# Interaction of driven “cold” electron plasma wave with thermal bulk mediated
by spatial ion inhomogeneity

Sanjeev Kumar Pandey[1] † and Rajaraman Ganesh[2,3] 1 India Center for Lab
Grown Diamonds (Incent-LGD), Material Science Research Center (MSRC),
Department of Physics, Indian Institute of Technology (IIT) Madras, Chennai
600036, India  2 Institute for Plasma Research (IPR), Bhat, Gandhinagar
382428, India  3 Homi Bhabha National Institute (HBNI), Mumbai, Maharashtra
400094, India  ${\dagger}$ Part of this work was done by one of the Authors
(SKP) during his PhD Thesis work at the Institute for Plasma Research,
Gandhinagar<EMAIL_ADDRESS>
###### Abstract

Using high resolution Vlasov - Poisson simulations, evolution of driven “cold”
electron plasma wave (EPW) in the presence of stationary inhomogeneous
background of ions is studied. Mode coupling dynamics between “cold” EPW with
phase velocity $v_{\phi}$ greater than thermal velocity i.e $v_{\phi}\gg
v_{thermal}$ and its inhomogeneity induced sidebands is illustrated as an
initial value problem. In driven cases, formation of BGK like phase space
structures corresponding to sideband modes due to energy exchange from primary
mode to bulk particles via wave-wave and wave-particle interactions leading to
particle trapping is demonstrated for inhomogeneous plasma. Qualitative
comparison studies between initial value perturbation and driven problem is
presented, which examines the relative difference in energy transfer time
between the interacting modes. Effect of variation in background ion
inhomogeneity amplitude as well as ion inhomogeneity scale length on the
driven EPWs is reported.

††: Phys. Scr.

  * •

July 2024

Keywords : Driven collisionless plasma systems, Electron plasma waves (EPW),
Non - Linear Landau damping, BGK mode, Wave - particle resonance interaction,
Wave - wave mode coupling interaction, Spatially non - uniform plasma system,
Vlasov - Poisson simulations.

##  1 Introduction

Electron plasma waves (EPWs) have been extensively researched since its
discovery in 1929 by Tonks and Langmuir [1]. Investigation of the mechanism
behind its excitation and dynamical response continues to be a fundamental
area of research. Several pioneering works have brought out various key
aspects associated with the EPWs [2, 3, 4, 5, 6, 7, 8, 9]. In the recent past,
numerous studies including kinetic or phase space simulations have been
carried out to understand intricate features of the driven dynamics at
electron and ion scales in homogeneous collisionless plasmas [10, 11, 12, 13,
14, 15, 16, 17, 18]. However, in most realistic scenarios like laboratories,
tokamaks, and astrophysical plasmas, equilibria are inhomogeneous in nature
and hence studies involving such inhomogeneities becomes a necessity.

In the above context, in the past, studies have been carried out involving
temporal evolution of long wavelength or high phase velocity EPWs in the
presence of ﬁnite amplitude ion density modulation in the background. These
studies demonstrates energy transfer from high phase velocity “cold” EPWs into
associated sidebands with much slower phase speeds and then into the particles
via efﬁcient off-resonant mode coupling mediated by inhomogeneous background
of ions [19, 20, 21]. Additionally, attempts were made to investigate the
evolution and energy transfer mechanism associated with EPWs in a bounded and
periodic inhomogeneous equilibria [22, 23, 24, 25]. In another work, using
warm plasma model [26] and kinetic model [27, 28] with initial value
perturbation (IVP) approach, authors illustrated damping of high phase
velocity EPWs via wave-wave mode coupling phenomenon initiated by
inhomogeneous stationary background of ions. This study [26] showed that an
EPW of mode number $k$ can interact with a non-uniform (i.e, spatially
periodic), stationary ion background of scale $k_{0}$ to produce coupled
secondary sideband modes with wavenumber $|k\pm Nk_{0}|$ (where $N$ is a
dimensionless coupling parameter), resulting into cascading energy transfer
from primary mode to sidebands upto $N^{th}$ sideband mode. In the process,
depending on the value of $N$, the primary perturbation mode may lose its
amplitude either fully or partially. These studies are proven useful in
various applications and help in understanding variety of plasma concepts such
as wave breaking in inhomogeneous equilibria [29, 30], laser plasma
interaction [31, 32] , laser absorption by ion acoustic turbulence [33],
plasma instabilities [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48], ionospheric turbulence [49, 50] etc.

In the present work, using high resolution OpenMP based kinetic (VPPM-OMP 1.0)
Vlasov - Poisson solver with constant frequency external electric field drive,
we address the energy exchange dynamics associated in both wave-wave mode
coupling as well as wave-particle interaction process. In this process, “cold”
primary perturbation is externally driven in the presence of background ion
inhomogeneity which in turn facilitates onset of coupled secondary sidebands
of the externally driven “cold” EPWs. At first, we demonstrate qualitative
agreement between the electron plasma waves excited via initial value
perturbation (IVP) and externally driven perturbation (DP) approaches. It is
found that despite having identical coupling parameter $N$, in IVP cases some
higher coupled sideband modes are involved in energy exchange interactions
compared to the DP cases. Also, the exact values of the time required for the
transfer of most of energy density from primary driven perturbation mode to
the coupled sideband modes differ slightly in both the IVP and DP cases. Next,
we illustrate the effect of increase in ion inhomogeneity amplitude (A) and
decrease in the ion inhomogeneity scale ($k_{0}$) on the driven dynamics of
EPWs, which indicates relatively quick energy transfer between the interacting
modes and generation of phase space structures corresponding to a coupled
single sideband mode via wave-particle interaction is reported.

This paper is organized as follows: In Sec. 2, we present the Vlasov - Poisson
model equations including the equations for stationary ion inhomogeneity and
external drive. Followed by numerical scheme in Sec. 3. In Sec. 4, we present
the simulation results for driven EPW cases with different inhomogeneity
amplitudes i.e $A=0.03$ (in Sec. 4.1) and $A=0.05$ (in Sec. 4.2) and finally
we conclude in Sec. 5.

