# Machine Learning Approaches for Principle Prediction in Naturally Occurring
Stories

Md Sultan Al Nahian 1, Spencer Frazier11footnotemark: 1 2, Brent Harrison1,
Mark Riedl2
1University of Kentucky
2Georgia Institute of Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>*Denotes equal contribution

###### Abstract

Value alignment is the task of creating autonomous systems whose values align
with those of humans. Past work has shown that stories are a potentially rich
source of information on human values; however, past work has been limited to
considering values in a binary sense. In this work, we explore the use of
machine learning models for the task of normative principle prediction on
naturally occurring story data. To do this, we extend a dataset that has been
previously used to train a binary normative classifier with annotations of
moral principles. We then use this dataset to train a variety of machine
learning models, evaluate these models and compare their results against
humans who were asked to perform the same task. We show that while individual
principles can be classified, the ambiguity of what "moral principles"
represent, poses a challenge for both human participants and autonomous
systems which are faced with the same task.

Machine Learning Approaches for Principle Prediction in Naturally Occurring
Stories

Md Sultan Al Nahian††thanks: *Denotes equal contribution 1, Spencer
Frazier11footnotemark: 1 2, Brent Harrison1, Mark Riedl2 1University of
Kentucky 2Georgia Institute of Technology<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

##  1 Introduction

As the capabilities of artificial intelligence systems expand, there has been
an increased need to create systems that can safely interact with human
operators. This is especially true for systems that are given degrees of
autonomy. One proposed means of creating safe, autonomous systems is to ensure
that the system’s values align with those of their human counterparts Soares
and Fallenstein (2014); Russell et al. (2015); Arnold et al. (2017). This
task, often referred to as value alignment, is a critical task as it is
assumed that this will guard against potential harmful interactions between
humans and AI systems Russell (2019); Moor (2006); Bostrom (2014).

One frequently encountered difficulty with value alignment is explicitly
defining what constitutes a value Soares (2015). Past approaches to AI value
alignment have leveraged learning from observations or other forms of
imitation learning Stadie et al. (2017); Wulfmeier (2019); Ho and Ermon
(2016), the idea being that one can circumvent the requirement of value
knowledge by learning to imitate human behavior instead. Learning knowledge
from demonstrations which generalize beyond the context of the observation is
difficult. Collecting sufficient demonstrations can be time consuming. Humans,
too, are not necessarily able to comprehensively define a full set of
principles or values even if asked to provide these examples.

Recent work has shown that stories are a promising potential source of value
information.In this work, authors use stories to learn a strong prior over
behaviors. The issue with this work is that it was limited to a coarse view of
value, opting to describe behavior as either being normative, aligning with
expected social norms, or non-normative, deviating from expected social norms.
Norms and the assessment of normative behavior can rarely be so neatly
categorized into positive and negative valences in every context. In addition,
it may be difficult to explain or remedy incorrect classification of normative
behavior if systems lack an understanding of the specific normative principle
that is being violated. As such, additional work is needed to identify
specific principles inherent in text-based descriptions and examples of
normative behavior to better inform agents and the humans that work with these
normative priors.

In this work, we seek to develop systems that have a more nuanced
understanding of descriptions of human behavior with respect to normative
principles. We define a normative principles as specific behavior tenets that
guide social normative behavior. An example of a potential normative principle
might be Be polite to others. To facilitate this work, we augment a dataset of
the children’s comic strip, Goofus and Gallant, to contain detailed
information about the principles being described in each frame. We then train
various machine learning models with the aim of predicting the principle being
either upheld or rejected based on images and text descriptions of the comics.
We evaluate our work by comparing the performance of our trained models
against humans who are tasked with performing a similar task.

##  2 Related Work

Value alignment encompasses the effort to have agents behave in accordance
with human values without explicit control of a system’s actions Bicchieri
(2005); Soares and Fallenstein (2014); Russell et al. (2015); Taylor et al.
(2016); Arnold et al. (2017); Abel et al. (2016). These values - in and of
themselves - are difficult to delineate. It can be confounding to attempt to
discuss and define what specifically constitute human values.

Various methods exist that attempt to instill moral reasoning or decision
making within agents. Many of these methods, such as (cooperative) inverse
reinforcement learning Ng et al. (2000); Hadfield-Menell et al. (2016),
learning through imitation Ho and Ermon (2016), and learning via expert
demonstration Schaal (1997); Ho et al. (2016), require substantial human input
to generate training data. Furthermore, because these approaches train on
limited and costly input, a strong prior distribution is necessary as is the
case in language transfer with sparse training data demonstrated by Zoph et al
Zoph et al. (2016).

Another approach that trains on natural language stories rather than
demonstrated actions is known as Learning from Stories Riedl and Harrison
(2016); Harrison and Riedl (2016). This work proved successful at
demonstrating the capability for agents to leverage reward signals from
structures derived from narratives and perform more normatively than models
which did not have similar priming.

Other work attempts to control the output of generative models; the language
model GPT was shown to be capable of generating sentences matching sentiment
of training data Ziegler et al. (2019). Though distinct from value alignment,
this work demonstrates training models from data instilled with human
characteristics and mannerisms.

One of the recent efforts to value alignment is Delphi Jiang et al. (2021), a
model trained on unified moral dataset that can make moral judgments on real-
world situations. Despite being able to make moral judgement, Delphi does not
give any notion of social norms or principles based on what it does make the
judgement. Moreover, the dataset they have created to train the model were
crowdsourced and from online platform i.e subreddit groups which consist of
many inappropriate and societal biased set of examples. Therefore Delphi often
makes improper and biased moral decisions. Another recent task on moral
reasoning is Moral Stories Emelin et al. (2020) which attempts to evaluate the
capabilities of natural language generative model on making moral reasoning on
social scenarios. The norms in Moral Stories are freeform sentences and very
specific to situations (e.g. "It is wrong to disparage a classmate" or "You
should return a jar of sugar once borrowed". This dataset was collected after
our analysis and completion of this research. In contrast, the principles in
our dataset are more general, broader and binned into a finite number of
classes which is a more tractable starting point to train a value aligned
agent.

##  3 Dataset

In order to create our models of normative principle understanding, we must
first find a suitable dataset for this task. To our knowledge, there is no
pre-existing dataset that contains naturally occurring stories that are
annotated with knowledge of normative principles. Thus, one of our primary
contributions in this work is the curation of such a dataset. To construct
this dataset, we use crowdsourcing to extend a previously collected dataset
that was used to train binary classification models on the children’s comic
strcip, Goofus & Gallant.

This comic strip has been published in the U.S. children’s magazine,
Highlights, since 1940 as a means to teach children socially acceptable
behavior. It features two main characters Goofus and Gallant; portraying them
in everyday common situations. The comic consistently portrays a pair of
situations: both a proper way and an improper way to navigate typical
situations that young adults may encounter. Thus Goofus & Gallant comic strip
is a natural corpus to categorize an action as normative or non-normative. But
for the purpose of our work, the dataset as it exists is not sufficient. There
is no specific identifying information that expands on what expectations the
children may or may not be adhering to. We need to know which social norms or
principles are violated or complied with by these actions.

Thus we use the crowdsourcing platform, Prolific, in order to enrich this
dataset. For each image-action text pair of the comic, we collect a detailed
text description of the scene in the image, refine these descriptions with
additional annotations, and separately prompt other annotators to provide - in
freeform text - a description of the social norm that is violated or upheld by
the character in the comic. A comprehensive discussion of the data collection
process is presented in the following section.

Figure 1: Dataset collection prompt/survey Figure 2: Exemplar Principles List
as provided in the prompt to crowd workers Figure 3: Instructions given for
the scene description task Figure 4: Prompt and exemplar for scene description
task survey Figure 5: Prompt and exemplar for "pick-and-rank 3" for 13 classes

###  3.1 Data Collection

Recall, the goal of this data collection is to use human annotators to
construct a dataset of Goofus & Gallant comics that are annotated with
normative principle information. To accomplish this, crowdworkers were
recruited using a service in the [redacted] metro area. The crowdworkers were
given a number of tasks, with no individual worker participating in more than
one phase of the data collection. The tasks were as follows: 1) validate the
original binary class in the Goofus and Gallant comic (i.e. - is this socially
acceptable or unacceptable) (see Figure 1), 2) provide a description - in
short, declarative sentences - of the comic image contents - but in a way that
excludes assumptions about the theory of mind of the characters in the comics
(e.g. not "The boy seems frustrated" only objectively observable information
in the image) (see Figure 3), 3) review in a secondary evaluation these
descriptions written by other crowdworkers and remove incorrect observations
or add missing descriptions and, finally, 4) Use the descriptions of the scene
and just the image, so as to not know whether the comic depicts Goofus or
Gallant, to describe which "social principle" is upheld or violated with light
prompting of what a "social principle" is. Crowdworkers received images in
tandem with the scene description and primary entity quote. In the comics,
often the quote indicates Goofus is talking or Gallant is talking. The phrases
were generalize to remove the identifying character (e.g. "’I’m bored,’ says
Goofus" becomes "I’m bored.)

