}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}(L_{\infty}^{+}+L_{\infty}^{-})\bigg{)}\frac{dU}{dr}-\frac{2U}{r}\bigg{(}\frac{L_{D}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}(L_{\infty}^{+}+L_{\infty}^{-})\bigg{)}-\frac{U}{\sigma
f(r)}\\\
&\times\bigg{(}\frac{L_{D}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}(L_{\infty}^{+}+L_{\infty}^{-})\bigg{)}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1},\end{split}$
(11)

where $L_{D}^{\infty}$ is the energy weighted correlation length for the 2D
residual energy $E_{D}^{\infty}$, and $E_{T}^{\infty}$ is the 2D total
turbulence energy. The term “S” refers to the turbulent shear source for the
2D outward and inward Elsässer energies, and the residual energy. Equation (8)
can be written in terms of $E_{T}^{\infty}=(\langle
z^{\infty+2}\rangle+\langle z^{\infty-2}\rangle)/2$, and the cross-helicity
$E_{C}^{\infty}=(\langle z^{\infty+2}\rangle-\langle z^{\infty-2}\rangle)/2$
as

$\begin{split}&U\frac{dE_{T}^{\infty}}{dr}=-\bigg{(}\frac{E_{T}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{D}^{\infty}\bigg{)}\frac{dU}{dr}-\frac{2U}{r}\bigg{(}\frac{E_{T}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{D}^{\infty}\bigg{)}-\frac{U}{\sigma
f(r)}\\\
&\times\bigg{(}\frac{E_{T}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{D}^{\infty}\bigg{)}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp\big{(}\frac{r-r_{a}}{\sigma}\big{)}+1}-\alpha\frac{|E_{T}^{\infty}+E_{C}^{\infty}|^{2}|E_{T}^{\infty}-E_{C}^{\infty}|^{1/2}}{L_{\infty}^{+}}\\\
&-\alpha\frac{|E_{T}^{\infty}-E_{C}^{\infty}|^{2}|E_{T}^{\infty}+E_{C}^{\infty}|^{1/2}}{L_{\infty}^{+}}+\frac{S^{\langle
z^{\infty+2}\rangle}+S^{\langle z^{\infty-2}\rangle}}{2};\end{split}$ (12)
$\begin{split}&U\frac{dE_{C}^{\infty}}{dr}=-\frac{E_{C}^{\infty}}{2}\frac{dU}{dr}-\frac{U}{r}E_{C}^{\infty}-\frac{U}{2\sigma
f(r)}E_{C}^{\infty}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp\big{(}\frac{r-r_{a}}{\sigma}\big{)}+1}\\\
&-\alpha\frac{|E_{T}^{\infty}+E_{C}^{\infty}|^{2}|E_{T}^{\infty}-E_{C}^{\infty}|^{1/2}}{L_{\infty}^{+}}+\alpha\frac{|E_{T}^{\infty}-E_{C}^{\infty}|^{2}|E_{T}^{\infty}+E_{C}^{\infty}|^{1/2}}{L_{\infty}^{+}}+\frac{S^{\langle
z^{\infty+2}\rangle}-S^{\langle z^{\infty-2}\rangle}}{2}.\end{split}$ (13)

Equation (12) can be written in the conservation form (Wang et al., 2022),

$\begin{split}&\frac{1}{r^{2}f(r)}\frac{d}{dr}\bigg{[}r^{2}f(r)U(E^{\infty}_{w}+P^{\infty}_{w})\bigg{]}=U\frac{dP_{w}^{\infty}}{dr}+\rho\bigg{[}-\alpha\frac{|E_{T}^{\infty}+E_{C}^{\infty}|^{2}|E_{T}^{\infty}-E_{C}^{\infty}|^{1/2}}{L_{\infty}^{+}}\\\
&-\alpha\frac{|E_{T}^{\infty}-E_{C}^{\infty}|^{2}|E_{T}^{\infty}+E_{C}^{\infty}|^{1/2}}{L_{\infty}^{+}}+\frac{S^{\langle
z^{\infty+2}\rangle}+S^{\langle z^{\infty-2}\rangle}}{2}\bigg{]},\end{split}$
(14)

where $E_{w}^{\infty}$ = $\rho E_{T}^{\infty}/2$ is the 2D turbulence energy
density, and

