# SIG Proceedings Paper in LaTeX Format

Extended Abstract

Ben Trovato  1234-5678-9012 Institute for Clarity in DocumentationP.O. Box
1212DublinOhio43017-6221<EMAIL_ADDRESS>,  G.K.M. Tobin  Institute
for Clarity in DocumentationP.O. Box 1212DublinOhio43017-6221
<EMAIL_ADDRESS>,  Lars Thørväld  The Thørväld Group1 Thørväld
CircleHeklaIceland<EMAIL_ADDRESS>,  Valerie Béranger  Inria Paris-
RocquencourtRocquencourtFrance ,  Aparna Patel  Rajiv Gandhi UniversityRono-
HillsDoimukhArunachal PradeshIndia ,  Huifen Chan  Tsinghua University30
Shuangqing RdHaidian QuBeijing ShiChina ,  Charles Palmer  Palmer Research
Laboratories8600 Datapoint DriveSan AntonioTexas78229<EMAIL_ADDRESS>,  John
Smith  The Thørväld Group<EMAIL_ADDRESS>and  Julius P. Kumquat  The
Kumquat Consortium<EMAIL_ADDRESS>
(2016)

###### Abstract.

This paper provides a sample of a LaTeX document which conforms, somewhat
loosely, to the formatting guidelines for ACM SIG Proceedings.111This is an
abstract footnote

ACM proceedings, LaTeX, text tagging

††copyright: rightsretained††doi: 10.475/123_4††isbn:
123-4567-24-567/08/06††conference: ACM Woodstock conference; July 1997; El
Paso, Texas USA††journalyear: 1997††price: 15.00††ccs: Computer systems
organization Embedded systems††ccs: Computer systems organization
Redundancy††ccs: Computer systems organization Robotics††ccs: Networks Network
reliability

##  1\. Introduction

Majority of existing wireless sensor nodes deployed for applications like
environmental monitoring (OTHMAN20121204) are battery powered and mostly
designed for short distance communication. As we scale to large deployments,
the manual replacement of batteries becomes a bottleneck. Battery replacements
can be mitigated using energy harvesting techniques that may vary based on the
environment of deployed sensor nodes. Sometimes, as a replacement to
batteries, sensing nodes utilize beefy super-capacitors which are prone to
current leakages after exceeding the charge cycle threshold (7356481). To
ensure a balance between the harvested energy and operational energy for a
sensing node, intelligent adaptive algorithms for power management should be
employed. Modular analysis of many such sensing systems clearly indicate that
the most power consumption is directly associated the communication
peripheral(either a radio or GPS). In this paper, we present the technique for
achieving Kilometer range communication on constrained devices that are
powered by harvested energy (Solar) for long term operation. The reason behind
this effort is due to the fact that batteryless devices that gather energy
from the environment and execute opportunistically usually promise a
sustainable, maintenance-free, and environmentally-friendly alternative
(flickr). However, these benefits don’t usually come without some challenges.
Furthermore, current state-of-the-art sensors for air quality monitoring are
both costly and power hungry.

In order to evaluate the efficiency and performance of our system, we design
and implement a custom PCB prototype for solar energy harvesting and
management. The board interface seamlessly with the MSP430FR5969 launchpad by
Texas Instrument. The federated energy storage-UFoP (Hester3) approach was
adopted in designing the board. Each peripheral (LoRa radio and Air quality
sensor) has a dedicated UFoP which allows automatic management of peripheral
charging. We evaluated the system using air quality monitoring as our
motivating applications. We carry out series of experiments to evaluate the
impact of transmit power on other characteristics like transmission range,
current consumption, data rate and line-of-sight (LOS) transmission. Also, a
comparative analysis was carried out between the performance of SX1276 based
RFM96W LoRa module and the popular CC1101 radio transceiver.

We were able to accomplish over 1KM transmission range in a pair-to-pair
communication mode using the RFM96W LoRa radios on 433MHz. The Arduino
Diecimila act as the TX node and Arduino Uno as the RX node. The arduino
platforms were used due the challenges faced while trying to get MSP430 to
work. The outcome of the comparison between LoRa and CC1101 shows that LoRa
communication achieves better Packet Delivery Ration (PDR) and transmission
range in a non-LOS setup with barriers and interference. We concluded by
stating the challenges faced and recommendations on achieving long range
communication on batteryless devices.

Contributions:  We explored the possibilities of implementing long range
communication on batteryless devices. The research effort was directed towards
understanding and harnessing the long range communication capability of the
LoRa radio module. We make the following contributions in this paper:

  1. (1)

Presents a novel approach of enabling Kilometer(km) range communication on
constraint devices powered solely by harvested energy (Solar).

  2. (2)

A custom hardware prototype design and implementation for energy harvesting
and management. This prototype is a breakout board that powers the system for
long range communication (LoRa) and it also interfaces seamlessly with the
MSP430FR5969 launchpad by TI.

  3. (3)

Comprehensive evaluation and juxtaposition of LoRa performance with
conventional CC1101 radio transceiver. This comparison cover metrics like
transmit power, current consumption, Packet Delivery Ratio (PDR) and
transmission range.

##  2\. Related Works

Existing research efforts have considered the possibility of enabling remote
battery-less sensing using commodity devices like Bluetooth Low Energy (BLE)
has dominated the space of mobile wireless communication (ble), especially for
devices that require low power operation. But it is not flexible enough for
networks that suffer from intermittent power failures and also lacks the
capability to transmit huge amount of data in a single packet (ble2). WiFi is
the technology of choice for wireless communication spectrum (wiki). Mostly
used on high end devices with considerable amount of energy. Although WiFi
supports high throughput and bandwidth (wiki), but a power consumption of
100mW at 20dB transmit power makes it unsuitable for use on low power devices
which manage to scramble micro Amps of current. ZigBee is popular because of
its lightweight, low-cost, low-speed, low-power protocol (omojokun). It can
support up to 128bytes of data and a data rate of 250Kbps operated in the
2.4GHz band (omojokun). However, it requires development of algorithms for
autonomous meshing of beacons (joshi), and also provides limited access to the
physical layer properties. Also, Zigbee communication is limited to few 100
meters.

Most WiFi, Bluetooth, Zigbee and other wireless chipsets use 2.4GHz, which is
great for high speed transfers (adafruit), but not suitable for low power long
range communication. LoRa technology eliminates this challenge by enabling
long range, low-power, low-bit rate wireless communication on end devices with
limited energy (aloy). The novelty of our work is inherent in the ability to
enable kilometer range communication on end devices that operate solely on
harvested energy. Our work brings the best of long range communication and
energy harvesting for enabling new applications that existing systems couldn’t
support. Our approach focus specifically on using LoRa radio (SX1276) for the
long range communication on 433MHz licence-free frequency. Our motivating
application uses the P2P LoRa communication mode. This enable us to configure
one node as a transmitter (TX) and the other as a receiver (RX).

##  3\. Background

Wireless Sensor and Actuator Networks (WSANs) plays a vital role of a
inseparable technology of the Internet of Things (IoT), WSANs have emerged as
a variation of Wireless Sensor Networks (WSNs). As an add on to the tasks
performed by WSNs, WSANs are capable of keeping an eye on physical phenomenons
(performing measurements e.g. vibration, humidity, radiation, article
concentration levels,…), processing data collected by sensors, making
decisions based on the sensed data and taking actions respectively (in our
case, send data to remotely located base station). For WSANs, IEEE 802.15.4
standard is the most widely-used communication scheme, which provides both
physical layer and Media Access Control (MAC) layer specifications (Figure 1).
This standard facilitates low-cost and low power transceivers, but have a
disadvantage over range of transmission that is only a few tens of meters. In
past few years, some wireless technologies enabling Long Range (LR)
communication of several kilometers with power usage similar to WSAN nodes
transceivers (Martinez) have emerged. One of those technologies in use is LoRa
(semtech), by the LoRa Alliance(Semtech). LoRa (short for long range) is a
spread spectrum modulation technique derived from chirp spread spectrum (CSS)
technology. LoRa operates in the 868/915 MHz ISM bands, promises great
communication distances: upto 2 km in urban and around 16 km in rural areas,
and a bit-rate in the range between 0.37 and 46.9 kbps
(goursaud:hal-01231221).

One of the main goals of this project is to acquire an easy, low power
solution, interconnecting nodes with the base station setup within long
distance (augustine) and in order to meet these requisites LoRa technology
seems like the best solution. The technology offers a mix of long range, low
power consumption and secure data transmission and is gaining significant
traction in IoT networks being deployed by wireless network operators. For our
project purpose as we focus more on comparative study of the LoRa technology
and CC1101 widely used long range, low power transceiver, we have made our
system to work on frequency level of 433MHz. Since, CC1101 operates on this
frequency, it’ll be meaningful to evaluate both the transceivers on this
frequency.

###  3.1. Communicating with LoRa

Figure 1. LoRaWAN Stack layers

LoRa technology is divided into two parts: LoRa physical layer and LoRaWAN.
LoRa physical layer enables the physical layer communication whereas LoRaWAN
is a low power, wide area networking protocol based on LoRa technology. It was
designed to wirelessly connect battery based or remotely placed systems to the
internet. The LoRaWAN takes advantage of the radio spectrum in Industrial,
scientific & medical ISM band. LoRaWAN also accounts for the power consumption
of the remotely located node, the capacity of the network, quality of service
and security.

Spread spectrum communication technique was opted to achieve better
communication distance ranges, this spread spectrum technique uses wideband
linear frequency modulated pulses whose frequency keeps changing over a
specific time interval to encode information. Opting this technology has two
benefits one of which being significant increase in receiver sensitivity due
to the processing gain of the spread spectrum technique and the other being
high tolerance to frequency misalignment between node and the base station.