For each of the tasks described, a workbook or template manual was provided
which gave example responses, clarified terminology in the online survey they
were instructed to take and explained the purpose of the experiment. The most
relevant and "leading" component of these manuals or pamphlets was a list of
exemplar "social principles" which were crowdsourced from our team (Figure 2).
The participants were never instructed to constrain their responses to this
list in particular. Their responses and the principles they provided were
always collected in freeform text.

We received 900 annotations from this data collection process. We remove
annotations where annotators could not reach a consensus with the original
Goofus & Gallant tags from the comics (that is- annotators were not provided
with text indicating the speaking character or who was being portrayed. The
resulting dataset contains 772 comics annotated with normative principles. We,
then, bin the freeform responses into 16 classes. These are then further
reduced to 13 by eliminating the lowest frequency bins. Authors of this paper
independently grouped responses together into categories. This process was
repeated until consensus was reached.

(a) Model architecture for image and text inputs

(b) Model architecture for text only inputs

Figure 6: Model architectures

##  4 Methods

The primary contribution of this paper is to explore how machine learning
models can be used to learn normative principles on a naturally occurring
story dataset.

To do this, we developed two machine learning models that predict the
normative principle involved in a Goofus & Gallant comic. In the first model,
we classify principles using both image and text inputs and in the second, we
inject only the text inputs into the model to investigate how
influential/helpful visual context is for classifying principles. In both
cases, we make use of proven transformer models - Vision Transformer
Dosovitskiy et al. (2021) and DistillBERT Sanh et al. (2019) \- as the basis
for the network architecture. Each architecture is shown in Figure 6. We
discuss in detail each of the model’s architecture in following sections.

###  4.1 Image-Text Model

For each comic image in the dataset, we now have the following: 1) text that
comes from the original comic strip, 2) a detailed description of the image in
simple declarative text as provided by annotators, and 3) the principle for
the corresponding scenario in the image. For the first model, we pass both
image and text information into the network. To give the network additional
context, we also provide information on whether the principle in question is
being violated or upheld int he form of a simplified binary vector. This
corresponds to whether the original comic depicted Goofus (violated) or
Gallant (upheld).

In this work we use the pre-trained Vision Transformer Dosovitskiy et al.
(2021) to create the feature vector of the images. To embed text inputs - the
detailed image description and comic sentence - we use a pre-trained
DistillBERT Sanh et al. (2019) model. We utilize the hidden representation of
the special classification (CLS) token of DistillBERT as the embedding
vectors. An identical layer is added on top of each pre-trained model which we
refer to as the projection layer. It consists of two linear layers followed by
activation , dropout and layer normalization after each layer. All the
embedding vectors are passed through these projection layers. The resultant
vectors from the three projection layers are concatenated and passed through
to another projection layer. The output vector of this layer is concatenated
with the embedding vector $p_{emb}$. $p_{emb}$ represents the information, if
the principle is violated or upheld in the current comic image. Finally this
output vector goes into the classification layer, which is comprised of a
linear and softmax layer. Figure 6(a) shows the overview of the network
architecture of this model.

###  4.2 Text Only Model

The detailed description of an image in our dataset contains comprehensive
descriptions of the scene and the state of the individual in the image. With
this second model, we omit the image and only give the network the detailed
scene description, the original comic text, and the vector indicating whether
the normative principle is violated or upheld as inputs. With this model, we
want to investigate the affect that the image has on predictive performance.
This model is similar to the model described previously, except that elements
related to learning image features have been removed. The overview of the
network architecture is shown in Figure 6(b).

##  5 Experiments

To evaluate the quality of our systems, we perform two sets of experiments: an
automated evaluation and a human subjects evaluation. Each experiment is
performed using both of our trained models. Each of these experiments will be
discussed in greater detail below.

Figure 7: Confusion matrix of the test data Table 1: Class Distribution and Test Accuracy for both Image-Text and Text-Only model with 13 Principles Dataset Class |  Number of Data Points (Train) |  Number of Data Points (Test) |  Accuracy (image+text) |  Accuracy (Text Only - top 1) |  Accuracy (Text Only - top 2) |  Accuracy (Text Only - top 3)
---|---|---|---|---|---|---
Humility |  35 |  11 |  $27.27$ |  $36.36$ |  $63.63$ |  $72.73$
Respect |  85 |  21 |  $23.81$ |  $9.52$ |  $38.1$ |  $42.86$
Law-abiding |  32 |  6 |  0 |  16.67 |  $33.3$ |  $50.0$
Sensibleness |  11 |  2 |  0 |  0 |  $0$ |  $0.0$
Friendliness |  103 |  27 |  $37.04$ |  $40.74$ |  $48.15$ |  $55.56$
Cleanliness |  64 |  21 |  47.62 |  52.38 |  66.67 |  66.67
Cooperation |  49 |  16 |  $12.5$ |  $18.75$ |  $25.0$ |  $31.25$
Self-care |  29 |  7 |  0.0 |  14.29 |  28.57 |  28.57
Caution |  27 |  10 |  50.0 |  70.0 |  80.0 |  80.0
Patience |  34 |  4 |  $25.0$ |  $25.0$ |  $50.0$ |  $50.0$
Assistiveness |  35 |  7 |  $28.57$ |  $57.14$ |  $85.71$ |  $85.71$
Politeness |  53 |  8 |  $12.5$ |  $25.0$ |  $37.5$ |  $37.5$
Attentiveness |  60 |  15 |  $20.0$ |  $20.0$ |  $40.0$ |  $46.67$
Totals/Averages |  617 |  155 |  27.1 |  32.26 |  48.39 |  52.9
Table 2: Class Distribution and Test Accuracy for both Image-Text and Text-Only model with 8 Principles Dataset Class |  Number of Data Points (Train) |  Number of Data Points (Test) |  Accuracy (image-text) |  Accuracy (Text Only - top 1) |  Accuracy (Text Only - Top 2) |  Accuracy (Text Only - Top 3)
---|---|---|---|---|---|---
Humility |  $88$ |  $19$ |  $21.05$ |  $36.84$ |  $52.63$ |  $57.89$
Respect |  $85$ |  $21$ |  $28.57$ |  $28.57$ |  $38.1$ |  $52.38$
Law-abiding |  $32$ |  $6$ |  $16.67$ |  $16.67$ |  $33.3$ |  $66.67$
Sensibleness |  $132$ |  $31$ |  32.26 |  38.71 |  41.94 |  48.39
Friendliness |  $103$ |  $27$ |  $44.44$ |  $40.74$ |  $48.15$ |  $66.67$
Cleanliness |  $64$ |  $21$ |  $57.14$ |  $42.86$ |  $52.38$ |  $71.43$
Cooperation |  $84$ |  $23$ |  $26.09$ |  $21.74$ |  $65.22$ |  $78.26$
Self-care |  $29$ |  $7$ |  14.29 |  42.86 |  42.86 |  57.14
Totals/Averages |  617 |  155 |  33.55 |  34.84 |  48.39 |  61.94
Table 3: Human classification (N=25) distribution and accuracy (Scene Description + Quote, No Image) Class |  Accuracy (13 classes) |  Accuracy (13-top2) |  Accuracy (13-top3) |  Accuracy (8 classes) |  Accuracy (8-top 2) |  Accuracy (8-top 3)
---|---|---|---|---|---|---
Humility |  $0\%$ |  $4\%$ |  $16\%$ |  $12\%$ |  $16\%$ |  $24\%$
Respect |  $16\%$ |  $28\%$ |  $40\%$ |  $28\%$ |  $60\%$ |  $80\%$
Law-abiding |  $4\%$ |  $8\%$ |  $32\%$ |  $28\%$ |  $36\%$ |  $48\%$
Sensibleness |  8% |  12% |  20% |  $4\%$ |  $16\%$ |  $28\%$
Friendliness |  $36\%$ |  $52\%$ |  68% |  $56\%$ |  $92\%$ |  96%
Cleanliness |  $0\%$ |  $4\%$ |  $4\%$ |  $0\%$ |  $12\%$ |  $12\%$
Cooperation |  $16\%$ |  $24\%$ |  $52\%$ |  $48\%$ |  $60\%$ |  $64\%$
Self-care |  $0\%$ |  $8\%$ |  $12\%$ |  $12\%$ |  $28\%$ |  $40\%$
Caution |  $32\%$ |  $56\%$ |  64% |  $-$ |  $-$ |  $-$
Patience |  $36\%$ |  $48\%$ |  $60\%$ |  $-$ |  $-$ |  $-$
Assistiveness |  $4\%$ |  $16\%$ |  $20\%$ |  $-$ |  $-$ |  $-$
Politeness |  $28\%$ |  $48\%$ |  $56\%$ |  $-$ |  $-$ |  $-$
Attentiveness |  $32\%$ |  $36\%$ |  $36\%$ |  $-$ |  $-$ |  $-$
Avg Accuracy |  13.923% |  26.461% |  36.923% |  23.500% |  40.000% |  49.000%

###  5.1 Automatic Evaluation Protocol

The first set of experiments involves evaluating our methods using automatic
performance metrics. We evaluate our models by holding out 20% as a test set.
We use top1, top2, and top3 accuracy for this evaluation. These metrics
describe the percentage of correct predictions that appear in the top 1, 2,
and 3 responses in terms of softmax probability, respectively.