$P_{w}^{\infty}=\frac{\rho}{2}\bigg{[}\frac{E_{T}^{\infty}}{2}+\bigg{(}2a-\frac{1}{2}\bigg{)}E_{D}^{\infty}\bigg{]}=\frac{E_{w}^{\infty}}{2}\bigg{[}1+2\bigg{(}2a-\frac{1}{2}\bigg{)}\sigma_{D}^{\infty}\bigg{]}$
(15)

is the 2D turbulence pressure. In the absence of the nonlinear term and
turbulence source terms, Equation (14) resemble the well-known Wentze-Kramers-
Brillouin (WKB) form. The terms in the square bracket on the left hand side
(lhs) of Equation (14) express the energy density flux vector, which describes
the amount of turbulence energy per unit time per unit area in a direction
perpendicular to the velocity (Landau & Lifshitz, 1987). Similarly, on the
lhs, the first term in […] describes the energy transmitted through the unit
surface area per unit time, and the second term describes the work done by the
turbulence pressure on the plasma in the surface. The first term on the right
hand side describes the rate of turbulence pressure gradient on the background
plasma flow.

The 1D steady-state transport equations for the energy in NI/slab forward
propagating modes and the corresponding energy weighted correlation length are
(Zank et al., 2017; Adhikari et al., 2020; Telloni et al., 2022b),

$\begin{split}(U-V_{A})\frac{d\langle{z^{*}}^{+2}\rangle}{dr}&=-\frac{1}{2}\frac{dU}{dr}\langle{z^{*}}^{+2}\rangle+(2b-1)\frac{U}{r}\langle{z^{*}}^{+2}\rangle_{+}\frac{V_{A}}{2\rho}\frac{d\rho}{dr}\langle{z^{*}}^{+2}\rangle+\frac{1}{2}(2b-1)\frac{U\langle{z^{*}}^{+2}\rangle}{\sigma
f(r)}\\\
&\times\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1}-2\alpha\frac{\langle{z^{*}}^{+2}\rangle\langle{z^{\infty}}^{-2}\rangle^{1/2}}{\lambda_{\infty}^{+}}+S^{\langle{z^{*}}^{+2}\rangle};\end{split}$
(16)
$\begin{split}(U-V_{A})\frac{dL_{*}^{+}}{dr}&=-\frac{1}{2}\frac{dU}{dr}L_{*}^{+}+(2b-1)\frac{U}{r}L_{*}^{+}+\frac{V_{A}}{2\rho}\frac{d\rho}{dr}L_{*}^{+}+\frac{1}{2}(2b-1)\frac{UL_{*}^{+}}{\sigma
f(r)}\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1}.\end{split}$
(17)

The parameter
$V_{A}(=({B_{0}}/{\sqrt{\mu_{0}\rho}})({r_{0}}/{r})^{2}(1/f(r))$, where
$B_{0}$ is the magnetic field at a reference point $r_{0}$, and $\mu_{0}$ is
the magnetic permeability) is the large-scale Alfvén velocity. The parameter
$S^{\langle{z^{*}}^{+2}\rangle}$ denotes the turbulent shear source for the
energy in NI/slab forward propagating modes. We use $b=0.26$ (see Zank et al.,
2012, 2017, for further discussion). Equation (16) can be written in terms of
the slab total turbulent energy $E_{T}^{*}(=\langle z^{*+2}\rangle/2\equiv
E_{C}^{*}$, $\langle z^{*-2}\rangle=0$) as

$\begin{split}&(U-V_{A})\frac{dE_{T}^{*}}{dr}=-\frac{1}{2}\frac{dU}{dr}E_{T}^{*}+(2b-1)\frac{U}{r}E_{T}^{*}+\frac{V_{A}}{2\rho}\frac{d\rho}{dr}E_{T}^{*}+\frac{1}{2}(2b-1)\frac{U}{\sigma
f(r)}E_{T}^{*}\\\
&\times\exp\bigg{(}\frac{r-r_{a}}{\sigma}\bigg{)}\frac{f_{m}-f(r)}{\exp(\frac{r-r_{a}}{\sigma})+1}-2\alpha\frac{E_{T}^{*}|E_{T}^{\infty}+E_{C}^{\infty}||E_{T}^{\infty}-E_{C}^{\infty}|^{1/2}}{L_{\infty}^{+}}+\frac{S^{\langle
z^{*+2}\rangle}}{2}.\end{split}$ (18)

