# On Dynamical Systems in the Étale Topology

Lars Andersen

###### Abstract.

We discuss $\mathcal{D}$-modules and dynamical systems in the étale topology.
We construct differential schemes associated to morphisms $f:X\to S$ of
schemes of the same dimension. We introduce differential inertia group
$I_{diff}^{i}$ which act trivially on the special fibers of these schemes. As
an application we discuss the problem of counting points on elliptic curves
which we show is connected to vanishing cycles of singularities and to closed
orbits of dynamical systems.

###### Contents

  1. 1 Introduction
  2. 2 Differential Schemes
    1. 2.1 The étale topology
    2. 2.2 Differential schemes
    3. 2.3 The differential inertia groups
      1. 2.3.1
      2. 2.3.2
    4. 2.4 The sheaf of differential cycles
  3. 3 Appendix: Collatz Conjecture

Classification: 14-XX, 94-XX

##  1\. Introduction

In these notes we make an assault on dynamical systems in positive
characteristics. In this framework a solution would be a geometric point of
the differential scheme of the dynamical system. There follows a discussion on
the number of points of elliptic curves which we formulate as being calculable
from the number of orbits of a dynamical system or equivalently as a sum of
Milnor numbers of a certain finite family of morphisms. A change in the number
of vanishing cycles would correspond to a bifurcation of the dynamical system.
The main result is in fact very trivial but in a sense pleasing theoretically
in that number theory, analysis and geometry are all present and connected.

######  Acknowledments.

My gratitude goes to Jan Stevens, Georges Comte and Michel Raibaut who were my
master thesis respectively doctoral thesis supervisor.

##  2\. Differential Schemes

###  2.1. The étale topology

Let $k$ be $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$. Let $Sch^{\'{e}t}(k)$
denote the category of étale schemes [1] over $k$. These are schemes as ringed
spaces in the sense of EGA but with the étale topology instead of the Zariski
topology. Let $X\in Sch^{\'{e}t}(k)$ be of finite type and irreducible. A
point of $X$ is the inclusion $\text{Spec}(\mathfrak{m})\to\text{Spec}(X)$
where $\mathfrak{m}\subset\mathcal{O}_{X}$ is a maximal ideal. A geometric
point of $X$ is a map $\text{Spec}(k^{alg})\to X$.

###  2.2. Differential schemes

Let $\mathcal{D}=A^{n}(k)$ be Weyl algebra consisting of differential
operators

$P=\sum_{\alpha}f_{\alpha}\partial_{\alpha}$

where $f_{\alpha}\in\mathcal{O}_{X}$ and where
$\partial_{\alpha}:\mathcal{O}_{X}\to\mathcal{O}_{X}$ are the partial
differential operators. A $\mathcal{D}$-module $M$ is a left $A^{n}(k)$-module
or equivalently a $\mathcal{O}_{X}$-module with a left $\mathcal{D}$-action.
Given an associative ring $R$ and a left $R$-module $M$ say that a proper
submodule $P\subset M$ is a prime submodule [4, 2] if $m(rR)\subseteq P$
implies $r\in P$ or $m\in Ann(M/P)$. Let $\text{Spec}(M)={N\subset
M|N\text{prime submodule of M}}$ be the _prime spectrum_ of $M$. Then there
exists a topology on $\text{Spec}(M)$ such that if $\nu(N)=ann(M/N)$ then

$D(N)=\text{Spec}(M)\setminus\nu(N)$

is a basis of open subsets. Furthermore $\nu:\text{Spec}(M)\to\text{Spec}(R)$
is continuous with respect to this topology which we will following[2] call
the Zariski topology on the prime spectrum of modules. We can remark here that
$\text{Spec}(M)$ might be empty. Suppose now that $R$ is such that there is a
choice of $N$ as above satisfying the left Ore conditions with respect to the
set of $\nu(N)$.

######  Definition 1.

Let $\Gamma(D(N),\mathcal{O}_{M})=M\otimes_{R}[\nu(N)^{-1}]R$ where
$[\nu(N)^{-1}]R$ is the left localisation of $R$ by $\nu(N)$.

Then $(M,\mathcal{O}_{M})$ is a ringed space.

######  Definition 2.

A differential affine scheme is any scheme isomorphic to $(M,\mathcal{O}_{M})$
as a ringed space where $M$ is a $\mathcal{D}$-module.

In the case where $M=\text{Spec}k[x_{1},\dots,x_{n}]$ one knows [3,
Proposition I.2] that $k[x_{1},\dots,x_{n}]$ is a simple module hence $(0)$ is
the only prime submodule. Then

$(\text{Spec}(k[x_{1},\dots,x_{n}]),[k[x_{1},\dots,x_{n}]\otimes_{\mathbb{D}}\text{Ann}(k[x_{1},\dots,x_{n}])^{-1}]\mathcal{D})=$
$((0),[k[x_{1},\dots,x_{n}]\otimes_{\mathbb{D}}\text{Ann}(k[x_{1},\dots,x_{n}])^{-1}]\mathcal{D})$

which already has a richer structure albeit that it consists of a point
topologically. Replacing the module in questions with algebraic formal power
series we would then get richer and richer generic points.
Note however that $\text{Spec}(\mathbb{D})$ is not a point; the Weyl algebrs
$\mathbb{D}$ is simple as a ring but not as a module over itself.

######  Definition 3.

A morphism of differential schemes is a morphism of ringed spaces.

###  2.3. The differential inertia groups

We propose to define an analogue of the inertia groups from number theory. In
number theory these groups gives information on the ramification of primes
such as wild ramification. In geometry they provide according to Deligne [6]
the number of vanishing cycles in the arithmetic case where one has to not
only take into account the usual vanishing cycles but also wild vanishing
cycles. The latter only appear when the residue characteristics is positive
whereas the tame vanishing cycles corresponds to the usual vanishing cycles.
The case to consider is when one has e.g.

$f(x,y)=y^{3}+x^{2}+x^{3},\qquad(x,y)\in(\mathbb{Z}/2\mathbb{Z})^{2}$

Then the origin is an isolated point but the Jacobian ideal is of dimension
$4$. Over the complex numbers the dimension is instead $2$. Thus, there are
$2$ wild and $2$ tame vanishing cycles. This is of course due to the behaviour
of tangent planes since the differential of a map depends on the residue
characteristics. For differential equations one has the same phenomenon yet
higher differentials plays a role. We therefore propose the following
definition.

######  Definition 4.

The differential inertia groups are the groups

$I_{diff}^{i}=\\{D\in\mathcal{D}_{X}/\mathcal{O}_{X}|D\partial^{k}\equiv
0\quad\forall k\leq i\\}$

where we write $\equiv$ to mean that only constant solutions to the
corresponding differential operator can be found.

Note that this means that there is only the trivial point (in the sense of the
functor of points) of the $\mathbb{D}$-module namely the trivial solution.

######  Example 1.

On $X=\text{Spec}\mathbb{Z}/p\mathbb{Z}[x]/x^{4}$ we have
$\partial^{2}\partial(1+x+x^{2}+x^{3})=6$ so $D=\partial$ is in $I_{diff}^{2}$
in any characteristics. If $p=3$ then $D\in G_{D}^{1}$. If $p=2$ then
$DD(1+x+x^{2}+x^{3})=D^{1}(1+3x^{2})=6x=0$ so $D\in G_{D}^{1}$.

Let $(M,\mathcal{O}_{M})$ be a differential scheme defined over
$(X,\mathcal{O}_{X})$. Then the differential inertia groups $I_{diff}^{i}$ act
on the constant sheaf $\mathcal{D}_{X}$ hence on the $\mathcal{D}_{X}$ module
$M$ and thus on $\Gamma(U,\mathcal{O}_{M})$ by construction. On the other hand
at a geometric point $\bar{x}=\text{Spec}(k(x)^{alg})$ whose local ring has
generic point $\eta$ and geometric generic point
$\bar{\eta}=\text{Spec}(k(\eta)^{alg})$ the inertia group $I$ is defined by

$1\to I\to\text{Gal}(k(\eta)^{alg}/k(\eta))\to\text{Gal}(k(s)^{alg}/k(s))\to
1$

The inertia group therefore acts trivially on $k(x)^{alg}$ and thus also
trivially on sheaves on $X$ localised at $\bar{x}$. Given a morphism of
schemes $f:X\to S$ write $X_{s}=X\times_{S}s$ for the scheme-theoretical fiber
over $s$.

######  Definition 5.

The sheaves of vanishing cycles of a morphism of schemes $f:X\to S$ at a point
$\bar{x}$ is the right derived functor

$R^{i}\Psi(K)_{\eta})=R^{i}\bar{i}^{\ast}\bar{j}_{\ast}(K_{\eta})$

where $\bar{i}:X_{\bar{s}}\to\bar{X}$ and $\bar{j}:X_{\bar{\eta}}\to\bar{X}$
are the inclusions of the geometric special fiber respectively geometric
generic fiber into $\bar{X}=X\times_{S}\bar{S}$ where $\bar{S}\to S$ is a
normalisation of $S$ at $k(\eta)$.

The above discussion means the action of the inertia group is trivial on the
sheaves of vanishing cycles.

Our goal is to connect these two different actions in the case where $f:X\to
S$ is a morphism of smooth schemes such that $f$ has an isolated critical
point in the origin $s$. We therefore propose to study

######  Definition 6.

The associated $\mathcal{D}_{S}$-module is $V=f-\partial/\partial s$

######  Definition 7.

The gradient associated $\mathcal{D}_{S}$-module is
$W=\text{grad}f-\partial/\partial s$.

The associated can be localised with localisations

$V_{s}=f_{s}-\partial/\partial s,\qquad V_{\eta}=f_{\eta}-\partial/\partial s$

and similarly for $W.$

#### 2.3.1.

The inertia group acts on the coefficients and acts trivially on
$V_{\bar{s}}$. The differential inertia group $I_{diff}^{i}$ on $S$ is
$I_{diff}^{i}=\\{D|\partial_{s}^{k}D\equiv 0,\quad\forall k\leq i\\}$ where
$\partial_{s}=\partial/\partial s$.

Let $k=0$. If $D=\partial_{s}\in I_{diff}^{0}$ then $\partial x/\partial s=0$
so there is only the constant solution. Let $k=1$. If $D=\partial_{s}\in
I_{diff}^{1}$ then $\partial^{2}x/\partial s^{2}=0$ and
$(\partial^{2}f/\partial x_{i}^{2})(\partial x_{i}/\partial s)^{2}=0$ which
gives $\partial^{2}f/\partial x_{i}c_{i}^{2}=0$.
Let $k=2$. If $D=\partial_{s}\in I_{diff}^{2}$ then $\partial^{3}x/\partial
s^{3}=0$ and

$\frac{\partial f}{\partial
x_{i}}(c_{i}t^{2}+c_{i}^{\prime}t+c_{i}^{\prime\prime})+3c_{i}(c_{i}t+c_{i}^{\prime})f=k_{i}(x^{\prime})+c(x^{\prime})$

with solutions

$f=-k_{i}(x^{\prime})\frac{c_{i}t^{2}+c_{i}^{\prime}t+c_{i}^{\prime\prime}}{9c_{i}^{2}(c_{i}t+c_{i}^{\prime})^{2}}$
$+k_{i}(x^{\prime})\frac{x_{i}}{9c_{i}(c_{i}t+c_{i}^{\prime})^{2}}+\frac{c(x^{\prime})}{3c_{i}(c_{i}t+c_{i}^{\prime})}$

which is clearly not a polynomial function anymore. We have proven

######  Lemma 0.1.

The differential inertia groups act trivially on $V_{s}$.

