# Energy spectra and fluxes of turbulent rotating Bose-Einstein condensates in
two dimension

Anirudh Sivakumar  Department of Physics, Bharathidasan University,
Tiruchirappalli 620 024, Tamil Nadu, India  Pankaj Kumar Mishra  Department of
Physics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam,
India  Ahmad A. Hujeirat  Interdisciplinary Center for Scientific Computing,
The University of Heidelberg, 69120 Heidelberg, Germany  Paulsamy Muruganandam
Department of Physics, Bharathidasan University, Tiruchirappalli 620 024,
Tamil Nadu, India

###### Abstract

We investigate the scaling of energy cascade in a harmonically trapped,
turbulent, rotating Bose-Einstein condensate in two dimensions. We achieve
turbulence by injecting a localized perturbation into the condensate and
gradually increasing its rotation frequency from an initial value to a
maximum. The main characteristics of the turbulent state depend on the initial
conditions, rotation frequency, and ramp-up time. We analyze the energy and
the fluxes of kinetic energy by considering initial profiles without vortices
and with vortex lattices. In the former case, we find the presence of
Kolmogorov-like scaling ($k^{-5/3}$) of the incompressible kinetic energy in
the inertial range, with a forward cascade of energy at smaller scales for a
high rotation rate. However, in the latter case, the energy spectrum follows
Vinen scaling ($k^{-1}$) at transient iterations and for cases of high
rotating frequencies shows Kolmogorov-like scaling at longer durations, with
positive kinetic energy fluxes implying a forward cascade of the energy.

##  I Introduction

Superfluid turbulence, also known as quantum turbulence (QT), has been a topic
of interest within the scientific community since the time of Feynman [1].
Feynman proposed that one could visualize QT as a chaotic tangle of quantized
vortex filaments. Building upon this, Vinen made an experimental observation
that superfluid turbulence, in general, can be sustained by the mutual
friction between quantized vortices and the normal fluid component of the
superfluid, driven by a constant temperature gradient in the horizontal
direction [2].

Vortices play a central role in understanding fluid turbulence, and a
comparative study between classical turbulence (CT) and QT could provide a
better understanding of the nature of vortices formed in these systems.
Specifically, in the case of 3D turbulence, both CT and QT exhibit similar
macroscopic and statistical properties. They both display a similar nature of
the energy cascades in the large length scale, particularly in the inertial
range, as confirmed numerically in earlier studies [3, 4]. However,
understanding the genesis of turbulence and further energy cascade from the
first principle in CT is relatively rigorous due to the continuous and chaotic
nature of the vortices. In contrast, vortices observed in QT possess vortex
cores with definite sizes and circulations, making them an ideal prototype for
investigating turbulence dynamics [5, 6, 7, 8].

Despite experimental studies on QT through superfluid helium, which show
significant promise in understanding the phenomena [9], manipulating and
studying these quantized vortices in such superfluids remains a challenging
experimental task. In the last few decades, Bose-Einstein condensates (BECs),
superfluids existing below a critical temperature and free of viscous effects,
have emerged as a novel platform for exploring the role of quantized vortices
in generating the QT. BECs offer advantages, such as compressibility, weak
atomic interactions, the ability to fine-tune atomic parameters, and the
availability of new experimental methods for probing and studying superfluid
flow [10].

Several studies in the past reveal chaotic dynamics of quantum vortices aided
by the complex nature of the vortex reconnection responsible for the
turbulence in QT [11, 12]. For instance, Aref investigated the influence of
vortex number on the transition of a quantum fluid from chaotic to turbulent
flow [13]. Cornell and co-workers have pioneered developing the cooling
technique that accelerates the rotation of an ultracold 87Rb gas and nucleates
vorticity in a Bose-Einstein condensate [14]. People have also achieved the
same effect by mechanically rotating an anharmonically confined condensate
[15, 16, 17]. However, Parker and Adams were the first to numerically
demonstrate the occurrence of turbulence, with direct energy scaling, upon
attainment of the crystallization of a vortex lattice [4]. Using the laser
beam stirring technique, Neely et al. demonstrated the first experimental
confirmation of quantum turbulence in BECs [18] through the observation of the
formation and evolution of a disordered distribution of vortices in highly
oblate annular BECs. Other methods for generating quantum turbulence in atomic
BECs include introducing a stirring potential in the condensate and evolving
the turbulent state into a vortex lattice through vortex-sound interactions
[4], combined-axis rotation [19], two counter-rotating disks [20], and phase-
imprinting a disordered vortex lattice onto a BEC [21].

A feature of 2D turbulence is the presence of energy cascades, in which
kinetic energy flows from larger to smaller length scales (direct energy
cascade) or from smaller to larger length scales (inverse energy cascade) in
an inertial range. The corresponding energy spectrum follows Kolmogorov
scaling, i.e. $\varepsilon_{\mathrm{kin}}^{i}(k)\sim k^{-5/3}$. In QT, both
inverse and direct cascades of the energy spectrum have been reported
depending on the initial configuration. This scaling has been observed in 3D
QT numerically by Kobayashi et al. [22, 23] and has also been observed in 3D
CT. Further similarities between QT and CT have been confirmed numerically by
solving the Gross-Pitaevskii equation [3, 4].

In a turbulent BEC, the energy cascade very much depends on the means through
which turbulence is being generated in the system. As an example, the
turbulent BECs in 2D generated using the rotating paddles show the forward
cascade of the energy with Kolmogorov-like scaling
($\varepsilon_{\mathrm{kin}}\sim k^{-5/3}$) [24]. Numasato et al. demonstrated
Kolmogorov-like scaling for BECs undergoing decaying homogeneous 2D quantum
turbulence generated by random phase initial conditions [25]. Their study also
revealed how vortices and sound waves introduced into the BECs can
significantly influence the observed spectra. However, the turbulence
generated through the stationary grid with obstacles in the BEC exhibits the
inverse cascade of the energy with Kolmogorov-like scaling [26]. Similar
features have been observed with decaying and unforced turbulence [27]. There
are some works that show the presence of Vinen-like scalings for the kinetic
energy spectra ($\varepsilon_{\mathrm{kin}}^{i}\sim k^{-1}$) for the fast
rotating BECs [28, 29]. Similar scalings have been reported for decaying
turbulence in which the cluster of vortices breaks into multiple ones [30,
31].

Previous studies on QT have primarily employed unstable initial condensates,
which tended to produce self-sustaining or decaying turbulence phenomena
during real-time evolution. Furthermore, earlier investigations into 2D and 3D
QT have typically focused on Gross-Pitaevskii (GP) equations without a
rotational term, with rotational forcing achieved through anharmonic and time-
varying potentials. For instance, Estrada and colleagues recently carried out
a study on 3D condensates in the rotational frame [28]. Given the substantial
disparities between dimensional regimes for QT in non-rotating GPEs, our
objective is to explore these differences by simulating rotational QT in 2D.

