# An Anisotropic Density Turbulence Model from the Sun to 1 au Derived From
Radio Observations

Eduard P. Kontar School of Physics & Astronomy, University of Glasgow,
Glasgow, G12 8QQ, UK  A. Gordon Emslie Department of Physics & Astronomy,
Western Kentucky University, Bowling Green, KY 42101, USA  Daniel L. Clarkson
School of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK
Xingyao Chen School of Physics & Astronomy, University of Glasgow, Glasgow,
G12 8QQ, UK  Nicolina Chrysaphi Sorbonne Université, École Polytechnique,
Institut Polytechnique de Paris, CNRS, Laboratoire de Physique des Plasmas
(LPP), 4 Place Jussieu, 75005 Paris, France  School of Physics & Astronomy,
University of Glasgow, Glasgow, G12 8QQ, UK  Francesco Azzollini School of
Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK  Natasha L.S.
Jeffrey Department of Mathematics, Northumbria University, Physics and
Electrical Engineering, Newcastle upon Tyne, NE1 8ST, UK  Mykola Gordovskyy
Department of Physics, Astronomy & Mathematics. University of Hertfordshire,
Hatfield AL10 9AB, UK

###### Abstract

Solar radio bursts are strongly affected by radio-wave scattering on density
inhomogeneities, changing their observed time characteristics, sizes, and
positions. The same turbulence causes angular broadening and scintillation of
galactic and extra-galactic compact radio sources observed through the solar
atmosphere. Using large-scale simulations of radio-wave transport, the
characteristics of anisotropic density turbulence from $0.1\,R_{\odot}$ to $1$
au are explored. For the first time, a profile of heliospheric density
fluctuations is deduced that accounts for the properties of extra-solar radio
sources, solar radio bursts, and in-situ density fluctuation measurements in
the solar wind at $1$ au. The radial profile of the spectrum-weighted mean
wavenumber of density fluctuations (a quantity proportional to the scattering
rate of radio-waves) is found to have a broad maximum at around
$(4-7)\,R_{\odot}$, where the slow solar wind becomes supersonic. The level of
density fluctuations at the inner scale (which is consistent with the proton
resonance scale) decreases with heliocentric distance as
$\langle\delta{n_{i}}^{2}\rangle(r)\simeq 2\times
10^{7}\,(r/R_{\odot}-1)^{-3.7}$ cm-6. Due to scattering, the apparent
positions of solar burst sources observed at frequencies between $0.1$ and
$300$ MHz are computed to be essentially cospatial and to have comparable
sizes, for both fundamental and harmonic emission. Anisotropic scattering is
found to account for the shortest solar radio burst decay times observed, and
the required wavenumber anisotropy is $q_{\parallel}/q_{\perp}=0.25-0.4$,
depending on whether fundamental or harmonic emission is involved. The deduced
radio-wave scattering rate paves the way to quantify intrinsic solar radio
burst characteristics.

interplanetary scintillation (828), interplanetary turbulence (830), radio
bursts (1339), solar corona (1483), solar wind (1534)

12ptrevtex4-1

##  1 Introduction

Radio-wave scattering affects the temporal characteristics, sizes, and
positions of both solar radio bursts and extra-solar sources observed through
the turbulent solar atmosphere. Solar radio bursts (such as Type I, Type II,
Type III, etc.) are produced predominantly via plasma mechanisms at
frequencies that are close to either the local plasma frequency or its double
(harmonic), and are thus particularly strongly affected by scattering (e.g.,
Fokker, 1965; Steinberg, 1972; Riddle, 1974; Bastian, 1994; Arzner & Magun,
1999; Thejappa & MacDowall, 2008; Kontar et al., 2017; Chrysaphi et al., 2018;
Krupar et al., 2018; McCauley et al., 2018; Gordovskyy et al., 2019; Kontar et
al., 2019; Chrysaphi et al., 2020; Krupar et al., 2020; Murphy et al., 2021;
Maguire et al., 2021; Mohan, 2021; Musset et al., 2021; Sharma & Oberoi,
2021). Similarly, extra-solar point sources observed through the solar
atmosphere and solar wind are noticeably broadened (e.g., Machin & Smith,
1952; Hewish, 1958; Dennison & Blesing, 1972; Anantharamaiah et al., 1994;
Sasikumar Raja et al., 2017) and scintillate (Hewish et al., 1964; Cohen et
al., 1967; Woo, 1977; Coles, 1978; Armand et al., 1987; Manoharan, 1993a;
Breen et al., 1999; Miyamoto et al., 2014; Sasikumar Raja et al., 2016;
Chhetri et al., 2018; Tyul’bashev et al., 2023). Overall, various observations
provide a qualitatively consistent picture of turbulence in the heliosphere.
These observations also strongly suggest that the density fluctuations are
anisotropic, with parallel wavenumbers $q_{\parallel}$ that are smaller than
the perpendicular wavenumbers $q_{\perp}$ (e.g., Dennison & Blesing, 1972;
Coles & Harmon, 1989; Armstrong et al., 1990), with stronger anisotropy
suggested at smaller scales. Observations of solar radio bursts at higher
frequencies ($\sim$30-50 MHz) originate closer to the Sun and usually require
an anisotropy $q_{\parallel}/q_{\perp}\simeq 0.15-0.3$ (Kontar et al., 2019;
Chen et al., 2020; Kuznetsov et al., 2020; Clarkson et al., 2021), while
simultaneous observations of solar radio bursts in interplanetary space using
multiple spacecraft require a somewhat larger ratio
$q_{\parallel}/q_{\perp}\simeq 0.3-0.4$ (Musset et al., 2021). In-situ
observations of the wavenumber anisotropy of density fluctuations are very
few: Celnikier et al. (1987) report an anisotropy factor between $1$ and $2$
with an uncertainty comparable to the measured values, while Roberts et al.
(2018) report an anisotropy factor between $1$ and $3$.

The scattering regimes for solar radio bursts and extra-solar point sources
are very different. Extra-solar sources are typically observed at frequencies
$\omega$ that are much greater than the plasma frequency of the solar medium
in which their emitted radiation propagates and scatters, i.e.,
$\omega/\omega_{pe}\gg 1$, where $\omega_{pe}=\sqrt{4\pi e^{2}n/m_{e}}$ (s-1)
is the local plasma frequency, $n$ (cm-3) is the local number density, and $e$
and $m_{e}$ are the electronic charge (esu) and mass (g), respectively. The
scattering mean free path $\lambda_{sc}$ is proportional to
$(\omega^{2}-\omega_{pe}^{2})^{2}$ and is typically much larger than the
distance $r$ traveled by the radio waves in the turbulent medium.
Consequently, such sources experience weak scattering and their broadening is
normally computed using a thin-screen approximation to deduce a wave structure
function that measures the spatial correlation of the turbulence at different
scales (see, e.g., Lee & Jokipii, 1975; Woo et al., 1977; Coles & Harmon,
1989). By contrast, solar radio burst emission is at frequencies that are
inherently close to the local plasma frequency (or its second harmonic), so
that $\omega/\omega_{pe}\simeq 1-2$. For values of $\omega$ so close to
$\omega_{pe}$, the mean free path $\lambda_{sc}$ can be much smaller than the
distance a wave propagates between the emission location and the observer, so
that the emitted radio waves are (at least close to their origin) subject to
multiple scatterings that randomize the wave propagation. Waves emitted in
solar radio bursts, at least close to the emitting region, thus propagate
diffusively (Bian et al., 2019; Kontar et al., 2019; Chen et al., 2020), with
the transport modeled using a stochastic random-phase description
(Chandrasekhar, 1952; Fokker, 1965; Steinberg et al., 1971; Krupar et al.,
2018; Thejappa & MacDowall, 2008) with inclusion of anisotropic effects
(Arzner & Magun, 1999; Kontar et al., 2019).

Regarding observations of the density fluctuations, solar corona sounding
experiments normally work well for heliocentric distances
$\sim(2-100)~{}R_{\odot}$ (e.g., Coles & Harmon, 1989), but become
progressively more difficult at smaller heliocentric distances due to the
strong radio emission of the quiet solar corona between $(1-2)$ $R_{\odot}$.
On the other hand, solar radio burst emissions originating between $(1-2)$
$R_{\odot}$ have brightness temperatures far greater than the corona and are
easily observed. As such, solar radio bursts, such as Type III bursts, are
better suited to provide constraints on scattering and density fluctuations
close to the Sun, i.e., below $2$ $R_{\odot}$ (Kontar et al., 2017; Chrysaphi
et al., 2018; Chen et al., 2020; Sharma & Oberoi, 2021; Mohan, 2021).

Since it is the same density turbulence that affects the properties of (1)
extra-solar sources, (2) solar sources, and (3) density fluctuations measured
in-situ in the solar wind, a common density turbulence model must self-
consistently explain all three sets of observations. Here, using the numerical
model of Kontar et al. (2019), we carry out a number of radio-wave propagation
simulations between $0.15-300$ MHz and we consider the results in light of the
veryu substantial array of extra-solar, solar, and in-situ observations
published in the literature. We show that the level of radio-wave scattering
necessary to account for the properties of non-solar sources matches the level
required to account for the source sizes and apparent positions of solar radio
bursts in the corona and in the heliosphere. Further, the burst decay times
and time durations predicted by such a turbulence model are consistent both
with typical Type III burst durations below $1$ MHz and with the shortest
radio bursts observed (such as spikes and Type IIIb striae) above $1$ MHz.
Intriguingly, the Type III burst striae that were observed by Parker Solar
Probe (PSP; Fox et al., 2016) show a transition from a decay time that is
proportional to $1/f$ at lower frequencies to a downshifted $1/f$ behavior at
higher frequencies, consistent with the trend predicted by the scattering
model.

In Section 2 we discuss the model of radio-wave propagation in a turbulent
medium with an anisotropic turbulence wavenumber spectrum
$S({{{\mathbf{\bm{q}}}}})$. In Section 3, we discuss the angular broadening of
extra-solar sources with ray paths that pass close to the turbulent atmosphere
of the Sun, and we determine the density and turbulence profiles required to
account for various source size observations. In Section 4 we discuss
simulations of solar radio bursts that use the turbulence profile (both
strength and anisotropy) established by the data on extra-solar source angular
broadening, and we compare the model predictions with observations of source
decay times (Section 4.1), sizes (Section 4.2), and apparent positions
(Section 4.3). In Section 5 we discuss the relation of the turbulence
wavenumber spectrum to in-situ measurements of the frequency spectrum of
density fluctuations, leading to further constraints on both the level of
turbulence and its degree of anisotropy. In Section 6 we use observations of
the turbulence level at the inner scale (the boundary between the inertial and
dissipative ranges of the turbulence spectrum) to further constrain the
turbulence amplitude, and we present results on the inferred variation of the
level of density fluctuations with solar distance. Finally, in Section 7 we
summarize the results obtained and use them to present a self-consistent model
of the interplanetary turbulence profile between the Sun and the Earth.

##  2 Radio-wave transport in an anisotropic turbulent medium

Density fluctuations in general are characterized by their three-dimensional
wavenumber spectrum $S({{\mathbf{\bm{q}}}})$, such that the density
fluctuation variance $\langle\delta n^{2}\rangle$ (cm-6), normalized by the
square of the local density, is

$\epsilon^{2}\equiv\frac{\langle\delta n^{2}\rangle}{n^{2}}=\int
S({{\mathbf{\bm{q}}}})\,\frac{d^{3}q}{(2\pi)^{3}}\,\,\,,$ (1)

where ${{{\mathbf{\bm{q}}}}}$ (cm-1) is the wavevector associated with a
density fluctuation, $\langle\cdots\rangle$ denotes an ensemble average, and
$n({{\mathbf{\bm{r}}}})$ is the background plasma density. A typical
wavenumber spectrum (e.g., Alexandrova et al., 2013; Tasnim et al., 2022) has
three main domains:

$S(q)=S(q_{i})\,\times\,\left\\{\begin{array}[]{lll}\left(\frac{q}{q_{o}}\right)^{-2-\gamma}\left(\frac{q_{o}}{q_{i}}\right)^{-11/3},&\qquad
q<q_{o}&{\rm(outer)}\cr\left(\frac{q}{q_{i}}\right)^{-2-5/3},&\qquad q_{o}\leq
q\leq q_{i}&{\rm(inertial)}\cr\left(\frac{q}{q_{i}}\right)^{-2-\delta},&\qquad
q>q_{i}&{\rm(dissipative)}\,\,\,,\end{array}\right.$ (2)

where the spectrum is normalized to the value $S(q_{i})$ at the inner scale
wavenumber $q=q_{i}$, the boundary between the inertial and dissipative
ranges. We note that the density fluctuation power spectrum shows evidence of
flattening between the inertial and dissipation scales (e.g., Celnikier et
al., 1987; Coles & Harmon, 1989; Šafránková et al., 2015), that we do not
consider here for simplicity. Qualitatively, since the wavenumber volume
element $d^{3}q=4\pi\,q^{2}\,dq$, the wavenumber spectrum (for an illustrative
isotropic density fluctuation spectrum) has three major parts (left panel of
Figure 1; see also Figure 2 of Tasnim et al., 2022):
$q^{2}\,S(q)\,q^{-\gamma}$ for large wavelengths ($q<q_{o}$), with values of
$\gamma$ between $0$ and $1$ (e.g., Marsch & Tu, 1990; Bird et al., 2002;
DeForest et al., 2018), the (Kolmogorov) inertial range $q^{2}\,S(q)\sim
q^{-5/3}$ between the outer scale $q_{o}$ and the inner scale $q_{i}$ (e.g.,
Marsch & Tu, 1990), and the dissipation range $q^{2}\,S(q)\sim q^{-\delta}$ at
small wavelengths ($q>q_{i}$; e.g., Celnikier et al., 1983).