For this experiment, we used two different versions of our dataset for
training. The first is the one described above containing 13 normative
principles for classification. We also used a second version of this dataset
that only contained 8 normative principles for classification. We have created
this dataset by merging some of the principles from 13 principles set. For
merging, we have mostly considered the principles which have relatively lower
number of data points and then merged them based on their context similarity.
For instance, Cooperation and assisting have similar meaning. Similarly, being
patient is also a showcasing of sensibleness, hence these concepts may
indicate similar situations. By merging principles that are related, we aim to
investigate how well our methods perform on a smaller set of principles that
are less likely to be correlated.

###  5.2 Human Subjects Evaluation Protocol

We also perform a human subjects experiment comparing the performance of our
models to the performance of humans on the task of predicting normative
principles based solely on text representations of scenes. The decision to use
only text was grounded in our belief that the task is sufficiently difficult
for humans even with slightly modified text from the comic still augmented
with more descriptive text containing what existed in the original image. This
better mirrors original attempts to use text-only transformers on the
classification task. In this experiment, crowd workers from Prolific were
presented with text-only descriptions of Goofus and Gallant comics drawn
paired with the original comic text. Comics were presented at random - one
representative image with a groundtruth principle tag - from our dataset
containing 13 normative principles. Workers were tasked with selecting and
ranking the top three principles from a list they were presented that
described the comic in question.

Workers completed 5 ranking tasks at a time - randomly selected but evenly
distributed among the 13 core examples. In total, across all images and
participants, the pick-and-rank task was completed 625 times (Figure 5 shows
the data collection interface). Each of the 13 examples description-quote
pairs received 25 rankings (that is, 65 participants chose their top 3
representative principles for the 5 images presented during their task set). A
total of 25 rankings per principle were collected as a result.

This experiment was repeated using our downselected dataset that contained
only 8 normative principles This experiment involved enough workers to achieve
25 rankings per principle as in our previous experiment.

Both experiments were evaluated using the top1, top2, and top3 accuracies, as
we did for our automatic evaluation. Here, these accuracy metrics describe how
often human participants correctly identified the normative principle for a
comic in their top 1, 2, and 3 responses respectively.

##  6 Results

###  6.1 Automatic Evaluation

The prediction results of the models for both 13 and 8 principles are shown in
the Table 1 and 2 respectively. Both tables show the accuracy of our two
models: 1) Image-Text model and 2) Text-Only model. From the table we can
observe, though injecting visual information into the model improves the
accuracy for some of the classes but the overall performance is decreased. It
indicates, the visual cues such as individual’s facial expression, surrounding
objects contribute very nominal in predicting social principles. Instead the
textual description of the scene and action dominantly influence in
understanding the principles. Thus in our later analysis we only refer to the
result of the Text-Only model.

Accuracy for some of the classes are relatively lower for 13 principles set as
they have smaller number of data points in the training set, for instance,
“Sensibleness” (Table 1). The accuracy of these classes increases
significantly after downsizing the classes. From Table 2, we can see that, the
model’s capability to predict "Self-care", "Sensibleness" improves
considerably than the model trained with 13 principles. But it is worth to
mention that, though downsizing the number of classes increases model’s
performance for some of the classes, the overall accuracy does not increase
largely.

###  6.2 Human Subject Evaluation

The results of the human subject evaluations on text-descriptions and quotes
(with no image) can be seen in Table 3. The results for 13 principles are
presented along side the results for 8 principles. A key observation we can
make is how this accuracy shifts when the principle list is reduced. Ambiguous
or more infrequent principles are absorbed and the annotators take less time
on the task, with fewer principles to deliberate between. It is worth noting
that only in the case of Sensibleness (aka "Sensibility") did accuracy
decline. In all other cases, the further binning of principles from 13 to 8
greatly increased the capability of human annotators with respect to correctly
identifying the principle.

##  7 Discussion

In this section, we discuss the results of both our automatic evaluation and
our human subjects evaluation.

###  7.1 Automatic Evaluation

The first thing to note about the results of our automatic evaluation is that
the performance, overall, of each model is relatively poor. The top1
performance for both the text only and text+image models are below 50%. An
interesting note is that the presence of the image did not improve prediction
performance. In fact, the machine learning model that utilized image features
did worse than the model that only had access to text features. It is for this
reason that we focused our evaluations on the text only model. One possible
explanation is how the comic image stylization has changed over the decades -
the dataset was already fairly sparse and so these differences likely had a
significant impact.

In addition, we see that our model’s overall average accuracy increases as we
consider wider rangers for accuracy. This lends support to the notion that
many of our principles may conceptually overlap with each other. The models
struggle to differentiate between principles that may occur in similar
situations (i.e. "Cooperation" and "Humility", can be seen from the confusion
matrix shown in Figure 7), making it less likely that the correct answer
appears as the top response, but more likely that it appears in the top 3
responses. This idea is further supported by the overall increase in
performance we see when moving to the downselected dataset. By merging certain
principles together, we enable the machine learning model to better
differentiate between principles, leading to overall better predictive
accuracy.

###  7.2 Human Subject Evaluation

Similar to the difficulty large scale language models faced when provided
text-only descriptions of the comics, human participants struggled to
accurately identify principles when given the same prompt. It is intuitive
that the accuracy improves when there are fewer principles to choose from. In
many cases, it may not be unreasonable to apply multiple principles to a given
situation or situational description or a quote from a peer. One
interpretation for the ambiguity of the results may also be the nature of the
original collection methods. The crowdworkers asked with attributing freeform
principles to the comics likely have significantly diverse mental templates,
expectations and memories of what "cooperation" may mean as opposed to
"assistiveness" as one example. When asking another set of participants to
select, even from a much reduced set of principles, this continues to be a
problem. But we do see with principles like "friendliness" or "caution" - two
which receive fairly high accuracy/consensus - that there are concepts,
situational descriptions and prompts which more clearly represent a subset of
the binned principles. Another explanation for "friendliness"’s high
performance across both bins may be that it becomes the default principle
participants choose when all others are confusing. Indeed, some principles may
be pre-requirements to others. If a person is effective at the other
principles, they are likely to be perceived as "friendly" in general.

Perhaps the most important thing to note is how our machine learning models
performed with respect to the human rankings. If one looks at average
accuracy, our models outperformed humans across all metrics on both the 13
principle dataset and the 8 principle dataset.

##  8 Conclusion

It is important and urgent to have more perspectives on how best to align
autonomous systems with human preferences, values and social norms. The task
remains difficult despite new datasets or other methods which focus on
debiasing or particular moral philosophies. The principles which were
collected can be applied to more than the western cultures specifically
depicted in the dataset which was extended. It may be useful to investigate
the mapping between descriptions of situations and principles that were
collected. We also see that the task of identifying socially-normative
principles is difficult for both human annotators as well as complex, state of
the art language models and custom architectures. It is not unreasonable to
assume additional context is needed; the theory of mind of entities being
assessed, previous actions and future outcomes of similar social normative
situations are part of ongoing work to probe even deeper into this problem. We
hope this facilitates discussion as to how best to expand existing datasets to
understand normative behavior in terms beyond simple "acceptability" and "non-
acceptable" behaviors or those outside traditional Western moral frameworks.

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# Is flare-ribbon fine structure related to tearing in the flare current
sheet?

P. F. Wyper  Department of Mathematical Sciences, Durham University, Durham,
DH1 3LE, UK<EMAIL_ADDRESS>D. I. Pontin  School of Mathematical
and Physical Sciences, University of Newcastle, University Drive, Callaghan,
NSW 2308, Australia  School of Science and Engineering, University of Dundee,
Dundee, DD1 4HN, UK<EMAIL_ADDRESS>
###### Abstract

Observations of solar flare ribbons show significant fine structure in the
form of breaking wave-like perturbations and spirals. The origin of this
structure is not well understood, but one possibility is that it is related to
the tearing instability in the flare current sheet. Here we study this
connection by constructing an analytical three-dimensional magnetic field
representative of an erupting flux rope with a flare current sheet below it.
We introduce small-scale flux ropes representative of those formed during a
tearing instability in the current layer, and use the squashing factor on the
solar surface to identify the shape of the presumed flare ribbons and fine
structure. Our analysis suggests there is a direct link between flare-ribbon
fine structure and flare current sheet tearing, with the majority of the
ribbon fine structure related to oblique tearing modes. Depending upon the
size, location and twist of the small-scale flux ropes, breaking wave-like and
spiral features within the hooks and straight sections of the flare ribbon can
be formed that are qualitatively similar to observations. We also show that
the handedness of the spirals/waves must be the same as the handedness of the
hooks of the main ribbon. We conclude that tearing in the flare current layer
is a likely explanation for spirals and wave-like features in flare ribbons.

Sun: corona; Sun: magnetic fields; Sun: jets; magnetic reconnection

##  1 Introduction

Flare ribbons below erupting coronal structures are understood to be the
chromospheric footprint of flare reconnection in the corona (e.g. Shibata &
Magara, 2011; Benz, 2017), with the surface flux swept out by flare ribbons
being directly related to the magnetic flux reconnected through the flare
current layer in that time (e.g. Forbes & Lin, 2000). Since direct imaging of
flare reconnection is highly challenging, flare ribbons are therefore often
used as a diagnostic for the flare reconnection process (e.g. Wang et al.,
2003; Kazachenko et al., 2017).