#### 2.3.2.

The inertia group $I$ acts on the coefficients and acts trivially on
$W_{\bar{s}}$. We have $\frac{\partial}{\partial s}\frac{\partial f}{\partial
x_{i}}(x_{i}(s))=\frac{\partial^{2}f}{\partial x_{i}^{2}}\frac{\partial
x_{i}}{\partial s}.$ Let $k=0$. If $D=\partial_{s}\in I_{diff}^{0}$ then
$\partial x/\partial s=0$ and the constant solution is the only solution. Let
$k=1$. If $D\in I_{diff}^{1}$ then $\partial^{2}x/\partial s^{2}=0$ and

$\frac{\partial^{2}f}{\partial x_{i}^{2}}\frac{\partial x_{i}}{\partial
s}=k_{i}$

and

$f=\frac{k_{i}}{m_{i}}x_{i}^{2}+d_{i}(x^{\prime})x_{i}+e_{i}(x^{\prime})$

for each $i=1,\dots,n$. By symmetry we find that

$f=\sum c_{i}x_{i}^{2}$

and in particular the differential inertia groups stabilise for $k\geq 2$. If
the characteristics divide the coefficients of $f$ we get more erratic
behaviour. Let $k=2$. If we consider the action of $I_{diff}^{k}$ on
$W_{\eta}$ instead then we reach the same conclusions as before for $k=0,1$.
If $k=2$ and if $D\in I_{diff}^{2}$ then $\partial^{3}x/\partial s^{3}=0$ and

$\frac{\partial^{3}f}{\partial
x_{i}^{3}}=\frac{k_{i}}{c_{i}(c_{i}t+l_{i})^{2}}$

as a map $\pi_{i}(k(\eta))\to k$. In particular the action is non-trivial in
general. We have proven

######  Lemma 0.2.

In the situation above, the differential inertia groups acts trivially on
$W_{s}$ for all $k\geq 1$ unless $f$ is a Morse singularity.

###  2.4. The sheaf of differential cycles

Consider $f:X\to S$ a morphism of smooth schemes of equal dimensions. A point
on a differential scheme $(\text{Spec}(M),\mathcal{O}_{M})$ where $M$ is a
$\mathcal{D}_{X}$-module is a morphism of ringed spaces
$\text{Spec}\mathcal{O}_{X,x}^{sh}\to\text{Spec}(M)$ where $x\in X$ and where
the ring of algebraic power series $\mathcal{O}_{X,x}^{sh}$ is endowed with
the structure of a $\mathcal{D}_{X}$-module by the standard differentials in
an étale neighborhood of $x$. We will write
$X_{(x)}=\text{Spec}\mathcal{O}_{X,x}^{sh}$ and
$\tilde{\eta}=\text{Spec}Frac\mathcal{O}_{X,x}^{sh}$. Recall [5] that the
étale Milnor fiber at $\bar{x}$ is $X_{(x)}\times_{\tilde{\eta}}\bar{\eta}$.
Let $TX$ be the tangent space of $X$ over $S$ and consider the commutative
diagram

${\mathcal{V}}$${X}$${S}$${TS}$$\scriptstyle{proj}$$\scriptstyle{(x_{1}^{-1},\dots,x_{n}^{-1})}$$\scriptstyle{f}$$\scriptstyle{\partial/\partial
s}$

with the diagonal morphism being $f\times_{S}\partial/\partial s$ and the
upper morphism coming from the inclusion $X\to TX$.

######  Definition 8.

The differential scheme of $f$ is $\mathcal{V}\in Sch^{\'{e}t}$ with fibers
$\mathcal{V}_{s}$ and $\mathcal{V}_{\eta}$ given by the forming commutative
diagrams as above but with $X$ replaced by $i:X_{s}\to X$ and $j:X_{\eta}\to
X$.

Let $x\in V(f)$ and let $s$ denote the origin in $S$. Consider the commutative
diagrams

${\mathcal{V}_{(s)}}$${X_{s}}$${S_{\bar{s}}}$${TS_{\bar{s}}}$$\scriptstyle{proj}$$\scriptstyle{(x_{1}^{-1},\dots,x_{n}^{-1})}$$\scriptstyle{f}$$\scriptstyle{\partial/\partial
s}$

and

${\mathcal{V}_{\bar{\eta}}}$${X_{\bar{\eta}}}$${S_{\bar{\eta}}}$${TS_{\bar{\eta}}}$$\scriptstyle{proj}$$\scriptstyle{(x_{1}^{-1},\dots,x_{n}^{-1})}$$\scriptstyle{f}$$\scriptstyle{\partial/\partial
s}$

In the fiber product $\mathcal{V}=X\times_{TS}S$ we have the lifts

$\bar{i}:X\times_{TS}S\times\bar{s}\to X\times_{TS}S$

and

$\bar{j}:X\times_{TS}S\times\bar{\eta}\to X\times_{TS}S$

of the inclusions $i:\text{Spec}(k(\bar{s}))\to TS$ and
$j:\text{Spec}(k(\bar{\eta}))\to TS$. Let $r:S\to k$ be
$r(x_{1},\dots,x_{n})=\sum x_{i}^{2}$ and consider instead
$i^{r}=r^{-1}\circ\tilde{i}:\text{Spec}(k(s^{r})\to S$ and
$j^{r}=r^{-1}\circ\tilde{j}:\text{Spec}(k(\eta^{r})\to S$ with
$\tilde{i},\tilde{j}$ denoting the inclusions of the closed and generic point
of the ground field.

######  Definition 9.

The differential sheaf of $K\in D(X)$ at $x$ is

$\zeta=\bar{i}^{r\ast}\bar{j}^{r}_{\ast}(K_{\eta})$

We will now define closed orbits as being $\mathbb{S}^{1}$-valued points of
$V_{x,s}=f_{x}\times_{S}\partial/\partial s$ over $s$ where $f_{x}$ denotes
the strict localisation of $f$. In detail

######  Definition 10.

The complex of differential cycles or closed orbits of $f$ at $x\in V(f)$ is
the derived complex of sheaves $R^{\ast}\zeta(K_{\eta})$.

The main theorem is then the following

######  Theorem 1.

Suppose that the characteristics is distinct from $2$. The stalks of
$R^{\ast}\zeta(K_{\eta})$ over $(x,s),x\in V(f)$ are generated by vanishing
cycles of $r\circ f$ at $x$. In particular their number is given by the
generalized Milnor number, by Deligne’s Theorem [6]. The stalks are also
generated by the periodic solutions of the dynamical system
$f_{s}=\partial/\partial s$.

###### Proof.

The stalks $R^{\ast}\zeta(K_{\eta})_{x,\eta^{r}}$ are

$H^{\ast}(r\circ f_{x})\times_{k}(r\circ\partial/\partial
s)\times_{k}\bar{\eta}^{r},K)=H^{\ast}((f_{x}-\partial/\partial
s)^{-1}(r^{-1}\eta^{r}))$

which if $char\neq 2$ are the periodic solutions to $f_{s}=\partial/\partial
s$ or the vanishing cycles of $r\circ f$ in the sense of [6]. ∎

######  Lemma 1.1.

The differential inertia groups act trivially on $R^{\ast}\zeta(K_{\eta}$

###### Proof.

See the discussions above and Corollary 0.1 . ∎

######  Example 2.

Consider the dynamical system

$dx/dt=y^{2}+ax^{3}+bx,\quad dy/dt=y-l$

over $k$ the $p$-adic numbers for $p$ a large prime. We let

$f_{i}:\text{Spec}\mathbb{Z}/p\mathbb{Z}[x,y]\to\mathbb{Z}/p\mathbb{Z},$
$f_{i}(x,y)=i^{2}+ax^{3}+bx$

with critical locus $\Sigma_{i}$ and let $l_{i}=\bigoplus_{j\in\Sigma_{i}}\dim
H^{0}(\mathcal{F}_{i}^{\'{e}t})$ denote the sum of dimensions of the homology
of the étale Milnor fibers of $f_{i}$. Then

$|E(\mathbb{Z}/p\mathbb{Z})|=\sum_{i=1}^{p}l_{i}+1$

where $E(\mathbb{Z}/p\mathbb{Z})$ denotes the _set_ of points of
$\mathcal{C}=\\{y^{2}+ax^{3}+bx=0\\}$. In fact since this set is finite,

$|E(\mathbb{Z}/p\mathbb{Z})|=\sum_{i=0}^{p}\mathcal{C}\cap\\{y=i\\}(\mathbb{Z}/p\mathbb{Z})+1=\sum_{i=0}^{p}l_{i}+1$

because at each point there is either one, two or three vanishing cycles
according to whether the cubic equation defining the summand has one, two or
three points so we get the result in light of the Theorem 1 . Here we added
the point at infinity.

##  3\. Appendix: Collatz Conjecture

An outline of a proof would be to consider Collatz function as a continuous
$2$-adic function $f:\mathbb{A}_{k}^{1}\to\mathbb{A}_{k}^{1}$. The associated
dynamical system has a stationary point at the origin $x_{1}=0$ and a
stationary point at $x_{2}=1$. Corresponding to $x_{2}$ is unique tame
vanishing cycle, since the local equation $3x+1$ has only tame ramification.
Corresponding to $x_{1}$ is a map singularity; the map $x\mapsto x/2$ isn’t
defined over the residue field $k(x_{1})=\mathbb{Z}/2\mathbb{Z}.$ Now, if
there is a closed orbit then there is a vanishing cycle at $x_{1}$. This
vanishing cycle would be in an étale neighborhood of the origin. Following [6]
it would be an element of

$f^{-1}(\text{Spec}\bigcup_{l}\text{Frac
}\mathbb{F}_{2}^{l}((x)))\cap\text{Spec}\bigcup_{l}\mathbb{F}_{2}^{l}[[x]]$

hence a solution in terms of a zero of an algebraic equation with
$\mathbb{Z}/2\mathbb{Z}$-coefficients. So one possible proof of the conjecture
would be to argue in detail that there can or cannot be any such solution due
to $f$ not being continuous reduced modulo $2$.

## References

  * [1] M. Artin, A. Grothendieck, and J-L. Verdier.  Theorie de Topos et Cohomologie Étale des Schemas I, II, III, volume 269, 270, 305 of Lecture Notes in Mathematics.  Springer, 1971.
  * [2] Janko Bračič.  The Prime Spectrum of a Module.  2000\.
  * [3] S. C. Coutinho.  A Primer of Algebraic D-Modules.  London Mathematical Society Student Texts. Cambridge University Press, 1995.
  * [4] J Dauns.  Prime Modules.  J. Reine Angew. Math., 298:156–181, 1978.
  * [5] D. S. Rim A. Grothendieck M. Raynaud P. Deligne, Michèle Raynaud.  Groupes de Monodromie en Geometrie Algebrique (Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 I).  Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 1972.
  * [6] N. M. Katz P. Deligne, N. Katz.  Groupes de Monodromie en Geometrie Algebrique (Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 II).  Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 03 1973.