As we shall show below (Equation (7); see also Kontar et al., 2019), the
diffusion coefficient for radio waves is proportional to the spectrum-weighted
mean wavenumber

$\overline{q\,\epsilon^{2}}=\frac{4\pi}{(2\pi)^{3}}\,\int
q\,S(q)\,q^{2}\,dq\,\,\,,$ (3)

which is the central quantity of interest in this work. This integral is
proportional to $\int q^{4}\,S(q)\,d\ln q$; the corresponding integrand is
shown schematically in the right panel of Figure 1 and is seen to peak near
$q_{i}$, corresponding to the inner scale. Quantitatively, the contribution to
$\int_{0}^{\infty}q^{3}\,S(q)\,dq\equiv[(2\pi)^{3}/4\pi]\,\overline{q\,\epsilon^{2}}$
from the long-wavelength outer range ($q<q_{o}$) is
$S(q_{i})\,q_{i}^{11/3}\,q_{o}^{\gamma-5/3}\,\int_{0}^{q_{o}}q^{1-\gamma}\,dq=[1/(2-\gamma)]\,S(q_{i})\,q_{i}^{4}\,(q_{o}/q_{i})^{1/3}$,
the contribution from the inertial range ($q_{o}<q<q_{i}$) is
$S(q_{i})\,q_{i}^{11/3}\int_{q_{o}}^{q_{i}}q^{-2/3}\,dq=3\,S(q_{i})\,q_{i}^{4}(1-[q_{o}/q_{i}]^{1/3})$,
and the contribution from the dissipative range ($q>q_{i}$) is
$S(q_{i})\,q_{i}^{\delta+2}\int_{q_{i}}^{\infty}q^{1-\delta}\,dq=[1/(\delta-2)]\,S(q_{i})\,q_{i}^{4}$.
The inertial range between $q_{0}$ and $q_{i}$ is typically about three orders
of magnitude (e.g., Alexandrova et al., 2013), and a typical dissipation range
spectral index $\delta$ varies from $2.3$ to $2.9$ (e.g., Celnikier et al.,
1987), so we adopt $\delta=2.5$. The contributions to
$\overline{q\,\epsilon^{2}}$ from the three ranges are therefore in the
approximate ratio 0.1: 3 : 2. Ignoring the small contribution at large scales
($q<q_{o}$), the spectrum-weighted mean wavenumber
$\overline{q\,\epsilon^{2}}$ is

$\overline{q\,\epsilon^{2}}\simeq\frac{4\pi}{(2\pi)^{3}}\left(3+\frac{1}{\delta-2}\right)\,S(q_{i})\,q_{i}^{4}=\frac{20\,\pi}{(2\pi)^{3}}\,S(q_{i})\,q_{i}^{4}\,\,\,,$
(4)

where the last equality uses $\delta=2.5$. Equation (4) and Figure 1
importantly highlight that $\overline{q\,\epsilon^{2}}$ is determined mostly
by density fluctuations close to the inner scale $q_{i}^{-1}$.

Figure 1: Left: Spectrum of density fluctuations showing the $q^{2}\,S(q)\sim
q^{-\gamma}$ spectrum at large scales $>q_{o}^{-1}$, the (Kolmogorov) inertial
range $q^{2}\,S(q)\sim q^{-5/3}$ between the outer scale $q_{o}^{-1}$ and the
inner scale $q_{i}^{-1}$, and the dissipation range $q^{2}\,S(q)\sim
q^{-\delta}$ (with $\delta>2$) at small scales $<q_{i}^{-1}$. Right: The $\ln
q$ space integrand required to compute the mean wavenumber
$\overline{q\,\epsilon^{2}}\propto\int q\,S(q)\,q^{2}\,dq=\int
q^{4}\,S(q)\,d\ln q$, highlighting the peak near the inner scale $q_{i}^{-1}$.
All axes are on logarithmic scales.

Observations often suggest the presence of anisotropy relative to the
direction of the magnetic field, suggesting a correspondingly anisotropic
wavenumber spectrum. Following previous studies (Dennison & Blesing, 1972; Woo
et al., 1977), we therefore consider a wavenumber spectrum of the form

$S({{\mathbf{\bm{q}}}})=S(\tilde{q})\,,\,\,\,\mbox{where}\;\;\;\tilde{q}=\sqrt{q_{\perp_{1}}^{2}+q_{\perp_{2}}^{2}+\frac{q_{\parallel}^{2}}{\alpha^{2}}}\,\,\,,$
(5)

which has axial symmetry around the $\parallel$ direction, i.e., along the
(assumed radial) magnetic field ${{\mathbf{\bm{B}}}}$. The wavenumber
diffusion tensor describing elastic scattering of radio waves with wavenumber
${{\mathbf{\bm{k}}}}$ in such an anisotropic medium can be written (e.g.,
Arzner & Magun, 1999; Bian et al., 2019; Kontar et al., 2019)

$D_{ij}=\frac{\omega_{pe}^{4}}{32\,\pi^{2}\,\omega\,c^{2}}\int
q_{i}\,q_{j}\,S({{\mathbf{\bm{q}}}})\,\,\delta({{\mathbf{\bm{q}}}}\cdot{{\mathbf{\bm{k}}}})\,{d^{3}q}\,\,\,,$
(6)

where $\omega({{\mathbf{\bm{k}}}})=\sqrt{\omega_{pe}^{2}+c^{2}k^{2}}$ is the
angular frequency and ${{\mathbf{\bm{k}}}}$ the wavevector of electromagnetic
waves in a plasma with local plasma frequency $\omega_{pe}(r)$. For the
spectrum given by Equation (5), we can obtain (Equation (16) of Kontar et al.,
2019) an explicit expression for the components of the wavenumber diffusion
tensor (cm-2 s-1):

$\displaystyle D_{ij}$ $\displaystyle=$
$\displaystyle\left[\frac{{A}^{-2}_{ij}}{({{\mathbf{\bm{k}}}}\textsf{{A}}^{-2}{{\mathbf{\bm{k}}}})^{1/2}}-\frac{(\textsf{{A}}^{-2}{{\mathbf{\bm{k}}}})_{i}(\textsf{{A}}^{-2}{{\mathbf{\bm{k}}}})_{j}}{({{\mathbf{\bm{k}}}}\textsf{{A}}^{-2}{{\mathbf{\bm{k}}}})^{3/2}}\right]\frac{\omega_{pe}^{4}}{32\,\pi\,\omega\,c^{2}}\,\alpha\int{\widetilde{q}}^{3}\,S(\widetilde{q})\,d\widetilde{q}$
(7) $\displaystyle=$
$\displaystyle\left[\frac{{A}^{-2}_{ij}}{({{\mathbf{\bm{k}}}}\textsf{{A}}^{-2}{{\mathbf{\bm{k}}}})^{1/2}}-\frac{(\textsf{{A}}^{-2}{{\mathbf{\bm{k}}}})_{i}(\textsf{{A}}^{-2}{{\mathbf{\bm{k}}}})_{j}}{({{\mathbf{\bm{k}}}}\textsf{{A}}^{-2}{{\mathbf{\bm{k}}}})^{3/2}}\right]\frac{\pi\,\omega_{pe}^{4}}{16\,\omega\,c^{2}}\,\overline{q\,\epsilon^{2}}\,\,\,,$
(8)

where ${\widetilde{\bf q}}=(q_{\perp,1},q_{\perp,2},q_{\parallel}/\alpha)$ is
the wavevector transformed into a space such that $S({\widetilde{q}})$ is
isotropic, ${\bf A}$ is the matrix ${\rm diag}\,(1,1,\alpha^{-1})$, and we
have used $d^{3}q=\alpha\,d^{3}\widetilde{q}$. The case $\alpha=1$ corresponds
to isotropic turbulence, while the typically observed case $\alpha<1$ has
power predominantly oriented in the perpendicular direction. In such a
situation, the density perturbations are elongated along the field lines, so
that the parallel length is larger than the perpendicular length:
$q_{\parallel}^{-1}>q_{\perp}^{-1}$, i.e., the perpendicular wavenumber is
larger than the parallel wavenumber: $q_{\parallel}<q_{\perp}$. Equation (7)
explicitly shows that the diffusion tensor is proportional to the quantity
$\overline{q\,\epsilon^{2}}$.

##  3 Angular Broadening of Extra-Solar Radio Sources

We consider the propagation of a radio wave from a distant extra-solar source
with a heliocentric angular separation of $\simeq 0.27\,r/R_{\odot}$ degrees,
where $r$ is the linear distance of closest approach of the ray path to the
center of the Sun, and $0^{o}.27$ is the angular radius of the Sun (see Figure
2). We define the $q_{\parallel}$ direction as that aligned with the (assumed
radial) solar magnetic field ${{\mathbf{\bm{B}}}}$, $q_{\perp_{1}}$ as the
perpendicular direction that is aligned with the wave propagation direction
${\hat{z}}$ at the point of closest approach, and $q_{\perp_{2}}$ as the
perpendicular direction ${\hat{y}}$ on the plane of the sky that is
perpendicular to ${{{\mathbf{\bm{B}}}}}$ (Figure 2). Evidently, radio waves
propagating from such a source will be affected by scattering in the turbulent
solar atmosphere and, for emission frequencies $\omega$ that are much larger
than the plasma frequencies $\omega_{pe}$ along the ray path though the
heliosphere, this can be considered in the weak-scattering limit. In this
limit, Equation (6) shows that the angular broadening rates per unit travel
distance $v_{gr}\,dt=(c^{2}\,k/\omega)\,dt\simeq c\,dt$, in the directions
$\parallel$ (i.e., radial) and $\perp_{2}$ (i.e., perpendicular to the radial
direction on the plane of the sky) can be written, using Equation (7) (see
also Kontar et al., 2019), as

$\frac{d\langle\theta^{2}_{\parallel\parallel}\rangle}{c\,dt}=\frac{2\,D_{\parallel\parallel}}{v_{\rm
gr}\,k^{2}}=\frac{\pi}{8}\,\frac{\omega_{pe}^{4}}{\omega^{4}}\,\,\alpha^{2}\,\overline{q\,\epsilon^{2}}$
(9)

and

$\frac{d\langle\theta^{2}_{\perp_{2}\perp_{2}}\rangle}{c\,dt}=\frac{2\,D_{\perp_{2}\perp_{2}}}{v_{\rm
gr}\,k^{2}}=\frac{\pi}{8}\,\frac{\omega_{pe}^{4}}{\omega^{4}}\,\,\overline{q\,\epsilon^{2}}\,\,\,,$
(10)

where ${{\mathbf{\bm{v}}}}_{\rm
gr}=\partial\omega/\partial{{{\mathbf{\bm{k}}}}}=c^{2}\,{{\mathbf{\bm{k}}}}/\omega$
is the group velocity of the radio wave and we have used the approximate
(high-frequency) dispersion relation $\omega=ck$. Since the radio waves
propagate along the $z$-direction, the broadening in $x$ and $y$ directions on
the plane of the sky become integrals along the line of sight :

$\langle\theta^{2}_{yy}\rangle=\int_{los}\frac{d\langle\theta^{2}_{\perp_{2}\perp_{2}}\rangle}{c\,dt}\,dz,$
(11)

and

$\langle\theta^{2}_{xx}\rangle=\int_{los}\left(\frac{d\langle\theta^{2}_{\parallel\parallel}\rangle}{c\,dt}\cos^{2}\chi+\frac{d\langle\theta^{2}_{\perp_{2}\perp_{2}}\rangle}{c\,dt}\sin^{2}\chi\right)\,dz\,\,\,,$
(12)

where $\chi(z)$ is the angle between the radial ($\parallel$) direction of the
magnetic field and the $x$-axis (see Figure 2). When the largest contribution
to $\langle\theta^{2}_{xx}\rangle$ comes from plasma near $z=0$ (a.k.a., the
thin-screen approximation),
$\langle\theta^{2}_{xx}\rangle\simeq\alpha^{2}\langle\theta^{2}_{yy}\rangle$,
so the source broadening in the $x$-direction is reduced by a factor $\alpha$
relative to that in the $y$-direction. Broadened sources are indeed observed
to be elliptical (e.g., Dennison & Blesing, 1972; Anantharamaiah et al.,
1994), and this requires that the scattering occurs in both directions in the
$x-y$ plane (rather than along $x$ or $y$ only), so that
$S({{\mathbf{\bm{q}}}})$ should be evaluated with $q_{\perp_{1}}=0$, i.e.,
with ${{\mathbf{\bm{q}}}}$ lying in the $\perp_{2}-\parallel$ plane). We note
that the observed form of source broadening is not consistent with a simple
2D-plus-slab model, in which
$S({{\mathbf{\bm{q}}}})=S_{\perp}\,\delta(q_{\parallel})+S_{\parallel}\,\delta({{\mathbf{\bm{q}}}}_{\perp})$
and the scattering tensor (Equation (6)) has a non-zero $\perp_{2}-\parallel$
component; such a model would result instead in “cross-like,” rather than the
observed elliptical, apparent source structures.

For a spherically symmetric corona, the apparent rms source sizes in the
perpendicular direction (along the $y$-axis in the Figure 2), observed at
frequency $f=\omega/2\pi$ (Hz), thus satisfy

$\langle\theta_{yy}^{2}\rangle^{1/2}\,(r)=\int_{los}\frac{d\langle\theta^{2}_{\perp_{2}\perp_{2}}\rangle}{c\,dt}\,dz\propto\frac{1}{f^{2}}\,\left[\int_{r}^{\infty}\overline{q\,\epsilon^{2}}(r^{\prime})\,n^{2}\,(r^{\prime})\,dr^{\prime}\right]^{1/2}\,\,\,,$
(13)

where the constant of proportionality depends111See Appendix B for a more
formal expression for Equation (13). on the distance of closest approach $r$
of the ray to the Sun, and we have used the relation $\omega_{pe}\propto
n^{1/2}$. At radial distance $r$, the mean (spectrum-averaged) wavenumber,
$\overline{q\,\epsilon^{2}}$, at the position $r$ is given by substituting
Equation (5) into Equation (1):

$\overline{q\,\epsilon^{2}}\,R_{\odot}=R_{\odot}\,\int
q\,S(q)\,\frac{d^{3}q}{(2\pi)^{3}}=\alpha\,R_{\odot}\,\int{\widetilde{q}}\,S(\widetilde{q})\,\frac{d^{3}\widetilde{q}}{(2\pi)^{3}}=\alpha\,\frac{4\pi}{(2\pi)^{3}}\,R_{\odot}\,\int_{0}^{\infty}{\widetilde{q}}^{3}\,S(\widetilde{q})\,d{\widetilde{q}}\,\,\,,$
(14)

where we have multiplied $\overline{q\,\epsilon^{2}}$ by the solar radius
$R_{\odot}$ to produce a dimensionless quantity
$\overline{q\,\epsilon^{2}}\,R_{\odot}$. The angular broadening rates in both
directions on the sky are thus, as expected from Equation (7), proportional to
the square root of the key quantity $\overline{q\,\epsilon^{2}}\,R_{\odot}$.