##  2 Mathematical model

In the kinetic theory framework, small amplitude externally driven electron
plasma waves (EPWs) in a 1D, collision-less, unmagnetized, spatially
inhomogeneous plasma system consisting of kinetic electrons and ions is
modeled using a set of coupled 1D Vlasov-Poisson (VP) equations [27, 28, 51,
52, 53, 54, 48, 55],

$\frac{\partial f_{e}}{\partial t}+v_{e}\frac{\partial f_{e}}{\partial
x}-E_{T}\frac{\partial f_{e}}{\partial v_{e}}=0$ (1) $\frac{\partial
f_{i}}{\partial t}+v_{i}\frac{\partial f_{i}}{\partial
x}+\left[\frac{E_{T}}{m_{r}}\right]\frac{\partial f_{i}}{\partial v_{i}}=0$
(2) $\frac{\partial E_{T}}{\partial x}=\int f_{i}dv_{i}-\int f_{e}dv_{e}$ (3)

where $f_{i}(x,v,t)$ and $f_{e}(x,v,t)$ is the ion and electron distribution
functions respectively, $m_{r}$ is the mass ratio of ions to electrons, i.e,
$m_{r}=M_{i}/M_{e}$, $E_{T}$ is the total electric field given as,

$E_{T}=E_{0}+E_{s}+E_{D}^{Ext}$ (4)
$E_{D}^{Ext}(x,t)=E_{D}^{0}\sin(kx\pm\omega_{D}^{Lan}t)$ (5)

where $E_{s}$ is the self-consistent electric field, $E_{0}$ is the
equillibrium electric field due to background ion non-uniformity and
$E_{D}^{Ext}$ is the driven external perturbation electric field to excite
EPW, $\omega_{D}^{Lan}$ is the Langmuir driving frequency, $k$ is the mode
number and $E_{D}^{0}$ is the amplitude of the external electric field drive.
In equation [1-4], time $t$ is normalized to electron plasma frequency
$\omega_{pe}^{-1}$, spatial coordinate $x$ is normalized to electron Debye
length $\lambda_{De}$, velocities to electron thermal velocity
$v_{the}=\lambda_{De}\omega_{pe}^{-1}$, electric ﬁeld to
$en_{0}\lambda_{De}/\epsilon_{0}$, and distribution function has been
normalized to $n_{0}/\lambda_{De}\omega_{pe}$ where $n_{0}$ is uniform plasma
density (i.e, $n_{0}$ is the mean density in the absence of inhomogeneity).

Using Maxwellian velocity distribution function $f_{Me}(v_{e})$ for electrons
with an arbitrary spatial inhomogeneity profile $n_{0e}(x)$, the equilibrium
solution of Vlasov equation requires (prime denotes spatial differentiation),

$E_{0}(x)=-\left[\frac{n_{0e}^{{}^{\prime}}(x)}{n_{0e}(x)}\right]$ (6)

where $E_{0}(x)$ is the equilibrium electric field resulting from equilibrium
inhomogeneity [24, 27]. Ions form a non-uniform, stationary
$(\partial/\partial t=0)$ background of number density $n_{0i}(x)$ in this
model which is determined by satisfying the equilibrium Poisson’s equation,

$\frac{\partial E_{0}(x)}{\partial x}=\int f_{0i}dv_{i}-\int f_{0e}dv_{e}$ (7)

Combining Eqs. 6, 7 and using equilibrium Maxwellian velocity distribution for
electrons and ions alongwith their respective spatial inhomogeneity profiles
i.e $n_{0e}(x)$ and $n_{0i}(x)$ one obtains,

$\frac{\partial}{\partial
x}\left[-\frac{n_{0e}^{\prime}(x)}{n_{0e}(x)}\right]=\int
n_{0i}(x)f_{Mi}(v_{i})dv_{i}-\int n_{0e}(x)f_{Me}(v_{e})dv_{e}$ (8)

which results in,

$n_{0i}(x)=n_{0e}(x)+\left[\frac{(n_{0e}^{{}^{\prime}}(x))^{2}-n_{0e}^{{}^{\prime\prime}}(x)n_{0e}(x)}{n_{0e}^{2}(x)}\right]$
(9)

Thus, using Eq. 1 \- 3, spatial inhomogeneous equilibrium density profile for
ions $n_{0i}(x)$ and for electrons $n_{oe}(x)$ are obtained self-consistently,
taking into account the equilibrium electric field variation given by Eq. 6 .
Consequently, one can presume any periodic density profile for $n_{0e}(x)$ and
obtain the corresponding $E_{0}(x)$ and $n_{0i}(x)$ from Eq. 6 and 9
respectively. Note that if
$n_{0e}^{{}^{\prime\prime}}(x)=n_{0e}^{{}^{\prime}}(x)=0$, from Eq. 6 and 9 we
get $E_{0}(x)=0$ and $n_{0i}(x)=n_{0e}(x)$ respectively [27, 54, 48, 28].

To construct an inhomoegeneous equilibrium, we consider a sinusoidal profile
for the spatially inhomogeneous electron density $n_{0e}(x)$ as,

$n_{0e}(x)=1+A\sin(k_{0}x)$ (10)

Using Eq. 6 , 9 and 10 one obtains $n_{0i}(x)$, $E_{0}(x)$ as,

$E_{0}(x)=-\left[\frac{(Ak_{0}\cos(k_{0}x))}{(1+A\sin(k_{0}x))}\right]$ (11)
$n_{0i}(x)=\left[\frac{(1+A\sin(k_{0}x))^{3}+A^{2}k_{0}^{2}+Ak_{0}^{2}\sin(k_{0}x)}{1+A^{2}\sin^{2}(k_{0}x)+2A\sin(k_{0}x)}\right]$
(12)

where $A$ is the strength of equilibrium inhomogeneity and $k_{0}$ is the
measure of equilibrium inhomogeneity scale. To enforce periodicity in the
system, which is essential and necessary condition for consistently addressing
periodic non-uniform profiles, every equilibrium inhomogeneity scale $k_{0}$
and all the perturbation $k$ scales in the simulation are expressed as integer
multiples of $k_{min}$ where $k_{min}=2\pi/L_{max}$. In the following, we
present numerical scheme used for solving the coupled Vlasov - Poisson
equations.

##  3 Numerical Scheme

An in-house developed OpenMP based Vlasov - Poisson solver i.e VPPM - OMP 1.0
[27, 54, 48, 28, 55] is used to solve the set of Vlasov - Poisson equations
Eq. 1 \- 3. VPPM - OMP 1.0 is capable of handling both electron and ions
dynamics simultaneously. It is based on Piecewise Parabolic Method (PPM)
advection scheme proposed by Colella and Woodward [56] and uses time stepping
method given by Cheng and Knorr [57]. A Fourier transform (FT) based method
has been implemented for solving Poisson equation.

Simulation domain in phase space is chosen to be
$D=[0,L_{max}]\times[-v_{e}^{max},v_{e}^{max}]$, where $L_{max}=2\pi/k_{min}$
is the system size and $v_{e}^{max}$ is the maximum electron velocity chosen
to be $v_{e}^{max}=6.0$. Periodic boundary conditions (PBC) have been
implemented in both spatial and velocity domains and the simulation domain is
discretized into $N_{x}$ grid points in spatial domain and $N_{v}$ grid points
in velocity domain for both ions and electrons.

