# FRBs from rapid spindown neutron stars

Dongzi Li1,2 and Ue-Li Pen3,4,5,6,7
1TAPIR, California Institute of Technology, Pasadena, CA 91125, USA
2 Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544, USA
3Institute of Astronomy and Astrophysics, Academia Sinica, Astronomy-
Mathematics Building, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan
4Canadian Institute for Theoretical Astrophysics, University of Toronto, 60
St. George Street, Toronto, ON M5S 3H8, Canada
5Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo,
ON, Canada N2L 2Y5
6Canadian Institute for Advanced Research, CIFAR program in Gravitation and
Cosmology
7Dunlap Institute for Astronomy & Astrophysics, University of Toronto, AB
120-50 St. George Street, Toronto, ON M5S 3H4, Canada E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

A fast radio burst (FRB) localized to a globular cluster (GC) challenges FRB
models involving ordinary young magnetars. In this paper, we examine the rapid
spindown millisecond neutron star (NS) scenario, which favours the dynamic
environment in GCs. Fast spindown corresponds to a larger magnetic field than
regular millisecond pulsars, which empirically favours giant pulse (GP)
emission. The kinetic energy in millisecond NSs can readily exceed the
magnetic energy in magnetars. The high inferred isotropic luminosity of most
FRBs is challenging to explain in spin-down powered pulsars. A recent
observation of a GP from the Crab pulsar, on the other hand, suggests highly
Doppler-beamed emission, making the required energy orders of magnitude
smaller than estimated with isotropic assumptions. Considering this strong
beaming effect, GPs from a recycled pulsar with a modest magnetic field could
explain the energetics and burst rates for a wide range of FRBs. The short
life span accounts for a paucity of bright FRBs in the Milky Way
neighbourhood. We point out that tidal disruption spin-up from a main sequence
star can provide sufficient accretion rate to recycle a NS with mild magnetic
field. It can also explain the observed source density and the spatial offset
in the GC for FRB 20200120E. Frequency variation in the scattering tail for
some of the brightest FRBs is expected in this scenario.

###### keywords:

globular clusters: general – pulsars:general – transients: fast radio bursts

††pubyear: 2023††pagerange: FRBs from rapid spindown neutron stars–FRBs from
rapid spindown neutron stars

##  1 Introduction

Fast radio bursts (FRBs) are enigmatic radio bursts with durations ranging
from microseconds ($\mu$s) to milliseconds (ms). Over the past decade,
substantial advances have been made in constraining the nature of FRBs. The
identification of repeating FRBs (e.g., Spitler et al., 2014, 2016; CHIME/FRB
Collaboration et al., 2019; Fonseca et al., 2020) suggests that at least a
subset of the FRB population originates from non-cataclysmic events. The
detection of exceptionally intense radio bursts from a galactic magnetar
(CHIME/FRB Collaboration et al., 2020b; Bochenek et al., 2020) has positioned
magnetars as a popular candidate progenitor for FRBs. However, it is unclear
whether all FRBs are produced by ordinary magnetars. Evidence is building for
a diverse population of FRBs. While magnetars are young and closely related to
the star-formation in the past $<10^{5}$ year, here is a growing variety of
FRB host galaxies including in elliptical galaxies with little star formation.
Moreover, the star-formation history among host galaxies exhibits a broad
delay-time distribution, covering a range from approximately 100 million years
(Myr) to around 10 billion years (Gyr)(Law et al., 2023), which are both much
longer than the magnetar life time. Two of the most active repeaters, FRB
20180916A and FRB 20121102B exhibit long-term periodicity (CHIME/FRB
Collaboration et al., 2020a; Rajwade et al., 2020), while no known magnetars
exhibit similar properties. The definite evidence supporting an alternative
formation channel is provided by the localization of FRB 20200120E (Majid et
al., 2021; Bhardwaj et al., 2021) to an old ($t\gtrsim 10\,$Gyr) globular
cluster (GC) in M81 (Kirsten et al., 2022). These old GCs have not experienced
massive star formation for billions of years, effectively ruling out the
possibility of a young magnetar born through a massive stellar collapse
Alternatively, Kirsten et al. (2022); Kremer et al. (2021a); Lu et al. (2022)
proposed that magnetars formed in accretion-induced collapse (e.g., Tauris et
al., 2013) or via massive white dwarf binary merger may provide a formation
mechanism for this repeater. Nevertheless, in simulations, there is an ongoing
debate regarding whether the mass can be retained to form magnetars or if it
will be ejected during the collision process.

In this paper we examine the scenario of giant pulses from rapid spindown
neutron stars as a mechanism for FRBs. In section 2 We show that it could
satisfy the key observation constraints: rate, spectral luminosity, and total
emitted energy. In section 3 we consider a generation mechanism due to tidal
disruption spinup. In section 4, we discuss the rate and spatial distribution
regarding the scenario. We summarize and discuss the proposed observation
tests in section 5.

Figure 1: P-$\dot{P}$ diagram of rapid spindown scenario. The thick orange
dashed line is the spin-up limit, which is similar to the abundance limit,
while the thick green solid line represents the observed mean energy at 1%
radiation efficiency. Dots represent known pulsars(Manchester et al., 2016).

##  2 Energetics

With the notable exception of magnetars, most pulsars are powered by their
spindown energy. The rapid rotation of the neutron star leads to magnetic
dipole radiation:

$P_{\mathrm{rad}}=\frac{2}{3c^{3}}(\mu\sin\alpha)^{2}\big{(}\frac{2\pi}{P}\big{)}^{4}\sim
10^{39}\mathrm{erg}/s\,B_{10}^{2}P_{-3}^{-4}$ (1)

where $B_{10}$ is the surface magnetic field in units of $10^{10}$ G, $\mu$ is
the magnetic dipole moment and $P_{-3}$ is the spin period in ms.

Typically radio emission only accounts for a small fraction of the spindown
energy, with most of the energy dissipated in a pair plasma wind, some of
which is visible by the reconnection processes at the light cylinder. We will
discuss below how this might change for giant pulse emission in the presense
of strong magnetic fields at the light cylinder. The peak flux of a radio
burst can be estimated with

$S_{\mathrm{pk}}=P_{\mathrm{rad}}f_{r}\Delta\nu^{-1}\Omega^{-1}d^{-2}N^{-1}$
(2)

where $f_{r}$ is the fraction of energy emitted in the radio frequency during
the burst; $\Delta\nu$ is the bandwidth of the burst; $\Omega$ is the solid
angle of the emission; $d$ is the distance to the Earth and $N$ is the number
of bursts emitted simultaneously.

The isotropic equivalent spectral luminosity for a single burst would be:

$\displaystyle L_{\nu}$ $\displaystyle=S_{\mathrm{pk}}4\pi
d^{2}=P_{\mathrm{rad}}f_{r}\Delta\nu^{-1}N^{-1}\frac{4\pi}{\Omega}$
$\displaystyle=10^{36}\mathrm{erg/s/Hz}\,B_{10}^{2}P_{-3}^{-4}f_{r,-3}\Delta\nu_{9}^{-1}N^{-1}\Omega^{-1}_{-8}$
(3)

Here we assume $B=10^{10}G$, millisecond period, $\Delta\nu=1$ GHz and radio
efficiency $f_{r}$ is $10^{-3}$, which is observed in the case of the galactic
magnetar SGR 1935+2154 FRB-like burst.

The GP scenario has been discussed previously by Cordes & Chatterjee (2019).
For a wide angle emission, ${\Omega\sim 4\pi}$, a milisecond neutron star with
an ordinary magnetic field will not produce enough energy for an FRB, which
all have spectral luminosity $L_{\nu}>10^{27}\mathrm{erg/s/Hz}$. However,
coherent radio radiation is expected to be strongly beamed. For radio pulsars,
the bulk motion of emitting plasma is expected to have Lorentz factors greater
than $10^{2}$(Melrose et al., 2021). Recent studies of pulsar scintillation
suggest highly relativistic motion in a bulk coherent flow in the emission
region of the Crab giant pulses. The measured Lorentz factors $\gamma\gtrsim
10^{4}$(Bij et al., 2021; Lin et al., 2022a) leading to beaming of giant
pulses into a tiny solid angle $\Omega\sim 1/\gamma^{2}$. We will assume the
beaming directions to be random. The measured high Lorentz factors can help
explain the radio emission with free electron lasers(Lyutikov, 2021) or moving
mirrors(Yalinewich & Pen, 2022). With $\gamma\sim 10^{4}$, we have forward
gain of $4\pi/\Omega\sim 10^{7}$. The nature of coherent emission, especially
that of giant pulses, is still not well understood. Giant pulses are
preferentially observed to occur in pulsars with strong magnetic fields at the
light cylinder $B_{\rm LC}>10^{5}$G(Knight et al., 2006). Extrapolating, rapid
spindown pulsars might dominate their spindown energy through the emission of
giant pulses. In this case, the spindown energy is able to account for most of
the FRBs with known redshift.

