# Non-Intrusive Driver Behavior Characterization From Road-Side Cameras

Pavana Pradeep Kumar, Krishna Kant and Amitangshu Pal Pavana Pradeep Kumar and
Krishna Kant are with the Department of Computer and Information Sciences,
Temple University, Philadelphia, PA, USA. (e-mail<EMAIL_ADDRESS>kkant@temple.edu)Amitangshu Pal is with the Computer Science and Engineering,
Indian Institute of Technology Kanpur, Kanpur, India. (e-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we demonstrate a proof of concept for characterizing vehicular
behavior using only the roadside cameras of the ITS system. The essential
advantage of this method is that it can be implemented in the roadside
infrastructure transparently and inexpensively and can have a global view of
each vehicle’s behavior without any involvement of or awareness by the
individual vehicles or drivers. By using a setup that includes
programmatically controlled robot cars (to simulate different types of
vehicular behaviors) and an external video camera set up to capture and
analyze the vehicular behavior, we show that the driver classification based
on the external video analytics yields accuracies that are within 1-2% of the
accuracies of direct vehicle-based characterization. We also show that the
residual errors primarily relate to gaps in correct object identification and
tracking and thus can be further reduced with a more sophisticated setup. The
characterization can be used to enhance both the safety and performance of the
traffic flow, particularly in the mixed manual and automated vehicle scenarios
that are expected to be common soon.

Keywords: Intelligent Transportation Systems, Reasoning, Event Logic, Smart
Cities, Cyber-Physical Systems.

##  I Introduction

Road accidents are responsible for $\sim$5 million severe injuries and
$\sim$50K deaths annually in the USA
(https://www.bankrate.com/insurance/car/car-crash-statistics/). According to
the National Highway Traffic Safety Administration (NHTSA), at least one of
the following risky behaviors was evident in 45% of fatal crashes involving
passenger vehicles: speeding, alcohol impairment, or not keeping a safe
distance, etc [1]. Intelligent Transportation Systems (ITS) aim to reduce
accidents and make traffic flow smoother by introducing capabilities to
monitor traffic and communicate with vehicles.

Characterizing driver behavior is essential for improving traffic safety and
performance. It should be noted that the term “driver” does not necessarily
imply a manually operated vehicle. As the market for automated vehicles
develops and becomes significant, it is anticipated that vehicles from
different manufacturers will have distinct personalities, influenced in part
by the design of the software and in part by the “personalization” that users
will desire [2]. Some users may be comfortable with automated driving
aggressively, while others prefer a more conservative driving style. This
aspect also depends on familiarity with automated driving and driving
conditions. Therefore, in this paper, driver behavior refers to vehicular
behavior, although the type of feedback provided will still be different for
manual vs. automated vehicles.

###  I-A Vehicular and Driver Behavior Characterization

There is a tremendous amount of work on vehicular behavior characterization,
mainly regarding monitoring vehicular kinematics parameters. Vehicular
behavior can be measured either from the sensors in the car through the OBD
(onboard diagnosis) port or a device carried/mounted in the car. Typical
behavioral measurements include acceleration, braking, distance to the next
car, speed and speed variability, lane change behavior, etc [3].

In addition, there is a large body of work specifically targeting the behavior
of the human driver in terms of direct actions (use of brake, accelerator, or
steering wheel), alertness, and physical health (e.g., eye or head movements,
facial expressions, lack of focus on the road, etc.) to determine if the
driver is tired, drowsy, etc. [4]. Other sensors may also be used for such
things as analyzing breathing patterns and breath smell (for alcohol), and may
even use contact sensors such as those for measuring heart rate, brain
signals, skin conductance, etc. With their increasing array of sensors and
wearable devices such as smartwatches, smartphones can easily measure many
vehicular and driver parameters. The measurements are generally used to train
a classifier to categorize the driver.

However, such methods have several inherent problems. First, the measurements
require either the deployment of special gadgets in the vehicles or depend on
the driver carrying or wearing devices with appropriate sensors. Using devices
carried by a driver for a driver or vehicular behavior is impractical and has
little value beyond a proof of concept. While the vehicles do monitor all of
the kinematic parameters and thus could estimate the driver behavior, this has
limitations since each vehicle only has a local view. The behavior of a group
of vehicles traveling together or in close proximity is interdependent, and
these interdependencies make it challenging to identify the cause-and-effect
relationships among the vehicles. For example, a sudden slowing of a vehicle
will cause the vehicle behind it to also slow suddenly. While the sudden
slowing of the front vehicle may be undesired (e.g., aggressive driving), that
of the vehicle behind is highly desired for maintaining safety. Another
serious problem with these methods is the awareness of the drivers that their
driving style or physical condition (e.g., drowsiness, drunkenness) is being
monitored. This intrusiveness would surely change driver behavior; thus, the
monitoring will be inaccurate. Furthermore, many drivers may refuse such
monitoring or try to defeat it.

###  I-B Our contributions:

In this paper, we demonstrate that it is possible to characterize vehicular
behavior very accurately using only the roadside cameras of the ITS system. To
the best of our knowledge, this is the first such study; its key advantage is
that it is entirely non-intrusive, inexpensive, and does not impact driving
behavior in any way. Also, the global view can deduce the more obvious cause-
and-effect relationships between adjacent vehicles (e.g., whether the car
decelerated on its own or due to diminishing distance from the vehicle ahead).
Furthermore, the minimal error rate of our video analytics system is
attributable to instances where the vehicular ID is not tracked correctly or
the camera is too far from the vehicle. These errors can be further minimized
by more elaborate feature-based tracking of vehicles (e.g., by color, shape,
etc.) and by having associations between multiple cameras. The monitoring can
be used to provide advice to the vehicles in terms of enhancing safety and
traffic performance. The scheme becomes even more effective with automated
vehicles since they can help smooth out the traffic by behaving in a specific
way.

###  I-C Paper outline:

The remainder of the paper is organized as follows. Section II discusses the
related work. Section III presents our proposed framework for driver behavior
characterization. Experimental evaluation and results are summarized in
section IV. Section V then concludes the paper.

##  II Related work

Different methods that have been studied for identifying driver behavior can
be classified into broadly two categories: using (a) in-vehicle sensor or
video data analysis and (b) analyzing driver’s physiological data. In the
following, we summarize them separately.

Sensor or video data analysis: Authors in [5] have developed a drunk driving
and alerting system using a mobile phone that can be installed inside a car;
the phone can record the acceleration samples from the phone sensors and
compare them with the drunk driving patterns. In [6], the authors have
proposed an aggressive driving behavior detection system that uses data from
multiple smartphone-based sensors (i.e., accelerometer, gyroscope,
magnetometer, GPS, video) and recognizes abnormal driving behaviors using the
Dynamic Time Warping (DTW) algorithm. Similarly, the authors in [7] have
developed a driver behavior classification and sharing system based on
vehicle-mounted acceleration samples. Authors in [8] have used a public
dataset named UAH-DriveSet [9] to analyze different in-vehicle sensory data
and classify driver behaviors using different machine learning based
classification methods. In [10], the authors have studied driver fatigue
detection using eye tracking, where distances between the intensity changes in
the eye area are measured to determine whether the eyes are closed or not.
Similar other studies [11, 12] have analyzed various in-vehicle video data
like driver’s facial expressions, eye or head movements, etc., for determining
driver’s behaviors.

Physiological data analysis: Psycho-physical states of drivers using
respiration rates and ECG signals can also be studied through wearable
devices, which also determine drivers’ stress or distraction levels. In [13],
the authors have shown that significant electroencephalographic and
psychological changes (like increased delta and theta activity, lower heart
rate, changes in blink rate, etc.) occur during fatigue. In [14], the authors
have used psychological features, such as EEG and ECG signals to classify the
driver’s behavior into four categories (alert, mild fatigue, deep fatigue, and
drowsiness) using a support vector machine (SVM). Similar studies on
physiological data analysis for driver’s state detection are reported in [15].

The existing works for driver behavior detection require either installation
of in-vehicle sensors and cameras or require on-body wearable sensors, which
bring additional installation overhead. As opposed to the existing literature,
our proposed framework characterizes driver behavior from roadside camera
frames, making the solution inexpensive, non-intrusive, and easily deployable.

Fig. 1: Driver Behavior Characterization Framework.

##  III Framework for Driver Behavior Prediction

###  III-A Categorizing Driver behavior

Driver behavior is important from the safety perspective, but there is no
standard way to characterize drivers. In this paper, we adopt a 3-way
classification, where a driver is designated as safe, aggressive, and
distracted [16]. Please note that our “safe” category includes any behavior
that is not considered aggressive or distracted; therefore, there is no
undetermined behavior. For this categorization, we define a small set of
“micro-behaviors” and then characterize the driver based on the combination of
micro-behaviors observed. The micro behaviors include: (a) Weaving, or driving
alternatively towards one side of the lane and then the other; (b) Sudden
Steer where the driver makes an abrupt redirection when driving along a
straight course; (c) Hard Braking; (d) Lane Drifting, or not keeping in the
center of the lane while driving; (e) Straddling, or driving while staying
close to one side of the lane; and (f) Over speeding, or driving significantly
above the speed limit. By performing object detection and estimating driving
parameters over traffic videos as well as in vehicular data, we observe these
micro-behaviors and classify the driver’s behavior into one of three
categories. For example, an aggressive driver can be identified by a
combination of the following behaviors: weaving through traffic, driving at a
higher speed (speeding), following other vehicles too closely with rapid
steering, and applying harsh braking.

In this paper, we formulate the driver behavior detection problem as a boolean
satisfiability problem so that the popular SMT (Satisfiability Modulo Theory)
based tools can be used along with suitable theories. We make use of a popular
framework called Event Calculus (EC) that introduces the concept of Events,
which are actions that occur at a specific point in time, and Fluents, which
are entities whose state changes in response to the occurrence of an event or
action. EC provides constructs to reason about situations, events, and changes
in time, allowing a precise specification of time relationships between
situations and events, which is essential for describing the various driving
style patterns exhibited by drivers. In addition, concurrent actions can also
be specified in EC.

