# RoadScan: A Novel and Robust Transfer Learning Framework for Autonomous
Pothole Detection in Roads

Guruprasad Parasnis  Sardar Patel Institute of Technology
Mumbai, India
<EMAIL_ADDRESS>Anmol Chokshi  Sardar Patel Institute of Technology
Mumbai, India
<EMAIL_ADDRESS>Kailas Devadkar  Sardar Patel Institute of Technology
Mumbai, India
<EMAIL_ADDRESS>
###### Abstract

This research paper presents a novel approach to pothole detection using Deep
Learning and Image Processing techniques. The proposed system leverages the
VGG16 model for feature extraction and utilizes a custom Siamese network with
triplet loss, referred to as RoadScan. The system aims to address the critical
issue of potholes on roads, which pose significant risks to road users.
Accidents due to potholes on the roads have led to numerous accidents.
Although it is necessary to completely remove potholes, it is a time-consuming
process. Hence, a general road user should be able to detect potholes from a
safe distance in order to avoid damage. Existing methods for pothole detection
heavily rely on object detection algorithms which tend to have a high chance
of failure owing to the similarity in structures and textures of a road and a
pothole. Additionally, these systems utilize millions of parameters thereby
making the model difficult to use in small-scale applications for the general
citizen. By analyzing diverse image processing methods and various high-
performing networks, the proposed model achieves remarkable performance in
accurately detecting potholes. Evaluation metrics such as accuracy, EER,
precision, recall, and AUROC validate the effectiveness of the system.
Additionally, the proposed model demonstrates computational efficiency and
cost-effectiveness by utilizing fewer parameters and data for training. The
research highlights the importance of technology in the transportation sector
and its potential to enhance road safety and convenience. The network proposed
in this model performs with a 96.12% accuracy, 3.89% EER, and a 0.988 AUROC
value, which is highly competitive with other state-of-the-art works.

###### Index Terms:

Transfer Learning, VGG16, Siamese Network, EER, Economy, Efficiency

##  I Introduction

In recent years, technology has played a crucial role in advancing automated
systems across different sectors. The introduction of autonomous systems has
significantly improved the convenience and efficiency of human lives. Among
various sectors benefiting from automation, transportation and surveillance
systems have witnessed remarkable progress. Within the realm of
transportation, roads hold a vital position as they form an extensive network.
Ensuring the safety of users is a top priority for autonomous systems, and one
of the significant threats on roads is the presence of potholes.

According to official data released by the Government of India, in 2022 alone,
1481 accidents resulted in loss of life, and 3103 individuals were injured due
to potholes. These statistics underscore the critical importance of road
maintenance. However, the COVID-19 pandemic and subsequent lockdowns have
adversely affected several sectors, including road maintenance. Consequently,
road conditions have deteriorated, necessitating the development of an
autonomous system capable of monitoring and detecting road conditions. This
paper proposes a pothole detection system that leverages Deep Learning and
Image Processing techniques. In recent times, numerous deep learning-based
object detection methods have emerged, utilizing Convolutional Neural Networks
(CNNs) for feature extraction. Various versions of pre-trained algorithms such
as YOLO, DenseNet, and RetinaNet have been trained on diverse datasets
encompassing both water-logged and dry potholes of varying shapes and sizes.
The performance of these models is evaluated using accuracy, precision,
recall, AUROC, and other relevant metrics. The models exhibit reasonable
accuracy in detecting a wide range of potholes. This paper proposes a novel
transfer learning approach to the detection of potholes with features
extracted from the VGG16 model and passed onto a unique, custom Siamese
network architecture that is efficient and economic and gives a state-of-the-
art performance. Overall, this research highlights the significance of
technology in the development of automated systems, particularly in the
transportation sector. By employing advanced Deep Learning and Image
Processing techniques, the proposed pothole detection system aims to mitigate
the risks associated with potholes, enhance road safety, and contribute to the
overall convenience and well-being of individuals utilizing road
transportation.

##  II Related Work

Various deep learning algorithms have been developed for the implementation of
pothole detection. Due to a shortage of real pothole data, various
augmentation techniques in combination with a deep CNN [1] have been proposed,
giving a strong 98% accuracy. The most famous pothole detection algorithm is
the YOLO (with its various versions like YOLOv3, YOLOv4, etc). [2], [3], [4],
[5] are some of the instances where YOLO is used to detect potholes. [2] gives
accuracy in the range of 33% to 69%, [3] gives an f1-score of 0.51, [4]
discusses results obtained with a modified YOLOv2 architecture, achieving
precisions and recalls of 0.89 and 0.87 respectively with 35 million
parameters, signifying that although YOLO is a good algorithm for pothole
detection, it largely depends on the data available and how that data is used.
In cases where it does perform very well, the number of parameters required is
huge, which tends to reduce the economy of the model. [6] proposes pothole
detection with a DenseNet121 architecture, achieving 99.3% accuracy, but
significantly increasing the number of parameters involved in the model. An
example of how sensory data can be integrated with a CNN is given by [7] but
does not give promising results with an 87.20% precision, 92.7% recall, and
89.9% F1-Score. Some works also include detecting areas of interest in the
pothole dataset proceeded by training a CNN architecture [8]. Experimental
results in this work give an overall accuracy of 90% for the computed area of
the pothole when compared with the actual measured area with ± 10% deviation.
Works that have various types of road data along with present potholes [9] and
use a CNN also have been tested with a high (96%) accuracy in detecting the
type of road but do not repeat this good performance in detecting potholes,
with an accuracy of 88%. [10] is a notable work that proposes a lightweight
model for the implementation of this task, but gives an accuracy of only
86.29%, which is a compromise on the efficiency for the purpose of reducing
the number of parameters in the network. The U-Net architecture proposed in
[11] gives a very good result with an accuracy of 97% which illustrates the
importance of deep learning networks in alleviating this task of detecting
potholes on the roads. There has also been an attempt at simulating the
detection of potholes in a virtual environment [12] which achieves stupendous
results, obtaining a (99.80 %) of Accuracy, Precision (100%), Recall (99.60%),
and F-Measure (99.60%) simultaneously. But, as these results are simulated in
a controlled environment, they cannot be emulated in real-world applications
directly. Finally, [13] gives another example of using some image processing
techniques, using the Single Shot Detector with a feed-forward neural network,
bagging a commendable 96.7% accuracy.

##  III Proposed Methodlogy

###  III-A Dataset

The dataset used in this paper has been obtained from Kaggle. It contains two
classes: normal and potholes. The normal instances have 352 images and the
pothole instances have 329 images. The paper uses 280 potholes and 280 normal
images for training and validation. The remaining 72 for normal and 49 for
potholes are used for testing.

###  III-B Data Pre-Processing

The data preprocessing involves resizing the images to a fixed shape
$(224,224,3)$ due to irregularities in the dataset. As the images are of
varied settings, taken under various conditions, the paper employs some image
processing techniques like thresholding with the help of Otsu’s method. Fig 1
showcases the output of Otsu’s thresholding on a sample image of the dataset
which contains a pothole.

Figure 1: Otsu’s Thresholding on a sample image

Otsu’s thresholding method is a popular image processing technique for
converting grayscale images to binary images by converting pixel values above
a certain threshold to white and those below it to black. Adaptive
thresholding is a type of thresholding in which the threshold value for each
pixel is adjusted based on the local intensity of its surroundings. This
technique is useful for our images with varying diversity in the roads,
objects and potholes.

###  III-C Feature Extraction

As Karen et.al [14] signifies, VGG16 is a useful model for the extraction of
features from an image and using these features to further employ in a
different model. This paper uses the VGG16 model to extract the textural
features, especially of the pothole, since a pothole is predominantly a
texture-based image. One of the reasons for resizing the input images to a
fixed size of $224x224$ is due to the fact that a VGG16 model works perfectly
with that shape. The feature extraction process occurs at the later
convolutional layers of VGG16, which capture higher-level and more abstract
features. These layers are deeper in the network and have larger receptive
fields. The earlier layers capture low-level features like edges and textures,
while the deeper layers capture high-level features like shapes and objects.
At each convolutional layer, the VGG16 model produces a set of feature maps,
also known as activation maps. These feature maps represent the activation of
each filter in the layer across the spatial dimensions of the input image.
They highlight regions in the pothole image that are particularly relevant to
the learned features. To obtain a fixed-length feature vector for the entire
image, a global average pooling operation is typically applied to the last
convolutional layer’s feature maps. This operation averages each feature map
spatially, resulting in a reduced-dimensional representation. The output of
the last global average pooling layer can be considered as the extracted
features from the VGG16 model. These features capture high-level information
about the input pothole image, which is then used for further fine-tuning the
pothole dataset.

Figure 2: Process Flow Diagram

###  III-D Proposed Deep Learning Network

This paper proposes a novel transfer learning network using a Siamese model.
Fig 2 provides the complete process flow diagram of the approach in this
paper. As described in the Feature Extraction section, the features are
extracted from the input images using the VGG16 pre-trained model, which are
in turn fed to a Siamese network. The Siamese model is an image comparison
model that takes two images as input and outputs a similarity score between
them. The Siamese network architecture is made up of two identical sub-
networks with the same weights. The input images are fed into each sub-
network, and the resulting feature vectors are compared using a distance
metric. The network is trained with triplet loss, which reduces the distance
between similar images while increasing the distance between dissimilar
images.

Two functions are defined as same-pair and diff-pairs to train our neural
network. The same-pair function creates the same pairs of images by selecting
two images at random from a batch of similar images, with no repeated pairs
allowed. The diff-pairs function generates unique image pairs by randomly
selecting two images from different batches, with no repeated pairs permitted.
The same pairs are labeled with a 1 and different pairs with a 0. The model is
fed with all possible same and different pairs which makes a combined 100000
pairs for training.

Following that, the extracted features are passed through a series of
convolutional layers that include batch normalization and max-pooling layers.
To prevent overfitting, the output of these layers is flattened and passed
through two fully connected layers with dropout regularization. To avoid
overfitting, the Siamese model is trained using triplet loss with a margin of
one and an early stopping technique. For the optimizer, the RMSprop optimizer
was employed. The model recognized similar textures in images containing
potholes with high accuracy. Based on the output given by this model, the
pothole images are classified as “positive” or “negative”. The model has also
been trained using different types of losses like the contrastive loss to get
a better insight into the results of the network. For one step of the Siamese
training, a total of three images are fed in. These images are: anchor image
(A); positive class image (P), which belongs to the same person as that of
anchor image; negative class image (N), which belongs to a person other than
that of the anchor image. The loss function as per which all the weights in
the embedding network E() are updated is given in equation (1). Fig 3
showcases how the images are fed into the proposed model.

