# Non-modal kinetic theory of the stability of the compressed-sheared plasma
flows generated by the inhomogeneous microscale turbulence in the tokamak edge
plasma

V. S. Mikhailenko<EMAIL_ADDRESS>Plasma Research Center, Pusan
National University, Busan 46241, South Korea  V. V. Mikhailenko
<EMAIL_ADDRESS>BK21 FOUR Information Technology, Pusan National
University, Busan 46241, South Korea  Hae June Lee<EMAIL_ADDRESS>Department of Electrical Engineering, Pusan National University, Busan 46241,
South Korea

###### Abstract

A nonmodal kinetic theory of the stability of the two-dimensional compressed-
sheared mesoscale plasma flows, generated by the radially inhomogeneous
electrostatic ion cyclotron parametric microturbulence in the pedestal plasma
with a sheared poloidal flow, is developed. This theory reveals that the
separate spatially uniform Fourier modes of the electrostatic responses of the
ions and of the electrons on the mesoscale convective flows are determined
only in the frames of references moved with velocities of the ion and electron
convective flows. In the laboratory frame, these modes are observed as the
compressed-sheared modes with time dependent wave numbers. The integral
equation, which governs the separate Fourier mode of the electrostatic
potential of the plasma species responses on the mesoscale convective flows,
is derived. In this equation, the effects of the compressing and shearing of
the convective flows are revealed as the time dependence of the finite ion
Larmor radius effect. The solution of this equation for the kinetic drift
instability displays the nonmodal transformation of the potential to the zero
frequency cell-like perturbation when time elapsed.

###### pacs:

52.35.Ra, 52.35.Kt

##  I Introduction

The linear theory of the interaction of the fast waves (FW) with tokamak
plasma predicts Ono ; Chiu  that the injection of FW may be the efficient
method for the electron heating and current drive to aid in steady-state non-
inductive tokamak operation. These predictions has been confirmed for the
propagation and absorption of FWs in the hot core tokamak plasma bounded by
the last closed flux surface (LCFS) on numerous tokamak devices over the past
half century. These experiments, however, demonstrated that the efficiency of
the FW heating and current drive reduces by the FW power lost, which occurs in
the near-antenna layer of the cold low density scrape-off layer (SOL) tokamak
plasma. The FW heating experiments on the National Spherical Torus eXperiment
(NSTX) showedPerkins ; Perkins1  that around 30% to more than 60% of the FW
energy lost directly in the SOL. The bursts of the ions with energy above 20
keV, experimentally observedPace  in SOL following FW injection, and the
development of the parametric instabilities in SOL Wilson , predicted earlier
theoreticallyPorkolab1 ; Porkolab2 , were considered as the main channels of
the FW absorption in SOL. The development of the ion cyclotron (IC) quasimode
decay instability was consideredPace ; Wilson  as the main nonlinear process
responsible for the absorption of the FW power in SOL plasma and of the
anomalous heating of ions in SOL. The analysis of the turbulent heating of
ions by the IC parametric turbulence, powered by the IC quasimode decay
instability, was given in Ref.Mikhailenko  on the base of the numerical
solution of the dispersion equation for the IC parametric instabilities driven
by FW, and on the base of quasilinear theory for ion distribution function,
which accounted for the interaction of ions with IC parametric turbulence
powered by the IC quasimode decay instability. The derived estimates for the
turbulent ion heating rates revealed that the absorption of the FW energy by
ions in SOL is a weak effect, which provides only negligibly small heating of
cold ions in SOL and can not be responsible for the observed generation of the
high energy ions.

The FW power loss in SOL was investigated in Refs. Bertelli ; Bertelli1  by
using the numerical full wave simulation code AORSA (all-orders spectral
algorithm)Jaeger ; Green , in which the edge plasma beyond LCFS is included in
the solution domain. This simulation displays that the dominant loss of the FW
power occurs in the edge of the tokamak plasma, where the plasma density and
the amplitude of the FW field strongly change on the radial intermediate
spatial scale (mesoscale) between the macroscale of the spatial inhomogeneity
length of the FW field in the bulk of the tokamak plasma, and the microscale
commensurable with the wavelength of the parametric instabilities of the IC
perturbations. The theory of the microscale parametric turbulence of the
inhomogeneous plasma driven by the strong inhomogeneous on the mesoscale FW,
was developed in Ref. Mikhailenko1 . This theory reveals the effect of the
formation of the radial and poloidal convective flows of such a plasma caused
by the mesoscale spatial inhomogeneity of the microscale IC or drift
turbulence. The radial and poloidal convective flow velocity components are
proportional to the gradient of the spectral intensity of the electric field
of the microturbulence. This result gives the possible explanation of the FW
heating experiment on the National Spherical Torus eXperiment (NSTX)Perkins ;
Perkins1 ; Hosea , where it was found that a significant part of the FW power
loss occurs due to the anomalous convective flow of the collisionless dense
hot plasma from the tokamak edge to the cold low density SOL plasma.

The pedestal region, where plasma density profile has largest radial gradient,
is the most preferable region in tokamaks for the development of the
convective flows, driven by the spatially inhomogeneous microturbulence. The
inherent component of the pedestal plasma is the sheared poloidal flow, in
which the drift type instabilities, responsible for the anomalous transport of
plasma, are suppressed when their growth rates are less than the flow velocity
shearing rateBurrell . It was proved in Refs. Mikhailenko2 ; Mikhailenko4 ;
Mikhailenko5  that the basic point in understanding the processes of the
instabilities and turbulence evolution in plasma sheared flows is the proper
treatment of the persistent deformation of the perturbations by the sheared
flows. This effect, which is completely ignored in the canonical stability
theory, where the perturbations are considered having a static structure of a
plane wave $\sim\exp\left(i\mathbf{k}\cdot\mathbf{r}-i\omega t\right)$, is
involved in the nonmodal kinetic theory, developed in Refs. Mikhailenko2 ;
Mikhailenko4 ; Mikhailenko5 , grounded on the methodology of the sheared
modes. It was foundMikhailenko2 ; Mikhailenko4  that in the sheared flow, the
separate spatial Fourier mode with a static spatial structure
$\sim\exp\left(ik_{x}x+ik_{y}y+ik_{z}z\right)$ can be determined only in the
frame convected with a sheared flow. In the laboratory frame, this mode is
observed as the sheared mode with time dependent structure resulted from the
continuous distortion with time the perturbation by the sheared flow. This
distortion grows with time and forms a time-dependent nonmodal process which
is investigated as the initial value problem.

In Ref.Mikhailenko3 , the Vlasov equations, which govern the ion and electron
mesoscale convective flows with radially inhomogeneous flow velocities in the
poloidal sheared flow were derived. These equations predict the generation of
the sheared poloidal convective flow and of the radial compressed flow with
radial flow velocity gradient. The hydrodynamic theory of the mesoscale
convective flows, derived as the moments of the obtained Vlasov equation,
reveals the radial compressed convective flow as the dominant factor in the
formation of the steep pedestal density profile with density gradient
exponentially growing with time. The focus of this paper is the development of
the nonmodal kinetic theory of the stability of the two-dimensional
compressed-sheared convective flow. In Sec. II, we present basic equations and
their transformations. In Sec. III, we develop the nonmodal approach to the
kinetic theory of the compressed-sheared convective flows. In this theory we
derived new spatial reference coordinates in which the distribution functions
of the unperturbed convective sheared-compressed flows is stationary. The
stability of such a distribution functions of the convected plasma species
against the development of the short scale instabilities is given in Sec. IV
employing the developed compressed-sheared modes approach. The Conclusions are
given in Sec. V.

##  II Basic equations and transformations

Our theory is based on the Vlasov-Poisson system in a slab geometry
approximation where $x,y,z$ directions are viewed as corresponding to the
radial, poloidal and toroidal directions, respectively, of the toroidal
coordinate system. Within this approximation, the Vlasov equation for the
velocity distribution function $F_{\alpha}$ of the poloidal sheared flow of
$\alpha$ plasma species ($\alpha=i$ for ions and $\alpha=e$ for electrons) in
the FW field with coordinates $\mathbf{r}=\left(x,y,z\right)$ has a form

$\displaystyle\displaystyle\frac{\partial
F_{\alpha}\left(\mathbf{v},\mathbf{r},t\right)}{\partial
t}+\mathbf{v}\frac{\partial
F_{\alpha}\left(\mathbf{v},\mathbf{r},t\right)}{\partial\mathbf{r}}$
$\displaystyle\displaystyle+\frac{e_{\alpha}}{m_{\alpha}}\left(\mathbf{E}_{0x}\left(x\right)+\mathbf{E}_{1}\left(x,t\right)+\tilde{\mathbf{E}}\left(\mathbf{r},t\right)\right.$
$\displaystyle\displaystyle\left.+\frac{1}{c}\left[\mathbf{v}\times\left(\mathbf{B}_{0}+\mathbf{B}_{1}\left(\mathbf{r},t\right)\right)\right]\right)\frac{\partial
F_{\alpha}\left(\mathbf{v},\mathbf{r},t\right)}{\partial\mathbf{v}}=0.$ (1)

This equation contains the inhomogeneous radial electric field
$\mathbf{E}_{0x}\left(x\right)$, which governs the basic poloidal sheared
flow, the FW electric field $\mathbf{E}_{1}\left(x,t\right)$, the electric
field $\tilde{\mathbf{E}}\left(\mathbf{r},t\right)$ of the self-consistent
plasma response on FW, the uniform plasma-confining magnetic field
$\mathbf{B}_{0}$ directed along coordinate $z$, and FW magnetic field
$\mathbf{B}_{1}\left(\mathbf{r},t\right)$. For the edge layer of the tokamak
plasma, this equation contains two disparate spatial inhomogeneity lengths,
which are introduced by the FW field and by plasma parameters. In the edge
plasma, the spatial inhomogeneity of $\mathbf{E}_{0x}\left(x\right)$ and of FW
fields are commensurable with a spatial inhomogeneity length of the pedestal
plasma density. These mesoscale spatial inhomogeneity lengths on order of the
pedestal width are much less than the the inhomogeneity scale lengths of FW
and of the plasma parameters in the plasma core, but are much larger than the
radial wavelengths of the IC parametric and drift microturbulence. Electric
field $\tilde{\mathbf{E}}\left(\mathbf{r},t\right)$, being the microscale
responc of the inhomogeneous plasma on the inhomogeneous FW and
$\mathbf{E}_{0x}\left(x\right)$ fields, contains micro- and meso- spatial
scales. Our theory of the mesoscale plasma evolution caused by the mesoscale
inhomogeneities of the microturbulence, involves the treatments on the micro-
and mesoscales. In our theoryMikhailenko3 , we introduced jointly with
variables $\mathbf{r}=\left(x,y,z\right)$ and time $t$ for the microscale fast
variations on time of the order of the FW period or of the period of the IC
microturbulence, the slow time $T=\varepsilon t$, and the slow spatial
variables $X=\varepsilon x$, $Y=\varepsilon y$, where the dimensionless
parameter $\varepsilon\ll 1$, for the description of the slow evolutionary
mesoscale processes in the pedestal region. With these microscale and
mesoscale variables, electric field $\mathbf{E}_{0x}$ depends only on $X$. The
FW fields $\mathbf{E}_{1}$, $\mathbf{B}_{1}$, determined as

$\displaystyle\displaystyle\mathbf{E}_{1}\left(X,t\right)=\mathbf{E}_{1x}\left(X\right)\cos\omega_{0}t+\mathbf{E}_{1y}\left(X\right)\sin\omega_{0}t,$
(2)
$\displaystyle\displaystyle\mathbf{B}_{1}\left(X,t\right)=\frac{c}{\omega_{0}}\frac{dE_{1y}\left(X\right)}{dX}\cos\omega_{0}t\,\mathbf{e}_{z}.$
(3)

depend on slow mesoscale $X$ and fast time $t$. The electric field
$\tilde{\mathbf{E}}\left(\mathbf{r},X,t\right)$ depends on the spatial micro-
and mesoscale variables and on the fast time. This field is determined by the
Poisson equation

$\displaystyle\displaystyle\nabla\cdot\tilde{\mathbf{E}}\left(\mathbf{r},X,t\right)=4\pi\sum_{\alpha=i,e}e_{\alpha}\int
f_{\alpha}\left(\mathbf{v},\mathbf{r},X,t\right)d{\bf v},$ (4)

in which $f_{\alpha}$ is the fluctuating part of the distribution function
$F_{\alpha}$, $f_{\alpha}=F_{\alpha}-F_{0\alpha}$, where $F_{0\alpha}$ is the
equilibrium distribution function.