# An optically thin view of the solar chromosphere from observations of the O
I 1355Å spectral line

Mats Carlsson Rosseland Centre for Solar Physics, University of Oslo, P.O. Box
1029 Blindern, NO-0315 Oslo, Norway  Institute of Theoretical Astrophysics,
University of Oslo, P.O. Box 1029 Blindern, NO-0315 Oslo, Norway  Bart De
Pontieu Lockheed Martin Solar & Astrophysics Laboratory, 3251 Hanover Street,
Palo Alto, CA 94304, USA  Rosseland Centre for Solar Physics, University of
Oslo, P.O. Box 1029 Blindern, NO-0315 Oslo, Norway  Institute of Theoretical
Astrophysics, University of Oslo, P.O. Box 1029 Blindern, NO-0315 Oslo, Norway

###### Abstract

The O I 1355Å spectral line is one of the only optically thin lines that are
both routinely observed and thought to be formed in the chromosphere. We
present analysis of a variety of observations of this line with the Interface
Region Imaging Spectrograph (IRIS), and compare it with other IRIS diagnostics
as well as diagnostics of the photospheric magnetic field. We utilize special
deep exposure modes on IRIS and provide an overview of the statistical
properties of this spectral line for several different regions on the Sun. We
analyze the spatio-temporal variations of the line intensity, and find that it
is often significantly enhanced when and where magnetic flux of opposite
polarities cancel. Significant emission occurs in association with
chromospheric spicules. Because of the optically thin nature of the O I line,
the non-thermal broadening can provide insight into unresolved small-scale
motions. We find that the non-thermal broadening is modest, with typical
values of 5-10 km/s, and shows some center-to-limb variation, with a modest
increase towards the limb. The dependence with height of the intensity and
line broadening off-limb is compatible with the line broadening being
dominated by the superposition of Alfvén waves on different structures. The
non-thermal broadening shows a modest but significant enhancement above
locations that are in between photospheric magnetic flux concentrations in
plage, i.e., where the magnetic field is likely to be more inclined with
respect to the line-of-sight. Our measurements provide strict constraints on
future theoretical models of the chromosphere.

Sun: chromosphere – Sun: transition region – Sun: magnetic fields –
magnetohydrodynamics (MHD)

††journal: ApJ

##  1 Introduction

The solar chromosphere is a highly dynamic and finely structured region of the
solar atmosphere that is sandwiched between the visible surface or photosphere
and the million-degree outer atmosphere or corona. All non-thermal energy that
drives the solar wind and the heating of the corona traverses this critical
region. Moreover, despite the only modest enhancement of the chromospheric
temperature compared to that of the photosphere, it requires several orders of
magnitude more non-thermal energy to drive the dynamics and energetics of the
chromosphere than the rest of the solar atmosphere combined. This is because
of the high chromospheric densities: the chromosphere contains more mass than
the region stretching from the transition region to the edges of the
heliosphere. Despite its obvious importance, it is relatively poorly
understood, with many open questions remaining regarding the physical
processes that drive the dynamics and energetics in the chromosphere (Carlsson
et al., 2019).

Start Date | End Date | Target | OBS-ID | Exp | Solar x | Solar y | $\theta$ | $\mu$
---|---|---|---|---|---|---|---|---
2015-09-09T07:59:58 | 2015-09-09T10:56:08 | AR 12412 | 3610091469 | 15 | -449 | -213 | 32 | 0.82-0.88
2015-09-23T00:09:43 | 2015-09-23T02:58:30 | AR 12920 | 3690092077 | 30 | -754 | 104 | 59 | 0.49-0.68
2015-09-23T20:22:31 | 2015-09-23T23:11:18 | AR 12920 | 3690092077 | 30 | -565 | 81 | 41 | 0.74-0.85
2015-09-24T17:22:13 | 2015-09-24T20:11:00 | AR 12920 | 3690092077 | 30 | -408 | 77 | 29 | 0.86-0.93
2015-09-25T21:00:43 | 2015-09-25T23:49:30 | AR 12920 | 3690092077 | 30 | -182 | 63 | 15 | 0.96-0.99
2015-09-26T21:23:41 | 2015-09-27T00:12:28 | AR 12920 | 3690092077 | 30 | 27 | 58 | 4 | 0.98-1.00
2015-09-28T17:08:41 | 2015-09-28T19:57:28 | AR 12920 | 3690092077 | 30 | 425 | 83 | 23 | 0.84-0.92
2015-10-02T00:54:43 | 2015-10-02T03:43:30 | AR 12920 | 3690092077 | 30 | 892 | 140 | 62 | 0.00-0.49
2015-10-02T20:47:20 | 2015-10-02T23:36:08 | AR 12920 | 3690092077 | 30 | 917 | 165 | 68 | 0.00-0.50
2016-03-04T10:34:35 | 2016-03-04T17:26:20 | QS | 3690094078 | 60 | -3 | 900 | 69 | 0.00-0.55
2016-03-05T10:14:03 | 2016-03-05T17:05:48 | QS | 3690094078 | 60 | 5 | -890 | 67 | 0.00-0.56
2016-03-05T17:17:07 | 2016-03-06T00:08:52 | QS | 3690094078 | 60 | 5 | -714 | 48 | 0.55-0.76
2016-03-06T00:20:11 | 2016-03-06T07:11:56 | QS | 3690094078 | 60 | 5 | -538 | 34 | 0.76-0.89
2016-03-06T10:41:36 | 2016-03-06T17:33:21 | QS | 3690094078 | 60 | 5 | -890 | 67 | 0.00-0.56
2016-03-06T23:01:56 | 2016-03-07T05:53:41 | QS | 3690094078 | 60 | -2 | 7 | 4 | 0.99-1.00
2016-03-07T06:05:00 | 2016-03-07T12:56:45 | QS | 3690094078 | 60 | -4 | 728 | 49 | 0.52-0.75
Table 1: For all IRIS observations used, we provide the start and end date,
target, IRIS OBS-ID, exposure time (s), solar x/y coordinates (″), $\theta$
(the viewing angle between the local vertical and line-of-sight vector, in
degrees), and $\mu=\cos\theta$.

One of the main reasons for our limited knowledge is the lack of unambiguous
diagnostics. Most spectral lines that emanate from the chromosphere are
optically thick and are subject to non-LTE radiative transfer effects such as
scattering, partial frequency redistribution, etc (e.g., Leenaarts et al.,
2013a, b). In addition, non-equilibrium ionization plays a key role in the
chromosphere and for some chromospheric diagnostics (e.g., Carlsson & Stein,
2002; Golding et al., 2014, 2016; Leenaarts et al., 2016; Golding et al.,
2017). This renders their interpretation difficult and dependent on inversion
techniques that often suffer from limiting assumptions and/or non-uniqueness,
despite major advances in techniques (de la Cruz Rodríguez et al., 2016; Sainz
Dalda et al., 2019). One of the few spectral lines that is optically thin and
formed in the chromosphere is the O I 1355.598 Å intersystem line
($2s^{2}\,2p^{3}\,3s\,^{5}\\!S_{2}-2s^{2}\,2p^{4}\,{}^{3}\\!P_{2}$), hereafter
called the O I 1355Å line. It is routinely observed with the Interface Region
Imaging Spectrograph (IRIS, De Pontieu et al., 2014). Lin & Carlsson (2015)
performed an analysis of the formation of this line in an advanced numerical
simulation calculated with the Bifrost code (Gudiksen et al., 2011) and
demonstrated that the line formation is optically thin.

In this paper we provide an overview of observational findings with IRIS
related to the O I 1355Å line. We describe the observations and analysis
techniques of the spectral line profiles in Section 2. We then describe the
statistical properties and center-to-limb variation of the intensity and line
broadening for various solar targets in Section 3. We also describe how the
properties of O I 1355Å are impacted by flux cancellation and the presence of
flux concentrations in, respectively, Sections 4 and 5. We finish with a
discussion and conclusions in Section 6.

##  2 Observations and analysis techniques

For the observational analysis we used several different datasets from IRIS.
For all of these observations we chose to use OBS-ID 3610091469 (for NOAA AR
12412), OBS-ID 3690092077 (for AR 12920) or 3690084078 (for quiet Sun), in
order to maximize the signal-to-noise (S/N) ratio while maintaining high
spectral resolution (to resolve the narrow O I 1355Å line) by introducing long
exposures (15s for AR 12412, 30s for AR 12920, 60s for QS), lossless
compression, and asymmetric summing (x2 summing spatially, no summing
spectrally). AR 12920 was followed over multiple days in September 2015 (see
Table 1), as the AR crossed the disk. The field-of-view of the observations
was 45″x 120″(AR 12412), 112″x 175″(AR 12920) and 140″x 175″(QS) with a
spatial sampling of 0.35″ by 0.33″. The full detector was read out, i.e.,
three different wavelength ranges covering wavelengths from, respectively,
1331.56–1358.40Å (FUV1), 1390.00–1406.79Å (FUV2), and 2782.56–2833.89Å (NUV),
with a spectral sampling of 12.98 mÅ in FUV1, 12.72 mÅ in FUV2, and 25.46 mÅ
in NUV. The nominal spatial resolution of IRIS is 0.33″ in the FUV and 0.4″ in
the NUV. Note that since we used spatially summed data, our spatial resolution
along the slit is Nyquist limited to twice the plate scale (0.66″, i.e., 2
times 0.33″). The spectral resolution (FWHM) of IRIS is dominated by Nyquist
sampling, i.e., 5.7 km/s in both FUV and NUV.

To determine the first moments of the O I spectral line, we used the following
steps: 1. remove the impact of cosmic rays on the detector (using the
clean_exposure.pro code in IDL SolarSoft), 2\. remove the impact of fixed
pattern noise in an extra dark current subtraction, 3\. take the mean of 3x3
pixels to increase the S/N. 4\. apply a single Gaussian fit to the spectral
line profile in each location.

The second step of this procedure is non-standard and is therefore described
in some detail here. The O I 1355Å line is rather weak and especially the
width determination is influenced by errors in the dark-level. We often see
horizontal stripes of increased width in the width-maps. These stripes are in
the same location over some time and are caused by fixed patterns in the dark-
level that are not removed in the standard reduction pipeline. When a ”hot”
pixel is present at a location in the wing of the O I line, a larger width
results from the Gaussian fit. We used the QS dataset from the center of the
disk (2016-03-06T23:01:56) and formed a mean intensity, $I(y,\lambda)$ by
taking the mean over scan position x of $I(x,y,\lambda)$ for all positions
where the line-core intensity was less than 15 DN. This mean will still have
an imprint of the spectral line. We removed that by subtracting at each
wavelength and pixel y the mean over $\pm$ 10 pixels. The fixed pattern noise
is not completely fixed in time so this procedure makes the most significant
improvement to the width determinations for the QS datasets, and not so much
for the AR 12920 datasets. The time-scales on which the fixed pattern changes
significantly appear to be days to months. This issue and potential fixes in
the pipeline are currently being studied in more detail by the IRIS
calibration team.

