# Evaluating Prompting Strategies for Grammatical Error Correction Based on
Language Proficiency

Min Zeng1∗ Jiexin Kuang1 Mengyang Qiu2 Jayoung Song3 Jungyeul Park1
1Department of Linguistics, The University of British Columbia, Canada
2Department of Psychology, Trent University, Canada
3Department of Asian Studies, Pennsylvania State University, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
*Equally contributed authors.

(
Preprint version. To appear in LREC-COLING 2024, short paper.
)

###### Abstract

This paper proposes an analysis of prompting strategies for grammatical error
correction (GEC) with selected large language models (LLM) based on language
proficiency. GEC using generative LLMs has been known for overcorrection where
results obtain higher recall measures than precision measures. The writing
examples of English language learners may be different from those of native
speakers. Given that there is a significant differences in second language
(L2) learners’ error types by their proficiency levels, this paper attempts to
reduce overcorrection by examining the interaction between LLM’s performance
and L2 language proficiency. Our method focuses on zero-shot and few-shot
prompting and fine-tuning models for GEC for learners of English as a foreign
language based on the different proficiency. We investigate GEC results and
find that overcorrection happens primarily in advanced language learners’
writing (proficiency C) rather than proficiency A (a beginner level) and
proficiency B (an intermediate level). Fine-tuned LLMs, and even few-shot
prompting with writing examples of English learners, actually tend to exhibit
decreased recall measures. To make our claim concrete, we conduct a
comprehensive examination of GEC outcomes and their evaluation results based
on language proficiency.

##  1 Introduction

Large language models (LLMs) like Generative Pre-trained Transformers (GPT)
have emerged as a transformative force in natural language processing (NLP)
and artificial intelligence. These models, boasting billions of parameters,
have been trained on an extensive corpus of internet text, making them highly
effective across a wide spectrum of language tasks, such as translation,
summarization, and question answering, often achieving state-of-the-art
results (Brown et al., 2020).

One such application of LLMs is Grammatical Error Correction (GEC). GEC is a
challenging task in NLP that involves detecting and correcting grammatical
mistakes in written text. LLMs like GPT have shown promising results in this
domain, with their ability to generate fluent, grammatically correct text
(e.g., Coyne et al., 2023; Loem et al., 2023). However, despite their
impressive performance, these models are not without limitations. For example,
LLMs have a tendency to overcorrect, leading to higher recall but lower
precision measures (Fang et al., 2023).

Grammatical Error Correction has been a pivotal task in NLP, with numerous
methodologies and systems being developed over the years to improve its
performance. Prior to the advent of LLMs, the most effective GEC systems have
predominantly adopted one of two paradigms: sequence-to-sequence Neural
Machine Translation (NMT)-based approaches and sequence tagging edit-based
approaches. The unique characteristic of GEC, notably the high overlap between
the source and target sentences, has led to the development of edit-based
approaches. These models employ a transformer-based architecture, akin to
their NMT-based counterparts. However, instead of predicting entire sentences,
they are trained to anticipate a sequence of editing operations, such as
delete, append, and replace, significantly enhancing the speed of inference
while preserving high performance (Omelianchuk et al., 2020).

The advent of LLMs has ushered in a new era for GEC. A notable example of is
the work by Rothe et al. (2021), where they leveraged the power of LLMs,
specifically the mT5 model with up to 11 billion parameters. Their work
establishes a new set of baselines for GEC and simplifies the typical GEC
training pipelines composed of multiple fine-tuning stages.

In addition to this fine-tuning approach, recent studies have begun to explore
the potential of the prompt-based approach in the application of LLMs for GEC,
which focuses more on the design of effective prompts that guide the model’s
generation of corrected sentences. For example, Loem et al. (2023)
investigated the impact of task instructions and the number of examples on the
performance of GPT-3 in GEC tasks. They found that instructive instructions
significantly improved GPT-3’s performance in both zero-shot and few-shot
settings, and the performance became more consistent as it received more
examples.

Another area which should be taken into account is L2 learners’ language
proficiency levels. Considering that there is a significant relationship
between learners’ language proficiency levels and types of errors they make
(Yuksel et al., 2017), having language proficiency as one of the variables in
the model might enhance the performance of the model. To be specific,
exploring the relationships between GEC using LLMs, especially, GPT, and
language proficiency levels could reduce the notable limitation of LLMs, that
it its tendency to overcorrection, leading to higher recall but lower
precision measures (Fang et al., 2023).

Building upon these observations, this paper intends to explore the
performance of LLMs in GEC by examining the interaction between LLMs’
performance and the language proficiency levels of the learners. We focus our
exploration on how prompting strategies and fine-tuning impact GEC
performance, with particular attention given to zero-shot and few-shot
prompting. Our goal is to provide a comprehensive understanding of the
strengths and limitations of LLMs in GEC, aiming to illuminate ways in which
their performance can be optimized for language learners of different
proficiency levels, which has hardly been explored thoroughly.

##  2 Language Proficiency

For prompting GEC using GPTs, we use the Cambridge English Write & Improve
(W&I) corpus, which is manually annotated with CEFR proficiency levels,
consisting of beginner level A, intermediate level B, and advanced level C
(Yannakoudakis et al., 2018). It was introduced at the Building Educational
Applications 2019 Shared Task: Grammatical Error Correction (BEA2019) (Bryant
et al., 2019). The text data was from writings of L2 English learners. It has
a propensity that sentences from data of higher proficiency are longer than
lower proficiency: average tokens per sentence in training data sets A, B, and
C are 17.538, 18.304, and 19.212, respectively. For a characteristic example
of proficiency A, the case of in in the ungrammatical sentence (2) is
corrected with In in its counterpart correction (2). It showcases a typical
replacement orthography error, to be more specific, a capitalization error. We
can also observe that the sentence contains an error with, which is corrected
with that (R:PREP). Although it is grammatically accurate to use agree with as
a transitive phrasal verb, an object clause of the verb in the example
sentence is not grammatical. In this case, the error annotation scheme
maintains the structure of the clause while replacing the preposition instead.

*in addition more and more scientists agree with alien really exist *In addition, more and more scientists agree that aliens really exist.

We analyze the error distribution in training data of different language
proficiency levels, in which the distribution of errors in the data sets of
proficiency levels B and C is similar: missing punctuation marks (M:PUNCT),
replacement prepositions (R:PREP), and missing determinants (M:DET) are the
most apparent types of errors. Additionally, proficiency A includes an extra
error type, replacement orthography (R:ORTH), which is defined for case or
whitespace errors. Table 1 shows the ratios of the most frequent error types
of training data in W&I, which we investigate thoroughly in $\S$4.

Proficiency A | Proficiency B | Proficiency C
---|---|---
M:PUNCT | 0.0933 | M:PUNCT | 0.1134 | M:PUNCT | 0.1183
R:ORTH | 0.0602 | R:PREP | 0.0589 | R:PREP | 0.0517
R:PREP | 0.0506 | M:DET | 0.0442 | M:DET | 0.0345
R:VERB:TENSE | 0.0455 | R:VERB | 0.0414 | R:VERB | 0.0323
R:VERB | 0.0419 | R:VERB:TENSE | 0.0393 | R:VERB:TENSE | 0.0273
Table 1: Most frequent errors and their ratio in W&I

##  3 Experimental Results

For experiments, we use the development data set of W&I from BEA2019, which
distinguishes language proficiency levels into A, B and C. We follow the
experimental setting described in Suzgun et al. (2022) for GPT-2 (gpt2-xl)
inferences, and we also adapt it to GPT-3.5 (text-davinci-003). Instead of
using the test data set for the BEA2019 (Bryant et al., 2019), we use the
development data set for evaluation to control proficiency levels. To evaluate
the performance of language proficiency levels A, B, and C, we report ERRANT
results (Bryant et al., 2017) as metrics that include true positive, false
positive, false negative, precision, recall, and more importantly, F0.5 scores
which emphasize precision than recall. Table 3 summarizes the prompting GEC
results for different language models, including GPT-2, GPT-3.5, and fine-
tuned GPT-2. We used default setting in Suzgun et al. (2022) for inference
parameters:

model | gpt2-xl
---|---
tokenizer | gpt2-xl
num_examplars | 0-4 shots
max_model_token_length | 256 if num_examplars is 0
| else 512
delimiter left and right | { }

1-shot | ungrammatical |  This is important thing.
---|---|---
A | grammatical |  This is an important thing.
2-shot | ungrammatical |  Water is needed for alive.
A | grammatical |  Water is necessary to live.
3-shot | ungrammatical |  And young people spend time more ther lifestile.
A | grammatical |  And young people spend more time on their lifestyles.
4-shot | ungrammatical |  Both of these men have dealed with situations in an unconventional manner and the results are with everyone to see.
A | grammatical |  Both of these men have dealt with situations in an unconventional manner and the results are plain to see.