At $t=0$, we initialize the simulation with normalized Maxwellian distribution
function for ions and electrons with respective spatial inhomogeneous
equilibrium density profiles $n_{0i}(x)$, $n_{0e}(x)$ obtained earlier (In
Eqs. 10 and 12),

$f_{e}(x,v_{e},t=0)=n_{0e}(x)f_{Me}(v_{e})$ (13)
$f_{i}(x,v_{i},t=0)=n_{0i}(x)f_{Mi}(v_{i})$ (14)
$f_{Me}(v_{e})=\frac{1}{\sqrt{2\pi}}exp\left[\frac{-v_{e}^{2}}{2}\right]$ (15)
$f_{Mi}(v_{i})=\frac{1}{\sqrt{2\pi}}\sqrt{\frac{m_{r}}{T_{r}}}exp\left[\frac{-m_{r}v_{i}^{2}}{2T_{r}}\right]$
(16)

where $f_{Mi}(v_{i})$, $f_{Me}(v_{e})$ are the normalized ion and electron
Maxwellian velocity distribution functions, $T_{r}=T_{i}/T_{e}$ is temperature
ratio of ions to electrons, $m_{r}=M_{i}/M_{e}$ is mass ratio of ions to
electrons. In the following, we present simulations results for driven EPW
cases in the presence of immobile non - uniform background of ions to
demonstrate off - resonant wave - wave mode coupling dynamics. Also, we
illustrate the comparative study between both IVP and DP cases for identical
parameter sets with inhomogeneity amplitude A as A=0.03.

##  4 Simulation Results

In this section, we present the simulation results of driven perturbation (DP)
studies with $A=0.03$ and qualitatively compare it with small amplitude
initial value perturbation (IVP) studies presented in [28], indicating the
existence of coupling of externally driven “cold” electron plasma wave (EPWs)
with phase velocity $v_{\phi}>>v_{thermal}:v_{\phi}>>v_{e}^{max}$ [where
$v_{\phi}$ is the phase velocity of the EPW, $v_{thermal}$ is the thermal
velocity of electrons and $v_{e}^{max}$ is the maximum sampled electron
velocity] due to the presence of background immobile ion non - uniformity
irrespective of chosen perturbation approaches. Also, we present the $A=0.05$
case, to observe the effect of increase in the inhomogeneity amplitude as well
as the effect of change in inhomogeneity scale $k_{0}$ on the plasma response
of DP cases.

###  4.1 Spatially non-uniform driven plasma case with $A=0.03$

Figure 1: Temporal evolution of primary perturbation $k/k_{min}=1$ mode and
coupled secondary interacting $|k\pm Nk_{min}|/k_{min}=3,5,7,9$ sideband
Fourier modes i.e $|\delta E_{k}|$ defined by Eq. 17 with $(k_{0},k):(4,1)$,
$k_{min}=0.1$, $A=0.03$, $\omega_{D}^{Lan}=1.0148$
($\omega_{D}^{Lan}=\sqrt{1+3k^{2}}$), $E_{D}^{0}=0.001$ and $\Delta t=100$
case. Solid black vertical line indicates the time when the external electric
field drive is switched off. Inset plot zooms $|\delta E_{k}|$ for initial
time $0$ to $100~{}\omega_{pe}^{-1}$ indicating the continuous increase in the
electric field amplitude value from $0$ to $0.10$ during the external drive.

As indicated earlier, numerical simulations are carried out in the limit of
immobile inhomogeneous background of periodic ions and kinetic electrons. In
this plasma system, two spatial scales i.e equilibrium inhomogeneity and
perturbation $(k_{0},k)$ scales respectively, are initiated and expressed as
an integer multiples of $k_{min}$ which is 0.1 such that
$L_{max}=2\pi/k_{min}=20\pi$. To begin with, external electric field drive of
the form $E_{D}^{Ext}=E_{D}^{0}sin(kx\pm\omega_{D}^{Lan}t)$ is applied for a
time period of $\Delta t=100~{}\omega_{pe}^{-1}$ with the other simulation
parameter set as
$v_{e}^{max}=6.0,~{}(k_{0},k):(4,1),~{}A=0.03,~{}\omega_{[D,k_{min}=0.1]}^{Lan}=\sqrt{1+3(0.1)^{2}}=1.0148,~{}E_{D}^{0}=0.001$.
Phase space grid discretization are set to $N_{x}\times N_{v}=[2048\times
10000]$. To avoid any recurrence effects in the solutions, we chose velocity
grid descretization large enough such that $t<T_{R}$, where $T_{R}$ is the
recurrence time expressed as $T_{R}=2\pi/k\Delta v:\Delta v=2v_{max}/N_{v}$
which is $T_{R}=10471.98$ [27, 52, 54, 55, 58, 59]. The simulation is advanced
till $t=3000~{}\omega_{pe}^{-1}$ in time. Evolution of Fourier mode amplitude
of electric field perturbation for mode $k:E_{k}(t)$ is shown in Fig. 1
obtained from,

$E(x,t)~{}=~{}\sum_{k}E_{k}(t)e^{-ikx}$ (17)

where $E(x,t)$ is the total electric field obtained by solving Poisson’s
equation (Eq. 3).

Figure 2: Variation of the $\tau_{FM}$ i.e time taken by the mode to reach the
first minimum of the energy density signature versus inhomogeneity amplitude
($A$) for the external driven perturbation case with $(k_{0},k):(4,1)$,
$k_{min}=0.1$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta
t=100$. Fig. 2 indicates a inversly porprotional relationship between
$\tau_{FM}$ and $A$.

Fig. 1 shows the temporal evolution of the primary $k/k_{min}=1$ and coupled
secondary interacting sideband Fourier mode $|\delta E_{k}|$ signatures i.e
$k/k_{min}=3,5,7,9$ with
$(k_{0},k):(4,1),~{}k_{min}=0.1,~{}A=0.03,~{}\omega_{D}^{Lan}=1.0148,E_{D}^{0}=0.001,\Delta
t=100$. It demonstrates the damping of the externally driven primary
perturbation mode $|\delta E_{k}|$ signature with phase velocity
$v_{\phi}=\omega_{D}^{Lan}/k\simeq
10.148>>v_{thermal}~{}:~{}v_{\phi}>>v_{e}^{max}$, where $v_{thermal}=1.0$ and
$v_{e}^{max}=6.0$, via resonant wave-particle interaction phenomenon
associated with the interacting coupled sideband modes $k/k_{min}=3,~{}5$ in
the presence of immobile inhomogeneous background of ions with inhomogeneity
amplitude $A=0.03$. For the comparative study, between driven perturbation
(DP) and initial value perturbation (IVP) cases, the amplitude of the external
electric field drive ($E_{D}^{0}$) and total time for applied external drive
($\Delta t=100$) is chosen such that the total electric field amplitude at the
time when external drive is switched off (indicated by solid black line in
Fig. 1) is equal to the amplitude of initial electric field
[$E_{0}=\alpha/k=0.01/0.1=0.10$] in the initial value perturbation (IVP) case
as presented in Sec 3.4 of Ref. [28]. Rest of the parameters in this study are
set identical to the initial value perturbation case. On comparing Fig 1 of
this work with Fig. 19 of Ref. [28] obtained via initial value perturbation
approach, one finds a qualitative agreement between both the cases. However,
considerable amount of energy transfer in the higher interacting coupled
sideband modes $k/k_{min}=7,~{}9$ can be seen in the IVP cases compared to
externally driven (DP) cases despite having the identical mode coupling
parameter which is given as $N\sim\sqrt{A/\gamma
k_{0}^{2}}\sim\sqrt{0.03/3(0.4)^{2}}\sim 0.25$ [27, 28].