###  III-B System Architecture

We assume that the camera deployment is such that all vehicles on the roadway
are visible without excessive image distortion. Given the increasing
processing power in smart cameras, each camera can do the object detection and
tracking tasks discussed here rather than sending the raw video feed to the
next level known as Road-Side Edge Controller (REC). In our earlier work, we
designed a lightweight object recognition and tracking algorithm called YLLO
(You Look Less than Once) that can run in the cameras and avoid transmission
of redundant frames to RECs [17]. The RECs receive video streams from multiple
cameras along a road segment and use them to monitor driver behaviors and
associated anomalies. Further processing, including perspective transformation
and estimation of orientations and speeds of the objects, may be done by the
camera itself or by the RECs. The REC can then build a spatio-temporal logic
model of the situation that includes all “facts” of different driver behaviors
and the conditions leading to near-miss accidents along with the supporting
“theories” (i.e., Newton’s laws, arithmetic, etc.)

Fig. 1 depicts our overall architecture, which is comprised of two different
frameworks. One framework is based on Deep Learning (DL) and reasoning that
runs on roadside cameras receiving traffic videos as input. The other is based
on logical reasoning that runs on individual vehicles receiving in-vehicle
parameters as inputs.

The initial stage of the roadside camera framework is a lightweight object
detection and tracking model based on a convolutional neural network (CNN)
model called YLLO that we have developed for video analysis [17]. YLLO
performs processing of video sequences in a highly efficient manner. The
second stage for each detected object (e.g., vehicles, pedestrians, etc.) is a
spatio-temporal logic-based reasoning system that captures the relative
movements of the objects in real-time to identify various driver behaviors.
The reasoning framework operating on individual vehicles consists of a formal
specification of driver behavior used by an event recognition tool.

###  III-C YLLO based Object Detection and Tracking

YLLO is a lightweight object detection technique based on YOLOv6 and is
optimized for continuous video streams by utilizing redundancy to identify the
“only” essential frames. The previous version of YLLO as in [17] was based on
YOLOv4; for this work, YLLO has been updated to a newer and more efficient
version among all versions of YOLO, YOLOv6 [18]. YLLO is a three-stage process
that begins with a scene change detection algorithm and progresses to object
detection via YOLOv6. The Simple Online and Real-time Tracking with a Deep
Association Metric (Deep-SORT) algorithm assign a tracker to each detected
object or multiple objects. YLLO decouples classification and regression tasks
to eliminate redundant objects between the frames. Additionally, before
sending frames to object detection, for the scene change detection, it
generates Color Difference Histograms (CDH) for edge orientations, where edge
orientations are determined using the Laplacian-Gaussian edge detection
framework.

###  III-D Spatio-Temporal Reasoning

In our work, we use an efficient dialect of the Event Calculus, termed “Event
Calculus for run-time Reasoning” (RTEC)[19]. RTEC is an open-source
implementation of the EC in Prolog and uses LTL (linear time logic) with
integer time points. RTEC implements novel techniques for identifying complex
events from a set of micro-behaviors and is scalable to large volumes of
complex events.

We can define micro-behavior in RTEC as rules that define the event instances
using the predicates “happens at” (hA) and “happens for” (hF). The fluents
that are time-varying properties and the effects of events on fluents are
defined using the inA and tA predicates. The value of fluents at any time
point is defined using the hoA and hoF predicates. If F is a variable ranging
over fluents, the term F=V denotes that variable F has a value V. There also
exists Boolean fluents with values true or false. Table I shows the predicates
used in the RTEC tool.

TABLE I: Table: Main predicates of RTEC Predicate | Meaning
---|---
hA(E, T) |  Event E happens at time T
hF(E, I) |  Event E happens for I intervals
in(F=V) |  Initial value of fluent F =V at time 0
hoA(F=V, T) |  Value of fluent F holds at V at time T
hoF(F=V, I) |  Value V of fluent F holds for I intervals continuously
inA(F=V, T) |  Fluent F with value V initiated at time T
tA(F=V, T) |  Fluent F with value V terminated at time T

The characterization of real-world behavior often involves fuzziness, whereas
a Boolean logic framework requires an assertion to be either true or false. It
is not straightforward to extend RTEC to fuzzy or probabilistic logic;
therefore, we model the fuzziness indirectly by introducing weights. Recall
that we classify driver behavior in terms of micro-behaviors, some of which
are essential for characteristics of certain driver behavior. For example,
sudden braking can be considered a key characteristic of aggressive driving.
Thus we consider certain micro-behavioral assertions as hard in that they must
hold, whereas others can be considered as soft or optional. Let $S_{k}$ denote
the set of soft assertions for driver behavior $k$. We define a weight,
denoted as $w_{ik}$ for each member $i$ of the set $S_{k}$. Then, the driver
behavior $k$ will be recognized as (a) all hard assertions (micro-behaviors)
holding, and (b) $\sum_{i\in S_{k}}iw_{ik}>W_{k}$ for some threshold $W_{k}$.
Note that the weights need not be static but depend on various spatio-temporal
factors and context (e.g., day vs. night time, roads with different speed
limits, etc.) A weight change would require pausing all current condition
evaluations, changing all weights that need to be changed simultaneously, and
then restarting the evaluation. To handle hard/soft conditions, we can use an
extension to the Boolean Satisfiability problem known as Weighted Partial
Maxsat (WPM2). In WPM2, each clause is designated as either hard or soft with
a given weight. We then have an optimization problem to find an assignment
that satisfies all hard clauses and minimizes the total weight of soft
clauses. The SAT core returned by the WPM2 solver confirms the presence of
driver behavior.

###  III-E Defining Events and Fluents in RTEC

In second stage of the framework, the RTEC tool receives input as event
calculus (EC) predicates representing time-stamped micro behaviors detected on
individual video frames or the recorded in-vehicular parameters as shown in
Fig 1. For example, the object’s bounding box coordinates can define the
appearance of a static object or multiple moving objects in each frame. We
also have the object’s angle/orientation and the direction in which they are
moving, as well as lane drifting, which indicates if the vehicle is not
staying in the center of the lane.

TABLE II: Table: Events and Fluents Defined Events Defined
---
proximityLeft(v) |  vehicle v is close to the left side of the lane
proximityRight(v) |  vehicle v is close to the right side of the lane
laneDrifting(v) |  vehicle v is not keeping the center of the lane
changeLane1(v) |  vehicle v changes to the lane1 to due to a steering maneuver
changeLane2(v) |  vehicle v changes to the lane2 to due to a steering maneuver
weaving(v) |  vechicle v driving alternatively on one side of the lane and then the other
suddenSteer(v) |  vehicle v applies a sudden steer
straddling(v) |  vehicle v is positioned over lane lines rather than between them
Fluents Defined |  Meaning
atLane1(v) |  the existence of vehicle v in lane 1 in the scenario
atLane2(v) |  the existence of vehicle v in lane 2 in the scenario
slowSpeed(v) |  vehicle v drops speed below the speed limit
overSpeed(v) |  vehicle v exceeds the speed limit in either lane
speed (v, $S_{v}$) |  momentary speed $S_{v}$ of vehicle v
acceleration (v, $A_{v}$) |  momentary acceleration $A_{v}$ of vehicle v
deceleration(v, $D_{v}$) |  momentary acceleration $D_{v}$ of vehicle v
hardBraking (v) |  vehicle v executes an emergency braking (hard braking deceleration (hbd) is -8 m/s2)
normalBraking(v) |  vehicle v executes normal braking (i.e., normal braking deceleration(nbd) is -3 m/s2 )
stopping (v) |  vehicle v reduces its speed after braking
safeDriving(v) |  vehicle v is a safe driver
distractedDriving(v) |  vehicle v is a distracted driver
aggressiveDriving(v) |  vehicle v is an aggressive driver

Micro-behaviors such as acceleration, braking, lane change to the left or
right, etc., are represented as events in EC that are defined along with their
associated timestamps, which indicate the point in time at which the activity
occurred. The hA predicate establishes this type of input. For instance,
hA(hardBreaking(id6, 60) indicates that an object(id6) engaged in hard or
sudden braking at video frame 60, which is determined by comparing the
deceleration value going beyond a predefined threshold. Some of the micro
behaviors are represented as fluents in the EC. We use the inA and tA
predicates for expressing the conditions in which these fluents initiate and
terminate a specific driver behavior described above.

The Micro-behaviors represented as EC events are defined mostly with hF
predicate, which can also compute the associated intervals. For example,
hF(laneDrifting(id3) = true, [(0, 60),(210, 280 )] indicates that object3 was
not keeping the center of the lane during the intervals (0, 60) and (210,
280). A few examples of events and fluents defined along with their meaning
are shown in Table II.

After defining events and fluents, we must define an initiation and
termination map for each defined fluent in the system, indicating which events
initiate and terminate which fluents. The next step is to specify the relation
between fluents and events in the form of rules. For example, the initiation
and termination map for fluent overSpeed(v) is shown in Definition 1. In the
definition, overSpeed, speed are input events and lane1 and lane2 are the
input fluents. th is an temporal predicate indicating numerical threshold of
driving parameters and in this case it represents the user-specified speeding
threshold. The ruleset defined states that overSpeed(v) is a Boolean fluent,
which is invoked when a speed event is detected. Further, the vehicle is
present in lane1 and not in lane2, which is detected by fluents lane1 and
lane2 respectively, and the momentary speed of the vehicle is more than the
user-specified speeding threshold. The event overSpeed(v) is terminated when
the vehicle vs. speed is smaller than the speeding threshold. The exact
definition applies when the vehicle is located in lane2, as indicated by
fluent lane2 and the negation of fluent lane1.

inA(overSpeed(v) = true, T) $\leftarrow$ hoA(speed(v, $S_{v}$), T) $\land$
hoA(atLane1(v), T) $\land$ $\neg$ hoA(atLane2(v), T) $\land$
th($os$, $S_{v}$ $>=$ $os$).

tA(overSpeed(v) = true, T) $\leftarrow$ hoA(speed(v, $S_{v}$), T) $\land$
hoA(lane1(v), T) $\land$ $\neg$ hoA(lane2(v), T) $\land$
th($os$, $S_{v}$ $<$ $os$).

List of Definitions 1 Initiation and Termination map of fluent overSpeed

###  III-F Characterizing Driver Behavior

As discussed in III-A, each category of driver behavior is a combination of
different types of micro-behaviors, wherein all micro-behaviors defining a
driver’s behavior must be satisfied. For instance, the micro-behaviors that
represent aggressive driver behavior as RTEC events or fluents include events
such as suddenSteer, weaving, as well as fluents such as laneChange, atLane1,
atLane2, overSpeed, and hardBraking. The aggressive driving behavior is
represented as a boolean fluent defined as shown in the Definition 2.
Similarly the distracted driver behavior is a boolean fluent defined as
conjunction of events like laneDrifting, straddling, and fluents like
laneChange, atLane1, atLane2, slowSpeed and normalBraking.

inA(aggressiveDriving(v) = true, T) $\leftarrow$ hoA(atLane1(v), T) $\land$
$\neg$ hoA(atLane2(v), T) $\land$ hoA(hardBraking(v), T) $\land$
hoA(laneChange(v), T) $\land$ hoA(overSpeed(v), T) $\land$
hA(weaving(v), T) $\land$ hA(suddenSteer(v), T) $\land$
$\neg$ hA(safeDriving(v), T) $\land$ $\neg$ hA(distractedDriving(v), T) .