###  3.2. Network Architecture

All remotely deployed systems want to transmit data as far as they can and to
achieve this most of them uses mesh network architecture. In a mesh network,
every specific node forwards information received from some other node to
increase the communication range. But, this approach adds complexity to the
network, reduces network capacity and also adds up to power consumption as all
the nodes have to stay awake all the time to receive and transmit data, that
too some nodes’ information might be irrelevant for that node.

Figure 2. LoRa P2P layout

A node in LoRa on the other hand is individually connected to one or more
gateways and sends information directly whenever needed and can go in sleep
mode when doesn’t have anything to do. Further, gateways forward information
to the cloud server. Server gets redundant data and then it’s job is to get
rid of that redundancy. This mechanism results in power consumption &
increased network capacity. In our project we opted star topology and have
created a test bed with one base station and one end node as shown in figure
2, to test P2P communication. Our system works on 433MHz instead of LoRa’s
proposed spectrum which is 868/915 MHz (depending upon the region of
operation) so as to get justifiable comparison with P2P communication between
two CC1101 transceivers.

###  3.3. Power Management

Powering up the device deployed at remote locations has been a big challenge
always, specially when the power supply is intermittent (Hester2). Researchers
have come up with solutions for energy harvesting (Hester) (Sharma) this work
will need something that can take decision on boost or not to boost the power
harvested, which will be made to relate with the characteristics of the
signals transmitted. Also, research and experiments have directed us towards
deciding that use of multiple capacitors specifically dedicated for a
peripheral unit(UFoP - the United Federation of Peripherals) is much better
then deploying one shared capacitor(Hester3).

##  4\. System Design

This section covers the various design decisions we made in order to
accomplish a system for adaptive long range communication on batteryless
things while leveraging the unique features of LoRa module. Some of our design
decisions were influenced by previous research efforts in the areas of
batteryless sensing and Wireless Sensor Networks (WSN). Majority of existing
research efforts either implement long range communication on a system with a
dedicated battery or tetherd power. Others use super capacitors in
collaboration with rechargeable batteries. We are the first to power a LoRa
based long range sensing and communication systems with harvested energy on
tiny ceramic capacitors.

Our system provide solutions to some of the major challenges encountered in
remote environmental monitoring applications like outdoor air quality
monitoring and algae bloom monitoring in fresh and marine water (airquality);
especially in constrained environments like space or arctic when human
intervention is highly limited. Despite the challenges attributed to the
intermittent nature of energy harvesting (Hester), our system appears to be
robust enough to handle variations in energy availability through dynamic
adaptation.

The system design is made up of three main sections: Energy Harvesting,
Computing Core and Peripherals. Since the MCU, LoRa radio and the air quality
sensor can operate between 1.8V and 3.6V, we designed the energy energy
harvesting section to supply 3.3v to all peripherals as well as the MCU. Our
initial design was based on 3V level appears insufficient for the RFM96W LoRa
breakout board. The breakout board already has a 3.3V regulator, thus making
it challenging to power the radio with 3V. The figure below shows simplified
block diagram of the system architecture.

Figure 3. System Block Diagram

The Energy section comprises of the energy harvesting circuitry which supplies
power to whole system. It interfaces the solar panel output with the federated
energy storage (UFoP) of the peripherals and the micro-controller unit. The
computation section comprises of the MCU, timing circuit as well as the power
sensing circuitry. The final stage (peripherals) consist of the UFoP circuitry
for both the LoRa radio and the air quality sensor.

##  5\. Implementation

In order to evaluate the performance and viability of long range communication
on batteryless devices, we designed a prototype sensor node with long range
communication and energy harvesting capabilities. The prototype was designed
as a breakout board which can be seamlessly interfaced with MSP430FR5969
launchpad by Texas Instrument. We used air-quality monitoring as the
motivating application for this work. The system design consist of three major
stages, which are energy harvesting, air-quality measurement, and long-range
communication which is based on LoRa module. In this section we describe the
main specifics of the hardware and software implementation of our prototype.

###  5.1. Energy Harvesting

This stage consist of components and modules needed for energy harvesting and
management. The federated energy storage-UFoP (Hester3)approach was adopted in
designing the energy harvesting section. UFoP enables automatic management of
peripheral charging and also allows for dynamic retasking and reprioritization
of energy resources at runtime (flickr). We assigned dedicated UFoP for each
of the peripherals on our prototype (LoRa and Air quality sensor). Energy
harvesting is done using sunboy new energy’s solar panel rated as 6V 70mA. The
solar panel supplies the main capacitor (100uf) which is protected by a 5.1V
Zener diode. An hysterisis chip MIC841 was introduced after the main capacitor
to ensure stable operation i.e. the system turns on above 3.38V and turns off
below 3.05V. The systems was designed to operate at 3V level, so we used 3V
S-1313 series voltage regulators.

###  5.2. SX1276 LoRa Radio Module

This stage contains the UFoP designed to power the radio module. We want the
radio module to start charging after all other peripherals have started
charging. A combination of voltage divider circuit and a TLV3691 comparator
was used to configure charging threshold for the radio UFoP as well as for
other peripherals. The Adafruit RFM96W LoRa Radio Transceiver-433 MHz was used
for the long range communication. The LoRa module was interfaced with the
microcontroller via SPI. The radio UFoP contains an 100uF capacitor which is
sufficient to hold substantial amount of charge needed to transmit radio
packets. The interrupt pin goes high when the capacitor is fully charged so
that the MCU can close the UFoP GATE in order to use the radio for
transmission. The RFM96W LoRa breakout board which is based on SX1276 LoRa
module exposes eight pins (MOSI, MISO, CS, CLK, RESET, DIO-0, VCC and GND) for
coordination with the MCU.

Figure 4. Radio UFoP designed to charge the capacitive storage for the LoRa
radio module from 3.3V and generate interrupt at 3.5V

###  5.3. Air-Quality Measurement

Figure 5. Air Quality Sensor output

This stage include the UFoP specifically designed for the air quality sensor.
The UFoP initiates the charging of the 22uF capacitor when the input voltage
gets to 3.2V and generates interrupt when it’s fully charged. We used the
Adafruit CCS811 Air Quality Sensor Breakout which can measure Volatile Organic
Compounds (VOC) and equivalent Carbon dioxide (eCO2) via I2C. The interrupt
pin goes high when the capacitor is fully charged and the MCU closes the gate
whenever air quality measurement is to be taken. The CCS811 air quality sensor
breakout board exposes some pins for I2C communication with the MCU and it is
based on ams unique micro-hotplate technology which enables a highly reliable
solution for gas sensors, very fast cycle times and a significant reduction in
average power consumption (as low as 1.2mW) (ams).

The prototype was designed as a breakout board that fits perfectly on the
MSP430FR5969 launchpad. The rational behind this results from need to minizie
the amount of components on the custom board by harnessing existing MCU on the
launchpad . This also helped to eliminate the need for extensive use of jumper
cables between the board and MCU. The MCU is powered externally from the
MCU_CC (3V) pin on the breakout board. This pin is connected to the VCC pin of
the jumper J12 and the J10 jumper (Power Select) is switched from debugger to
external. A 50 ohm SMA antenna connected to the LoRa radios ensures the radio
packets can travel long distance. We used ISL21080 voltage reference to
generate a 125mV signal which is compared with the output of the voltage
dividers to keep state.

Figure 6. The breakout board PCB interfaced with MSP430FR5969 launchpad.

###  5.4. Software

We utilized embedded C/C++ to implement all the necessary operations on the
MCU. Both the Energia platform as well as the Arduino platform were used
during code implementation. The MCU was programmed with the logic that
coordinates the operation of the UFoP as well as air quality sensing and LoRa
data transmission and receiving. Sensing and transmission takes place every 10
seconds. The MCU goes to sleep after each sensing and transmission cycle.
Whenever the UFoP capacitors are fully charged, the signal on the designated
interrupt pin goes high which allows the MCU to close the AIRQ_GATE for air
quality measurement and RADIO_GATE for long range transmission. The LoRa
module was configured to operate on 433MHz, this provides us with the
opportunity to perform comparative analysis between the LoRa module and the
CC1101 radio.

We implement the prototype design on Autodesk Eagle 9.0.1 and generate gerbers
files, which were later sent to OSHPARK for PCB printing. Majority of the
components used conforms with the 0805 package size.

##  6\. Experimentation

Our project was all about testing an idea and to see if it can work and if it
works how efficient it is in comparisons to the systems working out there on
harvested energy. As communication among wireless sensor network is one of the
areas where energy consumption is a very big issue specially when the system
is powered with intermittent energy, we believe in finding a solution for it
by testing our system in those real time scenarios.

Figure 7. LoRa Nodes

###  6.1. Current Consumption

We started the comparative study of our system with transceiver CC1101 with
this experiment, which was to measure the current consumed by it at different
transmission power levels. All experiments were done for P2P communication.
These measurements were taken by varying the transmission power of the LoRa
transceiver in the software part and checking current consumed by the system
for each of these configurations. In total 5 transmitting power configurations
were tested and readings were noted, this experiment was done by keeping the
receiver and transmitter at same distance throughout the experiment so that
the readings can be genuine and factors like orientation of radio and line of
sight doesn’t effect the readings.

Figure 8. Current Measurements with maximum, minimum and average currents for
transmitter

###  6.2. Range and Transmission Power

Our 2nd experiment was a vital one which gave us the idea about the range of
LoRa transceivers with respect to transmission power of LoRa radios. This
experiment was set up in two different settings first being the non line of
sight (LOS) setting in which the transmitter was fixed at a specific location
and the receiver was carried around a route which was kept same for all the
transmission power configurations and the second being the line of sight
setting where again transmitter was fixed at a specific location and receiver
was taken away from the transmitter in a straight path making sure both the
radios are visible to each other.