In our present work, we consider a stable initial condensate prepared via
imaginary-time iterations. We then generate the turbulence using a time-
dependent rotational frequency in a perturbed central barrier. We choose two
initial states, viz., the vortex-free and the vortex lattice. In the first
case, we identify that the incompressible kinetic energy exhibits Kolmogorov-
like scaling with a forward energy cascade. The Kolmogorov-like scaling
becomes more pronounced at higher rotational frequency. However, in the latter
case, we find the Vinen-like scaling for the energy spectrum with a forward
cascade at a lower rotational frequency. For both cases, the negative particle
number flux suggests the transfer of the particle from small to large length
scale, which becomes more significant at high rotational frequency.

The organization of this paper is as follows: In Sec. II, we present the
mathematical model for simulating rotating BECs with the details of the
perturbed central barrier used to generate turbulence, which also includes the
protocols used to introduce the angular frequency and detailed forms of the
relevant energy, such as incompressible and compressible energy spectra. We
use these spectra to characterize the energy cascade in the later section.
Sec. III deals with numerical simulation results that we obtained by solving
the mathematical model mentioned in Sec. II. In Sec. III.1, we discuss the
energy spectra and fluxes of the turbulent state, which we attain with an
initial state having no vortex present in the fluid, while in Sec. III.2, we
present the results for the situation when the vortices pinned on a lattice
are used as an initial condition to get the turbulent state. Finally, we
conclude our investigation in Sec. IV.

##  II Description of the model and energy spectra

We consider a quasi-two-dimensional condensate confined strongly in a
transverse direction rotating with an angular frequency $\Omega$. The
dynamical equation of the condensate in the non-dimensional form is given by

$\displaystyle\mathrm{i}\frac{\partial\psi}{\partial t}=$
$\displaystyle\left[-\frac{1}{2}\nabla^{2}+V(\mathbf{r})+g_{2D}\lvert\psi\rvert^{2}-\Omega(t)L_{z}\right]\psi,$
(1)

where $\psi\equiv\psi(\mathbf{r},t)$ denotes the condensate wave function,
with $\mathbf{r}\equiv(x,y)$, $\nabla^{2}$ is the two-dimensional Laplace
operator defined as $\nabla^{2}\equiv\partial_{x}^{2}+\partial_{y}^{2}$, and
$V(\mathbf{r})$ is the external potential, which includes the harmonic trap
along with the central circular barrier. The nonlinear term $g_{2D}=4\pi
aN/\sqrt{2\pi}d_{z}$ represents the interaction strength between the atoms,
where $N$ is the total number of atoms, $a$ is the s-wave scattering length,
and $d_{z}$ corresponds to the axial width of the trap. Here, $\Omega(t)$
represents the rotational frequency, and
$L_{z}=\mathrm{i}\hbar(y\partial_{x}-x\partial_{y})$ is the $z$ component of
angular momentum.

In Eq. (1), the unit length is measured in terms of the harmonic oscillator
length $l=\sqrt{\hbar/(m\omega)}$, time is measured in units of $\omega^{-1}$,
and trap frequency is measured in terms of $\omega$. Here, we consider two
types of initial configurations: (i) vortex-less and (ii) vortex lattice. For
the former case while we consider $N=1\times 10^{4}$ and $g_{2D}=100$, for the
later $N=2\times 10^{4}$ and $g_{2D}=200$ have been taken. With the choice of
the above parameters, we obtain the scattering length $a\approx 3.8a_{0}$ for
both cases. The BECs should have a strong axial confinement along $z$ axis so
as to restrict the dynamics within the $x-y$ plane by implementing trap
frequencies $\omega_{x}\sim 2\pi\times 33\mathrm{Hz}$, $\omega_{y}\sim
2\pi\times 33\mathrm{Hz}$, $\omega_{z}\sim 2\pi\times 1.5\mathrm{MHz}$. This
configuration provides a trap frequency of $\omega\sim 2\pi\times 116.4Hz$,
harmonic oscillator length $l\approx 1\mu$m and unit time $\omega^{-1}\approx
1.3$ms. The oscillator length along the $z$ direction would be $l_{z}\approx
0.28\mu$m. Experimentally, for a condensate of ${}^{87}\mathrm{Rb}$ atoms, the
desired scattering length can be accessible by tuning the magnetic fields
utilizing the Feshbach resonance [32, 33].

To induce turbulence in the rotating BEC, first, we prepare the initial state
of the condensate by considering a disordered configuration of the vortex
lattice, as implemented by Müller et al. [27]. Then, we evolve the condensate
using a gradually increasing rotational frequency via a perturbed barrier. The
external potential (in dimensionless form) is given by

$\displaystyle
V(\mathbf{r})=\frac{1}{2}\left(\gamma^{2}x^{2}+\nu^{2}y^{2}\right)+V_{\mathrm{B}}(\mathbf{r}),$
(2)

where $\gamma=1$ and $\nu=1$ are the aspect ratios of the harmonic trap along
the radial direction. The perturbed barrier ($V_{B}$) is given by

$\displaystyle
V_{\mathrm{B}}(\mathbf{r})=\begin{cases}V_{0},&r<R+A\left[\sin\left(\alpha\Theta\right)+\sin\left(\beta\Theta+\delta\right)\right]\\\
0,&\mathrm{otherwise},\end{cases}$ (3)

with $A$, $R$, $V_{0}$ are the perturbation amplitude, barrier radius, and
barrier height, respectively, $r=\lvert\mathbf{r}\rvert=\sqrt{x^{2}+y^{2}}$
and $\Theta=\arctan(y/x)$. For our simulations, we set the amplitude to be
$A=1.25\,l$, the barrier radius as $R=0.5\,l$, and we choose the barrier
height as $V_{0}=20\hbar\omega$ and $\alpha=10$. We consider two
configurations of perturbations: (i) periodic perturbation for which
$\beta=20$ and $\delta=0.4$ and (ii) quasiperiodic perturbation for which
$\beta=5\times\left(1+\sqrt{5}\right)$ and $\delta=0.4$ are chosen.

Figure 1: Plots showing central portion of the harmonic trap superimposed with
the perturbed barrier: (a) periodic perturbation with $\alpha=10$, $\beta=20$
and $\delta=0.4$ (b) quasiperiodic perturbation with $\alpha=10$,
$\beta=5\times\left(1+\sqrt{5}\right)$ and $\delta=0.4$.

In Figs. 1(a) and (b) we depict the central portion of the harmonic trap (2)
superimposed with the perturbed barrier (3) for periodic and quasiperiodic
perturbations, respectively.