Equation (18) can also be written in the conservation form (Wang et al.,
2022),

$\begin{split}&\frac{1}{r^{2}f(r)}\frac{d}{dr}\bigg{[}r^{2}f(r)\big{(}(U-V_{A})E_{w}^{*}+UP_{w}^{*}\big{)}\bigg{]}=U\frac{dP_{w}^{*}}{dr}+2E_{w}^{*}\bigg{(}4b\frac{u}{r}+2b\frac{U}{f(r)}\frac{df(r)}{dr}\bigg{)}\\\
&+\frac{\rho}{2}\bigg{[}-2\alpha\frac{E_{T}^{*}|E_{T}^{\infty}+E_{C}^{\infty}||E_{T}^{\infty}-E_{C}^{\infty}|^{1/2}}{L_{\infty}^{+}}+\frac{S^{\langle
z^{*+2}\rangle}}{2}\bigg{]},\end{split}$ (19)

where $E_{w}^{*}=\rho E_{T}^{*}/2$ is the slab turbulence energy density, and
$P_{w}^{*}=E_{w}^{*}/2$ is the slab turbulence pressure. Equation (19) also
resembles the WKR form in the absence of the mixing term, dissipation terms,
and the turbulence source term.

Equations (8)–(13) and (16) – (18) are a set of turbulence transport equations
describing the evolution of turbulence in the highly field-aligned flows (see
Adhikari et al., 2020, for a detailed discussion).

Similar to Telloni et al. (2022b), we use two forms of the turbulent shear
source, i) in region between the sonic surface and Alfvén surface, where the
sound speed $C_{S}$ is assumed to be the characteristic speed, and ii) beyond
the Alfvén surface, where the Alfvén speed is considered to be the
characteristic speed. The shear source of turbulence in the region between the
sonic and Alfvén surfaces can be written in the form (Telloni et al., 2022b)

$\begin{split}S^{s}_{\langle{z^{\infty,*}}^{\pm
2}\rangle}=C_{\infty,*}^{s+}\frac{r_{0}^{s}|\Delta U|C^{2}_{s}}{r^{2}}=\gamma
C_{\infty,*}^{s+}\frac{r_{0}^{s}|U-U_{0}^{s^{\prime}}|P}{\rho r^{2}};\\\ \quad
S^{s}_{\langle{z^{\infty}}^{-2}\rangle}=C_{\infty}^{s-}\frac{r_{0}^{s}|\Delta
U|C^{2}_{s}}{r^{2}}=\gamma
C_{\infty}^{s-}\frac{r_{0}^{s}|U-U_{0}^{s^{\prime}}|P}{\rho r^{2}};\\\ \quad
S^{s}_{E_{D}^{\infty}}=C_{\infty}^{sE_{D}}\frac{r_{0}^{s}|\Delta
U|C^{2}_{s}}{r^{2}}=\gamma
C_{\infty}^{sE_{D}}\frac{r_{0}^{s}|U-U_{0}^{s^{\prime}}|P}{\rho
r^{2}},\end{split}$ (20)

where $U_{0}^{s^{\prime}}$ is the solar wind speed at $r_{0}^{s}$, a position
above the sonic surface, and $C^{s\pm,E_{D}}_{\infty,*}$ denotes the strength
of the shear source of turbulence. Here we use $\Delta
U=|U-U_{0}^{s^{\prime}}|$, and $C_{s}^{2}=\gamma P/\rho$, which results in the
shear source of turbulence depending on distance $r$, solar wind speed $U$,
the thermal pressure $P$, and the proton mass density $\rho$. We use
$r_{0}^{s}=5.16$ R⊙, and $U_{0}^{s^{\prime}}=173.5$ kms-1.