The single Gaussian fit is made with three free parameters: the maximum
intensity, the Doppler shift of the profile and the width. A constant
background is first determined from a mean of the spectrum at blue-shifts of
between 200 and 40 km/s relative to the rest-wavelength of the O I 1355Å line
(a line-free region of the spectrum). This constant background is subtracted
from the spectrum before the single Gaussian fit. The fit is only including
the spectrum between -40 and 40 km/s relative to the rest-wavelength of the O
I 1355 Å line, in order to avoid influence from the C I line at 1355.843 Å
(corresponding to 54 km/s from the rest-wavelength of the O I 1355 Å line). A
fit is attempted for all pixels but the fit is judged unsuccessful if the
fitting algorithm (mpfitpeak.pro in IDL SolarSoft) does not converge or the
determined values are outside reasonable values ([0,1000] DN in maximum
intensity, [-30,30] km/s in Doppler shift, [2,30] km/s in width).

The reported line broadening is the 1/e half-width (henceforth referred to as
the 1/e width) as this provides the most probable velocity, which is useful
for interpretation in terms of physical mass motions in the solar atmosphere.
The line width has several contributions:
$w=\sqrt{w^{2}_{inst}+w^{2}_{th}+w^{2}_{nth}}$. We note that the instrumental
broadening $w_{inst}$ in the FUV channel of IRIS has been reported (De Pontieu
et al., 2014) as 25.85 mÅ (i.e., 5.7 km/s) full-width-half-maximum (FWHM),
which translates to 15.52 mÅ or 3.4 km/s 1/e width. The thermal broadening
$w_{th}$ of the O I 1355 Å line is somewhat uncertain given the large range of
chromospheric heights this line is formed over. If we assume that the
chromospheric temperature is within a range of 5,000 to 15,000 K, the thermal
broadening would range between 2.3 and 3.9 km/s (1/e width). Both the
instrumental and thermal broadening are smaller than most widths we report on
here and play only a minor role. For the Mg II k profiles we used a double
Gaussian fit similar to that used by Carlsson et al. (2015a) and Bryans et al.
(2016). The reported width is the 1/e width of the Gaussian fit to the profile
outside the central reversal. For the C II profiles we used a double Gaussian
fit similar to that used by Rathore et al. (2015b).

For some IRIS observations we also provide magnetogram data from HMI for
context. The magnetogram data is based on the circular polarization Stokes
signal in the Fe I 6173Å line and provides information on the line-of-sight
(LOS) magnetic field at photospheric levels. The HMI data has a pixel size of
0.5″. We use the full-disk data to obtain a field-of-view that is matched to
that of IRIS. The co-alignment of the IRIS and HMI magnetogram data is
accomplished through comparison of an IRIS spectroheliogram at 2800Å in the
photospheric wings of the Mg II h & k lines and the HMI magnetogram. This
alignment is greatly facilitated by the similarities between the bright points
in the 2800Å images and the flux concentrations that are detected with HMI.

For one of the IRIS observations, we also provide high-resolution context
magnetograms of the circular polarization in the Fe I 6173Å line using the
CRISP instrument (Scharmer, 2006) at the Swedish 1-m Solar Telescope (SST).
This data was obtained on 2015-09-09 from 09:04:12 UTC to 10:29:04 UTC with a
cadence of 67 seconds. The pixel size is 0.06″. The data was calibrated and
corrected for deformations because of seeing conditions introduced by the
Earth’s atmosphere in an identical manner as described in § 2.2 in Rouppe van
der Voort et al. (2020).

Figure 1: IRIS spectroheliograms of NOAA AR 12920 at 2015-09-26T21:23 UTC
showing in the top row 2800Å, O I 1355Å intensity, and Mg II k3 intensity, and
in the bottom row C II 1335Å intensity, O I 1355Å line broadening, and the Mg
II k line broadening. Black horizontal lines are fiducial marks. Black
vertical lines are data drop-outs. The O I 1355 Å intensity, Mg II k3
intensity, and C II 1335Å intensity are scaled, respectively, between 1 and 35
DN, 0 and 10586 DN, and 3 and 311 DN. Regions with bad fits and/or low peak
counts (below 6 DN) are masked out in the O I 1355 Å line width map. This
figure is accompanied by an animation that allows the reader to blink between
the different panels of the figure to see the various similarities and offsets
described in the text.

##  3 Properties of O I 1355 Å

###  3.1 Morphology and relationship to other spectral lines

Lin & Carlsson (2015) used numerical simulations of a region that aims to
mimic an enhanced network region in quiet Sun and found that the line forms
through a recombination cascade to the upper level of the O I 1355Å line,
followed by emission in the line under optically thin conditions. The
ionization balance of O I and O II is coupled to the ionization of hydrogen
through charge exchange. The O I 1355 Å line is thus formed in the
chromosphere, as O II ionization occurs towards the top of the chromosphere.
Since the line is optically thin, the line broadening is of particular
interest as it provides constraints on turbulent motions in the chromosphere,
a property otherwise only accessible through spectral line inversions of
optically thick chromospheric lines such as Mg II h 2803Å and Mg II k 2796Å
(e.g., de la Cruz Rodríguez et al., 2016). Since the line is chromospheric, it
is of interest to study the morphological properties of spectroheliograms or
maps of the O I 1355 Å intensity and width.

Figure 1 shows that in an active region at disk center, the O I 1355 Å line is
bright in plage regions and some parts of filaments. The morphology appears to
be a combination of dot-like and more extended features. The regions where O I
1355 Å is bright (top row, middle panel) slightly extend beyond the plage
perimeters that are seen in a photospheric 2800Å spectroheliogram (top row,
left panel). The extensions seem wispy and resemble the lower parts of loop-
like or fibril-like features.

There is overall a strong resemblance with the Mg II k3 intensity maps111The
Mg II k3 spectral feature occurs at the wavelength of the central reversal for
a double-peaked Mg II k spectral profile, or the wavelength of the peak for a
single-peaked Mg II k spectral profile (Leenaarts et al., 2013a). The main
difference between both maps is that the regions where Mg II k3 is bright
extend even further beyond the photospheric boundaries of the plage, possibly
because the signal-to-noise (S/N) of the O I 1355 Å line is significantly
lower. The other difference is that filaments are typically dark in Mg II k3
and most often bright in O I 1355 Å (but see the filament around $x=-10$″,
$y=60$″ which is dark in both). There are also significant morphological
similarities with the C II 1335Å intensity (bottom row, left panel). The main
difference here is that the latter shows more differences in brightness
between various plage regions, because it is more sensitive to transition
region conditions (Rathore et al., 2015a), which are, in turn, dominated by
the overlying coronal conditions. In summary, our comparison shows that in
plage regions the O I 1355 Å line does indeed look very chromospheric in
nature with significant similarity between the intensity patterns in the O I
1355 Å and Mg II k line.

Figure 2: Examples of O I 1355 Å profiles from four locations of Fig. 1. The
observed profile is given with symbols and the single Gaussian fit with a
solid line. The positions and reduced $\chi^{2}$ of the fit are given in the
upper left with the 1/e width and its 1$\sigma$ error in the upper right. The
increased intensity in the red part of the window is due to an emission line
from C I at 1355.8 Å. Figure 3: IRIS spectroheliograms of a quiet Sun region
at 2016-03-06T23:01 UTC showing in the top row 2800Å, O I 1355Å intensity, and
Mg II k3 intensity, and in the bottom row C II 1335Å intensity, O I 1355Å line
broadening, and the Mg II k line broadening. Black horizontal lines are
fiducial marks. Black vertical lines are data drop-outs. The O I 1355 Å
intensity, Mg II k3 intensity, and C II 1335Å intensity are scaled,
respectively, between 2 and 141 DN, 0 and 9563 DN, and 19 and 125 DN. Regions
with bad fits and/or low peak counts (below 8 DN) are masked out in the O I
1355 Å line width map. This figure is accompanied by an animation that allows
the reader to blink between the different panels of the figure to see the
various similarities and offsets described in the text.

The spatial patterns in the map of O I 1355 Å linewidths (bottom row, middle
panel) generally map, on large scales, those of the O I 1355 Å intensity (top
row, middle panel). The O I 1355 Å linewidth is enhanced in and around plage
regions, and very low in surrounding quiet Sun regions. The map of O I 1355 Å
line widths is remarkably homogeneous within each plage region, with only
relatively small deviations around the average value of about 7.5 km/s, as
previously remarked by Carlsson et al. (2015a). For each plage region, the
high values of O I 1355 Å widths spatially extend even further beyond the
photospheric plage boundaries than the O I 1355 Å intensity. The values of O I
1355 Å broadening in quiet Sun are very small, often insignificant, suggesting
that, in many quiet Sun locations, the line is not broadened beyond the
combination of instrumental and thermal broadening.

The spatial patterns of the line broadening in O I 1355 Å and Mg II k (bottom
row, right panel) show significant correspondence on very large scales
($\sim$10″): both lines are broader in plage regions and the immediate
vicinity, and narrower in the quiet Sun regions surrounding the plage.
However, the broadening values themselves are very different between these two
lines. Typical values for O I 1355 Å are of order 10 km/s or less, while the
Mg II k values are larger by a factor of $\sim$3\. This is in agreement with
the results from Carlsson et al. (2015a). It is not surprising given that O I
1355 Å is an optically thin line and its width is sensitive to velocity
variations along the LOS and turbulent motions, while Mg II k is an optically
thick line with velocity gradients, turbulent motions, and broadening as a
result of opacity all playing a role in the line width. On very small ($<1$″)
scales there is most often not a good match between O I 1355 Å broadening and
Mg II k broadening.

Figure 2 gives four examples of observed profiles together with the single
Gaussian fit of the O I 1355 Å line from four different regions seen in Fig.
1: Plage, a filament, internetwork and an unusually wide profile. Note that
the spectra have been averaged over 3x3 spatial pixels – the actual number of
photons per wavelength bin is thus nine times the given DN/pixel times four
photons per DN (the gain in the FUV passband222The gain in the NUV passband is
18 photons per DN., De Pontieu et al., 2014). The dominant error source is
photon-noise, proportional to the square-root of the number of photons for
Poisson statistics. For very low count-rates, a number of other error sources
come into play (readout-, digitization-, flat-fielding, fixed-pattern-noise,
straylight subtraction, etc, Wülser et al., 2018). We estimate the total error
as the square-root of the sum of the squares of a term not dependent on the
count-rate (estimated from the standard deviation of the count-rate in a line-
free region between -200 and -40 km/s) and the Poisson noise term. We
furthermore restrict our analysis to profiles with a line-centre count-rate
above a threshold (given in the respective figures). For the three first
profiles, the fit with a single Gaussian is very good. The unusually wide
profile has a markedly non-Gaussian shape. This is typical for the widest
profiles.