Figure 2:  Left: Schematic showing the coordinate system used and its relation
to the solar disk and the line of sight to an extra-solar point source with
separation distance $r$. The broadening of point sources is calculated as an
integral along $z$ in Appendix B. Right: Observed FWHM major-axis source sizes
of broadened point sources (Hewish, 1958; Slee, 1959; Högbom, 1960;
Gorgolewski et al., 1962; Hewish & Wyndham, 1963; Slee, 1966; Harries et al.,
1970; Blesing & Dennison, 1972; Woo, 1978; Bradford & Routledge, 1980; Narayan
et al., 1989; Anantharamaiah et al., 1994). The Hewish (1958) data shows
points where the detector baseline orientation was specified, and we take the
larger of the measurements as the assumed major axis dimension. The solid
black line shows the data fit $\theta=355(r/R_{\odot}-1)^{-2.15}$ arcsec, and
extrapolated by the dashed line. The colored lines show the simulated data for
various values of the anisotropy parameter $\alpha$, with the lower and upper
bounds of each region denoting values of
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ corresponding to $0.5\times$ and
$2\times$ the nominal values from Equation (15). Different observations are
scaled to a frequency of $1.5$ GHz using a frequency dependence $f^{-2}$, as
expected from Equation (10); see also Equation (B7) in Appendix B.

For a prescribed wave frequency $f$ and distance of closest approach $r$,
Equation (13) shows that the source rms angular size and shape depend only on
the profiles of plasma density $n$ (through the local plasma frequency
$\omega_{pe}$) and the spectrum-weighted mean wavenumber of density
fluctuations $\overline{q\,\epsilon^{2}}\,R_{\odot}$, at distances larger than
the closest ray-approach distance. The turbulence profile in turn depends on
the spectrum $S({\widetilde{q}})$ and its degree of anisotropy, characterized
by the parameter $\alpha$ (see Equation (5)). Further, Equation (13) shows
that the apparent angular extent of the source in the radial direction is
smaller than the apparent extent in the perpendicular direction on the plane
of the sky, by a factor $\alpha$. Hereafter we define the source “size” as the
major axis, i.e., the FWHM source extent in the $\perp_{2}$ direction,
perpendicular to the radial direction on the plane of sky.

If the density and turbulence profiles can be reasonably represented by power-
law forms ($n(r)\sim r^{-\eta}$ and $\overline{q\,\epsilon^{2}}(r)\sim
r^{-\zeta}$, respectively), then Equation (13) shows that
$\langle\theta^{2}\rangle^{1/2}\sim r^{-[\eta+(\zeta-1)/2]}$. The data on the
apparent sizes of extra-solar sources are presented in Figure 2; they show
that $\langle\theta^{2}\rangle^{1/2}\propto r^{-2.15}$ for $r\gg R_{\odot}$.
Thus, a radial density profile $n(r)\propto r^{-2.3}$ (valid at $r\gg
R_{\odot}$; see Appendix A), requires that $(\zeta-1)/2=2.15-2.3=-0.15$; i.e.,
$\zeta=0.7$. We therefore adopt the nominal turbulence profile

$\overline{q\,\epsilon^{2}}\,R_{\odot}=2\times
10^{3}\,\alpha\,\left(1-\frac{R_{\odot}}{r}\right)^{2.7}\left(\frac{R_{\odot}}{r}\right)^{0.7}\,\,\,,$
(15)

which is proportional to $\alpha$ (Equation (14)) and also includes an
additional factor $(1-{R_{\odot}}/{r})^{2.7}$ so that
$\overline{q\,\epsilon^{2}}$ decreases towards the solar surface as required
by solar burst observations (see next section). The later is required mostly
by $>20$ MHz solar radio burst observations discussed in the next section. The
scaling factor $2\times 10^{3}$ and the exponent $2.7$ in the additional term
are chosen to best match the simulation results with the observational data in
Figure 2.

##  4 Solar Radio Bursts

Radio waves emitted by solar sources (e.g., Type III bursts) suffer scattering
in the same turbulent environment that leads to the angular broadening of
extra-solar sources. Because such bursts are produced by plasma processes at
or near the plasma frequency $\omega_{pe}$ or its double (see, e.g., Ginzburg
& Zhelezniakov, 1958), the scattering process is much stronger than for extra-
solar sources with $\omega\gg\omega_{pe}$ and hence the weak-scattering
treatment of the previous section is not applicable. Here, using a code
developed by Kontar et al. (2019), we numerically simulate the propagation of
radio waves in the presence of a prescribed density fluctuation profile
$\overline{q\,\epsilon^{2}}\,$ as function of $r$ and anisotropy parameter
$\alpha$, which we take222In principle, $\alpha$ could be treated as a
function of $r$ in the simulations, but this would make the parameter space
too large to study effectively. to be a constant in our model. We performed
simulations with $\overline{q\,\epsilon^{2}}\,R_{\odot}$ scaled from its
nominal value in Equation (15) by a factor in the range $[1/4,4]$, and using
anisotropy parameters $\alpha$ in the range $[0.19,0.42]$. The results of
these simulations lead to predictions of solar radio source properties (source
size, apparent position, time profile) as seen from $1$ au, and are compared
with observations to ascertain the veracity of the
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ profile and anisotropy parameter
$\alpha$ used. For the simulations, we assume an initially isotropic
distribution of photons emitted near the plasma frequency (fundamental), or
twice the plasma frequency (harmonic). Assuming a spherically symmetric
corona, we ran numerical simulations of the wave propagation, taking into
account anisotropic scattering, large scale refraction, and free-free
absorption, using profiles for the ambient density profile $n(r)$ given in
Appendix A. We performed simulations for radio frequencies covering the range
from $\sim$0.15 MHz to $\sim$300 MHz, thus encompassing ground-based and
space-based (below the ionospheric cut-off) observations. We simulated burst
decay times (Subsection 4.1), angular source sizes (Subsection 4.2), and
source positions (Subsection 4.3), which are then compared with pertinent
observations to determine the spectrum-weighted mean wavenumber of density
fluctuations $\overline{q\,\epsilon^{2}}\,R_{\odot}$ as a function of $r$, and
anisotropy parameter $\alpha$, that together best account for the
observations.

###  4.1 Decay Time

The top panels of Figure 3 show predicted $1/e$ decay times for the range of
simulations discussed above, both for sources that emit at the fundamental
(left panel) and for those that emit at the second harmonic (right panel).

The decay time of solar radio bursts has been observed both from space and
from the ground, resulting in an extended set of decay-time observations. The
bottom panels of Figure 3 compare observations of characteristic decay times
(expanding the collection of decay times presented by Kontar et al., 2019)
with those obtained from our simulations, assuming emission either at the
fundamental (left panel) or at the harmonic (right panel). The vertical bars
in Figure 3 show the spread of measurements (where available) rather than
measurement uncertainties. The decay profile is normally well-fit by an
exponential form, with a generally well-measured characteristic decay time
(e.g., Krupar et al., 2018, 2020) both below $1$ MHz and above $20$ MHz, with
a gap in near-Earth data between a few MHz (due to the limited time-resolution
of space-based observations) and $\sim 10-20$ MHz (due to the ionospheric cut-
off for ground-based observations). Although there is a noticeable spread in
the observations, the grey area, representing a factor of $0.5\times$ to
$2\times$ the nominal $\overline{q\,\epsilon^{2}}\,R_{\odot}$ profile of
Equation (15), includes all data below $3$ MHz and is found close to the
observed decay times of the high-frequency spikes and Type IIIb fine
structures. This is convincing evidence that scattering effects account for
the shortest observed decay times. The decay time of an average Type III burst
at $>10$ MHz is normally quite a bit larger, and is principally determined by
the beam width and/or emission processes (e.g., Zhang et al., 2019; Chen et
al., 2020). We note that the simulated decay time is dependent on whether the
emission is fundamental or harmonic (Figure 3). If all observations are due to
harmonic emission, the simulations require $\alpha\simeq 0.4$, while the
assumption of emission at the fundamental hypothesis requires a smaller
anisotropy $\alpha\simeq 0.25$ to explain decay time observations.

Figure 3: Top panels: Simulated solar radio burst $1/e$ decay times versus
frequency. The simulations were performed for turbulence profiles
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ given by Equation (15) multiplied by
factors $[1/4,1/2,1,2,4]$, and values of the anisotropy parameter
$\alpha=[0.19,0.25,0.33,0.42]$, and for both fundamental (left panels) and
harmonic emission (right panels). Bottom panels: Observations of source decay
times. The solid black points show measurements of Type III decay times
(Alexander et al., 1969; Elgaroy & Lyngstad, 1972; Alvarez & Haddock, 1973a;
Barrow & Achong, 1975; Krupar et al., 2018; Reid & Kontar, 2018), while open
black points show spike decay times (McKim Malville et al., 1967; Barrow et
al., 1994; Shevchuk et al., 2016; Clarkson et al., 2023). The solid black
lines represent power-law fits to the spike data from Equation (3) of Clarkson
et al. (2023) and to the Type III data from Equation (51) of Kontar et al.
(2019). The light blue points show Type IIIb striae median decay times
observed by Parker Solar Probe between $0.95$ and $2.15$ MHz. The colored
lines represent simulated decay times from anisotropic scattering simulations
at the fundamental (left panel) and harmonic (right panel). The lower and
upper bound of each region denotes values corresponding to turbulence profiles
of $0.5\,\overline{q\,\epsilon^{2}}\,R_{\odot}$ and
$2\,\overline{q\,\epsilon^{2}}\,R_{\odot}$, respectively.

###  4.2 Angular Size

The top panels of Figure 4 show predicted apparent source sizes for the range
of simulations discussed above, both for sources that emit at the fundamental
(left panel) and for those that emit at the second harmonic (right panel).

We have expanded the collection of Type III solar radio burst angular extents
initially presented by Kontar et al. (2019). Source sizes at frequencies above
10 MHz are evaluated using ground-based interferometers, while those below $1$
MHz are determined using indirect techniques (e.g., spinning demodulation
and/or goniopolarimetric techniques; Alvarez, 1976; Bougeret et al., 2008;
Cecconi, 2014). Similarly to the decay time measurements of Figure 3, the bars
in Figure 4 show the spread of measurements (where available) rather than
measurement uncertainties. The bottom panels of Figure 4 show observed FWHM
sizes (given as the FWHM semi-major axis as observed at $1$ au) compared with
the predictions of the simulations, assuming emission either at the
fundamental (left panel) and at the harmonic (right panel). The simulated
source sizes are given in terms of their ${\rm FWHM}$ (for sources with
emission frequencies below $0.15$ MHz, the source is so large that it extends
all the way to the observer at $1$ au). One can see that the majority of
observations are within the range $[1/2,2]$ times the nominal
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ profile of Equation (15).

Figure 4: Top panels: Simulated FWHM source sizes of Type III bursts. The
simulations were performed for turbulence profiles
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ given by Equation (15), multiplied by
factors $[1/4,/1/2,1,2,4]$, for values of the anisotropy parameter
$\alpha=[0.19,0.25,0.33,0.42]$, and for both fundamental (left panels) and
harmonic emission (right panels). Bottom panels: average FWHM size
observations (Bougeret et al., 1970; Abranin et al., 1976; Alvarez, 1976;
Abranin et al., 1978; Chen & Shawhan, 1978; Dulk & Suzuki, 1980; Steinberg et
al., 1985; Saint-Hilaire et al., 2013; Krupar et al., 2014; Kontar et al.,
2017) and spikes (Clarkson et al., 2023). Simulated sizes from anisotropic
scattering simulations at the fundamental (left) and harmonic (right) are
overlaid with colored lines at various values of $\alpha$, with turbulence
levels ranging from $0.5\,\overline{q\,\epsilon^{2}}\,R_{\odot}$ (lower) to
$2\,\overline{q\,\epsilon^{2}}\,R_{\odot}$ (upper), where the nominal
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ profile is given by Equation (15). The
grey region shows the spread for $\alpha=0.25$, that is consistent with the
observations.

Publications of observations of Type III burst decay times and/or source sizes
rarely identify whether the observations correspond to emission at the
fundamental or at the second harmonic, and we therefore conservatively include
the possibility of either in our comparison of observed decay times and sizes
with simulation results. With that in mind, it is interesting to note that
both fundamental and harmonic sources have similar sizes over a wide range of
frequencies. This is because the scattering surfaces that principally
determine the source size are located at levels where both the plasma
frequency and its double are less than the observed frequency, so that whether
the emission is fundamental or harmonic is not a critical factor in
determining the apparent source size. This also explains why harmonic and
fundamental Type III burst source sizes measured at the same frequency are
nearly identical (Dulk & Suzuki, 1980; Kontar et al., 2017; Chen et al.,
2023).

###  4.3 Apparent Position

Since the early observations of Type III bursts, it was noted (e.g., Shain &
Higgins, 1959; Fainberg & Stone, 1974) that the source positions are not
coincident with the positions in the solar atmosphere and solar wind with
densities corresponding to emission at the plasma frequency (or its double);
equivalently, the observed burst frequencies required a higher plasma density
than that is normally observed by optical telescopes (e.g., Fainberg & Stone,
1974). However, as Figure 5 highlights, radio wave scattering substantially
shifts the burst emission away from the Sun, so that the density inferred by
associating the burst emission with the plasma frequency (or its double) could
be 4-5 times higher than the density at the apparent burst position (e.g.,
Chrysaphi et al., 2018). Figure 5 shows the variation of observed source
positions versus frequency, compared to the predictions of our simulations
under the assumptions that the emission is at the plasma frequency
(fundamental mode; left panel) or its second harmonic (right panel). We see
that the radio-wave scattering reproduces the observed positions over a wide
range of distances from the low corona into the solar wind, as shown by the
grey band; only a relatively small spread (factor of 4) in the density
fluctuation quantity $\overline{q\,\epsilon^{2}}\,R_{\odot}$ is required to
explain the available observations.