Similarly, the turbulent shear source above the Alfvén surface is,

$\begin{split}S^{A}_{\langle{z^{\infty,*}}^{\pm
2}\rangle}=C_{\infty,*}^{a+}\frac{r_{0}^{a}|\Delta
U|V^{2}_{A}}{r^{2}}=C_{\infty,*}^{a+}\frac{r_{0}^{a}|U-U_{0}^{a^{\prime}}|V^{2}_{A}}{r^{2}};\\\
\quad
S^{A}_{\langle{z^{\infty}}^{-2}\rangle}=C_{\infty}^{a-}\frac{r_{0}^{a}|\Delta
U|V^{2}_{A0}}{r^{2}}=C_{\infty}^{a-}\frac{r_{0}^{a}|U-U_{0}^{a^{\prime}}|V^{2}_{A}}{r^{2}};\\\
\quad S^{A}_{E_{D}^{\infty}}=C_{\infty}^{aE_{D}}\frac{r_{0}^{a}|\Delta
U|V^{2}_{A0}}{r^{2}}=C_{\infty}^{aE_{D}}\frac{r_{0}^{a}|U-U_{0}^{a^{\prime}}|V^{2}_{A}}{r^{2}},\end{split}$
(21)

where we use $\Delta U=|U-U_{0}^{a^{\prime}}|$. The parameter
$U_{0}^{a^{\prime}}$ is the solar wind speed at $r_{0}^{a}$, a position above
the Alfvén surface, and $C^{a\pm,E_{D}}_{\infty,*}$ denote the strength of the
shear source of turbulence. We use $r_{0}^{a}=9.22$ R⊙, and
$U_{0}^{a^{\prime}}=222.87$ kms-1.

##  3 Data analysis

We calculate the transverse turbulence energy and the transverse correlation
length using a method developed by Adhikari et al. (2022). A fluctuating
vector ${\bf a}(=a_{r}\hat{r}+a_{t}\hat{t}+a_{n}\hat{n}$, where $a_{r}$,
$a_{t}$, $a_{n}$ are the $R$, $T$, and $N$ components of a vector ${\bf a}$)
can be decomposed into parallel and perpendicular vectors relative to the mean
magnetic field ${\bf B}$ as

${\bf a}={\bf a}_{||}+{\bf a}_{\perp}=a_{||}\hat{b}+{\bf a}_{\perp},$

where $\hat{b}={\bf B}/|{\bf B}|$ is the unit vector, $|{\bf B}|$ is the
magnitude of the mean magnetic field ${\bf
B}(=B_{R}\hat{r}+B_{T}\hat{t}+B_{N}\hat{n}$, where $B_{R}$, $B_{T}$, and
$B_{N}$ denote the $R$, $T$, and $N$ components of the mean magnetic field),
and ${\bf a}_{\perp}$ and ${\bf a}_{||}(=a_{||}\hat{b})$ are the perpendicular
and parallel vectors, respectively. The parallel component $a_{||}$ can be
written as

$a_{||}={\bf a}\cdot\hat{b},$ (22)

and the perpendicular vector ${\bf a_{\perp}}$,

${\bf a}_{\perp}=-\hat{b}\times(\hat{b}\times{\bf a})=\frac{({\bf B}\times{\bf
a})\times{\bf B}}{|{\bf B}|^{2}},$ (23)

can be derived as (Adhikari et al., 2022),

$\begin{split}{\bf
a}_{\perp}&=\frac{C_{T}B_{N}-C_{N}B_{T}}{B_{R}^{2}+B_{T}^{2}+B_{N}^{2}}\hat{r}+\frac{C_{N}B_{R}-C_{R}B_{N}}{B_{R}^{2}+B_{T}^{2}+B_{N}^{2}}\hat{t}+\frac{C_{R}B_{T}-C_{T}B_{R}}{B_{R}^{2}+B_{T}^{2}+B_{N}^{2}}\hat{n},\end{split}$
(24)

where $C_{R}=B_{T}A_{N}-B_{N}A_{T}$, $C_{T}=B_{N}A_{R}-B_{R}A_{N}$, and
$C_{N}=B_{R}A_{T}-B_{T}A_{R}$. Using Equation (18), we calculate the
transverse Elsässer energies, fluctuating magnetic and kinetic energies,
normalized cross-helicity and residual energy, and the corresponding
transverse correlation lengths.