It is obvious that it is not possible to apply directly to the Vlasov-Poisson
system (1), (4) with spatially inhomogeneous oscillating FW fields the methods
of the solutions known for the investigations of the stability of a plasma in
static equilibrium. Any microscale perturbations of the ion and electron
densities are convected by FW field with inhomogeneous on the mesoscale
velocities, different for ions and electrons, and oscillating with frequency
of the FW field. It was found in Ref.Mikhailenko1  that the spatially
inhomogeneous FW field may be excluded from the Vlasov equation (1) by the
transformation of the velocity $\mathbf{v}$ and the position
$\mathbf{r}=\left(x,y,z\right)$ variables of the Vlasov equation (1) to new
velocity $\mathbf{v}_{i}$ and position
$\mathbf{r}_{i}=\left(x_{i},y_{i},z\right)$ variables determined in the
convected reference flow, which moves relative to the laboratory frame with
the velocity $\mathbf{V}_{i}\left(X_{i},t\right)$ of ion in the FW field,
given by the equation

$\displaystyle\displaystyle\frac{d\mathbf{V}_{i}\left(X_{i},t\right)}{dt}=\frac{e_{i}}{m_{\alpha}}\left(\mathbf{E}_{1}\left(X_{i}+\varepsilon
R_{ix}\left(X_{i},t\right),t\right)\right.$
$\displaystyle\displaystyle\left.+\frac{1}{c}\left[\mathbf{V}_{i}\times\mathbf{B}_{0}\right]+\frac{1}{c}\left[\mathbf{V}_{i}\times\mathbf{B}_{1}\left(X_{i}+\varepsilon
R_{ix}\left(X_{i},t\right),t\right)\right]\right),$ (5)

with initial value $\mathbf{V}_{i}\left(X_{i},t=t_{0}=0\right)=0$. The Vlasov
equation (1) with new variables $\mathbf{v}_{i}$, $X_{i}$, contains the
electric FW field only in terms on the order of $|R_{i}/L_{E}|\ll 1$, where
$|R_{i}|$ is the amplitude of the ion displacement in the spatially
inhomogeneous FW field with spatial inhomogeneity scale length $L_{E}$. These
terms are negligibly smallMikhailenko1  for the conditions of the FW tokamak
plasma heating and may be neglected. Without these terms, the Vlasov equation
in the frame convected with velocity $\mathbf{V}_{i}\left(X_{i},t\right)$ has
a form as for a static equilibria without the external FW field. That equation
for ions,

$\displaystyle\displaystyle\frac{\partial
F_{i}\left(\mathbf{v}_{i},\mathbf{r}_{i},X_{i},t\right)}{\partial
t}+\mathbf{v}_{i}\frac{\partial
F_{i}}{\partial\mathbf{r}_{i}}+\frac{e_{i}}{m_{i}c}\left[\mathbf{v}_{i}\times\mathbf{B}_{0}\right]\frac{\partial
F_{i}}{\partial\mathbf{v}_{i}}$
$\displaystyle\displaystyle+\frac{e_{i}}{m_{i}}\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)\frac{\partial
F_{i}\left(\mathbf{v}_{i},\mathbf{r}_{i},X_{i},t\right)}{\partial\mathbf{v}_{i}}=0,$
(6)

and similar equation for electrons, determined in the electron reference flow,
and the Poisson equation for the electric field

$\displaystyle\displaystyle\nabla\cdot\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)=4\pi\sum_{\alpha=i,e}e_{\alpha}\int
f_{\alpha}\left(\mathbf{v}_{\alpha},\mathbf{r}_{\alpha},X_{\alpha},t\right)d\mathbf{v}_{\alpha},$
(7)

determined in the ion reference flow, compose the system of equations for the
investigation of the microscale turbulence in FW field. The mesoscale
variables $X_{i}$ and $X_{e}$ are presented in this system as parameters.

At the time above the inverse growth rate of the IC instabilities,
$t\gg\gamma^{-1}\left(\mathbf{k}\right)>|\omega^{-1}\left(\mathbf{k}\right)|$
the microscale IC turbulence attains the steady state. At this state the
electric field $\tilde{\mathbf{E}}_{i}$ of the electrostatic two dimensional
IC parametric microturbulence, directed almost across the magnetic field
$\mathbf{B}_{0}$, may be presented in the ion reference flow in the form

$\displaystyle\displaystyle\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)=\sum\limits_{n}\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},n,t\right)$
$\displaystyle\displaystyle=\frac{1}{\left(2\pi\right)^{2}}\sum\limits_{n}\frac{1}{2}\int
d\mathbf{k}\left[\tilde{\mathbf{E}}_{i}\left(\mathbf{k},X_{i},n\right)e^{i\psi_{n}\left(\mathbf{r}_{i},X_{i},t\right)}\right.$
$\displaystyle\displaystyle\left.+\tilde{\mathbf{E}}_{i}^{\ast}\left(\mathbf{k},X_{i},n\right)e^{-i\psi_{n}\left(\mathbf{r}_{i},X_{i},t\right)}\right],$
(8)

where

$\displaystyle\displaystyle\psi_{n}\left(\mathbf{r}_{i},X_{i},t\right)=-i\omega_{n}\left(\mathbf{k}\right)t+i\mathbf{k}\mathbf{r}_{i}+i\theta\left(\mathbf{k}\right),$
(9)

i. e. as a linear superposition of the electric fields of IC perturbations
with frequencies
$\omega_{n}\left(\mathbf{k}\right)=n\omega_{ci}+\delta\omega_{n}\left(\mathbf{k}\right)$
with wave vectors $\mathbf{k}$ directed across the magnetic field and with
mesoscale position dependent amplitudes
$\tilde{\mathbf{E}}_{i}\left(\mathbf{k},X_{i},n\right)$ and with phases fast
changed with time on the microscales. Reality of
$\tilde{\mathbf{E}}_{i}\left(\mathbf{r}_{i},X_{i},t\right)$ is insured without
introducing negative frequencies by the addition of the complex conjugate
terms with amplitudes
$\tilde{\mathbf{E}}_{i}^{\ast}\left(\mathbf{k},X_{i},n\right)$. The
integration over $\mathbf{k}$ is performed over wave vectors of the linearly
unstable IC perturbations, and $\theta\left(\mathbf{k}\right)$ is their
initial phase. In the electron frame, oscillating relative to the ion frame,
this electric field is determined by Eq. (8) with species subscript $i$
changed on $e$. The electric field
$\tilde{\mathbf{E}}_{e}\left(\mathbf{r}_{e},X_{e},t\right)$ in variables
$\mathbf{r}_{i},X_{i}$ has a form

$\displaystyle\displaystyle\tilde{\mathbf{E}}_{e}\left(\mathbf{r}_{e},X_{e},t\right)=\sum\limits_{n}\tilde{\mathbf{E}}_{e}\left(\mathbf{r}_{e},X_{e},n,t\right)$
$\displaystyle\displaystyle=\frac{1}{\left(2\pi\right)^{2}}\sum\limits_{n}\frac{1}{2}\int
d\mathbf{k}\left[\tilde{\mathbf{E}}_{i}\left(\mathbf{k},X_{i},n\right)\sum\limits_{p=-\infty}^{\infty}J_{p}\left(a_{ie}\right)\right.$
$\displaystyle\displaystyle\left.\times
e^{i\psi_{n}\left(\mathbf{r}_{i},X_{i},t\right)-ip\left(\omega_{0}t+\delta_{ie}\left(\mathbf{k},X_{i}\right)\right)}\right.$
$\displaystyle\displaystyle\left.+\tilde{\mathbf{E}}_{i}^{\ast}\left(\mathbf{k},X_{i},n\right)\sum\limits_{p=-\infty}^{\infty}J_{p}\left(a_{ie}\right)\right.$
$\displaystyle\displaystyle\left.\times
e^{-i\psi_{n}\left(\mathbf{r}_{i},X_{i},t\right)+ip\left(\omega_{0}t+\delta_{ie}\left(\mathbf{k},X_{i}\right)\right)}\right],$
(10)

where $J_{p}\left(a_{ie}\right)$ is the first kind Bessel function of order
$p$ with argument $a_{ie}$. Parameters $a_{ie}\sim k\xi_{ie}$, where
$\xi_{ie}$ is the amplitude of the relative displacement of electrons relative
to ions in FW field, and $\delta_{ie}$ were determined in Ref.Mikhailenko .

##  III The kinetic theory of the mesoscale compressed-sheared convective
flows

For the investigation on the slow time scale $T$ the mesoscale evolution of
the poloidal plasma sheared flow with microscale turbulence not suppressed by
the sheared flow, we transform the Vlasov-Poisson system from the microscale
to the mesoscale variables using the relations
$x_{i}=\frac{1}{\varepsilon}X_{i}$, $y_{i}=\frac{1}{\varepsilon}Y_{i}$ and
$t=\frac{1}{\varepsilon}T$ variables. With mesoscale variables Eq. (6) becomes

$\displaystyle\displaystyle\varepsilon\frac{\partial
F_{i}\left(\mathbf{v}_{i},X_{i},Y_{i},T,\varepsilon\right)}{\partial
T}+\varepsilon v_{ix}\frac{\partial F_{i}}{\partial X_{i}}+\varepsilon
v_{iy}\frac{\partial F_{i}}{\partial Y_{i}}$
$\displaystyle\displaystyle+\frac{e_{i}}{m_{i}c}\left[\mathbf{v}_{i}\times\mathbf{B}_{0}\right]\frac{\partial
F_{i}}{\partial\mathbf{v}_{i}}$
$\displaystyle\displaystyle+\frac{e_{i}}{m_{i}}\tilde{\mathbf{E}}_{i}\left(X_{i},Y_{i},T,\varepsilon\right)\frac{\partial
F_{i}}{\partial\mathbf{v}_{i}}=0,$ (11)

The electric field
$\tilde{\mathbf{E}}_{i}\left(X_{i},Y_{i},T,\varepsilon\right)$ is determined
by Eq. (8), in which phase $\psi_{n}$ is determined in the form

$\displaystyle\displaystyle\psi_{n}\left(X_{i},Y_{i},T,\varepsilon\right)$
$\displaystyle\displaystyle=-i\frac{1}{\varepsilon}\left(\omega_{n}\left(\mathbf{k}\right)T-ik_{x}X_{i}-ik_{y}Y_{i}\right)+i\theta\left(\mathbf{k}\right).$
(12)

The similar transformations should be performed for the electron Vlasov
equation. On the nonlinear stage of the IC parametric microturbulence
evolution at the time above the inverse growth rate of the IC instabilities,
$t\gg\gamma^{-1}\left(\mathbf{k}\right)>|\omega^{-1}\left(\mathbf{k}\right)|$,
electric field (8) becomes the random function of the initial phase
$\theta\left(\mathbf{k}\right)$. The motion of ions and electrons in this
field has a form of the random scattering of particles by the turbulent
electric field and mimics to the thermal motion of particles.