In Ref. [28] Sec. 3.2, Authors draw our attention about the time taken by the
mode to reach the first minimum of the energy density signature $|\delta
E|^{2}$ which is denoted as $\tau_{FM}$. It is the time at which most of the
energy content is transferred to the coupled interacting sideband modes from
the primary perturbation mode. Fig. 2 illustrates variation of $\tau_{FM}$
versus inhomogeneity amplitude ($A$) for the externally driven perturbation
(DP) case with $(k_{0},k):(4,1)$, $k_{min}=0.1$, $\omega_{D}^{Lan}=1.0148$,
$E_{D}^{0}=0.001$ and $\Delta t=100$. It indicates that the $\tau_{FM}$ is
inversely proportional to the increment in the inhomogeneity amplitude values
i.e $\tau_{FM}\sim 1/A$ which is qualitatively similar to the finite
inhomogeneity amplitude effects in Ref. [28]. However, there is small
difference in the exact values of $\tau_{FM}$ obtained using driven
perturbation (DP) and initial value perturbation (IVP)cases i.e
$\tau_{FM}^{DP}=1131.91~{}\omega_{pe}^{-1}$ and
$\tau_{FM}^{IVP}=1099.0~{}\omega_{pe}^{-1}$ respectively for the identical
parameter sets $(k_{0},k):(4,1)$, $k_{min}=0.1$, $A=0.03$. This discrepancy in
$\tau_{FM}$ values can be related to the engagement of higher sideband modes
$k/k_{min}=7,~{}9$ during the energy exchange from the primary mode in the IVP
case compared to the DP case as indicated earlier.

Figure 3: Plot of spatially averaged distribution function $\widehat{f}(v,t)$
with respect to velocity $(v)$ at different times i.e
$t=0,~{}3000~{}\omega_{pe}^{-1}$ for $(k_{0},k):(4,1)$, $A=0.03$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ case. In the
inset plot, one can observe bump around phase velocity locations
$v_{\phi}^{k}=2.04002,~{}3.38270$ which corresponds to $k/k_{min}=3,~{}5$
sideband modes. Phase velocities corresponding each interacting mode is
tabulated in Table 1.

Resonance locations i.e phase velocities ($v_{\phi}^{k}=\omega_{k}/k$) of each
sideband modes is tabulated alongwith oscillation frequency ($\omega_{k}$)
obtained by 1D FFT (Fast Fourier Transform) analysis. From Table 1, one can
notice that the resonance location corresponding to the primary perturbation
mode is “outside” the velocity domain i.e
$v_{\phi}^{k_{min}=0.1}(10.1480)>v_{e}^{max}(6.0)$ which indicates that the
wave is “cold” and that there is absence of resonating wave-particle
interactions. However, we have observed wave-particle interaction signatures
such as vortex structure formation in the phase space $(x,v)$, plateau
creation in the spatially averaged distribution function etc, due to
interactions of the secondary sideband modes $k/k_{min}=3,~{}5,~{}7,~{}9$
interacting resonantly with the plasma bulk.

Table 1: Mode frequency $(\omega_{k})$ and Phase Velocity $(v_{\phi}^{k}=\omega_{k}/k)$ corresponding to the primary and secondary interacting sideband modes for $(k_{0},k):(4,1),~{}E_{D}^{0}=0.001,~{}A=0.03,~{}\omega_{D}^{Lan}=1.0148,~{}\Delta t=100,~{}k_{min}=0.1$ case. Mode No. | $[\omega_{k}]$ | $[v_{\phi}^{k}=\omega_{k}/k]$
---|---|---
$k=0.1$ | 1.01480 | 10.1480
$|k-k_{0}|=0.3$ | 1.01480 | 3.38270
$|k+k_{0}|=0.5$ | 1.02001 | 2.04002
$|k-2k_{0}|=0.7$ | 1.35509 | 1.93584
$|k+2k_{0}|=0.9$ | 1.01355 | 1.12616

Figure 4: Phase space portrait of electron distribution function
$f_{e}(x,v,t)$ at late time $t=3000~{}\omega_{pe}^{-1}$ for $(k_{0},k):(4,1)$,
$A=0.03$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$
case. In (b) and (c) subplots zoomed velocity ranges to shown to illustrate
phase space vortices corresponding to $k/k_{min}=3,~{}5$ sideband modes around
their corresponding phase velocities $v_{\phi}^{k}$ as tabulated in Table 1.
In (d) vortex phase space structures around $v=0$ is due to the chosen ion
equilibrium inhomogeneity.

Spatially averaged distribution function $\widehat{f}(v,t)$ acts as measure to
visualize exact state of distribution function around resonance locations and
is given as,

$\widehat{f}(v,t)=\frac{\int_{0}^{L_{max}}f(x,v,t)dx}{\int_{+v_{e}^{max}}^{-v_{e}^{max}}\int_{0}^{L_{max}}f(x,v,t)dxdv}$
(18)

where $f(x,v,t)$ is the electron distribution function. Fig. 3 shows the
signature of spatially averaged distribution function with respect to velocity
at different times i.e $t=0,~{}3000~{}\omega_{pe}^{-1}$ with
$(k_{0},k):(4,1)$, $A=0.03$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and
$\Delta t=100$. From Fig. 3 and the inset plot, we can observe formation of
plateau or hump around velocity locations $v\sim 2.04,~{}3.38$ which
corresponds to the phase velocity $v_{\phi}^{k}$ locations of sidebands
$k/k_{min}=5,~{}3$ respectively (Table 1). Formation of such plateau or hump
structures indicate the wave - particle resonance interaction and particle
trapping phenomenon around the phase velocity $v_{\phi}^{k}$ locations as
shown in Fig. 4. Fig. 4 [(a),(b),(c) and (d)] shows phase space plot of
electron distribution function at late time $t=3000~{}\omega_{pe}^{-1}$ for
$(k_{0},k):(4,1)$, $A=0.03$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and
$\Delta t=100$ DP case. In a coordinate system moving with the wave speed
$v_{\phi}^{k}$, these phase space portrait diagrams illustrates the phase
trajectories of the resonant electrons at phase velocity locations
$v_{\phi}^{k}$. One can observe in Fig. 4 [(b) and (c)], vortex structure
streams corresponding to sideband $k/k_{min}=5$ and $k/k_{min}=3$
respectively. Note that the phase space vortex chain at $v=0$ is due to
equilibrium inhomogeneity $k_{0}/k_{min}=4$ with four periodic vortices [Fig.
4 (d)]. It is interesting to note that in the phase space structure of IVP
case i.e we observe formation of vortex stream around $v_{\phi}=5.290$ which
belongs to $k/k_{min}=2$ sideband mode which is absent in the DP case despite
identical parameters as shown in Fig. 20 of Ref. [28] and Fig. 4 of this work.
Also, the same is confirmed by the spatially averaged distribution function
variations shown in Fig. 21 of Ref. [28] and Fig. 3.