####  6.2.1. Non-LOS Configuration

This was a very wild test in which nothing like the orientation, height or
position of the radios was considered and the receiver was just carried around
a fixed route which was randomly chosen. Experiment was started with keeping
the transmission power as 5dBm and the receiver was carried along the route,
one by one multiple transmission powers in the interval of 5dBm (lowest limit
of LoRa) - 23dBm (highest limit of LoRa) were chosen and the behaviour of the
communication channel was observed. For each transmission power, the current
consumption of the system was also monitored so as to get an idea if
transmitter put some extra efforts to transfer the packet successfully when
the receiver is taken a little far away from it.

####  6.2.2. LOS Configuration

This configuration was tested in a proper set up, where transmitter was fixed
at a position and receiver was carried in a straight path away from the
transmitter. Also, apart from the pre decided straight path some other factors
like the orientation of the antenna, height of the antenna all were maneuvered
so as to keep the radios in line of sight of each other. Similar to the prior
configuration in this configuration also the transmission power was varied
from 5dBm to 23dBm.

###  6.3. Power Supply

One of the important conclusion we wanted out of this project was if the LoRa
is suitable for batteryless field or not. To test this we conducted two
experiments, first one was with stable/continuous power supplies which were
table top power unit and USB power supply from a laptop and the second was
solar panel. We tried both to see how the results vary and what all problems
can arrive.

####  6.3.1. Continuous Power Supply

Under this experiment we supplied power to our system using table top power
supply and USB port of the laptop, both seemed to work well without any issue.
Also, we limited the current to 16mA and our system worked fine for 5dBm of
transmission power at which the maximum current consumption was around 15.6mA
and average consumption was around 11mA when our microcontroller was running
at 3.7V.

####  6.3.2. Intermittent Power Supply

For this test bed we tested multiple configurations of solar panels. We
started with using 3 solar panels each of 6V 70mA in indoor setting, they
worked but we had to flash light on them otherwise they were not harvesting
enough to suffice the current requirement of the system. As soon as we took
them outdoors in normal daylight they worked well and we were able to setup
proper connection between both the LoRa nodes. Then the number of solar panels
were decreased from three to two and we were still getting all the data
packets properly. Ultimately, we tested with one solar panel but it didn’t
harvested enough energy to start the communication though the system was
getting enough power to wake up but it got stuck in infinite booting loop. We
were able to make our system work perfectly with single solar panel also but
we had to put it directly under bright sunlight.

###  6.4. UFoP Bank Capacitor

Actual project was supposed to include MSP430FR5969 launchpad but we faced few
issues with it because our launchpad was designed using UFoP approach. In UFoP
approach each peripheral has it’s own power supply unit and the regulator in
it makes sure that the peripheral only gets the supply when there is enough
energy stored in bank capacitor to power it up.

The problem we faced was we made our UFoP to supply 3V to our LoRa radio(by
putting the 3V regulator), when we connected the LoRa radio to the peripheral
breakout board Vcc for radio the voltage dropped down to 1.9V which shouldn’t
have happened and this much voltage was not enough to power up the radio
nodes. In efforts to make it work we narrowed down the problem to two factors
that might have been responsible for the failure, either the bank capacitor of
the LoRa UFoP or the regulator that we were using to provide 3V supply to LoRa
radio. For solving the issue we started with using voltage booster but it
itself consumed some current and reduced the voltage. Next we tried changing
the bank capacitor to 100uF, 1000uF in parallel with the originally placed
100uF capacitor, none of them worked. Then we tried placing only 1000uF
capacitor instead of 100uF, that also didn’t give us results. One more thing
we wanted to try was higher rating voltage regulator but it was not readily
available in the time frame we had. So, we had to modify our system and use
the LoRa radios with arduino instead, which somewhere accounted for a little
more power consumption in our system as MSP430 boards work on relatively less
power than arduino.

Table 1. List of components used Module | Model | Availability | Cost($) | Count
---|---|---|---|---
Lora Radio | RFM96W | Adafruit | 19.95 | 2
Air Quality Sensor | CCS811 | Adafruit | 19.95 | 1
Voltage Regulator | SC82AB | Digikey | 0.851 | 1
MCU | MSP430FR5969 | TI | 15.99 | 2
Current Measurement | DS2470 | Mouser | 2.47 | 1
Hysteresis Chip | MIC841 | Digikey | 0.58 | 1
Zener Diode | SOD-523 | Digikey | 0.0.643 | 1
IC Comparator | SOT 353 | Digikey | 1.01 | 3
Solar Panel | 6V, 70mA | Amazon | 5.95 | 1

##  7\. Evaluation

We evaluate our system with ground truth results obtained from line-of-
sight(LOS) measurements and through-wall(Non-LOS) settings for measuring
parameters like range and current consumption for varying transmission powers.

###  7.1. Current Consumption

Figure 9. Current Consumption(mA) Comparison of Radio Modules with respect to
transmission power(dBm)

While measuring the current for receiver and transmitter it was clear that
steady state current (Mode0: when it neither transmits nor receive and goes to
sleep mode) for LoRa nodes is 1.8mA. Also, we kept an eye on the current
trends, the way they change, it was pretty much clear that they follow the
same trend of increasing current gradually and then reduction in the current
consumption. This variation in the current consumption was independent of the
transmission activity and also didn’t depend on the distance between receiver
and transmitter. We checked it by moving the receiver away from transmitter
and placing it side by side, movement of receiver apparently doesn’t matter.
As can be seen in Figure 8, the current consumption of LoRa radio is less than
the CC1101 transceiver on same transmission powers.

###  7.2. Range and Transmitter Power

Figure 10. Packet Delivery Ratio Comparison

####  7.2.1. Non-LOS Configuration

Figure 10 shows the relationship between the PDR and transmission range on
both the LoRa module and CC1101 radio. LoRa achieves 7% more PDR in a Non-LOS
settings while keeping the trasnmit power constant at 5dBm. This shows that
LoRa can tolerate more interference than CC1101 even in the presence of
obstacles like buildings.

####  7.2.2. LOS Configuration

Figure 11 shows that based on the readings we obtained for CC1101 and for LoRa
by our experiments CC1101 has a better range over LoRa at 433MHz. But, one
thing to be noted is CC1101 starts working at -30dbM and only works till 10dBm
whereas LoRa starts working at 5dBm and works till 23dBm. So, the transmission
power which is the starting point for LoRa is one of the highest transmitting
power of CC1101.

Figure 12 clearly describes about the Range of LoRa transmission which we
achieved at different transmitter power levels.

Figure 11. LOS Transmission Range Comparison at specific transmission power
levels Figure 12. Measured Approximate Range of LoS LoRa Communication

###  7.3. Power Supply

Continuous power supply doesn’t posses any problem and the system works fine.
Also, it’s safety can be insured by limiting the current. Whereas, in case of
USB power supply if somewhere in the system the connections are not proper and
if by mistake there is a shorted line, it can damage the USB port.

While intermittent power supply is a new approach for these kind of systems as
they may consume more power in comparison to other peripherals. Along with
being new it’s a tricky one too because if the system power is not managed
properly the system might go into infinite booting loop and it may not be
feasible to repair it if the node is placed somewhere remotely. Proper
Adaptive algorithms need to be implemented on it so that the system doesn’t
stay awake for a very long time and just wake up whenever the interrupt is
fired so as to conserve energy.

###  7.4. Current consumption at different stages of LoRa protocol

We tried to evaluate the current consumed by the LoRa transceiver for each
stage of operation. For this approach we used a current sense resistor along
with an Oscilloscope to measure the current waveform of the entire LoRa
communication. The main motive behind this experiment was to analyze the
behaviour of different LoRa stages with respect to current consumed. We used a
sense resistor of 1.5 ohms for our experiments and observed an interesting
characteristic of LoRa transceiver. Initially after we powered our transmitter
and set it to transit LoRa packets every 5 seconds, we noticed that without
the presence of an active receiver on the channel, the transmitter idle
current draw was significantly higher. We set-up the transmitter for a cold
start each time it is powered on, i.e. the it undergoes the initialization
process, handshaking modes, authentication and then transmission. This was
done in attempt to bifurcate the current consumed in each of these processes.
However, we couldn’t successfully evaluate time required for each of these
processes but could draw some useful conclusions. 1\. The transmitter would
consume significantly higher mA currents when idle in ”SEARCH MODE” than in an
active transmission mode when discrete link between TX-RX is established. 2\.
Current consumption for receiving an ACK from the receiver was significantly
lower than transmission currents. We need accurate nsec - msec timing
measurements to isolate each section of LoRa protocol to carry out in-depth
power measurement analysis. Also, the LoS and non LoS distance between the
transceiver and receiver might also play a huge role in packet time and
handshaking signals.

Figure 13. Current consumed vs Time for LoRa Communication

##  8\. Discussion and Future Work

###  8.1. Challenges & Recommendations

  1. (1)

We faced issues while taking line of sight readings, as we had to keep the
orientation of the radio pretty much same all the time otherwise we were
losing connection in between receiver and transmitter in case of long ranges.

  2. (2)

We couldn’t find a long LOS distance to test LoRa Radio to it’s limit. We
believe it would have worked much better if the readings were taken in an
large open field.

  3. (3)

The 100uF capacitor used in the radio UFoP was unable to store enough charge
to allow complete transmission of LoRa packet. We increased the values of the
capacitor up to 1000uF but still unable to achieve full packet transmission.

