# Characterizing Twitter users behaviour during the Spanish Covid-19 first
wave

Bernat Esquirol
Universitat de Barcelona
UBICS
Luce Prignano
Universitat de Barcelona
UBICS
&Albert Díaz-Guilera
Universitat de Barcelona
UBICS
&Emanuele Cozzo
Universitat de Barcelona
UBICS
Universitat Oberta de Catalunya
CNSC/IN3
<EMAIL_ADDRESS>

###### Abstract

People use Online Social Media to make sense of crisis events. A pandemic
crisis like the Covid-19 outbreak is a complex event, involving numerous
aspects of the social life on multiple temporal scales. Focusing on the
Spanish Twittersphere, we characterized users activity behaviour across the
different phases of the Covid-19 first wave.

Firstly, we analyzed a sample of timelines of different classes of users from
the Spanish Twittersphere in terms of their propensity to produce new
information or to amplify information produced by others. Secondly, by
performing stepwise segmented regression analysis and Bayesian switchpoint
analysis, we looked for a possible behavioral footprint of the crisis in the
statistics of users’ activity.

We observed that generic Spanish Twitter users and journalists experienced an
abrupt increment of their tweeting activity between March 9 and March 14, in
coincidence with control measures being announced by regional and State level
authorities. However, they displayed a stable proportion of retweets before
and after the switching point.

On the contrary, politicians represented an exception, being the only class of
users not experimenting this abrupt change and following a completely
endogenous dynamics determined by institutional agenda. On the one hand, they
did not increment their overall activity, displaying instead a slight
decrease. On the other hand, in times of crisis, politicians tended to
strengthen their propensity to amplify information rather than produce it.

_K_ eywords Online Human Behavior $\cdot$ Twitter $\cdot$ Covid-19 $\cdot$
Segmented Regression $\cdot$ Bayesian switchpoint

##  1 Introduction

The COVID-19 outbreak (1 December 2019 – present) is the first global pandemic
of the information age. A large-scale public debate is taking place in a
hybrid media ecosystem in which online social media (OSM) in general, and
Twitter in particular, play a central role [1, 2, 3]. The online debate
addresses all the aspects of the pandemic induced crisis that the public
opinion considers to be relevant: from the origin of the virus, to the
controversy surrounding vaccination campaigns; from the necessity (or lack of
it) for lockdowns and other social distancing measures, to the effectiveness
of the different national health systems. These are evidence that people use
Twitter to make sense of the pandemic crisis, i.e. they try to understand,
through interaction with others, an ongoing event that is new, uncertain, and
confusing [4]. However, unlike other crisis events, as for instance
earthquakes or terrorist attacks, a pandemic crisis is a complex phenomena
spanning different time scales. In particular, the first wave of a new
infectious disease is an even more peculiar phenomenon, being completely
unseen and unexpected. We can distinguish a priori three different phases for
the unfolding of the first wave in a given region: an early phase in which the
virus and information about it starts to spread, a phase in which the epidemic
hits a particular region and the debate about control and mitigation measures
arises, and a third phase in which these measures are finally implemented. Our
hypotheses, building on some recent findings [2, 5], are that online users
will react differently to different phases and that different type of users
will react in different moments.

In fact, it is well known that OSM react abruptly to external and internal
forces, eventually resulting in deep transformations without warning [6]. This
inherent volatility suggests that online behaviour will change in response to
crises and the associated interventions. In this sense, OSM can serve as an
early warning system for any kind of emergency worldwide, from natural
disasters to terrorist acts [7, 8].

Unfortunately, this useful property comes with an important drawback. The
efforts made in the last few years to measure and characterize human and
human-like activity in OSM mainly consist in the search for characteristic
patterns in users activity data. For instance, it has been shown that some
aspects of human communication are described by universal statistics which may
help distinguish among different types of human and human-like users [9, 10].
Nonetheless, when an emergency occurs, all the typical patterns of activity
are probably altered and usual classification algorithms are no longer
reliable. This is even more important if we take into account the
misinformation processes that accompany an ongoing crisis event, named
‘infodemics’ in the case of a pandemic [1]. In order to properly identify
harmful activity in the context of a crisis, either abrupt or prolonged, we
need to know how different classes of users behave in such circumstances.

In this work, we characterize online collective behaviour throughout the
different phases of the first wave of Covid-19, and, in particular, in
response to the prevention and control measures adopted, using Spain
Twittersphere as a case study. In this way, we will be able to exploit the OSM
intrinsic volatility to analyze at the same time the effect of the different
phases of the crisis on the collective behaviour and the long term effects of
that on the typical patterns of activity for the duration of the crisis.

To this end, we consider the whole tweeting activity, regardless of the topic
of the content, instead of only tracking tweets directly related to the
Covid-19 outbreak. We are thus able to compare online behaviors before and
during the crisis from an overall perspective. We partition the Spain
Twittersphere in three different classes of users: politicians, journalists,
and generic users, since we consider that they provide a good representation
of the division of roles in a networked public sphere [11].

Figure 1: Timeline of Covid-19 pandemic related events in Spain: restriction
measures, confirmed cases and deaths during the first wave. A interactive
version of the infographic is available at
https://public.flourish.studio/visualisation/4432020/

##  2 Methods

###  2.1 Data Collection

We collected Twitter users’ timelines, _i.e._ , the time stamped list of
tweets (including retweets) published by a user, of three different types of
Spain-based profiles: journalists, politicians, and random users. In the
following, we describe the methods used for the construction of the three set
of users.

Journalists’ accounts We observed that, usually, the news media follow on
Twitter the journalists working for them and journalists usually have some
variation of the word journalist in their bio. Leveraging on these two
elements, in order to sample Spanish journalists’ Twitter profiles, we
searched the Twitter accounts of the top news media in Spain (list in appendix
A) and got their friends (users being followed by the media account). Among
them, we retained those who have in their bio some version of the word
journalist. In this way, we obtained 1722 journalists’ profiles.

Politicians’ accounts We picked the members of the Congreso XIV Legistlatura
list that contains all the members in the Spanish congress during the February
to June 2020 period. There are 350 congresspersons in Spain, but not all of
them have a Twitter account: altogether, we retrieved 261 politicians
profiles.

Random users Getting unbiased random samples of Twitter users for a given
location is not an easy task given that geolocated tweets are sparse. We
developed a method with the aim to reduce as much as possible the bias toward
highly active users.

Our method works as follows:

  1. 1.