$(Y,D)=\frac{1}{2N}\sum_{i=1}^{N}Y\cdot D^{2}+(1-Y)\cdot\max(margin-D,0)^{2}$
(1)

These images are then accordingly fed into the loss function to evaluate how
close the input images are to the anchor image. This novel proposed network
has given efficient results.

Figure 3: Input through triplet loss Figure 4: Proposed Deep learning network

These are sample images from the dataset that are fed for training into the
model.

The network is structured as follows :

  1. 1.

The first layer has two input images owing to the Siamese network.

  2. 2.

From this layer, a custom CNN is embedded with Conv2D layers and Batch
Normalization along with a MaxPooling Layer

  3. 3.

The third layer is the Flatten Layer which transposes the input matrix into a
different channel configuration. and Dropout and Dense layers.

  4. 4.

The final output is obtained as a probability as to whether the fingerprints
are verified or not. Fig. 4 shows the entire schematic of the proposed
architecture.

###  III-E Performance Metrics

This section aims to explain the metrics used for evaluating the performance
of the proposed pothole detection system in a research paper. The experimental
approach involves modifying the network while keeping the other components of
the pipeline unchanged. To assess the verification performance, well-
established metrics such as Equal Error Rate (EER), Area Under Receiver
Operating Characteristics (AUROC), Area Under Precision-Recall (AUPR),
accuracy, precision, recall, and the F1-Score are employed.

By utilizing these metrics, the experimental evaluations focus exclusively on
the network’s performance and efficiency while keeping the remaining pipeline
components unchanged. The analysis of these metrics not only quantifies the
network’s performance but also assists in selecting and optimizing the most
suitable network for the pothole detection pipeline.

In this research, careful consideration is given to the number of actual
genuine pairs and imposter pairs that can be formed within the context of the
study. The total number of actual genuine pairs is determined by multiplying
the number of training images, validation images, and people in the dataset
(referred to as $UVP$). Similarly, the number of actual imposter pairs is
calculated by subtracting the number of actual genuine pairs from the total
number of pairs, which is $UVP^{2}-UVP$. Importantly, the ground truth
information regarding whether these pairs belong to the same class or not is
known.

To determine whether a pair is considered to belong to the same class, the
detection system compares the distance between their respective features to a
predetermined threshold value denoted as $T$. By sweeping across various
values of $T$, the precision-recall curve can be constructed, providing
insights into the system’s performance across different threshold settings.

To quantify the verification performance of the pothole detection system, two
widely recognized metrics are employed: the Area Under Precision-Recall (AUPR)
curve and the Equal Error Rate (EER). AUPR measures the overall performance of
the system by calculating the area under the precision-recall curve, which
reflects the trade-off between precision and recall. On the other hand, EER
represents the point at which the False Acceptance Rate (FAR) is equal to the
False Rejection Rate (FRR), serving as a pivotal point for evaluating system
performance.

By utilizing these metrics, the detection performance of the pothole detection
system can be effectively evaluated and analyzed. EER, AUROC, AUPR, precision,
recall, and F1-Score provide valuable insights into the system’s ability to
accurately identify genuine pairs and distinguish them from imposter pairs.
These evaluation measures contribute to a comprehensive assessment of the
system’s performance and assist in identifying the optimal threshold setting
for achieving desired verification outcomes.

To summarize, this research paper employs AUPR and EER values as quantitative
metrics for evaluating the detection and classification performance of the
proposed pothole detection system, while taking into account the modifications
made to the network.

Previous networks |  Dataset |  Algorithms |  Results
---|---|---|---
Manalo et al. [2] |  Pothole dataset from Kaggle |  Transfer Learning using YOLOv2 |  95.3% average precision with 33% to 69% accuracy
Arjapure et al. [8] |  Pothole dataset from Kaggle |  Region of Interest Extraction + CNN |  92% accuracy
Pereira et al. [11] |  Custom database collected from ASUS_z01RD phone |  U-Net architecture |  97% accuracy
Ping et al. [15] |  Custom database |  Single Shot Detector + YOLO |  82% accuracy
Rastogi et al. [16] |  Specially curated database from an Indian city Bangalore |  Modified YOLOv2 |  0.87 precision and 0.89 recall
Agrawal et al. [9] |  Kaggle Pothole database |  CNN based approach |  88% accuracy in detecting potholes
Karen et al. [14] |  Custom curated database |  Faster R-CNN using VGG16 |  0.79 precision and 0.80 recall
Proposed Network |  Kaggle Pothole database |  Novel VGG16 + Siamese network transfer learning |  3.89% EER, 96.1% accuracy, 0.985 AUPR, 0.988 AUROC
TABLE I: Comparison table for previous research based on dataset, processing
algorithms, and results

Network used | Trainable Parameters | AUROC | AUPR | EER | Accuracy | Precision | Recall | F1-Score
---|---|---|---|---|---|---|---|---
VGG16 + Siamese Network with a kernel regularizer | 265,120 | 0.981 | 0.981 | 6.04% | 93.7% | 0.939 | 0.936 | 0.937
VGG16 + Siamese Network with no kernel regularizer | 265,132 | 0.987 | 0.983 | 4.29% | 95.5% | 0.957 | 0.953 | 0.955
VGG16 + Siamese Network with no Dropout and only 1st block of CNN | 361,248 | 0.989 | 0.986 | 4.52% | 95.1% | 0.954 | 0.946 | 0.95
VGG16 + Siamese Network with only 1 CNN block and Dropout in the last layer | 361,248 | 0.98 | 0.973 | 5.16% | 94.7% | 0.948 | 0.946 | 0.947
VGG16 + Siamese Network with 3 CNN blocks and Dropout in the last layer | 196,272 | 0.972 | 0.964 | 5.67% | 94.7% | 0.948 | 0.946 | 0.947
VGG16 + Siamese Network with Dropout in all layers | 161,366 | 0.974 | 0.962 | 7.5% | 92.4% | 0.925 | 0.924 | 0.924
VGG16 + Siamese Network with only Flatten and Dense Layers | 64,55,680 | 0.973 | 0.958 | 7.29% | 92.7% | 0.927 | 0.927 | 0.927
VGG16 + Siamese Network with no kernel regularizer and 2x filters | 64,55,680 | 0.983 | 0.977 | 5.91% | 94.1% | 0.941 | 0.941 | 0.941
VGG16 + Siamese Network with no kernel regularizer and 2 extra blocks | 734,146 | 0.977 | 0.965 | 5.93% | 94.08% | 0.941 | 0.941 | 0.941
Proposed Network (RoadScan) | 372,448 | 0.988 | 0.985 | 3.89% | 96.12% | 0.961 | 0.961 | 0.961

TABLE II: Economy and Efficiency Analysis with Performance metrics

###  III-F Experimental Results and Tables

The proposed model was evaluated against previous state-of-the-art works and
it outperformed all the networks and results in terms of the performance
metrics. The proposed network performs with an unmatched 3.89% EER, 96.12%
testing accuracy, 0.985 AUPR and 0.988 AUROC values. The network proposed by
the paper, which includes a transfer learning technique of extracting features
from images using VGG16 and feeding those features to a Siamese network is a
novel architecture that gives a competitive performance surpassing state of
the art. Table I demonstrates the proposed network put up against previously
researched works and their performances. The model proposed in the paper
outweighs the other comparative networks by huge margins. Fig 5 provides a
graphical representation of the obtained results.

Figure 5: AUROC, AUPR and EER of the proposed network

###  III-G Economy and Efficiency Analysis of the System

This paper proposes not only an efficiently performing network but also a
state-of-the-art economic network. The proposed network is economical in the
sense that it contains a very less number of trainable parameters and still
provides excellent efficiency. Table II provides a detailed analysis of how
the network is tested in various conditions and how the pruning of the network
leads to lesser parameters and better performance, ultimately leading to an
optimal network that is the most efficient while maintaining a good economy.
The paper demonstrates different networks tried and tested to maintain a
robust system for pothole detection. 7 different performance metrics have been
obtained, which is not done by any of the cited works. All the values provide
evidence of the superior performance of the network. The proposed and optimal
approach gives a testing accuracy of 96.12%. The analysis demonstrates the
optimal nature of the network that is introduced in the paper. Not only does
it give as less as 372,448 parameters, it gives top-notch and efficacious
results to solidify its stature as a robust state-of-the-art system. In
contrast, heavy networks like YOLOv2 as seen from [2], [4] and [5] have lakhs
of parameters and still fail to provide a better result than the proposed
network. Networks that do perform very well in terms of performance metrics
like accuracy and precision [11] do not take into account the economy of the
model. In this paper, it has been experimentally demonstrated that the
proposed model not only maintains its economy by drastically reducing the
number of trainable parameters but also doesn’t compromise its efficiency in
performance.

##  IV Conclusion

In this paper, a novel network using a transfer learning approach: VGG16 to
extract features and a Siamese network with triplet loss, which we call
RoadScan, has been proposed. The proposed network performs with an unmatched
3.89% EER and 96.12% accuracy on the dataset used. The proposed model is
computationally efficient as the results obtained are comparable and better
than the state-of-the-art with the added advantage that it is economical as it
only utilizes much lesser parameters, and lesser data to train upon. Through
our experiments, the capability of our network to learn a custom generalized,
lightweight and economical model for pothole images is showcased. Using this
approach, our method has been able to successfully able to classify potholes
and normal roads and the obtained detection performance is better than that of
non-processed images and simple CNNs. As a part of future work, we will
explore the utility of other types of networks and other road obstacles such
as speedbumps.