These simulations also highlight that, similar to source sizes (but unlike
burst decay times), the simulated apparent source positions are relatively
independent of the assumption of whether fundamental or harmonic emission is
involved. If there were no scattering, at the same observation frequency,
sources corresponding to emission at the harmonic should be further outward
than sources corresponding to emission at the fundamental. However, in the
presence of strong scattering (Figure 5), the apparent source positions are
significantly further away from the Sun than the locations expected for either
emission at the harmonic mode or at the fundamental. This effect is noticeable
for all except the highest emission frequencies $f>100$ MHz, where the
apparent source position is close to the location expected from emission at
the harmonic. This shows clearly that measured source locations are dominated
by scattering effects and succinctly explains the observations reported by
Stewart (1972), showing that the radial distances for fundamental and harmonic
components at $80$ MHz are, on average, equal.

Figure 5: Observed source position versus frequency (Wild et al., 1959; Shain
& Higgins, 1959; Morimoto & Kai, 1961; Gurnett et al., 1978; Bougeret et al.,
1984a; Reiner et al., 1998, 2009; Chrysaphi et al., 2018). The colored lines
represent the observed positions from the simulations for the fundamental
(left) and harmonic (right) at various values of the anisotropy parameter
$\alpha$ and the turbulence profile $\overline{q\,\epsilon^{2}}\,R_{\odot}$,
multiplied by $1/4$ (lower bound) to $4$ (upper bound). The grey region shows
the spread across values for an anisotropy parameter $\alpha=0.25$. The black
dashed and dotted lines show the locations corresponding to the fundamental
(plasma frequency) and harmonic (double plasma frequency), respectively.

###  4.4 Remarks

Figure 6:  Left: Spectrum-averaged wavenumber
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ as a function of heliocentric distance
$r$, as given by Equation (15), for fundamental emission with an anisotropy
factor $\alpha=0.25$ (solid), and for harmonic emission with $\alpha=0.42$
(dashed). The grey region denotes the range from
$[1/2,2]\times\overline{q\,\epsilon^{2}}\,R_{\odot}$. The data from Coles &
Harmon (1989) are retrieved from the density spectrum in their Figure 4, with
a density profile $n(r)$ taken from equation (A1). The light grey points are
not well constrained due to a lack of observations at the (dominant; see
Section 2) inner scale wavenumber $q_{i}$. The in-situ density fluctuations
data at $1$ au (Celnikier et al., 1983, 1987; Kellogg & Horbury, 2005; Chen et
al., 2012) are retrieved using Equation (27), with the black points
corresponding to $\alpha=0.25$ (fundamental) and the red points corresponding
to $\alpha=0.42$ (harmonic). The data for Celnikier et al. (1987) are taken
from their Figure 7. We use the Kellogg & Horbury (2005) spectra for densities
(5–6) cm-3 and (10–15) cm-3. Right: As in the left panel but multiplied by
$n^{2}$, with $n(r)$ given by Equation (A1). The dotted line shows the
relation $\overline{q\,\epsilon^{2}}\,R_{\odot}\,n^{2}(r)=6.5\times
10^{14}\,(r/R_{\odot}-1)^{-5.17}$.

The observations of Type III burst decay time, source size and position
(Figures 3 through 5) show a noticeable data gap at frequencies between
$\sim$5 MHz and $\sim$15 MHz. Observations below the ionospheric cut-off are
made from space, while observations above $10-15$ MHz are ground-based. Parker
Solar Probe (PSP; Fox et al., 2016) has sufficiently high frequency resolution
to resolve Type IIIb burst striae (Pulupa et al., 2020). Our analysis of $10$
Type IIIb bursts ($58$ striae from April 2-9, 2019) provides interesting
measurements (blue symbols in Figure 3), suggesting that Type IIIb and Type
III bursts have nearly identical decay times below $1$ MHz, but the fine
structures in Type IIIb bursts decay faster at higher frequencies. This
suggests that the Type III decay is purely due to scattering below $1$ MHz and
that electron transport is of relevance at higher frequencies. In other words,
the scattering of Type III radio bursts is so large below $1$ MHz so that it
dominates the decay time.

The profiles of $\overline{q\,\epsilon^{2}}\,R_{\odot}$ (see Figure 6)
necessary to reproduce the observations of the source size, decay time and
source position are consistent with an anisotropic scattering model with
$\alpha\simeq 0.25$ (assuming emission at the fundamental) and $\alpha\simeq
0.42$ (assuming purely harmonic emission). We therefore conclude that
turbulent density fluctuations, with a magnitude profile consistent with
Equation (15) and an anisotropy parameter $0.2\,\lower 3.0pt\hbox{$\sim$}\hbox
to0.0pt{\hss\raise 2.0pt\hbox{$<$}}\,\alpha\,\lower 3.0pt\hbox{$\sim$}\hbox
to0.0pt{\hss\raise 2.0pt\hbox{$<$}}\,0.4$, are present in the solar corona and
heliosphere.

##  5 In-situ observations of density fluctuations

Density fluctuations are often measured in the solar wind (e.g., Celnikier et
al., 1983; Marsch & Tu, 1990) and can be compared to our density turbulence
model by converting the wavenumber spectrum into the frequency power spectrum
$P(f)$ (cm-6 Hz-1) measured by a spacecraft as the solar wind, with velocity
${{{\mathbf{\bm{V}}}}}_{\rm SW}$, advects the turbulent fluctuations past the
measuring instrumentation. $P(f)$ is related to the wavenumber spectrum
$S({{\mathbf{\bm{q}}}})$ of density fluctuations by (e.g., Fredricks &
Coroniti, 1976):

$P(f)=n^{2}\,\int
S({{{\mathbf{\bm{q}}}}})\,\delta\left(f-\frac{{{{\mathbf{\bm{q}}}}}\cdot{{{\mathbf{\bm{V}}}}}_{\rm
SW}}{2\pi}\right)\,\frac{d^{3}q}{(2\pi)^{3}}\,\,\,.$ (16)

In Appendix C, we evaluate $P(f)$ for several illustrative forms (both
isotropic and anisotropic) of $S({{{\mathbf{\bm{q}}}}})$. Here we explore the
reverse problem of determining the value of
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ from in-situ observations of $P(f)$.
We first transform the integration variable from
${{{\mathbf{\bm{q}}}}}=(q_{\perp 1},q_{\perp 2},q_{\parallel})$ to
$\widetilde{{{\mathbf{\bm{q}}}}}=(q_{\perp 1},q_{\perp
2},q_{\parallel}/\alpha)$ and define $\widetilde{{{\mathbf{\bm{V}}}}}_{\rm
SW}=(V_{{\rm SW}\,\perp\,1},V_{{\rm SW}\,\perp\,2},\alpha V_{{\rm
SW}\,\parallel})$, so that
$\widetilde{{{\mathbf{\bm{q}}}}}\cdot\widetilde{{{\mathbf{\bm{V}}}}}_{\rm
SW}={{{\mathbf{\bm{q}}}}}\cdot{{{\mathbf{\bm{V}}}}}_{\rm SW}$ and
$d^{3}q=\alpha\,d^{3}{\widetilde{q}}$. Then, since
$S(\widetilde{{{\mathbf{\bm{q}}}}})$ is (by construction333The analysis of
Equations (17) through (25) thus applies for the isotropic spectrum $P(f)$ of
Appendix C.1 and even for the anisotropic spectrum of Appendix C.2 (which can
be scaled into an isotropic form), but not for the spectrum of Appendix C.3,
which cannot be transformed into an isotropic form.) isotropic,

$P(f)=\frac{\alpha\,n^{2}}{(2\pi)^{2}}\,\int_{{\widetilde{q}}_{\perp}=0}^{\infty}\,\int_{{\widetilde{q}}_{\parallel}=-\infty}^{\infty}\,S\left(\sqrt{{\widetilde{q}}_{\perp}^{2}+{\widetilde{q}}_{\parallel}^{2}}\right)\,\delta\left(\frac{{\widetilde{q}}_{\parallel}\,{\widetilde{V}}_{\rm
SW}}{2\pi}-f\right)\,{\widetilde{q}}_{\perp}\,d{\widetilde{q}}_{\perp}\,d{\widetilde{q}}_{\parallel}\,\,\,,$
(17)

this can be written as

$\displaystyle P(f)$ $\displaystyle=$
$\displaystyle\frac{\alpha\,n^{2}}{2\pi\,{\widetilde{V}}_{\rm
SW}}\int_{{\widetilde{q}}_{\perp}=0}^{\infty}\,\int_{{\widetilde{q}}_{\parallel}=-\infty}^{\infty}\,S\left(\sqrt{{\widetilde{q}}_{\perp}^{2}+{\widetilde{q}}_{\parallel}^{2}}\right)\,\delta\left({\widetilde{q}}_{\parallel}-\frac{2\pi
f}{{\widetilde{V}}_{\rm
SW}}\right)\,{\widetilde{q}}_{\perp}\,d{\widetilde{q}}_{\perp}\,d{\widetilde{q}}_{\parallel}$
(18) $\displaystyle=$
$\displaystyle\frac{\alpha\,n^{2}}{2\pi\,{\widetilde{V}}_{\rm
SW}}\int_{{\widetilde{q}}_{\perp}=0}^{\infty}\,S\left(\,\left[{\widetilde{q}}_{\perp}^{2}+\left(\frac{2\pi
f}{{\widetilde{V}}_{\rm
SW}}\right)^{2}\right]^{1/2}\,\right)\,{\widetilde{q}}_{\perp}\,d{\widetilde{q}}_{\perp}$
(19) $\displaystyle=$
$\displaystyle\frac{\alpha\,n^{2}}{2\pi\,{\widetilde{V}}_{\rm
SW}}\int_{{\widetilde{q}}=2\pi f/{\widetilde{V}}_{\rm
SW}}^{\infty}\,{\widetilde{q}}\,S({\widetilde{q}})\,d{\widetilde{q}}\,\,\,,$
(20)

where in the last equality we have used the substitution
${\widetilde{q}}^{2}={\widetilde{q}}_{\perp}^{2}+(2\pi f/{\widetilde{V}}_{\rm
SW})^{2}$.

We can now construct the moments of the frequency spectrum of the observed
density fluctuations:

$\int_{0}^{\infty}f^{m}\,P(f)\,df=\frac{\alpha\,n^{2}}{2\pi\,{\widetilde{V}}_{\rm
SW}}\,\int_{f=0}^{\infty}f^{m}\,df\int_{{\widetilde{q}}=2\pi
f/{\widetilde{V}}_{\rm
SW}}^{\infty}{\widetilde{q}}\,S({\widetilde{q}})\,d{\widetilde{q}}\,\,\,,$
(21)

and reversing the order of integration gives

$\int_{0}^{\infty}f^{m}\,P(f)\,df=\frac{\alpha\,n^{2}}{2\pi\,{\widetilde{V}}_{\rm
SW}}\,\int_{{\widetilde{q}}=0}^{\infty}{\widetilde{q}}\,S({\widetilde{q}})\,d{\widetilde{q}}\,\int_{0}^{{\widetilde{q}}\,{\widetilde{V}}_{\rm
SW}/2\pi}f^{m}\,df\,\,\,.$ (22)

Two moments are of particular interest. First, for $m=0$,

$\int_{0}^{\infty}P(f)\,df=\frac{\alpha\,n^{2}}{(2\pi)^{2}}\,\int_{{\widetilde{q}}=0}^{\infty}{\widetilde{q}}^{2}\,S({\widetilde{q}})\,d{\widetilde{q}}=\alpha\,\frac{\langle\delta
n^{2}\rangle}{2}\,\,\,,$ (23)

where we have used Equation (1). The factor of 2 arises because only positive
frequencies are considered (e.g., for an isotropic distribution,
$\langle\delta
n^{2}\rangle=2\int_{o}^{\infty}P(f)\,df=\int_{-\infty}^{\infty}P(f)\,df$).
Second, for $m=1$,

$\int_{0}^{\infty}f\,P(f)\,df=\frac{\alpha\,n^{2}\,{\widetilde{V}}_{\rm
SW}}{2\,(2\pi)^{3}}\,\int_{{\widetilde{q}}=0}^{\infty}{\widetilde{q}}^{3}\,S({\widetilde{q}})\,d{\widetilde{q}}\,\,\,,$
(24)

and comparing this to the expression (14) for
$\overline{q\,\epsilon^{2}}\,R_{\odot}$, we immediately see that

$\overline{q\,\epsilon^{2}}\,R_{\odot}=4\,\frac{2\pi}{n^{2}{\widetilde{V}}_{\rm
SW}}\,R_{\odot}\,\int_{0}^{\infty}f\,P(f)\,df\,\,\,.$ (25)

Now, ${\widetilde{V}}_{\rm SW}=V_{\rm
SW}\,(\sin^{2}\theta_{B}+\alpha^{2}\cos^{2}\theta_{B})^{1/2}$, where
$\theta_{B}$ is the angle between the solar wind velocity and the (axis of
symmetry) magnetic field. Thus, if $\theta_{B}=0$,

$\overline{q\,\epsilon^{2}}\,R_{\odot}|_{\theta_{B}=0}=\frac{4}{\alpha}\,\frac{2\pi}{n^{2}\,V_{\rm
SW}}\,R_{\odot}\,\int_{0}^{\infty}f\,P(f)\,df\,\,\,,$ (26)

while, for a typical $1$ au angle $\theta_{B}\simeq 45^{o}$,

$\overline{q\,\epsilon^{2}}\,R_{\odot}|_{\theta_{B}=45^{\circ}}=4\,\sqrt{\frac{2}{1+\alpha^{2}}}\,\frac{2\pi}{n^{2}\,V_{\rm
SW}}\,R_{\odot}\,\int_{0}^{\infty}f\,P(f)\,df\,\,\,.$ (27)

The frequency-weighted integral over the measured fluctuation power spectrum
$P(f)$ at $1$ au can be readily evaluated numerically using the data from the
spacecraft measurements of Celnikier et al. (1983), Celnikier et al. (1987),
Kellogg & Horbury (2005), and Chen et al. (2012), allowing
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ to be determined at $1$ au. The
resulting values, for various values of $\alpha$, have been added to Figure 6,
and are consistent with the general form of
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ in Equation (15).