In quiet Sun, we find many of the same properties and correspondences, as
shown in Fig. 3. The O I 1355 Å is bright in and around quiet Sun network
regions and faint in the internetwork regions. At disk center, we see that the
O I 1355 Å line is too weak to detect in some of the internetwork regions,
despite the deep exposures and spatial summing in the special observing mode
for these observations. However, in much of the FOV there is sufficient signal
to determine the intensity and line width of the O I 1355 Å line. The findings
for QS are very similar to those described above for active regions. The
strongest signal can be found in and around network regions. A disk center
view shows that the regions where O I 1355 Å intensity is high again extend
beyond the spatial boundaries of the network, similar to what we see in plage
regions. The network-associated O I 1355 Å emission appears to be a
combination of dot-like features and more extended wispy structure that are
reminiscent of the spicules that are often seen to protrude from network flux
concentrations. The internetwork regions are significantly fainter. The O I
1355 Å intensity map shows strong similarity with the Mg II k3 intensity map.
There is, again, similarity with the C II 1335Å intensity map, although
somewhat less than with Mg II k3, with C II 1335Å showing higher contrast and
strongly reduced internetwork intensities.

The O I 1355 Å line width maps reveal extremely low values in the internetwork
regions. In many locations these values are so low that they appear compatible
with a lack of non-thermal broadening. The network regions show a significant
increase in the line width, with the regions of high broadening extending even
further from the network regions than the intensity itself. This is similar to
what we found for plage, although the line width is somewhat reduced in
network compared to plage. The O I 1355 Å broadening is much less than that
found in the Mg II k line, again similar to what we found for active regions.

Figure 4: IRIS spectroheliograms of NOAA AR 12920 as it traverses the disk
between 2015-09-22 and 2015-10-02. Top two rows show the O I 1355Å intensity,
while the bottom two rows show the O I 1355Å line broadening. The O I 1355 Å
intensity is scaled between 0 and 60 DN for all panels. Regions with bad fits
and/or low peak counts (below 4 DN) are masked out in the O I 1355 Å line
width maps.

To gain further insight into the morphology and formation region of O I 1355 Å
it is of interest to study the appearance of the O I 1355 Å intensity and line
width in solar targets that are closer to the limb, both for an active region
and a quiet Sun region. Fig. 4 shows NOAA AR 12920 as it is tracked over the
course of 9 days from close to the east limb to the west limb of the Sun, with
both the minimum and maximum of the color scale for both intensity and line
width identical for all images. It appears that the intensity is lowest close
to disk center, with a steady increase as we approach the limb. At the limb
itself the intensity is increased significantly, as is expected from an
optically thin line: the line-of-sight increases and captures more structures
as the viewing angle is more oblique. We also see localized regions of strong
intensity enhancements, in particular on 2015-09-23, 2015-09-24, 2015-09-25,
and 2015-10-02. We will describe these very bright regions in more detail in §
4.

Figure 5: IRIS spectroheliograms of NOAA AR 12920 at 2015-10-02T00:54 UTC
showing in the top row 2800Å, O I 1355Å intensity, and Mg II k3 intensity, and
in the bottom row C II 1335Å intensity, O I 1355Å line broadening, and the Mg
II k line broadening. Black horizontal lines are fiducial marks. Black
vertical lines are data drop-outs. The O I 1355 Å intensity, Mg II k3
intensity, and C II 1335Å intensity are scaled, respectively, between 0 and 92
DN, 0 and 13169 DN, and 3 and 224 DN. Regions with bad fits and/or low peak
counts (below 6 DN) are masked out in the O I 1355 Å line width map. This
figure is accompanied by an animation that allows the reader to blink between
the different panels of the figure to see the various similarities and offsets
described in the text.

The line width remains relatively independent of the viewing angle except
there is a clear increase for viewing angles close the limb (e.g., 2015-09-23,
2015-10-02). At the limb itself the line width increases in a step-like
fashion, which is to be expected since the integration length doubles at the
limb. Nevertheless the line widths remain relatively modest and the overall
variation is limited to just a few km/s on average, as described below using
histograms.

Figure 6: IRIS views of NOAA AR 12412 at 2015-09-09T07:59 UTC. The right
column shows spectroheliograms of Si IV 1394Å in the red wing (top), blue wing
(middle), and core (bottom). The left and middle columns show, respectively
the intensity and line broadening from a Gaussian fit for the O I 1355 Å line
(top), Si IV 1394Å (middle), and Cl I 1351Å (bottom). The color scale for the
line broadening is between 3 and 12 km/s for O I and Cl I, and between 5 and
40 km/s for Si IV. The O I 1355 Å intensity, Si IV red wing intensity, Si IV
intensity, Si IV blue wing intensity, Cl I intensity, and Si IV core
intensity, are scaled, respectively, between 2 and 30 DN, 5 and 500 DN, 10 and
500 DN, 5 and 500 DN, 10 and 250 DN, and 5 and 500 DN. Regions with bad fits
and/or low peak counts (below 5 DN) are masked out in the O I 1355 Å line
width map. This figure is accompanied by an animation that allows the reader
to blink between the different panels of the figure to see the various
similarities and offsets described in the text.

A more detailed view of NOAA AR 12920 on 2015-10-02 is provided in Fig. 5. At
this time the AR is close to the limb so that the FOV covers a wide range of
values for $\mu=\cos\theta$. At this extreme viewing angle it appears that O I
1355 Å intensity features (top row, middle panel) are slighly offset towards
the limb when compared with the photospheric plage footpoints (top row, left
panel). In addition, the O I 1355 Å intensity features that protrude towards
the limb from the photospheric footpoints protrude less (i.e., are shorter)
than the features in the Mg II k3 intensity maps. The protrusions are even
more extended towards the limb in the C II 1335Å images. They are strongly
reminiscent of spicules. This suggests that the O I 1355 Å line is indeed
formed in the chromosphere, but perhaps at lower heights than the upper
chromospheric and lower transition region features that are visible in the Mg
II k3 and C II 1335Å maps. This difference in apparent formation height is
perhaps even more clear when comparing the line width maps in O I 1355 Å and
Mg II k. The regions with enhanced O I 1355 Å line width around the plage
regions are relatively narrow, and definitely shorter than the equivalent
features in Mg II k. In addition, the O I 1355 Å line width map appears to
show significantly less enhancement (compared to the Mg II k linewidth map)
and the region where the line width is enhanced covers a smaller part of the
plage regions.

A comparison between Si IV 1402, Cl I 1352Å and O I 1355 Å spectroheliograms
of NOAA AR 12920 towards the limb is also illustrative (Fig. 6). The Cl I line
intensity333The moments of the Cl I line are calculated in the same way as for
O I., thought to be formed at similar heights in the chromosphere (Shine,
1983), shows a large degree of similarity to that of O I 1355 Å. That
correlation is also present to some extent for the line width of both lines.
For the Si IV this is different. Here we see a clear difference between the
limbward side and the disk-center side of the plage region. Towards the limb
there is a clear offset of order 1-2″, with the Si IV features offset towards
the limb. The Si IV brightenings are quite well understood (Skogsrud et al.,
2016), and are the TR counterparts to the magneto-acoustic shocks that
dominate the plage chromosphere, as can be determined from $\lambda-t$ plots.
This comparison suggests that the O I 1355 Å intensity features occur at lower
heights than the Si IV shocks. Such shocks are visible both in the blue and
red wing of the Si IV line, with the blue wing showing the upward phase and
the red wing the downward phase. These shocks have been shown to drive dynamic
fibrils (Skogsrud et al., 2016) and type II spicules (Rouppe van der Voort et
al., 2015). It seems that many O I 1355 Å intensity features are somehow
related to both, with the small round features possibly related to shocks (as
seen from above), and the wispy linear features possibly related to type II
spicules (when viewed from the side).

Figure 7: IRIS spectroheliograms of quiet Sun taken along the meridian from
the north pole (top left) to the south pole (bottom right). Top two rows show
the O I 1355Å intensity, while the bottom two rows show the O I 1355Å line
broadening. The O I 1355 Å intensity is scaled between 0 and 60 DN for all
panels. Regions with bad fits and/or low peak counts (below 4 DN) are masked
out in the O I 1355 Å line width maps.

The behavior of the O I 1355 Å line in quiet Sun regions appears to be
similar. We used observations that were obtained along the central meridian in
March 2016 covering several locations between the north and the south pole, as
shown in Fig. 7. We see very similar behavior, with the intensity lowest
around disk center, and significantly increased at or close to the limb, again
as expected from an optically thin line. The line width is enhanced around the
network regions, independent of the viewing angle, and lowest at disk center.
The increase of the line width exactly at the limb is significant but only a
few km/s. There is also a more gradual increase of the line width from disk
center towards the limb, but only by about 1 or 2 km/s.

Figure 8: IRIS spectroheliograms of a quiet Sun at 2016-03-06T23:01 UTC
showing in the top row 2800Å, O I 1355Å intensity, and Mg II k3 intensity, and
in the bottom row C II 1335Å intensity, O I 1355Å line broadening, and the Mg
II k line broadening. Black horizontal lines are fiducial marks. Black
vertical lines are data drop-outs. The O I 1355 Å intensity, Mg II k3
intensity, and C II 1335Å intensity are scaled, respectively, between 0 and
150 DN, 0 and 7865 DN, and 4 and 108 DN. Regions with bad fits and/or low peak
counts (below 8 DN) are masked out in the O I 1355 Å line width map. This
figure is accompanied by an animation that allows the reader to blink between
the different panels of the figure to see the various similarities and offsets
described in the text.

A detailed view at the polar limb again shows an offset towards the limb of
the bright O I 1355 Å intensity features when comparing to the photospheric
network concentrations (visible in 2800Å). The O I 1355 Å intensity features
appear to be shorter (in the direction towards the limb) and offset below the
features in Mg II k3 (upper chromosphere) and C II 1335Å (TR).

A coherent picture then appears in which, for both active regions and quiet
Sun, the O I 1355 Å formation height appears to be chromospheric in nature,
but somewhat lower and thus not quite identical to that of Mg II k.

Figure 9: Probability density function (PDF) of the O I 1355 Å line width as
function of the cosine of the viewing angle ($\mu$) for the AR 12290 datasets
(top panel) and the QS datasets (bottom panel). To facilitate comparison, a
quadratic least-squares fit to the QS width as function of $\mu$ is shown in
both panels as a dashed line. The color-scale is linear and profiles with a
central intensity below 10 DN/pixel are not included. Figure 10: Probability
density function (PDF) of the O I 1355 Å line core intensity (from the fit) as
function of the cosine of the viewing angle ($\mu$) for the AR 12290 datasets
(top panel) and the QS datasets (bottom panel). The color-scale is linear

###  3.2 Statistics and center to limb variation

Since the O I 1355 Å line is optically thin in the chromosphere, it is of
interest to study how the line broadening depends on the viewing angle. The
probability density function (PDF) of the line width as a function of the
cosine of the viewing angle is shown in Fig. 9 for the AR dataset (top panel)
and the QS dataset (bottom panel). There is, on average, a slightly larger
line width at the limb than at disk center, for both datasets. The line
broadening appears, on average, to increase by only a small amount towards the
limb, of order 1-2 km/s. The increase is larger for the AR 12920 dataset. The
PDF of the line-core intensity (from the fit) as a function of viewing angle
is shown in Fig. 10 for the two datasets. The intensity increases towards the
limb for both datasets. For QS, the core intensity increases from an average
of 0.3 DN/s at disk center to 2.7 DN/s at the limb. The corresponding numbers
for the AR dataset is an increase from 0.48 DN/s at disk center to 3.7 DN/s.
The AR dataset includes a larger variety of features and shows a larger spread
in intensity.