We carry out the simulation by gradually increasing the rotation frequency
from the initial value to the final one. Further, we have used the following
form of the temporal rotational frequency $\Omega(t)$:

$\displaystyle\Omega(t)=\begin{cases}\displaystyle\Omega_{0},+\left(\Omega_{\mathrm{f}}-\Omega_{0}\right)\sin^{2}\left(\frac{\pi
t}{2T_{r}}\right),&\mathrm{if}\;0\leq t\leq T_{r},\\\
\Omega_{\mathrm{f}},&\mathrm{if}\;t>T_{r},\\\ \end{cases}$ (4)

where $\Omega_{0}$ and $\Omega_{\mathrm{f}}$ represent the initial and final
angular frequencies, respectively, and $T_{r}$ represents the time interval
during which the rotation frequency is smoothly increased from an initial
value of $\Omega_{0}$ to a final value of $\Omega_{f}$. In the simulation we
have fixed $T_{r}=120$ and considered the different values of $\Omega_{f}$.

###  II.1 Spectra and Flux Calculations

The flow of incompressible kinetic energy across wavenumbers and its
relationship to vortex dynamics are central and crucial to understanding the
nature of the QT at different lengths and time scales. To obtain the scaling
laws for energy spectra, we decompose the condensate kinetic energy into
compressible and incompressible parts and analyze the distribution of kinetic
energy due to vortex lines and sound over length scales, as shown by Nore et
al. [3].

To obtain these spectra, we perform a Madelung transformation on the
wavefunction and obtain a density weighted velocity field as

$\displaystyle\mathbf{u}(\mathbf{r})=\sqrt{n}(\mathbf{r})\mathbf{v}(\mathbf{r}),$
(5)

where $n(\mathbf{r})=\left|\psi(\mathbf{r})\right|^{2}$ is the particle
density and $\mathbf{v}(\mathbf{r})=\hbar/m\nabla\theta(\mathbf{r})$ is the
superfluid velocity and $\theta(\mathbf{r})$ is the corresponding phase.

We decompose this weighted velocity field into the hydrodynamic and quantum
pressure components. This definition of the density-weighted velocity field
allows for a Helmholtz decomposition into the hydrodynamics components,
$\mathbf{u}(\mathbf{r})=\mathbf{u}(\mathbf{r})^{i}+\mathbf{u}(\mathbf{r})^{c}$.
The incompressible and compressible components satisfy

$\displaystyle\nabla\cdot\mathbf{u}^{i}(\mathbf{r})$ $\displaystyle=0,$ (6a)
$\displaystyle\nabla\times\mathbf{u}^{c}(\mathbf{r})$ $\displaystyle=0,$ (6b)

respectively. This decomposition is performed by applying a Fourier
transformation to the velocity fields. Apart from these hydrodynamic
components, the other relevant quantity quantum pressure velocity field is
defined as

$\displaystyle\mathbf{u}^{q}(\mathbf{r})=\frac{\hbar}{m}\nabla\sqrt{n(\mathbf{r})}.$
(7)

Quantum pressure becomes significant when the condensate density varies
sharply, such as near the vortex cores. For 2D condensates, it provides a more
useful quantitative measure for the vortex number. The quantum pressure
velocity field does not contribute to the physical velocity field but does
possess the dimensions of velocity and satisfies the condition
$\nabla\times\mathbf{u}^{q}=0$. Next, we express the total kinetic energy as

$\displaystyle
E_{\mathrm{kin}}=E_{\mathrm{kin}}^{i}+E_{\mathrm{kin}}^{c}+E_{\mathrm{kin}}^{q},$
(8)

where $E_{\mathrm{kin}}^{i}$, $E_{\mathrm{kin}}^{c}$ and
$E_{\mathrm{kin}}^{q}$ represent, respectively, incompressible, compressible,
and quantum pressure contributions of the kinetic energy. Following Parseval’s
theorem, different energy components can be represented as

$\displaystyle E^{\alpha}_{\mathrm{kin}}=\frac{m}{2}\int
d^{2}\mathbf{k}\lvert\tilde{\mathbf{u}}^{\alpha}(\mathbf{k})\rvert^{2},$ (9)

where $\alpha\in\\{i,c,q\\}$ and

$\displaystyle\tilde{\mathbf{u}}^{\alpha}(\mathbf{k})=\frac{1}{2\pi}\int
d^{2}\mathbf{r}e^{-i\mathbf{k}\cdot\mathbf{r}}\mathbf{u}^{\alpha}(\mathbf{r})$
(10)

Transforming Eq. (9) into the cylindrical coordinates in $k$ space for a 2D
condensate, we obtain

$\displaystyle
E_{\mathrm{kin}}^{\alpha}=\int_{0}^{\infty}dk\varepsilon_{\mathrm{kin}}^{\alpha}(k).$
(11)

Usually, the spectra $\varepsilon_{\mathrm{kin}}^{\alpha}$ are computed by
binning the data in $k$ space and further summing over an angular interval, an
approximation that holds good in small-$k$ regimes. We use the analytical
evaluation of $k$-space integrals and its numerical implementation developed
by Bradley and colleagues [34]. The method involves an angle-averaged Wiener-
Khinchin theorem relating the spectral densities to an associated correlation
function. So the spectra $\varepsilon_{\mathrm{kin}}^{\alpha}$ from Eq. (11)
can be written as [34]

$\displaystyle\varepsilon_{\mathrm{kin}}^{\alpha}(k)=\frac{m}{2}\int
d^{2}\mathbf{x}\Lambda_{2}(k,\lvert\mathbf{x}\rvert)C[\mathbf{u}^{\alpha},\mathbf{u}^{\alpha}](\mathbf{x}),$
(12)

where
$\Lambda_{2}(k,\lvert\mathbf{x}\rvert)=(1/2\pi)k^{2}\operatorname{sinc}(kr)$
is the 2D kernel function and
$C[\mathbf{u}^{\alpha},\mathbf{u}^{\alpha}](\mathbf{x})$ represents the two-
point auto-correlation function for a given velocity field. The above relation
implies that for any of the position-space fields $\mathbf{u}^{\alpha}$, there
exists a spectral density [see Eq. (12)], i.e., equivalent to an angle-
averaged two-point correlation in $k$ space.

