# A Chung-Fuchs type theorem for skew product dynamical systems

Xiong Jin  Department of Mathematics, University of Manchester, Oxford Road,
Manchester M13 9PL, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We prove a Chung-Fuchs type theorem for skew product dynamical systems such
that for a measurable function on such a system, if its Birkhoff average
converges to zero almost surely, and on typical fibres its Birkhoff sums have
a non-trivial independent random walk component, then its associated
generalised random walk oscillates, that is the supreme of the random walk
equals to $+\infty$ and the infimum equals to $-\infty$.

##  1\. Introduction

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space. Let
$\\{W_{n}\\}_{n\geq 1}$ be a sequence of independent and identically
distributed (i.i.d.) random variables in $\mathbb{R}$. Let

$S=\\{S_{n}=W_{1}+\cdots+W_{n}\\}_{n\geq 1}$

denote its associated random walk. We say that $S$ is recurrent if for all
$c>0$,

$\mathbb{P}(|S_{n}|\leq c\text{ for infinitely many }n)=1.$

We have the following well-known theorem of Chung and Fuchs [4]:

######  Theorem 1.1 (Chung and Fuchs 1951).

If $\mathbb{E}(|W_{1}|)<\infty$ and $\mathbb{E}(W_{1})=0$, then $S$ is
recurrent. Furthermore, if $\mathbb{P}(W_{1}\neq 0)>0$, then $S$ is
oscillating, that is

(1.1)
$\mathbb{P}(\liminf_{n\to\infty}S_{n}=-\infty)=\mathbb{P}(\limsup_{n\to\infty}S_{n}=+\infty)=1.$

In [4], the condition “$\mathbb{E}(|W_{1}|)<\infty$ and $\mathbb{E}(W_{1})=0$”
is to ensure that one can apply the strong law of large numbers to obtain

(1.2) $\mathbb{P}(\lim_{n\to\infty}\frac{S_{n}}{n}=0)=1.$

So Chung and Fuchs’ theorem can be stated in a slightly more general way
without $L^{1}$-integrable assumption that

$\eqref{sll}\ \Rightarrow\ S\text{ is recurrent}.$

For oscillation (1.1), Chung and Fuchs showed that the set of recurrent points
is a closed additive subgroup of $\mathbb{R}$, hence it is either empty, or
the whole space $\mathbb{R}$, or a lattice $a\cdot\mathbb{Z}$ for some
$a\in\mathbb{R}$. So if $S$ is recurrent, then the set of recurrent points is
non-empty, thus it is either $\mathbb{R}$ or $a\cdot\mathbb{Z}$ for some
$a\neq 0$ (since $\mathbb{P}(W_{1}\neq 0)>0$), hence either case implies
(1.1).

One may consider the Chung-Fuchs theorem for more general random walks: Let
$(X,\mathcal{B},T,\mu)$ be a measure-preserving dynamical system (m.p.d.s)
with $\mu(X)=1$. Let $f:X\to\mathbb{R}$ be a measurable function. Denote by

$S_{f}=\\{S_{n}f=f+f\circ T+\cdots+f\circ T^{n-1}\\}_{n\geq 1}$

its associated sequence of Birkhoff sums, or in other words, its associated
generalised random walk. We say that $S_{f}$ is recurrent if for all $c>0$,

$\mu(|S_{n}f|\leq c\text{ for infinitely many }n)=1.$

The following extension of Chung-Fuchs theorem to generalised random walks is
essentially due to Dekking [5] (see also [2] when assuming $f\in L^{1}(\mu)$).
For readers’ convenience we shall present a proof in the next section.

######  Theorem 1.2 (Dekking 1982).

If $\mu$ is ergodic and

(1.3) $\mu(\lim_{n\to\infty}\frac{S_{n}f}{n}=0)=1,$

then $S_{f}$ is recurrent.

For generalised random walks, the statement that (1.3) plus $f\not\equiv 0$
implies (1.1) is no longer true. A simple counter example is the following:
Let $X=\\{x_{0},x_{1}\\}$ be a space of two points and let the transformation
$T$ be $T(x_{0})=x_{1}$, $T(x_{1})=x_{0}$ and let
$\mu(\\{x_{0}\\})=\mu(\\{x_{1}\\})=\frac{1}{2}$. Take $f:X\to\mathbb{R}$ to be
$f(x_{0})=1$, $f(x_{1})=-1$. Then $S_{n}f$ is periodic and (1.3) holds, but
both liminf and limsup of $S_{n}f$ are bounded.

It is natural to ask for which settings between random walks and generalised
random walks it holds that (1.3) plus $f\not\equiv 0$ implies (1.1). In [3]
the authors studied the action of Mandelbrot cascades on ergodic measures, and
in the estimation of the degeneracy of this action it was sufficient to show
that

(1.4) $\mathbb{Q}(\limsup_{n\to\infty}S_{n}f+S_{n}F=+\infty)=1,$

where, in brief, the probability measure $\mathbb{Q}$ is a coupling of
$\mathbb{P}$ and $\mu$, $S_{n}f(\cdot)$ is a generalised random walk under
$\mu$ and $S_{n}F(x,\cdot)$ is a random walk under $\mathbb{Q}_{x}$ for
$\mu$-a.e. $x$, where $\mathbb{Q}_{x}$ is the disintegration of $\mathbb{Q}$
w.r.t. $\mu$. In the sub-critical case when $S_{n}f+S_{n}F$ has a positive
drift, that is $\mathbb{Q}$-a.s.
$\lim_{n\to\infty}\frac{S_{n}f+S_{n}F}{n}=c>0$, it is easy to deduce (1.4).
The critical case when
$\mathbb{Q}(\lim_{n\to\infty}\frac{S_{n}f+S_{n}F}{n}=0)=1$ corresponds to
(1.3) and it becomes more delicate to verify (1.4), which has led us to derive
the following result (a slightly more general and more abstract version of the
result in [3]).

### Main result

Consider the skew-product measure-preserving dynamical system

$(X\times\Omega,\hat{\mathcal{B}},\hat{T},\mathbb{Q}),$

where

  * -

$\hat{\mathcal{B}}=\mathcal{B}\otimes\mathcal{A}$.

  * -

$\hat{T}(x,\omega)=(T(x),g_{x}(\omega))$ is a skew-product transformation,
where for $x\in X$, $g_{x}:\Omega\to\Omega$ is $\mathcal{A}$-measurable and
for $A\in\mathcal{A}$, $x\to g_{x}(A)$ is $\mathcal{B}$-measurable.

  * -

$\mathbb{Q}$ is a coupling of $\mu$ and $\mathbb{P}$, that is
$\Pi_{X}^{*}(\mathbb{Q})=\mu$ and $\Pi_{\Omega}^{*}(\mathbb{Q})=\mathbb{P}$,
where $\Pi_{X}$ and $\Pi_{\Omega}$ are the projection from $X\times\Omega$
onto $X$ and $\Omega$ respectively.

Consider the measurable partition

$\eta=\\{\Pi_{X}^{-1}(x)=\\{x\\}\times\Omega:x\in X\\}$

and let $\mathbb{Q}^{\eta}_{x,\omega}$ denote the conditional measure of
$\mathbb{Q}$ w.r.t. $\eta$. For $\mathbb{Q}$-a.e. $(x,\omega)$,
$\mathbb{Q}^{\eta}_{x,\omega}$ is a probability measure carried by
$\\{x\\}\times\Omega$. Note that $\mathbb{Q}^{\eta}_{x,\omega}$ only depends
on $x$. Hence, equivalently, we may define a family of probability measures
$\mathbb{Q}_{x}=\Pi_{\Omega}^{*}(\mathbb{Q}^{\eta}_{x,\omega})$ on $\Omega$
for $\Pi_{X}^{*}(\mathbb{Q})=\mu$-a.e. $x\in X$ such that for any measurable
function $F:X\times\Omega\to\mathbb{R}$,

$\int_{X\times\Omega}F(x,\omega)\,\mathbb{Q}(\mathrm{d}(x,\omega))=\int_{X}\int_{\Omega}F(x,\omega)\,\mathbb{Q}_{x}(\mathrm{d}\omega)\mu(\mathrm{d}x).$

Let $F:X\times\Omega\to\mathbb{R}$ be a measurable function such that

  * (A)

for $\mu$-a.e. $x\in X$, the sequence

$\\{F(x,\cdot),F\circ\hat{T}(x,\cdot),F\circ\hat{T}^{2}(x,\cdot),\cdots\\}$

is an i.i.d. sequence of random variables under $\mathbb{Q}_{x}$ such that
there exist two disjoint intervals $I_{x},J_{x}\subset(0,\infty)$ such that
$\mathbb{Q}_{x}(F(x,\cdot)\in I_{x})\cdot\mathbb{Q}_{x}(F(x,\cdot)\in
J_{x})>0$.