Early 2D conceptual models of eruptive flares envisaged reconnection occurring
at an X-point beneath an erupting flux rope (O-point in 2D), forming two
parallel ribbons either side of the PIL; the CSHKP model (Carmichael, 1964;
Sturrock, 1966; Hirayama, 1974; Kopp & Pneuman, 1976). Later conceptual models
then incorporated the finite length of the erupting structure and the strong
guide field present within the filament channel (e.g. Moore et al., 2001;
Priest & Forbes, 2002). More recently the 3D evolution of flare reconnection
and its association with flare ribbons has been put on a firmer theoretical
footing through numerous numerical experiments and advances in our
understanding of 3D topology and reconnection in general (e.g. Longcope &
Beveridge, 2007; Isenberg & Forbes, 2007; Aulanier et al., 2012, 2013; Janvier
et al., 2013; Wyper et al., 2021).

Figure 1: Interface Region Imaging Spectrograph (IRIS) observation of the 2014
September X-class flare that exhibited spiral and wave-like motions of the
flare ribbons.

In 3D the X-point within the flare current layer is replaced with a Hyperbolic
Flux Tube (HFT) beneath a flux rope anchored to the surface at both ends.
Depending upon the eruption trigger scenario the HFT/flux rope pair can be
pre-existing (e.g. Torus Instability; Kliem & Török (2006)) or form during the
eruption (e.g. Magnetic Breakout; Antiochos et al. (1999)) and in general is
likely to exist in a state of flux between the two (Patsourakos et al., 2020).
However, most 3D eruption scenarios agree on this general topology once flare
reconnection is initiated.

As the flux rope is anchored at both ends the change in the field line mapping
is rapid, but not now discontinuous as one considers the connectivity of foot
points moving from the flare loops to the overlying arcade, or the overlying
arcade to the flux rope (e.g. Aulanier et al., 2012). This quasi-boundary
between these structures is formed by quasi-separatrix layers (QSLs; Titov,
2007), with the HFT formed by the intersection of two sheet-like QSLs that
wrap around the flux rope (Titov & Démoulin, 1999). The footprints of these
two QSLs form two ribbon-like straight/parallel sections beneath the flux
rope, each with a J-shaped hook at their end, outlining the intersection of
the flux rope with the surface. The HFT sits at the centre of the flare
current layer and so heat flux and non-thermal particles from the flare
reconnection follow the QSL field lines down to the solar surface. The
footprints of QSLs calculated on the solar surface have therefore become a
powerful tool for understanding the morphology of flare ribbons in eruptive
flares (e.g. Savcheva et al., 2015; Janvier et al., 2016; Zhao et al., 2016).

There is mounting evidence from both theory and observations that reconnection
within the flare current layer is fragmented and bursty in nature.
Observations of plasmoids in post CME rays, hard X-ray bursts, and
intermittent downflows (e.g McKenzie & Hudson, 1999; Riley et al., 2007; Kliem
et al., 2000; Asai et al., 2004; Cheng et al., 2018) all suggest that flare
reconnection is a fundamentally fragmented and bursty process. Linear theory
and numerical reconnection experiments show that for the high Lundquist
numbers characteristic of the solar corona, high aspect ratio current layers
form which then rapidly become unstable to tearing (e.g. Loureiro et al.,
2007; Bhattacharjee et al., 2009; Edmondson et al., 2010; Pucci & Velli, 2014;
Wyper & Pontin, 2014a). In 2D this produces multiple magnetic islands with
properties that follow power laws (e.g. Huang & Bhattacharjee, 2010). Indeed,
motivated by early reconnection experiments Shibata & Tanuma (2001) introduced
the idea of “fractal reconnnection” within the flare current layer whereby
repeated tearing and current sheet thinning leads to a fractal-like
distribution of plasmoids and current sheets. Plasmoid formation, sometimes in
this fractal-like manner, is a common feature of highly resolved flare
reconnection in 2.5D CME simulations (e.g. Bárta et al., 2011; Karpen et al.,
2012; Lynch & Edmondson, 2013; Lynch et al., 2016; Guidoni et al., 2016;
Hosteaux et al., 2018).

However, when extended to 3D the dynamics are considerably more complex.
Without a guide field, short, dynamic plasmoids with highly twisted field
lines form and evolve in a fully 3D manner (Edmondson et al., 2010; Nishida et
al., 2013). With the inclusion of a guide field linear theory and numerical
experiments show that oblique modes can form on multiple flux surfaces within
the current layer (e.g. Daughton et al., 2011; Baalrud et al., 2012; Wyper &
Pontin, 2014b; Huang & Bhattacharjee, 2016; Edmondson & Lynch, 2017; Stanier
et al., 2019). Such oblique modes form flux ropes at an angle to the guide
field which in the non-linear phase overlap and interact, sometimes leading to
a turbulent cascade (e.g. Huang & Bhattacharjee, 2016). Fully 3D CME
simulations with a realistically evolving guide field are only now beginning
to reach Lundquist numbers where such plasmoids are resolvable.

Flare ribbons provide a further potential piece of indirect evidence of the
fragmented/turbulent nature of flare reconnection. Flare ribbons often have
multiple kernels and generally exhibit a complex structure and evolution,
especially when viewed in close detail (e.g. Asai et al., 2002; Krucker et
al., 2003; Brannon et al., 2015; Jing et al., 2016; Li & Zhang, 2015; Li et
al., 2018). For instance, Brannon et al. (2015) analysed bright knots and
wave-like perturbations in a section of flare ribbon and Parker & Longcope
(2017) later tried to explain these findings based on a quasi-two-dimensional
tearing analysis involving velocity shear flows. Aside from wave-like
perturbations, spirals are also occasionally observed in flare ribbons. An
example is shown in Fig. 1 where an evolving spiral in the hook and wave-like
evolution of a straight section of the ribbon are highlighted. Dudík et al.
(2016) studied this event in detail (see also e.g. Cheng et al., 2015; Li &
Zhang, 2015; Zhao et al., 2016) and noted that the hook especially continually
evolved (“squirmed”) with similar spiral structures. Although in the past
spiral structures have been attributed to the Kelvin-Helmholtz instability
(Ofman & Thompson, 2011), in our view these are the most compelling signature
of flux rope formation within the current layer.

In our previous work on the fragmentation of current sheets at 3D null points,
we have shown that the small-scale flux ropes/plasmoids that form due to
tearing wrap up the separatrix surface which then maps to spirals on the
surface (or boundary of the domain) (Wyper & Pontin, 2014b; Pontin & Wyper,
2015). The presence of such plasmoids has been confirmed by high resolution
simulations and observations of null point reconnection in coronal jets
(Moreno-Insertis & Galsgaard, 2013; Wyper et al., 2016; Kumar et al., 2018,
2019). Moreover, studies of non-thermal particle acceleration indicate that
the photospheric particle impact patterns are also sensitive to the formation
of the plasmoids, being guided along the flux ropes that form the plasmoids
(Pallister et al., 2019, 2021). All of the above suggest that similar tearing-
induced structure within a flare current layer is a strong candidate for
explaining flare ribbon fine structure in eruptive two-ribbon flares.

In this work we explore this idea. Rather than simulating the self consistent
formation of a 3D flare current layer and its associated fragmentation, which
is a formidable task, we attack the problem analytically and consider a simple
model magnetic field that contains all the expected topological ingredients of
a flux rope eruption: a 3D flux rope above a current sheet formed at an HFT.
The simplicity of the model affords us control of where and in what way we
simulate tearing, which we do by introducing local regions of twist to form
small-scale flux ropes. We find that flare ribbon spirals/wave-like motions
are an expected feature of tearing in the current layer, but only when the
flux ropes are oblique. We conclude that flare ribbon structure is indeed
likely a result of tearing and flux ropes formed in the flare reconnection
region.

Figure 2: Contours of the vector potential without the current sheet (a) and
(b), and with it (c). Thick black lines outline the separatrices in these 2D
projections, which closely match the QSLs at $y=0$. (d) shows $B_{x}=0$ in the
$x=0$ plane in both fields. (e) field lines showing the core of the erupting
flux rope (yellow), flare loops (silver) and the flare current layer (semi-
transparent isosurface). $\log(Q)$ is shown on the surface in yellow/green.
The dashed red line shows the approximate position of the HFT. (f) $\log(Q)$
on the surface ($z=0$) scaled to the maximum value. The dashed line shows the
PIL.