Next, we consider the relevant quantities to characterize the energy cascade,
such as the kinetic energy and particle number fluxes of the turbulent state,
which provide a quantitative idea about the flow of the energy and the
particle from one scale to another. One can use these spectral densities to
compute the incompressible kinetic energy and particle fluxes. The
incompressible kinetic energy and density flux equations are given by [35]

$\displaystyle\Phi_{\varepsilon_{\mathrm{kin}}^{i}}(k)$
$\displaystyle=-\frac{d}{dt}\int_{k_{0}}^{k}\varepsilon_{\mathrm{kin}}^{i}(k^{{}^{\prime}})dk^{{}^{\prime}},$
(13a) $\displaystyle\Phi_{{n}}(k)$
$\displaystyle=-\frac{d}{dt}\int_{k_{0}}^{k}n(k^{{}^{\prime}})dk^{{}^{\prime}},$
(13b)

where $n(k)$ and $\varepsilon_{\mathrm{kin}}^{i}(k)$ correspond to the density
and incompressible kinetic energy spectra, respectively, and $k_{0}\sim
2\pi/L$ represents the largest length scale, with $L$ being the length of the
box. The positive nature of the flux indicates the transfer of energy from the
large to small scale, while the negative sign suggests the transfer from the
small to large scale of the turbulent state.

##  III Numerical simulations

We numerically solve the Gross-Pitaevskii (GP) equation (1) using the split-
step Crank-Nicolson method [36, 37, 38, 39, 40, 41]. We use imaginary time
propagation to generate the ground states with and without vortices and real-
time propagation to investigate the condensate dynamics. In all simulations
presented in the paper, we use a two-dimensional grid with $512\times 512$
points, where the space step is $dx=dy=0.05$ and the time step is $dt=0.001$.
This choice of grid and step sizes ensures numerical convergence and the
desired accuracy in our simulations. We have generated the ground state for
both the vortex-free and vortex-lattice cases using imaginary time propagation
and then evolved them further in real time for a finite angular frequency.

In this work, we intend to analyze the energy spectra and fluxes of the
turbulent state of the rotating condensate using various initial setups. In
the following sections, we present a systematic analysis of the effect of
rotational frequency on the turbulence attained through different initial
configurations of the vortices. We emphasize the role of the initial state in
achieving the various turbulent states in QT.

###  III.1 Energy transfer and fluxes for Vortex-Less initial state

We begin our analysis by considering the initial state without vortices and
demonstrate the appearance of quantum turbulence by increasing the rotation
beyond a threshold frequency. To accomplish this, we generate an initial
condensate profile of the rotating BECs using a perturbed barrier by evolving
the condensate with a constant rotating initial frequency of $\Omega_{0}$ in
imaginary time. We select the value of $\Omega_{0}$ based on the critical
rotating frequency, which determines the angular frequency above which
vortices start appearing in the condensate. As we consider $N=1\times 10^{4}$
atoms and nonlinearity strength $g_{2D}=100$, the critical frequency comes as
$\Omega_{c}\approx 0.4$ also considered in the Ref. [41]. Following this, we
have chosen $\Omega_{0}=0.3<\Omega_{c}$ to prepare a vortex-less initial
condensate ground state using the imaginary time propagation. Subsequently, we
employ a real-time evolution approach to delve into the dynamics of the
condensate for various values of the final angular frequency,
$\Omega_{\mathrm{f}}$. Furthermore, we utilize the evolved state to compute
the energy spectrum and fluxes of the turbulent condensate at various
rotational frequencies.

Figure 2: The time evolution of density profiles of the condensate with
vortex-less initial condition undergoing turbulence at
$\Omega_{\mathrm{f}}=1.0$ introduced $t=120$: (a) $t=0$, (b) $t=120$, (c)
$t=300$, (d) $t=600$.

To show the evolution of the vortex-less initial state to turbulent state in
Fig. 2, we depict the snapshots of the condensate density profiles at various
time instants as the final rotational frequency $\Omega_{\mathrm{f}}$ as
achieved at a time span of $t=120$. Figure 2(a) shows a vortex-less initial
condensate and the shape of the central barrier with a significant
perturbation. As the rotation frequency attains a value $\Omega_{\mathrm{f}}$
at $t=120$ vortices begin to appear in the condensate, though they are not yet
arranged in a vortex lattice formation as shown in Fig. 2(b). Subsequently, as
time progresses, the vortices start converging towards the central barrier and
organize into an ordered lattice, as illustrated in Figs. 2(c)-(d). The
density profiles clearly show a slow development of compressible turbulence
accompanied by consistent vortex generation beyond $t=120$.

To understand different transient states of the turbulence, we show the
temporal evolution of the mean angular momentum ($\langle L_{z}\rangle$) for
different final angular frequencies ($\Omega_{f}=0.6$, $0.8$, $1$, and $1.2$)
in Fig. 3. The steady-state value of $\langle L_{z}\rangle$ very much depends
upon the $\Omega_{\mathrm{f}}$; it increases with an increase in
$\Omega_{\mathrm{f}}$, indicating the increase in the number of vortices and
having more turbulent states at higher $\Omega_{f}$. For
$\Omega_{\mathrm{f}}<0.8$, there is no change in the angular momentum with
time, indicating no generation of vortices and thereby providing a lower bound
for $\Omega_{\mathrm{f}}\approx 0.8$ for our simulations below which we do not
have any vortex in the condensate.

Figure 3: Variation of the expectation value of angular momentum with respect
to time for different $\Omega_{\mathrm{f}}$ values, evaluated for a condensate
with a vortex-less initial condition. The condensate undergoes turbulence for
$\Omega_{\mathrm{f}}\geq 0.8$. The insets show condensate density profiles for
$t=550$ (red dotted line), $t=250$ (green dashed line), $t=120$ (black solid
line).

For $0.8\lesssim\Omega_{\mathrm{f}}\lesssim 1.0$, we find that the angular
momentum settles to a finite value after a brief period, and then it shows the
increasing trend. Once $\langle L_{z}\rangle$ saturates the corresponding
real-time density snapshot of the condensate density shows the clustering of
the vortices around the central barrier as shown in the insets for different
angular frequencies ($\Omega_{\mathrm{f}}=0.8,1$ and $1.2$). For final
rotation frequencies greater than the trap frequency
($\Omega_{\mathrm{f}}>1$), the angular momentum does not exhibit a tendency to
reach a steady value. One could attribute this to the continuous generation of
vortices in the condensate even after an extended period of time.

After characterizing the different turbulent states at various angular
frequencies, we then focus on analyzing the nature of the kinetic energy
spectra and their corresponding fluxes at different angular frequencies during
the time interval where the condensate exhibits the presence of vortices
clustered around the central barrier. We compute the temporal averages of the
spectra and fluxes for the different components of kinetic energy from the
instance when the final rotational frequency ($\Omega_{\mathrm{f}}$) is
achieved at $t=T_{r}=120$ until the end of the simulation at $t=600$. The
kinetic energy spectra provide insights into the characteristic structure and
arrangement of vortices and their overall effect in inducing turbulent
fluctuations in the condensate.