The observed burst rate can be parametrized:

$\displaystyle R_{\mathrm{obs}}=RN\frac{\Omega}{4\pi}$ (4)

where $R$ is the rate of the FRB bursts from this source in all directions.
The separation of the bursts has to be greater than the burst duration $W$ for
them to be separated in time. Therefore, the number of bursts emitted at the
same time $N\geq RW$. Together with Eq. 4, we have

$\displaystyle N\geq\sqrt{R_{\mathrm{obs}}W4\pi/\Omega}$ (5)

The total energy of the observed bursts accumulated over time $\Delta T$:

$\displaystyle E_{\mathrm{T}}=f_{r}P_{\mathrm{rad}}\Delta T$ (6)

The total fluence will be:

$\displaystyle F_{\mathrm{T}}=f_{r}P_{\mathrm{rad}}\Delta T/4\pi d^{2}$ (7)

After accumulating over time scales much longer than the burst separations,
the total fluence does not depend on the solid angle.

For FRB 20200120E detected in the GC of M81, the observed isotropic equivalent
spectrum luminosity $L_{\nu}\sim 10^{27}-10^{28}$ erg/Hz/s. As shown in
Equation 3, an ordinary recycled pulsar with a field strength of $10^{8}$ G
and a period of 10 ms should in principle have enough spin down energy to
produce the bursts given a beaming angle similar to the Crab pulsar.

For FRB 20201124A, energetic bursts of a few times $10^{34}$ erg/s/Hz have
been detected with more than 2000 hr observation with 25-32m telescopes. As
seen from Eq 3, the spindown energy from a milisecond neutron star with
$10^{10}$ G field is enough to produce the bursts in the highly beamed
scenario.

For FRB 20121102A, the observed isotropic equivalent spectrum luminosity
ranges $S_{\mathrm{pk}}\sim 10^{30}-10^{34}$ erg/Hz/s. The peak burst rate can
reach 122 hr-1 with the FAST observation, with the majority of the burst
$S_{\mathrm{pk}}\sim 10^{31}$. Assuming a highly beamed emission with
$\omega\sim 10^{-8}$ as discussed before, we expect N$\geq 300$ for the faint
bursts following Eq 5. N can be 1 for the energetic bursts. As seen from Eq 3,
the spindown energy from a milisecond neutron star with $10^{10}$ G field is
enough to produce the bursts in the highly beamed scenario. The total
isotropic energy emitted by the 1652 bursts detected in 59.5 hour spanning 47
days is $3.4\times 10^{41}$ erg, corresponding to a mean luminosity of $\sim
10^{36}$ erg/s. Following Eq 6, the total energy that can be provided by spin
down of a milisecond pulsar in 59.5 h is
$10^{45}\mathrm{erg}f_{r}B_{10}^{2}P_{-3}^{-4}$ which is enough to produce the
bursts assuming $f_{r}B_{10}\gtrsim 10^{-3}$ during the radio peak activity
time.

##  3 Spinup

As discussed in the last section, orders of magnitude larger magnetic field
than the typically observed millisecond pulsars is required to explain the
spectral luminosity of most of the FRBs. We discuss the physical mechanism to
generate this kind of short lived millisecond pulsars in globular clusters.
For simplicity, we consider a neutron star in the center of a GC tidally
partially disrupting a main sequence star in a close encounter. Lee et al.
(1996); Kremer et al. (2022) examined numerical simulations of such
disruptions, with the goal to explain isolated recycled pulsars in GCs.
Depending on impact parameter, a fraction of the close encounter star can be
disrupted, with most of the stripped off material bound to the neutron star. A
significant portion of this material will ultimately accrete onto the neutron
star, resulting in an increase of angular momentum. In an accretion disk,
angular momentum transfer occurs by inner mass transferring angular momentum
to outer mass. At the Alven radius $r_{A}$, the point of pressure balance
between the accretion disk and magnetic field, mass flows in directly,
transferring orbital angular momentum to the central object. While the details
in the accretion needs dedicated simulations, we discuss few basic limits
here.

For neutron star with large magnetic field, the required accretion rate is
large. Spinup only occurs while the Keplerian speed is higher than the co-
rotation speed at $r_{A}$. For materials accreting onto a magnetized neutron
star, the magnetic energy density balances the kinetic energy density
$B^{2}/8\pi=\rho v^{2}/2$ at the Alfven radius $r_{A}$. Assuming the matter
moves in spherical radial free fall, $\rho=\dot{M}/4\pi vr^{2}$ and
$v=\sqrt{2GM_{\mathrm{psr}}/r}$. The relevant magnetic field at the Alfven
radius is the dipole component $B=\mu/r^{3}$. Then the Alfven radius can be
estimated with:

$\displaystyle
r_{A}=\big{(}\frac{\mu^{4}}{2GM_{\mathrm{psr}}\dot{M}^{2}}\big{)}^{1/7}=25\mathrm{km}\,B_{10}^{4/7}\dot{M}_{-5}^{-2/7}$
(8)

where $\mu=BR^{3}$ is the magnetic dipole moment, $R$ is the neutron star
radius. $\dot{M}$ is in unit of $M_{\mathrm{sun}}/\mathrm{yr}$.

To spin-up the neutron star, the Alfven radius has to be smaller than the co-
rotation radius
$r_{\mathrm{co}}=(GM_{\mathrm{psr}}P^{2}/4\pi^{2})^{1/3}=20kmP_{-3}^{2/3}$. At
$r_{\mathrm{co}}$ the Keplerian angular velocity equals the spin angular
velocity. If material outside $r_{\mathrm{co}}$ interacts with the star via
the magnetic field, it will be spun up and repelled, slowing down the neutron
star. For millisecond pulsar, the co-rotation radius is close to the neutron
star radius. Therefore, a minimum accretion rate is required for the spin-up
to happen:

$\displaystyle\dot{M}_{\mathrm{min}}=\frac{1}{\sqrt{2}}\mu^{2}\big{(}\frac{2\pi}{P}\big{)}^{7/3}(GM_{\mathrm{psr}})^{-5/3}=10^{-5}M_{\mathrm{sun}}/\mathrm{yr}\,B_{10}^{2}P_{-3}^{-7/3}$
(9)

After the accretion, the angular momentum of the neutron star will be
increased by $\Delta J=\Delta MvR=\Delta M\sqrt{2GM_{\mathrm{psr}}R}$ and
therefore, it will be spun up to $P=2\pi I/\Delta J$. For a neutron star to
reach millisecond period, one needs

$\displaystyle\Delta
M=I\frac{2\pi}{P}(2GM_{\mathrm{psr}})^{-1/2}\mathrm{max}(R,r_{A})^{-1/2}\approx
0.16M_{\mathrm{sun}}\,P_{-3}^{-1}$ (10)

And all the material has to be accreted within

$\displaystyle T_{\mathrm{max}}=\frac{\Delta
M}{\dot{M}_{\mathrm{min}}}=10^{4}\mathrm{yr}\,B_{10}^{-2}P_{-3}^{4/3}$ (11)

The required $\dot{M}$ is substantial to create a milisecond pulsar with
moderate magnetic field. A significant fraction of the companion star has to
be accreted to the NS within less than $10^{4}$ year for $B=10^{10}$ G, and
less than a year for an ordinary pulsar of $B=10^{12}$ G, and within an hour
for a magnetar of $10^{14}$ G.

The upper limit of the accretion rate can be estimated with the shortest
possible time scale – the tidal disruption time. The neutron star has a close
encounter with the main sequence star at a timescale $T\approx d/v_{r}$. The
impact distance $d$ can be approximated with the radius of the main sequence
star $R_{\mathrm{star}}$ as the other relevant scales $r_{A}$ and $R$ are both
much smaller. $v_{r}=\sqrt{2GM_{\mathrm{star}}/R_{\mathrm{star}}}$ is the
relative velocity of the two object. Therefore

$\displaystyle T_{\mathrm{min}}\approx
d/v_{r}=\sqrt{2R_{\mathrm{star}}^{3}/GM_{\mathrm{star}}}=0.6\mathrm{hr}\,M_{\mathrm{star}}^{-1/2}R_{\mathrm{star}}^{3/2}$
(12)

where $M_{\mathrm{star}}$ and $R_{\mathrm{star}}$ are in the unit of solar
mass and solar radius respectively. Hence the maximum accretion rate will be:

$\displaystyle\dot{M}_{\mathrm{max}}$ $\displaystyle=\frac{\Delta
M}{T_{\mathrm{min}}}=I\frac{\pi}{P}(\frac{M_{\mathrm{star}}}{M_{\mathrm{psr}}})^{1/2}R_{\mathrm{star}}^{-3/2}\mathrm{max}(R,r_{A})^{-1/2}$
(13) $\displaystyle=2\times
10^{4}M_{\mathrm{sun}}/\mathrm{yr}\,M_{\mathrm{star}}^{1/2}R_{\mathrm{star}}^{-3/2}P_{-3}^{-1}$
(14)

To recycle a neutron star to millisecond
$\dot{M}_{\mathrm{max}}\gtrsim\dot{M}_{\mathrm{min}}$, hence,

$\displaystyle
B_{10}P_{-3}^{-2/3}<10^{4}M_{\mathrm{star}}^{1/4}R_{\mathrm{star}}^{-3/4}$
(15)

Therefore, it is not possible to recycle magnetars with $B\gtrsim 10^{14}$ to
milisecond. Actually, the accretion timescale can be much longer than the
tidal dynamical time scale. According to Kremer et al. (2022), the viscous
accretion time is approximately 1 to 5 days. In this case, magnetars with
$B\gtrsim 10^{13}$ G are difficult to be recycled to millisecond. However, for
most of the ordinary neutron stars with magnetic field $10^{12}$ G or lower,
it is possible to recycle them to millisecond in the tidal disruption events.