##  2 Methods

###  2.1 Background Field

Our background field is given by

$\mathbf{B}_{b}=\mathbf{B}_{fr}+\mathbf{B}_{cs}$ (1)

where $\mathbf{B}_{fr}$ produces a large-scale flux rope and HFT, and
$\mathbf{B}_{cs}$ a current layer. The flux rope field takes the form

$\displaystyle\mathbf{B}_{fr}$
$\displaystyle=\boldsymbol{\nabla}\times(A_{0}\hat{\mathbf{y}})+b_{0}\hat{\mathbf{y}},$
(2)

where

$\displaystyle A_{0}$ $\displaystyle=\frac{x^{2}}{2}+z$
$\displaystyle+\frac{16(z-z_{0})}{\left(1+y^{2}/L_{y}^{2}\right)\left(1+(z-z_{0})^{2}/L_{z}(y)^{2}\right)}$
$\displaystyle L_{z}(y)$ $\displaystyle=\beta[1-(y/L_{y})^{2}]+\gamma,$
$\displaystyle z_{0}$ $\displaystyle=z_{min}-\sqrt{3}\gamma.$ (3)

Throughout the majority of this work we set $L_{y}=20$, $\gamma=3$,
$z_{min}=5$ and $b_{0}=1.7$. This is a generalisation of the field first used
by Hesse et al. (2005) and then further explored by Titov et al. (2009), see
Appendix A for further details. At $y=0$, the contours of $A_{0}$ form an
O-point above an X-point, see Fig. 2(a). (Note that these contours do not
strictly show field lines but they do give an idea of the local field
structure projected onto this plane.). As $|y|$ increases, the X and O-points
approach each other; eventually merging at a cusp point when $|y|=L_{y}$ (Fig.
2(b)). The $z$ positions of the $X$ and $O$ points (corresponding to where
$B_{x}(0,y,z)=0$) are shown in Fig. 2(d) and are given by

$\displaystyle z_{O,X}$ $\displaystyle=z_{0}$
$\displaystyle+L_{z}\left\\{\pm\left[\frac{32}{\zeta(y)}\left(\frac{2}{\zeta(y)}-1\right)\right]^{1/2}+\frac{8}{\zeta(y)}-1\right\\}^{1/2},$
(4)

where $\zeta(y)=1+y^{2}/L_{y}^{2}$ (see Appendix A, for the general
expression). Near $y=0$, $z_{O}$ and $z_{X}$ closely follow the centre of the
flux rope and the HFT, respectively.

To form a strong current layer at the HFT consistent with the simulations and
observations discussed in the previous section, $\mathbf{B}_{cs}$ takes the
form

$\displaystyle\mathbf{B}_{cs}=\begin{cases}\nabla\times\left[c_{1}f(x)g(y)h(y,z)\hat{\mathbf{y}}\right]&(y/L_{y})^{2}\leq
1\\\ 0&\text{otherwise}\end{cases}$ (5)

where

$\displaystyle f(x)$
$\displaystyle=\ln\left[\cosh\left(x/l_{x}\right)\right]e^{-\left(x/k_{x}\right)^{2}},$
$\displaystyle g(y)$
$\displaystyle=\frac{[1-(y/L_{y})^{2}]^{2}}{[1+(y/L_{y})^{2}]^{2}},$
$\displaystyle h(y,z)$
$\displaystyle=\left[\tanh\left(\frac{z-z_{X}(y)+z_{c}}{l_{z}}\right)-\tanh\left(\frac{z-z_{X}(y)-z_{c}}{l_{z}}\right)\right]$
$\displaystyle\times\tanh\left(\frac{z}{k_{z}}\right).$ (6)

A full description of each function is given in Appendix B. Here we choose
$c_{1}=0.6,k_{x}=5,k_{z}=0.2,l_{x}=0.1,l_{z}=2.0$ and $z_{c}=6.0$. Since
$\mathbf{B}_{cs}(x=0)=\mathbf{B}_{cs}(z=0)=0$, the addition of this second
field does not affect the positions of $z_{X}$ and $z_{O}$ or the normal
magnetic field on the solar surface. Figure 2(c) shows contours of the
combined flux functions at $y=0$, while Fig. 2(e) shows a 3D volume rendering
of the current layer and field lines within the flux rope, respectively. The
current layer is fully 3D and stretched beneath the flux rope. The squashing
factor on the surface is shown in Fig. 2(e) and (f), showing that this field
has the typical J-shaped hooks and parallel strips of high-Q associated with
the flare ribbons of many eruptive flares.

###  2.2 Small-scale flux rope field

To this background field we add small-scale twists which model the local
conversion of magnetic shear to twist that occurs as plasmoids form within the
flare current layer. Their magnetic field takes the form

$\mathbf{B}_{I}=\boldsymbol{\nabla}\times(A_{I}\hat{\mathbf{y}}),$ (7)

where

$\displaystyle A_{I}=$
$\displaystyle\left\\{c_{2}-c_{1}\ln\left[\cosh\left(x/l_{x}\right)\right]\right\\}$
$\displaystyle\times$
$\displaystyle\,e^{-(x-x_{I})^{2}/r_{x}^{2}-(y-y_{I})^{2}/r_{y}^{2}-(z-z_{I})^{2}/r_{z}^{2}},$
(8)

and $c_{1}$ and $l_{x}$ are as in $\mathbf{B}_{cs}$. The $\log$ term creates a
cavity within the current layer where the magnetic shear is removed, while the
first term places a rotation into this cavity. The magnetic field structure
induced is consistent with those formed during the 3D tearing simulations
described in Section 1. For the oblique modes we considered small-scale flux
ropes which were rotatable about $(x_{I},y_{I},z_{I})$ in the $yz$-plane. For
simplicity in this case we set $r_{y}=r_{z}=r$, so that the perturbation field
takes the form:

$\mathbf{B}_{I}=\boldsymbol{\nabla}\times[\cos(\theta)A_{I}\hat{\mathbf{y}}+\sin(\theta)A_{I}\hat{\mathbf{z}}]$
(9)

where $\theta$ is the angle the axis of rotation makes with the horizontal
direction.

The final magnetic field is given by a combination of the background, current
sheet and island fields:

$\mathbf{B}=\mathbf{B}_{fr}+\mathbf{B}_{cs}+\sum_{m=1}^{n}\mathbf{B}_{I,m},$
(10)

where $n$ is the number of islands.

Figure 3: (a) Representative field lines showing the four main connectivity
types present in the field ${\bf B}={\bf B}_{fr}+{\bf B}_{cs}$. Red and blue:
arcade field lines on the flanks, in what would be the reconnection inflow
region in the standard two-ribbon flare picture. Black: field lines in the
sheath around the main flux rope. Lower yellow: flare loop field lines. (b)
log(Q) in the mid-plane ($y=0$) at the HFT where the four field line regions
meet. (c) colour map showing the different connectivities. The colours are
matched to the field lines in (a). Figure 4: Island centred on the HFT with
$c_{1}=1.0$. (a) wide view of current in mid-plane ($y=0$). Contour lines show
the $y$-component of the vector potential in the plane. The box shows where
the other panels are taken from. (b) current. (c) squashing factor $\log(Q)$
outlining the boundaries between the different flux systems. (d) colour map
showing the connectivity of each region.  Figure 5: (a) the first flux tube
pair formed by the first HFT bifurcation (red and green), $c_{2}=1.0$. (b) the
second pair (cyan and magenta), $c_{2}=1.25$. Note that the first pair are
also present, but not shown for clarity. Figure 6: Colour maps showing island
formation via increasing the twist of the flux rope (a)-(d), and island
ejection via varying the height of the island (e)-(h). Figure 7: $\log(Q)$
showing the sub-structure formed in the straight section of the flare ribbon.
(a) after the first bifurcation. (b) after the second. Foot points of arcade
field lines are above the ribbon, whereas flare loop foot points are below.

##  3 Results

As previously discussed, the HFT sits at the center of the flare current layer
and divides domains of different field line connectivity. Field lines
immediately next to the HFT in each of these domains are shown in Fig. 3
alongside a plot of $\log(Q)$ at the centre of the HFT (in the $y=0$ plane),
Fig. 3(b). To classify the field line connectivity we split the $z=0$ plane
(“photosphere”) into four regions, as shown in Fig. 3(a): regions $1$ and $4$
are located beyond the cusp points and near the hooks, while $2$ and $3$ are
between the cusps, near the parallel sections of the ribbon. The black field
line ($1$-$4$) is in the sheath of flux around the core of the flux rope and
loops underneath it. The yellow field line ($3$-$2$) is in the flare loops,
while the red ($1$-$2$) and blue ($3$-$4$) field lines form part of the
overlying arcades. For what follows it is useful to introduce a colour map
based on these four field line regions, shown in Fig. 3(c) where blue, red,
yellow and black correspond to regions of the plane threaded by flux
corresponding to the left and right arcades, the flare loops and the flux rope
sheath, respectively.

###  3.1 2D-like flux ropes

####  3.1.1 Structure within the current layer

First we consider the case most analogous to the start of tearing in a 2D
setting; the formation of a single small flux rope at the center of the HFT
with an axis parallel to the direction of current (the $y$ direction). Figure
4(a) and (b) shows the current in the mid-plane ($y=0$), along with contours
of the in-plane vector potential. Although these contours are not field lines,
they would become field lines in the limit of zero guide field ($b_{0}=0$) and
are often used in 2.5D studies to follow the flux evolution and are the most
readily comparable to the CSHKP model and its extensions to include plasmoids.
As previouly discussed, such studies have identified or inferred an often
near-fractal formation of islands within the flare current layer with chains
of X and O-points repeatedly forming and merging (e.g. Shibata & Tanuma, 2001;
Bárta et al., 2011; Lynch et al., 2016).