Figure 4: Time-averaged spectra of incompressible kinetic energy for the
vortex-less case, in the time range $t=120$ to $t=600$ (a)-(d) For
$\Omega_{\mathrm{f}}=0.85$, $\Omega_{\mathrm{f}}=0.95$,
$\Omega_{\mathrm{f}}=0.99$, $\Omega_{\mathrm{f}}=1.0$ respectively, it
exhibits $k^{-5/3}$ accompanied by $k^{-3}$ scaling at larger $k$ values.
Figure 5: Plots of the kinetic energy fluxes for the vortex-less case at
different instant of time as given in the color bar. Fluxes are computed for
the time range $t=200$ to $t=400$ at final rotation frequencies: (a)
$\Omega_{\mathrm{f}}=0.85$, (b) $\Omega_{\mathrm{f}}=0.9$, (c)
$\Omega_{\mathrm{f}}=0.95$, and (d) $\Omega_{\mathrm{f}}=1.0$.The various
scales are $k=2\pi/R$ (thick gray line), $k=1/\xi$ (thin gray line) and
$k=2\pi/\xi$ (dashed gray line). Thick dashed black line represents the time-
average of the flux values.  Figure 6: Time-averaged spectra of compressible
kinetic energy for the vortex-less case, in the time range $t=120$ to $t=600$
for the condensate rotating at: (a) $\Omega_{\mathrm{f}}=0.85$, (b)
$\Omega_{\mathrm{f}}=0.9$, (c) $\Omega_{\mathrm{f}}=0.99$, (d)
$\Omega_{\mathrm{f}}=1.0$. All cases display $k$ scaling, whose range extends
for higher rotation frequencies, indicating strong thermalization of the
condensate.

Figure. 4 illustrates the scaled incompressible kinetic energy spectra for
different rotational frequencies $\Omega_{\mathrm{f}}=0.85,0.95,0.99$ and
$1.0$ in the wavenumber space. For small frequency $\Omega_{\mathrm{f}}$ the
spectrum at large wave number ($k>1/\xi$) appears to fall with a scaling
$k^{-3}$ a typical behaviour for the energy spectrum due to the presence of
the vortex core [42]. However, for an infrared range (IR) ($2\pi/R<k<1/\xi$),
the energy spectrum appears to follow the Kolmogorov scaling and falls as
$k^{-5/3}$ as shown in Fig. 4(a). Upon increase in the rotation frequency, the
Kolmogorov scaling appears to fit better in the IR, indicating the chaotic
spatial distribution of the vortices [cf. Figs. 4(b)-(d)]. The observation of
Kolmogorov-like scaling for the incompressible kinetic energy is generally
associated with the faster decay of the vortex tangles in 3D superfluid
turbulence and equivalently transfer of energy towards the large scale during
the decay of the vortex tangle [43, 44]. However, establishing a similar kind
of feature for the 2D QT is quite a subtle task. As we look at the energy
spectra for $\Omega_{\mathrm{f}}>1.0$, we find a continuous generation of
vortices prohibits the system from achieving the steady state. Rotating at
such elevated frequencies leads to the disruption of confinement in the
condensate, facilitating its continuous expansion. This expanded state fosters
the creation of an environment conducive to the formation of additional
vortices. Without consistent turbulence dynamics and the continual breakdown
of vortices, this scenario imposes a constraint on the final rotational
frequency that can be achieved. In addition to spectral analysis of the
incompressible kinetic energy, we also examine the incompressible density
profiles of the kinetic energies in real space, as given in the Appendix A. It
suggests the localization of the incompressible kinetic energy at the outer
periphery of the condensate, which becomes more prominent at a higher angular
frequency as shown in Fig. A.1.

Figure 7: Plots of the average density spectra for the time range $t=120$ to
$t=600$ with final rotation frequencies (a) $\Omega_{\mathrm{f}}=0.85$, (b)
$\Omega_{\mathrm{f}}=0.9$, (c) $\Omega_{\mathrm{f}}=0.99$, (d)
$\Omega_{\mathrm{f}}=1.0$. All figures show $k^{-11/3}$ scaling, indicating
onset of Kolmogorov cascade.

In Fig. 5, we show the flux corresponding to the incompressible component of
the kinetic energy calculated using the expression defined in Eqs. (13a).
Figures 5(a)-(d) denote the flux for the rotational frequencies
$\Omega_{\mathrm{f}}=0.85,0.9,0.95$, and $1$, respectively, at different
instances where the pseudo colour bars indicate the time. We find that for all
the frequency ranges, the flux fluctuates between positive and negative with a
breathing frequency of $2.0\omega$. The flux in the IR range
($2\pi/R<k<1/\xi$) the amplitude of the flux does not change with the wave
number for a given instant of time, complementing the nature of cascade due to
the presence of the Kolmogorov scalings in this range. However, as we analyze
the variation of flux averaged over a time interval ($200<t<400$) with wave
number, we find that it appears to remain positive for all the final rotation
frequencies, as shown with the dashed black line. This feature indicates the
forward cascade of the incompressible part of the kinetic energy in the IR as
well as the UV range consistent with the earlier studies related to the energy
cascade in 2D quantum turbulence [25]. The role of rotation here is simply to
increase the rate of energy transfer from one scale to another.

In Fig. 6, we show the time-averaged scaled compressible kinetic energy in the
wave number space for different rotation frequencies
($\Omega_{\mathrm{f}}=0.85$, $0.9$, $0.99$, and $1$). For all the frequencies,
we find that the compressible spectrum behaves as $k$ for small wave numbers
($k<1/\xi$), while at large wave numbers, it exhibits $k^{-7/2}$ scaling. Note
that in general, the compressible kinetic energy attains the thermal
equilibrium scaling, i.e., $\varepsilon_{\mathrm{kin}}^{i}(k)\sim k$ for the
decaying turbulence where all the energy transferred to the compressible part
of the energy [25]. However, in the rotating turbulence case, we find that the
incompressible part of the kinetic energy dominates over the other components,
and thus there is the generation of the vortices. This is the reason that on
the scale below the healing length ($\xi$), we find the presence of the
$\varepsilon_{\mathrm{kin}}^{i}(k)\sim k^{-7/2}$. As we analyze the
distribution of the compressible density of the kinetic energies in real
space, we find that initially localized compressible kinetic energy near the
central barrier evolves and get distributed evenly in the condensates as shown
in the Fig. A.2.

Figure 8: Density flux profiles for the vortex-less case. The plots are taken
are averaged over the time range $t=200$ to $t=400$ at final rotation
frequencies: (a) $\Omega_{\mathrm{f}}=0.85$, (b) $\Omega_{\mathrm{f}}=0.9$,
(c) $\Omega_{\mathrm{f}}=0.99$, and (d) $\Omega_{\mathrm{f}}=1.0$.