##  4 Results

In this section, we compare the theoretical and observed results from the
extended solar corona to the very inner heliosphere. We select SPAN ion plasma
data (Kasper et al., 2016), and 1 minute resolution FIELDS data (Bale et al.,
2016) in a time interval 18:40 – 20:40 UT on January 2021 (interval #1) during
E7. We apply a boxcar method to the SPAN ion data to remove large spikes, and
smooth the data using 1 minute long intervals, and then merge the plasma data
with the magnetometer data. We calculate the transverse turbulence energy and
the transverse correlation length, and the solar wind parameters for interval
# 1. The Metis/SolO plasma data (solar wind speed and density) are obtained
from Telloni et al. (2022c). We use Runge-kutta fourth order method to solve
the solar wind + NI MHD turbulence transport equations from 3.3 – 30 R⊙. Table
1 shows the boundary conditions for the turbulence quantities and the solar
wind parameters at 3.3 R⊙. Table 2 shows the values of the strength of the
turbulent shear source. These boundary conditions are chosen so the
theoretical results are close to the observations. We note that the
theoretical results are similar with or without the turbulent shear source. In
this letter, we include the turbulent shear source, and compare the
theoretical results with the observed results.

Parameters | Values
---|---
$\langle{{z^{\infty}}^{\pm}}^{2}\rangle$ (km2s-2) | $10^{5}$
$E_{D}^{\infty}$ (km2s-2) | 2000
$L_{\infty}^{\pm}$ (km3s-2) | $3.03\times 10^{9}$
$L_{D}^{\infty}$ (km3s-2) | $1.6\times 10^{8}$
$\langle{{z^{*}}^{+}}^{2}\rangle$ (km2s-2) | 6000
$L_{*}^{+}$ (km3s-2) | $9.1\times 10^{7}$
$U$ (kms-1) | 45.13
$n$ (cm-3) | $8\times 10^{5}$
$T$ (K) | $7\times 10^{5}$
Table 1: Boundary values at 3.3 R⊙ for the turbulent quantities and the solar wind parameters.  Parameters | Values
---|---
$C_{\infty}^{s\pm}$ | 0.1
$C_{\infty}^{sE_{D}}$ | -0.1
$C_{*}^{s+}$ | 0.1
$C_{\infty}^{a\pm}$ | 0.1
$C_{\infty}^{aE_{D}}$ | -0.1
$C_{*}^{a+}$ | 0.1
Table 2: Assumed strengths for the shear driven quasi-2D and slab turbulence
in the region between the sonic and Alfvén surfaces, and above the Alfvén
surface.

Figure 1: Comparison between the theoretical solar wind speed and Alfvén
velocity (left), and the solar wind density (right) with the observed results
of the plasma parcel measured by Metis/SolO (obtained from Telloni et al.,
2022c) and PSP. The solid and dashed curves are the theoretical results. The
open red triangles are the observed speed and density from SolO/Metis, and the
cyan full triangles are the observed speed and density, and the blue full
triangle is the observed Alfvén speed measured by PSP during E7.

During the PSP – SolO Eastern-limb quadrature of mid-January, the same solar
wind plasma stream was observed simultaneously in both the extended corona and
the very inner heliosphere. The Metis coronagraph (Antonucci et al., 2020) on
board SolO imaged the $3.5$ to $6.3$ R⊙ altitude range remotely, which
corresponds to the coronal source region of the plasma flow impinging on PSP
at $0.11$ au (see Telloni et al., 2021, 2022c, for a detailed description of
the orbital geometry and connectivity during quadrature). The Metis instrument
is designed to observe the solar corona both in polarized brightness (pB) and
H I Ly$\alpha$ ultraviolet (UV) emission. This allows for the study of complex
coronal dynamics and structures. pB measurements are used to infer the
electron density by exploiting the inversion technique developed by van de
Hulst (1950). The UV light emitted by neutral hydrogen atoms is used as a
proxy to estimate the outflow velocity of the proton component of the solar
wind. The H I Ly$\alpha$ line emission at $121.6$ nm is mainly due to resonant
scattering processes of chromospheric radiation by coronal hydrogen atoms. It
follows that an outward motion of the coronal plasma causes a reduction in
intensity of the scattered Ly$\alpha$ line, since the incident radiation
profile appears to be Doppler-shifted in the rest frame of the scattering
atoms. This effect, known as Doppler dimming (Noci et al., 1987), can
therefore be used to infer the expansion velocity of the coronal plasma. Based
on a three-dimensional model of the large-scale solar corona (involving prior
knowledge of electron density and temperature, kinetic temperature of
scattering atoms, helium abundance, and temperature anisotropy, among others),
the speed of the coronal flows is estimated by comparing observed and
synthesized H I Ly$\alpha$ line intensity (see § $11.2$ in Antonucci et al.,
2020, for an exhaustive review of the Doppler dimming technique). Figure 1
displays the modeled plasma $U$ and Alfvén $V_{A}$ speed, and proton number
density, as a function of the altitude above the Sun. These profiles are
compared with observations from SolO/Metis (open red triangles) and PSP (full
triangles). Note that, accounting for a fully ionized plasma with $2.5\%$
helium (according to Moses et al., 2020), the electron density estimates
provided by Metis were multiplied by $0.95$ to obtain the corresponding proton
number density. The agreement between theory and observations is striking. The
Metis and PSP measurements obviously are unrelated. One is obtained remotely
and the other measured locally in situ. Evidently the model presented above
reproduces very well the joint observations of the extended corona and the
very inner heliosphere and the dynamic evolution of the solar wind plasma.