The average effect of the mesoscale inhomogeneity of the microscale IC
turbulence on the mesoscale evolution of the ion and electron distribution
functions of the poloidal sheared flow was considered in Ref. Mikhailenko3 .
The central point in this theory is the transformation of the velocity
$\mathbf{v}_{i}$ and coordinates $X_{i}$ and $Y_{i}$ to the new
microturbulence-associated velocity field $\tilde{\mathbf{v}}_{i}$ and
coordinates $\tilde{X}_{i}$, $\tilde{Y}_{i}$ determined by the
relationsMikhailenko3

$\displaystyle\displaystyle\tilde{\mathbf{v}}_{i}=\mathbf{v}_{i}-\tilde{\mathbf{V}}_{i}\left(X_{i},Y_{i},T,\varepsilon\right),$
(13)
$\displaystyle\displaystyle\tilde{X}_{i}=X_{i}-\int\limits^{t}_{t_{0}}\tilde{V}_{ix}\left(X_{i},Y_{i},T_{1},\varepsilon\right)dT_{1},$
(14)
$\displaystyle\displaystyle\tilde{Y}_{i}=Y_{i}-\int\limits^{T}_{T_{0}}\tilde{V}_{iy}\left(X_{i},Y_{i},T_{1},\varepsilon\right)dT_{1},$
(15)

or by their inverse,

$\displaystyle\displaystyle\mathbf{v}_{i}=\tilde{\mathbf{v}}_{i}+\tilde{\mathbf{U}}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right),$
(16) $\displaystyle\displaystyle
X_{i}=\tilde{X}_{i}+\tilde{R}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)=$
$\displaystyle\displaystyle=\tilde{X}_{i}+\int\limits^{T}_{T_{0}}\tilde{U}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T_{1},\varepsilon\right)dT_{1},$
(17) $\displaystyle\displaystyle Y_{i}=Y-V^{\prime}_{0}XT$
$\displaystyle\displaystyle=\tilde{Y}_{i}+\int\limits^{T}_{T_{0}}\tilde{U}_{iy}\left(\tilde{X}_{i},\tilde{Y}_{i},T_{1},\varepsilon\right)dT_{1},$
(18)

where $V^{\prime}_{0}$ is the velocity shear of the poloidal flow velocity
$V_{0}\left(X\right)=V^{\prime}_{0}X$ directed along coordinate $Y$. The
velocity $\tilde{\mathbf{V}}_{i}\left(X_{i},Y_{i},T,\varepsilon\right)$ is
determined by the Euler equation

$\displaystyle\displaystyle\varepsilon\frac{\partial\tilde{\mathbf{V}}_{i}}{\partial
T}+\varepsilon\tilde{V}_{ix}\frac{\partial\tilde{\mathbf{V}}_{i}\left(X_{i},Y_{i},T,\varepsilon\right)}{\partial{X}_{i}}$
$\displaystyle\displaystyle=\frac{e_{i}}{m_{i}}\left(\tilde{\mathbf{E}}_{i}\left(X_{i},Y_{i},T,\varepsilon\right)+\frac{1}{c}\left[\tilde{\mathbf{V}}_{i}\times\mathbf{B}_{0}\right]\right)$
(19)

as the velocity of an ion in the electric field
$\tilde{\mathbf{E}}_{i}\left(X_{i},Y_{i},T,\varepsilon\right)$ of the IC
parametric turbulence, where variables $X_{i}$ and $Y_{i}$ are determined in
the frame of references which moves with the velocity of an ion in the FW
field in the poloidal sheared flowMikhailenko3 . In variables
$\left(\tilde{X}_{i},\tilde{Y}_{i},T\right)$, the convective nonlinear part
$\tilde{V}_{ix}\frac{\partial}{\partial{X}_{i}}$ of the operator
$\frac{\partial}{\partial T}+\tilde{V}_{ix}\frac{\partial}{\partial{X}_{i}}$
of Eq. (19) vanishes and this operator is transformed to the linear one,
$\frac{\partial}{\partial T}$. Then, Eq. (19) becomes the ordinary
differential equation

$\displaystyle\displaystyle\varepsilon\frac{d}{dT}\tilde{\mathbf{U}}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)$
$\displaystyle\displaystyle=\frac{e_{i}}{m_{i}}\left(\tilde{\mathbf{E}}_{i}\left(\tilde{X}_{i}+\tilde{R}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right),\tilde{Y}_{i},T,\varepsilon\right)\right.$
$\displaystyle\displaystyle\left.+\frac{1}{c}\left[\tilde{\mathbf{U}}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)\times\mathbf{B}_{0}\right]\right).$
(20)

for
$\tilde{\mathbf{U}}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)=\tilde{\mathbf{V}}_{i}\left(X_{i},Y_{i},T,\varepsilon\right)$.
The solution to Eq. (20) may be easily derivedMikhailenko3  for the case of
the small displacement, $\left|\tilde{R}_{ix}\right|\ll L_{\tilde{E}}$, of an
ion in the inhomogeneous electric field $\tilde{\mathbf{E}}_{i}$. With the
approximation for the amplitude
$\tilde{\mathbf{E}}_{i}\left(\mathbf{k},X_{i},n\right)$ of the $n$-th harmonic
of the IC electric field in Eq. (20)

$\displaystyle\displaystyle\tilde{\mathbf{E}}_{i}\left(\mathbf{k},\tilde{X}_{i}+\tilde{R}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right),n\right)\approx\tilde{\mathbf{E}}_{i}\left(\mathbf{k},\tilde{X}_{i},n\right),$
(21)

the solution to Eq. (20) with the initial value
$\tilde{\mathbf{U}}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},T_{0}=0\right)=0$ is

$\displaystyle\displaystyle\tilde{U}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)=\frac{e_{i}}{\varepsilon
m_{i}}\sum\limits_{n}\int\limits^{T}_{0}dT_{1}$
$\displaystyle\displaystyle\times\left[\tilde{E}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},n,T,\varepsilon\right)\cos\frac{1}{\varepsilon}\omega_{ci}\left(T-T_{1}\right)\right.$
$\displaystyle\displaystyle\left.+\tilde{E}_{iy}\left(\tilde{X}_{i},\tilde{Y}_{i},n,T,\varepsilon\right)\sin\frac{1}{\varepsilon}\omega_{ci}\left(T-T_{1}\right)\right],$
(22)
$\displaystyle\displaystyle\tilde{U}_{iy}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)=\frac{e_{i}}{\varepsilon
m_{i}}\sum\limits_{n}\int\limits^{T}_{0}dT_{1}$
$\displaystyle\displaystyle\times\left[-\tilde{E}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},n,T,\varepsilon\right)\sin\frac{1}{\varepsilon}\omega_{ci}\left(T-T_{1}\right)\right.$
$\displaystyle\displaystyle\left.+\tilde{E}_{iy}\left(\tilde{X}_{i},\tilde{Y}_{i},n,T,\varepsilon\right)\cos\frac{1}{\varepsilon}\omega_{ci}\left(T-T_{1}\right)\right],$
(23)

With variables $\tilde{\mathbf{v}}_{i}$,
$\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T,\varepsilon$, where $Z_{i}=\varepsilon
z$, the Vlasov equation for the distribution function
$F_{i}\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T,\varepsilon\right)$
of ions in the sheared poloidal flow for time $T\gg\tau_{corr}\sim\gamma^{-1}$
becomes

$\displaystyle\displaystyle\varepsilon\frac{\partial F_{i}}{\partial
T}+\varepsilon\tilde{v}_{ix}\frac{\partial
F_{i}}{\partial\tilde{X}_{i}}+\varepsilon\left(\tilde{v}_{iy}-V^{\prime}_{0}T\tilde{v}_{ix}\right)\frac{\partial
F_{i}}{\partial\tilde{Y}_{i}}+\varepsilon v_{iz}\frac{\partial F_{i}}{\partial
Z_{i}}-\varepsilon\tilde{v}_{ix}\int\limits_{0}^{T}\frac{\partial}{\partial
X_{i}}\tilde{V}_{ix}\left(X_{i},Y_{i},T_{1},\varepsilon\right)dT_{1}\frac{\partial
F_{i}}{\partial\tilde{X}_{i}}$
$\displaystyle\displaystyle-\varepsilon\tilde{v}_{ix}\int\limits_{0}^{T}\frac{\partial}{\partial
X_{i}}\tilde{V}_{iy}\left(X_{i},Y_{i},T_{1},\varepsilon\right)dT_{1}\frac{\partial
F_{i}}{\partial\tilde{Y}_{i}}$
$\displaystyle\displaystyle-\varepsilon\tilde{U}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)\int\limits_{0}^{T}\frac{\partial}{\partial
X_{i}}\tilde{V}_{ix}\left(X_{i},Y_{i},T_{1},\varepsilon\right)dT_{1}\frac{\partial
F_{i}}{\partial\tilde{X}_{i}}$
$\displaystyle\displaystyle-\varepsilon\tilde{U}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)\left(V^{\prime}_{0}T+\int\limits_{0}^{T}\frac{\partial}{\partial
X_{i}}\tilde{V}_{iy}\left(X_{i},Y_{i},T_{1},\varepsilon\right)dT_{1}\right)\frac{\partial
F_{i}}{\partial\tilde{Y}_{i}}$
$\displaystyle\displaystyle+\omega_{ci}\tilde{v}_{iy}\frac{\partial
F_{i}}{\partial\tilde{v}_{ix}}-\omega_{ci}\tilde{v}_{ix}\frac{\partial
F_{i}}{\partial\tilde{v}_{iy}}$
$\displaystyle\displaystyle-\varepsilon\frac{e_{i}}{m_{i}}\left(\frac{\partial}{\partial\tilde{X}_{i}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)-V^{\prime}_{0}T\frac{\partial}{\partial\tilde{Y}_{i}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)\right)\frac{\partial
F_{i}}{\partial\tilde{v}_{ix}}$
$\displaystyle\displaystyle-\varepsilon\frac{e_{i}}{m_{i}}\frac{\partial}{\partial\tilde{Y}_{i}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},T\right)\frac{\partial
F_{i}}{\partial\tilde{v}_{iy}}-\varepsilon\frac{e_{i}}{m_{i}}\frac{\partial}{\partial
Z_{i}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)\frac{\partial
F_{i}}{\partial v_{iz}}=0.$ (24)

The electrostatic potential $\varphi_{i}$ in Eq. (24) depends on the micro-
and mesoscales and can be expressed in the form

$\displaystyle\displaystyle\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},\tilde{Y}_{i},t\right)=\tilde{\varphi}_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t\right)$
$\displaystyle\displaystyle+\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right),$
(25)

where $\tilde{\varphi}_{i}$ is the electrostatic potential of the microscale
turbulence,

$\displaystyle\displaystyle\tilde{\mathbf{E}}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)=-\nabla_{\mathbf{r}_{i}}\varphi_{i}\left(\tilde{\mathbf{r}}_{i}+\tilde{\mathbf{R}}_{i},\tilde{X}_{i},t\right),$
(26)

and $\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right)$ is the
potential which determines the electric field of the plasma response on the
mesoscale convective flows,

$\displaystyle\displaystyle\bar{\mathbf{E}}_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right)=-\nabla\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right).$
(27)

The Vlasov equation for the ion distribution function
$\bar{F}_{i}\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T,\varepsilon\right)$,
averaged over the microscale initial phases for a time
$t\gg\tau_{corr}\sim\gamma^{-1}$, is

$\displaystyle\displaystyle\frac{\partial\bar{F}_{i}}{\partial
T}+\tilde{v}_{ix}\frac{\partial\bar{F}_{i}}{\partial\tilde{X}_{i}}+\left(\tilde{v}_{iy}-V^{\prime}_{0}T\tilde{v}_{ix}\right)\frac{\partial\bar{F}_{i}}{\partial\tilde{Y}_{i}}-\bar{U}_{ix}\left(\tilde{X}_{i}\right)\frac{\partial\bar{F}_{i}}{\partial\tilde{X}_{i}}-\bar{U}_{iy}\left(\tilde{X}_{i}\right)\frac{\partial\bar{F}_{i}}{\partial\tilde{Y}_{i}}$
$\displaystyle\displaystyle+v_{iz}\frac{\partial\bar{F}_{i}}{\partial
Z_{i}}+\frac{1}{\varepsilon}\omega_{ci}\tilde{v}_{iy}\frac{\partial\bar{F}_{i}}{\partial\tilde{v}_{ix}}-\frac{1}{\varepsilon}\omega_{ci}\frac{\partial\bar{F}_{i}}{\partial\tilde{v}_{iy}}$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\left(\frac{\partial\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right)}{\partial\tilde{X}_{i}}-V^{\prime}_{0}T\frac{\partial\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right)}{\partial\tilde{Y}_{i}}\right)\frac{\partial
F_{i}}{\partial\tilde{v}_{ix}}$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\frac{\partial\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right)}{\partial\tilde{Y}_{i}}\frac{\partial
F_{i}}{\partial\tilde{v}_{iy}}-\frac{e_{i}}{m_{i}}\frac{\partial\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right)}{\partial
Z_{i}}\frac{\partial F_{i}}{\partial v_{iz}}=0,$ (28)

where the velocities $\bar{U}_{ix}\left(\tilde{X}_{i}\right)$ and
$\bar{U}_{iy}\left(\tilde{X}_{i}\right)$ are

$\displaystyle\displaystyle\bar{U}_{ix}\left(\tilde{X}_{i}\right)=\left\langle\tilde{U}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)\int\limits_{0}^{T}\frac{\partial}{\partial
X_{i}}\tilde{V}_{ix}\left(X_{i},Y_{i},T_{1},\varepsilon\right)dT_{1}\right\rangle,$
(29)
$\displaystyle\displaystyle\bar{U}_{iy}\left(\tilde{X}_{i}\right)=\left\langle\tilde{U}_{ix}\left(\tilde{X}_{i},\tilde{Y}_{i},T,\varepsilon\right)\int\limits_{0}^{T}\frac{\partial}{\partial
X_{i}}\tilde{V}_{iy}\left(X_{i},Y_{i},T_{1},\varepsilon\right)dT_{1}\right\rangle.$
(30)

The details of the calculation of $\bar{U}_{ix}\left(\tilde{X}_{i}\right)$ and
$\bar{U}_{iy}\left(\tilde{X}_{i}\right)$ for the arbitrary electric field
$\tilde{\mathbf{E}}_{i}$ are given in Ref. Mikhailenko1 . For the electric
field (8), the velocities $\bar{U}_{ix}\left(\tilde{X}_{i},T\right)$ and
$\bar{U}_{iy}\left(\tilde{X}_{i},T\right)$ are presented in the Appendix.