Using excess density fraction (EDF) $:\delta n/n_{0}$ calculations one can
quantify the trapped particle fractions which is given as,

$\frac{\delta n}{n_{0}}(x,t)~{}=~{}\frac{n(x,t)-n(x,t=0)}{n(x,t=0)}$ (19)

where $n(x,t)=\int f(x,v,t)dv$ is the density of the electrons i.e first
moment of the distribution function. Fig. 5 illustrate the variation of excess
density fraction at $x=L_{max}/4$ where $L_{max}=2\pi/k_{min}$ for
$(k_{0},k):(4,1)$, $A=0.03$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and
$\Delta t=100$ DP case. It indicates particle trapping and detrapping
modulations due to associated kinetic effects like wave - particle
interactions observed in interacting coupled sideband $k/k_{min}=5,~{}3$
modes.

Figure 5:  Temporal variation of excess density fraction (EDF)$:(\delta
n/n_{0})$ at $x=L_{max}/4$ where $L_{max}=20\pi=62.8318$ for
$(k_{0},k):(4,1)$, $A=0.03$, $[N_{x}\times N_{v}]=[2048\times 10000]$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP case. The
dotted black line indicates the time when the external drive is switched off.

Numerical consistency of the obtained solutions is demonstrated by plotting
the relative difference of kinetic energy $\Delta KE(t)=KE(t)-KE(0)$,
potential energy $\Delta PE(t)=PE(t)-PE(0)$ and total energy $\Delta
TE(t)=TE(t)-TE(0)$ defined by Eqs. 21, 22 with respect to time for
$(k_{0},k):(4,1)$, $A=0.03$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and
$\Delta t=100$ DP case in Fig. 6. It indicates good late time conservation and
saturation of relative change in the energies with chosen grid discretization
parameters as $N_{x}\times N_{v}=2048\times 10000$ for both electrons and ions
$(x,v)$ domains after the external electric field drive is switched off.

$TE(t)=KE(t)+PE(t)$ (20) $KE(t)=\int\int\frac{v^{2}}{2}f(x,v,t)dxdv$ (21)
$PE(t)=\int\frac{1}{2}E^{2}(x,t)dx$ (22)

Figure 6: Relative kinetic, potential and total energies $(\Delta KE,~{}\Delta
PE,~{}\Delta TE)$ variations with respect to time for $(k_{0},k):(4,1)$,
$A=0.03$, $[N_{x}\times N_{v}]=[2048\times 10000]$, $\omega_{D}^{Lan}=1.0148$,
$E_{D}^{0}=0.001$ and $\Delta t=100$ DP case. The solid vertical line
represents the time when the external electric field drive is switched off.

Figure 7: Temporal variation of the relative entropy $S_{rel}$ defined in Eq.
23 for $(k_{0},k):(4,1)$, $A=0.03$, $[N_{x}\times N_{v}]=[2048\times 10000]$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP case. The
dotted vertical line represents the time when the external electric field
drive is switched off.

To demonstrate that the obtained solutions are steady state solutions, we have
shown the temporal variation of the relative entropy defined by Eqs. 23 and 24
for $(k_{0},k):(4,1)$, $A=0.03$, $[N_{x}\times N_{v}]=[2048\times 10000]$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP case in
Fig. 7. It act as a measure to observe ‘information lost’ from the system as
entropy $S(t)$ has intrinsic tendency for monotonic increase with time due to
the ‘ﬁlamentation’ property of the Vlasov - Poisson system [51, 60]. The
distribution function generates small scale structures in phase space $(x,v)$
due to ﬁlamentation effect, which are dissipated when ﬁlamentation reaches the
phase space grid size $[N_{x}\times N_{v}]$ resolutions, leading to saturation
in numerical entropy with time as shown in Fig. 7. It indicates numerically
stable and high quality simulations as the information lost is small with
respect to time for chosen grid size resolutions after the external electric
field drive is switched off.

$S_{rel}(t)=\frac{S(t)-S(t=0)}{S(t=0)}$ (23)
$S(t)=~{}-\int_{0}^{L_{max}}\int_{-v_{e}^{max}}^{+v_{e}^{max}}f(x,v,t)\log
f(x,v,t)dxdv$ (24)

We have demonstrated the coupling of externally driven “cold” electron plasma
(EP) mode to the plasma bulk via immobile background of inhomogeneous ion
analogous to the fluid mode coupling with strong kinetic damping of initial
value perturbation (IVP) case presented in Ref. [28]. In the next section, we
study the effect of inhomogeneity amplitude on the externally driven plasma
cases.

###  4.2 Inhomogeneous driven plasma case with amplitude $A=0.05$

Figure 8: Temporal evolution of perturbed primary as well as coupled
interacting sideband Fourier mode amplitude $|\delta E_{k}|$ for externally
driven plasma case with $(k_{0},k):(4,1)$, $A=0.03$, $[N_{x}\times
N_{v}]=[2048\times 10000]$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and
$\Delta t=100$. Solid black vertical line indicates the time when the external
electric field drive is switched off. Inset plot zooms $|\delta E_{k}|$ for
initial time $0$ to $100~{}\omega_{pe}^{-1}$ illustrating the continuous
increase in the electric field amplitude value during the external drive.

In order to investigate the effect of increase in the background ion
inhomogeneous amplitude on the kinetic effects associated with coupled
interacting sideband modes as well as wave - wave coupled interactions between
primary perturbation and interacting sideband modes more prominently, we
present a case with increased background ion inhomogeneity amplitude i.e,
$A=0.05$, alongwith identical parameter sets used previously in Sec. 4.1 for
the driven electric field perturbation (DP) case.