##  6 Amplitude of density fluctuations

Here we provide further observational constraints on the profile of the
spectrum-weighted mean wavenumber of density fluctuations
$\overline{q\,\epsilon^{2}}\,R_{\odot}$, using both observations of the
(dominant – Section 2) inner scale at the boundary between the inertial and
dissipative ranges of the turbulence wavenumber spectrum (Equation (2)), and
published results related to scintillations of extra-solar point radio
sources.

###  6.1 The inner scale of the turbulent wavenumber spectrum

As noted in Section 2, $\overline{q\,\epsilon^{2}}\,R_{\odot}$ is mostly
determined by density fluctuations near the inner scale $q_{i}^{-1}$, where
$q_{i}$ is the boundary between the inertial and dissipative ranges of the
wavenumber spectrum $S(q)$ (Equation (2)). Observations of the inner
turbulence scale thus allow to infer the magnitude of the spectrum-weighted
mean wavenumber of density fluctuations
$\overline{q\,\epsilon^{2}}\,R_{\odot}$.

Figure 7: Left: Measured turbulence inner scales $q_{i}^{-1}$ (Yakovlev et
al., 1980; Scott et al., 1983; Coles & Harmon, 1989; Manoharan, 1993b;
Anantharamaiah et al., 1994; Yamauchi et al., 1996, 1998), and magnetic field
fluctuations (Bruno & Trenchi, 2014; Chiba et al., 2022; Lotz et al., 2023),
together with theoretical profiles of key plasma parameters: the proton
inertial length $d_{i}=c/\omega_{\mathrm{pi}}$, the proton gyroradius
$\rho_{i}=v_{\mathrm{Ti}}/\omega_{\mathrm{ci}}$, the proton resonance distance
$d_{r}=(v_{\mathrm{Ti}}+v_{A})/\omega_{\mathrm{ci}}$, calculated using the
plasma and magnetic field parameters in Appendix A. The dotted curve shows the
relation $q_{i}^{-1}=2.5\times 10^{4}\,(r/R_{\odot}-1)^{1.3}$ cm and the
dashed line shows the inner scale model $q_{i}^{-1}=10^{5}\,r/R_{\odot}$ cm by
Coles & Harmon (1989). Right: The same as the left panel, but for the inner
scale wavenumber $q_{i}$.

Figure 7 summarizes the pertinent observations of this scale, together with
the behavior of several plasma parameters as a function of distance from the
Sun. These parameters include the proton inertial length
$d_{i}=c/\omega_{\mathrm{pi}}$, the proton gyroradius
$\rho_{i}=v_{\mathrm{Ti}}/\omega_{\mathrm{ci}}$, the the resonance distance
$d_{r}=(v_{\mathrm{Ti}}+v_{A})/\omega_{\mathrm{ci}}$ for parallel-propagating
Alfvén waves (Leamon et al., 1998; Bruno & Trenchi, 2014), the proton thermal
velocity $v_{\rm{Ti}}=\sqrt{2k_{B}T_{i}/m_{p}}$, the proton plasma frequency
$\omega_{\mathrm{pi}}=\sqrt{4\pi n\,e^{2}/m_{p}}$, the proton gyrofrequency
$\omega_{\mathrm{ci}}=eB/m_{p}c$, and the Alfvén speed $v_{A}=B/\sqrt{4\pi
nm_{p}}$. The density, temperature, and magnetic field profiles used to
construct these parameter variations are given in Appendix A. Over most of the
range from the lower corona ($0.1\,R_{\odot}$) to $1$ au and beyond, the inner
scale $q_{i}^{-1}$ is more consistent with the Alfvén wave resonance distance
rather than either the proton gyroradius or the proton inertial length. Coles
& Harmon (1989) suggest that the inner scale between $2$ and $60$ $R_{\odot}$
is close to $3c/\omega_{ci}$ and that the data can be approximated by the
relation $q_{i}^{-1}=10^{5}\,r/R_{\odot}$ (cm) (see Figure 7). Near $1$ au
($r\simeq 200\,R_{\odot}$), the gyroradius $\rho_{i}$ is smaller than the
resonant scattering length $d_{r}$, so that finite gyroradius effects dominate
the scattering process. Observations by Šafránková et al. (2015) suggest that
the location of the density break point between these two domains is
controlled by the gyrostructure frequency near 1 au.

Figure 8:  Left: Normalization coefficient $C_{n}^{2}(r)$ of the density
spectrum given by Equation (30), with
$q_{i}=\omega_{\mathrm{ci}}/(v_{\mathrm{Ti}}+v_{\mathrm{A}})$ (see Figure 7).
As in Figure 6, the solid/dashed lines are for anisotropy factors
$\alpha=0.25$ (fundamental) and $0.42$ (harmonic), respectively, and the grey
region denotes the range from
$[1/2,2]\times\overline{q\,\epsilon^{2}}\,R_{\odot}$. The data from Coles &
Harmon (1989) are calculated using the values of
$\overline{q\,\epsilon^{2}}\,R_{\odot}$, with $q_{i}$ inferred from the break
at the larger wavenumber of their density spectrum. They grey points show
where the inner scale was not well defined in their data. The data from Marsch
& Tu (1990) are found via a power law fit of the form $A\,q^{-5/3}$ to the
density spectra in their Figure 6, where the slope corresponds to a power law
index of $-5/3$, and $C_{n}^{2}(r)=A\,n^{2}(r)/4\pi$, with $n$ taken from
Equation (A1). The dotted line shows the relation $C_{n}^{2}(r)=3.5\times
10^{3}\,(r/R_{\odot}-1)^{-4.7}$. Right: As in the left panel, but divided by
$n^{2}$, with $n$ given by Equation (A1).

###  6.2 Observations of Extra-Solar Radio Point Sources

Reports of radio observations of extra-solar radio point sources (Rickett,
1977; Coles & Harmon, 1989; Sakurai & Spangler, 1994) typically provide values
of the quantity $C^{2}_{n}\,(r)$, which is the normalization constant of the
Kolmogorov density power spectrum:

$\Phi(r;q)=C^{2}_{n}\,(r)\,q^{-11/3}$ (28)

between the outer and inner scales $q_{o}<q<q_{i}$, so that444note the
$(2\pi)^{3}$ difference between this normalization and that of Equation (1)
$\delta n^{2}(r)=\int\Phi(r;q)\,d^{3}q$. Noting that $q_{o}\ll q_{i}$, one
finds by comparing Equations (1), (2), (14), and (28) that the scaling
quantity $C_{n}^{2}\,(r)$ (cm-20/3) is related to the
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ profile by

$C^{2}_{n}\,(r)=\overline{q\,\epsilon^{2}}\,R_{\odot}\,\frac{q_{i}^{-1/3}}{12\,\pi\,R_{\odot}}\,n^{2}\,(r)\,\,\,,$
(29)

where $n(r)$ is the plasma density. Including the dissipative range above
$q_{i}$ increases $\overline{q\,\epsilon^{2}}\,R_{\odot}$ by a factor of
$\sim$5/3 due to power-law in dissipative range (see Equation (4)), so that a
more appropriate expression, applicable to the wavenumber spectrum in Equation
(2), is

$C^{2}_{n}\,(r)=\overline{q\,\epsilon^{2}}\,R_{\odot}\,\frac{q_{i}^{-1/3}}{20\,\pi\,R_{\odot}}\,n^{2}\,(r)\,\,\,.$
(30)

This $C_{n}^{2}\,(r)$ profile is plotted in the left panel of Figure 8, with
the right panel showing the quantity
$C^{2}_{n}\,(r)/n^{2}\,(r)=\overline{q\,\epsilon^{2}}\,R_{\odot}\,q_{i}^{-1/3}/20\,\pi\,R_{\odot}$.
We note that the right side of Equation (30) depends weakly ($\propto
q_{i}^{-1/3}$) on the inner scale wavenumber, so that even if the inner scale
is uncertain to within a factor of two, the value of $C_{n}^{2}$ changes by
only 26%.

Figure 9:  Left: Amplitude of the inner-scale density fluctuations
$\langle\delta{n_{i}}^{2}\rangle$ calculated from Equation (32). The grey
regions are the same as in Figure 6. The dotted line shows the relation
$\langle\delta{n_{i}}^{2}\rangle=2\times 10^{7}\,(r/R_{\odot}-1)^{-3.7}$. The
inner scale for Coles & Harmon (1989) is taken from the start of the
dissipation range visible in their model spectra, and the grey points denote
where the inner scale was not well observed in their data. The Marsch & Tu
(1990) values are calculated using Equation (31); for their results the inner
scale is not observed and instead we use the resonant distance $d_{r}$ as the
measure of the inner scale. As in Figure 6, the black and red data points
correspond to the assumption of fundamental and harmonic emission,
respectively. Right: As in the left panel, but divided by $n^{2}$, with $n(r)$
given by Equation (A1).

In addition to the model profile (30), Figure 8 shows values of $C^{2}_{n}$
deduced from observations of scintillations of extra-solar sources (Spangler
et al., 2002). The $C^{2}_{n}$ values from Spangler et al. (2002) appear to be
significantly smaller than those in our simulations and in the in-situ
observations at $0.4$ and $0.8$ au by Marsch & Tu (1990). Since Type III solar
radio bursts are generated by electrons from active regions/flares and thus
propagate not far from the ecliptic (e.g., Musset et al., 2021), our
measurements are more consistent with the larger $C^{2}_{n}$ values from the
slow dense solar wind (see similar discussion and conclusion by Spangler et
al., 2002). Sasikumar Raja et al. (2016) suggest that solar cycle variations
of $C^{2}_{n}$ might contribute to variations between observations at
different times.

The profiles of $\overline{q\,\epsilon^{2}}\,R_{\odot}$ (Equation (15)) and of
the inner scale $q_{i}^{-1}(r)$ (from Figure 7) also allows us to estimate the
amplitude of the density fluctuations at the dominant inner scale. Following,
e.g., Chandran et al. (2009) and Sasikumar Raja et al. (2016), we define the
squared fractional density perturbation amplitude at the inner-scale
wavenumber $q_{i}$ by

$\frac{\langle\delta n_{i}^{2}\rangle}{n^{2}}=4\pi
q_{i}^{3}\,\frac{S(q_{i})}{(2\pi)^{3}}=4\pi
q_{i}^{3}\,\frac{\Phi(q_{i})}{n^{2}\,(r)}=4\pi\,q_{i}^{-2/3}\,\frac{C^{2}_{n}\,(r)}{n^{2}\,(r)}\,\,\,,$
(31)

which also relates the various wavenumber spectrum normalizations
$\langle\delta n_{i}^{2}\rangle/n^{2}$ (Equation (1)), $S(q_{i})$ (Equation
(2)), and $C_{n}^{2}$ (Equation (28)). One also finds by using Equation (4)
that the inner-scale fractional density fluctuation is given by

$\frac{\langle\delta
n_{i}^{2}\rangle\,}{n^{2}(r)}=\frac{\overline{q\,\epsilon^{2}}\,R_{\odot}}{\left(3+\frac{1}{\delta-2}\right)\,q_{i}(r)\,R_{\odot}}\simeq\frac{\overline{q\,\epsilon^{2}}\,R_{\odot}}{5\,q_{i}(r)\,R_{\odot}}\,\,\,,$
(32)

which can be evaluated using the turbulence profile of Equation (15) and the
form of $q_{i}^{-1}(r)$ from Figure 7. This, plus the density profile from
Equation (A1), gives the profile of the inner-scale squared density
fluctuation values $\langle\delta n_{i}^{2}\rangle$ (cm-6). This quantity, and
its dimensionless fractional value $\langle\delta n_{i}^{2}\rangle/n^{2}$, are
shown as functions of $r$ in Figure 9.

##  7 Discussion and Summary

We have constructed a density fluctuation model that allows quantitative
analysis of radio-wave propagation in a medium that is characterized by an
anisotropic density turbulence, symmetric around the direction of magnetic
field ${{\mathbf{\bm{B}}}}$. The density turbulence is characterized by two
parameters: the spectrum-weighted mean wavenumber of density fluctuations
$\overline{q\,\epsilon^{2}}$ (Equation (14)) and the anisotropy measure
$\alpha\simeq q_{\parallel}/q_{\perp}$ (Equation (5)). We allow
$\overline{q\,\epsilon^{2}}$ to be a function of solar distance $r$ from the
low corona ($\sim$1.1 $R_{\odot}$) to $215\,R_{\odot}$ ($1$ au), but the
anisotropy parameter $\alpha$ is, for simplicity, considered a constant. The
inferred profile of the dimensionless quantity
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ is found to have a broad maximum with
a value of about $100$ located at around $\sim$(4-7) $\,R_{\odot}$, where the
slow solar wind becomes supersonic (Sittler & Guhathakurta, 1999).
Intriguingly, this is also the range of solar distances where Type III bursts
are observed to have the highest radio spectral flux density (note the broad
peak near $1-2$ MHz; Sasikumar Raja et al., 2022). The density turbulence
model allows quantitative analysis of radio-wave scattering that in turn could
be used to decouple the intrinsic properties of solar radio bursts and the
effects of radio-wave propagation.