We got the 100 most recent tweets from users located in Spain through the
Twitter Standard Search API

  2. 2.

We filtered those 100 tweets so that the creator of the tweet ($u_{0}$) had
more than a given threshold number of followers, friends, and actions (55,95,
and 1000 respectively).

  3. 3.

We got one follower and one friend of $u_{0}$ at random.

We repeated the process until we retrieved around 8.000 profiles in each
random set.

Step 2 is to filter out bots or extremely inactive users. Step 3 is to avoid a
bias towards high frequency users that we would have by taking user $u_{0}$
directly. In this way, we get two types of random users: random-friends are
those that we get sampling the friends of $u_{0}$, and random-followers those
obtained by sampling the followers. The first group is biased toward users
following many other accounts, and the other is biased towards users having
many followers. We used both sets separately and compared the results of the
analyses performed on them at each stage. The underlying idea is that,
whenever we identifies a trait common to both random-followers and random-
friends, such a trait can be regarded as a property of random users in
general.

The Twitter Search API allows us to retrieve 3200 tweets for each timeline,
starting from the most recent one. Hence, depending on the user’s activity, we
were able to cover a more or less large time period. The first data retrieval
was conducted on April 28, then we periodically collected new tweets from the
sampled users via the Twitter Search API.

###  2.2 Analysis of the activity

We characterized the the activity of an individual user both from an overall
(time-aggregated) perspective and analysing its evolution through time, during
the different phases of Covid-19 first wave.

First of all, we characterized individual profiles based on two variables. On
the one hand, we compared their tendency to publish their own contents and to
share other people’s tweets; on the other hand, we analysed how distributed or
concentrated their retweets are, in terms of the accounts that receive them
(sources). In this way, we could highlight differences between the four
samples of users under study, thus tracing a preliminary behavioral profiling
of the corresponding classes of accounts.

Secondly, we looked for possible abrupt jumps that the activity of a user may
have experienced in particular days. To identify and quantify such jumps, we
considered the mean activity before and after them, with respect to the
overall mean activity, and the associate probability that a jump actually
occurred that day. We performed two types of analysis and compared the
results.

####  2.2.1 Production VS Amplification of information

In this first preliminary time-aggregated analysis, we characterized each
individual users according to their propensity to produce new information,
_i.e._ , tweeting, or to amplify information produced by others, _i.e._ ,
retweeting. In addition, we quantify the diversity of the sources of the
amplified information. With this aim, we introduce for each user $u$ two
parameters: the fraction of replicated content $\rho_{u}$ and the mean entropy
of the sources $h_{u}$, a measure of how "generalist" a given user is in terms
of information/content sources. Given the number of total tweets including
retweets $M_{u}$ emitted by a users and the number of retweets $R_{u}$,

$\rho_{u}=\frac{R_{u}}{M_{u}}.$ (1)

The fraction of retweets of a given user to the same source $s$ is
$p^{s}_{u}=\frac{r^{s}_{u}}{R_{u}}$, where $r^{s}_{u}$ is the number of
retweets to the source $s$. Thus, the normalized entropy of the sources for
the user $u$ is defined as

$h_{u}=\frac{-\sum_{s}p^{s}_{u}\log p^{s}_{u}}{\log R_{u}},$ (2)

where the denominator is the entropy of the sources of a user that retweets
each source once.

In this way, we could compare each set of users with the others based on their
distribution in the $h-\rho$ plane. In particular, we use the Kullback-Leibler
(KL) divergence to measure the disparity between distributions of $\rho$ and
$h$ of different set of users. The KL divergence, given by the expression

$D_{KL}(P\mid\mid Q)=\sum_{i}p(x_{i})\log(\frac{p(x_{i})}{q(x_{i})}),$ (3)

is one of the most used disparity measures between probability distributions.
It is a measure of the information lost when $Q$ is used to approximate $P$,
being $P$ and $Q$ on the same finite support.

####  2.2.2 Stepwise segmented regression analysis

The first approach that we adopted for studying possible changes in the
activity of different types of Twitter users was a stepwise segmented
regression (SSR) analysis [6] of the timeline of each profile. Given the list
of timestamps of the tweets of a user, we optimized a piecewise linear fit
that divided the timeline into segments. In this way, for each timeline we
obtained a number of breakpoints – the end points of each segment – marking
the moments of abrupt change in the tweeting frequency of the user. The
optimal breakpoints were estimated by minimizing the residual sum of squares
[12], while their number was fixed by minimizing the Bayesian Information
Criterion of the model.

In this way, we could associate to each user $u$ the set of breakpoints
$\tau^{u}_{1},\tau^{u}_{2},...,\tau^{u}_{n}$, _i.e._ , the days in which their
activity changed abruptly, if any. Finally, we characterized users’
breakpoints by the relative jump

$J^{u}_{i}=\frac{L^{u}_{i}-L^{u}_{i-1}}{m^{u}}=\frac{\Delta L^{u}_{i}}{m^{u}}$
(4)

where $L_{i}^{u}$ is the tweeting frequency of user $u$ in the interval
$[\tau^{u}_{i},\tau^{u}_{i+1})$ and $m^{u}$ is the mean frequency on the whole
user u retrieved timeline.

We could hence describe the overall activity by summing separately positive
and negative jumps:

$J^{\pm}(\tau)=\sum_{u}J_{i}^{u}\theta(\pm
J_{i}^{u})\delta_{\tau}^{\tau_{i}^{u}}$ (5)

####  2.2.3 Bayesian switchpoint analysis

The second approach was based on Bayesian inference. We performed a Bayesian
switchpoint (BS) analysis, which looks for a single moment of change in the
rate at which events occurred. The model assumes, for each user, Poisson-
distributed tweeting events with constant but potentially different rates,
before and after a switchpoint.

$\displaystyle\lambda_{1}^{(0)}\ :\ \ \ $ Exponential (rate =$\alpha$) (6)
$\displaystyle\lambda_{2}^{(0)}\ :\ \ \ $ Exponential (rate =$\alpha$) (7)
$\displaystyle\tau\ :\ \ \ $ $\displaystyle\text{Uniform }[0,1)$ (8)

For $i=1\dots n$:

$\lambda_{i}=\begin{cases}\lambda_{1}^{(0)}\ \ \ \text{if}&\tau\leq i/n\\\
\lambda_{2}^{(0)}\ \ \ \text{if}&\tau>i/n\end{cases}$ (9)

Our prior for $\lambda_{(1,2)}$ was an exponential distribution with rate
$\alpha$ while our prior for $\tau$ was a uniform distribution over the whole
time span of the timeline.