The electrostatic potential
$\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right)$ of the plasma
response on the mesoscale convective flows is governed by the Poisson equation

$\displaystyle\displaystyle\frac{\partial^{2}\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T,\varepsilon\right)}{\partial\tilde{X_{i}^{2}}}+\frac{\partial^{2}\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T,\varepsilon\right)}{\partial\tilde{Y_{i}^{2}}}$
$\displaystyle\displaystyle+\frac{\partial^{2}\Phi_{i}\left(\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T,\varepsilon\right)}{\partial
Z_{i}^{2}}$
$\displaystyle\displaystyle=-4\pi\sum_{\alpha=i,e}e_{\alpha}n_{\alpha}\left(\tilde{X}_{\alpha},\tilde{Y}_{\alpha},Z_{\alpha},T,\varepsilon\right).$
(31)

in which
$n_{\alpha}\left(\tilde{X}_{\alpha},\tilde{Y}_{\alpha},Z_{\alpha},T,\varepsilon\right)=\int
d\tilde{\mathbf{v}}_{\alpha}\bar{f}_{\alpha}\left(\tilde{\mathbf{v}}_{\alpha},\tilde{X}_{\alpha},\tilde{Y}_{\alpha},Z_{\alpha},T,\varepsilon\right)$
is the density perturbation, and
$\bar{f}_{\alpha}\left(\tilde{\mathbf{v}}_{\alpha},\tilde{X}_{\alpha},\tilde{Y}_{\alpha},Z_{\alpha},T,\varepsilon\right)=\bar{F}_{\alpha}\left(\tilde{v}_{\alpha\perp},\phi,v_{z},\xi_{\alpha},\eta_{\alpha},Z_{\alpha},T,\varepsilon\right)-\bar{F}_{\alpha
0}$ is the perturbation of the equilibrium distribution function
$\bar{F}_{\alpha 0}$ of the convected plasma species $\alpha$.

The Vlasov equation for the average electron distribution
$\bar{F}_{e}\left(\tilde{\mathbf{v}}_{e},\tilde{X}_{e},\tilde{Y}_{e},Z_{e},T\right)$,
where $\tilde{X}_{e},\tilde{Y}_{e}$ are determined by Eqs. (14), (15) with ion
species subscript changed on the electron species subscript, has a form
similar to Eq. (28). The velocities $\bar{U}_{ex}\left(\tilde{X}_{e}\right)$
and $\bar{U}_{ey}\left(\tilde{X}_{e}\right)$ are determined in this equation
by Eqs. (29), (30) in which velocities
$\tilde{U}_{ex}\left(\tilde{\mathbf{r}}_{e},\tilde{X}_{e},T\right)$ and
$\tilde{U}_{ey}\left(\tilde{\mathbf{r}}_{e},\tilde{X}_{e},T\right)$ are
determined by Eqs. (22), (23) with the turbulent electric field
$\tilde{\mathbf{E}}_{e}\left(\tilde{\mathbf{r}}_{e},\tilde{X}_{e},T\right)$,
given by Eq. (10).

The Vlasov equations (28) for
$\bar{F}_{i}\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i},\tilde{Y}_{i},Z_{i},T\right)$,
and the similar equation for
$\bar{F}_{e}\left(\tilde{\mathbf{v}}_{e},\tilde{X}_{e},\tilde{Y}_{e},Z_{e},T\right)$,
and the Poisson equation (31) compose the Vlasov-Poisson system, which governs
the kinetic mesoscale evolution of a plasma under the average action of the
spatially inhomogeneous microturbulence.

As a first step to solution of the system of Eqs. (28), (31), we find the
characteristics of the operator

$\displaystyle\displaystyle\frac{D}{DT}=\frac{\partial}{\partial
T}-\bar{U}_{ix}\left(\tilde{X}_{i}\right)\frac{\partial}{\partial\tilde{X}_{i}}-\bar{U}_{iy}\left(\tilde{X}_{i}\right)\frac{\partial}{\partial\tilde{Y}_{i}},$
(32)

which are determined by the system

$\displaystyle\displaystyle
dT=-\frac{d\tilde{X}_{i}}{\bar{U}_{ix}\left(\tilde{X}_{i}\right)}=-\frac{d\tilde{Y}_{i}}{\bar{U}_{iy}\left(\tilde{X}_{i}\right)}.$
(33)

For deriving the simplest solution to system (33), which reveals the effects
of the spatial inhomogeneity of the convective flow velocities, we use in Eq.
(33) the expansions

$\displaystyle\displaystyle\bar{U}_{ix}\left(\tilde{X}_{i}\right)=\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}^{(0)}_{i}\right)\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right),$
(34)

and

$\displaystyle\displaystyle\bar{U}_{iy}\left(\tilde{X}_{i}\right)=\bar{U}^{(0)}_{iy}+\bar{U}^{\prime}_{iy}\left(\tilde{X}^{(0)}_{i}\right)\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)$
(35)

at the vicinity of an arbitrary coordinate $\tilde{X}^{(0)}_{i}$, and consider
the case of the uniform velocity compressing rate,
$\bar{U}^{\prime}_{ix}=const$, and of the uniform velocity shearing rate,
$\bar{U}^{\prime}_{iy}=const$. The solution to system (33) for this case has a
formMikhailenko3

$\displaystyle\displaystyle\check{X}_{i}=\frac{1}{\bar{U}^{\prime}_{ix}}\left[\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)\right)e^{\bar{U}^{\prime}_{ix}T}\right.$
$\displaystyle\displaystyle\left.-\bar{U}^{(0)}_{ix}\right],$ (36)

and

$\displaystyle\displaystyle\check{Y}_{i}=\tilde{Y}_{i}+\left(\bar{U}^{(0)}_{iy}-\bar{U}^{(0)}_{ix}\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\right)T$
$\displaystyle\displaystyle-\frac{\bar{U}^{\prime}_{iy}}{\left(\bar{U}^{\prime}_{ix}\right)^{2}}\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\left(\tilde{X}_{i}-\tilde{X}^{(0)}_{i}\right)\right),$
(37)

where $\check{X}_{i}$ and $\check{Y}_{i}$ are the integrals of system (33)
with expansions (34), (35). Note, that at $T=0$,
$\check{X}_{i}=\tilde{X}_{i}-\tilde{X}^{(0)}_{i}$. In what follows, we put
$X_{0}=0$ for simplicity.

With variables $\check{X}_{i},\check{Y}_{i},Z_{i},T,v_{z},\varepsilon$ and
with $\tilde{v}_{i\perp},\phi$, determined by relations
$\tilde{v}_{ix}=\tilde{v}_{i\perp}\cos\phi$ and
$\tilde{v}_{iy}=\tilde{v}_{i\perp}\sin\phi$, the Vlasov equation (28) obtains
the form, which does not contain the explicit dependence on $\tilde{X}_{i}$,

$\displaystyle\displaystyle\frac{\partial}{\partial
T}\bar{F}_{i}\left(\tilde{v}_{i\perp},\phi,v_{z},\check{X}_{i},\check{Y}_{i},Z_{i},T,\varepsilon\right)+\tilde{v}_{i\perp}\cos\phi\,e^{\bar{U}_{ix}^{\prime}T}\frac{\partial\bar{F}_{i}}{\partial\check{X}_{i}}$
$\displaystyle\displaystyle+\left(\tilde{v}_{i\perp}\sin\phi-\tilde{v}_{i\perp}\cos\phi\,\left(V^{\prime}_{0}T+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)\right)\frac{\partial\bar{F}_{i}}{\partial\check{Y}_{i}}+v_{iz}\frac{\partial\bar{F}_{i}}{\partial
Z_{i}}-\omega_{ci}\frac{\partial\bar{F}_{i}}{\varepsilon\partial\phi}$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\left(e^{\bar{U}_{ix}^{\prime}T}\frac{\partial\Phi_{i}\left(\check{X}_{i},\check{Y}_{i},Z_{i},T\right)}{\partial\check{X}_{i}}-\left(V^{\prime}_{0}T+\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}\right)\frac{\partial\Phi_{i}\left(\check{X}_{i},\check{Y}_{i},Z_{i},T\right)}{\partial\check{Y}_{i}}\right)\frac{\partial\bar{F}_{i}}{\partial\check{v}_{ix}}$
$\displaystyle\displaystyle-\frac{e_{i}}{m_{i}}\frac{\partial\Phi_{i}\left(\check{X}_{i},\check{Y}_{i},Z_{i},T\right)}{\partial\check{Y}_{i}}\frac{\partial\bar{F}_{i}}{\partial\tilde{v}_{iy}}-\frac{e_{i}}{m_{i}}\frac{\partial\Phi_{i}\left(\check{X}_{i},\check{Y}_{i},Z_{i},T\right)}{\partial
Z_{i}}\frac{\partial\bar{F}_{i}}{\partial v_{iz}}=0.$ (38)

In Eq. (38), the effects the spatial inhomogeneity of the sheared and
convected flows is transformed to the time domain. These effects are presented
by the linearly growing with time $V^{\prime}_{0}T$ coefficient originated
from the basic poloidal sheared flow, and of the exponentially growing with
time coefficient $e^{\bar{U}_{ix}^{\prime}T}$ originated from the compressed
convective flow. These coefficients reveal the effects of the continuous
distortion of the perturbations in the sheared and compressed flows
Mikhailenko2 ; Mikhailenko4 ; Mikhailenko5 . It was found in Ref. Mikhailenko3
that the compressing rate $\bar{U}^{\prime}_{ix}$ of the radial velocity of
the compressed flow and the shearing rate $V^{\prime}_{0}$ of the poloidal
sheared flow velocity are commensurable for the tokamak edge condition, and
both are much less than the ion cyclotron frequency $\omega_{ci}$. Equation
(38) displays that the exponentially growing with time effect of the
compressed flow is the dominant factor in the mesoscale temporal evolution at
time $T>\left(\bar{U}^{\prime}_{ix}\right)^{-1}$ of the plasma with a radially
inhomogeneous turbulence. Therefore the small parameter $\varepsilon$ in Eq.
(38), which determines a ratio of the micro- to meso- scales, is naturally to
define as equal to $\varepsilon=\frac{\bar{U}^{\prime}_{ix}}{\omega_{ci}}$.

The next substantial simplification of Eq. (38) arises with transformation of
the part of Eq. (38), which does not contain the electrostatic potential
$\Phi_{i}\left(\check{X}_{i},\check{Y}_{i},T\right)$, by employing the
characteristic equations

$\displaystyle\displaystyle dT=-\frac{\varepsilon
d\phi}{\omega_{ci}}=\frac{dZ_{i}}{v_{z}}=\frac{d\check{X}_{i}}{\tilde{v}_{i\perp}\cos\phi\,e^{\bar{U}_{ix}^{\prime}T}}$
$\displaystyle\displaystyle=\frac{d\check{Y}_{i}}{\tilde{v}_{i\perp}\sin\phi-\tilde{v}_{i\perp}\cos\phi\,\left(V^{\prime}_{0}T+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)}.$
(39)

The solutions to Eqs. (39) are given by the relations

$\displaystyle\displaystyle\check{X}_{i}=\xi_{i}-\frac{\tilde{v}_{i\perp}\varepsilon}{\omega_{ci}}e^{\bar{U}_{ix}^{\prime}T}\sin\left(\phi_{1}-\frac{1}{\varepsilon}\omega_{ci}T\right)$
$\displaystyle\displaystyle+O\left(\frac{\varepsilon\bar{U}^{\prime}_{ix}}{\omega_{ci}}\ll
1\right),$ (40)
$\displaystyle\displaystyle\check{Y}_{i}=\eta_{i}+\frac{\tilde{v}_{i\perp}\varepsilon}{\omega_{ci}}\cos\left(\phi_{1}-\frac{1}{\varepsilon}\omega_{ci}T\right)$
$\displaystyle\displaystyle+\frac{\tilde{v}_{i\perp}\varepsilon}{\omega_{ci}}\left(V^{\prime}_{0}T+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)\sin\left(\phi_{1}-\frac{1}{\varepsilon}\omega_{ci}T\right)$
$\displaystyle\displaystyle+O\left(\varepsilon\frac{V^{\prime}_{0}}{\omega_{ci}}\ll
1\right),$ (41)
$\displaystyle\displaystyle\phi=\phi_{1}-\frac{1}{\varepsilon}\omega_{ci}T,$
(42) $\displaystyle\displaystyle Z_{i}=Z_{i1}+v_{z}T.$ (43)