In Fig. 7, we show time-averaged density spectra for the different final
rotation frequencies ($\Omega_{\mathrm{f}}=0.85,0.9,0.99$ and $1.0$). As the
incompressible component of the kinetic dominates over the compressible
counterpart, we can have $k^{2}n(k)\sim\varepsilon^{i}_{\mathrm{kin}}(k)$,
which yields $k^{-11/3}$ and $k^{-5}$ scalings for the density in the IR and
UV range. For all the frequencies, we find that the particle density spectra
follow the scalings $k^{-11/3}$ in the IR range and $k^{-5}$ in the UV range
as depicted in Fig. 7.

To understand the detailed nature of the transfer of the particle from one
scale to another, in Fig. 8, we show the particle density flux
($\Phi_{\mathrm{n}}(k)$) for different instants of time where we use color
codes to illustrate the flux variation with wave number at that instant. Like
incompressible kinetic energy flux, density flux also oscillates between
positive and negative with time for a given range of the wavenumber. The
oscillation frequency is the same as that for the kinetic energy flux. The
density flux exhibits an increasing behaviour in the IR range while it attains
a constant value in the UV range. As we analyze the time-averaged density flux
variation in the spectrum, we find that it takes on negative values for all
rotational frequencies, as shown in the average dashed black lines of the
respective plots, indicating an inverse cascade of particles from small to
large scales. We can attribute these particular features to the aided features
of the rotation, which is quite evident from the increasing magnitude of the
time-averaged flux upon an increase in the rotational frequency.

###  III.2 Energy transfer and fluxes for Vortex-lattice initial state

After presenting the detailed nature of the spectrum and fluxes for the
vortex-less initial condition, we consider another type of initial state with
the presence of vortex lattices. This initial condensate has the same barrier
parameters as those considered for the vortex-less case. To introduce vortices
into the system, we first prepare an initial condensate in imaginary time with
an appropriate choice of $\Omega_{0}$ that yields the vortices arranged on the
lattice. An important difference from the vortex-less state here is the atom
number and interaction strengths, which we certainly need to increase for the
present case in comparison to those for the vortex-less case. Here, we
consider $N=2\times 10^{4}$ and $g_{2D}=200$, with a critical frequency of
$\Omega_{c}\approx 0.3$, above which vortices are generated and arranged on
the lattice in the condensate. Here, we have chosen a larger nonlinearity
strength ($g_{2D}$) than those for the vortex-less initial condition to
accommodate the vortex lattice in the condensate. To illustrate the dynamics
with a vortex-lattice initial profile clearly and consistently, we consider
the rotation frequency as $\Omega_{0}=0.6$ for the results presented below.

Figure 9: Snapshots of the condensate density at the different instants of
time: (a) $t=0$, (b) $t=120$, (c) $t=300$, (d) $t=600$, with the vortex-
lattice initial state as condensate starts rotating with
$\Omega_{\mathrm{fin}}=0.85$ at $t=120$. The other parameters are the same as
those in Fig. 2. Figure 10: Variation of the expectation value of angular
momentum with respect to time for different $\Omega_{\mathrm{f}}$ values,
evaluated for a condensate with vortex-lattice initial condition. The
condensate undergoes turbulence for $\Omega_{\mathrm{f}}\geq 0.8$. The insets
show condensate density profiles for $t=550$ (red dotted line), $t=430$ (green
dashed line), and $t=330$ (black solid line). Figure 11: Time-averaged
incompressible kinetic energy spectra for vortex-lattice initial condition:
(a) $t=280$ to $t=380$ at $\Omega_{\mathrm{f}}=0.8$, from $t=120$ to $t=220$
for (b) $\Omega_{\mathrm{f}}=0.825$, (c) $\Omega_{\mathrm{f}}=0.85$, (d)
$\Omega_{\mathrm{f}}=0.9$. All the plots display $k^{-1}$ scaling in an
inertial range, along with an enstrophy cascade $k^{-3}$. Figure 12: Kinetic
energy flux for the vortex-lattice case for (a) $t=360$ to $t=380$ at
$\Omega_{\mathrm{f}}=0.8$, for $t=200$ to $t=220$ at (b)
$\Omega_{\mathrm{f}}=0.825$, (c) $\Omega_{\mathrm{f}}=0.85$, and (d) at
$\Omega_{\mathrm{f}}=0.9$.  Figure 13: Time averaged compressible kinetic
energy spectra: (a) $t=280$ to $t=380$ at $\Omega_{\mathrm{f}}=0.8$, for
$t=120$ to $t=220$ at (b) $\Omega_{\mathrm{f}}=0.825$, (c)
$\Omega_{\mathrm{f}}=0.85$, (d) at $\Omega_{\mathrm{f}}=0.9$. At smaller
frequencies the plots show $k$ scaling at smaller $k$ and for larger
frequencies show $k^{-3/2}$ and $k^{-7/2}$ scaling at larger $k$ values.
Figure 14: Time averaged density spectra for the vortex-lattice case: (a)
$t=280$ to $t=380$ at $\Omega_{\mathrm{f}}=0.8$, for $t=120$ to $t=220$ at (b)
$\Omega_{\mathrm{f}}=0.825$, (c) $\Omega_{\mathrm{f}}=0.85$, (d)
$\Omega_{\mathrm{f}}=0.9$. Figure 15: Density flux profiles for the vortex-
lattice case: (a) $t=360$ to $t=380$ at $\Omega_{\mathrm{f}}=0.8$, for $t=200$
to $t=220$ at (b) $\Omega_{\mathrm{f}}=0.825$, (c) $\Omega_{\mathrm{f}}=0.85$,
(d) $\Omega_{\mathrm{f}}=0.9$.

In Fig. 9, we show the snapshots of the condensate at different instant of
time as the condensate was finally set to the final rotation with frequency
$\Omega_{f}=0.85$ at $t=120$. At $t=0$ [cf. Fig. 9(a)], we find the presence
of pinned vortices inhibiting them from reaching the condensate boundaries.
The condensate gets distorted at $t=120$. Beyond $t=120$, the sustained
rotation with frequency $\Omega_{\mathrm{f}}=0.85$ introduces more vortices in
the condensate, resulting in the enlargement of the condensate. During this
time, very few vortices get clustered around the central barrier, while the
rest of the vortices start spiralling around, thereby producing a disordered
lattice [see Figs. 9(c)-(d)].

Further, to get an insight into the transient states of the turbulence for the
vortex-lattice initial state, we illustrate the time evolution of the mean
angular momentum ($\langle L_{z}\rangle$) for different final angular
frequencies ($\Omega_{f}=0.6$, $0.8$, $1.0$, and $1.2$) in Fig. 10.