## References

  * [1] Priyanka Gupta and Manish Dixit  “Image-based Road Pothole Detection using Deep Learning Model”  In _2022 14th International Conference on Computational Intelligence and Communication Networks (CICN)_ , 2022, pp. 59–64  IEEE
  * [2] Jhon Michael C Manalo et al.  “A Transfer Learning-Based System of Pothole Detection in Roads through Deep Convolutional Neural Networks”  In _2022 International Conference on Decision Aid Sciences and Applications (DASA)_ , 2022, pp. 1469–1473  IEEE
  * [3] Madarapu Sathvik, G Saranya and S Karpagaselvi  “An Intelligent Convolutional Neural Network based Potholes Detection using Yolo-V7”  In _2022 International Conference on Automation, Computing and Renewable Systems (ICACRS)_ , 2022, pp. 813–819  IEEE
  * [4] R Kavitha and S Nivetha  “Pothole and object detection for an autonomous vehicle using yolo”  In _2021 5th International Conference on Intelligent Computing and Control Systems (ICICCS)_ , 2021, pp. 1585–1589  IEEE
  * [5] Pranjal A Chitale et al.  “Pothole detection and dimension estimation system using deep learning (yolo) and image processing”  In _2020 35th International Conference on Image and Vision Computing New Zealand (IVCNZ)_ , 2020, pp. 1–6  IEEE
  * [6] Necip Çınar and Mehmet Kaya  “An automated pothole detection via transfer learning”  In _2022 International Conference on Decision Aid Sciences and Applications (DASA)_ , 2022, pp. 1355–1358  IEEE
  * [7] Anup Kumar Pandey et al.  “Deep Neural Networks Based Approach for Pothole Detection”  In _2021 4th International Conference on Signal Processing and Information Security (ICSPIS)_ , 2021, pp. 1–4  IEEE
  * [8] Surekha Arjapure and DR Kalbande  “Deep learning model for pothole detection and area computation”  In _2021 International Conference on Communication information and Computing Technology (ICCICT)_ , 2021, pp. 1–6  IEEE
  * [9] Ravi Agrawal, Yash Chhadva, Sanjana Addagarla and Sheetal Chaudhari  “Road surface classification and subsequent pothole detection using deep learning”  In _2021 2nd International Conference for Emerging Technology (INCET)_ , 2021, pp. 1–6  IEEE
  * [10] Joyanta Jyoti Mondal, Md Farhadul Islam, Sarah Zabeen and Meem Arafat Manab  “InvoPotNet: Detecting Pothole from Images through Leveraging Lightweight Involutional Neural Network”  In _2022 25th International Conference on Computer and Information Technology (ICCIT)_ , 2022, pp. 599–604  IEEE
  * [11] Vosco Pereira, Satoshi Tamura, Satoru Hayamizu and Hidekazu Fukai  “Semantic segmentation of paved road and pothole image using U-net architecture”  In _2019 International Conference of Advanced Informatics: Concepts, Theory and Applications (ICAICTA)_ , 2019, pp. 1–4  IEEE
  * [12] Jung-Cheng Tsai et al.  “Learning pothole detection in virtual environment”  In _2020 International Automatic Control Conference (CACS)_ , 2020, pp. 1–5  IEEE
  * [13] Shebin Silvister et al.  “Deep learning approach to detect potholes in real-time using smartphone”  In _2019 IEEE pune section international conference (PuneCon)_ , 2019, pp. 1–4  IEEE
  * [14] Karen Simonyan and Andrew Zisserman  “Very deep convolutional networks for large-scale image recognition”  In _arXiv preprint arXiv:1409.1556_ , 2014
  * [15] Ping Ping, Xiaohui Yang and Zeyu Gao  “A deep learning approach for street pothole detection”  In _2020 IEEE Sixth International Conference on Big Data Computing Service and Applications (BigDataService)_ , 2020, pp. 198–204  IEEE
  * [16] Roopak Rastogi et al.  “A comparative evaluation of the deep learning algorithms for pothole detection”  In _2020 IEEE 17th India Council International Conference (INDICON)_ , 2020, pp. 1–6  IEEE


# Note on a Lyapunov-type inequality for a fractional boundary value problem
with Caputo-Fabrizio derivative

Zaid Laadjal

Department of Mathematics, ICOSI Laboratory,
University of Abbes Laghrour, Khenchela, 40000, Algeria.
E-mail<EMAIL_ADDRESS>
###### Abstract

In this short note, we present a Lyapunov-type inequality that corrects the
recently obtained result in [M. Kirane, B. T. Torebek: A Lyapunov-type
inequality for a fractional boundary value problem with Caputo-Fabrizio
derivative, J. Math. Inequal. 12, 4 (2018), 1005–1012].

Keywords: Caputo-Fabrizio derivative, Lyapunov-type inequality, boundary value
problem.

MSC (2010): 34A08, 34A40, 26A33.

##  1 Introduction

Recently, in [1] the authors discussed a Lyapunov-type inequality for the
following linear fractional boundary value problem:

$\left\\{\begin{array}[]{cc}{}^{CF}D_{a}^{\alpha}u(t)+q(t)u(t)=0,&0\leq
a<t<b,\\\ u(a)=u(b)=0,&\end{array}\right.$ (1)

where ${}^{CF}D_{a}^{\alpha}$ denotes the Caputo-Fabrizio derivative [2, 3] of
order $\alpha,(1<\alpha\leq 2),$ $q:[a,b]\rightarrow\mathbb{R}$ is a
continuous function. And they included the following result:

######  Theorem 1 ([1])

If the fractional boundary value problem (1) has a nontrivial solution, then

$\int_{a}^{b}|q(t)|ds>\frac{4(\alpha-1)(b-a)}{[(\alpha-1)(b-a)-2+\alpha]^{2}}.$
(2)

We have noticed that, the denominator in the inequality (2) is equal zero when
$b-a=\frac{2-\alpha}{\alpha-1}.$ So a mistake has been occured during the
previous result and some other results (Corrolary 3.4 and Corrolary 3.5 in
[1]) are also incorrect. These mistakes come from the main wrong in (Lemma 3.2
in [1] related to the calculations for the maximum value of the Green’s
function of the problem (1).

This work aims to show these mistakes and present the correct version of them.

We will also refer the interested reader in studying the Lyapunop-type
inequalities for a fractional boundary value problems to the works compiled in
chapter in the book [4], as well as some other papers published recently, for
example see [5]–[10] and the references cited therein.

##  2 Main results

The fractional boundary value problem (1) is equivalent to the integral
equation

$u(s)=\int_{a}^{b}G(t,s)q(s)u(s)ds,$ (3)

where $G(t,s)$ is called the Green’s function of the problem (1) and it’s
difened by

$G(t,s)=\left\\{\begin{array}[]{lll}g_{1}(t,s)=\frac{b-t}{b-a}[(\alpha-1)(s-a)-2+\alpha],&&a\leq
s\leq t\leq b,\\\ &&\\\
g_{2}(t,s)=\frac{t-a}{b-a}[(\alpha-1)(b-s)+2-\alpha],&&a\leq t\leq s\leq
b.\end{array}\right.$ (4)

See [1] for more details, (note that this function in (Lemma 3.1, [1]) is
written in wrong way. Although its proof is true).

The mistake alluded to in ([1], Section 3) is that the authors concluded that
the maximum value of the function $G(t,s)$ is obtained at the point

$t=s=\frac{1}{2}\left(b+a+\frac{2-\alpha}{\alpha-1}\right):=s^{\ast},$ (5)

where $s^{\ast}$ is defined by (equality (3.5) in [1]). However, this is wrong
for $(t,s)\in[a,b]\times[a,b]$ with $1<\alpha\leq 2$, as we will show nextly.

Let us start to discuss the previous value of $s^{\ast}$.
Note that, if $b-a<\frac{2-\alpha}{\alpha-1},$ we have

$\displaystyle\begin{array}[]{lll}b-a<\frac{2-\alpha}{\alpha-1}&\Longleftrightarrow&2b<b+a+\frac{2-\alpha}{\alpha-1}\end{array}$
$\displaystyle\begin{array}[]{lll}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
}&\Longleftrightarrow&b<\frac{1}{2}\left(b+a+\frac{2-\alpha}{\alpha-1}\right)\end{array}$
$\displaystyle\begin{array}[]{lll}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
}&\Longleftrightarrow&b<s^{\ast},\end{array}$

thus $s^{\ast}\notin[a,b].$ Then the maximum value of the function $G(t,s)$ is
not at $s^{\ast}$ when $b-a<$ $\frac{2-\alpha}{\alpha-1}.$
Now, for $a\leq s\leq t\leq b,$ with $b-a<\frac{2-\alpha}{\alpha-1},$ we have

$\displaystyle\begin{array}[]{ccc}b-a<\frac{2-\alpha}{\alpha-1}&\Longleftrightarrow&b<a+\frac{2-\alpha}{\alpha-1}\end{array}$
(10) $\displaystyle\begin{array}[]{ccc}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ }&\Longrightarrow&s<a+\frac{2-\alpha}{\alpha-1}\end{array}$ (12)
$\displaystyle\begin{array}[]{ccc}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
}&\Longrightarrow&(\alpha-1)(s-a)-2+\alpha<0,\end{array}$ (14)

on other hand, we have

$\frac{b-t}{b-a}\leq\frac{b-s}{b-a}.$ (15)

By the inequalities (14) and (15) we get

$g_{1}(t,s)\geq\frac{b-s}{b-a}[(\alpha-1)(s-a)-2+\alpha],\text{ \ }a\leq s\leq
b<a+\frac{2-\alpha}{\alpha-1}.$ (16)

Observe that the inequality (16) is contrary to (the inequality (3.4) in [1]).

###### Remark 2

Note that on a general interval $[a,b],0\leq a<b,$ we have

$h_{1}(s)\leq g_{1}(t,s)\leq 0,\text{ for\ }a\leq s\leq t\leq
a+\frac{2-\alpha}{\alpha-1},$ (17)

where the function $h_{1}$ is defined by

$h_{1}(s)=g_{1}(s,s)=\frac{b-s}{b-a}[(\alpha-1)(s-a)-2+\alpha],\text{ \
}s\in[a,b]$ (18)

We differentiate the function $h_{1}(s)$ to get

$h_{1}^{\prime}(s)=-\frac{2(\alpha-1)}{b-a}s+\frac{(\alpha-1)\left(b+a\right)+2-\alpha}{b-a}.$
(19)

We have $s^{\ast}$ is the unique solution of the equation
$h_{1}^{\prime}(s)=0,$ where $s^{\ast}$ is given by (5) but the value of
$s^{\ast}$ in some cases does not belong to the interval $[a,b]$ as we have
shown previously.