Analysis of the variation of the inner scale (i.e., the length associated with
the smallest eddies in the inertial range of the turbulence spectrum) with
solar distance presents a rather coherent picture. Radio measurements of
density inner scales (e.g., Coles & Harmon, 1989) are found to be in good
agreement with the inner scales deduced from magnetic fluctuations (e.g., Lotz
et al., 2023), supporting a close relation between magnetic fluctuations and
density fluctuations. It has been argued that kinetic Alfvén waves are a
compressive phenomenon that is responsible for both density, magnetic field,
and parallel electric field fluctuations near the break between inertial and
dissipation ranges (e.g., Chandran et al., 2009; Bian et al., 2010). Over a
wide range of distances, from the low solar corona into the solar wind, the
turbulence inner scale is comparable (within a factor of two) with the scale
of the resonant condition for protons
$(v_{\mathrm{Ti}}+v_{A})/\omega_{\mathrm{ci}}$, which is similar to the scale
of the break in the spectra of magnetic fluctuations (Bruno & Trenchi, 2014;
Lotz et al., 2023). Analysis of the inner scales (Coles & Harmon, 1989) at
(10-50) $R_{\odot}$ suggests that the proton inertial length
$c/\omega_{pi}=v_{A}/\omega_{ci}$ is a good approximation, while
$v_{Ti}/\omega_{pi}$ correlates with the break near $1$ au (Šafránková et al.,
2015). Since $(v_{\mathrm{Ti}}+v_{A})/\omega_{\mathrm{ci}}$ approaches
$v_{Ti}/\omega_{pi}$ at $r>50\,R_{\odot}$ and
$(v_{\mathrm{Ti}}+v_{A})/\omega_{\mathrm{ci}}$ is dominated by
$v_{A}/\omega_{ci}$ at $r>50\,R_{\odot}$, the resonance condition for protons
captures both sets of observations.

Figure 6 shows the profiles $\overline{q\,\epsilon^{2}}\,R_{\odot}$ inferred
from solar, non-solar and in-situ density fluctuation measurements; an
acceptable fit to the observations requires the anisotropy parameter to have a
value in the range $\alpha\simeq 0.2-0.4$. Unfortunately, observations of Type
III burst sizes and decay times do not systematically identify whether the
observed emission is at the fundamental frequency or its (second) harmonic,
and while this ambiguity does affect the best-fit value of the anisotropy
parameter $\alpha$, it has a much smaller effect on the inferred values of
$\overline{q\,\epsilon^{2}}\,R_{\odot}$. Assuming that the sources correspond
to emission at the fundamental suggests $\alpha\simeq 0.25$; however, if the
emission is at the harmonic then the anisotropy parameter is closer to $0.4$.
Assuming harmonic emission also requires a factor-of-two larger value of
$\overline{q\,\epsilon^{2}}\,R_{\odot}$, due to the $\alpha$-dependence in
Equation (15); see solid and dashed lines in Figure 6. Since the scattering
rate parallel to the magnetic field is a factor $\alpha^{2}\ll 1$ times the
perpendicular scattering rate (Equation (10)), the radio waves escape
predominantly parallel to the magnetic field, so that the degree of anisotropy
chiefly affects the duration and the decay time of the bursts, while the
source size is mostly controlled by the amplitude of the density fluctuations.
The high accuracy of the decay times of Type III bursts allows us to place
rather significant constraints on the anisotropy parameter; for example, any
value $\alpha>0.5$ is inconsistent with the available data.

Comparing the results of numerical simulations with observations of source
sizes and time profiles, over the range of frequencies $\sim$$0.1-300$ MHz, we
find that the turbulence profile $\overline{q\,\epsilon^{2}}\,R_{\odot}$ is,
within a factor of two, well represented by the empirical form Equation (15).
The spatial variation of the fractional density fluctuation amplitude at the
inner scale increases monotonically with heliocentric distance, with a
noticeable “knee” near the location corresponding to the maximum in
$\overline{q\,\epsilon^{2}}\,R_{\odot}$. The amplitude of the Kolmogorov
(1941) density turbulence spectrum varies with distance $r$ from the Sun as
$C_{n}^{2}\,(r)\simeq 3.5\times 10^{3}\,(r/R_{\odot}-1)^{-4.7}$ cm-20/3. The
quantity $C_{n}^{2}$ matches well with in-situ density fluctuation
measurements at $0.4$ and $0.8$ au for the slow solar wind and is generally
above the measurements of the fast solar wind (Marsch & Tu, 1990). However,
only the largest values of $C_{n}^{2}$ inferred from the slow solar wind
scintillation measurements of Spangler et al. (2002) are in agreement with our
model, possibly because Type III bursts propagate mostly in the ecliptic and
the scintillation measurements are mostly taken away from the ecliptic. The
normalized quantity $C_{n}^{2}/n^{2}$ demonstrates two distinct domains: it is
a constant (or close to a constant) for distances $>10R_{\odot}$, supporting
the conclusions of Spangler (2002), but exhibits a growth of three orders of
magnitude close to the Sun, reaching a maximum value near $(5-7)\,R_{\odot}$.

The variance of density fluctuations decreases with solar distance
approximately as $\langle\delta{n_{i}}^{2}\rangle\,(r)=2\times
10^{7}\,(r/R_{\odot}-1)^{-3.7}$ cm-6. The exponent in this expression is close
to $4$ and therefore is not inconsistent with the
$\langle\delta{n_{i}}^{2}\rangle\propto r^{-4}$ expectation (Zank et al.,
2017; Tasnim et al., 2022) associated with nearly-incompressible
magnetohydrodynamic turbulence. Comparison of the model with observations
does, however, suggest that a photosphere-rooted scaling (with a dependence on
the quantity $(r/R_{\odot}-1)$) could be a better rough symmetry than a simple
heliocentric symmetry (i.e., $\propto r^{-4}$), probably due to the turbulence
evolution along flux tubes rooted into the photosphere.

Figure 10: Frequency spectrum of density fluctuations (fundamental is solid,
harmonic is dashed line) in accordance with the density fluctuation model,
using Equation (C17) with the values of $\langle\delta n_{i}^{2}\rangle/n^{2}$
at $1$ au taken from Figure 9. The grey area shows the [1/2,2] multiplication-
factor interval used in the previous figures. The in-situ fluctuations data at
$1$ au (Celnikier et al., 1983, 1987; Kellogg & Horbury, 2005; Chen et al.,
2012, 2014) are shown for comparison.

The density fluctuations model presented in Equation (15) provides a unifying
picture for interpretation of solar radio bursts, extra-solar source
broadening, and in-situ measurements of density fluctuations in the solar
wind. As the model extends to $1$ au, we can predict (Appendix C.2; Equation
(C17)) the frequency spectrum and compare with those measured by spacecraft in
the slow solar wind. For a $1$ au value $\theta_{B}\simeq 45^{\circ}$, the
ratio of the spectral break frequency $\widetilde{f}_{i}$ to its value $f_{i}$
for an isotropic ($\alpha=1$) spectrum is $\sqrt{0.5(1+\alpha^{2})}$ (see
remark after Equation (C17)), which is weakly sensitive to $\alpha$ in the
range $0.25-0.4$, so that the spectral break point is about the same for all
$\alpha$ values in this range. Figure 10 demonstrates good agreement between
the model and observations near the break of the spectrum at $f\simeq f_{i}$.
The discrepancy below the break frequency is likely to be associated with
flattening of the density fluctuation spectra due to kinetic Alfvén waves
(Chandran et al., 2009) and has been observed close to the Sun before (e.g.,
Coles & Harmon, 1989). As discussed, this flattening is not included into our
model, but could be used for further improvement of the model.

An important feature of the model is that it provides a quantitative
description for solar burst source sizes, apparent positions, and decay times,
with the results summarized below:

Source sizes: Type III burst source sizes are predominantly determined by
radio-wave scattering over the entire range of frequencies and follow a $1/f$
trend (both in the simulations and observations) from $300$ MHz down to $0.1$
MHz. The source sizes do not depend on burst intensity (Saint-Hilaire et al.,
2013) and the source sizes for fine structures in Type III bursts and Type III
spikes are rather similar (e.g., Kontar et al., 2017; Clarkson et al., 2023).
At frequencies below $0.1$ MHz, the source sizes exceed $120$ degrees (see
Figure 2), comparable to half the sky, so that the source effectively
surrounds the observer.

Source positions: From Figure 5 we see that for high frequency sources between
$100$ and $300$ MHz the apparent source positions for both fundamental
($\omega=\omega_{pe}$) and harmonic ($\omega=2\,\omega_{pe}$) emission are at
a location corresponding to emission at double the plasma frequency. However,
for emission frequencies from $\sim$$100$ MHz down to $\sim$$0.1$ MHz, the
apparent source location is further away from the Sun than the location
corresponding to emission at the plasma frequency or even to the location
corresponding to emission at double the plasma frequency. This result
elegantly resolves the long-standing conundrum that source positions observed
at the fundamental and the harmonic are virtually co-spatial (Suzuki & Dulk,
1985): the apparent source position is moved away from the Sun by scattering
effects, which are largely independent of whether fundamental or harmonic
emission is involved. At low frequencies $\,\lower 3.0pt\hbox{$\sim$}\hbox
to0.0pt{\hss\raise 2.0pt\hbox{$<$}}\,0.2$ MHz, this effect is even more
pronounced: the source positions are shifted by 0.5 au, so that an observer at
$1$ au is actually embedded in the apparent source.

Type III burst decay times: The decay time of Type III solar radio bursts is
well approximated by an inverse-frequency ($1/f$) dependence between $0.1$ and
$300$ MHz. Our scattering simulations reveal a rather intricate picture, in
which scattering serves to provide a fundamental “floor” on the observed
duration of radio bursts emitted via plasma emission. Below $1$ MHz, the
average Type III burst decay time is consistent with the scattering
simulations, but at frequencies above a few MHz, the typical Type III burst
decay time is longer than the simulated value, suggesting that the decay time
for such bursts is determined by processes that operate on timescales longer
than scattering, such as electron propagation, electron injection, and/or
intrinsic emission time scales. The shortest features in the dynamic spectrum,
such as Type IIIb striae (e.g., Kontar et al., 2017; Sharykin et al., 2018),
drift pairs (Kuznetsov & Kontar, 2019), spikes (Clarkson et al., 2021), and
fine structures in type II solar radio bursts (Chrysaphi et al., 2018) are all
consistent with the scattering model. Statistical analysis of $\sim 1$$100$
fine structures (Clarkson et al., 2023) near $30$ MHz, with $0.02$ s
resolution, indeed shows that there are virtually no bursts shorter than $0.2$
s and that the typical burst decay time is about $0.3$ s, consistent with the
simulations in Figure 3.

In summary, solar burst shortest time profiles, source sizes, and positions
are determined mainly by propagation effects (mostly anisotropic scattering)
and not by intrinsic properties of the radio emission source. A detailed
knowledge of the scattering process paves the way to disentangling scattering
effects from observations and so better constraining the intrinsic properties
of solar radio burst sources. Since individual source sizes and decay times
are typically measured more accurately than the spread in measurements of
multiple sources, it is therefore likely that the spread in observational
properties is due to varying levels of turbulence and plasma density in
different events. Varying the magnitude of the
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ profile (Equation (15)) by a factor in
the range $0.5$ to $2$ covers the majority of the observations, while
extending this multiplicative factor by a further factor of two (to a range
between $0.25$ to $4$) covers virtually all observed data points (except for
some extreme outliers). We thus conclude that the
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ profile of Equation (15) is variable
within a factor of about two. The broadening of extra-solar point sources by
the turbulent solar atmosphere and solar radio burst measurements are
complementary data sets. We note the considerable data gap between ground-
based and space-based solar burst observations in the range $3-20$ MHz (where
extra-solar observations appear essential), and encourage the development of
observations to fill this gap and hence further constrain the level of
turbulence in the inner heliosphere. The Sun Radio Interferometer Space
Experiment (SunRISE) is likely to provide much needed data.

We thank P. Massa, Prasad Subramanian, Tim Bastian, Arnaud Zaslavsky, and
Yingjie Luo for helpful discussions. EPK, DLC and XC acknowledge financial
support from the STFC Consolidated Grant ST/T000422/1. AGE was supported by
NASA Kentucky under NASA award number 80NSSC21M0362. NC acknowledges funding
support from the Initiative Physique des Infinis (IPI), a research training
program of the Idex SUPER at Sorbonne University. We also acknowledge support
from the International Space Science Institute for the LOFAR
http://www.issibern.ch/teams/lofar/ and solar flare
http://www.issibern.ch/teams/solflareconnectsolenerg/ teams.

##  Appendix A Plasma density, magnetic field and temperature profiles

The density profile $n(r)$ (cm-3) used for the solar burst scattering
simulations by Kontar et al. (2019), Kuznetsov et al. (2020), and Chen et al.
(2020, 2023) is

$n(r)=4.8\times 10^{9}\left(\frac{R_{\odot}}{r}\right)^{14}+3\times
10^{8}\left(\frac{R_{\odot}}{r}\right)^{6}+1.4\times
10^{6}\left(\frac{R_{\odot}}{r}\right)^{2.3}\,\,\,.$ (A1)

This density profile (i) closely follows the Parker (1958) density model, (ii)
has an analytical form that can be easily differentiated to find the
derivatives that are useful in solving the ray-tracing equations, (iii) is
consistent at low heights with the higher densities observed in flaring active
regions that produce Type III bursts (e.g., Holman et al., 2011; Reid et al.,
2014), and (iv) is suitable for Type III burst modelling in the solar corona
and heliosphere (Mann et al., 1999; Kontar, 2001; Reid & Kontar, 2021). The
left panel of Figure 11 shows the model profile compared with different
density models frequently used in the literature.