The integrals $\xi_{i}$ and $\eta_{i}$ in Eqs. (40) and (41) are the guiding
center coordinates in the compressed-sheared convective flow. In coordinates
$\xi_{i},\,\eta_{i},\phi_{1},Z_{i1}$, Eq. (38) has a simple form

$\displaystyle\displaystyle\frac{\partial}{\partial
T}\bar{F}_{i}\left(\tilde{v}_{i\perp},\phi_{1},v_{z},\xi_{i},\eta_{i},Z_{i},T,\varepsilon\right)+\frac{e_{i}}{m_{i}}\frac{\omega_{ci}}{\varepsilon\tilde{v}_{i\perp}}\left(\frac{\partial\Phi_{i}}{\partial\phi_{1}}\frac{\partial\bar{F}_{i}}{\partial\tilde{v}_{i\perp}}-\frac{\partial\Phi_{i}}{\partial\tilde{v}_{i\perp}}\frac{\partial\bar{F}_{i}}{\partial\phi_{1}}\right)$
$\displaystyle\displaystyle+\frac{e_{i}\varepsilon}{m_{i}\omega_{ci}}e^{\bar{U}^{\prime}_{ix}T}\left(\frac{\partial\Phi_{i}}{\partial\xi_{i}}\frac{\partial\bar{F}_{i}}{\partial\eta_{i}}-\frac{\partial\Phi_{i}}{\partial\eta_{i}}\frac{\partial\bar{F}_{i}}{\partial\xi_{i}}\right)-\frac{e_{i}}{m_{i}}\frac{\partial\Phi_{i}}{\partial
Z_{i1}}\frac{\partial\bar{F}_{i}}{\partial v_{z}}=0.$ (44)

The solution to Eq. (44) for the ion distribution function $\bar{F}_{i}$ we
derive in the form
$\bar{F}_{i}\left(\tilde{v}_{i\perp},\phi,v_{z},\xi_{i},\eta_{i},Z_{i1},T,\varepsilon\right)=\bar{F}_{i0}+\bar{f}_{i}\left(\tilde{v}_{i\perp},\phi,v_{z},\xi_{i},\eta_{i},Z_{i1},T,\varepsilon\right)$,
where $\bar{F}_{i0}$ is the ion distribution function $\bar{F}_{i0}$ of the
unperturbed ion convective flow, and $\bar{f}_{i}$ is the perturbation of
$\bar{F}_{i0}$ caused by the ions respond on the plasma convective flows. The
equation for $\bar{F}_{i0}$ follows from Eq. (44), in which potential
$\Phi_{i}$ of the electrostatic response of plasma on the mesoscale convective
flows is excluded. This equation,

$\displaystyle\displaystyle\partial\bar{F}_{i0}/\partial T=0,$ (45)

reveals that with the guiding center coordinates $\xi_{i}$, $\eta_{i}$, the
unperturbed ion distribution function $\bar{F}_{i0}$ of the compressed-sheared
ion flow is stationary. The solution to Eq. (45) for the inhomogeneous ion
component along coordinate $\tilde{X}_{i}$ is an arbitrary function
$\bar{F}_{i0}=\bar{F}_{i0}\left(\tilde{\mathbf{v}}_{i},\xi_{i}\right)$. With
the initial Maxwellian distribution

$\displaystyle\displaystyle\bar{F}_{i0}\left(\tilde{\mathbf{v}}_{i},\tilde{X}_{i}\right)=\frac{n_{i0}\left(\tilde{X}_{i}\right)}{\left(2\pi
v^{2}_{Ti}\left(\tilde{X}_{i}\right)\right)^{3/2}}e^{-\frac{\tilde{v}^{2}_{i}}{2v^{2}_{Ti}\left(\tilde{X}_{i}\right)}},$
(46)

given for time $T=0$, at which $\xi_{i}\approx\check{X}_{i}=\tilde{X}_{i}$ (it
follows from Eq. (36) and from the estimate $\tilde{v}_{i}/\omega_{ci}\sim
v_{Ti}/\omega_{ci}\ll\left(L_{E},L_{n_{i}}\right)$) the solution for
$\bar{F}_{i0}$ will have a form (46), in which the ion equilibrium density and
the ion thermal velocity are equal to $n_{i0}\left(\xi_{i}\right)$ and
$v_{Ti}\left(\xi_{i}\right)$ respectively. Note, that with variable
$\tilde{X}_{i}$ the ion density $n_{i0}$ is the time dependent,

$\displaystyle\displaystyle
n_{i0}\left(\xi_{i}\right)=n_{i0}\left(\frac{1}{\bar{U}^{\prime}_{ix}}\left[\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\tilde{X}_{i}\right)e^{\bar{U}^{\prime}_{ix}T}\right.\right.$
$\displaystyle\displaystyle\left.\left.-\bar{U}^{(0)}_{ix}\right]\right).$
(47)

The same dependences on $\tilde{X}_{i}$ and $T$ has the ion thermal velocity
$v_{Ti}\left(\xi_{i}\right)$ in Eq. (46). Equation (46) reveals, that the time
dependent inhomogeneous ion density of the convective flow, as it is with
coordinate $\tilde{X}_{i}$, becomes the steady spatially inhomogeneous ion
density distribution along characteristic $\xi_{i}$ at any time $T$.

##  IV The compressed-sheared modes approach to the Stability theory of the
mesoscale convective flows

In this section, we consider the microscale respond of the ions and electrons,
determined by the functions $\bar{f}_{i}$ and $\bar{f}_{e}$ on the generation
of the mesoscale convective flows.

It follows from Eq. (44), that the Vlasov equation for the perturbation
$\bar{f}_{i}\left(\tilde{v}_{i\perp},\phi,v_{iz},\xi_{i},\eta_{i},Z_{i1},T,\varepsilon\right)$
of $\bar{F}_{i0}\left(\tilde{\mathbf{v}}_{i},\xi_{i}\right)$ has a simple form
in guiding center coordinates $\xi_{i}$ and $\eta_{i}$,

$\displaystyle\displaystyle\frac{\partial}{\partial
T}\bar{f}_{i}\left(\tilde{v}_{i\perp},\phi_{1},v_{iz},\xi_{i},\eta_{i},Z_{i1},T,\varepsilon\right)=-\frac{e_{i}}{m_{i}}\frac{\omega_{ci}}{\varepsilon\tilde{v}_{i\perp}}\frac{\partial\Phi_{i}}{\partial\phi_{1}}\frac{\partial\bar{F}_{i0}}{\partial\tilde{v}_{i\perp}}$
$\displaystyle\displaystyle+\frac{e_{i}\varepsilon}{m_{i}\omega_{ci}}e^{\bar{U}^{\prime}_{ix}T}\frac{\partial\Phi_{i}}{\partial\eta_{i}}\frac{\partial\bar{F}_{i0}}{\partial\xi_{i}}+\frac{e_{i}}{m_{i}}\frac{\partial\Phi_{i}}{\partial
Z_{i1}}\frac{\partial\bar{F}_{i0}}{\partial v_{iz}}.$ (48)

The Vlasov equation (48) for $\bar{f}_{i}$ with given unperturbed ion
distribution function $\bar{F}_{i0}$, the equation for the perturbation
$\bar{f}_{e}\left(\tilde{v}_{e\perp},\phi_{1},v_{z},\xi_{e},\eta_{e},Z_{e1},T,\varepsilon\right)$
of the electron distribution similar to Eq. (48), and the Poisson equation
(35) for the potential
$\Phi_{i}\left(\check{X}_{i},\check{Y}_{i},Z_{i1},T,\varepsilon\right)$ in
coordinates $\check{X}_{i},\,\check{Y}_{i}$

$\displaystyle\displaystyle
e^{2\bar{U}_{ix}^{\prime}T}\frac{\partial^{2}\Phi_{i}}{\partial\check{X}^{2}_{i}}-2e^{\bar{U}_{ix}^{\prime}T}\left(V^{\prime}_{0}T+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)\frac{\partial^{2}\Phi_{i}}{\partial\check{X}_{i}\partial\check{Y}_{i}}$
$\displaystyle\displaystyle+\left(1+\left(V^{\prime}_{0}T+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)^{2}\right)\frac{\partial^{2}\Phi_{i}}{\partial\check{Y}^{2}_{i}}+\frac{\partial^{2}\Phi_{i}}{\partial
Z^{2}_{i1}}$
$\displaystyle\displaystyle=-4\pi\left(e_{i}n_{i}\left(\check{X}_{i},\check{Y}_{i},Z_{i1},T;\varepsilon\right)\right.$
$\displaystyle\displaystyle\left.-|e|n_{e}\left(\check{X}_{e},\check{Y}_{e},Z_{e1},T;\varepsilon\right)\right).$
(49)

compose the system of equations for the investigations of the stability of the
mesoscale convective flows. In this section, we consider the stability of the
compressed-sheared flows against the development of the low frequency
microscale instabilities. For that goal, we transform Eqs. (48), (49) to the
microscale time $t=\frac{T}{\varepsilon}$ , the microscale spatial coordinates
$x_{i}=\frac{1}{\varepsilon}X_{i}$, $y_{i}=\frac{1}{\varepsilon}Y_{i}$ and to
microscale coordinates guiding center coordinates
$\check{\xi}_{i}=\frac{\xi_{i}}{\varepsilon}$,
$\check{\eta}_{i}=\frac{\eta_{i}}{\varepsilon}$. With time $t$ and microscale
coordinates $\check{\xi}_{i}$, $\check{\eta}_{i}$, the solution to Eq. (48)
with known distribution $\bar{F}_{i0}$ is

$\displaystyle\displaystyle\bar{f}_{i}=\frac{e_{i}}{m_{i}}\int\limits^{t}_{0}dt_{1}\left(\frac{e^{\bar{U}^{\prime}_{ix}t}}{\omega_{ci}}\frac{\partial\Phi_{i}}{\partial\check{\eta}_{i}}\frac{\partial\bar{F}_{i0}}{\partial\check{\xi}_{i}}\right.$
$\displaystyle\displaystyle\left.-\frac{\omega_{ci}}{\tilde{v}_{i\perp}}\frac{\partial\Phi_{i}}{\partial\phi_{1}}\frac{\partial\bar{F}_{i0}}{\partial\tilde{v}_{i\perp}}+\frac{\partial\Phi_{i}}{\partial
Z_{i1}}\frac{\partial\bar{F}_{i0}}{\partial v_{z}}\right),$ (50)

where the prime in $\bar{U}^{\prime}_{ix}$, $\bar{U}^{\prime}_{iy}$ and
$V^{\prime}_{0}$ denotes in this section the derivatives of $\bar{U}_{ix}$,
$\bar{U}_{iy}$ and $V_{0}$ with respect to the microscale coordinate
$\tilde{x}_{i}=\frac{\tilde{X}_{i}}{\varepsilon}$. Equation (50), as well as
Eq. (44), do not contain the spatial inhomogeneity originated from the
inhomogeneity of the convective flows velocities. Therefore, by the Fourier
transforming the potential
$\Phi_{i}\left(\check{x}_{i},\check{y}_{i},z_{i1},t\right)$ over the
microscale spatial coordinates $\check{x}_{i},\check{y}_{i}$,

$\displaystyle\displaystyle\Phi_{i}\left(\check{x}_{i},\check{y}_{i},z_{i1},t,\right)=\frac{1}{\left(2\pi\right)^{3}}\int
dk_{\check{x}_{i}}dk_{\check{y}_{i}}dk_{z}$
$\displaystyle\displaystyle\times\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)e^{i\left(k_{\check{x}_{i}}\check{x}_{i}+k_{\check{y}_{i}}\check{y}_{i}+k_{z}z_{i}\right)}$
(51)

we will derive from Eqs. (48) and (49) the equation for the separate spatial
Fourier mode
$\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)$ of the
microscale plasma response on the mesoscale compressed-sheared convective
flows. With coordinates $\check{\xi}_{i}$, $\check{\eta}_{i}$, used in Eq.
(50), potential
$\Phi_{i}\left(\check{\xi}_{i},\check{\eta}_{i},z_{i1},t\right)$ is determined
by the relation