Table 2: Prompt examples

We used the prompts described in Table 2, and the following setting for fine-
tuning parameters:

epochs | 5
---|---
using masked language modeling | False
block size (train) | 128
per_device_train_batch_size | 4
save_steps | 10000
save_total_limit | 2

|  | A | B | C | all
---|---|---|---|---|---
|  | TP | FP | FN | Prec | Rec | F0.5 | TP | FP | FN | Prec | Rec | F0.5 | TP | FP | FN | Prec | Rec | F0.5 | TP | FP | FN | Prec | Rec | F0.5
GPT-2 | zero-shot | 70 | 3944 | 2878 | 0.0174 | 0.0237 | 0.0184 | 45 | 5204 | 2453 | 0.0086 | 0.018 | 0.0096 | 28 | 4860 | 1058 | 0.0057 | 0.0258 | 0.0068 | 143 | 14008 | 6389 | 0.0101 | 0.0219 | 0.0113
| 1-shot | 86 | 3447 | 2862 | 0.0243 | 0.0292 | 0.0252 | 58 | 4240 | 2440 | 0.0135 | 0.0232 | 0.0147 | 28 | 3730 | 1058 | 0.0075 | 0.0258 | 0.0087 | 172 | 11417 | 6360 | 0.0148 | 0.0263 | 0.0163
| 2-shot | 103 | 4175 | 2845 | 0.0241 | 0.0349 | 0.0257 | 69 | 5442 | 2429 | 0.0125 | 0.0276 | 0.0141 | 30 | 4905 | 1056 | 0.0061 | 0.0276 | 0.0072 | 202 | 14522 | 6330 | 0.0137 | 0.0309 | 0.0154
| 3-shot | 140 | 4445 | 2808 | 0.0305 | 0.0475 | 0.0329 | 95 | 5710 | 2403 | 0.0164 | 0.038 | 0.0185 | 38 | 4979 | 1048 | 0.0076 | 0.035 | 0.009 | 273 | 15134 | 6259 | 0.0177 | 0.0418 | 0.02
| 4-shot | 133 | 4347 | 2815 | 0.0297 | 0.0451 | 0.0319 | 84 | 5422 | 2414 | 0.0153 | 0.0336 | 0.0171 | 31 | 4790 | 1055 | 0.0064 | 0.0285 | 0.0076 | 248 | 14559 | 6284 | 0.0167 | 0.038 | 0.0189
GPT-3.5 | zero-shot | 1203 | 3770 | 1740 | 0.2419 | 0.4088 | 0.2634 | 940 | 4693 | 1556 | 0.1669 | 0.3766 | 0.1878 | 407 | 4183 | 677 | 0.0887 | 0.3755 | 0.1047 | 2550 | 12646 | 3973 | 0.1678 | 0.3909 | 0.1894
| 1-shot | 1300 | 3086 | 1643 | 0.2964 | 0.4417 | 0.3173 | 1068 | 3562 | 1428 | 0.2307 | 0.4279 | 0.2541 | 472 | 3086 | 612 | 0.1327 | 0.4354 | 0.1541 | 2840 | 9734 | 3683 | 0.2259 | 0.4354 | 0.2499
| 2-shot | 1443 | 2983 | 1500 | 0.326 | 0.4903 | 0.3494 | 1116 | 3157 | 1380 | 0.2612 | 0.4471 | 0.2849 | 486 | 2592 | 598 | 0.1579 | 0.4483 | 0.1814 | 3045 | 8732 | 3478 | 0.2586 | 0.4668 | 0.2839
| 3-shot | 1477 | 2646 | 1466 | 0.3582 | 0.5019 | 0.38 | 1114 | 3164 | 1382 | 0.2604 | 0.4463 | 0.2841 | 479 | 2416 | 605 | 0.1655 | 0.4419 | 0.1891 | 3070 | 8226 | 3453 | 0.2718 | 0.4706 | 0.2969
| 4-shot | 1330 | 2328 | 1613 | 0.3636 | 0.4519 | 0.3784 | 1089 | 2424 | 1407 | 0.31 | 0.4363 | 0.329 | 457 | 1870 | 627 | 0.1964 | 0.4216 | 0.2199 | 2876 | 6622 | 3647 | 0.3028 | 0.4409 | 0.323
FT GPT-2 | zero-shot | 1118 | 1479 | 1830 | 0.4305 | 0.3792 | 0.4192 | 928 | 1203 | 1570 | 0.4355 | 0.3715 | 0.421 | 383 | 792 | 703 | 0.326 | 0.3527 | 0.331 | 2429 | 3474 | 4103 | 0.4115 | 0.3719 | 0.4029
| 1-shot | 1127 | 1668 | 1821 | 0.4032 | 0.3823 | 0.3989 | 925 | 1325 | 1573 | 0.4111 | 0.3703 | 0.4022 | 382 | 913 | 704 | 0.295 | 0.3517 | 0.3048 | 2434 | 3906 | 4098 | 0.3839 | 0.3726 | 0.3816
| 2-shot | 1107 | 1700 | 1841 | 0.3944 | 0.3755 | 0.3904 | 937 | 1359 | 1561 | 0.4081 | 0.3751 | 0.401 | 383 | 919 | 703 | 0.2942 | 0.3527 | 0.3043 | 2427 | 3978 | 4105 | 0.3789 | 0.3716 | 0.3774
| 3-shot | 1073 | 1860 | 1875 | 0.3658 | 0.364 | 0.3655 | 874 | 1596 | 1624 | 0.3538 | 0.3499 | 0.353 | 381 | 1168 | 705 | 0.246 | 0.3508 | 0.2616 | 2328 | 4624 | 4204 | 0.3349 | 0.3564 | 0.339
| 4-shot | 1032 | 1911 | 1916 | 0.3507 | 0.3501 | 0.3505 | 818 | 1815 | 1680 | 0.3107 | 0.3275 | 0.3139 | 359 | 1310 | 727 | 0.2151 | 0.3306 | 0.2313 | 2209 | 5036 | 4323 | 0.3049 | 0.3382 | 0.311
SOTA | gector | 1046 | 632 | 2054 | 0.6234 | 0.3374 | 0.533 | 785 | 458 | 1836 | 0.6315 | 0.2995 | 0.5169 | 315 | 208 | 845 | 0.6023 | 0.2716 | 0.4843 | 2146 | 1298 | 4735 | 0.6231 | 0.3119 | 0.5194
| t5 | 1338 | 741 | 1762 | 0.6436 | 0.4316 | 0.586 | 1018 | 620 | 1603 | 0.6215 | 0.3884 | 0.5549 | 377 | 351 | 783 | 0.5179 | 0.325 | 0.4629 | 2733 | 1712 | 4148 | 0.6148 | 0.3972 | 0.5541

Table 3: Prompting results using GPT-2 (gpt2-xl and ft = fine-tuned), GPT-3.5
(text-davinci-003) and SOTA results by models of gector (Omelianchuk et al.,
2020) and t5 (Rothe et al., 2021).