For $n\geq 1$ denote by

$S_{n}F=F+F\circ\hat{T}+\cdots+F\circ\hat{T}^{n-1}.$

Let $f:X\to\mathbb{R}$ be a measurable function on $X$. One may view $f$ as a
measurable function on $X\times\Omega$ by setting $f(x,\omega)=f(x)$. We have
the following extension of Chung and Fuchs theorem.

######  Theorem 1.3.

Assume (A). If $\mu$ is ergodic and

$\mu(\lim_{n\to\infty}\frac{S_{n}f}{n}=0)=\mathbb{Q}(\lim_{n\to\infty}\frac{S_{n}F}{n}=0)=1,$

then

$\mathbb{Q}(\limsup_{n\to\infty}S_{n}f+S_{n}F=+\infty)>0.$

######  Remark 1.4.

If we further assume that $\mathbb{Q}$ is $\hat{T}$-ergodic, then, since
$\\{\limsup_{n\to\infty}S_{n}f+S_{n}F=+\infty\\}$ is a $\hat{T}$-invariant
set, Theorem 1.3 implies
$\mathbb{Q}({\displaystyle\limsup_{n\to\infty}S_{n}f+S_{n}F=+\infty})=1$. If
we also assume $-F$ satisfies (A) then
$\mathbb{Q}({\displaystyle\liminf_{n\to\infty}S_{n}f+S_{n}F=-\infty})=1$.

######  Remark 1.5.

Assumption (A) holds for a more general class of coupling measures
$\mathbb{Q}$ than the Peyrière measure studied in [3]. In particular we are
not requiring that the law of $F(x,\cdot)$ under $\mathbb{Q}_{x}$ to be
independent of $x$.

######  Remark 1.6.

In the proofs we added more details comparing to [3]. First we provided a more
detailed proof of Dekking’s recurrence theorem for our need. Then we proved
Lemma 2.1 to ensure that under the recurrence assumption a “suspension flow”
induced from an infinite dynamical system is still a m.p.d.s.. We also added
an induced system argument to deduce that a measurable (not necessarily
integrable) coboundary $g=f-f\circ T$ with $f,g\geq 0$ is equal to zero almost
everywhere.

##  2\. Proofs

###  2.1. Proof of Theorem 1.2

###### Proof.

For $c>0$ and $x\in X$ define

$\tau_{c}^{0}(x)=\inf\\{n\geq 1:|S_{n}f(x)|\leq c\\},$

and for $k\geq 1$ define

$\tau_{c}^{k}(x)=\inf\\{n\geq\tau_{c}^{k-1}(x)+1:|S_{n}f(x)|\leq c\\}.$

It is equivalent to prove that for all $k\geq 0$ and $c>0$,
$\tau_{c}^{k}(x)<\infty$ for $\mu$-a.e. $x\in X$.

Set $\tau^{-1}_{c}(x)=0$. Hence for $k=0$, $\tau^{k-1}_{c}(x)<\infty$ for all
$c>0$ and $\mu$-a.e. $x\in X$.

We shall prove the statement by induction. Assume that for all $c>0$,
$\tau^{k-1}_{c}(x)<\infty$ for $\mu$-a.e. $x\in X$. Fix $a,b,c>0$ such that
$a+b=c$. For $n\geq 1$ denote by

$R_{n}(x)=\lambda\Big{(}\bigcup_{l=1}^{n}I_{a}(S_{\tau^{k-1}_{b}(x)+l}f(x))\Big{)},$

where $\lambda$ is the Lebesgue measure on $\mathbb{R}$ and
$I_{a}(u)=[u-a/2,u+a/2]$ stands for the interval of length $a$ centered at
$u\in\mathbb{R}$. By the transition invariance of Lebesgue measure we have
that $R_{n}$ is a subadditive function:

$\displaystyle R_{n+m}(x)\leq$
$\displaystyle\lambda\Big{(}\bigcup_{l=1}^{n}I_{a}(S_{\tau^{k-1}_{b}(x)+l}f(x))\Big{)}+\lambda\Big{(}\bigcup_{l=n+1}^{n+m}I_{a}(S_{\tau^{k-1}_{b}(x)+l}f(x))\Big{)}$
$\displaystyle=$
$\displaystyle\lambda\Big{(}\bigcup_{l=1}^{n}I_{a}(S_{\tau^{k-1}_{b}(x)+l}f(x))\Big{)}+\lambda\Big{(}\bigcup_{l=n+1}^{n+m}I_{a}(S_{\tau^{k-1}_{b}(x)+l}f(x)-S_{\tau^{k-1}_{b}(x)+n}f(x))\Big{)}$
$\displaystyle=$ $\displaystyle
R_{n}(x)+\lambda\Big{(}\bigcup_{l=1}^{m}I_{a}(S_{\tau^{k-1}_{b}(x)+l}f(T^{n}(x)))\Big{)}$
$\displaystyle=$ $\displaystyle R_{n}(x)+R_{m}(T^{n}x).$

Thus by Kingman’s subadditive ergodic theorem we have that for $\mu$-almost
every $x\in X$,

$\lim_{n\to\infty}\frac{R_{n}(x)}{n}=\inf_{n\geq
1}\frac{\mathbb{E}_{\mu}(R_{n})}{n}.$

Using again the transition invariance of Lebesgue measure we get that

$\displaystyle R_{n+1}(x)=$
$\displaystyle\lambda\Big{(}\bigcup_{l=1}^{n+1}I_{a}(S_{\tau^{k-1}_{b}f(x)+l}f(x))\Big{)}$
$\displaystyle=$
$\displaystyle\lambda\Big{(}I_{a}(S_{\tau_{b}^{k-1}(x)+1}(x))\cup\bigcup_{l=1}^{n}I_{a}(S_{\tau_{b}^{k-1}(x)+1}f(x)+S_{l}f(T^{\tau_{b}^{k-1}(x)+1}(x)))\Big{)}$
$\displaystyle=$
$\displaystyle\lambda\Big{(}I_{a}(0)\cup\bigcup_{l=1}^{n}I_{a}(S_{l}f(T^{\tau^{k-1}_{b}(x)+1}(x)))\Big{)}$
$\displaystyle=$
$\displaystyle\lambda\Big{(}I_{a}(0)\cup\bigcup_{l=1}^{n}I_{a}(S_{l}f(T_{b,k}(T(x))))\Big{)},$

where we denote by

$T_{b,k}(x)=T^{\tau^{k-1}_{b}(x)}(x).$

By the $T$-invariance of $\mu$ and Fubini’s theorem we deduce that

$\displaystyle\mathbb{E}_{\mu}(R_{n+1})-\mathbb{E}_{\mu}(R_{n})$
$\displaystyle=$
$\displaystyle\int_{X}\lambda\Big{(}I_{a}(0)\cup\bigcup_{l=1}^{n}I_{a}(S_{l}f(T_{b,k}(x)))\Big{)}-\lambda\Big{(}\bigcup_{l=1}^{n}I_{a}(S_{l}f(T_{b,k}(x)))\Big{)}\,\mu(\mathrm{d}x)$
$\displaystyle=$
$\displaystyle\int_{X}\lambda\Big{(}I_{a}(0)\setminus\bigcup_{l=1}^{n}I_{a}(S_{l}f(T_{b,k}(x)))\Big{)}\,\mu(\mathrm{d}x)$
$\displaystyle=$
$\displaystyle\int_{X}\int_{I_{a}(0)}\prod_{l=1}^{n}\mathbf{1}_{\\{y\not\in
I_{a}(S_{l}f(T_{b,k}(x)))\\}}\,\lambda(\mathrm{d}y)\,\mu(\mathrm{d}x)$
$\displaystyle=$
$\displaystyle\int_{X}\int_{I_{a}(0)}\prod_{l=1}^{n}\mathbf{1}_{\\{S_{l}f(T_{b,k}(x))\not\in
I_{a}(y)\\}}\,\lambda(\mathrm{d}y)\,\mu(\mathrm{d}x)$ $\displaystyle=$
$\displaystyle\int_{I_{a}(0)}\mu\big{(}x:S_{1}f(T_{b,k}(x))\not\in
I_{a}(y),\cdots,S_{n}f(T_{b,k}(x))\not\in
I_{a}(y)\big{)}\,\lambda(\mathrm{d}y).$

This means that $\mathbb{E}_{\mu}(R_{n+1})-\mathbb{E}_{\mu}(R_{n})$
decreasingly converges to