  4. (4)

Our UFoP for LoRa radio didn’t supply enough voltage so as to power it up.
Three ways this could have been possible were; firstly the system was not
getting enough current so voltage was just going down because of large current
consumption just to maintain the power supplied, second reason could have been
the bank capacitor was not getting enough time to charge back to full capacity
and before it can charge to full capacity the radio drew whole energy so it
was just staying at stable low voltage supply, third reason which came out to
be the actual problem was that one of our component can’t sustain the amount
of current that our radio needs and it’s just trimming it off. We realized it
because our radio was getting initialized but not sending packet as shown in
Figure 15 when we bypassed the voltage regulator and as soon as we bypassed
the hysteresis chip (Figure 14) it started sending packets.

Figure 14. Hysteresis Chip which was clipping current at 20mA Figure 15. LoRa
Radio initializing but not sending packet because of lack of current

We troubleshooted the problem by checking each component one by one that we
used on our UFoP, and our hysteresis chip could only allow 20mA whereas our
radio needs more than that often. So, the voltage was getting pulled down
because of lack of current supply.

###  8.2. Future Work

We wanted to have Stage 4 in our project which was focused at additional
features for our project involving but not limited to multiple nodes
communicating with LoRaWAN gateway, integrating water quality and quantity
sensors, exploring the frequency domain of 868 MHz frequency band for
potential LoRa communication and be able to perform adaptive data storage and
transmission. These goals can be termed as ambitious goals of our project. The
idea of implementing multiple sensor nodes communicating with a central
gateway for achieving LoRaWAN was our initial focus for this project. But
understanding the level of complexity involved in installing a LoRaWAN gateway
and designing a communication channel for multiple sensor nodes under given
time frame, we planned to achieve this functionality at a later stage. Knowing
the power requirements involved for P2P communication of our LoRa radio and
having a comparative analysis of energy requirements, we can then test the
gateway to single node and then multiple node challenges. The gray area of
working in 868 MHz frequency band is also what we would like to explore and
compare results with European counterparts using the same frequency for LoRa
communication.

Another fundamental functionality that we initially aimed at achieving was
water quality and quantity measurements to monitor algae bloom problems. After
a successful literature review of the available CO2 and dissolved O2 sensors,
we found that majority of the sensor work on 3.3V-12V ranges requiring current
draw of 100mA-2A range. These sensors are bulky and heavy and have very
restricted interfacing options. A majority of sensors are power hungry and
need SDI-12, MODBUS or 4-20mA communication standards. As the major niche of
this project is aimed towards achieving LoRa long distance communication on
federated energy storage, we have potentially planned on having air quality
monitoring as a first check-point from application point of view. Further work
and better circuit designs would be needed to meet water sensors energy demand
and we plan on interfacing them at a later stage in the project.

##  9\. Conclusion

Our project was aimed at accomplishing kilometer range long distance
communication for batteryless devices using the P2P LoRa communication
protocol. Based on the comparative tests, we found out that the SX1272 LoRa
Radio provided better transmit range and better packet delivery rate(PDR) as
compared to CC1101. But the transmit current for LoRa was significantly high
and thus some further work is needed to get it to work efficiently on long
distances. The area of LORAWAN is still to be exploited and studying behaviour
of multiple nodes with a gateway would be a potential field of research. We
would like to consider water quality measurement on power hungry sensors in
order to evaluate our system under extreme conditions. A breakthrough in this
area will enable new sets of interesting applications that current
technologies cannot support.


# Self-similar hierarchy of coherent tubular vortices in turbulence

Tomonori Tsuruhashi  Graduate School of Mathematical Sciences, University of
Tokyo, Komaba 3-8-1 Meguro, Tokyo 153-8914, Japan<EMAIL_ADDRESS>,
Susumu Goto  Graduate School of Engineering Science, Osaka University, 1-3
Machikaneyama, Toyonaka, Osaka 560-8531, Japan<EMAIL_ADDRESS>,
Sunao Oka  Graduate School of Engineering Science, Osaka University, 1-3
Machikaneyama, Toyonaka, Osaka 560-8531, Japan<EMAIL_ADDRESS>and  Tsuyoshi Yoneda  Graduate School of Economics, Hitotsubashi University,
2-1 Naka, Kunitachi, Tokyo 186-8601, Japan<EMAIL_ADDRESS>
###### Abstract.

Energy transfers from larger to smaller scales in turbulence. This energy
cascade is a process of the creation of smaller-scale coherent vortices by
larger ones. In our recent study (Yoneda, Goto and Tsuruhashi 2021), we
reformulated the energy cascade in terms of this stretching process and
derived the $-5/3$ law of the energy spectrum under physically reasonable
assumptions. In the present study, we provide a quantitative verification of
these assumptions by using direct numerical simulations. We decompose
developed turbulence in a periodic cube into scales by using the band-pass
filter and identify the axes of coherent tubular vortices by the low-pressure
method. Even when the turbulent kinetic energy and its dissipation rate
temporally fluctuate about their temporal means, the total length of the
vortices at each scale varies little with time. This result is consistent with
our assumption of the temporal stationarity on the vorticity decomposition.
The present numerical analysis also shows that the hierarchy of vortex axes is
self-similar in a wide range of scales, i.e. in the inertial range and a lower
part of the dissipation range and that the volume fraction occupied by the
tubular vortices at each scale is independent of the scale.

###### Key words and phrases:

Turbulence, Coherent Vortices, Energy Cascade, Self-similarity, Intermittency

##  1\. Introduction

Since the pioneering experiments by Corrsin [1], it has been well known that
turbulence is not random but consists of coherent structures. The notion of
coherent structures is powerful because it can describe, for example, the
sustaining mechanism of turbulence near a solid wall in terms of longitudinal
vortices and streaks [2, 3]. On the other hand, small-scale turbulence in the
region away from solid walls is sustained by the energy cascade process [4,
5], where the kinetic energy transfers from larger to smaller scales in a
scale-by-scale manner. Since the seminal direct numerical simulations (DNS)
[6, 7, 8] in the 1980s, the relationship between coherent structures and the
energy cascade has been studied by numerous authors; see Refs. [9, 10, 11, 12,
13, 14] for example. We also used the concept of coherent structures to
clarify the concrete energy cascade picture to demonstrate that vortices in
small scales away from solid walls are created by approximately twice as large
vortices in turbulence in a periodic cube [15, 16], turbulent boundary layer
[17] and turbulent channel flow[18]. Although such a process that the larger-
scale vortex stretches and creates smaller-scale ones has long been proposed
[4], recent DNS of developed turbulence at high Reynolds numbers make it
possible to show that developed turbulence is indeed composed of coherent
structures at various length scales. Furthermore, such DNS can capture
concrete energy-cascading events [15, 19] and quantify the scale-locality of
energy cascade due to vortex stretching [16, 20].

In Ref. [20], using this concrete picture [16] of energy cascade in terms of
the hierarchy of coherent vortices, we proposed a new regularity criterion for
a solution of the Navier-Stokes equation, and reformulated the energy cascade
in developed turbulence. In particular, we derived the $-5/3$ power law of the
energy spectrum from the Navier-Stokes equation without directly using the
Kolmogorov similarity hypothesis [21]. In Ref. [20], we decomposed turbulence
in a periodic cube into scales by using the band-pass filter and imposed
conditions on the interaction between the scales in the stretching process.
One of the most important assumptions on the interaction between scales is
that the vorticity at each scale is expressed by coherent vortices (see (13)
in §4(c) for the concrete expression). By using the decomposition, we
represented the interaction between large and small scale vortices to derive
the $-5/3$ power law. Note that we have imposed several assumptions on the
decomposition and the interactions between the scales. These assumptions are
physically reasonable and the scale-locality of the interaction was
numerically verified in Ref. [20]. However, there is no quantitative
verification of the assumptions on the decomposition itself. In this paper, we
verify the assumptions quantitatively by DNS. More concretely, we objectively
identify the axes of coherent tubular vortices by applying the low-pressure
method [22, 23, 16] to the scale-decomposed field. From the hierarchy of
coherent vortices obtained in this way, we provide a quantitative verification
of the assumptions in Ref. [20]. We emphasize that this gives not only a
verification of the mathematical assumptions but also physically meaningful
knowledge. For example, we show, in the following, that the total length of
tubular vortices at each scale is independent of the Reynolds number and
forcing, and has little temporal fluctuation. The universality of this
hierarchical structure of coherent vortices supports the assumptions in Ref.
[20]. In addition, we show that the volume fraction occupied by tubular
vortices at each scale is independent of the scale in a wide range. This
provides a new insight on the spatial intermittency of the energy dissipation
rate in turbulence, which gives a basis for the mathematical description of
the intermittency.

##  2\. Numerical methods

###  2.1. Direct numerical simulations

We consider the following Navier-Stokes equation,

(1) $\displaystyle\partial_{t}u+u\cdot\nabla u+\nabla p=\nu\Delta
u+f,\quad\nabla\cdot u=0\qquad\mathrm{in}\quad[0,\infty)\times\mathbb{T}^{3}$

where $u:[0,\infty)\times\mathbb{T}^{3}\rightarrow\mathbb{R}^{3}$ is a
velocity, $p:[0,\infty)\times\mathbb{T}^{3}\rightarrow\mathbb{R}$ is a
pressure, $f$ is an external force, $\nu$ is the kinematic viscosity of fluid,
and $\mathbb{T}^{3}=(\mathbb{R}/2\pi\mathbb{Z})^{3}$. We impose a suitable
initial condition on $u$. We also abbreviate $u(t,x)$ to $u(x)$, because we
mainly consider the case with time fixed.