Another challenge for the large accretion rate is the large radiation
pressure. As the mass accretes, it converts the gravitational potential energy
into thermal energy and exerts a radiation force $F_{\mathrm{rad}}=\kappa
mL/4\pi R^{2}c$ which opposes the gravity force $F_{\mathrm{grav}}=GMm/R^{2}$.
Here $\kappa$ is the opacity, which is defined as the cross-section per unit-
mass. For ionized hydrogen, $\kappa=\sigma_{T}/m_{p}$, where $\sigma_{T}$ is
the Thomson cross section and $m_{p}$ is the mass of the proton. $m$ is the
mass of the infalling material; $c$ is the speed of light. And the luminosity
$L$ can be estimated as a fraction $\epsilon$ of the total energy of the
accreted material $L=\epsilon\dot{M}c^{2}$. This gives the Eddington accretion
rate:

$\displaystyle\dot{M}_{\mathrm{EDD}}=\frac{4\pi GMm_{p}}{\epsilon
c\sigma_{T}}\approx 10^{-9}M_{\mathrm{sun}}/\mathrm{yr}\,\epsilon^{-1}$ (16)

Even for a relatively low magnetic field pulsar $B=10^{10}$ G, the required
accretion rate is still orders of magnitude higher than the Eddington rate.
This kind of “hypercritical" accretion has been considered in various fields,
including accretion during the supernovae and the common envelope evolution.
It can be featured in several ways. With sufficiently high mass inflow rate
(e.g. $\dot{M}_{\mathrm{cr}}\approx 10^{-4}M_{\mathrm{sun}}\mathrm{yr}^{-1}$
Fryer et al. 1996), the photons can be trapped and advected inwards, where the
neutrino cooling plays a role and allows for hypercritical
accretion(Chevalier, 1993, 1996). Apart from this, the accretion and radiation
do not occur in the same direction. The inward mass flow predominately occurs
in the plane of the disk while the accretion energy can flow out low-density
polar regions(Frank et al., 2002). Observationally, a high-mass X-ray binary
has been observed to accrete at 100 times Eddington accretion rateBachetti et
al. (2014).

In summary, more than 10% of the solar mass is needed to spin-up a neutron
star of $10^{10}$ – $10^{13}$ G magnetic field to millisecond. The required
large accretion rate is plausible to be achieved in the partial tidal
disruption events.

##  4 The rate and distribution

###  4.1 Fraction in the Overall Population

It is possible that a significant fraction of FRBs resides in this kind of
dynamic environment. Up to the distance of M81, there are around 1000 globular
clusters, and at least one of them has an active FRB source, giving a lower
limit of active source number density $\rho_{\mathrm{src}}=10^{-3}$GC-1.
Adopting a volumetric number density of GCs $\rho_{GC}=2.31$ GC Mpc-3
(Rodriguez et al., 2015), we can estimate the volumetric rate of the GC FRBs
from the rate of M81 bursts $R_{M81}=0.07$ hr-1 above an energy of
$E=F\Delta\nu D^{2}=2.5\times 10^{33}$ erg:

$\displaystyle R(>E)$
$\displaystyle=\rho_{GC}\rho_{\mathrm{src}}R_{M81}E_{33}^{-\gamma}$ (17)
$\displaystyle=10^{9}\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}E_{33}^{-\gamma}$ (18)

where $\gamma$ is the powerlaw index for the cumulative energy distribution.
For $\gamma=1.3$, $R(>E_{39})=20\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$, which is
much less than the number estimated for the CHIME population (Shin et al.,
2023). However, for $\gamma=0.5$ as in the case seen by non-repeaters and FRB
20201124A, $R(>E_{39})=10^{6}\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ which is more
than enough to explain the whole population.

###  4.2 The source density from the NS-MS TDE

The NS-MS TDE can provide enough active source that explains the existence of
the M81 FRB and the non-detection of similar events with the observation of
GCs in the Milky way.

We can estimate the number of active sources $N_{s}$ at a given distance d
with:

$\displaystyle N_{s}=R_{\mathrm{ns-ms}}N_{\mathrm{GC}}\tau_{c}$ (19)

where $R_{\mathrm{ns-ms}}$ is the rate of the close encounter;
$N_{\mathrm{GC}}$ is the cumulative number of globular clusters up to the
distance of $d$; and $\tau_{c}$ can approximate the life time. And $\tau_{c}$
can be estimated with the spin down time:

$\displaystyle\tau_{c}\sim\frac{P}{\dot{P}}\sim 5\times
10^{5}\mathrm{yr}\,P_{-3}^{2}B_{10}^{-2},$ (20)

where the period derivative is $\dot{P}=P_{\mathrm{rad}}P^{3}/4\pi^{2}I$, and
$I\sim 10^{45}$gcm2 is the moment of the inertia.

The NS-MS star collision rate per core-collapse globular cluster (CCGC) is
around $R_{\mathrm{ns-ms}}\approx 10^{-8}$ CCGC-1yr-1 calibrated to N-body
simulations(Kremer et al., 2021b). Therefore the number density of NS with
$10^{10}$G magnetic field that are spun up is $10^{-3}-10^{-2}$ Mpc-3K
$P_{-3}^{2}B_{10}^{-2}$, where K is the fraction of NS with $B_{10}$. And the
number of active source up to the distance of M81 with 1000 GCs (and probably
250 CCGC) will be approximately $1P_{-3}^{2}B_{10}^{-2}$. The number of active
source in the Milky way CCGC will be $1P_{-3}^{2}B_{10}^{-2}$. Therefore, it
is not surprising that we didn’t detect similar events with hundreds of hour
observation staring at few GCs in the Milky way.

In addition to producing FRBs, tidal disruption events (TDEs) have been
proposed as a mechanism to produce isolated neutron stars in GCs(Ye et al.,
2023). These can be rapidly spinning millisecond pulsars, or young-looking
pulsars. Isolated millsecond pulsars could also be produced by disruption of
binaries, or potentially electron capture supernovae after merger induced
collapse (MIC) of white dwarfs. Kremer et al. (2023a) promoted MIC for the
formation of young neutron stars and a young magnetar to explain GC FRB in
M81. This paper explores the alternative combination that TDEs lead to FRBs,
exploiting the observed trend of stronger giant pulses in pulsars with large
magnetic fields at the light cylinder. Our work examines the complementary FRB
energetics aspects of Ye et al. (2023). The dynamical properties of GCs allow
scaling of interaction rates from observables, though current modelling appear
to allow both TDE and MIC as mechanisms. There are four young pulsars observed
in the globular cluster, with spin-down ages between $10^{7}$ to $10^{8}$
years. Assuming that this population also comes from the close encounter of NS
and the main sequence, we expect to see 0.1-1 active source per CCGC. There
are around 150 GCs in the Milky way, and around 20% of them are CCGC.
Therefore, we expect a couple to dozens of young pulsars in the Milky way
CCGCs. And these young pulsars are indeed preferentially observed in the CCGC.

###  4.3 Location in the GC

A typical GC velocity dispersion $\sigma\sim 5$km/s is very small compared to
the escape velocity from the surface of a main sequence star $v_{\rm esc}\sim$
500km/s. The encounter is a nearly parabolic trajectory, with the main
sequency star imparting up to twice its asymptotic linear momentum onto the
neutron star. Taking a typical MS mass $M\sim 0.6M_{\odot}$ and neutron star
mass $M_{n}\sim 1.4M_{\odot}$, we expect a recoil velocity comparable to
$\sigma$. This places the neutron star onto a radial orbit going out to the
half mass radius, and objects are prefentially observed at apocentre, not at
the epicentre in the central core region which dominates the close encounter
cross section.

##  5 Summary and prediction

We have presented a scenario of FRBs powered by giant pulses of rapid spin
down neutron stars. Giant pulses are common in pulsars with large magnetic
fields at the light cylinder, for which these rapid spin down objects are the
extreme limit. we address the energetics challenge of FRBs with narrow beaming
angles. While it is not a standard assumption in emission mechanisms, the
narrow beaming angle has been observed in a bright Crab giant pulse(Bij et
al., 2021). This scenario naturally explains FRBs’ presence in old populations
including GCs and early type galaxies with rate consistent with the
observation. The short lifetime of the source accounts for the paucity in the
local group.