List of Definitions 2 Characterization of Aggressive Driver Behavior

To express the dependencies when modeling the driver behavior, we define
derived events, i.e., the events that occur due to the change in state or
value of another fluent and/or the occurrence of another event (e.g., the
effect of a normal or harsh braking on a vehicle’s speed). These events
indicate when specific actions occur in the traffic due to a combination of
particular conditions. If R denotes the rules passed to the WPM2 solver,
before invoking the WPM2 solver, we find relevant rules corresponding to
derived events based on current ongoing events [20]. We identify the rules or
relations $\textbf{R'}\subseteq\textbf{R}$ that lead to derived events based
on the current events, either directly or indirectly. The dependency is
expressed via a dependency graph $G$, where the vertices denote the
rules/relations, and the (directed) edges denote the dependency between them.
We then take the transitive closure of $G$ (say $G^{\prime}$) using the
Floyd–Warshall algorithm. Thus, in $G^{\prime}$ an edge $i\rightarrow j$
denotes that $j$ is directly or indirectly dependent on $i$. Finally, the
rules R’ expressed in CNF are then passed on to a WPM2 solver [21].

##  IV Experimental Evaluation

In this section, we evaluate our framework on our collected PiCarX dataset.
The effectiveness of our framework is determined by the following metrics: (a)
the accuracy and (b) the percentage of errors in driver behavior recognition.
The experiments were performed on a computer with Intel(R) Core(TM) i7-7700
CPU @ 3.60 GHz, 32 GB RAM, and 1 TB SSD and SWI-Prolog 8.2.3.

###  IV-A Modeling Vehicular Driving

The vehicular traffic on a roadway has been characterized by numerous models
starting with the early 1950s. The models can be microscopic (i.e., model the
behavior of each vehicle) or macroscopic (i.e., model the behavior of the
traffic as a whole). Numerous microscopic (or “car-following”) models exist,
which are reviewed in a recent article [22] that also examines the
incorporation of human factors into these models. Most models are continuous
time, continuous space type, and for a single lane, express acceleration of a
vehicle as a function of its current speed, distance to and speed of next
vehicle (and sometimes the previous vehicle as well), etc. Most models also
introduce some random slowdowns to model human behavior and to break the
strict vehicle following behavior. Generally, lane change behavior is tacked
on to the single-lane models with a set of rules concerning when lane change
can occur; however, such models quickly become very difficult to analyze
mathematically.

In our implementation, we used the so-called cellular automaton (CA) model,
which simplifies the introduction of complex rules and simulation
implementation by discretizing both time and space. In the CA model, a roadway
is seen as a sequence of cells of some fixed size. At any point in time, a
cell could be empty or occupied. That is, a cell can be occupied by only one
vehicle, although it is possible to model large vehicles that occupy multiple
consecutive cells. In each time step, a vehicle may move over some integral
number of cells depending on its speed and the availability of the cells
ahead. The CA model makes it easy to introduce complex rules that account for
various situations, including the presence of signals or other traffic control
mechanisms. Complex lane change rules can also be coded; for example, coding a
lane change decision based on the number of free cells ahead and behind in
both the source and target lanes. It is typical to assume that a lane change
always occurs in a single step, and the discrete model is well suited for this
kind of discontinuous change. CA model makes it easy to provide various forms
of driving personalities to a vehicle in terms of vehicle following, lane
change, safe/unsafe vehicle position in a cell, etc.

###  IV-B Experimental Setup and Dataset Collection

One significant difficulty in conducting this work was the scarcity of real
datasets that provide both the vehicular motion parameters and the external
videos that we can analyze.

This paper can be considered a proof of concept (PoC) of characterizing the
vehicular behavior remotely and non-intrusively without any involvement of the
vehicle/driver or deployment of any further instrumentation in the vehicles.
In particular, this paper aims to study how accurately this can be done by
comparing it against the ground truth. Unfortunately, it is impossible to
conduct such a study with actual vehicles on the road because of safety
concerns since we need the vehicles to perform unsafe maneuvers. Also, to
compare the video monitoring results against the ground truth, we need to tap
into the OBD and obtain detailed vehicular information.

In order to get around these difficulties, we set up an entire infrastructure
using the automated toy cars, known as “PiCarXs” [23] in an indoor basement
environment. Each PiCarX carries a raspberry pi board for programmatically
controlling the car. We assembled six such cars and used a logically
centralized controller to independently control each car’s behavior remotely
over the WiFi link. We also set up a “roadside” external camera to record the
videos of the cars independently and analyze those videos in real time to
determine and classify the behavior of each PiCarX. Each PiCarX used the
2-lane extension of the basic Cellular Automata (CA) car-following model [24]
and further updated the model to introduce different types of driver
behaviors. The software control of the vehicles can read their current speed
and other parameters via appropriate sensors. We can then analyze the driver
behavior for both in-vehicle sensing units and roadside camera units using a
mixture of CNN models and logic-based behavior analysis. Finally, we validate
our claim of analyzing driver behavior “only” from the roadside units by
correlating the in-vehicle data and the roadside data.

Thus, we collected such a dataset using our PiCarX setup. We captured High
Definition Video Streams (HDVS) at 30 FPS with PicarX cars imitating different
driver behavioral patterns on the road as discussed in section III-A. By
varying speed limits, we also capture the driving traffic patterns on
different road types, such as highways versus local roads. Along with
capturing the video streams, we also record the eight in-vehicular driving
parameters from all six robot cars, including vehicle orientation,
acceleration, deceleration, braking, steering angle, lateral position, and
lane change maneuvers to the left and right lanes.

Fig. 2(a) and (b) shows the front and side view of PicarX used in the
experiments, respectively. The experimental setup is shown in Fig. 3 and shows
two distinct lanes with a single direction flow of traffic. We use a tripod
and Sony FDR-AX33 Camcorder to record the simulation videos.

(a) Front View of PiCarX

(b) Side View of PiCarX

Fig. 2: PiCarX used in the experiments.

We conducted five video simulations for our controlled environment, with each
video being around 1-2 minutes long and averaging around 10,000 video frames.
In addition, to simulate the in-vehicle framework as discussed in section III,
we have approximately 2700 seconds of time-series data for all required in-
vehicle driving parameters as discussed above.

In addition to the simulated videos, we select approximately 48 real-world
traffic videos from our other dataset, TU_DAT dataset to evaluate the
effectiveness of our proposed work. TU_DAT dataset was used in our previous
work [25] to predict and resolve anomalous situations in CCTV traffic-videos,
which contain a diverse collection of accident videos collected in challenging
environments. Table III shows the details of the data collected and the number
of samples.

###  IV-C Implementation of Cellular Automata (CA) Model

We implement the CA car-following model in each PiCarX and then extend the
model to implement different driving behaviours. A classical CA model is a
uniform lattice of cells representing an identical finite automaton with a
state and a transition function; each cell takes its state and the states of a
set of neighboring cells defined by a time and space invariant geometrical
pattern. Starting with an initial condition, the CA evolves by sequentially
activating all transition functions simultaneously. We use a generic, multi-
layered and complete cellular automaton simulation engine in Python called
“Cellular Automata General Environment” (CAGE) [26]. The environment supports
multilayered grids, which provide several ways for the formation of
neighborhoods. It enables the definition of transition rules that can be
formulated algorithmically to reflect real-time driver behavior. Furthermore,
the rules can be specified at different spatial levels and can change as a
function of space and time, making it suitable for use alongside a logical
reasoning tool.

In our two-lane experimental setup, we implement abstractions of the CA model,
road topology, and neighborhood information using CAGE to represent the
various driver behaviors and a two-lane car-following model. An address in
CAGE is a tuple of one or more integers representing the location of a cell
within a given topology. Topologies determine the arrangements of cells in a
network, and neighborhoods encapsulate the translation of addresses to a list
of their neighbors, taking into account the topology they are connected with.
The Map is the high-level class used by the automaton to perform operations on
the cellular network. It is a combination of the topology and the Neighborhood
classes. For the purposes of our experiments, we implemented a line map
topology with a radial neighborhood for each lane having a total of 35 cells.

TABLE III: Statistics of Datasets Used Data Collected | Behavior | #Samples
---|---|---
Road-Side Camera Video Data (PicarX) | Safe | 410
Distracted | 340
Aggressive | 300
In-Vehicular Data (PicarX) | Safe | 125
Distracted | 115
Aggressive | 110
TU-DAT Dataset | Safe | 1200
Distracted | 720
Aggressive | 960

###  IV-D PicarX Object Detection

As discussed in III-C, the roadside camera or RECs use YLLO-based object
detection and tracking framework and a logical-reasoning-based method to
classify the driver’s behavior into one of three categories. The YLLO running
on the cameras must be able to detect and classify the robot cars since we are
using robot cars to simulate road traffic consisting of different driver
behaviors. To train YLLO to recognize the robot cars, we constructed our own
training and testing set by selecting positive samples from the recorded
videos. For negative samples, we have utilized the frames from our toy car
experiment videos used in our previous work [17]. Overall we have a total of
18,550 samples used to train the model. To reduce the workload of annotating
the dataset, we have used an auto annotation tool [27], which is based on a
semi-supervised architecture, where a model trained with a small amount of
labeled data is used to produce the new labels for the rest of the dataset.
Since YLLO is a CNN-based object detection model, it requires large amounts of
data. Hence, we have used various augmentation techniques to reach sufficient
data amounts, with most augmentations occurring at run-time using Keras built-
in functions. Keras offers several techniques for performing image
augmentation online, which means that the augmentation is done as each image
is processed by the network. Consequently, multiple augmentations can be
performed without the need to save each image separately on the computer. The
augmentations used were flipping, translation, shear, and rotation. The
trained model has an overall identification accuracy of 90.11%. Fig. 3 shows
the PiCarX detection and tracking results of the trained YLLO model. Due to
space limitations, the trained model’s results are omitted from this paper.

(a) Without object detection

(b) PicarX detection and tracking

Fig. 3: PiCarX object detection and tracking.