$\int_{I_{a}(0)}\mu\big{(}x:S_{l}f(T_{b,k}(x))\not\in I_{a}(y)\text{ for
}l=1,2,\cdots\big{)}\,\lambda(\mathrm{d}y)$

as $n\to\infty$. Therefore

$\displaystyle\inf_{n\geq 1}\frac{\mathbb{E}_{\mu}(R_{n})}{n}=$
$\displaystyle\inf_{n\geq
1}\frac{\sum_{m=0}^{n-1}\mathbb{E}_{\mu}(R_{m+1})-\mathbb{E}_{\mu}(R_{m})}{n}$
$\displaystyle=$
$\displaystyle\int_{I_{a}(0)}\mu\big{(}x:S_{l}f(T_{b,k}(x))\not\in I(y)\text{
for }l=1,2,\cdots\big{)}\,\lambda(\mathrm{d}y).$

Thus for $\mu$-almost every $x\in X$,

$\lim_{n\to\infty}\frac{R_{n}(x)}{n}=\int_{I_{a}(0)}\mu\big{(}x:S_{l}f(T_{b,k}(x))\not\in
I_{a}(y)\text{ for }l=1,2,\cdots\big{)}\,\lambda(\mathrm{d}y).$

By the assumption that $\lim_{n\to\infty}\frac{1}{n}S_{n}f(x)=0$ and
$\tau^{k-1}_{b}(x)<\infty$ for $\mu$-a.e. $x\in X$, we get that for
$\mu$-almost every $x\in X$,

$\displaystyle\lim_{n\to\infty}\frac{S_{\tau^{k-1}_{b}(x)+n}f(x)}{n}=$
$\displaystyle\lim_{n\to\infty}\frac{S_{\tau^{k-1}_{b}(x)+n}f(x)}{\tau^{k-1}_{b}(x)+n}$
$\displaystyle=$ $\displaystyle 0.$

Thus for any $\epsilon>0$ there exists $n_{\epsilon}(x)$ such that for all
$n\geq n_{\epsilon}(x)$, $S_{\tau^{k-1}_{b}(x)+n}f(x)\in[-\epsilon n,\epsilon
n]$. This means that for $n\geq n_{\epsilon}(x)$,

$R_{n}(x)\leq R_{n_{\epsilon}(x)}(x)+\lambda([-\epsilon n-a,\epsilon
n+a])=R_{n_{\epsilon}(x)}(x)+2\epsilon n+2a.$

Hence

$\lim_{n\to\infty}\frac{R_{n}(x)}{n}\leq 2\epsilon.$

Since $\epsilon$ is arbitrary, we get that for $\mu$-almost every $x\in X$,

$\lim_{n\to\infty}\frac{R_{n}(x)}{n}=0.$

This implies that for $\mu$-almost every $x\in X$,

$\int_{I_{a}(0)}\mu\big{(}x:S_{l}f(T_{b,k}(x))\not\in I_{a}(y)\text{ for
}l=1,2,\cdots\big{)}\,\lambda(\mathrm{d}y)=0.$

Note that for $y\in I_{a}(0)=[-a/2,a/2]$ we have
$I_{a}(y)=[y-a/2,y+a/2]\subset[-a,a]$. Furthermore, by the definition of
$\tau_{b}^{k-1}(x)$ we have $S_{\tau_{b}^{k-1}(x)}f(x)\in[-b,b]$. Then by

$S_{\tau^{k-1}_{b}(x)+l}f(x)=S_{\tau^{k-1}_{b}(x)}f(x)+S_{l}f(T_{b,k}x),$

where when $k=0$ we may use the convention that $S_{0}f(x)=0$, we deduce that

$\displaystyle 0=$
$\displaystyle\int_{I_{a}(0)}\mu\big{(}x:S_{l}f(T_{b,k}x)\not\in
I_{a}(y)\text{ for }l=1,2,\cdots\big{)}\,\lambda(\mathrm{d}y)$
$\displaystyle\geq$
$\displaystyle\lambda(I_{a}(0))\cdot\mu\big{(}x:S_{l}f(T_{b,k}x)\not\in[-a,a]\text{
for all }l=1,2,\cdots\big{)}$ $\displaystyle\geq$ $\displaystyle
a\cdot\mu\big{(}x:S_{n}f(x)\not\in[-a-b,a+b]\text{ for all
}n\geq\tau^{k-1}_{b}(x)+1\big{)}$ $\displaystyle=$ $\displaystyle
a\cdot\mu\big{(}x:S_{n}f(x)\not\in[-c,c]\text{ for all
}n\geq\tau^{k-1}_{b}(x)+1\big{)}$ $\displaystyle\geq$ $\displaystyle
a\cdot\mu\big{(}x:S_{n}f(x)\not\in[-c,c]\text{ for all
}n\geq\tau^{k-1}_{c}(x)+1\big{)}.$

This implies that $\mu\big{(}x:S_{n}f(x)\not\in[-c,c]\text{ for
}n\geq\tau^{k-1}_{c}(x)+1\big{)}=0$, or in other words, for $\mu$-almost every
$x\in X$,

$\tau_{c}^{k}(x)=\inf\\{n\geq\tau^{k-1}_{c}(x)+1:S_{n}f(x)\in[-c,c]\\}<\infty.$

∎

###  2.2. Proof of Theorem 1.3

Consider the transformation $T_{f}$ on $X\times\mathbb{R}$:

$T_{f}(x,t)=(Tx,t+f(x)).$

Recall that $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$. It is easy
to verify that the product measure $\theta=\mu\times\lambda$ is
$T_{f}$-invariant, thus
$(X\times\mathbb{R},\mathcal{B}\otimes\mathcal{B}(\mathbb{R}),T_{f},\theta)$
forms a measure-preserving dynamical system. But the measure $\theta$ is a
$\sigma$-finite measure with infinite mean, hence we need to induce this
system to a finite one. In order to do so we need the system to be
conservative. By Theorem 1.2 this is granted by (1.3).

######  Lemma 2.1.

If (1.3) holds and $\mu$ is ergodic then $T_{f}$ is conservative.

###### Proof.

It is enough to show that for set $B\in\mathcal{B}$ with $\mu(B)>0$ and
$(a,b)\subset\mathbb{R}$, for every $n\geq 1$ there exists $m\geq n$ such
that, writing $A=B\times(a,b)$,

$\theta(A\cap T_{f}^{-m}A)>0.$

Fix $B\in\mathcal{B}$ with $\mu(B)>0$. The induced system
$(B,\mathcal{B}|_{B},T^{B},\mu_{B})$ is measure-preserving and ergodic (see
[7, Proposition 1.4.1] for example), where $\mu_{B}=\frac{1}{\mu(B)}\mu|_{B}$
and $T^{B}(x)=T^{\rho_{B}(x)}(x)$ for $x\in B$, where

$\rho_{B}(x)=\inf\\{n\geq 1:T^{n}(x)\in B\\}$

is the first return time of $x\in B$. By Kac formula (see [7, Theorem 1.2.2]
for example) we have that $\rho_{B}$ is integrable and
$\int_{B}\rho_{B}\,\mathrm{d}\mu\leq 1$. Write
$S_{n}^{B}\rho_{B}(x)=\rho_{B}(x)+\cdots+\rho_{B}\circ(T^{B})^{n-1}(x)$. Then
Birkhoff ergodic theorem implies that

(2.1)
$\lim_{n\to\infty}\frac{S_{n}^{B}\rho_{B}(x)}{n}=\tau(B):=\int_{X}\rho_{B}(x)\,\mu_{B}(\mathrm{d}x)$

exists for $\mu_{B}$-a.e. $x\in B$ and $\tau(B)\leq\frac{1}{\mu(B)}<\infty$.

For $x\in B$ write $f^{\rho_{B}}(x)=f(x)+\cdots+f\circ T^{\rho_{B}(x)-1}(x)$
and

$S_{n}^{B}f^{\rho_{B}}(x)=f^{\rho_{B}}(x)+\cdots+f^{\rho_{B}}\circ(T^{B})^{n-1}(x).$

Then (1.3) and (2.1) implies that for $\mu_{B}$-a.e. $x\in B$,

$\lim_{n\to\infty}\frac{S_{n}^{B}f^{\rho_{B}}(x)}{n}=\lim_{n\to\infty}\frac{S_{n}^{B}f^{\rho_{B}}(x)}{S_{n}^{B}\rho_{B}(x)}\frac{S_{n}^{B}\rho_{B}(x)}{n}=0.$

In other words, the measurable function $f^{\rho_{B}}$ satisfies (1.3) in
terms of the induced system $(B,\mathcal{B}|_{B},T^{B},\mu_{B})$. Hence by
Theorem 1.2 the generalised random walk $\\{S_{n}^{B}f^{\rho_{B}}\\}_{n\geq
1}$ is recurrent in this induced system.