Future precision VLBI localization of nearby FRBs(Lin et al., 2022b) will
enable clear distinction between various scenarios (Kremer et al., 2023b). A
recycled pulsar population tends to trace older stellar populations(Kulkarni &
Narayan, 1988; Kulkarni et al., 1990). Other than this scenario, the
dynamically-formed magnetars from white dwarf mergers are so far the only
alternative that can explain the presence of the GC FRB and its observed
rate(Kremer et al., 2021a). The fundamental element of this scenario hinges on
the presence of strong Doppler beaming. Plasma lensing near the source can
provide requisite spatial resolution to substantiate this hypothesis. As
demonstrated in Bij et al. (2021), lensing at a small angle $\delta$ will
undergo a Doppler boost in frequency, with a change approximately given by
$\Delta f/f\approx\gamma\delta$, where $\gamma$ represents the Lorentz factor
of the emitting particle. Consequently, FRBs characterized by an intrinsic
narrow frequency profile may exhibit a frequency shift in the scattering tail.
This shift may manifest as an upward or downward drift or a broadening of
frequency, depending upon the geometric configuration. Furthermore, as the
time delay $\tau\propto\delta^{2}$, the frequency change against time is
likely $|\Delta f/f|\propto\gamma\sqrt{\tau}$, where $\tau$ can be
approximated as the delay corresponding to the peak of the burst. This type of
frequency variation in the scattering tail is anticipated to be observable in
the most luminous bursts, which, as per the hypothesis, are strongly Lorentz
beamed.

## Acknowledgements

D. Z. Li thanks E. Sterl Phinney and Bing Zhang for helpful discussion. Ue-Li
Pen thanks Claire Ye, Vicky Kalogara, Kazumi Kashiyama and Paolo Freire for
helpful discussions, and Sujin Lee for help with plotting. Ue-Li Pen receives
support from Ontario Research Fund–research Excellence Program (ORF-RE),
Natural Sciences and Engineering Research Council of Canada (NSERC) [funding
reference number RGPIN-2019-067, CRD 523638-18, 555585-20], Canadian Institute
for Advanced Research (CIFAR), the National Science Foundation of China
(Grants No. 11929301), Thoth Technology Inc, Alexander von Humboldt
Foundation, and the National Science and Technology Council (NSTC) of Taiwan
(111-2123-M-001-008-, and 111-2811-M-001-040-). Computations were performed on
the SOSCIP Consortium’s [Blue Gene/Q, Cloud Data Analytics, Agile and/or Large
Memory System] computing platform(s). SOSCIP is funded by the Federal Economic
Development Agency of Southern Ontario, the Province of Ontario, IBM Canada
Ltd., Ontario Centres of Excellence, Mitacs and 15 Ontario academic member
institutions.

## References

  * Bachetti et al. (2014) Bachetti M., et al., 2014, Nature, 514, 202
  * Bhardwaj et al. (2021) Bhardwaj M., et al., 2021, ApJ, 910, L18
  * Bij et al. (2021) Bij A., et al., 2021, ApJ, 920, 38
  * Bochenek et al. (2020) Bochenek C. D., Ravi V., Belov K. V., Hallinan G., Kocz J., Kulkarni S. R., McKenna D. L., 2020, Nature, 587, 59
  * CHIME/FRB Collaboration et al. (2019) CHIME/FRB Collaboration et al., 2019, ApJ, 885, L24
  * CHIME/FRB Collaboration et al. (2020a) CHIME/FRB Collaboration et al., 2020a, Nature, 582, 351
  * CHIME/FRB Collaboration et al. (2020b) CHIME/FRB Collaboration Andersen B. C., et al., 2020b, Nature, 587, 54
  * Chevalier (1993) Chevalier R. A., 1993, ApJ, 411, L33
  * Chevalier (1996) Chevalier R. A., 1996, ApJ, 459, 322
  * Cordes & Chatterjee (2019) Cordes J. M., Chatterjee S., 2019, ARA&A, 57, 417
  * Fonseca et al. (2020) Fonseca E., et al., 2020, ApJ, 891, L6
  * Frank et al. (2002) Frank J., King A., Raine D. J., 2002, Accretion Power in Astrophysics: Third Edition
  * Fryer et al. (1996) Fryer C. L., Benz W., Herant M., 1996, ApJ, 460, 801
  * Kirsten et al. (2022) Kirsten F., et al., 2022, Nature, 602, 585
  * Knight et al. (2006) Knight H. S., Bailes M., Manchester R. N., Ord S. M., Jacoby B. A., 2006, ApJ, 640, 941
  * Kremer et al. (2021a) Kremer K., Piro A. L., Li D., 2021a, ApJ, 917, L11
  * Kremer et al. (2021b) Kremer K., Piro A. L., Li D., 2021b, ApJ, 917, L11
  * Kremer et al. (2022) Kremer K., Ye C. S., Kıroğlu F., Lombardi J. C., Ransom S. M., Rasio F. A., 2022, ApJ, 934, L1
  * Kremer et al. (2023a) Kremer K., Fuller J., Piro A. L., Ransom S. M., 2023a, MNRAS, 525, L22
  * Kremer et al. (2023b) Kremer K., Li D., Lu W., Piro A. L., Zhang B., 2023b, ApJ, 944, 6
  * Kulkarni & Narayan (1988) Kulkarni S. R., Narayan R., 1988, ApJ, 335, 755
  * Kulkarni et al. (1990) Kulkarni S. R., Narayan R., Romani R. W., 1990, ApJ, 356, 174
  * Law et al. (2023) Law C. J., et al., 2023, arXiv e-prints, p. arXiv:2307.03344
  * Lee et al. (1996) Lee H. M., Kim S. S., Kang H., 1996, Journal of Korean Astronomical Society, 29, 19
  * Lin et al. (2022a) Lin R., van Kerkwijk M. H., Main R., Mahajan N., Pen U.-L., Kirsten F., 2022a, arXiv e-prints, p. arXiv:2211.05209
  * Lin et al. (2022b) Lin H.-H., et al., 2022b, PASP, 134, 094106
  * Lu et al. (2022) Lu W., Beniamini P., Kumar P., 2022, MNRAS, 510, 1867
  * Lyutikov (2021) Lyutikov M., 2021, ApJ, 922, 166
  * Majid et al. (2021) Majid W. A., et al., 2021, ApJ, 919, L6
  * Manchester et al. (2016) Manchester R. N., Hobbs G. B., Teoh A., Hobbs M., 2016, VizieR Online Data Catalog, p. B/psr
  * Melrose et al. (2021) Melrose D. B., Rafat M. Z., Mastrano A., 2021, MNRAS, 500, 4530
  * Rajwade et al. (2020) Rajwade K. M., et al., 2020, MNRAS, 495, 3551
  * Rodriguez et al. (2015) Rodriguez C. L., Morscher M., Pattabiraman B., Chatterjee S., Haster C.-J., Rasio F. A., 2015, Phys. Rev. Lett., 115, 051101
  * Shin et al. (2023) Shin K., et al., 2023, ApJ, 944, 105
  * Spitler et al. (2014) Spitler L. G., et al., 2014, ApJ, 790, 101
  * Spitler et al. (2016) Spitler L. G., et al., 2016, Nature, 531, 202
  * Tauris et al. (2013) Tauris T. M., Sanyal D., Yoon S. C., Langer N., 2013, A&A, 558, A39
  * Yalinewich & Pen (2022) Yalinewich A., Pen U.-L., 2022, MNRAS, 515, 5682
  * Ye et al. (2023) Ye C. S., Kremer K., Ransom S. M., Rasio F. A., 2023, arXiv e-prints, p. arXiv:2307.15740


Base Case:

$i=2$. In this case we prove that the claim is true for $i=2$ and every
$j\in\\{1,\ldots,\min\\{k-k^{\star},k^{\star}-1\\}\\}$. We again use induction
on $j$.

Base Case:

$j=1$. We first observe that for $i=2$ and $j=1$,
$T_{k^{\star}+2},\ldots,T_{k}$ are zero polynomials. Let $y^{S}$ be a monomial
in $T_{k^{\star}+1}$ with Hamming weight $s$. By the construction of
monomials, there exists monomial $y^{S^{\prime}}$ in $T_{k^{\star}}$ whose
Hamming weight is $s^{\prime}$ and $y^{S^{\prime\prime}}$ in
$P_{2}^{\mathcal{H}(S^{\prime\prime})}$ whose hamming weight is
$s^{\prime\prime}$ such that $s=s^{\prime}+s^{\prime\prime}$ and
$y^{S}=y^{S^{\prime}}\times y^{S^{\prime\prime}}$. Since
$\mathcal{H}(S^{\prime}+S^{\prime\prime})=s=s^{\prime}+s^{\prime\prime}$ and
$\mathcal{H}(S^{\prime})+\mathcal{H}(S^{\prime\prime})=s^{\prime}+s^{\prime\prime}$,
due to Observation 5.1, vectors $S^{\prime}$ and $S^{\prime\prime}$ are
disjoint. Due to Claim 5.2.1, there exists $k^{\star}$ pairwise disjoint
districts won by $c_{1}$ uniquely, say $Y_{1},\ldots,Y_{k^{\star}}$, such that
$\chi(Y_{1}\cup\ldots\cup Y_{k^{\star}})=S^{\prime}$. Since
$y^{S^{\prime\prime}}$ is a monomial in $P_{2}^{s^{\prime\prime}}$, there
exists a district in $\mathcal{F}_{2}$ of size $s^{\prime\prime}$ won by
$c_{2}$ uniquely, say $Y$, such that $\chi(Y)=S^{\prime\prime}$. Since
$S^{\prime}$ and $S^{\prime\prime}$ are disjoint characteristic vectors, sets
$Y$ and $Y_{1}\cup\ldots\cup Y_{k^{\star}}$ are disjoint. Therefore, due to
Proposition 5.1, $\chi(Y_{1}\cup\ldots\cup Y_{k^{\star}}\cup
Y)=\chi(Y_{1}\cup\ldots\cup
Y_{k^{\star}})+\chi(Y)=S^{\prime}+S^{\prime\prime}=S$. Therefore,
$Y_{1},\ldots,Y_{k^{\star}},Y$ are $k^{\star}+1$ districts satisfying all the
properties mentioned in the claim.