When evaluating the efficacy of few-shot strategies on GPT-2 and GPT-3.5, it
is evident that the few-shot prompting method exhibits better performance
compared to the zero-shot prompting method. For instance, in the all data set
which combines corpus of three language proficiency levels, we observe that
the 4-shot F0.5 scores for GPT-2 and GPT-3.5 are 0.0495 and 0.323
respectively, which are higher than the zero-shot F0.5 scores for GPT-2 and
GPT-3.5. It is also noticeable that the 4-shot approach consistently yields
higher F0.5 scores in comparison to the zero-shot approach. However, this
trend is not observed for the fine-tuned GPT-2 model on different language
proficiency levels. For example, in the all data set, the F0.5 score for the
4-shot approach is lower than the F0.5 score for the zero-shot approach.
Therefore, based on our experimental findings, it is feasible to conclude that
few-shot techniques may not have a significant impact on fine-tuned GPT-2
models.

In addition, GPT-2 exhibits a large decreasing rate of recall as the language
proficiency levels increase from A to C. Specifically, there is a notable
increase in the dropping rate of precision from 50.57% (0.0174 in A versus
0.0086 in B) to 33.72% (0.0086 in B versus 0.0057 in C). However, the fine-
tuned GPT-2 shows a better trend for the precision rate. From proficiency
level A to proficiency level C, the precision score increases from 0.4305 in A
to 0.4355 in B (+1.16%) and then drops to 0.326 (-25.14%) in C. It indicates
the fine-tuned model is more robust for different proficiency level data sets.

##  4 Analysis and Discussion

Unless specified otherwise, our analysis and discussion are based on results
of the fine-tuned gpt2-xl using zero-shot which we achieve the best results.

### Label-by-label evaluation approach

We implement a label-by-label evaluation method. As Bryant et al. (2017)
suggested, we provide edit operation-based and POS-based errors as well as
detailed breakdown composed errors (m—r—u with POS) to investigate further the
relationship between GEC and different proficiency levels. For example, Table
4 shows different types of error evaluation results. When comparing correcting
missing operation errors with all errors, it has higher F0.5 scores where it
suggests that GEC using GPT performs better in the specific missing error
regardless of language proficiency. M:PUNCT (missing punctuation marks) is the
most frequent error among all error types in three language proficiency, which
outperforms the entire results for all proficiency levels. This reflects the
general characteristics of the performance of GEC using GPT. R:VERB (replacing
verbs) consistently performs poorly compared to the entire results, and this
has the same tendency for all r edit errors where the proficiency C achieves
especially lower results. We observed that GEC using GPT contradicts to the
problem of over-correction for lower proficiency levels because of the much
higher numbers of FN in A and B.

|  | TP | FP | FN | Prec | Rec | F0.5
---|---|---|---|---|---|---|---
M:PUNCT | A | 189 | 171 | 134 | 0.525 | 0.5851 | 0.536
| B | 203 | 132 | 133 | 0.606 | 0.6042 | 0.6056
| C | 95 | 96 | 80 | 0.4974 | 0.5429 | 0.5059
R:VERB | A | 21 | 60 | 113 | 0.2593 | 0.1567 | 0.2293
| B | 17 | 55 | 113 | 0.2361 | 0.1308 | 0.2033
| C | 6 | 43 | 51 | 0.1224 | 0.1053 | 0.1186
m | A | 318 | 436 | 372 | 0.3703 | 0.3571 | 0.1691
| B | 336 | 347 | 344 | 0.4919 | 0.4941 | 0.2458
| C | 157 | 222 | 168 | 0.4142 | 0.4830 | 0.2180
Table 4: Detailed breakdown evaluation results for the most frequent errors,
and missing operation errors (FT GPT2, zero-shot).

### Is recall higher than precision in prompting GPT for the GEC task?

Consistent higher recall compared to precision showcases a tendency of over-
correction in prompting GPT for the GEC task. We have observed that
proficiency levels A and B, however, do not exhibit such a propensity. It
holds true even for GPT-3.5, where recall consistently surpasses precision.
Nevertheless, the difference between precision and recall measurements in
levels A and B is considerably smaller compared to level C.

### Results using various F-scores

Table 5 shows results of FT GPT-2 and GPT-3.5 obtained with different
F-scores, where $\beta=$ 0.5, 1, and 2. The result implies that FT GPT-2 is
less prone to over-correction in comparison to GPT-3.5 because the F2 scores
are mostly higher in GPT-3.5. In traditional approaches in GEC, such as SOTA
results in Table 3, where the total numbers of TP and FP are relatively small,
F0.5 would be relevant to measure GEC results. Since recent approaches by
prompting GPT in the GEC task illustrate much higher numbers, especially FP,
it appears that the F1-score proves to be a more effective indicator in GEC
results.

| FT GPT-2 | GPT-3.5
---|---|---
| F0.5 | F1 | F2 | F0.5 | F1 | F2
A | 0.4192 | 0.4032 | 0.3885 | 0.3784 | 0.4030 | 0.4310
B | 0.4210 | 0.4010 | 0.3827 | 0.3291 | 0.3625 | 0.4034
C | 0.3310 | 0.3388 | 0.3470 | 0.2199 | 0.2680 | 0.3430
all | 0.3907 | 0.4029 | 0.3792 | 0.3590 | 0.3230 | 0.4040
Table 5: Different F-scores with F0.5, F1 and F2. FT GPT-2 results are based
on 0-shot, while GPT-3.5 (text-davinci-003) results are based on 4-shot.

### Comparison between prompting GPT and SOTA

State-of-the-art (SOTA) results continue to demonstrate superior performance
compared to prompting GPT in the GEC task in all aspects of results including
precision and recall measures regardless of proficiency levels. Our assumption
is primarily based on the fact that SOTA models are usually subjected to
extensive fine-tuning processes.

### Discussion

In this section, we present the evaluation outcomes using our own
implementation to count the numbers of TP, FP, and FN, which are different
from the ERRANT scores. We leave it as future work to further investigate and
explore potential improvements.

Additionally, while we examine a correlation between proficiency level C and
native in prompting GPT in GEC as shown in Table 6, we are unable to identify
any comparable behavior in prompting GPT in GEC for native-like proficiency C
and native proficiency. Hawkins and Buttery (2010) analyze that some error
types are more notable in B1 and B2 levels than C1 and C2 levels, such as
missing preposition and form of determiner. For example, there are more errors
like missing preposition (M:PREP) or replacement of determiners (R:DET) in B
than in C, which confirm what the previous work proposes. Table 7 shows a
behavior of prompting GPT in the GEC task proficiency specific errors, in
which finding their correlation could be excessively challenging because of
the performance of GEC for proficiency level C. We consider results of the
proficiency level C as unnatural behavior, which deviates significantly from
what is considered typical prompting GPT in GEC. We also leave it as future
work.

| TP | FP | FN | Prec | Rec | F0.5
---|---|---|---|---|---|---
C | 383 | 792 | 703 | 0.326 | 0.3527 | 0.331
N | 2429 | 3474 | 4103 | 0.4115 | 0.3719 | 0.4029
Table 6: Results between proficiency level C and native |  | TP | FP | FN | Prec | Rec | F0.5
---|---|---|---|---|---|---|---
M:PREP | B | 24 | 29 | 31 | 0.4528 | 0.4364 | 0.4494
| C | 9 | 23 | 17 | 0.2812 | 0.3462 | 0.2922
R:DET | B | 15 | 30 | 41 | 0.3333 | 0.2679 | 0.3178
| C | 7 | 12 | 23 | 0.3684 | 0.2333 | 0.3302
Table 7: Detailed breakdown evaluation results for M:PREP and R:DET

##  5 Conclusion

In this paper, we investigated the strengths and limitations of prompting GPT
for the GEC task based on different language proficiency levels. We used our
own implementations to calculate relevant metrics for label-by-label analysis,
which are different from the current standard ERRANT scores by using m2 files.
We observed a tendency of over-correction in prompting GPT, and it is more
obvious in the recent version of GPTs, where recall consistently surpasses
precision. Additionally, since prompting GPT generates much higher false
positive numbers in results, the F1-score, rather than the F0.5-score, would
be a more effective measure in GEC results.