Fig. 8 demonstrates the variation of Fourier mode signature i.e $|\delta
E_{k}|$ of primary and coupled sideband modes with
$A=0.05,~{}(k_{0},k):(4,1),~{}\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$
driven upto $\Delta t=100~{}\omega_{pe}^{-1}$. It shows damping of the primary
perturbation mode initially and late time modulations of the $|\delta E_{k}|$
signature which does not satisfy resonance condition as phase velocity of the
mode is around $v_{\phi}^{k}=\omega_{D}^{Lan}/k=10.148$ which falls beyond the
velocity domain considered in the problem ($v_{e}^{max}=6.0$). It is important
to note that the occurrence of electric field amplitude damping in the primary
perturbed mode $k/k_{min}=1$ is mainly due to the wave - wave interaction
leading to energy exchange between primary $k/k_{min}=1$ and secondary
interacting sideband modes $k/k_{min}=3,~{}5$ which undergo kinetic damping
due to wave - particle resonance interactions (as phase velocities of
generated sideband modes $v_{\phi}^{k/k_{min}=3,~{}5}$ falls in the bulk of
the plasma) analogous to the previous case in Sec 4.1. Primary perturbed mode
$k/k_{min}=1$ does not involve directly in any kinetic phenomenon such as wave
- particle interaction as $v_{\phi}^{k/k_{min}=1}>>v_{e}^{max}$. Additionally,
from Figs. 1, 2 and 8, one can conclude that with the increase in the ion
inhomogeneity amplitude, time required for the energy transfer between primary
and sideband modes i.e $\tau_{FM}$ reduces rapidly. However this reduction in
$\tau_{FM}$ values is small for larger $A$ values.

Figure 9: 2D power spectrum $(\omega,k)$ contour plot for $(k_{0},k):(4,1)$,
$[N_{x}\times N_{v}]=[2048\times 10000]$, $\omega_{D}^{Lan}=1.0148$,
$E_{D}^{0}=0.001$, $\Delta t=100$ driven perturbation with (a) homogeneous
$A=0$ and (b) inhomogeneous $A=0.05$ cases. Plot (b) shows the mode coupling
signatures between primary perturbation $k/k_{min}=1$ and sideband modes
$k/k_{min}=3,~{}5$ in case of $A\neq 0$. The peak power of the primary mode
$k/k_{min}=1$ lies around $\omega/\omega_{min}=1615.1262$ indicating
$\omega=1.0148$ as $\omega_{min}=0.00062831$ consistent with 1D Fourier
Transform (FFT) analysis.

Fig. 9 demonstrates the 2D power $(\omega,k)$ spectrum for (a) homogeneous
$A=0$ and (b) inhomogeneous $A=0.05$ cases for $(k_{0},k):(4,1)$, $N_{x}\times
N_{v}=[2048\times 10000]$, $k_{min}=0.1$, $\omega_{D}^{Lan}=1.0148$,
$E_{D}^{0}=0.001$, $\Delta t=100$ driven perturbation (DP) case. The peak
power of the primary perturbation mode $k/k_{min}=1$ in both (a) homogeneous
($A=0$) and (b) inhomogeneous ($A=0.05$) cases lies around
$\omega/\omega_{min}=1615.1262$ indicating $\omega=1.0148$ as
$\omega_{min}=6.2831\times 10^{-4}$ consistent with 1D Fourier Transform (FFT)
analysis shown in Fig. 10 and tabulated in Table 2. Also, Fig. 9 illustrates
the mode coupling dynamics between perturbed primary $k/k_{min}=1$ and
interacting coupled secondary sideband modes $k/k_{min}=3,~{}5,~{}7,~{}9$ in
the presence of non - uniform background of ions with non - zero amplitude.
Also, in Fig. 9 (a) there is absence of sideband modes and all the power is
concentrated in $k/k_{min}=1$ primary perturbation mode, meanwhile in Fig. 9
(b), we observe the power transfer and generation of frequencies corresponding
to the coupled sideband modes i.e $k/k_{min}=3,~{}5$ due to mode coupling
interaction with the $k/k_{min}=1$ primary perturbed mode.

In Table 2, we have tabulated the oscillation frequency $(\omega_{k})$ and
phase velocity $(v_{\phi}^{k}=\omega_{k}/k)$ corresponding to the primary and
secondary interacting sideband modes for
$(k_{0},k):(4,1),~{}E_{D}^{0}=0.001,~{}A=0.05,~{}\omega_{D}^{Lan}=1.0148,~{}\Delta
t=100,~{}k_{min}=0.1$ DP case shown in Fig. 10. Also, the frequency values
obtained through 1D Fast Fourier Transformation (FFT) analysis matches well
with the obtained 2D $(\omega,k)$ power spectrum analysis. We can observe that
the phase velocities $v_{\phi}^{k}$ values corresponding to sideband
$k/k_{min}=3,~{}5,~{}7,~{}9$ modes lies well within $v_{e}^{max}=6.0$
indicating that the resonance locations of these interacting coupled sideband
modes lies in the plasma bulk. Hence, energy exchange takes place from primary
$k/k_{min}=1$ mode to these sideband modes $k/k_{min}=3,~{}5,~{}7,~{}9$ via
mode coupling wave - wave interaction at first, followed by transfer of energy
from the coupled sideband modes to the bulk plasma particles via wave -
particle interaction i.e Landau damping which finally, causes damping of the
externally driven primary $k/k_{min}=1$ mode.

Figure 10: Portrait of power $log(|E_{k}(\omega)|^{2})$ versus frequency $(\omega)$ using 1D Fast Fourier Transform (FFT) analysis for $(k_{0},k):(4,1)$, $N_{x}\times N_{v}=[2048\times 10000]$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$, $\Delta t=100$ DP case with $k_{min}=0.1$. Oscillation frequency of maximum amplitude for primary perturbation mode $k/k_{min}=1$ is $\omega_{k=0.1}=1.02290$ and corresponding frequencies for secondary modes are tabulated in Table 2 . Table 2: Oscillation frequency $(\omega_{k})$ and Phase Velocity $(v_{\phi}^{k}=\omega_{k}/k)$ corresponding to the primary and secondary interacting sideband modes for $(k_{0},k):(4,1),~{}E_{D}^{0}=0.001,~{}A=0.05,~{}\omega_{D}^{Lan}=1.0148,~{}\Delta t=100,~{}k_{min}=0.1$ case. Mode No. | $[\omega_{k}^{max}]$ | $[v_{\phi}^{k}=\omega_{k}^{max}/k]$
---|---|---
$k=0.1$ | 1.02290 | 10.2290
$|k-k_{0}|=0.3$ | 1.02290 | 3.4096
$|k+k_{0}|=0.5$ | 1.02290 | 2.0458
$|k-2k_{0}|=0.7$ | 1.00984 | 1.4426
$|k+2k_{0}|=0.9$ | 1.01034 | 1.1226

Figure 11: Variation of spatially averaged distribution function
$\widehat{f}(v,t)$ with respect to velocity $(v)$ at times
$t=0,~{}10000~{}\omega_{pe}^{-1}$ for $(k_{0},k):(4,1)$, $A=0.05$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP case. In
the inset plot, we can observe bump around phase velocity locations
$v_{\phi}^{k}=3.4096,~{}2.0458$ which corresponds to the coupled
$k/k_{min}=3,~{}5$ sideband modes respectively. Phase velocities corresponding
each interacting mode is tabulated in Table 2