Induction Step:

Suppose that the claim is true for $j=j^{\prime}-1$. Next, we prove the claim
for $j=j^{\prime}$. Note that in the algorithm, we add monomials in $T_{t}$,
where $t\in\\{k^{\star}+1,\ldots,k\\}$, constructed in the previous iteration.
Let $\tilde{T}_{k^{\star}+1},\ldots,\tilde{T}_{k}$ be the polynomials
constructed for $i=2$, $j=j^{\prime}-1$. Let $T_{t}$, where
$t\in\\{k^{\star}+1,\ldots,k\\}$, has a monomial $y^{S}$ with Hamming weight
$s$ which is not in $\tilde{T}_{t}$. By the construction of monomials
$y^{S}=y^{S^{\prime}}\times y^{S^{\prime\prime}}$, where $y^{S^{\prime}}$ is a
monomial in $\tilde{T}_{t-1}$ with Hamming weight $s^{\prime}$ and
$y^{S^{\prime\prime}}$ is a monomial in
$P_{2}^{\mathcal{H}(S^{\prime\prime})}$ with Hamming weight $s^{\prime\prime}$
such that $s=s^{\prime}+s^{\prime\prime}$. As argued above the characteristic
vectors $S^{\prime}$ and $S^{\prime\prime}$ are disjoint. Using inductive
hypothesis, there exists $t-1$ pairwise disjoint districts, say
$Y_{1},\ldots,Y_{t-1}$, such that $\chi(Y_{1}\cup\ldots\cup
Y_{t-1})=S^{\prime}$, $c_{1}$ wins in $k^{\star}$ districts uniquely, $c_{2}$
wins in $j^{\prime}-1$ districts uniquely, and all the other candidates do not
win any district in $Y_{1},\ldots,Y_{t-1}$. Since $y^{S^{\prime\prime}}$ is a
monomial in $P_{2}^{s^{\prime\prime}}$, there exists a district in
$\mathcal{F}_{2}$ of $s^{\prime\prime}$ won by $c_{2}$ uniquely, say $Y$, such
that $\chi(Y)=S^{\prime\prime}$. Since $S^{\prime}$ and $S^{\prime\prime}$ are
disjoint characteristic vectors, sets $Y$ and $Y_{1}\cup\ldots\cup Y_{t-1}$
are disjoint. Therefore, due to Proposition 5.1, $\chi(Y_{1}\cup\ldots\cup
Y_{t-1}\cup Y)=\chi(Y_{1}\cup\ldots\cup
Y_{t-1})+\chi(Y)=S^{\prime}+S^{\prime\prime}=S$. Thus,
$Y_{1},\ldots,Y_{t-1},Y$ are $t$ districts satisfying all the properties
mentioned in the claim.

Induction Step:

Suppose that the claim is true for $i=i^{\prime}-1$. We next prove it for
$i=i^{\prime}$. Again we need to prove the claim for $i=i^{\prime}$ and every
$j\in\\{1,\ldots,\min\\{k-k^{\star},k^{\star}-1\\}\\}$. The proof is same as
above using induction on $j$. We give here the proof for completeness.

Base Case:

$j=1$. Let $\tilde{T}_{k^{\star}+1},\ldots,\tilde{T}_{k}$ be the polynomials
constructed for $i=i^{\prime}-1$, $j=\min\\{k-k^{\star},k^{\star}-1\\}$. Let
$T_{t}$, where $t\in\\{k^{\star}+1,\ldots,k\\}$, has a monomial $y^{S}$ with
Hamming weight $s$ which is not in $\tilde{T}_{t}$. As argued above
$y^{S}=y^{S^{\prime}}\times y^{S^{\prime\prime}}$, where $y^{S^{\prime}}$ is a
monomial in $\tilde{T}_{t-1}$ with Hamming weight $s^{\prime}$ and
$y^{S^{\prime\prime}}$ is a monomial in
$P_{i^{\prime}}^{\mathcal{H}(S^{\prime\prime})}$ with Hamming weight
$s^{\prime\prime}$ such that $s=s^{\prime}+s^{\prime\prime}$. As argued above
vectors $S^{\prime}$ and $S^{\prime\prime}$ are disjoint. Using inductive
hypothesis, there exists $t-1$ pairwise disjoint districts, say
$Y_{1},\ldots,Y_{t-1}$, such that $\chi(Y_{1}\cup\ldots\cup
Y_{t-1})=S^{\prime}$, $c_{1}$ wins in $k^{\star}$ districts uniquely, for
$q\in\\{2,\ldots,i^{\prime}-1\\}$, $c_{q}$ wins in at most $k^{\star}-1$
districts uniquely, and all the other candidates do not win in any district in
$Y_{1},\ldots,Y_{t-1}$. Since $y^{S^{\prime\prime}}$ is a monomial in
$P_{i^{\prime}}^{s^{\prime\prime}}$, there exists a district in
$\mathcal{F}_{i^{\prime}}$ of size $s^{\prime\prime}$ won by $c_{i^{\prime}}$
uniquely, say $Y$, such that $\chi(Y)=S^{\prime\prime}$. Since $S^{\prime}$
and $S^{\prime\prime}$ are disjoint characteristic vectors, sets $Y$ and
$Y_{1}\cup\ldots\cup Y_{t-1}$ are disjoint. Therefore, as argued above, due to
Proposition 5.1, $\chi(Y_{1}\cup\ldots\cup Y_{t-1}\cup Y)=S$. Thus,
$Y_{1},\ldots,Y_{t-1},Y$ are $t$ districts satisfying all the properties
mentioned in the claim.

Induction Step:

Suppose that the claim is true for $j=j^{\prime}-1$. Next, we prove it for
$j=j^{\prime}$. Let $\tilde{T}_{k^{\star}+1},\ldots,\tilde{T}_{k}$ be the
polynomials constructed for $i=i^{\prime}$, $j=j^{\prime}-1$. Let $T_{t}$,
where $t\in\\{k^{\star}+1,\ldots,k\\}$, has a monomial $y^{S}$ with Hamming
weight $s$ which is not in $\tilde{T}_{t}$. As argued above
$y^{S}=y^{S^{\prime}}\times y^{S^{\prime\prime}}$, where $y^{S^{\prime}}$ is a
monomial in $\tilde{T}_{t-1}$ with Hamming weight $s^{\prime}$ and
$y^{S^{\prime\prime}}$ is a monomial in
$P_{i^{\prime}}^{\mathcal{H}(S^{\prime\prime})}$ with Hamming weight
$s^{\prime\prime}$ such that $s=s^{\prime}+s^{\prime\prime}$. As argued above
vectors $S^{\prime}$ and $S^{\prime\prime}$ are disjoint. Using inductive
hypothesis, there exists $t-1$ pairwise disjoint districts, say
$Y_{1},\ldots,Y_{t-1}$, such that $\chi(Y_{1}\cup\ldots\cup
Y_{t-1})=S^{\prime}$, $c_{1}$ wins in $k^{\star}$ districts uniquely, for
$q\in\\{2,\ldots,i^{\prime}-1\\}$, $c_{q}$ wins in at most $k^{\star}-1$
districts, $c_{i^{\prime}}$ wins in $j^{\prime}-1$ districts uniquely, and all
the other candidates do not win any district in $Y_{1},\ldots,Y_{t-1}$. Since
$y^{S^{\prime\prime}}$ is a monomial in $P_{i^{\prime}}^{s^{\prime\prime}}$,
there exists a district in $\mathcal{F}_{i^{\prime}}$ of size
$s^{\prime\prime}$ won by $c_{i^{\prime}}$ uniquely, say $Y$, such that
$\chi(Y)=S^{\prime\prime}$. Since $S^{\prime}$ and $S^{\prime\prime}$ are
disjoint characteristic vectors, sets $Y$ and $Y_{1}\cup\ldots\cup Y_{t-1}$
are disjoint. Therefore, as argued above, $\chi(Y_{1}\cup\ldots\cup
Y_{t-1}\cup Y)=S$. Thus, $Y_{1},\ldots,Y_{t-1},Y$ are $t$ districts satisfying
all the properties mentioned in the claim. ∎

If the algorithm returns Yes, then we know that there is a monomial
$y^{\chi(V(G))}$ in $T_{k}$. Therefore, due to Claim 5.2.2, there are $k$
districts such that $c_{1}$ wins in $k^{\star}$ districts and all the
candidates win in at most $k^{\star}-1$ districts. ∎

Next, we will analyze the running time of the algorithm.