##  6 Ethics Statement

To confirm Behavioural Research Ethics at the University of British
Columbia,111https://ethics.research.ubc.ca/behavioural-research-ethics authors
have obtained a certificate of the Tri-Council Policy Statement: Ethical
Conduct for Research Involving Humans (TCPS 2): Course on Research Ethics
(CORE-2022).222https://tcps2core.ca

## Acknowledgement

This work was supported in part by Oracle Cloud credits and related resources
provided by Oracle for Research.

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# Transverse MHD waves as signatures of braiding-induced magnetic reconnection
in coronal loops

A. Ramada C. Sukarmadji Department of Mathematics, Physics and Electrical
Engineering, Northumbria University
Newcastle upon Tyne, NE1 8ST, UK  Patrick Antolin Department of Mathematics,
Physics and Electrical Engineering, Northumbria University
Newcastle upon Tyne, NE1 8ST, UK

###### Abstract

A major coronal heating theory based on magnetic reconnection relies on the
existence of braided magnetic field structures in the corona. In this small-
angle reconnection scenario, numerical simulations indicate that the
reconnected magnetic field lines are driven sideways by magnetic tension and
can overshoot from their new rest position, thereby leading to low-amplitude
transverse MHD waves. This provides an efficient mechanism for transverse MHD
wave generation, and the direct causality also constitutes substantial
evidence of reconnection from braiding. However, this wave-generation
mechanism has never been directly observed. Recently, the telltale signature
of small-angle reconnection in a sheared coronal structure has been identified
through nanojets, which are small, short-lived, and fast jet-like bursts in
the nanoflare range transverse to the guide-field. We present for the first
time IRIS and SDO observations of transverse MHD waves in a coronal loop that
directly result from braiding-induced reconnection. The reconnection is
identified by the presence of nanojets at the loop apex which release
nanoflare-range energy. We find that the oscillations have an energy flux on
the order of $10^{6}-10^{8}$ erg cm-2 s-1, which is within the budget to power
active region loops. The estimated kinetic and thermal energy from the
nanojets is also sufficient to power the transverse waves and sustain the
observed heating at the loop apex. This discovery provides major support to
(a) existing theories that transverse MHD waves can be a signature of
reconnection, (b) the existence of braiding in coronal structures and (c) the
coronal reconnection scenario identified by nanojets.

The Sun (1693) — Solar corona (1483) — Solar magnetic fields (1503) — Solar
coronal waves (1995) — Magnetohydrodynamics (1964)

##  1 Introduction

Magnetohydrodynamic (MHD) waves and magnetic reconnection are the two leading
theories for the coronal heating problem, which has been a subject of
investigation for decades. MHD waves can carry large amounts of energy (Uchida
& Kaburaki, 1974), making them a suitable candidate for heating through their
dissipation (Wentzel 1979; Klimchuk 2006; Van Doorsselaere et al. 2020). This
was supported by the discovery of transverse oscillations in loop-like
structures (Aschwanden et al. 1999; Nakariakov et al. 1999), and followed up
by many other works reporting similar oscillations (Aschwanden et al. 2002;
Verwichte et al. 2004; Tomczyk et al. 2007; Tomczyk & McIntosh 2009; Van
Doorsselaere et al. 2009; Okamoto et al. 2015; Anfinogentov et al. 2015; Li &
Long 2023). However, direct observational evidence of wave-based heating
remains scarce (Van Doorsselaere et al., 2020).

Among transverse oscillations, the kink oscillations (Nakariakov et al., 2021)
are the most prominently observed and often found to be decayless (Tian et al.
2012; Wang et al. 2012; Nisticò et al. 2013; Anfinogentov et al. 2013). Many
external driving mechanisms have been proposed for these decayless
oscillations, including footpoint driving (Nisticò et al. 2013), quasi-steady
flows (Nakariakov et al. 2016), or by Alfvénic vortex shedding (Karampelas &
Van Doorsselaere 2021). Proposed internal mechanisms include the combination
of resonant absorption and the Kelvin-Helmholtz instability (Antolin et al.
2016; Antolin & Van Doorsselaere 2019) or coronal rain (Kohutova & Verwichte
2017), but there is still a lack of observational evidence.

Another major coronal heating candidate is the Parker nanoflare theory
(Parker, 1988), which conjectures the existence of myriad energy bursts at the
order of $10^{24}$ erg generated by small-scale magnetic reconnection events
driven by braiding. The braided state of a loop is thought to be the result of
slow footpoint motions at photospheric level (Pontin & Priest, 2022). Energy
releases within the nanoflare range have been previously reported by e.g.
Testa et al. (2013) and Testa et al. (2014), and Chitta et al. (2018) have
also suggested that chromospheric reconnection from flux cancellation in loop
footpoints may facilitate nanoflare-sized energy release in loops. However, a
direct link to coronal reconnection could not be established in these heating
events.

The discovery of nanojets by Antolin et al. (2021) provided direct evidence of
nanoflare-based heating driven by small-scale component reconnection. Nanojets
are small-scale and short-lived bursts, around 500 km in width and 1000 km in
length on average, that last no longer than 15 s on average. They are a result
of very fast transverse motion of reconnected field lines driven by magnetic
tension, combined with localised heating (the nanoflare). In Antolin et al.
(2021), they were observed in a loop-like structure and driven by the loss of
stability of a nearby prominence. This was then followed by observations of
nanojets in loop-like structures with coronal rain Sukarmadji et al. (2022),
with the Kelvin-Helmholtz instability (KHI) and Rayleigh-Taylor instability
(RTI) as the likely underlying drivers. The different observations of nanojets
in a variety of environments with different drivers further suggests that they
may be common, and could contribute significantly to the heating of the solar
corona.

It has been long known that magnetic reconnection can produce all kinds of MHD
waves (e.g. Petschek 1964, Parker 1991, Kigure et al. 2010), however, it is
unclear which waves would be predominantly observed in the Parker nanoflare
theory. The kink instability as a trigger of reconnection and driver of
coronal heating has been extensively studied through numerical simulations of
twisted magnetic fields (Browning et al. 2008; Hood et al. 2009; Bareford et
al. 2013; Hood et al. 2016; Reid et al. 2018; Reid et al. 2020). In
particular, Hood et al. (2009), Hood et al. (2016), Reid et al. (2018), and
Reid et al. (2020) proposed the existence of twisted coronal braids or
strands, some of which would become unstable thereby setting a cascade of
nanoflare-sized reconnection events affecting neighboring stable strands.
Although not investigated in detail, these works show the generation of
transverse MHD waves during the reconnection process. Observationally,
Kohutova et al. (2020) have detected torsional Alfvén waves produced from a
reconnection event, although the configuration leading to the reconnection
corresponds to the presence of 2 separate coronal structures and not a single
braided structure. To date, there are no direct observations of small-angle
reconnection events leading to kink waves. Yet, as mentioned previously, kink
waves are ubiquitous in the solar corona and their origin is highly debated.