$\displaystyle\displaystyle\Phi_{i}\left(\check{x}_{i},\check{\eta}_{i},z_{i1},t\right)=\frac{1}{\left(2\pi\right)^{3}}\int
dk_{\check{x}_{i}}dk_{\check{y}_{i}}dk_{Z}$
$\displaystyle\displaystyle\times\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)e^{i\left(k_{\check{x}_{i}}\check{\xi}_{i}+k_{\check{y}_{i}}\check{\eta}_{i}+k_{z}z_{i1}\right)}$
$\displaystyle\displaystyle\times\exp\left(-i\frac{k_{i\bot}\left(t\right)\tilde{v}_{i\perp}}{\omega_{ci}}\sin\left(\phi-\omega_{ci}t-\delta\left(t\right)\right)\right)$
$\displaystyle\displaystyle=\int
dk_{\check{x}_{i}}dk_{\check{y}_{i}}dk_{z}\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)$
$\displaystyle\displaystyle\times
e^{i\left(k_{\check{x}_{i}}\check{\xi}_{i}+k_{\check{y}_{i}}\check{\eta}_{i}+k_{z}z_{i1}\right)}$
$\displaystyle\displaystyle\times\sum\limits_{n=-\infty}^{\infty}J_{n}\left(\frac{k_{i\bot}\left(t\right)\tilde{v}_{i\perp}}{\omega_{ci}}\right)$
$\displaystyle\displaystyle\times
e^{-in\left(\phi_{1}-\omega_{ci}t-\delta\left(t\right)\right)},$ (52)

in which $J_{n}$ is the Bessel function of the first kind of the order $n$ and
the wave number component $k_{i\bot}\left(t\right)$ across the magnetic field
grows with time due to the distorting of the wave structure by the compressed
and sheared flows,

$\displaystyle\displaystyle
k^{2}_{i\bot}\left(t\right)=\left(k_{\check{x}_{i}}e^{\bar{U}^{\prime}_{ix}t}-k_{\check{y}_{i}}\left(V^{\prime}_{0}t+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)\right)^{2}+k^{2}_{\check{y}_{i}},$
(53)

and

$\displaystyle\displaystyle\tan\delta_{i}\left(t\right)=k_{\check{y}_{i}}\left(k_{\check{x}_{i}}e^{\bar{U}^{\prime}_{ix}t}-k_{\check{y}_{i}}\left(V^{\prime}_{0}t+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)\right)^{-1}.$
(54)

The solution (50) with potential (52) is

$\displaystyle\displaystyle\bar{f}_{i}\left(\tilde{v}_{i\perp},\phi_{1},v_{iz},\check{\xi}_{i},\check{\eta}_{i},z_{i1},t\right)=i\frac{e_{i}}{m_{i}}\int\limits^{t}_{0}dt_{1}$
$\displaystyle\displaystyle\times\int
dk_{\check{x}_{i}}dk_{\check{y}_{i}}dk_{z}\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t_{1}\right)$
$\displaystyle\displaystyle\times
e^{i\left(k_{\check{x}_{i}}\check{\xi}_{i}+k_{\check{y}_{i}}\check{\eta}_{i}+k_{z}z_{i}\right)}$
$\displaystyle\displaystyle\times\sum\limits_{n=-\infty}^{\infty}J_{n}\left(\frac{k_{i\bot}\left(t_{1}\right)\tilde{v}_{i\perp}}{\omega_{ci}}\right)e^{-in\left(\phi_{1}-\omega_{ci}t_{1}-\delta\left(t_{1}\right)\right)}$
$\displaystyle\displaystyle\times\left(\frac{k_{\check{y}_{i}}}{\omega_{ci}}e^{\bar{U}^{\prime}_{ix}t_{1}}\frac{\partial\bar{F}_{i0}}{\partial\check{\xi}_{i}}+\frac{n\omega_{ci}}{\tilde{v}_{i\perp}}\frac{\partial\bar{F}_{i0}}{\partial\tilde{v}_{i\perp}}+k_{z}\frac{\partial\bar{F}_{i0}}{\partial
v_{iz}}\right).$ (55)

In what follows, we consider the stability of the microscale perturbations of
the convective flows with wavelength much less than the plasma inhomogeneity
scale length $L_{\check{X}_{i}}$, for which $|k_{i\bot}L_{\check{x}_{i}}|\gg
1$ and the Fourier transform of $\bar{f}_{i}$ over $\check{x}_{i}$ can be
performed in the local approximation. The Fourier transformed microscale
perturbation of the ion density, determined within this approximation, is

$\displaystyle\displaystyle
n_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)=i\frac{2\pi
e_{i}}{m_{i}}\sum\limits_{n=-\infty}^{\infty}\int\limits^{t}_{0}dt_{1}\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t_{1}\right)\int\limits^{\infty}_{-\infty}dv_{iz}\int\limits_{0}^{\infty}d\tilde{v}_{i\perp}\tilde{v}_{i\perp}$
$\displaystyle\displaystyle\times\sum\limits_{n=-\infty}^{\infty}J_{n}\left(\frac{k_{i\bot}\left(t\right)\tilde{v}_{i\perp}}{\omega_{ci}}\right)J_{n}\left(\frac{k_{i\bot}\left(t_{1}\right)\tilde{v}_{i\perp}}{\omega_{ci}}\right)e^{-ik_{z}v_{iz}\left(t-t_{1}\right)-in\left(\omega_{ci}\left(t-t_{1}\right)-\delta\left(t\right)+\delta\left(t_{1}\right)\right)}$
$\displaystyle\displaystyle\times\left(\frac{k_{\check{y}_{i}}}{\omega_{ci}}e^{\bar{U}^{\prime}_{ix}t}\frac{\partial\bar{F}_{i0}}{\partial\check{\xi}_{i}}+\frac{n\omega_{ci}}{\tilde{v}_{i\perp}}\frac{\partial\bar{F}_{i0}}{\partial\tilde{v}_{i\perp}}+k_{z}\frac{\partial\bar{F}_{i0}}{\partial
v_{iz}}\right).$ (56)

For the Maxwellian distribution
$\bar{F}_{i0}\left(\tilde{\mathbf{v}}_{i},\check{\xi}_{i}\right)$ of ions with
initial value (46) in the case of the uniform ion temperature Eq. (56) gives

$\displaystyle\displaystyle
n_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)=i\frac{n_{0i}\left(\check{\xi}\right)e_{i}}{T_{i}}$
$\displaystyle\displaystyle\times\sum\limits_{n=-\infty}^{\infty}\int\limits^{t}_{0}dt_{1}\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t_{1}\right)$
$\displaystyle\displaystyle\times
I_{n}\left(k_{i\bot}\left(t\right)k_{i\bot}\left(t_{1}\right)\rho^{2}_{i}\right)e^{-\frac{1}{2}\rho^{2}_{i}\left(k_{i\bot}^{2}\left(t\right)+k_{i\bot}^{2}\left(t_{1}\right)\right)}$
$\displaystyle\displaystyle\times
e^{-\frac{1}{2}k^{2}_{z}v_{z}\left(t-t_{1}\right)^{2}-in\left(\omega_{ci}\left(t-t_{1}\right)-\delta\left(t\right)+\delta\left(t_{1}\right)\right)}$
$\displaystyle\displaystyle\times\left(k_{\check{y}_{i}}v_{di}e^{\bar{U}^{\prime}_{ix}t}-n\omega_{ci}+ik^{2}_{z}v^{2}_{Ti}\left(t-t_{1}\right)\right).$
(57)

where $v_{d\alpha}=(cT_{\alpha}/e_{\alpha}B_{0})d\ln
n_{0}(\check{x}_{i})/d\check{x}_{i}$ is the ion $\left(\alpha=i\right)$ and
electron $\left(\alpha=e\right)$ diamagnetic velocity, $\rho_{i}$ is the
thermal ion Larmor radius, and $I_{n}$ is the modified Bessel function of the
first kind and order $n$. The Fourier transform
$n_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t\right)$ of the
microscale perturbation of the electron density, performed in the electron
frame with coordinates $\check{x}_{e},\check{y}_{e},z_{e1},t$, is determined
in the same way as it is given by Eqs. (34)-(58) for
$n_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)$ with changed
ion on electron species subscripts. The derived ion and electron density
perturbations are employed in the Poisson equation, Fourier transformed over
the variables $\check{x}_{i},\check{y}_{i}$. Therefore
$n_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t\right)$ for the
Poisson equation should be recalculated in the variables
$\check{x}_{i},\check{y}_{i}$ of the ion frame. For this goal, we derive the
relations between variables $\check{x}_{i},\check{y}_{i}$ and
$\check{x}_{e},\check{y}_{e}$. Because the difference between $\tilde{x}_{i}$
and $\tilde{x}_{e}$, as well as between $\tilde{y}_{i}$ and $\tilde{y}_{e}$,
are on the order of the microscale displacements of the ions relative to
electrons in FW field, which are on the order of or less than the wavelength
of the microscale perturbations, we can use relations
$\tilde{x}_{i}=\tilde{x}_{e}$, and $\tilde{y}_{i}=\tilde{y}_{e}$ with Eqs.
(40), (41) and obtain on this way the relations

$\displaystyle\displaystyle\check{x}_{e}\left(\check{x}_{i},t\right)=\frac{1}{\bar{U}^{\prime}_{ex}}\left\\{e^{\bar{U}^{\prime}_{ex}t}\left[\bar{U}^{(0)}_{ex}+\frac{\bar{U}^{\prime}_{ex}}{{\bar{U}^{\prime}_{ix}}}\left[\left(\bar{U}^{(0)}_{ix}+\bar{U}^{\prime}_{ix}\check{x}_{i}\right)e^{-\bar{U}^{\prime}_{ix}t}-\bar{U}^{(0)}_{ix}\right]\right]-\bar{U}^{(0)}_{ex}\right\\},$
(58)
$\displaystyle\displaystyle\check{y}_{e}\left(\check{y}_{i},\check{x}_{i},t\right)=\check{y}_{i}-\left[\left(\bar{U}^{(0)}_{iy}-\bar{U}^{(0)}_{ix}\frac{\bar{U}^{\prime}_{iy}}{{\bar{U}^{\prime}_{ix}}}\right)-\left(\bar{U}^{(0)}_{ey}-\bar{U}^{(0)}_{ex}\frac{\bar{U}^{\prime}_{ey}}{{\bar{U}^{\prime}_{ex}}}\right)\right]t$
$\displaystyle\displaystyle+\frac{\bar{U}^{(0)}_{ix}}{\bar{U}^{\prime}_{ix}}\left(\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}-\frac{\bar{U}^{\prime}_{ey}}{\bar{U}^{\prime}_{ex}}\right)e^{-\bar{U}^{\prime}_{ix}t}+\frac{\bar{U}^{\prime}_{ey}}{\bar{U}^{\prime}_{ex}}\left(\frac{\bar{U}^{(0)}_{ix}}{\bar{U}^{\prime}_{ix}}-\frac{\bar{U}^{(0)}_{ex}}{\bar{U}^{\prime}_{ex}}\right)+\left(\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}-\frac{\bar{U}^{\prime}_{ey}}{\bar{U}^{\prime}_{ex}}\right)\check{x}_{i}e^{-\bar{U}^{\prime}_{ix}t}.$
(59)

Note, the relations for $\check{x}_{i}\left(\check{x}_{e},t\right)$ and for
$\check{y}_{i}\left(\check{y}_{e},\check{x}_{e},t\right)$ are derived by
changing species subscripts $i\leftrightarrows e$ in Eqs. (58), (59).