######  Lemma 5.3.

The algorithm in Section 5 runs in $2^{n}(n+m)^{{\mathcal{O}}(1)}$ time.

###### Proof.

In the algorithm, we first construct a family of districts, $\mathcal{F}_{i}$
for each candidate $c_{i}\in\mathcal{C}$. Since we check all possible sets of
$V(G)$ for the potential candidate of the family $\mathcal{F}_{i}$, this step
of the algorithm takes ${\mathcal{O}}(2^{n}(n+m))$ time. Then, for each
family, we construct at most $n$ polynomials. Since $|\mathcal{F}_{i}|\leq
2^{n}$, the number of terms in every polynomial is ${\mathcal{O}}(2^{n})$.
Thus, these polynomials can be constructed in ${\mathcal{O}}(2^{n}(n+m))$
time. Then, for every pair of integers, $i,j$, where $i\in\\{2,\ldots,m\\}$,
$j\in\\{1,\ldots,\min\\{k^{\star}-1,k-k^{\star}\\}\\}$, we multiply
polynomials at most $nk$ times. Since every polynomial has degree at most
$2^{n}$, using Proposition 5.2, every polynomial multiplication takes
${\mathcal{O}}(2^{n}n)$ time. Hence, the algorithm runs in
$2^{n}(n+m)^{{\mathcal{O}}(1)}$ time. ∎

Hence, Theorem 5.1 is proved.

##  6 In Conclusion

We have shown that GM on paths is NP-complete, thereby resolving an open
question in [21]. This gives parameterized intractability for parameters such
as maximum degree of a vertex in the graph. Furthermore, we have presented FPT
algorithms for paths when parameterized by the number of districts. We also
give an FPT algorithm running in time $2^{n}(n+m)^{{\mathcal{O}}(1)}$ on
general graphs.

We conclude with a few directions for further research:  (i) Does there exist
a ${\mathcal{O}}(c^{n})$ algorithm for W-GM when there are possibly multiple
winners in a district?;  (ii) Is W-GM on paths FPT parameterized by the number
of candidates?;  (iii) Is W-GM on trees FPT parameterized by the number of
districts?

## References

  * [1] N. Betzler, J. Guo, and R. Niedermeier.  Parameterized computational complexity of Dodgson and Young elections.  Inf. Comput., 208(2):165–177, 2010.
  * [2] N. Betzler and J. Uhlmann.  Parameterized complexity of candidate control in elections and related digraph problems.  Theor. Comput. Sci., 410(52):5425 – 5442, 2009.
  * [3] A. Björklund, T. Husfeldt, P. Kaski, and M. Koivisto.  Narrow sieves for parameterized paths and packings.  J. Comput. Syst. Sci., 87:119–139, 2017.
  * [4] Brian Brubach, Aravind Srinivasan, and Shawn Zhao.  Meddling metrics: the effects of measuring and constraining partisan gerrymandering on voter incentives.  In Proceedings of EC’20, pages 815–833, 2020.
  * [5] E. Clough.  Talking locally and voting globally: Duverger’s law and homogeneous discussion networks.  Political Res. Q., 3(60):531–540, 2007.
  * [6] A. Cohen-Zemach, Y. Lewenberg, and J. S. Rosenschein.  Gerrymandering over graphs.  In Proceedings of AAMAS’18, pages 274–282, 2018.
  * [7] M. Cygan, F. V. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh.  Parameterized Algorithms.  Springer, 2015.
  * [8] M. Cygan and M. Pilipczuk.  Exact and approximate bandwidth.  Theor. Comput. Sci., 411(40-42):3701–3713, 2010.
  * [9] P. Dey.  Gerrymandering: A briber’s perspective.  arXiv:1909.01583, 2019.
  * [10] P. Dey, N. Misra, and Y. Narahari.  Parameterized dichotomy of choosing committees based on approval votes in the presence of outliers.  Theor. Comput. Sci., 783:53–70, 2019.
  * [11] R. Diestel.  Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics.  Springer, 2012.
  * [12] R. G. Downey and M. R. Fellows.  Fundamentals of Parameterized Complexity.  Texts in Computer Science. Springer, 2013.
  * [13] E. Eiben, F. V. Fomin, F. Panolan, and K. Simonov.  Manipulating districts to win elections: Fine-grained complexity.  In Proceedings of AAAI’20, 2020.
  * [14] J. Ellenberg.  How computers turned gerrymandering into a science.  New York Times, October 2017.
  * [15] R. S. Erikson.  Malapportionment, gerrymandering, and party fortunes in congressional elections.  Am. Political Sci. Rev., 4(66):1234–1245, 1972.
  * [16] P. Faliszewski, E. Hemaspaandra, L. A. Hemaspaandra, and J. Rothe.  Copeland voting fully resists constructive control.  In Proceedings of AAIM’08, volume 5034, pages 165–176. Springer, 2008.
  * [17] P. Faliszewski, E. Hemaspaandra, L. A. Hemaspaandra, and J. Rothe.  Llull and Copeland voting computationally resist bribery and constructive control.  JAIR, 35:275–341, 2009.
  * [18] B. Fleiner, B. Nagy, and A. Tasnádi.  Optimal partisan districting on planar geographies.  Cent. Eur. J. Oper. Res., 25(4):879–888, 2017.
  * [19] F. V Fomin, D. Lokshtanov, F. Panolan, and S. Saurabh.  Efficient computation of representative families with applications in parameterized and exact algorithms.  J. ACM, 63(4):1–60, 2016.
  * [20] S. Issacharoff.  Gerrymandering and political cartels.  Harvard Law Review, pages 593–648, 2002.
  * [21] T. Ito, N. Kamiyama, Y. Kobayashi, and Y. Okamoto.  Algorithms for gerrymandering over graphs.  In Proceedings of AAMAS’19, pages 1413–1421, 2019.
  * [22] M. Zuckerman L. Xia, A. D. Procaccia, V. Conitzer, and J. S. Rosenschein.  Complexity of unweighted coalitional manipulation under some common voting rules.  In Proceedings of IJCAI’19, pages 348–353, 2009.
  * [23] V. B. Le and F. Pfender.  Complexity results for rainbow matchings.  Theor. Comput. Sci., 524:27–33, 2014.
  * [24] D. Marx.  A parameterized view on matroid optimization problems.  Theor. Comput. Sci., 410(44):4471–4479, 2009.
  * [25] R. T. Moenck.  Practical fast polynomial multiplication.  In Proceedings of SYMSAC’76, pages 136–148, 1976.
  * [26] B. Monien.  How to find long paths efficiently.  In North-Holland Mathematics Studies, volume 109, pages 239–254. Elsevier, 1985.
  * [27] R. Neidermeier.  Invitation to fixed-parameter algorithms.  Springer, 2006.
  * [28] C. Puppe and A. Tasnádi.  Optimal redistricting under geographical constraints: Why “pack and crack” does not work.  Econ. Lett., 105(1):93 – 96, 2009.
  * [29] N. Talmon.  Structured proportional representation.  Theor. Comput. Sci., (708):58–74, 2018.
  * [30] R. Williams.  Finding paths of length $k$ in ${O}^{\star}(2^{k})$ time.  Inf. Process. Lett., 109(6):315–318, 2009.
  * [31] M. Wines.  What is gerrymandering? and how does it work?  New York Times, June 2019.
  * [32] O. Lev Y. Lewenberg and J. S. Rosenschein.  Divide and conquer: Using geographic manipulation to win district-based elections.  In Proceedings of AAMAS’17, 2017.
  * [33] M. Zuckerman, A. D. Procaccia, and J. S. Rosenschein.  Algorithms for the coalitional manipulation problem.  JAIR, 173(2):392–412, 2009.


# Inducing intermittency in the inverse cascade
of two dimensional turbulence by a fractal forcing

George Sofiadis  Department of Mechanical Engineering, University of West
Attica, 250 Thivon & P. Ralli Str., Egaleo, 122 44, Athens, Greece  Ioannis E.
Sarris  Department of Mechanical Engineering, University of West Attica, 250
Thivon & P. Ralli Str., Egaleo, 122 44, Athens, Greece  Alexandros Alexakis
<EMAIL_ADDRESS>Laboratoire de Physique de l’Ecole normale supérieure,
ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005
Paris, France

###### Abstract

We demonstrate that like in the forward cascade of three dimensional
turbulence that displays intermittency (lack of self-similarity) due to the
concentration of energy dissipation in a small set of fractal dimension less
than three, the inverse cascade of two-dimensional turbulence can also display
lack of self-similarity and intermittency if the energy injection is
constrained in a fractal set of dimension less than two. A series of numerical
simulations of two dimensional turbulence are examined, using different
forcing functions of the same forcing length-scale but different fractal
dimension $D$ that varies from the classical $D=2$ case to the point vortex
case $D=0$. It is shown that as the fractal dimension of the forcing is
decreased from $D=2$, the self-similarity is lost and intermittency appears,
with the scaling of the different structure functions $\langle|\delta
u_{\|}^{p}|\rangle\propto r^{\zeta_{p}}$ differs from the dimensional analysis
prediction $\zeta_{p}=p/3$. The present model thus provides a unique example
that intermittency is controlled and can thus shed light and provide test beds
for multi-fractal models of turbulence.