Fig. 11 shows the logarithmic change of spatially averaged distribution
function $\widehat{f}(v,t)$ defined by Eq. 18 with respect to velocity $(v)$
at initial and final times i.e $t=0,~{}10000~{}\omega_{pe}^{-1}$ for
$(k_{0},k):(4,1)$ with
$E_{D}^{0}=0.001,~{}A=0.05,~{}k_{min}=0.1,\omega_{D}^{Lan}=1.0148,~{}\Delta
t=100$ DP case. Inset plot shows the zoomed variation of $\widehat{f}(v,t)$
around the resonance velocity $v_{\phi}^{k}$ locations. Formation of plateau
or hump region i.e “local flattening” around the resonance location
$v_{\phi}^{k}=3.4096,~{}2.0458$ corresponding to the coupled sideband modes
$k/k_{min}=3,~{}5$ indicating the resonating wave - particle energy exchange
interaction leading to the particle trapping structures. It is also important
to note that we have observed plateau formation around sideband like
$k/k_{min}=2$ in the IVP case [Sec. 3.4 of Ref. [28]], whereas despite the
increasing the ion inhomogeneity amplitude value which is a major factor in
mode coupling dynamics between interacting modes [27, 54, 48, 28], we are
unable to engage the $k/k_{min}=2$ sideband in the DP case [See Figs. 12 and
13].

Figure 12: Phase space portrait of electron distribution function
$f_{e}(x,v,t)$ at late time $t=10000~{}\omega_{pe}^{-1}$ for (a) homogeneous
$A=0$ and (b) inhomogeneous $A=0.05$ with $(k_{0},k):(4,1)$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP case. In
(b) we can observe the formation of phase space vortices due to particle
trapping around resonance locations $v_{\phi}^{k}=3.4096,~{}2.0458$
corresponding to $k/k_{min}=3,~{}5$ respectively as tabulated in Table 2.

Figure 13: Phase space portrait of electron distribution function
$f_{e}(x,v,t)$ at different time (a) $t=0~{}\omega_{pe}^{-1}$, (b)
$t=4000~{}\omega_{pe}^{-1}$, (c) $t=7000~{}\omega_{pe}^{-1}$ and (d)
$t=10000~{}\omega_{pe}^{-1}$ for $A=0.05,~{}(k_{0},k):(4,1)$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP case. One
can observe temporal evolution of vortex structures around resonance locations
$v_{\phi}^{k}=3.4096,~{}2.0458$ corresponding to $k/k_{min}=3,~{}5$ sideband
modes respectively as tabulated in Table 2.

Figure 14: Variation of excess density fraction (EDF)$:(\delta n/n_{0})$ at
$x=L_{max}/4$ where $L_{max}=20\pi=62.8318$ with respect to time for
$(k_{0},k):(4,1)$, $A=0.05$, $[N_{x}\times N_{v}]=[2048\times 10000]$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP case. The
dotted black line indicates the time when the external drive is switched off.

In Fig. 12, phase space portrait of the electron distribution function
$f_{e}(x,v)$ at late time $t=10000~{}\omega_{pe}^{-1}$ is shown for (a)
homogeneous $A=0$ and (b) $A=0.05$ inhomogeneous plasma cases with
$(k_{0},k):(4,1)$, $A=0.05$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and
$\Delta t=100$ DP case. It demonstrates that for the externally driven plasma
systems in the absence of ion inhomogeneity i.e for homogeneous $A=0$ case,
one can observe absence of phase space vortex structures [Fig. 12 (a)], due to
lack of wave - wave mode coupling and wave - particle interactions, whereas,
in the presence of background ion inhomogeneity i.e $A\neq 0$ cases, energy
exchange via wave - wave mode coupling interactions between primary
perturbation $k/k_{min}=1$ mode and the coupled sideband $k/k_{min}=3,~{}5$
modes takes place which leads to the formation of phase space vortices as a
result of resonant energy exchange interactions between particles and sideband
modes at phase velocity $v_{\phi}^{k}$ locations as tabulated in Table. 2.
Fig. 13 shows the phase space portrait of electron distribution function
$f_{e}(x,v)$ at different time (a) $t=0~{}\omega_{pe}^{-1}$, (b)
$t=4000~{}\omega_{pe}^{-1}$, (c) $t=7000~{}\omega_{pe}^{-1}$ and (d)
$t=10000~{}\omega_{pe}^{-1}$ for $A=0.05,~{}(k_{0},k):(4,1)$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP case,
illustrating the temporal evolution of the electron distribution function in
phase space $(x,v)$. In Fig. 13 [(b), (c) and (d)], vortex structure streams
with vortivity 3 and 5 belongs to coupled sideband modes $k/k_{min}=3,~{}5$
respectively indicating wave - particle resonance interactions at
$v_{\phi}^{k}=3.4096,~{}2.0458$ locations [Table 2]. Also, it signifies the
non-participation of the sideband modes other than $k/k_{min}=3,~{}5$ in DP
cases which are seen in the IVP cases [28].

Figure 15: Temporal variation of the (a) relative kinetic, potential and total
energies $(\Delta KE,~{}\Delta PE,~{}\Delta TE)$ defined in Eqs. 21, 22
respectively, and (b) relative entropy $S_{rel}$ defined in Eq. 23 for
$(k_{0},k):(4,1)$, $A=0.05$, $k_{min}=1$, $[N_{x}\times N_{v}]=[2048\times
10000]$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$ DP
case. The dotted vertical line at $t=100~{}\omega_{pe^{-1}}$ represents the
time when the external electric field drive is switched off.

We show the excess density fraction (EDF) $:\delta n/n_{0}$ variations in Fig.
14 which demonstrates the variation of EDF at $x=L_{max}/4$ where
$L_{max}=2\pi/k_{min}$ for $(k_{0},k):(4,1)$, $A=0.05$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$, $\Delta t=100$ DP case
indicating the particle trapping and detrapping phenomenon due to resonant
interactions between particles and coupled sideband modes. From Figs. 5 and
14, it is obvious that increase in the ion inhomogeneity amplitude $A$ leads
to the frequent particle trapping and detrapping modulations. Fig. 15 shows
the (a) relative kinetic, potential and total energies $(\Delta KE,~{}\Delta
PE,~{}\Delta TE)$ defined in Eqs. 21, 22 respectively, and (b) relative
entropy $S_{rel}$ defined in Eq. 23 for $(k_{0},k):(4,1)$, $A=0.05$,
$k_{min}=1$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$ and $\Delta t=100$
DP case with grid resolution $N_{x}\times N_{v}=[2048\times 10000]$ in $(x,v)$
domains for both the electrons and ions. The dotted vertical line at
$t=100~{}\omega_{pe}^{-1}$ represents the time when the external electric
field drive is switched off. As already mentioned in sec. 4.1, time invariance
shown in these plots for energy and entropy calculations when the external
electric field drive is switched off, act as an indicator of numerically
stable, high resolution, steady state simulations with the chosen grid
discretizations $[N_{x}\times N_{v}]$ in space and velocity $(x,v)$ domains.