The equation for
$n_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t\right)$, similar to
(56) for $n_{i}$, contains the Fourier transform
$\Phi_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t_{1}\right)$ of the
potential $\Phi_{e}\left(\check{x}_{e},\check{y}_{e},z,t_{1}\right)$. The
connection relation of
$\Phi_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t_{1}\right)$ with
$\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t_{1}\right)$ follows
from the relation

$\displaystyle\displaystyle\Phi_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t_{1}\right)=\int
d\check{x}_{e}\int
d\check{y}_{e}\Phi\left(\check{x}_{e},\check{y}_{e},k_{z},t_{1}\right)e^{-i\left(k_{\check{x}_{e}}\check{x}_{e}+k_{\check{y}_{e}}\check{y}_{e}\right)}$
$\displaystyle\displaystyle=\frac{1}{\left(2\pi\right)^{2}}\int
dk_{\check{x}_{i}}\int
dk_{\check{y}_{i}}\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{Y}_{i}},k_{z},t_{1}\right)\int
d\check{x}_{i}\int
d\check{y}_{i}\frac{\partial\left(\check{x}_{e},\check{y}_{e}\right)}{\partial\left(\check{x}_{i},\check{y}_{i}\right)}$
$\displaystyle\displaystyle\times
e^{i\left(k_{\check{x}_{i}}-k_{\check{x}_{e}}\right)\check{x}_{i}+i\left(k_{\check{y}_{i}}-k_{\check{y}_{e}}\right)\check{y}_{i}-ik_{\check{x}_{e}}\left(\check{x}_{e}-\check{x}_{i}\right)-ik_{\check{y}_{e}}\left(\check{y}_{e}-\check{y}_{i}\right)}$
$\displaystyle\displaystyle=\Phi_{i}\left(k_{\check{x}_{e}}+k_{\check{x}_{e}}b_{1x}\left(t_{1}\right)+k_{\check{y}_{e}}b_{1y}\left(t_{1}\right),k_{\check{y}_{e}},k_{z},t_{1}\right)e^{\left(\bar{U}^{\prime}_{ex}-\bar{U}^{\prime}_{ix}\right)t_{1}}e^{-ik_{\check{x}_{e}}b_{0x}\left(t_{1}\right)-ik_{\check{y}_{e}}b_{0y}\left(t_{1}\right)}.$
(60)

In Eq. (60),

$\displaystyle\displaystyle\frac{\partial\left(\check{x}_{e},\check{y}_{e}\right)}{\partial\left(\check{x}_{i},\check{y}_{i}\right)}=e^{\left(\bar{U}^{\prime}_{ex}-\bar{U}^{\prime}_{ix}\right)t_{1}}$
(61)

is the Jacobian of the transformation $\check{x}_{e},\check{y}_{e}$ to
$\check{x}_{i},\check{y}_{i}$, and the relations

$\displaystyle\displaystyle\check{x}_{e}\left(\check{x}_{i},t_{1}\right)-\check{x}_{i}=b_{0x}\left(t_{1}\right)+b_{1x}\left(t_{1}\right)\check{x}_{i}$
(62)

and

$\displaystyle\displaystyle\check{y}_{e}\left(\check{y}_{i},\check{x}_{i},t_{1}\right)-\check{y}_{i}=b_{0y}\left(t_{1}\right)+b_{1y}\left(t_{1}\right)\check{x}_{i},$
(63)

where

$\displaystyle\displaystyle
b_{0x}\left(t_{1}\right)=\frac{1}{\bar{U}^{\prime}_{ex}}e^{\bar{U}^{\prime}_{ex}t_{1}}\left[\bar{U}^{(0)}_{ex}+\bar{U}^{(0)}_{ix}\frac{\bar{U}^{\prime}_{ex}}{{\bar{U}^{\prime}_{ix}}}\left(e^{-\bar{U}^{\prime}_{ix}t_{1}}-1\right)\right]$
$\displaystyle\displaystyle-\frac{\bar{U}^{(0)}_{ix}}{{\bar{U}^{\prime}_{ix}}},$
(64) $\displaystyle\displaystyle
b_{1x}\left(t_{1}\right)=\left(e^{\left(\bar{U}^{\prime}_{ex}-\bar{U}^{\prime}_{ix}\right)t_{1}}-1\right),$
(65) $\displaystyle\displaystyle
b_{0y}\left(t_{1}\right)=\left[\left(\bar{U}^{(0)}_{ey}-\bar{U}^{(0)}_{ex}\frac{\bar{U}^{\prime}_{ey}}{{\bar{U}^{\prime}_{ex}}}\right)-\left(\bar{U}^{(0)}_{iy}-\bar{U}^{(0)}_{ix}\frac{\bar{U}^{\prime}_{iy}}{{\bar{U}^{\prime}_{ix}}}\right)\right]t_{1}$
$\displaystyle\displaystyle+\frac{\bar{U}^{(0)}_{ix}}{\bar{U}^{\prime}_{ix}}\left(\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}-\frac{\bar{U}^{\prime}_{ey}}{\bar{U}^{\prime}_{ex}}\right)e^{-\bar{U}^{\prime}_{ix}t_{1}}$
$\displaystyle\displaystyle+\frac{\bar{U}^{\prime}_{ey}}{\bar{U}^{\prime}_{ex}}\left(\frac{\bar{U}^{(0)}_{ix}}{\bar{U}^{\prime}_{ix}}-\frac{\bar{U}^{(0)}_{ex}}{\bar{U}^{\prime}_{ex}}\right),$
(66) $\displaystyle\displaystyle
b_{1y}\left(t_{1}\right)=\left(\frac{\bar{U}^{\prime}_{iy}}{\bar{U}^{\prime}_{ix}}-\frac{\bar{U}^{\prime}_{ey}}{\bar{U}^{\prime}_{ex}}\right)e^{-\bar{U}^{\prime}_{ix}t_{1}},$
(67)

were used.

Now we determine the relation between the Fourier transform
$n_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t\right)$ of the
electron density perturbation
$n_{e}\left(\check{x}_{e},\check{y}_{e},z_{e},t\right)$, performed in the
electron frame with variables $\check{x}_{e},\check{y}_{e}$, with the Fourier
transform
$n^{(i)}_{e}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)$ of
$n_{e}\left(\check{x}_{e},\check{y}_{e},Z_{e},t\right)$, performed in the ion
frame with variables $\check{x}_{i},\check{y}_{i}$.

$\displaystyle\displaystyle
n^{(i)}_{e}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)=\int
d\check{x}_{i}\int
d\check{y}_{i}e^{-i\left(k_{\check{x}_{i}}\check{x}_{i}+k_{\check{y}_{i}}\check{y}_{i}\right)}n_{e}\left(\check{x}_{e},\check{y}_{e},k_{z},t\right)$
$\displaystyle\displaystyle=\int d\check{x}_{e}\int
d\check{y}_{e}n_{e}\left(\check{x}_{e},\check{y}_{e},k_{z},t\right)\frac{\partial\left(\check{x}_{i},\check{y}_{i}\right)}{\partial\left(\check{x}_{e},\check{y}_{e}\right)}e^{-ik_{\check{x}_{i}}\check{x}_{e}-ik_{\check{y}_{i}}\check{y}_{e}-ik_{\check{x}_{i}}\left(\check{x}_{i}-\check{x}_{e}\right)-ik_{\check{y}_{i}}\left(\check{y}_{i}-\check{y}_{e}\right)}$
$\displaystyle\displaystyle=\int d\check{x}_{e}\int
d\check{y}_{e}n_{e}\left(\check{x}_{e},\check{y}_{e},k_{z},t\right)e^{\left(\bar{U}^{\prime}_{ix}-\bar{U}^{\prime}_{ex}\right)t}$
$\displaystyle\displaystyle\times
e^{-ik_{\check{x}_{i}}\check{x}_{e}-ik_{\check{y}_{i}}\check{y}_{e}-ik_{\check{x}_{i}}\left(a_{0x}\left(t\right)+a_{1x}\left(t\right)\check{x}_{e}\right)-ik_{\check{y}_{i}}\left(a_{0y}\left(t\right)+a_{1y}\left(t\right)\check{x}_{e}\right)}$
$\displaystyle\displaystyle=e^{\left(\bar{U}^{\prime}_{ix}-\bar{U}^{\prime}_{ex}\right)t}e^{-ik_{\check{x}_{i}}a_{0x}\left(t\right)-ik_{\check{y}_{i}}a_{0y}\left(t\right)}n_{e}\left(k_{\check{x}_{i}}\left(1+a_{1x}\left(t\right)\right)+k_{\check{y}_{i}}a_{1y}\left(t\right),k_{\check{y}_{i}},k_{z},t\right),$
(68)

where the relations

$\displaystyle\displaystyle\check{x}_{i}\left(\check{x}_{e},t\right)-\check{x}_{e}=a_{0x}\left(t\right)+a_{1x}\left(t\right)\check{x}_{e}$
(69)

and

$\displaystyle\displaystyle\check{y}_{i}\left(\check{y}_{e},\check{x}_{e},t\right)-\check{y}_{e}=a_{0y}\left(t\right)+a_{1y}\left(t\right)\check{x}_{e},$
(70)

where used. The functions $a_{0x}\left(t\right)$, $a_{1x}\left(t\right)$,
$a_{0y}\left(t\right)$, and $a_{1y}\left(t\right)$ are determined by the
functions $b_{0x}\left(t\right)$, $b_{1x}\left(t\right)$,
$b_{0y}\left(t\right)$, and $b_{1y}\left(t\right)$, respectively, by changing
species subscripts $i\leftrightarrows e$ in Eqs. (59) -(64).

By replacing $k_{\check{x}_{e}}$ and $k_{\check{y}_{e}}$ in Eq. (64) on
$k_{\check{x}_{i}}\left(1+a_{1x}\left(t\right)\right)+k_{\check{y}_{i}}a_{1y}\left(t\right)$
and $k_{\check{y}_{i}}$, which, as it follows from Eq. (64), are the new wave
numbers conjugate with coordinates $\check{x}_{e}$ and $\check{y}_{e}$ in
$n^{(i)}_{e}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)$, we
derive the following relation for $n^{(i)}_{e}$:

$\displaystyle\displaystyle
n^{(i)}_{e}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)=\frac{2i\pi
e}{m_{e}}e^{-ik_{\check{x}_{i}}a_{0x}\left(t\right)-ik_{\check{y}_{i}}a_{0y}\left(t\right)}\int\limits^{t}_{0}dt_{1}e^{\left(\bar{U}^{\prime}_{ix}-\bar{U}^{\prime}_{ex}\right)\left(t-t_{1}\right)}$
$\displaystyle\displaystyle\times\Phi_{i}\left(\left(k_{\check{x}_{i}}\left(1+a_{1x}\left(t\right)\right)+k_{\check{y}_{i}}a_{1y}\left(t_{1}\right)\right)\left(1+b_{1x}\left(t_{1}\right)\right)+k_{\check{y}_{i}}b_{1y}\left(t_{1}\right),k_{\check{y}_{i}},k_{z},t_{1}\right)$
$\displaystyle\displaystyle\times
e^{-i\left(k_{\check{x}_{i}}\left(1+a_{1x}\left(t\right)\right)+k_{\check{y}_{i}}a_{1y}\left(t_{1}\right)\right)b_{0x}\left(t_{1}\right)-ik_{\check{y}_{i}}b_{0y}\left(t_{1}\right)}$
$\displaystyle\displaystyle\times\int\limits^{\infty}_{0}dv_{e\bot}v_{e\bot}\int\limits_{-\infty}^{\infty}dv_{ez}e^{-ik_{z}v_{ez}\left(t-t_{1}\right)}\left(\frac{k_{\check{y}_{i}}}{\omega_{ce}}e^{\bar{U}^{\prime}_{ex}t_{1}}\frac{\partial\bar{F}_{e}}{\partial\check{\xi}_{e}}+k_{z}\frac{\partial\bar{F}_{e}}{\partial
v_{ez}}\right).$ (71)

Equation (71) is valid for the perturbations with frequency much less than the
electron cyclotron frequency and with wavelength across the magnetic field
much larger than the thermal electron Larmor radius.

The Poisson equation (49) for $\Phi_{i}$, Fourier transformed over
$\check{x}_{i}$ and $\check{y}_{i}$,

$\displaystyle\displaystyle\left(k^{2}_{\check{x}_{i}}e^{2\bar{U}_{ix}^{\prime}t}-2e^{\bar{U}_{ix}^{\prime}t}\left(V^{\prime}_{0}t+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)k_{\check{x}_{i}}k_{\check{y}_{i}}+\left(1+\left(V^{\prime}_{0}t+\frac{\bar{U}_{iy}^{\prime}}{\bar{U}_{ix}^{\prime}}\right)^{2}\right)k^{2}_{\check{y}_{i}}+k^{2}_{z}\right)\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)$
$\displaystyle\displaystyle=4\pi\left(e_{i}n_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)-|e|n^{(i)}_{e}\left(k_{\check{x}_{e}}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},t\right),k_{\check{y}_{i}},k_{z},t\right)\right),$
(72)

where $n_{i}$ and $n^{(i)}_{e}$ are determined by Eqs. (54) and (71)
respectively, is the equation which determines the temporal evolution of the
single spatial Fourier mode
$\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)$ in the
compressed-sheared flow.

Now we consider the particular cases for Eq.(72), in which $n_{i}$ and
$n^{(i)}_{e}$ are determined by Eqs. (57) and (71).