Take $c>0$ such that $a+c<b-c$. For $(x,t)\in B\times(a+c,b-c)$ and $n\geq 1$,
by recurrence for $\mu_{B}$-a.e. $x\in B$ one can find $k(x)\geq 1$ such that
$S_{k(x)}^{B}\rho_{B}(x)\geq n$ and $S_{k(x)}^{B}f^{\rho_{B}}(x)\in(-c,c)$.
Since $\mu(B)>0$, there exists $m\geq n+1$ such that
$\mu_{B}(S_{k(x)}^{B}\rho_{B}(x)=m)>0$. Therefore

$\displaystyle\theta(A\cap T_{f}^{-m}A)$ $\displaystyle\,\geq\theta((x,t)\in
B\times(a+c,b-c):S_{k(x)}^{B}\rho_{B}(x)=m)$ $\displaystyle\,\geq\mu(x\in
B:S_{k(x)}^{B}\rho_{B}(x)=m)\cdot(b-a-2c)>0.$

∎

Now fix $E=X\times[-c,c]$ for some $c>0$. By conservativeness (Lemma 2.1) the
induced system $(E,\mathcal{B}_{E},T_{f}^{E},\theta_{E})$ is a m.p.d.s. with
$\theta_{E}(E)=1$ (see [1, Proposition 1.5.3] for example). The measure
$\theta_{E}$ takes the form

$\theta_{E}=\frac{1}{\theta(E)}\theta\times\lambda|_{E}=\mu\times\lambda_{c},$

where we denote by $\lambda_{c}=\frac{1}{2c}\lambda|_{[-c,c]}$ the normalised
Lebesgue measure on $[-c,c]$. We may now define the following induced skew-
product dynamical system

$(\overline{X},\overline{\mathcal{B}},\overline{T},\overline{\mathbb{Q}}),$

where

  * -

$\overline{X}=X\times\Omega\times[-c,c]$ and
$\overline{B}=\mathcal{B}\otimes\mathcal{A}\times\mathcal{B}([-c,c])$.

  * -

$\overline{T}(x,\omega,t)=(\hat{T}^{\rho_{E}(x,t)}(x,\omega),t+f(x)+\cdots+f\circ
T^{\rho_{E}(x,t)-1}(x))$.

  * -

$\overline{\mathbb{Q}}=\mathbb{Q}\times\lambda_{c}$.

It is easy to verify that $\overline{\mathbb{Q}}$ is $\overline{T}$-invariant.
Note that the projection of $\overline{\mathbb{Q}}$ onto $E=X\times[-c,c]$ is
$\theta_{E}$ and the disintegration of $\overline{\mathbb{Q}}$ with respect to
$\theta_{E}$ is $\overline{\mathbb{Q}}_{x,t}=\mathbb{Q}_{x}$ for
$\theta_{E}$-a.e. $(x,t)\in E$.

###### Proof of Theorem 1.3.

We shall use the filling scheme [6]. For $g\in\mathbb{R}$ denote by
$g^{+}=\max\\{g,0\\}$ and $g^{-}=\max\\{-g,0\\}$.

We may view both $f$ and $F$ as functions on $\overline{X}$ by taking
$f(x,\omega,t)=f(x)$ and $F(x,\omega,t)=F(x,\omega)$. For $n\geq 1$ define

$G_{n}:=\max_{1\leq k\leq n}\overline{S}_{k}F,$

where we denote by

$\overline{S}_{k}F=F+F\circ\overline{T}+\cdots+F\circ\overline{T}^{k-1}.$

Then we have

$G_{n+1}^{+}-G_{n}^{+}\circ\overline{T}=F+G_{n+1}^{-}.$

Let $G=\lim_{n\to\infty}G_{n}$. Assume that
$\overline{\mathbb{Q}}(G<\infty)=1$. Then we may write

$G^{+}-G^{+}\circ\overline{T}=F+G^{-}.$

Let

$\kappa:=\inf\\{n\geq 1:F\circ\overline{T}^{n-1}>0\\}.$

Then, since for all $n\geq 1$, $G_{n}\circ\overline{T}^{\kappa-1}\geq
F\circ\overline{T}^{\kappa-1}>0$, we get that
$G\circ\overline{T}^{\kappa-1}\geq F\circ\overline{T}^{\kappa-1}>0$ hence
$G^{+}\circ\overline{T}^{\kappa-1}=G\circ\overline{T}^{\kappa-1}$ and
$G^{-}\circ\overline{T}^{\kappa-1}=0$. Denote by
$H(x,\omega,t)=G^{\kappa(x,\omega,t)-1}(x,\omega,t)$ and
$h(x,\omega,t)=F^{\kappa(x,\omega,t)-1}(x,\omega,t)$. Then we deduce that

$H-H\circ\overline{T}=h.$

For $u\in\mathbb{R}$ denote by

$\Phi_{u}(x):=\mathbb{E}_{\overline{\mathbb{Q}}_{x,t}}(e^{iuH(x,\cdot,t)})$

and

$\phi_{u}(x):=\mathbb{E}_{\overline{\mathbb{Q}}_{x,t}}(e^{iuh(x,\cdot,t)}).$

Then by (A) we have

$\Phi_{u}(x,t)=\phi_{u}(x,t)\cdot\Phi_{u}\circ T_{f}^{E}(x,t).$

Write $f(x,t)=-\log|\Phi_{u}(x,t)|$ and $g(x,t)=-\log|\phi_{u}(x,t)|$. Then we
have that $f,g\geq 0$ and

$f-f\circ T_{f}^{E}=g.$

Take $N>0$ large enough such that

$E_{N}=E\cap\\{f\leq N\\}$

has positive $\theta_{E}$-mass. Let $(E_{N},\mathcal{B}_{N},T_{N},\theta_{N})$
denote the induced dynamical system by $E_{N}$. We get $f-f\circ T_{N}=g_{N}$,
where

$g_{N}=g+g\circ T_{f}^{E}+\cdots+g\circ(T_{f}^{E})^{\rho_{E_{N}}(x,t)-1}.$

Now on $E_{N}$ we have that $0\leq f\leq N$, $0\leq f\circ T_{N}\leq N$ and
$0\leq g_{N}\leq f\leq N$ are all bounded functions hence integrable,
therefore by the $T_{N}$-invariance of $\mu_{N}$ and Birkhorff ergodic
theorem,

$\mathbb{E}_{\theta_{N}}(g_{N}\,|\,\mathcal{I}_{N})(x,t)=0$

for $\theta_{N}$-a.e. $(x,t)\in E_{N}$, where $\mathcal{I}_{N}$ is the
$T_{N}$-invariant $\sigma$-algebra of $E_{N}$. Since $g$ is non-negative
function, $g_{N}$ is also nonnegative, we deduce that $g_{N}(x,t)=0$ and
therefore $g(x,t)=0$ for $\theta_{N}$-a.e. $(x,t)\in E_{N}$. Since $N$ is
arbitrary, we deduce that $g(x,t)=0$ or equivalently $|\phi_{u}(x,t)|=1$ for
$\theta$-a.e. $(x,t)\in E$. This implies that for $\theta_{E}$-a.e. $(x,t)\in
E$, $h(x,\cdot,t)=F^{\kappa(x,\cdot,t)-1}(x,\cdot,t)$ is
$\mathbb{Q}_{x,t}$-a.s. a constant. But this is not possible under (A), the
reason is the following: Recall the disjoint intervals $I_{x}$ and $J_{x}$
from (A). We may choose the intervals a bit smaller if necessary so that
$a_{x}=\inf I_{x}\cup J_{x}>0$. Then we have $\mathbb{Q}_{x}(F(x,\cdot)\geq
a_{x})>0$. Note that when $F(x,\omega)\geq a_{x}$, we have
$\kappa(x,\omega,t)=1$ and $F^{\kappa(x,\omega,t)-1}(x,\omega,t)=F(x,\omega)$.
Then by the fact that both $\mathbb{Q}_{x}(F(x,\cdot)\in I_{x})>0$ and
$\mathbb{Q}_{x}(F(x,\cdot)\in J_{x})>0$, as well as $I_{x}$ and $J_{x}$ are
disjoint, $F^{\kappa(x,\cdot,t)-1}(x,\cdot,t)$ can not be
$\mathbb{Q}_{x,t}$-a.s. a constant.