By the discussion above, we can conclude that the maximum value of the
function $G(t,s)$ lays in the following two cases:

Case 1. $b-a<$ $\frac{2-\alpha}{\alpha-1}.$

Because $b-a<$ $\frac{2-\alpha}{\alpha-1},$ so then $s\leq b<s^{\ast}$ (here
$a+\frac{2-\alpha}{\alpha-1},s^{\ast}\notin[a,b])$, we obtain
$h_{1}^{\prime}(s)\geq$ $0$ and $h_{1}(s)\leq 0$ for $s\leq
b<a+\frac{2-\alpha}{\alpha-1}.$ So by (17) and the contunuity of the function
$h_{1}$ we conclude that

$\displaystyle\max_{a\leq s\leq t\leq
b<a+\frac{2-\alpha}{\alpha-1}}\left|g_{1}(t,s)\right|$ $\displaystyle=$
$\displaystyle\max_{a\leq s\leq
b<a+\frac{2-\alpha}{\alpha-1}}\left|h_{1}(s)\right|$ (20) $\displaystyle=$
$\displaystyle-h_{1}(a)$ $\displaystyle=$ $\displaystyle 2-\alpha.$

Next, for $a\leq t\leq s\leq b.$ Obviously,

$0\leq g_{2}(t,s)\leq\frac{s-a}{b-a}[(\alpha-1)(b-s)+2-\alpha].\text{ }$ (21)

We define a function $h_{2}$ by

$h_{2}(s)=g_{2}(s,s)=\frac{s-a}{b-a}[(\alpha-1)(b-s)+2-\alpha],\ a\leq s\leq
b.$ (22)

Differentiating the function $h_{2}(s)$

$h_{2}^{\prime}(s)=-\frac{2(\alpha-1)}{b-a}s+\frac{(\alpha-1)\left(a+b\right)+\left(2-\alpha\right)}{b-a},$
(23)

which implies that the function $h_{2}^{\prime}(s)$ has a unique zero, at the
point $s^{\ast},$ but $s^{\ast}>b$ (i.e. $s^{\ast}\notin[a,b]$). Because
$h_{2}^{\prime}(s)>0$ for all $s\in[a,b]$ and $g_{2}(t,s)\geq 0$ for $a\leq
t\leq s\leq b,$ and $h_{2}(s)$ is continuous function, then

$\displaystyle\underset{a\leq t\leq s\leq
b<a+\frac{2-\alpha}{\alpha-1}}{\max}\left|g_{2}(t,s)\right|$ $\displaystyle=$
$\displaystyle\max_{a\leq s\leq b<a+\frac{2-\alpha}{\alpha-1}}h_{2}(s)$ (24)
$\displaystyle=$ $\displaystyle h_{2}(b)$ $\displaystyle=$ $\displaystyle
2-\alpha.$

By (24) and (20) we get

$\underset{a\leq t,s\leq
b<a+\frac{2-\alpha}{\alpha-1}}{\max}\left|G(t,s)\right|=2-\alpha.$ (25)

Case 2. $b-a\geq$ $\frac{2-\alpha}{\alpha-1}.$

From the inequality $b-a\geq$ $\frac{2-\alpha}{\alpha-1}$ we get
$a+\frac{2-\alpha}{\alpha-1}\in[a,b]$ and
$s^{\ast}\in[a+\frac{2-\alpha}{\alpha-1},b],$ we obtain

$\left\\{\begin{array}[]{l}0\leq g_{1}(t,s)\leq h_{1}(s),\text{ \
}a+\frac{2-\alpha}{\alpha-1}\leq s\leq b.\\\ \\\ h_{1}(s)\leq g_{1}(t,s)\leq
0,\text{ \ }a\leq s\leq
a+\frac{2-\alpha}{\alpha-1}<s^{\ast}.\end{array}\right.$ (26)

Because $h_{1}(s)$ is continuous function, and
$h_{1}(a+\frac{2-\alpha}{\alpha-1})=h_{1}(b)=0,$ and $h_{1}(a)=-(2-\alpha),$
we conclude that

$\displaystyle\max_{a\leq s\leq t\leq a+\frac{2-\alpha}{\alpha-1}\leq
b}\left|g_{1}(t,s)\right|$ $\displaystyle=$ $\displaystyle\max_{a\leq s\leq
a+\frac{2-\alpha}{\alpha-1}\leq b}\left|h_{1}(s)\right|$ (27) $\displaystyle=$
$\displaystyle\max\left\\{-h_{1}(a),h_{1}(s^{\ast})\right\\}$ $\displaystyle=$
$\displaystyle\max\left\\{2-\alpha,\frac{[(\alpha-1)(b-a)-2+\alpha]^{2}}{4(\alpha-1)(b-a)}\right\\}.$

On other hand, we have

$\displaystyle\underset{a\leq t\leq s\leq a+\frac{2-\alpha}{\alpha-1}\leq
b}{\max}\left|g_{2}(t,s)\right|$ $\displaystyle=$ $\displaystyle\max_{a\leq
s\leq a+\frac{2-\alpha}{\alpha-1}\leq b}h_{2}(s)$ (28) $\displaystyle=$
$\displaystyle h_{2}(s^{\ast})$ $\displaystyle=$
$\displaystyle\frac{\left[(\alpha-1)\left(b-a\right)+\left(2-\alpha\right)\right]^{2}}{4(\alpha-1)\left(b-a\right)}.$

By (27) and (28) we get

$\displaystyle\max_{a\leq t,s\leq a+\frac{2-\alpha}{\alpha-1}\leq
b}\left|G(t,s)\right|$ $\displaystyle=$
$\displaystyle\max\left\\{2-\alpha,\frac{[(\alpha-1)(b-a)-\left(2-\alpha\right)]^{2}}{4(\alpha-1)(b-a)},\right.$
$\displaystyle\text{\ \ \ \ \ \ \ \ \ \ \ \
}\left.\frac{[(\alpha-1)(b-a)+\left(2-\alpha\right)]^{2}}{4(\alpha-1)(b-a)}\right\\}$
$\displaystyle=$
$\displaystyle\max\left\\{2-\alpha,\frac{[(\alpha-1)(b-a)+\left(2-\alpha\right)]^{2}}{4(\alpha-1)(b-a)}\right\\},$

using the inequality $\frac{(A+B)^{2}}{4}\geq AB,$ with $A=(\alpha-1)(b-a)$
and $B=\left(2-\alpha\right),$ we obtain

$\max_{a\leq t,s\leq a+\frac{2-\alpha}{\alpha-1}\leq
b}\left|G(t,s)\right|=\frac{[(\alpha-1)(b-a)+\left(2-\alpha\right)]^{2}}{4(\alpha-1)(b-a)}.$
(29)

Thus we conclude the follwing result.

###### Proposition 3

The Green’s function $G$ defined by (4), has the following properties:
$i\mathbf{).}$ If $b-a<$ $\frac{2-\alpha}{\alpha-1},$ then

$\underset{(t,s)\in[a,b]\times[a,b]}{\max}\left|G(t,s)\right|=2-\alpha,$ (30)

$ii).$ If $b-a\geq$ $\frac{2-\alpha}{\alpha-1},$ then

$\max_{(t,s)\in[a,b]\times[a,b]}\left|G(t,s)\right|=\frac{[(\alpha-1)(b-a)+\left(2-\alpha\right)]^{2}}{4(\alpha-1)(b-a)}.$
(31)

Hence we have the following Lyapunov-type inequality.

###### Theorem 4

If the fractional boundary value problem (1) has a nontrivial solution. Then

$\int_{a}^{b}|q(t)|ds\geq\left\\{\begin{array}[]{l}\frac{1}{2-\alpha.},\text{
\ if \ }b-a<\frac{2-\alpha}{\alpha-1},\\\ \\\
\frac{4(\alpha-1)(b-a)}{[(\alpha-1)(b-a)+\left(2-\alpha\right)]^{2}},\text{ \
if \ }b-a\geq\frac{2-\alpha}{\alpha-1}.\end{array}\right.$ (32)

Proof. Since the proof is well-known so that the reader can easily check it
on, where it’s used in [1] but in here we get into details in two cases
related the properties (30) and (31).

By using Theorem 4, the reader can smoothly correct (Corrolary 3.4 and
Corrolary 3.5 in [1]), but should seperate each of them in two cases $b-a<$
$\frac{2-\alpha}{\alpha-1},$ and $b-a\geq$ $\frac{2-\alpha}{\alpha-1}$.

## References

  * [1] M. Kirane and B. T. Torebek, A Lyapunov-type inequality for a fractional boundary value problem with Caputo-Fabrizio derivative, J. Math. Inequal. 12, 4 (2018), 1005–1012.
  * [2] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular Kernel, Progr. Fract. Differ. Appl. 1, 2 (2015), 1–13.
  * [3] J. Losada and J. J. Nieto, Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1, 2 (2015), 87–92.
  * [4] K. S. Ntouyas, B. Ahmad, T. P. Horikis,  Recent Developments of Lyapunov-Type Inequalities for Fractional Differential Equations. In: D. Andrica, T. Rassias (eds), Differential and Integral Inequalities, Springer Optimization and Its Applications, vol. 151, Springer, Cham, (2019).
  * [5] M. Jleli, M. Kirane, B Samet,  Lyapunov-type inequalities for a fractional p-Laplacian system, Fract. Calc. Appl. Anal. 20, 6 (2017), 1485–1506.
  * [6] M. Jleli, B. Samet, Y. Zhou,  Lyapunov-type inequalities for nonlinear fractional differential equations and systems involving Caputo-type fractional derivatives, J. Inequal. Appl. 2019, 19, (2019), 15 pages.
  * [7] Y. Wang, Q. Wang, Lyapunov-type inequalities for nonlinear fractional differential equation with Hilfer fractional derivative under multi-point boundary conditions, Fract. Calc. Appl. Anal. 21, 3 (2018), 833–843.
  * [8] Q. Ma, C. Ma and J. Wang, A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequal. 11, 1 (2017), 135–141.
  * [9] Z. Laadjal, N. Adjeroud, Q. Ma,  Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a,b], J. Math. Inequal. 13, 3 (2019), 789–799.
  * [10] M. Jleli, M. Kirane, B. Samet, On Lyapunov-type inequalities for a certain class of partial differential equations, Appl. Anal. 99, 1, (2020), 40-49.


0}}{B_{0}\bar{\iota}_{0}}\right)^{2}\right]-\frac{3}{2}|\eta|\frac{(1-\bar{\alpha})+(\bar{\alpha}+1)\bar{F}}{(\bar{\alpha}+1)\bar{F}-(3+\bar{\alpha})}\delta+\dots,$
(74a)
$B_{22}^{C}=-\frac{2\mu_{0}p_{2}}{B_{0}^{2}}\frac{\sqrt{\bar{\alpha}}}{(\bar{\alpha}+1)\bar{F}-(\bar{\alpha}+3)}\left(\frac{\eta
B_{\bar{\alpha}0}}{B_{0}\bar{\iota}_{0}}\right)^{2}-\frac{|\eta|}{2}\frac{(3-\bar{\alpha})+(\bar{\alpha}+1)\bar{F}}{(3+\bar{\alpha})-(\bar{\alpha}+1)\bar{F}}\delta+\dots,$
(74b)

where $\bar{\alpha}=(\eta/\kappa(0))^{4}$ and the dots denote terms
independent of second order choices that we shall not focus on.