Figure 2: Radial evolution of the total (2D+slab), 2D, and slab turbulence
energy from the extended coronal region to the very inner heliosphere. Panels
a), b), c), d), and e) describe the outward Elsässer energy, fluctuating
magnetic energy, fluctuating kinetic energy, normalized cross-helicity, and
normalized residual energy, respectively. The solid curve represents the 2D
component, the dashed curve the slab component, and the dashed-dotted-dashed
curve the 2D+slab component. The cyan triangle denotes the corresponding PSP
measurement.

Zank et al. (2018a) argued that the magnetic carpet on the photosphere
continuously pumps out the 2D structures along with a minority population of
Alfvén waves (i.e., slab turbulence) above the photosphere. The 2D structures
advect through the chromosphere, across the transition region, and into the
solar corona. Since advected 2D structures do not reflect at the transition
region, there should be no abrupt and significant decrease in the 2D flux at
the transition region, as is expected of outward propagating Alfvén waves
(Zank et al., 2018a). Zank et al. (2018a) argued that the mechanism for
heating the solar corona is the same for fast and slow solar wind flow. Recall
that the slow solar wind studied in this letter emerges near the equatorial
region, and may be due to the liberation of hot loop material into open field
regions by interchange reconnection above the photosphere (Fisk, 2003). The
slow solar wind in the equatorial region accelerates rapidly within $4$ R⊙,
similar to the fast solar wind in open field regions (Adhikari et al., 2020;
Telloni et al., 2022b). Telloni et al. (2007) found observationally that the
solar wind accelerates rapidly within 2 – 4 R⊙, consistent with Figure 1
(left). The Alfvén velocity increases initially to a peak value of $\sim
4\times 10^{2}$ kms-1, decreases gradually, forming the Alfvén surface at
$\sim 9.22$ R⊙, and is similar to the in situ PSP-observed Alfvén speed (blue
full triangle).

Figure 2 presents the evolution of the basic turbulence quantities. As we see
in Figure 2a, the 2D outward Elsässer energy $\langle z^{\infty+2}\rangle$
(solid curve) is dissipated rapidly in the extended coronal region, after
which it decreases more gradually. By contrast, the minority slab energy in
outward propagating modes $\langle z^{*+2}\rangle$ (dashed curve) increases in
the extended solar corona, and decreases beyond the Alfvén surface. The
initial increase of $\langle z^{*+2}\rangle$ is due to the presence of the
solar wind density gradient term, which acts as source term. The heating of
the solar corona by 2D turbulence is different from that due to counter-
propagating Alfvén waves (Matthaeus et al., 1999; Verdini et al., 2010), which
assumes that a large outwardly propagating Alfvénic flux produces reflected
Alfvén waves that interact nonlinearly to produce 2D modes that dissipates.
This physics is in fact incorporated in the slab turbulence equations that
comprised NI MHD. The theoretical $\langle z^{*+2}\rangle$ is similar to PSP
measurements, and smaller than the theoretical $\langle z^{\infty+2}\rangle$.
This can be understood by recognizing that the angle between the mean magnetic
field and solar wind speed in the interval 18:40 – 20:40 UT is measured to be
$\theta_{UB}\sim 165^{\circ}$, indicating that PSP measures the slab
component, but not the 2D component. The theoretical 2D + slab Elsässer energy
(dashed-dotted-dashed curve) decreases gradually with increasing distance.