This contradiction implies $\overline{\mathbb{Q}}(G<\infty)<1$, or in other
words, for $\theta_{E}$-positive $(x,t)\in E$, for $\mathbb{Q}_{x,t}$-positive
$\omega\in\Omega$,

$\limsup_{n\to\infty}\overline{S}_{n}F(x,\omega,t)=+\infty.$

On the other hand, note that by construction, for $(x,t)\in E=X\times[-c,c]$,
when $T_{f}^{E}(x,t)$ returns to $E$, we have

$t+f(x)+\cdots+f\circ T^{\rho_{E}(x,t)-1}(x)\in[-c,c].$

Thus for all $n\geq 1$,

$|\overline{S}_{n}f(x,\omega,t)|\leq 2c.$

Since $\theta_{E}=\mu\times\lambda_{c}$, we obtain that for $\mu$-positive
$x\in X$, for $\mathbb{Q}_{x}$-positive $\omega$, there exists a subsequence
$n_{k}=n_{k}(x,\omega)\to\infty$ such that $S_{n_{k}}f(x)\in[-2c,2c]$ for all
$k\geq 1$ and $\limsup_{k\to\infty}S_{n_{k}}F(x,\omega)=+\infty$, therefore

$\limsup_{n\to\infty}S_{n}f(x)+S_{n}F(x,\omega)=+\infty.$

∎

## References

  * [1] J. Aaronson.  An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50).  American Mathematical Society, Providence, RI, 1997.
  * [2] G. Atkinson.  Recurrence of co-cycles and random walks.  J. London Math. Soc. 13(2): 486–488, 1976.
  * [3] J. Barral and X. Jin  On the Action of Multiplicative Cascades on Measures.  Int. Math. Res. Not., 2022(18): 13857–13896, 2022.
  * [4] K. L. Chung and W. H. J. Fuchs.  On the distribution of values of sums of random variables.  Mem. Amer. Math. Soc., 6:1–12, 1951.
  * [5] F. M. Dekking.  On transience and recurrence of Generalized random walks.  Z. Wahrsch. Verw. Gebiete, 61:459–465, 1982.
  * [6] Y. Derriennic.  Ergodic theorem, reversibility and the filling scheme.  Colloq. Math., 118(2):599–608, 2010.
  * [7] M. Viana and K. Oliveira  Foundations of Ergodic Theory (Cambridge Studies in Advanced Mathematics).  Cambridge: Cambridge University Press, 2016.


# Understanding the variability of helium abundance in the solar corona using
three-fluid modeling and UV observations

Leon Ofman NASA Goddard Space Flight Center, Greenbelt, MD, 20771, USA  The
Catholic University of America, Washington, DC 20064, USA  Visiting, Tel Aviv
University, Tel Aviv, Israel  Yogesh NASA Goddard Space Flight Center,
Greenbelt, MD, 20771, USA  The Catholic University of America, Washington, DC
20064, USA  Silvio Giordano INAF – Astrophysical Observatory of Torino, Via
Osservatorio 20, 10025 Pino Torinese, Italy

###### Abstract

The variability of helium abundance in the solar corona and the solar wind is
an important signature of solar activity, solar cycle, solar wind sources, as
well as coronal heating processes. Motivated by recently reported remote
sensing UV imaging observations by Helium Resonance Scattering in the Corona
and Heliosphere (HERSCHEL) payload sounding rocket of helium abundance in
inner corona on 14-Sep-2009 near solar minimum, we present the results of the
first three-dimensional three-fluid (electrons, protons, alpha particles)
model of tilted coronal streamer belt and slow solar wind that illustrate the
various processes leading to helium abundance differentiation and variability.
We find good qualitative agreement between the three-fluid model and the
coronal helium abundances variability reported from UV observations of
streamers, providing insight on the effects of the physical processes, such as
heating, gravitational settling and inter-species Coulomb friction in the out-
flowing solar wind that produce the observed features. The study impacts our
understanding of the origins of the slow solar wind.

Solar wind — Sun: abundances — Sun: heliosphere — Sun: corona — Sun: magnetic
fields

##  1 Introduction

The solar wind consists typically of approximately 95% protons ($H^{+}$),
making them the principal constituent by mass, and around 5% helium
(predominantly $He^{++}$ or $\alpha$ particles) as the second major
constituent, and the corresponding number of electrons stemming from charge
neutrality. Here, the abundance of helium in the solar wind is defined
relative to protons as $A_{He}$ = ($n_{He}/n_{p}$)$\times$100, where $n_{He}$
and $n_{p}$ are the concentrations of helium and protons in the solar wind.
Understanding variations in helium abundance in the solar wind and solar
corona can provide important clues to coronal processes, such as the coronal
sources of the slow and fast solar wind, as well as heating and acceleration
processes of the solar wind (Neugebauer et al., 1996; Kasper et al., 2007,
2012; Abbo et al., 2016).

The $A_{He}$ was determined with remote sensing of UV emission in the inner
solar corona by using Sounding-rocket Coronagraph Experiment (SCORE), the
Helium Coronagraph (HeCOR) and Extreme Ultraviolet Imaging Telescope (HEIT) in
the he Helium Resonance Scattering in the Corona and Heliosphere (HERSCHEL)
payload sounding rocket, launched from White Sands, NM on 14-Sep-2009 (Moses
et al., 2020). In the photosphere, $A_{He}$ is nearly 8.5% (Grevesse & Sauval,
1998; Asplund et al., 2009). However, it remains about 4–5% in the solar
corona due to the First Ionization Potential (FIP) effects (Laming & Feldman,
2001, 2003). The $A_{He}$ can range from 0.1% to more than 10% in the solar
wind depending on sources, coronal, and interplanetary modulations and due to
the heavy mass of $He$ can account for significant fraction of the solar wind
mass and energy fluxes (Kasper et al., 2007; Alterman & Kasper, 2019; Yogesh
et al., 2021, 2022, 2023). At times, helium can account for the bulk of the
solar wind mass flux at the high end limit of $A_{He}$. The long-term
variability of $A_{He}$ follows the solar cycle (SC) and varies with solar
wind velocity (Kasper et al., 2007; Alterman & Kasper, 2019; Yogesh et al.,
2021). It can increase up to 30% in coronal mass ejections (Borrini et al.,
1982; Yogesh et al., 2022, and references therein). Although the above results
are mainly observed in the solar wind at 1AU, the variation in helium
abundance in the inner solar corona was studied using remote sensing
observations (Moses et al., 2020). Previous 2.5D three-fluid models of the
slow solar wind that include electrons, protons, and helium as separate
interacting fluids demonstrate the variability of A(He) associated with
streamer structures, heating and coronal sources of the solar wind (Ofman,
2004a, b; Li et al., 2006; Giordano et al., 2007; Ofman & Kramar, 2010). These
studies modeled the proton and helium streamer structures using 2.5D three-
fluid models and predicted their expected signatures long before the
feasibility of such UV observations (see, e.g., Figure 4 in Giordano et al.
(2007)). Despite these studies, the basic processes that result in the non-
uniformity in $A_{He}$ in the solar corona are not well understood in detail
so far, due to the complexity of the slow solar wind formation processes (e.g.
Abbo et al., 2016). Here, we use an idealized 3D multi-fluid model to
demonstrate the roles of the main ion differentiation physical processes that
affect the coronal helium abundance variation in a coronal streamer belt. This
is an important problem, as understanding coronal helium abundances provides
important insights into the formation and coronal sources of the slow solar
wind. In Section 2 we present the observational motivation, in Section 3 we
present the 3D multi-fluid model, Section 4 is devoted to numerical results,
with the discussion and conclusions in Section 5.

##  2 Remote sensing observations of coronal helium abundance

Recently, Moses et al. (2020) reported the global helium abundance variability
in the solar corona, analyzing the Helium Resonance Scattering in the Corona
and Heliosphere (HERSCHEL) sounding rocket observations launched on 14
September 2009, at the extended solar activity minimum of cycle 23. The
HERCHEL payload was composed of the Sounding-rocket Coronagraph Experiment
(SCORE), the Helium Coronagraph (HeCOR) and the HERSCHEL Extreme Ultraviolet
Imaging Telescope (HEIT). SCORE coronagraph is designed to observe the H I
(121.6 nm), He II (30.4 nm) and white light, simultaneously from 1.5 $R_{s}$
to $2.2R_{s}$. HeCOR coronagraph was designed to observe He II (30.4 nm) image
from 1.3 $R_{s}$ to 3 $R_{s}$ (where $R_{s}$ is the solar radius). The disk
images of He II were observed by HEIT. The observations from HERCHEL were
complemented by Solar and Heliospheric Observatory (SOHO) Extreme ultraviolet
Imaging Telescope (EIT), the Large Angle and Spectrometric Coronagraph (LASCO)
and the Ultraviolet Coronagraph Spectrometer (UVCS) instruments. In
particular, the SCORE was used to obtain the H I and He II emission images of
the west limb of the solar corona. Moses et al. (2020) found that the
morphology of the H I corona is markedly different from that of the He II
corona, owing to significant spatial variations in helium abundance. Figure 1
shows the observations of the H I and He II as reported in Moses et al.
(2020). Figures 1a and 1b show the images of the H I and He II emission
respectively, observed by SCORE. Figure 1b shows the EIT image of the stray
light–corrected photometric He II channel is superposed inside the inner
radius of the SCORE field of view. Figure 1c shows the He II emission image
from HeCOR superposed with the EIT image inside the HeCOR field of view. Since
these observations were focused on the equatorial region in the range of
distances from 1.3 to 4$R_{s}$, the contribution of other ions such as Si XI
emission to the streamer structure observed in He II is negligible (Fineschi
et al., 2003).