With these expressions in place, we may then rewrite the effects of second-
order on the trapped-particle precession (noting that we assumed $\eta>0$ in
the adiabatic invariant calculation in this paper),

$\omega_{\alpha,1}=\frac{H\eta}{B_{0}}\left[\eta\tilde{\mathcal{G}}-\frac{\mu_{0}p_{2}}{\eta
B_{0}^{2}}\mathcal{G}_{p2}+\frac{\delta}{2}\mathcal{G}_{\delta}\right],$ (75)

and write,

$\mathcal{G}_{p_{2}}=\mathcal{G}_{20}(k)+\left(\frac{\eta B_{\alpha
0}}{B_{0}\bar{\iota}_{0}}\right)^{2}\frac{2\sqrt{\bar{\alpha}}}{(\bar{\alpha}+3)-(\bar{\alpha}+1)\bar{F}}[2\mathcal{G}_{20}(k)-\mathcal{G}_{22}^{C}(k)]$
(76a)
$\mathcal{G}_{\delta}=\frac{3(1-\bar{\alpha})\mathcal{G}_{20}(k)-(3-\bar{\alpha})\mathcal{G}_{22}^{C}(k)}{(3+\bar{\alpha})-(\bar{\alpha}+1)\bar{F}}+\frac{(\bar{\alpha}+1)[3\mathcal{G}_{20}(k)-\mathcal{G}_{22}^{C}(k)]\bar{F}}{(3+\bar{\alpha})-(\bar{\alpha}+1)\bar{F}}.$
(76b)

These two expressions describe the effect of the pressure gradient and the
cross-section triangularity on the trapped-particle precession as a function
of the class of trapped particle. Note that $\tilde{\mathcal{G}}$ is
independent of the pressure gradient and triangularity both. As such, when
investigating dependencies on these parameters (at fixed $\eta$ and axis), we
may safely ignore contributions of $\tilde{\mathcal{G}}$. The derivation may
be checked with computer algebra, as provided in the repository associated to
this paper.

### Relation to the large-aspect ratio circular tokamak limit

Here we relate the found results to the large-aspect ratio tokamak limit
investigated by Connor et al. (1983). Let us start comparing the pressure
dependence of precession, and as such let us define
$\alpha_{p}=-4\mu_{0}rp_{2}/|\eta|B_{0}^{2}\bar{\iota}_{0}^{2}$. One may then
write

$\displaystyle-\frac{\mu_{0}rp_{2}}{|\eta|B_{0}^{2}}\mathcal{G}_{p2}$
$\displaystyle=\frac{\alpha_{p}\bar{\iota}_{0}^{2}}{4}\mathcal{G}_{p2}$ (77)
$\displaystyle=-\frac{\alpha_{p}\bar{\iota}_{0}^{2}}{2}+\frac{\alpha_{p}}{2}\left(\frac{\eta
B_{\alpha
0}}{B_{0}}\right)^{2}\frac{4\sqrt{\bar{\alpha}}}{(\bar{\alpha}+3)-(\bar{\alpha}+1)\bar{F}}\left(-1-\frac{\mathcal{G}_{22}^{C}}{4}\right).$

Specialising to the circular tokamak case (i.e. $\bar{\alpha}=1$ and
$\bar{F}=0$), we find

$-\frac{\mu_{0}rp_{2}}{|\eta|B_{0}^{2}}\mathcal{G}_{p2}=-\frac{\alpha_{p}\bar{\iota}_{0}^{2}}{2}-2\alpha_{p}\cdot\underbrace{\frac{1}{4}\left[\frac{E(k)}{K(k)}\left(2k^{2}-\frac{E(k)}{K(k)}\right)+\frac{3}{2}-k^{2}\right]}_{\stackrel{{\scriptstyle\cdot}}{{=}}G_{\alpha,p}(k)}.$
(78)

In the circular limit then, we may write,

$\omega_{\alpha,\mathrm{circle}}=\frac{H\eta}{B_{0}r}\left(G(k)-\frac{\alpha_{p}\bar{\iota}_{0}^{2}}{2}-2\alpha_{p}G_{\alpha,p}(k)\right),$
(79)

acknowledging that we are mixing terms of different order in $r$, and not
keeping all relevant terms consistently to the right order.

This expression has a similar form to Eq. (9) in Connor et al. (1983). The
first two terms are exactly identical, while the latter is different from,

$G_{\alpha,\mathrm{Connor}}(k)=\frac{2}{3}\left(\frac{E(k)}{K(k)}\left(2k^{2}-1\right)+1-k^{2}\right).$
(80)

We thus see that, though the functional form is similar, it is not the same,
especially near deeply trapped particles. These differences correspond to the
difference between the models considered and the meaning of changing a certain
parameter thereof. In the approach of Connor, the field is being constructed
in a way that locally, at finite radius, equilibrium is satisfied (as
explicitly shown in Roach et al. (1995)). Such a method requires the shape of
the flux surface and radial variations of various geometric and equilibrium
quantities to fully specify the local equilibrium conditions. Furthermore,
many of these parameters are treated independently, and one requires
subsidiary assumptions in order to set these values (in particular, the
‘artifice’ (Connor et al., 1983) of ordering the pressure in a particular
form). Though this does allow one to investigate the effect of such parameters
independently, it is not guaranteed that there exists a global solution
adhering to the set of local conditions. The near-axis approach presented,
although asymptotic in nature, is valid in some region near the axis, and as
such can perhaps be thought of a ‘more global’ solution. This translates into
a larger degree of coupling between the geometry and the magnitude of the
field (due to the singular nature of the axis) compared to the radially local
approach, which ends up reducing the number of free parameters while it
retains realism.

The difference is especially noticeable in the involvement of the magnetic
shear, which appears explicitly in Connor et al. (1983), but becomes a higher
order effect in the current treatment. In fact, we may formally obtain a sense
of the involvement of the magnetic shear by considering the next order in the
expansion for the precession, and focusing on the variation of the field line
length due to the change in $\iota$ when taking the derivative of
$\mathcal{J}_{\parallel}$ with respect to $\psi$, it can straightforwardly be
shown to give,

$\displaystyle\omega_{\alpha,\mathrm{shear}}=$
$\displaystyle-4\frac{H\eta}{B_{0}r}\frac{r\partial_{r}\bar{\iota}}{\bar{\iota}_{0}}\left(\frac{E(k)}{K(k)}+(k^{2}-1)\right)$
(81) $\displaystyle\stackrel{{\scriptstyle\cdot}}{{=}}$ $\displaystyle
4s\frac{H\eta}{B_{0}r}\underbrace{\left(\frac{E(k)}{K(k)}+(k^{2}-1)\right)}_{\stackrel{{\scriptstyle\cdot}}{{=}}G_{s}(k)},$

where we have defined the magnetic shear as
$s=-r\partial_{r}\bar{\iota}/\bar{\iota}_{0}$, and the term is clearly a
second order effect. This term has precisely the same form as in (Connor et
al., 1983). To arrive at such term we considered the magnetic shear separate
from other elements in the model (an independence that is partially correct
given the freedom in the toroidal current profile), but hides connections to
higher order considerations within the near-axis framework (Rodríguez et al.,
2022), especially its ties to the third order harmonics of $|\mathbf{B}|$,
which we would expect to modify this dependence at least partially.

##  Appendix E Critical value estimates for plasma $\beta$ and triangularity

In the main text we presented some estimates for the magnitude of plasma
$\beta$ and cross-section triangularity that made their effect on precession
of deeply trapped particles and available energy significant. In this Appendix
we present the values for these estimates for the sample near-axis QS
configurations in Fig. 3.

Table 2 includes the following parameters. First, the critical plasma $\beta$
for reversal of the deep-particle precession at the edge of the configuration,
Eq. (31),

$\beta_{\mathrm{crit}}=a|\eta|\frac{2}{\mathcal{G}_{p_{2}}(0)},$ (82)

with $a$ the minor axis, and taking $r=a$. The second is the critical beta but
for the available energy (see Appendix F and Sec. 4),

$\beta_{\mathrm{crit}}^{\mathrm{AE}}=-a|\eta|\frac{2}{\mathcal{R}_{20}(\mathcal{G}_{p_{2}}(0)+1)},$
(83)

where the denominator is given in Eq. (45).

For the critical triangularity, the value of $(r\delta)$ to revert the
precession of deeply trapped particles, we consider the definition in Eq.
(33),

$(r\delta)_{\mathrm{crit}}=\frac{2}{\mathcal{G}_{\delta}(0)}.$ (84)

Similarly, for the available energy, we shall use the same expression but for
the reinterpretation of $\mathcal{G}_{\delta}$ with

$\mathcal{G}_{\delta}^{\mathrm{AE}}=3\mathcal{R}_{20}\frac{(1-\bar{\alpha})+(1+\bar{\alpha})\bar{F}}{(3+\bar{\alpha})-(\bar{\alpha}+1)\bar{F}},$
(85)

which follows directly from Eq. (45). Finally, for a reference, we shall
compare this critical triangularity to $r_{c}\delta$, the maximum attainable
triangularity in a given near-axis configuration without incurring in
unphysical flux surface shapes.

| PQA | PQAw | PQH | PQHw | 22QA | 22QH2 | 22QH3v | 22QH3b | 22QH4l | 22QH4w
---|---|---|---|---|---|---|---|---|---|---
$\beta_{\mathrm{crit}}$ | 0.06 | 0.05 | 0.14 | 0.12 | 0.05 | 0.06 | 0.10 | 0.11 | 0.11 | 0.11
$\beta_{\mathrm{crit}}^{\mathrm{AE}}$ | 0.02 | 0.02 | 0.07 | 0.06 | 0.02 | 0.02 | 0.04 | 0.05 | 0.05 | 0.06
$(r\delta)_{\mathrm{crit}}$ | 0.96 | 1.09 | 1.47 | 1.42 | 1.05 | 0.99 | 1.22 | 1.32 | 1.40 | 1.36
$(r\delta)_{\mathrm{crit}}^{\mathrm{AE}}$ | 0.47 | 0.61 | 1.43 | 1.25 | 0.57 | 0.50 | 0.81 | 1.00 | 1.19 | 1.09
$r_{c}\delta$ | 2.02 | 3.46 | 1.60 | 2.23 | 3.64 | 0.09 | 1.08 | -0.29 | 1.18 | 5.13
$\Delta_{\mathrm{QS}}$ | 0.06 | 0.20 | 0.34 | 0.99 | 0.18 | 0.51 | 0.001 | 0.16 | 0.09 | 0.0003

| 22QH4m | 22QH7 | ARIESCS | GAR | HSX* | QHS48
---|---|---|---|---|---|---
$\beta_{\mathrm{crit}}$ | 0.16 | 0.14 | 0.06 | 0.05 | 0.15 | 0.11
$\beta_{\mathrm{crit}}^{\mathrm{AE}}$ | 0.06 | 0.05 | 0.03 | 0.02 | 0.07 | 0.06
$(r\delta)_{\mathrm{crit}}$ | 0.78 | 1.44 | 1.02 | 0.87 | 1.40 | 1.51
$(r\delta)_{\mathrm{crit}}^{\mathrm{AE}}$ | 0.33 | 1.30 | 0.53 | 0.40 | 1.19 | 1.57
$r_{c}\delta$ | -0.91 | 0.13* | 0.47 | 1.90 | 0.04 | 0.48
$\Delta_{\mathrm{QS}}$ | 0.02 | 0.0005 | 0.70 | 0.53 | 8.05 | 0.05

Table 2: Critical plasma $\beta$ and triangularity in QS configurations. The
table gathers critical plasma $\beta$, triangularity $r\delta$ and QS measure
$\Delta_{\mathrm{QS}}$ values for a number of near-axis QS configurations,
configurations used in Fig. 3. The short names for the configurations follow
straightforwardly from their full labels in Fig. 3. The starred triangularity
corresponds to the triangularity for an aspect ratio 10 consideration (as
there is difficulty in computing $r_{c}$). An aspect ratio of 10 was assumed
to evaluate the $\beta_{\mathrm{crit}}$ values. One should take the case of
HSX sceptically, as the near-axis description is far from being quasisymmetric
(a clearly indicated by $\Delta_{\mathrm{QS}}$; in fact, it also has a very
small $r_{c}$, hence the small value of attainable triangularity). One could
attempt modifying the near-axis configuration to make it more quasisymmetric
and a better description, but we shall not do this here.