Figure 11: Left panel: Coronal density model (black solid line) given by
Equation (A1). Other coronal density models (Allen, 1947; Newkirk, 1961;
Alvarez & Haddock, 1973b; Saito et al., 1977; Bougeret et al., 1984b; Bird et
al., 1994; Leblanc et al., 1998; Mancuso & Spangler, 2000) are also shown,
within the ranges of their respective validity. The models from Bird et al.
(1994) and Mancuso & Spangler (2000) are as used in Spangler (2002). Right
panel: Magnetic field model (black solid line) given by Equation (A2),
together with observations (Burlaga & Ness, 1968; Villante & Mariani, 1975;
Behannon, 1976; Rosenberg et al., 1978; Dulk & McLean, 1978; Patzold et al.,
1987; Sakurai & Spangler, 1994; Lin et al., 2000; Spangler, 2005; Ramesh et
al., 2010; Gopalswamy et al., 2012; Kolotkov et al., 2018; Spangler, 2020).
The red line shows the model of Dulk & McLean (1978)

The magnetic field $B(r)$ (G) model, shown in the right panel of Figure 11, is
given by

$B(r)=0.5\,\frac{\left(\frac{r}{R_{\odot}}-1\right)^{-1.5}}{\,\left(\frac{r}{10\,R_{\odot}}+1\right)}+1.18\,\left(\frac{R_{\odot}}{r}\right)^{2}\,\,\,,$
(A2)

where the first term is introduced to accommodate the results of Dulk & McLean
(1978), applicable for near-solar distances $r\,\lower 3.0pt\hbox{$\sim$}\hbox
to0.0pt{\hss\raise 2.0pt\hbox{$<$}}\,10\,R_{\odot}$, into the form of the
interplanetary magnetic field for $r\gg 10\,R_{\odot}$ (see discussion by
Patzold et al., 1987).

The ion temperature $T_{i}$ (K) is given by

$T_{i}(r)=2.4\times
10^{6}\,\left(\frac{r}{10\,R_{\odot}}+1\right)^{-0.74}\,\mbox{ K}\,\,\,,$ (A3)

where the interplanetary dependency $r^{-0.74}$ (with $T_{i}=2.5\times 10^{5}$
K at $1$ au $=215\,R_{\odot}$) is taken from Hellinger et al. (2011) and has
been adjusted to accommodate the range $r\,\lower 3.0pt\hbox{$\sim$}\hbox
to0.0pt{\hss\raise 2.0pt\hbox{$<$}}\,10\,R_{\odot}$ in accordance with the
results of Dulk & McLean (1978).

##  Appendix B Angular broadening of extra-solar point sources

The overall angular broadening of a point source at infinity can be calculated
by integrating the angular broadening rate along the path ($z$-direction)
traversed by the radio waves. For the most significant broadening, i.e., that
along the $\perp_{2}$ direction perpendicular to the (radial) magnetic field
in the plane of the sky, the rate of diffusive broadening is given by Equation
(10), repeated here:

$\frac{d\,\langle\theta_{\perp}^{2}\rangle}{dz}=\frac{\pi}{8}\,\frac{\omega_{pe}^{4}}{\omega^{4}}\,\,\overline{q\,\epsilon^{2}}\,\,\,\,.$

If $r$ is the distance of closest approach to the Sun of a ray from a distance
point source (Figure 2), then the apparent source, observed at $1$ au, will be
broadened by an amount

$\langle\theta_{\perp}^{2}\rangle=\frac{\pi}{8\,\omega^{4}}\,\int_{-\infty}^{1~{}{\rm
au}}\omega_{pe}^{4}\left(\sqrt{r^{2}+z^{2}}\right)\,\,\overline{q\,\epsilon^{2}}\left(\sqrt{r^{2}+z^{2}}\right)\,dz=\frac{2\,\pi^{3}\,e^{4}}{m_{e}^{2}\,\omega^{4}}\,\int_{-\infty}^{1~{}{\rm
au}}n^{2}\left(\sqrt{r^{2}+z^{2}}\right)\,\,\overline{q\,\epsilon^{2}}\left(\sqrt{r^{2}+z^{2}}\right)\,dz\,\,\,,$
(B1)

where we have used $\omega_{pe}=\sqrt{4\pi ne^{2}/m_{e}}$. With the
substitution $z=r\tan\chi$, this can be written

$\langle\theta_{\perp}^{2}\rangle=\alpha\frac{2\,\pi^{3}\,e^{4}}{m_{e}^{2}\,\omega^{4}}\,\left(\frac{r}{R_{\odot}}\right)\,\int_{-\pi/2}^{\tan^{-1}(215\,R_{\odot}/r)}n^{2}(r\sec\chi)\,\,\overline{q\,\epsilon^{2}}\,R_{\odot}(r\sec\chi)\,\sec^{2}\chi\,d\chi\,\,\,.$
(B2)

Equation (B2) provides the formal expression corresponding to the approximate
proportionalities in Equation (13). Using the form of
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ from our nominal model (15) and the
density model from Equation (A1), we obtain

$\displaystyle\langle\theta_{\perp}^{2}\rangle$ $\displaystyle=$
$\displaystyle 2\times
10^{3}\,\alpha\,\times\,\left(\frac{e^{4}}{8\pi\,m_{e}^{2}\,f^{4}}\right)\,\left(\frac{r}{R_{\odot}}\right)^{0.3}\,\times$
(B3)
$\displaystyle\int_{-\pi/2}^{\tan^{-1}(215\,R_{\odot}/r)}\,\left[4.8\times
10^{9}\left(\frac{R_{\odot}}{r}\right)^{14}\,\cos^{14}\chi+3\times
10^{8}\left(\frac{R_{\odot}}{r}\right)^{6}\,\cos^{6}\chi+1.4\times
10^{6}\left(\frac{R_{\odot}}{r}\right)^{2.3}\,\cos^{2.3}\chi\right]^{2}\,\times$
(B5)
$\displaystyle\qquad\qquad\times\,\left(1-\frac{R_{\odot}\,\cos\chi}{r}\right)^{2.7}\,\frac{d\chi}{\cos^{1.3}\chi}\,\,\,.$

At large closest-approach distances $r\gg R_{\odot}$, the density profile (A1)
is well approximated by the last term in the square brackets, i.e., by the
simple power-law $n(r)\propto r^{-2.3}$, and the term
$(1-R_{\odot}\cos\chi/r)^{2.7}\simeq 1$. In this situation, the FWHM size
assumes the relatively simple form

${\rm FWHM}_{\perp}=2.35\,\sqrt{\alpha}\times\sqrt{2\times
10^{3}}\times(1.4\times 10^{6})\times\,\left[\frac{{\rm B}\,(1;2.15,0.5)+{\rm
B}\,(\psi;2.15,0.5)}{16\pi}\,\right]^{1/2}\,\left(\frac{e^{2}}{m_{e}}\right)\,\left(\frac{r}{R_{\odot}}\right)^{-2.15}\,\times\frac{1}{f^{2}}\,\,\,,$
(B6)

where ${\rm B}(\psi;u,v)$ are the (incomplete if $\psi<1$) beta functions
corresponding to the integrals $2\,\int_{-\pi/2}^{0}\cos^{3.3}\chi\,d\chi$ and
$2\,\int_{0}^{\tan^{-1}(215R_{\odot}/r)}\cos^{3.3}\chi\,d\chi$, respectively,
and $\psi=[1+(r/215\,R_{\odot})^{2}]^{-1}$. Scaling to the nominal frequency
$f=1.5$ GHz (Figure 2), this evaluates to

$\text{FWHM}_{\perp}\simeq 4\times
10^{-3}\,\sqrt{\alpha}\,\left(\frac{r}{R_{\odot}}\right)^{-2.15}\left(\frac{1.5\,\text{GHz}}{f}\right)^{2}\,{\rm
radians}\,\simeq\,800\,\sqrt{\alpha}\,\left(\frac{r}{R_{\odot}}\right)^{-2.15}\left(\frac{1.5\,\text{GHz}}{f}\right)^{2}\,{\rm
arcseconds}\,\,\,.$ (B7)

Equation (B7) provides a simple, but accurate, analytical approximation for
FWHM⟂, valid for $r\gg R_{\odot}$.

##  Appendix C Forms of the in situ density fluctuation frequency spectra

The in situ frequency power spectrum $P(f)$ of density fluctuations measured
by a spacecraft is related to the wavenumber spectrum
$S({{{\mathbf{\bm{q}}}}})$ at a single location in the solar wind frame
through the relation (16), repeated here:

$P(f)=\frac{n^{2}}{(2\pi)^{3}}\,\int
S({{{\mathbf{\bm{q}}}}})\,\delta\left(\frac{{{{\mathbf{\bm{q}}}}}\cdot{{{\mathbf{\bm{V}}}}}_{\rm
SW}}{2\pi}-f\right)\,d^{3}q\,\,\,.$

In this Appendix we explore the forms of $P(f)$ for different forms of the
wavenumber spectrum $S({{\mathbf{\bm{q}}}})$.

###  C.1 Isotropic Wavenumber Spectrum

Our first example is the simplest case of an isotropic spectrum:
$S({{{\mathbf{\bm{q}}}}})=S(q)$. For such a case, Equation (16) may be written

$\displaystyle P(f)\,$ $\displaystyle=$
$\displaystyle\,\frac{n^{2}}{(2\pi)^{2}}\,\int_{q_{\perp}=0}^{\infty}\,\int_{q_{\parallel}=-\infty}^{\infty}S\left(\sqrt{q_{\parallel}^{2}+q_{\perp}^{2}}\right)\,\delta\left(\frac{q_{\parallel}V_{\rm
SW}}{2\pi}-f\right)\,q_{\perp}\,dq_{\perp}\,dq_{\parallel}$ (C1)
$\displaystyle=$ $\displaystyle\,\frac{n^{2}}{2\pi\,V_{\rm
SW}}\,\int_{q_{\perp}=0}^{\infty}S\left(\sqrt{q_{\perp}^{2}+\left(\frac{2\pi
f}{V_{\rm SW}}\right)^{2}}\right)\,q_{\perp}\,dq_{\perp}\,\,\,,$ (C2)

Here, we evaluate this for a specific form of $S(q)$, viz. the broken power-
law around $q_{i}$ (cf. Equation (2)), with an outer scale $q_{o}^{-1}\gg
q_{i}^{-1}$, so that the contribution to
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ from the large scales $>q_{o}^{-1}$ is
negligible:

$S(q)=S(q_{i})\,\times\,\begin{cases}\left(\frac{q}{q_{i}}\right)^{-11/3};\qquad
q\leq q_{i}\cr\left(\frac{q}{q_{i}}\right)^{-2-\delta};\qquad
q>q_{i}\,\,\,.\end{cases}$ (C3)

Evaluation of the fluctuation frequency spectrum $P(f)$ requires consideration
of two domains:

(i) $f\leq f_{i}=\frac{q_{i}V_{\rm SW}}{2\pi}$

Here the resonant $q_{\parallel}=2\pi f/V_{\rm SW}<q_{i}$, so that for values
of $q_{\perp}^{2}<q_{i}^{2}-q_{\parallel}^{2}$ the lower (inertial) branch of
the wavenumber spectrum applies, while for values of
$q_{\perp}^{2}>q_{i}^{2}-q_{\parallel}^{2}$ the upper (dissipative) branch of
the wavenumber spectrum is relevant. The integral is thus composed of two
parts:

$\displaystyle P(f)\\!\\!\\!\\!$ $\displaystyle=\frac{n^{2}}{2\pi\,V_{\rm
SW}}\,S(q_{i})$
$\displaystyle\\!\\!\\!\\!\left[q_{i}^{11/3}\,\int_{q_{\perp}=0}^{\sqrt{q_{i}^{2}-(2\pi
f/V_{\rm SW})^{2}}}\left(q_{\perp}^{2}+\left(\frac{2\pi f}{V_{\rm
SW}}\right)^{2}\right)^{-11/6}\,q_{\perp}\,dq_{\perp}\,\right.+$ (C5)
$\displaystyle\qquad+\left.\,q_{i}^{\delta+2}\,\int_{q_{\perp}={\sqrt{q_{i}^{2}-(2\pi
f/V_{\rm SW})^{2}}}}^{\infty}\left(q_{\perp}^{2}+\left(\frac{2\pi f}{V_{\rm
SW}}\right)^{2}\right)^{-1-\frac{\delta}{2}}\,q_{\perp}\,dq_{\perp}\right]$
$\displaystyle\,\,\,\,=\frac{n^{2}}{4\pi^{2}\,f_{i}}\,q_{i}^{3}\,S(q_{i})$
$\displaystyle\,\left[\frac{3}{5}\left(\left(\frac{2\pi f}{q_{i}\,V_{\rm
SW}}\right)^{-5/3}-1\right)+\frac{1}{\delta}\right]$ (C6)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{\langle\delta
n_{i}^{2}\rangle}{2\,f_{i}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[\frac{3}{5}\left(\left(\frac{f}{f_{i}}\right)^{-5/3}-1\right)+\frac{1}{\delta}\right]\,\,\,,$
(C7)

where in the last equality we have used Equation (31).