An Hamiltonian Monte Carlo method [13] as implemented in TensorFlow [14] was
used for sampling from the relevant posterior distributions of $\lambda_{1,2}$
and $\tau$ .

Thus, we could associate to each user a probability distribution for the
switchpoint and a probability distribution for the two different tweeting
rates, before and after the switch respectively. To carry out an overall
analysis of the behavior of the Spanish Twittersphere, as for the SSR
analysis, we needed to characterize the switchpoints. By approximating the
distributions of the $\lambda$s with a delta centered in their mode, we
obtained the following expression for the relative jump:

$J^{u}(\tau_{i})=P(\tau^{u}=\tau_{i})\frac{\tilde{\lambda^{u}_{2}}-\tilde{\lambda^{u}_{1}}}{m^{u}}=\frac{\Delta\lambda^{u}}{m^{u}}$
(10)

where $\tilde{\lambda}_{1,2}$ is the mode of the respective posterior
distribution. We could then characterize the overall activity by summing
separately positive and negative jumps of each user weighted by the jump
probability:

$J^{\pm}(\tau)=\sum_{u}J^{u}(\tau_{i})\Theta(\pm
J^{u}(\tau_{i}))\delta^{\tau_{i}}_{\tau}$ (11)

##  3 Results

###  3.1 Overall activity of different classes of users

Comparing the different sets of users with respect to their tendency to
produce new information rather than amplify already existing contents ($\rho$)
and how they distributed their retweets among different sources ($h$), we
observed that the two sets of random profiles – followers and friends – were
practically indistinguishable (Fig. 2, upper panels). They both showed a
nearly flat distribution of $\rho$ (vertical axe), covering equitably all the
spectrum from pure replicator ($\rho=1$) to pure content producer ($\rho=0$).
For what concerns $h$ (horizontal axe), both groups displayed a markedly right
skewed distribution, a clear indication that most of them tended to distribute
their retweets almost uniformly among different sources. Journalists (Fig. 2,
lower right panel) appeared to be quite similar to random users, but avoiding
extreme values of $\rho$, in line with the expectation that a journalist will
always tend to retweet her media and colleagues as well as produce new
contents. Finally, politicians (Fig. 2, lower left panel) displayed the most
peculiar activity patterns. They are more similar to journalists in $\rho$ but
showing a complete different distribution of $h$ (symmetric, with the mean and
the mode falling slightly above $0.5$).

(a) Random Followers

(b) Random Friends

(c) Journalists

(d) Politicians

Figure 2: Distribution of the fraction of retweets $\rho$ and normalized
entropy $h$ for each one of the considered classes of users.

Such differences and similarities are quantitatively described by the KL
divergence for the distributions of $\rho$ and $h$, represented as heatmap
matrices in Fig. 3.

KL divergence for $\rho$ distributions

KL divergence for $h$ distributions

Figure 3: Kullback-Leibler (KL) divergence of $\rho$ (left panel) and $h$
(right panel) distribution between classes of users.

###  3.2 Time evolution

Once we clarified that random users differed from politicians and, although to
a lesser extent, journalists in terms of their overall (re)tweeting activity,
the next step was to compared the evolution of their activity rate during the
first wave of Covid-19.

For a preliminary general inspection, we considered all the users together,
regardless their class. From both the SSR and BS analyses (Fig. 4), we
obtained a clear indication that in the week between March 9 and March 14 the
majority of Spanish twitter users abruptly changed their tweeting behaviour,
starting to tweet with higher frequency. These days coincided with the first
control measures being introduced at regional level on Monday 9 and the
declaration of the state of alarm by the prime minister of Spain on Friday 13,
which implied enforcement of a state level lockdown.

The BS analysis also detected a noticeable decrease in the tweeting activity
in those days, but much less intense than the opposite behaviour.

As a final note, the SSR and BS analyses were discordant near the boundary of
the considered time window. However, this fact can be easily explained
considering that these two models are biased in opposite ways in these
regions. The analysis had hence to be regarded as inconclusive for what
concerns both the end of 2019 and May 2020.

Figure 4: Ratio of the total daily negative activity jump to the total daily
positive activity jump calculated by performing stepwise segmented regression
(SSR, upper panel) and Bayesian switchpoint analysis (BS, lower panel) for the
whole set of users.

We repeated the analyses for the four sets of profiles separately (Fig. 5). In
this case, we could observe that the politicians were the only group that did
not display any change in their activity in the week of the lockdown
declaration. Contrarily, both analyses indicate a positive jump in the
activity near January 7, the day on which the two parties forming the
coalition government found an agreement, as well as a pronounced negative jump
near February 12, day in which the composition of the government was announced
by the prime minister. Finally, journalists also showed an interesting
behaviour. Relative positive jumps concentrate in the aforementioned week, but
they also presented a secondary peak on March 2, the day before the first
Covid-19 death was officially confirmed in Spain.

Figure 5: Ratio of the total daily negative activity jump to the total daily
positive activity jump calculated by performing stepwise segmented regression
(SSR, upper panels) and Bayesian switchpoint analysis (BS, lower panels) for
each individual class of users.

Finally, we compared the distribution of $h$ and $\rho$ for the four classes
of users before and after March 9 and starting on the first of January (Fig.
6). Contrary to what we observed for the activity, the only type of users that
change their behaviour are politicians. In particular, we observed a steep
increase in their propensity to amplify information produced by others instead
of publishing their own content.

Figure 6:  $h$ (left panel) and $\rho$ (right panel) statistics before (blue)
and after (red) March 9 of, from left to right, journalists, politicians,
random followers, and random friends. Whiskers are set at the 5th and 95th
percentiles. Dots represent outliers. Notches represent 95% CI of the medians.
Median values obtained by bootstraping data 1000 times.

##  4 Conclusions

In this work, we have shown how a statistical analysis of individual users’
activity on online social media can highlight the relation between crisis
events and abrupt changes in behaviours. In particular, our results indicate
that the debate about the government of the crisis triggers collective
participation more than events itself. This is in line with findings in [5],
in which the authors demonstrated a robust large increase in the use of words
related to control measures only since the first week of March. Generic Covid-
related words, instead, started to be increasingly used since January.