In order to investigate the influence of the type of external forces, we
examine two cases with different kinds of external forces. The first force
$f_{I}$ is time-dependent but statistically homogeneous isotropic, which is
expressed in the Fourier space as

(2)
$\widehat{f_{I}}(t,k)=\left\\{\begin{split}&\frac{P}{2E_{f}(t)}\>\widehat{u}(t,k)&\quad&\text{if}\
\ 0<|k|\leq k_{f},\\\ &0&\quad&\text{otherwise}.\end{split}\right.$

Here, $\widehat{f_{I}}$ and $\widehat{u}$ are the Fourier coefficients of
$f_{I}$ and $u$, respectively. The parameter $P$ denotes the energy input rate
and $k_{f}$ the maximum forcing wavenumber, which are set as $P=0.05$ and
$k_{f}=2.5$. In (2), $E_{f}$ is the kinetic energy in the forcing range
defined by

(3) $E_{f}(t)=\sum_{k\in\mathbb{Z}^{3},\ |k|\leq
k_{f}}\frac{1}{2}|\widehat{u}(t,k)|^{2}.$

The second kind of forcing $f_{V}$ is expressed by

(4) $f_{V}(x)=(-\sin x_{1}\cos x_{2},\cos x_{1}\sin x_{2},0)$

with $x=(x_{1},x_{2},x_{3})$. Note that $f_{V}$ is steady but anisotropic
forcing and the forcing wavenumber of $f_{V}$ is $k_{f}=\sqrt{2}$.

We conduct DNS by numerically solving the Navier-Stokes equation (1) by the
standard Fourier spectral method. The nonlinear terms are evaluated by using
the fast Fourier transform, where we remove the aliasing errors by the phase
shift method. For time integration of (1), we use the fourth order Runge-
Kutta-Gill scheme.

We use the external forces described in (2) and (4). For each force, we change
the Reynolds number by changing $\nu$ with $f$ fixed. We set the number
$N^{3}$ of the Fourier modes so that the dissipative scale, i.e. the
Kolmogorov length scale $\eta(t)=\nu^{3/4}\epsilon(t)^{-1/4}$ can be resolved.
Here, $\epsilon(t)$ is the spatial average of the energy dissipation rate.
More concretely, in our DNS with $f_{I}$ and $f_{V}$, we impose the condition
$k_{\mathrm{max}}\langle\eta\rangle=1.5$ and about $1.3$, respectively, by
appropriately setting $\nu$ and $N$. Here, $k_{\mathrm{max}}=\sqrt{2}N/3$ is
the maximum wavenumber and $\langle\cdot\rangle$ denotes the temporal average.
For the temporal integration, we choose the time increment satisfying the CFL
condition.

To evaluate the development of simulated turbulence, we use the Taylor-length
based Reynolds number $R_{\lambda}=u^{\prime}\lambda/\nu$. Here, $u^{\prime}$
and $\lambda$ denote the root mean square of a component of the velocity and
the Taylor length, respectively. In the case of homogeneous isotropic
turbulence, we can write

(5) $R_{\lambda}(t)=\sqrt{\frac{20}{3\nu\epsilon(t)}}\>K(t)$

where $K(t)$ is the kinetic energy per unit mass111In the case with $f_{V}$,
(4), the flow is accompanied by inhomogeneous anisotropic mean flow and
turbulence is neither statistically homogeneous nor isotropic. Therefore, (5)
is inaccurate, but we use it to evaluate $R_{\lambda}$ as we can calculate it
without the temporally averaged velocity field. We used (5) to evaluate
$R_{\lambda}$ with the forcing $f_{V}$ also in the previous studies [16, 20]..
By using $256^{3}$, $512^{3}$, $1024^{3}$ and $2048^{3}$ Fourier modes, we
have simulated turbulence with $\langle R_{\lambda}\rangle=130,210,350$ and
$520$ with $f_{I}$, and $\langle R_{\lambda}\rangle=180,290,500$ and $670$
with $f_{V}$. Recall that when $R_{\lambda}>140$ turbulence is considered
developed [24] in the sense that the forcing scale significantly $2\pi/k_{f}$
separates from the Kolmogorov length scale $\eta$.

###  2.2. Identification of tubular vortices in different scales

It is straightforward to numerically simulate turbulence at high Reynolds
numbers by the method described in the previous subsection. However, in
general, there are two difficulties when we analyze coherent vortical
structures in different scales in the developed turbulence.

The first difficulty stems from the fact that we cannot extract multiple-scale
features of coherent vortices by vorticity magnitude or the second invariant
(i.e. the $Q$ value) of the velocity gradient tensor. Since smallest-scale
flow structures predominantly determine the velocity gradient tensor, we only
observe the smallest-scale structures in the visualizations by using the
vorticity magnitude or the $Q$ value. This is the reason why we need a scale
decomposition to capture the multiple-scale structures in turbulence. For this
purpose, when we consider turbulence in a periodic box, a filter of the
Fourier modes can be used [15, 25, 19, 16] and here we also use the technique.
More precisely, to extract the hierarchy of vortices, we use the band-pass
filter,

(6)
$\displaystyle\mathbb{P}(k_{c})\>g(x)=\sum_{k\in\mathbb{Z}^{3}}\,\chi(k,k_{c})\,\widehat{g}(k)\,e^{ik\cdot
x}$

where $g$ is a function, $k_{c}$ is the highest wavenumber in the band,
$\widehat{g}$ denotes the Fourier component of $g$, and $\chi(k,k_{c})$ is the
characteristic function defined by

(7) $\chi(k,k_{c})=\left\\{\begin{split}&1&\quad&\text{if}\ \
k_{c}/2\leq|k|<k_{c},\\\ &0&\quad&\text{otherwise}.\end{split}\right.$

We use the phrase a large or small scale in accordance with the value of
$k_{c}$ because we can extract larger (or smaller) structures with smaller (or
larger) $k_{c}$. The band-pass filter (7) is simple but powerful to extract
multiple-scale coherent structures in spatially periodic turbulence [16].
Incidentally, for inhomogeneous turbulence, we can use real-space filters [26,
27, 17, 18] for the same purpose.

The above mentioned method can decompose a simulated turbulent velocity field
into different scales. However, we encounter another difficulty to objectively
identify coherent vortices at each scale. Although the simplest method is to
define vortices in a given scale by the regions where the scale-decomposed
vorticity magnitude is larger than a certain threshold, such a method is not
objective because it requires a threshold. Several methods such as
$\Delta$-method [28] and $\lambda_{2}$-method [29] were proposed as threshold-
free methods. Here, we employ the low-pressure method[22, 23]. This method is
based on the prerequisite that the pressure takes the minimum value at the
center of the swirling in a cross-section of a vortex. Although the low-
pressure method was originally proposed for dissipative-scale structures, it
was shown in Ref. [16] that this method was applicable at any scale. For
completeness, in the following, we briefly explain this method.

The concrete procedure of the low pressure method is as follows. First, we
search candidates to construct the axes of tubular vortices. Let $p_{c}$ be a
pressure satisfying the Poisson equation,

(8) $\Delta p_{c}=-\nabla(u_{c}\cdot\nabla u_{c})$

with $u_{c}(x,k_{c})=\mathbb{P}(k_{c})\>u(x)$. We assume that the pressure
around each grid point $a$ is expressed by

(9) $p_{c}(x)=\sum_{|\alpha|\leq
2}\frac{D^{\alpha}p_{c}(a)}{\alpha!}(x-a)^{\alpha}.$

Here, although we have assumed the pressure around each grid point is
estimated by the second order Taylor polynomial, this assumption holds in
practice due to the sufficient smoothness of $p_{c}$. The quadratic form
$(\ref{quad})$ can be written as the following normal form by changing the
coordinate system induced by a suitable transformation:

(10)
$p_{c}=p_{\mathrm{min}}+\sum_{j=1}^{3}\lambda^{(j)}(x^{\prime}_{j}-b_{j})^{2}$

where $x^{\prime}=(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3})$ denotes a
new coordinate system and $\lambda^{(1)}\geq\lambda^{(2)}\geq\lambda^{(3)}$
are the eigenvalues of the Hessian matrix of the pressure $p_{c}$. We denote
by $c$ a foot of a perpendicular lowered from $a$ to a line, which goes
through $b$ and is parallel to the eigenvector associated with
$\lambda^{(3)}$. If $\lambda^{(2)}>0$, we regard the point $c$ as being
located near the axis of a tubular vortex; otherwise we discard $c$. The
practical procedure to obtain $c$ is provided in Appendix A. For each grid
point $a$, we obtain the sequence $\\{a_{i}\\}$ by repeating this procedure as
follows: set $a_{0}=a$, apply this procedure by replacing $a$ with $a_{i}$ and
let $a_{i+1}$ be the obtained point corresponding to $c$. We terminate this
iteration at step $m$ $(1\leq m\leq 20)$ when $|a_{m}-a_{m-1}|<0.01d$. Here,
$d$ denotes the grid width. In this case, we record the point $a_{m}$ as a
candidate point on a vortex axis. However, we do not regard such a point as a
candidate point if there exists $i$ $(i\leq m)$ such that
$|a_{i}-a_{0}|>\sqrt{3}d/2$ or if the swirl condition [23] is not satisfied.
Note that the linear interpolation is used to evaluate the coefficients in (9)
at $a_{i}$.