(ii) $f>f_{i}=\frac{q_{i}V_{\rm SW}}{2\pi}$

Here the resonant $q_{\parallel}=2\pi f/V_{\rm SW}>q_{i}$, so that for all
values of $q_{\perp}$ only the upper branch of the wavenumber spectrum is
relevant. Thus

$\displaystyle P(f)\,$ $\displaystyle=$
$\displaystyle\,\frac{n^{2}}{2\pi\,V_{\rm
SW}}\,S(q_{i})\,q_{i}^{\delta+2}\,\int_{q_{\perp}=0}^{\infty}\left(q_{\perp}^{2}+\left(\frac{2\pi
f}{V_{\rm SW}}\right)^{2}\right)^{-1-\frac{\delta}{2}}\,q_{\perp}\,dq_{\perp}$
(C8) $\displaystyle=$
$\displaystyle\,\frac{n^{2}}{4\pi^{2}\,f_{i}}\,q_{i}^{3}\,S(q_{i})\,\frac{1}{\delta}\,\left(\frac{2\pi
f}{q_{i}\,V_{\rm SW}}\right)^{-\delta}$ (C9) $\displaystyle=$
$\displaystyle\,\frac{\langle\delta
n_{i}^{2}\rangle}{2\,f_{i}}\frac{1}{\delta}\,\left(\frac{f}{f_{i}}\right)^{-\delta}$
(C10)

Combining Equations (C5) and (C8),

$P(f)=\frac{\langle\delta
n_{i}^{2}\rangle}{2\,f_{i}}\,\times\,\begin{cases}\frac{3}{5}\,\left[\left(\frac{f}{f_{i}}\right)^{-5/3}\,-\,\left(1-\frac{5}{3\delta}\right)\,\right]\,\,\,\,\,\,\,\,\,;\qquad
f\leq
f_{i}\cr\frac{1}{\delta}\,\left(\frac{f}{f_{i}}\right)^{-\delta}\qquad\qquad\qquad\qquad\,\,;\qquad
f>f_{i}\,\,\,.\end{cases}$ (C11)

As a check on this result, we can evaluate (using algebra) the quantity

$\int_{0}^{\infty}f\,P(f)\,df=\frac{\langle\delta
n_{i}^{2}\rangle}{2\,f_{i}}\,\left(3+\frac{1}{\delta-2}\right)\,\frac{f_{i}^{2}}{2}=\frac{1}{4}\,\left(3+\frac{1}{\delta-2}\right)\,\langle\delta
n_{i}^{2}\rangle\,f_{i}\,\,\,.$ (C12)

so that, using the general result (25), with $\alpha=1$ for an isotropic
spectrum, the value of $\overline{q\,\epsilon^{2}}\,R_{\odot}$ is

$\overline{q\,\epsilon^{2}}\,R_{\odot}=\frac{8\pi}{n^{2}\,V_{\rm
SW}}\,R_{\odot}\,\int_{0}^{\infty}f\,P(f)\,df=\left(3+\frac{1}{\delta-2}\right)\,(q_{i}\,R_{\odot})\,\frac{\langle\delta
n_{i}^{2}\rangle}{n^{2}}\,\,\,,$ (C13)

in agreement with Equation (32).

The local power-law index of the frequency power spectrum is the negative of
its logarithmic derivative, viz.

$\delta_{\rm loc}=\,-\,\frac{d\ln P(f)}{d\ln
f}\equiv-\,\frac{f}{P}\,\frac{dP}{df}\,=\begin{cases}\frac{5/3}{1-\left(1-\frac{5}{3\delta}\right)\left(\frac{f}{f_{i}}\right)^{5/3}}\,\,\,\,\,\,\,;\qquad
f\leq f_{i}\cr\delta\,\,\,\qquad\qquad\qquad\,\,;\qquad
f>f_{i}\,\,\,.\end{cases}$ (C14)

At low frequencies $f\ll f_{i}$, the frequency spectrum has a power-law index
$\delta_{\rm loc}=5/3$; as $f$ increases, the spectrum gradually steepens to
$\delta_{\rm loc}=\delta$ as $f$ approaches $f_{i}$, and continues with that
value thereafter. Thus a double-power law spectrum in wavenumber does not
correspond to a double power-law spectrum in frequency, neither does a double-
power law in frequency correspond to a double power-law in wavenumber space.
Interestingly, the variation in the spectrum of magnetic fluctuations reported
by Sioulas et al. (2023, their Fig. 2) near the break frequency $f_{i}$, is
broadly consistent with Equation (C14).

###  C.2 Anisotropic density turbulence Spectrum with anisotropic spectral
break points

For anisotropic turbulence with a spectral break point that is also
anisotropic (see left panel of Figure 12),

$S(q)=S(q_{i})\,\times\,\begin{cases}\left(\frac{\sqrt{q_{\perp}^{2}+\frac{q_{\parallel}^{2}}{\alpha^{2}}}}{q_{i}}\right)^{-11/3};\qquad\sqrt{q_{\perp}^{2}+\frac{q_{\parallel}^{2}}{\alpha^{2}}}\leq
q_{i}\cr\left(\frac{\sqrt{q_{\perp}^{2}+\frac{q_{\parallel}^{2}}{\alpha^{2}}}}{q_{i}}\right)^{-2-\delta};\qquad\sqrt{q_{\perp}^{2}+\frac{q_{\parallel}^{2}}{\alpha^{2}}}>q_{i}\,\,\,.\end{cases}$
(C15)

The basic result (16) can be transformed from the wavevector
${{{\mathbf{\bm{q}}}}}$ to the variable $\widetilde{{{\mathbf{\bm{q}}}}}$ (cf.
Equation (17)), giving

$P(f)=\alpha\frac{n^{2}}{(2\pi)^{3}}\,\int
S(\widetilde{{{\mathbf{\bm{q}}}}})\,\delta\left(\frac{\widetilde{{{\mathbf{\bm{q}}}}}\cdot\widetilde{{{\mathbf{\bm{V}}}}}_{\rm
SW}}{2\pi}-f\right)\,d^{3}\widetilde{q}\,\,\,,$ (C16)

where $\widetilde{{{\mathbf{\bm{V}}}}}_{\rm SW}=(V_{\perp},\alpha
V_{\parallel})$. Since $S({\widetilde{{{\mathbf{\bm{q}}}}}})$ is isotropic, a
similar analysis to that of Appendix C.1 yields the spectrum

$P(f)=\alpha\,\frac{\langle\delta
n_{i}^{2}\rangle}{2\,\widetilde{f}_{i}}\,\times\,\begin{cases}\frac{3}{5}\,\left[\left(\frac{f}{\widetilde{f}_{i}}\right)^{-5/3}\,-\,\left(1-\frac{5}{3\delta}\right)\,\right]\,\,\,\,\,\,\,\,\,;\qquad
f\leq\widetilde{f}_{i}\cr\frac{1}{\delta}\,\left(\frac{f}{\widetilde{f}_{i}}\right)^{-\delta}\qquad\qquad\qquad\qquad\,\,;\qquad
f>\widetilde{f}_{i}\,\,\,,\end{cases}$ (C17)

where $\widetilde{f}_{i}=q_{i}{\widetilde{V}}_{\rm SW}/2\pi$, with
$\widetilde{V}_{\rm SW}=V_{\rm
SW}\left(\sin^{2}\theta_{B}+\alpha^{2}\cos^{2}\theta_{B}\right)^{1/2}$,
$\theta_{B}$ being the angle between the solar wind velocity
${{{\mathbf{\bm{V}}}}}_{\rm SW}$ and the magnetic field
${{{\mathbf{\bm{B}}}}}$. The form of the local power-law spectral index
$\delta_{\rm loc}$ is similar to Equation (C14), but with $f_{i}$ replaced by
$\widetilde{f}_{i}$. In the case where the solar wind velocity
${{{\mathbf{\bm{V}}}}}_{\rm SW}$ is parallel to the magnetic field
${{{\mathbf{\bm{B}}}}}$, ${\widetilde{V}}_{\rm SW}=\alpha V_{\rm SW}$,
${\widetilde{f}}_{i}=\alpha\,f_{i}$, and so

$P(f)=\frac{\langle\delta
n_{i}^{2}\rangle}{2\,f_{i}}\,\times\,\begin{cases}\frac{3}{5}\,\left[\left(\frac{f}{\alpha\,f_{i}}\right)^{-5/3}\,-\,\left(1-\frac{5}{3\delta}\right)\,\right]\,\,\,\,\,\,\,\,\,;\qquad
f\leq\alpha\,f_{i}\cr\frac{1}{\delta}\,\left(\frac{f}{\alpha\,f_{i}}\right)^{-\delta}\qquad\qquad\qquad\qquad\,\,;\qquad
f>\alpha\,f_{i}\,\,\,,\end{cases}$ (C18)

Figure 12: Forms of the parallel and perpendicular wavenumber spectra
corresponding to the two cases described in Appendices C.2 (Equation (C15);
left) and C.3 (Equation (C19); right). In the former case, the spectral break
points are different for parallel and perpendicular directions, while in the
latter case the spectral break points for both directions are the same. In
both plots, both axes are on logarithmic scales.

###  C.3 Anisotropic spectrum with isotropic break point

It has been argued by Armstrong et al. (1990) that a wavenumber spectrum that
breaks at a prescribed value of $q=\sqrt{q_{\parallel}^{2}+q_{\perp}^{2}}$ in
all directions (despite the anisotropy in the wavenumber distribution) is a
better fit to the observations. This corresponds to the following density
turbulence spectrum (see right panel of Figure 12):

$S({{\mathbf{\bm{q}}}})=S(q_{i})\,\times\,\begin{cases}\left(\frac{\sqrt{q_{\perp}^{2}+\frac{q_{\parallel}^{2}}{\alpha^{2}}}}{q_{i}}\right)^{-11/3};\qquad\sqrt{q_{\perp}^{2}+q_{\parallel}^{2}}\leq
q_{i}\cr\left(\frac{\sqrt{q_{\perp}^{2}+\frac{q_{\parallel}^{2}}{\alpha^{2}}}}{q_{i}}\right)^{-2-\delta};\qquad\sqrt{q_{\perp}^{2}+q_{\parallel}^{2}}>q_{i}\,\,\,.\end{cases}$
(C19)

Note that the argument of $S$ reflects the anisotropy in
$S({{{\mathbf{\bm{q}}}}})$ and so is different than the isotropic case of of
Appendix C.1 (Equation (C3)). However, the spectral break point is the same as
in Appendix C.1, and so differs from Appendix C.2, where the spectral break
point occurs at different values of $q$ for different directions (Equation
(C15)).

As before, the resonance occurs when $q_{\parallel}=2\pi f/V_{\rm SW}$, and so
we have, similar to Equation (C1), but with an additional factor of $\alpha$
appearing in the argument of $S$,

$P(f)=\frac{n^{2}}{2\pi\,V_{\rm
SW}}\,\int_{q_{\perp}=0}^{\infty}S\left(\sqrt{q_{\perp}^{2}+\left(\frac{2\pi
f}{\alpha\,V_{\rm SW}}\right)^{2}}\right)\,q_{\perp}\,dq_{\perp}\,\,\,.$ (C20)

The derivation of $P(f)$ proceeds similarly to Appendix C.1, but recognizes
the appearance of the factor $\alpha$ in the second term in the argument of
$S$:

(i) $f\leq f_{i}=\frac{q_{i}V_{\rm SW}}{2\pi}$

$\displaystyle P(f)$ $\displaystyle=\frac{n^{2}}{2\pi\,V_{\rm SW}}\,S(q_{i})$
$\displaystyle\,\left[q_{i}^{11/3}\,\int_{q_{\perp}=0}^{\sqrt{q_{i}^{2}-(2\pi
f/V_{\rm SW})^{2}}}\left(q_{\perp}^{2}+\left(\frac{2\pi f}{\alpha\,V_{\rm
SW}}\right)^{2}\right)^{-11/6}\,q_{\perp}\,dq_{\perp}\,\right.+$ (C22)
$\displaystyle\qquad+\left.\,q_{i}^{\delta+2}\,\int_{q_{\perp}={\sqrt{q_{i}^{2}-(2\pi
f/V_{\rm SW})^{2}}}}^{\infty}\left(q_{\perp}^{2}+\left(\frac{2\pi
f}{\alpha\,V_{\rm
SW}}\right)^{2}\right)^{-1-\frac{\delta}{2}}\,q_{\perp}\,dq_{\perp}\right]$
$\displaystyle=\frac{n^{2}}{4\pi^{2}\,f_{i}}\,q_{i}^{3}\,S(q_{i})$
$\displaystyle\,\left[\frac{3}{5}\left\\{\left(\frac{2\pi
f}{\alpha\,q_{i}\,V_{\rm SW}}\right)^{-5/3}-\left(1-\left(\frac{2\pi
f}{q_{i}\,V_{\rm
SW}}\right)^{2}\,\left(1-\frac{1}{\alpha^{2}}\right)\right)^{-5/6}\right\\}\right.+$
(C24) $\displaystyle\qquad+\left.\frac{1}{\delta}\,\left(1-\left(\frac{2\pi
f}{q_{i}\,V_{\rm
SW}}\right)^{2}\,\left(1-\frac{1}{\alpha^{2}}\right)\right)^{-\delta/2}\right]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{\langle\delta
n_{i}^{2}\rangle}{2\,f_{i}}\,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left[\frac{3}{5}\left\\{\left(\frac{f}{\alpha
f_{i}}\right)^{-5/3}-\left(1-\left(\frac{f}{f_{i}}\right)^{2}\,\left(1-\frac{1}{\alpha^{2}}\right)\right)^{-5/6}\right\\}+\frac{1}{\delta}\,\left(1-\left(\frac{f}{f_{i}}\right)^{2}\,\left(1-\frac{1}{\alpha^{2}}\right)\right)^{-\delta/2}\right]\,\,\,.$
(C25)

(ii) $f>f_{i}=\frac{q_{i}V_{\rm SW}}{2\pi}$

$\displaystyle P(f)\,$ $\displaystyle=$
$\displaystyle\,\frac{n^{2}}{2\pi\,V_{\rm
SW}}\,S(q_{i})\,q_{i}^{\delta+2}\,\int_{q_{\perp}=0}^{\infty}\left(q_{\perp}^{2}+\left(\frac{2\pi
f}{\alpha\,V_{\rm
SW}}\right)^{2}\right)^{-1-\frac{\delta}{2}}\,q_{\perp}\,dq_{\perp}$ (C26)
$\displaystyle=$ $\displaystyle\,\frac{n^{2}}{4\pi\,V_{\rm
SW}}\,S(q_{i})\,\frac{2}{\delta}\,q_{i}^{\delta+2}\left(\frac{2\pi
f}{\alpha\,V_{\rm SW}}\right)^{-\delta}$ (C27) $\displaystyle=$
$\displaystyle\,\frac{\langle\delta
n_{i}^{2}\rangle}{2\,f_{i}}\frac{1}{\delta}\,\left(\frac{f}{\alpha\,f_{i}}\right)^{-\delta}$
(C28)

Results (C22) and (C26) match at $f=f_{i}$ and also reduce to the isotropic
result (C11) when $\alpha=1$. However, the results (17) through (25) relating
$\overline{q\,\epsilon^{2}}\,R_{\odot}$ and $\int_{0}^{\infty}f\,P(f)\,df$ do
not hold in the general case $\alpha\neq 1$, because the wavenumber spectrum
$S(\widetilde{q})$ in Equation (C19) is not, and cannot be reduced to, an
isotropic form.

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