The analysis of the individual activity of users in the Spanish Twittersphere
shows that an abrupt change occurs for a large part of them in coincidence
with the days on which the control and mitigation measures were announced in
Spain, at regional and State level. In this sense, we can claim that it is not
the epidemic outbreak that drives the change in the users’ activity. Indeed,
the main driver is rather the political reactions to the crisis. Interestingly
enough, we do not observe any major activity change at the beginning of the
lockdown period, suggesting that there is no further effect on the online
activity of the change in the physical social behaviour. The only observable
consequence of the enforcement of the “Stay at home” recommendations is
precisely the absence of a second change, which under more ordinary
circumstances would have reestablished the overall activity at the previous
level. Differently from the case of a usual highly participated political
debate, the discussion around the introduction of the state of alarm and the
related measures, determined a persistent increase in the online activity.

It is worth mentioning that the phase in which we observe major changes in
users activities roughly matches the transition between two different phases
of the associated social epidemics [15] as shown in [16]. In particular,
extrapolating results from [16], jumps in activities should be located between
the suspended reality phase, in which people express anger about the looming
feeling that things were about to change, and the acceptance phase, started
after the authorities imposed physical-distancing measures, in which people
adjust to a "new normality". This observation calls for a deeper understanding
of the relation between online activity and psychological states during a
crisis.

Additionally, we have been able to collect evidence that points to the fact
that politicians’ activity has a dynamics that is mostly endogenous, while
journalists anticipate trends. Politicians did not follow the general pattern
described above. On the contrary, their activity responds rather to the
institutional agenda itself. With respect to the type of activity – production
or amplification of contents –, only politicians change their attitude, while
for other classes, the increased activity is not accompanied by a change in
the type of activity. Summarizing, policymakers tended to suppress their
individual voices, likely in the attempt to provide a more coherent view on
the topic, while reducing their involvement in other debates. On the contrary,
ordinary citizens probably just added a new persistent theme to their usual
activity, without changing quantitatively their content consumption habits.
Among them, journalists took an expected leadership role.

##  Appendix A List of news media twitter handlers

el_pais, JotDownSpain, eldiarioes, elespanolcom, revistamongolia, la_ser,
_infoLibre, EFEnoticias, elmundoes, elconfidencial, indpcom, ctxt_es,
publico_es, ondacero_es, cuatro, LaVanguardia, europapress, laSextaTV, rtve

##  Appendix B Overall activity

In fig. 7, we represent the mean number of tweets per day for each week. We
observe that the overall activity increases in the week after the one in which
we observe the majority of individual changes.

Figure 7: Activity per week for all our users.

##  Appendix C Distributions of the number of breakpoints in the Stepwise
Segmented Regression Analysis

In fig. 8, we show the fraction of the number of optimal breakpoints for each
user type. We can see that the distribution is quite similar among different
user types. We can observe that the maximum of the distribution is always
centered at 2 breakpoints, and that few users reach the 5 breakpoints. This
serves as an a posteriori validation of the maximum number of breakpoints we
set in our model.

Figure 8: Number of breakpoints for each user type.

##  Appendix D Bayesian Switchpoint Analysis in presence of multiple switches.

Users timeline may show more than one switch, while our model assumes there is
just one. We show some examples of the results we get when synthetic timelines
are generated with multiple switches.

Figure 9: Syntetic data with $\lambda$ changing from 1 to 5 at the 2019-01-12
and from 5 to 2 at the 2020-03-02. For the model we obtain a p-value=0.46
Figure 10: Syntetic data with $\lambda$ changing from 1 to 5 at the 2019-10-12
and from 5 to 2 at the 2020-03-02. For the model we obtain a p-value=0.50

In both examples we can see how in case of multiple switches the model will
obviously get only one of them. We can conclude that we can use the p-value as
a way of knowing if there are some existing breakpoints in the data that we
missed. When the p-value is low, it does not mean that the model has to be
rejected, but rather that it is choosing one of the breakpoints present in the
data, while other breakpoints probably exist.

##  Appendix E Bayesian Switchpoint Analysis with non-Poissonian models.

For comparison, we also analysed other models for a given example.

###  E.1 Sigmoidal

We analyze here the sigmoidal model. Instead of a sudden switch, we now have a
smooth transition between the two $\lambda$s. While in this way we can reduce
the uncertainty in $\tau$, in general the two models give very similar
results.

Figure 11: Comparison between Poissonian and Sigmoidal model

###  E.2 Multistate model

As in the case of the breakpoints model in SSR, we can allow for multiple
states or levels represented by more than two $\lambda$s. The maximum number
of levels being ($L_{\text{MAX}}$).

We will have several realization of the model with different
($L_{\text{MAX}}$) and get the optimal model minimizing the Bayesian
Information Criterion.

Figure 12: Multistate model for different states

This model has the advantage of fitting the data more accurately, but the
computation time and the simplicity of the result are worse than that of the
linear breakpoints model. For example, we can see in fig. 12 that the 4-state
model has a small spike at a given day, this is useful to detect anomalies in
the activity, but when we look for changes that persist through time, the
linear breakpoints model introduces less noise.

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# Locality of vortex stretching for the 3D Euler equations

Yuuki Shimizu  Graduate School of Mathematical Sciences, University of Tokyo,
Komaba 3-8-1 Meguro, Tokyo 153-8914, Japan<EMAIL_ADDRESS>and
Tsuyoshi Yoneda  Graduate School of Economics, Hitotsubashi University, 2-1
Naka, Kunitachi, Tokyo 186-8601, Japan<EMAIL_ADDRESS>
###### Abstract.

We consider the 3D incompressible Euler equations under the following
situation: small-scale vortex blob being stretched by a prescribed large-scale
stationary flow. More precisely, we clarify what kind of large-scale
stationary flows really stretch small-scale vortex blobs in alignment with the
straining direction. The key idea is constructing a Lagrangian coordinate so
that the Lie bracket is identically zero (c.f. the Frobenius theorem), and
investigate the locality of the pressure term by using it.

###### Key words and phrases:

Euler equations, vortex stretching, Frenet-Serret formulas, Frobenius theorem

###### 2010 Mathematics Subject Classification:

Primary 35Q31; Secondary 76F99

##  1\. Introduction

The most important features of the Navier-Stokes turbulence is that turbulence
is not random but composed of vortex stretching behavior. More precisely,
recent DNS [2, 3, 12, 13] of the Navier-Stokes turbulence at sufficiently high
Reynolds numbers have reported that there exists a hierarchy of vortex
stretching motions. In particular, Goto-Saito-Kawahara [3] clearly observed
that turbulence at sufficiently high Reynolds numbers in a periodic cube is
composed of a self-similar hierarchy of antiparallel pairs of vortex tubes,
and it is sustained by creation of smaller-scale vortices due to stretching in
larger-scale strain fields. This observation is further investigated by
Y-Goto-Tsuruhashi [15] (see also [14]). Thus we could conclude physically that
local-scale energy transfer is mainly induced by vortex stretching (see also
[5, 6, 7] for the related mathematical results). Therefore as the sequence of
these studies, our next study will be clarifying the vortex stretching
dynamics precisely.