In this way, we obtain the candidates. We allocate each candidate to the
nearest grid point. If there exists a grid point allocated more than one
candidate, we replace these candidates with a single candidate at the mass
center of them. These procedures give us the final candidates to construct the
axes of tubular vortices. Next we connect them. Each candidate is connected
with its nearest-neighbors in the adjacent grid cells in accordance with the
direction of the eigenvector associated with each $\lambda^{(3)}$. More
precisely, when we denote by $e^{(3)}$ a normalized vector obtained from such
an eigenvector for the candidate $q$, we search the nearest-neighbors in the
right circular cone $\\{x|e^{(3)}\cdot(x-q)/|x-q|>\mathrm{cos}20^{\circ}\\}$
where we choose the direction of $e^{(3)}$ so that the angle with the
vorticity becomes smaller.

Note that $d$ and $\mathrm{cos}20^{\circ}$ are artificial parameters. We set
$d$ as the numerical grid width in our DNS, which is very fine compared with
coherent vortices in the inertial range and sufficiently fine even in those in
the dissipative range. Since the connection between the candidates $q$ can be
different for different angle conditions ($\cos 20^{\circ}$), the number of
tubular vortices may depend on it. This is why we focus on the total length of
tubular vortices in the following.

These axes are composed of a connected series of line segments. We define the
length of each tubular vortex as the length of the polygonal chain. After
identifying the axes of tubular vortices, we calculate the length of each
vortex. Then, we evaluate the total length $L(k_{c})$ of the vortices in the
velocity field $u_{c}(x,k_{c})=\mathbb{P}(k_{c})u(x)$. In the next section, we
examine the functional form of $L(k_{c})$ to show the existence of the self-
similar hierarchy of tubular vortices.

##  3\. Numerical results

Figure 1. Vortex axes identified by applying the low-pressure method to the
scale-decomposed velocity fields obtained by the band-pass filter (7) in the
range $[k_{c}/2,k_{c})$ of the turbulence ($\langle R_{\lambda}\rangle=350$)
driven by the external force $f_{I}$. We show the whole domain in (a) and only
a $(1/4)^{3}$ part of the whole domain in (b). Red curves, vortex axes for
$k_{c}=8k_{f}/5$; blue, $k_{c}=32k_{f}/5$; gray, $k_{c}=128k_{f}/5$.

###  3.1. Self-similar hierarchy of vortex axes

First, let us observe the self-similar hierarchy of coherent vortices. We
visualize in figure 1 vortex axes, which we identify by the method described
in the preceding section, in turbulence driven by the isotropic force $f_{I}$.
In this figure, we show vortices at three different scales. The red curves are
identified vortex axes in the largest scale ($k_{c}=8k_{f}/5$), blue curves
are in the middle scale ($k_{c}=32k_{f}/5$), and gray curves are in the
smallest scale ($k_{c}=128k_{f}/5$) of the three scales. Note that the red
curves are at the length scale 16 times larger than the gray curves. Note also
that figure 1(a) shows the whole periodic box, whereas figure 1(b) shows a
$(1/4)^{3}$ part of the whole domain. We can see a qualitative similarity in
these two panels. More specifically, the number density ratio of red to blue
vortices in figure 1(a) seems similar to the ratio of blue to gray ones in
figure 1(b). More quantitative arguments in the following will show that this
is indeed the case. The similarity observed in these two panels implies the
self-similarity of the turbulence.

###  3.2. Dimension of the hierarchy

To quantify the self-similarity demonstrated in figure 1, we show the total
length $L(k_{c})$ of vortex axes in the scale $\ell={k_{c}}^{-1}$ in figures 2
(a) and (b) for the turbulence driven by $f_{I}$ and $f_{V}$, respectively. In
these figures, the darker symbols show the results for higher Reynolds
numbers. We show the results with changing the wavenumber $k_{c}$ of the band-
pass filter (7) as $k_{c}=1.2^{i}\times(8k_{f}/5)$ ($i=0,1,\cdots$). Note that
the ratio of the lowest and highest wavenumber of the bands, $[k_{c}/2,k_{c})$
is kept to be $2$.

Figure 2. The total length $L(k_{c})$ of the vortex axes in the wavenumber
range $[k_{c}/2,k_{c})$ in turbulence driven by (a, c, e) $f_{I}$ and (b, d,
f) $f_{V}$. We plot, in (c–f), $L(k_{c})/{k_{c}}^{2}$ as functions of $k_{c}$
normalized by (c, d) $k_{f}$ and (e, f) $\langle\eta\rangle^{-1}$. Darker
symbols are for higher Reynolds numbers: (a, c, e) $\langle
R_{\lambda}\rangle=130$ ($\circ$), $210$ ($\bullet$), $350$ ($\circ$), $520$
($\bullet$), (b, d, f) $\langle R_{\lambda}\rangle=180$ ($\circ$) $290$
($\bullet$), $500$ ($\circ$), $670$ ($\bullet$). Solid straight lines in (a,
b) indicate the power law with the exponent being $2$. The vertical lines in
(a, b) indicate $k_{c}=1/(3\langle\eta\rangle)$, and blue vertical lines in
(e, f) indicate $k_{c}=0.03/\langle\eta\rangle$. Figure 3. (a, b) The energy
spectra $E(k)$ of turbulence driven by (a) $f_{I}$ and (b) $f_{V}$. Solid
lines indicate the $-5/3$ power law, and the vertical lines are at
$k=1/(3\langle\eta\rangle)$. (c, d) The compensated spectrum
$\widehat{E}(k)\>\widehat{k}^{5/3}$, where
$\widehat{E}=E/(\langle\epsilon\rangle^{1/4}\nu^{5/4})$ and
$\widehat{k}=k/\langle\eta\rangle^{-1}$. Solid lines indicate the slope
$-0.12$. In all panels, the vertical black dashed lines indicate
$k=1/(3\langle\eta\rangle)$, whereas vertical blue dashed lines indicate the
boundary ($k=0.03/\langle\eta\rangle$) between the T (tilted) and B (bump)
ranges.

It is remarkable in figures 2(a) and (b) that $L(k_{c})$ obeys a quite clear
power-law function of $k_{c}$ in a surprisingly wide wavenumber range. We
emphasize that $L(k_{c})$ is estimated by a single snapshot and no temporal
average is taken. Although the data deviate from a straight line in a low
wavenumber range in this logarithmic plot, $L(k_{c})$ seems to obey a power
law for $k_{c}\gtrsim 5k_{f}$. The upper cut-off wavenumber of the scaling
depends on the Reynolds number. To demonstrate that it is determined by the
viscous scale, we show $1/(3\langle\eta\rangle)$ for each Reynolds number by
the vertical dashed lines. The data always deviate from the power law around
these vertical lines, which implies that the upper cut-off wavenumber is
proportional to $\langle\eta\rangle^{-1}$ irrespective of the Reynolds number
and forcing. Incidentally, $L(k_{c})$ rapidly decreases for $k_{c}\gtrsim
1/\langle\eta\rangle$.

Next, we evaluate the dimension of the hierarchy of the vortices by using the
statistics of $L(k_{c})$. The solid straight lines in figures 2 (a) and (b)
indicate the slope of $2$. It seems that the exponent $D_{L}$ of the power law
is approximately $2$. For more accurate arguments we show
$L(k_{c})/{k_{c}}^{2}$ in figures 2 (c–f). Panels (c) and (e) are results with
the forcing $f_{I}$, whereas (d) and (f) are those with $f_{V}$. Looking at
(c) and (d), where $k_{c}$ is normalized by $k_{f}$, $L(k_{c})/{k_{c}}^{2}$
takes a constant value for $k_{c}\gtrsim 10k_{f}$ in both cases of the
forcing. Since a clear plateau of $L(k_{c})/{k_{c}}^{2}$ is observed in this
semi-logarithmic plot, we may conclude that

(11)
$L(k_{c})=C_{L}\>\left(\frac{k_{c}}{k_{f}}\right)^{D_{L}}\qquad\text{(with
$C_{L}\approx 7$ and $D_{L}=2$)}\>.$

Here, we assume that the radius of tubular vortices is proportional to
$\ell={k_{c}}^{-1}$. Then, the scaling (11) of the total length of vortex
tubes implies that the volume fraction occupied by the vortex tubes in the
band $[k_{c}/2,k_{c})$ is proportional to
$L(k_{c})\ell^{2}\propto{k_{c}}^{0}$, i.e. the same volume fraction
independent of the scale. In other words, the dimension ($D_{L}+1$) of the
hierarchy of tubular vortices is $3$. Although this conclusion seems to be
inconsistent with the classical picture that turbulent eddies distribute
intermittently in space, we emphasize that we do not take into account the
vortex intensity in the identification of vortices. More concretely, we use
the low-pressure method, where we identify vortex axes on the basis of the
local pressure distribution and streamline pattern. This is the reason why the
identified vortices are always space-filling irrespective of the scale. In the
energy cascading process, these vortices at each scales are intensified or
weakened by the inter-scale nonlinear interactions (i.e. vortex stretching
process).

###  3.3. Upper cut-off wavenumber of the hierarchy

Here, we evaluate the upper cut-off wavenumber of the self-similarity (11) of
the hierarchy. For this purpose, we plot $L(k_{c})/{k_{c}}^{2}$ as a function
of the wavenumber normalized by $\langle\eta\rangle^{-1}$ in figures 2 (e) and
(f). We can see that $L(k_{c})/{k_{c}}^{2}$ is constant for $k_{c}\lesssim
1/(10\langle\eta\rangle)$. In summary, combining the observation in figures 2
(c) and (d) for the lower cut-off wavenumber, the power law (11) holds in the
wavenumber range

(12) $10k_{f}\lesssim k_{c}\lesssim 0.1\langle\eta\rangle^{-1}.$

Note that this wavenumber range (12) exists only when $R_{\lambda}$ is larger
than about $150$. As described in §2, we have conducted DNS with completely
different kinds of forcing; $f_{I}$ is statistically homogeneous and isotropic
but time-dependent, whereas $f_{V}$ is steady but inhomogeneous and
anisotropic. The shown results on the hierarchy of vortices are independent of
the forcing for $k_{c}\gtrsim 10k_{f}$.