Finally, to include a sense of the QS quality, we introduce the QS measure
motivated in Appendix B, Eq. (67),

$\Delta_{\mathrm{QS}}=\left|\frac{B_{20,\mathrm{max}}^{\prime}}{\bar{\iota}_{0}}\right|,$
(86)

where a vanishing value indicates an exactly QS configuration to 2nd order.
This measure shows that some near-axis configurations (especially that of HSX)
lie far from ideal QS. In the special case of HSX, the near-axis model is
constructed to reproduce the leading harmonic content of $|\mathbf{B}|$; small
variations can however lead to significant effects especially at second order
in the near-axis expansion (that is, where we asssess QS or compute
triangularity). Because we are using these as illustrating examples, we do
however not consider a further refinement of these configurations.

The values in Table 2 show that physically relevant finite beta effects can
have significant effect on the leading order behaviour of both deeply trapped
particle precession and Æ. This effect is stronger in quasi-axisymmetric
configurations, and we should remind ourselves, on the outermost surface. As
the magnetic axis is approached, the critical $\beta$ needed will grow.
Interestingly, Æ is more susceptible than the precession itself. The case of
triangularity shows that a significant degree of shaping is required to affect
either precession or Æ. This level of shaping is in many configurations
present or exceeded, as the relative comparison of the critical $r\delta$ to
$r_{c}\delta$ shows. Once again, these effects become strongest as the outer
surface is approached.

##  Appendix F Details of asymptotic available energy integral

In this appendix we provide mathematical and algebraic details concerning the
asymptotic evaluation of Æ in the two considered regimes. Note that these
considerations may be extended simply to other approximation schemes, such as
a large aspect ratio model.

###  F.1 The weakly driven regime

To perform the available energy integral in this regime, we showed how it is
the popuation of trapped particles that have an almost vanishing precession
that contribute. Thus, we first re-define the zero crossing of the trapped
particle precession, $k_{0}$,

$G(k_{0})=0.$ (87)

To make further progress with the integral, we ought to transform the
integration variable $\lambda$ into $k$, and thus require
$\mathrm{d}\lambda/\mathrm{d}k$. Details of this derivation are included in
Appendix C, and to leading order it reduces to

$\frac{\mathrm{d}\lambda}{\mathrm{d}k}\approx-4r\eta k.$ (88)

The final component needed for the integral is the normalized bounce-time
$\hat{G}^{1/2}\approx\int(1-\lambda\hat{B})^{-1/2}\mathrm{d}\ell/L$, for which
the leading order expression reduces to

$\hat{G}^{1/2}(k)=\frac{2\sqrt{2}K(k)}{\sqrt{r\eta}}\frac{B_{\alpha
0}}{B_{0}\bar{\iota}_{0}L},$ (89)

which may be calculated analytically or verified via the expressions for the
bounce-time given in Appendix C.

To explit the existence of a narrow contributing band, we will then make a
local approximation of the integrand about $k_{0}$. The region can be shown to
be small,

$\hat{\omega}_{\alpha}\approx\frac{\Delta\psi\eta}{rB_{0}}G^{\prime}(k_{0})(k-k_{0})\sim
O(1)\implies(k-k_{0})\sim r,$ (90)

so that conveniently using $c_{1}$ as integration variable,

$k=\frac{\hat{\omega}_{*,0}^{T}}{\frac{\Delta\psi\eta}{rB_{0}}G^{\prime}(k_{0})c_{1}}+k_{0}\longrightarrow\frac{\mathrm{d}k}{\mathrm{d}c_{1}}=-\frac{1}{c_{1}^{2}}\frac{rB_{0}\hat{\omega}_{*,0}^{T}}{\Delta\psi\eta
G^{\prime}(k_{0})}.$ (91)

the remaining integral becomes,

$\int_{0}^{\infty}\frac{\mathcal{F}(c_{1})}{c_{1}^{2}}\mathrm{d}c_{1}=\frac{\sqrt{\pi}}{6}.$
(92)

In doing so, we approximated the integration domain
$c_{1}\in\left(\hat{\omega}_{*,0}^{T}B_{0}r/\Delta\psi\eta|G^{\prime}(k_{0})|k_{0},\infty\right]$
as $(0,\infty]$. Doing so only introduces an error of order $\sqrt{r}$ on the
integral, which is the asymptotic value of $\mathcal{F}$ at small argument.

Collecting all the factors involved, the leading order expression of the Æ
simply becomes

$\widehat{A}\approx\frac{4\sqrt{2\pi}}{3\mathcal{V}}\frac{B_{\alpha
0}N_{\mathrm{wells}}}{B_{0}\bar{\iota}L}\frac{k_{0}K(k_{0})}{|G^{\prime}(k_{0})|}\left(\hat{\omega}_{*,0}^{T}\right)^{3}\frac{rB_{0}}{\Delta\psi}\sqrt{\frac{r}{\eta}}.$
(93)

Computing the dimensionless factor $\mathcal{V}$ to leading order,

$\mathcal{V}=\int\frac{2\sqrt{\pi}}{\hat{B}}\frac{\mathrm{d}\ell}{L}\approx
4\pi^{3/2}\frac{B_{\alpha 0}N_{\mathrm{wells}}}{B_{0}\bar{\iota}_{0}L}.$ (94)

and easing notation by approximating

$\frac{k_{0}K(k_{0})}{|G^{\prime}(k_{0})|}=0.666834\dots\approx\frac{2}{3}\pm
0.03\%,$

we find the result

$\widehat{A}\approx\frac{2\sqrt{2}}{9\pi}\left(\hat{\omega}_{*,0}^{T}\right)^{3}\frac{rB_{0}}{\Delta\psi}\sqrt{\frac{r}{\eta}}.$

One may now retrieve the result stated in the main text by imposing that
$\Delta\psi=C_{r}rB_{0}\rho$, $\varrho=r/a$, $\rho_{*}=\rho/a$, and
$-\partial_{\varrho}\ln n=\hat{\omega}_{n}$, resulting in

$\widehat{A}\approx\frac{2\sqrt{2}}{9\pi}C_{r}^{2}\rho_{*}^{2}\left(\frac{\hat{\omega}_{n}}{\varrho}\right)^{3}\frac{\varrho^{3}\sqrt{\varrho}}{\sqrt{a\eta}}.$
(95)

This concludes the analysis of the weakly driven asymptotic regime.

###  F.2 The strongly driven regime

The integral in the strongly driven regime is straightforward. Our starting
point is the full integral given in (38),

$\widehat{A}=\frac{(\hat{\omega}^{T}_{*,0})^{2}}{\mathcal{V}}\int\mathrm{d}\lambda\sum_{\mathrm{wells(\lambda)}}\hat{G}^{1/2}\mathcal{F}(c_{1})\Theta(\omega_{\alpha}).$
(38)

We again employ the fact that $\mathrm{d}\lambda/\mathrm{d}k\approx-4kr\eta$
to write

$\widehat{A}=\frac{2\sqrt{2}}{\pi^{3/2}}\left(\hat{\omega}_{*,0}^{T}\right)^{2}\sqrt{r\eta}\int
kK(k)\mathcal{F}\left(c_{1}\right)\Theta(\hat{\omega}_{\alpha})\mathrm{d}k$
(96)

In the strongly driven limit, the main assumption is that $c_{1}\gg 1$ for all
$k$, and thus we may replace $\mathcal{F}$ with its asymptotic form for large
$c_{1}$. Using (Olver et al., 2020, Secs. 7.2, 7.12),
$\mathcal{F}(c_{1})\approx 3\sqrt{\pi}/4c_{1}$, which yields

$\widehat{A}=\frac{3\Delta
r}{\pi\sqrt{2}}(\hat{\omega}_{*,0}^{T})\eta\sqrt{r\eta}\int_{0}^{k_{0}}kK(k)G(k)\mathrm{d}k.$
(97)

One can numerically evaluate the integral to find

$\int_{0}^{k_{0}}kK(k)G(k)\mathrm{d}k\approx 0.386842.$ (98)

Introducing the dimensionless variables as we did in the weak regime, we find

$\widehat{A}=1.1605~{}\frac{(C_{r}\rho_{*})^{2}}{\pi\sqrt{2}}\hat{\omega}_{n}(a\eta)^{3/2}\sqrt{\varrho},$
(99)

which is our final result.

### Dependence of Æ on $B_{20}$ and $B_{22}^{C}$

Before concluding this appendix, we show how to proceed to assess the effect
of $B_{22}^{C}$ and $B_{20}$ on the expression of available energy. The main
idea here is that we would like to evaluate these contributions without having
to evaluate all the consistent asymptotic dependence of Æ. For instance, we
are not interested in the order $r^{1/2}$ correction to the weak $\widehat{A}$
in Eq. (41) due to the finite extent of $\mathcal{F}$. To see how to proceed
let us write the relevant parts of the available energy integral (setting
overall factors aside for simplicity of notation),

$\widehat{A}\sim\sum_{\mathrm{wells}}\int\mathrm{d}c_{1}\frac{\mathrm{d}\lambda}{\mathrm{d}k}\frac{\mathrm{d}k}{\mathrm{d}c_{1}}\tau_{b}\mathcal{F}(c_{1}).$
(100)

To evaluate higher order corrections to the integral in the weak regime, we
remind the reader first that the calculation involves the local expansion of
the integrand. In that spirit, the changes due to second order on the first
factor in the integral is straightforward, and may be simply read off Eq.
(68b). It gives as a correction on the leading order integral

$\mathcal{R}_{1}=\frac{4rB_{22}^{C}}{\eta}(2k_{0}^{2}-1)$ (101)

where we have evaluated it at the original $k_{0}$ to leading order. A
similarly simple correction arises from the corrections to the bounce time
$\tau_{b}$, which may simply be read off Eq. (70),

$\mathcal{R}_{2}=\frac{2rB_{22}^{C}}{\eta}\left(\frac{E(k_{0})}{K(k_{0})}-k_{0}^{2}\right).$
(102)