In this paper we mathematically consider the locality of small-scale vortex
dynamics in the 3D incompressible Euler equations. More precisely, we consider
the inviscid flow under the following situation: small-scale vortex blob being
stretched by a prescribed large-scale stationary flow, and we clarify what
kind of large-scale stationary flows really stretch smaller-scale vortex blobs
in alignment with the straining direction. Now let us describe the
incompressible Euler equations (inviscid flow) as follows:

(1)
$\displaystyle\partial_{t}u+(u\cdot\nabla)u=\left(\partial_{t}(u\circ\Phi)\right)\circ\Phi^{-1}=\partial_{t}^{2}\Phi\circ\Phi^{-1}=-\nabla
p,\quad\nabla\cdot u=0\quad\text{in}\quad\mathbb{R}^{3},$ $\displaystyle\quad
u|_{t=0}=u_{0},$

where $\Phi$ is the associated Lagrangian flow given by

$\begin{split}\partial_{t}\Phi(t,x)=u(t,\Phi(t,x))=:u\circ\Phi\quad\text{with}\quad\Phi(0,x)=x\in\mathbb{R}^{3}.\end{split}$

Let $u^{S}:(-\epsilon,\epsilon)\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ be the
flow of a small-scale vortex blob and $u^{L}:\mathbb{R}^{3}\to\mathbb{R}^{3}$
be a prescribed large-scale stationary flow. Then the associated Lagrangian
flows $\eta^{L}$ and $\eta^{S}$ satisfying
$\Phi(t,x)=\eta^{L}(t,\eta^{S}(t,x))=\eta^{L}\circ\eta^{S}$ are given by

$\begin{split}&\partial_{t}\eta^{L}(t,x)=u^{L}(\eta^{L}(t,x))=:u^{L}\circ\eta^{L}\quad\text{with}\quad\eta^{L}(0,x)=x\in\mathbb{R}^{3},\\\
&\partial_{t}\eta^{S}(t,x)=u^{S}(t,\eta^{S}(t,x))=:u^{S}\circ\eta^{S}\quad\text{with}\quad\eta^{S}(0,x)=x\in\mathbb{R}^{3}.\end{split}$

Let us assume $\eta^{S}(t,x_{0})=\eta^{S}(0,x_{0})=x_{0}\in\ell$, where

(2) $\ell:=\bigcup_{t\in(-\epsilon,\epsilon)}\eta^{L}(t,x_{*})\quad\text{for
some}\quad x_{*}\in\mathbb{R}^{3}.$

$\ell$ represents the rotating axis of the small-scale vortex blob which
aligns with the large-scale straining flow. Based on DNS of homogeneous
isotropic turbulence, Hamlington-Schumacher-Dahm [4] showed vorticity tends to
align with the stretching direction of the background strain. Note that, in
their study, the background strain means the strain rate induced by the
vorticity beyond radius $\sim 12\eta$ ($\eta$ is the Kolmogorov scale). See
also [3]. With the aid of this physical observation, we constructed the
mathematical model (2).

In general, $\eta^{S}$ is strongly affected by the large-scale straining flow
$\eta^{L}$ when $\Phi=\eta^{L}\circ\eta^{S}$ comes from an Euler flow, thus,
in this study, we need to clarify the nonlinear interaction even partially.

######  Remark 1.

For readers’ convenience, in this remark, we state a typical vortex stretching
as an example. Let $x_{0}\in\\{x_{1}=x_{2}=0\\}=\ell$, and let
$(r,\theta,x_{3})$ be the cylindrical coordinate such that
$(x_{1},x_{2})=(r\cos\theta,r\sin\theta)$ with

$e_{r}:=\frac{(x_{1},x_{2},0)}{\sqrt{x_{1}^{2}+x_{2}^{2}}},\quad
e_{\theta}:=\frac{(-x_{2},x_{1},0)}{\sqrt{x_{1}^{2}+x_{2}^{2}}}\quad\text{and}\quad
e_{x_{3}}:=(0,0,1).$

First we give the typical straining flow
$(u_{r}^{L},u^{L}_{\theta},u_{x_{3}}^{L})$ as follows:

$u^{L}_{r}:=u^{L}\cdot e_{r}=-r,\quad u^{L}_{x_{3}}:=u^{L}\cdot
e_{x_{3}}=2x_{3}\quad\text{and}\quad u^{L}_{\theta}:=u^{L}\cdot e_{\theta}=0.$

In this case the corresponding Lagrangian flow is

$(\eta^{L}_{r},\eta^{L}_{\theta},\eta^{L}_{x_{3}})=(e^{-t}r_{0},\theta_{0},e^{2t}x_{3,0}),$

where $(r_{0},\theta_{0},x_{3,0})$ represents the initial position. For
$\omega:=\nabla\times u$, let $\omega^{S}:=\nabla\times u^{S}$ be axisymmetric
vorticity depending only on $r$ variable, such that

$\omega^{S}_{\theta}(t,r):=\omega^{S}\cdot
e_{\theta},\quad\omega^{S}_{x_{3}}:=\omega^{S}\cdot
e_{x_{3}}=0\quad\text{and}\quad\omega^{S}_{r}:=\omega^{S}\cdot e_{r}=0.$

Then we have the following explicit solution (vorticity) to the Euler
equations:

(3)
$\omega_{\theta}^{S}(t,\eta^{L}_{r}(t))=e^{t}\omega^{S}_{0,\theta}(e^{-t}r_{0})\quad\text{and}\quad\omega^{S}_{0,\theta}=\omega^{S}_{\theta}(t)|_{t=0}.$

With the aid of the typical stretching motion in Remark 1, we rigorously
define the meaning of “stably stretching” as follows.

######  Definition 1.

(Stably stretching.) $\eta^{L}$ is said to be stably stretching along the
rotating axis $\ell\subset\mathbb{R}^{3}$, if the following two conditions
hold.