We evaluate the performance of the proposed driver behavior detection system
using both roadside cameras and an in-vehicle framework. We evaluate the
classification performance of the driver behavior using standard performance
metrics such as precision, recall, and accuracy.

###  IV-E Results and Discussion

Estimating various vehicular parameters, such as speed, vehicle orientation,
etc., forms the basis of driver behavior analysis [28]. For this, we need to
first calibrate the camera so that it is possible to correct the inherent
perspective distortion in the images. The perspective effect relates 3D points
on the road (world) coordinate system to 2D pixels on the image plane
differently. This effect assigns distinct informational contents to different
image pixels. The objective of Inverse Perspective Mapping (IPM) is to invert
the perspective effect, thereby imposing a uniform distribution of information
across the image plane. To map the front-view image smoothly into a bird’s-eye
view for videos captured with PicarX cars, we employ the IPM technique.

TABLE IV: Performance Results using PicarX Dataset Data Collected |  Driving Behavior |  Precision |  Recall |  Accuracy
---|---|---|---|---
Road-Side Camera |  Safe |  100 |  98.78 |  99.52
Distracted |  95.95 |  98.22 |  98.09
Aggressive |  97.98 |  97.0 |  98.57
In-Vehicular Data |  Safe |  100 |  99.2 |  99.71
Distracted |  98.28 |  99.13 |  99.14
Aggressive |  99.09 |  99.09 |  99.43

The performance of the proposed roadside camera-based driver behavior analysis
framework is evaluated by first detecting a set of micro behaviors, which are
then composed to classify the driver behavior into safe, distracted, and
aggressive driving behaviors. We validate the proposed framework against the
logic-based reasoning framework operating in individual vehicles, which
provides the actual values of driving parameters such as speed, orientation,
distance from the car in front, distance from the left or right side of the
lane, etc. Table IV shows the results, and it can be seen that the roadside
camera framework achieves an average precision and recall of 97.98% and
98,005%, respectively, and an average accuracy of 98.73% which are averaged
over five runs of the experiments. The difference between these performance
metrics values and those of the in-vehicle framework is quite small, ranging
between 1-2 percent.

In addition, we analyzed the errors that may arise when characterizing driver
behavior via roadside cameras.

Fig. 4: Error analysis.

The outcome of this analysis is depicted in Fig. 4, where the x-axis
represents the types of errors and the y-axis represents the percentage of
errors across all runs of the experiments described above. It can be seen that
object detection and tracking errors are approximately 0.57%, and driving
signal parameter estimation errors in comparison to the ground truth values
are approximately 1.33%. Errors are nearly negligible, demonstrating the
effectiveness of roadside camera-based driver behavior characterization. Based
on our analysis, we conclude that the proposed framework can be used
effectively to determine a driver’s non-intrusive behavior without requiring
the installation of additional sensors within the vehicles.

TABLE V: Performance Results using TU_DAT (without IPM) Dataset | Driving Styles | PRECISION | RECALL | ACCURACY
---|---|---|---|---
TU-DAT | Safe | 95.0 | 95.0 | 95.83
Distracted | 83.5 | 84 | 91.7
Aggressive | 93.75 | 93.75 | 95.8

In addition to the recorded videos using PicarX, we have evaluated the
proposed mechanism using the TU_DAT dataset. Since this dataset is comprised
of CCTV traffic videos of anomalous situations from various parts of the
world, these cameras’ intrinsic and extrinsic parameters may be unknown or
different from one another due to mounting setups and camera types. Therefore,
individual calibration of the captured videos is not possible. Table V shows
the performance results of the proposed roadside camera framework on TU_DAT,
it can be seen that the average precision and recall values are 90.694%, and
the accuracy is approximately 94.5%. It is important to note that the worse
accuracy here is primarily due to lack of perspective correction; if the
camera position and angles were known, we believe that the accuracies here
would be similar to those obtained using PiCarX.

TABLE VI: Contribution of Micro-behavior to Aggressive/Distracted Driving after and before driver feedback Micro Behaviors | Aggressive | Distracted
---|---|---
Close to left side of the lane | 0.74 | 0.73
Close to right side of the lane | 0.77 | 0.73
Drifting from center of the lane | 0.7 | 0.59
Too near to a vehicle in the same lane | 0.71 | 0.69
Too near to a vehicle in the adjacent lane | 0.7 | 0.84
Speeding | 0.61 | N/A
Harsh braking | 0.69 | N/A
Changing Lanes | 0.64 | 0.62
Sudden steering | 0.66 | N/A
Slow Speed | N/A | 0.62

Table. VI shows how the change in micro-behavior by a driver as a result of
feedback can reduce the instances of aggressive and distracted driving. Each
row in this table represents the situation where the driver is informed about
its undesired micro-behavior, and as result, the driver reduces that behavior
by 50%. (This is done one at a time for each row, not cumulatively). The
columns show the aggressive and distracted driving occurrences that include
the said micro-behavior. The reported quantity is the ratio of the instances
after and before the feedback and thus shows the impact of the feedback. The
Not Applicable (N/A) reference indicates that the corresponding micro-behavior
is not considered when classifying a driver as aggressive or distracted. The
table demonstrates that the most effective micro-behavior improvements for
aggressive and distracted drivers are speed control and maintaining the center
of the lane. Thus, such an analysis can tell us the most effective feedback
for manual or automated vehicles.

##  V Conclusions and Discussion

The ITS support involves, at a minimum, roadside traffic monitoring
infrastructure with monitoring nodes deployed on the electric poles and
usually has cameras and some computing and backhaul communications capability
to send raw or processed video data to a central entity such as a cloud. Such
a minimal setup does not require any communications capability in the vehicles
themselves, and we demonstrate in this paper, through emulation of such an
environment, that such a capability is adequate to monitor and classify
behaviors of various drivers in the camera view without any involvement of or
awareness by either the car or the driver. As such, this behavior
characterization can only be exploited for improving the signage on the road
that warns the drivers or is used the change the requirements (e.g., speed
limit, lane following requirements, traffic control options, etc.).

Our proof of concept using automated robot cars demonstrates that driver
behavior characterization can be done very accurately by only using the view
from the roadside cameras. We believe this is a significant result and paves
the way for further development of the technique. Two notable additions would
be tracking vehicles through a rich feature set and fusing views from multiple
cameras to cover areas that cannot be covered well by a single camera. Another
piece is the use of characterization in providing appropriate feedback to
drivers and automated vehicles, although the form of feedback would be very
different and needs to be studied. This obviously requires V2I (vehicle to
infrastructure) communications capability, which is being implemented already.
Automated vehicles, when fed with driving behavior information about nearby
vehicles, can use it to reduce the traffic waves and other inefficiencies due
to manually driven vehicles [29]. The analysis can also be extended for more
general environments, such as those with more than two lanes, bi-directional
traffic, traffic signals, sharp curves, poor road conditions, etc.

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# Superfluid Rivers in Spinning-down Neutron Stars

Yuri Levin  Physics Department and Columbia Astrophysics Laboratory, Columbia
University, 538 West 120th Street New York, NY 10027, USA  Center for
Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY
10010, USA  Department of Physics and Astronomy, Monash University, Clayton
VIC 3800, AUSTRALIA  Bennett Link  Department of Physics, Montana State
University, Bozeman, MT 59717, USA

###### Abstract

We study the motion of neutron superfluid vortices in a spinning-down neutron
star, assuming axisymmetry of the flow and ignoring motion of vortices about
the rotation axis. We find that the vortex array, if initially rectilinear, is
soon substantially deformed as the star spins down; vortices are swept outward
by the Magnus force, accumulating in regions of the inner crust where they
pin, accompanied by significant bending of the vortex array. As the star spins
down to below a spin rate of $\sim 20$ Hz (twice the spin rate of the Vela
pulsar), the Magnus and pinning forces gradually compress the vortex array
into dense sheets that follow spherical shells. In some cases, the vortex
array bends on itself and reconnects, forming one or more tori of vortex rings
that contain superfluid “rivers” with significant angular momentum. Vortex
sheets are likely to form near the base of the inner crust, in the regime of
nuclear pasta.

##  1 Introduction

The manner in which the neutron superfluid (nSF) spins down in a neutron star
(NS) is likely related to various observed spin irregularities. Spin glitches
in NSs are generally attributed to sudden transitions, driven by stellar spin-
down, between different metastable states of pinned vorticity in the inner
crust (Anderson & Itoh, 1975) or the outer core (Ruderman et al. 1998; see
Haskell & Melatos 2015 for a review). Pulsars also exhibit long-timescale
stochastic spin irregularities, known as timing noise (see Hobbs et al. 2010
and references therein). Timing noise could arise from a number of superfluid-
related processes such as turbulence (Greenstein, 1970; Melatos & Link, 2014;
Lasky et al., 2015), stochastic transport of vorticity (Alpar et al., 1985;
Jones, 1990a), vortex avalanches (Cheng et al., 1988; Warszawski & Melatos,
2008), and magnetospheric processes unrelated to superfluid dynamics (Cheng,
1987a, b; Urama et al., 2006; Lyne et al., 2010).

Knowledge of the structure of the superfluid flow - in particular, columnar
and laminar versus turbulent - is essential for the development of realistic
models of NS spin-down and timing irregularities. Ruderman & Sutherland (1974)
argued long ago that the array of superfluid vortices in a spinning-down NS is
effectively very rigid and so remains rectilinear despite the presence of
forces on vortices that vary with position in the star; they concluded that
the vortex array will evolve without significant bending or tangling - a
superfluid version of the Taylor-Proudman theorem of classical hydrodynamics -
precluding the development of superfluid turbulence. As a result of the work
of Ruderman & Sutherland (1974), and also perhaps as an understandable
simplification of a complex problem, much work on the hydrodynamics of the NS
interior has assumed that the vortex array remains rectilinear as the NS
evolves (e.g., Alpar et al. 1984a, 1985; Link & Epstein 1996; Larson & Link
2002; Antonelli & Pizzochero 2017). Other work has assumed that the nSF finds
itself in a turbulent state, and has explored the consequences of this
assumption (e.g., Greenstein 1970; Andersson et al. 2007; Lasky et al. 2013;
Melatos & Link 2014; Haskell et al. 2020). A number of interesting simulations
of superfluid turbulence in NSs introduce laminar-turbulent transitions by
hand (e.g., Peralta et al. 2006).