It can be seen from Figure 1 that the H I emission is nearly uniformly
distributed in nearby equatorial regions, whereas the He II emission is dimmed
near equator as compared to higher and lower latitudes. It was found that the
corresponding helium abundance was low ($\leq$ 3 %) within $\pm$ $15^{o}$
across the equator and the minimum observed $A_{He}$ was 0.6% near equator. By
using potential (‘vacuum’) magnetic field approximation, Moses et al. (2020)
showed that the center of the region exhibiting the lowest $A_{He}$, is
situated between two closed-loop systems. Beyond the latitude range of $\pm
15^{o}$ and beyond the distance of 1.7 $R_{s}$ where magnetic field lines are
mostly open, the relative helium abundance increases. They attributed these
changes in helium abundance to the Coulomb drag which depends on the velocity
of solar wind, and gravitational settling in the closed field region. In open
field line regions, lesser expansion factors cause higher speed resulting in
the higher $A_{He}$. In Figure 2 we show for context the Carrington rotation
obtained from NISP NSO Integrated Synoptic Program centered on HERSCHEL
observation date with PFSS field where the open and closed field regions and
the tilted streamer belt structure are evident, consistent with the modeling
results discussed in Section 4 below. In Figure 2 of Moses et al. (2020) it is
apparent that the Helium intensity and abundance latitudinal dependence minima
at heights 1.7, 1.9, and 2.1$R_{s}$ are a few degrees below the 270o latitude.
The potential magnetic field extrapolation of the double streamer structure
presented in Figure 3 of Moses et al. (2020) is not well consistent with the
overall tilted streamer belt structure at solar minimum shown in Figure 2 and
the closed magnetic field in the equatorial region. Since the gravitational
settling of $He^{++}$ is taking place in closed field of a quiescent streamer
and the Coulomb friction enhancement is taking place in open field as
demonstrated in past multi-fluid models, the single streamer belt structure
provides more consistent magnetic field structure with the observed helium
emission.

Figure 1: Results from the HERSCHEL experiment on 14-Sep-2009. Photometric
images were acquired in (a) the SCORE H I $Ly\alpha$, (b) SCORE He II channels
spanning the range from 1.5$R_{s}$ to 2.2$R_{s}$. The white dots in panel (a)
denote the positions corresponding to the coordinated SOHO UVCS observations.
(c) A photometric image was captured in the HeCOR He II channel, covering the
range from 1.28 to 3$R_{s}$. The EIT image of the stray light–corrected
photometric data of He II channel are shown in (b) and (c). Reproduced with
permission from Moses et al. (2020). Figure 2: Representation of global
magnetic field topology from Potential-field Source-surface (PFSS) model (Wang
& Sheeley, 1992) for the Carrington rotation centered on HERSCHEL observation
date (model available at NISP NSO Integrated Synoptic Program). The closed
field lines are plotted in blue, Regions of open positive and negative flux
are represented by green and red, respectively. The neutral line is drawn in
black.

In their Figure 2 Moses et al. (2020) show the latitudinal variation of the He
abundance at the west solar limb derived from SCORE measurements at several
heights and found that the He++ relative abundance decreases by an order of
magnitude at the center of the streamer belt compared to the open field
regions, i.e., in the closed field regions in which the plasma is mainly
confined. The corresponding variation of H I in this region was found to be
small with overall increase and some dips in intensity. Thus, it is likely
that the difference in proton and He++ abundance variability arises from ion-
dependent physical processes such as gravitational scale-height dependent
($H=k_{B}T_{i}/m_{i}g_{s}$, where $k_{B}$ is Boltzmanns’ constant, $T_{i}$ is
ion temperature, $m_{i}$ is the ion mass, and $g_{s}$ is the solar
gravitational acceleration) settling in the plasma confined to closed magnetic
field regions. The observed depletion of helium abundance in the slow solar
wind was studied recently using Wind data at 1AU and PSP data at perihelia in
Yogesh et al. (2024), where they suggest that quiescent streamer cores are the
preferred locations for gravitational settling to occur. This settling can
cause a very low helium abundance, even below 1% in the solar wind protons.
The role of these physical processes is demonstrated using the 3D model three-
fluid model in upcoming sections.

##  3 Description of the 3D multi-fluid model

In previous multi-fluid studies, the streamer model incorporated a heavy ion
population as an additional fluid alongside plasma electrons and protons (see
the review, Abbo et al., 2016). The plasma is characterized using a three-
fluid approximation encompassing electrons, protons, and heavier ions, treated
as coupled fluids influenced by collisional and electromagnetic interactions.
This approximation extends well beyond the capabilities of single-fluid
magnetohydrodynamics (MHD) description. The model assumes quasi-neutrality of
the plasma and neglects electron inertia, employing $m_{e}\ll m_{p}$ to solve
the electron momentum equation for the electric field (i.e., generalized Ohm’s
law). The electron density is derived from the charge neutrality condition.
Notably, viscosity and explicit thermal conduction terms are disregarded in
this model. The 2.5D three-fluid model was first developed for streamers with
$O^{5+}$, and $He^{++}$ ions (e.g., Ofman, 2000, 2004a; Li et al., 2006; Ofman
et al., 2011). The 3D three-fluid model with $O^{5+}$ ions of a slow solar
wind in a tilted streamer belt was presented in Ofman et al. (2015), while
here we include $He^{++}$ ions as the third fluid. The normalized equations
employed in the three-fluid model and the detailed description are given in
Ofman et al. (2015). Below, we review the main model parameters relevant to
the present study.

In the model equations shown in Ofman et al. (2015), the index $k=e$, $p$, $i$
represents electrons, protons, or heavier ions (in the present study
$He^{++}$), where the $n_{k}$ denotes the number density, $V_{k}$ is the
velocity vector, $T_{k}$ is the temperature for each species, $S_{k}$ is the
empirical heating term (only used for $He^{++}$ ions), $S_{r,e}$ is the
electron radiative loss term, $\delta_{k,e}$ is the Kronecker delta, $C_{kjl}$
is the energy coupling term between the species (Ofman, 2004a), and
$\gamma_{k}$ is the polytropic index of each species. Empirical polytropic
indices of $\gamma_{k}$ = 1.05 are assumed to model the effects of coronal
heating and heat conduction loses for electrons, protons and He++ ions
consistent with global coronal models, e.g., Riley et al. (2006). These
empirical values of $\gamma_{k}>1$ allow realistic multifluid modeling of the
coronal streamer constituents temperatures in agreement with observations
(e.g., Ofman et al., 2011). The terms proportional to ion gyrofrequency are
neglected in the low-frequency limit, and the time is normzlized in units of
the Alfvén time $\tau_{A}=R_{s}/V_{A}$, where the Alfvén speed is
$V_{A}=B_{0}/\sqrt{4\pi m_{p}n_{e0}}$ defined here with the proton mass,
$B_{0}$ is the normalization magnetic field strength (in the present study we
set $B_{0}$ = 7G), $m_{p}$ is the proton mass, and $n_{e0}$ is the
normalization value of the electron number density. Using $n_{e0}=5\times
10^{8}$ ${cm}^{-3}$, we get $V_{A}$ = 683 ${km\ s}^{-1}$. For $He^{++}$ we
have $A_{He^{++}}=4$, and $Z_{He^{++}}=2$. The other model parameters are $S$
the Lundquist number (here, we set $S=10^{4}$, typical numerical resolution-
limited value in MHD models that does not significantly affect the result);
electron and proton Euler number $Eu_{\mathrm{e},\mathrm{p}}=$
$\left(k_{\mathrm{b}}T_{0,\mathrm{e},\mathrm{p}}/m_{\mathrm{p}}\right)V_{\mathrm{A}}^{-2}$;
ion Euler number
$Eu_{\mathrm{i}}=\left(k_{\mathrm{b}}T_{0,\mathrm{i}}/m_{\mathrm{i}}\right)V_{\mathrm{A}}^{-2}$;
and Froude number $F_{r}=V_{\mathrm{A}}^{2}R_{\mathrm{S}}/GM_{\mathrm{S}}$,
where $G$ is the universal gravitational constant and $M_{\mathrm{S}}$ is the
solar mass, $R_{\mathrm{S}}$ is the solar radius, $b=cB_{0}/\left(4\pi
en_{\mathrm{e}0}R_{\mathrm{S}}V_{\mathrm{A}}\right)$ the normalization
constant, $k_{\mathrm{b}}$ is the Boltzmann constant, and ${F}_{k{,coul}}$ is
the Coulomb friction (or drag) terms proportional to the velocity differences
between the species and the collision frequencies (see,e.g., Braginskii, 1965;
Geiss et al., 1970; Ofman & Davila, 2001, for details of these terms). The
empirical heating term for $\mathrm{He}^{++}$ ions is the function
$S_{\mathrm{i}}=S_{\mathrm{i}0}(r-$ 1) $e^{-(r-1)/\lambda_{\mathrm{i}0}}$,
where the constants $S_{\mathrm{i}0}$ and $\lambda_{\mathrm{i}0}$ determine
the magnitude and the decrease of the heating rate with distance $r$. In the
present model we use the empirical values $S_{\mathrm{i}0}=7.25$ and
$\lambda_{\mathrm{i}0}=0.5R_{s}$ constrained by radial density and temperature
structure of He in open field regions (e.g., Ofman, 2004b). The initial
temperatures were uniform $T_{e}=T_{p}=1.6$ MK, $T_{He}=4$ MK, and evolve
self-consistently. For simplicity, the possible effects of ion temperature
anisotropy are neglected in this model.