  * •

The rotation axis $\ell$ is stretched along $\partial_{t}\eta^{L}$ direction,
that is

(4) $\partial_{t}|\partial_{t}\eta^{L}(t,x_{0})|>0\quad\text{for each}\quad
x_{0}\in\ell.$

  * •

Let

$x_{0}^{\perp}:=\\{x\in\mathbb{R}^{3}:(x_{0}-x)\cdot\partial_{t}\eta^{L}(0,x_{0})=0,\
|x_{0}-x|<\delta\\}\quad\text{for each}\quad x_{0}\in\ell.$

The time evolution of the surface $x_{0}^{\perp}$ (accompanied by the fluid
particles) is always perpendicular to the stretching direction at
$\eta^{L}(t,x_{0})$ ($t>0$), that is,

(5) $\partial_{t}\eta^{L}(t,x_{0})\perp\Phi(t,x_{0}^{\perp})\quad\text{for
each}\quad x_{0}\in\ell.$

######  Remark 2.

(3) satisfies (4) and (5).

To state our theorem, we need to prepare “curvature”. First let us choose a
point $x_{0}\in\ell$ (reference point) and fix it. Identifying $\frac{d}{dz}$
with $\partial_{z}$, we define $t(z)$ as

(6)
$\partial_{z}t>0,\quad|\partial_{z}\eta^{L}|:=|\partial_{z}\eta^{L}(t(z),x_{0})|=1\quad\text{and}\quad
t(0)=0.$

In this case we immediately have $\partial_{z}t=|\partial_{t}\eta^{L}|^{-1}$
and its inverse is $\partial_{t}z=|\partial_{t}\eta^{L}|$ (with the variables
omitted). Then define the unit tangent vector $\tau$ as

$\tau(z)=\partial_{z}\eta^{L}(t(z),x_{0}),$

the unit curvature vector $n$ as $\kappa n=\partial_{z}\tau$ with a curvature
function $\kappa(z)>0$, the unit torsion vector $b$ as : $b(z):=\tau(z)\times
n(z)$, where $\times$ is the exterior product. Without loss of generality, we
can assume the torsion function $T(z)$ is positive by choosing the orientation
of the torsion vector $b(z)$. Then we now state our main theorem.

######  Theorem 1.

Let
$(\partial_{t}\Phi)\circ\Phi^{-1}=\partial_{t}(\eta^{L}\circ\eta^{S})\circ(\eta^{L}\circ\eta^{S})^{-1}$
be a solution to the Euler equations. If $\eta^{L}$ is stably stretching along
$\ell\subset\mathbb{R}^{3}$, then we have

(7) $\partial_{z}(\kappa|\partial_{t}\eta^{L}|^{2})=0\quad\text{for each}\quad
x_{0}\in\ell,$

in particular, we do not need any information of $\eta^{S}$,

######  Remark 3.

Since $\partial_{t}|\partial_{t}\eta^{L}|>0$, we obtain a necessary condition
$\partial_{z}\kappa(z)\leq 0$.

Finally, we examine a typical straining flow $(-x_{1},x_{2},0)$ whether or not
it satisfies the formula (7). Note that we could also examine the straining
flow $(-x_{1},-x_{2},2x_{3})$ which is already appeared in Remark 1, but in
this case the formula becomes much more complicated, thus we leave it as a
reader’s exercise.

######  Corollary 2.

If $u^{L}$ is a straining flow such that $u^{L}=(-x_{1},x_{2},0)$, then we
have

$\partial_{z}(\kappa|\partial_{t}\eta^{L}|^{2})=-\frac{\tanh 2t}{\cosh
2t}\quad\text{with}\quad t=\frac{1}{2}\log\left(\frac{x_{1}}{x_{2}}\right).$

This means that, if $\ell\not\subset\\{x_{1}=0\\}$ and is in the stretching
region: $\ell\subset\\{x:|x_{2}|>|x_{1}|\\}$, then the pair of $\eta^{L}$ and
$\Phi=\eta^{L}\circ\eta^{S}$ does not satisfy (5), which implies that
$\eta^{L}$ is “unstably” stretching along $\ell$.

##  2\. Proof of Theorem 1

First we rephrase the initial flat plane $x_{0}^{\perp}$ such that

(8)
$x_{0}^{\perp}=\\{x_{0}+r_{1}n(0)+r_{2}b(0):\sqrt{r_{1}^{2}+r_{2}^{2}}<\delta\\}.$

For any initial particle on the plane $x=x_{0}+r_{1}n(0)+r_{2}b(0)\in
x_{0}^{\perp}$, $\Phi(t,x)$ is uniquely expressed as (we omit the change of
variables)

(9) $\begin{split}\Phi(t,x)&=:\Phi(z,r_{1},r_{2})\\\
&=\eta^{L}(t(z),x_{0})+Z(z,r_{1},r_{2})\tau(z)+R_{1}(z,r_{1},r_{2})n(z)+R_{2}(z,r_{1},r_{2})b(z),\end{split}$

for sufficiently small $r_{1}$ and $r_{2}$, with $Z(z,0,0)=0$,
$Z(0,r_{1},r_{2})=0$, $R_{1}(0,r_{1},r_{2})=r_{1}$ and
$R_{2}(0,r_{1},r_{2})=r_{2}$.

######  Remark 4.

We can rephrase (5) as follows:

(10)
$\begin{split}\partial_{r_{1}}Z|_{r_{1}=r_{2}=0}=\partial_{r_{2}}Z|_{r_{1}=r_{2}=0}=0\quad\text{for}\quad
z>0.\end{split}$

This is due to the fact that

$\Phi(t,x_{0}^{\perp})=\bigcup_{\sqrt{r_{1}^{2}+r_{2}^{2}}<\delta}\Phi(z(t),r_{1},r_{2})$

by (8) and (9), and

$\partial_{t}\eta^{L}(t,x_{0})\cdot\partial_{r_{1}}\Phi(z(t),r_{1},r_{2})|_{r_{1}=r_{2}}=\partial_{t}\eta^{L}(t,x_{0})\cdot\partial_{r_{2}}\Phi(z(t),r_{1},r_{2})|_{r_{1}=r_{2}}=0.$

Since the corresponding Jacobian
$\frac{\partial(R_{1},R_{2})}{\partial(r_{1},r_{2})}$ is clearly nonzero for
sufficiently small $z>0$, so, by the inverse function theorem (for each
$z>0$), we can rewrite the equality (9) as follows: Let $R_{1}$ and $R_{2}$ be
variables, and $r_{1}$ and $r_{2}$ be the corresponding inverse functions. For
any particle $x=x_{0}+r_{1}(z,R_{1},R_{2})n(0)+r_{2}(z,R_{1},R_{2})b(0)\in
x_{0}^{\perp}$,