Fig. 11 illustrates the scaled incompressible kinetic energy spectra for the
vortex lattice initial state at different rotational frequencies
$\Omega_{\mathrm{f}}=0.85,0.825,0.85$ and $0.9$ in the wavenumber space. Like
the vortex less initial state for all the frequency $\Omega_{\mathrm{f}}$ the
spectrum at large wave number ( $k>1.25\pi/\xi$) follows $k^{-3}$ a typical
behaviour for the energy spectrum due to the presence of the vortex core [42].
In the infra-red ($2\pi/R<k<1/\xi$), the scaled spectrum fits well with
$k^{-1}$ a typical feature of the Vinen-like turbulence and indicative of
vortices clustering together without combining to form a larger vortex [31].
This $k^{-1}$ scaling is also shown to be accompanied by 2D weak wave
turbulence confirmed by the compressible kinetic energy spectra (see Fig. 13),
which consistently show $k^{-3/2}$ and $k^{-7/2}$ scaling laws. The Vinen or
ultraquantum scaling of $k^{-1}$ could possibly indicate that there is no
accumulation of kinetic energy at large scales and a direct cascade of energy
from $1/\xi$ onward.

Figure 12 shows the flux of the incompressible component of the kinetic energy
for the vortex-lattice initial state. Panels (a)-(e) represent the flux for
the condensate as it is rotated with the rotation frequency
$\Omega_{\mathrm{f}}=0.85,0.825,0.85$ and $0.9$, respectively. The different
colors represent the instants at which we plot the flux as indicated in the
color bar. Like the vortex-less initial state, this case also exhibits
oscillation in the flux profile with frequency $2.0\omega$ for all rotational
frequencies of the condensate. However, the time-averaged flux shown as the
thick dashed black line exhibits a constant positive flux in the UV range,
indicating the forward cascade of incompressible kinetic energy.

The compressible spectra of this the vortex initial state as shown in Fig. 13
exhibits $k^{1}$ scaling at small wave number while exhibits $k^{-3/2}$ and
$k^{-7/2}$, respectively in the IR and UV ranges for all the rotation
frequency $\Omega_{\mathrm{f}}=0.85,0.825,0.85$ and $0.9$.

In Fig. 14, we plot the density spectra for the vortex-lattice initial state
for the, which scales as $k^{-3}$ for $k^{-1}$ scaling in incompressible
spectra.

For all the final rotating frequencies ($0.8<\Omega_{\mathrm{f}}<0.9$), which
display scaling in the transient regime, it does not show any significant and
sustained turbulence, even when the condensate energy settles. This is due to
the perturbations not extending all the way to the condensate boundary and
being unable to develop significant compressible turbulence. However, when
rotated at sufficiently higher frequencies for longer time periods, we observe
consistent and sustained turbulent behaviour. We notice strong quantum
turbulence with a pronounced scaling law for the final rotating frequency of
$\Omega_{\mathrm{f}}=1.0$, as shown in Fig. 16(b), where the $k^{-5/3}$ and
$k^{-3}$ scales indicate the onset of the Kolmogorov cascade accompanied by an
enstrophy cascade. We also note that the accompanying density spectra, as
shown in Fig. 16(c), display $k^{-11/3}$ scaling. However, the compressible
spectra shows linear inclearsing behaviour with $k$ in the IR range while
$k^{-7/2}$ in the UV range same as those observed with the vortex less cases
[see Fig. 16(d)]. Further we complement the chaotic observation of the
condensate through the energy density representation in the real space in the
Fig. A.3-A.4. Note that Compared to the Kolmogorov scaling observed in the
vortex-less initial wave profile, the scaling behaviour, in this case, appears
at larger $k$ values. This is further confirmed by the kinetic energy and
density flux profiles in Fig. 17, which display positive and negative average
values, respectively, over the $k$ range, indicating a direct energy cascade.

Figure 16: Plots of (a) density and spectral profiles of (b) incompressible
kinetic energy, (c) density, and (d) compressible kinetic energy averaged over
the time interval $t=3600$ to $t=4800$ for vortex-lattice rotating at
$\Omega_{\mathrm{f}}=1.0$. Figure 17: Plots of (a) incompressible kinetic
energy and (b) density fluxes for vortex-lattice rotating at
$\Omega_{\mathrm{f}}=1.0$ for the time interval $t=3600$ to $t=4800$.

To further confirm the influence of the perturbation barrier on generating the
turbulence in the condensate, we have also considered the condensate along
with the quasiperiodic central barrier as illustrated in Fig. 1(b). We find
that the presence of a quasiperiodic barrier does not bring much change in the
dynamics of the condensate with both vortex-less and vortex-lattice initial
states. With the quasiperiodic perturbation, we do not find any signature of
turbulent fluctuation in the condensate. Both incompressible and compressible
components of the kinetic energy also do not exhibit any scaling laws for
various final rotation frequencies. However, one interesting observation is
made at higher rotation frequencies closer to $\Omega_{\mathrm{f}}=1.0$. For
this, we notice the presence of strong turbulence in the condensate, which is
quite evident from the power law behaviour of the incompressible kinetic
energy in the IR range that exhibits Kolmogorv-like scaling ($E(k)\sim
k^{-5/3}$) for both the initial state which is similar to that those a
periodically perturbed barrier.

##  IV Summary and Conclusions

In this paper, we investigated the impact of the perturbed central barrier on
rotating Bose-Einstein condensates confined in harmonic traps. We found that a
strong perturbation can trigger the formation of an ordered vortex lattice
from a vortex-free condensate, showing a direct energy cascade with Kolmogorov
scaling, and, in contrast, the initial wave profile with a vortex-lattice gets
distorted during transient stages and exhibits a non-Kolmogorov scaling law
within the inertial range and displays no consistent turbulence. But at a
longer duration under higher rotation frequencies, we observe turbulent
behaviour following Kolmogorov scaling.

For both cases, we have confined our studies to a range of initial and final
rotation frequencies mainly based on the appearance of scaling laws and the
onset of consistent turbulent dynamics. More generally, for rotating
frequencies greater than the trap frequency, a condensate undergoing rotation
will undergo constant expansion followed by an uncontrolled generation of
vortices. This region presents with no interesting spectral profiles as the
vortex core scaling of $k^{-3}$ is totally absent. Although the angular
momentum expectation values provide an idea regarding the dynamical behaviour
in both cases, we confirm and quantify its exact nature via spectral analysis,
energy density and flux profiles of the kinetic energy components.

The vortex-less profile, when subjected to turbulent rotation, creates an
ordered and symmetric vortex lattice whose onset and size depend on the final
rotating frequency $\Omega_{\mathrm{f}}$. With faster rotation, the condensate
can sustain more vortices, which eventually become attracted to the central
barrier. This breakdown of vortices, accompanied by strong compressible
turbulence, is indicated by the scalings of $k$, $k^{-11/3}$, and $k^{-5/3}$
for compressible, density, and incompressible spectra, respectively. To
further confirm the nature of vortex breakdown, the time-derivative profiles
of incompressible kinetic energy show the kinetic energy transfer to smaller
scales, indicating a direct energy cascade.