In this paper we address the evolution of the superfluid vortex array in
axisymmetry, accounting for vortex pinning in the inner crust. We find that
for realistic vortex pinning, the Taylor-Proudman theorem is strongly
violated. Instead, in regions where the pinning strength has strong gradients,
vortices accumulate into dense spherical sheets. For some distributions of the
pinning force, the spherical sheets bend and reconnect to make tori with with
superfluid rivers flowing inside them. Generally, we suspect that the vortex
array evolves into complex configurations that could become tangled, with an
accompanying superfluid flow that could be turbulent.

The plan of this paper is as follows. In §2 we derive the equations of motion
for superfluid vorticity. In §3, we derive a superfluid version of Taylor-
Proudman Theorem, and show that the theorem is typically violated when pinning
forces are present. In §4 we describe the motion of a single vortex. In §5 we
describe the global dynamics in axisymmetry. In §6 we present a simple global
model for vortex mobility that is easily solved. In §7 we present our
simulations. In §8 we discuss our results.

##  2 Superfluid Dynamics

A superfluid rotates by establishing an array of quantized vortices of
microscopic cross section. The circulation about a vortex is fixed according
to

$\oint\mathbf{v}_{s}(\mathbf{r},t)\cdot\mathbf{d}l=\kappa,$ (1)

where $\mathbf{v}_{s}(\mathbf{r},t)$ is the superfluid velocity at location
$\mathbf{r}$ at time $t$; $\kappa=2\pi\hbar/m$ is the circulation where $m$ is
the mass of the fundamental boson in the system, twice the neutron mass in the
case of a neutron superfluid. An important distinction between a superfluid
and a classical fluid is that vortices in a superfluid are physical entities
that can move at their own velocity distinct from that of the superfluid,
interact with their environment, and pin or experience physical drag from
other parts of the system. Regardless of how superfluid vortices move, each
vortex always carries a quantum of circulation according to eq. 1.

Consider an incompressible superfluid. The momentum equation in the inertial
frame is

$\frac{\partial\mathbf{v}_{s}}{\partial
t}+\mathbf{v}_{s}\cdot\nabla\mathbf{v}_{s}=-\nabla\mu-\nabla\phi+\rho_{s}^{-1}\mathbf{f},$
(2)

where $\mu$ is the chemical potential, $\phi$ is the gravitational potential,
$\mathbf{f}$ is the force per unit volume acting on the superfluid from vortex
pinning, mutual friction, and vortex tension, and $\rho_{s}$ is the mass
density of the superfluid. It is convenient to work in the frame corotating
with the crust at angular velocity $\mbox{\boldmath$\Omega$}(t)$ and we denote
quantities in this frame with the superscript *. The velocity is

$\mathbf{v}^{*}_{s}(\mathbf{r},t)=\mathbf{v}_{s}(\mathbf{r},t)-\mbox{\boldmath$\Omega$}\times\mathbf{r}.$
(3)

In the rotating frame, the momentum equation is

$\frac{\partial\mathbf{v}^{*}_{s}}{\partial
t}+\mathbf{v}^{*}_{s}\cdot\nabla\mathbf{v}^{*}_{s}=-\nabla\left(\mu+\phi-\frac{1}{2}(\mbox{\boldmath$\Omega$}\times\mathbf{r})^{2}\right)-2\mbox{\boldmath$\Omega$}\times\mathbf{v}^{*}_{s}-\frac{d\mbox{\boldmath$\Omega$}}{dt}\times\mathbf{r}+\rho_{s}^{-1}\mathbf{f}.$
(4)

Taking the curl, and using a vector identity111$\frac{1}{2}\nabla
v^{2}=\mathbf{v}\times\mbox{\boldmath$\omega$}+\mathbf{v}\cdot\nabla\mathbf{v}$.,
gives

$\frac{\partial\mbox{\boldmath$\omega$}}{\partial
t}=\nabla\times\left(\mathbf{v}^{*}_{s}\times\mbox{\boldmath$\omega$}+\rho_{s}^{-1}\mathbf{f}\right).$
(5)

Here $\omega$ is the total vorticity measured in the inertial frame

$\mbox{\boldmath$\omega$}=\nabla\times\mathbf{v}_{s}=2\mbox{\boldmath$\Omega$}+\nabla\times\mathbf{v}^{*}_{s}=2\mbox{\boldmath$\Omega$}+\mbox{\boldmath$\omega$}^{*},$
(6)

and $\mbox{\boldmath$\omega$}^{*}\equiv\nabla\times\mathbf{v}^{*}_{s}$ is the
vorticity in the rotating frame. In a quantum fluid, vortices are persistent
structures to which we can assign a local velocity $\mathbf{v}^{*}_{v}$ in the
rotating frame. Vortex motion obeys the conservation law,

$\frac{\partial\mbox{\boldmath$\omega$}}{\partial
t}=\nabla\times\left(\mathbf{v}^{*}_{v}\times\mbox{\boldmath$\omega$}\right).$
(7)

The force $\mathbf{f}$ contains contributions from mutual friction and bending
forces of the vortex array. Comparing eqs. 5 and 7 gives the vortex equation
of motion

$\rho_{s}^{-1}\mathbf{f}=(\mathbf{v}^{*}_{v}-\mathbf{v}^{*}_{s})\times\mbox{\boldmath$\omega$},$
(8)

up to the gradient of an arbitrary scalar potential, which we absorb into the
definition of $\mu$. The right-hand side is the Magnus force (per unit mass).
For $\mathbf{f}=0$, the vortex lines move with the superfluid, while for
$\mathbf{f}\neq 0$, vortices are forced to move locally at a different
velocity $\mathbf{v}^{*}_{v}\neq\mathbf{v}^{*}_{s}$.

We can write eq. 8 as an equation of motion for a single vortex by defining
the areal density of vortices $n_{v}$ related to the vorticity through

$n_{v}\mbox{\boldmath$\kappa$}=\nabla\times{\bf
v}_{s}\equiv\mbox{\boldmath$\omega$},$ (9)

where $\kappa$ has magnitude $\kappa$ in the direction of the local vorticity
$\omega$. Noting that the force per unit length on a single vortex is
$\mathbf{F}_{v}=n_{v}^{-1}\mathbf{f}$, gives

$\rho_{s}\mbox{\boldmath$\kappa$}\times(\mathbf{v}^{*}_{v}-\mathbf{v}^{*}_{s})+\mathbf{F}_{v}=0.$
(10)

##  3 Superfluid Taylor-Proudman Theorem

For steady rotation of a classical fluid in the limit that the Coriolis force
is much larger than the inertial force, the fluid flow does not depend on
position along the axis of rotation. If a solid body is slowly pulled through
the fluid, the fluid must flow around the body, and a vertical Taylor column
is established along the axis of rotation. In a rotating superfluid the
analogous effect is seen experimentally as vortices tend to become straight
and align with the rotation axis in a rapidly rotating vessel. We review the
derivation of the Taylor-Proudman for a rotating fluid. We then show that the
theorem is violated in the presence of pinning forces in a quantum fluid.

In a NS, the spin-down torque acts on the crust and is communicated to the nSF
over a long time scale $t_{\rm sd}$, so that $\Omega t_{\rm sd}\ll 1$. We
therefore consider very slow variations in $\omega$ when a spatially-dependent
force $\mathbf{f}$ is present. Eq. 7 gives

$(\mbox{\boldmath$\omega$}\cdot\nabla)\mathbf{v}^{*}_{s}-(\mathbf{v}^{*}_{s}\cdot\nabla)\mbox{\boldmath$\omega$}=-\nabla\times(\rho_{s}^{-1}\mathbf{f}).$
(11)

The Taylor-Proudman theorem applies when the Coriolis force dominates both the
inertial force in eq. 4 and $\rho_{s}^{-1}\mathbf{f}$. In terms of vorticity,
the Coriolis force dominates the inertial force when
$2\Omega\gg|\mbox{\boldmath$\omega$}^{*}|$. In this limit, with
$\mbox{\boldmath$\Omega$}=\hat{z}\Omega$ where $\hat{z}$ is a unit vector
along the rotation axis, eq. 11 becomes

$\frac{\partial\mathbf{v}^{*}_{s}}{\partial
z}=-\frac{1}{2\Omega}\,\nabla\times(\rho_{s}^{-1}\mathbf{f}).$ (12)

Typically in a classical fluid $\rho_{s}^{-1}\mathbf{f}$ is determined by the
properties of, for example, an obstacle affecting the flow. In the limit of
fast rotation the right-hand side goes to zero and we arrive at the Taylor-
Proudman theorem; all three components of $\mathbf{v}^{*}_{s}$ become
independent of $z$. For a quantum fluid $\rho_{s}^{-1}\mathbf{f}$ might be due
to mutual friction, which scales as the vortex density and hence as $2\Omega$
in the limit of fast rotation; then $\partial\mathbf{v}^{*}_{s}/\partial z$
does not tend to zero as in a classical fluid. In cylindrical coordinates
$(\varrho,\phi^{*},z)$, and assuming axisymmetry, eq. 5 becomes in the limit
of fast rotation

$\displaystyle\frac{\partial\omega_{\varrho}}{\partial
t}=\frac{\partial}{\partial t}\left(\frac{\partial v^{*}_{\phi^{*}}}{\partial
z}\right)=-2\Omega\frac{\partial v^{*}_{\varrho}}{\partial
z}+\frac{\partial\tilde{f}_{\phi^{*}}}{\partial z}$ (13)
$\displaystyle\frac{\partial\omega_{\phi^{*}}}{\partial
t}=\frac{\partial}{\partial t}\left(\frac{\partial{v^{*}_{\varrho}}}{\partial
z}-\frac{\partial v^{*}_{z}}{\partial\varrho}\right)=2\Omega\frac{\partial
v^{*}_{\phi^{*}}}{\partial z}+\frac{\partial\tilde{f}_{\varrho}}{\partial
z}-\frac{\partial\tilde{f}_{z}}{\partial\varrho}$ (14)
$\displaystyle\frac{\partial\omega_{z}}{\partial t}=\frac{\partial}{\partial
t}\left(\frac{1}{\varrho}\frac{\partial}{\partial\varrho}(\varrho
v^{*}_{\phi^{*}})\right)=2\Omega\frac{\partial v_{z}}{\partial
z}+\frac{1}{\varrho}\frac{\partial}{\partial\varrho}(\varrho\tilde{f}_{\phi^{*}}),$
(15)

where $\tilde{\mathbf{f}}\equiv\rho_{s}^{-1}\mathbf{f}$ and all quantities are
functions of the coordinates $(\varrho,z)$. If $f_{\varrho}=f_{z}=0$,
$v^{*}_{\phi^{*}}$ is independent of $z$ implying that the vortex array
remains rectilinear as it moves (Ruderman & Sutherland, 1974); the array does
not bend. Generally, though, $f_{\varrho}\neq 0$, and $v^{*}_{\phi^{*}}$
becomes dependent upon $z$; the array bends as it moves. The final term in eq.
14 becomes non-zero as the array bends. From eq. 14, we can estimate the angle
by which the vortex lattice deviates locally from straight:

$\delta\theta\sim\frac{v^{*}_{\phi^{*}}}{\varrho\Omega}\sim\frac{\rho_{s}^{-1}f_{\varrho}}{2\varrho\Omega^{2}}.$
(16)

The deformation of the vortex lattice scales as $\Omega^{-1}$ for a force that
is proportional to the vortex density. Conversely, lattice deformations can be
large in the limit of slow rotation, as the simulations we present later in
this paper show.