Figure 16: Phase space portrait of electron distribution function
$f_{e}(x,v,t)$ at time $t=10000~{}\omega_{pe}^{-1}$ for
$A=0.10,~{}(k_{0},k):(2,1)$, $\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$,
$\Delta t=100$ DP case. In (b), vortex chain at $v=0$ is due to chosen
equilibrium inhomogeneity scale $k_{0}/k_{min}=2$. In (c), we can observe two
separate streams of phase space vortex structures corresponding to
$k/k_{min}=3$ mode. In (d), phase space vortex structure stream corresponding
to $k/k_{min}=5$ sideband mode is shown.

To illustrate the effect of inhomogeneity scale $k_{0}$ and perturbation scale
$k$ dynamics in spatially non - uniform driven plasma systems, with rest of
the parameter sets identical to the DP cases, we simulated a case where
$k_{0}/k_{min}=2$ instead of $k_{0}/k_{min}=4$ along-with inhomogeneity
amplitude $A=0.10$. We only demonstrate Fig. 16 [(a), (b), (c) and (d)] which
shows phase space portrait of electron distribution function $f_{e}(x,v,t)$ at
time $t=10000~{}\omega_{pe}^{-1}$ for $A=0.10,~{}(k_{0},k):(2,1)$,
$\omega_{D}^{Lan}=1.0148$, $E_{D}^{0}=0.001$, $\Delta t=100$ DP case. In Fig.
16 (b), phase space vorteex chain at $v=0$ is due to choosen equilibrium
inhomogeneity $k_{0}/k_{min}=2$ with two periodic vortices. On detailed
inspection of this case, we draw the conclusion that physics behind the
damping and associated energy exchange mechanisms for this out of phase
primary driven perturbation $k/k_{min}=1$ mode is analogous to the previous
$A\neq 0$ driven inhomogeneous cases. Meanwhile, the sidebands created for
this case will govern according to $|k\pm Nk_{0}|$ [26, 27] where $N$ is the
mode coupling parameter defined in Sec. 4.1. Also, in Fig. 16 (c), we see the
existence of two different vortex structure streams belonging to the same
sideband $k/k_{min}=3$ which is quite different from the previous $A\neq 0$ DP
plasma cases but analogous to some extent to the IVP case shown in Ref. [28].

##  5 Discussion and Conclusion

In this work, we have presented an investigation on the interaction of a
primary “cold” ($v_{\phi}\gg v_{thermal}$) electron plasma (EP) mode coupling
as well as energy exchange dynamics created by low amplitude, constant
frequency (Langmuir frequency $\omega_{D}^{Lan}$) external electric field
drive, in the presence of stationary background of inhomogeneous ions. We have
also compared the solutions obtained from two different perturbation
approaches i.e initial value perturbation (IVP) approach and driven
perturbation (DP) approach for the identical simulation parameters. The
results are summarized as follows $:\longrightarrow$

  * •

We have demonstrated that externally driven electron plasma mode with phase
velocity $v_{\phi}\gg v_{thermal}:v_{\phi}\gg v_{e}^{max}$ can be coupled to
the plasma bulk via stationary background of non - uniform ions through wave -
wave and wave - particle interaction phenomenon analogous to initial value
perturbation (IVP) cases.

  * •

In both IVP and driven perturbation (DP) cases, from simulation data we infer
that time taken by the modes to reach the first minimum of the field signature
i.e $\tau_{FM}$ is inversely proportional to ion inhomogeneity amplitude
[Figs. 1, 2 and 8].

  * •

In Fig. 20 of Ref. [28], we found two vortex structures in phase space, around
$v_{\phi}=5.290$ due to $k/k_{min}=2$ sideband mode, which is absent in the DP
case as shown in Fig. 4. Also, it is important to note that with the increased
ion inhomogeneity amplitude DP case with $A=0.05$, we are unable to engage
$k/k_{min}=2$ sideband mode.

  * •

From Fig. 12, by illustrating the 2D $(\omega.k)$ power spectrum for $A=0.05$
DP case, we clearly demonstrate the distinct role of wave-wave mode coupling
dynamics between driven primary perturbation $k/k_{min}=1$ mode and
interacting coupled sideband $k/k_{min}=3,~{}5,~{}7,~{}9$ modes in the
inhomogeneous plasma cases which is absent in the homogeneous plasma $A=0$ DP
cases.

  * •

As we understand, following is the energy exchange mechanisms occurring
between driven $k/k_{min}=1$ wave, sidebands $k/k_{min}=3,~{}5,~{}7,~{}9$ and
particles which leads to the damping of out of phase primary perturbation mode
with $v_{\phi}\gg v_{thermal}$ : $v_{\phi}\gg v_{e}^{max}$ $\longrightarrow$

    * –

At first, energy exchange between primary perturbed $k/k_{min}=1$ and
sidebands $k/k_{min}=3,~{}5,~{}7,~{}9$ occurs via wave - wave mode coupling
interactions.

    * –

Next, phase velocities corresponding to the sideband modes
$k/k_{min}=3,~{}5,~{}7,~{}9$ falls inside the plasma bulk i.e
$v_{\phi}^{k}\sim v_{thermal}$ leading to energy exchange between sidebands
and particles through resonance wave - particle interactions (Landau damping).

    * –

Finally, energy from the externally driven primary perturbation $k/k_{min}=1$
mode is transferred into the plasma.

  * •

In the present work, kinetic phenomena such as Landau damping, wave -wave mode
coupling energy exchange interactions, formation of phase space vortex
structure or BGK waves etc, are demonstrated for the externally driven plasma
system in the presence of non-uniform background of stationary ions.

Our findings brings out several key features related to externally driven
systems in the presence of background spatial ion non-uniformity which may be
crucial in understanding the phenomena such as wave-wave mode coupling
dynamics and collisionless plasma turbulence etc, relevant to the plasmas in
laboratory as well as astrophysical settings. Our future investigations will
be based on the study of plasma systems consisting of several modes
interacting due to non-linearity and mode coupling in the presence of a
spectrum of ion waves excited with various types of external drivers.

## Acknowledgments

All the computational results of this paper were obtained using the ANTYA HPC
Linux cluster at Institute for Plasma Research (IPR) Gandhinagar, India. The
authors would like to thank the Data Center staff at IPR.

## Data availability statement

The data that support the findings of this study are available upon reasonable
request from the authors.

## References

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