In the case of condensate comprising a vortex lattice (containing a sufficient
number of vortices), the influence of the perturbed barrier on the lattice
structure leads to distinct scaling behaviour. The presence of the vortex
lattice is equivalent to initiating a condensate with energy injected at
smaller scales, a phenomenon substantiated by the emergence of prominent peaks
in the incompressible kinetic energy spectra at $k\approx 1/\xi$. According to
Barenghi [45], this constitutes a recipe for generating a disordered vortex
lattice, characterized by $\varepsilon_{\mathrm{kin}}^{i}(k)\sim k^{-1}$
scaling in the incompressible kinetic energy spectra. The $k^{-1}$ scaling in
the incompressible spectra and $k^{-3}$ scaling in density spectra are
indicative of weak wave turbulence, a fact reaffirmed by the presence of
$k^{-3/2}$ and $k^{-7/2}$ scaling in the compressible spectra [31]. These
scaling tendencies are observed for final rotational frequencies within the
range $0.8<\Omega_{\mathrm{f}}<0.9$.

In contrast to the scenario without vortices, choosing $\Omega_{0}$ value to
be greater than the critical frequency $\Omega_{\mathrm{c}}$ of the condensate
causes the condensate to expand and introduce vortices, reducing the
perturbation amplitude relative to the condensate size. As a result, the
perturbations are unable to generate sufficient compressible turbulence.
Despite the appearance of $k^{-1}$ scaling in the and $k^{-3/2}$,$k^{-7/2}$
scaling behaviour in the incompressible and compressible spectra respectively
in the transient time iterations, the lack of sufficient turbulence shows no
consistent scaling behaviour at longer durations, where the condensate energy
settles. By limiting the perturbation amplitude in such a manner, we only
observe turbulence by rotating the condensate at higher frequencies for longer
durations, where the condensate displays $k^{-5/3}$ Kolmogorov scaling.

In conclusion, our study highlights the behaviour of a perturbed barrier when
subjected to rotation. In addition to playing a crucial role in generating
turbulence, the perturbed barrier also significantly influences the nature of
this turbulence, depending on the choice of initial wave profiles, especially
in the presence of vortices. Under the influence of these perturbations
combined with a centered axis of rotation, vortices tend to cluster around the
barrier. Forcing vortices into the initial condensate causes the barrier to
behave differently by distorting the vortex lattice rather than creating a
settled vortex cluster. Our method provides a novel mechanism to generate
turbulence and alter the behaviour of the condensate in a rotating frame with
a barrier whose geometry and strength can be fine tuned. This study can be
simulated and experimentally realized for a given BEC, provided the barrier
has sufficient perturbation amplitude, barrier radius, and strength.

Finally, our present study is relevant for exploring the dynamical evolution
of fluids inside boundary layers between the superfluid cores inside pulsars
and the ambient normal matter, where the observed glitch phenomena are said to
occur. Once the rotating superfluid core jumps promptly into the next lower
energy state abruptly, a certain amount of rotational energy is ejected into
the ambient medium, setting the medium in the boundary layer in a turbulent
mode. Following Refs. [46, 47], such events are predicated to reoccur several
billion times during a pulsar’s lifetimes; hence, the core serves as a forcing
term for generating turbulence, as alluded to in the present study.

###### Acknowledgements.

A.S. acknowledges MoE RUSA 2.0 for financial support. The work of P.M. is
supported by DST-SERB under Grant No. CRG/2019/004059 and MoE RUSA 2.0.

##  Appendix A Incompressible and compressible energy densities

In addition to the spectral profiles discussed in subsections III.1 and III.2,
it is worth examining the incompressible and compressible density profiles of
kinetic energies in real space. In this appendix, we illustrate the
incompressible and compressible density profiles of kinetic energies for both
the vortex-less and vortex lattice cases.

In Fig. A.1, we show the snapshot of the density profile of the incompressible
kinetic energy in real space at initial ($t=120$) and final ($t=600$) instant
of time for $\Omega_{\mathrm{f}}=0.85$ and $\Omega_{\mathrm{f}}=1$. We can
observe that the incompressible energy, initially localized at the barrier, is
relatively small [See Figs. A.1(a), (c)].

Figure A.1:  Snapshots of the incompressible energy density for the vortex-
less case: $\Omega_{\mathrm{f}}=0.85$ at (a) $t=120$ and (b) $t=600$, and
$\Omega_{\mathrm{f}}=1.0$ at (c) $t=120$ and (d) $t=600$.

As turbulence sets in, the energy becomes concentrated around the edges of the
barrier, indicating the presence of vortices. For higher rotation frequencies
[see Figs. A.1(c)-(d)], the density distribution follows a similar trend, but
at later times, there is a band of high-energy density, indicating the
presence of several vortices clustered around the barrier. The initial time
snapshots illustrate the localization of compressible energy near the barrier
edge. This emphasizes the role of the barrier in inducing turbulence as it
compresses the condensate between the perturbations. As time progresses, the
compressible turbulence becomes evenly distributed within the condensate
surrounding the barrier [See Fig. A.2(b) and (d)]. This observation shows
significant phonon activity and thermalization, also confirmed by the
compressible energy spectra.

Figure A.2: Snapshots of the compressible energy density for the vortex-less
case: $\Omega_{\mathrm{f}}=0.85$ at (a) $t=120$ and (b) $t=600$, and
$\Omega_{\mathrm{f}}=1.0$ at (c) $t=120$ and (d) $t=600$.

This observation confirms what we have noticed in the spectral profiles and
their time derivatives as the energy becomes more concentrated toward the
healing length or vortex-size scales. In Fig. A.2, we show the snapshots of
the compressible kinetic energy densities for $\Omega_{\mathrm{f}}=0.85$
(upper panel) and $\Omega_{\mathrm{f}}=1.0$ (lower panel) at time $t=120$ and
$t=600$.

Figure A.3: Snapshots of the incompressible energy density for the vortex-
lattice case: $\Omega_{\mathrm{f}}=1.0$ at (a) $t=0$ and (b) $t=4800$ Figure
A.4: Snapshots of the compressible energy density for the vortex-lattice case:
$\Omega_{\mathrm{f}}=1.0$ at (a) $t=0$ and (b) $t=4800$

The density profiles of the kinetic energy components further confirm the
disordered nature of the turbulence. The incompressible kinetic energy
densities shown in Fig. A.3, for $\Omega_{\mathrm{f}}=1.0$ show an ordered
vortex lattice at the initial stage, which, when rotated at such a high
frequency develops considerable turbulence with the incompressible density
being similar to that of the vortex-less case but larger and more pronounced.
The compressible density profiles shown in Fig. A.4 for the same parameters
show no significant development of compressible turbulence at initial stages
but later distribute itself more evenly at the highly turbulent state.

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