It is instructive to consider the Taylor-Proudman limit for a quantum fluid in
terms of vortex motion. The vortex motion in the steady state satisfies, from
eq. 7

$(\mbox{\boldmath$\omega$}\cdot\nabla)\mathbf{v}^{*}_{v}-\mbox{\boldmath$\omega$}(\nabla\cdot\mathbf{v}^{*}_{v})-(\mathbf{v}^{*}_{v}\cdot\nabla)\mbox{\boldmath$\omega$}=0.$
(17)

In the limit of fast rotation with $2\Omega\gg|\mbox{\boldmath$\omega$}^{*}|$,

$2\Omega\left(\frac{\partial\mathbf{v}^{*}_{v}}{\partial
z}-\hat{z}\,\nabla\cdot\mathbf{v}^{*}_{v}\right)\rightarrow 0.$ (18)

In steady state, the vortex motion becomes independent of the axial coordinate
$z$, the lateral vortex velocity is nondivergent and independent of the axial
coordinate, and the axial velocity decouples from the lateral velocity (see
also Holm 2001). Hence, in this steady-state limit of fast rotation, the
vortices straighten and become parallel to the rotation axis.

The Taylor-Proudman theorem generally does not hold if vortices are
immobilized in parts of the fluid volume (as opposed to only at boundaries) by
pinning forces. In this case the pinning force follows from eq. 8 with
$\mathbf{v}^{*}_{v}=0$:

$\rho_{s}^{-1}\mathbf{f}=\mbox{\boldmath$\omega$}\times\mathbf{v}^{*}_{s}.$
(19)

Now $\partial\mbox{\boldmath$\omega$}/\partial t=0$ because the vortices are
immobilized and the vortices cannot adjust their configuration to make
$\mathbf{v}^{*}_{s}$ independent of $z$ for an arbitrary pinning force. If
$\rho_{s}^{-1}\mathbf{f}_{\varrho}$ varies along $z$, for example, so will
$\mathbf{v}^{*}_{\phi^{*}}$, implying that the vortex array is not
rectilinear. This is the situation we explore further in this paper.

Finally, we note that the force $\mathbf{f}$ contains a contribution from
rigidity of the vortex lattice, which is typically tiny compared to pinning
forces; see next section.

##  4 Motion of a Single Vortex

Let the vortex shape, measured with respect to the $z$ axis, be given by the
two-dimensional displacement vector
$\mathbf{u}(z,t)=\hat{x}u_{x}+\hat{y}u_{y}$, so the vortex velocity is
$\mathbf{v}_{v}^{*}=\partial\mathbf{u}/\partial t$. For small bending angles,
so that $|\partial\mathbf{u}/\partial z|\ll 1$, eq. 10 can be written (see,
also, Schwarz 1977, 1978)

$\rho_{s}\mbox{\boldmath$\kappa$}\times\left(\frac{\partial\mathbf{u}}{\partial
t}-\mathbf{v}^{*}_{s}\right)+T_{v}\frac{\partial^{2}\mathbf{u}}{\partial
z^{2}}+\mathbf{F}_{L}=0.$ (20)

The first term is the Magnus force. The second term is the force (per unit
length) from bending the vortex; $T_{v}$ is the vortex tension. The last term
$\mathbf{F}_{L}$ is the force due to interaction of the vortex with the
nuclear lattice. The second and third terms are $\mathbf{F}_{v}$ appearing in
eq. 10. The tension is typically

$T_{v}\simeq\frac{\rho_{s}\kappa^{2}}{4\pi}\Lambda,$ (21)

where $\Lambda$ is a factor of 3-10; $T_{v}\sim 1$ MeV fm. As explained below,
vortex tension is important against pinning forces only over length scales
below $\sim 10^{3}$ fm.

The force of the lattice on the vortex has two contributions: (i) a static,
conservative contribution and (ii) a velocity-dependent, damping force due to
the excitation of lattice phonons by the moving vortex that carry energy away
from the vortex (Epstein & Baym, 1992; Jones, 1992). To treat these two
contributions, we take the simple form

$\mathbf{F}_{L}=-\nabla_{\perp}V-\eta\mathbf{v}_{v}^{*},$ (22)

where $\mathbf{v}^{*}_{c}$ is the velocity of the crust. The first term gives
the static force; $V$ is the potential, and the gradient is perpendicular to
the vortex. The second term is the drag force, and $\eta$ is the drag
coefficient. A non-dissipative, velocity-dependent contribution to
$\mathbf{f}_{L}$ is possible, but shall not be considered here; see Sedrakian
et al. (1999).

Numerical solutions of eqs. 20 and 22 for various approximations to the
nuclear lattice show that the vortex often responds with a slip-stick
character to an applied force
$\rho_{s}\mbox{\boldmath$\kappa$}\times\mathbf{v}^{*}_{s}$ (Link & Levin,
2022). For zero applied force, the vortex settles into a pinned configuration
whose solution is given by

$T_{v}\frac{\partial^{2}\mathbf{u}}{\partial z^{2}}-\nabla_{\perp}V=0.$ (23)

The response of a pinned vortex to a slowly increasing applied force depends
on the properties of the lattice, its orientation, whether the vortex-lattice
interaction is attractive or repulsive, and the drag coefficient $\eta$. We
now summarize the characteristics of vortex motion in the case of low drag, in
the sense $\eta/\rho_{s}\kappa\ll 1$. If the lattice is regular and repulsive,
for most lattice orientations the vortex is able to move through the
propagation of kinks. As the force becomes large, the vortex enters a
ballistic regime in which the conservative contribution to the lattice force
and the tension contribution average nearly to zero. In this limit, the vortex
motion is given by

$\rho_{s}\mbox{\boldmath$\kappa$}\times\mathbf{v}^{*}_{v}-\eta\mathbf{v}^{*}_{v}=0.$
(24)

For some lattice orientations, the kinks hang up and the vortex remains
pinned. As the force is increased further for this case, the vortex unpins
suddenly at a critical value of the force, with Kelvin waves propagating along
its length. If the force is then reduced, the vortex eventually repins quite
suddenly, but typically at a value of the force that is smaller than the value
of the force at which the vortex unpinned. This hysteretic attribute of vortex
motion is due to the fact that the vortex is an extended object that supports
internal vibrational degrees of freedom. Impurities in the lattice tend to
prevent the propagation of kinks, giving a well-defined transition from pinned
to ballistic at a critical value of the force. If the lattice is regular and
attractive, there is a well-defined transition from pinned to ballistic for
most lattice orientations. If the lattice is strongly disordered, or if the
pinning sites are randomly positioned as in a glass, then a well-defined
transition from pinned to ballistic occurs whether the lattice is attractive
or repulsive. Typical values for the critical force are $10^{16}$ dyn cm-1 to
$10^{17}$ dyn cm-1, corresponding to critical superfluid velocities of $v_{\rm
crit}=|\mathbf{v}^{*}_{s}|$ of $10^{-5}c$ to $10^{-4}c$.

The motion in the ballistic regime is given by solution to eq. 24:

$\mathbf{v}^{*}_{v}=\cos\theta\,(\mathbf{v}^{*}_{s}\cos\theta+[\mathbf{v}^{*}_{s}\times\hat{\kappa}]\,\sin\theta)\qquad\tan\theta\equiv\frac{\eta}{\rho_{s}\kappa}.$
(25)

Locally, the vortices move at an angle $\theta$ with respect to the vector
$\mathbf{v}^{*}_{s}$; $\theta$ is sometimes referred to as the “dissipation
angle”. In the limit of low drag, $\eta/\rho_{s}\kappa<<1$ and
$\theta\simeq\eta/\rho_{s}\kappa$, giving

$\mathbf{v}^{*}_{v}\simeq\mathbf{v}^{*}_{s}+(\mathbf{v}^{*}_{s}\times\hat{\kappa})\,\theta.$
(26)

The vortex moves nearly with the superfluid, with slow drift along
$\mathbf{v}^{*}_{s}\times\hat{\kappa}$. In the limit of high drag,
$\eta/\rho_{s}\kappa>>1$ and $\theta\rightarrow\pi/2$, giving

$\mathbf{v}^{*}_{v}\simeq\left(\frac{\rho_{s}\kappa}{\eta}\right)\mathbf{v}^{*}_{s}\times\hat{\kappa}.$
(27)

The solutions described above apply for low drag. In the limit of high drag,
an unpinned vortex moves nearly with the crust, with slow drift along
$\mathbf{v}^{*}_{s}\times\hat{\kappa}$. We refer to this state of unpinned
vortex motion as “vortex drift” to distinguish the motion from that of the
low-drag, ballistic regime.

For vortex motion in the inner crust, the drag coefficient is typically
$\gamma\equiv\eta/\rho_{s}\kappa\sim 10^{-3}$ (Epstein & Baym, 1992; Jones,
1992), and depends on velocity.222For low-velocity motion ($v^{*}_{s}<10^{3}$
cm s-1), Jones (1990b) finds $\gamma\sim 10^{-5}-10^{-4}$. That calculation
assumes that vortices cannot pin, while Link & Levin (2022) find that vortices
pin strongly in this regime. In the outer core, SFn vortices are predicted to
acquire magnetization due to entrainment of protons by the neutrons that
circulate around the vortices (Alpar et al., 1984b). Electrons scatter with
the magnetic moments of the SFn vortices. The relaxation time for relative
motion between the neutrons and electron-proton plasma is (Alpar et al.,
1984b)

$\tau_{v}\sim P(\mbox{s})\mbox{ s},$ (28)

where $P(\mbox{s})$ is the spin period of the system in seconds. The
dissipation angle is related to the relaxation time at spin rate frequency
$\Omega$ through (Link, 2014)

$\sin
2\theta=(\Omega\tau_{v})^{-1}\longrightarrow\sin\theta\simeq(4\pi)^{-1}.$ (29)

According to this estimate, vortex motion in the core is in the low-drag
regime. However, neutron vortices are also expected to pin to the far more
numerous flux tubes of the outer core (see, e.g., Srinivasan et al. 1990;
Ruderman et al. 1998; Jones 1991), restricting the mobility of the neutron
vortices; in order to move, the vortices must cut through flux tubes; the
dissipation associated with this process is poorly understood.