Figure 2b shows that the theoretical 2D fluctuating magnetic energy (solid
curve) is larger than the theoretical slab component (dashed curve), and both
decrease gradually with increasing distance. The theoretical $\langle
B^{*2}\rangle$ is consistent with the PSP-observed $\langle B^{2}\rangle$ for
the reason given above. Similarly, the theoretical NI/slab fluctuating kinetic
energy $\langle u^{*2}\rangle$ is consistent with that measured by PSP. The
theoretical $\langle u^{*2}\rangle$ increases initially to a peak value, and
then decreases slowly. The theoretical $\langle u^{\infty 2}\rangle$ decreases
rapidly, indicating that it is strongly dissipated.

In this model, the slab energy in backward propagating modes $\langle
z^{*-2}\rangle$ is 0, leading to a value of 1 the slab normalized cross-
helicity $\sigma_{c}^{*}$ (dashed curve in Figure 2c), and is close to the
observations. Note that the observed solar wind speed between 3.5 – 6.3 R⊙
ranges from $96-201$ kms-1, and that at 23.4 R⊙ is 219.34 kms-1. This is an
Alfvénic slow solar wind with a high cross-helicity value ($\sigma_{c}\sim
1$). As shown in Figure 2a, the 2D outward and inward Elsässer energies are
equal, leading to a zero 2D normalized cross-helicity (solid curve in Figure
2d). The total normalized cross-helicity (dashed-dotted-dashed curve)
increases gradually until the Alfvén surface, and then remains approximately
constant. Due to the assumption that the slab turbulent kinetic and magnetic
energies are equipartitioned, the slab normalized residual energy (dashed
curve) is zero as a function of distance, and is close to the observed value.
The 2D normalized residual energy (solid curve) decreases rapidly initially,
and then tends to -1, i.e., the dominant 2D component is almost entirely
composed of magnetic fluctuations.

Figure 3: Radial evolution of the 2D (solid curve) and slab (dashed curve)
correlation lengths corresponding to the outward Elsässer energy (a),
fluctuating magnetic energy (b), and fluctuating kinetic energy (c). The cyan
triangle denotes the PSP measurements.

The correlation length is an important parameter because it determines the
turbulence heating rate (Adhikari et al., 2021a). Similar to the 2D outward
and inward Elsässer energies, the corresponding 2D correlation lengths are
equal (solid black and dashed curves in Figure 3a). The theoretical NI/slab
correlation length for the energy in outward propagating modes (dashed-dotted-
dashed curve) increases with distance, and is similar to PSP observations. The
2D correlation length of the magnetic field and velocity fluctuations (solid
curves in Figures 3b and 3c, respectively) increases much more rapidly than
the corresponding slab correlation lengths, the latter of which are in accord
with PSP measurements that can measure only the slab fluctuations as discussed
above.

Figure 4: The 2D (solid curve) and slab (dashed curve) turbulence pressure as
a function of distance. The cyan triangle is the PSP-observed turbulence
pressure.

Figure 4 displays the turbulence pressure as a function of heliocentric
distance. The solid curve denotes the 2D turbulence pressure and the dashed
curve the slab turbulence pressure. Initially, the 2D turbulence pressure is
larger than the slab turbulence pressure, however, the prior decreases more
rapidly than the latter. The rapid decrease of the 2D turbulence pressure is
due to the dominance of the 2D turbulent magnetic energy (see Figure 2e). The
theoretical slab turbulence pressure is similar to that observed by PSP at
23.2 R⊙.

From a turbulence perspective, the dissipation of turbulence energy heats the
coronal/solar wind plasma. The solar wind proton temperature is assumed to be
$7\times 10^{5}$ K at 3.3 R⊙, which increases to a peak value of $\sim
1.2\times 10^{6}$ K, and then decreases gradually but non-adiabatically in the
expanding supersonic solar wind (Figure 5). The theoretical and PSP-observed
proton temperature at $\sim 23$ R⊙ are very similar.

Figure 5: The solar wind proton temperature as a function of distance. The
cyan triangle is the PSP-observed proton temperature.