The contributions left come from the $c_{1}$ dependent terms,
$\mathcal{F}(c_{1})\mathrm{d}k/\mathrm{d}c_{1}$. Here the change in the zero-
crossing of precession due to second order corrections on $|\mathbf{B}|$
contribute to the available energy. We must thus assess where the new zero is.
With the precession defined as
$\omega_{\alpha}=\omega_{\alpha,0}+\omega_{\alpha,1}$, Eq. (24), we rewrite it
as

$\omega_{\alpha}=(\eta/rB_{0})[G(k)+(r/\eta)\hat{\mathcal{G}}(k)],$ (103)

ignoring every term that does not depend on $B_{20}$ or $B_{22}^{C}$, and
recalling that
$\hat{\mathcal{G}}=B_{20}\mathcal{G}_{20}+B_{22}^{C}\mathcal{G}_{2c}$, Eq.
(27). Then, define the new $\omega_{\alpha}(k^{\star})=0$ point with
$k^{\star}=k_{0}+r\delta k$, so that

$\delta k\approx-\frac{\hat{\mathcal{G}}(k_{0})}{\eta G^{\prime}(k_{0})}.$
(104)

With this “displaced” $k^{\star}$, we may assess the change in the available
energy by simply considering the perturbation of the leading order $k_{0}$
dependence, namely $k_{0}K(k_{0})/G^{\prime}(k_{0})$, noting that the
denominator comes from $\omega_{\alpha}^{\prime}$ and thus it has an
additional contribution. Thus, the third correction may be written as,

$\mathcal{R}_{3}=-\frac{r}{\eta}\frac{1}{G^{\prime}(k_{0})}\left[\left(1+\frac{K^{\prime}(k_{0})}{K(k_{0})}-\frac{G^{\prime\prime}(k_{0})}{G^{\prime}(k_{0})}\right)\hat{\mathcal{G}}(k_{0})+\hat{\mathcal{G}}^{\prime}(k_{0})\right].$
(105)

Evaluating all these terms numerically,191919There is a closed form for the
numerical factor in this correction. However, this is a complicated function
of $k_{0}$ and elliptic integrals, and thus we do not present the expression
which does not add much to the picture. the total correction due to
$B_{22}^{C}$ and $B_{20}$ is,

$\mathcal{R}=\sum_{i}\mathcal{R}_{i}\approx\frac{r}{\eta}\left(1.36782B_{20}-0.0316789B_{2c}\right),$
(106)

where this factor should be understood to be a relative modification of the
leading order available energy $\widehat{A}=\widehat{A}_{0}(1+\mathcal{R})$.
That is, the correction to available energy due to $B_{20}$ and $B_{22}^{C}$,
which we denote as $\widehat{A}_{1}$, is equal to

$\widehat{A}_{1}=\widehat{A}_{0}\frac{r}{\eta}\left(\mathcal{R}_{20}B_{20}-\mathcal{R}_{22}^{C}B_{22}^{C}\right),$
(107)

where $\mathcal{R}_{20}\approx 1.37$ and $\mathcal{R}_{22}^{C}\approx 0.03$.

In the case of the strongly driven scenario, the integrals are over the whole
$k$-space, but a term by term analysis can be performed in an analogous way to
that above and the various terms numerically evaluated. The result of the
calculation is,

$\widehat{A}_{1}=-\widehat{A}_{0}\mathcal{R}_{20}^{\mathrm{s}}\frac{rB_{20}}{\eta}$
(108)

where $\mathcal{R}_{20}^{\mathrm{s}}=3.91$. Astonishingly, the $B_{22}^{C}$
contribution completely drops out. In terms of dependence on $B_{22}^{C}$, the
two limits of the Æ are not that different, considering that
$\mathcal{R}_{20}\gg\mathcal{R}_{22}^{C}$ for the former case (dropping it
makes approximately $\sim 2\%$ error). For the remaining analysis we shall
thus only consider dependence on $B_{20}$.

As was the case in the discussion of the trapped electron precession, it is
most physical to express these effects at second order in terms of the
triangularity of a cross-section and the pressure gradient. This is explained
in more detail in Appendix D. Splitting $B_{20}$ up as

$B_{20}\approx-\frac{\mu_{0}p_{2}}{B_{0}^{2}}\mathcal{P}+\frac{\delta|\eta|}{2}\mathcal{T},$
(109)

where we define

$\displaystyle\mathcal{P}=\left[1+\frac{4\sqrt{\bar{\alpha}}}{(\bar{\alpha}+3)-(\bar{\alpha}+1)\bar{F}}\left(\frac{\eta
B_{\alpha 0}}{B_{0}\bar{\iota}_{0}}\right)^{2}\right],$ (110a)
$\displaystyle\mathcal{T}=3\frac{(1-\bar{\alpha})+(\bar{\alpha}+1)\bar{F}}{(3+\bar{\alpha})-(\bar{\alpha}+1)\bar{F}},$
(110b)

the correction in the weakly driven regime reduces to

$\left.\frac{\widehat{A}_{1}}{\widehat{A}_{0}}\right|_{\mathrm{weak}}=\mathcal{R}_{20}\left(-\frac{r}{\eta}\frac{\mu_{0}p_{2}}{B_{0}^{2}}\mathcal{P}+\frac{r\delta}{2}\mathcal{T}\right),$
(111)

and in the strongly driven regime,

$\left.\frac{\widehat{A}_{1}}{\widehat{A}_{0}}\right|_{\mathrm{strong}}=-\mathcal{R}_{20}^{\mathrm{s}}\left(-\frac{r}{\eta}\frac{\mu_{0}p_{2}}{B_{0}^{2}}\mathcal{P}+\frac{r\delta}{2}\mathcal{T}\right),$
(112)