We first discuss the center-to-limb behavior of the line broadening on the
disk (i.e., not including the off-limb region). The changed viewing angle
towards the limb could, in principle, cause two effects on the O I 1355 Å line
broadening. First, it could mean that the measurements closer to the limb are
more sensitive to the motions perpendicular to the magnetic field. This would
be the case if one assumes that, on average, the magnetic field is typically
more vertically (rather than horizontally) oriented in the line formation
region of O I 1355 Å. This is likely a reasonable assumption for a plage or
network region, but may not be a good assumption for other regions in the FOV,
such as those regions just adjacent to network or plage where fibril or
spicule-like features often appear more inclined, or for internetwork regions
where the chromospheric field is poorly known and may include more horizontal
fields. Secondly, the longer line-of-sight implies more superposition of
different structures within each pixel. If these locations have different LOS
velocities, this can broaden the line as well. If this were the case, one
would also expect to see a significant increase in intensity, depending on the
velocity gradients in the FOV and/or along the line-of-sight.

Disentangling these two effects is not straightforward. In addition, the
likely different field orientations between network or plage on the one hand,
and the other regions on the other hand, can easily render interpretation of a
center to limb variation plot such as that in Fig. 9 muddled. By mixing both
types of regions into one plot, any center-to-limb variation of each sub-
region may be hidden since the two different types of regions may have
oppositely signed center-to-limb variations. Another key aspect that should be
taken into account is that the line formation region of O I 1355 Å is very
large, covering a region from the low chromosphere all the way to the top of
the chromospheric spicules. We have already discussed the presence of spicule-
like features in the O I 1355 Å maps of Figs. 1 and 3. The likely field-
aligned motions and the amplitude of, e.g., Alfvén-wave associated motions
perpendicular to the field, are expected to vary very significantly between
the dense lower chromosphere and the top of spicules, because of the very
large density differences between those two regions. Assuming mass
conservation for field-aligned flows, or constant energy flux density for
Alfvén waves implies a strong increase of the flows from the low chromosphere
to the top of spicules. In addition, there are most likely significant
differences in intensity between the chromosphere and the top of spicules,
since the intensity is proportional to the square of the electron density for
this optically thin line (Lin & Carlsson, 2015). The relative contribution of
these two components (chromosphere vs. spicules) within an IRIS resolution
element is also expected to vary between disk center and the limb. This is
because at disk center, spicules will more often originate and be suspended
straight above the bright network or plage regions, whereas toward the limb
their greater height will show them more spatially offset from the network or
plage.

Figure 11: Probability density function (PDF) of the O I 1355 Å line width as
function of the cosine of the viewing angle ($\mu$) for quiet Sun internetwork
areas (top panel), network areas (middle panel) and for areas 1″ limbward of
network areas (bottom panel). In all panels the least-squares quadratic fit is
shown with a solid line and the fits for the two other area-types as dashed
lines.

In an attempt to disentangle these effects we have applied masks to isolate
the chromospheric from the spicular contributions as much as possible. We
perform this only for the quiet Sun dataset since the magnetic field in active
regions is more complex. Fig. 11 shows the center-to-limb variation of quiet
Sun network, regions adjacent to the network, and the internetwork. The masks
are created by first making a least-squares cubic polynomial fit to the core
intensity (from the fit) as a function of the cosine of the viewing angle
($\mu$). The network mask includes all pixels with a core intensity more than
22 DN above the center-to-limb fit. The ”adjacent to the network” mask
includes all pixels less than one arcsecond limb-ward from the network pixels,
excluding network pixels. The internetwork mask includes all the other pixels.
The regions adjacent to the network are typically dominated by fibrils or
spicules. We see that the regions adjacent to the network show an increased
line broadening. Because of the morphology, we believe that this is caused by
the larger motions in the low density environment of spicules. It is possible
that this effect also contributes to the modest increase of the line width by
1-2 km s-1 increase towards the limb that is seen in Fig. 9. The fainter
spicular signal (with its increased LOS motions) becomes more apparent towards
the limb because the spatial offset from the brighter network is increased as
the viewing angle changes. Another effect that may help explain the modest
increase of line width toward the limb is the increased superposition along
the line-of-sight of different structures with different LOS velocities.
However, it is also possible that towards the limb the LOS is more
perpendicular to the magnetic field direction, and that the modest increase of
line width towards the limb is in part caused by stronger motions
perpendicular to the magnetic field direction. This slight anisotropy (with
respect to the magnetic field direction) of turbulent motions likely plays a
significant role, as we illustrate in what follows below.

We now turn our attention to the off-limb behavior of the O I 1355 Å intensity
and line broadening in a quiet Sun dataset as a function of distance above the
limb (positive values). This is shown in Fig. 12. The distance from the limb
is based on a calculation of the solar radius for the date of the observation
and the header information in the IRIS data. The latter has an accuracy of
order $\sim 0.6$″ or better, now that cross-correlation of IRIS FUV SJI images
and AIA 1700Å data is automatically applied in the IRIS level 2 data pipeline.
The top panel shows the sharp drop-off of the intensity in the photospheric
wing of Mg II k at 2800Å, which starts about 0.5″ above the solar limb. This
is a reasonable number since this photospheric wing emission is formed in the
upper photosphere, confirming that the limb distance is accurately calculated.

The second panel from the top in Fig. 12 shows that the O I 1355 Å intensity
increases from the photospheric limb outward until it peaks at about 1.5″from
the limb. The intensity then rapidly drops off with increasing distance from
the limb. It is very interesting to note that between the limb and the 1.5″
distance where the O I 1355 Å intensity peaks, the line width increases (for
increasing distance from the limb) only very modestly to a value of about 9 km
s-1. This changes drastically after the peak intensity is reached: the line
width then increases rapidly to values of 15 km s-1 until the intensity drops
(at distances of 5 ″) to very low values where the Gaussian fits can no longer
be performed reliably.

What causes this puzzling spatial offset of about 3″ of the off-limb peak of
the intensity and the line width? If one assumed that the off-limb properties
of this optically thin line were caused by line-of-sight superposition of
intensity features with a random distribution of LOS velocities, the peaks of
the intensity and line width should be co-located.

Let us, instead, examine a different scenario that is inspired by our findings
in § 3. We assume that there are two major contributors to the O I 1355 Å
signals off-limb: a contribution from the chromosphere proper, and one from
spicules. Let us further assume that, at the limb, the line width is mostly
determined by motions perpendicular to the magnetic field, as expected from
Alfvén wave associated motions, and that the field is mostly vertical. As we
have seen in § 3, there are indications that the line width increases along
the spicule-like structures emanating from the network. It is natural to
assume that beyond a certain height, these spicular features dominate the
signal.

Let us now assume that the energy flux density $F$ of Alfvén waves propagating
from the lower chromosphere to the top of the spicules is roughly constant:

$F=\rho\delta v^{2}v_{A}\sim\sqrt{\rho}\delta v^{2}B=k$ (1)

in which $\rho$ is the mass density of the plasma, $\delta v$ is the Alfvén
wave amplitude, $v_{A}$ is the Alfvén speed, with $v_{A}\sim B/\sqrt{\rho}$,
and $B$ the magnetic field, and $k$ a constant. This would then imply:

$\delta v\sim\rho^{-{1\over 4}}{B}^{-{1\over 2}}\sim n_{e}^{-{1\over
4}}{B}^{-{1\over 2}}\sim I_{peak}^{-{1\over 8}}{B}^{-{1\over 2}}$ (2)

in which we assume that $\rho\sim n_{e}$ (the electron density), and
$I_{peak}\sim n_{e}^{2}$ (based on Lin & Carlsson, 2015).

Figure 12: Probability density functions (PDF) for the intensity at 2800Å
(upper photosphere, top), O I 1355 Å intensity $I_{peak}$ (second from top), O
I 1355 Å line width (third from top), and a quantity $A$ that is proportional
to $I_{peak}^{-1\over 8}$, all as a function of distance from solar limb, with
positive values of the latter for off-limb positions. The vertical dashed line
is the location of the photospheric limb, as determined from the solar radius
and IRIS headers. The solid vertical lines show the range of distances from
the limb for which $I_{peak}$ decreases with height. The value of $A$ is
scaled so that it is equal to the line broadening at the distance from the
solar limb indicated by the leftmost solid vertical line.

In the bottom panel of Fig. 12, we show the variation of the component
$I^{-{1\over 8}}$, normalized to the same line width values at the distance
where the O I 1355 Å line width peaks (about 1.5″). We find that the increase
of line width from 1.5″ to 5″is well reproduced qualitatively. Note that we
expect that the average magnetic field decreases with increasing distance from
the limb. Such a decrease in magnetic field $B$ would lead to an even larger
increase of the line width. It seems that our relatively simplistic model thus
reproduces the observed behavior very well. Note also that our approach
appears to also be compatible with the observed slight increase of line width
between the solar limb and the distance of peak O I 1355 Å intensity. While
the $I_{peak}^{-1\over 8}$ component predicts a slight decrease in line width
in this region, we note that it is precisely here (at heights between the
photosphere and low chromosphere) that we expect the largest drop in average
magnetic field strength as flux tubes expand with height from the photosphere
into the low chromosphere. If the field were to drop by a factor of 3, this
could reverse the predicted decrease of line width (from the intensity alone)
to the modest increase we actually see for increasing distance from the solar
limb.

Our results thus support a scenario in which the spatial variation of the O I
1355 Å linewidth is related to the superposition of Alfvénic wave motions or
turbulence along structures that show a density that significantly decreases
with height. Naturally reality is most likely more complex. However, the fact
that this scenario can easily explain the spatial offset between the peaks of
O I 1355 Å intensity and line width, and shows an increase of line width that
is self-consistent with the observed intensities and expected magnetic field
variations, both suggest that this scenario plays a role in the observed
behavior.

Figure 13: IRIS spectroheliograms of NOAA AR 12920 as it traverses the disk
between 2015-09-22 and 2015-09-24. Top row shows the O I 1355Å intensity,
while the bottom row shows the line-of-sight magnetic field as deduced from
HMI magnetograms stitched together to replicate the IRIS raster timings and
locations. The red circles indicate locations with neutral lines between
opposite polarities and increased O I 1355 Å intensities, as described in the
text. Figure 14: IRIS spectroheliograms of NOAA AR 12920 as it traverses the
disk between 2015-09-25 and 2015-09-28. Top row shows the O I 1355Å intensity,
while the bottom row shows the line-of-sight magnetic field as deduced from
HMI magnetograms stitched together to replicate the IRIS raster timings and
locations. The red circles indicate locations with neutral lines between
opposite polarities and increased O I 1355 Å intensities, as described in the
text.