The bending force per unit length - the first term in eq. 20 \- is $\sim
T_{v}/R_{v}$, where $R_{v}$ is the radius of curvature of a bent vortex. For a
pinning force of $10^{17}$ dyn cm-1 $\simeq 10^{-3}$ MeV fm-2, the pinning
force will dominate the bending force for
$R_{v}\hbox{${{}_{\displaystyle>}\atop{}^{\displaystyle\sim}}$}10^{3}$ fm =
$10^{-10}$ cm. Vortex tension is therefore completely negligible for the
problem of macroscopic fluid flow in a NS, though it plays an essential role
in pinning and slip-stick dynamics, as described in more detail in Link &
Levin (2022).

##  5 Global Dynamics

Consider a vortex that is initially pinned in a spinning-down NS with a nSF
flow velocity $\mathbf{v}_{s}^{*}$ in the frame of the crust. As the stellar
crust spins down, the superfluid flow speed past the pinned vortex will
approach the local critical value for unpinning. At zero temperature, the
vortex unpins only when the flow speed becomes critical. For low-drag vortex
motion, the vortex moves nearly with $\mathbf{v}_{s}^{*}$ when the vortex is
unpinned, while for high-drag vortex motion, the vortex drifts slowly
transverse to the $\mathbf{v}_{s}^{*}$ (along
$\mathbf{v}_{s}^{*}\times\hat{\kappa}$; see eq. 27).

At finite temperature, thermal excitations cause a pinned vortex to slowly
“creep” when the flow velocity is slightly subcritical (Alpar et al., 1984a;
Link et al., 1993; Sidery & Alpar, 2009; Link, 2014). The dissipation angle
for motion through creep can differ from the value given by
$\tan\theta=\eta/{\rho_{s}\kappa}$ in the ballistic regime. As we show in a
forthcoming publication (Link & Levin, 2023), for low-drag thermal creep the
vortex has comparable velocity components along and transverse to
$\mathbf{v}_{s}$, while for high-drag thermal creep the vortex motion is
nearly transverse to $\mathbf{v}_{s}^{*}$, as for zero temperature.

If vortex motion has a significant component along $\mathbf{v}_{s}^{*}$, the
vortex array will become substantially twisted as the NS spins down. For both
illustration and simplicity, we henceforth neglect vortex motion along
$\mathbf{v}_{s}^{*}$, and consider axisymmetric superfluid flow. We assume
that, in the rotating frame, the vortices are always poloidally-directed. This
condition is rigorously satisfied in the limit of high drag, but is not
satisfied in the limit of low drag if stellar spin down causes the vortex
array to make transitions between a pinned state and a ballistic state. The
limitations of our treatment are discussed §8.

We use cylindrical coordinates in the inertial and rotating frames with common
origin and $z$ axes that coincide:

$\displaystyle\mathbf{r}$ $\displaystyle=\varrho\hat{\varrho}+z\hat{z}$ (30)
$\displaystyle\mathbf{r}$ $\displaystyle=\varrho\hat{\varrho}^{*}+z\hat{z}.$
(31)

The rotation vector of the crust $\Omega$ is along $\hat{z}$. It is convenient
to work with the total circulation, defined as

$\Phi(\varrho,z)\equiv 2\pi\varrho v_{s}(\varrho,z).$ (32)

The quantity $\Phi/\kappa$ is equal to the total number of vortices enclosed
by a horizontal circle of radius $\varrho$ at height $z$.

We now establish a theorem for the time evolution of the circulation: so long
as $v^{*}_{s}$ exceeds the local threshold for unpinning, the Magnus force
will drive a local reduction in the circulation. Eqs. 7 and 8 give, by
Stokes’s Theorem,

$\frac{\partial}{\partial
t}\Phi(\varrho,z,t)=-2\pi\varrho\,\mathbf{v}^{*}_{v}\cdot(\hat{\phi}^{*}\times\mbox{\boldmath$\omega$})=2\pi\varrho\,\rho_{s}^{-1}f_{\phi^{*}}.$
(33)

The rate of change of the circulation is equal to the flux of the total
vorticity vector $\omega$ through the horizontal circle centered on the axis
of symmetry, whose points have fixed poloidal coordinates $(\varrho,z)$. If
the superfluid rotates faster than the crust (viz., $v_{s}^{*}>0$), the
azimuthal component of drag force, which opposes the azimuthal motion of
vortices, is along $-\hat{\phi^{*}}$, so $f_{\phi^{*}}$ is negative. From eq.
33, the Magnus force then drives a local reduction of the circulation; this
remains valid even if the vortices are directed opposite to the direction of
stellar rotation.

##  6 Global Model for Vortex Mobility

Based on eq. 33, we consider a simple model for the mobility of vortices: the
vortex is mobile only when the Magnus force exceeds a certain position-
dependent critical value, that is, when $v^{*}_{s}$ exceeds $v_{\rm
crit}(\mathbf{r})$. The critical velocity $v_{\rm crit}$ is a function of the
mass density, which has a spherically-symmetric distribution, so $v_{\rm
crit}$ is a function of the spherical radius $r$. When this unpinning
threshold is exceeded, vortex motion reduces the circulation locally.
Returning to the inertial frame, where the crust velocity is
$\varrho\Omega(t)\hat{\phi}$ and the superfluid velcocity is
$\mathbf{v}_{s}=(\varrho\Omega(t)+v_{s}^{*})\hat{\phi}$, we have the following
prescription for how the speed $v_{s}$ evolves with time:

  * •

So long $v_{s}(\varrho,z)<\varrho\Omega(t)+v_{\rm crit}(r)$, the constant-
circulation surface is pinned ; $\Phi(\varrho,z)$ and $v_{s}(\varrho,z)$
remain fixed.

  * •

Once $v_{s}(\varrho,z)\geq\varrho\Omega(t)+v_{\rm crit}(r)$, the constant-
circulation surface (representing vortices) becomes unpinned and moves under
the action of the Magnus force. This motion reduces $v_{s}(\varrho,z)$. We
assume that the timescale for the unbalanced vortex motion is much shorter
than that of the pulsar spindown; in that case $v_{s}$ reduces to the value
very close to $\varrho\Omega(t)+v_{\rm crit}(r)$ and the vortices/constant
circulation surfaces become pinned again.

We consider steady spin-down of the crust, and therefore once
$v_{s}(\varrho,z)$ reaches $\varrho\Omega+v_{\rm crit}(r)$, it will track this
quantity as $\Omega$ further decreases. Hence,

$v_{s}(\varrho,z,t)=\min\left[\varrho\Omega_{0},\varrho\Omega(t)+v_{\rm
crit}(r)\right].$ (34)

Here $\Omega_{0}$ is the initial angular velocity of the star, and it is
assumed that the superfluid was initially co-rotating with the star. The
circulation is

$\Phi(\varrho,z,t)=2\pi\varrho\,\min\left[\varrho\Omega_{0},\varrho\Omega(t)+v_{\rm
crit}(r)\right].$ (35)

The circulation is straightforward to calculate. The contours of constant
circulation trace out vortices.

The remarkable feature of eq. 34 is that the superfluid velocity is determined
locally, irrespective of the history of the pulsar spindown and of the pinning
strengths at other location. This feature is a consequence of the assumed
axisymmetry with negligible vortex motion around the rotation axis. The
neighbouring constant circulation surfaces do not feel attraction or repulsion
to each other; the motion of one does not change the superfluid velocity at
the other. One immediate corollary is that the strength of pinning in the core
does not affect the motion of nSF vortices in the crust.

In the initial stages of NS spin down, the star is spinning down quickly and
$\varrho\Omega_{0}>\varrho\Omega(t)+v_{\rm crit}(r)$ everywhere in star except
very close to the rotation axis. Then

$v_{s}(\varrho,z,t)\rightarrow\varrho\Omega(t)+v_{\rm crit}(r).$ (36)

The vorticity vector can be expressed as
$\mbox{\boldmath$\omega$}=\hat{\kappa}n_{v}\equiv\kappa\mathbf{n}_{v}$, where
$\mathbf{n}_{v}$ is the vector areal density of the vortex array. The vortex
density $\mathbf{n}_{v}$ at location ($r$, $\theta$) in spherical coordinates
is given by (except close to the rotation axis)

$\kappa\mathbf{n}_{v}\equiv\hat{\kappa}n_{v}=\nabla\times\mathbf{v}_{s}=\hat{r}\left[2\Omega(t)\cos\theta+\frac{\cot\theta}{r}v_{\rm
crit}(r)\right]-\hat{\theta}\left[2\Omega(t)\sin\theta+\frac{1}{r}\frac{d}{dr}\left\\{rv_{\rm
crit}(r)\right\\}\right].$ (37)

As described in the next section, radial gradients in the pinning force can be
large in the inner crust. In such regions (except close to the rotation axis)

$\kappa\mathbf{n}_{v}\simeq\hat{z}2\Omega(t)-\hat{\theta}\frac{1}{r}\frac{d}{dr}\left\\{rv_{\rm
crit}(r)\right\\}.$ (38)

At high spin rate, $\mathbf{n}_{v}$ is uniform and points in the $z$
direction. As $\Omega(t)$ becomes sufficiently small, the vorticity vector
begins to point along $\pm\hat{\theta}$ in regions where the radial gradient
in the pinning force is large. This transition begins when
$2\Omega(t)\hbox{${{}_{\displaystyle<}\atop{}^{\displaystyle\sim}}$}r^{-1}d\\{rv_{\rm
crit}(r)\\}/dr$, corresponding to a spin period of

$P\hbox{${{}_{\displaystyle>}\atop{}^{\displaystyle\sim}}$}\frac{4\pi\sigma_{p}}{v_{\rm
crit}^{\rm max}}\simeq
0.2\,\left(\frac{\sigma_{p}}{0.05R}\right)\left(\frac{v_{\rm crit}^{\rm
max}}{3\times 10^{6}\,\mbox{\hbox{\rm\hskip 3.50006ptcm
s}${}^{-1}$}}\right)^{-1}\mbox{ s}.$ (39)

where $\sigma_{p}$ is the characteristic dimension over which the pinning
forces varies, $0.05R$ is a fiducial value for the thickness of the crust, and
$v_{\rm crit}^{\rm max}$ is the maximum value of the critical velocity for
pinning. Hence, as the star spins down the vorticity becomes compressed into
dense sheets along $\pm\hat{\theta}$, even if the vortex array is uniform
initially; a sheet will be along $-\hat{\theta}$ in a region with $d(rv_{\rm
crit})/dr>0$ and along $\hat{\theta}$ in a region with $d(rv_{\rm
crit})/dr<0$.