(11) $\begin{split}\Phi(t,x)&=:\Phi(z,R_{1},R_{2})\\\
&=\eta^{L}(t(z),x_{0})+Z(z,r_{1}(z,R_{1},R_{2}),r_{2}(z,R_{1},R_{2}))\tau(z)+R_{1}n(z)+R_{2}b(z)\end{split}$

for sufficiently small $|R_{1}|,|R_{2}|,z>0$. Let us recall the Frenet-Serret
formulas:

$\frac{d}{dz}\begin{pmatrix}\tau\\\ n\\\
b\end{pmatrix}=\begin{pmatrix}0&\kappa&0\\\ -\kappa&0&T\\\
0&-T&0\end{pmatrix}\begin{pmatrix}\tau\\\ n\\\ b\end{pmatrix}.$

By combining the Frenet-Serret formulas, (10) and (11), we have

(12)
$\begin{cases}\partial_{z}\Phi&=\tau+R_{1}(Tb-\kappa\tau)-R_{2}Tn+\mathcal{O}(R_{1}^{2},R_{2}^{2}),\\\
\partial_{R_{1}}\Phi&=n+\mathcal{O}(R_{1},R_{2}),\\\
\partial_{R_{2}}\Phi&=b+\mathcal{O}(R_{1},R_{2}),\end{cases}$

where $\mathcal{O}$ is Landau’s notation. A direct calculation yields

$\begin{split}\partial_{z}^{2}\Phi=&\kappa
n-R_{1}(T^{2}+\kappa^{2})n+R_{1}((\partial_{z}T)b-(\partial_{z}\kappa)\tau)\\\
&\
-R_{2}T(-\kappa\tau+Tb)-R_{2}(\partial_{z}T)n+\mathcal{O}(R_{1}^{2},R_{2}^{2})\end{split}$

and then

$\begin{split}&\partial_{R_{1}}\partial_{z}\Phi|_{R_{1}=R_{2}=0}=Tb-\kappa\tau,\\\
&\partial_{R_{1}}\partial_{z}^{2}\Phi|_{R_{1}=R_{2}=0}=-(T^{2}+\kappa^{2})n+((\partial_{z}T)b-(\partial_{z}\kappa)\tau),\\\
&\partial_{R_{2}}\partial_{z}\Phi|_{R_{1}=R_{2}=0}=-Tn,\\\
&\partial_{R_{2}}\partial_{z}^{2}\Phi|_{R_{1}=R_{2}=0}=-T(-\kappa\tau+Tb)-(\partial_{z}T)n.\end{split}$

Thus

$(\partial_{R_{1}}\partial_{z}^{2}\Phi)\cdot\tau|_{R_{1},R_{2}=0}=-\partial_{z}\kappa\quad\text{and}\quad(\partial_{R_{1}}\partial_{z}\Phi)\cdot\tau|_{R_{1},R_{2}=0}=-\kappa.$

On the other hand, by the Leibniz rule, we see

$\partial_{t}^{2}\Phi=\partial_{z}^{2}\Phi(\partial_{t}z)^{2}+\partial_{z}\Phi\partial_{t}^{2}z\quad\text{with
the variables omitted.}$

Combining the facts $\partial_{t}^{2}z=\partial_{t}|\partial_{t}\eta^{L}|$ and
$\partial_{t}z=|\partial_{t}\eta^{L}|$, we have

(14) $\displaystyle-\partial_{R_{1}}(\nabla
p\cdot\tau)=\partial_{R_{1}}(\partial_{t}^{2}\Phi\cdot\tau)=(\partial_{R_{1}}\partial_{t}^{2}\Phi)\cdot\tau=-\kappa\partial_{t}|\partial_{t}\eta^{L}|-\partial_{z}\kappa|\partial_{t}\eta^{L}|^{2},$
$\displaystyle-\partial_{R_{2}}(\nabla
p\cdot\tau)=\partial_{R_{2}}(\partial_{t}^{2}\Phi\cdot\tau)=(\partial_{R_{2}}\partial_{t}^{2}\Phi)\cdot\tau=+T\kappa|\partial_{t}\eta^{L}|^{2}$

for $R_{1}=R_{2}=0$. Next we derive other formulae by using the Euler
equations. By the Leibniz rule, we see

$\kappa
n=\partial_{z}^{2}\eta^{L}(t(z),x_{0})=\partial_{z}(\partial_{t}\eta^{L}\partial_{z}t)=\partial_{t}^{2}\eta^{L}(\partial_{z}t)^{2}+\partial_{t}\eta^{L}\partial_{z}^{2}t.$

Combining
$\partial_{z}^{2}t=\partial_{z}|\partial_{t}\eta^{L}|^{-1}=-|\partial_{t}\eta^{L}|^{-2}\partial_{z}|\partial_{t}\eta^{L}|=-|\partial_{t}\eta^{L}|^{-3}\partial_{t}|\partial_{t}\eta^{L}|$,
we have

$\partial_{t}^{2}\eta^{L}=|\partial_{t}\eta^{L}|^{2}\kappa
n+\partial_{t}|\partial_{t}\eta^{L}|\tau.$

By using the Euler equations, we have

$\displaystyle-\nabla p\cdot\tau$ $\displaystyle=$
$\displaystyle\partial_{t}^{2}\Phi\cdot\tau=\partial_{t}|\partial_{t}\eta^{L}|,$
$\displaystyle-\nabla p\cdot n$ $\displaystyle=$
$\displaystyle\partial_{t}^{2}\Phi\cdot n=\kappa|\partial_{t}\eta^{L}|^{2},$
$\displaystyle-\partial_{z}(\nabla p\cdot n)$ $\displaystyle=$
$\displaystyle\partial_{z}\kappa|\partial_{t}\eta^{L}|^{2}+2\kappa\partial_{t}|\partial_{t}\eta^{L}|,$
$\displaystyle-\nabla p\cdot b$ $\displaystyle=$ $\displaystyle 0$

for $R_{1}=R_{2}=0$ with the change of variables $\circ\eta^{L}$ omitted
again. On the other hand, from (12), we have the following inverse matrix:

$\begin{pmatrix}\tau\\\ n\\\ b\\\ \end{pmatrix}=\begin{pmatrix}(1-\kappa
R_{1})^{-1}&R_{2}T(1-\kappa R_{1})^{-1}&-R_{1}T(1-\kappa R_{1})^{-1}\\\
0&1&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}\partial_{z}\Phi\\\
\partial_{R_{1}}\Phi\\\ \partial_{R_{2}}\Phi\end{pmatrix}$

with the higher order terms omitted since we finally take $R_{1},R_{2}\to 0$.
Then we see

$\begin{split}\nabla p\cdot\tau=&(1-\kappa R_{1})^{-1}(\nabla
p\cdot\partial_{z}\Phi)\\\ &+R_{2}T(1-\kappa R_{1})^{-1}(\nabla
p\cdot\partial_{R_{1}}\Phi)-R_{1}T(1-\kappa R_{1})^{-1}(\nabla
p\cdot\partial_{R_{2}}\Phi)\\\ =&(1-\kappa
R_{1})^{-1}\partial_{z}(p\circ\Phi)\\\ &+R_{2}T(1-\kappa
R_{1})^{-1}\partial_{R_{1}}(p\circ\Phi)-R_{1}T(1-\kappa
R_{1})^{-1}\partial_{R_{2}}(p\circ\Phi).\end{split}$

and then (omit the variable $\circ\Phi$)

$\displaystyle-\partial_{R_{1}}(\nabla p\cdot\tau)|_{R_{1}=R_{2}=0}$
$\displaystyle=$
$\displaystyle\left(-\kappa\partial_{z}p-\partial_{R_{1}}\partial_{z}p-T\partial_{R_{2}}p\right)|_{R_{1}=R_{2}=0}$
$\displaystyle(\text{commute}\ \partial_{R_{1}}\ \text{and}\ \partial_{z})$
$\displaystyle=$ $\displaystyle\left(-\kappa(\nabla
p\cdot\tau)-\partial_{z}(\nabla p\cdot n)-T(\nabla p\cdot
b)\right)|_{R_{1}=R_{2}=0}$ $\displaystyle=$ $\displaystyle
3\kappa\partial_{t}|\partial_{t}\eta^{L}|+\partial_{z}\kappa|\partial_{t}\eta^{L}|^{2}.$

Combining (14), we have the desired formula.

######  Remark 5.

We can rephrase the commutativity of $\partial_{R_{1}}$ and $\partial_{z}$ as

$[\partial_{z},\partial_{R_{1}}]=\partial_{z}\partial_{R_{1}}-\partial_{R_{1}}\partial_{z}=0,$

where $[\cdot,\cdot]$ is the Lie braket (c.f. the Frobenius theorem, see
Chapter 19 in [8] for example). For the previous studies using this property,
see Chan-Czubak-Y [1, Section 2.5] and Lichtenfelz-Y [9], more originally, see
Ma-Wang [10, (3.7)].

######  Remark 6.

Since $\nabla p\cdot b=\partial_{R_{2}}p\equiv 0$, then

$\begin{split}-\partial_{R_{2}}(\nabla
p\cdot\tau)&=-\partial_{R_{2}}\partial_{z}p-T\partial_{R_{1}}p\\\
(\text{commute}\ \partial_{R_{2}}\ \text{and}\ \partial_{z})&=-T(\nabla p\cdot
n)=T\kappa|\partial_{t}\eta^{L}|^{2}\end{split}$

for $R_{1}=R_{2}=0$. However this formula is useless, since it coincides with
(14).

##  3\. Proof of Corollary 2

For any $x\in\\{x:|x_{2}|>|x_{1}|\\}\cap\\{x_{1}\not=0\\}$, let us set

$\eta^{L}(t,x)=\begin{pmatrix}re^{t+t_{0}}\\\ re^{-(t+t_{0})}\\\
x_{3}\end{pmatrix},$

where $r:=\sqrt{x_{1}x_{2}}$ and $t_{0}=\frac{1}{2}\log(x_{1}/x_{2})$. In this
case we see

$\partial_{z}t:=|\partial_{t}\eta^{L}|^{-1}=\frac{1}{r(e^{2t}+e^{-2t})^{1/2}}=\frac{1}{r\sqrt{2}(\cosh
2t)^{1/2}}$

and

$\partial_{z}^{2}t=-\frac{(\sinh 2t)\partial_{z}t}{r\sqrt{2}(\cosh
2t)^{3/2}}=-\frac{\sinh 2t}{2r^{2}(\cosh 2t)^{2}}=-\frac{\tanh 2t}{2r^{2}\cosh
2t}.$

On the other hand,

$\begin{split}\kappa
n&=\partial_{t}^{2}\eta^{L}(\partial_{z}t)^{2}+\partial_{t}\eta^{L}\partial_{z}^{2}t\\\
&=\frac{1}{2r^{2}\cosh 2t}\begin{pmatrix}re^{t}\\\ re^{-t}\\\
0\end{pmatrix}-\frac{\tanh 2t}{2r^{2}\cosh 2t}\begin{pmatrix}re^{t}\\\
-re^{-t}\\\ 0\end{pmatrix}.\end{split}$

Thus

$\begin{split}\kappa^{2}&=\frac{1}{2r^{2}\cosh 2t}-\frac{(\tanh
2t)^{2}}{r^{2}\cosh 2t}+\frac{(\tanh 2t)^{2}}{2r^{2}\cosh
2t}=\frac{1}{2r^{2}(\cosh 2t)^{3}}\end{split}$

and then

$\begin{split}\kappa&=\frac{1}{\sqrt{2}r(\cosh
2t)^{3/2}},\quad\partial_{z}\kappa=-\frac{3\tanh 2t}{2r^{2}(\cosh
2t)^{2}},\end{split}$ $\begin{split}|\partial_{t}\eta^{L}|^{2}=2r^{2}\cosh
2t\quad\text{and}\quad\partial_{t}|\partial_{t}\eta^{L}|=\sqrt{2}r(\tanh
2t)^{1/2}(\sinh 2t)^{1/2}.\end{split}$

Combining the above calculations, we have the following desired formula:

$\begin{split}\partial_{z}(\kappa|\partial_{t}\eta^{L}|^{2})=2\kappa\partial_{t}|\partial_{t}\eta^{L}|+\partial_{z}\kappa|\partial_{t}\eta^{L}|^{2}=-\frac{\tanh
2t}{\cosh 2t}.\end{split}$

Acknowledgments. Research of YS was partially supported by Grant-in-Aid for
JSPS Fellows, Japan Society for the Promotion of Science (JSPS). Research of
TY was partly supported by the JSPS Grants-in-Aid for Scientific Research
20H01819.

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