Here the contours of the flux function depict a similar formation of an
O-point along with two X-points. However, $\log(Q)$ (Fig. 4(c) and (d))
reveals the true 3D evolution; that in fact the original HFT has bifurcated
into three (note that any point where the four colours meet defines the axis
of an HFT). This bifurcation is a fully 3D reconnection effect and occurs when
field lines from the flare loop region become folded over those from the CME
sheath region, which appears in the colour map (Fig. 4(c)) as the formation of
a new black and yellow region bordered by each of the three HFTs. The field
lines within these new regions are shown in red and green in Fig. 5(a) and
show that part of the “in flow” arcade field which originally connected the
surface regions $1$ to $2$ and $3$ to $4$ has been converted to sheath
($1$-$4$) and flare loop ($3$-$2$) field lines as the twist develops within
the plasmoid/small flux rope.

The more twisted the small flux rope that forms, the greater the number of
bifurcations and subsequent wrapping regions of field lines. Fig. 5(b) shows
field lines in flux regions formed by the next bifurcation. These flux tubes
wrap one half turn more before diverging from each other and connecting to the
surface. They now connect to the same surface regions as the original arcade
field did ($1$-$2$ and $3$-$4$). However, they are distinct from the nearby
arcade field as they wrap around each other once within the plasmoid. Each
subsequent bifurcation continues in this way and produces a new pair of HFTs
alongside a pair of nested flux ropes with a half turn more twist and a
similar jump back and forth of surface connectivity ($1$-$2\to 1$-$4$ and
$3$-$4\to 3$-$2$ and back again). Colour maps in Fig. 6(a) to (d) show the
formation of these new nested flux regions as the twist in the flux rope is
increased; recall the HFTs are at the locations where the four colours meet.

The formation of these new flux tubes shows the sensitive coupling between
local effects within the current layer and the global connectivity of the
system. It also highlights the difficulty in interpreting a fully 3D magnetic
field evolution using a 2D slice. Locally, the formation of the plasmoid is a
simple local increase in twist. However, the global consequences of this are
large jumps in the distant foot points that the plasmoid field lines map to as
the twist of the plasmoid varies.

While our model in static, it is well established that bifurcations such as
those discussed above also occur in dynamic evolutions. HFT bifurcations
occurred in the MHD simulations performed by Wilmot-Smith & De Moortel (2007).
There, as here, the field within the current layer had an O-point structure in
a cross-sectional slice, while connectivity plots revealed this region
contained multiple dynamically formed HFTs. Furthermore, HFTs can be thought
of as the continuous analogue of separators connecting null points and several
numerical and analytical models have shown that a local O-point structure in
the current layer is associated with the dynamic bifurcation of separators and
the formation of similar new flux tubes/flux systems as we have found here
(Parnell et al., 2010; Wilmot-Smith & Hornig, 2011; MacTaggart & Haynes, 2014;
Pontin & Wyper, 2015). We conclude that it is highly likely that such HFT
bifurcations would occur dynamically due to the onset of tearing, although the
exact nature of this formation is beyond the scope of this investigation.

####  3.1.2 Flare ribbon signatures

Turning now to the effect of the plasmoids on the flare ribbons, the field
lines in Fig. 5 demonstrate that these flux ropes have four relevant
footpoints; two in the straight main sections and two in the hooked ends.
Figure 7 shows the localised nested loops that form in $Q$ at a foot point
within a straight section of the ribbon where the connectivity of each field
line domain has been highlighted. The loop shapes adjacent to the main ribbon
outline the quasi-boundary between the newly formed flux tubes and their
surroundings. In particular, the loop in Fig. 7(a) shows the quasi-boundary
between the flux within the green flux tube shown in Fig. 5(a) and the
adjacent arcade field. The inner loop in Fig. 7(b) shows the quasi-boundary
between the magenta flux tube shown in Fig. 5(b) and the green flux system
which has one half turn less. Again, we have found analogous nested flux tube
formation in models of separator reconnection (Pontin & Wyper, 2015). This
closed-loop structure forms directly as a result the plasmoid being located at
the junction of four magnetic flux domains (the HFT axis), which facilitates
the formation of new flux systems with distinct connectivity. This is
fundamentally different from the spiral structure discussed in the following
sections that results from twisting up the boundary of two flux domains.

Given that the flux tubes that form them map to the center of the flare
current sheet, its not clear if only the high $Q$ regions would brighten in
the flare ribbon, or the entire nested region. However, as Fig. 6(e) to (f)
shows the formation of the multiple HFTs and nested flux regions is easily
destroyed if the flux rope is moved slightly off the axis of the original HFT.
Based on the sensitivity of their positioning, i.e. that they need to form
perfectly along a _line in space_ (the centre of the HFT), tearing modes of
this type seems an unlikely candidate for flare ribbon structure in general. A
much more plausible candidate are flux ropes forming on the _surfaces_ of high
$Q$ defining the borders of the four flux regions. These correspond to oblique
tearing modes.

Figure 8: Field lines within three oblique flux ropes formed within the
current layer below the HFT. The dashed red line shows the approximate
position of the HFT. Figure 9: $\log(Q)$ showing the sub-structure formed in
the straight section of the flare ribbon by multiple oblique islands ejected
downwards from the current layer. Foot points of arcade field lines are above
the ribbon, whereas flare loop foot points are below. An animation of panels
(a)-(e) is available online showing the evolution. The animation is $1$s long
and begins at $Z_{i}=Z_{x}-0.5$ and ends at $Z_{i}=Z_{x}-1.5$. Between each
frame, $Z$ changes by $\Delta Z=0.1$. Figure 10: $v_{\text{drift}}$ cartoon
showing how as the oblique flux rope is ejected downwards, it slips on to
lower field lines leading to a drift of the spiral along the flare ribbon.
Figure 11: Field lines within three oblique flux ropes formed within the
current layer above the HFT. The dashed red line shows the approximate
position of the HFT. Figure 12: $\log(Q)$ showing the sub-structure formed in
the hooked section of the flare ribbon by multiple oblique islands that are
ejected upwards from the current layer. Foot points of field lines from the
sheath around the main erupting flux rope are towards the top left. Figure 13:
$\log(Q)$ for sinistral (a) and dextral (f) flux ropes. (b) to (e) show close
up views of the spirals in the boxes shown in (a). Similarly, (g) to (j) show
close up views of the spirals in (f). Note in some of the close up views the
aspect ratio has been stretched to show the shape more clearly. Arrows show
the handedness of the hooks/spirals while the dashed line shows the PIL.

###  3.2 Oblique flux ropes

####  3.2.1 Oblique flux ropes on the arcade boundary

In a classic zero-guide field current layer, the only resonant surface where
tearing can occur is the surface where the field reverses, the scenario
considered above. However, once a guide field is introduced the current layer
becomes a rotational discontinuity where any flux surface within the current
layer can support tearing (e.g. Daughton et al., 2011; Huang & Bhattacharjee,
2016; Edmondson & Lynch, 2017). Flux ropes then form aligned to the local
field direction within the resonant flux surface, at some angle to the guide
field direction.

In localised studies of tearing these flux surfaces have no special
significance. However, in the context of the global field structure associated
with an erupting flux rope, the flux surfaces associated with the two sheet-
like QSLs that cross at the HFT are directly associated with the flare
ribbons. And since oblique modes form aligned to the _local_ field direction,
if the flux ropes form on QSL flux surfaces they will twist them up; forming
_spirals_ in the ribbons themselves.

Consider the scenario where small flux ropes form on a flux surface within the
current layer that is not exactly in its center (i.e., at $x\neq 0$). Then the
local field direction with which the flux ropes are aligned when they form is
tilted with respect to the guide field (since $B_{z}\neq 0$ for $x\neq 0$).
Let us first consider the case where these flux ropes form with $x>0$ and
below the HFT, straddling the QSL flux surface that divides the flare loops
from the overlying arcade, i.e. the lower right “leg” of the high-$Q$ X shown
in Fig. 3(b). Figure 8 shows three such flux ropes. Here we choose
$x_{I}=0.05$, $z_{I}=z_{X}(y=0)-1$, with $y_{I}=\pm 1,0$ and $c_{2}=-2$. At
this position the magnetic field is locally aligned at an angle of $\approx
73^{\circ}$ to the $y$-direction, so we set $\theta=73^{\circ}$ to align the
axis of flux rope twist to the local field direction. We note that this angle
is somewhat high compared to theoretically expected values (e.g. Daughton et
al., 2011). We chose this for illustrative purposes, and would expect similar
features for smaller angles corresponding to flux ropes forming nearer the
center of the layer.

As expected, the field lines that thread through the small flux ropes form two
bundles: the flare loops (yellow) and arcade field lines (magenta), which wrap
around one another where they meet below the HFT. Thus, the high-$Q$ layer at
the boundary between the two is twisted into a spiral as it maps from the HFT,
through the flux ropes and down to the surface. This spiral structure appears
in the straight section of the ribbon and is shown in Fig. 9(c) for the above
choice of parameters. With less twisted flux ropes (i.e., smaller $c_{2}$),
rather than spirals we could recover “breaking wave”-like ribbon features
similar to those shown in the top right of Fig. 1.