††preprint: APS/123-QED

##  I  Introduction

Turbulence is pervasive in natural and industrial flows. In his first
statistical description of turbulence Kolmogorov [1] argued that energy in
turbulent flows cascades to smaller and smaller scales in such a way that
there is a constant flux of energy from the large scales where energy is
injected to the small viscous scales where energy is dissipated. Assuming
further that this process is self-similar lead to the prediction that the
different moments of velocity differences

$S_{p}(r)\equiv\left\langle\left|\frac{\bf r}{r}\cdot({\bf u}({\bf x}+{\bf
r})-{\bf u}({\bf x}))\right|^{p}\right\rangle$ (1)

separated by a distance $r=|{\bf r}|$ scale like $S_{p}\propto r^{p/3}$ with
the case $p=3$ being an exact result (without the absolute value in eq.1).
There is a mass of evidence however from the past years that this result is
not exact; self-similarity is broken and the powers of velocity differences
scale with different exponents $S_{p}(r)\propto r^{\zeta_{p}}$ where
$\zeta_{p}\neq p/3$. This breaking of self-similarity is referred to as
intermittency. It appears because as the cascade develops towards smaller
scales, energy is concentrated in a set that occupies a smaller and smaller
fraction of the domain volume so that finally energy dissipation is
concentrated in a fractal set of dimension smaller than three [2, 3]. Modern
theory of turbulence attempts to understand quantitatively the origin of
intermittency and predict these exponents.

In two dimensions on the other hand, due to the presence of a second
invariant, enstrophy, the energy cascades in an inverse way from small to
large scales [4]. This behavior was first predicted by
Kraichnan–Leith–Batchelor (KLB) theory [5, 6, 7]. What was equally interesting
was that the inverse cascade of energy in two dimensions is in fact self-
similar so that all moments of velocity differences scale with $r$ with
exponents $\zeta_{p}=p/3$ [8]. This is explained by the fact that larger
eddies extract energy from an ensemble of smaller eddies averaging out this
way any extreme events. As energy moves up in scale it is not restricted in a
set of dimension other than two. This behavior however does not always have to
be the case as we argue in this work. If the energy injection in two
dimensional turbulence is not space filling but is restricted in a set of
dimension $D$ smaller than two then as energy moves up in scale it can occupy
larger and larger area fraction so that only at the largest scale it is
concentrated in a two-dimensional set.

Fractal forcing is not just an abstract construction. In engineering it has
been employed extensively in three dimensional turbulence with the use of
fractal grids in simulations and experiments in order to enhance turbulence
[9, 10, 11, 12, 13] but not as far as we know in two dimensions. In nature,
atmospheric and oceanic flows are close to two-dimensional. When driven by
winds over rough topography [14, 15] resemble two-dimensional turbulence
driven by a fractal forcing. Furthermore, quasi-two-dimensional flows are
believed to transition to an inverse cascade in a critical manner [3]. In such
flows the energy injected in the two-dimensional manifold appears in a set of
smaller dimension occupying a fraction of the domain area that approaches zero
as criticality is approached [16, 17].

In this work we show using an extensive set of numerical simulations that
indeed intermittency can appear in the inverse cascade of energy when the
energy-injection mechanism is restricted to a set of fractal dimension $D<2$.
This model does not only give new insights in two-dimensional turbulence but
also provides a unique example that intermittency is controlled and can thus
provide test beds for multi-fractal models of turbulence.

##  II Numerical simulations

We begin by considering the incompressible flow in a double periodic square
domain of side $2\pi L$. In terms of the vorticity $\omega$ the two
dimensional Navier-Stokes equation can be written as

$\partial_{t}\omega+{\bf
u}\cdot\nabla\omega=\nu\nabla^{2}\omega+\alpha\nabla^{-2}\omega+f_{\omega}$
(2)

where the velocity ${\bf u}$ is linked to $\omega$ by $\omega=\nabla\times{\bf
u}$, $\nu$ is the viscosity and $\alpha$ is a hypo-viscosity used to absorbs
energy arriving at the largest scales at a rate
$\epsilon_{\alpha}=\langle|\nabla^{-1}{\bf u}|^{2}\rangle$. The curl of the
forcing is given by $f_{\omega}$ that injects energy at a rate $\epsilon$ at a
lengthscale $\ell_{f}$. Given $f_{w}$ there are three independent non-
dimensional control numbers: the Reynolds number
$Re=\epsilon^{1/3}\ell_{f}^{4/3}/\nu$, the hypo-viscous Reynolds number
$Re_{\alpha}=\epsilon^{1/3}\ell_{f}^{-8/3}/\alpha$ and the domain to forcing
scale ratio $\lambda=L/\ell_{f}$. This system of equations was solved
numerically using the pseudo-spectral code ghost [18] with 2/3 de-aliasing and
second order Runge-Kutta method for the time advancement. Since we are
interested in the inverse cascade the Reynolds numbers was kept fixed to small
value $Re=10$. This value of $Re$ is sufficiently large to allow for the
development of the inverse cascade but suppresses the forward enstrophy
cascade and any intermittency related to it. As a result the smallest scales
in the system are given by $\ell_{f}$ and energy in these scales is
concentrated close to the forcing. The hypo-viscous Reynolds number
$Re_{\alpha}$ was set to $Re_{\alpha}=10\lambda^{8/3}$ so that the large scale
dissipation lengthscale $\ell_{\alpha}$ remains fixed and close to the domain
size $\ell_{\alpha}\simeq L$. Five different resolutions $N$ were used varying
$\lambda$ as given in the table 1.

$N$ | 512 |  | 1024 |  | 2048 |  | 4096
---|---|---|---|---|---|---|---
$\lambda$ | 16 |  | 32 |  | 64 |  | 128
$Re_{\alpha}$ | $2.2\cdot 10^{4}$ |  | $1.4\cdot 10^{5}$ |  | $8.9\cdot 10^{5}$ |  | $5.6\cdot 10^{6}$
Table 1:  Resolution $N$, scale separation $\lambda=L/\ell_{f}$ and hypo-
viscous Reynolds number $Re_{\alpha}$. Figure 1:  Demonstration of how a
fractal set of dimension 1/2 is formed. At every step (down) the initial set
is split in 4 equal sub-sets out of which subset 1 and 3 are disregarded.

Figure 2:  Top panels: Random forcing function for different dimension $D$ at
resolution $N=512$. Lower panels: Vorticity of the flow for the same cases as
above at steady state.

Finally, five different forcing functions of different fractal dimension $D$
are considered. The first one corresponding to $D=2$ is the classical random
forcing where all Fourier modes of wave-vectors ${\bf k}$ satisfying $|{\bf
k}|\simeq 1/\ell_{f}$ are forced with random phases. The $D=1$ forcing
corresponds to four vertical and four horizontal vortex lines with Gaussian
profiles of width $\ell_{f}$ randomly placed in the domain. Similarly $D=0$
corresponds to eight point-vortexes with Gaussian profile of width $\ell_{f}$
randomly placed in the domain. The $D=3/2$ and $D=1/2$ correspond to Cantor
sets that are constructed as follows. For $D=3/2$ a dense set of horizontal
and vertical vortex-lines are uniformly placed in the domain. This set is
split in four equal subsets from which subset one and three are removed. The
remaining sets are then split again in four from which again subset one and
three are removed and so on, as demonstrated in figure 1, until no further
splitting can be done. The resulting box-counting dimension is $D=3/2$ [19].
For $D=1/2$ we start with point-vortexes placed along one vertical and one
horizontal line and we follow the same procedure leading this time to a box-
counting dimension $D=1/2$. For all forcing functions the the forcing
lengthscale $\ell_{f}$ was fixed so that the peek of the forcing spectrum was
around similar wavenumber $k_{f}\simeq 1/\ell_{f}$. Furthermore, in all cases
the amplitude of the forcing function was varied randomly delta-correlated in
time fixing thus the energy injection rate $\epsilon$. A color plot of the
forcing functions for the five forcing functions is shown in the upper panels
of figures 2. For this figure we used the smallest $\lambda$ (largest
$\ell_{f}$) so that the point-vortexes in the left panel are clearly visible.

Figure 3:  Top: Energy spectra for the highest resolution runs. The dashed
lines give the forcing spectrum for the two extreme cases $D=2$ and $D=0$. The
straight dashed line gives the $k^{-5/3}$ scaling. Bottom: Energy flux for the
same cases.

##  III Results

All forcing functions lead to an inverse cascade of energy marked by an energy
spectrum close to $k^{-5/3}$ (top panel of figure 3) and a negative flux of
energy $\Pi(k)=\langle{\bf u}^{<}_{k}\cdot({\bf u}\cdot\nabla{\bf u})\rangle$
(where ${\bf u}^{<}_{k}$ indicates the field ${\bf u}$ filtered so that only
wavenumbers with $|{\bf k}|\leq k$ are kept) shown in the lower panel of
figure 3. The flux is approximately constant for the $D\geq 1$ cases but due
to the relative small $Re$ and the fact that the peek of the forcing spectrum
for the cases $D=3/2$ and $D=2$ is slightly higher, the flux is affected by
finite $Re$ effects displaying a decrease close to the forcing scales.