##  5 Discussion and Conclusions

Telloni et al. (2022c) studied the PSP – SolO quadrature, combining remote
imaging by the SolO Metis coronagraph, and in situ plasma data by PSP. We
present a first comparison of the theoretical model and the joint PSP – SolO
observations. We solved a solar wind + NI MHD turbulence model (Adhikari et
al., 2020; Telloni et al., 2022b) from 3.3 – 30 R⊙, and compared i) the
theoretical solar wind speed and density with the corresponding remote
observations from $3.5-6.3$ R⊙ (Telloni et al., 2022c) and in situ PSP
measurements at 23.2 R⊙ or 0.11 au, and ii) the theoretical turbulence energy
and correlation length with PSP measurements. We used the PSP magnetometer and
SPAN ion plasma data from 18:40 – 20:40 UT on January 18, 2021 (interval #1 in
Telloni et al. (2022c)), and calculated the transverse energy in forward
propagating modes, fluctuating magnetic energy, fluctuating kinetic energy,
normalized residual energy and cross-helicity, and the corresponding
correlation lengths using the method developed by Adhikari et al. (2022). We
found that the $\theta_{UB}$, angle between the mean solar wind flow and mean
magnetic field for the selected interval is $165^{\circ}$, indicating that PSP
observed primarily the slab turbulence component in this highly field-aligned
flow. We found very good agreement between theory and observations. We
summarize our findings as follows.

  1. 1.

The theoretical solar wind speed and density are consistent with those
measured by SolO/Metis from $3.5-6.3$ R⊙ and PSP at 23.2 R⊙. The theoretical
and observed solar wind speed increases rapidly within 3.3 – 4 R⊙ ranging from
96 – 201 kms-1, becoming supersonic at $\sim 5.16$ R⊙. Thereafter, the
theoretical solar wind speed increases gradually with distance, and is
consistent with the PSP speed of 219.34 kms-1 measured at 23.2 R⊙. PSP and
Metis/SolO measured a slow solar wind stream emerging from the southern
coronal hole near the equatorial region (Telloni et al., 2022c). The
theoretical Alfvén velocity increases initially to a peak value $\sim 4\times
10^{2}$ kms-1, and then decreases gradually to be consistent with that
measured by PSP. In this model, the Alfvén surface is located at $\sim 9.22$
R⊙.

  2. 2.

The theoretical 2D outward Elsässer energy and fluctuating magnetic energy are
larger than the corresponding slab components, the latter being close to the
corresponding PSP-observed results. Similarly, the theoretical slab
fluctuating kinetic energy is also consistent with the PSP-observed kinetic
energy at 23.2 R⊙.

  3. 3.

The theoretical slab normalized cross-helicity is close to the PSP-observed
cross-helicity ($\sigma_{c}=0.96$), indicating that PSP observed highly
Alfvénic slow solar wind in the inner heliosphere (e.g., D’Amicis et al.,
2019). The theoretical normalized slab residual energy is similar to the PSP-
observed residual energy ($\sigma_{D}\sim-0.07$).

  4. 4.

The theoretical 2D correlation lengths corresponding to outward Elsässer
energy, and magnetic field and velocity fluctuations exceed the theoretical
slab correlation lengths, and the slab correlation lengths are consistent with
those observed by PSP.

  5. 5.

We derived the two sets of equations in a conservation form, including the
super-radial expansion, from the 2D + NI/slab turbulence transport equations
that were derived for the unidirectional Alfvén waves (Adhikari et al., 2020;
Telloni et al., 2022b). Both sets of equations resemble the WKB form in the
absence of dissipation term, mixing term, and the turbulence source (Wang et
al., 2022). We calculated the theoretical 2D and slab turbulence pressures,
and both decrease with increasing distance. The theoretical slab turbulence
pressure is similar to that observed by PSP at 23.2 R⊙.

  6. 6.

The proton temperature is assumed to be $7\times 10^{5}$ K at 3.3 R⊙,
increases to a maximum value of $\sim 1.2\times 10^{6}$ K, and then decreases
gradually with the expanding solar wind. The PSP-measured temperature and the
predicted temperature at 23.2 R⊙ are very similar.

Our theoretical results successfully describe the slow solar wind stream
measured by Metis/SolO and PSP from the extended corona to the very inner
heliosphere. Future combined studies using combined PSP, SolO, and BeliColombo
measurement will be of great value.

We acknowledge the partial support of a Parker Solar Probe contract SV4-84017,
an NSF EPSCoR RII-Track-1 cooperative agreement OIA-1655280, and NASA awards
80NSSC20K1783 and 80NSSC21K1319. The SWEAP Investigation and this study are
supported by the PSP mission under NASA contract NNN06AA01C.

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