##  7 Pinning Models and Simulations

There has been substantial disagreement over the magnitude and sign of the
vortex–nucleus interaction in the inner crust. Quantum calculations (using a
mean-field Hartree–Fock–Bogoliubov formulation) show a repulsive interaction
with an energy of up to $\sim 3$ MeV (Avogadro et al., 2007, 2008) throughout
the inner crust, while semiclassical calculations (using a local density
approximation) show a repulsive interaction of $\sim 1-2$ MeV below an average
baryon density of $\sim 10^{-2}$ fm-3 ($\sim 2\times 10^{13}$ g cm-3) which
turns attractive with a strength of $\sim 5$ MeV at higher densities
(Pizzochero et al., 1997; Donati & Pizzochero, 2003, 2004, 2006). A step
toward resolving the controversy was made by Wlazłowski et al. (2016), using
density-functional theory (in principle, an exact approach); they find that
the vortex–nucleus interaction is always repulsive in the average baryon
density range 0.02 fm-3 ($3\times 10^{13}$ g cm-3) to 0.04 fm-3 ($7\times
10^{13}$ g cm-3), with a force of $\sim 1$ MeV fm-1 over a range of $\sim 4$
fm, corresponding to an interaction energy per nucleus of $\sim 4$ MeV. The
strength of pinning also depends on the symmetry of the nuclear lattice (Link
& Levin (2022); recall discussion of §4). Pinning is relatively strong in a
regular, attractive lattice. Pinning occurs less readily in a regular,
repulsive lattice, with no sharp transition to pinning for many lattice
orientations. The presence of impurities in an otherwise regular lattice, such
as dislocations, nuclei with different charge than their neighbors, or mono-
vacancies (missing nuclei), generally enhance pinning. Pinning in nuclear
glass is generally strong. Despite these uncertainties, it is clear that the
basic energy scale in the pinning interaction is $\sim 1$ MeV, with spatial
variations over length scales much shorter than the crust thickness of $\sim
0.05$ the stellar radius. Vortices are also predicted to pin to the far more
numerous flux tubes of the outer core (see, e.g., Srinivasan et al. 1990;
Ruderman et al. 1998; Jones 1991).

Though the strength and distribution of the pinning force is uncertain, we can
proceed with two illustrative examples for the distribution of the pinning
force from which we can draw some general conclusions. In both cases we make
the pinning region deeper in the star and somewhat thicker than in a real NS
crust for the sake of visibility. This choice has little effect on our chief
results.

Example Model 1: spherically-symmetric Gaussian.

$v_{\rm crit}(r)=v_{\rm crit}^{\rm max}\,{\rm
e}^{-(r-r_{0})^{2}/2\sigma^{2}},$ (40)

where $r=\sqrt{\varrho^{2}+z^{2}}$ is the spherical radius coordinate, $r_{0}$
is the coordinate at which $v_{\rm crit}$ takes its largest value, and
$\sigma$ is the width of the distribution.

Example Model 2: spherically-symmetric Gaussian with sinusoidal oscillations.
The purpose of this model is to show the effects of large spatial gradients in
pinning strength as indicated by the calculations of Pizzochero et al. (1997)
(see, also, Donati & Pizzochero 2003, 2004, 2006).

$v_{\rm crit}(r)=v_{\rm crit}^{\rm max}\,{\rm
e}^{-(r-r_{0})^{2}/2\sigma^{2}}(1-\sin kr),$ (41)

where $k$ is the wave number of spatial oscillations.

We fix $v_{\rm crit}^{\rm max}=10^{-4}c$ as a typical value for the inner
crust estimated by Link & Levin (2022). We fix the initial spin rate to be 1
kHz (an unimportant choice).

Assuming axisymmetry of the vorticity field, and neglecting vortex motion in
the azimuthal direction, the solution to the flow problem is given entirely by
eq. 34. As discussed at the end of §4, the effects of vortex tension are
negligible at global scales (but important at mesoscopic and microscopic
scales) and we neglect them. The subsequent flow depends only on the spin rate
of the crust $\Omega(t)$ for specified $v_{\rm crit}(r)$.

Fig. 1 shows contours of constant circulation $\Phi$ at different values of
the crust spin rate for Example Model 1. These contours follow the vortex
lines. The contour interval is chosen so that a uniform vortex distribution
gives equally-spaced contours. All dimensions are in units of the NS radius
$R\simeq 10$ km. Fig. 2 shows an animation of the simulation of Fig. 1. As the
star spins down, the vortex array is bent into tori of vortex rings, and the
vortices are compressed into dense sheets at constant spherical radius. Within
the pinning shell, vortex rings shrink and disppear. At late times a dense
torus of vortices remains that sustains a flow - a superfluid “river” - about
the vertical axis in the equatorial plane.

Fig. 3 shows Example Model 2 with $k=100R^{-1}$, giving three pinning maxima
within a spherical shell. An animation is shown in Fig. 4. The vortex array
becomes significantly deformed before a relatively short spin period of 0.03 s
(the spin period of the Crab pulsar) is reached. Now vortices become
compressed into multiple sheets. Shrinking tori are created in multiple
locations. Bent vortices at large polar radius connect to make shrinking tori
while their outer parts reconnect and exit the star. At late times, the vortex
distribution freezes into multiple tori that carry flow around the rotation
axis. A modest value of $k=100R^{-1}$ was selected to make easily readable
plots; pinning calculations indicate that configurations with stronger
gradients of $v_{\rm crit}$ are possible (Pizzochero et al., 1997; Donati &
Pizzochero, 2003, 2004, 2006), which would entail the formation of vortex
sheets at high rotation rates.

|  |
---|---|---
Figure 1: Contours of constant circulation for Model 1 for different spin
rates. The spin axis is in the vertical direction. All dimensions are in units
of the NS radius, $\simeq 10$ km. The pinning region is centered on a
spherical shell at $r=r_{0}=0.8$ with thickness $\sigma=0.05$.

Click here to animate

Figure 2: Animation of the simulation of Fig. 1. Note the formation of tori
within the shell that shrink and disappear. At late times (slow rotation), a
dense torus of vortices exists that sustains a flow about the vertical axis in
the equatorial plane.

|
---|---

|
---|---

Figure 3:  Contours of constant circulation for Model 2 for different spin
rates. The spin axis is in the vertical direction. The parameters are
$r_{0}=0.8$, $\sigma=0.05$, $k=100$. The horizontal scale has been stretched.
Note the formation of tori in multiple regions that shrink and disappear. At
late times (slow rotation), distinct regions exist that sustain flow about the
vertical axis in the equatorial plane. The three thin regions with light
coloring in the two lower-most figures are nearly devoid of vortices. For this
example, significant deformation of the vortex lattice exists even for a
relatively short spin period of 0.03 s (Crab).

Click here to animate

Figure 4: Animation of the simulation of Fig. 3

##  8 Discussion

As a NS spins down, the combined effects of the pinning force (which is always
comparable to the Coriolis force) and the Magnus force lead to significant
bending of the vortex array. Bending of the vortex array arises generally
because pinning forces hold vortices largely stationary in parts of the NS,
while the Magnus force drives vortex motion where the vortex array is not
pinned.

Despite the uncertainties of the distribution and magnitude of vortex pinning
in a NS, we can draw a number of robust conclusions. First, the vortex
distribution will deviate significantly from rectilinear by the time the spin
rate goes below $\sim 20$ Hz (about twice the spin rate of the Vela pulsar).
Later, the vortices become compressed into dense sheets that follow surfaces
of constant spherical radius. The number of sheets is equal to the number of
radii in the star at which $|d(rv_{\rm crit})/dr|$ is large. If $d(rv_{\rm
crit})/dr$ is first large and positive at some radius, and then large and
negative at larger radius, the vortex distribution evolves into a torus that
spans the two regions.

A region of particular interest is the base of the crust, where the nuclear
lattice makes a transition from a lattice of spherical nuclei to nuclear
pasta, finally dissolving at the boundary between the crust and the outer
core. Pinning forces must vary significantly in this region, so we expect
vortex sheets to be present. Dense sheets of opposing vorticity attract one
another, possibly leading to reconnection of vortex sheets. If sheet
reconnection does occur, the superfluid flow could become turbulent at length
scales comparable to the typical sheet thickness.

Our treatment depends on the assumption that the vortex array remains poloidal
in the rotating frame, thus avoiding the complications of winding of the
vortex array around the rotation axis of the star. As discussed in §4,
calculations of vortex drag indicate $\sin\theta\sim 10^{-3}$ in the inner
crust and $\sim 0.1$ in the outer core. If vortices are forced by stellar
spin-down to move in the ballistic regime, there will be significant winding
of the vortex array, coupling the poloidal and toroidal motions of the array.
It is more likely that vortices are never forced into the ballistic regime,
but instead move through thermally-activated vortex creep. As we show in a
forthcoming publication (Link & Levin, 2023), for low-drag thermal creep the
vortex has comparable velocity components along and transverse to
$\mathbf{v}_{s}$, while for high-drag thermal creep the vortex motion is
nearly transverse to $\mathbf{v}_{s}^{*}$ as for zero temperature. We expect
that the basic picture given here of the concentration of vorticity into sense
sheets still holds, but with significant twisting in the azimuthal direction.
We will address this problem in a future publication.

Our simplified treatment shows continuous evolution of a vortex array. In
reality, the vortex distribution might undergo abrupt changes. Abrupt changes
in the vortex distribution at large scales could be responsible for spin
glitches, while smaller, stochastic changes could be related to timing noise.

YL’s research on this subject is supported by Simons Investigator Grant
827103. We thank Andrei Beloborodov, Ashley Bransgrove, Andrei Gruzinov, and
Malvin Ruderman for discussions.

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