Figure 4:  The probability distribution function of $\delta u(r)$ normalized
by its variance $\sigma_{u}=S_{2}(r)^{1/2}$ for different values of $r$ and
for $D=2$ left and $D=0$ right. The curves have been shifted vertically for
clarity.  Figure 5: The top panel shows $K(r)$ for the flows at highest
$\lambda$ for the five different forcing functions. The lower panel shows
$K(r)$ for $D=0$ and four different values of $\lambda$.

Although all the five cases have an inverse cascade they do not have the same
turbulent statistical behavior. This can already be seen in the vorticity
plots shown in the lower panels of figure 2. Turbulence, marked by intense
vorticity regions, is uniformly spread in the domain for the $D=2$ case but as
$D$ is decreased intense vorticity regions occupy a smaller area fraction but
with larger intensity. In order to quantify this observation we plot in figure
4 the probability distribution function (p.d.f.) of velocity differences for
the two extreme cases $D=2$ and $D=0$ for different values of $r$ starting
from the largest $r=L/4$ to the forcing scale $r=L/128=\ell_{f}$. In the $D=2$
case the p.d.f.s are close to Gaussian for all examined $r$. Furthermore, no
significant change is observed as $r$ is varied: ie they are self-similar. In
the $D=0$ case on the other hand the p.d.f.s deviate from the Gaussian
distribution having large tails. Most importantly, as smaller values of $r$
are considered the deviations from Gaussianity become stronger with the
distribution becoming more peeked and with stronger tails. In other words
self-similarity is lost for the $D=0$ case.

A quantitative way to measure this lack of self-similarity, is to measure the
kurtosis $K(r)=S_{4}(r)/S_{2}^{2}(r)$. Kurtosis gives a measure of how heavy-
tailed is the distribution of $\delta u$. $K(r)=3$ corresponds to a Gaussian
distributed field while larger values correspond to fields of wider
distribution. If the distribution is self-similar $K(r)$ will be independent
of $r$. In figure 5 we plot the Kurtosis for different cases. The top panel
shows $K(r)$ for the flows at highest $\lambda=128$ (highest resolution) for
the five different forcing functions. For $D=2$, $K(r)$ is almost flat and
close to $3$, indicating that $\delta u$ follows a self-similar, nearly-
Gaussian distribution. As the dimension of the forcing is decreased $K(r)$
takes larger and larger values in the small $r$ range, with the $D=0$ case
having two orders of magnitude larger $K(\ell_{f})$ than a Gaussian field. The
lower panel shows the case $D=0$ for the different values of $\lambda$. As
$\lambda$ and $Re_{\alpha}$ are increased the non-self-similar behavior
extends to a larger range of $r$ with the deviation from Gaussianity
increasing. Therefore this amounts to a phenomenon that persists and extends
as the infinite $Re_{\alpha}$ and infinite box-size limit is reached.

Figure 6:  Scaling exponents $\zeta_{p}$ for the five different $D$ considered
calculated using extended-self-similarity.

The result of figure 5 already indicates that $\zeta_{2}$ and $\zeta_{4}$ can
not follow the scaling $\zeta_{p}=p/3$ that would imply an $r$ independent
Kurtosis. In figure 6 we plot the exponents $\zeta_{p}$ for all cases. The
exponents were measured using the extended self-similarity assumption [20] to
extend the range that a power-law behavior is observed where $S_{p}(r)$ is
plotted as a function of $S_{3}(r)$ and assuming the theoretically predicted
linear scaling of the third moment $S_{3}(r)\propto r$. The exponents up to
$p=6$ were calculated except for the $D=0$ case that showed a particular slow
convergence for the high $p$ moments. For $D=2$ the exponents follow the
linear scaling $\zeta_{p}=p/3$. As $D$ is decreased, the exponents with $p<3$
increase while exponents with $p>3$ decrease. The flow thus becomes more
intermittent as $D$ is decreased with the high order moments being dominated
by few but strong events in the tail of the distribution.

##  IV Conclusions

In this work we have shown, that the inverse cascade of energy can display
intermittent features provided that the forcing function injects energy in
fractal a set of dimension smaller than two. Intermittency is demonstrated by
long tails in the distribution of the velocity differences at small scales
caused by the forcing, that as larger scales are approached they flatten out
becoming closer to Gaussian. This behavior was shown to persist as larger
domains (larger $\lambda$) are considered and become stronger as the dimension
$D$ is decreased.

Here we employed only mono-fractal forcing. Bi-fractal, or multi-fractal
forcing can also be considered by adding with appropriate weight different
fractal forcing functions. This could lead to an intermittent behavior that is
closer to the three dimensional cascade, (albeit in the opposite direction).
More generalized fractal forcing functions that dynamically evolve could also
be considered. Such functions could be examined in future work.

Most importantly the strength of intermittency caused by the fractal forcing
is determined by its dimension and thus it provides a way to create a cascade
for which the intermittency can be varied from self-similar to strongly
intermittent. Ideas, about the behavior and origin of intermittency can thus
be put in the test using this simple model. Finally one can ask how a fractal
forcing can affect intermittency properties in three dimensional turbulence.

###### Acknowledgements.

This work was granted access to the HPC resources of MesoPSL financed by the
Région Île-de-France and the project EquipMeso (project no.
ANR-10-EQPX-29-01), of the HPC resources of GENCI-TGCC & GENCI-IDRIS (project
no. A0110506421) and the HPC facility ARIS from the Greek Research and
Technology Network (GRNET). AA is supported by the project Dysturb (project
no. ANR-17-CE30-0004) finnanced by the Agence Nationale pour la Recherche
(ANR).

## References

  * Kolmogorov [1941] A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers, Cr Acad. Sci. URSS 30, 301 (1941).
  * Frisch [1995] U. Frisch,  _Turbulence: the legacy of AN Kolmogorov_ (Cambridge university press, 1995).
  * Alexakis and Biferale [2018] A. Alexakis and L. Biferale, Cascades and transitions in turbulent flows, Physics Reports 767, 1 (2018).
  * Boffetta and Ecke [2012] G. Boffetta and R. E. Ecke, Two-dimensional turbulence, Annual review of fluid mechanics 44, 427 (2012).
  * Batchelor [1969] G. K. Batchelor, Computation of the energy spectrum in homogeneous two-dimensional turbulence, The Physics of Fluids 12, II (1969).
  * Kraichnan [1967] R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, The Physics of Fluids 10, 1417 (1967).
  * Leith [1968] C. E. Leith, Diffusion approximation for two-dimensional turbulence, The Physics of Fluids 11, 671 (1968).
  * Boffetta  _et al._ [2000] G. Boffetta, A. Celani, and M. Vergassola, Inverse energy cascade in two-dimensional turbulence: Deviations from gaussian behavior, Physical Review E 61, R29 (2000).
  * Hurst and Vassilicos [2007] D. Hurst and J. Vassilicos, Scalings and decay of fractal-generated turbulence, Physics of Fluids 19, 035103 (2007).
  * Nagata  _et al._ [2013] K. Nagata, Y. Sakai, T. Inaba, H. Suzuki, O. Terashima, and H. Suzuki, Turbulence structure and turbulence kinetic energy transport in multiscale/fractal-generated turbulence, Physics of Fluids 25, 065102 (2013).
  * Seoud and Vassilicos [2007] R. Seoud and J. Vassilicos, Dissipation and decay of fractal-generated turbulence, Physics of fluids 19, 105108 (2007).
  * Laizet and Vassilicos [2011] S. Laizet and J. C. Vassilicos, Dns of fractal-generated turbulence, Flow, turbulence and combustion 87, 673 (2011).
  * Mazzi and Vassilicos [2004] B. Mazzi and J. C. Vassilicos, Fractal-generated turbulence, Journal of Fluid Mechanics 502, 65 (2004).
  * Bretherton and Haidvogel [1976] F. P. Bretherton and D. B. Haidvogel, Two-dimensional turbulence above topography, Journal of Fluid Mechanics 78, 129 (1976).
  * Vallis and Maltrud [1993] G. K. Vallis and M. E. Maltrud, Generation of mean flows and jets on a beta plane and over topography, Journal of physical oceanography 23, 1346 (1993).
  * Benavides and Alexakis [2017] S. J. Benavides and A. Alexakis, Critical transitions in thin layer turbulence, Journal of Fluid Mechanics 822, 364 (2017).
  * Seshasayanan  _et al._ [2014] K. Seshasayanan, S. J. Benavides, and A. Alexakis, On the edge of an inverse cascade, Physical Review E 90, 051003 (2014).
  * Mininni  _et al._ [2011] P. D. Mininni, D. Rosenberg, R. Reddy, and A. Pouquet, A hybrid mpi–openmp scheme for scalable parallel pseudospectral computations for fluid turbulence, Parallel computing 37, 316 (2011).
  * Triebel [2010] H. Triebel,  _Fractals and spectra: related to Fourier analysis and function spaces_ (Springer Science & Business Media, 2010).
  * Benzi  _et al._ [1993] R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, Extended self-similarity in turbulent flows, Physical review E 48, R29 (1993).