##  4 Relationship to magnetic flux emergence and cancellation

The very bright features in O I 1355 Å in the active region (Fig. 4) appear to
be related to the effects of flux emergence and/or cancellation. A detailed
comparison of the locations in O I 1355 Å and magnetograms from HMI (Fig. 13,
14) shows that the O I 1355 Å intensity is often enhanced around the neutral
line between opposite polarities. To allow a proper comparison between the
IRIS rasters and HMI (or SST) magnetograms, we create, from the HMI (or SST)
data, synthetic rasters with identical field-of-view as the IRIS raster, in
which each raster position includes a vertical strip of HMI (or SST) data at
the same time as the time of each IRIS raster position.

On the largest scales ($\sim 20$″), we see that filaments, which most often
occur above the neutral line between opposite polarities, are often bright in
O I 1355 Å e.g., at (-166″, 42″) (left), and at (53″, 70″) (middle) in Fig.
14. However, the correlation between O I 1355 Å intensity and neutral lines is
also strong on smaller scales ($\sim 5$″), at locations where flux
concentrations of opposite polarities are in very close proximity, i.e.,
touching or almost touching. This is clearly illustrated, in Fig. 13 (red
circles), by the bright region at (-770″, 145″) in the left column, at (-526″,
33″) in the middle column, and (-367″, 44″) in the right column. Similarly,
Fig. 14 (red circles) shows examples at (-235″, 65″) (left), (-5″, 70″)
(middle), and (420″, 170″) (right).

When we saturate HMI magnetograms to enhance the visibility of weak flux of
opposite polarity, we find that on arsecond scales very weak flux immediately
adjacent to the dominant polarity flux similarly often leads to enhanced
brightenings in O I 1355 Å. Examples of that can be found in Fig. 15 (red
circles), at (-435″, 110″), (-435″, 92″), (-412″, 90″), (-424″, 60″), (-426″,
48″), (-420″, 10″). There are, however, also some locations in which opposite
polarity flux appears in close proximity but the O I 1355 Å intensity does not
appear to be significantly enhanced (e.g., the weak concentrations at (-370″,
55″)).

Figure 15: IRIS spectroheliograms of NOAA AR 12920 on 2015-09-24. Left column
shows the O I 1355Å intensity, while the right column row shows the line-of-
sight magnetic field as deduced from HMI magnetograms stitched together to
replicate the IRIS raster timings and locations. The red circles indicate
locations with neutral lines between opposite polarities and increased O I
1355 Å intensities, as described in the text.

To investigate this correlation on sub-arcsecond scales, we also analyze a
time series of very high-resolution magnetograms obtained at the SST (Fig.
16). These magnetograms have much higher spatial resolution ($\sim
0.1\arcsec$) and greater magnetic sensitivity. They similarly reveal locations
of enhanced O I 1355 Å brightenings where flux of the minority polarity is in
the proximity of stronger dominant polarity concentrations, e.g., at (4″,
17″). However, there are also locations with very weak minority flux where
this correlation is not as clear (e.g., at (12″, 35″)) or not apparent (e.g.,
at (18″, 56 ″)). Such locations of reduced or no correlation appear to be more
common in the SST dataset than in the HMI datasets. This perhaps suggests that
a minimum flux size is required for significant O I 1355 Å emission, or that
the O I 1355 Å intensity increase is perhaps shorter lived for very weak
minority flux concentrations, and missed by the slow cadence of the IRIS
rasters. The latter is driven by the required deep exposures to detect the
faint O I 1355 Å line.

The animation associated with Fig. 16 reveals that, in addition to the
correlation between neutral lines and O I 1355 Å intensity, which could be
associated with flux cancellation, there is another process that is clearly
associated with increased O I 1355 Å intensity. Flux emergence is present in
the region around $x=5-15\arcsec$, $y=45-55\arcsec$, and $x=38-43\arcsec$,
$y=40-45\arcsec$. These are the two regions that show the brightest O I 1355 Å
emission. This can also be seen in Fig. 6, with the brightest emission around
(-465″, -210″) occurring at a location where flux has emerged into a pre-
existing field configuration. The region of enhanced intensities in O I 1355 Å
(also seen in Si IV and Cl I) appears to outline a dome-like structure that
separates the two flux systems, possibly a quasi-separatrix layer (QSL).

The above findings appear to be compatible with a scenario in which
significant ionization and heating of the plasma, in response to reconnection
or currents associated with the interaction between field concentrations of
opposite polarity, leads to enhanced electron densities, which, based on
theoretical work, the O I 1355 Å intensity is proportional to (Lin & Carlsson,
2015). Our observational results indicate that such heating could occur in
association with flux cancellation or flux emergence. Out of all IRIS
observables the O I 1355 Å intensity appears to be the most sensitive to the
effects of cancellation and emergence, possibly because it is formed at low
enough heights that it is sensitive to heating, even from small-scale flux
concentrations whose fields do not reach into the upper chromosphere. Another
key aspect is that O I 1355 Å is optically thin and uniquely sensitive to the
electron density, thereby picking up any enhancement caused by heating or
ionization. It appears that O I 1355 Å may be acting as a canary in the coal
mine for solar atmospheric effects of cancellation or emergence.

One complication of our analysis is that the O I 1355 Å line is faint and
requires deep exposures, leading to low cadence observations. Given the
dynamic and ephemeral nature of heating associated with flux emergence it is
thus quite possible that some heating signals are simply missed in the O I
1355 Å rasters, when it occurs either before or after the IRIS slit has passed
the site of cancellation or emergence. Alternatively, it may be that not all
cancellations or emergence lead to significant heating at chromospheric
heights. Detailed comparisons with numerical simulations are required to
address this.

One final observational finding is that the locations of enhanced line
broadening do not appear to show a significant correlation with flux
cancellation or emergence, as illustrated by Fig. 16. In fact, regions of
active emergence often appear to show significantly reduced broadening (e.g.,
at (5″, 52″) in Fig. 16).

Figure 16: Comparison between SST Fe I 6173Å magnetogram raster (middle) and
IRIS O I intensity (left) and line width (right), taken on 2015-09-09. The
magnetogram synthetic raster has been constructed from a magnetogram
timeseries by matching with IRIS raster timings and locations. The O I 1355 Å
intensity is scaled between 2 and 30 DN. Regions with bad fits and/or low peak
counts (below 5 DN) are masked out in the O I 1355 Å line width map. This
figure is accompanied by an animation with the same layout as the figure,
except it shows in the middle panel a time sequence of magnetograms. Short
vertical bars at the top and bottom of each panel in the animation indicate,
for each time step, the location of the IRIS raster step (left and right
panels) at the time of the magnetogram shown in the middle panel. The time in
the animation is expressed in seconds after 9-Sep-2015 09:00:00 UTC.

##  5 Flux concentrations

Our analysis of the O I 1355 Å measurements and the underlying photosphere in
plage regions has revealed another intriguing finding. In particular, on sub-
arcsecond spatial scales, there is an anti-correlation between locations of
enhanced line broadening and the locations of strong flux concentrations in
the photosphere. This is illustrated in Fig. 17, which shows the O I 1355 Å
line broadening (top), a spectroheliogram at 2800Å (middle), and the O I 1355
Å intensity (bottom).

While the range of values for line broadening is not enormous, there is a
clear difference between non-plage regions and plage regions, with the former
(e.g., $x=-10$ to 15″, $y=80$ to 90″), showing significantly reduced line
broadening (less than 5 km/s). The plage region itself shows a relatively
narrow range of values for the line broadening, between 7 and 11 km/s, as
already remarked upon by Carlsson et al. (2015b). The top panel of Fig. 17
nevertheless reveals a spatial pattern of enhanced values for the line
broadening (3 to 4 km/s higher than the rest of the plage), occurring in
contiguous regions showing coherence on 0.5 to 2″spatial scales. These regions
of enhanced line width do not seem to correlate with locations of enhanced O I
1355 Å brightness. Similar, they do not occur where the photospheric wing
emission at 2800Å occurs. Instead, they preferentially avoid the locations of
bright points in the photosphere. The latter are indicated with black
contours, which are determined by setting a threshold brightness in the 2800Å
spectroheliogram. As can be seen in the top panel, most often the regions of
enhanced line broadening in O I 1355 Å occur in between the black contours.

Figure 17: For NOAA AR 12902 on 2015-09-26, the O I 1355 Å line width in km/s,
the intensity at 2800Å (formed in the photosphere), and the O I 1355Å
intensity. Black contours are based on intensity thresholding of photospheric
bright points in the 2800Å spectoheliogram. The O I 1355 Å intensity is scaled
from 3 to 40 DN.

This anti-correlation is further illustrated by Fig. 18, which shows the joint
probability density function (JPDF) of the O I 1355 Å line broadening and the
intensity at 2800Å. Only the locations where the O I 1355 Å intensity is
larger than 10 DN are included, within the plage region that is within the red
contours in the bottom panel of Fig. 18. By introducing the threshold for O I
1355 Å intensity we ensure that the resulting line widths included in the JPDF
are not significantly affected by noise.

As we can see, the locations of highest broadening occur at lower values in
photospheric intensity. The locations with highest photospheric intensity
(bright points) show moderate levels of broadening. The lowest broadening
occurs at the edge of the plage regions, as seen in the top panel of Fig. 17.
The scatter plot thus supports our finding that the locations of highest line
broadening occur in locations with the lowest values of 2800Å intensity within
plage regions, i.e., those in between photospheric bright points. We note that
the broader range of values for the O I 1355 Å broadening at lower 2800Å
intensities is not a signature of noise, as the noise in determining the line
width depends on the O I 1355 Å intensity (not the photospheric intensity). If
noise were behind this issue, then we would expect a correlation between
decreased O I 1355 Å intensity and decreased photospheric intensity. That is
not the case, as can be seen in the bottom two panels of Fig. 17.

It is also noteworthy that the O I 1355 Å line broadening map does not show a
good correlation or anti-correlation with any other width or intensity measure
of other chromospheric or TR lines. There are perhaps a few locations where
there is some correspondence with Mg II k line broadening, but this is not the
general trend (see Figs. 1 and 3).

This begs the question what causes this anti-correlation. Several scenarios
come to mind. First, the area between photospheric bright points in plage is
the region where the magnetic field canopy is expected to occur, as seen in
both observations (e.g., de la Cruz Rodríguez, 2019) and numerical simulations
(e.g., Hansteen et al., 2006). While the magnetic field is expected to be
mostly vertical directly above photospheric flux concentrations, it is
expected to be significantly more inclined with respect to the vertical in the
regions where the canopy forms, i.e., the locations where we find the highest
O I 1355 Å line broadening. This is particularly the case at low chromospheric
heights where the transition occurs from high to low plasma $\beta$. The O I
1355 Å line is thought to be sensitive to the electron density, with the
formation height covering the low chromospheric regions in particular.

But why would a heavily inclined magnetic field lead to enhanced O I 1355 Å
line broadening? The line width is determined by three contributors. The
instrumental broadening, which does not vary significantly across the FOV; the
thermal broadening, which depends on the local temperature, and the unresolved
motions along the line-of-sight (”non-thermal” broadening). If increased
heating (i.e., higher temperatures) were to occur in the chromospheric canopy
region between photospheric bright points, it would be expected to lead to
enhanced electron densities. This in turn would cause enhanced O I 1355 Å
intensities, since this optically thin line scales with the square of the
electron density. However, such a correlation is not observed in our data (top
and bottom panels of Fig. 17).