We present in this paper first observations of transverse oscillations driven
by small-angle reconnection events in a coronal loop, where the reconnections
are identified by the presence of nanojets. We will look into the properties
of the waves produced, and discuss the heating contributed from the observed
event. In Section 2, we present the observation and the present structures.
Section 3 discusses the reconnection nanojets, followed with a discussion of
the transverse waves produced in Section 4. We will discuss the energy budget
in Section 5 and provide conclusions in Section 6.

##  2 Observations

An observation of AR 12192 was taken by the Interface Region Imaging
Spectrograph (IRIS; De Pontieu et al. 2014) on the 29th of October 2014
between 08:37:04-13:43:35 UT, observing in the SJI 1330 filtergram with
spatial resolution, temporal cadence, and exposure time of $0.16\arcsec$, 9.6
s, and 8 s respectively. This is a large coarse 8-step raster observation
centered at (x,y) = ($956\arcsec$,$-262\arcsec$) with a field-of-view (FOV) of
$119\arcsec\times 119\arcsec$. We use the level 2 data for our analysis, along
with coaligned observations from the Atmospheric Imaging Assembly (AIA; Lemen
et al. 2012). During the observing period, the area produced a series of C to
M-class flare events with a number of surges and quiescent and flaring coronal
rain.

Our focus will be on a hot loop shown in Figure 1. We initially observe a
diffuse bright loop-like region at 11:49:37 UT only visible in AIA 94 and 131
and faintly in the SJI 1330 (the Fe XXI emission), which disappears after
13:18:01 UT. Following this disappearance, the loop is seen to form at
13:23:37 UT in SJI 1330, AIA 304, 171, 193, 211, and faintly in 131 and 335,
suggesting catastrophic cooling and the appearance of coronal rain. The time
range of interest starts at 13:30:19 UT, when the cool loop exhibits a
secondary heating event and is seen in all changes but only faintly in AIA 94,
335 and 1600. The structure remains visible until the end of the observing
period of IRIS, and until 14:19:07 UT in AIA. Unfortunately, the IRIS slit
does not cross our loop of interest, and therefore we can only obtain plane-
of-sky (POS) values for all of our measurements.

The loop appears first at a measured height at the apex of $29\pm 5$ Mm as
measured in the POS from the solar surface, and a length of $120\pm 20$ Mm in
the POS. Within the loop, we observe coronal rain flowing with POS velocities
of 20-32 km s-1 at the apex and 100-120 km s-1 along the legs of the loop. The
rain strands have widths of $600\pm 45$ km, and the apparent width of the loop
at the apex seen in the upper transition region or coronal AIA passbands is
$4059\pm 489$ km.

We used the Basis Pursuit Method, based on the Differential Emission Measure
(DEM) Analysis (Cheung et al. 2015) to estimate the emission distribution with
respect to temperature. The DEM weighted electron number density of the loop
$\langle n_{e}\rangle_{DEM}$ depends on the emission $DEM(T)$ in the LOS $l$
for a given temperature bin

$DEM(T)=n_{e}n_{H}\frac{dl}{dT},$ (1)

where $n_{e}$ and $n_{H}$ are the electron and hydrogen number densities,
respectively. $\langle n_{e}\rangle_{DEM}$ follows

$\langle n_{e}\rangle_{DEM}=\sqrt{\frac{\int_{T}DEM(T)_{loop}dT}{1.2l}},$ (2)

where we assume a fully ionised plasma with $10\%$ Helium abundance and $l$ is
the length of the emitting plasma in the loop along the LOS.
$DEM(T)_{loop}=DEM(T)_{LOS}-DEM(T)_{env}$ with $DEM(T)_{LOS}$ is the DEM at a
given temperature bin for a LOS crossing the loop, and $DEM(T)_{env}$ is the
DEM for a LOS neighboring the loop in the same temperature bin. We thus assume
that the foreground and background along this neighboring LOS is the same as
that crossing the loop so that the subtraction gives the DEM of the loop. We
averaged the EM values from the DEM calculation that has converged, for pixels
contained in the loop for each temperature bin. The temperature bins used are
$log(T)=5.5-6.4$ as they show emission from the loop.

We assume that $l$ is similar to the POS width of the loop to obtain an
electron number density of $4.7\pm 0.5\times 10^{9}$ cm-3, which corresponds
to the optically thin hot plasma surrounding the coronal rain strands. The
number density of the cool rain strands is estimated through pressure balance
and taking the peak temperature response for SJI 1330 ($10^{4.3}$ K), to find
$3.8\pm 0.8\times 10^{11}$ cm-3. This matches previous measurements of coronal
rain densities in observations (Antolin et al. 2015; Froment et al. 2020) and
numerical simulations (Li et al. 2022; Antolin et al. 2022a) .

##  3 Reconnection Nanojets

Figure 1: First two rows shows the snapshots of the loop at the time when the
nanojets are most visible in SJI 1330, AIA 131, 171, 193, 211, and 304. The
bottom row shows snapshots for selected emission bins from Log(T) = 5.8, 6.2
and 6.4. Three of the clearest nanojets (left to right, N1, N2, and N3) are
marked with the white arrows. An animation of this figure is available online
showing the time evolution of the loop.  Figure 2: The two panels of the left
column show the same snapshot of the apex of the loop with nanojets. In the
top panel, the three most visible nanojets are marked, and in the bottom
panel, slices along the trajectory of the nanojets A, B, and C are taken to
produce the time-distance diagrams on the first three panels of the right
column. The nanojets in the time-distance diagrams are marked with N1-N6. The
white vertical lines in the time-distance diagrams mark the time of the
snapshot from the left column. The region contained by the white contour line
in the bottom panel of the left row shows the region used to produce the
light-curve plot shown in the bottom panel of the right column. The light
curves are constructed by summing over the intensity values within the contour
at a given timestamp and then normalised. In the bottom we show a schematic
for our interpretation of how a nanojet forms from reconnection due to
misalignments between $\vec{B}_{1}$ and $\vec{B}_{2}$. The resulting
configuration is likely to be affected by the tension from the reconnection,
which overshoots the resting configuration and therefore produces
oscillations, as seen at $t=t_{3}$. An animation of this figure without the
schematic is available online, showing the nanojet formation in the SJI, and
the white vertical line in the time distance diagrams and light curves
following the timestamp of the SJI image. The 7 other observed nanojets are
also marked in the animation with white arrows (nanojets pointing upwards) and
cyan arrows (nanojets pointing downwards).

The reconnection is identified by the presence of nanojets at the apex of the
loop, shown in Figure 1 and the top left panel of Figure 2. After 13:35:35 UT,
we begin to observe around 10 small jet-like structures characteristic of
nanojets forming at the loop apex with 7 of them oriented upwards (away from
the solar surface) and the remaining oriented downwards (towards the solar
surface). Following Antolin et al. (2021) and Sukarmadji et al. (2022), we
identify these structures as nanojets if they have the small, transverse (to
the loop) jet-like features in the image and the running difference image
(current image subtracted with image from previous timestamp) with widths and
lengths of around 500 km and 1000 km, respectively, and are accompanied with a
transverse motion fairly perpendicular to the loop and with short timescales
of less than $25$ s. We investigated 3 of the clearest ones (N1, N2, and N3)
to have a measure of their properties. The mean of their POS lengths and
widths are 1500 km and 341 km, with values ranging from $838-2781$ km and
$320-405$ km respectively. The lifetimes of all three nanojets are $19\pm 5$
s. We also measured the DEM weighted electron number densities and
temperatures of the nanojets. The electron number density is measured through
Equation 2, using the emission from the pixel containing the nanojet and
assuming that $l$ is similar to the POS width of the nanojet. The nanojet’s
temperature is measured through

$T=T_{t_{0}}+\Delta T,$ (3)

averaged over all the nanojet pixels, where $T_{t_{0}}$ is the DEM weighted
temperature of the region before the nanojet forms, following

$T_{t_{0}}=\frac{\int DEM(T)_{t_{0}}TdT}{\int DEM(T)_{t_{0}}dT},$ (4)

and $\Delta T$ is the temperature change at the nanojet timestamp, measured by
the variation of the DEM:

$\Delta T=\frac{\int\Delta DEM(T)TdT}{\int\Delta DEM(T)dT},$ (5)

where $\Delta DEM(T)=DEM(T)_{nanojet}-DEM(T)_{t_{0}}$. $DEM(T)_{nanojet}$ is
the DEM at the nanojet timestamp and $DEM(T)_{t0}$ is the DEM at a timestamp
before the nanojet forms. We find a mean number density and temperature of
$1.2\times 10^{10}$ cm-3 and 2.3 MK, with values ranging from $0.9-1.4\times
10^{10}$ cm-3 and $2.2-2.4$ MK, respectively. The nanojets have an average POS
speed of 156 km s-1, ranging from $91-290$ km s-1.