which are in the form presented in the main text

## References

  * Anderson et al. (1995) Anderson, F. Simon B., Almagri, Abdulgader F., Anderson, David T., Matthews, Peter G., Talmadge, Joseph N. & Shohet, J. Leon 1995 The helically symmetric experiment, (hsx) goals, design and status. Fusion Technology 27 (3T), 273–277.
  * Bader et al. (2021) Bader, A, Anderson, DT, Drevlak, M, Faber, BJ, Hegna, CC, Henneberg, S, Landreman, M, Schmitt, JC, Suzuki, Y & Ware, A 2021 Modeling of energetic particle transport in optimized stellarators. Nuclear Fusion 61 (11), 116060.
  * Bauer et al. (2012) Bauer, Frances, Betancourt, Octavio & Garabedian, Paul 2012 Magnetohydrodynamic equilibrium and stability of stellarators. Springer Science & Business Media.
  * Bernardin et al. (1986) Bernardin, M. P., Moses, R. W. & Tataronis, J. A. 1986 Isodynamical (omnigenous) equilibrium in symmetrically confined plasma configurations. The Physics of Fluids 29 (8), 2605–2611.
  * Boozer (1981) Boozer, Allen H 1981 Plasma equilibrium with rational magnetic surfaces. The Physics of Fluids 24 (11), 1999–2003.
  * Boozer (1983) Boozer, Allen H 1983 Transport and isomorphic equilibria. The Physics of Fluids 26 (2), 496–499.
  * C. Mercier & N. Luc (1974) C. Mercier & N. Luc 1974 Report No. EUR-5127e 140 (Commission of the European Communities, Brussels, 1974). Tech. Rep..
  * Calvo et al. (2017) Calvo, Iván, Parra, Felix I, Velasco, José Luis & Alonso, J Arturo 2017 The effect of tangential drifts on neoclassical transport in stellarators close to omnigeneity. Plasma Physics and Controlled Fusion 59 (5), 055014.
  * Cary & Shasharina (1997) Cary, John R. & Shasharina, Svetlana G. 1997 Omnigenity and quasihelicity in helical plasma confinement systems. Physics of Plasmas 4 (9), 3323–3333.
  * Chu et al. (1978) Chu, KR, Ott, Edward & Manheimer, Wallace M 1978 Effects of cross-sectional elongation on trapped electron modes. The Physics of Fluids 21 (4), 664–668.
  * Connor et al. (1983) Connor, JW, Hastie, RJ & Martin, TJ 1983 Effect of pressure gradients on the bounce-averaged particle drifts in a tokamak. Nuclear fusion 23 (12), 1702.
  * Dewar & Hudson (1998) Dewar, R.L. & Hudson, S.R. 1998 Stellarator symmetry. Physica D: Nonlinear Phenomena 112 (1), 275–280, proceedings of the Workshop on Time-Reversal Symmetry in Dynamical Systems.
  * D’haeseleer et al. (2012) D’haeseleer, William D, Hitchon, William NG, Callen, James D & Shohet, J Leon 2012 Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory. Springer Science & Business Media.
  * Dimits et al. (2000) Dimits, Andris M, Bateman, G, Beer, MA, Cohen, BI, Dorland, W, Hammett, GW, Kim, C, Kinsey, JE, Kotschenreuther, M, Kritz, AH & others 2000 Comparisons and physics basis of tokamak transport models and turbulence simulations. Physics of Plasmas 7 (3), 969–983.
  * Freidberg (2014) Freidberg, Jeffrey P 2014 ideal MHD. Cambridge University Press.
  * Garbet et al. (2004) Garbet, X, Mantica, P, Ryter, F, Cordey, G, Imbeaux, F, Sozzi, C, Manini, A, Asp, E, Parail, V, Wolf, R & others 2004 Profile stiffness and global confinement. Plasma physics and controlled fusion 46 (9), 1351.
  * Garren & Boozer (1991a) Garren, D. A. & Boozer, A. H. 1991a Existence of quasihelically symmetric stellarators. Physics of Fluids B: Plasma Physics 3 (10), 2822–2834.
  * Garren & Boozer (1991b) Garren, D. A. & Boozer, A. H. 1991b Magnetic field strength of toroidal plasma equilibria. Physics of Fluids B: Plasma Physics 3 (10), 2805–2821.
  * Greene (1997) Greene, John M 1997 A brief review of magnetic wells. Comments on Plasma Physics and Controlled Fusion 17, 389–402.
  * Greene & Johnson (1962) Greene, John M & Johnson, John L 1962 Stability criterion for arbitrary hydromagnetic equilibria. The Physics of Fluids 5 (5), 510–517.
  * Hall & McNamara (1975) Hall, Laurence S. & McNamara, Brendan 1975 Three-dimensional equilibrium of the anisotropic, finite-pressure guiding-center plasma: Theory of the magnetic plasma. The Physics of Fluids 18 (5), 552–565.
  * Hegna (2000) Hegna, CC 2000 Local three-dimensional magnetostatic equilibria. Physics of Plasmas 7 (10), 3921–3928.
  * Hegna (2015) Hegna, CC 2015 The effect of three-dimensional fields on bounce averaged particle drifts in a tokamak. Physics of Plasmas 22 (7), 072510.
  * Helander (2014) Helander, Per 2014 Theory of plasma confinement in non-axisymmetric magnetic fields. Reports on Progress in Physics 77 (8), 087001.
  * Helander (2017) Helander, Per 2017 Available energy and ground states of collisionless plasmas. Journal of Plasma Physics 83 (4), 715830401.
  * Helander (2020) Helander, Per 2020 Available energy of magnetically confined plasmas. Journal of Plasma Physics 86 (2), 905860201.
  * Helander et al. (2013) Helander, Per, Proll, Josefine Henriette Elise & Plunk, Gabriel Galad 2013 Collisionless microinstabilities in stellarators. i. analytical theory of trapped-particle modes. Physics of Plasmas 20 (12), 122505.
  * Helander & Sigmar (2005) Helander, Per & Sigmar, Dieter J 2005 Collisional transport in magnetized plasmas, , vol. 4. Cambridge university press.
  * Hirshman & Whitson (1983) Hirshman, S. P. & Whitson, J. C. 1983 Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria. The Physics of Fluids 26 (12), 3553–3568.
  * Kadomtsev & Pogutse (1967) Kadomtsev, BB & Pogutse, OP 1967 Plasma instability due to particle trapping in a toroidal geometry. Sov. Phys. JETP 24, 1172–1179.
  * Kim et al. (2021) Kim, Patrick, Jorge, Rogerio & Dorland, William 2021 The on-axis magnetic well and mercier’s criterion for arbitrary stellarator geometries. Journal of Plasma Physics 87 (2).
  * Kolmes et al. (2020) Kolmes, EJ, Helander, P & Fisch, NJ 2020 Available energy from diffusive and reversible phase space rearrangements. Physics of Plasmas 27 (6), 062110.
  * Landreman (2021) Landreman, Matt 2021 Figures of merit for stellarators near the magnetic axis. Journal of Plasma Physics 87 (1), 905870112.
  * Landreman (2022) Landreman, Matt 2022 Mapping the space of quasisymmetric stellarators using optimized near-axis expansion. Journal of Plasma Physics 88 (6), 905880616.
  * Landreman & Catto (2012) Landreman, Matt & Catto, Peter J. 2012 Omnigenity as generalized quasisymmetry. Physics of Plasmas 19 (5), 056103.
  * Landreman & Jorge (2020) Landreman, Matt & Jorge, Rogerio 2020 Magnetic well and mercier stability of stellarators near the magnetic axis. Journal of Plasma Physics 86 (5), 905860510.
  * Landreman & Paul (2022) Landreman, Matt & Paul, Elizabeth 2022 Magnetic fields with precise quasisymmetry for plasma confinement. Physical Review Letters 128 (3), 035001.
  * Landreman & Sengupta (2019) Landreman, Matt & Sengupta, Wrick 2019 Constructing stellarators with quasisymmetry to high order. Journal of Plasma Physics 85 (6), 815850601.
  * Li & Kishimoto (2002) Li, Jiquan & Kishimoto, Y 2002 Role of collisionless trapped electron mode in tokamak plasmas with electron internal transport barrier. Plasma physics and controlled fusion 44 (5A), A479.
  * Lortz & Nührenberg (1978) Lortz, D & Nührenberg, J 1978 Ballooning stability boundaries for the large-aspect-ratio tokamak. Physics Letters A 68 (1), 49–50.
  * Mackenbach et al. (2023a) Mackenbach, RJJ, Duff, JM, Gerard, MJ, Proll, JHE, Helander, P & Hegna, CC 2023a Bounce-averaged drifts: Equivalent definitions, numerical implementations, and example cases. arXiv preprint arXiv:2305.01225 .
  * Mackenbach et al. (2023b) Mackenbach, R.J.J., Proll, J.H.E., Snoep, G. & Helander, P. 2023b Available energy of trapped electrons in miller tokamak equilibria [to be submitted]. Journal of Plasma Physics .
  * Mackenbach et al. (2023c) Mackenbach, R.J.J., Proll, J.H.E., Wakelkamp, R. & Helander, P. 2023c The available energy of trapped electrons: a nonlinear measure for turbulent transport [submitted]. Journal of Plasma Physics .
  * Mackenbach et al. (2022) Mackenbach, RJJ, Proll, Josefine HE & Helander, P 2022 Available energy of trapped electrons and its relation to turbulent transport. Physical Review Letters 128 (17), 175001.
  * Marinoni et al. (2019) Marinoni, A, Austin, ME, Hyatt, AW, Walker, ML, Candy, J, Chrystal, C, Lasnier, CJ, McKee, GR, Odstrčil, T, Petty, CC & others 2019 H-mode grade confinement in l-mode edge plasmas at negative triangularity on diii-d. Physics of Plasmas 26 (4), 042515.
  * Mercier (1962) Mercier, C 1962 Critère de stabilité d’un système toroïdal hydromagnétique en pression scalaire. Nucl. Fusion Suppl 2, 801.
  * Mercier (1974) Mercier, Claude 1974 Lectures in plasma physics: the magnetohydrodynamic approach to the problem of plasma confinement in closed magnetic configurations. Luxembourg Commission European Communities .
  * Merlo et al. (2015) Merlo, G, Brunner, Stephan, Sauter, Olivier, Camenen, Y, Görler, T, Jenko, F, Marinoni, A, Told, D & Villard, Laurent 2015 Investigating profile stiffness and critical gradients in shaped tcv discharges using local gyrokinetic simulations of turbulent transport. Plasma Physics and Controlled Fusion 57 (5), 054010.
  * Merlo & Jenko (2023) Merlo, Gabriele & Jenko, Frank 2023 Interplay between magnetic shear and triangularity in ion temperature gradient and trapped electron mode dominated plasmas. Journal of Plasma Physics 89 (1), 905890104.
  * Miller et al. (1998) Miller, RL, Chu, MS, Greene, JM, Lin-Liu, YR & Waltz, RE 1998 Noncircular, finite aspect ratio, local equilibrium model. Physics of Plasmas 5 (4), 973–978.
  * Nemov et al. (2008) Nemov, VV, Kasilov, SV, Kernbichler, Winfried & Leitold, GO 2008 Poloidal motion of trapped particle orbits in real-space coordinates. Physics of plasmas 15 (5).
  * Nührenberg & Zille (1988) Nührenberg, J. & Zille, R. 1988 Quasi-helically symmetric toroidal stellarators. Physics Letters A 129 (2), 113 – 117.
  * Olver et al. (2020) Olver, F. W. J., Daalhuis, A. B. Olde, Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., B. R. Mille and, B. V. Saunders, Cohl, H. S. & M. A. McClain, eds. 2020 Nist digital library of mathematical functions. http://dlmf.nist.gov/, Release 1.0.26 of 2020-03-15.
  * Proll et al. (2012) Proll, Josefine Henriette Elise, Helander, Per, Connor, John William & Plunk, GG 2012 Resilience of quasi-isodynamic stellarators against trapped-particle instabilities. Physical Review Letters 108 (24), 245002.
  * Qi et al. (2019) Qi, Lei, Kwon, Jae-Min, Hahm, TS, Yi, Sumin & Choi, MJ 2019 Characteristics of trapped electron transport, zonal flow staircase, turbulence fluctuation spectra in elongated tokamak plasmas. Nuclear Fusion 59 (2), 026013.
  * Roach et al. (1995) Roach, CM, Connor, JW & Janjua, S 1995 Trapped particle precession in advanced tokamaks. Plasma physics and controlled fusion 37 (6), 679.
  * Rodriguez (2022) Rodriguez, Eduardo 2022 Quasisymmetry. PhD thesis.
  * Rodriguez & Bhattacharjee (2021) Rodriguez, Eduardo & Bhattacharjee, Amitava 2021 Solving the problem of overdetermination of quasisymmetric equilibrium solutions by near-axis expansions. i. generalized force balance. Physics of Plasmas 28 (1), 012508.
  * Rodriguez et al. (2020) Rodriguez, Eduardo, Helander, Per & Bhattacharjee, Amitava 2020 Necessary and sufficient conditions for quasisymmetry. Physics of Plasmas 27 (6), 062501.
  * Rodriguez et al. (2022a) Rodriguez, Eduardo, Paul, EJ & Bhattacharjee, Amitava 2022a Measures of quasisymmetry for stellarators. Journal of Plasma Physics 88 (1), 905880109.
  * Rodriguez et al. (2022b) Rodriguez, Eduardo, Sengupta, Wrick & Bhattacharjee, Amitava 2022b Phases and phase-transitions in quasisymmetric configuration space. Plasma Physics and Controlled Fusion 64 (10), 105006.
  * Rodríguez et al. (2022) Rodríguez, Eduardo, Sengupta, Wrick & Bhattacharjee, Amitava 2022 Weakly quasisymmetric near-axis solutions to all orders. Physics of Plasmas 29 (1), 012507.
  * Rodríguez (2023) Rodríguez, Eduardo 2023 Magnetohydrodynamic stability and the effects of shaping: a near-axis view for tokamaks and quasisymmetric stellarators. Journal of Plasma Physics 89 (2), 905890211.
  * Rosenbluth & Sloan (1971) Rosenbluth, M. & Sloan, M. L. 1971 Finite-$\beta$ stabilization of the collisionless trapped particle instability. Physics of Fluids 14 (8), 1725–1741.
  * Rosenbluth (1968) Rosenbluth, Marshall N. 1968 Low-frequency limit of interchange instability. The Physics of Fluids 11 (4), 869–872, arXiv: https://aip.scitation.org/doi/pdf/10.1063/1.1692009.
  * Shafranov (1963) Shafranov, VD 1963 Equilibrium of a toroidal pinch in a magnetic field. Soviet Atomic Energy 13 (6), 1149–1158.
  * Shafranov (1983) Shafranov, VD 1983 Magnetohydrodynamic theory of plasma equilibrium and stability in stellarators: survey of results. The Physics of Fluids 26 (2), 357–364.
  * Solov’ev & Shafranov (1970) Solov’ev, L. S. & Shafranov, V. D. 1970 Reviews of Plasma Physics 5. New York - London: Consultants Bureau.
  * Velasco et al. (2021) Velasco, JL, Calvo, I, Mulas, S, Sánchez, E, Parra, FI, Cappa, A & others 2021 A model for the fast evaluation of prompt losses of energetic ions in stellarators. Nuclear Fusion 61 (11), 116059.
  * Velasco et al. (2023) Velasco, JL, Calvo, I, Sánchez, E & Parra, FI 2023 Robust stellarator optimization via flat mirror magnetic fields. arXiv preprint arXiv:2306.17506 .
  * Wesson (2011) Wesson, John 2011 Tokamaks; 4th ed.. International series of monographs on physics . Oxford: Oxford Univ. Press.
  * White & Chance (1984) White, RB t & Chance, MS 1984 Hamiltonian guiding center drift orbit calculation for plasmas of arbitrary cross section. The Physics of fluids 27 (10), 2455–2467.