Here, we further investigate the implication of the scaling range (12) in more
detail. We plot the energy spectrum $E(k)$ of the turbulence driven by $f_{I}$
and $f_{V}$ in figures 3 (a) and (b), respectively. For the visibility, we
only plot $E(k)$ for the highest Reynolds number in each case of forcing. As
widely known, the energy spectrum has a bump in wavenumbers between the
inertial range, where the energy flux is constant, and the dissipation rage,
where the spectrum decays exponentially. In order to compare the scaling
ranges of $E(k)$ and $L(k_{c})$, we plot the compensated energy spectrum
$E(k)\>k^{5/3}$ in figure 3(c) and (d). Ishihara et al. [30] showed by the DNS
at much higher Reynolds numbers (up to $R_{\lambda}=2297$) than ours that
there are distinct wavenumber ranges; namely, from the higher wavenumbers, the
B (bump) range where $E$ has a bump, the T (tilted) range where $E(k)\propto
k^{-5/3-\mu}$ (with the intermittency exponent $\mu\approx 0.12$), and the F
(flat) range where the compensated spectrum $E(k)k^{5/3}$ exhibits a plateau
indicating the Kolmogorov spectrum without intermittency effects. According to
Ref. [30], the boundary between the F and T ranges is about
$0.005\langle\eta\rangle^{-1}$ and that between T and B is about
$0.02\langle\eta\rangle^{-1}$. The energy spectrum in our DNS also shows
similar behaviors (figure 3), although there is no F range because of the
smallness of $R_{\lambda}$. More specifically, the boundary between the T and
B ranges is located around $0.03\langle\eta\rangle^{-1}$ (figure 3(c, d)).
Note that this value is slightly larger than the value
$0.02\langle\eta\rangle^{-1}$ suggested in Ref. [30]. Recalling the range (12)
of the self-similarity of the vortex hierarchy (figure 2), we conclude that
the self-similarity holds in T range and a lower part of the B range.

Before closing this subsection, it is worth mentioning the relationship shown
above and the energy cascading process. In Ref. [20], we have shown the self-
similar energy transfer due to vortex stretching; that is, vortices at given
length sales acquire the energy from about 1.7 times larger vortices, and
transfer it to about $1/1.7$ times smaller vortices. We have also shown that
this self-similarity is valid in the wavenumber range smaller than
$0.03\langle\eta\rangle^{-1}$; namely, in the T range. In the B range,
vortices transfer their energy to $1/1.7$ times smaller vortices, but the size
of mother vortices changes because of the attenuation of vortices by the
viscous effects; see figure 1(c) in Ref. [20].

###  3.4. Steadiness of the hierarchy

It is known that $E$ can temporally fluctuate about its temporal mean because
the intensity of vortices transfers from larger scales to smaller ones [31,
32]. In contrast, $L(k_{c})$ always obeys a clear power law. To demonstrate
this feature, we plot the temporal evolution of the spatially-averaged kinetic
energy $K(t)$ and average dissipation rate $\epsilon(t)$ in figure 4(a). This
is a result from the DNS with the forcing $f_{V}$. As was shown in Refs. [32,
16], these quantities in turbulence driven by the steady force $f_{V}$ evolves
quasi-periodically with significant amplitude with respect to their temporal
means. Within this time period, we take six instants with a fixed time
interval ($\Delta t/T=4.5$) to plot $L(k_{c})$ in figure 4(b) with six
different symbols. Despite the significant fluctuations in $K(t)$ and
$\epsilon(t)$, $L(k_{c})$ almost perfectly steady in the high wavenumber range
($k_{c}\gtrsim 10k_{f}$). Recall that the vortex axes are identified by the
low-pressure method without any assumption on the vortex strength. Hence,
these results imply that the number of tubular vortices are almost independent
of time even if the vortex strength changes. This result supports one of the
assumptions in Ref. [20], which we will discuss in more detail in the next
section.

Figure 4. (a) Time series of the spatially averaged kinetic energy $K(t)$ and
average dissipation rate $\epsilon(t)$ in the turbulence sustained by $f_{V}$
at the Reynolds number $\langle R_{\lambda}\rangle=500$. Time is normalized by
the turnover time $T$ of the largest eddies. We observe a quasi-periodic
behavior with a significant magnitude. (b) $L(k_{c})$, normalized by
${k_{f}}^{-1}$ at six different instants at $t/T=4.6$, $9.1$, $14$, $18$, $23$
and $27$ in the period shown in (a). We show six data with different symbols,
all of which are almost perfectly coincide for $k_{c}\gtrsim 10k_{f}$. Solid
line indicates the power law with the exponent $2$.

##  4\. Discussions

###  4.1. Justification of our energy cascade model

On the basis of the numerical results shown in the preceding section, we
discuss the assumption in Ref. [20]. By using the notation in this paper, the
vorticity decomposition, which is one of the most important assumptions in
Ref. [20], can be written as follows:

(13) $\displaystyle\mathbb{P}(k_{c})\
\omega(t,x)=A(k_{c})\sum_{j=1}^{N(k_{c})}W_{j,k_{c}}(x)$

where $A(k_{c})$ is the amplitude of this vorticity, $W_{j,k_{c}}$ is the
vorticity of each tubular vortex and $N(k_{c})$ is the number of tubular
vortices at this scale. There are three points to note in this assumption.
First, this decomposition is independent of time. In particular, $A(k_{c})$
and $N(k_{c})$ are independent of time. Second, the vortices at the same scale
have a constant intensity, and the intensities at each scale are all
represented by $A(k_{c})$. Third, this decomposition does not depend on the
kinematic viscosity $\nu$.

These assumptions are partially justified by our DNS. We have evaluated the
total length $L(k)$ of tubular vortices at each scale, corresponding to the
left-hand side of (13). The temporal stationarity of $L(k)$ in figure 4
implies that the right-hand side of (13) is also stationary. In particular, it
means that $N(k_{c})$ is independent of time if we consider the vortices at
the same scale have a characteristic length proportional to
$\ell={k_{c}}^{-1}$. Note that it is impossible to discuss the time dependence
of $A(k_{c})$ with this calculation alone. However, the assumption on
$A(k_{c})$ is regarded as a way to simplify the problem, since only the
structure of coherent vortices is taken into account in Ref. [20]. As already
shown in figure 2, the robustness for $\nu$ ($R_{\lambda}$ in our DNS) is also
verified.

In our DNS, it is shown that $L(k_{c})$ obeys the power law (11) irrespective
of time and the Reynolds number. These results are consistent with the
derivation of the $-5/3$ law in Ref. [20]. If we want to evaluate the
deviation from the $-5/3$ law in more detail, we need to consider the strength
of vortices, which we have not dealt with so far. This issue is discussed in
the following subsections.

###  4.2. Self-similar hierarchy of strong vortices

The numerical results in §4 show that the number density of coherent tubular
vortices at different scales is independent of the scale. This means that no
intermittency effects are observed in the number density. However, it is well
known that the energy dissipation rate in turbulence is intermittent. It has
been considered that the spatial intermittency of the energy dissipation rate
originates from the accumulation of the energy flux through a scale-by-scale
energy cascade process. Here, we investigate this intermittency effect of the
energy flux by evaluating the activity of the energy cascade in terms of the
band-pass-filtered vorticity $\omega_{\ell}$ in the range $[k_{c}/2,k_{c})$
with $\ell={k_{c}}^{-1}$ on each vortex axis. In practice, we add another
criterion to the identification method described in §2; we discard the
candidates at which the enstrophy $\Omega_{\ell}=|\omega_{\ell}|^{2}$ is
smaller than $\alpha\overline{\Omega_{\ell}}$, where
$\overline{\Omega_{\ell}}$ is the spatial average of $\Omega_{\ell}$. Then, we
estimate the total length $L^{(\alpha)}(k_{c})$ of the axes of those strong
vortices.

First, we show $L^{(\alpha)}(k_{c})$ with $\alpha=1$, $2$, $3$ and $4$ at
three different instances of the turbulence at $\langle
R_{\lambda}\rangle=350$ in figure 5(a). Note that we plot
$L^{(\alpha)}/{k_{c}}^{1.92}$. We have heuristically found this exponent 1.92
so that we can demonstrate $L^{(2)}(k_{c})\propto{k_{c}}^{1.92}$. This figure
shows that the hierarchy of stronger tubular vortices is indeed intermittent,
though the dimension $1.92+1=2.92$ of the hierarchy is not far from $3$. It is
also clear that $L^{(\alpha)}(k_{c})$ is also independent of time as in
$L(k_{c})$ (figure 2). Next, we plot $L^{(\alpha)}(k_{c})$ at different
Reynolds numbers in figure 5(b) to show that $L^{(\alpha)}(k_{c})$ is
independent of $R_{\lambda}$ except the low-wavenumber range. Incidentally,
carefully looking at the curves for different $\alpha$ in figure 5, we notice
that the scaling range expands to a higher wavenumber range for larger
$\alpha$. This reminds us of multifractal models of intermittency, in which
the upper bound of the inertial range depends on the order of the structure
function.

The results in figure 5 indeed indicate intermittency effects, but the effects
are observed in the dissipation range, or more precisely, in a lower part of
the B range ($0.02\langle\eta\rangle^{-1}\lesssim k_{c}\lesssim
0.1\langle\eta\rangle^{-1}$). In other words, the fractal nature observed in
figure 5 does not reflect spatial intermittency in the inertial range (i.e.
the T range with $k\lesssim 0.02\langle\eta\rangle^{-1}$).