Since these small flux ropes form beneath the HFT, they would be expected to
be ejected downwards as part of the downflowing plasma from the layer towards
the flare loops. We can approximately model this by varying the height of the
flux ropes within the current layer – from very close to the HFT to a distance
further below it – as shown in Fig. 9(a) to (e). This reveals that the flare
ribbon appears to twist up and then untwist as the flux ropes first encounter
the QSL and then move beyond it. Figure 9 also shows that the spirals have a
slight drift along the ribbon. The reason for this drift in our model is shown
by the cartoon in Fig. 10. As the small flux ropes are ejected downwards from
the current layer they sample field lines that shift the spiral position
progressively further forward. In reality, the “background” flare ribbon and
the growth of twist within the flux ropes themselves will also be time-
dependent, permitting the scenario where spirals form in the ribbon directly
(via a small-scale flux rope forming on the QSL directly) before drifting off
it. However, if the formation and ejection of oblique flux ropes are still
relatively fast then the above predicts drifting spiral structure in the
straight sections of the flare ribbons due to oblique modes forming below the
HFT.

####  3.2.2 Oblique flux ropes on the erupting flux rope boundary

Consider now the alternative scenario where oblique small flux ropes form
above the pre-existing HFT. Such flux ropes can form straddling the section of
the QSL flux surface that divides the sheath around the erupting flux rope
from the arcade field, i.e. the upper right “arm” of the high $Q$ X-shape
shown in Fig. 3(b). Figure 11 shows field lines within three such small flux
ropes within the current layer with axes of twist aligned to the local field
direction (here we choose $x_{I}=0.05$, $z_{I}=z_{X}(y=0)+1$, with $y_{I}=\pm
3,0$, $c_{2}=-2$ and $\theta=80^{\circ}$). Field lines within the flux ropes
from the sheath region are coloured red and those from the arcade are shown in
cyan. Forming above the HFT these field lines wrap around one another above
the HFT, and consequently the spirals in the flare ribbons occur in the hooked
end section of the ribbon far from the flare reconnection site itself. Figure
12 shows the associated spirals in $Q$, which also demonstrate a drift and
twisting/untwisting as the flux ropes are increased in height, modeling the
upward ejection of the flux ropes along with upflowing plasma into the
underside of the large-scale erupting flux rope. Again, if our assumption that
the flux ropes form and are ejected relatively quickly from the flare current
layer is reasonable, then even in a self consistent and evolving field we
would expect similar drifting spiral flare ribbon structure to form in the
hooked ends of the flare ribbons due to oblique modes above the HFT. The drift
in this case is away from the straight section of ribbon, towards the end of
the hook.

####  3.2.3 Relation between chirality of spirals and hooks

Finally, one further key prediction relating spirals in flare ribbons to flare
current sheet structure is how the chirality of the erupting flux rope relates
to the handedness (sense of rotation) in the spirals. To be consistent with
the global field reversal across the flare current layer, small-scale flux
ropes formed due to tearing must have the same sign of twist as the large-
scale erupting flux rope above them (hence our choice of a negative value for
$c_{2}$). As a result the spirals (or breaking waves) have the same sense of
rotation as the hooked ends of the ribbon, the nature of which is determined
by the chirality of the large-scale erupting flux rope. This is true for
spirals both in the straight sections and the hooks themselves. In Fig. 13 we
show the full $Q$ footprint of a sinistral ($b_{0}=1.7$) and dextral
($b_{0}=-1.7$) flux rope with three small flux ropes added above and below the
HFT on each side of the current layer. Note, here we have spaced the small
flux ropes equally in $y$ in this case (with $y_{I}=\pm 3,0$ for each) but use
the same parameter choices as above otherwise. The figure shows that in the
sinistral case the ribbons curve clockwise around the foot point of the
erupting flux rope with clockwise oriented spirals, whereas in the dextral
case the opposite is true, i.e. the spirals/waves match the hooks. Returning
to the observation shown in Fig. 1, we can see that both the spirals/waves and
the hooks do indeed have the same handedness; anti-clockwise in this case.

##  4 Discussion

###  4.1 Realism of the model

In this work we have explored the relation between flare-ribbon fine structure
and tearing using a parameterised analytical model. It is important to
consider the extent to which the model is representative of a snapshot of a
true, dynamic evolution. Despite not being formed by a dynamic evolution, the
topology of our background state is exactly that of an eruptive two ribbon
flare, i.e. a large-scale flux rope above a current sheet formed at an HFT
(Aulanier et al., 2012). Based on theoretical studies of reconnection in
current sheets with and without a large guide field (e.g. Daughton et al.,
2011; Huang & Bhattacharjee, 2016; Edmondson & Lynch, 2017) we introduced
small-scale flux ropes both in the center of the current layer – as only
occurs in 2D current sheets with zero guide field – and also on flux surfaces
still within the current layer, but not directly in its center so as to
approximate oblique tearing modes. The former led to multiple bifurcations of
the HFT and the formation of new (quasi-) flux systems, while the latter
twisted up the boundary between the different field line regions creating
spiral/breaking wave-like structure in the Q ribbons.

We emphasise that the analysis presented here explores the possible
topological states that could occur as a result of such tearing in the flare
current layer. The states which are dynamically realizable will depend on the
self consistent evolution both of the large-scale erupting flux rope field and
the nature of the instability within the flare current sheet. The multiple
bifurcations of the HFT for instance may well only occur transiently as in
previous MHD simulations (Wilmot-Smith & De Moortel, 2007). However, we
believe the spiral structure associated with oblique modes is more robust.
Relying only on the formation of small-scale flux ropes at an angle to the
guide field within the flare current sheet, which are a ubiquitous feature of
the non-linear phase of essentially all simulations of 3D guide field
reconnection (e.g. Daughton et al., 2011; Wyper & Pontin, 2014a, b; Huang &
Bhattacharjee, 2016; Edmondson & Lynch, 2017; Stanier et al., 2019).

To get an idea of how the evolution of the small-scale flux ropes might by
related to ribbon dynamics we also moved the flux ropes vertically to simulate
their ejection from the current layer. Here we make the assumption that the
plasmoids would be ejected rapidly compared to the evolution of the large-
scale background field. This is clearly a crude approximation of the actual
ejection process, but nonetheless it captures some of the salient features.
2.5D simulations of eruptive flares typically show plasmoids forming and being
ejected rapidly compared to the large-scale evolution of the system (e.g.
Karpen et al., 2012). In such 2.5D simulations, once the plasmoids grow wider
than the current layer they behave ideally as they are advected by the net
flow they find themselves in (e.g. Guidoni et al., 2016). However, in 3D the
small-scale flux ropes have a finite extent. The main body of the flux rope
will become advected by the net flow it finds itself in, however the field
lines that map from it then map through the current layer and as a result can
continue to change their connections. We have seen similar small-scale flux
rope evolution with drifting foot points in our previous simulations of
tearing in 3D null point current sheets (Wyper & Pontin, 2014b; Wyper &
DeVore, 2016). In fact, in 3D theoretically field line connections can change
within the flux rope itself as well (e.g. Hornig & Priest, 2003). Therefore,
although crude, we believe that the changes in connections associated with the
plasmoid ejection via a simple vertical displacement as we have used should
not be too different from what one would find in a true dynamic evolution.

###  4.2 Interpretation of observations

The analysis of section 3.2 suggests that oblique 3D tearing modes can
contribute to fine structure in flare ribbons. As 3D tearing is known to
naturally produce a turbulent spectrum of flux ropes, our analysis also
suggests a similar turbulent spectrum of flare ribbon fine structure should
exist with the largest most coherent spirals forming the least frequently,
perhaps explaining why large spirals are relatively rarely observed. Indeed
very large spirals may be associated with the rare “monster plasmoids”
predicted by non-linear tearing theory (Uzdensky et al., 2010). Furthermore,
the magnetic flux within the spirals/wave-like regions is related to the flux
within the small-scale flux ropes. Hence, ribbons traversing regions of
intense surface magnetic field strength are likely to have smaller, less
observable spiral/wave-like fine structure. The important role played by the
guide field in forming oblique tearing modes (e.g. Daughton et al., 2011;
Edmondson & Lynch, 2017; Leake et al., 2020) also suggests that flare ribbon
structure will vary throughout the life of a flare, as the guide field at the
main HFT reduces throughout the eruption.

How then does our model compare with other scenarios put forward to explain
flare ribbon fine structure and evolution? Dudík et al. (2016); Janvier et al.
(2013) have made the case that the fast motion of bright points/kernels moving
along the flare ribbons are signatures of foot point slipping due to
reconnection at the HFT. Such slippage is a natural result of the three
dimensional flare reconnection process and would occur both in a laminar or a
fragmented flare current sheet. However, the preferential brightening of
particular foot points/kernels over others implies there must be some
inhomogeneity in the reconnection process, either within the reconnection
region itself or associated with the coronal loops that are reconnected. As we
have shown, spiral or wave-like sub-structure within flare ribbons is a likely
observational signature of plasmoids/small flux ropes forming within the flare
current sheet. When both clear spiral sub-structure and fast moving bright
points/kernels are observed together, this suggests that fragmentation of the
current layer is likely the modulating factor in producing the distinct bright
kernels associated with the foot point slippage process. That is to say, the
two elements of ribbon fine structure are likely intrinsically linked to the
fact that the flare current sheet is fragmented. However, particularly in
regions of high surface magnetic field strength as discussed above, it is not
clear with current observations that all kernels are distinct observational
features and do not also include under-resolved spirals. Future high
resolution, high time cadence observations from for instance DKIST may be able
to test whether some or all bright kernels are simply unstructured loop foot
points or contain further spiral structure.