To obtain a measure of the kinetic and thermal energy, the kinetic energy
($E_{K}$) and thermal energy ($E_{T}$) is calculated through
$E_{K}=\frac{1}{2}1.27V\langle n_{e}\rangle_{DEM}m_{p}v^{2}$ and
$E_{T}=\frac{3}{2}2.09V\langle n_{e}\rangle_{DEM}kT$, where $V$ is the nanojet
volume, $m_{p}$ is the proton mass, $k$ is the Boltzmann constant, and $T$ is
the average nanojet temperature (following Equation 3). The factors 1.27 and
2.09 comes from assuming a 10% Helium abundance and a highly ionized plasma
111Although coronal rain corresponds to partially ionised plasma, numerical
simulations show that its ionisation fraction is relatively high (Antolin et
al. 2022b).. We assume that the nanojet has a cylindrical structure with
radius and length set by the observed mean values, and $v$ is set equal to the
mean POS velocity. This gives us energy releases within the nanoflare range,
with a kinetic and thermal energy release average of $7.8\times 10^{24}$ erg
and $1.4\times 10^{25}$ erg per nanojet, with values of $0.8-21.7\times
10^{24}$ erg and $0.9-2.3\times 10^{25}$ erg, respectively. The average total
energy per nanojet is therefore $2.2\times 10^{25}$ erg, and as we observe
around 10 nanojets, the total estimated energy released from all nanojets is
$2.2\times 10^{26}$ erg. From these measured cases, the nanojets have similar
morphologies, dynamics, and energy release as in Antolin et al. (2021) and
Sukarmadji et al. (2022), although slightly smaller in size. We also observe
local brightenings in the hot AIA channels in the regions where we observe
nanojets, as shown in in Figure 1, supporting the presence of localised
heating in the loop-like structure. The nanojets are particularly seen in AIA
131, 171, 193, 211, 304, and the DEM bin Log(T) = 5.8 and 6.4.

These signatures are clearer in a light curve plot integrating over the loop
region containing the nanojets, shown in Figure 2 (see figure caption for
methods), where the occurrence of nanojets coincides with peaks of the average
loop apex intensity in various passbands (all except 335). The light curve
intensity for the SJI 1330, AIA channels 211, 193, 171, 131 ramps up to two
intensity peaks as the nanojets form, with a minor first peak when N5 and N6
form, and a major peak when N1-N4 form (shown in the time-distance diagrams).
The intensity in all channels starts to decrease at 13:41:58 UT (or $t=383$ s
in the light-curve) after the nanojets have disappeared. This is with the
exception for AIA 304, where there is continuous increase afterwards likely
due to the hot Si XI emission (1.5 MK) within the 304 passband, as suggested
by the temperature variation.

##  4 Transverse Oscillations

Figure 3: The IRIS observation of our data taken in the SJI 1330 filtergram.
The two panels in the left column show the loop-like structure of interest
(top), and the zoomed-in portion from the top figure’s box with slices 1-5
across the loop (bottom). The 5 slices produce the time-distance diagrams
shown in the right column, where the solid white vertical lines indicate the
time of the left column’s snapshots. Time distances 1 and 2 show the slopes
indicating the transverse velocities of the strands. After $t=150$ s, we can
see oscillating structures in the time-distance diagrams. An animation of this
figure is available online, showing the evolution of the loop-like structure
where the white vertical line in the time distance diagrams follows the
timestamp of the panels showing the SJI images. The end of the time distance
diagram is also the end of the IRIS observing period.

After 13:37:59 UT (around $t=150$ s in the time-distance diagrams), the
nanojets that form are followed by the upward motion of several nearby rain
strands, initiating a transverse motion as seen in the time-distance diagrams
of Figure 2 and Figure 3 between t = $150-300$ s. The initial upward motion of
the strands originating from the nanojets are identified by the diagonal
bright slopes across the loop, where the slopes indicate velocities of $40-48$
km s-1 (two examples are marked in Figure 3). The moving strands surpass the
upper edges of the loop at $t=290$ s. After $t=290$ s, it can be seen that the
strands start an oscillatory motion while continuing to move upward at
gradually slower speeds until the end of the IRIS observation. A schematic of
these dynamics is shown in the bottom panel of Figure 2, where the nanojets’
transverse motion overshoots the resting configuration resulting in an
oscillatory field motion. These oscillations have a measured period of $97\pm
4$ s.

Figure 4: Top row shows the SJI 1330 and AIA 304 image with the slice (white
line) for the time distance diagram plots in the second and third row. The
white vertical line in the time distance diagrams marks the time of the SJI
and AIA image. The time-distance diagrams have the white dots marking the
position of the oscillations which are selected based on an intensity
threshold that separates the loop from its surrounding. The third row shows
the combined oscillation data from the SJI 1330 (first 232 seconds, in green)
and AIA 304 (remaining 232 seconds, in blue) and the calculated upward moving
trend (red dots). Each point in the upward moving trend is calculated by
averaging a given point’s value with 8 of its neighbouring points. The IRIS
and AIA points are subtracted with the global trend points to recover the
detrended oscillations, shown in the bottom plot. We observe 2 clear periods
before it damps.

The signatures of multiple strands oscillating can also be seen from the
presence of multiple waves in Figure 3’s time-distance diagrams, and an
example is shown in panel 5 where two waves are labeled as A and B. Note that
these 2 waves are out-of-phase with each other, suggesting that each wave in
the time-distance diagram comes from individual strands. Despite not being in
phase with one another, all of the waves have similar periods with only 2-5
seconds differences, suggesting similar conditions across the strands. In
total, we observe around $7\pm 1$ strands that oscillates from the SJI images
and time-distance diagrams. The oscillations are also most visible in the AIA
304 and 171, and faintly in 131, 193, 211, and 1600.

We show time-distance diagrams of the oscillations in selected channels in
Figures 4 and 5. Note that the sinusoidal shape of the oscillations in the AIA
is not as clear as it is in the SJI, due to the small displacement and low
spatial resolution of AIA. This means that it may be a challenge to identify
them as oscillations had there been no accompanying observations from IRIS.

In Figures 4 and 5, there appears to be a damping of the oscillation at the
extended observing time of AIA after IRIS’s observing time has ended. From
Figure 4, we can estimate the wave properties by combining the observed
oscillations in IRIS and AIA to obtain the de-trended oscillations (see
text/caption for further details). From the de-trended oscillations, two
periods are seen with a measured maximum POS displacement ($x_{max}$) $321\pm
30$ km. We fitted these values in a sinusoidal function of
$x(t)=x_{max}\sin(\frac{2\pi t}{T})$ using the period $T$ to obtain a velocity
profile of $v(t)=w\cos(2\pi t/T)$, with an amplitude $w=\frac{2\pi
x_{max}}{T}$ of $21\pm 2$ km s-1. The loop appears to oscillate for two
periods before it damps, but it must be noted that the oscillations that we
managed to recover well are only from the IRIS data thanks to its higher
spatial resolution.