1\. In the case of the currentless compressed-sheared flow
$\bar{U}^{(0)}_{ix}=\bar{U}^{(0)}_{ex}$,
$\bar{U}_{ix}^{\prime}=\bar{U}_{ex}^{\prime}$, and
$\bar{U}^{(0)}_{iy}=\bar{U}^{(0)}_{ey}$,
$\bar{U}_{iy}^{\prime}=\bar{U}_{ey}^{\prime}$. It follows from Eqs. (62)-(67)
that in this case $b_{0x}\left(t_{1}\right)=a_{0x}\left(t\right)=0$,
$b_{1x}\left(t_{1}\right)=a_{1x}\left(t\right)=0$, and
$b_{0y}\left(t_{1}\right)=a_{0y}\left(t\right)=0$,
$b_{1y}\left(t_{1}\right)=a_{1y}\left(t\right)=0$. Therefore in this case
$\check{x}_{i}=\check{x}_{e}$, $\check{y}_{i}=\check{y}_{e}$, and
$\Phi_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t\right)=\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)$
and
$n^{(i)}_{e}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)=n_{e}\left(k_{\check{x}_{e}},k_{\check{y}_{e}},k_{z},t\right)$.
Equation (72) in this case has a form

$\displaystyle\displaystyle\lambda^{2}_{Di}\left(k^{2}_{i\bot}\left(t\right)+k^{2}_{z}\right)\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t\right)=\sum\limits_{n=-\infty}^{\infty}\int\limits^{t}_{t_{0}}dt_{1}\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t_{1}\right)I_{n}\left(k_{i\bot}\left(t\right)k_{i\bot}\left(t_{1}\right)\rho^{2}_{i}\right)$
$\displaystyle\displaystyle\times
e^{-\frac{1}{2}\rho^{2}_{i}\left(k_{i\bot}^{2}\left(t\right)+k_{i\bot}^{2}\left(t_{1}\right)\right)}\left(ik_{\check{y}_{i}}v_{di}e^{\bar{U}^{\prime}_{ix}t_{1}}-in\omega_{ci}-k^{2}_{z}v^{2}_{Ti}\left(t-t_{1}\right)\right)e^{-\frac{1}{2}k^{2}_{z}v^{2}_{Ti}\left(t-t_{1}\right)^{2}-in\left(\omega_{ci}\left(t-t_{1}\right)-\delta\left(t\right)+\delta\left(t_{1}\right)\right)}$
$\displaystyle\displaystyle=\frac{T_{i}}{T_{e}}\int\limits^{t}_{t_{0}}dt_{1}\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t_{1}\right)e^{-\frac{1}{2}k^{2}_{z}v^{2}_{Te}\left(t-t_{1}\right)^{2}}\left[ik_{\check{y}_{i}}v_{de}e^{\bar{U}^{\prime}_{ex}t_{1}}-k^{2}_{z}v^{2}_{Te}\left(t-t_{1}\right)\right].$
(73)

where $t_{0}\geq 0$, $\lambda_{Di(e)}$ is the ion (electron) Debye length, and

$\displaystyle\displaystyle
A_{in}\left(t,t_{1}\right)=I_{n}\left(k_{i\bot}\left(t\right)k_{i\bot}\left(t_{1}\right)\rho^{2}_{i}\right)$
$\displaystyle\displaystyle\times
e^{-\frac{1}{2}\rho^{2}_{i}\left(k_{i\bot}^{2}\left(t\right)+k_{i\bot}^{2}\left(t_{1}\right)\right)}.$
(74)

Equation (73) was derived for the first time in Ref.Mikhailenko4  for the
poloidal sheared plasma flow without the convective flows (i. e. for the case
$\bar{U}^{\prime}_{ix}=\bar{U}^{\prime}_{ex}=0$ and
$\bar{U}^{\prime}_{iy}=\bar{U}^{\prime}_{ey}=0$). In that case, the poloidal
velocity shear manifests as a time- dependence of $A_{in}\left(t,t_{1}\right)$
function which determines effect of the finite ion Larmor radius. The solution
of Eq. (73), derived for the kinetic drift instability in the poloidal sheared
flow, displaysMikhailenko4  the nonmodal effect of the reduction with time of
the frequency and of the growth rate of this instability caused by the flow
velocity shear. By the integration by parts of the first term on the right
part of Eq. (73), this equation for the low frequency perturbations, for which
$d\Phi/dt\ll\omega_{ci}\Phi$, may be presented in the form similar to Eq. (25)
of Ref. Mikhailenko4 ,

$\displaystyle\displaystyle\int\limits_{t{0}}^{t}dt_{1}\frac{d}{dt_{1}}\left[\Phi_{i}\left(k_{\check{x}_{i}},k_{\check{y}_{i}},k_{z},t_{1}\right)\left(1+\frac{T_{i}}{T_{e}}-A_{in}\left(t,t_{1}\right)\right)\right]$
$\displaystyle\displaystyle-i\int\limits^{t}_{t_{0}}dt_{1}\Phi_{i}k_{\check{y}_{i}}v_{di}e^{\bar{U}^{\prime}_{ex}t_{1}}A_{in}\left(t,t_{1}\right)$
$\displaystyle\displaystyle=\frac{T_{i}}{T_{e}}\int\limits^{t}_{t_{0}}\left(\frac{d\Phi_{i}}{dt_{1}}+ik_{\check{y}_{i}}v_{de}e^{\bar{U}^{\prime}_{ex}t_{1}}\Phi_{i}\right)e^{-\frac{1}{2}k^{2}_{z}v^{2}_{Te}\left(t-t_{1}\right)^{2}}$
(75)

We found that in the time domain $\left(t,t_{0}\right)$, in which
$k_{i\bot}\left(t_{1}\right)\rho_{i}\gg 1$, the temporal evolution of the
potential $\Phi_{i}$ in the compressed flow, predicted by the solution to Eq.
(75), resembles the temporal evolution of the potential $\Phi_{i}$ in the
poloidal sheared flow: the potential $\Phi_{i}$ gradually becomes a zero-
frequency cell-like perturbation when time elapsed.

##  V Conclusions

A nonmodal kinetic theory of the stability of the two-dimensional compressed-
sheared mesoscale plasma flows, generated by the radially inhomogeneous
electrostatic ion cyclotron parametric microturbulence in the pedestal plasma
with a sheared poloidal flow, is developed. This theory reveals that the
separate spatially uniform Fourier modes of the electrostatic responses of the
ions and of the electrons on the mesoscale convective flows are determined
only in the frames of references moved with velocities of the ion and electron
convective flows. In the laboratory frame, these modes are observed as the
compressed-sheared modes with time dependent wave numbers. The integral
equation, which governs the separate Fourier mode of the electrostatic
potential of the plasma species responses on the mesoscale convective flows,
is derived. In this equation, the effects of the compressing and shearing of
the convective flows are revealed as the time dependence of the finite ion
Larmor radius effect. The solution of this equation for the kinetic drift
instability displays the nonmodal transformation of the potential to the zero
frequency cell-like perturbation when time elapsed.

###### Acknowledgements.

This work was supported by National R&D Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of Education, Science and
Technology (Grant No. NRF-2018R1D1A3B07051247) and BK21 FOUR, the Creative
Human Resource Education and Research Programs for ICT Convergence in the 4th
Industrial Revolution.

DATA AVAILABILITY

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

##  Appendix A Velocities $\tilde{U}_{ix}\left(\tilde{X}_{i}\right)$ and
$\tilde{U}_{iy}\left(\tilde{X}_{i}\right)$ of the convective flows

For the electric field $\tilde{\mathbf{E}}_{i}$, given by Eq. (8), velocities
$\bar{U}_{ix}\left(\tilde{X}_{i}\right)$ and
$\bar{U}_{iy}\left(\tilde{X}_{i}\right)$, after long calculation similar to
performed in Ref.Mikhailenko1 , are determined by the following relations:

$\displaystyle\displaystyle\bar{U}_{ix}\left(\tilde{X}_{i}\right)=\frac{1}{2\omega_{ci}}\frac{e_{i}^{2}}{m^{2}_{i}}\frac{1}{4\pi^{2}}\sum_{n}\int
d\mathbf{k}\left[a_{i1}\left(\mathbf{k},n\right)\tilde{E}_{ix}\left(\mathbf{k},\tilde{X}_{i},n\right)\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}^{\ast}_{iy}\left(\mathbf{k},\tilde{X}_{i},n\right)\right)\right.$
$\displaystyle\displaystyle\left.+a_{i2}\left(\mathbf{k},n\right)\tilde{E}_{iy}\left(\mathbf{k},\tilde{X}_{i},n\right)\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}^{\ast}_{ix}\left(\mathbf{k},\tilde{X}_{i},n\right)\right)\right]$
(76)

and

$\displaystyle\displaystyle\bar{U}^{(0)}_{iy}\left(\tilde{X}_{i}\right)=-\frac{1}{4\omega_{ci}}\frac{e_{i}^{2}}{m^{2}_{i}}\frac{1}{4\pi^{2}}\sum_{n}\int
d\mathbf{k}\left[a_{i1}\left(\mathbf{k},n\right)\frac{\partial}{\partial\tilde{X}_{i}}\left|\tilde{E}_{ix}\left(\mathbf{k},\tilde{X}_{i},n\right)\right|^{2}-a_{i2}\left(\mathbf{k},n\right)\frac{\partial}{\partial
X_{i}}\left|\tilde{E}_{iy}\left(\mathbf{k},\tilde{X}_{i},n\right)\right|^{2}\right]$
(77)

where the asterisk in Eq. (76) implies the operation of complex conjugate. The
coefficients $a_{i1}\left(\mathbf{k},n\right)$ and
$a_{i2}\left(\mathbf{k},n\right)$ are determined as

$\displaystyle\displaystyle
a_{i1}\left(\mathbf{k},n\right)=\left[\frac{\omega_{ci}}{\omega_{n}\left(\mathbf{k}\right)\left(\omega_{ci}+\omega_{n}\left(\mathbf{k}\right)\right)^{2}}+\frac{\omega_{ci}}{\omega_{n}\left(\mathbf{k}\right)\left(\omega_{ci}-\omega_{n}\left(\mathbf{k}\right)\right)^{2}}+\frac{1}{\left(\omega^{2}_{ci}-\omega_{n}^{2}\left(\mathbf{k}\right)\right)}\right],$
(78)

and

$\displaystyle\displaystyle
a_{i2}\left(\mathbf{k}\right)=\left[\frac{1}{\left(\omega_{ci}+\omega_{n}\left(\mathbf{k}\right)\right)^{2}}+\frac{1}{\left(\omega_{ci}-\omega_{n}\left(\mathbf{k}\right)\right)^{2}}+\frac{1}{\left(\omega^{2}_{ci}-\omega_{n}^{2}\left(\mathbf{k}\right)\right)}\right].$
(79)

The velocities of electrons $\bar{U}_{ex}\left(\tilde{X}_{i}\right)$ and
$\bar{U}_{ey}\left(\tilde{X}_{i}\right)$ in the ion frame, with electric field
$\mathbf{E}_{e}\left(\mathbf{\hat{r}}_{i},\tilde{X}_{i},t\right)$ determined
by Eq. (10), are

$\displaystyle\displaystyle\bar{U}_{ex}\left(\tilde{X}_{i}\right)\approx\frac{c^{2}}{B^{2}_{0}}\frac{1}{4\pi^{2}}\sum_{n}\int
d\mathbf{k}\tilde{E}_{ix}\left(\mathbf{k},\tilde{X}_{i},n\right)$
$\displaystyle\displaystyle\times\frac{\partial}{\partial\tilde{X}_{i}}\left(\tilde{E}^{\ast}_{iy}\left(\mathbf{k},\tilde{X}_{i},n\right)\right)\sum\limits_{p=-\infty}^{\infty}J^{2}_{p}\left(a_{ie}\left(\mathbf{k},\tilde{X}_{i}\right)\right)$
$\displaystyle\displaystyle\times\frac{1}{\Omega_{p}\left(\mathbf{k},\tilde{X}_{i}\right)},$
(80)

and

$\displaystyle\displaystyle\bar{U}_{ey}\left(\tilde{X}_{i}\right)\approx-\frac{1}{2}\frac{c^{2}}{B^{2}_{0}}\frac{1}{4\pi^{2}}\sum_{n}\int
d\mathbf{k}\frac{\partial}{\partial\tilde{X}_{i}}\left|\tilde{E}_{ix}\left(\mathbf{k},\tilde{X}_{i},n\right)\right|^{2}$
$\displaystyle\displaystyle\times\sum\limits_{p=-\infty}^{\infty}J^{2}_{p}\left(a_{ie}\left(\mathbf{k},\tilde{X}_{i}\right)\right)\frac{1}{\Omega_{p}\left(\mathbf{k},\tilde{X}_{i}\right)},$
(81)

where limit $|\omega_{ce}|\gg\Omega_{p}\sim\omega_{ci}$ was used.

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