These equations are evolved numerically in a 3-D spherical domain using 4th
order integration method until a quasi-streamer belt is formed (see, Ofman et
al., 2015, for details). The ranges for spatial coordinates are $1R_{s}\leq
r\leq 7R_{s}$, $0<\theta<\pi$ and $0<\phi<2\pi$, with a small circular region
around the poles removed to avoid singularity of the coordinate system. The
computation started with a tilted dipole (with a tilt of $10^{o}$) coronal
magnetic field model (Ofman et al., 2015). The following boundary conditions
are implemented in the simulation. At the inner boundary $r=1R_{s}$, the
magnetic field components are fixed (line-tied), and the values of density,
temperature, and velocity components at the boundary grid cells are
extrapolated from the first interior cells with zero gradients, approximating
outgoing characteristics (Steinolfson & Nakagawa, 1976). At the outer boundary
$r=7R_{s}$ open boundary conditions are set for all variables. The outer
boundary distance is chosen to satisfy supersonic slow solar wind transition.
The presented results were obtained with a uniform resolution of $160\times
128\times 508$ grid cells in the $\theta$, $\phi$ and $r$ domains,
respectively. The results were also tested for convergence with higher
resolutions (albeit shorter duration) run and no significant changes were
observed with higher resolution solutions.

##  4 Numerical Results

Here, we present the results of our 3D three-fluid model of streamer belt with
$He^{++}$ for the titled dipole initial state and the parameters given in the
previous section. The solutions are obtained by running the model to quasi-
steady state, where the initial tilted dipole field evolves to a streamer belt
solution (as in Ofman et al. (2015) but with $O^{5+}$). In Figure 3 we show
the section of the streamer belt in the $r$-$\theta$ plane cut at $\phi=0.76$
Rad in the radial extent $1.3R_{s}\leq r\leq 4R_{s}$, similar to the radial
extent covered by HERSCHEL observations. The while lines mark several closed
and open fieldlines in this plane. It is evident that the slow solar wind
outflow velocity $v_{r}$ is significant outside the streamer core, and
velocities are small inside the streamer core for both, protons and $He^{++}$.
The corresponding density structure of the $He^{++}$ dips significantly at the
closed-field core of the streamer, and peaks in the open-field streamer
flanks. The proton density structure shows the opposite latitudinal
dependence, and peaks in the streamer core. This streamers density structure
is in good qualitative agreement with observation in the region where $A_{He}$
was determined, shown in Figure 1.

Figure 3: The results of the 3D three-fluid model of the tilted streamer belt
shown in the $r-\theta$ plane at longitude $\phi=0.76$ Rad. The distance is in
$R_{s}$ and angles are in Rad. The densities are normalized in terms of
$n_{e0}=5\times 10^{8}$ cm-3. The while lines mark several fieldlines. (a) The
normalized density of the $He^{++}$ ions, (b) the normalized proton density
(blue), (c) the $He^{++}$ radial velocity in km s-1, (d) the proton radial
velocity in km s-1.

In Figure 4 we show the density structure of the tilted streamer belt in the
$\phi-\theta$ plane (i.e., analogous to a Carrington map but for density) at
the height of $r=1.5R_{s}$. The apparent sinusoidal structure of the streamer
belt is due to the $10^{o}$ tilt projected onto the plane, and the higher
proton density in the streamer core, anti-correlated with the $He^{++}$
density is evident in the tilted streamer belt structure. The modeled tilted
streamer belt structure is consistent with the Carrington map of the streamer
belt for the solar minimum conditions during HERSCHEL observations shown in
Figure 2.

Figure 4: The results of the 3D three-fluid model of the tilted streamer belt
shown in the $\theta-\phi$ [Rad] plane at the height of $r=1.5R_{s}$. The
densities are normalized in terms of $n_{e0}=5\times 10^{8}$ cm-3. Left panel:
the normalized proton density. Right panel: the normalized density of the
$He^{++}$ ions. The dashed green line marks the longitude of the $\theta-r$
plane cuts in Figure 3.

Figure 5 is devoted to the latitudinal dependence of the variables across the
streamer belt shown at a height of $1.75R_{s}$. The radial velocities $v_{r}$
of protons (solid line) and $He^{++}$ (dashes) are shown in Figure 5a. The
outflow velocities of the two ions are close due to the effects of the Coulomb
friction terms with magnitude that is proportional to the velocity difference
of the electrons, protons, and He++ ions (the $F_{k}$ terms in the momentum
equation). The effect of the Coulomb friction is to reduce the differential
flow of the three fluids. The modeled solar wind velocity outside the streamer
is consistent with observational values Abbo et al. (2010); Ofman et al.
(2011). It is evident that inside the streamer belt the velocities are very
low so that the plasma can be considered nearly static. The small downflow
velocity of $He^{++}$ is associated with gravitational settling. The anti-
correlated normalized density structures of protons and $He^{++}$ ions are
shown in Figure 5b. The densities are normalized with the corresponding
densities at $\theta=3$ Rad. The temperatures of protons, $He^{++}$ ions, and
electrons obtained from the three fluid model are shown in Figure 5c. It is
evident that the $He^{++}$ ions remain hotter than the other fluids due to the
overall effect of the ion heating term. However, in the core of the quiescent
streamer the $He^{++}$ temperature is larger than in the open-field region,
due to the thermal energy exchange and cooling by the protons and electrons.
It is also evident that in the core the quiescent streamer the electron and
proton temperature are nearly identical due to the thermal energy exchange
terms that tend to minimize the temperature difference between the species
when collision are significant.

Figure 5: The cross section cut of the variables at $r=1.75R_{s}$ for the
model run shown in Figure 3 (solid: protons, dashes: $He^{++}$ ions, dot-
dashes: electrons). (a) The outflow velocities $V_{r}$, (b) the densities
normalized by the values at maximal $\theta$. (c) Proton, $He^{++}$, and
electron temperatures.

##  5 Discussion and Conclusions

The variability of helium abundance in the solar wind and the solar corona is
a puzzling phenomenon that provides clues on the solar cycle, type of the
solar wind (fast or slow), and possibly on the acceleration processes and on
the origins of the solar wind (e.g., Alterman & Kasper, 2019). However, the
processes that lead to the variability of helium in the corona are not well
understood so far. The observation of UV helium emission is not routinely
available in coronal streamers. The only available observations so far,
provided UV imaging of streamers in both, H I and He II emission using the
HERSHEL rocket experiment reported recently by Moses et al. (2020). It is
evident from the imaging data of the inner corona that the emission of
hydrogen and helium is overall anti-correlated, suggesting that different
processes shape the helium abundance in streamers than hydrogen, as predicted
by past three-fluid models. Moreover, the potential magnetic field
extrapolation double streamer magnetic structures shown in Moses et al. (2020)
does note account well for the gravitational settling of $He^{++}$ in a
quiescent streamer core at the equatorial region, nor with the overall
magnetic structure of an equatorial streamer belt at solar minimum.