##  5 Energy Budget of the Reconnection Event and Waves

As the oscillations occur at the apex of the loop, the fundamental mode is the
most likely to be excited. The wave can either be a standing mode or a
propagating wave. Following the nanojets, we initially observe strands that
oscillate out-of-phase with each other, but eventually appear to oscillate
collectively in the upper part of the loop. For this case, we can then assume
that all strands oscillate with a global kink mode. However, we can also
consider a multiple kink wave scenario, in which individual strands oscillate
with their own kink mode (e.g. in Figure 3). The SJI time-distance diagrams in
Figures 2 and 3 also show 7 strands that oscillate individually. We therefore
have four possible cases: A global kink mode in which the strands oscillate as
a whole, multiple kink modes guided by individual strands, and for each of
these two cases we have either a standing (fundamental mode) or a propagating
wave.

In the case of a fundamental kink mode, the period $P$ of the fundamental mode
is

$P=\frac{2L}{c_{k}},$ (6)

where L is the length of the loop and the phase speed $c_{k}$ is

$c_{k}=\sqrt{\frac{\rho_{i}v_{A_{i}}^{2}+\rho_{e}v_{A_{e}}^{2}}{\rho_{i}+\rho_{e}}}.$
(7)

$c_{k}$ depends on the number density inside $\rho_{i}$ and outside $\rho_{e}$
the waveguide (the loop or the strand in the global kink mode or individual
strands cases, respectively), and the corresponding Alfvén speeds
$v_{A}=\frac{B}{\sqrt{\mu\rho}}$ inside and outside are written as $v_{A_{i}}$
and $v_{A_{e}}$, where the magnetic field strength B is expected to vary
little under coronal conditions. The energy flux of kink modes in a bundle of
loop $E_{flux}$ can be calculated from Equation 8 (Van Doorsselaere et al.,
2014), using the transverse velocity amplitude $w$ measured from the
oscillations and the filling factor $f$, following

$E_{flux}=\frac{1}{2}f(\rho_{i}+\rho_{e})w^{2}c_{k}.$ (8)

The total energy can also be estimated following Van Doorsselaere et al.
(2014) through

$E_{total}=\pi
R^{2}L\left(\frac{1}{2}(\rho_{i}+\rho_{e})w^{2}-f\frac{1}{4}\rho_{e}\frac{c^{2}_{k}+v^{2}_{Ae}}{c^{2}_{k}}w^{2}\right).$
(9)

R is the radius of the loop (we have used half of the apparent width of the
loop apex from Section 2), and $L$ is the length of the loop portion that is
oscillating.

For the global kink mode and propagating wave case, the filling factor can be
estimated from the area occupied by the observed number of strands in IRIS
within the oscillating loop’s cross-section observed in AIA (Van Doorsselaere
et al., 2014). Observationally, we observe 6 strands inside the loop portion
(we observe 7 oscillating strands, but 1 has dampen by the time the
oscillating portion forms), but it must be noted that this is a lower bound
since there may be other strands that overlap one another. The oscillating
loop width that contains all the oscillating strands is $3272\pm 386$ km,
whereas an individual strand has a measured width of $600\pm 45$ km. Assuming
a circular geometry, if we fill the 6 strands in the loop we will have a
filling factor of $0.20\pm 0.05$. We will also assume that the external number
density (outside the loop, $\rho_{e}$) is $10^{8}$ cm-3, and use the internal
number density of the loop (surrounding the strands observed in the SJI,
$\rho_{i}$) from the DEM analysis of $4.7\pm 0.5\times 10^{9}$ cm-3 for this
case.

Using the values for $\rho_{i}$ and $\rho_{e}$ obtained above, the measured
loop length of $120\pm 20$ Mm in the POS for $L$, the wave period of $97\pm 4$
s, we have $c_{k}=2474\pm 425$ km s-1 for the global standing kink mode.
Assuming that the magnetic field inside the loop and outside the loop are
similar, the estimated magnetic field is $B=66\pm 11$ G to match the observed
period for the fundamental mode. The $E_{flux}$ and $E_{total}$ for this case
are $1.2\pm 0.5$ $\times 10^{6}$ erg cm-2 s-1 and $2.3\pm 1.0\times 10^{25}$
erg, respectively. Whereas for the global propagating mode case, we have used
the measured minimum phase speed $v_{ph}$ for $c_{k}$ and the length of the
loop portion that appears to oscillate of $16800\pm 720$ km for $L$, to obtain
$B=32\pm 17$ G. The $E_{flux}$ and $E_{total}$ for this case is $0.6\pm 0.4$
$\times 10^{6}$ erg cm-2 s-1 and $3.3\pm 1.5\times 10^{24}$ erg, respectively.

For the multiple kink mode case, the filling factor is 1 since we are
resolving the strands with IRIS, and we will use the coronal rain number
density of $3.4\pm 0.7\times 10^{11}$ cm-3 for $\rho_{i}$ and the DEM weighted
number density for $\rho_{e}$. For the standing waves case we find that
$B=555\pm 96$ G, and $E_{flux}$ and $E_{total}$ for a single strand are
$4.3\pm 1.3\times 10^{8}$ erg cm-2 s-1 and $4.4\pm 1.3\times 10^{25}$ erg,
respectively. We have $7\pm 1$ strands oscillating, so the total energy
released is $3.1\pm 1.7\times 10^{26}$ erg. For the multiple propagating kink
modes case (using $v_{k}=c_{k}$), we obtain $B=274\pm 145$ G, and $E_{flux}$
and $E_{total}$ of $2.1\pm 1.2\times 10^{8}$ erg cm-2 s-1 and $6.1\pm
1.5\times 10^{24}$ erg, respectively. For 7 strands, $E_{total}$ is $4.3\pm
2.2\times 10^{25}$ erg.

The above are lower bound estimates since all measurements are only in the
POS. Assuming that the Doppler velocity component is of the same order as the
POS component, then v increases by $\sqrt{2}$. Taking account these
considerations, this leads to energy flux ranges of $1.2\pm 0.4$ $\times
10^{6}$ erg cm-2 s-1 to $8.6\pm 1.3$ $\times 10^{8}$ erg cm-2 s-1 from all
four cases. Whereas the estimated wave energy will range between $6.5\pm
1.5\times 10^{24}$ erg to $6.1\pm 2.5\times 10^{26}$ erg.

We can also calculate the total thermal energy released from the reconnection
events between a given time $t_{0}$ (right before any nanojet occurrence) and
$t$. This value $\Delta TE(t)$ can be calculated using the DEM values, for a
portion of the loop that contains nanojets and is oscillating using

$\Delta TE(t)=\frac{3}{2}2.09V\langle n_{e}\rangle_{DEM}k_{B}\langle\Delta
T(t)\rangle_{DEM}$ (10)

with the assumption of 10% Helium abundance and a highly ionized plasma.
$\langle{n}_{e}\rangle_{DEM}$ is the DEM weighted electron number density of
the loop (from Section 2), $V=\pi R^{2}L_{osc}$ is the loop portion’s volume
assuming a cylindrical structure and the length of the oscillating loop
$L_{osc}$ of $16800\pm 720$ km. $\langle\Delta T(t)\rangle_{DEM}$ is the
average temperature variation of the loop following Equation 5. This isolates
the temperature change from the reconnection events associated with the
nanojets that contributes to the thermal energy. We calculate the DEM for the
time period starting from 13:35:11 UT - 13:41:59 UT (meaning that $t_{0}$ =
13:35:11 UT). Figure 5 plots the $\Delta TE(t)$ for the loop portion, and we
find that there is a continuous increase in the thermal energy to a maximum of
$4.0\times 10^{25}$ erg.