Parker & Longcope (2017) also attempted to explain the wave-like evolution of
some flare ribbons (Brannon et al., 2015) using a 2D model of tearing with an
added velocity shear to create a drift. For the tearing in their model to map
to the surface, they envisaged tearing occurring within a current sheet formed
along the legs of a flare loop rather than in the flare current layer itself.
This is similar to what our model suggests for oblique modes formed on the
arcade/flare loop boundary, but with the key difference that the flare current
layer in their model is separate to the layer within which they modeled
tearing to occur. Here we have shown that when the full three-dimensional
structure of the erupting flux rope and flare current sheet are accounted for,
tearing modes within the flare current layer itself are able to reproduce the
wave-like ribbon structure and potentially also its drift. However, we note
that a shortcoming of our model is that our current layer is planar, whereas
simulations reveal it to stretch along the QSLs to the flare ribbons (e.g.
Janvier et al., 2013), so in reality both scenarios may occur. Indeed, in a
similar manner for fine structure occurring in the hooks, small flux ropes
need only form somewhere on the QSL surface wrapping over the flux rope. This
could be in the flare current layer as we have modeled, or in secondary
current layers formed dynamically around the erupting flux rope (Aulanier &
Dudík, 2019). The basic premise however is the same.

##  5 Conclusions

The analysis described above leads us to conclude that at least some flare-
ribbon fine structure is likely to be related to tearing within the flare
current layer. It reveals a direct link between fine structure in the QSLs
that align with the flare ribbons, and flare current layer tearing. Our
analysis suggests that the dominant contribution to this fine structure comes
from oblique tearing modes and that where these modes form in the flare
current layer is directly related to where they appear in the flare ribbon.
Plasmoids/flux ropes formed in the reconnection upflow region and that
straddle the erupting flux rope/arcade boundary create spiral or breaking
wave-like structure in the hooked ends of the ribbons. By contrast,
plasmoids/flux ropes formed in the downflow region and that straddle the
arcade/flare loop boundary form similar structures in the parallel straight
sections of the ribbons. Furthermore, our model predicts that the handedness
of the spirals/waves matches the direction of the hooks in the main ribbon
itself.

On the assumption that the timescale for plasmoid ejection from the flare
current layer is short compared to the timescale of the large-scale flux rope
eruption, we also varied the position of the plasmoid flux ropes to crudely
model their ejection from the current layer. The spiral evolution produced is
remarkably similar to the squirming and breaking wave-like evolution seen in
some flare ribbons (e.g. Brannon et al., 2015; Dudík et al., 2016), with the
spirals/waves exhibiting a drift away from the surface footpoints of the HFT
field line; towards the end of the hook in the hooked section, and away from
the hook in the straight section. However, we note that the rate of drift
would change if we were to relax our simple assumption of a purely vertical
ejection.

The next step is clearly to test these ideas and the predictions of the model
against full MHD simulations and observations, work that is currently
underway. Such studies may provide a framework for deducing certain properties
of the reconnection process on the basis of ribbon fine-structure and its
evolution. In this exciting time of high resolution flare ribbon observations
from for example IRIS, the New Vacuum Solar Telescope (NVST) and now also the
Daniel K. Inouye Solar Telescope (DKIST) it is hoped that this model will spur
on further investigation of transient wave-like and spiral flare ribbon
structures.

We would like to thank Joel Dahlin and Peter Young for stimulating discussions
regarding flare reconnection and ribbon signatures. We also thank Peter Young
for providing the IRIS movie used to make Fig. 1. We would also like to thank
the anonymous referee for their insightful comments which helped improve our
manuscript. DP acknowledges financial support from STFC through grants
ST/N000714 and ST/S000267.

##  Appendix A Generalised flux rope field

The original field from Titov et al. (2009) is given by

$\displaystyle\mathbf{B}_{fr}=\boldsymbol{\nabla}\times(A_{0}\hat{\mathbf{y}})+b_{0}\hat{\mathbf{y}},$
(A1)

with

$\displaystyle
A_{0}=\frac{x^{2}}{2}+z+\frac{\epsilon(t)z}{(1+y^{2}/L_{y}^{2})\left(1+z^{2}/L_{z}^{2}\right)},$
(A2)

where $L_{y}$ and $L_{z}$ are constants. They chose $b_{0}=0.2$ and varied
$\epsilon(t)$ so that a flux rope formed dynamically. The cusps in this field
however are well above the solar surface and the shape of $Q$ on the surface
doesn’t closely resemble that typically seen in eruptive flares. This field
can be generalised such that

$\displaystyle
A_{0}=\frac{x^{2}}{2}+z+\frac{\epsilon(t)(z-z_{0}(y,t))}{(1+y^{2}/L_{y}^{2})\left[1+(z-z_{0}(y,t))^{2}/L_{z}(y,t)^{2}\right]},$
(A3)

where $L_{y}$ is constant and the functions $L_{z}(y,t)$ and $z_{0}(y,t)$ can
be chosen so as to curve the flux rope upwards in its centre and downwards at
its ends to place the cusps near the surface and to produce a $Q$ distribution
that better resembles the morphology of observed flare ribbons. In this case
the $X$ and $O$-point heights are given by

$\displaystyle
z_{O,X}=z_{0}(y,t)+L_{z}(y,t)\left\\{\pm\left(\frac{2\epsilon(t)}{1+y^{2}/L_{y}^{2}}\left(\frac{\epsilon(t)}{8(1+y^{2}/L_{y}^{2})}-1\right)\right)^{1/2}+\frac{\epsilon(t)}{2(1+y^{2}/L_{y}^{2})}-1\right\\}^{1/2}.$
(A4)

As in Titov et al. (2009), real values for $z_{O,X}$ occur when
$\epsilon(t)\geq 8$ and $|y|<y_{max}$, where

$\displaystyle y_{max}(t)=L_{y}\left(\frac{\epsilon(t)}{8}-1\right)^{1/2}.$
(A5)

$\epsilon(t)=8$ is therefore the threshold value beyond which a flux rope and
HFT form. When $\epsilon(t)>8$ the cusps are then situated at

$(x_{c},y_{c},z_{c})=(0,\pm
y_{max}(t),z_{0}(y_{max}(t),t)+\sqrt{3}L_{z}(y_{max}(t),t)).$ (A6)

Here we consider a static flux rope for simplicity, so that $\epsilon$ is
constant and we choose

$\displaystyle L_{z}(y,t)$ $\displaystyle=\beta(1-(y/y_{max})^{2})+\gamma,$
(A7) $\displaystyle z_{0}(y,t)$ $\displaystyle=z_{min}-\sqrt{3}\gamma,$ (A8)

where $z_{min}$, $\gamma$ and $\beta$ are constant. $z_{0}$ is a constant
displacement which sets the positions of the cusp points to be

$\displaystyle(x_{c},y_{c},z_{c})$ $\displaystyle=(0,\pm y_{max},z_{min}),$
(A9)

while $L_{z}$ curves the flux rope upwards in its center, above the cusps. We
then choose $\epsilon=16$, which sets $y_{max}=L_{y}$ to give the form shown
in equation 3.

##  Appendix B Current sheet field

Figure 14: (a) contours of the vector potential for the current sheet field in
the $y=0$ plane. (b) $f(x)$ and its derivatives. (c) $h(y,z)$ evaluated at
$y=0$. (d) $g(y)$.

$\mathbf{B}_{cs}$ is derived by taking the curl of

$\mathbf{A}_{1}=c_{1}f(x)g(y)h(y,z)\hat{\mathbf{y}},$

where $f(x)$, $g(y)$ and $h(y,z)$ are given in equation (6). Contours of the
vector potential ($|\mathbf{A}_{1}|$) are shown in Fig. 14(a), evaluated in
the $y=0$ plane. Broadly, this field takes the form of two large rotations of
like sign with the HFT sandwiched between them. $f(x)$ controls the $x$
variation of the rotations, with the width of the strong gradient region
between the rotations (and therefore the width of the current layer)
controlled with $l_{x}$, and the large-scale extent of the rotations
controlled via $k_{x}$. The line plots in Fig. 14(b) show the sharp but
continuous gradient in the derivative of $f$ ($f_{x}$), which sets the field
reversal across the current sheet alongside the sharply localised peak in the
double derivative of $f$ ($f_{xx}$), which sets the current within the current
sheet. $h(y,z)$ is composed of two displaced $\tanh$ profiles centered on
$z=z_{X}(y)$ but offset by $z_{c}$. $z_{c}$ therefore defines the vertical
length ($\approx 2z_{c}$) of the current sheet, while $l_{z}$ sets the rate of
drop off of current at the sheet ends through the steepening/flattening of the
$\tanh$ profiles. The factor of $\tanh(z/k_{z})$ was introduced to guarantee
no perturbation of $B_{z}$ on the surface. The combined profile of $h(y,z)$ at
$y=0$ is shown in Fig. 14(c). Finally, the factor $g(y)$ (shown in Fig. 14(d))
modulates the strength of the twists so that it peaks at $y=0$ and decays to
zero at the cusp points situated at $y=\pm L_{y}=\pm 20$.

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