To further investigate intermittency, we estimate the probability density
function (PDF) of the band-pass-filtered velocity components $\Delta u_{\ell}$
on vortex axes. If we normalize $\Delta u_{\ell}$ by $\ell^{1/3}$, then the
PDF seems identical in the T range (figure is omitted), though the range is
quite narrow in our DNS. In contrast, in the B range, the tail of the PDF is
more pronounced for larger $k_{c}$ (i.e. smaller $\ell$). Therefore,
intermittency effects in the inertial range are rather weak in turbulence at
the moderate Reynolds numbers ($R_{\lambda}=500$–$600$) and those in the
dissipation range are much more significant.

Figure 5. (a) The total length $L^{(\alpha)}(k_{c})$ ($\alpha=1$, $2$, $3$ and
$4$) of strong vortices with enstrophy being $\alpha$ times larger than its
spatial mean. To show the scaling $L^{(2)}(k_{c})\propto{k_{c}}^{1.92}$, we
plot $L^{(\alpha)}(k_{c})/{k_{c}}^{1.92}$ as functions of $k_{c}$. The results
at three different instances of turbulence driven by $f_{I}$ at the Reynolds
number $\langle R_{\lambda}\rangle=350$. (b) The Reynolds-number dependence of
$L^{(\alpha)}$. Different symbols correspond to different $\langle
R_{\lambda}\rangle$; see figure 2.

###  4.3. Multifractal model

We have developed arguments on the energy cascade in terms of coherent
vortices in our previous studies [15, 19, 16, 17, 18, 20], and we have
objectively identified the hierarchy of their axes in the present study.
Therefore, we may construct a concrete cascade model to explain the spatial
intermittency of the energy flux and the dissipation rate. Here, we discuss
the probabilistic formulation of the multifractal model (see §8.5.4 of Ref.
[5]). First, suppose that we can estimate the velocity difference $\Delta
u_{\ell}$ on each vortex axis in a given scale $\ell={k_{c}}^{-1}$. In
practice, this can be done by an interpolation of the velocity field the
Fourier-filtered velocity field in the wavenumber band (7). Then, we can
estimate the $p$-th order moment of $\Delta u_{\ell}$ according to the
multifractal analysis of the hierarchy of vortex axes as follows. In a given
scale $\ell$ in the hierarchy, we denote by $dP_{\ell}(h)$ the probability
density for $\Delta u_{\ell}$ to be $v_{0}(\ell/\ell_{0})^{h}$. Here, $v_{0}$
and $\ell_{0}$ are constants and $h$ denotes the Hölder exponent. Then, we can
estimate the $p$-th order moment of $\Delta u_{\ell}$ as

(14) $\overline{\Delta
u_{\ell}^{p}}=\int{v_{0}}^{p}\>\left(\frac{\ell}{\ell_{0}}\right)^{hp}\>dP_{\ell}(h)\>.$

It is important that we may numerically estimate $dP_{\ell}(h)$ by evaluating
the total length $L_{h}(\ell)$ of the parts of vortex axes where the velocity
difference $\Delta u_{\ell}$ is $v_{0}(\ell/\ell_{0})^{h}$. Here, we expect
that $L_{h}$ obeys a power law,
$L_{h}(\ell)\propto{k_{c}}^{\widetilde{D}(h)}\propto{\ell}^{-\widetilde{D}(h)}$,
when $\ell$ is in the inertial range. Then, $dP_{\ell}(h)$ is expressed as

(15) $dP_{\ell}(h)\propto
A(h)\>\left(\frac{\ell}{\ell_{0}}\right)^{3-D(h)}dh=A(h)\>\left(\frac{\ell}{\ell_{0}}\right)^{2-\widetilde{D}(h)}dh$

where $D(h)=\widetilde{D}(h)+1$ and $A(h)$ is a constant depending on $h$.
Here, we have assumed again that the radius of vortex tubes are independent of
their intensity and proportional to $\ell={k_{c}}^{-1}$. Thus, substituting
(15) to (14) and using the steepest descent method, we can express the
exponent $\zeta_{p}$ of $\overline{\Delta
u_{\ell}^{p}}\propto\ell^{\zeta_{p}}$ in terms of $\widetilde{D}(h)$, which
may be evaluated by using DNS data.

Since we have shown the power-law behavior (figures 2 and 5) of the total
length $L$ and the length of strong parts of the vortex axes, we expect that
we can evaluate $D(h)$ by estimating $L_{h}$ in a similar manner. This kind of
analysis is attractive because we can reveal the physical origin of
intermittency in terms of the realistic coherent structures. Unfortunately,
however, this is impossible with our DNS data. As discussed in §3, in order to
discuss the intermittency effects (e.g. the deviation $\mu$ from the
Kolmogorov spectrum), we have to evaluate $D(h)$ using the data in the T range
(i.e. $k\lesssim 0.03\langle\eta\rangle^{-1}$). However, as discussed in the
previous subsection, the fractal nature observed in figure 5 is mainly in the
B range, which is contaminated by viscous effects. In other words, the T range
in our DNS data at $R_{\lambda}\approx 500$–$600$ is too narrow to evaluate
intermittency effects in the inertial range.

Detailed multifractal analysis of the T range with DNS data at higher Reynolds
numbers is a future interesting study. It is also an interesting issue to
understand the physical and mathematical origin of the boundary between F and
T ranges of $E(k)$ around $k\approx 0.005\langle\eta\rangle^{-1}$ [30].

##  5\. Conclusion

In our previous study [20], by using the notion of the hierarchy of coherent
tubular vortices, we mathematically reformulated the energy cascade process
and derived the $-5/3$ power law of the energy spectrum from the Navier-Stokes
equation without directly using the Kolmogorov similarity hypothesis. In the
present article, we have numerically examined the assumptions in the
formulation. To this end, we have applied the band-pass filter (7) to the
turbulent velocity fields obtained by DNS, and applied the low-pressure method
[22, 23, 16] to the scale-decomposed fields so that we can objectively
identify the skeletal structures of coherent tubular vortices at each scale.
The identified hierarchy (figure 1) of vortex axes is self-similar. To
evaluate the dimension of the hierarchy, we have estimated the total length
$L(k_{c})$ of the vortex axes in the wavenumber band $[k_{c}/2,k_{c})$. As
shown in figure 2, $L(k_{c})$ obeys the power law (11), even without time
averaging, in a significantly wide range of $k_{c}$. More precisely, this
power law holds in the wavenumber range (12); namely, not only the T range
(within the inertial range) but also a lower part of the B range (within the
dissipation range), where the energy spectrum is accompanied by a bump. See
figure 3 for the definitions of the T and B ranges. This result implies that
there exists the energy cascade process due to vortex stretching even in the
dissipation range, though the vorticity is attenuated by the viscosity in the
range. In fact, this result is also consistent with the observation (figure
1(c) of Ref. [20]) of the energy transfer in the B range. In other words, the
number of child vortices in the cascading process is constant irrespective of
the scale in the range (12). We have also shown that $L$ is robust and
independent of Reynolds number and forcing (figure 2), which implies the
universality of turbulence. It is particularly important for the mathematical
formulation of the energy cascade that $L$ is time independent (figure 4) even
when the kinetic energy and its dissipation rate significantly fluctuate. This
result supports the assumption in Ref. [20]; see §4(a). The results in figure
2 also show that the intermittency of the energy dissipation rate is not due
to the increase of volume fraction of smaller scale vortices but due to the
accumulation of the energy flux in smaller scales; see §4(b). Therefore, we
may further improve our mathematical formulation of the energy cascade [20],
by taking into account the multifractal nature (§4(c)), to describe the
deviation of the energy spectrum from the $-5/3$ law.

##  Appendix A Practical procedure to obtain a candidate

Here, $\bm{e}^{(i)}$ denotes the unit eigenvector associated with
$\lambda^{(i)}$ and symbols in the main text are shown in vector notation. We
assume $\lambda^{(2)}>0$. For the grid point $\bm{a}$, we put
$\bm{c}=\bm{a}+\bm{\xi}$. Note that $\bm{\nabla}p_{c}(c)$ is parallel to
$\bm{e}^{(3)}$. Since $p_{c}$ is approximated by its second-order Taylor
polynomial, we have

(19)
$\displaystyle\left\\{\begin{array}[]{l}0=\bm{e}^{(1)}\cdot\bm{\nabla}p_{c}(a)+\lambda^{(1)}\bm{e}^{(1)}\cdot\bm{\xi},\\\
0=\bm{e}^{(2)}\cdot\bm{\nabla}p_{c}(a)+\lambda^{(2)}\bm{e}^{(2)}\cdot\bm{\xi},\\\
0=\bm{e}^{(3)}\cdot\bm{\xi}.\end{array}\right.$

Hence, in order to obtain $\bm{c}$, it suffices to solve the following
equation:

(20)
$\left[\bm{e}^{(1)},\bm{e}^{(2)},\bm{e}^{(3)}\right]\left[\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\\\
\xi_{3}\end{array}\right]=\left[-\frac{\bm{e}^{(1)}\cdot\bm{\nabla}p_{c}}{\lambda_{1}},-\frac{\bm{e}^{(2)}\cdot\bm{\nabla}p_{c}}{\lambda_{2}},0\right].$

Acknowledgments. Research of SG was partly supported by the JSPS Grants-in-Aid
for Scientific Research 20H02068 and 20K20973. TY was partly supported by the
JSPS Grants-in-Aid for Scientific Research 20H01819.

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