Figure 5: Top row shows snapshots of the loop apex in IRIS SJI 1330, AIA 304,
and AIA 171. The white line in each panel is the slice taken for the
respective time-distance diagrams shown in the next three rows, where t = 0 s
corresponds to 13:35:11 UT. The time-distance diagrams have the white dots
marking the position of the oscillations which are selected based on an
intensity threshold that separates the loop from its surrounding. The vertical
white line in the time-distance diagram shows the time of the snapshots in the
respective panels above. In the time-distance diagrams, the green vertical
lines marks the period where we observe clear oscillations in IRIS. The plot
in the bottom row shows the thermal energy release change from $t=t_{0}$, for
the region of the loop that appears to be oscillating, estimated from the DEM
for the region bounded by the white contour in the AIA images.

##  6 Discussions and Conclusions

Our observations suggest that the transverse waves are produced by small-angle
reconnection events within the structure, where the reconnection signatures
can be identified by the nanojets. Prior to the nanojets, the loop did not
show any oscillations as seen in the time-distance diagrams of Figures 3 and
5. It is only when the nanojets form that we observe the separation of
individual strands and their subsequent oscillation, which is then followed by
multiple strands that eventually appear to oscillate collectively after all of
the nanojets have formed. This indicates that the nanojets and the transverse
MHD waves share the same generation mechanism, i.e. magnetic reconnection, and
that nanojets reflect the energy available to power the oscillations.

In the small-angle reconnection scenario, the reconnected magnetic field lines
are driven sideways by magnetic tension but overshoot from their new rest
position, thereby leading to transverse waves. This scenario suggests an
efficient mechanism for transverse MHD wave generation. If common (as
conjectured by Parker), it therefore provides an alternative explanation to
the observed ubiquity of small-amplitude transverse MHD waves in the corona.

The period of our observed oscillation is $97\pm 4$ s with a maximum
displacement and amplitude of $321\pm 30$ km and $21\pm 2$ km s-s
respectively. We have considered four cases, depending on whether the observed
kink mode is a global kink mode (case in which the strands oscillate in phase
on average) or a multiple kink mode (case in which each strand oscillates
independently). Furthermore, we consider 2 cases for the modes: standing or
propagating modes. The estimated magnetic field strengths are $32-66$ G for
the global cases, and $274-555$ G for the individual strands cases. These
values are considerably high but still expected from an active region
producing a series of C to M-class flares (e.g. Asai et al. 2001; Landi et al.
2021; Wei et al. 2021).

The produced oscillations also have similar periods suggesting similar
conditions across the strands, and we observe that they occur for 2 periods
before they damp. The fact that the kink waves strongly damp in a loop that is
visible in the hot AIA channels throughout the event strongly suggest wave
dissipation and heating. Based on the measured density, filling factor and
wave properties, we estimate that the energy flux from the waves is on the
order of $10^{6}-10^{8}$ erg cm-2 s-1 for all cases, which is sufficient to
balance the energy losses for active regions (Withbroe & Noyes 1977). These
values are lower bound estimates since we only measure projected velocities,
but they indicate that braiding-induced reconnection has enough energy to
power active region coronal loops.

If the dynamics at the origin of the nanojets are what triggers the kink mode,
then we may expect that the kinetic energy from the waves should match the
kinetic energy of the total number of nanojets. We divide the wave’s total
energy released ($E_{total}$) with the average kinetic energy released by a
nanojet of $7.8\times 10^{24}$ erg to obtain an estimate of how many nanojets
are required to match the wave energy for each case: For the global standing
kink mode, we only require 2-3 nanojets, which is less than the 10 clearest
nanojets observed by eye. For the global propagating wave case, we need less
than 1 nanojet’s worth of kinetic energy.

For the multiple standing kink mode, we require around 39-40 nanojets, which
is substantially more than the observed number of nanojets. We did observe
around 10-12 other nanojet-like features that were too small or faint,
suggesting that such numbers are indeed possible. Whereas for the multiple
propagating waves, we only require around 5-6 nanojets.

An expected feature from nanojets is the strand separation that accompanies
the small-angle reconnection (Antolin et al. 2021; Sukarmadji et al. 2022),
which would overshoot the resting field configuration. However, this strand
separation may not always lead to transverse oscillations. For example, if two
internal misalignments trigger nanojets that have opposite directions, the
resulting oscillation may be a sausage mode rather than a kink mode. Also, if
the separation is accompanied by a displacement of the footpoints then minimal
or no overshoot is produced. This suggests that the reconnection events may
need to have very specific conditions to produce sufficient overshoot to
trigger transverse waves.

The thermal energy increase from the DEM values at the apex also shows an
increase on the order of $10^{25}$ erg, with a maximum value of $4.0\times
10^{25}$ erg just after the nanojets have stopped forming. Part of the kinetic
energy of the nanojets is also likely converted into heat, and the thermal
energy increase is on the same order of magnitude than the nanojet’s average
total energy release of $2.2\times 10^{25}$ erg. If we assume that the thermal
energy increase comes from the nanojet’s total (kinetic and thermal) energy,
this would mean that we only need around 2 nanojets in total. This means that
only a few nanojets is required to sustain the heating seen at the apex of the
loop.

We observe strands appearingly misaligned to one another, similar to the loops
observed in Sukarmadji et al. (2022). Furthermore, the rain flows along the
legs of the loop also appear to be misaligned, suggesting a braided structure.
The entire event starts with a few nanojets, which produce transverse motion
and likely create more misalignments triggering the following nanojet clusters
that occur over the next five minutes. This is similar to an MHD avalanche,
which is expected from previous MHD simulations of braided structures, that
produce bursty nanoflare-sized heating (Hood et al. 2009; Hood et al. 2016;
Reid et al. 2018; Reid et al. 2020).

The event from this work is evidence that kink waves can be a signature of
braiding-induced magnetic reconnection, and that the generated kink waves can
be used as a diagnostic of the energy released through reconnection. It is
likely that a large proportion of heating is still undetected through AIA: The
fact that the oscillations are barely resolved in the AIA channels may wrongly
suggest that there is very little wave energy in the corona. The oscillations
and nanojets are only clear in IRIS, and were also only clearly detected
thanks to the presence of coronal rain in the strands.

A major open question is how often the small-angle reconnection leads to kink
waves, and whether a constant generation of nanojets could support the
decayless kink oscillations commonly observed. If this is indeed the case,
then braided field lines should be expected in oscillating loops as we require
numerous misalignments to consistently produce nanojets that would sustain a
decayless oscillation. However, the kink waves observed in this event damp
very quickly, leading to a question of whether unresolved reconnection
processes power decayless oscillations.

P. A. acknowledges funding from his STFC Ernest Rutherford Fellowship (No.
ST/R004285/2). 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
Centre. SDO is part of NASA’s Living With a Star Program. All data used in
this work are publicly available through the websites of the respective solar
missions.

##  7 Appendix 1

Figure 6: Left column, top to bottom: The SJI snapshot with the nanojets
marked, the running difference image (current snapshot subtracted with the
previous snapshot), and the SJI image showing the slices A, B, and C used for
the time-distance diagrams from the SJI images and the running difference
images on the right column. The nanojets in the time-distance diagrams are
marked with N1-N6. The white vertical lines in the time-distance diagrams mark
the time of the snapshots from the left column. The causality between the
nanojets and the oscillations is seen from how the strands only move upwards
following a nanojet, which is then followed by an oscillation. An example of
this trajectory is marked by the white dots in the time-distance diagram B,
where it starts with nanojet N6, and is followed by a transverse upward motion
that surpasses the upper part of the loop which then initiates the oscillatory
motion at t = 300 s of the diagram. An animation of this figure is available
online, showing the nanojet formation in the SJI, and the white vertical line
in the time distance diagrams following the timestamp of the SJI images.

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