We use the 3D three-fluid model with electron, proton, and $He^{++}$ ion
fluids to compute the tilted streamer belt structure in the inner corona in
the region observed by HERSHEL instruments. We find good qualitative agreement
of the observed streamer structure of protons and $He^{++}$ ions, that also
agrees with previous more simplified 2.5D three-fluid models of the slow solar
wind in coronal streamers, as well as 3D model with $O^{5+}$ ion as the third
fluid with depleted equatorial streamer belt core helium abundance and
enhanced helium abundance in the open field regions. Specifically, Moses et
al. (2020) computed the helium relative abundance shown in their Figure 3 from
the ratio of the H I and He II emissions, and we find qualitative agreement
with our three-fluid modeling results.

The present model shows that the main processes that shape the distinct helium
streamer structure and slow solar wind helium abundance are gravitational
settling of $He^{++}$ in the core of the quiescent streamer closed-field
region as well as collisional and thermal coupling of $He^{++}$ ions to the
electrons and protons. The increase $He^{++}$ relative abundance in open field
region is produced by the velocity-dependent Coulomb friction process with the
out-flowing electrons and protons, that tend to minimize differences between
the fluids and as a result ’drags-out’ the heavier $He^{++}$ ions. It is
interesting to note that the observed depleted $He^{++}$, and the enhanced
protons abundance in the streamer belt core is is captured by the idealized
tilted streamer belt structure appropriate for solar minimum conditions, and
does not require a complex multiple streamer configuration suggested in the
past (Noci et al., 1997; Moses et al., 2020). However, reproducing finer
details of the observation would indeed require more realistic magnetic field
structure in the 3D multi-fluid model, as well as detailed emission
calculations and line-of-sight integration, and these are left for future
studies. Our modeling results that find very low helium abundance in the core
of streamer are in agreement with recent in-situ observations of low $A_{He}$
at 1AU traced back to coronal streamer structures (Yogesh et al., 2024). The
present results extend the previous three-fluid modeling studies with
$He^{++}$ to more realistic 3D tilted streamer belt structure, appropriate for
solar minimum conditions.

##  6 Acknowledgements

We would like to acknowledge discussions with Dr. Lucia Abbo. L.O. and Y.
acknowledge support by NSF grant AGS-2300961, L.O. acknowledges support by
NASA LWS grant 80NSSC20K0648. Resources supporting this work were provided by
the NASA High-End Computing (HEC) Program through the NASA Advanced
Supercomputing (NAS) Division at Ames Research Center. We would like to
acknowledge high-performance computing support from Cheyenne
(doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information
Systems Laboratory, sponsored by the National Science Foundation.

## References

  * Abbo et al. (2010) Abbo, L., Antonucci, E., Mikić, Z., et al. 2010, Advances in Space Research, 46, 1400, doi: 10.1016/j.asr.2010.08.008
  * Abbo et al. (2016) Abbo, L., Ofman, L., Antiochos, S. K., et al. 2016, Space Sci. Rev., 201, 55, doi: 10.1007/s11214-016-0264-1
  * Alterman & Kasper (2019) Alterman, B. L., & Kasper, J. C. 2019, ApJ, 879, L6, doi: 10.3847/2041-8213/ab2391
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481, doi: 10.1146/annurev.astro.46.060407.145222
  * Borrini et al. (1982) Borrini, G., Gosling, J. T., Bame, S. J., & Feldman, W. C. 1982, J. Geophys. Res., 87, 7370, doi: 10.1029/JA087iA09p07370
  * Braginskii (1965) Braginskii, S. I. 1965, Reviews of Plasma Physics, 1, 205
  * Fineschi et al. (2003) Fineschi, S., Antonucci, E., Romoli, M., et al. 2003, Proc. SPIE, 4853, 162. doi:10.1117/12.460431
  * Geiss et al. (1970) Geiss, J., Hirt, P., & Leutwyler, H. 1970, Sol. Phys., 12, 458, doi: 10.1007/BF00148028
  * Giordano et al. (2007) Giordano, S., Fineschi, S., Ofman, L., Mancuso, S., & Abbo, L. 2007, in ESA Special Publication, Vol. 641, Second Solar Orbiter Workshop, ed. E. Marsch, K. Tsinganos, R. Marsden, & L. Conroy, 31
  * Grevesse & Sauval (1998) Grevesse, N., & Sauval, A. J. 1998, Space Sci. Rev., 85, 161, doi: 10.1023/A:1005161325181
  * Kasper et al. (2012) Kasper, J. C., Stevens, M. L., Korreck, K. E., et al. 2012, ApJ, 745, 162, doi: 10.1088/0004-637X/745/2/162
  * Kasper et al. (2007) Kasper, J. C., Stevens, M. L., Lazarus, A. J., Steinberg, J. T., & Ogilvie, K. W. 2007, ApJ, 660, 901, doi: 10.1086/510842
  * Laming & Feldman (2001) Laming, J. M., & Feldman, U. 2001, ApJ, 546, 552, doi: 10.1086/318238
  * Laming & Feldman (2003) —. 2003, ApJ, 591, 1257, doi: 10.1086/375395
  * Li et al. (2006) Li, B., Li, X., & Labrosse, N. 2006, J. Geophys. Res., 111, A08106, doi: 10.1029/2005JA011303
  * Moses et al. (2020) Moses, J. D., Antonucci, E., Newmark, J., et al. 2020, Nature Astronomy, 4, 1134, doi: 10.1038/s41550-020-1156-6
  * Neugebauer et al. (1996) Neugebauer, M., Goldstein, B. E., Smith, E. J., & Feldman, W. C. 1996, J. Geophys. Res., 101, 17047, doi: 10.1029/96JA01406
  * Noci et al. (1997) Noci, G., Kohl, J. L., Antonucci, E., et al. 1997, in ESA Special Publication, Vol. 404, Fifth SOHO Workshop: The Corona and Solar Wind Near Minimum Activity, ed. A. Wilson, 75
  * Ofman (2000) Ofman, L. 2000, Geophys. Res. Lett., 27, 2885, doi: 10.1029/2000GL000097
  * Ofman (2004a) —. 2004a, Advances in Space Research, 33, 681, doi: 10.1016/S0273-1177(03)00235-7
  * Ofman (2004b) —. 2004b, J. Geophys. Res., 109, A07102, doi: 10.1029/2003JA010221
  * Ofman et al. (2011) Ofman, L., Abbo, L., & Giordano, S. 2011, ApJ, 734, 30, doi: 10.1088/0004-637X/734/1/30
  * Ofman & Davila (2001) Ofman, L., & Davila, J. M. 2001, ApJ, 553, 935, doi: 10.1086/320960
  * Ofman & Kramar (2010) Ofman, L., & Kramar, M. 2010, in Astronomical Society of the Pacific Conference Series, Vol. 428, SOHO-23: Understanding a Peculiar Solar Minimum, ed. S. R. Cranmer, J. T. Hoeksema, & J. L. Kohl, 321, doi: 10.48550/arXiv.1004.4847
  * Ofman et al. (2011) Ofman, L., Abbo, L., & Giordano, S. 2011, ApJ, 734, 30. doi:10.1088/0004-637X/734/1/30
  * Ofman et al. (2015) Ofman, L., Provornikova, E., Abbo, L., & Giordano, S. 2015, Annales Geophysicae, 33, 47, doi: 10.5194/angeo-33-47-2015
  * Riley et al. (2006) Riley, P., Linker, J. A., Mikić, Z., et al. 2006, ApJ, 653, 1510, doi: 10.1086/508565
  * Steinolfson & Nakagawa (1976) Steinolfson, R. S., & Nakagawa, Y. 1976, ApJ, 207, 300, doi: 10.1086/154494
  * Wang & Sheeley (1992) Wang, Y. M., & Sheeley, N. R., J. 1992, ApJ, 392, 310, doi: 10.1086/171430
  * Yogesh et al. (2021) Yogesh, Chakrabarty, D., & Srivastava, N. 2021, MNRAS, 503, L17, doi: 10.1093/mnrasl/slab016
  * Yogesh et al. (2022) —. 2022, MNRAS, 513, L106, doi: 10.1093/mnrasl/slac044
  * Yogesh et al. (2023) —. 2023, MNRAS, 526, L13, doi: 10.1093/mnrasl/slad112
  * Yogesh et al. (2024) Yogesh, Gopalswamy, N., Chakrabarty, D., et al. 2024, ApJ, submitted.