This leaves the possibility that enhanced non-thermal motions, either in the
form of unresolved macroscopic motions along the line-of-sight, or in the form
of microscopic turbulence, are the cause of these enhancements in line
broadening. It is tempting to speculate that this may be caused by one (or
both) of two scenarios. The first is one in which strong Alfvénic wave and/or
vortical motions (i.e., perpendicular to the magnetic field direction) are
ubiquitous in plage and register in the O I 1355 Å line broadening only when
the magnetic field direction is more inclined from the line-of-sight vector.
In a disk center observation like the one considered here, such enhanced
vortical motions or Alfvénic wave power would thus be most visible in between
photospheric bright points. This is an intriguing scenario given the evidence
from the center-to-limb variation of line broadening that Alfvénic waves play
a key role in explaining the strong increase of line broadening off-limb.
These observational findings may also be compatible with recent suggestions
from numerical simulations that vortical motions in the photosphere are
ubiquitous and often propagate into the low atmosphere (e.g., Moll et al.,
2011; Yadav et al., 2021; Breu et al., 2023). However, it is not fully clear
whether these recent modeling results are fully compatible with our
observations. In particular, the lack of correlation between locations of
enhanced line width and O I 1355 Å intensity in observations does, at first
blush, not seem to be fully compatible with a scenario in which the vortices
lead to increased heating in the plage chromosphere (Yadav et al., 2021). This
is because one would expect that such heating leads to enhanced O I 1355 Å
densities and thus intensities. Detailed studies of synthetic O I 1355 Å
profiles are required to further investigate this.

An alternative possibility is that there are strong velocity gradients at the
interface of flux concentrations. These could, for example, occur because of
the LOS overlap between upward propagating shocks on neighboring flux
concentrations. Such shocks are ubiquitous in plage and drive strong flows
along dynamic fibrils (Hansteen et al., 2006). Perhaps a combination of both
effects plays a role. Detailed comparisons with advanced numerical simulations
of the chromosphere are required to further investigate these scenarios.

Figure 18: For NOAA AR 12902 on 2015-09-26, the JPDF (top panel) between the O
I 1355 Å line width in km/s, and the intensity at 2800Å (formed in the
photosphere) for a subset of the plage region studied in Fig. 17. The JPDF
only includes locations for which the O I 1355 Å intensity is larger than 10
DN (to avoid noisy profiles with uncertain or inaccurate line width) and for
locations that are within the red contours shown in the spectroheliogram at
2800 shown in the bottom panel.

##  6 Conclusions

We have analyzed the properties of the O I 1355 Å spectral line, which is
unique in that it is regularly observed with IRIS, optically thin, and formed
in the chromosphere. We find that the line shows properties that are different
from other spectral lines formed in the chromosphere. We find that intensities
are strongest in plage and network regions and in the proximity of neutral
lines where magnetic fields of opposite polarity are in close contact. Our
data suggests that the O I 1355 Å intensity is often increased and appears to
be very sensitive to the effects of cancellation and emergence of magnetic
flux. Because of the optically thin nature of the O I 1355 Å line, this
indicates that electron densities are locally enhanced, a signature of heating
in the chromosphere. We also see a significant increase of O I 1355 Å
intensities at the solar limb, as expected for an optically thin spectral line
from the increased line-of-sight superposition of different structures. We
find evidence for O I 1355 Å intensity structures being associated with shocks
in plage and network, as well as spicules, with some similarities to
counterparts at low TR temperatures that are visible in Si IVlines.

The O I 1355 Å line width offers a unique view on the unresolved or non-
thermal motions in the chromosphere, a quantity that is otherwise difficult to
directly determine, requiring inversions of optically thick spectral lines
whose formation is subject to non-LTE radiative transfer effects. We find, for
both active regions and quiet Sun, that the line width modestly increases
towards the limb, and along spicule-like structures that protrude away from
plage and network flux concentrations. The modest center-limb-variation
suggests that there are unresolved motions both along the magnetic field and
across the magnetic field, with the latter being stronger.

Off the limb the line broadening rapidly increases, compatible with a scenario
in which turbulent, Alfvén wave or vortical motions perpendicular to the
magnetic field dominate the line width, as they propagate upward with a
roughly constant flux along spicular structures in which the density decreases
with height.

The presence of strong vortical or Alfvénic motions in plage are further
supported by a curious but significant enhancement of line width in between
photospheric flux concentrations, when viewed close to disk center. These
results are compatible with the combined impact of inclined canopy fields and
enhanced motions perpendicular to the magnetic field on the O I 1355 Å
linewidth. Such inclined fields are expected to occur in between photospheric
flux concentrations, while enhanced motions perpendicular to the field are
predicted by various numerical simulations of vortices and Alfvén waves in
plage. However, the lack of obvious correlation between enhanced line width
and O I 1355 Å intensity is not immediately compatible with the enhanced
heating expected from vortices.

Our observations provide strict constraints on models of the chromosphere.

The authors are grateful to the observers at the SST who obtained the SST data
and to Luc Rouppe van der Voort who calibrated and processed the SST
timeseries. B.D.P. gratefully acknowledge support by NASA contract NNG09FA40C
(IRIS). This research is supported by the Research Council of Norway through
its Centres of Excellence scheme, project number 262622. To analyze the data
we have used IDL. Data are courtesy of IRIS. IRIS is a NASA small explorer
mission developed and operated by LMSAL with mission operations executed at
NASA Ames Research Center and major contributions to downlink communications
funded by ESA and the Norwegian Space Agency.

## References

  * Breu et al. (2023) Breu, C., Peter, H., Cameron, R., & Solanki, S. K. 2023, arXiv e-prints, arXiv:2305.03769, doi: 10.48550/arXiv.2305.03769
  * Bryans et al. (2016) Bryans, P., McIntosh, S. W., De Moortel, I., & De Pontieu, B. 2016, ApJ, 829, L18, doi: 10.3847/2041-8205/829/1/L18
  * Carlsson et al. (2019) Carlsson, M., De Pontieu, B., & Hansteen, V. H. 2019, ARA&A, 57, 189, doi: 10.1146/annurev-astro-081817-052044
  * Carlsson et al. (2015a) Carlsson, M., Leenaarts, J., & De Pontieu, B. 2015a, ApJ, 809, L30, doi: 10.1088/2041-8205/809/2/L30
  * Carlsson et al. (2015b) —. 2015b, ApJ, 809, L30, doi: 10.1088/2041-8205/809/2/L30
  * Carlsson & Stein (2002) Carlsson, M., & Stein, R. F. 2002, ApJ, 572, 626
  * de la Cruz Rodríguez (2019) de la Cruz Rodríguez, J. 2019, A&A, 631, A153, doi: 10.1051/0004-6361/201936635
  * de la Cruz Rodríguez et al. (2016) de la Cruz Rodríguez, J., Leenaarts, J., & Asensio Ramos, A. 2016, ApJ, 830, L30, doi: 10.3847/2041-8205/830/2/L30
  * De Pontieu et al. (2014) De Pontieu, B., Title, A., & Carlsson, M. 2014, Science, 346, 315, doi: 10.1126/science.346.6207.315
  * Golding et al. (2014) Golding, T. P., Carlsson, M., & Leenaarts, J. 2014, ApJ, 784, 30, doi: 10.1088/0004-637X/784/1/30
  * Golding et al. (2016) Golding, T. P., Leenaarts, J., & Carlsson, M. 2016, ApJ, 817, 125, doi: 10.3847/0004-637X/817/2/125
  * Golding et al. (2017) —. 2017, A&A, 597, A102, doi: 10.1051/0004-6361/201629462
  * Gudiksen et al. (2011) Gudiksen, B. V., Carlsson, M., Hansteen, V. H., et al. 2011, A&A, 531, A154+, doi: 10.1051/0004-6361/201116520
  * Hansteen et al. (2006) Hansteen, V. H., De Pontieu, B., Rouppe van der Voort, L., van Noort, M., & Carlsson, M. 2006, Apj, 647, L73, doi: 10.1086/507452
  * Leenaarts et al. (2016) Leenaarts, J., Golding, T., Carlsson, M., Libbrecht, T., & Joshi, J. 2016, A&A, 594, A104, doi: 10.1051/0004-6361/201628490
  * Leenaarts et al. (2013a) Leenaarts, J., Pereira, T. M. D., Carlsson, M., Uitenbroek, H., & De Pontieu, B. 2013a, ApJ, 772, 89, doi: 10.1088/0004-637X/772/2/89
  * Leenaarts et al. (2013b) —. 2013b, ApJ, 772, 90, doi: 10.1088/0004-637X/772/2/90
  * Lin & Carlsson (2015) Lin, H.-H., & Carlsson, M. 2015, ApJ, 813, 34, doi: 10.1088/0004-637X/813/1/34
  * Moll et al. (2011) Moll, R., Cameron, R. H., & Schüssler, M. 2011, A&A, 533, A126, doi: 10.1051/0004-6361/201117441
  * Rathore et al. (2015a) Rathore, B., Carlsson, M., Leenaarts, J., & De Pontieu, B. 2015a, ApJ, 811, 81, doi: 10.1088/0004-637X/811/2/81
  * Rathore et al. (2015b) Rathore, B., Pereira, T. M. D., Carlsson, M., & De Pontieu, B. 2015b, ApJ, 814, 70, doi: 10.1088/0004-637X/814/1/70
  * Rouppe van der Voort et al. (2015) Rouppe van der Voort, L., De Pontieu, B., Pereira, T. M. D., Carlsson, M., & Hansteen, V. 2015, ApJ, 799, L3, doi: 10.1088/2041-8205/799/1/L3
  * Rouppe van der Voort et al. (2020) Rouppe van der Voort, L. H. M., De Pontieu, B., Carlsson, M., et al. 2020, A&A, 641, A146, doi: 10.1051/0004-6361/202038732
  * Sainz Dalda et al. (2019) Sainz Dalda, A., de la Cruz Rodríguez, J., De Pontieu, B., & Gošić, M. 2019, ApJ, 875, L18, doi: 10.3847/2041-8213/ab15d9
  * Scharmer (2006) Scharmer, G. B. 2006, A&A, 447, 1111, doi: 10.1051/0004-6361:20052981
  * Shine (1983) Shine, R. A. 1983, ApJ, 266, 882, doi: 10.1086/160835
  * Skogsrud et al. (2016) Skogsrud, H., Rouppe van der Voort, L., & De Pontieu, B. 2016, ApJ, 817, 124, doi: 10.3847/0004-637X/817/2/124
  * Wülser et al. (2018) Wülser, J. P., Jaeggli, S., De Pontieu, B., et al. 2018, Sol. Phys., 293, 149, doi: 10.1007/s11207-018-1364-8
  * Yadav et al. (2021) Yadav, N., Cameron, R. H., & Solanki, S. K. 2021, A&A, 645, A3, doi: 10.1051/0004-6361/202038965